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Inventory and pricing models for perishable products
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Inventory and pricing models for perishable products
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INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. U M 1 films the text directly from the original or copy submitted. Thus, som e thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand corner and continuing from left to right in equal sections with small overlaps. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6” x 9” black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. ProQuest Information and Learning 300 North Zeeb Road, Ann Arbor, M l 48106-1346 USA 800-521-0600 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. INVENTORY AND PRICING MODELS FOR PERISHABLE PRODUCTS bv Alper §en A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (BUSINESS ADMINISTRATION) August 2000 Copyright 2000 Alper §en R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 3018124 ___ ® UMI UMI Microform 3018124 Copyright 2001 by Bell & Howell Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. Bell & Howell Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. UNIVERSITY OF SOUTHERN CALIFORNIA THE GRADUATE SCHOOL UNIVERSITY PARK LOS ANGELES. CALIFORNIA 90007 This dissertation, written by A jj> £ r £ en under the direction of hf.$........ Dissertation Committee, and approved by all its members* has been presented to and accepted by The Graduate School, in partial fulfillment of re quirements for the degree of DOCTOR OF PHILOSOPHY Dean of Graduate Studies Date . . . A t i j g x x s t _ _ _ 3 ^ 2 0 0 . o DISSERTATION COMMITTEE Chairperson R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Alper §en Alex X. Zhang IN V E N T O R Y A N D P R IC IN G M ODELS FO R P E R ISH A B L E PR O D U C T S In many industries, the demand for services and goods is perishable and uncertain. Companies in these industries often have inflexible supply processes for such products, as a consequence of overseas manufacturing, economies of scale or long term capacity decisions. Examples include fash ion retailing, airlines and hospitality industries. In such industries, effective decision making in procurement, allocation and pricing can help to reduce the mismatches between supply and demand, and can make the difference between a profitable and an unprofitable company. The main objective of this dissertation is to develop analytical models to support decision-makers in these critical decisions. The dissertation starts with an overview of opera tions and recent trends and an assessment for the need for research and use of analytical models in the apparel industry. The first model studies the procurement decisions when the perishable product can be sold in different markets with varying prices. It is assumed R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. th at the retailer or the manufacturer has only one chance to place an order. This leads to approaching the problem as a multiple market variant of the Newsboy problem. Two cases are distinguished: prices can go down, as in apparel, or up, as in airlines. The second model addresses the pricing decisions assuming that the one time procurement decision is already made. The model allows a systematic resolution of demand uncertainty based on actual sales; in particular, early sales information is used to update the probability distribution of the demand using a Bayesian procedure. This model is used to study the impact of cost, variance and price elasticity of demand and strategic procurement decisions on revenues. The third model studies the impact of competition in pricing decisions using a game theoretic analysis. The model assumes that there are two companies th a t sell fixed stocks of similar products over a selling season and derives the times that these companies switch from one price to another under a unique Nash equilibrium. It is shown th a t the equilibrium is a function of the degree of demand dependency and the difference in each company’s initial stocking levels. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Dedication Biricik anneme... (To my mom...) R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Contents D ed ication ii List O f Tables vi List O f Figures v ii 1 Introduction 1 1.1 A Model for Ordering D e cisio n s................................................................... 6 1.2 A Pricing Model with Dem and L e a rn in g ................................................... 8 1.3 A Pricing Model with C o m p e titio n ............................................................. 9 2 T h e Apparel Industry 11 2.1 U.S. Textile and Apparel Complex .................................................................11 2.1.1 Fiber and Yarn Production ................................................................12 2.1.2 Fabric P roduction....................................................................................13 2.1.3 Apparel M anufacture.............................................................................13 2.1.4 R e t a i l ....................................................................................................... 15 2.2 Apparel Manufacture and Retail O p e ra tio n s ................................................ 16 2.2.1 Manufacturing O p e ra tio n s............................................................... 16 2.2.2 Retail O p eratio n s................................................................................... 21 2.3 Trends in Apparel Manufacture and R e ta il................................................... 31 2.3.1 Retail Consolidation, Vertical Integration and Emerge of Pri vate L a b e ls ..............................................................................................31 2.3.2 Import Penetration and Production S h a r in g ....................................34 2.3.3 Quick Response S y s te m s ...................................................................... 36 2.3.4 Supplier selection: off-shore versus domestic s o u r c in g ...................42 2.4 Paugal In d u strie s..................................................................................................48 2.4.1 Operations Throughout the Year ..................................................... 50 2.4.2 Recent Problems faced by Paugal and other smaller manufac turers ....................................................................................................... 52 2.5 The Fashion Buying for R e ta ilin g ....................................................................53 2.5.1 Merchandise O rg a n iz a tio n ...................................................................54 2.5.2 Timing in the Merchandise Life C y cle...............................................55 iii R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. l O JO 2.5.3 The B u y e r s .............................................................................................. 57 2.5.4 Deciding W hat to B u y ..........................................................................58 2.5.5 Whom to Buy From: The Vendors ..................................................59 2.5.6 The Quantitative Decisions: How Much to B u y ...........................61 2.5.7 Information T e c h n o lo g y .......................................................................62 2.5.8 Inventory and D istrib u tio n ...................................................................62 2.5.9 P ricin g ........................................................................................................64 2.5.10 Mark-downs and S a le s ..........................................................................65 2.6 Concluding R e m a rk s ...........................................................................................66 3 A M odel for Ordering D ecisions 68 3.1 Literature S u rv e y ..................................................................................................72 3.2 The Model with Decreasing P ric es.................................................................... 76 3.2.1 The Effect of Aggregating Dem and C lasses.....................................80 3.3 A Model with Increasing Prices: Two Demand Classes ........................... 85 3.4 Concluding R e m a rk s ...........................................................................................98 4 A Pricing M odel with. D em and L earning 100 4.1 Literature S u rv e y ............................................................................................... 103 4.2 M o d e l................................................................................................................... 109 4.2.1 Demand M o d e l.....................................................................................109 4.2.2 Pricing M o d e l........................................................................................ 114 4.3 Properties of the Optim al Solution ..............................................................116 4.4 Computational S tu d y .........................................................................................118 4.4.1 The design of the study .................................................................... 123 4.4.2 The impact of starting inventory .....................................................123 4.4.3 The impact of price sen sitiv ity ............................................................127 4.4.4 The impact of demand v a ria n c e ........................................................ 131 4.4.5 The three period p r o b le m .................................................................. 132 4.5 Inventory flexibility............................................................................................136 4.6 Concluding R e m a rk s ........................................................................................ 149 5 A Pricing M odel w ith C om p etition 151 .1 Introduction and Literature S u r v e y ............................................................... 151 .2 The Mark-down Problem ........................................................................156 5.2.1 The Linear Demand M odel.................................................................158 5.2.2 The Case of Independent Demand: Optimal Mark-down Time for a M o n o p o ly ......................................................................................160 5.2.3 Case of Dependent Demand: Equilibrium Switching Times for a D u o p o ly ................................................................................................163 5.3 The M ark-up P ro b le m .....................................................................................174 5.3.1 Optimal M ark-up Time for a M o n o p o ly ......................................175 5.3.2 Equilibrium M ark-up Times for D uopoly......................................175 iv R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.3.3 Discount Fare Allocation P ro b le m .....................................................177 5.4 C onclusion.............................................................................................................180 6 C onclusions 183 B ibliography 191 A ppendix A Proofs for Chapter 3 ................................................................................................... 199 A .l Proof of Proposition 3 . 1 ..................................................................................199 A.2 Proof of Proposition 3 . 3 ..................................................................................200 A.3 Proof of Proposition 3 . 4 ..................................................................................203 A.4 Proof of Lemma 3 .1 ........................................................................................... 205 A ppendix B Proofs for Chapter 5 ................................................................................................... 207 B .l Proof of Theorem 5 . 1 ........................................................................................ 207 B.2 Proof of Theorem 5 . 2 ........................................................................................ 218 v R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. L ist O f Tables 2.1 Market Share of Im ports in Apparel in 1997 ........................................... 44 3.1 Optimal and heuristic order q u a n titie s ........................................................ 82 3.2 Optimal X and P as a function of s ............................................................... 94 3.3 Optimal X and P as a function of p o ............................................................95 4.1 Market share and cost advantage of imports for selected apparel . . . 148 5.1 Demand Rates for Two Competing F irm s................................................... 157 VI R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. List O f Figures 2.1 Average domestic and im port prices for m en’s and women’s swim wear (1992-1999)............................................................................................. 46 2.2 Market share of domestic and import men’s and women’s swim-wear (1992-1999)....................................................................................................... 47 3.1 Value of Q j .......................................................................................................... 77 3.2 Performance of the "separate newsboys” h e u ris tic .....................................83 3.3 Value of Q2 for the derivation of E[Q2] ........................................................ 88 3.4 The optimal profit 7 t(AT, P(AT)) for given X ..................................................95 3.5 The profit function tt(X , P) with optimally chosen X = X * .................... 96 4.1 Optimal revenues as a function of starting inventory............................... 124 4.2 Optimal first and second prices as a function of starting inventory (variance = 3 / i ) .................................................................................................. 125 4.3 Comparison of optimal revenues with and without learning as a func tion of starting inventory................................................................................. 126 4.4 Comparison of switching times with and without learning as a func tion of starting inventory (7 = 2/i, variance=2/z).......................................127 4.5 Optimal revenues as a function of price elasticity (7) 128 4.6 Optimal first and second prices as a function of demand elasticity (variance = 3 / i ) .................................................................................................. 129 4.7 Comparison of optimal revenues with and without learning as a func tion of demand e la stic ity ................................................................................. 130 4.8 Comparison of switching times with and without learning as a func tion of demand elasticity (variance = 1.5/i, starting inventory = /i) . 130 4.9 Optimal revenues as a function of v a ria n c e ............................................... 131 4.10 Optimal first and second prices as a function of variance (starting inventory = 1 .5 /i)...............................................................................................132 4.11 Comparison of revenues with and without learning as a function of demand v a ria n c e ...............................................................................................133 4.12 Comparison of switching times with and without learning as a func tion of demand variance (7 = 3/i, starting inventory = 1/ i ) ...................133 4.13 Comparison of prices w ith and without mark-down restriction (7 = 2, variance = 3 / i ) ...............................................................................................134 vii R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.14 Optimal revenues as a function of starting inventory (7 = 2) .............135 4.15 Optimal revenues as a function of variance (7 = 2, starting inventory = I.o/l l ) ................................................................................................................136 4.16 Comparison of off-shore, domestic and blended strategies (7 = 2) . . 142 4.17 Comparison of off-shore, domestic and blended strategies (7 = 2) . . 144 4.18 Comparison of off-shore, domestic and blended strategies as a func tion of price s e n s itiv ity .................................................................................. 145 4.19 Trade-off curves for off-shore and domestic production.......................... 146 4.20 Trade-off between domestic and off-shore production .............................147 A.l The left-hand-side probabilities of equation ( 3 .1 0 ) .................................204 v iii R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 1 Introduction One of the challenges today’s businesses face is demand uncertainty coupled with supply inflexibility. It is now more difficult to predict demand with continuously changing consumer needs, tastes and preferences, increasing variety and dynamic competitive structure. While there are advances in production, distribution and in formation technologies for a more responsive supply chain, factors such as economies of scale and overseas suppliers prevent companies from achieving complete flexibility in their supply. Effective solutions for production, ordering, allocation and pricing problems of such companies can help to reduce the mismatches between their supply and demand and can make the difference between a profitable and an unprofitable company. Consider the retailers of fashion goods. Popularity of colors, patterns and fabrics changes every year. Consumers now prefer casual clothing with fewer well-accepted dress guidelines: the textile industry respond with increased product variety enabled by the introduction of new electronic knitting technologies. In such an environment, 1 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. it is difficult to forecast demand and its sensitivity to price before any actual sales information becomes available. On the other hand, pressure to reduce costs has led retailers to work with global suppliers enjoying lower labor rates. Prices have decreased significantly, but lead times are now longer. Pashigian (1988) reports that the mean lead time for an apparel order from a Far Eastern country was 34.7 weeks in 1988. Therefore, retailers have to schedule their orders much in advance of the season when the demand uncertainty is at high levels. The selling season for such fashion goods can be as short as two months and is at most six months, limiting replenishment opportunities during the season. While stock-outs of popular items are inevitable in this uncertain and inflexible environment, excess stock of slow moving, unpopular items have to be marked down. Fisher et. al (1994) note that in 1990, one quarter of all departm ent stores sales were mark-down sales. Frazier (1986) reports th at the apparel industry’s total revenue loss due to the supply and demand mismatches was as much as $25 billion in 1986. In an effort to cut down these costs, the U.S. textile and apparel industry initiated a series of business practices and technological innovations, called Quick Response. Quick Response aims to enhance the industry’s responsiveness by shortening lead times through improvements in production and information technology. A detailed discussion of operations and recent issues in the U.S. apparel industry supply chain is provided in Chapter 2. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. An alternative or complement to these efforts is to improve the quality of deci sions prior to and during the season. Once the ordering decision is made, the lack of replenishment opportunities reduces the problem to selling a fixed stock of items over a finite horizon. Since the fashion goods have little value when the season is over, the only tools left to the retailers are mark-down pricing and diverting stock to secondary outlets. Effective pricing decisions that incorporate competition and use early sales information may make a considerable impact on profits of these retailers working with thin margins. T he possibility of such secondary markets and price adjustments should also be reflected in initial ordering decisions. Many service companies w ith fixed capacities also face the problem of selling a fixed stock of items over a finite horizon. Among them are airlines selling seats for a particular flight and hotels renting rooms for a particular night. The type of aircraft for a specific flight and the number of rooms in a hotel is fixed for the sales horizon. These companies would first differentiate the product in a costless manner enabling price discrimination. Different fares are created for the very same seats or rooms by attaching different restrictions such as Saturday night stay or cancellation penalties. The objective is to be able to charge full fare to business travelers, while also attracting price-sensitive leisure travelers with discount fares. Typically, airlines and hotels have more than enough capacity for full fare customers, but not enough capacity for full fare and discount fare customers combined. Thus, the sales in the discount fare should be restricted in order not to deny any business 3 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. customers who are willing to pay higher fares. Airlines and hotels allocate their capacity by opening and closing discount fares during the selling season based on observed sales, remaining capacity and tim e-to-go. An early closing will lead to empty seats or rooms, while a late closing will result in lost sales in full fare classes. The tim ing of such decisions is im portant in defining the success of the service. Known as Revenue Management, such efforts have proven to be very successful in enhancing the profitability of many service companies. Sm ith et. al (1992) reports that American Airlines increased its revenues by $ 1.4 billion over a three year period. Again, further increases are possible with decision tools that take into account similar services offered by competing companies or additional demand information from observed sales. We note the similarity between the pricing decisions in fashion retailing and the allocation decisions in service industries with multiple demand classes differentiated with price. We argue th at charging a particular price is equivalent to accepting the requests of all customers in the demand classes with prices higher than this particular price and rejecting all customers in the demand classes with prices lower than this particular price. Thus, opening and closing fares in services can in fact be described by simple price adjustments. Airlines are now using bid price controls for accepting or rejecting customers. Bid price controls prove to be effective, particularly when one considers the fact that each flight may consist of multiple legs. In such a situation, a seat for a particular flight is sold, only if the fare exceeds the sum of 4 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. the bid prices of legs in the flight. Fashion retailing and service industries differ in the way the willingness to pay changes during the season. Customer’s willingness to pay tend to go down over time in fashion retailing, but build up in services as the end of the season gets closer. Fashion retailers respond with mark-downs, while service companies respond with m ark-ups (closing discount fares) over the horizon. This dissertation presents three models that deal with procurement-production and pricing-allocation decisions of a company selling a stock of items over a finite horizon. The models are applicable in the above contexts and in others including the pricing of products with short life cycles such as high technology goods and allocation of limited manufacturing capacity or parts inventory to orders that can generate different revenues. Basically, we focus on single period models reflecting the short term inflexibility in supply and capacity. Thus, there are no resupply or capacity expansion opportunities during the sales horizon. We assume that the demand is price sensitive and once the season starts, the only tool left to the company is managing demand through price adjustm ents. We also assume that the goods or services of concern are perishable, i.e., they have little or no value when the season ends. Each model focuses on an im portant and previously unexplored aspect of the problem described above. However, we note that the models can constitute the parts of a hierarchical solution methodology for companies selling a stock of goods or services over a finite horizon. Our first model studies the stocking decisions in 5 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. this context. In the existence of different market segments demanding the same product, this model can be used to optimize the starting stock level or capacity of a single company. Assuming that the stocking and capacity decisions are al ready made, our second model studies the dynamics of price adjustments during the horizon considering observed sales, remaining inventory, and remaining time in the horizon. Distinct from the existing pricing models in the literature, this model incorporates learning from observed early sales. O ur third model refines the price adjustments in the second model to account for the actions of competing companies. The demand is assumed to be stochastic in the first two models, while it is assumed to be deterministic in the third model. 1.1 A Model for Ordering Decisions We first note th at ordering and capacity decisions of companies in this context should take into account the differences in customers1 valuation of the product. While some customers of fashion goods are willing to pay a premium for earlier availability, some others are willing to wait until the merchandise is marked down at the end of the season. Similarly, while business travelers are willing to pay a premium for flexibility, leisure travelers are willing to face a lot of restrictions for a discounted fare. Ideally, not only should the price sensitivity of customers or the m anner in which customers are segmented, but also the possible price changes or possible allocation changes be reflected in ordering and capacity decisions. Our 6 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. first study deals with ordering decisions of companies facing demand from multiple demand classes differentiated with price. We assume th at the demands are realized sequentially over time. Assuming stochastic demand, the problem is an extension of the well known single item newsboy problem. We analyze two cases where the prices are decreasing and increasing over time; the former case applies to fashion retailing, while the latter to service companies. In the decreasing price case, we find the optimal order quantity to maximize the expected profit with independent multiple demands. We show numerically that using optim al solutions (which are easily computable) may have huge im pact on revenues of companies th at otherwise have to use heuristics such as aggregating the demand and ordering as if there is a single demand class, or treating each demand class independently and ordering aggregated order quantities. In the increasing price case, we analyze a two-demand class model in which a fraction of the unsatisfied lower fare demand diverts to the high fare class, thus causing dependent sales. This helps us to incorporate travelers who may be willing to pay the full fare, if their discount fare requests are rejected. We follow a policy of protecting the sales in the higher fare class by lim iting the sales in the lower fare class. We derive both the fare allocation limit and the initial capacity, and discuss managerial implications. For both models, we give bounds on the optim al order quantity. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.2 A Pricing Model with Demand Learning Once the ordering or the capacity decision is made, the company can only manage demand to maximize its revenues. Demand management is usually through price adjustm ents considering observed sales, stock levels, remaining tim e in the season, com petitors’ actions and uncertainty in the environment. Our second study con siders an im portant aspect of the pricing problem: demand learning. We note th at a significant amount of demand uncertainty can be resolved using the early sales information. We use a Bayesian approach to update parameters of the demand process. A periodic model and a multiplicative dem and function are used to sum marize sales information and pricing history in an efficient way. This summary is used periodically to update the demand distribution in future periods. The pricing problem is modeled using a stochastic dynamic program which could be efficiently solved. We obtain structural properties of the optim al solution such as whether higher sales in earlier periods always lead to higher prices in future periods. We conduct computational studies to show the impact of demand learning on expected revenues. Additionally, we study how the accuracy and the degree of uncertainty of the initial demand estimates, the starting stock levels and the price sensitivity of the customers affect the expected revenues. Finally, we use our model to explore the possible benefits and trade-offs of having the possibility to re-order during the season. This helps us to understand the circumstances under which a flexible supply chain should be preferred over a lower cost supply chain for perishable products. 8 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.3 A Pricing Model with Competition Our third model studies the impact of competition in pricing decisions of companies that are selling fixed stocks of similar products over a finite horizon. We model as a duopoly two firms selling their fixed stocks of similar items in a given period of time. Our model differs from existing single-firm inventory models in that we explicitly consider demand interactions through the pricing decisions by two competing firms. Both firms start with the same initial price, and each firm has the option to decrease its price once during the selling season. The demand for the product at one firm depends on the prices offered by both firms. The price levels and the corresponding demand rates are known in advance. The problem for each firm is to decide when to decrease its price (mark-down) in order to maximize its revenue. We find the unique equilibrium switching times for both firms. We show that the higher stocked firm will decrease its price first but each firm will sell off its inventory at the end of the horizon. We also analyze the effect of demand interaction on price switching decisions. As the demand for one firm becomes more dependent on the price offered by the other firm, the higher stocked firm will tend to lower its price later, while the lower stocked firm will tend to lower price its price earlier, than they would without competition. Furthermore, with increased demand interaction, the revenue of the higher stocked firm will increase while the revenue of the lower stocked firm will decrease. Our results hold in the context of a linear demand model where the demand rates can be non-symmetric. We then extend our model to the mark-up 9 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. problem in which either company has one opportunity to increase its price to a preset level. We apply the m ark-up model to a discount fare allocation problem for airlines. This thesis is organized as follows. Chapter 2 reviews the U.S. apparel industry supply chain. Chapter 3 presents our first model that studies ordering decisions. Our pricing models with demand learning and competition are presented in Chapters 4 and 5. Each section independently reviews related literature. Conclusions follow in Chapter 6. 10 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 2 The Apparel Industry Our models are directly m otivated by the apparel industry, although they may be also applicable in other industries offering perishable services and products such as airlines, hotels and electronics manufacturers and retailers. Below, we present a more detailed description of the apparel industry in general and the results of our interviews with an apparel m anufacturer and an apparel retailer to highlight the business environment and the needs for research. 2.1 U.S. Textile and Apparel Complex The textile and apparel supply chain consists of about 25,000 companies (excluding retailing channels) employing about 2 million people in roughly four segments. At the top of the chain, there are are fiber producers using either natural or “man- made” (synthetic) materials. Raw fiber is spun, woven or knitted into fabric by the second segment, usually called textile mills. The third segment of the supply chain 11 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. is the appaxel manufacturers or manufacturers of industrial textile products. The final segment is the retailers which makes the apparel and other textile products available to consumers. Below, we briefly outline each segment. The discussion for this section is based on U.S. International Trade Commission (1999). Brown and Rice (1998), Ostic (1997), Hammond and Kelly (1991) and National Academy of Engineering (1983). 2.1.1 F ib er and Yarn P ro d u ctio n Fibers are usually classified into two groups: natural and man-made. Natural fibers include plant fibers such as cotton, linen, jute and cellulosic fibers and animal fibers such as wool th at are produced by agricultural firms. Agricultural firms are scat tered all around the U.S. and are usually small in size. Synthetic fibers include nylon, polyester and acrylic. Synthetic fiber production usually requires consider able capital and knowledge, and thus synthetic fiber producers, such as DuPont and Celanese, are large and sophisticated and very few in number (about 50 in U.S.). Top ten U.S. producers share 90% of the U.S. synthetic fiber production. Natural and synthetic fibers of short lengths are converted into yam by spinners, throwsters and texturizers. This conversion is also a capital intensive manufacturing process which is considerably different for each type of fiber. Blending different fibers may need further sophistication. A typical fiber plant can manufacture about 1 million pounds of fiber per day, supporting approximately 100 fabric plants. 12 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.1.2 Fabric P rod u ction This segment of the chain transforms the yam into fabric by weaving, knitting or a non-woven process. In weaving, yarns are interlaced lengthwise and widthwise at right angles. Yarn may be woven by a simple procedure to produce "greige” goods and then dyed for a specific fabric. Alternatively, dyed yarns may be woven. In knitting, yarn is inter-looped by latched and spring needles. The process may output rolls of knitted fabric or may specialize in a particular apparel such as sweaters. Non-woven processes involve compression and interlocking fibers by mechanical, thermal, chemical or fluid methods. This segment of the chain consists of about 5,000 companies in a dualistic nature. Thousands of small and medium companies are engaged in production of limited range of fabrics and a small number of huge firms such as Burlington and J.P. Stevens produce a wide range of fabrics. On average, a fabric plant manufactures about 1 million square yards of fabric every week supporting approximately 4 apparel manufacturers. 2.1.3 A pparel M anufacture More manufacturing companies are involved with this part of the supply chain. Apparel manufacturing starts with the design of the garment to be made. Patterns are made from the design which is then used to cut the fabric. The cut fabric is usually assembled into garments, tagged and shipped. The apparel segment is the most labor-intensive and fragmented segment of the supply chain. Capital and 13 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. knowledge requirements are not significant m aking it attractive for new entries. There are currently about 18,000 companies in this segment. The firms in the women’s and girl’s categories tend to be smaller, while firms in the less fashion sensitive men’s and boy’s clothing, knit-wear and underwear categories can utilize economies of scale and tend to be larger in size. Average number of employees in men’s apparel companies is about 122, compared to only 35 in women’s apparel companies. Apparel companies usually specialize in narrower product categories and hardly produce garm ents of both genders. Traditional manufacturers (and integrated knitting mills for knit-wear) are en gaged in all phases of apparel manufacturing: product design, material sourcing, production of apparel in house and marketing of the finished goods. Jobbers per form all of these activities except for production of garments. The production is contracted out to contractors either in U.S. or overseas. Contractors are engaged in manufacturing of garments and are not responsible for sourcing raw m aterial or the design and marketing of these garments. T he distinction between m anufactur ers and contractors is not very clear as manufacturers may contract out their work or perform contract work for other manufacturers, and contractors sometimes may start their own private labels. Some U.S. manufacturers cut fabrics in U.S. and send cuts to a low wage country to be assembled. The assembled garments are then shipped back to U.S. for finishing. Manufacturers pay tariff only on the value added outside the U.S. with this type of production, which is often called 807 sourcing. A 14 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. profitable choice for such production sharing is Caribbean Basin region countries be cause of their proximity to U.S. market. Mexico is further advantaged with reduced tariffs under NAFTA and more control over delivery with ground transportation options. 2.1.4 R etail Apparel products are made available to consumers in a variety of retail channels. Specialty stores, such as The Limited and The Gap, offer a limited range of ap parel products and related accessories specializing in a particular m arket segment. Specialty stores accounted for 22 percent of all retail sales in 1997. Another 19 percent of the apparel sales took place in discounters or mass merchandisers such as W al-mart, Kmart and Target. These retailers offer a variety of hard and soft goods in addition to apparel using an "everyday low prices" strategy. Departm ent stores, such as Nordstrom and Bloomingdales’ offer a large number of national brands in both hard and soft goods categories. The market share of these stores in apparel amounts 19 percent. Apparel chains offering a wider range of products such as J.C. Penney and Sears command a market share of 17 percent. Off-price stores, such as Marshalls and T.J. Maxx buy excess stock of designer-label and branded ap parel from manufacturers and other retailers and are able to offer considerably low prices but with incomplete assortments. The m arket share of these stores is about 1 5 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7 percent. The remaining 17 percent of the apparel sales is shared by mail order companies, factory outlets and other retail channels. 2.2 Apparel Manufacture and Retail Operations This section aims to give an overview of important issues and decision making in the last two segments of the textile and apparel supply chain. We analyze the manufac turing and retailing operations separately, although vertical integration taking place in the recent years makes it difficult to distinguish the retailers from manufacturers (see Section 2.3.1). 2.2.1 M anufacturing O p eration s Domestic apparel market can be divided into three different categories (U.S. Office of Technology Assessment 1987). • “Fashion” products, with a 10-week product life-approximately 35 percent of the market. • “Seasonal” products, with a 20-week product life-approximately 45 percent of the market. • “Basic” products, sold throughout the year-approximately 20 percent of the market. 16 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. The men’s and children’s merchandise usually fall into the basic category, while women’s merchandise dominate seasonal and fashion categories, showing the im portance of fashion and resulting frequent design changes in the women’s market. A similar categorization is made in Abernathy et. al (1995). The manufacturing companies usually specialize in narrower product categories. The type of product the company focuses on not only defines the manufacturing cycle and the intensity of the design in its operations, but also the manufacturing strategy as suggested by Fisher (1997). Companies manufacturing basic products can utilize larger batches and tend to be larger in size. Cost reduction is a priority for these companies. Com panies manufacturing fashion products have to live with smaller batches and tend to be smaller in size. Flexibility is the key to success for such companies (Taplin 1997). Companies’ involvement in apparel manufacturing vary. Traditional manufac turers are responsible for all phases of manufacturing. But most of the industry is organized in the form of jobbers and contractors; jobbers being responsible for the design, cutting and marketing and contractors being responsible for the sewing and assembly. Design is either completed in-house or commissioned to smaller design com panies. The first step in design is analyzing the consumer which the company is targeting. The design process is influenced by the works of other designers presented in collections in cities like Paris, Milan and New York, or trade shows of the earlier 17 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. seasons. Fashion prediction consultancies and magazines may also be an important input for the design efforts (Bohdanowicz and Clamp 1994). More im portant is the feedback gained from the sales of the sim ilar products that are developed earlier, which requires a collaboration between the retailers and the manufacturer. Usually, prototype garments are made for internal decision making. These tasks take consid erable amount of time. The design process usually starts while the previous year's garments are still retailed. The design process at Sport Obermeyer, a m ajor ski- wear manufacturer, starts as early as 19 m onths before the season and takes up to 8 months (Hammond and Raman 1994). Responsiveness may be greatly enhanced by reducing the tim e required for design development. Computer-aided design (CAD) systems are recently being used for such reduction efforts. Besides reductions in the actual design time, CAD systems also reduce the time for making the pattern and enable electronic storage of the design which makes later modifications and transmissions easy (Blackburn 1991). Levi Straus & : Co., the San Francisco based jeans-wear company, recently reduced its product development time for its Red Tab line from 53 weeks to 36 weeks (Women W ear Daily 1999). The next step after the design in the fashion calendar is the production of sam ples. The samples are shown to the retailers by market representatives at major trade shows or at the retailer sites. Some m ajor customers may be also invited for on-site exhibitions. Most small companies such as Paugal Industries (see Section 2.4) accumulate all of their orders and then proceed with the production. As a 18 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. result of capacity constraints in peak periods and recent trend of retailers willing to order much closer to and even during the selling season, some other companies have to commit themselves to some or all of their production volume prior to gathering all their actual orders. Sport Obermeyer’s initial production order before any order collection is as much as half of its annual production (Hammond and Ram an 1994). A strategic question for the apparel producers at this point is where to carry out the manufacturing operations. Some companies operate their own facilities for manufacturing. Some others use contractors. The trade-offs for this decision are typical of any m anufacturing operation. Some of them are: more control over quality and time, fewer communication problems with in-house production; less capital investment and more flexibility with out-sourcing (Brown and Rice 1998, Page 3). W hether this decision be out-sourcing or in-house production, another important issue is the venue of the production. Now the m ajor trade-off is between the responsiveness and cost efficiency. Increasing number of apparel producers are choosing lower cost off-shore production in Asia and Latin America. Among them are m ajor apparel m anufacturers such as VF Corp., Fruit of the Loom and Oxford Industries th at plan to produce majority of their garments offshore (US International Trade Commission 1998). K urt Salmon Associates reports that some companies are pursuing blended sourcing strategies (Apparel Industry Magazine 1997). Domestic production is used for fashion items, while basic products are produced in off-shore facilities. The report also includes an example of an apparel manufacturer which uses 19 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. three contractors for the very same product. A low cost high cycle time (90 days) Far East contractor, a medium cost medium cycle time (21 days) Latin American contractor and a high cost low cycle time (3-5 days) domestic contractor. The new designs are used to make patterns by which the fabric is cut. An efficient layout of the patterns on fabric is crucial in reducing the wasted material. CAD systems may be used for pattern layout and be further integrated to computer-aided cutting systems (Abernathy et. al 1995). The later stages of apparel manufacturing are quite labor intensive as they are not appropriate for any kind of automation. W hether it is in a large or small manufacturing facility, garment is usually assembled using the progressive bundle system (PBS). In PBS, or batch production with its general name, the work is delivered to individual work stations from the cutting room in bundles. Sewing machine operators then systematically process them in batches. The supervisors direct and balance the line activities and check quality. The result of such a system is of course large work-in-process inventories and minimal flexibility (Taplin 1997). In order to move the apparel faster through the successive sewing operations, some apparel producers began to use Unit Production Systems (UPS) which reduce the buffer sizes between the operations. Another way is to use modular assembly systems which allow a small group of sewing operators to assemble the entire garment (Abernathy et. al 1995, Blackburn 1991). Assembled garments are labeled, packaged and usually shipped to a warehouse. The garments are then shipped to the retailers’ warehouses. In an effort to compress 20 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. the time from placement of the retailer order to the consumer’s purchase of the ap parel, several practices are gaining popularity. First, there are increased automation and use of electronic processing in the warehouses of both manufacturers and retail ers. Manufacturers are assuming responsibility in many functions, once considered to be part of retailers’ services. Among them are labeling products with retailer’s price tags, preparing them on hangers and shipping them directly to stores. 2.2.2 R etail O p eration s A retailing organization is responsible for the following tasks: • buying merchandise for sale in stores • operating stores for the selling of merchandise • operating warehouses and trucks for receiving, storage and transshipment of merchandise in addition to the usual tasks such as finance, marketing and personnel management. Most large retailers are organized in a way that these three tasks are separated: a general merchandise manager responsible for buying, a manager of stores responsible for store operations and an operations manager responsible for logistics (Bell 1994). It should be noted that a close contact between the buying and sales organizations is required to better understand the point of view of the customers and merchandise assortments accordingly. 21 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Mass merchandisers, departm ent stores and specialty stores are the m ajor out lets for apparel. Merchandising practices vary depending on the type of outlet and the fashion content of the apparel. Large organizations manifest different levels of centralization in their buying organizations. Competitive deals with the vendors are possible with consolidated buying. However, a decentralized buying better addresses the different tastes and different size needs of the customers in different geographic areas. Nordstrom, for example, used this strategy to expand its operations in 1970s (Parpia 1995). Macy’s, on the other hand, has a centralized buying organization for all its stores in the Western U.S. (see the section on departm ent store buying). Federated Department Stores, which now owns Macy’s along with other departm ent stores such as Bloomingdale’s, Stem ’s and Goldsmith’ s, is trying to further consol idate its buying through its merchandising organization Federated Merchandising (Chain Store Age 1992). J.C. Penney uses a mixed strategy. National buyers work at the wholesale level and provide a huge assortment for the stores. Using a satellite broadcasting system, the store buyers order from this assortment finalizing the mer chandise plan (Blasberg and Wylie 1996). However, the inability to order on spot and lags encountered during the store orders result in some inefficiency and J.C. Pen ney is revising its system to allow for more direct centralized buying (Stores 1998). J.C. Penny plans to consolidate 60% of the purchasing at its headquarters, leaving remaining 40% of its merchandise to be bought by individual stores (Business Week 1998). 22 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Merchandising activities start as early as the end of a comparable season in the previous year. The planning process at J.C. Penney starts with the estimation of individual store sales for the next year (Blasberg and Wylie 1998). Initial whole sale purchase quantities are then established by buyers. Fashion direction for the season is developed based on a variety of sources including the past records of the organization, competition, market research, fashion and trade shows and magazines (Bohdanowicz and Clamp 1994, page 95). About nine months before the start of the season buyers shop at major markets and start developing their merchandise plan. Five months before the season buyers visit the markets and make their preliminary orders with the vendors. The contact with the vendors often takes place at trade show's. Prior personal contacts and recommendations also play an important role. Most larger retailers have strategic alliances vdth their vendors and buy a huge va riety of products in large quantities (Chain Store Age 1996). Some buyers (e.g., a Macy’s buyer) are only responsible for buying merchandise from one vendor (e.g., Liz Claiborne). The buyer’s decisions are controlled by a budget set by the merchandise man agers. The maximum amount of funds the buyer can allocate for new purchases 2 3 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. is often called o p e n -to -b u y . Open-to-buy (OTB) is calculated using the following formula: OTB = budgeted closing stock -+ - budgeted sales + budgeted reductions (mark-downs, thefts) — opening inventory — purchases already received — purchase orders placed but not yet received The budgeted components of OTB are derived before the start of the season from the corporate merchandising budget (first, dem and forecasts are used to determine budgeted sales, which is then used to calculate budgeted closing stock level which will maintain a specific inventory to sales ratio). During the season, the opening inventory is updated by the flow of merchandise that occurred since the start of the season. This updates the OTB figure which drives the new purchases, sales or re ductions (Goodwin 1992). The purpose of the system is to control the sales in order to keep the inventory levels in budgeted levels. Such a system has two potential problems. First, the calculation of OTB (which in effect determines the purchase quantities) uses only the point estim ate of demand (i.e., budgeted sales), ignoring the uncertain nature of the apparel industry. Second, most retailers do not update their budgeted sales (thus budgeted closing stocks) during the season. Therefore, especially when the pre-season forecast is conservative, service level deteriorates as new orders are placed only if OTB becomes available. W ith an empirical study, 24 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Goodwin verifies that OTB system constrains the performance of buyers and sug gests that it should incorporate the updates in demand forecasts. Goodwin also suggests that mark-downs should be based on sales activity rather than budgeted prior to season. At the start of the season, some buyers choose to spend all of their OTB. Some others choose to hold back some of their OTB for opportunistic buys after they know more about the popular styles, colors and fabrics of the current season. For example, suit buyers spend 80 percent up-front and keep the remaining 20 percent for on-season buys (Daily News Record 1993). A traditional exercise is to receive all bought merchandise before the season. However, recently, more and more retailers are using different windows of delivery through the season. This helps to maintain a fresh look of the store over the entire season. Suit retailers, for example, typically use two or three delivery window's (Daily News Record 1993). For J.C. Penney, preliminary orders constitute 50% to 75% of the anticipated total orders. The selections are reviewed by the stores and the merchandise com mitment is finalized after collecting individual store orders 2.5 months prior to the season. Above merchandising cycle is typical for nationally branded apparel. For companies that are marketing their own private labels, the merchandising activities are more complicated and may involve the coordination of manufacturing activities such as design and fabric sourcing. For such companies, buyers and merchandise managers have to work closely with brand managers responsible for the private label 25 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. apparel, in order to m aintain a profitable mix of branded-private label merchandise in their assortments. While this merchandising cycle is repeated for each season for fashion apparel, basic items are subject to longer life cycles and are mostly on autom atic replen ishment plans. EDI systems enabling these plans are gaining popularity as the vendors are compressing their cycle times by Quick Response systems. These plans are usually based on strategic alliances and are taking over the responsibilities of buyers. One such system is Liz Claiborne’s LizRim which has been a success for both retailers and the company itself (Blasberg 1997). Ideally, past sales data should be a major factor in buying and re-ordering de cisions. Recent advances make enormous amount of point-of-sales d ata available to buyers. However, this is not quite the fact, as the CEO of Federated’s Logistics and Operations division states “Where we have made little progress, ..., is in changing the way our buyers go to market and buy. I don’t see them using this data nearly as much as I expected.” In some departm ents, buyers are far away from efficient use of sales data in their merchandise selections, ending up with inventory turns less than once per year. This supports the old truism that top 20% of SKUs represent 80% of the business, while the bottom 28% of SKUs represent only 4% of the business. The result is a huge number of SKUs, most of them moving fairly slow. Macy’s Herald square store carries 3.5 million SKUs (Apparel Industry Magazine 1998). 26 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Apparel retailers usually employ cost based pricing techniques for the initial prices for their merchandise. Typically, the initial price is the cost of the product plus a percentage mark-on. This mark-on percentage is such that the revenue ob tained from the sales will be adequate to cover all expenses incurred in the business plus a reasonable profit. R ather than detailed item specific pricing based on ex pected sales activity, most retailers choose to follow company specific simple rules, or other retailers in the same category (i.e., department store, discount store and specialty store) offering similar merchandise. Mark-on percentages may also depend on the volume of the sales. As an example, custom printed and embroided sporting merchandisers are called to mark-on 100-150 percent for quantities of under two dozen pieces and 80-100 percent for quantities two to six dozen (Sporting Goods Business 1998). Also as a general rule, fashion items with higher risks and items with small volume command higher mark-ons (Bohdanowicz and Clamp 1994, page 110). Some retailers (usually discounters) try to group different styles around differ ent prices and charge the same price for the styles in the same group (price lining). Some retailers such as One Price Clothing Stores (Discount Merchandiser 1997) go as far as charging a single price for all of its merchandise (sin gular pricing). Overall, initial pricing is a part of retailer’s marketing strategy rather than micro-managed at the product level. In fact, retailers in the same category tend to follow similar pricing strategies (in the case of discount stores, price alone is the reason for catego rization) . Department stores have been known to charge high initial prices and offer 2 7 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. deep mark-downs later in the season. This is contrary to apparel specialty stores, offering fair prices throughout the season. According to W. J. Salmon, professor of retailing at the Harvard Business School, pricing policies of department stores which he refers to them as “usurious prices followed by illegitimate sales” , is one of the major reasons for department stores’ declining performance in early 1990s (Discount Merchandiser 1994). Realizing this, Dillard’s Department Stores began to practice every day low pricing or every day fair pricing (EDLP/ED FP) (Chain Store Age 1994). Most retailers change the prices of their merchandise during the season usually by offering discounts. Several factors distinguish apparel industry from other industries in pricing decisions. First, value of fashion merchandise deteriorates in an enormous speed. Left-over merchandise would have little or no value at the end of the season. Second, there is a considerable amount of uncertainty involved in consumer taste, hence in demand for a particular fashion merchandise. Part or all of this uncertainty can be resolved as the retailer starts to observe the sales after the start of season. Finally, retail space is highly competitive in fashion industry. Ideally, retailer should consider all of these factors in its sales decisions, maximizing its revenues over the entire season, preferably selling all inventory by the end of the season to allocate the entire retail space for fresh merchandise of the new season. 2 8 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. The sales fall into three categories: pre-season sales, within-season promotional sales and end-of-season clearance sales (Pashigian 1995). In some merchandise cat egories, retailers charge introductory low prices for a short period of time before the start of season. Resulting increased store traffic allows the retailer to gather infor m ation about the popular colors, styles and garm ents early enough for appropriate replenishments within season. W ithin-season promotional sales, on the other hand, use discounts on particular merchandise to increase store traffic for improving sales not only on discounted items, but also on other slow moving items. The most common form of sales is end-of-season clearance sales aimed to liq uidate all stocks before the end of season. Clearance sales are comparatively more tactical in nature and should be based on detailed analysis of individual item ’s sales activity. The timing and depth of these mark-downs are crucial decisions as early and deep mark-downs may result in revenue losses, and late and not sufficiently deep mark-dowms may result in obsolete inventory at the end of the season. Fac tors such as customers substituting regular priced items by marked down items (cannibilization) should also be considered. Despite the importance of this difficult problem, mark-down decisions in practice do not follow any scientific rule. This is again in spite of the fact th at required point-of-sales data is easily available to decision makers. Mark-downs are usually subject to buyer’s budget, limiting the responsiveness of these decisions to sales activity (Chain Store Age 1999, Goodwin 1992, Women Wear Daily 1999). For some companies, mark-downs are completely 29 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. sales driven and autom ated. In Filene’s Basement Store, all merchandise not sold within 14 days is marked down by 25 percent; remaining merchandise after 21 days is marked dowm by additional 25 percent and remaining merchandise after 28 days is marked down by another additional 25 percent. Finally, if there are still left-over merchandise after 35 days, it is given to the charity (Stores 1994). While this policy is easy to implement, it is questionable th a t it gives the maximum profit across all merchandise categories. A more rigorous analysis should include a probabilistic treatm ent of demand and allow dynamic pricing based on remaining inventory and time before the end of the season. Recently, fashion industry has seen some efforts to implement scientific methods for mark-down decisions. In one of these efforts, Mantrala and Tandon developed a model which assists the departm ent store buyer in selecting the optimal price from a set of permissible price levels for each period of the season (Stores 1994). The probabilistic periodic pricing model is implemented via a computer program called MARK. Technology Strategy Inc., a software and consulting firm in Cambridge, Massachusetts, developed several software solutions to assist apparel retailers in their mark-down decisions (Stores 1998, The Boston Globe 1998). One of them, named Endgame, uses massive historical d ata to determine the timing and amount of mark-downs which maximizes Gross Profit. TSI’s solutions are already in place in retailers like Gymboree. Dayton Hudson, the company that owns retailers Day ton’s, Hudson’s, Target and Mervyn’s California, works with Santa Clara University 30 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. researchers to come up with a m ark-down strategy for their fashion items (Chain Store Age 1999, Women Wear Daily 1999). Given the time of mark-down, the new system determines the depth of m ark-down so that the merchandise stock is cleared by a specific date. The system plans for a single mark-down as Dale Achabal, Santa Clara University professor argues: “there is no combination of two mark-downs that is more profitable than one mark-down". These recent efforts are pioneers in apparel industry as a business consultant working for Dayton Hudson states: “For many of our merchants, this is the first exposure they've had to intelligent systems” . 2.3 Trends in Apparel Manufacture and Retail 2.3.1 R etail C onsolidation, V ertical In tegration and E m erge o f P rivate L abels The retailing space per capita increased from 8 sq. ft. to 19 sq. ft. in the last 20 years reflecting the increased demand created by baby boomers. However, aging of the same population, changing consumer priorities and introduction of non-traditional retailing outlets decreased the consumer interest in many sub-sectors of the U.S. retailing industry. The over-stored U.S. retailing industry in general has faced con siderable number of bankruptcies and acquisitions in the recent years. As a result, the total U.S. retail sales are concentrated in a few m ajor retail companies. The top three retailers, Wal-Mart, Km art and Sears account for $ 175 billion annual sales 31 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. in 1997, almost 8% of all sales in the industry (Leamon 1998). The scene is not very different for apparel retailing. The apparel sales are far away from sustaining a productive use of retail space as a result of consumers preference for comfort over fashion and a casual work place. As a result, apparel and accessory stores have ex perienced very high failure rates, highest among all retail sub-sectors (Standard and Poors 1998). Domestic apparel market is dominated by 12 m ajor retail groups, now representing almost two thirds of the sales (U.S. Department of Commerce 1999). Retail consolidation shifted the industry power from apparel manufacturers to large and powerful retailers. Fewer and stronger retail firms are in a position to man date favorable terms in their contracts with manufacturers involving price, service, delivery' and product diversification and differentiation (U.S. International Trade Commission 1995). Mass retailers are willing to order closer to the actual sales and shrink their inventories by continuous supplier replenishment throughout their sell ing seasons. The result is higher inventory risk assumed by manufacturers. Several services once considered to be part of retailer operations, such as, pre-ticketing the retailer’s price tags and storing the apparel on hangers are now part of m anufactur ers’ operations (U.S. International Trade Commission 1998). The financial penalties in case of errors and failure to meet vendor compliance standards further increase the costs of manufacturers. As an example of these standards, since 1994, Sears expects all of its vendors to use EDI and bar-code shipping labels; ship merchan dise as close to floor ready as possible meeting the company’s Floor Ready Product 32 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. (FRP) standards (Apparel Industry Magazine 1998). These and other requirements are difficult to satisfy for sm all and medium sized companies. Consolidation also helped the retailers to reach the economies of scale for par ticipating in manufacturing activities. Mass merchandisers, departm ent stores, spe cialty retailers and up-scale retailers are now offering private labels with competitive prices. For example, Sears sells casual apparel under its private label, Canyon Rivers Blues, as well as national branded apparel such as Levi’s and Wrangler. Besides cost reductions through the elimination of intermediaries, retailers with manufacturing operations are able to respond quicker to changes in consumer demand and have a better control on the quality of products that they sell. Uniqueness of private- label apparel also helped to a ttrac t consumers who have been complaining about the sameness of the merchandise in different retail outlets. Retailers can also exploit their closeness to the consumers in the design and m arketing of their private labels. Private labels especially helped department stores to regain the market share they lost over the past several years. Most of the retailers with private labels are likely to source their private-label apparel overseas, eliminating the need for U.S. agents to develop marketing expertise in foreign markets and to improve their responsiveness to consumer demands. Im ports of private-label apparel accounts for 15 percent of U.S. apparel market in 1997, up from 12 percent in 1995 (US International Trade Commission 1998). 3 3 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. The industry also experiences a forward vertical integration of large manufactur ers. In an effort to increase efficiency, eliminate intermediary and better understand the consumer needs, increasing number of textile mills and apparel firms are in volved in retailing. Some of these companies only operate factory outlets where they dispose their excess or second-quality merchandise without damaging their brand image with merchandise sold in the off-price retailers. 2.3.2 Im p ort P en etration and P rod u ctio n Sharing Limited capital requirements and labor intensity of the apparel and other textile products manufacturing have made the industry a primary industry in low waged underdeveloped and developing countries starting early 1960s. Changes in trade regulations, advances in transportation and communication helped to increase the global trade in apparel. As a result, increasing portion of apparel production is mov ing to less developed countries and apparel industries in developed countries are ex periencing increasing import penetration in almost all apparel categories. The trend is similar and substantial amount of restructuring is taking place in almost ail de veloped countries including U.K., Japan, Germany and Italy (Taplin and W interton 1997). U.S. apparel industry was not immune to such globalization. Apparel imports reached $50.3 billion in 1997, up from $27.7 billion in 1991. The imported apparel now constitutes more than half of the $ 100 billion industry (U.S. Department of 34 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Commerce 1999). With reduced trade regulation with the North American Trade Agreement and progressive phase-out of M ulti-Fiber Agreement, we expect to see the same increasing trend in all apparel categories. Traditional suppliers of imported material to the U.S.: Taiwan, Hong Kong and Korea are losing their market share in the U.S. market since the beginning of 1990s, as the companies are seeking even lower cost production in countries such as China, India and Bangladesh. A relatively new trend is production sharing in the Central and South Ameri can countries. Chapter 98 of the Harmonized Tariff Schedule of the United States (formerly item 807 of the tariff schedule) permits cut fabric to be shipped to low waged countries and returned back to the United States with duty applied only to the value added part of the production. Such imports accounted for 26 percent of all apparel imports in 1997, up from 9 percent in 1990. Under the NAFTA agree ment, there are no duties for apparel cut in the U.S. and assembled in Mexico. This and proximity to the U.S. m arkets further advantaged Mexico making it the second largest supplier of U.S. im ports following China. Generally, U.S. apparel im ports concentrate on basic styles and fabrics for which design changes are minimal from one season to the other. Market share of imported apparel is especially high for all men’s and boy’s clothing, knit-wear, and women’s coats and jackets (U.S. Census Bureau 1999a). U.S. apparel imports under Chapter 98 of IiTSUS from Central and South American countries are concentrated in fewer products, with high, but unskilled labor content. The m ajor apparel categories 35 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. that are manufactured through production sharing operations include trousers and shorts, shirts and blouses, foundation garments, underwear, and coat and jackets (US International Trade Commission 1995). While U.S. manufacturers are mostly importing from Central and South American countries through production shar ing operations, U.S. retailers tend to import the full package from Asian countries since they do not have the expertise to coordinate manufacturing processes (US International Trade Commission 1998). Retailers and manufacturers are still restructuring themselves to increase their foreign sourcing. For example, V.F. Corporation, producer of Wrangler and Lee jeans, Vanity Fair intim ate apparel, source 50 percent of its sales globally in 1998, and planning to increase this to 80 percent of sales in the near future. Fruit of the Loom, a leading producer of underwear and basic casual family apparel plans to have 95 percent of its production in the western hemisphere (US International Trade Commission 1998). 2.3.3 Q uick R esp o n se S ystem s Consolidation, vertical integration and low cost imports in the apparel industry began to eliminate the weaker players in the apparel manufacturing industry. Al though there are not significant barriers to enter and expand in the industry with low capital requirements and use of contractors, remaining competitive is becoming extremely difficult. The failure rate for apparel and other textile manufacturing 36 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. businesses was 136 out of 10,000 in 1997, the highest rate among all other manu facturing sub-sectors. 364 businesses th a t failed in 1997 had about § 1 billion of liabilities (The Dun and Bradstreet Corp 1999). The total number of employees in apparel and other textile products manufacturing dropped to 840,000 in 1997, down from 985,000 in 1992 (U.S. Census Bureau 1999a). In order to compete with foreign manufacturers that are able to meet the increas ing demands of big and powerful retailers, the industry initiated a series of techno logical innovations and business practices called Quick Response in 1985 (Hammond and Kelly 1991). Quick Response intends to tie the apparel and textile manufactur ing and retailing operations to provide the flexibility to quickly respond to consumer needs in a volatile industry. In 1986, K urt Salmon Associates estimated that the inefficiencies in the supply chain cost the industry about 24% of net retail apparel sales annually or $ 25 billion in the form of forced mark-downs, excess inventory and stock-outs (Frazier 1986). As a result of various process changes that link the retailing and manufacturing operations, responsiveness can be used to effectively substitute for fashion sense, forecasting ability and/or inventory required for oper ating under uncertainty (Richardson 1996). Ideally, a quick response system would enable the m anufacturer to adjust the production of different styles, colors and sizes in response to retail sales during the season. The immediate objective is to reduce 3 7 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. the cycle times and be able to produce as close to the consumer need as possi ble decreasing risks and inventories at each stage of manufacturing and retailing operations. A number of technologies are used to help to reduce the cycle tim es in manu facturing and retailing. CAD/CAM equipment are used to reduce the cycle time from design to production. Point-of-sale (POS) scanners at the checkout counters read the bar code attached to each item and record the merchandise sales by its price, style, color and size. Electronic D ata Interchange (EDI) systems then can be used to transfer this real time information to different stages of the supply chain facilitating automatic reordering or even allowing the manufacturer to manage its retailers’ inventories. A successful quick response implementation also depends on substantial information sharing and coordination between the m anufacturer and the retailer. Hunter (1990) lists the principal practices required by a firm operating under an ideal Quick Response system under these different categories: • Information Technology — Uniform product codes — Point-of-sales data tracking — Electronic data interchange — Continuous updating of consumer demand 38 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. — Frequent orders — Computer automated product design — Infrastructure information network • Logistics — Frequent, small lot shipments — Just-in-time shipping policies — Pre-ticketing and drop shipment • Manufacturing — Flexible, short-run processing — High-speed manufacturing — Automated material handling — Rigorous quality control — Modular production concepts Abernathy et. al (1995) reports that a Quick Response retailer should be able meet the following standards • Track sales in individual styles, colors and sizes on a store-level and real-time basis. • Replenish products at the store quickly. 39 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. • Hold minimal excess inventories at the store level beyond w hat is on the sales floor. • Proride logistical support for the above practices • Create manufacturer performance standards for replenishable products, speci fying standards for order-to-replenishment lead times, shipment accuracy, and delivery information, and setting out penalties for noncompliance. These standards will then establish the following standards for the Quick Re sponse manufacturers • Label units, track sales, and respond in real time to product orders at specified style, color and size levels. • Exchange electronic information concerning current sales and related informa tion with retailers. • Provide goods to retailer distribution centers in ways th at allow good to be moved efficiently to stores for distribution (for example, boxes marked with computer-scannable symbols concerning contents; shipments of products ready for display in retail stores. W hile these standards are currently met mostly by increased inventory levels of finished goods, further manufacturing responsiveness may be achieved by establish ing or improving the following internal practices at the manufacturer level 40 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. • The ability to forecast and plan future production needs based on sales data provided by the retailer. • Distribution centers capable of providing logistical support to efficiently pro cess shipments to multiple retailers. • Manufacturing practices adapted to producing a variety of styles, sizes, and colors under shorter lead-time requirements. • Agreement with key suppliers to provide shorter procurement lead times and smaller minimum orders for textiles and other suppliers to accommodate chang ing demand requirements. Abernathy et. al (1995) reports th at between 1988 and 1992 there is a sub stantial growth in the number of retailers requiring suppliers to meet their Quick Response related standards such as bar-coding, EDI and autom ated distribution centers. More and more manufacturers are now changing their internal practices related to manufacturing and performing activities such as bar-coding, preparing the merchandise for selling and distribution to retail outlets that are not once con sidered the responsibilities of manufacturers. Kurt Salmon Associates notes the 10 years of Quick Response implementation a major success saving $13 billion through a combination of excess stocks being removed from the system and wider and more accurate assortments easily available (Bobbin 1997b). 4 1 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 .3 .4 Supplier selection: off-shore versus d o m estic sourcing Retailers and manufacturers consider a number of factors when deciding where to supply their merchandise. First group of factors includes the production or purchase costs, inventory storage costs and transportation costs. These are related to the efficiency of the supply chain. Fisher (1997) classifies them to be the physical costs of the supply chain. The other group is related to the responsiveness of the supply chain: how accurate and fast supply is able to match demand. If supply exceeds demand, the merchandise has to be marked down, and sold at a price possibly less than the cost. If supply is less than demand, the company loses sales opportunities and dissatisfies its customers. Fisher calls resulting costs m arket m ediation costs. For products that satisfy basic needs, with long life cycles, and thus stable demand, (fu n ctio n a l products as called by Fisher) physical costs should be the focus. For products with high fashion content, short life cycles and thus hard to predict demand (in n o v a tiv e products as called by Fisher), companies should rather try to minimize market mediation costs. Apparel market consists of many products with varying levels of fashion content (innovation or functionality). Fashion content not only defines the season length, but also affects where retailers or manufacturers source their merchandise. Basic ap parel merchandise generally have longer selling seasons and physical costs are likely to represent a major part of potential total costs. Like most labor intensive-low 42 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. technology industries in U.S., a natural choice of production venue for basic prod ucts is developing or underdeveloped countries where wages are substantially lower. Fashion products, on the other hand, have generally shorter life cycles and market mediation costs play a m ajor role. As lead times are still long for apparel im ports, it is only domestic manufacturing which can provide the responsiveness demanded by apparel retailers. An individual retailer’s choice may be to source its particular mer chandise from overseas or from a m anufacturer in U.S., whichever minimizes its total costs (physical and market mediation). For a particular merchandise category, these individual decisions may be aggregated in one statistic: market share of im ports (or domestic manufacturing). Table 2.1 lists the market share of imports in different apparel categories and material in 1997. In almost all categories, imports capture more than half of the market share. Notable exceptions are hosiery, women’s dresses, suits and swim-wear. It may be surprising to see that market share of im ports in hosiery is only 11% as consumer demand is fairly predictable in hosiery, making it a solid candidate for import penetration. However, hosiery manufacturing is highly autom ated and labor content is fairly limited which eliminates the cost advantage of imports. In fact, in 1997, average im port price (cost-t-import+freight) per dozen of hosiery was $10.27 (The Hosiery Association 1999), while average value of domestic shipments was only 14% more: $11.72 per dozen (value data compiled from U.S. Census Bureau 1999b-c, quantity data taken from The Hosiery Association 1999). 4 3 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 2.1: Market Share of Imports in A pparel in 1997 C otton (%) M aterial M M F (%) WooI(%) Total (%) Women’s and girls’ apparel Brassieres and other body support garments 74.0 45.7 n /a 50.2 Coats 78.8 71.8 60.7 71.0 Dresses 53.7 41.7 27.6 45.2 Knit shirts/blouses 66.8 69.8 n /a 68.5 Not knit shirts/blouses 65.1 53.6 75.8 59.9 Skirts 77.1 55.4 39.1 60.2 Suits n /a 23.6 27.7 25.2 Swim-wear n /a n /a n /a 40.9 Trousers/slacks/shorts 58.2 58.1 50.7 58.1 M en’ s and boys’ apparel Knit shirts 46.9 64.2 n /a 50.9 Not knit shirts 84.2 51.7 75.8 75.2 Other coats 51.8 70.8 54.6 65.3 Suit-type coats 48.1 38.7 66.9 49.8 Suits n /a 55.4 62.3 55.9 Swim-wear n /a n /a n /a 97.8 Trousers/breeches/shorts 53.0 58.9 66.4 54.7 Other apparel (women’s and m en’s) Coveralls n /a n /a n /a 77.2 Dressing gowns, robes 82.4 49.4 n /a 67.4 Hosiery 8.6 13.2 8.2 11.0 Night-wear/pajamas 89.2 52.9 n /a 70.8 Play-suits n /a n /a n /a 65.0 Underwear 49.1 62.1 n /a 50.8 Sweaters 55.6 65.9 96.3 68.8 Babies’ garments and clothing accessories n /a n /a n /a 80.7 M M F: Man made fiber, n/a: N ot available or not applicable. D ata com piled from U.S. International Trade Adm inistration (1998) and U .S. Census Bureau (1992-1999) R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. The low market share of imports for women’s dresses, suits and swim-wear re flects the importance of fashion in women’s apparel. Contrasting women’ s suits and swim-wear with their counterparts in men’s apparel shows how fashion con tent affects retailers’ and manufacturers’ sourcing decisions. Import penetration in women’s suits and swim-wear was only 25.2% and 40.9%, respectively, while im ports dominate men’ s suits and swim-wear by 55.9% and 97.8%, respectively. The differences can mostly be attributed to different levels of responsiveness required in industry for women’s and men’s suits and swim-wear as imports’ cost advantage over domestic production is essentially the same for these categories. Figure 2.1 shows the import and domestic unit prices for men’s and women’s swim-wear over the years 1991-1998. For both categories, imports have a substantial cost advantage over domestic production. In both categories, domestic prices increased gradually, while imports m aintain a steady average price. Figure 2.2 shows the domestic pro duction, imports and domestic market for men’s and women’ s swim-wear over the years 1991-1998 (domestic market is derived by subtracting exports from the sum of domestic production and imports). The market share of domestic production in men’s swim-wear is virtually disappeared. While expanding market in women’s swim-wear is exploited predominantly by imports, domestic manufacturers w r ere able to maintain their production volume in spite of considerable increases in their prices. An individual company’s sourcing decision is a result of the performance mea sure it uses in evaluating different supplier alternatives. A traditional measure has 45 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 2.1: Average domestic and import prices for men’s and women’s swim-wear ( 1 9 9 2 - 1 9 9 9 ) Average Domestic and Import Prices for Men's Swimwear domestic orjce ----- import'pnce~'~«— - 14 12 10 1993 1996 1997 1998 1991 1992 1994 1995 Years Average Domestic and Import Prices for Women's Swimwear domesticprice ----- imgarfpnfce--^*— 14 12 6 1993 1995 1996 1997 1998 1991 1992 1994 Years Data com piled from U.S. Census Bureau (1992-1999) 46 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 2.2: Market share of domestic and import m en’s and women’s swim-wear ( 1 9 9 2 - 1 9 9 9 ) Domestic Production. Imports and Domestic Market for Men's Swimwear 70000 domestic market ----- domestic production----- imports 60000 tn c 3 50000 o cn ■ a c c a cn 3 O x r o- 40000 30000 c ca 3 a 20000 10000 1994 1995 1996 1997 1991 1992 1993 1998 Years Domestic Production, Imports and Domestic Market for Women's Swimwear 90000 domesticmaiket ■ domestip^foduction----- imports 80000 tn 70000 c 3 o tn * o c 60000 ca tn 3 O £ T; 40000 c - 30000 50000 3 a 20000 10000 1995 1996 1991 1993 1994 1997 1998 1992 Years D ata compiled from U.S. Census Bureau (1992-1999) 4 7 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. been the gross margin to sales ratio which has put the focus on low cost imports. However, this measure totally ignores the costs associated with holding inventory. Advocates of Quick Response systems suggest the use of gross m argin return on investm ent (GMROI) as a performance measure, which is basically the gross mar gin to average inventory ratio (Bobbin 1995). Frequent replenishments advantage domestic manufacturing over imports in this measure especially when seasons are long. These two measures only capture physical costs of the supply chain. Measures capturing the market m ediation costs include service level: percentage of times a customer finds his or her first-choice SKU; lost sales: percent of customers finding none of their SKU preferences; sell-through: proportion of a season’s merchandise that sells at first price; and job bed-off: percentage of units remaining at the end of season which must be disposed off. A computer simulation model developed at North Carolina State University concludes th at Quick Response strategy outper forms offshore sourcing strategy in these four measures and GMROI, but falls short of generating higher gross margin to sales ratio in all the scenarios created (Bobbin 1997a). The same results are also reported in Hunter et al (1996). 2.4 Paugal Industries To better understand the decision problems of the components of the apparel supply chain, we contacted a manufacturer, Paugal Industries, which is located in the Fashion District in downtown Los Angeles. Mr. Pierre Levy, originally from France, 48 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. founded Paugal in 1983, after working as a sales representative for a large apparel retail chain where he accumulated intimate knowledge of the design, manufacture and retailing of apparel. Paugal is a women’s apparel m anufacturer specializing in products in the ’ ’fashion” category" characterized by product life cycle of about 10 weeks. Like many companies in the women’s category, Paugal is a small company. In 1999, Paugal employed 18 regular employees. Currently, Paugal has two type of operations. In the first category, Paugal de signs and develops women’ s sweaters under the name Ultraknits. Ultraknits has two brands; Fifi, targeting younger consumers and Loop, targeting consumers looking for distinctive fashion. All production in this category is performed by indepen dent contractors. Currently, Paugal contracts its production out to four factories in China and Bangladesh. Major customers of Paugal in this category include depart ment stores such as Nordstrom and specialty chain stores such as Urban Outfitters. Production volume for sweaters is about 40,000 units per month. In the second category, Paugal acts as an intermediary between the local con tractors and mail order companies for women’s dresses under the brand name Olive. Paugal is not responsible for the design of these dresses. Currently, Paugal uses two contractors which are both located in the Los Angeles area. All of the six customers in this category are mail order companies. Production volume for women’ s dresses is about 5,000 units per month. 4 9 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 2.4.1 O perations T h ro u g h o u t th e Year Historically, Paugal is a maker of women’s apparel in the fashion category. There are five seasons for fashion merchandise. The names and the times the merchandise is delivered to the retailers are given below. Season D e liv e ry tim e s to re ta ile rs Fall 1 July August Fall 2 September October Holiday October mid November Spring late January March Summer March mid April The time between design to delivery amounts to nine months in Paugal. The design efforts for Fall 1 (1999) merchandise to be delivered to retailers at the end of July 1999 should start as early as early October 1998. At this time, the design ers working for Paugal are able to observe any particular trends popular with the consumer in the Fall 1 (1998) season. Design takes place until January 1999 and sample production begins in Paugal’s own facilities. The first samples are produced and approved by mid February and final samples are ready by mid March. In a typical year, Paugal develops 20 groups of sweaters each of which includes three or four different styles. In April, Paugal’s salespeople are busy with marketing the new designs. Major sources of orders used to be the trade shows (e.g., Las Vegas Magic Show), but the emphasis now is the individual contacts with the retailers. Paugal’s current customers are large retailers including Nordstrom, which still uses decentralized buyers to better match the consumer needs in each geographic area. The Nordstrom 50 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. buyers that Paugal is in contact with are responsible for 12 stores. O ther customers of Paugal are specialty chains, the largest being Urban Outfitters. Order quantities from retailers are usually economically feasible. However, even if a particular retailer asks for a non-economic quantity of a particular design, the tendency is to accept the order, considering the long term relationships with the retailers. Paugal plans to expand its customer base to smaller retailers (three to four stores), which will complicate the sales and m arketing functions considerably. Fourth week of April is usually the time that Paugal checks to see whether the cumulative orders in each style exceeds minimum production quantities. Rarely, Paugal has to cancel the orders, if the cumulative dem and in a particular style is not enough to carry out a cost efficient production. Trading off the cost of such cancellations against the cost of failing to capture enough market share, Paugal has to plan its initial merchandise assortment (samples to be shown to the retailers) very carefully. Note that the customer (and thus the retailer) preferences are highly unpredictable when Paugal decides its assortment and starts to collect its customer orders. This is probably the only stochastic problem faced by Paugal in its operations. When the collection of orders is complete, cumulative orders in each style is assigned to one of four contractors in China and Bangladesh. The assignment is usually based on the production volume of each style. Two factories are used for high volume merchandise amounting 28,000 sweaters per month while other two factories are responsible for medium to smaller sized orders amounting 12,000 sweaters per 51 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. month. For all of these factories the production and transportation lead time is about 3 months. The finished merchandise is delivered to retailers at the end of July. All of PaugaFs manufacturing is based on the actual customer orders and thus does not use any forecasting. The buyers in the "fashion" category of women’s apparel usually buy only once for a season, because of the long lead times and short seasons evidenced here (10 weeks season versus 12 weeks lead time). The operations of Paugal are similar for sweaters for Fall 2, Holiday, Spring and Summer sessions. The operations for dresses include only the manufacturing stages as Paugal is not involved in any design work for this category. 2.4.2 R ecen t P rob lem s faced by P au gal and o th er sm aller m anufacturers Paugal also struggled with the restructuring and increased im port penetration in the apparel industry (see section 2.3). The company’s production volume decreased substantially. Pierre Levy, the chairman of the company, explains th at the retailers now are the price setters in the industry. Before even arranging an appointment with a large retailer, Paugal has to show the proof of its financial stability, its costs and sources. Pierre Levy says that these are the kind of information he would never reveal ten years ago. Using these crucial information, some retailers now began to remove the intermediaries between them and the contractors. Mr Levy explains that a major retailer initiated a direct business with a factory he put a lot of effort to 52 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. find in Mexico, after a year of business with Paugal. Increased pressures to decrease costs force Paugal to use contractors in China, Bangladesh and Mexico. Moreover, the company has to focus on sweaters, where a great deal of expertise is required for production and dresses for large size women where the consumers are still price insensitive. 2.5 The Fashion Buying for Retailing In order to better understand the business environment and the practice of pur chasing, inventory and pricing decisions in fashion retailing, particularly women’s apparel retailing, we talked to a former buyer for a major retail chain: Ms. Jennings and a buyer/owner of an independent boutique: Ms. Massoudian. Ms. Jennings worked for six years as an assistant buyer, department manager, group sales man ager, cosmetics and fragrance manager, and operations manager for Macy’s, and two years as store manager for The Gap. Ms. Massoudian owned an independent high-end women’s apparel store in Palos Verdes, California and were mostly involved with purchasing decisions. We selected buyers for our research contact, since the buyer is the person who directly makes the decisions for what to buy, whom to buy from, how much to buy, how much to price, when to mark-dowm, and how much to mark-down, whereas the store manager of a store in a chain has responsibilities in the daily maintenance of the store operations (both personnel and merchandise), 53 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. and the chain executive is more concerned with financial control and administra tive policy making. Below we summarize the operations in fashion retailing. We mostly focus on the operations in Macy’s as the sales volume necessitates a more structured approach. W hen necessary, we contrast the operations at Macy’s with the operations at the smaller independent apparel retailer. 2.5.1 M erch an d ise O rgan ization In a large departm ent store like Macy’s, the buying and pricing decisions are decen tralized based on “departm ents” . Men’s and women’s apparel follows very different sales patterns driven by different consumer behavior and therefore in most apparel retailing, the men’s and women’s apparel is under separate departm ents. Many smaller boutique stores specialize in women’s wear only. In Macy’s, the women’s apparel is subdivided into several categories; Ms. Jennings was responsible as an as sistant buyer for women’s apparel in the categories Junior’s (Misses), Casual, Career separates (brands like Jones New York and Liz Claiborne). The apparel buying is often done by “groups” (rather than by individual style items), especially in women’s categories: a group may consist of a shirt, a sweater, a pair of pants, and a skirt, all from the same vendor. Vendors also follow this approach and usually offer their items in groups. This can sometimes facilitate the buyer’s decisions (by reducing the buyer’s choice); other times, however, a vendor may attem pt to bundle its less popular items with a hot selling item . Depending on 54 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. the retailer’s buying power, a buyer may demand only a single item from a weaker vendor. W ithin a group, the purchasing quantities for different items are usually balanced. W ith a group consisting of a shirt and two skirts, a buyer will therefore place a purchase order for 400 shirts, 200 long skirts and 200 short skirts: the total number of long skirts and short skirts equals the number of shirts, based on the belief th at a store customer will buy either a short or a long skirt, but not both, to go with the shirt. In other apparel categories such as coats and swim-wear, such a bundling is not a typical practice. In Macy’s, buying for the same apparel category (like Junior’s) for all stores in the Western U.S. is centralized. As we note earlier, this consolidation is in an effort to get competitive deals from vendors and is in contrast with decentralized buying practices (as in Nordstrom) which try to understand and meet different consumer demands across different geographical areas. 2.5.2 T im ing in th e M erch an d ise Life C ycle For planning decisions, the year is divided into several seasons; the number of sea sons varies depending on the chain and the merchandise categories. At Macy’s for women’s clothing, there are four seasons: Spring, Summer, Fall and Winter. Many categories also have special sales seasons such as Christmas. For more fashionable apparel, there can be more seasons shorter in duration, while for career clothing with R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. more stable year-round demand, there can be only two seasons (Spring/Summer and Fall/W inter). The buying decisions are made well in advance, usually 6-9 m onths before the start of each selling season. For example, vendors are contacted and contracts are signed in March for the Christmas season. Such a long lead time is due to the long production lead times at the vendor’s production facilities, m any of which are located outside United States. As discussed in Section 2.3.4, industry-wide data indicate that im port penetration (overseas production) percentage is higher for men’s wear than for women’s wear: this reflects the fact that apparel with a higher fashion content demands a greater supply flexibility and shorter supply lead times. Despite the recent efforts of “Quick Response” , “Accurate Response" and “Efficient Response” , supply lead times for domestically sourced items are also still long. Throughout a selling season, merchandise display on the store floor is periodically updated, with two or three apparel groups marked as new arrivals at one time. There can be 8-10 apparel groups in total within a selling season. Arrangements are made, especially with domestic suppliers, to have the merchandise delivered to the stores (or the retail chain’s distribution center) monthly. Such a staggered schedule has two m ajor effects: (1) smoothing out production for the vendors; and (2) keeping the retail store constantly refreshed in merchandise display with new items. This is done to capture shoppers’ attention who are usually attracted to newly arrived 56 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. items being put on prominent display; a shopper who sees the same few items on display would assume that the store has nothing new to offer and would therefore quickly lose interest in the store. 2.5 .3 T he B u yers In Macy’s, about 10 buyers cover women’s wear. Each buyer is responsible for a separate category of apparel items and all merchandise under a buyer’s responsibility would potentially generate similar profit margins. This system ensures that the buyers would not only buy those items th at are believed to generate more margins as most of their compensation is determined by the total margin of their purchases. Successful buyers are generally among the highest paid employees of the chain, rivaling executives and store managers. An adm inistrator manages a group of buyers. Along with many other managerial functions, the adm inistrator sets a “budget” for each buyer. A buyer’s budget is usually updated each season based on his/her performance and consumer trends in the apparel line he/she is buying. Sometimes a buyer would set aside some budget dollars for special purchases that either a vendor m ay offer or a store manager may sponsor. The buyers have tremendous power in representing the chain to the vendors and are responsible for a large portion of the chain’s profit. Not only do the buyers decide what to buy, whom to buy from, they also set the initial selling prices and 57 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. the subsequent mark-downs. They can also “move” item s from a particular store to another store in the chain. Since the buyer’s performance evaluation criterion (and his or her bonus) is the total profitability (total margins) of the apparel lines he or she buys (which depends on the purchase cost, the initial selling price, the subsequent mark-downs, and the units sold under each price point), it is in the buyer’s interest to ensure th at she buys the right items generating the best financial results for the chain as a whole. 2.5.4 D ecid in g W h a t to B u y Although there are staple items in women’s apparel like wool suits for the career women, most apparel items contain a fashion content. And when it comes to fashion, the buyer’s intuition, hunch and to a lesser extent, experience, still play a large role. Here, guess work is the norm and the quantitative models are perhaps least likely to be helpful. Ms. Massoudian says that the apparel buying consumers in the United States are heavily influenced by TV, movies and magazines. Ms. Jennings believes that although the fashion shows in cities such as Paris, Milan or New York each year have definite influence on w hat’s hot, the teenager girl in United States is often going after what the leading actresses wear in a hit TV show or movie. An example is pink short skirts, which the Spice Girls worn in their performances, and suddenly every teenager girl in town wanted a pink short skirt also by rushing to the stores. In that situation, the buyer picks up the phone to call the vendor of pink skirts and 58 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. begged for whatever the vendor can ship quickly that is something similar to a pink skirt. Because TV shows such as “Beverly Hills 90210” and “Melrose Place” have such a profound impact on the buying behavior of young women, the buyers would watch every episode of the shows on a particular weekday night and pay attention to what the actresses wear. Ms. Massouidian says that the color is perhaps the most im portant thing to consider for fashion buying. Since fashion buying is so treacherous, a particular means to help the buyers decide what to buy is the frequently employed “pre-season” sales. At Macy’s, the pre-season sale for the W inter season is held in late August, and each garment is marked 25% off regular price, or comes with two price tags: one with the regular in season price, another with a 25% marked down price but with a purchase date limitation. The pre-season sales are usually displayed at the store’s most prominent window or floor area; the apparel items on this same display can come from different groups from the same vendor, or different vendors, or different buyers. The consumer reaction to these items through sales data gives the buyers advance advice so that the buyers can plan the regular purchases and can still do so in time. 2.5.5 W h o m to B u y From: T h e V endors Although the buyers decide what actually is brought to the stores, the vendors offer what the buyers can buy. The vendors also lim it the choices available to vendors by bundling (product groups) and often suggest a retail price, which small independent 59 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. retailers follow in order to avoid price competition among themselves, as the same brand shirt selling at $29.99 and $39.99 at two outlets in the same mall is viewed as undesirable by the retailers and the vendor. To avoid such competitive scenarios from appearing, vendors often supply slightly different groups based on the same design to different retailers. W hen asked about what criteria are used to select vendors, Ms. Jennings says that foremost criterion is business relationship. A buyer spends a significant portion of his or her time talking with and visiting the vendors. A buyer at Macy’s deals with varying number of vendors from a few to more than 25, and has intimate knowledge of the vendors’ designs, qualities, costs, and delivery capabilities. The most im portant aspect in building a relationship seems to be the vendor’s track record in keeping with promises in delivery and quality. For new vendors, it can be very difficult to establish a business relationship with a powerful retail chain. Ms. Jennings tells stories of stores rejecting a vendor’s shipment either due to unacceptable garment quality or late shipment. Only rarely do the buyers omit the im portant step of personally touching the vendor’s sample garment (this occurs only for the few established vendors) before placing a purchase order or entering a contract. Timing is another key aspect in fashion retailing - a shipment of only one day late can cause tremendous lost sales to the store due to the short selling season. Imagine a shipment of red and green sweaters for Christmas arriving in the middle of the selling reason - store employees would be told to return the merchandise 60 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. immediately back to the vendor. Somewhat surprisingly, price does not seem to be a significant factor in choosing a vendor. 2.5.6 T h e Q u an titative D ecision s: H ow M u ch to B u y Buyers rely on both d ata (past sales history) and intuition in deciding how much to order. For apparel item s that are considered “regular” like business suits for women, the dem and history provides good forecast of the future. For more fashionable items, the buyer exercises judgement. However, there are many lim itations on the order quantities such as minimum orders. Usually, the total number of pieces is decided first, then this quantity is broken by color and by size. Not much quantitative forecasting techniques are used. For small boutique stores like the one owned and managed by Ms. Massoudian, the order quantities are constrained to be small (such as 3 pieces of the same style, color and size), for an additional reason that customers who shop at boutiques do not like to see anyone else wearing the same clothes. For fashionable items, the sales history within a selling season can also vary significantly. Ms. Jennings says that a poor selling first week in the season for an item usually indicates poor sales also in the second and third weeks; but a good selling week initially does not guarantee good sales in subsequent sales. Ms. Massoudian disagrees on this point. She has observed items with weak initial sales followed by stronger sales. 61 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 2.5.7 In form ation T ech n ology W ith the widespread availability of point-of-sales data, the buyer’s job is becoming more information intensive. Every morning, a buyer would spend 3 or 4 hours ana lyzing screens after screens of data - previous week’s and day’s sales and inventory. W hen a problem or a trend is spotted, the buyer calls the stores or the vendors. For Macy’s, EDI (Electronic D ata Interchange) is used for regular and stable products like men’s shirts, but the telephone is still the most used media of communication and ordering for more fashion intensive items. As stated earlier, the point-of-sales data is not used in an efficient and structured way disappointing their CEO (see Section 2.2.2). 2.5.8 In ven tory and D istrib u tio n For many companies like Macy’s, a buyer is responsible for many stores in a par ticular sales region (like Western United States). W hen a buyer decides on what to buy and how much to buy, the buyer is deciding for all the stores in aggregate. A “planner” works closely with the buyer to distribute the assortment and purchasing quantities across different stores, accounting for differences in store locations such as area income level or demography. The “allocations” across stores are generally not even; a store may not “get” any allocation of a particular item at all. When a vendor delivers a batch of garments, the shipm ent can go to a central warehouse or distribution center first and then be broken down and re-shipped to 62 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. individual stores. Alternatively, the shipment can go directly to individual stores without ever entering a central warehouse or distribution center (this is called drop- ship); in this mode, the vendor’s garments must be “floor ready” (complete with the proper labels and price tags and hangers). A buyer sometimes moves an item from one store to another store; when this occurs, a direct transshipment between the two stores may not always occur. At Macy’s, a transfer between two stores has to go to a distribution center, for it was found that the potential costs of miscounts and mishandling of goods in direct transshipment between two stores can offset the additional transportation cost of moving merchandise through the DC. The receipt of merchandise from a vendor or the distribution center is usually planned such th at it will directly go to the store shelves. For Macy’s, very rarely do truck-loads of apparel have to be stored before they are put on the store display. This practice requires great reliability of delivery from the vendors. For small stores like Ms. Massoudian’s, however, shipments from European suppliers arrive on fixed and infrequent delivery schedules. Often, the store will have to temporarily stock the shipment before the selling season starts. The inventory at a store is usually kept near the display shelves and racks; more inventory is stored behind doors marked “For Employees Only” . The department manager’s office is often located in the middle of mountains of garments. It is the 63 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. job of some store employee (sometimes a sales person.) to move inventory stored in cabinets and back-rooms to constantly replenish the display shelves and racks. 2.5.9 P ricin g As mentioned earlier, pricing is subject to a few artificial restrictions (practical con siderations) although in most cases the retailer can theoretically set prices to any amount. Ms. Massoudian says that a store can be seriously hurt by a competitor’s price cuts; many customers are surprisingly well aware of the prices of many apparel items at many stores. The pricing impact in such a highly competitive environ ment is immediate. M any independent stores therefore wish to follow the vendor’s suggested retail price. In Macy’s, the buyer sets the initial price, but more or less based on a company- wide price schedule. Corporate management uses pricing guides and schedules to achieve control and uniformality of items bought by different buyers. Small stores sometimes multiple their cost by 2 or 2.2 as a general rule in setting prices. Usually, the national brand names do not bring a good profit margin to the store due to the higher prices from the vendor, whereas private label garments have higher margins. 64 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 2 .5 .1 0 M ark -d ow n s an d Sales There are two kinds of mark-downs: (1) Temporary (point-of-sales) discount (like 25% or 50% off) at the cash register. The POS mark-downs occur, for instance, during "This Saturday Only” sales. (2) Perm anent price change, in which the price tag is physically changed to reflect the new, decreased dollar amount. Almost never will the stores mark the price up, no m atter how hot an item is selling. We can conjecture that if the retailers had practiced m ark-ups during the season, the initial prices would not be this high, a problem well-known in departm ent stores. The “no-m ark-up” rule applies also to vendors. If a vendor has a “hit” item and the buyer screams for more of it, the vendor will usually charge the same price, but may bundle the item with some less popular items or attach some other sales conditions. Similar to pricing, there are company-wide guidelines as to when and how much to mark-down. At Macy’s, there are a few pre-set mark-down levels: 25%, 30%, 50% and 60%. The first mark-down occurs in approximately the sixth week. Once an item is marked down, its shelve space is consolidated (for example from two shelves to one shelve) or it is moved to a less prominent display area. For “hard” mark-downs like 50% or 60% off, the garments are moved to special racks organized by garment size so th at a shopper can go directly to the right rack to find all styles of 65 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. her size. Since two new groups arrive every two weeks throughout the same season, the stores can constantly refresh its display and inventory. While mark-downs are quite frequent, store-wide sales are relatively infrequent as it is feared th at consumers may act strategic and “wait for the sales” . Store— wide sales are twice a year at Macy’s, one of them after Christmas. At those events, the whole store space is organized for the sales (whereas for smaller sales, the items marked down are put in special corners or back space). Sales also tend to occur before annual or sem i-annual inventory counts, so that the store personnel have fewer items to count. 2.6 Concluding Remarks U.S. apparel industry has been in a transition over the last 20 years. Im ports from lowr er wage countries and retail consolidation forced U.S. manufacturers to look for other ways to remain competitive: quality and flexibility. Physical proximity and advances in information and manufacturing technologies enabled U.S. manufacturers to accept retailer orders closer to the season and replenish their stocks frequently during the season. However, retailers continue to source more and more of their merchandise from overseas with the cost of having to make risky inventory decisions. Advances in information technology also made an enormous amount of point-of- sales data available to decision makers in retailing. Decision makers (m ostly buyers) are able to learn more about consumer preferences over styles and colors, their size 66 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. distribution and the dynamics of sales. However, re-quoting the CEO of Federated: “Where we have made little progress, ..., is in changing the way our buyers go to market and buy. I don’t see them using this data nearly as much as I expected” , this had little effect on their ordering and pricing decision. These decisions are still considered as a form of art. While most of retailers are still foreign to quantitative models, some have only recently started to explore such models. Again re-quoting a business consultant working on an apparel pricing implementation: “For many of our merchants, this is the first exposure they’ve had to intelligent systems” . The result of such ignorance is obviously not impressive. Inventory turnovers in some departments are less than once a year. While we expect that point-of-sales data would have the easier impact on figuring out the size choices of the customer base given the geographical area, we continue to see only excessively small or large sizes on clearance racks. There is an apparent and urgent need for practical quantitative models that effectively use already available data. Noting the lack of such models in practice, we argue that the models described in Chapters 2-4 intend to model the way the retailers should do their business. We note that these models can also be used in other industries with perishable products and inflexible supply chains. Possible applications of these models are given in subsequent chapters. 67 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Chapter 3 A Model for Ordering Decisions Many inventory systems operate in segmented markets represented by customer classes that are differentiated by price. For example, fashion and sporting goods may have two or more market segments: regular and discount, where excess supply in the regular market is subsequently disposed of in the discount market. Many perishable items in retailing also follow the pattern of multiple demand classes with non-increasing prices: the first demand class consists of customers who buy the product on the first day (or week) when the commodity is freshest; other customers may constitute the second and third day (or week) demand classes, who pay less for less fresh products. On the other hand, some non-conventional inventory items such as airline seats and special event tickets have demand classes th at are increasing in price over time. For example, a 21-day advance purchase is generally less expensive than a 3-day advance purchase; an advance reservation ticket of a special event may cost less than a ticket purchased at the door. Pfeifer (1989) and Bodily and Weatherford (1995) give extensive examples of items with time sensitive pricing. 68 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. In these contexts, there is often only one inventory replenishment opportunity which exists before the selling season; there is no second replenishment opportunity should the subsequent sales turn out to exceed the amount of the stock purchased. This is usually due to the long replenishment cycle relative to the sales season (as often exhibited in fashion goods systems) or due to the fixed replenishment capacity (such as in airline seats). In many situations, however, the replenishment quantity can be set as a decision variable. It would be interesting to evaluate the value of the ability to choose the initial capacity optimally, and hence the value of additional units of capacity. In this chapter, we study the problem of determining the optim al replenishment quantity to maximize the expected profit. We consider the dem and classes as seg mented by time and price; each dem and class has a single price th a t is exogenously determined. The demand quantities are stochastic with known probability distribu tions. Thus, the proverbial newsboy orders his papers early in the morning and sells them at different prices in the m orning and in the afternoon, where the demands are realized sequentially. We will first assume that demands are independent and later examine the cases with dependencies created by the returning of a fraction of the unsatisfied demand in a price class to another demand class. An example of this situation is th at a fraction of the “low fare” buyers will join the “high fare” customer class (perhaps unwillingly) if their initial request is not m et at the lower price. This occurs frequently when the prices are increasing over time. While the 69 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. most general situation would be one in which the prices fluctuate in either direc tion, we will examine two special cases with monotonically increasing or decreasing prices. As it turns out, the model structures between these two cases are drastically different. We assume that demand quantities, often for two or more periods within a fixed selling horizon, are realized sequentially over time. This simplification, which is utilized in most Perishable Asset Revenue Management (PARM) literature, could be restrictive in a general yield management scenario. In airlines, different demand classes might be viewed as different products (from the customer’s perspective), each having its own restrictions, penalties and market characteristics; this may cause demands with different prices for essentially the same airline seat to co-exist simultaneously. However, we note that this effect is not very significant in airline ap plications, mainly due to the price sensitive travel and advance purchase restrictions (Belobaba 1989). We note th at while newsboy type models fit the very nature of buying decisions for apparel retailers, our analysis in Sections 2.2.2 and 2.5 show that retailers hardly employ such models in practice. Buyers usually use point estimates of demand ig noring the probabilistic nature of the problem. This shifts the focus from minimizing the expected profits to keeping the inventory to budgeted levels by controlling new purchases via open-to-buy systems. We insist that retailers should use newsboy type probabilistic models in their decisions and our model helps to remove one of 70 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. the shortcomings of these models: the assumption th at the retailer charges single price throughout the life of the merchandise. When the regular fashion season is over, many retailers sell their excess merchandise to discounters (e.g., Marshalls, Ross). Our decreasing model will be a good approximation for their buying de cisions and help them to optimize their profits considering their own retail sales (primary market) and sales to the discounters (secondary market). For this initial approximation, retailers can use the average price over the regular fashion season. They can capture some part of the dynamics of price changes during the regular fashion season by estimating demand separately around preset price levels (initial price, mark-down price). Once the order quantity is determined using such an ap proximation or otherwise, within season profits can be optimized via the pricing model discussed in Chapter 4. This chapter is organized as follows. In Section 3.1, we review the literature on both the single period stochastic inventory (newsboy) problem and the related mod els in PARM. In Section 3.2, we analyze the case with decreasing prices, and examine the effect of applying single demand newsboy model in multiple demands. In Sec tion 3.3, we examine the case with increasing prices and consider a simple, "booking limit” type of policy structure. Through an example, we discuss the the results and managerial implications. We give some concluding remarks in Section 3.4. 71 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 3.1 Literature Survey The newsboy problem has attracted numerous researchers since the pioneering pa pers by Arrow, Harris and Marschak (1951), and Morse and Kimball (1951). Some notable extensions follow the lines of considering alternative objectives other than maximizing the expected profit, and relaxing the requirements on the demand dis tribution. For example, Ismail and Louderback (1979) and Lau (1980) consider the satisfying objective of maximizing the probability of achieving a given profit level: Eeckhoudt, Gollier and Schlesinger (1995) consider the risk averse newsboy. Reyniers (1990) discusses delayed observation of sales: Gallego and Moon (1993) study the case in which the distribution of the demand is unknown. O ur decreasing price model extends the newsboy problem from a single demand to m ultiple demands with different selling prices. A few papers are related to ours. On the supply side, Jucker and Rosenblatt (1985) study the newsboy problem with a single demand but quantity discounts for purchasing costs. On the demand side, Khouja (1995) considers the newsboy problem with progressive discounts to sell off excess inventory. The amount of inventory sold (demand) at each discounted price is completely determined by (a fraction of) the amount of inventory sold at the original price; hence the multiple demands are perfectly correlated. Khouja presents a model formulation to maximize the expected profit and the probability of achieving a target profit. Khouja (1996) further extends his earlier model to multiple discounts from the supplier side. The problem setting th at we study differs from Khouja (1995) 72 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. in th at the demands are independent across price classes in our decreasing price model, and dependent through residual demands in our increasing price model. O ur treatm ent of demand independency and dependency tends to make our models more applicable. Lau and Lau (1988) examine the price-dependent demand distribution and present efficient solution procedures for finding optim al order quantity and price under different optimization objectives. Although Lau and Lau optimize both the order quantity and the price, it is a single demand model where the demand distribution depends on price. We assume th at prices are exogenous in each market segment. This allows us, without losing much realism, to obtain analytical results which are relatively simple and which allow for easy interpretations. O ther related models include the newsboy problem with multiple products. These products may share a given space or budget constraint (as in Silver and Peterson 1985, Chapter 10.3). Li, Lau and Lau (1991) study a two-product news boy problem with independent exponential demands to maximize the probability of achieving a targeted (total) profit. However, products are not substitutable so that each of the products has one demand. Our decreasing price model assumes a single product which can be used to satisfy multiple demand classes. Kouvelis and Gutier rez (1997) further consider the newsboy problem with a single product but in two markets (locations) and introduce differential ordering (production) costs, exchange risk and transportation cost. Our model differs from Kouvelis and Gutierrez as we 73 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. assume one-tim e production for multiple markets and we focus on the determina tion of the optimal production quantity. Our model for the increasing price case also applies when the secondary market has a higher price so that it may be profitable to ship the product from the primary market to the secondary market although there may be sufficient demand in the primary market (the protection policy, as we will discuss in Section 3.3). For the second case with increasing prices, our model is related to the PARM (Perishable Asset Revenue Management) models, which have been formulated mostly with applications in airline seat inventory control. Belobaba (1987), Weatherford and Bodily (1992) give a taxonomy and overview' of research in PARM. Most of the published w'ork utilizes probabilistic decision models to decide the optim al book ing limit for the discount demand (e.g. Belobaba 1989), w'hile the initial capacity, equivalent to the order quantity in the newsboy problem, is assumed to be fixed. These models also differ by their treatm ent of dependencies of demands and di version where a customer purchases a seat in a different fare class than he or she is originally willing to pay (upgrade or downgrade), as described by Belobaba and Weatherford (1996). Most of the models assume independent demands for the fare classes; one example is the 2-class demand model by Pfeifer (1989) who considers setting an allocation of a given initial seat capacity to the lowr fare class (i.e. choosing a booking limit) and who also points out the analogy to the newsboy problem. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Two papers that consider dependent demands are Brumelle et al. (1990) who treat two-class dependent demands via a bivariate normal distribution, and Belob aba and Weatherford (1996) who use a probability (/52) to represent the diversion effect (/?2 is then equal to 1— the upgrade percent from low fare to high fare) and numerically compare decision rules incorporating diversion. We model the demand dependency through the diversion fraction, similar to Belobaba and Weatherford: however, we also consider the selection of initial seat capacity, and we derive analyt ical results. Gerchak, Parlar and Yee (1985) study a 2-class model sim ilar to ours, although they use a discrete time, discrete demand unit formulation (ours is aggre gated demand) with backward-recursive dynamic programming. They also consider selecting the optimal initial order quantity. Later, Lee and Hersh (1993) extend the work of Gerchak et al. to multiple classes and multiple seat bookings; however, in both papers, demand dependencies, likely created by upgrading or diversion, are not considered. Our model combines all three aspects (1) fair allocation or booking limit decisions; (2) selection of initial seat capacity; and (3) explicit consideration of demand dependency through diversion. We are then able to examine the behavior of the booking limit. Our model enables us to derive economic interpretations for the optimal order quantity using marginal analysis, and extending the marginal rule in setting protection limits to incorporate the capacity decision. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 3.2 The Model with Decreasing Prices We shall label the first dem and to realize by class 1; and the last dem and by class n. The demand in class j, Dj , has a pdf We assume that Dj, j = 1, 2, - - •, n. are independent. The selling price in class j is Pj, with p\ > p2 > ■ ■ - > pn > 0. We also assume pi > c, for otherwise the optim al order quantity will be zero. W ith the decreasing prices, clearly the optimal allocation rule for existing stock is of the “no reservation” type: i.e., the available stock is allocated, to the fullest extent and without any reservation for future demand classes, to Di first: the remaining stock, if there is any, is then fully allocated to Do: and so on. We wish to point out th at this model also applies to a situation in which all n demands are realized simultaneously before the allocation decision is made, such as in a distribution system where the demand classes are divided geographically. We also assume that beyond the n classes, any unsold units have zero salvage value. This is not a real lim itation of our model, since any positive salvage value can be modeled as a last dem and class with price pn and a very large mean demand. Let the unit purchase or production cost be c, and the order quantity be X (the decision variable). Then the expected profit maximization problem is maxTr(X) = 'f^pjE[Qj ] - cX, j = i 76 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. where Qj is the sales units in demand class j. Next, we obtain E[Qj\. We have, Qi Qj m infD !, X}, min{L>y, X — Ei=t A } , 0, if E C l A < X, if E C i D t > X , = 9 Hence, rJC ro o E[Qi}= D lf l (Dl)dDl + X h iD J d D L J o J x For j > 2, let Tj = YljZi and gj(-) be the pdf, and Gj(-) the cdf, of Tj. Then Qj depends on two random variables Tj and Dj, as shown in Figure 3.1. By integrating over the three regions, we have, Figure 3.1: Value of Qj D. 7 7 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. £{Qi] = f * £ D j g j i T ^ d T M D ^ d D j + f X f X (X - T M T J d T i M D J d D j J 0 J X —Dj rX noo + ( X - T jlM D Jd D jgj^ d T j Jo J X = f X DjGjiX - DM ADfidDj Jo + f X f X (X - Tj)gj(Tj)dTjfj(Dj)dDj Jo J x - D j + f X (X - 2))[1 - F ^ X ^ T ^ d T , . (3.1) In Appendix A .l, we show that 7r(A') is concave so th at the optimal X is given by Proposition 3.1. P ro p o sitio n 3.1 The optimal X satisfies px P r{Dx > X } + Y . Pj |P r { £ Di < X } - P r { £ A < X } = c, (3.2) j = 2 L i = l i= l or equivalently, y > i - Vi*i) P r ( E Di > X } + p„ Pr{ £ Di > X } = c. (3.3) j= 1 1 = 1 1 = 1 Equations (3.2) and (3.3) allow for a straightforward marginal cost and revenue interpretation. The left-hand-side of both equations is the expected marginal rev enue with an extra unit of stock which might be sold in one of the n demand classes; 78 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. the right-hand-side is of course the unit marginal cost. Observe that the solution to the equation (3.3) is the classical newsboy solution, with the total demand dis tribution, if pj = p, y = 1 ,..., n. Also, setting Dn = oo (so Pr{2Z” =1 Dx > X } = 1) guarantees a unit salvage value pn. When n = 2 and Dn = oo, equation (3.2) reduces to the standard newsboy formula where p2 is now the salvage value. The next result shows that the optimal order quantity is bounded by the two newsboy quantities with the highest and lowest prices across all market segments. P ro p o s itio n 3.2 The optimal order quantity satisfies < X ' < ), (3.4) P n P i where Gn+i(-) is the cdf of H"=1 D j . P ro o f: From (3.3), we have pn Pr-j^iL i Z)t - > X*} < c, hence X * > G~\_x(P n p~c)- Replacing Pr{£;{=1 Di > X} by a larger quantity Pr{5Z” =1 Di > X } in (3.3), we obtain C < Y,(Pj - Pi+ 0 P r ( E D i > X '} + p „ P r { £ D, > X ’} = p, P r { £ D, > X '} , j —l i = l i=L i= i 79 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. hence X* < If we define Ps as the probability of shortage (i.e. Pr(5I{=i A > X }), then Proposition 3.2 implies c/pi < Ps < c/pn. This compares with the standard newsboy model with Ps = cfp. 3.2.1 T h e E ffect o f A gg reg a tin g D em a n d C lasses We now consider the effect of aggregating the demand classes in order to use the sin gle demand newsboy model. When the multiple demand class model is not available, a first heuristic approach might be to treat the market as an aggregated demand with some averaged price. Thus, the order quantity of this approach, X c, is given by P r { £ A > X '} = c/p, i= L where the average price p can be weighted by the mean demands: p = £?=i fiiPi/12i= i J “i- A second heuristic th at we consider is to treat each demand class in a separate newsboy model with price pj, cost c and demand cdf T}(-). A newsboy quantity is computed for each dem and class; these individual newsboy quantities are then 80 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. summed and used as the singie order quantity X . Thus, the order quantity for this additive approach, X s, is the sum of Xi where X,, i = 1,2, - • -, n, is given by P r { A > Xi} = c/Pi. To compare the optimal solution with the above two heuristics, we consider the two-demand problem (n = 2). From the optimality condition (3.3), the optimal order quantity X* is given by the following equation: (px - p 2) P r{D l > X*} +p2 Pr{£>! + D 2 > X*} = c. (3.5) For a numerical comparison, we normalize c = 1 and py = 1. The relative size of the second market, p2 is expressed as a multiple of px; (p2/Pi) has three values 0.5, 1, and 2. The selling price in the first market pi has 4 values, 1.2, 2, 3 and 5; the selling price in the second market p2 is expressed as a fraction of pi, with p2/p\ taking four values 0.2, 0.4, 0.6 and 0.8. We consider normally distributed demands; the coefficient of variation (cv), same for Dy and D2, has five values 0.1, 0.2, 0,3, 0.4 and 0.5. In total, we solve 3 x 4x 4x 5 = 240 problems. The relative sub-optimality of each heuristic, computed as [(optimal profit) — (heuristic profit)]/(optim al profit), is tabulated (Table 3.1). To reduce the size of the table, we only show the case with cv fixed at 0.5. Later we graphically demonstrate the impact of cv on the performance of the heuristics. 81 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Table 3.1: Optimal and heuristic order quantities c = 1; p i = 1 ; coefficient of variation cv = 0.5 M 2/Ml Pi P2/PI Optimal Average Price Heuristic Separate N ewsboys Heuristic X ' x x J T l Si A'2 T T o 62 0.5 1.2 0.2 0.5724 0.0642 0.0000 0.0000 100.00% 0.5163 0.0633 1.39% 0.4 0.6409 0.0755 0.0000 0.0000 100.00% 0.5163 0.0714 5.36% 0.6 0.7262 0.0900 0.5112 0.0791 12.19% 0.5163 0.0795 11.66% 0.8 0.8323 0.1095 0.8058 0.1093 0.20% 0.5163 0.0876 19.98% 2 0.2 1.0853 0.6717 1.2357 0.6553 2.44% 1.0000 0.6663 0.81% 0.4 1.1833 0.7464 1.3219 0.7334 1.74% 1.0000 0.7230 3.13% 0.6 1.2904 0.8359 1.3915 0.8293 0.80% 1.2581 0.8352 0.09% 0.8 1.3988 0.9414 1.4499 0.9397 0.18% 1.4203 0.9411 0.03% 3 0.2 1.3212 1.6082 1.5638 1.5553 3.29% 1.2154 1.5968 0.71% 0.4 1.4358 1.7735 1.6176 1.7450 1.60% 1.4735 1.7722 0.07% 0.6 1.5493 1.9631 1.6640 1.9518 0.58% 1.6804 1.9483 0.75% 0.8 1.6523 2.1743 1.7047 2.1719 0.11% 1.7680 2.1624 0.55% 5 0.2 1.5529 3.6349 1.8380 3.5508 2.31% 1.4208 3.6120 0.63% 0.4 1.6836 3.9920 1.8771 3.9533 0.97% 1.9208 3.9351 1.42% 0.6 1.7986 4.3846 1.9116 4.3708 0.32% 2.0285 4.3305 1.23% 0.8 1.8935 4.8028 1.9425 4.8000 0.06% 2.0894 4.7613 0.86% 1 1.2 0.2 0.5480 0.0653 0.0000 0.0000 100.00% 0.5163 0.0640 1.96% 0.4 0.6745 0.0788 0.0000 0.0000 100.00% 0.5163 0.0728 7.70% 0.6 0.8051 0.0988 0.0000 0.0000 100.00% 0.5163 0.0816 17.48% 0.8 1.0114 0.1316 0.9775 0.1313 0.21% 0.5163 0.0903 31.34% 2 0.2 1.1242 0.6892 1.3159 0.6655 3.45% 1.0000 0.6786 1.54% 0.4 1.3002 0.7986 1.5998 0.7358 5.61% 1.0000 0.7478 6.37% 0.6 1.5432 0.9558 1.7747 0.9324 2.44% 1.5163 0.9554 0.04% 0.8 1.8001 1.1744 1.9012 1.1695 0.41% 1.8407 1.1736 0.06% 3 0.2 1.3936 1.6614 1.9012 1.4939 10.08% 1.2154 1.6334 1.68% 0.4 1.6515 1.9323 2.0422 1.8506 4.23% 1.7317 1.9286 0.19% 0.6 1.9374 2.3020 2.1488 2.2766 1.10% 2.1455 2.2774 1.07% 0.8 2.1536 2.7494 2.2340 2.7451 0.16% 2.3206 2.7310 0.67% 5 0.2 1.6927 3.7906 2.3046 3.5582 6.13% 1.4208 3.7152 1.99% 0.4 2.0480 4.4459 2.4002 4.3679 1.75% 2.4208 4.3590 1.95% 0.6 2.3103 5.2485 2.4769 5.2274 0.40% 2.6362 5.1717 t.46% 0.8 2.4778 6.1222 2.5407 6.1187 0.06% 2.7581 6.0581 1.05% 2 1.2 0.2 0.5871 0.0648 0.0000 0.0000 100.00% 0.5163 0.0633 2.21% 0.4 0.6859 0.0785 0.0000 0.0000 100.00% 0.5163 0.0715 8.92% 0.6 0.8426 0.1003 0.0000 0.0000 100.00% 0.5163 0.0796 20.61% 0.8 1.1673 0.1436 1.0224 0.1400 2.55% 0.5163 0.0878 38.88% 2 0.2 1.1434 0.6953 0.0000 0.0000 100.00% 1.0000 0.6818 1.94% 0.4 1.3916 0.8251 1.9184 0.7377 10.60% 1.0000 0.7540 8.61% 0.6 1.9881 1.0689 2.4714 1.0258 4.04% 2.0326 1.0686 0.03% 0.8 2.6441 1.5301 2.7813 1.5248 0.35% 2.6814 1.5298 0.02% 3 0.2 1.4442 1.6865 2.3673 1.3452 20.24% 1.2154 1.6460 2.40% 0.4 2.0388 2.0693 2.8438 1.9374 6.50% 2.2479 2.0598 0.46% 0.6 2.8440 2.8177 3.1277 2.7919 0.91% 3.0757 2.8005 0.61% 0.8 3.2353 3.7657 3.3280 3.7621 0.10% 3.4258 3.7507 0.40% 5 0.2 1.8784 3.8967 3.2013 3.4674 11.02% 1.4208 3.7573 3.58% 0.4 3.0001 5.1292 3.4816 5.0477 1.59% 3.4208 5.0667 1.22% 0.6 3.4816 6.8017 3.6760 6.7837 0.26% 3.8515 6.7385 0.93% 0.8 3.7541 8.6001 3.8232 8.5974 0.03% 4.0953 8.5387 0.71% 82 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Figure 3.2: Performance of the “separate newsboys” heuristic Sub-opOmaiSy of 'separate new sboys' heunsoc (mu2/mul -T . p2fp1»0.4) Sub-optimafity a t fte ’separate newsboys* heuristic (mu2/mul «l.cv»0-5) p t- t.2 I 5. Sub-optwnakty a t 'separate new sboys' heurabc (muZ/muf «1.p2/p1»0.4) pl«2 _S pi -3 C v 8 3 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. We first note that the “average price” heuristic would give a zero order quantity (and hence 100% sub-optim ality) when the average price (p) is less than the cost. This happens, for example, when pi is greater than but close to c. The “separate newsboys” heuristic which approximates the optimal order quantity by treating each demand class with a separate newsboy order quantity is found to be more robust in a broad range of problems. We therefore focus on the separate newsboys heuristic in our subsequent discussions. Figure 3.2 shows the performance gap between the optimal and the separate newsboys heuristic as pi takes increasing values: as the ratio P 2 /P 1 increases from 0.2 to 1; and as the demand cv increases from 0.1 to 0.5, all with the pi/p .2 ratio fixed at 1. These results indicate th at the separate newsboy heuristic performs poorly when pi > c > p2 but all three param eters take on close values. As seen from Table 3.1, whenp! = 1.2, P2/P 1 = 0.8, c = 1, (and p,i = p? — 1, cv = 0.5), the separate newsboys heuristic is 31.34% off from optimality. W hen pt is close to c, both heuristics perform poorly. These cases are not at all unrepresentative of practical situations; in these situations, the optimal order quantity given by our decreasing price model should be used over the heuristics. W hen the dem and cv is small, the separate newsboys heuristic tends to perform well. 8 4 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 3.3 A Model with Increasing Prices: Two Demand Classes Now we consider the case pi < p2 < - - - < pn with pn > c. Again, these prices are pre— announced or fixed under exogenous market forces. As in Pfeifer (1989) and many others, we will consider the simplest case with n = 2. We will also assume that the demands are positive with probability 1. This type of demand realization with increasing prices does not allow for a trivial inventory allocation scheme across demand classes. Here, we adopt a static alloca tion policy of the following type: the maximum units sold at demand class 1 (low fare) is set to a pre-specified level P (the booking limit); any unsold units after the realization of class 1 demand will be made available to class 2 (high fare) demand. This corresponds to the closing of the discount fare class in airlines. Our model setting resembles that of Pfeifer (1989) in th at the low fare demand precedes the full fare demand and that once the low fare class is closed it is never re-opened (a common assumption in PARM). The order quantity (for both classes) is X . Under this scheme, a portion of the initial stock X is protected or reserved for the higher priced demand class, as the units available to the higher price demand will be at least X — P. We term X —P the p rotection level for the high fare class. Our problem here is to find the optimal X and P to maximize the expected profit. Our model 8 5 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. differs from the previous research in this field in that the initial order quantity (or capacity), X , is a decision variable in our model. A reasonable assumption in this environment is that some of the customers in the lower price class may be ready to pay the price of the next higher class (upgrading), if they are not able to buy the product at the price they have requested. This is described as diversion in its general sense by Belobaba and Weatherford (1 9 9 6 ), although the term diversion is often used to describe downgrading in fare classes. We model diversion by assuming that a fixed portion, s, of the unsatisfied lower price demand will join the higher price demand. For now, we assume 0 < s < 1. Thus, the proverbial newsboy sells papers at a higher price in the afternoon than in the morning; he then deliberately sets a limit to the amount of papers to sell in the morning and reserves some papers for the afternoon. Perhaps some of the customers who want to buy papers in the morning at the lower price will return in the afternoon. Under these assumptions, the units sold at the low and high fare classes are given by Qi = min{.Di, A ',P}, Q2 = m i n { X - Q i ,D 2 + s(Di — Q i)}, where s(Z?i — Qi) is the diverted demand from class 1 (low fare) to class 2 (high fare). We assume that D\ and D2 are independent: however, with diversion, the 86 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. effective dem and (and sales) in class 2 is dependent on the class 1 realization through the policy param eter P . The problem to maximize the expected profit can now be expressed as max7r(A\ P) = J2P jE [Qj] ~ cX j= i Define F(x) = 1 — F{x). Obviously, X > P > 0, which leads us to rewrite Q i = min{jDi, P}. Therefore, for X > P, dX dEjQ,] dP Our following derivations concern E[Q2]. V V e write Q2 = min{A’ — D 1.D 2 }, if P > D L , min{X — P, D2 + sD i — sP }, if P < D\. The value of Qo is shown in Figure 3.3. Integrating over the disjoint regions of Q2 in Figure 3.3, we obtain r P roc E[Q2] = / I ( X - D J M D J d D M D J d D i 0 J X —D1 + f P [ X~ °l D2f 2(D2)dD2f 1(D1)dD1 Jo Jo Y p X — P -j-sP — Do + [ 3 C D2 + s D l - s P ) M D 1)dD1M D 2)dD2 Jo Jp 87 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Figure 3.3: Value of Q2 for the derivation of E[Q2] X-Dl X-P X-P D2 0 1 X p r X —P roo + I (x ~ P ) M D l)dD,MD2)dD2 roo r OO + / / (X - P )fl(Dl)dDlf 2(D2)dD2. J x - p J P Taking the derivative w.r.t. X and after considerable algebraic simplification. dE[Q2\ dX = f F2(X - D l) f l(Dl)dDl Jo + F \(P )F ^ (X -P ) x ~p — ,X - P + s P - Do -I- Fl(- ) f 2{D2)dD2. (3 Similarly, taking the derivative w.r.t. P and simplifying, dE[Q2] = - SF\(P) - (1 - s)F\{P)F^X - P) - (1 - s) J X P~F\(X P + $P ~ D2-)f2(D2)dD2. Jo s dP (3 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. It can be shown that the second partial derivative of 7r(X, P ) w.r.t. X is negative, hence 7 r(X, P) is concave in X for any given P. However, tt(X, P ) is not necessarily concave in P (and hence is not jointly concave in (X, P)); this causes a slightly more complicated solution procedure. We discuss the optimal solution in three cases. C ase 1: 0 < P < X The first-order necessary conditions for an interiorly optim al (X , P) are dE[Qi] dE[Q2] _ Pl d X P2 d X dE[Qx\ , dE[Q2] __ n Pl dP P2 dP which, using (3.6) and (3.7), can be written as P 2 J qP F2(X - Dx) f x(Dx)dDx + Fl(P)F2{X - P ) + + [ X~P F\(X ~ P J r s P ~ D2)f,{D2)dD21 = c, (3.8) Jo s (Pl - sp2)Fl( P ) - p 2( 1 - s)Fi (P)F2(X - P ) - x ~p ^ - , X - P + sP - Do x — J-' 4 - s P — D o - p 2{ 1 - s) / F,{---------- - )f2{D2)dD2 = 0, (3.9) Jo s or equivalently in probability expressions, which we will use later to facilitate a marginal revenue discussion, P 2 [Pr{£>! - )-D 2 > X ,D x < P } + Pr{£>! > P} P r{D2 > X - P} + 89 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 4- Pr{s(£>i - P) + D2 > X - P ,D 2 < X - P} = c, (3.10) (pi - sp2) P r { A > P } - p2(l - s) Pr{£>! > P } Pr{P>2 > X - P} - - p2(l - s) Pr{s(£>! - P) + D2 > X - P, £> 2 < A' - P} = 0. (3.11) For given fi{-),f2(-) and parameters p\,p2,c and s. simultaneous equations (3.8) and (3.9) can now be solved in a standard numerical procedure for X and P . Note that pi is missing from (3.8) and (3.10) (partial derivative w.r.t. X ); this implies that once the booking limit P is fixed, the optimal X does not depend on the selling price pi. This is simply because an additional order quantity (or capacity) beyond P will affect only the sales in class 2. However, the optim al X given P still depends on the demand distribution in class 1, as the am ount of stock available to meet class 2 demand depends on the realization of class 1 demand. It is easy to see that the solution X from equations (3.10) and (3.11) will be non-negative, as the left-hand-side of equation (3.10) will be equal to p2 > c at X = 0 (for any P > 0). The next proposition can be used to determine if a solution from equations (3.8) and (3.9) is a local maximum. P ro p o sitio n 3.3 If {X, P) satisfies s + (1 — s)F2(X — P ) < pi/p2, then ir(X, P) is concave at (X , P ) . Proof: See Appendix A.2. 90 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. C ase 2: P = 0 In this case, the optim al X* is solved from equation (3.10), which reduces to p2 P r{sDi + D 2 > X*} = c. That is, X* is the newsboy quantity with price p2, cost c, and dem and sD x + Do- C ase 3: P = X This is the “unprotected” inventory allocation policy (all available units are open to class 1). We then have The above is identical to our previous decreasing price model described in Sec tion 3.2, except here pi < p2- The diversion fraction s does not affect the model in any way. The first-order necessary condition for optim al X is therefore given by (3.5). It is easy to see th a t the optimal X > 0 when px > c, since when X = 0, the left-hand-side of equation (3.5) is equal to px. We note, however, th a t in this case equation (3.5) is not sufficient to guarantee an optimal X , since 7r(X) may not be min{A' — D\, D2}, if < X if D x > X . 9 1 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. concave. The second order sufficient condition needs to be checked to ensure a local maximum. Now for given problem parameters, we first consider case (1) and solve equations (3.8) and (3.9) numerically to obtain (X,P). Proposition 3.3 is then used to deter mine if (X, P) is a local maximum. Any solution P < 0 or P > X is discarded. Next we consider the cases (2) and (3) and solve for optimal X respectively. Finally the best solution from the three cases is chosen as the global optimum. In all three cases, the solution X will be non-negative if P2 > Pi > c. We next give upper bounds on the optimal initial capacity X and the optimal protection level X — P. The proof of Proposition 3.4 is given in Appendix A.3. P ro p o s itio n 3.4 (a) The optimal X* satisfies X* < (?^“L (I — cjpf). where Gi(-) is the cdf of Di -I- Z?2- (b) The optimal X* and P* satisfies X* — P* < C?7l(l — c/pf), where Gs(-) is the cdf of sDi -F D2- The condition under which it is optimal to close the low fare class (Case 2) deserves a closer look. We next give two conditions for the optimal P to be zero. L em m a 3.1 If, for a given X , ^ < 0 at P = 0, then ^ f or any P > 0. P ro o f: See Appendix A.4. 92 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. P ro p o s itio n 3.5 For a fix e d X , th e co rresp o n d in g o p tim a l P = 0 i f a n d o n ly i f p 2{ 1 - s) Pr{s£>i + D 2 < X } < p 2 ~Pi- P ro o f: The partial derivative of ir(X, P) w.r.t. P at P = 0 is given by p2{l — s ) Pr{sZPi + D2 < X} — {p2 — pi) (see (A.l) in Appendix A.4). By Lemma 3.1, if p2( 1 — s) Pr{sDi + D2 < A'} < p2 — pi, then the partial derivative of tt(X.P) w.r.t. P is non-positive for all P > 0. Hence the optimal P = 0. On the other hand, if the optim al P = 0, then the partial derivative of tt(X , P) w.r.t. P should be non-positive at P = 0. P ro p o s itio n 3.6 I f p \ / p 2 < s, th e n o p tim a l P = 0. P ro o f: The partial derivative of tt(A, P) w.r.t. P is given by the LHS of equation (3.9) or (3.11). If Pi/p2 < s, then <9tt(X,P )fd P < 0 for all A " and P > 0. Hence P* = 0 . We note th at for the special case s = 0, it can be shown th at the last term in the left-hand-sides of equations (3.8), (3.9), (3.10) and (3.11) will disappear. Then, (3.11) can be rewritten simply as Pr{Z?2 > X - P } = Pl/ P2. (3.12) It can be shown that Proposition 3.3 will be satisfied from equation (3.12) so that any solution satisfying equations (3.8) and (3.9) will be a local maximum (in Case 93 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 1). Once the capacity, X, is fixed, equation (3.12) is known to be Littlewood’s (1972) rule. This is generalized for more than two classes as the Expected Marginal Seat Revenue (EMSR) model by Belobaba (1989). When s = 1, clearly an optim al solution is P = 0 and X = the newsboy quantity with price p2, cost c and demand Di + Do (Case 2). In this case, class 1 is closed to all customers so th at all demands are diverted to class 2. E x am p le. Consider demands D \ and Do that are uniformly distributed between 0 and 20. Letpi = 2 , p2 = 3, c = 1. For s values ranging from 0 to 1, we compute the optimal policy values in Table 3.2. A dash line (—) in the table indicates that the solution from the first-order conditions (3.8) and (3.9) does not satisfy 0 < P < X or that the solution is not a local maximum. Table 3.2: O ptim al X and P as a function of s Parameters: pi = 2, po = 3, c = L Case 1 (0 < P < X) Case 2 (P = 0) Case 3 I I O ptim al s A P >r(A\P) X * ( x :,P) A' ir(A\P) A" P' 7 T 0 23.33 16.67 20.93 13.33 13.33 23.67 20.90 23.33 16.67 20.93 0.1 23.12 15.48 20.97 14.33 15.31 23.67 20.90 23.12 15.48 20.97 0.2 22.75 13.80 21.05 15.33 17.23 23.67 20.90 22.75 13.80 21.05 0.3 22.12 11.29 21.23 16.33 19.11 23.67 20.90 22.12 11.29 21.23 0.4 20.92 7.26 21.64 17.33 20.93 23.67 20.90 20.92 7.26 21.64 0.5 — — — 18.33 22.71 23.67 20.90 18.33 0 22.71 0.6 — — — 19.33 24.44 23.67 20.90 19.33 0 24.44 0.7 — — — 20.34 26.11 23.67 20.90 20.34 0 26.11 0.8 — — — 21.39 27.77 23.67 20.90 21.39 0 27.77 0.9 — — — 22.51 29.31 23.67 20.90 22.51 0 29.31 1.0 — — — 23.67 30.90 23.67 20.90 23.67 0 30.90 Next, we arbitrarily fix s = 0.3 and vary p2. The results are shown Table 3.3. We note that, by comparing Case (1) and Case (3) (the “unprotected case”), the policy that limits sales in the lower priced class, thus allowing “reserved” units 94 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Table 3.3: Optimal X and P as a function of p2 Parameters: p i = 2, c = 1, s = 0.3 Case 1 (0 < P < X) Case 2 (P = 0) Case 3 (P = A') Optimal P2 X P *(X,P) X ~ (X ,P ) X *-(X ,P ) X' P - 7 T 2.0 20.00 20.00 13.33 13.00 7.85 20.00 13.33 20.00 20.00 13.33 2.2 20.85 17.95 14.72 13.91 9.98 20.93 14.72 20.85 17.95 14.72 2.5 21.67 15.25 16.98 15.00 13.31 22.11 16.95 21.67 15.25 16.98 3.0 22.12 11.29 21.23 16.33 19.11 23.67 20.90 22.12 11.29 21.23 3.5 21.61 7.49 26.02 17.29 25.10 24.88 25.10 21.61 7.49 26.02 4.0 20.18 3.41 31.38 18.00 31.200 25.86 29.43 20.18 3.41 31.38 5.0 — — — 19.00 43.63 27.35 38.41 19.00 0 43.63 6.0 — — — 19.67 56.22 28.45 47.70 19.67 0 56.22 8.0 — — — 20.52 81.67 30.00 66.63 20.52 0 81.68 Figure 3.4: The optimal profit 7r(X, P (X )) for given X P r o f i t . 20 15 1 0 1 0 20 25 for the future, higher priced demands, can generate higher expected profit. This is true even if no low fare demand is diverted to the high fare class ( 5 = 0). To evaluate the value of the ability to choose the initial capacity optimally, we compare the optimal 7 r* with the resulting maximum tt(X, P (X )) where X is given and P(X) is optimized for the given X . We set the problem param eters as = 2, P2 = 3 , .s = 0.3. Then X * = 22.12, P * = 11.29 and 7 r* = 21.228. Figure 3.4 shows the function ir(X, P(X)) when X is given in a range from 0 to 30. The difference between a point on the curve and the peak point is the value of choosing the capacity optimally. 95 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Figure 3.5: The profit function tt(X, P ) with optimally chosen X = X* P r o f i t : 20 15 10 5 20 0 1 0 3 Finally, it would be interesting to compare the sensitivity of profit w.r.t. P when X is chosen optimally (X* = 22.12). In Figure 3.5, we plot the function tt(X*, P). The profit curve here w.r.t P is flatter than the curve w\r.t. X (in Figure 3.4). Comparing Figures 3.4 and 3.5 suggests th at the profit is more sensitive to the choice of X than to P : i.e., the gain in profit with a well chosen initial capacity is greater than the gain in profit with an optimized booking limit. D iscussion o f R esu lts and. M anagerial Im plications The behavior of the optim al booking limit P in the increasing price model has clear implications. P = 0 corresponds to the closing of the lower fare class. The example indicates that the optim al P value is sensitive to the diversion fraction s. The s value at which the low fare class is closed for all customers can be indeed small. When the fare difference between the two classes increases, the critical s value to close the low fare class is further reduced. From Proposition 3.6, the critical value of s to close the low fare class is no greater than the fare ratio P1/P2 - This result has an intuitive expected m arginal cost interpretation: by closing the low fare class, we 96 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. forego a revenue of pi for each customer we reject, but gain sp2 from each customer diverted to the high fare class. The example also shows that the optim al P value decreases when the diversion fraction s increases, thus forcing more demand units into the higher fare class. Also, for given s, the optim al P decreases when the fare difference p2 — Pi increases. The protection level X — P for the high fare class is also sensitive to the diversion fraction s and the fare difference p2 — Pi- The optim al protection level increases in s and in p2 — Pi, as shown in our example. If there were a second replenishment opportunity for the high fare class, the "protection” level X — P would be equal to the newsboy quantity with unit revenue p2 and the effective demand smax{£>i — P, 0} + D2. As formally stated in Proposition 3.4b, the protection level X — P never exceeds the newsboy quantity with revenue p2 and the maximum effective Class 2 demand sDi + D2. So far, our model has focused on the static decision problem where permanent allocation of the seat inventory is determined for the two fare classes. In many appli cations, P can be recomputed based on continuously updated (remaining) capacity X and demand. In these cases, Proposition 3.5 gives the optimal closing limit for the low fare class, when Di and D2 are interpreted as the remaining demands during the two sales intervals (and X as the remaining capacity). The value of P determines if the next arriving customer should be granted a low fare. Proposition 3.5 implies that there exists an A 0 such that P* = 0 if (the remaining) A' < A T 0, and P* > 0 97 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. if X > X Q . Again, Proposition 3.5 allows for a simple expected marginal revenue (EMR) interpretation: rewriting the condition in Proposition 3.5, we have P2 P r{sD i + D 2 > X } 4- P2S Pr{sDi + D 2 — X } ■ > p \ . (3.13) The right-hand-side of the above is the marginal revenue to increase P from 0 to 1 unit (which will be sold since we assume Di > 0 with probability 1). The left- hand-side of the above is the expected marginal revenue lost when the total effective high-fare demand is greater than X (in which case we would lose a revenue of P2 ) or when the total effective high-fare demand is less than X (in which case we would lose only the diversion revenue P2 • s). We also note that (3.13) is a more general rule than a rule used in Belobaba and Weatherford (1996, equation (3)). 3.4 Concluding Remarks In both cases with decreasing and increasing prices, our results show that the opti mal X never exceeds the single newsboy quantity with the best possible unit revenue and the total demand (Proposition 3.2 and Proposition 3.4a). Moreover, all the op timality conditions allow for an interpretation in expected marginal cost and revenue when the revenues are properly weighted by the probabilities. These interpretations are consistent with those found in the PARM literature. Our numerical examples show that approxim ating the optim al order quantity by applying a single newsboy 98 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. model on an aggregated demand (the “average price” heuristic) or summing separate newsboy order quantities for each demand class (the “separate newsboys” heuristic) may lead to poor performances. We wish to point out that the alternative objective of maximizing the probability of achieving a targeted profit T (e.g. Lau 1980, Li, Lau and Lau 1991; Khouja 1995) results in X* > T [ (pL — c) in the decreasing price model with X * = T/(px — c) if Pi > c> p2 > ■ ■ ■ > pn- Finally, a possible direction for future research might be to extend the 2-class model with increasing prices to an n-class model. We expect the same managerial implications from the 2-class model; however, the mathematical expressions will be much more complex, since the sales in class j now depends recursively on the sales in all lower numbered classes. 99 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Chapter 4 A Pricing Model with Demand Learning Fashion goods such as ski-apparel or sunglasses are characterized by high degrees of demand uncertainty. Most of the merchandise in this category are new designs. Although some of the demand uncertainty may be resolved using sales history of similar merchandise offered in previous years, most of the uncertainty still remains due to the changing consumer tastes and economic conditions every year. Retailers of these items face long lead times and relatively short selling seasons that force them to order well in advance of the sales season with limited replenishment opportunities during the season. Demand and supply mismatches due to this inflexible and highly uncertain environment result in forced mark-downs or shortages. Frazier (1986) estimates that the forced mark-downs average 8 percent of net retail sales in apparel industry, which he states is also an indication of as much as 20 percent in lost sales from stock-outs. He estimates that the overall resulting revenue losses of the industry may be as much as $25 billion. 100 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. In 1985, U.S. textile and apparel industry initiated a series of business practices and technological innovations, called Quick Response, to cut down these costs and to be able to compete with foreign industry enjoying lower labor rates. Quick Response aims to shorten lead times through improvements in production and information technology. As a result, production and ordering decision can be shifted closer to the selling season which will help to resolve some uncertainty. Moreover, additional replenishment opportunities during the season may be created. See Hammond and Kelly (1991) and Section 2.3.3 for a review of Quick Response. Despite the efforts of domestic manufacturers to remain competitive in this in dustry, retailers are using more and more imports to source their apparel, preferring cost advantage over flexibility (see Sections 2.3.2 and 2.3.4). For most im ported apparel and some domestic apparel, managing demand through price adjustments is often the only tool left to retailers once the buying decisions took place. These adjustments are usually in the form of mark-downs in the apparel industry. Fisher (1994) notes that 25 % of all merchandise sold in department stores in 1990 was sold with mark-downs. Systems th at can intelligently decide the timing and mag nitude of such mark-downs may help balance the supply and dem and and improve the profits of these companies operating with thin margins. As discussed in Section 2.2.2, despite enormous amount of data made available to decision makers, such intelligent systems have found limited use in the apparel industry. Recent academic research such as Gallego and van Ryzin (1994) and Bitran and Mondschein (1997) 101 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. successfully model dynamic pricing of a given stock of items when the demand is probabilistic and price sensitive. These studies assume th at the retailer’s estimate of the demand does not change over the course of the season. However, substantial amount of uncertainty about the demand process can be resolved using the early sales information. The purpose of this chapter is to develop a dynamic pricing model that incor porates demand learning. By demand learning, we mean learning using the early sales information during the selling season as opposed to improving forecasts over time before the start of the season. A considerable portion of demand uncertainty can be eliminated by observing early sales in the apparel industry. A consultant at Dayton Hudson Corp. states “a week after an item hits the floor, a merchant knows whether it’s going to be a dog or a best-seller” (Chain Store Age 1999). For our pricing only model, we assume that ordering decision has already been made with the best use of pre-season information and no further replenishment opportunities are available to the retailer. Basically, the model uses a Bayesian approach to update retailer’s estim ate of a demand param eter. Our model enables us to summarize sales and price history in a direct way to set the problem as a computationally feasible dynamic program. We observe structural properties of the optimal policy and run computational tests to study the following issues. First, we discover that the price in future periods is an increasing function of observed sales. Second, we study how the accuracy and degree of uncertainty of the initial demand estimates, starting stock 102 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. levels and price sensitivity of customers impact optim al price paths and expected revenues. We are also interested in finding the conditions under which earlier sales information has the most impact on revenues and whether it is always optimal to use this information. Further, we explore the trade-off between more information and early control in pricing decisions. Finally, we extend the model to account for the possibility of re-ordering during the selling season. This helps us to understand the possible trade-offs for using quicker but more costly domestic manufacturing to achieve such flexibility. Next, we review literature on Bayesian learning in inventory control and dynamic pricing of fashion goods. We present our basic model in Section 4.2. We describe the structure of the optim al solution in Section 4.3. Our computational analysis is in Section 4.4. Section 4.5 studies the effects of inventory flexibility during the horizon. 4.1 Literature Survey Inventory models that incorporate the updating of demand forecasts have been studied by many researchers. Most of these models utilize a Bayesian approach to update demand parameters of a periodic inventory model. Demand in one period is assumed to be random with a known distribution but with an unknown param eter (or unknown parameters). This unknown param eter has a prior probability distribution which reflects the initial estimates of the decision maker. Observed 103 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. sales are then used to find a posterior distribution of the unknown param eter using Bayes’ rule. As more observations become available, uncertainty is resolved and the distribution of the demand approaches its true distribution. The prior distribution of the unknown parameter should be such th at the posterior distribution is similar to the prior which could be calculated easily. In addition, the demand distribution and the distribution of the unknown param eter should enable the decision maker to summarize information such that a dynamic program to solve the problem is computationally feasible. See DeGroot (1970, chapter 9) for such distributions. Demand learning in inventory theory using a Bayesian approach is first studied by Scarf (1959). He studies a simple periodic inventory problem in which at the beginning of each period the problem is how much to order with the assum ption of linear inventory holding, shortage and ordering costs and an exponential family of demand distributions with an unknown param eter. The distribution of the unknown param eter is updated after each period using Bayes’ rule. He formulates the problem as a stochastic dynamic program and among other results, shows that the optimal policy is to order up to a critical level and the critical level for each period is an increasing function of the past cumulative demand. Iglehart (1964) extends the results of Scarf (1959) to account for a range family of distributions and convex inventory holding and shortage costs. Azoury and Miller (1984) show th at in most cases non-Bayesian order quantities are greater than Bayesian order quantities, but also state that this may not always be true. The dynamic programs used in these 104 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. studies have two-dimensional state spaces, one for the starting inventory level and one for the cumulative sales. Scarf (1960) and Azoury (1985) show that the two- dimensional dynamic program can be reduced to one-dimensional for some specific demand distributions. A particular form of Bayesian approach, to demand learning is assuming Poisson demand with an unknown rate in each period. The unknown demand rate’s prior distribution is assumed to be Gamma, resulting in an unconditional prior distribu tion of demand which can be shown to be Negative Binomial. Posterior distribu tions are also Gamma and Negative Binomial whose param eters can be calculated by using only cumulative dem and. These specific distributions are used to model inventory decisions of aircraft spare parts by Brown and Rogers (1973). Popovic (1987) extends the model to account for non-constant demand rates. Demand learning models are most valuable to inventory problems of style goods th at are characterized with m oderate to extreme degrees of demand uncertainty that is resolvable significantly by observing early sales. M urray and Silver (1966) use a Bayesian model in which the purchase probability of homogeneous customers is unknown but distributed priorly with a Beta distribution. This distribution is updated after each period to optimize inventory levels in succeeding periods. Chang and Fyfee (1971) present an alternative approach to demand learning. Their model defines the demand in each period as a noise term plus a fraction of total demand which is a random variable whose distribution is revised once the sales information 105 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. becomes available each period. Bradford and Sugrue (1990) use Negative Binomial demand model described earlier to derive optimal inventory stocking policies in a two-period style-goods context. Fisher and Raman (1994) propose a production planning model for fashion goods which uses early sales information to improve forecasts. Their model, which is called Accurate Response, also considers the constraints in the production systems such as production capacity and minimum production quantities. Iyer and Bergen (1997) study the Quick Response systems, where the retailers have more information about upcoming demand due to the decreased lead times. They use Bayesian learning to address whether the retailer or the manufacturer wins under such systems. Iyer and Eppen (1997a) develop a different methodology for Bayesian learning of demand. The demand process is assumed to be one of a set of pure demand processes with discrete prior distribution. This distribution is updated periodically based on Bayes’ rule. This demand model is used in a dynamic programming formulation to derive the initial inventory levels and how much to divert periodically to a secondary outlet for a catalogue merchandiser. Iyer and Eppen (1997b) use the same demand model to study the impact of backup agreements on expected profits and inventory levels for fashion goods. Gurnani and Tang (1999) study the effect of forecast updating on ordering of seasonal products. Their model allows the retailer to order at two instants before the selling season. The forecast quality may be improved in the second instance, but the cost may either decrease or increase probabilistically. 1 0 6 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. All of the studies above ignore one crucial aspect of the problem: pricing. In economics literature, Lazear (1986) studies clearance sales where he uses Bayesian learning to update the reservation price distribution after observing early sales in the season. However, his model considers the initial and the mark-down prices of a single item and thus lacks the dynamics of price adjustments for a stock of items. Balvers and Casimano (1990) incorporate Bayesian learning in pricing models, but they assume a completely flexible supply and ignore inventories that link the pricing decisions. Style goods, on the other hand, face supply inflexibility as a result of short seasons, long lead times and limited production capacities. This characteristic of the problem gave rise to models such as those in Gallego and van Ryzin (1994) and Bitran and Mondschein (1997) that dynamically price the perishable good over the selling season. Both of these models assume th at there is no replenishment opportunity and the only decisions to be made are the timing and magnitude of price changes over the course of the season. Gallego and van Ryzin (1994) use a Poisson process for demand where the demand rate depends on the price of the product. Monotonicity results as a function of the remaining stock level and remaining time in the selling season are derived via a dynamic continuous-time model. Among other results, they show th at the optim al profit of the deterministic problem, in which demand rates are assumed to be constant, gives an upper bound for the optimal expected profit. For the continuous price case, fixed-price heuristics are shown to be asymptotically optimal. For the discrete price case, a deterministic solution can 107 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. be used to develop again asymptotically optimal heuristics. Feng and Gallego (1995) derive the optimal policy for the two price case. In Bitran and Mondschein’s (1997) model, the purchase process for a given price is determined by a Poisson process for the store arrival and a reservation price distribution. They show th at the model is equivalent to the model in Gallego and van Ryzin (1994). They also show that the loss associated with preferring a discrete-time rather than a continuous-time model is small. Smith and Achabal (1998) study clearance pricing in retailing. Their model is deterministic, but incorporates impact of reduced assortm ent and seasonal changes on demand rates. Recently, two closely related papers discuss Bayesian learning in pricing of style goods. Subrahmanyan and Shoemaker (1996) develop a general periodic demand learning model to optimize pricing and stocking decisions. As in Eppen and Iyer (1997a,b), they use a set of possible demand distribution functions for each period and a discrete prior distribution that tabulates the probability of these possible demand distributions being the true demand distribution. This discrete distribution is updated after each period using the Bayes: rule. The information requirements are extremely large in a general model as updating requires the history of sales, inventory levels and prices in each period. They present com putational results on specific dem and and price parameters. Bitran and Wadhwa (1996) consider only the pricing decisions utilizing the two-phased demand model and discrete-tim e dynamic programming formulation in Bitran and Mondschein (1997). A Poisson process for 108 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. store arrival and a reservation price distribution are used to define the purchase process. They assume that uncertainty is involved in a param eter of this reservation price distribution. An updating procedure on this param eter is proposed such that the rate of the purchase process has Gam ma priors and posteriors. The methodology allows them to summarize all sales and price information in two variables. They present computational tests to show the im pact of dem and learning on prices and expected profits. 4.2 Model 4 .2 .1 D em an d M o d el Assume that there are N points in time that the pricing decisions can be made. W ithout loss of generality, assume that each period in consideration is of unit length. The demand in each period has a Poisson distribution. The demand rate is separable and consists of two separate components: a base demand rate A , and a multiplier m(p) for the charged price p. The Poisson rate is equal to A(p) = m(p) A . 109 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. W ithout loss of generality, we assume m (l) = 1. Although our model does not depend on a particular demand function, we use exponential price sensitivity A(p) = ae-7P to find m (p ) = (4-1) in our computational analysis. Exponential price sensitivity and multiplicative de mand functions are widely used in practice and research (see Sm ith and Achabal (1998) and Smith et. al (1998) for examples). The distribution of demand given the price and base rate is given by, / ( l |A, p) = £ Z ^ ! W , for . = 0, 1, 2, . . . X l We assume that there is uncertainty only in the magnitude of demand, but not in the functional form of the price demand relationship (e.g., price elasticity). T hat is, we assume that we have perfect information about the function m(p) and that the uncertainty of the demand rate for a given price can be totally characterized by the uncertainty in the base demand rate A . Consequently, observing sales will facilitate learning on the magnitude of demand only. We assume that A is distributed as Gamma with parameters a and /?. The probability density function for Gam ma is given by, 8a AQ-1e-/3A A > °- 1 1 0 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Then, the prior distribution (unconditional of A) of demand in the first period will have Negative Binomial distribution: D l ~ N B ( a ^ if the price for period 1 is pi. The probability function is given by, ( \ a + x — 1 \ x / 0 Y f »*(P.) V _ for r = 0, 1, 2, . . . ^P + m(pi)J \P + m(pi)' This uses the fact that if A is distributed w ith Gamma with parameters a and , 8, kX is distributed with Gam m a with param eters a and (3/k for any m ultiplier k. An unconditional distribution that is Negative Binomial is consistent w ith the high uncertainty involved in fashion goods as the variance to mean ratio of a Negative Binomial random variable is always greater than 1. Additional support is provided in Nahmias and Smith (1994) where they discuss the suitability of Negative Binomial demand for a retailer system. If the realized demand in period 1 is x\, Bayes’ rule implies that the posterior distribution of A will be again Gamma w r ith parameters a+X]_ and (3 + m(j>i). Then, given that the price in second period is p% , the prior distribution of dem and in the second period will have a Negative Binomial form: D t - N B f c c + xt, /3 + m(p0 P + m(pi) +m{p2) J 111 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Likewise, when the realized demand is x l 5x2, ... ,x n_i and prices charged are P i,p 2, - - - ,pn- 1 in periods 1 ,..., n — 1, the prior distribution of demand in the nth period will also have a Negative Binomial distribution: „ atr> ( . ^ + Dn ~ N B a + 2^ Xi, „ t - y , V p + E ? = 1 m{pi) J given that the price in nth period is pn. Denote cumulative sales prior to period n as X n_! = x, and cumulative price multipliers prior to period n as Mn_i = ”r(Pi)- - A T n_i and iV /n_i summarize all the information in periods 1, . . . ,n — 1 and are called the sufficient statistics for estimating demand in period n, for a given price pn. The unconditional demand distribution for the nth period will have a mean of p r m (a + E?=il xi)m{pn) E [ D A = fl + Z ST .'m fe) which basically means that the sales rate in the nth period is a linear function of sales rate in the earlier n — 1 periods. This is in fact not surprising. Carlson (1983) has studied sales data of apparel merchandise from a m ajor department store to see whether the sales rate after a mark-down is predictable. Given an initial price and a mark-down percentage, he has shown th a t past mark-down sales rate is in fact a linear function of pre mark-down sales rate. Our model completely agrees with this empirical result. 112 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. It is also worthwhile to see how the mean and variance of the unconditional distribution of demand behaves as n increases. For simplicity of the exposition, assume that the price is equal to 1 throughout the season so that m(pi) = 1 for all i. The expected value and variance of the unconditional demand are given by, Q + E £ jl Xj (3 + n — 1 (tt + EfcTi gi)(/?+ w) (f3 + n — I )2 It is easy to see that as n approaches infinity, both the mean and variance approach x. average of xt -, which is the true rate of the Poisson process. We note th at the con vergence is faster if is smaller. This corresponds to higher degrees of uncertainty in the decision maker’s initial estimate of demand rate, and thus more reliance on actual sales information in estim ating future demand. While our analysis so far assumes that the periods are identical except for the prices charged, our model allows us to permit seasonality and any other extensions as long as the multiplicative nature of the dem and function is preserved. T hat is, as long as we can state the demand rate in period i as Ai(p,-, n) = m(pi, Ti)X, (where m now is a more general function of price pt - and seasonality factor r,-) our model is applicable. Also, uneven period lengths are easily accountable. In a 113 E[Dn] = V[Dn] = R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. situation where the periods have varying lengths and seasonality, the state variable Mn_1 can be represented as T l— 1 M„_1 = m (#> i=l where ti is the length of the period i. The details of the derivation for two periods are given in Section 4.4. 4.2.2 P ricin g M o d el The problem is determining prices in periods 1... . . N so that a fixed stock of Iq items is sold with maximum expected revenue. For simplicity of the presentation, We assume that the inventory holding costs within the selling season are negligible. We note th a t the it is very easy to relax this assumption in the context of our model. We use a discrete-time dynamic programming model. Let Vn(/n_i, J n_i, ik/n-i) be the maximum expected revenue from period n through N when the initial inventory is / n_ L and the cumulative sales and cumulative price multipliers are A'„_i and Mn_i, respectively. Mote that /„_! = max{0, 10 - Xn_!}, 1 1 4 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. and can be dropped from the formulation. But we keep /n_i in our formulation for ease of exposition. Also let ps be the salvage value for any inventory left unsold beyond period N . Backward recursion can be w ritten as Vn{In- UX n- UM n-{) = max E P n > P s pn min{£>n, I n- i} + Vn+l - Dn)+, X n- i + Dn, M n- i + m(pn)j (4.2) X n- u Mn- i + m(pn) Boundary conditions are V,v+i(/jv; Ajv, AAv) = PsI m, for all 7/v, JV,v, A/iv, (4-3) Vn(07X n- i ;M n-i) = 0, for all n, X n_L , Mn_i- (4.4) First condition states that any left over merchandise has only salvage value when the season ends at the end of period N . The second conditions states th at the future expected profits are zero, when there is no merchandise left in stock since re-ordering is not allowed. This property also allows us to avoid the problem with censoring of demand information due to unsatisfied demand. Because, in case of excess demand (when the inventory is exhausted), there are no further decisions 1 1 5 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. to be made and no further information about demand is required. The dynamic program can be solved by starting with the iVth period and proceeding backwards. 4.3 Properties of the Optimal Solution We solved many problems with different sets of parameters to investigate the struc tural properties of the optimal policy. The main observation we have made is that higher sales in earlier periods always translate into higher prices in future periods. The intuition behind this result is the following. First, higher sales in earlier periods m ean (stochastically) higher demand in future periods because of the Bayesian na ture of the demand distributions. Second, higher sales in earlier periods also mean lower left-over inventory for future periods since there are no further replenishment opportunities. Thus, higher sales in earlier periods inflate the expected demand while decreasing the available supply in future periods. This allows the seller to charge higher prices to balance the demand and supply. These results is stated in the following conjecture, which we have not found an analytical proof. C o n je c tu re 4.1 Given the cumulative price multipliers, M n- i = m(pi), the optimal price p* is a non-decreasing function of past cumulative sales, X n-i = W hen there is a lim ited number of allowable prices, we also observe the following which is closely related to the observation above. For a discrete price set V, we see 116 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. that there is, at most, one range of previous cumulative sales where each price is optimal. Namely: Conjecture 4.2 Given the cumulative price multipliers iV/„_L = l m(jpi), there exists a subset S of V that consists of feasible prices. Then, there exists [< S | — 1 non-negative integers ai < a 2 < ... < &|s|-i such that, P n = P O a i —l < X n < O i if p is the ith smallest price in S, ao = 0 and a|s| = oo. This helps the retailer tabulate the price it will charge for each period based on cumulative sales up to that period. In order to show how the optim al policy works, we include the following example. Exam ple: The retailer has 30 units to sell in a season of length 1. When the price is set to 1, the demand is Poisson with a rate distributed with Gamma with parameters a = 10 and d = 0.5. Thus, mean demand is a/,3 = 20 a nd variance of the demand is a(j3 -F l )//?2 = 60. W ithin the season, there are two periods of equal length. The retailer can charge different prices in these periods from a discrete set P = { 0 .50,0.55,.., 0.95,1.00}. The price affects the demand in an exponential m an ner with 7 = 3 described in equation 4.1. The mean total demand is given as follows with the prices in V. 1 1 7 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. price 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 mean demand 89.6 77.2 66.4 57.2 49.2 42.3 36.4 31.4 27.0 23.2 20.0 The problem is to find the price in the first period and the form of the pricing policy in the second period so as to maximize the total revenues. We solve the prob lem with the dynamic program given in equations (4.2-4.4). The optim al policy is to charge 1 in the first period and then charge the following prices in the second period based on the dem and realization in the first period. 1st prd. sales Xi 0-5 6 7-8 9-10 11-12 13-14 15 16-17 18 19-30 2nd prd. opt. pr. P 2 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 The resulting optim al expected revenue is 23.248, about 0.775 per unit. 4.4 Computational Study We first note that although pricing through a demand learning model is the best a retailer can do, it is not necessarily optimal. The optimal policy depends on the true value of underlying base demand rate. The optimal prices can be computed by using a dynamic programming formulation which uses the true Poisson demand dis tribution. The performance of the demand learning model is based on how accurate the retailer’s initial dem and estimate is and how fast the retailer can learn about the true demand process. Note that prior to the start of the season, the retailer 118 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. assumes th at the base demand rate is distributed with Gamma with parameters a and 8. The expected value and variance of this random variable are given by, E iM = | and V'ar[A] = Hence, a//3 defines the accuracy of the point estimate. Given a fixed ratio a /8, the magnitude of /? (or a) defines the variance of the initial estimate, and hence the decision m aker’s reliance on his prior beliefs about demand. When (3 (or a) is large, the retailer is confident about its initial estimate, and it hardly updates its demand estimate based on observed sales. As j3 (or a) gets smaller, more weight is given to the observed sales in estimating future demand. We note that a and /? are the only information available to the retailer before the season and thus we use the Bayesian distributions while calculating the expected revenues and prices. Our computational study aims to uncover how expected revenues and optimal price paths are affected by initial forecast uncertainty, as well as the starting stock level IQ (and how far it is from the point estimate of the demand, a / (8) and price sensitivity of the customers characterized by 7 . We also explore the conditions under which the early sales information has the most impact on revenues by comparing the solution with that of a model which does not use demand learning. Note that, in such a no-learning model, the retailer assumes th at the base demand rate is distributed with Gamma with parameters a and /3 independent of the observed sales. 119 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Often the cost and time associated with price adjustm ents (e.g., changing the price tags) force the retailers to make limited number of such adjustments. Also it is believed th at frequent price changes deteriorate customer trust. In these cases, the timing of price changes are im portant. As the retailers defer their limited pricing adjustments, they have a chance to learn more about the demand process and thus are better able to predict the future demand. However, as the retailers delay their action, they are in effect decreasing the amount of tim e they have control over the process. Earlier pricing decisions may be crucial in balancing the demand with supply in a profitable manner. Therefore, we also include the timing of price changes as a decision variable in our computational study. We compare the optimal switching times from the learning model to the optimal switching times from a no-learning model. We are interested in characterizing the conditions which drive the retailers to defer their pricing decisions in an effort to learn more about the demand process. For the purposes of computational study, we assume th a t there is only one chance to change the price. The resulting model is a special two-period case of the model described earlier. In addition, we assume th at the salvage value is zero and we relax the assumption that the periods are of equal length. This model is briefly described below. Demand in one period is again Poisson with rate A (p) = m(p) A , 120 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. where m(p) = e 7^ p Assume again that A is unknown, but distributed with t and the length of the second period is 1 — t. Then prior distribution of demand in the first period is Negative Binomial: given that the price is pi in the first period. If the realized demand in period 1 is xi and the charged price is pi, p osterio r distribution of A would be again Gamma with parameters a ■ + - xi and ,8 + m(pi)£. Then the priori distribution of demand in the second period is Negative Binomial: given that the price is P 2 in the second period. The problem is to set prices pi and p2 such that an initial stock of Iq items are sold with maximum expected revenue. The problem again can be solved by a dynamic program for each t. Given the price and observed demand in the first period, the second period problem is Gamma with parameters a and (3. We assume that the length of the first period is (4.5) (4.6) ^ ( / i,Xi,pi) = m a x £ p2 min P2>0 |_ 1 2 1 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. where the distribution of D 2 given x Xlpx.p2 is given by equation (4.6). Note also that Ix = max{/o — xx, 0} and V2{Q,xu ci) = 0. The problem of the first period is V i(/0) = m a xE pi min{I0,D x} 4- V2( (/0 - Dx)+,D x,px) px where the distribution of D x given px is shown in equation (4.5). When there is no learning, the demand distribution for the second periods does not depend on the price or the sales in the first period. While the first period’s demand distribution is still given by equation (4.5), the second period’ s demand distribution has different parameters although it is still Negative Binomial: Then, the dynamic program for the no learning model can be constructed as: where the distribution of D 2 given p2 is given by equation (4.7). Note again that Ix = max{ /0 — x x, 0} and V2(Q) = 0. The problem of the first period is (4.7) ^ ( A ) = maxi? p2 m in{/1, D2} p2 , Vi(/0) = m a x E px min{/0, D x} 4-V2[ (/0 — T>i)+ p i > o L V (< P i , where the distribution of D x given px is given by equation (4.5). 1 2 2 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 4.4.1 T h e d esign o f th e s tu d y We set the mean of the demand for the whole season to be 20 if the price is set to 1, the base price. Denoted by p, this is the only fixed param eter of our study and often it is used to normalize other variables and revenues. Starting inventory level runs from 10 to 30 (0.5 p to 1.5 p) with 2 units increments in each problem. While the mean of the demand a / 8 is fixed at p = 20, a runs from 10 to 40 with two units increments (thus 8 runs from from 0.5 to 2). The variance (a2) in our analysis is the prior estimate of the whole season's dem and if the price is set to 1. a2 is given by a(/3 + l)/fi2 and runs from 60 to 30 (3 p to 1.5 p). The price sensitivity measure 7 takes 11 values: from 1 (inelastic demand) to 3 with increments of 0.2 units. Thus, we have solved 1,936 problems. The decision variables, the initial price pi and the second price P 2 based on the demand realization in first period can take on 20 values, between 0.55 and 1.50 with increments of 0.05. We assume that the season is of unit length and the pricing decisions can be made at 0.2, 0.3, 0.4, 0.5, 0.6, 0.7 and 0.8. Hence, each problem in fact is solved 7 times with different switching times and the best solution is picked. For each problem, we have computed the optimal revenues, optimal first price and optim al second price policy with and without learning. 4.4.2 T h e im pact o f sta rtin g in ven tory We study the effect of starting inventory on the optim al revenues, optim al price paths and performance of learning model over the no-learning model. We use a 123 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Figure 4.1: O ptim al revenues as a function of starting inventory 1.3 1.2 < D 3 C < D > © 0.9 a . o T 3 © M ( O E 0.8 o c 0.7 var=1.5mu,gamma=1.5mu ----- var=1.5mu,gamma=3.0mu -e— var=3.0mu,gamma=1.5mu -h — var=3.0mu,gamma=3.0mu -e— 0.6 0.5 0.6 0.8 1 normalized starting inventory 1.2 1.4 range for the starting inventory of [10. 30] or [0.5 /i,1.5 fj\. Optimal revenues are also normalized to 20 which is the maximum revenue that can be obtained from 20 units of stock by charging the base price 1. Figure 4.1 depicts the impact of starting inventory on optimal revenues for 4 different variance-7 pairs. For all parameters, optimal revenues increase with in creasing starting stock level. However, marginal revenues from each additional is decreasing as the customer base is not changing with the starting stock level. When the stock level is low as compared to demand, the retailer is able to charge the higher prices for all of its inventory. As the starting stock level increases, the retailer has to introduce lower prices earlier in the season to be able to deplete its inventory by the end of the horizon. O ptim al first price and expected optimal second price are 124 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Figure 4.2: Optim al first and second prices as a function of starting inventory (vari ance = 3 fj.) p1A * (gamma=1.5mu) ----- E[p2A *] (gamma=1.5mu) -a— p2A * (gamma=3.0mu) -i— E[p2A * ] (gamma=3.0mu) -e— 1.4 1.3 1.2 tn < D U a. 0.9 0.8 0 .6 0 .8 1 .2 1.4 1 normalized inventory shown in Figure 4.2 for cr2=3 {jl and 7 = 1.5 or 7 = 3. The initial price is always higher than the expected second price, while both decrease with the starting stock level. Figure 4.3 depicts the impact of learning on optimal revenues for 4 a2- j param eter pairs. Learning has less value when the starting stock level is extremely low. The retailer can simply charge same higher prices in both periods, without any need to gather additional information (note that prices are constrained to be between 0.55 and 1.50). As the starting inventory gets higher reaching a moderate level, the demand information becomes more valuable in the selection of the second price and the performance of learning model improves. This behavior can be seen for variance = 3 fj. and 7 = 3. A similar argument holds when the stock level is extremely high; 125 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.3: Comparison of optimal revenues with, and without learning as a function of starting inventory 1.012 var=1.5mu,gamma=1,5mu ----- var=1.5mu,gamma=3.0mu -a— var=3.0mu,gamma=1.5mu -*— ■var=3.0mu,gamma=3.0mu -o— < 3 > C c 1.01 3 C O > o 1.008 to e Q . 2 1-006 O ) c ( O © 5 1.004. o 3 C © > o < a H 1.002 a . o 1 .2 0.8 1.4 0 .6 1 normalized inventory in this case the retailer can charge lower prices in both periods. We observe both behaviors for variance = 3 n and 7 = 3. The optim al switching times with learning and no-learning models are shown in Figure 4.4. While the switching times for the no-learning model are quite stable, the learning model’s switching time decreases w ith the starting stock level. The intuition behind this result is the following. In this particular case, as the starting stock level increases, there is need for an earlier action to balance the supply and the demand. The learning model gets stronger signals showing that the demand will be short of the available supply and calls for earlier switching times. 126 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.4: Comparison of switching times with and without learning as a function of starting inventory ( 7 = 2fi, variance=2 /z) 0.8 0.75 3 £ c o i t c n E E to a t 0.7 [ * ■ 0.65 3 E co » © o c 0 .6 n j < 5 > tn © E 0.55 cn c . e o 0.5 learning model no learning model 0.45 0.4 0 .6 0 .8 1 .2 1 normalized inventory 1.4 4.4.3 T h e im p act o f price sen sitiv ity We analyze how price sensitivity of customers may affect the optimal revenues, optimal price paths, performance of the learning model and optim al switching times. The measure for price sensitivity is 7 which takes values from 1.2 (which corresponds to a fairly inelastic demand) to 3 (which corresponds to a fairly elastic demand). Figure 4.5 shows the normalized optimal revenues for 4 starting inventory-variance pairs. First note th at, if the supply was unlimited, optim al revenues would increase with the price sensitivity of demand, as the retailer is b etter able to manipulate the demand. We observe this behavior in our study when the starting inventory is high comparative to the expected demand with base price (when starting inventory = 127 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.5: Optimal revenues as a function of price elasticity (7 ) 1.25 starting_inv=1.0mu,variance=1.5mu ----- starting_inv=1.0mu,variance=3.0mu -a— starting_inv=1.5mu,variance=1.5mu -h — startingjnv=1.5mu,variance=3.0mu - e— 1.15 CO CD D e CD > CD 1.05 T3 C D N C D E o c 0.95 0.9 0.85 1 .2 1 .6 1.4 1 .8 2.4 2 .6 2 2 .2 2 .8 3 gamma 1.5 n). However, when the starting inventory is not high, we observe that optimal revenues do not necessarily increase with price sensitivity of demand. This behavior is observed when the starting inventory equals 1 1. Optimal first period prices and expected optimal second period prices are shown in Figure 4.6 for starting inventory of 1.5 fj, and 1.0 fi and for a2 = 3 1 1. As 7 increases all prices go down, until 7 reaches a critical level. For moderate levels of price sensitivity, the retailer uses moderate discounts to optimize its profits. However, as the price sensitivity gets larger, the retailer is able to match its demand with its supply by small changes in its price, resulting in higher prices after a critical level of 7 . Figure 4.7 shows the comparison of optimal revenues with and without learning for 4 different starting inventory- variance pairs. As 7 increases, the performance of the learning model also increases. 128 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.6: Optimal first and second prices as a function of dem and elasticity (vari ance = 3 fi) 1 .2 p1A * (starting_inv=1.5mu) ----- E[p2A * ] (starting_inv=1.5mu) -e— p1A * (starting_inv=1.0mu) -> — E(p2A * ] (starting_inv=1.0mu) -o— 1.15 1.05 cn CD u Q . 0.95 0.9 0.85 0 .8 2 .8 2.4 2 .6 3 1 .8 2 2 .2 1 .2 1.4 1 .6 gamma However, when the demand is highly elastic (7 is high), the retailer can manipulate the demand with small changes in price. Since the price set is discrete, this limits the retailer’s choice to only a few prices. Thus, information gathered by the learning model can not be used as effectively, hence the performance of the learning model deteriorates for higher 7 . Figure 4.8 shows the optim al switching times for learning and no-learning models for varying price sensitivity levels. As 7 increases, switching times also increase for both models. The intuition is the following. As 7 increases, the retailer is able to match the dem and with supply in shorter tim e and with smaller price changes. This allows the retailer to keep its price high towards the end of the selling season. 129 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.7: Comparison of optimal revenues w ith and without learning as a function of dem and elasticity 1.012 var=1.5mu,starting_inv=1 .Omu var=1.5rnu,starting Jnv=1,5mu var=3.0mu,starting_inv=1 .Omu var=3.0mu.startinq inv=1.5mu cn c c 1.01 < a jd 0 1 © © c © > © 1.008 c o E a . ° 1.006 a t c e co © * 1.004 © 3 C © > © ■ < 5 J 1.002 a . o 2.6 2.8 3 1.8 2 2.4 1.2 1.4 1.6 gamma Figure 4.8: Comparison of switching times with and without learning as a function of dem and elasticity (variance = 1.5/i, starting inventory = fj.) 0.75 learning ----- no-learning -b— 0.7 0.65 cn 0 ) E a > c J C . 0 1 cn 0.6 0.55 0.5 0.45 2.8 3 1.8 2.2 24 2.6 1.2 1.4 1.6 2 gamma 130 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.9: O ptim al revenues as a function of variance 1.3 gamma=3.0mu,startingi_inv=1 .O m u ----- gamma=3.0mu,starting_inv=! .5mu -a— gamma=1.5mu,starting_inv=1 .O m u -< — - gamma=1.5mu,starting_inv=1.5mu -o— 1.25 1.15 cn < D 3 C O > © T3 © ~ c a £ o c 1.05 0.95 0.9 0.85 2.2 2.4 variance measured in mu 3 2.8 1.8 2 2.6 1.6 4 .4 .4 T h e im p act o f dem and variance We study how demand uncertainty affects the optimal revenues, optimal price paths, optimal switching times and performance of the learning model. Note that the variance in our analysis is the prior estimate of the demand variance, i.e., cr2 = a(/3 + 1)/(32. We normalize this with the mean of the demand and the normalized variance takes the values in the range [1.5 /z, 3 /z] (or [30,60]). Figure 4.9 shows how the optimal revenues change with variance for 4 pairs of starting inventory-7 values. In all cases, optim al revenues decrease with demand variance. The optimal first period and expected second period prices are depicted in Figure 4.10. However, we were not able to pick any meaningful pattern in any of our cases, presumably due to complex interactions of multiple factors in the model. Figure 4.11 shows the 131 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. Figure 4.10: Optimal first and second prices as a function of variance (starting inventory = 1.5/z) p1A * (gamma=3.0mu) ----- E[p2A * ] (gamma=3.0mu) -e— p1A * (gamma=1.5mu) -h — . E[p2A * ] (gamma=1.5mu) -e— 0.92 0.9 0.88 0 1 < D O Q . 0.86 0.84 0.82 0.8 2.8 2.6 2.4 1.6 2.2 variance measured in mu performance of the learning model as a function of demand variance. Although with some noise, we observe that learning model performs better as the demand becomes more uncertain. This shows the importance of early sales information for products that has high fashion content, (e.g., women’s apparel). We could not observe any particular pattern in the switching times for the learn ing and no-learning models (Figure 4.12.) 4.4.5 T h e th ree p eriod problem In this section we would like to see whether our observations for the two period problem hold for the three period problem. In addition, we also study the impact of mark-down restrictions (that the prices can not go up over the season) on optimal 132 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.11: Comparison of revenues with and without learning as a function of demand variance 1.014 gamma=3.0mu,starting_inv=1 .O m u ----- gamma=3.0mu,starting_inv=1.5mu -a— gamma=1.5mu,starting_inv=l.0mu h— gamma=1.5mu,starting_inv=1.5mu -e— at c c 1.012 (5 a t 1.01 3 C < D > © 1 1.008 Q . O c £ 1.006 c o © © 3 C O > C D 1.004 2.2 2.4 variance measured in mu 2.8 1.6 2.6 3 1.8 Figure 4.12: Comparison of switching times with and without learning as a function of demand variance (7 = 3//, starting inventory = 1/i) 0.75 learning model ----- no learning model -a— 0.7 1 ! ■ 0.65 at a t E at c £ .I C J 0.6 S at 1 5 E Q . O 0.55 0.5 0.45 2 2 2.4 variance measured in mu 2 8 1.6 1.8 2 2.6 3 133 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.13: Comparison of prices with and without mark-down restriction ( 7 = 2 , variance = 3/z) pv 0 .9 5 0 .9 S. 0.85 0.8 markdown restriction no restriction 0.75 0.7*— 0.5 0.7 0.6 0.8 1.2 1.3 1.5 0.9 1 1 .1 1.4 normalized starting inventory revenues and optimal price paths. The design of the study is similar to that of the two period problem, except that we now do not optimize the switching times and simply set all period lengths to 1/3. For the three period model, we also set the upper bound for the price to be 1. Figure 4.13 shows the optimal first, expected optimal second and third period prices with and without mark-down restrictions as a function of starting inventory. First, all prices decrease as the starting stock level increases, verifying our conjecture and earlier results. Second, the expected prices decrease over time regardless of the model. Third, the initial and second price of the mark-down restriction model are always higher than the initial and second price of the no-restriction model. W ith higher prices in the first and second periods, the retailer has more flexibility in the im portant third period. The expected price for 134 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.14: Optimal revenues as a function of starting inventory ( 7 = 2) 1.05 .2 0-95 0.9 markdown restriction no restriction 0.85 0.8 0.9 12 . normalized starting inventory 1.4 1.5 1.3 the restrictive model falls below the expected price of the non-restrictive model only in the third period. Clearly, the optimal revenue of the no-restriction model is higher than that of the restriction model. Resulting optimal revenues are shown in Figure 4.14 for 7 = 2 and cr2=1.5 /j- or cr2=3 fx. Optimal revenues increase with starting inventory level for both models as was the case for the two period problem. The gap between the restrictive and non-restrictive case increases with starting inventory level, but is hardly significant for this particular case (the retailer would lose 0.7% at most by imposing one-directional price changes). Optimal revenues as a function of variance are shown in Figure 4.15. Optimal revenues decrease with the variance as was the case for the two period model. 135 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.15: Optimal revenues as a function of variance (7 = 2, starting inventory = 1.5/z) 1 .1 2 no restriction H. 1.09 c 1.08 markdown restriction 1.07 1.06 1.5 2.5 variance (measured in mu) 4.5 Inventory flexibility The analysis so far assumes that there are no further replenishment opportunities available once the selling season starts. In the apparel industry, this corresponds to the case when the retailer orders from overseas and is not able to order during the season because of the long lead times relative to the selling seasons. Obviously, this limits the retailer’s control during the selling season to pricing only, which sharply diminishes its responsiveness. As a result, some retailers are willing to use domestic suppliers and to be able to order frequently, even though domestic suppliers are more costly. W ith domestic suppliers, the retailer is also able to make its initial order much closer to the season, when there is more information, hence less variance, 1 3 6 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. about the demand process. See Section 2.3.4 for products for which retailers prefer domestic suppliers over imports. Some companies are using two (or sometimes even three) different suppliers for the very same product: an off-shore low-cost supplier for the initial large orders, and a domestic high-cost supplier for replenishments during the selling season (Apparel Industry Magazine 1997). We study the value of this additional flexibility in the context of our pricing model. In a related study, Gumani and Tang (1999) study the impact of forecast improvements by having the flexibility to order at two instances, one of them being closer to the season. Their model differs from ours as they do not consider the possibility of ordering during the season by utilizing a structured learning from observed sales. Also they do not consider any pricing during the season. While their model allows the cost to go up or down as the merchandise is ordered closer to the season, we always assume that the ordering later is more costly reflecting the reality in the apparel industry. In our model, the off-shore strategy' will allow the company to order only once, but possibly with a low unit cost c°. The domestic strategy will allow the company to order before and during the selling season, but possibly with a high unit cost cd. The blended strategy, on the other hand, will allow the company to make its initial order at a unit cost c°, but later replenishments at the unit cost cd. We assume that there are no other costs involved, the pricing and inventory decisions are made 137 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. simultaneously at the start of the each period; period lengths are equal for each strategy and the lead time is zero for all strategies. To be able to compare these three strategies, we need to extend our pricing model to allow for inventory decisions. We suggest the following model. The problem is determining prices and stock levels in periods 1, N so that total expected profit is maximized. We use a discrete-tim e dynamic programming model. Let Vn(/n_i, W i-i: be the maximum expected profit from period n through N where the starting inventory is In- i and the cumulative sales and cumulative price multipliers are X n-i and Mn-i, respectively. Also let B n be the starting inventory level for period n, after the retailer receives its orders. Thus, the retailer acquires B n — In-i new units in the beginning of period n. Let pn be the price set in period n and let an be the acquisition cost per unit in period n. Backward recursion can be w ritten as L n ( / n — 1 j X n - l - , M n _ i ) = max E Pn > p s iBn ^ A i— I Cn(Bn Pn min{Dn, B n + ^ri+i — B n)+, X n_i + Dn, M n- 1 4- m(pn)S j X n —lz M n —1) Pn 1 3 8 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Boundary conditions are Fiv-rI (Tv: X , M ) — psI L y. for all Iry. X tf. Mtf. X 0 = M q = Tq = 0 . The first condition states that any left over merchandise has only salvage value (ps) when the season ends at the end of period JV . The dynamic program can be solved by starting with the JVth period and proceeding backwards. For the off-shore strategy, the model can be used with the following acquisition costs. Ci = c°, Cn = oo, n = 2 , . . . , N. For the domestic strategy, we simply have Cn = cd, n = 1 , . . . , N. For the blended strategy, we have, Ci = c°, Cn = cd , n = 2 , . . . ,N . 139 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Let V°(c°,cd),V d(c°,cd) and V b(c°,cd) be the optimal profits for the off-shore, do mestic and blended strategies respectively. Without any analytical derivations, it is easy to see the following. O b se rv a tio n 4.1 When the off-shore cost is higher than or equal to the domestic cost (which is not likely), domestic strategy outperforms the off-shore strategy. That is, for c° > cd, V°{c°,cd) < V d(c°, cd). In tu itio n : A domestic policy can simply imitate the the optim al off-shore policy by ordering as much as the optim al off-shore policy does in the first period and ordering zero units in later periods. Since the acquisition costs are lower for the domestic orders, this policy generates more profit than the optim al off-shore policy. O b se rv a tio n 4.2 When the off-shore cost is lower than the domestic cost (which is typical), blended strategy outperforms both strategies. That is, for c° < cd, V b{c°,cd) > V°(c°,cd) and V b{c°,cd) > V d{c°,cd). In tu itio n : A blended policy can im itate the optimal off-shore policy by simply ordering as much as the off-shore policy does in the first period and ordering zero units in later periods. Since the acquisition costs are the same for blended and off-shore strategies in the first period, this policy generates the same profit with the optimal off-shore profit. Likewise, another blended policy can imitate the optimal domestic policy by simply ordering as much as the optimal domestic policy does in each period. Since first period’s acquisition costs are lower for the blended strategy, this policy generates more profit than the optimal domestic policy. 140 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. While these comparisons are trivial, a question of interest is under which other circumstances the retailers should favor domestic policies over off-shore policies and under which circumstances the gap between the blended and domestic and off shore policies are minimal. This is im portant as acquisition costs may not be the only concern for a retailer. For example, using an additional supplier may involve additional fixed setup costs and complicate the coordination of the sourcing process which disadvantage the blended strategies. Also, in our study we do not consider the inventory holding and other logistics costs th a t may be incurred within the selling season. Inclusion of inventory holding costs to the model may advantage domestic and blended strategies against the off-shore strategy as domestic purchases may be used for frequent replenishments and m ay reduce inventory levels. However, if unit inventory holding costs are proportional to the unit cost and domestic cost is excessively higher than the import cost, inventory reduction effect will be less apparent. We use the computational design in Section 4.4 to answer above questions. Again, the mean demand is a / 0 = 20 and we have two periods of equal length. Different from the analysis in Section 4.4, the starting inventory level is optimized for all strategies. We assume th at the maximum price to charge is 1. We set the off-shore acquisition cost to 0.5 and vary the domestic acquisition cost to study the effect of acquisition costs on different strategies. Figure 4.16 shows the optimal profits of off-shore, domestic and blended strategies when 7 = 2 and when variance 141 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Figure 4.16: Comparison of off-shore, domestic and blended strategies ( 7 = 2) 1.3 Wended (variances .5 mu) blended (variance =3 mu) 1.2 a - 1 .1 H.0.9 off-shore (variance=3 mu) off-shore (variance=l .5 mu) 0.8 0.7 5 0.6 domestic (variance =3 mu) 2 0.5 domestic (variance=1.5 mu) 0.4 0.3 1.5 1.6 1.4 1.3 normalized domestic cost equals 1.5// or 3//. The optimal profits are normalized with the profit of the optimal off-shore policy when variance equals to 3 //. We first note that the optimal off shore profits do not vary with the domestic acquisition costs. Blended strategies, as shown above, outperform the domestic and off-shore strategies. Clearly, opti mal blended and domestic profits decrease with acquisition costs. However, optimal blended profit curves are rather flat, as blended strategies prefers to order more from the off-shore supplier as the domestic supplier becomes more expensive. In fact, optim al blended profits approach optimal off-shore profits as domestic acqui sition costs increase. The reduction in profits is more dram atic for domestic policies as they have to live with the expensive domestic suppliers. While domestic policies 142 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. outperform off-shore policies lor low domestic acquisition costs, off-shore policies are favorable as the domestic suppliers become more costly. For this particular example, domestic and off-shore profit curves intersect when the normalized domestic cost is 1.1 for a2 = 3/j.. This means that the "break even” point where off-shore profit equals domestic profit is when the unit domestic acquisition cost is 10% more than the unit off-shore acquisition cost. Any unit domestic acquisition cost 10% more than the unit off-shore acquisition cost will lead the retailers to source their merchandise off-shore. Another im portant factor for the efficiency of the off-shore, domestic and blended policies is the variance of the demand process. Typically, apparel retailers choose domestic suppliers for their high fashion content-high variance merchandise, while standard low fashion content- low variance merchandise can be sourced overseas. Also, the policy itself may help to reduce the variance as the domestic strategies can order closer to the season. Figure 4.17 shows the optimal profits for domestic, off-shore and blended strategies for three different domestic acquisition costs (1.00, 1.06 or 1.12 times the off-shore acquisition cost). Again, the profits are normalized with the optimal off-shore profit for cr2 = 3fi. Note that blended and domestic policies are equivalent when the cost equals 1.00. These policies outperform any other policy. As the variance increases, all profits decrease. The off-shore optim al profit curve is steeper as off-shore policies are subject to more variance as they order only once. Optimal domestic policy when the cost equals 1.06 is inferior to 143 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Figure 4.17: Comparison of off-shore, domestic and blended strategies ( 7 = 2) blended (cost-1.06) s 1.15 domestic (cost-i .06) § 1 .05 off-shore domestic (cost-1.12) 0 .9 5 2.5 1.5 variance (measured in mu) the off-shore policy for low variance levels, but becomes favorable as the variance increases. Note again that the base line is the optimal off-shore profit when the variance is 3 n . If we can reduce the variance to 1.75 /i by using a domestic policy, even the domestic acquisition costs of 1.06 can be desirable. Finally, the price uncertainty of the customers also affects the relative efficiency of these policies. We expect the off-shore strategy to be more sensitive to price sensitivity (7 ) as pricing is the only control for such strategy once the season starts. Figure 4.18 shows the optimal profits for varying levels of 7 . Note that when 7 = 1, the demand is inelastic and it is optim al to keep the price at its maximum. Thus, optim al profits at 7 = 1 represent the optimal profits when the only control over the process is through inventory. All profits increase, as the price sensitivity increases. As expected, optimal off-shore profit increases faster with the price sensitivity. The 144 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Figure 4.18: Comparison of off-shore, domestic and blended strategies as a function of price sensitivity domestic&blended (cost=1) blended (cost=1.06) 1.2 £ 1.15 domestic (cost=1.06) re 1.05 off-shore 1.6 2 2.2 1 .8 2.8 1 2.6 3 gamma off-shore strategy outperforms the domestic strategy with cost 1.06 when 7 is close to 3. Combining these ideas, we generate the regions in which one strategy is favor able to the other. Figure 4.19 shows the trade— off curves for off-shore and domestic strategies for 7 = 2 and 7 = 3. For variances and domestic costs on these lines, off-shore and domestic strategies generate the same profit. As domestic cost in creases and/or variance decreases, off-shore strategy becomes more desirable and vice versa. Mote that, the region for which off-shore strategy is more profitable is larger when 7 = 3, reflecting the increased strength of off-shore strategies with price sensitive demand. In general, as 7 increases, the trade-off curve moves to southeast. Although it is difficult to detect visually, we observe that the trade-off curves are 145 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Figure 4.19: Trade-off curves for off-shore and domestic production 1 .1 off-shore is profitable 1.09 « 1.08 5 1-07 © 1.06 S 1-05 e 1.04 1.03 domestic is profitable 1.02 1.5 2.5 variance concave in variance, possibly becoming flatter as variance gets larger. This means that if the variance gets excessively high, variance differences would have less im pact and supplier selection decisions would depend more on cost differences and price sensitivity of demand. The above analysis is based on the fact that the products are subject to same level of uncertainty under both strategies. However, in most cases, the choice of strategy itself may affect the level of uncertainty. As the retailers are able to order closer to the season with domestic strategies, they are able to know more about the consumer tastes that will shape the demand in the coming season and hence they face a more stable demand when they make their ordering decisions. To incorporate the possible reductions in variance, we choose a base case which is an off-shore 146 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Figure 4.20: Trade-off between domestic and off-shore production off-shore is profitable 12 s S e o o 3 ■ o O cl o u 5 © c o ■ o $ I S o o c (Z > c n C O C D U c domestic is profitable 15 20 25 30 35 reductions in variance with domestic production (%) 45 40 strategy with acquisition cost equals 0.5 and variance (a2) equals 3 /i. The trade-off curves in Figure 4.20 shows the increases in cost and reductions in variance with domestic strategy for which domestic and off-shore strategies generate equal profits. For example, for 7 = 3, if the domestic cost is about 7% higher than off-shore cost, the domestic strategy will still result in higher profits, if the variance is reduced by more than 30% as a result. Alternatively, if the variance is reduced about 30%, the domestic strategy will generate higher profits only if the cost does not increase by more than 7%. Again, optim ality region is larger for off-shore strategy for more price sensitive demand. Each retailer faces its own trade-off curve for each apparel item it offers and makes its decision to source it overseas or domestically. An aggregation of these 147 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Table 4.1: Market share and cost advantage of imports for selected apparel domestic off-shore ratio off-shore cost cost penetration Product (cd) (c°) (.C dfc°) (96) Swim-wear (men’s) 199.80 41.52 4.81 97.8 Swim-wear (women’s) 201.60 68.64 2.94 40.9 Hosiery (men’s and women’s) 11.72 10.27 1.14 11.0 Sweaters (men’s and women’s) 164.46 132.00 1.25 68.8 Prices are per dozen in U.S. dollars. Off-shore cost includes purchase cost, insurance and freight. Data is for the year 1997 and compiled from U .S. International Trade Adm inistration (1998) and U .S. Census Bureau (1992-1999) individual decisions determines the market share of imports and domestic production in the domestic market. Observing this, we note th at the conclusions from our analytical model can be supported by an analysis similar to the one in Section 2.3.4. In Table 4.1, we provide the average unit import and domestic costs and the m arket share of imports for selected apparel. We observe that for both men’ s and women’ s swim— wear, im ports have a substantial cost advantage. While this translates into a perfect market dom ination of im ports in men’s swim-wear, domestic manufacturers still control 60% of the market for women’s swim-wear. This shows that the cost advantage is not the only factor in supplier selection. Although it is very difficult to find an aggregate measure for the variance of demand in apparel, we are certainly aware of the importance of fashion in women’s apparel. Popular styles and colors change every year, making it very difficult to forecast demand for a particular SKU. From both our computational analysis and industry data, we see that predictability of demand plays a considerable role in sourcing decisions. When the variance effect is less apparent (as in hosiery 148 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. and sweaters), we observe that cost difference is the main driver for such decisions. Imports have only minor advantage over domestic production in hosiery (imports are only 12% cheaper than domestic merchandise). Resulting import penetration is only 11.0% in this category. Competing against relatively cheaper imports, domestic sweater manufacturers were able to m aintain only 31.2% of the market share. 4.6 Concluding Remarks In this chapter, we study the pricing decisions of a perishable products retailer in the existence of demand learning. This is one of the first studies th at incorporate “structured” Bayesian updating in the context of pricing for perishable products. The resulting model is computationally feasible and easy to understand and imple ment. We claim that our model is most useful for apparel retailers, as this industry is identified with high levels of uncertainty, most of which can be resolved after observing sales during the earlier weeks of the selling season. Moreover, information required for the application of our model is readily available through point-of-sales scanners. Through our computational study, we are able to understand the economics of pricing in this context. First, we observe that the optimal price in future periods is a non-decreasing function of sales in the earlier periods when demand learning takes place. Second, we pinpoint the circumstances under which this learning based 149 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. on observed sales has the most value. Basically, demand learning is most benefi cial when the dem and/supply mismatches necessitates price changes and when the sensitivity of demand to price is at moderate levels. Furthermore, we analyze the effect of various factors on optimal revenues. We see that optimal revenues increase as inventory levels increase, but marginal returns of additional inventory are di minishing. We also see th at profits increase as the demand becomes more sensitive to demand, but decrease as the demand becomes harder to predict. Finally, we study the impact of an opportunity to procure merchandise during the season, in addition to the up-front procurement before the selling season. This helps us to see how supplier selection decisions are affected by the volatility and price-sensitivity of demand and the procurement costs. We support our conclusions with aggregate data from the apparel industry. While our model was very easy to interpret and use in a computational study, we were not able to prove our conclusions analytically. We also note th a t our model with inventory flexibility can be extended to incorporate inventory holding costs and set-ups (cost and/or time) that may be attached to each purchase. We refrain from doing so, as their effects on costs are fairly trivial and their inclusion may complicate the presentation of the model. 150 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 5 A Pricing Model with Competition In the previous two chapters, we have studied the initial ordering decisions before the selling season and the pricing decisions during the selling season for a single decision maker. However, we have not explicitly considered the interactions of two competing retailers in the sam e market. In this chapter, we use a game theoretic model to study the impact of competition on pricing decisions during the selling season. 5.1 Introduction and Literature Survey Selling a fixed amount of inventory over a finite horizon is a problem faced by many companies. Examples include retailers selling fashion or seasonal goods such as ski-wear, airlines selling a fixed number of airplane seats, and hotels selling a fixed number of rooms. One frequently cited reason for the lack of replenishment opportunities during the selling season has been the relatively long replenishment 151 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. lead times as compared to the length of the selling season. Examples include five m onths lead tim e for Liz Claiborne apparel (Dalby and Flaherty 1990) and six to eight months lead time for a m ajor fashion retailer (Gallego and van Ryzin 1994). Pashigian (1988) reports that the mean lead time for an apparel order from a Far Eastern country is 34.7 weeks. Selling season, on the other hand, is at most six months for these fashion goods. Airlines, hotels and many other service companies have fixed capacities (seats and rooms) and share the same characteristic problems as style goods retailing: the fixed costs in acquiring and disposing additional capacities are prohibitive while the variable costs associated with providing these services and products are very low. This problem is also relevant to companies selling products near the end of their life cycles, for which no further production is planned. In all of these problem contexts, the inventory units unsold at the end of the horizon have little value, or in the case of airline seats and hotel rooms, no value at all. When the dem and is price sensitive, price adjustm ents are made to maximize a firm’s revenue over the horizon. Price adjustments can take the form of pure tem poral price discrimination, as is the case for retailers that use m ark-ups or m ark- downs. Pashigian (1988) reports th at the dollar value of total mark-downs as a percentage of dollar sales on all merchandise sold in departm ent stores increased over the years, reaching 16.8% in 1984. Fisher (1994) notes th at in 1990 this percentage went up to 25% in 1990. 152 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. An indirect way of price adjustm ent is to first differentiate the product in a costless manner, thereby creating different customer classes. The fixed capacity is then allocated to these customer classes by opening or closing classes as time advances. This latter form of price adjustment is known as yield management and is utilized by many service companies. Smith et. al. (1992) reports that American Airlines increased its revenues by $ 1.4 billion over a three year period, attributable to yield management. We refer to Weatherford and Bodily (1992) for a review of yield management. In the economics literature, clearance sales as a means of price adjustment (mark-dowm) has received some attention. Lazear (1986) develops a model of retail ing, in which the decision is the price of the product over a finite number of periods to maximize profit. It is assumed that the consumers are homogeneous (they have the same valuation of the item) and that the consumers shop at once when the price is declared, which makes the length of each period immaterial. Pashigian (1988) extends Lazear’s model to allow for industry equilibrium and shows that fashion and product variety are the leading reasons for increasing mark-down ratios in re cent years in fashion retailing. Pashigian and Bowen (1991) show empirically that uncertainty and price discrimination explain the mark-downs in fashion retailing, van Praag and Bode (1992) incorporate cost of clearance sales to Lazear’s model. Gallego and van Ryzin (1994) study the pricing decisions of a firm selling a fixed stock of items over a finite horizon with demand being a controlled Poisson 153 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. process. They show that the optimal profit of the deterministic problem, in which demand rates are assumed to be constant, gives an upper bound for the optimal expected profit. For the continuous price case, fixed-price heuristics are shown to be asymptotically optimal. For the discrete price case, a deterministic solution can be used to develop again asymptotically optim al heuristics. Bitran and Mondschein (1997) present a similar model. Feng and Gallego (1995) derive optim al policy for the two price case. Gallego and van Ryzin (1997) study the m ulti-product case: they suggest two asymptotically optimal heuristics and apply them in network yield management problems. While the marketing and economics literature contains much research on the ef fects of price competition, it has generally assumed that instantaneous and infinitely available supply is always possible, thus ignoring the role of limited inventory supply. On the other hand, the inventory literature has examined single— firm stocking and pricing decisions, but the effect of demand interactions in a multi-firm, competitive market has not been studied. In this study, we use a demand structure similar to those used in Gallego and van Ryzin (1994, 1997) and Feng and Gallego (1995). Our primary objectives and contribution in this research are to understand how competition impacts the pricing decisions of firms selling fixed inventories, and how increasing degrees of demand interaction affect these decisions. We use a game theoretic formulation to derive the equilibrium policies for two competing firms. Under competition, the demand rate 154 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. for one product depends on its own price and the price of the other product offered by the competing firm. We assume that these two products are not necessarily identical, and hence even if the prices are different, the company offering a higher price may still face some demand. Varian (1980) makes a distinction between the consumers who have complete information about the prices offered by different firms (the “informed” consumers) and the consumers who know nothing about the price distribution (the “uninformed” consumers) to explain the price dispersion even when the products are identical. We use a linear demand model to determine the demand rates for each firm given both the prices of both firms. Specifically, we use the linear model that is presented in McGuire and Staelin (1983). We describe the details of this model in Section 5.2.1. We assume th at both firms start with the same price, and that they have the option to decrease the price (mark-down) or increase the price (m ark-up) over the selling horizon. The mark-down problem is typical in retailing, while the mark up problem frequently occurs in service industries such as airlines and hotels. The price levels and their associated demand rates are publicized and known in advance. The problem for each firm then is to find the timing of its price change so as to maximize its (expected) profits. We model the problem as a non-cooperative game. In Section 5.2, we describe our mark-down model in detail. First, we describe the demand- price relationship using the linear demand model. We then study the single-firm (monopoly) mark-down problem and analyze the problem in the case of 155 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. two competing firms (duopoly). Q ualitative properties of the equilibrium are also derived. The results are compared with the case of no demand interaction between the firms. Section 5.3 analyzes the m ark-up model and describes the transformation of a fare allocation problem to this m ark-up model. We present the conclusions and implications in Section 5.4. 5.2 The Mark-down Problem In this section, we study the case when firms start with a high initial price and switch to a lower price during the selling season. Our model is an initial attem pt to understand the effects of competition on pricing decisions of firms, and we make a few simplifying assumptions about the structure of the problem based on the previous results of Feng and Gallego (1995) and Gallego and van Ryzin (1994, 1997). First, although the magnitude of these mark-downs might be equally im portant, we focus only on the tim ing of the mark-downs. We assume that the initial price and the mark-down price are publicized and known in advance. The pre-specified prices may be a result of price lining practiced by some retailers or a result of an industry level consensus. We allow each company to make at most one price change. This restriction may be justified when the costs associated with price changes are considered. Moreover, Gallego and van Ryzin (1994) have shown that a single price change system is as effective as a more flexible pricing system, especially when the sales volume is high and price changes are costly. 156 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 5.1: Demand Rates for Two Competing Firms Firm B Pi P2 stock-out Pi P2 stock-out A* A? a£ , a® C k Afn a F i a L A 2 4,A f Af/2 o o - c i° d - < -< Because of the complex structure of the stochastic solution for even the single firm case (Feng and Gallego 1995), and given our main purpose to study the effects of competition, we use deterministic demand rates in this work. Gallego and van Ryzin (1994, 1997) have shown that solutions to the deterministic problem may be used to construct asymptotically optimal heuristics for the stochastic problem. Now, consider two competing firms, A and B. Assume that firm .4 has a starting inventory nA and firm B has a starting inventory nB. Both firms start with the same price pi and each firm has the option to decrease the price to p2 at some time within a horizon of length t. The problem for company A and company B is to find the price switching times sA and s B, respectively, such that their profits over the entire selling season are optimized. We assume that pi and p2 are fixed and known in advance. We now give the representation of demand rates based on the prices charged by each firm (in the order of firm A, and firm B) in Table 5.1 . 157 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. The demand structure in Table 5.1 models price completion. Firm A receives a demand rate of Af if both firms offer the high price p\ and a demand rate of Af if both firms offer the low price P2 - Similarly, firm B receives a demand rate of Af if both firms offer the high price, and a demand rate of A ® if both firms offer the low price. Subscripts L and F are used to indicate the price leader and the price follower. If firm A has already lowered its price, while firm B is still charging the high price, firm A and firm B receive the demand rates Af and A f, respectively. If firm B is the leader in price change instead, firms A and B receive the demand rates Af and A f, respectively . When one firm B is out of stock, firm A observes a demand rate of Afn if the competitor is charging price p\, and Af/2 if it is charging price pa- Similarly, firm B observes the demand rates Afn and Af/2 when firm A is out of stock. We first outline how these demand rates are determ ined in the context of the linear demand model. Then we describe our model and results for the case when demands are independent and each firm acts as a monopolist to optimize its own profits. Then, we present our results for the two com peting firm (duopoly) case. 5.2.1 T h e Linear D em an d M o d el Throughout this chapter, we assume th at the two prices charged by firms impact each company’s demand rate in a linear fashion. Linear demand systems have been 158 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. widely used in the m arketing and economics literature. In particular, we use the linear demand model in McGuire and Staelin (1983). Assume that there are two firms, firm A and firm B that are offering two products (most likely, but not necessarily substitutable). If the prices are pA and pB, the demand rates for two products can be expressed by, * A(pA ,P B) = - j z r § p A + Y Z qP 8 ') > t5 -1) AB (pA , p B) = 0 - - P ) s ( l + ~ T qPA ~ Y Z T § P B\ (5 -2) where O < / / < 1 , O < 0 < 1 , and and S are positive constants. In this repre sentation, S is a scale factor which equals the total demand rate when both prices are zero. The demand for both products increase linearly as S increase, p captures the absolute difference in demand, i.e., if both firms charge the same price, firms A and B get the 100 p percent and 100 (1 — p) percent of the total available demand, respectively. 6 is a measure of product substitutability. When 0 = 0, the demands are independent, and each firm behaves as a monopolist. (3 is a measure for the sensitivity of demand to price changes. For this model, an arbitrary selection of parameters may result in negative de mand rates. To guarantee non-negative demand rates, we set the following bounds on the prices p " 1 < ^ - ^ + 0 p B , ( 5 . 3 ) 159 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. PB < ^ ^ 9^ + 6pA. (5.4) If a firm sets its price at its bound, the demand rate for th at firm simply drops to zero, and the demand rate for the competitor reaches its maximum that is attainable with its current price. To ensure that the total industry demand does not increase with an increase in either price, the parameter p for absolute difference in demand should be bounded as follows, 6 1 < P < Z a- (o.o; 1 + 0 “ ^ “ 1 + 0 5.2.2 The C ase o f In d ep en d en t D em and: O ptim al M ark down T im e for a M on op oly Assume that the products offered by the two firms are not substitutable and thus their demands are independent. In our specific model, this means that the param eter for substitutability (6) is zero. Using equations (5.1) and (5.2), A A{p-\pB) = p S ( l - ( 3 p A) AB{p-\pB) = (1 - /i)5 (l - (3pB) 160 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. As expected, when the products are not substitutable, the dem and rate for one product is totally independent of the price charged for the other product. In par ticular, the demand rates in Table 5.1 are as follows for the independent demand case: X f = Xp = Xfn = pS( 1 — /3pi), Af = Af = Afn = M l - S K l - t o ) , X ~ 2 = A^ = A'ij2 = pS( 1 — pP'i) ; A f= A £ = Af,2 = p( l - S ) ( l - / ? p 2). Independency leads the firms to act as monopolists when making their pricing de cisions. Consider one of these firms, firm i which faces the problem of selling a fixed stock of nl units in a given horizon of length t. Let the starting price be pi which generates a demand rate of A^. The problem is to find the switching time s after which the price is p2 and the demand rate is X\- We assume that p\ and p2 are fixed and known in advance. As pi > p2l the linear demand model ensures that A 2 > X\. W ithout loss of generality, we also assume that any unsold unit beyond time t is worthless. The objective of the firm is to maximize its total revenue from sales with price pi and sales with price p2, which is R u {8) = Pi m infn1 , iV^O, s1 )} + p2 min{nl — min-fn1 , ^ ( 0 , sl)}, iV 2(sl, £)} 161 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. where Nj(a,b) is the units sold between time a and b: we use subscript M to denote monopoly. For the deterministic demand case, Nj(a, 6) = At(6 —a): hence the payoff function equals r m (s1) = (Pi - P2) m in{n\ A^s*} + p 2 min{n*, A\s l + A\{t - s1 )}- Note that, in order for the firm to have an incentive to change its price, revenue rate should be increasing with the price change; i.e., we must have < P2A 9. We also exclude the trivial case, n1 < A\t, for which the firm is able to finish off its inventory with the high original price. L em m a 5.1 The optimal switching time for a single firm is given by, 4 = max{0, ^ }• (5-6) The solution is very simple. The monopoly firm will change the price at the latest possible time th a t it can still sell all of its inventory. Hence, the switching time is such that the demand exactly equals the total inventory. If the horizon is not long enough to finish the inventory even with the low price (nl > A\t), the firm simply 162 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. sets the price to P2 at the beginning of the selling season (s£f = 0). The firm’ s optim al revenue is, R m = P ^ \ s 1 m + P * W - S m ) = P iK s m + P2(nl - A[s 1 m ). For a stochastic treatm ent of the problem where Nj (a, b) is a Poisson random vari able w ith rate Xj(b — a), see Feng and Gallego (1995). Note that these results are valid for any demand price relationship satisfying P i^A x < P2A 2- Our linear demand model satisfies this condition if px -(- p2 > 1 //? and 5.2.3 C ase o f D ep en d en t D em and: E quilibrium S w itch in g T im es for a D u o p o ly When the demands are not independent, the firms can no longer optimize their profits by only considering their own actions. We use a game theoretic model to find the equilibrium price switching times for the two firms. The game is non- cooperative with complete information. T hat is, the ru les of the game are common knowledge. Each firm knows its own and its com petitor’s starting inventory levels, specifically, we have the following specific switching times for firms A and B : sf; = max{0, nA — /j.S( 1 — PP2 ) fj.S/3(p2 - P i ) }, sf? = max{0, n B - ( l - p ) S ( l /dp2) - 1 r ^ (1 - p ) S p ( p 2 - P i ) 1 6 3 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. demand rates and payoff functions as well as the length of the selling season. W ith computerized and widely accessible reservation systems for the service industry and huge amounts of online data in the retailing industry, we believe th at firms have the capability to have good estimates of their com petitor’s inventory levels, demand rates and motivations. Using equations (5.1) and (5.2), the demand rates in Table 5.1 can be derived. We first derive the demand rates when both firms charge the same price. Note that these demand rates are equal to the demand rates that these firms observe when the demands are totally independent. Using equations (5.1-5.2) and (5.8 -5.9), the demand rates when the firms charge different prices can be derived as follows: = /m S( 1 - 0pi), A j = i*S( I - fa ) , (5.8) A? = (1 - fj)S( 1 - /?px), Af = (1 - fi)S( 1 - 0p2), (5.9) 1 \ A _ M __________ & \ B ( 1 - 0 ) 2 ( l - M) ( l - 0 ) 11 (1 - 0) 1 ( 1 - / * ) 0 } A ■ 1 M ( 1 - 0 ) 2 + ( l - 9 ) 1 6 4 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. The demand rates for a firm when the its competitor is out of stock can be found by setting its competitor’s price to its maximum prescribed by equations (5.3) and (5.4). Then, ^ivn = + 0)(1 — PPi) = (1 + 0)Af, = fj£(l + 6)(1 — PP2) = {1 + 0)^2 i x f n = M (i- s ) ( i + fl)(i-to) = (i + »Mf. AB2 = ,i(l —S)(l + 0)(1 —/3p2) = (l+ 0 )A ,B. Having described the demand rates in terms of A f, A f, Af and A®, we observe the following, Af _ Af _ Af _ Af _ Afn Aff2 fj. . . Af Af ” Af Af Afn Af/2 (1 -ax) ' In other words, for all pairs of states (prices charged or inventory positions), firm .4’s demand rate is always the same multiple of the demand rate that firm B would get if the states were reversed. This helps us to rescale the problem such that the firms .4 and B are identical except for their starting inventory levels. In order to do so, we first rescale the starting inventory positions so th at, m A = n A S/M’ m B = n B 5(1 - / i ) (5.11) 165 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. While nA and n B define the absolute startin g inventory levels for firm A and f?, m A and m B measure firm A’s and firm B ’s starting inventory positions relative to their market potential (Remember that S is the total demand rate if both firms charge zero price). We also rescale the demand rates for firm A and firm B with Sp, and S(1 — p), respectively, so th at we have the same demand rates for both firms. Namely, Ai, A 2, A £,, A p. A,vn and A jv/2- Note also that the following properties hold for the linear demand model: Xc — b Af = A 2 + Ai, (5-12) A i - A f = (As - A I) | l ± | l = / 3 ( P2- p l) | i ± i | . (5.13) Equation (5.12) states that the sum of the demand rates for the leader and the follower in the price change is equal to the dem and rates that these firms would get if they were completely independent. Equation (5.13) states that the difference in leader’s and follower’s demand rates increases as the magnitude of price change and the rate of substitution (i.e., demand correlation) gets substantial. W hen the demand for two products are completely independent (i.e., 6 = 0), A 2 = Xl and Ai = A p . We assume that both firms are over-stocked with respect to the demand with the initial price (i.e., m A > Ait and m B > XL £), for otherwise, the problem is trivial and both firms will charge pi throughout the selling season. W ithout loss of generality, assume that m A > m B, i.e., the firm A is over-stocked relative to the firm B. We are 166 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. now ready to present our main result. We first describe the equilibrium switching times and then go on to formally state and prove that these switching times are unique Nash equilibrium. We show that, in equilibrium, firm A (being more over-stocked) lowers its price earlier than firm B does, and price switching times (sA* ,sB*) satisfy the following conditions: AxSA * + Al (sb* — sA *) -f- A 2(t — sB*) = m A, (5-14) AxSA * + A f (s b* — sA*) 4- A 2(£ — sB*) = vnB. (5.15) The solution to (5.14) and (5.15) indicates that, in equilibrium, each firm switches its price at a point such that it runs out of stock right at the end of the horizon. Closed form solution to (5.14) and (5.15) is the following, a* A2(A l — A F)t — (A c — X2) m B — (A2 — \ F) m A S = --------------------( A , - A l ) ( A t - A P )------------------- ' ( ° ' 1 6 ) b* ^2 (Al - AF)t - (XL - X i ) m B - (Ax - AF) m A S = --------------------- (A2 — Ax)(At — Ap)-------------------- ' (0'17) T h e o re m 5.1 (sA*,sB*) is a Nash equilibrium, if p\ + p 2 > 1//?- P ro o f: See Appendix B.l. T h e o re m 5.2 (sA*,sB*) is unique, ifp i + P 2 > 1 //?- P ro o f: See Appendix B.2. 167 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. The only condition for the unique Nash equilibrium (pi + po > l/{3) states that the revenue rate should be increasing (i.e., P2 X2 > PiXt) with the mark-down even if both firms are totally independent of each other. We now study the qualitative properties of the equilibrium. We first observe that the relatively over-stocked firm switches its price before its competitor does. W ithout competition, this is quite intuitive since the more over-stocked firm will need more time during which it faces a higher demand rate to liquidate its inventory and thus will lower its price earlier. Our results show that, even in the existence of demand interactions, the more over-stocked firm still lowers its price earlier than its competitor does. From (5.16) and (5.17), the length of the period during which the firms charge different prices equals, Note that the formula describes two forces in action. While the difference in in ventory positions (m A — m B) compels the firms to act individually, the intensity of demand interaction (A^ — A/r or 9) forces the firms to act together in their price switching decisions. Note again that 9 is a measure of demand dependency. As the dependency increases (9 increases), firms tend to follow each other’s decisions, and switch almost simultaneously when 9 gets close to 1. If there is no interaction n A — m B _ (m A — m B) (1 — 9) Xl — Xf? (pi — P2)P (1 + 9) (5.18) 1 6 8 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. (0 = 0), the firms switch at monopolistic switching times. These two forces are sim plifications of two general phenomena. W hile the differences in the structures and operations of different firms lead the firms to follow different strategies, the product substitutability and hence competition lim its them in doing so. At the other extreme, the products are not substitutable, and hence the demand rates of the firms depend on their own prices only. As described in section 5.2.2, when 6 = 0, firm .4 (or B ) would observe the demand rate Ai (or A2) with price px (or p2), regardless of its com petitor’s price. W hen this is the case, the firms command their own markets to act like monopolists and switch their prices at the monopoly switching times described by Lemma 5.1 From equation (5.7), firm A changes its price at time sA ( = L , and firm B changes its price at time sB ( = ™ 1 - From the discussion above, with demand interaction, it is clear that the length of the tim e period during which firms charge different prices is smaller, relative to the case of no demand interaction. Of particular interest is the switching tim e of each firm in the existence of demand interaction compared to its switching tim e when the demands are independent. The next lemma makes this comparison. L e m m a 5.2 The m o re o v e r s to c k e d f ir m c h a n g e s th e p r ic e la te r, a n d th e le ss ovei— s to c k e d f i r m c h a n g e s the p ric e earlier, th a n th e y w o u ld c h a n g e i f th ere was n o d e m a n d in te r a c tio n (a n d h en ce, n o c o m p e titio n ). Proof: If we replace m B in equation (5.16) by m A , we obtain Thus, sA* > sAf . If we replace m A in equation (5.17) by m B , we obtain s Bf . Thus, sB* < s Bf . 169 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Intuitively, the more over-stocked firm will m ark its price down first, and hence, during the tim e it charges a lower price than its competitor, will enjoy a higher demand rate than it would if there was no dem and interaction. Therefore it needs less time to deplete its inventory and can use more time to sell at a higher price, and hence should change the price later. O n the other hand, the less over-stocked firm wall m ark its price down second, and hence, during the time it charges a higher price than its competitor’s, it will face a dem and rate lower than it would face if there was no demand interaction. Therefore, it can sell fewer units than it would normally sell during the tim e it charges the high price, and so should change the price earlier. Now, we compare the revenue of each firm under competition to its revenue when there is no dem and interaction (and hence, no competition). L em m a 5.3 With demand interaction, the more overstocked firm has a higher payoff and the less overstocked firm has a lower payoff (both compared to the payoffs when there is no demand interaction). The total industry payoff increases with demand interaction (compared to the total payoff with no demand interaction) if and only if [x\i > (1 — 1 7 0 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. P ro o f: Let R A ; and RB ; be the optimal profits for firms A and B , respectively, if there is no demand interaction. Also let R * A and R * B be the optimal profits for firm A and firm B, respectively, with demand interaction. Then, R'm = + P*X2{t - 8$)], r b ; = (i-n )s\p lxls f;+ p 2x2(t-sf;)], Ra* = p S f a X ^ * + p 2[XL(sB* - s A*)] + X2( t - s B*)], RB * = ( l - p ) S \ p lXls A^-h p l [XF(sB* - s An + X 2( t - s B^}}. Let (m A — m B) d = (X2 — Xi)(Xi, — XF) It can be shown that * - s t; = t ( x L - a2), s i; - s B* = s ( x l - x F), which leads to RA* - R t; = ptS(pi-p2)6X1(XL - X 2), r b* - r b; = ( i - p ) S ( p 2 - p l )sx 2(x l - x F), r a* + r b* - r a; - r b; = 5 (p 1 - p 2) ^ a 1(a l - a 2) - ( i - ^ ) a 2(a 1 - a f )] 1 7 1 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Define ; _ S (m A — m B)(pi — p2)0 _ S {m A — m B)9 ■ = (A2 - A!)(l + «) = ,8(1 + 9) ’ One can show that, 6 Al — A2 = Ai — A p = -- j (A2 — A^, which leads to, RA* - R i T = ibp Xl RB* - R f r * = 4>(fi-l)\2 r a* + r b* - r a; - r b; = ^[pX x - (i - ^ )a 2] The results follow. For the linear model, as the dem and interaction becomes more intense (9 gets closer to 1), the payoff for the more over-stocked firm increases, the payoff for the less over-stocked firm decreases. This result may have an advertising implication: to increase its payoffs, the more over-stocked firm should expend in informing con sumers in the industry about its price and emphasize the similarity of its product to other products offered by competing firms. The less over-stocked firm, on the other hand, should emphasize the distinctive properties of its product. Remember that p, stands for the market share of the more over-stocked firm. The total payoff 172 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. increases if only if /zAi — (1 — /z) A 2 > 0, or fj > A 2/(Ai + A 2). T hat is, the competition helps the industry to improve its revenues, if and only if the more over-stocked firm is sufficiently large, as this firm is the only one that benefits from the competition. We note th at the above analysis does not compare the payoffs of two firms when they compete to their payoffs when they cooperate. Rather, we compare their payoffs with a specified level of demand interaction to their payoffs when there is no demand interaction. A n Exam ple Assume that S = 70 and /z = 0.4. T hat is, firm A and firm B have the market shares 40% and 60%, respectively, and if both firms charge zero price, the maximum total demand rate for the two firms is 70 per day. Assume that firm -4 has a stock of n A = 1,280 units and firm B has a stock of n B = 1,440 units to be sold in a season of 100 days. After rescaling the starting stock levels with the market share (/z and 1 — /z) and the total maximum demand rate (S'), we obtain m A = 320/7 and m B = 240/7. Note th at although the firm .4 has lower starting inventory, it is more over-stocked relative to the firm B. Let (3 = 1/14 and for now assume that the demands are independent, i.e., 9 = 0. Assume also th at pi = 10 and p2 = 6. In this case, A ^ = 2/7 and A 2 = 4/7. As the products are not substitutable, each firm acts as a monopolist: firm .4 switches its price on day 40 and firm B switches its price on day 80. At the end, the payoffs for firm A and firm B are $8,960 and $12,480, respectively, with a total industry payoff of $21,440. 173 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Now assume that the demands are dependent such that 9 = 1/3. Then Xp = 1/7, A £ , = 5 /7 and Ami = 8/21, Am2 = 16/21. In equilibrium, firm A changes the price on day 50 and firm B changes the price on day 70. Being more over-stocked, firm A achieves its maximum payoff of $9,280, while firm B has to react to firm A and can only get a payoff of $11,520. Demand interaction helped firm A increase its payoff while firm B's payoff decreased w ith the demand interaction. Note th at firm A changes its price 10 days later, and firm B changes its price 10 days earlier than they would if there was no demand interaction. W ith competition, the time period during which the firms charge differential prices is only 20 days. The total payoff has also decreased to $20,800. 5.3 The Mark-up Problem We develop a similar model when the firms actually m ark the prices up. This kind of price adjustm ent can be observed in yield management for airlines and hotels. The assumptions of the m ark-up model are exactly the same as those made for the m ark-down model in Section 5.2. We first describe our model for the single firm (monopoly) case. Then, we present our model for the two firm (duopoly) case. Finally, we discuss the discount fare allocation problem, and show how the results of our m ark-up model can be used via a transformation. In this section, we assume that the starting inventory levels and demand rates are already rescaled by each 174 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. market share and thus we can treat the firms act as if they are symmetric except for their starting inventory levels. 5.3.1 O p tim al M ark -u p T im e for a M o n o p o ly Consider the same setting as in Section 5.2.2. We now assume p2 > Pi and thus A 2 < X\. Note that we now have the restriction p2A| < or pi + p 2 < 1//3- since otherwise the firm would mark the price up at time 0. We also exclude the trivial case. A \t < nl, in which the firm is not able to sell off all inventory even with the low price. The optimal time to switch is, . r n l — A \ t , sM = m m {t, A, _ _ A V > - The equation states that if the demand rate with the low price is not large enough to deplete all the inventory, the firm would not switch the price at all. The firm now switches the price at the earliest (as opposed to the latest in Section 5.2.2) time that will liquidate all of its inventory. 5.3.2 E quilibrium M ark -u p T im es for D u o p o ly We use a model similar to the one described in Section 5.2.3. We now have p2 > pt and pxAx > p2A2. Also assume now that Ait > m A > m B. 175 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. The equilibrium is sim ilar to that of the mark-dowu problem. Again, each firm depletes its inventory right at the end of its selling season. But, this time the less over-stocked firm (firm B ) switches its price first at time b* _ X%{XL — AF ) t — ( \ L — X 2) m A — (A 2 — X F ) m 3 S ~ (A2 — A i)( A f, — X f ) ’ ' j followed by the firm with a higher inventory (firm A) switching at time .4, _ A2 ( A l ~ AF )t ~ (A c — Ai)7rr4 — (Ax — A f ) m g (A2 — X i ) ( X L — X p ) (5.20) The solution is exactly the same as the solution for the mark-down problem except that the m A and m B are interchanged in the formulae. Now', the only condition under which (sA*, sB*) is a unique Nash equilibrium is P1A1 > p2A 2 or pi +po < 1//?- The difference in switching times, s-4* —sB*. is now equal to _T~- The results of the mark-dowm problem can easily be translated into the m ark-up problem. The less over-stocked firm marks its price up later, and the more over-stocked firm marks its price up earlier, than they would without demand interaction. Thus, the time period for which the firms charge different prices is shorter w rhen demand interaction or competition is present. Also, with demand interaction, the profit of the more over-stocked firm increases, while profit of the less over-stocked firm decreases. 1 7 6 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. We now consider the same example discussed earlier in Section 5.2, only with reversed demand and price parameters. T hat is, pi = S6, po = §10, Xi = 4/7, A2 = 2/7, XL = 1/7 and Ap = 5/7. We still have nA =1,280, n B =1,440 and t = 100. The equilibrium is a time-reversed version of the equilibrium in the mark down problem. T hat is, firm B switches at tim e sA* = 30(= 100 — 70) and firm -4 switches at time s B“ = 50(= 100 — 50). 5.3.3 D iscou n t Fare A llo ca tio n P ro b lem Models similar in spirit can be used to model com petition in allocation of seats to regular and discount customers for service companies. O ptim al C losing T im e of D iscount Fares for a M onopoly Consider a firm which faces the problem of selling a fixed stock of n units (airline seats) to two different type of customers (regular and discount) with different prices. The regular demand over the entire horizon is less than n, while the regular and discount demand combined is more than n. The objective of the firm should be to meet as much regular demand as possible, while also being able to sell all its stock. The allocation can be done by closing the discount fare at some point in the selling horizon, leaving all remaining stock to the remaining regular demand. To formalize the model, again let t be the length of the selling horizon, pD be the price for the discount tickets, pR be the price for the regular tickets. Let XRi and Api 177 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. be the demand rates for regular and discount customers, respectively, before closing the discount fare; let Xpa be the regular demand rate after closing the discount fare. Note that Xr2 may be greater than Affl, because of diversion, or upgrading of originally discount customers to regular fare. For the deterministic problem, the closing time of the discount class should be such that the firm sells the maximum number of regular tickets while also depletes all of its stock. Thus, the optim al closing time for a monopoly is such that (Aiji + Adi)s*v/ + A /*2(t — s * v/) = n, which gives * _ n ~ A fg t M ADl + Afti — A /Z 2 If the total demand is less than the capacity ((ADl + ARi)t < n), it is optimal not to close the discount class at all. The solution can be also obtained by transforming the fare allocation problem into a m ark-up problem, by using average price and aggregate demand parameters. To find the equilibrium discount fare closing times, the following transformation can be used: Ai A2 178 — A 0 1 + Xr i , = A R 2 , R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. P i = (p d ^ di + P r ^ r i ) / ( ^ di + A K1) , P 2 = PR- E quilibrium C losing T im es o f D iscou n t Fares for a D u opoly Assume that firm A and firm B have starting inventories of n A and n B, respectively. Discount demand rates for both firms are: Firm B ’s discount fare Firm A/s discount fare open close open close Adi> ^D1 0, A df A df, 0 0,0 Similarly, regular demand rates are: Firm S ’s discount fare Firm A/s discount fare open close open close Am, Aftx A R L , A r f A RF, A r l Ar2, \ R 2 We did not identify the demand rates for each firm when its competitor is sold out, since these demand rates do not affect the equilibrium switching times as long as they are within reasonable bounds described in Theorem 5.2. Once again, the problem can be converted into a competitive m ark-up problem by using the following transformation: A i = A/ji + A a2 = A/J2, A L = A RL, 1 7 9 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. XF = Xr f + Xq f - However, we now will have four average prices: one for the case when both firms have their discount classes open, one for the leader, one for the follower and one for the case when both firms have their discount classes closed. But, we have seen in Section 5.3.2 th a t the analysis does not depend on the prices as long as the demands are deterministic and the assumptions still hold (p\Xi > P2 X2 , and A 2£ < nB < n At). W ith this transformation, the results of Section 5.3.2 can be extended to the discount fare allocation problem. Our model suggests that, with competition, the firm with a smaller capacity closes its discount class first. W ith increased demand interaction, the firm with a smaller capacity would close its discount class later and the firm with a larger capacity would close its discount earlier, which results in a shorter period of time during which the competing firms have different levels of fare availability. 5.4 Conclusion In this section, we have studied the effects of competition on firms in a market with perfect information. The two competing firms face sym m etric demand rates: the only differentiation between the tw'o firms is their initial inventories or capacities. We have used the results of previous researchers on the single-firm problem to simplify the structure of the two-firm problem. Using a game theoretic model, we show th a t a 180 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. unique Nash, equilibrium exists under mild conditions, and we derive the equilibrium price switching times. A major result of our model is that a com peting firm will choose to change its price at a time within the selling season so th a t it will just sell off all its inventory. In the m ark-down problem, the more over-stocked firm will act as a price leader by lowering its price first, whereas in the m ark-up problem the less over-stocked firm will raise its price first. We have shown th at, while differences in the starting inventory positions compel the competing firms to switch their prices at different times in the selling season, an increased degree of demand interaction forces them to follow each other’s actions closely. The length of time for the two firms to charge different prices is shown to be proportional to the difference in initial inventory levels but inversely proportional to the difference in demand rates under the two prices. That is, the stocking difference between the two firms as measured by initial inventories or capacities tends to increase the gap in price switching times, while the demand differential under the two potentially different prices tends to force the competing firms to follow each other’s pricing decisions. As a result of the competition, the more over-stocked firm changes its price later than it would wnthout competition, and the less over-stocked firm changes its price earlier than it would without competition. The behavior predicted by our model is not difficult to observe in practice. 1 8 1 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Another m ajor result is that the more over-stocked firm benefits from compe tition while the less over-stocked firm stands to lose from increased competition. This result might not be surprising if we note that in the mark-down problem the more over-stocked firm lowers its price later than it would when there is no com petition, and thus sells more units with the high price. Hence, competition benefits the more over-stocked firm. On the other hand, the less over-stocked firm lowers its price earlier than it would when there is no competition, and thus sells fewer units with the high price. Hence, competition hurts the less over-stocked firm. A similar argument applies for the m ark-up problem. There are several directions for future research. First, we may allow for multiple price changes. Moreover, the price levels may be decision variables. Introducing uncertainty in demand rates or incomplete information available to the parties of the game may further strengthen out results. However, all these extensions will complicate the game theoretic analysis, which may make it harder to identify and interpret the equilibrium. 1 8 2 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Chapter 6 Conclusions In this thesis, we attack procurement and pricing problems for companies selling a fixed stock of items over a finite horizon. Companies in many industries face such problems as they have inflexibilities in their supply, while their products have unpre dictable demand. Examples include an airline which has to manage the demand for 186 identical coach seats in a DFW-LAX flight by choosing from at least 7 possible fares each day over a 331 day period before departure. As thoroughly investigated in the thesis, another example is an apparel retailer which has to make procurement and pricing decisions for a fashion garment whose demand is highly unpredictable. The procurement decision has to be made as early as 9 months before the garment is put on the store racks when there is little known about the demand. Once the procurement decision is made, pricing is often the only tool left to the retailers and it has to be very responsive to actual sales and competitive factors in the market place. The main objective of the thesis is to develop quantitative models to solve problems th at arise in circumstances exemplified above: settings with uncertain 183 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. demand, inflexible supply and perishable inventories. In particular, we study the one-tim e procurement decision when the perishable inventory can be sold at differ ent prices and the post-procurement pricing decisions with a focus on structured learning based on early season sales and duopolistic competition. Individual models developed in this thesis can be customized and enhanced to support operational decisions in a firm. However, the thesis also aims to enhance the understanding of the economics of the procurement and pricing strategies in this context. Chapter 2 reviews the operations and recent trends in the apparel industry where our models are most applicable. We interviewed an owner of a small apparel manu facturer, a buyer for Macy’s department stores, and an owner/m anager of a specialty apparel retail store. We conclude that the nature of problems necessitates and the presence of information resources provides the means for the use of quantitative models in this industry. However we observe th a t even the simplest of such mod els are hardly applied; experience, intuition and ad-hoc procedures still play m ajor roles, especially in procurement and pricing decisions. This adm itted lack of execu tion results in inefficiencies (i.e., the unfortunate co-existence of low inventory turns and low service levels) across the supply chain, especially at the retailers. Given the current practice, as most studies in the literature are, our research is towards developing prescriptive models rather than simple improvements to existing practice in the apparel industry. We note that other industries facing similar problems in nature are far better off in using such quantitative decision tools for improving their 184 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. profitability (e.g., airlines). In Chapter 2, we also verified the assumptions in our analytic models. Chapter 3 studies the procurement decisions when the perishable product can be sold in different markets with varying prices. We assume that the retailer or the manufacturer has only one chance to place a purchase or production order: a fairly reasonable assumption for ordering decisions in the U.S. apparel industry as most of the manufacturing takes place overseas, prohibiting replenishments during the short fashion seasons. This leads us to approach the problem as a multiple market variant of the single period stochastic inventory problem (i.e., the Newsboy problem). We distinguish two cases: as the time marches forward, the price can go down or up: the former reflecting the case in apparel, the latter reflecting the case in service industries. For both cases, we obtain practicable and optimal solutions that also permit an easy marginal analysis interpretation. For the decreasing price case, we also study the added value of optimizing the procurement decision. For the increas ing price case, we enrich our analysis by incorporating the demand dependencies of different market segments. Chapter 4 studies the pricing decisions assuming that the one time procurement decision is already made. In industries w ith perishable products and inflexible sup ply chains, demand information is valuable more than anything else. Our research in this chapter develops a model for systematic resolution of demand uncertainty based 185 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. on actual sales; in particular, early sales information is used to update the proba bility distribution of the current period using a Bayesian procedure. Conducting a computational study using this model, we first find out that this kind of demand learning makes the most impact when the retailer’s stocking level necessitates prices that are not extremely low or high, and when the dem and is moderately sensitive to price changes. Then, we observe th at when there is fixed supply, mark-down price is an increasing function of sales in the early season. We also observe that retailer profits increase as dem and becomes more predictable and price sensitive. Finally, we study the value of having additional ordering opportunities during the season which helps us to analyze the factors affecting the supplier selection decisions; i.e., a low-cost versus a flexible supplier. Our results from the computational study are supported by empirical data from the apparel industry. Chapter 5 studies yet another important dimension of the problems in this con text: competition. Competition has a considerable impact on pricing and inventory (or capacity) decisions in the apparel industry and service industries such as airlines and hotels. At the strategic level, companies position themselves in the market place by focusing on a particular group of customers, which in turn shapes their long term pricing and inventory decisions. At the tactical level, companies make temporal pricing decisions (with m ark-ups and most likely with mark-downs) or temporal inventory allocations (e.g., opening or closing discount fares) to maximize their profits from a fixed inventory in the short run. While we note th at competition 186 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. has a definite impact at both levels, we study competition only at the tactical level using a game theoretic analysis. Specifically, we model price switching times for two companies that are selling fixed stocks of substitutable items over a finite selling season. We show the existence of price switching times th at form an equilibrium; none of the companies will increase its profits by unilaterally changing its switching time. For the case when both companies are over-stocked (as compared to their sales potential with initial price), we first show that the company with higher inventory to sales ratio reduces its price first. Then, we show that the amount of time during which the firms charge different prices is inversely proportional to the correlation of their demand (i.e., substitutability of the two products they offer), but proportional to the difference in their over-stocking. Finally, we extend our model to the case when both companies are under-stocked and apply it in the context of a discount fare allocation decision for airlines. Overall, the thesis enhances our understanding of pricing and inventory decisions for perishable products and prescribes practical solutions to improve profitability in manufacturing and merchandising operations involving such products. The thesis also sets the background for our future research in this area. We also note that each model developed in this thesis has more or less a separate focus. The first model studies the inventory decision assuming a forward look a t the m arket segmentation that is facilitated by company’s price changes. The second model, on the other hand, assumes th at the inventory decision is already made and focuses on optimizing the 187 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. magnitude and t iming of these price changes. Finally, the third model studies the impact of com petition on such pricing decisions. An obvious direction for future studies would be a more unified approach for pricing and inventory decisions in this context. Namely, an optimized inventory decision that considers the dynamics of price changes and competition within season, followed by a dynamic pricing mecha nism that reacts to consumer and competitor actions on a day-to-day basis during the season. Clearly, modeling the decision process and analytical derivation of the optimal decisions and mechanisms are challenging and we believe that any economic analysis is lim ited to computational studies. In fact, in Section 4.5, we develop a model that allows us to make inventory decisions that take dynamic price changes into account rather than assuming a segmented market as studied in Chapter 3. However, our results in Section 4.5 is based on a computational study, while we are able to reach a better understanding of the problem by deriving the qualitative properties of the optimal solution analytically in Chapter 3. Despite the modeling and analytical difficulties, our future studies will focus on the composite effects of stock levels, pricing and competition by integrating our individual models. First in our agenda is a game theoretic model in which retailers also compete on the start ing stock levels. This will strengthen our results especially in the context of fashion industry. We will also allow the magnitude of the price change as well as the timing of the pricing changes as decision variables. We will still strive for analytical results, but will perform computational analysis on a number of practical issues. 188 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. The three models developed in. this thesis are directly aimed at solving practical decision problems in industries with perishable goods and services. These models also make significant contributions to the existing literature on inventory and pricing theories. C hapter 3 is a unique and fundamental extension of the stochastic single period inventory problem. Rather than assuming that the demand is a single random variable, we assume that it is composed of n demand classes with different prices, each of them an independent random variable realizing sequentially over the single period. For both decreasing and increasing prices, optimal inventories are derived for general probability distributions and they allow for an easy marginal cost/marginal revenue interpretation. Chapter 4 develops a Bayesian updating procedure for a Poisson process whose rate is a separable function of a base rate which is unknown and a price multiplier which is a decision variable. The resulting procedure forms the basis of a dynamic program that is used for periodic pricing of a fixed inventory over a finite horizon. It is shown that the state space for the current period can be reduced to have only two variables: cumulative price multipliers and cumulative demand prior to the current period. This allows a tractable computational solution of the dynamic program which is used to explore the economics of the pricing problems faced by manufacturers and retailers of perishable products. Chapter 5 develops a game theoretic model to study the pricing problems of two competing firms with fixed stocks of items to be sold over a finite horizon. This is one of the first models that studies the duopolistic pricing competition when the supply is fixed for both 189 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. firms. Assuming that each firm changes the price only once during the horizon and prices are known in advance, the problem is first reduced to one in which firms compete on price switching times. Using the popular linear demand model, we first show th at the problem can be transformed into one for which the demands are symmetric. Our major contribution in this chapter is then to show the existence and uniqueness of an equilibrium such that both firms run out of stock right at the end of the horizon. Various economic interpretations are derived using the properties of the equilibrium switching times. 1 9 0 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Bibliography Abernathy, F. H., J. T. Dunlop, J. H. Hammond and D. Weil. 1995. The infor mation integrated channel: a study of the U.S. apparel industry in transition. Brookings Papers on Economic Activity: Microeconomics. 175-246. Apparel Industry Magazine. 1997. Quick response mandates today: an assessment of the industry’s statu s and needs after 10 years of implementing QR. March, 45-48. Apparel Industry Magazine. 1998. Supply chain links bolster retailer-vendor strengths. August, 66-70. Arrow, K. A., T. E. Harris, and J. Marschak. 1951. Optimal inventory policy. Econometrica, 19, 250-272. Azoury, K. S. and B. .L. Miller. 1984. A comparison of the optim al ordering levels of Bayesian and non-Bayesian inventory models Management Science, 30, 993-1003. Azoury, K. S. 1985. Bayes solution to dynamic inventory models under unknown demand distributions. Management Science, 31, 1150-1161. Balvers, R. J. and T. F. Cosimano. 1990. Actively learning about demand and dynamics of price adjustm ent. The Economic Journal, 100, 882-898. Bell, D. E. 1994. Note on retail organization. Harvard Business School Teaching Note #9-595-009. Belobaba, P. P. 1987. Airline yield management: an overview of seat inventory control. Transportation Science, 20, 63-73. Belobaba, P. P. 1989. Application of a probabilistic decision model to airline seat inventory control. Operations Research, 37, 183-197. Belobaba, P. P. and L. R. Weatherford. 1996. Comparing decision rules that incorporate customer diversion in perishable asset revenue management situ ations. Decision Sciences, 27, 343-363. 191 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Bikhchandani, S. and J. W . Mamer. 1993. A duopoly model of pricing for inventory liquidation. European Journal o f Operational Research, 69, 177-186. Bitran, G. R. and S. .V. Mondschein. 1997. Periodic pricing of seasonal products in retailing. Management Science, 43, 64— 79. Bitran, G. R. and H. K. Wadhwa. 1996. A methodology for demand learning with an application to the optimal pricing of seasonal products. M.I.T. Sloan School of Management, Working Paper #3896-96. Blackburn, J. D. 1991. The quick response movement in the apparel industry: a case study in time-compressing supply chains, in Time-based competition, J. D. Blackburn (Ed.) Homewood, IL: Business One/Irwin. Blasberg, J. and D. Wylie. 1996. J.C. Penney. Harvard Business School Case #9-596-102. Blasberg, J. 1997. Liz Claiborne, Inc. Harvard Business School Case #9-597-010. Bobbin. 1995. Time is money: understanding the product velocity advantage. March, 50— 55. Bobbin. 1997. Quick Response beats im porting in retail sourcing analysis. March, 22-27. Bobbin. 1997. A $ 13 billion success story: quick response. June, 46-48. Bodily, S. E. and L. R. Weatherford. 1995. Perishable asset revenue management: generic and multiple-price yield management with diversion. Omega, 23, 173- 185. Bohdanowicz, J. and L. Clamp. 1994. Fashion marketing. New York, NY: Rout- ledge. The Boston Globe. 1998. Looking back to fashion’ s future: firm helps retailers develop merchandising plans. Emerging Business Feature, October 7. Bradford, J. W. and P. K. Sugrue. 1990. A Bayesian approach to the two-period style-goods inventory problem with single replenishment and heterogeneous Poisson demands. Journal of the Operational Research Society, 41, 211-218. Brown, P. K. and J. Rice. 1998. Ready-to-wear apparel analysis. Upper Saddle River, NJ: Merrill. Brown, G. F. and W. F. Rogers. 1973. A Bayesian approach to demand estimation and inventory provisioning. Naval Research Logistics Quarterly, 19, 607-624. 1 9 2 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Brumelle, S. L., J. I. McGill, T. H. Oum, K. Sawaki, and M. W. Tretheway. 1990. Allocation of airline seats between stochastically dependent demands. Transportation Science, 24, 183-192. Business Week. 1998. One more face-lift for Penney. March 23, 1998. Carlson, P. G. 1983. Fashion retailing: the sensitivity of rate of sale to mark-down. Journal of Retailing, 59, 67-76. Chain Store Age. 1992. Federated looks to technology. June, page 46. Chain Store Age. 1994. Buying: different strokes for different folks. January, page 10MH. Chain Store Age. 1996. Retail distribution and logistics: managing the supply chain. October, section two. Chain Store Age. 1999. Managing mark-down madness. March 1, page 108. Chang, S. H. and D. E. Fyfee. 1971. Estimation of forecast errors for seasonal- style-goods sales. Management Science, 18, B-89-B-96. Dalby, J. S. and M. T. Flaherty. 1990. Liz Claiborne, Inc. and Ruentex Industries, Ltd. Harvard Business School, Case # 9-690-048, Cambridge, MA, USA. Daily News Record. 1993. Computer technology drives suit ordering: stores high on EDI for their O-T-B. September 15, 1993. Discount Merchandiser. 1994. Reinventing the department store. May, 54-56. Discount Merchandiser. 1997. Raising the roof. March, 74-76. DeGroot, M. H. 1970. Optimal statistical decisions. McGraw-Hill Book Company. The Dun & Bradstreet Corp. 1999. Business failures by industry. Business Failure Record: 1996 final - 1997 preliminary, 6-9. (http://w w w .dnb.com) Eeckhoudt, E., C. Gollier, and H.‘ Schlesinger. 1995. The risk-averse (and prudent) newsboy. Management Science, 41, 786-794. Eppen, G. D. and A. .V. Iyer. 1997. Improved fashion buying with Bayesian updates. Operations Research, 45, 805-819. Eppen, G. D. and A. .V. Iyer. 1997. Backup agreements in fashion buying - The value of upstream flexibility. Management Science, 43, 1469-1484. Feng, Y. and G. Gallego. 1995. Optimal starting times for end-of-season sales and optimal stopping times for promotional fares. Management Science, 41, 1371-1391. 193 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Fisher, M. L. 1997. W hat is the right supply chain for your product? Harvard Business Review, 74 (2), 105-116. Fisher, M. L., J. H. Hammond, W. R. Obermeyer and A. Raman. 1994. Making supply meet demand in an uncertain world. Harvard Business Review, 72, M ay-June, 83-93. Fisher, M. and A. Raman. 1996. Reducing the cost of demand uncertainty through accurate response to early sales. Operations Research, 44, 87-99. Frazier, R. M. 1986. Quick Response in Soft Lines. Discount Merchandiser, Jan uary, 40-46. Friedman, J. W. 1977. Oligopoly and the theory of games. North-Holland, Ams terdam. Gallego, G. and I. Moon. 1993. The distribution free newsboy problem: review and extensions. Journal of the Operational Research Society, 44, 825-834. Gallego, G. and G. van Ryzin. 1994. Optim al dynamic pricing of inventories with stochastic demand over finite horizons. Management Science, 40, 999-1020. Gallego, G. and G. van Ryzin. 1997. A m ulti-product dynamic pricing problem and its applications to network yield management. Operations Research, 45, 24-41. Gerchak, Y., M. Parlar, and T. K. M. Yee. 1985. Optimal rationing policies and production quantities for products with several demand classes. Canadian Journal of Administrative Sciences, 2, 161-176. Goodwin, D. R. 1992. The open-to-buy system and accurate performance mea surement. International Journal o f Retail and Distribution Management, 20, 2, 16-23. Gurnani, H. and C. S. Tang. 1999. Optim al ordering decisions with uncertain cost and dem and forecast updating. Management Science, 45, 1456-1462. Hammond, J. H. and M. Kelly. 1991. Quick response in the apparel industry. Harvard Business School Teaching Note #9-690-038. Hammond, J. H. and A. Raman. 1996 Sport Obermeyer, Ltd. Harvard Business School Case #9-695-022. The Hosiery Association. 1999. Annual Statistics (h ttp://nahm .com / annual- stats.htm l). 1 9 4 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Hunter, N. A. 1990. Quick response in apparel manufacturing. Manchester, U.K.: The Textile Institute. Hunter, N. A., R. E. King and H. L. W. Nuttle. 1996. Journal of the Textile Institute,87, P art 2, N o.l, 42-55. Ismail, B. and J. Luderback. 1979. Optimizing and satisfying in stochastic cost-volume profit analysis. Decision Sciences, 10, 205-217. Jucker, J. V. and M. J. Rosenblatt. 1985. Single-period inventory models with demand uncertainty and quantity discounts: behavioral implications and a new solution procedure. Naval Research Logistics, 32, 537-550. Khouja, M. 1995. The newsboy problem under progressive multiple discounts. European Journal of Operational Research, 84, 458-466. Khouja. M. 1996. The newsboy problem with multiple discounts offered by suppliers and retailers. Decision Sciences, 27, 589-599. Kouvelis, P. and G. Gutierrez. 1997. The newrs-vendor problem in a global mar ket: optimal centralized and decentralized policies for a two-market stochastic inventory system. Management Science, 43, 571-585. Lau, H. The newsboy problem under alternative optimization objectives. Journal of Operational Research Society, 31, 525-535. Lau, A. H. and H. Lau. 1988. The newsboy problem with price-dependent demand distribution. H E Transactions, 20, 168-175. Lazear, E. P. 1986. Retail pricing and clearance sales. American Economic Review, 76, 14-32. Leamon, A. 1998. Note on the retailing industry. Harvard Business School teach ing note #9-598-148. Lee, T. C. and M. Hersh. 1993. A model for dynamic airline seat inventory control with multiple seat bookings. Transportation Science, 27, 252-265. Littlewood, K. 1972. Forecasting and control of passenger bookings. AGIFORS Symp. Proc., 95-117. Li, J., H. Lau, and A.H. Lau. 1991. A two-product newsboy problem w ith satis ficing objective and independent exponential demands. H E Transactions, 23, 29-39. McGuire, T. W and R. Staelin. 1983. An industry equilibrium analysis of down stream vertical integration. Marketing Science, 2, 161-191. 195 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Morse, M. P. and G. E. Kimball. 1951. Methods of Operations Research, M.I.T. Press. Murray, G. R. and E. A. Silver. 1966. A Bayesian analysis of the style goods inventory problem. Management Science, 12, 785-797. Nahmias, S. and S. A. Smith. 1994. Optimizing inventory' levels in a two-echelon retailer system with partial lost sales. Management Science, 40, 582-596. National Academy of Engineering. 1983. The competitive status of the U.S. fibers, textiles and apparel complex. Washington, D.C.: National Academy Press. Ostic, K. O. 1997. An introduction to USITC enterprise analysis. Demand Acti vated Manufacturing Architecture (DAMA)-AMTEX research paper (http://w w w . tc2.sandia.gov). Parpia, M. 1995. Nordstrom. Harvard Business School case #9-579-218. Pashigian, B. P. 1988. Demand uncertainty and sales: a study of fashion and m ark-dow n pricing. American Economic Review, 78, 936-953. Pashigian, B. P. 1995. Price theory and applications. McGraw-Hill, New York, NY. Chapter 15: Pricing under uncertainty. Pashigian, B. P., and B. Bowen. 1991. Why are products sold on sale: Explanations of pricing regularities. The Quarterly Journal o f Economics, 106, 1014-1038. Pfeifer, P. E. 1989. The airline discount fare allocation problem. Decision Sciences, 20, 149-157. Popovic, J. B. 1987. Decision making on stock levels in cases of uncertain demand rate. European Journal of Operational Research, 32, 276-290. Reyniers, D. 1990. A high-low search algorithm for a newsboy problem with delayed information feedback. Operations Research, 38, 838-846. Richardson, J. 1996. Vertical integration and rapid response in fashion apparel. Organization Science, 7 (4), 400-412. Scarf, H. 1959. Bayes solutions to the statistical inventory problem. Annals of Mathematical Statistics, 30, 490-508. Scarf, H. 1960. Some remarks on Bayes solutions to the inventory problem. Naval Research Logistics Quarterly, 7, 591-596. Silver, E. A. and R. Peterson. 1985. Decision Systems for Inventory Management and Production Planning, second edition, Chapter 10.3, pp.406-410. Wiley. 196 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Smith, S. A. and D. D. Achabal. 1998. Clearance pricing and inventory policies for retail chains. Management Science, 44, 285-300. Smith, B., J. Leimkuhler, R. Darrow, and J. Samuels. 1992. Yield management at American Airlines. Interfaces, 22, 8— 31. Smith, S. A., N. Agrawal and S. H. McIntyre. 1998. A discrete optimization model for seasonal merchandise planning. Journal of Retailing, 74, 193-221. Sporting Goods Business. 1998. Pricing in the ballpark. March, 8S-10S. Standard & Poor’s. 1998. Industry Surveys: Apparel & Footwear. Stores. 1994. An implementable approach for optimizing departm ent store m ark down decisions. April, RR1-RR6. Stores. 1998. J.C. Penney puts product flow on FAST track. May, 49-51. Stores. 1998. Merchants try complex m athem atical tools to improve inventory decisions. November, 74-76. Subrahmanyan, S. and R. Shoemaker. 1996. Developing optimal pricing and inventory policies of retailers who face uncertain demand. Journal of Retailing, 72, 7-30. Taplin, I. M. 1997. Backwards into the future: new technologies and old work organization in the U.S. clothing industry, in Taplin and Winterton (eds) Re thinking global production, Brookfield, VT: Ashgate. Taplin, I. M. and J. W interton. 1997. Rethinking global production: a comparative analysis of restructuring in the clothing industry. Brookfield, VT: Ashgate. U.S. Census Bureau. 1992-1999. Current Industrial Reports. Apparel. MQ23A. (http: / / www.census.gov/cir/www/mq23a.html) U.S. Census Bureau. 1999. Advance comparative statistics for the United States (1987 SIC Basis):1997. the Official Statistic, 1997 Economic Census. U.S. Census Bureau. 1999. 1997 Economic Census, Manufacturing, Industry Se ries, Sheer Hosiery Mills (EC97M-3151A). U.S. Census Bureau. 1999. 1997 Economic Census, Manufacturing, Industry Se ries, Other Hosiery and Sock Mills (EC97M-3151B). U.S. Department of Commerce. 1999. U.S. Industry and Trade Outlook ’99, Chap ter 33: Apparel and fabricated textile products. 1 9 7 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. U.S. International Trade Administration. 1998. U.S. imports, production, mar kets, import production rations and domestic market shares for textile and apparel product categories, Quarterly Report for Fourth Q uarter 1997, Office of Textiles and Apparel, September. U.S. International Trade Commission. 1995. Forces behind restructuring in U.S. apparel retailing and its effect on the U.S. apparel industry. Industry, Trade, and Technology Review, March, 23-28. U.S. International Trade Commission. 1995. Industry & Trade Summary: Apparel, USITC publication 2853, January. U.S. International Trade Commission. 1998. Apparel sourcing strategies for com peting in the U.S. market. Industry, Trade, and Technology Review, USITC publication 3153, December, 31-40. U.S. International Trade Commission. 1999. Industry & Trade Summary: Apparel, USITC publication 3169, March. U.S. Office of Technology Assessment. 1987. The U.S. textile and apparel industry: a revolution in progress. Washington, D.C. Varian, H. R. 1980. A model of sales. American Economic Review, 70, 651-659. van Praag, B., and B. Bode. 1992. Retail pricing and the cost of clearance sales. European Economic Review, 36, 945-962. W eatherford, L. R. and S. E. Bodily. 1992. A taxonomy and research overview of perishable asset revenue management: yield management, over-booking, and pricing. Operations Research, 40, 831-844. Women Wear Daily. 1999. Levi’s looking beyond basics. March 11, page 15. W omen Wear Daily. 1999. Systemizing the mark-down. January 27, page 16. 1 9 8 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Appendix A Proofs for Chapter 3 A .l Proof of Proposition 3.1 Taking derivative w.r.t. X in (3.1) and simplifying, = £ Djgj ( X - D j )fj (Dj )dDj + 0 + C \C „ 9j(Tj)dTj + 0 - DM (X - D ^ U iD ^ d D j + J (J L«/ A — L/j + f \ x - Tj )gj (Tj )dTj f j ( X ) + Jo + C [1 - FjiXftgjiTjIdTj - f X (X - r3)/j (X )a,(T ,)< £ Z ’ + 0 «/ 0 JQ = r C „ ®(Ti ) dTi M D i ) d D i + / A[l - Fj (X)]gi (Tj )dTj J 0 J A — D j J 0 = P r{Tj + Dj > X, Tj < X , Dj < X} + Pr{7) < X, D3 > X} = Pr{Ty < X } - Pr{Ty + Dj < X} = Pr{Tj + Dj > X} - Pr{Tj > X}. 1 9 9 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Together with = 1 - F ^ X ) = Pr{£>, > X}, we obtain dir(X) P = p , P r { D , > X} + j ; P j [ P r £ A < X } - P r { £ C ( < X} L j =2 L i=l t=l dX n —1 — C = - Pi+.) P r { E ° i > A '} + P" Pr( E a > A'} - j=l 1= 1 t=l Taking the second derivative, we obtain ~fV ~ — = - 2 - C P j - P j + l ) 9j { A ) - P n ^ n ( A ) , c l a . j=i which is always non-positive since pj > Pj+\ and pn > 0. This proves that expected profit is concave in X so that the solution to equation (3.3) is the global maximum. A.2 Proof of Proposition 3.3 d2ir(X.P) n d2ir(X,P) n We need to show ---- < 0 ,----- < 0, and d27i(X, P) d X 2 d P 2 d2ir(X,P)]2 d2Tr(X,P) dXdP d X 2 dP 2 ’ From (3.6), we have d2E[Q2] d X 2 = [ P[ -f2{X - D ^ M D J d D , Jo 2 0 0 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. + Fi(P)[-m x - p)\ r X -P r „ , X - P + s P - D o + r [~ /i( — - ) } ( ! / s)f2(D2)dD2 Jo s + F\(P)f2( X - P ) = - f j M X - D J M D J d D r - - ( 1 /5 ) j X~P M X ~ P + s P ~ D2)f2(D2)dD2, J 0 5 hence fE B s l < o a x 2 a x 2 ~ ' From (3.7), we have = s M P ) - ( i - s ) l - M P ) m x - P ) - ( i - s ) 7 T ( P ) [ - / , ( A '- P ) ] ( - i ) . . rX—P X — P -f- sP — Do —1 “F 5 \ - , _ — (1 — s) / [ ~ /l( ----------------------- “)]( ) f2(D-2)dD2 Jo s \ s J - ( l - s ) F \ ( P ) M X - P ) ( - l ) = sfi(P) + (1 — s)fi(P)7^(X — P) — (1 - s)2 f X - P X - P + s P - D2 r f l i :- - - - - - - - - - £ J L £ - - - - - - - - - - - - - - - -- ) f 2(Do)dDo. Jo s 2 0 1 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Since d2E[Qi]/dP2 = — /i(.P ), we have dir (X,P) ^ d2E[Q1} d2E[Q2\ dP 2 Pl dP 2 P2 dP2 = —[Pl - sp2 - (1 - s)P2K ( X - P)]fi(P) - (1 - s)2 f X - P c , X - P + sP - Do, Vi- r A — ± 4 - S ± — / / o / h i - — ---------l)h {D 2)dD2. Jo s W hen s -f- (1 — .s)ir2(X — P) < P1/P2, the first term of the above is negative; hence the whole expression is negative. Next, consider ^ - From (3.6), we have a 2£ [Q 2] = F2( X - P ) h { P ) a x d p + [~h(P)W i(x - P) + F l( P ) [ - u x - P)](-i) fX— P X. — P - t - s P — Do . . f — 1 - f - s \ \ j j-i + [—f i i ------------------------ ) ] (----------)fi{D2)dDo Jo s V s / + F\(P)f2(X - P ) ( - 1) _ ~ s % \ f x ~P X — P -+ - sP — D2^ c ( n ~ Jo / l ( ------------;------------) M D 2)dD2. Let A = /0 A P h ( X P+*P D' )fi{D2)dD2. Then we have ahr(x,p) _ _ a2^ ] , _ a2£[Q 2] _ f i - s \ A dX dP Pl d X d P P2 dX dP P2V s 7* ' 2 0 2 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. It remains to prove a 27r(X, P) 12 dh r(X , P) d2w(X, P) d X dP d X 2 dP2 . Notice th at the integral . , . d2ir(X , P ) 32tt(X , P) , . L . . . n , term A appears m both ---- — — ~ a n d -= r= ---- (with a negative sign): all other a X l aP terms in both expressions are negative. Hence dX 2 dP2 - d27r(X. P) d X d P This proves th at the function tt(X, P) is jointly concave in X and P. A.3 Proof of Proposition 3.4 For Part (a), it is necessary and sufficient to prove p2 P r{ P i + Do > X*} > c. We consider three cases. In Case 1 (0 < P < X), the optim al X * satisfies equation (3.8) or (3.10). As shown in Figure A .l, a key observation is that the three probability terms on the left-hand-side of equation (3.10) correspond to regions I. II and III; whereas the probability term Pr-fZh + D2 > X } corresponds to a region wdiich includes regions I, II and III as sub-regions. Therefore, 2 0 3 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Figure A.l: The left-hand-side probabilities of equation (3.10) : = PTf o t*D ,> XI i : Pr(»D XI P X Pr{£>! + D2 > X, Dl < P} 4- P r{Dl > P } Pr{£>2 > X - P} + + Pr{s(T»x - P ) + D 2 > X - P , D 2 < X - P} ^ Pr{Z?i -I- D 2 > A }-. From (3.10), we have p2 Pr{T>i + D2 > X*} > c. In Case 2 (P = 0), the optimal X* satisfies p2Pr{sDi + D2 > X*} = c. Since 0 < s < 1, p2 Pr{-Dx + D2 > X*} > p2 PrlsD x + Do > X"*} = c. In Case 3 (P = X ), the optimal X* satisfies equation (3.5), from which we write p2 Pr{L>i + D2 > X*} = c + (p2 — pi) Pr{P>! > X*} > c. For Part (b), we need to prove p2Pv{sDi + D2 > X* — P*} > c. In Case 1 (0 < P < X ), we observe from Figure A .l that regions I, II and III are sub-regions 204 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. of the region represented by the probability term Pr{s£?i + D2 < X — P }. In Case 2 and Case 3, the proposition holds trivially. A.4 Proof of Lemma 3.1 diriX, P) Again referring to Figure A .l, we can rew rite pp (the left-hand-side of equa tion (3.11)) as M g p P) = p2( l - s ) M s ( D 1- P ) + D 2 < X - P , D l > P } -(p2 -P i)P r{£ > i> /’}- (A.1) Q'ftfX P) If — p p —- < 0 at P = 0, then from the above (setting P = 0), we have p2(l - s) P rls P ! + D 2 < X } < p2 - p i - (A.2) It can be easily shown that for two independent random variables A and B, if Pr{B > 6} > 0, then Pr{.4 + B <a B > b} < Pr{A + B < a} for all a. Hence Pr{s(Dl - P) + D2 < X — P, Di > P} = Pr{sDl + D 2 < X - P + s P \ D l > P} P r f A > P} < Pr{s£>! + D2 < X - P A- sP} Pv{Di > P}. 2 0 5 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Substituting inequality (A.2) into (A .l), we have, for any P > 0, < p2( l- s ) P r { s ( D l - P ) + D2 < X - P :D l > P } - p2{ 1 - s) Pr{s£>! + D2 < X } Pt{D1 > P} < p a(l - s) Pr{s£>! + D2 < X - P + sP} Pr{D x > P } - p2( 1 - s) Pv{.sDl + D 2 < X } P r{Dl > P} < 0 . 2 0 6 dnr(X, P) dP R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix B Proofs for Chapter 5 B .l Proof of Theorem 5.1 Let i?A(sA, s B) be the payoff for firm A if it switches its price at time s A and if the firm B switches the price at time s B. Likewise, let R B (sA, s B ) be the payoff for firm B for the switching times sA and s B. We divide the lengthy proof into several parts. P a r t I: We have to show, R a (s a* ,s b *) > R A (sA, s B*), for 0 < s A < t. A P a r t 1.1: R a (s a*,s b *) > R A (sA , s B*), for 0 < s A < s 2 0 7 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Note that R A(sAm, s Bm) = Pl\ lSA* +p2(mA - A1s-4*), R a (s a , s b * ) = pi^iSA + p2{mA — A i S ' 4 ) , since for both cases when s A = s A* and when s A < s A*. firm A is able to deplete all of its inventory. Then, R a ( s a \ s b * ) - R a {s a , s b *) = ( P ! — p2)Ai(sA* - s-4), which is always positive. Part 1.2: R a (s a *, s b ~ ) > R a (s a , s B*), for s-4* < s-4 < ss *. In this case, firm B will be stocked out, say at time £, before the end of the horizon. Then, Ais-4 + \ p ( s B* - s A) + X2(£ ~ s B*) = m B. (B.l) Recall th at we already have A i5a* + Af ( s b * - s A*) + A2{t - s B*) = m B. (B.2) 2 0 8 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. At this point, we need to give the conditions under which firm A is not able to deplete all of its inventory by postponing its switching time. Otherwise, firm A can definitely increase its payoff thus (sA*,sB*) is no longer an equilibrium. Let us find the minimum XM2 at which firm A can also deplete its inventory before the end of the horizon. We should have a solution for the minimum XM2 from the following equation: Using (B.l) and (B.3), (XL — Xf)(.sB* — sA) + A M2(t — Z) = m A — m B. Noting that (A ^, — A/r)(s'4* — sB*) = mA — m B, we have (Ai — Af')(s'4* — .s'4) + X\[2(t — £) = 0. Using (B.l) and (B.2), we have Als + Al(sb* - sA) + X2(i - sBn + XM2{t - t ) = m A. (B.3) (AF - Ax)(sA - s'4*) + A 2(£ - t) = 0. (B.4) And we have, (B.5) 2 0 9 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. T hat is, firm A will be able to deplete all its inventory at time t only if XM2 is given by equation (B.5). For Xl ~ Ai' (B.6) firm A is not able to finish its inventory, if its switching tim e exceeds s* 4*. Now assuming that equation (B.6) is satisfied, we compare the payoff when s A = sA* and the payoff when s A* < s A < s B * . R ( s , s ) = piXis- + P 2 [ X l(s b * — s ) + X2(t — s B* ) ], RA(sA, s b *) = P iX isA + P2 [X l(s B* — s A) + A2(£ — sB*) + Xm 2 {t — £)]> R A(sA*,sB*) - RA(sA,s Bn = (p1A1 - p 2Ai )(S4‘ - sA) + p 2(AM2- A 2) ( £ - t ) . Using equation (B.4). R a {sa\ s b * ) - R a {s a : s b *) = (sA* - s A) (PlXi — P2XL.) +P2(Xm2 — ^2) (A i— XF) A o which is positive only if Xm2 < A 2 1 + P2.Xl — P1A1' Pz{Xi — Ap). (B.7) P art 1.3: R a {sa\ s b *) > R a (s a , sB*), for s B* < s A < t. 2 1 0 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Assume sA is such that firm B still has some inventory, when firm A changes the price. Then, AisB* H - Al(sa - sB*) 4- X2(i - sA) = m B. Also using AlSB* + Af {sb* - sA*) + A2{t - sB*) = m B, We have, (At - Af )(sb* - sA*) + (XL - A 2)(s-4 - sB*) - A 2(f - i ) = Q . (B.8) On the other hand, RA(sA, sb*) = Pl[\lSB* + Xf{sa - .sB*)] + p2[X2(e - s A) + AM2(t - £)], R a (s a * , s b * ) - R a ( s a , s b *) = (p2XL — p i X i ) ( s B * — s ' 4 *) + (P2X2 — PlXF) (sa — sB ) — ( P 2 A j V / 2 — P2X0) (t — t). Using equation (B.8), R a {sa*,s b *) - R a (s a , s b*) = 2 1 1 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. (P2^L ~ P l ^ l ) ~ (p2A 2 ~ Pi^f) ~ {P2^M2 ~ P2^2)(^l — Xp) a2 (P2Aji/2 ~ P2^2){^L ~ ^2) \o (sB* - sA~) (sA - sBn, which is positive only if 1 + {P2^L ~ PlAl) p2(A 1 — Xp-) (B-9) and Part 1.4: X A /0 < A 2 1 + (p2 A 2 — P \ X p ) P2{Xl — A 2) . Ra(sa*, sb*) > R a(sa, sb*), for s"* < sA < t. B* „ .-l (B.10) Assume that firm B's inventory is depleted before firm A changes the price. From Part 1.3, we know that R A(sA*, sB*) > R A(sA: sB*) for sA — I. Now assume sA > t. Then, RA{sA, sB*) = pi[XiSB* + Xp(£ — s B*) + X\fi(sA — t}\ + P 2^m2(^ — - s‘ 4) But, RA(e, sB*) = Pl[xlS^ + A F(e - ss *)] + P2XM2(t - i) s* RA(i, sB*) - R a(s-\ sb*) = (p2XM2 - P i X M l )(e - sA). 2 1 2 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. For P2^M2 — P lA /v/1 > 0 , ( B . l l ) R A(i,sB*) - R A(sA,sB*) > 0 and R A(sA*, sB*) > RA{sA,sB*). Part II: We need to show, R b (sa\ sb*) > R b (sa*, s B), for 0 < sB < t. P art 1.1: R b (sa % sb*) > R b ( s a *, sB), for 0 < sB < s'4* R b (s a *. s b ) = piXi-sB + p2(ma — A iS B ), # B (s '4‘ , s b *) = pi.[AiS*4* 4- A F ( s B* - s'4*)] + p 2 [mB - A is A* — XF(sB* — s'4* )], R b (sa *, s b *) - i? B (s'4*, s B ) = ( p i - P 2 ) [ A i( s a * - sB ) 4- Xf ( s b * - s'4* )] > 0. P art II.2: R b (sa *, sb*) > R b ( s a *, s b ), for s'4* < sB < sB* R b (s a *, s B) = pi[Ai.sA * + Af (sb — s'4*)] + p 2[mB — AiS'4* — XF {sB — s'4*)], R b (sa *. s b *) = pt[AiSA * 4- A f ( s b * — s'4*)] + p 2[mB — Axs'4* — AF(sB* — s'4*)], R b (sa * ,s b*) - R b (sa\ s b ) = (p! — p2)AF(sB* — s B)] > 0. 213 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. P a r t I I . 3 : R b (sa *, s B*) > R b (sa \ s b ) , for sB* < s B <t. In this case, firm A will be stocked out before the end of the horizon. First assume th at firm B will switch the price before firm A is stocked out. Then, A ^'4* + A c(sB — .sA*) + A2(£ — sB) = m A. (B.12) Remember, we already have Ais-4* + A l ( s b * - sA*) + A2(t - sB*) = m A (B.13) where £ is the time that the firm A is stocked out. At this point, we need to give the conditions under which firm B is not able to deplete all of its inventory by postponing its switching time. Otherwise, firm B can definitely increase its payoff so that (s-4 *,sB*) is no longer an equilibrium. Let us find the minimum Xmi that firm B can deplete its inventory before the end of the horizon. We should have a solution for the minimum AA /2 from the following equation: AiS'4* + Xf {sb — s4*) + A 2(£ — s B) + Xm i(£ — = • (B.14) 214 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Using (B.12) and (B.14), (A/, — Xp)(sB — s'4*) + Aa/2(£ — 0 — mA Noting that (A/, — Af ) ( s a * — s B * ) = m A — m B , we have (Al - Af )(sb - s3-) - AM2(t - 0 = 0 . m Using (B.12) and (B.13), we have (At - A2) ( s b - s 3 * ) - A2 {t - £) = 0. (B.15) Aa/2 = A 2 + A 2A 2 4-F - (B.16) Then we have, A a /2 = A o + A 2 . . That is, firm B will be able to deplete all its inventory at time t only if Aa/2 is given by equation (B.16). For A„2 < A 2(l + ^ — ^ ) , (B.17) Firm B is not able to finish its inventory, if its switching tim e exceeds s s *. 215 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Now assuming that equation (B.17) is satisfied, we compare the payoff when s B = s B* and the payoff when s B* < s B < i. R b {s a * ,sb*) R b {s a \ s b*) R a {s a *, s B*) - R a (s a *, ss ) = = Pl[AiS‘ * d- Ap(s — S * )] d“ ~ S )) = PifAiS'"1* -F A f ( s b — s'4*)] d“ P2[A2(^ — SB ) -I- A ji/2(£ — £)\', (P2 X2 - P lAf ) ( s s - S B *) d- P2(Aa/2 — A 2)(£ — £)■ Using equation (B.lo), R a (sa *, s b *) - R A (sA* ,s B) = (P2 A2 — PlA/r) A 9 (Ar, — A 2) — P2 ( Aa/2 — A 2) ( t - 0 which is positive only if A A /2 < A 2 1 + P2A2 — PiAf P2(Al — A 2) . (B-18) P art II.4: R b (s a *, s B*) > i? B (s '4*, s B), fo r s B* < s * < t . 2 1 6 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Now, assume th at firm B will switch the price after firm A’s inventory is depleted. From Part II.3, we know that R B{sA*. sB*) > R A(sA *, sB) for sB = I. Now assume sB > I. Then, R A(sA*, sB) = P ip q s4* Xp(sB — s'4* ) -F Xmi(,sb — £)] poXhteit — s B ). But, Ra (sa*, £) = Pi[AiS'4* 4- Ap(i — s'4*)] + P2XM -2.it — £), i?'4(s'4*, £) - Ra{sa\ s b) = (p2XM2 - PlXMl)(sB - £). For P2 Aa/2 — Pi Ami > 0, (B.19) i?-4(s'4*, £) — f?'4(s'4*, sB) > 0 and i?'4(s'4*, s s *) > R A(sA*. sB). Combining the con ditions (B.7), (B.9), (B.10), (B .ll), (B.17), (B.18) and (B.19) all together, {sA\.sB*) is a Nash equilibrium if \ ^ \ (a , - cP^-Xl P1A1 P2A 2 — PiAf \ , D , A „ , < A 2(! +mm{^x— ^ }) and (B.20) PiAmi < P2AM2- (B.21) 217 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. For the linear demand model, one can show th at Xi -F X2 = Xp + XL in a symmetric linear model, which also leads to, At < < A,. Then, \ f i , - r P 2 ^ L ~ P l X i P 2 ^ 2 ~ P i X f ~,\ ^ p d xA l + m m { p , ^ - ^ r P^ - x 2) }) * 2 M - (B22) But Xm2 = (1 + 9)X2 in a linear model, and Xm2 < 2Ao, since 9 < 1, and condition (B.20) is always satisfied. Since pi -fp 2 > 1 //?- we also have PiX,\n < p2X ^2, which completes the proof. Note that conditions (B.20) and (B.21) are the general conditions for the exis tence of the Nash equilibrium. Any demand model that satisfies these conditions will have a Nash equilibrium. B.2 Proof of Theorem 5.2 To show the uniqueness of the equilibrium, we use Theorem 7.7. in Friedman (1977). We need to show that the response functions for both firms, rA and rB are contrac tions, i.e., K ( Sf ) - r ^ ( Sf)| < \sB - s B\, and | r * ( ^ ) - r * ( s y 4 ) | < y 218 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Note th a t each firm responds to its com petitor’s switching time in a way that it depletes its inventory right at time £, and response functions are piecewise linear functions. Below we focus on the response function for firm A. Five situations may be valid when firm A is responding to firm B. Note th at only one of these cases will be valid, based on the parameters of the problem and the switching tim e of firm B. 1. A’s response is in such a way that it switches before B and B is not able to finish its inventory before t. The response function for A can be written as: _ A 2£ - m A \ L - A 2 b {s xL - x x + xL - x , s • 2. A’s response is in such a way that it switches before B and makes B run out of stock before £. The response function for A can be w ritten as: .4 / B \ X m o ttib 4- X2{jnA — m B ) — A 2Aaj2£ Aa/2(A2 — Xp) — X2 ( X l — A^) B Aw2(Ai — A p) — A 2(A z, — Xp) A ,v/2(A ]. — Xp) — X2(Xc — A p) 3. A’s response is in such a way that it switches after B and B runs out of stock after A switches and before t. The response function for A can be written as: .4 / B\ _ AA/2m g + A2 (m '4 — m B ) — A2A A a / 2(A^ — At ) — A2(A^ — Ap) B Aa/2(Ai, — A2) — A2(A l — X p) Aa/2(A/, — A2) — A2(A£ — A p) 219 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4. A’s response is in such a way th at it switches after B and B runs out of stock before A switches and before t. The response function for A can be written as: A B _ Ac m A — (Am i — X p ) m B + XL\ Mot A i(A & — AF ) — Ami (Al — A L ) B Ai,(Am2 — Ami) Xl(Xm2 — Ami) 5. A’s response is in such a way th at it switches after B and B is not able to finish its inventory before t. The response function for A can be w ritten as: r* (s S) = Xf ™A + X> ~ FsB. Ao — A p Ao — A f To prove that rA is a contraction, we only need to show that each possible piece of rA has a slope whose absolute value is less than 1. The absolute value of each slope is less than 1, if the following conditions hold: \ ^ \ jl. ■ r ^ l — ( A l + A 2 ) ( A t + A 2 ) — 2 A f 1 ^ M 2 < A 2 1 1 + m i n ^ ( A l + A 2 ) - 2 A f ’ 2A £. - (A x + A o ) V ’ ( ^ A x ( A £ — Xp) — XMi (Xl — A x ) < 1. (B.24) Ai,(Am2 — Ami) Note that not all pieces will be feasible for a firm. Similar functions and conditions are required for firm B and equations (B.23) and (B.24) are the sufficient conditions for (sA*,sB*) to be a unique equilibrium. For the linear demand model, Am2 = (1-F0)A2 and A x +A 2 = A^ + A F and (B.23) is reduced to 9 < 1 which is always true. We also have, Xp — Xp = y if (^2 — Ax) 220 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. and Xl — A x = (A2 — Ai) which reduces the left hand side of equation (B.24) to 0, which completes the proof. Note that conditions (B.23) and (B.24) are the general conditions for the unique ness for any dem and model. The sufficiency bounds of Al V /i and Xm2 for uniqueness may be tighter than the conditions (B.20) and (B.21) for equilibrium given in The orem 5.1. 2 2 1 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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Sen, Alper
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Core Title
Inventory and pricing models for perishable products
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Graduate School
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Doctor of Philosophy
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Business Administration
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business administration, management,engineering, industrial,OAI-PMH Harvest,operations research
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Rajagopalan, Sampath (
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