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Essays on supply chain management

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Content ESSAYS ON SUPPLY CHAIN MANAGEMENT by Hiroshi Ochiumi 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 2008 Copyright 2008 Hiroshi Ochiumi Table of Contents List of Figures iv List of Tables vi Acknowledgements vii Abstract viii Part I Compensation Schemes for Forecasters 1 Chapter 1 Introduction 2 Chapter 2 Literature 7 Chapter 3 Linking Forecasts with Firm Performance 13 3.1 Refinement-based Ordering . . . .... ... .... .... ... .. 15 3.1.1 Scalable Family of Distributions . . . .... .... ... .. 17 3.1.2 Bayesian Update . . . . .... ... .... .... ... .. 17 Chapter 4 Compensation Schemes when Costs are Negligible 19 4.1 Compensation Schemes for Normally Distributed Forecasts . . . . . . 26 Chapter 5 Costly Forecasts 31 5.1 Normal Distributions . . . . . . .... ... .... .... ... .. 33 Chapter 6 Forecasting with Effort-Dependent Demand 37 Chapter 7 Summary 47 Part II Optimal Dynamic Capacity Allocation and Order Acceptance Policies in a Make-to-Order Environment 49 Chapter 8 Introduction 50 ii Chapter 9 Literature 52 Chapter 10 Model and Analysis 55 10.1 Two-Class Demand Problem . . .... ... .... .... ... .. 58 10.1.1 Week 1 . . .... ... .... ... .... .... ... .. 59 10.1.2 Week 2 . . .... ... .... ... .... .... ... .. 59 10.2 Discussion . . . . . .... ... .... ... .... .... ... .. 96 10.2.1 Acceptance Region . . . .... ... .... .... ... .. 96 10.2.2 Value of Shifting . . . . .... ... .... .... ... .. 98 10.2.3 Frequency of Early Production . . . . .... .... ... .. 101 10.2.4 Impact of Group Arrivals.... ... .... .... ... .. 102 10.3 Three-Class Demand Problem . .... ... .... .... ... .. 105 10.3.1 Week 1 . . .... ... .... ... .... .... ... .. 105 10.3.2 Week 2 . . .... ... .... ... .... .... ... .. 105 10.3.3 Week 3 . . .... ... .... ... .... .... ... .. 113 10.4 Discussion . . . . . .... ... .... ... .... .... ... .. 115 Chapter 11 Summary 119 Part III Conclusions and Future Directions 120 References 124 Appendices 128 A.1 Truth-Telling Compensation Schemes . . . . .... .... ... .. 128 A.2 Proofs .... ... .... ... .... ... .... .... ... .. 129 iii List of Figures 6.1 Separation is impossible . . . . .... ... .... .... ... .. 41 6.2 Forecasts are uniform but with different parameters in each regime. . . 42 6.3 Regimes with uniform and normal distributions and monotone com- pensations. . . . . .... ... .... ... .... .... ... .. 43 6.4 Regimes with uniform and normal distributions and monotone com- pensations. . . . . .... ... .... ... .... .... ... .. 46 10.1 The Model (N=3) . .... ... .... ... .... .... ... .. 56 10.2 After observing demand on Day 1 . . . . . . .... .... ... .. 57 10.3 Type 2 Acceptance Region . . . .... ... .... .... ... .. 92 10.4 A typical value function . . . . . .... ... .... .... ... .. 96 10.5 When arrivals are slow . . . . . .... ... .... .... ... .. 97 10.6 Value functions . . .... ... .... ... .... .... ... .. 98 10.7 Value functions in Week 5 . . . .... ... .... .... ... .. 99 10.8 Acceptance region in Week 2 . . .... ... .... .... ... .. 99 10.9 Acceptance regions in Week 2 . .... ... .... .... ... .. 100 10.10Value of being able to produce earlier: p 1 =0.3,p 2 =0.4,R 1 = 350 . 101 10.11Value of being able to produce earlier: p 1 =0.3,p 2 =0.4,R 1 = 350, R 2 = 100 .. ... .... ... .... ... .... .... ... .. 102 10.12Value of being able to produce earlier: p 1 =0.1,p 2 =0.133 ... .. 103 10.13Value of being able to produce earlier when there are 6 total accepted orders .... ... .... ... .... ... .... .... ... .. 104 iv 10.14Value functions . . .... ... .... ... .... .... ... .. 116 10.15Value functions in Week 6 . . . .... ... .... .... ... .. 117 10.16Acceptance boundaries on Day 5, Week 5 . . .... .... ... .. 118 v List of Tables 1.1 Scenarios . . . . . .... ... .... ... .... .... ... .. 5 10.1 Summary of Notation . . . . . . .... ... .... .... ... .. 55 10.2 Frequencies of Early Production .... ... .... .... ... .. 101 10.3 Accumulated Revenue during Week 3-5 (in dollars) . .... ... .. 103 11.1 Parameters . . . . . .... ... .... ... .... .... ... .. 129 vi Acknowledgements I wish to thank my advisor, Prof. Sriram Dasu, for his support and guidance, especially the freedom he gave me both academically and otherwise. He created a pleasant work atmosphere which made this study even more rewarding. I owe a debt of gratitude to all the members of my Ph.D. dissertation committee; Prof. Yehuda Bassok for introducing me to the area of Operations Management and encouraging me to pursue my goals; Prof. S. Rajagopalan for all his insightful comments and suggestions, from which this thesis benefited significantly; Prof. Richard B. Chase for always providing me with behavioral perspectives that pushed me to think in new ways; and Prof. Randolph W. Hall for his comments and insights which broadened the way I approach problems. I would also like to thank my colleagues at USC; Wayne Johannson, Mahesh Nagara- jan, Chunyang Tong, Feng Chen, Nan Xia, Levent Kocaga, and Jason Niggley. They made the Ph.D. program at USC stimulating and enjoyable. I truly appreciate the encouragement and support I received from my father Masao, mother Ayako, and my sisters Yasuko and Teruyo. Finally, I wish to thank my wife Sachiko to whom this thesis is dedicated. It is her love and support that made this possible. vii Abstract In the first essay, we study truth-revealing compensation schemes for forecasters employed by a firm. We begin by studying properties of forecast distributions that result in higher payoffs for the firm. We partition the space of forecast distributions into refinement level sets, which can be rank ordered based on profit potential. The detailed struc- ture of the compensation schemes depends on: (i) whether or not the forecaster incurs costs in the short term, and (ii) whether or not the forecaster influences the outcome. When forecasters do not incur costs, refinement levels can be used to rank them. In all the cases truth-revealing schemes are a menu of contracts that are hyperplanes to some convex functional in density space. Next, we develop compensations when fore- cast accuracy depends on the forecaster’s costly effort. We show that the decision of the firm to invest in multiple forecasters depends on how rapidly the cost of improving forecasts increases. Finally, we extend traditional salesforce compensation literature by developing compensation schemes for salespersons who also forecast. Unlike the tradi- tional salesforce literature, we permit the firm’s profits to depend on all the moments of the distribution, not just the mean. We assume that market conditions influence forecast uncertainty and the salesperson’s productivity. We show that if potential forecasts in each market regime belong to a refinement level then the firm can cause the salesperson to reveal the true regime and exert the appropriate level of effort. viii In the second essay, we study a capacity allocation and order acceptance problem faced by a make-to-order business that receives orders from customers who differ in their lead time requirement. Customers who desire quicker deliver are willing to pay more. We assume that deliveries occur at discrete time periods such as at the end of a working day. We formulate this problem as a dynamic rolling knapsack. We show that threshold policies are optimal. When the number of customer classes is only two a nested threshold policy is optimal. For problems with three or more classes, a nested policy need not be optimal. ix Part I Compensation Schemes for Forecasters 1 Chapter 1 Introduction Most operations management models assume that demand distributions are known, and they develop optimal decision rules based on the forecast (e.g., Clark and Scarf, 1960). Clearly, a firm’s performance depends not only on the quality of the decision rules but also on the quality of forecasts. The impact of poor forecasts can be quite substantial. Lakenan et al. (2001) discuss how companies such as Sony, Compaq, and Cisco failed to correctly anticipate demand, and, as a result, incurred large costs. Forecast accuracy depends on models that relate observable factors to the outcome of interest, knowledge of forecasting techniques, and access to data. To develop good forecasts, managers have to combine different types of knowledge, use private infor- mation, exert effort, and incur expenses. Given the same data, different forecasters may develop very different models. Hogarth and Makridakis (1981) discuss individual differences that exist in the acquisition and processing of information. 1 A key manage- rial contribution is the judgment and private information that underlie forecasts. The degree to which judgment is needed depends on the availability of appropriate historical data. Even when historical data are available and standard statistical techniques can be employed, a certain amount of judgment is unavoidable. It is easy to see that judgment plays an even greater role if past data are not available – e.g., when forecasting demand for a product in a new category. We argue that, in general, subjective judgment and private information play an important role. The importance of judgment is likely to be 1 A case in point is the article in the Los Angeles Times (Vincent, 2006) in which two forecasters, both of whom specialize in real estate forecasting and have access to similar data, not surprisingly, arrived at very different predictions. 2 greater in decisions that are more consequential because such decisions are likely to be strategic, have a longer time horizon, and involve new situations. In this paper we study compensation schemes that ensure the development of accu- rate forecasts. We are concerned solely with forecasters who work in a firm and assume that they are risk-averse rational agents of the firm. In addition to ensuring utilization of private information and effort, a firm also needs to measure and rank forecasters’ expertise. Firms typically engage a number of fore- casters, and they need to know the accuracy of each forecast in order to combine these forecasts (Winkler, 1968). If we can measure the expertise of forecasters and rank them, each member will know his or her standing in the group. If higher-ranked members receive higher compensation, it creates an incentive for everyone to improve their skills. Firms utilize many different types of forecasters, and forecasts can be classified in numerous ways (Chambers et al., 1971). For the purpose of this paper, there are two main factors to consider when analyzing forecasts: (i) whether or not costs are being incurred by the forecasters in the short term, and (ii) whether or not the forecaster influ- ences the variable being forecasted. Buyers in department stores (Mantrala and Rao, 2001, Fisher and Raman, 1996) estimate demand based on their intrinsic forecasting abilities, and cannot alter their precision in the short term either through significant effort or costs. Bhattacharya and Pfleiderer (1985) provide another example of a similar situation in the context of financial portfolio management. There are other situations where the precision of the forecasts can be altered by incurring costs. Managers may have to invest in data collection, forecasting tools, forecasting expertise, or informa- tion technology, to generate more accurate forecasts. Precision of the forecasts can also be altered through costly activities such as change in the structure of the supply chain, redesign of products, etc. 3 While financial portfolio managers and buyers in department stores do not influence the forecasted variable, a salesperson does influence the outcome. The salesforce is in direct contact with the customers and has valuable information about the effort required to generate demand and the level of uncertainty in the market. It is well known that inducing the salesforce to reveal this information truthfully is challenging. As Chen (2005) points out, firms face a conflict between getting the salespersons to exert effort and to reveal market conditions. In the traditional salesforce management literature (Coughlan, 1993, Basu et al., 1985, Lal and Staelin, 1986, Chen, 2005) the firm depends on salesforce to reveal the first moment of the demand distribution, while all other moments are common knowledge. Firm’s payoff depends only on the first moment of the true demand distribution. In our work the firm depends on the salesforce to pro- vide information about all the moments of the demand distribution. We allow market conditions to influence not only the effort required to generate sales but also the ability to estimate demand. To incorporate the cost associated with mis-matches in supply and demand, we let the firm’s profits depend on all the moments of the demand distribu- tion. For these more general settings, we determine conditions under which a firm can induce the salesperson to reveal the market regime and exert the targeted effort level. We believe that our work is the first one to explicitly model forecast uncertainty while developing optimal salesforce compensation schemes. In summary in this paper we study compensation schemes for forecaster employed by a firm. We consider four different situations depending on whether or not the fore- caster (i) incurs costs and (ii) influences the outcome. Table 1.1 summarizes the scenar- ios that we consider. Our truth-revealing compensation schemes are not based on any assumptions about the demand distributions. If there are multiple forecasters, then each of them can generate a forecast from a different family of probability distributions. 4 Table 1.1: Scenarios Forecast accuracy Expertise is intrinsic. depends on cost. Forecaster cannot affect outcomes. Chapter 4 Chapter 5 Forecaster can affect outcomes. Chapter 6 Chapter 6 The rest of the paper proceeds as follows. In Chapter 3, we develop the relationship between forecast distributions and firm’s payoff. We develop this relationship for a gen- eral optimization problem, and we show that forecast distributions that are refinements (DeGroot and Fienberg, 1983) of other forecasts result in higher profits for the firm. In Chapter 4, we develop compensations for forecasters who do not influence the outcome, and when the cost of forecasting is negligible. We know (Savage, 1971, Hen- drickson and Buehler, 1971) that there exist truth-telling compensation schemes even when there are no restrictions on the possible forecast distributions. This scheme is a menu of contracts that are linear in density space. We present approaches for construct- ing truth-telling contracts that are hyperplanes to a convex function in functional space, and identify conditions under which profit sharing results in truth-telling. The analysis in this chapter serves as a basis for the remainder of the paper. In Chapter 5, we show how the schemes developed in Chapter 4 can be modified to accommodate the cost of forecasting. Here, we assume that the relationship between the cost function and the forecast precision is common knowledge. The effort, however, is not observable by the firm. We also explore whether or not a firm should invest in multiple forecasters. We show that this decision depends on how rapidly the cost of improving the forecast precision increases for any single forecaster. Below a particular threshold rate of growth, the firm is better off investing in only one forecast. In Chapter 6 we consider forecasts made by salespersons. We identify conditions under which the salesperson will reveal the true state of the market, exert the right level 5 of effort, and reveal her true forecast. It is desirable for the compensation paid to the salesperson to be monotone increasing in realized demand. We also identify conditions under which such schemes are feasible. Our analysis does not require any assumptions with regard to the density functions. However, given the wide application of normal distributions, in several sections we specifically explore compensation schemes for normally distributed forecasts. Chapters 4, 5, and 6 present general results followed by analysis of normal cases or examples using normal distributions. We present our analysis with the assumption that the fore- casted variable is demand and that it is a continuous variable. Our conclusions are presented in Chapter 7. 6 Chapter 2 Literature Truth-telling payoff schemes for forecasters have been studied in the statistical decision theory literature (Savage, 1971, DeGroot and Fienberg, 1983, Osband and Reichelstein, 1985). 1 The classic work is due to Savage (1971). He studies compensation schemes when events are discrete and shows that every truth-telling compensation scheme must be a hyperplane tangential to some convex function in probability space. Since his model assumes that a forecaster does not affect future outcomes and that there is no cost associ- ated with developing a forecast, it is related to Chapter 4 of our work. Hendrickson and Buehler (1971) extend the concepts of convexity and sub-gradients to infinite dimen- sional spaces formed by probability density functions for continuous random variables. Employing results from these two papers we present truth-revealing compensation func- tions that are based on differentiable convex functionals. We present schemes that are based solely on the precision of the forecast and not the mean or the location of the distribution. We also derive conditions under which profit sharing is truth revealing. Osband (1989) points out that the traditional statistical decision theory literature ignores questions of optimality by assuming that the size of the payment to a forecaster is large enough for the forecaster to reveal the truth, and yet so small that the firm does not have to explicitly worry about the payment. Osband (1989) addresses this limitation by incorporating cost of forecasting explicitly, and his work is closely related to Chapter 5 of the current paper. But our work differs from Osband’s in several important ways. He requires the forecaster to provide an estimate of the mean of the distribution, while 1 In statistical decision theory, these payoff functions are usually called scoring rules. 7 we require the forecaster to reveal the entire density. He assumes that the firm makes a decision based on the mean of the forecast and the cost of an error is quadratic in the difference between the estimated mean and the realized outcome. For his error function, the variance of the forecast distribution completely characterizes the quality of the forecast. We consider general objective functions, and allow the firm’s decision to depend on all the moments of the forecast distribution, not just the mean. Under these general conditions we identify criteria for ordering forecast distributions and develop compensation schemes. Another paper that is closely related to Chapter 5 is due to Clemen (2002). He also formulates an agency-theoretic problem when the forecaster incurs cost to collect data. The firm’s problem is to design compensation schemes so as to maximize its expected profit, net of the forecaster’s salary. Clemen links forecaster’s effort and forecast preci- sion to the sample size employed by the forecaster. He derives conditions under which the principal can offer contracts that will cause the forecaster to truthfully select any sample size that the principal requires. Our approach focuses on determining the most profitable effort level, which is a continuous variable in our model, and then designing a truth-telling scheme that ensures that the desired effort level is selected. As stated above our objective function is more general. We show that if all the forecasts are normally distributed, it suffices to offer contracts that are linear in deviation. We also explore whether or not a firm should employ multiple forecasters. There is a body of literature that analyzes optimal compensation schemes for manag- ing a salesforce. For a comprehensive review the reader is referred to Coughlan (1993). Work by Basu et al. (1985) and Lal and Staelin (1986) are representative of research in this field that is related to our work. Basu et al. (1985) assume that the sales of a product depend on the salesperson’s costly effort. They assume homogeneity in the salesforce and information symmetry. Lal and Staelin (1986) extend Basu et al. (1985) 8 by permitting heterogeneity and information asymmetry. In their work, demand (x i )is a random variable that is expressed as a sum of a deterministic term (η i (t)) and random shock ( i ). Here i represents the salesperson type, t is the effort level, and η i (·) is a monotone increasing deterministic function. There are two types of salesperson – high type and low type. Lal and Staelin allow the distribution of the demand, not just the mean, to vary with salesperson type. But they impose a restriction by assuming that for any given effort level t, the realized demand from a high type is stochastically larger than that from a low type. 2 Their focus is on difference in sales capabilities and not forecast precision. They are interested in compensation schemes that optimize the mix of salespersons employed by the firm. They can exclude a low type from participating but otherwise cannot separate the two types. Also Lal and Staelin (1986), like the rest of the salesforce compensation literature (Coughlan, 1993), assume that the firm’s payoff depends only on the expected sales. They are not concerned with inventory costs or other costs associated with matching supply and demand. These costs will depend on the demand distribution and not just the expected sales. We account for these costs. Given our interest in forecast precision, we assume that there is only one salesper- son and that the demand distribution, and the salesperson’s productivity depend on the market regime. We let the firm’s profits depend not just on the expected demand, but on the entire demand distribution. This allows us to account for costs that arise from imbalances in supply and demand. Under these general conditions we identify relation- ships among the demand distributions realized in different regimes that enable the firm to induce the salesperson to truthfully reveal the market regime and pick the appropriate effort level. 2 If i are Normally distributed then the standard deviation must be the same for both the high and the low type. This is because they require demand for the high type to be stochastically larger than that for the low type. Consequently forecast precision is the same for high and low type salespersons. 9 There are a few papers in the operations management literature that address sales- force incentives. An interesting paper that examines the impact of salesforce compen- sation on inventory and production costs is Chen (2000). He develops incentives that induce the salesforce to smooth demand over time. The paper, however, is not concerned with forecast accuracy, as it assumes common information. Huff et al. (2003) examine the interactions between stocking decisions and salesforce compensation in a newsven- dor environment. In their model, demand distribution is stochastically increasing both in the firm’s effort and in the salesperson’s effort. Their main interest is to address the issue of double-sided moral hazard under various scenarios, and one of their scenarios is similar to our model in Chapter 6. They find that the firm can delegate stocking deci- sions to its risk-neutral salesforce without loss of profit. Our model in Chapter 6 is an extension of their model in several ways. First, their agent is risk-neutral while ours is risk-averse. Second, Huff et al. assume that both the principal and the agent have the same information about the future demand whereas we assume information asymme- try. Third, in their model the agent partially bears the overage/underage costs. In our model, the agent is compensated solely based on submitted forecast distributions and is not responsible for quantity decisions. Chen (2005) extends the salesforce compensation literature by explicitly incorpo- rating the costs of matching supply and demand. Like our work, he assumes that a firm employs a single salesperson. This person may be one of two types. He adopts a demand model that is very similar to that of Lal and Staelin (1986), by letting demand: x =a+θ i +. Herea is the agent’s selling effort,θ i is a constant that varies by type, and is a normal random variable. The error term does not depend on the type. The pay- off to the firm depends not only on the realized demand but also on over-stocking and under-stocking levels, just as in the classic newsvendor problem. For this model he com- pares the performance of a menu of linear contracts with that of the schemes proposed 10 by Gonik (1978), whose work is often cited as an example of compensation schemes that have been implemented. Chen (2005) finds that a menu of linear contracts domi- nates piece-wise linear schemes proposed by Gonik. Our work extends Chen’s work by generalizing the cost function. The penalty costs that arise in the newsvendor problems are a special case of our cost function. Although Chen incorporates the cost of matching supply and demand, he is not concerned with forecast precision and forecasting abilities of salespersons. The forecast distribution in his analysis is normally distributed with a fixed standard deviation. We permit the demand distribution to vary by regime and this enables us to link forecast precision with market conditions. We study optimal contracts that take into account forecast distributions, while he analyzes the performance of linear contracts. While our work is concerned with forecasters working for a firm, there are many papers that have studied the value of sharing demand information among different firms in a supply chain (Lee et al., 1997, Aviv, 2002, Cachon and Lariviere, 2001), and many papers have explored incentive problems that arise around information sharing in supply chains (Cachon and Netessine, 2004). The work by Cachon and Lariviere (2001) is one of the first papers in this area to permit asymmetry in demand information. In their paper, a supplier has to commit to building capacity based on demand information he receives from his buyer, a manufacturer. The manufacturer’s initial order consists of firm orders and options. The authors identify how a manufacturer can credibly convey demand information to the supplier, thereby inducing the supplier to build capacity. The main focus of this body of literature is on how specific contracting instruments such as options, firm orders, wholesale prices, option prices, and buy-back prices can be used to credibly convey information and coordinate the channel. Supply chain coordination literature has also studied effort-dependent demand. Krishnan et al. (2003) consider both ex-post effort and ex-ante effort. Ex-post effort 11 refers to the retailer’s effort that is induced after all the uncertainty is resolved. Ex-ante effort, on the other hand, is what the retailer exerts before basic demand is realized. We only deal with ex-ante effort in their terminology. They show that a buy-back contract alone cannot coordinate the channel. They also show that if effort costs are verifiable, effort-sharing contracts will coordinate the channel. Since we are dealing with the sales- force in this paper, it is hard to argue that the ex-ante effort level is verifiable. One of our objectives in this paper is to rank forecasters in terms of their expertise levels. In order to do that, we have to define the expertise of a forecaster, or the “good- ness” of a forecast distribution. Murphy (1993) identifies three criteria for judging a weather forecast. The first criterion is consistency; a forecast is considered good if the expressed forecast reflects the forecaster’s internal knowledge. The second criterion is quality; the quality of a forecast, the author argues, is multifaceted in nature, and promi- nent examples include the mean absolute deviation and the mean squared error. The last criterion is value; a forecast distribution is considered good if the user of the forecast benefits or incurs less loss by using the forecast. These criteria are not, however, equiv- alent. Ridder et al. (1998), for instance, discuss situations in which a distribution with a smaller standard deviation leads to a larger cost. Thereby, they show that Murphy’s second and third criteria are not consistent. In Chapter 3, we define the “goodness” of a forecast in ways that are more specific to operational decision-making problems and are consistent with Murphy’s third criterion. 12 Chapter 3 Linking Forecasts with Firm Performance Throughout the paper, we let d denote the forecasted variable, f(·) the forecaster’s true forecast, and ˆ f(·) the distribution that the forecaster announces to the firm. In all instances the events occur in the following sequence. (i) The firm announces a menu of contracts ξ( ˆ f(·),d 0 ), as a function of the announced distribution ˆ f(·) and the realized outcomed 0 . (ii) Each forecaster selects a contract by announcing a distribution ˆ f(·). (iii) Once the forecast(s) are received, the firm makes its decision Γ ∗ based on the received forecast distribution(s). (iv) The outcome d 0 is observed, and payment ξ is made according to the chosen contract. Since in many cases the forecasted variable is demand, which may not be fully observable in some cases, some discussion is in order. In the retail industry where con- sumers have direct access to inventory, lost sales are difficult to observe. There are many other situations in which the consumer does not have access to inventories. Examples include catalogue based sales, Internet based sales, sales from manufacturers to distrib- utors, and sales of industrial supplies and raw materials that are procured on the basis of purchase orders. In all these cases, the firm observes demand. As a concrete example, 13 consider Carhartt, a manufacturer of jackets worn by industrial workers. A supplier to Carhartt that we have studied anticipates orders from Carhartt and produces the fabric in anticipation. Garment production is initiated only after receiving a purchase order. If the order size is larger than anticipated then additional fabric is produced on an expe- dited basis. In operations management, particularly in inventory theory, there is a long tradition of assuming that lost sales are back-logged. Demand forecasting can be used in manufacturing and service industries to determine staff levels also. High levels of demand may result in lost sales or merely in longer waiting times. Up to a point, the firm may have the option to bring on board, on an emergency basis, flexible resources. Demand forecasting facilitates such planning. Call center operators such as LiveOps employ a highly flexible geographically dispersed work force that they can quickly tap into. These firms have the ability to monitor lost calls. The menu of contracts has to be such that it (i) forces the forecaster to truthfully reveal her forecast, (ii) ensures that the right effort (if any is needed) is expended by the forecaster, and (iii) enables the firm to maximize expected earnings net of any forecast- ing costs. In order to put this into practice we have to establish the relationship between the characteristics of the forecast and the firm’s profitability. We employ Murphy’s third criterion, and order forecasts based on expected profits. We assume that after the firm receives a forecast it makes an optimal decision. Firm’s profits depend on this decision and the outcome of the forecasted variable. The value of a forecaster to the firm is measured in terms of the realized profits and the cost of generating the forecast. Accordingly, we rank forecasters using the following criterion. Definition 1 Let Π(d 0 ,γ) be the payoff when the decision is γ and the outcome is d 0 . 1 Let Γ ∗ (·) be a function that maps a forecast to an optimal decision, that is, Γ ∗ (f)= argmax γ Π(d,γ)f(d)dd. To rank forecasters we use the expected profits. 1 In a newsvendor problemγ would be the stock level and d 0 would be the realized demand. 14 If the forecasts are made a large number of times, so that the outcome for thej-th trial is d j , forecast density of forecaster i for the j-th trial is f i,j (·), and C(i) is the cost of generating the forecast. Forecaster 2 is considered to have greater expertise than forecaster 1, if: lim n→∞ 1 n n j=1 Π(d j ,Γ ∗ (f 1,j (·)))−C(1) < lim n→∞ 1 n n j=1 Π(d j ,Γ ∗ (f 2,j (·)))−C(2) . Notice that the optimal decision mapping Γ ∗ is the same for both players. Any difference in the expected profits is due only to different forecast distributions and the costs associated with generating them. To develop compensation schemes, we need to understand how differences in fore- cast distributions translate to differences in profitability. 3.1 Refinement-based Ordering Suppose there are two discrete outcomes. The default or “ignorance prior” is to assign equal probabilities to each outcomes (Fox and Clemen, 2005). At the other extreme, we would expect a forecaster with “complete” information to assign a probability of1 to the true outcome. Additional information allows a forecaster to refine her belief, and this should move the mass away from the “middle” to one of the possibilities. For example, for a baseball player with a batting average of 0.300 we would predict a 30% chance of a hit. If, however, we know that his batting average against left-handed pitchers is 0.325 and against right-handed pitchers is 0.285, we would alter our forecasts if we knew whether he was facing a right- or left-handed pitcher. Forecasts from the set {0.325, 0.285} are considered refinements of forecasts from the set{0.300}. Observe that 0.300 can be expressed as a convex combination of 0.325 and 0.285. DeGroot and Fienberg (1983) present the concept of refinement when outcomes are binary. We 15 assume forecasters are asked to announce a density function and apply the concept of refinement to density functions. We partition the space of forecast distributions into sets that correspond to different levels of refinement. The possible set of forecasts is given by F. F is such that if f(d)∈F, thenf(d)≥ 0,∀d, and f(d)dd=1. We partitionF intorefinementsets χ(i)={f s i (·)}, wherei is a refinement level ands is an index, such that: (a): All the members of the setχ(i) are linearly independent. (b): IfH>L, then for anyf l (d)∈χ(L), there exists a densityα(s) such thatf l (d)= f s h (d)α(s)ds, wheref s h ∈χ(H),∀d. 2 Noteχ(i)∩χ(j)=∅ ifi=j. For a given partition, a forecaster is considered to be at refinement level H if potential forecasts belong only to the set χ(H), and forecasts are well calibrated in the sense defined in Dawid (1982) and DeGroot and Fienberg (1983). In summary, a forecaster is well calibrated if the long run relative frequency of a measurable outcome is the same as the probability assigned to that outcome, holding the information available constant. Lemma 1 A forecaster with a higher refinement level H generates greater profit in expectation than a forecaster with a lower refinement levelL. Proofs are provided in Appendix unless otherwise noted. Lemma 1 does not assume any particular form of decision problem or objective function. For any problem with any form of objective function or optimal decision mapping, we can use Lemma 1 to rank order forecasters. The set of probability distribution functions can be partitioned into refinements lev- els in a number of different ways. Given a partitioning scheme, we can always determine 2 We use the phrase “f l is a convex combination off s h ,” although convex combination normally implies convex combination of a finite number of elements. 16 the refinement level of any given distribution, but there may not be any parameters of the density function that allow us to easily do so. There are, however, a few families of distributions in which one or two parameters can be used to define the level of refine- ment. 3.1.1 Scalable Family of Distributions Probability distributions corresponding to two random variablesX andY belong to the same scalable family (Cachon and Lariviere, 2001), ifY =ϑX+M, whereϑ is the scale parameter, M is the location parameter, and ϑ and M are real numbers. Examples of scalable distributions include normal, shifted exponential, and uniform. The following proposition states thatϑ represents refinement levels. Proposition 1 (a): Any density function within a scalable family can be expressed as a convex combination of densities with a lower scale parameter. (b): Density functions within the scalable family with the same scale parameter are linearly independent. Therefore,ϑ is a measure of refinement of the densities. For normal distributionsϑ is the standard deviation. 3.1.2 Bayesian Update Bayesian techniques are often used for combining forecasts (Winkler, 1981, Clemen, 1989). Letf p (.) be the distribution of demand prior to incorporating a new forecast. A forecaster observes a signals, that is a function of the true demand and a noise term that has a distributiong(·). The posterior distribution of demandf P that is obtained after the new information is incorporated is given by (Winkler, 1981): 17 f P (d|s)= f p (d)·g(s|d) f p (y)·g(s|y)dy . (3.1) The noise term is often either additive or multiplicative. If the noise is additive, g(s|d)=g(s−d), and if it is multiplicative,g(s|d)=g(s/d). In either caseg(s|d) is of the formg(h(s,d)), whereh : R× R→ R. Under the assumption that the forecasts are unbiased and the functionh is common knowledge, the firm can determineg(s|d) from the forecast. The following proposition shows that refinement ordering is preserved by Bayesian updating. Proposition 2 Suppose we have two sets of signal densitiesχ(1) ={g 1,a } andχ(2) = {g 2,b }, wherea,b∈ R are indices. Then there exits a set of refinement levels for poste- riors;χ(i)={f P 1,a (·|s)}, andχ(j)={f P 2,a (·|s)},i> 1,j> 2 andi<j. The proposition shows that when we merge forecasts, a forecast with a higher level of refinement results in a posterior that has a higher level of refinement. The posteriors, not surprisingly, are refinements of the prior. For families of probability distributions that have a conjugate likelihood function, the prior and the posterior belong to the same family (DeGroot, 1970). Distributions that have conjugate priors include normal, gamma, beta, and Pareto. For each of these distribution Bayesian updating alters a few parameters and these parameters can be used to infer the level of refinement. For example, for normal distributions this parameter is the standard deviation and the posterior will always have a smaller standard deviation. 18 Chapter 4 Compensation Schemes when Costs are Negligible Next, we develop compensation schemes that (i) ensures that forecasters reveal their true forecasts, and (ii) provide higher compensation to those who generate better forecasts according to the definition developed in the previous section. We begin with the case in which the forecaster does not influence the outcome and the cost of forecasting is neg- ligible. Because the forecaster does not incur a cost, we assume that the compensation payments are small from the perspective of the firm. Such situations occur when exper- tise is intrinsic to each forecaster. Forecasters in Sport Obermeyer (Fisher and Raman, 1996) are an example. The results in this section are also applicable to relatively short- term performance evaluation, assuming that the long-term investment decisions have already been made. Here, revealing the truth is at least a weakly dominant strategy because the agent does not incur any costs to develop forecasts. Thus one could argue that a flat salary should be adequate. This would be true if there were no long-term implications. Our compensation schemes allow us to rank forecasters and therefore cre- ate an incentive for improvement in the longer term. For instance, at Hilti, a construction tool company, purchasing analysts’ forecast accuracies impact their bonuses. They pro- vide a point estimate for each product and after periodic sales are observed, they receive bonuses based on how far their estimates were from the realized demand. Forecasters at Hilti do not announce their forecasting capabilities, but their bonuses depend on how 19 accurate their estimates were. Therefore those analysts have incentive to improve over time. Recall that the firm first offers a payment scheme ξ as a function of the declared distribution and outcome. The forecaster announces a forecast distribution ˆ f(·), and an outcomed 0 is observed. The forecaster receivesξ( ˆ f(·),d 0 ) in monetary units, which is U(ξ( ˆ f(·),d 0 )) in utility. We assumeU is concave increasing. Suppose there are several forecasters estimating the likelihood of outcomes. Can we, without making any assumption about the family of underlying distributions, provide a compensation scheme that will induce each of them to provide their true estimates? ClearlyU(ξ( ˆ f(·),d 0 )) must be such that: E f U(ξ(f(·),·))>E f U(ξ( ˆ f(·),·)) for all ˆ f =f. (4.1) Let H be a vector space whose members are either continuous functions or functions that are right continuous with left hand limits. All functions in H map R→ R, are twice differentiable except at a finite number of points, and are bounded. We also require that for allf ∈ H, f p (x)dx <∞, for 1<p<N, whereN> 2. All probability density functions generated by the forecasters belong to H. Theorem 1 Condition 4.1 is satisfied if and only if the compensation scheme U(ξ( ˆ f(·),d 0 )) is a supporting hyperplane to a strictly convex functional H→ R. Hendrickson and Buehler (1971) consider a space that consists only of a convex class of probability densities. They require the compensation schemes to be based on convex functionals that are homogeneous. In our space, which is a vector space that includes probability densities, the convex functional does not have to be homogeneous. We can base our analysis on differentiable convex functions in R. This makes it easier to 20 develop compensation functions. Next we present approaches for constructing convex functionals that have linear supports at all points of interest. Letg(·): R→ R be a strictly convex, bounded, twice differentiable function. Let G(f)= g(f(d))dd. (4.2) We assume thatg is such thatG(f) is bounded for allf ∈ H. Becauseg(αf 1 (d)+ (1−α)f 2 (d))<αg(f 1 (d))+(1−α)g(f 2 (d)),∀f 1 ,f 2 ,wehaveG(αf 1 +(1−α)f 2 )< αG(f 1 )+(1−α)G(f 2 ). It follows thatG(·): H→ R is a strictly convex functional. The directional derivative forG(·) atf is given by: ∂G(f)h = lim λ→0+ g(f(d)+λh(d))dd− g(f(d))dd λ ,f,h∈ H. The integral is finite and we can interchange the limit and the integration operation. Therefore, ∂G(f)h = h(d) lim λh(d)→0+ g(f(d)+λh(d))−g(f(d)) λh(d) dd = h(d) lim µ →0+ g(f(d)+µ )−g(f(d)) µ dd = h(d)g (f(d))dd. The above integral is finite and g (·) also belongs to H. ∂G(f)h is a linear oper- ator from H to R. Because the derivatives exist and are linear, the sub-gradients of G(·) are unique (Corollary 7.17 of Aliprantis and Border, 2007). The vector g (f(x)) is the Gˆ ateaux derivative or the sub-gradient at the point f(x) for the functional G(·). The unique supporting hyperplane for G(·) at f is G(f)− g (f(d))f(d)dd + 21 g (f(d))h(d)dd, forh∈ H. We can use this supporting hyperplane to design a truth- telling payoff function. Lemma 2 If the firm offers U(ξ( ˆ f(·),d 0 )) =G( ˆ f)− g ( ˆ f(d)) ˆ f(d)dd+g ( ˆ f(d 0 )), then the forecaster will reveal the true distribution. The functionalG(·) is such that iff 1 (x)= f 2 (x−c),f 1 ,f 2 ∈ H, wherec is a con- stant, thenG(f 1 )= G(f 2 ). This implies that for distributions such as normal, uniform, triangular, and shifted exponential, the compensation depends on the second and higher moments, and not the mean. This is a useful property when forecasters do not influence outcome. To implement such compensation schemes, the firm has to be able to observe the outcome. Depending on the nature of the forecasted variable, however, it may or may not be easy to implement. If the forecasted variable is the stock price, for instance, the outcome is always observable. So the compensation schemes discussed above can be easily implemented. If, however, the forecasted variable is the demand, the outcome may not be fully observed. If demand exceeds the available stock, for instance, the actual demand can not be observed. From a modeling point of view, we can assume that there exits an emergency action such as special expedited production to meet customer demand, so that demand is always observable, as Chen (2005) assumes. Example (Quadratic Compensation Scheme): Letg(d)=d 2 in (4.2). By Lemma 2, U(ξ( ˆ f,d 0 )) =− ˆ f 2 (d)dd+2 ˆ f(d 0 ), will ensure truth-telling. For this scheme if the offered forecast is uniformly dis- tributed: 22 f(d)=        1 u−l if l ≤d≤ u, 0 otherwise, then the forecaster’s payoff is 1 U(u,l,d 0 )) =        1 u−l if l ≤d 0 ≤ u, − 1 u−l otherwise. All uniform distributions with the same range (u−l) will receive the same utility in expectation, regardless of the mean. Also notice that the expected utility decreases as the range increases. If the forecast is symmetric triangular: f(d)=        4 u−l (d− l) if d≤ u+l 2 , −4 u−l (d− u) if d≥ u+l 2 . When the outcome isd 0 , the agent receives, in utility, U(ξ(u,l,d 0 )) =        − 4 9 (u− l)+2 4 u−l (d 0 − l) if d 0 ≤ u+l 2 , − 4 9 (u− l)+2 −4 u−l (d 0 − u) if d 0 ≥ u+l 2 . The first term can be considered the base salary and the second term a correction that is based on the outcome. If the agent sets tighter bounds then the base payoff gets larger as does the correction rate. 1 Since truth telling is ensured, we do not use separate sets of notations for the true and the announced parameters. 23 If the forecast is Normal (N(µ,σ )), the payoff function, in utility, is: U(ξ(µ,σ,d 0 )) =− 1 √ 4π + 2 2πσ 2 exp − 1 2 d 0 −µ σ 2 . The above payoff function is also location independent. Notice again, in all the three examples above, that the forecaster is required to characterize the entire distribution, not just the mean. Also, better forecasters earn more in expectation. After receiving a truthfully reported forecast distribution, the firm can solve the opti- mization problem of any kind. In order for the preceding compensation schemes to work, we must have (i) the outcome is observable and (ii) the firm’s action does not alter the distribution. In the preceding analysis, we constructed a convex functional G : H → R using a convex function g : R → R. More generally, we can use a convex functional with functional derivatives to construct compensation schemes. Lemma 3 Let G(·): H → R be a convex functional whose derivatives exist at every point that corresponds to a probability density. A forecaster with forecast f(·) ∈ H who announces ˆ f(·) and receives a payoffξ( ˆ f(·),d 0 ) when outcomed 0 is realized, will reveal the truth if: U(ξ( ˆ f(·),d 0 )) =G( ˆ f)− δG δ ˆ f x=d ˆ f(d)dd+ δG δ ˆ f x=d 0 . Here δG δ ˆ f x=d is the functional derivative ofG at the pointd for the function ˆ f. The only requirement we have is that the functionalG is strictly convex and differen- tiable. Thus, a variety of compensation schemes are possible. As an illustration, another approach for constructing compensation functions is to use bounded twice differentiable functions g ∗ (x,y) that map R× R to R. We need g ∗ (x,·) to be convex. Let g ∗ y (x,·) 24 denote the partial derivative of g ∗ (·,·) with respect to y.If G(f)= g ∗ (d,f(d))dd, then U(ξ( ˆ f(·),d 0 )) =G( ˆ f)− g ∗ ˆ f (d, ˆ f(d)) ˆ f(d)dd+g ∗ ˆ f (d 0 , ˆ f(d 0 )), will ensure truth-telling. For example,g ∗ (x,f(x)) =xf(x)+f 2 (x). Theorem 1 affords considerable latitude in constructing compensation schemes. There are, however, a few restrictions. The outcomed must be observable and the firm should not influence the forecasted variable. Some of the other implications that follow from Theorem 1 are given below. Proposition 3 All the forecasts that are linearly independent can have the same highest expected payoff. Proof directly follows from the theorem of space separating hyperplanes (Aliprantis and Border, 2007). This proposition implies that the firm can make payment based solely on precision, and not mean. The example above that was based on a quadratic compensation function illustrates this proposition. Corollary 1 Forecasts that are convex combinations of other forecasts must have payoff less than the highest. Corollary 1 is consistent with Lemma 1. Forecasters with higher levels of refinement have to receive higher payments in expectation under any truth-revealing compensation scheme. Another interesting implication of Theorem 1 is that profit-sharing compensa- tions are weakly truth-revealing, provided some conditions are met. Lemma 4 Profit sharing is truth-revealing provided (i) the firm’s decision is optimal for the given forecast, (ii) the decision does not affect the forecasted outcome, and (iii) the forecaster is risk-neutral. 25 Profit sharing is an appealing compensation scheme, and is easy to implement. Unfortunately, Lemma 4 also points out the limitations of profit sharing. For one, profit sharing is truth-revealing if the firm acts solely on the forecast of interest. This will not be the case when there are multiple forecasts and a decision is made after combining these forecasts. Profit may also be affected by other uncertainties that are forecasted by other entities. Further, profits can be affected by the actions of those the forecaster does not influence, so there is a possibility of double-sided moral hazard. Consequently, in general, profit sharing may not be truth-revealing even when forecasting costs are negligible. Finally, profits are a function of the mean and the precision of the forecast, but the forecaster may not influence the mean. As the preceding examples illustrate, we can offer compensations that focus solely on precision, the element that is actually influenced by the forecaster. 4.1 Compensation Schemes for Normally Distributed Forecasts It is widely believed that normally distributed errors describe many real-world processes reasonably well. Even if the data are not normally distributed, analysts often try to trans- form data into normal forms (Mickey et al., 2004). Regression models generally assume that errors are distributed normally (see, for example, Rice, 1995). Each forecast, there- fore, is going to be a normal distribution. Given the general applicability of normal distributions, we derive a class of linear rules that are easy to implement. To gain insights into the structure of the compensation scheme, and also to accom- modate the Bayesian paradigm, we slightly change the sequence of events. First, the principal offers a compensation functionξ(ˆ σ,ˆ µ,d 0 ), where d 0 is the realized outcome. The agent announces her standard deviation ˆ σ before observing a noisy signal, and the 26 mean of the distribution ˆ µ after observing the signal. The standard deviation is the mea- sure of forecasting expertise if all forecasts are normally distributed. The compensation is paid out after the outcome is observed. We consider the following compensation function, which we refer to as a linear symmetric compensation scheme. ξ(ˆ σ,ˆ µ,d 0 )=w(ˆ σ)−β(ˆ σ)·|d 0 − ˆ µ |, where both w and β are positive, monotonic, and differentiable functions. Since β is positive, the forecaster’s payoff is decreasing in deviation. The forecaster receivesw as a function of her precision, and then gets penalized based on the deviation. The severity of the penalty depends on the announced precision. We find it insightful to first develop the compensation schemes for risk-neutral fore- casters. Proposition 4 If the compensation scheme is of the form: ξ(ˆ σ,ˆ µ,d 0 )= w(ˆ σ)−β(ˆ σ)· |d 0 − ˆ µ |, wherew andβ are positive, monotonic, and differentiable functions such that w (σ)− 2 π ·σ·β (σ)=0, w (σ)− 2 π ·δ·β (σ)<0, ∀σ,δ∈ Σ, where Σ is the set of all possible standard deviations, then a risk neutral agent reveals her true mean and standard deviation. Next, we consider a risk-averse forecaster maximizing her expected utility that is given by U, a concave increasing function; i.e., U > 0 and U < 0. The agent can potentially lie about either mean or standard deviation. We first prove that for any announced standard deviation, the forecaster reveals the true mean. 27 Proposition 5 Under a symmetric compensation scheme a risk-averse forecaster reveals the true mean, regardless of the announced standard deviation. This is true because the payoff function is symmetric. Now we show that the risk- averse forecaster does not report the true standard deviation if the firm offers a compen- sation scheme that satisfies the conditions in Proposition 4. If, furthermore, w and β satisfy certain conditions, then we can tell whether the forecaster inflates or deflates her standard deviation. Proposition 6 Suppose the firm offers a compensation scheme that satisfies conditions in Proposition 4 to a risk-averse forecaster. The forecaster deflates her standard devia- tion ifw ,β > 0 and inflates ifw ,β < 0. Although a risk-averse forecaster always reveals the true mean, she may inflate or deflate the standard deviation, depending on the functions w and β.If w , and β > 0, then a risk-averse forecaster will inflate her standard deviation. To understand this intuitively, consider w(ˆ σ) the “fixed” part of the payoff function and β(ˆ σ)|d− ˆ µ | the “variable” penalty part. By deflating the standard deviation, the forecaster is settling for a lower fixed payment because it reduces the severity of the penalty. Further, reporting a smaller standard deviation reduces the variability of the payoff. Similarly, ifw ,β < 0, then the forecaster will inflate the standard deviation. We now show that a mixed strategy can be deployed to force truth revelation. Let: 28 ξ(ˆ σ,ˆ µ,d 0 )=        w 1 (ˆ σ)−β 1 (ˆ σ)·|d 0 − ˆ µ | with probabilityα, w 2 (ˆ σ)−β 2 (ˆ σ)·|d 0 − ˆ µ | with probability1−α, where α∈ [0,1], w i (σ)− 2 π ·σβ i (σ)=0, w i (σ)− 2 π ·δβ i (σ)< 0 i=1,2, w 1 (σ)< 0,β 1 (σ)< 0, w 2 (σ)> 0,β 2 (σ)> 0, ∀σ,δ∈ Σ, where Σ is the set of all possible standard deviations. The firm offers a pair of schemes, under one of which the agent will inflate the precision, and under the other deflate it. By randomizing the two, the agent is incentivized to reveal the truth. Proposition 7 If the firm is aware of the forecasters utility function then it can deter- mine α ∈ [0,1], and force the forecaster to truthfully reveal her forecast precision by employing the following randomized scheme: ξ I (ˆ σ,ˆ µ,d 0 )=w 1 (ˆ σ)−β 1 (ˆ σ)·|d 0 − ˆ µ | with probabilityα, ξ D (ˆ σ,ˆ µ,d 0 )=w 2 (ˆ σ)−β 2 (ˆ σ)·|d 0 − ˆ µ | with probability1−α. The structure of the compensation scheme for normal distributions is rooted in the linear relationship among normally distributed random variables. If X and Y are nor- mally distributed, thenY =ϑX+M. We can use the compensation schemes developed for normal distributions for any other scalable family of distributions. 29 Proposition 8 Let the family of distributions be such that if X is a random variable from this family, then Y = ϑX +M is also a random variable from the same family. Suppose the forecaster first declares her precision level ϑ, and announces a location parameterM after observing the signal. When all forecasts belong to the same family, the following compensation scheme is truth-revealing. Let G(ϑ) be a strictly convex differentiable decreasing function. If a forecaster selects a precision level ϑ, location value M, and the outcome is d 0 , the payoff, in utility, is: U(ξ(ϑ,M,d 0 )) =w(ϑ)+β(ϑ)[c 1 (M−d 0 ) + +c 2 (d 0 −M) + ], where: w(ϑ)=G(ϑ)−G (ϑ)ϑ, β(ϑ)= G (ϑ) I(c 1 ,c 2 ) , I(c 1 ,c 2 )=c 1 F −1 X ( c 2 c 1 +c 2 ) 0 (M X −a)dF X (a)+c 2 ∞ F −1 X ( c 2 c 1 +c 2 ) (a−M X )dF X (a). (4.3) F X (·) is the CDF for the random variableX, andM X = F −1 X ( c 1 c 1 +c 2 ). The location parameterM that the forecaster will declare isF −1 Y ϑ ( c 1 c 1 +c 2 ). The expected earning of the forecaster isG(ϑ). We know from Proposition 1 that higher values of ϑ result in lower profits. Since G(·) is a decreasing function, the above scheme offers higher compensation to better forecasts. 30 Chapter 5 Costly Forecasts Let us now consider situations in which forecasting requires the agent to expend effort or resources. These expenditures may be due to investments in forecasting systems, data collection, data analysis, or investment in supply chain redesign. We have estab- lished in Lemma 1 that refinement in forecast distributions leads to higher expected profits. Here we consider situations in which costly effort is needed to generate higher levels of refinement. Because the concept of refinement is consistent with acquisition of additional information, cost of increased refinement can also be due to the cost of addi- tional information. 1 We make the conventional assumption that the relationship between effort and the resulting precision is common knowledge, while the actual effort level is not observed by the firm. We present the analysis for a single forecaster. If there are multiple forecasts, we have to understand how to combine their forecasts. In any specific context, if the com- bined distribution can be analytically represented, such as for normal distributions, then our analysis can be extended. We discuss the case of normal distributions in Section 5.1. As we saw in Lemma 4, the firm’s optimal expected profit is convex in density space. Let this expected profit function beEΠ(f(·)), when the forecast isf(·). Note that this profit function does not include the cost of forecasting. We assume that the firm and the forecasters share a prior distributionf 0 (·). For convenience, let the quality level of f 0 (·) be0; i.e.,f 0 (·)∈χ(0). Without loss of generality assume that the only member of 1 There is a nice correspondence between data acquisition to increase the level of refinement and the structure of a decision tree in which a decision branch corresponds to costly information gathering. 31 χ(0) isf 0 (·). Let the cost of achieving levelq forecasts,q≥ 0,beV(q), in utility units. Given the setχ(q), there exists a unique densityα q (s) such that f 0 (d)= fs(·)∈χ(q) f s (d)α q (s)ds. (5.1) The densityα q (s) corresponds to the probability that the forecast will bef s (·) if refine- ment level q is reached. Equation (5.1) establishes the relationship between the firm’s prior and the likely forecasts. α q (·) is unique because all distributions in a level set are, by definition, linearly independent. As is usually assumed in the literature (Holm- str¨ om, 1979; Basu et al., 1985), we assume the agent’s utility is additively separable. In other words, if the agent expends effort to achieve levelq forecasts and earnsξ, then her overall utility isU(ξ)−V(q). The firm’s problem is: max q ξ(·) EΠ(f s (·))− ξ(f s (·),d)f s (d)dd α q (s)ds subject to: U(ξ(f s (·),d))f s (d)dd α q (s)ds≥V(q), U(ξ(f s (·),d))f s (d)dx α q (s)ds−V(q) ≥ U(ξ(f r (·),d))f r (d)dd α r (s)ds−V(r) ∀r≥ 0, U(ξ(f(·),d))f(d)dd≥ U(ξ(g(·),d))f(d)dd ∀f(·),g(·)∈F. (5.2) The first constraint ensures the forecaster’s participation at the desired effort level. The second constraint states that the forecaster’s expected payoff net of effort cost is maximized at the desired effort levelq. The third ensures truth-telling. Strictly speaking, 32 the formulation should distinguish between the true and the announced distribution. To simplify the presentation, we did not explicitly use different notations. As far as the third constraint is satisfied, this will not cause a problem. We now slightly abuse the notation and letV(f(·)),f(·)∈χ(q), also denote the cost of achieving levelq forecasts. IfV(·) is twice differentiable and convex, the following proposition characterizes the optimal effort level and compensation scheme. Proposition 9 If V(·) is convex, then the optimal desired effort level q ∗ is such that E f(·)∈χ(q) [EΠ(f(·))−U −1 (V(q))] is maximized. The optimal compensation is: U(ξ(f(·),d 0 )) =V(f(·))− δV δf x=d f(d)dd+ δV δf x=d 0 for f(·)∈χ(q ∗ ), (5.3) whered 0 is the realized outcome. Proof directly follows from Theorem 1 and the fact that the agent’s expected payoff isV(f(·)). 5.1 Normal Distributions For normal distributions we can use standard deviationσ as a measure of the refinement. In the case that there is only one forecaster, the firm has to determine the standard deviation level that is best for the firm, after accounting for payments to be made for the forecast. The corresponding optimization problem is identical to the general case in the previous subsection except that distributions are characterized byµ andσ, andV is now a function ofσ. Letξ(ˆ σ, ˆ µ,d 0 ) denote the compensation function when the agent announces ˆ σ and ˆ µ , and the outcome isd. We assumeV(σ), the cost incurred by the forecaster to generate a forecast with standard deviationσ, is a strictly convex decreasing function that is twice 33 differentiable. We know from Proposition 4 that if the payment scheme is of the form a+b|d− ˆ µ |, the forecaster always reveals the true mean. Proposition 10 Letσ ∗ =Argmax σ E[Π(σ,µ,d )]−U −1 (V(σ)). The following con- tract is optimal: U(ξ(ˆ µ,d 0 )) =V(σ ∗ )−σ ∗ V (σ ∗ )+ π 2 V (σ ∗ )|d 0 − ˆ µ |, whereV (·) is the derivative of the effort function. What if there are multiple forecasters? We have to determine the level of precision we require from each of the forecasters, and how to combine different forecasts, because the precision of the combined forecasts will determine the profit for the firm. We can partition the overall problem into two sub-problems. The first sub-problem determines the optimal precision level for the firm, and the second determines the minimum cost required to attain that precision level. Before we formulate the sub-problems, we show how to combine forecasts. These results are from Winkler (1981). It is very likely that forecasts will be positively cor- related. Let ρ denote the correlation coefficient. For simplicity and without loss of generality, we assume a diffuse, or an improper flat prior density.Winkler (1981) has shown that, if forecaster i, i=1,2 predicts N(µ i ,σ i ), the combined forecast is also normal with the following parameters. Mean : µ c = (σ 2 2 −ρσ 1 σ 2 )µ 1 +(σ 2 1 −ρσ 1 σ 2 )µ 2 σ 2 1 +σ 2 2 −2ρσ 1 σ 2 Variance : σ 2 c = (1−ρ 2 )σ 2 1 σ 2 2 σ 2 1 +σ 2 2 −2ρσ 1 σ 2 To proceed further, we need to make some assumptions about effort functions. Often the variance depends on the amount of information processed, or the sample size. So 34 we can expect the cost to be inversely proportional to some function of the standard deviation of the forecast. We model this simply as V(σ)= K σ k , where K and k are constants. The overall problem can be formulated as: (P I ): max σc E d [Π(σ c ,d)]−T(σ c ) subject to σ c ≥ 0. (P II ): T(σ c )=min σ 1 ,σ 2 E d [ξ 1 (µ 1 ,d)+ξ 2 (µ 2 ,d)], subject to: U(ξ i (µ i ,d 0 )) =V(σ i )−σ i V (σ i )+ π 2 V (σ i )|d 0 −µ i | i=1,2, σ 2 c = (1−ρ 2 )σ 2 1 σ 2 2 σ 2 1 +σ 2 2 −2ρσ 1 σ 2 . Here expectation is taken over all the possible outcomes given a certain standard deviation. P I determines the optimal combined standard deviationσ c , whileP II deter- mines the lowest cost to achieveσ c and correspondingσ 1 andσ 2 . By Proposition 10, the first constraint ensures truth-telling. Now we turn to the question of whether the firm should invest in only one forecast or in multiple forecasts. Proposition 11 If the cost of forecasting is given byV(σ)= K σ k , then (i) ifk ≤ 2,it is optimal for the firm to invest only in one forecaster, and (ii) ifk> 2, it may be optimal for the firm to invest in multiple forecasters. The optimal number of forecasters to be employed depends on the rate of growth of costs. The proposition identifies the threshold level at which we transition from sin- gle to multiple forecasts. If V(σ)= K σ k , To gain analytic insights, we have assumed a symmetric system. In practice, firms may have to use multiple forecasts because differ- ent agents may have access to different information sources (or samples). The cost of accessing these information sources may vary by agent. If multiple agents are needed, 35 once the desired precision level of each source is determined, an incentive scheme given by Proposition 10 can be used for each agent. 36 Chapter 6 Forecasting with Effort-Dependent Demand In previous chapters the forecaster did not impact the forecasted variable. In this chapter we assume the forecaster is a salesperson forecasting demand. As Chen (2005) argues, firms rely on salespersons to understand customer needs and the market they serve. Such market knowledge is critical for the firm’s inventory planning and new product development. Given this information asymmetry, the firm wants to offer a menu of compensation schemes such that the salespersons will reveal their private information that is useful to the firm (Lal and Staelin, 1986). A simple model would be to allow the salesperson to “generate” a demand distribution. Here, the forecaster influences the entire demand density. Let the cost of attaining a demand distributionf(·) beV(f(·)). If V(·) is known, then, as in the previous chapter, we can solve for the optimal truth- revealing compensation functions. The firm’s problem is: max f(·),ξ EΠ(f(·))− ξ(d)f(d)dd subject to: U(ξ(d))f(d)dd≥V(f(·)), U(ξ(d))f(d)dd−V(f(·))≥ U(ξ(d))g(d)dd−V(g(·)) ∀f(·),g(·)∈F, (6.1) 37 whereEΠ(f(·)) is the expected profit given the firm’s optimal decision based on the densityf(·). The two constraints are for individual rationality and incentive compatibil- ity, respectively. Furthermore, if V is convex and differentiable, then the optimal truth-revealing scheme is a tangent plane toV(f(·)). Proposition 12 If V is convex, then the optimal desired density f ∗ (·) is such that EΠ(f(·))−U −1 (V(f(·))) is maximized. The optimal compensation is: U(ξ(d 0 )) =V(f ∗ (·))− δV δf ∗ x=d f ∗ (d)dd+ δV δf ∗ x=d 0 , (6.2) whered 0 is the realized outcome. Proof is omitted because it immediately follows from Theorem 1 and Proposition 9. A more realistic model should allow forecast precision, sales productivity, and effort disutility to depend on market conditions. In some situations the salesperson can antic- ipate demand accurately, whereas in other cases demand is more uncertain. In prac- tice, forecasting may be more challenging when there are significant changes in macro- economic conditions or in the competitive landscape, and easier conditions might arise in stable markets. Market conditions can also influence a salesperson’s ability to induce demand. Furthermore, given the same level of effort, the salesperson’s disutility may be different in different market conditions. The market conditions are observed by the salesperson, and the firm has to provide incentives that ensure that the salesperson reveals the true regime and makes the appropriate effort level. Accordingly, we assume that there are multiple market regimes. Associated with each regime are a base forecast distribution, a sales productivity function, and the sales- person’s disutility function. In market regime i, the base density is f i (d). When the effort level ist and the market regimei is realized, demand is distributed asf i (d|t), and 38 the salesperson incurs disutility ofV i (t). The firm is aware of the possible regimes. The salesperson and the firm share a common prior distribution about the likelihood of each regime, but only the salesperson observes the true regime. Thus far we have not specified the firm’s objective function. We show below how our model incorporates operational costs that depend on forecast uncertainty. Traditional salesforce compensation literature has not taken these costs into account. Recall that given a forecastf(·) the firm makes a decisionΓ(f(·)). Let us assume that this decision is measured in the same units as the forecasted variable. The objective function of decision problems encountered in operations management can frequently be parsed into two components. One component depends on the realization of the forecasted variable and the second component is a penalty for not correctly anticipating the state of nature. Π(d 0 ,Γ(f(·))) =π(d 0 )−κ e ([Γ(f(·))−d 0 ] + )−κ s ([d 0 −Γ(f(·))] + ), (6.3) whereκ e (·) andκ s (·) are monotone non-decreasing functions. The set of problems where this decomposition applies includes inventory problems, capacity planning, man- power scheduling, and production planning. We will formulate the salesforce incentive problem using this objective. This is done purely to illustrate the link to operational problems, and the need for considering higher moments of forecast distributions. The firm’s problem is to design profit maximizing compensation schemesξ i (·), for each regimei, that ensure truth-telling. For ease of exposition, we present our analysis assuming two regimes, i=1 and 2, occurring with probabilitiesα and 1−α, respec- tively. Extension of the analysis to multiple regimes is relatively straight-forward. 39 As in Chapter 5, we formulate the firm’s problem as a two-stage optimization prob- lem: P III :max t 1 ,t 2 α Π(d,Γ(f 1 (·|t 1 )))f 1 (d|t 1 )dd +(1−α) Π(d,Γ(f 2 (·|t 2 )))f 2 (d|t 2 )dd−T(t 1 ,t 2 ) P IV : T(t 1 ,t 2 )= min ξ 1 (·),ξ 2 (·) α ξ 1 (d)f 1 (d|t 1 )dd +(1−α) ξ 2 (d)f 2 (d|t 2 )dd Subject to: U(ξ 1 (d))f 1 (d|t 1 )dx≥V 1 (t 1 ), (6.4) U(ξ 2 (d))f 2 (d|t 2 )dx≥V 2 (t 2 ), (6.5) U(ξ 1 (d))f 1 (d|t)−V 1 (t)≤ U(ξ 1 (d))f 1 (d|t 1 )dd−V 1 (t 1 ) ∀t, (6.6) U(ξ 2 (d))f 2 (d|t)dd−V 2 (t)≤ U(ξ 2 (d))f 2 (d|t 2 )−V 2 (t 2 ) ∀t, (6.7) U(ξ 2 (d))f 1 (d|t)dd−V 1 (t)≤ U(ξ 1 (d))f 1 (d|t 1 )dd−V 1 (t 1 ) ∀t, (6.8) U(ξ 1 (d))f 2 (d|t)−V 2 (t)≤ U(ξ 2 (d))f 2 (d|t 2 )−V 2 (t 2 ) ∀t. (6.9) The first two constraints ensure the agent’s participation. Constraints (6.6) and (6.7) place the salesperson in a worse position if she does not make the target effort. Due to the last two constraints, (6.8) and (6.9), the salesperson is in a worse position if she does not announce the true regime. Given a pair of effort levelst 1 and t 2 , P IV determines the lowest cost at which the firm can influence the salesperson to make the effort levelt i when the regime isi. The key here is to determine whether P IV is feasible for all possible (t 1 ,t 2 ) combinations. If P IV is always feasible, then the firm can find the overall optimal solution. As the 40 Utility Effort Prob[event1] V1 V2 t1 t2 a b c d 01 Figure 6.1: Separation is impossible following example illustrates, it may not always be possible for the firm to provide incentives that will cause the salesperson to make the designated effort levels. Example: There are two potential outcomes and two potential demand regimes, 1 and2. The salesperson’s disutility function isV i (·) under regimei, and the salesperson’s effort increases the probability of the favorable outcome. Figure 6.1 depicts the effort curves and the effort levelst 1 andt 2 , or equivalently the probability levels that the firm wants in regimes1 and2, respectively. This combination cannot be induced. From The- orem 1 we know that the compensations must be such that the expected payoff curves are straight lines with slopesV 1 (t 1 ) andV 2 (t 2 ). Lines (a,b) and (c,d) represent one pos- sible set of payoff functions, but under this scheme the salesperson does not reveal the true regime. Any other scheme will result in lines that are parallel to (a,b) and (c,d). It is clear that shifting these lines cannot induce the salesperson to make the desired level of effort and reveal the true regime. 1 Although we cannot guarantee truth-telling in general, in the following proposition we show that if all forecasts in a regime belong to the same refinement level, then the salesperson can be incented to reveal the true regime and select any given effort level. 1 One can easily construct examples in which the two disutility curves do not cross. 41 Figure 6.2: Forecasts are uniform but with different parameters in each regime. Theorem 2 If each forecast in regimei belongs to refinement setsχ(J i ), andχ(J 1 ) = χ(J 2 ), then there exists a set of truth-revealing compensation schemes that ensures that the salesperson reveals the true regime and selects a given effort level t i , in regime i=1,2. Theorem 2 shows that if regimes differ in difficulty of forecasting, then firms can ensure that each regime is truthfully identified by the salesperson. Notice that the only requirement is that distributions in a regime belong to the same refinement level. We do not place any restrictions on the effort function. The effect of effort can be additive, mul- tiplicative, or otherwise. The theorem also accommodates situations in which the shape of the distribution changes as the salesperson’s effort increases. The base distributions in different regimes, f i (·|t=0), i=1,2, do not have to have the same mean. We do not place any restrictions on the disutilities associated with effort (V i (t)). Technically, V i does not even have to be non-decreasing. Further, regimes with forecast distributions that have a higher level of refinement will receive higher payments in expectation. The ordering of forecast distributions in this theorem is consistent with that in Lemma 1. 42 Figure 6.3: Regimes with uniform and normal distributions and monotone compensa- tions. Figures 6.2, 6.3, and 6.4 illustrate solutions to P IV when the forecast distributions are either uniform or normally distributed. 2 In Figure 6.2, there are two regimes. In both regimes the forecasts are uniformly distributed but with different spreads. Potential distributions in regime 1 lie in the convex hull of the distributions in regime 2. Effort disutility functions are the same in both regimes. In Figure 6.2-(a), the target effort levels are the same in both regimes, that is,t 1 =t 2 . Consequently, the expected demand would be the same in both regimes. In Figure 6.2-(b), the target effort levels are set such that the agent expends greater effort in regime 2 than in regime 1, that is,t 1 <t 2 . Notice the compensation schemes are not monotone non-decreasing functions of demand. In the salesforce literature, compensation schemes often are non-decreasing functions (Basu et al., 1985, Lal and Staelin, 1986, Chen, 2005). 2 The details are in Appendix A.1. 43 Although Lal and Staelin (1986) and Chen (2005) employ non-decreasing schemes, they do not establish the overall optimality of their contracts. Lal and Staelin (1986) only consider non-decreasing compensation functions. Chen (2005) finds that the optimal menu of linear contracts outperforms the optimal menu of piece-wise linear functions proposed by Gonik. So neither of these papers claims that their solutions are optimal among all the possible compensation functions. Basu et al. (1985) assume df(d|t) dt /f(d|t) is strictly increasing, in order to show that optimal compensations are non-decreasing. They use gamma and binomial distribu- tions. For gamma and binomial distributions the mean and the standard deviation are determined by the effort level. In other words, precision and location of the distribution are connected via effort level. We do not assume any relationships between the location and the shape of the distribution. Also, we do not make any assumption about how the distributions shift as the agent changes her effort level. Therefore we do not have, in gen- eral, that df(d|t) dt /f(d|t) is strictly increasing. Consequently the compensation schemes we present may not be monotone. We have implicitly assumed that the agent does not observe sales until she finishes exerting all the intended effort. In some situations, the outcome of the sales effort could be partially observed prior to the end of the period. If we incorporate this intertemporal aspect, then non-decreasing compensations may be optimal. 3 In Figure 6.3, we show the result of adding a constraints toP IV that forces the pay- offs to be monotone increasing. 4 Figure 6.3-(a) and 6.3-(b) show the results of adding this constraints to the problem depicted in Figure 6.2. In Figures 6.3-(c) and 6.3-(d), the underlying distributions are normal. The standard deviation in regime 2 is twice as that in regime 1. Figure 6.4 shows the case in which there are three demand regimes. 3 Lal and Srinivasan (1993) study such intertemporal behavior of a salesperson. 4 The details are in Appendix A.1. 44 In all of the examples, the cost of effort is modeled as µ k i , where µ is the mean of the distribution andi is the regime. In our numerical examples we find that as the difference among thek i ’s increases, monotone increasing policies cease to be feasible. Although monotone increasing compensation schemes are desirable we cannot always achieve truth-telling if we restrict the policies to be monotone increasing. Our numerical experiments suggest that the feasibility depends on the effort functions. In the proposition below, we identify sufficient conditions under which monotone poli- cies will be truth-revealing. We now need to assume that the base demand distribution in each regime, f i (·|t=0), has the same mean, that the effort effect is additive, i.e., f i (d|t)= f i (d−η i (t)), where η i (·) is the effort productivity function in regimei, that η 1 =η 2 , and that the sales disutility function remains the same:V 1 =V 2 . Lemma 5 If the cost of effort does not depend on the regime, and if the mean of the distribution given an effort level is the same for two regimes, then we can achieve truth- telling even if we restrict the incentive schemes to be monotone increasing. Further, in each regime the salesperson will be paid in expectation an amount exactly equal to the cost of her effort. Thus, if the only difference between the regimes is the level of forecast refinement, then we can restrict the incentive schemes to be monotone increasing. The firm pays only the reservation utility level in each case regardless of the choice of effort levels in each regime. In practice, salespeople receive some combination of salary and commission (Basu et al., 1985). Since salespeople are not responsible for any excess or shortage, they tend to inflate the demand estimate whenever they are asked. It is rarely the case that salespeople are compensated based on their forecasting capabilities. However, a few attempts have been made to alleviate this problem. Gonik (1978) is one such example 45 Figure 6.4: Regimes with uniform and normal distributions and monotone compensa- tions. that was implemented in IBM Brazil. By offering a menu of piece-wise linear con- tracts, IBM Brazil successfully extracted important information about the sales prospect of each sales territory, and use the information to plan their production and logistics. Gonik’s schemes are by no means optimal. His focus was on extracting crucial infor- mation about the local sales environment from the salesforce, not on designing optimal contracts. Our compensation schemes discussed above are a little more involved than those proposed by Gonik, but if the cost of developing such compensation schemes is justified, given Gonik’s scheme’s success, our compensation schemes have a potential that allows a firm to understand the market better and become more profitable. 46 Chapter 7 Summary In this part, we discuss how to elicit probability estimations from managers. We contend that, very often, crucial managerial judgment comes in the form of forecasts. These can be judgments about prices, costs, demand, or the duration of projects. The optimiza- tion algorithms that follow these estimates can be validated, and hence may not repre- sent either hidden effort or private information. We develop truth-telling compensation schemes that rank forecasters and pay more for forecasts that are preferred by the firm. In the process, we provide incentives for improvement in forecasts. Improvements can come immediately if effort or costs can be expended right away. In other cases, there may be no opportunity for improvement in the short term, but the payment schemes create an incentive to improve in the long term. In many situations, the forecaster does not directly influence the outcome. Examples include buyers in department stores, financial portfolio managers, and purchasers of commodities. For these situations, our compensation schemes focus on the precision of the forecast and not on the mean. Because our schemes focus on factors that the forecaster can influence, they should be preferred to profit-sharing schemes. We propose refinement as a basis of ordering forecasters. Refinement ordering is intimately linked to convex sets in density space. Because incentive schemes are also based on convex functions in density space we are able to demonstrate nice connec- tions between refinement ordering and compensation schemes. Forecasters with higher refinement level will receive higher compensation in expectation under any incentive compatible scheme, and will generate higher expected profits for the firm. Even when 47 the forecaster is a salesperson, we find that if the regimes differ on the basis of forecast refinement we can induce the salesperson to reveal the true regime and exert the given effort level. Refinement ordering is also intuitively appealing because it is consistent with acquisition of additional information. We present our analysis assuming that the forecasted variable is demand. Our paper can be extended to other forecasts as well. One interesting application would be a project manager estimating the duration of a project. Like salespersons, project managers can affect the forecasted variable – project duration. 48 Part II Optimal Dynamic Capacity Allocation and Order Acceptance Policies in a Make-to-Order Environment 49 Chapter 8 Introduction The problem studied in this part was motivated by a credit card printing firm. Credit cards are required on short notice if either a new patron enrolls or an existing one needs a replacement because a card was stolen or lost. Credit cards are also printed as replace- ments for ones that have expired. The demand for these types of cards can be anticipated in advance. Credit cards are delivered to the issuing banks or are mailed directly to the credit card holder by the printer on a daily basis. Thus deliveries occur only at discrete time points. The banks pay a premium for cards that have to be printed on a short notice. To maximize expected revenue, the printer has to determine how much capacity to set aside on a daily basis for urgent orders. Similar problems are encountered in the package delivery industry. Here too cus- tomers vary in their delivery time requirements, and quicker deliveries fetch higher fees. Some shippers have limited capacity on certain routes, and they have to decide how much to set aside for urgent orders. In many cases deliveries occur once a day. Once again, we have discrete delivery times. Motivated by these problems we study optimal capacity allocation and order accep- tance policies in a make-to-order business that serves multiple customer classes. Cus- tomers differ only in terms of their lead time preferences. Our analysis is restricted to three customer classes and we make a number of simplifications. We assume that deliv- ery times are discrete and divide the time line into buckets of equal size. We will refer to each time bucket as a “week.” The week in turn is divided into a shorter time period that we refer to as “days.” Each week may have more than seven days. Each day at 50 most one new customer arrives and the firm must decide whether or not it should accept this order. Each customer requires only one unit of the product and specifies a due date. A rejected order is lost forever. We analyze a finite horizon model that consists of a finite number of weeks. A week, in addition to being the delivery frequency, is also the transportation time, or the production lead time depending on the setting. 1 At the end of a week, the firm needs to decide the set of orders it will process in the next week. The firm has the option to process orders prior to their respective due dates. The number of units that can be processed in a week is capacitated. Because due dates are discrete, we model this problem as a knapsack problem. Each week is a knapsack. The state space for our problem is given by the day, week, and the number of orders of each customer class that have been accepted. We show that threshold policies are optimal. If it is optimal to accept an order for a given state space, then it is also optimal to accept the order if, everything else being equal, the number of accepted orders for any customer class is smaller. We also find that if there are only two customer classes, a nested policy is optimal. There is a threshold level such that if the sum of all accepted orders is less than this threshold, it is optimal to accept any order. Otherwise we accept only urgent orders. If the number of classes is three or more, nested policies need not be optimal. It may be optimal to accept an order with a longer delivery time and not one with a shorter delivery time. The rest of the paper proceeds as follows. In Chapter 9, we relate our work with the existing literature. We describe the problem in Chapter 10. The 2-class model is discussed in Section 10.1, and the 3-class model in Section 10.3. Our conclusions and future research direction are presented in Chapter 11 1 This assumption can easily be relaxed to permit production/transportation lead times that are multiple weeks. 51 Chapter 9 Literature An extension of the classical knapsack problem (Martello and Toth, 1990,Kellerer et al., 2004) that is related to our model is the dynamic stochastic knapsack problem. In a dynamic stochastic knapsack problem, objects with different payoffs and weights arrive sequentially. The decision maker has to decide whether or not to accept an object as it arrives. Dynamic stochastic knapsack problems in our parlance are single period (“week”) problems, and the weights correspond to required capacity. Papastavrou et al. (1996) study discrete-time finite-horizon dynamic stochastic knap- sack problems. Order arrival is uncertain, but all orders are due at the end of the hori- zon. We extend their model by allowing multiple periods and incorporating multiple customer classes with different lead times. In their model, orders differ in weight and reward. They too show that the optimal decision rule is a threshold type. An order is accepted if the reward exceeds a threshold that is a function of weight of the job, available capacity, and time. Kleywegt and Papastavrou (1998) study the dynamic stochastic knapsack problem in continuous time. Their model allows the process to stop at any moment. The stop- ping time can be interpreted as the dispatch of a truck. The firm incurs a waiting cost for orders that have not been shipped, therefore it may be optimal to dispatch prior to the end of the time horizon. They compare the dynamic stochastic knapsack problem with its Markovian decision process counterpart to gain insights into optimal policies. Once again, the optimal acceptance rule is a threshold policy. If the revenue from an arriv- ing order exceeds the threshold that is a function of the remaining capacity, then it is 52 optimal to accept the order. They assume all the orders are of equal size. Kleywegt and Papastavrou (2001) extend the model to accommodate orders with different weights. A similar problem is analyzed by van Slyke and Young (2000) in a continuous- time, finite horizon setting. They study three different scenarios of an airline seat inven- tory management problem – single-leg, multi-hop, and origin-destination network prob- lems. The multi-hop and origin-destination network problems are modeled as multi- dimensional knapsack problems. Each dimension corresponds to each hop. For the single-leg problem, they too show that the threshold policy is optimal. The dynamic knapsack problems are closely related to perishable asset revenue man- agement (Weatherford and Bodily, 1992) models. From a revenue management perspec- tive, knapsack problems arise in quantity-based revenue management schemes (Talluri and van Ryzin, 2005). Littlewood (1972) studies a single-resource rationing policy with two classes of customers. Littlewood’s model assumes that low-paying customers arrive before high-paying customers and shows that the optimal policy is of a threshold type. This model would correspond to a 2-class, single-week problem in our frame- work. Belobaba (1989) develops a heuristics called the expected marginal seat revenue model (EMSR), that is similar to Littlewood’s rule, and uses the heuristics for airline seat inventory control. Heuristics such as EMSR and its variants are widely used in the revenue management industry (Talluri and van Ryzin, 2005). Our work differs from this literature in two ways. We consider multiple time periods, and a novel feature of this work is that we allow orders to be processed prior to their due date. Another stream of literature that is related to our work is concerned with due date management. In this literature, due dates are the key decision variables. They develop an optimization problem, usually with a tardiness related objective function, subject to service level constraints. There is usually no guarantee that an order will be delivered as quoted. An exception is the work by Kapuscinski and Tayur (2007). They consider 53 the due date management problem with a guarantee that an order will be produced and delivered as promised. For a comprehensive review of due date management literature, the reader is referred to Keskinocak and Tayur (2004). Our approach in this paper is different from those in due date management literature. A typical due date management paper assumes that order acceptance decisions are exogenously given, and then solve for a due date setting policy and/or a scheduling policy. In this study, we assume that due dates are fixed depending on customer class and find an optimal acceptance policy. The problem of scheduling jobs to meet due dates on a dynamic basis has also been studied by queuing theorists. Work by Shanthikumar and Yao (1992) and Maglaras (2006) illustrate this approach. The main difference between their approach and ours is that we do not have to make any assumptions about the arrival process. Arrival rates do not have to be stationary, nor do we have to assume that the system operates at utilizations close to one. Also in our model, the due dates are guaranteed. Our model assumes that due dates are customer-dependent, and the firm makes two types of decisions: (a) accept or reject an order, and (b) determine jobs that are processed each week. Our approach of modeling this problem as a dynamic rolling knapsack is a novel approach that is more versatile than traditional queuing models. It also extends the traditional revenue management models to multi-period systems where orders can be processed earlier than they are needed. This option is not available for airlines carrying passengers, but it is certainly possible for cargo shipments and production systems such as the credit card manufacturers. There are two important limitations to our work. For one, we assume that all orders are of the same size. Second, we ignore any scale economies that may exist in the production process. 54 Chapter 10 Model and Analysis As stated above we use the term “day” to denote a short period of time in which at most one order arrives. We use the term “week” for the delivery periods. All due dates are at the end of a week. We assume there are T days in a week. Production starts at the beginning of a week and finishes at the end of the week. In our dynamic programming formulation, we label the days and weeks backwards. The last day of a week, therefore, is “Day 1,” and the last week is ”Week 1.” The index for a day denotes the number of days till the end of the week, and the index for a week denotes the number of weeks till the end of the horizon. The production plan for a week is determined at the end of the previous week. There are N different types of orders. A Type n order, 1 ≤ n ≤ N, arrives with probability p n . A Typen order, if accepted, has to be produced and delivered withinn weeks. Each day, if an order arrives, the firm has to decide whether or not to accept that order (See Figure 10.1). A Type n order generates a profit of R n , and R m >R n , for allm<n. Table 10.1: Summary of Notation B Capacity in a period, i.e., weekly capacity; p n Probability of a Typen order arriving in a day; R n Reward or profit generated by a Typen order; a n Number of Typen orders already accepted, 1≤n≤N; t Time to go by the end of the week, 1≤t≤T ; V w (t,a 1 ,a 2 ,...,a N ) Expected payoff if optimal policy is followed on Dayt of Weekw,w≥ 2; A w (t,a 1 ,a 2 ,...,a N ,n) Optimal action when the state is, (w,t,a 1 ,a 2 ,...,a N ) and a Typen order arrives,w≥ 2; Possible action is either (A)ccept or (R)eject. 55 BB B a2 a1 a3 Day t Type 1 order Type 2 order Type 3 order Weekly Capacity Figure 10.1: The Model (N=3) We leta n denote the number of orders of Typen that have been accepted but have not been produced. We require n i=1 a i ≤nB, for all1≤n≤N. Noticea n may be larger thanB for somen> 1. Table 10.1 summarizes the notation. We do not assumeR n andp n are homogeneous. In other words, rewards and prob- abilities can differ from one day to another. Although it would be more appropriate to use the notation R w,t,n and p w,t,n , we use R n and p n for ease of exposition. The week and day can readily be inferred from the context. Furthermore, we assume R n and p n are independent of one another. The only assumption we have here is thatR n andp n are known. LetA denote the class of policies that prescribe whether the firm should accept or reject an order. Sincep n andR n , 1≤ n≤ N, are all history independent, and the state is completely characterized by the week, the day, and the number of orders accepted for each type, we restrict our attention to the class of policies that are history independent. Let V w (t,a 1 ,a 2 ,...,a N−1 ,a N | ˜ A) denote the expected accumulated reward if ˜ A∈A is implemented. Let V w (t,a 1 ,a 2 ,...,a N−1 ,a N ) denote the optimal expected accumulated reward: V w (t,a 1 ,a 2 ,...,a N−1 ,a N )=sup ˜ A∈A V w (t,a 1 ,a 2 ,...,a N−1 ,a N | ˜ A). 56 BB B a2 a1 a3 Day 1 Figure 10.2: After observing demand on Day 1 Consequently, ˜ A∈A is a function that maps a state to the action space. We let A w (t,a 1 ,a 2 ,...,a N ,n) denote the optimal acceptance policy when the state is (w, t, a 1 , a 2 ,..., a N ) and a Type n order arrives. This optimal acceptance function A w (t,a 1 ,a 2 ,...,a N ,n) returns either (A)ccept or (R)eject. Fort> 1, the optimal action is A w (t,a 1 ,a 2 ,...,a N ,n)=              (A)ccept if R n +V w−1 (t−1,a 1 ,a 2 ,...,a n +1,...,a N ) ≥V w−1 (t−1,a 1 ,a 2 ,...,a n ,...,a N ), (R)eject otherwise. At the end of Week w, after observing Day 1’s demand, the firm has to freeze the production schedule of the following week. Figure 10.2 illustrates a case in which the firm has the option to produce less urgent orders. We note parenthetically that the pro- duction decisions depicted in Figure 10.2 need not be optimal. Suppose that, at the end of Weekw, there are ˜ a 1 , ˜ a 2 , ..., ˜ a N orders accepted. The firm must produce at least ˜ a 1 units in the following week. If ˜ a 1 <B, then the firm may want to produce more. The optimal production quantity for the next week will be to produce as many as possible. This will enable the firm to accept more orders in the future. Which of those less urgent orders to produce earlier, however, is not clear. The objective of the ensuing analysis is to determine which orders the firm should produce in advance, given that the current 57 capacity is not fully utilized by orders that are due immediately. Let (a 1 ,a 2 , ...,a N−1 )be the next week’s starting state on DayT . (Notea N =0.) The firm’s production decisions must take into account due dates and capacity constraints. max a 1 ,a 2 ,...,a N−1 V w−1 (T,a 1 ,a 2 ,...,a N−1 ,0) subject to: n i=1 a i ≤ n+1 i=2 ˜ a i ∀n,1≤n≤N−1 ˜ a 1 ≤ N i=1 ˜ a i − N i=1 a i ≤B. We show in the following sections that it is optimal to prioritize production on the basis of customer class. Customers with shorter lead time requirements are preferred over those with longer lead time requirements. For example, if˜ a 1 <B and˜ a 1 +˜ a 2 >B, then it is optimal to produce B units next week and start the new week with a 1 = 2B−(˜ a 1 +˜ a 2 ),a 2 =˜ a 3 , ...,a N−1 =˜ a N , anda N =0. 10.1 Two-Class Demand Problem We first study the case of N =2 in this section. Since there are only two classes, the optimal policy will accept any Type 1 urgent order subject to availability of capacity. We only have to determine whether or not it is optimal to accept Type 2 orders. We simplify the notation by using A w (t,a 1 ,a 2 ), instead of A w (t,a 1 ,a 2 ,2), to denote the optimal acceptance decision when a Type 2 order arrives. As is usually the case, we work backwards. We start our analysis with Week 1, which is the terminal week. 58 10.1.1 Week 1 In Week 1, the problem is a standard stochastic knapsack. LetV 1 (b) denote the optimal value function when there are b units of capacity remaining at the beginning of Week 1. We know from Kleywegt and Papastavrou (1998) that the value function is concave increasing inb, that is,V 1 (b)−V 1 (b−1)≥V 1 (b+1)−V 1 (b). 10.1.2 Week 2 Optimal Shifting at the end of Week 2 After accepting or rejecting orders, if any, on Day 1 of Week 2, the firm has to freeze the production schedule for Week 1. Intuitively, such a policy should process as many orders as possible. The following proposition establishes this result. Proposition 13 Suppose there are a 1 Type 1 orders and a 2 Type 2 orders accepted at the end of Week 2, then the firm’s optimal production schedule for Week 1 is to produce Min(a 1 +a 2 ,B). Therefore, at the beginning of Week 1, the firm’s available capacity isMin(2B−(a 1 +a 2 ),B). Proof follows from the fact thatV 1 is increasing in remaining capacity. A Threshold Policy is Optimal in Week 2 Given the above optimal shifting, we next prove, by induction, that on any day of Week 2, a threshold type action is optimal. This is true becauseV 1 (b) is concave increasing in b, whereb is the remaining capacity. Proposition 14 IfA 2 (t,a 1 ,a 2 )=(A) for somet, a 1 , a 2 , thenA 2 (t,a 1 − 1,a 2 )=(A) andA 2 (t,a 1 ,a 2 −1) = (A). 59 Proof: We first show that if A 2 (1,a 1 ,a 2 )=(A), then A 2 (1,a 1 − 1,a 2 )=(A) and A 2 (1,a 1 ,a 2 −1) = (A). SinceA 2 (1,a 1 ,a 2 )=(A), using Proposition 13: R 2 +V 1 (min(2B−(a 1 +a 2 +1),B)) ≥V 1 (min(2B−(a 1 +a 2 ),B)). SinceV 1 is concave, it follows that R 2 +V 1 (min(2B−(a 1 +a 2 ),B)) ≥V 1 (min(2B−(a 1 +a 2 −1),B)). This implies bothA 2 (1,a 1 −1,a 2 )=(A) andA 2 (1,a 1 ,a 2 −1) = (A). Next we establish this property for all t, 1 <t ≤ T . We want to show if A 2 (t,a 1 ,a 2 )=(A), then A 2 (t,a 1 − 1,a 2 )= (A) and A 2 (t,a 1 ,a 2 − 1) = (A) for allt. NoticeA 2 (t,a 1 ,a 2 )=(A) is equivalent to: R 2 +V 2 (t−1,a 1 ,a 2 +1) ≥V 2 (t−1,a 1 ,a 2 ). A 2 (t,a 1 −1,a 2 )=(A) andA 2 (t,a 1 ,a 2 −1) = (A) are equivalent to: 60 R 2 +V 2 (t−1,a 1 −1,a 2 +1) ≥V 2 (t−1,a 1 −1,a 2 ), and R 2 +V 2 (t−1,a 1 ,a 2 ) ≥V 2 (t−1,a 1 ,a 2 −1), respectively. It is now easy to see that it is sufficient to show: V 2 (t,a 1 −1,a 2 +1)−V 2 (t,a 1 ,a 2 +1) ≥V 2 (t,a 1 −1,a 2 )−V 2 (t,a 1 ,a 2 ), (10.1) and V 2 (t,a 1 ,a 2 )−V 2 (t,a 1 ,a 2 +1) ≥V 2 (t,a 1 ,a 2 −1)−V 2 (t,a 1 ,a 2 ), (10.2) for all1≤t≤T anda 1 ,a 2 , such thata 1 ≥ 1 anda 1 +a 2 +1≤ 2B. 1 We show the above two inequalities using induction. First we show (10.1) holds for t=1, i.e., we want to show: 1 We say “for all admissible t, a 1 , and a 2 ” hereafter. 61 V 2 (1,a 1 −1,a 2 +1)−V 2 (1,a 1 ,a 2 +1) ≥V 2 (1,a 1 −1,a 2 )−V 2 (1,a 1 ,a 2 ). (10.3) Each of the four terms can be expanded: V 2 (1,a 1 −1,a 2 +1) =p 1 [R 1 +V 1 (min(2B−(a 1 +a 2 +1),B))] +p 2 ·max[R 2 +V 1 (min(2B−(a 1 +a 2 +1),B)),V 1 (min(2B−(a 1 +a 2 ),B))] +(1−p 1 −p 2 )V 1 (min(2B−(a 1 +a 2 ),B)) V 2 (1,a 1 ,a 2 +1) =p 1 [R 1 +V 1 (min(2B−(a 1 +a 2 +2),B)] +p 2 ·max[R 2 +V 1 (min(2B−(a 1 +a 2 +2),B)),V 1 (min(2B−(a 1 +a 2 +1),B))] +(1−p 1 −p 2 )V 1 (min(2B−(a 1 +a 2 +1),B) V 2 (1,a 1 −1,a 2 ) =p 1 [R 1 +V 1 (min(2B−(a 1 +a 2 ),B)] +p 2 ·max[R 2 +V 1 (min(2B−(a 1 +a 2 ),B)),V 1 (min(2B−(a 1 +a 2 −1),B))] +(1−p 1 −p 2 )V 1 (min(2B−(a 1 +a 2 −1),B)) 62 V 2 (1,a 1 ,a 2 ) =p 1 [R 1 +V 1 (min(2B−(a 1 +a 2 +1),B))] +p 2 ·max[R 2 +V 1 (min(2B−(a 1 +a 2 +1),B)),V 1 (min(2B−(a 1 +a 2 ),B))] +(1−p 1 −p 2 )V 1 (min(2B−(a 1 +a 2 ),B)). SinceV 1 is concave, we have, forp 1 terms: V 1 (min(2B−(a 1 +a 2 +1),B))−V 1 (min(2B−(a 1 +a 2 +2),B)) ≥V 1 (min(2B−(a 1 +a 2 ),B))−V 1 (min(2B−(a 1 +a 2 +1),B)), and for (1−p 1 −p 2 ) terms, V 1 (min(2B−(a 1 +a 2 ),B))−V 1 (min(2B−(a 1 +a 2 +1),B)) ≥V 1 (min(2B−(a 1 +a 2 −1),B))−V 1 (min(2B−(a 1 +a 2 ),B). What is left is to examinep 2 terms. We know optimal policy is of a threshold type whent=1, so there are six cases to examine: Case I: A 2 (1,a 1 −1,a 2 +1)=(A) A 2 (1,a 1 ,a 2 +1)=(A) A 2 (1,a 1 −1,a 2 )=(A) A 2 (1,a 1 ,a 2 )=(A). 63 SinceV 1 is concave, we have R 2 +V 1 (min(2B−(a 1 +a 2 +1),B))−(R 2 +V 1 (min(2B−(a 1 +a 2 +2),B))) ≥R 2 +V 1 (min(2B−(a 1 +a 2 ),B))−(R 2 +V 1 (min(2B−(a 1 +a 2 +1),B))), and (10.3) holds. Case II: A 2 (1,a 1 −1,a 2 +1)=(A) A 2 (1,a 1 ,a 2 +1)=(R) A 2 (1,a 1 −1,a 2 )=(A) A 2 (1,a 1 ,a 2 )=(A). We have R 2 +V 1 (min(2B−(a 1 +a 2 +1),B))−V 1 (min(2B−(a 1 +a 2 +1),B)) =R 2 +V 1 (min(2B−(a 1 +a 2 ),B))−V 1 (min(2B−(a 1 +a 2 ),B)) ≥R 2 +V 1 (min(2B−(a 1 +a 2 ),B))−(R 2 +V 1 (min(2B−(a 1 +a 2 +1),B)). So (10.3) holds. Case III: 64 A 2 (1,a 1 −1,a 2 +1)=(R) A 2 (1,a 1 ,a 2 +1)=(R) A 2 (1,a 1 −1,a 2 )=(A) A 2 (1,a 1 ,a 2 )=(A). We have V 1 (min(2B−(a 1 +a 2 ),B))−V 1 (min(2B−(a 1 +a 2 +1),B)) =R 2 +V 1 (min(2B−(a 1 +a 2 ),B))−(R 2 +V 1 (min(2B−(a 1 +a 2 +1),B))). So (10.3) holds. Case IV: A 2 (1,a 1 −1,a 2 +1)=(R) A 2 (1,a 1 ,a 2 +1)=(R) A 2 (1,a 1 −1,a 2 )=(A) A 2 (1,a 1 ,a 2 )=(R). We have V 1 (min(2B−(a 1 +a 2 ),B))−V 1 (min(2B−(a 1 +a 2 +1),B)) ≥R 2 +V 1 (min(2B−(a 1 +a 2 +1),B))−V 1 (min(2B−(a 1 +a 2 +1),B)) =R 2 +V 1 (min(2B−(a 1 +a 2 ),B))−V 1 (min(2B−(a 1 +a 2 ),B). So (10.3) holds. 65 Case V: A 2 (1,a 1 −1,a 2 +1)=(A) A 2 (1,a 1 ,a 2 +1)=(R) A 2 (1,a 1 −1,a 2 )=(A) A 2 (1,a 1 ,a 2 )=(R). R 2 +V 1 (min(2B−(a 1 +a 2 +1),B))−V 1 (min(2B−(a 1 +a 2 +1),B)) =R 2 +V 1 (min(2B−(a 1 +a 2 ),B))−V 1 (min(2B−(a 1 +a 2 ),B)). So (10.3) holds. Case VI: A 2 (1,a 1 −1,a 2 +1)=(R) A 2 (1,a 1 ,a 2 +1)=(R) A 2 (1,a 1 −1,a 2 )=(R) A 2 (1,a 1 ,a 2 )=(R). SinceV 1 is concave, V 1 (min(2B−(a 1 +a 2 ),B))−V 1 (min(2B−(a 1 +a 2 +1),B)) ≥V 1 (min(2B−(a 1 +a 2 −1),B))−V 1 (min(2B−(a 1 +a 2 ),B)). 66 So (10.3) holds. Having shown that we have the desired result fort=1, we want to show that it also holds for allt. We want to show that if V 2 (t,a 1 −1,a 2 +1)−V 2 (t,a 1 ,a 2 +1) ≥V 2 (t,a 1 −1,a 2 )−V 2 (t,a 1 ,a 2 ) ∀a 1 ,a 2 (10.4) for somet≥ 1, then V 2 (t+1,a 1 ,a 2 +1)−V 2 (t+1,a 1 ,a 2 +1) ≥V 2 (t+1,a 1 −1,a 2 )−V 2 (t+1,a 1 ,a 2 ). (10.5) Each term can be expanded: V 2 (t+1,a 1 −1,a 2 +1) =p 1 [R 1 +V 2 (t,a 1 ,a 2 +1)] +p 2 ·max[R 2 +V 2 (t,a 1 −1,a 2 +2),V 2 (t,a 1 −1,a 2 +1)] +(1−p 1 −p 2 )V 2 (t,a 1 −1,a 2 +1), V 2 (t+1,a 1 ,a 2 +1) =p 1 [R 1 +V 2 (t,a 1 +1,a 2 +1)] +p 2 ·max[R 2 +V 2 (t,a 1 ,a 2 +2),V 2 (t,a 1 ,a 2 +1)] +(1−p 1 −p 2 )V 2 (t,a 1 ,a 2 +1), 67 V 2 (t+1,a 1 −1,a 2 ) =p 1 [R 1 +V 2 (t,a 1 ,a 2 )] +p 2 ·max[R 2 +V 2 (t,a 1 −1,a 2 +1),V 2 (t,a 1 −1,a 2 )] +(1−p 1 −p 2 )V 2 (t,a 1 −1,a 2 ), V 2 (t+1,a 1 ,a 2 ) =p 1 [R 1 +V 2 (t,a 1 +1,a 2 )] +p 2 ·max[R 2 +V 2 (t,a 1 ,a 2 +1),V 2 (t,a 1 ,a 2 )] +(1−p 1 −p 2 )V 2 (t,a 1 ,a 2 ). It is easy to verify, using (10.4), for the terms involvingp 1 and (1−p 1 −p 2 ), V 2 (t,a 1 ,a 2 +1)−V 2 (t,a 1 +1,a 2 +1) ≥V 2 (t,a 1 ,a 2 )−V 2 (t,a 1 +1,a 2 ), and V 2 (t,a 1 −1,a 2 +1)−V 2 (t,a 1 ,a 2 +1) ≥V 2 (t,a 1 −1,a 2 )−V 2 (t,a 1 ,a 2 ). Forp 2 terms, there are six cases: Case I: 68 A 2 (t,a 1 −1,a 2 +1)=(A) A 2 (t,a 1 ,a 2 +1)=(A) A 2 (t,a 1 −1,a 2 )=(A) A 2 (t,a 1 ,a 2 )=(A). Due to (10.4), R 2 +V 2 (t,a 1 −1,a 2 +2)−(R 2 +V 2 (t,a 1 ,a 2 +2)) ≥R 2 +V 2 (t,a 1 −1,a 2 +1)−(R 2 +V 2 (t,a 1 ,a 2 +1)). So we have (10.5). Case II: A 2 (t,a 1 −1,a 2 +1)=(A) A 2 (t,a 1 ,a 2 +1)=(R) A 2 (t,a 1 −1,a 2 )=(A) A 2 (t,a 1 ,a 2 )=(A). Then R 2 +V 2 (t,a 1 −1,a 2 +2)−V 2 (t,a 1 ,a 2 +1) ≥V 2 (t,a 1 −1,a 2 +1)−V 2 (t,a 1 ,a 2 +1) =R 2 +V 2 (t,a 1 −1,a 2 +1)−(R 2 +V 2 (t,a 1 ,a 2 +1)). 69 So we have (10.5). Case III: A 2 (t,a 1 −1,a 2 +1)=(R) A 2 (t,a 1 ,a 2 +1)=(R) A 2 (t,a 1 −1,a 2 )=(A) A 2 (t,a 1 ,a 2 )=(A). V 2 (t,a 1 −1,a 2 +1)−V 2 (t,a 1 ,a 2 +1) =R 2 +V 2 (t,a 1 −1,a 2 +1)−(R 2 +V 2 (t,a 1 ,a 2 +1)). So we have (10.5). Case IV: A 2 (t,a 1 −1,a 2 +1)=(A) A 2 (t,a 1 ,a 2 +1)=(R) A 2 (t,a 1 −1,a 2 )=(A) A 2 (t,a 1 ,a 2 )=(R). 70 R 2 +V 2 (t,a 1 −1,a 2 +2)−V 2 (t,a 1 ,a 2 +1) ≥V 2 (t,a 1 −1,a 2 +1)−V 2 (t,a 1 ,a 2 +1) =R 2 +V 2 (t,a 1 −1,a 2 +1)−(R 2 +V 2 (t,a 1 ,a 2 +1)) ≥R 2 +V 2 (t,a 1 −1,a 2 +1)−V 2 (t,a 1 ,a 2 ). So we have (10.5). Case V: A 2 (t,a 1 −1,a 2 +1)=(R) A 2 (t,a 1 ,a 2 +1)=(R) A 2 (t,a 1 −1,a 2 )=(A) A 2 (t,a 1 ,a 2 )=(R). V 2 (t,a 1 −1,a 2 +1)−V 2 (t,a 1 ,a 2 +1) =R 2 +V 2 (t,a 1 −1,a 2 +1)−(R 2 +V 2 (t,a 1 ,a 2 +1)) ≥R 2 +V 2 (t,a 1 −1,a 2 +1)−V 2 (t,a 1 ,a 2 ). So we have (10.5). Case VI: 71 A 2 (t,a 1 −1,a 2 +1)=(R) A 2 (t,a 1 ,a 2 +1)=(R) A 2 (t,a 1 −1,a 2 )=(R) A 2 (t,a 1 ,a 2 )=(R). Due to (10.4), V 2 (t,a 1 −1,a 2 +1)−V 2 (t,a 1 ,a 2 +1) ≥V 2 (t,a 1 −1,a 2 )−V 2 (t,a 1 ,a 2 ). So we have (10.5). Similarly, we can prove (10.2) as follows. First we want to show (10.2) holds fort=1, that is: V 2 (1,a 1 ,a 2 )−V 2 (1,a 1 ,a 2 +1) ≥V 2 (1,a 1 ,a 2 −1)−V 2 (1,a 1 ,a 2 ). (10.6) Each term can be expanded: V 2 (1,a 1 ,a 2 ) =p 1 [R 1 +V 1 (min(2B−(a 1 +a 2 +1),B))] +p 2 ·max[R 2 +V 1 (min(2B−(a 1 +a 2 +1),B)),V 1 (min(2B−(a 1 +a 2 ),B))] +(1−p 1 −p 2 )V 1 (min(2B−(a 1 +a 2 ),B)), 72 V 2 (1,a 1 ,a 2 +1) =p 1 [R 1 +V 1 (min(2B−(a 1 +a 2 +2),B))] +p 2 ·max[R 2 +V 1 (min(2B−(a 1 +a 2 +2),B)),V 1 (min(2B−(a 1 +a 2 +1),B))] +(1−p 1 −p 2 )V 1 (min(2B−(a 1 +a 2 +1),B)), V 2 (1,a 1 ,a 2 −1) =p 1 [R 1 +V 1 (min(2B−(a 1 +a 2 ),B))] +p 2 ·max[R 2 +V 1 (min(2B−(a 1 +a 2 ),B)),V 1 (min(2B−(a 1 +a 2 −1),B))] +(1−p 1 −p 2 )V 1 (min(2B−(a 1 +a 2 −1),B)). SinceV 1 is concave, we have, forp 1 terms: V 1 (min(2B−(a 1 +a 2 +1),B))−V 1 (min(2B−(a 1 +a 2 +2),B)) ≥V 1 (min(2B−(a 1 +a 2 ),B))−V 1 (min(2B−(a 1 +a 2 +1),B)), and for (1−p 1 −p 2 ) terms, V 1 (min(2B−(a 1 +a 2 ),B))−V 1 (min(2B−(a 1 +a 2 +1),B)) ≥V 1 (min(2B−(a 1 +a 2 −1),B))−V 1 (min(2B−(a 1 +a 2 ),B)). What is left is to examinep 2 terms. We know optimal policy is of a threshold type whent=1, so there are four cases to examine: Case I: 73 A 2 (1,a 1 ,a 2 )=(A) A 2 (1,a 1 ,a 2 +1)=(A) A 2 (1,a 1 ,a 2 −1) = (A). SinceV 1 is concave, we have R 2 +V 1 (min(2B−(a 1 +a 2 +1),B))−(R 2 +V 1 (min(2B−(a 1 +a 2 +2),B))) ≥R 2 +V 1 (min(2B−(a 1 +a 2 ),B))−(R 2 +V 1 (min(2B−(a 1 +a 2 +1),B))), and (10.6) holds. Case II: A 2 (1,a 1 ,a 2 )=(A) A 2 (1,a 1 ,a 2 +1)=(R) A 2 (1,a 1 ,a 2 −1) = (A). R 2 +V 1 (min(2B−(a 1 +a 2 +1),B))−V 1 (min(2B−(a 1 +a 2 +1),B)) =R 2 +V 1 (min(2B−(a 1 +a 2 ),B))−V 1 (min(2B−(a 1 +a 2 ),B)) ≥R 2 +V 1 (min(2B−(a 1 +a 2 ),B))−(R 2 +V 1 (min(2B−(a 1 +a 2 +1),B))). So (10.6) holds. Case III: 74 A 2 (1,a 1 ,a 2 )=(R) A 2 (1,a 1 ,a 2 +1)=(R) A 2 (1,a 1 ,a 2 −1) = (A). V 1 (min(2B−(a 1 +a 2 ),B))−V 1 (min(2B−(a 1 +a 2 +1),B)) ≥R 2 +V 1 (min(2B−(a 1 +a 2 +1),B))−V 1 (min(2B−(a 1 +a 2 +1),B)) =R 2 +V 1 (min(2B−(a 1 +a 2 ),B))−V 1 (min(2B−(a 1 +a 2 ),B)). So (10.6) holds. Case IV: A 2 (1,a 1 ,a 2 )=(R) A 2 (1,a 1 ,a 2 +1)=(R) A 2 (1,a 1 ,a 2 −1) = (R). SinceV 1 is concave, V 1 (min(2B−(a 1 +a 2 ),B))−V 1 (min(2B−(a 1 +a 2 +1),B)) ≥V 1 (min(2B−(a 1 +a 2 −1),B))−V 1 (min(2B−(a 1 +a 2 ),B)). So (10.6) holds fort=1. Next we show (10.6) holds for anyt by induction. 75 Assume: V 2 (t,a 1 ,a 2 )−V 2 (t,a 1 ,a 1 +1) ≥V 2 (t,a 1 ,a 1 −1)−V 2 (t,a 1 ,a 2 ) ∀a 1 ,a 2 (10.7) for somet. Now we want to show: V 2 (t+1,a 1 ,a 2 )−V 2 (t+1,a 1 ,a 1 +1) ≥V 2 (t+1,a 1 ,a 1 −1)−V 2 (t+1,a 1 ,a 2 ). (10.8) This result is derived below. V 2 (t+1,a 1 ,a 2 ) =p 1 [R 1 +V 2 (t,a 1 +1,a 2 )] +p 2 ·max[R 2 +V 2 (t,a 1 ,a 2 +1),V 2 (t,a 1 ,a 2 )] +(1−p 1 −p 2 )V 2 (t,a 1 ,a 2 ), V 2 (t+1,a 1 ,a 2 +1) =p 1 [R 1 +V 2 (t,a 1 +1,a 2 +1)] +p 2 ·max[R 2 +V 2 (t,a 1 ,a 2 +2),V 2 (t,a 1 ,a 2 +1)] +(1−p 1 −p 2 )V 2 (t,a 1 ,a 2 +1), 76 V 2 (t+1,a 1 ,a 2 −1) =p 1 [R 1 +V 2 (t,a 1 +1,a 2 −1)] +p 2 ·max[R 2 +V 2 (t,a 1 ,a 2 ),V 2 (t,a 1 ,a 2 −1)] +(1−p 1 −p 2 )V 2 (t,a 1 ,a 2 −1). It is easy to verify, using (10.7), for the terms involvingp 1 and (1−p 1 −p 2 ), V 2 (t,a 1 +1,a 2 )−V 2 (t,a 1 +1,a 2 +1) ≥V 2 (t,a 1 +1,a 2 −1)−V 2 (t,a 1 +1,a 2 ), and V 2 (t,a 1 ,a 2 )−V 2 (t,a 1 ,a 2 +1) ≥V 2 (t,a 1 ,a 2 −1)−V 2 (t,a 1 ,a 2 ). Forp 2 terms, there are four cases: Case I: A 2 (t,a 1 ,a 2 )=(A) A 2 (t,a 1 ,a 2 +1)=(A) A 2 (t,a 1 ,a 2 −1) = (A). By (10.7), 77 R 2 +V 2 (t,a 1 ,a 2 +1)−(R 2 +V 2 (t,a 1 ,a 2 +2)) ≥R 2 +V 2 (t,a 1 ,a 2 )−(R 2 +V 2 (t,a 1 ,a 2 +1)). So we have (10.8). Case II: A 2 (t,a 1 ,a 2 )=(A) A 2 (t,a 1 ,a 2 +1)=(R) A 2 (t,a 1 ,a 2 −1) = (A). R 2 +V 2 (t,a 1 ,a 2 +1)−V 2 (t,a 1 ,a 2 +1) =R 2 +V 2 (t,a 1 ,a 2 )−V 2 (t,a 1 ,a 2 ) ≥R 2 +V 2 (t,a 1 ,a 2 )−(R 2 +V 2 (t,a 1 ,a 2 +1)). So we have (10.8). Case III: A 2 (t,a 1 ,a 2 )=(R) A 2 (t,a 1 ,a 2 +1)=(R) A 2 (t,a 1 ,a 2 −1) = (A). 78 V 2 (t,a 1 ,a 2 )−V 2 (t,a 1 ,a 2 +1) ≥R 2 +V 2 (t,a 1 ,a 2 +1)−V 2 (t,a 1 ,a 2 +1) =R 2 =R 2 +V 2 (t,a 1 ,a 2 )−V 2 (t,a 1 ,a 2 ). So we have (10.8). Case IV: A 2 (t,a 1 ,a 2 )=(R) A 2 (t,a 1 ,a 2 +1)=(R) A 2 (t,a 1 ,a 2 −1) = (R). Due to (10.7), V 2 (t,a 1 ,a 2 )−V 2 (t,a 1 ,a 2 +1) ≥V 2 (t,a 1 ,a 2 −1)−V 2 (t,a 1 ,a 2 ). This completes the proof. Finally, we show the following to connect with Week 3 onward. Proposition 15 V 2 (t,a 1 −1,0)−V 2 (t,a 1 ,0) ≥V 2 (t,a 1 ,0)−V 2 (t,a 1 +1,0), (10.9) 79 for all admissiblet,a 1 , anda 2 . Proof: Clearly, (10.9) holds when t=1, since V 1 is concave increasing. Suppose (10.9) holds for somet≥ 1. We want to show: V 2 (t+1,a 1 −1,0)−V 2 (t+1,a 1 ,0) ≥V 2 (t+1,a 1 ,0)−V 2 (t+1,a 1 +1,0). (10.10) Note: V 2 (t+1,a 1 −1,0) =p 1 [R 1 +V 2 (t,a 1 ,0)] +p 2 ·max[R 2 +V 2 (t,a 1 −1,1),V 2 (t,a 1 −1,0)] +(1−p 1 −p 2 )V 2 (t,a 1 −1,0), V 2 (t+1,a 1 ,0) =p 1 [R 1 +V 2 (t,a 1 +1,0)] +p 2 ·max[R 2 +V 2 (t,a 1 ,1),V 2 (t,a 1 ,0)] +(1−p 1 −p 2 )V 2 (t,a 1 ,0), 80 V 2 (t+1,a 1 +1,0) =p 1 [R 1 +V 2 (t,a 1 +2,0)] +p 2 ·max[R 2 +V 2 (t,a 1 +1,1),V 2 (t,a 1 +1,0)] +(1−p 1 −p 2 )V 2 (t,a 1 +1,0). Clearly, p 1 and (1− p 1 − p 2 ) terms satisfy (10.10). For p 2 term, there are four possibilities. Case I: A 2 (t+1,a 1 −1,0) = (A) A 2 (t+1,a 1 ,0) = (A) A 2 (t+1,a 1 +1,0) = (A). Clearly (10.10) is satisfied. Case II: A 2 (t+1,a 1 −1,0) = (A) A 2 (t+1,a 1 ,0) = (A) A 2 (t+1,a 1 +1,0) = (R). 81 R 2 +V 2 (t,a 1 −1,1)−(R 2 +V 2 (t,a 1 ,1)) ≤R 2 +V 2 (t,a 1 ,1)−(R 2 +V 2 (t,a 1 +1,1)) ≤R 2 +V 2 (t,a 1 ,1)−V 2 (t,a 1 +1,1). So (10.10) holds. Case III: A 2 (t+1,a 1 −1,0) = (A) A 2 (t+1,a 1 ,0) = (R) A 2 (t+1,a 1 +1,0) = (R). R 2 +V 2 (t,a 1 −1,1)−V 2 (t,a 1 ,0) ≤V 2 (t,a 1 −1,1)−V 2 (t,a 1 ,0) ≤V 2 (t,a 1 ,1)−V 2 (t,a 1 +1,1). So (10.10) holds. Case IV: A 2 (t+1,a 1 −1,0) = (R) A 2 (t+1,a 1 ,0) = (R) A 2 (t+1,a 1 +1,0) = (R). It is clear that (10.10) holds. The proof is complete. 82 By Proposition 15, using induction, all the properties that hold in Week 2 also hold in Weekw≥ 3. Acceptance Region is Decreasing int. The dynamic stochastic knapsack with equal size has an optimal policy that is a thresh- old type, and (Kleywegt and Papastavrou, 1998) show that the optimal threshold level is decreasing in the time remaining. A decreasing threshold implies that the firm becomes less selective if there is less time available to accept orders. We expect this property to be inherited by the rolling knapsack problem. We want to show that the optimal accep- tance policy is such that if there is less time available, then the firm will becomes less selective. Equivalently as we get closer to the end of a week, the size of the acceptance region increases. In other words, if A 2 (t+1,a 1 ,a 2 )=(A), thenA 2 (t,a 1 ,a 2 )=(A). We first show that this holds fort=1, and then use induction to show that it holds for allt. Proposition 16 IfA 2 (t+1,a 1 ,a 2 )=(A), thenA 2 (t,a 1 ,a 2 )=(A) for allt<T . Proof: First we show that ifA 2 (2,a 1 ,a 2 )=(A), thenA 2 (1,a 1 ,a 2 )=(A). NoticeA 2 (2,a 1 ,a 2 )=(A) is equivalent to R 2 +V 2 (1,a 1 ,a 2 +1) ≥V 2 (1,a 1 ,a 2 ), andA 2 (1,a 1 ,a 2 )=(A) is 83 R 2 +V 1 (2B−(a 1 +a 2 +1)) ≥V 1 (2B−(a 1 +a 2 )). It is sufficient to show, therefore: V 1 (2B−(a 1 +a 2 +1))−V 2 (1,a 1 ,a 2 +1) ≥V 1 (2B−(a 1 +a 2 ))−V 2 (1,a 1 ,a 2 ). Since V 2 hasp 1 , p 2 , and (1−p 1 −p 2 ) terms, we restateV 1 (2b− (a 1 +a 2 +1)) as follows: V 1 (2B−(a 1 +a 2 +1)) =p 1 V 1 (2B−(a 1 +a 2 +1)) +p 2 V 1 (2B−(a 1 +a 2 +1)) +(1−p 1 −p 2 )V 1 (2B−(a 1 +a 2 +1)). The other three are V 2 (1,a 1 ,a 2 +1) =p 1 [R 1 +V 1 (2B−(a 1 +a 2 +2))] +p 2 max[R 2 +V 1 (2B−(a 1 +a 2 +2)),V 1 (2B−(a 1 +a 2 +1))] +(1−p 1 −p 2 )V 1 (2B−(a 1 +a 2 +1)), 84 V 1 (2B−(a 1 +a 2 )) =p 1 V 1 (2B−(a 1 +a 2 ))) +p 2 V 1 (2B−(a 1 +a 2 ))) +(1−p 1 −p 2 )V 1 (2B−(a 1 +a 2 ))), V 2 (1,a 1 ,a 2 ) =p 1 [R 1 +V 1 (2B−(a 1 +a 2 +1))] +p 2 max[R 2 +V 1 (2B−(a 1 +a 2 +1)),V 1 (2B−(a 1 +a 2 ))] +(1−p 1 −p 2 )V 1 (2B−(a 1 +a 2 )). SinceV 1 is concave,p 1 and(1−p 1 −p 2 ) terms satisfy the necessary inequality. For p 2 terms, we know that an optimal policy is a threshold type in Week 1, so we have only to consider the following three cases: Case I: A 2 (1,a 1 ,a 2 +1)=(A) A 2 (1,a 1 ,a 2 )=(A). (10.11) Then V 1 (2B−(a 1 +a 2 +1))−[R 2 +V 1 (2B−(a 1 +a 2 +2))] ≥V 1 (2B−(a 1 +a 2 ))−[R 2 +V 1 (2B−(a 1 +a 2 +1))]. 85 So we have the desired result. Case II: A 2 (1,a 1 ,a 2 +1)=(R) A 2 (1,a 1 ,a 2 )=(A). Then V 1 (2B−(a 1 +a 2 +1))−V 1 (2B−(a 1 +a 2 +1)) ≥0 ≥V 1 (2B−(a 1 +a 2 ))−[R 2 +V 1 (2B−(a 1 +a 2 +1))]. So we have the desired result fort=1. Next we employ induction to show that the result is true for allt; i.e., ifA 2 (t+1,a 1 ,a 2 )=(A), thenA 2 (t,a 1 ,a 2 )=(A). SinceA 2 (t+1,a 1 ,a 2 )=(A), R 2 +V 2 (t,a 1 ,a 2 +1)≥V 2 (t,a 1 ,a 2 ). We want to show R 2 +V 2 (t−1,a 1 ,a 2 +1)≥V 2 (t−1,a 1 ,a 2 ). Clearly, it is sufficient to show, for allt: V 2 (t+1,a 1 ,a 2 )−V 2 (t+1,a 1 ,a 2 +1) ≥V 2 (t,a 1 ,a 2 )−V 2 (t,a 1 ,a 2 +1), ∀a 1 ,a 2 . 86 We know the above holds for t=1. Assume it is true for some t, and we need to show that the property also holds fort+1, that is: V 2 (t+2,a 1 ,a 2 )−V 2 (t+2,a 1 ,a 2 +1) ≥V 2 (t+1,a 1 ,a 2 )−V 2 (t+1,a 1 ,a 2 +1), ∀a 1 ,a 2 . Notice V 2 (t+2,a 1 ,a 2 ) =p 1 [R 1 +V 2 (t+1,a 1 +1,a 2 )] +p 2 max[R 2 +V 2 (t+1,a 1 ,a 2 +1),V 2 (t+1,a 1 ,a 2 )] +(1−p 1 −p 2 )V 2 (2B−(t+1,a 1 ,a 2 ), V 2 (t+2,a 1 ,a 2 +1) =p 1 [R 1 +V 2 (t+1,a 1 +1,a 2 +1)] +p 2 max[R 2 +V 2 (t+1,a 1 ,a 2 +2),V 2 (t+1,a 1 ,a 2 +1)] +(1−p 1 −p 2 )V 2 (t+1,a 1 ,a 2 +1), V 2 (t+1,a 1 ,a 2 ) =p 1 [R 1 +V 2 (t,a 1 +1,a 2 )] +p 2 max[R 2 +V 2 (t,a 1 ,a 2 +1),V 2 (t,a 1 ,a 2 )] +(1−p 1 −p 2 )V 2 (2B−(t,a 1 ,a 2 ), 87 V 2 (t+1,a 1 ,a 2 +1) =p 1 [R 1 +V 2 (t,a 1 +1,a 2 +1)] +p 2 max[R 2 +V 2 (t,a 1 ,a 2 +2),V 2 (t,a 1 ,a 2 +1)] +(1−p 1 −p 2 )V 2 (t,a 1 ,a 2 +1). Therefore, terms corresponding top 1 and (1−p 1 −p 2 ) terms satisfy the induction requirements. Next consider terms corresponding to p 2 . We have the following six cases. By assumption, if we accept a Type 2 order at timet+2, then we accept a Type 2 order at timet+1. Case I: A 2 (t+2,a 1 ,a 2 )=(A) A 2 (t+2,a 1 ,a 2 +1)=(A) A 2 (t+1,a 1 ,a 2 )=(A) A 2 (t+1,a 1 ,a 2 +1)=(A). By assumption, we have R 2 +V 2 (t+1,a 1 ,a 2 +1)−[R 2 −V 2 (t+1,a 1 ,a 2 +2)] ≥R 2 +V 2 (t,a 1 ,a 2 +1)−[R 2 −V 2 (t,a 1 ,a 2 +2)]. Case II: 88 A 2 (t+2,a 1 ,a 2 )=(A) A 2 (t+2,a 1 ,a 2 +1)=(R) A 2 (t+1,a 1 ,a 2 )=(A) A 2 (t+1,a 1 ,a 2 +1)=(A). Since A 2 (t+1,a 1 ,a 2 )=(A),wehaveR 2 +V 2 (t,a 1 ,a 2 +2) ≥ V 2 (t,a 1 ,a 2 +1). Therefore we get: R 2 +V 2 (t+1,a 1 ,a 2 +1)−V 2 (t+1,a 1 ,a 2 +1) =R 2 ≥V 2 (t,a 1 ,a 2 +1)−V 2 (t,a 1 ,a 2 +2) =R 2 +V 2 (t,a 1 ,a 2 +1)−[R 2 −V 2 (t,a 1 ,a 2 +2)]. Case III: A 2 (t+2,a 1 ,a 2 )=(A) A 2 (t+2,a 1 ,a 2 +1)=(R) A 2 (t+1,a 1 ,a 2 )=(A) A 2 (t+1,a 1 ,a 2 +1)=(R). 89 R 2 +V 2 (t+1,a 1 ,a 2 +1)−V 2 (t+1,a 1 ,a 2 +1) =R 2 =R 2 +V 2 (t,a 1 ,a 2 +1)−[R 2 −V 2 (t,a 1 ,a 2 +2)]. Case IV: A 2 (t+2,a 1 ,a 2 )=(R) A 2 (t+2,a 1 ,a 2 +1)=(R) A 2 (t+1,a 1 ,a 2 )=(A) A 2 (t+1,a 1 ,a 2 +1)=(R). SinceA 2 (t+2,a 1 ,a 2 )=(R),wehaveR 2 +V 2 (t+1,a 1 ,a 2 +1)≤V(t+1,a 1 ,a 2 ). Consequently, V 2 (t+1,a 1 ,a 2 )−V 2 (t+1,a 1 ,a 2 +1) ≥R 2 =R 2 +V 2 (t,a 1 ,a 2 +1)−V 2 (t,a 1 ,a 2 +1). Case V: 90 A 2 (t+2,a 1 ,a 2 )=(R) A 2 (t+2,a 1 ,a 2 +1)=(R) A 2 (t+1,a 1 ,a 2 )=(A) A 2 (t+1,a 1 ,a 2 +1)=(A). SinceA 2 (t+1,a 1 ,a 2 +1)=(A),wehaveR 2 +V 2 (t,a 1 ,a 2 +2)≥V 2 (t,a 1 ,a 2 +1), yielding, V 2 (t+1,a 1 ,a 2 )−V 2 (t+1,a 1 ,a 2 +1) ≥R 2 +V 2 (t+1,a 1 ,a 2 +1)−V 2 (t+1,a = ,a 2 +1) ≥R 2 ≥R 2 +V 2 (t,a 1 ,a 2 +1)−V 2 (t,a 1 ,a 2 +2). Case VI: A 2 (t+2,a 1 ,a 2 )=(R) A 2 (t+2,a 1 ,a 2 +1)=(R) A 2 (t+1,a 1 ,a 2 )=(R) A 2 (t+1,a 1 ,a 2 +1)=(R). This is trivially true. 91 Accept a Type 2 order Reject a1 a2 Figure 10.3: Type 2 Acceptance Region Nested Policy is Optimal. In addition to monotonicity, the optimal policy for a 2-class problem turns out to be a nested policy. For each Day t, there exists a constantk(t) such that the optimal policy is to accept a Type 2 order if a 1 + a 2 ≤ k. In other words, whether or not an order is accepted depends only on the total number of orders accepted. Nested policies are appealing because they are very easy to implement. Figure 10.3 illustrates the policy. If the nested policy is optimal, then the only thing that matters is the total number of orders accepted and not how these orders are split between the classes. The following proposition shows that a nested policy is optimal. Proposition 17 (A nested policy is optimal.) If A 2 (t,a 1 ,a 2 )= (A) then A 2 (t,a 1 − 1,a 2 +1)=(A), and ifA 2 (t,a 1 ,a 2 )=(A) thenA 2 (t,a 1 +1,a 2 −1) = (A). 92 Proof: We establish only the first if-then clause. The proof for the second implication is identical. A 2 (t,a 1 ,a 2 )= A 2 (t,a 1 −1,a 2 +1) is obviously true ift=1.Fort> 1,we want to prove that if R 2 +V 2 (t− 1,a 1 ,a 2 +1) ≥ V 2 (t− 1,a 1 ,a 2 ), then R 2 +V 2 (t− 1,a 1 −1,a 2 +2)≥V 2 (t−1,a 1 −1,a 2 +1). It is sufficient to show: V 2 (t,a 1 −1,a 2 +2)−V 2 (t,a 1 ,a 2 +1) ≥V 2 (t,a 1 −1,a 2 +1)−V 2 (t,a 1 ,a 2 ) ∀t,a 1 ,a 2 . (10.12) Suppose, for somet, we have 10.12. We want to show V 2 (t+1,a 1 −1,a 2 +2)−V 2 (t+1,a 1 ,a 2 +1) ≥V 2 (t+1,a 1 −1,a 2 +1)−V 2 (t+1,a 1 ,a 2 ). As we have seen above, V 2 (t+1,a 1 −1,a 2 +2) =p 1 [R 1 +V 2 (t,a 1 ,a 2 +2)] +p 2 max[R 2 +V 2 (t,a 1 −1,a 2 +3),V 2 (t,a 1 −1,a 2 +2)] +(1−p 1 −p 2 )V 2 (t,a 1 −1,a 2 +2), 93 V 2 (t+1,a 1 ,a 2 +1) =p 1 [R 1 +V 2 (t,a 1 +1,a 2 +1)] +p 2 max[R 2 +V 2 (t,a 1 ,a 2 +2),V 2 (t,a 1 ,a 2 +1)] +(1−p 1 −p 2 )V 2 (t,a 1 ,a 2 +1), V 2 (t+1,a 1 −1,a 2 +1) =p 1 [R 1 +V 2 (t,a 1 ,a 2 +1)] +p 2 max[R 2 +V 2 (t,a 1 −1,a 2 +2),V 2 (t,a 1 −1,a 2 +1)] +(1−p 1 −p 2 )V 2 (t,a 1 −1,a 2 +1), and V 2 (t+1,a 1 ,a 2 ) =p 1 [R 1 +V 2 (t,a 1 +1,a 2 )] +p 2 max[R 2 +V 2 (t,a 1 ,a 2 +1),V 2 (t,a 1 ,a 2 )] +(1−p 1 −p 2 )V 2 (t,a 1 ,a 2 ). Terms corresponding to p 1 and (1−p 1 −p 2 ) meet our requirement. For p 2 terms, we need to analyze the following three cases. Case I: 94 A 2 (t+1,a 1 −1,a 2 +2)=(A) A 2 (t+1,a 1 ,a 2 +1)=(A) A 2 (t+1,a 1 −1,a 2 +1)=(A) A 2 (t+1,a 1 ,a 2 )=(A). Case II: A 2 (t+1,a 1 −1,a 2 +2)=(R) A 2 (t+1,a 1 ,a 2 +1)=(R) A 2 (t+1,a 1 −1,a 2 +1)=(A) A 2 (t+1,a 1 ,a 2 )=(A). Case III: A 2 (t+1,a 1 −1,a 2 +2)=(R) A 2 (t+1,a 1 ,a 2 +1)=(R) A 2 (t+1,a 1 −1,a 2 +1)=(R) A 2 (t+1,a 1 ,a 2 )=(R). Cases I and III clearly satisfy the property. For case II, notice that both sides are equal. So overall, we have (10.12). Thus, we see that if there are only two customer classes, we end up with a policy that is easy to implement. 95 0 500 1000 1500 2000 2500 3000 3500 4000 4500 1234 5 671 2345 6712 3456 7123 4567 Day Vw(t, 3, 3) Week 2 Week 3 Week 4 Week 5 Figure 10.4: A typical value function 10.2 Discussion 10.2.1 Acceptance Region In this subsection we use numerical examples to illustrate the structure of the optimal policy and the reward function. We also study the impact of the assumption that all orders are of unit size. Figure 10.4 shows the relationship between the optimal expected reward and time. Probability and reward parameters arep 1 =0.3,p 2 =0.4,R 1 = 350, and R 2 = 100, respectively. The probabilities and rewards are time homogeneous. Weekly capacity isB=5. As we can expect, the value of capacity is monotone increas- ing in the time remaining till the end of the horizon. Figure 10.4, shows that there is a sharp “jump” in the value at the beginning of each week. The start of a week results in an additional batch, or an additional week worth of capacity becoming available. The magnitude of this jump depends on the arrival rate. Lower arrival rates decrease the incremental value. In Figure 10.5, p 1 =0.1 and p 2 =0.1 and rewards are identical to 96 0 200 400 600 800 1000 1200 1400 1600 1800 12345671234567 12345671234567 Day Vw(t, 3, 3) Week 2 Week 3 Week 4 Week 5 Figure 10.5: When arrivals are slow the previous example; i.e., R 1 = 350 and R 2 = 100. Observe a decrease in the jump size. We also expect value function to be monotone in capacity available. Figure 10.6 illustrates this by comparing two situations. Probabilities and rewards are the same as in Figure 10.4. In one of the two cases, there are three Type 1 and three Type 2 orders accepted. In the other, there are three Type 1 and four Type 2 orders. The gap between the two graphs determines whether or not we accept Type 2 orders. Clearly, if the gap between these two graphs is less thanR 2 , then it is optimal to accepts a Type 2 orders. Figure 10.7 focuses on Week 5. Notice that the gap between the two graphs is increasing in time. As we go further away from the end of the horizon the gap increases. This implies the optimal policy is more likely to reject a type 2 order as we go further away from the end of the horizon. 97 0 500 1000 1500 2000 2500 3000 3500 4000 4500 1234567123456712345671234567 Day Dollars Vw(t, 3, 3) Vw(t, 3, 4) Week 2 Week 3 Week 4 Week 5 Figure 10.6: Value functions Figure 10.8 shows the area in which a Type 2 order is accepted at the beginning of Week 2. Parameters we use here are p 1 =0.3, p 2 =0.4, R 1 = 200, and R 2 = 180. Figure 10.9 shows how the acceptance region changes as time goes by. On Days 6 and 7, since there is more time, it is optimal to accept a Type 2 order ifa 1 +a 2 ≤ 4. As time elapses, the acceptance region expands. On Days 1, 2, and 3, it is optimal to accept a Type 2 order ifa 1 +a 2 ≤ 7. Observe that this policy has a simple structure and is easy to implement. 10.2.2 Value of Shifting An interesting feature of our model as compared to traditional yield management models such as that of Littlewood (1972), is that we can produce ahead of time. In Littlewood’s model, a Type 2 order must be produced only in the week it is due. Figure 10.10 presents the value of having an option to produce earlier. Parameters used in this example are: p 1 =0.3, p 2 =0.4, R 1 = 350, and R 2 = 100. All the data points assume that there are three Type 1 orders and three Type 2 orders accepted. Figure 10.11 focuses on the 98 3700 3750 3800 3850 3900 3950 4000 4050 4100 4150 4200 1234567 Day Dollars V5(t, 3, 3) V5(t, 3, 4) Figure 10.7: Value functions in Week 5 0 1 2 3 4 5 01 23 45 Number of Type 1 orders accepted Number of Type 2 orders acceptd Figure 10.8: Acceptance region in Week 2 99 Day 1, 2, and 3 Day 4 and 5 Day 6 and 7 0 1 2 3 4 5 6 7 8 012 3 45 Number of Type 1 orders accepted Number of Type 2 orders accepted Figure 10.9: Acceptance regions in Week 2 difference between these two functions on Day 7 of Weeks 2, 3, 4, and 5, respectively. As the horizon becomes longer, there are more opportunities to exercise this option, leading to larger differences between the two expected reward functions. As the arrival rate decreases, however, this benefit diminishes. Figure 10.12 shows the value of shifting whenp 1 =0.1 andp 2 =0.133, while the other parameters remain the same. We also find that if the total number of accepted orders is held constant, as the number of Type 2 orders increases, so does the value of being able to shift production. Figure 10.13 illustrates this when p 1 =0.3, p 2 =0.4, R 1 = 350, R 2 = 100, and the total number of accepted orders is 6. As the number of accepted Type 2 order increases, the value of being able to shift production increases. 100 0 500 1000 1500 2000 2500 3000 3500 4000 4500 1234567123456712345671234567 Day Expected accumulated reward No shifting Shifting Week 2 Week 3 Week 4 Week 5 Figure 10.10: Value of being able to produce earlier: p 1 =0.3,p 2 =0.4,R 1 = 350 Table 10.2: Frequencies of Early Production Starting state Week 3 Week 4 Week 5 BType 1 orders accepted = 0 BType 2 orders accepted = 0 0.415 0.321 0 BType 1 orders accepted = 2 BType 2 orders accepted = 3 0.545 0.565 0.691 10.2.3 Frequency of Early Production We employed Monte Carlo simulations to understand how often “shifting” occurs. We find that higher utilization levels result in more frequent early production. Table 10.2 summarizes the result of the simulations. We continue to use the set of parameters p 1 =0.3,p 2 =0.4,R 1 = 350, andR 2 = 100. In one experiment, no orders have been accepted at the start of the horizon. In the other experiment we assume that two Type 1 and three Type 2 orders have been accepted at the beginning of the horizon. 101 0 50 100 150 200 250 300 350 400 450 23 45 #weeks Difference ($) Figure 10.11: Value of being able to produce earlier: p 1 =0.3, p 2 =0.4, R 1 = 350, R 2 = 100 10.2.4 Impact of Group Arrivals One crucial assumption we have is that orders arrive one at a time. We use simulation to study the significance of this assumption. We compute the optimal policy assuming that all orders are of unit size, but in the simulation we permit arriving customers to order more than one unit. If the number of accepted orders is such that the optimal policy permits a type 2 policy to be accepted, we permit the firm to accept a Type 2 order even if it is for more than one unit, provided the capacity constraint is not violated. This distortion of the optimal policy is consequential only if we accept more type 2 orders than the optimal policy would permit us to accept if all orders were of unit size. The parameters employed in this simulation are as follows: p 1 =0.3, p 2 =0.4,R 1 = 350, andR 2 = 100. We first employ simulations to determine the average revenue generated during the weeks of 3 to 5, using the optimal acceptance policy, assuming all the orders are of the same size. We next allow group arrivals. We assume that each Type 2 order is for either one or two units. In the new scenario we continue to assume thatp 1 is 0.3, but we alterp 2 . We letp 2 =0.2828. Furthermore, if a Type 2 order arrives, there isp 2 102 0 200 400 600 800 1000 1200 1400 1600 1800 123456712 3456712345 671234567 Day Expected accumulated reward No shifting Shifting Week 2 Week 3 Week 4 Week 5 Figure 10.12: Value of being able to produce earlier: p 1 =0.1,p 2 =0.133 Table 10.3: Accumulated Revenue during Week 3-5 (in dollars) Starting state Equal size Batch Percentage assumption assumption loss BType 1 orders accepted = 0 BType 2 orders accepted = 0 3,004 2,899 3.5% BType 1 orders accepted = 1 BType 2 orders accepted = 1 2,819 2,789 1% BType 1 orders accepted = 3 BType 2 orders accepted = 3 2,293 2,255 1.6% BType 1 orders accepted = 4 BType 2 orders accepted = 4 1,743 1,721 1.2% probability that its size is two and1−p 2 that it is one. If the firm accepts a Type 2 order of the larger size, it generates twice the revenue and consumes twice the capacity. By alteringp 2 this way, we ensure the arrival rate in measured in dollar value remains the same. Surprisingly, as we can see in Table 10.3, our policy performs very well even if we allow group arrivals. There are two types of inefficiency that may occur by using the optimal policy for the problem in which we assume all the orders are of the same size. First, we may accept 103 0 200 400 600 800 1000 1200 1400 1600 1800 2000 012 3 45 Number of Type 2 orders accepted Difference ($) Figure 10.13: Value of being able to produce earlier when there are 6 total accepted orders an order that forces the state space to go beyond the boundary of the acceptance region. We keep track of the number of times we exceed the acceptance region. The other inefficiency comes from cases in which we cannot accept a larger sized order because of the capacity constraint. We may have to reject the entire order and not accept just one unit. We also keep track of the number of times we reject a larger sized order because of capacity constraints. We simulated a 3-week problem. We find that that in 40% of the cases neither of these violations is observed. In cases in which the acceptance region is violated, the violation occurs on average about twice over a 3 week horizon. Furthermore, in our simulation the capacity constraint is never violated. We conjecture that the severity of these assumptions depends on the size of an arrival batch relative to the total capacity and the overall utilization rate. We next analyze the 3-class problem. Although the nested policy is not optimal, the optimal policy continues to be a threshold policy and the thresholds are monotone in time. 104 10.3 Three-Class Demand Problem We now let V w (t,a 1 ,a 2 ,a 3 ) denote the optimal expected payoff if there are a 1 ,a 2 ,a 3 orders on Day t of Week w, and A w (t,a 1 ,a 2 ,a 3 ,n) denote the corresponding optimal action when a Type n order arrives. Possible action is either (A)ccept or (R)eject. We continue to assumeR n andp n are non-homogeneous. 10.3.1 Week 1 In Week 1, the problem is essentially a standard stochastic knapsack. LetV 1 (b) denote the optimal value function when there areb units of capacity remaining at the beginning of Week 1. We know the value function is concave increasing in remaining capacity, that is,V 1 (b+1)−V 1 (b)≥V 1 (b+2)−V 1 (b+1),∀b≥ 0. 10.3.2 Week 2 On Day 1 of Week 2, after either accepting or rejecting an order, if any, the firm produces any Type 2 and 3 orders that it can produce, and start Week 1 with capacity min(2B− (a 1 +a 2 +a 3 ),B). Proposition 18 Suppose there are a 1 Type 1 orders, a 2 Type 2 orders, and a 3 Type 3 orders accepted at the end of Week 2, then the firm’s optimal production schedule for Week 1 is to produceMin(a 1 +a 2 +a 3 ,B). Therefore, at the beginning of Week 1, the firm’s available capacity isMin(2B−(a 1 +a 2 +a 3 ),B). The above proposition is true because V 1 is concave increasing in the remaining capacity. Next, we show that the optimal policy is a threshold policy. These conditions establish that the acceptance policies remain the same if the available capacity is either more or the same. For example the first condition states that if it is optimal to accept a 105 Type 2 order for a particular state space, then it is also optimal to accept a type 2 order when the firm has fewer type 2 orders on its books. The third and the fourth condition shows that the acceptance policy for Type 2 orders depends only on the total number of Type 1 and Type 2 orders accepted to date, if we hold the number of Type 3 orders fixed. So we have partial nesting. Proposition 19 An optimal policy has the following threshold properties. (i) IfA 2 (t,a 1 ,a 2 ,a 3 ,2) = (A), thenA 2 (t,a 1 ,a 2 −1,a 3 ,2) = (A). (ii) IfA 2 (t,a 1 ,a 2 ,a 3 ,2) = (A), thenA 2 (t,a 1 ,a 2 ,a 3 −1,2) = (A). (iii) IfA 2 (t,a 1 ,a 2 ,a 3 ,2) = (A), thenA 2 (t,a 1 +1,a 2 −1,a 3 ,2) = (A). (iv) IfA 2 (t,a 1 ,a 2 ,a 3 ,2) = (A), thenA 2 (t,a 1 −1,a 2 +1,a 3 ,2) = (A). (v) IfA 2 (t,a 1 ,a 2 ,a 3 ,2) = (A), thenA 2 (t,a 1 −1,a 2 ,a 3 +1,2) = (A). (vi) IfA 2 (t,a 1 ,a 2 ,a 3 ,2) = (A), thenA 2 (t,a 1 ,a 2 −1,a 3 +1,2) = (A). Proof: We first show (i) fort=1: IfR 2 +V 1 (min(2B−(a 1 +a 2 +a 3 +1),B))≥V 1 (min(2B−(a 1 +a 2 +a 3 ),B)), thenR 2 +V 1 (min(2B−(a 1 +a 2 +a 3 ),B))≥V 1 (min(2B−(a 1 +a 2 +a 3 −1),B)). This is clearly true sinceV 1 is concave. Similarly, (ii)-(vi) hold fort=1. Next step is to show all the above hold for any day of Week 2. For (i) we want to show that ifR 2 +V 2 (t,a 1 ,a 2 +1,a 3 )≥ V 2 (t,a 1 ,a 2 ,a 3 ), thenR 2 +V 2 (t,a 1 ,a 2 ,a 3 )≥ V 2 (t,a 1 ,a 2 −1,a 3 ), for allt. Clearly, it is sufficient to show: V 2 (t,a 1 ,a 2 ,a 3 )−V 2 (t,a 1 ,a 2 +1,a 3 ) ≥V 2 (t,a 1 ,a 2 −1,a 3 )−V 2 (t,a 1 ,a 2 ,a 3 ) ∀a 1 ,a 2 , 106 for allt. LetV 2 (0,a 1 ,a 2 ,a 3 )≡V 1 (min(2B−(a 1 +a 2 +a 3 ),B)). The above holds fort=0. Assume that the above inequality holds for some t, and we would like to show that it holds fort+1, i.e., V 2 (t+1,a 1 ,a 2 ,a 3 )−V 2 (t+1,a 1 ,a 2 +1,a 3 ) ≥V 2 (t+1,a 1 ,a 2 −1,a 3 )−V 2 (t+1,a 1 ,a 2 ,a 3 ) ∀a 1 ,a 2 . Notice: V 2 (t+1,a 1 ,a 2 ,a 3 ) =p 1 [R 1 +V 2 (t,a 1 +1,a 2 ,a 3 )] +p 2 max[R 2 +V 2 (t,a 1 ,a 2 +1,a 3 ),V 2 (t,a 1 ,a 2 ,a 3 )] +p 3 max[R 3 +V 2 (t,a 1 ,a 2 ,a 3 +1),V 2 (t,a 1 ,a 2 ,a 3 )] +(1−p 1 −p 2 −p 3 )V 2 (t,a 1 ,a 2 ,a 3 ), V 2 (t+1,a 1 ,a 2 +1,a 3 ) =p 1 [R 1 +V 2 (t,a 1 +1,a 2 +1,a 3 )] +p 2 max[R 2 +V 2 (t,a 1 ,a 2 +2,a 3 ),V 2 (t,a 1 ,a 2 +1,a 3 )] +p 3 max[R 3 +V 2 (t,a 1 ,a 2 +1,a 3 +1),V 2 (t,a 1 ,a 2 +1,a 3 )] +(1−p 1 −p 2 −p 3 )V 2 (t,a 1 ,a 2 +1,a 3 ), 107 and V 2 (t+1,a 1 ,a 2 −1,a 3 ) =p 1 [R 1 +V 2 (t,a 1 +1,a 2 −1,a 3 )] +p 2 max[R 2 +V 2 (t,a 1 ,a 2 ,a 3 ),V 2 (t,a 1 ,a 2 −1,a 3 )] +p 3 max[R 3 +V 2 (t,a 1 ,a 2 −1,a 3 +1),V 2 (t,a 1 ,a 2 −1,a 3 )] +(1−p 1 −p 2 −p 3 )V 2 (t,a 1 ,a 2 −1,a 3 ). By assumption, we have, for thep 1 terms, R 1 +V 2 (t,a 1 +1,a 2 ,a 3 )−[R 1 +V 2 (t,a 1 +1,a 2 +1,a 3 )] ≥R 1 +V 2 (t,a 1 +1,a 2 −1,a 3 )−[R 1 +V 2 (t,a 1 +1,a 2 ,a 3 )]. Forp 2 terms, by Proposition 19, there are four cases to examine. Case I: A 2 (t+1,a 1 ,a 2 ,a 3 ,2) = (A) A 2 (t+1,a 1 ,a 2 +1,a 3 ,2) = (A) A 2 (t+1,a 1 ,a 2 −1,a 3 ,2) = (A). By assumption, R 2 +V 2 (t,a 1 ,a 2 +1,a 3 )−R 2 +V 2 (t,a 1 ,a 2 +2,a 3 ) ≥R 2 +V 2 (t,a 1 ,a 2 ,a 3 )−[R 2 +V 2 (t,a 1 ,a 2 +1,a 3 )]. Case II: 108 A 2 (t+1,a 1 ,a 2 ,a 3 ,2) = (A) A 2 (t+1,a 1 ,a 2 +1,a 3 ,2) = (R) A 2 (t+1,a 1 ,a 2 −1,a 3 ,2) = (A). SinceR 2 ≥V 2 (t,a 1 ,a 2 ,a 3 )−V 2 (t,a 1 ,a 2 +1,a 3 ), by assumption, we have: R 2 +V 2 (t,a 1 ,a 2 +1,a 3 )−V 2 (t,a 1 ,a 2 +1,a 3 ) =R 2 ≥V 2 (t,a 1 ,a 2 ,a 3 )−V 2 (t,a 1 ,a 2 +1,a 3 ) =R 2 +V 2 (t,a 1 ,a 2 ,a 3 )−[R 2 +V 2 (t,a 1 ,a 2 +1,a 3 )]. Thus we have the desired result. Case III: A 2 (t+1,a 1 ,a 2 ,a 3 ,2) = (R) A 2 (t+1,a 1 ,a 2 +1,a 3 ,2) = (R) A 2 (t+1,a 1 ,a 2 −1,a 3 ,2) = (A). V 2 (t,a 1 ,a 2 ,a 3 )−V 2 (t,a 1 ,a 2 +1,a 3 ) ≥R 2 +V 2 (t,a 1 ,a 2 +1,a 3 )−V 2 (t,a 1 ,a 2 +1,a 3 ) =R 2 =R 2 +V 2 (t,a 1 ,a 2 ,a 3 )−V 2 (t,a 1 ,a 2 ,a 3 ). 109 Case IV: A 2 (t+1,a 1 ,a 2 ,a 3 ,2) = (R) A 2 (t+1,a 1 ,a 2 +1,a 3 ,2) = (R) A 2 (t+1,a 1 ,a 2 −1,a 3 ,2) = (R). By assumption we have V 2 (t,a 1 ,a 2 ,a 3 )−V 2 (t,a 1 ,a 2 +1,a 3 ) ≥V 2 (t,a 1 ,a 2 −1,a 3 )−V 2 (t,a 1 ,a 2 ,a 3 ). Thus the p 2 terms also have the desired property. Next we consider the p 3 terms. Once again there are four cases. Case I: A 2 (t+1,a 1 ,a 2 ,a 3 ,3) = (A) A 2 (t+1,a 1 ,a 2 +1,a 3 ,3) = (A) A 2 (t+1,a 1 ,a 2 −1,a 3 ,3) = (A). By assumption, R 3 +V 2 (t,a 1 ,a 2 ,a 3 +1)−[R 3 +V 2 (t,a 1 ,a 2 +1,a 3 +1)] ≥R 3 +V 2 (t,a 1 ,a 2 −1,a 3 +1)−[R 3 +V 2 (t,a 1 ,a 2 ,a 3 +1)]. So we have the desired property. 110 Case II: A 2 (t+1,a 1 ,a 2 ,a 3 ,3) = (A) A 2 (t+1,a 1 ,a 2 +1,a 3 ,3) = (R) A 2 (t+1,a 1 ,a 2 −1,a 3 ,3) = (A). Using Proposition 19, we have R 3 +V 2 (t,a 1 ,a 2 ,a 3 +1)−V 2 (t,a 1 ,a 2 +1,a 3 ) ≥R 3 +V 2 (t,a 1 ,a 2 −1,a 3 +1)−V 2 (t,a 1 ,a 2 ,a 3 ) ≥R 3 +V 2 (t,a 1 ,a 2 −1,a 3 +1)−[R 3 +V 2 (t,a 1 ,a 2 ,a 3 +1)]. Case III: A 2 (t+1,a 1 ,a 2 ,a 3 ,3) = (R) A 2 (t+1,a 1 ,a 2 +1,a 3 ,3) = (R) A 2 (t+1,a 1 ,a 2 −1,a 3 ,3) = (A). Using Proposition 19, V 2 (t,a 1 ,a 2 ,a 3 )−V 2 (t,a 1 ,a 2 +1,a 3 ) ≥R 3 +V 2 (t,a 1 ,a 2 ,a 3 +1)−V 2 (t,a 1 ,a 2 +1,a 3 ) ≥R 3 +V 2 (t,a 1 ,a 2 −1,a 3 +1)−V 2 (t,a 1 ,a 2 ,a 3 ). Case IV: 111 A 2 (t+1,a 1 ,a 2 ,a 3 ,3) = (R) A 2 (t+1,a 1 ,a 2 +1,a 3 ,3) = (R) A 2 (t+1,a 1 ,a 2 −1,a 3 ,3) = (R). By assumption, V 2 (t,a 1 ,a 2 ,a 3 )−V 2 (t,a 1 ,a 2 +1,a 3 ) ≥V 2 (t,a 1 ,a 1 −1,a 3 )−V 2 (t,a 1 ,a 2 ,a 3 ). And we have the desired result. For (1−p 1 −p 2 −p 3 ) terms, we have, V 2 (t,a 1 ,a 2 ,a 3 )−V 2 (t,a 1 ,a 2 +1,a 3 ) ≥V 2 (t,a 1 ,a 1 −1,a 3 )−V 2 (t,a 1 ,a 2 ,a 3 ). This completes the proof of (i) for allt. Similarly, we can show (ii)-(vi) for allt. Next consider Type 3 orders. As the following proposition shows, the optimal accep- tance policy for Type 3 orders is analogous to that for Type 2 orders. Proposition 20 An optimal policy has the following threshold properties. (i) IfA 2 (t,a 1 ,a 2 ,a 3 ,3) = (A), thenA 2 (t,a 1 ,a 2 −1,a 3 ,3) = (A) for allt. (ii) IfA 2 (t,a 1 ,a 2 ,a 3 ,3) = (A), thenA 2 (t,a 1 ,a 2 ,a 3 −1,3) = (A) for allt. (iii) IfA 2 (t,a 1 ,a 2 ,a 3 ,3) = (A), thenA 2 (t,a 1 +1,a 2 −1,a 3 ,3) = (A) for allt. (iv) IfA 2 (t,a 1 ,a 2 ,a 3 ,3) = (A), thenA 2 (t,a 1 −1,a 2 +1,a 3 ,3) = (A) for allt. 112 (v) IfA 2 (t,a 1 ,a 2 ,a 3 ,3) = (A), thenA 2 (t,a 1 +1,a 2 ,a 3 −1,3) = (A) for allt. (vi) IfA 2 (t,a 1 ,a 2 ,a 3 ,3) = (A), thenA 2 (t,a 1 ,a 2 +1,a 3 −1,3) = (A) for allt. 10.3.3 Week 3 To complete the analysis, we need to show that the properties hold for Week 3 and beyond. It turns out that all the properties from Week 2 carry over to Week 3. For Type 2 orders, we have the following proposition. Proposition 21 An optimal policy has the following threshold properties. (i) IfA 3 (t,a 1 ,a 2 ,a 3 ,2) = (A), thenA 3 (t,a 1 ,a 2 −1,a 3 ,2) = (A), for allt. (ii) IfA 3 (t,a 1 ,a 2 ,a 3 ,2) = (A), thenA 3 (t,a 1 ,a 2 ,a 3 −1,2) = (A) for allt. (iii) IfA 3 (t,a 1 ,a 2 ,a 3 ,2) = (A), thenA 3 (t,a 1 +1,a 2 −1,a 3 ,2) = (A) for allt. (iv) IfA 3 (t,a 1 ,a 2 ,a 3 ,2) = (A), thenA 3 (t,a 1 −1,a 2 +1,a 3 ,2) = (A) for allt. (v) IfA 3 (t,a 1 ,a 2 ,a 3 ,2) = (A), thenA 3 (,a 1 −1,a 2 ,a 3 +1,2) = (A) for allt. (vi) IfA 3 (t,a 1 ,a 2 ,a 3 ,2) = (A), thenA 3 (t,a 1 ,a 2 −1,a 3 +1,2) = (A) for allt. Proof: First we show that (i) holds for t=1, that is, if A 3 (1,a 1 ,a 2 ,a 3 ,2) = (A), then A 3 (1,a 1 ,a 2 −1,a 3 ,2) = (A). There are three cases to consider. Case I: Ifa 1 +a 2 >B, then we want to show: 113 If R 2 +V 2 (T,a 1 +a 2 +1−B,a 3 ,0) ≥V 2 (T,a 1 +a 2 −B,a 3 ,0), then R 2 +V 2 (T,a 1 +a 2 −B,a 3 ,0) ≥V 2 (T,a 1 +a 2 −1−B,a 3 ,0). This is clearly true from Proposition 19. Case II: Ifa 1 +a 2 =B, then we want to show: If R 2 +V 2 (T,1,a 3 ,0) ≥V 2 (T,0,a 3 ,0), then R 2 +V 2 (T,0,a 3 ,0) ≥V 2 (T,0,max(a 3 −1,0),0). Ifa 3 ≥ 1, the above statement is true. (Note: ifR 2 +V 2 (T,1,a 3 ,0)≥V 2 (T,0,a 3 ,0), thenR 2 +V 2 (T,0,a 3 +1,0)≥V 2 (T,0,a 3 ,0).) Ifa 3 =0, then it is obvious. Case III: Ifa 1 +a 2 <B, then we want to show: If R 2 +V 2 (T,0,max(a 1 +a 2 +a 3 +1−B,0),0) ≥V 2 (T,0,max(a 1 +a 2 +a 3 −B,0),0), then R 2 +V 2 (T,0,max(a 1 +a 2 +a 3 −B,0),0) ≥V 2 (T,0,max(a 1 +a 2 +a 3 −1−B,0),0). The above is true because of Proposition 19. So (i) holds fort=1. It is straightfor- ward now to show the rest since the proof is essentially identical to Week 2 analysis. 114 By repeating the Week 2 analysis, we have, for Type 3 orders, Proposition 22 An optimal policy has the following threshold properties. (i) IfA 3 (t,a 1 ,a 2 ,a 3 ,3) = (A), thenA 3 (t,a 1 ,a 2 −1,a 3 ,3) = (A) for allt. (ii) IfA 3 (t,a 1 ,a 2 ,a 3 ,3) = (A), thenA 3 (t,a 1 ,a 2 ,a 3 −1,3) = (A) for allt. (iii) IfA 3 (t,a 1 ,a 2 ,a 3 ,3) = (A), thenA 3 (t,a 1 +1,a 2 −1,a 3 ,3) = (A) for allt. (iv) IfA 3 (t,a 1 ,a 2 ,a 3 ,3) = (A), thenA 3 (t,a 1 −1,a 2 +1,a 3 ,3) = (A) for allt. (v) IfA 3 (t,a 1 ,a 2 ,a 3 ,3) = (A), thenA 3 (t,a 1 +1,a 2 ,a 3 −1,3) = (A) for allt. (vi) IfA 3 (t,a 1 ,a 2 ,a 3 ,3) = (A), thenA 3 (t,a 1 ,a 2 +1,a 3 −1,3) = (A) for allt. Now it is easy to see we have the above two propositions hold for all weeksw ≥ 4 as well. Thus the threshold type policies that appear to be partially nested continue to be valid for all finite horizons. 10.4 Discussion Based on our analysis in previous sections, we can now solve a three-class problem recursively. Figure 10.14 illustrate typical value functions. Here probability and reward parameters are; p 1 =0.2, p 2 =0.3, p 3 =0.4, R 1 = 400, R 2 = 200, and R 3 = 100. These parameters do not depend on time. B is assumed to be 5. One graph in Figure 10.14 shows the value function when three orders of each type have been accepted. The other shows the value function when a 1 =3, a 2 =3, and a 3 =4. Of course, as the number of orders accepted increases,the available capacity decreases and the optimal value function decreases. 115 0 1000 2000 3000 4000 5000 6000 12 345 6 7 1 2 345 67 12 345 67 12 345 67 Day Dollars V3(t, 3, 3, 3) V3(t, 3, 3, 4) Week 3 Week 4 Week 5 Week 6 Figure 10.14: Value functions Figure 10.15 focuses on Week 6. As was the case in two-class problems, the differ- ence between the two functions is decreasing as time till the end of the horizon becomes shorter. This, again, implies that the optimal acceptance policy becomes lenient as time elapses. The two value functions in figure 10.14 differ only by one Type 3 order. The difference between the two curves is less thanR 3 , then it is optimal to accept a Type 3 order. In two-class problems, optimal policies are always nested. Whenever you accept a Type 2 order, you also accept a Type 1 order. We have seen in the previous section that the optimal policy need not be nested when there are three or more classes. Figure 10.16 illustrates a case in which nested policies are not optimal. Our extensive experiments suggest that we tend to see these “non-nested” optimal policies if the arrival rate of urgent orders is small and the rewards for less urgent orders is large. For the policy depicted in Figure 10.16the parameters are as follows: p 1 =0.1, p 2 =0.1, p 3 =0.4, R 1 = 400, R 2 = 200, and R 3 = 190. The number of accepted Type 1 orders is fixed at a 1 =5. The figure shows, on Day 5 of Week 5, that when a 2 =3 and a 3 =1, the 116 4900 4950 5000 5050 5100 5150 5200 5250 5300 5350 5400 12 345 6 7 Day Dollars V6(t, 3, 3, 3) V6(t, 3, 3, 4) Figure 10.15: Value functions in Week 6 optimal policy accepts a Type 3 order but not a Type 2 order. This type of non-nested policies are frequently observed in traditional yield management problems as well. 117 0 1 2 3 4 5 6 7 012345 a2 a3 Type 2 Acceptance Boundary Type 3 Acceptance Boundary Figure 10.16: Acceptance boundaries on Day 5, Week 5 118 Chapter 11 Summary In this section we analyzed the problem of dynamic capacity allocation and order accep- tance policies in a make-to-order environment. We modeled this problem as dynamic stochastic rolling knapsack problem. We show that if there are two classes of demand, then an optimal policy is of a nested type. These policies are very easy to implement. If we have three classes, however, the optimal policy need not be a nested policy. The optimal policy for a 3-class problem is more complex than that for the 2-class problem. Nevertheless, the 3-class problem is a threshold type policy. We contend that this work presents a novel approach to modeling order acceptance policies in a make-to-order environment. As we stated earlier, it differs from and has several advantages over queuing models that have traditionally been employed to model these environments. The biggest advantage is that we do not have to assume that the arrival process is stationary. We extend the yield management literature by allowing orders to be processed in advance. We show that threshold type policies are always opti- mal. This suggests that threshold policies that are simpler in structure than the optimal policies may perform well. In particular, it will be interesting to investigate the loss in optimality if we restrict ourselves to nested policies. We conjecture, just as in the revenue management problems, optimality of nested policies depends on the value of different orders and the relative arrival rates of different customer classes. 119 Part III Conclusions and Future Directions 120 In the first essay, we study truth-revealing compensation schemes for forecasters employed by a firm. We identify a scheme for partitioning the space of forecast dis- tributions into sets, using the notion of refinement. These sets can be rank ordered based on profit potential and we call these sets refinement levels. Probability distri- butions belonging to a refinement level are linearly independent and lie in the convex hull of distributions that are of higher refinement level. Because we are able to estab- lish nice connections between refinement ordering and the firm’s profitability, we can design compensation schemes in which forecasters with higher refinement level will receive higher expected payoffs. The detailed structure of the compensation schemes depends on: (i) whether or not the forecaster incurs costs in the short term, and (ii) whether or not the forecaster influ- ences the outcome. In all the cases truth-revealing schemes are a menu of contracts that are hyperplanes to some convex functional in density space. When forecasters do not incur costs, refinement levels can be used to rank them. We identify the limitations of profit sharing mechanisms in this case. Next, we develop compensations when forecast accuracy depends on the forecaster’s costly effort. We show that the decision of the firm to invest in multiple forecasters depends on how rapidly the cost of improving forecasts increases. Finally, we extend traditional salesforce compensation literature by developing com- pensation schemes for salespersons who also forecast. Unlike the traditional saleforce literature we permit the firm’s profits to depend on all the moments of the distribution, not just the mean. We assume that market conditions influence forecast uncertainty and the salesperson’s productivity. We show that if potential forecasts in each market regime belong to a refinement level then the firm can cause the salesperson to reveal the true regime and exert the appropriate level of effort. We also identify sufficient conditions in which the firm can offer a set of non-decreasing payoff functions to achieve truth-telling. 121 It will be highly interesting to investigate how the firm’s optimal compensation, salespersons’ corresponding effort levels, and expected payoffs are going to change in different scenarios. For example, our preliminary result shows that if the salesperson’s disutility functions and productivity functions are identical in both regimes and the only thing that is different is the forecast uncertainties, then the firm’s optimal compensation scheme is such that the salesperson makes the same level of effort in either regime. Although we assumed throughout the essay that the forecasted variable is demand. Our analysis can be extended to other forecasts as well. One interesting application would be a project manager estimating the duration of a project. Like salespersons, project managers can affect the forecasted variable – project duration. In the second essay, we study a capacity allocation and order acceptance problem faced by a make-to-order business that receives orders from customers who differ in their lead time requirement. Customers who desire quicker deliveries are also willing to pay more. We assume that deliveries occur at discrete time periods such as at the end of a working day/week. We formulate this problem as a dynamic rolling knapsack. We show that threshold policies are always optimal. When the number of customer classes is only two a nested threshold policy is optimal. For problems with three or more classes a nested policy need not be optimal. By modeling the problem as a rolling knapsack, we present a novel approach to modeling order acceptance policies in a make-to-order environment. Our advantage over traditional models is that we do not have to assume that the arrival process is stationary. We also extend the revenue management literature by allowing orders to be processed before their due dates. In our analysis we used simulation techniques to demonstrate the value of an early production option. We may be able to show analytically how this value depends on 122 parameters such as arrival rates, revenue associated with each type, and the number of accepted orders of each type. Also we showed, by simulated experiments, the effect of group arrivals. We would like to compare how the optimal policy for the current problem performs relative to the optimal policy for the problem with combinatorial consideration. We conjecture that if the arrival size is small compared to the periodic capacity, then our policy will perform well. As we stated in the essay, we are currently restricted to 2- and 3-class problems. Extending this to generalN-class problems is certainly of interest. We conjecture all the analytical results will hold in an N-class problem, but we may find additional insights in terms of value of early production. Another potential extension would be relaxing some of the assumptions we have here in the current model. 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Management Sci. 27(4) 479–488. 127 Appendices A.1 Truth-Telling Compensation Schemes In all the examples the cost of attaining a forecast distribution equals (mean of distribution) k i , where k i is a constant for regime i. The firm specifies the average demand to be achieved in each regime. Figures 6.2-(a), 6.2-(b), 6.3-(a), and 6.3-(b) assume uniform distributions (discrete outcomes) and two regimes. If regime 1 is realized, the probabilities are (0.2, 0.2, 0.2, 0.2, 0.2). In regime 2 the probabilities are (0.25, 0.25, 0.25, 0.25). In Figures 6.3-(c) and 6.3-(d), the underlying distributions are normal. Standard deviation of the distribution is 2 in regime 1 whereas 4 in regime 2. We discretize the normal distribution to solve for the compensation schemes. In Figure 6.4, there are three demand regimes. The underlying distributions are nor- mal with standard deviations of 2, 4, and 5, respectively. Table 11.1 gives the remaining parameters for each of the figures. 128 Table 11.1: Parameters Average Average Average Figure# Demand 1 Demand 2 Demand 3 k 1 k 2 k 3 2(a) 12 12 1.2 1.2 2(b) 5 15 1.2 1.2 3(a) 12 12 1.2 1.2 3(b) 5 15 1.2 1.2 3(c) 12 12 1.2 1.2 3(d) 18 12 1.2 1.2 4(a) 20 20 20 1.2 1.2 1.2 4(b) 15 20 15 1.2 1.2 1.2 4(c) 15 20 15 1.1 1.2 1.3 4(d) 15 20 15 1.3 1.1 1.1 A.2 Proofs Proof of Lemma 1 Let f H,s (d) and f L,t (d), s,t ∈ R, denote density functions in χ(H) and χ(L), respec- tively. If the forecasters are both well-calibrated, their forecasts are, on average, the same. So we have f H,s (d)g H (s)ds = f L,t (d)g L (t)dt ∀d, (A-1) where g H (·) and g L (·) are density functions for the event that the forecast has a density f H,s (·) and f L,t (·), respectively. In addition, by definition of level sets, there exists another densityα t (s), for anyt, such that f L,t (d)= f H,s (d)α t (s)ds. (A-2) Plugging (A-2) into (A-1), 129 f H,s (d)g H (s)ds = f H,s (d)α t (s)ds g L (t)dt = f H,s (d) α t (s)g L (t)dt ds, for alld. Since allf H,s (·)’s are linearly independent, it follows that g H (s)= α t (s)g L (t)dt. (A-3) Now, letEΠ(H) andEΠ(L) denote the optimal expected profit when forecasts are from χ(H) and χ(L), respectively, and π(d,γ) the profit when the outcome is d given the firm’s decisionγ. EΠ(L)= max γ π(d,γ)f L,t (d)dd g L (t)dt = max γ π(d,γ) f H,s (d)α t (s)dsdd g L (t)dt ≤ max γ π(d,γ)f H,s (d)dd α t (s)dsg L (t)dt = max γ π(d,γ)f H,s (d)dd α t (s)g L (t)dtds = max γ π(d,γ)f H,s (d)dd α t (s)g L (t)dtds = max γ π(d,γ)f H,s (d)dd g H (s)ds =EΠ(H), where the second from the last equality follows from (A-3). The proof is complete. 130 Proof of Proposition 1 We first show that ifY =ϑX,ϑ> 1, then there exists a densityα(m) such that f Y (d)= f X (d−m)α(m)dm. Without loss of generality, assumeEX =0. It is known that X is smaller than Y in the convex order ifY = ϑX,ϑ> 1. (See, for instance, Theorem 2.A.18 of Shaked and Shanthikumar, 2007.) By Theorem 2 of Rothschild and Stiglitz (1970), there exists another random variable, call itM, such that Y = ST X +M. Therefore we have f Y (x)= f X (x−m)α(m)dm, where α(m) is the density of M. So we have that the density of Y is a convex combination of densities of lowerϑ. Next we show that for a fixedϑ, densities with differentM are linearly independent. Considerf Y (x−m),m∈ R the densities with a fixedϑ and differentm. If they are not linearly independent, there exists a measurable functionα(m) such that f Y (x−m)α(m)dm=0 ∀x. (A-4) This implies f Y (x−z)α(z)dz=0 ∀z. (A-5) 131 Becausef Y (x)≥ 0 and f Y (z)dz=1, condition (A-5) will be true only ifα(m)= 0 for allm, and this completes the proof. Proof of Proposition 2 Using Bayes’ rule, f P 1,a (h(s,d)) = g 1,a (h(s,d))f p (d) g 1,a (h(s,y))f p (y)dy . (A-6) Becauseg 1,a lies in the convex hull generated by the familyg 2,b ,b∈ R, for eachg 1,a , there exists a densityα a (b) such that g 1,a (h(s,d)) = b α a (b)g 2,b (h(s,d))db. Using the above, f P 1,a (h(s,d)) = α a (b)g 2,b (h(2,b))dbf p (d) g 1,a (h(s,y))f p (y)dy = b α a (b) g 2,b (h(s,d))f p (d) g 2,b (h(s,y))f p (y)dy · g 2,b (h(s,y))f p (y)dy g 1,a (h(s,y))f p (y)dy db. Letβ a (b,s)= R g 2,b (h(s,y))fp(y)dy R g 1,a (h(s,y))fp(y)dy . Noticeβ a (b,s)≥ 0. f P 1,a (h(s,d)) = α a (b)β a (b,s) g 2,b (h(s,d))f p (d) g 2,b (h(s,y))f p (y)dy db = α a (b)β a (b,s)g 2,b (h(s,d))db. Becauseg 1,a andg 2,b are densities ofd, by integrating both sides we get 132 b α a (b)β a (b,s)db=1. It is easy to verify, using Equation (A-6), thatf P 1,a are linearly independent iff 1,a are linearly independent. Proof of Theorem 1 For truth-revealing, we must have E f U(ξ(f(·),·))>E f U(ξ( ˆ f(·),·)) for all ˆ f =f. (A-7) For a given announced forecast, a compensation scheme maps an outcome to a pay- off in utility. SoU(ξ( ˆ f,d)) defines a hyperplane in the infinite dimensional space. Let G(f(·)) =Max ˆ f U(ξ( ˆ f,d))f(d)dd. (A-8) By (A-7), G(f(·))> U(ξ( ˆ f,d))f(d)dd ∀ ˆ f =f. (A-9) Notice the right hand side is the supporting hyperplane evaluated at f. Also, for convexity ofG, 133 αG(f 1 (·))+(1−α)G(f 2 (·))−G(αf 1 (·)+(1−α)f 2 (·)) =αmax ˆ f(·) U(ξ( ˆ f(·),d))f 1 (d)dd+(1−α)max ˆ f(·) U(ξ( ˆ f(·),d))f 2 (d)dd −max ˆ f(·) U(ξ i ( ˆ f(·),d))(αf 1 (d)+(1−α)f 2 (d))dd =max ˆ f(·) α U(ξ( ˆ f(·),d))f 1 (d)dd+max ˆ f(·) (1−α) U(ξ( ˆ f(·),d))f 2 (d)dd −max ˆ f(·) α ξ( ˆ f(·),d)f 1 (d)dd+(1−α) ξ( ˆ f(·),d)f 2 (d)dd ≥0. Therefore the compensation function is a supporting hyperplane ofG. Conversely, if there exists a strictly convex functionalG(f(·)) and if a compensation functionU(ξ( ˆ (f),d))is a supporting hyperplane ofG,wehave G(f(·))> U(ξ( ˆ f(·),d))f(d)dd. (A-10) Therefore, truth-telling is ensured. Proof of Lemma 3 Expected payoff of a forecaster who believesf(·) and announces ˆ f(·) is U(ξ( ˆ f(·),d))f(d)dd =G( ˆ f(·))− δG δ ˆ f x=d ˆ f(d)dd+ δG δ ˆ f x=d f(d)dd =G( ˆ f(·))+ δG δ ˆ f x=d (f(d)− ˆ f(d))dd. By convexity ofG, 134 G(f(·))>G( ˆ f(·))+ δG δ ˆ f x=d (f(d)− ˆ f(d))dd ∀ ˆ f(·)=f(·). Conversely, for truth-telling, we must have U(ξ(f(·),d))f(d)dd> U(ξ( ˆ f(·),d))f(d)dd ∀ ˆ f(·)=f(·). Let G(f(·)) = max ˆ f(·) U(ξ( ˆ f(·),d))f(d)dd. Now we must show thatG is convex. Forα∈ [0,1] andf 1 (·),f 2 (·)∈F, αG(f 1 (·))+(1−α)G(f 2 (·))−G(αf 1 (·)+(1−α)f 2 (·)) =αmax ˆ f(·) U(ξ( ˆ f(·),d))f 1 (d)dd+(1−α)max ˆ f(·) U(ξ( ˆ f(·),d))f 2 (d)dd −max ˆ f(·) U(ξ i ( ˆ f(·),d))(αf 1 (d)+(1−α)f 2 (d))dd =max ˆ f(·) α U(ξ( ˆ f(·),d))f 1 (d)dd+max ˆ f(·) (1−α) U(ξ( ˆ f(·),d))f 2 (d)dd −max ˆ f(·) α ξ( ˆ f(·),d)f 1 (d)dd+(1−α) ξ( ˆ f(·),d)f 2 (d)dd ≥0. SoG is convex, and this completes the proof. Proof of Lemma 4 Let Π(d,γ) be the profit function when the outcome is d given the firm’s decision γ. Assume the density ofd isf(d).By (i), the firm solves 135 max γ Π(d,γ)f(d)dd. Notice f(d) is independent of γ because of (ii). Let G(f(·)) = αmax γ Π(d,γ)f(d)dd, where α ∈ [0,1] is a share of the forecaster. It is easy to see G is convex in F. Noting that the forecaster is risk-neutral, by Theorem 1, it is optimal to reveal the truth. Proof of Proposition 4 First, we show that the agent announces the true mean regardless of her announced standard deviation. Expected payoff of the agent as a function of ˆ µ , given the agent announced ˆ σ as standard deviation, is Eξ(ˆ µ )= ˆ µ 0 [w(ˆ σ)−β(ˆ σ)·(ˆ µ −d)]f(d)dd+ ∞ ˆ µ [w(ˆ σ)−β(ˆ σ)·(d− ˆ µ )]f(d)dd, where f(d) is the density of the true distribution N(µ,σ 2 ). Taking the derivative with respect to the announced mean, we get the first order condition. w(ˆ σ)·f(ˆ µ )+ ˆ µ 0 −β(ˆ σ)f(d)dd−w(ˆ σ)·f(ˆ µ )+ ∞ ˆ µ β(ˆ σ)f(d)dd=0 −β(ˆ σ)·F(ˆ µ )+β(ˆ σ)·(1−F(ˆ µ )) = 0. Since β(ˆ σ) > 0, we get F(ˆ µ )=0.5. Of course, we have to check to see if the second derivative is negative. 136 ∂ 2 Eξ ∂ˆ µ 2 =−β(ˆ σ)·f(ˆ µ )+β(ˆ σ)·(−f(ˆ µ ))< 0. So the expected payoffEξ is concave in ˆ µ , and the agent sets ˆ µ =µ ; that is, the agent reveals the true mean of the posterior regardless of her announced standard deviation. Now that we know that the agent tells the true mean, her expected payoff, which is now a function of the true and announced standard deviations, is given by: Eξ(ˆ σ,σ)=w(ˆ σ)−β(ˆ σ)· ∞ 0 |d−µ |f(d)dd =w(ˆ σ)−β(ˆ σ)· µ 0 (µ −d)f(d)dd+ ∞ µ (d−µ )f(d)dd =w(ˆ σ)−β(ˆ σ)· 2 π ·σ. Since we have the first and second-order truth-revealing conditions for standard devi- ation, this completes the proof. Proof of Proposition 5 Given the agent’s announced standard deviation, letf(d) be the true density. The agent’s expected utility, if she announces ˆ µ ,is EU(ξ(ˆ µ )) = ˆ µ 0 U(w−β·(ˆ µ −d))f(d)dd+ ∞ ˆ µ U(w−β·(d− ˆ µ ))f(d)dd. The derivatives are, 137 ∂EU(ξ) ∂ˆ µ = ˆ µ 0 U (w−β·(ˆ µ −d))·(−β)f(d)dd + ∞ ˆ µ U (w−β·(d− ˆ µ ))·βf(d)dd. ∂ 2 EU(ξ) ∂ˆ µ 2 =U (w)·(−β)·f(ˆ µ )+ ˆ µ 0 U (w−β(ˆ µ −d))·(−β) 2 f(d)dd −U (w)·β·f(ˆ µ )+ ∞ ˆ µ U (w−β(d− ˆ µ ))·β 2 f(d)dd. (A-11) Since U > 0, U < 0, and f(d) > 0, the second derivative is negative and thus EU(ξ) is concave. Plugging ˆ µ = µ to (A-11), if it turns out to be zero, then the agent will reveal the true mean. ∂EU(ξ) ∂ˆ µ | ˆ µ =µ = µ 0 U (w−β·(µ −d))·(−β)f(d)dd+ ∞ µ U (w−β·(d−µ ))·βf(d)dd=0. Notice thatU (w−β·|µ −d|) is symmetric with respect tod = µ ,asisf(d). The agent tells the true mean. Proof of Proposition 6 Let f(d) denote the true density N(µ,σ 2 ). Because we know from the previous sec- tion that the agent will reveal µ in stage 2, expected utility is a function of announced standard deviation. LetE(U(ξ(ˆ σ))) denote this expected utility function. 138 EU(ξ(ˆ σ)) = U(w(ˆ σ)−β(ˆ σ)·|d−µ |)f(d)dd = µ −∞ U(w(ˆ σ)−β(ˆ σ)·(µ −d))f(d)dd + ∞ µ U(w(ˆ σ)−β(ˆ σ)·(d−µ ))f(d)dd =2 ∞ 0 U(w(ˆ σ)−β(ˆ σ)·σz)φ(z)dz, whereφ(z) is the standard normal density. Taking the derivative, ∂EU(ξ) ∂ˆ σ =2 ∞ 0 U (w(ˆ σ)−β(ˆ σ)·σ·z)·(w (ˆ σ)−β (ˆ σ)·σ·z)φ(z)dz. Let us plug in ˆ σ =σ and see if∂EU(ξ)/∂ˆ σ is positive or negative. ∂EU(ξ) ∂ˆ σ | ˆ σ=σ =2 ∞ 0 U (w(σ)−β(σ)·σ·z)·(w (σ)−β (σ)·σ·z)φ(z)dz =2 √ 2 π 0 U (w(σ)−β(σ)·σ·z)·(w (σ)−β (σ)·σ·z)φ(z)dz + ∞ √ 2 π U (w(σ)−β(σ)·σ·z)·(w (σ)−β (σ)·σ·z)φ(z)dz . If bothw andβ are negative, using the conditions in Proposition 4, 139 ∂EU(ξ) ∂ˆ σ | ˆ σ=σ =2 √ 2 π 0 U (w(σ)−β(σ)·σ·z)·(w (σ)−β (σ)·σ·z)φ(z)dz + ∞ √ 2 π U (w(σ)−β(σ)·σ·z)·(w (σ)−β (σ)·σ·z)φ(z)dz >2 √ 2 π 0 U (w(σ)−β(σ)·σ· 2 π )·(w (σ)−β (σ)·σ·z)φ(z)dz + ∞ √ 2 π U (w(σ)−β(σ)·σ· 2 π )·(w (σ)−β (σ)·σ·z)φ(z)dz =2U (w(σ)−β(σ)·σ· 2 π )· ∞ 0 [w (σ)−β (σ)·σ·z]φ(z)dz =2U (w(σ)−β(σ)·σ· 2 π )· 1 2 w (σ)− 1 √ 2π ·σ·β (σ) =U (w(σ)−β(σ)·σ· 2 π )·(w (σ)− 2 π ·σβ (σ)) =0. This implies that if EU(ξ) is concave the risk-averse agent inflates the precision; that is, she sets ˆ σ>σ. Similarly, if both w and β are positive, the risk-averse agent deflates the precision. Now, let’s make sure that the second derivative is negative. ∂ 2 EU(ξ) ∂ˆ σ 2 =2 ∞ 0 U (w(ˆ σ)−β(ˆ σ)·σ·z)·(w (ˆ σ)−β (ˆ σ)·σ·z) 2 +U (w(ˆ σ)−β(ˆ σ)·σ·z)·(w (ˆ σ)−β (ˆ σ)·σ·z) φ(z)dz. The first term is negative so if the second term is negative after integration, then EU(ξ) is concave. This happens, for example, whenβ > 0.Ifβ > 0, 140 ∂ 2 EU(ξ) ∂ˆ σ 2 = ∞ 0 U (w(ˆ σ)−β(ˆ σ)·σ·z)·(w (ˆ σ)−β (ˆ σ)·σ·z)φ(z)dz = z 0 0 U (w(ˆ σ)−β(ˆ σ)·σ·z)·(w (ˆ σ)−β (ˆ σ)·σ·z)φ(z)dz + ∞ z 0 U (w(ˆ σ)−β(ˆ σ)·σ·z)·(w (ˆ σ)−β (ˆ σ)·σ·z)φ(z)dz <U (w(ˆ σ)−β(ˆ σ)·σ·z 0 )· ∞ 0 (w (ˆ σ)−β (ˆ σ)·σ·z)φ(z)dz =U (w(ˆ σ)−β(ˆ σ)·σ·z 0 )·( 1 2 w (ˆ σ)−β (ˆ σ)·σ· 1 √ 2π ) <0, wherez 0 =max(0,z 1 ), andz 1 is such thatw (ˆ σ)−β (ˆ σ)·σ·z 1 =0. Proof of Proposition 7 Because expected utility of the agent is a convex combination of two expected utilities, it follows from the previous analysis that the agent tells the true mean. Also notice that at ˆ σ = σ, ∂EU(ξ I )/∂ˆ σ is positive and ∂EU(ξ D )/∂ˆ σ is negative. And since we can compute the first derivative of each expected profit function, we can find α such that ∂EU(ξ T )/∂ˆ σ| ˆ σ=σ =0, whereEU(ξ T )= αEU(ξ I )+(1−α)EU(ξ D ). Because both EU(ξ I ) andEU(ξ D ) are concave, so isEU(ξ T ). Proof of Proposition 8 Let the true precision beϑ and the announced precision ˆ ϑ. Independent of the announced precision, once the forecaster observes a signal she needs to solve a newsvendor problem 141 to determine the location parameter M. The forecaster will set M = F −1 Y ϑ ( c 2 c 1 +c 2 ). The expected payoff for the forecaster is: w( ˆ ϑ)+β( ˆ ϑ)· c 1 M 0 (M−a)dF Y ϑ (a)+c 2 ∞ M (a−M)dF Y ϑ (a) =w( ˆ ϑ)+β( ˆ ϑ)·ϑ·I(c 1 ,c 2 ) =G( ˆ ϑ)−G ( ˆ ϑ) ˆ ϑ+G ( ˆ ϑ)ϑ ≤G(ϑ) =G(ϑ)−G (ϑ)ϑ+G (ϑ)ϑ =w(ϑ)+β(ϑ)· c 1 M 0 (M−a)dF Y ϑ (a)+c 2 ∞ M (a−M)dF Y ϑ (a) Proof of Proposition 10 By Proposition 5, the forecaster always reveals the true mean. Hence the expected payoff if the forecaster selects an effort level corresponding to a standard deviation ofσ is V(σ ∗ )−σ ∗ V (σ ∗ )+σV (σ ∗ ). Due to the convexity of V(·), V(σ ∗ )−σ ∗ V (σ ∗ )+ σV (σ ∗ )<V(σ) forσ = σ ∗ . On the other hand, if the forecaster selectsσ ∗ , the payoff isV(σ ∗ ). Proof of Proposition 11 Fork=2, suppose the optimal solution is to set the standard deviation level of forecaster i at σ i . Consider an alternative forecasting system in which forecasters are drawing samples from a normal population with known varianceK in order to estimate the mean. Forecasteri drawsn i samples, withn i = K σ 2 i . Each sample costs 1 unit. However, there are some samples that are common to both forecasters. The number of common samples 142 is given byn c , andn c = ρ √ n 1 n 2 . The variance of forecasteri’s estimate is K σ 2 i , and the correlation between the two estimates is nc √ n 1 n 2 =ρ. The combined variance is given by σ 2 c = (1−ρ 2 )σ 2 1 σ 2 2 σ 2 1 +σ 2 2 −2ρσ 1 σ 2 . The cost of the two forecasts is n 1 +n 2 = K σ 2 1 + K σ 2 2 . If there is no overlap among the two samples, the pooled variance is: σ 2 1 σ 2 2 σ 2 1 +σ 2 2 . Eliminating the overlap improves the information content, therefore: σ 2 1 σ 2 2 σ 2 1 +σ 2 2 < (1−ρ 2 )σ 2 1 σ 2 2 σ 2 1 +σ 2 2 −2ρσ 1 σ 2 If only one forecaster was engaged, and she was given the entire budget, the variance of the resulting forecast would have been K n 1 +n 2 = σ 2 1 σ 2 2 σ 2 1 +σ 2 2 . It therefore follows that, in the original problem, the firm is better off with one forecaster. For arbitrary values ofk in the equationV(σ)= K σ k , we employ a similar argument. We let the cost of a sample in the alternate system be given byσ 2−k . The cost of attaining a precision level ofσ 2 i remains K σ k i in both systems. The cost of attaining a precision level of σ 2 1 σ 2 2 σ 2 1 +σ 2 2 by allocating the entire budget to one forecaster in the original system or by eliminating overlaps in the alternate is K(σ 2 1 +σ 2 2 ) 0.5k σ k 1 σ k 2 . The result follows because: K σ k 1 + K σ k 2 > K(σ 2 1 +σ 2 2 ) 0.5k σ k 1 σ k 2 for K < 2 K σ k 1 + K σ k 2 < K(σ 2 1 +σ 2 2 ) 0.5k σ k 1 σ k 2 for K > 2 Proof of Theorem 2 LetC 1 ⊂ H be a set of density functions such that iff ∈C 1 , thenf can not be expressed as a convex combination of other members of the setC 1 . We can construct an arbitrary convex function: Claim: Given a set of scalarsk f ∈ R,f ∈C 1 , there is a convex functionalG : H→ R, such thatG(f)=k f . 143 Proof: For eachg∈C 1 , there exists a vectora g ands∈ R, such that: a g ·g +s =k g a g ·f +s≤k f , forf ∈C 1 \g. (A-12) By the theorem of the alternative (Corollary 5.84 of Aliprantis and Border, 2007), if there is no vectora g satisfying these constraints, then there existsW f ≥ 0 andW g in R, such that: W h k h <0 W h h(x)=0, ∀x W h =0 (A-13) This would mean thatg can be expressed as a convex combination of other members ofC 1 . The functionalG(f)=sup h a h ·f is a convex functional withG(f)=k f . (Alternatively we haveW g as a scalar anddW(h).) Below claim shows that we can lower values of a convex function in the interior while maintaining values on the boundary. Claim: Let G : H → R be a proper convex functional with linear supports on a closed convex set C ⊂ F ⊂ H. Let ∂C be the boundary of C. There exists another convex functional ˜ G(·) such that G(f)= ˜ G(f) for f ∈ ∂C, and ˜ G(f) ≤ G(f) for f ∈C 0 . Proof: C is a convex set in a Hilbert space with an interior C 0 . Let ˜ g : R → R + be such that ˜ g(x)dx is finite. Let f ∈ ∂C and ¯ g = f+˜ g. The linear functional generated by ¯ g andf supportC at f because it cannot contain any probability density 144 function other thanf. Thus for any pointf on the boundary the convex setC is properly supported by a linear operatorL f (Lemma 7.37 of Aliprantis and Border, 2007). L f (·) will be such that L f ·f<L f · g for g ∈ C\f, where · denotes the inner product. Let S g be the supporting hyperplane for the convex functional G(·) at g ∈ ∂C. Let ˜ G(·)=sup h S h ·f−L h ·f +L h ·h. Proof of Lemma 5 Letf i (d|t)∈χ(i) denote distributions that are realized by exerting levelt∈T effort in regimei,i=1,2. Without loss of generality, assume df 1 (d|t)dd = df 2 (d|t)dd =t, (A-14) andV(t) is strictly convex increasing int. To prove the lemma, we need the follow- ing: C1: The payoff function for regime 1,ξ 1 (d), is increasing. C2: The payoff function for regime 2,ξ 2 (d), is increasing. C3: Given the agent announced regime 1 truthfully, she makest 1 level of effort. C4: Given the agent announced regime 2 truthfully, she makest 2 level of effort. C5: If regime 1 is realized, the agent declares regime 1. C6: If regime 2 is realized, the agent declares regime 2. LetEU(ξ i (t)) denote the expected utility that the agent earns when she announces regimei truthfully and makes effortt. EU(ξ i (t))≡ U(ξ i (d))f i (d|t)dd (A-15) 145 Let us note thatf i (d|t)≤ ST f i (d|t+ε), for allε> 0,i=1,2, where≤ ST denotes the usual stochastic order. 1 It is known that U(ξ i (d)) is increasing in d if and only if E(ξ i (t)) is increasing in t. 2 Furthermore, since U −1 is increasing, if U(ξ i (d)) is increasing in d,so is ξ i (d). Therefore C1 and C2 can be satisfied by ensuring that EU(ξ i (t)) is increasing in t, i=1,2. In what follows, we specifyEU(ξ i (t)) and not ξ i (d) explicitly. Let’s start with regime 2, the regime in which more accurate forecasts are possible. Since V is strictly convex increasing, there always exists a strictly concave increasing function that is tangential toV att 2 . Let this beEU(ξ 2 (t)). ThenU(ξ 2 (d)) is increasing and so isξ 2 (d). We have C2 and C4. Let us remember that the agent earns zero att 2 . Similarly, we can designEU(ξ 1 (t)) that satisfies C1 and C3. What is left now is to make sure we have C5 and C6. By assumption any member of χ(1) can be expressed as a convex combination of distributions inχ(2). f 1 (d|t)= f 2 (d|s)α t (s)ds ∀d, (A-16) whereα t (s) is a density. Likewise, any member ofχ(2) can be expressed as a linear combination of distributions inχ(1). f 2 (d|t)= f 1 (d|s)α t (s)ds ∀d, (A-17) whereα t (s) is such that α t (s)ds=1. Now, suppose the agent announces regime 2 when the truth is regime 1. Then the agent will solve the following problem to maximize her expected net utility: 1 Usual stochastic order is referred to as first order stochastic dominance in the economics literature. 2 See, for instance, Theorem 1.2.8. of M¨ uller and Stoyan (2002). 146 max t U(ξ 2 (d))f 1 (d|t)dd−V(t). Using (A-15) and (A-16), U(ξ 2 (d))f 1 (d|t)dd−V(t) = U(ξ 2 (d)) f 2 (d|s)α t (s)ds dd−V(t) = α t (s) U(ξ 2 (d))f 2 (d|s)dd ds−V(t) = α t (s)EU(ξ 2 (s))ds−V(t). To evaluate the first term, notice, from (A-14) and (A-16), t = df 1 (d|t)dd = d α t (s)f 2 (d|s)ds dd = α t (s) df 2 (d|s)dd ds = α t (s)sds. Since α t (s) is a density and EU(ξ 2 ) is strictly concave, α t (s)EU(ξ 2 (s))ds < EU(ξ 2 (t))≤V(t) for allt. The agent is worse-off if he lies. We now have C2. Alternately, suppose the agent announces regime 1 when the truth is regime 2. The agent’s problem is: max t U(ξ 1 (d))f 2 (d|t)dd−V(t). 147 Again, using (A-15) and (A-17), U(ξ 1 (d))f 2 (d|t)dd−V(t) = α t (s)EU(ξ 1 (s))ds−V(t). Sinceα t (s) is not a density, consider a straight lineτ(t) that is tangential toV(t) at t 1 . τ(t)=V(t 1 )+(t−t 1 )V (t 1 ). Obviously,EU(ξ 1 (t))≤τ(t)≤V(t),∀t. Since α t (s)τ(s)ds =τ(t), wehaveC2. This completes the proof. 148

Abstract (if available)

Abstract In the first essay, we study truth-revealing compensation schemes for forecasters employed by a firm. We begin by studying properties of forecast distributions that result in higher payoffs for the firm. We partition the space of forecast distributions into refinement level sets, which can be rank ordered based on profit potential. The detailed structure of the compensation schemes depends on: (i) whether or not the forecaster incurs costs in the short term, and (ii) whether or not the forecaster influences the outcome. When forecasters do not incur costs, refinement levels can be used to rank them. In all the cases truth-revealing schemes are a menu of contracts that are hyperplanes to some convex functional in density space. Next, we develop compensations when forecast accuracy depends on the forecaster's costly effort. We show that the decision of the firm to invest in multiple forecasters depends on how rapidly the cost of improving forecasts increases. Finally, we extend traditional salesforce compensation literature by developing compensation schemes for salespersons who also forecast. Unlike the traditional salesforce literature, we permit the firm's profits to depend on all the moments of the distribution, not just the mean. We assume that market conditions influence forecast uncertainty and the salesperson's productivity. We show that if potential forecasts in each market regime belong to a refinement level then the firm can cause the salesperson to reveal the true regime and exert the appropriate level of effort.

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Creator Ochiumi, Hiroshi (author)

Core Title Essays on supply chain management

School Marshall School of Business

Degree Doctor of Philosophy

Degree Program Business Administration

Publication Date 07/31/2008

Defense Date 05/08/2008

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag Forecasting,game theory,incentives,OAI-PMH Harvest,salesforce management

Language English

Advisor Dasu, Sriram (committee chair), Bassok, Yehuda (committee member), Chase, Richard B. (committee member), Hall, Randolph W. (committee member), Rajagopalan, Sampath (committee member)

Creator Email ochiumi@marshall.usc.edu,ochiumi@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m1490

Unique identifier UC1190022

Identifier etd-Ochiumi-2139 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-92323 (legacy record id),usctheses-m1490 (legacy record id)

Legacy Identifier etd-Ochiumi-2139.pdf

Dmrecord 92323

Document Type Dissertation

Rights Ochiumi, Hiroshi

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu