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New approaches for routing courier delivery services

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New approaches for routing courier delivery services

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Content New Approaches for Routing
Courier Delivery Services

By

Chen Wang
__________________________________________________________________

A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(INDUSTRIAL AND SYSTEMS ENGINEERING)
Dec 2012

Copyright 2012 Chen Wang

ii

Acknowledgements

Pursing a PhD is a long journey that is not only full of struggle and work, but also
full of joy and gratitude.
I want first to thank my advisor, Dr. Maged Dessouky. He might not have noticed
how many times he has rescued me during my PhD study time. Any time I feel confused,
or lost, he is always there to help. His intelligence and kindness does not only enlighten
me on my research and study, but also in my work and in my life. All these are precious
treasure that will accompany me through my entire life.
I also thank my co-advisor, Dr. Fernando Ordonez, for all the support he gave me
on all aspects. He inspires me on my research, teaches me tools, guides me to think, and
helps me work on details. This thesis would have been impossible without his supports.
I thank my other committee members, Dr. Petros Ioannou, Dr. James E. Moore II,
and Dr. Alejandro Toriello. They all have provided insightful comments and ideas, which
are of great value for me to make improvement on my dissertation and academic papers.
I thank our former department chair, Dr. Stan Settles, current department chair, Dr.
Julie Higle, and staffs in my department, Evelyn Felina, Georgia Lum, Aven Yam, Rick
Scott, Mary Ordaz, and Norma E. Orduna. As a PhD student, I feel being cared and
valued. It is a great pleasure to have studied in such a nice environment.
I thank all my lab-mates, Huayu Xu, Xiaoqing Wang, Christine Nyhen, Lunce Fu,
Shi Mu, and Yingtao Ren. They are all smart and lovely folks. I enjoy the time we were

iii

together to learn, to discuss, to help, and to share. Special thank is given to my best friend,
Zhihong (Iris) Shen. She is the one to whom I dump all the trashes, and from whom I
absorb new ideas. I also thank all other friends that are with me during the five years. I
feel so lucky to have them all.
I thank every single member in my family. I thank my parents for giving birth to
me and raising me up. I thank them for supporting me to do whatever I like in every
possible way. I am very proud to be their daughter. I thank all my grand-parents. From
them, I learned diligent, trust, and love. I thank my parents in-law and my grand-parents
in law, for all their supports, financially and spiritually. A big thank is given to my
husband, Liye, for his accompanying and supports all the time. He is the one who gives a
pair of wings to my dreams and lets them fly. Last but not least, I want to thank daughter,
Kelly. She is the biggest motivation for me to complete the thesis. This thesis is dedicated
to them all.

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Abstract

Courier delivery services deal with the problem of routing a fleet of vehicles from
a depot to service a set of customers that are geographically dispersed. In many cases, in
addition to a regular uncertain demand, the industry is faced with sporadic, tightly
constrained, urgent requests. An example of such an application is the transportation of
medical specimens, where timely, efficient, and accurate delivery is crucial in providing
high quality and affordable patient services.
In the first part of this study, we propose to develop better vehicle routing
solutions that can efficiently satisfy random demand over time and rapidly adjust to
satisfy these sporadic, tightly constrained, urgent requests. We formulate a multi-trip
vehicle routing problem using mixed integer programming. We use stochastic
programming with recourse for daily plans to address the uncertainty in customer
occurrence. The recourse action considers a multi-objective function that maximizes
demand coverage, maximizes the quality of delivery service, and minimizes travel cost.
Because of the computational difficulty for large size problems, we devise an insertion
based heuristic in the first phase, and then use Tabu Search to find an efficient solution to
the problem. Simulations have been done on randomly generated data and on a real data
set provided by a leading healthcare provider in Southern California. Our approach has
shown significant improvement in travel costs as well as in quality of service as measured
by route similarity than existing methods.

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In the second part of this thesis, we study a method to cluster the customers into
groups without generating the actual routes. There is uncertainty in customer occurrence
in the problem studied; we therefore develop a new objective function based on expected
distance that includes the occurring probability of customers. We show the
appropriateness of the new objective, and propose a mixed integer programming
formulation for customer clustering based on it. A Tabu search based heuristic approach
is developed for solving the problem, and we conduct experiments on randomly
generated data. The new approach shows improvement in expected distance of routes
over a clustering method based on building routes for instances where the depot is in a
corner.

vi

Table of Contents

Acknowledgements ............................................................................................................. ii
Abstract .............................................................................................................................. iv
List of Tables ................................................................................................................... viii
List of Figures .................................................................................................................... ix
1. Introduction ................................................................................................................. 1
1.1 Background .......................................................................................................... 1
1.2 Problem Description ............................................................................................. 4
1.2.1 Courier Delivery Service with Urgent Demand ................................................. 4
1.2.2 Customer Clustering with Customer Uncertainty .............................................. 5
1.3 Motivation ............................................................................................................ 6
1.3.1 Courier Delivery Service with Urgent Demand ................................................. 6
1.3.2 Customer Clustering with Customer Uncertainty ............................................ 10
1.4 Organization of the Thesis ...................................................................................... 11
2. Literature Review ...................................................................................................... 12
2.1 Healthcare Logistics & Vehicle Routing Problem .................................................. 12
2.2 Multi-trip VRP ........................................................................................................ 15
2.3 Stochastic VRP ........................................................................................................ 17
2.4 Customer Service .................................................................................................... 20
2.5 Customer Clustering ................................................................................................ 22
2.6 Research Gap ........................................................................................................... 23
3. Vehicle Routing with Urgent Requests ..................................................................... 25
3.1 Model Formulation .................................................................................................. 25
3.2 A Sample Problem .................................................................................................. 33
3.3 Heuristic .................................................................................................................. 35
3.2.1 Insertion ............................................................................................................ 36
3.2.2 Tabu Search ...................................................................................................... 42

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3.2.3 Master Routes ................................................................................................... 45
3.2.4 Daily Plans with Urgent Requests .................................................................... 47
3.4 Experimental Results............................................................................................... 50
3.4.1 Data Generation and Input Parameters ............................................................. 50
3.4.2 Simulations and Results ................................................................................... 52
3.5 Simulations and Results with Actual Data .............................................................. 78
4. A New Clustering Approach ..................................................................................... 86
4.1 The Expected Travel Distance ................................................................................ 89
4.2 Discussion and Example ......................................................................................... 96
4.3 Problem Formulation for Customer Clustering ....................................................... 98
4.4 Tabu Heuristic ....................................................................................................... 101
4.5 Experiments and Results ....................................................................................... 107
5. Conclusions and Future Work ................................................................................. 118
Bibliography ................................................................................................................... 121

viii

List of Tables

Table 3.1: Customer Information for the Sample Problem ............................................... 34
Table 3.2: Time Windows of Regular and Urgent Requests ............................................ 51
Table 3.3: Simulation Results with 50 Customers ............................................................ 55
Table 3.4: Simulation Results with100 Customers ........................................................... 56
Table 3.5: Simulation Results with 500 Customers .......................................................... 57
Table 3.6: Simulation Results with Actual Data ............................................................... 80
Table 4.1: for the Three Clustering of Customers in the Small Example ................. 98
Table 4.2: Simulation Parameters for Experiment Set A ................................................ 109
Table 4.3: Simulation Results for Experiment Set A ...................................................... 110
Table 4.4: Simulation Parameters for Experiment Set B ................................................ 114
Table 4.5: Simulation Results for Experiment Set B ...................................................... 115

ix

List of Figures

Figure 3.1: Customers, Depot, and Lab ............................................................................ 26
Figure 3.2: The Optimal Routing Solution for Day 0 / 1 .................................................. 34
Figure 3.3: Insertion of Customer Request i ..................................................................... 40
Figure 3.4: Pickup is followed directly by delivery .......................................................... 41
Figure 3.5: Pickup is followed directly by delivery .......................................................... 41
Figure 3.6: -interchange Operator ................................................................................... 44
Figure 3.7: 2-opt Exchange Operator................................................................................ 45
Figure 3.8: City Size and Customer Locations ................................................................. 50
Figure 3.9: Travel Time with 50 Customers ..................................................................... 58
Figure 3.10: Taxi Cost with 50 Customers ....................................................................... 59
Figure 3.11: Dissimilarity with 50 Customers .................................................................. 60
Figure 3.12: Number of Taxi Trips with 50 Customers.................................................... 61
Figure 3.13: Travel Time per Request with 50 Customers ............................................... 62
Figure 3.14: Total Daily Cost with 50 Customers ............................................................ 63
Figure 3.15: Travel Time with 100 Customers ................................................................. 64
Figure 3.16: Taxi Cost with 100 Customers ..................................................................... 65
Figure 3.17 : Dissimilarity with 100 Customers ............................................................... 66
Figure 3.18: Number of Taxi Trips with 100 Customers.................................................. 67
Figure 3.19: Travel Time per Request with 100 Customers ............................................. 68
Figure 3.20: Total Daily Cost with 100 Customers .......................................................... 69
Figure 3.21: Travel Time with 500 Customers ................................................................. 70
Figure 3.22: Taxi Cost with 500 Customers ..................................................................... 71
Figure 3.23: Dissimilarity with 500 Customers ................................................................ 72
Figure 3.24: Number of Taxi Trips with 500 Customers.................................................. 73
Figure 3.25: Travel Time per Request with 500 Customers ............................................. 74
Figure 3.26: Total Daily Cost with 500 Customers .......................................................... 75
Figure 3.27: Average Travel Time of a Vehicle ............................................................... 82
Figure 3.28: Average Taxi Cost of a Vehicle ................................................................... 82
Figure 3.29: Average Dissimilarity for the Fleet .............................................................. 83
Figure 3.30: Average Number of Taxi Trips for the Fleet ................................................ 83
Figure 3.31: Average Travel Time per Request ................................................................ 84
Figure 3.32: Average Total Cost ....................................................................................... 84
Figure 4.1: Customers Scattered Around Depot ............................................................... 87
Figure 4.2: Customers Clustered, Depot at a Corner ........................................................ 88
Figure 4.3: Customers Scattered Around Depot ............................................................... 88
Figure 4.4: Four Customers in the City ............................................................................ 96
Figure 4.5: Location of Customers in Experiment Set A ................................................ 108
Figure 4.6: Full Route Length of the Clusters in Experiment Set A ............................... 111
Figure 4.7: Expected Route Length of the Clusters in Experiment Set A ...................... 111

x

Figure 4.8: Location of Customers in Experiment Set B ................................................ 113
Figure 4.9: Full Route Length of the Clusters in Experiment Set B ............................... 116
Figure 4.10: Expected Route Length of the Clusters in Experiment Set B .................... 116

1

1. Introduction

1.1 Background
The vehicle routing problem (VRP) is a problem of designing optimal routes of
collection or delivery from one or several depots to a number of geographically dispersed
customers. This type of problem is faced by many industries such as courier services (e.g.,
UPS, Federal Express, and Overnight United States Postal Service) and local trucking
companies. In recent years, these types of services have experienced tremendous growth.
For example, both UPS and Federal Express have shown a steady increase in annual
revenue in the past decade, both exceeding $30 billion annually.
These routing applications not only have to schedule efficient routes for uncertain
demands, they also have to handle sporadic, tightly constrained, and urgent requests. For
example, typical courier services have a deadline (e.g. 5 pm) for overnight delivery
service. Requests for overnight services that are received after the deadline are not
accommodated, although it is possible for these packages to be delivered through some
other re-routing process.
Another example application is the transportation of clinical specimens, which is
pervasive in the healthcare industry. On a daily basis, millions of specimens are delivered
in the United States from dispersed hospitals and clinics to centralized laboratories for

2

testing and reporting. Timely and efficient transportation of specimens is crucial in
providing high-quality and affordable patient service in the healthcare industry. The
current situation, however, is far from ideal, where lost or delayed delivery of specimen
is the most common problem jeopardizing patient safety (Astion et al, 2003). Barenfanger
et al. (1999) report that shorter turnaround time (TAT) of microbiological procedures is
correlated with improved clinical outcomes and financial return. For the cause of
excessive TAT, Steindel and Novis (1999) found in their research that specimen
transportation problems account for 56.3% of delays in the collection-to-receipt phase.
And according to Steindel and Howanitz (2001) and Holland et al. (2005), the percentage
of excessive laboratory test TAT is significantly correlated with delay in treatment and
increased average length of stay in emergency departments. The cost on the
transportation of clinical specimens is a significant burden to healthcare systems,
especially for urgent cases which require prompt courier services.
There are several unique characteristics in the medical specimen routing problem
that determines the significance of the proposed research. One of them is the nature of the
demand. The clinical specimens generally fall into two kinds of delivery time windows in
terms of testing and reporting. The urgent ones typically need to be transported within an
hour, and regular ones have several hours of turnaround time. Urgent requests occur at
random times throughout the day in a laboratory service area. In the current practice,
many of these urgent requests are delivered by an outsourced courier service, such as
taxis. For mid-to-large scale laboratory systems, the cost of handling urgent demands by

3

taxis is significant; therefore, an opportunity of cost reduction is presented by
incorporating these urgent demands into the routine specimen routing system.
Another characteristic of the specimen routing system is the two types of the
facilities that the testing requests come from, namely hospitals and clinics. Hospitals
normally operate around the clock, whereas clinics typically do not require service during
nights and weekends. For this reason, optimal routing of courier service will need to take
into account the changing demand levels at different time periods.
Additionally, the medical specimen routing problem includes random customer
demands in the healthcare industry, which comes from uncertain requests and the strict
testing requirements. Also, because most specimens are perishable, the courier must
strictly follow the delivery time windows.
Last but not least, high level of customer service is required for providing high
level of healthcare delivery service. In terms of routing, high level of customer service
can be represented by the similarity of daily routes, which can be boosted by having the
same driver visiting the same location each time.
In this thesis, we study the approaches for routing problems with the above
characteristics, namely a routing problem with random customers, varying multi-period
demand and urgent requests with tight time windows. We first focus on approaches for
routing courier deliveries with random urgent requests and then discuss a new customer
clustering approach when there is uncertainty in customer occurrence.

4

1.2 Problem Description
In this thesis, we study two problems for the routing of courier delivery services,
whose application is especially important for the medical specimen routing problems.
The first one is routing courier delivery service with random urgent demand. The second
one is customer clustering in the presence of customer uncertainty.

1.2.1 Courier Delivery Service with Urgent Demand
In this section, we will describe the courier delivery service with urgent demand
problem with a real-life example. A leading healthcare provider in the Southern
California region operates about 200 medical facilities. The healthcare provider
continuously delivers medical samples, lab-specimens, mails, x-rays, and documents etc.
between various medical facilities and a central lab for testing. The medical facilities are
located throughout Southern California, where the travel time between facilities is
comparable to a complete trip length. This makes this routing problem one in which there
are typically few locations visited per trip. The healthcare provider has about 50 vehicles
to carry out the deliveries. Because of the random nature of the demand in the healthcare
industry, the requests may occur at any time during the day. As most medical samples are
perishable and should be processed in a short time period, the demands have time
windows for pickup and for delivery. The deterministic routine requests typically have to
be delivered to the lab in 4 hours after being collected from the customer; the random

5

urgent requests always have to be delivered to the lab within an hour after being collected.
There are no capacity constraints, because the sizes of the samples are small compared to
the capacity of the vehicles. The vehicles travel through multiple urban areas in several
consecutive trips each day to serve the requests. The vehicle depot is located at the
central lab where all routes start and end a trip. Third-party couriers (i.e. taxis) are
introduced to serve the unmet demands of the regular fleet.
In this work we propose a multi-trip VRP formulation, with deterministic routine
requests and random urgent requests, to represent the real world healthcare routing
problem described above. Particularly, we are developing a best possible plan for a
horizon of several days, with a number of vehicles, to service the customers that send out
deterministic and stochastic requests that follow certain time window constraints.

1.2.2 Customer Clustering with Customer Uncertainty
The problem of customer clustering when there is customer uncertainty can be
described by the following example. Suppose there are 20 customers in the area, and we
have two vehicles to service the requests from the customers. There is uncertainty in the
customer occurrence, meaning that we do not know in advance which customer will
request service on any given day. However, we know the probability that any customer
will request for service, based on historical information. Our study focuses on the
approach of finding the best customer clustering without generating actual routes.

6

In this thesis, we propose a new measure for customer clustering based on
expected distance, and show the appropriateness of it. Meanwhile, we build a mixed
integer programming formulation to obtain the best clustering, and propose a heuristic to
solve the problem. Based on the given locations of customer and the occurring
probability of them, we show a better way for clustering the customers such that the total
distance that the fleet travels during the planning horizon is minimized. The experiments
show that the new probability aware clustering method is better than a route based
clustering method for asymmetric instances.

1.3 Motivation
1.3.1 Courier Delivery Service with Urgent Demand
Even though there are a number of studies and published results in the routing
literature, the scheduling of urgent requests for medical specimen deliveries is still a
manual process in practice. This basically is because of the nature of the demand. Besides
the routine demands, there is a significant amount of urgent demands that occur randomly
throughout the day and have tight time window constraints. The key issue in this problem
is how to integrate these uncertain demands into the delivery schedule for the routine
demands. This could be achieved by handling each type of demand by a different system:
a regular system for the routine demands and a taxi system for the random urgent ones.
However, this can become extremely costly, especially for a mid-to-large size system

7

with a large number of urgent requests. The second reason for the manual process of
specimen delivery is the continuous nature of the demand. Because of the random nature
of customer demands in the healthcare industry, a request for delivery may occur at any
time of the day. The third reason for the phenomenon is that most medical specimens are
perishable, and therefore must be processed within a short time window. The tight time
window requires the algorithms for the routing problem be capable of handling multiple
trips.
In this work, we propose to address the gap by developing routing methods
considering these specific requirements of routing clinical specimens. When modeling
the specimen delivery system for the healthcare industry, the following aspects must be
taken into consideration: healthcare network configuration, nature of delivery requests,
the objective of quality of healthcare service, and the cost of unmet demands.
In the current practice, the healthcare delivery service runs fixed daily routes in
the planning horizon and services all random requests that cannot be accommodated
using taxi. Having fixed daily routes keeps the similarity of routes, basically having the
customers visited by the same vehicle at roughly the same time every day. Such stability
with master routes is required in repeating systems where the quality of service is
important. If the routes used every day are similar to each other, then drivers become
more familiar with the area and it becomes easier to adjust to local changes each day due
to the familiarity of the drivers (Groë r et al. 2008, Sungur et. al 2010). However, rigid
routing strategies, with constant routes, also have a drawback in that it is inefficient when
there are plenty of random urgent demands. Currently, most of the random urgent

8

specimen bypass the routing system and use outsourced vehicles such as taxis, which
introduce a substantial additional cost. The ability to adapt the routing solutions in
response to these urgent requests can make a fundamental difference in customer service
and operational costs, which is essential in the industry. It should be mentioned that, as an
abstract concept, route similarity is a user defined measurement of how a route resembles
another. It can correspond to the number of customers that are visited by the same
vehicle in different days, or the number of arcs that are repeated (Sungur et. al 2010), or
measured in terms of the area that is covered by each vehicle (Zhong et. al 2007). We
consider route similarity a benefit, and the other objectives (travel distance and taxi usage)
in this study are considered as costs. We create a new measure to represent the degree of
similarity in the routes created. This measure, which we refer to as “dissimilarity”
increases by 1 every time a customer is visited by different vehicles in two consecutive
days.
Simply including the urgent requests as part of the possible demand is not
straightforward. If a chance constraint model or a robust optimization approach is used,
either the unlikely requests are ignored or the solution considering them is at a high cost.
We use an approach called stochastic programming with recourse to handle the urgent
requests. This approach requires a massive number of scenarios, leading to large scale
routing problems. To solve this problem, we develop a model with a multi-period time
horizon to compare the frequency of urgent requests with that of regular ones. The
overall idea is to understand whether we can sacrifice some optimality with regard to

9

regular demand to free some capacity or to obtain more flexible routes, which could
accommodate more urgent requests at a lower cost.
In this research, we build a model for this vehicle routing problem, and solve it
using heuristic algorithms. The model and the heuristic algorithms take into account the
following characteristics of the healthcare delivery application: continuous demand,
urgent requests, and multiple objectives. The work is built based on the assumption that it
may be possible to satisfy the regular demands in a way that the slack of the vehicles can
be used to address urgent requests. We build a multi-trip formulation and use stochastic
programming with recourse for the master and daily routes. When formulating the master
plan, it is desired that the master plan is similar to the daily plans that have uncertainty in
customer occurrence. We use an approach that forms the master plans that would require
little modification when adapted to daily schedules. Both the master plan and the recourse
action for each daily schedule consider a multi-objective function that minimizes the
delivery cost, minimizes taxi usage, and maximizes the quality of the healthcare customer
services.
Besides the modeling and the heuristic algorithms, we investigate how different
uncertainty modeling decisions impact the quality of the routing solutions. Given the
nature of the healthcare delivery problem, we evaluate the quality of the planned master
and daily routes under different demand loads in terms of routing efficiency and route
similarity. We compare the performance of the routing solutions through simulation
under different settings and uncertainty scenarios.

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In summary, there are three major contributions we make with this research:
1) Propose a routing model suitable for the healthcare industry courier delivery
problem.
2) Develop new heuristic algorithms to solve the problem.
3) Establish recommendations of best practices via simulations.
Even though the stated application is for the healthcare delivery problem, the
methods we develop are applicable to any routing problems with urgent and stochastic
demand in general.

1.3.2 Customer Clustering with Customer Uncertainty
In this research, we study strategies of building master plans that can be adapted
to daily plans. In order to generate efficient master routes given the uncertainty of
customers, we propose a new method to group customers into clusters, without
generating actual routes. Instead of constructing routes on the territory locations when
clustering the customers into groups, we propose to cluster customers guided by a new
measure that is based on expected distance, which takes into account the probability of
occurrence of customers. Specifically, there are four contributions we make on customer
clustering with this thesis:
1) Propose a new measure and objective for customer clustering based on expected
distance.

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2) Propose a new mixed integer formulation model for obtaining the best customer
clustering with the new objective function.
3) Develop a new heuristic to solve the problem.
4) Evaluate the benefit of this new clustering method with experiments on artificial
instances.
With the experiments on artificial instances, we find that this clustering method is
superior to a clustering method based on building routes when instances are not
symmetric.

1.4 Organization of the Thesis
The rest of the thesis is organized as follows. In chapter 2, a literature review of
the relevant vehicle routing problems is presented. Chapter 3 introduces the new
approach for routing courier delivery services. In this chapter, a problem formulation is
presented; a heuristic for solving the problem is proposed; experimental results of the
proposed heuristic on randomly generated data and on real industry data are presented
and discussed. Chapter 4 introduces a new approach for clustering the customers into
groups without generating actual routes, when there is customer uncertainty. In Chapter 4,
the new objective function is presented; a problem formulation for customer clustering is
proposed; a heuristic for solving the problem and experiments on randomly generated
data are presented. Conclusion for the study and discussion on future research are
covered in Chapter 5.

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2. Literature Review

In this section, we review the literature relevant to our research. First of all, we
give a brief review on the studies in the healthcare logistics and the general vehicle
routing problem (VRP). Generally speaking, existing research on healthcare logistics
does not collectively consider multi-trip delivery, randomness, and urgency in the nature
of the demand. To overcome these limitations, the model for the courier delivery problem
of medical specimens should take these aspects into consideration. In the following two
subsections, we focus on the relevant literature in the class of VRP: multi-trip VRP
(MtVRP) and stochastic VRP (SVRP). Next, we briefly review the literature on customer
services and customer clustering in the vehicle routing problem. In the last part, we
address the gap we fill with this research.

2.1 Healthcare Logistics & Vehicle Routing Problem
Since the most important application of the proposed work is in the logistics and
the supply chain systems of the healthcare system, we first review the related papers in
healthcare logistics. The research in the logistic and supply chain systems of the
healthcare industries has primarily been focused on the pharmaceutical industry (i.e.,
Papagergiou et al. 2001; Shah 2004; Meller et al. 2009). These models focus on
optimizing the inventory systems or medication repackaging options. Other examples

13

studying healthcare system logistics include Nicholson et al. (2004) who study
outsourcing inventory management decisions, and Jarrett (2006) who investigates the
implementation of just-in-time systems; Vissers and Beech (2005) study the management
of patient flows between organizations etc.
Another relevant research area for healthcare logistics is home healthcare (HHC)
service. HHC is a service that provides nursing assistance to patients, especially to the
elderly, in their homes. Usually a HHC service operates a fleet of vehicles that are used to
drive the nurses to the patients, where the nurses perform specific jobs. Begur et al (1997)
develop an integrated spatial decision support system for scheduling HHC nurses. Both
Bertels and Fahle (2006) and Steeg and Schroeder (2007) study the heuristics for home
healthcare problems that is related to the vehicle routing problem and the nurse rostering
problem. Ganesh and Narendran (2007) present a multistage heuristic for a vehicle
routing problem that involves a single item pickup, delivery under time window
constraints; this problem can be applied to blood delivery for a public healthcare system.
Hemmelmayr et al. (2009) study the delivery of blood products, analyzing the potential
value of switching to a vendor-managed inventory system (VMI); they present solution
approaches with integer programming and variable neighborhood search. Bachouch et al.
(2009) study the drug delivery problem for homecare, using mixed integer programming.
The courier delivery problem for medical specimens studied by this research falls
under the class of the vehicle routing problem (VRP), which was first introduced by
Dantzig and Ramser in 1959. Being a fundamental problem in transportation, distribution,
and logistics, VRP studies the scheduling a fleet of vehicles to satisfy a set of

14

geographically dispersed demands at minimum cost. General review of the VRP can be
found in a number of literatures, such as Toth and Vigo (2002), Fisher (1995), and
Laporte and Osman (1995). To the best of our knowledge, there has been no research
studying courier delivery in multiple trips with stochastic urgent requests, especially for
the healthcare industry.
The vehicle routing problem is known to be NP-hard, because the travelling
salesman problem (TSP), a special case of the VRP, is NP-hard. To solve the vehicle
routing problem, a number of approaches are proposed in the literature. Exact algorithms
(i.e. dynamic programming, branch and bound, branch and cut, branch and price), which
can solve the VRP optimally, are only applicable to small-size problems. To solve
moderate-size problems, heuristics are proposed and utilized in practice. Heuristics
include constructive heuristic (e.g., Clarke and Wright, 1964), two phase heuristics (e.g.,
Gillett and Miller, 1974), and improvement methods (e.g., Thompson and Psaraftis,
1993). In the past two decades, several metaheuristics (e.g.,Tabu search, genetic
algorithms, simulated annealing, neural networks) have been proposed to solve the
vehicle routing problems. Gendreau et al. (1994) propose a Tabu search heuristic to solve
the vehicle routing problem with route length and capacity restrictions. Baker and
Ayechew (2003) develop a genetic algorithm for the basic vehicle routing problem with
weight limit and travel distance limit on the vehicles. Breedam (1995) proposes
simulated-annealing based improvement heuristics for the vehicle routing problems.
Modares et al. (1999) address several algorithms for the routing problems based on a self-
organizing neural network approach. These metaheuristics typically perform a thorough

15

exploration of the solutions, allowing deteriorating even infeasible intermediate ones;
some of the metaheuristics maintain a pool of good solutions, which can be recombined
to produce better ones.

2.2 Multi-trip VRP
Multi-trip VRP (MtVRP), as a variant of the VRP, has gained little attention in
the literature. In the MtVRP, vehicles can be used more than once during the planning
horizon. Taillard et al. (1996) are the first researchers who studied the problem. They
suggest that assigning more routes to a vehicle is a more practical solution in real life.
They design an algorithm based on Tabu search, and their algorithm tries to avoid
obtaining a local minimum.
The study of Brandao and Mercer (1997) made an improvement to that of Taillard
et al. (1996). This article does not only consider multi-trip VRP, using Tabu search, it
also includes the delivery time window and the capacity of the vehicles. Moreover, this
article assumes the flexible hiring of vehicles. Later, a simplified version of the paper is
published by Brandao and Mercer (1998), with comparison of their study to Taillard’s
algorithm.
Petch and Salhi (2003) integrate the approaches proposed by Taillard et al. (1996)
and Brandaoand Mercer (1997 & 1998). Azi et al. (2006) first describe an exact
algorithm for solving a multi trip VRP problem for a single vehicle with time windows.

16

Salhi and Petch (2007) provide a comprehensive literature review on the multi-trip VRP,
and present a genetic algorithm based on a heuristic for the solution of MtVRP.
Another variant of the VRP study which considers periodicity of the usage of
vehicles is the periodic VRP, which customers have to be visited once or several times in
the planning horizon (Angelelli and Speranza, 2002). PVRP extends the classic planning
horizon to several days (Hemmelmayr et al., 2009). Angelelli and Speranza (2002)
propose a Tabu search based heuristic for the solution of a PVRP with intermediate
facilities, where vehicles can renew their capacities. Francis and Smilowitz (2006)
present a continuous approximation for service choice of a PVRP with capacity
constraints. Hemmelmayr et al. (2009) propose a new heuristic for solving PVRP as well
as a Periodic Travelling Salesman Problem, based on a neighborhood search. The paper
of Alonso et al. (2008) extends the classic VRP to a periodic and multi-trip VRP with
site-dependency and proposes a Tabu search based algorithm to solve the problem.
Besides the models with multiple shifts or trips, overtime can also be an important
strategy when a multi-trip model is constructed. Overtime has been widely used as an
effective option in production planning and scheduling; it is however rarely used in the
study of vehicle routing and scheduling problems. Sniezek and Bodin (2002) propose “a
Measure of Goodness” criteria for their cost models, which includes capital cost of a
vehicle, salary cost of the driver, overtime time, mileage cost, and cost of capacity
renewal at the disposal facilities, to solve their Capacitated Arc Routing Problem with
Vehicle/Site Dependencies (CARP-VSD). This model confirms that using overtime does
help in generating less expensive routes because of the saving in the capital cost of

17

vehicles. In recent years, Zapfel and Bogla (2008) provide a study of a multi-trip vehicle
routing and crew scheduling with overtime and outsources options. Ren et al. (2010)
introduce the usage of shifts into the VRP, and study a new variant of the VRP, which is
with time windows, multi-shifts, and overtime. The results show that the shift dependent
heuristics has significant cost savings. However, the proposed Tabu search based
algorithm applies only to deterministic cases.

2.3 Stochastic VRP
The stochastic VRP (SVRP) introduces uncertainty in the parameters. A general
review of the SVRP can be found in Gendreau et al. (1996). The stochastic VRP can be
classified based on the following criteria:
(1) Uncertainty in the problem: The uncertainty can be present in several parts of
the vehicle routing problem, i.e., the presence of a customer, the level of demand, and the
travel and service times. Generally, the related variants of the problem include VRP with
stochastic customers (VRPSC), VRP with stochastic demand (VRPSD), and with
stochastic service and travel times (VRPSSTT).
(2) Modeling method: There are several modeling methods prevailing in the
literature for solving the SVRP. The most common one is stochastic programming, which
can be further divided into chance constraint programming (CCP) and stochastic
programming with recourse (SPR).

18

(3) Solution technique: Similar to the classic VRP, the solution techniques for the
SVRP generally fall into two categories: exact methods and heuristic methods.
When the customer demand follows a given probability distribution, the problem
is referred to as VRPSD, which consists of routing the vehicles to minimize expected
total distance travelled such that all demands are served. Early contributions on the
VRPSD include Stewart and Golden (1983) who apply chance constraint programming
and recourse methods in the modeling, and Dror and Trudeau (1986) who illustrate the
impact of the direction of a designed route on the expected cost.
When customers are associated with demand that has a probability of being
present, the vehicle routing problem becomes VRP with stochastic customers (VRPSC)
(also called probabilistic VRP in the literature), which was initially studied by Jezequel
(1985) and Jaillet (1988). The routing problems considering both stochastic customers
and demands are typically classified as the VRP with stochastic customers and demands
(VRPSCD), and it is a combination of the VRPSC and VRPSD.
In the recent literature, VRPSD, VRPSC, and VRPSCD have been studied under
two distinct approaches, the “a-priori optimization” approach and the “re-optimization”
approach (Secomandi, 2001). Bertsimas (1992) proposes “a-priori sequence” solutions,
which define a visiting sequence in advance that includes all the demand and skipping of
the nodes or routes which are known to have no demands. Bertsimas and Simchi-levi
(1996) survey the development for the VRPSCD with emphasis on the proposed
algorithms. In these variants of the routing problems, a number of models and solutions
allow for recourse actions to adjust an “a-priori solution” after the uncertainty is revealed.

19

The recourse actions proposed in the literature include skipping non-occurring customers,
returning to the depot when capacity is exceeded, or complete rescheduling for occurring
customers (Jaillet 1988; Bertsimas et al, 1990; Waters 1989).
With respect to the re-optimization approach, routing is dynamic in a sense that it
occurs concurrently with service and no a-priori tours are followed (Secomandi, 2001).
Dror et al. (1989, 1993) propose a Markov decision process for a single-stage and multi-
stage stochastic model to investigate the VRPSD. However, the algorithms for the re-
optimization approach are limited in the literature because of the computational difficulty
with this approach. Recent papers include Secomandi (2001) and Secomandi and Margot
(2009), in which a re-optimization routing policy and a rollout algorithm are developed.
Another class of the SVRP is the VRP with Stochastic Travel Time and Service
Time (VRPSSTT), which has received relatively less attention in the VRP literature
compared to the VRPSC, VRPSD, and VRPSCD. In the VRPSSTT, the traffic condition
on the roads as well as the service time is uncertain. In other words, the travel time
between two locations is not deterministic, but rather depends on the congestion situation
on the roads; the service time for each request is not deterministic, but depending on the
vehicle that is performing the service. Kao (1978) first studies the Travelling Salesman
Problem with Stochastic Travel Time (TSPSTT) and proposes heuristics based on
dynamic programming and implicit enumeration. Carraway et al. (1989) use a
generalized dynamic programming methodology to solve the TSPSTT. Laporte et al.
(1992) study the VRPSSTT problem and proposes a chance constrained model, a 3-index
recourse model, and a two-index recourse model. A branch-and-cut algorithm is proposed

20

for the three models. Besides the above applications on VRP, the VRPSSTT model is
also applied to a banking problem and solved with adapting the savings algorithm
(Lambert et al. 1993).
Robust optimization, introduced by Ben-Tal and Nemirovski (1998), has also
been used in solving vehicle routing problems. Sungur et al. (2008) solve a capacitated
VRP problem with uncertain demand on a fixed set of demand nodes. They use the robust
optimization technique to formulate a new method for solving the problem, the Robust
Vehicle Routing Problem (RVRP). Shen et al. (2009) study a routing problem for
minimizing unmet demand with uncertain demand and travel time. They present a chance
constraint model and compare it to a robust optimization approach. Sungur et al. (2010)
study a Courier Delivery Problem (CDP), which is a Vehicle Routing Problem with Time
Windows Problem (VRPTW) with uncertain service times and customers. After
formulating the problem, the authors proposed a two-phase heuristic based on insertion
and Tabu search. Robust optimization is used to construct a worst-case service time for
the master plan.

2.4 Customer Service
The healthcare courier delivery problem differs from the classical vehicle routing
problem in a few ways. An important one is that it has a high requirement on the quality
of customer service. In the problem we are considering for example, the clinics and
hospitals prefer the samples or specimens to be delivered by the same driver in repetitive

21

days. This would not only guarantee the promptness in the processing of the requests, but
also warrant the familiarity of the delivery, both of which are key factors for efficient
healthcare logistical systems.
Some recent work has included customer service in the models for fixed route
delivery systems under stochastic demand (Haughton and Stenger 1998). Haughton (2000)
develops a framework for quantifying the benefits of route re-optimization, also under
stochastic customer demands. Zhong et al. (2007) propose an efficient way of designing
driver service territories, considering uncertainty in customer locations and demand.
Their method uses a two stage model: in the strategic level, core service territories are
constructed; in the operational level, customers in the non-core territories are assigned on
a daily basis to adapt to uncertainty. This approach however does not consider customer
time windows. Groer et al. (2008) introduce the Consistent VRP (ConVRP) model. The
objective is to obtain routes such that the customers are visited by the same driver at
roughly the same time on each day. They develop an algorithm, ConRTR (ConVRP
Record-to-Record travel), which first generates a template and then generates daily
schedules from the template by skipping non-occurring customers and inserting new
customers. Sungur et al. (2010) introduce the concept of “route similarity” as the number
of customers of the daily routes that are within a given distance of any customer on the
master plan route, and use it as a key measure for developing optimal routing strategies.

22

2.5 Customer Clustering
The customer clustering problem considers the consolidation of customer orders
into one vehicle. Mulvey and Beck (1984) develop heuristics for solving the capacitated
clustering problem (CCP). Koskosidis and Powell (1992) extend the study of Mulvey and
Beck (1984) and Fisher and Jaikumar (1981), and propose optimization-based heuristic
algorithms for the CCP. Ioachim et al. (1995) study the request clustering algorithm for
door-to-door handicapped transportation. They develop a new optimization-based
clustering approach which globally generates a set of mini-clusters. This approach
consists of solving a multi-vehicle Pickup and Delivery Problem with Time Windows (m-
PDPTW) using column generation. Kelly and Xu (1998) develop a generic Tabu search
heuristic for solving the capacitated vehicle routing problem (CVRP) with multiple
vehicles. The heuristic algorithm explores the advantage of local search, improvement
heuristics, and a complex meta-heuristic. The solution generated by the heuristics are
further selected and assembled by a set-partitioning model to produce better solutions.
Another variant of the customer clustering problem considers coordinating a fleet
of vehicles such that all demands on a territory are serviced and that the workload of a
vehicle is evenly distributed. For example, Haugland et al. (2007) and Carlsson (2011)
use a strategy that first divides the service territory into sub-regions and require that each
vehicle be only responsible for the demand occurring in its own sub-region. This heuristic
has the advantage that drivers become more effective at serving their territory and
customers over time. Additionally, Carlsson and Delage (2011) consider a partitioning

23

problem that the client locations are unknown at the time of partitioning the territory.
Their approach suggests partitioning the region regarding the worst-case distribution, that
satisfies first and second order moments.

2.6 Research Gap
Our work is different from the previous research in a few ways. The primary
distinction in the domain of multi-trip VRP is that the earlier research on multi-trip VRP
has equal length operation period for all vehicles with the routing in one period
independent of the next. In this work we allow continuous operation of non-equal length
periods for different vehicles. For example, in a planning horizon of one day, a MtVRP
may require the customers to be visited twice in two trips in a workday, with the length of
a trip fixed. Or a PVRP may have all the customers be visited in one trip each workday
during the planning horizon of a week, where the length of a trip of a vehicle is 8 hours
per day. In our problem, the vehicles operate in multiple trips each workday during the
planning horizon, which is usually multiple days; the length of the trips for each vehicle
is not pre-defined, but will be flexible according to the time window of the demands.
There are multiple trips with varying length during the planning horizon because when
we have a vehicle to visit a customer for pickup of a medical specimen, it is required that
the specimen should be delivered to the lab by the same vehicle on the same trip. This
variant of the VRP that has continuous multiple trips with varying length has not yet been
studied in the literature.

24

Another distinction of our work from the previous work is that the latter has
mainly focused on developing daily independent routes without considering the
integration of regular demand with random urgent requests. This requires the formation
of master routes that have the flexibility to integrate a high number of urgent requests that
have tight time windows and that may randomly occur any time of the day.
Additionally, the prior work focuses on an objective of minimizing travel cost,
e.g., the total travel distance and vehicle costs. While in the healthcare domain, customer
service is another important factor that needs to be taken into consideration. As
emphasized by the current practice, a similar daily plan is a representation of a high
quality of customer service, with which we will have the same driver visiting the same
customer such that the promptness and accuracy of delivery is desired. We develop a
model that has both the cost of vehicles and taxis in the objective function, but also
includes route similarity as a measure for the quality level of customer service.
Furthermore, there is limited research focusing on customer clustering based on
expected distance including both customer locations and the occurring probability at the
same. In this research, we aim to fill the gap by developing a new approach for customer
clustering based on expected distance without generating actual routes.

25

3. Vehicle Routing with Urgent Requests

3.1 Model Formulation
We formulate a multi-trip vehicle routing model for the healthcare industry
courier delivery problem, taking into account the efficient scheduling of regular and
urgent requests, as well as route similarities. In this section, we provide a mixed integer
programming formulation of this multi-trip VRPTW with stochastic clients.
Assume we are making a routing schedule for a healthcare courier delivery
service provider. There are potential customers (hospitals, clinics) in the region that
must be visited during a planning horizon by a fleet of identical vehicles. Each day, some
hospitals and clinics out of the potential customers send out a request that patients’
specimen should be picked up at the customer location and delivered to the lab, where
both the pick-up and the delivery have to follow certain time windows. The locations of
all the potential customers are known. However, the information of which customers
have requests is only revealed on the day the requests are made.
There is one depot (node ) located at the central lab. Each vehicle should leave
the depot at the beginning of the day, and return to the depot at the end of the day. It can
also return to the lab anytime during the day when required (i.e., when there are urgent
requests that need samples delivered by a certain time at the lab.). As each vehicle has
multiple trips, we assume a dummy depot (represented by node ) located also at the

26

central lab to keep track of which trip the request is on. An example can be found in
Figure 3.1 where there are five customers. Nodes 1 to 5 are used to represent the
customers; node 0 and 6 are used to represent the depot and the dummy one, both of
which located at the central lab.
The notation of the model formulation is as follows.

Figure 3.1: Customers, Depot, and Lab

The routing parameters:
: set of days in the planning horizon.
: set of customers, .
: set of vehicles.
: set of daily trips of a vehicle, .

1
2
3
4
5
0

6

27

The cost parameters:

: minimum travel time between node and j.

: unit travel cost, dollars per mile.

: unit outsource cost, dollars per taxi trip.

: unit dissimilarity cost, dollars for each count of dissimilarity.

The stochastic parameters:

: set of occurring customer requests on day d.

: service time of customer request i on day d.

: the earliest time that the customer can be visited for request i on day d.

: the latest time that the customer can be visited for request i on day d.

: the latest time that the customer request i can be delivered to the lab on day d.

Other parameters:
: a sufficiently large number.

The routing variables:

{

{

{

28

: the time vehicle k arrives at customer on day d.

: the time that vehicle k leaves the depot for its trip w on day d.

: the time that vehicle k returns to depot from its trip won day d.

The auxiliary demand variables:

{

{

{

Before we present the mathematical formulation of the model, some clarification
on the parameters and decision variables need to be made.
1) The planning horizon has a length of | | days; and is used to represent the
planning for the master routes.
2) The maximum number of trips each vehicle can make in a day is . We allow
artificial trips that do not deal with any customers, but just “move” from the depot
to the lab and back to the depot without spending any actual time.
3)

is the minimum travel time between node and j. Particularly,

is the
minimum travel distance between the depot and node i ;

is the minimum
travel time between node and the lab.

29

4)

is defined as the measure of dissimilarity, with mathematical expression

∑

∑

.

equals to 1 if customer i is visited by vehicle
k either on day d or on day 0, but not both.

equals to 0 if customer is visited
by vehicle both on day d and on day 0, or on neither days. In other words the
dissimilarity is counted as one if a customer is visited by a different vehicle than
in the master plan.

Problem formulation:
Minimize

∑ ∑(∑ ∑

∑ ∑

∑ ∑

)

∑ ∑

∑ ∑ ∑

(3.1)

Subject to:
Routing constraints:

∑ ∑

∑ ∑

(3.2)

∑

∑

∑

∑

∑

(3.3)

∑

∑

(3.4)

30

∑

∑

(3.5)

(

)

(3.6)

(

)

(3.7)

(

)

(3.8)

(3.9)

(3.10)

(

)

(

)

(3.11)

(

)

(3.12)

∑

∑

(3.13)

Domain constraints:

(3.14)

(3.15)

(3.16)

(3.17)

(3.18)

(3.19)

(3.20)

(3.21)

(3.22)

31

As previously described, the healthcare courier delivery problem should focus not
only on plans with minimum travelling cost, but also those with high level of customer
service. Therefore, the objective function of our model, as shown in Equation (3.1), is to
minimize a total cost, that is composed of traveling cost, outsourcing cost, and route
dissimilarity cost. The travel cost is represented by

∑ ∑ ( ∑ ∑

∑ ∑

∑ ∑

), which is the total distance
traveled by all the vehicles in the planning horizon. The outsourcing cost is represented
by

∑ ∑

, which is the total number of trips that a taxi is used to handle the
demands unmet by the regular fleet. It should be noted that this term could easily include
the total taxi distance if we change it to

∑ ∑

. The route dissimilarity
is measured by

∑ ∑ ∑

, the total number of customers in the planning
horizon, that are serviced by a vehicle different from the one servicing it in the master
plan.
There are two groups of constraints in our model, namely routing constraints and
domain constraints. Constraint (3.2) assures on each day that each customer should be
visited directly from the depot, right after a vehicle services customer , or by a taxi when
the regular fleet is unavailable. Constraint (3.3) assures that each vehicle must leave the
customer after visiting it. It also addresses the fact that a customer has to be visited by a
vehicle in one of its trips in a day. Constraint (3.4) ensures that each individual trip
should start with leaving the depot and end by returning to the depot. Constraint (3.5)
enforces the usage of early trips as much as possible, which force the empty trips close to
the end of the day instead of at the beginning of the day. Constraint (3.6) assures the

32

relationship of arrival times at customers and , when customer is visited right after
is visited. Constraint (3.7) expresses the relationship of arrival time to customer , when
is the first customer request a vehicle handles in a trip. Constraint (3.8) expresses the
relationship of arrival time to customer , when is the last customer request a vehicle
handles in a trip. Constraint (3.9) enforces that the finish time of a trip of a vehicle should
be no later than the start time of the next trip of the vehicle. Constraint (3.10) enforces the
arrival time of a vehicle at a customer to be in the required time window for handling the
customer request. Constraint (3.11) requires that the arrival time at a customer on a trip
should be between the start time and the end time of the trip. Constraint (3.12) requires
that each vehicle should visit the lab before the drop-off deadline of each specimen
collected by a vehicle on a trip. Constraint (3.13) is another representation of our
expression for dissimilarity

∑

∑

. It removes the usage of the
absolute value in the expression, so that the system is linearized. Constraints (3.14) –
(3.22) are the domain constraints.
There are a few observations we make from this model.
1. Multi trip of a vehicle is used because of the time window constraint of the
customer requests. In other words, a trip of a vehicle needs to be finished so that
all the medical specimens can be delivered to the lab on time.
2. It is optimal to combine some customer requests into one trip of a vehicle, such
that the summation of the travelling time of the vehicle is minimized.
3. It is optimal to use the same vehicle to visit the same customers on different days
so that the route similarity is increased.

33

4. When the cost of introducing a third party vehicle (e.g., taxi) is comparatively
high, it is more beneficial to use the regular fleet instead of a third party vehicle.

3.2 A Sample Problem
To illustrate this model, consider the following small example. There is one day
(Day 1) in the planning horizon, and another day (Day 0) is used to represent the master
plan. There are five customers in the small problem. All of them request for delivery
service on Day 1, and all of them will be included in the master plan. The location of the
customers, the pickup time window, and the deadline for drop-off at the lab are shown in
Table 3.1. The vehicle is assumed travelling at a speed of 50 km/hour. The other
coefficients are

,

, and

. In this small example, we use a high
value for taxi cost, to discourage the use of taxis and we can focus on the optimal routes
generated with the vehicles by the healthcare provider.
The small instance uses c1 to c5 to represent customer 1 to customer 5 requesting
service, and we use c0 and c6 to represent the depot and the lab. Notation d0 and d1 are
used to represent the day for the master plan and day 1, and k1 and k2 are used to
represent the two vehicles operated by the healthcare provider to handle the service.
Notation w1 to w5 are used to represent the five trips that a vehicle can make.

34

Table 3.1: Customer Information for the Sample Problem

x y Earliest Pickup Latest pickup Latest Drop-off
Day 1
Customer 1 110 70 16 18 22
Customer 2 113 73 13 15 19
Customer 3 113 70 6 8 12
Customer 4 110 73 9 11 15
Customer 5 108 68 9 11 15

Figure 3.2: The Optimal Routing Solution for Day 0 / 1

35

The optimal solution can be illustrated by Figure 3.2. Both in the master (day 0)
and the daily (day 1) plan, only vehicle k1 is used with two trips. For the first trip, it
travels from the depot to customer 3, then to customer 4, then customer 5, and then back
to the depot. It travels from the depot to customer 2, then to customer 1, then back to the
depot for its second trip. The optimal objective value of the small sample problem is 21.6
km. No taxi is introduced in this problem and there is no cost on route dissimilarity as the
daily plan is the same as the master plan. In this example, artificial trips (trips between
node 0 and node 6) are observed because in the model we assume the maximum number
of trips each vehicle can make is the number of customer requests. In the optimal solution,
however, several customer requests can be combined and handled in one trip of a vehicle.

3.3 Heuristic
As discussed in earlier chapters exact solution methods will only be able to solve
small size instances of this problem. As there are days during the planning horizon,
and on each vehicle there is trips (including real and artificial trips), then solving a
problem with n customers and k vehicles is equivalent to solving a routing problem with
customers with vehicles. Therefore, heuristic algorithms need to be
constructed, in order to solve large size problems. In this section, we present a heuristic to
solve this courier delivery problem with urgent requests. The heuristic can be divided into
four parts. The first part is the insertion algorithm, which is used repeatedly when
constructing master and daily routes. The second part is Tabu search, which is used to

36

obtain a near-optimal solution for the routing solutions. Insertion and Tabu search are
generic techniques for this problem and are used to obtain efficient routes. The third part
is the construction of the master routes, which can be used to build daily routes. The last
part is the construction of the daily routes, basically focusing on the handling of the
urgent requests. The construction of master routes against the daily routes is a specific
separation of the problem in order to be able to manage the problem size.

3.2.1 Insertion
Insertion techniques have been widely used as an efficient method for solving
vehicle routing and scheduling problems. Insertion heuristics are popular because they
are fast, easy to implement, and produce good solutions, and they are easy to extend to
handle complicating constraints. A comprehensive review of insertion heuristics can be
found in Campbell et al. (2004).
Our heuristic uses the insertion technique as a basic cell, and builds the master
routes and the daily plans by calling the insertion heuristic. The insertion heuristics used
for constructing master routes and daily routes are different due to the objective function
considered in each. The insertion for constructing master routes needs only to consider
the efficiency in travel distance. The insertion for constructing daily routes, however,
needs to consider travel distance, as well as cost for taxis and route dissimilarity. One
reason behind this strategy is not to use taxi in the master routes as long as it is feasible to
use the fleet of vehicles. The other reason is that route dissimilarity is measured on daily

37

routes against master routes, and it is only meaningful to include dissimilarity in the cost
function for inserting requests into daily routes.

Algorithm 1: Insertion of request to form master routes
Input: the scheduled routes; a request to insert.
Output: the updated routes or taxi cost.

for all the positions in all the activated routes
find the feasible insertion positions with minimum insertion cost;
if the insertion is feasible then
update the routes;
else if there is a vehicle to activate then
put the request on the new vehicle;
else update the taxi cost;

An insertion heuristic for building master routes is introduced (Algorithm 1). On a
daily basis, for the customer requests that are not in the master routes, the insertion
algorithm changes to Algorithm1.1. In this updated algorithm, the insertion cost is the
summation of the travel cost and the dissimilarity cost when inserting the request into the
regular fleet; the taxi cost is the summation of the travelling cost and the dissimilarity
cost when using the taxi service.

38

The basic procedure of an insertion is illustrated in Figure 3.3. In Algorithm 1 and
Algorithm 1.1, to check the feasibility of an insertion, we need to “update the arrival
times” after we tentatively insert a pickup or delivery of a request. The arrival time at
each node can be calculated as

, where node and
node are the two nodes consecutively visited by a vehicle.

is the arrival time at node
,

is the earliest time a vehicle can visit node , and

is the travel time between
node and node .

39

Algorithm 1.1: Insertion of a daily request not in the master routes
Input: the scheduled routes; the master routes; a request to insert.
Output: the updated routes.

for all the positions in all routes
find the feasible insertion positions with minimum insertion cost;
calculate taxi cost;
if minimum insertion cost is smaller than taxi cost
then use fleet;
else use taxi;
if use fleet
then update the routes;
if use taxi or infeasible to insert
then update the taxi cost;

In these algorithms, the feasibility of an insertion can be confirmed by checking if

for all the nodes in the route, and

is the latest time that node can be visited. It
should be noted that each customer request corresponds to the handling of a pickup and a
delivery request pair. For the pickup of a customer request,

is the earliest pickup time
of the specimen and

is the latest pickup time of the specimen. For the delivery of the
customer request,

is set to 0 (the earliest time the vehicle can return to the lab for

40

delivery) and

is the latest time the specimen has to be delivered at the lab. The
insertion of a customer request is feasible, if both the pickup and the delivery of the
request are feasible.

Figure 3.3: Insertion of Customer Request i

Before Insertion:

Vehicle 1 Depot Depot

…

Vehicle k Depot Depot

…

Vehicle K Depot Depot

After Insertion of

:

Vehicle 1 Depot Depot

…

Vehicle k Depot

Depot

…

Vehicle K Depot Depot

After Insertion of

:

Vehicle 1 Depot Depot

…

Vehicle k Depot

Depot

…

Vehicle K Depot Depot

Pickup of customer request i

Delivery of customer request i

The cost on the distance traveled can be calculated as follows. If the pickup and
delivery of a request are inserted as two consecutive nodes, i.e., if the pickup is inserted
as node and the delivery is inserted as node in a route (see Figure 3.4), then the

41

insertion cost can be calculated as

. If the pickup and
delivery of a request are inserted not next to each other, i.e., if the pickup is inserted as
node and the delivery is inserted as node ( ) (see Figure 3.5), then the
insertion cost can be calculated as

.

Figure 3.4: Pickup is followed directly by delivery

Pickup Delivery

i-2 i-1 i i+1

Figure 3.5: Pickup is followed directly by delivery

Pickup

Delivery

i-2 i-1 i i+a-1 i+a i+a+1

The taxi cost is made up of two parts in the algorithms. One is a fixed pickup cost,
which is proportional to the number of trips. The other is the variable cost, which is in
proportion to the distance from the pickup location to the delivery location.
The cost for dissimilarity is calculated by comparing the scheduled routes to the
master routes. If a request is serviced by the same vehicle, then the dissimilarity is 0;
otherwise, it is 1. It should be noted that we assume the dissimilarity cost is always 1,
when a customer is visited by a taxi.

42

It should be noted that the described insertion heuristic is a parallel insertion
procedure. However, to construct the master routes, we first use the activated vehicle for
the handling of the delivery requests. We activate a new vehicle when it is not feasible to
handle the request with a currently activated vehicle. This approach is favored for less
usage of vehicles in the master routes, which is another factor of cost reduction for the
healthcare provider. This approach tends to use less vehicles to handle the requests.

3.2.2 Tabu Search
As described above, insertion heuristic algorithms are used to build initial
solutions for the master and the daily routes. However, in order to obtain efficient
solutions, a Tabu search algorithm (Algorithm 2) is developed as the post phase
improvement for the master and the daily routes. The implementation of the Tabu search
considers the neighborhoods obtained from the standard 2-opt exchange move (Lin, 1965)
and the -interchange move (Osman, 1993). The -interchange operators are generated
by randomly selecting two requests from two different routes, and exchanging the
requests by interchanging the pickup and the delivery of each request (see Figure 3.6).
As the problem requires the pickup and delivery of a request handled by the same vehicle,
it must be assured that the pickup and the delivery of a request stay on the same vehicle.
The 2-opt exchange operator is generated by randomly selecting two nodes (pickup or
delivery) on a randomly selected vehicle (see Figure 3.7). As a package can only be

43

delivered after it is picked up, it must be assured that the delivery of any request is
located after the pickup of the request.

Algorithm 2: Tabu Search Algorithm
Input: a master plan or a daily plan to improve
Output: improved master plan or daily plan

repeat
randomly chose two routes from the solution
generate

neighbors from -interchange operator
generate

neighbors from 2-opt operator
choose the best solution and make the move;
randomly generate tabu tenure from a uniform distribution U (

);
if the move is -interchange then
set the tabu for moving the exchanged requests for iterations;
else
set the tabu for moving the exchanged nodes for iterations;
until no improvement in

iterations;
calculate the objective and save the current solution;

In each iteration, the Tabu search generates

-interchange neighbors and

2-opt neighbors of the current solution. The number of Tabu iterations is a

44

random number uniformly distributed in (

). The Tabu search at each iteration
moves to the best neighbor. A temporary move to a worse solution is allowed to escape
from the local minimum. The Tabu status is overwritten if the new solution improves
from the best solution. The algorithm terminates if there is no improvement in

iterations.
The Tabu search algorithm is applied to both the master routes and the daily
routes. When it is applied on master routes, the objective is to minimize the total distance
traveled, as to have more slack time to accommodate the random requests. When it is
applied on daily routes, the objective is to minimize the cost including total distance
traveled, taxi cost, and route dissimilarity.

Figure 3.6: -interchange Operator
Before the Move:

Route 1 Pickup 1 Delivery 1

Route 2 Pickup 2 Delivery 2

After the Move:

Route 1 Pickup 2 Delivery 2

Route 2 Pickup 1 Delivery 1

45

Figure 3.7: 2-opt Exchange Operator
Before the Move:

Route Pickup 1 Delivery 1 Pickup 2 Delivery 2

After the Move:

Route Pickup 1 Pickup 2 Delivery 1 Delivery 2

3.2.3 Master Routes
When forming master routes, we need to consider the following conflicting
objectives: an efficient template to satisfy regular demands for routine business, and be
able to rapidly adapt to random and urgent requests that arise throughout the day.
Therefore we face two main challenges when determining the master plan for the courier
routing problem:
1. To determine which requests to include in the master plan.
2. How to obtain large scale, multi-shift routing solutions under uncertainty.
The extreme cases in classifying regular and urgent requests are comparatively
easy. A customer that requests service every day usually has wide time windows and
should be considered a regular request and be included in the master plan. A customer
request in a matter of life and death (i.e. testing of compatibility of donor organs) should
be considered urgent, and should not be included in the master route as it occurs rarely.

46

The problem is how to classify routing requests that have wide windows and
occur randomly. If the frequency is high, they could be considered regular requests.
Requests of this sort should be scheduled in the master routes and skipped in the day they
do not appear. If the request occurs rarely then they should not be included in the master
plan, but should rather be handled in the most efficient way, such as to be included in
some master route or use a separate dedicated vehicle.
We study different configurations of the courier delivery problem to identify how
to balance the benefits of master routes and recourse actions to better service urgent
requests. Specifically, if the master routes are built to service a large amount of regular
requests, most demand points could be satisfied efficiently; however, there will be less
slack time left to handle urgent requests which arise later, and will therefore drive up the
outsourcing costs. On the other hand, if the master plans are built to service very few
regular requests, then it will be less efficient to service most daily requests and
determining the daily routes will be more difficult, but more slack will be left in the
master routes to handle urgent request. In this situation, it is possible that some vehicles
would not be used in the master routes but be used as a dedicated vehicle to handle urgent
requests only.
In order to obtain efficient master routes for the courier delivery problem,
algorithm 3 is developed as the solution procedure. The idea is to include the customers
that have a high probability of occurrence. An insertion algorithm is used to construct an
initial solution for master routes. Tabu search is used to improve the efficiency in travel
distance so that more slack is obtained for more random urgent requests. The simulation

47

results on the comparison of the configuration of the master routes can be found in
section 5.

Algorithm 3: Formation of a Master Plan
Input: All the customers to insert; the probability of a customer to request service in a
day; a threshold for probability of customer occurring
Output: Master routes

for all the customers do
if the occurring probability of a customer is larger than the threshold then
include the customer into the master plan by calling Algorithm 1.1;
end for
improve the master routes with Tabu search by calling Algorithm 2;

3.2.4 Daily Plans with Urgent Requests
As described earlier, in the first stage, we obtain the solution of an effective
master plan, and in the second stage, we adjust the planned routes to handle the urgent
requests occurred. The objective of the second stage is to accommodate as many of the
urgent requests as possible with the existing resources, including the slack time of the

48

vehicles for the master routes and the dedicated vehicles for urgent requests. In this
second stage, we need to quickly modify the master plan to service the updated requests.
An ideal benchmark solution is obtained by solving the problem once the
uncertainty is revealed; however, it is impossible to implement because of the size of a
real problem and the limitation in the current computational power. Alternative recourse
actions to implemente must have the three objectives:
1. Easy and quick to compute
2. Obtain a high quality solution
3. Easy to execute with small deviations from the planned routes
If the recourse action allows skipping customers then the problem can be
approximated by a knapsack problem (Kellerer et al., 2004). The recourse strategy is
inspired by the classic recourse strategy (strategy b) in Bertsimas (1992), which assumes
the demand will be revealed before the vehicle leaves the depot to service the customer.
Therefore, a customer will be skipped if it does not request service in a particular day.
In our strategy, we also make the same assumption that the travel time and the
actual demand on each day are known before the vehicle departs from the depot. The
recourse action in each day includes skipping the customers in the master routes that do
not request service from the master plan and inserting the customers who request service
into the existing routes if possible.
The heuristic algorithm for building daily plans by adapting the master plan using
recourse action can be found in Algorithm 4. In the next section, we compare our
heuristics to other benchmark approaches.

49

Algorithm4: Formation of Daily Plans
Input: the master plan; daily requests
Output: the daily plans

for each day do
take the master plan (generated by Algorithm 3) as the initial daily plan;
for all the requests in the master plan
if the request does not occur on the day then
drop the request from the daily plan;
end for
for all the requests on the day do
if a request is NOT included in the master plan then
insert the request into the daily plan by calling algorithm 1.1;
end for
improve the daily plan with Tabu search by calling Algorithm 2;
for all the requests serviced by taxi do
try inserting the request into the daily plan again by calling algorithm 1.1;
end for

50

3.4 Experimental Results
3.4.1 Data Generation and Input Parameters
We test our model and heuristic using simulation on the following randomly
generated data set and input parameters. We assume a service area of a square plane.
Consider a city with a two-dimensional coordinate system, the boundary of the city is
from -10 to 10 miles in both the x-axis and the y-axis. The depot and the only lab where
all the vehicles start and end their services every day are located at the center of the city,
that is (0, 0) on the two-dimensional plane. (see Figure 3.8)

Figure 3.8: City Size and Customer Locations

51

The locations of all the potential customers are known a priori, and the potential
customers, in each experiment, are uniformly distributed in the city (see Figure 3.8).
Some customers request service at a fixed time every day (deterministic requests), while
others only request services at a fixed time on some of the days (random requests). Each
random request has a probability of occurring on each day where is sampled from a
uniform [0, 1] distribution. The earliest pickup time (the earliest time a customer can be
visited) of a request is uniformly distributed from 9 am to 5 pm on each day. The latest
pickup time (the latest time a customer can be visited) of the request is 30 minutes after
the corresponding earliest pickup time. Each request has a latest drop-off time (a deadline
by which the sample has to be delivered to the lab); the latest drop-off time for regular
requests is 2 hours after its earliest pickup time, and the latest drop-off time for urgent
requests is 1 hour after its earliest pickup time (see Table 3.2).

Table 3.2: Time Windows of Regular and Urgent Requests
Earliest Pickup Time
(hours)
Latest Pickup Time (hours) Latest Drop-off Time
(hours)
Regular Request [9, 17] Uniformly 0.5+Earliest Pickup Time 2 + Earliest Pickup Time
Urgent Request [9, 17] Uniformly 0.5 + Earliest Pickup Time 1 + Earliest Pickup Time

We assume a given number of vehicles to service the requests, which might be
different in each experiment. And the vehicles drive at an average speed of 30 miles per
hour to service the requests.

52

3.4.2 Simulations and Results
In the section, we show the simulation results with the above assumptions and
data inputs. In each experiment, we assume a fixed number of potential requests, a fixed
proportion of deterministic requests among all the requests, and a fixed number of
available vehicles to handle the requests. The result of each experiment is taken by
averaging the results of 10 replications, each of which takes the average result of 10 days.
In each replication, a random request customer is assigned a probability p of occurring,
where p is a sample from a uniform [0, 1]. In each day of a replication, we determine the
occurrence of each request by sampling based on the probability .
In each experiment, we compare the following four strategies in terms of average
travel distance, average taxi cost, average route dissimilarity, average number of taxi trips,
average travel distance per requests, and average total cost, on a daily basis.
A. TAXI: schedule all the deterministic requests as master routes using the insertion
heuristic algorithm; use a third party courier, i.e., taxi, for all the random requests.
(Apply Algorithm 3with a customer occurrence probability threshold of 1 to build
the master routes; handle all the random requests by taxi.)
B. IND: form a schedule independently for each day, using the insertion heuristic.
(Use Algorithm 1 to build daily routes independently.)
C. MFIX: schedule the deterministic requests as master routes, and insert the random
requests into the scheduled routes on each day. Use taxi if it is infeasible or more

53

expensive to insert the random request into the scheduled routes. (Use Algorithm
3 to build the daily plans with a customer occurrence probability threshold of 1.)
D. MHALF: schedule the deterministic requests and high occurring probability
requests (those who have an occurrence probability of 0.5 or higher) as master
routes. In the daily schedules, skip the non-occurring customers and insert the
unscheduled random requests into the scheduled routes. Use a taxi if it is
infeasible or more expensive to insert the random request into the scheduled
routes. (Use Algorithm 3 to build the daily plans with a customer occurrence
probability threshold of 0.5.)
The parameters we use in the experiments for the Tabu search algorithms are

,

,

,

, and

. Table 3.3, Table 3.4,
and Table 3.5 summarize the simulation results with 50, 100, and 500 customers
respectively. Simulations have been done with different combinations on the number of
vehicles and cost parameters. In these tables, “#Customers” gives the number of potential
customers; “#Vehicle” shows the number of vehicles used in the simulation.

is the unit
cost per hour traveled.

is the fixed cost per trip of taxi.

is the varying cost per
hour the taxi traveled.

is the unit cost per count of dissimilarity. Column “Proportion
Fix” shows the proportion of deterministic customers among all the potential customers.
Column “Strategy” lists the four strategies we are comparing. Column “Travel” shows
the total distance that a vehicle travels per day on average. Column “Taxi Cost” shows
the average daily taxi cost. Column “Dissimilarity” shows the average dissimilarity,
which is the total number of vehicles used in the daily routes that is different than the one

54

in the master routes. If a taxi is used, then the dissimilarity is increased by one, as we
assume that a different taxi will come to service a different request. Moreover, as there is
no master routes generated for independent scheduling, the dissimilarity is calculated by
comparing the daily routes to the master routes generated in strategy “master fix”.
Column “#Taxi Trips” shows the total number of daily taxi trips introduced on average.
Column “Travel/Requests” shows the distance that a vehicle travels to service a request
on a daily basis on average. Column “Total Cost” shows the average daily total cost
including travel cost, taxi cost, and cost on dissimilarity. It is the summation of each type
of costs weighted by the unit cost of that type.
The results are also shown in Figure 3.9 – Figure 3.26 as follows.

55

Table 3.3: Simulation Results with 50 Customers

#Customers: 50; #Vehicles: 4; αt=1, αo_f=100, αo_v=0.5, αs=0.01; #Customers: 50; #Vehicles: 2; αt=1, αo_f=100, αo_v=0.5, αs=0.01;
Proporti
on Fixed
Strategy Travel
Taxi
Cost
Dissmila
rity
# Taxi
Trips
Travel/R
equest
Total
Cost
Proporti
on Fixed
Strategy Travel
Taxi
Cost
Dissmila
rity
# Taxi
Trips
Travel/R
equest
Total
Cost
0.8 TAXI 2.89 424.53 4.24 4.24 0.28 436.13 0.8 TAXI 5.61 694.87 6.94 6.94 0.3 706.15
IND 3.05 0 21.58 0 0.27 12.43 IND 6.03 294.4 17.23 2.94 0.29 306.63
MFIX 3.14 0 6.08 0 0.28 12.61 MFIX 6.02 300.41 7.46 3 0.29 312.52
MHALF 3.05 0 2.34 0 0.27 12.22 MHALF 6.12 209.28 6.58 2.09 0.29 221.58
0.6 TAXI 2.23 1012.32 10.11 10.11 0.29 1021.34 0.6 TAXI 4.46 1082.41 10.81 10.81 0.3 1091.43
IND 2.94 0 23.32 0 0.29 12.01 IND 5.79 188.27 21.69 1.88 0.3 200.06
MFIX 2.93 0 12.55 0 0.29 11.86 MFIX 5.72 186.29 11.76 1.86 0.3 197.84
MHALF 2.98 0 5.4 0 0.29 11.99 MHALF 5.85 169.25 6.8 1.69 0.3 181.01
0.4 TAXI 1.69 1391.76 13.9 13.9 0.32 1398.66 0.4 TAXI 3.38 1391.76 13.9 13.9 0.32 1398.66
IND 2.71 0 22.84 0 0.31 11.08 IND 5.35 126.18 21.65 1.26 0.32 137.1
MFIX 2.73 0 17.07 0 0.31 11.1 MFIX 5.24 121.19 15.73 1.21 0.31 131.82
MHALF 2.72 0 7.8 0 0.31 10.95 MHALF 5.35 153.21 8.23 1.53 0.32 163.99
0.2 TAXI 1.01 2005.57 20.03 20.03 0.38 2009.81 0.2 TAXI 2.04 2005.57 20.03 20.03 0.38 2009.84
IND 2.62 0 23.94 0 0.34 10.73 IND 5.15 54.08 23.58 0.54 0.34 64.61
MFIX 2.61 0 22.5 0 0.34 10.65 MFIX 5.09 68.1 21.83 0.68 0.34 78.51
MHALF 2.59 0 8.13 0 0.34 10.44 MHALF 5.15 60.09 7.53 0.6 0.34 70.46
#Customers: 50; #Vehicles: 4; αt=1, αo_f=100, αo_v=0.5, αs=100; #Customers: 50; #Vehicles: 2; αt=1, αo_f=100, αo_v=0.5, αs=100;
Proporti
on Fixed
Strategy Travel
Taxi
Cost
Dissmila
rity
# Taxi
Trips
Travel/R
equest
Total
Cost
Proporti
on Fixed
Strategy Travel
Taxi
Cost
Dissmila
rity
# Taxi
Trips
Travel/R
equest
Total
Cost
0.8 TAXI 2.88 424.53 4.24 4.24 0.28 860.06 0.8 TAXI 5.61 694.87 6.94 6.94 0.3 1400.09
IND 3.09 0 24.17 0 0.28 2429.37 IND 5.98 222.31 21.22 2.22 0.28 2356.27
MFIX 3.2 0 4.24 0 0.28 436.79 MFIX 6.01 324.45 6.94 3.24 0.29 1030.48
MHALF 3.11 0 1.18 0 0.28 130.44 MHALF 6.22 230.33 5.34 2.3 0.29 776.77
0.6 TAXI 2.22 1012.32 10.11 10.11 0.29 2032.19 0.6 TAXI 4.48 1082.41 10.81 10.81 0.3 2172.38
IND 2.97 0 24.22 0 0.29 2433.89 IND 5.78 155.23 21.72 1.55 0.3 2338.78
MFIX 3.09 0 10.11 0 0.31 1023.37 MFIX 5.77 225.34 10.81 2.25 0.3 1317.87
MHALF 3.09 0 2.38 0 0.31 250.37 MHALF 5.94 191.28 4.75 1.91 0.31 678.15
0.4 TAXI 1.67 1391.76 13.9 13.9 0.32 2788.44 0.4 TAXI 3.37 1391.76 13.9 13.9 0.32 2788.49
IND 2.72 0 23.73 0 0.31 2383.89 IND 5.35 124.18 21.66 1.24 0.32 2300.89
MFIX 2.87 0 13.9 0 0.33 1401.5 MFIX 5.32 147.23 13.9 1.47 0.32 1547.87
MHALF 2.84 0 4.01 0 0.32 412.35 MHALF 5.42 165.24 5.72 1.65 0.32 748.07
0.2 TAXI 1.02 2005.57 20.03 20.03 0.38 4012.64 0.2 TAXI 2.02 2005.57 20.03 20.03 0.38 4012.62
IND 2.6 0 24.38 0 0.34 2448.4 IND 5.1 69.1 24 0.69 0.34 2479.31
MFIX 2.73 0 20.03 0 0.36 2013.93 MFIX 5.29 92.14 20.03 0.92 0.36 2105.71
MHALF 2.68 0 5.04 0 0.35 514.7 MHALF 5.21 66.1 5.49 0.66 0.35 625.52
#Customers: 50; #Vehicles: 4; αt=1, αo_f=0.5, αo_v=0.5, αs=100; #Customers: 50; #Vehicles: 2; αt=1, αo_f=0.5, αo_v=0.5, αs=100;
Proporti
on Fixed
Strategy Travel
Taxi
Cost
Dissmila
rity
# Taxi
Trips
Travel/R
equest
Total
Cost
Proporti
on Fixed
Strategy Travel
Taxi
Cost
Dissmila
rity
# Taxi
Trips
Travel/R
equest
Total
Cost
0.8 TAXI 2.91 2.65 4.24 4.24 0.29 438.28 0.8 TAXI 5.63 4.34 6.94 6.94 0.3 709.61
IND 2.84 1.43 24.37 2.08 0.27 2449.79 IND 5.37 2.74 23.02 4.05 0.26 2315.49
MFIX 3.13 0.36 4.24 0.52 0.28 436.9 MFIX 5.89 2.17 6.94 3.43 0.28 707.96
MHALF 3.07 0.07 1.18 0.1 0.27 130.35 MHALF 6.13 1.84 5.34 2.91 0.29 548.09
0.6 TAXI 2.22 6.37 10.11 10.11 0.29 1026.24 0.6 TAXI 4.41 6.81 10.81 10.81 0.3 1096.63
IND 2.6 1.99 24.83 2.89 0.28 2495.39 IND 5.03 2.82 22.77 4.14 0.28 2289.87
MFIX 2.85 0.91 10.11 1.31 0.29 1023.33 MFIX 5.41 2.13 10.81 3.23 0.29 1093.95
MHALF 2.99 0.25 2.38 0.37 0.3 250.22 MHALF 5.88 1.33 4.75 2.06 0.31 488.09
0.4 TAXI 1.7 8.71 13.9 13.9 0.32 1405.49 0.4 TAXI 3.41 8.71 13.9 13.9 0.32 1405.54
IND 2.34 1.99 23.51 2.9 0.29 2362.34 IND 4.47 2.7 22.15 3.95 0.29 2226.65
MFIX 2.57 1.22 13.9 1.77 0.31 1401.5 MFIX 4.92 1.89 13.9 2.81 0.3 1401.74
MHALF 2.78 0.26 4.01 0.38 0.32 412.4 MHALF 5.33 1.27 5.72 1.95 0.32 583.92
0.2 TAXI 1.01 12.58 20.03 20.03 0.38 2019.62 0.2 TAXI 2.02 12.58 20.03 20.03 0.38 2019.62
IND 2.13 2.18 24.55 3.16 0.31 2465.69 IND 4.21 2.34 24.19 3.41 0.31 2429.76
MFIX 2.31 1.81 20.03 2.62 0.33 2014.05 MFIX 4.5 2.17 20.03 3.19 0.33 2014.18
MHALF 2.6 0.34 5.04 0.49 0.35 514.75 MHALF 5.04 0.73 5.49 1.09 0.34 559.81

56

Table 3.4: Simulation Results with100 Customers

#Customers: 100; #Vehicles: 8; αt=1, αo_f=100, αo_v=0.5, αs=0.01; #Customers: 100; #Vehicles: 4; αt=1, αo_f=100, αo_v=0.5, αs=0.01;
Proporti
on Fixed
Strategy Travel
Taxi
Cost
Dissmila
rity
# Taxi
Trips
Travel/R
equest
Total
Cost
Proporti
on Fixed
Strategy Travel
Taxi
Cost
Dissmila
rity
# Taxi
Trips
Travel/R
equest
Total
Cost
0.8 TAXI 2.31 969.30 9.68 9.68 0.23 987.90 0.8 TAXI 4.64 999.34 9.98 9.98 0.23 1017.99
IND 2.61 0.00 55.72 0.00 0.23 21.46 IND 5.25 45.07 54.09 0.45 0.23 66.61
MFIX 2.63 0.00 13.84 0.00 0.23 21.17 MFIX 5.23 104.17 13.74 1.04 0.23 125.21
MHALF 2.63 0.00 5.57 0.00 0.23 21.06 MHALF 5.20 64.09 6.60 0.64 0.23 84.97
0.6 TAXI 1.91 2060.73 20.58 20.58 0.25 2076.23 0.6 TAXI 3.88 2070.74 20.68 20.68 0.25 2086.45
IND 2.54 0.00 55.64 0.00 0.25 20.90 IND 5.11 42.07 54.46 0.42 0.25 63.07
MFIX 2.54 0.00 26.60 0.00 0.25 20.58 MFIX 5.11 49.09 25.92 0.49 0.25 69.77
MHALF 2.50 0.00 8.92 0.00 0.24 20.06 MHALF 4.98 39.07 9.94 0.39 0.25 59.08
0.4 TAXI 1.45 2844.79 28.41 28.41 0.27 2856.64 0.4 TAXI 2.89 2844.79 28.41 28.41 0.27 2856.64
IND 2.38 0.00 51.23 0.00 0.27 19.58 IND 4.75 20.03 51.38 0.20 0.27 39.56
MFIX 2.39 0.00 33.86 0.00 0.27 19.47 MFIX 4.70 28.05 33.82 0.28 0.26 47.20
MHALF 2.38 0.00 11.32 0.00 0.27 19.13 MHALF 4.68 23.04 12.52 0.23 0.26 41.90
0.2 TAXI 0.94 3765.88 37.61 37.61 0.33 3773.76 0.2 TAXI 1.87 3765.88 37.61 37.61 0.33 3773.75
IND 2.21 0.00 49.71 0.00 0.29 18.19 IND 4.41 12.02 49.94 0.12 0.29 30.14
MFIX 2.22 0.00 42.08 0.00 0.29 18.17 MFIX 4.38 22.04 41.62 0.22 0.29 39.99
MHALF 2.17 0.00 14.84 0.00 0.29 17.48 MHALF 4.34 12.02 15.18 0.12 0.29 29.52
#Customers: 100; #Vehicles: 8; αt=1, αo_f=100, αo_v=0.5, αs=100; #Customers: 100; #Vehicles: 4; αt=1, αo_f=100, αo_v=0.5, αs=100;
Proporti
on Fixed
Strategy Travel
Taxi
Cost
Dissmila
rity
# Taxi
Trips
Travel/R
equest
Total
Cost
Proporti
on Fixed
Strategy Travel
Taxi
Cost
Dissmila
rity
# Taxi
Trips
Travel/R
equest
Total
Cost
0.8 TAXI 2.33 969.30 9.68 9.68 0.23 1955.90 0.8 TAXI 4.70 999.34 9.98 9.98 0.23 2016.13
IND 2.65 0.00 58.21 0.00 0.23 5842.18 IND 5.29 53.08 55.02 0.53 0.23 5576.24
MFIX 2.69 0.00 9.68 0.00 0.24 989.48 MFIX 5.30 116.21 9.98 1.16 0.24 1135.41
MHALF 2.71 0.00 2.35 0.00 0.24 256.66 MHALF 5.29 95.14 4.06 0.95 0.24 522.31
0.6 TAXI 1.93 2060.73 20.58 20.58 0.25 4134.14 0.6 TAXI 3.77 2070.74 20.68 20.68 0.25 4153.84
IND 2.55 0.00 55.80 0.00 0.25 5600.41 IND 5.08 49.08 54.13 0.49 0.25 5482.38
MFIX 2.69 0.00 20.58 0.00 0.26 2079.51 MFIX 5.34 49.09 20.68 0.49 0.26 2138.43
MHALF 2.59 0.00 4.50 0.00 0.25 470.72 MHALF 5.06 75.13 5.70 0.75 0.25 665.36
0.4 TAXI 1.44 2844.79 28.41 28.41 0.27 5697.30 0.4 TAXI 2.93 2844.79 28.41 28.41 0.27 5697.51
IND 2.38 0.00 52.34 0.00 0.27 5253.03 IND 4.71 12.02 51.40 0.12 0.26 5170.85
MFIX 2.54 0.00 28.41 0.00 0.28 2861.33 MFIX 4.99 40.07 28.41 0.40 0.28 2901.03
MHALF 2.49 0.00 7.36 0.00 0.28 755.89 MHALF 4.88 26.04 8.08 0.26 0.27 853.57
0.2 TAXI 0.93 3765.88 37.61 37.61 0.32 7534.31 0.2 TAXI 1.86 3765.88 37.61 37.61 0.32 7534.34
IND 2.21 0.00 50.30 0.00 0.29 5047.70 IND 4.41 13.02 49.81 0.13 0.29 5011.65
MFIX 2.36 0.00 37.61 0.00 0.31 3779.86 MFIX 4.69 15.02 37.61 0.15 0.31 3794.77
MHALF 2.31 0.00 10.10 0.00 0.30 1028.46 MHALF 4.60 14.03 10.20 0.14 0.30 1052.44
#Customers: 100; #Vehicles: 8; αt=1, αo_f=0.5, αo_v=0.5, αs=100; #Customers: 100; #Vehicles: 4; αt=1, αo_f=0.5, αo_v=0.5, αs=100;
Proporti
on Fixed
Strategy Travel
Taxi
Cost
Dissmila
rity
# Taxi
Trips
Travel/R
equest
Total
Cost
Proporti
on Fixed
Strategy Travel
Taxi
Cost
Dissmila
rity
# Taxi
Trips
Travel/R
equest
Total
Cost
0.8 TAXI 2.35 6.14 9.68 9.68 0.23 992.92 0.8 TAXI 4.70 6.33 9.98 9.98 0.23 1023.15
IND 2.44 3.23 56.29 4.68 0.23 5651.76 IND 4.88 3.57 55.51 5.19 0.23 5574.11
MFIX 2.57 0.83 9.68 1.21 0.23 989.41 MFIX 5.13 1.22 9.98 1.81 0.23 1019.73
MHALF 2.72 0.13 2.35 0.19 0.24 256.90 MHALF 5.31 0.73 4.06 1.13 0.24 427.95
0.6 TAXI 1.89 13.02 20.58 20.58 0.25 2086.16 0.6 TAXI 3.77 13.08 20.68 20.68 0.25 2096.15
IND 2.31 3.51 55.44 5.08 0.24 5566.03 IND 4.63 3.75 54.28 5.44 0.24 5450.26
MFIX 2.57 1.15 20.58 1.66 0.26 2079.67 MFIX 5.02 1.58 20.68 2.31 0.25 2089.68
MHALF 2.53 0.20 4.50 0.29 0.25 470.48 MHALF 5.10 0.56 5.70 0.84 0.25 590.95
0.4 TAXI 1.44 18.00 28.41 28.41 0.27 2870.55 0.4 TAXI 2.88 18.00 28.41 28.41 0.27 2870.52
IND 2.12 3.32 52.04 4.82 0.25 5224.27 IND 4.24 3.34 51.71 4.85 0.25 5191.29
MFIX 2.30 1.85 28.41 2.69 0.27 2861.27 MFIX 4.63 2.01 28.41 2.92 0.27 2861.53
MHALF 2.38 0.51 7.36 0.74 0.27 755.58 MHALF 4.63 0.79 8.08 1.16 0.26 827.31
0.2 TAXI 0.91 23.68 37.61 37.61 0.32 3791.96 0.2 TAXI 1.85 23.68 37.61 37.61 0.32 3792.10
IND 1.94 3.12 49.73 4.52 0.28 4991.61 IND 3.85 3.17 49.74 4.60 0.28 4992.58
MFIX 2.07 2.57 37.61 3.73 0.29 3780.16 MFIX 4.13 2.65 37.61 3.84 0.29 3780.16
MHALF 2.23 0.59 10.10 0.85 0.30 1028.39 MHALF 4.46 0.71 10.20 1.02 0.30 1038.56

57

Table 3.5: Simulation Results with 500 Customers

#Customers: 500; #Vehicles: 20; αt=1, αo_f=100, αo_v=0.5, αs=0.01; #Customers: 500; #Vehicles: 10; αt=1, αo_f=100, αo_v=0.5, αs=0.01;
Proporti
on Fixed
Strategy Travel
Taxi
Cost
Dissmila
rity
# Taxi
Trips
Travel/R
equest
Total
Cost
Proporti
on Fixed
Strategy Travel
Taxi
Cost
Dissmila
rity
# Taxi
Trips
Travel/R
equest
Total
Cost
0.8 TAXI 3.19 4963.36 49.57 49.57 0.16 5027.59 0.8 TAXI 6.35 5724.57 57.17 57.17 0.16 5788.62
IND 3.64 0.00 349.67 0.00 0.16 76.35 IND 7.11 968.58 347.14 9.67 0.16 1043.20
MFIX 3.60 0.00 64.94 0.00 0.16 72.62 MFIX 7.00 1210.95 72.18 12.09 0.16 1281.69
MHALF 3.61 0.00 32.00 0.00 0.16 72.48 MHALF 7.14 1047.74 52.76 10.46 0.16 1119.64
0.6 TAXI 2.64 10102.10 100.89 100.89 0.17 10155.81 0.6 TAXI 5.13 10132.15 101.19 101.19 0.17 10184.44
IND 3.58 0.00 329.18 0.00 0.18 74.89 IND 7.03 775.28 327.38 7.74 0.18 848.89
MFIX 3.41 0.00 113.19 0.00 0.17 69.26 MFIX 6.86 713.18 112.56 7.12 0.17 782.92
MHALF 3.57 0.00 53.07 0.00 0.18 72.00 MHALF 7.05 870.47 59.88 8.69 0.18 941.55
0.4 TAXI 1.94 14873.09 148.54 148.54 0.19 14913.35 0.4 TAXI 3.70 14873.09 148.54 148.54 0.18 14911.57
IND 3.42 0.00 302.63 0.00 0.19 71.40 IND 6.81 499.83 300.63 4.99 0.19 570.91
MFIX 3.28 0.00 162.81 0.00 0.18 67.16 MFIX 6.53 327.55 162.14 3.27 0.19 394.43
MHALF 3.36 0.00 58.97 0.00 0.19 67.80 MHALF 6.72 456.81 68.69 4.56 0.19 524.69
0.2 TAXI 1.16 19846.46 198.21 198.21 0.23 19871.59 0.2 TAXI 2.22 19846.46 198.21 198.21 0.22 19870.60
IND 3.21 0.00 271.83 0.00 0.22 66.97 IND 6.40 182.31 271.85 1.82 0.22 248.99
MFIX 3.13 0.00 211.62 0.00 0.21 64.71 MFIX 6.32 218.37 208.59 2.18 0.21 283.67
MHALF 3.12 0.00 71.77 0.00 0.21 63.11 MHALF 6.22 219.40 78.57 2.19 0.21 282.40
#Customers: 500; #Vehicles: 20; αt=1, αo_f=100, αo_v=0.5, αs=100; #Customers: 500; #Vehicles: 10; αt=1, αo_f=100, αo_v=0.5, αs=100;
Proporti
on Fixed
Strategy Travel
Taxi
Cost
Dissmila
rity
# Taxi
Trips
Travel/R
equest
Total
Cost
Proporti
on Fixed
Strategy Travel
Taxi
Cost
Dissmila
rity
# Taxi
Trips
Travel/R
equest
Total
Cost
0.8 TAXI 3.21 4963.36 49.57 49.57 0.16 9984.51 0.8 TAXI 6.41 5724.57 57.17 57.17 0.16 11505.69
IND 3.63 0.00 356.59 0.00 0.16 35731.55 IND 7.14 1138.83 354.36 11.37 0.16 36646.26
MFIX 3.58 0.00 49.57 0.00 0.16 5028.68 MFIX 7.03 1384.21 57.17 13.82 0.16 7171.46
MHALF 3.71 0.00 12.72 0.00 0.17 1346.11 MHALF 7.24 1371.22 35.98 13.69 0.17 5041.67
0.6 TAXI 2.63 10102.10 100.89 100.89 0.17 20243.74 0.6 TAXI 5.25 10132.15 101.19 101.19 0.17 20303.69
IND 3.53 0.00 331.84 0.00 0.18 33254.60 IND 6.98 701.17 328.71 7.00 0.18 33641.96
MFIX 3.52 0.00 100.89 0.00 0.17 10159.39 MFIX 6.90 828.33 101.19 8.27 0.18 11016.37
MHALF 3.78 0.00 23.79 0.00 0.19 2454.62 MHALF 7.19 1099.84 41.48 10.98 0.18 5319.76
0.4 TAXI 1.87 14873.09 148.54 148.54 0.18 29764.56 0.4 TAXI 3.79 14873.09 148.54 148.54 0.18 29765.04
IND 3.42 0.00 304.67 0.00 0.19 30535.34 IND 6.72 454.77 302.92 4.54 0.19 30814.00
MFIX 3.42 0.00 148.54 0.00 0.19 14922.31 MFIX 6.69 437.73 148.54 4.37 0.19 15358.64
MHALF 3.56 0.00 36.02 0.00 0.20 3673.26 MHALF 6.99 796.36 47.27 7.95 0.20 5593.22
0.2 TAXI 1.14 19846.46 198.21 198.21 0.23 39690.36 0.2 TAXI 2.25 19846.46 198.21 198.21 0.23 39689.96
IND 3.20 0.00 272.97 0.00 0.21 27360.92 IND 6.38 224.38 272.34 2.24 0.22 27522.17
MFIX 3.27 0.00 198.21 0.00 0.22 19886.45 MFIX 6.46 239.41 198.21 2.39 0.22 20125.01
MHALF 3.31 0.00 48.36 0.00 0.22 4902.20 MHALF 6.52 359.63 53.54 3.59 0.22 5778.80
#Customers: 500; #Vehicles: 20; αt=1, αo_f=0.5, αo_v=0.5, αs=100; #Customers: 500; #Vehicles: 10; αt=1, αo_f=0.5, αo_v=0.5, αs=100;
Proporti
on Fixed
Strategy Travel
Taxi
Cost
Dissmila
rity
# Taxi
Trips
Travel/R
equest
Total
Cost
Proporti
on Fixed
Strategy Travel
Taxi
Cost
Dissmila
rity
# Taxi
Trips
Travel/R
equest
Total
Cost
0.8 TAXI 3.21 31.14 49.57 49.57 0.16 5052.31 0.8 TAXI 6.26 36.16 57.17 57.17 0.16 5815.79
IND 3.45 8.62 351.70 12.53 0.16 35247.71 IND 6.73 12.99 353.80 19.03 0.16 35460.30
MFIX 3.55 2.37 49.57 3.43 0.16 5030.38 MFIX 6.87 9.45 57.17 14.19 0.16 5795.20
MHALF 3.67 0.50 12.72 0.72 0.16 1345.81 MHALF 7.28 9.17 35.98 13.88 0.17 3679.95
0.6 TAXI 2.70 63.55 100.89 100.89 0.18 10206.48 0.6 TAXI 5.14 63.74 101.19 101.19 0.17 10234.18
IND 3.30 10.27 327.90 14.94 0.17 32866.21 IND 6.46 12.53 328.73 18.27 0.17 32950.16
MFIX 3.42 4.35 100.89 6.34 0.17 10161.67 MFIX 6.76 8.77 101.19 13.03 0.17 10195.34
MHALF 3.72 0.81 23.79 1.17 0.19 2454.25 MHALF 7.16 7.91 41.48 11.88 0.18 4227.55
0.4 TAXI 1.90 93.37 148.54 148.54 0.18 14985.33 0.4 TAXI 3.73 93.37 148.54 148.54 0.18 14984.68
IND 3.13 10.89 302.23 15.87 0.18 30296.40 IND 6.15 11.64 301.84 16.97 0.18 30257.15
MFIX 3.11 7.31 148.54 10.68 0.18 14923.56 MFIX 6.33 8.88 148.54 13.03 0.19 14926.16
MHALF 3.49 0.89 36.02 1.30 0.20 3672.69 MHALF 6.92 5.71 47.27 8.50 0.20 4801.94
0.2 TAXI 1.14 124.57 198.21 198.21 0.23 19968.27 0.2 TAXI 2.21 124.57 198.21 198.21 0.22 19967.66
IND 2.86 11.64 271.72 16.95 0.20 27240.90 IND 5.65 11.87 271.81 17.30 0.20 27249.35
MFIX 2.92 10.51 198.21 15.30 0.21 19889.82 MFIX 5.82 10.35 198.21 15.09 0.21 19889.57
MHALF 3.23 1.68 48.36 2.43 0.22 4902.25 MHALF 6.39 3.19 53.54 4.67 0.22 5421.10

58

Figure 3.9: Travel Time with 50 Customers
# Vehicle =4 # Vehicle =2

59

Figure 3.10: Taxi Cost with 50 Customers
# Vehicle =4 # Vehicle =2

60

Figure 3.11: Dissimilarity with 50 Customers
# Vehicle =4 # Vehicle =2

61

Figure 3.12: Number of Taxi Trips with 50 Customers
# Vehicle =4 # Vehicle =2

62

Figure 3.13: Travel Time per Request with 50 Customers
# Vehicle =4 # Vehicle =2

63

Figure 3.14: Total Daily Cost with 50 Customers
# Vehicle =4 # Vehicle =2

64

Figure 3.15: Travel Time with 100 Customers
# Vehicle =8 # Vehicle =4

65

Figure 3.16: Taxi Cost with 100 Customers
# Vehicle =8 # Vehicle =4

66

Figure 3.17 : Dissimilarity with 100 Customers
# Vehicle =8 # Vehicle =4

67

Figure 3.18: Number of Taxi Trips with 100 Customers
# Vehicle =8 # Vehicle =4

68

Figure 3.19: Travel Time per Request with 100 Customers
# Vehicle =8 # Vehicle =4

69

Figure 3.20: Total Daily Cost with 100 Customers
# Vehicle =8 # Vehicle =4

70

Figure 3.21: Travel Time with 500 Customers
# Vehicle =20 # Vehicle =10

71

Figure 3.22: Taxi Cost with 500 Customers
# Vehicle =20 # Vehicle =10

72

Figure 3.23: Dissimilarity with 500 Customers
# Vehicle =20 # Vehicle =10

73

Figure 3.24: Number of Taxi Trips with 500 Customers
# Vehicle =20 # Vehicle =10

74

Figure 3.25: Travel Time per Request with 500 Customers
# Vehicle =20 # Vehicle =10

75

Figure 3.26: Total Daily Cost with 500 Customers
# Vehicle =20 # Vehicle =10

76

From the simulation results, we make the following observations:
A. Strategy TAXI has the smallest travel distance, because of its inability to use
the slack times to accommodate the random requests, thus resulting servicing
fewer requests. When handling the same amount of requests on the fleet,
strategy MFIX and MHALF have similar travel distance to that of strategy
IND, a near-optimal routing solution, suggesting that our master plans
provide efficient routing solutions. This is further illustrated by the results for
travel per request, in which cases the solutions from strategy TAXI, IND,
MFIX, and MHALF are close.
B. Strategy TAXI has the largest taxi cost and number of taxi trips, again
because of its inflexibility of inserting the random requests into the slack of
the fleet of vehicles.
C. Strategy IND has the largest dissimilarity; strategy MHALF has the lowest
dissimilarity. If we schedule the routing for each day independently without a
master plan, the routes become dissimilar from day to day. Even though we
get a near optimal solution in terms of routing efficiency as measured in
travel distance and taxi cost, the quality of service as measured in route
dissimilarity is poor. If we form master routes with the deterministic requests
and a number of random requests of high probability of occurrence, daily
routes are created, which are similar from day to day, without scarifying
much in routing efficiency.

77

D. When the unit cost for route dissimilarity increases (from 0.01 to 100), while
the other parameters remain the same, the dissimilarity for strategies with
master plans decreases and the routing efficiency (travel distance and/or taxi
cost) increases. This is because when we give a higher weight on
dissimilarity, the routing solution favors less dissimilarity, and trades that
with less routing efficiency. The change in the unit cost for route
dissimilarity does not have a significant impact on the solutions of strategies
TAXI and IND. The reason is that for strategy TAXI, the dissimilarity is
contributed by the random requests handled by taxi, which remain the same
with any set of parameters; for strategy IND, there is no master plan to use to
construct daily routes, but the dissimilarity is measured against the master
plan from strategy MFIX, hence the dissimilarity with IND might even
increase when the unit cost of dissimilarity increases.
E. When the fixed unit taxi cost decreases (from 100 to 0.5), while all the other
parameters remain the same, there is more taxi use represented by number of
taxi trips. This implies that as taxi usage become inexpensive, it becomes a
more economical solution to use taxi rather than rerouting or picking up
packages by the regular fleet of vehicles.
F. With the same pool of potential customers and the same amount of vehicles,
as the proportion of fixed customers increase, the total travel time and the
travel time per request increase, because there are more expected customers
to service. For the same reason, the taxi cost or number of taxi trips for IND,

78

MFIX or MHAL remains at 0 or increase. The taxi cost or number of taxi
trips for TAXI decreases because there are less random customers for the
TAXI strategy to accommodate. The dissimilarity for TAXI, MFIX, and
MHALF decrease because there are more customers included in the master
plan. The dissimilarity for IND increases because there is no master plan but
there are more expected customers to service.
G. With the same number of potential customers, and when the fleet size
increases, the average per vehicle travel time, the taxi cost, and the taxi trips
decrease because there are more vehicles to handle the requests. The
dissimilarity for MHALF decreases because more customers can be included
into the master plan. The other trends of the results discussed above remain
the same with different fleet size.
H. When the size of the customer pool increases, the problem size increases,
resulting larger total cost. The pattern of the above results remains the same
for different size of pool of potential customers.

3.5 Simulations and Results with Actual Data
We tested our routing approach also using real-life data collected from a leading
healthcare provider in Southern California. There are two types of requests in the data set.
One is regular daily requests, which needs to be visited every day at a specific time. The

79

other is random requests that need to be handled by taxis. We have compared three
strategies with this set of data.
1) MD Routes: Include a customer into the master plan if it is a deterministic
request or if the pickup and delivery location of a request has a probability of
occurring higher than a threshold (e.g., 10%). Recourse action (drop the non-
occurring requests and insert the occurring requests) is taken for daily plans.
2) Industry Reroute: Take the existing master plan from the healthcare provider
as the simulated master routes. Recourse action (drop the non-occurring
requests and insert the occurring requests) is taken for daily plans.
3) Industry Taxi: Take the existing master plan from the healthcare provider as
the daily routes. Use Taxi for all the random requests.
It should be noted that Industry Taxi is the current practice of this healthcare
provider. In the following simulation of a horizon of 30 days, there are 85 deterministic
requests and 100 potential random requests on each day. On a daily basis, 14 vehicles are
used to handle all the requests. The simulation results are shown in Table 3.6. As with the
random data,

is the unit fleet cost per hour traveled;

is the fixed cost per trip of
taxi;

is the varying cost per hour the taxi traveled;

is the unit cost per count of
dissimilarity. Column “Strategy” lists the three strategies we are comparing. Column
“Travel” shows the overall time that a vehicle travels per day on average. Column “Taxi
Cost” shows the daily taxi cost. Column “Dissimilarity” shows the average daily
dissimilarity, which is the daily total number of vehicles used in the daily routes that is
different than the one for the master routes. If a taxi is used, then the dissimilarity is

80

increased by one. Column “#Taxi Trips” shows the total number of daily taxi trips
introduced on average. Column “Travel/Requests” shows the distance that a vehicle
travels to service a request on a daily basis on average. Column “Total Cost” shows the
average daily total cost including travel cost, taxi cost, and cost on dissimilarity. It is the
summation of each type of costs weighted by the unit cost of that type.

Table 3.6: Simulation Results with Actual Data
αt=1, αo_f=100, αo_v=0.5, αs=0.01;

Strategy
Travel
(hours/day)
Taxi Cost
($/day)
Dissimilarity
(counts/day)
# Taxi Trips
(trips/day)
Travel/Request
(hours/day)
Total Cost
($/day)
MD Routes 8.78 5221.10 148.00 52.10 0.03 5345.50
Industry Reroute 8.36 6223.40 164.00 62.10 0.03 6342.10
Industry Taxi 7.24 10023.80 200.00 100.00 0.04 10127.20
αt=1, αo_f=100, αo_v=0.5, αs=100;

Strategy
Travel
(hours/day)
Taxi Cost
($/day)
Dissimilarity
(counts/day)
# Taxi Trips
(trips/day)
Travel/Request
(hours/day)
Total Cost
($/day)
MD Routes 8.82 5291.13 143.40 52.77 0.03 19754.60
Industry Reroute 8.47 6333.60 157.23 63.17 0.03 22175.51
Industry Taxi 7.24 10023.77 200.00 100.00 0.04 30125.17
αt=1, αo_f=0.5, αo_v=0.5, αs=100;

Strategy
Travel
(hours/day)
Taxi Cost
($/day)
Dissimilarity
(counts/day)
# Taxi Trips
(trips/day)
Travel/Request
(hours/day)
Total Cost
($/day)
MD Routes 8.13 43.60 134.23 54.57 0.03 13580.76
Industry Reroute 8.08 49.79 152.93 64.20 0.03 15456.24
Industry Taxi 7.24 73.77 200.00 100.00 0.04 20175.17

The simulation results on the industry data are also plotted in Figure 3.27 – Figure
3.32. From Figure 3.27, we see that strategy “Industry Taxi” has the shortest average
travel time. Figure 3.28 and Figure 3.30 show that the taxi cost and the number of taxi
trips of strategy “Industry Taxi” are significantly higher than those of the strategies that
have recourse actions (MD Routes and Industry Reroute). This implies that with recourse

81

technique, we are able to better utilize the slack time on the vehicles to reduce taxi cost.
Meanwhile, even though the average total travel time of a vehicle is higher with the
strategies with recourse actions, the average travel time spent for each customer request is
lower with these strategies, and this is confirmed by Figure 3.31. From these figures, we
also see that the proposed strategy – MD Routes has the smallest taxi cost and average
travel time per request. This shows that our strategy not only better utilizes the slack time
to reduce taxi cost, but is also an efficient routing solution with the least travel time spent
on each request.
Besides the reduction in taxi cost, MD Routes significantly reduced the route
dissimilarity, as shown in Figure 3.29. In general, any strategy with a rerouting technique
has smaller dissimilarity as the location visited in the master plan is going to be more
frequently repeated in the daily plans. And the strategy we propose is the best in
generating similar routes. This is achieved by having the proposed strategy “MD Routes”
including the high probabilistic customers into the master routes; whereas the strategy
“Industry Reroute” has only the deterministic customers in the master plan.

82

Figure 3.27: Average Travel Time of a Vehicle

Figure 3.28: Average Taxi Cost of a Vehicle

0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00
αt=1,
αo_f=100,
αo_v=0.5,
αs=0.01;
αt=1,
αo_f=100,
αo_v=0.5,
αs=100;
αt=1,
αo_f=0.5,
αo_v=0.5,
αs=100;
Hours
Travel Time
MD Routes
Industry Reroute
Industry Taxi
0.00
2000.00
4000.00
6000.00
8000.00
10000.00
12000.00
αt=1,
αo_f=100,
αo_v=0.5,
αs=0.01;
αt=1,
αo_f=100,
αo_v=0.5,
αs=100;
αt=1,
αo_f=0.5,
αo_v=0.5,
αs=100;
$
Taxi Cost
MD Routes
Industry Reroute
Industry Taxi

83

Figure 3.29: Average Dissimilarity for the Fleet

Figure 3.30: Average Number of Taxi Trips for the Fleet

0.00
50.00
100.00
150.00
200.00
250.00
αt=1,
αo_f=100,
αo_v=0.5,
αs=0.01;
αt=1,
αo_f=100,
αo_v=0.5,
αs=100;
αt=1,
αo_f=0.5,
αo_v=0.5,
αs=100;
Counts
Dissimilarity
MD Routes
Industry Reroute
Industry Taxi
0.00
20.00
40.00
60.00
80.00
100.00
120.00
αt=1,
αo_f=100,
αo_v=0.5,
αs=0.01;
αt=1,
αo_f=100,
αo_v=0.5,
αs=100;
αt=1,
αo_f=0.5,
αo_v=0.5,
αs=100;
Counts
# Taxi Trips
MD Routes
Industry Reroute
Industry Taxi

84

Figure 3.31: Average Travel Time per Request

Figure 3.32: Average Total Cost

0.00
0.01
0.01
0.02
0.02
0.03
0.03
0.04
0.04
0.05
αt=1,
αo_f=100,
αo_v=0.5,
αs=0.01;
αt=1,
αo_f=100,
αo_v=0.5,
αs=100;
αt=1, αo_f=0.5,
αo_v=0.5,
αs=100;
Hours
Travel/Request
MD Routes
Industry Reroute
Industry Taxi
0.00
5000.00
10000.00
15000.00
20000.00
25000.00
30000.00
35000.00
αt=1,
αo_f=100,
αo_v=0.5,
αs=0.01;
αt=1,
αo_f=100,
αo_v=0.5,
αs=100;
αt=1,
αo_f=0.5,
αo_v=0.5,
αs=100;
$
Total Cost
MD Routes
Industry Reroute
Industry Taxi

85

The results of the analysis with real-life data shows that our heuristic can improve
the routing solution by decreasing the taxi and dissimilarity costs. With the current
resource of vehicles, the current deterministic requests, and sampling on current data set,
our heuristic beats the current industry solution by reducing taxi cost by 45%-48%and
reducing dissimilarity by 26%-33%. If we compare with the daily routes obtained by
applying recourse actions on a master plan taken from the current practice industry, our
heuristic reduces taxi cost by 16%-17% and it reduces dissimilarity by 9%-12%.

86

4. A New Clustering Approach

In a previous chapter, we discussed a heuristic method to generate master plans,
based on a stochastic programming approach. This method includes the high occurring
probabilistic customers into the master plan to obtain high route similarity and a low taxi
usage at the same time. We consider only customer locations when formulating the
master plan, such that the generated master plan has the smallest travel distance for all the
vehicles. This master plan forms in effect a clustering of the most frequently occurring
customers.
In this chapter, we directly address the problem of how to cluster the customers
into groups when there is uncertainty in customer occurrence. We consider not only
customer locations, but also occurring probabilities of customers, when formulating the
master plan. Instead of using a stochastic programming approach as in the previous
chapter, we propose a new metric based on expected distance. In order to simplify the
study, we assume all the customers are handled by the fleet, and there is no taxi usage.
Here, we compare grouping methods for problems with uncertain customers for
two scenarios. In one scenario, customers are scattered around the depot, as shown in
Figure 4.1; in another one, the customers are clustered such that the depot is in a corner,
as shown in Figure 4.2. Clustering customer into groups is comparatively more
straightforward when customers are scattered around the depot. One can group them
based on their physical locations. For example, if there are two vehicles to handle the

87

customers in Figure 4.1, the customers could simply be clustered by a line across the
depot, as shown in Figure 4.3. However, if the customers are clustered such that the depot
is not at the center, the clustering is not as straightforward. In this chapter, we discuss the
grouping methods for both scenarios. We propose a new customer grouping approach
which considers not only the locations of customers, but also their probability of
occurrence.

Figure 4.1: Customers Scattered Around Depot

88

Figure 4.2: Customers Clustered, Depot at a Corner

Figure 4.3: Customers Scattered Around Depot

89

4.1 The Expected Travel Distance
In a previous chapter of this dissertation, a stochastic programming approach has
been proposed to generate master and daily plans. The mixed integer formulation of the
stochastic programming approach has an objective function of minimizing the expected
average cost (travelled distance, taxi, route similarity) of the daily plans. In a similar
manner, we can measure the expected distance the fleet travels by first looking at the
distance under each possible scenario of customer occurrence, weighting each scenario
by the probability of the scenario, and then summing up the weighted value of all the
scenarios. In order to present the expected distance, we define the notations as follows.

: cluster index, .
: customer index, .

: set of customers on vehicle , where .

: updated set of customers on vehicle after a new customer is inserted.

: set of occurring customers on vehicle in combination

, where

is the
index for combinations for vehicle , and

.

: set of occurring customers in combination , where .

: updated set of occurring customers in scenario after a new customer is
inserted, where .

: the probability that all the customers in set

request service.

90

: the shortest tours starting from the depot, visiting all the nodes in

, and
ending at the depot.

: the total length of the TSP tours visiting all the nodes in set

.

: the probability that customer requests service.
: the probability that a new customer requests service.

: the expected distance of cluster on a total of customers.

: the updated expected distance of cluster on a total of customers.

We use the following example to explain this notation. Suppose there are three
customers 1, 2, 3, and there are two vehicles. When we assign customer 1 and 2 to
vehicle 1, and customer 3 to vehicle 2, we will have

and

. As
there are two customers on vehicle 1, there are four possible combinations on vehicle 1.
Specifically, we have

,

,

, and

. As there is one
customer on vehicle 2, there are two possible combinations on vehicle 2, that is

,

. For the same reason,

,

{ },

{ },

{ } ,

{ } ,

{ } ,

{ } ,

{ } .
Moreover, a cluster is a partition of the customers into disjoint groups. For this example,
it is

.
Based on the above definitions, we see that the expected distance cluster with
customers is written as follows:

91

∑

(4.1)

If there are no clusters and the n clients are random the number of possible
outcomes for is is

. But if there are clusters then we can number the clusters of

is

. We next show that the
expected travel distance in Equation (4.1) can be simplified.
Proposition: The expected travel distance in Equation (4.1) can be written as:

∑ ∑ ∏

∏ (

)

(4.2)
We now prove the correctness of the above equation by recursion. It can be seen
that Equation (4.2) holds for the case of , where customers 1 and 2 is on vehicle 1
and customer 3 is on vehicle 2. From Equation (4.1), we see that we can write the
expected distance as

. After rewriting the formula, the
expected distance can be written as

, which conforms with Equation (4.2).
Suppose Equation (4.2) holds for customers, which are serviced by no more
than vehicles. When there is one more customer to be handled by the fleet, the new
customer can only be added to one of the vehicles. Assume the new customer is added
to vehicle , where the set of all the nodes on the vehicle is

before adding the new

92

customer to the vehicle. After inserting the new customer to vehicle , the updated
expected travel distance for the fleet can be written as:

(

) (∑

)
(4.3)
, where ∑

is the expected distance of the scenarios where the new
customer requests service. It can be seen that (

) is the weighted distance of
the scenarios where the new customer does not show up, and ( ∑

) is
the weighted distance of the scenarios where the new customer occurs.
Plugging Equation (4.2) into Equation (4.3), one gets the following equation.

(∑ ∑ ∏

∏ (

)

)
(∑

)

(4.4)
When we separate the expression for vehicle t which is used for the new customer,
we get the following expression for the first part of the right hand side of Equation (4.4).

93

(∑ ∑ ∏

∏ (

)

)
(∑ ∑ ∏

∏ (

)

)
( ∑ ∏

∏

)

(4.5)

Plugging Equation (4.2) into Equation (4.1), one can rewrite ( ∑

) as:
(∑

)
(∑ ∑ ∏

∏ (

)

)
( ∑ ∏

∏

)

(4.6)

Plugging Equation (4.5) and (4.6) into (4.4), we get the following equation.

94

(∑ ∑ ∏

∏ (

)

)
( ∑ ∏

∏

)
(∑ ∑ ∏

∏ (

)

)
( ∑ ∏

∏

)

(4.7)

The following equation holds for the vehicles to which the new customer is not
inserted.

(∑ ∑ ∏

∏ (

)

)
(∑ ∑ ∏

∏ (

)

)
∑ ∑ ∏

∏ (

)

(4.8)
The following equation holds for the vehicle to which the new customer is
inserted.

95

( ∑ ∏

∏

)
( ∑ ∏

∏

)
∑ ∏

∏

(4.9)

Rewriting Equation (4.7) with the presence of Equation (4.8) and (4.9), one gets
the following equation.

∑ ∑ ∏

∏ (

)

∑ ∏

∏

(4.10)

Rewriting Equation (4.10), we get the following equation.

∑ ∑ ∏

∏ (

)

(4.11)
The above expression conforms with the presumed expression, as shown in
Equation (4.2), with customers; therefore, we claim that Equation (4.2) is correct.
□

96

4.2 Discussion and Example
In the previous section, we have shown a new measure of grouping of customers
based on an expected distance, including the locations as well as the occurring
probabilities of customers. In this section, we use a small example to show the meaning
of customer grouping based on the expected distance, and compare it with the grouping
method we used in the previous chapter for generating master plans, which has only
distance in the objective function. We calculate Equation (4.2) in the above section for
each strategy. The one which yields a lower

in the equation is the one that is more
effective in generating a better grouping based on expected distance. We show this
explicitly in the following small example.

Figure 4.4: Four Customers in the City

97

Suppose there are four customers located in the city. For illustration purpose, we
choose an extreme case by assuming the customers are located at (0, 1), (1, 0), (0, -1), (-1,
0) as shown in Figure 4.4. The depot is located at the center of the city (0, 0). On each
day, the probability that customer requests service is

, where . To further
simplify the problem, we assume that there are two vehicles to service these four
customers. The speed of each vehicle is 1. The common deadline for each vehicle is 4,
which makes that each vehicle should service exactly two customers. Based on these
assumptions, there are three possible ways of clustering customers, as illustrated in
Figure 4.4:
1. (A B) (C D)
2. (A C) (B D)
3. (A D) (B C)
The first clustering, for example, means that we group A and B on one vehicle,
and customer C and D on the other. According to the analysis from the last section, we
see that the clustering of customers, which has the lowest measure in

as shown in
Equation (4.2), is the best clustering that can generate routes with lowest expected
distance. In this simple example, it can easily be calculated that

for
,

√ , and

.
With the calculated TSP lengths, the expected distance

for the three ways of
clustering can be calculated using Equation (4.2) and the results are shown in Table 4.1.

98

Table 4.1:

for the Three Clustering of Customers in the Small Example
Clusters

(A B) (C D)

√

(A C) (B D)

(A D) (B C)

√

From the above table, it is obvious that the clustering of (A C) (B D) has the
largest expected distance, compared to the other two ways of clustering. Clustering of (A
B) (C D) will have a smaller expected distance than (A D) (B C), if

. Therefore, if

, clustering of (A B) (C D) is
preferred with smaller expected distance. In other words, if

and

, or if

and

, clustering of (A B) (C D) is preferred. For the same reason, if

and

, or if

and

, clustering of (A D) (B C) is preferred.
This small example shows that a good clustering will consider distance first. If
there is less difference in distance, the better way is to cluster customers with high
occurring probabilities grouped on the same vehicle.

4.3 Problem Formulation for Customer Clustering
From the above discussion, we know that Equation (4.2) can be used to compare
which groups are better in generating routes with short expected traveling distances. In

99

this section, we present a mathematical formulation for obtaining an efficient clustering
of customers, so that each customer is visited by the same vehicle during the planning
horizon, and in the meantime, the efficiency in the overall length of the routes are
maintained. To achieve this goal, we formulate the following integer programming
problem.
We define a group as a cluster of customers which are on the same vehicle. For
example, to have customer 1 and 2 on one vehicle is a group; to have customer 1, 2, and 3
on one vehicle is another group. Thus, determining which customers are assigned to
which vehicles gives the grouping of the customers. The model is presented as follows.

Parameters:
g: group index, .
i: customer index, .

{

: the “cost” associated with each group, and

∑ ∏

∏ (

)

(4.12)

: the set of occurring customers in group in combination

,
where

is the index for combinations of group and

.

: the set of all customers in group .

100

: the shortest path starting at the depot, visiting all the nodes in group
, and ending at the depot.

: the probability that customer request service.
: vehicle index, .
: the length of the common deadline for all vehicles.

Variables:

{

The problem can then be written as follows:

∑

(4.13)

∑

(4.14)

∑

(4.15)

(4.16)

(4.17)

101

The objective function (4.13) minimizes the total expected routing cost.
Constraint (4.14) ensures that each customer belongs to one group. Constraint (4.15)
limits the number of groups, so that it is not higher than the number of vehicles.
Constraint (4.16) limits the length of the routes for each group. Constraint (4.17) is the
standard binary restriction on the decision variable.
The above formulation is typically described as a set-partitioning problem in the
literature (Ioachim et al., 1995). The disadvantage of this formation is that it contains an
exponential number of group variables (

) and an exponential number of constraints.
This clustering problem or the set-partitioning problem is typically solved heuristically
by neighborhood search in the literature (Koskosidis and Powell, 1992; Kelly and Xu,
1999). We propose a Tabu search based heuristic to solve the above problem.

4.4 Tabu Heuristic
In this section, we propose a Tabu search based heuristic to solve the problem in
the previous section. The Tabu search heuristic can be used to obtain the best grouping of
customers based on the locations and the occurring probabilities of the customers,
without generating the actual routes for each day. In order to compare the difference
between the new approach and the old approach, we use another Tabu search to get the
best routes for the customers in each group. Initial routes for the given customers are
formed randomly. Then the two methods of clustering the customers and routing are
compared in each experiment.

102

Method 1 (M1): Tabu search (Algorithm 5) with an objective of minimum
overall length of routes to obtain the master routes. This method is the same procedure as
the method previously discussed in this thesis. This method generates complete routes
instead of just the grouping of customers.
Method 2 (M2): the first phase determines the groups and the second phase
determines the routes for the given grouping. Specifically, phase 1 (Algorithm 5.1) uses
an objective of minimizing the weighted TSP tours to get the best customer clustering.
Phase 2 (Algorithm 5.2) uses an objective of minimizing the overall length of routes to
get the shortest routes for the clusters of customers obtained by phase 1.

103

Algorithm 5: Tabu Search Algorithm for Master Routes
Input: an initial master plan to improve
Output: an improved master plan

repeat
randomly chose two routes from the solution
generate

neighbors from -interchange operator
generate

neighbors from 2-opt operator
choose the best solution and make the move;
randomly generate tabu tenure from a uniform distribution U (

);
if the move is -interchange then
set the tabu for moving the exchanged requests for iterations;
else
set the tabu for moving the exchanged nodes for iterations;
until no improvement in

iterations;
calculate the objective and save the current solution;

104

Algorithm 5.1: Tabu Search Algorithm for Clustering
Input: an initial grouping
Output: an improved grouping

repeat
randomly chose two groups from the solution
generate

neighbors from -interchange operator
choose the best solution and make the move;
randomly generate tabu tenure from a uniform distribution U (

);
set the tabu for moving the exchanged requests for iterations;
until no improvement in

iterations;
calculate the objective and save the current solution;

105

Algorithm 5.2: Tabu Search Algorithm for Routing for Clusters
Input: a grouping of customers with randomly generated routes
Output: improved routes for the grouping

repeat
randomly chose one route from the solution
generate

neighbors from 2-opt operator
choose the best solution and make the move;
randomly generate tabu tenure from a uniform distribution U (

);
set the tabu for moving the exchanged nodes for iterations;
until no improvement in

iterations;
calculate the objective and save the current solution;

In each iteration of Algorithm 5, Algorithm 5.1, or Algorithm 5.1, the Tabu search
generates

-interchange neighbors and / or

2-opt neighbors of the current
solution. The number of Tabu iterations is a random number uniformly distributed in
(

). The Tabu search at each iteration moves to the best neighbor. A temporary
move to a worse solution is allowed to escape from the local minimum. The Tabu status
is overwritten if the new solution improves from the best solution. The algorithm
terminates if there is no improvement in

iterations.

106

Discussion on M1: It should be noted that, Algorithm 5, applied in M1, is almost
the same as Algorithm 4 in the previous chapter. The only difference is that in the last
chapter, the Tabu search algorithm is applied on both the master routes, and on the daily
routes. When it is applied on master routes, the objective is to minimize the total distance
traveled, as to have more slack time to accommodate the random requests. When it is
applied on daily routes, the objective is to minimize the cost including total distance
traveled, and route dissimilarity, as to improve the overall efficiency of the final routes.
In M1, however, Algorithm 5 is only used to improve the master routes, with an objective
function of minimizing the total distance of the routes.
Discussion on M2: M2 is the method that we particularly focus on and propose
with this chapter. Specifically, with this method, the Tabu search algorithm is broken into
two parts. The first phase (Algorithm 5.1) has only inter-cluster exchange (lambda
interchange) of customers to get a good partition, and the second phase (Algorithm 5.2)
has only intra-route exchange (2-opt exchange) of customers to get short routes for each
partition. Each weighted TSP tour is calculated using Equation (4.12), which needs the
calculation of a TSP tour including all the customers in the group. As the TSP problem is
NP-hard, an approximation technique is used to calculate the length of a TSP tour. In this
study, the length of a TSP tour is approximated by the following equation.

∑

∑ ∑

(4.18)

where N is the set of customers on the vehicle, and

is the distance between customer i
and customer j.

107

In summary, method 1 focuses on obtaining short routes based purely on the
location of customers; method 2 aims to get good clustering of customers based not only
on the location, but also the occurring probability of customers. We do two sets of
experiments to test the effectiveness of M1 and M2.

4.5 Experiments and Results
The first set of experiments (set A) is for the cases where the customers are
randomly located in a squared city and the depot is located at the center of the city. In the
two dimensional coordinate system, the four corners of the city are (1, 1) (-1, 1) (-1, -1)
and (1, -1). The depot is located at (0, 0). The distance a vehicle travels between the
nodes is the Euclidean distance in the coordinate system. The vehicle is assumed to travel
at a speed of 1. Therefore, the longest time that one vehicle can service a single node in a
single trip from the depot and back to the depot is
√

; this amount of time is
needed when the single node is located at the one of the corners. The locations of the 20
sample customers are shown in Figure 4.5.

108

Figure 4.5: Location of Customers in Experiment Set A

For experiment set A, the parameters in the Tabu search in both Method 1 and
Method 2 are

,

,

,

, and

. The
simulation parameters of experiment set A are listed in Table 4.2. In the first column of
the table “Experiment”, the index name of each experiment is shown. In column “#
Customers”, the number of customers to be serviced in the corresponding experiment is
listed. The number of vehicles used to service these customers is shown in column “#
Vehicles”. In column “# Days”, the number of days in the planning horizon is shown.
The maximum length of a vehicle trip of a day is shown in column “Common Deadline”.
The column “Customer Probability” lists the occurring probabilities of the customers on

109

each day. For example, in Experiment A.1, customer 1 to customer 4 has an occurring
probability of 0.8, and customer 5 to customer 8 has an occurring probability of 0.1. The
locations of the customers are pre-determined for the 30 days and only the demand is
sampled on each day according to the probability of occurring. The experiments are
constructed this way, such that we can create instances where the routes created by the
new method would be different from those created by the old method, as the new
methods tends to group the customers with high probabilities of occurrence together
while the old customer considers only distance while grouping the customers.

Table 4.2: Simulation Parameters for Experiment Set A
Experiment
#
Customers
#
Vehicles
#
Days
Common
Deadline
Customer
Probability
A.1 8 2 30 5
1-4: 0.8
5-8: 0.1
A.2 10 2 30 6
1-5: 0.8
6-10: 0.1
A.3 15 2 30 7
1-5: 0.8
6-15: 0.1
A.4 20 2 30 10
1-5: 0.8
6-20: 0.1

The simulation results of experiment set A are shown in Table 4.3. For each
experiment, “R1” and “R2” are the resulting routes from approach M1 and M2. “Full
Route Length” is the total length of R1 and R2. “Expected Route Length” is the expected
length is for 30 realizations of the demand.

110

Table 4.3: Simulation Results for Experiment Set A
Experiment Clustering and Routes with M1 Clustering and Routes with M2
A.1 R1 3, 7, 5 R1 3, 5, 4, 6, 2, 1
R2 8, 1, 2, 6, 4 R2 7, 8
Full Route Length 6.00 Full Route Length 6.73
Expected Route
Length
3.24 Expected Route
Length
3.26
A.2 R1 7, 3 R1 3, 5, 4, 6, 2, 1,
10, 8
R2 5, 4, 6, 2, 1, 9,
10, 8
R2 9, 7
Full Route Length 6.08 Full Route Length 7.86
Expected Route
Length
3.99 Expected Route
Length
4.07
A.3 R1 3, 7, 11, 5, 15, R1 9, 10, 12, 8,
11, 7, 14
R2 8, 12, 10, 9, 13,
1, 2, 6, 4, 14
R2 13, 1, 6, 2, 4,
15, 5, 3
Full Route Length 7.20 Full Route Length 9.41
Expected Route
Length
4.56 Expected Route
Length
4.51
A.4 R1 7, 11, 5, 19 R1 16, 6, 2, 4, 19,
5, 15, 3, 20, 8,
13, 1, 18, 17,
7
R2 14, 15, 3, 20,
17, 8, 12, 10, 9,
13, 1, 18, 16, 6,
2, 4
R2 9, 10, 12, 11,
14
Full Route Length 8.24 Full Route Length 12.58
Expected Route
Length
4.58 Expected Route
Length
5.08

The full route length of the four experiments in set A with M1 and M2 are plotted
in Figure 4.6. The expected total distance of all the daily routes in the 30 days for the four
experiments in set A with M1 and M2 are plotted in Figure 4.7.

111

Figure 4.6: Full Route Length of the Clusters in Experiment Set A

Figure 4.7: Expected Route Length of the Clusters in Experiment Set A

0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
A.1 A.2 A.3 A.4
Full Route Length
Full Route Length - Set A
M1
M2
0.00
1.00
2.00
3.00
4.00
5.00
6.00
A.1 A.2 A.3 A.4
Expected Route Length
Expected Route Length - Set A
M1
M2

112

From the results of this set of experiments, we see that the new approach of
clustering always generate master routes that have longer length than the master routes
generated by the old method. The new approach tends to group the customers with high
occurring probability together, and the old approach considers only distance. In terms of
resulting expected route length, the new approach for clustering the customers sometimes
is better, and sometimes is worse than the old approach. In experiment A.4, for example,
the expected route length with the new approach M2 is much worse than that with the old
approach. We notice that for the experiment A.4, the resulting routes for the clusters with
M2 includes customers on opposite side of the depot, meaning that the vehicle has to
“traverse” the depot to service the customers. This suggests that even though we have the
high probabilistic customers grouped on a vehicle, the daily routes do not save distance
from repeating the master routes.
The second set of experiments (set B) is for the cases where the customers are
randomly located in a squared city and the depot is located at one corner of the city. In
the two dimensional coordinate system, the four corners of the city are (0, 0) (2, 0) (2, 2)
and (0, 2). The depot is located at (0, 0). The distance a vehicle travels between the nodes
is still the Euclidean distance in the coordinate system. The vehicle is assumed to travel at
a speed of 1. Therefore, the longest time that one vehicle can service a single node in a
single trip from the depot and back to the depot is
√

. The locations of the 20
sample customers are shown in Figure 4.8.

113

Figure 4.8: Location of Customers in Experiment Set B

For experiment set B, the parameters in the Tabu search in both Method 1 and
Method 2 are the same as the previous setting. As the parameters for experiment set A are
shown in Table 4.2, the experiment parameters for experiment set B are shown in Table
4.4. The experiment parameters are mostly the same, except that the “Common Deadline”
for Experiment B.1 is different from the “Common Deadline” for Experiment A.1 for
feasibility reason.

114

Table 4.4: Simulation Parameters for Experiment Set B
Experiment # Customers
#
Vehicles
# Days
Common
Deadline
Customer
Probability
B.1 8 2 30 6
1-4: 0.8
5-8: 0.1
B.2 10 2 30 6
1-5: 0.8
6-10: 0.1
B.3 15 2 30 7
1-5: 0.8
6-15: 0.1
B.4 20 2 30 10
1-5: 0.8
6-20: 0.1

The simulation results are shown in Table 4.5. The full route length of the four
experiments in set B with M1 and M2 are plotted in Figure 4.9. The expected route length
of the daily routes over 30 days for the four experiments in set B with M1 and M2 are
plotted in Figure 4.10.

115

Table 4.5: Simulation Results for Experiment Set B
Experiment Clustering and Routes with M1 Clustering and Routes with M2
B.1 R1 1, 2, 6, 4 R1 1, 2, 6, 4, 3
R2 8, 7, 5, 3 R2 5, 7, 8
Full Route Length 8.78 Full Route Length 9.32
Expected Route
Length
6.49 Expected Route
Length
5.03
B.2 R1 9, 1, 2, 6, 4,
10
R1 8, 7, 6, 10
R2 8, 7, 5, 3 R2 9, 3, 5, 4, 2, 1
Full Route Length 8.88 Full Route Length 10.58
Expected Route
Length
8.23 Expected Route
Length
6.00
B.3 R1 12, 8, 14, 15,
5, 11, 7, 3, 9,
13
R1 3, 11, 5, 4, 6,
2, 1, 10
R2 10, 4, 6, 2, 1 R2 13, 9, 14, 15,
7, 8, 12
Full Route Length 10.12 Full Route Length 10.62
Expected Route
Length
8.07 Expected Route
Length
6.71
B.4 R1 12, 10, 9, 1,
18, 16, 4, 6,
2, 13
R1 20, 17, 3, 5, 4,
6, 2, 16, 18, 1,
10
R2 8, 20, 17, 3,
7, 11, 5, 19,
15, 14
R2 8, 7, 11, 19,
15, 14, 9, 13,
12
Full Route Length 10.33 Full Route Length 12.03
Expected Route
Length
8.09 Expected Route
Length
6.52

116

Figure 4.9: Full Route Length of the Clusters in Experiment Set B

Figure 4.10: Expected Route Length of the Clusters in Experiment Set B

0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
B.1 B.2 B.3 B.4
Full Route Length
Full Route Length - Set B
M1
M2
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
B.1 B.2 B.3 B.4
Expected Route Length
Expected Route Length - Set B
M1
M2

117

The same as experiment set A, the new approach of clustering always generate
master routes that have longer length than the master routes generated by the old method.
Applying the new approach tends to group the customers with high occurring probability
together. However, as for resulting expected route length, the new approach for clustering
the customers is always better than the old approach, because the old approach uses travel
distance as the objective function when formulating master plans, while the new
approach takes expected distance as the objective function. Compared with the old
approach, when the depot is in a corner of a group of customers as in experiment set B,
the expected distance is more efficient in generating master routes with overall short
travel distances, because when clustering the customers that have high occurring
probabilities together, we will have the master routes repeated when they are applied on
daily plans. However, with the old method, more modification will be made to the master
plans; therefore, we do not gain much from the master routes.

118

5. Conclusions and Future Work

In this thesis, we first consider a real-life Courier Delivery Problem (CDP), a
variant of the Multi-trip Vehicle Routing Problem (MtVRP) with uncertainty in customer
occurrence and urgency in customer demands. We present a problem formulation with
mixed integer programming for an example application of the transportation of medical
specimens. We develop an efficient heuristic based on insertion and Tabu search. Our
model represents the probabilistic nature of customer occurrence using scenario-based
stochastic programming with recourse. We benefit from the simplicity and flexibility of a
master plan with daily recourse actions.
Our first model includes a master plan problem which represents the uncertainty
in the customer occurrence by the probabilities customers are likely to appear and
addresses the urgency in delivery time windows by use of the fleet of vehicles in multiple
trips. We then define a recourse action of partial rescheduling of routes by omitting non-
occurring customers and rescheduling new customers. The created master routes consider
efficiency in routing, to represent slack time for accommodating random requests. The
created daily plans take into account the efficiency in routing, efficiency in alternative
third party courier, as well as route similarities to boost the quality of service. To solve
large size problems of the model, we develop a heuristic based on insertion and Tabu
search.

119

We explore experimentally the sensitivity of our heuristic on randomly generated
problems and a real-life data collected from industry. Experiments on randomly
generated problems include sensitivity analysis in varying problem size, customer
uncertainty scenarios, resource availability and cost parameters. We compare the quality
of the solution with independent daily scheduling, and to an industry standard solution. In
the experiments with real-life data, we compare the quality of solution with current
industry solution with and without recourse action. Sensitivity analysis on varying cost
parameters shows that our heuristic produces a better solution then the current practice by
significantly reducing the cost on taxi use and improving route similarity.
In this thesis, we also study a customer clustering approach which considers not
only the customer locations but also the customer occurring probabilities. In summary, in
the chapter 3, we present a stochastic programming approach based on total distance to
formulate master plans; and chapter 4, we propose to use expected distance as the new
objective function when formulating master plans. We show the significance of studying
this problem with a small sample example. An integer programming formulation is
presented. A Tabu search based heuristic is proposed to solve the problem. Experiments
have been made on the newly proposed heuristic approach; comparison has been made to
the old approach discussed in the previous chapter. Improvement with the new approach
has been discovered on the data set where the customers are clustered such that the depot
is located at a corner. The improvement is gained by clustering the customers with high
occurring probabilities together, so that short master routes get repeated when applied on
daily basis.

120

There are several interesting topics for the future research.
1) Analytical analysis with the new measure for customer clustering given the
uncertainty in customer demand.
2) Experimental analysis with lager size data on the heuristics for solving the
new problem.
3) Extend the study from static cases to dynamic cases.

121

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Abstract (if available)

Abstract Courier delivery services deal with the problem of routing a fleet of vehicles from a depot to service a set of customers that are geographically dispersed. In many cases, in addition to a regular uncertain demand, the industry is faced with sporadic, tightly constrained, urgent requests. An example of such an application is the transportation of medical specimens, where timely, efficient, and accurate delivery is crucial in providing high quality and affordable patient services. ❧ In the first part of this study, we propose to develop better vehicle routing solutions that can efficiently satisfy random demand over time and rapidly adjust to satisfy these sporadic, tightly constrained, urgent requests. We formulate a multi-trip vehicle routing problem using mixed integer programming. We use stochastic programming with recourse for daily plans to address the uncertainty in customer occurrence. The recourse action considers a multi-objective function that maximizes demand coverage, maximizes the quality of delivery service, and minimizes travel cost. Because of the computational difficulty for large size problems, we devise an insertion based heuristic in the first phase, and then use Tabu Search to find an efficient solution to the problem. Simulations have been done on randomly generated data and on a real data set provided by a leading healthcare provider in Southern California. Our approach has shown significant improvement in travel costs as well as in quality of service as measured by route similarity than existing methods. ❧ In the second part of this thesis, we study a method to cluster the customers into groups without generating the actual routes. There is uncertainty in customer occurrence in the problem studied

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Asset Metadata

Creator Wang, Chen (author)

Core Title New approaches for routing courier delivery services

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Industrial and Systems Engineering

Publication Date 11/14/2012

Defense Date 10/17/2012

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

Tag clustering,multi-trip,OAI-PMH Harvest,stochastic vehicle routing,tabu search,time window

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Dessouky, Maged M. (committee chair), Ordóñez, Fernando (committee chair), Petros, Ioannou (committee member)

Creator Email chenwang.ch@gmail.com,wang20@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-110992

Unique identifier UC11288842

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