Heuristic and exact algorithms for vehicle routing
problems
Stefan Ropke
December 2005

Preface
This Ph.D. thesis has been prepared at the Department of Computer Science at the University of
Copenhagen (DIKU), during the period November 2002 to December 2005. The work has been
supervised by Professor David Pisinger.
The thesis consists of four introductory chapters: Chapters 1, 2, 3 and 7, and five research
papers: Chapters 4, 5, 6, 8 and 9. The five research papers have been written in collaboration with
coauthors which are mentioned in the beginning of each paper. The four introductory chapter
have been written solely by the undersigned. The five research papers are relatively self-contained.
Note that each research paper contains it own bibliography and sometimes an appendix. The
bibliography for the introductory chapters are found at the end of this thesis.
The thesis contains three parts. The first part contains the introduction and is split into two
chapters. The next part deals with heuristic and contains one introductory chapter and three
research papers about heuristics. These papers are “technical report versions” that contains more
results than the papers that have been submitted to journals. These extra results are placed in
the appendix of each paper. The last part is about exact methods and contains one introductory
chapter and two research papers.
The thesis started out being solely about heuristics, but after having worked with heuristics
for four or five years, first as a graduate student, then in the industry and as a Ph.D. student I
felt it was time to learn something new and started studying exact methods more intensively in
2004. This has certainly been interesting and I hope the knowledge I have obtained will allow me
to design even better heuristics in the future.
Chapter 9 is the only one of the five papers that has not been submitted to a journal yet. In its
current state it is not ready to be submitted either - it is clearly too long and contains too much
material. We do plan to submit a condensed version. The rest of this section is going to describe
how the paper could be condensed. To understand this, one needs to have read chapter 9.
One way of condensing the paper would be to focus on the SP1 and SP2 relaxations and leave
the SP3 and SP4 relaxations out as well as the addition of valid inequalities. The contribution of
this paper would be
1. Improvements of domination criteria for ESPPTWCPD.
2. The computational comparison of SP1 and SP2.
3. The new pricing heuristics and experiments. More experiments could be carried out.
4. Introduction of standard test instances for exact solution of the PDPTW.
For this paper it would be nice if the issues with algorithms SP1* and SP2* were worked out. The
simplest way of doing this would be to use algorithms SP1*/SP2* to get a lower bound. If the
linear relaxation solution turns out to be fractional then one should switch to algorithms SP1/SP2
to perform branching.
A better approach would be to implement a branching rule that is compatible with the strongest
domination criteria. Branching on time windows as proposed in the paper would be a good
candidate. An alternative is to find a way of perturbing the (dij) matrix such that dij + djk ≥ dik
i

PREFACE ii
always holds when j is a delivery node, even when general cuts have been added to the master
problem. Valid perturbations of the (dij) matrix include subtracting a constant αi from all edges
leaving pickup node i and adding αi to all edges leaving node n+ i. This is valid as a path in the
ESPPTWCPD and SPPTWCPD that visits a pickup must visit the corresponding delivery and
vice versa. This would allow us to add cuts in the original variables to the master problem and
would thereby make the current branching rule work with with SP1*/SP2*.
A second paper could describe the SP3 and SP4 relaxations and incorporate the valid in-
equalities in the branch-and-price algorithm. This paper could also include the strengthened SP4
relaxation that is described in the conclusion of the paper.
Acknowledgments
I would first of all like to thank my supervisor, Professor David Pisinger for encouragement,
countless discusions and for his help with writing the thesis. Without David I would not had taken
on the task of doing a Ph.D. study. Associate Professor Jean-François Cordeau and Professor
Gilbert Laporte also deserves great thank for making my visits to the University of Montreal
possible and for taking time to work with me while I have been visiting. The input I received
from my advisory group, Professor Jacques Desrosiers and Professor Oli Madsen, is also greatly
appreciated.
I would also like to thank the guys at the Algorithmics and Optimization Group at DIKU
for encouragement and for making the average work day more fun and interesting. Similarly I
would like to thank the people I met at the Centre for Research and Transportation in Montreal,
especially “the gang”, for making me feel welcome in a foreign country. I also wish to thank Irina
Dumitrescu for her patience with me when I have postponed working on our joint projects because
of the work involved in finishing this thesis.
Finally I would like to thank friends and family for their support. I especially wish to thank
my parents for their love and support throughout my life. And to my girlfriend Alice: Thank you
for lifting my mood on the days when I have been feeling down in the last couple of months, for
helping me improving the language in the thesis and for being you!
Copenhagen, December 2005, Stefan Røpke
Updated version, June 2006
A number of typos and errors have been corrected in this version of the thesis. Since December
2005 I have spent time working on the research paper presented in chapter 9. This work has
lead to resolution of the most of the issues discussed above and mentioned in the chapter 9: A
transformation of the distance matrix has been found that makes it possible to use SP1*/SP2*
after adding cuts in the master problem and it has been shown that many of the cuts that seemed
worthless in the computational experiements in fact are implied by the strongest set partition
relaxations. These developments have not been included in the updated version of the thesis, but
are described in Ropke and Cordeau [2006].
Let me use the opportunity to thank my opponents: Stefan Irnich, Daniele Vigo and Martin
Zachariasen at the Ph.D. defense, for evaluating the thesis within a short time and for valuable
comments that has lead to several improvements in this updated version of the thesis.
Montreal, June 2006, Stefan Røpke

CONTENTS ccxxxix
6 A general heuristic for vehicle routing problems 100
III Exact methods 145
7 Introduction to exact methods 146
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.2 Linear programming based lower bounds . . . . . . . . . . . . . . . . . . . . . . . . 146
7.2.1 Cutting plane algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7.2.2 Branch-and-cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
7.3 Introduction to branch-and-price . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.3.1 Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.3.2 Column generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.3.3 Branch-and-price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
7.3.4 Branch-and-cut-and-price . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.3.5 Further topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
8 Models and a Branch-and-Cut Algorithm for Pickup and Delivery Problems
with Time Windows 158
9 Branch-and-Cut-and-Price for the Pickup and Delivery Problem with Time
Windows 186
IV Conclusion 239
10 Conclusion 240
11 Summary (in Danish) 242

CHAPTER 7. INTRODUCTION TO EXACT METHODS 154
simplex algorithm. Column generation can be used for any linear programming problem, but it is
particularly useful for linear programs with a huge number of variables as the ones arising from
the extensive models obtained in section 7.3.1. In this section we are going to see how column
generation works for the linear relaxation of (7.20)–(7.23), we refer to Sigurd [2004] for a more
throughout exposition.
The linear relaxation of (7.20)–(7.23) is:
min
∑
w∈Ω
cωλω (7.36)
subject to
∑
ω∈Ω
aωλω ≥b (7.37)
∑
ω∈Ω
λω =1 (7.38)
0 ≤ λω ≤1 ∀ω ∈ Ω (7.39)
Thus the linear relaxion considers a linear combination of the elements from X . Consider looking
at a reduced set of columns Ω¯ ⊆ Ω such that ∣∣Ω¯∣∣ is much smaller that |Ω|. Ω¯ must be chosen such
that the linear program
min
∑
w∈Ω¯
cωλω (7.40)
subject to
∑
ω∈Ω¯
aωλω ≥b (7.41)
∑
ω∈Ω¯
λω =1 (7.42)
0 ≤ λω ≤1 ∀ω ∈ Ω¯ (7.43)
has a feasible solution. If it is difficult to select a subset of columns such that the linear program
has a feasible solution, then one can generate one or more dummy columns that has very high
cost, but constitute a feasible solution. In order to proceed one first need to consider how the
simplex algorithm solves a linear program like
min cTx (7.44)
subject to
Ax ≥ b (7.45)
x ∈Rn+ (7.46)
The simplex algorithm maintains a basic feasible solution that as the name implies is a feasible
solution to the LP, but not necessarily optimal. In each iteration of the simplex algorithm a new
column is chosen to enter the basis. If the new column should have a chance of improving the
basic feasible solution, it must have negative reduced cost cpii
cpii = ci − piAi
where pi is the current dual variables associated with the constraints (7.45) and Ai is the ith
column in A. The typical approach is to select the column with minimum reduced cost, that is,
the column
arg min
i∈{1,...,n}
{ci − piAi}