Close
About
FAQ
Home
Collections
Login
USC Login
Register
0
Selected
Invert selection
Deselect all
Deselect all
Click here to refresh results
Click here to refresh results
USC
/
Digital Library
/
University of Southern California Dissertations and Theses
/
Continuous approximation formulas for cumulative routing optimization problems
(USC Thesis Other)
Continuous approximation formulas for cumulative routing optimization problems
PDF
Download
Share
Open document
Flip pages
Contact Us
Contact Us
Copy asset link
Request this asset
Transcript (if available)
Content
CONTINUOUS APPROXIMATION FORMULAS FOR CUMULATIVE ROUTING
OPTIMIZATION PROBLEMS
by
Ying Peng
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)
May 2023
Copyright 2023 Ying Peng
Dedication
I dedicate this thesis to Beardwood-Halton-Hammersley (BHH),
for their never ending mysteries.
ii
Acknowledgements
Throughout my Ph.D. years at University of Southern California, I have been so fortunate to be supported
by many amazing people. I would like to give my deepest gratitude to my advisor, Professor John Gunnar
Carlsson, to whom I am indebted forever; I could not have done any of this without his guidance. I was
always amazed by his astonishing research abilities, great personality, impressive mentorship, and enthu-
siasm towards science and life. His passion for research and life has been very inspirational for me. I feel
so fortunate to meet Professor Carlsson and complete my Ph.D with him.
I would like to thank my amazing committee members - Professor Maged M. Dessouky, Professor
Meisam Razaveyin, Professor Andres Gomez Escobar, and Professor Vishal Gupta. They provided very
insightful suggestions during my qualifying exam, which greatly helped on this dissertation.
It was my privilege to be a part of Professor Carlsson’s research group for my Ph.D life. This group is
full of talented and fun people who always gave me useful advice and left me with unforgettable moments
throughout the years. I would like to thank all my labmates; Bo Jones, Jiachuan Chen, Javad Azizi, Shannon
Sweitzer, Haochen Jia, and Julien Yu. I also appreciate the aid and help from other ISE students and friends
that I met. With them, life has been so meaningful and joyful.
I also thank my apartment staffs and coordinators, specially Shelly Lewis, Roxanna Carter, and Grace
Owh, for their support during my Ph.D years. They were always my reference for all the academic and
official matters.
iii
Last but not least, I would like to take this opportunity to thank my parents and all my families, who
have been so supportive and encouraging all the way along. It is their unconditional love and care that
helped me go through all the ups and downs, to which I am indebted forever.
iv
TableofContents
Dedication. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Chapter 2: Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1 Research on TSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Research on Cumulative TSP (CTSP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Research on Cumulative CVRP (CCVRP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 Research on continuous approximation models . . . . . . . . . . . . . . . . . . . . . . . . . 8
Chapter 3: Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.1 Problem Assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Chapter 4: Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.1 General preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.2 Preliminaries related to the CVRP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.2.1 Sparse Subset Vehicle Routing Problem (SSVRP) . . . . . . . . . . . . . . . . . . . . 23
Chapter 5: Analysis of the Cumulative TSP whenn→∞ . . . . . . . . . . . . . . . . . . . . . . . 31
5.1 The “most dense to Least dense” Rule Explained . . . . . . . . . . . . . . . . . . . . . . . . 31
5.2 Upper Bounds for CTSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.2.1 X
i
i.i.d
∼ U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.2.2 X
i
i.i.d
∼ P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.3 Lower Bound for CTSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.3.1 X
i
i.i.d
∼ U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.3.2 X
i
i.i.d
∼ P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Chapter 6: Analysis of the Cumulative CVRP whenn→∞ . . . . . . . . . . . . . . . . . . . . . . 43
6.1 Choice ofc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
v
6.2 Upper Bound for CCVRP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6.3 Lower Bound for CCVRP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Chapter 7: Multiple Vehicle Cumulative Routing Problem: m-CTSP . . . . . . . . . . . . . . . . . 52
7.1 Upper Bound for m-CTSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
7.1.1 X
i
i.i.d
∼ U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
7.1.2 X
i
i.i.d
∼ P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
7.2 Lower Bound for m-CTSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
7.2.1 X
i
i.i.d
∼ U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
7.2.2 X
i
i.i.d
∼ P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Chapter 8: Multiple Vehicle Cumulative Routing Problem: m-CCVRP . . . . . . . . . . . . . . . . 61
8.1 Upper Bound for m-CCVRP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
8.2 Lower Bound for m-CCVRP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Chapter 9: Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
9.1 Single Vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
9.1.1 Simulated Instances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
9.2 Multiple Vehicles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
9.2.1 Simulated Instances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
9.2.2 Experiments with Road Network Data . . . . . . . . . . . . . . . . . . . . . . . . . 68
9.3 Managerial Insights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Chapter 10: Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Chapter 11: Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
vi
ListofTables
3.1 Notational Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5.1 Feasible Starting positions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
vii
ListofFigures
5.1 Figure (a) shows a step functionφ that presumably satisfies the conditions (darker - i.e.,
denser-regions are smaller, reflecting the assumption that a
i
Area(⊡ i
) = 1/s) for all i) (b)
shows the TSP tour of a collection of independent samplesY
1
,...,Y
n
of φ whose length
differs from that of a collection of tour within each component (c) by a constant. (d) shows
the rescaled component⊡ ′
i
=Ψ(⊡ i
) and the pointsY
′
1
,...,Y
′
n
. . . . . . . . . . . . . . . . 32
9.1 Ratio of uniform customers’ cumulative waiting time to
√
An
3/2
. . . . . . . . . . . . . . . 65
9.2 Ratio of normal customers’ cumulative waiting time to
√
An
3/2
. (Top Left) figure depot
is at (500, 500) with standard deviation 500. (Top Right) figure depot is at (750, 750) with
standard deviation 500. (Bottom Left) figure depot is at (500, 500) with standard deviation
700. (Bottom Right) figure depot is at (750, 750) with standard deviation 700. . . . . . . . . 72
9.3 Ratio of normal customers’ cumulative waiting time to
√
An
3/2
(left) heatmap for
corresponding region (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
9.4 Ratio of uniform customers’ cumulative waiting time to ¯ rn
3/2
. . . . . . . . . . . . . . . . 74
9.5 Ratio of normal customers’ cumulative waiting time to ¯ rn
3/2
. (Top Left) figure depot is
at (500, 500) with standard deviation 500. (Top Right) figure depot is at (750, 750) with
standard deviation 500. (Bottom Left) figure depot is at (500, 500) with standard deviation
700. (Bottom Right) figure depot is at (750, 750) with standard deviation 700. . . . . . . . . 75
9.6 Ratio of normal customers’ cumulative waiting time to ¯ rn
3/2
(left) heatmap for Circle and
Pentagram region (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
9.7 (Top Left) Ratio of uniform customers’ cumulative waiting time to
√
An
3/2
. (Top Right)
Log ratio of uniform customers’ cumulative waiting time to
√
An
3/2
. (Bottom Left) 500
data points’ heatmap for rectangular and uniform case. (Bottom Right) Tour schedule for
4 vehicles following m-CTSP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
9.8 (Left) Ratio of uniform customers’ cumulative waiting time to
√
An
3/2
. (Center) Log ratio
of customers’, following normal distribution, cumulative waiting time to
√
An
3/2
. (Right)
Tour schedule for 4 vehicles following m-CTSP. (Top) Depot is at (500, 500) with standard
deviation 500. (Bottom) Depot is at (500, 500) with standard deviation 700. . . . . . . . . . 78
viii
9.9 (Left) Ratio of uniform customers’ cumulative waiting time to
√
An
3/2
. (Center) Log ratio
of customers’, following normal distribution, cumulative waiting time to
√
An
3/2
. (Right)
Tour schedule for 4 vehicles following m-CTSP. (Top) Depot is at (750, 750) with standard
deviation 500. (Bottom) Depot is at (750, 750) with standard deviation 700. . . . . . . . . . 78
9.10 (Left 1) Ratio of customers’, who follow the normal distribution, cumulative waiting time
to
√
An
3/2
. (Left 2) Log ratio of customers’, following normal distribution, cumulative
waiting time to
√
An
3/2
. (Left 3) heatmap for different regions. (Left 4) Tour schedule for
4 vehicles following m-CTSP. (Top) Circle Region. (Middle) Triangular Region. (Bottom)
Pentagram Region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
9.11 (Top Left) Ratio of uniform customers’ cumulative waiting time to ¯ rn
3/2
. (Top Right) Log
ratio of uniform customers’ cumulative waiting time to ¯ rn
3/2
. (Bottom Left) 500 data
points’ heatmap for rectangular and uniform case. (Bottom Right) Tour schedule for 4
vehicles following m-CCVRP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
9.12 (1
st
and 3
rd
column) Ratio of customers’, sampled from normal distribution. cumulative
waiting time to ¯ rn
3/2
. 1
st
column with standard deviation 500 and 3
rd
column with
standard deviation 700. (2
nd
and 4
th
column) Log ratio of customers’ cumulative waiting
time to ¯ rn
3/2
. (Top) Depot is at (500, 500). (Bottom) Depot is at (750, 750). . . . . . . . . . . 81
9.13 (Left) Ratio of customers’, who follow the normal distribution, cumulative waiting time to
¯ rn
3/2
. (Center) Log ratio of customers’, following normal distribution, cumulative waiting
time to ¯ rn
3/2
. (Right) Tour schedule for 4 vehicles following m-CCVRP. (Top) Circle
Region. (Bottom) Pentagram Region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
9.14 Prediction Cost: (Up) CTSP. (Bottom) CCVRP . . . . . . . . . . . . . . . . . . . . . . . . . . 82
9.15 Prediction Cost: (Left) m-CTSP. (Right) m-CCVRP . . . . . . . . . . . . . . . . . . . . . . . 83
ix
Abstract
Traditional objectives in vehicle routing problems (VRPs) include time, distance, or profitability. When one
adopts a “customer-centric” perspective, different objectives emerge. For example, instead of minimizing
the total cost of performing service (which causes some commodities/customer to be served significantly
later than others), this thesis studies a family ofCumulativeRoutingProblems in which the goal is to min-
imize the total waiting time of all customers to be served. This thesis analyzes the asymptotic behavior of
these problems, including theCumulativeTravellingSalesmanProblem and theCumulativeCapacitatedVe-
hicleRoutingProblem, which we call the CTSP and CCVRP respectively. For each problem, we derive lower
and upper bounds that characterize the cost of these problems in an asymptotic limit as demand becomes
large, under the assumption that all point-to-point distances are Euclidean, and all points are independent
and identical samples of a probability density on a compact planar region. We also describe the impact
of vehicle capacities on the final cost. Next, we extend our analysis to the case where multiple vehicles
provide coordinated service. Numerical simulations confirm that our analysis provides good predictions
when applied to simulations in the Euclidean plane and on road network data, where we are able to predict
total costs within 5% of the ground truth.
x
Chapter1
Introduction
Thetravelingsalesmanproblem (TSP) is an NP-hard problem in computer science and combinatorial opti-
mization. It is concerned with finding the optimal Hamiltonian cycle which has the minimum sum of edge
weights. There are many problems in the literature that are based on the TSP, such as the traveling re-
pairman problem (TRP), vehicle routing problem (VRP), and the traveling purchaser problem (TPP), which
all have important applications in engineering and computer science problems such as optimal routing in
communication networks, planning, logistics, manufacturing of microchips, transportation, and delivery
systems.
However, as shown in recent publications, this kind of classical cost-minimizing problem may not
properly reflect the need for fast service, or equity [60]. One specific example is the procurement of
humanitarian aid in the context of natural disasters, such as tsunamis or earthquakes [60, 62, 22, 61].
When natural disasters strike, humanitarian supply chains (unlike commercial supply chains, which focus
on quality and profitability) are focused on minimizing loss of life and suffering and should have higher
priority and thus require the definition of more customer-centric or service-based objective functions.
There are many performance measures that can be used to define such functions. Minimizing the average
arrival time, or minimizing the sum of arrival times, are common. Apart from humanitarian aid after a
1
natural disaster, there exist many other practical applications such as distribution, machine scheduling and
power control and receiver optimization in wireless telecommunication systems [16].
The preceding examples suggest that a “customer-centric” problem should be considered, so as to bet-
ter reflect priorities and ensure equity and fairness [22]. Instead of minimizing the total travel cost, which
caused some commodities or customers to be served significantly later than others, this new problem’s
goal is to minimize the sum of customers’ waiting times. This new problem is a variant of classical routing
problems called the Cumulative Routing Problem. In this thesis, we are interested in the Cumulative Trav-
elling Salesman Problem (CTSP) and the Cumulative Capacitated Vehicle Routing Problem (CCVRP), where
CCVRP is a generalization of the CTSP.
Bianco, Mingozzi, and Ricciardelli [17] considered minimizing the sum of all distances traveled from the
origin to all other cities in 1993. Later, Ngueveu, Prins, and Wolfler Calvo [60] took vehicle capacities into
consideration and first introduced the Cumulative Vehicle Routing Problem (CVRP). Despite considerable
research focus on this kind of problem and the generation of efficient heuristics and exact algorithms to
attack it, they are still not flexible enough for real-world applications. One of the issues is that they cannot
handle large-scale points distributed by different probability laws, which is the main characteristics of the
real-world problems. For instance, the postal system relies on continuous approximations of tour length
to partition the service territory [38]. Based on this motivation, this thesis studies the asymptotic analysis
of this family of problems.
There is a long history of studies on asymptotic and probabilistic bounds over various graph structures.
These works study many graph structures over Euclidean points in the context of the Beardwood-Halton-
Hammersley (BHH) theorem and its extensions. This theorem is originally stated in [13] and later further
developed by Steele in [76]. Instead of finding the optimal (shortest) tour length over random sample
points embedded in Euclidean spaces, our goal of finding optimal (shortest) cumulative tour lengths adds
an additional problem characteristic that compounds its difficulty.
2
This thesis is primarily concerned with the asymptotic behavior of both the Cumulative Travelling
Salesman Problem (CTSP) and the Cumulative Capacitated Vehicle Routing Problem (CCVRP). We perform
the analysis in the Euclidean plane, where points are uniformly and non-uniformly distributed respectively.
We also describe the impact of vehicle capacities on the final cost. Furthermore, we extend our analysis to
the case where multiple vehicles are involved, and we explore how coordinated service can affect the final
solution. Finally, we demonstrate a practical application of our cumulative model by using the CTSP and
CCVRP to predict the total waiting cost of a population, even when we only have access to a subset of the
data. This application highlights the potential of our models to address real-world optimization problems,
where incomplete information is often a significant challenge.
3
Chapter2
LiteratureReview
2.1 ResearchonTSP
The traveling salesman problem (TSP) is one of the most famous NP-hard combinatorial optimizations
problems, which has captured the attention of mathematicians and computer scientists. There are many
practical applications for this problem such as printed circuit board design [4], X-Ray Crystallography
[18], Overhauling Gas Turbine Engines [63], the Order-Picking Problem in Warehouses [65], Computer
Wiring [66], Vehicle Routing [12], Mask Plotting in PCB Production [34], control robots [51] and so on.
This problem’s statement is very simple: there aren cities on a map, a salesman wants to start from a home
city, visit all the other cites exactly once, and come back to the home city, with the minimum amount of
distance travelled.
One of the earliest studies on the TSP was conducted by Held and Karp [46]. They proposed an integer
linear programming formulation for this problem and demonstrated its effectiveness through numerical
experiments on various benchmark instances. They also proposed a dynamic programming algorithm that
could optimally solve small instances of the problem. However, finding exact solution for large instances
is infeasible due to its computational complexity. Therefore, researchers have turned to heuristics for
TSP. These algorithms depend on an initial starting solution, and although they may not find the optimal
solution, they can often find high-quality “sub-optimal” solutions much faster than exact approaches. One
4
of the most widely used heuristic algorithms for the TSP is the Lin-Kernighan algorithm, developed by Lin
and Kernighan [52]. Other approaches include genetic algorithms proposed by Grefenstette et al. [44], a
hybrid heuristic algorithm combining genetic algorithms and local search techniques proposed by Bektaş
and Laporte [14], simulated annealing introduced by Aarts, Korst, and Laarhoven [1], tabu search used by
Fiechter [36], and an effective evolutionary algorithm proposed by Nagata [57].
In many applications, assuming a probability distribution on the sample points of the graph offers
insights. There is a long history of studies on asymptotic bounds over various graph structures. One of
the famous studies is the Beardwood-Halton-Hammersley (BHH) theorem and its extensions [13]. This
analysis is to develop an effective approximation of the tour cost with a small computational price. Many
graph structures over Eucilidean sample points have been studied in the context of BHH theorem.
2.2 ResearchonCumulativeTSP(CTSP)
TheTime-DependentTravelingSalesmanProblem (TDTSP) is one of the most well-known variants of TSP.
Given a graphG=(V,A) whereV ={1,2,...,n} andA={(i, j):i, j=1,...,n,i̸= j}, this problem tries
to find a minimum cost Hamiltonian circuit, where arc costs depend on its position in the tour [43].
One of the special case of TDTSP is called the Cumulative Traveling Salesman Problem (CTSP) also
known as the traveling delivery problem [41], the The Traveling Repairman Problem (TRP) [56, 2, 9, 39],
and the the Minimum Latency Problem (MLP) [7, 8, 19, 42, 73]. Its goal is to find a path, initiated at the
depot and visiting every customer exactly once, such that the sum of the times required to reach every
customer, along the path, is minimal.
Such problems are often used when a fairness criterion about visiting clients needs to be enforced. One
of the application is delivering pizzas, which requires that the total time to visit all customers be minimal
[37]. This problem can also be used in the area of computer networks [9]. Simchi-Levi and Berman [72]
show how to use this problem to find the routing of automated guided vehicles through cells in a flexible
5
manufacturing system. Arora and Karakostas[8] applied this problem in disk-head scheduling. However,
even CTSP is a variant of TSP, it is an NP-hard problem and much harder to solve and approximate [19].
The CTSP problem is similar to TSP, and many different mathematical programming formulations have
been proposed. Fischetti, Laporte, and Martello [37] used Integer Linear Programming to solve it. Méndez-
Dıéaz, Zabala, and Lucena [56] exploited connections between TDP and the Linear Ordering Problem
(LOP). Wu [80] used a dynamic programming algorithm to solve it, and Branch-and-Bound algorithm,
which is used in TSP, can still be used to solve this problem [81]. However, due to the NP-hardness of this
problem, all the approaches above cannot be used to solve medium- and large-scale instances.
Due to the NP-hardness of CTSP, researchers have turned to heuristic algorithms to obtain more
tractable time complexity guarantees. Some promising approaches have been proposed. Dewilde et al.[32]
described a tabu search algorithm with multiple neighborhoods that can quickly generate high-quality
solutions. Beraldi et al.[15] formulated the problem via a non-linear model and used a beam search heuris-
tic to solve it heuristically. Bruni, Beraldi, and Khodaparasti[21] presented a new heuristic based on
General Variable Neighborhood Search that combines multiple neighborhoods in an effective way, out-
performing previous heuristics in experiments. Salehipour et al.[70] developed a new approach called
GRASP+VND/VNS, a multi-start method that consists of a greedy randomized construction phase and a
variable neighborhood descent or search improvement phase in each iteration.
2.3 ResearchonCumulativeCVRP(CCVRP)
The Vehicle Routing Problem (VRP), a classic combinatorial optimization problem introduced by Dantzig
and Ramser[31], has practical applications in resource allocation for electric power distribution [82], phar-
maceutical distribution [49], and waste collection [64].
By considering additional constraints on route constructions, different Vehicle Routing Problems (VRPs)
have been formulated. One of the practical and central variants is the Capacitated Vehicle Routing Problem
6
(CVRP), which aims to design vehicle routes from a depot to a set of geographically-scattered customers,
with capacity limit constraints. This problem’s objective is to minimize the total travel cost. In such case,
some customers may be served later than others, which may (in some contexts) violate equity and fairness
[33]. To better reflect priorities and to ensure equity and fairness, it has been argued that the waiting time
of a service system from the customer’s point of view should be considered [22]. Based on this need, a new
variant of the classical CVRP, the cumulative capacitated vehicle routing problem (CCVRP), has arisen.
The CCVRP was first introduced by Ngueveu, Prins, and Wolfler Calvo [60]. In contrast to the tradi-
tional VRP, whose objective function is cost-based, CCVRP’s objective goal can be viewed as service-based.
Its aims to minimize the sum of arrival times at all intermediate customers. Campbell, Vandenbussche, and
Hermann [22] states that the CCVRP’s and traditional VRP’s optimal solutions are significantly different.
Ngueveu, Prins, and Wolfler Calvo [60] presented a mathematical model based on a cumulative vehicle
routing problem with time windows [47] and developed a memetic algorithm (MA) as a solution method
[60]. It is assessed using instance whose size range from 50 to 199 nodes. Rivera, Murat Afsar, and Prins
[69] used mixed integer linear programs, a flow-based model and a set partitioning model for small in-
stances with 20 sites. Their exact algorithm outperforms a commercial MIP solver on small instances and
can solve cases with 40 sites to optimality. Mattos Ribeiro and Laporte [55] proposed an Adaptive Large
Neighborhood Search (ALNS) for this problem. Chen, Dong, and Niu [26] provided a comparison with MA
and ALNS. They showed that ALNS outperformed MA in terms of computational time and quality of the
solution. Later, Ke and Feng [48] developed a two-phase metaheuristic and its result is better than MA and
ALNS. In 2014, Lysgaard and Wøhlk [54] investigated the first exact algorithm for the CCVRP based on
Branch-and-cut-and-price procedure (BCP). This algorithm is capable of solving instances with up to 150
customers in reasonable computational time.
When one has multiple depots for this cumulative routing problem, Rivera, Murat Afsar, and Prins
[69] transformed it into a resource-constrained shortest path problem where each node corresponds to
7
one trip and the sites to visit become resources. This transferred problem can be solved via an adaptation
of Bellman–Ford algorithm to a directed acyclic graph with resource constraints and a cumulative objective
function. Rivera, Afsar, and Prins [68] compared a mixed integer linear program (MILP), a dominance rule,
and a hybrid metaheuristic: a multi-start iterated local search (MS-ILS) calling a variable neighborhood
descent withO(1) move evaluations. On three sets of instances, MS-ILS obtains good solutions.
2.4 Researchoncontinuousapproximationmodels
There exist numerous discrete models for routing problems which have also been developed to address
issues at the operational level in both deterministic and stochastic environments [11, 28, 50]. However,
there exist many drawbacks. First, they generally have a relatively complex formulation structure that may
hinder one’s understanding of problem properties and managerial insights [5]. Often, the routing problems
belong to the class of NP-hard problems, and hence solving large-scale instances would require enormous
computational efforts, which likely increase exponentially with the problem instance size. Hence, it is not
practical to solve large-scale logistics problems to optimality. All of this drawbacks are compounded further
when we need to make decisions in stochastic, time-varying, competitive and coupled environment.
Thecontinuousapproximation paradigm was proposed as a means of partially addressing the preceding
challenges. Continous approximation approaches were first proposed by Newell[58] and Newell [59]. It
features continuous representations of input data and decision variables as density functions over time
and space, and the key idea is to approximate the objective into a functional (e.g. and integral) of localized
functions that can be optimized by relatively simple analytical operations. Each localized function approx-
imates the cost structure of a local neighborhood with nearly homogeneous settings. Such homogeneous
approximations enable mapping otherwise high-dimensional decision variables into a low-dimensional
space, which allows the optimal design for this neighborhood to be obtained with simple calculus, even
when spatial stochastic, temporal dynamics and other operational complexities are present.
8
One of the famous applications of continuous approximations is in routing problem, which determines
the most economic routes for vehicles to deliver or pickup commodities or people across a continuous
space. In routing problems, one of the most famous such theorems is the aforementioned BHH theo-
rem[13]. It states that when the number of customer points approaches infinity on a compact area, the
optimal tour length can be approximated by a simple analytic expression.
TSP(n)∼ k
TSP
√
An
wherek
TSP
is a constant, whose value is determined by the distance metric (typically Euclidean).
It is known that in practice, the BHH theorem result underestimates the tour length when the area is
an elongated shape [30]. In order to address this issue, Daganzo [30] introduced a strip strategy method,
which can efficiently compute the optimal tour length in different shapes. These two methods [13, 30] gave
rise to several extensions. Later, Webb [78], Christofides and Eilon [27] [35] introduced the approximation
to length of capacitated vehicle routing problem, which shows that the length is related to three terms:
capacity, number of customers, and the average distance between the customers and the depot and area.
Daganzo [29] shows when the depot is not necessarily located in the area that contains the customers, the
tour length admits the following approximation:
CVRP(n)∼ 2¯ rn
c
+k
CVRP
√
nA
Where ¯ r represent the average distance between the customers,c corresponds to the capacity limit per
vehicle, andA is the area.
9
Results of this kind inspired many subsequent studies. For example, Rifki et al. [67] focused on an
asymptotic approximation of the traveling salesman problem with uniform non-overlapping time win-
dows. Instead of visiting all points, Aldous and Krikun [3] formalized the idea of minimum average edge-
length in a path linking some infinite subset of points of a Poisson process.
Based on the literature review, we conclude that both CTSP and CCVRP have been extensively studied,
but have not yet been analyzed from the continuous approximation perspective. In this paper, we try to
connect the bridge between both and fill the gap.
10
Chapter3
ProblemStatement
3.1 ProblemAssumption
We begin with the following assumptions about our four problems of interest, the Cumulative TSP (CTSP),
the Cumulative Capacitated VRP (CCVRP) and their generalizations, which are the multiple Vehicle Cu-
mulative TSP (m-CTSP) and the multiple vehicle Cumulative Capacitated VRP (m-CCVRP) respectively:
• The travel speed is a constant, which is proportional to the distance between points. Therefore,
“distance” and “time” are essentially interchangeable.
• Regarding the Cumulative Capacitated Vehicle Routing Problem (CCVRP), we restrict our analysis
to the scenario where a single vehicle is available, and it must periodically return to the depot as a
result of its capacity constraints. It should be noted that the generalized version of CTSP and CCVRP
will not be subject to this assumption.
3.2 Notation
The notational conventions for this thesis are summarized in table 3.1. We will also make use of some
standard conventions in asymptotic analysis:
11
• We say that f(x)∈O(g(x)) if there exists a constantc and a valuex
0
such that f(x)≤ c· g(x) for all
x≥ x
0
.
• We say that f(x)∈Ω(g(x)) if there exists a constantc and a valuex
0
such that f(x)≥ c· g(x) for all
x≥ x
0
, and
• We say that f(x)∼ g(x) if lim
x→∞
f(x)/g(x)=1.
and we also make the following definition:
Definition 3.2.1. Let φ(x)≡ ∑
s
i=1
a
i
1(x∈⊡ i
) be a step density function with compact supportR such
that a
1
≥···≥ a
s
and a
i
Area(⊡ i
)=
1
s
for all i (so that Area(R) = 1). Define Ψ:R→R
2
be the union of
allΨ
i
, where{Ψ
i
:⊡ i
→⊡ ′
i
|Ψ
i
(y)=
√
a
i
y+ξ
i
}. ξ
i
is chosen to make:
• Each⊡ ′
i
disjoint;
• Area(⊡ ′
i
)=
1
s
for alli;
• Points in⊡ ′
i
are uniform distribution.
12
Table 3.1: Notational Conventions
R An area that customers showing up having Area(R)=A
n Number of clients inR
x
0
Depot
X
i
i’th customer whereX
i
i.i.d
∼ P ,i={1,...,n}
c Each vehicle’s capacity
d
ij
Euclidean distance between customeri and customer j
r
i
Euclidean distance between customeri and depot
¯ r Average euclidean distance between customers and depot over alln customers
⊡ A rectangle inR
2
1{E} Indicator function of eventE
φ A step density function defined on R
f An absolutely continuous probability distribution defined on R
TSP(X
1
,...,X
n
) Length of shortest TSP tour that visits alln points
VRP(X
1
,...,X
n
;c) Length of shortest VRP tour that visits alln points with vehicle capacityc
L(X
1
,...,X
n
;pn) length of the shortest tour that visits pn out of alln points, where p∈(0,1)
CumL(X
1
,...,X
n
) Length of the shortest cumulative TSP tour that visits alln points
CumL(X
1
,...,X
n
;c) Length of the shortest cumulative VRP tour that visits alln points
with vehicle capacityc
CumL(m;X
1
,...,X
n
) Length of the shortest cumulative TSP tour that visits alln points
usingm different vehicles
CumL(m;X
1
,...,X
n
;c) Length of the shortest cumulative VRP tour that visits alln points
with vehicle capacityc usingm different vehicles.
13
Chapter4
Preliminaries
This section consists of preliminary results that we will make of in our subsequent analysis; the first com-
ponent consists of both well-known results (such as the Borel-Cantelli lemma or Stirling’s approximation),
or routine technical exercises such as application of the union bound or Cavalieri’s principle. The second
component deals with results that are particularly relevant to the capacitated VRP.
4.1 Generalpreliminaryresults
Theorem 1 is the famous Beardwood-Halton-Hammersley (BHH) [13] theorem.
Theorem1 (BHH theorem). There is a constantβ
d
such that, for almost any sequence of independent vari-
ables{X
i
} sampled from an absolutely continuous density f onR
d
with compact support, we have
lim
n→∞
TSP(X
1
,...,X
n
)
n
(d− 1)/d
=β
d
Z
R
d
f(x)
(d− 1)/d
dx
with probability one.
For d = 2, although the exactβ
2
is unknown, numerical computations suggestβ
2
≈ 0.714 [6].
Lemma2 (Borel-Cantelli). Let{E
n
}beasequenceofeventsinasamplespace. Thenif∑
∞
n=1
Pr(E
n
)<∞,we
have
14
Pr(E
n
occurs infinitely often )=0
which is equivalent to
Pr(limsup
n→∞
E
n
)=Pr(∩
∞
n=1
∪
∞
m=n
E
m
)=0
Lemma 3 (Approximation with a step function). Let f be a probability density function with compact
supportR⊂ R
2
whose level sets have Lebesgue measure zero. Define
P(x):=Pr(f(X)≤ f(x))=
Z
x
′
:f(x
′
)≤ f(x)
f(x
′
)dx
′
For anyε >0, there exists a step density functionφ(x):=∑
s
i=1
a
i
1(x∈⊡ i
) and corresponding
Π(x):=Pr(φ(X)≤ φ(x))=
Z
x
′
:φ(x
′
)≤ φ(x)
φ(x
′
)dx
′
such that the following conditions hold:
1.
R
R
|φ(x)− f(x)|dx≤ ε,
2. |1(P(X)≥ p)− 1(Π(X))≥ p)|≤ ε ∀x∈R,
3. All of the components ofφ have the same mass, i.e. a
i
Area(⊡ i
)=1/s.
Proof. Even though the level sets having measure zero is not necessary, in order to keep notation consis-
tent, we keep it (this requirement is violated when f is a uniform distribution, which is our base case for all
of the instances in this paper). For a large integerq, define contour sets S
i
={x:(i− 1)/q≤ P(x)≤ i/q}.
15
For eachS
i
, we can approximate the restriction of f toS
i
(i.e., f(x)1(x∈S
i
)) to arbitrary precisionε
′
by
a step functionψ
i
(x)=∑
j
a
ij
1{x∈⊡ ij
} (This is the classical result of measure theory, see e.g., Theorem
2.4 (ii) of [77], i.e., step functions are dense in L
1
(R
d
)). Based on the claim shown in Appendix, for all i
and j, we can assume without loss of generality that
• ⊡ ij
⊂S
i
for alli and j (i.e., the support ofψ
i
is contained inS
i
),
• a
ij
0, we setq=⌈1/ε
′
⌉ andε
′
=ε/q in the above construction. The functionψ :=∑
q
i=1
ψ
i
is therefore a step density approximation of f whose aggregate error overR is at most ε, so condition 1
is satisfied. If we define Π
′
(x)=
R
x
′
:ψ(x
′
)≤ ψ(x)
φ(x)dx, then condition 2 is satisfied as well (using triangular
inequality and add subtractΠ
′
).
For ease of notation, we now re-index all of the components of ψ (i.e., we disregard the fact that ψ
decomposes into a sum of ψ
i
’s) so that we simply have ψ(x)=∑
j
b
j
1(x∈⊡ j
), where b
j
and Area(⊡ j
)
are rational. If we take δ to be the lowest common denominator over all b
j
Area(⊡ j
), then we can write
b
j
Area(⊡ j
) =z
j
δ, withz
j
a positive integer. To satisfy the last condition, all that remains is to decompose
each⊡ j
intoz
j
pieces of equal area, and letφ denote the step function resulting thereof, which completes
the proof.
Lemma4 (Super- and sub-additivity of the TSP). LetR⊂ R
2
be a compact Lebesgue measurable set, par-
titioned into piecesP
1
,...,P
m
whose common boundaries have finite length. There exists a constant C that
depends only on the partition such that, for any set of pointsX =X
1
,...,X
n
⊂R , we have:
− C+
m
∑
i=1
TSP(X∩P
i
)≤ TSP(X)≤ C+
m
∑
i=1
TSP(X∩P
i
)
16
Proof. The proof is shown in Lemma 2.3.1 of [75].
Lemma5. Let f :R→R be a real-valued function and letB
d
(r)⊂ R
d
be a ball of radiusr centered about
the origin. We have
Z
B
d
(r)
f(∥x∥)dx=
Z
r
0
S
d− 1
(t)f(t)dt
WhereS
d− 1
(t) is the surface area of a(d− 1)-sphere of radiust, which is given by
S
d− 1
(t)=
2π
d/2
Γ(d/2)
t
d− 1
Lemma6. The volume of a d-dimensional ball of radiusr is
π
d/2
r
d
Γ(d/2+1)
Lemma 7 (Stirling’s formula). The gamma function Γ(x) satisfies logΓ(x+1) = xlogx− x+
1
2
logx+
1
2
log2+
1
2
logπ+O(
1
x
) asx→∞.
Lemma 8. Let l > 0 and letD⊂ R
dn
denote the set of all n-tuple (u
1
,...,u
n
) of points inR
d
such that
∑
n
i=1
∥u
i
∥≤ l. The volume ofD, Vol(D), satisfies
Vol(D)=
2π
d/2
Γ(d/2)
!
n
Γ(d)
n
Γ(dn+1)
· l
dn
(4.1)
Ford=2, this equation reduces to
Vol(D)=
(2πl
2
)
n
(2n)!
17
Proof. This proof is straightforward:
Vol(D)=
Z
B
d
(l)
Z
B
d
(l−∥ u
n
∥)
··· Z
B
d
(l− ∑
n
i=2
∥u
i
∥)
1du
1
du
2
...du
n
and apply Theorem 5.
Corollary 8.1. Let l > 0 and D
′
⊂ R
dn
be the set of all n-tuples (x
1
,...,x
n
) of points in R
d
such that
∥x
1
∥+∑
n
i=2
∥x
i
− x
i− 1
∥≤ l. The volume ofD
′
, Vol(D
′
) satisfies
Vol(D
′
)=
2π
d/2
Γ(d/2)
!
n
Γ(d)
n
Γ(dn+1)
· l
dn
Proof. Apply Cavalieri’s principle to Theorem 8.
Lemma9. LetX
0
betheorigininR
d
andletX
1
,...,X
n
beacollectionofindependent,uniformsamplesdrawn
from a regionR of unit volume inR
d
. Then
Pr(TSP(X
0
,X
1
,...,X
n
)≤ l)≤ Γ(n+1)·
2π
d/2
Γ(d/2)
!
n
Γ(d)
n
Γ(dn+1)
· l
dn
Ford=2, this inequality reduces to
Pr(TSP(X
0
,X
1
,...,X
n
)≤ l)≤ n!
(2πl
2
)
n
(2n)!
18
Proof. We can regard the samples X
1
,...,X
n
as being a single sample drawn uniformly fromR
dn
. Us-
ing Corollary 8.1, it is easy to see that the probability∥X
1
∥+∑
n
i=2
∥X
i
− X
i− 1
∥≤ l should be equal to
2π
d/2
Γ(d/2)
n
Γ(d)
n
Γ(dn+1)
· l
dn
. Since Vol(D
′
∩R
n
)≤ Vol(D
′
), the formulation becomes:
Pr(∥X
1
∥+
n
∑
i=2
∥X
i
− X
i− 1
∥≤ l)≤
2π
d/2
Γ(d/2)
!
n
Γ(d)
n
Γ(dn+1)
· l
dn
Finally, we note that there exist n!=Γ(n+1) different permutations of X
1
,...,X
n
, which we multiply by
the right-hand side of the above.
Theorem 10 (Uniform sparse subset TSP). Let X
1
,...,X
n
be independent uniform samples drawn from a
region of areaA inR
2
. LetL(X
1
,...,X
n
;pn) denote the length of the shortest tour that visits pn of the points
X
1
,...,X
n
, for fixed 0< p<1. We have
liminf
n→∞
L(X
1
,...,X
n
;pn)
p
√
n
≥ λ
1
√
A
with probability one, whereλ
1
=0.2935.
Proof. LetE
n
be the event thatL(X
1
,...,X
n
;pn)≤ bp
√
An for fixed b. Using Theorem 9 we have:
Pr(E
n
)≤
n
⌈pn⌉
Γ(⌈pn⌉+1)·
2π
d/2
Γ(d/2)
!
⌈pn⌉
Γ(d)
⌈pn⌉
Γ(d⌈pn⌉+1)
· (bp
√
An)
d⌈pn⌉
Here,
n
⌈pn⌉
represent the number of possible subsets choosing⌈pn⌉ points fromn. Since we are only
interested in the cased=2 in this paper, the above equation reduces to
Pr(E
n
)≤
n
⌈pn⌉
Γ(⌈pn⌉+1)·
2π
Γ(1)
⌈pn⌉
Γ(2)
⌈pn⌉
Γ(2⌈pn⌉+1)
· (bp
√
An)
2⌈pn⌉
=
n
⌈pn⌉
Γ(⌈pn⌉+1)
Γ(2⌈pn⌉+1)
(2πb
2
p
2
An)
⌈pn⌉
19
Since
n
⌈pn⌉
=
n!
⌈pn⌉!(n−⌈ pn⌉)!
=
Γ(n+1)
Γ(⌈pn⌉+1)Γ(n−⌈ pn⌉+1)
Combining it into the above equation, we can get:
Pr(E
n
)≤ Γ(n+1)
Γ(⌈pn⌉+1)Γ(n−⌈ pn⌉+1)
Γ(⌈pn⌉+1)
Γ(2⌈pn⌉+1)
(2πb
2
p
2
An)
⌈pn⌉
=
Γ(n+1)
Γ(2⌈pn⌉+1)Γ(n−⌈ pn⌉+1)
(2πb
2
p
2
An)
⌈pn⌉
It is easy to verify thatΓ(x+1)≤ Γ(⌈x⌉+1)≤ Γ((x+1)+1)=(x+1)Γ(x+1)≤ (x+1)Γ(⌈x⌉+1).
The above formulation becomes:
Pr(E
n
)≤ (n− pn)Γ(n+1)
Γ(2pn+1)Γ(n− pn+1)
(2πb
2
p
2
An)
⌈pn⌉
Takinglog of both sides and applying Theorem 7, we have:
logPr(E
n
)≤ (plog(1− p)+2plogb− plog2+plogπ+plogA− log(1− p)+p)n
− 1
2
logp+
1
2
logn+
1
2
log(1− p)− log2− 1
2
logπ+O(
1
n
)
Since n dominates in the above formulation, when n→∞, if we want logPr(E
n
)→− ∞, we should
require that its coefficient be negative, i.e.:
20
plog(1− p)+2plogb− plog2+plogπ+plogA− log(1− p)+p<0
log(1− p)+2logb− log2+logπ+logA− 1
p
log(1− p)+1<0
logb
2
<
1− p
p
log(1− p)+log
2
Aπe
b<
r
2
Aπe
(1− p)
(1− p)/p
This is a convex function and increasing in p∈(0,1), so we can get:
lim
p→0
+
r
2
Aπe
(1− p)
(1− p)/p
≥ √
2Aπ
e
>0.2935
√
A=λ
1
√
A
This guarantees that∑
∞
n=1
Pr(E
n
)<∞ because Pr(E
n
)≤ a
− n
for somea>1. Applying Theorem 2, we
get:
λ
1
≤ liminf
n→∞
L(X
1
,...,X
n
;pn)
p
√
An
with probability one, which completes the proof.
Corollary10.1. (Tourlengthinasubsetwithuniformdemand)LetX
1
...,X
n
beindependentuniformsamples
drawn from a compact regionR⊂ R
2
with areaA and letS⊂R with Area(S) =q. Then
liminf
n→∞
L(X
1
,...,X
n
∩S;pn)
p
√
n
≥
λ
1
√
A if p≤ q
∞ otherwise
whereλ
1
=0.2935.
21
Proof. Ifp>q, then the Law of Large Numbers shows that|X
1
,...,X
n
∩S|/n→q asn→∞ with probability
one, soL(X
1
,...,X
n
∩S;pn) does not exist.
If p≤ q, it is obvious that
L(X
1
,...,X
n
∩S;pn)≥ L(X
1
,...,X
n
;pn)
Now we apply Theorem 10 to conclude the proof.
4.2 PreliminariesrelatedtotheCVRP
This section presents some preliminary results that are necessary for our analysis of the Cumulative Ca-
pacitated VRP (CCVRP). They are fairly standard, albeit lengthy, and are merely a probabilistic limiting
interpretation of a seminal result due to [45]:
Theorem11 ( [45]). Let{X
1
,...,X
n
}beasetofcustomers(pointsinaEuclideanPlane). Theoptimallength
of the shortest tour that visits all these n points with the vehicle capacity c, written as VRP
∗ (X
1
,...,X
n
;c),
satisfies the following:
max{
2n
c
r,TSP(X
1
,...,X
n
)}≤ VRP
∗ (X
1
,...,X
n
;c)≤ 2
l
n
c
m
r+(1− 1
c
)TSP(X
1
,...,X
n
)
The purpose of this section is to apply Theorem 11 to randomly distributed demand in order to discern
its asymptotic behavior. The upper and lower bounds of interest are as follows:
Claim 1. Let f(x) be a density function with compact supportR⊂ R
2
. Consider a VRP with capacities c
that vary relative ton via the relationshipc=k
c
√
n. We have
22
limsup
n→∞
VRP(X
1
,...,X
n
;k
c
√
n)
√
n
≤ Z
x∈R
(β
2
p
f(x)+
2
k
c
f(x)∥x∥
2
)dx
and
liminf
n→∞
VRP(X
1
,...,X
n
;k
c
√
n)
√
n
≥ 1
2
Z
x∈R
(β
2
p
f(x)+
2
k
c
f(x)∥x∥
2
)dx
Proof. See Section 4 of [25]. Note that the upper and lower bounds are within a factor of2 of one another.
4.2.1 SparseSubsetVehicleRoutingProblem(SSVRP)
For this section, we let VRP(X
1
,...,X
n
;c;pn) denote the length of the shortest tour that visits pn of points
fromX
1
,...,X
n
with capacity limitc. To save wear and tear on floors and ceilings, when pn is non-integer,
we round it up.
Theorem 12 (Tour length from a step density). Let φ(x)≡ ∑
s
i=1
a
i
1(x∈⊡ i
) be a step density function
with compact supportR such thata
1
≥···≥ a
s
anda
i
Area(⊡ i
)=
1
s
for alli (so that Area(R) = 1). Consider
capacityc=k
c
√
n. For all fixed 0< p<1, we have:
liminf
n→∞
VRP(X
1
,...,X
n
;k
c
√
n;pn)
√
n
≥ 1
2
Z
x∈R
(λ
1
p
φ(x)+
2
k
c
φ(x)∥x∥
2
)1(N(x)≥ p)dx
with probability one, where
N(x)=Pr(ν(X)≤ ν(x))=
Z
x
′
:ν(x
′
)≤ ν(x)
φ(x
′
)dx
′
and
23
ν(x)≡ Z
x∈R
λ
1
p
φ(x)+
2
k
c
φ(x)∥x∥
2
dx
Proof. Letm= pn, so that we have two cases:
1. m≤ k
c
√
n: If this is the case, this problem immediately becomes solvable as a Sparse Subset TSP
problem because of the triangle inequality. Since we are taking about n→∞, we can pay attention
to the second case.
2. m>k
c
√
n: Under this case, we have:
VRP(X
1
,...,X
n
;k
c
√
n;m) = min
|S|=m
VRP(S;k
c
√
n)
= min
|S|=m
s
∑
i=1
VRP(S∩⊡ i
;k
c
√
n)
=min
q∈Q
s
∑
i=1
VRP(S∩⊡ i
;k
c
√
n;q
i
)
whereq is a vector denoting the number of points from each⊡ i
that are selected:
Q={q∈Z
m
+
:
s
∑
i=1
q
i
=m, q
i
≤| S∩⊡ i
|, ∀i}
Combined with Theorem 11, we can get:
liminf
n→∞
VRP(X
1
,...,X
n
;k
c
√
n;m) =liminf
n→∞
min
q∈Q
s
∑
i=1
VRP(S∩⊡ i
;k
c
√
n;q
i
)
≥ liminf
n→∞
min
q∈Q
s
∑
i=1
max{TSP(S∩⊡ i
,q
i
),
2
k
c
√
n
∑
i:i∈S∩⊡ i
r
i
}
=liminf
n→∞
min
q∈Q
s
∑
i=1
max{TSP(S∩⊡ i
,q
i
),
2q
i
k
c
√
n
r
S∩⊡ i
}
≥ liminf
n→∞
min
q∈Q
s
∑
i=1
max{TSP(S∩⊡ i
,q
i
),
2q
i
k
c
√
n
r
⊡ i
}
≥ 1
2
liminf
n→∞
min
q∈Q
(
s
∑
i=1
TSP(S∩⊡ i
,q
i
)+
s
∑
i=1
2q
i
k
c
√
n
r
⊡ i
)
24
wherer
⊡ i
is the average distance to depot for⊡ i
andr
S∩⊡ i
is the average distance to depot for those
picked point within⊡ i
. Using Definition 3.2.1, we can reconstruct:
TSP(S∩⊡ i
)=
1
√
a
i
TSP(S
′
∩⊡ ′
i
)
for all subsetS and points in each⊡ i
,∀i follows uniform distribution. It is easy to get:
min
q∈Q
s
∑
i=1
TSP(S∩⊡ i
;q
i
) =min
q∈Q
s
∑
i=1
1
√
a
i
TSP(S
′
∩⊡ ′
i
;q
i
)
≥ min
e q∈
e
Q
s
∑
i=1
1
√
a
i
TSP(S
′
∩⊡ ′
i
;e q
i
)
min
q∈Q
s
∑
i=1
2q
i
k
c
√
n
r
⊡ i
≥ min
e q∈
e
Q
s
∑
i=1
2e q
i
k
c
√
n
r
⊡ i
Where
e
Q is a “lower bounding set ” ofQ defined as follows: fix ε and letξ(t)=ε⌊t/ε⌋, which in par-
ticular tells us that0≤ t− ξ(t)≤ ε for allt. The set
e
Q is the image of(⌊nξ(q
1
/n)⌋,...,⌊nξ(q
s
/n)⌋)
for all feasible vectorsq∈Q. The detail is shown in [23].
Combined with the result in corollary 10.1, we can get:
liminf
n→∞
min
e q∈
e
Q
s
∑
i=1
1
√
a
i
TSP(S
′
∩⊡ ′
i
;e q
i
) ≥ liminf
n→∞
min
e q∈
e
Q
s
∑
i=1
1
√
a
i
TSP(S
′
∩⊡ ′
i
;e q
i
)
≥ min
t∈p∆
s− 1
liminf
n→∞
s
∑
i=1
1
√
a
i
TSP(S
′
∩⊡ ′
i
;(t
i
− ε)n)
limsup
n→∞
min
e q∈
e
Q
s
∑
i=1
2e q
i
k
c
√
n
r
⊡ i
≥ liminf
n→∞
min
e q∈
e
Q
s
∑
i=1
2e q
i
k
c
√
n
r
⊡ i
≥ min
t∈p∆
s− 1
liminf
n→∞
s
∑
i=1
2(t
i
− ε)
√
n
k
c
r
⊡ i
It is easy to show thate q
i
≥ (t
i
− ε)n bye q
i
=⌊nε⌊
q
i
nε
⌋⌋ andt=t
i
=q
i
/n in0≤ t− ξ(t)≤ ε. Now by
corollary 10.1, we have
25
liminf
n→∞
TSP(S
′
∩⊡ ′
i
;(t
i
− ε)n)
√
n
≥
λ
1
(t
i
− ε) ift
i
− ε≤ 1/s
∞ otherwise
and so ultimately, as this is a minimization problem, we can bound it below in terms of the fraction
of points in each cell:
min
t
1
2
[λ
1
√
n
s
∑
i=1
1
√
a
i
(t
i
− ε)+
2
√
n
k
c
n
∑
i=1
r
⊡ i
(t
i
− ε)]
s.t
n
∑
i=1
t
i
= p
0≤ t
i
≤ 1/s ∀i
(4.2)
For clarity, we can rewrite the objective function as
1
2
s
∑
i=1
(
λ
1
√
n
√
a
i
+
2
√
nr
⊡ i
k
c
)(t
i
− ε)
By monotonicity, it is easy to see that the optimal solution is achieved by settingt
1
=··· =t
⌊ps⌋
=1/s
and t
⌈ps⌉
= p−⌊ ps⌋/s (see e.g. exercises 4.8(e) of Boyd [20]). We can disregard the t
⌊ps⌋
term for
notational convenience and use the fact thatx∈∪
⌊ps⌋
i=1
⊡ i
if and only ifN(x)≥⌊ ps⌋/s. Therefore, the
objective function of eq. (4.2) is at least
26
1
2
⌊ps⌋
∑
i=1
(
λ
1
√
n
√
a
i
+
2
√
nr
⊡ i
k
c
)(1/s− ε) =
1
2
[
√
n
s
⌊ps⌋
∑
i=1
(
λ
1
√
a
i
+
2r
⊡ i
k
c
)− ε
⌊ps⌋
∑
i=1
(
λ
1
√
n
√
a
i
+
2
√
nr
⊡ i
k
c
)]
≥ 1
2
[
√
n
s
⌊ps⌋
∑
i=1
(
λ
1
√
a
i
+
2r
⊡ i
k
c
)− ε
s
∑
i=1
(
λ
1
√
n
√
a
i
+
2
√
nr
⊡ i
k
c
)]
=
√
n
2
(λ
1
R
R
p
φ(x)1(N(x)≥ p)dx+
2
k
c
R
R
φ(x)∥x∥
2
1(N(x)≥ p)dx)
− ε
2
√
n(λ
1
s
R
R
p
φ(x)dx+
2
k
c
R
R
φ(x)∥x∥
2
dx)
=
√
n
2
R
R
(λ
1
p
φ(x)+
2
k
c
φ(x)∥x∥
2
)1(N(x)≥ p)dx
− ε
√
n
2
R
R
(λ
1
s
√
φx+
2
k
c
φ(x)∥x∥
2
)dx
which completes the proof.
Theorem 13 (Tour length from a general distribution). Let f,R be as in the notational conventions. The
capacity satisfies c=k
c
√
n With probability one, we have
liminf
n→∞
VRP(X
1
,...,X
n
;k
c
√
n;pn)
√
n
≥ 1
2
Z
x∈R
(λ
1
p
f(x)+
2
k
c
f(x)∥x∥
2
)1(ϒ(x)≥ p)dx
where
ϒ(x)=Pr(υ(X)≤ υ(x))=
Z
x
′
:υ(x
′
)≤ υ(x)
υ(x
′
)dx
′
and
υ(x)≡ Z
x∈R
λ
1
p
f(x)+
2
k
c
f(x)∥x∥
2
dx
27
Proof. Let φ be the approximation of f from theorem 12. By a standard coupling argument (for example
theγ coupling of [53]), there is a joint distribution for random variables(X,Y) such thatX has density f ,
Y has densityφ and then Pr(X̸=Y)≤ ε for anyε. We have
VRP(X
1
,...,X
n
;k
c
√
n;pn) >L(X
1
,...,X
n
:X
i
=Y
i
;k
c
√
n;pn)
=VRP(Y
1
,...,Y
n
:X
i
=Y
i
;k
c
√
n;pn)
≥ VRP(Y
1
,...,Y
n
;k
c
√
n;pn)− VRP(Y
1
,...,Y
n
:X
i
̸=Y
i
;k
c
√
n;pn)−O (1)
VRP(Y
1
,...,Y
n
:X
i
̸=Y
i
;k
c
√
n;pn)≤ 2⌈p
√
n⌉
k
c
r+(1− 1
k
c
√
n
)TSP(Y
1
,...,Y
n
:X
i
̸=Y
i
;k
c
√
n;pn)
Since the convergence result in [79] is almost surely forTSP(Y
1
,...,Y
n
:X
i
̸=Y
i
;k
c
√
n;pn), we have
limsup
n→∞
TSP(Y
1
,...,Y
n
:X
i
̸=Y
i
;k
c
√
n;pn)
√
n
≤ α
2
p
Area(R)εpn
√
n
Combining them, we get:
liminf
n→∞
VRP(X
1
,...,X
n
;k
c
√
n;pn)
√
n
≥ liminf
n→∞
VRP(Y
1
,...,Y
n
;k
c
√
n;pn)
√
n
− VRP(Y
1
,...,Y
n
:X
i
̸=Y
i
;k
c
√
n;pn)
√
n
≥ 1
2
R
x∈R
(λ
1
p
φ(x)+
2
k
c
φ(x)∥x∥
2
)1(N(x)≥ p)dx
−
2εp
k
c
r+(1− 1
k
c
√
n
)α
2
p
Area(R)εpn
=
λ
1
2
R
x∈R
p
φ(x)1(N(x)≥ p)dx− (1− 1
k
c
√
n
)α
2
p
Area(R)εp
+
1
k
c
R
x∈R
φ(x)∥x∥
2
1(N(x)≥ p)dx− 2εpr
(4.3)
We can select our approximationϒ,N arbitrarily closely so that
28
ε ≥ max
x
|1(ϒ≥ p)− 1(N≥ p)|+
R
R
|φ− f|dx
≥ R
φ|1(ϒ≥ p)− 1(N≥ p)|+
R
|φ− f|1(ϒ≥ p)
≥ R
|φ 1(N≥ p)− φ 1(ϒ≥ p)+φ 1(ϒ≥ p)− f 1(ϒ> p)|
=
R
|φ 1(N≥ p)− f 1(ϒ≥ p)|
and furthermore, we have
p
Area(R)pε ≥ R
p
|φ 1(N≥ p)− f 1(ϒ≥ p)|≥|
R√
φ 1(N≥ p)− R√
f 1(P≥ p)|
⇒
R
p
φ 1(N≥ p) ≥ R√
f 1(ϒ≥ p)− p
pεArea(R)
εpr ≥ R
|φ 1(N≥ p)− f 1(ϒ≥ p)|∥x∥
2
⇒
R
φ 1(N≥ p)∥x∥
2
≥ R
f 1(ϒ≥ p)∥x∥
2
− εpr
Therefore, we ultimately find that
λ
1
2
R
x∈R
p
φ(x)1(N(x)≥ p)dx− (1− 1
k
c
√
n
)α
2
p
Area(R)εp
≥ λ
1
2
R
x∈R
p
f(x)1(ϒ(x)≥ p)dx− p
pεArea(R)
− (1− 1
k
c
√
n
)α
2
p
Area(R)εp
=
λ
1
2
R
x∈R
p
f(x)1(ϒ(x)≥ p)dx− (
λ
1
2
+(1− 1
k
c
√
n
))
p
Area(R)εp
| {z }
(∗ )
R
x∈R
φ(x)∥x∥
2
1(N(x)≥ p)dx− 2εpr
≥ R
x∈R
f(x)1(ϒ(x)≥ p)∥x∥
2
dx− εpr− 2εpr
=
R
x∈R
f(x)1(ϒ(x)≥ p)∥x∥
2
dx− 3εpr
|{z}
(∗∗ )
Where (*) and (**) shrink to 0 by choosing sufficiently small values of ε.
Putting them into eq. (4.3), we have
29
liminf
n→∞
VRP(X
1
,...,X
n
;k
c
√
n;pn)
√
n
=
1
2
R
x∈R
(λ
1
p
f(x)+
2
k
c
f(x)∥x∥
2
)1(ϒ(x)≥ p)dx
which completes the proof.
30
Chapter5
AnalysisoftheCumulativeTSPwhenn→∞
This chapter presents a comprehensive probabilistic analysis on the upper and lower bounds for the Cu-
mulative TSP where the number of points tends to infinity. The analysis begins by considering uniform
distributions, followed by non-uniform distributions. The objective of this analysis is to provide a thorough
understanding of the behavior of the bounds and their relationship to the distribution of the points, which
is a crucial step towards the development of efficient algorithms for large-scale problems. The results of
this analysis have significant implications for a range of applications, from logistics and transportation to
network optimization and facility location.
5.1 The“mostdensetoLeastdense”RuleExplained
In analyzing the non-uniform case of CTSP, we will repeatedly refer to a routing strategy that we call the
“most dense to least dense”. It proceeds as follows:
From Theorem 3, we know that step density functionφ can be used as the approximation of f , due to
a standard coupling argument. Define Ψ:R→R
2
be the union of allΨ
i
, where{Ψ
i
:⊡ i
→⊡ ′
i
|Ψ
i
(y)=
√
a
i
y+ξ
i
}. ξ
i
is chosen to make:
• Each⊡ ′
i
disjoint;
• Area(⊡ ′
i
)=
1
s
for alli;
31
• Points in⊡ ′
i
are uniform distribution;
The rule is shown in Figure 5.1. The idea is that the tour should start in the darkest (i.e., densest and
smallest) areas to lighter (larger and sparser) areas.
Figure 5.1: Figure (a) shows a step functionφ that presumably satisfies the conditions (darker - i.e., denser-
regions are smaller, reflecting the assumption that a
i
Area(⊡ i
) = 1/s) for all i) (b) shows the TSP tour of
a collection of independent samplesY
1
,...,Y
n
of φ whose length differs from that of a collection of tour
within each component (c) by a constant. (d) shows the rescaled component⊡ ′
i
=Ψ(⊡ i
) and the points
Y
′
1
,...,Y
′
n
Compared to the traditional TSP problem, this assumption is a strong one, which is decided by the
different objective functions.
TSP: min
n
∑
i=1
n
∑
j=1
C
ij
X
ij
CTSP: min
n
∑
i=1
n
∑
j=1
C
ij
Y
ij
32
Where
X
ij
=
1, if the repairman travels arc (i, j)
0, otherwise
Y
ij
=
n− k+1, if arc (i, j) appears in position k on the Hamiltonian tour
0, if arc (i, j) is not used
While the TSP is order-invariant and allows for multiple optimal tours, the CTSP introduces the addi-
tional constraint of time, with the cost of an arc depending on its position in the tour. As such, minimizing
the cost of the tour requires considering both arc costs and position, prioritizing the placement of arcs with
smaller costs earlier in the tour. In the context of continuous approximation formulas, the tour should be
constructed by traveling from the most dense to the least dense points.
5.2 UpperBoundsforCTSP
The upper bounding of the CTSP involves the key concept that the waiting time of each customer, repre-
sented by the distance travelled by the vehicle until it reaches the customer, should not exceed the waiting
time of the previously visited customer. This approach ensures an effective upper bounding strategy of
the CTSP, with each customer being visited within an appropriate time frame.
5.2.1 X
i
i.i.d
∼ U
Theorem14 (Upper bound; Uniform CTSP). LetX
1
,...,X
n
be independent uniform samples drawn from a
region ofR with areaA inR
2
. We have
limsup
n→∞
CumL(X
1
,...,X
n
)
n
3/2
≤ β
2
√
A
2
(5.1)
33
with probability one whereβ
2
≈ 0.714.
Proof. The main objective of this proof is to establish a feasible solution for upper bounding the routing
problem, where the optimal value of the problem should not exceed this feasible solution. To achieve this,
we consider the TSP tour as a feasible tour, where the earlier position of a customer carries more weight
in the CTSP objective. Since the optimal starting point is unknown, we take each node as a starting node
and compute the average cost as the upper bound to avoid worst-case scenarios. The aggregate feasible
solution costs are presented in the table 5.1, demonstrating the effectiveness of this approach.
Starting Position Cumulative Tour Value
Starting atX
1
nr
1
+(n− 1)d
12
+(n− 2)d
23
+··· +d
(n− 1)n
+0d
n0
Starting atX
2
nr
2
+(n− 1)d
23
+(n− 2)d
34
+··· +d
n1
+0d
12
... ...
Starting atX
n
nr
n
+(n− 1)d
n1
+(n− 2)d
12
+··· +d
(n− 2)(n− 1)
+0d
(n− 1)n
Table 5.1: Feasible Starting positions
Adding them together, we have:
n
∑
r
i
+[(n− 1)+(n− 2)+··· +0]TSP(X
1
,...,X
n
)
=n
n
∑
i=1
r
i
+
n(n− 1)
2
TSP(X
1
,...,X
n
)
As the minimum is smaller than the average, combined with Theorem 1, the upper bound will be:
limsup
n→∞
CumL(X
1
,...,X
n
)
n
3/2
≤ limsup
n→∞
∑
i:x
i
∈R
r
i
n
3/2
+
n− 1
2n
3/2
TSP(X
1
,...,X
n
)
=
β
2
√
A
2
+ lim
n→∞
r
√
n
− β
2
√
A
2n
=
β
2
√
A
2
For the second line, asn→∞,
1
√
n
→0 and
1
n
→0, which completes the proof.
34
5.2.2 X
i
i.i.d
∼ P
Theorem15 (Upper bound; Step function CTSP). Let X
1
,...,X
n
be independent samples, where X
i
follows
thestepdensityfunctionφ(x)=∑
s
i=1
a
i
1(x∈⊡ i
)withcompactsupportR⊂ R
2
suchthata
1
≥···≥ a
s
and
a
i
Area(⊡ i
)=
1
s
for alli (so that Area(R)=1). Suppose thatΠ(x) is as defined in Lemma 3. We have
limsup
n→∞
CumL(X
1
,...,X
n
)
n
3/2
≤ β
2
Z
x∈R
p
φ(x)Π(x)dx (5.2)
with probability one whereβ
2
≈ 0.714.
Proof. As in the uniform case, we begin by finding a feasible solution. For each ⊡ i
, we can reconstruct⊡ i
to⊡ ′
i
using a functionψ
i
(x)=
√
a
i
x+ξ
i
to make:
• Each⊡ ′
i
disjoint;
• Area(⊡ ′
i
)=
1
s
for alli;
• Points in⊡ ′
i
are uniform distribution;
Basic scaling arguments tell us that TSP(⊡ i
)=
1
√
a
i
TSP(⊡ ′
i
)
Since now each point in this area is uniformly distributed, we can apply the BHH theorem:
lim
n→∞
TSP(⊡ ′
i
)
√
n
=β
2
r
1
s
r
1
s
=
β
2
s
⇒ lim
n→∞
TSP(⊡ i
)=
1
√
a
i
TSP(⊡ ′
i
)=
β
2
s
√
a
i
Since each⊡ i
is related to a
i
, one of the possible ways to travel to all points is to follow “most dense
to least dense” rule, which merely means that we travel from⊡ 1
,...,⊡ s
. (From the definition of our step
35
function, the densest part of the distribution is equivalent to the smallest area⊡ , which is⊡ 1
). Using
Theorem 4, for pointX
k
∈⊡ j
, its waiting time should be less than or equal to :
TSP(⊡ 1
)+ TSP(⊡ 2
)+··· + TSP(⊡ j
)=
j
∑
i=1
β
2
s
r
n
a
i
=β
2
√
n
Z
x
′
:φ(x
′
)≥ φ(x)
p
φ(x
′
)dx
′
asn→∞.
Using the Law of Large Numbers,
|X
1
,...,X
n
∩⊡ i
|/n→
1
s
⇒|X
1
,...,X
n
∩⊡ i
|→
n
s
We find that, as n→∞, each set will have
n
s
points. Therefore, combining both terms together, we can
get the cumulative waiting time in each⊡ i
for alli:
• cumulative waiting time in⊡ 1
≤ n
s
L(⊡ 1
)
• cumulative waiting time in⊡ 2
≤ n
s
[L(⊡ 1
)+L(⊡ 2
)].
• . . .
• cumulative waiting time in⊡ s
≤ n
s
[L(⊡ 1
)+··· +L(⊡ s
)].
Summing all entries, we find:
CumL(X
1
,...,X
n
)≤ s
∑
j=1
n
s
j
∑
i=1
L(⊡ i
)=n
s
∑
j=1
a
j
a
j
s
j
∑
i=1
L(⊡ i
)
=n
Z
x∈R
φ(x)
j
∑
i=1
L(⊡ i
)dx
and therefore, from our previous formulation, we have:
36
limsup
n→∞
CumL(X
1
,...,X
n
)
n
3/2
≤ β
2
Z
x∈R
φ(x)
Z
x
′
:φ(x
′
)≥ φ(x)
p
φ(x
′
)dx
′
dx
=β
2
Z
x∈R
Z
x
′
:φ(x
′
)≥ φ(x)
φ(x)
p
φ(x
′
)dx
′
dx
=β
2
Z
x
′
∈R
Z
x:φ(x)≤ φ(x
′
)
φ(x)
p
φ(x
′
)dxdx
′
=β
2
Z
x
′
∈R
p
φ(x
′
)
Z
x:φ(x)≤ φ(x
′
)
f(x)dx
dx
′
=β
2
Z
x
′
∈R
p
φ(x
′
)Π(x
′
)dx
′
=β
2
Z
x∈R
p
φ(x)Π(x)dx
which completes the proof.
Theorem 16 (Upper bound; non-uniform CTSP). Let X
1
,...,X
n
be independent samples from a region of
R⊂ R
2
, whereX
i
follows the density function f,∀i. P(x) is as defined in Lemma 3. We have
lim
n→∞
CumL(X
1
,...,X
n
)
n
3/2
≤ β
2
Z
x∈R
p
f(x)P(x)dx (5.3)
with probability one whereβ
2
≈ 0.714.
Proof. The key point is the same as what we did in step density function: Theorem 15. Based on Theorem 3,
f can be approximated usingφ. Based on the “Most dense to Least dense ” Rule, for point x, the amount
of time he/she has to wait is:
β
2
√
n
Z
x
′
:f(x
′
)≥ f(x)
p
f(x
′
)dx
′
When traveling alln points, the length is
37
β
2
√
n
Z
x∈R
f(x)
Z
x
′
:f(x
′
)≥ f(x)
p
f(x
′
)dx
′
dx
=β
2
√
n
Z
x∈R
Z
x
′
:f(x
′
)≥ f(x)
f(x)
p
f(x
′
)dx
′
dx
=β
2
√
n
Z
x
′
∈R
Z
x:f(x)≤ f(x
′
)
f(x)
p
f(x
′
)dxdx
′
=β
2
√
n
Z
x
′
∈R
p
f(x
′
)
Z
x:f(x)≤ f(x
′
)
f(x)dx
dx
′
=β
2
√
n
Z
x
′
∈R
p
f(x
′
)P(x
′
)dx
′
=β
2
√
n
Z
x∈R
p
f(x)P(x)dx
Each points’ traveling time should be less than or equal to the total tour length, so we get:
lim
n→∞
CumL(X
1
,...,X
n
)
n
3/2
≤ β
2
Z
x∈R
p
f(x)P(x)dx
which completes the proof.
5.3 LowerBoundforCTSP
This section presents a lower bounding approach for the tour length of CTSP, which involves dividing the
optimal tour ofn points intom consecutive sets and obtaining lower bounds for each point by considering
the shortest possible tour that visitsn/m points out ofn points to travel. The starting time for each set is
determined by adding the travelling time of all previous visited sets. By summing up the lower bounds for
each point, we can obtain a lower bound for the CTSP tour length.
38
5.3.1 X
i
i.i.d
∼ U
Theorem17 (Lower bound; Uniform CTSP). LetX
1
,...,X
n
beindependentuniformsamplesdrawnfroma
region of areaR inR
2
. We have
liminf
n→∞
CumL(X
1
,...,X
n
)
n
3/2
≥ λ
1
2
√
A
with probability one, whereλ
1
=0.2935.
Proof. Fix p∈(0,1). Divide alln points into
1
p
sets. As introduced in Section 4, letL(X
1
,...,X
n
;pn) denote
the length of the shortest tour that visits pn points out of X
1
,...,X
n
Each customer’s waiting time in the
ith set should be at least(i− 1)L(X
1
,...,X
n
;pn). Since each set has pn points, then the cumulative waiting
time in this set is (i− 1)pnL(X
1
,...,X
n
;pn). Based on this analysis, the total waiting time should be at
least:
CumL(X
1
,...,X
n
)≥ 1
p
∑
i=1
(i− 1)pnL(X
1
,...,X
n
;pn)
=
1
2
(
1
p
− 1)nL(X
1
,...,X
n
;pn)
Applying Theorem 10, we see that
liminf
n→∞
CumL(X
1
,...,X
n
)
n
3/2
≥ 1
2
(1− p)λ
1
√
A
This holds for all p, and
1
2
(1− p)λ
1
√
A increases as p decreases. The tightest lower bound will be
reached when we choose p→0
+
. Therefore,
liminf
n→∞
CumL(X
1
,...,X
n
)
n
3/2
≥ λ
1
2
√
A
39
which completes the proof.
5.3.2 X
i
i.i.d
∼ P
Theorem 18 (Lower bound; step function CTSP). Let X
1
,...,X
n
be independent samples, where X
i
fol-
lows the density function φ(x) =∑
s
i=1
a
i
1(x∈⊡ i
) with compact support R such that a
1
≥···≥ a
s
and
a
i
Area(⊡ i
)=
1
s
for alli (so that Area(R)=1).Π(x) is defined in Theorem 3. We have
liminf
n→∞
CumL(X
1
,...,X
n
)
n
3/2
≥ λ
1
Z
x∈R
p
φ(x)Π(x)dx
with probability one, whereλ
1
=0.2935.
Proof. For each⊡ i
, using the same mapping procedure as shown in Section 5.1 to reconstruct L(⊡ i
)=
1
√
a
i
L(⊡ ′
i
). Applying the same reasoning as in the uniform analysis, suppose that we want to visit pn
points out of n in total as cheaply as possible. Suppose we pick p
i
n points in each⊡ ′
i
so that
s
∑
i=1
p
i
= p.
Using the result in Corollary 10.1, we see that
L(X
1
,...,X
n
;pn)=
s
∑
i=1
1
√
a
i
L(⊡ ′
i
;p
i
n)≥ λ
1
√
n
s
∑
i=1
p
i
√
a
i
asn→∞, where L(⊡ ′
i
;p
i
n) represents that picking p
i
n points from⊡ ′
i
It is straightforward to verify that the tightest lower bound can be achieved if we visit as many points as
possible in the denser part (wherea
i
is large). Therefore, based on the Law of Large Numbers:|X
1
,...,X
n
∩
⊡ ′
|/n→
1
s
asn→∞, the maximum number of points we can visit in each⊡ ′
i
is
n
s
.
The optimal p
i
values that minimize the above expression are to set p
1
=··· = p
⌊ps⌋
=1/s and p
⌈ps⌉
=
p−⌊ ps⌋/s. Under this assignment, the equation above changes to:
40
λ
1
√
n
s
∑
i=1
p
i
√
a
i
=λ
1
√
n(
⌊ps⌋
∑
i=1
p
i
√
a
i
+
p
⌈ps⌉
√
a
⌈ps⌉
)≥ λ
1
√
n
⌊ps⌋
∑
i=1
p
i
√
a
i
=λ
1
√
n
⌊ps⌋
∑
i=1
1
s
√
a
i
=λ
1
√
n
Z
x∈R
p
φ(x)1(Π(X)≥⌈ ps⌉/s)dx
≥ λ
1
√
n
Z
x∈R
p
φ(x)1(Π(X)≥ p)dx
as n→∞. Using the Law of Large Numbers,|X
1
,...,X
n
∩⊡ i
|/n→
1
s
⇒|X
1
,...,X
n
∩⊡ i
|→
n
s
. Finally,
we observe that these bounds enable us to state the following:
• cumulative waiting time in⊡ 1
≥ 0.
• cumulative waiting time in⊡ 2
≥ n
s
L(⊡ 1
,pn)≥ n
s
L(X
1
,...,X
n
,pn).
• . . .
• cumulative waiting time in⊡ s
≥ n
s
L(∪
s− 1
i=1
⊡ i
,pn)≥ n
s
L(X
1
,...,X
n
,pn).
CumL(X
1
,...,X
n
) ≥ s
∑
j=1
n
s
L(X
1
,...,X
n
,pn)=n
s
∑
j=0
1
s
L(X
1
,...,X
n
,pn)
=n
R
1
0
L(X
1
,...,X
n
,pn)dp
≥ n
R
1
0
λ
1
√
n
R
x∈R
p
φ(x)1(Π(X)≥ p)dxdp
=λ
1
n
3/2
R
1
0
R
x∈R
p
φ(x)1(Π(X)≥ p)dx
=λ
1
n
3
2
R
x∈R
R
1
0
p
φ(x)1(Π(x)≥ p)dpdx
=λ
1
n
3
2
R
x∈R
p
φ(x)
R
1
0
1(Π(x)≥ p)dp
dx
=λ
1
n
3
2
R
x∈R
p
φ(x)Π(x)dx
⇒ liminf
n→∞
CumL(X
1
,...,X
n
)
n
3/2
≥ λ
1
R
x∈R
p
φ(x)Π(x)dx
41
The preceding step function analysis yields the following result by standard coupling arguments:
Theorem19 (Lower bound; non-uniform CTSP). LetX
1
,...,X
n
beindependentsamplesdrawnfromaregion
ofR⊂ R
2
, whereX
i
follows the density function f,∀i. P(x) is defined in Lemma 3. We have
liminf
n→∞
CumL(X
1
,...,X
n
)
n
3/2
≥ λ
1
Z
x∈R
p
f(x)P(x)dx
with probability one, whereλ
1
=0.2935.
42
Chapter6
AnalysisoftheCumulativeCVRPwhenn→∞
In this chapter, we present a comprehensive probabilistic analysis of the upper and lower bounds for the
cumulative capacitated vehicle routing problem (CCVRP) as the number of points tends to infinity. Start-
ing with uniform distributions, we then extend our analysis to non-uniform distributions. We begin by
examining the scenario where there is only one vehicle, and in the following chapter, we will general-
ize our approach to consider multiple vehicles. This analysis will provide a deeper understanding of the
performance of the algorithms designed to solve the CCVRP, which is an essential problem in the field of
operations research.
6.1 Choiceofc
The core idea to find the necessary bounds for CCVRP is the same as our approach for the CTSP. In
Chapter 5, we use a probabilistic bound for the TSP (the BHH theorem), and for this section, it is necessary
to find a related bound for the CVRP.
The bound that we will use for CVRP is proven in Section 4.2. In a nutshell, we will find that local cost at
a pointx can be approximated byβ
2
p
f(x)+
2
√
n
c
f(x)∥x∥
2
. Since we are interested in asymptotic behavior,
we will assume thatc varies relative ton via the relationshipc=k
c
√
n. This is because ifc=o(
√
n), then
43
the radial cost term
2
√
n
c
f(x)∥x∥
2
dominates, and if c=ω(
√
n), then the TSP term β
2
p
f(x) dominates;
both of these are easier problems to analyze.
As in our preceding analysis, the key for our analysis of the non-uniform CCVRP relies on an approx-
imation of the true density f with a step function:
Lemma 20 (Approximation with a step function). Let f be a probability density function with compact
supportR⊂ R
2
whose level set have Lebesgue measure zero. Define
ϒ(x,c,k
p
)=Pr(υ(X,c,k
p
)≤ υ(x,c,k
p
))=
Z
x
′
:υ(x
′
,c,k
p
)≤ υ(x,c,k
p
)
f(x
′
)dx
′
Where
υ(x,c,k
p
)=k
p
p
f(x)+
2
√
n
c
f(x)∥x∥
2
For anyε >0, there exists a step density functionφ(x):=∑
s
i=1
a
i
1(x∈⊡ i
) and corresponding
I(x,c,k
p
)=Pr(ι(X,c,k
p
)≤ ι(x,c,k
p
))=
Z
x
′
:ι(x
′
,c,k
p
)≤ ι(x,c,k
p
)
φ(x
′
)dx
′
Where
ι(x,c,k
p
)=k
p
p
φ(x)+
2
√
n
c
φ(x)∥x∥
2
such that the following conditions hold:
1.
R
R
|φ(x)− f(x)|dx≤ ε,
2. |1(ϒ(X,c)≥ p)− 1(I(X,c)≥ p)|≤ ε ∀x∈R,
3. All of the components ofφ have the same mass, i.e. a
i
Area(⊡ i
)=1/s.
Proof. The proof is essentially identical to the step function approximation from Theorem 3 and we omit
it for brevity; we merely replace
p
f(x) withυ(x,c,k
p
).
44
The functionsϒ(·) and υ(·) serve as substitutes for the functions P(x) and
p
f(x) which we used in
our analysis of the CTSP; equivalently, the functionsI(·) andι(·) serve as substitutes forΠ(x) and
p
φ(x),
i.e., the step function approximations ofP(x) and
p
f(x).
The necessary machinery to bound the CCVRP under the assumption that c= k
c
√
n follows nearly
identical logic to our analysis of the CTSP; in the interest of brevity, we will present our bounds here
without additional proofs.
6.2 UpperBoundforCCVRP
Theorem21 (Upper bound; uniform CCVRP). LetX
1
,...,X
n
be independent uniform samples drawn from
regionR inR
2
, whose area isA and vehicle capacity isk
c
√
n. We have
limsup
n→∞
CumL(X
1
,...,X
n
;k
c
√
n)
n
3/2
≤ r
k
c
+
β
2
√
A
2
Proof. Inspired by the Upper bound in Theorem 11:
• We construct a feasible solution coming from Optimal Tour Partition (OPT)[45] to this routing prob-
lem whose value is at most equal to the upper bound.
• Cumulative length of the n solutions comes from two parts: leaving and entering depot plus tour
length.
We start from shortest travelling salesman’s tour, breaking it into l=⌈
√
n
k
c
⌉ disjoint segments so that
each segment cannot contain more than k
c
√
n customers, and connecting endpoints of each segment to
the depot.
To improve on this result, we start from an arbitrary orientation of the travelling salesman’s tour,
repeating the above construction by moving the endpoints of the originall paths 1, 2,... up ton positions
45
in the direction of this orientation. The value of the best resulting solution will be less than the average
value founded.
W.L.O.G, suppose that{X
1
,X
2
,...,X
n
} are the shortest travelling salesman’s tour we consider. For
leaving and coming back to the depot, the summation is
n+2[(n− k
c
√
n)+··· +(n− (
√
n
k
c
− 1)k
c
√
n)]
∑
r
i
=[n+n(
√
n
k
c
− 1)]
∑
r
i
=
n
3/2
k
c
∑
r
i
The sum of cumulative tour length is the similar as what we did in Theorem 14:
1. For the starting sub-tour L(X
1
,...,X
c
), its cumulative tour length is(n− 1)d
12
+(n− 2)d
23
+··· +
1d
(n− 1)n
+0d
n1
2. For the starting sub-tour L(X
2
,...,X
c
+1), its cumulative tour length is (n− 1)d
23
+(n− 2)d
34
+
··· +1d
n1
+0d
12
3. . . .
4. For the starting sub-tourL(X
n
,...,X
c− 1
), its cumulative tour length is(n− 1)d
n1
+(n− 2)d
12
+··· +
1d
(n− 2)(n− 1)
+0d
(n− 1)n
The sum of all the terms above is
n(n− 1)
2
L(X
1
,...,X
n
)
Combining them together, the average value we found is:
46
lim
n→∞
CumL(X
1
,...,X
n
;k
c
√
n)
n
3/2
≤ ¯ r
k
c
+
β
√
A
2
− β
√
A
2n
≤ ¯ r
k
c
+
β
√
A
2
which completes the proof.
Theorem 22 (Upper bound; Step function CCVRP). Let X
1
,...,X
n
be independent samples where X
i
fol-
lows the density function φ(x)=∑
s
i=1
a
i
1(x∈⊡ i
) with compact supportR⊂ R
2
such that a
1
≥···≥ a
s
and a
i
Area(⊡ i
)=
1
s
for all i (so that Area(R)= 1). Suppose capacity is k
c
√
n and that ι(x,k
c
√
n,β
2
) and
I(x,k
c
√
n,β
2
) are as defined in Lemma 20. We have
limsup
n→∞
CumL(X
1
,...,X
n
;k
c
√
n)
n
3/2
≤ Z
x∈R
ι(x,k
c
√
n,β
2
)I(x,k
c
√
n,β
2
)dx
Proof. As in the uniform case, we begin by finding a feasible solution. For each ⊡ i
, we can reconstruct⊡ i
to⊡ ′
i
. The details are shown in Theorem 15 Proof.
Starting from⊡ 1
and the cumulative waiting time in each⊡ i
for alli is:
• cumulative waiting time in⊡ 1
≤ n
s
VRP(⊡ 1
;k
c
√
n).
• cumulative waiting time in⊡ 2
≤ n
s
[VRP(⊡ 1
;k
c
√
n)+ VRP(⊡ 2
;k
c
√
n)].
• ...
• cumulative waiting time in⊡ 1
≤ n
s
[VRP(⊡ 1
;k
c
√
n)+··· + VRP(⊡ s
;k
c
√
n)].
Summing all entries, we find:
47
CumL(X
1
,...,X
n
;k
c
√
n)≤ s
∑
j=1
n
s
j
∑
i=1
VRP(⊡ i
;k
c
√
n)=n
s
∑
j=1
a
j
a
j
s
j
∑
i=1
VRP(⊡ i
;k
c
√
n)
=n
Z
x∈R
φ(x)
j
∑
i=1
VRP(⊡ i
;k
c
√
n)dx
and therefore, combined the result in Remark 1, we have:
limsup
n→∞
CumL(X
1
,...,X
n
;k
c
√
n)
n
3/2
≤ Z
x∈R
φ(x)
Z
x
′
:ι(x
′
,k
c
√
n,β
2
)≥ ι(x,k
c
√
n,β
2
)
ι(x
′
,k
c
√
n,β
2
)dx
′
dx
=
Z
x∈R
Z
x
′
:ι(x
′
,k
c
√
n,β
2
)≥ ι(x,k
c
√
n,β
2
)
φ(x)ι(x
′
,k
c
√
n,β
2
)dx
′
dx
=
Z
x
′
∈R
Z
x:ι(x,k
c
√
n,β
2
)≤ ι(x
′
,k
c
√
n,β
2
)
φ(x)ι(x
′
,k
c
√
n,β
2
)dxdx
′
=
Z
x
′
∈R
ι(x
′
,k
c
√
n,β
2
)
Z
x:ι(x,k
c
√
n,β
2
)≤ ι(x
′
,k
c
√
n,β
2
)
φ(x)dx
dx
′
=
Z
x
′
∈R
ι(x
′
,k
c
√
n,β
2
)I(x
′
,k
c
√
n,β
2
)dx
′
=
Z
x∈R
ι(x,k
c
√
n,β
2
)I(x,k
c
√
n,β
2
)dx
which completes the proof.
The preceding step function analysis yields the following result by standard coupling arguments:
Theorem 23 (Upper bound; non-uniform CCVRP). Let X
1
,...,X
n
be independent samples drawn from a
regionofR⊂ R
2
,whereX
i
followsthedensityfunction f,∀i. Supposecapacityisk
c
√
nandthatυ(x,k
c
√
n,β
2
)
andϒ(x,k
c
√
n,β
2
) are as defined in Lemma 20. We have
limsup
n→∞
CumL(X
1
,...,X
n
;k
c
√
n)
n
3/2
≤ Z
x∈R
υ(x,k
c
√
n,β
2
)ϒ(x,k
c
√
n,β
2
)dx
48
6.3 LowerBoundforCCVRP
Theorem24 (Lower bound; uniform CCVRP). LetX
1
,...,X
n
be independent uniform samples drawn from
a region of areaA inR
2
and vehicle capacity isk
c
√
n. We have
liminf
n→∞
CumL(X
1
,...,X
n
;k
c
√
n)
n
3/2
≥ max{
1
2
λ
1
√
A,
1
k
c
¯ r}
Proof. From Theorem 11, it is easy to see that the optimal CVRP should be larger than or equal to both
TSP result and leaving and returning depot distances. This conclusion can apply to CCVRP as well, which
is the optimal CCVRP should be larger than or equal to both CTSP result and leaving and returning depot
distances summation.
From Theorem 17, we can get:
liminf
n→∞
CumL(X
1
,...,X
n
;k
c
√
n)
n
3/2
≥ liminf
n→∞
CumL(X
1
,...,X
n
)
n
3/2
≥ 1
2
λ
1
√
A
Consider the subsetX
j
={X
j
1
,X
j
2
,...,X
j
q
} of customers, visited by the jth tour. Ignoring any tours
before this tour, It’s easy to know the time for this tour is at least2max
X
i
∈X
j
r
i
To make it easy to illustrate, we define:
e r
j
≡ ∑
X
i
∈X
j
r
i
r
mj
≡ max
X
i
∈X
j
r
i
49
We have
r
mj
= max
X
i
∈X
j
r
i
≥ ∑
X
i
∈X
j
r
i
|X
j
|
≥ 1
k
c
√
n
e r
j
r
mj
≤ max
i=1,...,n
{r
i
}
So the total time is :
CumL(X
1
,...,X
n
;k
c
√
n)≥ 2
√
n/k
c
∑
i=1
(n− (i− 1)k
c
√
n)r
mi
=2(
n
3/2
k
c
− n
2
(
√
n
k
c
− 1))r
mi
=
n
3/2
k
c
r
mi
+nr
mj
≥ n
3/2
k
c
¯ r+n¯ r
⇒liminf
n→∞
CumL(X
1
,...,X
n
;k
c
√
n)
n
3/2
≥ 1
k
c
¯ r
which completes the proof.
Theorem 25 (Lower bound; Step function CCVRP). Let X
1
,...,X
n
be independent samples where X
i
fol-
lows the density function φ(x)=∑
s
i=1
a
i
1(x∈⊡ i
) with compact supportR⊂ R
2
such that a
1
≥···≥ a
s
and a
i
Area(⊡ i
)=
1
s
for all i (so that Area(R)= 1). Suppose capacity is k
c
√
n and that ι(x,k
c
√
n,λ
1
) and
I(x,k
c
√
n,λ
1
) are as defined in Lemma 20. We have
liminf
n→∞
CumL(X
1
,...,X
n
;k
c
√
n)
n
3/2
≥ 1
2
Z
x∈R
ι(x,k
c
√
n,λ
1
)I(x,k
c
√
n,λ
1
)dx
Proof. Similar as uniform case, for each⊡ i
, we can reconstruct⊡ i
to⊡ ′
i
. The details are shown in Theo-
rem 15 Proof. With the result in Theorem 12, each⊡ i
cumulative waiting time (set the start time in⊡ i
as
zero) satisfies:
50
• cumulative waiting time in⊡ 1
≥ 0.
• cumulative waiting time in⊡ 2
≥ n
s
VRP(⊡ 1
;k
c
√
n;pn)≥ n
s
VRP(X
1
,...,X
n
;k
c
√
n;pn).
• . . .
• cumulative waiting time in⊡ s
≥ n
s
VRP(∪
s− 1
i=1
⊡ i
;k
c
√
n;pn)≥ n
s
VRP(X
1
,...,X
n
;k
c
√
n;pn).
CumL(X
1
,...,X
n
) ≥ s
∑
j=1
n
s
VRP(X
1
,...,X
n
;k
c
√
n;pn)=n
s
∑
j=0
1
s
VRP(X
1
,...,X
n
;k
c
√
n;pn)
=n
R
1
0
VRP(X
1
,...,X
n
;k
c
√
n;pn)dp
liminf
n→∞
CumL(X
1
,...,X
n
;k
c
√
n)
n
3/2
≥ R
1
0
1
2
R
x∈R
(λ
1
p
φ(x)+
2
k
c
φ(x)∥x∥
2
)1(N(x)≥ p)dxdp
=
1
2
R
1
0
R
x∈R
ι(x,k
c
√
n,λ
1
)1(N(x)≥ p)dxdp
=
1
2
R
x∈R
ι(x,k
c
√
n,λ
1
)
R
1
0
1(N(x)≥ p)dpdx
=
1
2
R
x∈R
ι(x,k
c
√
n,λ
1
)I(x,k
c
√
n,λ
1
)dx
which completes the proof.
The preceding step function analysis yields the following result by standard coupling arguments:
Theorem 26 (Lower bound; non-uniform CCVRP). Let X
1
,...,X
n
be independent samples drawn from a
regionofR⊂ R
2
,whereX
i
followsthedensityfunction f,∀i. Supposecapacityisk
c
√
nandthatυ(x,k
c
√
n,λ
1
)
andϒ(x,k
c
√
n,λ
1
) are as defined in Lemma 20. We have
liminf
n→∞
CumL(X
1
,...,X
n
;k
c
√
n)
n
3/2
≥ 1
2
Z
x∈R
υ(x,k
c
√
n,λ
1
)ϒ(x,k
c
√
n,λ
1
)dx
51
Chapter7
MultipleVehicleCumulativeRoutingProblem: m-CTSP
In this chapter, we aim to expand our analysis from the single-vehicle case to multiple vehicles in the con-
text of cumulative vehicle routing problems. Our focus is on providing a probabilistic analysis of upper and
lower bounds for the multiple vehicles’ cumulative traveling salesman problem (m-CTSP) and the multiple
vehicles’ cumulative capacitated vehicle routing problem (m-CCVRP), taking into consideration the sce-
nario where the number of points tends to infinity. Our analysis will begin with the uniform distribution
case and subsequently move on to non-uniform distributions.
7.1 UpperBoundform-CTSP
7.1.1 X
i
i.i.d
∼ U
Theorem27 (Upper bound; Uniform m-CTSP). LetX
1
,...,X
n
beindependentuniformsamplesdrawnfrom
a region ofR with areaA inR
2
. There exist m vehicles in total. We have
limsup
n→∞
CumL(m;X
1
,...,X
n
)
n
3/2
≤ β
2
√
A
2m
2
(7.1)
with probability one whereβ
2
≈ 0.714.
52
Proof. Divide the regionA intom equal-area sub-regions, such that each vehicle visits one sub-region. The
optimal value of this problem should be less than or equal to the value generated by this feasible solution.
For each sub-region, we can use Theorem 14 to finish the proof. The number of each sub-region’s
points converge to
n
m
whenn→∞. Since all vehicles start tour at the same time, each vehicle’s cumulative
traveling time is equal to this cumulative problem’s traveling time:
limsup
n→∞
CumL(m;X
1
,...,X
n
)
(n/m)
3/2
≤ β
2
2
r
A
m
= ⇒limsup
n→∞
CumL(m;X
1
,...,X
n
)
n
3/2
≤ β
2
√
A
2m
2
7.1.2 X
i
i.i.d
∼ P
Theorem 28 (Upper bound; Step function m-CTSP). Let X
1
,...,X
n
be independent samples, where X
i
follows the density functionφ(x)=∑
s
i=1
a
i
1(x∈⊡ i
) with compact supportR⊂ R
2
such that a
1
≥···≥ a
s
anda
i
Area(⊡ i
)=
1
s
for alli (so that Area(R)=1). Suppose thatΠ(x) is as defined in Lemma 3. There exist
m vehicles in total. We have
limsup
n→∞
CumL(m;X
1
,...,X
n
)
n
3/2
≤ β
2
m
2
Z
x∈R
p
φ(x)Π(x)dx (7.2)
with probability one whereβ
2
≈ 0.714.
Proof. As in the uniform case, we begin by finding a feasible solution. For each ⊡ i
, we can reconstruct⊡ i
to⊡ ′
i
using a functionψ
i
(x)=
√
a
i
x+ξ
i
to make:
• Each⊡ ′
i
disjoint;
53
• Area(⊡ ′
i
)=
1
s
for alli;
• Points in⊡ ′
i
are uniform distribution;
Basic scaling argument tells us that TSP(⊡ i
)=
1
√
a
i
TSP(⊡ ′
i
)
Similar as uniform case proof before, we can divide each cell⊡ ′
i
into m sub-cell and every vehicle
travels one sub-cell. Since now each point in the sub-cell is uniformly distributed, we can apply the BHH
theorem:
lim
n→∞
TSP(⊡ i
′
;m
)
√
n
=β
2
1
sm
⇒ lim
n→∞
TSP(⊡ i
)=
1
√
a
i
TSP(⊡ ′
i
)=
β
2
sm
√
a
i
where⊡ i
′
;m
represents themth sub-cell related to⊡ i
′ cell.
All m vehicles start at the same time, so one vehicle’s TSP can represent this region’s TSP. Each⊡ i
is
related to a
i
, one of the possible ways to travel to all points is to follow “most dense to least dense” rule,
which merely means that we travel from⊡ 1
,...,⊡ s
. (From the definition of our step function, the densest
part of the distribution is equivalent to the smallest area⊡ , which is⊡ 1
). Using Theorem 4, for point
X
l
∈⊡ j
, its waiting time should be less than or equal to :
TSP(⊡ 1,m
)+ TSP(⊡ 2,m
)+··· + TSP(⊡ j,m
)=
j
∑
i=1
β
2
sm
r
n
a
i
=
β
2
m
√
n
Z
x
′
:φ(x
′
)≥ φ(x)
p
φ(x
′
)dx
′
asn→∞.
Using the Law of Large Numbers,
|X
1
,...,X
n
∩⊡ i,m
|/n→
1
sm
⇒|X
1
,...,X
n
∩⊡ i
|→
n
sm
54
We find that, as n→∞, each set’s sub-region will have
n
sm
points. Therefore, combining both terms
together, we can get the cumulative waiting time in each⊡ i
for alli:
• cumulative waiting time in⊡ 1,m
≤ n
sm
L(⊡ 1
)
• cumulative waiting time in⊡ 2,m
≤ n
sm
[L(⊡ 1,m
)+L(⊡ 2,m
)].
• . . .
• cumulative waiting time in⊡ s,m
≤ n
sm
[L(⊡ 1,m
)+··· +L(⊡ s,m
)].
Summing all entries, we find:
CumL(m;X
1
,...,X
n
)≤ s
∑
j=1
n
sm
j
∑
i=1
L(⊡ i,m
)=
n
m
s
∑
j=1
a
j
a
j
s
j
∑
i=1
L(⊡ i,m
)
=
n
m
2
Z
x∈R
φ(x)
j
∑
i=1
L(⊡ i
)dx
and therefore, from our previous formulation, we have:
limsup
n→∞
CumL(m;X
1
,...,X
n
)
n
3/2
≤ β
2
m
2
Z
x∈R
φ(x)
Z
x
′
:φ(x
′
)≥ φ(x)
p
φ(x
′
)dx
′
dx
=
β
2
m
2
Z
x∈R
Z
x
′
:φ(x
′
)≥ φ(x)
φ(x)
p
φ(x
′
)dx
′
dx
=
β
2
m
2
Z
x
′
∈R
Z
x:φ(x)≤ φ(x
′
)
φ(x)
p
φ(x
′
)dxdx
′
=
β
2
m
2
Z
x
′
∈R
p
φ(x
′
)
Z
x:φ(x)≤ φ(x
′
)
f(x)dx
dx
′
=
β
2
m
2
Z
x
′
∈R
p
φ(x
′
)Π(x
′
)dx
′
=
β
2
m
2
Z
x∈R
p
φ(x)Π(x)dx
55
which completes the proof.
Theorem29 (Upper bound; non-uniform m-CTSP). LetX
1
,...,X
n
beindependentsamplesfromaregionof
R⊂ R
2
, whereX
i
follows the density function f,∀i. P(x) is as defined in Lemma 3. There exist m vehicles in
total. We have
lim
n→∞
CumL(m;X
1
,...,X
n
)
n
3/2
≤ β
2
m
2
Z
x∈R
p
f(x)P(x)dx (7.3)
with probability one whereβ
2
≈ 0.714.
Proof. The key point is the same as what we did in step density function: Theorem 28. Based on Theorem 3,
f can be approximated using φ. Again, since we have m vehicles, each vehicle will handle
n
m
points as
n→∞. Based on “Most dense to Least dense ” Rule, for pointx, the amount of time he/she has to wait is:
β
2
√
n
m
Z
x
′
:f(x
′
)≥ f(x)
p
f(x
′
)dx
′
When traversing
n
m
points, the length is
β
2
√
n
m
Z
x∈R
f(x)
Z
x
′
:f(x
′
)≥ f(x)
p
f(x
′
)dx
′
dx
=β
2
√
n
m
Z
x∈R
Z
x
′
:f(x
′
)≥ f(x)
f(x)
p
f(x
′
)dx
′
dx
=β
2
√
n
m
Z
x
′
∈R
Z
x:f(x)≤ f(x
′
)
f(x)
p
f(x
′
)dxdx
′
=β
2
√
n
m
Z
x
′
∈R
p
f(x
′
)
Z
x:f(x)≤ f(x
′
)
f(x)dx
dx
′
=β
2
√
n
m
Z
x
′
∈R
p
f(x
′
)P(x
′
)dx
′
=β
2
√
n
m
Z
x∈R
p
f(x)P(x)dx
56
Each points’ traveling time should be less than or equal to the total tour length and each tour length,
there exists
n
m
points, so we get:
lim
n→∞
CumL(m;X
1
,...,X
n
)
n
3/2
≤ β
2
m
2
Z
x∈R
p
f(x)P(x)dx
which completes the proof.
7.2 LowerBoundform-CTSP
7.2.1 X
i
i.i.d
∼ U
Theorem30 (Lower bound; Uniform m-CTSP). LetX
1
,...,X
n
beindependentuniformsamplesdrawnfrom
a region of areaR with areaA inR
2
. There exist m vehicles in total. We have
liminf
n→∞
CumL(m;X
1
,...,X
n
)
n
3/2
≥ λ
1
2m
2
√
A
with probability one, whereλ
1
=0.2935.
Proof. Each vehicle will travel
n
m
as n→∞, so we can fix p∈ (0,1). Divide all n points into
m
p
sets.
As introduced in Section 4, letL(X
1
,...,X
n
;pn/m) denote the length of the shortest tour that visits pn/m
points out ofX
1
,...,X
n
Each customer’s waiting time in theith set should be at least(i− 1)L(X
1
,...,X
n
;pn).
Since each set has pn/m points, then the cumulative waiting time in this set is(i− 1)
pn
m
L(X
1
,...,X
n
;pn).
Based on this analysis, the total waiting time for each vehicle should be at least:
CumL(m;X
1
,...,X
n
)≥ 1
p
∑
i=1
(i− 1)
pn
m
L(X
1
,...,X
n
;pn/m)
=
1
2
(
1
p
− 1)
n
m
L(X
1
,...,X
n
;pn)
57
Applying Theorem 10, we see that
liminf
n→∞
CumL(m;X
1
,...,X
n
)
n
3/2
≥ 1
2
(1− p)
1
m
2
λ
1
√
A
This holds for all p, and
1
2m
2
(1− p)λ
1
√
A increases as p decreases. The tightest lower bound will be
reached when we choose p→0
+
. Therefore,
liminf
n→∞
CumL(m;X
1
,...,X
n
)
n
3/2
≥ λ
1
2m
2
√
A
which completes the proof.
7.2.2 X
i
i.i.d
∼ P
Theorem 31 (Lower bound; step function m-CTSP). Let X
1
,...,X
n
be independent samples, where X
i
fol-
lows the density function φ(x) =∑
s
i=1
a
i
1(x∈⊡ i
) with compact support R such that a
1
≥···≥ a
s
and
a
i
Area(⊡ i
)=
1
s
for alli (so that Area(R)=1).Π(x) is defined in Theorem 3. There exist m vehicles in total.
We have
liminf
n→∞
CumL(m;X
1
,...,X
n
)
n
3/2
≥ λ
1
m
2
Z
x∈R
p
φ(x)Π(x)dx
with probability one, whereλ
1
=0.2935.
Proof. For each⊡ i
, using the same mapping procedure as shown in Section 5.1 to reconstruct L(⊡ i
)=
1
√
a
i
L(⊡ ′
i
). Applying the same reasoning as in the uniform analysis, suppose that we want to visit pn/m
points out of n/m in total for each vehicle as cheaply as possible. Suppose we pick p
i
n/m points in each
⊡ ′
i
so that
s
∑
i=1
p
i
= p. Using the result in Corollary 10.1, we see that
L(X
1
,...,X
n
;pn/m)=
s
∑
i=1
1
√
a
i
L(⊡ ′
i
;p
i
n/m)≥ λ
1
√
n
s
∑
i=1
p
i
m
√
a
i
58
asn→∞, where L(⊡ ′
i
;p
i
n/m) represents that picking p
i
n/m points from⊡ ′
i
It is straightforward to verify that the tightest lower bound can be achieved if we visit as many points as
possible in the denser part (wherea
i
is large). Therefore, based on the Law of Large Numbers:|X
1
,...,X
n
∩
⊡ ′
|/n→
1
s
asn→∞, the maximum number of points we can visit in each⊡ ′
i
is
n
s
.
The optimal p
i
values that minimize the above expression are to set p
1
=··· = p
⌊ps⌋
=1/s and p
⌈ps⌉
=
p−⌊ ps⌋/s. Under this assignment, the equation above changes to:
λ
1
√
n
m
s
∑
i=1
p
i
√
a
i
=λ
1
√
n
m
(
⌊ps⌋
∑
i=1
p
i
√
a
i
+
p
⌈ps⌉
√
a
⌈ps⌉
)≥ λ
1
√
n
m
⌊ps⌋
∑
i=1
p
i
√
a
i
=λ
1
√
n
m
⌊ps⌋
∑
i=1
1
s
√
a
i
=λ
1
√
n
m
Z
x∈R
p
φ(x)1(Π(X)≥⌈ ps⌉/s)dx
≥ λ
1
√
n
m
Z
x∈R
p
φ(x)1(Π(X)≥ p)dx
as n→∞. Using the Law of Large Numbers,|X
1
,...,X
n
∩⊡ i
|/n→
1
s
⇒|X
1
,...,X
n
∩⊡ i
|→
n
s
. Each
vehicle only needs to travel
n
sm
out of all
n
s
points. Finally, we observe that these bounds enable us to state
the following:
59
CumL(m;X
1
,...,X
n
) ≥ s
∑
j=1
n
ms
L(X
1
,...,X
n
,pn/m)=
n
m
s
∑
j=0
1
s
L(X
1
,...,X
n
,pn/m)
=
n
m
R
1
0
L(X
1
,...,X
n
,pn/m)dp
≥ n
m
R
1
0
λ
1
√
n
m
R
x∈R
p
φ(x)1(Π(X)≥ p)dxdp
=λ
1
n
3/2
m
2
R
1
0
R
x∈R
p
φ(x)1(Π(X)≥ p)dx
=λ
1
n
3/2
m
2
R
x∈R
R
1
0
p
φ(x)1(Π(x)≥ p)dpdx
=λ
1
n
3/2
m
2
R
x∈R
p
φ(x)
R
1
0
1(Π(x)≥ p)dp
dx
=λ
1
n
3/2
m
2
R
x∈R
p
φ(x)Π(x)dx
⇒ liminf
n→∞
CumL(m;X
1
,...,X
n
)
n
3/2
≥ λ
1
m
2
R
x∈R
p
φ(x)Π(x)dx
The preceding step function analysis yields the following result by standard coupling arguments:
Theorem 32 (Lower bound; non-uniform m-CTSP). Let X
1
,...,X
n
be independent samples drawn from a
region ofR⊂ R
2
, where X
i
follows the density function f,∀i. P(x) is defined in Lemma 3. There exist m
vehicles in total. We have
liminf
n→∞
CumL(m;X
1
,...,X
n
)
n
3/2
≥ λ
1
m
2
Z
x∈R
p
f(x)P(x)dx
with probability one, whereλ
1
=0.2935.
60
Chapter8
MultipleVehicleCumulativeRoutingProblem: m-CCVRP
In this chapter, we build upon our previous analysis by introducing vehicle constraints to our models.
Specifically, we consider the cumulative capacitated vehicle routing problem (CCVRP) with the assumption
that the relationship between the vehicle capacityc and the number of pointsn is given byc=k
c
√
n. We
will investigate the probabilistic upper and lower bounds for this problem, focusing on both uniform and
non-uniform distributions.
8.1 UpperBoundform-CCVRP
Theorem 33 (Upper bound; uniform m-CCVRP). Let X
1
,...,X
n
be independent uniform samples drawn
from regionR inR
2
, whose area isA and vehicle capacity isk
c
√
n. There existm vehicles in total. We have
limsup
n→∞
CumL(m;X
1
,...,X
n
;k
c
√
n)
n
3/2
≤ 1
m
2
(
r
k
c
+
β
2
√
A
2
)
Proof. Divide the regionA intom equal-area sub-regions, such that each vehicle visits one sub-region. This
problem reduces to CCVRP with region
A
m
. Using the result shown in Theorem 21 and proofs in m-CTSP,
we can complete the proof.
61
By reducing all cases to a CCVRP that restricts each vehicle to a sub-region, we obtain the non-uniform
result as:
Theorem 34 (Upper bound; Step function m-CCVRP). Let X
1
,...,X
n
be independent samples where X
i
follows the density functionφ(x)=∑
s
i=1
a
i
1(x∈⊡ i
) with compact supportR⊂ R
2
such that a
1
≥···≥ a
s
and a
i
Area(⊡ i
)=
1
s
for all i (so that Area(R)= 1). Suppose capacity is k
c
√
n and that ι(x,k
c
√
n,β
2
) and
I(x,k
c
√
n,β
2
) are as defined in Lemma 20. There exist m vehicles in total. We have
limsup
n→∞
CumL(m;X
1
,...,X
n
;k
c
√
n)
n
3/2
≤ 1
m
2
Z
x∈R
ι(x,k
c
√
n,β
2
)I(x,k
c
√
n,β
2
)dx
Theorem35 (Upper bound; non-uniform m-CCVRP). LetX
1
,...,X
n
beindependentsamplesdrawnfroma
regionofR⊂ R
2
,whereX
i
followsthedensityfunction f,∀i. Supposecapacityisk
c
√
nandthatυ(x,k
c
√
n,β
2
)
andϒ(x,k
c
√
n,β
2
) are as defined in Lemma 20. There exist m vehicles in total. We have
limsup
n→∞
CumL(m;X
1
,...,X
n
;k
c
√
n)
n
3/2
≤ 1
m
2
Z
x∈R
υ(x,k
c
√
n,β
2
)ϒ(x,k
c
√
n,β
2
)dx
8.2 LowerBoundform-CCVRP
Theorem36 (Lower bound; uniform CCVRP). LetX
1
,...,X
n
be independent uniform samples drawn from
a region of areaA inR
2
and vehicle capacity isk
c
√
n. There existm vehicles in total. We have
liminf
n→∞
CumL(m;X
1
,...,X
n
;c)
n
3/2
≥ 1
m
2
max{λ
1
√
A,
1
k
c
(2¯ r− max
i=1,...,n
{r
i
})}
62
Proof. Divide the region A into m equally sub-regions, each vehicle visits one sub-region. This problem
reduces to CCVRP with region
A
m
. Using the result shown in Theorem 24 and proofs in m-CTSP, we can
complete the proof.
By reducing all cases to sub-region CCVRP, we can get the non-uniform result as:
Theorem 37 (Lower bound; Step function m-CCVRP). Let X
1
,...,X
n
be independent samples where X
i
follows the density functionφ(x)=∑
s
i=1
a
i
1(x∈⊡ i
) with compact supportR⊂ R
2
such that a
1
≥···≥ a
s
and a
i
Area(⊡ i
)=
1
s
for all i (so that Area(R)= 1). Suppose capacity is k
c
√
n and that ι(x,k
c
√
n,λ
1
) and
I(x,k
c
√
n,λ
1
) are as defined in Lemma 20. There exist m vehicles in total. We have
liminf
n→∞
CumL(m;X
1
,...,X
n
;k
c
√
n)
n
3/2
≥ 1
2m
2
Z
x∈R
ι(x,k
c
√
n,λ
1
)I(x,k
c
√
n,λ
1
)dx
Theorem38 (Lower bound; non-uniform m-CCVRP). LetX
1
,...,X
n
beindependentsamplesdrawnfroma
regionofR⊂ R
2
,whereX
i
followsthedensityfunction f,∀i. Supposecapacityisk
c
√
nandthatυ(x,k
c
√
n,λ
1
)
andϒ(x,k
c
√
n,λ
1
) are as defined in Lemma 20. There exist m vehicles in total. We have
liminf
n→∞
CumL(m;X
1
,...,X
n
;k
c
√
n)
n
3/2
≥ 1
2m
2
Z
x∈R
υ(x,k
c
√
n,λ
1
)ϒ(x,k
c
√
n,λ
1
)dx
63
Chapter9
ExperimentalResults
In this section, we present a comprehensive evaluation of the proposed bounds for simulated instances and
real-world datasets. To illustrate the effectiveness of our approach, we provide several examples demon-
strating the tightness of the bounds in simulated instances. Furthermore, we apply subsets of real data to
predict the optimal waiting time. To obtain the optimal solutions, we utilize LKH-3, a heuristic method,
for solving the cumulative Traveling Salesman Problem, and Google OR-Tools for solving the cumulative
Vehicle Routing Problems. The experimental results provide insightful observations on the efficacy of our
proposed bounds in practice.
9.1 SingleVehicle
9.1.1 SimulatedInstances
In this section, we explore two different scenarios: regular and irregular regions. In the regular case, we
generate instances from a uniform distribution and Normal distribution with variancesσ
2
=490000 and
σ
2
=250000, centered at(500,500) and(750,750), respectively, within a square area of side length 1000.
For our first instance, we analyze cumulative TSP experiments with uniform sample points.
Figure 9.1 illustrates the results of the uniform distribution experiment, where we show the ratio of
cumulative waiting time to
√
AN
3/2
for 10 runs of eachn∈{50,100, 150,...,1000}. The red dashed line
64
Figure 9.1: Ratio of uniform customers’ cumulative waiting time to
√
An
3/2
represents the upper bound we proved before. The results show that asn increases, the ratio converges to
a value close to[0.34,0.35], which is consistent with our upper bound.
For the non-uniform distribution experiment, we used a bi-variate Gaussian distribution. Figure 9.2(left)
displays the results when the depot is located at the center of the square, while Figure 9.2(right) shows the
results when the depot is located at an offset center. We also experimented with two different standard
deviations, 500 and 700, as shown in the top and bottom panels, respectively. The results indicate that
the variance only has an impact at the beginning when the data points are sparse. As the size of the data
increases, having the same depot location, the results converge to the same range. The different depot
locations converge to different ranges, but their difference is negligible.
Another case we consider is an irregularly-shaped region. For this part, we shift our focus to two con-
vex and one non-convex scenario. Both of them take a bi-variate Gaussian distribution. Figure 9.3 (left)
shows the results of this experiment when the region is a circle, triangular and pentagram, with corre-
sponding heatmaps (right). We observe that the ratio of cumulative waiting time is primarily determined
by the density function of the sample points.
65
For the CCVRP with uniformly distributed sample points, we choose the vehicle capacity to be
√
n.
The upper bound for this problem isO(1.914n
3/2
). Figure 9.4 presents the results of the experiment, which
shows the ratio of cumulative waiting time ¯ rn
3/2
for 10 runs of each n∈{50,100,...,1000} for uniform
sample points. We observe that the ratio converges to[1.7,1.8], which is consistent with the theoretical
upper bound.
Figure 9.5 is the result when all points follow bi-variate Gaussian distribution. The left figures show
the experiment when the depot is in the center of the square and the right figures have the depot location
at an offset from the center. In addition, the top figures follow standard deviation 500 and the bottom
figures have 700 instead. We can figure out that variance only has impact at the beginning when the data
points are few. When we increase the size of data, having the same depot location, they will converge to
the same ratio range. Different depot locations have impact on the ratio range.
Still same as CTSP, we did the experiments considering another convex region: circle and non-convex
case: pentagram, whose points follow the bi-variate Gaussian distribution. The result is shown in Fig-
ure 9.6. It still presents that the ratio only relates to the density distribution.
Overall, the results of our experiments for both CTSP and CCVRP indicate that the ratio of cumulative
waiting time for large n is close to the upper bounds we proved for uniform and non-uniform distribu-
tions. Furthermore, we have demonstrated that the upper bounds hold for different types of regions and
distributions.
9.2 MultipleVehicles
Instead of considering only one vehicle, we investigate the effect of having multiple vehicles in this section.
Specifically, we consider four different cases where we have 2, 4, 8, and 16 vehicles, respectively. Our
investigation aims to deepen our understanding of the m-CTSP and m-CCVRP under different scenarios
and provide insights for practical applications.
66
9.2.1 SimulatedInstances
As in the preceding single-vehicle situations, we still consider the vehicle with infinite and finite capacity
respectively. For each scenario, both regular and irregular boundary will be considered. As a service
region, we use a square area with side length 1000.
In this section, we extend our investigation to the Multi-Vehicle Capacitated Traveling Salesman Prob-
lem (m-CTSP) by examining experiments with both uniform and non-uniform sample points. The top left
of Figure 9.7 illustrates the ratio of cumulative waiting time to
√
AN
3/2
for eachn∈{50,100,150,...,1000},
where the points follow the uniform distribution. To assess the impact of varying numbers of vehicles, we
consider four different cases: 2, 4, 8, or 16 vehicles in the system, with the single-vehicle case serving as
the baseline for comparison. Remarkably, our findings demonstrate that, similar to the single-vehicle case,
regardless of the number of vehicles in the system, the ratio converges to a constant value of
1
m
2
[0.34,0.35],
where m denotes the number of vehicles. This result closely aligns with the upper bound we previously
provided, further validating the correctness of our approach. Moreover, doubling the number of vehicles
leads to a reduction of the cumulative waiting time to a quarter of its original value. To clarify this effect,
we take the logarithm of the ratio results, shown in the top right of Figure 9.7. We further examine the
tours of multiple vehicles by sampling 500 data points, which must be visited by one of four vehicles. The
resulting heatmap and tour plan are depicted in the bottom of Figure 9.7. The tour results confirm the
validity of our proof process, with the entire region divided into four equal sub-regions, and each vehicle
assigned to take charge of one.
For the non-uniform distribution, we consider a bi-variate Gaussian distribution, with two cases: the
depot location at the center (Figure 9.8) and the offset center (Figure 9.9), similar to the single-vehicle
case. Our findings indicate that doubling the number of vehicles reduces the cumulative waiting time to
a quarter of its initial value. Moreover, from the tour plans on the right in Figure 9.8 and Figure 9.9, we
67
observe that minimizing the cumulative waiting time yields an optimal solution that follows our previously
established rule, progressing from the most dense to the least dense regions.
In addition, we examine the effect of region irregularity on the performance of the m-CTSP algorithm,
considering three different scenarios: two convex regions and one non-convex region. Each region is
assumed to follow a bi-variate Gaussian distribution. Figure 9.10 presents the experimental results for
circular, triangular, and pentagram regions, respectively. The outcomes demonstrate that the algorithm
performs as expected in each scenario, thereby providing further evidence of the algorithm’s robustness
and versatility.
In the subsequent experiments, we aim to examine the performance of the Capacitated Vehicle Routing
Problem (CVRP) with multiple vehicles, considering both uniform and non-uniform sample points. Similar
to the CCVRP case, we adopt the square root of the number of nodes, denoted as
√
n, as the vehicle capacity.
For uniformly distributed points, Figure 9.11 verifies that double the number of vehicles will decrease
the cumulative waiting time to a quarter.
Figure 9.12 is the result when all points follow bi-variate Gaussian distribution. We also did the ex-
periments whose region is not regular, which is shown in Figure 9.13. Here, we only consider one convex
(Circle) and one non-convex (pentagram) case.
9.2.2 ExperimentswithRoadNetworkData
The dataset we are using is driving times using HERE Maps API for 1500 points sampled in a rectangle in
downtown Los Angeles. We consider CTSP and CCVRP respectively, using subsets of data, from 50 points
to 1450 data points, to predict the cumulative waiting time. The predicted rule is as follows:
1. Randomly selectn points fromN data points where|n|∈{50,100,...,1500}.
2. If this is CCVRP case, calculate the average distance between picked points and depot.
68
3. Put previous equation in theorem to calculate ratios:
(a) ratio
CTSP
=
CumL(x
1
,...,x
n
)
n
3/2
(b) ratio
CCVRP
=
CumL(x
1
,...,x
n
)
¯ rn
3/2
4. Use ratio got from 3, multipleN
3/2
(or
¯
RN
3/2
) to predict the simulated cost for 1500 data points.
5. For the multiple vehicles’ case, we need to multiple
N
3/2
m
2
(or
¯
RN
3/2
m
2
), wherem is the number of vehicles
to predict the simulated cost for 1500 data points.
Figure 9.14 presents the results of our study. The bold red line represents the average cost when we
travel to all 1500 data points under CTSP and CCVRP situations. From the left figure, we observe that as
we increase the number of data points, the predicted cost becomes closer to the real cost. Moreover, the
right-hand side of the figure shows that using only 100 data points is already sufficient to obtain predictions
close to the real results.
Figure 9.15 is the result showing what’s happening if we have multiple vehicles. Same, the bold red
line represents the average cost when we travel to all 1500 data points under CTSP and CCVRP situations
when we have one vehicle. Still, it verifies our result.
9.3 ManagerialInsights
Based on the experiment results and continuous approximation we proved, we can get the following man-
agerial insights:
• CTSP
– When the demand is uniform, as shown from the experiment results that optimal CTSP is close
to our upper bound. Since we generated the upper bound using TSP, this indicates that CTSP
and TSP when the demand is uniform do not have much difference.
69
– However, when the demand is non-uniform, the TSP can be dramatically worse than CTSP. As
shown in our analysis process, the tour that we generated for CTSP needs to follow the “most
dense to least dense” rule. However, the optimal TSP tour does not have any such notion in
its objective, so there is no particular incentive to visit denser regions earlier in the tour. In
fact, an extreme situation could occur when the TSP tour is travelled from the totally opposite
direction, which is “Least dense to most dense”, which can be arbitrarily poor.
– Combined these two observations, we conclude that a good cumulative TSP tour is a good TSP
tour, but not vice versa.
– Analysis suggests that districting strategies are not viable for non-uniform CTSP (but they are
popular for TSP), because all vehicles should move simultaneously from the densest regions to
less dense regions (see [24] for more details).
• CCVRP
– From both the experiment results and our analysis process, we figure out unlike CTSP, the
depot location has significant impact on CCVRP.
– For CCVRP, if capacitiesc are not of the same order as
√
n, then either the returns to the depot
or driving between customers dominates
• Multiple Vehicle (m-CTSP and m-CCVRP)
– As shown from the experiment result that we can easily divide the region into multiple small
non-overlap sub-region and assign one vehicle to travel.
• Classical Vehicle Routing Problem vs Cumulative Vehicle Routing Problem
– It is easy to see that the Cumulative Problem is at least as hard as the Classical Problem. For
example, given a set of points on which we want to minimize the length of the TSP tour, one
70
can augment the set of points with N points at “infinity” (for a large number N) (see BHH
theorem results, O(n)), so that the cumulative problem on the augmented set of points will
have to minimize the length of the TSP on the original set of points (see our result, O(n
3/2
)).
This connection shows that the Cumulative problem is NP-hard even in the case where the
metric space is a plane.
71
Figure 9.2: Ratio of normal customers’ cumulative waiting time to
√
An
3/2
. (Top Left) figure depot is at
(500, 500) with standard deviation 500. (Top Right) figure depot is at (750, 750) with standard deviation
500. (Bottom Left) figure depot is at (500, 500) with standard deviation 700. (Bottom Right) figure depot is
at (750, 750) with standard deviation 700.
72
Figure 9.3: Ratio of normal customers’ cumulative waiting time to
√
An
3/2
(left) heatmap for corresponding
region (right)
73
Figure 9.4: Ratio of uniform customers’ cumulative waiting time to ¯ rn
3/2
74
Figure 9.5: Ratio of normal customers’ cumulative waiting time to ¯ rn
3/2
. (Top Left) figure depot is at (500,
500) with standard deviation 500. (Top Right) figure depot is at (750, 750) with standard deviation 500.
(Bottom Left) figure depot is at (500, 500) with standard deviation 700. (Bottom Right) figure depot is at
(750, 750) with standard deviation 700.
75
Figure 9.6: Ratio of normal customers’ cumulative waiting time to ¯ rn
3/2
(left) heatmap for Circle and
Pentagram region (right)
76
Figure 9.7: (Top Left) Ratio of uniform customers’ cumulative waiting time to
√
An
3/2
. (Top Right) Log
ratio of uniform customers’ cumulative waiting time to
√
An
3/2
. (Bottom Left) 500 data points’ heatmap
for rectangular and uniform case. (Bottom Right) Tour schedule for 4 vehicles following m-CTSP.
77
Figure 9.8: (Left) Ratio of uniform customers’ cumulative waiting time to
√
An
3/2
. (Center) Log ratio of
customers’, following normal distribution, cumulative waiting time to
√
An
3/2
. (Right) Tour schedule for
4 vehicles following m-CTSP. (Top) Depot is at (500, 500) with standard deviation 500. (Bottom) Depot is
at (500, 500) with standard deviation 700.
Figure 9.9: (Left) Ratio of uniform customers’ cumulative waiting time to
√
An
3/2
. (Center) Log ratio of
customers’, following normal distribution, cumulative waiting time to
√
An
3/2
. (Right) Tour schedule for
4 vehicles following m-CTSP. (Top) Depot is at (750, 750) with standard deviation 500. (Bottom) Depot is
at (750, 750) with standard deviation 700.
78
Figure 9.10: (Left 1) Ratio of customers’, who follow the normal distribution, cumulative waiting time to
√
An
3/2
. (Left 2) Log ratio of customers’, following normal distribution, cumulative waiting time to
√
An
3/2
.
(Left 3) heatmap for different regions. (Left 4) Tour schedule for 4 vehicles following m-CTSP. (Top) Circle
Region. (Middle) Triangular Region. (Bottom) Pentagram Region.
79
Figure 9.11: (Top Left) Ratio of uniform customers’ cumulative waiting time to ¯ rn
3/2
. (Top Right) Log
ratio of uniform customers’ cumulative waiting time to ¯ rn
3/2
. (Bottom Left) 500 data points’ heatmap for
rectangular and uniform case. (Bottom Right) Tour schedule for 4 vehicles following m-CCVRP.
80
Figure 9.12: (1
st
and3
rd
column) Ratio of customers’, sampled from normal distribution. cumulative waiting
time to ¯ rn
3/2
. 1
st
column with standard deviation 500 and 3
rd
column with standard deviation 700. (2
nd
and 4
th
column) Log ratio of customers’ cumulative waiting time to ¯ rn
3/2
. (Top) Depot is at (500, 500).
(Bottom) Depot is at (750, 750).
Figure 9.13: (Left) Ratio of customers’, who follow the normal distribution, cumulative waiting time to
¯ rn
3/2
. (Center) Log ratio of customers’, following normal distribution, cumulative waiting time to ¯ rn
3/2
.
(Right) Tour schedule for 4 vehicles following m-CCVRP. (Top) Circle Region. (Bottom) Pentagram Region.
81
Figure 9.14: Prediction Cost: (Up) CTSP. (Bottom) CCVRP
82
Figure 9.15: Prediction Cost: (Left) m-CTSP. (Right) m-CCVRP
83
Chapter10
Conclusion
We conducted asymptotic analysis of the cumulative tour problems of n points. For this problem, we
repeatedly refer to a routing strategy that we call it “most dense to least dense”. This rule forces that
vehicles travel based on the probability density. Later, our simulated experiments show that the optimal
solution closes to tour solution generated via this rule.
We start from Cumulative TSP, wheren→∞ and without vehicle capacity’s limit. We prove
λ
1
≤ liminf
n→∞
CumL(X
1
,...,X
n
)
n
3/2
R
x∈R
p
f(x)P(x)dx
≤ limsup
n→∞
CumL(X
1
,...,X
n
)
n
3/2
R
x∈R
p
f(x)P(x)dx
≤ β
2
where f(x) is the probability density function and
P(x):= Pr(f(X)≤ f(x))=
Z
x
′
:f(x
′
)≤ f(x)
f(x
′
)dx
′
The results we provide tell us that the cumulative tour lengthCumL(X
1
,...,X
n
)=O(n
3/2
).
In the next section, we added capacity constraints to the vehicles, which suggests a particular scaling
factor for modelling vehicle capacity size. Considering capacityc=
√
n, asn→∞ , we found that:
84
1
2
≤ liminf
n→∞
CumL(X
1
,...,X
n
;k
c
√
n)
n
3/2
R
x∈R
υ(x,k
c
√
n,λ
1
)ϒ(x,k
c
√
n,λ
1
)dx
≤ limsup
n→∞
CumL(X
1
,...,X
n
;k
c
√
n)
n
3/2
R
x∈R
υ(x,k
c
√
n,β
2
)ϒ(x,k
c
√
n,β
2
)dx
≤ 1
Definitions about υ(x,k
c
√
n,_) andϒ(x,k
c
√
n,_) are shown in 20.
The results provide a constant boundary.
Until now, the above results have a strong assumption that we only have a single vehicle, which is
unrealistic. From Chapter 7, we extend our analysis from single vehicle to multiple vehicles. We divide
the original region into multiple sub-regions and assign one vehicle to travel one sub-region. According
to this strategy, for multiple CTSP case, we get
λ
1
m
2
≤ liminf
n→∞
CumL(m;X
1
,...,X
n
)
n
3/2
R
x∈R
p
f(x)P(x)dx
≤ limsup
n→∞
CumL(m;X
1
,...,X
n
)
n
3/2
R
x∈R
p
f(x)P(x)dx
≤ β
2
m
2
For multiple CCVRP case, still, we consider the vehicle size is
√
n, we have
1
2m
2
≤ liminf
n→∞
CumL(m;X
1
,...,X
n
;k
c
√
n)
n
3/2
R
x∈R
υ(x,k
c
√
n,λ
1
)ϒ(x,k
c
√
n,λ
1
)dx
≤ limsup
n→∞
CumL(m;X
1
,...,X
n
;k
c
√
n)
n
3/2
R
x∈R
υ(x,k
c
√
n,β
2
)ϒ(x,k
c
√
n,β
2
)dx
≤ 1
m
2
Both m-CTSP and m-CCVRP cases, the conclusions are similar to single vehicle, but when we double
the vehicle size, the boundary will decrease to quarter.
In Chapter 9, we conducted numerical experiments on CTSP and CCVRP, considering single vehicle
and multiple vehicles, using artificially generated data and real map data. The numerical results align with
our theoretical analysis, we verify that our upper bounds are tight to the optimal solution. In addition,
our results are suitable for any region, no matter convex or not. Depot locations may influence final
converge result, but their difference can be negligible. In addition, we show that our analysis provides a
85
good prediction when applied to simulations in the Euclidean plan and on road network data, where we
are able to predict the total costs within 5 % of the ground truth.
86
Chapter11
FutureWork
The Cumulative Routing Optimization Problem, which is considered as a “customer-centric” optimization
problem, has attracted more and more attention. It reflects the need for fast service and equity. In this
thesis, we have proved a few theoretical results, however, there still have a lot of topics waiting for us to
explore, both in the theoretical and practical aspects.
One area that we aim to improve is the expansion of our current work. At present, all of our findings
are time-independent, meaning that data points can be accessed at any given time. However, in the real
world, time is a crucial factor that must be taken into account. For instance, in the event of a disaster,
minimizing the loss of life becomes the top priority, and people with varying degrees of injury need to be
grouped and treated differently. In other words, every person has a specific time window within which
they need to be served. The addition of a time window is a vital variation of the Vehicle Routing Problem
(VRP), known as the VRP with Time Windows (VRPTW), which extends the Capacitated VRP (CVRP) by
requiring that each customer be visited within a specified time interval, or time window [10]. As previously
mentioned, the VRPTW also focuses on minimizing total travel distance to ensure economic sustainability
[40]. Therefore, we hope to develop a method that combines both probability density function and time to
group people, generate the visiting rule, and identify the boundary, which could prove to be a promising
solution.
87
A second direction is also related to another famous problem: General Pickup and delivery Problems
(GPDPs), in which objects or people have to be transported between a start location and an end location
[71]. Currently, our work is just a special case of GPDP: All points have the same destination. In the future,
we will extend our solution and consider the situation when we have multiple origins and destinations.
How this will influence current tour length.
Obviously, another part that is worth significant efforts is the fairness problem. For a customer-centric
problem, we aim to maximize the overall revenue. In this thesis, we designed a strategy, which forces the
vehicle travels from “most dense to least dense”. however, this rule may tend to privilege “convenient”
customers typically located in the most dense area and close the depot. This may lead to a systematic
discrimination of customers located in rural areas, in our case, the least dense area [74]. For our future
work, we need to focus on it. We need to find the balance between fairness and optimal solution. One of
the promising method is to set the same service time for different regions and travel them back and forth.
88
Bibliography
[1] Emile HL Aarts, Jan HM Korst, and Peter JM van Laarhoven. “A quantitative analysis of the
simulated annealing algorithm: A case study for the traveling salesman problem”. In: Journal of
Statistical Physics 50 (1988), pp. 187–206.
[2] Foto Afrati, Stavros Cosmadakis, Christos H Papadimitriou, George Papageorgiou, and
Nadia Papakostantinou. “The complexity of the travelling repairman problem”. In:
RAIRO-Theoretical Informatics and Applications-Informatique Théorique et Applications 20.1 (1986),
pp. 79–87.
[3] David Aldous and Maxim Krikun. “Percolating paths through random points :” in: (2005), pp. 1–28.
url: http://arxiv.org/abs/math/0509492.
[4] Mircea Ancău. “The optimization of printed circuit board manufacturing by improving the drilling
process productivity”. In: Computers & Industrial Engineering 55.2 (2008), pp. 279–294.
[5] Sina Ansari, Mehmet Başdere, Xiaopeng Li, Yanfeng Ouyang, and Karen Smilowitz.
“Advancements in continuous approximation models for logistics and transportation systems:
1996–2016”. In: Transportation Research Part B: Methodological 107 (2018), pp. 229–252.
[6] D Applegate, W Cook, DS Johnson, and NJA Sloane. “Using large-scale computation to estimate
the Beardwood-Halton-Hammersley TSP constant”. In: Presentation at 42 (2010).
[7] Aaron Archer and David P Williamson. “Faster approximation algorithms for the minimum
latency problem”. In: SODA. Vol. 3. 2003, pp. 88–96.
[8] Sanjeev Arora and George Karakostas. “Approximation schemes for minimum latency problems”.
In: SIAM Journal on Computing 32.5 (2003), pp. 1317–1337.
[9] Giorgio Ausiello, Stefano Leonardi, and Alberto Marchetti-Spaccamela. “On salesmen, repairmen,
spiders, and other traveling agents”. In: Italian Conference on Algorithms and Complexity. Springer.
2000, pp. 1–16.
[10] Roberto Baldacci, Aristide Mingozzi, and Roberto Roberti. “Recent exact algorithms for solving the
vehicle routing problem under capacity and time window constraints”. In: European Journal of
Operational Research 218.1 (2012), pp. 1–6.
89
[11] Roberto Baldacci, Paolo Toth, and Daniele Vigo. “Recent advances in vehicle routing exact
algorithms”. In: 4OR 5.4 (2007), pp. 269–298.
[12] Nikhil Bansal, Avrim Blum, Shuchi Chawla, and Adam Meyerson. “Approximation algorithms for
deadline-TSP and vehicle routing with time-windows”. In: Proceedings of the thirty-sixth annual
ACM symposium on Theory of computing. 2004, pp. 166–174.
[13] Jillian Beardwood, J. H. Halton, and J. M. Hammersley. “The shortest path through many points”.
In: Mathematical Proceedings of the Cambridge Philosophical Society 55.4 (1959), pp. 299–327.issn:
14698064.doi: 10.1017/S0305004100034095.
[14] Tolga Bektaş and Gilbert Laporte. “The pollution-routing problem”. In: Transportation Research
Part B: Methodological 45.8 (2011), pp. 1232–1250.
[15] Patrizia Beraldi, Maria Elena Bruni, Demetrio Laganà, and Roberto Musmanno. “The risk-averse
traveling repairman problem with profits”. In: Soft Computing 23.9 (2019), pp. 2979–2993.
[16] Livio Bertacco, Lorenzo Brunetta, and Matteo Fischetti. “The linear ordering problem with
cumulative costs”. In: European Journal of Operational Research 189.3 (2008), pp. 1345–1357.
[17] Lucio Bianco, Aristide Mingozzi, and Salvatore Ricciardelli. “The traveling salesman problem with
cumulative costs”. In: Networks 23 (1993), pp. 81–91.
[18] Robert G Bland and David F Shallcross. “Large travelling salesman problems arising from
experiments in X-ray crystallography: a preliminary report on computation”. In: Operations
Research Letters 8.3 (1989), pp. 125–128.
[19] Avrim Blum, Prasad Chalasani, Don Coppersmith, Bill Pulleyblank, Prabhakar Raghavan, and
Madhu Sudan. “The minimum latency problem”. In: Proceedings of the twenty-sixth annual ACM
symposium on Theory of computing. 1994, pp. 163–171.
[20] Stephen Boyd, Stephen P Boyd, and Lieven Vandenberghe. Convex optimization. Cambridge
university press, 2004.
[21] Maria Elena Bruni, Patrizia Beraldi, and Sara Khodaparasti. “A hybrid reactive GRASP heuristic for
the risk-averse k-traveling repairman problem with profits”. In: Computers & Operations Research
115 (2020), p. 104854.
[22] Ann Melissa Campbell, Dieter Vandenbussche, and William Hermann. “Routing for relief efforts”.
In: Transportation Science 42.2 (2008), pp. 127–145.issn: 15265447.doi: 10.1287/trsc.1070.0209.
[23] John Gunnar Carlsson. “Continuous Approximation for Selection Routing Problems”. In: Available
at SSRN 4286865 (2021).
[24] John Gunnar Carlsson. “Dividing a territory among several vehicles”. In: INFORMS Journal on
Computing 24.4 (2012), pp. 565–577.
[25] John Gunnar Carlsson and Mehdi Behroozi. “Worst-case demand distributions in vehicle routing”.
In: European Journal of Operational Research 256.2 (2017), pp. 462–472.
90
[26] Ping Chen, Xingye Dong, and Yanchao Niu. “An Iterated Local Search Algorithm for the
Cumulative Capacitated Vehicle Routing Problem”. In: Technology for Education and Learning.
Ed. by Honghua Tan. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012, pp. 575–581.isbn:
978-3-642-27711-5.
[27] Nicos Christofides and Samuel Eilon. “Expected Distances in Distribution Problems”. In: OR 20.4
(Oct. 1969), pp. 437–443.issn: 14732858.doi: 10.2307/3008762.
[28] Jean-François Cordeau, Gilbert Laporte, Martin WP Savelsbergh, and Daniele Vigo. “Vehicle
routing”. In: Handbooks in operations research and management science 14 (2007), pp. 367–428.
[29] Carlos F Daganzo. “The Distance Traveled to Visit N Points with a Maximum of C Stops per
Vehicle: An Analytic Model and an Application”. eng. In: Transportation science 18.4 (1984),
pp. 331–350.issn: 0041-1655.doi: 10.1287/trsc.18.4.331.
[30] Carlos F. Daganzo. “The length of tours in zones of different shapes”. In: Transportation Research
Part B 18.2 (1984), pp. 135–145.issn: 01912615.doi: 10.1016/0191-2615(84)90027-4.
[31] G. B. Dantzig and J. H. Ramser. “The Truck Dispatching Problem Author ( s ): G . B . Dantzig and J .
H . Ramser Published by : INFORMS Stable URL : http://www.jstor.org/stable/2627477
REFERENCES Linked references are available on JSTOR for this article : You may need to log in to
JSTOR t”. In:ManagementScience 6.1 (1959), pp. 80–91.url: https://www.jstor.org/stable/2627477.
[32] Thijs Dewilde, Dirk Cattrysse, Sofie Coene, Frits CR Spieksma, and Pieter Vansteenwegen.
“Heuristics for the traveling repairman problem with profits”. In: Computers & Operations Research
40.7 (2013), pp. 1700–1707.
[33] Karen M. Douglas and Robbie. M. Sutton. “Kent Academic Repository”. In: European Journal of
Social Psychology 40.2 (2010), pp. 366–374.issn: 1939-1285.
[34] Ekrem Duman and I Or. “Precedence constrained TSP arising in printed circuit board assembly”.
In: International Journal of Production Research 42.1 (2004), pp. 67–78.
[35] Samuel Eilon, Carl Donald Tyndale Watson-Gandy, Nicos Christofides, and Richard de Neufville.
“Distribution management-mathematical modelling and practical analysis”. In: IEEE Transactions
on Systems, Man, and Cybernetics 6 (1974), p. 589.
[36] C-N Fiechter. “A parallel tabu search algorithm for large traveling salesman problems”. In: Discrete
Applied Mathematics 51.3 (1994), pp. 243–267.
[37] Matteo Fischetti, Gilbert Laporte, and Silvano Martello. “The delivery man problem and
cumulative matroids”. In: Operations Research 41.6 (1993), pp. 1055–1064.
[38] Anna Franceschetti, Ola Jabali, and Gilbert Laporte. “Continuous approximation models in freight
distribution management”. In: Top 25.3 (2017), pp. 413–433.issn: 18638279.doi:
10.1007/s11750-017-0456-1.
[39] A Garcıéa, Javier Tejel, and Pedro Jodrá Esteban. “A note on the travelling repairman problem.” In:
Pre-publicaciones del Seminario Matemático" Garcıéa de Galdeano" 3 (2001), pp. 1–16.
91
[40] T Gocken and M Yaktubay. “Comparison of different clustering algorithms via genetic algorithm
for VRPTW”. In: (2019).
[41] Maria Teresa Godinho, Luis Gouveia, and Pierre Pesneau. “Natural and extended formulations for
the time-dependent traveling salesman problem”. In: Discrete Applied Mathematics 164 (2014),
pp. 138–153.
[42] Michel Goemans and Jon Kleinberg. “An improved approximation ratio for the minimum latency
problem”. In: Mathematical Programming 82.1 (1998), pp. 111–124.
[43] Luis Gouveia and Stefan Voß. “A classification of formulations for the (time-dependent) traveling
salesman problem”. In: European Journal of Operational Research 83.1 (1995), pp. 69–82.
[44] John Grefenstette, Rajeev Gopal, Brian Rosmaita, and Dirk Van Gucht. “Genetic algorithms for the
traveling salesman problem”. In: Proceedings of the first International Conference on Genetic
Algorithms and their Applications. Psychology Press. 2014, pp. 160–168.
[45] M. Haimovich and A. H.G. Rinnooy Kan. “Bounds and Heuristics for Capacitated Routing
Problems.” In: Mathematics of Operations Research 10.4 (1985), pp. 527–542.issn: 0364765X.doi:
10.1287/moor.10.4.527.
[46] Michael Held and Richard M Karp. “A dynamic programming approach to sequencing problems”.
In: Journal of the Society for Industrial and Applied mathematics 10.1 (1962), pp. 196–210.
[47] Brian Kallehauge, Jesper Larsen, and Oli B.G. Madsen. “Lagrangian duality applied to the vehicle
routing problem with time windows”. In: Computers and Operations Research 33.5 (2006),
pp. 1464–1487.issn: 03050548.doi: 10.1016/j.cor.2004.11.002.
[48] Liangjun Ke and Zuren Feng. “A two-phase metaheuristic for the cumulative capacitated vehicle
routing problem”. In: Computers and Operations Research 40.2 (2013), pp. 633–638.issn: 03050548.
doi: 10.1016/j.cor.2012.08.020.
[49] Raphael Kramer, Jean-François Cordeau, and Manuel Iori. “Rich vehicle routing with auxiliary
depots and anticipated deliveries: An application to pharmaceutical distribution”. In:
Transportation Research Part E: Logistics and Transportation Review 129 (2019), pp. 162–174.
[50] Gilbert Laporte. “Fifty years of vehicle routing”. In: Transportation science 43.4 (2009), pp. 408–416.
[51] Anh Vu Le, Prabakaran Veerajagadheswar, Phone Thiha Kyaw, Mohan Rajesh Elara, and
Nguyen Huu Khanh Nhan. “Coverage Path Planning Using Reinforcement Learning-Based TSP for
hTetran—A Polyabolo-Inspired Self-Reconfigurable Tiling Robot”. In: Sensors 21.8 (2021), p. 2577.
[52] Shen Lin and Brian W Kernighan. “An effective heuristic algorithm for the traveling-salesman
problem”. In: Operations research 21.2 (1973), pp. 498–516.
[53] Torgny Lindvall. Lectures on the coupling method. Courier Corporation, 2002.
92
[54] Jens Lysgaard and Sanne Wøhlk. “A branch-and-cut-and-price algorithm for the cumulative
capacitated vehicle routing problem”. In: European Journal of Operational Research 236.3 (2014),
pp. 800–810.issn: 03772217.doi: 10.1016/j.ejor.2013.08.032.
[55] Glaydston Mattos Ribeiro and Gilbert Laporte. “An adaptive large neighborhood search heuristic
for the cumulative capacitated vehicle routing problem”. In: Computers and Operations Research
39.3 (2012), pp. 728–735.issn: 03050548.doi: 10.1016/j.cor.2011.05.005.
[56] Isabel Méndez-Dıéaz, Paula Zabala, and Abilio Lucena. “A new formulation for the traveling
deliveryman problem”. In: Discrete applied mathematics 156.17 (2008), pp. 3223–3237.
[57] Yuichi Nagata. “New EAX crossover for large TSP instances”. In: Parallel Problem Solving from
Nature-PPSN IX: 9th International Conference, Reykjavik, Iceland, September 9-13, 2006, Proceedings.
Springer. 2006, pp. 372–381.
[58] Gordon F Newell. “Dispatching policies for a transportation route”. In: Transportation Science 5.1
(1971), pp. 91–105.
[59] Gordon Frank Newell. “Scheduling, location, transportation, and continuum mechanics: some
simple approximations to optimization problems”. In: SIAM Journal on Applied Mathematics 25.3
(1973), pp. 346–360.
[60] Sandra Ulrich Ngueveu, Christian Prins, and Roberto Wolfler Calvo. “An effective memetic
algorithm for the cumulative capacitated vehicle routing problem”. In: Computers and Operations
Research 37.11 (2010), pp. 1877–1885.issn: 03050548.doi: 10.1016/j.cor.2009.06.014.
[61] Pamela C Nolz, Karl F Doerner, Walter J Gutjahr, and Richard F Hartl. “A bi-objective
metaheuristic for disaster relief operation planning”. In: Advances in multi-objective nature inspired
computing. Springer, 2010, pp. 167–187.
[62] Włodzimierz Ogryczak. “Inequality measures and equitable approaches to location problems”. In:
European Journal of Operational Research 122.2 (2000), pp. 374–391.
[63] Robert D Plante, Timothy J Lowe, and R Chandrasekaran. “The product matrix traveling salesman
problem: an application and solution heuristic”. In: Operations Research 35.5 (1987), pp. 772–783.
[64] Masoud Rabbani, Hamed Farrokhi-Asl, and Bahare Asgarian. “Solving a bi-objective location
routing problem by a NSGA-II combined with clustering approach: application in waste collection
problem”. In: Journal of Industrial Engineering International 13.1 (2017), pp. 13–27.
[65] H Donald Ratliff and Arnon S Rosenthal. “Order-picking in a rectangular warehouse: a solvable
case of the traveling salesman problem”. In: Operations research 31.3 (1983), pp. 507–521.
[66] Gerhard Reinelt. The traveling salesman: computational solutions for TSP applications. Vol. 840.
Springer, 2003.
93
[67] Omar Rifki, Thierry Garaix, Christine Solnon, Omar Rifki, Thierry Garaix, Christine Solnon,
Omar Rifki, Thierry Garaix, and Christine Solnon. “An asymptotic approximation of the traveling
salesman problem with uniform non-overlapping time windows To cite this version : HAL Id :
hal-03270043 An asymptotic approximation of the traveling salesman problem with uniform
non-overlapping time windows”. In: (2021).
[68] Juan Carlos Rivera, H. Murat Afsar, and Christian Prins. “A multistart iterated local search for the
multitrip cumulative capacitated vehicle routing problem”. In: Computational Optimization and
Applications 61.1 (2015), pp. 159–187.issn: 15732894.doi: 10.1007/s10589-014-9713-5.
[69] Juan Carlos Rivera, H. Murat Afsar, and Christian Prins. “Mathematical formulations and exact
algorithm for the multitrip cumulative capacitated single-vehicle routing problem”. In: European
Journal of Operational Research 249.1 (2016), pp. 93–104.issn: 03772217.doi:
10.1016/j.ejor.2015.08.067.
[70] Amir Salehipour, Kenneth Sörensen, Peter Goos, and Olli Bräysy. “Efficient GRASP+ VND and
GRASP+ VNS metaheuristics for the traveling repairman problem”. In: 4or 9.2 (2011), pp. 189–209.
[71] Martin WP Savelsbergh and Marc Sol. “The general pickup and delivery problem”. In:
Transportation science 29.1 (1995), pp. 17–29.
[72] David Simchi-Levi and Oded Berman. “Minimizing the total flow time of n jobs on a network”. In:
IIE TRANSACTIONS 23.3 (1991), pp. 236–244.
[73] René Sitters. “The minimum latency problem is NP-hard for weighted trees”. In: International
conference on integer programming and combinatorial optimization. Springer. 2002, pp. 230–239.
[74] Ninja Soeffker, Marlin W Ulmer, and Dirk C Mattfeld. “On fairness aspects of customer acceptance
mechanisms in dynamic vehicle routing”. In: Proceedings of logistikmanagement 2017 (2017),
pp. 17–24.
[75] J Michael Steele. Probability theory and combinatorial optimization. SIAM, 1997.
[76] J Michael Steele. “Subadditive Euclidean functionals and nonlinear growth in geometric
probability”. In: The Annals of Probability (1981), pp. 365–376.
[77] Elias M Stein and Rami Shakarchi. Real analysis. Princeton University Press, 2009.
[78] M H J Webb. “Cost Functions in the Location of Depots for Multiple-Delivery Journeys”. In:
Journal of the Operational Research Society 19.3 (1968), pp. 311–320.issn: 1476-9360.doi:
10.1057/jors.1968.74.
[79] Eric W Weisstein. “Traveling Salesman Constants”. In: https://mathworld. wolfram. com/ (2004).
[80] Bang Ye Wu. “Polynomial time algorithms for some minimum latency problems”. In: Information
Processing Letters 75.5 (2000), pp. 225–229.
[81] Bang Ye Wu, Zheng-Nan Huang, and Fu-Jie Zhan. “Exact algorithms for the minimum latency
problem”. In: Information Processing Letters 92.6 (2004), pp. 303–309.
94
[82] Angel Augusto Agudelo Zapata, Eduardo Giraldo Suarez, and Jairo Alberto Villegas Florez.
“Application of vrp techniques to the allocation of resources in an electric power distribution
system”. In: Journal of Computational Science 35 (2019), pp. 102–109.
95
Abstract (if available)
Abstract
Traditional objectives in vehicle routing problems (VRPs) include time, distance, or profitability. When one adopts a "customer-centric" perspective, different objectives emerge. For example, instead of minimizing the total cost of performing service (which causes some commodities/customer to be served significantly later than others), this thesis studies a family of Cumulative Routing Problems in which the goal is to minimize the total waiting time of all customers to be served. This thesis analyzes the asymptotic behavior of these problems, including the Cumulative Travelling Salesman Problem and the Cumulative Capacitated Vehicle Routing Problem, which we call the CTSP and CCVRP respectively. For each problem, we derive lower and upper bounds that characterize the cost of these problems in an asymptotic limit as demand becomes large, under the assumption that all point-to-point distances are Euclidean, and all points are independent and identical samples of a probability density on a compact planar region. We also describe the impact of vehicle capacities on the final cost. Next, we extend our analysis to the case where multiple vehicles provide coordinated service. Numerical simulations confirm that our analysis provides good predictions when applied to simulations in the Euclidean plane and on road network data, where we are able to predict total costs within 5% of the ground truth.
Linked assets
University of Southern California Dissertations and Theses
Conceptually similar
PDF
Continuous approximation formulas for location and hybrid routing/location problems
PDF
Continuous approximation for selection routing problems
PDF
A continuous approximation model for the parallel drone scheduling traveling salesman problem
PDF
Dynamic programming-based algorithms and heuristics for routing problems
PDF
New approaches for routing courier delivery services
PDF
Models and algorithms for pricing and routing in ride-sharing
PDF
Models and algorithms for the freight movement problem in drayage operations
PDF
An online cost allocation model for horizontal supply chains
PDF
Routing for ridesharing
PDF
Train scheduling and routing under dynamic headway control
PDF
Asymptotic analysis of the generalized traveling salesman problem and its application
PDF
Computational geometric partitioning for vehicle routing
PDF
Routing problems for fuel efficient vehicles
PDF
Package delivery with trucks and UAVs
PDF
Applications of explicit enumeration schemes in combinatorial optimization
PDF
The warehouse traveling salesman problem and its application
PDF
Cost-sharing mechanism design for freight consolidation
PDF
Vehicle routing and resource allocation for health care under uncertainty
PDF
The robust vehicle routing problem
PDF
Information design in non-atomic routing games: computation, repeated setting and experiment
Asset Metadata
Creator
Peng, Ying
(author)
Core Title
Continuous approximation formulas for cumulative routing optimization problems
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Industrial and Systems Engineering
Degree Conferral Date
2023-05
Publication Date
05/05/2023
Defense Date
04/18/2023
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
CCVRP,continuous approximation,CTSP,cumulative routing problem,m-CCVRP,m-CTSP,OAI-PMH Harvest
Format
theses
(aat)
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Carlsson, John (
committee chair
), Dessouky, Maged (
committee member
), Gupta, Vishal (
committee member
)
Creator Email
yingpeng@usc.edu,yingpeng0221@gmail.com
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-oUC113099968
Unique identifier
UC113099968
Identifier
etd-PengYing-11780.pdf (filename)
Legacy Identifier
etd-PengYing-11780
Document Type
Dissertation
Format
theses (aat)
Rights
Peng, Ying
Internet Media Type
application/pdf
Type
texts
Source
20230505-usctheses-batch-1038
(batch),
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Access Conditions
The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the author, as the original true and official version of the work, but does not grant the reader permission to use the work if the desired use is covered by copyright. It is the author, as rights holder, who must provide use permission if such use is covered by copyright.
Repository Name
University of Southern California Digital Library
Repository Location
USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA
Repository Email
cisadmin@lib.usc.edu
Tags
CCVRP
continuous approximation
CTSP
cumulative routing problem
m-CCVRP
m-CTSP