the vehicle routing problem: latest …...the vehicle routing problem: latest advances and new...

THE VEHICLE ROUTING PROBLEM: LATEST ADVANCES AND NEW CHALLENGES

OPERATIONS RESEARCH/COMPUTER SCIENCE INTERFACESProfessor Ramesh Sharda Prof. Dr. Stefan Voß Oklahoma State University Universität Hamburg Bierwirth / Adaptive Search and the Management of Logistics Systems Laguna & González-Velarde / Computing Tools for Modeling, Optimization and Simulation Stilman / Linguistic Geometry: From Search to Construction Sakawa / Genetic Algorithms and Fuzzy Multiobjective Optimization Ribeiro & Hansen / Essays and Surveys in Metaheuristics Holsapple, Jacob & Rao / Business Modelling: Multidisciplinary Approaches — Economics,

Operational and Information Systems Perspectives Sleezer, Wentling & Cude/Human Resource Development and Information Technology: Making Global

Connections Voß & Woodruff / Optimization Software Class Libraries Upadhyaya et al / Mobile Computing: Implementing Pervasive Information and Communications

Technologies Reeves & Rowe / Genetic Algorithms—Principles and Perspectives: A Guide to GA Theory Bhargava & Ye / Computational Modeling and Problem Solving In the Networked World: Interfaces in

Computer Science & Operations Research Woodruff / Network Interdiction and Stochastic Integer Programming Anandalingam & Raghavan / Telecommunications Network Design and Management Laguna & Martí / Scatter Search: Methodology and Implementations in C Gosavi/ Simulation-Based Optimization: Parametric Optimization Techniques and Reinforcement

Learning Koutsoukis & Mitra / Decision Modelling and Information Systems: The Information Value Chain Milano / Constraint and Integer Programming: Toward a Unified Methodology Wilson & Nuzzolo / Schedule-Based Dynamic Transit Modeling: Theory and Applications Golden, Raghavan & Wasil / The Next Wave in Computing, Optimization, And Decision Technologies Rego & Alidaee/ Metaheuristics Optimization via Memory and Evolution: Tabu Search and Scatter

Search Kitamura & Kuwahara / Simulation Approaches in Transportation Analysis: Recent Advances and

Challenges Ibaraki, Nonobe & Yagiura / Metaheuristics: Progress as Real Problem Solvers Golumbic & Hartman / Graph Theory, Combinatorics, and Algorithms: Interdisciplinary Applications Raghavan & Anandalingam / Telecommunications Planning: Innovations in Pricing, Network Design

and Management Mattfeld / The Management of Transshipment Terminals: Decision Support for Terminal Operations in

Finished Vehicle Supply Chains Alba & Martí/ Metaheuristic Procedures for Training Neural Networks Alt, Fu & Golden/ Perspectives in Operations Research: Papers in honor of Saul Gass’ 80th Birthday Baker et al/ Extending the Horizons: Adv. In Computing, Optimization, and Dec. Technologies Zeimpekis et al/ Dynamic Fleet Management: Concepts, Systems, Algorithms & Case Studies Doerner et al/ Metaheuristics: Progress in Complex Systems Optimization Goel/ Fleet Telematics: Real-time management & planning of commercial vehicle operations Gondran & Minoux/ Graphs, Dioïds and Semirings: New models and algorithms Alba & Dorronsoro/ Cellular Genetic Algorithms Golden, Raghavan & Wasil/ The Vehicle Routing Problem: Latest advances and new challenges Raghavan, Golden & Wasil/ Telecommunications Modeling, Policy and Technology

THE VEHICLE ROUTING PROBLEM: LATEST ADVANCES AND NEW CHALLENGES

Edited by

BRUCE GOLDEN University of Maryland

S. RAGHAVAN University of Maryland

EDWARD WASIL American University

123

Editors Bruce Golden S. Raghavan University of Maryland University of Maryland College Park, MD, USA College Park, MD, USA Edward Wasil American University Washington, DC, USA Series Editors Ramesh Sharda Stefan Voß Oklahoma State University Universität Hamburg Stillwater, Oklahoma, USA Germany

ISBN: 978-0-387-77777-1 e-ISBN: 978-0-387-77778-8

Library of Congress Control Number: 2008921562

© 2008 Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now know or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if the are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com

DOI: 10 .1007/ 978-0-387-77778-8

Preface

Theoretical research and practical applications in the field of vehicle routingstarted in 1959 with the truck dispatching problem posed by Dantzig andRamser [1]: find the “. . . optimum routing of a fleet of gasoline delivery trucksbetween a bulk terminal and a large number of service stations supplied bythe terminal.” Using a method based on a linear programming formulation,their hand calculations produced a near-optimal solution with four routes toa problem with twelve service stations. The authors proclaimed: “No practicalapplications of the method have been made as yet.”

In the nearly 50 years since the Dantzig and Ramser paper appeared,work in the field has exploded dramatically. Today, a Google Scholar searchof the words vehicle routing problem (VRP) yields more than 21,700 entries.The June 2006 issue of OR/MS Today provided a survey of 17 vendors ofcommercial routing software whose packages are currently capable of solvingaverage-size problems with 1,000 stops, 50 routes, and two-hour hard-timewindows in two to ten minutes [2]. In practice, vehicle routing may be thesingle biggest success story in operations research. For example, each day103,500 drivers at UPS follow computer-generated routes. The drivers visit7.9 million customers and handle an average of 15.6 million packages [3].

While much has been documented about the VRP in major studies thathave appeared from 1971 (starting with Distribution Management by Eilon,Watson-Gandy, and Christofides) to 2002 (ending with The Vehicle RoutingProblem by Toth and Vigo), there are important advances and new challengesthat have been raised in the last five years or so due to technological innova-tions such as global positioning systems, radio frequency identification, andparallel computing. The portfolio of techniques for modeling and solving thestandard, capacitated VRP and its many variants has advanced significantly.Researchers and practitioners have developed faster, more accurate solutionalgorithms and better models that give them the ability to solve large-scaleproblems.

The papers in this edited volume seek to build on the legacy of publishedVRP studies in three ways. They summarize the most significant results for

VI Preface

the VRP and its variants since 2000. They present significant methodologicaladvances or new approaches for solving existing vehicle routing problems.They present novel problems that have arisen in the vehicle routing domainand highlight new challenges for the field.

This volume is organized into three sections: overviews and surveys (ninepapers), new directions in modeling and algorithms (eleven papers), and prac-tical applications (five papers). We hope that the academic community (es-pecially new and young researchers entering the field) and practitioners inindustry will find all twenty-five papers in this volume interesting, informa-tive, and useful.

We thank all of the authors for their participation in producing a first-ratevolume. We also thank Gary Folven, senior editor at Springer, and RameshSharda and Stefan Voß, series editors, for their encouragement and support.

College Park, MD and Washington, DC Bruce GoldenNovember 2007 S. Raghavan

Edward Wasil

References

1. G. Dantzig and J. Ramser. The truck dispatching problem. Management Science,6:80–91, 1959.

2. R. Hall. On the road to integration. OR/MS Today, 33(3):50–57, 2006.3. UPS. Driving success: Why the UPS model for manag-

ing 103,500 drivers is a competitive advantage. Available athttp://pressroom.ups.com/mediakits/popups/factsheet/0,1889,1201,00.html.

Contents

Part I Overviews and Surveys

Routing a Heterogeneous Fleet of VehiclesRoberto Baldacci, Maria Battarra, Daniele Vigo . . . . . . . . . . . . . . . . . . . . . 3

A Decade of Capacitated Arc RoutingSanne Wøhlk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Inventory RoutingLuca Bertazzi, Martin Savelsbergh, Maria Grazia Speranza . . . . . . . . . . . . 49

The Period Vehicle Routing Problem and its ExtensionsPeter M. Francis, Karen R. Smilowitz, Michal Tzur . . . . . . . . . . . . . . . . . . 73

The Split Delivery Vehicle Routing Problem: A SurveyClaudia Archetti, Maria Grazia Speranza . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Challenges and Advances in A Priori RoutingAnn Melissa Campbell, Barrett W. Thomas . . . . . . . . . . . . . . . . . . . . . . . . . 123

Metaheuristics for the Vehicle Routing Problem and ItsExtensions: A Categorized BibliographyMichel Gendreau, Jean-Yves Potvin, Olli Braysy, Geir Hasle, ArneLøkketangen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

Parallel Solution Methods for Vehicle Routing ProblemsTeodor Gabriel Crainic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

Recent Developments in Dynamic Vehicle Routing SystemsAllan Larsen, Oli B.G. Madsen, Marius M. Solomon . . . . . . . . . . . . . . . . . 199

VIII Contents

Part II New Directions in Modeling and Algorithms

Online Vehicle Routing Problems: A SurveyPatrick Jaillet, Michael R. Wagner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

Modeling and Solving the Capacitated Vehicle RoutingProblem on TreesBala Chandran, S. Raghavan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

Using a Genetic Algorithm to Solve the GeneralizedOrienteering ProblemXia Wang, Bruce L. Golden, Edward A. Wasil . . . . . . . . . . . . . . . . . . . . . . 263

An Integer Linear Programming Local Search for CapacitatedVehicle Routing ProblemsPaolo Toth, Andrea Tramontani . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

Robust Branch-Cut-and-Price Algorithms for Vehicle RoutingProblemsArtur Pessoa, Marcus Poggi de Aragao, Eduardo Uchoa . . . . . . . . . . . . . . . 297

Recent Models and Algorithms for One-to-One Pickup andDelivery ProblemsJean-Francois Cordeau, Gilbert Laporte, Stefan Ropke . . . . . . . . . . . . . . . . 327

One-to-Many-to-One Single Vehicle Pickup and DeliveryProblemsIrina Gribkovskaia, Gilbert Laporte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

Challenges and Opportunities in Attended Home DeliveryNiels Agatz, Ann Melissa Campbell, Moritz Fleischmann, MartinSavelsbergh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379

Chvatal-Gomory Rank-1 Cuts Used in a Dantzig-WolfeDecomposition of the Vehicle Routing Problem with TimeWindowsBjørn Petersen, David Pisinger, Simon Spoorendonk . . . . . . . . . . . . . . . . . 397

Vehicle Routing Problems with Inter-Tour ResourceConstraintsChristoph Hempsch, Stefan Irnich . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

From Single-Objective to Multi-Objective Vehicle RoutingProblems: Motivations, Case Studies, and MethodsNicolas Jozefowiez, Frederic Semet, El-Ghazali Talbi . . . . . . . . . . . . . . . . . 445

Contents IX

Part III Practical Applications

Vehicle Routing for Small Package Delivery and PickupServicesRichard T. Wong . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475

Advances in Meter Reading: Heuristic Solution of the CloseEnough Traveling Salesman Problem over a Street NetworkRobert Shuttleworth, Bruce L. Golden, Susan Smith, Edward Wasil . . . . . 487

Multiperiod Planning and Routing on a Rolling Horizon forField Force Optimization LogisticsNathalie Bostel, Pierre Dejax, Pierre Guez, Fabien Tricoire . . . . . . . . . . . 503

Health Care Logistics, Emergency Preparedness, and DisasterRelief: New Challenges for Routing Problems with a Focuson the Austrian SituationKarl F. Doerner, Richard F. Hartl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527

Vehicle Routing Problems and Container TerminalOperations – An Update of ResearchRobert Stahlbock, Stefan Voß . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551

Part I

Overviews and Surveys

Routing a Heterogeneous Fleet of Vehicles

Roberto Baldacci, Maria Battarra, and Daniele Vigo

DEIS, University of Bolognavia Venezia 52, 47023 Cesena, Italy{rbaldacci, mbattarra, dvigo}@deis.unibo.it

Summary. In the well-known Vehicle Routing Problem (VRP) a set of identicalvehicles, based at a central depot, is to be optimally routed to supply customerswith known demands subject to vehicle capacity constraints.

An important variant of the VRP arises when a fleet of vehicles characterizedby different capacities and costs is available for distribution activities. The problemis known as the Mixed Fleet VRP or as the Heterogeneous Fleet VRP.

This chapter gives an overview of approaches from the literature to solve het-erogeneous VRPs. In particular, we classify the different variants described in theliterature and, as no exact algorithm has been presented for any variants of hetero-geneous VRP, we review the lower bounds and the heuristic algorithms proposed.Computational results, comparing the performance of the different heuristic algo-rithms on benchmark instances, are also discussed.

Key words: Heterogeneous vehicle routing problem.

1 Introduction

The Vehicle Routing Problem (VRP) is one of the most studied combinatorialoptimization problems and is concerned with the optimal design of routesto be used by a fleet of vehicles to serve a set of customers. Since it wasfirst proposed by Dantzig and Ramser [15], hundreds of papers have beendevoted to the exact and approximate solution of the many variants of thisproblem, such as the Capacitated VRP (CVRP), in which a homogeneousfleet of vehicles is available and the only constraint is the vehicle capacity,or the VRP with Time Windows (VRPTW), where customers may be servedwithin a specified time interval and the schedule of the vehicle trips needs tobe determined.

More recently, greater attention has been devoted to more complex vari-ants of the VRP, sometimes named “rich” VRPs, that are closer to the prac-tical distribution problems that the VRP models. In particular, these variantsare characterized by multiple depots, multiple trips to be performed by the

B. Golden et al. (eds.), The Vehicle Routing Problem,doi: 10.1007/978-0-387-77778-8 1, c© Springer Science+Business Media, LLC 2008

4 Baldacci, Batarra, and Vigo

vehicles, multiple vehicle types or other operational issues such as loadingconstraints. Trying to systematize such a huge literature is a challenging anduseful activity that has attracted considerable efforts in the scientific commu-nity. Among the various surveys on the VRP are the book by Toth and Vigo[55] and the more recent update by Cordeau et al. [14]. Specific surveys ofrich VRPs may be found in Braysy et al. [6].

This chapter considers an important variant of the VRP, in which a fleetof vehicles characterized by different capacities and costs, is available for thedistribution activities. The problem is known as the Mixed Fleet VRP or asthe Heterogeneous Fleet VRP and was first considered in a structured way inGolden et al. [30].

We examine the basic problem including capacity constraints only, whichhas received greater attention in the literature, as well as the more recentlystudied variants including time window constraints. Moreover, we briefly re-view a related variant known as the Site-Dependent VRP (SDVRP), wherethere are compatibility relations between customers and vehicle types. Ad-ditional case studies and applications related to the solution of Heteroge-neous VRPs can be found in Semet and Taillard [48], Rochat and Semet [45],Brandao and Mercer [5], Prins [43], Wu et al. [57] and Tavakkoli-Moghaddamet al. [53]. In addition, Engevall et al. [19] use a game-theoretic approach tomodel the problem of allocating the cost of the heterogeneous fleet to thecustomers.

This chapter is organized as follows. The next section introduces the nota-tion used throughout the chapter and describes the variants of heterogeneousVRPs with capacity constraints studied in the literature. Section 3 presentsthe main integer programming formulations and discusses lower bounding ap-proaches. Section 4 reviews heuristics and metaheuristics and reports on theirperformances. Finally, the last section offers conclusions and suggestions forfuture research.

2 Notation and Problem Variants

A directed graph G = (V, A) is given, where V = {0, 1, . . . , n} is the set ofn + 1 nodes and A is the set of arcs. Node 0 represents the depot, while theremaining node set V ′ = V \{0} corresponds to the n customers.

Each customer i ∈ V ′ requires a supply of qi units from the depot (weassume q0 = 0). A heterogeneous fleet of vehicles is stationed at the depot andis used to supply the customers. The vehicle fleet is composed by m differentvehicle types, with M = {1, . . . , m}. For each type k ∈ M , mk vehicles areavailable at the depot, each having a capacity equal to Qk. Each vehicle typeis also associated with a fixed cost, equal to Fk that models rental or capitalamortization costs. In addition, for each arc (i, j) ∈ A and for each vehicletype k ∈ M , a non-negative routing cost, ck

ij , is given.

Routing a Heterogeneous Fleet of Vehicles 5

A route is defined as the pair (R, k), where R = (i1, i2, . . . , i|R|), withi1 = i|R| = 0 and {i2, . . . , i|R|−1} ⊆ V ′, is a simple circuit in G containingthe depot, and k is the type of vehicle associated with the route. R is usedto refer both to the visiting sequence and to the set of customers (includingthe depot) of the route. A route (R, k) is feasible if the total demand of thecustomers visited by the route does not exceed the vehicle capacity Qk (i.e.,|R|−1∑h=2

qih≤ Qk). The cost of a route corresponds to the sum of the costs of

the arcs forming the route, plus the fixed cost of the vehicle associated with

it (i.e.,|R|−1∑h=1

ckihih+1

+ Fk).

The most general version of the Heterogeneous VRP consists of designinga set of feasible routes with minimum total cost, such that:

i) each customer is visited by exactly one route;ii) the number of routes performed by vehicles of type k ∈ M is not greater

than mk.

Two versions of the problem naturally arise: the symmetric one, whenckij = ck

ji, for every pair i, j of nodes and for each vehicle type k ∈ M , and theasymmetric version, otherwise. In addition, several variants of these generalproblems have been presented in the literature, depending on the availablefleet and the type of costs. In particular, the following problem characteristicswere modified:

i) the vehicle fleet is composed by an unlimited number of vehicles for eachtype, i.e., mk = +∞, ∀k ∈ M ;

ii) the fixed costs of the vehicles are not considered, i.e., Fk = 0, ∀k ∈ M ;iii) the routing costs are vehicle-independent, i.e., ck1

ij = ck2ij = cij , ∀k1, k2 ∈

M , k1 �= k2, and ∀(i, j) ∈ A.

A related problem that has received some attention in the literature is theSite-Dependent VRP (SDVRP), in which there is a limited heterogeneousfleet available for the service, no vehicle fixed costs are considered, routingcosts are vehicle-independent, and each customer may include restrictions onthe vehicle types that may visit it. Observe that the SDVRP is a special caseof the general Heterogeneous VRP described above, where the routing costckij of all arcs entering node j is set to infinity for all vehicles types k that are

incompatible with node j.Table 1 summarizes the different problem variants that have been consid-

ered in the literature, together with the corresponding references. The differ-ent problem variants have been referred to in the literature using differentnames. However, there is a certain homogeneity towards calling HeterogenousVRPs the variants with limited number of vehicles, and Fleet Size and Mixthose with unlimited ones. Therefore, we adopt a unified naming convention,that uses two acronyms (HVRP and FSM) and adds two letters that indicate


whether fixed or routing costs are considered. We used “F” for fixed costs and“D” for vehicle dependent routing costs.

Thus, in this chapter, we refer to the problem variants as follows (seeTable 1):

(a) Heterogeneous VRP with Fixed Costs and Vehicle Dependent RoutingCosts (HVRPFD);

(b) Heterogeneous VRP with Vehicle Dependent Routing Costs (HVRPD);(c) Fleet Size and Mix VRP with Fixed Costs and Vehicle Dependent Routing

Costs (FSMFD);(d) Fleet Size and Mix VRP with Vehicle Dependent Routing Costs (FSMD);(e) Fleet Size and Mix VRP with Fixed Costs (FSMF).

For the SDVRP we kept the original acronym that is used consistently inthe literature. Moreover, for each problem, the variants with time windowsare denoted by adding TW to the acronym of the specific problem. All theproblems described above are NP-hard as they are natural generalizations ofthe Traveling Salesman Problem (TSP).

For the FSMF, a usual assumption on the vehicle types in M imposesthat they are undominated, i.e., ordered so that Q1 < Q2 < . . . < Qm andF1 < F2 < . . . < Fm.

3 Mathematical Formulations and Lower Bounds

In this section, we describe some of the mathematical formulations and lowerbounds presented in the literature for heterogeneous VRPs. As far as we areaware, no exact algorithm has ever been developed for any of the differentversions of the heterogeneous VRPs described in the previous section and forthe SDVRP.

Most integer programming formulations of the basic VRP use binary vari-ables as vehicle flow variables to indicate if a vehicle travels between twocustomers in the optimal solution. In this way, decision variables combineassignment constraints, modeling vehicle routes, with commodity flow con-straints, modeling movements of goods. Formulations of this type were firstproposed by Garvin et al. [22] to model an oil delivery problem and laterextended by Gavish and Graves [24].

Gheysens et al. [27] formulate the FSMF using three-index binary variablesxk

ij as vehicle flow variables that take value 1 if a vehicle of type k travelsdirectly from customer i to customer j, and 0 otherwise. In addition, flowvariables yij specify the quantity of goods that a vehicle carries when it leavescustomer i to service customer j. The formulation, for the HVRPFD which isthe most general variant, is as follows:


Table

1.

Variants

ofth

eV

RP

that

hav

ebee

nco

nsi

der

edin

the

lite

ratu

re.

Pro

ble

mFle

et

Fix

ed

Routi

ng

Refe

rences

Vari

ant

Siz

eC

ost

sC

ost

s

HV

RP

FD

Lim

ited

Consi

der

edD

epen

den

tLiet

al.

[32]

HV

RP

DLim

ited

Notco

nsi

der

edD

epen

den

tTailla

rd[5

0],

Gen

dre

au

etal.

[25],

Prins

[43],

Tara

n-

tilis

etal.

[51],

Tara

ntilis

etal.

[52],

Liet

al.

[32]

SD

VR

PLim

ited

Notco

nsi

der

edSite-

Dep

enden

tN

ag

etal.

[37],

Chao

etal.

[8],C

ord

eau

and

Laport

e[1

2],

Pisin

ger

and

Ropke

[42]

FSM

FD

Unlim

ited

Consi

der

edD

epen

den

tFer

land

and

Mic

hel

on

[20],

Teo

doro

vic

etal.

[54],

Choi

and

Tch

a[9

]

FSM

DU

nlim

ited

Notco

nsi

der

edD

epen

den

tTailla

rd[5

0],

Gen

dre

au

etal.

[25],

Wass

an

and

Osm

an

[56],

Choiand

Tch

a[9

]

FSM

FU

nlim

ited

Consi

der

edIn

dep

enden

tG

hey

senset

al.

[27],

Gold

enet

al.

[30],

Ghey

senset

al.

[28],

Des

roch

ers

and

Ver

hoog

[17],

Salh

iand

Rand

[47],

Osm

an

and

Salh

i[4

1],

Tailla

rd[5

0],

Och

iet

al.

[38],

Och

iet

al.

[39],

Gen

dre

au

etal.

[25],

Liu

and

Shen

[34],

Wass

an

and

Osm

an

[56],

Dullaer

tet

al.

[18],

Ren

aud

and

Boct

or

[44],

Choiand

Tch

a[9

],Y

am

an

[58],

Del

l’A

mic

oet

al.

[16]


(F1) z(F1) = Min∑k∈M

Fk

∑j∈V ′

xk0j +

∑k∈M

∑i,j∈Vi�=j

ckijx

kij (1)

s.t.∑k∈M

∑i∈V

xkij = 1, ∀j ∈ V ′ (2)

∑i∈V

xkip −

∑j∈V

xkpj = 0, ∀p ∈ V ′, ∀k ∈ M (3)

∑j∈V ′

xk0j ≤ mk, ∀k ∈ M (4)

∑i∈V

yij −∑i∈V

yji = qj , ∀j ∈ V ′ (5)

qjxkij ≤ yij ≤ (Qk − qi)xk

ij , ∀i, j ∈ V, i �= j, ∀k ∈ M (6)

yij ≥ 0, ∀i, j ∈ V, i �= j (7)

xkij ∈ {0, 1} , ∀i, j ∈ V, i �= j, ∀k ∈ M (8)

In the above formulation, constraints (2) and (3) ensure that a customeris visited exactly once and that if a vehicle visits a customer, it must alsodepart from it. The maximum number of vehicles available for each vehicletype is imposed by constraints (4). Constraints (5) are the commodity flowconstraints: they specify that the difference between the quantity of goods avehicle carries before and after visiting a customer is equal to the demandof that customer. Finally, constraints (6) ensure that the vehicle capacity isnever exceeded.

Golden et al. [30] proposed a formulation for the FSMF similar to formu-lation F1 where the capacity and subtour elimination constraints are modeledwith an extension of the Miller-Tucker-Zemlin (MTZ) inequalities for the TSP(see Miller et al. [35]). Other mathematical formulations for FSMF were pre-sented by Yaman [58], who described six different formulations: the first fourbased on the use of MTZ inequalities to model subtour elimination and thelast two based on flow variables.

Another important type of formulation for Heterogeneous VRPs can beobtained by extending the Set Partitioning (SP) model of the VRP, origi-nally proposed by Balinski and Quandt [2], which associates a binary variablewith each feasible route. The formulation, again written for HVRPFD, canbe described as follows.

Let Rk be the index set of all feasible routes for a vehicle of type k ∈ M .Each route � ∈ Rk has an associated cost d�k. Let Bik ⊂ Rk be the indexsubset of the routes for a vehicle of type k covering customer i ∈ V ′. Inthe following we will use R� to indicate the subset of vertices (i.e., R� ={0, i1, i2, . . . , ih}, {i1, i2, . . . , ih} ⊆ V ′) visited by route �.

Let ξ�k be a binary variable that is equal to 1 if and only if route � ∈ Rk

belongs to the optimal solution. The set partitioning model is as follows:


(F2) z(F2) = min∑k∈M

∑�∈Rk

d�kξ�k (9)

s.t.∑k∈M

∑�∈Bik

ξ�k = 1, ∀i ∈ V ′ (10)

∑�∈Rk

ξ�k ≤ mk, ∀k ∈ M (11)

ξ�k ∈ {0, 1}, ∀� ∈ Rk, ∀k ∈ M. (12)

Constraints (10) specify that each customer i ∈ V ′ must be covered exactlyby one route and constraints (11) require that at most mk routes are selectedfor a vehicle of type k ∈ M .

Note that in the case of FSMF, each route �1 ∈ Rk1 is dominated byanother route �2 ∈ Rk2 , if k1 > k2 and R�1 = R�2 . This happens since forFSMF we have an unlimited number of vehicles of type k2 and d�2k2 < d�1k1 .Thus, sets {Rk} can be redefined as the sets of undominated feasible routes.

Mathematical formulations for the time windows variants of the problemwere described in Ferland and Michelon [20], Dell’Amico et al. [16] and Braysyet al. [7]. Moreover, we are not aware of specific models, lower bounding proce-dures, or exact algorithms for the SDVRP, although, as previously mentioned,the SDVRP may be modeled as a special case of the HVRPD.

3.1 Lower Bounds

Lower bounds for the FSMF were proposed by Golden et al. [30], Yaman [58]and Choi and Tcha [9]. These latter authors also described lower bounds forthe FSMFD and the FSMD. In this section, we present the lower bound ofGolden et al. [30], and we briefly examine those proposed by Choi and Tcha[9] and by Yaman [58].

Let us consider the FSMF problem and suppose (without loss of generality)that the customers are numbered according to decreasing distance from thedepot (i.e., c01 ≥ c02 ≥ . . . ≥ c0n). Given a route (R, k), the pivot of a routeis defined as the vertex i∗ ∈ R such that c0i∗ = max

j∈R\{0}{c0j} (i.e., i∗ is the

customer of the route located farthest from the depot). In those cases wheremore than one vertex produces the maximum of the expression, we call thepivot of route (R, k) the vertex having the smallest index.

Using the definition of a pivot, the set of routes Rk can be partitioned asR1k∪R2k∪. . .∪Rnk, where Rik is the index set of all routes having as a pivotthe customer i ∈ V ′ and using a vehicle of type k ∈ M . Let us denote thecost of a route � ∈ Rik as di

�k. Moreover, let Bjik ⊂ Rik be the index subsetof the routes for a vehicle of type k, for the pivot i and covering customerj ∈ V ′. Finally, let ξi

�k be a binary variable that is equal to 1 if and only ifroute � ∈ Rik belongs to the optimal solution. Starting from F2, the FSMFcan be formulated as follows:


(F3) z(F3) = min∑k∈M

∑i∈V ′

∑�∈Rik

di�kξi

�k (13)

s.t.∑k∈M

∑i∈V ′

∑�∈Bjik

ξi�k = 1, ∀j ∈ V ′ (14)

ξi�k ∈ {0, 1}, ∀� ∈ Rik, ∀k ∈ M, ∀i ∈ V ′. (15)

Note that in the case of the FSMF, constraints (11) of formulation F2 areredundant as mk = +∞, ∀k ∈ M .

If the cost matrix {cij} is symmetric and satisfies the triangle inequality, alower bound to the FSMF can be obtained from formulation F3 by computingthe cost di

�k as di�k = 2c0i +Fk, i.e., by approximating the route cost with the

radial component associated with the pivot of the route, plus the fixed costof the vehicle assigned to it.

The above observation leads to the following relaxation of formulationF3. Let ξik be a binary variable which is equal to 1 if and only if a routefor the pivot i, using a vehicle of type k, is in the solution, and 0 otherwise.In addition, let xijk, with j ≥ i, be a binary variable which is equal to 1 ifand only if customer j is served by a route having pivot i and vehicle typek. Then, the optimal solution of the following mixed integer programmingproblem gives a valid lower bound to the FSMF:

(LB1) z(LB1) = min∑k∈M

∑i∈V ′

(2c0i + Fk)ξik (16)

s.t.∑k∈M

∑i∈V ′

xijk = 1, ∀j ∈ V ′ (17)

∑j∈V ′

qjxijk ≤ Qkξik, ∀i ∈ V ′, ∀k ∈ M (18)

xijk ∈ {0, 1}, ∀i, j ∈ V ′, j ≥ i, ∀k ∈ M (19)ξik ∈ {0, 1}, ∀i ∈ V ′, ∀k ∈ M. (20)

Constraints (17) state that each customer must be assigned to a pivot, whileconstraints (18) impose the vehicle capacities. Note that the definition ofvariables {xijk} implies ξ1k = 1, for some k ∈ M .

A relaxation of lower bound LB1 can be obtained if the integrality con-straints (19) are relaxed and variables {ξik} are assumed to be general integer(i.e., a customer can be a pivot of more than one route). In addition, let sk

be the sum of demands of customers for which vehicle type k is the smallestone that can service the demand, and define sm+1 = 0. Then, the optimalsolution of the following mixed integer programming problem gives a validlower bound to the FSMF:

(LB2) z(LB2) = min∑k∈M

∑i∈V ′

(2c0i + Fk)ξik (21)

s.t. (17), (18) and


∑j∈V ′

qjxijk ≥ sk −m∑

h=k+1

(∑j∈V ′

qjxijh − sh), ∀k ∈ M (22)

0 ≤ xijk ≤ 1, ∀i, j ∈ V ′, j ≥ i, ∀k ∈ M (23)ξik ≥ 0 integer , ∀i ∈ V ′, ∀k ∈ M. (24)

Constraints (22) impose that the sum of the customer demands served by allthe vehicles of type k ∈ M must be greater than or equal to sk, minus thedemand which can be served by all vehicles having capacities greater thanQk. Note that the demand of a customer can be split among the pivots whichare selected in the solution of lower bound LB2.

Lower bound z(LB2) can be efficiently computed by using the procedureproposed by Golden et al. [30] which can be described as follows.

Let Dtot =∑

i∈V ′ qi be the total customer demand, and let G = (V , A)be a directed graph where V = {0, 1, . . . , Dtot}. Associate with each vertexq ∈ V \ {Dtot} the cost C(q) = 2c0h, where the index h is such that thefollowing inequalities are satisfied:

h−1∑j=0

qj ≤ q <

h∑j=0

qi. (25)

The set A of arcs of graph G is composed by the following arcs: for each vehicleof type k ∈ M , there is an arc from node q ∈ V to min{Dtot, q+Qk} with costequal to C(q) + Fk if and only if q ≥

∑m+1h=k+1 sh. Note that in graph G, the

first sm vertices represent the demands that require the largest vehicle type.On the other hand, the first q1 vertices represent the demand of the farthestcustomer.

The cost of the shortest path in graph G from vertex 0 to vertex Dtot giveslower bound z(LB2). Note that, each arc used in the shortest path correspondsto a pivot i and to a vehicle type k, i.e., to a variable ξik of formulation LB2.

Yaman [58] proposed several lower bounds based on cutting-plane tech-niques used to strengthen the LP relaxation of six mathematical formulationsof the FSMF. Also in Yaman [58], a comparison among the LP relaxation ofthe different mathematical formulations is reported. The following families ofvalid inequalities were considered to improve the lower bounds given by thedifferent LP relaxations: covering type inequalities, subtour elimination in-equalities, generalized large multistar inequalities and valid inequalities basedon the lifting of the MTZ constraints.

Choi and Tcha [9] proposed lower bounds for the FSMFD, the FSMD andthe FSMF, based on the set partitioning formulation F2, which were computedusing a column generation technique. More precisely, the lower bounds werederived from the relaxation of the partitioning formulation F2 into a coveringformulation, where the set partitioning columns correspond to the set of q-routes (see Christofides et al. [10]), where a q-route is a (not necessarily simple)


Table 2. Lower bounds for the FSMF using the G12 instances.

Golden et al. [30] Yaman [58] Choi and Tcha [9]Problem n UB LB %LB LB %LB LB %LB

3E 20 961.03 791.91 82.40 931.05 96.88 951.61 99.024E 20 6437.33 6265.26 97.33 6387.76 99.23 6369.15 98.945E 20 1007.05 876.23 87.01 957.70 95.10 988.01 98.116E 20 6516.47 6385.47 97.99 6466.94 99.24 6451.62 99.00

13E 50 2406.36 2118.49 88.04 2365.78 98.31 2392.77 99.4414E 50 9119.03 8873.58 97.31 8943.94 98.08 8748.57 95.9415E 50 2586.37 2327.46 89.99 2503.61 96.80 2544.84 98.3916E 50 2720.43 2440.41 89.71 2650.76 97.44 2685.92 98.7317E 75 1744.83 1380.03 79.09 1689.93 96.85 1709.85 98.0018E 75 2371.49 2001.71 84.41 2276.31 95.99 2342.84 98.7919E 100 8661.81 8290.01 95.71 8574.33 98.99 8431.87 97.3520E 100 4039.49 3607.86 89.31 3931.79 97.33 3995.16 98.90

Avg. 89.86 97.52 98.38

circuit covering the depot and a subset of customers, whose total demand isequal to q.

The computational testing for the FSMF, is generally performed by usinga set of 20 symmetric instances proposed by Golden et al. [30] (in the follow-ing, referred to as the G20 instances), that are extensions to the FSMF ofclassical VRP test instances. In addition, some authors considered only the12 instances that are defined by using Euclidean distances (referred to as theG12 instances).

Table 2 reports a comparison on the quality of the lower bounds obtainedfor the FSMF by Golden et al. [30], Yaman [58] and Choi and Tcha [9].The table reports the lower bounds on the G12 set, which were used in bothYaman [58], and Choi and Tcha [9]. In particular, in order to make a faircomparison among the different lower bounds, we computed the lower boundof Golden et al. [30] using real-valued distance data, and the lower boundvalues produced by the best formulation proposed by Yaman [58]. In the table,columns labeled LB report the lower bound values, while columns labeled%LB report the percentage ratio of lower bounds computed with respect tothe best upper bound known in the literature, reported in column UB (i.e.,%LB = 100 LB/UB).

Table 2 shows that on average the Choi and Tcha [9] lower bound is best.On the G12 instances, the lower bound of Yaman [58] dominates the lowerbound of Golden et al. [30] and it is not dominated by the lower bound ofChoi and Tcha [9] (see instances 4E, 6E, 14E and 19E). Furthermore, thelower bound of Golden et al. [30] is not dominated by the lower bound ofChoi and Tcha [9] (see instance 14E).


4 Heuristic Algorithms

Due to the intrinsic difficulty of this family of routing problems, all solutionapproaches presented so far in the literature are heuristic algorithms. In addi-tion, they generally are adaptations or extensions of the methods proposed inthe last decades for the basic VRP variants, such as the CVRP and the VRPwith Time Windows.

In this section, we briefly review the main contributions to the approx-imate solution of Heterogeneous VRPs. We separately examine traditionalconstruction heuristics and metaheuristics and we provide, where available,information about their computational performance. To this end, for eachapproach we report the average percentage gaps of the published heuristicsolution values with respect to the current best-known values, as well as theaverage computing time required, expressed in seconds of various CPUs. Thepublished results are summarized in Tables 3 to 7.

The detailed results for the instances proposed by Golden et al. [30] arereported in two tables: Table 3 collects all the results obtained by consid-ering integer-valued distances, whereas Table 4 gives results for real-valueddistances.

As to the variants with vehicle-dependent routing costs, namely theHVRPD and the FSMD, the computational testing is generally performedby using an adaptation of eight instances of the G12 set, as proposed byTaillard [50], referred to as the T8 instance set. The detailed results for in-stance set T8 can be found in Tables 5 and 6. The data of all instances andmore details on the published results for the HVRP variants can be found athttp://or.ingce.unibo.it/research/hvrp.

Finally, SDVRP testing is performed by considering three set of test in-stances. The first one includes the 6 instances proposed by Nag [36]. Thesecond set is described in Chao et al. [8] and is made up by 12 new regularlyshaped instances and 5 instances derived from standard CVRP instances,whereas the last set, proposed by Cordeau and Laporte [12], includes 12 in-stances with time windows constraints. The detailed results for this problemare reported in Table 7.

Tables 3 to 7 report for each instance, the problem name, the number ofcustomers n, the number of vehicle types m and the value of the best solutionfound in the literature. The last two lines of each table report the averagepercentage gap and the number of the best-known solutions found by thecorresponding heuristic, respectively. Given the value z of a heuristic solutionand the best upper bound known zbest for the corresponding instance, thepercentage gap is computed as 100(z − zbest)/zbest.

4.1 Construction Heuristics

The first comprehensive study of Heterogeneous VRPs (especially the FSMF)is due to Golden et al. [30]. They presented both the formulation and the lower

http://or.ingce.unibo.it/research/hvrp


Table

3.

Com

pariso

nofth

ebes

t-know

nre

sults

for

the

FSM

Fw

ith

inte

ger

dista

nce

s.

Gold

enet

al.

(1984)

Ghey

sens

etal.

(1984)

Des

roch

ers

and

Salh

iand

Ver

hoog

(1991)

Rand

(1993)

Pro

ble

mn

mB

est

RO

S-γ

MG

T+

Or-

opt

LB

(5)+

VR

PPen

alty

RO

M-ρ

RO

M-γ

1S

12

3602

618

622

618

634

606

602

614

2S

12

3722

768

722

722

722

730

722

722

3E

20

5966

1009

966

968

966

990

980

1003

4E

20

36447

6937

6930

6451

6473

6547

6891

6447

5E

20

51013

1048

1013

1030

1023

1040

1032

1015

6E

20

36516

6522

6974

6518

6953

6517

6517

6516

7S

30

57354

7452

7389

7354

7372

7421

7444

7402

8S

30

42362

2468

2367

2362

2370

2387

2389

2367

9S

30

52209

2266

2220

2261

2226

2231

2231

2209

10S

30

42370

2424

2370

2388

2371

2393

2387

2377

11S

30

44763

4953

4763

4788

4805

4862

4911

4819

12S

30

64092

4221

4136

4133

4248

4254

4248

4092

13E

50

62437

2566

2438

–2437

2525

2508

2493

14E

50

39132

9178

9132

9156

9132

9155

9196

9153

15E

50

32621

2763

2640

2621

2640

2622

2642

2623

16E

50

32765

2894

2822

––

2809

2868

2765

17E

75

41767

1958

1783

––

1877

1877

1767

18E

75

62432

2520

2432

––

2489

2489

2439

19E

100

38700

8741

8721

––

8700

8700

8751

20E

100

34187

4293

4195

––

4248

4280

4187

Aver

age

%3.7

91.2

10.6

81.3

51.7

02.0

10.5

9#

ofbes

tso

l.0

74

41

38


Table

4.

Com

pariso

nofth

ebes

t-know

nre

sults

for

the

FSM

Fw

ith

realdista

nce

s.

Osm

an

and

Salh

i(1

996)

Tailla

rdG

endre

au

Liu

and

Ren

aud

and

Wass

an

and

Choiand

(1999)

at

al.

(1999)

Shen

(1999)

Boct

or

(2002)

Osm

an

(2002)

Tch

a(2

006)

Pro

ble

mn

mB

est

MR

PE

RT

TSV

FM

1S

12

3602.0

0606.0

0602.0

0–

–602.0

0602.0

0602.0

0–

2S

12

3722.0

0722.0

0722.0

0–

–722.0

0722.0

0722.0

0–

3E

20

5961.0

3971.9

5971.2

4961.0

3961.0

3972.0

4963.6

1961.0

3961.0

34E

20

36437.3

36447.8

06445.1

06437.3

36437.3

36444.7

26437.3

36437.3

36437.3

35E

20

51007.0

51015.1

31009.1

51008.5

91007.0

51014.0

51007.9

61007.0

51007.0

56E

20

36516.4

76516.5

66516.5

66516.4

76516.4

76516.4

76537.7

46516.4

76516.4

77S

30

57273.0

07377.0

07310.0

0–

–7313.3

37346.0

07273.0

0–

8S

30

42346.0

02352.0

02348.0

0–

–2347.0

02347.0

02346.0

0–

9S

30

52209.0

02209.0

02209.0

0–

–2214.9

62211.0

02209.0

0–

10S

30

42355.0

02377.0

02363.0

0–

–2368.0

02362.0

02355.0

0–

11S

30

44755.0

04787.0

04755.0

0–

–4777.7

74761.0

04755.0

0–

12S

30

64087.0

04092.0

04092.0

0–

–4101.0

04092.0

04087.0

0–

13E

50

62406.3

62462.0

12471.0

72413.7

82408.4

12465.0

32406.4

32422.1

02406.3

614E

50

39119.0

39141.6

99125.6

59119.0

39119.0

39132.0

09122.0

19119.8

69119.0

315E

50

32586.3

72600.3

12606.7

22586.3

72586.3

72608.0

02618.0

32586.3

72586.3

716E

50

32720.4

32745.0

42745.0

12741.5

02741.5

02808.9

62761.9

62730.0

82720.4

317E

75

41744.8

31766.8

11762.0

51747.2

41749.5

01806.0

51757.2

11755.1

01744.8

318E

75

62371.4

92439.4

02412.5

62373.6

32381.4

32415.9

42413.3

92385.5

22371.4

919E

100

38661.8

18704.2

08685.7

18661.8

18675.1

68684.0

08687.3

18665.7

48664.2

920E

100

34039.4

94166.0

34188.7

34047.5

54086.7

64148.0

44094.5

44061.6

44039.4

9

Aver

age

%0.9

00.6

80.1

40.2

40.9

60.4

70.1

40.0

04

#ofbes

tso

l.2

46

63

313

11


bounding procedure described in Section 3. Moreover, Golden et al. [30] pro-posed constructive heuristics that adapted to the FSMF the savings and giant-tour based approaches for VRP (see Clarke and Wright [11] and Beasley [4],respectively).

As to savings approaches, four different expressions were proposed to in-corporate the heterogeneous fleet concept in the savings computation andenhance the algorithm performance. As reported by Golden et al. [30], thedirect adaptation of the Clarke and Wright [11] algorithm (CW) producessolutions with an average percentage gap equal to 14.31% with respect tothe current best-known integer solutions for the G20 set. Combined Savings(CS) includes in the savings formula the variation of the fixed costs associ-ated with the route merging: the resulting average improvement on CW is4.25%. Optimistic and Realistic Opportunity Savings (denoted as OOS andROS, respectively) add to CS two different terms that favor the opportunityof having residual capacity on the vehicle used to service the merged routes.In this case, the improvement to CW is 1.15% and 5.75%, respectively. Fi-nally, ROS-γ adds to ROS the route shape parameter proposed by Gaskell[23], and Yellow [59]. This latter approach is used in a multi-start fashion, byconsidering 31 different values of γ parameter between 0 and 3. The result-ing percentage gap with respect to the best-known solution values is equalto 3.79%, corresponding to an improvement in CW as high as 8.18% (seeTable 3).

Desrochers and Verhoog [17] further extended savings-based approachesto the FSMF by adopting the matching-based savings heuristic proposed byAltinkemer and Gavish [1] for the VRP. The basic savings expression consid-ers the cost difference of the TSPs associated with the routes involved in thecurrent merging, rather than the simpler classical one. Various extensions ofthe savings formula, similar to those proposed by Golden et al. [30], are con-sidered. At each iteration, the pair of routes to be merged is chosen as thatcorresponding to the largest savings in the solution of a matching problemover the current savings matrix. As reported in Table 3, the two proposedsavings expressions, called ROM-ρ and ROM-γ, produced solutions with anaverage percentage gap equal to 1.70% and 2.01% on the G20 instances, re-spectively, corresponding to improvements to the CW results by 10.05% and9.80%, respectively.

The giant-tour based approaches proposed by Golden et al. [30] are two-phase algorithms. First a TSP over all the nodes is, heuristically solved toobtain an uncapacitated tour. In the second phase, this tour is partitionedinto the final capacitated set of routes. Two different ways of defining theinitial giant tour were adopted, namely with and without the depot in thetour. In the latter case, which on average produced slightly better results, thepartitioning is obtained by solving a suitably defined shortest path problem.Also in this case, a multi-start framework is obtained by applying the parti-tioning step to different initial tours. The best obtained solutions were refinedby using 2-opt and OR-opt procedures (see Lin [33] and Or [40]). As shown in


Table 3, this approach, called MGT+Or-opt, produced solutions within 1.21%from the best integer known ones on the G20 test instances.

A different giant-tour based approach for the FSMF was introduced byGheysens et al. [26], in which a penalty function that allows for a limitedcapacity violation in the tour partitioning step is used, so as to favor thepresence in the routes of some empty space that may be possibly exploited inthe refining step. This method has been tested on the 16 smallest instancesof the G20 set allowing for an average percentage gap of 1.35% (see column“Penalty” of Table 3). More recently, Teodorovic et al. [54] used a giant-tourapproach to solve the stochastic version of the HFVRP, where customer’sdemand may vary stochastically and the initial tour is obtained through theBartholdi and Platzman [3] spacefilling curves heuristic for the TSP.

Gheysens et al. [27] and Gheysens et al. [28] developed an extension tothe FSMF of the well-known Fisher and Jaikumar [21] algorithm for VRP,where the initial fleet is determined through the lower bounding procedure ofGolden et al. [30]. Gheysens et al. [27] report an average percentage gap of0.68% on the 15 smallest instances of the G20 set (see column “LB(5)+VRP”of Table 3).

Ferland and Michelon [20] introduced three different heuristic methods tosolve the FSMFD with Time Windows. The first one directly uses the three-index mathematical formulation of the problem and simplifies it by discretiz-ing the time windows, so as to obtain a possibly solvable integer problem. Thetwo remaining heuristics are construction approaches in which the solution isobtained by iteratively assigning customers to the routes through the solutionof either a matching or a transportation problem. No computational testingof the proposed methods was reported by the authors.

Salhi and Rand [47] described a heuristic for the FSMF that starts from asolution obtained by heuristically solving a VRP with a single vehicle capacity,selected among the available ones. This starting solution is then iterativelyimproved by several procedures that attempt, in turn, to change the vehicletype assigned to each route, merging or removing routes and moving customersfrom one route to another. The average percentage gap of this method is equalto 0.59%, whereas the average computing time of a VAX 8700 computer isequal to 2 seconds. Osman and Salhi [41] extended the heuristic proposedby Salhi and Rand [47] by (i) enlarging the neighborhood size by allowingmoves which can lead to a utilization of a larger-sized vehicle and (ii) usinga multi-start technique to restart the heuristic with the best solution foundat the end of the previous iteration. Osman and Salhi [41] used real-valueddistances, hence their results cannot be compared directly with those obtainedby Salhi and Rand [47]. The average percentage gap is equal to 0.90% (seecolumn “MRPERT” of Table 4) and the average computing time is equal to5.65 seconds on a VAX 4500 computer.

Taillard [50] proposed a heuristic column generation method for the FSMF,the FSMD and the HVRPD. In this approach, a large set of routes is initiallyobtained by solving homogeneous fleet VRPs for each vehicle type. Then the


final set of routes is selected by solving a set partitioning problem to ensurethat each customer is served by exactly one route. This method, tested for theFSMF on the 8 largest instances in G20, produces very good results: in factthe average percentage gap is equal to 0.14% and the average computing timeover five runs is 2648 seconds on a Sun Sparc work station (50 MHz). Theresults obtained with the T8 set for the HVRPD, show an average percentagegap of 0.93%, and an average computing time around 2000 seconds on thesame machine. For the FSMD, the percentage gap is equal to 0.77%, requiringabout the same amount of time. A similar approach is used by Renaud andBoctor [44] to solve the FSMF, where the set of routes is obtained by usingproblem-specific extensions of the sweep algorithm for the VRP (see Gillettand Miller [29]). The average percentage gap of the best solutions found outof several runs on set G20 is equal to 0.47%.

More recently, Choi and Tcha [9] proposed a heuristic approach based on acolumn generation technique, to derive high quality heuristic solutions for theFSMD and the FSMF. More precisely, they (i) computed a lower bound to theFSMD and to the FSMF as the optimal solution value of the LP relaxation ofthe covering relaxation of formulation F2 and (ii) solved, using a branch-and-bound algorithm, a restricted Set Partitioning problem obtained by limitingthe set of all feasible routes to the set of routes generated by the columngeneration algorithm in computing the lower bound. The FSMD method hasbeen tested on the T8 instances obtaining the best-known solutions, requiringon average 81 seconds on a Pentium IV 2.6GHz processor. With respect tothe FSMF, this method produces on the G12 instances an average percentagegap equal to 0.004%, with an average computing time of 150 seconds on thesame machine.

We now briefly examine the heuristics proposed for the time window vari-ant of the FSMF, denoted by FSMFTW. We consider the two constructionheuristics that were developed by Liu and Shen [34]. They proposed a two-phase algorithm in which an initial solution is obtained through a savingsalgorithm that evaluates the insertion of complete routes in all possible in-sertion places of the other routes, and also takes into account the vehiclescheduling component associated with the time windows. In the second phase,an improvement procedure is then applied to several best fleet solutions foundduring the first stage: intra-route customer shifting and inter-route customerexchanges are performed. Computational results were performed on a set of168 test instances, hereafter called LS168, derived from the Solomon [49]VRPTW test set. The proposed algorithm was also used to solve the G20FSMF instances and obtained an average percentage gap equal to 0.96%.

Dullaert et al. [18] extended to the FSMFTW the sequential insertion algo-rithm proposed by Solomon [49] for the VRPTW. In particular, the adoptedinsertion criterion combines standard insertion cost evaluations used in theVRPTW with a new term that incorporates the Golden et al. [30] modifiedsaving expressions. Dullaert et al. [18] reported in their tables only the com-ponent of the objective function relative to the schedule times. Thus, their


Table 5. Comparison of the best-known results for HVRPD with real distances.

Taillard Tarantilis Tarantilis Li et al.(1999) et al.(2003) et al.(2004) (2006)

Problem n m Best LBTA

13E 50 6 1517.84 1518.05 1519.96 1519.96 1517.8414E 50 3 607.53 615.64 612.51 611.39 607.5315E 50 3 1015.29 1016.86 1017.94 1015.29 1015.2916E 50 3 1144.94 1154.05 1148.19 1145.52 1144.9417E 75 4 1061.96 1071.79 1071.67 1071.01 1061.9618E 75 6 1823.58 1870.16 1852.13 1846.35 1823.5819E 100 3 1117.51 1117.51 1125.64 1123.83 1120.3420E 100 3 1534.17 1559.77 1558.56 1556.35 1534.17

Average % 0.93 0.79 0.62 0.03# of best sol. 1 0 1 7

results cannot be compared with the results of the heuristics reporting thefixed cost component.

The first attempt to solve the SDVRP is due to Nag et al. [37] who de-veloped four different heuristics. The first one, considers one vehicle type ata time and constructs the routes by means of a sweep algorithm. In this step,route capacity is slightly enlarged. Later, the route feasibility is obtained bymoving customers to other compatible routes by using a savings criterion.The remaining three heuristics are adaptations of the Fisher and Jaikumar[21] approach differing in the seed selection criteria adopted in each of them.The average percentage gap of the best of these heuristics on the six Naginstances is equal to 14.64%.

Chao et al. [8] proposed a heuristic for the SDVRP in which an initialassignment of customers to vehicle types is performed by solving a relaxedILP with the objective of minimizing the total load fraction for each vehicletype, and vehicle routes are determined through a savings algorithm. Then,seed customers are extracted from the routes by considering geographical andload considerations. The relaxed ILP is then run again, with the additionalobjective of minimizing the routing cost in the assignment of each customerto the seeds and again routes are determined by using a savings algorithm.This process is iterated by perturbing the choice of seed points and by usingthree local search improvement procedures. The average percentage gap ofthis algorithm over the Nag and Chao instances is equal to 5.76 % with anaverage computing time of 449 seconds on a 166MHz Pentium PC.

4.2 Metaheuristics

Since the 1990s metaheuristic approaches started to be applied to the solutionof heterogeneous VRPs as well.


Table 6. Comparison of the best-known results for the FSMD with real distances.

Taillard Gendreau Wassan and Choi and(1999) et al.(1999) Osman (2002) Tcha (2006)

Problem n m Best

13E 50 6 1491.86 1494.58 1491.86 1499.69 1491.8614E 50 3 603.21 603.21 603.21 608.57 603.2115E 50 3 999.82 1007.35 999.82 999.82 999.8216E 50 3 1131.00 1144.39 1131.00 1131.00 1131.0017E 75 4 1038.60 1044.93 1038.60 1047.74 1038.6018E 75 6 1801.40 1831.24 1801.40 1814.11 1801.4019E 100 3 1105.44 1110.96 1105.44 1108.98 1105.4420E 100 3 1530.43 1550.36 1541.19 1530.43 1530.43

Average % 0.77 0.09 0.41 0.00# of best sol. 1 7 3 8

One of the first such algorithms is the genetic approach proposed by Ochiet al. [38] for the FSMF that creates an initial population by means of asweep-based heuristic. The same algorithm is tested in a parallel frameworkin Ochi et al. [39], but both papers do not report details of the computationaltesting.

Tabu search approaches for this problem family were developed by Osmanand Salhi [41], Gendreau et al. [25], and Wassan and Osman [56]. All thesealgorithms were extensions to the FSMF and to the HVRPD of approachesalready proposed for the VRP, using the problem-specific feasibility check andobjective function evaluation.

In particular, Osman and Salhi [41] used a 1-interchange neighborhoodtogether with a simple tabu list mechanism, whereas Wassan and Osman[56] mix several effective strategies to improve the overall quality of the al-gorithm. Reactive and variable-neighborhood search mechanisms based on λ-interchange neighborhoods are combined with efficient data management tech-niques for handling tabu lists and hashing functions. Finally, the tabu searchof Gendreau et al. [25] embeds a classical algorithm based on the GENIUSneighborhoods with the adaptive memory mechanism of Rochat and Taillard[46]. The Osman and Salhi [41] algorithm performance on the G20 instancesshows an average percentage gap of about 0.68% (see column “TSVFM” ofTable 4) with respect to the best-known solutions. The Gendreau et al. [25]algorithm was tested on the G12 test instances, obtaining an average percent-age gap of 0.24%, with an average computing time of 765 seconds on a SunSparcstation 10. The algorithm was also tested on the T8 instances, obtainingan average percentage gap of 0.09% within an average computing time of 1151seconds. The Wassan and Osman [56] tabu search produces good results: onthe G20 test bed, the average percentage gap is 0.41% and on T8 this gap is0.47%, but the average computing time increases to 1215 seconds and to 2098seconds, respectively, using a Sun Sparc server 1000.


Table 7. Comparison of the best-known results for the SDVRP and the SDVRPTW.

Nag et al. Chao et al. Cordeau & Pisinger &(1988) (1999) Laporte Ropke

Problem n m Best (2001) (2007)

1Nag 50 3 640.32 746.4 668.58 642.66 640.322Nag 50 2 598.1 624.3 610.04 598.1 598.13Nag 75 3 957.04 1169.6 1002.45 959.36 957.044Nag 75 2 854.43 944.4 900.93 854.43 854.435Nag 100 3 1003.57 1203.4 1071.54 1020.22 1003.576Nag 100 2 1028.52 1175.2 1080.89 1036.02 1028.52

7Chao 27 3 391.3 – 391.3 391.3 391.38Chao 54 3 664.46 – 664.46 664.46 664.469Chao 81 3 948.23 – 948.23 948.23 948.23

10Chao 108 3 1218.75 – 1252.57 1223.88 1218.7511Chao 135 3 1463.33 – 1526.6 1464.98 1463.3312Chao 162 3 1678.4 – 1854.82 1695.67 1678.413Chao 54 3 1194.18 – 1205.53 1196.73 1194.1814Chao 108 3 1960.62 – 2092.68 1962.66 1960.6215Chao 162 3 2685.09 – 2966.77 2751.45 2685.0916Chao 216 3 3396.36 – 3710.96 3491.18 3396.3617Chao 270 3 4085.61 – 4441.53 4230.96 4085.6118Chao 324 3 4755.5 – 5085.28 4929.71 4755.519Chao 100 3 846.07 – 878.58 850.39 846.0720Chao 150 3 1030.78 – 1126.8 1046.14 1030.7821Chao 199 3 1271.75 – 1420.85 1337.83 1271.7522Chao 120 3 1008.71 – 1150.13 1012.87 1008.7123Chao 100 3 803.29 – 837.98 818.75 803.29

1Cordeau 48 4 1380.77 – – 1384.15 1380.772Cordeau 96 4 2311.54 – – 2320.97 2311.543Cordeau 144 4 2602.13 – – 2623.31 2602.134Cordeau 192 4 3474.01 – – 3500.79 3474.015Cordeau 240 4 4416.38 – – 4479.34 4416.386Cordeau 288 4 4444.52 – – 4546.79 4444.527Cordeau 72 6 1889.82 – – 1955.11 1889.828Cordeau 144 6 2977.50 – – 3082.32 2977.59Cordeau 216 6 3536.20 – – 3664.22 3536.2

10Cordeau 288 6 4648.76 – – 4739.43 4648.7611Cordeau 1008 4 12719.65 – – 13227.96 12719.6512Cordeau 720 6 9388.07 – – 9621.99 9388.07

Average % 14.64 5.76 1.48 0# of best sol. 0 3 5 35


Tarantilis et al. [51, 52] developed two list-based threshold accepting meta-heuristics for the HVRPD: both methods start with an initial solution gen-erated by a construction heuristic, followed by an iterative threshold accept-ing phase. In this phase, a solution is iteratively randomly generated in theneighborhood of the current one and its threshold value, defined as the rela-tive improvement with respect to the current solution value, is computed. Thenew solution is accepted as new current one by comparing its threshold valuewith those stored in a list which includes the M best threshold values foundduring the search. Different ways of updating the threshold list are consideredin the two papers. The method in Tarantilis et al. [51] was tested on the T8test instances and produces results on average 0.79% (see column “LBTA”of Table 5) from the best-known solutions within an average of 223 secondson a Pentium III, 550 MHz PC. The results using the method in Tarantiliset al. [52] are slightly better on the T8 instances with an average gap of 0.62%within an average computing time of 607 seconds on a Pentium II/400 PC.

Li et al. [32] considered a similar approach based on a record-to-recordalgorithm: a deterministic variant of the simulated annealing metaheuristic.Their method was tested on T8 instances and on five large-scale instanceswith 200 to 360 customers from Golden et al. [31]. On the T8 problems, theyobtained an average percentage gap of 0.03% and an average computing timeof 286 seconds on an Athlon 1 GHz Pc.

Dell’Amico et al. [16] proposed a ruin-and-recreate approach for theFSMFTW. In particular, a parallel insertion procedure is used both to obtainthe initial solution and to possibly complete partial ones that are producedduring the ruin step. This step is performed by selecting a target route to beruined according to a criterion which is inspired by one used in metaheuristicsthat solve bin packing problems. The proposed approach outperformed boththe Liu and Shen [34] and Dullaert et al. [18] algorithms on the LS168 testinstances.

Braysy et al. [7] proposed a new deterministic annealing metaheuristic forthe FSMFTW. The metaheuristic is based on three phases: (i) initial solutionsare generated by means of a savings-based heuristic combining diversificationstrategies with learning mechanisms, (ii) an attempt is made to reduce thenumber of routes in the initial solution with a new local search procedure and(iii) the solution from the second phase is further improved by a set of fourlocal search operators that are embedded in a deterministic annealing frame-work to guide the improvement process. The computational experiments onthe LS168 benchmark instances show that the suggested method outperformsthe previously published results and improves almost all the best-known so-lutions.

Finally, we can mention two metaheuristics for the SDVRP both basedupon its transformation into more general routing problems. The first oneis the tabu search procedure proposed by Cordeau and Laporte [12] for thevariant including time windows constraints. The problem is reduced to a Pe-riod Vehicle Routing Problem (PVRP) by associating each vehicle type to a

the vehicle routing problem: latest …...the vehicle routing problem: latest advances and new...

Documents