European Journal of Operational Research 240 (2015) 382–392

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European Journal of Operational Research

journal homepage: www.elsevier .com/locate /e jor

Discrete Optimization

The mixed capacitated general routing problem under uncertainty

http://dx.doi.org/10.1016/j.ejor.2014.07.0230377-2217/� 2014 Elsevier B.V. All rights reserved.

⇑ Corresponding author.E-mail addresses: beraldi@deis.unical.it (P. Beraldi), mebruni@deis.unical.it

(M.E. Bruni), demetrio.lagana@unical.it (D. Laganà), musmanno@unical.it(R. Musmanno).

Patrizia Beraldi, Maria Elena Bruni, Demetrio Laganà ⇑, Roberto MusmannoDepartment of Mechanical, Energy and Management Engineering, University of Calabria, 87036 Arcavacata di Rende, CS, Italy

a r t i c l e i n f o a b s t r a c t

Article history:Received 29 July 2013Accepted 20 July 2014Available online 30 July 2014

Keywords:Routing problemMixed graphNeighborhood searchProbabilistic constraints

We study the General Routing Problem defined on a mixed graph and with stochastic demands. The prob-lem under investigation is aimed at finding the minimum cost set of routes to satisfy a set of clientswhose demand is not deterministically known. Since each vehicle has a limited capacity, the demanduncertainty occurring at some clients affects the satisfaction of the capacity constraints, that, hence,become stochastic. The contribution of this paper is twofold: firstly we present a chance-constrainedinteger programming formulation of the problem for which a deterministic equivalent is derived. Theintroduction of uncertainty into the problem poses severe computational challenges addressed by thedesign of a branch-and-cut algorithm, for the exact solution of limited size instances, and of a heuristicsolution approach exploring promising parts of the search space. The effectiveness of the solutionapproaches is shown on a probabilistically constrained version of the benchmark instances proposedin the literature for the mixed capacitated general routing problem.

� 2014 Elsevier B.V. All rights reserved.

1. Introduction

An important operative issue in the context of the distributivelogistics consists in planning the delivery routes performed by afleet of vehicles to satisfy the requests of a set of elements of a net-work, namely required vertices, edges and arcs. In mathematicalterms, the problem is modeled as Mixed Capacitated General Rout-ing Problem (MCGRP): it basically consists in finding a set of routeson a mixed graph, beginning and ending at the same vertex(depot), with minimum total cost, satisfying demands located atlinks and vertices and with a capacity restriction on the demandsatisfied by each route. The MCGRP generalizes many vehicle rout-ing problems that have been widely studied in the last forty yearsand for which hundreds of papers have been written, either to giveexact or heuristic procedures for their resolution or to providelower bounds. Despite the practical importance of the mixed gen-eral routing problem, relatively few studies have been publishedon it. Most works deal with the uncapacitated case. Corberán,Letchford, and Sanchis (2001, 2003, 2005) studied the feasiblepolyhedron starting from an integer programming formulationsolved through an efficient cutting-plane algorithm. Blais andLaporte (2003) proposed a different approach based on the trans-formation of the original problem into an equivalent Traveling

Salesman Problem or Rural Postman Problem which are solved inturn through available exact algorithms.

With respect to the capacitated case, Bosco, Laganà, Musmanno,and Vocaturo (2013) proposed a novel integer programming for-mulation and a branch-and-cut algorithm (B&C) where surrogateinequalities, introduced for the Capacitated Arc Routing Problem,are extended to the MCGRP polyhedron.

The aim of this paper is the introduction of the uncertainty issuein this latter and more involved case, where each vehicle has a lim-ited capacity. In effect, most of the real-world applications mod-eled as MCGRP are characterized by some uncertainty whichaffects the customers’ demand. For example, the operational planof pickup routes in solid waste collection systems implies model-ing the service by the means of required arcs or edges wheneverthe collection points are distributed along the streets, while somevertices are required if the collection is concentrated around spe-cific points (e.g., hospitals, schools, and supermarkets). For gener-ality, we shall also assume that the requests of random elementsmight be correlated to faithfully represent real situations. Forexample, in the garbage collection, the geographical nearness ofsome customers within the same regional district or along thesame street suggests to consider a statistical correlation amongtheir garbage productions.

Following these considerations, we bring the stochasticity intothe MCGRP by adopting the paradigm of the probabilistic con-straints defined within the general Stochastic Programming (SP)framework (Birge & Louveaux, 1997). This modeling paradigm isappropriate in many situations, where an operational plan is

P. Beraldi et al. / European Journal of Operational Research 240 (2015) 382–392 383

periodically updated over a long planning horizon, and hence,becomes crucial to design a set of a priori routes that will coverthe uncertain requests with a high reliability level. In particular,we formally introduce a stochastic formulation of the MCGRPwhere the stochastic capacity constraints are re-formulated interms of probabilistic constraints. The explicit inclusion of theuncertainty within an already proved NP-hard problem, posesadditional challenges, calling for the design of tailored solutionapproaches. This represents the second core contribution of thepresent paper. We develop a branch-and cut (B&C) algorithm forsolving small instances and we design a large neighborhood searchheuristic for the solution of instances of larger size where the B&Calgorithm is used, in turn, to perform an exact local search on aportion of the overall feasible region.

To put our contribution in the right perspective, we should pre-cise that the adoption of the SP framework to model routing prob-lems under uncertainty is not completely new. For an extensivesurvey, the readers are referred to Dror and Trudeau (1986) andGendreau, Laporte, and Seguin (1996).

Within this stream, most of the contributions rely on the two-stage paradigm and different recourse policies have been proposedin the literature. Bertsimas (1992) and Bertsimas and Simchi-Levi(1994) focused their researches on simple recourse policies thatare separable by vehicle. A different policy has been presented byAk and Erera (2007), that proposed a two-vehicle sharing recoursepolicy. During the last decades various heuristic and exact optimi-zation approaches have been proposed and analyzed for construct-ing a set of tours minimizing expected costs given this recoursepolicy. Gendreau, Laporte, and Seguin (1995) proposed an exactsolution for an a priori optimization model based on an integerL-shaped method. Laporte, Louveaux, and van Hamme (2002) pre-sented an improved method where strong lower bounds at the rootnode contribute significantly to speed up the solution times.Gendreau, Laporte, and Seguin (1996) applied local search con-cepts embedded into a tabu search scheme to solve the a priorimodel presented in Gendreau et al. (1995). More recently,Laporte, Musmanno, and Vocaturo (2010) studied the capacitatedarc-routing problem with stochastic demands in the context of gar-bage collection and proposed an adaptive large neighborhoodsearch heuristic.

Scant attention has been devoted to the formulation of routingproblems with probabilistic constraints. Stewart and Golden(1983) presented a model able to find minimum cost routes witha threshold constraint on the probability of a route failure, whereasLaporte, Louveaux, and Mercure (1989) proposed a chance-con-strained model for location-routing problems. A chance con-strained version of the vehicle routing problem, solved tooptimality by algorithms similar to those developed for the deter-ministic case, has been presented in Dror, Laporte, and Louveaux(1993).

Besides the stochastic programming approach, the robust opti-mization framework has been adopted to deal with routing prob-lems involving uncertain parameters where the probabilitydistributions are not known. Amongst the recent contributions,we cite Sungur, Ordóñez, and Dessouky (2008), who analyze thecase of uncertain customer demands and travel times. The goal isto determine vehicle routes which satisfy the capacity constraintsand the specified time windows if all the uncertain parametersattain the worst case realizations simultaneously. The problemcan be simplified to a deterministic model, which is attractive froma computational standpoint. Gounaris, Wiesemann, and Floudas(2013) (see also the references therein) investigate the case ofcapacitated vehicle routing problem. Robust optimization counter-parts of several deterministic formulations of the problem arederived and numerically compared. Robust rounded capacityinequalities are developed, which can be separated efficiently for

two broad classes of demand supports. Finally, the authors analyzethe relation between the robust models and the chance con-strained counterparts. Lee, Lee, and Park (2004) considered twotypes of uncertainty sets for the possible realizations of traveltimes and demands. The authors propose a column generationalgorithm which encapsulates the robustness in the pricing prob-lem cast as a robust version of the shortest path with resourceconstraints.

In this paper we study the Mixed Capacitated General RoutingProblem with Probabilistic Constraints MCGRPPC. In Section 2, weintroduce the problem and we provide a chance-constrained inte-ger linear programming formulation for the MCGRPPC. In Section 3we define the B&C algorithm for solving small instances of theMCGRPPC. In Section 4 we present a tailored heuristic search tosolve larger MCGRPPC instances. In Section 5, we present theresults of our computational study. Finally, in Section 6, we giveour conclusions and discuss future perspectives in this area.

2. Problem description

The MCGRPPC is defined over a mixed graph G ¼ ðV ;A; EÞ, whereV ¼ f1; . . . ;ng represents the set of vertices, where vertex 1represents the depot, and A ¼ fði; jÞ # V � Vg is the set of arcs,whereas E ¼ fði; jÞ # V � V : i < jg is the set of edges.

In the following, we shall denote by L ¼ A [ E, the set of linksand we shall indicate by cij a non-negative cost coefficient associ-ated with each link ði; jÞ. We assume that the service activity mayoccur at some vertices VR # V , named required vertices, arcsAR # A and/or edges ER # E, named required arcs and requirededges, respectively. Thus, LR ¼ AR [ ER denotes the set of requiredlinks of G and all the required vertices and links will be referredto as required elements and indicated by R.

For each subset S � V of vertices, or its complementary setSðS ¼ V n SÞ, we define the following sets:

(a) dþðSÞ ¼ fði; jÞ 2 A : i 2 S ^ j 2 Sg,(b) d�ðSÞ ¼ fði; jÞ 2 A : i 2 S ^ j 2 Sg,(c) dþAR

ðSÞ ¼ fði; jÞ 2 AR : i 2 S ^ j 2 Sg,(d) d�AR

ðSÞ ¼ fði; jÞ 2 AR : i 2 S ^ j 2 Sg,(e) dðSÞ ¼ fði; jÞ 2 E : i 2 S ^ j 2 S; or i 2 S ^ j 2 Sg,(f) dER ðSÞ ¼ fði; jÞ 2 ER : i 2 S ^ j 2 S; or i 2 S ^ j 2 Sg,(g) dLðSÞ ¼ dþðSÞ [ d�ðSÞ [ dðSÞ,(h) dLR ðSÞ ¼ dþAR

ðSÞ [ d�ARðSÞ [ dER ðSÞ,

(i) SR ¼ S \ VR,(j) ARðSÞ ¼ fði; jÞ 2 AR : i 2 S ^ j 2 Sg,(k) ERðSÞ ¼ fði; jÞ 2 ER : i 2 S ^ j 2 Sg,(l) RðSÞ ¼ ARðSÞ [ ERðSÞ [ SR.

The previous notation remains valid as long as S is replaced by v,and S by v , or V n fvg. We denote by GR the graph induced on G byall the required links and vertices. Generally, this graph is non-con-nected. The vertex sets corresponding to connected components ofGR are called R-sets. The subgraphs of G induced by the R-setsdefine the so-called R-connected components of G. An isolatedrequired vertex represents itself an R-connected component of G.

In real settings, the service demand associated with all but asubset of required elements is seldom, if ever, known at the timeroutes have to be designed. Thus, with the aim of more realisticallymodeling general routing problems, one should deal with the sto-chastic nature of the input parameters. In the following, we shallassume that the set of required elements is partitioned into twosubsets RC and RU to differentiate between elements with knownand uncertain demands, respectively. Following the stochastic pro-gramming modeling framework, we shall assume that the uncer-tain demands are represented in terms of random variables

384 P. Beraldi et al. / European Journal of Operational Research 240 (2015) 382–392

defined on a given probability space ðX; F;PÞ. Thus, we shall denoteby diðxÞ and dijðxÞ the random demands associated with the ‘‘sto-chastic’’ required vertices and links, respectively. Deterministicdemands are denoted analogously but without explicating thedependence on x. The introduction of the random variables withinthe mathematical formulation makes the classical criterion of fea-sibility, introduced for the deterministic case, no longer valid, sinceit may happen that the stochastic capacity constraints are satisfiedfor some elements x 2 X, but not for all. To deal with this issue, weadopt the paradigm of probabilistic constraints, determining a setof a priori routes of minimum cost satisfying the random demandswith a fixed reliability level a. In the next subsection, we shallintroduce the mathematical formulation.

2.1. The probabilistically constrained formulation

For each required link ði; jÞ 2 LR and each vehicle k ¼ 1; . . . ;m,we denote by xk

ij the binary variable equal to 1 if ði; jÞ is servicedby vehicle k which travels from vertex i to vertex j and 0 otherwise,and by yk

ij the non-negative variable representing the number ofdeadheading traversals from vertex i to vertex j by vehicle k, i.e.,the number of times that link ði; jÞ 2 LR is traversed from vertex ito vertex j by vehicle k without being serviced.

Moreover, for each required vertex i 2 VR and each vehicle k, wedenote by xk

i the binary variable equal to 1 if i is serviced by k and 0otherwise.

The following constraints hold:

Xm

k¼1

ðxkij þ xk

jiÞ ¼ 1; ði; jÞ 2 ER ð1aÞ

Xm

k¼1

xkij ¼ 1; ði; jÞ 2 AR ð1bÞ

Xm

k¼1

xki ¼ 1; i 2 VR ð1cÞ

PX

ði;jÞ2ER\RU

dijðxÞ xkijþxk

ji

� �þ

Xði;jÞ2AR\RU

dijðxÞxkijþ

Xi2VR\RU

diðxÞxki

þX

ði;jÞ2ER\RC

dij xkijþxk

ji

� �þ

Xði;jÞ2AR\RC

dijxkijþ

Xi2VR\RC

dixki 6Q

!Pa; k¼1; .. . ;m

ð1dÞ

Xj:ði;jÞ2dþ

ARðiÞ

xkij þ

Xj:ði;jÞ2dþðiÞ

ykij �

Xj:ðj;iÞ2d�AR

ðiÞxk

ji �X

j:ðj;iÞ2d�ðiÞyk

ji ¼X

j:ði;jÞ2dERðiÞ

xkji

þX

j:ði;jÞ2dðiÞyk

ji �X

j:ði;jÞ2dERðiÞ

xkij �

Xj:ði;jÞ2dðiÞ

ykij; k ¼ 1; . . . ;m; i 2 V ð1eÞ

Xði;jÞ2dþ

ARðSÞ

xkijþ

Xðj;iÞ2d�AR

ðSÞxk

jiþX

ði;jÞ2dERðSÞðxk

ijþxkjiÞþ

Xði;jÞ2dþðSÞ

ykijþ

Xðj;iÞ2d�ðSÞ

ykji

þXði;jÞ2dðSÞ

ðykijþyk

jiÞP

2ðxkuvþxk

vuÞ; ðu;vÞ2ERðSÞ;

2xkuv ; ðu;vÞ2ARðSÞ;

2xkh; h2SR;

8>>>><>>>>:

k¼1; . . . ;m; S # V nf1g ð1fÞ

xkij 2 f0;1g; k ¼ 1; . . . ;m; ði; jÞ 2 AR [ ER ð1gÞ

xkji 2 f0;1g; k ¼ 1; . . . ;m; ði; jÞ 2 ER ð1hÞ

ykij 2 Zþ; k ¼ 1; . . . ;m; ði; jÞ 2 A [ E ð1iÞ

ykji 2 Zþ; k ¼ 1; . . . ;m; ði; jÞ 2 E ð1jÞ

xki 2 f0;1g; k ¼ 1; . . . ;m; i 2 VR: ð1kÞ

Constraints (1a)–(1c) ensure that each request is servicedexactly once by exactly one vehicle (assignment constraints). Con-straints (1d) are the probabilistic capacity constraints imposing, foreach route, that the probability of not exceeding the vehicle capac-ity should be greater than or equal to a reliability level a. Inequal-ities (1e) represent flow constraints. They model the symmetryconditions at each vertex. Note that, together with the integralityconditions, such constraints also imply parity conditions at eachvertex.

Constraints (1f) are connectivity constraints. They impose thatfor each subset of vertices (excluding the depot) containing arequired link or vertex serviced by a vehicle, at least two links inci-dent to the subset must be used to visit it (deadheaded or ser-viced); they also eliminate subtours not connected with the depot.

The selection of the optimal route is guided by the minimizationof the total cost, expressed by the sum of the total service cost f1and the total deadheading cost f2:

Min f ¼ f1 þ f2 ð2Þ

f1 ¼Xm

k¼1

Xði;jÞ2ER

cij xkij þ xk

ji

� �þXm

k¼1

Xði;jÞ2AR

cijxkij

f2 ¼Xm

k¼1

Xði;jÞ2E

cij ykij þ yk

ji

� �þXm

k¼1

Xði;jÞ2A

cijykij

The feasibility region of model (1) can be rewritten in a compactform as:

X ¼ x 2 f0;1gðjAR jþ2jER jþjVR jÞ�m; y 2 ZðjAjþ2jEjÞ�m

þ jx; y 2 X 0;nPðDðxÞxk

6 QÞP a; k ¼ 1; . . . ;mo:

Here X0 is the feasible set defined by the deterministic constraints(1a)-(1f), where D denotes a row vector having the followingstructure:

D ¼ dijðxÞ . . .|fflfflfflfflfflffl{zfflfflfflfflfflffl}ði;jÞ2ER\RU

dijðxÞ . . .|fflfflfflfflfflffl{zfflfflfflfflfflffl}ði;jÞ2AR\RU

diðxÞ . . .|fflfflfflfflffl{zfflfflfflfflffl}i2VR\RU

������� dij . . .|fflffl{zfflffl}ði;jÞ2ER\RC

dij . . .|fflffl{zfflffl}ði;jÞ2AR\RC

di . . .|ffl{zffl}i2VR\RC

:

0B@

1CA

The formulation introduced above imposes individual chance con-straints on each vehicle. This condition provides a guarantee thatany individual route is feasible (with a reliability level a), but it doesnot account for the performance of the entire fleet. Such an issuecould be addressed by means of the joint probabilistic constraints,posing additional theoretical and computational challenges. Ourformulation may constitute a safe approximation of the joint caseby setting appropriately the values of the reliability parameter.Indeed, by Bonferroni’s inequality, a sufficient condition for ensur-ing feasibility in the joint chance constrained problem, is to dividethe joint probability level � among the m individual chance con-straints and letting, for instance, a ¼ �=m (Nemirovski & Shapiro,2006).

Notwithstanding the individual nature of the chance con-straints, the proposed model poses several challenges since itbelongs to the class of integer problems under probabilistic con-straints, for which the literature is rather scarce. Problems involv-ing discrete random variables, arising either directly or asempirical approximation of the continuous ones, have been stud-ied by Beraldi and Ruszczynski (2002), Beraldi and Ruszczynski(2002), Beraldi and Ruszczynski (2005), Beraldi and Bruni (2010),Beraldi, Bruni, and Violi (2012), and Bruni, Beraldi, and Laganà(2013). Here both exact and heuristic approaches have been pro-posed and tested.

P. Beraldi et al. / European Journal of Operational Research 240 (2015) 382–392 385

The case of general continuous random variables is even lessinvestigated. In Fortz and Poss (2010) deterministic reformulationsare analyzed for the case of independently distributed randomvariables, whereas in Klopfenstein (2010) the author studies validinequalities for the problem with individual probabilistic con-straints with uncertainty in both sides.

In this paper, we assume that the random variables follow amultivariate normal distribution. While this assumption mightappear restrictive, it often provides an accurate approximation ofdifferent probabilistic assumption, because of the well-known Cen-tral Limit Theorem and its variants. Under the above assumption,the chance constraints can be equivalently rewritten as second-order cone constraints. In the next subsection, we introduce ourderivations.

2.2. The deterministic equivalent formulation

Let de; e 2 RU denote the random demands which assume nor-mally distributed with mean le, variance r2

e , and let he;e0 denote thecovariance between e and e0. The probabilistic constraints can berewritten as (see, e.g. Section 4 of Fortz & Poss (2010))Xe2RU

lezke þ

Xe2RC

dezke þU�1ðaÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXðe;e0 Þ2R2

Uhe;e0zk

ezke0

r6 Q ð3Þ

where

zke ¼ xk

ij þ xkji if e ¼ ði; jÞ 2 ER;

zke ¼ xk

ij if e ¼ ði; jÞ 2 AR;

zke ¼ xk

i if e ¼ i 2 VR:

ð4Þ

By assuming that de ¼ le and re ¼ 0 whenever e 2 RC , and he;e0 ¼ 0when either e 2 RC or e0 2 RC , we can simplify the above relationas follows:Xe2R

lezke þU�1ðaÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXðe;e0 Þ2R2 he;e0zk

ezke0

q6 Q ; ð5Þ

that is equivalent to:Pe2Rlezk

e 6 Q ;

U�1ðaÞh i2 P

e2Rr2e zk

e þPðe;e0 Þ2R2 je–e0he;e0zk

ezke0

� �6 Q �

Pe2Rlezk

e

� �2:

8<:Since zk

e , are binary variables, the latter inequality can be linearizedusing classical techniques (see also Section 4 in Beraldi &Ruszczynski (2005)), so that (3) can be rewritten as:

Xk ¼ fzk 2 f0;1gjRjXe2R

lezke 6 Q

����� ;

Xe2R

U�1ðaÞh i2

r2e þ 2Q � le

� �le

zk

e 6 ð6Þ

Q 2 þX

ðe;e0 Þ2R2 je–e0

lele0 � U�1ðaÞh i2

he;e0

wk

e;e0 ;

wke;e0 6 zk

e ; wke;e0 6 zk

e0 ; wke;e0 P zk

e þ zke0 � 1; ðe; e0Þ 2 R2; e – e0;

zke ¼ xk

ij þ xkji if e ¼ ði; jÞ 2 ER;

zke ¼ xk

ij if e ¼ ði; jÞ 2 AR;

zke ¼ xk

i if e ¼ i 2 VRg: ð7Þ

where the binary variables wke;e0 act to linearize zk

ezke0 . As a result, an

equivalent deterministic integer programming formulation of theMCGRPPC is obtained by replacing each probabilistic constraint(1d) with the set of additional variables and constraints given by

(6). More precisely, it is ðjRj2�jRjÞ2 þ jRj variables and 2þ 3

2 ðjRj2 � jRjÞ

þ jRj constraints. The complexity of the above model increasesquickly with the number of vehicles and the number of uncertainrequired elements.

Remark 1. The reformulation (5) and the successive derivationsalso hold for the significant class of radial distribution (Calafiore &El Ghaoui, 2006), for which the probability constraints can beconverted explicitly into convex second-order cone constraints oftype (5). It is worth to highlight also the close relation that existsbetween the constraints (5) and the explicit deterministic coun-terparts of distributionally robust chance constraints, which areenforced over an entire family of probability distributions. Inparticular, for the family composed of all distributions havinggiven mean and covariance, the distributionally robust constraintcan be converted, once again, into an explicit second-order coneconstraint of the type (5). This broadens the applicability of theapproach presented in this paper to several interesting contexts.

In the following section we present a branch-and-cut algorithmto optimally solve instances with a small number of vehicles and alimited number of required elements.

3. A branch-and-cut algorithm

In this section, we present the branch and cut algorithmdesigned to optimally solve the MCGRPPC. We mainly focus onthe crucial aspects of (i) the determination of an initial feasiblesolution to use as upper bound and (ii) the definition of validinequalities. The outline of the algorithm is provided by Algorithm1 in Appendix A.

3.1. Initial solution

In order to design an efficient branch-and-cut algorithm, it isalso important also to have a good initial solution, that we builton the basis of the ‘‘partition-first-route-next’’ paradigm. A firstattempt to obtain a good feasible partition relies on solving a prob-abilistic version of the capacitated concentrator location-basedproblem (see Gouveia & Saldanha-da Gama, 2006; Correia,Gouveia, & Saldanha-da Gama, 2010) with stochastic demands, inwhich several required elements are selected as concentrator loca-tions, named ‘‘seeds’’, and the remaining required elements areaggregated around each seed, while respecting, amongst the otherdeterministic constraints reported in Gouveia and Saldanha-daGama (2006) also the probabilistic capacity constraints. The num-ber of required elements selected as concentrators must be equal

to m, initially set to dT ðaÞQ

l m, where dTðaÞ represents the a-quantile

of the random variables representing the total demand of therequired elements. If the model is infeasible because m is inade-quate, then m is increased to ensure feasibility.

The goal consists in minimizing the total assignment cost of allthe required elements to the selected seeds computed as the aver-age of all the costs associated with the shortest paths linking theendpoints of two required elements.

The solution of the above model returns a set of clusters defin-ing a partition of all the required elements. Each cluster representsan instance of the mixed general routing problem that is solvedthrough the B&C algorithm.

This partitioning model does not adequately take into accountthe routing cost associated with each cluster. To improve the rout-ing cost estimation an iterative scheme is designed, in which a set

386 P. Beraldi et al. / European Journal of Operational Research 240 (2015) 382–392

of diversification constraints is added dynamically to the partition-ing model with the aim of selecting other seeds around which dif-ferent clusters are generated and possible better routes are built.Such diversification constraints aim at exploring more promisingportions of the search space according to the classical VariableNeighborhood Search (VNS) scheme (see, e.g. Fischetti, Polo, &Scantamburlo, 2004). More precisely, given a feasible partition ofall the required elements, we identify the seed generating the clus-ter with the highest routing cost and we impose that such a seedcannot be selected for the next f consecutive iterations. We solveagain the probabilistic version of the capacitated concentratorlocation-based problem with this additional constraint and repeatthe diversification procedure by exiting with the best cost solutionobtained within a maximum number of iterations, say I . Observethat all the required elements that are prevented to become a seedcan be selected again as a concentrator after f iterations. In ourimplementation we set I as the minimum between the numberof all the required elements and a threshold equal to 10, whilef ¼ I

2

� �.

Alternatively, a feasible partition can be built heuristically asfollows:

(a) a set of seeds is defined by identifying at each step the onethat is the unclustered required element farthest from thedepot and the other seeds;(b) a cluster of required elements closest to each seed is gener-ated in such a way that the overall demand associated with thiscluster satisfies the probabilistic capacity constraint.

The routing of the required elements collected in each cluster isobtained by using the B&C algorithm. In order to define a startingnumber of vehicles, we set m equal to the number of vehicles inthe minimum cost solution chosen between the best solutionsreturned by both heuristic procedures.

3.2. Valid inequalities

The initial LP relaxation includes Eqs. (1a)–(1c), (6), (1e), oneconnectivity inequality (1f) for each R-set, and some additionalconnectivity and R-odd cut inequalities that are identified accord-ing to the Sequence of edge cutsets procedure described by Belenguerand Benavent (2003). The separation problem associated with con-nectivity inequalities is solvable heuristically through a modifica-tion of the heuristic procedure presented by Fischetti, Salazar,and Toth (1997), and to optimality by means of polynomial timemax-flow calculations.

Besides these well-known cuts, we also derive some additionalcuts exploiting the reformulation of the problem, which take as abasis the capacity inequalities proposed by Belenguer andBenavent (2003).Xði;jÞ2dLðSÞ

gij P 2DðRðSÞ; dLR ðSÞÞ

Q

�� jdLR ðSÞj; S # V n f1g; ð8Þ

where DðRðSÞ; dLR ðSÞÞ ¼P

e2RðSÞ [ dLRðSÞle with

gij ¼

Xm

k¼1

ðykij þ yk

jiÞ if ði; jÞ 2 E

Xm

k¼1

ykij if ði; jÞ 2 A:

8>>>><>>>>:

These inequalities can be adapted to our problem as follows:

P 2DðRðS;xÞ;dLR ðS;xÞÞ

Q

�6

Xði;jÞ2dLðSÞ

gijþjdLR ðSÞj !

Pa; S # V nf1g;

ð9Þ

where DðRðS;xÞ; dLR ðS;xÞÞ represents the demand associated withthe stochastic and deterministic required elements inside S denotedby RðS;xÞ, and dLR ðS;xÞ indicates the stochastic and deterministicrequired links with one endpoint in S and the other outside of S.

To the best of our knowledge, no polynomial algorithm exists toexactly solve the separation problem (8). However, a max-flowalgorithm can be used to solve the separation problem of the so-called fractional capacity inequalities (see Belenguer & Benavent,2003). In our context,we have to deal with the separation problemof the following probabilistic fractional capacity inequalities:

P 2DðRðS;xÞ; dLR ðS;xÞÞ

Q6

Xði;jÞ2dLðSÞ

gij þ jdLR ðSÞj !

P a; S # V n f1g; ð10Þ

that can be transformed in the following way:Xði;jÞ2dLðSÞ

gij P 2DðRðSÞ; dLR ðSÞÞ

Q� jdLR ðSÞj; and ð11aÞ

4Q 2 U�1ðaÞh i2 X

e2RðSÞ [ dLRðSÞr2

e þX

ðe;e0 Þ2ðRðSÞ [ dLRðSÞÞ2 je–e0

he;e0

0B@

1CA

62 jdLR ðSÞj�2DðRðSÞ;dLR ðSÞÞ

Q

Xði;jÞ2dLðSÞ

gij

þ 2DðRðSÞ;dLR ðSÞÞ

Q�jdLR ðSÞj

2

þX

ði;jÞ2dLðSÞgij

!2

; S # V nf1g ð11bÞ

The following property holds.

Property 3.1. For S # V n f1g, if the following linear inequalitieshold:

ðl1ÞPði;jÞ2dLðSÞgij P 2

DðRðSÞ;dLRðSÞÞ

Q � jdLR ðSÞj, and

ðl2Þ 4Q2 U�1ðaÞh i2 P

e2RðSÞ [ dLRðSÞr2

e þPðe;e0 Þ2ðRðSÞ [ dLR

ðSÞÞ2 je–e0he;e0

� �6

2 jdLR ðSÞj � 2DðRðSÞ;dLR

ðSÞÞQ

� �Pði;jÞ2dLðSÞgij þ 2 2

DðRðSÞ;dLRðSÞÞ

Q � jdLR ðSÞj� �2

,

then (10) is valid for S.

Proof. From (l1), (l2), and for any S # V n f1g, it follows that:

4

Q 2 U�1ðaÞh i2 X

e2RðSÞ [ dLRðSÞr2

e þX

ðe;e0Þ2ðRðSÞ [ dLRðSÞÞ2 je–e0

he;e0

0B@

1CA

6 2 jdLR ðSÞj � 2DðRðSÞ;dLR ðSÞÞ

Q

Xði;jÞ2dLðSÞ

gij

þ2 2DðRðSÞ;dLR ðSÞÞ

Q� jdLR ðSÞj

2

¼ 2 jdLR ðSÞj �2DðRðSÞ;dLR ðSÞÞ

Q

�X

ði;jÞ2dLðSÞgij þ 2

DðRðSÞ;dLR ðSÞÞQ

� jdLR ðSÞj 2

þ 2DðRðSÞ;dLR ðSÞÞ

Q� jdLR ðSÞj

2

6 2 jdLR ðSÞj � 2DðRðSÞ;dLR ðSÞÞ

Q

�X

ði;jÞ2dLðSÞgij þ2

Xði;jÞ2dLðSÞ

gij

!2

; ð12Þ

that is inequality (11b). h

Observe that (l1) and (l2) are sufficient but not necessary condi-tions to ensure that the probabilistic inequality (10) holds. Anexample of violation of the condition is provided in Appendix A.

P. Beraldi et al. / European Journal of Operational Research 240 (2015) 382–392 387

Observe that, whenever all the demands are deterministic, theprobabilistic inequality (10) reduces to the deterministic fractionalcapacity inequality expressed by (11a). In such a case, condition(l1) is necessary and sufficient to ensure that (10) holds.

The following theorem states that conditions expressed byProperty (3.1) are sufficient to ensure that inequality (10) holds.Hence, no violation of (10) occurs if no violations of (l1) and (l2)are checked.

Theorem 3.1. For any S # V nf1g, let �¼2DðRðSÞ;dLR

ðSÞÞQ �jdLR ðSÞjP 0

be the fractional minimum number of deadheading traversals in thecutset dLR ðSÞ to service the overall average demand in RðSÞ [ dLR ðSÞ,then the probabilistic inequality (10) is valid for the MCGRPPC ifconditions ðl1Þ and ðl2Þ are satisfied.

Proof. From condition (l2), it follows that:

4Q 2 U�1ðaÞh i2 X

e2RðSÞ [ dLRðSÞr2

e þX

ðe;e0 Þ2ðRðSÞ [ dLRðSÞÞ2 je–e0

he;e0

0B@

1CA

6 �2�X

ði;jÞ2dLðSÞgij þ 2�2:

Condition ðl1Þ implies thatPði;jÞ2dLðSÞgij P �P 0, therefore the

following inequalities hold

4Q 2 U�1ðaÞh i2 X

e2RðSÞ [ dLRðSÞr2

e þX

ðe;e0 Þ2ðRðSÞ [ dLRðSÞÞ2 je–e0

he;e0

0B@

1CA

6 �2�X

ði;jÞ2dLðSÞgij þ 2�2

6 �2�X

ði;jÞ2dLðSÞgij þ �2 þ

Xði;jÞ2dLðSÞ

gij

!2

:

Hence, inequality (10) is valid for the MCGRPPC. h

If � < 0, then ðl2Þ is satisfied for a given value ofPði;jÞ2dLðSÞgij.

A heuristic procedure to identify violations of the probabilisticcapacity inequalities is applied at every node of the branch andcut tree, consisting in identifying S� such that (l1) is checked forpossible violations. If a violation occurs, then �gij; ði; jÞ 2 dLðS�Þ iscomputed according to the LP optimal solution, and the condition(l2) verified. Hence, the following inequality is added to theLP relaxation:

Xði;jÞ2dLðS�Þ

gij P 2DðS�Þ

Q

& ’� jdLR ðS

�Þj; ð11mÞ

where

DðS�Þ ¼X

e2RðS�Þ [ dLRðS�Þ

le þU�1ðaÞ

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX

e2RðS�Þ [ dLRðS�Þ

r2e þ

Xðe;e0 Þ2ðRðS�Þ [ dLR

ðS�ÞÞ2 je–e0he;e0

r:

As usual in the branch-and-cut method, a cut pool is maintainedwith some cuts generated so far in the algorithm. When the cutpool has grown to a certain size (50 cuts in our implementation),it is permanently deleted.

4. The heuristic approach

In this section we describe a neighborhood search algorithm forthe MCGRPPC. It relies on an outer diversification scheme, wheredifferent types of diversification methods are tested for differentrates, followed by both heuristic and exact intensification phases.The diversification rate indicates the number of required elements

modified in the current solution, while the diversification typespecifies how to diversify the solution. After a solution is foundin the outer loop, new solutions are explored through an intensifi-cation phase, after which an exact local search procedure based onthe B&C algorithm is performed with the aim of finding the bestsolution in a given neighborhood.

In the next subsections we provide a detailed description of themain procedures briefly discussed above. Algorithm 2 reported inAppendix A provides a pseudocode of the overall heuristic scheme.

4.1. The diversification strategy

The diversification phase consists of removing some requiredelements from a solution s, and reinserting them to the modifiedsolution ~s. The rationale is that the solutions obtained by diversifi-cation represent the entry point to explore more promising portionof the feasible space.

A lot of freedom arises in designing the strategy for selectingthe customers to remove from the solution, and reallocating theminto ~s (see. e.g. Shaw, 1998; Russell, 1995; Franceschi, Fischetti, &Toth, 2006; Pisinger & Ropke, 2007).

We implemented some of the removal and insertion heuristicspresented by Pisinger and Ropke (2007), but without using statis-tics from the search to guide the choice. A detailed description ofthe removal and insertion procedures is provided below:

� randomRemoveðp; sÞ selects p% required elements at randomand removes them from the solution.� worstRemoveðp; sÞ removes p% of the required elements from

solution s by selecting the ones whose removal returns themaximum saving.� demandOrientedRemoveðp; sÞ removes p% of the required ele-

ments from solution s according to a distance defined on thebasis of the demands of these elements. More precisely, letDij ¼ kdi � djk be the metric associated with the required ele-ments i and j with demands di and dj, respectively. (for randomelements we have considered the mean value). The firstrequired element to remove is randomly chosen, while the sec-ond is selected as the nearest to the first, the third as the nearestto the second and so on until p% of the required elements areremoved from s.

All the removal procedures return the list C of the required ele-ments removed from s (see Algorithm 3 in Appendix A).

Concerning the insertion heuristics, the following proceduresare used in combination with the removal ones:

� randomInsert ðC;~sÞ randomly inserts in ~s the elementsextracted from C. More precisely, for each element e 2 C, aroute r 2 ~s is selected randomly among the routes such thatthe total demand associated with the already serviced requiredelements satisfies (1d). A random position p is chosen inside rwhere e will be inserted and serviced. Whenever e is alreadydeadheaded in r, no attempt is made to select its position.� regretInsert ðC;~sÞ inserts into ~s the elements extracted from C,

with the aim of maximizing the regret associated with each ele-ment. We use the regret objective proposed by Pisinger andRopke (2007), with k ¼ 3, and verify (1d).

4.2. The intensification phase

Current solutions provided by the diversification phase repre-sent a good starting point to address the search towards the con-struction of improved MCGRPPC solutions. Several neighborhoodsare explored according to the basic Variable Neighborhood Descent(VND) strategy described by Hansen, Mladenovic, and Gerad

388 P. Beraldi et al. / European Journal of Operational Research 240 (2015) 382–392

(2003) until no further improvement can be found. All these neigh-borhoods are searched within a mechanism allowing a restart ofthe search from the first neighborhood at each improving solutionfound. The main features of the neighborhood moves are summa-rized as follows:

(m1) merge: for each couple of routes r and r0 in the currentsolution s, an attempt is made in order to merge these routesby servicing all the required elements of r0 after the requiredelements serviced in r.(m2) k-interchange: this move is similar to the one defined byOsman (1993), and the CROSS-exchange move proposed byTaillard, Badeau, Gendreau, Guertin, and Potvin (1997). In thecomputational experiments, we used k 2 f1;2;3;4;5g.(m3) interchangeInterRoute: this move is the same introducedby Savelsbergh (1992) and consists of reinserting a singlerequired element at a time in an alternate route.(m4) interchangeInterRouteNewPath: this move creates a newempty route in which a single required element is inserted.(m5) twoOptPlusInterRoute: this move corresponds to the 2-opt⁄ move defined by Potvin and Rousseau (1995).

Each move is applied with the best-accept strategy. Whenever amove generates a solution with one or more empty routes, suchroutes are dropped from the solution. In order to escape from localoptima, we accept also infeasible solutions, penalized in the objec-tive function by a constraint violation penalty factor (see Cordeau,Gendreau, & Laporte, 1997; Toth & Vigo, 2003).

More precisely, solutions with a number of vehicles larger thanm are m-infeasible (alternatively, m-feasible), whereas the ones inwhich one route at least (alternatively, no route) violates (1d) areQ-infeasible (alternatively, Q-feasible). Given a solution, letnm ¼ ½mðsÞ �m�þ be the number of routes in solution s exceedingm, and

nQ ¼XmðsÞ

k¼1lk � Q� �þ þ ½U�1ðaÞ�2r2

k � ðQ � lkÞ2

h iþ ;

Table 1Computational results with a ¼ 0:95.

Name m jV j jAj jEj jVRj jARj jERj jVRU j jARU j jERU j Branch-and-cut

UB CON PCAP cost

mggdbsd1 10 12 34 5 6 12 3 4 4 1 443 408 805 443mggdbsd2 13 12 40 6 6 15 4 3 10 1 566 391 422 566mggdbsd3 11 12 34 5 7 12 3 2 7 2 456 37,302 6915 456mggdbsd4 10 11 30 4 4 11 3 0 8 2 494 11,811 2097 494mggdbsd5 11 13 40 6 5 15 4 3 11 2 566 17,566 3598 547mggdbsd6 10 12 34 5 6 12 3 2 5 2 473 937 362 473mggdbsd7 8 12 34 5 5 12 3 2 6 3 449 420 181 433mggdbsd8 17 27 70 11 11 26 8 7 13 6 530 9726 783 530mggdbsd9 14 27 78 72 9 29 9 2 13 4 430 7435 2547 430mggdbsd10 6 12 38 6 4 14 4 2 7 2 309 7825 5196 309mggdbsd11 7 22 68 11 8 25 8 4 12 6 422 14,162 7337 422mggdbsd12 9 13 36 5 6 13 3 3 7 1 610 13,644 6525 610mggdbsd13 8 10 42 7 6 15 5 2 8 2 455 355 526 455mggdbsd14 6 7 32 5 5 12 3 3 6 0 112 270,485 37,157 108mggdbsd15 5 7 32 5 5 12 3 3 7 1 61 165 19 55⁄

mggdbsd16 8 8 42 7 5 15 5 3 5 4 108 148 52 108mggdbsd17 7 8 42 7 5 15 5 3 9 3 77 1761 710 71⁄

mggdbsd18 6 9 54 9 6 20 6 3 11 2 169 4 4 169mggdbsd19 4 8 18 2 3 6 1 1 1 0 61 1063 140 61⁄

mggdbsd20 6 11 34 5 5 12 3 1 7 1 139 471 11 130mggdbsd21 10 11 50 8 7 18 6 5 11 3 172 63 0 168mggdbsd22 11 11 66 11 6 24 8 3 11 4 181 197 46 176mggdbsd23 17 11 84 13 8 31 9 4 20 5 208 267 121 208

Avg. 322.70NumB. 3

where ½:�þ ¼maxf0; :g;lk and r2k are the mean and variance of the

random demand associated with the required elements servicedby the kth vehicle, respectively. Then, the penalty cost associatedwith such a solution is defined as tmnm þ tQ nQ , where tm and tQ

are the unit penalties associated with the m-infeasibility or Q-infea-sibility, or both. The modified objective function becomes~f ¼ f þ tmnm þ tQnQ . Penalty tm is decreased by settingtm ¼max tmin

m ; tm � t�m� �

if, after s consecutive iterations, all thegenerated solutions are m-feasible, whereas it is increased bysetting tm ¼min tmax

m ; tm � tþm� �

whenever these solutions arem-infeasible. Similarly, penalty tQ is decreased by setting

tQ ¼max tminQ ; tQ � t�Q

n o, and it is increased by setting

tQ ¼min tmaxQ ; tQ � tþQ

n oin the opposite case. The overall algorithm

is detailed in Algorithms 4 and 5 reported in Appendix A.

4.3. Exact local search using integer programming

The B&C algorithm presented in Section 3 cannot be used tosolve real-life instances directly because of prohibitive computa-tional times, but it can effectively solve instances with a numberof vehicles not greater than six. This suggests that the B&C algo-rithm may be used successfully to improve portions of the solution.In particular, a route optimization search relying on integer pro-gramming is used as a neighborhood search scheme. With respectto other applications of this idea (see Savelsbergh & Song, 2008;Song & Furman, 2013), in our implementation the B&C is used torecover the feasibility of the schedules of some required elements.We select such schedules in a subset of couples and triplet ofroutes of the solution returned by the intensification phase, andwe solve a restricted problem, where the schedules of the remain-ing routes are kept fixed. Several strategical decisions are adoptedto make efficient the proposed approach. How do we select the twoor three vehicles whose routes define the neighborhood? Which isthe criterion to stop the neighborhood search? We choose a simple

HSGR

GAP Veh. Time Avg. cost Avg. veh. Best cost Best veh. Avg. gap Avg. time

0.46 10 – 421.2 10 414 10 �4.92 1224.4270.46 13 – 482.6 11 475 11 �14.73 907.760.46 11 – 443.4 11 438 11 �2.76 326.350.45 10 – 442 8 442 8 �10.53 497.670.37 11 – 501.6 9 495 9 �8.30 1703.240.41 10 – 472.4 10 470 10 �0.13 288.690.37 8 – 374 7 374 7 �13.63 364.310.51 17 – 455 15 448 15 �14.15 4207.790.43 14 – 396.8 14 393 14 �7.72 6913.150.18 6 – 291 6 291 6 �5.83 834.790.24 7 – 382 7 380 7 �9.48 9.511,250.36 9 – 573.4 9 563 9 �6.00 741.500.31 8 – 423 8 423 8 �7.03 1037.620.01 6 – 108.4 6 108 6 0.37 137.110.00 5 4.76 55 5 55⁄ 5 0.00 4.500.13 8 – 100.4 8 100 8 �7.04 314.180.00 7 926.41 71 7 71⁄ 7 0.00 230.340.21 6 – 149 6 149 6 �11.83 9.526,750.00 4 16.44 61 4 61⁄ 4 0.00 7.000.14 6 – 122.4 6 121 6 �5.85 302.3060.22 10 – 156.4 10 156 10 �6.90 437.140.16 11 – 168.8 11 168 11 �4.09 2015.750.16 17 – 197.8 17 196 17 �4.90 1090.50

0.26 12,562.94 297.77 295.35 �6.32 1.853,2223

Table 2Computational results with a ¼ 0:85.

Name m jV j jAj jEj jVRj jARj jERj jVRU j jARU j jERU j Branch-and-cut HSGR

UB CON PCAP cost GAP Veh. Time Avg. cost Avg. veh. Best cost Best veh. Avg. gap Avg. time

mggdbsd1 8 12 34 5 6 12 3 4 4 1 434 47,556 12,944 414 0.43 8 – 370 8 368 8 �10.63 952.00mggdbsd2 10 12 40 6 6 15 4 3 10 1 483 381 576 483 0.38 10 – 427.4 9 415 9 �11.51 2553.54mggdbsd3 9 12 34 5 7 12 3 2 7 2 391 33,399 8355 391 0.39 9 – 364.8 8 360 8 �6.70 618.75mggdbsd4 8 11 30 4 4 11 3 0 8 2 439 184,122 29,825 432 0.37 8 – 373.6 6 371 6 �13.52 964.54mggdbsd5 9 13 40 6 5 15 4 3 11 2 503 16,376 5055 503 0.29 9 – 483.8 9 477 9 �3.82 2016.51mggdbsd6 8 12 34 5 6 12 3 2 5 2 430 63,895 20,708 419 0.32 10 – 373.4 8 367 8 �10.88 551.04mggdbsd7 9 12 34 5 5 12 3 2 6 3 450 402 206 433 0.35 9 – 417.6 8 409 8 �3.56 364.31mggdbsd8 13 27 70 11 11 26 8 7 13 6 514 19,213 3531 453 0.46 17 – 408.8 13 402 13 �9.76 3068.50mggdbsd9 13 27 78 72 9 29 9 2 13 4 399 7534 2313 399 0.41 13 – 363.4 13 356 13 �8.92 6885.3mggdbsd10 6 12 38 6 4 14 4 2 7 2 319 471 29 302 0.18 6 – 288.8 6 288 6 �4.37 7937.19mggdbsd11 6 22 68 11 8 25 8 4 12 6 429 1925 438 402 0.21 6 – 369.2 6 368 6 �8.16 8278.05mggdbsd12 8 13 36 5 6 13 3 3 7 1 610 11,089 10,848 589 0.37 8 – 529.8 8 527 8 �10.05 1032.44mggdbsd13 8 10 42 7 6 15 5 2 8 2 456 304 725 455 0.21 8 – 417.6 8 408 8 �8.22 1668.81mggdbsd14 6 7 32 5 5 12 3 3 6 0 115 8347 4525 107⁄ 0.00 6 1164.20 108 6 108 6 0.93 1669.98mggdbsd15 5 7 32 5 5 12 3 3 7 1 57 54 15 55⁄ 0.00 4 3.85 55 4 55⁄ 4 0.00 7.43mggdbsd16 7 8 42 7 5 15 5 3 5 4 112 60 26 112 0.17 7 – 100 7 98 7 �10.71 661.97mggdbsd17 6 8 42 7 5 15 5 3 9 3 83 1761 710 71⁄ 0.00 6 926.41 71 6 71⁄ 6 0.00 755.68mggdbsd18 6 9 54 9 6 20 6 3 11 2 170 20 2 170 0.23 6 – 146.8 5.4 144 5 �13.65 7704.14mggdbsd19 4 8 18 2 3 6 1 1 1 0 61 362 63 59⁄ 0.00 4 24.03 59 4 59⁄ 4 0.00 6.25mggdbsd20 5 11 34 5 5 12 3 1 7 1 136 684 41 117⁄ 0.00 5 13,790.76 117.6 5 117⁄ 5 0.51 200.42mggdbsd21 9 11 50 8 7 18 6 5 11 3 171 177 21 165 0.21 9 – 154 9 153 9 �6.67 613.96mggdbsd22 10 11 66 11 6 24 8 3 11 4 177 235 104 170 0.17 10 – 167.2 10 167 10 �1.65 876.62mggdbsd23 15 11 84 13 8 31 9 4 20 5 205 211 101 203 0.25 15 – 192 14 192 14 �5.42 409.364

Avg. 300,17 0.24 11,961.27 276.47 273.04 �6.38 2165.08NumB. 5 23

P. Beraldi et al. / European Journal of Operational Research 240 (2015) 382–392 389

and straightforward scheme. Let LddðkÞ be the set of links dead-headed by vehicle k in route rk. Among the subsets of two vehiclesthat have not been selected before, we select the couple of vehiclesðk1; k2Þ with the following rules:

(a) Lddðk1Þ \ Lddðk2Þ��� ��� is maximum. In case of multiple couples,

we select(b) the couple for which the maximum violation of (1d) ischecked. If no violation of (1d) takes place, then we select thetwo vehicles with the maximal residual capacity. If morecouples still exist, then(c) we choose randomly the couple of vehicles with the maxi-

mum value of Lddðk1Þ \ Lddðk2Þ��� ��� and the maximum violation of

(1d) or the maximal residual capacity.

We stop after evaluating 30% of all the couples. This scheme hasbeen evaluated on a subset of tuning instances with a number ofvehicles equal to or less than six, and it was able to find the optimalsolution provided by the B&C algorithm, or a near optimal solution.However, to deal with instances with a number of vehicles largerthan six, we enlarged the neighborhood size by considering theschedules of the required elements serviced by three vehicles,and optimizing the triple selected in accordance with the aboverules. To maintain tractability with respect to the computationaltimes, we optimized a triple every factor � ncd e couples of routes,where factor ¼ 0:2 and nc is the total number of couples to beexamined.

5. Computational experiments

In order to test the efficiency of the proposed approaches, weused a dataset of randomly generated instances with probabilitylevel a 2 f0:85;0:95g. Each instance is built starting from thecorresponding mggdb instance with b ¼ 0:25 of the datasetdesigned by Bosco et al. (2013) for the MCGRP. For each requiredelement e 2 R with demand de, a random binary number isextracted to decide if de is affected by uncertainty or not. Then,

a random value from the discrete uniform distribution overf1; . . . ; coef � deg; coef ¼ 5, is selected as variance of the randomvariable deðxÞ. All the stochastic required elements in the neigh-borhood of the end vertices of each stochastic required link areconsidered to define a negative or positive correlation with thislink. Correlation q is selected randomly in the interval ½�1;1�.Whenever the correlation between e; e0 2 R; e – e0, is not zero,then the relevant entry in the covariance matrix is set ashe;e0 ¼ qrere0 . Finally, if no non-zero correlation occurs, then another attempt is made by forcing some stochastic required ele-ments close to the end vertices of the stochastic links to have acovariance with such links. The new instances are named mggdbsd,where sd stands for stochastic demand.

Computational experiments have been carried out on a PCequipped with 2 Intel Xeon Quad Core CPUs @3.0 gigahertz, with6 gigabytes RAM. The heuristic and the B&C algorithm have beencoded in java. The B&C has been implemented by using ILOG CPLEXlibrary, release 12.2, where all the standard CPLEX cuts are acti-vated. We have run each heuristic for 5 times by using the sameparameter configuration.

In the tuning phase, we experimented a large range of diversi-fication rates. We removed from 5 up to 90% of all the requestsin each iteration. Due to the weakness of the insertion heuristics,the improvements obtained with the lowest and highest rates werevery limited. Consequently we decided to reduce the range from 15up to 60. More precisely, we observed that for small instances thebest efficiency in removing requests relies on rates chosen in theset f15;25;35g, while for larger instances the best rates are theones selected in the set f50;60g. To ensure consistency, we usedthe same set of rates for all the instances, that isf15;25;35;50;60g.

5.1. Results

The comparison between the results achieved by the proposedheuristic, named HSGR, and those provided by the B&C algorithm(with a time limit of four hours) is reported in Tables 1 and 2.The column headings are defined as follows:

Fig. 1. G ¼ ðV ;A; EÞ.

390 P. Beraldi et al. / European Journal of Operational Research 240 (2015) 382–392

� Name: instance name;� m: number of vehicles in the upper bound solution;� jV j: number of vertices;� jAj: number of arcs;� jEj: number of edges;� jVRj: number of required vertices;� jARj: number of required arcs;� jERj: number of required edges;� jVRU j: number of vertices with stochastic demand;� jARU j: number of arcs with stochastic demand;� jERU j: number of edges with stochastic demand;� UB: upper bound corresponding to the initial solution value;� CON: number of added connectivity inequalities;� CAP: number of added capacity inequalities (11m);� cost: cost of the best solution provided by the B&C algorithm

within a time limit of four hours (optimal values are markedby an asterisk);� GAP: percentage gap between upper and lower bounds at the

termination of the B&C algorithm;� veh.: number of vehicles in the best solution returned by the

B&C;� time: computational time (in seconds) of the B&C algorithm

(the time limit is denoted by –);� avg. cost: average solution value provided by the heuristic (over

5 experiments);� avg. veh.: average number of vehicles provided by HSGR (over 5

experiments).� best cost: cost of the best solution provided by heuristic (over 5

experiments). Whenever this cost is equal to the optimal cost, itis labeled with an asterisk, while the costs in bold are those out-performing the upper bounds returned by the B&C within thetime limit.� best veh.: number of vehicles of the best solution returned by

the heuristic (over 5 experiments);� avg. gap: percentage improving gap computed as 100½avg:cost—cost�

cost ;� avg. time: computational time (in seconds) spent by HSGR (over

5 experiments) to perform one experiment.

The row Avg. averages key columns, while NumB indicates thenumber of best solutions found by the B&C and HSGR, respectively.

Several observations may be pointed out according to differentcriteria based on the number of vehicles, the prevalence in the totalnumber of required links with respect to the total number of therequired vertices, the distribution of the uncertainty within thestochastic required elements. Looking at the number of vehiclesonly, HSGR outperforms the B&C when the number of vehiclesincreases for the growing difficulty in solving the mathematicalmodel. This emerges from instances in which the number of

vehicles is greater than 6, as comfirmed by the percentage gapsreported in columns GAP in Tables 1 and 2. From routing pointof view, we observe that when the number of required links makesthe problem more ‘‘arc’’ than ‘‘node’’ routing based, the perfor-mance of the B&C is very poor compared with the proposed heuris-tic, and deteriorates quickly with the increase in the reliabilitylevel and the number of vehicles, as shown in instance mggdbsd2in which the number of required links is more than twice the num-ber of required vertices, the number of vehicles is greater than 10and the improvement achieved by HSGR increases with a. Such abehavior is motivated by the increase in the performance of theexact route optimization strategy that has more chances toimprove portions of the feasibility region due to the link-basednearness between the couple of routes to be optimized. Moreover,when the uncertainty affects only links, as for instance mggdbsd4,HSGR performs much better than the B&C regardless the reliabilitylevel, confirming our conjecture.

Looking at Tables 1 and 2, only in three instances the B&C pro-vides an optimal solution within a limited computational time forthe considered reliability levels. This happens when the number ofrequired edges is less than the number of required arcs and verti-ces, since modeling a required edge through pairs of binary vari-ables increases the difficulty in solving the mathematical model.Finally, we observe that the performance of HSGR varies with thedistribution of uncertainty within the same number of requiredelements. For example, if we consider instances mggdbsd1 andmggdbsd6 having the same total number of stochastic elementsbut a different number of stochastic vertices and links, we remarkthat HSGR produces better results when the number of stochasticvertices is comparable with the number of stochastic links. Thisbehavior seems reducing when the reliability level decreases asshown in Table 2. We argue that, in general, the reliability levelsimpact on the hardness of the problem regardless of the distribu-tion of the uncertainty amongst the required elements.

6. Conclusions

We studied the mixed capacitated general routing problemunder uncertainty and we proposed a probabilistically constrainedformulation where capacity constraints are imposed to hold with agiven reliability value. We provided a deterministic equivalent for-mulation under the assumption that the uncertain parameters fol-low a normal distribution. We designed a B&C algorithm for theoptimal solution of instances of limited size and an efficient heuris-tic approach. Extensive numerical experiments have been carriedout on a set of stochastic instances of benchmark test problemsproposed in the literature for the deterministic case. The analysisof the numerical results has shown the efficiency of the proposedapproaches. The proposed approach can be extended to deal withother general routing problems in which operative constraintscome into play.

Appendix A

Hereafter is reported an example showing that conditions (l1)and (l2) are not necessary to ensure that the probabilistic inequal-ity (10) holds. Consider, for example the instance defined by themixed graph G depicted in Fig. 1, where all the required elementsare marked in bold and their demands are defined through pairsle;r2

e

� �that represent the corresponding mean and variance,

respectively. Suppose U�1ðaÞ ¼ 1:645 and Q ¼ 3. Edge costs arenot shown as they are not relevant to the example. Let S ¼ f6;7gbe a subset of vertices not connected with the depot such thatDðRðSÞ; dLR ðSÞÞ ¼ 2; jdLR ðSÞj ¼ 1, and

Pðe;e0 Þ2ðRðSÞ [ dLR

ðSÞÞ2 je–e0he;e0 ¼ 0:5.

P. Beraldi et al. / European Journal of Operational Research 240 (2015) 382–392 391

The minimum number of deadheading traversals in the cutset dLðSÞto satisfy (10) must be equal to 3. In fact:

Xði;jÞ2dLðSÞ

gij ¼ 3 > 0:333 ¼ 2DðRðSÞ; dLR ðSÞÞ

Q� jdLR ðSÞj;

the left hand side of inequality (11b) is equal to:

4Q 2 U�1ðaÞh i2 X

e2RðSÞ [ dLRðSÞr2

e þX

ðe;e0 Þ2ðRðSÞ [ dLRðSÞÞ2 je–e0

he;e0

0B@

1CA ¼ 3:006;

while the right hand side becomes:

2 jdLR ðSÞj � 2DðRðSÞ; dLR ðSÞÞ

Q

Xði;jÞ2dLðSÞ

gij

þ 2DðRðSÞ; dLR ðSÞÞ

Q� jdLR ðSÞj

2

þX

ði;jÞ2dLðSÞgij

!2

¼ 7:111:

On the other hand, since

2 jdLR ðSÞj � 2DðRðSÞ; dLR ðSÞÞ

Q

Xði;jÞ2dLðSÞ

gij

þ 2 2DðRðSÞ; dLR ðSÞÞ

Q� jdLR ðSÞj

2

¼ �1:777;

condition ðl2Þ is violated.

Algorithm 1. The general scheme of the heuristic

diversification rates ¼ f15;25;35;50;60g.diversification types ¼ fRANDOM;WORST;

DEMAND ORIENTEDg.Initial solution: �s. Best solution: sB ¼ ;. Current solution:

s ¼ �s.if �s is feasible then

sB ¼ �s.end iffor rate 2 diversificationRates do

ITERATIONLocal best solution: sL ¼ ;.for type 2 diversificationTypes do

sD ¼ diversificationðrate; type; sÞif sD is feasible and ~f ðsDÞ < ~f ðsBÞ then

sB ¼ sD.end ifsI ¼ IntensificationphaseðsDÞ.s�I ¼ ExactLocalSearchðsI Þ.if s�I is feasible and ~f ðs�I Þ < ~f ðsBÞ then

sB ¼ s�I .end if

if sL ¼ ; or ~f ðs�I Þ < ~f ðsLÞ thensL ¼ s�I

end ifend fors ¼ sL.

end forreturn sB .

Algorithm 2. Diversification procedure: diversificationðrate;type;sÞ

if type ¼ RANDOM thenC = randomRemoveðrate; sÞ.s0 = randomInsertðC;~sÞ.return s0.

end ifif type ¼WORST then

C = worstRemoveðrate; sÞ.s0 = regretInsertðC;~sÞ.return s0.

end ifif type ¼ DEMAND ORIENTED then

C = demandOrientedRemoveðrate; sÞ.s0 = regretInsertðC;~sÞ.return s0.

end if

Algorithm 3. IntensificationphaseðsÞ Part I

moves ¼ fmerge; k-intechange, with k ¼ 1; . . . ;5,interchangeInterRoute,

interchangeInterRouteNewPath, twoOptPlusInterRoute7gSet s0 ¼ s. Set cm ¼ 0; cQ ¼; s ¼ 10; tmin

m ¼ 1; tminQ ¼ 1,

tmaxm ¼ 1000,

tmaxQ ¼ 1000; t�m ¼ 0:7; t�Q ¼ 0:7; tþm ¼ 1:5; tþQ ¼ 1:5.

Set NS iterations ¼ 0.for all i ¼ 0; i < moves:size; i ¼ iþ 1 do

Apply the ith move to s:~s ¼moveðs0;moves½i�Þ;if ~s is m�infeasible then

cm ¼ cm þ 1.end ifif ~s is Q-infeasible then

cQ ¼ cQ þ 1.end if

NS iterations ¼ NS iterationsþ 1.if NS iterations P s then

Update unit penalties:if cm ¼ 0 then

tm ¼ max tminm ; tm � t�m

� �.

end if

Algorithm 4. IntensificationphaseðsÞ Part II

if cm ¼ NS iterations thentm ¼ min tmax

m ; tm � tþm� �

.end ifif cQ ¼ 0 then

tQ ¼max tminQ ; tQ � t�Q

� �.

end ifif cQ ¼ NS iterations then

tQ ¼min tmaxQ ; tQ � tþQ

� �.

end ifNS iterations ¼ 0.cm ¼ 0.cQ ¼ 0.

(continued on next page)

392 P. Beraldi et al. / European Journal of Operational Research 240 (2015) 382–392

end if

if ~f ð~sÞ < ~f ðs0Þ thens0 ¼ ~s.Restart from the first move: i ¼ �1.

end ifend forreturn s0.

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