European Journal of Operational Research 220 (2012) 349–360

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European Journal of Operational Research

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

Production, Manufacturing and Logistics

The close–open mixed vehicle routing problem

Ran Liu, Zhibin Jiang ⇑Department of Industrial Engineering & Logistics Management, Shanghai Jiao Tong University, Shanghai 200240, PR China

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

Article history:Received 24 January 2011Accepted 31 January 2012Available online 22 February 2012

Keywords:MetaheuristicsClose and openVehicle routingMemetic algorithm

0377-2217/$ - see front matter � 2012 Elsevier B.V. Adoi:10.1016/j.ejor.2012.01.061

⇑ Corresponding author. Address: Department of Intics Management, School of Mechanical Engineering, S800 Dong Chuan Road, Shanghai 200240, China. Tel./f

E-mail address: zbjiang@sjtu.edu.cn (Z. Jiang).

We consider the Close–Open Mixed Vehicle Routing Problem (COMVRP) in this paper. The COMVRP dif-fers from the classical vehicle routing problems because simultaneously considering open and closeroutes in the solution of the problem. The objective of the problem is to minimize the fixed and variablecosts for operating the open and close routes. No attention was devoted to this problem. A mix integerprogramming (MIP) model and an effective metheuristic, i.e., memetic algorithm, are established forthe COMVRP. Computational experiments are conducted. The results of experiments show that the pro-posed metheuristic algorithm is able to produce satisfied solutions within an acceptable running time,and outperforms the robust MIP solver CPLEX.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction the classical VRP, but contrary to the classical VRP a route ends

The Vehicle Routing Problem (VRP) is one of the most famousand widely studied problems in combinatorial optimization. Itwas introduced by Dantzig and Ramser (1959) about 50 yearsago, and holds a central place in the fields of physical distributionand logistics (Bodin, 1990; Cordeau et al., 2002). The classical VRPcan be defined as follows. There is a depot and a set of customerswith specified demand for goods. A vehicle fleet is located at thedepot. Every vehicle has a specified capacity and an operating cost.The traveling cost between the depot and the customers, as well asbetween any pair of customers is known. Each customer must beserved by a single vehicle and no vehicle can serve a set of custom-ers whose demand exceeds its capacity. The objective of the classi-cal VRP is to find a set of vehicle routes of minimum cost, whereeach vehicle used leaves from and returns to the depot. There ex-ists a very rich scientific literature on the VRP, including exact algo-rithms, classical heuristics, and metaheuristics, because of VRP’sconsiderable difficulty as well as because of its practical relevance.In practice, a number of variants of the classical VRP exist, e.g., theVRP with time windows (Azi et al., 2010; Braysy and Gendreau,2005a,b; Homberger and Gehring, 2005; Solomon, 1987), the mul-ti-depot VRP (Cordeau et al., 1997; Crevier et al., 2007; Salhi andSari, 1997), because of the diversity of operating rules and con-straints encountered in real-life applications. Meanwhile, fromthe early this century, a kind of problem called the Open VehicleRouting Problem (Open-VRP) was studied by the researchers(Sariklis and Powell, 2000). The Open-VRP is closely related to

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as soon as the last customer has been served as the vehicles donot need to return to the depot. There are also many practicalapplications in which the Open-VRP naturally arise (Li et al.,2007; Repoussis et al., 2009, 2010; Russell et al., 2008; Tarantiliset al., 2005). For example, the Open-VRP operational frameworkis faced by company which either does not own a vehicle fleet atall, or its fleet is inappropriate or inadequate to satisfy the demandof its customers. Therefore, the company is obliged to contract partor all of its product distribution to external couriers. The hiredvehicles will be assigned to routes in which they are not obligedto return to the depot. Because in the classical VRP each vehicleroute starts and ends at the same depot and forms a closed circuit,in this paper the classical VRP is called the Close-VRP, in order todifferentiate from the Open-VRP.

In this paper, we tackle a special optimization problem calledthe Close–Open Mixed Vehicle Routing Problem (COMVRP). TheCOMVRP can be seen as a combination of the Close-VRP andOpen-VRP. Both the Close-VRP and Open-VRP are concerned atdesigning optimal routes used by a fleet of vehicles, located at a de-pot, to serve a set of customers. In the former, each route starts andends at exact one depot. While in the latter, each route starts at thedepot and finishes at one of the customer node. The important fea-ture of the COMVRP, which distinguishes it from the Close-VRP andthe Open-VRP, is that some vehicles are not required to return tothe depot, while the rest vehicles must returns back to the depotafter serving the customers. That is to say, there exist both theopen and closed vehicle routes in the solution of the COMVRP.The COMVRP has significant applications in transportation system.For example, in Shanghai, China, most chemicals plants have theirinternal fleet. Each chemical plant provides the hazardous materi-als shipment tasks to its private fleet. However, since sometimesthe capacity of a special type of vehicle (e.g., for vitriol shipment)

350 R. Liu, Z. Jiang / European Journal of Operational Research 220 (2012) 349–360

is not sufficient, some vehicles are hired from the external carriersto complete the hazardous materials shipments. In this case, thevehicles of internal fleet start from the depot (or warehouse), thenserve many customers and return to the depot in the respect ofsecurity and maintenance. The vehicles of external carrier also loadthe hazardous materials in the depot, and then deliver goods to aset of customers. However, these vehicles need not return to thestarting depot and thus they require efficient open paths that arenot concerned with returning to the depot. Logistics managersmust make a selection between the private vehicle and externalcarrier vehicle for each customer. Meanwhile, the best vehicleroutes followed by internal fleet and external carrier should bespecified, with the objective of minimizing the total costs.

To the best of our knowledge, the COMVRP has not been pre-sented before. All the literature on the Close-VRP only considersthe closed routes in the problem and designs the corresponding ap-proaches. Similarly, all the researches on the Open-VRP are built onthe assumption that the vehicles do not return to the starting de-pot. As stated in the literature on the Open-VRP, the Open-VRPshould not be solved as the classical Close-VRP by deleting thelargest edge between the depot and the adjacent customers, sincesuch solution are worse than using a specific algorithm (Brandão,2004; Sariklis and Powell, 2000). Similarly, we also should designthe specific optimization approach to solve the COMVRP to getgood solution to the problem.

However, designing a sophisticated algorithm for the COMVRPis not easy. Both the Close-VRP and the Open-VRP have beenproved to be NP-hard (Brandão, 2004; Laporte, 2009). The COMVRPis also NP-hard since it reduces to the Close-VRP when the numberof the vehicles which perform the open routes is zero, and reducesto the Open-VRP when all the routes are open. Since the basicClose-VRP and the Open-VRP are very difficult to solve (Brandão,2004; Laporte, 2009), the COMVRP introduced in this paper is morecomplex and even harder to be tackled than the Close-VRP and theOpen-VRP. Because of the difficulties in solving instances of practi-cal interest, most research to the Close-VRP and Open-VRP resortto the heuristics. Genetic algorithm (Holland, 1975) is a famousmetaheuristics, which has been widely adopted in the field of com-binational optimization. In recent years, it has been proven that ge-netic algorithm performs well on the Close-VRP, Open-VRP andsome other VRP variants. For some VRP problems, genetic algo-rithm even can get the best solutions and outperform other meta-heuristics. All these powerful genetic algorithms have been appliedin conjunction with local search methods. This is usually achievedby improving the offspring through local search. Such kind of hy-brid genetic algorithm is also called the memetic algorithm (MA).In this paper, we also design an effective memetic algorithm tosolve the real-world large-scale COMVRP.

The rest of this paper is organized as follows. Section 2 intro-duces the relevant literature. A mathematical programming formu-lation of the COMVRP is developed In Section 3. Section 4 proposesthe heuristic algorithm for solving the COMVRP. Computational re-sults on benchmark instances are reported in Section 5. Finally,conclusions and future work are presented in Section 6.

2. Literature review

The COMVRP has never been studied before. However, theClose-VRP and Open-VRP have been studied in the existing litera-ture. There exists a wide board literature on the Close-VRP. Formore details on the state of the art in the Close-VRP research, read-ers can refer to the survey (Cordeau et al., 2007; Laporte, 1992;Laporte, 2009; Toth and Vigo, 2002). Some side constraints are con-sidered for the Close-VRP, such as capacity restrictions, time

restrictions and vehicle travel distance restrictions. Since capacityrestrictions are basic and practical, the capacity constrained Vehi-cle Routing Problem (CVRP) literature is abundant. Toth and Vigo(2002) reviewed the branch and bound algorithms proposed forthe symmetric and asymmetric CVRP. Simultaneously some pow-erful exact branch and cut and branch and price methods are ap-plied to the CVRP (Achuthan et al., 2003; Baldacci et al., 2008;Fukasawa et al., 2006; Letchford and Salazar-González, 2006; Lysg-aard et al., 2004). The dimension of the largest instance solved bythese exact methods has been reached to 135 customers. Since theCVRP is so hard to solve exactly and algorithmic behavior is highlyunpredictable, a great deal of effort has been invested on the de-sign of heuristics. Compared with classical heuristics (Clarke andWright, 1964; Fisher and Jaikumar, 1981; Laporte and Semet,2002), metaheuristics perform a much more thorough search ofthe solution space, and has experienced a formidable growth overthe past 10 years (Cordeau et al., 2002, 2005; Derigs and Kaiser,2007; Ergun et al., 2006; Gendreau et al., 1994; Laporte, 2009;Mester and Braysy, 2007; Nagata, 2007; Pisinger and Ropke,2007; Prins, 2004). Compared with CVRP, the distance constrainedVRP (DVRP) has received relatively less attention in the literature.Laporte et al. (1984) present the exact algorithms for DVRP. Li et al.(1992) present a heuristic for DVRP, which provides a good worstcase result when the number of vehicles used is relatively small.

The distance and capacity constrained VRP (DCVRP) is the com-bination of CVRP and DVRP, which considers both the distance con-straints and vehicle capacity limits. Laporte et al. (1985) give theinteger liner programming algorithm for DCVRP and present twoversions of exact algorithms. Several effective metaheuristics, suchas tabu search (Taillard, 1993), ant system (Reimann et al., 2004)and active guided evolution strategy (Mester and Bräysy, 2005)are developed for DCVRP. Recently Kek et al. (2008) solve twonew distance-constrained capacitated vehicle routing problems.Albareda-Sambola et al. (2009) proposes a Capacity and DistanceConstrained Plant Location Problem, addresses different modelingaspects of the problem and describes a tabu search algorithm forthe problem.

Another important related problem Open-VRP is studied fromSariklis and Powell (2000), who developed a two-phase construc-tion heuristic for the Open-VRP with capacitated vehicles andunlimited route lengths. Letchford et al. (2006) gives an exact opti-mization method based on branch and cut algorithm for solvingOpen-VRP. Meanwhile, many heuristics have been proposed forthe Open-VRP. Brandão (2004) present a tabu search procedurewhich makes use of customer insertion and swap local search oper-ators. Tarantilis et al. (2005) present a single-parameter approachfor this problem, which exploits a list of threshold values to intelli-gently guide an advanced local search method. Li et al. (2007)develops a local search metaheuristic which uses the concept of re-cord-to-record travel, and test the performance of the metaheuris-tic on several large-scale OVRP instances. Pisinger and Ropke (2007)use adaptive large neighborhood search method to solve some VRPvariants, including the Open-VRP. Fleszar et al. (2009) propose aneffective variable neighborhood search heuristic for this problem,whose neighborhoods are based on reversing segments of routesand exchanging segments between routes. Li et al. (2009) presenta hybrid metaheuristic based on ant colony optimization and tabusearch for the Open-VRP. Finally, Repoussis et al. (2010) presentsa hybrid evolution strategy, Salari et al. (2010) present a heuristicimprovement procedure for OVRP based on integer linear program-ming techniques, and Zachariadis and Kiranoudis (2010) design anoriginal OVRP method which examines rich neighborhood struc-tures formed by exchanging large customer sequences.

Another related stream of research focuses on the VRP with pri-vate fleet and common carrier (VRPPC) (Bolduc et al., 2007, 2008;

Table 1Glossary of mathematical symbols.

Symbol Explanation

G Basic completed graphN Customers setV = {0,N} Vertex set, 0 represents the depotqi Demand of customer i

R. Liu, Z. Jiang / European Journal of Operational Research 220 (2012) 349–360 351

Côté and Potvin, 2009; Chu, 2005; Liu et al., 2010). In this problem,each customer can be served by one of the vehicles of an internalfleet or by an external common carrier. When a customer is as-signed to the common carrier, a penalty cost is incurred, whichrepresents all costs associated with this assignment. The major dif-ference between the VRPPC and COMVRP is that in the former theroutes associated with the external carrier fleet are not considered.

Xk2K

cij Travel distance, and travel cost between two nodes i and jK Set of vehicle types. K = {0,1}, represents the private

vehicle and hired vehicle, respectivelyQk Capacity of the type k vehicleHk Distance span of the vehicle of type kFk Fixed cost of using a vehicle of type k, i.e., when a vehicle

of type k starts from the depot and serves the customers, afixed cost Fk occurs

Nu Maximal number of the private vehiclesNv Lower bound for the minimal number of vehicle used in

the solutionS = (s1, . . . ,sn) A chromosome in the MA, si represents a customer nodeG0 Acyclic auxiliary graph used for splitting the chromosome

to solution to the problemV0 Vertex set in G0

A0 Directed arc set in G0

wi,j,k Weight (distance) of arc (i, j,k) 2 A0

Pi Path from node 0 to node i satisfying time constraints inG0

ðTmi ;W

mi Þ Label set associated with node i 2 V0 , m is the index of the

path which starts from node 0 and ends at node iC(i) = {jj(i, j) 2 A0} Set of successors of node i in G0

EFF(Ri) Efficient labels of Ri, Ri is the labels already in node i, i 2 V0

Pop Population (set of chromosomes) in MAps Number of chromosome in Population PopPopi A chromosomes in the population, 1 6 i 6 pschs Child chromosome selected from crossover results (two

child chromosomes)

3. Problem description and formulation

The COMVRP is described as follows. Let G = (V,A) be a com-pleted graph, where V = {0} [ N is the vertex set and A is the arcset. Vertex 0 represents the depot and N = {1, . . . ,n} correspond tothe set of customers. Each customer is associated with a determin-istic demand qj to be delivered (the depot is assigned a demandq0 = 0). A non-negative distance cij is associated with each arc(i, j) 2 A and represents the travel cost from node i to node j. Thecost matrix is symmetric, i.e., for all i, j 2 V, cij = cji. The use of theloop arc (i, i) is not allowed and defining cii = +1 for all i 2 V.

A fleet of vehicles composed of two types are given: the vehicleof the private fleet (type 0) and the vehicles from external collab-orative carrier (type 1). The number of the vehicles of the privatefleet (type 0) is Nu. The second type of vehicles (type 1) is assumedto be available with infinite supply. Each vehicle of type k(k = 0.1)has the capacity Qk, the travel distance span Hk. Suppose a fixedcost Fk is incurred each time when a vehicle of type k is used,and the vehicle variable cost is equivalent to its traveling distance.The operating cost of a vehicle equals the sum of its fixed cost andvariable cost. Some constraints must be respected as follows.

(1) Each route performed by the private vehicles starts and endsat the depot.

(2) Each route performed by the vehicle from external collabo-rative carrier starts from the depot, and ends at any cus-tomer node.

(3) Each customer is visited exactly once by exactly one vehicle.(4) Each route is assigned to exactly one vehicle.(5) The total demand of all customers served in a route cannot

exceed the capacity of the vehicle assigned to that route.(6) The total traveling distance for a route cannot exceed the

distance span of the vehicle assigned to that route.(7) The number of the private vehicles used in the solution can-

not exceed Nu.

The objective of the problem is to determine the optimal com-bination of the vehicles and the corresponding routes, so as to min-imize the operating cost of all the vehicles, i.e., the sum of the totalvehicle fixed costs and of the total variable costs. To give a cleardefinition to the notations in the paper, Table 1 lists a quick refer-ence for the notation. Some symbols in this table are used in themathematical formulation, while the others concern the algo-rithms introduced in Section 4.

The exact MIP formulation for the COMVRP is given as follows.

� Decision variables:xk

ij if a vehicle of type k travels directly from customer i to cus-tomer j, xk

ij ¼ 1; otherwise xkij ¼ 0.

ui the continuous variable, be an upper bound on the load of vehi-cle upon leaving node i.hi the continuous variable, be an upper bound on the travel dis-tance of vehicle upon leaving node i.� Objective:

MinX

k2K

Fk

Xi2N

xk0i þ

Xk2K

Xi2V

Xj2V

cijxkij �

Xi2N

ci0x1i0 ð1Þ

� Constraints:

Xi2Vi–j

xkij ¼ 1 8j 2 N ð2Þ

Xi 2 V

i–j

xkij ¼

Xi 2 V

i–j

xkji 8j 2 V ; k 2 K

ð3Þ

ui þ qj � 1� xkij

� �M 6 uj 8i; j 2 N; k 2 K; i – j ð4Þ

qi 6 ui 6Xk2K

Xj2V

xkji � Qk 8i 2 N ð5Þ

h0 ¼ 0 ð6Þ

hi þ cij � 1� xkij

� �M 6 hj 8i 2 V ; j 2 N; k 2 K; i – j ð7Þ

hi þ ci0 � 1� x0i0

� �M 6 H0 8i 2 N ð8Þ

0 6 hi 6Xk2K

Xj2V

xkji � Hk 8i 2 N ð9Þ

Xi2N

x00i 6 Nu ð10Þ

xkij 2 f0;1g 8i 2 V ; j 2 V ; i – j ð11Þ

In the objective formulation (1), since each vehicle performs oneroute in the solution the term

Pi2Nxk

0j equals the number of typek vehicle used in the solution. Meanwhile, since for each vehicleof type k its fixed cost Fk occurs when it is used, the first double-sum

Pk2K Fk

Pi2Nxk

0i gives the total fixed costs in the solution. Wecan find if we assume all the vehicles must begin and return tothe depot to complete a route, then the second term representsthe total traveling costs (i.e., the total variable costs), and the firsttwo terms represent the operating cost of all the vehicles. However,

352 R. Liu, Z. Jiang / European Journal of Operational Research 220 (2012) 349–360

because each vehicle hired from the external carrier (i.e., type 1)ends at a customer node and does not return to the depot, the trav-eling costs from the customer nodes back to depot of type 1 vehicles(i.e., the last term

Pi2Nci0x1

i0) need to be subtracted from the firsttwo terms. In other words, the last two terms of objective (1) de-scribe the total variable costs (i.e., traveling distance) of the vehiclesof types 1 and 2. The whole objective (1) is to minimize the sum oftotal fixed costs and variable costs. Constraints (2) state that all thecustomers must be visited by exactly one vehicle. Constraints (3)ensure that if a vehicle arrives at a customer, it must depart fromthe customer. Simultaneously, constraints (3) imply the vehiclearriving and departing balance for the depot node. The next threesets of constraints (4) and (5) are lifted Miller–Tucker–Zemlin(MTZ) subtour elimination constraints for classical VRP, which arefirst proposed by Desrochers and Laporte (1991), and corrected byKara et al. (2004). In our problem these constraints ensure the sub-tour elimination and also account for the vehicle capacity limita-tion. The distance span constraints are given in (6)–(9), where Mstands for a big positive number. These four sets of constraints en-sure that each route cannot exceed the distance span of the vehicleassigned to that route. Constraints (10) imply the number of privatevehicles in the solution is limited by Nu.

4. Memetic algorithm for the problem

MA is also called hybrid GA. It is a population-based approachthat basically combines global crossover operators with localsearch. MA has been widely used to tackle the combinational opti-mization problems. In the section, we design the MA for tacklingthe COMVRP effectively.

4.1. Chromosomes and fitness

The chromosome used in the proposed MA is a sequence of allthe customer nodes, without trip delimiters. A chromosome maybe interpreted as the order in which a vehicle must visit all cus-tomers, just like a solution of the classical TSP. Such a chromosomeis very simple but does not express a COMVRP solution directly. Itmust be partitioned into many sub-routes, each of which corre-sponds to a vehicle route. Prins (2004) provides an elegant ‘Split’algorithm for partitioning the chromosome when tackling the clas-sical VRP. Liu et al. (2009) improve the split algorithm to solve thefleet size and mix vehicle routing problem. However, the algo-rithms of Prins (2004), Liu et al. (2009) cannot be adopted directlyfor the COMVRP due to: (1) the number of private fleet vehicles islimited; (2) the open vehicle route is not considered in the existingliterature. Therefore, an improved split algorithm is proposed asfollows.

Given a chromosome S = (s1, . . . , sn), we build an acyclic auxiliarygraph G0 = (V0,A0), where V0 = {0,1, . . . ,n} is the vertex set and A0 isthe directed arc set. Set A0 contains one arc (i, j,k) 0 6 i, j 6 n,k 2 K, if a vehicle of type k serving customers si+1–sj is feasible interms of following two sets of constraints, which represent theconstraints of vehicle capacity and traveling distance span in ourCOMVRP, respectively:

Xj

r¼iþ1

qSr 6 Qk ð12Þ

Hk Pc0;siþ1 þ

Pj�1

r¼iþ1ðcsr;srþ1Þ þ csj;0 if k ¼ 0

c0;siþ1 þPj�1

r¼iþ1ðcsr;srþ1Þ if k ¼ 1

8>>><>>>:

ð13Þ

Note that there may be more than one arc from i to j in graph G0.The weight wi,j,k of arc (i, j,k) is equal to the serving cost by a vehicleof type k from customer nodes si+1 to sj, i.e., the fixed cost plus thevariable cost:

wi;j;k ¼c0;siþ1 þ

Pj�1

r¼iþ1ðcsr;srþ1Þ þ csj;0 þ F0 if k ¼ 0

c0;siþ1 þPj�1

r¼iþ1ðcsr;srþ1Þ þ F1 if k ¼ 1

8>>><>>>:

ð14Þ

Then, in auxiliary graph G0, let each arc (i, j, 0) has a travel time tij = 1,and each arc (i, j,1) has a travel time tij = 0. The optimal partition of achromosome is a constrained shortest path problem (CSP) in G0: defin-ing the shortest path from node 0 to node n, while the maximumallowable time to transverse all the paths is Pv, which equals thenumber of the private fleet vehicles. Then, the best solution for theCOMVRP respecting the customer sequence of (s1, . . . ,sn) can beobtained for the shortest path in G0, and the fitness of the chromo-some S = (s1, . . . ,sn) is the length of the shortest path in G0. To moreclearly describe the chromosome partition method in the COMVRP,an example is shown in Fig. 1. The top of Fig. 1 shows a sequence ofthree customer nodes: (a,b,c) with Q0 = Q1 = 2, H0 = 30, H1 = 20,F0 = 5, F0 = 50, Nu = 1 and all customer demands are 1. In the middlepart of Fig. 1, each arc in the auxiliary graph G0 represents a possibleroute. The dashed arcs A1–A4 corresponds to all possible externalcarrier vehicle routes and the solid arcs A5–A8 represent all possibleroutes served by private vehicles. The traveling distance, total cost(i.e., the sum of traveling distance and fixed cost), and traveling timefor each arc is shown in the bracket. The shortest path with travelingtime no more than 1 from node 1 to node 4 in G0 consists of arcs A7and A4. Note that traveling time of the shortest path must be nomore than 1, because there is only one private vehicle can be used.Here, arc A7 represents the route (depot ? a ? b ? depot) and A4represents the open route (depot ? c). Finally, the low part inFig. 1 shows the resulting COMVRP solution with two routes.

The CSP is NP-hard. However, in general the CSP can be solvedfairly quickly even for large problem instances. In this paper, basedon the method of Desrosiers et al. (1983), a labeling algorithm isdesigned to tackle the CSP in the forgoing auxiliary graph G0.

With each path Pi from node 0 to node i satisfying the time con-straints (i.e., time consuming is no more than Nu) is associated atwo dimensional labels (time, cost) corresponding to the arrivingtime at node i and the cost of path Pi, respectively. A set of labelsTm

i ;Wmi

� �ðm ¼ 1;2; . . .Þ are associated with each node i, which

indicate the characteristics of mth path from node 0 to node i.The indices m and i may be dropped when the context is unambig-uous. The labels are calculated iteratively along the pathPi = (i0, i1, . . . , iL):

ðTi0 ;Wi0Þ ¼ ð0;0ÞðTil ;Wil Þ ¼ ðTil�1

þ til�1; il ;Wil�1þwil�1; il Þ; l ¼ 1;2 . . . L

Definition 1. Let T1i ;W

1i

� �and T2

i ;W2i

� �at a node i be two

different labels from two paths from node 0 to node i, the first labeldominates the second one, i.e., T1

i ;W1i

� �� T2

i ;W2i

� �iff T1

i ;W1i

� ��

T2i ;W

2i

� �6 ð0;0Þ.

Definition 2. A label (Ti,Wi) at a node i is said to be efficient if noother labels at i dominate it. Its corresponding path from 0 to i iscalled an efficient path of node i.

Let Ri to be the set of all labels of node i, we define EFF(Ri) to bethe efficient labels among the set of Ri of node i. For each node, weonly keep the labels undominated by any other labels of this node.The shortest path from 0 to i satisfying the time constraints is ob-tained directly from the least cost label among the set of efficientlabels at node i.

R. Liu, Z. Jiang / European Journal of Operational Research 220 (2012) 349–360 353

Let C(i) = {jj(i, j) 2 A0} be the set of successors of node i, andRi ¼ [m Tm

i ;Wmi

� �� �be the set of efficient labels of node i. The basic

operation in labeling algorithm is the treatment and comparison ofthe labels. It consists of creating new labels at nodej 2 C(i) by add-ing arc (i, j) to the path from 0 to i associated with label Tm

i ;Wmi

� �.

The new label for each j 2 C(i) is:

fði;jÞ Tmi ;W

mi

� �¼ Tm

i þ tij;Wmi þwij if Tm

i þ tij 6 Pv

£ otherwise

�

Thus, the set of new efficient labels at node j is given by[mfði;jÞ Tm

i ;Wmi

� �. Let Rj represent the set of already existing labels of

node j, the new set of efficient labels at each node j 2 C(i) is given by:

EFFð[mfði;jÞ Tmi ;W

mi

� �[ RjÞ

The labeling algorithm for chromosome partition can be de-scribed as Algorithm 1. For the given chromosome S = (s1, . . . ,sn),the algorithm does not generate the auxiliary graph explicitly.The optimal partition result of S is the least cost among the labelsat node sn.

Algorithm 1. Splitting algorithm for the COMVRP

T0s0¼ 0; W0

s0¼ 0;

for i:¼1 to n,the label for each node i: Rsi ¼£

end forfor i:¼1 to n

dis0 = 0;dis1 = 0;cost0 = 0;cost1 = 0;load = 0;j = i;doload = load + q(j)

if (i==j)cost0 ¼ c0;sj

þ csj ;0 þ F0; dis0 ¼ c0;sjþ csj;0

cost1 ¼ c0;sjþ F1; dis1 ¼ c0;sj

elsecost0 ¼ cost0 � csj�1;0 þ csj�1;sj

þ csj;0; dis0 ¼ dis0�csj�1;0 þ csj�1;sj

þ csj ;0

cost1 ¼ cost1 þ csj�1;sj; dis1 ¼ dis1 þ csj�1;sj

end ifm = 1

While (label Tmsi;Wm

si

� �is efficient)

do

f 1ðsi ;sjÞ Tm

si;Wm

si

� �¼ f 12ðsi ;sjÞ Tm

si;Wm

si

� �¼£

if (load 6 Q0) and (dis0 6 H0) and Tmsiþ 1 6 Nu

� �

f 1ðsi ;sjÞ Tm

si;Wm

si

� �¼ Tm

siþ 1;Wm

siþ cost0

� �end ifif (load 6 Q1) and (dis1 6 H1)

f 2ðsi ;sjÞ Tm

si;Wm

si

� �¼ Tm

si;Wm

siþ cost1

� �end if

Rsj ¼ EFF f 1ðsi ;sjÞ Tm

si;Wm

si

� �[ f 12ðsi ;sjÞ Tm

si;Wm

si

� �[ Rsj

� �m ++

end doj ++;

While (j 6 n) and ((load 6 Q0) or (load 6 Q1)) and((dis0 6 H0) or (dis1 6 H1)))

end for

4.2. Population structure and initialization

The population is an array Pop of ps chromosomes sorted inincreasing cost order. The best solution is denoted as Pop1 andthe worst one is Popps. First, the initial population is filled by savingalgorithm solutions and random permutations of customers. Thesaving algorithm is first proposed for the classical Close-VRP(Clarke and Wright, 1964). Golden et al. (1984) extend the conceptof savings to include vehicle fixed costs. Meanwhile, some modi-fied saving algorithms are used to generate the initial solutionsin the initialization phase of metaheuristics for the Closed VRPand the Open VRP (Repoussis et al., 2010). The basic saving algo-rithm (Clarke and Wright, 1964) cannot be adopted for solvingthe COMVRP directly. Therefore, we create the first two chromo-somes as following steps.

Step 1. Assuming that in the COMVRP the number of hired vehi-cles is 0 and private vehicles are supplied infinitely, theproblem becomes the Close-VRP;

Step 2. We solve this Close-VRP with the extended saving algo-rithm, which extends the concept of saving to include thevehicle fixed costs. First, each customer is served by a ded-icated vehicle. When two routes (0, . . . , i,0) and (0, j, . . . ,0),are merged into a new route (0, . . . , i, j, . . . ,0), a correspond-ing saving value ci0 + c0j + F0 is occurred. We combine thetwo routes which are associated with the maximal savingvalue, when the vehicle capacity and traveling distance arenot broken by the new route.

Step 3. Repeat the routes combination iteration until we cannotcombine any two routes, getting a solution consists ofmany closed routes.

Step 4. All the closed routes are concatenated into the firstchromosome.

Step 5. Suppose that in the COMVRP the number of hired vehiclesis not limited and the number of private vehicles is 0, theproblem is transformed into an Open-VRP.

Step 6. Solve the resulting Open-VRP with the similar saving algo-rithm. Two open routes (0, . . . , i) and (0, j, . . . , l) can bemerged into a new route (0, . . . , i, j, . . . , l) to get a savingvalue c0j � cij + F1. As in Step 2, we repeat combining thetwo routes associated with the maximal saving value, untilwe cannot combine any two routes.

Step 7. Get a solution to the resulting Open-VRP, and concatenateall the routes in the Open-VRP solution into the secondchromosome.

Besides two chromosomes gotten by the simple heuristic, eachother chromosome is initialized as a random permutation of cus-tomers. When all the initial chromosomes are generated, the splitalgorithm (described in above section) is applied to each chromo-some s to get the best COMVRP solution and corresponding solu-tion cost, respecting the customer sequence in s. Meanwhile, thefitness of chromosome s is also identified, which equals the corre-sponding solution cost.

It is shown that perverting the diversity of MA population isnecessary, since it can diminish the risk of premature convergence(Hertz and Widmer, 2003). Many methods were proposed to con-trol the solutions diversity (Sörensen and Sevaux, 2006; Camposet al., 2005). In this research, a simple dispersal rule in the objec-tive space is adopted, i.e., the fitness of any two chromosomesmust be different. This rule is enforced in the initial populationand for each new child chromosome. After generating the initialchromosomes, we check the diversity of the population. If therequirement is not satisfied, the violated chromosomes arereplaced by randomly generated chromosomes. The procedure is

Parent 1 : 5 7 2 9 3 1 4 6 8 10

Parent 2 : 8 9 1 3 2 6 5 10 4 7

Child 1 : 8 6 2 9 3 1 4 5 10 7

Child 2 : 7 9 1 3 2 6 5 4 8 10

Fig. 2. Example of OX procedure in the MA.

10

10 10

15

10a

b

c

depot

4321

A1(20,80,0)

A2(10,60,0) A3(10,60,0)

A4(10,60,0)

A5(20,25,1)A6(20,25,1)

A7(30,35,1)A8(20,25,1)

a

b

cprivate vehicledistance=30

cost=35hired vehicledistance=10

cost=60

depot

Fig. 1. Example of split algorithm in MA.

354 R. Liu, Z. Jiang / European Journal of Operational Research 220 (2012) 349–360

repeated until all chromosomes in the population satisfy the diver-sity requirement.

4.3. Selection and crossover

Since we do not use route delimiters in the chromosome to en-code COMVRP solution, the classical crossover methods can beadopted directly in the proposed MA. In the genetic algorithmsfor solving the traveling salesman problem (TSP), several crossoveroperators have been proposed and widely used, such as crossoversof linear order crossover (LOX) (Falkenauer and Bouffouix, 1991),

order crossover (OX) (Oliver et al., 1987), partial-mapped crossover(PMX) (Goldberg and Robert Lingle, 1985). In these crossovermethods, OX is designed for circular permutations. In our studythe chromosome likes a giant circular trip visiting all the customernodes. Thus, OX should be better than other methods, such as LOXwhich is designed for linear chromosomes. We tested severalcrossover methods, i.e., OX, LOX and PMX, in the proposed MA.Our preliminary experiments confirmed that OX is slightly supe-rior to other classical crossovers.

In the OX crossover, two chromosomes and are selected to beparents based on the tournament selection. In the tournamentselection, firstly, several chromosomes are chosen randomly fromthe population. Then, the best (i.e., the least cost) chromosome isselected to be the first parent chromosome. The procedure is re-peated to select the second one. With the increase in tournamentsize, weak chromosomes will have smaller chance to be selectedin the tournament selection. In this study, the tournament sizeequals to 2.

Then, OX constructs the child chromosomes from parents. Asshown in Fig. 2, a substring is selected from the first parent chro-mosome randomly. The substring is copied into the correspondingpositions in the first child, and those genes in the substring are de-leted from the second parent chromosome. The resulting genes inthe second parent chromosome form a sequence and are placed

i

i+1

i

j

i+1j+1

j

j+1

i

i+1

j

j+1

Fig. 3. Example of 2-opt procedure.

R. Liu, Z. Jiang / European Journal of Operational Research 220 (2012) 349–360 355

into the empty positions of the first child from left to right. At last,two parents are exchanged and the other child is produced by thesame procedure.

One child chromosome chs is selected at random from OX re-sults, and undergoes local search methods (described in Section4.4). Then, this chromosome replaces a mediocre chromosomePopm in the population iff: (1) the cost of this child is smaller thanthe worst chromosome in the population; (2) this child does notviolate the diversity of the population. Popm is randomly selectedin the worst 50% of the population Pop.

4.4. Local search as mutation

The local search (LS) is employed to the child chromosome cs

with a probability q. LS operates on a COMVRP solution with routedelimiters and not on the chromosome itself. In the interest ofeffectiveness and efficiency, three simple procedures are adoptedin this study. They are applied to one and two routes.

1. 2-opt: the 2-opt move is implemented in one route (closed oropen route), or applied to two routes simultaneously (twoclosed routes, or two open routes, or the combination of closedand open routes). The descriptions and pictorial illustrations of2-opt for one or two closed routes can be found in Irnich et al.(2006), and the 2-opt for one or two open routes can be found inTarantilis et al. (2005). The example in Fig. 3 shows how 2-opt isapplied to a close route and an open route. As shown in the left-hand part of Fig. 3, arc (j, j + 1) and arc (i, i + 1) are two arcsbelonging to close route r1 and open route r2, respectively.The middle-hand and right-hand of Fig. 3 illustrate two possibleimprovement cases: arcs (i, i + 1) and (j, j + 1) are replaced byarcs (j, i) and (j + 1, i + 1), and two arcs are replaced by arcs(i, j + 1) and (j, i + 1). For the first case, some segments of theroutes are reversed.

2. 1–1 Exchange: two nodes are swapped.3. 1–0 Exchange: one node is deleted and reinserted into after the

second node.

Note for 1–1 exchange and 1–0 exchange, two nodes maybe be-long to the same route, or two different routes. While one feasibleand less cost solution is found during the LS procedure, it isadopted as the new seed solution for repeating the LS. The LS stopsuntil no additional improvement can be obtained.

4.5. General structure of MA

In this section, the general structure of the proposed MA is de-scribed in Algorithm 2. The initial dispersal population is built inlines 2–6. The MA performs nc main phases, as shown in the lines7–30. Each main phase consists of a number of short phases (lines9–29), followed by a partial replacement procedure (line 29). Ineach iteration of the short phase, MA first sort the chromosomesin increasing order of costs, then select two parent chromosomes,crossover them and get a child chs (lines 11–14). Chromosomechs is improved by LS algorithms with the probability q (lines

15–20), and inserted into the Pop, iff chs is better than the worstchromosome in the Pop and do not break the population diversity(lines 21–25). Each short phase in MA stops after a given number ofiterations na or after a given nb times without improving the bestsolution. When a short phase of the MA terminates, a partial re-newal of the population is executed: two best chromosomes arekept in incumbent population, while the other are replaced bynew random chromosomes. Finally, the best solution of the MAis extracted from the first chromosome.

Algorithm 2. General structure of Memetic Algorithm

1: Input Algorithm parameters: population size ps, iterationtime na, nb, nc, and q

2: set population Pop: = £

3: Pop = Pop[ two chromosomes generated by savingalgorithm

4: repeat5: Pop = Pop[ an random chromosome6: until (jPopj==ps) and ("1 6 i, j 6 ps and

i – j, jcost(Popi) � cost(Popj)j > 0)7: for i:¼1 to nc8: i = 0 and j = 09: repeat10: i = i + 111: sort Pop in increasing cost order12: select two different parent chromosomes from Pop

by tournament selection13: apply OX operator to parent chromosomes,

randomly select a chromosome chs from two offspringchromosomes

14: apply split algorithm to chs, to calculate its costcost(chs)

15: if a random number between 0 and 1 is less than qthen

16: extract chromosome chs to solution s17: improve s with LS methods18: converted s into the mutated chromosomes ch0s19: chs :¼ ch0s20: end if21: randomly select a chromosomes Popw in the worst

50% of the population22: if (cost(chs) < cost(Popps)) and ("1 6 i 6 ps and

i – w, jcost(chs) � cost(Popi)j > 0)23: replace Popw by chs

24: j = 025: else26: j = j + 127: end if28: Until (i==na) or (j==nb)29: Partially replace the population Pop30: End for31: Extract the best solution from the first chromosome in Pop

356 R. Liu, Z. Jiang / European Journal of Operational Research 220 (2012) 349–360

5. Computational experiments

Computational experiments were performed to assess the per-formance of the proposed MA. First the algorithm implementationand the test instances used in these experiments are described.This is followed by the presentation and discussion of the experi-mental results.

5.1. Implementation and test instances

All the algorithms were implemented in C and tested on an IntelXeon E552 2.26 GHz computer with 32 GB memory, running Linux.Since there is no any prior work on the COMVRP, new test in-stances are created based on several well-known benchmarks,i.e., (Augerat et al., 1995) Set A, B, P Instances and Christofidesand Eilon Set E Instances for the CVRP (download from http://branc-handcut.org/vrp/data), and Daniele Vigo Set D Instances for theDCVRP (download from http://www.coin-or.org/symphony/branc-handcut/vrp/data). The proposed MA is implied in some of theabove instances (6 from Set A, 5 from Set B, 5 from Set E, 2 fromSet P, and 4 from Set D). According to the number of the customers,the instances chosen from first four sets are small-scale (contains20–80 customers), while the instances picked up from Set D In-stances are large-scale instances (contains 101–361 customers).

Based on each set of original instances, we created two sets ofinstances for the COMVRP. For example, Augerat et al. (1995) pro-vides the node locations, customer demands, vehicle capacity QA,and the number of routes Nr for Set A instances in its solutions.For the first new instance for the COMVRP, called Set A-1, we setQ0 = Q1 = QA, F0 = 50, F1 = 150, Nu = dNr/2e, and the vehicle distancespan is defined as:

Hk ¼ 0:2� 2�maxi2Vfc0ig þ 0:8� rcw k ¼ 0; 1 ð15Þ

where rcw is the maximum route distance in saving algorithm solu-tion (i.e., the method for the first initial chromosome, described inSection 4.2). All the new instances generated according to this pro-cedure are called Type1 instances.

Meanwhile, based on Augerat et al. (1995) Set A instances, weset Q0 = Q1 = +1, F0 = 50, F1 = 150, Nu = 1, and

Hk ¼ 1:5�maxi2Vfc0ig k ¼ 0;1 ð16Þ

Table 2Computational result for Type1 instances based on two formulations.

Instance H Nu Nv COMVRPF1

LBF1 ObjF

A-n32-k5-1 246.89 3 5 443.40 10A-n33-k5-1 176.68 3 5 448.88 10A-n33-k6-1 175.83 3 6 478.49 12A-n34-k5-1 201.73 3 5 477.05 11A-n36-k5-1 236.53 3 5 470.00 12B-n50-k8-1 232.11 4 8 491.67 19B-n51-k7-1 186.69 4 7 274.24 15B-n52-k7-1 158.17 4 7 205.21 12B-n56-k7-1 186.22 4 7 217.39 12B-n57-k7-1 197.65 4 7 241.01 18E-n22-k4-1 110.47 2 4 538.55 7E-n30-k3-1 215.12 2 3 343.29 7E-n33-k4-1 262.48 2 4 375.79 11E-n51-k5-1 135.77 3 5 425.58 11E-n76-k10-1 129.29 5 10 541.76 18P-n21-k2-1 132.10 1 2 313.19 3P-n22-k8-1 109.71 5 8 623.43 11D101-09c-1 230 5 8 631.36 18D151-14b-11 200 7 12 699.71 29D281-08k-1 1500 4 7 7485.30 104D361-09k-1 1300 5 8 8623.64 123Average 1159.47 22

a Asterisks denote optimal solution.

Then, we create a new instance Set A-2. All the new instancesgenerated according to this method are called Type2 instances.Note that Set D instances provide the vehicle travel distance spans,which are adopted directly in the new instances Set D-1 and Set D-2.

5.2. Computational results

To assess the performance of MA, the COMVRP formulation de-scribed in Section 3, called COMVRPF1 was implemented in CPLEX12.2 with a time limit of 48 h. It is found that CPLEX cannot reachan optimal solution even for any small-scale instance. Further-more, we find that for each instance the CPLEX lower bound isfar away from its best solution result. To evaluate the quality ofMA results, we improve the performance of CPLEX by adding aset of valid constraints.

Observe following three sets of inequalities (17)–(19) are validfor the COMVRP. Inequalities (17) bound the number of vehiclesi.e., private and hired vehicles, in the aspects of vehicle capacityconstraints. Inequalities (18) and (19) bound the number of privateand hired vehicles in the aspects of distance span, respectively.X

i2N

xk0i P

Xi2V

Xj2V

qi � xkij=Q k k 2 K ð17Þ

Xi2N

x00i P

Xi2V

Xj2V

cij � x0ij=H0 ð18Þ

Xi2N

x10i P

Xi2V

Xj2N

cij � x1ij=H1 ð19Þ

Let Nv be a lower bound for the minimum number of vehicles, itcan be obtained from the formulation, called COMVRPNv: followingobjective (20) subject to constraints (2)–(10), (17)–(19) and linearrelaxation of (11).

MinXk2K

Xi2N

xk0i ð20Þ

Obviously, Nv can be set to be dP

j2Nxdje. Then, we find thatCPLEX provides much better results for the new formulation COM-VRPF2: objective (1) subject to constraints (2)–(11) and followingconstraints (21).

COMVRPF2

1 GapF1 (%) LBF2 ObjF2 GapF2 (%)

63.53 58.31 1026.91 1063.53 3.4411.15 55.61 906.42 1011.15 10.3615.98 60.65 1071.42 1171.35 8.5330.31 57.79 972.29 1109.61 12.3824.60 61.62 963.03 1106.55 12.9702.69 74.16 1386.47 1756.27 21.0689.18 82.74 1211.32 1586.59 23.6579.72 83.96 940.85 1263.50 25.5416.02 82.12 921.14 1172.21 21.4203.90 86.64 1009.31 1801.78 43.9822.19 25.43 700.94 700.94a 0.0038.60 53.52 622.46 733.24 15.1103.78 65.95 944.27 1076.90 12.3246.88 62.89 886.47 933.37 5.0376.43 71.13 1566.21 1847.55 15.2383.46 18.33 383.46 383.46a 0.0094.56 47.81 1183.87 1183.87a 0.0068.57 66.21 1365.38 1476.68 7.5474.16 76.47 1856.21 2699.22 31.2357.02 28.42 8418.28 10198.94 17.4662.90 30.25 9682.37 12208.27 20.6998.36 59.52 1810.43 2213.57 14.66

Table 3Computational result for Type2 instances based on two formulations.

Instance H Nu Nv COMVRPF1 COMVRPF2

LBF1 ObjF1 GapF1 (%) LBF2 ObjF2 GapF2 (%)

A-n32-k5-2 152.12 1 3 473.58 1079.90 56.15 839.06 1079.90 22.30A-n33-k5-2 110.03 1 3 453.93 1210.95 62.52 782.60 1074.22 27.15A-n33-k6-2 107.77 1 4 525.54 1115.35 52.88 926.39 1115.35 16.94A-n34-k5-2 114.24 1 4 496.45 1142.05 56.53 951.99 1136.46 16.23A-n36-k5-2 159.23 1 3 496.65 988.52 49.76 835.17 901.45 7.35B-n50-k8-2 169.53 1 2 408.67 997.71 59.04 556.55 894.88 37.81B-n51-k7-2 107.55 1 3 289.79 1179.30 75.43 696.01 1178.24 40.93B-n52-k7-2 106.68 1 2 225.34 1037.53 78.28 413.75 951.53 56.52B-n56-k7-2 136.85 1 2 219.11 738.63 70.34 426.59 658.03 35.17B-n57-k7-2 143.01 1 2 229.11 1313.55 82.56 443.00 917.97 51.74E-n22-k4-2 74.05 1 3 344.19 866.04 60.26 644.25 861.51 25.22E-n30-k3-2 103.15 1 3 357.44 1040.63 65.65 704.54 989.33 28.79E-n33-k4-2 178.30 1 2 385.15 909.60 57.66 622.14 898.24 30.74E-n51-k5-2 65.90 1 6 425.89 1843.28 76.89 1216.98 1481.70 17.87E-n76-k10-2 64.90 1 8 538.55 2457.61 78.09 1626.67 1896.53 14.23P-n21-k2-2 65.90 1 3 194.22 627.90 69.07 555.82 627.90 11.48P-n22-k8-2 74.05 1 3 408.71 861.51 52.56 643.76 861.51 25.28D101-09c-2 230 5 3 631.57 1623.31 61.09 794.07 1343.51 40.90D151-14b-2 200 7 3 701.49 2394.44 70.70 863.46 1570.72 45.03D281-08k-2 1500 4 5 7484.84 13485.79 44.50 8071.02 10323.93 21.82D361-09k-2 1300 5 7 8644.67 15742.30 45.09 9506.73 14910.39 36.24Average 1139.76 2507.42 63.10 1529.55 2174.92 29.04

R. Liu, Z. Jiang / European Journal of Operational Research 220 (2012) 349–360 357

Xk2K

Xi2N

xk0i P Nv ð21Þ

Table 2 shows the CPLEX results for Type1 instances based on for-mulation COMVRPF1 and COMVRPF2 within a time limit of 48 h. Ta-ble 3 shows the same CPLEX results for Type2 instances. In Tables2 and 3, the four first columns give the test instance, the vehicle dis-tance span H (for these test instances, H = H0 = H1), the number ofprivate vehicles and the lower bound for the number of vehicles ob-tained by formulation COMVRPNv. Columns 5–7% present the CPLEXresults of the COMVRP based on formulation COMVRPF1, i.e., thelower bound LBF1, the best solution ObjF1, and the percentage gapGapF1 between them. Columns 8–10 provide the corresponding re-sults based on formulation COMVRPF2. As shown in Tables 2 and 3,we find that although CPLEX cannot reach an optimal solution formost test instances, obviously formulation COMVRPF2 gives much

Table 4Computational results for CPLEX and MA for small-scale instances of Type1.

Instance Q H Nu Nv CPLEX

LB Obj

A-n32-k5-1 100 246.89 3 5 1026.91 1063.5A-n33-k5-1 100 176.68 3 5 906.42 1011.1A-n33-k6-1 100 175.83 3 6 1071.42 1171.3A-n34-k5-1 100 201.73 3 5 972.29 1109.6A-n36-k5-1 100 236.53 3 5 963.03 1106.5B-n50-k8-1 100 232.11 4 8 1386.47 1756.2B-n51-k7-1 100 186.69 4 7 1211.32 1586.5B-n52-k7-1 100 158.17 4 7 940.85 1263.5B-n56-k7-1 100 186.22 4 7 921.14 1172.2B-n57-k7-1 100 197.65 4 7 1009.31 1801.7E-n22-k4-1 6000 110.47 2 4 700.94 700.9E-n30-k3-1 4500 215.12 2 3 622.46 733.2E-n33-k4-1 8000 262.48 2 4 944.27 1076.9E-n51-k5-1 160 135.77 3 5 886.47 933.3E-n76-k10-1 140 129.29 5 10 1566.21 1847.5P-n21-k2-1 160 132.10 1 2 383.46 383.4P-n22-k8-1 3000 109.71 5 8 1183.87 1183.8Average 1170.7

a Asterisks denote optimal solution.

better results compared to COMVRPF1. The lower bounds for allthe test instances are greatly improved. In terms of CPLEX solutioncosts, formulation COMVRPF2 are more promising than COMVRPF1,with 18 better solutions out of 22 (including three optimal solu-tions) for Type1 instances, with 18 better solutions for Type2instances.

Tables 4 and 5 compares the solution costs obtained by the pro-posed MA and CPLEX (with formulation COMVRPF2 and within atime limit of 48 h) for Type 1 instances. The MA performs withthe parameters: Ps = 30, Na = 5000, Nb = 3000, Nc = 6 and q = 0.2.Because MA belongs to the class of metaheuristics and is adaptiveprocedures inspired by principles of natural selection and survivalof the fittest, for some hard problems it cannot always get its ‘best’solution at each execution. To measure the accuracy of the solu-tion, the consistency and fluctuation of the accuracy, the wholeMA was executed 10 times on each instance. The MA best solutionvalue (MAb), the MA average solution value (MAavg), and average

MA Gap (%)

MAB MAAvg time Gap1 Gap2 Gap3

3 1063.53 1066.24 14.8 3.44 0.00 0.255 1004.49 1009.90 16.2 9.76 0.66 0.545 1170.88 1177.26 15.3 8.49 0.04 0.541 1102.98 1109.30 19.5 11.85 0.60 0.575 1096.35 1099.12 21.6 12.16 0.92 0.257 1739.95 1747.07 40.9 20.32 0.93 0.419 1494.54 1502.15 42.1 18.95 5.80 0.510 1208.93 1213.80 46.9 22.17 4.32 0.401 1169.84 1176.06 58.1 21.26 0.20 0.538 1620.51 1659.12 59.2 37.72 10.06 2.334a 700.94a 701.98 6.0 0.00 0.00 0.154 733.24 741.35 12.1 15.11 0.00 1.090 1065.46 1067.11 14.6 11.37 1.06 0.157 929.70 936.76 42.8 4.65 0.39 0.755 1750.26 1764.70 121.9 10.52 5.27 0.826a 383.46a 383.46 4.1 0.00 0.00 0.007a 1183.87a 1187.37 7.9 0.00 0.00 0.290 1142.29 1149.57 32.0 12.22 1.78 0.56

Table 5Computational results for CPLEX and MA for large-scale instances of Type1.

Instance Q H Nu Nv CPLEX MA Gap (%)

LB Obj MAB MAAvg time Gap1 Gap2 Gap3

D101-09c-1 200 230 5 8 1365.38 1476.68 1476.33 1481.11 290.0 7.52 0.02 0.32D151-14b-1 200 200 7 12 1856.21 2699.22 2057.53 2058.95 788.1 9.78 23.77 0.07D281-08k-1 900 1500 4 7 8418.28 10198.94 9041.52 9063.66 1887.3 6.89 11.35 0.24D361-09k-1 900 1300 5 8 9682.37 12208.27 11104.70 11116.27 3419.6 12.81 9.04 0.10Average 5920.02 5930.00 5920.02 5930.00 1596.3 9.25 11.05 0.19

358 R. Liu, Z. Jiang / European Journal of Operational Research 220 (2012) 349–360

running time (in seconds) are gotten from these 10 runs andreported in Tables 4 and 5. Note that for each instance, CPLEXran until finding an optimal solution or until exhausting a prede-termined computation time of 48 h. The last three columns showthe gaps between the MA solution value and CPLEX result, andthe gaps between MA best solution value and average solution va-lue: Gap1 represents the gap between MAb and CPLEX lower boundLB, which is computed as Gap1 = 100% � (MAb � LB)/MAb. Gap2 isthe gap between MAb and CPLEX result, computed as Gap2 = 100 �(Obj �MAb)/Obj. Gap3 is the gap between MAavg and MAb, computedas Gap3 = 100 � (MAavg �MAb)/MAavg. Similarly, the computationalresults for the second type instances are shown in Tables 6 and 7,where the MA parameters for the first type instances are usedagain.

From the figures in foregoing Tables 4–7, we can assert thefollowing:

1. As shown in Tables 4 and 6, we first find that for small-scale testinstances in all cases the MA dominates the CPLEX. For small-scale test instances, the CPLEX can get optimal solutions forthe three smallest instances (E-n22-k4-1, P-n21-k2-1 and P-n22-k8-1) within the time limit, where the objective valuesreach the lower bounds. Meanwhile, the proposed MA obtainsthe optimal solutions for these three instances within a shortrunning time as well. The CPLEX objective values cannot reachthe lower bounds for all other test instance, while MA identifiesbetter or equal solutions quickly. MA and CPLEX find five equalsolutions for the first type instances, including three optimalsolutions, and 12 equal solutions for the second type instances.For the first type small-scale test instances the average devia-tion between MA best solution and CPLEX objective values is1.78%; and for the second type small-scale instances the

Table 6Computational results for CPLEX and MA for small-scale instances of Type2.

Instance H Nu Nv CPLEX

LB Obj

A-n32-k5-2 152.12 1 3 839.06 1079.90A-n33-k5-2 110.03 1 3 782.60 1074.22A-n33-k6-2 107.77 1 4 926.39 1115.35A-n34-k5-2 114.24 1 4 951.99 1136.46A-n36-k5-2 159.23 1 3 835.17 901.45B-n50-k8-2 169.53 1 2 556.55 894.88B-n51-k7-2 107.55 1 3 696.01 1178.24B-n52-k7-2 106.68 1 2 413.75 951.53B-n56-k7-2 136.85 1 2 426.59 658.03B-n57-k7-2 143.01 1 2 443.00 917.97E-n22-k4-2 74.05 1 3 644.25 861.51E-n30-k3-2 103.15 1 3 704.54 989.33E-n33-k4-2 178.30 1 2 622.14 898.24E-n51-k5-2 65.90 1 6 1216.98 1481.70E-n76-k10-2 64.90 1 8 1626.67 1896.53P-n21-k2-2 65.90 1 3 555.82 627.90P-n22-k8-2 74.05 1 3 643.76 861.51Average 1030.87

average deviation is 1.23%. For all 34 small-scale test instancesthe average gap between MA best solution and CPLEX objectivevalues is 1.50%.

2. For large-scale test instances, the CPLEX only provides poorsolutions. But MA performs still well, whose solutions are muchbetter than the CPLEX’s. For the first and the second type testinstances, the average gaps between MA best solution andCPLEX objective values are 11.50% and 26.78%, respectively.These results show that for large-scale test instances, there isa significant gap between MA and CPLEX solutions with respectto the solution quality. Furthermore, we find the solutionsobtained by MA are reasonably since the gaps between MAsolutions and CPLEX lower bounds are acceptable. For example,for first type large-scale test instances, the maximal and aver-age gaps between MA best solution and lower bounds (providedby CPLEX) are 12.68% and 9.25%. For the second type large-scaletest instances, the average gap between MA best solution andlower bounds is 12.81%.

3. In term of running time, MA increases rapidly with the increaseof customer nodes in the test instances. But even for largest-scale instances, MA proposed in this paper can provide high-quality solutions within a reasonable computing time span. Asshown in above Tables 4–7, for small-scale test instances theaverage running time of the MA is 22.2 s. And for all large-scaleinstances the average MA running time is 27.5 min. Clearly, allthese MA computing CPU times are much shorter that the run-ning times of CPLEX, i.e., a time limit of 48 h.

Therefore, it is evident that the proposed MA outperforms theCPLEX solver in terms of the quality of solutions and computingtime. The superiority of MA is more apparent in large-scale testinstances.

MA Gap (%)

MAB MAAvg time Gap1 Gap2 Gap3

1079.90 1083.10 5.0 22.30 0.00 0.301074.22 1074.22 6.0 27.15 0.00 0.001115.35 1117.93 6.0 16.94 0.00 0.231136.46 1146.60 6.2 16.23 0.00 0.88

901.45 901.70 10.0 7.35 0.00 0.03894.88 912.36 19.3 37.81 0.00 1.92

1176.92 1177.66 15.9 40.86 0.11 0.06875.22 880.28 18.7 52.73 8.02 0.57656.86 660.31 28.6 35.06 0.18 0.52846.81 848.91 25.0 47.69 7.75 0.25861.51 861.51 2.1 25.22 0.00 0.00989.33 1021.19 5.7 28.79 0.00 3.12855.12 871.40 6.6 27.25 4.80 1.87

1481.70 1492.55 14.7 17.87 0.00 0.731896.53 1921.57 38.2 14.23 0.00 1.30

627.90 729.10 2.2 11.48 0.00 13.88861.51 862.00 2.1 25.28 0.00 0.06

1019.51 1033.08 12.5 26.72 1.23 1.51

Table 7Computational results for CPLEX and MA for large-scale instances of Type2.

Instance H Nu Nv CPLEX MA Gap (%)

LB Obj MAB MAAvg time Gap1 Gap2 Gap3

D101-09c-2 230 5 3 794.07 1343.51 894.52 905.16 303.9 11.23 33.42 1.18D151-14b-2 200 7 3 863.46 1570.72 1054.55 1058.99 871.9 18.12 32.86 0.42D281-08k-2 1500 4 5 8071.02 10323.93 8933.59 8988.47 1856.4 9.66 13.47 0.61D361-09k-2 1300 5 7 9506.73 14910.39 10830.44 10954.36 3800.1 12.22 27.36 1.13Average 7037.14 5428.28 5476.75 1708.1 12.81 26.78 0.83

R. Liu, Z. Jiang / European Journal of Operational Research 220 (2012) 349–360 359

6. Conclusions and future work

This paper investigates a special optimization problem, Close–Open Mixed Vehicle Routing Problem, in transportation system.The COMVRP can be seen as the combination of Closed-VRP andOpen-VRP. The COMVRP has significant applications in transporta-tion system, especially when a company used both the private andhired vehicles to serve customers. In such case, each private vehi-cle’s route is closed and the hired vehicle’s route is open. The COM-VRP has not been studied in the existing literature.

A mathematical formulation is presented for the COMVRP. Aneffective MA is developed to solve the problem. In the proposedMA, saving algorithm is adopted to provide two good initial solu-tions. The chromosome is encoded without trip delimiters. Theprocedure of evaluating the chromosome is transformed into theconstrained shortest path problem, which can be solved quicklyin practice. The MA performance is tested in a range of test in-stances, including small-scale instances and large-scale ones. Forthese instances the proposed MA always outperforms the CPLEXSolver applied to an improved COMVRP formulation. For large-sizeinstances, CPLEX may fail to get a feasible solution. However, theproposed MA performs well both in solution quality and computa-tion time.

For the future research, one important direction is to work onthe similar but more difficult variant, the COMVRP incorporatingwith soft or hard time window, i.e. time windows must be obeyed,or that can be violated at a cost. Obviously, the COMVRP with timewindow is very complex and more similar to the real-life scenarios.Efficient algorithm needs to be designed for this challengingproblem.

Acknowledgment

This work was supported by Research Grant from National Nat-ural Science Foundation of China (No. 70872077), and NationalNatural Science Foundation of China/ Research Grants Council ofHong Kong joint research projects (No. 70831160527).

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