Transportation Research Part E 46 (2010) 1071–1085

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Transportation Research Part E

journal homepage: www.elsevier .com/locate / t re

Task selection and routing problems in collaborativetruckload transportation

Ran Liu, Zhibin Jiang *, Xiao Liu, Feng ChenDepartment of Industrial Engineering and Logistics Management, School of Mechanical Engineering, 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 29 August 2009Received in revised form 8 December 2009Accepted 18 March 2010

Keywords:Collaborative transportationFull truckloadTask selectionRoutingMemetic algorithm

1366-5545/$ - see front matter � 2010 Elsevier Ltddoi:10.1016/j.tre.2010.05.003

* Corresponding author. Address: Department of InUniversity, 800 Dong Chuan Rd., 200240 Shanghai,

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

This paper introduces the task selection and routing problem in collaborative transporta-tion in which a truckload carrier receives tasks from shippers and other partners andmakes a selection between a private vehicle and an external carrier to serve each task.The objective is to minimize the variable and fixed costs for operating the private fleet plusthe total costs charged by the external carrier. The mathematical formulation and thelower bound are established. A memetic algorithm is developed to solve the problem.The computational results show that the proposed algorithm is effective and efficient.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Nowadays, in the trucking industry more and more carriers adopt a new transportation model called collaborative trans-portation (CT) to improve logistics performance, reduce system-wide inefficiencies and cut down operational costs. In the CT,many collaborative carriers use practices such as group purchasing and capacity and information sharing to increase eachpartner’s profit (Ergun et al. 2007a,b; Agarwal et al., 2009; Liu et al., 2010). For example, transportation tasks are exchangedamong various carriers instead of each carrier only using internal vehicle to serve his own tasks. In such situations, a carriermay receive two types of transportation tasks, with one given by shippers, while the other by external carriers. Each firsttype task (i.e., given by shippers) must be served by one of vehicles of the internal fleet or by an external collaborative carrier.When it is assigned to a collaborative carrier, a penalty cost is incurred which represents all costs associated with this assign-ment. For each second type task (i.e., given by external partners), the carrier can accept or reject it. Similarly, when an exter-nal carrier’s task is accepted and served by a vehicle of the private fleet, a compensative payment is given by the externalcarrier. Thus, a carrier’ manager has to decide which tasks are selected to be served by private fleet and to route the internalvehicles. Meanwhile, we find that in most practical CT applications, the carriers are required to move goods between spec-ified pairs of nodes with full truckload, i.e., picking up the goods at one node and delivering the goods at the destination.Therefore, we assume that each carrier provides only full truckload transport in this study. The problem can thus be calledthe task selection and routing problem with full truckload. In this paper, we address this special optimization problem. Theobjective is to develop a heuristic algorithm to make a selection of tasks and to route the private vehicles by minimizing atotal cost function.

. All rights reserved.

dustrial Engineering and Logistics Management, School of Mechanical Engineering, Shanghai Jiao TongPR China. Tel./fax: +86 21 34206065.

1072 R. Liu et al. / Transportation Research Part E 46 (2010) 1071–1085

Although the task selection and routing problem with full truckload is practical and important in the CT, the problem hasnot been previously studied in the literature. The closely related reference we are aware of is on a VRP with private fleet andcommon carrier (VRPPC) introduced by Chu (2005), who modeled the problem and solved it through a savings-based heu-ristic, followed by intra-route and inter-route exchanges. Bolduc et al. (2007) proposed a simple heuristic and generated bet-ter results. Bolduc et al. (2008) presented two formulations for the VRPPC, showing that the VRPPC could be formulated as aheterogeneous vehicle routing problem. Meanwhile, they developed a powerful metaheuristic for the VRPPC, which used aperturbation procedure in the construction and improvement phases, and performed a streamlined family of edge ex-changes. Experiments results indicated that the proposed algorithm dominated two previous methods. Recently, Côté andPotvin (2009) described a tabu search algorithm for the VRPPC, whose numerical experimental results showed that the tabusearch performed well on a set of benchmark instances.

The VRPPC studied in these papers differs from our setting with respect to two main issues. First, in the VRPPC the carrierallocates part of customers to the external carrier, and does not receive outside carriers’ tasks. Second, in the VRPPC the car-rier serves node-customers, and thus the problem is formulated as a variant of the node routing problem. In this paper, weassume that the carriers are required to move goods between specified pairs of nodes with full truckload. The correspondingproblem should be formulated as a variant of the Arc Routing Problem (ARP).

The task selection and routing problem with full truckload is NP-hard since it reduces to the rural postman problem (RPP),another NP-hard problem (Lenstra and Rinnooy Kan, 1976), when all the tasks are served by a private vehicle. Many exactalgorithms (Christofides et al., 1986; Corberán and Sanchis, 1994; Letchford and Eglese, 1998; Ghiani and Laporte, 2000) andheuristics (Frederickson, 1979; Pearn and Wu, 1995; Fernández de Córdoba et al., 1998; Corberán et al., 2000; Ghiani et al.,2006; Holmberg, 2010) have been proposed for the RPP and its variations. Although these algorithms have been designed forthe RPP, they cannot be applied directly for our problem because of two major differences between the problems. First, in theRPP all the tasks (required arcs or edges) must be served by private vehicle, while in our problem we have to decide whichtasks are to be entrusted to external carriers, and which tasks from them will be accepted or rejected. Meanwhile, the RPP isa single-vehicle version of our problem, and it only can be considered as a particular case of our problem.

This paper presents a new heuristic algorithm, based on memetic algorithm (MA), for the solution of the task selectionand routing problem with full truckload. MA has been proved to be a successful technique for the solution of related prob-lems such as CVRP (Prins, 2004) and CARP (Lacomme et al., 2004). The proposed algorithm was tested on a range of randomlygenerated instances and showed being able to tackle practical large-scale problem instances with up to hundreds of trans-portation tasks.

The rest of this paper is organized as follows. In Section 2, the problem is defined and transformed into an equivalent noderouting problem, followed by the resulting mathematical formulation presented in Section 3. Section 4 analyses the lowerbound for the problem. Section 5 describes the MA algorithm that was used to solve the problem. Section 6 presents the re-sults from the computational testing of randomly generated problem instances. Section 7 presents our conclusions and fu-ture work.

2. Problem definition and transformation

The task selection and routing problem with full truckload is defined as follows. Let G = (V, A) be a directed Euclideangraph, where V = 0 [ N is the vertex set and A is the arc set. Vertex 0 represents the depot, and the other vertices are thestart-points and end-points of the arcs. Each arc a e A has a non-negative travel distance la. Two subsets A.1 � A andA.2 � A are two types of tasks needed to be served. Each arc (i, j) e A.1, i, j e N represents a task given by shippers, i.e., loadingthe goods at node i, traveling to node j directly and delivering the goods, which is characterized by a penalty cost gij when it isassigned to an external carrier. Each arc (i, j) e A.2, i, j e N represents a task outsourced by other carrier, which is characterizedby a compensative payment eij when it is served by the private vehicle. A fleet of identical private vehicles, initially located atthe depot, is available to serve the tasks. Each vehicle has a maximal distance span H, and an operating cost, i.e., fixed costsplus variable costs. Suppose a fixed cost f is incurred each time when a vehicle is used, and the vehicle variable costs areequivalent to its travel distance. Each private vehicle starts from the depot, serves many tasks and return to the depot.The objective of the task selection and routing problem with full truckload is to serve all required arcs (tasks) at minimaltotal cost, i.e., the sum of fixed vehicle costs, variable costs, penalty costs and subtracting compensative payments.

The proposed approach is based on transforming the original problem in graph G into an equivalent node routing problemin graph G0 = (V0, A0). This type of transformation is of interest since it permits to use state-of-art models and algorithms forthe VRP to solve the ARP (Baldacci and Maniezzo, 2006; Longo et al., 2006; Tagmouti et al., 2007). In our case, the detailedtransformation is the following. First, each arc a e A.1 [ A.2 corresponds to a node a0 in graph G0 with a travel distance la. Eachpair of node (i, j) e G0 is connected by an arc (i, j) e A0 with travel distance cij. The value of cij is calculated in graph G, equalingthe distance from the end node of the first required arc to the start node of the second required arc and includes the secondrequired arc. Let set N0 stands for all the nodes in graph G0. Then, the depot node d0 is added into graph G0. Let set V0 = d0 . [ N0

represents all the nodes in G0. The distance from the depot d0 to node a0 e N0 equals the distance from the depot to the startnode of required arc in graph G, and plus the distance of the required arc. The distance from node a0 e N0 to d0 is equivalent tothe distance from the end node of required arc to the depot node in graph G. Finally, each node a e N0 in graph G0 is associatedwith a pair of cost (ea, ga), where ea and ga are the compensative payment and the penalty cost of node a, respectively. If node

R. Liu et al. / Transportation Research Part E 46 (2010) 1071–1085 1073

a is transformed from an arc (i, j) e A.1 in graph G (i.e., node a is given by shippers), then ea = 0 and ga = gij, which means ifnode a be served by the private fleet, the external carrier will not give compensative payment (ea = 0), and if node a is servedby external carrier, a penalty cost ga (ga = gij) is incurred. Similarly, If node a is transformed from an arc (i, j) e A.2 in graph G,ea = eij and ga = 0.

3. Model formulation

After the transformation, each task is replaced by a customer node in graph G0. The goal of the problem is to find a set ofroutes for the private vehicles, such that:

� Each route performed by the private vehicle starts and ends at the depot.� Each route is assigned to exactly one vehicle.� Each customer node in graph G0 is visited exactly once by one private vehicle, or entrusted to an external carrier.� The distance of a route performed by the private vehicle must not exceed a given distance span H.� The sum of travel costs, fixed vehicle costs, penalty costs, and subtracting compensative payments is minimized.

The exact formulation for the proposed problem is given in this section. The following decision variables are used in theformulation:

xij if a private vehicle travels from node i to node j, xij = 1, else xij = 0; i, j e V0, i – j.zi if node i is assigned to an external carrier, zi = 1, else zi = 0; i e N0.hi the upper bound on the travel distance of a private vehicle upon leaving customer i; i e V0.

Objective:

minX

i

2 N0f x0i þXi2V 0

Xj2V 0

i–j

cijxij þXi2N0

gizi �Xi2N0

eið1� ziÞ ð1Þ

Constraints:

Xi2V 0

i–j

xij ¼Xi2V 0

i–j

xji 8j 2 V 0 ð2Þ

zi þXj2V 0

xji ¼ 1 8i 2 N0 ð3Þ

hd ¼ 0 ð4Þ

hi þ cij � ð1� xijÞM 6 hj 8i 2 V 0; j 2 N0; i–j ð5Þ

hi þ cid � ð1� xidÞM 6 H 8i 2 N0 ð6Þ

0 6 hi 6 H 8i 2 V 0 ð7Þ

xij 2 f0;1g i; j 2 V 0; i–j ð8Þ

zi 2 f0;1g i 2 N0 ð9Þ

In this formulation, the objective function (1) minimizes the carrier’s total costs, where first two terms represent the totaltravel costs (i.e., the variable costs), and following three terms represent the total fixed costs, the penalty costs and compen-sative payments given by external carrier, respectively. Constraints (2) impose the degree balance for each node, i.e., if avehicle enters a node, it must leave this node. Constraints (3) ensure that each customer node is served by exactly one pri-vate vehicle or by an external carrier. Constraints (4)–(7) impose the subtour elimination and ensure that the given distancespan is not exceeded, where M stands for a big positive number. Note that after the problem transformation, our formulationis similar to the model of Bolduc et al. (2008), and the differences come from three main facts. First, in Bolduc et al. (2008) aprivate customer may be assigned to an external carrier, but the private fleet cannot accept and serve the external customer.Second, Bolduc et al. (2008) consider the vehicle capacity constraints, while in this paper we impose the constraints on thetotal distance traveled by each vehicle. And last, Bolduc et al. (2008) specify that the number of the private fleet vehicles usedin the solution is limited. While in our formulation, the private vehicles are supposed to be abundant.

1074 R. Liu et al. / Transportation Research Part E 46 (2010) 1071–1085

4. Lower bound

Since the task selection and routing problem with full truckload is NP-hard, a heuristic algorithm is designed to solve theproblem. However, this approach may lead to suboptimal solutions of a priori unknown quality. To assess the performance ofthe heuristic algorithm, the formulation described in Section 3 was implemented in one of the most robust MIP solversCPLEX 10.0, with a time limit of three hours. Unfortunately, it is found that even for small-scale instance (e.g., 20 requiredtasks in the instance) CPLEX cannot reach an optimal solution and its lower bound is poor. Therefore, we replace inequalities(4)–(7) in the foregoing exact formulation by following inequalities to get a lower bound for the proposed problem.

Proposition 1. The inequalities

Xi2N0

H � xdi þXi2N0

zi � li PXi2V 0

Xj2V 0ðcij � ljÞ � xij þ

Xi2N0

li ð10Þ

are valid for the task selection and routing problem.

Proof. Let M0 # N0 contains all nodes served by the private vehicles, and let M00 ¼ N0 nM0 contains all nodes served by exter-nal carriers. Thus, if i 2 M00; zi = 1, otherwise, zi = 0. Then the left-hand side of inequalities (10) is equal to

X

i2N0H � xdi þ

Xi2M0

zi � li þXi2M00

zi � li ¼Xi2N0

H � xdi þXi2M00

li

The right-hand side of inequalities (10) is equal to

Xi2V 0

Xj2V 0ðcij � ljÞ � xij þ

Xi2M0

li þXi2M00

li

Thus, we should show that

Xi2N0

xdi PXi2V 0

Xj2V 0ðcij � ljÞ � xij þ

Xi2M0

li

0@

1A,H

In above inequalities, the left-hand side represents the private vehicles used in the solution. In the right-hand side,PieV0P

jeV0(cij � lj) � xij andP

ieM0 li represent empty and productive travel distance of the private vehicles, respectively.Clearly, the above inequalities bound the minimum number of private vehicles with respect to the maximum travel distanceof private vehicle, so they are satisfied. h

The new formulation, i.e., objective (1) constrained by (2), (3) and (8)–(10), is called LBF. We find that CPLEX can solve LBFquickly for small-scale instances. For most large-scale instances with more than 100 tasks, CPLEX fails to get LBF optimumwithin the time limit. Hopefully, CPLEX is able to provide tight lower bounds for these large-scale instances, where the gapsbetween the objective values and lower bounds are small. Therefore, the CPLEX lower bounds for the LBF can be used directlyas the lower bound for our proposed problem. As shown in the following computational experiments on large-scale in-stances, the heuristic results are assessed by comparing to the CPLEX lower bounds for LBF.

5. Memetic algorithm

In this section, a memetic algorithm is designed to tackle the problem. The proposed algorithm is based on the frameworkof Prins (2004). The basic components of the MA, e.g., chromosome representation, initial population, selection and cross-over, mutation by local search and stop criteria are described in the following.

5.1. Representation and fitness function

The chromosome used in the proposed MA is a sequence of all the nodes, without trip delimiters. Such kind of chromo-some coding is very simple and straightforward, used in some excellent MAs for solving the VRP (Prins, 2004) and its variantproblems (Prins, 2009; Liu et al., 2009). In these cases, a chromosome is interpreted as the order in which a vehicle must visitall customer nodes. Each chromosome is partitioned into feasible routes by the splitting procedure based on the shortest pathmethod (Beasley, 1983). In this study, the chromosome is also a sequence of all the nodes, which are composed of two parts.Supposed there are n nodes in the chromosome, the first m nodes performed by the private vehicles, and the last n–m nodesserved by the external carrier. A revised splitting algorithm is used to optimally determine the value of m and partition thefirst m nodes into routes to extract the solution for the problem. The splitting algorithm is explained as follows.

Given a chromosome S = (s1,. . .,sn) and an acyclic auxiliary graph T = (X, Y), where vertex set X contains nodes indexedfrom 0 to n, and Y is the directed arc set. Set Y contains one arc (i, j), i < j, iff a vehicle serving customers si+1 to sj is feasiblein terms of following constraints (11):

R. Liu et al. / Transportation Research Part E 46 (2010) 1071–1085 1075

cd;siþ1 þXj

k¼iþ1

csk;skþ1 þ csj;d 6 H ð11Þ

and, the cost of the arc (i, j) is:

cd;siþ1 þXj

k¼iþ1

csk;skþ1 þ csj;d þ f ð12Þ

First, we assume that: (1) all the nodes must be served by private vehicles; (2) for each node the compensative paymentand penalty costs both equal zero. Under these two assumptions, the optimal partition of the chromosome S corresponds to ashortest path problem: defining the minimal cost path from node 0 to node n in graph T. Based on Prins (2004) splitting algo-rithm, the minimal costs from node 0 to other nodes are gotten in 0(n) time. Let v0i represents the cost of min-cost path fromnode 0 to node i (0 6 i 6 n) in graph T.

Next, we determine how many customer nodes are entrusted to external carrier. If first m nodes are served by privatevehicles, the total solution cost respecting the sequence of chromosome S is:

qm ¼ v0m �Xm

k¼1

esk þXn

k¼mþ1

gsk; 0 6 m 6 n ð13Þ

Therefore, for the proposed problem the optimal solution respecting the sequence of chromosome S is Min (qm),8 0 6 m 6 n: It is adopted as the fitness of chromosome S.

To describe chromosome partition method more clearly, an example is shown in Fig. 1. The top of Fig. 1 shows a sequenceof demand nodes (a, b, c) with H = 50, f = 5 and compensative payment, penalty costs for each node in brackets. Note thatsince the illustration is based on graph G0, each task is transformed into a node, and the distances between two nodes areasymmetric.

In the second part of Fig. 1, we assume that all the nodes are served by private vehicles, compensative payments and pen-alty costs equal zero. Each arc in the auxiliary graph T represents a possible vehicle route, with the label of travel distanceand total cost. The minimal costs from node 0 to other nodes are shown in brackets.

a

b

c

depot

privatevehicle external carrier

q3=50 q2=45 q1=55 q0=50

a: 25,30 b:25,30

ab: 40,45

c:25,30

bc:45,50

15

15 10

20

10a (0,25)

b (0,25)

c (25, 0)

depot

(v00=0) (v01=30) (v02=45) (v03=75)

10

15

15

Fig. 1. Example of chromosome partition in MA.

1076 R. Liu et al. / Transportation Research Part E 46 (2010) 1071–1085

Then, in the third part, the solution costs qm (0 6 m 6 3) in expression (13) are given. For example, q3 is calculated as:q3 = v03 � e1 � e2 � e3 = 75 � 0 � 0 � 25 = 50. We find the minimal cost (i.e., the optimal solution) is q2 = 45, correspondingto an arc a–b and the last isolated node.

Finally, the lower part of Fig. 1 gives the resulting problem optimal solution, where nodes a and b are served by privatevehicle and node c is entrusted to external carrier.

5.2. Population structure and Initialization

The MA population is stored in a table P of ps chromosomes, always sorted in an ascending cost order. The initial MA pop-ulation is filled by Saving algorithm solutions (Golden et al., 1984), Nearest Neighbor algorithm (NN) (Cirasella et al., 2001)solutions, Randomized Arbitrary Insertion algorithm (RAI) (Brest and Zerovnik, 2005) solutions and random permutationsof required nodes.

Saving algorithm is the most famous and simple algorithm for the VRP (Clarke and Wright, 1964), which is based on thenotion of saving. Golden et al. (1984) extend the concept of savings to include vehicle fixed costs. Since saving algorithm can-not be adopted for solving our problem directly, we assume that all the nodes are served by the private vehicles. Then theproblem can be solved, i.e., the two routes (0, . . . , i,0) and (0, j, . . . , 0) are merged into a new route (0, . . . , i, j, . . . , 0) accordingto the largest saving that can be generated. We use generalized savings of the form:

saveij ¼ ci0 þ cj0 � c � cij þ f ð14Þ

where c is a positive parameter. When various c are used in (14), different solutions are generated. The routes of each solu-tion are concatenated into a chromosome.

NN algorithm and RAI algorithm are proposed for solving the traveling salesman problem (TSP). The principle of the NN isto start with the first customer chosen randomly, then go to the nearest unvisited customer node until all customer nodesare covered. RAI algorithm starts by generating an initial route randomly. Then, some nodes are randomly chosen and re-moved from the route, and then they are inserted into the route in the cheapest possible way. The removing and insertingprocedure is repeated until the stop criterion is satisfied (e.g., repeating n2 times, where n is the number of nodes). For allrequired nodes in graph G0, NN algorithm and RAI algorithm are adopted to generate the sequence of nodes. Various se-quences are generated when changing the start node in NN algorithm, or repeating the RAI algorithm.

Besides the chromosomes generated by three simple heuristic algorithms, other initial chromosomes are built randomly,each of which is a random sequence of nodes. All the initial chromosomes are partitioned by partition algorithm mentionedin Section 5.1 in order to define the exact fitness.

It has been proven that perverting the diversity of MA population is necessary since it can diminish the risk of prematureconvergence (Hertz and Widmer, 2003). Many methods were proposed to control the solutions diversity (Campos et al.,2005; Sörensen and Sevaux, 2006). In Liu et al. (2009), a simple rule is imposed, i.e., the costs of any two chromosomes mustbe different. If several chromosomes have the same costs, only one of them is reserved. In our research, this rule can beadopted when the test instance is not large, e.g., no more than 40 required nodes. In such case, three chromosomes p1,p2, p3 are computed using simple heuristic algorithms. The rest chromosomes are randomly generated. Then, the diversityof the population is checked. If the requirement is not satisfied, the violated chromosomes are removed from the population.The population is filled with new randomly generated chromosomes and is checked again. The procedure is iterated until allchromosomes satisfy the diversity requirement. In the sequel, the population is called well spaced. Commonly we can get awell spaced population within trying up to 5–10 times. However, when dealing with large-scale instances (e.g., more than100 tasks), and, for each task, penalty cost is close to its travel distance (e.g., penalty cost is twice the value of travel dis-tance), this method may fail to get an well spaced population quickly since the random permutations of nodes always breakthe diversity of P. In such cases, we try to generate more initial chromosomes using heuristic algorithms, and then try fillingthe population with random chromosomes more times (e.g., 50 times). If it is still not possible to get a satisfied population,the resulting population is used even it does not satisfy the diversity requirement.

5.3. Selection and crossover

In the proposed MA, two parent chromosomes P1 and P2 are selected based on the tournament selection. First, a few chro-mosomes are chosen randomly from the population. Then, the best (i.e., the least solution cost) chromosome is selected to beP1. The procedure is repeated to get the second parent P2. With the increase in tournament size, weak chromosomes willhave smaller chance to be selected in the tournament selection. In this study, the tournament size equals 2.

In the research on the TSP, several crossover operators have been proposed, such as partial-mapped crossover (PMX)(Goldberg and Lingle, 1985), Order Crossover (OX) (Oliver et al., 1987) and position based crossover (PBX) (Syswerda,1989). OX is used in the proposed MA to perform the permutation representation, see Fig. 2 as an example. First, two differ-ent parents P1 and P2 are chosen from population. A substring is selected from P1 randomly. The substring is copied into thecorresponding positions in the first child C1, and those genes in the substring are deleted from P2. The resulting genes form asequence and are placed into the empty positions of C1 from left to right. At last, two parents are exchanged and the otherchild C2 is produced by the same procedure.

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 crossover in MA.

R. Liu et al. / Transportation Research Part E 46 (2010) 1071–1085 1077

One child chromosome c is selected at random from OX results, and undergoes local search methods (described in Section5.4). Then, this chromosome replaces a mediocre chromosome Pw in the population iff: (1) the cost of this child is smallerthan the worst chromosome in the population; (2) this child does not violate the diversity of the population. Pw is randomlyselected in the worst 50% of the population.

5.4. Local search as mutation

The local search (LS) procedure is employed to the child chromosome c with a probability q. LS procedure operates on aproblem solution with route delimiters and not on the chromosome itself. In the interest of effectiveness and efficiency,three simple procedures are adopted in this study, i.e., 2-opt move, 1–1 exchange and 1–0 exchange.

1. 2-opt: the 2-opt move is implemented in one private vehicle route or in two private vehicle routes simultaneously.2. 1–1 exchange: this procedure consists of three situations, i.e., exchanging two nodes of one private route, exchanging two

nodes of different private routes, and exchanging two nodes that are served by the private vehicle and external carrierrespectively.

3. exchange: this procedure is the relocation of one node, i.e., transferring a node from its position in one route to anotherposition either the same or a different route. When this procedure is applied two different routes, they may be performedby two private vehicles, or served by a private vehicle and an external carrier, respectively.

The descriptions and pictorial illustrations of 2-opt, 1–1 exchange and 1–0 exchange for one or two private routes can befound in (Tarantilis and Kiranoudis, 2002; Irnich et al., 2006; Prins, 2009). The example in Fig. 3 shows how 1–1 exchange isapplied to nodes served by the private vehicle and the external carrier. In the left-hand part of Fig. 3, node i is served byexternal carrier; nodes j and j + 1 are performed by private vehicle. The right-hand part of Fig. 3 illustrates that node iand j are swapped (1–1 exchange). Similarly, Fig. 4 gives one example of 1–0 exchange for nodes served by the private vehi-cle and the external carrier, where node i is relocated into the private route from the external carrier.

During the LS procedure, once a feasible and less cost solution is found, it is adopted as the new seed for repeating the LS.The LS stops until no additional improvement can be obtained. Note that the mutation result is converted into a chromosome

j

depot

j+1i

j

depot

j+1 i

Fig. 3. The 1–1 exchange between private and external routes.

j

depot

j+1i

j

j+1 i

depot

Fig. 4. The 1–0 exchange between private and external routes.

1078 R. Liu et al. / Transportation Research Part E 46 (2010) 1071–1085

after the local search procedure. The Splitting algorithm is applied to this resulting chromosome since it may find a betterpartition into routes for the same chromosome.

5.5. General structure of MA

Outline of the proposed MA is given in Algorithm 1. Initially, a population is built from three simple heuristics and ran-dom selections. The population is accepted when it is well spaced or we try filling the population with random chromosomesup to vc�max times. Then, two parent chromosomes p1 and p2 are selected with the binary tournament method from the pop-ulation P, and they are subjected to the crossover operator. Two offspring chromosomes are created and one child c is ran-domly selected. Then, chromosome c is improved by local search operators with the probability q. Assuming that themaximal cost chromosome in the population is cmax, a chromosome pw randomly selected from the worst half of the popu-lation is replaced by c iff cost (c) is less than cost (cmax) and c does not break the diversity of P/pw. Otherwise, c is discarded. Avariable vt is used to provide the stopping criteria of MA, recording the algorithm iteration. Once a pair of offspring chromo-somes is created, vt increases by 1. The MA stops when vt reaches the limitation vt�max.

Algorithm 1: Memetic algorithm for the task selection and routing problem

1:

input MA parameters: population size ps, vc�max, vt�max and q; set vt = 0, vc = 0; 2: set population P: = £

3:

P = P [ Chromosomes generated by saving, NN, RAI algorithms and randomized algorithm 4: calculate each chromosome’s cost, sort the chromosomes in increasing cost order 5: check the diversity of P:

if 81 6 i; j 6 ps; i–j; ::jcostðPiÞ � costðPjÞj > 0, then mark = 0; otherwise, mark = 1;

6: while mark = =1 and vc< vc�max do 7: delete the chromosomes, which violate the diversity of P 8: fill each empty chromosome in P with a random permutation of required nodes 9: vc ++

10:

if the diversity of P is satisfied, mark = 0; otherwise, mark = 1 11: end while 12: repeat 13: sort P in increasing cost order 14: select two different parent chromosomes p1 and p2 from P by tournament selection 15: apply OX operator to p1 and p 2, select one chromosome c from two offspring chromosomes 16: if a random number between (0, 1) is less than q then 17: extracted the problem solution s from chromosome c 18: improve s with local search methods 19: convert s into mutated chromosomes c0, c = c0

20:

end if 21: apply split algorithm to c, get cost(c) 22: vt ++ 23: randomly select a chromosomes pw from the worst half of P 24: if (cost(c) < cost (pps)) and (81 6 i 6 ps and i–w; ijcostðcÞ � costðpiÞj > 0) then 25: pw: = c 26: end if 27: until vt = vt�max

28:

output the best solution in the P

6. Computational experiments

Computational experiments were performed to assess the performance of the proposed MA. In this section, the algo-rithms implementation and the test instances used in these experiments are described, followed by the presentation anddiscussion of the experimental results.

6.1. Experimental data

All the algorithms were implemented in C and tested on an Intel Core2 E7200 2.53 GHz PC with 2 GB memory under Win-dows XP. Since we are not aware of any prior test instance for the task selection and routing problem with full truckload, theproposed algorithm was tested on a range of randomly generated instances.

Test problems were created randomly in two steps. First, a complete Euclidean digraph was created where a depot and aset of nodes were included. Then, a set of tasks was generated, each of which is specified by a pickup node and a deliverynode, both randomly chosen from the set of nodes.

R. Liu et al. / Transportation Research Part E 46 (2010) 1071–1085 1079

The test problems are divided into two sets (i.e., set1 and set2) with respect to the geographical data of the nodes. All theproblems within a set have an identical graph structure, with the only difference coming from the number of the tasks, rang-ing from 15 to 400. In set1 problems, 200 nodes are placed in a rectangular region (i.e., basic graph). All the nodes are equallydivided into 20 clusters. Clusters are introduced to represent geographical concentrations of nodes, each of which is a circu-lar of radius 1. The clusters are uniformly distributed in the rectangular region, and the nodes within each cluster are ran-domly generated. Let short task be arc connecting nodes within a cluster, and long task be arc connecting nodes betweendifferent clusters. We randomly let 80% of the tasks are long and 20% of the tasks are short. Set2 problems have 500 nodesrandomly located in a rectangular region, and tasks are randomly created between two different nodes. For notational con-venience, a problem is named as the problem’s type with the number of the tasks. For example, the first set problem asso-ciated with 100 tasks is referred to as Set1–100.

Each problem combined with a price parameter (k; e, g) is called a test instance, where k describes how many tasks areprovided by the shippers, e and g are the penalty and compensative prices for each task, respectively. For example, whenk = 60%, e = 1.5 and g = 2, we randomly choose 60% tasks and assume that these tasks are given by shippers (if a non-integernumber is gotten, it is rounded downwards to the nearest integer), and the rest tasks are provided by external carriers. Foreach task i from shippers, the penalty costs equal 2 � li and the compensative payment is 0, where li is the travel distance ofthe task. For each task provided by external carriers, the penalty cost equals 0 and the compensative payment equals 1.5 � li.In our experiment, three instances are generated from each problem with three price parameters, as shown in Table 1. Fornotational convenience, an instance within a problem is named as the problem’s label with its price parameter (e.g., an in-stance in problem Set1–100 with price 1 is denoted by Set1–100-1). The vehicle fixed costs and the vehicle distance span aredefined in transformed graph G0:

Table 1Random

Para

nn

nt

w �

depo

(k; e

F ¼ a�Xi2N0

li

H ¼ b�maxi2N0fcdi þ cidg

For small-scale instances, i.e., the total number of tasks ranges from 15–40, a = 0.1 and b = 5; for large-scale instances with100–400 tasks, a = 0.05 and b = 10. Further details of the random problems and instances generation are shown in Table 1.

6.2. Computational results for small-scale instances

We first test the proposed MA on 36 small-scale instances, with 15–40 required tasks. The setting of MA parameters isexplained as follows. The population size is 30, the stopping criteria vt�max equals 40000, and the LS mutation probabilityq equals 0.25. When generating the initial population, vc�max equals 5. Each heuristic provides an initial chromosome, wherec equals 1.0 in the saving algorithm. The MA was run 10 times on each instance. The best results, the average results andaverage running time are gotten from these 10 runs. The computational results are presented in Tables 2 and 3.

In Tables 2 and 3, the first column presents the instance number. The columns with the header ‘Saving’, ‘NN’ and ‘RAI’refer to the best solution costs in the initial population obtained by saving, NN and RAI algorithms over 10 runs, respectively.Columns 5–7 contain the computational results and running time using the proposed MA. The columns headed ‘MAb’ and‘MAavg’ present the best and average MA solution costs over 10 runs. Some detailed information of the best MA solution(e.g., the number of tasks served by private fleet, as well as the number of tasks exchanged between them) is provided inthe Appendix A. The column headed ‘Sec’ gives the average MA running time in seconds. Column 8 with the header ‘LB’ pre-sents the lower bound of the instance, which is the CPLEX solution value for the formulation LBF proposed in Section 4. Notethat for each small-scale instance, CPLEX can solve the formulation LBF optimally. The last four Columns contain the percent-

ized generation scheme of test problems and instances.

meter Explanation Values

Number of the nodes set 1 problems: 200; set 2 problems: 500;Number of the tasks For set 1 problems:

15, 20, 25, 30, 35, 40, 100, 150, 200, 250, 300, 350, 400For set 2 problems:15, 20, 25, 30, 35, 40, 100, 150, 200, 250, 300, 350, 400

h Dimensions of the rectangle (the basic graph) For set 1 problems:x coordinate: [0, 100], y coordinate: [0, 100]For set 2 problems:x coordinate: [0, 50], y coordinate: [0, 20]

t Coordinates of the depot For set 1 problems: (x = 50, y = 50);For set 2 problems: (x = 1.5, y = 1.5);

, r) Coefficients to be used in the task price parameters For each problem, three parameters are tested:Type 1: (k=50%, e = g = 1.5)Type 2: (k=50%, e = g = 2.0)Type 3: (k=70%, e = g = 3.0)

Table 2Computational results on the first set instances with small scales.

GAP

MA Saving NN RAI MA

Instance Saving NN RAI MAb MAavg Sec. LB (%) (%) (%) (%)

Set1-15-1 415.21 399.73 399.73 383.71 385.51 1.3 381.07 8.22 4.67 4.67 0.69Set1-20-1 496.93 496.93 485.31 464.05 467.45 2.0 459.49 7.53 7.53 5.32 0.98Set1-25-1 596.34 583.38 581.69 476.31 486.44 3.3 476.31 20.13 18.35 18.12 0.00Set1-30-1 1079.91 1011.08 1062.33 901.55 902.98 5.0 901.55 16.52 10.83 15.13 0.00Set1-35-1 1187.35 1128.42 1183.46 1033.89 1035.42 6.9 1033.80 12.93 8.39 12.65 0.01Set1-40-1 1070.45 1007.39 1070.45 932.00 937.82 10.2 927.78 13.33 7.90 13.33 0.45Set1-15-2 310.51 299.61 299.53 272.86 272.86 1.4 272.74 12.16 8.97 8.94 0.04Set1-20-2 435.10 383.05 394.95 320.38 326.95 2.1 320.28 26.39 16.39 18.91 0.03Set1-25-2 368.45 435.02 379.17 325.03 325.60 3.4 325.03 11.78 25.28 14.28 0.00Set1-30-2 935.67 734.21 793.42 613.73 614.11 4.9 612.65 34.52 16.56 22.78 0.18Set1-35-2 1138.77 962.78 1028.98 822.38 825.15 7.5 820.72 27.93 14.76 20.24 0.20Set1-40-2 931.34 864.62 835.49 655.47 660.03 10.6 652.82 29.91 24.50 21.86 0.40Set1-15-3 401.26 390.10 384.84 363.36 363.36 1.4 363.23 9.48 6.89 5.62 0.04Set1-20-3 430.52 442.77 395.62 361.01 361.30 2.1 360.50 16.26 18.58 8.88 0.14Set1-25-3 71.77 154.41 90.36 56.90 57.08 3.2 56.89 20.73 63.16 37.04 0.02Set1-30-3 1200.67 1118.31 1176.80 918.46 920.52 5.4 917.07 23.62 18.00 22.07 0.15Set1-35-3 1270.66 1057.32 1207.75 904.90 928.14 7.1 898.58 29.28 15.01 25.60 0.70Set1-40-3 1046.33 1168.80 993.36 844.70 848.31 11.5 841.65 19.56 27.99 15.27 0.36Average 5.0 18.91 17.43 16.15 0.24

1080 R. Liu et al. / Transportation Research Part E 46 (2010) 1071–1085

age deviation of the heuristics saving, NN and RAI to LB, and the percentage deviations between MAb and LB, respectively. Thepercentage deviation between saving algorithm cost and LB is calculated by: 100 � (saving–LB)/saving. Similar formulas areused for calculating NN, RAI and MAb.

Three remarkable conclusions can be drawn from these experimental results. First, the proposed MA is able to providehigh-quality solutions for small-scale instances. As shown in Tables 2 and 3, for each instance, the solution cost of MA ismuch better than that of the simple heuristics, and is very close to the lower bound. For 18 test instances generated fromthe first set problems, the average deviation between MAb and LB is only 0.24%; while for test instances generated fromthe second set problems, the average deviation is 0.70%. Second, we find that the proposed MA is very robust with respectto different combinations of problems and price parameters. Finally, we see that MA can obtain these good solutions in a veryshort computational time.

6.3. Computational results for large-scale instances

In Tables 4 and 5, we present the experimental results on 42 large-scale instances, where each instance has 100–400tasks. For most of the large-scale instances, the lower bound formulation LBF cannot be solved optimally by CPLEX due to

Table 3Computational results on the second set instances with small scales.

GAP

MA Saving NN RAI MA

Instance Saving NN RAI MAb MAavg Sec. LB (%) (%) (%) (%)

Set2-15-1 88.24 86.31 88.31 86.14 86.14 1.1 86.14 2.38 0.20 2.46 0.00Set2-20-1 187.63 178.84 182.87 159.67 161.41 2.3 159.10 15.21 11.04 13.00 0.36Set2-25-1 177.81 167.57 177.81 153.06 153.11 3.1 151.26 14.93 9.73 14.93 1.18Set2-30-1 274.67 248.09 262.85 226.29 229.85 5.5 224.98 18.09 9.32 14.41 0.58Set2-35-1 242.96 239.92 242.96 201.62 212.81 9.1 200.62 17.43 16.38 17.43 0.50Set2-40-1 326.18 323.00 326.18 282.22 284.71 11.2 281.53 13.69 12.84 13.69 0.24Set2-15-2 92.45 87.91 71.62 66.36 66.39 1.0 66.04 28.57 24.88 7.79 0.48Set2-20-2 175.24 144.41 141.42 120.15 120.19 2.1 119.71 31.69 17.10 15.35 0.37Set2-25-2 161.87 114.81 152.07 95.69 95.96 3.4 94.04 41.90 18.09 38.16 1.72Set2-30-2 203.90 162.52 177.72 151.25 151.54 5.4 150.27 26.30 7.54 15.45 0.65Set2-35-2 237.93 197.05 187.31 153.78 155.37 8.8 153.36 35.54 22.17 18.13 0.27Set2-40-2 328.04 255.17 239.06 202.92 204.69 11.6 200.89 38.76 21.27 15.97 1.00Set2-15-3 112.79 107.13 91.95 86.70 86.74 1.2 86.37 23.42 19.38 6.07 0.38Set2-20-3 198.15 186.52 151.14 128.21 130.43 2.1 126.91 35.95 31.96 16.03 1.01Set2-25-3 173.00 132.97 127.32 105.73 105.87 3.6 104.24 39.75 21.61 18.13 1.41Set2-30-3 166.98 138.32 147.91 114.29 114.69 5.3 113.35 32.12 18.05 23.37 0.82Set2-35-3 235.19 174.74 183.01 142.16 144.24 9.0 141.77 39.72 18.87 22.53 0.27Set2-40-3 377.07 280.34 263.19 229.58 230.82 12.1 226.54 39.92 19.19 13.93 1.32Average 5.4 27.52 16.65 15.93 0.70

Table 4Computational results on the first set instances with large scales.

GAP

MA GA Saving NN RAI MA Dev.

Instance Saving NN RAI MAb MAavg Minutes GAb GAavg Minutes LB (%) (%) (%) (%) (%)

Set1-100-1 2252.67 1922.92 2162.93 1678.59 1690.82 1.6 1822.23 1880.16 0.2 1627.67 27.74 15.35 24.75 3.03 7.88Set1-150-1 4095.79 3557.12 3817.13 3176.68 3203.79 4.6 3522.71 3553.52 0.5 3103.47 24.23 12.75 18.70 2.30 9.82Set1-200-1 5971.68 5274.25 5592.93 4730.50 4769.36 9.6 5174.96 5233.49 1.2 4572.45 23.43 13.31 18.25 3.34 8.59Set1-250-1 7167.18 6257.43 6979.44 5866.07 5917.82 18.2 6206.83 6254.64 2.1 5606.99 21.77 10.39 19.66 4.42 5.49Set1-300-1 9321.99 8351.63 9126.66 8137.00 8169.88 19.3 8347.76 8379.90 3.8 7873.02 15.54 5.73 13.74 3.24 2.52Set1-350-1 10649.42 10376.39 10645.07 10041.60 10211.39 23.3 10345.00 10377.90 5.8 9693.64 8.97 6.58 8.94 3.47 2.93Set1-400-1 12130.85 12029.82 12130.85 12019.37 12041.60 33.1 12155.20 12162.70 9.2 11801.11 2.72 2.71 2.72 1.82 1.12Set1-100-2 1317.83 1274.95 1272.39 989.20 1003.94 1.6 1134.30 1144.72 0.1 894.63 32.11 29.83 29.69 9.56 12.79Set1-150-2 2865.00 2628.19 2427.67 2121.87 2128.90 4.3 2396.31 2425.66 0.3 1945.90 32.08 25.96 19.84 8.29 11.45Set1-200-2 4398.08 3830.30 3862.08 3247.13 3285.37 9.2 3754.72 3771.27 1.2 3036.38 30.96 20.73 21.38 6.49 13.52Set1-250-2 5572.11 4309.06 4700.24 3804.68 3839.38 17.2 4210.03 4257.01 2.1 3595.53 35.47 16.56 23.50 5.50 9.63Set1-300-2 8408.85 6755.25 7464.76 6220.77 6263.78 27.5 6535.72 6561.11 3.8 5836.87 30.59 13.60 21.81 6.17 4.82Set1-350-2 10329.00 8181.26 8706.38 7540.88 7584.42 40.0 8080.03 8168.28 5.7 7371.00 28.64 9.90 15.34 2.25 6.67Set1-400-2 12608.99 10574.02 11173.15 9497.42 9626.32 61.5 10315.46 10377.46 9.1 9019.46 28.47 14.70 19.28 5.03 7.93Set1-100-3 1947.75 1819.82 1755.53 1552.90 1554.81 1.6 1698.20 1733.21 0.1 1462.69 24.90 19.62 16.68 5.81 8.56Set1-150-3 3337.48 3373.76 2963.50 2521.46 2537.07 4.3 2811.70 2915.73 0.2 2433.65 27.08 27.87 17.88 3.48 10.32Set1-200-3 5638.75 5310.08 5245.45 4636.00 4888.83 9.3 5221.79 5238.97 0.4 4433.65 21.37 16.51 15.48 4.36 11.22Set1-250-3 6501.86 5698.81 5638.66 5136.30 5145.11 17.7 5538.51 5580.01 0.6 4587.14 29.45 19.51 18.65 10.69 7.26Set1-300-3 9608.74 8151.50 8180.75 7618.31 7625.57 27.5 7915.87 8009.29 1.5 7049.27 26.64 13.52 13.83 7.47 3.76Set1-350-3 11816.09 9778.58 9963.38 9152.48 9162.19 36.2 9331.97 9391.99 2.2 8648.20 26.81 11.56 13.20 5.51 1.92Set1-400-3 14378.07 12251.26 12493.44 11051.38 11088.90 55.8 12105.32 12170.10 4.0 10737.76 25.32 12.35 14.05 2.84 8.71Average 20.2 2.9 24.97 15.19 17.49 5.00 7.47

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Table 5Computational results on the second set instances with large scales.

GAP

MA GA Saving NN RAI MA Dev.

Instance Saving NN RAI MAb MAavg Minutes GAb GAavg Minutes LB % % % % %

Set2-100-1 898.46 672.73 763.53 613.33 620.06 1.6 652.83 657.01 0.2 580.14 35.43 13.76 24.02 5.41 6.05Set2-150-1 1279.39 1050.49 1225.65 983.97 989.39 5.2 1042.98 1047.00 0.5 919.01 28.17 12.52 25.02 6.60 5.66Set2-200-1 1837.99 1606.1 1801.66 1521.75 1531.26 11.0 1559.34 1569.11 1.2 1407.75 23.41 12.35 21.86 7.49 2.41Set2-250-1 2189.77 1945.3 2187.75 1867.08 1872.26 17.8 1932.02 1943.03 2.2 1737.02 20.68 10.71 20.60 6.97 3.36Set2-300-1 2595.78 2343.84 2595.78 2274.79 2289.29 24.5 2338.29 2339.35 3.8 2120.83 18.30 9.51 18.30 6.77 2.72Set2-350-1 3268.22 3061.95 3268.25 3000.37 3010.09 26.7 3083.49 3092.45 6.0 2862.64 12.41 6.51 12.41 4.59 2.70Set2-400-1 3471.09 3397.51 3394.09 3312.12 3321.74 44.3 3390.09 3398.72 9.3 3157.14 9.04 7.07 6.98 4.68 2.30Set2-100-2 493.36 384.95 461.33 327.07 338.66 1.3 368.80 376.18 0.1 283.18 42.60 26.44 38.62 13.42 11.32Set2-150-2 843.34 695.54 745.88 572.96 600.73 4.6 662.55 673.11 0.2 499.41 40.78 28.20 33.04 12.84 13.52Set2-200-2 1485.06 1168.03 1277.85 1052.90 1064.73 10.2 1158.93 1175.73 1.0 951.96 35.90 18.50 25.50 9.59 9.15Set2-250-2 1739.74 1419.31 1597.64 1255.40 1281.82 17.2 1403.23 1408.84 2.2 1106.92 36.37 22.01 30.72 11.83 10.53Set2-300-2 2103.11 1572.39 1972.1 1468.76 1484.41 26.3 1547.04 1564.17 3.8 1309.06 37.76 16.75 33.62 10.87 5.06Set2-350-2 2925.15 2329.53 2797.54 2157.55 2174.00 37.9 2287.45 2310.07 6.0 1964.49 32.84 15.67 29.78 8.95 5.68Set2-400-2 3449.15 2835.78 3217.22 2539.38 2569.03 60.3 2786.05 2812.12 9.4 2249.16 34.79 20.69 30.09 11.43 8.85Set2-100-3 701.19 530.34 617.35 485.96 494.23 1.5 519.58 527.10 0.1 424.78 39.42 19.90 31.19 12.59 6.47Set2-150-3 856.49 714.25 752.67 608.46 613.25 4.1 690.82 709.83 0.2 554.79 35.23 22.33 26.29 8.82 11.92Set2-200-3 1716.44 1357.65 1416.73 1218.36 1232.00 10.3 1309.13 1335.78 0.4 1092.74 36.34 19.51 22.87 10.31 6.93Set2-250-3 1810.07 1558.9 1591.33 1282.60 1287.25 19.5 1475.96 1494.81 0.7 1156.49 36.11 25.81 27.33 9.83 13.10Set2-300-3 2382.44 1869.21 2190.71 1762.31 1793.85 25.7 1855.97 1868.22 1.3 1523.91 36.04 18.47 30.44 13.53 5.05Set2-350-3 3234.59 2725.61 3026.35 2479.45 2489.10 53.9 2587.92 2608.96 2.0 2184.90 32.45 19.84 27.80 11.88 4.19Set2-400-3 3579.35 3257.93 3375.49 2784.67 2814.85 55.5 3170.09 3188.49 3.9 2541.70 28.99 21.98 24.70 8.73 12.16Average 21.9 2.6 31.10 17.55 25.77 9.39 7.10

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Table 6Information of the best solutions on small-scale instances.

Instance No Ne No�e Ne�o MAb Instance No Ne No�e Ne�o MAb

Set1-15-1 8 7 5 6 383.71 Set2-15-1 15 0 0 8 86.14Set1-20-1 10 10 6 6 464.05 Set2-20-1 17 3 1 8 159.67Set1-25-1 16 9 6 9 476.31 Set2-25-1 16 9 6 10 153.06Set1-30-1 15 15 7 7 901.55 Set2-30-1 15 15 6 6 226.29Set1-35-1 15 20 10 8 1033.89 Set2-35-1 21 14 7 10 201.62Set1-40-1 16 24 12 8 932.00 Set2-40-1 19 21 10 9 282.22Set1-15-2 14 1 1 8 272.86 Set2-15-2 15 0 0 8 66.36Set1-20-2 12 8 5 7 320.38 Set2-20-2 19 1 1 10 120.15Set1-25-2 16 9 4 8 325.03 Set2-25-2 19 6 5 12 95.69Set1-30-2 26 4 3 14 613.73 Set2-30-2 30 0 0 15 151.25Set1-35-2 31 4 1 15 822.38 Set2-35-2 32 3 1 16 153.78Set1-40-2 31 9 3 14 655.47 Set2-40-2 40 0 0 20 202.92Set1-15-3 15 0 0 5 363.36 Set2-15-3 15 0 0 5 86.70Set1-20-3 18 2 2 6 361.01 Set2-20-3 19 1 1 6 128.21Set1-25-3 25 0 0 8 56.90 Set2-25-3 25 0 0 8 105.73Set1-30-3 26 4 3 8 918.46 Set2-30-3 30 0 0 9 114.29Set1-35-3 33 2 1 10 904.90 Set2-35-3 34 1 1 11 142.16Set1-40-3 40 0 0 12 844.70 Set2-40-3 40 0 0 12 229.58

Table 7Information of the best solutions on large-scale instances.

Instance No Ne No�e Ne�o MAb Instance No Ne No�e Ne�o MAb

Set1-100-1 69 31 17 36 1678.59 Set2-100-1 87 13 4 41 613.33Set1-150-1 119 31 18 62 3176.68 Set2-150-1 134 16 4 63 983.97Set1-200-1 161 39 16 77 4730.50 Set2-200-1 137 63 37 74 1521.75Set1-250-1 209 41 17 101 5866.07 Set2-250-1 186 64 14 75 1867.08Set1-300-1 222 78 45 117 8137.00 Set2-300-1 242 58 32 124 2274.79Set1-350-1 233 117 50 108 10041.60 Set2-350-1 239 111 56 120 3000.37Set1-400-1 149 251 114 63 12019.37 Set2-400-1 142 258 107 49 3312.12Set1-100-2 98 2 2 50 989.20 Set2-100-2 98 2 2 50 327.07Set1-150-2 149 1 1 75 2121.87 Set2-150-2 149 1 1 75 572.96Set1-200-2 196 4 2 98 3247.13 Set2-200-2 183 17 14 97 1052.90Set1-250-2 246 4 1 122 3804.68 Set2-250-2 223 27 5 103 1255.40Set1-300-2 268 32 18 136 6220.77 Set2-300-2 295 5 4 149 1468.76Set1-350-2 333 17 11 169 7540.88 Set2-350-2 337 13 11 173 2157.55Set1-400-2 374 26 14 188 9497.42 Set2-400-2 370 30 7 177 2539.38Set1-100-3 100 0 0 30 1552.90 Set2-100-3 100 0 0 30 485.96Set1-150-3 150 0 0 45 2521.46 Set2-150-3 150 0 0 45 608.46Set1-200-3 197 3 3 60 4636.00 Set2-200-3 200 0 0 60 1218.36Set1-250-3 250 0 0 75 5136.30 Set2-250-3 250 0 0 75 1282.60Set1-300-3 300 0 0 90 7618.31 Set2-300-3 300 0 0 90 1762.31Set1-350-3 350 0 0 106 9152.48 Set2-350-3 350 0 0 105 2479.45Set1-400-3 399 1 1 120 11051.38 Set2-400-3 400 0 0 120 2784.67

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excessive memory requirements or excessive running time requirements. Therefore, we set a time limit of 24 h on CPLEX. Foreach instance, CPLEX ran with default settings until finding an optimal solution, or until exhausting the memory or prede-termined maximum computation time. Then, CPLEX handle this instance with the settings of strong branching again. Thebetter CPLEX lower bound is adopted to yield a proper estimation of the MA performance. We also implemented some otherCPLEX settings, such as depth-first search strategy, but such settings did not lead to performance improvement.

Meanwhile, the performance of the proposed MA is compared with that of the classical GA under the same conditions.The structure of the classical GA considered in this paper is the same as the structure of the proposed MA. The differencesbetween MA and classical GA are as follows: (1) Two children chromosomes obtained from crossover operator are both un-dergo mutation with a fixed probability q0: (2) The LS methods are replaced by simple mutation mechanism, i.e., randomlyswapping 10% tasks in the children chromosomes.

For these large-scale instances, the common parameters for MA and GA include: each of them has been run with vc�max of50, vt�max of 20000 and a population size of 30. The MA has been applied to each instance 10 times with the mutation prob-ability q of 0.2. The GA has been run 20 times for each instance with a relatively high mutation rate q0 of 0.8. For large-scaleinstances associated with the first and the second price parameter, we find that the random permutations of tasks often leadto identical solution cost (i.e., the solution obtained by split algorithm indicates that all tasks are served by external carrier).Therefore, in these cases we let each simple heuristic obtains three chromosomes in the initial population, where the savingalgorithm has been run under three different values of parameter c, i.e., 0.6, 1.0 and 1.4, and the NN and RAI algorithms havebeen executed three times. Then, if the population is not well spaced when vc�max reaches 50, the unsatisfied population isaccepted, since it may be time consuming to getting a well spaced initial population.

1084 R. Liu et al. / Transportation Research Part E 46 (2010) 1071–1085

Besides the columns in Tables 2 and 3, four columns are added into Tables 4 and 5. Columns 8–10 (with the headers ‘GAb’,‘GAavg’ and ‘Minutes’) present GA best and average solution costs, and average computational time in minutes over 20 inde-pendent runs. The last column headed ‘Dev’ shows the relative deviation between the best solution values obtained by theMA and the GA, which is calculated as: Dev = 100 � (GAb �MAb)/GAb.

As shown in Tables 4 and 5, we first find that in all cases the MA dominates the simple heuristics and the classical GA. Forthe first set problem instances, the maximum and average gaps between GAb and MAb are 13.52% and 7.47%, respectively.For the second set problem instances, the maximum and average deviations between GAb and MAb are 13.52% and 7.10%. Forall 42 large-scale instances, the average gap between GAb and MAb is 7.29%. Furthermore, for each test instance, the averageMA solution cost (MAavg) over 10 runs is better than the best solution cost provided by the classical GA (GAb). The resultspoint out that there is a significant gap between these two modern heuristic algorithms with respect to the solution quality.In general, it is proven that the classical GA cannot achieve very good solution for the task selection and routing problemwithout incorporating a powerful local search engine.

Also, the improvement in the solution quality is obtained by elapsing a larger CPU time. We find the running time of theGA is less than that of the MA. This is because compared with the simple mutation operator used in the GA, the LS methodsadopted in the MA are more complex. However, you may find that the computing time of the MA is still reasonable andacceptable. Furthermore, if we increase the running time of the GA to the same with the MA for each instance, the MA stillalways outperforms the GA, leading to a maximum gap of 13.12% (against 13.52%) and average gap of 6.72% (against 7.29%).

Meanwhile, we find the gap between MA solution cost and lower bound is not large. The average percentage deviationbetween MA best solution and lower bound for the first set problem instances is 5.00%. For the second set problem instances,the average percentage deviation is 9.39%. These results prove that the proposed MA can obtain good solutions for differenttypes of large-scale instances. However, you may find these gaps are larger than that of small-scale instances. This may bedue to three reasons. First, the problems become more complex with the number of tasks increasing. Second, here the MIPtool CPLEX cannot solve the lower bound formulation LBF optimally, which leads to the lower bound provided for large-scaleinstances are worse than the ones for small-scale instances. The gaps will be decreased when the MA solutions are comparedwith tighter lower bounds. And last, we reduce the value of vt�max in the MA when solving such large-scale instances. Thereduction of the vt�max yields a computing time saving but also has a negative effect on solution quality.

7. Conclusions and future research

In the present paper, we investigate the task selection and routing problem, an extension of the classical arc routing prob-lem. The problem is of interest because of its theoretical complexity and of the many practical applications in collaborativetransportation. An effective memetic algorithm is developed to solve the problem. In the proposed MA, three classical heu-ristic algorithms are adopted to provide good initial solutions. Many powerful local search methods are utilized as mutationoperator. The splitting algorithm that computes the fitness allows simple chromosomes and crossovers for the TSP.

This paper makes three main contributions to the literature. First, we introduce a new and important problem in the col-laborative transportation: the task selection and routing problem with full truckload. The important feature of this newproblem, which distinguishes it from the classical VRP and the VRPPC (Chu, 2005; Bolduc et al., 2007, 2008; Côté and Potvin,2009) is that the carrier not only decides which customers are to be served by external carriers, but also decides which cus-tomers from external carriers will be accepted or rejected. As far as we are aware, there is no published research on thisproblem. Second, a mathematical model for this problem is developed. Meanwhile, a tight lower bound is proposed, whichis essential for evaluating the quality of the heuristic solutions. Third, we extend the classical memetic algorithm (Prins,2004, 2009) to the optimization problem proposed in this paper. Our memetic algorithm can provide high-quality solutionsin reasonable computing times.

The computational performance of the proposed MA is tested on a range of randomly generated test instances. The com-putational results on small-scale instances indicate that the proposed MA obtains very good results, since their solution val-ues are very close to the lower bounds. Furthermore, the performance of the proposed MA is verified by many large-sizeinstances and compared with that of the classical genetic algorithm. The results obtained indicate that our proposed MAclearly outperforms the classical genetic algorithm in terms of the solution quality. It can provide good solutions in a rea-sonable computation time.

The problem can be extensively studied in the future. Its formulation could be extended to cover more operational con-straints, such as time windows and heterogeneous fleet of vehicles. From the methodological point of view, one goal is tospeed up the local search in the proposed MA to reduce the total running time. Meanwhile, it would be interesting to seeif other intelligent optimization techniques, such as tabu search and simulated annealing can be modified to solve this prob-lem and even provide better results in a shorter time.

Acknowledgments

This work was partly supported by Research Grant from National Natural Science Foundation of China (No. 70872077)and National Natural Science Foundation of China/Research Grants Council of Hong Kong joint research projects (No.

R. Liu et al. / Transportation Research Part E 46 (2010) 1071–1085 1085

70831160527). The authors are indebted to the editor and two anonymous referees for their valuable and constructive com-ments on this paper.

Appendix A

The detailed information of MA best solution for each test instance is presented in Tables 6 and 7. The column headed‘Instance’ presents the instance name. The column headed ‘No’ lists the number of tasks served by private fleet in MA bestsolution, while column ‘Ne’ shows the number of tasks served by the external carries. The column headed ‘No�e’ gives thenumber of tasks which are provided by shippers and entrusted to the external carriers. The column headed ‘Ne�o’ givesthe number of accepted external tasks, which are served by private fleet. The column headed ‘MAb’ gives the MA best solu-tion cost.

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