Fourth party logistics routing problem with fuzzy duration time

Int. J. Production Economics ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Int. J. Production Economics

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Fourth party logistics routing problem with fuzzy duration time

Min Huang a, Yan Cui a,n, Shengxiang Yang b, Xingwei Wang a

a College of Information Science and Engineering, Northeastern University; State Key Laboratory of Synthetical Automation for Process Industries (Northeastern University),Shenyang, Liaoning 110819, Chinab School of Computer Science and Informatics,De Montfort University, Leicester LE1 9BH, UK

a r t i c l e i n f o

Article history:Received 31 January 2012Accepted 7 March 2013

Keywords:Fourth party logistics routing problemFuzzy duration timeCredibility theoryGenetic algorithmK-th shortest path algorithm

73/$ - see front matter & 2013 Elsevier B.V. Ax.doi.org/10.1016/j.ijpe.2013.03.007

esponding author at: Tel.: þ86 24 83691272;ail address: [email protected] (Y. Cui).

e cite this article as: Huang, M., et auction Economics (2013), http://dx.d

a b s t r a c t

As fourth party logistics (4PL) has the power to integrate the supply chain, from the beginning of the 21stcentury, it has attracted more and more attention in many fields. As one of the most important aspects in4PL, the fourth party logistics routing problem (4PLRP) is very difficult to solve because many issues, suchas the selection of third party logistics, cost and time, need to be considered. In this paper, a 4PLRP withfuzzy duration time (4PLRPF) model is presented, which is to find a route of the minimum cost withconstraints under uncertain environments, and fuzzy numbers are used to denote the uncertainty of theduration time on each node and arc. Fuzzy programming model is established according to theuncertainty theory. In order to solve the modeled 4PLRPF, a two-step genetic algorithm with the fuzzysimulation is designed to find approximate optimal solutions. Numerical experiments are carried out toinvestigate the performance of the proposed algorithm on a set of 4PLRPF instances. The experimentalresults show that the proposed method is a valuable tool for making decisions for the 4PLRPF.

& 2013 Elsevier B.V. All rights reserved.

1. Introduction

In the last century, many companies have outsourced theirlogistics to special logistics providers. The logistics providers areknown as third party logistics (3PL). Trebilcock (2002) showedthat nearly 75% of the Fortune 500 companies have relied on 3PLservice since 1996. And 83% of the top 500 manufacturers in theUnited States had already adopted 3PL by 2003 (Lieb and Bentz,2004).

3PL has been widely accepted by many companies. But most ofthe 3PL providers only provide transportation and warehousingservices. How to integrate, operate and monitor the complexresources of the supply chain and how to maximize the currentand long-term benefits of the supply chain members is still aquestion.

The initial concept of fourth party logistics (4PL) was intro-duced by Accenture. It is an integrator that assembles theresources, capability, and technology of its own organization andother organizations to design, build and run comprehensivesupply chain solutions (Bauknight and Miller, 1999). Since theessence and core superiority of 4PL lies in its ability to integratethe supply chain resources, it encourages strategic alliance among3PLs, and manages the logistic process within the entire supplychain member. Many papers, such as Bade (2010), Bumstead and

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l., Fourth party logistics rouoi.org/10.1016/j.ijpe.2013.03

Kempton (2002), Foster (1999), Love (2004), and Marino (2002)discussed this concept and reveal that 4PL can exceed 3PL andbecome the next logistics service mode in the following years.

As 4PL plays an important role in modern logistics, manyresearches have been conducted on 4PL ever since it was intro-duced. These researches can be classified into two classes. Oneclass is about the perspective of 4PL, which focuses on its macro-framework, such as the historical inevitability, the advantages, andthe development prerequisites, etc., of 4PL (Bumstead andKempton, 2002; Han, 2003; Love, 2004; Tierney, 2004); the otherclass is to optimize operational problems under the guidance of4PL, such as the 4PL routing problem (4PLRP) (Chen et al., 2003;Huang et al., 2006, 2008), the selection of 3PL suppliers (Zhanget al., 2004), and the operational problem in the 4PL supply chain(Lau et al., 2002; Li et al., 2003; Wang et al., 2006), etc.

The routing problem is one of the most important issues inlogistics and it has attracted an extensive attention from research-ers in the past several decades. When studying the routingproblem in 4PL, researchers found that more issues, such as theselection of 3PL, cost factors, time, and capacity and reputation of3PL providers, should be considered. These conditions make the4PL network become a multi-graph, which is more difficult thanthe simple graph in 3PL network. For this reason, the 4PLRP hasattracted more and more attention from researchers. There havebeen publications in the literature that apply different intelligentmethods for the 4PLRP. Chen et al. (2003) proposed a geneticalgorithm (GA) for solving the 4PLRP with ten nodes. Li et al.(2003) introduced some main optimization processes in the

ting problem with fuzzy duration time. International Journal of.007i

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Fig. 1. Multi-graph of a 4PLRPF.

M. Huang et al. / Int. J. Production Economics ∎ (∎∎∎∎) ∎∎∎–∎∎∎2

logistics operation, including routing optimization, job decompos-ing, and job assignment, and depicted the relationship betweenthese processes. Huang et al. (2006) considered the node-edgeproperty with three different scales and developed a nonlinearinteger programming to solve the 4PLRP model. In order toenhance the capability of seeking optimum solutions, Huanget al. (2008) also proposed a hybrid immune algorithm bycombining the memory base with the immunities operator, whichwas shown to provide better results than that in Huang et al.(2006).

There have been researches devoted to the 4PLRP. But in theprevious researches the parameters are fixed and deterministic,which is not realistic. In a real-world situation, driving time couldbe an effect on many factors like weather, human factors and soon. In the traditional way, researchers tend to use the stochasticoptimization methods. But in some conditions, it is hard todescribe the parameters of the problem as random variablesbecause there is not enough data to analyze. For example, theduration time from one point to another is often lack of historicaldata. Sometimes it can be “between 30 and 40 min”, “around 1 h”,act. Generally, we can use fuzzy variables to deal with theseuncertainty parameters (Liu, 2009).

Fuzzy theory is widely used in many problems, such as vehiclerouting problem (Cao and Lai, 2010; Sheng et al., 2006; Tang et al.,2009; Zheng and Liu, 2006), job-shop scheduling problem (Kurodaand Wang, 1996; Lei, 2010; Niu et al., 2008; Sakawa and Kubota,2000), and so on. However, it is still a new field in the 4PLRP. Inthis paper, a real-world 4PLRP model with fuzzy duration time(denoted 4PLRPF) is built by using the credibility theory (Liu,2009). In 4PLRPF, there are more than one 3PL provider betweentwo nodes, so it can be considered as a constrained shortest pathproblem (CSPP) with fuzzy parameters in a multi-graph.

The CSPP is a kind of shortest path problem that includes someconstraints on the route. Though CSPP and fuzzy shortest pathproblem (FSPP) are studied over recent years, there has been noattempt to study them together, not to mention fuzzy constrainedshortest path problem (FCSPP) in multi-graph. The primary reasonmay be the difficulty to deal with the infeasibility of a path and thecomparison between fuzzy numbers. One approach to solvingCSPP is to utilize a K-th shortest path algorithm, terminating withthe first path that satisfies the constraint (Gabriel and Israel, 1980).But this approach is impractical when the terminal value of k islarge. The second approach is to use dynamic programming, whichis usually used in the simple graph (Bolanda et al., 2006; Gao,2011; Ghoseiri and Nadjari, 2010; Glinas et al., 1998; Klein, 1991;Mahdavi et al., 2009; Moungla et al., 2010; Righini and Salani,2008). For the multi-graph, take label setting method as anexample, there are usually more than one efficient edge betweentwo nodes, so it will be more than one edge that could be extendto the same label. For the 4PLRPF, it will not be effective when thenumber of nodes increases to 1000 (see Section 4.3). Therefore, theabove two methods are not very effective to solve our problem.

Evolutionary algorithm is another way to solve path problems.Mohamed et al. (2010) proposed a genetic algorithm (GA)approach to solve the bicriteria shortest path problem. Ghoseiriand Nadjari (2010) provided an ant colony algorithm for bi-objective SPP. Liu et al. (2012) analyzed the difficulty in CSPPand adopted an oriented spanning tree based GA for solving multi-criteria SPP as well as multi-criteria CSPP. However, in ourproblem, the 3PL providers need to satisfy the customers' require-ments and there are usually more than one arc (3PL provider) thatcould be selected. It means that we need to get the shortest path ina multi-graph with fuzzy parameters and constraints. Theseconditions will easily result the route infeasible. In this paper, areal world 4PLRPF model is built by using the credibility theory(Liu, 2009). In order to solve it, a standard GA (SGA) with fuzzy

Please cite this article as: Huang, M., et al., Fourth party logistics rouProduction Economics (2013), http://dx.doi.org/10.1016/j.ijpe.2013.03

simulation is designed. For getting higher quality individuals inthe initialization process, K-th shortest path algorithm (Bellman,1958) is embedded in the initialization process, denoted KGA (K-thshortest path algorithm embedded GA). For further improving thequality of the result got by KGA, the algorithm employs the secondstep GA to adjust the 3PL providers (edges) on the route whichhave the same path as the leader, called KTGA (two-step geneticalgorithm based on the K-th shortest path algorithm). Numericalexperiments are carried out to investigate the performance of theproposed algorithms. The results show that KTGA can get the samesolution as the label setting method with n¼500, and that it is abetter algorithm to solve the 4PLRPF in comparison with the SGAand KGA.

The rest of this paper is organized as follows. The next sectionpresents the description of the 4PLRPF. Section 3 describes thefuzzy simulation process and the proposed KTGA for solving4PLRPF in detail. Section 4 presents the numerical study ofinvestigating the performance of the proposed algorithm on sometest 4PLRPFs. Finally, Section 5 concludes this paper with somediscussions on relevant future work.

2. The modeled 4PLRPF

2.1. Problem description

The 4PLRPF can be defined as selecting logistics companies fora 4PL provider to optimize the supply chain on purchase, sale anddistribution logistics in diverse phases and locations (Chen et al.,2003). One key problem that 4PL providers should consider is howto optimize the route of transporting goods from a source to adestination so that it can minimize the cost and satisfy therelevant constraints.

Assume that there is a network of 4PLRPF as shown in Fig. 1. Insuch a network, there are more than one arc (3PL suppliers) thatcould be selected between any two nodes. Each node has itstransit cost, time and capacity. When a 4PL provider wants totransport goods from the source to the destination, they need toselect nodes and 3PL suppliers on the network. The 3PL supplierswho want to undertake the job between two nodes will give theirservice information including the transportation cost, time andcapacity to the 4PL provider, and the 4PL provider will evaluate the3PL providers' reputation through the previous cooperation. Thenthe 4PL provider could select which 3PL supplier can be usedaccording to their information and the customers' requirements.Furthermore, the time given by the transit node and the 3PLsupplier is not precise enough, so we use fuzzy variables to dealwith it.





M. Huang et al. / Int. J. Production Economics ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 3

As shown in Fig. 1, the 4PLRPF can be represented by a multi-graph GðV ,EÞ, where V ðjV j ¼ nÞ is the set of nodes, which repre-sents cities or ports, and E is the set of edges, which connects pairsof nodes and represents different 3PL suppliers that are availableto provide the transportation service between connected nodes.The variables used in the representation of the 4PLRPF are definedas follows:

rij: The number of edges (3PL suppliers) between node i andnode j (i.e., between node i and node j).

eijk: The K-th edge between node i and node j ði,j¼ 1,2,…,nÞ.xijk: 0–1 variables that represent that, when there is a need to

transport goods from the source to the destination, whether edgeeijk undertakes the transportation task between node i and node jif both node i and node j are selected. If eijk undertakes thecorresponding task, xijk≔1; otherwise, xijk≔0.

yi: 0–1 variables that represent whether node i undertakes thetransportation task. If node i undertakes the corresponding task,yi≔1; otherwise, yi≔0.

Cijk, Pijk, Aijk: The cost, transportation capacity, and reputation ofthe 3PL supplier represented by eijk.

ξijk: The fuzzy duration time given by the K-th 3PL supplierwhen they transport between i and j.

C′j: The cost of node j, including the cost of processing,inventory, loading, unloading, etc.

ηj: The fuzzy duration time of node j, including the time ofprocessing, loading and unloading, changing vehicle, storage, etc.

P′j: The processing capacity of node j, including the capacitiesof processing, manufacturing, storage, etc.

P,A: The transportation capacity and reputation that theselected 3PL suppliers are required by the customer to satisfy.

P′: The transportation capacity that the selected nodes arerequired by the customer to satisfy.

T: The due date of the task required by the customer.With the above definition of variables, the mathematical model

for the 4PLRPF can be built up. The objective of this problem is tofind a route from the source node (1) to the destination node (n)that has the minimum cost and is subject to constraints on thefinal delivery due date, 3PL supplier's reputation, and load require-ment given by the customer. The mathematical model for the4PLRPF can be presented as follows:

Z ¼min ∑n

i ¼ 1∑n

j ¼ 1∑rij

k ¼ 1Cijkxijkþ ∑

n

i ¼ 1C′iyi

( )ð1Þ

s.t.

∑n

i ¼ 1∑n

j ¼ 1∑rij

k ¼ 1ξijkxijkþ ∑

n

i ¼ 1ηiyi≤T ð2Þ

∑rij

k ¼ 1∑n

i ¼ 1xijk− ∑

rij

k ¼ 1∑n

i ¼ 1xjik

¼−1 if j¼ 11 if j¼ n

0 otherwise

8><>: , j∈f1,2,…,ng ð3Þ

∑n

i ¼ 1∑rij

k ¼ 1xijk ¼ yj, j∈f2,3,…,ng ð4Þ

∑n

j ¼ 1∑rij

k ¼ 1xijk ¼ yi, i∈f1,2,…,n−1g ð5Þ

Pxijk≤Pijk, i,j∈f1,2,…,ng, k∈f1,2,…,rijg ð6Þ

P′yi≤P′i, i∈f1,2,…,ng ð7Þ

Axijk≤Aijk, i,j∈f1,2,…,ng, k∈f1,2,…,rijg ð8Þ


xijk ¼ 0 or 1,i,j∈f1,2,…,ng, k∈f1,2,…,rijg ð9Þ

yi ¼ 0 or 1, i∈f1,2,…,ng ð10ÞIn the formulation, Eq. (1) is the objective function, i.e.,

minimizing the sum of costs; Eq. (2) represents that the timeneeded on the route should not be more than the due date T that isrequired by the customer; Eq. (3) means to keep a balance of thenetwork flow; Eqs. (4) and (5) ensure that the selected nodes andedges should made up of routes from the source to the destina-tion; Eq. (6) ensures that the capacity constraint of the selected3PL supplier is not less than the transportation capacity P requiredby the customer; Eq. (7) ensures that the capacity of any node onthe route is not less than P′, which is required by the customer;Eq. (8) ensures that the reputation of the selected 3PL supplier isnot less than A, which is required by the customer; Eqs. (9) and(10) represent that xijk and yi are 0–1 decision variables,respectively.

In practice, some pre-processing actions can be taken beforethe simulation. According to the requirement of the customer, theinformation of 3PL providers who cannot satisfy the constraints inEqs. (6)–(8) can be easily eliminated. Then, the model can besimplified and only the objective function with constraints in Eqs.(2), (3), (4), (5), (9) and (10) will be left.

2.2. Fuzzy duration time

When the duration time ξijk and ηi are fuzzy variables, the timeneeded on a route is also a fuzzy variable (Dubois and Prade, 1978).In order to rank them, a credibility measure (Liu, 2009) isemployed to mathematically model the 4PLRPF.

According to the characteristic of the problem, a confidencelevel α ð0oα≤1Þ is provided by the decision-maker at which thedecision is allowed not to satisfy the constraints all the time butalso at the defined confidence level α. In the framework of thecredibility theory, the constraint in Eq. (2) has no practicalsignificance with fuzzy variables. According to the credibilitytheory, the chance constrained programming can be employed torepresent the time constraint in Eq. (2) as follows:

Crf ∑n

i ¼ 1∑n

j ¼ 1∑rij


n

i ¼ 1ηiyi≤Tg≥α ð11Þ

This means that the credibility that the time needed on theroute is not more than the time T required by the customer is notless than α.

3. Algorithm designed for the 4PLRPF

The 4PLRPF described above is a difficult combinational opti-mization problem. In this paper, KTGA with the fuzzy simulation isproposed to solve it as follows. Fig. 2 shows the flow chart of theproposed algorithm, which is briefly described as follows:

Step1: Initialize the linkage matrix according to themodel graph.

Step2: Generate the initial population of PS individuals by theK-th shortest path algorithm, where PS denotes thepopulation size.

Step3: Calculate the fitness function of the individuals withfuzzy simulation.

Step4: Select individuals by the roulette wheel selectionscheme.

Step5: Perform crossover operations on pairs of selected indi-viduals with a crossover probability Pc.

Step6: Perform mutation operations to individuals with amutation probability Pm.





Second Step GA

First Step GA Generate the initial population bythe K-th shortest path algorithm

?I NG

YES

Calculate the fitness of individualswith fuzzy simulation

Select individual by roulette wheel

Crossover pairs of individuals

Mutate individuals

I:=1

NO

Expand to Q PS individuals

I:=1

2?I NG

YES

NO

I:=I+1

Output the best solution accordingto the fitness function

I:=I+1

I:=I+1

Calculate the fitness of individualswith fuzzy simulation

Select individual by roulette wheel

Crossover pairs of individuals

Mutate individuals

Fig. 2. Flowchart of the fuzzy simulation based KTGA.


Step7: Check whether the maximum allowable number ofgenerations, denoted NG1, for the first step GA has been reached.If it is false, go to Step 3; otherwise, stop the first step GA andoutput the solutions in the final population for the second step GA.

Step8: Expand solutions obtained by the first step GA to QnPSsolutions by selecting 3PL suppliers of the best individualrandomly.

Step9: Calculate the fitness function of the individuals withfuzzy simulation.

Step10: Select individuals by the roulette wheel selectionscheme.

Step11: Perform crossover operation on pairs of the selected3PL suppliers' array of the obtained Q � PS individuals with acrossover probability PC.

Step12: Perform the second step GA on the 3PL suppliers' arrayof the obtained QnPS individuals for NG2 generations.


Step13: Stop the process and output the best solution accordingto the fitness function.

In the following sections, fuzzy simulation process and majorcomponents of KTGA will be described in detail, respectively.

3.1. Fuzzy simulation

In our problem, the constraint in Eq. (11) contains fuzzyparameters. In this section, we use fuzzy simulation (Liu, 2009)to estimate it, which is given as follows:

Suppose x and y are two decision vectors, which are composedof xijk and yi ði,j∈f1,2,…,ng,k∈f1,2,…,rijgÞ, respectively.

Let

f ðx,y,ξ,ηÞ ¼ ∑n

i ¼ 1∑n

j ¼ 1∑rij


n

i ¼ 1ηiyi ð12Þ

where ξ and η are fuzzy vectors composed of ξijk andηi ði,j∈f1,2,…,ng,k∈f1,2,…,rijgÞ, respectively. Denote that μξijk isthe membership function of ξijk and μηi is the membership functionof ηi, respectively.

In the following we will show how to simulate the followingfuzzy function:

U : x-minff jCrff ðx,y,ξ,ηÞ≤f g≥αg ð13Þwhere 0oα≤1.

Firstly, generate θqijk and sqi from the ε-level sets of fuzzy variables

ξijk and ηi ði,j∈f1,2,…,ng,k∈f1,2,…,rijgÞ, respectively, where ε is asufficiently small positive number, q¼ 1,2,…,N, and N is a sufficientlylarge number. Set νq ¼mini,j,kfμξijk ðθ

qijkÞg⋀minifμηi ðs

qi Þgðq¼ 1,2,…,NÞ.

According to the concept of credibility measure, for any number r, weset (Liu, 2009)

LðrÞ ¼ 12

max1≤q≤N

fνqjf qðx,yÞ≤rg�

þ min1≤q≤N

f1−νqjf qðx,yÞ > rg�

It follows from monotonicity that we may employ bisection search tofind the maximal value r such that LðrÞ≥α. This value is an estimation

of f . The process can be run as follows:Step1: Generate θqijk and sqi from the ε-level sets of fuzzy

variables ξijk and ηi ði,j∈f1,2,…,ng,k∈f1,2,…,rijgÞ, respectively,where ε is a sufficiently small positive number, and q¼ 1,2,…,N.

Step2: Set νq ¼mini,j,kfμξijk ðθqijkÞg⋀minifμηi ðs

qi Þgðq¼ 1,2,…,NÞ for

q¼ 1,2,…,N.Step3: Find the minimal value r such that LðrÞ≥α.Step4: Return r.Set f ¼ r, then Eq. (11) could be rewritten as follows:

f ≤T ð14Þ

3.2. Two-step genetic algorithm

3.2.1. Adaptive double arrays encodingAccording to the characteristic of the 4PLRPF, an encoding

based on two arrays is adopted to represent a chromosome inKTGA, where the first array consists of nodes on a route encodedby natural number and the second array consists of 3PL suppliers(edges) on a route encoded by integer number. The length of achromosome is variant. Suppose the number of nodes in the graphis n (i.e., jV j ¼ n) and the index number is allocated to each nodeand edge (as shown in Fig. 1). Then, the encoding of a chromosomeis shown as follows:

path : ½v1,v2,…,vm−1,vm�PL : ½ev1v2k1 ,…,evm−1vmkm−1

� ð15Þ






where m≤n, kh∈f1,2,…,rvhvhþ 1 g,h¼ 1,2,…,m−1. The path array is apermutation of the nodes on the route. v1 is the start and vm is thedestination. v2,…,vm−1 are the intermediate nodes between v1 andvm. This array means that when we want to transport goods fromv1 to vm, we need to undergo v2,…,vm−1 as ports. The PL array is alist of 3PL providers for the above transportation. For example, ifev1v2k1 ¼ 2, it means that when goods undergo the route from node

Initialization Process

I:=1

I<=PS?

Represent the multi-graph with a linkagematrix D of dimension n n

YES

Convert the multi-graph into a simple graph

Generate a route from the simple graph

Calculate the time and cost of the route

I:=I+1

Output PS solutions

NO

Fig. 3. The flowchart of the initialization process.

0 2 3 0 0 0 0 00 0 2 1 0 1 2 00 1 0 3 2 2 1 00 1 2 0 1 3 2 00 0 2 1 0 2 2 00 1 2 2 1 0 1 20 1 2 1 3 1 0 20 0 0 0 0 0 0 0

F

0 0 3 00 0 2 10 0 0 30 0 2 00 0 2 10 0 2 20 0 2 10 0 0 0

21 2

21 2

0 0 3 0 0 0 0 00 0 2 0 0 1 0 00 0 0 0 2 2 0 00 0 2 0 1 3 0 00 0 2 0 0 2 0 00 0 2 0 1 0 0 20 0 2 0 3 1 0 20 0 0 0 0 0 0 0

0 0 3 00 0 2 00 0 0 00 0 2 00 0 2 00 0 2 00 0 2 00 0 0 0

2 2 1 11 2 7 4 5

2 21 2 7

Fig. 4. The process of constr


v1 to node v2, the 3PL supplier with index 2 between the twonodes is selected.

The two arrays together form an individual on which geneticoperators will be applied. Each individual corresponds to a path asa solution to the key problem. But not all the combinations arefeasible, because we may not find a 3PL provider to connect tworandomly chosen nodes (Li et al., 2003). So, in this paper, in orderto avoid infeasible individuals using the above encoding method, acorresponding linkage matrix is constructed in the initializationphase, as described below.

3.2.2. InitializationThe initialization of the population for the first step GA involves

several steps. Fig. 3 shows the flow chart of this initializationprocess, and much briefly description is as follows:

Step1: Represent the multi-graph with a linkage matrix D ofdimension n� n with respect to a source node and a destinationnode. Each element dij∈D represents whether node i is linked tonode j directly in the multi-graph. If there is no edge betweennode i and node j, dij is set to zero; otherwise, dij is set to thenumber of edges between node i and node j. Furthermore, duringthe transportation, goods are not allowed to return to the sourcenode after they leave from it, and they will terminate theirtraveling after they arrive at the destination. Take Fig. 1 as anexample, the corresponding linkage matrix D with node 1 ðv1 ¼ 1Þas the source and node 8 (vm¼8) as the destination is given asfollows:

D¼ ðdijÞ8�8 ¼

0 3 4 0 0 0 0 00 0 2 3 0 2 2 00 2 0 3 3 2 2 00 3 3 0 2 3 2 00 0 3 2 0 2 3 00 2 2 3 2 0 2 30 2 2 2 3 2 0 30 0 0 0 0 0 0 0

266666666666664

377777777777775

ð16Þ

0 0 0 00 1 2 02 2 1 01 3 2 00 2 2 01 0 1 23 1 0 20 0 0 0

0 0 3 0 0 0 0 00 0 2 1 0 1 0 00 0 0 3 2 2 0 00 0 2 0 1 3 0 00 0 2 1 0 2 0 00 0 2 2 1 0 0 20 0 2 1 3 1 0 20 0 0 0 0 0 0 0

27

2 2 11 2 7 4

0 0 0 00 1 0 00 2 0 00 3 0 00 2 0 00 0 0 20 1 0 20 0 0 0

0 0 3 0 0 0 0 00 0 2 0 0 0 0 00 0 0 0 0 0 0 00 0 2 0 0 0 0 00 0 2 0 0 0 0 00 0 2 0 0 0 0 20 0 2 0 0 0 0 20 0 0 0 0 0 0 0

1 1 24 5 6

2 2 1 1 2 21 2 7 4 5 6 8

ucting a feasible route.






Step2: Convert the multi-graph into a simple graph (i.e., there isat most one edge between any two nodes) as follows: an edge israndomly selected from available edges between any adjacentnodes in the multi-graph so that a sub-graph G′ðV ′,E′Þ, wherejV ′j ¼ jV j ¼ n, E′ is the set of edges in the sub-graph, is obtained,which can be described as a matrix F(F ¼ ðf ijÞn�n). Take the elementd12 ðd12 ¼ 3Þ in the matrix D as an example, we first generate aninteger randomly in the range of 1 to d12, say 2 is generated. Then,set fij to 2, which means in the simple graph G′ when goods aretransported from node 1 to node 2, the second 3PL supplier ischosen to complete the task, and so on, then we can get the matrixF ðF ¼ ðf ijÞ8�8Þ shown in Fig. 4(a).

Step3: Generate an initial route from the simple graph G′ asfollows. Starting from the source node v1, we choose an adjacentnode randomly as the second node on the route, record its index,mark it as “used”, which means we cannot use the node any moreon this route, and set the elements of the corresponding column inthe current simple graph to 0. Then, we randomly choose an un-used node from those nodes that are adjacent to the second nodeon the current route, record its index, mark it as “used”, andupdate the current simple graph accordingly. This process con-tinues until the destination node is reached. If a node which ischosen has no other nodes adjacent and it is not the destination,then stop and re-start from the source node. Fig. 4 illustrates theprocess of generating a feasible route from the simple graph Fconverted from the multi-graph D in Eq. (16).

Step4: Calculate the time and cost of the corresponding route.Step5: Repeat Steps 2–4 until PS solutions are generated.

3.2.3. The K-th shortest path algorithmThe quality of an initialization population has a direct impact

on the algorithm. The method proposed in Section 3.2.2 can get aninitialization population, but it is not effective when the number ofnodes increases (see Section 4.3). K-th shortest path (KSP) algo-rithm is an effective way to solve constrained shortest pathproblem, but its application is limited to the simple graph. Forthe multi-graph, we can convert the multi-graph into simplegraphs, then use KSP algorithm to solve every simple graph andget the shortest one. But it will be a large number if we enumerateall the possibility results. In order to get high quality solutions, weuse KSP algorithm to get the initial individuals, which is employedin Step 3 during the initialization phase. According to the algo-rithm presented in Bellman (1958), this process is shown asfollows:

Step1: Find the shortest path p1 that links from v1 to vm in thesimple graph G′ðV ′,E′Þ generated in Section 3.2.2. Record itsadjacent matrix as F (F ¼ ðf ijÞn�n, i,j¼ 1,2,…,n) and the elementfij is the 3PL supplier selected to do the transportation if both i andj are on the route. Set k≔1, pk≔fvk1,…,vkmg (vki ði¼ 1,2,…,mÞ is the i-th node on pk, m is the number of nodes on pk), P≔fp1g and ii≔1.

Step2: Calculate the time Tpk needed on the route pk. If Tpk isless than the time T requested by the customer, terminate thealgorithm and output pk as the best individual; otherwise, go toStep 3.

Step3: Set Ee≔E′−{ðvkii,vkiiþ1Þjðvkii,vkiiþ1Þ is an edge of pk}, then findthe shortest path ppii in G′ðV ′,EeÞ.

Step4: If iiom−1,ii≔iiþ1,Ee≔E′ and turn to Step 3; otherwise,Ee≔E′ and record a series of the shortest paths got in Step 3 as W,where W ¼ fppiijii¼ 1,…,m−1g.

Step5: P≔P∪W−fpkg and k≔kþ1. Order the elements of Paccording to the cost from low to high and record the route withthe lowest cost in P as the K-th shortest path pk.

Step6: If k > K , terminate the algorithm and output pk as anindividual in the initial population; otherwise turn to Step 2.


3.2.4. Fitness functionFitness function is usually designed according to the objective

function and constrains of the problem, which can reflect therequirements of the customers and the consideration of the 4PLprovider. In this paper, the time constraint is estimated by thefuzzy simulation, so when it is used as a punishment added to theobjective function, by the theory mentioned in Section 3.1, thefitness function can be represented by

f ¼ ∑n

i ¼ 1∑n

j ¼ 1∑rij

k ¼ 1Cijkxijkþ ∑

n

i ¼ 1C′iyiþβnmaxð0,f −TÞ ð17Þ

where f is estimated in Section 3.1 and β ðβ≥1Þ is a punishingfactor, which is set according to a particular problem. Here β is setto be a value and it is defined in Section 4.2.

After evaluating the fitness of the PS individuals in the popula-tion according to Eq. (17), we sort the individuals in the order ofincreasing fitness value (i.e., the better the chromosome is, thesmaller the ordinal number it has). Then select them as anevaluation function, denoted by evalðIiÞ, i¼ 1,…,PS, to assign aprobability of reproduction to each chromosome Ii so that itslikelihood of being selected is proportional to its fitness relative tothe other chromosomes in the population. That is, the chromo-somes with higher fitness will have more chance to produceoffspring by using roulette wheel selection.

Now let parameter γ∈ð0,1Þ in the genetic system be given, thenthe rank-based evaluation function of calculating the fitness ofeach chromosome Ii is given as follows:

evalðIiÞ ¼ γð1−γÞi−1, i¼ 1,…,PS ð18Þ

3.2.5. Selection mechanismThe selection process is based on the roulette wheel selection

scheme, which is a fitness-proportional selection. We spin theroulette wheel PS times. Each time we select a single individualinto the mating pool to undergo crossover and mutation. Theselection process is described as follows:

Step1: Calculate the cumulative probability qi for each indivi-dual Ii (i¼ 1,…,PS) as follows:

q0 ¼ 0,qi ¼ ∑j ¼ i

j ¼ 1evalðIjÞ, i¼ 1,…,PS ð19Þ

Step2: Generate a random number r∈ð0,qPS�.Step3: Select the individual Ii such that qi−1or≤qi ði∈f1,…,PSgÞ.Step4: Repeat Steps 2 and 3 PS times to select PS individuals.

3.2.6. Crossover operationIn every generation, crossover operation is applied on parents

with a crossover probability Pc. This probability makes theexpected number PcnPS of individuals undergo the crossoveroperation.

From i¼1 to PS: generating a random number r from theinterval [0,1], the individual Bi is selected as a parent if roPc.Denote the selected parents by B′1,B′2,…. Then divide them intothe following pairs and delete the last one if the number of them isodd:

ðB′1,B′2Þ,ðB′3,B′4Þ,ðB′5,B′6Þ,…Let us illustrate the crossover operator on each pair by ðB′1,B′2Þ.

Firstly, we determine the number of nodes on B′1 and B′2, andrecord the less one as L. Secondly, we generate a random crossoverpoint c between 2 and L−2. Thirdly, the crossover operator on B′1and B′2 will produce two children BB′1 and BB′2 by exchanging thegenetic materials after the crossover point c. An example of thecrossover operation is given below. We assume the following two





Table 1The data of nodes.

Node Cost Time

1 6 (1.0, 1.5, 2.0)2 3 (0.4, 0.6, 0.7)3 3 (0.5, 0.6, 0.7)4 5 (0.8, 1.2, 1.5)5 10 (2.0, 4.0, 6.0)6 9 (1.0, 2.0, 3.0)7 8 (1.0, 1.5, 3.0)8 7 (0.5, 1.0, 2.0)


parents B′1 and B′2:

B′1 : path½v1,v2,…,vc−1,⋮vc ,…,vm−1,vm�

PL½ev1v2k1 ,…,evc−2vc−1kc−2 ,⋮evc−1vckc−1 ,…,evm−1vmkm−1�

B′2 : path½v′1,v′2,…,v′c−1,⋮v′c ,…,v′t−1,v′t �

PL½ev′1v′2k′1 ,…,ev′c−2v′c−1k′c−2 ,⋮ev′c−1v′ck′c−1 ,…,ev′t−1v′t k′t−1 �We calculate L¼minðm,tÞ and generate a random integerc∈½2,L−2�. Then, two children BB′1 and BB′2 are generated asfollows:

BB′1 : path ½v1,v2,…,vc−1,⋮v′c ,…,v′t−1,v′t �

PL½ev1v2k1 ,…,evc−2vc−1kc−2 ,⋮ev′c−1v′ck′c−1 ,…,ev′t−1 ,v′t k′t−1 �

BB′2 : path ½v′1,v′2,…,v′c−1,⋮vc,…,vm−1,vm�

PL½ev′1v′2k′1 ,…,ev′c−2v′c−1k′c−2 ,⋮evc−1vckc−1 ,…,evm−1vmkm−1�

After the crossover operation, we need to check the feasibilityof each child before accepting it. Take BB′1 as an example, if it isstill a route from the source to the destination, we replace theparent B′1 with BB′1. If not, try to do some reparation to the childas follows.

If there are same nodes on the route, delete the part betweenthem, remove one duplicate node, and calculate the time and coston the new route; otherwise, we examine whether the nodebefore the crossover point is connected with the other nodesbehind it. If that happens, we delete the part between them andrandomly add a corresponding 3PL supplier, and replace theparent B′1 with BB′1; otherwise, quit with failure. Finally, if thechild is infeasible, we use the corresponding parent instead.

3.2.7. Mutation operationMutation is carried out on the PL array of individuals according

to a probability Pm, as follows.In a similar manner to the process of selecting parents for

crossover operation, we repeat the following steps from i¼1 to PS:generating a random number r from the interval [0,1], theindividual Bi is selected as a parent for mutation if roPm.

For each selected parent, we mutate it in the following way.Firstly, we calculate the total number of nodes in the PL array,denoted as P1. Secondly, generate an integer p randomly from theinterval ½1,P1�. And last, choose another 3PL supplier randomly toreplace the one between node p−1 and node p of the individual.

3.2.8. Second step GA used for 3PL suppliersWhen the first step GA finishes, it can obtain a satisfactory

solution, but may not be able to find the optimum (see Table 4,KGA). Considering the characteristic of individuals, we found thatafter several generations, 99.5% of the best solutions' paths in thepopulation will be the same as the previous leader with only somechanges of the 3PL suppliers' array. For this reason, the second stepGA is employed to adjust the routes which have the same pathwith the leader. In order to enhance the effect of optimization, weexpand the individuals with the same path as the best individualof the population to QnPS individuals. The detailed description isgiven below.

Step1: Find individuals in the final population obtained by thefirst step GA which have the same path as the best individual Ibestin the population, and denote the total number of such individualsas nbest.

Step2: Expand the best individual Ibest to QnPS individuals byselecting 3PL suppliers randomly on the route as follows: if s and tare two adjacent nodes on the route represented by Ibest, select aninteger randomly from 1 to dst (dst is the number of 3PL suppliers


between node s and node t) as the 3PL supplier chosen toundertake the transportation from s to t. This process is repeateduntil all the edges of the route are assigned a 3PL supplier andhence a new individual with the same path as Ibest is generated.

Step3: Repeat Step 2 until QnPS−nbest new individuals areproduced. These new individuals and the original nbest individualsobtained by the first step GA together form the initial populationfor the second step GA.

Step4: Select individuals for variation by the roulette wheelscheme.

Step5: Crossover the selected parents' 3PL array with a prob-ability Pc.

Step6: Mutate the offspring' 3PL array with a probability Pm.Step7: Calculate the fitness for each offspring and update the

elitism if a better solution arises.Step8: Repeat Steps 4–7 for NG2 cycles.Step9: Output the best solution obtained as the final solution.

4. Numerical experiments

In this section, we present three groups of experiments toinvestigate the performance of KTGA for the 4PLRPF. The firstgroup is to introduce two kinds of 4PLRPF instances as the test bedto test the solution quality of the proposed KTGA. The secondgroup is to test the effect of α on the decision. The third group is tocompare the performance of KTGA with the standard GA (SGA), aone step GA with the KSP algorithm (KGA) and label setting (LS)method for the 4PLRPF. All the investigated algorithms were codedin Matlab 7.0 and run on a Core 2 2.83GHZ PC.

4.1. The 4PLRPF examples

We use two kinds of 4PLRPF as the test examples in this study.For the first 8-node 4PLRP, as shown in Fig. 1, we assume that a4PL company undertakes transnational transportation from theNode 1 to the Node 8 and there are six intermediate nodesbetween them. Each node has the properties of cost, fuzzy time,and capacity. If there is any business between two nodes, an edgeis added. Since it may exist several 3PL supplier companies fortransportation between any two nodes, there may be multipleedges between them (one edge stands for a 3PL supplier). Thisresults in a multi-graph, where each edge and node has propertiesof cost, fuzzy time, capacity, and reputation. Suppose the fuzzytime is represented by triangular fuzzy numbers, then the data ofnodes and edges are given in Tables 1 and 2, respectively.

For another kinds of 4PLRPFs, they are respectively small scale4PLRPF with 15, 30 nodes and relatively large scale M-S 4PLRPswith 100, 500, 1000, 2000, 3000 and 5000 nodes. All of them aregenerated randomly. We use H≔½a,b� to mean that H is an integernumber randomly generated between a and b. And the cost andtime in the following are respectively recorded by the unit of100Yuan and Day. Suppose the fuzzy time used in these examples





Table 2The data of edges, where “S” and “E” mean the start and end nodes, respectively.

S E Edge Cost Time S E Edge Cost Time

1 2 1 1 (0.2, 0.3, 0.4) 3 6 2 8 (16, 17, 21)1 2 2 2 (0.1, 0.25, 0.3) 3 7 1 9 (16, 19, 20)1 2 3 3 (0.1, 0.15, 0.3) 3 7 2 10 (15, 17, 19)1 3 1 3 (1.3, 1.5, 1.6) 4 5 1 6 (9, 10, 14)1 3 2 5 (1.1, 1.3, 1.5) 4 5 2 7 (8, 9, 12)1 3 3 4 (1.2, 1.4, 1.6) 4 6 1 13 (19, 22, 24)1 3 4 6 (1.1, 1.2, 1.3) 4 6 2 14 (18, 20, 23)2 3 1 5 (2, 2.2, 2.5) 4 6 3 12 (21, 23, 25)2 3 2 6 (1.8, 2, 2.3) 4 7 1 14 (22, 24, 26)2 4 1 10 (2, 3, 4) 4 7 2 15 (24, 26, 28)2 4 2 8 (2, 4, 6) 5 6 1 8 (3, 3.5, 4)2 4 3 6 (2, 5, 7) 5 6 2 7 (2.9, 3.2, 3.7)2 6 1 10 (18, 20, 21) 5 7 1 10 (3.5, 3.8, 4)2 6 2 11 (18, 19, 22) 5 7 2 8 (3.8, 4.1, 4.4)2 7 1 11 (20, 22, 24) 5 7 3 11 (3.2, 3.5, 3.8)2 7 2 12 (18, 20, 23) 6 7 1 2 (0.8, 1.1, 1.2)3 4 1 5 (1.2, 1.5, 1.8) 6 7 2 3 (0.7, 0.9, 1.0)3 4 2 4 (1.5, 1.8, 2) 6 8 1 1 (0.5, 0.6, 0.7)3 4 3 3 (1.4, 1.6, 1.7) 6 8 2 2 (0.4, 0.5, 0.7)3 5 1 9 (9, 11, 14) 6 8 3 3 (0.3, 0.4, 0.6)3 5 2 7 (10, 13, 15) 7 8 1 3 (0.2, 0.3, 0.4)3 5 3 8 (9, 12, 14) 7 8 2 1 (0.3, 0.5, 0.6)3 6 1 9 (14, 16, 18) 7 8 3 2 (0.3, 0.4, 0.6)


are also triangular fuzzy numbers, then the parameters are set asfollows:

The number of 3PL suppliers rij between node i and node j,where i,j¼ 1,2,…,n, is generated as follows: if ji−jjo4, i≠n, andi≠j, then rij≔random½2,5�; otherwise, rij≔0.

The cost of the first 3PL supplier between node i and node j isCij1≔random½2,30�. The cost of the K-th 3PL supplier between nodei and node j is Cijk≔Cijðk−1Þ þrandom½1,3�, where k¼ 2,…,rij. Set

ξijk ¼ ðξLijk,ξMijk,ξUijkÞ, ηi ¼ ðηLi ,ηMi ,ηUi Þ, where i,j¼ 1,2,…,n and k¼ 1,2,…,

rij. Then the most likely duration time of the first 3PL supplier

between node i and node j is ξMij1≔random½4,20�. The optimisticduration time of the first 3PL supplier between node i and node j isξLij1≔ξMij1−random½1,3�. The pessimistic duration time of the first 3PL

supplier between node i and node j is ξUij1≔ξMij1þrandom½1,3�. Thefuzzy duration time of the K-th 3PL supplier between node i andnode j is ξLijk≔ξLijðk−1Þ−random½1,3�, ξMijk≔ξMijðk−1Þ−random½1,3�, and

ξUijk≔ξUijðk−1Þ−random½1,3�, where k¼ 2,…,rij. The most likely dura-

tion time at node i is ηMi ≔random½4,9�.The optimistic duration timeat node i is ηLi≔ηMi −random½1,3�, and the pessimistic duration timeat node i is ηLi≔ηMi þrandom½1,3�. The cost at node i isC′i≔random½6,15�.

Table 3Comparison of KTGA with different α, where “n” means the number of nodes and“C” means the cost of the best solution, whose Path and 3PL arrays are as shown.

α n T C Path 3PL ~T

0.8 8 26 40 [1,3,6,8] [1,1,3] (18.6,23,28)0.7 8 26 38 [1,3,7,8] [1,2,2] (19.6,23.6,29)0.6 8 26 37 [1,3,6,8] [1,2,1] (20.8,24.2,31.5)0.8 15 75 116 [1,3,7,14,15] [3,2,3,2] (39,61,84)0.7 15 75 111 [1,3,7,14,15] [2,2,1,2] (43,66,88)0.6 15 75 107 [1,3,7,14,15] [2,1,1,1] (46,70,93)0.8 30 136 190 [1,5,9,13,16,19,22,26,30] [2,2,3,3,3,1,3,1] (62,118,148)0.7 30 136 186 [1,5,9,13,16,19,23,26,30] [1,1,2,1,3,3,2,1] (68,112,156)0.6 30 136 181 [1,5,9,13,16,19,23,26,30] [2,2,3,3,3,3,3,1] (79,128,158)

4.2. The effect of α on the decision

In this section, the effect of α on the decision is tested with thesmall scale examples, which have 8, 30 and 50 nodes, respectively.Recorded the average time taken by the algorithm for one run. Theparameters considered are the number of generation (NG), PS, K,Pc, and Pm. The algorithms were run 100 times and the bestsolution (Best), the worst solution (Bad), the average value (Avg)and the mean square deviation of T (msd), were recorded. And Avgand msd are computed by

AvgðT Þ ¼ 1II

∑II

i ¼ 1Ti


and

msdðT Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

II−1∑II

i ¼ 1ðTi−AvgðT ÞÞ22

s

where II is the number of runs and Ti is the best solution in theith run.

The parameters are set as follows:

β¼ 20, K ¼ 4, γ ¼ 0:05, PS¼ 20,

NG1¼NG2¼ 15, Q ¼ 2, Pc¼ 0:5, Pm¼ 0:2

The experimental results are shown in Table 3. Take 8-node4PLRPF as an example, if the due date is set to 26 day and thecredibility parameter α is set to 0.8, we can see that the total cost is40 and Crf ~T≤Tg ¼ 0:8. In the 4PL operation, this means that whengoods are to be transported from Node 1 to Node 8 within 26 daywith a credibility 0.8, the best choice is to undergo Node 3 andNode 6 as two interim ports, and the 3PL suppliers with index 1, 1,and 3 are chosen on the three links from Node 1 to Node 3, fromNode 3 to Node 6, and from Node 6 to Node 8, respectively. And itcosts the 4PL company 40 to finish the whole job.

From Table 3, it can also be seen that given a due date (T) foreach route, when α is changed, the most suitable routes aredifferent. Given the same due date, the cost needed on the routeswill be less with the requirement of a lower credibility level. So thecustomers could select which α is suitable to choose according totheir time requirements.

4.3. The effect of the problem scale on the algorithm

In this section, we compare the performance of KTGA withother two variants of GAs, i.e., SGA and KGA , and LS (Bolanda et al.,2006). The purpose of the experiments is to evaluate the efficiencyof the designed KTGA for solving 4PLRPF with different scaleexamples.

The experiments include the above three small scale problemand six randomly generated large scale problem with the numberof (Nodes, Edges) equal to (100,3104), (500,15700), (1000, 31345),(2000,63250), (3000, 94428) and (5000,157352). The first fourexamples are executed 50 runs and the other examples areexecuted 20 runs. In order to compare the running time of eachalgorithm, the Time we computed in the following are notincluded the time for generating the network. And the parametersof the test algorithms were set based on the operator msd.

The experimental results are presented in Table 4. From Table 4,it can be seen that for the n¼8,15,30,100 and 500 problems, thethree intelligent algorithms (SGA, KGA, KTGA) can get the samebest results as those got by LS. When n¼1000, LS cannot get aresult in an acceptable time, but the intelligent algorithms can stillget the same best result within a few second. The same best resultcontinues until n¼2000. When n¼3000, SGA cannot get the same





Table 4Comparison of the algorithms with different population sizes.

Algorithm Nodes T PS NG Best Bad msd Time

LS 8 26 – – 40 – – o1SGA 8 26 40 20 40 40 0 o1KGA 8 26 40 20 40 40 0 o1KTGA 8 26 5 20 40 40 0 o1LS 15 75 – – 116 – – o1SGA 15 75 50 30 116 116 0 o1KGA 15 75 50 30 116 116 0 o1KTGA 15 75 10 30 116 116 0 o1LS 30 136 – – 190 – – o1SGA 30 136 80 40 190 196 2.2 o1KGA 30 136 60 40 190 194 0.9 o1KTGA 30 136 15 40 190 193 0.5 o1LS 100 450 – – 408 – – 9SGA 100 450 120 60 408 427 9.6 2KGA 100 450 80 60 408 410 4.4 39KTGA 100 450 30 60 408 413 1.6 4LS 500 1800 – – 1146 – – 189SGA 500 1800 300 150 1146 1237 41.2 10KGA 500 1800 100 150 1146 1178 15.1 73KTGA 500 1800 50 150 1146 1162 7.3 25LS 1000 3000 – – – – – –

SGA 1000 3000 400 200 2259 2385 58.6 30KGA 1000 3000 150 200 2259 2315 21.8 97KTGA 1000 3000 80 200 2259 2293 13.4 45LS 2000 7000 – – – – – –

SGA 2000 7000 800 400 5041 5216 88.2 165KGA 2000 7000 250 400 5041 5154 38.6 596KTGA 2000 7000 100 400 5041 5177 22.1 267LS 3000 10 000 – – – – – –

SGA 3000 10 000 1000 800 7994 8266 123.2 363KGA 3000 10 000 300 800 7961 8136 63.5 1241KTGA 3000 10 000 150 800 7961 8079 41.8 588LS 5000 17 000 – – – – – –

SGA 5000 17 000 1500 1000 14 870 15 334 225.4 640KGA 5000 17 000 500 1000 14 832 15 142 94.9 2479KTGA 5000 17 000 200 1000 14 817 14 981 68.0 1052


best result as that got by KGA and KTGA, and its msd is more than100. When n¼5000, either SGA or KGA could get the same bestresult as that got by KTGA, and KTGA's msd is much better thanthe other two algorithms'. Generally speaking, with the increase ofthe number of nodes, KGA performs better than SGA. However, thecomplex initialization will also lead to more time consumption.KTGA cannot only achieve a good solution but also reduce therunning time due to the use of the second step. So the proposedKTGA can achieve the optimal solution with a higher probabilityand does not increase the computational time quickly with theincrease of the problem size. The LS can guarantee the optimumsolution when the number of nodes is not more than 500, but itcannot obtain the solution in an acceptable time when theproblem size larger. Although SGA can solve the problem quickly,it can only find the optimal solution in a big msd. At the sametime, KGA can get the optimal solution in an acceptable time witha comparable msd. When we further improve KGA by introducinga second step GA, the obtained KTGA algorithm is able toefficiently improve the computational performance on the 4PLRPF.

5. Conclusion

With the increasing intense of market competition, more andmore enterprises begin to realize the importance of fourth partylogistics and pay more attention to the fourth party logisticsrouting problem (4PLRP). In this paper, the node-to-node, singletask 4PLRP with fuzzy duration time (4PLRPF) is discussed, whichcan be described as a selection of the shortest path problem withconstraints of fuzzy variants in a multi-graph. A fuzzy


programming model is built up for this problem based on thecredibility theory and a two-step genetic algorithm with fuzzysimulation (Liu, 2009) is proposed to solve the modeled 4PLRPF. Inthe proposed KTGA, the K-th shortest path algorithm is employedfor generating high-quality initial individuals. Then, the first stepGA is employed to search for satisfactory solutions based on thedouble arrays encoding. After the first step GA finishes, the bestsolutions obtained are expanded by randomly selecting 3PLsuppliers to obtain more solutions and the second step GA isapplied to expand solutions but only on the 3PL supplier array ofthe encoding scheme.

Numerical experiments were carried out to investigate theperformance of the proposed KTGA. The numerical experimentsshow that KTGA can obtain the same best result as the labelsetting (LS) method (within n¼500) and that KTGA is a betteralgorithm to solve the 4PLRPF in comparison with the standard GA(SGA) and the one step GA with the K-th shortest path algorithm(KGA). The experimental results indicate that the proposed KTGAis a new efficient method to solve the 4PLRPF.

In the future work, other fuzzy factors, such as fuzzy cost couldbe discussed. Moreover, more realistic problem, such as multi-source problem could be discussed.

Acknowledgments

This work is supported by the National Natural Science Foun-dation of China under Grant nos. 71071028, 71021061, 70931001and 61070 162, the National Science Foundation for DistinguishedYoung Scholars of China under Grant no. 61225012; the Specia-lized Research Fund of the Doctoral Program of Higher Educationfor the Priority Development Areas under Grant no.20120042130003; Specialized Research Fund for the DoctoralProgram of Higher Education under Grant nos. 20110042110024and 20100042110025; the Specialized Development Fund for theInternet of Things from the Ministry of Industry and InformationTechnology of the PR China; the Fundamental Research Funds forthe Central Universities under Grant no. N110204003.

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Fourth party logistics routing problem with fuzzy duration time