A Graph-based Fuzzy Evolutionary Algorithm for …...A Graph-based Fuzzy Evolutionary Algorithm for Solving Two-Echelon Vehicle Routing Problems Xueming Yan, Han Huang, Zhifeng Hao

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IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL., NO., 2019 1

A Graph-based Fuzzy Evolutionary Algorithm forSolving Two-Echelon Vehicle Routing Problems

Xueming Yan, Han Huang, Zhifeng Hao and Jiahai Wang

Abstract—Two-echelon vehicle routing problem is a challeng-ing problem that involves both strategic and tactical planning de-cisions on both echelons. The satellite locations and the customerdistribution affect the cost of different components on the secondechelon, thus the possibilities of satellite-to-customer assignmentcomplicates the problem. In this study, we propose a graph-basedfuzzy evolutionary algorithm for solving 2E-VRP. The proposedmethod integrates a graph-based fuzzy assignment scheme intoan iteratively evolutionary learning process to minimize thetotal cost. To resolve the possibilities of the satellite-to-customerassignment, graph-based fuzzy operator is used to take advantageof population evolution and avoid excessive fitness evaluationsof unpromising moves in different satellites. Each offspring isproduced via graph-based fuzzy assignment procedure out of anassignment graph from parent individuals, and fuzzy local searchprocedure is used to further improve the offspring. Experimentalresults on the public test sets demonstrate the competitiveness ofthe proposed method.

Index Terms—evolutionary algorithm, fuzzy subset, an assign-ment graph , two-echelon vehicle routing problem.

I. INTRODUCTION

THE TWO-ECHELON vehicle routing problem (2E-VRP)is a combinatorial optimization problem [1] [2]. As a

result of its wide real-world applications, including the con-cepts of city logistics [3] and freight transportation producingcongestion [4], 2E-VRP has been intensively investigated inthe past few decades. The freight in 2E-VRP which begins ata central depot, is brought first to so-called satellite facilities,and is then transported to the final customer by smallervehicles. 2E-VRP aims to determine a series of routes witha minimal total routing cost and to satisfy server constraints.

When the satellite-to-customer assignment relationship isgiven, 2E-VRP can be maximally divided into ns + 1 ca-pacitated vehicle routing problems, of which one on the first

Manuscript received xx, 2018; revised xx, 2018 and and xx, 2019; acceptedApril 7, 2019. This work is supported in part by National Natural ScienceFoundation of China under Grant 61370102, Grant 61876207, and Grant61673403, in part by Guangdong Natural Science Funds for DistinguishedYoung Scholar under Grant 2014A030306050, in part by NSFC-GuangdongJoint Found under Grant U1501254, in part by International CooperationProject of Guangzhou under Grant 201807010047 and in part by Guangzhouscience and technology project under Grant 201804010276. (Correspondingauthor: H. Huang.)

X. Yan is with Eastern Language Processing Centre, the School of In-formation Science and Technology & School of Cyber Security, GuangdongUniversity of Foreign Studies, Guangzhou, China

H. Huang is with the School of Software Engineering, South China Uni-versity of Technology, Guangzhou 510000, China (email:[email protected])

Z. Hao is with School of Mathematics and Big Data, Foshan University,Foshan, P. O. Box, China

J. Wang is with Department of Computer Science, Sun Yat-sen University,Guangzhou, China

This paper has supplementary downloadable material available athttp://ieeexplore.ieee.org, provided by the author.

echelon and at most ns ones on the second echelon becauseeach satellite should at least serve one customer to be countedas a capacitated vehicle routing problem (ns is the numberof satellites) [5]. Most studies have focused on separating thefirst and second echelon routing problems for 2E-VRP, andmore and more interest has been attracted in analysing therelationship in routing subproblems on the second echelon [6].

One direction aims to find an optimal solution without asatellite-to-customer assignment. During the last century, allkinds of valid inequalities of 2E-VRP were often adopted fortheir ability to overcome symmetry issues [7] [8]. In addition,the exact method available for 2E-VRP was introduced basedon dynamic programming and a dual ascent method [9].Although those two kinds of methods can generate bettersolutions for small instances, they are less valid when dealingwith a subset of large-scale instances [1]. More recently,researchers have shifted their attention to heuristic algorithmsbased on large-neighborhood explorations for 2E-VRP [10][11]. However, those methods may make excessive fitnessevaluations of unpromising moves in different satellites.

Another direction is an approach based on satellite-to-customer assignment. A family of multistart heuristics [5]has been proposed based on changing the satellite-to-customerassignment iteratively to improve the solution. A hybrid al-gorithm [12] has been proposed that combines a greedy ran-domized adaptive search procedure (GRASP) algorithm with apath relinking procedure. A mixed integer linear programmingformulation [13] has been proposed based on the observa-tion that the 2E-VRP can be decomposed into subroutingproblems when the customer’s assignment to the satellite isgiven. However, the focus of most researchers was not onmodelling the real relationship of routing subproblems [14].In the earliest analyses, the customer-to-satellite assignmentconfiguration was obtained by cluster analysis [15]. After that,Crainic et al. [16] investigated the impact of parameters ontotal cost, including different locations of the depot and thesatellites, the distribution of the customers and the number ofsatellites. The routing subproblems on the first and secondechelon affect the different components of the generalizedcost [17]. The first echelon routing subproblems are formedon the basis of the second echelon routing subproblems.Especially, as it is not easy to completely separate differentrouting subproblems on the second echelon only by distanceinformation or other factors, satellite-to-customer assignmentof 2E-VRP is a difficult task [1]. Moreover, the number ofthe routing subproblems on the second echelon is uncertain,because not all satellites need to be started for 2E-VRP. Thus,regarding the mutual relationships in routing subproblems,




it becomes more significant to consider possible satellite-to-customer assignments.

In this study, we propose a graph-based fuzzy evolutionaryalgorithm (GFEA) based on satellite-to-customer assignmentto seek robust solutions. The proposed algorithm is graph-based population evolution method combined with a fuzzyassignment procedure. At each generation, a new population ofindividuals is produced via a series of graph-based fuzzy oper-ators, and an assignment graph with fuzzy weight probabilitycan be used to represent uncertain assignment relationship for2E-VRP, which is produced by using the representation ofthe individuals. For uncertain satellite-to-customer assignmentin the solution, each customer can be served by differentsatellites and each satellite can be chosen to deliver goods todifferent customers. There may exist inconsistencies betweencustomer preferences and satellite preferences under constraintconditions; for example, more closely related customers shouldbe served by the same satellite, and satellites should be usedto serve more customers who are close to them. To solve theunstable satellite-to-customer problem of 2E-VRP [17], weuse the relevant theory of fuzzy sets [18] to construct graph-based fuzzy operators between satellite set and customer set,and then analyze the routing subproblems for getting bettersolutions.

The contribution of this study is mainly in the followingfour aspects. First, a graph-based representation of individualscombined with estimated fuzzy assignment values is proposedfor the adaptation of evolutionary strategies to 2E-VRP. Sec-ond, graph-based fuzzy operator are developed to balancethe assignment preferences of satellite and customer sets.Third, a fuzzy local search procedure is proposed for routinggeneration of each satellite. Fourth, a multi-parent effectiveupdating of assignment graph is a process of evolutionarylearning [19]. Compared to state-of-art methods, the proposedalgorithm produces high-quality results, especially of large-scale problem instances where the number of the satellites issignificantly increased.

The remainder of this paper is organized as follows. SectionII introduces the 2E-VRP problem. Section III proposes thedetails of the GFEA. Section IV describes the computationalexperiments on benchmark sets. Finally, conclusions are drawnin Section V.

II. PROBLEM FORMULATION

2E-VRP can be formally defined as follows. There is aweighted undirected graph G(V,E), where V and E standfor the set of vertices and edges, respectively. Set V consistsof the depot node v0, the subset Vs of m satellites and thesubset Vc of n customers. Set E = E1 ∪ E2 is dividedinto two parts representing the first and second echelonsrespectively. Set E1 = 〈i, j〉 : i, j ∈ v0

⋃Vs comprises

the edges connecting the depot to the satellites on the firstechelon, as well as those connecting pairs of satellites. SetE2 = 〈i, j〉 : i, j ∈ Vs

⋃Vc, 〈i, j〉 /∈ Vs × Vs includes the

edges connecting the satellites to the customers on the secondechelon, as well as those connecting pairs of customers. Eachedge between node i and node j has the traveling cost cij . The

objective of 2E-VRP is to determine a set of vehicle routes onthe first and second echelons that minimize the total cost of theroutes by consolidating the freight through the satellites [1].The customer’s demands cannot be directly delivered from thedepot but must be consolidated in a satellite. The deliveries ofthe satellite’s facilities can be split on the first echelon, but thedemand di of each customer Vi cannot be split on the secondechelon.

A solution of 2E-VRP can be represented by a set of routeson the first and second echelons. Two examples of feasiblesolutions are shown in Fig. 1. Each route on the first echelonstarts and ends at the depot (e.g., r1 = (V0, S1, S3, V0) in Fig.1a), and each route on the second echelon starts and ends atthe satellite (e.g., r4 = (S1, c5, c4, c3, S1) in Fig. 1b). Giveneach route rj ∈ R1 ∪ R2, all possible routes are denoted bySet R1 on the first echelon, and all possible routes are definedby Set R2 on the second echelon. Let qj denote the distancecost of each route rj ∈ R1

⋃R2. The decision variable xj

(xj ∈ 0, 1) is equal to 1 only if the route rj is in thesolution. The objective of 2E-VRP can be stated as

c(pi) = Min∑

rj∈R1

⋃R2

qjxj (1)

The constraints for 2E-VRP can be defined as follows.1) Freight balanced constraint: The freight demand of each

satellite on the first echelon needs to be equal to the totaldelivery demand of customers served by the same satellite onthe second echelon. For example, the total customer demandon the routes r5 and r6 on the second echelon should beequal to the total demand of Satellite S3 on the first echelonin Fig. 1a. A change in satellite-to-customer assignment maymore or less affect the first echelon route. In a certain sense,the satellite-to-customer assignment determines the depot-to-satellite assignment [13] under the freight balanced constraint.

2) Vehicle capacity constraint: The total delivery demandof each route rj on the first echelon should not exceed thevehicle capacity (e.g., the total demand of satellites on theroute r1 cannot be greater than the vehicle capacity Q1 inFig. 1a). Also, the total delivery demand of each route rj onthe second echelon (e.g., the total demand of customers on theroute r4 cannot be greater than the smaller vehicle capacityQ2 in Fig. 1b).

3) Vehicle number constraint: The number of vehicles fora depot on the first echelon should not exceed the allowedmaximum vehicle number b1, as well as the number of vehiclesb2 for a satellite on the second echelon. As shown the feasiblesolution in Fig. 1a, there are two routes (r1, r2) on thefirst echelon and five routes (r3, r4, r5, r6, r7) on the secondechelon. Each route needs a vehicle to serve; thus, b1 shouldbe greater than or equal to 2, and b2 should be greater thanor equal to 5.

4) Customer visited constraint: Each customer is visitedexactly once.

The possibilities of the satellite-to-customer assignmentleads to an increase in the number of possible routes onthe second echelon, which causes difficulty choosing properroutes under the constraints for each satellite. As shown in




Fig. 1. Two examples of feasible solutions of an instance, which can berepresented by a set of routes on the first and second echelons for 2E-VRP.There exists a significant difference that Customer c12 is assigned to differentsatellites between the two feasible solutions, and the feasible solution changesfrom Fig. 1a to Fig. 1b with a lower cost.

Fig. 1, there exists a significant difference when Customerc12 is assigned to different satellites between the two feasiblesolutions. Considering the correlation between Customer c12and Customer c13, the feasible solution changes from Fig. 1ato Fig. 1b with a lower cost. Actually, customer correlationon the second echelon affects the relationship of the routingsubproblems for 2E-VRP [20]. Thus, customer correlationshould be considered in the satellite-to-customer assignmentprocedure.

III. PROPOSED ALGORITHM FOR 2E-VRP

Evolutionary algorithms (EAs) [21] are regarded as a largeclass of problem-solving methods that imitate the process ofnatural evolution. The main object of EAs is the evolutionof a population guided by a set of operators. EAs have beenused for routing optimization problems, including uncertaincapacitated arc routing problems [22], multiobjective vehiclerouting problems [23] and vehicle routing problems with timewindows [24]. By information exchange between individuals,a population of individuals evolves by producing new genera-tions of offspring, which requires an iterative learning processuntil some convergence criteria are met [25].

2E-VRP can be transformed to a permutation optimizationproblem and solved by EAs. However, because of the differentrouting subproblems, basic and simple EAs do not have aparticularly significant advantage over well-established EAsfor uncertain optimization problems [26] [27]. To enhance theperformance of EAs, a natural idea is to integrate them withmutual relationships among different routing subproblems. In2E-VRP, due to the uncertain satellite-to-customer assignmentproblem, it is difficult to obtain satellite-to-customer assign-ments only by distance information or other factors. Thus, wepropose a graph-based fuzzy satellite-to-customer assignmentscheme for 2E-VRP and couple it with an evolutionary algo-rithm, forming a new approach we call GFEA.

The pseudocode of the proposed algorithm is given in Algo-rithm 1. The main ingredients including solution representationand initiation, a graph-based fuzzy assignment procedure, afuzzy local search procedure, and other components of theproposed algorithm are shown in detail in Section III, A–D.

In addition, the computational complexity of one generationof GFEA is described in Section III-E.

A. Representation and Initialization

1) Representation: We consider an assignment graph G∗ =(N,E∗) for each generation among depot set, satellite set andcustomer set. N is the vertices set, E∗ is the assignment edgeset including the edges between depot and satellites and theedges between satellites and customers.

With an assignment structure graph for each generation, wedefine a cross list L = (L0, L1, ..., Li, ..., Lm) consisting of apermutation of customers or satellites in different routes foran individual [29] [30], where Li(0 < i ≤ m) represents thesequence of the routes through Satellite Vi on the second ech-elon, and L0 represents the vehicle routes on the first echelon.For example, Fig. 2 represents a graph-based representationof the solutions in Fig. 1b. Fig. 2a represents an estimationassignment graph for the individual in the current generation,and Fig. 2b represents a graph-based cross list storage ofthe individual in Fig. 1b. The depot-to-satellite assignmentand satellite-to-customer assignment for the individual canbe directly obtained in this encoding scheme. At the sametime, when an assignment graph G∗ is changed for the nextgeneration, the corresponding element of the individual can berapidly modified.

To evaluate the performance of the individual pi, we con-sider the objective function

C(pi) = c(pi) +5g(pi)

g(pi), (2)

where the objective function C(pi) is used to evaluate the effectiveness of the individual pi including the total route cost c(pi) in Eq. (1) and the penalty terms associated with fuzzy assignment metrics g(pi) in Eq. (14) (in Section IV–E) for unreasonable satellite-to-customer assignment. 5g(pi)is the difference of g(pi) from the previous generation. Inparticular, g(pi) is set at 0 for the initial population, andthe objective function C(pi) is equal to the total route costc(pi) for the initial generation. As the number of iterationsincreases, the relative fuzzy assignment metrics value 5g(pi)g(pi)for two successive generations will also be less and less, andthe objective function is getting closer to c(pi).

2) Initialization: The initial solutions for each individualare generated as in [10]. First, every customer is assigned toa satellite facility by a roulette wheel selection mechanismbased on the distance between the customer and the satelliteunder concerned constraints. Then, the VRPs for the secondechelon are solved by the savings algorithm [28] for everysatellite. Finally, with the given demand at each satellite, thefirst-echelon routes are constructed in the same way as thoseof the second echelon. If the initial solution is not a feasibleone, it would be removed in the initialization process.

Moreover, an assignment graph is generated based on theinitial solutions, and the weights on the assignment graph G∗

need to be calculated on the basis of the current individuals.We consider the selection of depot nodes, satellite nodesand customer nodes on the assignment graph G∗ as three




Algorithm 1 Pseudocode of GFEAInput:

a 2E-VRP instance; the number of customer fuzzy subsets, denoted as Ncf ; the number of individuals N ; customer setVc; satellite set Vs.

Output: The best-so-far solution sr//Initialization:

1) sr = ∅.2) Construct the initial population pop randomly, generate an assignment graph G∗.

// Fuzzy Assignment Scheme:while Stopping criteria are not satisfied do1) Calculate fuzzy equivalent matrix R based on the best half of the individuals of pop.2) Decompose the fuzzy equivalent matrix R over customer set Vc to obtain the fuzzy subsets Ak, k = 1, 2, ..., Ncf .3) Match the fuzzy subsets Ak, k = 1, 2, ..., Ncf with satellite set Vs.4) Set i = 1.5) Assign the routes in different satellites based on fuzzy subset and the assignment graph G∗ for pi (pi ∈ pop) on the

second echelon by Algorithm 2. /*Section III-B*/6) Apply a fuzzy local search procedure to generate a new individual p

′

i. /*Section III-C*/7) Construct the routes with fuzzy neighbourhood selection procedure on the first echelon according to Ref. [28].

/*Section III-D-1*/8) Update sr with p

′

i, if C(p′

i) > C(sr).9) Set i← i+ 1. if i ≤ N , go back to (4).

10) Sort the solutions in pop, and then select N/2 better individuals and generate N/2 new individuals to update popand the assignment graph G∗.

// In Steps (1) – (3), some common characteristics of the individuals with high-quality solutions are obtained duringpopulation evolution. Step (5) adopts indeterministic mutation operators in a fuzzy assignment procedure for eachindividual, while a local search based on the fuzzy subsets for the second echelon and a fuzzy neighbourhood searchfor the first echelon are mainly driven by mutation in Step (6) and Step (7) respectively. Step (10) creates new offspringon the basis of parental individuals.

end while

different fuzzy events Z0, Z1,and Z2 with the sample spaceΩ1 = D0, Ω2 = Si|1 ≤ i ≤ m and Ω3 = cj |1 ≤ j ≤n, separately. The weights probability including depot-to-satellite assignment and satellite-to-customer assignment canbe obtained in Bayesian formula[31].

P (Z1i |Z0

0 ) =P (Z1

i )P (Z00 |Z1

i )∑mi=1 P (Z1

i )P (Z00 |Z1

i )(3)

P (Z2j |Z1

i ) =P (Z2

j )P (Z1i |Z2

j )∑nj=1 P (Z2

j )P (Z1i |Z2

j )(4)

P (Z1i |Z0

0 ) stands for the posterior assignment probabilitywhere Satellite Si is also assigned under the condition thatDepot D0 is selected in the solution, and P (Z2

j |Z1i ) stands

for posterior assignment probability where Customer cj is alsoassigned under the condition that Satellite Si is selected inthe solution. P (Z1

i ) = 1/m stands for the post assignmentprobability where Satellite Si is assigned in the solution, andP (Z2

j ) = 1/n stands for the post assignment probability whereCustomer cj is assigned in the solution. Since not all satellitesare required to be selected in depot-to-customer assignment,P (Z0

0 |Z1i ) refers to the probability of satellite selection in

the solution on the first echelon. Since all customers mustbe served, P (Z1

i |Z2j ) refers to the probability that Customer

cj is served through Satellite Si in the solution on the secondechelon. The probability of P (Z0

0 |Z1i ) and P (Z1

i |Z2j ) can be

counted with fuzzy statistical method [32] by analyzing thecross list of the current individuals. Fuzzy statistical methodis a way to determine membership function of fuzzy eventwith the ideas in probability statistics, where the membershipfrequency tends to be a stable value in a large number ofexperiments. More importantly, the stable value approximatesthe probability of occurrence of fuzzy events, which can becalculated by the observational results of experiments. Eachindividual in the population represents a random experiment,so P (Z0

0 |Z1i ) =

∑Nk=1 θi(pk)

N stands for the fuzzy conditionalprobability of satellite being opened for the current generation,where N is the number of the individuals, and θi(pk) isequal to 1 when Depot V0 and Satellite Si are linked inthe first row list for the individual pk, otherwise, θi(pk) isequal to 0. Similarly, P (Z1

i |Z2j ) =

∑Nk=1 ϑij(pk)

N stands forthe fuzzy conditional probability that Customer cj which hasbeen served needs to go through Satellite Si for the currentgeneration, where N is the number of the individuals, andϑij(pk) is equal to 1 when cj and Satellite Si are linked inthe same row list for the individual pk, otherwise, ϑij(pk) isequal to 0.

B. Graph-based Fuzzy Assignment Procedure

The goal of the fuzzy assignment procedure is to generatepromising solutions with an assignment graph. Compared tosimple depot-to-satellite assignment, the influences of possible




(a) (b)

Fig. 2. Graph-based representation of the solutions in Fig. 1. (a) An assignment graph among depot set Vd, satellite set Vs and customer set Vc for twosolutions in Fig. 1, and the weights in this graph represent the probability of assignments (b) Cross list storage of the feasible solution in Fig. 1b. (Note thatL1 = (r3, r4) represents the sequence of the routes through Satellite V1 on the second echelon. )

Fig. 3. An example of graph-based fuzzy satellite-to-customer assignmenton the second echelon, where fuzzy subsets are estimated by decomposingthe customer correlation and are applied to analyze the relationship in routingsubproblems.

satellite-to-customer assignment deserve more considerations[7], so we focus on the satellite-to-customer assignment ofan assignment graph in this study. Considering the customer’scorrelation, an assignment graph based on the population isregarded as the synthesis of the relationship between the cus-tomer set and the fuzzy subset combined with the relationshipbetween the fuzzy subset and the satellite set. Based on fuzzysubsets, which are estimated by decomposing the customer’scorrelation, a series of routes associated with satellites areconstructed to solutions of 2E-VRP. An example of the fuzzyassignment procedure is shown in Fig. 3, where fuzzy subsetsare applied to analyze the relationships in routing subproblems.

1) Fuzzy Subsets: Several factors often result from this“fuzzyfying” [33] of customer preferences in 2E-VRP. Thechoice of Customer i over Customer j is modeled by a simplecomplete binary relation R over Vc × Vc (where Vc is theset of customer nodes). We construct a fuzzy relation to thecustomer set Vc to analyze preference structures of customers.In each generation, the relationship of customers is reflected bya fuzzy relation matrix R, which is realized in the probability

statistics method based on customer correlations. We considerthe fuzzy similarity matrix Rs over the customers set Vc. rij isthe estimated entry of Row i and Column j in fuzzy similaritymatrix Rs.

rij =

n∑k=1

minyik, yjkn∑k=1

√yikyjk

, if yik 6= 0, yjk 6= 0 (5)

where n is the number of customers. In particular, rij is equalto 1 when Customer ci is equal to Customer cj . yik stands forstatistical probability of Customer ci associated with Customerck.

yik =

N∑l=1

ωl(i, k)

N(6)

where N is the number of individuals in each generation.ωl(i, k) is equal to 1 when there exists an edge 〈i, k〉 connect-ing Customer ci to Customer ck in individual pl, otherwise,ωl(i, k) is equal to 0. Besides, yjk follows the same schemeas yik.

The fuzzy equivalent matrix R is the transitive closureof fuzzy similar matrix Rs, which is also the fuzzy binaryrelation matrix over the customer set Vc. We consider thedecomposition of a fuzzy relation R as a series of fuzzy setsdefined in Vc, such that

R =

Ncf⋃k=1

(Ak ×Ak) (7)

Where a set Ak is assumed to be the element of Vc that denotesthe kth pair of fuzzy subsets, Ncf is the number of fuzzysubsets, and the membership functions of the above expressionare stated as follows:

rij =Ncfmaxk=1

(a1(i, j), a2(i, j), ..., ak(i, j), .., aNcf(i, j)) (8)




ak(i, j) = min1≤i≤n,1≤j≤n

(ai(k), aj(k)), k = 1, 2, ..., Ncf (9)

where a(k) =< a1(k), ..., ai(k), ..., an(k) > stands for themembership grades of the kth fuzzy subset. The performanceindex of the theoretical analysis for max-min aggregation[34] is regarded as a sum of squared error according to theexpression

Q =n∑i=1

n∑j=1

(rij − rij)2 (10)

The fuzzy subsets Ak = cl|cl ∈ Vc,al(k) ∈ ai(t)|ai(t) ≥1 − 0.05,1 ≤ i ≤ n, t = k,1 ≤ k ≤ Ncf , are establishedby minimizing decomposition squared error Q with the aidof a standard gradient-based learning method [35]. We usea threshold of 0.05 meaning that the membership gradesover 0.05 indicate significant dominant factors, and groupsof customers are identified by these Ncf decomposing fuzzysubsets.

Owing to the transitivity of the fuzzy binary relations, eachfuzzy subset Ak contains customers with relatively strong cor-relation. A customer that occurs to various fuzzy subsets maybe related to different groups. Evidently, the decompositionquality in terms of the customer fuzzy relation is closelyrelated to the satellite-to-customer assignments to 2E-VRP.

2) Assigning Routes Based on Fuzzy Subset: This sectionexplains the role of fuzzy subset for satellite-to-customerassignment. Remarkably, the customers in the same fuzzysubset may meet transitivity, so they may be served in thesame satellite or in the same route of the same satellite. Inaddition, these customers that belong to more than two fuzzysubsets may also be attributed to different routes.

The satellites are selected to match each fuzzy subset inthe assignment graph G∗ for each generation. The subsets’preference to each satellite is defined as

Fj(Ak) =1

nc

∑i∈Ak

didmax

e−β−1 hij−hmin

hmax−hmin , k = 1, 2, ..., Ncf

(11)where nc is the number of customers in a fuzzy subset, di isthe demand of customer i, and dmax is the maximum demandover the customers in the fuzzy subset Ak. hij is the distancebetween Satellite Sj and Customer ci. hmin and hmax are theminimum and maximum distances in the fuzzy subset Ak. βis set to be as the weight probability between Satellite Sj andCustomer ci in an assignment graph G∗ .

Moreover, the similarity between route ri and fuzzy subsetAk is defined as

sim(ri, Ak) =σ

num(ri) · |Ak|(12)

where σ is the number of common customers in route riand fuzzy subset Ak, num(ri) is the number of customersin the route, and |Ak| is the number of customers in the fuzzysubset Ak. If the route rj has the maximum similarity withthe fuzzy subset Ak, the route rj is seen as the route derivedfrom the fuzzy subset Ak. When applying fuzzy subsets toestimate the routes for each satellite, some issues need to

be addressed. First, the customers in a fuzzy subset shouldbe likely to be served by a satellite with the highest fuzzysubset preferences. The subset preference gives a measureof assignment relationship between customers in a fuzzysubset and the satellite. Customers with larger assignmentweight probability have a greater impact on the correspondingsatellite, reflecting the fuzzy subset’s potentially more crucialrole in the satellite-to-customer assignment. Moreover, thenumber of the overlapping customers in the route and fuzzysubset is largest, and the route can be seen as a variant of theroute that can be generated from the customers of the fuzzysubset. Hence, the customers in the same fuzzy subset may beserved in the same satellite or in the same route of the samesatellite.

Based on the characteristics of fuzzy subsets describedabove, we define the customers set B = b1, b2, ...bl whichrepresents the l customers that belong to more than twofuzzy subsets. The customers in B are essential for fuzzyassignments between satellites and customers, because thesecustomers make an accurate assignment [12] difficult. Basedon the maximum similarity principle of routes and fuzzy sub-sets, each customer in B needs to be removed from the currentroute and be assigned to the other route which is derived fromthe corresponding fuzzy subset under the capacity constraints.In particular, moves of customers in different routes can occurin different satellites. The routes on the second echelon arereassigned, and simultaneously the customers are assigned tothe corresponding satellites. A series of routes are attributedto respective satellites with the fuzzy subsets.

To summarize, the graph-based fuzzy assignment procedureworks as follows. The similarity to routes and fuzzy subsetsfor each individual is calculated with an assignment graph,and the routes are attributed to respective satellites with thesubsets preference. Subsequently, the customers that belongto different fuzzy subsets are assigned to related routes withmaximum similarities. Algorithm 2 shows the method ofmatching routes based on customer fuzzy subset.

C. Fuzzy Local Search Procedure

To determine whether the customer’s fuzzy relation is usedto improve the quality of the solution, the customer fuzzysubset is adapted to a fuzzy local search procedure for eachsatellite, which is implemented with move operators for eachsatellite.

1) Fuzzy Removal(ci, rj , Ak): Remove selected cus-tomers ci from the current route rj if the current route rjof the customer ci is the one most associated to the fuzzysubset Ak in which the customer ci exists.

2) Merge(ri, rj , Ak): Merge the route ri and the route rjwhich are attributed to the same fuzzy subset Ak.

3) Route Generation(rk): To maintain the diversity ofthe individuals, randomly select some removal customers toform a new route rk for the satellite when the route rk meetsthe capacity constraint.

4) Fuzzy Repair(ci, rj): The fuzzy repair operation isapplied with a simplified insertion heuristic. The remainingremoval customers(e.g.,ci) are inserted into the route rj which




Algorithm 2 Matching Routes Based on Fuzzy SubsetsInput:

An individual pi, i = 1, 2, ...N , a series of fuzzy subsets Ak, k = 1, 2, ..., Ncf , and the customers set B

Output:The new individual pi

1: begin2: obtain the route set Ri2 which includes all routes of the

individual pi on the second echelon3: calculate the similarity between fuzzy subset Ak and the

route set Ri2 by Eq. (11), and find the derived routes for each fuzzy subset from route set Ri2.

4: Select the fuzzy subset Ak with maximum preference tosatellites by Eq. (12), and attribute the derived routes of the fuzzy subset Ak to corresponding satellites.

5: for each bi ∈ B do6: for each fuzzy subset Ak do7: if bi ∈ Ak then8: move Customer bi from the current route to the

other route which is derived from the fuzzy subsetAk under the capacity constraints

9: end if10: end for11: end for12: return pi13: end

has the maximum similarity with this subset (any fuzzy subsetto which the remaining customer ci belongs is selected).Moreover, the greedy heuristic insertion [10] is also takeninto account when the fuzzy repair operation cannot meetthe vehicle capacity constraints. After that, fuzzy equivalentmatrix R is regarded as heuristic information for each route[36] and for every pair of routes [37].

To avoid excessive computation between different satellites,fuzzy local search operations are carried out for each satellite.A perturbation to a partial individual based on a customer’sfuzzy relation will change the order of the subsequence ofcustomers, affecting the optimal solution obtained by the fuzzylocal search procedure.

D. Other Components of GFEA

1) Fuzzy neighbourhood selection Strategy: As describedin the previous sections, an individual is first partitioned intotwo echelons to make full use of an assignment graph. Asdepot-to-satellite assignment on the first echelon is not ascomplex as satellite-to-customer assignment on the secondechelon, the simple saving algorithm [28] can find an optimalor near-optimal solution on the first echelon for small-scaleinstances. However, the distance information and the savingvalue need to be constantly neighborhood searched for large-scale instances, which directly deteriorate the performanceof the saving algorithm, so the rationality of neighborhoodselection needs to be improved.

On the basis of an assignment graph, we proposed a fuzzyneighbourhood selection strategy that fuzzy weight probability

was used for searching radius in order to avoid too manyrepetitive search operators on the first echelon. For the currentSatellite Si (or Depot V0), the fuzzy neighborhood radius h

′

is as follows.

h′

= ||ξ1hmax − ξ2hmin|| (13)

hmax and hmin refer to the nearest Satellite Sj′ and thefarthest Satellite Sj′′ from the current Satellite Si (or DepotV0). ξ1 is set to be as the fuzzy assignment weight betweenDepot V0 and Satellite Sj′ , and ξ2 is set to be as thefuzzy assignment weight between Depot V0 and Satellite Sj′′ .Neighborhood search radius is a variable based on fuzzycombinatorial weight of depot-to-satellite assignment, only thedirectional valid saving value and distance information needto be preserved in fuzzy neighbourhood selection strategy. Inparticular, if the choice of any satellites in the searching radiusrange causes overload of vehicle capacity on the first echelon,the satellite with the max saving value in the neighborhoodsearch area can be chosen, and the satellite capacity needsto be split in part to make vehicle capacity just saturated onthe current route. In general, fuzzy neighbourhood selectionstrategy can effectively avoid excessive repetitive searching ofsatellites on the first echelon, thus improving the efficiency ofsaving algorithm for large-scale instances of 2E-VRP.

2) Stopping Criterion: The GFEA stops when a maximalnumber of fitness evaluations, fen, is met.

3) Update of Solutions: The main effort during the se-lection of half of individuals at each generation is to countobjective function in terms of solution quality and level ofgraph-based fuzzy assignment, in order to avoid prematureconvergence and to generate new half of individuals to updatepop. In particular, the weights on the assignment graph G∗

need to be updated for the next generation. The selectionscheme is elitist toward high-quality individuals and alsofavors highly evaluated individuals adequately with the as-signment graph G∗.

E. Computational Complexity of One GenerationThe max-min decomposition of a population of size N

can gain Ncf fuzzy subsets, and the computational timecomplexity of the max-min decomposition [35] is O(N).Graph-based fuzzy operations in Algorithm 2 would needO(NNcf ) computations. Thereafter, the computational timecomplexity of the fuzzy local search procedure and fuzzyneighbourhood selection strategy in the worst case are O(N).All operations in Algorithm 1 associating with N populationsrequire O(N + N2Ncf + N2) in the worst case for eachgeneration. According to the operational rules of the symbolO(·), the worst time complexity of GFEA can be simplifiedas O(N2). Because an assignment graph takes the advantageof population evolution, and sequences of customers of anindividual are handled on the basis of fuzzy subsets, GFEAcan reduce the computational complexity of the evolutionaryalgorithms to O(N2) for each generation.

IV. EXPERIMENT RESULTS

To evaluate the effectiveness of the GFEA, we did empiricalstudies to compare it with state-of-the-art approaches. We also




TABLE IIPARAMETERS OF GFEA

Name Description Valuepsize Population size 100Ncf Number of the fuzzy subset ns + 2fen Maximum number of fitness evaluation 500,000

conducted further empirical analysis to assess the contributionof the graph-based fuzzy assignment scheme to the GFEA.

A. Benchmark Instances

The characteristics of the test sets and source are given inTable I. It lists instances of the benchmark test sets concerningthe number of satellites m and the number of customers n. Letb1 and b2 be maximum numbers of vehicles for the first andsecond echelons. Q1 and Q2 are the capacities available onthe first and second echelons.

Perboli’s Set 2 and Set 3 [7], Set 4 [16] and Set 5 test setswere selected to evaluate GFEA. Perboli’ test sets (Set 2, Set3) include all instances with no more than 50 customers andtwo or four satellites, each instance containing no more than200 potential assignment relations. Set 4 has 54 instances withthe number of the potential assignment relations ranging from100 to 250. Set 5 [10] consists of 18 instances with 500 to2000 potential assignment relations. According to the numberof the potential assignment relations, the test sets of 2E-VRPcan be classified into small (Set 2 and Set 3), medium (Set 4),and large scale (Set 5) instances. Although GFEA is proposedspecifically for solving large scale instances, one may still beinterested in its performance on the small and medium scaleinstances.

B. Competitors

To evaluate our approach, we compared it with LNS-2E[11], GRASP [12], ALNS [10] and BMRW [9]. LNS-2E is oneof the state-of-the-art heuristic approaches for 2E-VRP, wheredestroy-and-repair principles are applied to obtain the finalsolution. GRASP is based on satellite-to-customer assignment,where customers are assigned according to a less-distance-based rule. ALNS is a typical metaheuristic with a largeneighborhood search strategy. BMRW is one of the state-of-the-art exact methods with mathematical formulation of lowerbounds for 2E-VRP.

C. Parameter Setting

Experiments were conducted on a workstation computerwith an Intel Xeon E5 2.4GHz CPU with 32GB of memory.All involved algorithms were implemented in JAVA. Forcompetitions, the parameter settings reported in the originalpublications were used. For a fair comparison, the algorithms[10] [11] [12] were assigned to the same fitness evaluationtimes fen, and all the experiments went through 30 indepen-dent runs. Table II summarizes the parameter setting for theGFEA.

D. Comparing the GFEA with State-of-the-art algorithms

Because satellite-to-customer assignment cannot be solvedoptimally for large benchmark instances [1], we are moreinterested in the performance in large scale instances (e.g.,Set 5) to 2E-VRP. The large number of potential assignmentrelations in large-scale instances raises the possibilities ofsatellite-customer assignment for 2E-VRP. For the large scalebenchmark Set 5, the computational results of GFEA and theexact method BMRW [9] are described in Table III, and thedetailed results compared with the heuristic algorithms[10][11] [12] are reported in Table IV. Because of the page limit,more detailed results can be found in Tables VI–VIII in thesupplementary material.

Table III shows that a simpler but more effective satellite-to-customer assignment scheme like the GFEA can find a bettersolution than the exact method BMRW with an acceptablecomputation time. The column “Best” shows the best costfor BMRW and GFEA, respectively. The column “T1(s)”shows the execution time limited to two days for a feasiblesolution found by BMRW, and the column “T2(s)” showsthe execution time for the best solutions by GFEA in 30independent runs. The column “GAP ” shows the percentagegap between the solution of BMRW c(pi) and the solution ofGFEA c(pj), which can be computed as (

c(pi)−c(pj)c(pi)

)× 100.From Table III, it can be found that GFEA was more likelyto obtain better solutions on all instances except one (200-10-1b), which is significantly better than BMRW. In particular,from the gap of the solution of BMRW to that of GFEAover large-scale instances, GFEA is capable of achieving bettersolutions than BMRW. More importantly, BMRW managed tofind feasible solutions to 17 instances with more executiontime. However, there is still no feasible solution found in oneinstance (200-10-3) within the limited time. All of the aboveobservations are evidences that GFEA has more advantagesthan BMRW for large-scale instances. As the numbers ofsatellites and customers were relatively large on Set 5, thesatellite-to-customer assignment problem of 2E-VRP becamemore complicated, which also demonstrates the need for thesatellite-to-customer assignment scheme such as the GFEA weintroduce in this study.

Table IV shows that the GFEA outperformed the LNS-2E,GRASP, and ALNS in most instances with an acceptable com-putation time. The columns “Average”, “Std” and “T(s)” showthe average total cost, the standard deviation of the cost, andthe average execution time respectively over 30 independentruns. The results of the other three algorithms were comparedto those of the GFEA by a nonparametric estimation test witha significance level of 0.05. For LNS-2E, GRASP, and ALNS,the highlighted average results indicate that the correspondingalgorithm was statistically significantly better than the GFEA.Underlined results indicate that the corresponding algorithmwas statistically significantly worse than the GFEA. Resultswithout a symbol indicate that the difference between GFEAand other three algorithms were insignificant. In Table IV,the average cost of the GFEA (1565.46) is marked with anasterisk, denoting that the GFEA was the algorithm with thebest performance for instance 100-5-1. The average cost of




TABLE ICHARACTERISTICS AND SOURCES OF INSTANCE SETS

Set Instances m n b1 b2 Q1 Q2

2 [7] 6 2 21 3 4 15000 60002 6 2 32 3 4 20000 80002 6 2 50 3 5 400 1602 3 4 50 4 5 400 1603 6 2 21 3 4 15000 60003 [7] 6 2 32 3 4 20000 80003 6 2 50 3 5 400 1604 [16] 18 2 50 3 6 12500 50004 18 3 50 3 6 12500 50004 18 5 50 3 6 12500 50005 [10] 6 5 100 5 [15,32] [520,528] [70,150]5 6 10 100 5 [17,35] [512,537] [70,150]5 6 10 200 5 [30,63] [1026,1034] [70,150]

LNS-2E (1566.87) is underlined, which indicates that GFEAperformed significantly better than LNS-2E in instance 100-5-1. For instance 100-10-3, the average cost of GRASP(1043.57) is in bold, indicating that the GRASP performedsignificantly better than the GFEA on 100-10-3. The averagecost of ALNS (1055.88) is not highlighted, which indicatesthat there is no statistically significant difference betweenthe GFEA and ALNS in instance 100-10-3. When comparedwith ALNS, the GFEA outperformed ALNS in 16 out of 18instances of Set 5. The GFEA outperformed GRASP in almostall instances. The GFEA was also as competitive as LNS-2E in 12 instances except 5 instances (100-5-2b, 100-5-3b,100-10-1, 100-10-2b, and 100-10-3) with small gaps (0.93,0.45, 0.12, 0.01 and 0.10). Furthermore, the GFEA was alsocompetitive in terms of standard deviations for all instances inSet 5. Average standard deviation of instances is just 2.35 forGFEA, while those obtained by LNS-2E, GRASP, and ALNSwere 4.52, 6.20 and 6.46, respectively. These observationsillustrate the advantage of the GFEA in terms of solutionstability. In addition, Fig. 4 presents the box plots of theGFEA, GRASP, LNS-2E, and ALNS in the instances of Set 5.It can be observed that the proposed algorithm can achieve asolution with smaller cost fluctuations in most instances whencompared with other algorithms. For example, compared toLNS-2E, the GFEA performs better in 12 instances and worsein the other 6 instances. The inferior performance of the GFEAis understandable, because the GFEA obtained lower costs inthe instances. The reason might be that graph-based fuzzyoperators avoid unpromising moves in different satellites, soit can be useful to improve the stability of the solution.

Moreover, nonparametric tests were also conducted to com-pare the performance of the GFEA, LNS-2E, and ALNS.Detailed results for instances of other test sets (Set 2, Set 3and Set 4) are shown in Tables VI –VIII in the supplementarymaterial, respectively. The p-value of nonparametric tests isshown in the “p-value” column of Table IV and Tables VI– VIII. It can be seen that the compared methods weresignificantly different at a 95% confidence interval of all theset instances except for six instances ( E-n51-k5-s11-19-27-47,E-n51-k5-s12-18, Instance50-s3-27.dat, Instance50-s3-31.dat,100-10-1 and 200-10-1). Simultaneously, the number of “ w-d-l ” of the GFEA versus other approaches is shown in the lastrows of Table IV and Tables VI – VIII. The results in column “w-d-l ” of Tables IV show that GFEA significantly outperforms

TABLE IIICOMPARISON WITH THE EXACT METHOD BMRW ON

SET 5

Instances BMRW GFEA GAPBest T1(s) Best T2(s)100-5-1 1564.46 612 1564.46 34 0100-5-1b 1142.53 912 1108.62 32 3100-5-2 1016.32 851 1016.32 41 0100-5-2b 796.53 799 782.35 33 2100-5-3 1045.29 167 1045.29 21 0100-5-3b 833.94 1671 828.98 21 1100-10-1 1137.00 965 1124.93 55 1100-10-1b 928.01 2141 916.25 64 1100-10-2 1009.49 1065 990.58 79 2100-10-2b 773.58 460 768.65 56 1100-10-3 1055.28 1014 1043.65 73 1100-10-3b 850.92 938 853.12 52 0200-10-1 1574.12 1654 1556.79 68 1200-10-1b 1201.75 812 1209.62 78 -1200-10-2 1374.74 601 1365.74 89 1200-10-2b 1018.57 918 1002.85 100 2200-10-3 1788.03 98200-10-3b 1218.94 1910 1120.62 121 8mean 1115.93 61.9

Best: the cost of the feasible solution for BMRW, and the costof the best solution for GFEAT1(s): the execution time (seconds) limited to two days for afeasible solution found by BMRWT2(s): the execution time (seconds) for the best solutions byGFEA in 30 independent runsGAP: the percentage gap between the feasible solution ofBMRW c(pi) and the best solution of GFEA c(pj), which

can be computed as (c(pi)−c(pj)

c(pi))× 100

mean: the average values of all instances on Set 5.

GRASP and ALNS on more than half of the instances, andsignificantly outperforms LNS-2E on most of the instances.The results in column “ w-d-l ” of Tables VI – VIII show thatGFEA significantly outperforms the state-of-the-art algorithmson most of the instances.

E. Comparative Analysis on Satellite-to-Customer Assignment

1) Fuzzy assignment metrics: For a better understandingof the graph-based fuzzy assignment procedure, a perspectivefor further analysis of our proposed algorithm is provided. Thefuzzy assignment metrics function g(pi) is defined to evaluatethe effect of the graph-based fuzzy assignment scheme on thebasis of fuzzy subsets. For the individual pi, g(pi) is stated as

g(pi) =∑

1≤j≤m

∑0≤t≤b2

|Fj(Lj(pi, rt))− Fj(Ak)|2 (14)




Fig. 4. Box plots in the instances of Set 5 in terms of the worst-case solution costs in four different algorithms, where 1, 2, 3 and 4 in x-axis stand forGFEA, LNS-2E, GRASP and ALNS, respectively.

TABLE IVRESULTS ON THE LARGER SET 5 IN TERMS OF THE WORST-CASE SOLUTION COSTS

Instances GFEA LNS-2E GRASP ALNS p-valueAverage Std T(s) Average Std T(s) Average Std T(s) Average Std T(s)

100-5-1 1565.46∗ 1.80 34 1566.87 0.00 65 1569.42 5.27 140 1578.4 4.28 235 0.000100-5-1b 1108.62∗ 2.30 32 1111.93 2.13 60 1111.98 3.56 132 1118.95 1.34 155 0.000100-5-2 1016.32∗ 2.70 41 1017.94 5.27 79 1017.34 6.41 178 1016.36 7.51 183 0.000100-5-2b 783.18 4.90 33 782.25∗ 3.56 45 786.04 7.92 130 785.02 8.59 130 0.000100-5-3 1045.29∗ 3.90 20 1045.61 6.41 31 1046.67 9.18 99 1046.17 10.10 124 0.000100-5-3b 828.98 0.01 21 828.54∗ 7.92 42 829.87 4.92 92 828.99 3.12 99 0.000100-10-1 1132.23 3.98 45 1132.11∗ 1.18 39 1132.24 12.31 120 1133.17 8.37 169 0.000100-10-1b 917.01∗ 2.56 64 922.85 4.92 68 917.05 6.53 210 917.35 9.12 205 0.000100-10-2 990.58∗ 4.09 79 991.61 8.31 92 991.78 1.05 180 997.42 3.02 204 0.001100-10-2b 768.65∗ 1.28 56 786.66 6.53 103 777.98 4.08 160 773.56 7.14 174 0.000100-10-3 1043.65 3.57 78 1043.67 1.05 78 1043.55∗ 2.97 98 1055.88 6.03 648 0.000100-10-3b 853.12∗ 3.05 45 858.72 4.08 89 867.32 4.35 120 863.42 4.39 205 0.000200-10-1 1575.79∗ 3.01 68 1598.46 6.97 132 1587.12 4.38 187 1697.83 5.01 220 0.200200-10-1b 1209.62∗ 2.58 78 1217.23 1.35 148 1210.18 5.12 189 1225.61 6.10 189 0.000200-10-2 1375.74∗ 0.19 89 1376.16 4.38 155 1389.94 4.17 170 1419.94 3.22 173 0.000200-10-2b 1004.15∗ 1.09 100 1016.05 5.12 145 1057.90 9.01 147 1018.83 5.12 147 0.000200-10-3 1788.03∗ 0.08 98 1789.44 4.17 210 1792.49 8.24 111 1799.76 4.5 7 625 0.000200-10-3b 1201.92∗ 1.26 120 1206.85 8.01 180 1203.61 12.16 164 1208.61 17.36 194 0.000mean 1122.69 2.35 61 1127.38 4.52 97.83 1129.58 6.20 145.94 1138.07 6.46 226.61w-d-l 12-3-3 11-6-1 13-5-0

Average: the average value over 30 independent executions;Std: standard deviation over 30 independent executions;T(s): average execution time (seconds) over 30 independent executions;p-value: p-values of nonparametric test for comparing GFEA, LNS-2E, GRASP and ALNS;The minimum average results are with “*”. Bold(Underlined) results indicate that the corresponding algorithm is better (worse) than GFEA based onnonparametric test with the level of significance α = 0.05;w-d-l: the number of ‘win-draw-lose’ of GFEA versus other algorithms;mean: the average values of all instances on Set 5.




Fig. 5. Box plots of correlation coefficients between the best solution andthe fuzzy assignment procedure on four test sets

where m is the number of satellites, and Fj(∗) is the fuzzypreference value for each satellite according to Eq. (9),Lj(pi, rt) represents the set of customers on Route rt servedby Satellite Vj for the individual pi. Route rt is derivedfrom the fuzzy subset Ak according to Eq. (10). The fuzzyassignment metrics g(pi) is used to assess the impact of graph-based fuzzy operators on satellite-to-customer assignment.

To investigate the contribution of the graph-based fuzzyassignment scheme to the GFEA, a sample correlation co-efficient [38] [39] was used to analyze the relevance betweenthe best solution and the fuzzy assignment metrics value for30 independent runs. We have one vector f ′1, ..., f

′

l on thereciprocal of the best solution (the actual cost according toEq. (1) and another vector g1, ..., gl on the fuzzy assignmentmetrics value; then, a measure of the linear correlation for ρis

ρ =

l∑i=1

(f′

i − f′i )(gi − gi)√

l∑i=1

(f′i − f

′i )

2

√l∑i=1

(gi − gi)2(15)

where l is the independent execution times and f′

i is thereciprocal of the best solution, defined as

f′

i =1

c(pi)(16)

Fig. 5 gives the box plots on four test sets in terms ofcorrelation coefficient ρ between the best solution and thefuzzy assignment procedure. For each instance, all values arethe results of 30 independent runs. It can be observed that thebest solution generally coincided with the fuzzy assignmentvalue for the GFEA. To be specific, the best solution wasstrongly associated with the fuzzy assignment procedure whenthe value of ρ was greater than 0.6 in most instances of Set 4and Set 5. Meanwhile, the best solution was associated withthe fuzzy assignment procedure when the value of ρ wasgreater than 0.4 in most instances of Set 2 and Set 3. Thisindicates that the fuzzy assignment procedure is more likelyto find the better solutions in larger instances.

Fig. 6. Average cost of the after-assignment individuals obtained by GFEAand GRASP versus generations

2) Further analysis of fuzzy satellite-to-customer assign-ment: To further analyze the influence of the graph-basedfuzzy assignment scheme; that is the calculation of the g(pi)value to evaluate the assignment on the basis of fuzzy subsets,we compared the proposed algorithm with a less-distance-based assignment rule of GRASP [12]. Two improved currentbest solutions were generated and compared with each other.Fig. 6 shows the cost of the “after-assignment” solution versusthe iteration times for two typical instances. GRASP areexecuted on the best solution of the initial individuals of theGFEA to make a fair comparison.

The results in instance (E-n33-k4-s24-28) indicate thatGRASP obtained solutions with lower costs than the GFEAin the early phase. But the GFEA kept up with GRASP inthe later phase. The reason is that the quality of the g(pi)value could greatly affect the performance of the satellite-to-customer assignment. Results in instance (200-10-3b) canbe divided into three phases. The first two phases indicate asimilar trend to that in E-n33-k4-s24-28. The line of GRASPwas higher than that of GFEA. A possible reason might bethat the GFEA took advantage of the population evolution inan iterative learning process, where the value of g(pi) learnedfrom the population enhanced the correlation of customers inthe same fuzzy subsets, on the basis of population informationexchange with the graph-based fuzzy satellite-to-customerassignment scheme.

F. Parameter Sensitivity of Ncf

The number of fuzzy subsets Ncf is a user-defined parame-ter. The relationship between fuzzy subsets and satellites playsan important role in the fuzzy satellite-to-customer assignmentscheme; thus, the number of satellites affects the parameterNcf . We set the parameter Ncf of the GFEA from ns to ns+3to investigate how it influences the performance of GFEA. Theproposed algorithm was then run on all the instances for 30times. Table V summarizes the experiment results, where thebest and average results obtained are given. For the sake ofbrevity, only the average cost for each test set is given. Thebest results are highlighted in bold. We may see clearly thatGFEA basically performed the best when Ncf = ns+2. If we




(a) (b) (c) (d)

Fig. 7. Convergence curves of four different Ncf values in instances E-n22-k4-s8-14, E-n33-k4-s24-28, Instance50-s3-30.dat and 200-10-3b, where Ncf =ns, ns + 1, ns + 2 and ns + 3, ns is the number of the satellites on instances, respectively.

TABLE VAVERAGE COST OF GFEA ON THE TEST SETS WITH DIFFERENT VALUE OF “Ncf ”

Best of 30 runs Average of 30 runsTest setNcf = ns Ncf = ns + 1 Ncf = ns + 2 Ncf = ns + 3 Ncf = ns Ncf = ns + 1 Ncf = ns + 2 Ncf = ns + 3

Set 2 565.50 565.50 565.50 565.50 565.61 565.53 565.50 565.51Set 3 632.59 632.49 632.49 632.49 632.76 632.59 632.57 632.59Set 4 1398.91 1399.21 1396.13 1396.13 1401.12 1399.21 1397.01 1398.01Set 5 1123.92 1123.61 1121.30 1121.69 1129.32 1127.10 1122.69 1122.90

consider the average results, GFEA with Ncf = ns + 2 led toenhanced results on all test sets. In summary, experimentalresults show that a user-defined parameter Ncf is indeedcrucial for the proposed algorithm and Ncf = ns + 2 is areasonably good default choice if no other prior knowledge issuggested.

Convergence curves are plotted for typical instances in Fig.7. The sensitivity analysis was conducted on four instances,including E-n22-k4-s8-14, E-n33-k4-s24-28, Instance50-s3-30.dat and 200-10-3b. As a rule of thumb, Ncf is suggestedto take the value ns + 2 that will lead to a better solutionwith an increase in iteration times. However, due to its fastconvergence, it could converge to a local optimum (e.g., onE-n33-k4-s24-28) when the parameter value Ncf is not rea-sonable. To improve the performance of the next generation,graph-based fuzzy operators need to set effective parameterNcf for GFEA. Therefore, as the scale of the benchmark testsets becomes larger, the relationship between satellites andfuzzy subsets would need to be further considered, which willbe a topic of our future research.

V. CONCLUSION

In this paper, we have presented a graph-based fuzzyevolutionary algorithm (GFEA) for 2E-VRP. First, GFEA hasan advantage in solving 2E-VRP for instances with a largenumber of satellites and customers. It avoids excessive fitnessevaluations of unpromising movements in different satelliteswithout reducing the performance. Second, the GFEA takesadvantage of the population evolution in an iterative learningprocess, on the basis of information exchange in individualswith an assignment graph. Finally, fuzzy satellite-to-customerschemes for 2E-VRP may yield a significant improvementin performance. Despite the fact that the GFEA can obtainhigh quality and efficient solutions, it has some disadvantages.For example, the graph-based fuzzy operators are sensitiveto the number of fuzzy subsets, and the impact of fuzzyassignment metrics on the solution evaluation needs to be

further discussed. One option is to establish the evolutionaryevaluation model with fuzzy theory [40] to analyze the impactsbetween fuzzy subsets and routes to the solutions.

A research direction worth pursuing is the investigationof different graph-based assignment operators for 2E-VRP inan iterative learning process. Furthermore, one may incorpo-rate evolutionary learning mechanisms [41] [42] to constrainthe search in different satellites for either intensification orsatellite-to-customer assignment diversification purposes. Pos-sible future research topics include a theoretical analysis ofthe GFEA and applications to other variants of the routingproblem [43] [44], multimodal multi-objective optimizationproblems[45] [46] and fuzzy optimal control problem [47].

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Xueming Yan received the Ph.D. degree in Comput-er Science from the South China University of Tech-nology (SCUT), Guangzhou, in 2017. She is current-ly a lecturer at School of Information Science andTechnology & School of Cyber Security, GuangdongUniversity of Foreign Studies, Guangzhou, China.His current research interest includes computationalintelligence and its applications.

Han Huang received the B.Man. degree in Informa-tion Management and Information System, in 2002,and the Ph.D. degree in Computer Science fromthe South China University of Technology (SCUT),Guangzhou, in 2008. Currently, he is a professorat School of Software Engineering in SCUT. Hisresearch interests include evolutionary computation,swarm intelligence and their application. Dr. Huangis a senior member of CCF and member of IEEE.




Zhifeng Hao received the Ph.D. degree in Mathe-matics from Nanjing University, in 1995. He is nowa Professor and Ph.D. supervisor. His main researchinterests include design and analysis of algorithms,mathematical foundation of bio-inspired algorithms,combinatorial optimization and algebra.

Jiahai Wang received the Ph.D. degree from U-niversity of Toyama, Toyama, Japan, in 2005. In2005, he joined Sun Yat-sen University, Guangzhou,China, where he is currently a Professor with theDepartment of Computer Science. His main researchinterests include computational intelligence and itsapplications.

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A Graph-based Fuzzy Evolutionary Algorithm for …...A Graph-based Fuzzy Evolutionary Algorithm for Solving Two-Echelon Vehicle Routing Problems Xueming Yan, Han Huang, Zhifeng Hao