Expert Systems with Applications 39 (2012) 6949–6954

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Expert Systems with Applications

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A variable neighborhood search for the multi-depot vehicle routing problemwith loading cost

Yiyo Kuo a,⇑, Chi-Chang Wang b

a Department of Marketing and Logistics Management, Hsing Kuo University of Management, Tainan 709, Taiwanb Department of Mechanical and Computer-Aided Engineering, Feng Chia University, Taichung 407, Taiwan

a r t i c l e i n f o

Keywords:Vehicle routing problemMultiple depotsVariable neighbourhood searchLoading cost

0957-4174/$ - see front matter Crown Copyright � 2doi:10.1016/j.eswa.2012.01.024

⇑ Corresponding author. Tel.: +886 6 2873523; fax:E-mail address: yiyo@mail.hku.edu.tw (Y. Kuo).

a b s t r a c t

The purpose of this paper is to propose a variable neighbourhood search (VNS) for solving the multi-depot vehicle routing problem with loading cost (MDVRPLC). The MDVRPLC is the combination ofmulti-depot vehicle routing problem (MDVRP) and vehicle routing problem with loading cost (VRPLC)which are both variations of the vehicle routing problem (VRP) and occur only rarely in the literature.In fact, an extensive literature search failed to find any literature related specifically to the MDVRPLC.The proposed VNS comprises three phases. First, a stochastic method is used for initial solution genera-tion. Second, four operators are randomly selected to search neighbourhood solutions. Third, a criterionsimilar to simulated annealing (SA) is used for neighbourhood solution acceptance. The proposed VNS hasbeen test on 23 MDVRP benchmark problems. The experimental results show that the proposed methodprovides an average 23.77% improvement in total transportation cost over the best known results basedon minimizing transportation distance. The results show that the proposed method is efficient and effec-tive in solving problems.

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1. Introduction

The basic vehicle routing problem (VRP) is concerned with find-ing a set of routes to serve a given number of customers and min-imize the total travel cost (Ren, Dessouky, & Ordo9ñez, 2010). VRPhas been studied extensively because it is found to be widely appli-cable to many real-world situations (Ho, Ho, Ji, & Lau, 2008). Basedon the structure of the VRP, several variant types of problem havebeen developed in literature. These variations include the capacity-limited vehicle routing problem (CVRP), vehicle routing problemwith time window (VRPTW), periodic vehicle routing problem(PVRP), split delivery vehicle routing problem (SDVRP), vehiclerouting problem with pickup and delivery (VRPPD), timedependent vehicle routing problem (TDVRP) and so on. CVRP addsthe restriction of the capacity of vehicle loading (Mester, Bräysy, &Dullaert, 2007). VRPTW restricts vehicular services to all thecustomers to a given interval (Yu, Yang, & Yao, 2011). PVRP consid-ers the service time as a period other than a day (Francis & Smilo-witz, 2006). SDVRP allows customers’ orders to be split amongseveral vehicles on different routes (Gulczynski, Golden, & Wasil,2010). TDVRP emphasizes travel speeds that change when vehiclestravel at different times (Kuo, 2010).

Multi-depot VRP (MDVRP) is another variant type of VRP and itis the main research focus of this paper. MDVRP considers cases

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where there is more than one depot. Each vehicle departs from adepot to serve customers, follow its route and finally returns tothe depot where they started. Every customer is to be served byone vehicle on one occasion, and the loading of each vehicle cannotexceed a specified limit. In this research an extended MDVRP ispresented, more fully described as MDVRPLC, which takes loadingcost (LC) into account in the objective function when optimizingthe vehicle routes.

Since the VRP is an NP-hard problem, all the variant forms of theVRP are also NP-hard problems. Metaheuristics are general solu-tion strategies rather than heuristic rules designed for a specifictype of problem (Chen, Huang, & Dong, 2010). Many metaheuris-tics have been applied to optimizing problems that are derivedfrom the VRP. Those metaheuristics include the genetic algorithm(GA), tabu search (TS), simulated annealing (SA), ant colony opti-mization (ACO), scatter search (SS) and so on (Bell & McMullen,2004; Ho et al., 2008; Kuo, Wang, & Chuang, 2009; Tang, Zhang,& Pan, 2010; Tavakkoli-Moghaddam, Safaei, & Gholipour, 2006).Variable neighbourhood search (VNS) is another metaheuristics,and it uses systematic changes of neighbourhood within a possiblyrandomized local search algorithm to produce a simple and effec-tive metaheuristic for combinatorial and global optimization (Han-sen & Mladenovic, 2001). In this research, a VNS is proposed as amethod for solving the multi-depot vehicle routing problem withloading cost (MDVRPLC).

The remainder of this paper is organized as follows. In Section 2,the literature relating to the MDVRP and VRPLC is reviewed. Then,

ights reserved.

6950 Y. Kuo, C.-C. Wang / Expert Systems with Applications 39 (2012) 6949–6954

in Section 3, a VNS is proposed as a metaheuristic for solving theMDVRPLC. Computational results are reported in Section 4, fol-lowed by a summary and some concluding remarks in Section 5.

2. Literature review

Literature related to MDVRP is rare. In recent years, Crevier,Cordeau, and Laporte (2007) proposed a three-phase methodologyfor solving MDVRP in which vehicles may be replenished at inter-mediate depots along their route. The methodology is based onadaptative memory and tabu search for the generation of a set ofroutes, and on integer programming in the execution of a set par-titioning algorithm for the determination of least cost feasible rota-tions. Ho et al. (2008) proposed a hybrid genetic algorithm (HGA)for the MDVRP. The proposed HGA generates initial solution bytwo methods. The first method generates the initial solutions ran-domly, while the second incorporates Clarke and Wright’s savingmethod and the nearest neighbour heuristic to generating initialsolutions. Ho et al. also mentioned that no research was foundwhich applies HGA to the solution of the MDVRP. Mirabi, FatemiGhomi, and Jolai (2010) proposed three hybrid heuristics to solvethe MDVRP. The three hybrid heuristics combine elements fromboth constrictive heuristic search and improvement techniques.The improvement techniques are deterministic, stochastic andsimulated annealing, respectively. Liu, Jiang, Fung, Chen, and Liu(2010) proposed a two-phase greedy algorithm for solving practi-cal larger-scale MDVRP to minimize empty vehicle movements.In the first phase, a set of directed cycles is created to fulfil thetransportation orders. In the second phase, chains that are com-posed of cycles are generated. Finally, a set of local search strate-gies are employed to improve the initial results.

The literature related to the VRPLC is even rarer. Taveares, Zaig-raiova, Semiao, and da Graca Carvalho (2008) used optimizationsoftware to optimize the route for a solid waste collection andtransportation process. When calculating fuel consumption, theymentioned that the fuel consumption is based on distance trav-elled, road gradient and vehicle load. Kuo (2010) proposed a modelfor calculating total fuel consumption for the TDVRP. In the model,the fuel consumption takes transportation distance, transportationspeed and loading weight into consideration. Then an SA algorithmis proposed for finding the vehicle routing with the lowest totalfuel consumption. Tang et al. (2010) proposed an SS for the VRPLCto minimize total cost. The total travel cost is the sum of dispatch-ing costs, which depends on the number of vehicles dispatched,travel cost, which depend on the distance travelled, and the load-ing cost, which depends on the loading weight and the correspond-ing distance travelled. With the exception of the literature above,no report related to VRPLC was found.

Fig. 1. An example of customer assignment.

3. Variable neighbourhood search

Variable neighbourhood search (VNS) is a metaheuristic, or aframework for building heuristics, based upon systematic changesof neighbourhoods that combine a descent phase, to find a localminimum, and a perturbation phase, to emerge from the corre-sponding valley (Hansen, Mladenovic, & Pérez, 2010). The pro-posed VNS starts from an initial solution, and then uses multiplesystematic search methods to find neighbourhood solutions. If aneighbourhood solution with better performance than the initialsolution is found, that neighbourhood solution will be accepted.If no improved solution is found after a certain number of neigh-bourhood solutions have been examined, then the VNS starts to ac-cept new neighbourhood solutions that are found, even if there is apossibility that they represent a worse solution. The neighbour-hood of the accepted solution will then be the basis for the next

neighbourhood search using multiple systematic methods. Theprocess continues until the stop criterion is satisfied.

The proposed VNS for the MDVRPLC comprises three phases:the generation of an initial solution, neighbourhood search andneighbourhood solution acceptance. The details of these phasesare described in the following sections.

3.1. Initial solution generation

The initial solution for the MDVRPLC is generated by followingthree steps. The first step is called customer assignment. Each cus-tomer is assigned to one of the depots. If a customer is assignedto a certain depot, that means the order from the customer willbe transported from that specific depot. After all customers havebeen assigned to depots, the second step, called customer grouping,then separates the customers who have been assigned to the samedepot into several groups. For customers in the same group, thecorresponding order will be transported by the same vehicle. Thethird step, called customer sequencing, then decides the service se-quence of customers in each group. Following the above threesteps, the routes of all vehicles can be generated and form the ini-tial solution of the MDVRPLC. The details of these three steps areintroduced in the following paragraphs.

3.1.1. Customer assignmentThis step assigns customers to the depot which is conveniently

close to them. This follows the method adopted by Ho et al. (2008),who assign customers to the depot which is nearest them. How-ever, in order to make the initial solutions more flexible, the pres-ent research assigns customers to depots based on a probability.Suppose that d(Ci,Da) indicates the distance between customer i(Ci) and depot a (Da), and dðCi;DÞ indicates the average distance be-tween customer i and all depots. The probability of customer ibeing assigned to depot a, p(Ci,Da), and is calculated by Eq. (1).

pðCi;DaÞ ¼max dðCi;DÞ � dðCi;DaÞ;0

n oPA

a¼1 max dðCi;DÞ � dðCi;DaÞ;0n o ð1Þ

Based on the probability calculated using Eq. (1), a customermay still be assigned to a depot which the one that is closest tothem. In Fig. 1, for example, the distance between Customer 1and Depots 1, 2 and 3 are 10, 6 and 5 respectively. The average dis-tance between Customer 1 and all three depots is 7. The probabilitythat Customer 1 is assigned to Depots 1, 2 and 3 are 0.00%, 33.33%and 66.77% respectively. Although Depot 3 is closest to Customer 1,there is a chance that Customer 1 will be assigned to Depot 2.

Fig. 2. An example of CS.

Y. Kuo, C.-C. Wang / Expert Systems with Applications 39 (2012) 6949–6954 6951

3.1.2. Customer groupingAfter the step of customer assignment, when a proportion of

the deliveries to customers have been assigned to the same depot,the corresponding orders of those customers will be delivered fromthe same depot. However, the orders of those customers may notbe transported by a single vehicle on its delivery route, becausevehicles have fixed capacities that must not be exceeded. There-fore, the operation of each depot can be view as a capacity-limitedvehicle routing problem (CVRP) (Chen et al., 2010). The CVRP issimilar to the VRP, but all vehicles have a capacity restriction.

In the present research, a clockwise search (CS) method pro-posed by Kuo et al. (2009) is adopted for customers grouping.The CS selects a customer at random and then searches for othercustomers in the group in a clockwise direction until the capacityof a vehicle will be exceeded by adding the order of the next cus-tomer, or until all the customers have been selected. In Fig. 2, forexample, there are 8 customers assigned to Depot 1. The ordersfor each customer are shown in the right hand side of the circle.If the capacity of vehicle is 20 units, and Customer 3 is selectedfirstly by a random method, then following in the clockwise direc-tion, Customers 3 to 7 will form one group.

3.1.3. Customer sequencingThis step decides the service sequence of customers in one

group. It can be view as a travelling salesman problem (TSP), whichis a special case of the VRP, when the number of vehicles is equal toone. The saving method (SM) proposed by Clarke and Wright(1964) is adopted for this step. Suppose that na indicate the num-ber of customer assigned to Depot a. SM is based on the notion thatone vehicle only serves each customer, and that there are na vehi-cle routes for Depot a. The saving value, Sij, is the distance that canbe saved if the route for customer i is combined with the route forcustomer j. The two routes which have the largest saving valuehave the highest priority to be combined (Lin, Lee, Ying, & Lee,2009). For Depot a, the saving values between the routes for cus-tomer i and customer j are calculated by Eq. (2).

(a)

Fig. 3. An example of cu

Sij ¼ dðCi;DaÞ þ dðCj;DaÞ � dðCi;CjÞ for all i; j and i – j ð2Þ

After calculating all the saving values, the pair of customerswith the biggest saving value are selected firstly and the ordersof these two customers will be transported in one vehicle routesequentially. From the remaining customers, the one with thehighest saving value, Sij, in relation to one of the customers whois either the first or last in the delivery sequence is then added tothe delivery sequence at either the beginning or the end, as appro-priate. Adding the sequence Ci–Cj to the original transportation se-quence produces a new sequence. Finally, when all customers areselected, the depot is added at both the beginning and the end ofthe transportation sequence to form a vehicle route.

For example, a group including 5 customers, with details of thelocation (coordinates) of the customers and the depot, are shownin Fig. 3(a). Using Eq. (2) the saving value of all pairs of customersare calculated and the results are shown in Table 1. Because Sij = Sji,Table 1 only shows the saving value for cases where i < j. In Table 1,the biggest value is S5,7, so the orders of Customer 5 and 7 are cho-sen to be transported sequentially (denoted as C5–C7). For theremaining saving values, S4,5 is the biggest one. Moreover the Cus-tomer 4 is unselected and Customer 5 is at one end of the transpor-tation sequence C5–C7. Combining sequence C5–C7 with C4–C5, thesequence C4–C5–C7 is constructed. In the same way, combiningthe sequence C3–C4 with C4–C5–C7 produces the sequence C3–C4–C5–C7. After constructing the sequence C3–C4–C5–C7, S4,7 and S5,6

are the biggest and second biggest of the remaining saving values.However, customer 4 and 7 are both in the sequence C3–C4–C5–C7,and customer 5 is not in the beginning or the end of the sequence.The third biggest saving value S6,7 is then taken into consideration.Combining sequence C6–C7 with C3–C4–C5–C7 produces the se-quence C3–C4–C5–C7–C6. Because all customers in the group arein that sequence, the depot is added at the beginning and theend of the transportation sequence to construct the vehicle routeD1–C3–C4–C5–C7–C6–D1 as shown in Fig. 3(b).

For VRP the route between D1–C3–C4–C5–C7–C6–D1 and D1–C6–C7–C5–C4–C3–D1 are the same. However, when loading cost is takeninto consideration the customers with the largest orders shouldhave their deliveries made first to reduce the loading cost of theremaining delivery route. In the present research, the route inwhich the order of the first customer served is bigger will be se-lected as part of the initial solution. For the result in the exampleshown in Fig. 3(b), if the order of Customer 3 is bigger than thatof Customer 6, then the route D1–C3–C4–C5–C7–C6–D1 will beselected.

3.2. Neighbourhood search

When an initial solution has been generated, it contains multi-ple vehicle routes. In the present research, four operations are pro-posed for the search for neighbourhood solutions. They are node

(b)

stomer sequencing.

Table 1Saving values of customer sequencing example.

Sij

C3 C4 C5 C6 C7

C3 – 11.80 9.57 6.84 7.63C4 – 13.42 8.44 11.25C5 – 9.88 14.78C6 – 9.69C7 –

Fig. 4. An example of insertion operator.

Fig. 5. An example of node exchange.

Fig. 6. An example of arc exchange.

6952 Y. Kuo, C.-C. Wang / Expert Systems with Applications 39 (2012) 6949–6954

insertion, node exchange, arc exchange and path exchange. Theproposed heuristic for searching the neighbourhood for a moreeconomical solution (neighbourhood decent) randomly selectsone of the four operations to search the neighbourhood solutions.The use of multiple operations can make the solution space searchmore extensive (Chen et al., 2010). The details of the four opera-tions are introduced in the following sections.

3.2.1. Node insertionThe node insertion operator deletes a customer at random from

a vehicle route, and then inserts the deleted customer in betweentwo other customers who occupy successive positions in a deliveryroute. The new position of the transferred node (customer) may bein the same delivery route, or in a different one. An example of thenode insertion operator is shown in Fig. 4.

3.2.2. Node exchangeThe node exchange operator selects two customers at random

and then exchange their positions. Like the node insertion opera-tor, this may involve two customers in different routes, or two cus-tomers in the same route. An example of the node exchangeoperator is shown in Fig. 5.

3.2.3. Arc exchangeThe arc exchange operator is similar to node exchange. It ran-

dom selects two pairs of two successive customers and then ex-changes their positions. Like the node insertion and the nodeexchange operator, the arc exchange operator may involve two dif-ferent routes or two arcs within a single route. An example of thearc insertion operator is shown in Fig. 6.

3.2.4. Section exchangeThe section exchange operator selects two routes at random,

and cuts a section (which either starts or ends at the depot) fromeach. The two selected sections are then exchanged to form twonew routes. An example of the section exchange operator is shown

in Fig. 7. Unlike the three previous operators, the section exchangeonly selects two routes that both start and end at the same depot.

3.3. Solution acceptance

First improvement and best improvement are the two basicdescent methods of VNS (Hansen & Mladenovic, 2001). The firstimprovement method searches neighbourhood solutions in a ran-dom way. If an improved solution is found, the solution will be ac-cepted for descent. The best improvement method searches allneighbourhood solutions and only the best one, which is also bet-ter than the current solution, is accepted for descent. However, it ispossible that neither method will find a better solution in theneighbourhood. In the proposed VNS the first improvement meth-od is adopted. However, if no improved solution is found after acertain number of neighbourhood solutions are searched, theVNS will start to accept new neighbourhood solutions, with a prob-ability based on their performance, even if the solution is worsethan the established solution. This is similar to the method of sim-ulated annealing and is designed to avoid the solution searchbecoming trapped in a local minimum (Hansen and Mladenovic,2001; Kirkpatrick, Gelatt, & Vecchi, 1983). The probability ofaccepting a neighbourhood solution (x’) is calculated by Eq. (3).

Pðx0Þ ¼1; if f ðx0Þ 6 f ðxÞ

ef ðxÞ�f ðx0 Þ

Tk

� �; if f ðx0Þ > f ðxÞ

8<: ð3Þ

In Eq. (3), P(x’) is the probability of accepting neighbourhoodsolution x’ over current solution x, and f(x) and f(x’) are the perfor-mance of current solution x and its neighbourhood solution x’

Fig. 7. An example of section exchange.

Fig. 9. The flow chart of proposed VNS.

Y. Kuo, C.-C. Wang / Expert Systems with Applications 39 (2012) 6949–6954 6953

respectively. Tk is a gradually reducing value in the search processin which k is the index of search iterations. In this research, Eq. (4)is used for updating Tk.

Tkþ1 ¼ aTk þ ð1� aÞð1� bÞTe ð4Þ

In Eq. (4), Tk+1 is the T value in iteration k + 1, Tk is the T value initeration k, Te is the final value of T in the last iteration of the VNSprocess. b is an adjusted parameter for controlling the speed atwhich the T value reduces. Therefore, a can be calculated by Eq. (5).

a ¼ bTe

Ts � ð1� bÞTe

� �ð1=KÞ

ð5Þ

In Eq. (5), Ts is the initial value of T in the first iteration of theVNS process. K is the maximum number of iteration of the VNS.Fig. 8 gives examples of the different rates at which the T value re-duces for different values of b. It shows when b is smaller, the T va-lue reduces more quickly.

According to the proposed VNS presented above, if no improvedsolution is found after Y neighbourhood solutions are searched,then the VNS starts to accept new neighbourhood solutions basedon P(x’). The relationship between initial solution generation,neighbour search and solution acceptance are illustrated in Fig. 9.

4. Computational experiments

In this section, the performance of the proposed VNS in solving theMDVRPLC is examined. The webpage http://neo.lcc.uma.es/radi-aeb/WebVRP/index.html?/Problem_Instances/instances.html presents a range of VRP related instances, including CVRP, MDVRP, PVRP,SDVRP, and VRPPD. The present research takes the MDVRP instances

β =0.001

β =0.01

β =1

β =0.1

401

Iteration number

Val

ues

of T

11 21 31 41 51 61 71 81 91

90

140

Fig. 8. The reducing speed of T values with different b (Ts = 150, Te = 50 andK = 100).

from that website for benchmarking purposes. The first 23 MDVRPexamples from the webpage were used to test the proposed method.The corresponding best known solutions, based on minimizing totaltravel distance, are also taken from the webpage.

Unlike the MDVRP, the proposed MDVRPLC takes the loadingcosts into consideration. In this research it is assumes that the costof vehicle movement is 1.5 per unit-distance, the cost of deliveringproduct is 0.2 per unit-weight and per unit-distance, and the costof dispatching a vehicle is 100. These costs are determined accord-ing to an investigation conducted in a local transportation com-pany (Tang et al., 2010). Moreover the parameters Ts, Te, Y, K andb are set as 5000, 5, 50, 100,000 and 0.1, respectively. Using an IntelCore 2 1.86 GHz personal computer, every test was completedwithin 30 min. The search for each solution was conducted fivetimes, and the best results are shown in Table 2.

In Table 2 the second and third columns are the results of opti-mizing using the proposed VNS. The fourth and fifth columns arethe performance values of the best known solutions based on min-imizing total travel distance which are provided on the webpage. Itcan be seen from Table 2 that a vehicle route with the minimumtotal travel distance cannot guarantee the minimum total trans-portation cost. In Table 2 the routes optimized using the proposedVNS provide a 23.77% improvement in total transportation costover the best known results which are based on minimizing trans-portation distance.

5. Conclusions

There are several variant types of VRP. The multi-depot vehiclerouting problem with loading cost (MDVRPLC), which combines themulti-depot vehicle routing problem (MDVRP) and vehicle routingproblem with loading cost (VRPLC) has not been addressed in the lit-erature. A variable neighbourhood search (VNS) algorithm, whichcomprises three phases, is proposed for solving the MDVRPLC. The firstphase uses a stochastic method to generate an initial solution. Based

Table 2Experimental results.

Problems Proposed method Best Know Improvement by total cost (%)

Total cost Total distance Total cost Total distance

P01 5242.72 716.01 6230.48 567.87 15.85P02 5571.41 606.44 8343.71 473.53 33.23P03 7452.93 810.54 10044.68 641.19 25.80P04 10114.33 1157.04 12873.22 1001.59 21.43P05 10691.40 915.01 15504.44 750.03 31.04P06 8920.60 1056 10821.29 876.50 17.56P07 9373.05 1035.23 11100.58 885.80 15.56P08 178315.93 6045.79 222170.34 4420.95 19.74P09 144177.41 6265.93 200350.76 3900.22 28.04P10 139951.56 5346.30 182944.23 3663.02 23.50P11 141778.45 5355.14 174969.91 3554.18 18.97P12 6577.79 1675.45 9763.85 1318.95 32.63P13 6482.43 1655.79 9946.57 1318.95 34.83P14 6829.59 1708.48 10257.73 1360.12 33.42P15 13782.39 3508.14 19133.69 2505.42 27.97P16 14176.28 3552.85 19834.31 2572.23 28.53P17 14082.71 3553.45 20092.66 2709.09 29.91P18 21575.81 5350.25 29250.33 3702.85 26.24P19 23900.27 6149.01 29612.38 3827.06 19.29P20 25014.72 6545.20 31348.06 4058.07 20.20P21 37637.81 10244.89 43513.84 5474.84 13.50P22 38342.81 10300.02 43234.94 5702.16 11.32P23 38623.05 10421.27 47215.44 6095.46 18.20

Average 39505.02 4085.84 50806.85 2668.70 23.77

6954 Y. Kuo, C.-C. Wang / Expert Systems with Applications 39 (2012) 6949–6954

on the initial solution the second phase than selects one of four oper-ators to search neighbourhood solutions. If the solution that is foundby the neighbourhood search is better, then it will replace the originalsolution. However, in phase three a worse neighbourhood solution willstill have chance to replace the original solution. 23 examples of theMDVRP provided on a specialist webpage are tested for benchmarkingpurposes. For all 23 instances the proposed VNS finds vehicle routeswith lower transportation costs than the best known results whichare based on minimizing transportation distance. Moreover the finalvehicle routes are found in no more than 30 min. Therefore the pro-posed method is efficient and effective in solving problems.

Acknowledgments

This work was supported, in part, by the National Science Coun-cil of Taiwan, Republic of China, under Grant NSC-100-2221-E-432-001.

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