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Transportation Research Part E journal homepage: www.elsevier.com/locate/tre

The vehicle routing problem with uncertain demand at nodes Chang-Shi Liu a,b, Ming-Yong Lai b,* Economics and Trade College, Hunan University, Hunan, Changsha 410079, PR China Key Laboratory of Logistics Information and Simulation Technology of Hunan Province, Hunan, Changsha 410079, PR China

a r t i c l e

i n f o

Article history: Received 1 April 2008 Received in revised form 17 August 2008 Accepted 1 November 2008

Keywords: Sweeping algorithm Fuzzy arithmetic rule Vehicle routing problem Vehicle coordinated strategy

1. Introduction

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The vehicle routing problem with uncertain demand at nodes is considered. The approximate reasoning algorithm is developed to determine the preference strength to send the vehicle to next node, and the improved sweeping algorithm with vehicle coordinated strategy is originally proposed to determine a set of vehicle routes that minimizes costs. Finally the computational results are presented to show the high effectiveness and performance of the solution approaches. Crown Copyright Ó 2008 Published by Elsevier Ltd. All rights reserved.

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Usually the basic input data to solve the problem of routing vehicles when demand at the nodes is uncertain are the probability density functions of the random variables representing demand at the nodes. However, in order to verify the probability density functions of demand at the nodes, demand must be ‘‘recorded” over a long period of time and a detailed statistical analysis of the collected data must be made. On the other hand, actually the information about vehicle demand at some nodes is often not precise enough. For example, based on experience, it can be concluded that demand at a node is ‘‘around 50 units”, ‘‘between 80 and 120 units”, ‘‘approximately 70 units”, etc. Thus, there is often uncertainty regarding the amount of demand at some nodes. Using methods from the theory of fuzzy sets, it is possible to successfully model problems which contain an element of uncertainty, subjectivity, ambiguity and vagueness. To date, most studies on vehicle routing problem (VRP) are based on vehicle uncoordinated strategy, in which each vehicle serves their customers independently, and there is no coordination between any two vehicles. Liu Xing (2005) have recently reported that vehicle uncoordinated strategy may result in immerse wasting of resource, but vehicle coordinated strategy not only takes full advantage of vehicle’s capacity on the way, but also shortens service time, cuts down the transport cost and improves service quality. To the author’s knowledge, at present there is little literature researched VRP based on vehicle coordinated strategy. Shang and Cuff (1996) examined a multi-criteria pickup and delivery problem allowing transfer of hospital documents between vehicles provided no additional travel time is violated. They developed a look-ahead heuristic allowing users to input the number of delivery items to look-ahead, the number of time windows to look-ahead, and the minimum number of new routes to be constructed. Each new item is inserted into the existing schedule or into the best new route. New routes are evaluated according to a multi-criteria weighted score method considering four performance measures (number of items that can be transported, amount of travel time saved, number of unscheduled items that can use the route as origin or * Corresponding author. Tel./fax: +86 731 8684625. E-mail address: [email protected] (M.-Y. Lai). 1366-5545/$ - see front matter Crown Copyright Ó 2008 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.tre.2008.11.002

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destination, and the number of items that could be transferred between the new route and existing routes). The route with the largest score is selected into the existing set. Alan Laurence Erera (2000) studied a VRP based on vehicle coordinated strategy by continuous approximation model. Liu Xing (2005) proposed two vehicle-paired strategy and a simple general vehicle coordinated strategy. Lin (2008) studied the vehicle coordinated strategy with single or multiple uses of vehicles and vehicles are allowed to travel to transfer items to another vehicle returning to the depot, provided no time window constraint and capacity constraint are violated. However their investigations are on the basis of customers are uniform distribution in certain region, and they all assume that demands at nodes are certain, of course their approaches can not solve the actual problem. In this paper we consider a vehicle routing problem regarding uncertain demands at the nodes. We provide rules of fuzzy arithmetic for deciding whether send vehicle to next node, present heuristic sweeping algorithm to design vehicle routing for the proposed problem, and the solution is improved by vehicle coordinated strategy. Finally we present computational results that show the effectiveness and performance of the solution approaches. The remainder of this paper is organized as follows. In the following section, we provide problem assumptions. In Section 3, we provide solution for proposed problem. In Section 4, we design two vehicle-paired loop strategy. In Section 5 we present the results of some computational tests of the performance of the heuristic approaches. Finally, in Section 6, we discuss our on-going and future research directions. 2. Problem assumptions

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We assume that there are n nodes in the network to be served. Vehicles set out from depot C, serve a number of nodes and upon completion of their service, return to the depot (Fig. 1). We also assume that the demand at each node is only approximately known. Such demand can be represented by a triangular fuzzy number (Fig. 2). Fig. 2 presents the membership function of triangular fuzzy number D representing demand at the node. Triangular fuzzy number D = (d1, d2, d3) is described by its left boundary d1 and its right boundary d3. Thus, the dispatcher or analyst studying the problem can subjectively estimate, based on his experience and intuition and/or available data, that demand at the node will not be less than d1 or greater than d3. The value of d2 corresponding to a grade of membership of 1 can also be determined by a subjective estimate. The problem discussed in this paper can be deﬁned as follows: for known locations of the depot and nodes to be served, known vehicle capacity, and demand at the nodes which is only approximate known (represented by triangular fuzzy numbers), design a set of vehicle routes that minimizes costs. 3. Solution to the proposed problem

3.1. Deciding whether sending vehicle to next node

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When demand at the nodes is deterministic, it is easy to calculate whether the vehicle is able to serve the next node after completing service at one node. But when demand at the nodes is uncertain and only characterized by triangular fuzzy numbers, it is not easy to decide whether the vehicle should serve the next node or return to the depot. It is clear that the greater the vehicle’s remaining capacity and the lesser the demand at the next node, the greater the vehicle’s ‘‘chances” of being able to serve the next node. In this paper, we solve the proposed problem based on the algorithms developed by Teodorovic and Pavkovic (1996). For simple we assume that service is provided by vehicles of the same size. We will denote vehicle capacity by Q and the fuzzy number representing demand at the ith node by Di. After serving the ﬁrst k nodes, the available capacity of the vehicle, Ak, will equal:

Ak ¼ Q

k X

Di

i¼1

Fig. 1. Depot C and n nodes to be served.

ð1Þ

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Fig. 2. Membership function of triangular fuzzy number D presenting demand at the node.

The capacity of vehicle Q can also be presented as a triangular fuzzy number Q = (Q, Q, Q). Using the rules of fuzzy arithmetic proposed by Kaufmann and Gupta (1985), we can easily show that Ak is also a triangular fuzzy number, i.e.

Q

i¼1

d1i ; Q

k X

d2i ; Q

i¼1

k X

! d3i

i¼1

D

Ak ¼

k X

ð2Þ

0 6 pk 6 1; k ¼ 1; 2; . . . ; n:

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where d1i = left boundary of fuzzy number Di; d2i = value of fuzzy number Di corresponding to a grade of membership of 1; d3i = right boundary of fuzzy number Di. It is clear that the ‘‘strength” of our preference for the vehicle to serve the next node after it has served k nodes depends on available capacity Ak. Our preference can be, in this paper, ‘‘low”, ‘‘medium” or ‘‘high”. We will denote by Pk the preference index that expresses the strength of our preference to send the vehicle to the next node after it has served k nodes. Let the preference index be between 0 and 1, i.e.

ð3Þ

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If Pk = 1, we are absolutely certain that we want the vehicle to serve the next node; If Pk = 0, we are completely sure that the vehicle should return to the depot. The linguistic expressions ‘‘low preference”, ‘‘medium preference” and ‘‘high preference” can also be represented by corresponding fuzzy sets. Fig. 3 presents the membership functions of fuzzy sets ‘‘low”, ‘‘medium” and ‘‘high” preference. Available capacity Ak can also be subjectively estimated as ‘‘small”, ‘‘medium” and ‘‘large”. We denote respectively, by S, M and L. The fuzzy sets denoting small, medium and large available vehicle capacity Ak. The membership functions of these fuzzy sets are shown on Fig. 4.

Fig. 3. Fuzzy set describing preference strength.

Fig. 4. Fuzzy set describing available capacity.

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Approximate reasoning algorithm is developed in this paper to determine preference strength regarding sending a vehicle to the next node. This algorithm assumes that the strength of our preference Pk depends exclusively on available capacity Ak. This approximate reasoning algorithm reads: rule 1: if Ak = S, then Pk = low; rule 2: if Ak = M, then Pk = medium; rule 3: if Ak = L, then Pk = large. For known available capacity Ak that remains after serving k nodes, it is possible to use the approximate reasoning rules to determine the strength of our preference to send the vehicle to next node. As noted by Mamdani and Assilian (1999), every rule represents a fuzzy relation. In our case, every rule represents a fuzzy relation between available capacity Ak and preference strength Pk. Let us denote by R the fuzzy relation between available capacity and preference strength. Membership function lR(x, y) of this fuzzy relation equals:

lR ðx; yÞ ¼ minflM ðxÞ; lmedium ðyÞg

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x

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maxfmin½lAk ðxÞ; minðlM ðxÞ; lmedium ðyÞÞg 8x; y

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Now we are able to approximately answer the following question: should we send the vehicle to the next node or return it to the depot after completing service to k nodes? In other words, now we can use known available capacity Ak to calculate the preference strength P k to send the vehicle to the next node. The membership function values of fuzzy set Pk are calculated as:

ð5Þ

This approximate reasoning procedure is that for known capacity Ak, the ﬁrst step is to establish how much this available data satisﬁes the premises of the rules. In other words, ﬁrst we must determine whether available capacity Ak is ‘‘small”, ‘‘medium” or ‘‘large”; then we use the approximate reasoning rules proposed above to calculate the strength of preference Pk. Once the membership function of the preference index has been determined, defuzziﬁcation must take place, i.e. one speciﬁc preference index must be chosen. The value corresponding to the highest grade of membership or to the center of gravity is usually chosen. In this paper the preference index chosen was the center of gravity (As demonstrated by Runkler and Glesner, 1993a,b). Let the chosen strength preference index equal Pk . Based on this value, a decision must be made whether to send the vehicle to the next node or return it to the depot. In this paper, the decision was made as follows: the vehicle should be sent to the next node if the following relation if fulﬁlled:

Pk P p

ð6Þ

where p* is a parameter indicating threshold value in the interval [0, 1]. When

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Pk < p

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the vehicle should be returned to the depot. Lower values of parameter p* express our endeavor to use the vehicle capacity as much as possible. On the other hand, when parameter p* is low, there is a rise in the number of cases in which the vehicle arrives at the next node and is not able to carry out planned service due to small available capacity. 3.2. Solving problem with improved sweeping algorithm In the past three decades, a large number of different heuristic algorithms have been developed to route vehicles. One of the simplest is the sweeping algorithm proposed by Gillet and Miller (1974). This algorithm is applied to polar coordinates and the depot is considered to be the origin of the coordinate system. Then the depot is joined with an arbitrarily chosen point which is called the seed point. All other points are joined to the depot and then aligned by increasing angles which are formed by the segments which connect the points to the depot and the segment which connects the depot to the seed point. The route starts with the seed point and then the points aligned by increasing angles are included, respecting given constraints all the while. When a point cannot be included in the route since this would violate a certain constraint, this point becomes the seed point of a new route, etc. The algorithm developed in this paper to route vehicles when demand at the nodes is uncertain based on sweeping algorithm proposed by Gillet and Miller (1974) consists of the following algorithmic steps: Step 1: Express demand at the nodes in the form of triangular fuzzy numbers. Assign a value to parameter p*. Step 2: If all nodes have been assigned to routes, end the algorithm. Otherwise, go to Step 3. Step 3: Choose a seed point. Link this seed point to the depot. Step 4: First include the seed point in a route. Then include nodes in the route by increasing order of the angles that are made by the lines that link the nodes with the depot and the line that links the seed point to the depot. Before deciding to

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include a node into the route, ﬁrst use the approximate reasoning algorithm to calculate the preference index. If the calculated preference index is greater than or equal to the previously assigned value of p*, include the node in the route. Otherwise, this node must be a new seed point and must link it to the depot. Return to Step 2. Step 5: Improvement with node-exchange The solution obtained by sweeping algorithm can be further improved by the node-exchange heuristic. This is an optional improvement, and it is not required for the general solution. In our computational analysis, we test the performance of the proposed improved sweeping heuristic with and without node-exchange improvement. As we report in our results in Section 5, the node-exchange heuristic helps to improve the quality of the solutions generated by the heuristic. The node-exchange heuristic starts with the solution provided by the improved sweeping algorithm. In this solution, we randomly select a pair of vehicles. Then, by exchanging a pair of nodes with each other, we obtain a new solution. If this solution is better than the current solution, i.e. the costs of the new solution is less than the value of the current solution, we continue selecting vehicles. Otherwise, we consider the current solution as the improved solution. We repeat the node-exchange heuristic on every solution in the solution obtained by the improved sweeping algorithm.

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4. Improving the algorithm by vehicle coordinated strategy

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The real value of demand at a node is only known when the vehicle reaches the node. On the other hand, the vehicle routes are designed in advance by applying the proposed algorithm. Due to the uncertainty of demand at the nodes, a vehicle might not be able to service a node once it arrives there due to insufﬁcient capacity. In such situations the vehicle must return to the depot because there is no coordination between any two vehicles, empties what it has picked up thus far, then return to the node where it had a ‘‘failure” and continue service along the rest of the planned route (Fig. 5). When evaluating the planned route, the ‘‘additional distance” that the vehicle makes due to ‘‘failure” arising in some nodes along the route must be taken into consideration, so there also is additional ‘‘loading times” at the depot. If the improved sweeping algorithm is improved by vehicle coordinated strategy, vehicle capacity will be used more fully and total serve time will be more less, such that the cost is decreased while the serve quality is kept or even improved (as demonstrated by Liu Xing, 2005). In this paper, two vehicle-paired loop strategy is originally designed for the proposed problem. As a rule, ﬁrst each vehicle serves its outer customers, then serves its inner customers. In this way, the ‘‘failure nodes” are near to depot, thus in favor of vehicle coordination. Two vehicle-paired loop strategy consists of the following rules:

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Rule 1: If one vehicle complete its task and there is no available capacity, it return to depot. Rule 2: If one vehicle has a ‘‘failure” at node i on the way, it return to depot, and give the other vehicle ‘‘failure information” (Fig. 6a). Rule 3: If one vehicle complete its task and there is available capacity, wait for the other vehicle’s information. Rule 4: if the vehicle receive the ‘‘failure information”, it go to the node where it had a ‘‘failure” and continue service along the rest of the planned route of the other vehicle (Fig. 6b). Rule 5: If the vehicle can not complete the rest of task, the third vehicle is called for completing the residual task (Fig. 6c). Now we can use the proposed vehicle coordinated strategy to solve the proposed problem. Step 1: Use the improved sweeping algorithm to design vehicle routes. Step 2: If all vehicles have been combined, end the algorithm. Otherwise go to step 3. Step 3: Combine vehicle with another vehicle which is nearest to it, and use the two vehicle-paired loop strategy between them. Return to step 2.

Fig. 5. ‘‘Failure” instance in the planned route.

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Fig. 6. Vehicle coordinated rules.

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5. Computational results

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Of course, there are some deﬁciencies of the two vehicle-paired loop strategy. For instance, it must take human behavior into consideration, for example, constrained wait. The vehicle which completes its task and there is available capacity, must wait for the other vehicle’s information. It may wait very long time under the circumstances of trafﬁc time is variant or operating time is not identical, thus increase the vehicle time cost. Also it is not easy to ﬁnd the nearest vehicle for the appointed vehicle in large-scale logistics problem, for example, there are more than 100 identical vehicles in one depot. We will discuss the vehicle coordinated strategy for large transportation in our next paper.

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The algorithm developed was tested on a large number of different numerical examples. The locations of the nodes requiring service and of the depot were generated in a random manner. Demand at the nodes was also arbitrarily determined. After concluding the computer experiments, the ﬁrst question to be answered was the efﬁciency of the proposed algorithm. CPU times are shown in Table 1. All computer experiments were done on a PC computer (486 processor, 512M memory). Very small CPU times were achieved. Bearing in mind the fact that the problem did not have to be solved in ‘‘real time”, the achieved CPU times were absolutely acceptable. In Table 2, we summarize of the results for the improved sweeping heuristic with, and without, the node-exchange heuristic improvement. In this experiment, the number of node to be served with in the interval of 200–1000 with a step of 200, each instance is solved ﬁve times, we pick the best of the ﬁve runs for each instance as the heuristic solution. For small and medium problem instances, node number is less than 600, the average gap between the optimum solution and the proposed heuristic without improvement is less than 8%. For large instances, the average gap can be as high as 10%. It is clear that using node-exchange heuristic improvement to solve the proposed problem provides better solution Quality.

Table 1 CPU times.

Number of nodes

100

200

300

400

500

600

700

800

CPU time (s)

2

4

5

8

11

14

18

21

Table 2 Results for improved sweeping heuristic. Number of nodes

200 400 600 800 1000

Without node-exchange

With node-exchange

TP length

TP length

2836.47 4573.27 6734.52 8901.64 11247.64

2754.32 4352.98 6329.84 7956.23 10075.28

Average gap

0.03 0.05 0.06 0.11 0.11

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C.-S. Liu, M.-Y. Lai / Transportation Research Part E 45 (2009) 517–524 Table 3 Total length of planned routes, additional distance due to failures, total expected length of all routes. Serving under vehicle coordinated strategy

TP length

AD length

TE length

TP length

AD length

TE length

1700.61 1780.11 1790.25 1795.36 1806.24 1826.35 1856.42 1879.48 2100.71 2054.79 2314.30 2500.38 2554.37 2876.43 3534.12 5738.95 5738.95 5738.95 5738.95 5738.95 5738.95

423.41 459.78 571.49 582.13 684.17 695.32 733.35 756.58 554.21 446.78 417.62 312.46 210.35 156.30 85.24 0.00 0.00 0.00 0.00 0.00 0.00

2124.02 2239.89 2361.74 2377.49 2490.41 2521.67 2589.77 2636.06 2454.92 2501.57 2731.92 2812.84 2764.72 3032.73 3619.36 5738.95 5738.95 5738.95 5738.95 5738.95 5738.95

1700.61 1780.11 1790.25 1795.36 1806.24 1826.35 1856.42 1879.48 2100.71 2054.79 2314.30 2500.38 2554.37 2876.43 3534.12 5738.95 5738.95 5738.95 5738.95 5738.95 5738.95

423.41 405.21 500.46 512.90 604.34 610.61 620.71 635.21 331.16 190.50 150.33 116.45 64.62 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

2124.02 2185.32 2290.71 2308.26 2410.58 2436.96 2477.13 2514.69 2431.87 2245.29 2464.63 2616.83 2618.99 2876.43 3534.12 5738.95 5738.95 5738.95 5738.95 5738.95 5738.95

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0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00

Serving under vehicle uncoordinated strategy

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p*

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As already mentioned, lower values of parameter p* express our desire to use vehicle capacity the best we can. These values correspond to routes with shorter total distances. However, lower values of parameter p* increase the number of cases in which vehicles arrive at a node and are unable to service it, thereby increasing the total distance they cover due to the ‘‘failure”. Higher values of parameter p* are characterized by less utilization of vehicle capacity along the planned routes and less additional distance to cover due to failures. The problem logically arises of determining the value of parameter p* which will result in the least total sum of planned route lengths and additional distance covered by vehicles due to failure. It was decided in this paper to use simulation to reach the value of parameter p* corresponding to least total distance. It was noted earlier that node demand in this paper is considered to be a triangular number. In other words, all possibility laws that refer to node demand have a triangular form. In this paper, node demand was simulated by the manner proposed by Kaufmann and Gupta (1985). In other words, demand at each node is a deterministic amount that is obtained by simulation. Moving along the route designed by the approximate reasoning algorithm and accumulating the amounts picked up at each node, it was easy to determine the nodes where failures occurred and to calculate the additional distance that the vehicles had to make. Five hundred simulation experiments were made for the set of routes considered, resulting in 500 additional distances that the vehicles made due to failures. In other words, the total additional distance covered by the vehicle due to failures was treated as a random variable with a corresponding empirical frequency distribution. The values of parameter p* varied within the interval of 0–1 with a step of 0.05. As parameter p* rose, a general rising tendency was noted in the total length of planned routes with a decrease in the unplanned distances that vehicles had to make due to failures at the nodes. Table 3 shows the lengths of the planned routes, additional distances covered due to failures at the nodes, and the total expected distance that the vehicles were to cover (the total distance is the sum of the total lengths of the planned routes and the additional distances that are covered due to failures at the nodes). Table 3 refers to a hypothetical example with 100 nodes. In Table 3, we also summarize of the results for the serving under vehicle uncoordinated strategy and serving under vehicle coordinated strategy. ‘‘TP length” is total planned length of all routes, ‘‘AD length” is additional distance due to failures, ‘‘TE length” is total expected length of all routes. We note from Table 3 that the least total expected distance is realized when 0:40 6 p 6 0:50. Additional distance length arising by vehicle coordinated strategy is less than 10% the values of serving under vehicle uncoordinated strategy when 0.05 < p* < 0.35, and it is above 20% for large p* values, thus cut down the ‘‘loading times” at depot. These results show that using vehicle coordinated strategy to solve the proposed problem provides better performance, both in terms of solution quality and the ‘‘loading times”. It is signiﬁcance for the logistics which loading cost is great, or for the logistics merchandise which can not be reloaded. 6. Conclusions This paper solved the vehicle routing problem when demand at the nodes is uncertain. The proposed approximate reasoning algorithm enabled a quick determination of the preference strength to send the vehicle to next node. The value of parameter p* had to be determined before using the proposed algorithms. The choice of this parameter greatly inﬂuences the length of the planned routes and the ‘‘additional distances” covered by vehicles due to failures at the nodes. In this paper, the ‘‘best” value of parameter p* was found by simulation.

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To date, in general, studies about VRP have not considered vehicle coordinated strategy, each vehicle serves its customers independently and there is no coordination between any two vehicles, in which vehicle capacity’s usage is very low on the way thus the total cost is very high. The contribution of this paper to the literature is that we originally designed the improved sweeping algorithm with vehicle coordinated strategy making full use of the vehicle capacity on the way to cut down the ‘‘additional distance” and ‘‘loading times” that the vehicle makes due to ‘‘failure” arising in some nodes because of the insufﬁcient capacity. The computational results show that the better solution quality is provided by the proposed strategy. It is signiﬁcance for the logistics which loading cost is great, or for the logistics merchandise which can not be reloaded. More complex routing and scheduling include those in which there is uncertainty about demand at the nodes, travel time between the nodes, desired service start-up at the nodes, multi-vehicle coordinated strategy, etc. Further research should be oriented towards the development of new approximate reasoning algorithms and new vehicle coordinated strategy to solve problems from the ﬁeld of routing and scheduling problems. Acknowledgements

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The authors are grateful to the constructive comments of the Editor-in-Chief and two anonymous referees. This work was supported by Project supported by the National High Technology Joint Research Program of China (Grant No. 2006BAJ07B03) and Project supported by the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20050532029).

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