Mathematical Problems in Engineering

Volume 2016, Article ID 9071394, 10 pages

http://dx.doi.org/10.1155/2016/9071394

## Vehicle Coordinated Strategy for Vehicle Routing Problem with Fuzzy Demands

^{1}School of Management, Hunan University of Commerce, Changsha 410205, China^{2}Mobile E-Business Collaborative Innovation Center of Hunan Province, Hunan University of Commerce, Changsha 410205, China^{3}Key Laboratory of Hunan Province for Mobile Business Intelligence, Hunan University of Commerce, Changsha 410205, China^{4}Institute of Big Data and Internet Innovation, Hunan University of Commerce, Changsha 410205, China^{5}School of Business Administration, Southwestern University of Finance and Economics, Chengdu 611130, China

Received 24 December 2015; Accepted 8 August 2016

Academic Editor: Thomas Hanne

Copyright © 2016 Chang-shi Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The vehicle routing problem with fuzzy demands (VRPFD) is considered. A fuzzy reasoning constrained program model is formulated for VRPFD, and a hybrid ant colony algorithm is proposed to minimize total travel distance. Specifically, the two-vehicle-paired loop coordinated strategy is presented to reduce the additional distance, unloading times, and waste capacity caused by the service failure due to the uncertain demands. Finally, numerical examples are presented to demonstrate the effectiveness of the proposed approaches.

#### 1. Introduction

The vehicle routing problem (VRP) was first proposed by Dantzig and Ramser [1]. The VRP has since been the topic of many operational studies. VRP consists of designing efficient routes to serve a number of nodes with a fleet of vehicles. Each node is visited exactly once by one vehicle. The activity of the vehicle is bounded by certain constraints. Each vehicle starts at the depot and returns to the same depot after completing its task. Most VRP studies employ the vehicle uncoordinated strategy; that is, there is no coordination between the vehicles, and each vehicle completes only its own task. There are many significant VRP results based on this case, including those of Clarke and Wright [2], Solomon [3], Laporte [4], Figliozzi [5], Sprenger and Mönch [6], Pillac et al. [7], Kou et al. [8], and Kou et al. [9].

The vehicle routing problem with fuzzy demands (VRPFD) is an extension of the VRP; that is, the demand of each node is uncertain, subjective, ambiguous, and/or vague [10]. The VRPFD is widely employed for many real applications due to their numerous uncertainties, including garbage collection systems, product recall systems, and raw milk collection systems (collecting raw milk from dairy farmers for milk powder production enterprises). There are also several classical studies that refer to the VRPFD, such as Bertsimas [11], Cao and Lai [12], Kuo et al. [13], Kou and Lin [14], Kou et al. [15], Allahviranloo et al. [16], and Hu et al. [17]. The VRPFD typically assumes that the real value of a node’s demand is known when the vehicle reaches the node, whereas the vehicle’s route is planned in advance. After serving nodes, the vehicle might not be able to service the node once it arrives due to insufficient capacity. In such situations, if the vehicle uncoordinated strategy is employed, the vehicle must return to the depot and unload what it has picked up thus far and then return to the node where it had a “service failure” and continue to serve the remaining nodes. Thus, “additional distance” and “additional unloading times” are introduced due to the “service failure.” However, there are also vehicles with surplus capacity after completing their own tasks, introducing “waste capacity.” All of these cases result in increasing logistics cost. To the authors’ knowledge, few researchers have addressed the problem of minimizing the “additional distance” and “waste capacity,” let alone “additional unloading times,” in the VRPFD.

In this paper, vehicle coordinated strategy (VCS) is defined such that each vehicle finishes its own assigned task first; then, if there is a vehicle with surplus capacity, the vehicle must help any vehicle that has not completed its own task according to the specified vehicle coordination rules [18, 19]. Only a few VRP studies have considered VCS. Shang and Cuff [20] considered a multiobjective vehicle routing heuristic for a pickup and delivery problem. They assumed that the fleet size is not predetermined and that customers are allowed to transfer between vehicles. These transfers can occur at any location and between any two vehicles. Yang et al. [21] proposed a mixed-integer programming formulation for the offline version of the real-time VRP and compared five rolling horizon strategies for the real-time version. To some extent their work is relevant to vehicle coordination. Liu et al. [18] proposed a simple general VCS for the VRP with deterministic demands. Lin [19] designed a VCS with single or multiple vehicle uses. The VCS is defined as allowing vehicles to travel to transfer items to another vehicle returning to the depot, provided that no time window constraints are violated. Sprenger and Mönch [6] studied a methodology to solve a cooperative transportation planning problem motivated by a real-world scenario found in the German food industry. Several manufacturers with the same customers but complementary food products share their vehicle fleets to deliver to their customers. They designed a heuristic to solve the problem. The results of extensive simulation experiments demonstrated that the cooperative setting outperforms the noncooperative one. Hu et al. [22] presented a feasible routing solution to accommodate the changes (such as customer’s demand changes, delivery time window changes, disabled roads induced by traffic accidents or traffic jams, and vehicle breakdowns) and to minimize the negative impacts on the existing distribution process in real-time VRP. They handle these disruptions by readjusting vehicle routes in real time to improve vehicles’ efficiency and enhance service quality. To a certain extent their work is relevant to vehicle coordination. However, all investigations assumed that the customers were uniformly distributed in certain regions and that the demands were deterministic. To the authors’ knowledge, few studies have employed the VCS in the VRPFD.

Thus, in this paper, the fuzzy reasoning constrained program model for VRPFD is formulated, and the hybrid ant colony algorithm is designed to minimize total travel distance. In particular, the two-vehicle-paired loop coordinated strategy (TVPLCS) is presented to reduce the “additional distance,” “additional unloading times,” and “waste capacity” caused by the service failure due to the uncertain demands. Finally, numerical examples are presented to demonstrate the effectiveness of the proposed approaches.

The remainder of this paper is organized as follows. In Section 2, the fuzzy reasoning constrained program model for VRPFD is formulated. In Section 3, we design a hybrid ant colony algorithm for VRPFD. In Section 4, in particular, we present the TVPLCS to minimize the “additional distance,” “additional unloading times,” and “waste capacity”. In Section 5, we give numerical examples to demonstrate the effectiveness of the proposed approaches. Finally, we summarize the contributions of this paper.

#### 2. VRPFD Assumptions and Model

##### 2.1. VRPFD Assumptions

In this paper, the VRPFD assumes that there is only one depot denoted by , and there are nodes with fuzzy demands served by vehicles. The locations of the depot and nodes are known. The fuzzy demand of each node is uncertain and only characterized by a triangular fuzzy number , . is the minimum of the demand of node , is the maximum of the demand of node , and is the most likely value. The distance between nodes and node is known. Each node is served exactly once by one vehicle. For simplicity, the capacity of each vehicle is the same, and the activity of the vehicle is only bounded by capacity constraints. Each vehicle starts at the depot and returns to the same depot after completing its task. The objective is to design a set of vehicle routes that minimizes the total logistics costs.

##### 2.2. Deciding whether the Vehicle Serves the Next Node or Returns to the Depot

When the demand of each node is deterministic, it is easy for us to decide whether the vehicle is able to serve the next node after serving nodes. However, while the demand at each node is uncertain and only characterized by a triangular fuzzy number , it is difficult for us to decide whether the vehicle should serve the next node or return to the depot. We only know 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 this problem by triangular fuzzy number theory proposed by Liu [23], which described as follows.

The membership function of triangular fuzzy number is defined as

Let be the occurrence possibility of event . For triangular fuzzy number and , is defined as

Now, we can deduce that the occupied capacity of the vehicle which had served nodes is

Also, the capacity of each vehicle can be presented as a triangular fuzzy number . So, the remaining capacity of the vehicle is

Thus, the possibility , which means that the demand of node is less than the remaining capacity, is

Let be the decision maker’s preference. A large value of indicates that the decision maker is risk averse, and the decision maker aims to ensure service. In this case, . The possibility of “service success” is relatively high. In contrast, a small value of means that the decision maker has an insatiable appetite for risk and tries to serve more nodes with each vehicle. In that case, . The possibility of “service success” is relatively low.

Now, after serving nodes, we can make a decision whether the vehicle should serve the next node or return to the depot . The decision was made as follows.

If , the vehicle should serve the next node ; else, the vehicle should return to the depot .

##### 2.3. VRPFD Model

The notations used in the formulation of the VRPFD are described as follows: : node index ( stands for the depot). : number of nodes. : set of nodes. : nonempty proper subset of the set . : fuzzy demand of each node , . : distance between node and node . : vehicle index. : number of vehicles. : set of vehicles. : capacity of the vehicle. : decision maker’s preference, .

The decision variables used in the formulation of the VRPFD are described as follows: : if node is served by vehicle , ; otherwise, . : if vehicle moves from node to node , ; otherwise, .

Thus, the fuzzy reasoning constrained program model of the VRPFD is mathematically formulated as follows:

The object of the proposed VRPFD is to minimize the total distance. Constraint (7) ensures that all nodes are served within the vehicle’s capacity at the values of the decision maker’s preference. Constraint (8) ensures that each node is visited by one vehicle. Constraints (9) and (10) define the relationships between and , respectively. Constraint (11) guarantees that a vehicle must enter and leave each node exactly once. Constraint (12) ensures that at most vehicles are used. Constraint (13) ensures that vehicle routes start from the depot and terminate at the same depot. Constraint (14) represents the subtour elimination constraint where stands for the cardinality of set .

#### 3. Hybrid Ant Colony Algorithm for VRPFD

The ant colony algorithm (ACA) is one of the most popular swarm-inspired methods in the field of computational intelligence. The first ACA was developed by Clolrni et al. [24]. It was successfully applied to the traveling salesman problem. The first ant system for the VRP was proposed by Bulleneimer et al. [25]. Doerner et al. [26] further improved this ant colony system using a savings-based heuristic. Recently, ACA has been applied to the VRP with different constraints, for example, Ellabib et al. [27], Gajpal and Abad [28], Yu et al. [29], and Fleming et al. [30]. By looking at success of above hybridised ant colony algorithms on VRP, we decided to develop hybrid ant colony algorithm (HACA) for VRPFD too.

Let be the set of all candidate nodes in the dataset, let be the set of nodes yet to be served by ant , and let be the set of nodes already served by ant .

##### 3.1. Transfer Probabilities

The probability that ant chooses to serve node having served node is given by where is the pheromone density of edge ; is the visibility of edge ; is the relative influence of the pheromone trails; and is the relative influence of the visibility.

##### 3.2. Pheromone Updating

###### 3.2.1. Local Pheromone Updating Rule

*Definition 1 (ant attraction). *The ant attraction of edge is the ratio of number of ants that have travelled edge to the number of ants that have visited node .

Each ant leaves constant quantity of pheromone on the edge it travels, and larger ant attraction of edge results in greater amount of ant travel on edge . Thus, more frequent local pheromone updating will result in a larger pheromone quantity between all edges. The global searching of the ACO algorithm represents a handicap. To address this problem a local pheromone updating rule has been designed.

Let be the number of ants that visited node before arrival of ant at node , and let be the number of ants that travelled edge before ant travelled edge . The local pheromone update quantity of edge caused by ant can be calculated as

Let be the local pheromone quantity of edge before updating; let be the local pheromone quantity of edge after updating; and let be the pheromone volatilization coefficient. Then, the local pheromone updating rule can be defined as

###### 3.2.2. Global Pheromone Updating Rule

If ant has already served all nodes, the global pheromone updating rule is employed. It is defined as

##### 3.3. The Steps of HACA

The steps of the proposed HACA are depicted below.

*Step 1 (algorithm initialization). *(1) Set the values of the current iteration number , the maximum iteration number , the capacity value , the ant number , and the decision maker’s preference . (2) Set all ants at the central depot , and let each ant start from the depot. (3) Let be the initial occupied load of ant , and the remaining capacity .

*Step 2 (route construction). *(1) Calculate the transfer probability . (2) Select node according to the sequence of arranged in decreasing order. (3) If , , ant must move to node from the current node , and the current node of ant is changed to be , , , and the occupied load ; otherwise, ant should return to the depot, , and move to the next node . (4) Repeat this selection until .

*Step 3 (pheromone updating). *If , that is, ant has already served all nodes, the global pheromone updating rule is employed; otherwise, the local pheromone updating rule is employed.

*Step 4 (judgment). *If the total number of searching ants is smaller than , return to step 2; otherwise, find the best solution by the path set obtained with .

*Step 5 (the 2-opt local search). *(1) The obtained route is broken at random into three segments. (2) The middle segment must not contain the depot. (3) The route is then reconstructed by reversing the middle segment. (4) The route is updated whenever there is an improvement. (5) The process is repeated until there is no further improvement in the solution [28]. (6) If the new solution is better than the current solution, the new solution will replace the current solution.

*Step 6 (termination rule of the algorithm). *If , , return to step 2, and repeat the above steps; otherwise, terminate the HACA.

#### 4. Two-Vehicle-Paired Loop Coordinated Strategy

As mentioned above, the VRPFD typically assumes that the “actual” value of a node’s demand is known when the vehicle reaches the node, and the vehicle route is planned in advance. After serving nodes, a vehicle might not be able to service the node once it arrives due to its insufficient capacity. In such situations, if the vehicle uncoordinated strategy is employed, the vehicle must return to the depot, unload what it has picked up thus far, return to the node where it had a “service failure,” and continue to service the remaining nodes (e.g., in raw milk collection systems). Thus, “additional distance” and “additional unloading times” are introduced due to the “service failure” (Figure 1). In contrast, there are also vehicles with surplus capacity after completing their own tasks; thus, “waste capacity” is created. All of these cases increase the logistics costs. To the authors’ knowledge, few studies have considered the problem of how to effectively minimize the “additional distance” and “waste capacity,” let alone reduce “additional unloading times,” in the VRPFD.