A mathematical model and a solving procedure for multi-depot vehicle ...

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Aug 6, 2014 - Abstract. One of the most important and yet complex decision-making problems in the area of transportation programming issues is vehicle ...
Int J Adv Manuf Technol (2014) 75:793–802 DOI 10.1007/s00170-014-6141-8

ORIGINAL ARTICLE

A mathematical model and a solving procedure for multi-depot vehicle routing problem with fuzzy time window and heterogeneous vehicle Mehdi Adelzadeh & Vahid Mahdavi Asl & Mehdi Koosha

Received: 4 September 2012 / Accepted: 3 July 2014 / Published online: 6 August 2014 # Springer-Verlag London 2014

Abstract One of the most important and yet complex decision-making problems in the area of transportation programming issues is vehicle routing problem. There are various exact, heuristic, and metaheuristic methods presented for solving different vehicle routing problems. In this manuscript, a mathematical model and a new heuristic solution method are proposed for solving multi-depot vehicle routing problem with time windows and different types of vehicles. In this problem, depots must serve customers between their fuzzy time windows with vehicles having different capacities, velocities, and costs. For this purpose, the mathematical model for multi-depot routing problem is developed to consider the mentioned circumstances. The objectives of this model is travel distance reduction and customers’ service level increscent which leads to cost and service time reduction. For complexity of this problem and much computational time of exact solutions of developed model, a heuristic approach is proposed. This systematic approach has some steps as: customer clustering, routing, vehicle type determination, scheduling, and routes improvement using simulated annealing and customer service level improvement. The efficiency of the proposed method is analyzed by a case study in ISACO Co. Results show that the method is efficient and applicable in industries.

M. Adelzadeh Department of Industrial Engineering, Sharif University, Tehran, Iran V. Mahdavi Asl (*) Department of Industrial Engineering, Yazd University, Yazd, Iran e-mail: [email protected] M. Koosha Faculty of Industrial Engineering, Iran University of Science and Technology, Tehran, Iran

Keywords Multi-depot vehicle routing problem . Fuzzy time window . Service level . Multi-objectives . Simulated annealing

1 Introduction Vehicle routing problem (VRP) is a generic name referring to a class of combinatorial optimization problems in which customers are to be served by a number of vehicles. The vehicles leave the depot to serve customers in the network and after completion of their routes return to the depot. Each customer is described by a certain demand. In other words, VRP involves the design of a set of routes for a fleet of vehicles, starting and ending at a depot and serving a number of customers with predetermined demands. Each customer must be served by one of these routes, and the objective is to minimize the global cost of the set of routes [21]. VRP has extensive variants, including the periodic VRP (PVRP) in which the customers are served in a period of time rather than 1 day, VRP with pickup and delivery (VRPPD) in which the customers may both receive and send products, and VRP with time windows (VRPTW) in which the vehicles must reach the customers before the latest arrival time, while arriving before the earliest arrival time results to waiting, etc. [1]. Many researches have been performed in the area of VRP. For example, Farhang Moghaddam et al. [7] considered VRP with uncertain demands and extended an advanced particle swarm optimization (PSO) to solve it. Also, Goksal et al. [11] developed a hybrid discrete PSO for VRP with simultaneous pickup and delivery. Common point of aforementioned researches is that they are all based on one depot. Thus, they can be regarded as single-depot VRPs. Although the single-depot VRPs have attracted so much attention, they are not suitable for some cases where a company has more than one depot [6]. Hence, this paper focuses on the multi-depot VRP (MDVRP) in

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which more than one depot is considered. Because there are additional depots for storing the products, the decision-makers also have to determine which customers are served by which depots, that is, the grouping problem is prior to the routing and scheduling problems. Obviously, this type of problem is more challenging and sophisticated than the single-depot VRP. Moreover, the multi depot vehicle routing problem (MDVRP) is classified as NP hard, which means that an efficient algorithm to optimize the problem is unavailable. Therefore, solving the problem by an exact algorithm needs large computational time and is computationally intolerable [4]. In practice, transportation often involves routing vehicles according to customer-specific time windows, which are highly relevant to customers’ satisfaction level. The VRP with time windows (VRPTW) and its versions have been studied by academic researchers, and a number of models and algorithms have been proposed [18, 19]. There exist many real-life applications where time windows can be violated. In one propane gas application, wholesale supply tanks have to receive deliveries three times a day in order to meet the peak demand for retail sales. The three deliveries should be placed spaced apart enough to unload a full tanker without overfilling the storage tank. Thus, delivery has to occur approximately in the morning, at midday, and in the afternoon. In this situation, deliveries should arrive at time intervals that prevent being out-of-stock most of the time. If the delivery arrives outside the predetermined time window, the wholesale will have a large chance of running out of stock, and its customer satisfaction level will drop [5]. Another application is found in the flight ticket sales business in China. The development of the Chinese airline business has brought fierce competition between flight ticket sales service companies. To win a greater market share, some service companies have started to provide a free shuttle service to the airport for customers who buy flight tickets from them. Everyday, a flight ticket sales service company needs to make a decision as to when and how to assign buses or cars to pick up which customers in what order, according to the time windows specified by the customers. However, a little tardiness is acceptable to a customer if the customer can still catch his/ her flight. As shown in the above cases, the customers’ time windows are frequently violated in practice [22]. Wang and Chen [24] proposed a genetic algorithm for the simultaneous delivery and pickup problems with time window. To deal with the issues pertaining to the violation of time windows, researchers have proposed the concept of “soft time windows.” In the vehicle routing problem with soft time windows (VRPSTW), a penalty cost is added once a time window is violated, and the penalty cost is often assumed to follow a linear relationship with the amount of variation from the predetermined time window. The VRPSTW successfully models many scenarios where time window constraints do not hold strictly [3]. Sexton and Choi [17] were among the first to

Int J Adv Manuf Technol (2014) 75:793–802

consider soft time windows in the literature of a single-vehicle problem involving pickups and deliveries. They used a Bender’s decomposition approach to solve the problem. Ferland and Fortin [8] applied a heuristic approach for solving a related sliding time window problem in which the time windows of pairs of customers are adjusted in order to achieve a lower cost solution. Balakrishnan [1] used simple route construction heuristics to develop fast solutions to VRPSTW problems. Calvete et al. [3] modeled and solved the VRPSTW using goal programming techniques. Giosa et al. [10] considered multi depot vehicle routing problem with time windows (MDVRPTW) and proposed six heuristic algorithms for assigning costumers to depots and then compared their performance. Gillet and Miller [9] have improved sweeping algorithm for solving MDVRP. This algorithm has two stages including assignment of customers to depots and solving some simple VRP problems. Each VRP is solved by sweeping algorithm. Tillman and Cain [23] solved MDVRP with saving algorithm. The first step of their algorithm is assigning customers to depots and second the one is solving simple VRP problems with saving algorithm. Ho et al. [12] proposed a hybrid genetic algorithm to solve MDVRP. In their paper, the initial population is generated by two ways which consist of randomization and neighborhood search (NS). The solutions of NS are better and needs less computational time than randomization. Sur et al. [20] considered the MDVRP by using genetic algorithm. In this model, the constraint of spent time to deliver the product to each customer is being mentioned. Generally, the objective of the MDVRP is to minimize the total delivery distance or time spent in serving all customers. Shorter delivery time results in higher level of customer satisfaction. The ultimate goal of the MDVRP is to increase the efficiency of the delivery [14]. Balseiro et al. [2] proposed an Ant Colony algorithm hybridized with insertion heuristics for the time-dependent vehicle routing problem with time windows. Hong [13] proposed an improved LNS algorithm for real-time vehicle routing problem with time windows. In a feasible solution to a traditional VRPTW, the service time (delivery time) must fall within each customer’s time window. This kind of time window is usually called a crisp or hard time window. In real-life transportation problems, however, the time windows may be violated out of several practical considerations. Relaxing the constraints of hard time windows can result in better solutions in terms of the number of vehicles, distance, and time. Sometimes, no feasible solution can be found if all time window constraints need to be satisfied; a little earliness or lateness to some customers is acceptable in order to obtain an executable delivery plan. Usually, customers demand a “narrower” time window than they need, while in fact, a little bit deviation from the specified time window is acceptable to them [22]. However, “soft time windows” have one disadvantage. In some cases, violation of time windows does not directly incur any penalty cost,

Int J Adv Manuf Technol (2014) 75:793–802

although this violation can drop the service level of suppliers and, as a result, has destructive effect on the satisfaction levels of customers and lead to profit loss in the long term. As known, the supplier’s service level greatly influences customer satisfaction. Modeling these service level issues seems to be necessary. Hence, Fuzzy set theory which is first proposed by Zadeh [26] and found to be a useful tool to describe subjective opinions that should be applied in VRPTW problems. In the field of network routing, there are two important papers that successfully apply fuzzy theory to the routing problem. Wang and Wen [25] applied fuzzy theory to the Chinese postman problem with time windows. Their paper was the first one to study network routing using fuzzy theory, and they considered two cases in the fuzzy time-constrained Chinese postman problem. In the first one, the arrival time at a node may be fuzzy to describe the degrees of possibility, and in the second one, the upper and lower bounds of time paper written by Zheng and Liu [27] considers a VRP in which the traveling times are assumed to be fuzzy variables and proposes a hybrid solution algorithm. Salhi and Sari [15] proposed a hybrid heuristic algorithm to solve multi-depot vehicle fleet mix problem with multi-levels. This model contains three levels which consist of customer classification, route generation, and improvement. Until now, there is no published paper that applies fuzzy theory to implement the service level in the MDVRPTW with heterogeneous vehicles. This paper analyzes how to use fuzzy concept to describe the service levels associated with time windows in the MDVRP, and with the concept of fuzzy time windows, the MDVRP is formulated as a multi-objective model, and a three-stage algorithm is developed to break down the problem to some common VRPs. After that, a heuristic approach is proposed to provide an initial solution. Moreover, a simulated annealing based algorithm is developed to improve the solutions. The rest of this paper is classified as follows: Relationship between customer satisfaction and time window is analyzed in Section 2. Formulation of MDVRP with fuzzy time window (MDVRPFTW) with heterogeneous vehicles as the multiobjective programming problem is defined in Section 3. A three-stage algorithm is developed in order to decompose MDVRPFTW into three subproblems and solve them sequentially in Section 4. Moreover, a simulated annealing (SA)based method is proposed in this section to solve the proposed model. Section 5 is dedicated to the case study and model’s validation. Finally, the results and future research suggestions are described in Section 6.

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the customer would like to be served. The lower bound ei represents the earliest time at which the service of customer i should start. Similarly, the upper bound Li represents the latest time at which the service of customer i should finish. When a customer gives service in a determined time window, service level of the supplier is good, otherwise it is bad. On the basis of this fact, the quantity of each customer satisfaction can be shown with a binary variable. Customer satisfaction level will be 1 if service time is in a determined time window and 0 otherwise [16]. The service level function of the customer with hard time window is depicted in Fig. 1. Sometimes, hard time windows cannot be satisfied due to economic and operational reasons, but there are bounds and constraints for the acceptable value of earliness or tardiness. EST is the earliest service time that a customer can endure when a service starts earlier than ei, and LST is the latest service time that the customer can endure when a service finishes later than Li. Service times out of time window [ei, Li] have some earliness or tardiness. Amounts of earliness or tardiness have a direct relationship with supplier service level. The response of a customer satisfaction level to a given service time may not be simply “good” or “bad”; instead, it may be between good and bad. For example, the customer might say, “it’s all right” to be served within [EST, ei] or [Li, LST]. In either case, the service level cannot be described by two states. For problems involving personal human feelings, fuzzy set theory is a strong tool [26]. An example is given to describe the relationship of ei, Li, EST, and LST. A factory needs some kind of raw material for its daily production. Everyday, the factory opens at 8:00 and production starts at 10:00. The raw material is shipped from an upstream supplier, and the process of unloading the raw material requires 30 min. The factory specifies its preferred delivery time window to be [8:30, 9:00] because materials delivered within that time window can be directly moved to the workshop without any tardiness. However, the factory is not operating in a just-in-time mode; the delivery can be a little earlier or later than the specified time window. A reasonable combination of EST and LST could be [8:00, 9:30]. If the materials are delivered within [8:00, 8:30], then instead of being moved directly into the workshop, they must be moved Service Level

1

2 Customer satisfaction and fuzzy time window In a traditional VRPTW, a feasible solution must satisfy all time windows. Customer i has the service duration time interval [ei, Li] that indicates the moment of the day at which

0

e

L

Fig. 1 The service level function with hard time window

Time

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Int J Adv Manuf Technol (2014) 75:793–802

in to the warehouse to wait due to limited space in the workshop. This is of course not what the manager of the factory wants to see, but he/she can accept it. If the materials are delivered within [9:00, 9:30], no inventories have to be held; however, this demands that the execution of the production plan have higher accuracy, which reduces the robustness of the production operations in the factory. Since the factory opens at 8:00, deliveries before 8:00 must wait outside the factory; since the production procedure starts at 10:00, delivery after 9:30 is totally unacceptable because of the 30-min unloading process. Although the manager of the factory will be happiest to be served within [8:30, 9:00], he/she will also be reasonably satisfied if served within [8:00, 8:30] or [9:00, 9:30]; however, the consequence is that the customer’s satisfaction will go down, and deliveries made before 8:00 or after 9:30 are not acceptable. Similar scenarios also appear in dial-a-ride problems. Service level of the supplier for each customer can be defined by a fuzzy membership function as Eq. 1.   0 t < ESTi   ei ðt Þ ESTi ≤ t < ei  S i ðt Þ ¼  1 ð1Þ e i ≤ t < Li  Li ðt Þ Li ≤ t < LSTi   0 LSTi ≤ t In this paper, ei(t) and Li(t) are ascending and descending functions, respectively. The function Si(t) denotes the service levels of different service times, and this time window is called a fuzzy time window. In the simplest version of fuzzy time windows, ei(t) and Li(t) have linear relationships as Eqs. 2 and 3. ei ð t Þ ¼

t−ESTi ei −ESTi

ð2Þ

Li ðt Þ ¼

LSTi −t LSTi −Li

ð3Þ

The corresponding service level function can be described by Fig. 2.

3 MDVRP with fuzzy time window and heterogeneous vehicles

Service Level

1

0

EST

LST

Time

Fig. 2 The service level function of fuzzy time window

the operational costs and satisfy customers, the supplier must select routes in which vehicle services its customers with respect to time window and least service level. The mathematical model for MDVRPFTW with multiple objectives will be proposed in this section. In this model, different customers compete to each other for using limited capacity of vehicles. In this model, there are some depots, and demand of customers will be satisfied by one of these depots. Lack of inventory or backorder is not allowed. Demand of each customer is fixed. Total demand of each customer will be supplied by one vehicle. The parameters and the decision variables include the following: i=1,2,…,n: Set of all customers j=1,2,…,m: Set of all depots k=1,2,…,K: Set of all vehicles Ckij: Travel cost between nodes i and j with vehicle k (Ckij =Ckji)(i,j∈setofi∪j) k t ij: Travel time between nodes i and j with vehicle k (tkij =tkji)(i,j∈setofi∪j) di: Demand of customer i Qk: Capacity of vehicle k Tik: Service duration time that vehicle k starts to give service to customer i Zkij: A binary variable will be 1 if vehicle k travels from node i to node j and 0 otherwise (i,j∈setofi∪j) Wik: Service time of vehicle k at customer i MDVRPFTW can be formulated as follows: m X n X k X

Min

i¼1 In MDVRPFTW, each customer has his/her own desirable time window [ei, Li] and an acceptable time window [ESTi, LSTi], but it is not correct for a distributor to have a time table in which some customers have been serviced too early and some other customers have been serviced too late because it will harm the relationship between the customer and the supplier, and as a result, it can be a cause of loss. A practical method for this issue is determining the lowest service level αi(i=1,2,…,n) for each customer in each zone. αi is the least service level that can satisfy customer i. In order to decrease

L

e

n X

Max

C k ij :Z kij

ð4Þ

j¼1 k¼1

S i ðt Þ

i¼1

m X K X j¼1

! Z kij W ki

ð5Þ

k¼1

St: m X k X j¼1

k¼1

Z kij ¼ 1

i ¼ 1; 2; …; n

ð6Þ

Int J Adv Manuf Technol (2014) 75:793–802

X

Z kij −

i∪ j n X

X

Z kij ¼ 0 i≠ j; k ¼ 1; 2; …; K

797

ð7Þ

i∪ j

di

X

Z kij ≤ Qk k ¼ 1; 2; …; K

ð8Þ

depot is Euclidian distance between customer and depots. Customers are assigned to the nearest depot. For example, if we have two depots (dA and dB) in MDVRP problem, customer i (Ci) should be allocated to exactly one depot. The selection is based on the following calculation:

i∪ j

i¼1



&



Z kij W ki þ t kij þ T ki −W kj ≤ 0 i ¼ 1; 2; …; n k ¼ 1; 2; …; K j ¼ 1; 2; …; m

S i ðt Þ

m X

K X

j¼1

k¼1

ð9Þ

&

! Z kij W ki ≥ αi

&

i ¼ 1; 2; …; n

ð10Þ

Z kij ¼ f0; 1g i ¼ 1; 2; …; n ; j ¼ 1; 2; …; m ; k ¼ 1; 2; …; K

ð11Þ W ki ≥ 0 i ¼ 1; 2; …; n ; k ¼ 1; 2; …; K

If distance (Ci, dA)