The open vehicle routing problem with fuzzy demands

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Abstract

According to the open vehicle routing problem (OVRP), a vehicle is not required to return to the distribution depot after servicing the last customer on its route. In this paper, the open vehicle routing problem with fuzzy demands (OVRPFD) is considered. A fuzzy chance-constrained program model is designed based on fuzzy credibility theory. Stochastic simulation and an improved differential evolution algorithm are integrated so as to use a hybrid intelligent algorithm to solve the OVRPFD model. The influence of the decision-maker’s preference on the final outcome of the problem is analyzed using stochastic simulation, and the range of possible preferences is calculated.

Introduction

The open vehicle routing problem (OVRP) analyzes efficient routes with minimum total cost for a fleet of vehicles that serve some commodity to a given number of customers. Each customer is visited exactly once by one vehicle, while vehicle activity is bounded by capacity constraints, duration constraints and time constraints. Either each route is a sequence of customers that starts at the depot and finishes at one of the customers to whom goods are delivered, or each route is a sequence of customers that begins at a certain customer and ends at the distribution depot, where goods are gathered. The OVRP differs from the well-known vehicle routing problem (VRP) (Dantzing & Ramser, 1959) in that the vehicles do not necessarily return to their original locations after delivering goods to customers; if they do, they must visit the same customers in the reverse order.

The OVRP is encountered in practice in many contexts, such as home delivery of packages and newspapers. Contractors who are not employees of the delivery company use their own vehicles and do not return to the depot; the “missing” node on a route can be driver’s home or a parking lot where the vehicle stays overnight. The major difference in theory between the OVRP and the VRP is that the routes in the OVRP consist of Hamiltonian paths originating at the depot and terminating at one of the customers, while the routes in the VRP are Hamiltonian cycles. In other words, the best Hamiltonian path is NP-hard, since the Hamiltonian path problem is equivalent to the traveling salesperson problem, which is known to be NP-hard (Reinelt, 1991). The best Hamiltonian path problem with a fixed source node must be solved for each vehicle in the OVRP, and OVRP solutions involve finding the best Hamiltonian path for each set of customers assigned to a vehicle. Consequently, the OVRP is also a NP-hard problem.

The OVRP has received sparse attention in the literature compared to the VRP. While the earliest description of OVRP offered by Schrage (1981) appeared in the literature over 20 years ago, OVRPs have just recently attracted the attention of practitioners and researchers.

From the early 1980s to the late 1990s, the OVRP received very little attention in the operations research literature. However, since 2000, several researchers have used various heuristics and meta-heuristics to solve the OVRP with some success. Brandão, 2004, Fu et al., 2005 implemented a tabu search (TS) heuristic to solve the OVRP with constraints on vehicle capacity and maximum route length. Tarantilis and Kiranoudis (2002) considered a real-life fresh meat distribution problem, formulating it as a multi-depot OVRP; the problem was solved using a metaheuristic method called the list-based threshold-accepting (LBTA) algorithm. Tarantilis, Ioannou, Kiranoudis, and Prastacos (2005) solved the OVRP by adopting the LBTA algorithm they proposed for the solution of the multi-depot OVRP (Tarantilis & Kiranoudis, 2002). Pisinger and Ropke (2007) presented an adaptive large-neighborhood search heuristic to solve an OVRP. Li, Golden, and Wasil (2007) proposed a record-to-record travel heuristic and a deterministic variant of simulated annealing to solve the OVRP.

However, traditional studies of the OVRP as well as the VRP have assumed that all information is deterministic, including customer information, vehicles information, roads information, and dispatcher information. As such, the proposed algorithms were only used to solve deterministic circumstances. That is, all the above-mentioned papers assumed that the demands of all customers visited on its route by any vehicle were deterministic.

Actually, in some new systems, it is hard to describe the parameters of the VRP as deterministic, because there exists much uncertainty in data such as customer demands, traveling time and the set of customers to be visited. We call these problems non-deterministic VRPs. Given this aspect of VRPs, a consideration of stochastic vehicle routing problems (SVRP) and fuzzy vehicle routing problems (FVRP) may be useful. SVRPs arise whenever some elements of a given problem are random. Common examples include stochastic demands and stochastic travel times. Sometimes, the set of customers to be visited is not known with certainty. In this case, each customer has a certain probability of being visited. Researchers have developed many models and algorithms for SVRPs (Bertsimas, 1992, Dror et al., 1989, Gendreau et al., 1996, Liu and Lai, 2002). Alternatively, FVRPs arise whenever some elements of a given problem involve uncertainty, subjectivity, ambiguity and vagueness (Teodorovic & Pavkovic, 1996). For instance, in OVRPFDs, information about each customer’s demand is often not adequately precise. For example, based on experience, a customer’s demand can be approximated as “around 50 units,” “between 20 and 60 units,” and so on. Generally, we can use fuzzy variables to deal with these uncertain parameters. In fact, Teodorovic and Pavkovic (1996)used fuzzy variables with regard to VRPs, and Cheng and Gen (1995) used a genetic algorithm to solve VRPs with fuzzy due time. Finally, moreover, Lai, Liu, and Peng (2003) modeled VRPs with fuzzy travel times using fuzzy programming with a possibility measure; they then adopted a genetic algorithm to solve the model. Zheng and Liu (2006)researched VRPs with fuzzy travel times and presented a chance-constrained program (CCP) model with a credibility measure; they then integrated fuzzy simulation and a genetic algorithm (GA) to design a hybrid intelligent algorithm to solve the model. Note that FVRPs differ from their deterministic counterparts in several fundamental respects. The concept of a solution is different, as several fundamental properties of deterministic VRPs no longer hold in FVRPs. Thus, solution methodologies are considerably more intricate. To the best of our knowledge, no research has considered non-deterministic information in an OVRP framework. In this paper, we consider the situation in which the demands of customers are fuzzy variables in OVRP. We model the OVRPFD by using fuzzy programming with a credibility measure and adopt a hybrid differential algorithm to solve the model.

Differential evolution (DE) is a novel evolutionary technique that was originally developed for continuous optimization (Storn, 1996, Storn and Price, 1995). DE is a population-based, global evolutionary algorithm, which uses both a simple operator to create new candidate solutions and a one-to-one competition scheme to select new candidates greedily. Due to its simple structure, easy implementation, quick convergence, and robustness, DE has been one of the best evolutionary algorithms for solving continuous problems in a variety of fields. Nevertheless, due to the fact that DE is continuous, the research on DE for combinatorial optimization is very limited. Obviously, it is difficult to apply DE to problems other than the continuous optimization on which its inventors originally focused. Recently, some researchers have used DE to address machine layout problems (Nearchou, 2006, Storn, 1996) and to solve manufacturing problems with mixed-integer, discrete variables (Lampinen & Zelinka, 1999). But to the best of our knowledge, there is no work on VRPs that uses differential evolution. In this paper, we will first adopt the differential evolution algorithm to solve routing problems; the proposed differential evolution algorithm also can solve other deterministic aspects of the OVRPFD insofar as it considers more complicated constraints.

This paper is organized as follows: in Section 2, we explain some basic concepts regarding fuzzy theory. In Section 3, we introduce open vehicle routing problems with fuzzy demands and present a chance-constrained program (CCP) model with which we measure fuzzy events with credibility. Then in Section 4, we integrate both stochastic simulation and a differential evolution algorithm to design a hybrid intelligent algorithm to solve the CCP model. In Section 5, we discuss two experiments in order to demonstrate the effectiveness of the hybrid intelligent algorithm. In the final section, we summarize the contributions of this paper.

Section snippets

Fuzzy credibility measure theory

The concept of a fuzzy set was first discussed by Zadeh (1965) with regard to membership functions. Since then, the concept of fuzzy sets has been well developed and applied in a wide variety of real problems. In order to measure a fuzzy event, the term fuzzy variable was introduced by Kaufman (1975), while possibility measure theory of fuzzy variable was proposed by Zadeh (1978). Liu (2004) recently developed credibility theory.

In this section, we introduce briefly some basic concepts in fuzzy

The fuzzy chance-constrained program model of the OVRPFD

We assume that: (a) each vehicle has a container with a physical limitation so that the total loading of each vehicle can not exceed its capacity C; (b) each vehicle has maximum distance constraints so that the total distance traveled by each vehicle can not exceed L; (c) a vehicle will be assigned for only one route on which there may be more than one customer; (d) a customer will be visited by one and only one vehicle; (e) each route begins at the distribution depot (0) and ends at the last

A hybrid intelligent algorithm

In this paper, we design a hybrid intelligent algorithm integrating stochastic simulation and a differential evolution algorithm to solve the above fuzzy chance-constrained problem. For a given value of dispatcher preference index Cr, we adopt a triangular fuzzy number within vehicle capacity to represent demand at each customer; the real value of demand at a given customer is a real number within fuzzy boundaries that is made known when the vehicle reaches the customer. First, we apply

Numerical experiments

Now we will offer some examples to illustrate the models that we have just discussed and show how the hybrid intelligent algorithm works. Two types of experimental conditions are created based on the size of the problem (in this case, the number of customers). We assume that there are 30 customers and one depot for the small-size problem and 100 customers and one depot for the large-size problem. In each experiment, the coordinates of all customers and the depot are randomly generated in [100 × 

Conclusion

This paper contributed to understanding vehicle routing problem with fuzzy demands in the following respects: (a) a chance-constrained model of OVRPFD was proposed based on credibility theory; (b) stochastic simulation and differential evolution algorithm were integrated to design a hybrid intelligent algorithm used to solve this problem by focusing on minimizing total distance; (c) the dispatcher preference index greatly influenced the length of planned routes and the additional distances

Acknowledgment

This research was supported by the Specialized Research Fund for Doctoral Program of Higher Education of China under Grant 20050532029.

References (29)

  • L.A. Zadeh

    Fuzzy sets

    Information and Control

    (1965)
  • Y. Zheng et al.

    Fuzzy vehicle routing model with credibility measure and its hybrid intelligent algorithm

    Applied Mathematics and Computation

    (2006)
  • D.J. Bertsimas

    A vehicle routing problem with stochastic demand

    Operational Research

    (1992)
  • R. Cheng et al.

    Vehicle routing problem with fuzzy due-time using genetic algorithm

    Japanese Journal of Fuzzy Theory and Systems

    (1995)
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