Computing, Artificial Intelligence and Information ManagementFuzzy facility location-allocation problem under the Hurwicz criterion
Introduction
The facility location-allocation (FLA) problem is widely used in practical life, such as building an emergency service systems and constructing a telecommunication net works, etc. In recent years, people pay significant attention to it especially in supply chain management because of its effect on key operational performance measures, e.g., lead time, inventory, responsiveness, etc.
FLA problem was initially studied by Cooper [5] in 1963, and then Hakimi [12], [13] applied it in network design as a powerful tool. In 1982, Murtagh and Niwattisyawong [34] proposed the capacitated FLA problem, which is considered as one of the most important researches in this field, specially focusing on facilities which have capacity constraints. Meanwhile, considerable research has been carried out in the field of FLA theory [1], [14]. It is proved that the FLA problem is strongly NP-hard [36], and thus a large amount of solution approaches for different models have been proposed in the past decades [18], [33]. A series of heuristic algorithms, including hybrid algorithms, have also been developed to solve complicated FLA problems. For instance, Ernst and Krishnamoorthy [10] combined the simulated annealing and random descent method, and Gong et al. [11] utilized the Lagrange relaxation method and genetic algorithm to approximate the optimal solutions.
In the real world, it is usually very hard to present the precise demands of customers and thus they are estimated from historical data. Furthermore, researchers tried to describe FLA problem under stochastic environment. Logendran and Terrell [19] considered the uncapacitated FLA problem in the presence of price sensitive stochastic demands to maximize the expected net profits. Then Zhou [38] proposed the expected cost model, α-cost model and probability maximum model for uncapacitated FLA problem with stochastic demands. However, in many cases, facilities have a constraining upper limit on the amount of demand. For example, in telecommunications and power transmission networks, there are hard constraints on the capacities of concentrators, switches, and transformers. Thus the expected cost model, α-cost model and probability maximum model were given for capacitated FLA problem with stochastic demands [39].
Although stochastic models can cater for a variety of cases, they are not sufficient to describe many other situations, where the probability distribution of customers demands may be unknown or partially known. For example, we want to build some new water companies in new regions to service some new customers whose demands can neither be given precisely nor from history data. But those demands can be described by the natural language such as large, little or general, etc. In these cases, fuzzy set theory may do better in dealing with ambiguous information. Fuzzy set theory was initialized by Zadeh [42] and has been widely applied in many real problems. It has been proved to be a useful tool to solve problems with uncertainty. In the past decades many researchers have introduced fuzzy theory into FLA problem. Various facility location problems by fuzzy logic methods were discussed [2], [4], [8]. Apart from the above-mentioned papers, the capacitated FLA problem with fuzzy demands of customers were formulated by Zhou and Liu [40], [41].
The most familiar criteria to us is the optimistic criterion and pessimistic criterion which are widely used in uncertain environment. Several other criteria are proposed, and a list of properties of rationality and consistency were set forth as a set of axioms to be obeyed by a rational criterion [32], [6]. The most well-known criteria is the Hurwicz criterion, suggested by Hurwicz [15], [16] in 1951, which attempts to find a middle ground between the extremes posed by the optimistic and pessimistic criteria. Instead of assuming total optimism or pessimism, the Hurwicz criterion incorporates a measure of both by assigning a certain percentage weight λ to optimism and to pessimism, . Many researchers use the optimistic criterion or pessimistic criterion to model the FLA problem, which are both extreme cases. In order to give a balance between these two models, we employ the Hurwicz criterion to model the FLA problem.
This paper is organized as follows. In Section 2, we give some basic concepts and conclusions on the fuzzy variables. In Section 3, the problem and symbols are introduced. A new model named α-cost model under the Hurwicz criterion is proposed in Section 4. In order to solve this fuzzy model, we integrate the simplex algorithm, fuzzy simulations and genetic algorithm to design a powerful hybrid intelligent algorithm in Section 5. Finally, Section 6 provides some numerical examples to illustrate the performance and the effectiveness of the proposed algorithm.
Section snippets
Fuzzy variable
In this section, we will state some basic concepts and results on fuzzy variables. These results are crucial for the remainder of this paper. An interested reader may consult Liu [25], [27] where important properties of fuzzy variables are recorded.
Let Θ be a nonempty set, and the power set of Θ. For any , Liu and Liu [31] presented a credibility measure Cr{A} to express the chance that fuzzy event A occurs. Li and Liu [20] proved that a set function Cr{·} is a credibility measure if
Facility location-allocation problem
The capacitated continuous FLA problem is to find the locations of n facilities in continuous space in order to serve customers at m fixed points as well as the allocation of each customer to the facilities so that total transportation costs are minimized. In order to model the capacitated FLA problem, firstly we make some assumptions:
- 1.
Each facility has a limited capacity. Thus we need to select locations and decide the amount from each facility i to each customer j.
- 2.
The path between any customer
Fuzzy FLA problem
This paper considers fuzzy FLA problem in which we use to denote the fuzzy demand of customer j, . Similar to Section 2, we can give the feasible allocation set byThen for each θ, we can give the transportation cost with the best allocation z,If is an empty set for some , we can defineAs the demands of
Hybrid intelligent algorithm
The α-cost model under the Hurwicz criterion is uncertain programming which is difficult to solve by traditional methods. A good way is to design some hybrid intelligent algorithms for solving them [21], [22], [23], [24], [28], [29], [30]. In this paper, we integrate the simplex algorithm, fuzzy simulations and genetic algorithms to produce a hybrid intelligent algorithm for solving α-cost model under the Hurwicz criterion.
Numerical examples
In order to illustrate the effectiveness of the hybrid intelligent algorithm, we give some numerical examples. Example 1 Consider a company that wishes to locate four new facilities within the square region . Assume that there are 20 customers whose demands ξi are trapezoidal fuzzy variables. The location and the fuzzy demand ξi of the customer i, are given in Table 1. The capacities si of the four facilities are 40, 50, 60 and 70, respectively. In practical problems, the
Conclusion
In this paper, we have contributed to the research area of the FLA problem in the following three aspects.
- (i)
The α-cost minimization model under the Hurwicz criterion was proposed. This model can deal with more practical problems by varying the value λ which reflects the balance between the optimistic value and the pessimistic value of the total transportation cost.
- (ii)
To solve the models efficiently, we integrated the simplex algorithm, fuzzy simulations and genetic algorithm to produce a hybrid
Acknowledgment
This work was supported by National Natural Science Foundation of China (No. 60425309).
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