The k- product and t-state uncapacitated facility location problem with fuzzy random costs

In this paper, the -product and t-state uncapacitated facility location problem is investigated. To be more precise, it is assume that each customer can request  different products in a -state network. First, the mathematical formulation for the -product and t-state uncapacitated facility location problem with certain costs is proposed. Also, it is shown that this paoblem is NP-hard. Since in most real-world problems the input of data are often ambiguous and uncertain, we study the -product and -state uncapacitated facility location problem in which the facility set-up costs and customer service costs are fuzzy random variables. Using three criteria, ‎probability-possibility‎, ‎probability-necessity‎ and ‎probability-credibility, the -product and‌ -state uncapacitated facility location problem is formulated as a quadratic programming. Finally, a practical example is given to illustrate the efficiency of the proposed approaches.