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Recently, Yang et al. (Computers & Industrial Engineering, 131: 157-171, 2019) proposed a solution approach to the two-stage fuzzy location problem under the Value-at-Risk (VaR) criterion, which significantly reduced the computational complexity compared with the other approximation treatments. However, in their work, the approach was developed for cases with regular fuzzy variables, which renders it useless when dealing with other types of fuzzy parameters such as discrete fuzzy variables and trapezoidal fuzzy numbers. In this note, for the general case involving arbitrary types of fuzzy parameters, we show that the VaR of a location decision can be determined exactly by solving a corresponding deterministic linear programming. Consequently, a similar approach is developed for the generalized case. Utilizing the extended approach, we show that the discrete case presented by Yang et al. can be solved directly rather than by enumerating all possible scenarios. Furthermore, a numerical example with regular fuzzy variables studied in their work is extended to the continuous but irregular case to illustrate our extension.

A Note on Two-Stage Fuzzy Location Problems Under VaR Criterion With Irregular Fuzzy Variables