Multistage stochastic programming with fuzzy probability distribution

In this paper, we introduce the multistage stochastic program with fuzzy probability distribution. We focus on the case where fuzzy probability distribution is defined by (triangular) fuzzy numbers. We extend Ben Abdelaziz and Masri [Stochastic programming with fuzzy linear partial information on probability distribution, European Journal Operational Research 162 (2005) 619–629] solution strategy, for the two-stage stochastic program with fuzzy probability distribution, to solve the multistage model. The proposed solution strategy is based on two transformation steps. In the first step, the fuzzy transformation step, we propose to use the α-cut defuzzification technique. The level α relates to the DM credibility degree on information sources. This step ends with a certainty equivalent program. In the second step, the stochastic transformation step, we decompose the certainty equivalent program based on a minimax approach. The obtained problem is then solved using a modified version of the nested decomposition method. The modification on the nested decomposition method concerns the way in which we generate optimal constraints. The modified nested decomposition algorithm may be used to solve the multistage problem with interval probability distribution.