Integration of fuzzy optimization and stochastic programming in multi-commodity network flow problems

This study analyzes the problem of transporting vital commodities and emergency personnel during the response stage of a disaster relief operation by integrating fuzzy linear optimization and stochastic programming. This transportation system is treated as a probabilistic, multi-commodity, multi-modal network flow problem and first a fuzzy model that allows the satisfaction of demand requirements through different supply nodes over a fuzzy network is developed. Since the available information in case of disasters is generally vague and imprecise about the values of model parameters, fuzzy linear optimization is employed to deal with this uncertainty and to generate different scenarios depending upon the preferences of the decision maker. Then these scenarios are handled within the framework of stochastic programming by defining a scenario as a joint realization of the uncertain parameters.
Multi-commodity network flow problem is decomposed into uncapacitated single-commodity network flow problems by using Lagrangian relaxation. An exact algorithm is developed for each subproblem, and a sequential structure is designed to maintain consistency and feasibility between the solutions of these subproblems.