Mean and CV reduction methods on Gaussian type-2 fuzzy set and its application to a multilevel profit transportation problem in a two-stage supply chain network

The transportation problem (TP) is an important supply chain optimization problem in the traffic engineering. This paper maximizes the total profit over a three-tiered distribution system consisting of plants, distribution centers (DCs) and customers. Plants produce multiple products that are shipped to DCs. If a DC is used, then a fixed cost (FC) is charged. The customers are supplied by a single DC. To characterize the uncertainty in the practical decision environment, this paper considers the unit cost of TP, FC, the supply capacities and demands as Gaussian type-2 fuzzy variables. To give a modeling framework for optimization problems with multifold uncertainty, different reduction methods were proposed to transform a Gaussian type-2 fuzzy variable into a type-1 fuzzy variable by mean reduction method and CV reduction method. Then, the TP was reformulated as a chance-constrained programming model enlightened by the credibility optimization methods. The deterministic models are then solved using two different soft computing techniques—generalized reduced gradient and modified particle swarm optimization, where the position of each particle is adjusted according to its own experience and that of its neighbors. The numerical experiments illustrated the application and effectiveness of the proposed approaches.