Entropy based solid transportation problems with discounted unit costs under fuzzy random environment

Abstract

This paper investigates some multi-objective fixed charge solid transportation problems (FCSTPs) under fuzzy-random environment with fuzzy-random fixed charges and transportation times. Here objectives are total transportation cost and time which are minimized. Unit transportation costs are crisp and three types of discount i.e. All Unit Discount (AUD), Incremental Quantity Discount (IQD), IQD within AUD are applied upon these costs. In the first model (Model-I) supplies, demands and capacities of conveyances are assumed to be fuzzy-random in nature. These quantities are fuzzy in the second model (Model-II). Imprecise objectives of the models are transformed into equivalent crisp ones using random expectation and fuzzy possibility/necessity measures on fuzzy event. Imprecise constraints are reduced to equivalent crisp constraints using two different fuzzy-random chance constraint methods. Similarly constraints of Model-II are reduced to equivalent crisp constraints with the use of possibility and necessity measures on fuzzy events. Models are also formulated with and without entropy function. Finally transformed constrained multi-objective deterministic problems are solved using a multi-objective genetic algorithm (MOGA) based on arithmetic crossover and boundary mutation. Moreover, Model-IA1 is converted to a single objective non-linear programming (SONLP) problem following fuzzy non-linear programming (FNLP) technique and the reduced Model-IA1S is solved using generalized reduced gradient (GRG) method (using LINGO software). Numerical examples are used for illustration and comparison of the models.