Multi-objective capacitated transportation problem with mixed constraint: a case study of certain and uncertain environment

TLDR: A new model of multi-objective capacitated transportation problem (MOCTP) with mixed constraints is formulated and fuzzy goal programming approach has been applied for solving this resultant MOCTP model for obtaining the optimum order quantity.

Abstract: In this paper, we have formulated a new model of multi-objective capacitated transportation problem (MOCTP) with mixed constraints. In this model, some objective functions are linear and some are fractional and are of conflicting in nature with each other. The main objective of this paper is to decide the optimum order of the product quantity which is to be shipped from source to the destination subject to the capacitated restriction on each route. Here the two situations have been discussed for the MOCTP model. In the first situation, we have considered that all the input information of the MOCTP model is exactly known and therefore a fuzzy goal programming approach have been directly used for obtaining the optimum order quantity of the product. While in the second situation the input information of the MOCTP model are uncertain in nature and this uncertainty have been studied and handled by the suitable approaches like trapezoidal fuzzy numbers, multi-choices, and probabilistic random variables respectively. Due to the presence of all these uncertainties and conflicting natures of objectives functions, we cannot solve this MOCTP directly. Therefore firstly we converted all these uncertainties into deterministic forms by using the appropriate transformation techniques. For this, the vagueness in MOCTP defined by trapezoidal fuzzy numbers has been converted into its crisp form by using the ranking function approach. Multichoices in input information parameters have been converted into its exact form by the binary variable transformation technique. Randomness in input information is defined by the Pareto probability distribution, and for conversion into deterministic form chance constrained programming has been used. After doing all these transformations, we have applied fuzzy goal programming approach for solving this resultant MOCTP model for obtaining the optimum order quantity. A case study has been done to illustrate the computational procedure.