Goal programming technique for solving fully interval-valued intuitionistic fuzzy multiple objective transportation problems

Abstract

In transportation problems, the cost depends on various irresistible factors like climatic conditions, fuel expenses, etc. Consequently, the transportation problems with crisp parameters fail to handle such situations. However, the construction of the problems under an imprecise environment can significantly tackle these circumstances. The intuitionistic fuzzy number associated with a point is framed by two parameters, namely membership and non-membership degrees. The membership degree determines its acceptance level, while the non-membership measures its non-belongingness (rejection level). However, a person, because of some hesitation, instead of giving a fixed real number to the acceptance and rejection levels, may assign them intervals. This new construction not only generalizes the concept of intuitionistic fuzzy theory but also gives wider scope with more flexibility. In the present article, a balanced transportation problem having all the parameters and variables as interval-valued intuitionistic fuzzy numbers is formulated. Then, a solution methodology based on goal programming approach is proposed. This algorithm not only cares to maximize the acceptance level of the objective functions but simultaneously minimizes the deviational variables attached with each goal. To tackle the interval-valued intuitionistic fuzzy constraints corresponding to each objective function, three membership and non-membership functions, linear, exponential and hyperbolic, are used. Further, a numerical example is solved to demonstrate the computational steps of the algorithm, and a comparison is drawn amidst linear, exponential and hyperbolic membership functions.