Using necessarily weak efficient solutions for solving a biobjective transportation problem with fuzzy objective functions coefficients

This paper considers a biobjective transportation problem with various fuzzy objective functions coefficients. Fuzzy coefficients can be of different types such as triangular, trapezoidal, (semi) $L-R$, or flat (semi) $L-R$ fuzzy numbers. First, we convert the problem to a parametric interval biobjective transportation problem using $gamma$-cuts of fuzzy coefficients. Then, we consider a fix $gamma$-cut and obtain a necessarily weak efficient solution to the yielded interval biobjective program by a new algorithm. It uses basic feasible solutions and the parametric simplex algorithm. Furthermore, we suggest another algorithm for finding a reasonable solution, called $gamma^*$-necessarily weak efficient, to the main biobjective transportation problem. To illustrate the validity and performance of the proposed algorithms, we present some numerical examples.