In this paper, we establish a proof for  a  necessary condition for  multiple objective fractional programming. In order to derive the set of necessary conditions, we employ an equivalent parametric problem. Also, we  present the related semi parametric model.

Traditionally, supplier selection should simultaneously take into account numerous heterogeneous criteria, and then is a tedious task for the purchasing decision makers. It becomes especially complicated when quantity discounts are considered at the same time. Under such manner, most studies often formulate such a problem as a Multi-Objective Linear Programming (MOLP) problem, and then scale it...

optimal allocation of water resources is an essential service in agriculture that must be considered by farmers. one of the most significant factors in optimal allocation of water resources in agriculture is to define optimal farm cropping pattern. in this study, in order to determine optimal cropping pattern and water resources allocation in central district of mashhad city (toos village), the...

In practice, there are many problems which decision parameters are fuzzy numbers, and some kind of this problems are formulated as either possibilitic programming or multi-objective programming methods. In this paper, we consider a multi-objective programming problem with fuzzy data in constraints and introduce a new approach for solving these problems base on a combination of the multi-objecti...

Linear programming problems with alternate solutions are challenging due to the choice of multiple strategiesresulting in the same optimal value of the objective function. However, searching for these solutions is atedious task, especially when using mixed integer linear programming (MILP), as previously applied tometabolic models. Therefore, judgment on plurality of optimal m...