Iterative algorithms for determining optimal solution set of interval linear fractional programming problem

Abstract

Determining the optimal solution (OS) set of interval linear fractional programming (ILFP) models is generally an NP-hard problem. Few methods have been proposed in this field which have only been able to obtain the optimal value of the objective function. Thus, there is a need for an appropriate method to determine the OS set of the ILFP model. In this paper, we introduce three algorithms to obtain the OS of ILFP. In the first and second algorithms, using the definition of strong and weak feasible solutions, the objective function of ILFP has been transformed to a linear objective function on the largest feasible region (LFR) and we obtain the OS of ILFP. These two algorithms, only introduce one point as the feasible OS. Since ILFP is an interval model, we seek an algorithm, where for the first time a solution set is obtained as the OS set by solving two sub-models. Hence, we transform the ILFP model into two pessimistic and optimistic sub-models, as one is in the smallest feasible region (SFR) and the other on the LFR. We add constraints to the optimistic model to ensure that the OS set is feasible. Then, we introduce pessimistic and modified optimistic model (PMOM) algorithm. In this algorithm, each PMOM is solved separately. The OSs obtained from these two models give the OS set so that this OS set is feasible. Note that the union of feasible OSs obtained from the proposed algorithms will be a more complete feasible OS set.