Determining maximal efficient faces in multiobjective linear programming problem

Finding an efficient or weakly efficient solution in a multiobjective linear programming (MOLP) problem is not a difficult task. The difficulty lies in finding all these solutions and representing their structures. Since there are many convenient approaches that obtain all of the (weakly) efficient extreme points and (weakly) efficient extreme rays in an MOLP, this paper develops an algorithm which effectively finds all of the (weakly) efficient maximal faces in an MOLP using all of the (weakly) efficient extreme points and extreme rays. The proposed algorithm avoids the degeneration problem, which is the major problem of the most of previous algorithms and gives an explicit structure for maximal efficient (weak efficient) faces. Consequently, it gives a convenient representation of efficient (weak efficient) set using maximal efficient (weak efficient) faces. The proposed algorithm is based on two facts. Firstly, the efficiency and weak efficiency property of a face is determined using a relative interior point of it. Secondly, the relative interior point is achieved using some affine independent points. Indeed, the affine independent property enable us to obtain an efficient relative interior point rapidly.