Optimization over the efficient set of a parametric multiple objective linear programming problem

Cone Characterizations of Approximate Solutions in Real Vector Optimization

“…Over the years, many authors have shown interest in the problem of optimizing over the efficient set of a multiobjective programming problem, see Benson and Sayin (1994), Tu (2000) and Jorge (2005), among others. Here we consider the particular problem of maximizing over the set of ε-minimal elements while retaining to the same underlying ordering cone C. Motivation was provided by Engau and Wiecek (2005a,b) who described applications of multiobjective programming problems which require the generation of suboptimal solutions in practical decision making situations.…”

Optimal and strongly optimal solutions for linear programming models with variable parameters

“…Moreover, model (1) can be used for optimizing a function over the efficient set of a multiobjective LP with parameters in the coefficient matrix, right hand side vector, and cost matrix (see [6]). Also, this can be useful for solving some classes of knapsack problems [5].…”

Determining maximal efficient faces in multiobjective linear programming problem

“…Numerous studies and applications of multicriteria problems and MOLP have been reported in the literature in literally hundreds of books, monographs, articles and chapters in books. For an overview of these studies and applications, see, for instance [1,7,9,12,16,19] and references therein. An MOLP problem is to minimize or maximize several linear objective functions subject to a set of linear constraints.…”

“…In the case when the decision maker's preference is quantifiable and given explicitly in the form of a function f : R n → R, then the problem for finding a most preferred efficient solution can be stated in the form "optimize { f (x) | x ∈ X E }" where X E is the efficient set of MOLP [19]. Since the efficient set is not convex it is a difficult task to optimize f over this set directly; thus the most convenient way for optimizing f over the efficient set is to decompose it into some convex sets.…”

“…So, in order to optimize f over X E it is sufficient to maximize f over all maximal efficient faces and obtain an overall optimal efficient solution. In the case when f is a linear function the above mentioned optimization problem can be solved using finite number of linear programming (LP) problems corresponding to efficient maximal faces; therefore, obtaining the maximal efficient faces is the most critical task in MOLP [19].…”