GoNDEF: an exact method to generate all non-dominated points of multi-objective mixed-integer linear programs

Abstract

Most real-world problems involve multiple conflicting criteria. These problems are called multicriteria/multi-objective optimization problems (MOOP). The main task in solving MOOPs is to find the nondominated (ND) points in the objective space or efficient solutions in the decision space. A ND point is a point in the objective space with objective function values that cannot be improved without worsening another objective function. In this paper, we present a new method that generates the set of ND points for a multiobjective mixed-integer linear program (MOMILP). The Generator of ND and Efficient Frontier (GoNDEF) for MOMILPs finds that the ND points represented as points, line segments, and facets consist of every type of ND point. First, the GoNDEF sets integer variables to the values that result in ND points. Fixing integer variables to specific values results in a multi-objective linear program (MOLP). This MOLP has its own set of ND points. A subset of this set establishes a subset of the ND points set of the MOMILP. In this paper, we present an extensive theoretical analysis of the GoNDEF and illustrate its effectiveness on a set of instance problems.

An adaptive patch approximation algorithm for bicriteria convex mixed-integer problems

“…It is a randomly generated synthetic problem. We follow here roughly the presentation in [39] and restrict it to the case of two criteria to simplify notation. Additionally, we change the optimization direction to minimization.…”

Design and Analysis of Novel Hybrid Multi-Objective Optimization Approach for Data-Driven Sustainable Delivery Systems

“…The main idea behind selection of these two solution methodologies is that although proposed solution methodology shares exact solution approach, how Pareto solution sets will be changed if another exact solution algorithm is used or apart from the exact solution algorithms, if one of the meta heuristics approaches, in which nondominated sorting technique is used to provide the solution as close to the Pareto-optimal solution as possible, is used. One of these chosen methodologies (GoNDEF) proposes an exact method to generate all non-dominated points of multiobjective mixed-integer linear programs that is provided by Rasmi and Turkay [44]. This method handles multi-objective problems in a combination of some sub-problems and creates potential sub-Pareto frontiers by testing some predefined integer solution sets and eliminating dominated solutions from non-dominated ones.…”

A multi-criteria decision analysis to include environmental, social, and cultural issues in the sustainable aggregate production plans

“…A small number of studies exist in the literature with exact solution approach for finding the ND points of MOMILPs when the number of objective functions is greater than two (see Vincent et al (2013), Belotti et al (2013), Stidsen et al (2014), Boland et al (2015), Soylu and Yıldız (2016), and Fattahi and Turkay (2018) for solving bi-objective MILPs). We use the Generator of ND and Efficient Frontier (GoNDEF) method that ensures finding all or a large proportion of ND points for MOMILPs (Rasmi and Türkay, 2019). This method also enables generation of the ND points such that each of them is associated with a different binary solution.…”

“…Let EIS be the set of explored potentially efficient binary solutions. We set EIS := ∅ at the beginning of the algorithm and indicate binary solutions by x B (see Rasmi and Türkay (2019) for detailed discussion).…”