A linear optimization problem to derive relative weights using an interval judgement matrix

• Published in: European journal of operational research Vol. 201; no. 2; pp. 537 - 544
• Main Authors: Conde, Eduardo, de la Paz Rivera Pérez, María
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01-03-2010; Elsevier; Elsevier Sequoia S.A
• Series: European Journal of Operational Research
• Subjects: Applied sciences; Comparative analysis; Decision analysis; Decision analysis Multi-objective optimization; Decision making models; Decision theory. Utility theory; Exact sciences and technology; Matrix; Multi-objective optimization; Operational research and scientific management; Operational research. Management science; Optimization; Studies; Decision analysis; Multi-objective optimization; Pareto optimum; Decision support system; Hierarchical classification; Decision making; Geometrical properties; Multiobjective programming; Linear programming; Interval arithmetic; Linear model; Optimization; Efficiency; Preference; Problem solving; Interval matrix
• ISSN: 0377-2217; 1872-6860
• DOI: 10.1016/j.ejor.2009.03.029

In this paper, we propose a new Decision Making model, enabling to assess a finite number of alternatives according to a set of bounds on the preference ratios for the pairwise comparisons between alternatives, that is, an “
interval judgement matrix”. In the case in which these bounds cannot be achieved by any assessment vector, we analyze the problem of determining of an efficient or Pareto-optimal solution from a multi-objective optimization problem. This multi-objective formulation seeks for assessment vectors that are
near to simultaneously fulfil all the bound requirements imposed by the interval judgement matrix. Our new model introduces a linear optimization problem in order to define a
consistency index for the interval matrix. By solving this optimization problem it can be associated a weakly efficient assessment vector to the consistency index in those cases in which the bound requirements are infeasible. Otherwise, this assessment vector fulfils all the bound requirements and has geometrical properties that make it appropriate as a
representative assessment vector of the set of feasible weights.