Incomplete pairwise comparison and consistency optimization

Abstract

This paper proposes a new method for calculating the missing elements of an incomplete matrix of pairwise comparison values for a decision problem. The matrix is completed by minimizing a measure of global inconsistency, thus obtaining a matrix which is optimal from the point of view of consistency with respect to the available judgements. The optimal values are obtained by solving a linear system and unicity of the solution is proved under general assumptions. Some other methods proposed in the literature are discussed and a numerical example is presented

Multiplicative consistency of intuitionistic reciprocal preference relations and its application to missing values estimation and consensus building

“…An issue to address in GDM problems with IRPRs is the lack of information, a problem extensively studied in the case of RPRs [14,20]. In this context, consistency based methods to 'estimate' the missing values from known ones have been proposed in [1,2,27], which later were extended to the GDM framework [28,29].…”

Consistent completion of incomplete judgments in decision making using AHP

“…The method uses the multiplicative condition, as proposed by Saaty, instead of the additive condition as in [17]; and last but not least, it produces an entire consistent comparison matrix, instead of just the priority vector as in [7]. Obtaining the entire consistent comparison matrix is crucial, for example if the aggregation of individual judgments, as opposed to aggregation of individual priorities, is necessary within a given participative process.…”

“…A matrix R = (r ij ) is said to be reciprocal in the additive sense if (r ik −0.5)+(r kj −0.5) = r ij −0.5 for any i, j, k. The purpose of [17] is to minimize the global inconsistency index defined by ρ = ijk (r ik + r kj − r ij − 0.5) 2 considering the missing entries as variables. This solution involves solving a linear system.…”

“…In [17], instead of managing with multiplicative properties (Saaty's treatment), the emphasis is on additive properties of the given matrices. A matrix R = (r ij ) is said to be reciprocal in the additive sense if (r ik −0.5)+(r kj −0.5) = r ij −0.5 for any i, j, k. The purpose of [17] is to minimize the global inconsistency index defined by ρ = ijk (r ik + r kj − r ij − 0.5) 2 considering the missing entries as variables.…”

“…Several authors [17,7] have addressed the problem of producing preference data generated from incomplete information using various techniques that mainly involve optimization applied to different objective functions. For example, in [7] an approximation to the priority vector is obtained from an incomplete judgement.…”