A New Step-by-Step (SBS) Algorithm for Inconsistency Reduction in Pairwise Comparisons

Abstract

Pairwise comparisons constitute a popular and important tool for multiple-criteria decision making. Consistency is one particular aspect of pairwise comparisons that has been studied thoroughly in recent decades since human judgements are seldom fully consistent. The aim of this paper is to propose a new simple and intuitive iterative Step-by-Step (SBS) algorithm for inconsistency reduction of a (multiplicative) pairwise comparison matrix based on step-by-step changes of the most inconsistent matrix element, where the inconsistency is measured by a modified Koczkodaj's inconsistency index. A comparison of the proposed SBS algorithm with other algorithms for inconsistency reduction is provided as well.

A Numerical Comparison of Iterative Algorithms for Inconsistency Reduction in Pairwise Comparisons

“…Since the algorithm of Mazurek et al [35] implements Koczkodaj's inconsistency index and a notion of a triad inconsistency, here we provide the necessary notation.…”

“…Definition 6. [35] Let A = (a ij ) be a pairwise comparison matrix of the order n. Let T {(a ij , a jk , a ik )} be the set of all triads, i < j < k, ∀i, j, k ∈ {1, ..., n}. Let T KI(a ij , a jk , a ik ) denote Koczkodaj's inconsistency of a triad (a ij , a jk , a ik ).…”

“…Definition 7. [35] Let A = (a ij ) be a pairwise comparison matrix, then E(A) denotes the set of all a ij ∈ A, i < j, ∀i, j ∈ {1, ..., n}.…”

“…Definition 8. [35] Let A be a set of pairwise comparison matrices of the order n, and let µ denote a measure of inconsistency such that: µ : A → R 0+ . Then an algorithm whose main objective is to transform the PC matrix A ∈ A with µ(A) = m into the matrix A ∈ A so that µ(A ) < µ(A) and µ(A ) ≤ ε, ε > 0, is called an algorithm for inconsistency reduction (AIR).…”

“…The former algorithms adjust one or more (or all) matrix elements in each iteration so that the inconsistency of the whole pairwise comparison (PC) matrix gradually decreases, and stops when the level of inconsistency falls below a given threshold. The algorithms of Cao et al [11], Kou et al [25], Mazurek et al [35], Szybowski [44], or Xu and Wei [47] fall into this category. Non-iterative algorithms are represented, for instance, by the INSITE algorithm by Abel et al [1] that applies a multiple-objective linear programming method, or algorithms by Bozoki et al [6], [7] which are based on non-linear mixed-integer optimization.…”