Measuring inconsistency and deriving priorities from fuzzy pairwise comparison matrices using the knowledge-based consistency index

Highlights

• A new consistency index (KCI) for fuzzy pairwise comparison matrix is proposed.

• A comparison study shows that KCI performs faster than Ohnishi’s index.

• An optimal solution of the MCDM problem is computed using fuzzy AHP.

• Decision makers can save a substantial amount of time by adopting KCI.

Abstract

The fuzzy analytic hierarchy process (AHP) is a widely applied multiple-criteria decision-making (MCDM) technique, making it possible to tackle vagueness and uncertainty arising from decision makers, especially in a pairwise comparison process. Indeed, as the human brain reasons with uncertain rather than precise information, pairwise comparisons may involve some degree of inconsistency, which must be correctly managed to guarantee a coherent result/ranking. Several consistency indexes for fuzzy pairwise comparison matrices (FPCMs) have been proposed in the literature. However, some scholars argue that most of these fail to be axiomatically grounded, which may lead to misleading results. To overcome this lack of an axiomatically grounded index, a new index is proposed in this paper, referred to as the knowledge-based consistency index (KCI). A comparative study of the proposed index with an existing one is carried out, and the results show that KCI contributes to substantially reducing the computation time. In addition, the different fuzzy weights derived from the initial FPCM (for KCI computation purposes) can also be employed to find a crisp set of weights that corresponds to an optimal solution to the MCDM problem according to the decision maker’s viewpoint and expertise.

Introduction

According to some specialists in human judgments, it has been proven that the human brain can only consider a limited amount of information at one time [1], making it unreliable for making decisions regarding complex problems (i.e., with multiple and conflicting parameters). Hence, multiple-criteria decision-making (MCDM) methods have been introduced to help decision makers overcome this issue. MCDM methods can be classified into two categories [2]: multi-attribute decision-making (MADM) and multi-objective decision-making (MODM). Unlike MODM, MADM techniques involve considerable human participation. One of the most widely employed MADM techniques is the analytic hierarchy process (AHP), initially introduced by Saaty [3]. Its main strengths lie in its objective and logical ranking system, and its flexibility to be jointly used with other techniques such as fuzzy logic, neural networks and SWOT (strengths, weaknesses, opportunities, threats) analysis [4], [5]. However, this technique requires the decision makers to express their knowledge in a consistent manner. Indeed, in AHP knowledge assessment is performed by carrying out pairwise comparisons between a set of items (criteria or alternatives). More specifically, decision makers must specify by “how many more times item $i$ is preferred to item $j$”. Because human beings typically reason using “local” information (i.e., one pairwise comparison at the same time) rather than with global information (i.e., taking into account the whole set of pairwise comparisons at a time), such a process may introduce some degree of inconsistency [6], [7]. To overcome this problem, Saaty introduced a consistency ratio (CR) that aims to measure the degree of inconsistency for a given pairwise comparison matrix. When this ratio exceeds $10\text{\%}$, a judgment often needs reexamination. According to [8], consistency indexes can be classified into two categories: (i) “intra” expert consistency, which focuses on a single decision maker/matrix [9], [10], and (ii) “inter” expert consistency, which focuses on inconsistency analyses resulting from a group of decision makers [8], [11].

Fuzzy logic has been introduced in AHP, in an approach more commonly known as fuzzy AHP (FAHP), as a way to cope with uncertainty and vagueness arising in knowledge assessment. This has found significant applications in recent years [5], [12]. Unlike the classical set theory, fuzzy logic enables the gradual assessment of the membership of elements in relation to a set [13]. The first FAHP method was introduced by van Laarhoven and Pedrycz [14] in 1983. Since then, many other methods have been introduced, as reviewed in a recent state-of-the-art survey of FAHP applications [5].

Similarly to AHP, the consistency also has to be quantified in FAHP. In 1985, Buckley [15] proposed a first consistency index for fuzzy pairwise comparison matrices (FPCMs). Several similar indexes have since been introduced, including the fuzzy logarithmic least squares consistency [14], the feasible region consistency [16], the fuzzy preference-programming consistency [17], the additive consistency [6], [18], [19], and the geometric consistency [20], [21]. Despite their various advantages and disadvantages, several theoretical calculation problems and questions have been raised concerning the introduction of fuzzy sets in AHP, especially with regard to the axiomatic foundation of the approach [22], [23]. Dubois [22] argues that fuzzy sets have often been incorporated in to existing methods, such as PROMETHEE and ELECTRE, without clear benefits. He also adds that fuzzy sets in AHP must be considered, first and foremost, at the “axiomatic” level, not simply at the technical one. Looking more closely at the reasons behind such criticisms, one may find that a major problem lies in the difficulty of successfully satisfying the transitivity and reciprocal axioms [24], [25], [26].

Existing consistency indexes are more thoroughly reviewed and discussed in Section 2, with discussions spanning from their evolution over time to their pros and cons. Following this literature review, a new index, referred to as “Knowledge-based Consistency Index” (KCI), is introduced in Section 3. This is evaluated and compared with a known consistency index in Section 4. Finally, conclusions are provided in Section 5. All acronyms used in this paper are summarized in Table A.2. Let us note that a preliminary version of this research work was presented to the IEEE International Conference on Fuzzy Systems (FUZZ-IEEE) in Naples, July 2017 [27]. The present article extends this work through (i) a more in-depth literature review of existing consistency indexes, (ii) a new section detailing the mathematical formulation of crisp FPCM’s weight derivation, and (iii) a more complete evaluation and comparison study considering a wider range of FPCMs ($237$ in this article against $48$ in the conference version).