Analytic Hierarchy Process (AHP) in Group Decision Making and its Optimization With an Allocation of Information Granularity

In group decision making, one strives to reconcile differences of opinions (judgments) expressed by individual members of the group. Fuzzy-decision-making mechanisms bring a great deal of flexibility. By admitting membership degrees, we are offered flexibility to exploit different aggregation mechanisms and navigate a process of interaction among decision makers to achieve an increasing level of consistency within the group. While the studies reported so far exploit more or less sophisticated ways of adjusting/transforming initial judgments (preferences) of individuals, in this paper, we bring forward a concept of information granularity. Here, information granularity is viewed as an essential asset, which offers a decision maker a tangible level of flexibility using some initial preferences conveyed by each individual that can be adjusted with the intent to reach a higher level of consensus. Our study is concerned with an extension of the well-known analytic hierarchy process to the group decision-making scenario. More specifically, the admitted level of granularity gives rise to a granular matrix of pairwise comparisons. The granular entries represented, e.g., by intervals or fuzzy sets, supply a required flexibility using the fact that we select the most suitable numeric representative of the reciprocal matrix. The proposed concept of granular reciprocal matrices is used to optimize a performance index, which comes as an additive combination of two components. The first one expresses a level of consistency of the individual pairwise comparison matrices; by exploiting the admitted level of granularity, we aim at the minimization of the corresponding inconsistency index. The second part of the performance index quantifies a level of disagreement in terms of the individual preferences. The flexibility offered by the level of granularity is used to increase the level of consensus within the group. Given an implicit nature of relationships between the realizations of the granular pairwise matrices and the values of the performance index, we consider using particle swarm optimization as an optimization vehicle. Two scenarios of allocation of granularity among decision makers are considered, namely, a uniform allocation of granularity and nonuniform distribution of granularity, where the levels of allocated granularity are also subject to optimization. A number of numeric studies are provided to illustrate an essence of the method.