Introduction

In a real life decision making, the experience and judgment of experts (or decision makers) represent an effective way to resolve non-structural, or qualitative, or combined qualitative and quantitative decision problems (Saaty, 1980; Xu, 1988). Pairwise comparisons, also known as preference relations, are often used by decision makers to compare a set of decision alternatives with respect to a pre-determined criterion. An ideal process of decision making might wish that each individual reason rationally and the group of all the decision makers cooperates harmonically. In other words, the ideal scenario is that the judgment of each individual decision maker is consistent, and the group of the decision makers reaches good consensus. Then, on the basis of the ideal consistency and consensus, the total amount of information of the individual judgments is aggregated so that the optimized decision can be obtained. However, in reality there is so much complex, uncertain information involved, while the available knowledge is limited and human rationality is bounded. Therefore, it is hard for experts to present precise, consistent judgments; and it is difficult for them to achieve an optimized decision making. Consequently, experts have to rely on incomplete, uncertain, approximately consistent judgments to reach their locally optimal decisions instead of possibly global optimal alternatives. Because of this reason, the theory of expert decision making under uncertainty represents an important scientific focus in the area of decision analysis.