Introduction

Multi-criteria appear in many modern technological tasks such as medical diagnosis, information retrieval, financial decision making and pattern recognition [1-5]. Collectively we shall refer to these as multi-criteria decision problems. Professor Janusz Kacprzyk has made important contributions this field [6-9]. In multi-criteria decision problems our interest is in selecting from some set of alternatives the one that best satisfies the criteria. Since it is generally difficult to rank alternatives based on their satisfaction’s to multiple individual criteria a standard approach is to aggregate an alternative’s satisfaction to the individual criteria to obtain a single scalar value corresponding to the alternative’s overall satisfaction to the collection of criteria. These scalar values can then be used to rank the alternatives and enable a choice to be made. The aggregation of these multi-criteria satisfactions generally requires the use of some information regarding the importance of the individual criteria. The classic approach to this aggregation is to take a weighted average of an alternative’s satisfaction to the individual criteria, the weights in this approach being the importance of the individual criteria. Implicit in this approach is an assumption that the individual criteria importance weights are additive. That is, for example, the importance of criteria weight of criteria one and two together is simply the addition of the two individual criteria importance weights. More generally, this assumes that the importance of any group of criteria taken together is simply the sum of the importance of the individual criteria. In many cases of decision making this simplifying assumption is not valid. For example, when selecting an employee the situation where the criteria of having a good education or considerable experience are interchangeable doesn’t justify this assumption. More generally the situation in which the satisfaction of any one of a group of criteria is all that is needed does not satisfy the assumption of an additive relationship between individual criteria importance. To model more complex relationships about the importance of subsets of criteria recent interest has focused on the use of a fuzzy measure [10-13]. In this approach, the additivity of the individual criteria importance’s has been replaced by a monotonicity condition, if A and B are subsets of criteria such that A contains all the criteria in B then it is assumed that the importance of collection the A is at least as large as the collection B of criteria. The use of this more general measure structure to represent our information about the importance of subsets of criteria complicates the process of aggregating the satisfactions of the individual criteria based on the importance information. The use of the simple weighted average of individual satisfactions does not always work. Here we show that the Choquet integral [14-18] provides an approach to the aggregation of the individual criteria satisfactions which generalizes the simple weighted average approach for additive weights to the case where the importance information is carried by a measure. In some applications of multi-criteria decision making the criteria can be categorized, these categories can then used for expressing the information about criteria importance (see Zadeh [19]). Here, the collection of criteria in the same category shares a given amount of importance.