Enabling Agents to Perform Prioritized Multi-Criteria Aggregation

Here the weights correspond to the importance of the Many agent based decisions involve an aggregation of individual of the criteria. the satisfaction to multiple criteria. We consider the It is easy to see that this type of aggregation is situation where there exists a prioritization relationship monotonic, if Ci(x) ≥ Ci(y) for all i then C(x) ≥ C(y). over the criteria. In this situation increase in satisfaction It is bounded Mini[Ci(x)] ≤ C(x) ≤ Max[Ci(x)). It is to a lower priority criteria doesn't readily compensate for a also idempotent, if all Ci(x) = a then C(x) = a. Because decrease in satisfaction in higher priority criteria. of these properties this is an averaging operator. Typically the relationship between safety and cost Central to this type of aggregation is the ability to manifests this property. We suggest that prioritization trade off between criteria. In particular wk wi is the between criteria can be modeled by making the weights associated with a criteria dependent upon the satisfaction of relation between criteria Ci and Ck. In this type of the higher priority criteria. We discuss a number of aggregation we can compensate for a decrease of ∆ in methods for obtaining the associated weights. satisfaction to criteria Ci by gain wk wi ∆ in satisfaction Introduction to criteria Ck. In many important applications we do not want to Many agent based decision problems involve the allow this kind of compensation between criteria. selection of an alternative based upon its satisfaction to Consider the situation in which we are selecting a a collection of criteria. In these problems we have a bicycle for our child based upon the criteria of safety and collection of criteria C = {C1, ..., Cn}, a set of cost. In this situation we may not want a benefit with alternatives X = {x1, ..., xm} and the availability of a respect to cost to compensate for a loss in safety. Here we have a kind of prioritization of the criteria. measure of the satisfaction of criteria Ci by each Safety has a higher priority. More generally decisions alternative, Ci(x) ∈ [0, 1]. One commonly used involving issues related to safety and homeland security approach is to calculate for each alternative x a score often manifest this type of prioritization of criteria C(x) as an aggregation of the Ci(x), (Popp and Yen 2006). In organizational decision C(x) = F(C1(x), ......, Cn(x)) making criteria desired by superiors generally have a and then order the alternatives using these scores. Since higher priority then those of their subordinates. the score for any alternative x just depends on only x's In this work we suggest an averaging type satisfaction to each of criteria, it is pointwise, this type aggregation operator that allows for the inclusion of of approach satisfies the Arrow condition of priority between the criteria. Central to our approach independence to irrelevant alternatives (Arrow 1951, will be the modeling of priority by using a kind of Luce and Raiffa 1967) that is adding a new alternative importance weight in which the importance of a lower will not effect the score of x priority criteria will be based on its satisfaction to the The form for F depends upon the agents desired higher priority criteria. As we shall see this results in a imperative for performing this multi-criteria aggregation situation in which importance weights will not be the (Yager 1988). A commonly used form for F is the same across the alternatives. weighted average of the Ci(x). In this case we calculate C(x) = ∑ i = 1 n wi Ci(x) where the weights satisfy wi ∈ [0, 1] and sum to one. PRI-AVE: A Prioritized Averaging This equation along with the fact that T1 = S0 = 1 Operator gives a recursive definition at Ti. We now see that for all Cij ∈ Hi we have uij = Ti. In the following we assume that we have a Using this we obtain for each Cij a weight wij with collection of criteria partitioned into q distinct respect to alternative x such that categories, H1, H2, ..., Hq such that Hi = {Ci1, Ci2, wij = uij uij ∑ j = 1 ni ∑ i = 1 q ..., ini ). Here Cij are the criteria in category Hi. We assume a prioritization between these categories H1 > H2, ... > Hq. More precisely we should use the notation wij(x) The criteria in class Hi have a higher priority than those however for simplicity we use wij. in Hk if i < k. The total collection of criteria is C = Since uij = Ti.this simplifies to wij = Ti niTi ∑ i = 1 q . Hi ∪ i = 1 q . We assume n = ∑ i = 1 q ni the total number of