Optimization of the ordinal and cardinal consistency of a preference matrix in decision making

In multi-criteria decision problems the relative importance of alternatives is computed from preference matrices, which come from experience and can possibly be inconsistent. Two consistency types of preferences are studied in the paper. The ordinal consistency preserves the order in which the alternatives are arranged, and it does not allow cycles. The term ‘cyclic consistency’ is also used for this type. The second type is the cardinal consistency, when not only the order, but also the exact values of the relative importance must be consistent. In this paper efficient algorithms for computing consistent approximations of both types for a given preference matrix are described. The main result is an algorithm which combines the advantages of both particular types and computes the optimal consistent approximation of a given preference matrix in the ordinal and in the cardinal sense. The described algorithm can also be used for processing preference matrices with missing data. The performance of the algorithm is illustrated by numerical examples.