Abstract: For multiple attribute decision making with preference dependence relationship (MADMPDR), a problem is that the decision maker is required to estimate too many attribute-set capacities, which is also called the exponential complexity problem of attribute-set capacity determination. For this problem, the two fuzzy measures namely the λ-measure and the k-order measure, as well as attribute-set capacity calculation (ASCC) models based on these two measures, have been presented in literature. However, they are suffered from impracticality in many decision cases due to arbitrary hypotheses on preference dependence relationship. The impracticality embodies in two aspects. One is the unfeasibility of measurement pattern, the other is the inaccurateness in capacity calculation. To overcome the impracticality of the λ-measure and the k-order measure, a new measurement pattern on attribute-set capacity, called sandwich measurement pattern (SMP), is proposed based on such a strategy of balancing the feasibility of attribute-set capacity determination with the accurateness of ASCC. Then, a linear programming model for ASCC corresponding to SMP, shorted as LPM-SMP, is presented. In specific, the model determines an attribute-set capacity through restricting it between the minimum and maximum of attribute-set capacities of the same order, and squeezing it into a particular range by theoretical quantitative relationships of attribute-set capacities. These relationships include μ(Ø)=0, μ(N)=1, and μ(A∪{C_i})-μ(A) ≥ 0, where N represents the attribute set of attributes C₁,…,C_n, μ(·) does the capacity of the attribute-set denoted by "·", {C_i} does the singleton of attribute C_i, and A does a subset of N except {C_i}. Besides, the attribute-set capacity given by LPM-SMP satisfies other constraints built on the judgment information of the decision maker. The judgment information is of three types. The first is on the attribute-set capacity ranks, the second is on the numerical values of low-order attribute-set capacities, and the third is on the minimum and maximum attribute-set capacities within a high-order capacity rank. SMP as well as LPM-SMP has two advantages. First, the number of attribute-set capacities to be estimated is C_n¹+C_n²+2(n-3) for a decision with n attributes, which is very acceptable for decision makers since it is only approximate to that when the 2-order measure is adopted. Second, the given attribute-set capacities are more rational than those given by ASCC models based on the λ-measure and k-order measure, since arbitrary hypotheses like those adopted in the measures mentioned above are no longer made in SMP. Data simulation analysis shows that SMP is not only more feasible than the k-order measure, but also greatly superior in calculation accurateness to the λ-measure and the k-order measure. As a result, SMP is more applicable to real-world decisions of MADMPDR.

Key words: multiple attribute decision making, preference correlated attributes, attribute-set capacity, λ fuzzy measure, k-order additive fuzzy measure, sandwich measurement pattern