A method for discrete stochastic MADM problems based on the ideal and nadir solutions

• First, the probability distributions of the ideal and nadir variates are defined.
• Second, the rationale of the ideal and nadir variates is proved.
• Third, the distance between two discrete stochastic variables is defined.
• Fourth, some properties of the metric are discussed.
• Fifth, the normalizations of attribute values with different scales are given.

Many real life decision making problems can be modeled as discrete stochastic multi-attribute decision making (MADM) problems. A novel method for discrete stochastic MADM problems is developed based on the ideal and nadir solutions as in the classical TOPSIS method. In a stochastic MADM problem, the evaluations of the alternatives with respect to the different attributes are represented by discrete stochastic variables. According to stochastic dominance rules, the probability distributions of the ideal and nadir variates, both are discrete stochastic variables, are defined and determined for a set of discrete stochastic variables. A metric is proposed to measure the distance between two discrete stochastic variables. The ideal solution is a vector of ideal variates and the nadir solution is a vector of nadir variates for the multiple attributes. As in the classical TOPSIS method, the relative closeness of an alternative is determined by its distances from the ideal and nadir solutions. The rankings of the alternatives are determined using the relative closeness. Examples are presented to illustrate the effectiveness of the proposed method. Through the examples, several significant advantages of the proposed method over some existing methods are discussed.