The following ranking problem is considered.
(1) There exist m candidates to be ranked with respect to s criteria.
(2) Each candidate has s characteristics and each one corresponds to each criterion. Based on the value of the characteristics, candidates are ranked with respect to the corresponding criterion. But this ranking may change due to the change of value of the characteristics. This change occurs according to the some probability. Therefore total change with respect to all criteria occurs with scenarios.
(3) Our theme is to make the total ranking among candidates considering not only the importance of criteria but the total change of ranking about candidates with respect to all criteria. That is, how to make a consensus formation is the main problem.
First for the fixed scenario case, that is, the fixed preference matrices, we define the preference matrix of candidates with respect to each criterion and distance measure of Cook [W.D.Cook. EJOR 172 (2006) 369-385] based on the matrix. Next we extend the Cook distance measure toward the weighted distance measure and then formulate the equivalent assignment problem. The third we solve the assignment problem and based on the optimal assignment, that is, optimal ranking of candidate is found. The fourth we struggle to aggregate the result of each scenario case and make the final suitable ranking considering the risk. We also show our procedure can apply other ranking model, especially the case that preference matrix includes ambiguity in place of randomness. Finally we conclude our results and discuss the further research problem including the efficient solution method for the assignment problem corresponding to the extended weighted distance.
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