Goal programming models for incomplete interval additive reciprocal preference relations with permutations

Within the framework of the fuzzy analytic hierarchy process, the importance of alternatives could be expressed as interval-valued comparison matrices to model some uncertainty experienced by decision makers (DMs). Owing to the complexity of the considered problem and the lack of DMs' knowledge, the given interval-valued comparison matrix may be incomplete. It is of importance to develop mathematical models for incomplete interval-valued comparison matrices, so that the missing entries can be estimated. In this study, it is considered that a missing value may be one of the bounds of interval-valued entries. Acceptable incomplete interval additive reciprocal preference relations (IARPRs) are redefined and the existing drawback is overcome. By considering the random behavior of decision makers in comparing alternatives, the novel definitions of multiplicative and additive approximate consistency for incomplete IARPRs are introduced. Then, we propose two goal programming models to estimate the missing values of incomplete IARPRs. The permutations of alternatives are incorporated into the proposed mathematical models. It is found that when the permutations of alternatives are different, the estimated preference values could be different. Some numerical results are reported to illustrate the given algorithms for solving decision-making problems with incomplete IARPRs. The observations reveal that the proposed mathematical models can be used to capture the randomness experienced by decision makers in comparing alternatives.