Abstract

     A tolerance framework is developed to address a group multiple criteria ranking problem with indirect preference information, which is referred to as the interaction, importance and tolerance of criteria as well as pairwise comparisons among alternatives and criteria. Choquet integral preference model is employed to capture the interaction, importance and tolerance of criteria, all of which are specific to decision makers (DMs). Some mandatory/sufficient requirements concerning criteria which are global or local, also called the tolerance attitudes of DMs, are quantified as tolerability constraints. Preference disaggregation analysis is extended to solve this type of tolerability constraints for preference elicitation. Confronted with the inconsistency issue (the feasibility of the whole preference constraints translated from indirect preference information), cause oriented strategy and consequence oriented strategy are established by regression-based mixed 0–۱ integer linear programs with the objectives of the most credible minimal inconsistent preference constraints and the most credible maximal preference constraints in the context of group decision making. Considering a wide range of the minimal unsatisfied subsets of preference constraints responsible for the inconsistency and the maximum satisfied subsets of preference constraints as a consistent result, to reach a robust decision, stochastic multicriteria acceptability analysis (SMAA)-like simulation analysis is generated to examine the whole instances of compatible preference models in the modified feasible preference polyhedron and compute the result in a probabilistic form. Simulation experiment is conducted to investigate the influence of global tolerance attitudes of DMs on preference elicitation in conservative and radical scenarios. Finally, the application of the proposed approach to a credit ranking of small and medium-sized enterprises (SMEs) and the comparison analysis with objective methods are presented and discussed for the effectiveness of the proposed tolerance framework.

Introduction

     Many real-world decision problems in different fields can be formulated as multiple criteria decision making (MCDM) as the performance of alternatives can be decomposed as the evaluation of alternatives on considered criteria and ranked by the comprehensive score which is a straightforward intuition for decision makers (DMs) to comprehend. The preference model in MCDM would be paid more attention to the property that is consistent with observable properties of human decision process (i.e., intuitive reasoning, common sense, and expert knowledge), otherwise it would deteriorate a justifiable decision. The theory of fuzzy measures and integrals Liginlal and Ow (2006) has emerged to characterize the preference of DMs with the realistic hypothesis about preferential dependence among criteria and give the opportunity to represent and interpret the typical human decision process, which is such a predominant character that can address numerous practical preference modelling.

     The DMs have distinguished aggregation behaviors in context of group MCDM, also called the tolerance attitudes of DMs in some literatures. From a mathematical perspective, the tolerance attitudes of DMs are equivalent to the tolerance (or intolerance) of criteria represented by the aggregation operator (Li, Yao, Sun, & Wu, 2018). In a strategic investment decision problem, when searching for most qualified candidates among prospective alternatives, every candidate failing at most k criteria would be rejected. It implies k-intolerance attitudes of the DMs which can be perceived as a selection policy. In the field of fuzzy aggregation operator field, ordered weighted averaging (OWA) operator was studied which used the global tolerance of criteria to measure the optimism degree of DMs (Kim and Ahn, 2018, Zeng et al., 2018). Based on the k-tolerance (or k-intolerance) properties of the Bonferroni mean aggregation operator (Beliakov & James, 2013), Dutta, Figueira, and Das (2019) studied orness measure to define the or‐like degree of the Bonferroni mean and its variants. Considering non-additive value function, previous researchers constructed the optimization model using fuzzy measure to capture tolerance attitudes required from DMs without any post optimality analysis (Li et al., 2018, Yao et al., 2018). It is doubted that the decision from the inferred preference model is robust in case of the existence of the multiple feasible near-optimal solutions. As noted by Dujmović and Larsen (2007), the requirements (or expectations) on criteria combining by preference model cannot be neglected during preference aggregation. Subsequently, concise mathematical formulas for them (Hudec and Mesiar, 2020, Marichal, 2004) were defined to express a global or local tolerance degree of the DMs.

Conclusion

     In this paper, under a robust perspective, we present a tolerance framework to address a group multiple criteria ranking problem with indirect preference information such as the interaction, importance and tolerance of criteria as well as pairwise comparison among alternatives and criteria. Based on indirect preference information, Choquet integral preference model is elicited to capture the interaction, importance and tolerance of criteria for the final decision. With respect to the tolerance attitudes of DMs such as the mandatory/sufficient requirements on criteria for specific alternatives, they are quantified as tolerability constraints for preference elicitation. Once inconsistency issue (the feasibility of the whole preference constraints) in preference constructive learning process occurs, in the tolerance framework, two developed strategies including cause oriented strategy and consequence oriented strategy are applied. Then identify the minimal unsatisfied subsets of preference constraints responsible for the inconsistency or the maximum satisfied subsets of preference constraints as a specific set of consistent preference information by regression-based mixed 0-1 integer linear programs. By following the idea of treating all possible sets of preference parameters in line with indirect preference information, SMAA-like simulation algorithm is constructed to generate the rank result in the probabilistic form, which is used to generate a solution considered by exploring the whole instances of the compatible preference model. The proposed method can deal with indirect preference information involved with not only less cognitive effort but also the tolerance attitudes on criteria for the DMs, which has been paid insufficient attention in the literatures. From the illustration and application, it suggested that the proposed unified framework for MCDA has two characteristics: expressiveness of the underlying preference model which can reconstruct indirect preference information provided by the DMs and robustness of the final result which address uncertainties and imprecision observed in the actual decision support processes.