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

  • Mao-Jie Huang
  • Fang LiuEmail author
  • Ya-Nan Peng
  • Qin Yu
Original Paper


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.


Incomplete interval additive reciprocal preference relation (incomplete IARPR) Goal programming model Approximate consistency Analytic hierarchy process (AHP) Permutation 



The authors would like to express their thanks to the anonymous referees for the valuable comments and suggestions for improving the paper. The work was supported by the National Natural Science Foundation of China (Nos. 71571054, 71871072), the Guangxi Natural Science Foundation for Distinguished Young Scholars (No. 2016GXNSFFA380004). 2017 Guangxi high school innovation team and outstanding scholars plan, and the Innovation Project of Guangxi Graduate Eduation (2019).


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Mathematics and Information ScienceGuangxi UniversityNanningChina

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