A Decision Making Model Based on Intuitionistic Multiplicative Preference Relations With Acceptable Consistency

Xuan Yang,Zhou-Jing Wang

doi:10.1109/access.2019.2933457

Abstract

Two categories of hesitancy often exist in an intuitionistic multiplicative preference relation (IMPR). One is inconsistency among three or more preferences in an IMPR, and the other is hesitancy of a non-diagonal intuitionistic multiplicative judgment itself. Capturing such hesitancy plays an important role for deriving priority weights of IMPRs and measuring inconsistency degrees of IMPRs. This article first analyzes a recent consistency definition of IMPRs and reveals its deficiency. A normalized intuitionistic multiplicative weight vector is introduced and utilized as a benchmark of different intuitionistic multiplicative weight vectors with equivalency. A new notion of consistent IMPRs is then proposed by building the link between an IMPR and its normalized intuitionistic multiplicative weights. In order to find the closest IMPR with consistency to an original IMPR, the paper establishes a minimization model, in which the hesitancy difference between the two IMPRs is regarded as a constraint and the optimal goal function value is used to determine an inconsistency index of the original IMPR. The minimization model is further transformed into a least square model and its analytical solution is discovered by the Lagrangian multiplier method. Based on the analytical solution, an intuitionistic multiplicative judgment based index is devised to measure inconsistency of IMPRs, and an approach with checking acceptable consistency is developed for multi-criteria decision making with IMPRs. The presented models are illustrated and validated by three numerical examples including a hierarchical multi-criteria decision making problem.

Highlights

Multi-criteria decision making (MCDM) is often occurred in business management, industrial engineering, and our daily lives [1]–[3]
It is obvious that this result is in conflict with w−1 ≤ w+1, implying that there does not exist an intuitionistic multiplicative weight vector such that the non-diagonal intuitionistic multiplicative judgments in the consistent intuitionistic multiplicative preference relation (IMPR) A 2 are totally expressed by wi (i = 1, 2, 3)
This result shows that the intuitionistic multiplicative preferences in the consistent IMPR A 1 cannot be totally expressed and captured by any of the priority weight vectors ω Xu, W Xia, ω Re, W Jin, ω Zh1 and W Zh2 derived from Xu [35], Xia et al [15], Ren et al [8], Jin et al [38] and

Summary

INTRODUCTION

Multi-criteria decision making (MCDM) is often occurred in business management, industrial engineering, and our daily lives [1]–[3]. A new framework of normalized intuitionistic multiplicative weights is presented and used to put forward consistency of IMPRs. Thirdly, an analytical solution of normalized intuitionistic multiplicative weights is found for IMPRs. Lastly, an intuitionistic multiplicative judgment based CI is devised and employed to propose an analytical solution based decision making model with checking acceptable consistency of IMPRs. The paper analyzes the consistency definition [37] and illustrates its deficiency by a numerical example. A geometric-mean-based index is defined to measure hesitancy of IMPRs. A Euclidean-metric-based function and the hesitancy difference between an IMPR and its closest IMPR with consistency are introduced to establish a minimization model for finding normalized intuitionistic multiplicative weights from IMPRs. In order to obtain its analytical solution by the Lagrangian multiplier method, the minimization model is transformed into a least square model.

PRELIMINARIES

CONSISTENCY OF INTUITIONISTIC MULTIPLICATIVE PREFERENCE RELATIONS

AN ANALYTICAL SOLUTION BASED

ILLUSTRATIVE EXAMPLES AND COMPARATIVE

CONCLUSION