A chi-square method for priority derivation in group decision making with incomplete reciprocal preference relations

Abstract

a b s t r a c tThis paper proposes a chi-square method (CSM) to obtain a priority vector for group decision making (GDM) problems where decision-makers' (DMs') assessment on alternatives is furnished as incomplete reciprocal preference relations with missing values. Relevant theorems and an iterative algorithm about CSM are proposed. Saaty's consistency ratio concept is adapted to judge whether an incomplete reciprocal preference relation provided by a DM is of acceptable consistency. If its consistency is unacceptable, an algorithm is proposed to repair it until its consistency ratio reaches a satisfactory threshold. The repairing algorithm aims to rectify an inconsistent incomplete reciprocal preference relation to one with acceptable consistency in addition to preserving the initial preference information as much as possible. Finally, four examples are examined to illustrate the applicability and validity of the proposed method, and comparative analyses are provided to show its advantages over existing approaches.

A least deviation method for priority derivation in group decision making with incomplete reciprocal preference relations

“…. w n ) T is able to precisely fit the reciprocal preference relation C (k) , then |d Table 2, it's shown that the LDM achieves an identical ranking as LSM [14], CSM [37], and LLSM [44], while the ranking derived by GPM [47] is slightly different. The GPM fails to discriminate x 2 and x 4 .…”

“…For this single incomplete reciprocal preference relation, it can also be solved by the EM [46], NRAM [38], LSM [14], LLSM [44], CSM [37] and GPM [47]. The results are shown in Table 5, from which we can see that LDM achieves the same ranking x 2 x 4 x 5 x 3 x 6 x 1 as LLSM, GPM, NRAM and CSM.…”

Missing values estimation and consensus building for incomplete hesitant fuzzy preference relations with multiplicative consistency

“…In Example 1, Zhang 16 obtained the multiplicative consistent HFPR, in which the element mh 23 = {0.3132, 0.3184, 0.2949}. It is obvious that the values in mh 23 are not arranged in ascending order, which is inconsistent with his definition. But if the value of mh 23 is arranged in ascending order, mH is not a multiplicative consistent HFPR according his definition.…”

“…(9) In mH,mh 23 = {0.3132, 0.3184, 0.2949}, its elements do not arrange in ascending order, because 0.2949 is smaller than 0.3132 or 0.3184, which do not meet the requirement h…”

“…There are so many researches have been developed for the complete hesitant preference relations. However, in many real decision making problems, due to time pressure, lack of knowledge, and DM's limited expertise related with the problem domain, the DMs may offer a preference relation with incomplete information 23,24,25 , i.e., some of the pairwise comparison information is missing. Xu et al 17 called the HFPR with some entries are unknown incomplete HFPR, and developed two goal programming models to derive the priority weights from an incomplete HFPR based on multiplicative and additive consistency respectively.…”