Group Decision Making with Dual Hesitant Fuzzy Preference Relations

Abstract

Background/Introduction

Due to the complexity and uncertainty of socioeconomic environments and cognitive diversity of group members, the cognitive information over alternatives provided by a decision organization consisting of several experts is usually uncertain and hesitant. Hesitant fuzzy preference relations provide a useful means to represent the hesitant cognitions of the decision organization over alternatives, which describe the possible degrees that one alternative is preferred to another by using a set of discrete values. However, in order to depict the cognitions over alternatives more comprehensively, besides the degrees that one alternative is preferred to another, the decision organization would give the degrees that the alternative is non-preferred to another, which may be a set of possible values. To effectively handle such common cases, in this paper, the dual hesitant fuzzy preference relation (DHFPR) is introduced and the methods for group decision making (GDM) with DHFPRs are investigated.

Methods

Firstly, a new operator to aggregate dual hesitant fuzzy cognitive information is developed, which treats the membership and non-membership information fairly, and can generate more neutral results than the existing dual hesitant fuzzy aggregation operators. Since compatibility is a very effective tool to measure the consensus in GDM with preference relations, then two compatibility measures for DHFPRs are proposed. After that, the developed aggregation operator and compatibility measures are applied to GDM with DHFPRs and two GDM methods are designed, which can be applied to different decision making situations.

Results and Conclusions

Each GDM method involves a consensus improving model with respect to DHFPRs. The model in the first method reaches the desired consensus level by adjusting the group members’ preference values, and the model in the second method improves the group consensus level by modifying the weights of group members according to their contributions to the group decision, which maintains the group members’ original opinions and allows the group members not to compromise for reaching the desired consensus level. In actual applications, we may choose a proper method to solve the GDM problems with DHFPRs in light of the actual situation. Compared with the GDM methods with IVIFPRs, the proposed methods directly apply the original DHFPRs to decision making and do not need to transform them into the IVIFPRs, which can avoid the loss and distortion of original information, and thus can generate more precise decision results.

Appendix

According to the fact that the envelope of the DHFE d = (h, g) is the IVIFV ([γ ⁻, γ ⁺], [η ⁻, η ⁺]), where γ ⁻ and η ⁻ are the minimum elements in h and g, respectively, and γ ⁺ and η ⁺ are the corresponding maximum elements [21], the DHFPRs U ⁽¹⁾–U ⁽³⁾ are transformed into the following IVIFPRs $\widetilde{U}^{(1)} - \widetilde{U}^{(3)}$:

$$ \widetilde{U}^{(1)} = \left( {\begin{array}{*{20}c} {([0.5,0.5],[0.5,0.5])} & {([0.2,0.5],[0.3,0.5])} & {([0.2,0.4],[0.1,0.6])} & {([0.2,0.7],[0.1,0.2])} & {([0.2,0.5],[0.1,0.5])} & {([0.1,0.8],[0.1,0.2])} \\ {([0.3,0.5],[0.2,0.5])} & {([0.5,0.5],[0.5,0.5])} & {([0.25,0.8],[0.1,0.2])} & {([0.3,0.3],[0.1,0.7])} & {([0.3,0.5],[0.15,0.4])} & {([0.2,0.3],[0.2,0.7])} \\ {([0.1,0.6],[0.2,0.4])} & {([0.1,0.2],[0.25,0.8])} & {([0.5,0.5],[0.5,0.5])} & {([0.3,0.6],[0.1,0.4])} & {([0.2,0.5],[0.1,0.5])} & {([0.3,0.6],[0.2,0.4])} \\ {([0.1,0.2],[0.2,0.7])} & {([0.1,0.7],[0.3,0.3])} & {([0.1,0.4],[0.3,0.6])} & {([0.5,0.5],[0.5,0.5])} & {([0.1,0.3],[0.2,0.6])} & {([0.4,0.6],[0.1,0.3])} \\ {([0.1,0.5],[0.2,0.5])} & {([0.15,0.4],[0.3,0.5])} & {([0.1,0.5],[0.2,0.5])} & {([0.2,0.6],[0.1,0.3])} & {([0.5,0.5],[0.5,0.5])} & {([0.55,0.8],[0.05,0.2])} \\ {([0.1,0.2],[0.1,0.8])} & {([0.2,0.7],[0.2,0.3])} & {([0.2,0.4],[0.3,0.6])} & {([0.1,0.3],[0.4,0.6])} & {([0.05,0.2],[0.55,0.8])} & {([0.5,0.5],[0.5,0.5])} \\ \end{array} } \right), $$

$$ \widetilde{U}^{(2)} = \left( {\begin{array}{*{20}c} {([0.5,0.5],[0.5,0.5])} & {([0.3,0.7],[0.15,0.3])} & {([0.4,0.6],[0.3,0.4])} & {([0.2,0.7],[0.1,0.3])} & {([0.1,0.4],[0.1,0.6])} & {([0.3,0.9],[0.05,0.1])} \\ {([0.15,0.3],[0.3,0.7])} & {([0.5,0.5],[0.5,0.5])} & {([0.1,0.7],[0.1,0.3])} & {([0.15,0.4],[0.05,0.6])} & {([0.1,0.6],[0.2,0.4])} & {([0.1,0.45],[0.2,0.55])} \\ {([0.3,0.4],[0.4,0.6])} & {([0.1,0.3],[0.1,0.7])} & {([0.5,0.5],[0.5,0.5])} & {([0.3,0.5],[0.2,0.5])} & {([0.3,0.6],[0.15,0.4])} & {([0.1,0.7],[0.1,0.3])} \\ {([0.1,0.3],[0.2,0.7])} & {([0.05,0.6],[0.15,0.4])} & {([0.2,0.5],[0.3,0.5])} & {([0.5,0.5],[0.5,0.5])} & {([0.1,0.2],[0.3,0.7])} & {([0.4,0.6],[0.1,0.2])} \\ {([0.1,0.6],[0.1,0.4])} & {([0.2,0.4],[0.1,0.6])} & {([0.15,0.4],[0.3,0.6])} & {([0.3,0.7],[0.1,0.2])} & {([0.5,0.5],[0.5,0.5])} & {([0.5,0.8],[0.15,0.2])} \\ {([0.05,0.1],[0.3,0.9])} & {([0.2,0.55],[0.1,0.45])} & {([0.1,0.3],[0.1,0.7])} & {([0.1,0.2],[0.4,0.6])} & {([0.15,0.2],[0.5,0.8])} & {([0.5,0.5],[0.5,0.5])} \\ \end{array} } \right), $$

$$ \widetilde{U}^{(3)} = \left( {\begin{array}{*{20}c} {([0.5,0.5],[0.5,0.5])} & {([0.1,0.65],[0.1,0.35])} & {([0.1,0.3],[0.25,0.65])} & {([0.2,0.7],[0.05,0.3])} & {([0.15,0.35],[0.5,0.65])} & {([0.25,0.8],[0.1,0.2])} \\ {([0.1,0.35],[0.1,0.65])} & {([0.5,0.5],[0.5,0.5])} & {([0.3,0.8],[0.15,0.2])} & {([0.2,0.2],[0.1,0.8])} & {([0.1,0.4],[0.3,0.6])} & {([0.1,0.3],[0.1,0.7])} \\ {([0.25,0.65],[0.1,0.3])} & {([0.15,0.2],[0.3,0.8])} & {([0.5,0.5],[0.5,0.5])} & {([0.2,0.6],[0.3,0.4])} & {([0.3,0.3],[0.05,0.7])} & {([0.1,0.6],[0.1,0.4])} \\ {([0.05,0.3],[0.2,0.7])} & {([0.1,0.8],[0.2,0.2])} & {([0.3,0.4],[0.2,0.6])} & {([0.5,0.5],[0.5,0.5])} & {([0.2,0.3],[0.2,0.7])} & {([0.1,0.4],[0.2,0.6])} \\ {([0.5,0.65],[0.15,0.35])} & {([0.3,0.6],[0.1,0.4])} & {([0.05,0.7],[0.3,0.3])} & {([0.2,0.7],[0.2,0.3])} & {([0.5,0.5],[0.5,0.5])} & {([0.15,0.6],[0.05,0.4])} \\ {([0.1,0.2],[0.25,0.8])} & {([0.1,0.7],[0.1,0.3])} & {([0.1,0.4],[0.1,0.6])} & {([0.2,0.6],[0.1,0.4])} & {([0.05,0.4],[0.15,0.6])} & {([0.5,0.5],[0.5,0.5])} \\ \end{array} } \right). $$

Then by Eq. (12) in Ref. [12], the transformed IVIFPRs are aggregated into the following collective IVIFPR:

$$ \begin{aligned} \widetilde{U} & = \left( {\begin{array}{*{20}l} {([0.5,0.5],[0.5,0.5])} \hfill & {([0.2,0.6165],[0.1836,0.3835])} \hfill & {([0.2336,0.4336],[0.2165,0.5498])} \hfill \\ {([0.1836,0.3835],[0.2,0.6165])} \hfill & {([0.5,0.5],[0.5,0.5])} \hfill & {([0.2165,0.7666],[0.1166,0.2334])} \hfill \\ {([0.2165,0.5498],[0.2336,0.4336])} \hfill & {([0.1166,0.2334],[0.2165,0.7666])} \hfill & {([0.5,0.5],[0.5,0.5])} \hfill \\ {([0.0834,0.2665],[0.2,0.7])} \hfill & {([0.0833,0.6998],[0.2168,0.3002])} \hfill & {([0.1997,0.4334],[0.2668,0.5666])} \hfill \\ {([0.2326,0.5831],[0.1501,0.4169])} \hfill & {([0.2164,0.4663],[0.1669,0.5002])} \hfill & {([0.1001,0.5329],[0.2665,0.4671])} \hfill \\ {([0.0833,0.1666],[0.2165,0.8334])} \hfill & {([0.1668,0.6499,[0.1335,0.3501])} \hfill & {([0.1335,0.3666],[0.1669,0.6334])} \hfill \\ \end{array} } \right. \\ & \quad \left. {\begin{array}{*{20}l} {([0.2,0.7],[0.0834,0.2665])} \hfill & {([0.1501,0.4169],[0.2326,0.5831])} \hfill & {([0.2165,0.8334],[0.0833,0.1666])} \hfill \\ {([0.2168,0.3002],[0.0833,0.6998])} \hfill & {([0.1669,0.5002],[0.2164,0.4663])} \hfill & {([0.1335,0.3501],[0.1668,0.6499])} \hfill \\ {([0.2668,0.5666],[0.1997,0.4334])} \hfill & {([0.2665,0.4671],[0.1001,0.5329])} \hfill & {([0.1669,0.6334],[0.1335,0.3666])} \hfill \\ {([0.5,0.5],[0.5,0.5])} \hfill & {([0.1332,0.2666],[0.2334,0.6665])} \hfill & {([0.3005,0.5337],[0.1332,0.3661])} \hfill \\ {([0.2334,0.6665],[0.1332,0.2666])} \hfill & {([0.5,0.5],[0.5,0.5])} \hfill & {([0.4007,0.7337],[0.0834,0.2663])} \hfill \\ {([0.1332,0.3661],[0.3005,0.5337])} \hfill & {([0.0834,0.2663],[0.4007,0.7337])} \hfill & {([0.5,0.5],[0.5,0.5])} \hfill \\ \end{array} } \right). \\ \end{aligned} $$

Furthermore, by Eq. (11) in Ref. [12], the compatibility degrees $c(\widetilde{U}^{(k)} ,\widetilde{U})$ of $\widetilde{U}^{(k)} ,k = 1,2,3$ and $\widetilde{U}$ are computed:

$$ c(\widetilde{U}^{(1)} ,\widetilde{U}) = 0.9871,\quad c(\widetilde{U}^{(2)} ,\widetilde{U}) = 0.9677,\quad c(\widetilde{U}^{(3)} ,\widetilde{U}) = 0.9502. $$

Since all $c(\widetilde{U}^{(k)} ,\widetilde{U}) \ge 0.93,k = 1,2,3$, then by Eq. (14) in Ref. [12], the overall preference values $\widetilde{{r_{i} }},i = 1,2, \ldots ,6$ corresponding to the alternatives a _i , i = 1, 2, …, 6 are derived:

$$ \begin{aligned} \widetilde{{r_{1} }} & = ([0.2501,0.5834],[0.2166,0.4083]),\quad \widetilde{{r_{2} }} = ([0.2362,0.4668],[0.2139,0.5277]),\quad \widetilde{{r_{3} }} = ([0.2556,0.4917],[0.2306,0.5055]), \\ \widetilde{{r_{4} }} & = ([0.2167,0.45],[0.2584,0.5166]),\quad \widetilde{{r_{5} }} = ([0.2805,0.5804],[0.2167,0.4029]),\quad \widetilde{{r_{6} }} = ([0.1834,0.3859],[0.2863,0.5974]). \\ \end{aligned} $$

Finally, by Eq. (15) in Ref. [12], the closeness coefficients of the overall preference values are calculated:

$$ c(\widetilde{{r_{1} }}) = 0.541,\quad c(\widetilde{{r_{2} }}) = 0.4925,\quad c(\widetilde{{r_{3} }}) = 0.5022,\quad c(\widetilde{{r_{4} }}) = 0.4788,\quad c(\widetilde{{r_{5} }}) = 0.5479,\quad c(\widetilde{{r_{6} }}) = 0.4383 $$

by which the ranking orders of the alternatives a ₁, a ₂, …, a ₆ are achieved: a ₅ ≻ a ₁ ≻ a ₃ ≻ a ₂ ≻ a ₄ ≻ a ₆.

In what follows, Liao et al.’s method [13] will be adopted to resolve the GDM problem described in “Description of the Problem and the Analysis Process” section. Before doing so, it should be pointed out that as Liao et al. [13] measured the consensus of each individual IVIFPR by calculating the distance between the IVIFPR and the collective IVIFPR, and the smaller the distance, the better the consensus of the IVIFPR, here the threshold value is assumed as τ ^* = 1 − 0.93 = 0.07 for facilitating comparisons with other decision making methods.

Firstly, by Algorithm 2 in Ref. [13], the following multiplicative consistent IVIFPRs $\widetilde{U}_{*}^{(k)}$ from $\widetilde{U}^{(k)} ,k = 1,2,3$ are constructed:

$$ \begin{aligned} \widetilde{U}_{*}^{(1)} & = \left( {\begin{array}{*{20}l} {([0.5,0.5],[0.5,0.5])} \hfill & {([0.2,0.5],[0.3,0.5])} \hfill & {([0.0769,0.8],[0.0455,0.2])} \hfill \\ {([0.3,0.5],[0.2,0.5])} \hfill & {([0.5,0.5],[0.5,0.5])} \hfill & {([0.25,0.8],[0.1,0.2])} \hfill \\ {([0.0455,0.2],[0.0769,0.8])} \hfill & {([0.1,0.2],[0.25,0.8])} \hfill & {([0.5,0.5],[0.5,0.5])} \hfill \\ {([0.0237,0.6044],[0.0968,0.3956])} \hfill & {([0.0122,0.1429],[0.125,0.8571])} \hfill & {([0.1,0.4],[0.3,0.6])} \hfill \\ {([0.0287,0.419],[0.054,0.4663])} \hfill & {([0.0182,0.4833],[0.0593,0.4615])} \hfill & {([0.027,0.5],[0.0455,0.3913])} \hfill \\ {([0.0211,0.3333],[0.1196,0.6101])} \hfill & {([0.0145,0.2325],[0.2172,0.7134])} \hfill & {([0.0084,0.2109],[0.2281,0.75])} \hfill \\ \end{array} } \right. \\ & \quad \left. {\begin{array}{*{20}l} {([0.0968,0.3956],[0.0237,0.6044])} \hfill & {([0.054,0.4663],[0.0287,0.419])} \hfill & {([0.1196,0.6101],[0.0211,0.3333])} \hfill \\ {([0.125,0.8571],[0.0122,0.1429])} \hfill & {([0.0593,0.4615],[0.0182,0.4833])} \hfill & {([0.2172,0.7134],[0.0145,0.2325])} \hfill \\ {([0.3,0.6],[0.1,0.4])} \hfill & {([0.0455,0.3913],[0.027,0.5])} \hfill & {([0.2281,0.75],[0.0084,0.2109])} \hfill \\ {([0.5,0.5],[0.5,0.5])} \hfill & {([0.1,0.3],[0.2,0.6])} \hfill & {([0.4,0.6],[0.1,0.3])} \hfill \\ {([0.2,0.6],[0.1,0.3])} \hfill & {([0.5,0.5],[0.5,0.5])} \hfill & {([0.1196,0.6316],[0.013,0.2727])} \hfill \\ {([0.1,0.3],[0.4,0.6])} \hfill & {([0.013,0.2727],[0.1196,0.6316])} \hfill & {([0.5,0.5],[0.5,0.5])} \hfill \\ \end{array} } \right), \\ \end{aligned} $$

$$ \begin{aligned} \widetilde{U}_{*}^{(2)} & = \left( {\begin{array}{*{20}l} {([0.5,0.5],[0.5,0.5])} \hfill & {([0.3,0.7],[0.15,0.3])} \hfill & {([0.0455,0.8448],[0.0192,0.1552])} \hfill \\ {([0.15,0.3],[0.3,0.7])} \hfill & {([0.5,0.5],[0.5,0.5])} \hfill & {([0.1,0.7],[0.1,0.3])} \hfill \\ {([0.0192,0.1552],[0.0455,0.8448])} \hfill & {([0.1,0.3],[0.1,0.7])} \hfill & {([0.5,0.5],[0.5,0.5])} \hfill \\ {([0.0306,0.3956],[0.1282,0.6044])} \hfill & {([0.027,0.3],[0.0455,0.7])} \hfill & {([0.2,0.5],[0.3,0.5])} \hfill \\ {([0.0514,0.3345],[0.0674,0.6244])} \hfill & {([0.0206,0.5],[0.0297,0.433])} \hfill & {([0.0968,0.7],[0.0455,0.2])} \hfill \\ {([0.026,0.2178],[0.0825,0.7375])} \hfill & {([0.0145,0.1841],[0.0516,0.7617])} \hfill & {([0.0286,0.1695],[0.2592,0.75])} \hfill \\ \end{array} } \right. \\ & \quad \left. {\begin{array}{*{20}l} {([0.1282,0.6044],[0.0306,0.3956])} \hfill & {([0.0674,0.6244],[0.0514,0.3345])} \hfill & {([0.0825,0.7375],[0.026,0.2178])} \hfill \\ {([0.0455,0.7],[0.027,0.3])} \hfill & {([0.0297,0.433],[0.0206,0.5])} \hfill & {([0.0516,0.7617],[0.0145,0.1841])} \hfill \\ {([0.3,0.5],[0.2,0.5])} \hfill & {([0.0455,0.2],[0.0968,0.7])} \hfill & {([0.2592,0.75],[0.0286,0.1695])} \hfill \\ {([0.5,0.5],[0.5,0.5])} \hfill & {([0.1,0.2],[0.3,0.7])} \hfill & {([0.1,0.5],[0.0703,0.3684])} \hfill \\ {([0.3,0.7],[0.1,0.2])} \hfill & {([0.5,0.5],[0.5,0.5])} \hfill & {([0.5,0.8],[0.15,0.2])} \hfill \\ {([0.0703,0.3684],[0.1,0.5])} \hfill & {([0.15,0.2],[0.5,0.8])} \hfill & {([0.5,0.5],[0.5,0.5])} \hfill \\ \end{array} } \right), \\ \end{aligned} $$

$$ \begin{aligned} \widetilde{U}_{*}^{(3)} & = \left( {\begin{array}{*{20}l} {([0.5,0.5],[0.5,0.5])} \hfill & {([0.1,0.65],[0.1,0.35])} \hfill & {([0.0455,0.8814],[0.0192,0.1186])} \hfill \\ {([0.1,0.35],[0.1,0.65])} \hfill & {([0.5,0.5],[0.5,0.5])} \hfill & {([0.3,0.8],[0.15,0.2])} \hfill \\ {([0.0192,0.1186],[0.0455,0.8814])} \hfill & {([0.15,0.2],[0.3,0.8])} \hfill & {([0.5,0.5],[0.5,0.5])} \hfill \\ {([0.0403,0.6202],[0.027,0.3533])} \hfill & {([0.0703,0.1429],[0.0968,0.8571])} \hfill & {([0.3,0.4],[0.2,0.6])} \hfill \\ {([0.0218,0.6029],[0.0322,0.379])} \hfill & {([0.0158,0.7],[0.0968,0.3])} \hfill & {([0.0968,0.6087],[0.0588,0.3913])} \hfill \\ {([0.0232,0.5133],[0.0187,0.4724])} \hfill & {([0.0226,0.5],[0.0287,0.5])} \hfill & {([0.0169,0.555],[0.0438,0.445])} \hfill \\ \end{array} } \right. \\ & \quad \left. {\begin{array}{*{20}l} {([0.027,0.3533],[0.0403,0.6202])} \hfill & {([0.0322,0.379],[0.0218,0.6029])} \hfill & {([0.0187,0.4724],[0.0232,0.5133])} \hfill \\ {([0.0968,0.8571],[0.0703,0.1429])} \hfill & {([0.0968,0.3],[0.0158,0.7])} \hfill & {([0.0287,0.5],[0.0226,0.5])} \hfill \\ {([0.2,0.6],[0.3,0.4])} \hfill & {([0.0588,0.3913],[0.0968,0.6087])} \hfill & {([0.0438,0.445],[0.0169,0.555])} \hfill \\ {([0.5,0.5],[0.5,0.5])} \hfill & {([0.2,0.3],[0.2,0.7])} \hfill & {([0.0423,0.3913],[0.013,0.6087])} \hfill \\ {([0.2,0.7],[0.2,0.3])} \hfill & {([0.5,0.5],[0.5,0.5])} \hfill & {([0.15,0.6],[0.05,0.4])} \hfill \\ {([0.013,0.6087],[0.0423,0.3913])} \hfill & {([0.05,0.4],[0.15,0.6])} \hfill & {([0.5,0.5],[0.5,0.5])} \hfill \\ \end{array} } \right). \\ \end{aligned} $$

Then, by the symmetric interval-valued intuitionistic fuzzy weighted averaging operator in Ref. [13], the multiplicative consistent IVIFPRs are aggregated into the following collective IVIFPR:

$$ \begin{aligned} \widetilde{U}_{ * } & = \left( {\begin{array}{*{20}l} {([0.5,0.5],[0.5,0.5])} \hfill & {([0.1861,0.6197],[0.1692,0.3803])} \hfill & {([0.0543,0.8448],[0.0257,0.1552])} \hfill \\ {([0.1692,0.3803],[0.1861,0.6197])} \hfill & {([0.5,0.5],[0.5,0.5])} \hfill & {([0.2007,0.7697],[0.1147,0.2303])} \hfill \\ {([0.0257,0.1552],[0.0543,0.8448])} \hfill & {([0.1147,0.2303],[0.2007,0.7697])} \hfill & {([0.5,0.5],[0.5,0.5])} \hfill \\ {([0.0308,0.5407],[0.0707,0.45])} \hfill & {([0.0287,0.186],[0.0826,0.814])} \hfill & {([0.1856,0.4329],[0.2639,0.5671])} \hfill \\ {([0.0319,0.45],[0.0491,0.4902])} \hfill & {([0.0181,0.5643],[0.0557,0.3959])} \hfill & {([0.0638,0.6057],[0.0495,0.3193])} \hfill \\ {([0.0233,0.345],[0.0581,0.6126])} \hfill & {([0.0168,0.2897],[0.0712,0.6667])} \hfill & {([0.016,0.2893],[0.1442,0.6595])} \hfill \\ \end{array} } \right. \\ & \quad \left. {\begin{array}{*{20}l} {([0.0707,0.45],[0.0308,0.5407])} \hfill & {([0.0491,0.4902],[0.0319,0.45])} \hfill & {([0.0581,0.6126],[0.0233,0.345])} \hfill \\ {([0.0826,0.814],[0.0287,0.186])} \hfill & {([0.0557,0.3959],[0.0181,0.5643])} \hfill & {([0.0712,0.6667],[0.0168,0.2897])} \hfill \\ {([0.2639,0.5671],[0.1856,0.4329])} \hfill & {([0.0495,0.3193],[0.0638,0.6057])} \hfill & {([0.1442,0.6595],[0.016,0.2893])} \hfill \\ {([0.5,0.5],[0.5,0.5])} \hfill & {([0.1269,0.2636],[0.2303,0.6681])} \hfill & {([0.0804,0.5085],[0.023,0.4106])} \hfill \\ {([0.2303,0.6681],[0.1269,0.2636])} \hfill & {([0.5,0.5],[0.5,0.5])} \hfill & {([0.3757,0.7429],[0.0731,0.2571])} \hfill \\ {([0.023,0.4106],[0.0804,0.5085])} \hfill & {([0.0731,0.2571],[0.3757,0.7429])} \hfill & {([0.5,0.5],[0.5,0.5])} \hfill \\ \end{array} } \right). \\ \end{aligned} $$

Furthermore, by Eq. (34) in Ref. [13], the deviation between each multiplicative consistent IVIFPR and the collective IVIFPR is computed:

$$ d(\widetilde{U}_{*}^{(1)} ,\widetilde{U}_{ * } ) = 0.0501,\quad d(\widetilde{U}_{*}^{(2)} ,\widetilde{U}_{ * } ) = 0.0582,\quad d(\widetilde{U}_{*}^{(3)} ,\widetilde{U}_{ * } ) = 0.0717. $$

Since $d(\widetilde{U}_{*}^{(3)} ,\widetilde{U}_{ * } ) > 0.07$, then by Eqs. (35–38) in Ref. [13] (where η = 0.5), the IVIFPR $\widetilde{U}_{*}^{(3)}$ is modified as:

$$ \begin{aligned} \widetilde{U}_{1*}^{(3)} & = \left( {\begin{array}{*{20}l} {([0.5,0.5],[0.5,0.5])} \hfill & {([0.1375,0.635],[0.1308,0.365])} \hfill & {([0.0497,0.8641],[0.0222,0.1359])} \hfill \\ {([0.1308,0.365],[0.1375,0.635])} \hfill & {([0.5,0.5],[0.5,0.5])} \hfill & {([0.247,0.7852],[0.1313,0.2148])} \hfill \\ {([0.0222,0.1359],[0.0497,0.8641])} \hfill & {([0.1313,0.2148],[0.247,0.7852])} \hfill & {([0.5,0.5],[0.5,0.5])} \hfill \\ {([0.0352,0.581],[0.044,0.4007])} \hfill & {([0.0451,0.1633],[0.0894,0.8367])} \hfill & {([0.2381,0.4163],[0.2304,0.5837])} \hfill \\ {([0.0263,0.5271],[0.0398,0.4338])} \hfill & {([0.0169,0.6348],[0.0737,0.3464])} \hfill & {([0.0788,0.6072],[0.054,0.3545])} \hfill \\ {([0.0233,0.4271],[0.0332,0.5434])} \hfill & {([0.0195,0.3897],[0.0455,0.5858])} \hfill & {([0.0165,0.4161],[0.0808,0.5548])} \hfill \\ \end{array} } \right. \\ & \quad \left. {\begin{array}{*{20}l} {([0.044,0.4007],[0.0352,0.581])} \hfill & {([0.0398,0.4338],[0.0263,0.5271])} \hfill & {([0.0332,0.5434],[0.0233,0.4271])} \hfill \\ {([0.0894,0.8367],[0.0451,0.1633])} \hfill & {([0.0737,0.3464],[0.0169,0.6348])} \hfill & {([0.0455,0.5858],[0.0195,0.3897])} \hfill \\ {([0.2304,0.5837],[0.2381,0.4163])} \hfill & {([0.054,0.3545],[0.0788,0.6072])} \hfill & {([0.0808,0.5548],[0.0165,0.4161])} \hfill \\ {([0.5,0.5],[0.5,0.5])} \hfill & {([0.1601,0.2815],[0.2148,0.6843])} \hfill & {([0.0585,0.4492],[0.0173,0.51])} \hfill \\ {([0.2148,0.6843],[0.1601,0.2815])} \hfill & {([0.5,0.5],[0.5,0.5])} \hfill & {([0.2458,0.6755],[0.0605,0.3245])} \hfill \\ {([0.0173,0.51],[0.0585,0.4492])} \hfill & {([0.0605,0.3245],[0.2458,0.6755])} \hfill & {([0.5,0.5],[0.5,0.5])} \hfill \\ \end{array} } \right). \\ \end{aligned} $$

Then, the obtained new collective IVIFPR $\widetilde{U}_{1 * }$ is

$$ \begin{aligned} \widetilde{U}_{1 * } & = \left( {\begin{array}{*{20}l} {([0.5,0.5],[0.5,0.5])} \hfill & {([0.205,0.6145],[0.1838,0.3855])} \hfill & {([0.0559,0.8379],[0.027,0.1621])} \hfill \\ {([0.1838,0.3855],[0.205,0.6145])} \hfill & {([0.5,0.5],[0.5,0.5])} \hfill & {([0.1868,0.7643],[0.1096,0.2357])} \hfill \\ {([0.027,0.1621],[0.0559,0.8379])} \hfill & {([0.1096,0.2357],[0.1868,0.7643])} \hfill & {([0.5,0.5],[0.5,0.5])} \hfill \\ {([0.0294,0.5271],[0.0825,0.4666])} \hfill & {([0.0246,0.194],[0.0804,0.806])} \hfill & {([0.1703,0.4384],[0.2756,0.5616])} \hfill \\ {([0.034,0.4248],[0.0526,0.509])} \hfill & {([0.0185,0.5402],[0.0507,0.4129])} \hfill & {([0.0595,0.6052],[0.0481,0.308])} \hfill \\ {([0.0234,0.3195],[0.0698,0.6347])} \hfill & {([0.016,0.2601],[0.0824,0.6917])} \hfill & {([0.0158,0.2527],[0.173,0.6915])} \hfill \\ \end{array} } \right. \\ & \quad \left. {\begin{array}{*{20}l} {([0.0825,0.4666],[0.0294,0.5271])} \hfill & {([0.0526,0.509],[0.034,0.4248])} \hfill & {([0.0698,0.6347],[0.0234,0.3195])} \hfill \\ {([0.0804,0.806],[0.0246,0.194])} \hfill & {([0.0507,0.4129],[0.0185,0.5402])} \hfill & {([0.0824,0.6917],[0.016,0.2601])} \hfill \\ {([0.2756,0.5616],[0.1703,0.4384])} \hfill & {([0.0481,0.308],[0.0595,0.6052])} \hfill & {([0.173,0.6915],[0.0158,0.2527])} \hfill \\ {([0.5,0.5],[0.5,0.5])} \hfill & {([0.1173,0.2579],[0.2357,0.6626])} \hfill & {([0.0892,0.5281],[0.0253,0.3788])} \hfill \\ {([0.2357,0.6626],[0.1173,0.2579])} \hfill & {([0.5,0.5],[0.5,0.5])} \hfill & {([0.4244,0.7631],[0.0777,0.2369])} \hfill \\ {([0.0253,0.3788],[0.0892,0.5281])} \hfill & {([0.0777,0.2369],[0.4244,0.7631])} \hfill & {([0.5,0.5],[0.5,0.5])} \hfill \\ \end{array} } \right). \\ \end{aligned} $$

By Eq. (34) in Ref. [13], the deviations between the IVIFPRs $\widetilde{U}_{*}^{(1)}$,$\widetilde{U}_{*}^{(2)}$,$\widetilde{U}_{1*}^{(3)}$ and the new collective IVIFPR $\widetilde{U}_{1 * }$ are computed:

$$ d(\widetilde{U}_{*}^{(1)} ,\widetilde{U}_{1 * } ) = 0.0432,\quad d(\widetilde{U}_{*}^{(2)} ,\widetilde{U}_{1 * } ) = 0.0493,\quad d(\widetilde{U}_{1*}^{(3)} ,\widetilde{U}_{1 * } ) = 0.0489. $$

As $d(\widetilde{U}_{*}^{(1)} ,\widetilde{U}_{1 * } ) \le 0.07$, $d(\widetilde{U}_{*}^{(2)} ,\widetilde{U}_{1 * } ) \le 0.07$ and $d(\widetilde{U}_{1*}^{(3)} ,\widetilde{U}_{1 * } ) \le 0.07$, then by the symmetric interval-valued intuitionistic fuzzy averaging operator in Ref. [13], all the interval-valued intuitionistic fuzzy preference values corresponding to the alternatives a _i , i = 1, 2, …, 6 are fused into the overall preference values $\widetilde{{r_{i} }},i = 1,2, \ldots ,6$:

$$ \begin{aligned} \widetilde{r}_{1} & = ([0.118,0.6052],[0.0686,0.3752]),\quad \widetilde{{r_{2} }} = ([0.1431,0.6063],[0.0733,0.3768]),\quad \widetilde{{r_{3} }} = ([0.134,0.3944],[0.1064,0.5791]), \\ \widetilde{r}_{4} & = ([0.1013,0.397],[0.1452,0.5711]),\quad \widetilde{{r_{5} }} = ([0.1271,0.5878],[0.0983,0.3638]),\quad \widetilde{{r_{6} }} = ([0.0466,0.319],[0.1777,0.6399]). \\ \end{aligned} $$

Finally, according to the comparison laws of IVIFVs [13], the ranking orders of $\widetilde{{r_{1} }},\widetilde{{r_{2} }}, \ldots ,\widetilde{{r_{6} }}$ are achieved: $\widetilde{{r_{2} }} \succ \widetilde{{r_{1} }} \succ \widetilde{{r_{5} }} \succ \widetilde{{r_{3} }} \succ \widetilde{{r_{4} }} \succ \widetilde{{r_{6} }}$. Thus, the ranking orders of the alternatives a ₁, a ₂, …, a ₆ are a ₂ ≻ a ₁ ≻ a ₅ ≻ a ₃ ≻ a ₄ ≻ a ₆.