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The consistency method of linguistic information and other four preference information in group decision-making

Data publikacji: 25 May 2022
Tom & Zeszyt: AHEAD OF PRINT
Zakres stron: -
Otrzymano: 19 Jul 2021
Przyjęty: 26 Sep 2021
Informacje o czasopiśmie
License
Format
Czasopismo
eISSN
2444-8656
Pierwsze wydanie
01 Jan 2016
Częstotliwość wydawania
2 razy w roku
Języki
Angielski
Abstract

The transformation method between the linguistic judgement matrix and other four forms of preference information is researched in this paper. The four forms of preference information are preference ordering, utility value, reciprocal judgement matrices and complementary judgement matrices. First, based on the definition of preference information, the mutual transformation methods of linguistic judgement matrix and other four forms of preference information are given. Then new transformation equations are obtained. Next, it is proved that when the linguistic judgement matrix has complete consistency, the other four forms of preference information will have complete consistency too. Finally, a numerical analysis is offered to show that these methods are feasible and effective.

Keywords

Introduction

Owing to differences in cultural background, psychological quality and life experience, different decision-makers in a group may have different individual preferences for the same decision-making problems, and their importance or authority may be different. The heterogeneity of decision-making groups in decision-making practice cannot be ignored. Therefore, the quest for the means to design a scientific, effective and rapid mechanical method for the different preference information given by heterogeneous groups is an urgent basic theoretical problem to be solved. The content of decision-makers’ perception of the external environment is diverse and the information involved is complex. There is also uncertainty in the process of different decision-makers’ perception of information. At the same time, the information itself also has a certain degree of randomness and conflict; so, it is necessary to unify different preference information.

With the development of group decision-making, different forms of preference information have arisen, including the initial preference order, utility value, reciprocal judgement matrix and complementary judgement matrix, but this preference information cannot meet the needs of decision-makers. Since Zadeh and Yager put forward the concept of using linguistic information to represent evaluation results [1, 2, 3, 4, 5, 6, 7], the linguistic judgement matrix has assumed importance as a vital mode of information provided by decision-makers to compare the two alternatives. Due to the complexity and fuzziness of decision-makers’ knowledge background and decision objects, decision-makers may give different forms of preference information for the same decision-making problem. In order to aggregate different preference information given by decision-makers, it is necessary to unify different forms of preference information. The research in this field has attracted extensive attention of scholars and achieved certain results [8, 9, 10, 11, 12, 13, 14, 15, 16]. The aggregation method of natural language and numerical preference information was given in Delgado et al. [8]; the conversion method of four kinds of preference information in group decision-making was given in the study by Wujiang [9]; Xiao et al. [10] study the conversion method of two kinds of preference information with reciprocal judgement matrix and fuzzy complementary judgement matrix in group decision-making; Huayou and Chunlin [11] give three kinds of preference information: order relation value, reciprocal judgement matrix and fuzzy complementary judgement matrix. The effectiveness of information transformation is calculated in the form of value. Most of these studies analyse the mutual transformation of four kinds of preference information, and seldom systematically study the mutual transformation between linguistic information and among them. Yan-Wu and Hua-You [12] transform linguistic judgement matrix into derived matrix, and prove that derived matrix is complementary judgement matrix, which is also equivalent to transforming linguistic judgement matrix into complementary judgement matrix. When the preference information given by experts consists of fuzzy complementary judgement matrix, interval value, positive reciprocal matrix, order relation value and utility value, the preference information of different forms is transformed into fuzzy complementary judgement matrix, and then the ranking value of each scheme is obtained according to the fuzzy complementary judgement matrix in the study by Yangjing [13].

A GDM framework based on consistency driven and consistency driven optimisation model of personalised normalisation method is proposed to manage complete and incomplete probabilistic linguistic preference relation in Tian et al. [15].

This paper presents the conversion formula between linguistic judgement matrix and four kinds of preference information, which mainly solves the problem of mutual conversion between linguistic information and among them, and theoretically proves the rationality of the conversion formula and the consistency after conversion. Finally, an example is used to verify the rationality and effectiveness of the conversion formula.

Preliminqries

I = {1, 2, … , n} and U = {0, 1, 2, … , T} are two sets; the following is a brief description of the linguistic judgement matrix [13, 14, 15, 16]. X = {x1, x2, … , xn} is the set of alternatives and D = {d1, d2, … , dm} is the set of decision-makers. The preference information of pairwise comparison given by decision-makers can be described by a matrix P = (pij)n×n. The objects in the matrix are selected from the linguistic term set S = {si |iU} as the evaluation results of xi and xj. The number of elements in the set S = {si |iU} is called granularity of linguistic term set. For example, a linguistic term set with 13 granularity can be described as S = {s0 = DD = absolute difference, s1 = V HD = quite poor, s2 = HD = very poor, s3 = MD = weak, s4 = LD = Poor, s5 = V LD = slightly poor, s6 = AS = equivalent, s7 = V LP = slightly better, s8 = LP = better, s9 = MP = good, s10 = HP = very good, s11 = V HP = quite good, s12 = DP = absolutely good}. A linguistic term set should have odd elements and the following properties:

Orderliness: when i < j, there is sisj or sjsi, namely, it is inferior si to sj or sj better than si;

Inverse operation neg: neg(si) = sj, j = Ti;

Maximisation operation: if sisj, max {si, sj} = si;

Minimisation operation: if sisj, min {si, sj} = sj;

SL={s0,s1,,sT21} {S^L} = \left\{{{s_0},\,\,{s_1}, \cdots,\,{s_{{{\rm{T}} \over 2} - 1}}} \right\} , SU={sT2+1,,sT} {S^{\rm{U}}} = \left\{{{s_{{{\rm{T}} \over 2} + 1}},\, \cdots,\,{s_T}} \right\} , ST2L={s0,s1,,sT2} S_{{{\rm{T}} \over 2}}^L = \left\{{{s_0},\,\,{s_1}, \cdots,\,{s_{{{\rm{T}} \over 2}}}} \right\} , ST2U={sT2,,sT} S_{{{\rm{T}} \over 2}}^{\rm{U}} = \left\{{{s_{{{\rm{T}} \over 2}}},\, \cdots,\,{s_T}} \right\} are four linguistic term sets.

Definition 1

[13, 14, 15] Let P = (pij)n×n be a matrix, where pij satisfy all the following properties for all i, jI, pijS;pii=sT2;pij=sk,pji=neg(sk) {p_{ij}} \in S;{p_{ii}} = {s_{{T \over 2}}};{p_{ij}} = {s_k},{p_{ji}} = neg\left({{s_k}} \right)

P = (pij)n×n is called linguistic judgement matrix.

The definitions of linguistic information and four kinds of preference information are given as follows.

(1) Language information [1, 3]

The preference information given by the decision-maker can be expressed by linguistic information. The preference information can be described by a matrix P = (pij)n×n, corresponding membership functions µP : X × XS. The element pij is selected from a predefined set of linguistic term set S = {si |iU} as the evaluation result of the comparison between alternatives xi and xj.

(2) Preference order relation [19]

The decision-maker directly gives the order of the decision alternative set according to the individual preference: Ok = {ok(1), ok(2), … , ok(n)} where Ok is a substitution of {1, 2, … , n}. ok(i) indicates the position order of the decision alternative xi. Generally, the smaller the decision alternative ok(i), the better the alternative xi.

(3) Utility value [20]

For the alternative set X, the decision-maker gives the utility value set U = {u1, u2, … , un} of the scheme according to the preference of the scheme, in which ui ∈ [0,1] represents the ranking utility value of the scheme given by the decision-maker. Generally, the larger the utility value ui, the better the alternative xi.

(4) Reciprocal judgement matrix [21]

The decision-maker compares the alternatives in the alternative set X and gives the reciprocal judgement matrix A = aijn×n. aij, indicating the relative importance of the alternative xi to the alternative xj, and 19aij9 {1 \over 9} \le {a_{ij}} \le 9 , aij=1aji {a_{ij}} = {1 \over {{a_{ji}}}} , aii = 1, ∀i, j.

(5) Complementary judgement matrix [21]

The decision-maker compares the alternatives in the alternative set and gives complementary judgement matrix B = (bij)n×n. bij, indicating the degree to which the alternative xi is superior to the alternative xj, and 0 ≤ bij ≤ 1, bij + bji = 1, bii = 0.5,∀i, j.

The consistency method of linguistic information and other four preference information

In the case of strict preference relationship between each alternative, this paper discusses the consistency method of linguistic judgement matrix with the other four kinds of preference information, and assumes that linguistic judgement matrix has satisfactory consistency.

Transformation between linguistic judgement matrix and preference order
Linguistic judgement matrix is transformed into preference order
Definition 2

[13, 14] S = {sα |α ∈ [0q]} is a linguistic term set, let IT:SR,IT(si)=i {I_T}:S \to R,{I_T}\left({{s_i}} \right) = i then the function IT : SR is the subscript function corresponding to the linguistic phrase evaluation set. order IT1:RS,IT1(i)=si,0iq \matrix{{I_T^{- 1}:R \to S,} & {I_T^{- 1}\left(i \right) = {s_i},} & {0 \le i \le q} \cr} then the function IT1:RS I_T^{- 1}:R \to S is called the linguistic information function of the decision-maker on the real number set.

Definition 3

Let C = (cij)n×n be a matrix, where cij={1ifpijST2U0ifpijSL {c_{ij}} = \left\{{\matrix{1 & {{\rm{if}}\,{p_{ij}} \in S_{{T \over 2}}^U} \cr 0 & {{\rm{if}}\,{p_{ij}} \in {S^L}} \cr}} \right. C = (cij)n×n is called the preference relation matrix of the linguistic judgement matrix.

Definition 4

Let C = (cij)n×n be the preference relation matrix of linguistic judgement matrix P = (pij)n×n, where ai=j=1nqij,bj=i=1nqij, \matrix{{{a_i} = \sum\limits_{j = 1}^n {{q_{ij}},}} \cr {{b_j} = \sum\limits_{i = 1}^n {{q_{ij}},}} \cr} where ai is called the row preference value of the row of the preference relation matrix and bj is called the column preference value of the column of the preference relation matrix.

In this way, the row preference value of each alternative can be obtained, and the preference order of the alternative s can be determined according to the size of the row preference value. If calculated by column, the result is just the opposite.

Preference order is transformed into linguistic judgement matrix

The smaller the value ok(i) in the preference order, the higher the position of the alternative i in the preference order, and the better it is compared with other alternatives. Assuming the granularity of the linguistic phrase evaluation set on which the converted linguistic judgement matrix rests to be T + 1, then the conversion formula can be stated as follows: IT(pij)=[T2(1ok(i)ok(j)n1)] {I_T}\left({{p_{ij}}} \right) = \left[ {{T \over 2}\left({1 - {{{o^k}\left(i \right) - {o^k}\left(j \right)} \over {n - 1}}} \right)} \right] where IT (pij) is the subscript function of the evaluation set of language phrases, [] denotes rounding and the following similar statements will not be repeated. Then the linguistic judgement matrix is given by the following: P=(pij)n×n=(IT1([T2(1ok(i)ok(j)n1)]))n×n. P = {\left({{p_{ij}}} \right)_{n \times n}} = {\left({I_T^{- 1}\left({\left[ {{T \over 2}\left({1 - {{{o^k}\left(i \right) - {o^k}\left(j \right)} \over {n - 1}}} \right)} \right]} \right)} \right)_{n \times n}}.

Transformation between linguistic judgement matrix and utility value
Linguistic judgement matrix is transformed into utility value

When calculating the utility value of an alternative i, we use the subscript function to add the linguistic phrase subscripts obtained from comparing all schemes with the scheme together, which is called the subscript sum of the alternative i. If there are n alternatives, we divide by nT, which is the utility value of the alternative i. At the same time, we also know that the better the alternative, the greater the utility value, and the utility value is between [0,1].

Definition 5

Let IT : SR, IT (si) = i be the subscript function of the linguistic judgement matrix P = (pij)n×n, where i=j=1nIT(pij) i' = \sum\limits_{j = 1}^n {{I_T}\left({{p_{ij}}} \right)} and i′ is called the subscript sum of the scheme in the linguistic judgement matrix.

Then the conversion formula of the utility value obtained from the linguistic judgement matrix can be stated as follows: ui=inT {u_i} = {{i'} \over {nT}}

Utility value is transformed into linguistic judgement matrix

The utility value is the preferred utility value given by the decision-maker. The larger the utility value, the better the alternative. The conversion formula of utility value into linguistic judgement matrix is given as follows: IT(pij)=[T2(uiuj)+T2] {I_T}\left({{p_{ij}}} \right) = \left[ {{T \over 2}\left({{u_i} - {u_j}} \right) + {T \over 2}} \right]

Then, the linguistic judgement matrix is expressed as: P=(pij)n×n=(IT1([T2(uiuj)+T2]))n×n. P = {\left({{p_{ij}}} \right)_{n \times n}} = {\left({I_T^{- 1}\left({\left[ {{T \over 2}\left({{u_i} - {u_j}} \right) + {T \over 2}} \right]} \right)} \right)_{n \times n}}.

Transformation between linguistic judgement matrix and reciprocal judgement matrix
Linguistic judgement matrix is transformed into reciprocal judgement matrix

The elements of the reciprocal judgement matrix satisfy [1/9, 9] and aijaji = 1. Assuming that the granularity of the evaluation set of linguistic phrases of linguistic judgement matrix is T + 1, the subscript function representing linguistic phrases is IT (pij). The conversion formula of linguistic judgement matrix into reciprocal judgement matrix can be stated as follows: aij=92TIT(pij)1 {a_{ij}} = {9^{{2 \over T}{I_T}\left({{p_{ij}}} \right) - 1}}

Proof. Let IT (pij) = i, aijaji=92TIT(pij)1×92TIT(pji)1=92Ti1×92T(TI)1=92Ti1+2T(Ti)1=92T(i+Ti)2=90=1 {a_{ij}}{a_{ji}} = {9^{{2 \over T}{I_T}\left({{p_{ij}}} \right) - 1}} \times {9^{{2 \over T}{I_T}\left({{p_{ji}}} \right) - 1}} = {9^{{2 \over T}i - 1}} \times {9^{{2 \over T}\left({T - I} \right) - 1}} = {9^{{2 \over T}i - 1 + {2 \over T}\left({T - i} \right) - 1}} = {9^{{2 \over T}\left({i + T - i} \right) - 2}} = {9^0} = 1 ;

There is pii=sT2 {p_{ii}} = {s_{{T \over 2}}} , aii=92TIT(pii)1=92T×T21=90= {a_{ii}} = {9^{{2 \over T}{I_T}\left({{p_{ii}}} \right) - 1}} = {9^{{2 \over T} \times {T \over 2} - 1}} = {9^0} = .

So, the condition of reciprocal judgement matrix is satisfied.

Reciprocal judgement matrix is transformed into linguistic judgement matrix

When the reciprocal judgement matrix is transformed into the linguistic judgement matrix, the inverse function expressed in Eq. (8) is used as the conversion formula. IT(pij)=[T2log9aij+T2] {I_T}\left({{p_{ij}}} \right) = \left[ {{T \over 2}\log _9^{{a_{ij}}} + {T \over 2}} \right]

Then, the linguistic judgement matrix P=(pij)n×n=(IT1([T2log9aij+T2]))n×n P = {\left({{p_{ij}}} \right)_{n \times n}} = {\left({I_T^{- 1}\left({\left[ {{T \over 2}\log _9^{{a_{ij}}} + {T \over 2}} \right]} \right)} \right)_{n \times n}} .

Definition 6

Let the granularity of the phrase evaluation set on the linguistic judgement matrix P = (pij)n×n be T + 1, and δ(pij)=IT(pij)T \delta \left({{p_{ij}}} \right) = {{{I_T}\left({{p_{ij}}} \right)} \over T}

δ (pij) is called the derivation function of linguistic preference information pij given by the decision-maker, and qij=δ(pij) {q_{ij}} = \delta \left({{p_{ij}}} \right)

Q = (qij)n×n is called the derivation matrix of linguistic judgement matrix P = (pij)n×n.

Definition 7

[13, 14] If the linguistic judgement matrix satisfies δ(pij)=δ(pik)+δ(pkj)0.5,i,j{1,2,,n}, \matrix{{\delta \left({{p_{ij}}} \right) = \delta \left({{p_{ik}}} \right) + \delta \left({{p_{kj}}} \right) - 0.5,} & {\forall i,j \in \left\{{1,2, \cdots,n} \right\},} \cr} then the linguistic judgement matrix P = (pij)n×n is called complete consistency linguistic judgement matrix.

Definition 8

[21] Let A = (aij)n×n be a positive reciprocal judgement matrix, if aij=aikajk,i,j,kI, {a_{ij}} = {{{a_{ik}}} \over {{a_{jk}}}},\,\forall i,j,k \in I,

Then A is called consistent positive reciprocal judgement matrix.

Theorem 1

if the linguistic judgement matrix is completely consistent, then the transformed reciprocal judgement matrix will also be completely consistent.

Proof

If the linguistic judgement matrix is completely consistent, then it is satisfied that δ(pij)=δ(pik)+δ(pkj)0.5,δ(pij)=IT(pij)T,aij=92TIT(pij)1=92(IT(pik)T+IT(pkj)T0.5)1=92TIT(pik)1+2TIT(pkj)1=92TIT(pik)1×92TIT(pkj)1=aikakj. \matrix{{\delta \left({{p_{ij}}} \right)} \hfill & = \hfill & {\delta \left({{p_{ik}}} \right) + \delta \left({{p_{kj}}} \right) - 0.5,\,\,\delta \left({{p_{ij}}} \right) = {{{I_T}\left({{p_{ij}}} \right)} \over T},} \hfill \cr {\,\,\,\,\,\,\,\,\,{a_{ij}}} \hfill & = \hfill & {{9^{{2 \over T}{I_T}\left({{p_{ij}}} \right) - 1}} = {9^{2\left({{{{I_T}\left({{p_{ik}}} \right)} \over T} + {{{I_T}\left({{p_{kj}}} \right)} \over T} - 0.5} \right) - 1}} = {9^{{2 \over T}{I_T}\left({{p_{ik}}} \right) - 1 + {2 \over T}{I_T}\left({{p_{kj}}} \right) - 1}} = {9^{{2 \over T}{I_T}\left({{p_{ik}}} \right) - 1}} \times {9^{{2 \over T}{I_T}\left({{p_{kj}}} \right) - 1}} = {a_{ik}}{a_{kj.}}} \hfill \cr}

It is concluded that the reciprocal judgement matrix is also consistent.

Transformation between linguistic judgement matrix and complementary judgement matrix
Linguistic judgement matrix is transformed into complementary judgement matrix

Let the granularity of the linguistic phrase evaluation set of the linguistic judgement matrix be T + 1 and have satisfactory consistency, and the elements of the complementary judgement matrix B = (bij)n×n satisfy bij ∈ [0.1,0.9], bij + bji = 1; then the conversion formula of the linguistic judgement matrix into the complementary judgement matrix can be stated as follows: bij=45IT(pij)T+110. {b_{ij}} = {4 \over 5}{{{I_T}\left({{p_{ij}}} \right)} \over T} + {1 \over {10}}.

Proof. Let IT (pij) = i, bij+bji=45IT(pij)T+110+45IT(pij)T+110=45iT+110+45TiT+110=45Ti+iT+210=1 {b_{ij}} + {b_{ji}} = {4 \over 5}{{{I_T}({p_{ij}})} \over T} + {1 \over {10}} + {4 \over 5}{{{I_T}({p_{ij}})} \over T} + {1 \over {10}} = {4 \over 5}{i \over T} + {1 \over {10}} + {4 \over 5}{{T - i} \over T} + {1 \over {10}} = {4 \over 5}{{T - i + i} \over T} + {2 \over {10}} = 1 . bii=45IT(pii)T+110=45(Ti+i)T+210=25+110=0.5 {b_{ii}} = {4 \over 5}{{{I_T}({p_{ii}})} \over T} + {1 \over {10}} = {4 \over 5}{{(T - i + i)} \over T} + {2 \over {10}} = {2 \over 5} + {1 \over {10}} = 0.5 .

So the condition of complementary judgement matrix is satisfied.

Complementary judgement matrix is transformed into linguistic judgement matrix

The inverse function expressed in Eq. (14) is used as the conversion function of the complementary judgement matrix to the linguistic judgement matrix, which is qij=[10TbijT8] {q_{ij}} = \left[ {{{10T{b_{ij}} - T} \over 8}} \right]

Then the linguistic judgement matrix P=(pij)n×n=(IT1([10TbijT8]))n×n P = {\left({{p_{ij}}} \right)_{n \times n}} = {\left({I_T^{- 1}\left({\left[ {{{10T{b_{ij}} - T} \over 8}} \right]} \right)} \right)_{n \times n}} .

Definition 9

[23] The complementary judgement matrix B = (bij)n×n satisfies the following conditions: bij=bik+bkj0.5,i,j{1,2,,n} \matrix{{{b_{ij}} = {b_{ik}} + {b_{kj}} - 0.5,} \hfill & {\forall i,j \in \left\{{1,2, \cdots,n} \right\}} \hfill \cr}

Then the matrix is said to be completely consistent.

Theorem 2

If the linguistic judgement matrix has complete consistency, the transformed complementary judgement matrix also has complete consistency.

Proof

If the linguistic judgement matrix is completely consistent, then δ(pij)=IT(pij)T \delta \left({{p_{ij}}} \right) = {{{I_T}\left({{p_{ij}}} \right)} \over T} , bij=45IT(pij)T+110=45(IT(pik)T+IT(pkj)T0.5)+110=45IT(pik)T+45IT(pkj)T25+110=45IT(pik)T+110+45IT(pkj)T+1100.5=bik+bkj0.5 {b_{ij}} = {4 \over 5}{{{I_T}\left({{p_{ij}}} \right)} \over T} + {1 \over {10}} = {4 \over 5}\left({{{{I_T}\left({{p_{ik}}} \right)} \over T} + {{{I_T}\left({{p_{kj}}} \right)} \over T} - 0.5} \right) + {1 \over {10}} = {4 \over 5}{{{I_T}\left({{p_{ik}}} \right)} \over T} + {4 \over 5}{{{I_T}\left({{p_{kj}}} \right)} \over T} - {2 \over 5} + {1 \over {10}} = {4 \over 5}{{{I_T}\left({{p_{ik}}} \right)} \over T} + {1 \over {10}} + {4 \over 5}{{{I_T}\left({{p_{kj}}} \right)} \over T} + {1 \over {10}} - 0.5 = {b_{ik}} + {b_{kj}} - 0.5

So theorem 2 is proved.

Case analysis

Example 1. Five decision-makers give the preference information about four alternatives as follows: d1:O={3,1,4,2};d2{0.5,0.7,0.9,0.1} d3:A=[11/93791821/31/8191/71/21/91 ];d4:B=[0.50.10.60.70.90.50.80.60.40.20.50.90.30.40.10.5 ];d5:P=[s4s2s5s6s6s4s6s5s3s2s4s6s3s3s2s4 ]. \eqalign{& \matrix{{{d_1}:O = \left\{{3,\,1,\,4,\,2} \right\};} & {{d_2}\left\{{0.5,\,0.7,\,0.9,\,0.1} \right\}} & {} \cr} \cr & \matrix{{{d_3}:A = \left[ {\matrix{1 & {1/9} & 3 & 7 \cr 9 & 1 & 8 & 2 \cr {1/3} & {1/8} & 1 & 9 \cr {1/7} & {1/2} & {1/9} & 1 \cr}} \right];} & {{d_4}:B = \left[ {\matrix{{0.5} & {0.1} & {0.6} & {0.7} \cr {0.9} & {0.5} & {0.8} & {0.6} \cr {0.4} & {0.2} & {0.5} & {0.9} \cr {0.3} & {0.4} & {0.1} & {0.5} \cr}} \right];} & {{d_5}:P = \left[ {\matrix{{{s_4}} & {{s_2}} & {{s_5}} & {{s_6}} \cr {{s_6}} & {{s_4}} & {{s_6}} & {{s_5}} \cr {{s_3}} & {{s_2}} & {{s_4}} & {{s_6}} \cr {{s_3}} & {{s_3}} & {{s_2}} & {{s_4}} \cr}} \right].} \cr} \cr} and according to the above preference information, give the order of the four alternatives.

This paper studies the mutual conversion between linguistic judgement matrix and other four kinds of preference information. To judge the order of alternatives, we transform the other decision preference information into the linguistic judgement matrix to judge the order of alternatives. The fifth decision-maker gives a linguistic judgement matrix based on a linguistic phrase evaluation set with a granularity of 9. The granularity of the linguistic phrase evaluation set used to transform other preference information into linguistic judgement matrix is also 9. According to the above transformation formula, the preference information is transformed into linguistic judgement matrix. P1=[s4s1s5s2s7s4s8s5s3s0s4s1s5s3s7s4 ];P2=[s4s3s2s6s5s4s3s6s6s5s4s7s2s2s1s4 ]P3=[s4s0s6s8s8s4s8s5s2s0s4s8s0s3s0s4 ];P4=[s4s0s5s6s8s4s7s5s3s1s4s8s2s3s0s4 ]. \matrix{{{P^1} = \left[ {\matrix{{{s_4}} & {{s_1}} & {{s_5}} & {{s_2}} \cr {{s_7}} & {{s_4}} & {{s_8}} & {{s_5}} \cr {{s_3}} & {{s_0}} & {{s_4}} & {{s_1}} \cr {{s_5}} & {{s_3}} & {{s_7}} & {{s_4}} \cr}} \right];} & {{P^2} = \left[ {\matrix{{{s_4}} & {{s_3}} & {{s_2}} & {{s_6}} \cr {{s_5}} & {{s_4}} & {{s_3}} & {{s_6}} \cr {{s_6}} & {{s_5}} & {{s_4}} & {{s_7}} \cr {{s_2}} & {{s_2}} & {{s_1}} & {{s_4}} \cr}} \right]} \cr {{P^3} = \left[ {\matrix{{{s_4}} & {{s_0}} & {{s_6}} & {{s_8}} \cr {{s_8}} & {{s_4}} & {{s_8}} & {{s_5}} \cr {{s_2}} & {{s_0}} & {{s_4}} & {{s_8}} \cr {{s_0}} & {{s_3}} & {{s_0}} & {{s_4}} \cr}} \right];} & {{P^4} = \left[ {\matrix{{{s_4}} & {{s_0}} & {{s_5}} & {{s_6}} \cr {{s_8}} & {{s_4}} & {{s_7}} & {{s_5}} \cr {{s_3}} & {{s_1}} & {{s_4}} & {{s_8}} \cr {{s_2}} & {{s_3}} & {{s_0}} & {{s_4}} \cr}} \right].} \cr}

Using Eq. (3), the above linguistic judgement matrix is transformed into the following preference relation matrix: Q1=[1010111100101011 ];Q2=[1001110111110001 ]Q3=[1011111100110001 ];Q4=[1011111100110001 ];Q5=[1011111100110001 ]. \matrix{{{Q^1} = \left[ {\matrix{1 & 0 & 1 & 0 \cr 1 & 1 & 1 & 1 \cr 0 & 0 & 1 & 0 \cr 1 & 0 & 1 & 1 \cr}} \right];} & {{Q^2} = \left[ {\matrix{1 & 0 & 0 & 1 \cr 1 & 1 & 0 & 1 \cr 1 & 1 & 1 & 1 \cr 0 & 0 & 0 & 1 \cr}} \right]} & {} \cr {{Q^3} = \left[ {\matrix{1 & 0 & 1 & 1 \cr 1 & 1 & 1 & 1 \cr 0 & 0 & 1 & 1 \cr 0 & 0 & 0 & 1 \cr}} \right];} & {{Q^4} = \left[ {\matrix{1 & 0 & 1 & 1 \cr 1 & 1 & 1 & 1 \cr 0 & 0 & 1 & 1 \cr 0 & 0 & 0 & 1 \cr}} \right];} & {{Q^5} = \left[ {\matrix{1 & 0 & 1 & 1 \cr 1 & 1 & 1 & 1 \cr 0 & 0 & 1 & 1 \cr 0 & 0 & 0 & 1 \cr}} \right].} \cr}

We arrange the alternatives according to the number of 1 in the preference relation matrix. The order of d1 is x2x4x1x3; the order of d2 is x3x2x1x4; the order of d3 is x2x1x3x4; the order of d4 is x2x1x3x4. Let the weight of the decision-maker be the same; the weighted average operator of the ranking of the schemes is x2x1x3x4 according to the weighted average operator.

Conclusions

In this paper, the transform formula between linguistic judgement matrix and other four preference information (preference order, utility value, reciprocal judgement matrix and complementary judgement matrix) is given systematically, and the rationality of the transform formula is theoretically proved. This method provides the transformation of different preference information forms in group decision-making. Further, more detailed and accurate transform methods between preference information and its application need to be studied further.

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