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

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.

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, … , x_n} is the set of alternatives and D = {d₁, d₂, … , d_m} is the set of decision-makers. The preference information of pairwise comparison given by decision-makers can be described by a matrix P = (p_ij)_n×n. The objects in the matrix are selected from the linguistic term set S = {s_i |i ∈ U} as the evaluation results of x_i and x_j. The number of elements in the set S = {s_i |i ∈ U} 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 s_i ≺ s_j or s_j ≻ s_i, namely, it is inferior s_i to s_j or s_j better than s_i;

SL={s0, s1,⋯, sT2−1} {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.

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 = (p_ij)_n×n, corresponding membership functions µ_P : X × X → S. The element p_ij is selected from a predefined set of linguistic term set S = {s_i |i ∈ U} as the evaluation result of the comparison between alternatives x_i and x_j.

(2) Preference order relation [19]

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

(3) Utility value [20]

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

(4) Reciprocal judgement matrix [21]

The decision-maker compares the alternatives in the alternative set X and gives the reciprocal judgement matrix A = a_ijn×n. a_ij, indicating the relative importance of the alternative x_i to the alternative x_j, and 19≤aij≤9 {1 \over 9} \le {a_{ij}} \le 9 , aij=1aji {a_{ij}} = {1 \over {{a_{ji}}}} , a_ii = 1, ∀i, j.

(5) Complementary judgement matrix [21]

The decision-maker compares the alternatives in the alternative set and gives complementary judgement matrix B = (b_ij)_n×n. b_ij, indicating the degree to which the alternative x_i is superior to the alternative x_j, and 0 ≤ b_ij ≤ 1, b_ij + b_ji = 1, b_ii = 0.5,∀i, j.

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.

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 x₂ ≻ x₄ ≻ x₁ ≻ x₃; the order of d₂ is x₃ ≻ x₂ ≻ x₁ ≻ x₄; the order of d₃ is x₂ ≻ x₁ ≻ x₃ ≻ x₄; the order of d₄ is x₂ ≻ x₁ ≻ x₃ ≻ x₄. Let the weight of the decision-maker be the same; the weighted average operator of the ranking of the schemes is x₂ ≻ x₁ ≻ x₃ ≻ x₄ according to the weighted average operator.

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.