Model for selection of hospital constructions with probabilistic linguistic GRP method

1 Introduction

In practical life, making decision in sufficient and precise numerical value may be difficult due to the lack of information for decision makers [1–10]. In daily life, decision makers usually use linguistic terms to express which they prefer and give probabilities to them respectively [11–14]. When giving expression to preferences of qualitative information, some possible probabilities linguistic terms may be considered simultaneously [15]. Thus, the conception of probabilistic linguistic term sets was first propose by Pang, Wang and Xu [16], who came up with some basic operational laws and aggregation for PLTSs. Zhang, Dong and Xu [17] defined the concept of distribution assessments in a linguistic term set, and investigated the operational laws of linguistic distribution assessments. Zhang, Yu, Martinez and Gao [18] used the linguistic distribution-based approach to manage the multigranular unbalanced hesitant fuzzy linguistic information in multi-attribute large-scale GDM. Dong, Wu, Zhang and Zhang [19] developed the concept of linguistic distribution assessments with interval symbolic proportions under multi-granular unbalanced linguistic contexts. Zhang, Xiao, Palomares, Liang and Dong [20] used the linguistic distribution-based optimization approach for large-scale GDM with comparative linguistic information with application on the selection of wastewater disinfection technology. Wu, Dong, Qin and Pedrycz [21] used the linguistic distribution and priority based approximation to flexible linguistic expressions in decision-making. Wu, Zhang, Kou, Zhang, Chao, Li, Dong and Herrera [22] defined the distributed linguistic representations in decision making. Zhang, Guo and Martinez [23] defined the computational model based on the use of extended linguistic hierarchies, which not only could be used to operate with multigranular linguistic distribution assessments but also could give interpretable linguistic results to DMs. Tang, Peng, Zhang, Pedrycz and Yang [24] defined the consistency and consensus-driven models to personalize individual semantics of linguistic terms for supporting GDM updates with distribution linguistic preference relations. Gou and Xu [25] redefined a few logical operational laws for PLTS built on two equivalent transformation function, which can avoid the excessive or lack of the probability information. Zhai, Xu and Liao [26] proposed the conception of probabilistic linguistic vector-term set, which improved the accuracy of multi-granular linguistic information and extended the application of it. The existing approaches associated with PLTSs are limited or highly complex in real applications. So Bai, Zhang, Qian and Wu [27] used a possibility degree formula for ranking PLTSs and came up with a more efficient way to handle with MCDM problems. Bai, Zhang, Shen, Huang and Fan [28] introduced interval-valued probabilistic linguistic term set, and it reflecting the indeterminacy and disparity of decision makers which the linguistic information is incomplete. Li, Wang, Xiong and Liu [29] integrated fuzzy petri nets and probabilistic linguistic term sets to propose an effect analysis approach and a novel failure mode. And the PLTSs were used to capture the uncertainty of Failure mode and effect analysis team members’ subjective judgments, which is reliable and reasonable. In this paper, the GRP method utilized by us to handle probabilistic linguistic term sets.

The CRITIC method, proposed by Diakoulaki, Mavrotas and Papayannakis [30], which is an effective way which determines the weights of attributes in solving MCDM problems. The weights contain not only contrast intensity but also conflict. Thus, this method extracts all information contained in the evaluation criteria by evaluating the relative strength and the conflict of the criterion [31]. Combined with other methods, CRITIC method can be applied in many fields. Miao, Teng, Wang and Zhou [32] integrated the CRITIC method and GRA method to assess the population vulnerability of geological disasters, which helped get the ranking of evaluation results and come to relevant conclusions. Abdel-Basset and Mohamed [33] proposed the new TOPSIS-CRITIC model for sustainable supply chain risk management, which measured the uncertainty of the risk significantly. Xu, Liu and Zhang [34] combined the CRITIC method and GRA method to estimate simplified likelihood of ship total loss, which identifying the inter-correlation of three factors mentioned in the paper in relation to ship total loss. In this paper, we took advantage of CRITIC and GRP method to optimize the site selection of hospital constructions. The CRITIC method evaluated the weight of the attributes that we considered, and it helped us with the following method design.

The grey relational analysis was first developed by Deng [35] to solve the problem of multiple attribute decision making, and it was proposed to cope with uncertain and incomplete systems [15, 36]. The grey relational analysis can depict multiple attributes with physical measurement units, score attributes and different value types, consisting of those attributes with limited dependability and veracity [37]. The grey relational analysis is appropriate for coming up with issues along with sophisticated interrelationships among multiple factors and could be lend itself to solve the evaluation problems of multi-objective [38–40]. The grey relational analysis obtains the difference between a reference series and each compared series. Thus, applying the grey relational analysis method can sort different alternatives. The grey relational analysis method has been utilized in coping with a plenty of MADM issues. Zheng, Jing, Huang and Gao [41] proposed grey relational projection method on the basis of the grey relational analysis method and vector projection. The projection value is depicted built on the product of the norm and the cosine of the angle between the given alternatives and the ideal alternative. The main merits of the grey relational projection method are that the results are built on the original data, and it’s easy and reliable to calculate. Built on intuitionistic trapezoidal fuzzy number, Zhang, Jin and Liu [42] defined a grey relational projection method for multiple attribute decision making, and they took advantages of an illustrative example to demonstrate its practicality and effectiveness. Liu, You, Fan and Lin [43] researched on the failure mode and effects analysis by utilized the grey relational projection method and D numbers. They utilized the grey relational projection method to determine the risk priority order of the failure modes that have been identified. With interval-valued dual hesitant fuzzy information, Zang, Sun and Han [44] devised the grey relational projection method for multi-attribute decision making problem. Their proposed method is clear and simple, and is easy to be operated, which provides a new idea to solve the fuzzy MADM problem. With hesitant intuitionistic fuzzy linguistic information, Zang, Zhao, Li and Nazir [45] defined the grey relational bidirectional projection method for MADM, which was demonstrated flexibility and effectiveness.

In this paper, to solve the problems of multiple attribute group decision making in probabilistic linguistic environment, the probabilistic linguistic grey relational projection method, a new extension of the projection measure, was extended by us. The results are built on the original data, and it’s easy and reliable to calculate. So it provides a novel idea to deal with fuzzy MAGDM problem. Hence, the following summarized are the principal purposes of this paper: (1) to present a GRP measure in the form of PLTSs; (2) to introduce CRITIC method in the form of PLTSs to count the attribute weights; (3) to present a probabilistic linguistic MAGDM method built on the GRP and CRITIC method; (4) a case study for site selection of hospital constructions is applied to demonstrate the extended method; (5) the comparative studies are made with the extend TOPSIS method, MABAC method, TODIM method and PLWA operator demonstrating the availability of the PL-GRP method.

The following is the general structure of this paper. Section 2 supplies a few conception correlated with PLTSs. Section 3 proposes probabilistic linguistic grey relational projection method for MAGDM with weight determined by CRITIC method. In Section 4, a case study for site selection of hospital constructions is applied to demonstrate the extended method, and then the extend TOPSIS method and PLWA operator which are used to do comparative studies demonstrate the availability of the PL-GRP method. Finally, the study ends up with conclusions in section 5.

2 Preliminaries

In this section, some basic concepts relevant to probabilistic linguistic term sets are reviewed briefly.

3 GRP method for PL-MAGDM with weight determined by CRITIC method

To solve MAGDM problems, a novel probabilistic linguistic grey relational projection (PL-GRP) method which the weight information is unknown is projected in this section. The following are relevant notations. Let E ={ E₁, E₂, . . . , E _e  }

be a group of expert, H ={ H₁, H₂, . . . , H _m  } be a discrete number of alternatives, and W ={ w₁, w₂, . . . , w _n  } be the weight vector of attributes P ={ P₁, P₂, . . . , P _n  }. Suppose the experts are qualified and give the values of n qualitative attributes. Their values are denoted as linguistic expressions information l ij k ( i = 1 , 2 , . . . , m ; j = 1 , 2 , . . . , n ; k = 1, 2, . . . , # L (p)).

Then, the extend GRP method with weight determined by CRITIC method is proposed (see the flowchart).

The following are the specific calculating steps:

Step 1. Transform the linguistic information l ij k ( i = 1 , 2 , . . . , m ; j = 1 , 2 , . . . , n ; k = 1 , 2 , . . . , # L ( p ) ) into probabilistic linguistic information L ij ( k ) ( p ij ( k ) ) , k = 1, 2, . . . , # L (p), then the probabilistic linguistic decision matrix L = (L _ij  (p)) _m×n is expressed as L i j ( p ) = { L i j ( k ) ( p i j ( k ) ) | k = 1 , 2 , ... , # L i j ( p ) } (i = 1, 2, . . . , m ; j = 1, 2, . . . , n).

Step 2. Transform L ij ( k ) ( p ij ( k ) ) into L ˜ = ( L ˜ ij ( p ) ) m × n , L ˜ ij ( p ) = { γ ˜ ij ( k ) ( p ij ( k ) ) | k = 1 , 2 , . . . , # L ˜ ( p ) } by Definition 3 and Definition 4 respectively. Then the alternative H _i  ∈ H concerning all the attributes P can be expressed as PLA i = { L ˜ i 1 ( k ) ( p i 1 ( k ) ) , L ˜ i 2 ( k ) ( p i 2 ( k ) ) , . . . , L ˜ in ( k ) ( p in ( k ) ) | k = 1 , 2 , . . . , # L ˜ ( p ) } .

Step 3. Compute the weight values by CRITIC method.

Criteria Importance Through Intercriteria Correlation (CRITIC) [30] method are used to calculate the weights of evaluation criteria, which extracts all message comprised in the attributes [31]. On the one hand, the standard deviation σ _j can be used to evaluate the relative strength of the evaluation criteria, and then the information of the evaluation index can be fully considered. On the other hand, the correlation coefficient ρ _jt can be used to evaluate the conflict between evaluation criteria, and then coordination of evaluation criteria can be fully considered. So the CRITIC method is a valid way to determine weights of evaluation criteria.

Firstly, shift the positive attributes into negative attributes. If S ={ s _α |α = - τ, . . . , -1, 0, 1, . . . , τ } is a LTS, the positive attributes value is s _α and the negative attributes value is s_-α. Then transform the probabilistic linguistic group decision matrix L ˜ = ( L ˜ ij ( p ) ) m × n by converting the linguistic interval [- τ, τ] into numerical interval [0, 1] according to definition 5, and it is expressed as L ̈ = ( L ̈ ij ( p ) ) m × n .

Secondly, computing the correlation coefficient as follow: ρ j t = ∑ i = 1 m ( ∑ k = 1 # L ( p ) γ ¨ i j ( k ) p ¨ i j ( k ) - γ ¯ j ( k ) p ¯ j ( k ) ) ( ∑ k = 1 # L ( p ) γ ¨ i t ( k ) p ¨ i t ( k ) - γ ¯ t ( k ) p ¯ t ( k ) ) ∑ i = 1 m ( ∑ k = 1 # L ( p ) γ ¨ i j ( k ) p ¨ i j ( k ) - γ ¯ j ( k ) p ¯ j ( k ) ) 2 ∑ i = 1 m ( ∑ k = 1 # L ( p ) γ ¨ i t ( k ) p ¨ i t ( k ) - γ ¯ t ( k ) p ¯ t ( k ) ) 2(7) where γ ¯ j ( k ) ( p ¯ j ( k ) ) = ∑ i = 1 m γ ¨ i j ( k ) m ( ∑ i = 1 m p ¨ i j ( k ) m ) and γ ¯ t ( k ) ( p ¯ t ( k ) ) = ∑ i = 1 m γ ¨ i t ( k ) m ( ∑ i = 1 m p ¨ i t ( k ) m )

Thirdly, formulating the standard deviation as follows: σ j = 1 m − 1 ∑ i = 1 m ( ∑ k = 1 # L ( p ) γ ¨ i j ( k ) p ¨ i j ( k ) - γ ¯ j ( k ) p ¯ j ( k ) ) 2(8)

Then, the index C _j is calculated as follows: C j = σ j ∑ k = 1 n ( 1 - ρ jt )(9)

Finally, the vector of evaluation criteria weight W ={ w₁, w₂, . . . , w _n  } is counted: w j = C j ∑ j = 1 n C j(10)

It is concluded that the higher value of w _j , the more preferred the attribute is.

Step 4. Definite the probabilistic linguistic positive ideal solution(PLPIS) and probabilistic linguistic negative ideal solution(PLNIS) [49]: PLPIS = ( PLPIS 1 , PLPIS 2 , . . . , PLPIS n )(11) PLNIS = ( PLNIS 1 , PLNIS 2 , . . . , PLNIS n )(12) where PLPIS j = ( γ j ( k ) p j ( k ) | k = 1 , 2 , . . . , # L ( p ) ) , E ( PLPIS j ) = { max i E ( L ij ( p ) ) } PLNIS j = ( γ j ( k ) p j ( k ) | k = 1 , 2 , . . . , # L ( p ) ) , E ( PLNIS j ) = { min i E ( L ij ( p ) ) }

Step 5. Calculate the Hamming distance between L ̈ ij ( p ) and PLPIS which expresses as D ij + , and the Hamming distance between L ̈ ij ( p ) and PLNIS which expresses as D ij - respectively by definition 8.

Step 6. Formulate the grey relational coefficient of each alternative from the PLPIS and PLNIS: g ij + = min i min j D ij + + θ max i max j D ij + D ij + + θ max i max j D ij +(13) g ij - = min i min j D ij - + θ max i max j D ij - D ij - + θ max i max j D ij -(14) where θ ∈ [0, 1] represents the resolution coefficient which is given by the decision makers.

Step 7. Construct the grey relational coefficient matrix GRCM + = [ g ij + ] m × n and GRCM - = [ g ij - ] m × n as follow: GRCM + = [ g 11 + g 12 + ⋯ g 1 n + g 21 + g 22 + ⋯ g 2 n + ⋮ ⋮ ⋱ ⋮ g m 1 + g m 2 + ⋯ g mn + ] , GRCM - = [ g 11 - g 12 - ⋯ g 1 n - g 21 - g 22 - ⋯ g 2 n - ⋮ ⋮ ⋱ ⋮ g m 1 - g m 2 - ⋯ g mn - ]

And the grey relational coefficient between PLPIS and PLPIS, PLNIS and PLNIS are shown as: GRCM 0 + = ( g 01 + g 02 + ⋯ g 0 n + ) = 1 1 ⋯ 1 ︸ n(15) GRCM 0 - = ( g 01 - g 02 - ⋯ g 0 n - ) = 1 1 ⋯ 1 ︸ n(16)

Step 8. Construct the weighted grey relational coefficient matrix WGRCM + = [ w j g ij + ] m × n and WGRCM - = [ w j g ij - ] m × n as follow: WGRCM + = [ w 1 g 11 + w 2 g 12 + ⋯ w n g 1 n + w 1 g 21 + w 2 g 22 + ⋯ w n g 2 n + ⋮ ⋮ ⋱ ⋮ w 1 g m 1 + w 2 g m 2 + ⋯ w n g mn + ] WGRCM - = [ w 1 g 11 - w 2 g 12 - ⋯ w n g 1 n - w 1 g 21 - w 2 g 22 - ⋯ w n g 2 n - ⋮ ⋮ ⋱ ⋮ w 1 g m 1 - w 2 g m 2 - ⋯ w n g mn - ]

And the weighted grey relational coefficient between PLPIS and PLPIS, PLNIS and PLNIS are shown as: WGRCM 0 + = ( w 1 g 01 + w 2 g 02 + ⋯ w n g 0 n + ) = ( w 1 w 2 ⋯ w n )(17) WGRCM 0 - = ( w 1 g 01 - w 2 g 02 - ⋯ w n g 0 n - ) = ( w 1 w 2 ⋯ w n )(18)

Step 9. Refer to the projection formula [43, 44], and the weighted grey relational coefficient matrix WGRCM + = [ w j g ij + ] m × n and WGRCM - = [ w j g ij - ] m × n , the weighted grey relational projection(WGRP) of alternative H _i onto PLPIS or PLNIS could be obtained as shown: WGRP i + = | WGRCM i + | · cos ( WGRCM i + , WGRCM 0 + ) = ∑ j = 1 n ( w j g ij + ) 2 · ∑ j = 1 n ( w j g ij + · w j ) ∑ j = 1 n ( w j g ij + ) 2 · ∑ j = 1 n w j 2 = ∑ j = 1 n ( w j g ij + · w j ) ∑ j = 1 n w j 2 = ∑ j = 1 n w ¯ j g ij +(19) WGRP i - = | WGRCM i - | · cos ( WGRCM i - , WGRCM 0 - ) = ∑ j = 1 n ( w j g ij - ) 2 · ∑ j = 1 n ( w j g ij - · w j ) ∑ j = 1 n ( w j g ij - ) 2 · ∑ j = 1 n w j 2 = ∑ j = 1 n ( w j g ij - · w j ) ∑ j = 1 n w j 2 = ∑ j = 1 n w ¯ j g ij -(20) where w ¯ j = w j 2 / ∑ j = 1 n w j 2

Step 10. For alternative H _i , the bigger value of WGRP i + is, the closer it is to PLPIS, whereas the smaller value of WGRP i - is, the closer it is to PLNIS, and vice versa. Hence, for fusing the grey relational projection values across-the-board and choosing the most ideal alternative, we could build the relative closeness formula in line with the TOPSIS. RC i = WGRP i + WGRP i + + WGRP i - = ∑ j = 1 n w ¯ j g ij + ∑ j = 1 n w ¯ j g ij + + ∑ j = 1 n w ¯ j g ij -(21)

Distinctly, the larger RC _i is, the better H _i is, and vice versa.

5 Conclusion

In this paper, a novel GRP method was designed for PL-MAGDM, in which the attribute values are expressed by PLTSs and the weight of attributes are uncharted. Firstly, the basic concept of the PLTSs were reviewed. Then, the PL-GRP method derived from the Hamming distance measure was designed and the CRITIC method was built to obtain the weight of attributes. Finally, the feasibility and validity of the designed method is interpreted by a numerical example for site selection of hospital constructions. The PL-GRP method provides a novel idea to deal with fuzzy MAGDM problem which is effective and easy to be operated. We will put the designed method and algorithms into use for some other practical problems in the near future. For instance, the aspect of supplier selection, image reconstruction, vehicle routing problem, logistics services providers selection, pattern recognition and boundary detection [57–62] can use the new PL-GRP method to come up with relevant issues. We will also be committed to the extension of the designed algorithms for other uncertain MAGDM issues [63–71] and consensus analysis [72, 73].