Model for selection of hospital constructions with probabilistic linguistic GRP method

Abstract

Probabilistic linguistic term sets are used to express uncertain decision information in multiple attribute group decision making problems. For probabilistic linguistic multiple attribute group decision making (MAGDM) with weight determined by CRITIC (Criteria Importance Through Intercriteria Correlation) method, the probabilistic linguistic grey relational projection method is proposed in this paper. Firstly, the correlation coefficient among attributes and standard deviation of each attribute are utilized to compute the attributes weights. Then the most ideal alternative is decided by means of counting the grey relational projection (GRP) from probabilistic linguistic positive ideal solution and probabilistic linguistic negative ideal solution. In the end, a numerical example for site selection of hospital constructions is applied to further account for the extended method. The result demonstrates the availability of the proposed method and it can be used in other fields which refers to problems of selection.

Keywords

Multiple attribute group decision making;probabilistic linguistic term sets CRITIC method grey relational projection method site selection of hospital constructions

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.

2.1 The concept of PLTSs

The concept called probabilistic linguistic term set was proposed by Pang, Wang and Xu [16] to depict qualitative information.

Definition 1 [16]. Let S = {s_-τ, . . . , s_-1, s₀, s₁, . . . , s_τ} be a LTS, a PLTS can be defined as:

$\begin{matrix} L (p) = {L^{(k)} (p^{(k)}) | L^{(k)} \in S, p^{(k)} ⩾ 0, \\ k = 1, 2, . . ., # L (p), \sum_{k = 1}^{# L (p)} p^{(k)} ⩽ 1} \end{matrix}$ (1) where L^(k) (p^(k)) is the linguistic term L^(k) associated with the probability p^(k), and # L (p) is the number of all different linguistic term in L (p).

If $\sum_{k = 1}^{# L (p)} p^{(k)} = 1$ , then the decision maker provides complete assessment information of all possible linguistic terms; if $0 ⩽ \sum_{k = 1}^{# L (p)} p^{(k)} ⩽ 1$ , then the decision maker offers only partial assessment information because current knowledge is not enough for decision maker to provide complete assessment information and it is a very common case in the practical decision making problems; if $\sum_{k = 1}^{# L (p)} p^{(k)} = 0$ , then the decision maker does not offer any assessment.

Definition 2 [16]. Given a PLTS, L (p) ={ L^(k) (p^(k)) |L^(k) ∈ S, p^(k) ⩾ 0, k = 1, 2, . . . , # L (p) }, and γ^(k) is the subscript of linguistic term L^(k). L (p) is called an ordered PLTS, if the linguistic terms L^(k) (p^(k)) (k = 1, 2, . . . , # L (p)) are arranged according to the values of γ^(k) (k = 1, 2, . . . , # L (p)) in ascending order.

2.2 The normalization of PLTSs

For calculating easily, Pang, Wang and Xu [16] normalized the PLTS.

Definition 3 [16]. Given a PLTS L (p) with $\sum_{k = 1}^{# L (p)} p^{(k)} < 1$ , then the associated PLTS $\tilde{L} (p)$ is defined by $\tilde{L} (p) = {L^{(k)} ({\tilde{p}}^{(k)}) | k = 1, 2, ..., # L (p)}$ , where ${\tilde{p}}^{(k)} = p^{(k)} / \sum_{k = 1}^{# L (p)} p^{(k)}$ .

Now, the superiorities of the PL-GRP method compared with other existed methods are summarized as follows: (1) The GRP method, which could connect the impact of the whole attribute space and avert the monodirectional deviation, merges the GRA with the projection by comparison to the ordinary projection method. That is, it could shun the deviations triggered by the comparison between the single attribute value and each alternative. Consequently, it could represent the impact of the whole attribute space across the board. (2) The weight is identified derived from the CRITIC method, which can cut down the impact of the lack of information, and make it come much nearer to the real intention of DMs. (3) As we know, the PLTSs is a generalization of the hesitant fuzzy linguistic sets. Therefore, the new PL-GRP method could be utilized to settle not only MAGDM with PLTSs but also the MAGDM with the hesitant fuzzy linguistic set and other fuzzy sets and systems. (4) By comparison with the hesitant fuzzy linguistic MADM or MAGDM methods studied in Ref. [52 –56], the PL-GRP method doesn’t take the magnitude of the angle cosine or distance measure as the criterion of alternative ranking. The GRP method links the correlation coefficient with the distance, which could depict the close degree between the given alternatives and the ideal solution.

Definition 4 [16]. Let L₁ (p) and L₂ (p) be any two PLTSs, $L_{1} (p) = {L_{1}^{(k)} (p_{1}^{(k)}) | k = 1, 2, . . ., # L_{1} (p)}$ and L₂ (p) = ${L_{2}^{(k)} (p_{2}^{(k)}) | k = 1, 2, . . ., # L_{2} (p)}$ , and let # L₁ (p) and # L₂ (p) be the numbers of linguistic terms in L₁ (p) and L₂ (p) respectively. If # L₁ (p) ⩾ # L₂ (p), then # L₁ (p) - # L₂ (p) linguistic terms are added to L₂ (p) until the number of linguistic terms L₁ (p) and L₂ (p) are equal. The added linguistic terms are the smallest ones in L₂ (p), and the probabilities of all the linguistic terms are zero.

Definition 5 [16]. Let S = {s_t|t = - τ, . . . , -2, - 1, 0, 1, 2, . . . , τ} be a finite and ordered discrete LTS, then the linguistic variable s_t which can express the equivalent information to the membership degree γ is obtained with the following function g: $g : [- τ, τ] \to [0, 1] g (s_{t}) = \frac{t + τ}{2 τ} = γ$ (2) where t is the subscript of a linguistic term in a PLE and γ is a positive value with [0, 1].

2.3 The comparison of PLTSs

In order to simplify the calculation, we choose the comparison methods put forward by Pang, Wang and Xu [16] to design our proposed method in this paper.

Definition 6 [16]. Let L (p) = {L^(k) (p^(k)) |k = 1, 2, . . . , # L (p)} be a PLTS, and γ^(k) be the subscript of linguistic term L^(k). Then the score of L (p) is: $E (L (p)) = s_{\bar{α}}$ (3) where $\bar{α} = \sum_{k = 1}^{# L (p)} γ^{(k)} p^{(k)} / \sum_{k = 1}^{# L (p)} p^{(k)}$ .

For two PLTSs L₁ (p) and L₂ (p), if E₁ (L (p)) ⩾ E₂ (L (p)), then L₁ (p) > L₂ (p); if E₁ (L (p)) ⩽ E₂ (L (p)), then L₁ (p) < L₂ (p).

Definition 7 [16]. Let L (p) ={ L^(k) (p^(k)) |k = 1, 2, . . . , # L (p) } be a PLTS, and γ^(k) be the subscript of linguistic term L^(k), and $E (L (p)) = s_{\bar{α}}$ , where $\bar{α} = \sum_{k = 1}^{# L (p)} γ^{(k)} p^{(k)} / \sum_{k = 1}^{# L (p)} p^{(k)}$ . Then the deviation degree of L (p) is: $σ (L (p)) = {(\sum_{k = 1}^{# L (p)} (p^{(k)} {(γ^{(k)} - \bar{α})}^{2}))}^{1 / 2} / \sum_{k = 1}^{# L (p)} p^{(k)}$ (4) For two PLTSs L₁ (p) and L₂ (p) with E₁ (L (p)) , E₂ (L (p)), if σ₁ (L (p)) > σ₂ (L (p)), then L₁ (p) < L₂ (p); if σ₁ (L (p)) < σ₂ (L (p)), L₁ (p) > L₂ (p).

2.4 The distance of PLTSs

Definition 8 [16]. Let $L_{1} (p) = {L_{1}^{(k)} (p_{1}^{(k)}) | k = 1, 2, . . ., # L_{1} (p)}$ and $L_{2} (p) = {L_{2}^{(k)} (p_{2}^{(k)}) | k = 1, 2, . . ., # L_{2} (p)}$ be two PLTSs with # L₁ (p) = # L₂ (p), then the Hamming distance d_HD (L₁ (p) , L₂ (p)) between L₁ (p) and L₂ (p) is defined as follows: $d_{HD} (L_{1} (p), L_{2} (p)) = \frac{\sum_{k = 1}^{# L (p)} | γ_{1}^{(k)} p_{1}^{(k)} - γ_{2}^{(k)} p_{2}^{(k)} |}{# L (p)}$ (5)

2.5 The operations of PLTSs

Definition 9 [46]. Let L (p) = {L^(k) (p^(k)) |k = 1, 2, . . . , # L (p)} be a PLTSs, then the probabilistic linguistic weighted averaging (PLWA) operator is: $\begin{matrix} PLWA (L_{1} (p), L_{2} (p), . . ., L_{n} (p)) = w_{1} L_{1} (p) \oplus w_{2} L_{2} (p) \oplus \dots \oplus w_{n} L_{n} (p) \\ = ⋃_{L_{1}^{(k)} \in L_{1} (p)} {w_{1} p_{1}^{(k)} L_{1}^{(k)}} \oplus ⋃_{L_{2}^{(k)} \in L_{2} (p)} {w_{2} p_{2}^{(k)} L_{2}^{(k)}} \oplus \dots \oplus ⋃_{L_{n}^{(k)} \in L_{n} (p)} {w_{n} p_{n}^{(k)} L_{n}^{(k)}} \end{matrix}$ (6)

2.6 The operations of LTSs

Definition 10 [47, 48]. Let $\bar{S} = {s_{α}, s_{1} < s_{α} < s_{t}, α \in [1, t]}$ be a continuous linguistic term set, where S = { s_i } (i = 1, . . . , t) denotes a finite and order discrete term set and $s_{α}, s_{β} \in \bar{S}$ . Then: s_α ⊕ s_β = s_β ⊕ s_α = s_α+β

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 $\tilde{L} = ({\tilde{L}}_{ij} (p))_{m \times n}$ , ${\tilde{L}}_{ij} (p) = {{\tilde{γ}}_{ij}^{(k)} (p_{ij}^{(k)}) | k = 1, 2, . . ., # \tilde{L} (p)}$ by Definition 3 and Definition 4 respectively. Then the alternative H_i ∈ H concerning all the attributes P can be expressed as ${PLA}_{i} = {{\tilde{L}}_{i_{1}}^{(k)} (p_{i_{1}}^{(k)}), {\tilde{L}}_{i_{2}}^{(k)} (p_{i_{2}}^{(k)}), . . ., {\tilde{L}}_{in}^{(k)} (p_{in}^{(k)}) | k = 1, 2, . . ., #$ $\tilde{L} (p)}$ .

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 $\tilde{L} = {({\tilde{L}}_{ij} (p))}_{m \times n}$ by converting the linguistic interval [- τ, τ] into numerical interval [0, 1] according to definition 5, and it is expressed as $\ddot{L} = {({\ddot{L}}_{ij} (p))}_{m \times n}$ .

Secondly, computing the correlation coefficient as follow: $ρ_{j t} = \frac{\sum_{i = 1}^{m} (\sum_{k = 1}^{# L (p)} {\ddot{γ}}_{i j}^{(k)} {\ddot{p}}_{i j}^{(k)} - {\bar{γ}}_{j}^{(k)} {\bar{p}}_{j}^{(k)}) (\sum_{k = 1}^{# L (p)} {\ddot{γ}}_{i t}^{(k)} {\ddot{p}}_{i t}^{(k)} - {\bar{γ}}_{t}^{(k)} {\bar{p}}_{t}^{(k)})}{\sqrt{\sum_{i = 1}^{m} {(\sum_{k = 1}^{# L (p)} {\ddot{γ}}_{i j}^{(k)} {\ddot{p}}_{i j}^{(k)} - {\bar{γ}}_{j}^{(k)} {\bar{p}}_{j}^{(k)})}^{2} \sum_{i = 1}^{m} {(\sum_{k = 1}^{# L (p)} {\ddot{γ}}_{i t}^{(k)} {\ddot{p}}_{i t}^{(k)} - {\bar{γ}}_{t}^{(k)} {\bar{p}}_{t}^{(k)})}^{2}}}$ (7) where ${\bar{γ}}_{j}^{(k)} ({\bar{p}}_{j}^{(k)}) = \frac{\sum_{i = 1}^{m} {\ddot{γ}}_{i j}^{(k)}}{m} (\frac{\sum_{i = 1}^{m} {\ddot{p}}_{i j}^{(k)}}{m})$ and ${\bar{γ}}_{t}^{(k)} ({\bar{p}}_{t}^{(k)}) = \frac{\sum_{i = 1}^{m} {\ddot{γ}}_{i t}^{(k)}}{m} (\frac{\sum_{i = 1}^{m} {\ddot{p}}_{i t}^{(k)}}{m})$

Thirdly, formulating the standard deviation as follows: $σ_{j} = \sqrt{\frac{1}{m - 1} \sum_{i = 1}^{m} {(\sum_{k = 1}^{# L (p)} {\ddot{γ}}_{i j}^{(k)} {\ddot{p}}_{i j}^{(k)} - {\bar{γ}}_{j}^{(k)} {\bar{p}}_{j}^{(k)})}^{2}}$ (8)

Then, the index C_j is calculated as follows: $C_{j} = σ_{j} \sum_{k = 1}^{n} (1 - ρ_{jt})$ (9)

Finally, the vector of evaluation criteria weight W ={ w₁, w₂, . . . , w_n } is counted: $w_{j} = \frac{C_{j}}{\sum_{j = 1}^{n} C_{j}}$ (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 ${\ddot{L}}_{ij} (p)$ and PLPIS which expresses as $D_{ij}^{+}$ , and the Hamming distance between ${\ddot{L}}_{ij} (p)$ 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: $g_{ij}^{+} = \frac{min_{i} min_{j} D_{ij}^{+} + θ max_{i} max_{j} D_{ij}^{+}}{D_{ij}^{+} + θ max_{i} max_{j} D_{ij}^{+}}$ (13) $g_{ij}^{-} = \frac{min_{i} min_{j} D_{ij}^{-} + θ max_{i} max_{j} D_{ij}^{-}}{D_{ij}^{-} + θ max_{i} max_{j} D_{ij}^{-}}$ (14) where θ ∈ [0, 1] represents the resolution coefficient which is given by the decision makers.

Step 7. Construct the grey relational coefficient matrix ${GRCM}^{+} = {[g_{ij}^{+}]}_{m \times n}$ and ${GRCM}^{-} = {[g_{ij}^{-}]}_{m \times n}$ as follow: $\begin{matrix} {GRCM}^{+} = [\begin{matrix} g_{11}^{+} & g_{12}^{+} & \dots & g_{1 n}^{+} \\ g_{21}^{+} & g_{22}^{+} & \dots & g_{2 n}^{+} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ g_{m 1}^{+} & g_{m 2}^{+} & \dots & g_{mn}^{+} \end{matrix}], \\ {GRCM}^{-} = [\begin{matrix} g_{11}^{-} & g_{12}^{-} & \dots & g_{1 n}^{-} \\ g_{21}^{-} & g_{22}^{-} & \dots & g_{2 n}^{-} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ g_{m 1}^{-} & g_{m 2}^{-} & \dots & g_{mn}^{-} \end{matrix}] \end{matrix}$

And the grey relational coefficient between PLPIS and PLPIS, PLNIS and PLNIS are shown as: ${GRCM}_{0}^{+} = (\begin{matrix} g_{01}^{+} & g_{02}^{+} & \dots & g_{0 n}^{+} \end{matrix}) = \underset{n}{\underset{︸}{\begin{matrix} 1 & 1 & \dots & 1 \end{matrix}}}$ (15) ${GRCM}_{0}^{-} = (\begin{matrix} g_{01}^{-} & g_{02}^{-} & \dots & g_{0 n}^{-} \end{matrix}) = \underset{n}{\underset{︸}{\begin{matrix} 1 & 1 & \dots & 1 \end{matrix}}}$ (16)

Step 8. Construct the weighted grey relational coefficient matrix ${WGRCM}^{+} = {[w_{j} g_{ij}^{+}]}_{m \times n}$ and ${WGRCM}^{-} = {[w_{j} g_{ij}^{-}]}_{m \times n}$ as follow: ${WGRCM}^{+} = [\begin{matrix} w_{1} g_{11}^{+} & w_{2} g_{12}^{+} & \dots & w_{n} g_{1 n}^{+} \\ w_{1} g_{21}^{+} & w_{2} g_{22}^{+} & \dots & w_{n} g_{2 n}^{+} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ w_{1} g_{m 1}^{+} & w_{2} g_{m 2}^{+} & \dots & w_{n} g_{mn}^{+} \end{matrix}]$ ${WGRCM}^{-} = [\begin{matrix} w_{1} g_{11}^{-} & w_{2} g_{12}^{-} & \dots & w_{n} g_{1 n}^{-} \\ w_{1} g_{21}^{-} & w_{2} g_{22}^{-} & \dots & w_{n} g_{2 n}^{-} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ w_{1} g_{m 1}^{-} & w_{2} g_{m 2}^{-} & \dots & w_{n} g_{mn}^{-} \end{matrix}]$

And the weighted grey relational coefficient between PLPIS and PLPIS, PLNIS and PLNIS are shown as: $\begin{matrix} {WGRCM}_{0}^{+} = (\begin{matrix} w_{1} g_{01}^{+} & w_{2} g_{02}^{+} & \dots & w_{n} g_{0 n}^{+} \end{matrix}) \\ = (\begin{matrix} w_{1} & w_{2} & \dots & w_{n} \end{matrix}) \end{matrix}$ (17) $\begin{matrix} {WGRCM}_{0}^{-} = (\begin{matrix} w_{1} g_{01}^{-} & w_{2} g_{02}^{-} & \dots & w_{n} g_{0 n}^{-} \end{matrix}) \\ = (\begin{matrix} w_{1} & w_{2} & \dots & w_{n} \end{matrix}) \end{matrix}$ (18)

Step 9. Refer to the projection formula [43, 44], and the weighted grey relational coefficient matrix ${WGRCM}^{+} = {[w_{j} g_{ij}^{+}]}_{m \times n}$ and ${WGRCM}^{-} = {[w_{j} g_{ij}^{-}]}_{m \times n}$ , the weighted grey relational projection(WGRP) of alternative H_i onto PLPIS or PLNIS could be obtained as shown: $\begin{matrix} {WGRP}_{i}^{+} = | {WGRCM}_{i}^{+} | \cdot \cos ({WGRCM}_{i}^{+}, {WGRCM}_{0}^{+}) \\ = \sqrt{\sum_{j = 1}^{n} {(w_{j} g_{ij}^{+})}^{2}} \cdot \frac{\sum_{j = 1}^{n} (w_{j} g_{ij}^{+} \cdot w_{j})}{\sqrt{\sum_{j = 1}^{n} {(w_{j} g_{ij}^{+})}^{2}} \cdot \sqrt{\sum_{j = 1}^{n} w_{j}^{2}}} \\ = \frac{\sum_{j = 1}^{n} (w_{j} g_{ij}^{+} \cdot w_{j})}{\sqrt{\sum_{j = 1}^{n} w_{j}^{2}}} = \sum_{j = 1}^{n} {\bar{w}}_{j} g_{ij}^{+} \end{matrix}$ (19) $\begin{matrix} {WGRP}_{i}^{-} = | {WGRCM}_{i}^{-} | \cdot \cos ({WGRCM}_{i}^{-}, {WGRCM}_{0}^{-}) \\ = \sqrt{\sum_{j = 1}^{n} {(w_{j} g_{ij}^{-})}^{2}} \cdot \frac{\sum_{j = 1}^{n} (w_{j} g_{ij}^{-} \cdot w_{j})}{\sqrt{\sum_{j = 1}^{n} {(w_{j} g_{ij}^{-})}^{2}} \cdot \sqrt{\sum_{j = 1}^{n} w_{j}^{2}}} \\ = \frac{\sum_{j = 1}^{n} (w_{j} g_{ij}^{-} \cdot w_{j})}{\sqrt{\sum_{j = 1}^{n} w_{j}^{2}}} = \sum_{j = 1}^{n} {\bar{w}}_{j} g_{ij}^{-} \end{matrix}$ (20) where ${\bar{w}}_{j} = w_{j}^{2} / \sqrt{\sum_{j = 1}^{n} w_{j}^{2}}$

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. ${RC}_{i} = \frac{{WGRP}_{i}^{+}}{{WGRP}_{i}^{+} + {WGRP}_{i}^{-}} = \frac{\sum_{j = 1}^{n} {\bar{w}}_{j} g_{ij}^{+}}{\sum_{j = 1}^{n} {\bar{w}}_{j} g_{ij}^{+} + \sum_{j = 1}^{n} {\bar{w}}_{j} g_{ij}^{-}}$ (21)

Distinctly, the larger RC_i is, the better H_i is, and vice versa.

4 A case study and comparative analysis

4.1 A case study

Site selection plays a very important role in formulating the operation strategies and goal, because of its stationarity and long-term. Once a position is established, it is difficult to change. For a service enterprise, its position has a great impact on the survival of the enterprise. As for the hospital, its problem of location deserves more attention. The location of a hospital is closely related to the rational allocation of medical resources and the vital interests of the people. In recent years, the harmonious coexistence of project constructions and environment attract people’s attention, and the interaction between the two has been a cause for concern. As a result, the continuous development and improvement of “green constructions” and “sustainable development” provide a new perspective for site selection of hospital constructions. A lot of factors may be involved when it comes to site selection of hospital constructions. Thus, a numerical example for site selection of hospital constructions is put forward by us in this section to exemplify the PL-GRP method. Suppose that there are five sites of hospital construction H ={ H₁, H₂, H₃, H₄, H₅ } to select, and five experts E ={ E₁, E₂, E₃, E₄, E₅ } select four attribute P ={ P₁, P₂, P₃, P₄ } (Suppose that the attributes’ weight are completely unknown) to evaluate the four possible sites: (1) P₁ is the ecological security condition; (2) P₂ is the traffic condition; (3) P₃ is the noisy environment; (4) P₄ is the communications condition, where P₃ is negative attribute and the rest are positive attributes. The five possible sites H ={ H₁, H₂, H₃, H₄, H₅ } are evaluated by the linguistic term set $S = {\begin{matrix} s_{- 3} = extremely poor, s_{- 2} = very poor, s_{- 1} = poor, s_{0} = medium, \\ s_{1} = good, s_{2} = very good, s_{3} = extremely good \end{matrix}}$ (22) and the original decision matrices are listed in Tables1 –5.

Table 1

Linguistic decision matrix by the first expert E₁

Alternatives	P₁	P₂	P₃	P₄
H₁	s₂	s₃	s₀	s_–1
H₂	s_–1	s₃	s₃	s_–2
H₃	s₃	s₁	s₂	s₂
H₄	s₁	s₃	s₂	s₁
H₅	s_–3	s₂	s₀	s₁

Table 2

Linguistic decision matrix by the second expert E₂

Alternatives	P₁	P₂	P₃	P₄
H₁	s₂	s₁	s₃	s₀
H₂	s_–3	s₂	s₀	s₁
H₃	s₀	s_–1	s₂	s₁
H₄	_S2	s_–1	s₁	s₁
H₅	s_–1	s₂	s₃	s₂

Table 3

Linguistic decision matrix by the third expert E₃

Alternatives	P₁	P₂	P₃	P₄
H₁	s₁	s₃	s₂	s₃
H₂	s_–2	s_–1	s₃	s_–1
H₃	s₀	s₁	s_–1	s₂
A₄	S1	S–1	S1	S2
H₄	s₁	s_–1	s₁	s₂
H₅	s₃	s₃	s₂	s₂

Table 4

Linguistic decision matrix by the fourth expert E₄

Alternatives	P₁	P₂	P₃	P₄
H₁	s₃	s₁	s₀	s₃
H₂	s_–1	s₂	s₃	s_–2
H₃	s₂	s₂	s₁	s₂
H₄	s₁	s₁	s₂	s₂
H₅	s₃	s_–1	s₀	s₃

Table 5

Linguistic decision matrix by the fifth expert E₅

Alternatives	P₁	P₂	P₃	P₄
H₁	s₂	s₂	s₃	s_–1
H₂	s_–3	s₃	s₂	s_–1
H₃	s₀	s_–1	s₁	s₁
H₄	s₂	s₁	s₁	s₃
H₅	s_–3	s₂	s₂	s₁

In the following, PL-GRP method are used to select the best site of hospital constructions.

Step 1. Transform the negative attribute into positive attributes (Tables6 –10).

Table 6

Linguistic decision matrix by the first expert E₁

Alternatives	P₁	P₂	P₃	P₄
H₁	s₂	s₃	s₀	s_–1
H₂	s_–1	s₃	s₃	s_–2
H₃	s₃	s₁	s₂	s₂
H₄	s₁	s₃	s₂	s₁
H₅	s_–3	s₂	s₀	s₁

Table 7

Linguistic decision matrix by the second expert E₂

Alternatives	P₁	P₂	P₃	P₄
H₁	s₂	s₁	s₃	s₀
H₂	s_–3	s₂	s₀	s₁
H₃	s₀	s_–1	s₂	s₁
H₄	s₂	s_–1	s₁	s₁
H₅	s_–1	s₂	s₃	s₂

Table 8

Linguistic decision matrix by the third expert E₃

Alternatives	P₁	P₂	P₃	P₄
H₁	s₁	s₃	s₂	s₃
H₂	s_–2	s_–1	s₃	s_–1
H₃	s₀	s₁	s_–1	s₂
A₄	S1	S–1	S1	S2
H₄	s₁	s_–1	s₁	s₂
H₅	s₃	s₃	s₂	s₂

Table 9

Linguistic decision matrix by the fourth expert E₄

Alternatives	P₁	P₂	P₃	P₄
H₁	s₃	s₁	s₀	s₃
H₂	s_–1	s₂	s₃	s_–2
H₃	s₂	s₂	s₁	s₂
H₄	s₁	s₁	s₂	s₂
H₅	s₃	s_–1	s₀	s₃

Table 10

Linguistic decision matrix by the fifth expert E₅

Alternatives	P₁	P₂	P₃	P₄
H₁	s₂	s₂	s₃	s_–1
H₂	s_–3	s₃	s₂	s_–1
H₃	s₀	s_–1	s₁	s₁
H₄	s₂	s₁	s₁	s₃
H₅	s_–3	s₂	s₂	s₁

Steps 2. Transform the linguistic information into probabilistic linguistic decision matrix (See Table11).

Table 11

Probabilistic linguistic decision matrix

Alternatives	P₁	P₂
H₁	{s₁ (0.2) , s₂ (0.6) , s₃ (0.2)}	{s₁ (0.4) , s₂ (0.2) , s₃ (0.4)}
H₂	{s_-3 (0.4) , s_-2 (0.2) , s_-1 (0.4)}	{s_-1 (0.2) , s₂ (0.4) , s₃ (0.4)}
H₃	{s₀ (0.6) , s₂ (0.2) , s₃ (0.2)}	{s_-1 (0.4) , s₁ (0.4) , s₂ (0.2)}
H₄	{s₁ (0.6) , s₂ (0.4)}	{s_-1 (0.4) , s₁ (0.4) , s₃ (0.2)}
H₅	{s_-3 (0.4) , s_-1 (0.2) , s₃ (0.4)}	{s_-1 (0.2) , s₂ (0.6) , s₃ (0.2)}
Alternatives	P₃	P₄
H₁	{s₀ (0.4) , s₂ (0.2) , s₃ (0.4)}	{s_-1 (0.4) , s₀ (0.2) , s₃ (0.4)}
H₂	{s₀ (0.2) , s₂ (0.2) , s₃ (0.6)}	{s_-2 (0.4) , s_-1 (0.4) , s₁ (0.2)}
H₃	{s_-1 (0.2) , s₁ (0.4) , s₂ (0.4)}	{s₁ (0.4) s₂ (0.6)}
H₄	{s₁ (0.6) , s₂ (0.4)}	{s₁ (0.4) , s₂ (0.4) , s₃ (0.2)}
H₅	{s₀ (0.4) , s₂ (0.4) , s₃ (0.2)}	{s₁ (0.4) , s₂ (0.4) , s₃ (0.2)}

Step 3. Calculate the normalized probabilistic linguistic decision matrix (See Table12).

Table 12

Normalized probabilistic linguistic decision matrix

Alternatives	P₁	P₂
H₁	{s₁ (0.2) , s₂ (0.6) , s₃ (0.2)}	{s₁ (0.4) , s₂ (0.2) , s₃ (0.4)}
H₂	{s_-3 (0.4) , s_-2 (0.2) , s_-1 (0.4)}	{s_-1 (0.2) , s₂ (0.4) , s₃ (0.4)}
H₃	{s₀ (0.6) , s₂ (0.2) , s₃ (0.2)}	{s_-1 (0.4) , s₁ (0.4) , s₂ (0.2)}
H₄	{s₁ (0) , s₁ (0.6) , s₂ (0.4)}	{s_-1 (0.4) , s₁ (0.4) , s₃ (0.2)}
H₅	{s_-3 (0.4) , s_-1 (0.2) , s₃ (0.4)}	{s_-1 (0.2) , s₂ (0.6) , s₃ (0.2)}
Alternatives	P₃	P₄
H₁	{s_-3 (0.4) , s_-2 (0.2) , s₀ (0.4)}	{s_-1 (0.4) , s₀ (0.2) , s₃ (0.4)}
H₂	{s_-3 (0.6) , s_-2 (0.2) , s₀ (0.2)}	{s_-2 (0.4) , s_-1 (0.4) , s₁ (0.2)}
H₃	{s_-2 (0.4) , s_-1 (0.4) , s₁ (0.2)}	{s₁ (0) , s₁ (0.4) , s₂ (0.6)}
H₄	{s_-2 (0) , s_-2 (0.4) , s_-1 (0.6)}	{s₁ (0.4) , s₂ (0.4) , s₃ (0.2)}
H₅	{s_-3 (0.2) , s_-2 (0.4) , s₀ (0.4)}	{s₁ (0.4) , s₂ (0.4) , s₃ (0.2)}

Step 5. Determine the PLPIS and PLNIS by Equation (11-12) (See Tables17-18):

Table 13

The correlation coefficientamong attributes by Eq.7

Alternatives	P₁ × P₂	P₁ × P₃	P₁ × P₄	P₂ × P₃	P₂ × P₄	P₃ × P₃
ρ _jk	-0.3268	0.6615	0.6846	-0.7042	-0.5768	0.8787

Table 14

The standard deviation of each attribute by Eq. 8

Alternatives	P₁	P₂	P₃	P₄
σ _j	0.0882	0.0406	0.0253	0.0663

Table 15

The standard deviation of each attribute by Eq. 8

Alternatives	P₁	P₂	P₃	P₄
c _j	0.1746	0.1872	0.0548	0.1335

Table 16

The weight of attributes by Equation 10

Alternatives	P₁	P₂	P₃	P₄
w _j	0.3174	0.3403	0.0996	0.2427

Table 17

E (L_ij (p))

Alternatives	P₁	P₂	P₃	P₄
H₁	0.8333	0.8333	0.2333	0.6333
H₂	0.1667	0.8	0.1333	0.3333
H₃	0.6667	0.5667	0.3333	0.7667
H₄	0.7333	0.6	0.2667	0.8
H₅	0.4667	0.7667	0.2667	0.8

Table 18

PLPIS and PLNIS

	P₁	P₂
PLPIS	{s_0.6667 (0.2) , s_0.8333 (0.6) , s₁ (0.2)}	{s_0.6667 (0.4) , s_0.8333 (0.2) , s₁ (0.4)}
PLNIS	{s₀ (0.4) , s_0.1667 (0.2) , s_0.3333 (0.4)}	{s_0.3333 (0.4) , s_0.6667 (0.4) , s_0.8333 (0.2)}
	P₃	P₄
PLPIS	{s_0.1667 (0.4) , s_0.3333 (0.4) , s_0.6667 (0.2)}	{s_0.6667 (0.4) , s_0.8333 (0.4) , s₁ (0.2)}
PLNIS	{s₀ (0.6) , s_0.1667 (0.2) , s_0.5 (0.2)}	{s_0.1667 (0.4) , s_0.3333 (0.4) , s_0.6667 (0.2)}

Step 4. Count the weight values of attributes.

Then we can get the weight of attributes as follows by Equation 10: (See Table16)

Step 6. Calculate the distances $d_{HD} ({\ddot{L}}_{i} (p), PLPIS)$ and $d_{HD} ({\ddot{L}}_{i} (p), PLNIS)$ by Equation (5) (See Tables19-20):

Table 19

$d_{HD} ({\ddot{L}}_{ij} (p), PLPIS)$

Alternatives	P₁	P₂	P₃	P₄
H₁	0	0	0.0778	0.1889
H₂	0.2222	0.1222	0.0667	0.1556
H₃	0.1667	0.1556	0	0.2111
H₄	0.1222	0.1444	0.0667	0
H₅	0.2556	0.2444	0.0667	0

Table 20

$d_{HD} ({\ddot{L}}_{ij} (p), PLNIS)$

Alternatives	P₁	P₂	P₃	P₄
H₁	0.2222	0.1556	0.0333	0.1222
H₂	0.0556	0.2667	0.0444	0.1111
H₃	0.2222	0.1889	0.1111	0.2556
H₄	0.2444	0.2000	0.0889	0.2667
H₅	0.1556	0.2556	0.0889	0.2667

Step 7. Formulate the grey relational coefficient of each alternative from the PLPIS and PLNIS by Equations (13-14) (See Tables21-22):

Table 21

$g_{ij}^{+}$

Alternatives	P₁	P₂	P₃	P₄
H₁	1	1	0.6216	0.4035
H₂	0.3651	0.5111	0.6571	0.4510
H₃	0.4340	0.4510	1	0.3770
H₄	0.5111	0.4694	0.6571	1
H₅	0.3333	0.3433	0.6571	1

Table 22

$g_{ij}^{-}$

Alternatives	P₁	P₂	P₃	P₄
H₁	0.4688	0.5769	1.0000	0.6522
H₂	0.8824	0.4167	0.9375	0.6818
H₃	0.4688	0.5172	0.6818	0.4286
H₄	0.4412	0.5000	0.7500	0.4167
H₅	0.5769	0.4286	0.7500	0.4167

Step 8. Formulate the alternative H_i of weighted GRP onto PLPIS and PLNIS by Equations (19-20) (See Table23):

Table 23

${WGRP}_{i}^{+}$ and ${WGRP}_{i}^{-}$

	${WGRP}_{i}^{+}$	${WGRP}_{i}^{-}$
H₁	0.4614	0.3039
H₂	0.2416	0.3493
H₃	0.2398	0.2605
H₄	0.3206	0.2515
H₅	0.2597	0.2616

Step 9. Compute the index RC_i by Equation (21) (See Table24):

Table 24

RC _i

	RC _i
H₁	0.6029
H₂	0.4088
H₃	0.4793
H₄	0.5604
H₅	0.4982

Table 25

The calculating results by utilizing PLWA method

PLWA method	Calculating results
	Z₁1 (w) = {s_0.1654, s_0.2430, s_0.3166}Z₂ (w) = {s_0.0389, s_0.1597, s_0.2208}
Z_i (w) (i = 1, 2, 3, 4, 5)
	Z₃ (w) = {s_0.1472, s_0.2216, s_0.2548}Z₄ (w) = {s_0.1101, s_0.3052, s_0.2423}
	Z₅ (w) = {s_0.0874, s_0.2789, s_0.2635}
E (Z_i (w))	E(Z₁ (w)) = s_0.2417 E(Z₂ (w)) = s_0.1398 E(Z₃ (w)) = s_0.2079
Ordering	E(Z₄ (w)) = s_0.2192 E(Z₅ (w)) = s_0.2099
	H₁ > H₄ > H₅ > H₃ > H₂

Table 26

The calculating results by utilizing PL-TOPSIS method

TOPSIS method	Calculating results
The distances of each alternative from PLPIS	$d_{1}^{+} = 0 . 0548, d_{2}^{+} = 0 . 1888, d_{3}^{+} = 0.1816, d_{4}^{+} = 0 . 0969, d_{5}^{+} = 0 . 1834$
The distances of each alternative from PLPIS	$d_{1}^{-} = 0.1907, d_{2}^{-} = 0 . 0495, d_{3}^{-} = 0.1240, d_{4}^{-} = 0.1299, d_{5}^{-} = 0.1442$
Closeness coefficients	CI₁ = 0 . 0000, CI₂ = -3.1840, CI₃ = -2.6621, CI₄ = -1.0862, CI₅ = -2.5877
Ordering	H₁ > H₄ > H₅ > H₃ > H₂

Table 27

The calculating results by utilizing PL-TODIM method

TODIM method	Calculating results
The dominance degree of each alternative	δ₁ = .0.0300, δ₂ = .14.9889, δ₃ = .6.4363, δ₄ = .2.6839, δ₅ = .3.5159
The overall dominance degree	ϕ (r₁) = 1, ϕ (r₂) = 0 ϕ (r₃) = 0.5717 ϕ (r₄) = 0.8226 ϕ (r₅) = 0.7670
Ordering	H₁ > H₄ > H₅ > H₃ > H₂

Table 28

The calculating results by utilizing PL-MABAC method

MABAC method	Calculating results
The total distances from the border approximate area	PSLV₁ = -0.0883, PSLV₂ = -0.1967, PSLV₃ = -0.0969,
Ordering	PSLV₄ -0.1346, PSLV₅ = -0.1313
	H₁ > H₄ > H₅ > H₃ > H₂

Step 10. According to the RC_i, we can get the ranking as H₁ > H₄ > H₅ > H₃ > H₂. As a result, the best alternative is H₁.

4.2 Comparative analysis

In this chapter, we make a comparison between our proposed PL-GRP method and PLWA operator [16], PL-TOPSIS method [16], PL-TODIM method [50] as well as PL-MABAC method [51]. The result of comparative analysis are listed in Tables 25–28.

4.2.1 Compared with PLWA operator

Firstly, we make a comparison between PLWA operator [16] and the PL-GRP methods proposed by us. So we can get the same result that the best alternative is H₁.

4.2.2 Compared with PL-TOPSIS operator

Then, we make a comparison with extend TOPSIS which is proposed by Pang, Wang and Xu [16]. We can obtain the calculating results showed in Table 26. Therefore, we get the same result that the optimal alternative is H₁.

4.2.3 Compared with PL-TODIM operator

What’s more, the PL-TODIM method [50] is made a comparison with the PL-GRP by us, and the best alternative is H₁ as well.

4.2.4 Compared with PL-MABAC operator

In addition, we compare our method with PL-MABAC method [51]. Obviously, the most ideal alternative is H₁.

On that basis, the optimal solution of above four methods is H₁. Furthermore, we can reach the same conclusion that the best alternative is H₁, and the result demonstrates the availability of the PL-GRP method.

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].

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