1 Introduction

We mostly interact with multi-criteria group decision-making (MCGDM) problems in our routine life, which are used to choose the best alternatives among different alternatives. Often we deal with dubious situations, and due to the ambiguous nature of MCGDM problems, the fuzzy method may be considered as a valid technique to define the characteristics like linguistic variables [1–3], interval values [4, 5] and hesitant fuzzy numbers [6]. We have to deal with uncertainty at different levels. The two most important approaches introduced to investigate, interrogate, and tackle this type of uncertainty are fuzzy set theory and probability theory. For the first time, Zadeh [7] characterized the fuzzy linguistic techniques to manage the decision-making problems, which utilizes linguistic factors whose contents are not quantities, but phrases or terms, to address subjective assessments. After the development of the linguistic approach, many researchers took an interest in this area and applied it to MCGDM problems [8]. Over time, researchers feel some limitation in the linguistic variable: it can only express the membership degree. There are many situations where consideration of membership and non-membership functions is necessary.

For some dynamic issues, it is simple for decision-makers or experts to communicate their assessments in phonetic or linguistic terms, for example, “low” cost, “high” quality, “great” execution. The linguistic variable upgrades the possibility and adaptability of decision models, and the advancement of analyzing linguistic variables have prompted a new research territory named computing with words [7]. However, the customary linguistic methodology uses a one-word linguistic term to evaluate a linguistic variable. This way cannot represent the hesitant impression of decision-makers. For example, while assessing a research proposal, a decision-maker may state, “it is between good and excellent”. Also, in a group decision-making process, the gathering’s appraisals could be uncertain if there are various assessments among many individuals. To overthrow the inefficiency of the linguistic process, Rodríguez et al. [9] introduced the concept of hesitant fuzzy linguistic term set (HFLTS), where decision-makers can give their ambiguous evaluations using several linguistic terms. Consequently, the HFLTS has pulled in numerous scholars’ attention [10, 11].

Herrera and Martins [12, 13] proposed a 2-tuple linguistic representation model that can deal with verbal and numerical data viably in decision-making without mutilating and harming data. This helpful model is basically founded on the symbolic translation of linguistic variables and has been generally utilized in different MCDM problems [14]. In light of the difficulty of items and dubiousness of individuals thinking, it is increasingly suitable to give an evaluation by using linguistic data rather than numerical numbers [15, 16]. Accordingly, the number of analysts dealt with linguistic data for taking care of MCGDM issues and characterized various techniques to explore the linguistic assessment components. There are mainly three parts of the linguistic information model [16, 17], procedure based on membership degree, procedure based on linguistic symbols, and the procedure based in linguistic 2-tuple. However, the initial two techniques have a few restrictions that they played out their interpretation procedure as an estimation procedure to assess the finding in the original space, thus invigorating the absence of exactness and precision [18]. Contrarily, the precision of linguistic terms, calculation, and interpretation of results can be improved with the assistance of the 2-tuple linguistic model. In light of its recognized quality and viability in managing the linguistic judgment of experts, the 2-tuple linguistic models have been generally utilized in decision-making problems [19–21]. Zhang et al. [22] examined the transitivity of 2-tuple to tackle the issue of group decision-making problems. Apart from this, the idea of a hesitant 2-tuple fuzzy linguistic set is proposed by Wei and Liao [23]. Beg and Rashid [14] extended fuzzy TOPSIS for hesitant 2-tuple linguistic term sets with the assessment of some experts about the properties of alternatives. Wang et al. [24] built up a hesitant 2-tuple linguistic aggregation operator for structuring a new type of group decision-making approach to deal with addressing the data combination, including the interrelationship between aggregated terms and the prioritization relationship among experts under hesitant 2-tuple linguistic circumstance.

At whatever point we talk about the strength of the relationship between two variables, the correlation coefficient is a suitable measurement tool. In all estimation procedures, the correlation coefficient is an effective apparatus and broadly utilized in statistical analysis to quantify the strength or level of relationship between two variables [25–27]. However, it is conspicuous that the idea constructed on daily life problems is uncertain. So as to handle this issue, correlation is expanded by different analysts to fuzzy correlation [28–30]. Tyagi [31] broadened the idea of the hesitant fuzzy set to dual hesitant fuzzy set and built up a correlation coefficient between dual hesitant fuzzy sets. They applied it to assess the correlation coefficient estimation among various water parameters of four distinct lakes. A correlation coefficient based on the central interval is proposed by Saneifard [27]. Chen et al. [32] outlined a correlation coefficient for hesitant fuzzy sets and applied it to clustering analysis in a hesitant fuzzy environment. Moreover, they likewise proposed some interval-valued hesitant fuzzy sets, introduced their correlation coefficients, and determined their applications in clustering. Arora and Garg [33] took inspiration from the soft set theory and built up the correlation coefficient to measure the relationship between dual hesitant fuzzy soft sets, named the novel correlation coefficient.

MCDM is a developing field of study that helps experts distinguish the best alternative from a set of alternatives assessed through a set of criteria weights [34]. A basic portion of this issue is the arrangement of the significance of criteria weights. Various methods have been developed to recognize these criteria, including [35–37], SMARTS and SMARTER, analytic hierarchy process. BWM is also linked to them, designed by Rezaie [38] that depends on an organized pairwise comparison framework for an optimal model to get the ideal weights of criteria. This method has fascinated scholars’ attention from different fields since its presentation. Hadi et al. [39] presented z-number BWM to deal with the uncertain situation of MCGDM problems. Mi et al. [40] presented some questions relating to BWM in the form of why what, what for, and what next. Muhammad & Razaei [41] introduced a Bayesian BWM to find the aggregated final criteria weights of models for a group of decision-makers. Furthermore, many other works are available in the literature on BWM, including multiplicative BWM [42], BWM MULTIMOORA [43], BWM VIKOR [44], and others [45].

Adopting inspiration from the existing approaches [24, 46], a new type of correlation coefficient between H2FLSs is proposed in this paper. Numerous methodologies exist in the literature that works in information energy and develops a recipe for computing the correlation coefficient. However, there was no technique for estimating the distance-based correlation coefficient from a statistics point of view in the context of H2FLSs. The proposed correlation coefficient can determine how strong the relationship between the two alternatives/variables is. It reveals information about the direction of the relationship and can also display data behaviour that is difficult to study by simple observation. However, drawbacks are also related, such as the inability to determine the curvilinear relationship. Furthermore, it is unclear which alternatives/variables are most affected in the correlation process and which ones are dependent and independent alternatives/variables. It also has no control over the third alternatives/variables influencing the relationship.

2 Preliminaries

In this section, we discuss some basic notions of linguistic term set (LTS), hesitant fuzzy LTS, 2-tuple linguistic model and H2FLS.

Definition 1 [47, 48] Suppose, the LTS S = {u₀, u₁, …, u_g}, has an odd cardinality, where g is even and u_i ∈ [0, g]. The set S has the given two characteristics:

1. Negation operator: Neg(u_i) = u_j, such that i + j = g;
2. Ordered set: u_i ≤ u_j ⇔ i ≤ j. thus, there exist a maximum and minimum operators i.e.
   1. max(u_i, u_j) = u_i, if u_j ≤ u_i;
   2. min(u_i, u_j) = u_i, if u_i ≤ u_j.

The continuous LTS is defined by Xu [49, 50], which is an extension of the discrete term set S where . If u_k ∈ S, u_k is called the original linguistic term and will be used to assess the alternatives throughout a decision process. If u_k ∉ S, then u_k is called the virtual linguistic term of S that will appear only in the calculations process.

Definition 2 [9] Let x_i ∈ X, i = 1, 2, ⋯, N, be fixed and S = {s_t|t = −τ, ⋯, −1, 0, 1, ⋯, τ} be a LTS. A hesitant fuzzy LTS on X, H_S, is in the form of where with L(x_i) being the number of LTS in h_S(x_i) and φ_l ∈ [−τ, τ].

Herrera and Martínez [12] defined 2-tuple linguistic model to represent the linguistic type of data by 2-tuple (u_i, a), where u_i ∈ S and a ∈ [−0.5, 0.5). It means that, a function is transformed into 2-tuple linguistic term and a numerical number.

Definition 3 [12] Let be a LTS and β ∈ [0, g] is a numerical value indicating the result of a symbolic aggregation operator. Then, △ : [0, g] → S× [-0.5, 0.5) shows a 2-tuple linguistic function that presenting the same information to β is defined as follows: △ operator is a one-to-one function and an inverse function is ▽((u_i, a)) = i + a.

The 2-tuple linguistic model ought to be intelligible and predefined, by which each decision-maker can additionally qualify potential alternatives for specific criteria. However, because of the complication of objects and vagueness of people thinking, the 2-tuple linguistic model can not wholly describe the hesitating circumstances. Therefore, Wei and Liao [23] presented the hesitant 2-tuple fuzzy linguistic data model to deal with the situations in which data portrayed is in linguistic terms, and the decision-maker has some hesitation in choosing its conceivable linguistic interpretations. It can deal with the information viably in decision-making without harming data and more accurate results. Wei and Liao [23] defined the H2FLS as follows.

Definition 4 [23] Let S = {u₀, u₁, …, u_g} be a LTS. A H2FLS on X, H_S, is in the form of H_S = {< x_i, h(x_i) > ; x_i ∈ X)}, where h(x_i) = {β_i = (u_i, α_i)|i = 1, 2, …, #h(x_i)} and α ∈ [−0.5, 0.5). For convenience h(x_i) is called a H2FLE.

In our next study, this hesitant 2-tuple fuzzy linguistic data model will be used in our research for the construction of correlation coefficient for H2FLEs.

3 Mean and variance of H2FLEs

In this section, we established the mean and variance for H2FLEs. The developed mean and variance will further use in the construction of the correlation coefficient for H2FLSs. It can be defined as follows:

Definition 5 Let S = {u₀, u₁, …, u_g} be a LTS and h(x) = β_i, be a H2FLEs. The mean of h(x) is defined as: (1) where β_i = (u_i, α_i) and i = 1, 2, …, #h. The variance of h(x) is defined as: (2)

Example: For a LTS S = {u₀, u₁, …, u₆}, suppose h¹ = ((u₃, 0.2487), (u₁, 0.2752)), h² = ((u₂, 0.0477), (u₁, 0.1427), (u₁, 0.2752)) and h³ = ((u₄, 0.2487), (u₁, 0.2251)) are three H2FLEs. By using Eqs 1 and 2, the mean and variance of h¹, h² and h³ are computed as , , Var(h¹) = 0.9734, Var(h²) = 0.1592 and Var(h³) = 2.2855.

4 Method to determine the criteria weights

Experts have weighed the measurements in some circumstances, but sometimes it just does not. For the case, we must determine the weight of each criterion where the weight vector is not exact. In a situation where there is no evidence to confirm differences in criteria, we can assume that all criteria are considered equal. Otherwise, we must face a strategy to determine the weight of the criteria. We should handle the strategy to measure the criteria weights in the position where there is no evidence to confirm the difference in patterns. This part applies BWM to solve this issue. Here, we shortly discuss the procedure of BWM that can be utilized to obtain the weights of criteria [52]. Fig 1 shows summary of this procedure.

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Fig 1. The procedure of BWM used to obtain the weights of criteria.

https://doi.org/10.1371/journal.pone.0270414.g001

1. Step 1. Determine a set of decision criteria: In this progression, the experts recognizes n criteria {C₁, C₂, …, C_n} that are utilized to create a decision.
2. Step 2. Decide the best (for example most attractive, most significant) and the worst (for example least attractive, least significant) criteria.
3. Step 3. Decide the preference of the best criteria over all the others, H2FLEs of best criterion to all the other criteria is performed and then transformed to numbers according to (1). The consequent best-to-other vector would be: Q_B = (q_B1, q_B2, …, q_Bn), where q_Bj shows the preference of the best criterion B over criterion j. Obviously q_BB = 1
4. Step 4. Decide the preference of the considerable number of criteria over the worst criteria, utilizing H2FLEs and then transformed to numbers according to (1). The consequent others-to worst vector would be: Q_W = (q_1B, q_2B, …, q_nB)^T, where q_jW Shows the preference of the criterion j over the worst criterion W. Obviously q_WW = 1
5. Step 5. Determine the feasible weights [52].

To derive the above weights presented in Step 5, we minimize the maximum among the set of {|P_wB − q_BjP_wj|, |P_wj − q_jWP_wW|}. Hence it can be formulated as follows: subject to (9) for all j.

Model (9) can be transferred to the following linear programming model: subject to (10) for all j

By solving problem (10), the optimal weights and ζ^L are obtained.

4.1 MCGDM problem solving approach based on the proposed correlation coefficient for H2FLSs

In the current subsection, a method of MCGDM for decision-making purposes in the environment of H2FLSs is analyzed. Assume, there is a set of n alternatives A = {A₁, A₂, …, A_n}, where alternatives are assessed under m criteria, represented by C = {C₁, C₂, …, C_m}. The H2FLSs estimation of A_i under C_j is represented by A_ij. Let, there is a set of k experts D = {D₁, D₂, …, D_K} and they are proposed to keep confidential their conclusions to get a progressively exact outcome. The criterion weight C_j is w_j, where w_j ≥ 0 (j = 1, 2, …, m) and Σw_j = 1. The decision among different alternatives contain the steps which are summarized in the Fig 2.

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Fig 2. An algorithm for solving the problem using the proposed corr. coeff.

https://doi.org/10.1371/journal.pone.0270414.g002

1. Step 1. The configuration of H2FLSs judgment matrices R(t) (t = 1, …, K) established in the authoritative perspective on the efficiency of A_i (i = 1, …, n) on criteria C_j (j = 1, …, m) using the LTS S. (see [24]).
2. Step 2. The configuration of H2FLSs as summation findings of matrix R. (see [24]).
3. Step 3. Check out even if the weight vector of criteria is presented by the decision-makers or not. If present, proceed to Step 5. If not, go to the next step.
4. Step 4. Set up the weight vector by linear model of BWM [52] (section 4), proceed to the next step.
5. Step 5. Computation of correlation coefficient between different alternatives with an ideal alternative. An ideal alternative has the maximum membership degree and minimum non-membership degree.
6. Step 6. Ranking of the alternatives on the basis of measurement of the correlation coefficient. The best alternative has the largest correlation coefficient.

5 Numerical example (with known & unknown criteria weights)

To feature the practicality and viability of the planned decision-making method, two models are utilized from some relevant literature [24, 53, 54].

Assume an organization needs to pick a provider with the most grounded far quality for improving its administration levels. There involve four suppliers A_i = {A₁, A₂, A₃, A₄}. Three assessment criteria are considered for four goods providers, i.e., quality (C₁), supply capacity and flexibility (C₂), and price (C₃) with ω = (0.504, 0.335, 0.161). Experts give their views based on the LTS, S = {u₀ =nothing, u₁ = very poor, u₂ = poor, u₃ = moderate, u₄ = good, u₅ = very good, u₆ = perfect}.

The H2FLSs aggregated judgment matrix is used the same as obtained in Wang et al. [24], given as follows:

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Table 1. H2FLSs aggregated judgment matrix.

https://doi.org/10.1371/journal.pone.0270414.t001

Step 1 & 2. Taking an ideal alternative for the calculation of correlation coefficients between the alternatives of Table 1 and ideal alternative. The ideal alternative for the given example is as follows:

Step 3 & 4. Weight vector is given by the decision-makers, so we omit the steps.

Step 5. The correlation coefficients of each alternative A_i (1 ≤ i ≤ 4) and A* with respect to their given weight are calculated as follows in Table 2.

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Table 2. Calculation results with respect to the given weights.

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Step 6. In response to the estimations of correlation coefficients in step 3, the alternatives are ranked. The ranking order is acquired as A₄ > A₁ > A₂ > A₃. Consequently, The organization will pick A₄ for their services.

In the given an example, it is seen that the weight vector is presented already by the decision-makers, but sometimes, it is likely that the weight vector is obscure. If so, we can utilize the BWM to determine the weights of the criteria. Here we again go back to step 4 only to show what to do if we lack this knowledge and set up the weight vector.

Step 4. The company recognizes Quality (C₁) for best and Price (C₃) for the worst criteria. Tables 3 and 4 demonstrate the comparison vectors. Due to the ambiguity and intangibility of decision-making, H2FLSs is used for optimal company selection.

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Table 3. Pairwise comparison vector for the best criterion.

https://doi.org/10.1371/journal.pone.0270414.t003

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Table 4. Pairwise comparison vector for the worst criterion.

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After the simplification of the given data presented in Tables 3 and 4 by using (1), we have the following results in the model (10) for this problem, as follows:

min ζ^L

subject to

Solving this model, we have (stage 5): and

Step 5. The correlation coefficients of each alternative A_i (1 ≤ i ≤ 4) and A* with respect to their weights are calculated as follows:

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Table 5. Calculation results with respect to the BWM.

https://doi.org/10.1371/journal.pone.0270414.t005

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Table 6. Calculation results of ordered weighted correlation coefficients with respect to the BWM.

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Step 6. As per the estimations of correlation coefficients in Tables 5 & 6, we come to know that the ρ_ω(A₄, A*) in Table 5 and in Table 6, has the maximum correlation.

6 Comparative analysis

The area of study being talked about here keep up a comparative study among the existing methods proposed by [24, 52, 53] with our work. Referring to Table 7 of the above example, the ranking order of the alternatives is as follows:

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Table 7. Comparison of obtained results.

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Wang et al. [24] dealt with a similar example utilizing hesitant 2-tuple linguistic Bonferroni mean operator and prioritized weighted hesitant 2-tuple linguistic Bonferroni mean operator, in which the prioritization relationship of specialists is considered. By analyzing the two ranking outcomes in Table 7, the proposed order weighted correlation, and Wang et al. method are similar. However, there exists a difference in positioning consequence of the first two alternatives A₁ and A₄ of the proposed weighted correlation with wang et al. alternatives ranking order. The reason is that the different prioritization relationships of experts considered by Wang et al. can cause changes in overall values. However, we can say that the overall ranking order is stable and adequate.

Ge & Wei [53] also coped with a similar example using hesitant 2-tuple weighted averaging operator and hesitant 2-tuple ordered weighted averaging operator, in which the prioritization relationship of specialists is not thought of, and criteria weights are given in advance. Wang et al. [54] likewise proposed a technique to deal with a similar illustration by utilizing hesitant 2-tuple linguistic prioritized weighted averaging operator and hesitant 2-tuple linguistic correlated averaging operator. In [53, 54] and our proposed approach, the most feasible option is A₄, which indicates the validity of the proposed work.

Similarly, it can also be seen that the ranking order obtained from the proposed method using BWM criteria weights is very similar to the ranking order of alternatives obtained in [24, 53, 54]. From the above comparisons, the obtained ranking orders of the alternatives show the efficiency of the proposed correlation coefficient. The proposed approach can be applied in every field where there is a relationship between two variables. Furthermore, the proposed approach is easy to calculate for H2FLSs compared to other methodologies mentioned in section 1, which are based on different approaches. As a result, we can conclude that our work is strong and capable of solving the MCGDM problem in the context of H2FLSs compared to the existing approaches’ computational difficulties.

7 Conclusion

The correlation measure of H2FLSs investigated in this paper, which lies in the [−1, 1], gives us more information than that the correlation presented by [26]. The presented correlation coefficient in the context of the hesitant 2-tuple fuzzy linguistic environment indicates the degree of relationship between two variables and investigates their linear change in a positive or negative direction. We first explored the ideas of the mean and variance of H2FLEs, and afterwards proposed a new correlation coefficient. Linked to current methodologies, the most important feature of the proposed method identifies the positive and negative associations for H2FLSs. We additionally defined the weighted and ordered weighted correlation coefficients of the H2FLSs. We have proved some desired properties for the covariance and the correlation coefficient. BWM has also been used to determine the weights of criteria and alternatives w.r.t different criteria. The validation of this work is strengthened by the parallel findings of the suggested method with the available ones. The proposed work is appropriate for H2FLSs and easy to calculate, also capable to holds the property of solving MCGDM problems. In the future, we plan to propose the correlation coefficient for intuitionistic H2FLSs and apply them to the MCGDM problems.