Group decision making based on hesitant fuzzy ranking of hesitant fuzzy preference relations

Abstract

Hesitant fuzzy sets (HFSs) play a dominant role in the decision making process. Different tools are developed to attract the decision makers (DMs) in making the effective decision, the hesitant fuzzy preference relation (HFPR) is one of the important implementation of them. Preference of an alternative over another alternative is a useful way to express the opinion of decision maker. In this paper, a hesitant fuzzy ranking (HFR) technique is established, constructed the hesitant fuzzy ranking from the HFPR in the group decision making situations. Secondly, a correlation between the alternatives is developed by using Spearman ranked correlation coefficient formula, which helps the DMs to identify the better alternative. The novelty of the proposed strategy is that it evades the need to compute the cooperative preference relations and approvals are generated for the individuals in their original domains.

Keywords

Hesitant fuzzy sets Fuzzy preference relations Hesitant fuzzy preference relation Fuzzy rankings Group decision making

1 Introduction

For the last few decades, fuzzy sets have become a useful means to describe the imprecision and hesitation of real-life information through precise mathematical symbols. The fuzzy set theory was first presented by Zadeh [50] in 1965, which is the extension of the classical notion of a set. It has been effectively applied in many areas, such as fuzzy decision making [2, 48], fuzzy control [9], fuzzy preferences [6], fuzzy clustering [21] and fuzzy granular computing [4, 11]. Also, we can see the applications of fuzzy sets in many fields like artificial intelligence, computer science, medicine, control engineering, decision theory, expert systems, management science, operations research, pattern recognition and robotics. Torra [33] introduced the concept of hesitant fuzzy sets (HFSs), an extension of fuzzy sets. HFSs are a new idea that stipulates connection within a set by giving some hesitant values to each of its candidate. Consequently, the HFSs can define ambiguous belongings more conveniently than other fuzzy sets, and have ultimately received increasing attention. Furthermore, the scholars some have proposed some extensions of HFSs, such as the dual hesitant fuzzy set [60], generalized hesitant fuzzy set [27], trapezoidal-valued hesitant fuzzy sets [28], multi-hesitant fuzzy sets [26] and intuitionistic-valued hesitant fuzzy set [35].

Decision making process is a useful way in which decision makers feel happy to present his/her preferences by associating each pair and then established a preference relation. It is an effective and common expression of data collection of preference values, each of which is provided by a DM to express his/her view over a pair of elements by means of a already defined scale. By using different types of scales, a number of preference relations have been suggested, for example, the fuzzy preference relation [32], the linguistic preference relation [10, 12], the intuitionistic fuzzy preference relation [42, 43], the intuitionistic multiplicative preference relation [44], the interval-valued fuzzy preference relation [39], the interval-valued intuitionistic fuzzy preference relation [45, 46], the interval-valued intuitionistic multiplicative preference relation [47] and the triangular fuzzy preference relation [40]. Kacprzyk [15] derived a solution in GDM directly from the individual fuzzy preference relations. Nevertheless, all these preference relations do not cover the hesitant fuzzy environment because it cannot give all the possible data provided by the decision makers while comparing pairwise alternatives or criteria. In order to solve this issue, being motivated by HFSs, Zhu et al. [63] presented the concept of hesitant fuzzy preference relations (HFPRs) and then explored its distinct characteristics. In HFPRs, hesitant fuzzy elements (HFEs) describes which alternative or criteria is preferred. For example, in order to provide the grade to which an alternative a₁ is preferred to another alternative a₂, the first DM gives 0.3, the second DM gives 0.4, and the third DM gives 0.6, then the degree to which a₁ is preferred to a₂ is represented by a hesitant fuzzy element {0.3, 0.4, 0.6}. This representation preserves the original data instead of [0.3, 0.6]. Since hesitant fuzzy sets reduce the information loss. Therefore, a minimum loss of data occurs during the transformation of HFPRs into HFR. Moreover, the heterogeneous preference structure [52] is a significant tool which minimizes the loss of information in GDM. It is an effective tool to represent the preference information over alternatives for a group of decision makers, particularly when obscurity is essential to keep DM’s privacy or evade influencing each other. Furthermore, Zhu and Xu [62] developed a regression method to convert hesitant fuzzy preference relations into fuzzy preference relations (FPRs). Liao et al. [16] explored the multiplicative consistency of HFPRs and its application in group decision making. Zhang and Guo [55] implemented the consistency and consensus models for GDM with uncertain 2-tuple linguistic preference relations. Zhang et al. [56] developed a technique to derive the priority weight from an incomplete HFPR on the basis of logarithmic least squares methodology. Later on, Zhang et al. [57] described the best, worst and the average additive consistency indices to measure the consistency level of an HFPR. Consequently, Zhang and Guo [58] observed the deficiency in the already existing work about the consistency definition of an intuitionistic multiplicative preference relations (IMPR) and then suggested a novel definition to overcome these flaws. Group decision making (GDM) is an issue resolute movement through which a group of DMs mutually select the best alternative among the several existing options. A group decision making procedure comprises the assessment of the options and the decision of the most acceptable one, considering every one of the elements and opposing necessities and as indicated by the preferences of the larger part of the included specialists. Group decision making has been generally contemplated because it has many uses in numerous areas. A few methodologies have been developed so far for the portrayal of specialists’ inclinations, their aggregation,and the determination of the greatest alternative. Group decision making (GDM) helps the DMs to select the best alternative from a set of possible alternatives in accordance with the preferences that are provided by a group of decision makers [6, 34]. In GDM, every decision maker (DM) compares each pair of alternatives and provides a preference value of one alternative over another and all of which make up a preference relation. Zhan et al. [18] explored the regression method and feedback mechanism to increase the consistency and consensus for hesitant multiplicative preference relation (HMPRs) and developed it for a group decision making model with HMPRs. In the continuity of the previous article, Zhan et al. [19] also focused on the consistency and consensus for probabilistic hesitant multiplicative preference relations (PHMPRs) with the help of multiplicative preference relations and apply it for GDM. Zhang et al. [53] developed a consensus among the DMs and analyzed its effect and failure mode under the linguistic framework.

Correlation is a statistical technique that uses to estimate and study the level of association between two parameters. Correlation plays a dominant role in statistics and engineering sciences. By correlation analysis, the connection of two parameters can be observed with the help of a measure of dependency of the two variables. It measures the linear relationship between two factors. Many researchers have presented the idea of correlation about fuzzy sets. For instance, Yu [49] introduced the technique to find the correlation of fuzzy numbers, Nguyen et al. [23] presented the basics of statistics with fuzzy data. Hong and Hwang [13] developed the correlation coefficient of intuitionistic fuzzy sets in probability space by using the generalization of fuzzy sets. Chiang and Lin [7] claimed that belonging levels are tangible observational values based on the belonging functions of fuzzy sets to define fuzzy correlation coefficients. Chaudhuri and Bhattacharya [8] explored the correlation of two fuzzy sets on the same universal support. Hong [14], Ni and Cheung [22] recommended specific approaches for evaluating fuzzy correlations. Chen et al. [3] developed certain correlation coefficient formulae for HFSs and used these to cluster analysis under hesitant fuzzy situations. Arteaga et al. [1] not only expressed the quality of the connection between two hesitant fuzzy sets but also exhibited whether they are positively or negatively related. Zeng et al. [17] put forward a new correlation coefficient to evaluate the relationship between two HFSs and also introduced some new notions, for example, the average of a hesitant fuzzy elements (HFEs) and the variance of HFSs. Xia and Xu [38] described the properties of distance and correlation measures for hesitant fuzzy information in detail and then applied them in decision making problems. Recently, Zhou et al. [31] developed an innovative TOPSIS method based on hesitant fuzzy correlation coefficient in which they presented, two types of analogical factors. Based on these factors then discovered four kinds of relative closeness of the alternative and then implemented it to make the decision directly. Capuano et al. [5] presented the fuzzy ranking with respect to fuzzy preference relations (FPRs) which permits DMs to emphasis on two options at a time without reducing the entire problem so decreasing irregularities. In literature, many preference models already exist through which DMs can rank the options from top to lowest (ordinal ranking). Ordinal rankings are easy to use and help diminishing errors and irregularities. In some circumstances, specialists are unable to allocate an exact position in a level to the choices that are measured identical, or they might be essential to stipulate on what level an option is superior to the next or the ideal one.

Hesitant fuzzy preference relations (HFPRs) play a dominant role to select the best alternative from the set of options under the group decision making (GDM) situations, and a lot of work as stated in the present article, has been share to the literature by several scholars. The Spearman’s rank correlation coefficient is an important tool which gives the strength between the two ranks. Also, many experts have utilized it in their various articles under different environments as stated in the Section 7. The fuzzy rankings [5] permit the DMs to pay heed on two options simultaneously without changing the entire problem so that it can reduce the irregularities. Hence, inspired by the merits of the Spearman’s rank correlation coefficient, HFPRs and fuzzy ranking, we propose the hesitant fuzzy ranking model (the extension of fuzzy ranking [5]) for hesitant fuzzy preference relations. Since the existing techniques give the ordinal ranking of the alternatives of an HFPR, that is, from best to worst. Similar to normal rankings, DMs are requested to arrange options from top to the most inferior at the same time, utilizing particular separators. It is remarkable that decision making under the hesitant fuzzy framework got more attention and merited wide acknowledgment and further research. Moreover, by considering the merits of hesitant fuzzy relations and the fuzzy ranking model has motivated us to work with GDM based on hesitant fuzzy ranking under the environment of HFPRs. We can expand or reduce the gap among the consequent positions to strengthen the ordering connections. Similar data along with fractional rank is identified. The fractional ranks are valuable when a specialist cannot assess a few alternatives. Moreover, a fruitful discussion is presented to certify the efficiency and practicality of the said approach with the help of a numerical example.

The remaining part of the paper is organized in the following way. Section 2 introduces some basic knowledge of hesitant fuzzy preference relations and their consistency. Section 3 presents the related concepts of GDM with HFPRs. Sections 4 and 5 comprise the conversion algorithms from HFPRs to HFR and vice versa. A transformation of HFPRs into HFR is demonstrated in Section 6. Section 7 illustrates the Spearman’s rank correlation coefficient formula which is applied to find the strength between the alternatives. A numerical example illustrates the effectiveness and practicality of the said approach in Section 8. Finally, Section 9 points out some conclusions.

2 Some fundamental concepts

In the current segment, we present some fundamentals from the previous knowledge such as definition of fuzzy preference relations (FPRs), hesitant fuzzy sets and HFPRs.

Definition 2.1. [33]. Let X ={ x₁, x₂, . . . , x_n } be a fixed set, a hesitant fuzzy set B on X is defined in terms of a function b_B (x) that when applied to X returns a finite subset of [0, 1].

In order to understand easily, Xia and Xu [41] represent the HFS mathematically as:

$B = {〈 x, b_{B} (x) 〉 : x \in X}$ , where b_B (x) is a set of some distinct values in [0, 1] denoting the possible membership degrees of the element x ∈ X to the set B.

For simplicity, we can write b = b_B (x) a hesitant fuzzy element (HFE) and B the set of all HFEs. HFSs can be calculated with the help of HFEs by using some operational laws or aggregation techniques or decision methods.

Definition 2.2. Let b, b₁ and b₂ be three HFEs, and a be a positive real number, then the operations on HFEs b, b₁ and b₂ are given below [41, 61]:

$b_{1} \oplus b_{2} = \underset{r_{1} \in b_{1}, r_{2} \in b_{2}}{\cup} {r_{1} + r_{2}}$

$b ⊖ a = \underset{r \in b}{\cup} {r - a}$

$b_{1} \otimes b_{2} = \underset{r_{1} \in b_{1}, r_{2} \in b_{2}}{\cup} {r_{1} r_{2}}$

$b_{1} ø b_{2} = \underset{r_{1} \in b_{1}, r_{2} \in b_{2}}{\cup} {\begin{matrix} \frac{r_{1}}{r_{2}}, r_{1} \leq r_{2}, r_{2} \neq 0 \\ 1, otherwise . \end{matrix}$

Definition 2.3. [24]. The preference relation M = (m_ij) _n×n is said to be a fuzzy preference relation, if m_ij ∈[0, 1] such that m_ij + m_ji = 1, m_ii = 0.5 for all i, j ∈ N.

Definition 2.4. [32]. The preference relation M = (m_ij) _n×n is said to be an additive consistency fuzzy preference relation if the given below additive transitivity is satisfied,

$m_{ij} = m_{ik} - m_{kj} + 0.5 forall i, j, k \in N .$

Motivated by HFSs and FPRs, Xia and Xu [37] presented the concept of hesitant fuzzy preference relations (HFPRs) as follows:

Definition 2.5. Let X ={ x₁, x₂, . . . , x_n } be a set of alternatives, then an HFPR M on X is denoted by a square matrix $\bar{M} = ({\bar{m}}_{ij})_{n \times n} \subset X \times X$ , with ${\bar{m}}_{ij}$ = { ${\bar{m}}_{ij}^{α}$ : α = 1, . . ., $# {\bar{m}}_{ij}$ }, where $# {\bar{m}}_{ij}$ is the cardinality of ${\bar{m}}_{ij}$ , is an HFE representing all the possible values of preference degrees of the alternative x_i over x_j. Additionally, ${\bar{m}}_{ij}$ should verify the conditions below: ${\bar{m}}_{ij}^{α} \oplus {\bar{m}}_{ji}^{α} = 1, # {\bar{m}}_{ij} = # {\bar{m}}_{ji}$ for all i, j ∈ N, where ${\bar{m}}_{ij}^{α}$ is the α^th value in ${\bar{m}}_{ij}$ .

As in most of the information given by the DMs, the cardenality of HFEs is not same that is the numbers of distinct pairwise comparison elements in an HFPR may be not identical ( ${\bar{m}}_{1}$ $\neq {\bar{m}}_{2}$ ). To cope such issues Zhu et al. [63] presented a method called normalization to make that all the comparisons have the same number of elements. Hesitant fuzzy preference relations after normalization becomes normalized hesitant fuzzy preference relation (NHFPR).

Definition 2.6. Let $\bar{M} = ({\bar{m}}_{ij})_{n \times n}$ be an HFPRs, then a square matrix ${\bar{M}}^{*} = (s {\bar{m}}_{ij})_{n \times n}$ , where $s {\bar{m}}_{ij} = \frac{1}{# {\bar{m}}_{ij}} {\sum_{j = 1, j \neq i}}^{# {\bar{m}}_{ij}} ({\bar{m}}_{ij})$ is called a score value of HFPRs of $\bar{M}$ .

3 Description of GDM with HFPRs

A group decision making (GDM) issue is described by a group of decision makers D = {d₁, d₂, d₃, . . . , d_l}, each with his own data, thoughts, knowledge, and inspiration, that presents their inclinations on a limited arrangement of options X = {x₁, x₂, . . . , x_n} to accomplish a mutual solution by considering the HFPRs matrices M^l, where l denotes the number of DMs. As foreseen in Section 1, a few different manners to show and DMs’ model of inclinations have been offered. Amid these, we present underneath the most firmly identified with our work: normal or ordinal ranking and HFPR. Normal or ordinal ranking is one of the easiest inclination models. The normal positioning of the components of X may be represented as: x_σ(1) ≻ x_σ(2) ≻ . . . ≻ x_σ(n-1) ≻ x_σ(n) where σ : {1, …, n} → {1, …, k} is a permutation function. Otherwise it may be signified with a positioning assortment E = (e₁, e₂, . . . , e_n) where all the component e_i ∈ N represents the location of i^th option among the ranking. An HFPR stipulates the level to which each alternative x_i ∈ X is at least as better as any other alternative x_j by means of a hesitant fuzzy relation $\bar{M}$ , that is, a hesitant fuzzy set on X × X with a belonging function $ζ_{\bar{M}} : X \times X ⟶ {0, 1}$ such that:

$ζ_{\bar{M}} (x_{i}, x_{j}) = {\begin{matrix} 1 & if & x_{i} is certainly prioritize to x_{j}, \\ a \in [0.5, 1] & if & x_{i} is moderately prioritize to x_{j}, \\ 0.5 & if & x_{i} and x_{j} are equally prioritize, \\ b \in [0, 0.5] & if & x_{j} prioritize to x_{i}, \\ 0 & if & x_{j} is certainly prioritize to x_{i} . \end{matrix}$ (3.1)

An HFPR can be suitably denoted as a n × n matrix $\bar{M} = (\bar{m_{ij}})_{n \times n}$ where $\bar{m_{ij}} = ζ_{\bar{M}} (x_{i},$ x_j). An HFPR fulfill the additive reciprocity property that is, ${\bar{m}}_{ij}^{α} \oplus {\bar{m}}_{ji}^{α} = 1$ for all i, j ∈ N. An HFPR is said to be an additive consistent if it fulfills the additive transitivity property that is ${\bar{m}}_{ij}^{α} \oplus {\bar{m}}_{jk}^{α} \oplus {\bar{m}}_{ki}^{α} = 1.5 \forall i, j, k \in N$ .

Occasionally, because of space unpredictability, constrained aptitude or strain to settle on a decision, DMs characterize incomplete HFPRs. In order to find the missing HFPR values, an additive transitivity and reciprocity properties is applied. According to the additive transitivity’s definition, we can evaluate the missing element ${\bar{m}}_{ij}$ of options x_i over x_j with the help of equation below:

${\begin{matrix} η_{k, 1} ({\bar{m}}_{ij}) = {\bar{m}}_{ik} + {\bar{m}}_{kj} - 0.5; \\ η_{k, 2} ({\bar{m}}_{ij}) = {\bar{m}}_{kj} - {\bar{m}}_{ki} + 0.5; \\ η_{k, 3} ({\bar{m}}_{ij}) = {\bar{m}}_{ik} - {\bar{m}}_{jk} + 0.5 . \end{matrix}$ (3.2)

If $\bar{M}$ is consistent under addition, then $η_{k, 1} ({\bar{m}}_{ij}) = η_{k, 2} ({\bar{m}}_{ij}) = η_{k, 3} ({\bar{m}}_{ij}) \forall i, j, k \in {1, 2, . . ., n}$ . There are several measures exist in literature to estimate the dominance and non-dominance level [25]. Among the simplest measures motivated by [36], we proposed a net flow denoted by Ψ_nf (x_i) for the level of preference. Once the consistent HFPR $\bar{M}$ is acquired, the alternatives must be appraised partner a level of preference Ψ_nf (x_i) for all x_i ∈X depending on $\bar{M}$ .

$Ψ_{nf} (x_{i}) = \underset{j = 1, j \neq i}{\sum^{# {\bar{m}}_{ij}}} ({\bar{m}}_{ij}) ⊖ \underset{j = 1, j \neq i}{\sum^{# {\bar{m}}_{ij}}} ({\bar{m}}_{ji})$ (3.3) The first summation of Equation 3.3 shows the leaving flow, it means that, the complete level of preference x_i upon the other alternatives and the second summation shows the incoming stream, that is, the whole level of inclination of all the other options upon x_i. Subsequent to having evaluated the accessible choices with one of the depicted measures, the one with the most noteworthy level of inclination is the result of the GDM issue.

4 The hesitant fuzzy ranking model

A collection of symbols F = {⪢ , > , ≥ , ≈} (where ⪢ is used for far superior than, > is superior than, ≥ is a slightly superior than and ≈ for identical to) introduced to distinguished the level of preference between successive terms or alternatives. The hesitant fuzzy ranking on a set of alternatives X ={ x₁, x₂, . . . , x_n } can be defined as an arrangement $\tilde{R} = (x_{σ (1)} f_{1} x_{σ (2)} f_{2} . . . x_{σ (k - 1)} f_{k - 1} x_{σ (k)})$ where k ≤ n and f_i ∈ F, i ∈ {1, 2, . . . , k - 1}. The terms in the odd places in the arrangement denote a subset of the alternatives, while σ : {1, …, n} → {1, …, k} is a k-permutation function. The terms in even places are the elements belonging to F and describe a level of inclination between successive terms (where ⪢ is used for far superior than, > is superior than, ≥ is a slightly superior than and ≈ for identical to). For each option which appears maximum once in the positioning so repetitions are not permissible while fractional rankings are acknowledged. For example, the hesitant fuzzy ranking $\tilde{R} = (x_{2} ⪢ x_{4} \approx x_{5} \geq x_{1} > x_{3})$ defined on X ={ x₁, x₂, x₃, x₄, x₅ } shows that, according to DM’s judgment, the second alternative is far superior than the fourth one that, in turn, is similar to the fifth one, while both are a less superior than the first one that, in turn, is superior than the third one. From example, it turns out to be certain that, based on normal positioning, it can be seen that for a similar DM to determine his certainty so completely. Indeed, the normal raking or ordinal ranking R is x₂≻ x₄≻ x₅ ≻ x₁ ≻ x₃ that can be briefly written as E = (2, 4, 5, 1, 3), has a profoundly extraordinary semantics: similarities are not permitted so the proportionate choices x₄ and x₅ are misleadingly requested while the inclination gaps somewhere in the range of x₂ and x₄, x₅ and x₁, x₁ and x₃ appears to be similar in $\tilde{R}$ while they are altogether different in DM’s conviction.

5 From hesitant fuzzy ranking To HFPR

If a hesitant fuzzy ranking (HFR) $\tilde{R} = (x_{σ (1)} f_{1} x_{σ (2)} f_{2} . . . x_{σ (k - 1)} f_{k - 1} x_{σ (k)})$ is given, we can create the related HFPR matrix $\bar{M} = {\bar{m}}_{ij}$ with the help of two techniques. The 1^st technique comprises in partner a predefined preference level L (f_i) to every symbol f_i ∈ F and acquire HFPR elements as:

${\begin{array}{l} {\bar{m}}_{σ (i) σ (i + 1)} & = & L (f_{i}) \forall i \in {1, 2, \dots, k - 1}, \\ {\bar{m}}_{σ (i + 1) σ (i)} & = & 1 - L (f_{i}) \forall i \in {1, 2, \dots, k - 1}, \\ {\bar{m}}_{σ (i) σ (i)} & = & 0.5 \forall i \in {1, 2, \dots, k} . \end{array}$ (5.1) where the primary proclamation changes the levels of preference inserted in $\tilde{R}$ in estimations of $\bar{M}$ , wheats the 2nd and 3rd articulations are gone for guaranteeing the reciprocity of $\bar{M}$ as indicated by the definition given in Section 3. A plausible arrangement of values for the function L (f_i) is presented in Table 1.

Table 1

Realistic values for the preference level and the relative strength related to ranking symbols:

Symbol	Preference level L (f)	Relative strength \|R_s\|
⪢	[0.86, 1]	{2, 3}
>	[0.66, 0.85]	{1, 1.5}
≥	[0.51, 0.65]	{0.5, 0.75}
≈	[0, 0.5]	{0}

Example 5.1. If $\tilde{R} = (x_{4} ⪢ x_{5} \approx x_{2} \geq x_{3} > x_{1})$ is a HFR for the set of alternatives X ={ x₁, x₂, x₃, x₄, x₅ }, then a matrix $\bar{M}$ for HFPR can be obtained by using Equation 5.1 and preference level values from Table 1 as:

$\bar{M} = (\begin{matrix} {0.5} & - & {0.3, 0.35} & - & - \\ - & {0.5} & {0.58} & - & {0.5} \\ {0.65, 0.7} & {0.42} & {0.5} & - & - \\ - & - & - & {0.5} & {0.85} \\ - & {0.5} & - & {0.15} & {0.5} \end{matrix})$ The above matrix $\bar{M}$ can be completed by using Equation 3.2. $\bar{M} = (\begin{matrix} {0.5} & {0.22, 0.27} & {0.3, 0.35} & {0} & {0.22, 0.27} \\ {0.73, 0.78} & {0.5} & {0.58} & {0.15} & {0.5} \\ {0.65, 0.7} & {0.42} & {0.5} & {0.07} & {0.37, 0.47} \\ {1} & {0.85} & {0.93} & {0.5} & {0.85} \\ {0.73, 0.78} & {0.5} & {0.53, 0.63} & {0.15} & {0.5} \end{matrix})$

Another technique is constructed to get an HFPR from a HFR. For this a relative strength |R_s| is introduced for each symbol f_i ∈ F and given a HFR $\tilde{R}$ , a fractional rank f_r (x_i) is connected to each option such that: $f_{r} (x_{σ (i)}) = {1} \oplus \overset{i - 1}{\sum_{l = 1}} | R_{s_{l}} |$ (5.2) here i ∈ N and f_r (x_i) is not defined if σ (i) is not defined, it means that, if x_i is not a member of $\tilde{R}$ . The values of |R_s| should be chosen so that every symbol twice the strength of the next one. When the symbol > is used the HFR gives a normal positioning and Equation 3.3 produced the positioning arrangement as described in Section 3. The usage of the symbols ⪢ or ≥ in the place of >, respectively twice or $\frac{1}{2}$ the distance of the previous and following terms in the positioning whereas the usage of ≈ shows that the previous and following terms have the same position. So any pair of options x_i over x_j present in $\tilde{R}$ and belonging to the matrix $\bar{M}$ can be calculated with help of the formula given below: ${\bar{m}}_{ij} = {\frac{1}{2}} ({1} \oplus \frac{r (x_{j}) ⊖ r (x_{i})}{r_{max} ⊖ {1}})$ (5.3) where r_max = f_r (x_σ(k)) is the largest rank. In the case when, r_max = 1, it means that DM thoughts that all alternatives are similar that is, $\tilde{R} = (x_{1} \approx x_{2} \approx . . . \approx x_{k})$ . In order to handle such situation DM sets, ${\bar{m}}_{ij} = 0.5$ for any x_i over x_j present in $\tilde{R}$ .

Example 5.2. Let R = (x₄ ⪢ x₅ ≈ x₂ ≥ x₃ > x₁) be a hesitant fuzzy ranking for the set of options X ={ x₁, x₂, x₃, x₄, x₅ }, then with the help of relative strength values given in Table 1 and Equation 5.2, we can find the fractional ranks as:

f_r (x_σ(1)) = f_r (x_σ(i)) = {1} ⊕ ∑_j=1^i-1|R_{s
_j}| = {1} ⊕ {0} = {1}

r (x₄) = {1}

f_r (x_σ(2)) = {1} ⊕ ∑_j=1^i-1|R_{s
_j}| = {1} + |R_{s
₁}| = {1} ⊕ {2, 3}

r (x₅) = {3, 4}.

Similarly other values can be evaluated, r (x₂) = {3, 4} , r (x₃) = {3.5, 4.75} and r (x₁) = {4.5, 6.25}. By using Equation 5.1, a matrix $\bar{M}$ for HFPR is obtained:

$\begin{matrix} \bar{M} = (\begin{matrix} {0.5} & {0.18, 0.29} & [0.29, 0.36, 0.49} & {0} & {0.18, 0.29} \\ {0.71, 0.82} & {0.5} & {0.55, 0.57, 0.61} & {0.07, 0.22, 0.31} & {0.5} \\ {0.51, 0.64, 0.71} & {0.39, 0.43, 0.45} & {0.5} & {0.40, 0.15, 0.26} & {0.40, 0.43, 0.45} \\ {1} & {0.69, 0.78, 0.93} & {0.74, 0.85, 0.60} & {0.5} & {0.79, 0.69, 0.93} \\ {0.71, 0.82} & {0.5} & {0.55, 0.57, 0.60} & {0.07, 0.31, 0.21} & {0.5} \end{matrix}) \end{matrix}$

which is different from the Example 5.1, in which there is no need to complete the HFPR with the method given in Section 3. Furthermore we can see that the matrix $\bar{M}$ of HFPR is additive consistent.

6 Transformation of HFPRs To hesitant fuzzy ranking

At times, it tends to be valuable to decipher the preferences represented with HFPR $\bar{M}$ back to a hesitant fuzzy ranking. In this unit we will trnsform the preferences represented by an HFPR $\bar{M}$ to HFR. The given procedure can help DMs, D = {d₁, d₂, d₃, . . . , d_l} to achieve HFR, F = {⪢ , > , ≥ , ≈} of available set of alternatives X ={ x₁, x₂, . . . , x_n } from the given HFPRs $\bar{M} = ({\bar{m}}_{ij})_{n \times n}$ . In both cases it is expected to compute the level of inclination Ψ_nf (x_i) of every x_i ∈ X from $\bar{M}$ as discussed in Section 3. Then, a HFR $\tilde{R} = (x_{σ (1)} f_{1} x_{σ (2)} f_{2} . . . x_{σ (k - 1)} f_{k - 1} x_{σ (k)})$ is calculated where σ : {1, …, n} → {1, …, k} is a permutation function such that Ψ (σ (x_i)) ≥Ψ (σ (x_i+1)) , is taken and f_i ∈ F for every i ∈ {1, 2, . . . , k - 1}.

Two methodologies can be then constructed to distinguish the symbols f₁, f₂, . . . , f_n-1. According to first approach, for any two consecutive alternatives σ (x_i) and σ (x_i+1) in $\tilde{R}$ , evaluate symbol f_i between the two adjacent alternatives from preference ${\bar{m}}_{σ (i) σ (i + 1)}$ of $\bar{M}$ as given below: $f_{i} = {\begin{matrix} \approx & if & {\bar{m}}_{σ (i) σ (i + 1)} < 0.53 \\ \geq & if & 0.53 \leq {\bar{m}}_{σ (i) σ (i + 1)} < 0.61 \\ > & if & 0.61 \leq {\bar{m}}_{σ (i) σ (i + 1)} < 0.74 \\ ⪢ & if & {\bar{m}}_{σ (i) σ (i + 1)} \geq 0.74 \end{matrix}$ (6.1) for every i ∈ N, where the limit esteems 0.53, 0.61 and 0.74 are acquired by taking an average of each pair of consequent inclination level esteems from the the Table 1. The first technique doable only when the property of additive transitivity is hold. It means that each element of $\bar{M}$ is consistent with each others. Else, it might happen that at least one non-rational values are chosen and afterward incongruent ranking symbols produced.

Example 6.1. As Example 5.2, the matrix $\bar{M}$ of HFPR is additive consistent, based on Equation 3.3, we can determine the preference level of each option in the form of net flow as:

Ψ_nf (x₁) = {-2.76} , Ψ_nf (x₂) = {-0.03} , Ψ_nf (x₃) = {-0.83} , Ψ_nf (x₄) = {3.56} and Ψ_nf (x₅) = {-0.03}. Based on these values of the net flow, the possible ordinal ranking R is: x₄ ≻ x₂ ≻ x₅ ≻ x₃ ≻ x₁ and the corresponding HFR can be obtained by using Equation 6.1 as: ${\bar{M}}_{4, 2} = {0.85}, {\bar{M}}_{2, 5} = {0.5}, {\bar{M}}_{5, 3} = {0.53, 0.63} = {0.57}$ and ${\bar{M}}_{3, 1} = {0.65, 0.70} = {0.68}$ which is $\tilde{R} = x_{4} ⪢ x_{2} \approx x_{5} \geq x_{3} > x_{1}$ .

For any two consecutive alternatives x_σ(i) and x_σ(i+1) for a HFR $\tilde{R}$ , the second way evaluate its middle symbol f_i from the levels of preference Ψ (σ (x_i)) and Ψ (σ (x_i+1)) related to the options according to technique described in Section 3:

$\begin{matrix} f_{i} = \\ {\begin{matrix} \approx & if & Ψ (x_{σ (i + 1)}) - Ψ (x_{σ (i)}) < 0.32 \cdot Ω \\ \geq & if & 0.32 \cdot Ω \leq Ψ (x_{σ (i + 1)}) - Ψ (x_{σ (i)}) < 0.94 \cdot Ω \\ > & if & 0.94 \cdot Ω \leq Ψ (x_{σ (i + 1)}) - Ψ (x_{σ (i)}) < 1.88 \cdot Ω \\ ⪢ & if & Ψ (x_{σ (i + 1)}) - Ψ (x_{σ (i)}) \geq 1.88 \cdot Ω \end{matrix} \end{matrix}$ (6.2) where the limit esteems 0.32, 0.94 and 1.88 are acquired by taking the mean of each pair of consequent relative strength esteems from Table 1, and Ω is the mean difference between the levels of preference of two consequent options in the ranking and can be calculated by using the formula given given in Equation 6.3. $Ω = \frac{1}{n - 1} \underset{i = 1}{\sum^{n - 1}} (Ψ (x_{σ (i + 1)}) ⊖ Ψ (x_{σ (i)}))$ (6.3)

Example 6.2. From the HFPR obtained from Example 5.2, the ordinal ranking R is obtained as: x₄ ≻ x₂ ≻ x₅ ≻ x₃ ≻ x₁. The mean difference (Ω) can be evaluated by usjing Equation 6.3 as Ω = 1.48. Based on Equation 6.2, the HFR of the given alternatives can be obtained as: $\tilde{R} = x_{4} ⪢ x_{2} \approx x_{5} \geq x_{3} > x_{1}$ .

7 Spearman’s rank correlation coefficient

The Spearman’s rank correlation coefficient is the non-parametric statistical tool which use to contemplate the quality of relationship between the two positioned factors. This strategy is connected to the ordinal arrangement of numbers, which can be organized all together, i.e. in a steady progression with the goal that positions can be given to each. The Spearman’s rank relationship coefficient strategy is connected just when the underlying information is as positions, and number of observations are genuinely little, i.e. not more than 25 or 30.

Among the correlation coefficients proposed by Charles Spearman [29, 30] Spearman1,Spearman2 is a generally utilized nonparametric relationship measure that Maurice Kendall properly connected with Spearman’s name a fourth of a century later [20], and that is one of the most established insights in view of positions. The Spearman rank coefficient computed for a sample of data is typically designated as ρ. If the data has two or more identical fuzzy ranking values, the data is called an identical or similar data. The Spearman rank coefficient can be computed for such identical data as,

$ρ = \frac{\overset{n}{\sum_{k = 1}} (f_{ri} (x_{k}) - {\bar{r}}_{i}) (f_{rj} (x_{k}) - {\bar{r}}_{j})}{\sqrt{\overset{n}{\sum_{k = 1}} (f_{ri} (x_{k}) - {\bar{r}}_{i}) \cdot \overset{n}{\sum_{k = 1}} (f_{rj} (x_{k}) - {\bar{r}}_{j})}}$ (7.1) where ${\bar{r}}_{i} = {\sum_{k = 1}}^{n} f_{ri} (x_{k})$ is a mean value of fractional rank withdraw from ${\tilde{R}}_{i}$ and ${\bar{r}}_{j}$ is a mean value of fractional rank withdraw from ${\tilde{R}}_{j}$ .

8 Numerical example

Consider an organization wants to appoint a manager for a particular job. The competent authority gave this task to the five decision makers to select an outstanding candidate out of five.

Let D = {d₁, d₂, d₃, d₄, d₅} be the five DMs and X = {x₁, x₂, x₃, x₄, x₅} be the set of candidates which are considered as the alternatives. The five DMs provide their HFPRs matrices, M¹, M², M³, M⁴ and M⁵, respectively according to their expertise.

$\begin{matrix} M^{1} = (\begin{matrix} {0.5} & {0.3, 0.5} & {0.1, 0.2} & {0.2, 0.6} & {0.5} \\ {0.5, 0.7} & {0.5} & {0.6, 0.7} & {0.1} & {0.5} \\ {0.8, 0.9} & {0.3, 0.4} & {0.5} & {0, 0.1} & {0.3, 0.6} \\ {0.4, 0.8} & {0.9} & {0.9, 1} & {0.5} & {0.7, 0.8} \\ {0.5} & {0.5} & {0.7, 0.4} & {0.2, 0.3} & {0.5} \end{matrix}) \end{matrix}$

$\begin{matrix} M^{2} = (\begin{matrix} {0.5} & {0.4, 0.6} & {0.5, 0.3} & {0.3, 0.4} & {0.4} \\ {0.6, 0.4} & {0.5} & {0.3, 0.4} & {0.3, 0.5} & {0.4, 0.7} \\ {0.5, 0.7} & {0.6, 0.7} & {0.5} & {0.3, 0.5} & {0.5, 0.7} \\ {0.6, 0.7} & {0.5, 0.7} & {0.5, 0.7} & {0.5} & {0.2, 0.3} \\ {0.6} & {0.3, 0.6} & {0.3, 0.5} & {0.7, 0.8} & {0.5} \end{matrix}) \end{matrix}$

$\begin{matrix} M^{3} = (\begin{matrix} {0.5} & {0.4, 0.2} & {0.3, 0.4} & {0.4, 0.6} & {0.5} \\ {0.6, 0.8} & {0.5} & {0.5, 0.7} & {0.1, 0.5} & {0.5} \\ {0.6, 0.7} & {0.3, 0.5} & {0.5} & {0.2, 0.7} & {0.1, 0.3} \\ {0.4, 0.6} & {0.5, 0.9} & {0.3, 0.8} & {0.5} & {0.5, 0.6} \\ {0.5} & {0.5} & {0.7, 0.9} & {0.4, 0.5} & {0.5} \end{matrix}) \end{matrix}$

$\begin{matrix} M^{4} = (\begin{matrix} {0.5} & {0.2, 0.3} & {0.1, 0.3} & {0.4, 0.5} & {0.2, 0.4} \\ {0.7, 0.8} & {0.5} & {0.6, 0.7} & {0.3, 0.5} & {0.5} \\ {0.7, 0.9} & {0.3, 0.4} & {0.5} & {0.5, 0.7} & {0.2, 0.3} \\ {0.5, 0.6} & {0.5, 0.7} & {0.3, 0.5} & {0.5} & {0.9, 1.0} \\ {0.6, 0.8} & {0.5} & {0.7, 0.8} & {0, 0.9} & {0.5} \end{matrix}) \end{matrix}$

$\begin{matrix} M^{5} = (\begin{matrix} {0.5} & {0.6, 0.7} & {0.5, 0.6} & {0.3, 0.8} & {0.7} \\ {0.3, 0.4} & {0.5} & {0.4} & {0, 1} & {0.9, 1} \\ {0.4, 0.5} & {0.6} & {0.5} & {0.8, 0.9} & {0.7, 0.9} \\ {0.2, 0.7} & {1, 0} & {0.1, 0.2} & {0.5} & {0.6} \\ {0.3} & {0, 0.1} & {0.1, 0.3} & {0.4} & {0.5} \end{matrix}) \end{matrix}$

From the received HFPRs matrices, based on Equation 3.3, obtain the ordinal ranking R₁, R₂, …, R₅ as discussed in Section 6 and presented in Fig. 1. Then, formulate the hesitant fuzzy ranking (HFR), ${\tilde{R}}_{1}, {\tilde{R}}_{2}, \dots, {\tilde{R}}_{5}$ among the alternatives x₁, x₂, x₃, x₄ and x₅, respectively.

Fig.1

Interpretation of ordinal ranking of ${\tilde{R}}_{i}, i = 1, 2, 3, 4, 5$ .

To find the HFR, first of all find the ordinal ranking among the alternatives x_i, where i = 1, 2, …, 5 by using Equation 3.3 such that:

Ψ_nf (x₁) = {-2.4, - 1.8} = {-2.1} , Ψ_nf (x₂) = {-0.6, 0} = {0 - 0.3} , Ψ_nf (x₃) = {-1.2, - 0} = {-0.6} , Ψ_nf (x₄) = {1.8, 3} = {2.4} and Ψ_nf (x₅) = {-0.2, - 0.6} = {-0.4}. Then, the ordinal ranking R₁ is x₄ ≻ x₂ ≻ x₅ ≻ x₃ ≻ x₁. From the ordinal ranking, the HFR among the alternatives can be evaluated by using Definition 2.6 and Equation 6.1 as: $M_{4, 2}^{1} = {0.9}$ , $M_{2, 5}^{1} = {0.5}$ , $M_{5, 3}^{1} = {0.4, 0.7} = {0.55}$ , $M_{3, 1}^{1} = {0.8, 0.9} = {0.85}$ , we get, x₄ ⪢ x₂ ≈ x₅ ≥ x₃ > x₁. Therefore, HFR ${\tilde{R}}_{1}$ obtained from the HFPRs matrix given by the DM d₁ is ${\tilde{R}}_{1} = x_{4} ⪢ x_{2} \approx x_{5} \geq x_{3} > x_{1}$ . Now the fractional ranks of ${\tilde{R}}_{1}$ can be evaluated by using Equation 5.2 and Table 1 as:

f_r (x_σ(1)) = r (x₄) = {1} ⊕ ∑_j=1^i-1|R_{s
_j}| = {1} ⊕ {0} = {1},

f_r (x_σ(2)) = r (x₅) = {1} ⊕ |R_{s
₁}| = {1} ⊕ {2, 3} = {3, 4},

f_r (x_σ(3)) = r (x₂) = {1} ⊕ |R_{s
₁}| ⊕ |R_{s
₂}| = {1} ⊕ {2, 3} = {3, 4},

f_r (x_σ(4)) = r (x₃) = {1} ⊕ |R_{s
₁}| ⊕ |R_{s
₂}| ⊕ |R_{s
₃}| = {1} ⊕ {2, 3} ⊕ {0.5, 0.75} = {3.5, 4.75} and

f_r (x_σ(5)) = r (x₁) = {1} ⊕ |R_{s
₁}| ⊕ |R_{s
₂}| ⊕ |R_{s
₃}| ⊕ |R_{s
₄}| = {1} ⊕ {2, 3} ⊕ {0.5, 0.75} ⊕ {1, 1.5} = {4.5, 6.25}.

Similarly remaining ordinal ranking, HFR and their fractional ranks can be found and written in the Table 2.

Table 2

HFRs (R_i, i = {1, 2, . . . , 5}) and the fractional ranks of each alternative:

Ranking	f_ri (x₁)	f_ri (x₂)	f_ri (x₃)	f_ri (x₄)	f_ri (x₅)
${\tilde{R}}_{1} = x_{4} ⪢ x_{2} \approx x_{5} \geq x_{3} > x_{1}$	{4.5, 6.25}	{3, 4}	{3.5, 4.75}	{1}	{3, 4}
${\tilde{R}}_{2} = x_{5} > x_{4} \geq x_{3} \geq x_{2} \approx x_{1}$	{3, 4}	{3, 4}	{2.5, 3.25}	{2, 2.5}	{1}
${\tilde{R}}_{3} = x_{4} \geq x_{5} \approx x_{2} \geq x_{3} \geq x_{1}$	{4, 5.5}	{1.5, 1.75}	{3.5, 4.75}	{1}	{1.5, 1.75}
${\tilde{R}}_{4} = x_{4} ⪢ x_{5} \approx x_{2} \geq x_{3} > x_{1}$	{4, 5.5}	{3, 4}	{2.5,3.5}	{1}	{3, 4}
${\tilde{R}}_{5} = x_{3} \approx x_{1} \geq x_{2} ⪢ x_{4} \geq x_{5}$	{1}	{1.5, 1.75}	{1}	{3.5, 4.75}	{4, 5.5}

As the Spearman’s rank correlation coefficient is utilized to measure the deviation among the opinions of different decision makers. Therefore, based on Equation 7.1, we can find out the Spearman’s rank correlation coefficient ρ such that: $ρ ({\tilde{R}}_{1}, {\tilde{R}}_{2}) = {0.4102}$ (weak positive correlation), $ρ ({\tilde{R}}_{1}, {\tilde{R}}_{3}) = {0.8347}$ (strongly positive correlation), $ρ ({\tilde{R}}_{1}, {\tilde{R}}_{4}) = {0.9533}$ (very strongly positive correlation) and $ρ ({\tilde{R}}_{1}, {\tilde{R}}_{5}) = {- 0.6807}$ (moderate negative correlation). We see that Spearman’s rank correlation coefficient between ${\tilde{R}}_{1}$ and ${\tilde{R}}_{4}$ is much stronger than the others. Hence the consensus between ${\tilde{R}}_{1}$ and ${\tilde{R}}_{4}$ is superior.

From the Fig. 2, by comparing the five HFR, we can see that the hesitant fuzzy rankings, ${\tilde{R}}_{1}$ and ${\tilde{R}}_{4}$ are slightly identical. In fact, the preference gap between x₄ and x₅ or x₂ is perfectly same in ${\tilde{R}}_{1}$ and ${\tilde{R}}_{4}$ while the preference gap among x₅ or x₂, x₃ and x₁ is slightly differ in ${\tilde{R}}_{1}$ and ${\tilde{R}}_{4}$ , which shows that the alternative x₄ is the best choice.

Fig.2

Interpretation of HFR ${\tilde{R}}_{i}, i = 1, 2, 3, 4, 5$ .

9 Conclusion

The present work exhibits a novel preference model for group decision making named as hesitant fuzzy ranking, which consolidates the ease of use of the ordinal ranking model. Then, the correlation between the alternatives is developed by using Spearman’s rank correlation coefficient, which helps the DMs to identify the strength among the hesitant fuzzy rankings. Similar to HFPRs, HFR permit DMs to concentrate on the two options simultaneously. Actually, some important symbols have been inserted among the alternatives. Hesitant fuzzy rankings are common speculation of ordinal rankings and might be utilized in situations when a DM is not sure about exact position of the given object. Additionally, the introduced approach can be utilized in GDM, as rankings given by some specialists can be effortlessly changed over into hesitant fuzzy ranking which is to be further assessed. The principal distinction with deference our model exist in the manner in which the ranking idea is fuzzyfied. Rather than permitting a similar object belongs to numerous situations, in real, our model permits to expand or reduce the gap between consequent positions to strengthen or diminish the ordering connection. Recently, consensus reaching processes [51 , 59] and consensus measures [57, 54] have achieved much attention to develop a consensus between the DMs under the framework of fuzzy extensions. There are many prospects to work on HFR, using these concepts in the future.

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