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Abstract

As the rapidly progressing applications of uncertainty theories, the need for modifications to some of their existing mathematical tools or creating new tools to deal correctly with them in various environments is also exposed. Hesitant fuzzy numbers (HFNs), as a particular case of fuzzy numbers, are not an exception to this rule. Considering the necessity of determining the distance between given HFNs in many of their practical applications, this article shows that the existing methods either do not provide correct results or are not able to meet the needs of users. This paper aims to present new methods for distance measures of hesitant fuzzy numbers. To do them, three prevalent distance measures, i.e., the generalized distance measure, the Hamming distance measure, and the Euclidean distance measure, will be optimized into three distinct trinal categories. With the approach of reducing error propagation via reducing some unnecessary mathematical computations, new distance measures on HFNs will be introduced, first. The middle is the modification of the first category, which is more suitable when the given HFNs are equal-distance by the previous formula. Also, as the third category, the weighted form of these distance measures has been proposed, to be used where the real and membership parts of HFNs are not of equal importance. As an application of these, a TOPSIS-based technique for solving multi-attribute group decision-making problems with HFNs has been proposed. A numerical example will be implemented to describe the presented method. Finally, along with the validation of the proposed method, its numerical comparison with some other existing methods will be discussed in detail.

Keywords

Hesitant fuzzy numbers MAGDM Hamming distance Euclidean distance TOPSIS

1 Introduction

It is not in vain to call today’s world, which is the product of right/wrong, timely/untimely, and so on, decisions, a world full of successes and failures. In the process of solving practical problems, familiarity with the nature of the given problem is very significant in doing better in its first step (modeling of uncertainty) and achieving the successfulness. In general, the real world’ problems are divided into three categories [42]:

Problems with a very limited number of factors (maximum in levels 2, 3, and 4 factors), called organized simplicity problems;

Problems with a large but finite number of factors, called organized complexity problems;

Problems with an infinite number of factors, called disorganized complexity problems.

Many of the real problems have a large, but limited number of factors and fall into the category of organized complexity problems, that have remained untouched [42].

Uncertainty, as the main element in the decision-making process, plays an important role in achieving one of the two mentioned positions (success and failure), and with more mastery of this element, more success will be possible. So far, to model uncertainty as a branch of ignorance with ingredients of vagueness, probability, and ambiguity [37], various theories have been proposed. For example, after introducing the FSs theory and its many applications, in response to the need for modeling some other types of uncertainty problems, other extensions of it were introduced. Some of these are type-2 fuzzy sets [55], intuitionistic fuzzy sets [1], hesitant fuzzy sets (HFSs) [39], hesitant fuzzy numbers (HFNs) [17], generalized hesitant fuzzy numbers (GHFNs) [19], pythagorean fuzzy set [4], neutrosophic sets [62], neutrosophic fuzzy sets [2], etc. It is also possible to see many scientific researches that are trying to improve decision-making processes by introducing new tools or by combining existing scientific tools. Three-way decision theory [57, 60], probabilistic linguistic term sets [14-16], three-way consensus model based on regret theory [61], double hierarchy hesitant fuzzy linguistic term set [12, 59], linguistic preference orderings [13] are examples of such efforts.

HFSs, as one of the extensions of FSs, are suitable for situations where the hesitation degrees of the decision-maker (DM) have been given by a finite set of some values between 0 and 1. This has caused HFSs to receive a lot of attention, and find many applications in solving organized complexity problems [31]. For simplicity, what is used in practical applications of HFSs, is the set of hesitation degrees, called hesitant fuzzy elements (HFEs). Therefore, some mathematical concepts such as arithmetic operations and operation laws [24 , 58], distance/similarity/entropy measures and correlation coefficient [28 , 51], aggregation operators [44], and the comparison of hesitant fuzzy elements [7 , 27], have been investigated in various manners by scientists. In some other real applications, HFEs have been extended to HFNs, where each element is either defined as a fuzzy number in [0, 1] [33], or as trapezoidal fuzzy numbers that are different only in their heights [3].

However, due to the variety of sources of uncertainty in the real world, which is called the garden of uncertainty by Pollak [32], it cannot be claimed that the need for uncertainty modeling tools has been eliminated. For example, when a weightlifter requests a certain weight, say 200kg, there is no guarantee of his success. Let attendances’ comments about his success have been expressed with the HFE {.4, . 5, . 6, . 75, . 8, . 9, . 95}. As a result, we have two types of data for this athlete that may be modeled as an interval [180, 205], a trapezoidal fuzzy number [180, 187, 195, 205], and so on. It is caused that the domain of possible weights is extended from a singleton to an infinite set and that the cardinality of the membership set is also extended to infinite. This will change the nature of the problem from organized complexity to disorganized complexity problems, and then the original problem will be abandoned unresolved. Perhaps this approach has led Waever [42] to claim that a large part of real-world issues (i.e., organized complexity problems) remains intact.

In one of the latest generalizations, it is assumed that there is a predetermined crisp value p, or the DM uses a real value for whatever reason that hesitated and expressed by an HFE {γ₁, γ₂, …, γ_n} , γ_i ∈ [0, 1] (i = 1, …, n). In this case, preserving the nature of problem means preserving the originality of these two given data sets, as much as possible. It can be done by considering them as a unique concept as 〈p ; {γ₁, γ₂, …, γ_n} 〉. The ordered pair ${\tilde{p}}_{H} = 〈 p; {γ_{1}, γ_{2}, \dots, γ_{n}} 〉, γ_{i} \in [0, 1] (i = 1, \dots, n)$ including the real part p and the membership part {γ₁, γ₂, …, γ_n}, is called an HFN [17, 18]. So, in the above example, we will be able to express the selected weight value and its related assessment values as a HFN 〈200 ; {.4, . 5, . 6, . 75, . 8, . 9, . 95} 〉.

It should be mentioned that HFNs have existed in the real world and that before they were introduced, other types of modeling tools instead of them have been mistakenly used. Also, note that we encounter with three completely different definitions of HFNs in the literature, which are only common in the title. Deli [3] assumed that a trapezoidal fuzzy number [a, b, c, d] , a < b < c < d is not necessarily normal, and its height may be hesitated by the finite set {h₁, h₂, …, h_k} , h_i ∈ [0, 1]. It is displayed by 〈 [a, b, c, d] , {h₁, h₂, …, h_k} 〉 and called an HFN. The second definition has been introduced by Ranjbar et al. [33, 34], in which an HFN is an HFE whose elements are fuzzy numbers. The latest, which is discussed in this paper, defines the HFN as an ordered pair 〈p ; {γ₁, γ₂, …, γ_n} 〉 [18].

In order to apply the new concept of HFNs, the operation laws of HFNs, arithmetic operations of HFNs, the distance measure of HFNs, and the comparison method of HFNs were introduced [8, 17]. Recently, some mean-based aggregation operators of HFNs such as hesitant fuzzy weighted arithmetic averaging, hesitant fuzzy weighted geometric averaging, hesitant fuzzy ordered weighted arithmetic averaging, and hesitant fuzzy ordered weighted geometric averaging operators, have been discussed to aggregate HFNs [9, 17].

Also, if for the real part of the HFN 〈p ; {γ₁, γ₂, …, γ_n} 〉, i.e. p, more than one value and of course with a finite set of values such as {p₁, p₂, …, p_m} are provided, by generalizing this concept to generalized hesitant fuzzy numbers (GHFNs) 〈 {p₁, p₂, …, p_m} ; {γ₁, γ₂, …, γ_n} 〉, γ_i ∈ [0, 1] (i = 1, …, n), they can be used to model such situations [9 , 20].

Determining the distance between HFNs is one of the necessities in many of their practical applications. Although methods have been introduced for this task [8], there is evidence that in some cases do not lead to reliable results. Therefore, the purpose of this article is to find ways to calculate the distance more accurately. For this purpose, the existing methods have been examined and detailed pathology, it was observed that in the process of calculating the distance, each of the degrees of doubt in each HFN participated with all the degrees of doubt of the other HFN. In other words, for two given HFNs 〈p ; {γ₁, γ₂, …, γ_n} 〉 and 〈q ; {η₁, η₂, …, η_m} 〉, there exist m × n different combinations of hesitation degrees. Also, by dividing the resulting values by the number of performed operations, i.e., 1 + m × n in this case, as a result, the final obtained value is far from the real state. For example, let $\tilde{a} = 〈 20; {1, 1, 1, 1, 1, 1, 1} 〉$ and $\tilde{b} = 〈 2; {1, 1, 1, 1, 1, 1, 1} 〉$ be the HFNs forms of two real values 20 and 2. Based on the given Hamming distance in Equation (1), we have $d_{H} (\tilde{a}, \tilde{b}) = \frac{1}{1 + 6 \times 4} (| 20 - 2 | + 0) = 0.36$ , which is far from reality. While the given Hamming distance in Def. 7 gets us the real distance between 20 and 2, i.e., in this case we have, $d_{H} (\tilde{a}, \tilde{b}) = | 20 - 2 | + \frac{1}{7} \times (0) = 18$ .

Moreover, from the perspective of error analysis, when we work with approximate data, as the mathematical operations increase, so does the computational error in the final result. As the HFNs are approximate values to model a real uncertain situation, these formulas have also disadvantages from the point of view of error propagation due to the excessive use of degrees of doubt in determining the distance and in fact, due to great mathematical calculations. Therefore, in this paper, in order to solve the beforementioned defects, we have introduced improved distance functions in three separate categories. In fact, m × m different combinations of hesitation degrees have been reduced to m, (m is the cardinality of the membership set of the given adjusted HFNs). The common feature of all three categories is that, first, the number of calculations has been reduced without losing the available information. Secondly, the dispersion of membership degrees is considered in determining the distance.

In a special category of decision problems called multi-criteria decision-making (MADM), a finite number of options must be ranked according to a finite number of criteria. Some methods for solving these problems are as follows: analytic hierarchy process (AHP), analytic network process (ANP), Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS), Choquet integral (CI), simple additive weighting (SAW) [40], Vise Kriterijumsk Optimizacija Kompromisno Resenje (VIKOR) [6], ELimination and Choice Expressing REality (ELECTRE) [56], and so on. These problems are one of the areas that have the greatest variety of data uncertainty. Therefore, by generalizing the above existing methods from accurate to inaccurate environments, the various introduced theories can be extensively used to model uncertainty [30, 36]. For instance, some of the generalized methods to solve uncertain MADM problems with HFSs data are the hesitant fuzzy TOPSIS method [53], hesitant fuzzy VIKOR method [23, 29], hesitant fuzzy power average-based method [22], hesitant fuzzy COMET (Characteristic Objects Method) [5], hesitant fuzzy preference relation [27], hesitant fuzzy aggregation operators [24 , 58], some approaches to hesitant fuzzy MADM problems with incomplete weight information [45], and defining a class of hesitant fuzzy knowledge measure with its application in MADM [21].

Also, utilizing HFNs, a hybrid technique TOPSIS-CI based on combining CI and TOPSIS methods [8] and mean-based averaging methods [17] are proposed to solve multi-attribute group decision-making (MAGDM) problems.

In this paper, new distinct modifications of three popular distance measures –the generalized distance measure, Hamming distance measure, and Euclidean distance measure–will be proposed to determine the distance between any two given HFNs. These enable us to manage the error propagation by reducing the number of computations, and also, we can increase the efficiency of the proposed distance measures by considering the closeness and dispersion of input data. Based on these new distance measures, the most familiar and successful method in solving MADM problems, the TOPSIS method, will be updated to be applied with the decision matrix $\tilde{D} = [〈 d_{ij}, h_{ij} 〉]$ consisting of HFNs information.

In this way, the continuation of the article will be set as follows. HFSs, HFNs, and some concepts needed in other sections are given in Section 2. The new three distinct optimized trinal categories of hesitant distance measures of HFNs, called generalized distance measure (d_G), Hamming distance measure (d_H), and Euclidean distance measure (d_E), have been proposed in Section 3. The TOPSIS method will be extended to solve an MADM problem with the decision matrix $\tilde{D} = [〈 d_{ij}, h_{ij} 〉]$ consisting of HFNs, in Section 4. In Section 5, the proposed method is described with a numerical example, and the results will be analyzed and validated. The conclusion of the article will be given in Section 6.

2 HFSs and HFNs

HFSs, HFNs, and some other necessary concepts will be reviewed in this section.

Let X be a reference set. A subset E of X defined as E = {< x, h (x) > |x ∈ X}, is called an HFS, where h (x) = {γ₁, γ₂, …, γ_l} for γ_j ∈ [0, 1] (j = 1, …, l) is a set containing the membership degrees of x ∈ X to the set E [39, 48]. For simplicity, the HFE h (x) = {γ₁, γ₂, …, γ_l}, is used to model uncertainty in practical applications of HFSs. Therefore, many required computational and mathematical concepts, such as arithmetic operations, comparisons, distance determination, and so on, are developed on this basis [27 , 48]. It is clear that two different HFEs can be different in terms of the number of members. Consider two arbitrary HFEs h₁ and h₂, with |h₁| = k and |h₂| = l, where k < l. To adjust them, the HFE h₁ must be extended by repeating a value l - k times in it, which is defined as follows: In the optimistic state, it is the maximum element in h₁; in the pessimistic state, it is the minimum element in h₁; in the indifference state, it is 0.5; and otherwise, it is the power average of the available elements in h₁ [22, 26].

In some real applications of HFSs, we need to compute the distance between given HFEs. The Euclidean and Hamming distances, as two famous distance functions, have been extended to do it [38 , 50].

Definition 1. [38] Consider two adjusted HFEs h_i = {γ_i(j)|j = 1, 2, …, l} , i = 1, 2, in which γ_i(j) is the jth smallest value in h_i. Then $\begin{matrix} d_{HNH} (h_{1}, h_{2}) = \frac{1}{l} \sum_{j = 1}^{l} | γ_{1 (j)} - γ_{2 (j)} |, {% \end{matrix} d_{HNE} (h_{1}, h_{2}) = \sqrt{\frac{1}{l} \sum_{j = 1}^{l} | γ_{1 (j)} - γ_{2 (j)} |^{2}},$ are called hesitant normalized Hamming and Euclidean distances, respectively.

Arithmetic operations of HFEs have been also defined in different ways [24 , 58].

Definition 2. [27] Let h (x) = {γ₁, γ₂, …, γ_l}, let h_i = {γ_ij|j = 1, 2, …, l} , (i = 1, 2, …, n) be a collection of given HFEs, and let λ be a positive real number. Then

(1) h^λ = {(γ_j) ^λ|j = 1, 2, …, l},

(2) λh = {1 - (1 - γ_j) ^λ|i = 1, 2, …, l },

(3) h₁ ⊕ h₂ = {γ_1(j) + γ_2(j) - γ_1(j) γ_2(j)|j = 1, 2, …, l },

(4) h₁ ⊗ h₂ = {γ_1(j) γ_2(j)|j = 1, 2, …, l },

(5) $\oplus_{i = 1}^{n} h_{i} = {1 - Π_{i = 1}^{n} (1 - γ_{i (j)}) | j = 1, 2, \dots, l}$ ,

(6) $\otimes_{i = 1}^{n} h_{i} = {Π_{i = 1}^{n} γ_{i (j)} | j = 1, 2, \dots, l}$ ,

where γ_(j) and γ_i(j) are the jth smallest value in h and h_i, respectively.

In response to the great variety of real problems that necessitated new tools to solve them, HFNs were introduced by generalizing hesitant fuzzy elements, and they were applied to solve decision-making problems [3 , 33].

Definition 3. [17] Suppose that a positive value $a \in ℝ$ hesitated by an HFE h (a) = {γ₁, γ₂, …, γ_n} , γ_i ∈ [0, 1] is given. Then, combining these two pieces of information together as ${\tilde{a}}_{H} = 〈 a; h (a) 〉$ will give us what is called an HFN.

Definition 4. [17] For any two arbitrary HFNs ${\tilde{a}}_{H} = 〈 a, h (a) 〉$ and ${\tilde{b}}_{H} = 〈 b, h (b) 〉$ and real value λ > 0, the following notations are defined:

$(1) {\tilde{a}}_{H} \oplus {\tilde{b}}_{H} = 〈 a + b, h (a) \cup h (b) 〉$ , where h (a) ∪ h (b) = ⋃ _{γ₁∈h(a),γ₂∈h(b)} max {γ₁, γ₂},

$(2) λ {\tilde{a}}_{H} = 〈 λ a, h (a) 〉$ ,

$(3) ({\tilde{a}}_{H})^{λ} = 〈 a^{λ}, h (a) 〉$ ,

$(4) {\tilde{a}}_{H} \otimes {\tilde{b}}_{H} = 〈 a . b, h (a) \cap h (b) 〉$ , where h (a) ∩ h (b) = ⋃ _{γ₁∈h(a),γ₂∈h(b)} min {γ₁, γ₂}.

Definition 5. [8] Let there exist two HFNs ${\tilde{a}}_{H} = 〈 a, h (a) 〉$ and ${\tilde{b}}_{H} = 〈 b, h (b) 〉$ , where $a, b \in ℝ$ and h (a) and h (b) with cardinalities |h (a) | = m and |h (b) | = n, are two sets of some values in [0, 1]. Then the hesitant distance between them based on the Euclidean distance (d_E) and Hamming distance (d_H) are defined as follows:

$\begin{array}{l} d_{E} ({\tilde{a}}_{H}, {\tilde{b}}_{H}) \\ = \sqrt{\frac{1}{1 + m \times n} (| a - b |^{2} + \sum_{\underset{γ_{j} \in h (b)}{γ_{i} \in h (a),}} | γ_{i} - γ_{j} |^{2}),} \\ d_{H} ({\tilde{a}}_{H}, {\tilde{b}}_{H}) \\ = \frac{1}{1 + m \times n} (| a - b | + \sum_{\underset{γ_{j} \in h (b)}{γ_{i} \in h (a),}} | γ_{i} - γ_{j} |) . \end{array}$ (1)

3 New hesitant distance measures of HFNs

In this section, the generalized, Hamming, and Euclidean distances [10 , 35] will be updated to be used with HFNs.

Definition 7. Let ${\tilde{a}}_{H} = 〈 a; {γ_{1}, γ_{2}, \dots, γ_{m}} 〉$ and ${\tilde{b}}_{H} = 〈 b; {λ_{1}, λ_{2}, \dots, λ_{m}} 〉$ be two arbitrary given adjusted HFNs. Then, their generalized hesitant distance $d_{G} ({\tilde{a}}_{H}, {\tilde{b}}_{H})$ , Hamming hesitant distance $d_{H} ({\tilde{a}}_{H}, {\tilde{b}}_{H})$ , and Euclidean hesitant distance $d_{E} ({\tilde{a}}_{H}, {\tilde{b}}_{H})$ will be defined as follows:

$\begin{matrix} d_{G} ({\tilde{a}}_{H}, {\tilde{b}}_{H}) = [| a - b |^{ν} + \frac{1}{m} \sum_{j = 1}^{m} | γ_{(j)} - λ_{(j)} |^{ν}]^{\frac{1}{ν}}, \\ d_{H} ({\tilde{a}}_{H}, {\tilde{b}}_{H}) = | a - b | + \frac{1}{m} \sum_{j = 1}^{m} | γ_{(j)} - λ_{(j)} |, \\ d_{E} ({\tilde{a}}_{H}, {\tilde{b}}_{H}) = \sqrt{| a - b |^{2} + \frac{1}{m} \sum_{j = 1}^{m} | γ_{(j)} - λ_{(j)} |^{2}}, \end{matrix}$ (2)

Note 1 It is easy to see that, the generalized hesitant distance reduces to the Hamming hesitant distance for ν = 1, and for ν = 2, it is reduced to the Euclidean hesitant distance.

Example 1. Let ${\tilde{a}}_{H} = 〈 5; {. 4, . 5, . 2, . 9} 〉$ and ${\tilde{b}}_{H} = 〈 2; {. 5, . 8, . 1, 1} 〉$ . Then $d_{G} ({\tilde{a}}_{H}, {\tilde{b}}_{H}) = [3^{ν} + \frac{1}{4} (. 1^{ν} + . 1^{ν} + . 3^{ν} + . 1^{ν})]^{\frac{1}{ν}}$

$d_{H} ({\tilde{a}}_{H}, {\tilde{b}}_{H}) = 3 + \frac{1}{4} (. 1 + . 1 + . 3 + . 1) = 3.15$ ,

$d_{E} ({\tilde{a}}_{H}, {\tilde{b}}_{H}) = \sqrt{3^{2} + \frac{1}{4} (. 01 + . 01 + . 09 + . 01)} = 3.02$ .

Note 2 In Example 1, the distances between given HFNs ${\tilde{a}}_{H}$ and ${\tilde{b}}_{H}$ , using the Hamming and Euclidean distances in Eqs. (1), are 0.8289 and 0.4944, respectively.

Considering the obtained distances in both methods, the newly introduced method in this paper is more logical. Because, if we consider any real number to be an HFN with satisfaction degrees equal to 1, then the new definitions of distance functions are reduced to the distance between two real numbers. For example, the HFNs ${\tilde{x}}_{H} = 〈 5; {1, 1, 1, 1} 〉$ and ${\tilde{y}}_{H} = 〈 2; {1, 1, 1, 1} 〉$ are real values x = 5 and y = 2, with real distance d (x, y) = |5 - 2|=3. While the distance between their corresponding HFNs, i.e., ${\tilde{x}}_{H}$ and ${\tilde{y}}_{H}$ , using any of the optimized formulas in this article is equal to their distance as real numbers, but according to the given formula in Eqs. (1), different values 0.1765 (Hamming distance) and 0.7276 (Euclidean distance) are obtained.

Furthermore, from the point of view of error propagation and runtime, with increasing the number of calculations, the modeling error and runtime will also be increased. So, the applicability of the proposed distance measures to a real number and the reduction of mathematical calculations are the most important factors that make the new method superior to the previous ones.

Theorem. Suppose ${\tilde{a}}_{H} = 〈 a; {γ_{1}, γ_{2}, \dots, γ_{m}} 〉$ , ${\tilde{b}}_{H} = 〈 b; {λ_{1}, λ_{2}, \dots, λ_{m}} 〉$ , and ${\tilde{c}}_{H} = 〈 c; {ν_{1}, ν_{2}, \dots, ν_{m}} 〉$ , be three arbitrary given adjusted HFNs. Then $\begin{matrix} (i) d ({\tilde{a}}_{H}, {\tilde{a}}_{H}) = 0; \\ (ii) d ({\tilde{a}}_{H}, {\tilde{b}}_{H}) = d ({\tilde{b}}_{H}, {\tilde{a}}_{H}); \\ (iii) d ({\tilde{a}}_{H}, {\tilde{c}}_{H}) \leq d ({\tilde{a}}_{H}, {\tilde{b}}_{H}) + d ({\tilde{b}}_{H}, {\tilde{c}}_{H}); \end{matrix}$ where, the distance function d can be replaced with d_G, d_H, and d_E.

B (a). sed on the definitions of distance functions d_G, d_H, and d_E, the proof is obvious.

However, by reducing the number of calculations and error propagation, the revised distance measures in Def. 7 are able to provide better results than Eqs. (1), in some real situations especially when the given HFNs are equal distances by these formulas, they may give us incorrect results (see the following example). This is because they are only based on the given absolute values in each HFNs, and the dispersion or concentration of degrees of doubt in determining the distance between HFNs has not been taken into account. In this paper, modified distance measures are introduced to solve this problem in the cases where the two given distinct HFNs are equal distances.

Example 2. Let ${\tilde{a}}_{H}^{1} = 〈 124.25; {. 9, . 8, . 7} 〉, {\tilde{a}}_{H}^{2} = 〈 123.75; {. 3, . 4, . 7} 〉$ and ${\tilde{a}}_{H}^{3} = 〈 124; {. 7, . 65, . 6} 〉$ be three given HFNs.

Based on the proposed Hamming distance in Def. 7, we have $d_{H} ({\tilde{a}}_{H}^{1}, {\tilde{a}}_{H}^{3}) = d_{H} ({\tilde{a}}_{H}^{2}, {\tilde{a}}_{H}^{3}) = 0.4$ . It is shown that the HFN ${\tilde{a}}_{H}^{3}$ plays the role of middle point and belongs to both non-equal patterns ${\tilde{a}}_{H}^{1}$ and ${\tilde{a}}_{H}^{2}$ with $d_{H} ({\tilde{a}}_{H}^{1}, {\tilde{a}}_{H}^{2}) = 0.83$ . Apart from the significant distance of HFNs ${\tilde{a}}_{H}^{1}$ and ${\tilde{a}}_{H}^{2}$ , the HFN ${\tilde{a}}_{H}^{1}$ unlike the HFN ${\tilde{a}}_{H}^{2}$ has less dispersion in its membership part, which can be effective in determining the distance. Therefore, we cannot say that ${\tilde{a}}_{H}^{3}$ belongs to ${\tilde{a}}_{H}^{1}$ or ${\tilde{a}}_{H}^{2}$ patterns. In the following, the proposed distance measures will be modified to solve this issue, in which the given distinct HFNs are equal distances.

Definition 8. Consider two arbitrary given adjusted HFNs ${\tilde{a}}_{H} = 〈 a; {γ_{1}, γ_{2}, \dots, γ_{m}} 〉$ and ${\tilde{b}}_{H} = 〈 b; {λ_{1}, λ_{2}, \dots, λ_{m}} 〉$ . Then, new modifications of the Hamming hesitant distance $d_{MH} ({\tilde{a}}_{H}, {\tilde{b}}_{H})$ , Euclidean hesitant distance $d_{ME} ({\tilde{a}}_{H}, {\tilde{b}}_{H})$ , and generalized hesitant distance $d_{MG} ({\tilde{a}}_{H}, {\tilde{b}}_{H})$ will be defined as follows: $\begin{matrix} d_{MH} ({\tilde{a}}_{H}, {\tilde{b}}_{H}) = | a - b | + \frac{1}{m} \sum_{j} | \frac{γ_{(j)} - λ_{(j)}}{γ_{(j)} + λ_{(j)}} |, \\ d_{ME} ({\tilde{a}}_{H}, {\tilde{b}}_{H}) = \sqrt{| a - b |^{2} + \frac{1}{m} \sum_{j} | \frac{γ_{(j)} - λ_{(j)}}{γ_{(j)} + λ_{(j)}} |^{2}}, \\ d_{MG} ({\tilde{a}}_{H}, {\tilde{b}}_{H}) = [| a - b |^{ν} + \frac{1}{m} \sum_{j} | \frac{γ_{(j)} - λ_{(j)}}{γ_{(j)} + λ_{(j)}} |^{ν}]^{\frac{1}{ν}} . \end{matrix}$ (3) where ν > 0, γ_(t) and λ_(t) are the tth smallest value in {γ₁, γ₂, …, γ_m} and {λ₁, λ₂, …, λ_m} respectively.

Now, let us consider the given HFNs in Example 2, and compute their distances from each other using the proposed modified distance measures in Def. 8. Then, we will have $d_{MH} ({\tilde{a}}_{H}^{1}, {\tilde{a}}_{H}^{2}) = 0.786, d_{MH} ({\tilde{a}}_{H}^{1}, {\tilde{a}}_{H}^{3}) = 0.352,$ and $d_{MH} ({\tilde{a}}_{H}^{2}, {\tilde{a}}_{H}^{3}) = 0.41$ . It means that the HFN ${\tilde{a}}_{H}^{3}$ , as its minimum distance from the HFN ${\tilde{a}}_{H}^{1}$ , belongs to the pattern defined by the HFN ${\tilde{a}}_{H}^{1}$ .

In some situations, for any reason, during the process of distance detection, the two real and membership parts of given HFNs are not equally important. Then, we can weight them and propose weighted distance measures as follows.

Definition 9. The weighted distance measures for any two arbitrary given adjusted HFNs ${\tilde{a}}_{H} = 〈 a; {γ_{1}, γ_{2}, \dots, γ_{m}} 〉$ and ${\tilde{b}}_{H} = 〈 b; {λ_{1}, λ_{2}, \dots, λ_{m}} 〉$ , where their real and membership parts are weighted by 0 ≤ w ≤ 1 are defined as follows.

$\begin{matrix} d_{H} ({\tilde{a}}_{H}, {\tilde{b}}_{H}) \\ = w | a - b | + (1 - w) \frac{1}{m} \sum_{j = 1}^{m} | γ_{(j)} - λ_{(j)} |, \\ d_{E} ({\tilde{a}}_{H}, {\tilde{b}}_{H}) \\ = \sqrt{w | a - b |^{2} + (1 - w) \frac{1}{m} \sum_{j = 1}^{m} | γ_{(j)} - λ_{(j)} |^{2}}, \\ d_{G} ({\tilde{a}}_{H}, {\tilde{b}}_{H}) \\ = [w | a - b |^{ν} + (1 - w) \frac{1}{m} \sum_{j = 1}^{m} | γ_{(j)} - λ_{(j)} |^{ν}]^{\frac{1}{ν}}, \end{matrix}$ (4) and

$\begin{matrix} d_{MH} ({\tilde{a}}_{H}, {\tilde{b}}_{H}) \\ = w | a - b | + (1 - w) \frac{1}{m} \sum_{j} | \frac{γ_{(j)} - λ_{(j)}}{γ_{(j)} + λ_{(j)}} |, \\ d_{ME} ({\tilde{a}}_{H}, {\tilde{b}}_{H}) \\ = \sqrt{w | a - b |^{2} + (1 - w) \frac{1}{m} \sum_{j} | \frac{γ_{(j)} - λ_{(j)}}{γ_{(j)} + λ_{(j)}} |^{2}}, \\ d_{MG} ({\tilde{a}}_{H}, {\tilde{b}}_{H}) \\ = [w | a - b |^{ν} + (1 - w) \frac{1}{m} \sum_{j} | \frac{γ_{(j)} - λ_{(j)}}{γ_{(j)} + λ_{(j)}} |^{ν}]^{\frac{1}{ν}} . \end{matrix}$ (5)

4 Application of the proposed modified distance measures

In this section, the modified distance measures will be applied to solve a MADM problem using the TOPSIS method, in which HFNs are used as assessment values.

Consider a general form of a MADM problem with m options O₁, O₂, …, O_m, n weighted criteria c₁, c₂, …, c_n, and the weight vector W = (w₁, w₂, …, w_n). Suppose that the assessment value of ith alternative against to jth criterion is modeled by the HFN ${\tilde{d}}_{ij} = 〈 d_{ij}, h (d_{ij}) 〉$ and that all of them are arranged in a matrix, called hesitant decision matrix, $H = [{\tilde{d}}_{ij}]_{m \times n}$ .

The following steps display an extension of the TOPSIS method with HFNs. To begin the solving process, the elements of the hesitant decision matrix must be adjusted.

Step 1 (Normalization) Depending on the nature of the benefit (B) or cost (C) criteria, compute the weighted normalized hesitant decision matrix $NH = [{\tilde{nd}}_{ij}]_{m \times n}$ as follows: $\begin{matrix} {\tilde{nh}}_{ij} & = 〈 {nd}_{ij}, h (d_{ij}) 〉 \\ = {\begin{matrix} 〈 w_{j} \times \frac{d_{j}^{max} - d_{ij}}{d_{j}^{max} - d_{j}^{min}}, h (d_{ij}) 〉; j \in B, \\ 〈 w_{j} \times \frac{d_{ij} - d_{j}^{min}}{d_{j}^{max} - d_{j}^{min}}, h (d_{ij}) 〉; j \in C . \end{matrix} \end{matrix}$ Step 2 (Ideal solutions) Define two subjective options O⁺ and O^- as follows: $\begin{matrix} O^{+} = {{\tilde{d}}_{1}^{+}, {\tilde{d}}_{2}^{+}, \dots, {\tilde{d}}_{n}^{+}}; \\ O^{-} = {{\tilde{d}}_{1}^{-}, {\tilde{d}}_{2}^{-}, \dots, {\tilde{d}}_{n}^{-}} \end{matrix}$ where, $\begin{matrix} {\tilde{d}}_{j}^{+} = {\begin{matrix} 〈 max_{i} {nd}_{ij}, {1, 1, \dots, 1} 〉; & if j \in B, \\ 〈 min_{i} {nd}_{ij}, {\cup_{t} {min_{i} {γ_{ij}^{(t)}}}} | γ_{ij}^{(t)} \in h (d_{ij}) 〉; & if j \in C . \end{matrix} \\ {\tilde{d}}_{j}^{-} = {\begin{matrix} 〈 min_{i} {nd}_{ij}, {\cup_{t} {min_{i} {γ_{ij}^{(t)}}}} | γ_{ij}^{(t)} \in h (d_{ij}) 〉; & if j \in B, \\ 〈 max_{i} {nd}_{ij}, {1, 1, \dots, 1} 〉; & if j \in C . \end{matrix} \end{matrix}$ and $γ_{ij}^{(t)}$ is the tth smallest value from h (d_ij), which is defined as $γ_{ij}^{(1)} \leq γ_{ij}^{(2)} \leq \dots \leq γ_{ij}^{(| h (d_{ij}) |)}$ .

Step 3 (Positive and negative distances) For each alternative O_i, i = 1, 2, …, m, compute its distances from O⁺ and O^- using one of the proposed distance functions in this paper, as follows: $\begin{matrix} S_{i}^{+} = \sum_{j = 1}^{n} d_{G / H / E} ({\tilde{d}}_{j}^{+}, {\tilde{nh}}_{ij}), \\ S_{i}^{-} = \sum_{j = 1}^{n} d_{G / H / E} ({\tilde{d}}_{j}^{-}, {\tilde{nh}}_{ij}); \end{matrix}$ Step 4 (Relative distance) With respect to all alternatives, compute their relative distance $r_{i} = \frac{S_{i}^{-}}{S_{i}^{+} + S_{i}^{-}}, i = 1, 2, \dots, m$ , and rank them in the ascending order.

Step 5 (Options ranking) Sort the options as r_i, i = 1, 2, …, m, according to the ranking of their relative distances in Step 4.

5 Numerical example

A practical MADM problem using the TOPSIS method will be solved in this section.

Example 3. (Adapted from [8]) Consider an MADM problem with options O_i (i = 1, 2, …, 7) and criteria c_j, j = 1, 2, …, 6, which are weighted by the weight vector W = (.125, . 33, . 175, . 09, . 15, . 13). Note that O₅ is a test/virtual alternative with unrealistic self-assessment values, which helps us to better analyze the proposed method. The evaluation process would be done in two ways: self-assessment and non-governmental organization’ (NGO) assessment. In the self-assessment method, each organization must complete a pre-designed form and upload the related documents. In this case, the final score of the ith option against to the jth criterion is denoted by d_ij as the ijth element of decisiom matrix D = [d_ij] _7×6, i.e. $\begin{array}{l} D = \begin{array}{l} O_{1} \\ O_{2} \\ O_{3} \\ O_{4} \\ O_{5} \\ O_{6} \\ O_{7} \end{array} \overset{\begin{array}{l} c_{1} & c_{2} & c_{3} & c_{4} & c_{5} & c_{6} \end{array}}{(\begin{array}{l} 124 & 321 & 175 & 82 & 144 & 127 \\ 123 & 324 & 160 & 88 & 147 & 125 \\ 123 & 327 & 170 & 85 & 148 & 127 \\ 121 & 329 & 173 & 85 & 147 & 129 \\ 125 & 327 & 171 & 90 & 149 & 130 \\ 125 & 325 & 172 & 90 & 145 & 129 \\ 122 & 325 & 172 & 89 & 147 & 129 \end{array})} \end{array}$

In the second method, with respect to six criteria, six NGOs, as six DMs familiar with the specialized activities of the evaluated units, have been invited to evaluate the performance of the units and express their opinions in the form of a finite set of some values between 0 and 1 as ∪_i {γ_i}, wich are summed up in a hesitant decision matrix $\tilde{H}$ .

$\begin{matrix} \tilde{H} = (\begin{matrix} 〈 {. 2, . 3, . 4, . 5, . 5} 〉 & 〈 {. 1, . 4, . 7, . 8, . 9} 〉 & 〈 {. 2, . 4, . 5, . 6, . 6} 〉 \\ 〈 {. 3, . 5, . 6, . 8, . 9} 〉 & 〈 {. 3, . 5, . 5, . 6, . 9} 〉 & 〈 {. 9, . 9, . 9, . 9, . 9} 〉 \\ 〈 {. 3, . 5, . 6, . 7, . 9} 〉 & 〈 {. 1, . 5, . 6, . 9, . 9} 〉 & 〈 {. 3, . 5, . 6, . 7, . 9} 〉 \\ 〈 {. 7, . 8, . 9, . 9, . 9} 〉 & 〈 {. 1, . 3, . 7, . 8, . 9} 〉 & 〈 {. 2, . 4, . 5, . 6, . 7} 〉 \\ 〈 {. 1, . 1, . 1, . 1, . 1} 〉 & 〈 {. 1, . 1, . 1, . 2, . 2} 〉 & 〈 {. 1, . 1, . 2, . 2, . 3} 〉 \\ 〈 {. 8, . 8, . 8, . 9, . 9} 〉 & 〈 {. 7, . 8, . 8, . 9, . 9} 〉 & 〈 {. 2, . 2, . 3, . 4, . 5} 〉 \\ 〈 {. 4, . 4, . 5, . 6, . 9} 〉 & 〈 {. 5, . 6, . 6, . 8, . 9} 〉 & 〈 {. 3, . 5, . 6, . 6, . 6} 〉 \end{matrix} \\ \begin{matrix} 〈 {. 5, . 6, . 7, . 7, . 9} 〉 & 〈 {. 2, . 3, . 3, . 3, . 6} 〉 & 〈 {. 3, . 4, . 6, . 7, . 7} 〉 \\ 〈 {. 5, . 5, . 7, . 8, . 9} 〉 & 〈 {. 3, . 4, . 4, . 6, . 8} 〉 & 〈 {. 8, . 8, . 8, . 9, . 9} 〉 \\ 〈 {. 5, . 6, . 6, . 7, . 9} 〉 & 〈 {. 3, . 3, . 5, . 6, . 6} 〉 & 〈 {. 6, . 6, . 7, . 8, . 9} 〉 \\ 〈 {. 8, . 7, . 5, . 5, . 6} 〉 & 〈 {. 1, . 5, . 4, . 5, . 9} 〉 & 〈 {. 7, . 3, . 3, . 4, . 5} 〉 \\ 〈 {. 1, . 2, . 2, . 2, . 2} 〉 & 〈 {. 1, . 1, . 2, . 3, . 3} 〉 & 〈 {. 1, . 1, . 2, . 2, . 3} 〉 \\ 〈 {. 2, . 3, . 4, . 5, . 5} 〉 & 〈 {. 1, . 6, . 7, . 8, . 9} 〉 & 〈 {. 5, . 3, . 7, . 4, . 5} 〉 \\ 〈 {. 3, . 5, . 5, . 6, . 9} 〉 & 〈 {. 5, . 6, . 7, . 7, . 8} 〉 & 〈 {. 3, . 4, . 4, . 5, . 6} 〉 \end{matrix}) . \end{matrix}$

Then, these two assessment values about each option regarding each criterion will be mixed together to obtain an HFN and arranged in a hesitant decision matrix $\tilde{HD} = [〈 d_{ij}, {γ_{1}, \dots, γ_{5}} 〉]_{7 \times 6}$ . $\begin{matrix} \tilde{HD} = (\begin{matrix} 〈 124; {. 2, . 3, . 4, . 5, . 5} 〉 & 〈 321; {. 1, . 4, . 7, . 8, . 9} 〉 & 〈 175; {. 2, . 4, . 5, . 6, . 6} 〉 \\ 〈 123; {. 3, . 5, . 6, . 8, . 9} 〉 & 〈 324; {. 3, . 5, . 5, . 6, . 9} 〉 & 〈 160; {. 9, . 9, . 9, . 9, . 9} 〉 \\ 〈 123; {. 3, . 5, . 6, . 7, . 9} 〉 & 〈 327; {. 1, . 5, . 6, . 9, . 9} 〉 & 〈 170; {. 3, . 5, . 6, . 7, . 9} 〉 \\ 〈 121; {. 7, . 8, . 9, . 9, . 9} 〉 & 〈 329; {. 1, . 3, . 7, . 8, . 9} 〉 & 〈 173; {. 2, . 4, . 5, . 6, . 7} 〉 \\ 〈 125; {. 1, . 1, . 1, . 1, . 1} 〉 & 〈 327; {. 1, . 1, . 1, . 2, . 2} 〉 & 〈 171; {. 1, . 1, . 2, . 2, . 3} 〉 \\ 〈 125; {. 8, . 8, . 8, . 9, . 9} 〉 & 〈 325; {. 7, . 8, . 8, . 9, . 9} 〉 & 〈 172; {. 2, . 2, . 3, . 4, . 5} 〉 \\ 〈 122; {. 4, . 4, . 5, . 6, . 9} 〉 & 〈 325; {. 5, . 6, . 6, . 8, . 9} 〉 & 〈 172; {. 3, . 5, . 6, . 6, . 6} 〉 \end{matrix} \\ \begin{matrix} 〈 82; {. 5, . 6, . 7, . 7, . 9} 〉 & 〈 144; {. 2, . 3, . 3, . 3, . 6} 〉 & 〈 127; {. 3, . 4, . 6, . 7, . 7} 〉 \\ 〈 88; {. 5, . 5, . 7, . 8, . 9} 〉 & 〈 147; {. 3, . 4, . 4, . 6, . 8} 〉 & 〈 125; {. 8, . 8, . 8, . 9, . 9} 〉 \\ 〈 85; {. 5, . 6, . 6, . 7, . 9} 〉 & 〈 148; {. 3, . 3, . 5, . 6, . 6} 〉 & 〈 127; {. 6, . 6, . 7, . 8, . 9} 〉 \\ 〈 85; {. 8, . 7, . 5, . 5, . 6} 〉 & 〈 147; {. 1, . 5, . 4, . 5, . 9} 〉 & 〈 129; {. 7, . 3, . 3, . 4, . 5} 〉 \\ 〈 90; {. 1, . 2, . 2, . 2, . 2} 〉 & 〈 149; {. 1, . 1, . 2, . 3, . 3} 〉 & 〈 130; {. 1, . 1, . 2, . 2, . 3} 〉 \\ 〈 90; {. 2, . 3, . 4, . 5, . 5} 〉 & 〈 145; {. 1, . 6, . 7, . 8, . 9} 〉 & 〈 129; {. 5, . 3, . 7, . 4, . 5} 〉 \\ 〈 89; {. 3, . 5, . 5, . 6, . 9} 〉 & 〈 147; {. 5, . 6, . 7, . 7, . 8} 〉 & 〈 129; {. 3, . 4, . 4, . 5, . 6} 〉 \end{matrix}) . \end{matrix}$

Then, utilizing the TOPSIS method, the following steps will be taken.

Step 1 As the given data are scaleless, normalization is not necessary, but they must be weighted as in $\tilde{WHD}$ matrix.

$\begin{matrix} \tilde{WHD} = (\begin{matrix} 〈 15.500; {. 2, . 3, . 4, . 5, . 5} 〉 & 〈 105.93; {. 1, . 4, . 7, . 8, . 9} 〉 & 〈 30.625; {. 2, . 4, . 5, . 6, . 6} 〉 \\ 〈 15.375; {. 3, . 5, . 6, . 8, . 9} 〉 & 〈 106.92; {. 3, . 5, . 5, . 6, . 9} 〉 & 〈 28.000; {. 9, . 9, . 9, . 9, . 9} 〉 \\ 〈 15.375; {. 3, . 5, . 6, . 7, . 9} 〉 & 〈 107.91; {. 1, . 5, . 6, . 9, . 9} 〉 & 〈 29.750; {. 3, . 5, . 6, . 7, . 9} 〉 \\ 〈 15.125; {. 7, . 8, . 9, . 9, . 9} 〉 & 〈 108.57; {. 1, . 3, . 7, . 8, . 9} 〉 & 〈 30.275; {. 2, . 4, . 5, . 6, . 7} 〉 \\ 〈 15.625; {. 1, . 1, . 1, . 1, . 1} 〉 & 〈 107.91; {. 1, . 1, . 1, . 2, . 2} 〉 & 〈 29.925; {. 1, . 1, . 2, . 2, . 3} 〉 \\ 〈 15.625; {. 8, . 8, . 8, . 9, . 9} 〉 & 〈 107.25; {. 7, . 8, . 8, . 9, . 9} 〉 & 〈 30.100; {. 2, . 2, . 3, . 4, . 5} 〉 \\ 〈 15.250; {. 4, . 4, . 5, . 6, . 9} 〉 & 〈 107.25; {. 5, . 6, . 6, . 8, . 9} 〉 & 〈 30.100; {. 3, . 5, . 6, . 6, . 6} 〉 \end{matrix} \\ \begin{matrix} 〈 7.38; {. 5, . 6, . 7, . 7, . 9} 〉 & 〈 21.60; {. 2, . 3, . 3, . 3, . 6} 〉 & 〈 16.51; {. 3, . 4, . 6, . 7, . 7} 〉 \\ 〈 7.92; {. 5, . 5, . 7, . 8, . 9} 〉 & 〈 22.05; {. 3, . 4, . 4, . 6, . 8} 〉 & 〈 16.25; {. 8, . 8, . 8, . 9, . 9} 〉 \\ 〈 7.65; {. 5, . 6, . 6, . 7, . 9} 〉 & 〈 22.20; {. 3, . 3, . 5, . 6, . 6} 〉 & 〈 16.51; {. 6, . 6, . 7, . 8, . 9} 〉 \\ 〈 7.65; {. 8, . 7, . 5, . 5, . 6} 〉 & 〈 22.05; {. 1, . 5, . 4, . 5, . 9} 〉 & 〈 16.77; {. 7, . 3, . 3, . 4, . 5} 〉 \\ 〈 8.10; {. 1, . 2, . 2, . 2, . 2} 〉 & 〈 22.35; {. 1, . 1, . 2, . 3, . 3} 〉 & 〈 16.90; {. 1, . 1, . 2, . 2, . 3} 〉 \\ 〈 8.10; {. 2, . 3, . 4, . 5, . 5} 〉 & 〈 21.75; {. 1, . 6, . 7, . 8, . 9} 〉 & 〈 16.77; {. 5, . 3, . 7, . 4, . 5} 〉 \\ 〈 8.01; {. 3, . 5, . 5, . 6, . 9} 〉 & 〈 22.05; {. 5, . 6, . 7, . 7, . 8} 〉 & 〈 16.77; {. 3, . 4, . 4, . 5, . 6} 〉 \end{matrix}) . \end{matrix}$ Step 2 The subjective options are constructed as O⁺ and O^-. $\begin{matrix} O^{+} = {〈 15.625; {1, 1, 1, 1, 1} 〉, 〈 107.91; \\ {1, 1, 1, 1, 1} 〉, 〈 30.625; {1, 1, 1, 1, 1} 〉, \\ 〈 8.1; {1, 1, 1, 1, 1} 〉, 〈 22.35; {1, 1, 1, 1, 1} 〉, \\ 〈 16.9; {1, 1, 1, 1, 1} 〉}, \end{matrix}$

$\begin{matrix} O^{-} = {〈 15.125; {0.1, 0.1, 0.1, 0.1, 0.1} 〉, 〈 105.93; \\ {0.1, 0.2, 0.1, 0.2, 0.1} 〉, 〈 28; {0.2, 0.2, 0.3, 0.1, 0.1} 〉, \\ 〈 7.38; {0.1, 0.2, 0.2, 0.2, 0.2} 〉, 〈 21.6; {0.1, 0.2, 0.1, \\ 0.3, 0.3} 〉, 〈 16.25; {0.3, 0.1, 0.2, 0.2, 0.1} 〉} . \end{matrix}$

Step 3 The hesitant Hamming distances of alternatives O_i (i = 1, 2, …, 7) from O⁺ and O^-, that is, $S_{i}^{+}$ and $S_{i}^{-}$ , can be obtained as in the second and third columns of Table 1.

Step 4 Compute and rank relative distances R_i, i = 1, …, 7, that is, the fourth and fifth columns of Table 1.

Step 5 Arrange the options similar to ranking of their corresponding relative distances: $O_{4} ≻ O_{6} ≻ O_{3} ≻ O_{7} ≻ O_{5} ≻ O_{2} ≻ O_{1} .$

Table 1
Numerical results of solving Example 3 using the TOPSIS method

Alternatives Positive distance ( $S_{i}^{+}$ ) Negative distance ( $S_{i}^{-}$ ) Relative distance (R_i) Ranking

O ₁ 7.645 5.260 0.4076 7

O ₂ 7.555 5.330 0.4136 6

O ₃ 5.135 7.730 0.6010 3

O ₄ 4.310 8.595 0.6660 1

O ₅ 6.380 6.525 0.5060 5

O ₆ 5.115 7.790 0.6036 2

O ₇ 5.312 7.625 0.5890 4

Alternatives	Positive distance ( $S_{i}^{+}$ )	Negative distance ( $S_{i}^{-}$ )	Relative distance (R_i)	Ranking
O ₁	7.645	5.260	0.4076	7
O ₂	7.555	5.330	0.4136	6
O ₃	5.135	7.730	0.6010	3
O ₄	4.310	8.595	0.6660	1
O ₅	6.380	6.525	0.5060	5
O ₆	5.115	7.790	0.6036	2
O ₇	5.312	7.625	0.5890	4

5.1 Numerical analysis and comparative study

Given the dual parts (real part and HFEs part) of HFNs used in the hesitant decision matrix, in this subsection, we will use each of these parts alone, in the problem-solving process, firstly. In addition, we will solve the above problem by the TOPSIS-CI method [8], and we will analyze and compare the results.

I. Crisp data

Choose the real part of given HFNs and construct the real decision matrix D = [d_ij] _7×6, where d_ij are involved in the ranking process. Then, the final score of the ith option based on the SAW method is O₅ ≻ O₄ ≻ O₆ ≻ O₇ ≻ O₃ ≻ O₁ ≻ O₂, and utilizing the TOPSIS method results is O₅ ≻ O₃ ≻ O₄ ≻ O₆ ≻ O₇ ≻ O₁ ≻ O₂. In this case, based on the results of both methods, O₅, the option with unrealistic evaluations, is the best option.

II. HFEs data

If we consider only the decision matrix with HFEs, in which the option with the unrealistic evaluations has the least satisfaction, then based on the TOPSIS method, the options are arranged as follows:

O₂ ≻ O₆ ≻ O₃ ≻ O₇ ≻ O₄ ≻ O₁ ≻ O₅.

A comparison between the results of the crisp and hesitant fuzzy decision matrices shows that the worst and the best options in each are the best and the worst options in the other, respectively. This means that some judges’ opinions about organizations are so far from their self-assessment. It is not possible to abandon any of them in favor of the other, and a way must be found to use both such that the assessment is close to reality.

When two decision matrices are combined, and the decision matrix is obtained with HFNs, they are slightly modified. In this case, as can be seen in cases III, and IV, by changing the solving method or by changing some needed concepts in a single method, such as distance functions in the TOPSIS method, the obtained rankings are also different.

III. HFNs data and TOPSIS-CI method

In this case, we refer to what is obtained in [8]: O₆ ≻ O₄ ≻ O₃ ≻ O₇ ≻ O₂ ≻ O₁ ≻ O₅.

IV. HFNs data and the TOPSIS method based on given distance measures as in Equation (1)

Consider the given Hamming distance measure in Equation (1), and apply one of them with the proposed steps in the TOPSIS method in Section 5. Then based on Table 2 by the Hamming distance given in Equation (1), we have O₂ ≻ O₄ ≻ O₆ ≻ O₇ ≻ O₃ ≻ O₁ ≻ O₅.

Table 2
Solving Example 3 using Equation (1) with the TOPSIS method

Alternatives Positive distance ( $S_{i}^{+}$ ) Negative distance ( $S_{i}^{-}$ ) Relative distance (R_i) Ranking

O ₁ 2.928 2.087 0.416 6

O ₂ 2.044 3.086 0.611 1

O ₃ 2.857 2.731 0.489 5

O ₄ 2.547 2.644 0.509 2

O ₅ 4.879 0.574 0.105 7

O ₆ 2.601 2.643 0.504 3

O ₇ 2.586 2.544 0.496 4

Alternatives	Positive distance ( $S_{i}^{+}$ )	Negative distance ( $S_{i}^{-}$ )	Relative distance (R_i)	Ranking
O ₁	2.928	2.087	0.416	6
O ₂	2.044	3.086	0.611	1
O ₃	2.857	2.731	0.489	5
O ₄	2.547	2.644	0.509	2
O ₅	4.879	0.574	0.105	7
O ₆	2.601	2.643	0.504	3
O ₇	2.586	2.544	0.496	4

Comparing the given results in cases III and IV with what is obtained by the proposed method in this paper is interesting. As we know, the former versions of the TOPSIS and TOPSIS-CI methods use a similar distance measure, namely Eqs. (1), in which we have to do (2 + m × m) computations (m is the cardinality of the membership set of given adjusted HFNs). While in their modified versions, i.e., the proposed distance measures in Def. 7, these computations have been reduced to (1+m). From the point of view of error analysis, the TOPSIS method is preferable when the criteria are not interactive because it uses fewer calculations with approximate data than the TOPSIS-CI method. However, in both methods, the options that are in the top or bottom half of the ranking are similar but do not have exactly the same position. The given result in case IV is even more interesting because it introduces the option as the best option with the worst self-assessment result according to case I. This is due to the large effect of the skepticism degrees according to the type of used distance measure because it almost follows the pattern presented in case II. Given that we use both types of evaluations, namely self-evaluation and expert evaluation, in HFNs simultaneously. It is logical that the results of the evaluation are not similar to the case where only one of the evaluation methods is used. This important issue is observed in the method introduced in this article. In other words, the optimized expressed distance functions have been able to balance the huge differences between the two types of used evaluations in a single assessment.

5.2 Validity test

In this subsection, the feasibility of the proposed method will be checked via criteria tests [41]:

Test criterion 1 Replacing a non-optimal alternative with a worse one cannot change the best alternative, if the weight vector is constant.

Test criterion 2 For three alternatives A, B, and C, if A ≻ B and B ≻ C, then A ≻ C. (Transitivity property is true).

Test criterion 3 Breaking a decision-making problem into multiple subproblems does not affect total ranking.

We know, in TOPSIS method, the ranking order of alternatives is dependent on positive real values, which are their relative distance from PIS and NIS alternatives. Let O_b be the best option, and O_W is given as worse option than a non-optimal option O_n. Let R_b, R_w and R_n displayed their relative distances, respectively. It is easy to see that R_n < R_b, and R_w < R_n (the relative distance of worse option is less than the relative distance of any non-optimal option), then R_w < R_b. It means that the best option has a larger relative distance than the entering worse option, and the test criterion 1 is satisfied. Given the properties of real numbers, the other criteria tests can be considered in a similar manner easily.

Now, let the given problem decompose into some arbitrary sub-problems through its alternative set as {O₂, O₄, O₅, O₆, O₇}, {O₁, O₃, O₄, O₅, O₇}, {O₁, O₂, O₄, O₆, O₇} and {O₂, O₃, O₄, O₅, O₆}. It means we have new MADM problems with the same criteria. Solving these using the proposed method in this paper gives us the following ranking orders: O₄ ≻ O₆ ≻ O₇ ≻ O₅ ≻ O₂; O₄ ≻ O₃ ≻ O₇ ≻ O₅ ≻ O₁; O₄ ≻ O₆ ≻ O₇ ≻ O₂ ≻ O₁ and O₄ ≻ O₆ ≻ O₃ ≻ O₅ ≻ O₂. The overall ranking order will be obtained by integrating these partial orders: O₄ ≻ O₆ ≻ O₃ ≻ O₇ ≻ O₅ ≻ O₂ ≻ O₁.

Further, suppose the non-optimal alternative O₇ is replaced with $\begin{matrix} O = {〈 & 121; {0.3, 0.6, 0.2, 0.4, 0.5} 〉, 〈 323; {0.4, 0.3, 0.2, 0.1, 0.4} 〉, 〈 167; {0.1, 0.5, 0.4, 0.6, 0.5} 〉, \\ 〈 86.5; {0.3, 0.7, 0.1, 0.2, 0.2} 〉, 〈 142; {0.4, 0.6, 0.3, 0.2, 0.3} 〉, 〈 123; {0.1, 0.3, 0.4, 0.2, 0.4} 〉} . \end{matrix}$

It is easy to see that its distances from PIS and NIS are $S_{O}^{+} = 9.267$ and $S_{O}^{-} = 3.89$ , respectively. Hence, the best option will be unchanged, because of the smaller closeness degree R_O = 0.297 for the new option.

6 Conclusion

Optimization of HFNs distance functions along with their application in solving decision problems was discussed in this article. First, we showed that the existing distance measures (generalized, Hamming, Euclidean) suffer from calculation errors and it is necessary to be improved. Therefore, we introduced an optimized triple group of the above distance measures for determining the distance between HFNs. These have been modified, as another set of distance formulas, to be used when the previous formulas show equal distances. The third category of the presented formulas is suitable for the case that, in the manager’s opinion, the double parts of the HFNs do not have the same importance for any reason. In fact, in this paper, we presented methods that are able to correctly determine the distances between HFNs as much as possible. So, the resulting distance values are not affected by each of the components in the cases where their components are equally important. In addition, by improving the proposed distance functions, situations where the HFNs are quantitatively close to each other, or the degrees of importance of the components of the HFNs are not equal, were also managed by introducing new distance functions. In the end, with the development of the TOPSIS method based on the introduced distance measures, we solved a real group decision-making problem using the introduced tools, and analyzed the obtained answer, too. Considering the novelty of hesitant fuzzy numbers, future studies can focus on expanding their applications, especially in the fields of telemedicine, engineering, meta-heuristic algorithms, future studies and development planning, while generalizing some existing MADM’ techniques such as the ELECTRE methods, VIKOR method, MULTIMOORA ( (Multi-Objective Optimization on the basis of a Ratio Analysis plus the full MULTIplicative form) [47], and ORESTE (Organization, Rangement Et Synthese De Donnes Relationnelles) methods [46], solving transportation problems, neural networks.

Footnotes

Acknowledgment

We would like to thank Editor and Reviewers.

Conflict of interest

The authors allude that there exist no any conflict of interest between them.

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Optimized distance measures on hesitant fuzzy numbers: An application

Abstract

Keywords

1 Introduction

2 HFSs and HFNs

5 Numerical example

6 Conclusion

Footnotes

Acknowledgment

Conflict of interest

References