1 Introduction

Uncertainty and decision making (Denoeux 2014; Bellman and Zadeh 1970), as two widely used concepts of human daily life, get more complicated with the progression of communication and technology. Quantification of qualitative data is only one source of uncertainty that can be modeled using fuzzy sets theory and its generalizations (Zadeh 1965; Atanassov 1983; Karnik and Mendel 2001; Faizi et al. 2018; Muhammad et al. 2020; Ibrahim et al. 2021; Al-shami et al. 2022). Hesitant fuzzy sets (HFSs) (Torra 2010), as the latest extension of fuzzy sets, are suitable for situations in which decision makers (DMs) are hesitated between a finite set of some values from [0,1], that is called hesitant fuzzy elements (HFEs). What is used in practical applications of HFSs are HFEs. Therefore, the development of mathematical principles of HFEs, such as comparison methods (Liao et al. 2014; Liao and Xu 2017), arithmetic and aggregation operations (Torra 2010; Liao and Xu 2014; Verma and Sharma 2013; Verma 2015; Xia and Xu 2011; Xu et al. 2013; Wei 2012; Zhang 2013; Liao and Xu 2014, 2015, 2017; Zhang 2016), determining the distance between two HFEs (Xu and Xia 2011a, b; Tong and Yu 2016), correlation coefficient and entropy measure (Xu and Xia 2011b; Tong and Yu 2016; Liao et al. 2015; Xu and Xia 2012), etc., was considered by researchers. Recently, various definitions of hesitant fuzzy numbers (HFNs), as the newly extension of HFSs, have been proposed. Deli (2020) and Deli and Karaaslan (2021) defined each element of a HFE as a trapezoidal fuzzy number which are common in the real parameters defining the trapezoid, but each has different heights, and called it generalized trapezoidal hesitant fuzzy (GTHF) numbers. Ranjbar et al. (2020) applied a finite set of fuzzy numbers from [0, 1] than crisp values from this interval as the elements of a HFE and called it hesitant fuzzy number. Keikha (2021a) combined a crisp positive real value with a finite set of some values between 0 and 1 and called it hesitant fuzzy number. In this regard, a HFN is displayed by \(\langle a;\{\gamma _1,\gamma _2,\ldots ,\gamma _n\}\rangle \), where a is a positive real value and \(\gamma _i\in [0,1],~i=1,2,\ldots ,n\).

These are some of the introduced tools to model uncertainty. Meta-heuristic optimization algorithms (Abualigah et al. 2021c) such as reptile search algorithm (RSA) (Abualigah et al. 2022), the arithmetic optimization algorithm (Abualigah et al. 2021a), applications, deployments, and integration of internet of drones (IoD) (Abualigah et al. 2021b) are another research fields for solving uncertain practical problems. Perhaps the most fundamental question be,“Why is the need for so much diversity in uncertainty modeling tools?’ The answer is presented in the mixed language with delicacy of the eminent scholar (Pollack 2003). In referring to the diversity of sources of uncertainty, he refers to the “garden of uncertainty”. From this, it can be understood that as we explore more the around environment, newer sources of uncertainty may be discovered, that in the past were either not accepted as new sources, and were wrongly modeled using existing tools, or were not discussed at all. Another similar answer has been given by Klir (2006), when he classified the real-world problems into organized simplicity problems, organized complexity problems, and disorganized complexity problems. In this scientific classification, he refers to organized complexity problems as a group that, despite covering a huge volume of experimental issues, has received less attention. This, of course, also confirms the previous claim of the wrong modeling of some problems through existing mathematical tools.

Another question is, do we really need generalized hesitant fuzzy numbers? What are the disadvantages of using the available tools in this field? The answers to these questions and the advantages of the proposed method of the article are described by examining an example below.

Consider solving the simple problem of determining the number of homeless people in a state. In fact, the reference set X would be defined as \(X=\bigcup _i\{x_i\}\), in which \(x_i\) is the number of homeless people in ith city. Looking at the reports in this field does not indicate a definite number, and we will encounter a finite set of numbers as \(x_i=\{a_{i1},a_{i2},\ldots ,a_{im}\}\). The DMs are not able to sift through these values to remove all but one value, and therefore, accept all values with degrees of doubt that expressed as a HFE \(\{\gamma _{i1},\gamma _{i2},\ldots ,\gamma _{in}\}\), interpreted as different degrees of acceptance \(x_i\). It is showed that the given problem is an organized complexity problem, because it has finite factors. Variety of math tools from simple to complex are offered for modeling such values. Averaging is one of the simplest and of course the oldest methods in such situations. In this case, if we consider only the mean of the values, the problem is transferred from the inaccurate space to the exact one. If we simultaneously consider the mean of the values and the mean of the degrees of doubt, the result is a discrete fuzzy number, and the given organized complexity problem is transferred to the category of organized simplicity problems, too. Also, we can consider \(x_i\) as an interval \(x_i=[a_{i(1)},a_{i(m)}]\), in which \(a_{i(1)}=\min \{a_{i1},a_{i2},\ldots ,a_{im}\}\), and \(a_{i(m)}=\max \{a_{i1},a_{i2},\ldots ,a_{im}\}\). Then, we will have an infinite set of values, but with the same membership grades as 1. Furthermore, modeling can be done using type-1/interval type-2/intuitionistic fuzzy numbers. In all of these modeling methods, the two finite sets of values and degrees of membership are transformed into two infinitely corresponding sets. In the case of using interval values, and various types of fuzzy numbers, the problem space is changed from discrete to continuous, and a new problem is created and solved instead of the original. This transition, although it provides many tools for mathematical analysis, but in addition to increasing uncertainty, practically leaves the main problem unresolved. In other words, by changing the nature of the real-world’ problems from organized complexity problems to disorganized complexity problems, a large volume of them have been remained intact (Klir 2006).

To model such situations, where the DMs are hesitated between a finite set of positive real values with a finite set of hesitation degrees, the HFNs have been extended to generalized HFNs (GHFNs) (Keikha 2021b). Let \(h(A)=\{a_1,a_2,\ldots ,a_m\}\) be the finite set of possible values which are hesitated by \(mh(A)=\{\gamma _1,\gamma _2,\ldots ,\gamma _n\}\). By merging these two parts, a GHFN \(\tilde{\tilde{A}}=\langle h(A);mh(A)\rangle =\langle \{a_1,a_2,\ldots ,a_m\}; \{\gamma _1,\gamma _2,\ldots ,\gamma _n\}\) is obtained, in which h(A) called the real part of GHFN \(\tilde{\tilde{A}}\), and mh(A) is its membership part. For example, in homeless problem, the number of homeless people in ith city can be expressed by GHFN \(\tilde{\tilde{x_i}}=\langle \{a_{i1},a_{i2},\ldots ,a_{im}\};\{\gamma _{i1},\gamma _{i2},\ldots ,\gamma _{in}\}\rangle \). It can be seen that in this modeling method, i.e., applying GHFNs, the problem space did not change and practically the data in the same original format are used without the slightest change. On the other hand, another important feature of GHFNs is the conversion of the opinions of a group of DMs on a single issue in the form of a GHFN, which may themselves have some degrees of uncertainty, too. It is hoped that with the development of the mathematical methodology of generalized hesitant fuzzy numbers and their practical application, a large number of problems that in Klir’ belief have been remained untouched (Klir 2006) will be solved. GHFNs can be used in various fields, such as mechanism design, performance evaluation of employees/organizations/companies, future studies, stock markets, future markets, supply chain, society networks, cosmology, medical diagnostic and telemedicine (Glaz et al. 2021), decision making, and planning.

MADM problems are constructed of a finite set of alternatives/options, and a finite set of attributes/criteria. In many of the proposed methods for solving MADM problems, alternatives evaluation against to all criteria is the first step (Tzeng and Huang 2011; Zhang et al. 2020). Due to the fact that subjective judgments and evaluations always have some degrees of uncertainty, the development of these methods in inaccurate environments was considered by researchers, which led to the introduction of fuzzy-based methods (for different types of fuzzy sets) (Palczewski and Sałabun 2019; Garg et al. 2020; Nan and Zhang 2014; Xu and Zhang 2012; Sun and Ouyang 2015; Aggrawal 2021; Atef et al. 2021; Al-shami 2021). Therefore, the analytical hierarchy process (AHP) method, the analytical network process (ANP) method, TOPSIS method, Choquet integral (CI) method, viekriterijumsko kompromisno rangiranje (VIKOR) method, simple additive weighting (SAW) method, elimination and choice expressing reality (ELECTERE) method, etc. (Tzeng and Huang 2011) have been updated to be applied with the various uncertainty theories. As the hesitant extension of such methods we can refer to, hesitant fuzzy TOPSIS (HFTOPSIS) method (Xu and Zhang 2012), hesitant fuzzy VIKOR (HFVIKOR) method (Liao and Xu 2013), hesitant fuzzy power average-based method (Liao et al. 2018), hesitant fuzzy COMET (Faizi et al. 2018), hesitant fuzzy preference relation (Liao et al. 2014), hesitant fuzzy aggregation operators (Wei 2012; Zhang 2013; Liao and Xu 2014, 2015), some approaches to hesitant fuzzy MADM problems with incomplete weight information (Wei et al. 2014).

Also, utilizing HFNs, a hybrid technique TOPSIS-CI based on combining Choquet integral (CI) and TOPSIS methods (Garg et al. 2020), and mean-based averaging methods (Keikha 2021a), is proposed to solve multi-attribute group decision making (MAGDM) problems. As we know, the TOPSIS method is completely dependent to distance measures. So, in each expansion of it to inaccurate environment, it is necessary to develop the distance measures, firstly (Xu and Xia 2011a, b; Tong and Yu 2016; Li et al. 2015).

GHFNs may be used in some practical MADM problems to construct decision matrix or model the human’ assessments. We can replace them by one of the popular types of fuzzy numbers/intuitionistic fuzzy numbers/ interval numbers, etc., and then use an appropriate method for solving the given problem. As discussed earlier, converting the given data to one of the previous types will cause us to solve another problem instead of the main problem and practically leave the real problem unresolved. Therefore, the best way is to use the data directly in the solution process, which of course requires the development of existing methods and updating some of the required concepts. In this paper, TOPSIS method, as a widely used and popular method in other fields, will be extended to be used with GHFNs. To do this, we need to find a way to determine the distance between the GHFNs. Generalized distance measure, Hamming distance measure, and Euclidean distance measure are three successful distance measure which are used before along with other types of fuzzy numbers. These important distance measures will be generalized to determine the distances of GHNs, firstly. We will then update this method for use with generalized hesitant fuzzy numbers by redefining each step of the common TOPSIS method.

The main advantage of this method is that it does not change the nature of the data and solves exactly the current problem. But other previous methods, by being in such situations, first change the nature of the data through some common simplifications such as averaging or generalization such as interval/fuzzy numbers. Therefore, their obtained answer is not suitable for the current problem because it was obtained in a space other than the real space.

Also, converting and solving a MAGDM problem into a MADM problem will be done easily, by considering the group evaluations of experts from an option against a criterion as a GHFN.

Due to the many used abbreviation symbol in this article, Table 1 is provided with the necessary explanations for easier use.

Table 1 List of some used abbreviation symbols in this article
Full size table

So, the remainder of this article is organized as follows. HFSs, HFNs and GHFNs and some related concepts , such as distance measures of HFEs and HFNs, arithmetic operations of GHFNs, that are required later in this article, will be discussed in Sect. 2. Section 3 presents methods to determine the Euclidean distance (\(d_e\)), Hamming distance (\(d_h\)), generalized distance (\(d_g\)) between two given GHFNs. Section 4 consists the general structure of MADM problems, the traditional TOPSIS method, and proposed a novel TOPSIS method to be used with GHFNs. The proposed method will be illustrated by numerical examples, validated and analyzed comparatively in Sect. 5. The concluding remarks consisting of describing academic implications, major findings, and directions for future research as the conclusion section of this article are given in Sect. 6.

2 Hesitant fuzzy sets and hesitant fuzzy numbers

HFSs, HFNs and GHFNs are reviewed in this section.

HFSs theory has been proposed to model uncertainty, where decision maker is hesitated between a finite set of some values from [0, 1]. Let \(h(x)=\{\gamma _i|i=1,2,\ldots ,n;\gamma _i\in [0,1]\}\), and X be a reference set. Then, \(E=\{<x,h(x)>|x\in X\}\) is called a HFS, in which h(x) is called a hesitant fuzzy element (HFE) (Xia and Xu 2011). HFEs are not necessarily the same in length, and there exist some methods to adjusted them. Consider two HFEs \(h_1\) and \(h_2\), where \(|h_1|=m\) and \(|h_2|=n\) with \(m<n\). To adjust them, it is real to add \(n-m\) elements to the HFE \(h_1\). In the optimistic mode, the maximum element of \(h_1\), in the pessimistic approach its minimum element, in the indifference approach the value of 0.5 (Liao and Xu 2017), and otherwise the power average of the available elements in \(h_1\) (Liao et al. 2018), proposed to be iterated \(n-m\) times in the set \(h_1\).

As one of the most important and fundamental concepts in HFSs theory, we can refer to distance measure. Li et al. (2015) and Li et al. (2015) addressed some axiomatic definition and distance measures.

Definition 1

Let \(h_1,h_2\) and \(h_3\) be three arbitrary HFEs. A distance measure d for HFSs must be satisfied the following properties:

  1. (i)

    \(0\le d(h_1,h_2)\le 1\);

  2. (ii)

    \(d(h_1,h_2)=0~~~~\iff ~~~~h_1= h_2\);

  3. (iii)

    \(d(h_1,h_2)=d(h_2,h_1)\);

  4. (iv)

    \(d(h_1,h_2)\le d(h_1,h_3)+d(h_3,h_2)\);

  5. (v)

    \( h_1\preccurlyeq h_2\preccurlyeq h_3~~\Rightarrow d(h_1,h_2)\le d(h_1,h_3)~~ \& ~~ d(h_2,h_3)\le d(h_1,h_3)\).

Computing the distance between given HFSs has also discussed by many researchers, and there are many methods in this regard (Xu and Xia 2011a, b; Tong and Yu 2016; Li et al. 2015).

Definition 2

(Xu and Xia 2011a) Consider two adjusted HFEs \(h_1\) and \(h_2\). Let \(h_{j(i)}\) be the ith smallest value of \(h_j\) and l be the maximum length of the given HFEs, then

  1. (i)

    \(d_{hnh}(h_1,h_2)=\frac{1}{l}\sum _{i=1}^l|h_{1(i)}-h_{2(i)}|\);

  2. (ii)

    \(d_{hne}(h_1,h_2)=\sqrt{\frac{1}{l}\sum _{i=1}^l|h_{1(i)}-h_{2(i)}|^2}\);

  3. (iii)

    \(d_{hng}(h_1,h_2)=\big [\frac{1}{l}\sum _{i=1}^l|h_{1(i)}-h_{2(i)}|^\lambda \big ]^\frac{1}{\lambda },~~~\lambda >0\);

  4. (iv)

    \(d_{ghnh}(h_1,h_2)=\big [\max _i|h_{1(i)}-h_{2(i)}|^\lambda \big ]^\frac{1}{\lambda },~~~\lambda >0\);

are called hesitant normalized Hamming, Euclidean, generalized, and Hausdorff distances, respectively.

In some other uncertain problems, such as self-assessments, the DMs may be encounter with some crisp predetermined/recorded values which are not accepted completely and must be considered during evaluations. Hesitant fuzzy numbers, as the generalization of HFSs, are suitable to model such problems (Keikha 2021a; Garg et al. 2020). A HFN consists of two parts: real part, and HFE part, i.e., \(\tilde{E}=\langle a;\{\gamma _1,\gamma _2,\ldots ,\gamma _n\}\rangle \) is called a HFN, in which a is a positive real number, and \(\gamma _i\in [0,1]\) are hesitation/satisfaction degrees.

Definition 3

(Garg et al. 2020; Keikha 2021c) Let \(\tilde{a}_H=\langle a, h(a)\rangle \) and \(\tilde{b}_H=\langle b, h(b)\rangle \) be two arbitrary HFNs where, \(a,b\in \mathbb {R}\), h(a) with cardinality \(|h(a)|=k\) and h(b) with cardinality \(|h(b)|=l\) are two sets of some values in [0, 1]. Their Euclidean distance (\(d_e\)) and Hamming distance (\(d_h\)) were be defined as follows:

$$\begin{aligned}&d_{e}(\tilde{a}_H,\tilde{b}_H)=\sqrt{\frac{1}{1+k\times l}\bigg (|a-b|^2+\sum _{\begin{array}{c} \gamma _i\in h(a),\\ \gamma _j\in h(b) \end{array}}|\gamma _i-\gamma _j|^2\bigg )}, \\&d_{h}(\tilde{a}_H,\tilde{b}_H)=\frac{1}{1+k\times l}\bigg (|a-b|+\sum _{\begin{array}{c} \gamma _i\in h(a),\\ \gamma _j\in h(b) \end{array}}|\gamma _i-\gamma _j|\bigg ). \end{aligned}$$

In some other problems, it may be that the real part of a HFN is not a crisp value and expressed by a finite set of real values. In such cases, HFNs must be extended to what called GHFNs. In fact, \(\tilde{\tilde{A}}^H=\big \langle \{a_1,a_2,\ldots , a_n\}; \{\gamma _1,\gamma _2,\ldots ,\gamma _m\}\big \rangle =\langle h(A),mh(A)\rangle \), in which \(a_i,~i=1,2,\ldots ,n\) are positive real values, and \(\gamma _j\in [0,1],~j=1,2,\ldots ,m\) are membership/satisfaction degrees, is nominated as a GHFN. It is easy to see that a GHFN is called a HFN if \(|h(A)|=1\), or a discrete fuzzy number if \(|h(A)|=|mh(A)|=1\).

GHFNs \(\tilde{\tilde{A}}^H=\big \langle \{a_1,a_2,\ldots , a_m\}; \{\gamma _1,\gamma _2,\ldots \gamma _n\}\big \rangle \) and \(\tilde{\tilde{B}}^H=\big \langle \{b_1,b_2,\ldots , b_k\}; \{\lambda _1,\lambda _2,\ldots ,\lambda _l\}\big \rangle \) are called adjusted GHFNs (AGHFNs), if \(m=k\) and \(n=l\). If not so, without lose of generality let \(m<k\) or \(n<l\), \(a^{max}=\max \{a_1,a_2,\ldots , a_n\}, a^{min}=\min \{a_1,a_2,\ldots , a_n\}, \gamma ^{max}=\max \{\gamma _1,\gamma _2,\ldots ,\gamma _m\}, \text {and}~ \gamma ^{min}=\min \{\gamma _1,\gamma _2,\ldots ,\gamma _m\}\). Then, \(\tilde{\tilde{A}}^H\) must be extended: for optimistic DM, \(a^{max}\) or \(\gamma ^{max}\) will be repeated \(k-m\) or \(l-n\) times in \(\{a_1,a_2,\ldots , a_n\}\) or \(\{\gamma _1,\gamma _2,\ldots \gamma _m\}\), respectively. It is done by \(a^{min}\) or \(\gamma ^{min}\) for pessimistic DM, and \(\overline{a}=\frac{a_1+a_2+\cdots + a_m}{m}\) or 0.5 for indifference DM, similarly.

Definition 4

(Keikha 2021b) Consider two AGHFNs \(\tilde{\tilde{A}}^H=\big \langle \{a_1,a_2,\ldots , a_m\}; \{\gamma _1,\gamma _2,\ldots ,\gamma _n\}\big \rangle \) and \(\tilde{\tilde{B}}^H=\big \langle \{b_1,b_2,\ldots , b_m\}; \{ \lambda _1,\lambda _2, \ldots , \lambda _n\}\big \rangle \). Then, for any positive real value w we have:

  1. (i)

    \(w\tilde{\tilde{A}}^H=\big \langle \{wa_1,wa_2,\ldots ,w a_m\}; \{\gamma _1,\gamma _2,\ldots ,\gamma _n\}\big \rangle ,~~~~w>0\);

  2. (ii)

    \(\big (\tilde{\tilde{A}}^H\big )^w=\big \langle \{(a_1)^w,(a_2)^w,\ldots , (a_m)^w\}; \{\gamma _1,\gamma _2,\ldots ,\gamma _n\}\big \rangle ,~~~~w>0\);

  3. (iii)

    \(\tilde{\tilde{A}}^H\oplus \tilde{\tilde{B}}^H=\big \langle \cup _i \{a_{(i)}+b_{(i)}\};mh(A)\cup mh(B) \big \rangle \);

  4. (iv)

    \(\tilde{\tilde{A}}^H\otimes \tilde{\tilde{B}}^H=\big \langle \cup _i\{a_{(i)}b_{(i)}\};\cup _i\min \{\gamma _i,\lambda _i\}\big \rangle \);

where \(a_{(1)}, a_{(2)},\ldots ,a_{(m)}\) is a permutation of \(a_1,a_2,\ldots , a_m\) such that \(a_{(1)}\le a_{(2)}\le \cdots \le a_{(m)}\), and \(b_{(1)},b_{(2)},\ldots ,b_{(m)}\) is a permutation of \(b_1,b_2,\ldots , b_m\) such that \(b_{(1)}\le b_{(2)}\le \cdots \le b_{(m)}\).

GHFNs are at the starting point of a long and dark way, and like any new scientific subject, they need much attempts in both theoretical and application aspects. In order to propose a TOPSIS-based method for solving uncertain MADM problems that are classified in organized complexity problems, we need to define the distance measure of GHFNs. These process will be done in the next sections.

3 Distance functions of GHFNs

In some practical applications of GHFNs, such as decision-making problems, it is needed to determine the distances between them. Generalized, Hamming and Euclidean distances are the most famous which have been updated for different types of fuzzy numbers and are used (Guha and Chakraborty 2010b, a; Sang et al. 2014). These distance functions will be updated to be used with GHFNs, in this Section.

Definition 5

Let \(\tilde{\tilde{A}}^H=\big \langle \{a_1,a_2,\ldots , a_n\}; \{\gamma _1,\gamma _2,\ldots ,\gamma _m\}\big \rangle \), and \(\tilde{\tilde{B}}^H=\big \langle \{b_1,b_2,\ldots , b_n\}; \{\lambda _1,\lambda _2,\ldots ,\lambda _m\}\big \rangle \) be two arbitrary given AGHFNs. Then, their Generalized distance \(d_g\bigg (\tilde{\tilde{A}}^H,\tilde{\tilde{B}}^H\bigg )\), Hamming distance \(d_h\bigg (\tilde{\tilde{A}}^H,\tilde{\tilde{B}}^H\bigg )\), and Euclidean distance \(d_e\bigg (\tilde{\tilde{A}}^H,\tilde{\tilde{B}}^H\bigg )\) can be defined as follows:

$$\begin{aligned} \begin{array}{l} d_{g}\bigg (\tilde{\tilde{A}}^H,\tilde{\tilde{B}}^H\bigg )=\big [{\frac{1}{n}\sum \limits _{i=1}^n|a_{(i)}-b_{(i)}|^\lambda +\frac{1}{m}\sum \limits _{j=1}^m|\gamma _{(j)}-\lambda _{(j)}|^\lambda }\big ]^\frac{1}{\lambda },\\ d_{h}\bigg (\tilde{\tilde{A}}^H,\tilde{\tilde{B}}^H\bigg )=\frac{1}{n}\sum \limits _{i=1}^n|a_{(i)}-b_{(i)}|+\frac{1}{m}\sum \limits _{j=1}^m|\gamma _{(j)}-\lambda _{(j)}|,\\ \\ d_{e}\bigg (\tilde{\tilde{A}}^H,\tilde{\tilde{B}}^H\bigg )=\sqrt{\frac{1}{n}\sum \limits _{i=1}^n|a_{(i)}-b_{(i)}|^2+\frac{1}{m}\sum \limits _{j=1}^m|\gamma _{(j)}-\lambda _{(j)}|^2}, \end{array} \end{aligned}$$

where \(._{(t)}\) is the tth largest value in its corresponding set.

Example 1

Let \(\tilde{\tilde{A}}^H=\big \langle \{2,3,4,5\};\{.4,.6,.7,.9\}\big \rangle \) and \(\tilde{\tilde{B}}^H=\big \langle \{1,3,6,7\};\{.5,.6,.8,1\}\big \rangle \). Then,

$$\begin{aligned}&d_h(\tilde{\tilde{A}}^H,\tilde{\tilde{B}}^H)=\frac{1}{4}(1+0+2+2)+\frac{1}{4}(.1+0+.1+.1)\\&\qquad \qquad \qquad \quad =1.325, \\&d_e(\tilde{\tilde{A}}^H,\tilde{\tilde{B}}^H)\\&=\sqrt{\frac{1}{4}(1+0+4+4)+\frac{1}{4}(.01+0+.01+.01)}=1.503,\\&\text {Get } \lambda =3, d_g(\tilde{\tilde{A}}^H,\tilde{\tilde{B}}^H)=[\frac{1}{4}(1^3+0^3+2^3+2^3)\\&\qquad \qquad +\frac{1}{4}(.001+0+.001+.001)]^\frac{1}{3}=1.62. \end{aligned}$$

Remark

The generalized distance measure \(d_g\big (\tilde{\tilde{A}}^H,\tilde{\tilde{B}}^H\big )\) is reduced to Hamming distance measure \(d_h\big (\tilde{\tilde{A}}^H,\tilde{\tilde{B}}^H\big )\) if \(\lambda =1\), and for \(\lambda =2\) it is called Euclidean distance measure \(d_e\big (\tilde{\tilde{A}}^H,\tilde{\tilde{B}}^H\big )\), in which \(\tilde{\tilde{A}}^H\) and \(\tilde{\tilde{B}}^H\) are arbitrary GHFNs.

Theorem

Let \(\tilde{\tilde{A}}^H=\big \langle \{a_1,a_2,\ldots , a_n\}; \{\gamma _1,\gamma _2,\ldots ,\gamma _m\}\big \rangle \), \(\tilde{\tilde{B}}^H=\big \langle \{b_1,b_2,\ldots , b_n\}; \{\lambda _1,\lambda _2,\ldots ,\lambda _m\}\big \rangle \), and \(\tilde{\tilde{C}}^H=\big \langle \{c_1,c_2,\ldots , c_n\}; \{\nu _1,\nu _2,\ldots ,\nu _m\}\big \rangle \), be arbitrary given AGHFNs. Then,

$$\begin{aligned}&(i)~~ d(\tilde{\tilde{A}}^H,\tilde{\tilde{A}}^H)=0;\\&(ii)~~ d(\tilde{\tilde{A}}^H,\tilde{\tilde{B}}^H)=d(\tilde{\tilde{B}}^H,\tilde{\tilde{A}}^H);\\&(iii)~~ d(\tilde{\tilde{A}}^H,\tilde{\tilde{C}}^H)\le d(\tilde{\tilde{A}}^H,\tilde{\tilde{B}}^H)+d(\tilde{\tilde{B}}^H,\tilde{\tilde{C}}^H); \end{aligned}$$

where the distance function d can be interpreted as \(d_g,d_h\), and \(d_e\).

Proof

For any \(x,y,z\in \mathbb {R}\), it is easy to see that \(|x-y|=|y-x|\), and \(|x-y|\le |x-z|+|z-y|\). Using these properties, and regarding to the definitions of distance functions \(d_g,d_h\), and \(d_e\), the proof is obvious. \(\square \)

4 MADM problems and GHFNs

In this section, as a practical application of GHFNs, they will be used to model the uncertainty contained in a MADM problem. Then, the TOPSIS method, as a widely used method in solving MADM problem, will be generalized to the new type of hesitant fuzzy numbers, i.e., GHFNs.

Let there exist r candidates \(C_1,C_2,\ldots ,C_r\) to be ranked based on s attributes \(\mathsf {a}_1,\mathsf {a}_2,\ldots ,\mathsf {a}_s\), in ascending order. Usually each candidate is evaluated against to all attribute, and arranged in a matrix, called decision matrix, i.e., \(D=[d_{ij}]_{r\times s}\), in which \(d_{ij}\) means the evaluation value of ith candidate against to jth attribute. Because of existing uncertainty in any subjective judgment or other qualitative evaluations, they must be modeled to quantitative values for doing extra mathematical processes. In this paper, GHFNs have been used in the modeling process. Then, in the solving process, the TOPSIS method will be extended to do it.

4.1 The traditional TOPSIS method

The TOPSIS method (Tzeng and Huang 2011) is based on the following steps:

Step 1 Constructing the decision matrix \(D=[d_{ij}]_{r\times s}\) via the assessment of all candidates against to all attributes, are expressed by crisp values.

Step 2 Normalizing decision matrix to \(N=[n_{ij}]_{r\times s}\), called normalized decision matrix:

$$\begin{aligned} n_{ij} = \left\{ \begin{array}{ll} \frac{d_{ij}-d_{j}^{min}}{d_{j}^{max}-d_{j}^{min}},&{} j\in B;\\ \\ \frac{d_{j}^{max}-d_{ij}}{d_{j}^{max}-d_{j}^{min}},&{} j\in C.\end{array} \right. \end{aligned}$$

where, B is the set of attributes with positive aspect or benefit attributes (the larger is better), and C is the set of attributes with negative aspect or cost attributes (the smaller is better).

Step 3 Calculating weighted normalized decision matrix \(NW=[n_{ij}\times w_i]_{r\times s}=[v_{ij}]\), if the attributes are weighted by weight vector \(W=(w_1,w_2,\ldots ,w_n)\).

Step 4 Determining positive ideal solution (PIS), and negative ideal solution (NIS) as two subjective alternatives:

$$ \begin{aligned} A^{+}&=\{p^{+}_1,p^{+}_2,\ldots ,p^{+}_s\}\\&=\{(max_{i}v_{ij}|j\in B) \& (min_{i}v_{ij}|j\in C)\},\\ A^{-}&=\{p^{-}_1,p^{-}_2,\ldots ,p^{-}_s\}\\&=\{(min_{i}v_{ij}|j\in B) \& (max_{i}v_{ij}|j\in C) \}. \end{aligned}$$

Step 5 Calculating the distance of each alternative from subjective alternatives:

$$\begin{aligned} S_{i}^{+}= & {} \sqrt{\sum _{j=1}^{s}(v_{ij}-p^{+}_{j})^{2}},~~~~~~~~ S_{i}^{-}=\sqrt{\sum _{j=1}^{s}(v_{ij}-p^{-}_{j})^{2}},~~~~~~~~~~~~~\\&\qquad i=1,2,\ldots ,r. \end{aligned}$$

Step 6 Computing the relative distances \(R_{i}=\frac{S_{i}^{-}}{S_{i}^{+}+S_{i}^{-}},~~~i=1,2,\ldots ,r\), and rank them.

Step 7 Reorder the candidates \(C_i\) according to order of \(R_i\), (\(i=1,2,\ldots ,r\)).

4.2 Updating TOPSIS method with GHFNs

In this subsection, the traditional TOPSIS method will be extended to be applicable with GHFNs.

Step 1 Evaluation phase All options will be evaluated against to all criteria, and the assessment values of each option, which are allowed to be expressed by qualitative/linguistic terms, or uncertain values, are arranged in a decision matrix \(D=[d_{ij}]_{r\times s}\).

Step 2 Modeling phase GHFNs have been applied to model uncertain elements of decision matrix or quantify the qualitative assessments. Then, the decision matrix \(D=[d_{ij}]_{r\times s}\) converted to the hesitant decision matrix \(\tilde{\tilde{HD}}=[\tilde{\tilde{d_{ij}}}]_{r\times s}\).

Step 3 Adjusting phase The GHFNs containing in the hesitant decision matrix \(\tilde{\tilde{HD}}\) should be adjusted, as in \(\tilde{\tilde{d_{ij}}}=\bigg \langle \{a_{ij}^{(1)},a_{ij}^{(2)},\ldots ,a_{ij}^{(m)}\};\{\gamma _{ij}^{(1)},\gamma _{ij}^{(2)},\ldots ,\gamma _{ij}^{(n)}\}\bigg \rangle \), where \(a_{ij}^{(l)}, \text {and} \gamma _{ij}^{(l)}\) are lth largest value in their corresponding set.

Step 4 Normalizing phase The elements of hesitant decision matrix \(\tilde{\tilde{HD}}\) must be scale-less. It is done via the normalizing process, and normalized hesitant decision matrix \(\tilde{\tilde{NHD}}=[\tilde{\tilde{n}}_{ij}]_{r\times s}\), in which \(\tilde{\tilde{n}}_{ij}=\bigg \langle \{\alpha _{ij}^{(1)},\alpha _{ij}^{(2)},\ldots , \alpha _{ij}^{(m)}\};\{\gamma _{ij}^{(1)},\gamma _{ij}^{(2)},\ldots ,\gamma _{ij}^{(n)}\}\bigg \rangle \), will be resulted as:

$$\begin{aligned} \alpha _{ij}^{(l)} = \left\{ \begin{array}{ll} \frac{a_{ij}^{(l)}-a_{j}^{(l)min}}{a_{j}^{(l)max}-a_{j}^{(l)min}} &{} j\in B; \\ \\ \frac{a_{j}^{(l)max}-a_{ij}^{(l)}}{a_{j}^{(l)max}-a_{j}^{(l)min}} &{} j\in C.\end{array} \right. l=1,2,\ldots ,m \end{aligned}$$

where, \(a_{j}^{(l)min}=\min _i\{a_{ij}^{(l)}\}, a_{j}^{(l)max}=\max _i\{a_{ij}^{(l)}\}\).

Step 5 Weighted phase If the attributes are weighted by weight vector \(W=(w_1,w_2,\ldots ,w_n)\), the weighted normalized decision matrix \(NW=[w_i\times \tilde{\tilde{n}}_{ij}]_{r\times s}=[\tilde{\tilde{v}}_{ij}],~\Big (\tilde{\tilde{v}}_{ij}=\bigg \langle \{v_{ij}^{(1)},v_{ij}^{(2)},\ldots ,v_{ij}^{(m)}\};\{\gamma _{ij}^{(1)},\gamma _{ij}^{(2)},\ldots ,\gamma _{ij}^{(n)}\}\bigg \rangle \Big )\) will be constructed.

Step 6 Subjective options Two subjective alternatives PIS and NIS must be determined using the proposed ranking function in this paper:

$$ \begin{aligned} A^{+}&=\{\tilde{\tilde{p}}^{+}_1,\tilde{\tilde{p}}^{+}_2,\ldots ,\tilde{\tilde{p}}^{+}_s\}\\&=\{(max_{i}\tilde{\tilde{v}}_{ij}|j\in B) \& (min_{i}\tilde{\tilde{v}}_{ij}|j\in C)\},\\ \\ A^{-}&=\{\tilde{\tilde{p}}^{-}_1,\tilde{\tilde{p}}^{-}_2,\ldots ,\tilde{\tilde{p}}^{-}_s\}\\&=\{(min_{i}\tilde{\tilde{v}}_{ij}|j\in B) \& (max_{i}\tilde{\tilde{v}}_{ij}|j\in C) \}, \end{aligned}$$

in which,

$$\begin{aligned} max_{i}\tilde{\tilde{v}}_{ij}&=\bigg \langle \{max_{i}v_{ij}^{(1)},max_{i}v_{ij}^{(2)},\ldots ,max_{i}v_{ij}^{(m)}\};\\&\{max_{i} \gamma _{ij}^{(1)},max_{i}\gamma _{ij}^{(2)},\ldots ,max_{i}\gamma _{ij}^{(n)}\}\bigg \rangle , \end{aligned}$$

and

$$\begin{aligned} min_{i}\tilde{\tilde{v}}_{ij}&=\bigg \langle \{min_{i}v_{ij}^{(1)},min_{i}v_{ij}^{(2)},\ldots ,min_{i}v_{ij}^{(m)}\};\nonumber \\&\{min_{i} \gamma _{ij}^{(1)},min_{i}\gamma _{ij}^{(2)},\ldots ,min_{i}\gamma _{ij}^{(n)}\}\bigg \rangle . \end{aligned}$$
(1)

Step 7 Distancing phase Utilizing one of the proposed distance functions in this paper, the distance between each alternative and two subjective alternatives must be computed:

$$\begin{aligned} S_{i}^{+}= & {} \sum _{j=1}^{s}d(\tilde{\tilde{v}}_{ij}-\tilde{\tilde{p}}^{+}_{j}),~~~~~~~~ S_{i}^{-}=\sum _{j=1}^{s}d(\tilde{\tilde{v}}_{ij}-\tilde{\tilde{p}}^{-}_{j}),\\&\quad ~~~~~~~~~~~~~ i=1,2,\ldots ,r. \end{aligned}$$

Step 8 Relative distance Computing the relative distances \(R_{i}=\frac{S_{i}^{-}}{S_{i}^{+}+S_{i}^{-}},~~~i=1,2,\ldots ,r\), and rank them.

Step 9 Ranking phase Reorder the candidates \(C_i\) according to order of \(R_i\), (\(i=1,2,\ldots ,r\)).

5 Numerical example

Example 2

(Energy project selection) (Xu and Zhang 2012) Let there exist five energy projects (candidates) \(C_i (i=1,2,3,4,5)\) to be ranked based on attributes: \(\mathsf {a}_1\) (technological), \(\mathsf {a}_2\) (environmental), \(\mathsf {a}_3\) (socio-political), and \(\mathsf {a}_4\) (economic). It is clear that in the real world, different factors influence a decision. In this example, considering the different effects of such projects on the environment, people, research units and industries, in addition to the opinions of experts, the opinions of various social groups, universities and NGOs are also used. Given that historical data, different reports from different social/scientific/economical/NGOs (non-governmental organization) sources, and even existing documents state different values for each option against each criterion, the director not only wants to use all of them without eliminating some of them in decision making, but also asks the evaluators (experts) to express their opinions and even increase the number of these values, as \(h(A)=\{a_1,a_2,\ldots ,a_m\}\). Then, decision makers express their hesitation (satisfaction) degrees of these values with a finite set of values between zero and one, as in \(mh(A)=\{\gamma _1,\gamma _2,\ldots ,\gamma _n\}, \gamma _i\in [0,1]\). It means that each evaluation value is a GHFNs \(\tilde{\tilde{A}}=\big \langle \{a_1,a_2,\ldots ,a_m\}; \{\gamma _1,\gamma _2,\ldots ,\gamma _n\}\big \rangle \).

Thus, each element of the decision matrix, as shown in Table 2, has its own interpretation, and consists of two parts: the first part contains the values of direct evaluation, and the second part includes the degrees of skepticism of experts from the first part. For instance, the real part of \(\tilde{\tilde{A}}_{11}=\langle \{8,7,9,10\};\{0.3,0.4,0.5\}\rangle \) shows the technological advantages of the first energy project based on the available documents and reports provided by various units. Which, of course, after studying and evaluating by the experts, the allocation of low levels of satisfaction indicates that they do not welcome the reports. The other given information in Table 2 has also similar interpretations.

Table 2 Decision matrix of energy project selection problem
Full size table

Mathematical processing of these given GHFNs in Table 2 requires to adjust their both parts. This is done in pessimistic way in this article. Then, hesitant fuzzy subjective options PIS and NIS can be determined as

$$\begin{aligned} {\text {PIS}}&=\Big ( \langle \{8,8,8,8,9,10\}; \{0.6,0.6,0.6,0.7,0.9\}\rangle , \\&\quad \langle \{9,9,9,9,9,10\};\{0.6,0.6,0.7,0.8,0.9\}\rangle \\&\quad \langle \{4,4,5,6,8,9\};\{0.7,0.7,0.7,0.8,0.9\}\rangle , \\&\quad \langle \{5,5,5,6,7,10\};\{0.6,0.6,0.6,0.8,0.9\}\rangle \Big ), \\ {\text {NIS}}&=\Big ( \langle \{1,1,2,2,3,7\}; \{0.1,0.3,0.3,0.3,0.5\}\rangle ,\\&\quad \langle \{2,2,4,5,7,7\};\{0.1,0.1,0.2,0.4,0.7\}\rangle \\&\quad \langle \{1,1,1,1,3,8\};\{0.1,0.1,0.1,0.1,0.5\}\rangle , \\&\quad \langle \{1,1,2,3,4,5\};\{0.3,0.3,0.3,0.4,0.6\}\rangle \Big ). \end{aligned}$$

Now, utilizing the proposed distance measure in this paper, the distance of each candidate \(C_i, i=1,2,\ldots ,5,\) from both PIS and NIS options, which is displayed by \(d_i^+\) and \(d_i^-\), respectively, must be calculated:

$$\begin{aligned} S_1^+&=3.1135,~S_2^+=2.3991,~S_3^+=3.5197, \\ S_4^+&=2.7723,~S_5^+=1.8767,\\ S_1^-&=2.5738,~S_2^-=3.5154, \\ S_3^-&=2.0993,~S_4^-=3.2047,\\ S_5^-&=3.5605. \end{aligned}$$

By these values, the relative distance of all candidates is computed as

$$\begin{aligned} R_1=0.4525, R_2=0.5944, R_3=0.3736, R_4=0.5362,\\ R_5=0.6548. \end{aligned}$$

According to \(R_i\)’ values, candidates will be ranked as

$$C_3\prec C_1\prec C_4\prec C_2\prec C_5.$$

5.1 Validity discussion

Wang and Triantaphyllou (2008) proposed three criteria test to check and demonstrating the feasibility of the introduced MCDM methods: 1-Replacing the non-optimal option with a worse option has no effect on the superior option. 2- Transitivity property is satisfied in an effective MCDM method. 3- Integrating the resulted rankings by solving the obtained subproblems from decomposition of the initial problem is the same as the initial ranking of the main problem.

It is easy to discuss these criteria test: A worse option than a non-optimal one is more near to NIS solution than it, and simultaneously is more far from PIS solution. So, the best option will be unchanged. The final ranking orders of options are determined by ranking of their scores, which are positive real values and obtained at the end of TOPSIS method. Therefore, the criteria 2 and 3 are true due to the properties of the real numbers.

5.2 Comparison analysis

In this subsection, by considering different scenarios for given data in decision matrix and applying some other methods, the results will be compared with the method proposed by the article.

  1. I.

    Consider only the real part of the given GHFNs in Table 2.

    It means, each element of the resulting decision matrix is a finite set of real values. First, we calculate the average of each matrix element and replace it. In this way, we will have a new decision matrix based on definite numbers. Then, by applying the TOPSIS method on this matrix, we rank the options: \(C_1\prec C_5\prec C_3\prec C_4\prec C_2\).

  2. II.

    Consider only the membership part of the given GHFNs in Table 2.

    Hesitant fuzzy TOPSIS has been proposed by Xu and Zhang (2012) to solve MADM problems in which the uncertainty is modeled via the hesitant fuzzy sets. Xu and Zhang (2012), consider HFSs, i.e., only the second part of each element given in Table 2 and solved Example 2. They ranked candidates like: \(C_4\prec C_1\prec C_2\prec C_3\prec C_5\).

A direct comparison of these two types of rankings with that obtained based on applying GHFNs may not be the right thing to do, because the data they use are inherently different despite the similarities.

  1. III.

    Change the given GHFNs in Table 2to GHFNs with equal values in their real part.

    Let us pave the way for a better comparison. Suppose the real part of all the given GHFNs in Table 2 changed to be equal and solve the problem again using TOPSIS method. The new obtained ranking order in this case is quite consistent with what was previously obtained in Xu and Zhang (2012). It is showed that the proposed method in this article is an extension of the existing methods in a new environment.

Example 3

(Adapted from Keikha (2021a)) In this Example, we investigate a MADM problem, in which the evaluation value of options against criteria has been modeled through hesitant fuzzy numbers as in Table 3.

Table 3 The hesitant fuzzy numbers decision matrix of Example 3
Full size table
Table 4 The aggregated hesitant fuzzy decision matrix of Example 3
Full size table

Utilizing weighted hesitant arithmetic averaging (WHAA) operator, each row of Table 3 has been aggregated to a single HFN. Then, the derived HFNs have been ranked in ascending order due to their scores (see Table 4), and candidates ranked as \(C_3\prec C_1\prec C_5\prec C_2\prec C_4\). Since HFNs are tools for uncertainty modeling, they are not error free. From the point of view of error analysis, with increasing the number of mathematical operations, the accumulation and propagation of error also increases during the calculations. Therefore, methods that use fewer operations are more appropriate. The WHAA method, in addition to having a relatively large number of mathematical operations to align the row data, also depends on the score value of the resulted HFN, which is actually an approximation of it.

Let us consider each given HFN in Table 2 as a GHFN, in which its real part containing a single value. Utilizing TOPSIS method resulted the following ranking:

$$C_1\prec C_5\prec C_3\prec C_4\prec C_2.$$

As it is observed, these two ranking orders are completely different, but have similarities, like the options available in the top and bottom half of the rankings. It is natural to have differences in results because the methods used and the type of data processing are fundamentally different. However, the method proposed in this paper produces more reliable results due to the use of less arithmetic operations (only distance function without any extra approximation such as score value is used) in the TOPSIS method. What can be said about the difference between the proposed method and other similar methods is that in this method, decisions are made according to the existing reality. Because the use of generalized hesitant fuzzy numbers causes the data to enter the processing phase without any change, while the use of fuzzy numbers or similar ones leads to the use of infinite values and infinite membership degrees. Another important advantage of this approach is that it transforms a group decision problem into an individual decision problem without the need to change the nature of the problem. And even the greater participation of people in the group not only does not cause any challenge, but also leads to better decisions. But it should be noted that we are at the beginning of the road and as we will see in the conclusion section, many mathematical concepts still need to be developed.

6 Conclusion

This article desires to present an exquisite MADM method for solving the decision-making problems under the hesitant and uncertain environments. The uncertainty of the data in this paper is handled with the help of the generalized hesitant fuzzy information which consists of two disjoint finite sets: real and membership parts. In addition to keeping the nature of the information intact, this type of numbers enables us to represent the opinions of a group of experts on a single subject from the perspective of a particular criterion by means of a GHFN. In other words, a MAGDM problem can easily be expressed in the form of a typical MADM problem. Given that this concept is at the beginning of a long journey, the development of some mathematical tools is a prerequisite for the practical development of the GHFNs. Maintenance the advantages of it, this paper focuses around to proposed a way to compute the distances between GHFNs by utilizing three useful distance measures, i.e., Hamming distance, Euclidean distance, and generalized distance measures, individually. Next, using these definitions, we updated the common TOPSIS method for solving MADM problems with generalized hesitant fuzzy numbers. The advantage of the presented method is that it eschews the wrong decisions based on the small changes in entrance information. Finally, by solving numerical examples, we explained and compared the proposed method.

What is certain is that much work still needs to be done theoretically in this area, such as entropy measure, similarity measure, norm-based operators of GHFNs, and so on. In the future and in the field of application, many decision-making methods such as analytic hierarchy process AHP/ANP method, Choquet integral method, VIKOR method, ELECTRE method (Tzeng and Huang 2011) will have to be updated to use GHFNs. Mathematical development of linear programming methods, data envelopment analysis, graph theory, game theory and mechanism design (Marek and Peter 2021), brain hemorrhage patients(Garg and Kaur 2020), natural language processing (Nagarhalli et al. 2021), etc., can be considered by researchers, and the necessary conditions should be provided for applying GHFNs in these fields.