## 1. Introduction

For human activities and their problems, the Multi-Criteria Group Decision-Making (MCGDM) is an important tool [

1,

2]. In complex real-world conditions, it is not possible for a single Decision-Maker (DM) to recognize all of the relevant aspects of a decision-making problem [

3]. Thus, the decision-making procedure requires considering many DMs or experts from different fields. In many group decision-making problems, a group is established by various DMs from different fields, including work experience, education backgrounds and knowledge structure [

4]. It could be implemented to select the most suitable alternative from a given set of decision variants or their subset [

5,

6]. The essential prerequisite of the MCGDM is the combination of experts’ preferences and judgments about the candidate alternatives versus the conflicting criteria [

7], which is a popular trend of present research to develop new group MCDM methods [

8,

9,

10,

11].

In the decision-making, the problems of uncertainty and hesitancy usually turn out to be unavoidable. To express the DMs’ evaluation information more objectively, several tools have been developed, such as fuzzy set [

1,

12], interval-valued fuzzy set [

13,

14], linguistic fuzzy set [

15,

16,

17], which allow one to present an element’s membership function as a set denoted by a fuzzy number, an interval fuzzy number, a linguistic variable and a fuzzy set, respectively. Intuitionistic fuzzy set [

18] and fuzzy multiset [

19,

20] are another two generalizations of the fuzzy set. Whilst the former contains three types of information (the membership, the non-membership and the hesitancy), the latter permits the elements to repeat more than once.

In many practical problems, sometimes, it is difficult to define the membership grade of an element, because of a set of possible membership values [

21]. This issue is very important in MCGDM problems, when the DMs do not support the same membership grade for an element [

22,

23]. In this case, the difficulty of establishing a common membership grade is caused not by the margin of error (as happens in Intuitionistic Fuzzy Set (IFS)) or some possible distribution values (as happens in Type-2 Fuzzy Sets), but by the fact that several membership values are possible [

10]. To deal with these cases, the Hesitant Fuzzy Set (HFS) was introduced [

24] as a new generalization of fuzzy sets. Many MCDM methods have been extended by using the HFS theory, e.g., the ELECTRE family methods [

25], Viekriterijumsko Kompromisno Rangiranje (VIKOR) [

26] or prospect theory [

27]. There was also established a number of new methods [

13,

28,

29,

30] or aggregation operators [

31,

32], which are based on the HFS concept. Presently, group decision-making problems are solved for hesitant fuzzy sets and with aggregation operators in [

33,

34,

35,

36]. Interval-valued hesitant fuzzy sets have been used in the applications of group decision-making in [

28,

37,

38,

39,

40]. MCGDM with hesitant two-tuple linguistic information and by using trapezoidal valued HFSs is discussed in [

41,

42]. Yu [

43] gave the concept of triangular hesitant fuzzy sets and used it for the solution of decision-making problems. Unfortunately, all of the mentioned group decision-making methods are susceptible to the occurrence of the rank reversal phenomenon paradox, which lies at the heart of the main MCDM challenges.

The Characteristic Objects Method (COMET) is a useful technique in dealing with Multi-Criteria Decision-Making (MCDM) problems [

44,

45,

46,

47,

48]. It helps a DM to organize the structure of the problems to be solved and carry out the analysis, comparisons and ranking of the alternatives, where the complexity of the algorithm is completely independent of the alternatives’ number [

49,

50]. Additionally, comparisons between the Characteristic Objects (COs) are easier than comparisons between alternatives. However, the most important merit of the COMET method is the fact that this method is completely free of the rank reversals phenomenon [

51] because the final ranking is constructed based on COs and fuzzy rules.

In this study, we extend the COMET concept to develop a methodology for solving multi-criteria group decision-making problems under uncertainty. The proposed method allows a group of DMs to make their opinion independent of linguistic terms by using HFS. The proposed method is designed for modeling uncertainty from different sources, which are related to expert knowledge. The main motivation of this research is the fact that the presented extension is also completely free of the rank reversals paradox as the classical version.

The group version of the HFS COMET method can be used in various research fields and disciplines such as economics [

29,

30,

32], resource management [

51], production [

52], transport [

53], game theory (Nash equilibrium) [

54,

55,

56,

57,

58,

59,

60,

61,

62,

63], medical problems [

48,

64], sustainability manufacturing [

65] or web systems [

66]; especially in decision situations requiring the involvement of many experts [

67].

The rest of this paper is organized as follows. In

Section 2, we introduced some basic concepts related to the hesitant fuzzy sets, L-R-type Generalized Fuzzy Numbers (GFNs), the fuzzy rule, the rule base and the t-norm. In

Section 3, we established a group decision-making method based on COMET to deal with the uncertainty environment. In

Section 4, an illustrative example is given to demonstrate the practicality and effectiveness of the proposed approach. Finally, we conclude the paper and give some remarks in

Section 5.

## 3. COMET for MCGDM Using HFS

Consider an MCGDM problem in which the ratings of the alternative evaluations are expressed as HFSs. The solution procedure for the proposed MCGDM approach is described below.

Let ${A}_{j}\phantom{\rule{4pt}{0ex}}(j=1,2,...,m)$ be the set of alternatives and suppose a group of DMs $D=\{{d}_{1},{d}_{2},...,{d}_{k}\}$ is asked to evaluate the given alternatives with respect to several criteria ${C}_{i}\phantom{\rule{3.33333pt}{0ex}}(i=1,2,...,n).$ The ranking algorithm of the COMET has the following five steps:

Step 1: Define the space of the problem as follows:

Let $\mathcal{F}$ be the collection of all L-R-type GFNs and ${F}_{i}^{1\delta},{F}_{i}^{2\delta},...,{F}_{i}^{q\delta}$ be different families of subsets of $\mathcal{F}$ selected by a DM ${d}_{\delta}\phantom{\rule{4pt}{0ex}}(\delta =1,2,...,k)$ for each criterion ${C}_{i}\phantom{\rule{4pt}{0ex}}(i=1,2,...,n)$ where

${F}_{i}^{1\delta}=\{{F}_{i1}^{1\delta},{F}_{i2}^{1\delta},...,{F}_{i{c}_{i}}^{1\delta}\};$

${F}_{i}^{2\delta}=\{{F}_{i1}^{2\delta},{F}_{i2}^{2\delta},...,{F}_{i{c}_{i}}^{2\delta}\};$

$\vdots $

${F}_{i}^{q\delta}=\{{F}_{i1}^{q\delta},{F}_{i2}^{q\delta},...,{F}_{i{c}_{i}}^{q\delta}\}.$

In this way, the following result is obtained:

${C}_{1}=\left\{{F}_{11}^{b\delta},{F}_{12}^{b\delta},...,{F}_{1{c}_{1}}^{b\delta}\right\};$ |

${C}_{2}=\left\{{F}_{21}^{b\delta},{F}_{22}^{b\delta},...,{F}_{2{c}_{2}}^{b\delta}\right\};$ |

$\vdots $ |

${C}_{n}=\left\{{F}_{n1}^{b\delta},{F}_{n2}^{b\delta},...,{F}_{n{c}_{n}}^{b\delta}\right\};$ |

where

$1\le b\le q$ and

${c}_{1},{c}_{2},...,{c}_{n}$ are the numbers of fuzzy numbers in each family

${F}_{i}^{b\delta}\phantom{\rule{4pt}{0ex}}(1\le b\le q,1\le i\le n)$ for all criteria.

Initially, suppose each alternative is assessed by all DMs by means of

n criteria in the form of a single family of TFNs

${F}_{i}^{t}$ $(1\le i\le n)$ with their fuzzy semantics shown in

Figure 1,

Figure 2,

Figure 3,

Figure 4,

Figure 5 and

Figure 6. Suppose each DM further provides the hesitant information of an alternative for each criterion in the form of L-R-type GFNs. Note that, in this method, the observations already provided by all of the DMs for each criterion in the form of the single family of TFNs set

${F}_{i}^{t}$ $(1\le i\le n)$ is a necessary part of all of the family of the remaining L-R-type GFNs set during the computation. The core of each criterion is defined as the core of each

${F}_{i}^{t}$ $(1\le i\le n),$ i.e.,

$C({C}_{1})=\left\{C({F}_{11}^{t}),C({F}_{12}^{t}),...,C({F}_{1{c}_{1}}^{t})\right\};$ |

$C({C}_{2})=\left\{C({F}_{21}^{t}),C({F}_{22}^{t}),...,C({F}_{2{c}_{2}}^{t})\right\};$ |

$\vdots \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ |

$C({C}_{n})=\left\{C({F}_{n1}^{t}),C({F}_{n2}^{t}),...,C({F}_{n{c}_{n}}^{t})\right\}.$ |

Step 2: Generate the characteristic objects:

By using the Cartesian product of all TFNs cores, the COs can be obtained as follows:

$CO=C({C}_{1})\times C({C}_{2})\times ...\times C({C}_{n})$

As the result of this, the ordered set of all COs is obtained:

$C{O}_{1}=\left\{C({F}_{11}^{t}),C({F}_{21}^{t}),...,C({F}_{n1}^{t})\right\};$ |

$C{O}_{2}=\left\{C({F}_{11}^{t}),C({F}_{21}^{t}),...,C({F}_{n2}^{t})\right\};$ |

$\vdots \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ |

$C{O}_{s}=\left\{C({F}_{1{c}_{1}}^{t}),C({F}_{2{c}_{2}}^{t}),...,C({F}_{n{c}_{n}}^{t})\right\};$ |

where

$s={\displaystyle \prod _{i=1}^{n}}{c}_{i}$ is a number of COs.

Step 3: Rank and evaluate the characteristic objects:

A comparison of COs is obtained by adding the opinion of DMs. After this, determine the Matrix of Expert Judgment (MEJ) as follows:

$\mathbf{ME}J=\left[\begin{array}{cccc}{\tilde{h}}_{11}& {\tilde{h}}_{12}& \cdots & {\tilde{h}}_{1s}\\ {\tilde{h}}_{21}& {\tilde{h}}_{22}& \cdots & {\tilde{h}}_{2s}\\ \vdots & \vdots & \ddots & \vdots \\ {\tilde{h}}_{s1}& {\tilde{h}}_{s2}& \cdots & {\tilde{h}}_{ss}\end{array}\right]$ |

where

${\tilde{h}}_{\alpha \beta}=\{{\tilde{h}}_{\alpha \beta}^{\omega},\omega =1,2,...,{l}_{{\tilde{h}}_{\alpha \beta}}\}$ is the HFE containing preferences of all DMs and is obtained as a result of comparing

$C{O}_{\alpha}$ and

$C{O}_{\beta}$. The more preferred CO obtains a stronger preference degree, and the second object obtains a weaker one. If the preferences are balanced, then both objects obtain a preference degree denoted by HFE

${\tilde{h}}_{f}=\{0.5\}$. The selection of

${\tilde{h}}_{\alpha \beta}$ depends solely on the knowledge and opinion of the experts. Mathematically,

${\tilde{h}}_{\alpha \beta}$ should satisfy the following conditions:

${\tilde{h}}_{\alpha \beta}^{\sigma (\omega )}+{\tilde{h}}_{\beta \alpha}^{\sigma ({l}_{{\tilde{h}}_{\alpha \beta}}-\omega +1)}=1,\alpha ,\beta =l,2,...,s;$

${\tilde{h}}_{\alpha \alpha}=\{0.5\},\alpha =l,2,...,s;$

${l}_{{\tilde{h}}_{\alpha \beta}}={l}_{{\tilde{h}}_{\beta \alpha}},\alpha ,\beta =l,2,...,s.$

where the values in ${\tilde{h}}_{\alpha \beta}$ are assumed to be arranged in increasing order for convenience, and let ${\tilde{h}}_{\alpha \beta}^{\sigma (\omega )}(\omega =1,2,...,{l}_{{\tilde{h}}_{\alpha \beta}})$ denote the $\omega th$ smallest value in ${\tilde{h}}_{\alpha \beta}$ and ${l}_{{\tilde{h}}_{\alpha \beta}}$ the number of the values in ${\tilde{h}}_{\alpha \beta}$.

The last equation indicates that the sum of the

$\omega th$ smallest value in

${\tilde{h}}_{\alpha \beta}$ and the

$\omega th$ largest value in

${\tilde{h}}_{\beta \alpha}$ should be equivalent to one, which is the complement condition as introduced by Torra in [

24] (see Definition 2). In other words, if

${\tilde{h}}_{\alpha \beta}=\{{\tilde{h}}_{\alpha \beta}^{\omega},\omega =1,2,...,{l}_{{\tilde{h}}_{\alpha \beta}}\}$ is known, then we can obtain

${\tilde{h}}_{\beta \alpha}$, which is given by

${\tilde{h}}_{\beta \alpha}=\{1-{\tilde{h}}_{\alpha \beta}^{\omega},\omega =1,2,...,{l}_{{\tilde{h}}_{\alpha \beta}}\}$. The second equation indicates that the diagonal elements in

$\mathbf{MEJ}$ should be equivalent to

$\{0.5\}$, which implies the balanced preference degrees of

$C{O}_{\alpha}$ and

$C{O}_{\beta}$. The third equation indicates that the number of elements in

${\tilde{h}}_{\alpha \beta}$ and

${\tilde{h}}_{\beta \alpha}$ should be the same.

Suppose

${\tilde{H}}_{\alpha}={\oplus}_{\beta =1}^{s}{\tilde{h}}_{\alpha \beta},$ where each

${\tilde{H}}_{\alpha}$ is an HFE. Afterward, we get a vertical vector

$SJ$ of the summed judgments where

$S{J}_{\alpha}=Sc({\tilde{H}}_{\alpha})$ $=\frac{1}{{l}_{{\tilde{H}}_{\alpha}}}{\sum}_{\gamma \in {\tilde{H}}_{\alpha}}\gamma \phantom{\rule{4pt}{0ex}}$(see Definition 3). To assign the approximate value of preference to each CO, we use the same MATLAB code as used by Salabun in [

45]. As a result, we get a vertical vector

P, where the

$\alpha $-th component of

$P$ represents the approximate value of preference for

$C{O}_{\alpha}$.

Step 4: The rule base:

Each CO and value of preference is converted to a fuzzy rule as follows:

$IF$ $\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}C{O}_{\alpha}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}THEN\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{P}_{\alpha}$

$IF$ $\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}C({F}_{1\alpha}^{t})\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}AND\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}C({F}_{2\alpha}^{t})\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}AND\phantom{\rule{4pt}{0ex}}...\phantom{\rule{4pt}{0ex}}THEN\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{P}_{\alpha}$

In this way, the complete fuzzy rule base is obtained, which can be presented as follows:

$IF$ $\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}C{O}_{1}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}THEN\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{P}_{1}$

$IF$ $\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}C{O}_{2}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}THEN\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{P}_{2}$

$\vdots \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$

$IF$ $\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}C{O}_{s}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}THEN\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{P}_{s}$

Step 5: Inference in a fuzzy model and final ranking:

Each alternative activates the specified number of fuzzy rules, where for each one, the fulfillment degree of the conjunctive complex premise is determined. The fulfillment degrees of each activated rule corresponding to each element of ${F}_{i}^{b}\phantom{\rule{4pt}{0ex}}(1\le b\le q,1\le i\le n)$ always sum to one. Each alternative is a set of crisp numbers, corresponding to criteria ${C}_{1},{C}_{2},...,{C}_{n}$. It can be presented as follows:

${A}_{j}=\left\{{a}_{1j},{a}_{2j},...,{a}_{nj}\right\},$ where the following conditions must be satisfied:

${a}_{1j}\in [C({F}_{11}^{t}),C({F}_{1{c}_{1}}^{t})];$

${a}_{2j}\in [C({F}_{21}^{t}),C({F}_{2{c}_{2}}^{t})];$

$\vdots \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$

${a}_{nj}\in [C({F}_{n1}^{t}),C({F}_{n{c}_{n}}^{t})].$

To infer the final ranking of the alternatives corresponding to each criterion, we proceed as follows:

For each

$j=1,2,...,m,$${a}_{1j}\in [C({F}_{1{k}_{1}}^{t}),C({F}_{1({k}_{1}+1)}^{t})];$ |

${a}_{2j}\in [C({F}_{2{k}_{2}}^{t}),C({F}_{2({k}_{2}+1)}^{t})];$ |

$\vdots \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ |

${a}_{nj}\in [C({F}_{n{k}_{n}}^{t}),C({F}_{n({k}_{n}+1)}^{t})];$ |

where

${k}_{i}=1,2,...,({c}_{i}-1),\phantom{\rule{4pt}{0ex}}(1\le i\le n).$ The activated rules (COs), i.e., the group of those COs where the membership function of each alternative

${A}_{j}\phantom{\rule{4pt}{0ex}}(1\le j\le m)$ is non-zero, are:

$\left(C({F}_{1{k}_{1}}^{t}),C({F}_{2{k}_{2}}^{t}),...,C({F}_{n{k}_{n}}^{t})\right);$

$\left(C({F}_{1{k}_{1}}^{t}),C({F}_{2{k}_{2}}^{t}),...,C({F}_{n({k}_{n}+1)}^{t})\right);$

$\vdots $

$\left(C({F}_{1({k}_{1}+1)}^{t}),C({F}_{2({k}_{2}+1)}^{t}),...,C({F}_{n({k}_{n}+1)}^{t})\right).$

The number of COs are obviously ${2}^{n}$ and $1\le {2}^{n}\le s.$

Let

${p}_{1},{p}_{2},...,{p}_{{2}^{n}}$ be the approximate values of the preference of the activated rules (COs), which were already calculated in Step 3, where

${p}_{\eta}$’s

$(1\le \eta \le {2}^{n})$ are some values in

${P}_{\alpha}$’s

$(1\le \alpha \le s)$. We denote the HFE at the point

$x\in {A}_{j}\phantom{\rule{4pt}{0ex}}(1\le j\le m)$ provided by a DM

${d}_{\delta}\phantom{\rule{4pt}{0ex}}(\delta =1,2,...,k)$ as:

for each criterion

${C}_{i}\phantom{\rule{4pt}{0ex}}(i=1,2,..,n).$To aggregate the information in the form of HFEs from every DM, in order to achieve a single HFE, which summarizes all of the information provided by the different DMs, there are several aggregation operators that are available in the literature. However, in this paper, we simply use the average operator to get the average of the membership values obtained from LR-type GFNs provided by the DMs in the form of HFE corresponding to each

${a}_{ij}\in {A}_{j}\phantom{\rule{4pt}{0ex}}(1\le i\le n,1\le j\le m).$ Suppose

${h}_{ij}(x)$ is an HFE obtained as a result of aggregating the HFEs

${h}_{ij}^{\delta}(x),\phantom{\rule{4pt}{0ex}}(\delta =1,2,...,k)$ where:

Let ${\mathbf{A}}_{j}$ be HFE, which is computed as the sum of the products of all activated rules, as their fulfillment degrees and their values of the preference, i.e.,

${\mathbf{A}}_{j}$ $={p}_{1}\left({h}_{1{k}_{1}}({a}_{1j})\otimes {h}_{2{k}_{2}}({a}_{2j})\otimes \dots {h}_{n{k}_{n}}({a}_{nj})\right)\oplus {p}_{2}\left({h}_{1{k}_{1}}({a}_{1j})\otimes {h}_{2{k}_{2}}({a}_{2j})\otimes \dots {h}_{n({k}_{n}+1)}({a}_{nj})\right)$

$\oplus \dots {p}_{{2}^{n}}\left({h}_{1({k}_{1}+1)}({a}_{1j})\otimes {h}_{2({k}_{2}+1)}({a}_{2j})\otimes \dots {h}_{n({k}_{n}+1)}({a}_{nj})\right)$

The preference of each alternative

${A}_{j}\phantom{\rule{4pt}{0ex}}(1\le j\le m)$ can be found by finding the score of the corresponding HFE

${\mathbf{A}}_{j}$ $(1\le j\le m)$ as follows:

The final ranking of alternatives is obtained by sorting the preference of alternatives. The greater the preference value, the better the alternative ${A}_{j}\phantom{\rule{4pt}{0ex}}(1\le j\le m)$.

As the summary of this section,

Figure 1 presents the stepwise procedure of the proposed extension of the COMET method. After initiating the decision process, the procedure starts by modeling the structure of a considered decision problem. At this point, each expert determine generalized fuzzy numbers for each criterion. This is followed by generating characteristic objects in Step 2, evaluating the preferences of the characteristic objects in Step 3 and generating the fuzzy rule base in Step 4. The procedure ends by computing the assessment for each alternative from the considered set. The set of alternatives can be ranked according to the descending order of the computed assessments.

## 4. An Illustrative Example

In this section, an example is given to understand our approach. We used the method proposed in

Section 3 to get the most desirable alternative, as well as to rank the alternatives from the best to the worst or vice versa.

Let us consider a factory, whose maximum capacity of using mobile units is a total of 1000 per month, which intends to select a new mobile company. Four companies

${A}_{1}$,

${A}_{2}$,

${A}_{3}$ and

${A}_{4}$ are available, and three DMs are asked to consider two criteria

${C}_{1}\phantom{\rule{4pt}{0ex}}$(fixed line rent) and

${C}_{2}$ (rates per unit) to decide which mobile company to choose. The fixed line rent, rates per unit and the original ranking order of the feasible mobile companies are shown in

Table 1.

A set of TFNs and trapezoidal fuzzy numbers for both criteria

${C}_{1}$ and

${C}_{2}$ set by three DMs are shown in

Table 2 and

Table 3. The average of the membership values obtained from LR-type GFNs for both the criteria are shown in

Table 4.

The cores of the family of TFNs for both the criteria

${C}_{1}$ and

$\phantom{\rule{4pt}{0ex}}{C}_{2}$ are respectively

$\{30,200,300\}$ and

$\{1200,1800,2500\}.$ The solution of the COMET is obtained for different number of COs. The simplest solution involves the use of nine COs, which are presented as follows:

To rank and evaluate the COs, suppose the three DMs give their assessments by providing the HFEs as shown in

Table 5 and

Table 6, and therefore, the Matrix of Expert Judgment (MEJ) is as follows:

The vector

$SJ$ on the basis of

$MEJ$ is obtained as follows:

A vertical vector

P is obtained by using a MATLAB code (see [

45]) as follows:

Each component of the vector

$P$ represents the approximate values of the preference for the generated COs. Each CO and the value of preference

${p}_{i}$ is converted to a fuzzy rule, as follows:

With respect to Model 4, for the alternative ${A}_{1}=\{150,1500\}$, we have nine rules (COs), but the activated rules are $C{O}_{1},\phantom{\rule{4pt}{0ex}}C{O}_{2},\phantom{\rule{4pt}{0ex}}C{O}_{4},\phantom{\rule{4pt}{0ex}}C{O}_{5}.$ The approximate values of the preference of corresponding COs are ${p}_{{}_{1}}\sim 1,\phantom{\rule{4pt}{0ex}}{p}_{{}_{2}}\sim 0.7500,\phantom{\rule{4pt}{0ex}}{p}_{3}\sim 0.8750,\phantom{\rule{4pt}{0ex}}{p}_{4}\sim 0.5000.$ The HFE ${\mathbf{A}}_{1}$ and the preference value of the corresponding alternative ${A}_{1}$ are computed respectively as follows:

${\mathbf{A}}_{1}={p}_{{}_{1}}{h}_{11}(150)\otimes {h}_{21}(1500)\oplus {p}_{{}_{2}}{h}_{11}(150)\otimes {h}_{22}(1500)\oplus {p}_{3}{h}_{12}(150)\otimes {h}_{21}(1500)\oplus {p}_{4}{h}_{12}(150)\otimes {h}_{22}(1500)$

$Sc({\mathbf{A}}_{1})=\frac{1}{{l}_{{\mathbf{A}}_{1}}}{\sum}_{y\in {\mathbf{A}}_{1}}y=0.5725$

Similarly, we can find the preference values for the rest of the alternatives and their ranking, which are shown in

Table 7.

The best choice is the alternative ${A}_{3}$ followed by ${A}_{2},{A}_{1}$ and ${A}_{4}$. The worst choice is the alternative ${A}_{4}$. The extrema elements are consistent with the original ranking. However, the ranking obtained by the COMET method is not perfect. The main reason is that this problem was solved under an uncertain environment by a group of decision-makers. In other words, it is extremely difficult to make a reliable decision using uncertain data, but we believe that it is possible. This example also illustrates how hard it is to make a group decision under uncertainty. Notwithstanding, the COMET method shows the best and the worst decision.

The main contribution of the proposed approach can be expressed by the most important properties of this extension, i.e., the proposed approach is completely free of the rank reversal phenomenon and obtains not only a discrete value of priority, but the mathematical function, which can be used to calculate the priority for all alternatives from the space of the problem. Quantitative expression of efficiency is a very difficult task because a large number of assumptions is needed. Additionally, the reference ranking of the alternatives set is needed in this task, but the reference rank is almost always unknown. However, the problem of quantitative effectiveness assessment is a very important and interesting direction for further research.