Abstract

The utilization of linguistic variables is of vital importance for qualitative information processing in mathematical modelling and decision making under uncertainty. The primary purpose of our study is to introduce a novel structure by the fusion of Fermatean fuzzy sets and linguistic term sets. This hybrid structure is termed as the linguistic Fermatean fuzzy set which can be employed for dealing with decision-making problems involving qualitative information. Several fundamental operations of linguistic Fermatean fuzzy sets are introduced. Furthermore, we present four classes of linguistic Fermatean fuzzy Hamy mean operators, namely, the linguistic Fermatean fuzzy Hamy mean operator, the linguistic Fermatean fuzzy dual Hamy mean operator, the linguistic Fermatean fuzzy weighted Hamy mean operator, and the linguistic Fermatean fuzzy weighted dual Hamy mean operator. Some basic properties of these linguistic Fermatean fuzzy Hamy mean operators are examined as well. In addition, we develop a linguistic Fermatean fuzzy extension of the COPRAS method. We also propose a novel approach to multi-attribute group decision making with linguistic Fermatean fuzzy information. With our proposed methods, we solve two practical problems regarding food company ranking and green supplier selection, respectively. Finally, the efficacy of our developed method is validated through a comparative analysis with existing methods.

1. Introduction

Multi-attribute decision making (MADM) is a process of evaluating the superlative alternative supporting certain attributes. If a panel of decision experts is involved in this procedure, it is referred to as multi-attribute group decision making (MAGDM). The crisp set theory was only limited to exact information. Therefore, there is a need that a certain model should be assembled to tackle vague information. Zadeh [1] put forward the foundation of fuzzy sets (FSs). In FSs, the satisfaction level of an expert is asserted as degree of membership function and lies in unit interval. Fuzzy set theory (FST) is not confined to only decision making, but it is applicable in numerous fields, including medicine, business and administration, control system engineering, and facial pattern recognition.

Although FSs are capable of handling the vague information, they cannot deal with dissatisfaction level of human beings. Atanassov [2] put forward the idea of intuitionistic fuzzy sets (IFSs) to make up this lack for fuzzy set theory. In IFS, the membership degree (MD) and the non-membership degree (NMD) of an element can be described with the condition that their sum is less or equal to 1. IFSs were considered inadequate to handle the vague information when sum of MD and NMD exceeds 1. Yager [3] evolved the concept of Pythagorean fuzzy sets (PFSs) as a generalization of IFSs. In PFS, the sum of square of MD and NMD cannot exceed 1. However, PFSs have some limitations when membership and non-membership degree exceed 1. PFSs are more efficient to deal with imprecise information than IFSs. But when the MD is 0.8 and NMD is 0.7, PFS theory is not enough to handle this type of information. As a generalization of IFSs and PFSs, Senapati and Yager [4] gave the concept of Fermatean fuzzy sets (FFSs) in which sum of a cube of MD and a cube of NMD is less than or equal to 1. For example, is an intuitionistic fuzzy number (IFN) as , but if the NMD is 0.6, then it is a Pythagorean fuzzy number (PFN) but not a IFN. But if the NMD is 0.8, then is neither IFN nor PFN. So, decision makers prefer to use FFSs to handle such kind of situation. Yager [5] developed -rung orthopair fuzzy sets in which the sum of the -th power MD and the -th power of NMD cannot exceed 1. -rung orthopair fuzzy sets (-ROFSs) are more effective to deal with imprecise information than PFSs and IFSs. Certain applications of FFSs and -ROFSs have been discussed in [615].

Many aspects of real life cannot be handled in quantitative way but rather in qualitative way. In decision making, there are many criteria which cannot be dealt with real numbers. Therefore, there is a need to express the imprecise information in qualitative form. For example, to describe the quality of mobile phone, the decision makers use the term good or bad. Realizing this, Zadeh [16] introduced linguistic fuzzy sets (LFSs). Herrera and Verdegay [17] defined the operational laws for linguistic fuzzy sets. Normally, linguistic fuzzy set is based on seven terms which can be defined as . The quality of company considered as good (0.7) can be represented as , and the membership degree is taken as 0.8. But the linguistic fuzzy sets cannot describe the qualitative vague information in proper way. If one group of decision makers takes the membership degree of the quality of mobile phone as 0.7 and the other group of decision makers considers it as 0.5, then linguistic fuzzy set cannot deal with this type of information. Therefore, Chen and Hong [18] proposed linguistic intuitionistic fuzzy sets (LIFSs) by uniting LTSs and IFSs. LIFSs describe the linguistic membership (LMD) and linguistic non-membership degree (LNMD) of element in qualitative way in more effective manner. LIFSs are restricted to that information where the sum of LMD and LNMD is less than or equal to . If decision makers give their partiality in form of LIFN where s denotes the linguistic term in linguistic term set and , this preference cannot be solved under LIFN. To overcome this limitation of LIFSs, Garg [19] defined linguistic Pythagorean fuzzy sets (LPFSs) in which the sum of square of LMD and LNMD cannot exceed . But LPFSs cannot deal with that type of imprecise information where LMD and LNMD exceed . For other applications and notions, the readers can refer to [2023].

Following linguistic fuzzy set, a lot of aggregation operators were defined. Peng et al. [24] developed an LIFN-based decision-making scheme and defined the notion of Frank Heronian mean operators. Liu et al. [25] elongated it to partitioned Heronian mean operator. Liu and Qin [26] defined Maclaurin symmetric mean operator for LIFNs. Hara et al. [27] proposed the Hamy mean operator which considered the connection between different parameters by modifying their values. Qin [28] developed interval type-2 fuzzy Hamy mean operator. Deng et al. [29] defined 2-tuple linguistic fuzzy Hamy mean operator. Li et al. [30] defined the (weighted) Pythagorean fuzzy Hamy mean operator. Wei et al. [21] introduced dual hesitant Pythagorean fuzzy Hamy mean operator.

Zavadaskas et al. [31] proposed a methodology to deal with decision-making problems, named as COmplex PRoportional ASsessment (COPRAS) method. Wang et al. [32] utilized the COPRAS method to determine the ranking order of the failure modes which was recognized in failure mode and effect analysis (FMEA). Yazdani et al. [33] utilized the fuzzy COPRAS method to develop the risk analysis of critical infrastructures. Hajiagha et al. [34] developed a COPRAS method for group decision making under interval-valued intuitionistic fuzzy environment. Lu et al. [35] devised a COPRAS method under picture fuzzy environment.

The major motivation for the proposed model comes from the following observation:(i)Neither LIFSs nor LPFSs can handle the situation when the sum of squares of linguistic membership and non-membership degrees exceeds 1. Hence, it is necessary to develop a new model in this case.(ii)The Hamy mean operator was designed to describe the correlation with numerous parameters. It is necessary to broaden the concept of Hamy mean operators using LFSs.(iii)The COPRAS method was developed to determine the superlative alternative in classical decision making. Hence, this technique should be further extended to solve MAGDM problems involving linguistic term sets.

Motivated by the above facts, we introduce a novel concept termed as the linguistic Fermatean fuzzy set (LFFS). The main contributions can be summarized as follows:(i)We devise a mathematical approach for group decision making by the fusion of linguistic fuzzy sets and Fermatean fuzzy sets.(ii)The linguistic Fermatean fuzzy Hamy mean operators are constructed to capture interrelationships among different parameters.(iii)The classical COPRAS method is extended to handle MAGDM problems based on linguistic Fermatean fuzzy sets.

The remainder of this article is structured as follows. In Section 2, we define LFFNs and related operations, linguistic score and linguistic accuracy function, linguistic Fermatean fuzzy Hamy mean operator, and linguistic Fermatean fuzzy weighted Hamy mean operator and examine their properties. In Section 3, we present linguistic Fermatean fuzzy dual Hamy mean operator and linguistic Fermatean fuzzy weighted dual Hamy mean operator and investigate their properties. In Section 4, we develop an extension of the COPRAS method on the basis of LFFNs and apply it to a problem regarding food company ranking. In Section 5, we propose another approach to linguistic Fermatean fuzzy MAGDM and apply it to a problem regarding green supplier selection. In Section 6, we check the validity of our methods by comparing with existing methods. Section 7 presents conclusions and future directions.

2. Linguistic Fermatean Fuzzy Hamy Mean Operators

Definition 1. (see [4]). Letbe a universe of discourse. A Fermatean fuzzy set (FFS)inis an object of the formwhere and including the conditionfor all . and denote, respectively, the MD and NMD of element in the set . For any FFS and ,is called the degree of indeterminacy of to . The pair in is defined as Fermatean fuzzy number (FFN).

Definition 2. (see [36]). Let there exist a linguistic term set, wheredenotes a possible linguistic term for a linguistic variable and the cardinality of the setis odd. For instance, a linguistic term sethaving five terms can be described as follows:If , then the linguistic term set satisfies the following properties:(i), if and only if.(ii), if and only if.(iii), if and only if.(iv)Negsuch that.

Next, let us propose the notion of linguistic Fermatean fuzzy sets and some basic operations of linguistic Fermatean fuzzy numbers. For other notions and applications, the readers are suggested to [3751].

Definition 3. Suppose thatis a universal set andis a continuous linguistic term set (CLTS). A linguistic Fermatean fuzzy set (LFFS) is defined aswherestand for linguistic membership and non-membership degrees of the element. It is required thatwith. We refer toas the linguistic indeterminacy degree ofto, which is defined as. The pairrepresents a linguistic Fermatean fuzzy number (LFFN) in.

Definition 4. Letandbe two LFFNs. Some relations and operations of LFFNs can be defined as follows:(i)if.(ii).(iii).(iv).(v)if.

Definition 5. Letandbe two LFFNs. We can define the following operations of LFFNs:(i).(ii).(iii).(iv).

Definition 6. Letbe LFFS. The linguistic score function and the linguistic accuracy function of LFFS are represented aswhere and .

Definition 7. Letbe a collection of LFFNs. We can define LFFHM operator as follows:whereis a parameter andareintegers belonging to the setofinteger values,.

Theorem 1. Letbe a collection of LFFNs. Then, the aggregate result of them by the above definition is still a LFFN:

Proof. From Definition 5, we defineHence, (8) is established. We need to show that (8) is a LFFN. It satisfies the following two conditions:(1).(2).LetSince ,This means ; similarly, . So, condition (1) is verified. Since , we getThis means , so condition (2) is verified.

The LFFHM operator has three properties.

Theorem 2. Ifare equal, then

Proof.

Theorem 3. Letandbe two sets of LFFNs. Ifandhold for all u and v, then

Proof. Let and . Suppose that , and we obtainThereafter,Furthermore,This means . Similarly, we get .
If and ,If and ,

Theorem 4. Letbe a collection of LFFNs. Ifand, then

Proof. From Theorem 2,From Theorem 3,

Definition 8. Letbe a collection of LFFNs with weight vector, satisfyingand. Then, we can define LFFWHM operator as follows:

Theorem 5. Letbe a collection of LFFNs; then, the aggregate result of them by the above definition is still a LFFN:

Proof. From Definition 5, we defineHence, (24) is established. We need to show that (24) is a LFFN. The following two conditions are satisfied:(1).(2).LetSince ,This means ; similarly, . So, condition (1) is proved.
(2) Since ,This means , so condition (2) is proved.

Theorem 6. Letandbe two sets of LFFNs. Ifandhold for all i and v, then

Proof. Let and ; suppose that , and we can obtainThereafter,Furthermore,This means . Similarly, .
If and ,If and ,

Theorem 7. Letbe a set of LFFNs. Ifand, then

Proof. From Theorem 5,From Theorem 6,

3. Linguistic Fermatean Fuzzy Dual Hamy Mean Operators

Definition 9. Letbe a collection of LFFNs. The LFFDHM operator is defined as follows:where is a parameter and are integers belonging to the set of integer values, .

Theorem 8. Letbe a collection of LFFNs. Then, the aggregate result of them by the above definition is still a LFFN:

Proof. From Definition 5, we defineHence, (39) is established. We need to show that (39) is a LFFN. The following two conditions are satisfied:(1).(2).LetSince ,This means ; similarly, . So, condition (1) is verified.
(2) Since ,This means , so condition (2) is verified.

The LFFDHM operator has three properties.

Theorem 9. Ifare equal, then

Proof.

Theorem 10. Letandbe two sets of LFFNs. Ifandhold for all u and v, thenIf and ,If and ,

Theorem 11. Letbe a set of LFFNs. Ifand, then

Proof. From Theorem 9,From Theorem 10,

Definition 10. Letbe a collection of LFFNs with weight vector, satisfyingand. Then, we define LFFWDHM operator as follows:

Theorem 12. Letbe a collection of LFFNs. Then, the aggregate result of them by the above definition is still a LFFN:

Proof. From Definition 5, we define