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1. Introduction
The preferred information in actual decision-making situations is frequently imprecise, uncertain, and unpredictable. As a result, fuzzy decision making is a beneficial approach in a variety of fuzzy situations [1–6]. Since Smarandache’s neutrosophic set (NS) [7, 8] can adequately define imprecise, ambiguous, and inconsistent data. Several researchers have created a few subclasses of NS that may be used easily in real-world scientific and engineering problems. For instance, Wang et al. [9, 10] defined a single-valued neutrosophic set (SVNS) and an interval neutrosophic set (INS), as well as the set-theoretic operators and characteristics of SVNSs and INSs. After, the introduction of SVNSs and INSs, several researchers anticipated correlation coefficient [11–13], distance measures [14–16], and normalized Bonferroni mean [17–20] and apply these concepts to solve MADM problems under SVN and IN environments. For IN MADM problems, Zhang et al. [21] advanced the score, accuracy, and certainty functions of IN numbers (INNs) and created the INN weighted average (INNWA) operator and INN weighted geometric (INNWG) operator. Khan et al. [22] and Liu et al. [23] anticipated IN Dombi power Bonferroni mean operator, IN power Hamy mean operator and applied it to solve MADM, MAGDM problems under IN information.
In many real-life decision scenarios, however, experts may prefer to convey their alternatives using linguistic information rather than numerical numbers. As a result, linguistic term sets (LTS) [24] are studied extensively and used in the decision-making process to convey expertise’ preferred alternatives [25–27]. Motivated by INS and LTS, Ye [28] initiated the concept of IN linguistic sets (INLSs) and then proposed some basic aggregation operators (AOs) to deal with MAGDM problems under INL information. Further, Ye [29] merged SVNS with LTS and initiated the idea of SVN linguistic sets (SVNLS) and introduces an extended TOPSIS method to handle MADM problems under SVNL information. Ji et al. [30] initiated combined MABAC-ELECTRE for SVNLSs and applied it to solve MADM problems under SVNL information. Wang et al. [31] initiated a series of SVNL generalized Maclaurin symmetric mean (SVNLGMSM) operator and apply these AOs to solve MADM problems. Both the linguistic variable expressed by the decision maker’s assessment of the assessed entity and the quantitative performance value expressed by an INN as the credibility of the provided linguistic variable are contained in an INLS. When decision makers offer their judgments on characteristics in the form of INLNs in complicated decision-making situations, though, they might hesitate between a variety of possible interval values. To deal with such scenario, Ye [32] initiated the idea of hesitant interval neutrosophic sets (HINLSs) by merging INLS with HFS [33, 34]. Ye also introduced some basic operational laws and some weighted AOs to deal with MADM problems under HINL information.
AOs are extremely helpful tools for combining expert opinions in order to calculate the total value of each option. The power average (PA) operator, which was first developed by Yager [35], can decrease the detrimental influence of high expert evaluation values on final decision results. The power geometric (PG) operator and its weighted form were created by Xu and Yager [36], who were inspired by the notion of PA operators. Zhou et al. [37] merged PA operator with generalized average operator and initiated a new type of AOs, that is, generalized power average (GPA) operators. After the introduction of PA, PG, and GPA operators, several scholars extended these AOs for different types of fuzzy extensions. However, mostly these AOs are based on traditional operational laws, and it is unable to fulfill the various semantic needs of various experts. They cannot be used to aggregate HINLNs; as a result, the goal of this paper is to offer a number of novel generalized power AOs for integrating HINL data. The processing of language information is an essential topic that requires consideration in the research of linguistic decision-making methods. Several linguistic information processing methods have been suggested thus far, including the membership function transformation method [38, 39], the symbolic calculation method based on the subscripts of linguistic words [40–42], the cloud model transformation method, and the 2-tuple linguistic representation approach [43–45]. As the abovementioned decision making have certain advantages, but it cannot deal all types of decision-making problems. When evaluating an object, decision makers may believe that the semantic divergence between “acceptable” and “somewhat acceptable” is more or smaller than the semantic difference between “acceptable” and “completely acceptable.” That is, when the number of linguistic subscripts grows, the semantic divergence between adjacent linguistic words does not necessarily remain constant [46]. Decision makers may have various semantic criteria for established linguistic terms in numerous actual decision-making scenarios. Clearly, the existing linguistic technique fails to handle identical decision-making difficulties in the presence of HINL data. So, to overcome such drawbacks, in this article, we utilized LSF to redefine the operational laws for LTs and Schweizer-Sklar -norm and -conorm [47] for HINLNs. Then, we further initiate four generalized HINL PA operators to solve MADM problems.
As a result of the foregoing research inspirations, the following are the article’s aims and offerings:
(1) To define some novel operational laws for HINLNs based on Schweizer-Sklar -norm, Schweizer-Sklar -conorm and linguistic scale function
(2) Anticipating four types of generalized power aggregation operators based on these novel operational laws for HINLNs
(3) Inspecting core properties and specific cases of these generalized power aggregation operators with respect to generalized parameters
(4) Presenting a MADM technique under HINL environment which can not only remove the bad impact of high assessment values on the decision making results but also adjust to distinct semantic environment, fulfill semantic requirements of distinct experts, and make decision-making process flexible
To do so, the rest of the article is organized as follows: in Section 2, some basic ideas are examined briefly. In Section 3, based on LSF and SS -norm and -conorm, some core operational laws are initiated for HINLNs. In Section 4, based on these operational laws, various GPA are developed to aggregate HINNs, and various core properties and special cases are investigated. In Section 5, a MADM model is presented to deal with HINL information. In Section 6, a numerical example is given to show the effectiveness and practicality of the developed MADM approach. Finally, comparison with the existing approach is also discussed.
2. Preliminaries
2.1. The Interval Neutrosophic Linguistic Set
Definition 1 (see [28]).
Let be the finite set. An INLS in is identified by
where and , respectively, signifying the TMED, IND, and FLMD of an element in to the linguistic variable with the constraint for any .
For simplicity, the INLN is signified by .
Definition 2 (see [28]).
Let , and be three INLNs and , then, the core operations for INLNs are defined as follows:
2.2. Hesitant Fuzzy Set
Definition 3 (see [33, 34]).
Let be a fixed set, a HFS on is an object of the form
where is a group of finite values in [0, 1], signifying the possible MED of an element to . For ease, we shall inscribe as a replacement for and is called a hesitant fuzzy element.
Let and be HFEs, then, the core operational laws for HFEs are identified below:
2.3. Linguistic Scale Function
Linguistic scale functions (LSFs) apply various semantic values to linguistic scale under different conditions to make data more effective and to describe semantics more flexibly [46]. In practice, these functions are preferred because they are more versatile and may produce more predictable outputs based on varied meanings.
Definition 4 (see [46]).
If is a numeric value, then, the LSF that demeanor the mapping from to can be identified as follows:
(i) Consider
where Evidently, the symbol reflects the preference of the decision makers while they are using the linguistic term . Consequently, the function value in fact denotes the semantics of the linguistic terms.
On average, the assessment scale for the linguistic information presented above is divided.
(ii) Consider
The absolute deviation between neighboring linguistic subscripts rises as the length of the supplied linguistic term set is extended from the middle to both ends.
(iii) Consider
The absolute deviation between consecutive linguistic subscripts will decrease when the extension from the centre of the supplied linguistic term to both ends is increased.
The above function may be extended to keep all of the provided data and make the computation easier which satisfies and is a strictly monotonically increasing and continuous function. Therefore, the mapping from is one-to-one because of its monotonicity, and the inverse function of exists and is denoted by
2.4. Hesitant Interval Neutrosophic Linguistic Set
Definition 5 (see [32]).
Let be the domain set. Then, a HINLS in is characterized by the following mathematical form:
where is a group of INLNs, representing the possible INLNs of the element to the set is an INLN. For ease, we shall inscribe as a replacement for in . Here we identify a HINULE and is called an INLN. Then, is the group of all INLNs.
Definition 6 (see [32]).
Let and be any three HINLNs and Then, some core operational rules for HINLNs are described as follows:
Definition 7 (see [32]).
Let be a HINLN. Then, the score function is signified as follows:
where is the number INLNs in , ând is the cardinality of the linguistic term set .
The score and accuracy function identified by Ye [32] have some limitation in some special cases for comparing two HINLNs, and this can be shown in an example below.
Example 1.
Let , and be two HINLNs. Then, by utilizing the above score function defined by Ye [32], we have
Which shows that . However, is greater than
To overcome the above existing limitation identified in an example, we signified new score function based on linguistic scale functions for comparing HINLNs.
Definition 8.
Let be a HINLN. Then, the improved score function can be signified as
where the values of a indicate the decision-makers’ views, and a , , and denote the decision-makers’ levels of optimist, temperance, and pessimist. Furthermore, by using various linguistic scale functions, alternative scoring functions can be produced.
Definition 9.
Let and be two HINLNs. Then, the comparison rules for comparing two HINLNs are identified as follows:
(1) If , then
(2) If , then
(3) If , then
Now, utilizing the improve score function to solve Example 1 and assume that , we have
From the score values, we can observe that is greater than
Definition 10.
Let and be any two HINLNs. Then, the Hamming distance between and can be described as
2.5. The PA Operator
PA operator initiated by Yager [35] is one of the imperative AOs. The PA operator reduces a number of unconstructive influences of unreasonably high or unreasonably low arguments given by DMs. The conservative PA operator can only contract with real numbers and is identified as follows.
Definition 11 (see [35]).
Let be a group of positive real numbers. A PA operator is classified as follows:
where and are the support degree (SPD) for from satisfying the following axioms:
(1)
(2)
(3) if
Definition 12 (see [36]).
Let be a set of positive real numbers. A PG operator is described as follows:
where and are the SPD for from satisfying the above axioms.
Definition 13 (see [37]).
Let be a group of positive real numbers. A WGPA operator is described as follows:
where and are the SPD for from satisfying the following axioms.
3. SS Operational Laws for HINLNs
The SS operations [47] contain SS product and SS addition, which are exacting cases of Archemedean -norm and -conorm.
The SS -norm and -conorm are elucidated as follows:
where
Moreover, when , SS -norm and SS -conorm degenerate into algebraic -normn and -conorm.
Based on -norm and -conorm of SS operations, we can provide the following definition about SS operations for HINLNs.
Definition 14.
Let be any three HINLNS, and . Then, we initiate some core operational laws for HINLNs based on Schweizer-Sklar -norm and -co-norm.
Theorem 15.
Let and be any three HINLNs. Then
4. Some Generalized Power Aggregation Operators for HINLNs
In this part, we develop some generalized power AOs established on the initiated operational rules for HINLNs.
4.1. Weighted Generalized Hesitant Interval Neutrosophic Linguistic Schweizer-Sklar Power Aggregation Operator
In this subpart, we initiate generalized hesitant interval neutrosophic linguistic Schweizer-Sklar power average (GHINLSSPA) operator, weighted (WGHINLSSPA) operator and examine their enviable properties and various particular cases.
Definition 16.
For a collection of HINLNs GHINLSSPA operator is a function where , parameter and is the SPD for from with the following constraint:
(1) ;
(2) ;
(3) if , where is the distance measure among two HINLNs.
To write Equation (29) in unsophisticated form, we have
So, from Equation (30), Equation (29) becomes
Theorem 17.
Let be a set of HINLNs, then the value aggregated utilizing Definition 16 is still HINLN, and we have
Proof.
In the following, first, we prove
by exploiting mathematical induction on .
For .
From the operational laws explained for HINLNs in Definition 14, we have
and
Similarly,
Then,
If Equation (33) holds for
Then, when by the operational laws explained in Definition 14, we have
That is, Equation (33) is true for So Equation (33) is true for all Then,
Therefore,
Which completes the proof of the Theorem 17.
Theorem 18.
Commutativity: let be any permutation of , then
Proof.
Suppose that is any permutation of , then for each , there exists one and only one such that and vice versa. And . Then based on Theorem 17, we have
Note that the GHINLSSPA operators are neither idempotent nor monotonic.
Further, we shall discuss a few cases of the initiated AOs with respect to the parameter , which are listed below:
(1) If , then, the GHINLSSPA operator trim down to HINLSS power average (PA) operator:
(2) If and , then, the GHINLSSPA operator trim down to HINL PA operator based on algebraic operation. That is
(3) If , ( a constant) and , then, the GHINLSSPA operator trim down to HINL average operator based on algebraic operation. That is
Definition 19.
For a group of HINLNs a WGHINLSSPA operator is a function where is the weight vector for such that and parameter and is the support for from with the following constraint:
To write Equation (47) in uncomplicated way, we have
so, from Equation (49), Equation (47) becomes
Theorem 20.
Let be a group of HINLNs, then the value aggregated utilizing Definition 19 is still HINLN, and we have
Proof.
Proof of the Theorem 20 is same as Theorem 17.
Theorem 21.
For a group of HINLNs , is a parameter and . is weighting vector for , , and If then WGHINLSSPA trims down to generalized HINLSS PA operator:
where is the power weight vector. Further, we shall discuss a few cases of the initiated AOs with respect to the parameter and , which are listed below:
(1) If , then, the WGHINLSSPA operator reduces into HINLSS power average (PA) operator:
(2) If , , and , then, the WGHINLSSPA operator trims down to HINL PA operator based on algebraic operation. That is
(3) If , , ( a constant) and , then, the WGHINLSSPA operator reduces into HINL average operator based on algebraic operation. That is
4.2. Some Generalized Hesitant Interval Neutrosophic Schweizer-Sklar Power Geometric Aggregation Operators
In this subpart, we initiate generalized hesitant interval neutrosophic linguistic Schweizer-Sklar power geometric average (GHINLSSPWGA) operator, weighted GHINLSSPGA, discuss their desirable properties and some particular cases.
Definition 22.
For a group of HINLNs GHINLSSPGA operator is a function where , parameter and is the SPD for from with the following constraint:
(1) if , where is the distance measure among two HINLNs
To write Equation (56) in an uncomplicated way, we have
So, from Equation (58), Equation (56) becomes
Theorem 23.
Let be a group of HINLNs, then, the value aggregated utilizing Definition 22 is still HINLN, and we have
Proof.
In the following, first, we prove
To prove Equation (61), we utilize mathematical induction on .
For .
From the operational rules explained for HINLNs in Definition 14, we have
Similarly,
Then,
If Equation (50) holds for
Then, when by the operational rules given in Definition 14, we have
That is, Equation (61) is true for So Equation (61) is true for all Then,
Therefore,
This completes the proof of Theorem 23.
Further, we shall examine a few cases of the initiated AOs with respect to the parameter and , which are listed below:
(1) If , then, the GHINLSSPGA operator trims down to HINLSS PGA operator:
(2) and , then, the GHINLSSPGA operator reduces into HINL PGA operator based on algebraic operation. That is
(3) If , ( a constant) and , then, the GHINLSSPGA operator reduces into HINL GA operator based on algebraic operation. That is
The GIFSSPGA operator has the property of commutativity.
Definition 24.
For a group of HINLNs WGHINLSSPGA operator is a function where is the weight vector for such that and parameter and is the support for from with the following constraint:
(1) if , where is the distance measure among two HINLNs
To write Equation (72) in unsophisticated way, we have
So, from Equation (74), Equation (72) becomes
Theorem 25.
Let be a group of HINLNs, then, the value aggregated utilizing Definition 24 is still HINLN, and we have
Further, we shall discuss a few cases of the initiated AO with respect to the parameter and , which are listed below:
(1) If , then, the WGHINLSSPGA operator trims down to WHINLSS PA operator:
(2) If , , and , then, the WGHINLSSPGA operator reduces into HINL PGA operator based on algebraic operation. That is
(3) If , , ( a constant) and , then, the WGHINLSSPGA operator trims down to HINL GA operator based on algebraic operation. That is
5. An Application of Generalized Hesitant Interval Neutrosophic Linguistic Schweizer-Sklar Power Aggregation Operator to Group Decision Making
In this part, we pertain the aforementioned generalized hesitant interval neutrosophic linguistic Schweizer-Sklar power AOs to ascertain productive approaches for MADM under HINL environments. Let be the group of detached alternatives, the group of attributes is articulated by , and the weight vector of the attributes is symbolized by such that . In the process of decision making, the assessment information about the alternative with respect to the attribute is expressed by a HINL decision matrix , where is a HINLN.
Then, gamble on factual decision situations where the weight vector of attributes is entirely identified in advance. For that reason, we initiate MADM approaches established on the proposed GHINLSSPA operators.
5.1. MADM with Known Weight Vectors of Attributes
In this subsection, to deal with real decision situations in which the weighting vectors of attributes is totally known, we apply WGHINLSSPA operator and WGHINLSSPGA operator to establish the following approach to solve MADM problems under HINL environments. To do so, immediately go behind the steps below.
Step 1.
Find out support by the following formula;
where is the distance measure and is calculated by utilizing Equation (19).
Step 2.
Find out the support degree that HINLN collects from other HINLNs , where
Step 3.
Determine weighting vector associated with ,
Step 4.
Utilize WGHINLSSPA or WHINLSSPGA operators to collective all evaluation values into overall evaluation value matching to the alternatives
Or
Step 5.
Find out the scores for the overall HINLN of the alternatives by exploiting Definition 8.
Step 6.
Rank all alternatives and select the best one (s) with the ranking order.
6. Illustrative Example
In this section, an example of alternative selection taken from Ye [32] is utilized to demonstrate the usefulness of the anticipated decision-making process under a hesitant interval neutrosophic linguistic environment. An investment firm would like to put money into the best reasonable choice. A panel with four investment options (alternatives) is available to spend the money. The available options are, a car firm, is a food firm, is a computer firm, and is an arms firm. The investment firm must make a decision based on the three attributes, the risk , the growth , and the environmental impact . The weight vector of the attributes is The possible four alternatives are assess with respect to three attributes by three decision maker and provide the assessment values in the form of HINLNs under the linguistic term set . Thus, when the possible four alternatives are assessed by three decision makers with respect to the three attributes, and the HINL decision matrix are constructed as given in Table 1.
Step 1.
Find out the supports by utilizing formula (80). For simplicity, we shall denote by which means the supports between the th and the th columns of .
Step 2.
Utilizing Equation (81) to find out the weighted support degree that HINLN collects from other HINLNs . We express by .
Step 3.
Utilize Equation (82) to get the weights associated with . This is revealed as follows.
Step 4.
Utilize Equation (83) to amalgamate all the evaluation values in the th row of and acquire the inclusive evaluation value . This is revealed as follows. .
Or utilize Equation (84) to amalgamate all the evaluation values in the th row of and acquire the inclusive evaluation value . This is revealed as follows. .
Step 5.
Exploiting Definition 8 to find out the scores . This is revealed as follows:
Or
Step 6.
According to the score values the ranking order of the alternatives .
, or .
So, the best alternative is , and the worst alternative is .
Table 1
HINL decision matrix .
| | | |
| | | |
| | | |
| | | |
| | | |
6.1. The Effect of the LSFs on Ranking Results
In this subsection, other different kinds of LSFs are also used to the abovementioned decision-making process to obtain the ranking results to demonstrate the effect from other LSFs on the ranking results. The score values and final ranking orders are shown in Table 2.
Table 2
Score values and ranking orders of alternatives utilizing different LSFs.
| LSF | HINSSPWA operator | HINSSPWGA operator | Ranking order |
| | | or |
|
| | | or |
From Table 2, we can observe that when the LSF is utilized the ranking orders gained from both the aggregation operators remain the same as that gained from the first LSF. But when the second LSF is used, the ranking order acquired from the HINLSSPWA operator is the same as the ranking order gained from the first LSF; however, when the HINLSSPWA operator is used, the ranking order is modified. That is, the best alternative remains the same but the worst one is changed, which is . The major explanation for this variation is that three distinct forms of LSFs affect three different sorts of semantic circumstances. This might lead to a variety of semantic preferences and semantic deviations, resulting in a variety of ranking results. As a result, one of the benefits of our suggested technique is that it can adapt to various semantic decision-making environments and fulfill the semantic needs of various experts. So, experts can choose the suitable LSF in real-time decision-making based on their linguistic preferences.
6.2. Effect of the Parameteron Final Ranking Order
From Table 3, one can observe that for different values of the parameter , different score values are obtained, while utilizing WHINLSSPWA and WHINLSSPGA operators. We can also observe from Table 3, when the values of the parameter increases while exploiting WHINLSSPWA operator, the score values of the alternatives increases. Similarly, when utilizing WHINLSSPWG operator, the score values of the alternatives decreases, while the final ranking order remains the same at both the cases. This makes the decision-making process more flexible, so, the decision makers may choose the value of the parameter according to the actual need of the situations.
Table 3
Effect of the parameter on final ranking order.
| Parameter | WSSHINPWA operator | WSSHINPWG operator | Ranking order |
| | | or |
|
| | | or |
|
| | | or |
|
| | | or |
|
| | | or |
|
| | | or |
|
| | | or |
|
| | | or |
6.3. Effect of the Parameteron Decision Result
From Table 4, we can see that for different values of the parameter different score values are obtained, while utilizing WHINLSSPA and WHINLSSPGA operators. One can also observe from Table 4, when utilizing WHINLSSPA operator the ranking order remains the same, but when we utilized WHINLSSPGA operator different ranking orders are obtained. This makes the decision-making process more flexible, and the makers may use the value of the parameter according to the need of the actual situations.
Table 4
Effect of the parameter on decision result.
| Parameter | WSSHINPWA operator | WSSHINPGA operator | Ranking order |
| | | or |
|
| | | or |
|
| | | or |
|
| | | or |
|
| | | or |
|
| | | or |
|
| | | or |
|
| | | or |
6.4. Comparison of the Proposed MADM Method with Existing Method
In this subpart, comparison of the anticipated MADM method initiated on the newly proposed AOs with existing method is discussed.
From Table 5, we can see that the ranking order obtained from Ye [32] is the same as the ranking order obtained from the proposed approach. This shows that the initiated approach is valid. The initiated approach has several advantages over the approach developed by Ye [32]. The initiated approach can remove the bad impact of unreasonable data by power weight vector and also make the decision making process more flexible due to general parameters and fit in with distinct semantic scenarios. Therefore, the proposed technique for solving MADM problems is more practical then the existing one.
Table 5
Comparison with other approach.
| Approach | Score values | Ranking order |
| Ye [32] HINLWA operator | | |
| Ye [32] HINLWG operator | | |
| Proposed HINLSSGPWA operator | | |
| Proposed HINLSSGPWA operator | | |
7. Conclusion
Accessible information is frequently incomplete and incompatible in real decision-making, and the HINLS is a superior tool for indicating such information. In this article, merging LSFs, SS operational laws, and GPA operators, a technique is initiated to deal with HINL MADM problems and fit in with distinct semantic scenarios. Initially, a number of core operational laws for HINLNs are initiated based on LSF, SS -norm, and SS -conorm and some of its core properties are investigated. Second, limitations of the existing score function are discussed, and a new score function and distance measure are anticipated based on LSFs. Then, as standard PA operators cannot handle scenarios when expert assessment values are HINLNs, several novel generalized power AOs are proposed to aggregate HINLNs. The most significant characteristic of these operators are that they can also adapt to a variety of semantic situations while also reducing the detrimental impact of unreseasonably high or unreseasonably low evaluation values. Additionally, utilizing the newly instigated AOs, a novel MADM technique is suggested. Lastly, a numerical example is offered to reveal the potency of the initiated technique, along with comprehensive comparison with the existing approaches.
In future, we will explore LSFs and SS operational laws for other generalizations of INL and SVNL sets, such as hesitant bipolar valued neutrosophic sets [48], single valued spherical hesitant neutrosophic sets [49], interval valued neutrosophic vague sets [50], refined single valued neutrosophic sets [51], and initiate different AOs such as MSM operator, Muirhead mean operators, Hamy Mean operators and apply these AOs to solve MADM and MAGDM problems in different fields.
Acknowledgments
This work was funded by the Deanship of Scientific Research (DSR), King AbdulAziz University, Jeddah, under Grant no. D-714-150-1441.
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