Aczel Alsina t-norm and t-conorm-based aggregation operators under linguistic interval-valued intuitionistic fuzzy setting with application

This article uses the Aczel-Alsina t-norm and t-conorm to make several new linguistic interval-valued intuitionistic fuzzy aggregation operators. First, we devised some rules for how linguistic interval-valued intuitionistic fuzzy numbers should work. Then, using these rules as a guide, we created a set of operators, such as linguistic interval-valued intuitionistic fuzzy Aczel-Alsina weighted averaging (LIVIFAAWA) operator, linguistic interval-valued intuitionistic fuzzy Aczel-Alsina weighted geometric (LIVIFAAWG) operator, linguistic interval-valued intuitionistic fuzzy Aczel-Alsina ordered weighted averaging (LIVIFAAOWA) operator, linguistic interval-valued intuitionistic fuzzy Aczel-Alsina ordered weighted geometric (LIVIFAAOWG) operator, linguistic interval-valued intuitionistic fuzzy Aczel-Alsina hybrid weighted averaging (LIVIFAAHWA) operator and linguistic interval-valued intuitionistic fuzzy Aczel-Alsina hybrid weighted geometric (LIVIFAAHWG) operators are created. Several desirable qualities of the newly created operators are thoroughly studied. Moreover, a multi-criteria group decision-making (MCGDM) method is proposed based on the developed operators. The proposed operators are then applied to real-world decision-making situations to demonstrate their applicability and validity to the reader. Finally, the suggested model is contrasted with the currently employed method of operation.


INTRODUCTION
The concepts of uncertainty and ambiguity hold pivotal roles in the realms of science and mathematics, igniting recent fervour in mathematical circles.Originating in the late 19th century, uncertainty emerged as a significant concern, particularly evident in problems involving multi-criteria decision-making (MCDM).Throughout the 19th century, researchers encountered myriad obstacles in grappling with such challenges.Addressing the need for precise numerical values in resolving MCDM quandaries, Zadeh (1965) embarked on a pioneering investigation into the realm of uncertainty within practical fieldwork.Introducing the innovative notion of a fuzzy set, Zadeh sought to confront and mitigate the inherent uncertainty in MCDM scenarios.Unlike traditional sets with clearcut boundaries, a fuzzy set embodies ambiguity, allowing for degrees of membership rather than binary acceptance or rejection.Due to fuzzy set theory, we now have the opportunity to communicate fuzzy concepts in meaningful ways using everyday language (Wang & Parkan, 2005;Zhang et al., 2024).
However, many realistic conditions pose limitations, including inadequate knowledge ambiguity, challenges in information collection, and more (Wu et al., 2018;Li et al., 2023;Chen et al., 2023).Determining characteristics with exact numbers presents challenges (Li et al., 2018).In later years, researchers built on the idea of a fuzzy set and named new types of fuzzy sets, such as type-1, type-2, the intuitionistic fuzzy number, the interval value intuitionistic fuzzy number, the hesitant fuzzy number (Zuo et al., 2023), and the Pythagorean hesitant fuzzy number (Zeng et al., 2024).Atanassov (1999) was the first to introduce the intuitionist fuzzy set (IFS).It is a very effective tool for dealing with information and is defined by a degree of membership or non-membership.Researchers from many fields have shown great interest in the IFS, and several developments have been made, including the IFS entropy measure (Burillo & Bustince, 1996), distance measure, and similarity measure (Hung & Yang, 2008).Xu & Yager (2006) suggested using the ordered weighted aggregation operator to group the IF information by giving each input value a weight based on its position in the list.Additionally, several authors have devised extra ways to deal with unclear and wrong information when there are two or more sources of uncertainty simultaneously (Tan & Chen, 2010;Xu & Xia, 2011;Wan & Yi, 2015).Atanassov & Gargov (1989) eventually generalised it as an interval-valued intuitionist fuzzy set (IVIFS).IVIFS uses intervals instead of a single integer for its membership (MD) and non-membership (NMD) functions.There are several uses and implications of the IVIFS, notably in multi-criteria decision analysis (MCDA).Many research studies have used decision-making models and methods in the IVIFS environment.These have included operational laws (Luo, Xu & Gou, 2018), score functions (Dymova, Sevastjanov & Tikhonenko, 2013), accuracy values (Nayagam, Muralikrishnan & Sivaraman, 2011), preference linkages between sets (Chen & Xu, 2019), and different ways of choosing (Kumar & Garg, 2018) that depend on the sets.Even though IFSs and IVIFSs have grown in significance and acceptance within MCDA, they can only use numbers to represent the rating levels' real numbers.
In contrast, practical decision-making problems may use other variables to denote the rating values, of which the most important types are linguistic variables (Shen et al., 2018;Gupta et al., 2018;Mao et al., 2019).Chen, Liu & Pei (2015) suggested linguistic IFS (LIFS) as a way to expand IFS.In LIFS, linguistic variables stand for MD and NMD.Other similar techniques supporting Because of such articulation, several researchers have applied the LIFSs to address decision-making problems (DMP).For instance, Li, Liu & Qin (2017) used LIFSs to define entropy and some new operational laws, and Zhang et al. (2017) used LIFSs to present a method of extended outranking for multi-criteria DMP(MCDMP); similarly, Garg & Kumar (2018), Peng, Wang & Cheng (2018), Liu, Liu & Merigó (2018), and Teng & Liu (2019) used LIFSs and developed some average operators.Menger (1942) made the expression of LIFS even better and created linguistic interval valued intuitionistic fuzzy sets (LIVIFS), where linguistic variable intervals show the MD and NMD values.Because of the LIVIFS environment, experts can use intervals of linguistic phrases rather than just one to convey their preferences.i.e., with the LIVIFSs, experts have more freedom.With these types of features, Menger (1942) defined some based on operators using LIVIFNs and solved MAGDM problems using these operators, which involve a weighted average operator (WA), ordered WA operator (OWA), a hybrid average operator (HA), a weighted geometric operator (WG), an ordered WG operator (OWG), a hybrid geometric operator (HG).Garg (2016) was the first to present the idea of triangular norms (t-NMs).Which was later advanced by many researchers, for example, Einstein's t-norm and t-conorm (Garg, 2017), Lukasiewicz's t-norm and t-conorm (Venkatesan & Sriram, 2019), Archimedean t-norm and t-conorm (Garg & Kumar, 2019) etc. Wei et al. (2018) presented the characteristics and related features of t-norms.
. Examine the behaviour and characteristics of the operators.
. Create an algorithm to address problems with MADM in the LIVIF environment.
. Present a new MADM technique that incorporates the LIVIFAAWA and LIVIFAAWG operators.
. Demonstrate that the suggested method is both legitimate and better.
The structure of the rest of this article is as follows: "Preliminaries" delves into the basics of LIVIFS and Aczel-Alsina operators, providing insights into their characteristics, definitions, and operating principles."Aczel-Alsina Operational Rules for Linguistic Interval-Valued Intuitionistic Fuzzy Numbers" introduces Aczel-Alsina operations in the context of LIVIFNs."LIVIF Aczel-Alsina Average Aggregation Operators" examines several proposed operators, including the LIVIFAAWA operator, the LIVIFAAOWA operator, and the LIVIFAAHA operator."LIVIF Aczel-Alsina Geometric Aggregation Operators" introduces the remaining consistent operators: LIVIFAAWG, LIVIFAAOWG, and LIVIFAAHG operators.In "MADM Approach Based With LIVIF Information", we explore a MADM model using the LIVIFAAWA operator."Numerical Example" employs an example to illustrate the suggested method."Investigation on the Impact of Parameter À on DM Outcomes" investigates the impact of parameters on decision-making outcomes.
In "Comparison Analysis", we compare the suggested approach with various other similar techniques that support the chosen method.Finally, "Conclusions" presents the conclusion.

PRELIMINARIES
In the following part, some basic ideas about LIVIFSs, operational laws for LIVIFNs and a review of the Aczel-Alsina t-norm and t-conorm are presented.

Linguistic interval valued intuitionistic fuzzy set
Definition 1 (Garg & Kumar, 2019) If V ¼ fv 1 ; v 2 ; . . .; v k g represent the universal set while Ŝ½0;q ¼ fŝ l j0 l qg be a linguistic term set.The LIVIFS Z is defined as: where the intervals ½ŝ demonstrate the MD and NMD values, respectively, and for each v 2 V; the pair ½ŝ Definition 2 For any v 2 V the extent of the linguistic indeterminacy of v to Z is denoted by ŝp v ð Þ and defined as : For the comparison of LIVIFNs, Garg & Kumar (2019) defined the score and accuracy functions, as below.
Likewise, the accuracy function of Z is represented by e H Z ð Þ and defined as follows: In Teng & Liu (2019), the writers created the concept of entropy for IVLIFN.We adopt the same idea for Linguistic interval-valued intuitionistic fuzzy numbers: Definition 5 (Wei et al., 2018) An entropy function N : LIVIFS V ð Þ !½0; 1 which satisfies the following axioms: and is defined as;

T-NM, T-CNM, Aczel-Alsina T-NM
The concept of t-NM was initially presented by Schweizer & Sklar (1960) as an extension of Peng, Wang & Cheng (2018) theories.In the case of probabilistic metric spaces, they established their approach to generalise the triangle inequality of metrics; however, their approach has been tested in various fields, including fuzzy set theory.Originally investigated in the setting of probabilistic metric spaces, the t-conorms (dual operations to t-NMs) was subsequently expanded to include fuzzy disjunctions (Schweiser, 1961).
Definition 6 (Aczél & Alsina, 1982;Schweizer & Sklar, 1960) To distinguish t-NMs from t-CNMs with clarity, T will be used as usual t-norms, and S will be used t-conorms.A t-norms is represented by the function S : ½0; is the strongest (largest) t-NM using the notation from Garg (2016), while drastic product T D is the smallest t-NM, which vanishes on ½0; Þand two traditional t-NMs that make up the product t-NM TP play a significant role in both theory and applications.See Aczél & Alsina (1982) for more information and thorough findings.We'll look at some of the most unique t-NMs in this article.Like rigorous t-NMs, these are produced by reducing the bijective additive generators and invariant to the t-NM t : ½0; The product t-NM T P ; extremal t-NMs T M and T D , commute with the power functions, satisfying the equivalence Aczél & Alsina (1982) reported tnorm solutions aforementioned in the early 1980s (Garg, 2017), implying that t w ; w 20; 1½ provided by A are defined as: The extremal t-NMs, which are strictly increasing and continuous in parameter, are added to create the Aczel-Alsina family T w A ; w 2 ½0; 1 of t-NMs.
Due to their dual nature, t-CNMs may be used to make similar statements and provide examples.S M ¼ max(dual to T M ) is lowest tconorm and drastic S D is the greatest, which is always 1 on 0; The T L has a dual t-CNM denoted by S L and defined as and T P has a dual t-CNM denoted by S P is given by S Additionally, additive generators (which are growing) may be used to produce continuous Archimedean t-CNMs.If S is dual to a continuous Archimedean t-NM T produced by an additive generator, then S is produced by an additive generator t supplied by are defined as: When the extremal t-CNMs are added, the Aczel-Alsina family S w A ; w 2 0; 1 ½ of t-CNMs is obtained.

ACZEL-ALSINA OPERATIONAL RULES FOR LINGUISTIC INTERVAL-VALUED INTUITIONISTIC FUZZY NUMBERS
Next, we will apply the Aczel-Alsina t-norm and t-conorm to define Aczel-Alsina operational laws for LIVIFNs.
Now, we define LIVIFAAOWA operator.
Theorem 10 (Commutativity property) Let Z j and Z j where j ¼ 1; 2; . . .; k, be two sets of LIVIFNs, such that for each j; Z j Z j then LIVIFAAOWA Z 1 ; Z 2 ; . . .
Using Aczel-Alsina operations under LIVIFNs information, the following theorem can be deduced.
where j ¼ 1; 2; . . .; k is a set of LIVIFNs.Accumulated value acquired by LIVIFAAHA operator is again a LIVIFN, i.e., LIVIFAAHA Z 1 ; Z 2 ; . . .;  Proof: This theorem can easily be proved in the same manner as Theorem 2 is.
Theorem 12 The LIVIFAAWA and LIVIFAAOWA are special types LIVIVIFAAHA operator. Proof:
Theorem 21 (Commutativity property) Let Z j and Z j such that Z j Z j for all j, then LIVIFAAOWG Z 1 ; Z 2 ; . . .

MADM APPROACH BASED WITH LIVIF INFORMATION
Therefore in part, we will use a MADM issue in a LIVIF environment to apply the suggested operators.To give decision choice, MADM techniques analyse tradeoffs of alternative exhibits across many aspects.Attribute values or performance metrics are the essential data for the MADM approach.A MADM issue is a task that entails selecting a workable compromise choice from among all feasible possibilities based on numerous qualities.

MADM model for LIVIF information based on proposed operators
Using the LIVIFAAWA and LIVIFAAWG operators, we construct an approach for determining the optimal alternative(s) for the MADM problem, which consist of different steps below: Step 1. Considering MADM LIVIFNs, first, develop the LIVIF decision table mÂn , and each entry E m regarding the attribute ĉn .
Step 2. We make use of the following equation to convert the LIVIF decision matrix where Z c ij denote the complement of Z ij : Step 3. Utilize the LIVIFAAWA/LIVIFAAWG operator to aggregate the individual linguistic interval valued intuitionistic fuzzy decision matrices presented by the k DMs. ;  ; Step 4.During this stage, the Method Based on the Removal Effects of Criteria (MEREC) (Keshavarz-Ghorabaee et al., 2021) is employed to determine the weighted vector of attributes.Subsequently, the weighted vector of alternatives is computed following the subsequent steps.
a: Assess the comprehensive performance of alternatives S i by employing the subsequent equation: where s ij denotes the score values of the Z ij .
b: Determine the performance of alternatives by systematically eliminating individual criteria.Let S ij represent the overall performance of i-th alternative concerning the removal of j-th criterion.The subsequent equation is employed to perform the calculations in this step: c: Calculate the sum of absolute deviations.In this stage, determine the removal effect of the j-th criterion using the values derived from a and b.Denote the effect of removing the jth criterion as E j .Utilize the subsequent formula to derive the values of E j : d. Establish the ultimate weights for the criteria.In this stage, compute the objective weight of each criterion by utilizing the removal effects (E j ) derived from c. Subsequently, represent the weight of the j-th criterion as w j .The ensuing equation is employed to calculate w j : Step 5.The overall accumulated value E i under various attributes ĉj is obtained by applying the LIVIFAAWA operator to the decision matrix e T q created in Step 2. ;   Step 7. All of the e S e E i i ¼ 1; 2; . . .; m ð Þscore values are ranked in descending order and the most the desirable choice is chosen.

NUMERICAL EXAMPLE
In this case study, we consider the issue of health facilities in the hospitals of Peshawar city of kpk Pakistan.Since there are several main hospitals in Peshawar city, including Khyber teaching hospital, LRH Hospital and Peshawar health complex etc.These hospitals face a shortage of basic medical equipment and a lack of doctors and paramedical staff during the COVID-19 pandemic.Now the government wants these hospitals to be ranked according to their healthcare facilities, so the hospitals lacking health facilities can be improved.To overcome this issue, the following approach may be followed.
Suppose there are three decision makers K ¼ fK 1 ; K 2 ; K 3 g; who are tasked to evaluate the four alternatives' core competencies using these four criteria.The LIVIFN can be used to denote the evaluated value of alternative and the decision-maker K t t ¼ 1; 2; 3 ð Þusing criteria ĉj where j ¼ 1; 2; 3; 4: The three experts K 1 ; K 2 and K 3 are allowed to use the linguistic term set and offer their choices for each alternative E i on each attribute ĉj .The preferences of the three experts are shown in Tables 2-4 are in the following order: Step 1.The linguistic interval valued decision matrices presented by the DMs are presented in Tables 2-4.
Step 2. In this problem we consider ĉ1 as the cost attribute, and the remaining all other attributes are benefit attributes, so we use Eq. ( 4) to transform the LIVIF decision matrices q and e T 3 ð Þ q (Tables 5-7).

LIVIFAAWG operator procedure
Step 3. We use the LIVIFAAWG operator with À ¼ 1 and obtain the aggregate matrix T q of the three normalised LIVIF decision matrices, as shown in the following Table 9.
Step 4. Utilizing (Eqs.( 5)-( 8)) we obtain the weighted vector of each criteria as:  We will rank the alternatives using numerous estimations to characterise working MADM outcomes.The ranking alternatives E i i ¼ 1; 2; . . .; 4 ð Þfor the values of À in the range of 1 À 50 by using the LIVIFAAWA operator are presented in Table 10.
From Table 10, it can be observed that the LIVIFAAWA operator gives the same best choice i.e., E 3 , for different values of k.

COMPARISON ANALYSIS
An evaluation of the proposed operators and a few current techniques is presented in this section, such as linguistic intuitionistic fuzzy density hybrid weighted averaging (LIFDHWA) p;q operator (Teng & Liu, 2019), LIVIFHWA operator (Peng, Wang & Cheng, 2018), LIVIFHWG operator (Peng, Wang & Cheng, 2018), LIVIFWA operator (Garg & Kumar, 2019), LIVIFWG operator (Garg & Kumar, 2019).The following Table presents the score values for the options depending on the suggested operators and current techniques.

CONCLUSIONS
In summary, this study has made significant contributions to the field of decision-making by introducing a novel set of operating principles employing Aczel-Alsina t-norm and t-conorm operators in the context of LIVIF scenarios.These principles have facilitated the development of various new aggregation operators, including the LIVIFAAWA, LIVIFAAOWA, and LIVIFAAHA operators, along with their specialized counterparts, the LIVIFAAWG, LIVIFAAOWG, and LIVIFAAHG operators.By considering these innovative operators, we have established a comprehensive approach to address MADM challenges within the framework of LIVIF.The practicality of this approach is aptly demonstrated through a mathematical example, underscoring its efficacy in maintaining the quality of decision-making processes.Furthermore, an examination of Table 11 reveals that, through the utilization of the linguistic intuitionistic fuzzy density hybrid weighted averaging (LIFDHWA) p;q operator (Teng & Liu, 2019), LIVIFHWA operator (Peng, Wang & Cheng, 2018), LIVIFHWG operator (Peng, Wang & Cheng, 2018), LIVIFWA operator (Garg & Kumar, 2019), LIVIFWG operator (Garg & Kumar, 2019), and our proposed LIVIFAAWA and LIVIFAAWG operators, the same best alternative, denoted as E 3 , is achieved.Consequently, it may be inferred that the linguistic intuitionistic fuzzy density hybrid weighted averaging (LIFDHWA) p;q operator, LIVIFHWA operator, LIVIFHWG operator, LIVIFWA operator, and LIVIFWG operators are specific instances of the suggested LIVIFAAWA and LIVIFAAWG operators.As a result, our recommended operators demonstrate greater generality and applicability compared to existing approaches.
In future research, we plan to integrate insights from recent studies to advance the application of LIVIF aggregation operators in decision-making processes.Specifically, we aim to incorporate methodologies proposed in recent articles such as pricing policy for a dynamic spectrum allocation scheme (Wu, Jin & Yue, 2022), PAL-BERT, an improved question answering model (Zheng et al., 2024), support vector machine parameter optimization algorithm (Li & Sun, 2020), adapting feature selection algorithms (Liu et al., 2023), an inclusiveness-degree based Signed Deffuant-Weisbush model (Jiang et al., 2024), and others to enhance the adaptability, robustness, and performance of our aggregation operators across various real-world scenarios.By integrating insights from diverse domains, we seek to contribute to the broader research agenda of developing methodologies that bridge theoretical advancements with practical applications in decision support systems.

Table 10
Ranking order of alternatives E i in the range of 1 À 50.Table11Comparison analysis with a few prevailing techniques.