Interval Valued Picture Fuzzy Aczel–Alsina Aggregation
					Operators and Their Application by Using the Multiattribute Decision Making
					Problem

Jabeen, Khalida; Ullah, Kifayat; Akram, Maria; Haleemzai, Izatmand

doi:https://doi.org/10.1155/2023/1707867

Journal of Mathematics

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 1707867 | https://doi.org/10.1155/2023/1707867

Interval Valued Picture Fuzzy Aczel–Alsina Aggregation Operators and Their Application by Using the Multiattribute Decision Making Problem

Khalida Jabeen,¹Kifayat Ullah,¹Maria Akram,¹and Izatmand Haleemzai²

Academic Editor: Ghous Ali

Received29 Aug 2022

Revised12 Dec 2022

Accepted23 Dec 2022

Published14 Mar 2023

Abstract

The engineering and construction sector is vital to a country’s economic growth, financial activities, and development. These sectors generate opportunities for the unemployed, unskilled, and skilled workforce. Recently, a lot of researchers worked on the Aczel–Alsina t-norm (TN) and t-conorm (TCN), which are generalizing many other t-norms and producing reliable results. In this article, first, we developed some new aggregation operators (AOs) and fundamental operational laws of Aczel–Alsina operations, including Aczel–Alsina product, sum, and scalar multiplication based on the IVPF information. Furthermore, we introduced an innovative AOs in the form of IVPF Aczel–Alsina weighted averaging (IVPFAAWA) operators with some basic characteristics. Moreover, we also generalized Aczel–Alsina operations in the form of the IVPF Aczel–Alsina weighted geometric (IVPFAAWG) operator. For the solution of daily life problems by utilizing a multiattribute decision-making (MADM) approach, we also established an application under the system of engineering and construction sectors. We illustrated a numerical example to find the suitable construction material for the engineering and construction sectors. To find the validity and flexibility of our proposed AOs, we also studied a comprehensive comparative analysis, in which we compared the results of exiting AOs with the results of our current invented approaches. At the end, we sum up our whole article in a single paragraph.

1. Introduction

Decision making (DM) is an efficient apparatus to overcome the problem of vague and uncertain data in everyday problems. The multiattribute decision‐making (MADM) strategy has gotten various considerations from various researchers. MADM strategy is one of the valuable and certified systems, which is used in day‐to‐day life. The justification behind its existence is to rank the restricted course of action of DM issues concerning the fact that they are so sensitive to decision-makers. To deal with MADM problems under uncertainty, the theory of fuzzy sets (FSs) was developed by Zadeh [1], where human opinion can be expressed in terms of the membership value (MV) on a scale of unit intervals, i.e., . Atanassov [2] extended the concept of FS in the form of intuitionistic FSs (IFSs), where two aspects of uncertain information are expressed by using a MV and nonmembership value (NMV) with a restriction that their sum lies in the interval . The extension of IFS in the framework of interval-valued IFS (IVIFS) was discovered by Atanassov and Gargov [3]. The IVIFSs carried two aspects of MV and NMV as lower and upper terms of both MV and NMV. Cuong [4] introduced the new concept of picture fuzzy sets (PFS) which is an extension of FS and IFSs that lies in four values, i.e., MV, abstinence value (AV), NMV, and refusal value (RV). In PFSs, the sum of MV, NMV, and AV lies in the interval . Wei [5] extended the theory of PFSs under similarity measures and their applications. Cuong et al. [6] developed the concepts of TN and TCN based on PFSs. Dutta [7] explored the system of medical diagnosis under the PFSs. Khalil et al. [8] enlarged the theory of PFSs in the environment of IVPFSs. We studied work on fuzzy sets in a different environment by several researchers [9–11].

A triangular norm plays an important role to aggregate statistical information. Firstly, Menger [12] worked on triangular norms in the system of statistical information. He also introduced different AOs of t-norm (TN) and its associative t-conorm (TCN) such as algebraic TN and TCN, Einstein t- norm and TCN, frank TN and TCN, Archimedean TN and TCN, nilpotent TN and TCN, Hamacher TN and TCN, Dombi TN and TCN, Lukasiewicz TN and TCN. We studied previously existing AOs such as AOs of IFWA by the Yu and Xu [13], AOs of IFWG by the Xu and Yager [14], PF Maclaurin symmetric mean operators by Ullah [15], interval-valued Pythagorean fuzzy AOs by Garg [16], PF Dombi AOs by Jana et al. [17], AOs of T-spherical fuzzy Dombi weighted (TSFDW) averaging, and IVTSFDW geometric by the Ullah et al. [18]. Lin et al. [19] developed an interactional partitioned Heronian mean AOs based on PFSs. Riaz and Farid [20] provided a list of new AOs for the selection of logistics third parties under the system of PFSs. Mahmood [21] introduced innovative concepts of bipolar fuzzy sets in the form of bipolar complex fuzzy Maclaurin symmetric mean operators. Mahmood et al. [22] also discussed a series of new AOs of a complex PF soft set based on the theory of power AOs to solve real-life problems based under the system of MADM technique. Zhou et al. [23] explored the interactive recursive feature to promote and improve the structure of the network under the system of detection salient. Kovalnogov et al. [24] developed some new tools to solve initial value problems based on the Runge–Kutta pair technique and a variety of scalar autonomous issues. Ye et al. [25] proposed some new models of state damping method under the system of sliding mode control and adaptive control. Liu et al. [26] provided some new detection approaches to evaluate the reliability and effectiveness of the thermal flaws under the application of the vehicle cables. Zheng et al. [27] generalized theory of graphs in the framework of knowledge based embedding models to verify experimental information. Zheng and Jin [28] evaluated the performance of the offloading strategy under the stochastic models. Khalaf [29] established a few algorithms and optimal options for the selection of suitable power plants to improve the infrastructure of the energy system. The aggregation operators (AOs) play an important role to investigate fuzzy information in MADM problems. Some Bonferroni mean AOs of interval-valued TSFSs are developed by Akram et al. [30]. We also studied the previous research work given in [31, 32].

Klement et al. [33] generalized the idea of the triangular norm and its basic properties. Klement and Navara [34] introduced the characterization of Lukasiewicz TN and TCN by utilizing the binary operations of TN and TCNs. Wang [35] explored the theory of TN and TCN in the form of Drastic TN and TCN. Fodor [36] enlarged the idea of triangular norms in the frame of Nilpotent minimum TN and TCN. Gál et al. [37] developed the theory of TN and TCN in Hamacher product TN and TCN. Xu and Zhang [38] approached the concepts of Dombi TN and TCN in the environment of the q-rung orthopair fuzzy system. Navara [39] introduced nearly frank TN and TCN by using the basic operations of TN and TCN and discussed its properties. A lot of researchers utilized the idea of TN and TCN in the field of research given in [40–42].

In 1982, Aczél and Alsina (AA) [43] discovered new operations, namely, Aczel–Alsina t-norm (AA-TN) and AA-TCN with basic properties. Richman and Walker [44] examined the families of AA-TN and AA-TCN. The characteristics of AA-TN and AA-TCN are to analyze the information. Farahbod and Eftekhari [45] examined the best ones is AA-TN and AA-TCN by evaluating the family of TN and TCN. Senapati et al. [46] discovered new AOs of IFS AA by utilizing some basic operations of AA-TN and AA-TCN. Hussain et al. [47] enlarged the concepts of AA IFSs in the environment of PyFSs. Senapati [48] explored the concepts of PFSs and presented some new AOs with the help of Aczel–Alsina operations. Senapati et al. [49] extended the concepts of IFS AA in the environment of IVIFAA. He gave an application to check the effectiveness and flexibility of discussed AOs.

When the abovementioned existing AOs are utilized to handle the uncertain and vague information, sometimes a piece of information is lost during the investigation process. The purpose of this article is to cope with ambiguous and unreliable information by utilizing the operations of AA-TN and AA-TCN based on PFSs. We proposed some new AOs such as IVPFAAWA and IVPFAAWG. A MADM technique was used to select suitable concrete for the construction of a building. We established an application with the help of a numerical example to find the reliability and effectiveness of current proposed AOs. In the comparative analysis, we compared existing AOs with current approaches. In the end, we summarized this article.

This article is organized in the form: In Section 1, the author recalls the previous history to understand the background of this article. In Section 2, the authors define the notions of AA-TN and AA-TCN, PFS, IVIFS, IVPFS, and some basic operations. In Section 3, the author utilizes the basic operations of AA-TN and AA-TCN in the environment of PFNS. In Section 4, the author presents some AOs in the form IVPFAAWA and IVPFAAOWA and their properties in the form of Idempotency, boundedness, and monotonicity. In Section 5, the author elaborates the AOs of IVPFAAWA and IVPFAAWG with the help of AA-TN and AA-TCN. The author gave an application with the help of a numerical example to check the effectiveness and flexibility of proposed AOs.

2. Preliminaries

In this section, we recall some basic definitions of IVIFSs and IVPFSs. We also discussed the fundamental operations of AA-TN and AA-TCN.

Now we discuss the notion of IVIFS developed by Atanassov and Gargov [3].

Definition 1. see ([3]). An IVIFS on the universe is defined as follows:where are subintervals of the interval denoting MV and NMV of provided that and is called the hesitancy value (HV) of . Further, is known as an interval-valued intuitionistic fuzzy value (IVIFV).
In the following definition, we recall the concepts of IVPFS discovered by Cuong and Kreinovich [10].

Definition 2. see ([10]). An IVPFS on the universe is defined as follows:where are closed subintervals of the interval denoting the MV, abstinence value (AV), and NMV of respectively, provided that and is called the refusal value (RV) of . Further, is known as IVPF value (IVPFV).
Now, we discuss some basic operations of IVPFVs in Definition 3.

Definition 3. (see [50]). Let us consider three IVPFVs ,, and . Then(1) iff and .(2) iff and(3)(4)(5)In the following definitions, the idea of TN and TCN are discussed in the form:

Definition 4. (see [51]). A TN is a function that satisfies the below properties, i.e., :(1)(2)(3)(4)

Definition 5. (see [52]). A TCN is a function that satisfies the below properties, i.e., :(1)(2)(3)(4)The theory of AA-TN and AA-TCN is developed by Aczél and Alsina [43]. The AA-TN and TCN are more flexible and diverse in nature and generalizes many other TNs and TCNs. Farahbod and Eftekhari [45] investigated the application of several TNs and TCNs in a classification problem and concluded the AA-TN and AA-TCN as more appropriate. The AA-TN and AA-TCN are defined as follows:

Definition 6. (see [43]). For , the AA-TN and AA-TCN are defined, respectively, as follows:The concept of the score function of IVPFV was introduced by [50].

Definition 7. (see [50]). Suppose be the IVPFV. Then the score function of is given by

3. Aczel–Alsina (AA) Operations of IVPFVs

In this section, we will discuss the AA operations including product, sum, scalar multiplications, and power operations based on AA-TN and AA-TCN for IVPFVs. We also study their basic properties.

Suppose represents the AA-TN and represents the AA-TCN. Then the AA product represents the generalized intersection over two IVPFVs and the AA sum represents the generalized union over two IVPFVs is written as follows:

Definition 8. Suppose three IVPFVs, , and , and Then(1)(2)(3)(4)To support the above-discussed operations and to see their working, we present the following Example.

Example 1. Suppose three IVPFVs , , and , and by applying Definition 9, we have:(1)(2)(3)(4)Now, we have to prove some fundamental operations of AA sum and product under the system of IVPFVs by utilizing Definition 8.

Theorem 1. Suppose three IVPFVs , and , and . Then(1)(2)(3)(4)(5)(6)

Proof. Straightforward.

4. Interval Valued Picture Fuzzy Aczel–Alsina Averaging Aggregation Operators

In this section, we aim to introduce some AOs based on Definition 8. We introduce the notion of the IVPFAAWA operators and discuss the scenario where the ordered position of the alternatives does matter by introducing the IVPFAAOWA operator. From this section onward, by , we denote the weight vector of attributes with for .

Definition 9. Suppose be the set of IVPFVs. Then the IVPFAAWA operator is a function defined as follows:By using some basic fundamental operations defined in Definition 8, we proposed a result for the IVPFAAWA operator in the following Theorem 2.

Theorem 2. Suppose be the set of IVPFVs. Then, the aggregated value of the IVPFAAWA operator is also an IVPFV given by

Proof. Suppose be the set of IVPFVs, and we prove above theorem by using the mathematical induction technique.
For we haveNow, from the Definition 8 and Definition 9, we get the following equation:Hence, we have shown that equation satisfy all the condition for .
Suppose it is also true for we haveNow, we have to show that for then we obtain the following equation:Hence, proved.

Theorem 3 (Idempotency). If all are identical, i.e., . Then .

Proof. We know that , then we haveTherefore, .

Theorem 4. (Boundedness). Suppose be the set of IVPFVs. Suppose and Then

Proof. Suppose be the set of IVPFVs. Suppose and ThenTherefore, .

Theorem 5 (Monotonicity). Suppose be the set of IVPFVs. If . Then

Proof. Straightforward.

Now we discuss IVPFAA ordered weighted averaging (IVPFAAOWA) AO by using some basic operations defined in Definition 8 where we keep the ordered position of the alternatives in the aggregation process.

Definition 10. Suppose be the set of IVPFVs. The IVPFAAOWA is a function defined as:where be the permutation of such that
Now, we establish a relation for using the IVPFAAOWA operator on IVPFVs that depend upon AA operations.

Theorem 6. Suppose be the set of IVPFVs. The IVPFAAOWA is a function then:

The IVPFAAOWA operator satisfies the properties discussed in Theorems 3, 4, and 5 analogously.

Clearly, from Definitions 10 and 11, the IVPFAAWA operator and IVPFAAOWA operator weights only the IVPFVs and the ordered positions of IVPFVs respectively. By merging these two we get the IVPF AA hybrid averaging (IVPFAAHA) operator which is discussed as follows:

Definition 11. Suppose be the set of IVPFVs. Then IVPFAAHA operator of dimension is a function defined as:where be the weight vector of PFNs, with and with as a balancing coefficient.
Now, we can derive the theorem for the IVPFAAHA operator as follows.

Theorem 7. Suppose be the set of IVPFVs. Then, the aggregated value of the IVPFAAHA operator is also an IVPFV, and

Proof. This proof is similar to Theorem 2.

Now, we discuss some special cases of IVPFAAHA operators as follows.

Theorem 8. The IVPFAAWA and IVPFAAOWA operators are special cases of the IVPFAAHA operator.

Proof. Suppose , Then:Suppose . Then andThis completes the proof.

5. Interval Valued Picture Fuzzy Aczel–Alsina Geometric Aggregation Operators

In this section, we aim to introduce some AOs based on Definition 8. We introduce the notion of the IVPFAAWG, IVPFAAOWG, and IVPFAAHG operators.

Definition 12. Suppose be the set of IVPFVs. Then the IVPFAAWG operator of dimension is a function defined as follows:By using some basic fundamental operations defined in Definition 8 and Definition 12, we proposed a result for the IVPFAAWG operator in the following Theorem 14.

Theorem 9. Suppose be the set of IVPFVs. Then the aggregated value of the IVPFAAWG operator is also an IVPFV given by

Proof. Suppose be the set of IVPFVs, we prove this theorem by using mathematical induction technique.
For we haveNow, from the Definition 8 and Definition 12,Hence, it is true for
Suppose it is also true for we haveNow, we have to show that for the result holds:Hence proved.
Now we prove that the IVPFAAWG operator satisfies the basic properties of an aggregation function in Theorem 10–12.

Theorem 10 (Idempotency). If all are identical i.e. . Then .

Proof. Straightforward.

Theorem 11. (Boundedness). Suppose be the set of IVPFVs. Suppose and Then

Proof. Straightforward.

Theorem 12. (Monotonicity):
Suppose be the set of PFNs. If . Then

Proof. Straightforward.
Now, we discuss IVPF AA ordered weighted geometric (IVPFAAOWG) AO by using some basic operations defined in Definition 9 where we keep the ordered position of the alternatives in the aggregation process.

Definition 13. Suppose be the set of IVPFVs. The IVPFAAOWG is a function defined as follows:where be the permutation of such that
Now we establish a relation for using the IVPFAAOWG operator on IVPFVs that depend upon AA operations.

Theorem 13. Suppose be the set of IVPFVs. The IVPFAAOWG is a function then

The IVPFAAOWG operator satisfies the properties discussed in Theorem 15, 16, and 17 analogously.

Clearly, from the Definitions 12 and 13, the IVPFAAWG operator and IVPFAAOWG operator weights only the IVPFVs and the ordered positions of IVPFVs, respectively. By merging these two we get the IVPF AA hybrid geometric (IVPFAAHG) operator which is discussed as follows.

Definition 14. Suppose be the set of IVPFVs. Then IVPFAAHA operator is a function defined as follows:where be the weight vector of IVPFVs, with and with as the balancing coefficient.
Now, we can derive the theorem for the IVPFAAHG operator as follows.

Theorem 14. Suppose be the set of IVPFVs. Then the aggregated value of the IVPFAAHG operator is also an IVPFV, and

Proof. We can prove this theorem easily by using the steps of Theorem 2.

6. MADM Technique Based on Aczel–Alsina AOs

In this section, we shall discuss the method of MADM problems by using the IVPFAAWA and IVPFAAWG operators and apply them to a practical problem.

Suppose the alternatives to be ranked based on attributes and let be the weight vector as discussed earlier. Here, the uncertain information is discussed in the form of IVPFVs where the four aspects of the uncertain information are elaborated by MV, AV, NMV, and RV in a decision matrix given by the decision-makers. Each entry of the information of is denoted by PFNs that represents the alternatives’ evaluation values for criteria such that are the subintervals of and and denote the indexing terms.

The detailed steps of the algorithm based on IVPFAAWA and IVPFAAWG operators are given as follows: Step 1: This step involves obtaining information from experts about the alternatives based on IVPFVs in the form of a decision matrix as discussed earlier. Step 2: We get the normalization matrix of the decision matrix by the transformation . Where be the complement of the decision matrix . If all the attributes are identical (in nature) then it is not necessary to change them but when the attributes are distinct (in nature) then it’s necessary to make them identical i.e., to make all of them benefit attributes. Step 3: This step involves using the IVPFAAWA and IVPFAAWG operators discussed in Definition 9 and Definition 12, respectively, to aggregate the information obtained in the previous step. using the score formula defined in Definition 7. Step 5: To evaluate the most desirable alternative by using the method of ranking and ordering technique of obtained score values by our invented methodologies.

6.1. Applications in Construction Engineering

Engineering and construction are comprehensive businesses that play a vital role growth of the economy of a country. It provides millions of jobs opportunity and about a 15% growth rate in the GDP. Engineering assumes an important part in guaranteeing the development and improvement of a country’s economy as well as in working on the personal satisfaction of citizens inside the country. Its dispatch is far more extensive than just structures and bridges, crossing enhancements in sustainable power innovations and arrangements through to worldwide health challenges. Accordingly, there is a significant connection between a nation’s design limit and its monetary turn of events. In this part of the report, we set everything up at a wide macroeconomic level, featuring the manners by which designing can add to the financial turn of events before considering the particular markers that add to the Engineering Index in the remaining areas.

Engineering and construction are known as the backbone of economic growth, they play a major role to uplift economic development, financial development, and economic activities. The engineering and economic sector considered a big partner of the Development and construction industry assumes a significant part in the monetary inspire and improvement of the country. It very well may be viewed as an instrument of producing the business and extending open positions to a huge number of the unskilled, semiskilled, and skilled workforce. It likewise assumes a key part in creating pay in both formal and informal areas. It supplements the unfamiliar trade profit got from the exchange of development material and designing administrations.

6.2. Numerical Example

Several types of construction or building material are utilized to contract a building. At the time when we select construction materials, we have to ensure the safety and long life of a building. In this numerical example, we have to choose suitable construction material such as concrete from four different types of concrete on the based following characteristics .(1) represents the size and shape of concrete(2) about the quality of concrete (strength and hardness)(3) represents the affordable cost of concrete

We have to select the best concrete material according to the weight vectors of attributes are . These weight vectors are distributed by the decision-maker. The set of information in the form of IVPFVs is depicted in Table 1.

6.3. The Method Based on IVPFAAWA and IVPFAAWG Operators

We utilize the IVPFAAWA operator to build up MADM theory with IVPF data for the selection of the most desirable person which is discussed as follows: Step 1. Suppose that for . We calculate the alternative values of four applicants by using the IVPFAAWA operator and we have Step 2. Now we calculate the score values of IVPFVs. . Step 3. Ranking and order of four applicants corresponding to the score values of the IVPFVs. Step 4. The most desirable choice is .

In Table 2, we show the consequences of the alternatives by utilizing the AOs of IVPFAAWA and IVPFAAWG. We observe that the score values of the alternatives gradually increase and decrease if the magnitude of is increased. It means that the choice maker can decide the most suitable value according to their preferences.

In Figure 1, we show the score values of IVPFAAWA and IVPFAAWG operators as a graphical representation, which are depicted in Table 2.

6.4. The Impact of the Parameter ᵿ in This Technique

To classify the influence of the different sizes of the boundary , we see advantages of unmistakable boundary inside our referenced procedures to arrange the other options. The requested impacts of the other options in light of IVPFAAWA administrators are introduced in Table 3, and outlined graphically in Figure 2. It is obvious that when the greatness of ℵ increments for IVPFAAWA administrator, the score upsides of the choices increment step by step, however, the comparing requests is something very similar and it is . By seeing the similarities of the ranking and ordering of the score values of the different values of , we observe that this technique show the property of isotonicity, the decision‐makers can choose the reasonable worth as indicated by their inclinations.

In above Table 3, we observe that, when we take then the order of score values . If we take then upcoming all the values of the score function will remain the same. It means that is the stability point. It is also a highly observable thing when we take as even then there will be no aggregation result found. So, we must take as an odd number.

Figure 2 depicts the consequences of the IVPFAAWA operator when the magnitude value of is variate. From Figure 2, we analyze that after the consequences of proposed AOs gradually decrease.

In Table 4, we see that when we change the values of then the order of score values will be changed. When we take then remaining all the values of the score function will remain stable. It means that a stability point exists when . When we take the value of as an even number then the values of the score function do not exist.

Figure 3 shows a graphical representation of the score values which are obtained by the IVPFAAWG operator for the different values of the magnitude of .

7. Comparative Analysis

In this section, to observe the validity and reliability of our discussed AOs in the form of PFAAWA and PFAAWG operators. For this purpose we compare the experimental results of our proposed AOs with existing AOs given in [11, 17, 53, 54]. We applied all existing AOs on the information given by the decision-maker depicted in Table 1, AOs of PFDWA and PFDWG by Jana et al. [17] and IVIFWA and IVIFWG by Xu and Gou [54], are failed to aggregate the given information. AOs of IVPHFWA and IVPHFWG by Kamacı et al. [11], and AOs of IVPFFWA and IVPFFWG by Mahmood et al. [53] are also applied on the information given in Table 1. The consequences of current work and existing AOs are shown in Table 5.

The advantages of IVPFAOs over the existing AOs are as follows.

The proposed AOs can eliminate the impact of useless data on the decision results. The discussed AOs are very useful operators to calculate the wide range for assigning MV, AV, and NMV. Further, we applied AOs discovered by Kamacı et al. [11] and Mahmood et al. [53]. The consequences of the AOs discussed in [11, 43] are similar but our proposed AOs gave more suitable results than existing AOs.

The following graphical representation shows the consequences of existing methodologies with proposed methodologies in Figure 4.

8. Conclusion

In this article, we enhanced the idea of IVPFS in the framework of IVPFAA AOs by utilizing the basic operations of AA-TN and AA-TCN. We have to investigate new aggregation operators just as the IVPFAAWA operator, IVPFAAOWA operator, IVPFAAHA operator, IVPFAAWG operator, IVPFFAAOWG operator, and IVPFAAHG operator. We proved that the IVPFAAWA and IVPFAAWG operators satisfy the basic aggregation properties including idempotency, boundedness, and monotonicity. One of the most widely discussed applications of AOs is MADM, and we studied the application of the IVPAAWA and IVPFAAWG operator to analyze the selection of construction material, where the information is based on IVPFVs. We also studied the impact of variable parameters associated with IVPFAAWA and IVPFAAWG operators numerically. To see the reliability of our proposed AOs, we developed a comparative analysis of our proposed work with the existing work. In the near future, we aim to extend our proposed work in the environment of bipolar-valued hesitant fuzzy sets [55, 56]. Moreover, we will also extend our proposed techniques in the environment of complex q-rung orthopair graphs and spherical fuzzy sets [57–59].

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors are also thankful to the office of research, innovation, and commercialization (ORIC) of Riphah International University for supporting this study under the project: Riphah-ORIC-21-22/FEAS-20.

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Copyright © 2023 Khalida Jabeen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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