In this manuscript, the notion of a hesitant fuzzy soft fixed point is introduced. Using this notion and the concept of Suzuki-type (μ,ν)-weak contraction for hesitant fuzzy soft set valued-mapping, some fixed point results are established in the framework of metric spaces. Based on the presented work, some examples reflecting decision-making problems related to real life are also solved. The suggested method’s flexibility and efficacy compared to conventional techniques are demonstrated in decision-making situations involving uncertainty, such as choosing the best options in multi-criteria settings. We noted that the presented work combines and generalizes two major concepts, the idea of soft sets and hesitant fuzzy set-valued mapping from the existing literature.
Hesitant fuzzy soft set valued mappingSuzuki-type (μ, ν)-weak contractionfixed pointdecision making problemNational Science, Research and Innovation Fund (NSRF)King Mongkut’s University of Technology North Bangkok with ContractKMUTNB-FF-68-B-46Introduction
This part consists of two subsections. In the first section relevant literature is presented while the second part is about the motivation and research gap.
Literature Review
The real world is full of complex issues. Engineering, social science, management science, and economics are a few of the major real-world fields. Imprecision, vagueness, and uncertainty are the main issues we are dealing with in the aforementioned fields. These problems have a variety of uncertainties, which makes it impossible for us to solve them using traditional methods. The interval mathematics theory, theory of probability, fuzzy set theory, and rough sets theory are just a few of the numerous theories. To improve uncertainty modeling and provide more flexibility, Saeed et al. [1] provide refined fuzzy soft sets that offer a comprehensive approach to analyzing data while addressing the challenges of uncertainty and ambiguity in real-world data. A better representation of ambiguous and parameter-dependent data is possible with this hybrid approach, which combines the best features of both techniques. The concept of the fuzzy soft set was applied to the decision-making problems by different researchers (see, e.g., references [2–4] and the cited works within). To improve decision-making in situations when there is uncertainty among several membership values, Ali et al. [5] expanded the hesitant fuzzy soft set architecture. This method provides a deeper understanding of uncertainty, which offers a full range of information measures for hesitant fuzzy soft sets. It is more adaptable than earlier models and may be used for intricate and unpredictable problems. It develops decision-making scenarios by offering a more precise and flexible framework for dealing with these difficulties. Similarly, fuzzy and hesitant fuzzy concepts [6] manipulate data uncertainty modeling in a descent manner as well as a powerful tool for solving multi-criteria decision-making challenges raised in machine learning problems [7]. Recent developments in fuzzy and hesitant fuzzy sets have greatly enhanced frameworks for making decisions. For instance, an improved hesitant fuzzy model for multi-criteria decision-making was presented by Shen et al. [8]. More recently, Ali et al. [9] suggested a hybrid approach for election prediction scenarios that combine soft expert sets with dual hesitant fuzzy sets. Nevertheless, in spite of these developments, Suzuki-type (μ,ν)-weak contractions have not yet been integrated with hesitant fuzzy soft set-valued mappings, especially for scenarios with substantial ambiguity and hesitation. The purpose of this study is to close this gap.
Now, such scenarios can be manipulated more appropriately by the representation of mappings. So researchers are trying to model real-world complex issues using multi-valued mappings. So for this purpose, fuzzy and soft set valued mappings are employed to manage and arrive at decisions in scenarios including ambiguity, imprecision, or uncertainty. The main aim of the soft set-valued map is to assign items to sets rather than single values to address data uncertainty. Similarly fuzzy soft set valued mappings are used for managing even more complicated uncertainty. So, a family of fuzzy mapping that takes a broad view of the idea of set-valued mappings was introduced in 1981 by Heilpern [10] using the concept of fuzzy sets. During this procedure, he revealed the presence of the fuzzy fixed point theorem, advancing over Nadler’s fixed point theorem [11] and the Banach contraction theorem [12] for multi-valued mapping. Frigon et al. [13] enhanced the Heilpern fixed point theorem for the case in which it should be under observation that 1-level sets are not supposed to be compact and convex beneath a contractive reservation. Later, on metric and quasi-metric spaces, several researchers investigated the presence of fixed and joint fixed points of fuzzy mapping sustaining different kinds of contractive constraints (see, e.g., references [14–17] and the cited works within).
Motivation
Existing fuzzy set frameworks are inadequate for solving many decision-making problems that involve hesitation and ambiguity. Though their potential in conjunction with sophisticated contraction principles is still unexplored, hesitant fuzzy soft sets provide a novel method for modeling such issues. Finding fixed-point outcomes for hesitant fuzzy soft set-valued mappings using Suzuki-type (μ,ν)-weak contractions is the goal of this work. The goals are to produce theoretical results, show how they might be used, and compare them to current approaches. Suzuki-type weak contractions characterize metric completeness and generalize the Banach contractions [18]. Suzuki-type contractions can handle multi-valued mapping problems adequately [19–21]. For this reason, Suzuki-type weak contraction is chosen for this study. Because it considers the inherent uncertainties and hesitation in real-world data, the Suzuki-type (μ,ν)-weak contraction is especially well-suited for hesitant fuzzy soft set-valued mappings. Through its parameters μ and ν, it provides more flexibility than classical contraction principles, allowing for more precise control over the mapping process. This guarantees convergence to fixed points under more general circumstances and enables a more accurate depiction of intricate interactions. It is the perfect option for applications needing sophisticated decision-making frameworks because of these features [22]. Because it considers the two problems of uncertainty and hesitation that arise in real-world decision-making, the Suzuki-type (μ,ν)-weak contraction is especially well-suited for hesitant fuzzy soft set-valued mappings. The dual-parameter structure of Suzuki-type contractions provides for more precise control over mapping behavior than standard fuzzy set generalizations, which might not be able to capture multi-valued dependencies. Because of this, it is perfect for resolving difficult decision-making issues where conventional approaches are inadequate.
Research Gaps
The integration of hesitant fuzzy soft sets with sophisticated fixed-point theorems, such as Suzuki-type (μ,ν)-Weak Contractions, has not been discovered, even though they have been thoroughly examined in the context of uncertainty modeling. Current approaches, such as classical contraction principles, can not ensure convergence under more general circumstances and have limitations when dealing with multi-valued membership structures. Furthermore, although hesitant fuzzy sets have a wealth of research on their usage in decision-making, nothing is known about how they might be used with dual-parameter contraction techniques to address practical issues. By creating a thorough framework for hesitant fuzzy soft set-valued mappings and showing its usefulness in real-world scenarios, this work seeks to close these gaps. So, in this work, we study fixed point via Suzuki-type (μ,ν)-weak contraction under a new framework of study which is a hesitant fuzzy soft set-valued mapping. We extend the concept of Suzuki-type (μ,ν)-weak contraction in the context of hesitant fuzzy soft set-valued mapping and explore sufficient conditions for establishing a hesitant fuzzy soft fixed point for the mentioned contractions, inspired by the work of Shagari et al. [23]. In addition to expanding the scope of contraction principles, the suggested method tackles real-world decision-making difficulties, which advances fuzzy mathematics and its purposes. The established work generalizes the related notions of fuzzy sets, soft sets, and multi-valued maps within the framework of metric spaces. Furthermore, a decision-making problem is also addressed using the presented results. With, the help of a hesitant fuzzy soft fixed point we find out an optimal result in the example.
Proposed Decision-Making Application
A real-world decision-making problem involving multi-criteria group decision-making under uncertainty is used in this study to test the suggested framework. Hesitancy and competing preferences are considered when modeling the situation using hesitant fuzzy soft set-valued mappings. Suzuki-type (μ,ν)-weak contractions permit dual-parameter control, which allows for iterative solution refining and optimal decision-making. This method can be used, for instance, to choose the best option from a range of selections or allocate resources while considering preference hesitancy.
Overview of the Methodology
Following the definition of hesitant fuzzy soft set-valued mappings, Suzuki-type (μ,ν)-weak contraction principles are applied to construct fixed-point outcomes and conclude the approach. The hesitant fuzzy values are refined by an iterative process that guarantees convergence to a fixed point representing the best course of action. In order to validate the theoretical results and show how resilient the approach is, numerical examples and a real-world application are given. This research manuscript is split into four parts. In the first part, literature review and motivation are given, the second section consists of necessary definitions, the third section consists of our main work, and in the fourth section, applications of the presented results are discussed.
Preliminaries
In this part of the paper, we need some results and definitions from the existing literature. Throughout this manuscript, we represent a metric function by “D”, W be a non-empty set and the set of all hesitant fuzzy soft sets by H~(W)E.
Definition 1.[23] LetWbe a non-empty set. A mappingLˇ: W⟶Wis known as weak contraction if∃, a constantζ∈(0,1)andK≥0such thatD(Lˇw,Lˇb)≤ζD(w,b)+KD(b,Lˇw);∀,b,w∈W.
Definition 2.[23] Assume that (W, D) be a metric space andLˇ: W⟶CB(W) is a multi-valued mapping. ThenLˇis known as multi-valued weak contraction or multi-valued (ϑ, K)-weak contraction iff∃, constantsϑ∈(0, 1) andK≥0 such thatH(Lˇw,Lˇb)≤ϑD(w,b)+KD(b,Lˇw);∀,w,b∈W.
Definition 3.[22] A couple (G, E) be known as a soft set on (W) iffGis a function fromEinto theP(W).
Definition 4.[5] Assume that a hesitant fuzzy setBonWbe in the form of a mapping overW, gives us a subset of [0, 1], and mathematically be determined byB={⟨w,hB~(w)⟩∣w∈W}.
In the above set, hB~(w) represents the possible values of membership degrees in [0, 1] of ‘w’∈WtoB. For simplicity, hB~(w) is called a hesitant fuzzy element. So, all hesitant fuzzy elements are represented by a setH.
Definition 5.[23] Assume thatℏis a hesitant fuzzy element inB. Then its score function is determined bySc(ℏ)=(1N(ℏ))∑ς∈ℏς.
So, N(ℏ) shows number of objects in ℏ and actually Sc(ℏ) represents the average of the objects in ℏ and Sc(ℏ)∈[0,1].
Definition 6.[23] Given thatμ∈(0,1], then ordinary crisp subset ofWdetermined asℏμA={t∈W:Sc(ℏA(t))≥μ},
be known asμ-cut of the hesitant fuzzy softAonW.
Definition 7.[5] SupposeH~(W)is the power set of hesitant fuzzy set inW; a pair(G~,A)is said to be a hesitant fuzzy soft set overW, whereG~is a mapping given byG~:A→H~(W).
In this part of our manuscript we present some definitions and the main results. Now motivated by the Definition 6 we can defined μ-cut of the hesitant fuzzy soft set (G~,A) as follows:
Definition 8.Givenμ∈(0, 1] then the ordinary subset means crisp subset ofWcan be presented byℏμG~e={w∈W:Sc(hG~e(w))≥μ},
is known as theμ-cut of the hesitant fuzzy soft setG~onW.
Definition 9.The mappingLˇ:W→H~(W)Eis called hesitant fuzzy soft set valued mapping.
Remark 1.The above definition is the extended version of the hesitant fuzzy set-valued mapping of the paper [22]. Where, note that ifLˇis a hesitant fuzzy soft set valued mapping onW, thenLˇw: E→H~(W)is a hesitant fuzzy soft set, i.e., for each w∈Wand (Lˇw)(ë) = Lˇ(w;e)∈H~(W)is a hesitant fuzzy soft, so for each w∈W, Lˇ(w;e)(w)⊆[0,1]is a hesitant fuzzy element (HFE).
Definition 10.Assume thatLˇ:W→H~(W)Eis a hesitant fuzzy soft set valued mapping, w∈W, e∈DomLˇw andμ∈(0, 1]. So, theμ-level set ofLˇis represented by[Lˇ(w;e)]μ={r∈W:S([Lˇ(w;e)](r)≥μ}.
Remark 2.Definition 10 is an extended form ofμ-level set of hesitant fuzzy set valued mapping [22]. Here for w∈W, Lˇw H~(W)E, that is, Lˇw:E⟶H~(W). This represents that(Lˇw)(e)=Lˇ(w;e)∈H~(W). This implies thatLˇ(w;e):W→P(I), henceLˇ(w;e)(r)⊆I. So that forμ∈I, [Lˇ(w;e)]μis well-given.
Here I = [0, 1], P(I) represents collection of all subsets of I = [0, 1].
Definition 11.A point w is said to be hesitant fuzzy soft-fixed point ofLˇ:W⟶H~(W)E, if∃, e∈Eandμ∈(0, 1] for which w∈[Lˇ(w;e)]μ. The point w is known as a common hesitant fuzzy soft-fixed point ofSˇ, Lˇ:W⟶H~(W)Eif w∈[Sˇ(w;e)]μ∩[Lˇ(w;e)]μ.
Remark 3.The Definition 11 is an enhanced form of hesitant fuzzy fixed point of the paper [22].
Definition 12.A mappingLˇ:W→H~(W)Eis said to be a hesitant fuzzy soft set valued mapping. A pointw∈Wis known as an e-hesitant fuzzy soft-fixed point ofLˇifw∈[Lˇ(w;e)]μ, for somee∈Eandμ∈(0,1].It can be written asw∈[Lˇw]μ, for short. IfDom(Lˇw)=E, andw∈[Lˇ(u;e)]μ,∀,e∈E, then w is known as anE-hesitant fuzzy soft fixed point ofLˇ.
Here the set of all E-hesitant fuzzy soft fixed points of a hesitant fuzzy soft set valued mapping Lˇ is represented by E−HFSFix(Lˇ).
Definition 13.Assume thatW≠⧸0andEbe set of parameters andLˇbe a hesitant fuzzy soft set onW. Choosing(μ,ν)∈(0,1]×[0,1), the crisp subset ofWdetermined by(Lˇw;e)(μ,ν)={t∈W:Sc((Lˇw;e)(t))≥μ}∪{t∈W:Sc((Lˇw;e)(t))≤ν},
is known as the(μ,ν)-level set ofLˇ, and hereSc((Lˇw;e)(t))shows score function ofLˇ. So, a pointu^∈Wis known hesitant fuzzy soft fixed point ofLˇif∃,(μ,ν)∈(0,1]×[0,1)such thatu^∈(Lˇu^;e)(μ,ν).
Note that in the whole manuscript, the function r:(0,1)⟶(1/2,1] is defined by
r(μ)={1,if0<μ<1/2,1−μ,if1/2≤μ<1.
For w,b∈W and (μ,ν)∈(0,1]×[0,1), defining
∨(w,b)=max{D(w,b),D(w,(Lˇw;e)(μ,ν)),D(b,(Lˇb;e)(μ,ν)),D(w,(Lˇb;e)(μ,ν))+D(b,(Lˇw;e)(μ,ν))2,D(w,(Lˇw;e)(μ,ν))D(b,(Lˇb;e)(μ,ν))1+D(w,b)},∧(w,b)=min{D(w,(Lˇw;e)(μ,ν)),D(b,(Lˇw;e)(μ,ν)),D(w,(Lˇw;e)(μ,ν))D(b,(Lˇb;e)(μ,ν))1+D(w,b)}.
Remark 4.Definition 13 generalizes the Definition [9] of the paper [23].
Definition 14.A hesitant fuzzy soft set valued mappingLˇ:W⟶H~(W)Eis known as Suzuki-type(μ,ν)-weak contraction if∀,w,b∈Wwithw≠bande∈A⊆E,∃, some(μ,ν)∈(0,1]×[0,1)such thatr(μ)D(w,[Lˇ(w;e)](μ,ν))≤D(w,b).
Then we haveH((Lˇw;e)(μ,ν),(Lˇb;e)(μ,ν))≤μ∨(w,b)+ν∧(w,b).
Theorem 1.Assume that(W,D)is a complete metric space andLˇ:W⟶H~(W)Eis a Suzuki-type(μ,ν)-weak contraction. Suppose that∀,w∈W, ande∈A⊆E,∃,(μ,ν)∈(0,1]×[0,1)such that[Lˇ(w;e)](μ,ν)≠⧸0, closed and bounded subset ofW. ThenFix(Lˇ)≠ϕ.
Proof. Assume that (μ,ν)∈(0,1]×[0,1) be such that 0<μ≤μ1<1,w1∈W and ρ=1/μ. Assume that (Lˇw1;e)(μ,ν)≠⧸0. So, we can find w2∈W such that w2∈(Lˇw1;e)(μ,ν). Since ρ>1, choose w3∈(Lˇw2;e)(μ,ν) such that
D(w2,w3)≤ρH((Lˇw1;e)(μ,ν),(Lˇw2;e)(μ,ν)).
If w1=w2, then w1∈(Lˇw1;e)(μ,ν) for some (μ,ν)∈(0,1]×[0,1) and hence theorem is proved. Suppose w1≠w2. Since r(μ)≤1, Further suppose that
r(μ)D(w1,(Lˇw1;e)(μ,ν))≤D(w1,(Lˇw1;e)(μ,ν))≤D(w1,w2)
Then we have
D(w2,w3)≤ρH((Lˇw1;e)(μ,ν),(Lˇw2;e)(μ,ν))≤1μ(μ∨(w,b)+ν∧(w,b))≤μmax{D(w1,w2),D(w1,(Lˇw1;e)(μ,ν)),D(w2,(Lˇw2;e)(μ,ν)),D(w1,(Lˇw2;e)(μ,ν))+D(w2,(Lˇw1;e)(μ,ν))2,D(w1,(Lˇw1;e)(μ,ν))D(w2,(Lˇw2;e)(μ,ν))1+D(w1,w1)+νμminD(w1,(Lˇw1;e)(μ,ν)),D(w2,(Lˇw1;e)(μ,ν)),D(w1,(Lˇw1)(μ,ν))D(w2,(Lˇw2)(μ,ν))1+D(w1,w2)}≤μmax{D(w1,w2),D(w2,w3),D(w1,w3)+D(w2,w2)2,D(w1,w2)D(w2,w3)1+D(w1,w2)}+νμmin{D(w1,w2),D(w2,w2),D(w1,w2)D(w2,w3)1+D(w1,w2)}≤μmax{D(w1,w2),D(w2,w3),D(w1,w2)+D(w2,w3)2}≤μmax{D(w1,w2),D(w2,w3)}
If D(w1,w2)≤D(w2,w3), then from Eq. (3), we have
D(w2,w3)≤μD(w2,w3)<D(w2,w3),a contradiction. Hence, D(w1,w2)>D(w2,w3) and Eq. (3) becomes
D(w2,w3)≤μD(w1,w2)≤μ1D(w1,w2).
To continue the above process, we can make a progression {wn}n∈N in W such that
wn+1∈(Lˇwn;e)(μ,ν)and
D(wn,wn+1)≤μ1D(wn−1,wn).
So we obtain
∑n=1∞D(wn,wn+1)≤∑n=1∞(μ1)n−1D(w1,w2)<∞.
As a result, we came to know that {wn}n∈N be a Cauchy progression in W. So, completeness of W shows that ∃,u^∈W such that wn⟶u^ as n⟶∞. Assume: ∀,u^≠w, then we get
D(u^,(Lˇw;e)(μ,ν))≤μmax{D(u^,w),D(w,(Lˇw;e)(μ,ν))}
Since wn⟶u^ as n⟶∞,∃, a natural number k such that
5D(wn,u^)≤D(u^,w)∀,n≥k.
We are given that wn+1∈(Lˇwn;e)(μ,ν), we obtain
r(μ)D(wn,(Lˇwn;e)(μ,ν))≤D(wn,(Lˇwn;e)(μ,ν))≤D(wn,wn+1)≤D(wn,u^)+D(u^,wn+1)≤25D(u^,w)
Hence, for n≥k, we obtain
r(μ)D(wn,(Lˇwn;e)(μ,ν))≤25D(u^,w)≤D(u^,w)−15D(u^,w)≤D(u^,w)−D(u^,wn)≤D(wn,w).
Hence, applying Eq. (5) we get
D(wn+1,(Lˇw;e)(μ,ν))≤H((Lˇwn;e)(μ,ν),(Lˇw;e)(μ,ν))≤μmax{D(wn,w),D(wn,(Lˇwn;e)(μ,ν)),D(w,(Lˇw;e)(μ,ν)),D(wn,(Lˇw;e)(μ,ν))+D(w,(Lˇwn;e)(μ,ν))2,D(wn,(Lˇwn;e)(μ,ν))D(w,(Lˇw;e)(μ,ν))1+D(wn,w)}+νmin{D(wn,(Lˇwn;e)(μ,ν)),D(w,(Lˇwn;e)(μ,ν)),D(wn,(Lˇwn;e)(μ,ν))D(w,(Lˇw;e)(μ,ν))1+D(wn,w)}.
From Eq. (6), we have
D(wn+1,(Lˇw;e)(μ,ν))≤μmax{D(wn,w),D(wn,wn+1),D(w,(Lˇw;e)(μ,ν)),D(wn,(Lˇw;e)(μ,ν))+D(w,wn+1)2,D(wn,(Lˇwn;e)(μ,ν))D(w,(Lˇw;e)(μ,ν))1+D(wn,w)}+νmin{D(wn,wn+1),D(w,wn+1),D(wn,wn+1)D(w,(Lˇw;e)(μ,ν))1+D(wn,w).}.
Since we know that in (7) as n⟶∞, so we get
D(u^,(Lˇw;e)(μ,ν))≤μmax{D(u^,w),D(w,(Lˇw;e)(μ,ν)),D(u^,(Lˇw;e)(μ,ν))+D(u^,w)2}+νmin{0,D(u^,w)}≤μmax{D(u^,w),D(w,(Lˇw;e)(μ,ν))}.
Next we will show that u^∈(Lˇu^;e)(μ,ν), for this we assume the following two cases.
Case (1). 0 < μ < 1/2.
Assume that ∀,(μ,ν)∈(0,1]×[0,1) with μ<1,u^≠p,u^∉(Lˇu^;e)(μ,ν) and p∈(Lˇu^;e)(μ,ν) such that
D(p,u^)<D(u^,(Lˇu^;e)(μ,ν)).
Setting w=p in Eq. (4), we have
D(u^,(Lˇp;e)(μ,ν))≤μmax{D(u^,p),D(p,(Lˇp;e)(μ,ν))}
Then we have
D(p,(Lˇp;e)(μ,ν))≤H((Lˇu^;e)(μ,ν),(Lˇp;e)(μ,ν))≤μmax{D(u^,p),D(u^,(Lˇu^;e)(μ,ν)),D(p,(Lˇp;e)(μ,ν)),D(u^,(Lˇp;e)(μ,ν))+D(p,(Lˇu^;e)(μ,ν))2,D(u^,(Lˇu^;e)(μ,ν))D(p,(Lˇp;e)(μ,ν))1+D(u^,p)}+νmin{D(u^,(Lˇu^;e)(μ,ν)),D(p,(Lˇu^;e)(μ,ν)),D(u^,(Lˇu^;e)(μ,ν))D(p,(Lˇp;e)(μ,ν))1+D(u^,p)}.≤μmax{D(u^,p),D(p,(Lˇp;e)(μ,ν)),D(u^,(Lˇp;e)(μ,ν))2,D(u^,p)D(p,(Lˇp;e)(μ,ν))1+D(u^,p)}+νmin{D(u^,p),0,D(u^,p)D(p,(Lˇp;e)(μ,ν))1+D(u^,p)}≤max{D(u^,p),D(p,(Lˇp;e)(μ,ν)),D(u^,p)+D(p,(Lˇp;e)(μ,ν))2}≤μmax{D(u^,p),D(p,(Lˇp;e)(μ,ν))}.
Suppose that D(u^,p)≤D(p,(Lˇp;e)(μ,ν)), so we get from the above inequalities as below:
D(p,(Lˇp;e)(μ,ν))≤μD(p,(Lˇp;e)(μ,ν))<D(p,(Lˇp;e)(μ,ν)),which shows a contradiction. So,
D(u^,p)>D(p,(Lˇp;e)(μ,ν)),and
D(p,(Lˇp;e)(μ,ν))≤μD(u^,p)<D(u^,p).
So, our supposition gives a contradiction. Therefore, u^∈(Lˇu^;e)(μ,ν). Case (2). 1/2 ≤μ < 1. Now, we will prove that
H((Lˇw;e)(μ,ν),(Lˇu^;e)(μ,ν))≤μmax{D(,w),D(w,(Lˇw;e)(μ,ν)),D(u^,(Lˇu^;e)(μ,ν)),D(w,(Lˇu^;e)(μ,ν))+D(u^,(Lˇw;e)(μ,ν))2,D(w,(Lˇw;e)(μ,ν))D(u^,(Lˇu^;e)(μ,ν))1+D(u^,w)}+νmin{D(w,(Lˇw;e)(μ,ν)),D(u^,(Lˇw;e)(μ,ν)),D(u^,(Lˇu^;e)(μ,ν))D(w,(Lˇw;e)(μ,ν))1+D(u^,w)}.satisfied ∀,w∈W with w≠u^. Now, ∀,m∈N,∃,vm∈(Lˇw;e)(μ,ν) such that
D(u^,vm)≤D(u^,(Lˇw;e)(μ,ν))+15mD(u^,w).
Thus, we have
D(w,(Lˇw;e)(μ,ν))≤D(w,vm)≤D(w,u^)+D(u^,vm)≤D(w,u^)+D(u^,(Lˇw;e)(μ,ν))+15mD(w,u^).
Using Eq. (4), we have
D(w,(Lˇw;e)(μ,ν))≤D(u^,w)+μmax{D(u^,w),D(w,(Lˇw;e)(μ,ν))}+15m.
If D(w,u^)>D(w,(Lˇw;e)(μ,ν)), then from Eq. (10), we get
D(w,(Lˇw;e)(μ,ν))≤D(u^,w)+μD(u^,w)+15mD(u^,w)=[(1+μ)+15m]D(u^,w).from which we have
[11+μ]D(w,(Lˇw;e)(μ,ν))≤[1+15(1+μ)m]D(u^,w).
Using r(μ) = 1 − μ, we get
r(μ)D(w,(Lˇw;e)(μ,ν))=(1−μ)D(w,(Lˇw;e)(μ,ν))≤(11+μ)D(w,(Lˇw;e)(μ,ν))≤[1+15(1+μ)m]D(u^,w)
From the above inequalities as m⟶∞, we get
r(μ)D(w,(Lˇw;e)(μ,ν))≤D(u^,w).
From the other side, if
D(u^,w)<D(w,(Lˇw;e)(μ,ν)),then Eq. (10) gives
D(w,(Lˇw;e)(μ,ν))≤D(u^,w)+μD(w,(Lˇw;e)(μ,ν))+15mD(u^,w)
Which yields
(1−μ)D(w,(Lˇw;e)(μ,ν))≤(1+15m)D(u^,w).
So, in the above inequality when m⟶∞, we get
(1−μ)D(w,(Lˇw;e)(μ,ν))≤D(u^,w).
It means that
r(μ)D(w,(Lˇw;e)(μ,ν))≤D(u^,w).
Which is an output of the Definitions 13 and 14 collectively. Furthermore, since
wn+1≠wn∀,n∈N,thenu^≠wn+1.
Therefore, keeping wn = w in Eq. (9), we obtain
D(wn+1,(Lˇu^;e)(μ,ν))≤H((Lˇwn;e)(μ,ν),(Lˇu^;e)(μ,ν))≤μmax{D(wn,u^),D(wn,(Lˇwn;e)(μ,ν)),D(u^,(Lˇu^;e)(μ,ν)),D(wn,(Lˇu^;e)(μ,ν))+D(u^,(Lˇwn;e)(μ,ν))2,D(wn,(Lˇwn;e)(μ,ν))D(u^,(Lˇu^;e)(μ,ν))1+D(wn,u^)}+kmin{D(wn,(Lˇwn;e)(μ,ν)),D(u^,(Lˇwn;e)(μ,ν)),D(wn,(Lˇwn;e)(μ,ν))D(u^,(Lˇu^;e)(μ,ν))1+D(wn,u^)}≤μmax{(D(wn,u^),D(wn,wn+1),D(u^,(Lˇu^;e)(μ,ν)),D(wn,(Lˇu^;e)(μ,ν))+D(u^,wn+1)2,D(wn,wn+1)D(u^,(Lˇu^;e)(μ,ν))1+D(wn,u^)}+νmin{D(wn,wn+1),D(u^,wn+1),D(wn,wn+1)D(u^,(Lˇu^;e)(μ,ν))1+D(wn,u^)}
In the above inequalities as n⟶∞, we get
D(u^,(Lˇu^;e)(μ,ν))≤μmax{D(u^,(Lˇu^;e)(μ,ν)),D(u^,(Lˇu^;e)(μ,ν))2,0}≤μD(u^,(Lˇu^;e)(μ,ν)).
Since 1 − μ> 0, therefore, using Eq. (11), we get
D(u^,(Lˇu^;e)(μ,ν))=0,and as a result, u^∈(Lˇu^;e)(μ,ν).□
Corollary 1.Assume that (W, D) be a complete metric space andLˇ:W→H(W)Ebe a hesitant fuzzy soft mapping. Assume that forw,b∈Wwithw≠b,∃,(μ,ν)∈(0,1]×[0,1)such that(Lˇw;e)(μ,ν)≠⧸0,bounded and closed subset ofW. Ifr(μ)D(w,(Lˇw;e)(μ,ν))≤D(w,b)impliesH((Lˇw;e)(μ,ν),(Lˇb;e)(μ,ν))≤μ∨(w,b), thenFix(Lˇ)≠ϕ.
Further, we provide a local hesitant fuzzy fixed point theorem for Suzuki-type (μ,ν)−WC. First of all, we remember the definition of the open ball having radius r^> 0, centered at w0 in a metric space W, is represented by
Br^(w0)={w∈W:D(w,w0)<r^}.
Theorem 2.Assume that (W, D) be a complete metric space, Lˇ:Br^(w0)⟶H(W)Ebe a Suzuki-type (μ,ν)-WC. Suppose that∀,w∈W,∃,(μ,ν)∈(0,1]×[0,1)such that(Lˇw;e)(μ,ν)≠ϕ,closed and bounded subset ofW, andD(w0,(Lˇw0;e)(μ,ν))<(1−μ)r^. ThenFix(Lˇ)≠ϕ.
Proof. Assume that 0<γ<r^ is chosen in such a way that
0<(1−μ)(1+μ)≤1/(1+γ),Bγ∗(w0)⊂Br(w0)andD(w0,(Lˇw0;e)(μ,ν))<(1−μ)γ.Therefore,
(1−μ)γ−D(w0,(Lˇw0;e)(μ,ν))>0.
Choose γ=(1−μ)γ−D(w0,(Lˇw0;e)(μ,ν))>0; then ∃,w1∈(Lˇw0;e)(μ,ν) such that
D(w0,w1)<D(w0,(Lˇw0;e)(μ,ν))+γ.
Thus, D(w0,w1)<(1−μ)γ. Now, for ρμ=1 and
w1∈(Lˇw0;e)(μ,ν),∃,w2∈(Lˇw1;e)(μ,ν)such that
D(w1,w2)≤ρH((Lˇw0;e)(μ,ν),(Lˇw1;e)(μ,ν)).
Noting that r(μ)D(w0,(Lˇw0;e)(μ,ν))≤r(μ)D(w0,w1)≤D(w0,w1).
We have
D(w1,w2)≤ρH((Lˇw0;e)(μ,ν),(Lˇw1;e)(μ,ν))=1μH((Lˇw0;e)(μ,ν),(Lˇw1;e)(μ,ν))≤μ∨(w0,w1)+νμ∧(w0,w1)≤μmax{D(w0,w1),D(w0,(Lˇw0;e)(μ,ν)),D(w1,(Lˇw1;e)(μ,ν)),D(w0,(Lˇw1;e)(μ,ν))+D(w1,(Lˇw0;e)(μ,ν))2,D(w0,(Lˇw0;e)(μ,ν))D(w1,(Lˇw1;e)(μ,ν))1+D(w0,w1)}+νμmin{D(w0,(Lˇw0;e)(μ,ν)),D(w1,(Lˇw0;e)(μ,ν)),D(w0,(Lˇw0;e)(μ,ν))D(w1,(Lˇw1;e)(μ,ν))1+D(w0,w1)}≤μmax{D(w0,w1),D(w0,w1),D(w1,w2),D(w0,w2)+D(w1,w1)2,D(w0,w1)D(w1,w2)1+D(w0,w1)}+νμmin{D(w0,w1),D(w1,w1),D(w0,w1)D(w1,w2)1+D(w0,w2)}≤μmax{D(w0,w1),D(w1,w2),D(w0,w1)+D(w1,w2)2}+νμmin{D(w0,w1),0,D(w1,w2)}≤μmax{D(w0,w1),D(w1,w2)}
If D(w0,w1)≤D(w1,w2), then the above gives
D(w1,w2)≤μD(w1,w2)<D(w1,w2),a contradiction. Hence,
D(w0,w1)>D(w1,w2)and
D(w1,w2)<μD(w0,w1)μ(1−μ)γ.
So by our supposition and the triangle inequality, we have
D(w0,w2)≤D(w0,w1)+D(w1,w2)<(1−μ)γ+μ(1−μ)γ≤γ1+γ<γ
Hence, w2∈Bγ(w0). So by recursive process we make a progression {wn}n∈N such that
wn∈Bγ(w0);∀,n∈N;
wn∈(Lˇwn;e)(μ,ν);∀,n∈N;
D(wn,wn+1)≤(μ)n(1−μ)γ;∀,n∈N.
From (3), it is clearly known that {wn}n∈N is a Cauchy progression and shrinks to some u^∈Br(w0). Furthermore, after this step, we can follow the steps of the Theorem 1, as a result, we get that
Fix(Lˇ)≠ϕ.◻
Corollary 2.Assume that(W,D)be a complete metric space andΛ:W⟶CB(W)be a multi-valued mapping. Suppose that∀,w,b∈W,∃,(μ,ν)∈I+×[0,1)such thatr(μ)D(w,Λw)≤D(w,b)impliesH(Λw,Λb)≤μ∨(w,b)+ν∧(w,b),and the mappingr:(0,1)⟶I+1is determined byr(μ)={1,if0≤μ<1/21−μ,if1/2≤μ<1.
ThenFix(Λ)≠ϕ.
Proof. Assume that I+=(0,1] and P(I+) be the power set of I+. Suppose we have a map Ψ:W⟶P(I+) and a hesitant fuzzy soft map Lˇ:W⟶H~(W)E defined by
(Lˇw;e)(t¨)={Ψ(w)∖{1},ift¨∈Λw,{1},ift¨∉Λw.
Assume that, μ=inf{s((Lˇw;e)(t¨)):t¨∈W} and ν=0. Then,
(Lˇw;e)(μ,ν)={t¨∈W:s((Lˇw;e)(t¨))≥μ}∪{t¨∈W:Sc((Lˇw;e)(t¨))≤ν}=Λw.
Define a hesitant fuzzy soft mapping Lˇ:W⟶H~(W)E as follows:
(Lˇw;e1)t¨={{0.6,0.8,1},ift¨=4,{0.2,0.5,0.7},ift¨=5,{0,0.3,0.4},ift¨=6,(Lˇw;e2)t¨={{0.1,0.4,0.5},ift¨=4,{0.3,0.6,0.8},ift¨=5,{0.5,0.7,0.9},ift¨=6.
For w∈W and e1∈E, the score values of Lˇ with respect to t¨=4,5 and t¨=6, are respectively evaluated by Sc((Lˇw;e1)(t¨))=(0.6+0.8+1)3=0.8,Sc((Lˇw;e1)(t¨))=(0.2+0.5+0.7)3=0.47,Sc((Lˇw;e1)(t¨))=(0+0.3+0.4)3=0.23; and for w∈W, and e2∈E the score functions of Lˇ corresponding to t¨=4,5 and t¨=6, are respectively evaluated by Sc((Lˇw;e2)(t¨))=(0.1+0.4+0.5)3=0.33,Sc((Lˇw;e2)(t¨))=(0.3+0.6+0.8)3=0.57,Sc((Lˇw;e2)(t¨))=(0.5+0.7+0.9)3=0.7. Then, for μ=0.7 and ν=0.16, we get
(Lˇw;e)(μ,ν)={t¨∈W:Sc((Lˇw;e)(t¨))≥μ}∪{t¨∈W:Sc((Lˇw;e)(t¨))≤ν}={{4},ife=e1{6},ife=e2.
Case I.
For w∈W and for e∈E,D(w,(Lˇw;e)(μ,ν))=inf{D(w,b):b∈(Lˇw;e)(μ,ν)}=0.
Hence, for w = b, we have
r(μ)D(w,(Lˇw;e)(μ,ν))≤D(w,b).
Case II.
For w∈W∖{4} and for e1∈E,D(w,(Lˇw:e1)(μ,ν))=inf{D(w,b):b∈(Lˇw:e1)(μ,ν)}={1ifw=57/20ifw=6.
Now for w∈W∖{6} and for e2∈E,D(w,(Lˇw:e2)(μ,ν))=inf{D(w,b):b∈(Lˇw:e2)(μ,ν)}={7/20ifw=49/20ifw=5.
Therefore, for w=4,5 and w≠b, we get
r(μ)D(w,(Lˇw:e2)(μ,ν))=21/200≤D(w,b)
r(μ)D(w,(Lˇw:e2)(μ,ν))=27/200≤D(w,b), respectively. Hence, we concluded from the above two possibilities, it follows that ∃,(μ,ν)=(0.7,0.16)∈(0,1]×[0,1) such that
r(μ)D(w,(Lˇw;e)(μ,ν))≤D(w,b)we get
H((Lˇw;e)(μ,ν),(Lˇb;e)(μ,ν))≤μ∨(w,b)+ν∧(w,b).
In short, we can say that all the assumptions of Theorem 15 are satisfied to get 4∈(Lˇ4;e1)(0.7,0.16) and 6∈(Lˇ6;e2)(0.7,0.16).
For the supporting and generality of the Corollary 2 we present an example as follows:
Example 2.Assume thatW={x,y,z,m,n}andD:W×W⟶[0,∞)be defined byD(x,y)=D(x,z)=7,D(y,n)=D(z,m)=D(z,n)=D(y,z)=12,D(x,m)=D(x,n)=16,D(y,m)=14,D(m,n)=8,D(t¨,t¨)=0andD(t¨,s)=D(s,t¨);∀,t¨,s∈W.
Next, we see that ∃,t¨=x∈W such that t¨∈Λt¨. From the other side, we observe that if we set t¨=m and s=n, then
∨(m,n)=max{D(m,n),D(m,Λm),D(n,Λn),D(m,Λn)+D(n,Λm)2,D(m,Λm)D(n,Λn)1+D(m,n)}=max{8,14,12,12,18.6}=18.6,and H(Λm,Λn)=12>11.625=μ∨(m,n). Therefore, [[24], Theorem 2.1] and [[25], Theorem 3] can not be used at this movement to search out t¨∈Λt¨.
Algorithm: Decision-Making Using Suzuki-Type (<inline-formula id="ieqn-344"><mml:math id="mml-ieqn-344"><mml:mi mathvariant="bold-italic">μ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="bold-italic">ν</mml:mi></mml:math></inline-formula>)-Weak Contraction
Input
(i) A set of alternatives W={w1,w2,...,wn}
(ii) A set of criteria E={e1,e2,...,ek}
(iii) Hesitant fuzzy soft sets G(e), where G(e) maps each criterion e∈E to hesitant fuzzy membership values for alternatives in W
(iv) Parameters (μ,ν), where μ∈(0,1] and ν∈[0,1)
Output
(v) Optimal alternative(s) w∗∈W.
We provide a comprehensive decision-making case to illustrate the usefulness of the suggested Suzuki-type (μ,ν)-Weak Contraction framework. The effectiveness of hesitant fuzzy soft set-valued mappings in managing uncertainty and reluctance in practical situations is demonstrated by this example. Utilizing the flexibility offered by the dual-parameter structure (μ,ν), we seek to find optimal solutions through the use of the iterative contraction process. The following example demonstrates how well our method works when dealing with challenging multi-criteria decision-making issues.
Example 3.Based on three evaluation criteria, a commuter selects one of three routesW={A,B,C}.
E={TrafficCondition(TC),Safety(S),Time(T)}. In the following Table 1, each route’s hesitant fuzzy values represent hesitancy and doubt in the decision-making process.
The hesitant fuzzy membership values
Route
Time (T)
Safety (S)
Traffic condition (TC)
A
{0.7,0.8}
{0.6,0.7}
{0.5,0.6}
B
{0.6,0.7}
{0.7,0.8}
{0.6,0.7,0.8}
C
{0.5,0.6}
{0.7,0.8}
{0.6,0.7}
Note: Each hesitant fuzzy value represents hesitation in rating a route for a particular criterion.
Step 1: Initial Hesitant Fuzzy Membership Values
Step 2: Compute the Score Function Sc
For each hesitant fuzzy set h={v1,v2,...,vn}, we calculate the score function in the following Table 2:
Sc(h)=(1N(h))∑vi∈hvi
The score values
Route
Time score Sc(T)
Safety score Sc(S)
Traffic score Sc(TC)
Total score (Average)
A
0.7+0.82=0.75
0.6+0.72=0.65
0.5+0.62=0.55
0.75+0.65+0.553=0.65
B
0.6+0.72=0.65
0.7+0.82=0.75
0.6+0.7+0.83=0.7
0.65+0.75+0.73=0.7
C
0.5+0.62=0.55
0.7+0.82=0.75
0.6+0.72=0.65
0.55+0.75+0.653=0.65
Thus the Route B has the highest total score “0.7”.
Step 3: Define the (μ,ν)-Level Set
Choosing (μ,ν) Parameters
let us suppose μ = 0.6 and ν = 0.3.
Definition 12 states that the (μ,ν)-level set of Lˇ is
(Lˇw;e)(μ,ν)={t∈W:Sc((Lˇw;e)(t))≥μ}∪{t∈W:Sc((Lˇw;e)(t))≤ν}
It means that the route satisfying either Sc((Lˇw;e)(t))≥μ or Sc((Lˇw;e)(t))≤ν belongs to the (μ,ν) level set.
Step 4: Identify Routes in the (μ,ν)-Level Set
Routes with Sc((Lˇw;e)(t))≥μ=0.6 are
{A, B, C} and routes with Sc((Lˇw;e)(t))≤ν=0.3 are {}
Thus, the (μ,ν)-level set contains:
(Lˇw;e)(μ,ν)={A,B,C}
Step 5: Identify Hesitant Fuzzy Soft Fixed Points
A hesitant fuzzy soft fixed point u^ is a route such that:
u^∈(Lˇw;e)(μ,ν)
From the (μ,ν)-level set, we see that all routes satisfy the fixed-point condition, meaning: A, B, C are hesitant fuzzy soft fixed points. However, since B has the highest score (0.7), it is the most optimal choice.
Step 6: Decision Making
i) The (μ,ν)-level set contains all three routes.
ii) Since route B has the highest hesitant fuzzy score (0.7), it is the best choice for the selection.
iii) Thus, B is selected as the final route.
Final Conclusion
i) Definition 12 ensures that the best route is within the (μ,ν)-level set.
iii) B is the best route, as it has the highest hesitant fuzzy score and belongs to the level set.
The following example shows real world problem solved by Definitions 10 and 11.
Example 4.Let’s say that a cricket organisation t ravels the nation doing trials to find local talent. Three well-known Test and first-class cricket players, who are also among the top instructors in the sport, have led the tryouts. After 15 players have been chosen, it is agreed to hold a camp for training to choose the top 11 players before the cricket team is finalised. First of all, these three trainers have scheduled a coaching event. After the final trials, these trainers and instructors are asked to serve as assessors by offering their unbiased thoughts. With the use of the hesitant fuzzy soft set theory, let’s examine this event mathematically as follows:W={p¨1,p¨2,p¨3,p¨4,p¨5,p¨6,p¨7,p¨8,p¨9,p¨10,p¨11,p¨12,p¨13,p¨14,p¨15}(set of players) andM⊂Ebe given byM={bt,bl,bf},wherebt,bl, andbfrepresents batting, bowling, and fielding, respectively. The three trainers’ opinions, F1,F2, andF3, are represented by the following hesitant fuzzy soft set F:
Presume that the weights assigned to batting, bowling, and fielding are 5:4:3. In this manner, a player’s ultimate score is determined using the following formula. Score=5Sc(F(bt))+4Sc(F(bl))+3Sc(F(bf)).
This data is presented in Table 3.
The trainers’ survey results
Players
F(bt)
Sc(F(bt))
F(bl)
Sc(F(bl))
F(bf)
Sc(F(bf))
Score
p¨1
{0.8,0.7,0.9}
0.8
{0.4,0.5,0.6}
0.5
{0.7,0.8,0.6}
0.7
8.1
p¨2
{0.5,0.5,0.6}
0.53
{0.8,0.7,0.8}
0.76
{0.6,0.7,0.7}
0.66
7.67
p¨3
{0.8,0.7,0.9}
0.8
{0.4,0.5,0.6}
0.5
{0.9,0.7,0.8}
0.8
8.4
p¨4
{0.8,0.8,0.9}
0.83
{0.4,0.6,0.5}
0.5
{0.8,0.6,0.6}
0.66
8.13
p¨5
{0.5,0.5,0.6}
0.53
{0.5,0.4,0.5}
0.46
{0.6,0.4,0.5}
0.5
5.99
p¨6
{0.5,0.5,0.6}
0.53
{0.5,0.6,0.7}
0.6
{0.7,0.7,0.8}
0.73
7.24
p¨7
{0.7,0.8,0.6}
0.43
{0.7,0.8,0.6}
0.7
{0.7,0.7,0.8}
0.7
7.05
p¨8
{0.4,0.4,0.5}
0.43
{0.4,0.5,0.4}
0.43
{0.6,0.6,0.5}
0.56
5.55
p¨9
{0.5,0.5,0.6}
0.53
{0.5,0.6,0.7}
0.6
{0.6,0.6,0.7}
0.63
6.94
p¨10
{0.8,0.7,0.9}
0.8
{0.6,0.5,0.6}
0.56
{0.7,0.8,0.6}
0.7
8.34
p¨11
{0.8,0.7,0.9}
0.8
{0.5,0.4,0.6}
0.5
{0.5,0.4,0.6}
0.5
7.5
p¨12
{0.8,0.7,0.6}
0.7
{0.5,0.6,0.7}
0.6
{0.7,0.6,0.7}
0.66
7.88
p¨13
{0.5,0.6,0.7}
0.6
{0.5,0.4,0.5}
0.46
{0.4,0.4,0.6}
0.46
6.22
p¨14
{0.6,0.6,0.7}
0.63
{0.7,0.8,0.9}
0.8
{0.8,0.9,0.7}
0.8
8.75
p¨15
{0.4,0.4,0.4}
0.4
{0.5,0.6,0.4}
0.5
{0.6,0.7,0.6}
0.63
5.89
Note: bt, bl, bf are parameters used for batting, bowling and fielding.
A selection of this data is displayed in the following charts. Each player individual score is represented by Fig. 1. A cursory examination of these charts reveals that p¨1,p¨3,p¨4,p¨10,p¨11 are the best five batsman; p¨2,p¨7,p¨9,p¨12,p¨14 are the best five bowlers; and p¨3,p¨6,p¨14 are the best fielders.
Score for each individual
The following Fig. 2 represents the final score of each player.
Last tally
Here the mid-level decision rule and the mid-level threshold fuzzy set of ΔF is as mid
ΔF={(bt,0.62),(bl,0.5),(bf,0.65)}.
Furthering, we can get the mid-level soft set ΔF={(bt,{p1,p3,p4,p10,p11,p12,p14}),(bl,{p2,p6,p7,p9,p10,p12,p14})(bf,{p1,p2,p3,p4,p6,p7,p10,p12,p14})}.
Now let us define hesitant fuzzy soft set valued mapping Lˇ:W→H~(W)E by Lˇ(p)=F;∀,p∈W. For μ=0.62∈(0,1] we note that p14∈Lˇ[p14]0.62 be E-hesitant fuzzy soft fixed point of Lˇ, which is the best player of the team. The above mechanism of the problem is represented by the Fig. 3.
An illustrative representation of a hesitant fuzzy soft set-valued mapping F, where a universal set <inline-formula id="ieqn-546"><mml:math id="mml-ieqn-546"><mml:mrow><mml:mi mathvariant="double-struck">W</mml:mi></mml:mrow></mml:math></inline-formula> is mapped into hesitant fuzzy soft set within <inline-formula id="ieqn-547"><mml:math id="mml-ieqn-547"><mml:mrow><mml:mrow><mml:mover><mml:mi>H</mml:mi><mml:mo stretchy="false">~</mml:mo></mml:mover></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi mathvariant="double-struck">W</mml:mi></mml:mrow><mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant="double-struck">E</mml:mi></mml:mrow></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula>. The parameters <inline-formula id="ieqn-548"><mml:math id="mml-ieqn-548"><mml:mi>M</mml:mi><mml:mo>⊆</mml:mo><mml:mi>E</mml:mi></mml:math></inline-formula> influence the mapping, defining hesitant fuzzy decision components <italic>b<sub>t</sub></italic>, <italic>b<sub>l</sub></italic>, <italic>b<sub>f</sub></italic>, each associated with a power set transformation into P(I)
The above methodology can be summarised in the following Fig. 4. It is clearly shown from the results that hesitant fuzzy soft set mechanism gives us more prominent results as compared to the other mechanisms.
Decision-making methodology
Conclusion
Fixed points theory plays a significant role in decision-making problems, particularly in economics, game theory, and optimization. In the context of decision-making problems, fixed points are often used to identify stable solutions. Applications in decision-making issues like risk management and resource allocation show how helpful and reliable the suggested method is. We introduced the Suzuki-type (μ,ν)-weak contraction and hesitant fuzzy soft set valued-mapping. Further, using these concepts, we established some fixed point theorems in the framework of metric spaces. We also solved some decision-making problems based on the presented work. Moreover, these theorems only show the existence of the “hesitant fuzzy soft fixed point.”
The suggested approach assumes the completeness of the metric space and uses Suzuki-Type (μ,ν)-Weak Contraction for hesitant fuzzy soft set-valued mappings. In some real-world applications, especially those involving substantially non-convex or fragmented datasets, this assumption might not hold. To increase the method’s usefulness, future research could examine expansions to quasi-metric spaces or loosen the completeness assumption.
The authors Muhammad Sarwar and Kamaleldin Abodayeh are thankful to Prince Sultan University for the support of this manuscript through TAS research lab.
Funding Statement
This research was funded by National Science, Research and Innovation Fund (NSRF), and King Mongkut’s University of Technology North Bangkok with Contract No. KMUTNB-FF-68-B-46.
Author Contributions
Conceptualization, writing—review and editing, Muhammad Sarwar and Rafiq Alam; Supervision, project administration, formal analysis, Muhammad Sarwar and Kamaleldin Abodayeh; Visualization, investigation, funding acquisition, Saowaluck Chasreechai and Thanin Sitthiwirattham; Writing—original draft preparation, data curation, Rafiq Alam. All authors reviewed the results and approved the final version of the manuscript.
Availability of Data and Materials
Not applicable.
Ethics Approval
Not applicable.
Conflicts of Interest
The authors declare no conflicts of interest to report regarding the present study.
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