Time-Shadow Soft Set: Concepts and Applications

Ayman A. Hazaymeh

doi:10.5391/IJFIS.2024.24.4.387

Original Article

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International Journal of Fuzzy Logic and Intelligent Systems 2024; 24(4): 387-398

Published online December 25, 2024

https://doi.org/10.5391/IJFIS.2024.24.4.387

Time-Shadow Soft Set: Concepts and Applications

Ayman A. Hazaymeh

Department of Mathematics, Faculty of Science, Jadara University, Irbid, Jordan

Correspondence to :
Ayman Hazaymeh (aymanha@jadara.edu.jo)

Received: March 8, 2023; Revised: August 20, 2024; Accepted: September 30, 2024

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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Abstract
1. Introduction
2. Preliminaries
3. Time-Shadow Soft Set
4. AND and OR Operations
5. An Application of Time-Shadow Soft Set in Decision-Making
6. Conclusion
Acknowledgments
Conflict of Interest
Acknowledgments
Conflict of Interest
Tables
References
Biography

In this study, we investigate the effect of the time component affects the application of fuzzy soft sets by extending the theory of soft sets to introduce the notion of time-shadow soft sets (T_sh − SS). In addition, we define and examine the characteristics of the fundamental operations-complement, union intersection, “AND,” and “OR.” Finally, we extend this idea to decision-making issues.

Keywords: Soft sets, Fuzzy soft sets, Time fuzzy soft sets, Shadow soft sets, Time-shadow soft sets

1. Introduction

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Abstract
1. Introduction
2. Preliminaries
3. Time-Shadow Soft Set
4. AND and OR Operations
5. An Application of Time-Shadow Soft Set in Decision-Making
6. Conclusion
Acknowledgments
Conflict of Interest
Acknowledgments
Conflict of Interest
Tables
References
Biography

Most problems in different fields, such as engineering, medical science, economics, and the environment, involve various uncertainties. Molodtsov [1] defined soft set theory as a mathematical tool for addressing such uncertainties. Afterward, applications and soft-set operations were studied by Chen et al. [2], Maji et al. [3] and Maji et al. [4]. The concept of a fuzzy soft set, was introduced by Maji et al. [5] as a more general concept and as a combination of a fuzzy set and soft set, where they studied its properties. Furthermore, it exhibits control over factors that might affect membership values after application. Recently, researchers have begun researching the properties and applications of soft set theory. Several topics in the fuzzy relations of soft set theory that deal with uncertainties have also been studied in more detail by Shakhatreh and Qawasmeh [6]. Hazaymeh et al. [7] provided an overview of a fuzzy parameterized fuzzy soft expert set, which provides a membership value for each parameter in a set of parameters, and generalizes the fuzzy soft expert set. By integrating our research into different disciplines, we identified novel and significant subjects. For instance, we may incorporate the study of fuzzy soft sets into the findings of scholars such as [8–12]. The concept of soft and fuzzy soft expert sets, allows the user to know the opinions of all experts in a single model, as defined by [13] and [14]. The idea of a generalized fuzzy soft expert set with two and multiple opinions (four views) was presented by Hazaymeh et al. [15] as a generalization of fuzzy soft expert sets, which is more beneficial and effective. In addition, the authors of [16] presented the concepts of bipolar fuzzy soft sets and a domain of complex numbers. Bipolar complex fuzzy soft sets can translate bipolar fuzzy soft information into a mathematical formula while retaining the importance of information that may originate from different phases. To address decision-making issues, the notion of n-valued refined neutrosophic soft sets and their characteristics has been presented [17]. Roy and Maji [18] utilized this theory to solve decision-making problems. Furthermore, Hazaymeh [19] introduced the concept of a time-fuzzy soft set, which means that we take the component time value of the information into consideration when making decisions.

Hazaymeh [20] presented the concept of time-effective fuzzy soft sets as extensions of fuzzy soft sets. They also went over its basic workings and provided two examples of how this concept may be used in decision-making scenarios, and the second use that neurosophic membership provides. Subsequently, several scholars combined the idea of time fuzziness with fuzzy soft sets to create new concepts, as in [21] and [22]. The notation of the shadow soft set introduced in [23] is used to study its properties. In addition, they provided an example of the significance of shadow soft sets. This study introduces the notation of the time-shadow soft set, emphasizing its effectiveness in enhancing decision-making precision by incorporating the component time value of information. Furthermore, we define its basic operations, namely complement, union, and intersection, and investigate their properties. Finally, an application of this concept to decision-making problems is presented.

2. Preliminaries

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Abstract
1. Introduction
2. Preliminaries
3. Time-Shadow Soft Set
4. AND and OR Operations
5. An Application of Time-Shadow Soft Set in Decision-Making
6. Conclusion
Acknowledgments
Conflict of Interest
Acknowledgments
Conflict of Interest
Tables
References
Biography

In this section, some basic concepts of soft set theory are introduced. Molodtsov [?] defined soft set over U in the following way. Let U be a universe set, E be a set of parameters, P(U) denote the power sets of U and A ⊆ E.

Definition 2.1 [?]. Let us consider the mapping

F:V→P(U).

Then any a pair (F, V ) is called a soft set over U. In other words, a soft set over U is a parameterized family of subsets of the universal set U. Therefore, for ɛ ∈ V, F (ɛ) may be considered as the set of ɛ approximate elements of soft set (F, V ).

Definition 2.2 [5]. Let U be an initial universal set and E be a set of parameters. Let I^U denote the power set of all fuzzy subsets of U. Let V ⊆ E and F be mappings

F:V→IU.

A pair (F,E) is called a fuzzy soft set over U.

Definition 2.3 [5]. For two fuzzy soft sets (F, V ) and (G,W) over U, (F, V ) is called a fuzzy soft subset of (G,W) if

1. V ⊂ W and
2. ∀ɛ ∈ V, F (ɛ) is fuzzy subset of G(ɛ).

This relationship is denoted as (F, V ) ⊆̃ (G,W). Here, (G,W) is called the fuzzy soft superset of (F, V ).

Definition 2.4 [5]. The complement of a fuzzy soft set (F, V ) is denoted by (F, V )^c and is defined by (F, V )^c = (F^c, ⌉V ) where F^c : ⌉V → P(U) is a mapping given by

Fc(α)=c (F (⌉α)) ∀α∈⌉V,

where c denotes a fuzzy complement.

Definition 2.5 [5]. If (F, V ) and (G,W) are two fuzzy soft sets then “(F, V ) AND (G,W)” denoted by (F, V ) ∧ (G,W) is defined by

(F,V)∧(G,W)=(H,V×W),

such that H (α, β) = t (F (α) ,G(β)), ∀ (α, β) ∈ V × W, where t is any t-norm.

Definition 2.6 [5]. If (F, V ) and (G,W) are two fuzzy soft sets then “(F, V ) OR (G,W)” denoted by (F, V ) ∨ (G,W) is defined by

(F,V)∨(G,W)=(O,V×W),

such that O (α, β) = s (F (α) ,G(β)), ∀ (α, β) ∈ V × W, where s represents the s-norm.

Definition 2.7 [5]. The union of two fuzzy soft sets (F, V ) and (G,W) over a common universe U is the fuzzy soft set (H,Z) where Z = V ∪W, and ∀ɛ ∈ Z,

H (ɛ)={F (ɛ),if ɛ∈V-W,G (ɛ),if ɛ∈W-V,s (F (ɛ),G (ɛ)),if ɛ∈V∩W,

where s is any s-norm.

Definition 2.8 [5]. The intersection of two fuzzy soft sets (F, V ) and (G,W) over a common universe U is the fuzzy soft set (Z,C) where Z = V ∪W, and ∀ɛ ∈ Z,

H (ɛ)={F (ɛ),if ɛ∈V-W,G (ɛ),if ɛ∈W-V,s (F (ɛ),G (ɛ)),if ɛ∈V∩W.

Definition 2.9 [19]. Let U be an initial universal set and let E be a set of parameters. Let I^U denote the power set of all fuzzy subsets of U, V ⊆ E, and T be a set of times, where T = {t₁, t₂, ..., t_n}. V collection of pairs (F,E)_t ∀t ∈ T is called a time-fuzzy soft set (T-FSS) over U where F is the mapping given by

Ft:V→IU.

Definition 2.10 [19]. A time-fuzzy soft set (F, V )_t over U is said to be a semi-null T-FSS denoted by T_∼φ if ∀t ∈ T and F_t (e) = φ for at least one e.

Definition 2.11 [19]. A time-fuzzy soft set (F, V )_t over U is considered a null T-FSS denoted by T_φ if ∀t ∈ T, F_t (e) = φ ∀e.

Definition 2.12 [19]. A time-fuzzy soft set (F, V )_t over U is considered a semi-absolute T-FSS denoted by T_∼V if ∀t ∈ T and F_t (e) = 1̄ for at least one e.

Definition 2.13 [19]/ A time-fuzzy soft set (F, V )_t over U is considered an absolute T-FSS denoted by T_V if ∀t ∈ T and F_t (e) = 1̄ ∀e.

Definition 2.14 [19]. The complement of T-FSS (F, V )_t is denoted by c̃(F, V )_t ∀t ∈ T where c̃ is a fuzzy soft complement.

Definition 2.15 [23]. Let U = {x₁, x₂, ..., x_n} be the universal set of elements, (F,E) is a fuzzy soft set over U, where F is a mapping given by F : E → I^U, E = {e₁, e₂, ..., e_m} be the set of parameters, and let shdw = {(α₁, β₁), (α₂, β₂), ..., (α_m, β_m)} be the shadow parameters set that is related to E. Let shdw(U) is the set of all shadow subsets on U. A pair (F,E)_shdw is called a shadow soft set over U, where F is a mapping F₍_α_{_i}_,β_{_i)} : E → shdw(U), ∀i = 1, 2, ..., m. Thus, we define F₍_α_{_i}_,β_{_i)} as follows:

F(αi,βi)(ei)={xjfj (xj)},

∀i = 1, 2, ..., m, j = 1, 2, ..., n, where

fj (xj)={0,if μj (xj)≤α,1,if μj (xj)≥β,a,α<μj (xj)<β.

Definition 2.16 [23]. Let (F, V )_shdw and (G,W)_shdw be the two soft shadow sets over common universe U. (F, V )_shdw is considered a shadow-soft subset of (G,W)_shdw if A ⊆ B; and F(e)_shdw(x) ≤ G(e)_shdw(x); ∀e ∈ A, x ∈ U. This is denoted as (F, V )_shdw ⊆ (F, V )_shdw, where 0 < [0, 1] < 1.

Definition 2.17 [23]. A null shadow soft set (φ,E)_shdw over the common universe U is a shadow soft set with φ(e)_shdw(x) = 0, ∀e ∈ E; x ∈ U.

Definition 2.18 [23]. An absolute shadow soft set (Φ,E)_shdw over an the common universe U is a shadow soft set with Φ(e)_shdw(x) = 1, ∀e ∈ E; x ∈ U.

Definition 2.19 [23]. A completely shadowed soft set (Ω, E)_shdw over common universe U is a shadow soft set with Ω(e)_shdw(x) = [0, 1], ∀e ∈ E; x ∈ U.

3. Time-Shadow Soft Set

Go to

Abstract
1. Introduction
2. Preliminaries
3. Time-Shadow Soft Set
4. AND and OR Operations
5. An Application of Time-Shadow Soft Set in Decision-Making
6. Conclusion
Acknowledgments
Conflict of Interest
Acknowledgments
Conflict of Interest
Tables
References
Biography

This section defines a time-shadow soft set and describes its basic properties. Parameters may include time values from previous information, which must be considered in decision-making. The concept of time-fuzzy shadow sets is introduced for each fuzzy shadow set, with operations, properties, and illustrative examples discussed.

Definition 3.1. Let U be a universe, E be a set of parameters, I^U denote the power set of all fuzzy subsets of U, and T be a set of times, where T = {t₁, t₂, ..., t_n}. Then, the pairs (F_t_{_i},Z) ∀t ∈ T are called time-fuzzy soft sets where

Fti(σ)={ujμF(ujti)}, ujti∈U, μFti∈[0,1],

∀t ∈ T and σ ∈ Z = E × T, i = 1, 2, ..., m, j = 1, 2, ..., n.

Let (α_i, β_i) be given by an expert and define the time-shadow soft set (T_sh-ss in short) as a mapping

Fsh(αi,βi)ti:Z→{0, 1, [0,1]},

∀t ∈ T and σ ∈ E × T, i = 1, 2, ..., m, j = 1, 2, ..., n. Here,

Fsh(αi,βi)ti(σ)={0,if μF(ujti) (σ)≤αi,1,if μF(ujti) (σ)≥βi,[0,1],αi<μF(ujti) (σ)<βi.

Example 3.1. Let U = {u₁, u₂, u₃, u₄} be a set of universe, E = {e₁, e₂, e₃} is a set of parameters, and T = {t₁, t₂, t₃} is a set of times. Define a function

Ft:A→IU,

as follows:

F1 (e1)={u1t10.6,u2t10.3,u3t10.2,u4t10.4},F1 (e2)={u1t10.5,u2t10.3,u3t10.2,u4t10.7},F1 (e3)={u1t10.3,u2t10.6,u3t10.8,u4t10.9},F2 (e1)={u1t20.7,u2t20.4,u3t20.1,u4t20.3},F2 (e2)={u1t20.5,u2t20.3,u3t20.2,u4t20.7},F2 (e3)={u1t20.7,u2t20.8,u3t20.6,u4t20.4},F3 (e1)={u1t30.7,u2t30.8,u3t30.6,u4t30.4},F3 (e2)={u1t30.7,u2t30.5,u3t30.6,u4t30.7},F3 (e3)={u1t30.3,u2t30.6,u3t30.8,u4t30.9}.

Let shdw = {(0.3, 0.8), (0.4, 0.9), (0.2, 0.6)} be the shadow parameter set related to Z. Define a function

Ftsh:A→IU.

Subsequently, we can find the time-shadow soft sets (Fsh(αi,βi),Z) consisting of the following collection of approximations:

(Fsh(αi,βi),Z)={(e1,{u1[0,1],u20,u30,u4[0,1]})(0.3,0.8)t1,(e2,{u1[0,1],u20,u30,u4[0,1]})(0.4,0.9)t1,(e3,{u1[0,1],u21,u31,u41})(0.2,0.6)t1,(e1,{u1[0,1],u2[0,1],u30,u40})(0.3,0.8)t2,(e2,{u1[0,1],u20,u30,u4[0,1]})(0.4,0.9)t2,(e3,{u11,u21,u31,u4[0,1]})(0.2,0.6)t2,(e1,{u1[0,1],u21,u3[0,1],u4[0,1]})(0.3,0.8)t3,(e2,{u1[0,1],u2[0,1],u3[0,1],u4[0,1]})(0.4,0.9)t3,(e3,{u1[0,1],u21,u31,u41})(0.2,0.6)t3}.

Here, a = [0, 1]

Definition 3.2. ((F,A)sh(αi,βi),Z) and ((G,B)sh(αi,βi),Z) over U and shdw(U) = {(α₁, β₁), (α₂, β₂), ..., (α_m, β_m)}, ((F,A)sh(αi,βi),Z), is called the T_sh-SS subset of ((G,B)sh(αi,βi),Z) if

A ⊆ B,
∀t ∈ T, ε ∈ A, F_t (ε) is fuzzy soft subset of G_t (ε).

Definition 3.3. Two T_sh − SSs, ((F,A)sh(αi,βi),Z]) and ((G,B)sh(αi,βi),Z) over U is said to be equal if ((F,A)sh(αi,βi),Z) is a T_sh − SS subset of ((G,B)sh(αi,βi),Z) and ((G,B)sh(αi,βi),Z) is a T_sh − SS subset of ((F,A)sh(αi,βi),Z).

Example 3.2. Consider Example 3.1 and suppose that

((F,A)sh(αi,βi),Z)={(e1,{u11,u2[0,1],u30,u4[0,1]})(0.3,0.8)t1,(e2,{u1[0,1],u2[0,1],u30,u41})(0.4,0.9)t1,(e2,{u1[0,1],u20,u3[0,1],u4[0,1]})(0.4,0.9)t2,(e3,{u11,u21,u31,u41})(0.2,0.6)t2,(e1,{u1[0,1],u21,u3[0,1],u41})(0.3,0.8)t3,(e3,{u1[0,1],u21,u31,u41})(0.2,0.6)t3},((G,B)sh(αi,βi),Z)={(e2,{u1t10,u2t10,u3t10,u4t1[0,1]})(0.4,0.9)t1,(e3,{u1t21,u221,u3t21,u4t21})(0.2,0.6)t2,(e1,{u1t3[0,1],u2t31,u3t3[0,1],u4t30})(0.3,0.8)t3}.

Therefore ((G,B)sh(αi,βi),Z)⊆((F,A)sh(αi,βi),Z).

Definition 3.4. A time-shadow soft set (F,A)tsh(αi,βi) over U is said to be semi-null T_sh − SS denoted by T_sh_∼_φ, if ∀t ∈ T, F_t_{_sh} (e) = φ for at least one e.

Definition 3.5. A time-shadow soft set (F,A)tsh(αi,βi) over U is considered null T_sh − SS denoted by T_sh_{_φ}, if ∀t ∈ T, F_t_{_sh}(e) = φ ∀e.

Definition 3.6. A time-shadow soft set (F,A)tsh(αi,βi) over U is considered semi-absolute T_sh − SS denoted by T_sh_∼_A, if ∀t ∈ T, F_t_{_sh} (e) = 1̄ for at least one e.

Definition 3.7. A time-shadow soft set (F,A)tsh(αi,βi) over U is considered absolute T_sh − SS denoted by T_sh_{_A}, if ∀t ∈ T, F_tsh (e) = 1̄∀e.

Example 3.3. Consider Example 3.1. Let

(F,A)tsh(αi,βi)={(e1(0.3,0.8),{Φ}t1),(e2(0.4,0.9),{u1t10,u2t1[0,1],u3t10,u4t1[0,1]}),(e3(0.2,0.6),{u1t10,u2t1[0,1],u3t1[0,1],u4t1[0,1]}),(e1(0.3,0.8),{Φ}t2),(e2(0.4,0.9),{u1t2[0,1],u2t2[0,1],u3t2[0,1],u4t20.3})(e3(0.4,0.9),{u1t20,u2t20,u3t20,u4t2[0,1]}),(e1(0.3,0.8),{Φ}t3),(e2(0.4,0.9),{u1t30,u2t3[0,1],u3t30,u4t30}),(e3(0.2,0.6),{u1t31,u2t3[0,1],u3t30,u4t30})}.

Then (F,A)tsh(αi,βi)=Tsh~φ.

Let

(F,A)tsh(αi,βi)={(e1(0.3,0.8),{Φ}t1),(e2(0.4,0.9),{Φ}t1),(e3(0.2,0.8),{Φ}t1),(e1(0.3,0.8),{Φ}t2),(e2(0.4,0.9),{Φ}t2),(e3(0.2,0.6),{Φ}t2),(e1(0.3,0.8),{Φ}t3),(e2(0.4,0.9),{Φ}t3),(e3(0.2,0.6),{Φ}t3)}.

Then (F,A)tsh(αi,βi)=Tshφ.

(F,A)tsh (αi,βi)={(e1(0.3,0.8),{u1t11,u2t11,u3t11,u4t11}),(e2(0.4,0.9),{u1t11,u2t11,u3t11,u4t11}),(e3(0.2,0.6),{u1t11,u2t11,u3t11,u4t11}),(e1(0.3,0.8),{u1t21,u2t21,u3t21,u4t21}),(e2(0.4,0.9),{u1t21,u2t21,u3t21,u4t21}),(e3(0.2,0.6),{u1t21,u2t21,u3t21,u4t21}),(e1(0.3,0.8),{u1t3[0,1],u2t31,u3t31,u4t3[0,1]}),(e2(0.4,0.9),{u1t30,u2t3[0,1],u3t30,u4t30}),(e3(0.2,0.6),{u1t3[0,1],u2t3[0,1],u3t30.2,u4t30})}.

Then (F,A)tsh(αi,βi)=Tsh~A.

Let

(F,A)tsh(αi,βi)={(e1(0.3,0.8),{u1t11,u2t11,u3t11,u4t11}),(e2(0.4,0.9),{u1t11,u2t11,u3t11,u4t11}),(e2(0.4,0.9),{u1t21,u2t21,u3t21,u4t21}),(e3(0.2,0.6),{u1t21,u2t21,u3t21,u4t21}),(e1(0.3,0.8),{u1t31,u2t31,u3t31,u4t31}),(e3(0.2,0.6),{u1t31,u2t31,u3t31,u4t31})}.

Then (F,A)tsh(αi,βi)=TshA.

3.1 Basic Operations

Here, we introduce some basic operations on the time-shadow soft set, namely complement, union, and intersection, and provide some properties related to these operations.

Definition 3.8. Let (Fsh(αi,βi),Z) be a time-shadowed soft set over U. Then the complement of (Fsh(αi,βi),Z), denoted by (Fsh(αi,βi),Z)c and is defined by (Fsh(αi,βi),Z)c=c(Fsh(αi,βi),Z), where c denotes a time-shadow soft complement.

Example 3.4. Consider a time-shadow soft set (Fsh(αi,βi),Z) over U as in Example 3.1. Subsequently, we can find the complement of (Fsh(αi,βi),Z) as follows:

(Fsh(αi,βi),Z)c={(e1,{u1[0,1],u21,u31,u4[0,1]})(0.3,0.8)t1,(e2,{u1[0,1],u21,u31,u4[0,1]})(0.4,0.9)t1,(e3,{u1[0,1],u20,u30,u40})(0.2,0.8)t1,(e1,{u1[0,1],u2[0,1],u31,u41})(0.3,0.8)t2,(e2,{u1[0,1],u21,u31,u4[0,1]})(0.4,0.9)t2,(e3,{u10,u20,u30,u4[0,1]})(0.2,0.6)t2,(e1,{u1[0,1],u20,u3[0,1],u4[0,1]})(0.3,0.8)t3,(e2,{u1[0,1],u2[0,1],u3[0,1],u4[0,1]})(0.4,0.9)t3,(e3,{u1[0,1],u20,u31,u40})(0.2,0.6)t3}.

Proposition 3.1. Let (F,E)tsh(αi,βi) be a time-shadow soft set over U. Thus, the following holds.

((F,E)tsh(αi,βi)c)c=(F,E)tsh(αi,βi).

Proof. The proof of this is straightforward.

Definition 3.9. The union of two time-shadow soft sets ((F,A)sh(αi,βi),Z) and ((G,B)sh(αi,βi),Z) over a common universe U and a time-shadow parameters set is a time-shadow soft set ((H,C)sh(αi,βi),Z), denoted by ((F,A)sh(αi,βi),Z)∪˜((G,B)sh(αi,βi),Z), such that C = A ∪ B ⊂ Z and defined as follows:

((H,C)sh(αi,βi),Z) (ɛ)={Ftsh(αi,βi)(ɛ),if ɛ∈A-B,Gtsh(αi,βi)(ɛ),if ɛ∈B-A,Ftsh(αi,βi)(ɛ)∪˜Gtsh(αi,βi)(ɛ),if ɛ∈A∩B,

where Ũ denotes the time-shadow of the soft union.

Example 3.5. Consider Example 3.1. Suppose (F,A)_t_{_sh}and (G,B)_t_{_sh}are two time-shadow soft sets over U such that

(F,A)tsh(αi,βi)=(e1(0.3,0.8),{u1t1[0,1]u2t10u3t10,u4t1[0,1]}),(e2(0.4,0.9),{u1t1[0,1]u2t10,u3t10,u4t1[0,1]}),(e2(0.4,0.9),{u1t2[0,1]u2t20,u3t20,u4t2[0,1]}),(e3(0.2,0.6),{u1t3[0,1]u2t31u3t31,u4t31})},(G,B)t(αi,βi)={(e1(0.3,0.8),{u1t1[0,1],u2t1[0,1],u3t10,u4t10}),(e3(0.2,0.6),{u1t2[0,1],u2t21,u3t21,u4t2[0,1]}),(e3(0.2,0.6),{u1t3[0,1],u2t31,u3t31,u4t30})},(H,C)tsh(αi,βi)={(e1(0.3,0.8),{u1t1[0,1],u2t1[0,1],u3t10,u4t1[0,1]}),(e2(0.4,0.9),{u1t1[0,1],u2t10,u3t10,u4t1[0,1]}),(e2(0.4,0.9),{u1t2[0,1],u2t20,u3t20,u4t2[0,1]}),(e3(0.2,0.6),{u1t20.5,u2t20.6,u3t20.9,u4t20.4}),(e3(0.2,0.6),{u1t3[0,1],u2t31,u3t31,u4t31})}.

Proposition 3.2. If ((F,A)sh(αi,βi),Z),((G,B)sh(αi,βi),Z) and ((H,C)sh(αi,βi),Z) are three T_t_{_sh} − SSs over U. Subsequently, the following results hold.

1. ((F,A)sh(αi,βi),Z)∪˜((G,B)sh(αi,βi),Z)∪˜((H,C)sh(αi,βi),Z))=((F,A)tsh(αt,βt)∪˜(G,B)tsh(αi,βi))∪˜(H,C)tsh(αi,βi),
2. ((F,A)sh(αi,βi),Z)∪˜((F,A)sh(αi,βi),Z)=((F,A)sh(αi,βi),Z).
3. ((F,A)sh(αi,βi),Z)∪˜((G,B)sh(αi,βi),Z)=((G,B)sh(αi,βi),Z)∪˜((F,A)sh(αi,βi),Z).
4. ((F,A)sh(αi,βi),Z)∪φtsh=((F,A)sh(αi,βi),Z).

Proof. The proof of this is straightforward.

Definition 3.10. The intersection of two time-shadow soft sets ((F,A)sh(αi,βi),Z) and ((G,B)sh(αi,βi),Z) over a common universe U and a time-shadow parameters set is a time-shadow soft set ((H,C)sh(αi,βi),Z), denoted by ((F,A)sh(αi,βi),Z)∩˜((G,B)sh(αi,βi),Z), such that C = A ∩ B ⊂ Z and defined as follows:

((H,C)sh(αi,βi),Z) (ɛ)={Ftsh(αi,βi) (ɛ),if ɛ∈A-B,Gtsh(αi,βi) (ɛ),if ɛ∈B-A,Ftsh(αi,βi) (ɛ)∩˜Gtsh(αi,βi) (ɛ),if ɛ∈A∩B,

where ∩̃ denotes the time-shadow of the soft union.

Example 3.6. Consider Example 3.1. Suppose (F,A)_t_{_sh} and (G,B)_t_{_sh} are two time-shadow soft sets over U such that

(F,A)tsh(αi,βi)={(e1(0.3,0.8),{u1t1[0,1],u2t10,u3t10,u4t1[0,1]}),(e2(0.4,0.9),{u1t1[0,1],u2t10,u3t10,u4t1[0,1]}),(e2(0.4,0.9),{u1t2[0,1],u2t20,u3t20,u4t2[0,1]}),(e3(0.2,0.6),{u1t3[0,1],u2t31,u3t31,u4t31})},(G,B)tsh(αi,βi)={(e1(0.3,0.8),{u1t1[0,1],u2t1[0,1],u3t10,u4t10}),(e3(0.2,0.6),{u1t2[0,1],u2t21,u3t21,u4t2[0,1]}),(e3(0.2,0.6),{u1t3[0,1],u2t31,u3t31,u4t31})}.(H,C)tsh={(e1(0.3,0.8),{u1t1[0,1],u2t10,u3t10,u4t1[0,1]}),(e2(0.4,0.9),{u1t1[0,1],u2t10,u3t10,u4t1[0,1]}),(e2(0.4,0.9),{u1t2[0,1],u2t20,u3t20,u4t2[0,1]}),(e3(0.2,0.6),{u1t20.5,u2t20.6,u3t20.9,u4t20.4}),(e3(0.2,0.6),{u1t3[0,1],u2t31,u3t31,u4t31})}.

Proposition 3.3. If ((F,A)sh(αi,βi),Z),((G,B)sh(αi,βi),Z) and ((H,C)sh(αi,βi),Z) are three T_t_{_sh}–SSs over U. Then, the following results hold.

1. ((F,A)sh(αi,βi),Z)∩˜((G,B)sh(αi,βi),Z)∩˜((H,C)sh(αi,βi),Z))=((F,A)tsh∪˜(G,B)tsh)∪˜(H,C)tsh.
2. ((F,A)sh(αi,βi),Z)∩˜((F,A)sh(αi,βi),Z)=((F,A)sh(αi,βi),Z).
3. ((F,A)sh(αi,βi),Z)∩˜((G,B)sh(αi,βi),Z)=((G,B)sh(αi,βi),Z)∩˜((F,A)sh(αi,βi),Z).
4. ((F,A)sh(αi,βi),Z)∩φtsh((F,A)sh(αi,βi),Z).

Proof. The proof of this is straightforward.

Proposition 3.4. If ((F,A)sh(αi,βi),Z),((G,B)sh(αi,βi),Z) and ((H,C)sh(αi,βi),Z) are three T_sh − SSs over U, then

1. ((F,A)sh(αi,βi),Z)∪˜(((G,B)sh(αi,βi),Z)∩˜(H,C)tsh(αi,βi),Z)=((F,A)sh(αi,βi),Z)∪˜(∩˜((F,A)sh(αi,βi),Z)∪˜((H,C)sh(αi,βi),Z).
2. ((F,A)sh(αi,βi),Z)∩˜(((G,B)sh(αi,βi),Z)∪˜(H,C)tsh(αi,βi),Z)=(((F,A)sh(αi,βi),Z)∩˜((F,A)sh(αi,βi),Z)∪˜((F,A)sh(αi,βi),Z)∩˜((H,C)sh(αi,βi),Z).

Proof. The proof of this is straightforward.

Proposition 3.5. If ((F,A)sh(αi,βi),Z) and ((G,)sh(αi,βi),Z) are two T_sh − SSs over U, then

1. ((F,A)tsh(αi,βi),Z)∪˜((G,B)tsh(αi,βi),Z)c=((F,A)tshc(αi,βi),Z)∩˜((G,B)tshc(αi,βi),Z).
2. ((F,A)tsh(αi,βi),Z)∩˜((G,B)tsh(αi,βi),Z)c=((F,A)tshc(αi,βi),Z)∪˜((G,B)tshc(αi,βi),Z).

Proof. The proof of this is straightforward.

4. AND and OR Operations

Go to

Abstract
1. Introduction
2. Preliminaries
3. Time-Shadow Soft Set
4. AND and OR Operations
5. An Application of Time-Shadow Soft Set in Decision-Making
6. Conclusion
Acknowledgments
Conflict of Interest
Acknowledgments
Conflict of Interest
Tables
References
Biography

In this section, we introduce the definitions of “AND and OR” operations for T_sh − SSs, derive their properties, and provide some examples.

Definition 4.1. If ((F,A)sh(αi,βi),Z)and ((G,B)sh(αi,βi),Z) are two T_sh − SS values over the U then “ ((F,A)sh(αi,βi),Z) AND ((G,B)sh(αi,βi),Z) is denoted by ((F,A)sh(αi,βi),Z)∧((G,B)sh(αi,βi),Z) is defined as follows:

((F,A)sh(αi,βi),Z)∧((G,B)sh(αi,βi),Z)=(H,A×B)tsh(αi,βi),Z))

such that H(α,β)tsh(αi,βi)=F(α)tsh(αi,βi)∩˜G(β)tsh(αi,βi), ∀(α_i, β_i) ∈ A × B, where ∩̃ is a time-shadow soft intersection.

Example 4.1. Consider Example 3.1. Suppose (F,A)tsh(αi,βi) and (G,B)tsh(αi,βi) are two time-shadow soft sets over U such that

(F,A)tsh(αi,βi)={(e1(0.3,0.8),{u1t1[0,1],u2t10,u3t10,u4t1[0,1]}),(e2(0.4,0.7),{u1t1[0,1],u2t10,u3t10,u4t11}),(e2(0.4,0.7),{u1t20,u2t2[0,1],u3t21,u4t20}),e3(0.2,0.9),{u1t3[0,1],u2t3[0,1],u3t3[0,1],u4t31})},(G,B)tsh(αi,βi)=(e1(0.3,0.8),{u1t11,u2t1[0,1],andu3t10,u4t10}),(e3(0.2,0.9),{u1t2[0,1],u2t2[0,1],u3t2[0,1],u4t2[0,1]}),(e3(0.2,0.9),{u1t30,u2t2[0,1],u3t3[0,1],u4t31})}.

Then

(F,A)tsh(αi,βi)∧(G,B)tsh(αi,βi)=(H,A×B)tsh(αt,βt)={((e1t1,e1t1) (0.3,0.8),{u1t1,1[0,1],u2t1,10,u3t1,10,u4t1,10}),((e1t1,e3t2) (0.2,0.8),{u1t1,2[0,1],u2t1,2[0,1],u3t1,20,u4t1,2[0,1]}),((e1t1,e1t3) (0.2,0.8),{u1t1,30,u2t1,3[0,1],u3t1,30,u4t1,3[0,1]}),((e2t1,e1t1) (0.3,0.7),{u1t1,1[0,1],u2t1,10,u3t1,10,u4t1,10}),((e2t1,e3t2) (0.3,0.7),{u1t1,2[0,1],u2t1,20,u3t1,20,u4t1,21}),((e2t1,e3t3) (0.2,0.7),{u1t1,30,u2t1,3[0,1],u3t1,30,u4t1,31}),((e2t2,e1t1) (0.3,0.7),{u1t2,1[0,1],u2t2,1[0,1],u3t2,10,u4t2,10}),((e2t2,e3t2) (0.2,0.7),{u1t2,2[0,1],u2t2,2[0,1],u3t2,2[0,1],u4t2,2[0,1]}),((e2t2,e3t3) (0.2,0.7),{u1t2,30,u2t2,3[0,1],u3t2,31,u4t2,3[0,1]}),((e3t3,e1t1) (0.2,0.8),{u1t3,1[0,1],u2t3,1[0,1],u3t3,10,u4t3,10}),((e3t3,e3t2) (0.2,0.9),{u1t3,2[0,1],u2t3,20.6,u3t3,2[0,1],u4t3,2[0,1]}),((e3t3,e3t3) (0.2,0.3),{u1t3,30,u2t3,31,u3t3,31,u4t3,31})}.

Definition 4.2. If ((F,A)sh(αi,βi),Z) and ((G,B)sh(αi,βi),Z) over U then “((F,A)sh(αi,βi),Z) OR ((G,B)sh(αi,βi),Z),” which is denoted by ((F,A)sh(αi,βi),Z)∨((G,B)sh(αi,βi),Z), is defined by

((F,A)sh(αi,βi),Z)∨((G,B)sh(αi,βi),Z)=(H,A×B,)tsh(αi,βi,Z),

such that H(α,β)tsh(αi,βi)=F (α)tsh(αi,βi)∪˜G (β)tsh(αi,βi), ∀(α_i, β_i) ∈ A × B, where ũ is a time-shadow soft union.

Example 4.2. Consider Example 4.1 We have U then, “(F,A)sh(αi,βi) OR (G,B)sh(αi,βi)” denoted as (H,Csh(αi,βi))t=(F,A)sh(αi,βi)∨(G,B)sh(αi,βi) where

(H,C)sh(αi,βi)={((e1t1,e1t1) (0.3,0.8),{u1t1,11,u2t1,1[0,1],u3t1,10,u4t1,1[0,1]}),((e1t1,e3t2) (0.3,0.9),{u1t1,2[0,1],u2t1,2[0,1],u3t1,2[0,1],u4t1,2[0,1]})((e1t1,e3t3) (0.3,0.9),{u1t1,3[0,1],u2t1,3[0,1],u3t1,3[0,1],u4t1,31}),((e2t1,e1t1) (0.4,0.8),{u1t1,11,u2t1,1[0,1],u3t1,10,u4t1,1[0,1]}),((e2t1,e3t2) (0.4,0.9),{u1t1,2[0,1],u2t1,2[0,1],u3t1,2[0,1],u4t1,2[0,1]}),((e2t1,e3t3) (0.4,0.9),{u1t1,3[0,1],u2t1,3[0,1],u3t1,3[0,1],u4t1,31}),((e2t2,e1t1) (0.4,0.8),{u1t2,11,u2t2,1[0,1],u3t2,11,u4t2,10}),((e2t2,e3t2) (0.4,0.9),{u1t2,20,u2t2,2[0,1],u3t2,2[0,1],u4t2,2[0,1]}),((e2t2,e3t3) (0.4,0.9),{u1t2,30,u2t2,3[0,1],u3t2,3[0,1],u4t2,31}),((e3t3,e1t1) (0.3,0.9),{u1t3,1[0,1],u2t3,1[0,1],u3t3,1[0,1],u4t3,11}),((e3t3,e3t2) (0.2,0.9),{u1t3,2[0,1],u2t3,2[0,1],u3t3,2[0,1],u4t3,21}),((e3t3,e3t3) (0.2,0.9),{u1t3,30,u2t3,3[0,1],u3t3,3[0,1],u4t3,31})}.

Proposition 4.1. Let ((F,A)sh(αi,βi),Z) and ((G,B)sh(αi,βi),Z) be any two time-shadow soft sets. Subsequently, the following results hold.

1. (((F,A)sh(αi,βi),Z)∧((G,B)sh(αi,βi),Z))c=((F,A)sh(αi,βi),Z)c∨((G,B)sh(αi,βi),Z)c.
2. (((F,A)sh(αi,βi),Z)∨((G,B)sh(αi,βi),Z))c=((F,A)sh(αi,βi),Z)c∧((G,B)sh(αi,βi),Z)c.

Proof. Straightforward from Definitions 3.8, 4.1 and 4.2.

Proposition 4.2. Let ((F,A)sh(αi,βi),Z),((G,B)sh(αi,βi),Z) and ((H,C)sh(αi,βi),Z) be any three shadow soft sets. Subsequently, the following results hold.

1. (((F,A)sh(αi,βi),Z)∧(((G,B)sh(αi,βi),Z)∧((H,C)sh(αi,βi),Z)=(((F,A)sh(αi,βi),Z)∧((G,B)sh(αi,βi),Z)∧(((H,C)sh(αi,βi)),Z).
2. (((F,A)sh(αi,βi),Z)∨(((G,B)sh(αi,βi),Z)∨((H,C)sh(αi,βi),Z)=(((F,A)sh(αi,βi),Z)∨((G,B)sh(αi,βi),Z)∨(((H,C)sh(αi,βi)),Z)).
3. (((F,A)sh(αi,βi),Z)∨(((G,B)sh(αi,βi),Z)∧((H,C)sh(αi,βi),Z)=(((F,A)sh(αi,βi),Z)∨((G,B)sh(αi,βi),Z)∧(((F,A)sh(αi,βi)),Z)∨((H,C)sh(αi,βi),Z).
4. (((F,A)sh(αi,βi),Z)∧(((G,B)sh(αi,βi),Z)∧((H,C)sh(αi,βi),Z)=(((F,A)sh(αi,βi),Z)∧((G,B)sh(αi,βi),Z)∨(((F,A)sh(αi,βi)),Z)∧((H,C)sh(αi,βi),Z)).

Proof. Straightforward from Definitions 4.1 and 4.2.

5. An Application of Time-Shadow Soft Set in Decision-Making

Go to

Abstract
1. Introduction
2. Preliminaries
3. Time-Shadow Soft Set
4. AND and OR Operations
5. An Application of Time-Shadow Soft Set in Decision-Making
6. Conclusion
Acknowledgments
Conflict of Interest
Acknowledgments
Conflict of Interest
Tables
References
Biography

In this section, we apply the time-shadow soft set theory to a decision-making problem. Suppose that one of the broadcasting channels plans to invite experts to evaluate its show by discussing a controversial issue. The producers of the show used the following criteria to evaluate their findings: the four alternatives are U = {u₁, u₂, u₃, u₄}. Suppose there are five parameters E = {e₁, e₂, e₃, e₄, e₅}, that choose the experts for the programs. For i = 1, 2, 3, 4, 5 the parameters e_i (i = 1, 2, 3, 4, 5) stands for “this criteria to discriminate,” “this criterion is independent of the other criteria,” “this criteria measures one thing,” “the universal criteria,” and “the criteria that are important to some of the stakeholders.” T = {t₁, t₂, t₃} is a set of previous time periods, and let shdw = {(0.3, 0.8), (0.4, 0.7), (0.3, 0.5), (0.4, 0.6), (0.3, 0.8)} be the shadow parameter set related to E. Based on these findings, we determined the most suitable choice. After extensive discussion, the committee constructed the following time-shadow soft set:

(F,E)t={(e1,{u1t10.6,u2t10.3,u3t10.2,u4t10.4}),(e2,{u1t10.5,u2t10.3,u3t10.2,u4t10.7}),(e3,{u1t10.3,u2t10.6,u3t10.8,u4t10.9}),(e4,{u1t10.5,u2t10.4,u3t10.6,u4t10.8}),(e5,{u1t10.9,u2t10.2,u3t10.4,u4t10.8}),(e1,{u1t20.7,u2t20.4,u3t20.1,u4t20.3}),(e2,{u1t20.3,u2t20.1,u3t20.2,u4t20.6}),(e3,{u1t20.7,u2t20.8,u3t20.6,u4t20.4}),(e4,{u1t20.9,u2t20.3,u3t20.4,u4t20.7}),(e5,{u1t20.2,u2t20.7,u3t20.8,u4t20.5}),(e1,{u1t30.7,u2t30.8,u3t30.6,u4t30.4}),(e2,{u1t30.7,u2t30.5,u3t30.6,u4t30.4}),(e3,{u1t30.6,u2t30.4,u3t30.5,u4t30.7}),(e4,{u1t30.6,u2t30.7,u3t30.5,u4t30.3}),(e5,{u1t30.7,u2t30.2,u3t30.6,u4t30.3}),},(F,E)sh(αt,βt)={(e1(0.3,0.8),{u1t1[0,1],u2t10,u3t10,u4t1[0,1]}),(e2(0.4,0.7),{u1t1[0,1],u2t10,u3t10,u4t11}),(e3(0.3,0.5),{u1t10,u2t11,u3t11,u4t11}),(e4(0.4,0.6),{u1t1[0,1],u2t10,u3t11,u4t11}),(e5(0.3,0.8),{u1t11,u2t10,u3t1[0,1],u4t11}),(e1(0.3,0.8),{u1t2[0,1],u2t2[0,1],u3t20,u4t20}),(e2(0.4,0.7),{u1t20,u2t20,u3t20,u4t2[0,1]}),(e3(0.3,0.5),{u1t21,u2t21,u3t21,u4t2[0]}),(e4(0.4,0.6),{u1t21,u2t20,u3t20,u4t21}),(e5(0.3,0.8),{u1t20,u2t2[0,1],u3t21,u4t2[0,1]}),(e1(0.3,0.8),{u1t3[0,1],u2t31,u3t3[0,1],u4t3[0,1]}),(e2(0.4,0.7),{u1t31,u2t3[0,1],u3t3[0,1],u4t30}),(e3(0.3,0.5),{u1t31,u2t3[0,1],u3t31,u4t31}),(e4(0.4,0.6),{u1t31,u2t31,u3t3[0,1],u4t30}),(e5(0.3,0.8),{u1t3[0,1],u2t30,u3t3[0,1],u4t30})}.

5.1 Algorithm

1. Find the tabular representation of (F,E)_t as in Table 1.
2. Find the tabular representation of F (E) as in Table 2, where F (E) defined as follows:

F (e)={u∑i=1ntiFt(e)\n∑i=1nFt(e):u∈U,e∈E}

where n = |T|.
3. Find the tabular representation of (F,E)_t_{_sh} as in Table 3.
4. Find the score of each element in U as in Table 4.

Table 4 shows that choice u₃ has the highest acceptance score, whereas choice u₄ has the highest waiting score, and there is no rejection choice.

6. Conclusion

Go to

Abstract
1. Introduction
2. Preliminaries
3. Time-Shadow Soft Set
4. AND and OR Operations
5. An Application of Time-Shadow Soft Set in Decision-Making
6. Conclusion
Acknowledgments
Conflict of Interest
Acknowledgments
Conflict of Interest
Tables
References
Biography

This study introduced the time-shadow soft expert set concept and studied some of its properties. The complement, union, and intersection operations are defined on the time-fuzzy soft set. This theory is hypothetically applied to address decision-making problems.

Acknowledgments

Go to

Abstract
1. Introduction
2. Preliminaries
3. Time-Shadow Soft Set
4. AND and OR Operations
5. An Application of Time-Shadow Soft Set in Decision-Making
6. Conclusion
Acknowledgments
Conflict of Interest
Acknowledgments
Conflict of Interest
Tables
References
Biography

The authors acknowledge the financial support received from Jadara University.

Conflict of Interest

Go to

Abstract
1. Introduction
2. Preliminaries
3. Time-Shadow Soft Set
4. AND and OR Operations
5. An Application of Time-Shadow Soft Set in Decision-Making
6. Conclusion
Acknowledgments
Conflict of Interest
Acknowledgments
Conflict of Interest
Tables
References
Biography

No potential conflict of interest relevant to this article was reported.

Acknowledgments

Go to

Abstract
1. Introduction
2. Preliminaries
3. Time-Shadow Soft Set
4. AND and OR Operations
5. An Application of Time-Shadow Soft Set in Decision-Making
6. Conclusion
Acknowledgments
Conflict of Interest
Acknowledgments
Conflict of Interest
Tables
References
Biography

The authors acknowledge the financial support received from Jadara University.

Conflict of Interest

Go to

Abstract
1. Introduction
2. Preliminaries
3. Time-Shadow Soft Set
4. AND and OR Operations
5. An Application of Time-Shadow Soft Set in Decision-Making
6. Conclusion
Acknowledgments
Conflict of Interest
Acknowledgments
Conflict of Interest
Tables
References
Biography

No potential conflict of interest relevant to this article was reported.

Tables

Go to

Abstract
1. Introduction
2. Preliminaries
3. Time-Shadow Soft Set
4. AND and OR Operations
5. An Application of Time-Shadow Soft Set in Decision-Making
6. Conclusion
Acknowledgments
Conflict of Interest
Acknowledgments
Conflict of Interest
Tables
References
Biography

Table. 1.

Table 1. Tabular representation of (F,E)_t.

U	u₁	u₂	u₃	u₄
(e₁, t₁)	0.6	0.3	0.2	0.4
(e₁, t₂)	0.7	0.4	0.1	0.3
(e₁, t₃)	0.7	0.8	0.6	0.4
(e₂, t₁)	0.5	0.3	0.2	0.7
(e₂, t₂)	0.3	0.1	0.2	0.6
(e₂, t₃)	0.7	0.5	0.6	0.4
(e₃, t₁)	0.3	0.6	0.8	0.9
(e₃, t₂)	0.7	0.8	0.6	0.4
(e₃, t₃)	0.6	0.4	0.5	0.7
(e₄, t₁)	0.5	0.4	0.6	0.8
(e₄, t₂)	0.9	0.3	0.4	0.7
(e₄, t₃)	0.6	0.7	0.5	0.3
(e₅, t₁)	0.9	0.2	0.4	0.8
(e₅, t₂)	0.2	0.7	0.8	0.5
(e₅, t₃)	0.7	0.2	0.6	0.3

Table. 2.

Table 2. Tabular representation of F (E).

U	u₁	u₂	u₃	u₄
e₁	0.68	0.77	0.81	0.66
e₂	0.51	0.66	0.80	0.60
e₃	0.72	0.62	0.61	0.63
e₄	0.68	0.93	0.64	0.49
e₅	0.62	0.66	0.70	0.77

Table. 3.

Table 3. Tabular representation of (F,E)_t_{_sh}.

U	u₁	u₂	u₃	u₄
e₁	[0,1]	[0,1]	1	[0,1]
e₂	[0,1]	[0,1]	1	[0,1]
e₃	1	1	1	1
e₄	1	1	1	[0,1]
e₅	[0,1]	[0,1]	[0,1]	[0,1]

Table. 4.

Table 4. Score table.

U	r₁	r_[0,1]
u₁	2	3
u₂	2	3
u₃	4	1
u₄	1	4

References

Go to

Abstract
1. Introduction
2. Preliminaries
3. Time-Shadow Soft Set
4. AND and OR Operations
5. An Application of Time-Shadow Soft Set in Decision-Making
6. Conclusion
Acknowledgments
Conflict of Interest
Acknowledgments
Conflict of Interest
Tables
References
Biography

Molodtsov, D (1999). Soft set theory: first results. Computers & Mathematics with Applications. 37, 19-31. https://doi.org/10.1016/S0898-1221(99)00056-5
Chen, D, Tsang, ECC, Yeung, DS, and Wang, X (2005). The parameterization reduction of soft sets and its applications. Computers & Mathematics with Applications. 49, 757-763. https://doi.org/10.1016/j.camwa.2004.10.036
Maji, PK, Biswas, R, and Roy, AR (2003). Soft set theory. Computers & Mathematics with Applications. 45, 555-562. https://doi.org/10.1016/S0898-1221(03)00016-6
Maji, PK, Roy, AR, and Biswas, R (2022). An application of soft sets in a decision making problem. Computers & Mathematics with Applications. 44, 1077-1083. https://doi.org/10.1016/S0898-1221(02)00216-X
Maji, PK, Biswas, R, and Roy, AR (2001). Fuzzy soft sets. Journal of Fuzzy Mathematics. 9, 589-602.
Shakhatreh, M, and Qawasmeh, T . Associativity of maxmin composition of three fuzzy relations., Proceedings of the 28th International Conference of The Jangion Mathematical Society (ICJMS), 2015, Antalya, Turkey.
Hazaymeh, A, Abdullah, IB, Balkhi, Z, and Ibrahim, R (2012). Fuzzy parameterized fuzzy soft expert set. Applied Mathematical Sciences. 6, 5547-5564.
Hazaymeh, A, Qazza, A, Hatamleh, R, Alomari, MW, and Saadeh, R (). On further refinements of numerical radius inequalities. Axioms. 12, 2023. article no 807
Hazaymeh, A, Saadeh, R, Hatamleh, R, Alomari, MW, and Qazza, A (). A perturbed Milne’s quadrature rule for n-times differentiable functions with Lp-error estimates. Axioms. 12, 2023. article no 803
Abuhijleh, EA, Massa’deh, M, Sheimat, A, and Alkouri, A (2021). Complex fuzzy groups based on Rosenfeld’s approach. WSEAS Transactions on Mathematics. 20, 368-377. https://doi.org/10.37394/23206.2021.20.38
Hazaymeh, AAM 2013. Fuzzy soft set and fuzzy soft expert set: some generalizations and hypothetical applications. Ph.D. dissertation. Universiti Sains Islam Malaysia, Negeri Sembilan. Malaysia. pp.397.
Alsharo, D, Abuteen, E, Abd Ulazeez, MJS, Alkhasawneh, M, and Al-Zubi, FM (2024). Complex shadowed set theory and its application in decision-making problems. AIMS Mathematics. 9, 16810-16825. https://doi.org/10.3934/math.2024815
Alkhazaleh, S, and Salleh, AR (2011). Soft expert sets. Advances in Decision Sciences, 2011. article no 757868
Alkhazaleh, S, and Salleh, AR (2014). Fuzzy soft expert set and its application. Applied Mathematics. 5, 1349-1368. https://doi.org/10.4236/am.2014.59127
Hazaymeh, AA, Abdullah, IB, Balkhi, ZT, and Ibrahim, RI (2012). Generalized fuzzy soft expert set. Journal of Applied Mathematics, 2012. article no 328195
Alqaraleh, SM, Abd Ulazeez, MJS, Massa’deh, MO, Talafha, AG, and Bataihah, A (2022). Bipolar complex fuzzy soft sets and their application. International Journal of Fuzzy System Applications (IJFSA). 11, 1-23. https://doi.org/10.4018/IJFSA.285551
Alkhazaleh, S, and Hazaymeh, AA (2018). N-valued refined neutrosophic soft sets and their applications in decision making problems and medical diagnosis. Journal of Artificial Intelligence and Soft Computing Research. 8, 79-86.
Roy, AR, and Maji, PK (2007). A fuzzy soft set theoretic approach to decision making problems. Journal of computational and Applied Mathematics. 203, 412-418. https://doi.org/10.1016/j.cam.2006.04.008
Hazaymeh, AA (2025). Time fuzzy soft sets and its application in design-making. International Journal of Neutrosophic Science. 25, 37-50. https://doi.org/10.54216/IJNS
Hazaymeh, A (2024). Time effective fuzzy soft set and its some applications with and without a neutrosophic. International Journal of Neutrosophic Science (IJNS). 23, 129-149. https://doi.org/10.54216/ijns.230211
Hazaymeh, AA (2025). Time factor’s impact on fuzzy soft expert sets. International Journal of Neutrosophic Science. 25, 155-176. https://doi.org/10.54216/ijns.250315
Alkhazaleh, S (2016). Time-neutrosophic soft set and its applications. Journal of Intelligent & Fuzzy Systems. 30, 1087-1098. https://doi.org/10.3233/IFS-151831
Alkhazaleh, S (2022). Shadow soft set theory. International Journal of Fuzzy Logic and Intelligent Systems. 22, 422-432. https://doi.org/10.5391/IJFIS.2022.22.4.422

Biography

Go to

Abstract
1. Introduction
2. Preliminaries
3. Time-Shadow Soft Set
4. AND and OR Operations
5. An Application of Time-Shadow Soft Set in Decision-Making
6. Conclusion
Acknowledgments
Conflict of Interest
Acknowledgments
Conflict of Interest
Tables
References
Biography

Ayman A. Hazaymeh is an associate professor of mathematics at Jadara University in Jordan. He received his M.A. degree from the Utara University of Malaysia (UUM) and his Ph.D. from the University Sains Islam Malaysia (USIM). He specializes in operation research, fuzzy sets, soft fuzzy sets, neutrosophic set, neutrosophic soft set; is interested in topics related to uncertainty; and has conducted extensive research in this field. He is currently working as a faculty member at the College of Science.

Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2024; 24(4): 387-398

Published online December 25, 2024 https://doi.org/10.5391/IJFIS.2024.24.4.387

Time-Shadow Soft Set: Concepts and Applications

Ayman A. Hazaymeh

Department of Mathematics, Faculty of Science, Jadara University, Irbid, Jordan

Correspondence to:Ayman Hazaymeh (aymanha@jadara.edu.jo)

Received: March 8, 2023; Revised: August 20, 2024; Accepted: September 30, 2024

Abstract

Keywords: Soft sets, Fuzzy soft sets, Time fuzzy soft sets, Shadow soft sets, Time-shadow soft sets

1. Introduction

2. Preliminaries

Definition 2.1 [?]. Let us consider the mapping

F : V \to P (U) .

Definition 2.2 [5]. Let U be an initial universal set and E be a set of parameters. Let I^U denote the power set of all fuzzy subsets of U. Let V ⊆ E and F be mappings

F : V \to I^{U} .

A pair (F,E) is called a fuzzy soft set over U.

Definition 2.3 [5]. For two fuzzy soft sets (F, V ) and (G,W) over U, (F, V ) is called a fuzzy soft subset of (G,W) if

1. V ⊂ W and
2. ∀ɛ ∈ V, F (ɛ) is fuzzy subset of G(ɛ).

This relationship is denoted as (F, V ) ⊆̃ (G,W). Here, (G,W) is called the fuzzy soft superset of (F, V ).

Definition 2.4 [5]. The complement of a fuzzy soft set (F, V ) is denoted by (F, V )^c and is defined by (F, V )^c = (F^c, ⌉V ) where F^c : ⌉V → P(U) is a mapping given by

F^{c} (α) = c (F (⌉ α)) \forall α \in ⌉ V,

where c denotes a fuzzy complement.

Definition 2.5 [5]. If (F, V ) and (G,W) are two fuzzy soft sets then “(F, V ) AND (G,W)” denoted by (F, V ) ∧ (G,W) is defined by

(F, V) \land (G, W) = (H, V \times W),

such that H (α, β) = t (F (α) ,G(β)), ∀ (α, β) ∈ V × W, where t is any t-norm.

Definition 2.6 [5]. If (F, V ) and (G,W) are two fuzzy soft sets then “(F, V ) OR (G,W)” denoted by (F, V ) ∨ (G,W) is defined by

(F, V) \lor (G, W) = (O, V \times W),

such that O (α, β) = s (F (α) ,G(β)), ∀ (α, β) ∈ V × W, where s represents the s-norm.

Definition 2.7 [5]. The union of two fuzzy soft sets (F, V ) and (G,W) over a common universe U is the fuzzy soft set (H,Z) where Z = V ∪W, and ∀ɛ ∈ Z,

H (ɛ) = {\begin{array}{l} F (ɛ), & if ɛ \in V - W, \\ G (ɛ), & if ɛ \in W - V, \\ s (F (ɛ), G (ɛ)), & if ɛ \in V \cap W, \end{array}

where s is any s-norm.

Definition 2.8 [5]. The intersection of two fuzzy soft sets (F, V ) and (G,W) over a common universe U is the fuzzy soft set (Z,C) where Z = V ∪W, and ∀ɛ ∈ Z,

H (ɛ) = {\begin{array}{l} F (ɛ), & if ɛ \in V - W, \\ G (ɛ), & if ɛ \in W - V, \\ s (F (ɛ), G (ɛ)), & if ɛ \in V \cap W . \end{array}

F_{t} : V \to I^{U} .

Definition 2.10 [19]. A time-fuzzy soft set (F, V )_t over U is said to be a semi-null T-FSS denoted by T_∼φ if ∀t ∈ T and F_t (e) = φ for at least one e.

Definition 2.11 [19]. A time-fuzzy soft set (F, V )_t over U is considered a null T-FSS denoted by T_φ if ∀t ∈ T, F_t (e) = φ ∀e.

Definition 2.12 [19]. A time-fuzzy soft set (F, V )_t over U is considered a semi-absolute T-FSS denoted by T_∼V if ∀t ∈ T and F_t (e) = 1̄ for at least one e.

Definition 2.13 [19]/ A time-fuzzy soft set (F, V )_t over U is considered an absolute T-FSS denoted by T_V if ∀t ∈ T and F_t (e) = 1̄ ∀e.

Definition 2.14 [19]. The complement of T-FSS (F, V )_t is denoted by c̃(F, V )_t ∀t ∈ T where c̃ is a fuzzy soft complement.

F_{(α_{i}, β_{i})} (e_{i}) = {\frac{x_{j}}{f_{j} (x_{j})}},

∀i = 1, 2, ..., m, j = 1, 2, ..., n, where

f_{j} (x_{j}) = {\begin{array}{l} 0, & if μ_{j} (x_{j}) \leq α, \\ 1, & if μ_{j} (x_{j}) \geq β, \\ a, & α < μ_{j} (x_{j}) < β . \end{array}

Definition 2.17 [23]. A null shadow soft set (φ,E)_shdw over the common universe U is a shadow soft set with φ(e)_shdw(x) = 0, ∀e ∈ E; x ∈ U.

Definition 2.18 [23]. An absolute shadow soft set (Φ,E)_shdw over an the common universe U is a shadow soft set with Φ(e)_shdw(x) = 1, ∀e ∈ E; x ∈ U.

Definition 2.19 [23]. A completely shadowed soft set (Ω, E)_shdw over common universe U is a shadow soft set with Ω(e)_shdw(x) = [0, 1], ∀e ∈ E; x ∈ U.

3. Time-Shadow Soft Set

F_{t_{i}} (σ) = {\frac{u_{j}}{μ_{F} (u_{j}^{t_{i}})}}, u_{j}^{t_{i}} \in U, μ_{F_{t_{i}}} \in [0, 1],

∀t ∈ T and σ ∈ Z = E × T, i = 1, 2, ..., m, j = 1, 2, ..., n.

Let (α_i, β_i) be given by an expert and define the time-shadow soft set (T_sh-ss in short) as a mapping

F_{s h}^{{(α_{i}, β_{i})}_{t_{i}}} : Z \to {0, 1, [0, 1]},

∀t ∈ T and σ ∈ E × T, i = 1, 2, ..., m, j = 1, 2, ..., n. Here,

F_{s h}^{{(α_{i}, β_{i})}_{t_{i}}} (σ) = {\begin{array}{l} 0, & if μ_{F} (u_{j}^{t_{i}}) (σ) \leq α_{i}, \\ 1, & if μ_{F} (u_{j}^{t_{i}}) (σ) \geq β_{i}, \\ [0, 1], & α_{i} < μ_{F} (u_{j}^{t_{i}}) (σ) < β_{i} . \end{array}

Example 3.1. Let U = {u₁, u₂, u₃, u₄} be a set of universe, E = {e₁, e₂, e₃} is a set of parameters, and T = {t₁, t₂, t₃} is a set of times. Define a function

F_{t} : A \to I^{U},

as follows:

\begin{array}{l} F_{1} (e_{1}) = {\frac{{u_{1}}^{t_{1}}}{0.6}, \frac{{u_{2}}^{t_{1}}}{0.3}, \frac{{u_{3}}^{t_{1}}}{0.2}, \frac{{u_{4}}^{t_{1}}}{0.4}}, \\ F_{1} (e_{2}) = {\frac{{u_{1}}^{t_{1}}}{0.5}, \frac{{u_{2}}^{t_{1}}}{0.3}, \frac{{u_{3}}^{t_{1}}}{0.2}, \frac{{u_{4}}^{t_{1}}}{0.7}}, \\ F_{1} (e_{3}) = {\frac{{u_{1}}^{t_{1}}}{0.3}, \frac{{u_{2}}^{t_{1}}}{0.6}, \frac{{u_{3}}^{t_{1}}}{0.8}, \frac{{u_{4}}^{t_{1}}}{0.9}}, \\ F_{2} (e_{1}) = {\frac{{u_{1}}^{t_{2}}}{0.7}, \frac{{u_{2}}^{t_{2}}}{0.4}, \frac{{u_{3}}^{t_{2}}}{0.1}, \frac{{u_{4}}^{t_{2}}}{0.3}}, \\ F_{2} (e_{2}) = {\frac{{u_{1}}^{t_{2}}}{0.5}, \frac{{u_{2}}^{t_{2}}}{0.3}, \frac{{u_{3}}^{t_{2}}}{0.2}, \frac{{u_{4}}^{t_{2}}}{0.7}}, \\ F_{2} (e_{3}) = {\frac{{u_{1}}^{t_{2}}}{0.7}, \frac{{u_{2}}^{t_{2}}}{0.8}, \frac{{u_{3}}^{t_{2}}}{0.6}, \frac{{u_{4}}^{t_{2}}}{0.4}}, \\ F_{3} (e_{1}) = {\frac{{u_{1}}^{t_{3}}}{0.7}, \frac{{u_{2}}^{t_{3}}}{0.8}, \frac{{u_{3}}^{t_{3}}}{0.6}, \frac{{u_{4}}^{t_{3}}}{0.4}}, \\ F_{3} (e_{2}) = {\frac{{u_{1}}^{t_{3}}}{0.7}, \frac{{u_{2}}^{t_{3}}}{0.5}, \frac{{u_{3}}^{t_{3}}}{0.6}, \frac{{u_{4}}^{t_{3}}}{0.7}}, \\ F_{3} (e_{3}) = {\frac{{u_{1}}^{t_{3}}}{0.3}, \frac{{u_{2}}^{t_{3}}}{0.6}, \frac{{u_{3}}^{t_{3}}}{0.8}, \frac{{u_{4}}^{t_{3}}}{0.9}} . \end{array}

Let shdw = {(0.3, 0.8), (0.4, 0.9), (0.2, 0.6)} be the shadow parameter set related to Z. Define a function

F_{t_{s h}} : A \to I^{U} .

Subsequently, we can find the time-shadow soft sets $(F_{s h}^{(α_{i}, β_{i})}, Z)$ consisting of the following collection of approximations:

\begin{array}{l} (F_{s h}^{(α_{i}, β_{i})}, Z) \\ = {{(e_{1}, {\frac{u_{1}}{[0, 1]}, \frac{u_{2}}{0}, \frac{u_{3}}{0}, \frac{u_{4}}{[0, 1]}})}^{(0.3, 0.8) t_{1}}, \\ {(e_{2}, {\frac{u_{1}}{[0, 1]}, \frac{u_{2}}{0}, \frac{u_{3}}{0}, \frac{u_{4}}{[0, 1]}})}^{(0.4, 0.9) t_{1}}, \\ {(e_{3}, {\frac{u_{1}}{[0, 1]}, \frac{u_{2}}{1}, \frac{u_{3}}{1}, \frac{u_{4}}{1}})}^{(0.2, 0.6) t_{1}}, \\ {(e_{1}, {\frac{u_{1}}{[0, 1]}, \frac{u_{2}}{[0, 1]}, \frac{u_{3}}{0}, \frac{u_{4}}{0}})}^{(0.3, 0.8) t_{2}}, \\ {(e_{2}, {\frac{u_{1}}{[0, 1]}, \frac{u_{2}}{0}, \frac{u_{3}}{0}, \frac{u_{4}}{[0, 1]}})}^{(0.4, 0.9) t_{2}}, \\ {(e_{3}, {\frac{u_{1}}{1}, \frac{u_{2}}{1}, \frac{u_{3}}{1}, \frac{u_{4}}{[0, 1]}})}^{(0.2, 0.6) t_{2}}, \\ {(e_{1}, {\frac{u_{1}}{[0, 1]}, \frac{u_{2}}{1}, \frac{u_{3}}{[0, 1]}, \frac{u_{4}}{[0, 1]}})}^{(0.3, 0.8) t_{3}}, \\ {(e_{2}, {\frac{u_{1}}{[0, 1]}, \frac{u_{2}}{[0, 1]}, \frac{u_{3}}{[0, 1]}, \frac{u_{4}}{[0, 1]}})}^{(0.4, 0.9) t_{3}}, \\ {(e_{3}, {\frac{u_{1}}{[0, 1]}, \frac{u_{2}}{1}, \frac{u_{3}}{1}, \frac{u_{4}}{1}})}^{(0.2, 0.6) t_{3}}} . \end{array}

Here, a = [0, 1]

Definition 3.2. $({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z)$ and $({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z)$ over U and shdw(U) = {(α₁, β₁), (α₂, β₂), ..., (α_m, β_m)}, $({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z)$ , is called the T_sh-SS subset of $({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z)$ if

A ⊆ B,
∀t ∈ T, ε ∈ A, F_t (ε) is fuzzy soft subset of G_t (ε).

Definition 3.3. Two T_sh − SSs, $({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z])$ and $({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z)$ over U is said to be equal if $({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z)$ is a T_sh − SS subset of $({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z)$ and $({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z)$ is a T_sh − SS subset of $({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z)$ .

Example 3.2. Consider Example 3.1 and suppose that

\begin{array}{l} ({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z) \\ = {{(e_{1}, {\frac{u_{1}}{1}, \frac{u_{2}}{[0, 1]}, \frac{u_{3}}{0}, \frac{u_{4}}{[0, 1]}})}^{{(0.3, 0.8)}^{t_{1}}}, \\ {(e_{2}, {\frac{u_{1}}{[0, 1]}, \frac{u_{2}}{[0, 1]}, \frac{u_{3}}{0}, \frac{u_{4}}{1}})}^{{(0.4, 0.9)}^{t_{1}}}, \\ {(e_{2}, {\frac{u_{1}}{[0, 1]}, \frac{u_{2}}{0}, \frac{u_{3}}{[0, 1]}, \frac{u_{4}}{[0, 1]}})}^{{(0.4, 0.9)}^{t_{2}}}, \\ {(e_{3}, {\frac{u_{1}}{1}, \frac{u_{2}}{1}, \frac{u_{3}}{1}, \frac{u_{4}}{1}})}^{{(0.2, 0.6)}^{t_{2}}}, \\ {(e_{1}, {\frac{u_{1}}{[0, 1]}, \frac{u_{2}}{1}, \frac{u_{3}}{[0, 1]}, \frac{u_{4}}{1}})}^{{(0.3, 0.8)}^{t_{3}}}, \\ {(e_{3}, {\frac{u_{1}}{[0, 1]}, \frac{u_{2}}{1}, \frac{u_{3}}{1}, \frac{u_{4}}{1}})}^{(0.2, 0.6) t_{3}}}, \\ ({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z) \\ = {{(e_{2}, {\frac{{u_{1}}^{t_{1}}}{0}, \frac{{u_{2}}^{t_{1}}}{0}, \frac{{u_{3}}^{t_{1}}}{0}, \frac{{u_{4}}^{t_{1}}}{[0, 1]}})}^{(0.4, 0.9) t_{1}}, \\ {(e_{3}, {\frac{{u_{1}}^{t_{2}}}{1}, \frac{{u_{2}}^{2}}{1}, \frac{{u_{3}}^{t_{2}}}{1}, \frac{{u_{4}}^{t_{2}}}{1}})}^{(0.2, 0.6) t_{2}}, \\ {(e_{1}, {\frac{{u_{1}}^{t_{3}}}{[0, 1]}, \frac{{u_{2}}^{t_{3}}}{1}, \frac{{u_{3}}^{t_{3}}}{[0, 1]}, \frac{{u_{4}}^{t_{3}}}{0}})}^{(0.3, 0.8) t_{3}}} . \end{array}

Therefore $({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z) \subseteq ({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z)$ .

Definition 3.4. A time-shadow soft set ${(F, A)}_{t_{s h}}^{(α_{i}, β_{i})}$ over U is said to be semi-null T_sh − SS denoted by T_sh_∼_φ, if ∀t ∈ T, F_t_{_sh} (e) = φ for at least one e.

Definition 3.5. A time-shadow soft set ${(F, A)}_{t_{s h}}^{(α_{i}, β_{i})}$ over U is considered null T_sh − SS denoted by T_sh_{_φ}, if ∀t ∈ T, F_t_{_sh}(e) = φ ∀e.

Definition 3.6. A time-shadow soft set ${(F, A)}_{t_{s h}}^{(α_{i}, β_{i})}$ over U is considered semi-absolute T_sh − SS denoted by T_sh_∼_A, if ∀t ∈ T, F_t_{_sh} (e) = 1̄ for at least one e.

Definition 3.7. A time-shadow soft set ${(F, A)}_{t_{s h}}^{(α_{i}, β_{i})}$ over U is considered absolute T_sh − SS denoted by T_sh_{_A}, if ∀t ∈ T, F_tsh (e) = 1̄∀e.

Example 3.3. Consider Example 3.1. Let

\begin{array}{l} {(F, A)}_{t_{s h}}^{(α_{i}, β_{i})} \\ = {(e_{1 (0.3, 0.8)}, {Φ}^{t_{1}}), \\ (e_{2 (0.4, 0.9)}, {\frac{{u_{1}}^{t_{1}}}{0}, \frac{{u_{2}}^{t_{1}}}{[0, 1]}, \frac{{u_{3}}^{t_{1}}}{0}, \frac{{u_{4}}^{t_{1}}}{[0, 1]}}), \\ (e_{3 (0.2, 0.6)}, {\frac{{u_{1}}^{t_{1}}}{0}, \frac{{u_{2}}^{t_{1}}}{[0, 1]}, \frac{{u_{3}}^{t_{1}}}{[0, 1]}, \frac{{u_{4}}^{t_{1}}}{[0, 1]}}), \\ (e_{1 (0.3, 0.8)}, {Φ}^{t_{2}}), \\ (e_{2 (0.4, 0.9)}, {\frac{{u_{1}}^{t_{2}}}{[0, 1]}, \frac{{u_{2}}^{t_{2}}}{[0, 1]}, \frac{{u_{3}}^{t_{2}}}{[0, 1]}, \frac{{u_{4}}^{t_{2}}}{0.3}}) \\ (e_{3 (0.4, 0.9)}, {\frac{{u_{1}}^{t_{2}}}{0}, \frac{{u_{2}}^{t_{2}}}{0}, \frac{{u_{3}}^{t_{2}}}{0}, \frac{{u_{4}}^{t_{2}}}{[0, 1]}}), \\ (e_{1 (0.3, 0.8),} {Φ}^{t_{3}}), \\ (e_{2 (0.4, 0.9)}, {\frac{{u_{1}}^{t_{3}}}{0}, \frac{{u_{2}}^{t_{3}}}{[0, 1]}, \frac{{u_{3}}^{t_{3}}}{0}, \frac{{u_{4}}^{t_{3}}}{0}}), \\ (e_{3 (0.2, 0.6)}, {\frac{{u_{1}}^{t_{3}}}{1}, \frac{{u_{2}}^{t_{3}}}{[0, 1]}, \frac{{u_{3}}^{t_{3}}}{0}, \frac{{u_{4}}^{t_{3}}}{0}})} . \end{array}

Then ${(F, A)}_{t_{s h}}^{(α_{i}, β_{i})} = T_{s h ~} φ$ .

Let

\begin{array}{l} {(F, A)}_{t_{s h}}^{(α_{i}, β_{i})} \\ = {(e_{1 (0.3, 0.8)}, {Φ}^{t_{1}}), (e_{2 (0.4, 0.9)}, {Φ}^{t_{1}}), \\ (e_{3 (0.2, 0.8)}, {Φ}^{t_{1}}), (e_{1 (0.3, 0.8)}, {Φ}^{t_{2}}), \\ (e_{2 (0.4, 0.9)}, {Φ}^{t_{2}}), (e_{3 (0.2, 0.6)}, {Φ}^{t_{2}}), \\ (e_{1 (0.3, 0.8)}, {Φ}^{t_{3}}), (e_{2 (0.4, 0.9)}, {Φ}^{t_{3}}), \\ (e_{3 (0.2, 0.6)}, {Φ}^{t_{3}})} . \end{array}

Then ${(F, A)}_{t_{s h}}^{(α_{i}, β_{i})} = T_{s h φ}$ .

\begin{array}{l} {(F, A)}_{t s h}^{(α_{i}, β_{i})} \\ = {(e_{1 (0.3, 0.8)}, {\frac{{u_{1}}^{t_{1}}}{1}, \frac{{u_{2}}^{t_{1}}}{1}, \frac{{u_{3}}^{t_{1}}}{1}, \frac{{u_{4}}^{t_{1}}}{1}}), \\ (e_{2 (0.4, 0.9)}, {\frac{{u_{1}}^{t_{1}}}{1}, \frac{{u_{2}}^{t_{1}}}{1}, \frac{{u_{3}}^{t_{1}}}{1}, \frac{{u_{4}}^{t_{1}}}{1}}), \\ (e_{3 (0.2, 0.6)}, {\frac{{u_{1}}^{t_{1}}}{1}, \frac{{u_{2}}^{t_{1}}}{1}, \frac{{u_{3}}^{t_{1}}}{1}, \frac{{u_{4}}^{t_{1}}}{1}}), \\ (e_{1 (0.3, 0.8)}, {\frac{{u_{1}}^{t_{2}}}{1}, \frac{{u_{2}}^{t_{2}}}{1}, \frac{{u_{3}}^{t_{2}}}{1}, \frac{{u_{4}}^{t_{2}}}{1}}), \\ (e_{2 (0.4, 0.9)}, {\frac{{u_{1}}^{t_{2}}}{1}, \frac{{u_{2}}^{t_{2}}}{1}, \frac{{u_{3}}^{t_{2}}}{1}, \frac{{u_{4}}^{t_{2}}}{1}}), \\ (e_{3 (0.2, 0.6)}, {\frac{{u_{1}}^{t_{2}}}{1}, \frac{{u_{2}}^{t_{2}}}{1}, \frac{{u_{3}}^{t_{2}}}{1}, \frac{{u_{4}}^{t_{2}}}{1}}), \\ (e_{1 (0.3, 0.8)}, {\frac{{u_{1}}^{t_{3}}}{[0, 1]}, \frac{{u_{2}}^{t_{3}}}{1}, \frac{{u_{3}}^{t_{3}}}{1}, \frac{{u_{4}}^{t_{3}}}{[0, 1]}}), \\ (e_{2 (0.4, 0.9)}, {\frac{{u_{1}}^{t_{3}}}{0}, \frac{{u_{2}}^{t_{3}}}{[0, 1]}, \frac{{u_{3}}^{t_{3}}}{0}, \frac{{u_{4}}^{t_{3}}}{0}}), \\ (e_{3 (0.2, 0.6)}, {\frac{{u_{1}}^{t_{3}}}{[0, 1]}, \frac{{u_{2}}^{t_{3}}}{[0, 1]}, \frac{{u_{3}}^{t_{3}}}{0.2}, \frac{{u_{4}}^{t_{3}}}{0}})} . \end{array}

Then ${(F, A)}_{t_{s h}}^{(α_{i}, β_{i})} = T_{s h ~ A}$ .

Let

\begin{array}{l} {(F, A)}_{t_{s h}}^{(α_{i}, β_{i})} \\ = {(e_{1 (0.3, 0.8)}, {\frac{{u_{1}}^{t_{1}}}{1}, \frac{{u_{2}}^{t_{1}}}{1}, \frac{{u_{3}}^{t_{1}}}{1}, \frac{{u_{4}}^{t_{1}}}{1}}), \\ (e_{2 (0.4, 0.9)}, {\frac{{u_{1}}^{t_{1}}}{1}, \frac{{u_{2}}^{t_{1}}}{1}, \frac{{u_{3}}^{t_{1}}}{1}, \frac{{u_{4}}^{t_{1}}}{1}}), \\ (e_{2 (0.4, 0.9)}, {\frac{{u_{1}}^{t_{2}}}{1}, \frac{{u_{2}}^{t_{2}}}{1}, \frac{{u_{3}}^{t_{2}}}{1}, \frac{{u_{4}}^{t_{2}}}{1}}), \\ (e_{3 (0.2, 0.6)}, {\frac{{u_{1}}^{t_{2}}}{1}, \frac{{u_{2}}^{t_{2}}}{1}, \frac{{u_{3}}^{t_{2}}}{1}, \frac{{u_{4}}^{t_{2}}}{1}}), \\ (e_{1 (0.3, 0.8)}, {\frac{{u_{1}}^{t_{3}}}{1}, \frac{{u_{2}}^{t_{3}}}{1}, \frac{{u_{3}}^{t_{3}}}{1}, \frac{{u_{4}}^{t_{3}}}{1}}), \\ (e_{3 (0.2, 0.6)}, {\frac{{u_{1}}^{t_{3}}}{1}, \frac{{u_{2}}^{t_{3}}}{1}, \frac{{u_{3}}^{t_{3}}}{1}, \frac{{u_{4}}^{t_{3}}}{1}})} . \end{array}

Then ${(F, A)}_{t_{s h}}^{(α_{i}, β_{i})} = T_{s h A}$ .

3.1 Basic Operations

Here, we introduce some basic operations on the time-shadow soft set, namely complement, union, and intersection, and provide some properties related to these operations.

Definition 3.8. Let $(F_{s h}^{(α_{i}, β_{i})}, Z)$ be a time-shadowed soft set over U. Then the complement of $(F_{s h}^{(α_{i}, β_{i})}, Z)$ , denoted by ${(F_{s h}^{(α_{i}, β_{i})}, Z)}^{c}$ and is defined by ${(F_{s h}^{(α_{i}, β_{i})}, Z)}^{c} = c (F_{s h}^{(α_{i}, β_{i})}, Z)$ , where c denotes a time-shadow soft complement.

Example 3.4. Consider a time-shadow soft set $(F_{s h}^{(α_{i}, β_{i})}, Z)$ over U as in Example 3.1. Subsequently, we can find the complement of $(F_{s h}^{(α_{i}, β_{i})}, Z)$ as follows:

\begin{array}{l} {(F_{s h}^{(α_{i}, β_{i})}, Z)}^{c} \\ = {{(e_{1}, {\frac{u_{1}}{[0, 1]}, \frac{u_{2}}{1}, \frac{u_{3}}{1}, \frac{u_{4}}{[0, 1]}})}^{{(0.3, 0.8)}_{t_{1}}}, \\ {(e_{2}, {\frac{u_{1}}{[0, 1]}, \frac{u_{2}}{1}, \frac{u_{3}}{1}, \frac{u_{4}}{[0, 1]}})}^{{(0.4, 0.9)}_{t_{1}}}, \\ {(e_{3}, {\frac{u_{1}}{[0, 1]}, \frac{u_{2}}{0}, \frac{u_{3}}{0}, \frac{u_{4}}{0}})}^{{(0.2, 0.8)}_{t_{1}}}, \\ {(e_{1}, {\frac{u_{1}}{[0, 1]}, \frac{u_{2}}{[0, 1]}, \frac{u_{3}}{1}, \frac{u_{4}}{1}})}^{{(0.3, 0.8)}_{t_{2}}}, \\ {(e_{2}, {\frac{u_{1}}{[0, 1]}, \frac{u_{2}}{1}, \frac{u_{3}}{1}, \frac{u_{4}}{[0, 1]}})}^{{(0.4, 0.9)}_{t_{2}}}, \\ {(e_{3}, {\frac{u_{1}}{0}, \frac{u_{2}}{0}, \frac{u_{3}}{0}, \frac{u_{4}}{[0, 1]}})}^{{(0.2, 0.6)}_{t_{2}}}, \\ {(e_{1}, {\frac{u_{1}}{[0, 1]}, \frac{u_{2}}{0}, \frac{u_{3}}{[0, 1]}, \frac{u_{4}}{[0, 1]}})}^{{(0.3, 0.8)}_{t_{3}}}, \\ {(e_{2}, {\frac{u_{1}}{[0, 1]}, \frac{u_{2}}{[0, 1]}, \frac{u_{3}}{[0, 1]}, \frac{u_{4}}{[0, 1]}})}^{{(0.4, 0.9)}_{t_{3}}}, \\ {(e_{3}, {\frac{u_{1}}{[0, 1]}, \frac{u_{2}}{0}, \frac{u_{3}}{1}, \frac{u_{4}}{0}})}^{{(0.2, 0.6)}_{t_{3}}}} . \end{array}

Proposition 3.1. Let ${(F, E)}_{t_{s h}}^{(α_{i}, β_{i})}$ be a time-shadow soft set over U. Thus, the following holds.

{({(F, E)}_{t_{s h}}^{{(α_{i}, β_{i})}^{c}})}^{c} = {(F, E)}_{t_{s h}}^{(α_{i}, β_{i})} .

Proof. The proof of this is straightforward.

Definition 3.9. The union of two time-shadow soft sets $({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z)$ and $({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z)$ over a common universe U and a time-shadow parameters set is a time-shadow soft set $({(H, C)}_{s h}^{(α_{i}, β_{i})}, Z)$ , denoted by $({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z) \tilde{\cup} ({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z)$ , such that C = A ∪ B ⊂ Z and defined as follows:

\begin{array}{l} ({(H, C)}_{s h}^{(α_{i}, β_{i})}, Z) (ɛ) \\ = {\begin{array}{l} F_{t_{s h}}^{(α_{i}, β_{i})} (ɛ), & if ɛ \in A - B, \\ G_{t_{s h}}^{(α_{i}, β_{i})} (ɛ), & if ɛ \in B - A, \\ F_{t_{s h}}^{(α_{i}, β_{i})} (ɛ) \tilde{\cup} G_{t_{s h}}^{(α_{i}, β_{i})} (ɛ), & if ɛ \in A \cap B, \end{array} \end{array}

where Ũ denotes the time-shadow of the soft union.

Example 3.5. Consider Example 3.1. Suppose (F,A)_t_{_sh}and (G,B)_t_{_sh}are two time-shadow soft sets over U such that

\begin{array}{l} {(F, A)}_{t_{s h}}^{(α_{i}, β_{i})} \\ = (e_{1 (0.3, 0.8)}, {\frac{{u_{1}}^{t_{1}}}{[0, 1]} \frac{{u_{2}}^{t_{1}}}{0} \frac{{u_{3}}^{t_{1}}}{0}, \frac{{u_{4}}^{t_{1}}}{[0, 1]}}), \\ (e_{2 (0.4, 0.9)}, {\frac{{u_{1}}^{t_{1}}}{[0, 1]} \frac{{u_{2}}^{t_{1}}}{0}, \frac{{u_{3}}^{t_{1}}}{0}, \frac{{u_{4}}^{t_{1}}}{[0, 1]}}), \\ (e_{2 (0.4, 0.9)}, {\frac{{u_{1}}^{t_{2}}}{[0, 1]} \frac{{u_{2}}^{t_{2}}}{0}, \frac{{u_{3}}^{t_{2}}}{0}, \frac{{u_{4}}^{t_{2}}}{[0, 1]}}), \\ (e_{3 (0.2, 0.6)}, {\frac{{u_{1}}^{t_{3}}}{[0, 1]} \frac{{u_{2}}^{t_{3}}}{1} \frac{{u_{3}}^{t_{3}}}{1}, \frac{{u_{4}}^{t_{3}}}{1}})}, \\ {(G, B)}_{t}^{(α_{i}, β_{i})} \\ = {(e_{1 (0.3, 0.8)}, {\frac{{u_{1}}^{t_{1}}}{[0, 1]}, \frac{{u_{2}}^{t_{1}}}{[0, 1]}, \frac{{u_{3}}^{t_{1}}}{0}, \frac{{u_{4}}^{t_{1}}}{0}}), \\ (e_{3 (0.2, 0.6)}, {\frac{{u_{1}}^{t_{2}}}{[0, 1]}, \frac{{u_{2}}^{t_{2}}}{1}, \frac{{u_{3}}^{t_{2}}}{1}, \frac{{u_{4}}^{t_{2}}}{[0, 1]}}), \\ (e_{3 (0.2, 0.6)}, {\frac{{u_{1}}^{t_{3}}}{[0, 1]}, \frac{{u_{2}}^{t_{3}}}{1}, \frac{{u_{3}}^{t_{3}}}{1}, \frac{{u_{4}}^{t_{3}}}{0}})}, \\ {(H, C)}_{t_{s h}}^{(α_{i}, β_{i})} \\ = {(e_{1 (0.3, 0.8)}, {\frac{{u_{1}}^{t_{1}}}{[0, 1]}, \frac{{u_{2}}^{t_{1}}}{[0, 1]}, \frac{{u_{3}}^{t_{1}}}{0}, \frac{{u_{4}}^{t_{1}}}{[0, 1]}}), \\ (e_{2 (0.4, 0.9)}, {\frac{{u_{1}}^{t_{1}}}{[0, 1]}, \frac{{u_{2}}^{t_{1}}}{0}, \frac{{u_{3}}^{t_{1}}}{0}, \frac{{u_{4}}^{t_{1}}}{[0, 1]}}), \\ (e_{2 (0.4, 0.9)}, {\frac{{u_{1}}^{t_{2}}}{[0, 1]}, \frac{{u_{2}}^{t_{2}}}{0}, \frac{{u_{3}}^{t_{2}}}{0}, \frac{{u_{4}}^{t_{2}}}{[0, 1]}}), \\ (e_{3 (0.2, 0.6)}, {\frac{{u_{1}}^{t_{2}}}{0.5}, \frac{{u_{2}}^{t_{2}}}{0.6}, \frac{{u_{3}}^{t_{2}}}{0.9}, \frac{{u_{4}}^{t_{2}}}{0.4}}), \\ (e_{3 (0.2, 0.6)}, {\frac{{u_{1}}^{t_{3}}}{[0, 1]}, \frac{{u_{2}}^{t_{3}}}{1}, \frac{{u_{3}}^{t_{3}}}{1}, \frac{{u_{4}}^{t_{3}}}{1}})} . \end{array}

Proposition 3.2. If $({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z), ({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z)$ and $({(H, C)}_{s h}^{(α_{i}, β_{i})}, Z)$ are three T_t_{_sh} − SSs over U. Subsequently, the following results hold.

1. $({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z) \tilde{\cup} ({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z) \tilde{\cup} ({(H, C)}_{s h}^{(α_{i}, β_{i})}, Z)) = ({(F, A)}_{t_{s h}}^{(α_{t}, β_{t})} \tilde{\cup} {(G, B)}_{t_{s h}}^{(α_{i}, β_{i})}) \tilde{\cup} {(H, C)}_{t_{s h}}^{(α_{i}, β_{i})}$ ,
2. $({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z) \tilde{\cup} ({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z) = ({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z)$ .
3. $({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z) \tilde{\cup} ({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z) = ({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z) \tilde{\cup} ({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z)$ .
4. $({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z) \cup φ_{t_{s h}} = ({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z)$ .

Proof. The proof of this is straightforward.

Definition 3.10. The intersection of two time-shadow soft sets $({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z)$ and $({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z)$ over a common universe U and a time-shadow parameters set is a time-shadow soft set $({(H, C)}_{s h}^{(α_{i}, β_{i})}, Z)$ , denoted by $({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z) \tilde{\cap} ({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z)$ , such that C = A ∩ B ⊂ Z and defined as follows:

\begin{array}{l} ({(H, C)}_{s h}^{(α_{i}, β_{i})}, Z) (ɛ) \\ = {\begin{array}{l} F_{t_{s h}}^{(α_{i}, β_{i})} (ɛ), & if ɛ \in A - B, \\ G_{t_{s h}}^{(α_{i}, β_{i})} (ɛ), & if ɛ \in B - A, \\ F_{t_{s h}}^{(α_{i}, β_{i})} (ɛ) \tilde{\cap} G_{t_{s h}}^{(α_{i}, β_{i})} (ɛ), & if ɛ \in A \cap B, \end{array} \end{array}

where ∩̃ denotes the time-shadow of the soft union.

Example 3.6. Consider Example 3.1. Suppose (F,A)_t_{_sh} and (G,B)_t_{_sh} are two time-shadow soft sets over U such that

\begin{array}{l} {(F, A)}_{t_{s h}}^{(α_{i}, β_{i})} \\ = {(e_{1 (0.3, 0.8)}, {\frac{{u_{1}}^{t_{1}}}{[0, 1]}, \frac{{u_{2}}^{t_{1}}}{0}, \frac{{u_{3}}^{t_{1}}}{0}, \frac{{u_{4}}^{t_{1}}}{[0, 1]}}), \\ (e_{2 (0.4, 0.9)}, {\frac{{u_{1}}^{t_{1}}}{[0, 1]}, \frac{{u_{2}}^{t_{1}}}{0}, \frac{{u_{3}}^{t_{1}}}{0}, \frac{{u_{4}}^{t_{1}}}{[0, 1]}}), \\ (e_{2 (0.4, 0.9)}, {\frac{{u_{1}}^{t_{2}}}{[0, 1]}, \frac{{u_{2}}^{t_{2}}}{0}, \frac{{u_{3}}^{t_{2}}}{0}, \frac{{u_{4}}^{t_{2}}}{[0, 1]}}), \\ (e_{3 (0.2, 0.6)}, {\frac{{u_{1}}^{t_{3}}}{[0, 1]}, \frac{{u_{2}}^{t_{3}}}{1}, \frac{{u_{3}}^{t_{3}}}{1}, \frac{{u_{4}}^{t_{3}}}{1}})}, \\ {(G, B)}_{t_{s h}}^{(α_{i}, β_{i})} \\ = {(e_{1 (0.3, 0.8)}, {\frac{{u_{1}}^{t_{1}}}{[0, 1]}, \frac{{u_{2}}^{t_{1}}}{[0, 1]}, \frac{{u_{3}}^{t_{1}}}{0}, \frac{{u_{4}}^{t_{1}}}{0}}), \\ (e_{3 (0.2, 0.6)}, {\frac{{u_{1}}^{t_{2}}}{[0, 1]}, \frac{{u_{2}}^{t_{2}}}{1}, \frac{{u_{3}}^{t_{2}}}{1}, \frac{{u_{4}}^{t_{2}}}{[0, 1]}}), \\ (e_{3 (0.2, 0.6)}, {\frac{{u_{1}}^{t_{3}}}{[0, 1]}, \frac{{u_{2}}^{t_{3}}}{1}, \frac{{u_{3}}^{t_{3}}}{1}, \frac{{u_{4}}^{t_{3}}}{1}})} . \\ {(H, C)}_{t_{s h}} \\ = {(e_{1 (0.3, 0.8)}, {\frac{{u_{1}}^{t_{1}}}{[0, 1]}, \frac{{u_{2}}^{t_{1}}}{0}, \frac{{u_{3}}^{t_{1}}}{0}, \frac{{u_{4}}^{t_{1}}}{[0, 1]}}), \\ (e_{2 (0.4, 0.9)}, {\frac{{u_{1}}^{t_{1}}}{[0, 1]}, \frac{{u_{2}}^{t_{1}}}{0}, \frac{{u_{3}}^{t_{1}}}{0}, \frac{{u_{4}}^{t_{1}}}{[0, 1]}}), \\ (e_{2 (0.4, 0.9)}, {\frac{{u_{1}}^{t_{2}}}{[0, 1]}, \frac{{u_{2}}^{t_{2}}}{0}, \frac{{u_{3}}^{t_{2}}}{0}, \frac{{u_{4}}^{t_{2}}}{[0, 1]}}), \\ (e_{3 (0.2, 0.6)}, {\frac{{u_{1}}^{t_{2}}}{0.5}, \frac{{u_{2}}^{t_{2}}}{0.6}, \frac{{u_{3}}^{t_{2}}}{0.9}, \frac{{u_{4}}^{t_{2}}}{0.4}}), \\ (e_{3 (0.2, 0.6)}, {\frac{{u_{1}}^{t_{3}}}{[0, 1]}, \frac{{u_{2}}^{t_{3}}}{1}, \frac{{u_{3}}^{t_{3}}}{1}, \frac{{u_{4}}^{t_{3}}}{1}})} . \end{array}

Proposition 3.3. If $({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z), ({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z)$ and $({(H, C)}_{s h}^{(α_{i}, β_{i})}, Z)$ are three T_t_{_sh}–SSs over U. Then, the following results hold.

1. $({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z) \tilde{\cap} ({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z) \tilde{\cap} ({(H, C)}_{s h}^{(α_{i}, β_{i})}, Z)) = ({(F, A)}_{t_{s h}} \tilde{\cup} {(G, B)}_{t_{s h}}) \tilde{\cup} {(H, C)}_{t_{s h}}$ .
2. $({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z) \tilde{\cap} ({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z) = ({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z)$ .
3. $({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z) \tilde{\cap} ({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z) = ({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z) \tilde{\cap} ({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z)$ .
4. $({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z) \cap φ_{t_{s h}} ({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z)$ .

Proof. The proof of this is straightforward.

Proposition 3.4. If $({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z), ({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z)$ and $({(H, C)}_{s h}^{(α_{i}, β_{i})}, Z)$ are three T_sh − SSs over U, then

1. $({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z) \tilde{\cup} (({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z) \tilde{\cap} {(H, C)}_{t_{s h}}^{(α_{i}, β_{i})}, Z) = ({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z) \tilde{\cup} (\tilde{\cap} ({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z) \tilde{\cup} ({(H, C)}_{s h}^{(α_{i}, β_{i})}, Z)$ .
2. $({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z) \tilde{\cap} (({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z) \tilde{\cup} {(H, C)}_{t_{s h}}^{(α_{i}, β_{i})}, Z) = (({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z) \tilde{\cap} ({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z) \tilde{\cup} ({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z) \tilde{\cap} ({(H, C)}_{s h}^{(α_{i}, β_{i})}, Z)$ .

Proof. The proof of this is straightforward.

Proposition 3.5. If $({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z)$ and $({(G,)}_{s h}^{(α_{i}, β_{i})}, Z)$ are two T_sh − SSs over U, then

1. $({(F, A)}_{t_{s h}}^{(α_{i}, β_{i})}, Z) \tilde{\cup} {({(G, B)}_{t_{s h}}^{(α_{i}, β_{i})}, Z)}^{c} = ({(F, A)}_{t_{s h}}^{c}^{(α_{i}, β_{i})}, Z) \tilde{\cap} ({(G, B)}_{t_{s h}}^{c}^{(α_{i}, β_{i})}, Z)$ .
2. $({(F, A)}_{t_{s h}}^{(α_{i}, β_{i})}, Z) \tilde{\cap} {({(G, B)}_{t_{s h}}^{(α_{i}, β_{i})}, Z)}^{c} = ({(F, A)}_{t_{s h}}^{c}^{(α_{i}, β_{i})}, Z) \tilde{\cup} ({(G, B)}_{t_{s h}}^{c}^{(α_{i}, β_{i})}, Z)$ .

Proof. The proof of this is straightforward.

4. AND and OR Operations

In this section, we introduce the definitions of “AND and OR” operations for T_sh − SSs, derive their properties, and provide some examples.

Definition 4.1. If $({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z)$ and $({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z)$ are two T_sh − SS values over the U then “ $({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z)$ AND $({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z)$ is denoted by $({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z) \land ({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z)$ is defined as follows:

({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z) \land ({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z) = {(H, A \times B)}_{t_{s h}}^{(α_{i}, β_{i})}, Z))

such that $H {(α, β)}_{t_{s h}}^{(α_{i}, β_{i})} = F (α) t_{s h}^{(α_{i}, β_{i})} \tilde{\cap} G {(β)}_{t_{s h}}^{(α_{i}, β_{i})}$ , ∀(α_i, β_i) ∈ A × B, where ∩̃ is a time-shadow soft intersection.

Example 4.1. Consider Example 3.1. Suppose ${(F, A)}_{t_{s h}}^{(α_{i}, β_{i})}$ and ${(G, B)}_{t_{s h}}^{(α_{i}, β_{i})}$ are two time-shadow soft sets over U such that

\begin{array}{l} {(F, A)}_{t_{s h}}^{(α_{i}, β_{i})} \\ = {(e_{1} (0.3, 0.8), {\frac{{u_{1}}^{t_{1}}}{[0, 1]}, \frac{{u_{2}}^{t_{1}}}{0}, \frac{{u_{3}}^{t_{1}}}{0}, \frac{{u_{4}}^{t_{1}}}{[0, 1]}}), \\ (e_{2} (0.4, 0.7), {\frac{{u_{1}}^{t_{1}}}{[0, 1]}, \frac{{u_{2}}^{t_{1}}}{0}, \frac{{u_{3}}^{t_{1}}}{0}, \frac{{u_{4}}^{t_{1}}}{1}}), \\ (e_{2} (0.4, 0.7), {\frac{{u_{1}}^{t_{2}}}{0}, \frac{{u_{2}}^{t_{2}}}{[0, 1]}, \frac{{u_{3}}^{t_{2}}}{1}, \frac{{u_{4}}^{t_{2}}}{0}}), \\ e_{3} (0.2, 0.9), {\frac{{u_{1}}^{t_{3}}}{[0, 1]}, \frac{{u_{2}}^{t_{3}}}{[0, 1]}, \frac{{u_{3}}^{t_{3}}}{[0, 1]}, \frac{{u_{4}}^{t_{3}}}{1}})}, \\ {(G, B)}_{t_{s h}}^{(α_{i}, β_{i})} \\ = (e_{1} (0.3, 0.8), {\frac{{u_{1}}^{t_{1}}}{1}, \frac{{u_{2}}^{t_{1}}}{[0, 1]}, a n d \frac{{u_{3}}^{t_{1}}}{0}, \frac{{u_{4}}^{t_{1}}}{0}}), \\ (e_{3} (0.2, 0.9), {\frac{{u_{1}}^{t_{2}}}{[0, 1]}, \frac{{u_{2}}^{t_{2}}}{[0, 1]}, \frac{{u_{3}}^{t_{2}}}{[0, 1]}, \frac{{u_{4}}^{t_{2}}}{[0, 1]}}), \\ (e_{3} (0.2, 0.9), {\frac{{u_{1}}^{t_{3}}}{0}, \frac{{u_{2}}^{t_{2}}}{[0, 1]}, \frac{{u_{3}}^{t_{3}}}{[0, 1]}, \frac{{u_{4}}^{t_{3}}}{1}})} . \end{array}

Then

\begin{array}{l} {(F, A)}_{t_{s h}}^{(α_{i}, β_{i})} \land {(G, B)}_{t_{s h}}^{(α_{i}, β_{i})} \\ = {(H, A \times B)}_{t_{s h}}^{(α_{t}, β_{t})} \\ = {((e_{1}^{t_{1}}, e_{1}^{t_{1}}) (0.3, 0.8), {\frac{{u_{1}}^{t_{1, 1}}}{[0, 1]}, \frac{{u_{2}}^{t_{1, 1}}}{0}, \frac{{u_{3}}^{t_{1, 1}}}{0}, \frac{{u_{4}}^{t_{1, 1}}}{0}}), \\ ((e_{1}^{t_{1}}, e_{3}^{t_{2}}) (0.2, 0.8), {\frac{{u_{1}}^{t_{1, 2}}}{[0, 1]}, \frac{{u_{2}}^{t_{1, 2}}}{[0, 1]}, \frac{{u_{3}}^{t_{1, 2}}}{0}, \frac{{u_{4}}^{t_{1, 2}}}{[0, 1]}}), \\ ((e_{1}^{t_{1}}, e_{1}^{t_{3}}) (0.2, 0.8), {\frac{{u_{1}}^{t_{1, 3}}}{0}, \frac{{u_{2}}^{t_{1, 3}}}{[0, 1]}, \frac{{u_{3}}^{t_{1, 3}}}{0}, \frac{{u_{4}}^{t_{1, 3}}}{[0, 1]}}), \\ ((e_{2}^{t_{1}}, e_{1}^{t_{1}}) (0.3, 0.7), {\frac{{u_{1}}^{t_{1, 1}}}{[0, 1]}, \frac{{u_{2}}^{t_{1, 1}}}{0}, \frac{{u_{3}}^{t_{1, 1}}}{0}, \frac{{u_{4}}^{t_{1, 1}}}{0}}), \\ ((e_{2}^{t_{1}}, e_{3}^{t_{2}}) (0.3, 0.7), {\frac{{u_{1}}^{t_{1, 2}}}{[0, 1]}, \frac{{u_{2}}^{t_{1, 2}}}{0}, \frac{{u_{3}}^{t_{1, 2}}}{0}, \frac{{u_{4}}^{t_{1, 2}}}{1}}), \\ ((e_{2}^{t_{1}}, e_{3}^{t_{3}}) (0.2, 0.7), {\frac{{u_{1}}^{t_{1, 3}}}{0}, \frac{{u_{2}}^{t_{1, 3}}}{[0, 1]}, \frac{{u_{3}}^{t_{1, 3}}}{0}, \frac{{u_{4}}^{t_{1, 3}}}{1}}), \\ ((e_{2}^{t_{2}}, e_{1}^{t_{1}}) (0.3, 0.7), {\frac{{u_{1}}^{t_{2, 1}}}{[0, 1]}, \frac{{u_{2}}^{t_{2, 1}}}{[0, 1]}, \frac{{u_{3}}^{t_{2, 1}}}{0}, \frac{{u_{4}}^{t_{2, 1}}}{0}}), \\ ((e_{2}^{t_{2}}, e_{3}^{t_{2}}) (0.2, 0.7), {\frac{{u_{1}}^{t_{2, 2}}}{[0, 1]}, \frac{{u_{2}}^{t_{2, 2}}}{[0, 1]}, \frac{{u_{3}}^{t_{2, 2}}}{[0, 1]}, \frac{{u_{4}}^{t_{2, 2}}}{[0, 1]}}), \\ ((e_{2}^{t_{2}}, e_{3}^{t_{3}}) (0.2, 0.7), {\frac{{u_{1}}^{t_{2, 3}}}{0}, \frac{{u_{2}}^{t_{2, 3}}}{[0, 1]}, \frac{{u_{3}}^{t_{2, 3}}}{1}, \frac{{u_{4}}^{t_{2, 3}}}{[0, 1]}}), \\ ((e_{3}^{t_{3}}, e_{1}^{t_{1}}) (0.2, 0.8), {\frac{{u_{1}}^{t_{3, 1}}}{[0, 1]}, \frac{{u_{2}}^{t_{3, 1}}}{[0, 1]}, \frac{{u_{3}}^{t_{3, 1}}}{0}, \frac{{u_{4}}^{t_{3, 1}}}{0}}), \\ ((e_{3}^{t_{3}}, e_{3}^{t_{2}}) (0.2, 0.9), {\frac{{u_{1}}^{t_{3, 2}}}{[0, 1]}, \frac{{u_{2}}^{t_{3, 2}}}{0.6}, \frac{{u_{3}}^{t_{3, 2}}}{[0, 1]}, \frac{{u_{4}}^{t_{3, 2}}}{[0, 1]}}), \\ ((e_{3}^{t_{3}}, e_{3}^{t_{3}}) (0.2, 0.3), {\frac{{u_{1}}^{t_{3, 3}}}{0}, \frac{{u_{2}}^{t_{3, 3}}}{1}, \frac{{u_{3}}^{t_{3, 3}}}{1}, \frac{{u_{4}}^{t_{3, 3}}}{1}})} . \end{array}

Definition 4.2. If $({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z)$ and $({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z)$ over U then “ $({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z)$ OR $({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z)$ ,” which is denoted by $({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z) \lor ({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z)$ , is defined by

\begin{array}{l} ({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z) \lor ({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z) \\ = {(H, A \times B,)}_{t_{s h}}^{(α_{i}, β_{i}, Z)}, \end{array}

such that $H {(α, β)}_{t_{s h}}^{(α_{i}, β_{i})} = F {(α)}_{t_{s h}}^{(α_{i}, β_{i})} \tilde{\cup} G {(β)}_{t_{s h}}^{(α_{i}, β_{i})}$ , ∀(α_i, β_i) ∈ A × B, where ũ is a time-shadow soft union.

Example 4.2. Consider Example 4.1 We have U then, “ ${(F, A)}_{s h}^{(α_{i}, β_{i})}$ OR ${(G, B)}_{s h}^{(α_{i}, β_{i})}$ ” denoted as ${(H, C_{s h}^{(α_{i}, β_{i})})}_{t} = {(F, A)}_{s h}^{(α_{i}, β_{i})} \lor {(G, B)}_{s h}^{(α_{i}, β_{i})}$ where

\begin{array}{l} {(H, C)}_{s h}^{(α_{i}, β_{i})} \\ = {((e_{1}^{t_{1}}, e_{1}^{t_{1}}) (0.3, 0.8), {\frac{{u_{1}}^{t_{1, 1}}}{1}, \frac{{u_{2}}^{t_{1, 1}}}{[0, 1]}, \frac{{u_{3}}^{t_{1, 1}}}{0}, \frac{{u_{4}}^{t_{1, 1}}}{[0, 1]}}), \\ ((e_{1}^{t_{1}}, e_{3}^{t_{2}}) (0.3, 0.9), {\frac{{u_{1}}^{t_{1, 2}}}{[0, 1]}, \frac{{u_{2}}^{t_{1, 2}}}{[0, 1]}, \frac{{u_{3}}^{t_{1, 2}}}{[0, 1]}, \frac{{u_{4}}^{t_{1, 2}}}{[0, 1]}}) \\ ((e_{1}^{t_{1}}, e_{3}^{t_{3}}) (0.3, 0.9), {\frac{{u_{1}}^{t_{1, 3}}}{[0, 1]}, \frac{{u_{2}}^{t_{1, 3}}}{[0, 1]}, \frac{{u_{3}}^{t_{1, 3}}}{[0, 1]}, \frac{{u_{4}}^{t_{1, 3}}}{1}}), \\ ((e_{2}^{t_{1}}, e_{1}^{t_{1}}) (0.4, 0.8), {\frac{{u_{1}}^{t_{1, 1}}}{1}, \frac{{u_{2}}^{t_{1, 1}}}{[0, 1]}, \frac{{u_{3}}^{t_{1, 1}}}{0}, \frac{{u_{4}}^{t_{1, 1}}}{[0, 1]}}), \\ ((e_{2}^{t_{1}}, e_{3}^{t_{2}}) (0.4, 0.9), {\frac{{u_{1}}^{t_{1, 2}}}{[0, 1]}, \frac{{u_{2}}^{t_{1, 2}}}{[0, 1]}, \frac{{u_{3}}^{t_{1, 2}}}{[0, 1]}, \frac{{u_{4}}^{t_{1, 2}}}{[0, 1]}}), \\ ((e_{2}^{t_{1}}, e_{3}^{t_{3}}) (0.4, 0.9), {\frac{{u_{1}}^{t_{1, 3}}}{[0, 1]}, \frac{{u_{2}}^{t_{1, 3}}}{[0, 1]}, \frac{{u_{3}}^{t_{1, 3}}}{[0, 1]}, \frac{{u_{4}}^{t_{1, 3}}}{1}}), \\ ((e_{2}^{t_{2}}, e_{1}^{t_{1}}) (0.4, 0.8), {\frac{{u_{1}}^{t_{2, 1}}}{1}, \frac{{u_{2}}^{t_{2, 1}}}{[0, 1]}, \frac{{u_{3}}^{t_{2, 1}}}{1}, \frac{{u_{4}}^{t_{2, 1}}}{0}}), \\ ((e_{2}^{t_{2}}, e_{3}^{t_{2}}) (0.4, 0.9), {\frac{{u_{1}}^{t_{2, 2}}}{0}, \frac{{u_{2}}^{t_{2, 2}}}{[0, 1]}, \frac{{u_{3}}^{t_{2, 2}}}{[0, 1]}, \frac{{u_{4}}^{t_{2, 2}}}{[0, 1]}}), \\ ((e_{2}^{t_{2}}, e_{3}^{t_{3}}) (0.4, 0.9), {\frac{{u_{1}}^{t_{2, 3}}}{0}, \frac{{u_{2}}^{t_{2, 3}}}{[0, 1]}, \frac{{u_{3}}^{t_{2, 3}}}{[0, 1]}, \frac{{u_{4}}^{t_{2, 3}}}{1}}), \\ ((e_{3}^{t_{3}}, e_{1}^{t_{1}}) (0.3, 0.9), {\frac{{u_{1}}^{t_{3, 1}}}{[0, 1]}, \frac{{u_{2}}^{t_{3, 1}}}{[0, 1]}, \frac{{u_{3}}^{t_{3, 1}}}{[0, 1]}, \frac{{u_{4}}^{t_{3, 1}}}{1}}), \\ ((e_{3}^{t_{3}}, e_{3}^{t_{2}}) (0.2, 0.9), {\frac{{u_{1}}^{t_{3, 2}}}{[0, 1]}, \frac{{u_{2}}^{t_{3, 2}}}{[0, 1]}, \frac{{u_{3}}^{t_{3, 2}}}{[0, 1]}, \frac{{u_{4}}^{t_{3, 2}}}{1}}), \\ ((e_{3}^{t_{3}}, e_{3}^{t_{3}}) (0.2, 0.9), {\frac{{u_{1}}^{t_{3, 3}}}{0}, \frac{{u_{2}}^{t_{3, 3}}}{[0, 1]}, \frac{{u_{3}}^{t_{3, 3}}}{[0, 1]}, \frac{{u_{4}}^{t_{3, 3}}}{1}})} . \end{array}

Proposition 4.1. Let $({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z)$ and $({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z)$ be any two time-shadow soft sets. Subsequently, the following results hold.

1. ${(({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z) \land ({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z))}^{c} = {({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z)}^{c} \lor {({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z)}^{c}$ .
2. ${(({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z) \lor ({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z))}^{c} = {({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z)}^{c} \land {({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z)}^{c}$ .

Proof. Straightforward from Definitions 3.8, 4.1 and 4.2.

Proposition 4.2. Let $({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z), ({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z)$ and $({(H, C)}_{s h}^{(α_{i}, β_{i})}, Z)$ be any three shadow soft sets. Subsequently, the following results hold.

1. $(({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z) \land (({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z) \land ({(H, C)}_{s h}^{(α_{i}, β_{i})}, Z) = (({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z) \land ({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z) \land (({(H, C)}_{s h}^{(α_{i}, β_{i})}), Z)$ .
2. $(({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z) \lor (({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z) \lor ({(H, C)}_{s h}^{(α_{i}, β_{i})}, Z) = (({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z) \lor ({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z) \lor (({(H, C)}_{s h}^{(α_{i}, β_{i})}), Z))$ .
3. $(({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z) \lor (({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z) \land ({(H, C)}_{s h}^{(α_{i}, β_{i})}, Z) = (({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z) \lor ({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z) \land (({(F, A)}_{s h}^{(α_{i}, β_{i})}), Z) \lor ({(H, C)}_{s h}^{(α_{i}, β_{i})}, Z)$ .
4. $(({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z) \land (({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z) \land ({(H, C)}_{s h}^{(α_{i}, β_{i})}, Z) = (({(F, A)}_{s h}^{(α_{i}, β_{i})}, Z) \land ({(G, B)}_{s h}^{(α_{i}, β_{i})}, Z) \lor (({(F, A)}_{s h}^{(α_{i}, β_{i})}), Z) \land ({(H, C)}_{s h}^{(α_{i}, β_{i})}, Z))$ .

Proof. Straightforward from Definitions 4.1 and 4.2.

5. An Application of Time-Shadow Soft Set in Decision-Making

\begin{array}{l} {(F, E)}_{t} = {(e_{1}, {\frac{{u_{1}}^{t_{1}}}{0.6}, \frac{{u_{2}}^{t_{1}}}{0.3}, \frac{{u_{3}}^{t_{1}}}{0.2}, \frac{{u_{4}}^{t_{1}}}{0.4}}), \\ (e_{2}, {\frac{{u_{1}}^{t_{1}}}{0.5}, \frac{{u_{2}}^{t_{1}}}{0.3}, \frac{{u_{3}}^{t_{1}}}{0.2}, \frac{{u_{4}}^{t_{1}}}{0.7}}), \\ (e_{3}, {\frac{{u_{1}}^{t_{1}}}{0.3}, \frac{{u_{2}}^{t_{1}}}{0.6}, \frac{{u_{3}}^{t_{1}}}{0.8}, \frac{{u_{4}}^{t_{1}}}{0.9}}), \\ (e_{4}, {\frac{{u_{1}}^{t_{1}}}{0.5}, \frac{{u_{2}}^{t_{1}}}{0.4}, \frac{{u_{3}}^{t_{1}}}{0.6}, \frac{{u_{4}}^{t_{1}}}{0.8}}), \\ (e_{5}, {\frac{{u_{1}}^{t_{1}}}{0.9}, \frac{{u_{2}}^{t_{1}}}{0.2}, \frac{{u_{3}}^{t_{1}}}{0.4}, \frac{{u_{4}}^{t_{1}}}{0.8}}), \\ (e_{1}, {\frac{{u_{1}}^{t_{2}}}{0.7}, \frac{{u_{2}}^{t_{2}}}{0.4}, \frac{{u_{3}}^{t_{2}}}{0.1}, \frac{{u_{4}}^{t_{2}}}{0.3}}), \\ (e_{2}, {\frac{{u_{1}}^{t_{2}}}{0.3}, \frac{{u_{2}}^{t_{2}}}{0.1}, \frac{{u_{3}}^{t_{2}}}{0.2}, \frac{{u_{4}}^{t_{2}}}{0.6}}), \\ (e_{3}, {\frac{{u_{1}}^{t_{2}}}{0.7}, \frac{{u_{2}}^{t_{2}}}{0.8}, \frac{{u_{3}}^{t_{2}}}{0.6}, \frac{{u_{4}}^{t_{2}}}{0.4}}), \\ (e_{4}, {\frac{{u_{1}}^{t_{2}}}{0.9}, \frac{{u_{2}}^{t_{2}}}{0.3}, \frac{{u_{3}}^{t_{2}}}{0.4}, \frac{{u_{4}}^{t_{2}}}{0.7}}), \\ (e_{5}, {\frac{{u_{1}}^{t_{2}}}{0.2}, \frac{{u_{2}}^{t_{2}}}{0.7}, \frac{{u_{3}}^{t_{2}}}{0.8}, \frac{{u_{4}}^{t_{2}}}{0.5}}), \\ (e_{1}, {\frac{{u_{1}}^{t_{3}}}{0.7}, \frac{{u_{2}}^{t_{3}}}{0.8}, \frac{{u_{3}}^{t_{3}}}{0.6}, \frac{{u_{4}}^{t_{3}}}{0.4}}), \\ (e_{2}, {\frac{{u_{1}}^{t_{3}}}{0.7}, \frac{{u_{2}}^{t_{3}}}{0.5}, \frac{{u_{3}}^{t_{3}}}{0.6}, \frac{{u_{4}}^{t_{3}}}{0.4}}), \\ (e_{3}, {\frac{{u_{1}}^{t_{3}}}{0.6}, \frac{{u_{2}}^{t_{3}}}{0.4}, \frac{{u_{3}}^{t_{3}}}{0.5}, \frac{{u_{4}}^{t_{3}}}{0.7}}), \\ (e_{4}, {\frac{{u_{1}}^{t_{3}}}{0.6}, \frac{{u_{2}}^{t_{3}}}{0.7}, \frac{{u_{3}}^{t_{3}}}{0.5}, \frac{{u_{4}}^{t_{3}}}{0.3}}), \\ (e_{5}, {\frac{{u_{1}}^{t_{3}}}{0.7}, \frac{{u_{2}}^{t_{3}}}{0.2}, \frac{{u_{3}}^{t_{3}}}{0.6}, \frac{{u_{4}}^{t_{3}}}{0.3}}),}, \\ {(F, E)}_{s h}^{(α_{t}, β_{t})} = {(e_{1} (0.3, 0.8), {\frac{{u_{1}}^{t_{1}}}{[0, 1]}, \frac{{u_{2}}^{t_{1}}}{0}, \frac{{u_{3}}^{t_{1}}}{0}, \frac{{u_{4}}^{t_{1}}}{[0, 1]}}), \\ (e_{2} (0.4, 0.7), {\frac{{u_{1}}^{t_{1}}}{[0, 1]}, \frac{{u_{2}}^{t_{1}}}{0}, \frac{{u_{3}}^{t_{1}}}{0}, \frac{{u_{4}}^{t_{1}}}{1}}), \\ (e_{3} (0.3, 0.5), {\frac{{u_{1}}^{t_{1}}}{0}, \frac{{u_{2}}^{t_{1}}}{1}, \frac{{u_{3}}^{t_{1}}}{1}, \frac{{u_{4}}^{t_{1}}}{1}}), \\ (e_{4} (0.4, 0.6), {\frac{{u_{1}}^{t_{1}}}{[0, 1]}, \frac{{u_{2}}^{t_{1}}}{0}, \frac{{u_{3}}^{t_{1}}}{1}, \frac{{u_{4}}^{t_{1}}}{1}}), \\ (e_{5} (0.3, 0.8), {\frac{{u_{1}}^{t_{1}}}{1}, \frac{{u_{2}}^{t_{1}}}{0}, \frac{{u_{3}}^{t_{1}}}{[0, 1]}, \frac{{u_{4}}^{t_{1}}}{1}}), \\ (e_{1} (0.3, 0.8), {\frac{{u_{1}}^{t_{2}}}{[0, 1]}, \frac{{u_{2}}^{t_{2}}}{[0, 1]}, \frac{{u_{3}}^{t_{2}}}{0}, \frac{{u_{4}}^{t_{2}}}{0}}), \\ (e_{2} (0.4, 0.7), {\frac{{u_{1}}^{t_{2}}}{0}, \frac{{u_{2}}^{t_{2}}}{0}, \frac{{u_{3}}^{t_{2}}}{0}, \frac{{u_{4}}^{t_{2}}}{[0, 1]}}), \\ (e_{3} (0.3, 0.5), {\frac{{u_{1}}^{t_{2}}}{1}, \frac{{u_{2}}^{t_{2}}}{1}, \frac{{u_{3}}^{t_{2}}}{1}, \frac{{u_{4}}^{t_{2}}}{[0]}}), \\ (e_{4} (0.4, 0.6), {\frac{{u_{1}}^{t_{2}}}{1}, \frac{{u_{2}}^{t_{2}}}{0}, \frac{{u_{3}}^{t_{2}}}{0}, \frac{{u_{4}}^{t_{2}}}{1}}), \\ (e_{5} (0.3, 0.8), {\frac{{u_{1}}^{t_{2}}}{0}, \frac{{u_{2}}^{t_{2}}}{[0, 1]}, \frac{{u_{3}}^{t_{2}}}{1}, \frac{{u_{4}}^{t_{2}}}{[0, 1]}}), \\ (e_{1} (0.3, 0.8), {\frac{{u_{1}}^{t_{3}}}{[0, 1]}, \frac{{u_{2}}^{t_{3}}}{1}, \frac{{u_{3}}^{t_{3}}}{[0, 1]}, \frac{{u_{4}}^{t_{3}}}{[0, 1]}}), \\ (e_{2} (0.4, 0.7), {\frac{{u_{1}}^{t_{3}}}{1}, \frac{{u_{2}}^{t_{3}}}{[0, 1]}, \frac{{u_{3}}^{t_{3}}}{[0, 1]}, \frac{{u_{4}}^{t_{3}}}{0}}), \\ (e_{3} (0.3, 0.5), {\frac{{u_{1}}^{t_{3}}}{1}, \frac{{u_{2}}^{t_{3}}}{[0, 1]}, \frac{{u_{3}}^{t_{3}}}{1}, \frac{{u_{4}}^{t_{3}}}{1}}), \\ (e_{4} (0.4, 0.6), {\frac{{u_{1}}^{t_{3}}}{1}, \frac{{u_{2}}^{t_{3}}}{1}, \frac{{u_{3}}^{t_{3}}}{[0, 1]}, \frac{{u_{4}}^{t_{3}}}{0}}), \\ (e_{5} (0.3, 0.8), {\frac{{u_{1}}^{t_{3}}}{[0, 1]}, \frac{{u_{2}}^{t_{3}}}{0}, \frac{{u_{3}}^{t_{3}}}{[0, 1]}, \frac{{u_{4}}^{t_{3}}}{0}})} . \end{array}

5.1 Algorithm

1. Find the tabular representation of (F,E)_t as in Table 1.
2. Find the tabular representation of F (E) as in Table 2, where F (E) defined as follows:

$F (e) = {\frac{u}{\sum_{i = 1}^{n} t_{i} F_{t} (e) \ n \sum_{i = 1}^{n} F_{t} (e)} : u \in U, e \in E}$

where n = |T|.
3. Find the tabular representation of (F,E)_t_{_sh} as in Table 3.
4. Find the score of each element in U as in Table 4.

Table 4 shows that choice u₃ has the highest acceptance score, whereas choice u₄ has the highest waiting score, and there is no rejection choice.

6. Conclusion

Acknowledgments

The authors acknowledge the financial support received from Jadara University.

Conflict of Interest

No potential conflict of interest relevant to this article was reported.

Abstract
1. Introduction
2. Preliminaries
3. Time-Shadow Soft Set
4. AND and OR Operations
5. An Application of Time-Shadow Soft Set in Decision-Making
6. Conclusion
Acknowledgments
Conflict of Interest

Table 1 . Tabular representation of (F,E)_t.

U	u₁	u₂	u₃	u₄
(e₁, t₁)	0.6	0.3	0.2	0.4
(e₁, t₂)	0.7	0.4	0.1	0.3
(e₁, t₃)	0.7	0.8	0.6	0.4
(e₂, t₁)	0.5	0.3	0.2	0.7
(e₂, t₂)	0.3	0.1	0.2	0.6
(e₂, t₃)	0.7	0.5	0.6	0.4
(e₃, t₁)	0.3	0.6	0.8	0.9
(e₃, t₂)	0.7	0.8	0.6	0.4
(e₃, t₃)	0.6	0.4	0.5	0.7
(e₄, t₁)	0.5	0.4	0.6	0.8
(e₄, t₂)	0.9	0.3	0.4	0.7
(e₄, t₃)	0.6	0.7	0.5	0.3
(e₅, t₁)	0.9	0.2	0.4	0.8
(e₅, t₂)	0.2	0.7	0.8	0.5
(e₅, t₃)	0.7	0.2	0.6	0.3

Table 2 . Tabular representation of F (E).

U	u₁	u₂	u₃	u₄
e₁	0.68	0.77	0.81	0.66
e₂	0.51	0.66	0.80	0.60
e₃	0.72	0.62	0.61	0.63
e₄	0.68	0.93	0.64	0.49
e₅	0.62	0.66	0.70	0.77

Table 3 . Tabular representation of (F,E)_t_{_sh}.

U	u₁	u₂	u₃	u₄
e₁	[0,1]	[0,1]	1	[0,1]
e₂	[0,1]	[0,1]	1	[0,1]
e₃	1	1	1	1
e₄	1	1	1	[0,1]
e₅	[0,1]	[0,1]	[0,1]	[0,1]

Table 4 . Score table.

U	r₁	r_[0,1]
u₁	2	3
u₂	2	3
u₃	4	1
u₄	1	4

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SCOPUS 1

International Journal
of Fuzzy Logic and
Intelligent Systems

Original Article

Time-Shadow Soft Set: Concepts and Applications

Abstract

1. Introduction

2. Preliminaries

3. Time-Shadow Soft Set

3.1 Basic Operations

4. AND and OR Operations

5. An Application of Time-Shadow Soft Set in Decision-Making

5.1 Algorithm

6. Conclusion

Acknowledgments

Conflict of Interest

Acknowledgments

Conflict of Interest

Tables

References

Biography

Article

Original Article

Time-Shadow Soft Set: Concepts and Applications

Abstract

1. Introduction

2. Preliminaries

3. Time-Shadow Soft Set

3.1 Basic Operations

4. AND and OR Operations

5. An Application of Time-Shadow Soft Set in Decision-Making

5.1 Algorithm

6. Conclusion

Acknowledgments

Conflict of Interest

References

Article Tools

Metrics

Share this article on :

Related articles in IJFIS

New Soft Rough Set Approximations

Shawkat Alkhazaleh and Emad A. Marei
International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(2): 123-134 https://doi.org/10.5391/IJFIS.2021.21.2.123

New Soft Rough Set Approximations

International Journalof Fuzzy Logic andIntelligent Systems

Original Article

Time-Shadow Soft Set: Concepts and Applications

Abstract

1. Introduction

2. Preliminaries

3. Time-Shadow Soft Set

3.1 Basic Operations

4. AND and OR Operations

5. An Application of Time-Shadow Soft Set in Decision-Making

5.1 Algorithm

6. Conclusion

Acknowledgments

Conflict of Interest

Acknowledgments

Conflict of Interest

Tables

References

Biography

Article

Original Article

Time-Shadow Soft Set: Concepts and Applications

Abstract

1. Introduction

2. Preliminaries

3. Time-Shadow Soft Set

3.1 Basic Operations

4. AND and OR Operations

5. An Application of Time-Shadow Soft Set in Decision-Making

5.1 Algorithm

6. Conclusion

Acknowledgments

Conflict of Interest

References

Article Tools

Metrics

Share this article on :

Related articles in IJFIS New Soft Rough Set Approximations Shawkat Alkhazaleh and Emad A. Marei International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(2): 123-134 https://doi.org/10.5391/IJFIS.2021.21.2.123

New Soft Rough Set Approximations

International Journal
of Fuzzy Logic and
Intelligent Systems

Related articles in IJFIS

New Soft Rough Set Approximations

Shawkat Alkhazaleh and Emad A. Marei
International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(2): 123-134 https://doi.org/10.5391/IJFIS.2021.21.2.123