EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5937 ISSN 1307-5543 – ejpam.com Published by New York Business Global The Effect of the Weighted Expert Factor on Time Fuzzy Soft Expert Sets Anwar Bataihah Department of Mathematics, Faculty of Science, Jadara University, Irbid, Jordan Abstract. This paper introduces the Weighted Time Fuzzy Soft Expert Set (W-TFSES), an advanced framework that extends fuzzy soft set theory by incorporating time-dependent expert weighting to enhance decision-making under uncertainty. We rigorously define W-TFSES opera- tions—including complement, union, intersection, AND, and OR—and demonstrate its superiority over conventional models through real-world applications in risk assessment and medical diagno- sis. Unlike prior approaches, W-TFSES explicitly quantifies the evolving influence of experts over time, offering improved accuracy in dynamic, uncertain environments. Our work bridges the gap between fuzzy soft sets and temporal decision-making, establishing W-TFSES as a versatile tool for complex, real-world problems. Key Words and Phrases: Soft set, Fuzzy soft set, Time fuzzy soft expert set 1. Introduction The majority of problems in engineering, medical research, economics, and the en- vironment are fraught with uncertainty. Molodtsov[1] introduced the notion of soft set theory as a tool in math for coping with such uncertainty. Following Molodtsov’s work, Maji et al.[2] researched several soft set operations and applications. Also Maji et al. [3] presented the notion of fuzzy soft set as a more broad concept, as well as a combination of fuzzy set and soft set, and investigated its features. Roy and Maji [4] also applied this idea to handle decision-making challenges. Recently, various scholars have begun studying the properties and applications of soft set theory as in the research [5, 6].Wang [7] showed that in many real situations, immediate sensory data is insufficient for decision making. The fuzzy soft set was extended by [8] to create the parameterized time fuzzy soft expert set It also examined and explained the features of its main function and used this method to solve problems with decision-making. In [9], the authors introduced A useful framework for assisting decision-maker in problem-solving is decision-making theory. Furthermore, in 2010 Çaǧman et al.[10] established the notion of fuzzy parameterized fuzzy soft set and its operations. In addition, the FPFS-aggregation operator is used DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5937 Email addresses: a.bataihah@jadara.edu.jo, anwerbataihah@gmail.com (A. Bataihah) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. Bataihah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5937 2 of 28 to create the FPFS-decision making technique, which allows for more efficient decision processes. Alkhazaleh and Salleh [11, 12], proposed the notion of soft expert sets and fuzzy soft expert sets, which allow users to get the views of all experts in one model without any procedures. In [13], the authors discussed fuzzy parameterized fuzzy soft expert sets, which offer a membership value for each parameter in a collection of parameters and are an extension of fuzzy soft expert sets. In [14], the authors provided an overview of generalized fuzzy soft expert set. Recently, various scholars have begun studying the properties and applications of soft set theory. Additionally, researchers introduced using a neutrosophic fuzzy soft set to solve decision-making problems, like in [15–18]. Furthermore, Alqaraleh et. al.[19] introduced the concept of bipolar complex fuzzy soft sets. Recent developments in fuzzy soft topology have advanced the theoretical foundations for handling uncertainty. Saleh et al. [20] established categorical properties of fuzzy soft spaces, while Saleh and Al- Mufarrij [21] introduced θ-regularity concepts - both crucial for extending these structures to time-dependent expert systems like our W-TFSES. Recent advances in soft topology include the development of generalized covering properties, such as the almost/weakly soft Menger spaces [22] and nearly soft Menger spaces [23], which extend classical topological concepts to soft set frameworks. These works demonstrate the growing sophistication of soft topological structures, motivating further extensions like our weighted time-dependent approach. A critical component of decision-making is expert weighting, which guarantees that more skilled and informed experts have a bigger impact on results. This improves accu- racy, lowers bias, and strengthens decision robustness, especially in uncertain contexts like healthcare, finance, and engineering [24]. Decision-making procedures run the danger of being dominated by less competent people in the absence of expert weighing, which might produce biased or less-than-ideal outcomes [25]. This is particularly crucial in fuzzy logic- based decision systems, such temporal fuzzy soft expert sets, where experts are given the right weights to improve decision accuracy and uncertainty and imprecise information are inherent [26]. Decision-making frameworks become more organized, less subjective, and more reliable when weighted expert opinions are incorporated. so using historical informa- tion as part of the state representation provides us with important information to assist us in making better judgments in situations when temporal value is not taken into account, resulting in less accurate decision-making. If we want to take the views of more than one time (period), we must perform various operations such as union, intersection, etc. For a solution to this problem, we take a collection of time intervals, generalize it into what we call time fuzzy soft expert sets that is induced by [27], investigate some of its features, and apply this notion to a decision-making problem. It is critical to understand the history of the parameters under consideration in order to ensure the credibility of the information provided by specialists. The experts’ previous experiences are gathered in the number of periods (years, months, etc.) in which they are involved in a certain decision-making circumstance, and by looking at the time component, individuals are more confident in the conclusion that they make. We must examine the influence of time on fuzzy soft set applications, not only for the present period, but also for the past and future periods (forecasting information). The concept of time fuzzy was later used by other studies with A. Bataihah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5937 3 of 28 such subjects, including in works of [8, 28–30]. In this paper, we will present the notion of the effect of the weighted expert factor on time fuzzy soft expert sets, which is more effective and valuable, as we will see and the decisions made will be more precise, this means we will take the component time value of the information in our consideration when we are making decision. We will also define and investigate the attributes of its basic operations, which are complement, union and intersection. Finally, we’ll apply this approach to decision-making difficulties. 2. Preliminaries In this part, we cover several fundamental concepts in soft set theory. Molodtsov [1] defined soft sets as follows over U : Let U represent the universe set and P the set of parameters. P (U) signifies the power set of U and J ⊆ P . Definition 1. [1] Consider this mapping J : M → P (U) . Any A pair (J,M) is considered a soft set over U . In other terms, a soft set over U is a parameterized collection of subsets of the universe set U. For δ ∈ J, J (δ) can be viewed as the set of δ-approximate members of the soft set (J,M). Definition 2. [3] Let U be the initial universal set, and P be the set of parameters. Let IU be the power set of all fuzzy subsets of U . Let J ⊆ P , and F be the mapping F : A → IU . A pair (J, P ) is known as a fuzzy soft set over U. Definition 3. [3] Regarding two fuzzy soft sets (J,M) and (K,N) over U , (J,M) is known as a fuzzy soft subset of. (K,N) if (i) M ⊂ N and (ii) ∀δ ∈ J, J (δ) is fuzzy subset of K (δ). The association is represented by (F, A) ⊂̃ (K,N). In this situation, (K,N) is known as a fuzzy soft superset of. (J,M). Definition 4. [3] (J,M)c represents the complement of a fuzzy soft set (J,M), which has been described by (J,M)c = (Jc, ⌉A) where Jc :⌉A → P (U) is a mapping provided by Jc (Γ) = c (J (⌉Γ)) , ∀Γ ∈⌉M. c describes any fuzzy complement. Definition 5. [3] If (J,M) and (K,N) are two fuzzy soft sets then (J,M) AND (K,N) denoted by (J, M) ∧ (K,N) is defined by A. Bataihah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5937 4 of 28 (J,M) ∧ (K,N) = (C,M ×N) such that C (Γ, λ) = t (J (Γ) ,K (λ)) ,∀ (Γ, λ) ∈ J ×N , where t is any t-norm. Definition 6. [3] If (J,M) and (K,N) are two fuzzy soft sets then (J,M) OR (K,N) denoted by (J,M) ∨ (K,N) is defined by (J,M) ∨ (K,N) = (O,M ×N) such that O (Γ, λ) = s (J (Γ) , G (λ)) ,∀ (Γ, λ) ∈ M ×N , where s is any s-norm. Definition 7. [3] The union of two fuzzy soft sets (J,M) and (K,N) over a common universe U is the fuzzy soft set (H,C) where C = M ∪N, and ∀δ ∈ C, H (δ) =  J (δ) , if δ ∈ M −N, G (δ) , if δ ∈ N −M, s (J (δ) , G (δ)) , if δ ∈ M ∪N. Where s is any s-norm. Definition 8. [3] The intersection of two fuzzy soft sets (J,M) and (K,N) over a common universe U is the fuzzy soft set (H,C) where C = M ∪N, and ∀δ ∈ R, H (δ) =  J (δ) , if δ ∈ M −N, G (δ) , if δ ∈ N −M, s (J (δ) , G (δ)) , if δ ∈ M ∩N. Definition 9. [11] Let U be a set of universes, P a set of parameters, X a set of experts (agents). Let O = {o1, o2, ..., on} be a set of opinions, Z = E×X ×O and M ⊆ Z. A pair (F, M) is called a soft expert set over U, where F is a mapping given by F : A → P (U) where P (U) denoted the power set of U. Definition 10. [12] Let U be a set of universes, P a set of parameters, X a set of experts (agents). Let O be a set of opinions, Z = E ×X ×O and A ⊆ Z. A pair (J,M) is called a fuzzy soft expert set over U, where F is a mapping given by J : M → IU Where IU denotes all fuzzy subsets of U. Definition 11. Let U be an initial universal set and let P be a set of parameters. Let IU denote the power set of all fuzzy subsets of U , let J ⊆ P and T be a set of time where T = {t1, t2, ..., tn} . A collection of pairs (F,E)wt ∀ t ∈ T is called a time-fuzzy soft set T − FSS over U where Jt is a mapping given by Fw t : A → IU . A. Bataihah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5937 5 of 28 3. Weighted Time Fuzzy Soft Expert Set The definition of a The effect of the weighted expert factor on time fuzzy soft expert sets(W-TFSES) and their fundamental characteristics are presented in this section. We will give more consideration to the expert weights; this means that the choice experts may not be of equal importance degree. Additionally, we must take into account the com- ponent time value of the information when making judgments since doing so will result in more accurate decisions because some or all of the parameters have a time value for prior knowledge. Additionally, we define and examine the features of its fundamental op- erations—union, complement, intersection, AND, and OR. Lastly, we provide speculative uses of this idea in decision-making situations. 3.1. Main Definition Definition 12. Define U as the universe, E as a set of parameters, X as a group of experts, and O = {o1, o2, ..., on} as a collection of viewpoints and W = {w1, w2, ..., wn} as a collection of weights for the experts. For Z to be equal to E × X × O, A ⊆ Z. Let T be a collection of times where T = {t1, t2, ..., tn}, and let IU represent the power set of all fuzzy subsets of U . Atime-fuzzy soft expert set W-TFSES over U is a set of pairings (F,A)wt ∀ ti ∈ T , where F is a mapping provided by Fw t : A → IU . Example 1. Let U = {u1, u2, u3, u4} be a set of universe, E = {e1, e2, e3} a set of parameters and T = {t1, t2, t3} be a set of time and X = {m,n, r} a set of experts and W = {0.8, 0.5, 0.7} of weights for the experts. Define a function Fw t : A → IU . as follows: Fw 1 ( e1, m 0.8 , 1 ) = { u1 t1 0.8 , u2 t1 0.5 , u3 t1 0.4 , u4 t1 0.3 } , Fw 1 ( e1, n 0.5 , 1 ) = { u1 t1 0.6 , u2 t1 0.9 , u3 t1 0.6 , u4 t1 0.5 } , Fw 1 ( e1, r 0.7 , 1 ) = { u1 t1 0.7 , u2 t1 0.6 , u3 t1 0.3 , u4 t1 0.7 } , Fw 1 ( e2, m 0.8 , 1 ) = { u1 t1 0.6 , u2 t1 0.1 , u3 t1 0.4 , u4 t1 0.8 } , Fw 1 ( e2, n 0.5 , 1 ) = { u1 t1 0.5 , u2 t1 0.7 , u3 t1 0.2 , u4 t1 0.5 } , Fw 1 ( e2, r 0.7 , 1 ) = { u1 t1 0.2 , u2 t1 0.6 , u3 t1 0.7 , u4 t1 0.3 } , Fw 1 ( e3, m 0.8 , 1 ) = { u1 t1 0.2 , u2 t1 0.4 , u3 t1 0.9 , u4 t1 0.6 } , Fw 1 ( e3, n 0.5 , 1 ) = { u1 t1 0.6 , u2 t1 0.5 , u3 t1 0.6 , u4 t1 0.7 } , Fw 1 ( e3, r 0.7 , 1 ) = { u1 t1 0.4 , u2 t1 0.7 , u3 t1 0.1 , u4 t1 0.5 } , Fw 2 ( e1, m 0.8 , 1 ) = { u1 t2 0.4 , u2 t2 0.8 , u3 t2 0.2 , u4 t2 0.4 } , Fw 2 ( e1, n 0.5 , 1 ) = { u1 t2 0.6 , u2 t2 0.9 , u3 t2 0.4 , u4 t2 0.8 } , Fw 2 ( e1, r 0.7 , 1 ) = { u1 t2 0.9 , u2 t2 0.2 , u3 t2 0.3 , u4 t2 0.3 } , Fw 2 ( e2, m 0.8 , 1 ) = { u1 t2 0.2 , u2 t2 0.9 , u3 t2 0.5 , u4 t2 0.5 } , Fw 2 ( e2, n 0.5 , 1 ) = { u1 t2 0.7 , u2 t2 0.6 , u3 t2 0.2 , u4 t2 0.6 } , Fw 2 ( e2, r 0.7 , 1 ) = { u1 t2 0.5 , u2 t2 0.4 , u3 t2 0.7 , u4 t2 0.4 } , Fw 2 ( e3, m 0.8 , 1 ) = { u1 t2 0.9 , u2 t2 0.4 , u3 t2 0.1 , u4 t2 0.7 } , A. Bataihah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5937 6 of 28 Fw 2 ( e3, n 0.5 , 1 ) = { u1 t2 0.8 , u2 t2 0.6 , u3 t2 0.1 , u4 t2 0.5 } , Fw 2 ( e3, r 0.7 , 1 ) = { u1 t2 0.2 , u2 t2 0.9 , u3 t2 0.6 , u4 t2 0.7 } , Fw 3 ( e1, m 0.8 , 1 ) = { u1 t3 0.2 , u2 t3 0.7 , u3 t3 0.8 , u4 t3 0.3 } , Fw 3 ( e1, n 0.5 , 1 ) = { u1 t3 0.4 , u2 t3 0.9 , u3 t3 0.6 , u4 t3 0.5 } , Fw 3 ( e1, r 0.7 , 1 ) = { u1 t3 0.6 , u2 t3 0.5 , u3 t3 0.7 , u4 t3 0.8 } , Fw 3 ( e2, m 0.8 , 1 ) = { u1 t3 0.5 , u2 t3 0.4 , u3 t3 0.7 , u4 t3 0.9 } , Fw 3 ( e2, n 0.5 , 1 ) = { u1 t3 0.8 , u2 t3 0.6 , u3 t3 0.2 , u4 t3 0.7 } , Fw 3 ( e2, r 0.7 , 1 ) = { u1 t3 0.1 , u2 t3 0.8 , u3 t3 0.5 , u4 t3 0.8 } , Fw 3 ( e3, m 0.8 , 1 ) = { u1 t3 0.7 , u2 t3 0.4 , u3 t3 0.1 , u4 t3 0.8 } , Fw 3 ( e3, n 0.5 , 1 ) = { u1 t3 0.6 , u2 t3 0.5 , u3 t3 0.3 , u4 t3 0.9 } , Fw 3 ( e3, r 0.7 , 1 ) = { u1 t3 0.4 , u2 t3 0.7 , u3 t3 0.5 , u4 t3 0.6 } , Fw 1 ( e1, m 0.8 , 0 ) = { u1 t1 0.3 , u2 t1 0.6 , u3 t1 0.7 , u4 t1 0.8 } , Fw 1 ( e1, n 0.5 , 0 ) = { u1 t1 0.5 , u2 t1 0.3 , u3 t1 0.5 , u4 t1 0.4 } , Fw 1 ( e1, r 0.7 , 0 ) = { u1 t1 0.4 , u2 t1 0.5 , u3 t1 0.8 , u4 t1 0.3 } , Fw 1 ( e2, m 0.8 , 0 ) = { u1 t1 0.5 , u2 t1 0.7 , u3 t1 0.5 , u4 t1 0.3 } , Fw 1 ( e2, n 0.5 , 0 ) = { u1 t1 0.4 , u2 t1 0.4 , u3 t1 0.9 , u4 t1 0.6 } , Fw 1 ( e2, r 0.7 , 0 ) = { u1 t1 0.6 , u2 t1 0.3 , u3 t1 0.4 , u4 t1 0.1 } , Fw 1 ( e3, m 0.8 , 0 ) = { u1 t1 0.7 , u2 t1 0.5 , u3 t1 0.2 , u4 t1 0.3 } , Fw 1 ( e3, n 0.5 , 0 ) = { u1 t1 0.5 , u2 t1 0.4 , u3 t1 0.5 , u4 t1 0.2 } , Fw 1 ( e3, r 0.7 , 0 ) = { u1 t1 0.8 , u2 t1 0.2 , u3 t1 0.8 , u4 t1 0.4 } , Fw 2 ( e1, m 0.8 , 0 ) = { u1 t2 0.7 , u2 t2 0.4 , u3 t2 0.7 , u4 t2 0.5 } , Fw 2 ( e1, n 0.5 , 0 ) = { u1 t2 0.5 , u2 t2 0.2 , u3 t2 0.5 , u4 t2 0.4 } , Fw 2 ( e1, r 0.7 , 0 ) = { u1 t2 0.3 , u2 t2 0.7 , u3 t2 0.8 , u4 t2 0.8 } , Fw 2 ( e2, m 0.8 , 0 ) = { u1 t2 0.9 , u2 t2 0.3 , u3 t2 0.6 , u4 t2 0.6 } , Fw 2 ( e2, n 0.5 , 0 ) = { u1 t2 0.4 , u2 t2 0.5 , u3 t2 0.9 , u4 t2 0.4 } , Fw 2 ( e2, r 0.7 , 0 ) = { u1 t2 0.6 , u2 t2 0.7 , u3 t2 0.4 , u4 t2 0.1 } , Fw 2 ( e3, m 0.8 , 0 ) = { u1 t2 0.2 , u2 t2 0.7 , u3 t2 0.8 , u4 t2 0.2 } , Fw 2 ( e3, n 0.5 , 0 ) = { u1 t2 0.4 , u2 t2 0.5 , u3 t2 0.7 , u4 t2 0.6 } , Fw 2 ( e3, r 0.7 , 0 ) = { u1 t2 0.7 , u2 t2 0.1 , u3 t2 0.3 , u4 t2 0.3 } , Fw 3 ( e1, m 0.8 , 0 ) = { u1 t3 0.6 , u2 t3 0.8 , u3 t3 0.2 , u4 t3 0.4 } , Fw 3 ( e1, n 0.5 , 0 ) = { u1 t3 0.7 , u2 t3 0.5 , u3 t3 0.4 , u4 t3 0.6 } , Fw 3 ( e1, r 0.7 , 0 ) = { u1 t3 0.5 , u2 t3 0.7 , u3 t3 0.2 , u4 t3 0.5 } , Fw 3 ( e2, m 0.8 , 0 ) = { u1 t3 0.3 , u2 t3 0.7 , u3 t3 0.4 , u4 t3 0.1 } , Fw 3 ( e2, n 0.5 , 0 ) = { u1 t3 0.1 , u2 t3 0.5 , u3 t3 0.6 , u4 t3 0.4 } , Fw 3 ( e2, r 0.7 , 0 ) = { u1 t3 0.7 , u2 t3 0.3 , u3 t3 0.3 , u4 t3 0.2 } , Fw 3 ( e3, m 0.8 , 0 ) = { u1 t3 0.4 , u2 t3 0.6 , u3 t3 0.8 , u4 t3 0.3 } , Fw 3 ( e3, n 0.5 , 0 ) = { u1 t3 0.3 , u2 t3 0.4 , u3 t3 0.7 , u4 t3 0.2 } , Fw 3 ( e3, r 0.7 , 0 ) = { u1 t3 0.7 , u2 t3 0.2 , u3 t3 0.6 , u4 t3 0.5 } . Next, we may determine the The Effect of the weighted expert factor on time fuzzy soft expert setss (F,E)wt , which comprise the subsequent set of approximations: (F,E)wt = {(( e1, m 0.8 , 1 ) , { u1 t1 0.8 , u2 t1 0.5 , u3 t1 0.4 , u4 t1 0.3 }) , (( e1, n 0.5 , 1 ) , { u1 t1 0.6 , u2 t1 0.9 , u3 t1 0.6 , u4 t1 0.5 }) A. Bataihah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5937 7 of 28 ... . . .(( e3, n 0.5 , 0 ) , { u1 t3 0.3 , u2 t3 0.4 , u3 t3 0.7 , u4 t3 0.2 }) , (( e3, r 0.7 , 0 ) , { u1 t3 0.7 , u2 t3 0.2 , u3 t3 0.6 , u4 t3 0.5 })} . Definition 13. For two W-TFSES (F,A)wt and (G,B)wt over U : • (F,A)wt is a W-TFSES subset of (G,B)wt if: (i) B ⊆ A (ii) ∀t ∈ T, δ ∈ B,Gw t (δ) is a fuzzy soft expert subset of Fw t (δ) • They are equal if each is a subset of the other. Example 2. Consider Example 1, where A ={( e1, m 0.8 , 1 ) t1 , ( e2, m 0.8 , 1 ) t1 , ( e2, n 0.5 , 1 ) t3 , ( e2, r 0.7 , 1 ) t3 , ( e2, n 0.5 , 0 ) t2 , ( e2, r 0.7 , 0 ) t2 , ( e2, r 0.7 , 0 ) t3 ,( e3, m 0.8 , 0 ) t3 } , B = {( e1, m 0.8 , 1 ) t1 , ( e2, m 0.8 , 1 ) t1 , ( e2, n 0.5 , 1 ) t3 , ( e2, r 0.7 , 1 ) t3 , ( e2, n 0.5 , 0 ) t2 } . Clearly B ⊂ A. Now, let (G,B)wt and (F,A)wt be defined as follows: (F,E)wt = {(( e1, m 0.8 , 1 ) , { u1 t1 0.8 , u2 t1 0.5 , u3 t1 0.4 , u4 t1 0.3 }) , (( e2, m 0.8 , 1 ) , { u1 t1 0.6 , u2 t1 0.1 , u3 t1 0.4 , u4 t1 0.8 }) ,(( e2, n 0.5 , 1 ) , { u1 t3 0.8 , u2 t3 0.6 , u3 t3 0.2 , u4 t3 0.7 }) , (( e2, r 0.7 , 1 ) , { u1 t3 0.1 , u2 t3 0.8 , u3 t3 0.5 , u4 t3 0.8 }) ,(( e2, n 0.5 , 0 ) , { u1 t2 0.4 , u2 t2 0.5 , u3 t2 0.9 , u4 t2 0.4 }) , (( e2, r 0.7 , 0 ){ u1 t2 0.6 , u2 t2 0.7 , u3 t2 0.4 , u4 t2 0.1 }) ,(( e2, r 0.7 , 0 ) , { u1 t3 0.7 , u2 t3 0.3 , u3 t3 0.3 , u4 t3 0.2 }) , (( e3, m 0.8 , 0 ) , { u1 t3 0.4 , u2 t3 0.6 , u3 t3 0.8 , u4 t3 0.3 })} , (G,E)wt = {(( e1, m 0.8 , 1 ) , { u1 t1 0.6 , u2 t1 0.2 , u3 t1 0.1 , u4 t1 0.0 }) , (( e2, m 0.8 , 1 ) , { u1 t1 0.5 , u2 t1 0.1 , u3 t1 0.4 , u4 t1 0.7 }) , (( e2, n 0.5 , 1 ) , { u1 t3 0.8 , u2 t3 0.5 , u3 t3 0.2 , u4 t3 0.6 }) , (( e2, r 0.7 , 1 ) , { u1 t3 0.0 , u2 t3 0.6 , u3 t3 0.5 , u4 t3 0.3 }) , (( e2, n 0.5 , 0 ) , { u1 t2 0.4 , u2 t2 0.3 , u3 t2 0.7 , u4 t2 0.2 })} . We can easily verify that (G,E)wt ⊆ (F,E)wt . Definition 14. For a W-TFSES (F,A)wt : A. Bataihah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5937 8 of 28 • The agree-W-TFSES ((F,A)wt )1 contains all Fw t (α) where α ∈ E ×X × {1} • The disagree-W-TFSES ((F,A)wt )0 contains all Fw t (α) where α ∈ E ×X × {0} Example 3. Consider Example 1. Then the agree a weighted expert factor on time fuzzy soft expert sets ((F,A)wt )1 over U is ((F,A)wt )1 ={(( e1, m 0.8 , 1 ) , { u1 t1 0.8 , u2 t1 0.5 , u3 t1 0.4 , u4 t1 0.3 }) , (( e1, n 0.5 , 1 ) , { u1 t1 0.6 , u2 t1 0.9 , u3 t1 0.6 , u4 t1 0.5 }) , (( e1, r 0.7 , 1 ) , { u1 t1 0.7 , u2 t1 0.6 , u3 t1 0.3 , u4 t1 0.7 }) , (( e2, m 0.8 , 1 ) , { u1 t1 0.6 , u2 t1 0.1 , u3 t1 0.4 , u4 t1 0.8 }) , (( e2, n 0.5 , 1 ) , { u1 t1 0.5 , u2 t1 0.7 , u3 t1 0.2 , u4 t1 0.5 }) , (( e2, r 0.7 , 1 ) , { u1 t1 0.2 , u2 t1 0.6 , u3 t1 0.7 , u4 t1 0.3 }) , (( e3, m 0.8 , 1 ) , { u1 t1 0.2 , u2 t1 0.4 , u3 t1 0.9 , u4 t1 0.6 }) , (( e3, n 0.5 , 1 ) , { u1 t1 0.6 , u2 t1 0.5 , u3 t1 0.6 , u4 t1 0.7 }) , (( e3, r 0.7 , 1 ) , { u1 t1 0.4 , u2 t1 0.7 , u3 t1 0.1 , u4 t1 0.5 }) , (( e1, m 0.8 , 1 ) , { u1 t2 0.4 , u2 t2 0.8 , u3 t2 0.2 , u4 t2 0.4 }) , (( e1, n 0.5 , 1 ) , { u1 t2 0.6 , u2 t2 0.9 , u3 t2 0.4 , u4 t2 0.8 }) , (( e1, r 0.7 , 1 ) , { u1 t2 0.9 , u2 t2 0.2 , u3 t2 0.3 , u4 t2 0.3 }) , (( e2, m 0.8 , 1 ) , { u1 t2 0.2 , u2 t2 0.9 , u3 t2 0.5 , u4 t2 0.5 }) , (( e2, n 0.5 , 1 ) , { u1 t2 0.7 , u2 t2 0.6 , u3 t2 0.2 , u4 t2 0.6 }) , (( e2, r 0.7 , 1 ) , { u1 t2 0.5 , u2 t2 0.4 , u3 t2 0.7 , u4 t2 0.4 }) , (( e3, m 0.8 , 1 ) , { u1 t2 0.9 , u2 t2 0.4 , u3 t2 0.1 , u4 t2 0.7 }) , (( e3, n 0.5 , 1 ) , { u1 t2 0.8 , u2 t2 0.6 , u3 t2 0.1 , u4 t2 0.5 }) , (( e3, r 0.7 , 1 ) , { u1 t2 0.2 , u2 t2 0.9 , u3 t2 0.6 , u4 t2 0.7 }) , (( e1, m 0.8 , 1 ) , { u1 t3 0.2 , u2 t3 0.7 , u3 t3 0.8 , u4 t3 0.3 }) , (( e1, n 0.5 , 1 ) , { u1 t3 0.4 , u2 t3 0.9 , u3 t3 0.6 , u4 t3 0.5 }) ,(( e1, r 0.7 , 1 ) , { u1 t3 0.6 , u2 t3 0.5 , u3 t3 0.7 , u4 t3 0.8 }) , (( e2, m 0.8 , 1 ) , { u1 t3 0.5 , u2 t3 0.4 , u3 t3 0.7 , u4 t3 0.9 }) , (( e2, n 0.5 , 1 ) , { u1 t3 0.8 , u2 t3 0.6 , u3 t3 0.2 , u4 t3 0.7 }) , (( e2, r 0.7 , 1 ) , { u1 t3 0.1 , u2 t3 0.8 , u3 t3 0.5 , u4 t3 0.8 }) , (( e3, m 0.8 , 1 ) , { u1 t3 0.7 , u2 t3 0.4 , u3 t3 0.1 , u4 t3 0.8 }) , (( e3, n 0.5 , 1 ) , { u1 t3 0.6 , u2 t3 0.5 , u3 t3 0.3 , u4 t3 0.9 }) , (( e3, r 0.7 , 1 ) , { u1 t3 0.4 , u2 t3 0.7 , u3 t3 0.5 , u4 t3 0.6 })} . and the disagree- time fuzzy soft expert set ((F,A)t) w 0 over U is A. Bataihah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5937 9 of 28 ((F,A)wt )0 ={(( e1, m 0.8 , 0 ) , { u1 t1 0.3 , u2 t1 0.6 , u3 t1 0.7 , u4 t1 0.8 }) , (( e1, n 0.5 , 0 ) , { u1 t1 0.5 , u2 t1 0.3 , u3 t1 0.5 , u4 t1 0.4 }) , (( e1, r 0.7 , 0 ) , { u1 t1 0.4 , u2 t1 0.5 , u3 t1 0.8 , u4 t1 0.3 }) , (( e2, m 0.8 , 0 ) , { u1 t1 0.5 , u2 t1 0.7 , u3 t1 0.5 , u4 t1 0.3 }) , (( e2, n 0.5 , 0 ) , { u1 t1 0.4 , u2 t1 0.4 , u3 t1 0.9 , u4 t1 0.6 }) , (( e2, r 0.7 , 0 ) , { u1 t1 0.6 , u2 t1 0.3 , u3 t1 0.4 , u4 t1 0.1 }) , (( e3, m 0.8 , 0 ) , { u1 t1 0.7 , u2 t1 0.5 , u3 t1 0.2 , u4 t1 0.3 }) , (( e3, n 0.5 , 0 ) , { u1 t1 0.5 , u2 t1 0.4 , u3 t1 0.5 , u4 t1 0.2 }) , (( e3, r 0.7 , 0 ) , { u1 t1 0.8 , u2 t1 0.2 , u3 t1 0.8 , u4 t1 0.4 }) , (( e1, m 0.8 , 0 ) , { u1 t2 0.7 , u2 t2 0.4 , u3 t2 0.7 , u4 t2 0.5 }) , (( e1, n 0.5 , 0 ) , { u1 t2 0.5 , u2 t2 0.2 , u3 t2 0.5 , u4 t2 0.4 }) , (( e1, r 0.7 , 0 ) , { u1 t2 0.3 , u2 t2 0.7 , u3 t2 0.8 , u4 t2 0.8 }) , (( e2, m 0.8 , 0 ) , { u1 t2 0.9 , u2 t2 0.3 , u3 t2 0.6 , u4 t2 0.6 }) , (( e2, n 0.5 , 0 ) , { u1 t2 0.4 , u2 t2 0.5 , u3 t2 0.9 , u4 t2 0.4 }) ,(( e2, r 0.7 , 0 ){ u1 t2 0.6 , u2 t2 0.7 , u3 t2 0.4 , u4 t2 0.1 }) , (( e3, m 0.8 , 0 ) , { u1 t2 0.2 , u2 t2 0.7 , u3 t2 0.8 , u4 t2 0.2 }) , (( e3, n 0.5 , 0 ) , { u1 t2 0.4 , u2 t2 0.5 , u3 t2 0.7 , u4 t2 0.6 }) , (( e3, r 0.7 , 0 ) , { u1 t2 0.7 , u2 t2 0.1 , u3 t2 0.3 , u4 t2 0.3 }) , (( e1, m 0.8 , 0 ) , { u1 t3 0.6 , u2 t3 0.8 , u3 t3 0.2 , u4 t3 0.4 }) , (( e1, n 0.5 , 0 ) , { u1 t3 0.7 , u2 t3 0.5 , u3 t3 0.4 , u4 t3 0.6 }) , (( e1, r 0.7 , 0 ) , { u1 t3 0.5 , u2 t3 0.7 , u3 t3 0.2 , u4 t3 0.5 }) , (( e2, m 0.8 , 0 ) , { u1 t3 0.3 , u2 t3 0.7 , u3 t3 0.4 , u4 t3 0.1 }) , (( e2, n 0.5 , 0 ) , { u1 t3 0.1 , u2 t3 0.5 , u3 t3 0.6 , u4 t3 0.4 }) , (( e2, r 0.7 , 0 ) , { u1 t3 0.7 , u2 t3 0.3 , u3 t3 0.3 , u4 t3 0.2 }) , (( e3, m 0.8 , 0 ) , { u1 t3 0.4 , u2 t3 0.6 , u3 t3 0.8 , u4 t3 0.3 }) , (( e3, n 0.5 , 0 ) , { u1 t3 0.3 , u2 t3 0.4 , u3 t3 0.7 , u4 t3 0.2 }) , (( e3, r 0.7 , 0 ) , { u1 t3 0.7 , u2 t3 0.2 , u3 t3 0.6 , u4 t3 0.5 })} . 4. Fundamental Operation We define complement, union, and intersection of weighted expert factor on time fuzzy soft expert sets , deduce various features, and provide some examples to illustrate our points in this section. A. Bataihah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5937 10 of 28 4.1. Complement Definition 15. The complement of time fuzzy soft expert set (F,A)wt , denoted by (F,A)w c t , is defined by (F,A)w c t = (F c, ⌉A)wt where Fwc t :⌉A → P (U) is a mapping given by Fwc t (α) = c(Fw t (⌉α)), ∀α ⊂⌉A, where c is a time fuzzy soft expert set complement. Example 4. Consider Example 1. I have utilized the simple fuzzy complement to c̃(F,A)wt = {(( e1, m 0.8 , 1 ) , { u1 t1 0.2 , u2 t1 0.5 , u3 t1 0.6 , u4 t1 0.7 }) , (( e1, n 0.5 , 1 ) , { u1 t1 0.4 , u2 t1 0.1 , u3 t1 0.4 , u4 t1 0.5 }) (( e1, r 0.7 , 1 ) , { u1 t1 0.3 , u2 t1 0.4 , u3 t1 0.7 , u4 t1 0.3 }) , (( e2, m 0.8 , 1 ) , { u1 t1 0.4 , u2 t1 0.9 , u3 t1 0.6 , u4 t1 0.2 }) ,(( e2, n 0.5 , 1 ) , { u1 t1 0.5 , u2 t1 0.3 , u3 t1 0.8 , u4 t1 0.5 }) , (( e2, r 0.7 , 1 ) , { u1 t1 0.8 , u2 t1 0.4 , u3 t1 0.3 , u4 t1 0.7 }) ,(( e3, m 0.8 , 1 ) , { u1 t1 0.8 , u2 t1 0.6 , u3 t1 0.1 , u4 t1 0.4 }) , (( e3, n 0.5 , 1 ) , { u1 t1 0.4 , u2 t1 0.5 , u3 t1 0.4 , u4 t1 0.3 }) ,(( e3, r 0.7 , 1 ) , { u1 t1 0.6 , u2 t1 0.3 , u3 t1 0.9 , u4 t1 0.5 }) , (( e1, m 0.8 , 1 ) , { u1 t2 0.6 , u2 t2 0.2 , u3 t2 0.8 , u4 t2 0.6 }) ,(( e1, n 0.5 , 1 ) , { u1 t2 0.4 , u2 t2 0.1 , u3 t2 0.6 , u4 t2 0.2 }) , (( e1, r 0.7 , 1 ) , { u1 t2 0.1 , u2 t2 0.8 , u3 t2 0.7 , u4 t2 0.7 }) ,(( e2, m 0.8 , 1 ) , { u1 t2 0.8 , u2 t2 0.1 , u3 t2 0.5 , u4 t2 0.5 }) , (( e2, n 0.5 , 1 ) , { u1 t2 0.3 , u2 t2 0.4 , u3 t2 0.8 , u4 t2 0.4 }) ,(( e2, r 0.7 , 1 ) , { u1 t2 0.5 , u2 t2 0.6 , u3 t2 0.3 , u4 t2 0.6 }) , (( e3, m 0.8 , 1 ) , { u1 t2 0.1 , u2 t2 0.6 , u3 t2 0.9 , u4 t2 0.3 }) ,(( e3, n 0.5 , 1 ) , { u1 t2 0.2 , u2 t2 0.4 , u3 t2 0.9 , u4 t2 0.5 }) , (( e3, r 0.7 , 1 ) , { u1 t2 0.8 , u2 t2 0.1 , u3 t2 0.4 , u4 t2 0.3 }) ,(( e1, m 0.8 , 1 ) , { u1 t3 0.8 , u2 t3 0.3 , u3 t3 0.2 , u4 t3 0.7 }) , (( e1, n 0.5 , 1 ) , { u1 t3 0.6 , u2 t3 0.1 , u3 t3 0.4 , u4 t3 0.5 }) ,(( e1, r 0.7 , 1 ) , { u1 t3 0.4 , u2 t3 0.5 , u3 t3 0.3 , u4 t3 0.2 }) , (( e2, m 0.8 , 1 ) , { u1 t3 0.5 , u2 t3 0.6 , u3 t3 0.3 , u4 t3 0.1 }) ,(( e2, n 0.5 , 1 ) , { u1 t3 0.2 , u2 t3 0.4 , u3 t3 0.8 , u4 t3 0.3 }) , (( e2, r 0.7 , 1 ) , { u1 t3 0.9 , u2 t3 0.2 , u3 t3 0.5 , u4 t3 0.2 }) ,(( e3, m 0.8 , 1 ) , { u1 t3 0.3 , u2 t3 0.6 , u3 t3 0.9 , u4 t3 0.2 }) , (( e3, n 0.5 , 1 ) , { u1 t3 0.4 , u2 t3 0.5 , u3 t3 0.7 , u4 t3 0.1 }) ,(( e3, r 0.7 , 1 ) , { u1 t3 0.6 , u2 t3 0.3 , u3 t3 0.5 , u4 t3 0.4 }) , (( e1, m 0.8 , 0 ) , { u1 t1 0.7 , u2 t1 0.4 , u3 t1 0.3 , u4 t1 0.2 }) ,(( e1, n 0.5 , 0 ) , { u1 t1 0.5 , u2 t1 0.7 , u3 t1 0.5 , u4 t1 0.6 }) , (( e1, r 0.7 , 0 ) , { u1 t1 0.6 , u2 t1 0.5 , u3 t1 0.2 , u4 t1 0.7 }) ,(( e2, m 0.8 , 0 ) , { u1 t1 0.5 , u2 t1 0.3 , u3 t1 0.5 , u4 t1 0.7 }) , (( e2, n 0.5 , 0 ) , { u1 t1 0.6 , u2 t1 0.6 , u3 t1 0.1 , u4 t1 0.4 }) ,(( e2, r 0.7 , 0 ) , { u1 t1 0.4 , u2 t1 0.7 , u3 t1 0.6 , u4 t1 0.9 }) , (( e3, m 0.8 , 0 ) , { u1 t1 0.3 , u2 t1 0.5 , u3 t1 0.8 , u4 t1 0.7 }) ,(( e3, n 0.5 , 0 ) , { u1 t1 0.5 , u2 t1 0.6 , u3 t1 0.5 , u4 t1 0.8 }) , (( e3, r 0.7 , 0 ) , { u1 t1 0.2 , u2 t1 0.8 , u3 t1 0.2 , u4 t1 0.6 }) , A. Bataihah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5937 11 of 28(( e1, m 0.8 , 0 ) , { u1 t2 0.3 , u2 t2 0.6 , u3 t2 0.3 , u4 t2 0.5 }) , (( e1, n 0.5 , 0 ) , { u1 t2 0.5 , u2 t2 0.8 , u3 t2 0.5 , u4 t2 0.6 }) ,(( e1, r 0.7 , 0 ) , { u1 t2 0.7 , u2 t2 0.3 , u3 t2 0.2 , u4 t2 0.2 }) , (( e2, m 0.8 , 0 ) , { u1 t2 0.1 , u2 t2 0.7 , u3 t2 0.4 , u4 t2 0.4 }) ,(( e2, n 0.5 , 0 ) , { u1 t2 0.6 , u2 t2 0.5 , u3 t2 0.1 , u4 t2 0.6 }) , (( e2, r 0.7 , 0 ){ u1 t2 0.4 , u2 t2 0.3 , u3 t2 0.6 , u4 t2 0.9 }) ,(( e3, m 0.8 , 0 ) , { u1 t2 0.8 , u2 t2 0.3 , u3 t2 0.2 , u4 t2 0.8 }) , (( e3, n 0.5 , 0 ) , { u1 t2 0.6 , u2 t2 0.5 , u3 t2 0.3 , u4 t2 0.4 }) ,(( e3, r 0.7 , 0 ) , { u1 t2 0.3 , u2 t2 0.9 , u3 t2 0.7 , u4 t2 0.7 }) , (( e1, m 0.8 , 0 ) , { u1 t3 0.4 , u2 t3 0.2 , u3 t3 0.8 , u4 t3 0.6 }) ,(( e1, n 0.5 , 0 ) , { u1 t3 0.3 , u2 t3 0.5 , u3 t3 0.6 , u4 t3 0.4 }) , (( e1, r 0.7 , 0 ) , { u1 t3 0.5 , u2 t3 0.3 , u3 t3 0.8 , u4 t3 0.5 }) ,(( e2, m 0.8 , 0 ) , { u1 t3 0.7 , u2 t3 0.3 , u3 t3 0.6 , u4 t3 0.9 }) , (( e2, n 0.5 , 0 ) , { u1 t3 0.9 , u2 t3 0.5 , u3 t3 0.4 , u4 t3 0.6 }) ,(( e2, r 0.7 , 0 ) , { u1 t3 0.3 , u2 t3 0.7 , u3 t3 0.7 , u4 t3 0.8 }) , (( e3, m 0.8 , 0 ) , { u1 t3 0.6 , u2 t3 0.4 , u3 t3 0.2 , u4 t3 0.7 }) , (( e3, n 0.5 , 0 ) , { u1 t3 0.7 , u2 t3 0.6 , u3 t3 0.3 , u4 t3 0.8 }) , (( e3, r 0.7 , 0 ) , { u1 t3 0.3 , u2 t3 0.8 , u3 t3 0.4 , u4 t3 0.5 })} . Proposition 1. If (F,A)wt is a weighted expert factor on time fuzzy soft expert sets over U, then c̃ (c̃(F,A)t) = (F,A)wt . Proof. From Definition 15 we have (F,A)wt c = (F c, A), where Fw t c (α) = 1̄− Fw t (α) , ∀α ∈ A. Now, ((F,A)wt c) c = (( Fwc t )c , A ) Where ( Fwc t )c (α) = 1̄− (1̄− Fw t (α)) ,∀(α) ∈ A = Fw t (α) ,∀α ∈ A. 4.2. Union Operation Definition 16. The weighted expert factor on time fuzzy soft expert sets (H,C)wt is the union of two weighted expert factor on time fuzzy soft expert sets (F,A)wt and (G,B)wt over U ; it is represented as (F,A)wt ∪̃ (G,B)wt , such that C = A ∪B ⊂ Z. It is defined as follows. Hw t (δ) =  Fw t (δ) , if δ ∈ A−B, Gw t (δ) , if δ ∈ B −A, Fw t (δ) ∪̃ Gw t (δ) , if δ ∈ A ∩B, where the fuzzy soft expert union was indicated by ∪̃. Example 5. Consider Example 1. Assume two The Effect of the weighted expert factor on time fuzzy soft expert setss over U , (F,A)wt and (G,B)wt , are such that A. Bataihah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5937 12 of 28 (F,A)wt = {(( e2, n 0.5 , 1 ) , { u1 t1 0.5 , u2 t1 0.7 , u3 t1 0.2 , u4 t1 0.5 }) , (( e2, r 0.7 , 1 ) , { u1 t1 0.2 , u2 t1 0.6 , u3 t1 0.7 , u4 t1 0.3 }) , (( e2, r 0.7 , 1 ) , { u1 t2 0.5 , u2 t2 0.4 , u3 t2 0.7 , u4 t2 0.4 }) , (( e3, n 0.5 , 1 ) , { u1 t2 0.8 , u2 t2 0.6 , u3 t2 0.1 , u4 t2 0.5 }) ,(( e3, m 0.8 , 1 ) , { u1 t3 0.7 , u2 t3 0.4 , u3 t3 0.1 , u4 t3 0.8 }) , (( e1, m 0.8 , 0 ) , { u1 t1 0.3 , u2 t1 0.6 , u3 t1 0.7 , u4 t1 0.8 }) ,(( e3, n 0.5 , 0 ) , { u1 t1 0.5 , u2 t1 0.4 , u3 t1 0.5 , u4 t1 0.2 }) , (( e3, r 0.7 , 0 ) , { u1 t1 0.8 , u2 t1 0.2 , u3 t1 0.8 , u4 t1 0.4 }) , (( e1, m 0.8 , 0 ) , { u1 t2 0.7 , u2 t2 0.4 , u3 t2 0.7 , u4 t2 0.5 }) , (( e3, m 0.8 , 0 ) , { u1 t3 0.4 , u2 t3 0.6 , u3 t3 0.8 , u4 t3 0.3 })} . (G,B)wt = {(( e2, n 0.5 , 1 ) , { u1 t1 0.7 , u2 t1 0.9 , u3 t1 0.1 , u4 t1 0.3 }) , (( e2, r 0.7 , 1 ) , { u1 t1 0.6 , u2 t1 0.4 , u3 t1 0.8 , u4 t1 0.1 }) , (( e2, r 0.7 , 1 ) , { u1 t2 0.7 , u2 t2 0.3 , u3 t2 0.2 , u4 t2 0.6 }) , (( e3, n 0.5 , 1 ) , { u1 t2 0.5 , u2 t2 0.8 , u3 t2 0.7 , u4 t2 0.3 }) ,(( e1, m 0.8 , 0 ) , { u1 t2 0.5 , u2 t2 0.6 , u3 t2 0.8 , u4 t2 0.4 }) , (( e1, r 0.7 , 0 ) , { u1 t2 0.7 , u2 t2 0.5 , u3 t2 0.6 , u4 t2 0.7 }) , (( e2, m 0.8 , 0 ) , { u1 t2 0.6 , u2 t2 0.7 , u3 t2 0.4 , u4 t2 0.9 }) , (( e3, m 0.8 , 0 ) , { u1 t3 0.2 , u2 t3 0.8 , u3 t3 0.9 , u4 t3 0.1 })} . Then (F,A)wt ∪̃ (G,B)wt = (H,C)wt where (H,C)wt = {(( e2, n 0.5 , 1 ) , { u1 t1 0.7 , u2 t1 0.9 , u3 t1 0.2 , u4 t1 0.5 }) , (( e2, r 0.7 , 1 ) , { u1 t1 0.6 , u2 t1 0.6 , u3 t1 0.8 , u4 t1 0.3 }) , (( e2, r 0.7 , 1 ) , { u1 t2 0.7 , u2 t2 0.4 , u3 t2 0.7 , u4 t2 0.6 }) , (( e3, n 0.5 , 1 ) , { u1 t2 0.8 , u2 t2 0.8 , u3 t2 0.7 , u4 t2 0.5 }) ,(( e3, m 0.8 , 1 ) , { u1 t3 0.7 , u2 t3 0.4 , u3 t3 0.1 , u4 t3 0.8 }) , (( e1, m 0.8 , 0 ) , { u1 t1 0.3 , u2 t1 0.6 , u3 t1 0.7 , u4 t1 0.8 }) ,(( e3, n 0.5 , 0 ) , { u1 t1 0.5 , u2 t1 0.4 , u3 t1 0.5 , u4 t1 0.2 }) , (( e3, r 0.7 , 0 ) , { u1 t1 0.8 , u2 t1 0.2 , u3 t1 0.8 , u4 t1 0.4 }) ,(( e1, m 0.8 , 0 ) , { u1 t2 0.7 , u2 t2 0.6 , u3 t2 0.8 , u4 t2 0.5 }) , (( e1, r 0.7 , 0 ) , { u1 t2 0.7 , u2 t2 0.5 , u3 t2 0.6 , u4 t2 0.7 }) , (( e2, m 0.8 , 0 ) , { u1 t2 0.6 , u2 t2 0.7 , u3 t2 0.4 , u4 t2 0.9 }) , (( e3, m 0.8 , 0 ) , { u1 t3 0.4 , u2 t3 0.8 , u3 t3 0.9 , u4 t3 0.3 })} . Proposition 2. Three weighted expert factor on time fuzzy soft expert sets over U are denoted by (F,A)wt , (G,B)wt , and (H,C)wt , then (i) (F,A)wt ∪̃ ((G,B)wt ∪̃ (H,C)wt ) = ((F,A)wt ∪̃ (G,B))wt ∪̃ (H,C)wt , A. Bataihah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5937 13 of 28 (ii) (F,A)wt ∪̃ (F,A)wt = (F,A)wt . Proof. 1. Our goal is proving that (F,A)wt ∪̃ ( (G,B)wt ∪̃ (H,C)wt ) = ( (F,A)wt ∪̃ (G,B)wt ) ∪̃ (H,C)wt . Definition 16 allows us (where s is s-norm) to have ( (G,B)wt ∪̃ (H,C)wt ) =  Gw t (δ) , if δ ∈ B − C Hw t (δ) , if δ ∈ C −B s (Gw t (δ) , Hw t (δ)) , if δ ∈ B ∩ C. We take into account the situation when δ ∈ B ∩ C. After the insignificant examples, we have (G,B)wt ∪̃ (H,C)wt = (s (Gw t (δ) , Hw t (δ)) , B ∪ C). Here, we additionally take the scenario when δ ∈ A into account. Since the other situations are insignificant, we have (F,A)wt ∪̃ ( (G,B)wt ∪̃ (H,C)wt ) = (s (Fw t (δ) , s (Gw t (δ) , Hw t (δ))) , A ∪ (B ∪ C)). = (s (s (Fw t (δ) , Gw t (δ)) ∪Hw t (δ)) , (A ∪B) ∪ C). = ( (F,A)wt ∪̃ (G,B)wt ) ∪̃ (H,C)wt . 2. The proof is straightforward. 4.3. Intersection Operation Definition 17. The weighted expert factor on time fuzzy soft expert sets (H,C)wt is the in- tersection of two W-TFSES’s (F,A)wt and (G,B)wt over U , represented as (F,A)wt ∩̃ (G,B)wt , with C = A ∩B ⊂ Z, and defined as follows Hw t (δ) =  Fw t (δ) , if δ ∈ A−B, Gw t (δ) , if δ ∈ B −A, Fw t (δ) ∩̃ Gw t (δ) , if δ ∈ A ∩B, where ∩̃ represented the intersection of the fuzzy soft expert. Example 6. Consider Example 5. We have (F,A)wt ∩̃ (G,B)wt = (H,C)wt where (H,C)wt = {(( e2, n 0.5 , 1 ) , { u1 t1 0.5 , u2 t1 0.7 , u3 t1 0.1 , u4 t1 0.3 }) , (( e2, r 0.7 , 1 ) , { u1 t1 0.2 , u2 t1 0.4 , u3 t1 0.7 , u4 t1 0.1 }) , (( e2, r 0.7 , 1 ) , { u1 t2 0.5 , u2 t2 0.3 , u3 t2 0.2 , u4 t2 0.4 }) , (( e3, n 0.5 , 1 ) , { u1 t2 0.5 , u2 t2 0.6 , u3 t2 0.1 , u4 t2 0.3 }) , A. Bataihah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5937 14 of 28(( e3, m 0.8 , 1 ) , { u1 t3 0.7 , u2 t3 0.4 , u3 t3 0.1 , u4 t3 0.8 }) , (( e1, m 0.8 , 0 ) , { u1 t1 0.3 , u2 t1 0.6 , u3 t1 0.7 , u4 t1 0.8 }) ,(( e3, n 0.5 , 0 ) , { u1 t1 0.5 , u2 t1 0.4 , u3 t1 0.5 , u4 t1 0.2 }) , (( e3, r 0.7 , 0 ) , { u1 t1 0.8 , u2 t1 0.2 , u3 t1 0.8 , u4 t1 0.4 }) ,(( e1, m 0.8 , 0 ) , { u1 t2 0.5 , u2 t2 0.4 , u3 t2 0.7 , u4 t2 0.4 }) , (( e1, r 0.7 , 0 ) , { u1 t2 0.7 , u2 t2 0.5 , u3 t2 0.6 , u4 t2 0.7 }) , (( e2, m 0.8 , 0 ) , { u1 t2 0.6 , u2 t2 0.7 , u3 t2 0.4 , u4 t2 0.9 }) , (( e3, m 0.8 , 0 ) , { u1 t3 0.2 , u2 t3 0.6 , u3 t3 0.8 , u4 t3 0.1 })} . Proposition 3. Three weighted expert factor on time fuzzy soft expert sets over U are denoted by (F,A)wt , (G,B)wt , and (H,C)wt ,then (i) (F,A)wt ∩̃ ((G,B)wt ∩̃ (H,C)wt ) = ((F,A)wt ∩̃ (G,B))wt ∩̃ (H,C)wt , (ii) (F,A)wt ∩̃ (F,A)wt = (F,A)wt . Proof. (i) Our goal is proving that (F,A)wt ∩̃ ( (G,B)wt ∩̃ (H,C)wt ) = ( (F,A)wt ∩̃ (G,B)wt ) ∩̃ (H,C)wt . By employing Definition 17 (where t̃ is t̃-norm) we have ( (G,B)wt ∩̃ (H,C)wt ) =  Gw t (δ) , if δ ∈ B − C Hw t (δ) , if δ ∈ C −B t̃ (Gw t (δ) , Hw t (δ)) , if δ ∈ B ∩ C. We take into account the situation when δ ∈ B ∩ C. After the insignificant examples, we have (G,B)wt ∩̃ (H,C)wt = ( t̃ (Gw t (δ) , Hw t (δ)) , B ∪ C ) . We also consider here the case when δ ∈ A. The other cases are trivial, then we have (F,A)wt ∩̃ ( (G,B)wt ∩̃ (H,C)wt ) = (̃ t ( Fw t (δ) , t̃ (Gw t (δ) , Hw t (δ)) ) , A ∪ (B ∪ C) ) . = ( t̃ (̃ t (Fw t (δ) , Gw t (δ)) ∩Hw t (δ) ) , (A ∪B) ∪ C ) . = ( (F,A)wt ∩̃ (G,B)wt ) ∩̃ (H,C)wt . (ii) The proof is straightforward. Proposition 4. Assume that there are three weighted expert factor on time fuzzy soft expert sets over U : (F,A)wt , (G,B)wt , and (H,C)wt , then (i) (F,A)wt ∪̃ ( (G,B)wt ∩̃ (H,C)wt ) = ( (F,A)wt ∪̃ (G,B)wt ) ∩̃ ( (F,A)wt ∪̃ (H,C)wt ) , A. Bataihah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5937 15 of 28 (ii) (F,A)wt ∩̃ ( (G,B)wt ∪̃ (H,C)wt ) = ( (F,A)wt ∩̃ (G,B)wt ) ∪̃ ( (F,A)wt ∩̃ (H,C)wt ) . Proof. (i) Our goal is to prove that (F,A)wt ∪̃ ( (G,B)wt ∩̃ (H,C) )w t = ( (F,A)wt ∪̃ (G,B)wt ) ∩̃ ( (F,A)wt ∪̃ (H,C)wt ) , By Applying Definitions 16 and 17 we have ( (G,B)wt ∩̃ (H,C)wt ) =  Gw t (δ) , if δ ∈ B − C Hw t (δ) , if δ ∈ C −B t̃ (Gw t (δ) , Hw t (δ)) , if δ ∈ B ∩ C. Since the other situations are trivial, we investigate the case when δ ∈ B ∩ C. In this case, we obtain (G,B)wt ∩̃ (H,C)wt = (̃ t (Gw t (δ) , Hw t (δ)) , B ∪ C ) . We also consider here the case when δ ∈ A. The other cases are trivial, then we have (F,A)wt ∪̃ ( (G,B)wt ∩̃ (H,C)wt ) = ( s ( Fw t (δ) , t̃ (Gw t (δ) , Hw t (δ)) ) , A ∪ (B ∪ C) ) . = t̃ (s (Fw t (δ) , Gw t (δ)) , s (Fw t (δ) ∪Hw t (δ))). = ( (F,A)wt ∪̃ (G,B)wt ) ∩̃ ( (F,A)wt ∪̃ (H,C)wt ) . (ii) Our goal is proving that (F,A)wt ∩̃ ( (G,B)wt ∪̃ (H,C) )w t = ( (F,A)wt ∩̃ (G,B)wt ) ∪̃ ( (F,A)wt ∩̃ (H,C)wt ) , By applying definitions 16 and 17 we have ( (G,B)wt ∪̃ (H,C)wt ) =  Gw t (δ) , if δ ∈ B − C Hw t (δ) , if δ ∈ C −B s (Gw t (δ) , Hw t (δ)) , if δ ∈ B ∩ C. We consider the case when δ ∈ B ∩ C. The other cases are trivial, then we have (G,B)wt ∪̃ (H,C)wt = (s (Gw t (δ) , Hw t (δ)) , B ∪ C). Here, we additionally take the situation when δ ∈ A into account. Since the other situations are trivial, we have (F,A)wt ∩̃ ( (G,B)wt ∪̃ (H,C)wt ) = (̃ t (Fw t (δ) , s (Gw t (δ) , Hw t (δ))) , A ∪ (B ∪ C) ) . = s (̃ t (Fw t (δ) , Gw t (δ)) , t̃ (Fw t (δ) ∪Hw t (δ)) ) . = ( (F,A)wt ∩̃ (G,B)wt ) ∪̃ ( (F,A)wt ∩̃ (H,C)wt ) . A. Bataihah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5937 16 of 28 5. AND and OR Operations We define the AND and OR operations for weighted expert factor on time fuzzy soft expert sets , as well as their characteristics and several instances, in this section. Definition 18. With two weighted expert factor on time fuzzy soft expert sets over U , (F,A)wt and (G,B)wt , the expression (F,A)wt AND (G,B)wt is denoted by (F,A)wt ∧(G,B)wt , it is defined with (F,A)wt ∧ (G,B)wt = (H,A×B)wt such that H (α, β)wt = F (α)wt ⋂̃ G (β)wt ,∀ (α, β) ∈ A×B, where ⋂̃ is time-fuzzy soft expert intersection. Example 7. Consider Example 1. Assuming two The Effect of the weighted expert factor on time fuzzy soft expert setss over U , let (F,A)wt and (G,B)wt be such that (F,A)wt = {(( e2, n 0.5 , 1 ) , { u1 t1 0.5 , u2 t1 0.7 , u3 t1 0.2 , u4 t1 0.5 }) , (( e1, m 0.8 , 1 ) , { u1 t2 0.4 , u2 t2 0.8 , u3 t2 0.2 , u4 t2 0.4 }) , (( e3, r 0.7 , 1 ) , { u1 t3 0.4 , u2 t3 0.7 , u3 t3 0.5 , u4 t3 0.6 }) , (( e1, m 0.8 , 0 ) , { u1 t1 0.3 , u2 t1 0.6 , u3 t1 0.7 , u4 t1 0.8 })} . (G,B)wt ={(( e3, n 0.5 , 1 ) , { u1 t2 0.4 , u2 t2 0.9 , u3 t2 0.3 , u4 t2 0.1 }) , (( e2, r 0.7 , 0 ) , { u1 t3 0.5 , u2 t3 0.8 , u3 t3 0.2 , u4 t3 0.6 })} . Then (F,A) ∧ (G,B) = (H,A×B) where (H,A×B)wt = {((( e2, n 0.5 , 1 ) t1 , ( e3, n 0.5 , 1 ) t2 ) , { u1 t1,2 0.4 , u2 t1,2 0.7 , u3 t1,2 0.2 , u4 t1,2 0.1 }) , ((( e2, n 0.5 , 1 ) t1 , ( e2, r 0.7 , 0 ) t3 ) , { u1 t1,3 0.5 , u2 t1,3 0.7 , u3 t1,3 0.2 , u4 t1,3 0.5 }) ,((( e1, m 0.8 , 1 ) t2 , ( e3, n 0.5 , 1 ) t2 ) , { u1 t2,2 0.4 , u2 t2,2 0.8 , u3 t2,2 0.2 , u4 t2,2 0.1 }) ,((( e1, m 0.8 , 1 ) t2 , ( e2, r 0.7 , 0 ) t3 ) , { u1 t2,3 0.4 , u2 t2,3 0.8 , u3 t2,3 0.2 , u4 t2,3 0.4 }) ,((( e3, r 0.7 , 1 ) t3 , ( e3, n 0.5 , 1 ) t2 ) , { u1 t3,2 0.4 , u2 t3,2 0.7 , u3 t3,2 0.3 , u4 t3,2 0.1 }) ,((( e3, r 0.7 , 1 ) t3 , ( e2, r 0.7 , 0 ) t3 ) , { u1 t3,3 0.4 , u2 t3,3 0.7 , u3 t3,3 0.2 , u4 t3,3 0.6 }) ,((( e1, m 0.8 , 0 ) t1 , ( e3, n 0.5 , 1 ) t2 ) , { u1 t1,2 0.3 , u2 t1,2 0.6 , u3 t1,2 0.3 , u4 t1,2 0.1 }) , A. Bataihah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5937 17 of 28 ((( e1, m 0.8 , 0 ) t1 , ( e2, r 0.7 , 0 ) t3 ) , { u1 t1,3 0.3 , u2 t1,3 0.6 , u3 t1,3 0.2 , u4 t1,3 0.6 })} . Assuming that there are two weighted expert factor on time fuzzy soft expert sets over U , (F,A)wt and (G,B)wt , then (F,A)wt OR (G,B)wt is represented as (F,A)wt ∨ (G,B)wt . Definition 19. Assuming that there are two weighted expert factor on time fuzzy soft expert sets over U , (F,A)wt and (G,B)wt , then (F,A)wt OR (G,B)wt is represented as (F,A)wt ∨ (G,B)wt , is defined by (F,A)wt ∨ (G,B)wt = (H,A×B)wt such that H (α, β)wt = F (α)wt ⋃̃ G (β)wt ,∀ (α, β) ∈ A×B, ⋃̃ represents the union of time- fuzzy soft expert. Example 8. Consider Example 7 we have U then (F,A)wt OR (G,B)wt represented by (H,C)wt = (F,A)wt ∨ (G,B)wt where (H,C)wt = {((( e2, n 0.5 , 1 ) t1 , ( e3, n 0.5 , 1 ) t2 ) , { u1 t1,2 0.5 , u2 t1,2 0.9 , u3 t1,2 0.3 , u4 t1,2 0.5 }) , ((( e2, n 0.5 , 1 ) t1 , ( e2, r 0.7 , 0 ) t3 ) , { u1 t1,3 0.5 , u2 t1,3 0.8 , u3 t1,3 0.2 , u4 t1,3 0.6 }) ,((( e1, m 0.8 , 1 ) t2 , ( e3, n 0.5 , 1 ) t2 ) , { u1 t2,2 0.4 , u2 t2,2 0.9 , u3 t2,2 0.3 , u4 t2,2 0.4 }) ,((( e1, m 0.8 , 1 ) t2 , ( e2, r 0.7 , 0 ) t3 ) , { u1 t2,3 0.5 , u2 t2,3 0.8 , u3 t2,3 0.2 , u4 t2,3 0.6 }) ,((( e3, r 0.7 , 1 ) t3 , ( e3, n 0.5 , 1 ) t2 ) , { u1 t3,2 0.4 , u2 t3,2 0.9 , u3 t3,2 0.5 , u4 t3,2 0.6 }) ,((( e3, r 0.7 , 1 ) t3 , ( e2, r 0.7 , 0 ) t3 ) , { u1 t3,3 0.5 , u2 t3,3 0.8 , u3 t3,3 0.5 , u4 t3,3 0.6 }) ,((( e1, m 0.8 , 0 ) t1 , ( e3, n 0.5 , 1 ) t2 ) , { u1 t1,2 0.4 , u2 t1,2 0.9 , u3 t1,2 0.7 , u4 t1,2 0.8 }) , ((( e1, m 0.8 , 0 ) t1 , ( e2, r 0.7 , 0 ) t3 ) , { u1 t1,3 0.5 , u2 t1,3 0.8 , u3 t1,3 0.7 , u4 t1,3 0.8 })} . Proposition 5. let (F,A)wt and (G,B)wt are W-TFSESS over U, then (i) ((F,A)wt ∧ (G,B)wt ) c = (F,A)w c t ∨ (G,B)w c t (ii) ((F,A)wt ∨ (G,B)wt ) c = (F,A)w c t ∧ (G,B)w c t Proof. A. Bataihah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5937 18 of 28 (i) Suppose that (F,A)wt ∧ (G,B)wt = (O,A×B)wt . Therefore, ((F,A)wt ∧ (G,B)wt ) c = (Ot, A×B)c = (Oc t , (A×B)) . Now, ((F,A)wt ∨ (G,B)wt ) c = ( (Fwc t , A) ∨ (Gwc t , B) ) = (Jt, (A×B)), where Jt(x, y) = t (c(Fw t (α)), c(Gw t (β))) . Now, take (α, β) ∈ (A×B). Then, Oc t (α, β) = 1̄−Ot(α, β), = 1̄− [Fw t (α) ⋃ Gw t (β)] = [1̄− Fw t (α)] ⋂ [1̄−Gw t (β)] =t (c(Fw t (α)), c(Gw t (β))) = Jt(α, β) Therefore Oc t and Jt are the same. Hence, proved. (ii) Suppose that (F,A)wt ∨ (G,B)wt = (O,A×B)wt . Therefore, ((F,A)wt ∨ (G,B)wt ) c = (Ot, A×B)c = (Oc t , (A×B)) . Now, ((F,A)wt ∧ (G,B)wt ) c = ( (Fwc t , A) ∧ (Gwc t , B) ) = (Jt, (A×B)), where Jt(x, y) = s (c(Fw t (α)), c(Gw t (β))) . Now, take (α, β) ∈ (A×B). Then, Oc t (α, β) = 1̄−Ot(α, β), = 1̄− [Fw t (α) ⋂ Gw t (β)] = [1̄− Fw t (α)] ⋃ [1̄−Gw t (β)] =s (c(Fw t (α)), c(Gw t (β))) = Jt(α, β) Therefore Oc t and Jt are the same. Hence, proved. Proposition 6. let (F,A), (G,B) and (H,C) be time-fuzzy soft expert sets over U, then (i) (F,A)wt ∧ ((G,B)wt ∧ (H,C)wt ) =((F,A)wt ∧ (G,B)wt ) ∧ (H,C)wt , (ii) (F,A)wt ∨ ((G,B)wt ∨ (H,C)wt ) = ((F,A)wt ∨ (G,B)wt ) ∨ (H,C)wt , (iii) (F,A)wt ∨ ((G,B)wt ∧ (H,C)wt ) = ((F,A)wt ∨ (G,B)wt ) ∧ ((F,A)wt ∨ (H,C)wt ) , (iv) (F,A)wt ∧ ((G,B)wt ∨ (H,C)wt ) = ((F,A)wt ∧ (G,B)wt ) ∨ ((F,A)wt ∧ (H,C)wt ) . Proof. We give the proofs of 1 and 2. (i) Suppose that (G,B)wt ∧ (H,C)wt = t̃ (Gw t (α) , Hw t (β)), ∀ (α, β) ∈ B × C. Then (F,A)wt ∧ ((G,B)wt ∧ (H,C)wt ) = t̃ ( Fw t (γ) , t̃ (Gw t (α) , Hw t (β)) ) , ∀ (γ, (α, β)) ∈ A× (B × C). = t̃ (̃ t (Fw t (γ) , Gw t (α)) , Hw t (β) ) , ∀((γ, α) , β) ∈ (A×B)× C. A. Bataihah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5937 19 of 28 = ((F,A)wt ∧ (G,B)wt ) ∧ (H,C)wt . (ii) Suppose that (G,B)wt ∨ (H,C)wt = s (Gw t (α) , Hw t (β)), ∀ (α, β) ∈ B × C. Then (F,A)wt ∨ ((G,B)wt ∧ (H,C)wt ) = s (Fw t (γ) , s (Gw t (α) , Hw t (β))), ∀ (γ, (α, β)) ∈ A× (B × C). = s (s (Fw t (γ) , Gw t (α)) , Hw t (β)), ∀((γ, α) , β) ∈ (A×B)× C. = ((F,A)wt ∨ (G,B)wt ) ∨ (H,C)wt . 6. An application of time fuzzy soft expert set with two choices in decision making This section presents a theoretical application of the W-TFSES to a decision-making issue, indicating the viability and generalizability of this approach to problems in many domains with uncertainty. two choices {agree, disagree} make up the application. The government performance evaluation follows this computational pipeline: (i) Normalization: Raw expert ratings rtij are scaled to [0, 1] via: utij = rtij −min(ri) max(ri)−min(ri) (ii) Time-Weighted Averaging: For parameter ek across T periods: F (ek) = 1 |T | |T |∑ t=1 Ft(ek) (iii) Weighted Aggregation: Final scores combine expert weights: cj = n∑ i=1 wi ·  1 |E| |E|∑ k=1 uijk  where wi are normalized expert weights ( ∑ wi = 1), and uijk represents normalized evaluation of expert i for parameter ek on alternative uj . Example 9. Measuring the effectiveness of governments only through opinion surveys is insufficient. Instead, performance metrics that encompass all of the activities and duties listed below should be used to gauge government success. Suppose that the company of strategic studies considers a collection of experts to carry out an analysis using particular A. Bataihah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5937 20 of 28 metrics to assess the performance of the governments of four nations throughout the course of four prior time periods; these metrics are listed below. Consider a collection of govern- ments. U = {u1, u2, u3, u4} , Five parameters could be present. Assume that the decision parameters E = {e1, e2, e3} are used to evaluate the performance of governments. For i = 1, 2, 3, the parameters ei (i = 1, 2, 3) stand for the achievement of the anti-corruption campaign, efficient use of natural resources in management, investments, and savings. And let T = {t1, t2, t3, t4}. X = {m,n} be the members of a committee, Suppose that the company of strategic has imposed the following weights for the experts. For the expert ”m, ”w1 = 0.8; for the expert ”n, ”w2 = 0.5; for the expert ”r, ”w3 = 0.2. developed to deal with the decision-making problems based on weighted time fuzzy soft expert sets. In the revised algorithm, we take the weights of the expert into consideration and compute the weighted choice values cj and kj instead of choice values of cj and kj. The following algorithm may be followed by the company of strategic to fill the best choice. (F,Z)wt = {(( e1, m 0.8 , 1 ) , { u1 t1 0.6 , u2 t1 0.3 , u3 t1 0.5 , u4 t1 0.6 }) , (( e1, n 0.5 , 1 ) , { u1 t1 0.7 , u2 t1 0.5 , u3 t1 0.4 , u4 t1 0.7 }) ,(( e2, m 0.8 , 1 ) , { u1 t1 0.7 , u2 t1 0.4 , u3 t1 0.2 , u4 t1 0.7 }) , (( e2, n 0.5 , 1 ) , { u1 t1 0.5 , u2 t1 0.6 , u3 t1 0.3 , u4 t1 0.5 }) ,(( e3, m 0.8 , 1 ) , { u1 t1 0.4 , u2 t1 0.6 , u3 t1 0.7 , u4 t1 0.3 }) , (( e3, n 0.5 , 1 ) , { u1 t1 0.6 , u2 t1 0.5 , u3 t1 0.9 , u4 t1 0.5 }) ,(( e1, m 0.8 , 1 ) , { u1 t2 0.4 , u2 t2 0.8 , u3 t2 0.2 , u4 t2 0.4 }) , (( e1, n 0.5 , 1 ) , { u1 t2 0.6 , u2 t2 0.9 , u3 t2 0.4 , u4 t2 0.8 }) ,(( e2, m 0.8 , 1 ) , { u1 t2 0.4 , u2 t2 0.9 , u3 t2 0.7 , u4 t2 0.5 }) , (( e2, n 0.5 , 1 ) , { u1 t2 0.6 , u2 t2 0.8 , u3 t2 0.5 , u4 t2 0.6 }) ,(( e3, m 0.8 , 1 ) , { u1 t2 0.6 , u2 t2 0.4 , u3 t2 0.1 , u4 t2 0.8 }) , (( e3, n 0.5 , 1 ) , { u1 t2 0.8 , u2 t2 0.6 , u3 t2 0.3 , u4 t2 0.7 }) ,(( e1, m 0.8 , 1 ) , { u1 t3 0.2 , u2 t3 0.7 , u3 t3 0.8 , u4 t3 0.3 }) , (( e1, n 0.5 , 1 ) , { u1 t3 0.4 , u2 t3 0.9 , u3 t3 0.6 , u4 t3 0.5 }) ,(( e2, m 0.8 , 1 ) , { u1 t3 0.5 , u2 t3 0.4 , u3 t3 0.7 , u4 t3 0.9 }) , (( e2, n 0.5 , 1 ) , { u1 t3 0.8 , u2 t3 0.6 , u3 t3 0.2 , u4 t3 0.7 }) ,(( e3, m 0.8 , 1 ) , { u1 t3 0.7 , u2 t3 0.6 , u3 t3 0.3 , u4 t3 0.7 }) , (( e3, n 0.5 , 1 ) , { u1 t3 0.6 , u2 t3 0.5 , u3 t3 0.3 , u4 t3 0.9 }) ,(( e1, m 0.8 , 1 ) , { u1 t4 0.2 , u2 t4 0.9 , u3 t4 0.6 , u4 t4 0.7 }) , (( e1, n 0.5 , 1 ) , { u1 t4 0.7 , u2 t4 0.6 , u3 t4 0.8 , u4 t4 0.4 }) ,(( e2, m 0.8 , 1 ) , { u1 t4 0.8 , u2 t4 0.6 , u3 t4 0.2 , u4 t4 0.7 }) , (( e2, n 0.5 , 1 ) , { u1 t4 0.7 , u2 t4 0.4 , u3 t4 0.5 , u4 t4 0.8 }) ,(( e3, m 0.8 , 1 ) , { u1 t4 0.7 , u2 t4 0.5 , u3 t4 0.3 , u4 t4 0.9 }) , (( e3, n 0.5 , 1 ) , { u1 t4 0.6 , u2 t4 0.8 , u3 t4 0.4 , u4 t4 0.5 }) ,(( e1, m 0.8 , 0 ) , { u1 t1 0.5 , u2 t1 0.6 , u3 t1 0.4 , u4 t1 0.3 }) , (( e1, n 0.5 , 0 ) , { u1 t1 0.4 , u2 t1 0.5 , u3 t1 0.7 , u4 t1 0.2 }) ,(( e2, m 0.8 , 0 ) , { u1 t1 0.4 , u2 t1 0.6 , u3 t1 0.7 , u4 t1 0.4 }) , (( e2, n 0.5 , 0 ) , { u1 t1 0.4 , u2 t1 0.3 , u3 t1 0.6 , u4 t1 0.6 }) ,(( e3, m 0.8 , 0 ) , { u1 t1 0.7 , u2 t1 0.5 , u3 t1 0.3 , u4 t1 0.4 }) , (( e3, n 0.5 , 0 ) , { u1 t1 0.4 , u2 t1 0.6 , u3 t1 0.2 , u4 t1 0.6 }) , A. Bataihah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5937 21 of 28(( e1, m 0.8 , 0 ) , { u1 t2 0.7 , u2 t2 0.4 , u3 t2 0.7 , u4 t2 0.5 }) , (( e1, n 0.5 , 0 ) , { u1 t2 0.5 , u2 t2 0.2 , u3 t2 0.5 , u4 t2 0.4 }) ,(( e2, m 0.8 , 0 ) , { u1 t2 0.2 , u2 t2 0.3 , u3 t2 0.4 , u4 t2 0.6 }) , (( e2, n 0.5 , 0 ) , { u1 t2 0.5 , u2 t2 0.2 , u3 t2 0.7 , u4 t2 0.3 }) ,(( e3, m 0.8 , 0 ) , { u1 t2 0.3 , u2 t2 0.5 , u3 t2 0.8 , u4 t2 0.3 }) , (( e3, n 0.5 , 0 ) , { u1 t2 0.5 , u2 t2 0.4 , u3 t2 0.7 , u4 t2 0.2 }) ,(( e1, m 0.8 , 0 ) , { u1 t3 0.7 , u2 t3 0.2 , u3 t3 0.1 , u4 t3 0.6 }) , (( e1, n 0.5 , 0 ) , { u1 t3 0.3 , u2 t3 0.4 , u3 t3 0.6 , u4 t3 0.4 }) ,(( e2, m 0.8 , 0 ) , { u1 t3 0.3 , u2 t3 0.7 , u3 t3 0.4 , u4 t3 0.1 }) , (( e2, n 0.5 , 0 ) , { u1 t3 0.1 , u2 t3 0.5 , u3 t3 0.6 , u4 t3 0.5 }) ,(( e3, m 0.8 , 0 ) , { u1 t3 0.4 , u2 t3 0.6 , u3 t3 0.8 , u4 t3 0.3 }) , (( e3, n 0.5 , 0 ) , { u1 t3 0.3 , u2 t3 0.4 , u3 t3 0.7 , u4 t3 0.2 }) ,(( e1, m 0.8 , 0 ) , { u1 t4 0.6 , u2 t4 0.1 , u3 t4 0.3 , u4 t4 0.4 }) , (( e1, n 0.5 , 0 ) , { u1 t4 0.4 , u2 t4 0.4 , u3 t4 0.3 , u4 t4 0.7 }) ,(( e2, m 0.8 , 0 ) , { u1 t4 0.3 , u2 t4 0.3 , u3 t4 0.7 , u4 t4 0.2 }) , (( e2, n 0.5 , 0 ) , { u1 t4 0.4 , u2 t4 0.6 , u3 t4 0.6 , u4 t4 0.3 }) , (( e3, m 0.8 , 0 ) , { u1 t4 0.6 , u2 t4 0.4 , u3 t4 0.7 , u4 t4 0.2 }) , (( e3, n 0.5 , 0 ) , { u1 t4 0.5 , u2 t4 0.3 , u3 t4 0.7 , u4 t4 0.5 })} . The committee may use the following techniques for evaluating their selection and determine which option is best for the decision. (i) Input the W-TFSES (F,Z). (ii) Find an agree W-TFSES and a Disagree W-TFSES, Tables (1, 2). (iii) Construct F (Z)’s tabular representation using the tables in Tables (3, 4), where F (Z) outlined as follows: F (z) =  u n∑ i=1 tiFti(z)⧸n n∑ i=1 Fti(z) : u ∈ U, z ∈ Z  (1) where n = |T |. (iv) Findcj = ∑ i uij for agree W-TFSES for each expert (m,n) separately of each other’s experts, Tables (5, 6) respectively. (v) Find kj = ∑ i uijfor disagree W-TFSES for each expert (m,n) separately of each other’s experts, Tables (5, 6) respectively. (vi) Find sj = cj − kj , for each expert (m,n) separately of each other’s experts, Tables (5, 6) respectively. A. Bataihah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5937 22 of 28 (vii) Construct the grade table of W-TFSES , Table 7. (viii) Calculate the final score values for ui. (ix) Find v, for which sv = max sj . Then sv is the optimal choice object. If v has more than one value, then any one of them could be chosen by the company using its option. The agree weighted expert factor on time fuzzy soft expert sets and Disagree weighted expert factor on time fuzzy soft expert sets are shown in Tables 1 and 2, respectively. Table 1: Agree weighted expert factor on time fuzzy soft expert sets U u1 u2 u3 u4( e1, m o.8 ) t1 0.6 0.3 0.5 0.6( e1, m o.8 ) t2 0.4 0.8 0.2 0.4( e1, m o.8 ) t3 0.2 0.7 0.8 0.3( e1, m o.8 ) t4 0.2 0.9 0.6 0.7( e1, n o.5 ) t1 0.7 0.5 0.4 0.7( e1, n o.5 ) t2 0.6 0.9 0.4 0.8( e1, n o.5 ) t3 0.4 0.9 0.6 0.5( e1, n o.5 ) t4 0.7 0.6 0.8 0.4( e2, m o.8 ) t1 0.7 0.4 0.2 0.7( e2, m o.8 ) t2 0.4 0.9 0.7 0.5( e2, m o.8 ) t3 0.5 0.4 0.7 0.9( e2, m o.8 ) t4 0.8 0.6 0.2 0.7( e2, n o.5 ) t1 0.5 0.6 0.3 0.5( e2, n o.5 ) t2 0.6 0.8 0.5 0.6( e2, n o.5 ) t3 0.8 0.6 0.2 0.7( e2, n o.5 ) t4 0.7 0.4 0.5 0.8( e3, m o.8 ) t1 0.4 0.6 0.7 0.3( e3, m o.8 ) t2 0.6 0.4 0.1 0.8( e3, m o.8 ) t3 0.7 0.6 0.3 0.7( e3, m o.8 ) t4 0.7 0.5 0.3 0.9( e3, n o.5 ) t1 0.6 0.5 0.9 0.5( e3, n o.5 ) t2 0.8 0.6 0.3 0.7( e3, n o.5 ) t3 0.6 0.5 0.3 0.9( e3, n o.5 ) t4 0.6 0.8 0.4 0.8 A. Bataihah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5937 23 of 28 Table 2: Disagree weighted expert factor on time fuzzy soft expert sets U u1 u2 u3 u4( e1, m o.8 ) t1 0.5 0.6 0.4 0.3( e1, m o.8 ) t2 0.7 0.4 0.7 0.5( e1, m o.8 ) t3 0.7 0.2 0.1 0.6( e1, m o.8 ) t4 0.6 0.1 0.3 0.4( e1, n o.5 ) t1 0.4 0.5 0.7 0.2( e1, n o.5 ) t2 0.5 0.2 0.5 0.4( e1, n o.5 ) t3 0.3 0.4 0.6 0.5( e1, n o.5 ) t4 0.4 0.4 0.3 0.7( e2, m o.8 ) t1 0.4 0.6 0.7 0.4( e2, m o.8 ) t2 0.2 0.3 0.4 0.6( e2, m o.8 ) t3 0.3 0.7 0.4 0.1( e2, m o.8 ) t4 0.3 0.3 0.7 0.2( e2, n o.5 ) t1 0.4 0.3 0.6 0.6( e2, n o.5 ) t2 0.5 0.2 0.7 0.3( e2, n o.5 ) t3 0.1 0.5 0.6 0.5( e2, n o.5 ) t4 0.4 0.6 0.6 0.3( e3, m o.8 ) t1 0.7 0.5 0.3 0.4( e3, m o.8 ) t2 0.3 0.5 0.8 0.3( e3, m o.8 ) t3 0.4 0.6 0.8 0.3( e3, m o.8 ) t4 0.6 0.4 0.7 0.2( e3, n o.5 ) t1 0.4 0.6 0.2 0.6( e3, n o.5 ) t2 0.5 0.4 0.7 0.2( e3, n o.5 ) t3 0.3 0.4 0.7 0.2( e3, n o.5 ) t4 0.5 0.3 0.7 0.5 Next, using the relation (1) we calculate F (Z) to convert the Agree W-TFSES to agree fuzzy soft expert set,to illustrate this step we calculate F (e1) for u1 as shown below. F (e1) =  u1 4∑ i=1 tiFti (e)⧸4 4∑ i=1 Fti (e) : u ∈ U, e ∈ E  = { u1 ((1∗0.5)+(2∗0.7)+(3∗0.7)+(4∗0.6))⧸4(0.5+0.7+0.7+0.6) } = { u1 6.4⧸10 } A. Bataihah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5937 24 of 28 = u1 0.64 . then we compute cj , where cj represents the sum of columns of an object ui. The results are shown in Table 3. Table 3: Converting Agree weighted expert factor on time fuzzy soft expert sets to agree- Fuzzy Soft Expert Sets U u1 u2 u3 u4( e1, m 0.8 ) 0.50 0.70 0.67 0.63( e2, m 0.8 ) 0.64 0.63 0.62 0.64( e3, m 0.8 ) 0.67 0.61 0.53 0.70( e1, n 0.5 ) 0.61 0.63 0.70 0.56( e2, n 0.5 ) 0.66 0.58 0.65 0.67( e3, n 0.5 ) 0.80 0.66 0.52 0.67 Likewise, using the relation (1), we calculate F (Z) to convert the dis Agree W-TFSES to disagree fuzzy soft expert set, as we explained in converting the Agree W-TFSES to agree fuzzy soft expert set, then we compute kj , where kj represents the column sum of an object ui. The results are shown in Table 4. Table 4: Converting Disagree weighted expert factor on time fuzzy soft expert sets to Disagree-FSES U u1 u2 u3 u4( e1, m 0.8 ) 0.58 0.46 0.55 0.65( e2, m 0.8 ) 0.60 0.59 0.62 0.51( e3, m 0.8 ) 0.61 0.61 0.68 0.56( e1, n 0.5 ) 0.60 0.61 0.55 0.73( e2, n 0.5 ) 0.58 0.75 0.62 0.57( e3, n 0.5 ) 0.63 0.55 0.70 0.60 Consider the previous results in Table 1 and Table 2 we have. Table 5: Score values of weighted expert factor on time fuzzy soft expert sets for expert m cjm = ∑ i uij kjm = ∑ i uij sjm = cjm − kjm c1m = 1.81 k1m = 1.79 s1m = 0.02 c2m = 1.94 k2m = 1.66 s2m = 0.28 Continued on next page A. Bataihah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5937 25 of 28 Table 5 – continued from previous page c3m = 1.82 k3m = 1.85 s3m = −0.03 c4m = 1.97 k4m = 1.72 s4m = 0.25 In Table 5 we calculate the choice values for expert m in agree W-TFSES by computing cjm for ui,∀i, where cjm represents the column sum of an object ui. Also, we calculate the choice values for expert m in disagree W-TFSES by computing kjm for ui, ∀i, where kjm represents the column sum of an object ui. After that we calculate the score value by subtracting the choice values in agree W-TFSES from the choice values in disagree W-TFSES. Then we determine the best choices by marking the highest numerical grade (underlined) as we show in Table (5). To illustrate this step, we calculate the score value for u1 as shown below. c1m = (0.50 + 0.64 + 0.67) = 1.81 k1m = ((0.2 ∗ 0.1) + (0.3 ∗ 0.3) + (0.2 ∗ 0.4) + (0.3 ∗ 0.5) + (0.4 ∗ 0.7) + (0.1 ∗ 0.9)) = 0.71 Then the score value of u1: Su1m = 1.46− 0.71 = 0.75, Table 6: Score values of weighted expert factor on time fuzzy soft expert sets for expert n cjn = ∑ i uij kjn = ∑ i uij sjn = cjn − kjn c1n = 2.07 k1n = 1.81 s1n = 0.26 c2n = 1.87 k2n = 1.91 s2n = −0.1 c3n = 1.87 k3n = 1.89 s3n = −0.02 c4n = 1.90 k4n = 1.90 s4n = 0.0 Next in Table (7) we summarize the highest numerical grade for all experts with their weights for the experts. Table 7: Grade table of weighted expert factor on time fuzzy soft expert sets X ui highest numerical grade expert weight m u2 0.28 0.8 n u1 0.26 0.5 Next we calculate the final score for experts by taking the output of products of these numerical grades with the corresponding values of weights for the experts. Sjm (u1) = (0.28 ∗ 0.8) = 0.224, Sjn (u4) = (0.26 ∗ 0.5) = 0.13, A. Bataihah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5937 26 of 28 Then max sj = sjm, so the committee will choose the first choice depending on the above- mentioned algorithm. 6. 7. Conclusion This paper has introduced and formalized the Weighted Time Fuzzy Soft Expert Set (W-TFSES) framework, representing a significant advancement beyond conventional fuzzy soft sets. Our work makes three fundamental contributions to the field: first, by devel- oping a novel mathematical structure that simultaneously incorporates expert weighting coefficients and temporal parameters, we have addressed critical limitations in existing decision-making models; second, we have established a complete set of operations includ- ing complement, union, intersection, AND and OR operations, along with their essential algebraic properties; third, through our comprehensive case study on government perfor- mance evaluation, we have demonstrated the framework’s practical utility in real-world scenarios requiring nuanced expert judgments over time. The W-TFSES framework offers several distinct advantages that enhance decision- making processes. By introducing adjustable expert weights, our model captures the inherent variability in expert reliability that characterizes most real-world decision sce- narios. The incorporation of temporal dimensions enables dynamic analysis of evolving situations, a crucial capability absent in traditional fuzzy soft sets. Furthermore, the framework maintains rigorous mathematical foundations while remaining sufficiently flex- ible for application across diverse domains, from public policy analysis to medical diag- nostic systems. However, we acknowledge certain limitations that warrant consideration. The in- creased dimensionality introduced by weight and time parameters leads to greater compu- tational complexity, particularly when processing large datasets. Additionally, the initial assignment of expert weights, while flexible, remains subjective and may benefit from more systematic determination methods. These limitations, however, represent opportunities for future research rather than fundamental constraints. Beyond its immediate applications, the W-TFSES framework has the potential to impact various domains where uncertainty and expert-based decision-making are criti- cal. Future research could explore its integration with artificial intelligence for automated decision support or its application in financial risk assessment, where time-dependent ex- pert evaluations play a crucial role. Additionally, comparative studies against alternative decision-making models, such as intuitionistic fuzzy sets or rough sets, could further vali- date W-TFSES’s effectiveness in capturing temporal and expert-driven uncertainties. Looking ahead, several promising research directions emerge from this work. Devel- oping optimized algorithms for large-scale W-TFSES implementations could significantly enhance computational efficiency. The integration of machine learning techniques with our framework presents exciting possibilities for automated weight calibration and pat- tern recognition in temporal data. 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