1.1 Uncertainty

Classical mathematics is one of the most important requirements in real-life problems, some of which involve imprecise data. Thus, solving these problems involves using mathematical principles based on uncertainty (not crisp). Therefore, many scientists and engineers are interested in non-crisp modeling to describe and debrief useful information hidden in uncertain data. Many researchers have proposed a set of theories to deal with this uncertainty, such as fuzzy set theory [1], intuitionistic fuzzy set theory [2], vague set theory [3], theory of interval mathematics [4], theory of rough set [5], theory of shadow set [6], and neutrosophic set theory [7].

1.2 Soft Set

Soft set, which is a theory defined as a new mathematical tool for dealing with uncertainties that traditional mathematical tools cannot handle, was initiated in 1999 by Molodtsov [8]. Since then, researchers have raced to enrich this topic in all aspects, and the related research has rapidly. They have worked on a combination of soft set theory and other theories starting from Maji et al. [9], who introduced the concept of fuzzy soft set, a more general concept as a combination of fuzzy set and soft set, and studied its properties. Then, in 2003 [10], the theory of soft sets and their properties were studied. In 2001, they introduced an intuitionistic fuzzy soft set, a combination of intuitionistic fuzzy sets and soft sets. In addition, Xu et al. combined a vague set with a soft set to define a vague soft set in 2010.

Rough soft set theory is a combination concept defined by Feng et al. [11]. Yang et al. presented a combination of an interval-valued fuzzy set and soft set in 2009 [12]. A neutrosophic soft set is a new concept defined by Maji in 2013 [13]. Many researchers have studied soft set theory, attempted to develop it, and studied its applications in decision-making and medical diagnosis problems. For example, Alkhazaleh et al. [14] introduced the concept of soft multiset as a generalization of the soft set. They also generalized this fuzzy soft multiset theory concept in 2012 [15]. Majumdar and Samanta [16] introduced and studied a generalized fuzzy soft set, where the degree is attached to the parameterizations of fuzzy sets while defining a fuzzy soft set. Alkhazaleh et al. also defined fuzzy parameterized interval-valued fuzzy soft set [17] and possibility fuzzy soft set [18]. They provided their applications in decision-making and medical diagnosis problems. Alkhazaleh and Salleh [19] introduced the concept of a soft expert set, in which the user can know the opinions of all experts in one model without any operations. Even after any operation, the user can know the opinions of all experts. Then, they generalized this concept to a fuzzy soft expert set and established its application in decision-making. Adam and Hassan in 2014 [20] introduced the theory of Q–fuzzy soft set with its application in decision-making and medical diagnosis problems.

1.3 Shadow Set

The theory of shadowed sets [6] is one of several key contributors to uncertainty, and its direct application can be envisioned in several areas. Pedrycz and Vukovich [21] discussed the underlying theoretical underpinnings of shadowed sets that dwell primarily on the pillars of three-valued logic. They also presented several illustrative examples that help grasp the essence of the concept. De Morgan Brouwer Zadeh Many valued (BZMV^dm) algebras were introduced as an abstract environment to describe both shadowed and fuzzy sets by Cattaneo and Ciucci in 2003 [22]. Pedrycz [23, 24] demonstrated how shadowed sets help in data interpretation problems in fuzzy clustering by leading to the three-valued quantification of data structures that comprise core, shadowed, and uncertain structures. In 2014, a three-way, three-valued, or three-region approximation of a fuzzy set from a pair of thresholds (α, β) on the fuzzy membership function was developed by Deng and Yao [25]. Fuzzy sets with time-varying membership degrees are frequently used. Cai et al. [26] interpreted dynamic fuzzy sets using shadowed sets. They suggested an analytic solution for computing a pair of thresholds by searching for a balance of uncertainty in the framework of shadowed sets. In a previous study, Yao et al. [27] in 2017 generalized the definition of three-valued sets by using a set of three values {n, m, p} to replace {0, [0, 1], 1}. They introduced an optimization-based framework for constructing three-way approximations. In 2018, Ibrahim and William-West [28] proposed a new method for computing the threshold value of an α-cut for the induction of shadowed sets from fuzzy sets by modifying the formula for the balance of uncertainty. The authors described an algorithm to illustrate the proposed method. In addition, Wang et al. in 2018 [29] proposed a novel method for MADM problems under a linguistic-term environment combining shadowed sets and Pythagorean fuzzy sets. They defined Pythagorean shadowed numbers and described their operation and basic properties. In 2019, Zhao and Yao [30] proposed a general definition for a three-way fuzzy partition using two sets of properties. Considering all possible nonequivalent properties subsets, they obtained 21 classes of three-way fuzzy partitions and studied three classes of three-way fuzzy partitions. Zhou et al. [31] in 2019 analyzed the continuous and convex properties of Pedrycz’s optimization objective function to construct shadowed sets. Ibrahim et al. [32] studied the principle of uncertainty balance, which guarantees the preservation of uncertainty in the induced shadowed set and suggested an alternative formulation for determining the optimum partition thresholds of shadowed sets. They also introduced a five-region shadowed set that effectively dealt with the issue of uncertainty balance. Yang and Yao, in their paper [33], suggested two plausible semantics of soft sets for modeling uncertainty: multi-context and possible-world semantics. Under these two semantics, they investigated three-way decisions under uncertainty represented by soft sets. They also introduced a qualitative model of three-way decisions based on the core and support of a soft set. They formulated a quantitative model by transforming a soft set into a fuzzy set, transforming the resulting fuzzy set into a shadowed set and making a three-way decision using the shadowed set.

In this section, we recall the definitions related to this work, starting from the fuzzy set, the definition of the shadow set, where we define the shadow set, its properties, and basic operations. Thereafter, we recall the concepts related to soft set theory and fuzzy soft sets based on their properties and operations. Since Zadeh published his new classic paper almost 60 years ago, fuzzy set theory has received increasing attention from researchers in various scientific areas, especially in the past few years. The difference between a binary set and a fuzzy set is that in a “normal” set, every element is either a member or a non-member. Here, it either must be A or notA. In a fuzzy set, an element can be a member of a set to some degree and, simultaneously, a non-member of the same set to some degree. In classical set theory, the membership of elements in a set is assessed in binary terms according to a bivalent condition: an element either belongs or does not belong to the set. By contrast, fuzzy set theory permits the gradual assessment of the membership of elements in a set, which is described with a membership function valued in the closed unit interval [0, 1]. Fuzzy sets generalize classical sets because the indicator functions of classical sets are special cases of the membership functions of fuzzy sets if the latter only takes values 0 or 1. Therefore, a fuzzy set A in a universe of discourse X is a function A : X → [0, 1], usually referred to as the membership function, denoted by μ_A(x). Some mathematicians use the notation A(x) to denote: membership function instead of μ_A(x). Fuzzy set A is written symbolically in various

Definition 2.1 ([1])

Let X be a nonempty set. A fuzzy set A in X is characterized by its membership function

and μ_A(x) is interpreted as the degree of membership of element x in the fuzzy set A for each x ∈ X.

A is completely determined by the set of pairs

The family of all the fuzzy sets in X is denoted by F(X). If X = {x₁, …, x_n} is a finite set and A is a fuzzy set in X, we often use the notation

where the term μ_A(x_i), i = 1, …, n signifies that μ_i is the membership grade of x_i in A. Here, we use the following notation for the fuzzy set.

In what follows, we summarize the quintessence of shadowed sets, a concept presented by Pedrycz. [6]. Pedrycz introduced the concept of a shadow set using a different formula. In general, by considering a fuzzy set A, the main idea is to elevate all membership values that are high enough and reduce those that are viewed as substantially low, where the elevation and reduction components are very radical as we go for 1 and 0, respectively. Mathematically, we define the shadow set as follows:

Definition 2.2 ([6, 21])

Let X be a set of objects called the universe, and μ_A : X → [0, 1] be a fuzzy set over X. A shadowed set on X is any mapping B : X → {0, [0, 1], 1}, B : X → {0, a, 1}. by

The shadowed set B is a mapping defined in X into a three-element set, where the elements of X for which B(x) reaches 1 constitute its core. The zones of X, where B(x) = [0, 1] or B(x) = a form a shadow of this construct. Eventually, the elements with B(x) = 0 are excluded from B. Pedrycz presented a new way of understanding uncertainty. This definition has evolved and undergone some development and modernization, either through Pedrycz or other researchers. To easily demonstrate how to produce a shadow set, one way was to choose α to generate this set by selecting the excluded zone for all values less than α and shadowed set between α and α − 1 where the core zone of all values are greater than α − 1. In contrast, the other direction selects α and β for the same generation process. Let X = {x₁, x₂, …, x₁₀} be a set of universes, FX={x10.3,x20.4,x30.8,x40.5,x50.7,x60.9,x10,x80.6,x91,x100.2} be a fuzzy set over X. Suppose α = 0.3, then we can write the shadow set Shdw(X) as

Without loss of generality, we can indicate [0, 1] or a with a value 0.5.

Defining operations on shadowed sets as union, intersection, and complement; here, a = [0, 1].

Definition 2.3 ([6, 21])

The complement of the shadowed set Shdw(X) = {0, [0, 1], 1} is shadowed set Shdw(X)^c = {1, [0, 1], 0}.

Definition 2.4

The union of the two shadowed sets Shdw_A(X) and Shdw_B(X) over a common universe X is the shadowed set Shdw_C(X) presented in Table 1.

Definition 2.5

The intersection of the two shadowed sets Shdw_A(X) and Shdw_B(X) over a common universe X is the shadowed set Shdw_C(X) presented in Table 2.

Note that for any a in [0, 1] we have min([0, 1], a) = [0, a] and max([0, 1], a) = [a, 1].

Molodtsov defined a soft set as a mapping from a set of parameters to a crisp subset of the universe, where the approximate function of a soft set is defined from a crisp set of parameters to a crisp subset of a universal set. Let U be a universe and E be a set of parameters. Let P(U) denote the power set of U and A ⊆ E.

Definition 2.6 ([8])

A pair (F,A) is called a soft set over U, where F is a mapping

In other words, a soft set over U is a parameterized family of subsets of the universe U. For ɛ ∈ A, F (ɛ) may be considered as the set of ɛ-approximate elements of the soft set (F,A).

In 2001, Maji et al. [9] generalized the soft set to a fuzzy soft set. In the following, we review some definitions and properties related to fuzzy soft-set theory, which we use in our work. The following definitions and propositions were provided by Maji et al. [9]:

Definition 2.7

Let U be an initial universal set and E be a set of parameters. Let I^U denote all fuzzy subsets of U. Let A ⊆ E. A pair (F,E) be a fuzzy soft set over U where F is a mapping given by

Definition 2.8

The union of two fuzzy soft sets (F,A) and (G,B) over a common universe U is the fuzzy soft set (H,C) where C = A ∪ B, and ∀ɛ ∈ C,

where s denotes any s-norm.

Definition 2.9

The intersection of the two fuzzy soft sets (F,A) and (G,B) over a common universe U is the fuzzy soft set (H,C) where C = A ∪ B, and ∀ɛ ∈ C,

where t denotes any t-norm.

In this section, we describe soft shadow sets. In our definition, we introduced different shadow sets for each fuzzy set related to each parameter. Some operations and properties are presented and studied with some examples.

Definition 3.1

Let U = {x₁, x₂, …, x_n} be the universal set of elements, (F,E) be 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 parameter set related to E. Let shdw (U) be 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_(αi,βi) : E → shdw (U), ∀i = 1, 2, …, m. Thus, we can write F_(αi,βi) as

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

Example 3.1

Let U = {u₁, u₂, u₃, u₄, u₅} be the universal set of elements and E = {e₁, e₂, e₃, e₄} be the set of parameters

be the fuzzy soft set over U. Let

be the shadow parameters set that related to E. Then the shadow soft set (F,E)_shdw can be defined as follows:

Then, we can view the shadow soft set (F,E)_shdw, comprising the following collection of approximations:

Here, a = [0, 1].

Definition 3.2

Let (F,A)_shdw and (G,B)_shdw be two shadow soft sets over common universe U. (F,A)_shdw is said to be a shadow-soft subset of (G,B)_shdw if A ⊆ B; and F_(αi,βi) (e_i) (x) is a shadow subset of G_(αi,βi) (e_i) (x); ∀e_i ∈ A ∩ B; x ∈ U.. We denote this by (F,A)_shdw ⊆ (F,A)_shdw, where 0 < [0, 1] < 1,

Definition 3.3

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

Definition 3.4

An absolute shadow soft set (Ψ,E)_shdw over common universe U is a shadow soft set with Ψ(e)_shdw(x) = 1, ∀e ∈ E; x ∈ U.

Definition 3.5

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.

3.1 Basic Operations

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