Decision-making is a crucial aspect of daily life, and the ability to make informed and accurate choices is vital for success. However, most real-life problems are plagued by uncertainty, vagueness, and imprecision, making it difficult to determine optimal solutions. To address these issues, researchers have developed various mathematical tools such as the increasing (decreasing) soft rough approach.

The increasing (decreasing) soft rough approach generalizes the classic soft rough set model and is based on the concept of topologically ordered spaces introduced by Nachbin [1] in 1965. Subsequently, in 1982, Pawlak [2] introduced the concept of rough set theory, which provided a new approach to soft computing and has been widely applied in various fields, such as information systems, economics, medicine, engineering, and game theory. In 1999, Molodtsov [3] first introduced the notion of soft sets to overcome problems associated with vagueness, impreciseness, and incomplete data. Feng et al. [4] introduced the concept of soft rough sets, which can be considered as a generalization of the rough set model. Numerous researchers have studied this concept and combined it with different types of sets to handle uncertainties. The increasing (decreasing) soft rough approach builds on the foundations of previous models and offers a more specific and broader method for solving real-life problems. This is a modified version of the approach proposed by Alkhazaleh and Marei [5] and addresses some of the limitations of their method. These limitations include the modified soft lower approximation of the universe set not being equal to the universe, and the modified soft upper approximation of the empty set not being empty.

The proposed method, demonstrated and compared through diverse examples, exhibits significant advantages over the classic soft rough set model and other methodologies. It provides a practical and effective solution for decision-making in uncertain and vague scenarios and is a valuable tool for making informed choices in real-life problems.

This section introduces the core concepts and principles of soft sets, rough sets, soft rough sets, and increasing (decreasing) sets.

Definition 2.1 [6]

A binary relation ≤ on a set Ψ is called a partial order if it satisfies the following three properties: reflexivity, antisymmetry, and transitivity. The set (Ψ, ≤) is called a partially ordered set (POS). In addition, the equality relation on Ψ, represented by ▲, is the set of all pairs of the form (n, n) for each element n in Ψ.

Definition 2.2 [6]

A POS can also be classified as a linear ordered set (LOS) or directed ordered set (DOS) based on additional properties. An LOS is a POS wherein, for any two elements n, b ∈ Ψ, either n ≤ b, b ≤ n, or n = b. A DOS is a POS wherein, for any two elements n, b ∈ Ψ, there exists an element z ∈ Ψ such that z ≤ n and z ≤ b.

Definition 2.3 [1]

Let (Ψ, ≤) be a POS, n ∈ Ψ and G ⊆ Ψ. This is defined as follows:

• 1. i(n) = {b ∈ Ψ : n ≤ b},

• 2. d(n) = {b ∈ Ψ : b ≤ n},

• 3. i(G) = {b ∈ Ψ : n ≤ b for some n ∈ G} = ∪_n∈G(i(n)),

• 4. d(G) = {b ∈ Ψ : b ≤ n for some n ∈ G} = ∪_n∈G(d(n)).

If i(G) = G, G is considered an increasing set, and if d(G) = G, G is considered a decreasing set.

Definition 2.4 [7]

Quadruples—consisting of a set of finite objects (denoted as Ψ), a set of finite attributes (denoted as δ), a value set for each attribute (denoted as V), and an information function (denoted as f) that maps objects to their attribute values—are used in rough set theory. Notably, Ψ and δ must not be empty sets.

Definition 2.5 [2]

An equivalence class of element x in a set of data (denoted as Ψ), determined by the equivalence relation δ, is defined as

The lower approximation of set H (a subset of Ψ) with respect to the equivalence relation δ is defined as

The upper approximation of set H with respect to the equivalence relation δ is defined as follows:

The boundary approximation of set H with respect to an equivalence relation δ is defined as follows:

The accuracy measure of the degree of crispness for set H with respect to the equivalence relation δ is defined as follows:

if δ(H) ≠ ∅, and a set H is considered a crisp set if δ(H) = δ(H) with respect to δ; otherwise, it is considered a rough set.

Henceforth, Ψ denotes the universe set, and P denotes the fixed set of parameters. T ⊆ P and 2^Ψ denotes the power set of Ψ.

Definition 2.6 [3]

A soft set is a mathematical concept composed of two sets: L and T. L is a set of labels and T is a subset of a fixed set of parameters denoted by P. The function L : T → 2^Ψ links each element in T to a subset of the universe set Ψ.

Definition 2.7 [4]

Let M = (Ψ, N) be a soft approximation space, with N = (L, T) being a soft set over Ψ. The soft rough lower, upper, and boundary approximations for a set H in this space, by using the parameter set P, are defined as follows:

If ∪{L(k) : k ∈ T} = Ψ, (L, T) is called a full soft set.

Definition 2.8 [5]

In a soft approximation space M = (Ψ, N), where N = (L, T) is a soft set over the universe Ψ, the following definitions apply to any subset H ⊆ Ψ:

where H^c be the complement of H, bISRQ(H)=SR¯Q(H)-SR_Q(H),CISRQ(H)=∣SR_Q(H)∣∣SR¯Q(H)∣,H≠∅.

This section defines a partially ordered soft approximation space (POSA) and presents related results. These include increasing (decreasing) soft lower, increasing (decreasing) soft upper, and increasing (decreasing) soft boundary approximations. Finally, it contrasts this structure with prior structures.

Definition 3.1

A triple (Ψ, N, ≤) is called a POSA, where N = (L, T) is a soft set of Ψ and (Ψ, ≤) is a POS. The definitions of the increasing (decreasing) soft lower, increasing (decreasing) soft upper, and increasing (decreasing) soft boundary approximations for any subset H ⊆ Ψ are as follows:

• 1. ISR_inc(H) = ∪{i(L(k)), k ∈ T : L(k) ⊆ H},

• 2. ISR¯inc(H)=[ISR_inc(Hc)]c,

• 3. BISRinc(H)=ISR¯inc(H)-ISR_inc(H),

• 4. DSR_dec(H) = ∪{d(L(k)), k ∈ T : L(k) ⊆ H},

• 5. DSR¯dec(H)=[DSR_dec(Hc)]c,

• 6. BDSRdec(H)=DSR¯dec(H)-DSR_dec(H).

Remark 3.1

In Definition 3.1, the expressions ISR_inc(H) and DSR_dec(H) as defined by Alkhazaleh and Marei [5] in the following forms:

• 1. ISR_inc(H) = ∪{i(L(k)), k ∈ T : L(k) ⊆ H}; and

• 2. DSR_dec(H) = ∪{d(L(k)), k ∈ T : L(k) ⊆ H}

cannot be accepted mathematically.

The following example describes Remark 3.1.

Example 3.1

Consider a partial order ≤ defined over a set Ψ = {n₁, n₂, n₃, n₄, n₅, n₆} as ≤ = ▲ ∪ {(n₂, n₆), (n₃, n₄)}. Let P = {k₁, k₂, k₃} be a fixed of parameters such that L(k₁) = {n₁, n₃, n₄, n₅, n₆}, L(k₂) = {n₃, n₄}, and L(k₃) = {n₂, n₄}.

Let H = {n₂, n₄}. According to the definitions provided by Alkhazaleh and Marei [5], we calculated the increasing soft lower and decreasing soft lower of H as follows:

However, as noted in Remark 3.1, these expressions do not satisfy the mathematical requirements.

We modified the increasing (decreasing) soft rough set approximations introduced to solve these two problems.

Definition 3.2

A triple (Ψ, N, ≤) is called a POSA, where N = (L, T) is a soft set of Ψ and (Ψ, ≤) is a POS. The definitions of the increasing (decreasing) soft lower, increasing (decreasing) soft upper, and increasing (decreasing) soft boundary approximations for any subset H ⊆ Ψ are as follows:

• 1. ISR_inc(H) = ∪{i(L(k)), k ∈ T : L(k) ⊆ H} ∩ H,

• 2. ISR¯inc(H)=[ISR_inc(Hc)]c,

• 3. BISRinc(H)=ISR¯inc(H)-ISR_inc(H),

• 4. DSR_dec(H) = ∪{d(L(k)), k ∈ T : L(k) ⊆ H} ∩ H,

• 5. DSR¯dec(H)=[DSR_dec(Hc)]c,

• 6. BDSRdec(H)=DSR¯dec(H)-DSR_dec(H).

Definition 3.3

Given a soft set N = (L, T) over Ψ and a POSA (Ψ, N, ≤), the degree of crispness of any subset H ⊆ Ψ is represented by C_ISRinc (H) and is defined as follows:

Clearly, the value of C_ISRinc (H) is between 0 and 1. If ISR_inc(H)=ISR¯inc(H), H is a crisp set. Otherwise, H is considered as an increasingly soft rough set.

Proposition 3.1

Given a POSA (Ψ, N, ≤), let H, Y ⊆ Ψ. Then,

• 1. ISR_inc(H)⊆H⊆ISR¯inc(H).

• 2. ISR_inc(∅)=∅   and   ISR¯inc(Ψ)=Ψ.

• 3. H ⊆ Y ⇒ ISR_inc(H) ⊆ ISR_inc(Y).

• 4. H⊆Y⇒ISR¯inc(H)⊆ISR¯inc(Y).

• 5. ISR_inc(H ∩ Y) ⊆ ISR_inc(H) ∩ ISR_inc(Y).

• 6. ISR_inc(H) ∪ ISR_inc(Y) ⊆ ISR_inc(H ∪ Y).

• 7. ISR¯inc(H∩Y)⊆ISR¯inc(H)∩ISR¯inc(Y).

• 8. ISR¯inc(H)∪ISR¯inc(Y)⊆ISR¯inc(H∪Y).

• 9. ISR_inc(ISR_inc(H)) = ISR_inc(H).

• 10. ISR¯inc(ISR¯inc(H))=ISR¯inc(H).

• 11. ISR_inc(ISR¯inc(H))⊆ISR¯inc(H).

• 12. ISR_inc(H)⊆ISR¯inc(ISR_inc(H)).

• 13. ISR_inc(Hc)=[ISR¯inc(H)]c

Proposition 3.2

In POSA (Ψ, N, ≤), if H is a subset of the set of states Ψ, the following statements are true:

• 1. SR_Q(H) ⊆ ISR_inc(H), and they are equal if ≤ = ▲.

• 2. ISR¯inc(H)⊆SR¯Q(H), and they are equal if ≤ = ▲.

• 3. B_ISRinc (H) ⊆ b_ISRQ(H), and they are equal if ≤ = ▲.

Example 3.2

In the example given by Alkhazaleh and Marei [5], a soft set (L, P) was used to represent the condition of patients suspected of having the flu. Set P includes seven flu symptoms: difficulty breathing, headache, fever, cough, runny nose, sore throat, and lethargy, which were approved previously [8–10]. In this example, six patients were examined at a medical center, represented by a set. Ψ = {n₁, n₂, n₃, n₄, n₅, n₆} and P is a set of parameters, such as P = {k₁, k₂, k₃, k₄, k₅, k₆, k₇}, where k₁ = fever, k₂ = difficult breathing, k₃ = runny nose, k₄ = cough, k₅ = headache, k₆ = sore throat, k₇ = lethargy. Consider the mapping L : P → 2^Ψ given by L(k₁) = {n₁, n₃, n₄, n₅, n₆}, L(k₂) = {n₁, n₂}, L(k₃) = {n₁, n₂, n₄}, L(k₄) = {n₁}, L(k₅) = {n₃, n₄}, L(k₆) = {n₂, n₄}, L(k₇) = {n₁, n₃, n₅, n₆}.

For example, L(k₁) = {n₁, n₃, n₄, n₅, n₆} indicates that patients 1, 3, 4, 5, and 6 have a fever. This soft set can be represented in Boolean tabular form, as shown in Table 2. This example was used to automate the process of identifying flu symptoms in patients.

Let ≤ = ▲ ∪ {(n₂, n₆), (n₃, n₄)} be a partial-order relationship on Ψ. Subsequently, the increasing sets are i(L(k₁)) = {n₁, n₃, n₄, n₅, n₆}, i(L(k₂)) = {n₁, n₂, n₆}, i(L(k₃)) = {n₁, n₂, n₄, n₆}, i(L(k₄)) = {n₁}, i(L(k₅)) = {n₃, n₄}, i(L(k₆)) = {n₂, n₄, n₆}, and i(L(k₇)) = {n₁, n₃, n₄, n₅, n₆}. If H = {n₂, n₃, n₄, n₅, n₆}, then SR_Q(H) = {n₂, n₃, n₄} and ISR_inc(H) = {n₂, n₃, n₄, n₆}; clearly, SR_Q(H) ⊆ ISR_inc(H). If H = {n₃, n₄, n₅}. Then, SR¯Q(H)={n3,n4,n5,n6} and ISR¯inc(H)={n3,n4,n5} implies that ISR¯inc(H)⊆SR¯Q(H). Also, b_ISRQ(H) = {n₅, n₆} and B_ISRinc (H) = {n₅}, so that B_ISRinc (H) ⊆ b_ISRQ(H).

Table 3 compares the boundary and accuracy of the method proposed by Alkhazaleh and Marei [5, Table 5] and the current method (defined as Definition 3.2) when applied to increasing sets. The comparison was based on the partial-order relation ≤ = ▲ ∪ {(n₂, n₆), (n₃, n₄)} for Ψ.

This section describes the improvement in the precision and addressing of the limitations by converting POS to LOS (DOS and total directed ordered set [TDOS]). This should address issues such as those found in the approach by Alkhazaleh and Marei [5] approach, where the modified soft upper bound approximation of the universe set may not match the actual universe, and the modified soft upper bound approximation of the empty set may not be truly empty. The new approximations have specific properties, and counterexamples are presented. The relationship between this approach and that of Alkhazaleh and Marei [5] was also established.

Definition 4.1

Let Ψ be a POS. Ψ is considered a TDOS if, for any two elements n and b in Ψ and any z in Ψ, we have z ≤ n and z ≤ b. In other words, any two elements in Ψ have a common lower bound that is less than or equal to them for any choice of z.

Proposition 4.1

If (Ψ, ≤) is a TDOS in the soft set N = (L, T) over a nonempty set Ψ, then

• 1. i(L(k_j)) = Ψ, ∀k_j ∈ T, L(k_j) ≠ ∅.

• 2. ISR_inc(Ψ) = Ψ.

• 3. ISR¯inc(∅)=∅.

• 4. ∀H ⊆ Ψ, if ∃∅≠L(k)⊆H⇒ISR_inc(H)=H=ISR¯inc(H).

Example 4.1

In Example 3.2, if ≤ = ▲ ∪ {(n₂, n₁), (n₂, n₃), then: {(n₂, n₄), (n₂, n₅), (n₂, n₆), (n₃, n₁), (n₃, n₄), (n₃, n₅), (n₃, n₆), (n₄, n₁), (n₄, n₅), (n₄, n₆), (n₅, n₁), (n₅, n₆), (n₆, n₁)}. Clearly, (Ψ, ≤) is an LOS and the increasing sets are i(L(k₁)) = i(L(k₂)) = i(L(k₃)) = i(L(k₆)) = Ψi(L(k₄)) = {n₁}, i(L(k₅)) = i(L(k₇)) = {n₁, n₃, n₄, n₅, n₆}, and the result in the Table 4 show that several subsets are more exact (crisp) than those listed in Table 3.

Remark 4.1

The following example demonstrates that all subsets become crisp if POS (Ψ, ≤) is a DOS in an increasing number of sets.

Example 4.2

In Example 3.2, if ≤ = ▲ ∪, then: {(n₁, n₂), (n₁, n₃), (n₁, n₄), (n₁, n₅), (n₁, n₆), (n₂, n₁), (n₂, n₃), (n₂, n₄), (n₂, n₅), (n₂, n₆), (n₃, n₁), (n₃, n₂), (n₃, n₄), (n₃, n₅), (n₃, n₆). Clearly, (Ψ, ≤) is a DOS and the increasing sets are i(L(k₁)) = i(L(k₂)) = i(L(k₃)) = i(L(k₄)) = i(L(k₅)) = i(L(k₆)) = i(L(k₇)) = Ψ. As shown in Table 5, all subsets in the study by Alkhazaleh and Marei [5, Table 5] become exact (crisp) if POS(Ψ, ≤) is a DOS.

Example 4.3

In Example 3.2, if ≤ = ▲ ∪, then: {(n₁, n₂), (n₁, n₃), (n₁, n₄), (n₁, n₅), (n₁, n₆), (n₂, n₁), (n₂, n₃), (n₂, n₄), (n₂, n₅), (n₂, n₆), (n₃, n₁), (n₃, n₂), (n₃, n₄), (n₃, n₅), (n₃, n₆), (n₄, n₁), (n₄, n₂), (n₄, n₃), (n₄, n₅), (n₄, n₆), (n₅, n₁), (n₅, n₂), (n₅, n₃), (n₅, n₄), (n₅, n₆), (n₆, n₁), (n₆, n₂), (n₆, n₃), (n₆, n₄), (n₆, n₅)}. Clearly, (Ψ, ≤) is a TDOS and the increasing sets are i(L(k₁)) = i(L(k₂)) = i(L(k₃)) = i(L(k₄)) = i(L(k₅)) = i(L(k₆)) = i(L(k₇)) = Ψ. As per Table 5, all subsets in the study by Alkhazaleh and Marei [5, Table 5] become exact (crisp) if POS (Ψ, ≤) is a TDOS.

Remark 4.2

As indicated Tables 5 and 6, the results correspond to the case of Ψ is DOS and TDOS,; however, if we consider H = {n₄, n₅}, L(k₈) = {n₄} in Example 3.2, we have two cases:

• 1. If Ψ is DOS, then CISRinc(H)=12,

• 2. If Ψ is TDOS, then C_ISRinc (H) = 1.

Proposition 4.2

In POSA (Ψ, N, ≤), for any subset H, Y ⊆ Ψ, the following statements hold:

• 1. ISR_inc(H – Y) ⊆ ISR_inc(H) – ISR_inc(Y).

• 2. ISR¯inc(H-Y) (may not be)⊆ISR¯inc(H)-ISR¯inc(Y).

• 3. ISR¯inc(H)-ISR¯inc(Y) (may not be)⊆ISR¯inc(H-Y).

Example 4.4

Consider the scenario in Example 3.2 where ≤ = ▲, H = {n₁, n₂} and Y = {n₁}. Subsequently, ISR_inc (H) = {n₁, n₂}, ISR_inc(Y) = {n₁}, ISR_inc(H – Y) = ∅, ISR_inc(H)–ISR_inc(Y) = {n₂}, it follows that ISR_inc(H)– ISR_inc(Y) = {n₂} ⊈ ISR_inc(H–Y).

The following two examples demonstrate the validity of Propositions (2) and (3) in Proposition 4.2.

Example 4.5

Let H = {n₃} and Y = {n₁, n₄} from Example 4.2. Subsequently, ISR¯inc(H)={n3},ISR¯inc(Y)=Ψ,ISR¯inc(H-Y)={n3},ISR¯inc(H)-ISR¯inc(Y)=∅, Hence, ISR¯inc(H-Y)⊈ISR¯inc(H)-ISR¯inc(Y).

Example 4.6

Let H = {n₁, n₄} and Y = {n₄} from Example 4.2. We have ISR¯inc(H)=Ψ,ISR¯inc(Y)={n4},ISR¯inc(H-Y)={n1},ISR¯inc(H)-ISR¯inc(Y)={n1,n2,n3,n5,n6}. Hence, ISR¯inc(H)-ISR¯inc(Y)⊈ISR¯inc(H-Y).

In this section, we present soft concepts. Furthermore, by using the degree of accuracy of the modified soft roughness and increasing soft roughness methods, we compared these approaches.

Definition 5.1

Let N = (L, T) be a soft set over set Ψ, and let (Ψ, N, ≤) be a POSA. Then, H ⊆ Ψ is referred to as

• 1. Totally ISR_inc-definable (ISR_inc–exact) set, if ISR_inc(H)=H=ISR¯inc(H).

• 2. Internally ISR_inc-definable set, if ISR_inc(H) = H, H⊂≠ISR¯inc(H).

• 3. Externally ISR_inc-definable set, if ISR_inc(H) ⫋ H, H=ISR¯inc(H).

• 4. Totally ISR_inc-definable rough set, if ISR_inc(H) ⫋ H, H=ISR¯inc(H).

Theorem 5.1

Each ISR_Q-definable set is also a total ISR_inc-definable set.

Example 5.1

Consider H = {n₁} in Example 4.2. It follows that ISR_inc(H)=H=ISR¯inc(H), but SR¯Q(H)≠H.

Theorem 5.2

Every total ISR_inc-definable rough set is also an ISR_Q-definable rough set.

Remark 5.1

Internally, the ISR_inc-definable set is not equivalent to the ISR_Q-definable set.

Example 5.2 illustrates the statement in Remark 5.1.

Example 5.2

By considering H = {n₁, n₂, n₃} in Example 4.1, we obtain ISR_inc(H) = {n₁, n₂, n₃} and ISR¯inc(H)=Ψ, which shows that H is internally ISR_inc-definable set. However, SR_Q(H) ≠ H, making it not internally ISR_Q-definable set. In contrast, if we consider Y = {n₁, n₂}, we have SR_Q(Y) = {n₁, n₂} and SR¯Q(Y)={n1,n2,n5,n6}, which indicates that Y is internally ISR_Q-definable set. However, ISR¯inc(Y)={n1,n2}=Y, which is not internally ISR_inc-definable set.

Remark 5.2

Externally, the ISR_inc-definable set is not equal to the externally defined ISR_Q– set.

The following examples demonstrate Remark 5.2.

Example 5.3

Consider H = {n₃, n₄, n₅} from Example 3.2. Then, ISR_inc(H) = {n₃, n₄} and ISR¯inc(H)={n3,n4,n5}. However, SR¯Q(H)={n3,n4,n5,n5}≠H, which means H is externally ISR_inc-definable set but it is not externally ISR_Q-definable set.

Example 5.4

Consider Y = {n₃, n₄, n₅, n₆} from Example 4.1. Then, SR_Q(Y) = {n₃, n₄}, and SR¯Q(Y)={n3,n4,n5,n6}=Y which demonstrates that Y is externally ISR_Q-definable set. However, ISR_inc(Y) = {n₃, n₄, n₅, n₆} = Y, is not externally ISR_inc-definable.

Proposition 5.1

Let (Ψ, N, ≤) be a POSA and N = (L, T) be a soft set over Ψ. If H ⊆ Ψ, then H is the total ISR_inc-definable set if and only if C_ISRinc (H) = 1.

Definition 5.2

A soft set N = (L, T) over Ψ is referred to as a full soft-increasing set if (Ψ, N, ≤) is a POSA and (Ψ, ≤) is a TDOS.

Remark 5.3

If N = (L, T) is a full soft set, it is a fully soft increasing set.

Remark 5.3 is not necessarily true in all the cases, as demonstrated in Example 5.5.

Example 5.5

Example 3.7 in [5] shows that even if N = (L, T) is a fully soft increasing set, it may not be a full soft set.

Remark 5.4

The results obtained will be similar if “increasing sets” are replaced with “decreasing sets” in this search.

The concept of topologically ordered spaces was first proposed by Nachbin [1]. Rough set theory was introduced by Pawlak [2] in 1982 as a mathematical framework for handling uncertainty and vagueness in data analysis. In 1999, Molodtsov [3] introduced the concept of soft sets for extending rough sets to uncertain objects. Alkhazaleh and Marei [5] proposed a modified soft-rough-set model for 2021. This study considered increasing (decreasing) soft rough sets, solved problems by using the modified model, and demonstrated enhanced results as compared to the traditional approach. The novel method addresses the limitations associated with the method by Alkhazaleh and Marei [5] by converting partially ordered relations into linearly ordered relations such as directed or totally directed relations. This conversion ensures that every subset is crisp, resulting in improved accuracy in various domains.

Table 1. Tabular representation of the soft set (L, P).

Table 2. Boolean tabular representation of the soft set (L, P).

Table 3. Comparison of boundary and accuracy between the method by Alkhazaleh and Marei [5, Table 5] and the proposed method (Definition 3.2) using the partial order relation ≤ = ▲ ∪ {(n₂, n₆), (n₃, n₄)} on Ψ for increasing sets.