A study of bipolar fuzzy parameterized soft sets and their application in decision making

Abstract

In this work, bipolar fuzzy parameterized sets in conjunction with soft sets are studied. This research focuses on the application of bipolar fuzzy parameterized soft sets (BFPSS), arising from association of bipolar fuzzy parameterized sets with soft sets. Some useful operations and fundamental properties of BFPSS are presented. Mainly, we aim to design a novel and comparatively labour-saving algorithm to see the efficacy of BFPSS. We discuss an application of our algorithm in pharmaceutical decision making problem based on effectiveness and harmfulness of certain drugs. However, the algorithm is equally applicable in other decision making environments as well where BFPSS arise. By preserving the structural implication of BFPSS, we compare our technique with a most recent algorithm to prove the significance of our method.

Keywords

Fuzzy sets bipolar fuzzy sets soft sets bipolar fuzzy parameterized soft sets decision making

1 Introduction

Decision making is a process which stands on cogent human decree for identifying and selecting substitutes based on human predilections that mainly applied in the management level in any organization. Every decision environment is outlined as a pool of statistics, possible replacements, values, and preferences on hand at the time of the verdict. Since both information and their substitutes are constrained as the time and labor to gain statistics or identify alternatives are restricted, so findings made must be within this constrained setting.

Now a days, the foremost challenge of decision making is vagueness, and a major target of decision makers is to trim down haziness. As a result of such type of ambiguity and subjectivity involved in decision making, fuzziness came into picture. Handling decision making in such kind of blurred and elusive situations, (Zadeh, 1965) [1] introduced fuzzy set theory to develop thinking as a human normally does. The phrase “decision making in fuzzy environment” connotes a decision making procedure in which objectives and/or the conditions are fuzzy in nature. This signifies that the objectives and/or the constraints makeup classes of alternatives/substitutes whose boundaries are not eye-catchingly demarcated [2]. Fuzziness is a kind of imprecision associated with the usage of fuzzy sets i.e. the classification in which there is no clear-cut transition from non-membership to membership or vice versa.

As the time moved on, in order to deal with imprecisions and uncertainties in sociology, engineering, medical science etc., (Pawlak, 1982) introduced rough sets [3], (Atanassov, 1983) [4] intuitionistic fuzzy sets, (Gau, 1993) [5] vague sets but each notion has its own weaknesses as pointed out by Molodtsov. Then, (Molodtsov, 1999) [6] propositioned a new theory called it the soft sets theory to model elusiveness and uncertainty, which was possibly at that time free from the shortcomings of existing fuzzy algebraic structures. In the mean while, (Maji, 2001) presented two new concepts, one of fuzzy soft set [7] and other is of intuitionistic fuzzy soft set [8]. Due to the intrinsic nature of these structures, over the period of time, they came up as a handy and robust tool in almost every prevailing critical decision making situations arising in economics, engineering, medical and social sciences etc. Some notable contributions that used soft sets [9], intuitionistic fuzzy sets [10 –12], fuzzy soft sets [13 –18], and intuitionistic fuzzy soft sets [19 –28] in decision making problems.

Bipolarity of information is a crux factor to be taken in consideration while probing the information and creating mathematical models of real life challenges. Bipolarity enunciates about two opposite traits of information. The affirmative information exemplifies what is sure fired to be viable, while the adverse information symbolizes what is unviable, proscribed, or surely untrue. A wide range of human decisions is built on bipolar dire thinking, for example, hope and fear in future appraisal; sympathy and cruelty in social life; hate and love in psychoanalysis; contentment and disappointment in an event based activity; modesty and arrogance in self-appraisal; patience and frustration etc. The synchronization, balance and agreement of these twofold aspect are necessary for the sound footings of any social approach.

With the passage of time it is further clarified that the theory of fuzzy and soft sets is no more appropriate tool to control bipolarity; like, a food which is not sweet, need not be sour. To avoid these problems, the conceptualization of bipolar fuzzy sets is instigated by (Zhang, 1994) [29] as generality of fuzzy sets. Theory of fuzzy bipolar soft sets [30] has strong capability to cope with bipolarity besides fuzziness of the information. A lot of decision making in bipolar fuzzy soft sets environment is done by [31 –37].

Quite recently (Deli, 2020) [38] the concept of bipolar fuzzy parameterized soft sets. The authors used the soft level sets and proposed parameter reduction method for decision making. However they didn’t define any criterion to choose the threshold value for the k-level soft sets. Further, presentation of BFPSS using these k-level soft sets, as in [38], converts them in to crisp soft sets which limits the accuracy of decision and enhances computational labour. To overcome this issue, we propose a more precise and efficient algorithm for decision making, which relies on the structural uniqueness of BFPSS. Our technique is quite simple however it leads to more accurate solution. Besides this, we model a pharmaceutical problem based on effectiveness and harmfulness of certain drugs and relate the study with BFPSS. Then with the help of the proposed algorithm, we solve the under study problem by reaching the optimal solution based on the properties of BFPSS.

The formal layout of the presented material follows the order; Section 2 provides necessary background for this study. Section 3 presents some well-known operations on the bipolar fuzzy parameterized soft sets. In Section 4, application of BFPSS in decision making problems is discussed. In this regard, an algorithm is developed and applied for decision making regarding pharmaceutical composition of drugs. Besides this, a detailed comparative analysis is given in section 4. Lastly section 5 states the conclusion.

2 Preliminaries

This section presents detailed background material based on fuzzy sets and their generalization along with soft sets. We incorporate the requisite definitions related with the current framework. Throughout the article, χ will be universe of discourse, P (χ) will represent the set of all subsets of χ, and E will be set of parameters.

Definition 2.1. [6] Let E₁ ⊆ E. A soft set S over χ is described by $S = {(α, Γ (α)) : α \in E_{1}},$ where Γ : E₁ → P (χ) is termed as approximate function of S with respect to E₁. If Γ (α) = φ, then (α, Γ (α)) will not appear in expression for S.

Example 1. Let a set of 4 applicants be χ = {x₁, x₂, x₃, x₄} and E ₁ ⊆E is given by {α₁ = brilliant, α₂ = young, α₃ = experienced} then a soft set S is given by $S = {(α_{1}, {x_{2}, x_{4}}), (α_{2}, {x_{3}}), (α_{3}, {x_{1}, x_{3}, x_{4}})}$

Definition 2.2. [1] A fuzzy set (FS) A over χ is defined by $A = {(x, η (x)) : x \in χ},$ where η : χ → [0, 1] is known as membership map. The image of η for each x ∈ χ is called its degree of membership.

Example 2. Given χ = {x₁, x₂, x₃} then A = {(x₁, 0.5) , (x₂, 0.2) , (x₃, 0.7)} represents a fuzzy set with x₁, x₂ and x₃ having respective membership degrees 0.5, 0.2 and 0.7.

Definition 2.3. [29] A bipolar fuzzy set (BFS) is defined by $B = {(x, η^{+} (x), η^{-} (x)) : x \in χ},$ (1) here η⁺, η^- : χ → [0, 1] represent the positive and negative degrees of membership respectively.

Definition 2.4. [22] Let A be a fuzzy set over E₁ ⊆ E. We define a fuzzy parameterized soft set (FPSS) on χ as, $(Γ_{A}, E_{1}) = {(α, η (α), Γ_{A} (α)) : α \in E}$ where η : E₁ → [0, 1] and Γ_A : E₁ → P (χ) satisfies Γ_A (α) = φ if η (α) =0

Definition 2.5. [38] Let B be a bipolar fuzzy set over E₁ ⊆ E. A bipolar fuzzy parameterized soft set (BFPSS) over χ with respect to E₁ is defined as $ω_{E_{1}} = {(α, η^{+} (α), η^{-} (α), Γ_{B} (α)) : α \in E_{1}},$ (2) here η⁺ and η^- represent the positive and negative degrees of membership respectively. It is worth mentioning that η⁺ (α) stands for the degree to which α satisfies the property, however, η^- (α) measures the degree to which α satisfies the counter-property. Now-onwards, we will denote the set of all bipolar fuzzy parameterized soft sets over χ by Ω (χ).

Example 3. If χ = {x₁, x₂, x₃} and E ₁ ={α₁, α₂}⊆E.

Let B = {(α₁, 0.3, 0.5) , (α₂, 0.8, 0.2)} be a bipolar fuzzy set over E ₁ , as defined by Eq. (1). Then the following is a BFPSS $ω_{E_{1}} = {(α_{1}, 0.3, 0.5, {x_{1}, x_{3}}), (α_{2}, 0.8, 0.2, {x_{2}})}$ .

Definition 2.6. Let B be a bipolar fuzzy set over E₁ ⊆ E and ω_{E
₁} ∈ Ω (χ). Then complement of ω_{E
₁} (denoted by $ω_{E_{1}}^{c} \in Ω (χ)$ ) is defined by $ω_{E_{1}}^{c} = {(α, 1 - η^{+} (α), 1 - η^{-} (α), Γ_{B}^{c} (α)) : α \in E_{1}}$ (3)

Here $Γ_{B}^{c} (α) = χ - Γ_{B} (α)$ .

3 Some operations on bipolar fuzzy parameterized soft sets

This section deals with some of the primarily important and applicable operations, which are regarded as direct outcome of the definition of BFPSS. For the convenience of the readers, we state some of the operations and the most significant results below, one can see further details and proofs of the stated results in [38].

Proposition 3.1. Let ω_{E
₁} ∈ Ω (χ), then $(ω_{E_{1}}^{c})^{c} = ω_{E_{1}}$ .

Definition 3.2. Let ω_{E
₁} and ω_{E
₂} ∈ Ω (χ). We say ω_{E
₁} is a subset of ω_{E
₂} (ω_{E
₁} ⊆ ω_{E
₂}) iff E₁ ⊆ E₂, $η_{E_{1}}^{+} (α) \leq η_{E_{2}}^{+} (α), η_{E_{1}}^{-} (α) \geq η_{E_{2}}^{-} (α)$ , and Γ_{B
₁} (α) ⊆ Γ_{B
₂} (α) , ∀ α.

Example 4. Let χ = {x₁, x₂, x₃, x₄, x₅}, E = {α₁, α₂, α₃, α₄},

E₁ = {α₁, α₂}, E₂ = {α₁, α₂, α₃}, such that

ω_{E
₁} = {(α₁, 0.6, 0.4, {x₂, x₃}) , (α₂, 0.5, 0.4, {x₁})},

ω_{E
₂} = {(α₁, 0.7, 0.3, {x₁, x₂, x₃}) , (α₂, 0.6, 0.2, {x₁, x₅}) , (α₃, 0.5, 0.3, {x₂})} . Then ω_{E
₁} ⊆ ω_{E
₂}.

Proposition 3.3. Let ω_{E
₁}, ω_{E
₂}, ω_{E
₃} ∈ Ω (χ), then

i) ω_{E
₁} ⊆ ω_{E
₂} and ω_{E
₂} ⊆ ω_{E
₁} ⇔ ω_{E
₁} = ω_{E
₂}.

ii) ω_{E
₁} ⊆ ω_{E
₂} and ω_{E
₂} ⊆ ω_{E
₃}, then ω_{E
₁} ⊆ ω_{E
₃}.

Definition 3.4. Let B₁ and B₂ represent two bipolar fuzzy sets over E₁ ⊆ E and E₂ ⊆ E respectively. Let E₁ ∩ E₂ ≠ φ, and ω_{E
₁}, ω_{E
₂} ∈ Ω (χ) be the BFPSS corresponding to B₁ and B₂ respectively (as defined by Eq. (2)).

We define Union of ω_{E
₁} and ω_{E
₁} by $ω_{E_{1}} \cup ω_{E_{2}} = {(α, ν^{+} (α), ν^{-} (α), Γ_{B_{1} \cup B_{2}} (α))},$ where $\begin{matrix} (ν^{+} (α), ν^{-} (α)) = {\begin{matrix} (Max {η_{E_{1}}^{+} (α), η_{E_{2}}^{+} (α)}, Max {η_{E_{1}}^{-} (α), η_{E_{2}}^{-} (α)}) : α \in E_{1} \cap E_{2} \\ (η_{E_{1}}^{+} (α), η_{E_{1}}^{-} (α)) : α \in E_{1} \ E_{2} \\ (η_{E_{2}}^{+} (α), η_{E_{2}}^{-} (α)) : α \in E_{2} \ E_{1}, \end{matrix} \end{matrix}$ and $\begin{matrix} Γ_{B_{1} \cup B_{2}} (α) = {\begin{matrix} Γ_{B_{1}} (α) \cup Γ_{B_{2}} (α) : α \in E_{1} \cap E_{2} \\ Γ_{B_{1}} (α) : α \in E_{1} \ E_{2} \\ Γ_{B_{2}} (α) : α \in E_{2} \ E_{1} . \end{matrix} \end{matrix}$ Similarly Intersection of ω_{E
₁} and ω_{E
₁} is defined by

Definition 3.5. $ω_{E_{1}} \cap ω_{E_{2}} = {(α, υ^{+} (α), υ^{-} (α), Γ_{B_{1} \cap B_{2}} (α))},$ where $\begin{matrix} (υ^{+} (α), υ^{-} (α)) = {\begin{matrix} (Min {η_{E_{1}}^{+} (α), η_{E_{2}}^{+} (α)}, Min {η_{E_{1}}^{-} (α), η_{E_{2}}^{-} (α)}) : α \in E_{1} \cap E_{2} \\ (η_{E_{1}}^{+} (α), η_{E_{1}}^{-} (α)) : α \in E_{1} \ E_{2} \\ (η_{E_{2}}^{+} (α), η_{E_{2}}^{-} (α)) : α \in E_{2} \ E_{1}, \end{matrix} \end{matrix}$ and $\begin{matrix} Γ_{B_{1} \cap B_{2}} (α) = {\begin{matrix} Γ_{B_{1}} (α) \cap Γ_{B_{2}} (α) : α \in E_{1} \cap E_{2} \\ Γ_{B_{1}} (α) : α \in E_{1} \ E_{2} \\ Γ_{B_{2}} (α) : α \in E_{2} \ E_{1} . \end{matrix} \end{matrix}$

Remark 1. If the condition E₁ ∩ E₂ ≠ φ is removed then one can see there would be no difference in the definition of union and intersection of bipolar fuzzy parameterized soft sets.

Remark 2. If in above definitions, only α ∈ E₁ ∩ E₂ are taken into account, then this leads to the definition of restricted union/ intersection as stated below. $ω_{E_{1}} ⊔ ω_{E_{2}} = {(α, Max {η_{E_{1}}^{+} (α), η_{E_{2}}^{+} (α)}, Max {η_{E_{1}}^{-} (α), η_{E_{2}}^{-} (α)}, Γ_{B_{1}} (α) \cup Γ_{B_{2}} (α))}$ $ω_{E_{1}} ⊓ ω_{E_{2}} = {(α, Min {η_{E_{1}}^{+} (α), η_{E_{2}}^{+} (α)}, Min {η_{E_{1}}^{-} (α), η_{E_{2}}^{-} (α)}, Γ_{B_{1}} (α) \cap Γ_{B_{2}} (α))}$

Proposition 3.6. Let ω_{E
₁}, ω_{E
₂} and ω_{E
₃} ∈ Ω (χ). Then

i) ω_{E
₁} ∪ ω_{E
₂} = ω_{E
₂} ∪ ω_{E
₁}

ii) (ω_{E
₁} ∪ ω_{E
₂}) ∪ ω_{E
₃} = ω_{E
₁} ∪ (ω_{E
₂} ∪ ω_{E
₃})

iii) ω_{E
₁} ∩ ω_{E
₂} = ω_{E
₂} ∩ ω_{E
₁}

iv) (ω_{E
₁} ∩ ω_{E
₂}) ∩ ω_{E
₃} = ω_{E
₁} ∩ (ω_{E
₂} ∩ ω_{E
₃})

Proposition 3.7. Let ω_{E
₁} and ω_{E
₁} ∈ Ω (χ). Then for the union, intersection and their restrictions, De-Morgan’s laws hold, i.e.

i) $(ω_{E_{1}} \cup ω_{E_{2}})^{c} = ω_{E_{1}}^{c} \cap ω_{E_{2}}^{c}$

ii) $(ω_{E_{1}} \cap ω_{E_{2}})^{c} = ω_{E_{1}}^{c} \cup ω_{E_{2}}^{c}$

iii) $(ω_{E_{1}} ⊔ ω_{E_{2}})^{c} = ω_{E_{1}}^{c} ⊓ ω_{E_{2}}^{c}$

iv) $(ω_{E_{1}} ⊓ ω_{E_{2}})^{c} = ω_{E_{1}}^{c} ⊔ ω_{E_{2}}^{c}$

Example 5. Let χ = {x₁, x₂, x₃, x₄, x₅},

E = {α₁, α₂, α₃, α₄}, E₁ = {α₁, α₂}, E₂ = {α₁, α₃},

ω_{E
₁} = {(α₁, 0.6, 0.4, {x₂, x₃}) , (α₂, 0.5, 0.4, {x₁})},

ω_{E
₂} = {(α₁, 0.5, 0.3, {x₂, x₃, x₄}) , (α₃, 0.6 , 0.4 , {x₁, x₅})}.

Then

ω_{E
₁} ∪ ω_{E
₂} = {(α₁, 0.6, 0.4, {x₂, x₃, x₄}) , (α₂, 0.5, 0.4, {x₁}) , (α₃, 0.6, 0.4, {x₁, x₅})},

ω_{E
₁} ∩ ω_{E
₂} = {(α₁, 0.5, 0.3, {x₂, x₃}) , (α₂, 0.5, 0.4, {x₁}) , (α₃, 0.6, 0.4, {x₁, x₅})},

ω_{E
₁} ⊔ ω_{E
₂} = {(α₁ , 0.6, 0.4, {x₂, x₃, x₄})},

ω_{E
₁} ⊓ ω_{E
₂} = {(α₁, 0.5, 0.3 , {x₂, x₃})},

$ω_{E_{1}^{c}} = {(α_{1}, 0.4, 0.6, {x_{1}, x_{4}, x_{5}}), (α_{2}, 0.5, 0.6, {x_{2}, x_{3}, x_{4}, x_{5}})}$ ,

$ω_{E_{2}^{c}} = {(α_{1}, 0.5, 0.7, {x_{1}, x_{5}}), (α_{3}, 0.4, 0.6, {x_{2}, x_{3}, x_{4}})}$ ,

(ω_{E
₁} ⊔ ω_{E
₂}) ^c = {(α₁, 0.6, 0.4, {x₁, x₅})},

$ω_{E_{1}^{c}} ⊓ ω_{E_{2}^{c}} = {(α_{1}, 0.6, 0.4, {x_{1}, x_{5}})}$ ,

$\Rightarrow (ω_{E_{1}} ⊔ ω_{E_{2}})^{c} = ω_{E_{1}^{c}} ⊓ ω_{E_{2}^{c}}$ ,

Similarly,

(ω_{E
₁} ⊓ ω_{E
₂}) ^c = {(α₁, 0.5, 0.7, {x₁, x₄, x₅})}, and

$ω_{E_{1}^{c}} ⊔ ω_{E_{2}^{c}} = {(α_{1}, 0.5, 0.7, {x_{1}, x_{4}, x_{5}})}$ ,

$\Rightarrow (ω_{E_{1}} ⊓ ω_{E_{2}})^{c} = ω_{E_{1}^{c}} ⊔ ω_{E_{2}^{c}}$

Definition 3.8. Let ω_{E
₁}, ω_{E
₂} ∈ Ω (χ), then OR-Sum of ω_{E
₁} & ω_{E
₂} is

$ω_{E_{1}} \lor_{\oplus} ω_{E_{2}} = {(α, η_{E_{1}}^{+} (α) + η_{E_{2}}^{+} (α) - η_{E_{1}}^{+} (α) . η_{E_{2}}^{+} (α), η_{E_{1}}^{-} (α) . η_{E_{2}}^{-} (α), Γ_{B_{1} \cup B_{2}} (α))}$ .

Definition 3.9. Let ω_{E
₁}, ω_{E
₂} ∈ Ω (χ), then AND-Sum of ω_{E
₁} & ω_{E
₂} is

$ω_{E_{1}} \land_{\oplus} ω_{E_{2}} = {(α, η_{E_{1}}^{+} (α) + η_{E_{2}}^{+} (α) - η_{E_{1}}^{+} (α) . η_{E_{2}}^{+} (α), η_{E_{1}}^{-} (α) . η_{E_{2}}^{-} (α), Γ_{B_{1} \cap B_{2}} (α))}$ .

Proposition 3.10. Let ω_{E
₁}, ω_{E
₂}, ω_{E
₃} ∈ Ω (χ), then

i) ω_{E
₁} ∨ _⊕ω_{E
₂} = ω_{E
₂} ∨ _⊕ω_{E
₁}

ii) ω_{E
₁} ∧ _⊕ω_{E
₂} = ω_{E
₂} ∧ _⊕ω_{E
₁}

iii) (ω_{E
₁} ∨ _⊕ω_{E
₂}) ∨ _⊕ω_{E
₃} = ω_{E
₁} ∨ _⊕ (ω_{E
₂} ∨ _⊕ω_{E
₃})

iv) (ω_{E
₁} ∧ _⊕ω_{E
₂}) ∧ _⊕ω_{E
₃} = ω_{E
₁} ∧ _⊕ (ω_{E
₂} ∧ _⊕ω_{E
₃}).

Definition 3.11. Let ω_{E
₁}, ω_{E
₂} ∈ Ω (χ), then OR-Product of ω_{E
₁} & ω_{E
₂} is

$ω_{E_{1}} \lor_{\otimes} ω_{E_{2}} = {(α, η_{E_{1}}^{-} (α) + η_{E_{2}}^{-} (α) - η_{E_{1}}^{-} (α) . η_{E_{2}}^{-} (α), η_{E_{1}}^{+} (α) . η_{E_{2}}^{+} (α), Γ_{B_{1} \cup B_{2}} (α))}$ .

Definition 3.12. Let ω_{E
₁}, ω_{E
₂} ∈ Ω (χ), then AND-Product of ω_{E
₁} & ω_{E
₂} is

$ω_{E_{1}} \land_{\otimes} ω_{E_{2}} = {(α, η_{E_{1}}^{-} (α) + η_{E_{2}}^{-} (α) - η_{E_{1}}^{-} (α) . η_{E_{2}}^{-} (α), η_{E_{1}}^{+} (α) . η_{E_{2}}^{+} (α), Γ_{B_{1} \cap B_{2}} (α))}$ .

Proposition 3.13. Let ω_{E
₁}, ω_{E
₂}, ω_{E
₃} ∈ Ω (χ), then

i) ω_{E
₁} ∨ _⊗ω_{E
₂} = ω_{E
₂} ∨ _⊗ω_{E
₁}

ii) ω_{E
₁} ∧ _⊗ω_{E
₂} = ω_{E
₂} ∧ _⊗ω_{E
₁}

iii) (ω_{E
₁} ∨ _⊗ω_{E
₂}) ∨ _⊗ω_{E
₃} = ω_{E
₁} ∨ _⊗ (ω_{E
₂} ∨ _⊗ω_{E
₃})

iv) (ω_{E
₁} ∧ _⊗ω_{E
₂}) ∧ _⊗ω_{E
₃} = ω_{E
₁} ∧ _⊗ (ω_{E
₂} ∧ _⊗ω_{E
₃}).

4 Application of BFPSS in decision making

In this section we propose a detailed algorithm used for decision making based on bipolar fuzzy parameterized soft sets and state a practical example that illustrates the algorithm.

4.1 Decision making based on BFPSS

Let χ = {x_i : 1 ≤ i ≤ n} and E₁ ⊆ E be given by E₁ = {α_j : 1 ≤ j ≤ m}. The following steps are carried.

Step 1: Express data in the form of a BFPSS on χ, ω_{E
₁} ∈ Ω.

Step 2: Present numerical values (η⁺, η^-) corresponding to each x_i and α_j in tabular or matrix form.

Step 3: Calculate $(η_{rel}^{+}, η_{rel}^{-})$ for each of x_i such that $η_{rel}^{+} = \frac{\sum_{j = 1}^{m} η_{α_{j}}^{+}}{m} and η_{rel}^{-} = \frac{\sum_{j = 1}^{m} η_{α_{j}}^{-}}{m}$

Step 4: For each x_i calculate the score function $δ_{x_{i}} = η_{rel}^{+} (x_{i}) (η_{rel}^{+} (x_{i}) - η_{rel}^{-} (x_{i}))$

Step 5: Choose the item x_i with the optimal score δ (x_i).

Figure 1 depicting the whole process carried out through Step 1 to Step 5

Fig. 1

Process Flow Diagram.

Example 6. Suppose a pharmaceutical company makes three different drugs χ = {x_i : 1 ≤ i ≤ 3} to cure the same disease. The pharmaceutical composition of each of these drugs x_i uses different ingredients E = {α_j : 1 ≤ j ≤ 4}, preserving their effect, such that the drug x₁ uses α₁ and α₂. Drug x₂ uses α₁, α₂, α₃ and α₄. Drug x₃ uses α₂ and α₄. These α_j’s are effective to treat certain medical problem but at the same time have some harmful side effects too. A medical practitioner has to choose the best drug based on the effectiveness and harmfulness. Let η⁺ (α_j) ∈ [0, 1] and η^- (α_j) ∈ [0, 1] represent the degree of effectiveness and harmfulness respectively, of each of α_j. We can use the bipolar fuzzy parameterized soft set approach to make suitable and reliable decision in such problems as stated below.

Step 1: Data as BFPSS is given by $\begin{matrix} ω_{E} = & {(α_{1}, 0.6, 0.3, {x_{1}, x_{2}}), \\ (α_{2}, 0.5, 0.4, {x_{1} x_{2} x_{3}}), \\ (α_{3}, 0.7, 0.2, {x_{2}}), \\ (α_{4}, 0.6, 0.1, {x_{2}, x_{3}})} \end{matrix}$ (4)

Step 2: The complete data along with the degree of effectiveness and harmfulness of ingredients is expressed in Table 1.

Step 3: For x₁, $η_{rel}^{+} (x_{1}) = \frac{0.6 + 0.5 + 0 + 0}{4} = 0.275$ $η_{rel}^{-} (x_{1}) = \frac{0.3 + 0.4 + 0 + 0}{4} = 0.175$ In a similar way, we calculate $η_{rel}^{+}$ and $η_{rel}^{+}$ for x₂ and x₃ (see second last row of Table 1.).

Step 4: For each x_i the calculated values of the score function (shown in the last row of table 1) are δ (x₁) =0.0275, δ (x₂) =0.21 and δ (x₃) =0.04125.

Step 5: Since the highest value of the score function occurs corresponding to the item x₂, therefore drug x₂ is selected for optimal effectiveness and least harmfulness.

Table 1

Tabular Presentation of BFPSS

	x ₁	x ₂	x ₃
α ₁	(0.6,0.3)	(0.6,0.3)	(0,0)
α ₂	(0.5,0.4)	(0.5,0.4)	(0.5,0.4)
α ₃	(0,0)	(0.7,0.2)	(0,0)
α ₄	(0,0)	(0.6,0.1)	(0.6,0.1)
$(η_{rel}^{+}, η_{rel}^{-})$	(0.275, 0.175)	(0.6,0.25)	(0.275,0.125)
δ (x_i)	0.0275	0.21	0.04125

4.2 Detailed comparative analysis

In this subsection we discuss in detail the similarities and differences of our proposed method by comparing it with the existing techniques. At first level, we examine why the notion of BFPSS is important to be used in presenting this kind of data (as in Example 6) and what are the drawbacks if some previously studied presentations and algorithms are deployed. At second level, we prove the significance of our algorithm by comparing the outcome with a recent method based on BFPSS.

If we don’t use the concept the bipolar fuzzy parameterized soft sets, then the most relevant possible presentation of data as in Example 6 is in terms of bipolar fuzzy soft sets [39] as follows. $\begin{matrix} {F (x_{1}) = {(α_{1}, 0.6, 0.3), (α_{2}, 0.5, 0.4)}, \\ F (x_{2}) = {(α_{1}, 0.6, 0.3), (α_{2}, 0.5, 0.4), \\ (α_{3}, 0.7, 0.2), (α_{4}, 0.6, 0.1)}, \\ F (x_{3}) = {(α_{2}, 0.5, 0.4), (α_{4}, 0.6, 0.1)}} \end{matrix}$ (5) The concept of bipolar fuzzy soft sets is quite useful, however it fails to handle this kind of data, as discussed in Example 6, for decision making purpose. Since both the positive and negative information is available for α_i, for 1 ≤ i ≤ 4, rather the drugs x_j, for 1 ≤ j ≤ 3 (see Eq. (5)), the obvious drawback of this presentation is that instead of drugs, it will guide to the best ingredient. The decision making algorithm based on bipolar fuzzy soft sets [39] involves time consuming calculations as both the positive and negative memberships are treated separately. However these lengthy calculations end on a misleading situation, without drawing any concrete conclusion about the optimal choice of the drugs. Therefore it necessities to present the available information in terms of BFPSS.

In the literature, there is only one algorithm regarding decision making by BFPSS. Now we solve the same problem by applying the technique which is based on BFPSS. In this regard, we use the mid-level soft set, constructed by choosing the average values of η₊ and η_- from Eq. (4), as the threshold values (s_mid, t_mid) given by $\begin{matrix} (s_{mid}, t_{mid}) = & ((0.6 + 0.5 + 0.7 + 0.6) / \\ 4, (0.3 + 0.4 + 0.2 + 0.1) / 4) \\ = & (0.6, 0.25) \end{matrix}$

For the corresponding mid-level soft set, we assign 1 to each x₁, having the values (η⁺, η^-) greater than or equal to the threshold values (s_mid, t_mid) and 0 to the rest, as shown in Table 2. Calculate c_i by adding the entries corresponding to each x_i, i.e. c₁ = 1, c₂ = 1 and c₃ = 0. Now x_i with the optimal value of c_i is chosen as the best outcome. We therefore choose both x₁ and x₂ as the drugs with highest level of effectiveness and the least level of harmfulness.

Table 2

Parameter reduction method on BFPSS

	x ₁	x ₂	x ₃
α ₁	1	1	0
α ₂	0	0	0
α ₃	0	0	0
α ₄	0	0	0
c _i	1	1	0

If we compare both of the above techniques, we may clearly observe that in parameter reduction method, individual values of η⁺ (x_i) and η^- (x_i) are absolutely ignored and all the values lying above the threshold value are teated same by assigning 1 to all of them. Similarly, all the values lying below the threshold value are equally assigned 0. This causes confusion and might lead to a misperceiving conclusion. We may try other threshold values of max-level or min-level as well, again we come across the same confusing outcome. However, in our proposed method, the precision and accuracy can be witnessed in the decision at a very low computational cost.

5 Conclusion

Fuzzy sets and their applications have made a substantial contribution is almost all practical areas. Recently bipolar fuzzy sets and soft sets appeared as emerging tools to engender improved outcomes in theory of fuzzy sets. However, there are several real life situations (such as the one discussed in Example 6), that can’t be covered by the aforementioned notions and tools. This article instigates a novel application of bipolar fuzzy parameterized soft sets (BFPSS) and introduces a more applicable and proficient approach to deal with numerous decision making issues. We study some of the indispensable features and properties of such sets and establish a highly significant algorithm to introduce the application of BFPSS. A practical example of decision making is presented in pharmaceutical environment. The implication visibly exhibits that involvement of BFPSS helps reaching the appropriate decision without laborious calculations and tedious techniques. The proposed algorithm can help in handling complex engineering problems wherever a bipolarity effect is present. The study can be further expanded by deploying the proposed method in association with the existing structures and techniques to reach desirable outcomes in more efficacious applications related to the decision making challenges.

Footnotes

Acknowledgments

The research was supported by the Taif University Researches Supporting Project number (TURSP-2020/16), Taif university, Taif, Saudi Arabia.

References

Zadeh

L.A.

, Fuzzy sets, Information and Control 8(3) (1965), 338–353.

Bellman

R.E.

and Zadeh

L.A.

, Decision-making in a fuzzy environment, Man-agement Science 17(4) (1970), 141–164.

Pawlak

, Rough sets, International Journal of Computer &Information Sciences 11(5) (1982), 341–356.

Atanassov

, Intuitionistic fuzzy sets, central tech, Library, Bulgarian Academy Science, Sofia, Bulgaria, Rep (1697/84) (1983).

Gau

and Buehrer

D.J.

, Vague sets, 245 and Cybernetics 23(2) (1993), 610–614.

Molodtsov

, Soft set theory—first results, Computers& Mathematics with Applications 37(4-5) (1999), 19–31.

Maji

P.K.

, Biswas

and Roy

A.R.

, Fuzzy soft sets, Journal offuzzy mathematics 9(3) (2001), 589–602.

Maji

P.K.

, Biswas

and Roy

A.R.

, Intuitionistic fuzzy soft sets, Journal of Fuzzy Mathematics 9(3) (2001), 677–692.

Çağman

and Enginoğlu

, Soft set theory anduni–int decision making, European Journal of OperationalResearch 207(2) (2010), 848–855.

10.

S.K.

, Biswas

and Roy

A.R.

, An application of intuitionisticfuzzy sets 255 in medical diagnosis, Fuzzy sets and Systems 117(2) (2001), 209–213.

11.

Own

C.-M.

, Switching between type-2 fuzzy sets and intuitionisticfuzzy sets: an application in medical diagnosis, AppliedIntelligence 31(3) (2009), 283.

12.

and Xu

, Intuitionistic fuzzy two-sided matching model andits application to personnel-position matching problems, Journal of the Operational Research Society 71(2) (2020), 312–321.

13.

Roy

A.R.

and Maji

, A fuzzy soft set theoretic approach todecision making problems, Journal of Computational and AppliedMathematics 203(2) (2007), 412–418.

14.

Cagman

, Enginoglu

and Citak

, Fuzzy soft set theory and itsapplications, Iranian Journal of Fuzzy Systems 8(3) (2011), 137–147.

15.

Kong

, Wang

and Wu

, Application of fuzzy soft set indecision making problems based on grey theory, Journal ofComputational and Applied Mathematics 236(6) (2011), 1521–1530.

16.

Xiao

, Xia

, Gong

and Li

, The trapezoidal fuzzy soft setand its application in mcdm, Applied Mathematical Modelling 36(12) (2012), 5844–5855.

17.

Basu

T.M.

, Mahapatra

N.K.

and Mondal

S.K.

, A balanced solution of afuzzy soft set based decision making problem in medical science, Applied Soft Computing 12(10) (2012), 3260–3275.

18.

Ibrar

, Khan

and Abbas

, Fuzzy parameterized bipolarfuzzy soft expert set and its application in decision making, International Journal of Fuzzy Logic and Intelligent Systems 19(4) (2019), 234–241.

19.

Jiang

, Tang

and Chen

, An adjustable approach tointuitionistic fuzzy soft sets based decision making, AppliedMathematical Modelling 35(2) (2011), 824–836.

20.

Zhang

, A rough set approach to intuitionistic fuzzy soft setbased decision making, Applied Mathematical Modelling 36(10) (2012), 4605–4633.

21.

Agarwal

, Biswas

K.K.

and Hanmandlu

, Generalized intuitionisticfuzzy soft sets with applications in decision-making, AppliedSoft Computing 13(8) (2013), 3552–3566.

22.

Deli

and Çağman

, Intuitionistic fuzzy parameterizedsoft set theory and its decision making, Applied SoftComputing 28 (2015), 109–113.

23.

Deli

and Karataş

, Interval valued intuitionistic fuzzyparameterized soft set theory and its decision making, Journalof Intelligent & Fuzzy Systems 30(4) (2016), 2073–2082.

24.

Karaaslan

and Karataş

, Or and and-products ofifp-intuitionistic fuzzy soft sets and their applications indecision making, Journal of Intelligent & Fuzzy Systems 31(3) (2016), 1427–1434.

25.

Muthukumar

and Krishnan

G.S.S.

, A similarity measure ofintuitionistic fuzzy soft sets and its application in medicaldiagnosis, Applied Soft Computing 41 (2016), 148–156.

26.

Zhao

, Ma

and Sun

, A novel decision making approach based onintuitionistic fuzzy soft sets, International Journal ofMachine Learning and Cybernetics 8(4) (2017), 1107–1117.

27.

, Pan

, Yang

and Chen

, A group medical diagnosis modelbased on intuitionistic fuzzy soft sets, Applied SoftComputing 77 (2019), 453–466.

28.

Wang

and Li

, Pythagorean fuzzy interaction power bonferronimean aggregation operators in multiple attribute decision making, Journal of Intelligent Systems 35(1) (2020), 150–183.

29.

Zhang

W.R.

, Bipolar fuzzy sets and relations: a computational framework for cognitive modeling and multiagent decision analysis, in: NAFIPS/IFIS/NASA’94. Proceedings of the First International Joint Conference of The North American Fuzzy Information Processing Society Biannual Conference. The Industrial Fuzzy Control and Intelligence, IEEE, (1994), 305–309.

30.

Dubois

and Prade

, An introduction to bipolar representationsof information and preference, International Journal ofIntelligent Systems 23(8) (2008), 866–877.

31.

Ali

, Son

L.H.

, Deli

and Tien

N.D.

, Bipolar neutrosophic softsets and applications in decision making, Journal ofIntelligent & Fuzzy Systems 33(6) (2017), 4077–4087.

32.

Jana

and Pal

, Application of bipolar intuitionistic fuzzy softsets in decision making problem, (IJFSA) 7(3) (2018), 32–55.

33.

Akram

, Feng

, Borumand Saeid

and Leoreanu-Fotea

, A newmultiple criteria decision-making method based on bipolar fuzzy softgraphs, Iranian Journal of Fuzzy Systems 15(4) (2018), 73–92.

34.

Hashim

, Gulistan

and Smarandache

, Applications ofneutrosophic bipolar fuzzy sets in hope foundation for planning tobuild a children hospital with different types of similaritymeasures, Symmetry 10(8) (2018), 331.

35.

Karaaslan

and Çağman

, Bipolar soft rough sets andtheir applications in decision making, Afrika Matematika 29(5-6) (2018), 823–839.

36.

Alghamdi

, Alshehri

N.O.

and Akram

, Multi-criteriadecision-making methods in bipolar fuzzy environment, International Journal of Fuzzy Systems 20(6) (2018), 2057–2064.

37.

Malik

and Shabir

, Rough fuzzy bipolar soft sets andapplication in decision-making problems, Soft Computing 23(5) (2019), 1603–1614.

38.

Deli

and Karaaslan

, Bipolar fpss-tsheory with applications indecision making, Afrika Mathematika 31 (2020), 493–505.

39.

Abdullah

, Aslam

and Ullah

, Bipolar fuzzy soft sets and itsapplications in decision making problem, Journal of Intelligent& Fuzzy Systems 27(2) (2014), 729–742.