Sigmoid valued fuzzy soft set and its application to haze management

Abstract

The impetus of this paper is to broaden the structure of linguistic soft set (LSS) to a new domain namely sigmoid valued fuzzy soft set (SVFSS). Some operating laws on SVFSS are also provided. Using the complement concept on SVFSS we define maximum rejection. This maximum rejection paves a way for defining a new similarity measure on SVFSS termed as maximum likely ratio (MLR). A new MCGDM algorithm for SVFSS is proposed using MLR. An illustrative example of haze equipment problem on sigmoid valued fuzzy soft set setting is also given. A comparative analysis of our approach with the existing approaches are also presented to justify our work.

Keywords

Sigmoid valued fuzzy soft set maximum rejection maximum likely ratio generalized maximum likely ratio weighted maximum likely ratio

1 Introduction

Multi-expert phenomenon is utilized in group decision making process. Owing to ambiguity and uncertainty, certain decision making processes are difficult to handle. In the real world, the experts’ confusion, limitations and even vague information causes decision makers to struggle in giving exact figures in order to express their views. Responding to the linguistic setting decision problems can be achieved using word computing (CW) [37]. The 2-tuple linguistic fuzzy representation model (2-Tuple LFRM) [32] is a special type of linguistic model used for CW that provides a computation routine that contains linguistic information which includes no loss of knowledge. Xu [34] extended the linguistic term set (LTS) to uncertain LTS and provided specifications.

Numerical scale is the second key factor in decision making problems. Saaty’s 1-9 scale and [0, 1] scale are the two types of scales broadly used in decision making situations. The 2-tuple linguistic representation model (2-tuple LRM) is one such type, the linguistic hierarchy-based model and the numerical scale model [4] was built to pact with linguistic decision-making problems. Dong et al. [5] interrelated the 2-tuple linguistic model with hesistant unbalanced linguistic information. In dealing with fuzzy preference relations, Zhou & Xu [40] developed a new numerical scale for consistency namely asymmetric sigmoid numerical scale (ASNS) in hesistant fuzzy setting. This ASNS meets asymmetry, through requirements of utility, volatility and consistency. Ma & Xu [20] created the hyperbolic scales based on intuitionistic multiplicative relationships of choice and its implementation. Soft sets (SS), pioneered by Molodtsov [23] is free from all the problems prevailed in ([25, 36]).

Fuzzy set is generalized to soft set, fuzzy soft set (FSS) together with their union and intersection ([18, 35]), intuitionistic fuzzy soft set ([10 , 21]), rough soft set (RSS) ([24, 38]), hesistant fuzzy soft set [33], neutrosophic soft set ([2 , 15]), TOPSIS under fuzzy soft set ([6 , 9]) and belief interval-valued soft set [31]. The following papers exhibits about the recent development of soft set theory ([3 , 39]).

Our primary motivation in this paper is to extend the idea of linguistic soft set (LSS) to sigmoid valued fuzzy soft set (SVFSS). It is evident to note that if LTS is provided by the team of experts then the corresponding transformed numerical membership of FSS can be established in the form of SVSS to some suitable scale. This is the main objective of introducing sigmoid valued fuzzy soft set (SVFSS). Further the intention of this paper is to develop the notion of maximum likely ratio (MLR) namely generalized maximum likely ratio on sigmoid valued fuzzy soft set and to provide an application on it. SVFSS integrates the following functions namely sigmoid, linguistic variable and fuzzy soft set theories. It allows decision makers to use linguistic variables to evaluate an object and the linguistic variable converts to fuzzy values using sigmoid function to describe the corresponding degree of support for their decisions. The SVFSS relationship is based on this approach and few operations on this are established.

We summarize our contributions as follows. Initially we provide some definitions and results for the development of SVFSS from LSS in Section 2. The notion of sigmoidal valued fuzzy soft set (SVFSS) is given in Section 3 with suitable examples and theoretical developments. Generalized maximum likely ratio on SVFSS is given in Section 4. A novel MCGDM problem on SVFSS is provided in Section 5 along with algorithms and flow chart. Section 6 gives a comparative analysis of our work with the existing works. A concluding part is presented in Section 7.

2 Preliminaries

2.1 Linguistic fuzzy soft set

Definition 2.1. [28] Let L = {Ω_k|Ω_-t ≤ Ω_k ≤ Ω_t} be a compilation of all the linguistic variables. The following linguistic membership function. $F_{L} : U \to L$ (1) is termed as linguistic fuzzy set defined on U (universe set). It is denoted as LFS (U).

Definition 2.2. [28] A mapping $\bar{F} : A \to \bar{P} (U) .$ (2) is called as uncertain linguistic fuzzy soft set (ULFSS) over U(universe), where $\bar{P} (U)$ be the set of all uncertain linguistic fuzzy subsets of U and A ⊆ E (parameter set of U).

2.2 Generalized sigmoid function

Assume that L^ψ to be a numerical scale, ψ is a strictly increasing function on [- t, t] and G : L → L^ψwith G (Ω_k) = ψ (k) called as numerical scale value of Ω_k.

Definition 2.3. [40] A generalized sigmoid function L^ψ can be defined as: L^{ψ
^{θ
₁
θ
₂}} = {ψ^{θ
₁
θ
₂} (k) |k ∈ [- t, t]}, where $ψ^{θ_{1} θ_{2}} (k) = {\begin{matrix} \frac{e^{θ_{1} k}}{1 + e^{θ_{1} k}}, & k \in [0, t] \\ \frac{e^{θ_{2} k}}{1 + e^{θ_{2} k}}, & k \in [- t, 0] \end{matrix}$ (3) here θ₁ and θ₂ are two parameters with θ₁ ≥ 0 and θ₂ ≥ 0. Also

ψ^{θ
₁
θ
₂} is a generalized sigmoid curve with $ψ^{θ_{1} θ_{2}} (0) = \frac{1}{2}$ .

The negation of ψ^{θ
₁
θ
₂} (k) can be defined as $Neg (ψ^{θ_{1} θ_{2}} (k)) = ψ^{θ_{2} θ_{1}} (- k) = {\begin{matrix} \frac{1}{1 + e^{θ_{2} k}}, & k \in [0, t] \\ \frac{1}{1 + e^{θ_{1} k}}, & k \in [- t, 0] \end{matrix}$ (4) such that ψ^{θ
₁
θ
₂} (k) + Neg (ψ^{θ
₁
θ
₂} (k)) =1.

3 Sigmoid valued fuzzy soft set

It is evident that the consistency in a constant domain includes the numerical scales [0, 1]. This prompted a foundation to fuzzy preference relationship. By characterizing the idea of numerical scale, Dong et al. [4] found that setting various numerical scales produces a 2-tuple linguistic sequence. Therefore, the key drill of the projected 2-tuple patterns is the concept of a scale function, which defines the linguistic erudition and numerical values of one to one mapping between them. In this section, we continue the analysis of the linguistic representation model to fuzzy soft setting and propose sigmoid valued linguistic fuzzy soft setting, as well as some operations on sigmoid valued fuzzy soft set (SVFSS). Also we provide several interesting definitions and results on SVFSS are provided with suitable examples.

Definition 3.1. A mapping F_ψ : U → G (_k) is called as sigmoid valued fuzzy set (SVFS).

Example 3.2. Let U = {mt₁, mt₂, . . . , mt_n} be the universe set and let L^ψ = L⁴ = {S_-4 = T_p (Terriblypoor) , S_-3 = H_p (Highlypoor) , S_-2 = P (Poor) , S_-1 = M_p (Marginallypoor) , S₀ = I (Impartial) , S₁ = M_g (Marginallygood) , S₂ = G (Good) , S₃ = H_g (Highlygood) , S₄ = T_g (Terriblygood)}. Then $G (_{2}) = G = \frac{e^{2}}{1 + e^{2}} = 0.8808$ $G (_{- 4}) = T_{p} = \frac{e^{- 4}}{1 + e^{- 4}} = 0.0179$ Similarly, we can find other values. Consider the linguistic FS $F_{L} = {\frac{{mt}_{1}}{S_{3} (H_{g})}, \frac{{mt}_{2}}{S_{4} (T_{g})}, \frac{{mt}_{3}}{S_{- 3} (H_{p})}, \frac{{mt}_{4}}{S_{0} (I)}, \frac{{mt}_{5}}{S_{- 1} (M_{p})}},$ mt_i ∈ U and S_j ∈ L^ψ Then sigmoid valued FS can be written as $F_{ψ} = {\frac{{mt}_{1}}{0.9526}, \frac{{mt}_{2}}{0.9820}, \frac{{mt}_{3}}{0.0474}, \frac{{mt}_{4}}{0.5000}, \frac{{mt}_{5}}{0.2689}}$

Definition 3.3. A mapping $S^{F_{ψ}} : E \to P_{F_{ψ}} (U)$ (5) is called as SVFSS on U, with P_{F
_ψ} (U) representing the set of all subsets of SVFS on U. It is indicated by (S^{F
_ψ}, E).

Definition 3.4. A mapping ${PS}^{F_{ψ}} : U \to P_{F_{ψ}} (E)$ (6) is called as pseudo SVFSS, where P_{F
_ψ} (E) represents set of all subsets of E. It is indicated by (PS^{F
_ψ}, U).

Example 3.5. Many countries around the World face a major air pollution problem arising from fog and haze. The human health is impacted by the effects of haze emissions. Haze emission is not only harmful to human health but also has adverse atmospheric and climatic impacts. The children are typically affected by headache, cough, mucus and respiratory disease from air pollution. We model this in the form of SVFSS as follows. Let U = {mt₁, mt₂, mt₃, mt₄, mt₅} be the set of five major towns in the World that are affected by the air pollution and let E = {hh₁, hh₂, hh₃, hh₄} denotes the set of health problems arose for the children namely, ”hh₁ = headache”, ”hh₂ = cough”, ”hh₃ = mucus” and ”hh₄ = throat respiratory illness”. The linguistic fuzzy soft set can be written as $(F_{L}, E) = {({hh}_{1}, {\frac{{mt}_{1}}{T_{g}}, \frac{{mt}_{2}}{M_{g}}, \frac{{mt}_{3}}{I}, \frac{{mt}_{4}}{M_{p}}, \frac{{mt}_{5}}{P}}), ({hh}_{2}, {\frac{{mt}_{1}}{T_{g}}, \frac{{mt}_{2}}{M_{g}}, \frac{{mt}_{3}}{H_{p}}, \frac{{mt}_{4}}{I}, \frac{{mt}_{5}}{T_{p}}}), ({hh}_{3}, {\frac{{mt}_{1}}{G}, \frac{{mt}_{2}}{H_{g}}, \frac{{mt}_{3}}{T_{g}}, \frac{{mt}_{4}}{M_{p}}, \frac{{mt}_{5}}{H_{p}}}), ({hh}_{4}, {\frac{{mt}_{1}}{M_{g}}, \frac{{mt}_{2}}{T_{p}}, \frac{{mt}_{3}}{P}, \frac{{mt}_{4}}{T_{g}}, \frac{{mt}_{5}}{G}})} .$ The sigmoid valued fuzzy soft set (S^{F
_ψ}, E) can be written as $\begin{matrix} (S^{F_{ψ}}, E) = \\ {({hh}_{1}, {\frac{{mt}_{1}}{0.9820}, \frac{{mt}_{2}}{0.7311}, \frac{{mt}_{3}}{0.5000}, \frac{{mt}_{4}}{0.2689}, \frac{{mt}_{5}}{0.1192}}), \end{matrix}$ }} $({hh}_{2}, {\frac{{mt}_{1}}{0.9820}, \frac{{mt}_{2}}{0.7311}, \frac{{mt}_{3}}{0.0474}, \frac{{mt}_{4}}{0.5000}, \frac{{mt}_{5}}{0.0180}}),$ $({hh}_{3}, {\frac{{mt}_{1}}{0.8808}, \frac{{mt}_{2}}{0.9526}, \frac{{mt}_{3}}{0.9820}, \frac{{mt}_{4}}{0.2689}, \frac{{mt}_{5}}{0.0474}}),$ $({hh}_{4}, {\frac{{mt}_{1}}{0.7311}, \frac{{mt}_{2}}{0.0180}, \frac{{mt}_{3}}{0.1192}, \frac{{mt}_{4}}{0.9820}, \frac{{mt}_{5}}{0.8808}})}$ In other words $\begin{matrix} S^{F_{ψ}} ({hh}_{1}) = {\frac{{mt}_{1}}{0.9820}, \frac{{mt}_{2}}{0.7311}, \frac{{mt}_{3}}{0.5000}, \frac{{mt}_{4}}{0.2689}, \frac{{mt}_{5}}{0.1192}} \end{matrix}$ $\begin{matrix} S^{F_{ψ}} ({hh}_{2}) = {\frac{{mt}_{1}}{0.9820}, \frac{{mt}_{2}}{0.7311}, \frac{{mt}_{3}}{0.0474}, \frac{{mt}_{4}}{0.5000}, \frac{{mt}_{5}}{0.0180}} \end{matrix}$ and so on. The pseudo sigmoid valued fuzzy soft set (PS^{F
_ψ}, U) is ${PS}^{F_{ψ}} ({mt}_{1}) = {\frac{{hh}_{1}}{0.9820}, \frac{{hh}_{2}}{0.9820}, \frac{{hh}_{3}}{0.8808}, \frac{{hh}_{4}}{0.7311}}$ and so on.

Definition 3.6. Let (S^{F
_ψ}, E) be an SVFSS over U. The complement of (S^{F
_ψ}, E) over U is defined as (S^{F
_ψ}, E) ^C = (S^{F
_ψ}, E^C), where (S^{F
_ψ}, E^C) = (¬ hh_j, {F_ψ (U)} ^C). Here ${F_{ψ} (U)}^{C} = Neg {\frac{{mt}_{i}}{k}} = {\frac{{mt}_{i}}{- k}},$ mt_i ∈ U, Ω_k ∈ L^ψ, and $Ω_{- k} = \frac{1}{1 + e^{k}}$ .

Example 3.7. From Example. 3.5, the complement of (S^{F
_ψ}, E) can be written as (S^{F
_ψ}, E) ^C = (S^{F
_ψ}, E^C) = ${(\neg {hh}_{1}, {\frac{{mt}_{1}}{0.0180}, \frac{{mt}_{2}}{0.2689}, \frac{{mt}_{3}}{0.5000}, \frac{{mt}_{4}}{0.7311}, \frac{{mt}_{5}}{0.8808}}),$ $(\neg {hh}_{2}, {\frac{{mt}_{1}}{0.0180}, \frac{{mt}_{2}}{0.2689}, \frac{{mt}_{3}}{0.9526}, \frac{{mt}_{4}}{0.5000}, \frac{{mt}_{5}}{0.9820}}),$ $(\neg {hh}_{3}, {\frac{{mt}_{1}}{0.1192}, \frac{{mt}_{2}}{0.0474}, \frac{{mt}_{3}}{0.0180}, \frac{{mt}_{4}}{0.7311}, \frac{{mt}_{5}}{0.9526}}),$ $(\neg {hh}_{4}, {\frac{{mt}_{1}}{0.2689}, \frac{{mt}_{2}}{0.9820}, \frac{{mt}_{3}}{0.8808}, \frac{{mt}_{4}}{0.0180}, \frac{{mt}_{5}}{0.1192}})}$

Definition 3.8. The operations of two SVFSSs (S^{F
_{ψ
₁}}, R_A) and (S^{F
_{ψ
₂}}, R_B) are defined below.

Intersection: (S^{F
_{ψ
₁}}, R_A) ⋂ (S^{F
_{ψ
₂}}, R_B) = (S^{F
_{ψ
_M}}, K), where K = R_A ∩ R_B, ∀ R_A, R_B ⊆ E, and hh_j ∈ R_A ∩ R_B $S^{F_{ψ_{M}}} ({hh}_{j}) = S^{F_{ψ_{1}}} ({hh}_{j}) ⋂ S^{F_{ψ_{2}}} ({hh}_{j})$ (7)

Union: (S^{F
_{ψ
₁}}, R_A) ⋃ (S^{F
_{ψ
₂}}, R_B) = (S^{F
_{ψ
_G}}, W), where W = R_A ∪ R_B, ∀ R_A, R_B ⊆ E, and hh_j ∈ R_A ∪ R_B $S^{F_{ψ_{G}}} ({hh}_{j}) = {\begin{matrix} R_{A} ({hh}_{j}), & {hh}_{j} \in R_{A} - R_{B} \\ R_{A} ({hh}_{j}) ⋃ R_{B} ({hh}_{j}), & {hh}_{j} \in R_{A} \cap R_{B} \\ R_{B} ({hh}_{j}), & {hh}_{j} \in R_{B} - R_{A} \end{matrix}$ (8)

Extended intersection: (S^{F
_{ψ
₁}}, R_A) ⊓ (S^{F
_{ψ
₂}}, R_B) = (S^{F
_{ψ
_Q}}, Z), where Z = R_A ∪ R_B, ∀ R_A, R_B ⊆ E, and hh_j ∈ R_A ∪ R_B $S^{F_{ψ_{Q}}} ({hh}_{j}) = {\begin{matrix} R_{A} ({hh}_{j}), & {hh}_{j} \in R_{A} - R_{B} \\ R_{B} ({hh}_{j}), & {hh}_{j} \in R_{B} - R_{A} \\ R_{A} ({hh}_{j}) ⋂ R_{B} ({hh}_{j}), & {hh}_{j} \in R_{A} \cap R_{B} \end{matrix}$ (9)

Example 3.9. Let (S^{F
_{ψ
₁}}, R_A) and (S^{F
_{ψ
₂}}, R_B) are two SVFSSs over U. Let U = {mt₁, mt₂, mt₃, mt₄, mt₅}, R_A = {hh₁, hh₃, hh₄} and R_B = {hh₁, hh₂, hh₃}. Then (S^{F
_ψ}, R_A) $= {({hh}_{1}, {\frac{{mt}_{1}}{0.9820}, \frac{{mt}_{2}}{0.7311}, \frac{{mt}_{3}}{0.5000}, \frac{{mt}_{4}}{0.2689}, \frac{{mt}_{5}}{0.1192}}),$ $({hh}_{3}, {\frac{{mt}_{1}}{0.8808}, \frac{{mt}_{2}}{0.9526}, \frac{{mt}_{3}}{0.9820}, \frac{{mt}_{4}}{0.2689}, \frac{{mt}_{5}}{0.0474}}),$ $({hh}_{4}, {\frac{{mt}_{1}}{0.7311}, \frac{{mt}_{2}}{0.0180}, \frac{{mt}_{3}}{0.1192}, \frac{{mt}_{4}}{0.9820}, \frac{{mt}_{5}}{0.8808}})}$ and (S^{F
_ψ}, R_B) $= {({hh}_{1}, {\frac{{mt}_{1}}{0.9526}, \frac{{mt}_{2}}{0.8808}, \frac{{mt}_{3}}{0.5000}, \frac{{mt}_{4}}{0.1192}, \frac{{mt}_{5}}{0.2689}}),$ $({hh}_{2}, {\frac{{mt}_{1}}{0.9820}, \frac{{mt}_{2}}{0.5000}, \frac{{mt}_{3}}{0.1192}, \frac{{mt}_{4}}{0.7311}, \frac{{mt}_{5}}{0.0474}}),$ $({hh}_{3}, {\frac{{mt}_{1}}{0.5000}, \frac{{mt}_{2}}{0.9820}, \frac{{mt}_{3}}{0.9820}, \frac{{mt}_{4}}{0.1192}, \frac{{mt}_{5}}{0.0474}})}$

Intersection: (S^{F
_{ψ
_M}}, K) $= {({hh}_{1}, {\frac{{mt}_{1}}{0.9526}, \frac{{mt}_{2}}{0.7311}, \frac{{mt}_{3}}{0.5000}, \frac{{mt}_{4}}{0.1192}, \frac{{mt}_{5}}{0.1192}}),$ $({hh}_{3}, {\frac{{mt}_{1}}{0.5000}, \frac{{mt}_{2}}{0.9526}, \frac{{mt}_{3}}{0.9820}, \frac{{mt}_{4}}{0.1192}, \frac{{mt}_{5}}{0.0474}})}$

Union: (S^{F
_{ψ
_M}}, W) $= {({hh}_{1}, {\frac{{mt}_{1}}{0.9820}, \frac{{mt}_{2}}{0.8808}, \frac{{mt}_{3}}{0.5000}, \frac{{mt}_{4}}{0.2689}, \frac{{mt}_{5}}{0.2689}}),$ $({hh}_{2}, {\frac{{mt}_{1}}{0.9820}, \frac{{mt}_{2}}{0.5000}, \frac{{mt}_{3}}{0.1192}, \frac{{mt}_{4}}{0.7311}, \frac{{mt}_{5}}{0.0474}}),$ $({hh}_{3}, {\frac{{mt}_{1}}{0.8808}, \frac{{mt}_{2}}{0.9820}, \frac{{mt}_{3}}{0.9820}, \frac{{mt}_{4}}{0.2689}, \frac{{mt}_{5}}{0.0474}}),$ $({hh}_{4}, {\frac{{mt}_{1}}{0.7311}, \frac{{mt}_{2}}{0.0180}, \frac{{mt}_{3}}{0.1192}, \frac{{mt}_{4}}{0.9820}, \frac{{mt}_{5}}{0.8808}})}$

Extended intersection: (S^{F
_{ψ
_Q}}, Z) $= {({hh}_{1}, {\frac{{mt}_{1}}{0.9526}, \frac{{mt}_{2}}{0.7311}, \frac{{mt}_{3}}{0.5000}, \frac{{mt}_{4}}{0.1192}, \frac{{mt}_{5}}{0.1192}}),$ $({hh}_{2}, {\frac{{mt}_{1}}{0.9820}, \frac{{mt}_{2}}{0.5000}, \frac{{mt}_{3}}{0.1192}, \frac{{mt}_{4}}{0.7311}, \frac{{mt}_{5}}{0.0474}}),$ $({hh}_{3}, {\frac{{mt}_{1}}{0.5000}, \frac{{mt}_{2}}{0.9526}, \frac{{mt}_{3}}{0.9820}, \frac{{mt}_{4}}{0.1192}, \frac{{mt}_{5}}{0.0474}}),$ $({hh}_{4}, {\frac{{mt}_{1}}{0.7311}, \frac{{mt}_{2}}{0.0180}, \frac{{mt}_{3}}{0.1192}, \frac{{mt}_{4}}{0.9820}, \frac{{mt}_{5}}{0.8808}})}$

Theorem 3.10. Let (S^{F
_{ψ
₁}}, R_A) and (S^{F
_{ψ
₂}}, R_B) be two SVFSSs, then

((S^{F
_{ψ
₁}}, R_A) ⋂ (S^{F
_{ψ
₂}}, R_B)) ^C ≠ (S^{F
_{ψ
₁}}, R_A) ^C ⋃ (S^{F
_{ψ
₂}}, R_B) ^C.

((S^{F
_{ψ
₁}}, R_A) ⋃ (S^{F
_{ψ
₂}}, R_B)) ^C ≠ (S^{F
_{ψ
₁}}, R_A) ^C ⋂ (S^{F
_{ψ
₂}}, R_B) ^C.

Proof: Straight forward.

Example 3.11. To validate the above theorem we use same arguments as in Example 3.9.

((S^{F
_{ψ
₁}}, R_A) ⋂ (S^{F
_{ψ
₂}}, R_B)) ^C =

{(\neg {hh}_{1}, {\frac{{mt}_{1}}{0.0474}, \frac{{mt}_{2}}{0.2689}, \frac{{mt}_{3}}{0.5000}, \frac{{mt}_{4}}{0.8808}, \frac{{mt}_{5}}{0.8808}}),

(\neg {hh}_{3}, {\frac{{mt}_{1}}{0.5000}, \frac{{mt}_{2}}{0.0474}, \frac{{mt}_{3}}{0.0180}, \frac{{mt}_{4}}{0.8808}, \frac{{mt}_{5}}{0.9526}})}

(S^{F
_{ψ
₁}}, R_A) ^C =

{(\neg {hh}_{1}, {\frac{{mt}_{1}}{0.0180}, \frac{{mt}_{2}}{0.2689}, \frac{{mt}_{3}}{0.5000}, \frac{{mt}_{4}}{0.7311}, \frac{{mt}_{5}}{0.8808}}),

(\neg {hh}_{3}, {\frac{{mt}_{1}}{0.1192}, \frac{{mt}_{2}}{0.0474}, \frac{{mt}_{3}}{0.0180}, \frac{{mt}_{4}}{0.7311}, \frac{{mt}_{5}}{0.9526}}),

(\neg {hh}_{4}, {\frac{{mt}_{1}}{0.2689}, \frac{{mt}_{2}}{0.9820}, \frac{{mt}_{3}}{0.8808}, \frac{{mt}_{4}}{0.0180}, \frac{{mt}_{5}}{0.1192}})}

(S^{F
_ψ}, R_B) ^C =

{(\neg {hh}_{1}, {\frac{{mt}_{1}}{0.0474}, \frac{{mt}_{2}}{0.1192}, \frac{{mt}_{3}}{0.5000}, \frac{{mt}_{4}}{0.8808}, \frac{{mt}_{5}}{0.7311}}),

(\neg {hh}_{2}, {\frac{{mt}_{1}}{0.0180}, \frac{{mt}_{2}}{0.5000}, \frac{{mt}_{3}}{0.8808}, \frac{{mt}_{4}}{0.2689}, \frac{{mt}_{5}}{0.9526}}),

(\neg {hh}_{3}, {\frac{{mt}_{1}}{0.5000}, \frac{{mt}_{2}}{0.0180}, \frac{{mt}_{3}}{0.0180}, \frac{{mt}_{4}}{0.8808}, \frac{{mt}_{5}}{0.9526}})}

and (S^{F
_{ψ
₁}}, R_A) ^C ⋃ (S^{F
_{ψ
₂}}, R_B) ^C =

{(\neg {hh}_{1}, {\frac{{mt}_{1}}{0.0474}, \frac{{mt}_{2}}{0.2689}, \frac{{mt}_{3}}{0.5000}, \frac{{mt}_{4}}{0.8808}, \frac{{mt}_{5}}{0.8808}}),

(\neg {hh}_{2}, {\frac{{mt}_{1}}{0.0180}, \frac{{mt}_{2}}{0.5000}, \frac{{mt}_{3}}{0.8808}, \frac{{mt}_{4}}{0.2689}, \frac{{mt}_{5}}{0.9526}}),

(\neg {hh}_{3}, {\frac{{mt}_{1}}{0.5000}, \frac{{mt}_{2}}{0.0474}, \frac{{mt}_{3}}{0.0180}, \frac{{mt}_{4}}{0.8808}, \frac{{mt}_{5}}{0.9526}}),

(\neg {hh}_{4}, {\frac{{mt}_{1}}{0.2689}, \frac{{mt}_{2}}{0.9820}, \frac{{mt}_{3}}{0.8808}, \frac{{mt}_{4}}{0.0180}, \frac{{mt}_{5}}{0.1192}})}

Therefore, ((S^{F
_{ψ
₁}}, R_A) ⋂ (S^{F
_{ψ
₂}}, R_B)) ^C ≠ (S^{F
_{ψ
₁}}, R_A) ^C ⋃ (S^{F
_{ψ
₂}}, R_B) ^C. Similarly to prove, ((S^{F
_{ψ
₁}}, R_A) ⋃ (S^{F
_{ψ
₂}}, R_B)) ^C ≠ (S^{F
_{ψ
₁}}, R_A) ^C ⋂ (S^{F
_{ψ
₂}}, R_B) ^C.

Theorem 3.12. Let (S^{F
_{ψ
₁}}, R_A) and (S^{F
_{ψ
₂}}, R_B) be two SVFSSs, then

((S^{F
_{ψ
₁}}, R_A) ⊓ (S^{F
_{ψ
₂}}, R_B)) ^C = (S^{F
_{ψ
₁}}, R_A) ^C ⋃ (S^{F
_{ψ
₂}}, R_B) ^C.

((S^{F
_{ψ
₁}}, R_A) ⋃ (S^{F
_{ψ
₂}}, R_B)) ^C = (S^{F
_{ψ
₁}}, R_A) ^C ⊓ (S^{F
_{ψ
₂}}, R_B) ^C.

Proof: Straight forward.

Definition 3.13. The restricted intersection and restricted union of two SVFSSs (S^{F
_{ψ
₁}}, R_A) and (S^{F
_{ψ
₂}}, R_B) are defined below.

Restriced intersection: (S^{F
_{ψ
₁}}, R_A) ⋒ (S^{F
_{ψ
₂}}, R_B) = (S^{F
_{ψ
_X}}, H), where H = R_A ⋂ R_B, ∀ R_A, R_B ⊆ E, and hh_j ∈ R_A ⋂ R_B ≠ ϕ. $S^{F_{ψ_{H}}} ({hh}_{j}) = S^{F_{ψ_{1}}} ({hh}_{j}) ⋂ S^{F_{ψ_{2}}} ({hh}_{j}) \neq ϕ$ (10)

Restriced union: (S^{F
_{ψ
₁}}, R_A) ⋓ (S^{F
_{ψ
₂}}, R_B) = (S^{F
_{ψ
_Y}}, H), where H = R_A ⋂ R_B, ∀ R_A, R_B ⊆ E, and hh_j ∈ R_A ⋂ R_B ≠ ϕ. $S^{F_{ψ_{J}}} ({hh}_{j}) = S^{F_{ψ_{1}}} ({hh}_{j}) ⋃ S^{F_{ψ_{2}}} ({hh}_{j}) \neq ϕ$ (11)

Proposition 3.14.

(S^{F
_{ψ
₁}}, R_A) ⋒ (S^{F
_{ψ
₂}}, R_B) = (S^{F
_{ψ
₁}}, R_B) ⋒ (S^{F
_{ψ
₂}}, R_A).

(S^{F
_{ψ
₁}}, R_A) ⋓ (S^{F
_{ψ
₂}}, R_B) = (S^{F
_{ψ
₁}}, R_B) ⋓ (S^{F
_{ψ
₂}}, R_A).

4 Generalized maximum likely ratio on SVFSS

Similarity and distance measurements are common tools widely used to quantify the degrees of divergence and closeness of the various arguments. Majumdar & Samanta [22] developed the concept of matrix representation based distance and similarity measurements of soft sets. The definition of similarity used by them is based on some calculation, which is found to be void. Here we define a new measure of similarity based on complement method. First we define the maximum rejection value (complement) and then we define a new measure of similarity using the definition of maximum rejection values, called maximum likely ratio on sigmoid valued fuzzy soft sets.

Definition 4.1. Let (S^{F
_ψ}, E) be an SVFSS over U. The maximum rejection of mt_i is M_R (mt_i) = (1 - S^{F
_ψ} (hh_j)) (mt_i) , mt_i ∈ U and hh_j ∈ E.

Example 4.2. From Example 3.5. The maximum rejection of mt₁ is M_R (mt₁) = (1 - S^{F
_ψ} (hh₁)) (mt₁). That is $M_{R} ({mt}_{1}) = {\frac{{hh}_{1}}{0.0180}, \frac{{hh}_{2}}{0.0180}, \frac{{hh}_{3}}{0.1192}, \frac{{hh}_{4}}{0.2689}}$

Definition 4.3. Let (S^{F
_{ψ
₁}}, R_A) and (S^{F
_{ψ
₂}}, R_B) are two SVFSSs over U and let M_{R
₁} (mt_i) and M_{R
₂} (mt_i) be the two maximum rejection values of sigmoid valued fuzzy soft sets (S^{F
_{ψ
₁}}, R_A) and (S^{F
_{ψ
₂}}, R_B) respectively. Then we have the following.

0 ≤ M_R
₁ (mt_i) ≤1.

M_R
₁ (mt_i) = M_R
₂ (mt_i), then (S^{F
_{ψ
₁}} (hh_j)) = (S^{F
_{ψ
₂}} (hh_j)).

M_R
₁ (mt_i) ≤ M_R
₂ (mt_i), then (S^{F
_{ψ
₁}} (hh_j)) ≥ (S^{F
_{ψ
₂}} (hh_j))

Definition 4.4. Let (S^{F
_{ψ
₁}}, R_A) and (S^{F
_{ψ
₂}}, R_B) are two SVFSSs over U. Let M_{R
₁} (mt_i) and M_{R
₂} (mt_i) be the two maximum rejection values of sigmoid valued fuzzy soft sets (S^{F
_{ψ
₁}}, R_A) and (S^{F
_{ψ
₂}}, R_B) respectively. Then the maximum likely ratio (similarity) of M_{R
₁} (mt_i) (hh_j) and M_{R
₂} (mt_i) (hh_j) is defined as MLR {M_{R
₁} (mt_i) , M_{R
₂} (mt_i)} (hh_j) $= \frac{1}{1 + {\frac{1}{2} (M_{R_{1}} ({mt}_{i}) + M_{R_{2}} ({mt}_{i})) ({hh}_{j})}^{2}}$ (12) where mt_i ∈ U and hh_j ∈ E.

Definition 4.5. Let (S^{F
_{ψ
₁}}, R_A) and (S^{F
_{ψ
₂}}, R_B) are two SVFSSs over U. Let M_{R
₁} (mt_i) and M_{R
₂} (mt_i) be the two maximum rejection values of sigmoid valued fuzzy soft sets (S^{F
_{ψ
₁}}, R_A) and (S^{F
_{ψ
₂}}, R_B) respectively. Then the maximum dislikely ratio (distance) of M_{R
₁} (mt_i) (hh_j) and M_{R
₂} (mt_i) (hh_j) is defined as MDLR {M_{R
₁} (mt_i) , M_{R
₂} (mt_i)} (hh_j) $= 1 - {\frac{1}{1 + {\frac{1}{2} (M_{R_{1}} ({mt}_{i}) + M_{R_{2}} ({mt}_{i})) ({hh}_{j})}^{2}}}$ (13) where mt_i ∈ U and hh_j ∈ E.

Example 4.6. Let $(F_{L}, R_{A}) = [\begin{matrix} M_{g} & H_{g} & T_{g} & M_{g} & M_{p} & G & G \\ T_{p} & M_{p} & M_{p} & G & M_{g} & M_{g} & T_{p} \\ T_{g} & G & M_{g} & H_{p} & T_{p} & G & M_{g} \\ H_{p} & P & G & H_{p} & T_{g} & G & H_{p} \\ T_{g} & P & H_{p} & T_{g} & M_{g} & H_{g} & T_{g} \end{matrix}]$ and $(F_{L}, R_{B}) = [\begin{matrix} T_{g} & M_{g} & G & M_{g} & P & M_{g} & P \\ H_{p} & M_{g} & H_{g} & P & M_{p} & H_{g} & M_{p} \\ G & T_{g} & M_{g} & M_{p} & T_{g} & P & T_{p} \\ M_{g} & M_{p} & G & T_{g} & H_{p} & P & G \\ T_{g} & M_{p} & G & H_{p} & G & T_{g} & P \end{matrix}],$ be two linguistic fuzzy soft sets. Then the maximum rejection of MR (R_A) and MR (R_B) are $\begin{matrix} MR (R_{A}) = \\ [\begin{matrix} 0.2689 & 0.0474 & 0.0180 & 0.2689 & 0.7311 & 0.1192 & 0.1192 \\ 0.9820 & 0.7311 & 0.7311 & 0.1192 & 0.2689 & 0.2689 & 0.9820 \\ 0.0180 & 0.1192 & 0.2689 & 0.9526 & 0.9820 & 0.1192 & 0.2689 \\ 0.9526 & 0.8808 & 0.1192 & 0.9526 & 0.0180 & 0.1192 & 0.9526 \\ 0.0180 & 0.8808 & 0.9526 & 0.0180 & 0.2689 & 0.0474 & 0.0180 \end{matrix}] \end{matrix}$ and $\begin{matrix} MR (R_{B}) = \\ [\begin{matrix} 0.0180 & 0.2689 & 0.1192 & 0.2689 & 0.8808 & 0.2689 & 0.8808 \\ 0.9526 & 0.2689 & 0.0474 & 0.8808 & 0.7311 & 0.0474 & 0.7311 \\ 0.1192 & 0.0180 & 0.2689 & 0.7311 & 0.0180 & 0.8808 & 0.9820 \\ 0.2689 & 0.7311 & 0.1192 & 0.0180 & 0.9526 & 0.8808 & 0.1192 \\ 0.0180 & 0.7311 & 0.1192 & 0.9526 & 0.1192 & 0.0180 & 0.8808 \end{matrix}] \end{matrix}$ The maximum likely ratio (similarity) of MR (R_A) and MR (R_B) is calculted as $\begin{matrix} MLR {MR (R_{A}), MR (R_{B})} = \\ [\begin{matrix} 0.9798 & 0.9756 & 0.9953 & 0.9326 & 0.6062 & 0.9637 & 0.8000 \\ 0.5166 & 0.8000 & 0.8684 & 0.8000 & 0.8000 & 0.9756 & 0.5768 \\ 0.9953 & 0.9953 & 0.9326 & 0.5852 & 0.8000 & 0.8000 & 0.7188 \\ 0.7283 & 0.6062 & 0.9860 & 0.8094 & 0.8094 & 0.8000 & 0.7769 \\ 0.9997 & 0.6062 & 0.7769 & 0.8094 & 0.9637 & 0.9989 & 0.8320 \end{matrix}] \end{matrix}$

Definition 4.7. Let (S^{F
_{ψ
₁}}, R_{A
₁}) , (S^{F
_{ψ
₂}}, R_{A
₂}) , . . . , (S^{F
_{ψ
_r}}, R_{A
_r}) be the r (r ≥ 2) SVFSS over U. Let M_{R
₁} (mt_i) , M_{R
₂} (mt_i) , . . . , M_{R
_r} (mt_i) be the r maximum rejection values of SVFSS

(S^{F
_{ψ
₁}}, R_{A
₁}) , (S^{F
_{ψ
₂}}, R_{A
₂}) , . . . , (S^{F
_{ψ
_r}}, R_{A
_r}) respectively.

Then the generalized maximum likely ratio (similarity) of M_{R
₁} (mt_i) (hh_j) , M_{R
₂} (mt_i) (hh_j) , . . . ,

M_{R
_r} (mt_i) (hh_j) is defined as

GMLR {M_{R
₁} (mt_i) , M_{R
₂} (mt_i) , . . . , M_{R
_r} (mt_i)} (hh_j) $= \frac{1}{1 + {\frac{1}{r} (M_{R_{1}} ({mt}_{i}) + M_{R_{2}} ({mt}_{i}) + . . . + M_{R_{r}} ({mt}_{i})) ({hh}_{j})}^{r}}$ (14) where mt_i ∈ U and hh_j ∈ E.

5 A novel MCGDM problem on SVFSS

In this section, using the Definition 4.7. we suggest a method for evaluating the weight vectors for alternatives and decision makers. Let U = {mt₁, mt₂, . . . , mt_m} with ’m’ alternatives and ’n’ parameters expressed as E = {hh₁, hh₂, . . . , hh_n}. Let DM = {DM₁, DM₂, . . . , DM_t} be the group of decision makers who involved in the MCGDM process. The decision makers chose the linguistic terms form the LTS L = {Ω_k|k = - t, …, 0, …, t} to express their evaluation information over each alternatives mt_m concerning to each parameter hh_n.

5.1 Decision making algorithm and its flow chart

Find linguistic decision matrix given by the decision maker DM_t as $R_{t} = [r_{ij}^{t}]_{m \times n} = [\begin{matrix} r_{11}^{t} & r_{12}^{t} & \dots & r_{1 n}^{t} \\ r_{21}^{t} & r_{22}^{t} & \dots & r_{2 n}^{t} \\ ⋮ & ⋮ & \dots & ⋮ \\ r_{m 1}^{t} & r_{m 2}^{t} & \dots & r_{mn}^{t} \end{matrix}]$ where $r_{ij}^{t} \in L$ .

Find the maximum rejection matrix (M_R) of each decision makers.

Find the generalized maximum likely ratio (GMLR).

Find the weighted maximum likely ratio W_{(mt_i)} of each alternatives. That is

$\begin{matrix} W_{({mt}_{i})} = \frac{[\sum_{j = 1}^{n} GMLR ({hh}_{j})] ({mt}_{i})}{\sum_{i = 1}^{m} [\sum_{j = 1}^{n} GMLR ({hh}_{j})] ({mt}_{i})}, \\ i = 1, 2, \dots, m . \end{matrix}$ (15)

Rank the alternatives mt_i.

Conclusion.

A geometrical representation of the above algorithm is presented below (Fig. 1.) which depicts the clear view of the algorithm.

Fig. 1

Flow chart for the algorithm

5.2 MCGDM on the issue of haze equipment selection

Air (haze) pollution has worsened in recent years and as a result it affects people’s health. Haze is the result of specific climate change due to human activities. To a greater extent, many countries are affected by haze pollution. Strengthening the management of haze, thus reducing harm to human health as much as possible due to haze, is a key issue for these countries. Choosing the best solution for haze management is therefore the most wanted solution in recent days.

Let U = {en₁, en₂, en₃, en₄} be the set of four air purifier equipments and let E = {hh₁, hh₂, hh₃, hh₄, hh₅} be the set of five viruses. Here hh₁ = Variola (smallpox) virus, hh₂ = Corona virus (SARS), hh₃ = H1N1 (Swine Flu) virus, hh₄ = Influenza A (Bird Flu) virus, and hh₅ = Rhino virus (Common cold). The four experts R₁, R₂, R₃ and R₄ give the feedback on removal rate of virus on each companies, by using the LTS L^ψ = L⁴ = {T_p = Terriblypoor, H_p = Highlypoor, P = Poor, M_p = Marginallypoor, I = Impartial, M_g = Marginallygood, G = Good, H_g = Highlygood, T_g = Terriblygood} .

[Step 1.] The linguistic decision matrix of each experts are given below. $\begin{matrix} R_{1} & = [\begin{matrix} T_{g} & H_{g} & T_{g} & G & T_{g} \\ G & H_{g} & I & M_{p} & P \\ H_{g} & G & G & M_{g} & I \\ H_{p} & T_{p} & I & G & T_{g} \end{matrix}] \\ R_{2} & = [\begin{matrix} T_{g} & H_{g} & G & M_{g} & G \\ P & M_{g} & H_{g} & H_{p} & M_{p} \\ G & T_{g} & H_{g} & I & T_{g} \\ G & P & G & T_{g} & H_{p} \end{matrix}] \\ R_{3} & = [\begin{matrix} M_{p} & G & T_{g} & I & G \\ P & G & H_{p} & M_{p} & T_{p} \\ G & T_{p} & H_{g} & M_{g} & G \\ H_{p} & M_{p} & G & P & T_{g} \end{matrix}] \end{matrix}$ $\begin{matrix} R_{4} & = [\begin{matrix} H_{p} & G & T_{g} & P & H_{g} \\ P & H_{p} & I & G & T_{p} \\ G & P & H_{g} & G & T_{g} \\ H_{g} & P & T_{g} & T_{p} & H_{p} \end{matrix}] \end{matrix}$

[Step 2.] We find the maximum rejection matrix M (R_x) on each experts as follows. $M (R_{1}) = [\begin{matrix} 0.0180 & 0.0474 & 0.0180 & 0.1192 & 0.0180 \\ 0.1192 & 0.0474 & 0.5000 & 0.7311 & 0.8808 \\ 0.0474 & 0.1192 & 0.1192 & 0.2689 & 0.5000 \\ 0.9526 & 0.9820 & 0.5000 & 0.1192 & 0.0180 \end{matrix}]$

$M (R_{2}) = [\begin{matrix} 0.0180 & 0.0474 & 0.1192 & 0.2689 & 0.1192 \\ 0.8808 & 0.2689 & 0.0474 & 0.9526 & 0.7311 \\ 0.1192 & 0.0180 & 0.0474 & 0.5000 & 0.0180 \\ 0.1192 & 0.8808 & 0.1192 & 0.0180 & 0.9526 \end{matrix}]$

$M (R_{3}) = [\begin{matrix} 0.7311 & 0.1192 & 0.0180 & 0.5000 & 0.1192 \\ 0.8808 & 0.1192 & 0.9526 & 0.7311 & 0.9820 \\ 0.1192 & 0.9820 & 0.0474 & 0.2689 & 0.1192 \\ 0.9526 & 0.7311 & 0.1192 & 0.8808 & 0.0180 \end{matrix}]$ $M (R_{4}) = [\begin{matrix} 0.9526 & 0.1192 & 0.0180 & 0.8808 & 0.0474 \\ 0.8808 & 0.9526 & 0.5000 & 0.1192 & 0.9820 \\ 0.1192 & 0.8808 & 0.0474 & 0.1192 & 0.0180 \\ 0.0474 & 0.8808 & 0.0180 & 0.9820 & 0.9526 \end{matrix}]$

[Step 3.] We find the generalized maximum likely ratio matrix (GMLR)is as follows. $\begin{matrix} GMLR (R_{1}, R_{2}, R_{3}, R_{4}) \\ = [\begin{matrix} 0.9670 & 1.0000 & 1.0000 & 0.9632 & 1.0000 \\ 0.8149 & 0.9857 & 0.9412 & 0.8613 & 0.6102 \\ 1.0000 & 0.9412 & 1.0000 & 0.9930 & 1.0000 \\ 0.9329 & 0.6372 & 0.9987 & 0.9412 & 0.9474 \end{matrix}] \end{matrix}$

[Step 4.] We find the weighted maximum likely ratio W_{(en_i)}. $W_{({en}_{1})} = \frac{\sum_{j = 1}^{5} [GMLR (d_{j})] ({en}_{1})}{\sum_{j = 1}^{5} {\begin{matrix} [GMLR (d_{j})] ({en}_{1}) + [GMLR (d_{j})] ({en}_{2}) \\ + [GMLR (d_{j})] ({en}_{3}) + [GMLR (d_{j})] ({en}_{4}) \end{matrix}}} = 0.2660$ similarly, W_(en₂) = 0.2273; W_(en₃) = 0.2662; W_(en₄) =0.2405.

[Step 5.] The ranks are en₃ > en₁ > en₄ > en₂.

[Step 6.] Conclusion.

The best air purifier equipment is en₃.

6 Comparison with existing works

Our method concerns about DM’s preference and experience for haze management with several tangible constraints. Further our method paves a way to change DM’s preference and experience into interesting results. To validate our work, we compare our result with two works namely score-entropy based ELECTRE II method for edge node selection problem [19] and health information technology (HIT) problem [1].

6.1 Method. 1.

Considering Lin et al. [19] method we take five edge nodes to be the universe set U = {en₁, en₂, en₃, en₄, en₅} and seven factors as the parameters E = {fa₁, fa₂, fa₃,

fa₄, fa₅, fa₆, fa₇}

Step. 1. The linguistic matrices R₁, R₂, R₃ and R₄ based on the above preferences of five alternatives with seven criteria are presented below by using LTSs. $R_{1} = [\begin{matrix} M_{g} & H_{g} & T_{g} & M_{g} & M_{p} & G & G \\ T_{p} & M_{p} & M_{p} & G & M_{g} & M_{g} & T_{p} \\ T_{g} & G & M_{g} & H_{p} & T_{p} & G & M_{g} \\ H_{p} & P & G & H_{p} & T_{g} & G & H_{p} \\ T_{g} & P & H_{p} & T_{g} & M_{g} & H_{g} & T_{g} \end{matrix}]$ $R_{2} = [\begin{matrix} T_{g} & M_{g} & G & M_{g} & P & M_{g} & P \\ H_{p} & M_{g} & H_{g} & P & M_{p} & H_{g} & M_{p} \\ G & T_{g} & M_{g} & M_{p} & T_{g} & P & T_{p} \\ M_{g} & M_{p} & G & T_{g} & H_{p} & P & G \\ T_{g} & M_{p} & G & H_{p} & G & T_{g} & P \end{matrix}]$ $R_{3} = [\begin{matrix} M_{p} & G & M_{g} & M_{p} & G & M_{p} & P \\ M_{p} & G & P & H_{p} & T_{p} & G & T_{g} \\ G & T_{p} & G & M_{g} & T_{g} & P & M_{p} \\ M_{p} & M_{g} & G & M_{p} & H_{p} & M_{p} & G \\ H_{p} & M_{p} & G & T_{p} & T_{g} & H_{g} & M_{p} \end{matrix}]$ $R_{4} = [\begin{matrix} H_{p} & G & H_{g} & M_{p} & H_{g} & P & P \\ P & H_{p} & M_{p} & M_{g} & T_{p} & H_{p} & H_{g} \\ G & M_{p} & H_{g} & M_{g} & H_{g} & G & H_{p} \\ G & T_{p} & G & T_{p} & H_{p} & H_{p} & P \\ P & M_{p} & P & M_{g} & H_{g} & G & M_{g} \end{matrix}]$

Step. 2. The maximum rejection matrix (M_{R
_x}) on each experts are given below. $\begin{matrix} M (R_{1}) = \\ [\begin{matrix} 0.2689 & 0.0474 & 0.0180 & 0.2689 & 0.7311 & 0.1192 & 0.1192 \\ 0.9820 & 0.7311 & 0.7311 & 0.1192 & 0.2689 & 0.2689 & 0.9820 \\ 0.0180 & 0.1192 & 0.2689 & 0.9526 & 0.9820 & 0.1192 & 0.2689 \\ 0.9526 & 0.8808 & 0.1192 & 0.9526 & 0.0180 & 0.1192 & 0.9526 \\ 0.0180 & 0.8808 & 0.9526 & 0.0180 & 0.2689 & 0.0474 & 0.0180 \end{matrix}] \end{matrix}$ $\begin{matrix} M (R_{2}) = \\ [\begin{matrix} 0.0180 & 0.2689 & 0.1192 & 0.2689 & 0.8808 & 0.2689 & 0.8808 \\ 0.9526 & 0.2689 & 0.0474 & 0.8808 & 0.7311 & 0.0474 & 0.7311 \\ 0.1192 & 0.0180 & 0.2689 & 0.7311 & 0.0180 & 0.8808 & 0.9820 \\ 0.2689 & 0.7311 & 0.1192 & 0.0180 & 0.9526 & 0.8808 & 0.1192 \\ 0.0180 & 0.7311 & 0.1192 & 0.9526 & 0.1192 & 0.0180 & 0.8808 \end{matrix}] \end{matrix}$ $\begin{matrix} M (R_{3}) = \\ [\begin{matrix} 0.7311 & 0.1192 & 0.2689 & 0.7311 & 0.1192 & 0.7311 & 0.8808 \\ 0.7311 & 0.1192 & 0.8808 & 0.9526 & 0.9820 & 0.1192 & 0.0180 \\ 0.1192 & 0.9820 & 0.1192 & 0.2689 & 0.0180 & 0.8808 & 0.7311 \\ 0.7311 & 0.2689 & 0.1192 & 0.7311 & 0.9526 & 0.7311 & 0.1192 \\ 0.9526 & 0.7311 & 0.1192 & 0.9820 & 0.0180 & 0.0474 & 0.7311 \end{matrix}] \end{matrix}$ $\begin{matrix} M (R_{4}) = \\ [\begin{matrix} 0.9526 & 0.1192 & 0.0474 & 0.7311 & 0.0474 & 0.8808 & 0.8808 \\ 0.8808 & 0.9526 & 0.7311 & 0.2689 & 0.9820 & 0.9526 & 0.0474 \\ 0.1192 & 0.7311 & 0.0474 & 0.2689 & 0.0474 & 0.1192 & 0.9526 \\ 0.1192 & 0.9820 & 0.1192 & 0.9820 & 0.9526 & 0.9526 & 0.8808 \\ 0.8808 & 0.7311 & 0.8808 & 0.2689 & 0.0474 & 0.1192 & 0.2689 \end{matrix}] \end{matrix}$

Step. 3. The generalized maximum likely ratio matrix (GMLR) is $\begin{matrix} GMLR (R_{1}, R_{2}, R_{3}, R_{4}) = \\ [\begin{matrix} 0.9444 & 0.9996 & 0.9997 & 0.9412 & 0.9624 & 0.9412 & 0.8149 \\ 0.6181 & 0.9329 & 0.8869 & 0.9131 & 0.7684 & 0.9857 & 0.9624 \\ 0.9647 & 0.9562 & 0.9990 & 0.9131 & 0.9950 & 0.9412 & 0.7754 \\ 0.9329 & 0.7922 & 0.9998 & 0.8315 & 0.7892 & 0.8315 & 0.9329 \\ 0.9545 & 0.7414 & 0.9329 & 0.9131 & 0.9998 & 0.9999 & 0.9517 \end{matrix}] \end{matrix}$ Step. 4. The weighted maximum likely ratio W_{(en_i)} are W_(en₁) = 0.2075; W_(en₂) = 0.1907; W_(en₃) = 0.2057; W_(en₄) = 0.1920; W_(en₅) = 0.2041. Step. 5. The ranks are en₁ > en₃ > en₅ > en₄ > en₂. Step. 6. The node en₁ is satisfied the seven factors. The comparison of the edge node decision making Lin et al. [19] and our approaches are listed in Table 1.

Table 1
Comparative analysis of our proposed method with Lin et al. [19] approach

Ranking Best

Lin et al. [19] en₁ > en₃ > en₂ > en₅ > en₄ en ₁

Our approach en₁ > en₃ > en₅ > en₄ > en₂ en ₁

	Ranking	Best
Lin et al. [19]	en₁ > en₃ > en₂ > en₅ > en₄	en ₁
Our approach	en₁ > en₃ > en₅ > en₄ > en₂	en ₁

Though our approach and Lin et al. [19] give the same results, We have justified that our work is flexible for computation and understanding purpose rather than the former.

6.2 Method. 2

To further test the validity of the decision-making process of multiple attributes proposed in this paper, another comparative study is carried out using the idea of Agbodah & Darko [1] in health information technology (HIT) problem.

Let U = {hit₁, hit₂, hit₃, hit₄, hit₅} and

E = {cri₁, cri₂, cri₃, cri₄}.

The generalized maximum likely ratio matrix (GMLR) is $\begin{matrix} GMLR (R_{1}, R_{2}, R_{3}) = \\ [\begin{matrix} 0.9164 & 0.9926 & 0.9592 & 0.9896 \\ 0.9996 & 0.9853 & 0.9809 & 0.9952 \\ 0.8889 & 0.9297 & 0.9991 & 0.9970 \\ 0.9970 & 0.9506 & 0.9297 & 0.9970 \\ 0.9747 & 0.9998 & 0.9997 & 0.9896 \end{matrix}] \end{matrix}$ The weighted maximum likely ratio W_{(hit_i)} are W_(hit₁) = 0.1981; W_(hit₂) = 0.2034; W_(hit₃) = 0.1959; W_(hit₄) = 0.1990; W_(hit₅) = 0.2036. The ranks are hit₅ > hit₂ > hit₄ > hit₁ > hit₃. The hospital hit₅ is satisfied the all criteria.

Under the sigmoid valued fuzzy soft set method, to validate the effectiveness of our proposed methods. We carefully compute the decision results for [1]. Using SVFSS we compare our method with [1] and present below in the form of a table (Table 2).

Table 2
Comparative analysis of our proposed method with Agbodah & Darko [1] approaches

Ranking Best

WPLEA [1] hit₅ > hit₂ > hit₃ > hit₁ > hit₄ hit ₅

WPLEG [1] hit₅ > hit₃ > hit₄ > hit₁ > hit₂ hit ₅

Our approach hit₅ > hit₂ > hit₄ > hit₁ > hit₃ hit ₅

	Ranking	Best
WPLEA [1]	hit₅ > hit₂ > hit₃ > hit₁ > hit₄	hit ₅
WPLEG [1]	hit₅ > hit₃ > hit₄ > hit₁ > hit₂	hit ₅
Our approach	hit₅ > hit₂ > hit₄ > hit₁ > hit₃	hit ₅

Note: WPLEA - weighted probabilistic linguistic Einstein average and WPLEG - weighted probabilistic linguistic Einstein geometric operators.

Though our approach and Agbodah & Darko [1] give the same results, we have justified that our work is flexible for computation and understanding purpose rather than the former.

7 Conclusion

In this paper, we have proposed a new idea namely sigmoid valued fuzzy soft set for treating the haze management problem. We have used the principle of complement to determine each alternative’s overall rejection value. Based on the maximum rejection, we have defined a new similarity measure called maximum likely ratio of two sigmoid valued soft sets. Based on maximum likely ratio, we have defined a new weight called weighted maximum likely ratio and a multi-criteria decision making procedure is also provided. A comparative analysis is also done with our approach and the existing approaches. In future, we plan to extend this idea in soft rough fuzzy set (SRFS) and soft fuzzy rough set (SFRS) to handle the haze management problems. We further aim to develop this idea in hyperbolic scales based FSS and SRFS.

Acknowledgments

We express our heartfelt thanks to the reviewers for bringing the paper in the present form.

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Sigmoid valued fuzzy soft set and its application to haze management

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Linguistic fuzzy soft set

5.1 Decision making algorithm and its flow chart

6 Comparison with existing works

6.1 Method. 1.

Table 1 Comparative analysis of our proposed method with Lin et al. [19] approach Ranking Best Lin et al. [19] en1 > en3 > en2 > en5 > en4 en 1 Our approach en1 > en3 > en5 > en4 > en2 en 1

Table 2 Comparative analysis of our proposed method with Agbodah & Darko [1] approaches Ranking Best WPLEA [1] hit5 > hit2 > hit3 > hit1 > hit4 hit 5 WPLEG [1] hit5 > hit3 > hit4 > hit1 > hit2 hit 5 Our approach hit5 > hit2 > hit4 > hit1 > hit3 hit 5

Acknowledgments

References

Table 1
Comparative analysis of our proposed method with Lin et al. [19] approach

Ranking Best

Lin et al. [19] en₁ > en₃ > en₂ > en₅ > en₄ en ₁

Our approach en₁ > en₃ > en₅ > en₄ > en₂ en ₁

Table 2
Comparative analysis of our proposed method with Agbodah & Darko [1] approaches

Ranking Best

WPLEA [1] hit₅ > hit₂ > hit₃ > hit₁ > hit₄ hit ₅

WPLEG [1] hit₅ > hit₃ > hit₄ > hit₁ > hit₂ hit ₅

Our approach hit₅ > hit₂ > hit₄ > hit₁ > hit₃ hit ₅