Operations on complex intuitionistic fuzzy soft lattice ordered group and CIFS-COPRAS method for equipment selection process

Abstract

This paper introduces some new operations on complex intuitionistic fuzzy lattice ordered groups such as sum, product, bounded product, bounded difference and disjoint sum, and verifying its pertinent properties. The research exhibits the CIFS-COPRAS algorithm in a complex intuitionistic fuzzy soft set environment. This method was furthermore applied for the equipment selection process.

Keywords

Complex intuitionistic fuzzy soft sets complex intuitionistic fuzzy soft lattice ordered group COPRAS C-COPRAS equipment selection process

1 Introduction

In 1965, fuzzy sets [22] (FSs) pioneered by Zadeh, which have widely intensified decision-makers’ attention. In 1986, Atanassov [3] proposed intuitionistic fuzzy sets (IFSs) as an extension of FSs, by adding the non-membership function. IFSs have been widely implemented in different areas than FSs. In 1994, Zavadskas et al. [23] introduced a new approach called COmplex PRoportional ASsessment (COPRAS), an efficient framework for the decision process. Recently, Garg and Rishu Arora [8] developed the algorithm based on COPRAS to solve decision-making problems in possibility intuitionistic fuzzy soft sets (PIFSSs) environment. Also, Mishra et al. [19] proposed the multi-criteria COPRAS Method in IFSs environment to Green Supplier Selection process. Further, he demonstrated the SWARA-COPRAS [2] method based on the Stepwise Weight Assessment Ratio Analysis (SWARA) and COPRAS approach. However, these concepts are not applicable to model the partial ignorance and fluctuations of the data with the phase of time. In this regard, Ramot et al. [18] proposed the idea of complex fuzzy sets (CFSs). The CFSs [5 , 12] can handle both uncertainty and temporal features. In 2012, Alkouri and Salleh [1] established the concept of a complex intuitionistic fuzzy sets (CIFSs). In 2014, Kumar et al. [10] presented the concept of complex intuitionistic fuzzy soft sets (CIFSSs) and its distance and entropy measures. In 2019, Garg and Rani [6, 7] defined some aggregation operators in CIFSs and found the application in the decision-making process using the aggregation operators. In 2017, Jun Ma et al. [11] derived the C-COPRAS method to select the strategic cost control measure on supply chain downstream, an extension of the COPRAS method in a complex fuzzy environment. In the existing works, the uncertainties in the data are handled with membership and non-membership degrees, which are the subset of real numbers, which may lose some useful seasonality or periodicity information and, consequently, affect the decision results. Thus, motivated by this, here we introduced the Complex Intuitionistic Fuzzy Soft - COPRAS (CIFS-COPRAS) method. It can handle two-dimensional information.

In recent years, lattice ordered [4] algebraic structures on FSs have been a significant amount of research. Vimala et al. [14 , 21] developed lattice-ordered structures in a fuzzy soft environment. Recently, Rajareega et al. [16, 17] bestowed the concept of complex intuitionistic fuzzy soft lattice ordered group ( $CIFSL - G$ ) and studied its pertinent properties. In this regard, the research defined the various operations on $CIFSL - G$ and interpreted its properties. It also developed the CIFS-COPRAS algorithm to solve the decision-making problem. The main contribution of this study includes:

Operations on complex intuitionistic fuzzy soft lattice ordered groups is promoted, namely sum, product, bounded product, bounded difference and disjoint sum. Also, investigate that these operations should preserve the layout of complex intuitionistic fuzzy soft lattice ordered group.

The CIFS-COPRAS method is presented and apply this to medical equipment selection process.

CIFS-COPRAS is a generalization of the existing studies such as CFS [18], IFS [3], soft set [13].

The ranking for the alternative is evaluated by using the COPRAS approach.

A comparison study is presented to reveal the stability and validity of the proposed methodology.

The remnant of this manuscript is organized as follows. Section 2 overviews the concept of lattice ordered group, CIFSS and

CIFSL - G

. Section 3 presents the new operations on

CIFSL - G

s and verifies its properties. Section 4 develops an algorithm (CIFS-COPRAS) in a complex intuitionistic fuzzy soft set environment. Furthermore, this algorithm was applied in medical equipment selection process.

2 Preliminaries

This section reviews the concept of ℓ- group, complex intuitionistic fuzzy soft sets and $CIFSL - G$ .

Definition 2.1. [4] The system G = (G, ∘ , ≤) is said to be a lattice ordered group (ℓ - group) if,

(G, ∘) is a group

(G, ≤) is a lattice

x ≤ y ⇒ a ∘ x ∘ b ≤ a ∘ y ∘ b for all a, b, x, y ∈ G.

Definition 2.2. [18] A complex fuzzy set (CFS) A defined on U is characterized by μ_A (x) that gives a complex-valued grade of membership in A to any x ∈ U. By definition, all values of μ_A (x) belongs to the unit disc in the complex plane and are represented by μ_A (x) = r_A (x) e^iw_A(x), where $i = \sqrt{- 1}$ , r_A (x) and w_A (x) are both real-valued, r_A (x) ∈ [0, 1] and w_A (x) ∈ (0, 2π]. A CFS A is thus of the form, A = {(x, μ_A (x)) : x ∈ U} = {(x, r_A (x) e^iw_A(x)) : x ∈ U}.

Definition 2.3. [10] Let E be a set of parameters, CIFS(U) represents the collection of all complex intuitionistic fuzzy sets over U, and $\tilde{F}$ be a function from E to CIFS(U). Then the set of ordered pairs ${(e, \tilde{F} (e)) : e \in E, \tilde{F} (e) \in CIFS (U)}$ , represented by $(\tilde{F}, E)$ , is a complex intuitionistic fuzzy soft set (CIFSS) on U. For all e ∈ E, $\begin{matrix} \tilde{F} (e) = {(x, μ_{\tilde{F} (e)} (x), ν_{\tilde{F} (e)} (x)) : x \in U} \\ = & {(x, r_{\tilde{F} (e)} (x) e^{{iw}_{μ_{\tilde{F} (e)}} (x)}, k_{\tilde{F} (e)} (x) e^{{iw}_{ν_{\tilde{F} (e)}} (x)}) : x \in U} \end{matrix}$ where $μ_{\tilde{F} (e)} : U ⟶ {a | a \in ℂ, | a | \leq 1}$ and $ν_{\tilde{F} (e)} : U ⟶ {a^{'} | a^{'} \in ℂ, | a^{'} | \leq 1}$ , $| μ_{\tilde{F} (e)} (x) + ν_{\tilde{F} (e)} (x) | \leq 1$ .

2.1 Complex intuitionistic fuzzy soft lattice ordered group

Throughout this paper G represents the ℓ - group and ∨ and ∧ are the maximum and minimum operators respectively and

(i) μ ≤ ν, if both r ≤ τ and w ≤ ψ and

(ii) μ ≥ ν, if both r ≥ τ and w ≥ ψ, where μ = re^iw and ν = τe^iψ, with r, τ ∈ [0, 1] and w, ψ ∈ (0, 2π].

Definition 2.4. [16] Let $(\tilde{F}, E) \in CIFSS$ over G. Then $(\tilde{F}, E)$ is said to be a complex intuitionistic fuzzy soft lattice ordered group ( $CIFSL - G$ ) over G if for all $a \in ρ (\tilde{F}, E)$ , and x, y, e_G ∈ G,

$μ_{\tilde{F} (a)} ({xy}^{- 1})$ ≥ $μ_{\tilde{F} (a)} (x) \land μ_{\tilde{F} (a)} (y)$

$ν_{\tilde{F} (a)} ({xy}^{- 1})$ ≤ $ν_{\tilde{F} (a)} (x) \lor ν_{\tilde{F} (a)} (y)$

$μ_{\tilde{F} (a)} (x \lor e_{G})$ ≥ $μ_{\tilde{F} (a)} (x)$

$ν_{\tilde{F} (a)} (x \lor e_{G})$ ≤ $ν_{\tilde{F} (a)} (x)$ ,

where

ρ (\tilde{F}, E) = {a \in E : \tilde{F} (a) is non null}

is the support set of

(\tilde{F}, E)

and e_G is the identity element of G.

3 Operations on complex intuitionistic fuzzy soft lattice ordered group

This section defines some new operations on complex intuitionistic fuzzy soft lattice ordered group ( $CIFSL - G$ ) and verified that, whether $CIFSL - G$ can be preserved for these operations.

Definition 3.1. Let $({\tilde{F}}_{1}, E_{1})$ , $({\tilde{F}}_{2}, E_{2}) \in CIFSL - G (G)$ . Then their sum $({\tilde{F}}_{1}, E_{1}) \oplus ({\tilde{F}}_{2}, E_{2})$ is defined as $({\tilde{F}}_{1}, E_{1}) \oplus ({\tilde{F}}_{2}, E_{2}) = {(c, ({\tilde{F}}_{1} \oplus {\tilde{F}}_{2}) (c)) : c = (a, b) \in E_{1} \times E_{2}}$ , where

$\begin{matrix} μ_{({\tilde{F}}_{1} \oplus {\tilde{F}}_{2}) (c)} (x) = & sup_{x = p \lor q} {\min {r_{{\tilde{F}}_{1} (a)} (p), r_{{\tilde{F}}_{2} (b)} (q)} \\ e^{imin {w_{{\tilde{F}}_{1} (a)} (p), w_{{\tilde{F}}_{2} (b)} (q)}}} and \\ ν_{({\tilde{F}}_{1} \oplus {\tilde{F}}_{2}) (c)} (x) = & inf_{x = p \lor q} {\max {τ_{{\tilde{F}}_{1} (a)} (p), τ_{{\tilde{F}}_{2} (b)} (q)} \\ e^{i \max {ψ_{{\tilde{F}}_{1} (a)} (p), ψ_{{\tilde{F}}_{2} (b)} (q)}}}, \end{matrix}$ for all x ∈ G.

Theorem 3.2. The sum of two $CIFSL - G$ of G is again a $CIFSL - G$ of G.

Proof. Let $({\tilde{F}}_{1}, E_{1})$ and $({\tilde{F}}_{2}, E_{2}) \in CIFSL - G (G)$ . By Definition 3.1, $\begin{matrix} μ_{({\tilde{F}}_{1} \oplus {\tilde{F}}_{2}) (c)} (x) = & sup_{x = p \lor q} {\min {r_{{\tilde{F}}_{1} (a)} (p), r_{{\tilde{F}}_{2} (b)} (q)} \\ e^{i \min {w_{{\tilde{F}}_{1} (a)} (p), w_{{\tilde{F}}_{2} (b)} (q)}}} and \\ ν_{({\tilde{F}}_{1} \oplus {\tilde{F}}_{2}) (c)} (x) = & inf_{x = p \lor q} {\max {τ_{{\tilde{F}}_{1} (a)} (p), τ_{{\tilde{F}}_{2} (b)} (q)} \\ e^{i \max {ψ_{{\tilde{F}}_{1} (a)} (p), ψ_{{\tilde{F}}_{2} (b)} (q)}}}, \forall x \in G . \end{matrix}$ $\begin{matrix} Let x, y ∈ G and let min{ μ (F ˜ 1 ⊕ F ˜ 2) (c) (x), μ (F ˜ 1 ⊕ F ˜ 2) (c) (y-1)} = h . Then for any δ > 0, h - δ < & μ_{({\tilde{F}}_{1} \oplus {\tilde{F}}_{2}) (c)} (x) \\ = & sup_{x = p \lor q} {\min {k_{{\tilde{F}}_{1} (a)} (p), k_{{\tilde{F}}_{2} (b)} (q)} \\ e^{i \min {w_{{\tilde{F}}_{1} (a)} (p), w_{{\tilde{F}}_{2} (b)} (q)}}} \\ = & sup_{x = p \lor q} {\min {μ_{{\tilde{F}}_{1} (a)} (p), μ_{{\tilde{F}}_{2} (b)} (q)}} \\ h - δ < & μ_{({\tilde{F}}_{1} \oplus {\tilde{F}}_{2}) (c)} (y^{- 1}) \\ = & sup_{y^{- 1} = r \lor s} {\min {k_{{\tilde{F}}_{1} (a)} (r), k_{{\tilde{F}}_{2} (b)} (s)} \\ e^{i \min {w_{{\tilde{F}}_{1} (a)} (r), w_{{\tilde{F}}_{2} (b)} (s)}}} \\ = & sup_{y^{- 1} = r \lor s} {\min {μ_{{\tilde{F}}_{1} (a)} (r), μ_{{\tilde{F}}_{2} (b)} (s)}} . Then there exist x = p ∨ q such that h - δ & < \min {μ_{{\tilde{F}}_{1} (a)} (p), μ_{{\tilde{F}}_{2} (b)} (q)} \\ \Rightarrow h - δ & < μ_{{\tilde{F}}_{1} (a)} (p), h - δ < μ_{{\tilde{F}}_{2} (b)} (q) . So far y-1 = r ∨ s, we have h - δ < & \min {μ_{{\tilde{F}}_{1} (a)} (p), μ_{{\tilde{F}}_{1} (a)} (r \lor s)} \\ \leq & μ_{{\tilde{F}}_{1} (a)} (p (r \lor s)) \\ h - δ < & \min {μ_{{\tilde{F}}_{2} (b)} (q), μ_{{\tilde{F}}_{2} (b)} (r \lor s)} \\ \leq & μ_{{\tilde{F}}_{2} (b)} (q (r \lor s)) \\ [∵ ({\tilde{F}}_{1} & , E_{1}), ({\tilde{F}}_{2}, E_{2}) \in CIFSL - G (G)] \\ ∴ h - δ < & \min {μ_{{\tilde{F}}_{1} (a)} (p (r \lor s)), μ_{{\tilde{F}}_{2} (b)} (q (r \lor s))} . Note that G is a lattice ordered group, {xy}^{- 1} = & (p \lor q) (r \lor s) = (p (r \lor s)) \lor (q (r \lor s)) \\ μ_{({\tilde{F}}_{1} \oplus {\tilde{F}}_{2}) (c)} & ({xy}^{- 1}) = sup_{{xy}^{- 1} = (p (r \lor s)) \lor (q (r \lor s))} \\ min {μ_{{\tilde{F}}_{1} (a)} (p (r \lor s)), μ_{{\tilde{F}}_{2} (b)} (q (r \lor s))} \\ > mi & n {μ_{{\tilde{F}}_{1} (a)} (p (r \lor s)), μ_{{\tilde{F}}_{2} (b)} (q (r \lor s))} \\ > h - & δ Since δ is arbitrary and μ (F ˜ 1 ⊕ F ˜ 2) (c) (y) = μ (F ˜ 1 ⊕ F ˜ 2) (c) (y-1), μ_{({\tilde{F}}_{1} \oplus {\tilde{F}}_{2}) (c)} & ({xy}^{- 1}) > h \\ = \min {μ_{({\tilde{F}}_{1} \oplus {\tilde{F}}_{2}) (c)} (x), μ_{({\tilde{F}}_{1} \oplus {\tilde{F}}_{2}) (c)} (y)} (1) Now, Let x, e G ∈ G, where e G is the identity element of G and let min { μ (F ˜ 1 ⊕ F ˜ 2) (c) (x), μ (F ˜ 1 ⊕ F ˜ 2) (c) (eG)} =i . Then for any δ > 0, i - δ < & μ_{({\tilde{F}}_{1} \oplus {\tilde{F}}_{2}) (c)} (x) \\ = & sup_{x = p \lor q} {\min {μ_{{\tilde{F}}_{1} (a)} (p), μ_{{\tilde{F}}_{2} (b)} (q)}} \\ i - δ < & μ_{({\tilde{F}}_{1} \oplus {\tilde{F}}_{2}) (c)} (e_{G}) \\ = & sup_{e_{G} = r \lor s} {\min {μ_{{\tilde{F}}_{1} (a)} (r), μ_{{\tilde{F}}_{2} (b)} (s)}} Then there exist x = p ∨ q and e G = r ∨ s such that i - δ < & \min {μ_{{\tilde{F}}_{1} (a)} (p), μ_{{\tilde{F}}_{2} (b)} (q)} and \\ i - δ < & \min {μ_{{\tilde{F}}_{1} (a)} (r), μ_{{\tilde{F}}_{2} (b)} (s)} \\ \Rightarrow i - δ < & μ_{{\tilde{F}}_{1} (a)} (p), i - δ < μ_{{\tilde{F}}_{2} (b)} (q), \\ i - δ < & μ_{{\tilde{F}}_{1} (a)} (r), i - δ < μ_{{\tilde{F}}_{2} (b)} (s) Therefore, i - δ < & \min {μ_{{\tilde{F}}_{1} (a)} (p), μ_{{\tilde{F}}_{1} (a)} (r)} \leq μ_{{\tilde{F}}_{1} (a)} (p \lor r) \\ i - δ < & \min {μ_{{\tilde{F}}_{2} (b)} (q), μ_{{\tilde{F}}_{2} (b)} (s)} \leq μ_{{\tilde{F}}_{2} (b)} (q \lor s) \\ [∵ ({\tilde{F}}_{1}, & E_{1}), ({\tilde{F}}_{2}, E_{2}) \in CIFSL - G (G)] \\ ∴ i - δ < & \min {μ_{{\tilde{F}}_{1} (a)} (p \lor r), μ_{{\tilde{F}}_{2} (b)} (q \lor s)} Note that G is a lattice ordered group, x \lor e_{G} & = (p \lor q) \lor (r \lor s) = (p \lor r) \lor (q \lor s) \\ μ_{({\tilde{F}}_{1} \oplus {\tilde{F}}_{2}) (c)} (x \lor e_{G}) = sup_{x \lor e_{G} = (p \lor r) \lor (q \lor s)} \\ {\min {μ_{{\tilde{F}}_{1} (a)} (p \lor r), μ_{{\tilde{F}}_{2} (b)} (q \lor s)}} \\ > & \min {μ_{{\tilde{F}}_{1} (a)} (p \lor r), μ_{{\tilde{F}}_{2} (b)} (q \lor s)} \\ > & i - δ \\ Since δ is arbitrary, μ_{({\tilde{F}}_{1} \oplus {\tilde{F}}_{2}) (c)} (x \lor e_{G}) > i \\ = \min { & μ_{({\tilde{F}}_{1} \oplus {\tilde{F}}_{2}) (c)} (x), μ_{({\tilde{F}}_{1} \oplus {\tilde{F}}_{2}) (c)} (e_{G})} (2) Let x, y ∈ G and let max { ν (F ˜ 1 ⊕ F ˜ 2) (c) (x), ν (F ˜ 1 ⊕ F ˜ 2) (c) (y-1)} = j . Then for any δ > 0, \end{matrix}$ $\begin{matrix} j - δ > & μ_{({\tilde{F}}_{1} \oplus {\tilde{F}}_{2}) (c)} (x) \\ = & inf_{x = p \lor q} {\max {τ_{{\tilde{F}}_{1} (a)} (p), τ_{{\tilde{F}}_{2} (b)} (q)} \\ e^{i \max {ψ_{{\tilde{F}}_{1} (a)} (p), ψ_{{\tilde{F}}_{2} (b)} (q)}}} \\ = & inf_{x = p \lor q} {\max {ν_{{\tilde{F}}_{1} (a)} (p), ν_{{\tilde{F}}_{2} (b)} (q)}} \\ j - δ > & ν_{({\tilde{F}}_{1} \oplus {\tilde{F}}_{2}) (c)} (y^{- 1}) \\ = & inf_{y^{- 1} = r \lor s} {\max {τ_{{\tilde{F}}_{1} (a)} (r), τ_{{\tilde{F}}_{2} (b)} (s)} \\ e^{i \max {ψ_{{\tilde{F}}_{1} (a)} (r), ψ_{{\tilde{F}}_{2} (b)} (s)}}} \\ = & inf_{y^{- 1} = r \lor s} {\max {ν_{{\tilde{F}}_{1} (a)} (r), ν_{{\tilde{F}}_{2} (b)} (s)}} Then there exist x = p ∨ q such that j - δ & > \max {ν_{{\tilde{F}}_{1} (a)} (p), ν_{{\tilde{F}}_{2} (b)} (q)} \\ \Rightarrow j - δ & > ν_{{\tilde{F}}_{1} (a)} (p), j - δ > ν_{{\tilde{F}}_{2} (b)} (q) So far y-1 = r ∨ s, we have j - δ & > \max {ν_{{\tilde{F}}_{1} (a)} (p), ν_{{\tilde{F}}_{1} (a)} (r \lor s)} \\ \geq ν_{{\tilde{F}}_{1} (a)} (p (r \lor s)) \\ j - δ & > \max {ν_{{\tilde{F}}_{2} (b)} (q), ν_{{\tilde{F}}_{2} (b)} (r \lor s)} \\ \geq ν_{{\tilde{F}}_{2} (b)} (q (r \lor s)) \\ [∵ ({\tilde{F}}_{1}, & E_{1}), ({\tilde{F}}_{2}, E_{2}) \in CIFSL - G (G)] \\ ∴ j - δ & > \max {ν_{{\tilde{F}}_{1} (a)} (p (r \lor s)), ν_{{\tilde{F}}_{2} (b)} (q (r \lor s))} Note that G is a lattice ordered group, {xy}^{- 1} = & (p \lor q) (r \lor s) = (p (r \lor s)) \lor (q (r \lor s)) \\ ν_{({\tilde{F}}_{1} \oplus {\tilde{F}}_{2}) (c)} ({xy}^{- 1}) = inf_{{xy}^{- 1} = (p (r \lor s)) \lor (q (r \lor s))} \\ {\max {ν_{{\tilde{F}}_{1} (a)} (p (r \lor s)), ν_{{\tilde{F}}_{2} (b)} (q (r \lor s))} \\ < \max {ν_{{\tilde{F}}_{1} (a)} (p (r \lor s)), ν_{{\tilde{F}}_{2} (b)} (q (r \lor s))} \\ < j - δ & Since δ is arbitrary and ν (F ˜ 1 ⊕ F ˜ 2) (c) (y) = ν (F ˜ 1 ⊕ F ˜ 2) (c) (y-1), & ν_{({\tilde{F}}_{1} \oplus {\tilde{F}}_{2}) (c)} ({xy}^{- 1}) < j \\ = \max {ν_{({\tilde{F}}_{1} \oplus {\tilde{F}}_{2}) (c)} (x), ν_{({\tilde{F}}_{1} \oplus {\tilde{F}}_{2}) (c)} (y)} (3) Similarly we can get, ν (F ˜ 1 ⊕ F ˜ 2) (c) (x ∨ e G) \leq \max {ν_{({\tilde{F}}_{1} \oplus {\tilde{F}}_{2}) (c)} (x), ν_{({\tilde{F}}_{1} \oplus {\tilde{F}}_{2}) (c)} (e_{G})} (4) \end{matrix}$ From (1), (2), (3) and (4), $({\tilde{F}}_{1}, E_{1}) \oplus ({\tilde{F}}_{2}, E_{2}) \in CIFSL - G (G) .$ □

Definition 3.3. Let $({\tilde{F}}_{1}, E_{1})$ , $({\tilde{F}}_{2}, E_{2}) \in CIFSL - G (G)$ . Then their product $({\tilde{F}}_{1}, E_{1}) \otimes ({\tilde{F}}_{2}, E_{2})$ is defined as: $({\tilde{F}}_{1}, E_{1}) \otimes ({\tilde{F}}_{2}, E_{2}) = {(c, ({\tilde{F}}_{1} \otimes {\tilde{F}}_{2}) (c)) : c = (a, b) \in E_{1} \times E_{2}}$ , where $\begin{matrix} μ_{({\tilde{F}}_{1} \otimes {\tilde{F}}_{2}) (c)} (x) = & sup_{x \leq p \land q} {\min {k_{{\tilde{F}}_{1} (a)} (p), k_{{\tilde{F}}_{2} (b)} (q)} \\ e^{i \min {w_{{\tilde{F}}_{1} (a)} (p), w_{{\tilde{F}}_{2} (b)} (q)}}} and \\ ν_{({\tilde{F}}_{1} \otimes {\tilde{F}}_{2}) (c)} (x) = & inf_{x \leq p \land q} {\max {τ_{{\tilde{F}}_{1} (a)} (p), τ_{{\tilde{F}}_{2} (b)} (q)} \\ e^{i \max {ψ_{{\tilde{F}}_{1} (a)} (p), ψ_{{\tilde{F}}_{2} (b)} (q)}}}, \end{matrix}$ for all x ∈ G.

Theorem 3.4. The product of two $CIFSL - G$ of G is again a $CIFSL - G$ of G.

Proof. Proof for this theorem is similar to Theorem 3.2.□

Definition 3.5. Let $({\tilde{F}}_{1}, E)$ and $({\tilde{F}}_{2}, E) \in CIFSL - G (G)$ . The bounded product of these $({\tilde{F}}_{1}, E)$ and $({\tilde{F}}_{2}, E)$ is denoted as $({\tilde{F}}_{1}, E) ⊙ ({\tilde{F}}_{2}, E)$ and is defined as:

$\begin{matrix} μ_{({\tilde{F}}_{1} ⊙ {\tilde{F}}_{2}) (a)} (x) = & Max [0, (r_{{\tilde{F}}_{1} (a)} (x) + r_{{\tilde{F}}_{2} (a)} (x) - 1) \\ e^{i (w_{{\tilde{F}}_{1} (a)} (x) - w_{{\tilde{F}}_{2} (a)} (x))}] and \\ ν_{({\tilde{F}}_{1} ⊙ {\tilde{F}}_{2}) (a)} (x) = & Max [0, (τ_{{\tilde{F}}_{1} (a)} (x) + τ_{{\tilde{F}}_{2} (a)} (x) - 1) \\ e^{i (ψ_{{\tilde{F}}_{1} (a)} (x) - ψ_{{\tilde{F}}_{2} (a)} (x))}] \end{matrix}$ for all a ∈ E, x ∈ G.

Theorem 3.6. The bounded product of two $CIFSL - G$ of G is also a $CIFSL - G$ of G.

Proof. Let $({\tilde{F}}_{1}, E)$ and $({\tilde{F}}_{2}, E) \in CIFSL - G (G)$ . By Definition 3.5, for all a ∈ E, x ∈ G,

$\begin{matrix} μ_{({\tilde{F}}_{1} ⊙ {\tilde{F}}_{2}) (a)} (x \lor e_{G}) \\ = & Max [0, μ_{{\tilde{F}}_{1} (a)} (x \lor e_{G}) ⊙ μ_{{\tilde{F}}_{2} (a)} (x \lor e_{G})] \\ \geq & Max [0, (μ_{{\tilde{F}}_{1} (a)} (x)) ⊙ (μ_{{\tilde{F}}_{2} (a)} (x))] \\ = & Max [0, (r_{{\tilde{F}}_{1} (a)} (x) + r_{{\tilde{F}}_{2} (a)} (x) - 1) \\ e^{i (w_{{\tilde{F}}_{1} (a)} (x) - w_{{\tilde{F}}_{2} (a)} (x))}] \\ \geq & μ_{({\tilde{F}}_{1} ⊙ {\tilde{F}}_{2}) (a)} (x) \end{matrix}$

$\begin{matrix} ν_{({\tilde{F}}_{1} ⊙ {\tilde{F}}_{2}) (a)} (x \lor e_{G}) \\ = & Max [0, ν_{{\tilde{F}}_{1} (a)} (x \lor e_{G}) ⊙ ν_{{\tilde{F}}_{2} (a)} (x \lor e_{G})] \\ \leq & Max [0, (ν_{{\tilde{F}}_{1} (a)} (x)) ⊙ (ν_{{\tilde{F}}_{2} (a)} (x))] \\ = & Max [0, (τ_{{\tilde{F}}_{1} (a)} (x) + τ_{{\tilde{F}}_{2} (a)} (x) - 1) \\ e^{i (ψ_{{\tilde{F}}_{1} (a)} (x) - ψ_{{\tilde{F}}_{2} (a)} (x))}] \\ \leq & ν_{({\tilde{F}}_{1} ⊙ {\tilde{F}}_{2}) (a)} (x) \end{matrix}$

$\begin{matrix} μ_{({\tilde{F}}_{1} ⊙ {\tilde{F}}_{2}) (a)} ({xy}^{- 1}) \\ = & Max [0, μ_{{\tilde{F}}_{1} (a)} ({xy}^{- 1}) ⊙ μ_{{\tilde{F}}_{2} (a)} ({xy}^{- 1})] \\ \geq & Max [0, (μ_{{\tilde{F}}_{1} (a)} (x) \land μ_{{\tilde{F}}_{1} (a)} (y)) \\ ⊙ (μ_{{\tilde{F}}_{2} (a)} (x) \land μ_{{\tilde{F}}_{2} (a)} (y))] \\ \geq & Max [0, μ_{{\tilde{F}}_{1} (a)} (x) ⊙ μ_{{\tilde{F}}_{2} (a)} (x)] \\ \land Max [0, μ_{{\tilde{F}}_{1} (a)} (y) ⊙ μ_{{\tilde{F}}_{2} (a)} (y)] \\ \geq & μ_{({\tilde{F}}_{1} ⊙ {\tilde{F}}_{2}) (a)} (x) \land μ_{({\tilde{F}}_{1} ⊙ {\tilde{F}}_{2}) (a)} (y) \end{matrix}$

$\begin{matrix} ν_{({\tilde{F}}_{1} ⊙ {\tilde{F}}_{2}) (a)} ({xy}^{- 1}) \\ = & Max [0, ν_{{\tilde{F}}_{1} (a)} ({xy}^{- 1}) ⊙ ν_{{\tilde{F}}_{2} (a)} ({xy}^{- 1})] \\ \leq & Max [0, (ν_{{\tilde{F}}_{1} (a)} (x) \lor ν_{{\tilde{F}}_{1} (a)} (y)) \\ ⊙ (ν_{{\tilde{F}}_{2} (a)} (x) \lor ν_{{\tilde{F}}_{2} (a)} (y))] \\ \leq & Max [0, ν_{{\tilde{F}}_{1} (a)} (x) ⊙ ν_{{\tilde{F}}_{2} (a)} (x)] \\ \lor Max [0, ν_{{\tilde{F}}_{1} (a)} (y) ⊙ ν_{{\tilde{F}}_{2} (a)} (y)] \\ \leq & ν_{({\tilde{F}}_{1} ⊙ {\tilde{F}}_{2}) (a)} (x) \lor ν_{({\tilde{F}}_{1} ⊙ {\tilde{F}}_{2}) (a)} (y) \end{matrix}$

\Rightarrow ({\tilde{F}}_{1}, E) ⊙ ({\tilde{F}}_{2}, E) \in CIFSL - G .

□

Definition 3.7. Let $({\tilde{F}}_{1}, E)$ and $({\tilde{F}}_{2}, E) \in CIFSL - G (G)$ . The bounded difference of these $({\tilde{F}}_{1}, E)$ and $({\tilde{F}}_{2}, E)$ is denoted as $({\tilde{F}}_{1}, E) ⊖ ({\tilde{F}}_{2}, E)$ and is defined as:

$\begin{matrix} μ_{({\tilde{F}}_{1} ⊖ {\tilde{F}}_{2}) (a)} (x) = & Max [0, r_{{\tilde{F}}_{1} (a)} (x) - r_{{\tilde{F}}_{2} (a)} (x) \\ e^{i (w_{{\tilde{F}}_{1} (a)} (x) - w_{{\tilde{F}}_{2} (a)} (x))}] and \\ ν_{({\tilde{F}}_{1} ⊖ {\tilde{F}}_{2}) (a)} (x) = & Max [0, τ_{{\tilde{F}}_{1} (a)} (x) - τ_{{\tilde{F}}_{2} (a)} (x) \\ e^{i (ψ_{{\tilde{F}}_{1} (a)} (x) - ψ_{{\tilde{F}}_{2} (a)} (x))}], \end{matrix}$ for all a ∈ E, x ∈ G.

Theorem 3.8. The bounded difference of two $CIFSL - G$ of G is not a $CIFSL - G$ of G.

Example 1. Let $G = (ℤ, +, \land, \lor)$ be the ℓ - group and E = {a, b} be the parameters. Next define two $CIFSL - G$ of G as follows:

(i) $({\tilde{F}}_{1}, E)$ = ${{\tilde{F}}_{1} (a), {\tilde{F}}_{1} (b) : a, b \in E}$ , where ${\tilde{F}}_{1} (a)$ = ${\begin{matrix} (x, 0.6 e^{i (1.5 π)}, 0.4 e^{i (0.5 π)}), & if x = 2 k; k \in ℤ / {0} \\ (x, 0.8 e^{i (1.7 π)}, 0.2 e^{i (0.3 π)}), & x = 0 \\ (x, 0.45 e^{i π}, 0.55 e^{i (1.2 π)}), & otherwise, \end{matrix}$

${\tilde{F}}_{1} (b)$ = ${\begin{matrix} (x, 0.8 e^{i (1.2 π)}, 0.3 e^{i π}), & if x = 2 k; k \in ℤ / {0} \\ (x, 0.9 e^{i (1.5 π)}, 0.2 e^{i (0.5 π)}), & if x = 0 \\ (x, 0.2 e^{i (0.4 π)}, 0.8 e^{i (1.6 π)}), & otherwise \end{matrix}$

(ii) $({\tilde{F}}_{2}, E)$ = ${{\tilde{F}}_{2} (a), {\tilde{F}}_{2} (b) : a, b \in E}$ , where ${\tilde{F}}_{2} (a)$ $= {\begin{matrix} (x, 0.5 e^{i (1.2 π)}, 0.5 e^{i (0.8 π)}), & if x = 2 k; k \in ℤ / {0} \\ (x, 1 e^{i 2 π}, 0), & if x = 0 \\ (x, 0.4 e^{i (0.8 π)}, 0.6 e^{i π}), & otherwise, \end{matrix}$

${\tilde{F}}_{2} (b)$ $= {\begin{matrix} (x, 0.7 e^{i π}, 0.2 e^{i (0.8 π)}), & if x = 2 k; k \in ℤ / {0} \\ (x, 0.8 e^{i (1.3 π)}, 0.1 e^{i (0.3 π)}), & if x = 0 \\ (x, 0.7 e^{i (0.3 π)}, 0.3 e^{i (1.7 π)}), & otherwise \end{matrix}$

Then, the bounded difference of these $({\tilde{F}}_{1}, E)$ and $({\tilde{F}}_{2}, E)$ is $({\tilde{F}}_{1}, E) ⊖ ({\tilde{F}}_{2}, E)$ , where $({\tilde{F}}_{1} ⊖ {\tilde{F}}_{2}) (a)$ $= {\begin{matrix} (x, 0.1 e^{i (0.3 π)}, 0), & if x = 2 k; k \in ℤ / {0} \\ (x, 0, 0.2 e^{i (0.3 π)}), & if x = 0 \\ (x, 0.05 e^{i (0.2 π)}, 0), & otherwise, \end{matrix}$

$({\tilde{F}}_{1} ⊖ {\tilde{F}}_{2}) (b)$ $= {\begin{matrix} (x, 0.1 e^{i 0.2 π}, 0.1 e^{i (0.2 π)}), & if x = 2 k; k \in ℤ / {0} \\ (x, 0.1 e^{i (0.2 π)}, 0.1 e^{i (0.2 π)}), & if x = 0 \\ (x, 0, 0.5 e^{i (0.1 π)}), & otherwise \end{matrix}$ Now, let x ∈ G. Then there exist x^-1 ∈ G such that $μ_{({\tilde{F}}_{1} ⊖ {\tilde{F}}_{2}) (a)} ({xx}^{- 1}) \geq μ_{({\tilde{F}}_{1} ⊖ {\tilde{F}}_{2}) (a)} (x)$ $\Rightarrow μ_{({\tilde{F}}_{1} ⊖ {\tilde{F}}_{2}) (a)} (e_{G}) \geq μ_{({\tilde{F}}_{1} ⊖ {\tilde{F}}_{2}) (a)} (x)$ .

But $μ_{({\tilde{F}}_{1} ⊖ {\tilde{F}}_{2}) (a)} (e_{G}) ≱ μ_{({\tilde{F}}_{1} ⊖ {\tilde{F}}_{2}) (a)} (x), for all x \in G .$ $∴ ({\tilde{F}}_{1}, E) ⊖ ({\tilde{F}}_{2}, E) \notin CIFSL - G (G) .$

Definition 3.9. Let $({\tilde{F}}_{1}, E)$ and $({\tilde{F}}_{2}, E) \in CIFSL - G (G)$ . Then their disjoint sum $({\tilde{F}}_{1}, E) \tilde{\oplus} ({\tilde{F}}_{2}, E) = (\tilde{F}, E)$ is defined as:

$\begin{matrix} μ_{\tilde{F} (a)} (x) = & | [r_{{\tilde{F}}_{1} (a)} (x) - r_{{\tilde{F}}_{2} (a)} (x)] \\ e^{imin {w_{{\tilde{F}}_{1} (a)} (x), w_{{\tilde{F}}_{2} (a)} (x)}} | and \\ ν_{\tilde{F} (a)} (x) = & | [τ_{{\tilde{F}}_{1} (a)} (x) - τ_{{\tilde{F}}_{2} (a)} (x)] \\ e^{imax {ψ_{{\tilde{F}}_{1} (a)} (x), w_{{\tilde{F}}_{2} (a)} (x)}} |, \end{matrix}$ for all a ∈ E, x ∈ G .

Theorem 3.10. The disjoint sum of two $CIFSL - G$ of G is always a intuitionistic fuzzy soft set of G.

Proof. Let $({\tilde{F}}_{1}, E)$ and $({\tilde{F}}_{2}, E) \in CIFSL - G (G)$ . Then by Definition 3.9, $({\tilde{F}}_{1}, E) \tilde{\oplus} ({\tilde{F}}_{2}, E) = (\tilde{F}, E)$ .

$| μ_{\tilde{F} (a)} (x) | = | r_{\tilde{F} (a)} (x) e^{{iw}_{\tilde{F} (a)} (x)} | = | r_{\tilde{F} (a)} (x) |$ $| ν_{\tilde{F} (a)} (x) | = | τ_{\tilde{F} (a)} (x) e^{i ψ_{\tilde{F} (a)} (x)} | = | τ_{\tilde{F} (a)} (x) |$ $[∵ | e^{{iw}_{\tilde{F} (a)} (x)} | = 1 and | e^{i ψ_{\tilde{F} (a)} (x)} | = 1]$ , for all a ∈ E. Hence, $(\tilde{F}, E)$ is forms a intuitionistic fuzzy soft set of G.□

4 CIFS-COPRAS method

The COPRAS method is introduced by Zavadskas et al. [23], which is an effective method for the decision process. The main benefits of the COPRAS method are: (1) it is elementary and entirely explicable; (2) it includes the ratios of the beneficial and the non-beneficial solutions simultaneously; (3) the time duration is minimum. Here, the Complex Intuitionistic Fuzzy Soft - COPRAS (CIFS-COPRAS) method, an extension of C- Copras method [11], is proposed.

Let U = {x₁, x₂, . . . , x_n} be the set of alternatives and E = {a₁, a₂, . . . , a_m} be the set of criteria for a decision problem and W = {w₁, w₂, . . . , w_m} be the associated weights for E with $\sum_{j = 1}^{m} w_{j} = 1$ . Suppose X be the initial complex intuitionistic fuzzy soft decision matrix $\begin{matrix} X & = (\begin{matrix} x_{11} & x_{12} & \dots & x_{1 m} \\ x_{21} & x_{22} & \dots & x_{2 m} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ x_{n 1} & x_{n 2} & \dots & x_{nm} \end{matrix}) \end{matrix}$ where x_ij = (μ_{a
_j} (x_i) , ν_{a
_j} (x_i)) indicates the initial assessment value for the alternative x_i with respect to the criteria a_j in terms of CIFSS, for i = 1, 2, . . . , n and j = 1, 2, . . . , m. Firstly, the normalised value is found for x_ij as defined below, ${\bar{x}}_{ij} = (μ_{a_{j}} ({\bar{x}}_{i}), ν_{a_{j}} ({\bar{x}}_{i})) = (\frac{μ_{a_{j}} (x_{i})}{\sum_{i = 1}^{n} | μ_{a_{j}} (x_{i}) |}, \frac{ν_{a_{j}} (x_{i})}{\sum_{i = 1}^{n} | ν_{a_{j}} (x_{i}) |}) .$

Next, the weighted normalised values are constructed by equation 5 with its associated weights w_j of a_j, where $w_{j} \in ℝ$ and $\sum_{j = 1}^{m} w_{j} = 1$ , for j = 1, 2, . . . , m, $\begin{matrix} {\hat{x}}_{ij} = (μ_{a_{j}} ({\hat{x}}_{i}), ν_{a_{j}} ({\hat{x}}_{i})) = (μ_{a_{j}} ({\bar{x}}_{i}) . w_{j}, ν_{a_{j}} ({\bar{x}}_{i}) . w_{j}) . \end{matrix}$ (5) For the next step, let us assume A⁺ be the set of positive direction (beneficial) criteria and A^- be the negative direction (non-beneficial) criteria for the decision problem. Then for each x_i, determine the optimisation indexes $S_{i}^{+}$ and $S_{i}^{-}$ with respect to A⁺ and A^-, respectively, i = 1, 2, . . . , n. In this method, the optimisation indexes will be calculated by the vector sum of weighted normalised membership and non-membership values. Once, $S_{i}^{-}$ is calculated, then found $S_{\min}^{-}$ , which is the minimum value of $S_{i}^{-}$ , i = 1, 2, . . . , n. Then based on the standard COPRAS method, the priority value Q_i is calculated for each x_i by equation 6, $\begin{matrix} Q_{i} = S^{*} (S_{i}^{+}) + \frac{S^{*} (S_{\min}^{-}) \sum_{a_{j} \in A^{-}} S^{*} (S_{i}^{-})}{S^{*} (S_{i}^{-}) \sum_{a_{j} \in A^{-}} (\frac{S^{*} (S_{\min}^{-})}{S^{*} (S_{i}^{-})})} . \end{matrix}$ (6) In equation 6, $S^{*} (x) = \frac{1}{4} (S (x) + 2)$ is the normalised score function, where $S^{*} (x) \in [0, 1]$ and $S (x) = (r - τ) + \frac{1}{2 π} (w - ψ)$ is the score function for CIFN x = (re^iw, τe^iψ) which is defined in [7]. And then, calculate the ranking index N_i, for each x_i by $N_{i} = \frac{Q_{i}}{Q_{\max}} \times 100 %$ , where Q_max ≠ 0 is the maximum value of Q_i. Finally, make the decision based on the ranking index N_i. Based on the above discussion we summarize the CIFS-COPRAS method as below:

Compute the normalised assessment value to the initial CIFSs decision matrix.

Calculate the weighted assessment value by equation 5.

Determine the optimisation indexes $S_{i}^{+}$ and $S_{i}^{-}$ for A⁺ and A^-, respectively.

Find the priority value Q_i, for each x_i by equation 6.

Make the final decision based on ranking index N_i.

4.1 Case study

In this segment, the proposed methodology is to be applied to a medical equipment selection process for a health organization. Medical equipment is a crucial asset for the healthcare organization. There is a need to know its associated management methodology to assure its safety and effectiveness. Nevertheless, a typical life cycle approach is problematic. The main issues impacting medical equipment selection are safe to use, quality, technical support, amount of non-biodegradable waste, and device policies. The selection of medical equipment should not be based on the hasty or insufficient decision, and also it is based on the latest version of the equipment. A senior executive person of the health organization has appointed as the expert to handle the equipment selection process. The CIFS-COPRAS method is applied to select the best medical equipment among the four alternatives x₁, x₂, x₃, x₄.

First, define the criteria and weights, directions for the selection process in Table 1 and define the linguistic expressions and values for the criteria in Table 2 and 3, respectively. The initial assessment values about the alternatives in the CIFSS structure is represented in Table 4. In Table 4, the amplitude terms in CIFSS may represent an expert’s decision regarding the alternative of equipment. The phase terms may be used to represent the expert’s decision regarding equipment’s latest version.

Table 1
Criteria for the Decision Problem With Its Directions and Predefined Weights (’+’ = Positive, ’-’ = Negative)

Criteria Description Weights Direction

a ₁ Policies Consistent with ethical policies and procedures 0.1 +

a ₂ Cost factor Equipment acquistion cost, maintenance cost, installation cost 0.17 -

a ₃ Non- biodegradable waste Non- biodegradable waste is the long-term threat to the environment 0.15 -

a ₄ Technology and Quality Health benefits and reduces the risk injury to patients 0.25 +

a ₅ Technical Support Vendor support and other technical support for maintenance 0.20 +

a ₆ Risk Management Patients have harm or potential harm caused by the device 0.13 -

	Criteria	Description	Weights	Direction
a ₁	Policies	Consistent with ethical policies and procedures	0.1	+
a ₂	Cost factor	Equipment acquistion cost, maintenance cost, installation cost	0.17	-
a ₃	Non- biodegradable waste	Non- biodegradable waste is the long-term threat to the environment	0.15	-
a ₄	Technology and Quality	Health benefits and reduces the risk injury to patients	0.25	+
a ₅	Technical Support	Vendor support and other technical support for maintenance	0.20	+
a ₆	Risk Management	Patients have harm or potential harm caused by the device	0.13	-

Table 2

Linguistic expressions for the criteria

Criteria	x ₁	x ₂	x ₃	x ₄
a ₁	M	L	M	VH
a ₂	VL	H	L	L
a ₃	L	M	L	H
a ₄	H	H	VH	H
a ₅	L	H	M	M
a ₆	M	L	VL	M

Table 3

Corresponding values for the linguistic expressions in CIF

	r	w	μ = rcos (w) + irsin (w)	τ	ψ	ν = τcos (ψ) + iτsin (ψ)
Very Low (VL)	0.2	0.2π	0.162+ i 0.118	0.7	0.45π	0.110+i0.691
Low (L)	0.3	0.2π	0.176+ i 0.243	0.5	0.4π	0.155+i0.476
Medium (M)	0.5	0.35π	0.227+ i 0.446	0.3	0.3π	0.176+i0.243
High (H)	0.6	0.42π	0.149+ i 0.581	0.35	0.2π	0.283+i0.206
Very High (VH)	0.8	0.5π	0.000+ i 0.800	0.1	0.1π	0.095+i0.030

Table 4

Initial assessments in CIFSS

	x ₁	x ₂	x ₃	x ₄
a ₁	(0.227,0.446)(0.176,0.243)	(0.176,0.243)(0.155,0.476)	(0.227,0.446)(0.176,0.243)	(0.000,0.800)(0.090,0.030)
a ₂	(0.162,0.118)(0.110,0.691)	(0.149,0.581)(0.283,0.206)	(0.176,0.243)(0.155,0.476)	(0.176,0.243)(0.155,0.476)
a ₃	(0.176,0.243)(0.155,0.476)	(0.227,0.446)(0.176,0.243)	(0.176,0.243)(0.155,0.476)	(0.000,0.800)(0.090,0.030)
a ₄	(0.149,0.581)(0.283,0.206)	(0.149,0.581)(0.283,0.206)	(0.000,0.800)(0.095,0.030)	(0.149,0.581)(0.283,0.206)
a ₅	(0.176,0.243)(0.155,0.476)	(0.149,0.581)(0.283,0.206)	(0.227,0.446)(0.176,0.243)	(0.227,0.446)(0.176,0.243)
a ₆	(0.227,0.446)(0.176,0.243)	(0.176,0.243)(0.155,0.476)	(0.162,0.118)(0.110,0.691)	(0.227,0.446)(0.176,0.243)

Normalised assessment value in Table 5.

As the above discussion, for j = 1,

\sum_{i = 1}^{n} | μ_{a_{1}} (x_{i}) | = 2.1

\sum_{i = 1}^{n} | ν_{a_{1}} (x_{i}) | = 1.2

. Hence, the normalisation result of x₁₁ is (0.108,0.212)(0.147,0.203). Similarly, find the normalised values for other criteria.

Weighted normalized assessment values in Table 6 by equation 5.

Optimisation indexes and its normalized score values in Table 7.

Table 5

Normalized Assessment Values

	x ₁	x ₂	x ₃	x ₄
a ₁	(0.108,0.212)(0.147,0.203)	(0.084,0.116)(0.129,0.397)	(0.108,0.212)(0.147,0.203)	(0.000,0.381)(0.079,0.025)
a ₂	(0.116,0.084)(0.054,0.337)	(0.106,0.415)(0.138,0.101)	(0.126,0.174)(0.076,0.232)	(0.126,0.174)(0.076,0.232)
a ₃	(0.093,0.128)(0.111,0.340)	(0.120,0.235)(0.126,0.174)	(0.093,0.128)(0.111,0.340)	(0.000,0.421)(0.064,0.021)
a ₄	(0.057,0.224)(0.246,0.179)	(0.057,0.224)(0.246,0.179)	(0.000,0.308)(0.083,0.026)	(0.057,0.224)(0.246,0.179)
a ₅	(0.093,0.128)(0.107,0.328)	(0.078,0.306)(0.195,0.142)	(0.120,0.235)(0.121,0.168)	(0.120,0.235)(0.121,0.168)
a ₆	(0.151,0.297)(0.098,0.135)	(0.117,0.162)(0.086,0.264)	(0.108,0.079)(0.061,0.384)	(0.151,0.297)(0.098,0.135)

Table 6

Weighted Normalized Assessment Values

	x ₁	x ₂	x ₃	x ₄
a ₁	(0.011,0.021)(0.015,0.020)	(0.008,0.012)(0.013,0.040)	(0.011,0.021)(0.015,0.020)	(0.000,0.038)(0.008,0.003)
a ₂	(0.020,0.014)(0.009,0.057)	(0.018,0.071)(0.024,0.017)	(0.021,0.030)(0.013,0.039)	(0.021,0.030)(0.013,0.039)
a ₃	(0.014,0.019)(0.017,0.051)	(0.018,0.035)(0.019,0.058)	(0.014,0.010)(0.017,0.051)	(0.000,0.063)(0.010,0.003)
a ₄	(0.014,0.056)(0.062,0.045)	(0.014,0.056)(0.062,0.045)	(0.000,0.077)(0.021,0.007)	(0.014,0.056)(0.062,0.045)
a ₅	(0.019,0.026)(0.021,0.066)	(0.016,0.061)(0.039,0.028)	(0.024,0.047)(0.024,0.034)	(0.024,0.047)(0.024,0.034)
a ₆	(0.020,0.039)(0.013,0.018)	(0.015,0.021)(0.011,0.034)	(0.014,0.010)(0.008,0.050)	(0.020,0.039)(0.013,0.018)

Table 7

Optimisation indexes and its normalized score values

	$S_{i}^{+}$	$S_{i}^{-}$	$S^{*} (S_{i}^{+})$	$S^{*} (S_{i}^{-})$
x ₁	(0.1120e^i2π(0.1857), 0.1636e^i2π(0.1478))	(0.0900e^i2π(0.1476), 0.1319e^i2π(0.2022))	0.4966	0.4759
x ₂	(0.1345e^i2π(0.2044), 0.1605e^i2π(0.1243))	(0.1369e^i2π(0.1892), 0.0940e^i2π(0.1527))	0.5135	0.5187
x ₃	(0.1490e^i2π(0.2123), 0.0856e^i2π(0.1263))	(0.0767e^i2π(0.1397), 0.1451e^i2π(0.2078))	0.5374	0.4659
x ₄	(0.1460e^i2π(0.2081), 0.1247e^i2π(0.1142))	(0.1382e^i2π(0.2021), 0.0569e^i2π(0.1409))	0.5288	0.5356

Find priority values Q_i, for each x_i by equation 6.

Q₁ = 1.0118, Q₂ = 0.9920, Q₃ = 1.0701 and Q₄ = 0.9919.

Determine the ranking index N_i.

N₁ = 94.55%, N₂= 92.70%, N₃= 100% and N₄= 92.69%. Therefore, N₃ ≻ N₁ ≻ N₂ ≻ N₄. Hence, x₃ is the best option selected by the senior executive person, then followed by x₁, x₂ and x₄.

4.2 Advantages of the proposed method

This section highlights some advantages of the proposed method, which are as follows:

The CIFS-COPRAS is used to handle the two-dimensional information in a single set with amplitude and phase terms. Here, the phase term represents the range of alternative, and the amplitude term represents its latest version. This cannot be represented accurately using traditional FSs and IFSs. This is the main advantage of the CIFS-COPRAS method. In this method the normalized score function under CIFSS environment is used, which is a generalization of existing score functions. It includes the ratios of the beneficial and the non-beneficial solutions simultaneously.

In order to verify the validity of the ranking results of the CIFS-COPRAS, the results were compared with some other methods [8 , 23]. The ranking results are shown in Table 8.

Table 8
The ranking of alternatives with respect to the existing approaches

x ₁ x ₂ x ₃ x ₄

COPRAS [23] 2 3 1 4

PIFSS-COPRAS method

with possibility = 1 [8] 2 4 1 3

CIFWA operator [7] 2 4 1 3

C- COPRAS [11] 3 4 1 2

Proposed CIFS-COPRAS 2 3 1 4

	x ₁	x ₂	x ₃	x ₄
COPRAS [23]	2	3	1	4
PIFSS-COPRAS method
with possibility = 1 [8]	2	4	1	3
CIFWA operator [7]	2	4	1	3
C- COPRAS [11]	3	4	1	2
Proposed CIFS-COPRAS	2	3	1	4

5 Conclusion

This paper illustrates several operations on $CIFSL - G$ s, namely sum, product, disjoint sum, bounded difference and bounded product and proposes specific results on these operations. Further, the research defines CIFS- COPRAS method and provides a numerical application to demonstrate the CIFS-COPRAS method. In future, this method can be extended with some new criteria and applies it to solve real or simulation problems.

Footnotes

Acknowledgment

The article has been written with the joint financial support of RUSA-Phase 2.0 grant sanctioned vide letter No.F 24-51/2014-U, Policy (TN Multi-Gen), Dept. of Edn. Govt. of India, Dt. 09.10.2018, UGC-SAP (DRS-I) vide letter No.F.510/8/ DRS - I /2016(SAP-I) Dt. 23.08.2016, DST-PURSE 2nd Phase programme vide letter No. SR/PURSE Phase 2/38 (G) Dt. 21.02.2017 and DST (FST - level I) 657876570 vide letter No.SR/FIST/MS-I/2018/17 Dt. 20.12.2018.

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Operations on complex intuitionistic fuzzy soft lattice ordered group and CIFS-COPRAS method for equipment selection process

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Complex intuitionistic fuzzy soft lattice ordered group

3 Operations on complex intuitionistic fuzzy soft lattice ordered group

4 CIFS-COPRAS method

Table 8 The ranking of alternatives with respect to the existing approaches x 1 x 2 x 3 x 4 COPRAS [23] 2 3 1 4 PIFSS-COPRAS method with possibility = 1 [8] 2 4 1 3 CIFWA operator [7] 2 4 1 3 C- COPRAS [11] 3 4 1 2 Proposed CIFS-COPRAS 2 3 1 4

Footnotes

Acknowledgment

References

Table 8
The ranking of alternatives with respect to the existing approaches

x ₁ x ₂ x ₃ x ₄

COPRAS [23] 2 3 1 4

PIFSS-COPRAS method

with possibility = 1 [8] 2 4 1 3

CIFWA operator [7] 2 4 1 3

C- COPRAS [11] 3 4 1 2

Proposed CIFS-COPRAS 2 3 1 4