Application of a preference relationship in decision-making based on intuitionistic fuzzy soft sets

Abstract

Intuitionistic fuzzy soft set (IFSS) provides more efficient and flexible tool than ordinary fuzzy soft set (FSS) to express decision makers’ opinions in real-world decision problems. Until now, different researchers have designed different algorithms for ranking objects in order to select the best option in group decision-making problems under intuitionistic fuzzy soft environment. In this paper, an adjustable algorithm is developed to handle the problem of group decision-making based on a preference relationship of IFSSs. The effectiveness of this approach in comparison with some existing methods is also demonstrated. To achieve the higher level of compatibility with the real-world problems, the proposed algorithm here is extended to the weighted case.

Keywords

Fuzzy soft set intuitionistic fuzzy soft set weighted intuitionistic fuzzy soft set preference relationship multi-attribute group decision-making

1 Introduction

Multi-attribute group decision-making (MAG-DM) refers to a collective process for finding the best alternative based on some attributes/criteria/parameters. Uncertain and imprecise data in real-world decision problems highlights using different theories dealing with uncertainty and vagueness, such as fuzzy set theory [24], interval-valued fuzzy set theory [27], intuitionistic fuzzy set theory [23], and rough set theory [4]. This list has not been completed if we do not add the theory of soft set proposed by Molodtsov [13] in 1999. A typical soft set can be seen as a parameterized family of subsets of a universal set of objects. This makes the soft set theory as a flexible tool for dealing with different areas, such as set theory [1 , 35, 42], topology [11 , 44–46], algebra [18 , 52], decision-making [12 , 59] and so on.

However, the inherent limitation of soft set theory for modeling the real-life problems have been motivated researchers to combine this new concept with the former theories dealing with uncertainty in order to introduce some new hybrid notions. In this regard, various types of extended soft set, such as fuzzy soft set [5 , 58], rough soft set [14], multi-fuzzy soft set [53], interval-valued fuzzy soft set [47], intuitionistic fuzzy soft set [37 , 50], and interval-valued intuitionistic fuzzy soft set [48] can be considered. In particular, the concept of intuitionistic fuzzy soft set (IFSS) comes up by combining the soft set theory with the theory of intuitionistic fuzzy set [23].

In recent years, intuitionistic fuzzy soft-based decision-making methods have received great attraction and accordingly various approaches have been proposed. Jiang at al. [49] considered initially adapting of IFSS for decision-making problems. They developed the notion of α-level soft set, presented firstly in [15, 16], to (α, β)-level soft set of an IFSS. Then, they applied the choice value technique (see [38]) to propose an approach for solving decision-making problems. Authors in [20, 21] introduced a new concept called intuitionistic fuzzy parameterized soft set in order to handle situations in which criteria can be considered as some fuzzy sets. Using rough set methodology, Zhang et al. [60] constructed an upper and a lower approximation sets based on (α, β)-level soft set of IFSSs and then, checked the process of consensus by using the soft union operator over these sets. A rough approach was then applied to solve decision-making problems. Inspired from intuitionistic fuzzy aggregation operators, Mao et al. [22] discussed two kinds of aggregation operators: (1) weighted arithmetic average operator and (2) weighted geometric average operator under three different cases of experts’ weights called as completely known, partly known, and completely unknown. Then, they computed the amount of choice value for each object to rank data and select the optimal option. Çağman and Karataş [34] defined some operations on IFSSs and then applied them to solve decision-making problems. Agarwal et al. [25] designed a method for medical diagnosis problems based on IFSS. They extended the concept of IFSS to the generalized intuitionistic fuzzy soft set by adding the moderator’s opinion degree, which shows the validity of information. And then, used the intuitionistic fuzzy aggregation operators to reach the consensus among experts. However, later, Khalil [3] and Yang et al. [54] showed some assertions proposed in [25] are not correct in general. In [40], authors proposed a similarity measure between IFSSs to check the process of consensus in medical diagnosis problems. Das and Kar [43] added the degree of confident for different experts to the IFSS and then, used the product and addition operators of IFSS matrices in order to obtain a collective opinion among decision makers which was defined as the full agreement of all experts.

Until now, most existing methods have used two strategies known as choice value and level soft set to solve decision-making problems. But, recently, Zahedi et al. [10] pointed out that the idea of choice value and level soft set are not suitable for all decision situations. In order to solve the problem of consensus, they constructed a preference relationship of FSSs and then, defined a new overall score value for selecting the optimal option in group decision-making problems.

In this paper, we develop the proposed approach in [10] to IFSSs. Our approach overcomes the problem of consensus by applying a preference relationship which is constructed by checking the agreement of most experts not necessarily all of them. We also propose a new overall score value in order to attain the process of selection in group decision-making problems. To reach this aim, the rest of this paper is organized as follows. In Section 2, we review some basic concepts which will be used along this work. In Section 3, we present a preorder relation and an equivalence relationship based on intuitionistic fuzzy soft sets and then discuss some of their properties. In sequel, in Section 4, a new overall score value and subsequently, a preference relationship are constructed and then, we design an algorithm to MAGDM based on IFSSs. Moreover, we compare our algorithm with existing approaches to group decision-making based on IFSS. In Section 5, we extend our proposed algorithm for weighted case where each parameter has its own weight. And finally, in Section 6, we highlight the advantages of our approach in comparison with the existingmethods.

2 Preliminaries

In this section we will review some basic definitions and properties of soft sets, fuzzy soft sets, and intuitionistic fuzzy soft sets. The reader is referred to see [7–9 , 39] for further details and information.

Throughout this paper, let X be the set of objects and E be the set of parameters. Moreover, let us show the set of all intuitionistic fuzzy subsets of X by $IF (X)$ and the unit interval [0, 1] by I. The abbreviations “IFSS”, “iff”, and “w.r.t” are used for “intuitionistic fuzzy soft set”, “if and only if”, and “with respect to”, respectively. Notation ¬P, pronounced “not P”, is referred to negation or complement of proposition P.

Definition 2.1. [13] A soft set over the set X is a pair F_E or (F, E) where F : E → 2^X is the mapping that assigns a subset of X to each parameter e ∈ E.

Definition 2.2. [36] A fuzzy soft set over X is a pair f_E or (f, E) where f : E → I^X is the mapping that assigns a fuzzy subset of X with membership function f_e : X → [0, 1] to each parameter e ∈ E.

By introducing the concept of intuitionistic fuzzy set [23] into the theory of soft sets, Maji et al. [37, 39] proposed the concept of intuitionistic fuzzy soft set (IFSS) as follow.

Definition 2.3. [37, 39] An intuitionistic fuzzy soft set (IFSS) over X is a pair f_E = (f, E) such that $f : E \to IF (X)$ is the mapping that assigns an intuitionistic fuzzy subset of X to each parameter e ∈ E.

If X = {x₁, ⋯ , x_m} and E = {e₁, ⋯ , e_n} are finite universes of objects and parameters, respectively. Then, the IFSS f_E = (f, E) can be presented by a tabular form as shown in Table 1. If f_E and g_E are two IFSSs over common universe X. Then, the IFSS union, IFSS intersection, and the IFSS complement of them, indicated by $f_{E} \tilde{\lor} g_{E}$ , $f_{E} \tilde{\land} g_{E}$ , and $f_{E}^{c}$ respectively, are defined as below: $\begin{matrix} f (e) \lor g (e) \\ = {〈 x, μ_{f (e)} (x) \lor μ_{g (e)} (x), ν_{f (e)} (x) \\ \land ν_{g (e)} (x) 〉 : x \in X} \\ f (e) \land g (e) \\ = {〈 x, μ_{f (e)} (x) \land μ_{g (e)} (x), ν_{f (e)} (x) \\ \lor ν_{g (e)} (x) 〉 : x \in X} \\ f^{c} (e) = {〈 x, ν_{f (e)} (x), μ_{f (e)} (x) 〉 : x \in X} \\ = {〈 x, μ_{f (\neg e)} (x), ν_{f (\neg e)} (x) 〉 : x \in X} \end{matrix}$ where e ∈ E, ∨ shows the fuzzy union (supremum), ∧ shows the fuzzy intersection (infimum), and ¬e means parameter “not e”. Moreover, f_E is an IFSS subset of g_E, denoted by $f_{E} \tilde{\leq} g_{E}$ , if for all e ∈ E and any x ∈ X we have μ_f(e) (x) ≤ μ_g(e) (x) and ν_f(e) (x) ≥ ν_g(e) (x).

Table 1
Tabular form of IFSS

X e ₁ e ₂ ⋯ e _n

x ₁ (μ_f(e₁) (x₁) , ν_f(e₁) (x₁)) (μ_f(e₂) (x₁) , ν_f(e₂) (x₁)) ⋯ (μ_{f(e_n)} (x₁) , ν_{f(e_n)} (x₁))

x ₂ (μ_f(e₁) (x₂) , ν_f(e₁) (x₂)) (μ_f(e₂) (x₂) , ν_f(e₂) (x₂)) ⋯ (μ_{f(e_n)} (x₂) , ν_{f(e_n)} (x₂))

⋮ ⋮ ⋮ ⋮

x _m (μ_f(e₁) (x_m) , ν_f(e₁) (x_m)) (μ_f(e₂) (x_m) , ν_f(e₂) (x_m)) ⋯ (μ_{f(e_n)} (x_m) , ν_{f(e_n)} (x_m))

X	e ₁	e ₂	⋯	e _n
x ₁	(μ_f(e₁) (x₁) , ν_f(e₁) (x₁))	(μ_f(e₂) (x₁) , ν_f(e₂) (x₁))	⋯	(μ_{f(e_n)} (x₁) , ν_{f(e_n)} (x₁))
x ₂	(μ_f(e₁) (x₂) , ν_f(e₁) (x₂))	(μ_f(e₂) (x₂) , ν_f(e₂) (x₂))	⋯	(μ_{f(e_n)} (x₂) , ν_{f(e_n)} (x₂))
⋮	⋮	⋮	⋮
x _m	(μ_f(e₁) (x_m) , ν_f(e₁) (x_m))	(μ_f(e₂) (x_m) , ν_f(e₂) (x_m))	⋯	(μ_{f(e_n)} (x_m) , ν_{f(e_n)} (x_m))

Definition 2.4. Let Λ = (f, E) be an IFSS over the universe X. If (α, β) ∈ [0, 1] × [0, 1] is a given threshold value pair on membership and non-membership degrees, then (α, β)-level soft set of Λ is a soft set denoted by L (Λ ; α, β) = (f_(α,β), E) and defined by $\begin{matrix} f_{(α, β)} (e) & = & L (f (e); α, β) \\ = & {x \in X : μ_{f_{e}} (x) \geq α, ν_{f_{e}} (x) \leq β} \end{matrix}$ for any e ∈ E.

Note that, threshold value pair (α, β) can be presented as an intuitionistic fuzzy set γ : E → [0, 1] × [0, 1], called threshold intuitionistic fuzzy set, that assigns the value (α (e) , β (e)) ∈ [0, 1] × [0, 1] to each parameter e ∈ E.

3 Using a (α, β)-preorder relation and a (α, β)-equivalence relation for consensus

In this section, a preorder relation and subsequently an equivalence relationship are defined to attain an IFSS-based consensus method in group decision-making. To begin with, we recall the IFSS and the intuitionistic fuzzy relation are close.

Remark 3.1. [60] Let f_E be an IFSS over X. Then, f_E indicates a binary intuitionistic fuzzy relation R from E to X which is defined by R_f : E × X → I × I such that ∀ (e, x) ∈ E × X : (μ_{R
_f} (e, x) , ν_{R
_f} (e, x)) = (μ_f(e) (x) , ν_f(e) (x)). Conversely, if R is an intuitionistic fuzzy relation from E to X, then the set-valued mapping $f_{R} : E \to IF (X)$ defined by f_R (e) (x) = (μ_R (e, x) , ν_R (e, x)) , ∀ e ∈ E, ∀ x ∈ X is an IFSS over X.

If R is an intuitionistic fuzzy relation from E to X, induced by IFSS f_E, then by applying the concept of level set at any thresholds α, β ∈ I, we can define some binary relations between E and X as follow. $\begin{matrix} R_{α} & = & {(e, x) \in E \times X : μ_{f (e)} (x) \geq α} \\ R_{α^{+}} & = & {(e, x) \in E \times X : μ_{f (e)} (x) > α} \\ R^{β} & = & {(e, x) \in E \times X : ν_{f (e)} (x) \leq β} \\ R^{β^{+}} & = & {(e, x) \in E \times X : ν_{f (e)} (x) < β} \\ R_{α}^{β} & = & {(e, x) \in E \times X : μ_{f (e)} (x) \\ \geq & α, ν_{f (e)} (x) \leq β} \\ R_{α^{+}}^{β^{+}} & = & {(e, x) \in E \times X : μ_{f (e)} (x) \\ > & α, ν_{f (e)} (x) < β} \end{matrix}$

For example, R_α⁺ indicates all elements x ∈ X that are related to the parameter “e” more than α, while R^β⁺ shows all objects x ∈ X which are related to the parameter “not e” less than β. The below sets are also can be obtained. $\begin{matrix} R_{α} (e) & = & {x \in X : μ_{f (e)} (x) \geq α} \\ R_{α^{+}} (e) & = & {x \in X : μ_{f (e)} (x) > α} \\ R^{β} (e) & = & {x \in X : ν_{f (e)} (x) \leq β} \\ R^{β^{+}} (e) & = & {x \in X : ν_{f (e)} (x) < β} \\ R_{α}^{β} (e) & = & {x \in X : μ_{f (e)} (x) \geq α, ν_{f (e)} (x) \leq β} \\ R_{α^{+}}^{β^{+}} (e) & = & {x \in X : μ_{f (e)} (x) > α, ν_{f (e)} (x) < β} \end{matrix}$

It is clear that, each of these sets partitions the set X into some disjoint subsets. According to each partition, an equivalence relation can be induced over X as below. $\begin{matrix} {\tilde{R}}_{α} (e) & = & {(x, y) \in X \times X : {x, y} \subseteq R_{α} (e)} \\ {\tilde{R}}_{α^{+}} (e) & = & {(x, y) \in X \times X : {x, y} \subseteq R_{α^{+}} (e)} \\ {\tilde{R}}^{β} (e) & = & {(x, y) \in X \times X : {x, y} \subseteq R^{β} (e)} \\ {\tilde{R}}^{β^{+}} (e) & = & {(x, y) \in X \times X : {x, y} \subseteq R^{β^{+}} (e)} \\ {\tilde{R}}_{α}^{β} (e) & = & {(x, y) \in X \times X : {x, y} \subseteq R_{α}^{β} (e)} \\ {\tilde{R}}_{α^{+}}^{β^{+}} (e) & = & {(x, y) \in X \times X : {x, y} \subseteq R_{α^{+}}^{β^{+}} (e)} . \end{matrix}$

Proposition 3.1. Let f_E be an IFSS and R be an intuitionistic fuzzy relation from E to X induced by f_E. Let α, β ∈ [0, 1] are given. Then

If α = 0, β = 1, then we have the trivial case $R_{α}^{β} = R_{α} = R^{β} = E \times X$ , $R_{α}^{β} (e) = R_{α} (e) = R^{β} (e) = X$ , and ${\tilde{R}}_{α}^{β} (e) = {\tilde{R}}_{α} (e) = {\tilde{R}}^{β} (e) = X$ .

$R_{α}^{β} = R_{α} \cap R^{β}$ , $R_{α}^{β} (e) = R_{α} (e) \cap R^{β} (e)$ , and ${\tilde{R}}_{α}^{β} (e) = {\tilde{R}}_{α} (e) \cap {\tilde{R}}^{β} (e)$ .

$R_{α^{+}}^{β^{+}} = R_{α^{+}} \cap R^{β^{+}}, R_{α^{+}}^{β^{+}} (e) = R_{α^{+}} (e) \cap R^{β^{+}} (e)$ ,and ${\tilde{R}}_{α^{+}}^{β^{+}} (e) = {\tilde{R}}_{α^{+}} (e) \cap {\tilde{R}}^{β^{+}} (e)$ .

If α₁ ≤ α₂, then R_α₂ ⊆ R_α₁, R_α₂ (e) ⊆ R_α₁ (e), and ${\tilde{R}}_{α_{2}} (e) \subseteq {\tilde{R}}_{α_{1}} (e)$ . Similarly, $R_{α_{2}^{+}} \subseteq R_{α_{1}^{+}}$ , $R_{α_{2}^{+}} (e) \subseteq R_{α_{1}^{+}} (e)$ , and ${\tilde{R}}_{α_{2}^{+}} (e) \subseteq {\tilde{R}}_{α_{1}^{+}} (e)$ .

If β₁ ≤ β₂, then R^β₁ ⊆ R^β₂, R^β₁ (e) ⊆ R^β₂ (e), and ${\tilde{R}}^{β_{1}} (e) \subseteq {\tilde{R}}^{β_{2}} (e)$ . Similarly, $R^{β_{1}^{+}} \subseteq R^{β_{2}^{+}}$ , $R^{β_{1}^{+}} (e) \subseteq R^{β_{2}^{+}} (e)$ , and ${\tilde{R}}^{β_{1}^{+}} (e) \subseteq {\tilde{R}}^{β_{2}^{+}} (e)$ .

Proof 1. It is straightforward.

Proposition 3.2. Let R be an intuitionistic fuzzy relation from E to X which is induced by IFSSf_E and α, β ∈ [0, 1] are given. Then

If $f_{E} = \tilde{X}$ , then we have the trivial case $R_{α} = R_{α^{+}} = R^{β} = R^{β^{+}} = R_{α}^{β} = R_{α^{+}}^{β^{+}} = E \times X$ , $R_{α} (e) = R_{α^{+}} (e) = R^{β} (e) = R^{β^{+}} (e) = R_{α}^{β} (e) = R_{α^{+}}^{β^{+}} (e) = X$ , and ${\tilde{R}}_{α} (e) = {\tilde{R}}_{α^{+}} (e) = {\tilde{R}}^{β} (e) = {\tilde{R}}^{β^{+}} (e) = {\tilde{R}}_{α}^{β} (e) = {\tilde{R}}_{α^{+}}^{β^{+}} (e) = X$ .

If f_E = Φ, then we have $R_{α} = R_{α^{+}} = R^{β} = R^{β^{+}} = R_{α}^{β} = R_{α^{+}}^{β^{+}} = E \times \emptyset, R_{α} (e) = R_{α^{+}} (e) = {\overset{f}{R}}^{β} (e) = R^{β^{+}} (e) = R_{α}^{β} (e) = {\overset{f}{R}}_{α^{+}}^{β^{+}} (e) = \emptyset$ , and ${\tilde{R}}_{α} (e) = {\tilde{R}}_{α^{+}} (e) = {\tilde{R}}^{β} (e) = {\tilde{R}}^{β^{+}} (e) = {\tilde{R}}_{α}^{β} (e) = {\tilde{R}}_{α^{+}}^{β^{+}} (e) = \emptyset$ .

If ${f_{1}}_{E} \tilde{\leq} {f_{2}}_{E}$ , then R₁_α ⊆ R₂_α, R₁_α (e) ⊆ R₂_α (e), ${\tilde{R_{1}}}_{α} (e) \subseteq {\tilde{R_{2}}}_{α} (e)$ , R₁^β ⊆ R₂^β, R₁^β (e) ⊆ R₂^β (e), ${\tilde{R}}_{1}^{β} (e) \subseteq {\tilde{R}}_{2}^{β} (e)$ , ${R_{1}}_{α}^{β} \subseteq {R_{2}}_{α}^{β}$ , ${R_{1}}_{α}^{β} (e) \subseteq {R_{2}}_{α}^{β} (e)$ , and ${\tilde{R_{1}}}_{α}^{β} (e) \subseteq {\tilde{R_{2}}}_{α}^{β} (e)$ . Moreover, R₁_α⁺ ⊆ R₂_α⁺, R₁_α⁺ (e) ⊆ R₂_α⁺ (e), ${\tilde{R_{1}}}_{α^{+}} (e) \subseteq {\tilde{R_{2}}}_{α^{+}} (e)$ , R₁^β⁺ ⊆ R₂^β⁺, R₁^β⁺ (e) ⊆ R₂^β⁺ (e), ${\tilde{R_{1}}}^{β^{+}} (e) \subseteq {\tilde{R_{2}}}^{β^{+}} (e)$ , ${R_{1}}_{α^{+}}^{β^{+}} \subseteq {R_{2}}_{α^{+}}^{β^{+}}$ , ${R_{1}}_{α^{+}}^{β^{+}} (e) \subseteq {R_{2}}_{α^{+}}^{β^{+}} (e)$ , and ${\tilde{R_{1}}}_{α^{+}}^{β^{+}} (e) \subseteq {\tilde{R_{2}}}_{α^{+}}^{β^{+}} (e)$ .

Proof 2. It is straightforward.

Theorem 3.1. Let {f₁_E, …, f_k_E} be a family of IFSSs such that ${f_{1}}_{E} \tilde{\leq} {f_{2}}_{E} \tilde{\leq} \dots \tilde{\leq} {f_{k}}_{E}$ and (α, β) ∈ [0, 1] × [0, 1] be a threshold value pair.Then

R₁_α (e) ⊆ R₂_α (e) ⊆ … ⊆ R_k_α (e) and R₁_α⁺ (e) ⊆ R₂_α⁺ (e) ⊆ … ⊆ R_k_α⁺ (e).

R₁^β (e) ⊆ R₂^β (e) ⊆ … ⊆ R_k^β (e) and R₁^β⁺ (e) ⊆ R₂^β⁺ (e) ⊆ … ⊆ R_k^β⁺ (e).

${R_{1}}_{α}^{β} (e) \subseteq {R_{2}}_{α}^{β} (e) \subseteq \dots \subseteq {R_{k}}_{α}^{β} (e)$ and ${R_{1}}_{α^{+}}^{β^{+}} (e) \subseteq {R_{2}}_{α^{+}}^{β^{+}} (e) \subseteq \dots \subseteq {R_{k}}_{α^{+}}^{β^{+}} (e)$ .

Proof 3.

Let z ∈ R₁_α (e). Then μ_R
₁ (e, z) = μ_f₁(e) (z) ≥ α, so for all 1 ≤ s ≤ k, μ_{f_s(e)} (z) ≥ α. Thus for all 1 ≤ s ≤ k, z ∈ R_s_α (e) or R₁_α (e) ⊆ R₂_α (e) ⊆ … ⊆ R_k_α (e). Similarly, we have R₁_α⁺ (e) ⊆ R₂_α⁺ (e) ⊆ … ⊆ R_k_α⁺ (e).

By substituting R_s^β (e) for R_s_α (e) and R_s^β⁺ (e) for R_s_α⁺ (e) where 1 ≤ s ≤ k the second part can be proved similarly.

It is obvious by (i) and (ii).

According to each threshold value α and β, authors in [10] defined two preorder relations over the universal set X. Here, we develop the similar methodology to construct a preorder relation by using (α, β)-level soft set of IFSSs.

Theorem 3.2. Let X be the set of objects and E be the set of parameters. Suppose that (α, β) ∈ [0, 1] × [0, 1] be a given threshold value pair. Assume that {f_ι_E} _ι∈J be a family of IFSSs where J is an index set. Then $(X, \geq_{α}^{β} (e))$ such that $x \geq_{α}^{β} (e) y \Leftrightarrow [\forall ι \in J, y \in {R_{ι}}_{α}^{β} (e) \Rightarrow x \in {R_{ι}}_{α}^{β} (e)]$ (1) a preordered set.

Proof 4. It is clear that for all x ∈ X, $x \geq_{α}^{β} (e) x$ (reflexivity). Now let for x, y, z ∈ X, $x \geq_{α}^{β} (e) y$ and $y \geq_{α}^{β} (e) z$ . Then we have ${f_{ι} : y \in {R_{ι}}_{α}^{β} (e)} \subseteq {f_{ι} : x \in {R_{ι}}_{α}^{β} (e)}$ and ${f_{ι} : z \in {R_{ι}}_{α}^{β} (e)} \subseteq {f_{ι} : y \in {R_{ι}}_{α}^{β} (e)}$ . Thus, ${f_{ι} : z \in {R_{ι}}_{α}^{β} (e)} \subseteq {f_{ι} : x \in {R_{ι}}_{α}^{β} (e)}$ and so $x \geq_{α}^{β} (e) z$ . This shows the transitivity. Moreover, it is obvious that $\geq_{α}^{β} (e)$ is not a symmetric relation.

If for all x ∈ X, $α \leq inf_{ι, x} μ_{f_{ι} (e)} (x)$ and $β \geq sup_{ι, x} ν_{f_{ι} (e)} (x)$ , then ∀x, y ∈ X we have $x \geq_{α}^{β} (e) y$ . Specially, for case (α, β) = (0, 1) we have the trivial case $\geq_{α}^{β} (e) = X^{2}$ .

Corollary 3.1. Let {f_ι_E} _ι∈J, where J is an index set, be a family of IFSS and “ $\geq_{α}^{β} (e)$ ” be the preorder relation defined in Theorem 3. Then, we have the following properties.

$x \geq_{α}^{β} (e) y \Leftrightarrow {f_{ι} : y \in {R_{ι}}_{α}^{β} (e)} \subseteq {f_{ι} : x \in {R_{ι}}_{α}^{β} (e)}$ .

$x \geq_{α}^{β} (e) y \Leftrightarrow {f_{ι} : x \in {R_{ι}^{c}}_{β^{+}}^{α^{+}} (e)} \subseteq {f_{ι} : y \in {R_{ι}^{c}}_{β^{+}}^{α^{+}} (e)}$ .

where ${R_{ι}^{c}}_{β^{+}}^{α^{+}} (e)$ is used to show the partition ${R_{ι}^{c}}_{β^{+}}^{α^{+}} (e) = {x \in X : μ_{f_{ι}^{c} (e)} (x) > β, ν_{f_{ι}^{c} (e)} (x) < α}$ that is induced by ${f_{ι}}_{E}^{c}$ , the complement of IFSSf_ι_E, over the set X.

Proposition 3.3. Let {f₁_E, …, f_k_E} be a finite set of some IFSSs such that ${f_{1}}_{E} \tilde{\leq} {f_{2}}_{E} \tilde{\leq} \dots \tilde{\leq} {f_{k}}_{E}$ . Then

For all $x \in {R_{1}}_{α}^{β} (e)$ , $x \geq_{α}^{β} (e) y$ , ∀y ∈ X.

For all $x \in {R_{k}^{c}}_{β^{+}}^{α^{+}} (e)$ , $y \geq_{α}^{β} (e) x$ , ∀y ∈ X.

where ${R_{1}}_{α}^{β} (e)$ and ${R_{k}^{c}}_{β^{+}}^{α^{+}} (e)$ are used to show the partitions ${R_{1}}_{α}^{β} (e) = {x \in X : μ_{f_{1} (e)} (x) \geq α, ν_{f_{1} (e)} (x) \leq β}$ and ${R_{k}^{c}}_{β^{+}}^{α^{+}} (e) = {x \in X : μ_{f_{k}^{c} (e)} (x) > β, ν_{f_{k}^{c} (e)} (x) < α}$ over the set X.

Proof 5.

Let $x \in {R_{1}}_{α}^{β} (e)$ . So, μ_R
₁ (e, x) ≥ α and ν_R
₁ (e, x) ≤ β or equivalently μ_f₁(e) (x) ≥ α and ν_f₁(e) (x) ≤ β. Thus, for all s, 1 ≤ s ≤ k we have μ_{f_s(e)} (x) ≥ α and ν_{f_s(e)} (x) ≤ β result from ${f_{1}}_{E} \tilde{\leq} {f_{2}}_{E} \tilde{\leq} \dots \tilde{\leq} {f_{k}}_{E}$ (since for any z ∈ X, μ_f₁(e) (z) ≤ μ_f₂(e) (z) ≤ ⋯ ≤ μ_{f_k(e)} (z) and ν_f₁(e) (z) ≥ ν_f₂(e) (z) ≥ ⋯ ≥ ν_{f_k(e)} (z)). So, ∀s ; 1 ≤ s ≤ k, $x \in {R_{s}}_{α}^{β} (e)$ . Now, let y ∈ X such that for some t, 1 ≤ t ≤ k, $y \in {R_{t}}_{α}^{β} (e)$ . So, it is clear that $x \in {R_{t}}_{α}^{β} (e)$ . This implies that $x \geq_{α}^{β} (e) y$ .

By substituting ${R_{k}^{c}}_{β^{+}}^{α^{+}} (e)$ for ${R_{1}}_{α}^{β} (e)$ the second part can be proved similarly.

Theorem 3.3. Let X be the set of objects and E be the set of parameters. Suppose that (α, β) ∈ [0, 1] × [0, 1] be a given threshold value pair. Assume that {f_ι_E} _ι∈J be a family of IFSSs where J is an index set. Then, the binary relation “ $≃_{α}^{β} (e)$ ” defined by

x ≃_{α}^{β} (e) y \Leftrightarrow x \geq_{α}^{β} (e) y, y \geq_{α}^{β} (e) x

(2)

is an equivalence relation on X.

Corresponding to each parameter e ∈ E, there exists an associated partition $P_{e}^{α, β}$ of X where for any x ∈ X the equivalence classes of $P_{e}^{α, β}$ are described as $[x]_{e}^{α, β} = {z \in X : z ≃_{α}^{β} (e) x}$ .

Proof 6. It is an immediate result from Theorem 3.2.

It is clear that if $α \leq inf_{ι, x} μ_{f_{ι} (e)} (x)$ and $β \geq sup_{ι, x} ν_{f_{ι}^{c} (e)} (x)$ , then ∀x, y ∈ X, $x ≃_{α}^{β} (e) y$ . Specially, for case (α, β) = (0, 1) we have the trivial case $≃_{α}^{β} (e) = X^{2}$ and thus trivial partition {X} corresponding to each parameter e.

Corollary 3.2. Let {f_ι_E} _ι∈J, where J is an index set, be a family of IFSSs and “ $≃_{α}^{β} (e)$ ” be the equivalence relation defined in Theorem 3.3. Then $x ≃_{α}^{β} (e) y \Leftrightarrow {f_{ι} : y \in {R_{ι}}_{α}^{β} (e)} = {f_{ι} : x \in {R_{ι}}_{α}^{β} (e)}$

The intersection of all partitions $P_{e}^{α, β}$ among the parameter set E is again a partition over X that classifies the set of decision alternatives X into the blocks having the same description in terms of the whole set of parameters. This issue is given in the following definition.

Definition 3.1. Let X be the universal set of objects and E be a parameter set. Suppose that (α, β) ∈ [0, 1] × [0, 1] is given as a threshold value pair. Then $P_{E}^{α, β}$ , defined as below, is a partition of the set X. $P_{E}^{α, β} = ⋂_{e \in E} P_{e}^{α, β}$ (3)

So, we can say if the objects x and y of X are in the same classes of all partitions $P_{e}^{α, β}$ for any e ∈ E, then they are in the same class of the partition $P_{E}^{α, β}$ . It is clear that if $α \leq inf_{ι, x} μ_{f_{ι} (e)} (x)$ and $β \geq sup_{ι, x} ν_{f_{ι}^{c} (e)} (x)$ , then $P_{E}^{α, β} = {X}$ . Specially, for case (α, β) = (0, 1) we have the trivial case $P_{E}^{α, β} = {X}$ .

4 IFSS-based method to MAGDM

In this section, we apply the proposed binary relations in Theorems 3.2 and 3.3 to define a new overall score value and a preference relationship for ranking data and selecting the best option in MAGDM based on IFSSs. We then, design an adjustable approach for finding an optimal option in group decision-making problems based on intuitionistic fuzzy soft information. Additionally, some examples are given to illustrate the application of this method in MAGDM based on IFSSs.

4.1 Ordering and selecting with a preference relationship of IFSS

Let X = {x₁, x₂, ⋯ , x_m} and E = {e₁, e₂, ⋯ , e_n} be the universal sets of objects and parameters, respectively, and D = {d₁, d₂, …, d_k} be a set of k experts who evaluate the elements of X on the basis of parameter set E by IFSSs f₁_E, f₂_E, …, f_k_E.

4.1.1 Two comparison matrices

We utilize the binary relations “ $\geq_{α}^{β} (e)$ ” and “ $≃_{α}^{β} (e)$ ” in order to create two square matrices, known as comparison matrices, as follows.

Definition 4.1. The preorder relation “ $\geq_{α}^{β} (e_{t})$ ” and the equivalence relation “ $≃_{α}^{β} (e_{t})$ ” create comparison matrices $G_{α}^{β} (e_{t})$ and ${EQ}_{α}^{β} (e_{t})$ , respectively. These matrices are two square matrices where rows and columns of them are labeled by objects of X and are defined as below. $G_{α}^{β} (e_{t}) = [{g_{ij}}_{α}^{β} (e_{t})]_{m \times m}$ where ${g_{ij}}_{α}^{β} (e_{t}) = {\begin{matrix} 1 & if x_{i} \geq_{α}^{β} (e_{t}) x_{j} \\ 0 & otherwise \end{matrix}$ (4) and ${EQ}_{α}^{β} (e_{t}) = [{eq}_{ij}_{α}^{β} (e_{t})]_{m \times m}$ such that ${eq}_{ij}_{α}^{β} (e_{t}) = {\begin{matrix} 1 & if x_{i} ≃_{α}^{β} (e_{t}) x_{j} \\ 0 & otherwise \end{matrix}$ (5) for i, j = 1, ⋯ , m.

Proposition 4.1. Let $G_{α}^{β} (e_{t})$ and ${EQ}_{α}^{β} (e_{t})$ be the comparison matrices introducing in Definition 4.1. Then, for any e_t ∈ E, 1 ≤ t ≤ n,

All main diagonal entries are equal to 1: ${g_{ii}}_{α}^{β} (e_{t}) = 1$ and ${eq}_{ii}_{α}^{β} (e_{t}) = 1$ , 1 ≤ i ≤ m.

If ${g_{ij}}_{α}^{β} (e_{t}) = {g_{js}}_{α}^{β} (e_{t}) = 1$ , then ${g_{is}}_{α}^{β} (e_{t}) = 1$ . Similarly, if ${eq}_{ij}_{α}^{β} (e_{t}) = {eq}_{js}_{α}^{β} (e_{t}) = 1$ , then ${eq}_{is}_{α}^{β} (e_{t}) = 1$ .

If ${eq}_{is}_{α}^{β} (e_{t}) = {eq}_{js}_{α}^{β} (e_{t}) = 1$ , then ${eq}_{ij}_{α}^{β} (e_{t}) = 1$ . If ${eq}_{si}_{α}^{β} (e_{t}) = {eq}_{sj}_{α}^{β} (e_{t}) = 1$ , then ${eq}_{ij}_{α}^{β} (e_{t}) = 1$ .

${EQ}_{α}^{β} (e_{t})$ is a symmetric matrix, while $G_{α}^{β} (e_{t})$ is symmetric if the binary relation “ $\geq_{α}^{β} (e_{t})$ ” is symmetric.

Proof 7. It is straightforward.

4.1.2 Overall score value and preference relationship

In this subsection, we propose a new overall score value and a preference relationship for ranking objects.

Definition 4.2. The mapping $S : X \to ℝ$ defined by

$\begin{matrix} S (x_{i}) = S_{i} & = & \sum_{t = 1}^{n} r_{i} (e_{t}; α, β) \\ = & \sum_{t = 1}^{n} [\sum_{j = 1}^{m} {g_{ij}}_{α}^{β} (e_{t}) - \sum_{j = 1}^{m} {eq}_{ij}_{α}^{β} (e_{t})] \end{matrix}$ (6) assigns the overall score values S_i to all elements x_i of X.

Algorithm 1: MAGDM based on IFSS
Input:
Step 1.	n= Size of parameter set E, m= Size of object set X, k= Size of decision maker set D,
Matrix form of IFSSf_s_E, 1 ≤ s ≤ k.
Step 2.	Threshold intuitionistic fuzzy set γ : E → [0, 1] × [0, 1] such that μ_γ (e_t) = α_t and ν_γ (e_t) = β_t
(or threshold value pair (α, β) ∈ [0, 1] × [0, 1]).
Begin
For t = 1, 2, …, n, i = 1, 2, …, m, and s = 1, 2, …, k:
Step 3.	Compute R_s_γ (e_t) = {x ∈ X : μ_{f_s(e_t)} (x) ≥ μ_γ (e_t) , ν_{f_s(e_t)} (x) ≤ ν_γ (e_t)}
(or ${R_{s}}_{α}^{β} (e_{t}) = {x \in X : μ_{f_{s} (e_{t})} (x) \geq α, ν_{f_{s} (e_{t})} (x) \leq β}$ ).
Step 4.	Compute matrices $G_{α_{t}}^{β_{t}} (e_{t})$ and ${EQ}_{α_{t}}^{β_{t}} (e_{t})$ (or $G_{α}^{β} (e_{t})$ and ${EQ}_{α}^{β} (e_{t})$ ) (Definition 4.1).
Step 5.	Compute the amount of S_i, ∀i (Definition 4.2).
Step 6.	According to the values of S_i, rank the set of objects X (Theorem 4.1).
Step 7.	The optimum is to select object x_h such that $S_{h} = max_{i} S_{i}$ .
The alternative x_l such that $S_{l} = min_{i} S_{i}$ should not be selected.
Step 8.	If the number of elements that S_h is maximum is more than one, then any one of such objects may be chosen.
Output:	Optimum and worst objects.
End

The value r_i (e_t ; α, β) in (6) shows the score value of object x_i w.r.t the specific parameter e_t. Obviously, for any pair of elements x_i and x_j in X if $x_{i} ≃_{α}^{β} (e_{t}) x_{j}$ for some e_t ∈ E, then r_i (e_t ; α, β) = r_j (e_t ; α, β). Consequently, if for all e_t ∈ E, 1 ≤ t ≤ n, we have $x_{i} ≃_{α}^{β} (e_{t}) x_{j}$ , then S_i = S_j. This shows if objects x_i and x_j have the same descriptions based on all decision parameters e_t, then their utility level will be the same. A preference relationship can be also defined as below.

Theorem 4.1. The binary relation “≽_E” on the universal set X defined as $x_{i} ≽_{E} x_{j} \Leftrightarrow S_{i} \geq S_{j}$ (7) a preference relationship on X where S_i and S_j refer as the overall score value of objects x_i and x_j, respectively.

Proof 8. It is straightforward.

By Theorem 4.1, the notation x ≽ _Ey is used to show x is preferred to y based on a collective judgment received by agreement of most decision makers, while the symbol ∼_E is applied to present an indifference relation on X such that $x \sim_{E} y \Leftrightarrow [x ≽_{E} y, y ≽_{E} x]$

4.2 An adjustable algorithm for MAGDM based on IFSS

The main aim in a decision-making problem is to select the best option among a list of choices. Since each decision maker has his/her own opinion about the choice set X, the central goal in any MAGDM problems is to put all of these information together to come up with a collective preference relationship to decide about all alternatives. Such collective preference relation allows decision makers to order the choice set X, from the most preferred object to the least preferred object, and to choose the bestoption.

In what follows, we shall utilize the preference relationship ≽_E which is determined by $x_{i} ≽_{E} x_{j} \Leftrightarrow S_{i} \geq S_{j}$ to design an adjustable algorithm for solving MAGDM based on IFSSs. For simplicity, we assume that E is a set of positive parameters. By developing the proposed algorithm in [10] to IFSSs, the following algorithm, called Algorithm 1, is designed.

Algorithm 1

There are three remarks here.

Firstly, in Algorithm 1, we utilize a rational weak preference relationship, instead of using a linear ordering system, to rank objects from the highest amount of utility to the lowest.

Secondly, at step 8 of Algorithm 1, we can go back to the step 2 and enter different amount for the threshold value pair (α, β) in order to adjust the final decision.

Thirdly, by taking ν_{f_s(e_t)} (x) =1 - μ_{f_s(e_t)} (x), Algorithm 1 can be applied successfully for FSSs. To illustrate the idea of Algorithm 1, let us to consider the following example.

Example 4.1. Suppose that the mathematics department of university X wants to fill an academic position known as “senior lecturer”. Let 5 applicants, having Ph.D. in Mathematics and denoted by A = {a₁, a₂, a₃, a₄, a₅}, applied for this post. The list of qualifications includes “teaching skill”, “knowledge and skill in database programs”, “strong publication record in top journals”, and “research ability in mathematics” is characterized by the set of parameters E = {e₁, e₂, e₃, e₄}. Three experts from mathematics department, f₁, f₂, and f₃, are invited to qualify these applicants for this post. These three experts provide the following three IFSS matrices f₁_E, f₂_E, and f₃_E to demonstrate their opinion about the list of candidates (Step 1). $\begin{matrix} {f_{1}}_{E} & = & (\begin{matrix} (0.6, 0.2) & (0.7, 0.1) & (0.3, 0.3) & (0.7, 0.2) \\ (0.5, 0.4) & (0.6, 0.4) & (0.8, 0.1) & (0.5, 0.2) \\ (0.7, 0.2) & (0.2, 0.4) & (0.5, 0.2) & (0.8, 0.2) \\ (0.7, 0.1) & (0.7, 0.2) & (0.4, 0.5) & (0.7, 0.1) \\ (0.5, 0.4) & (0.8, 0.1) & (0.5, 0.3) & (0.5, 0.2) \end{matrix}) \\ {f_{2}}_{E} & = & (\begin{matrix} (0.6, 0.1) & (0.5, 0.2) & (0.3, 0.2) & (0.7, 0.1) \\ (0.5, 0.2) & (0.6, 0.3) & (0.7, 0.3) & (0.6, 0.3) \\ (0.5, 0.4) & (0.3, 0.3) & (0.5, 0.3) & (0.7, 0.2) \\ (0.6, 0.3) & (0.7, 0.2) & (0.5, 0.4) & (0.6, 0.3) \\ (0.6, 0.3) & (0.6, 0.4) & (0.3, 0.6) & (0.8, 0.1) \end{matrix}) \\ {f_{3}}_{E} & = & (\begin{matrix} (0.7, 0.2) & (0.5, 0.3) & (0.3, 0.4) & (0.8, 0.2) \\ (0.6, 0.3) & (0.4, 0.4) & (0.6, 0.2) & (0.4, 0.2) \\ (0.5, 0.1) & (0.2, 0.5) & (0.4, 0.6) & (0.6, 0.2) \\ (0.8, 0.2) & (0.7, 0.2) & (0.5, 0.3) & (0.5, 0.3) \\ (0.8, 0.2) & (0.6, 0.2) & (0.4, 0.3) & (0.7, 0.1) \end{matrix}) \end{matrix}$

Based on the knowledge and experience of this committee, the pair of threshold values (0.6, 0.3) is given (Step 2).

Step 3. Take α = 0.6 and β = 0.3, then compute matrices ${R_{1}}_{0.6}^{0.3} (e_{t})$ , ${R_{2}}_{0.6}^{0.3} (e_{t})$ , and ${R_{3}}_{0.6}^{0.3} (e_{t})$ for t = 1, 2, 3, 4.

$\begin{matrix} {R_{1}}_{0.6}^{0.3} (e_{t}) & = & (\begin{matrix} 1 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 \\ 1 & 1 & 0 & 1 \\ 0 & 1 & 0 & 0 \end{matrix}), \\ {R_{2}}_{0.6}^{0.3} (e_{t}) & = & (\begin{matrix} 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 1 \\ 1 & 1 & 0 & 1 \\ 1 & 0 & 0 & 1 \end{matrix}), \\ {R_{3}}_{0.6}^{0.3} (e_{t}) & = & (\begin{matrix} 1 & 0 & 0 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 1 \end{matrix}) \end{matrix}$

Step 4. Using Equations 4 and 5, we get the comparison matrices $G_{0.6}^{0.3} (e_{t})$ and ${EQ}_{0.6}^{0.3} (e_{t})$ , t = 1, 2, 3, 4, as below. $\begin{matrix} G_{0.6}^{0.3} (e_{1}) & = & (\begin{matrix} 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 & 1 \end{matrix}), \\ {EQ}_{0.6}^{0.3} (e_{1}) & = & (\begin{matrix} 1 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix}) \\ G_{0.6}^{0.3} (e_{2}) & = & (\begin{matrix} 1 & 0 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 \\ 1 & 0 & 1 & 0 & 1 \end{matrix}), \\ {EQ}_{0.6}^{0.3} (e_{2}) & = & (\begin{matrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix}) \\ G_{0.6}^{0.3} (e_{3}) & = & (\begin{matrix} 1 & 0 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 & 1 \end{matrix}), \\ {EQ}_{0.6}^{0.3} (e_{3}) & = & (\begin{matrix} 1 & 0 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 & 1 \end{matrix}) \\ G_{0.6}^{0.3} (e_{4}) & = & (\begin{matrix} 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 1 \end{matrix}), \\ {EQ}_{0.6}^{0.3} (e_{4}) & = & (\begin{matrix} 1 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix}) \end{matrix}$

Step 5. According to Equation 6, the overall score values of each alternative are computed asfollows: $\begin{matrix} S_{1} = (5 - 2) + (2 - 1) + (4 - 4) + (5 - 2) = 7, \\ S_{2} = (1 - 1) + (2 - 1) + (5 - 1) + (1 - 1) = 5, \\ S_{3} = (1 - 1) + (1 - 1) + (4 - 4) + (5 - 2) = 3, \\ S_{4} = (5 - 2) + (5 - 1) + (4 - 4) + (2 - 1) = 8, \\ S_{5} = (2 - 1) + (3 - 1) + (4 - 4) + (2 - 1) = 4 . \end{matrix}$

Step 6. Therefore, using formula in (7), the following preferences over the set C: a₄ ≽ _Ea₁ ≽ _Ea₂ ≽ _Ea₅ ≽ _Ea₃ is obtained.

Step 7. Hence, the applicant a₄ can be considered as the best applicant for this post, while the third applicant, i.e., a₃, should not be selected.

So, the acceptance region is {a₄}, the rejection region is {a₃} and boundary region is {a₁, a₂, a₅}, however, a₁ is the sub-optimal choice after a₄.

The following Fig. 1 shows the overall score values S_i for the candidates a_i ∈ A of above example.

Fig.1

Overall score value.

The proposed algorithm here, in comparison with Algorithm 1 in [10], does not need to compute some topological spaces but only deal with a preorder relation and an equivalence relation derived from (α, β)-level soft sets of IFSSs. This makes the new algorithm simpler and easier for application. Moreover, due to using the threshold value pair (α, β) instead of applying threshold values α and β, the number of comparison matrices and thus computational steps is less than Algorithm 1 in [10]. To illustrate the difference between Algorithm 1 in [10] and the proposed algorithm here, let us reconsider Example 1 in [10].

Example 4.2. Consider Example 1 in [10] including fuzzy soft set S_P. Since the given data in that example is in standard fuzzy soft form, we apply ν_{S
_{p
_t}} (x) =1 - μ_{S
_{p
_t}} (x), 1 ≤ t ≤ 7, in order to present the fuzzy soft set S_P as an intuitionistic fuzzy soft set given in Table 2 (Step 1).

Table 2

Tabular representation of S_P in [10]

S _P	p ₁	p ₂	p ₃	p ₄	p ₅	p ₆	p ₇
o ₁	(0.3,0.7)	(0.1,0.9)	(0.4,0.6)	(0.4,0.6)	(0.1,0.9)	(0.1,0.9)	(0.5,0.5)
o ₂	(0.3,0.7)	(0.3,0.7)	(0.5,0.5)	(0.1,0.9)	(0.3,0.7)	(0.1,0.9)	(0.5,0.5)
o ₃	(0.4,0.6)	(0.3,0.7)	(0.5,0.5)	(0.1,0.9)	(0.3,0.7)	(0.1,0.9)	(0.6,0.4)
o ₄	(0.7,0.3)	(0.4,0.6)	(0.2,0.8)	(0.1,0.9)	(0.2,0.8)	(0.1,0.9)	(0.3,0.7)
o ₅	(0.2,0.8)	(0.5,0.5)	(0.2,0.8)	(0.3,0.7)	(0.5,0.5)	(0.5,0.5)	(0.4,0.6)
o ₆	(0.3,0.7)	(0.5,0.5)	(0.2,0.8)	(0.2,0.8)	(0.4,0.6)	(0.3,0.7)	(0.3,0.7)

Step 2. Let we apply the threshold value vector (α, β) that is given in Example 1 of [10] as below. (α, β) = ((0.7, 0.2) , (0.5, 0.1) , (0.5, 0.2) , (0.4, 0.1) , (0.5, 0.1) , (0.5, 0.1) , (0.6, 0.3)).

Step 3. By taking the threshold vector (α, β), the following matrix $R_{α}^{β} (p_{t})$ is computed. $R_{α}^{β} (p_{t}) = (\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix})$

Step 4. Using Equations 4 and 5, the comparison matrices $G_{α}^{β} (p_{t})$ and ${EQ}_{α}^{β} (p_{t})$ , t = 1, ⋯ , 7, are obtained as below. $\begin{matrix} G_{0.7}^{0.2} (p_{1}) & = & (\begin{matrix} 1 & 1 & 1 & 0 & 1 & 1 \\ 1 & 1 & 1 & 0 & 1 & 1 \\ 1 & 1 & 1 & 0 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 0 & 1 & 1 \\ 1 & 1 & 1 & 0 & 1 & 1 \end{matrix}), \\ {EQ}_{0.7}^{0.2} (p_{1}) & = & (\begin{matrix} 1 & 1 & 1 & 0 & 1 & 1 \\ 1 & 1 & 1 & 0 & 1 & 1 \\ 1 & 1 & 1 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 & 1 & 1 \\ 1 & 1 & 1 & 0 & 1 & 1 \end{matrix}) \\ G_{0.5}^{0.1} (p_{2}) & = & G_{0.5}^{0.2} (p_{3}) = G_{0.4}^{0.1} (p_{4}) = G_{0.5}^{0.1} (p_{5}) \\ = & G_{0.5}^{0.1} (p_{6}) = G_{0.6}^{0.3} (p_{7}) = {(1)}_{6 \times 6} . \\ {EQ}_{0.7}^{0.2} (p_{2}) & = & {EQ}_{0.5}^{0.2} (p_{3}) = {EQ}_{0.4}^{0.1} (p_{4}) \\ = & {EQ}_{0.5}^{0.1} (p_{5}) = {EQ}_{0.5}^{0.1} (p_{6}) \\ = & {EQ}_{0.6}^{0.3} (p_{7}) = {(1)}_{6 \times 6} . \end{matrix}$

Step 5. According to Equation 6, the overall score values of each alternative are computed asfollows: $\begin{matrix} S_{1} = 0, S_{2} = 0, S_{3} = 0, S_{4} = 5, S_{5} = 0, \\ S_{6} = 0, S_{7} = 0 . \end{matrix}$

Step 6. Therefore, we have the following preferences over the set C: o₄ ≽ _Eo₁ ∼ _Eo₂ ∼ _Eo₃ ∼ _Eo₅ ∼ _Eo₆.

Step 7. Thus, the object o₄ should be selected as the best option.

However, if Algorithm 1 of [10] is applied to solve this example, then object o₅ should be selected since: o₅ ≽ _Eo₃ ≽ _Eo₂ ≽ _Eo₆ ≽ _Eo₄ ≽ _Eo₁.

4.3 Comparison analysis with existing group decision-making methods

In this part, we compare the numerical Example 4.1 with some existing methods for group decision-making problems (see [22 , 60]).

Authors in [22 , 60] proposed a linear ordering system of objects based on their choice values in level-sets of IFSSs. These methods can be used widely in some applications, however there are certain situations, in which they do not result in the right decision.

Example 4.3. (Example 4.1 continued) Let us reconsider the decision making problem involving the IFSSs f₁_E, f₂_E and f₃_E with their matrix representations which is discussed earlier in Example 4.1 where A = {a₁, a₂, a₃, a₄, a₅} is the universal set and E = {e₁, e₂, e₃, e₄} is the parameter set.

Jiang et al.’ method [49]

Suppose that Algorithm 1 of [49] is applied to solve Example 4.1. In [49], objects are ranked based on their choice values obtained from (0.6, 0.3)-level soft sets of IFSSs f₁_E, f₂_E and f₃_E which are given in Table 3 (see also matrices ${R_{1}}_{0.6}^{0.3} (e_{t})$ , ${R_{2}}_{0.6}^{0.3} (e_{t})$ , and ${R_{3}}_{0.6}^{0.3} (e_{t})$ ). However, this method cannot be used to deal with group decision-making basedon IFSS.

Table 3
Table of (0.6, 0.3)-level soft set of f_i_E, i = 1, 2, 3

Applicants e ₁ e ₂ e ₃ e ₄ Choice

Value c_i

a ₁ 1 1 0 1 3

a ₂ 0 0 1 0 1

a ₃ 1 0 0 1 2

a ₄ 1 1 0 1 3

a ₅ 0 1 0 0 1

a ₁ 1 0 0 1 2

a ₂ 0 1 1 1 3

a ₃ 0 0 0 1 1

a ₄ 1 1 0 1 3

a ₅ 1 0 0 1 2

a ₁ 1 0 0 1 2

a ₂ 1 0 1 0 2

a ₃ 0 0 0 1 1

a ₄ 1 1 0 0 2

a ₅ 1 1 0 1 3

Applicants	e ₁	e ₂	e ₃	e ₄	Choice
a ₁	1	1	0	1	3
a ₂	0	0	1	0	1
a ₃	1	0	0	1	2
a ₄	1	1	0	1	3
a ₅	0	1	0	0	1
a ₁	1	0	0	1	2
a ₂	0	1	1	1	3
a ₃	0	0	0	1	1
a ₄	1	1	0	1	3
a ₅	1	0	0	1	2
a ₁	1	0	0	1	2
a ₂	1	0	1	0	2
a ₃	0	0	0	1	1
a ₄	1	1	0	0	2
a ₅	1	1	0	1	3

Solving Example 4.1 by using method [49], the ranking of alternatives is generated as a₁ ∼ a₄ ≻ a₃ ≻ a₂ ∼ a₅ in terms of f₁_E, a₂ ∼ a₄ ≻ a₁ ∼ a₅ ≻ a₃ in terms of f₂_E and a₅ ≻ a₁ ∼ a₂ ∼ a₄ ≻ a₃ in terms of f₃_E.

Zhang’s method [60]

In [60], objects are ranked according to their choice values computed based on a collective IFSS formulated by soft union of (0.6, 0.3)-level soft sets of f₁_E, f₂_E and f₃_E.

By applying method [60] for Example 4.1, the collective table ${f_{1}}_{0.6}^{0.3} \tilde{\lor} {f_{2}}_{0.6}^{0.3} \tilde{\lor} {f_{3}}_{0.6}^{0.3}$ is computed as given in Table 4. Thus, using given method in [60] (see Table 20 of [60]) the following ranking of applicants is obtained as a₂ ≻ a₁ ∼ a₄ ∼ a₅ ≻ a₃.

Table 4

Table of ${f_{1}}_{0.6}^{0.3} \tilde{\lor} {f_{2}}_{0.6}^{0.3} \tilde{\lor} {f_{3}}_{0.6}^{0.3}$

Applicants	e ₁	e ₂	e ₃	e ₄	Choice Value c_i
a ₁	1	1	0	1	3
a ₂	1	1	1	1	4
a ₃	1	0	0	1	2
a ₄	1	1	0	1	3
a ₅	1	1	0	1	3

Mao et al.’ method [22]

In method [22], authors took a weight vector for the set of decision makers. Here, without loos of generality, let the weight vector of experts is given as $ω = (\frac{1}{3}, \frac{1}{3}, \frac{1}{3})^{T}$ , which means the weights of all experts are equal. Solving Example 4.1 by Algorithm 2 in [22], the aggregation operators: $f_{ω} = {(1 - \prod_{k = 1}^{3} (1 - μ_{f_{k} (e_{j})} (x_{i}))^{ω_{k}}, \prod_{k = 1}^{3} (ν_{f_{k} (e_{j})} (x_{i}))^{ω_{k}})}_{5 \times 4}$ and $g_{ω} = {(\prod_{k = 1}^{3} {(μ_{f_{k} (e_{j})} (x_{i}))}^{ω_{k}}, 1 - \prod_{k = 1}^{3} {(1 - ν_{f_{k} (e_{j})} (x_{i}))}^{ω_{k}})}_{5 \times 4}$

are computed as Tables 5 and 6.

Table 5

Table of f_ω

f _ω	e ₁	e ₂	e ₃	e ₄
a ₁	(0.6365758814,0.1587401052)	(0.5782836673,0.1817120593)	(0.3,0.2884499141)	(0.7379258606,0.1587401052)
a ₂	(0.5358411166,0.2884499141)	(0.542114303,0.3634241186)	(0.7115500859,0.1817120593)	(0.5067575851,0.2289428485)
a ₃	(0.5782836673,0.2)	(0.2348275269,0.3914867641)	(0.4686707154,0.3301927249)	(0.7115500859,0.2)
a ₄	(0.7115500859,0.1817120593)	(0.7,0.2)	(0.4686707154,0.3914867641)	(0.6085132359,0.2080083823)
a ₅	(0.6580048107,0.2884499141)	(0.6825197896,0.2)	(0.4056078047,0.377976315)	(0.6892767494,0.125992105)

Table 6

Table of g_ω

g _ω	e ₁	e ₂	e ₃	e ₄
a ₁	(0.6316359598,0.1679664708)	(0.559344471,0.2041885584)	(0.3,0.304794671)	(0.731861142,0.1679664708)
a ₂	(0.5313292846,0.304794671)	(0.5241482788,0.3683640402)	(0.695205329,0.2041885584)	(0.4932424149,0.2348275269)
a ₃	(0.559344471,0.2440473701)	(0.2289428485,0.4056078047)	(0.4641588834,0.3926822056)	(0.695205329,0.2)
a ₄	(0.665205329,0.2041885584)	(0.7,0.2)	(0.4641588834,0.4056078047)	(0.5943921953,0.2388337389)
a ₅	(0.6214465012,0.304794671)	(0.6603854498,0.2440473701)	(0.3914867641,0.4191214266)	(0.654213262,0.1346502578)

After applying the threshold value pair (0.6, 0.3), we have the following Tables 7 and 8.

Table 7

Table of aggregation operator ${f_{ω}}_{0.6}^{0.3}$

Applicants	e ₁	e ₂	e ₃	e ₄	Choice Value c_i
a ₁	1	0	0	1	2
a ₂	0	0	1	0	1
a ₃	0	0	0	1	1
a ₄	1	1	0	1	3
a ₅	1	1	0	1	3

Table 8

Table of aggregation operator ${g_{ω}}_{0.6}^{0.3}$

Applicants	e ₁	e ₂	e ₃	e ₄	Choice Value c_i
a ₁	1	0	0	1	2
a ₂	0	0	1	0	1
a ₃	0	0	0	1	1
a ₄	1	1	0	0	2
a ₅	0	1	0	1	2

Algorithm 2: MAGDM based on weighted IFSS
Input:
Step 1.	n=Size of parameter set E, m=Size of object set X, k=Size of decision maker set D,
Matrix form of IFSSf_s_E, 1 ≤ s ≤ k.
Step 2.	Weight vector ω = (ω₁, ω₂, ⋯ , ω_n) ^T.
Step 3.	Threshold intuitionistic fuzzy set γ : E → [0, 1] × [0, 1] such that μ_γ (e_t) = α_t and ν_γ (e_t) = β_t
(or threshold value pair (α, β) ∈ [0, 1] × [0, 1]).
Begin
For t = 1, 2, …, n, i = 1, 2, …, m, and s = 1, 2, …, k:
Step 4.	Compute R_s_γ (e_t) (or ${R_{s}}_{α}^{β} (e_{t})$ ).
Step 5.	Compute matrices $G_{α_{t}}^{β_{t}} (e_{t})$ and ${EQ}_{α_{t}}^{β_{t}} (e_{t})$ (or $G_{α}^{β} (e_{t})$ and ${EQ}_{α}^{β} (e_{t})$ ).
Step 6.	Compute the amounts of ${\bar{S}}_{i}$ , ∀i (Definition 5).
Step 7.	According to the values of ${\bar{S}}_{i}$ , rank the set X.
Step 8.	The optimum is to select object x_h that ${\bar{S}}_{h} = max_{i} {\bar{S}}_{i}$ .
The alternative x_l such that ${\bar{S}}_{l} = min_{i} {\bar{S}}_{i}$ should not be selected.
Step 9.	If the number of elements that ${\bar{S}}_{h}$ is maximum is more than one, then any one of such objects may be chosen.
Output:	Optimum and worst objects.
End

Thus, we have the ranking of objects as a₄ ∼ a₅ ≻ a₁ ≻ a₂ ∼ a₃ based on ${f_{ω}}_{0.6}^{0.3}$ and the following ranking a₁ ∼ a₄ ∼ a₅ ≻ a₂ ∼ a₃ based on ${g_{ω}}_{0.6}^{0.3}$ .

Table 9 and Fig. 2 show the comparison results for the methods [22 , 60] and the proposed Algorithm 1 in this current paper for Example 4.1.

Table 9

Comparison Table

Applicants	${f_{1}}_{0.6}^{0.3}$	${f_{2}}_{0.6}^{0.3}$	${f_{3}}_{0.6}^{0.3}$	c _i	${f_{ω}}_{0.6}^{0.3}$	${g_{ω}}_{0.6}^{0.3}$	S _i
	[49]	[49]	[49]	[60]	[22]	[22]
a ₁	3	2	2	3	2	2	7
a ₂	1	3	2	4	1	1	5
a ₃	2	1	1	2	1	1	3
a ₄	3	3	2	3	3	2	8
a ₅	1	2	3	3	3	2	4

Fig.2

Comparison chart.

5 Extended to MAGDM based on Weighted IFSS

In Section 4, a novel approach to decision-making problems based on intuitionistic fuzzy soft sets was investigated. However, the weight of parameters was not emphasized. In real decision-making situations, usually, there is an associating value with each parameter for presenting the importance of different parameters. So, some further respective values ω_t ∈ [0, 1] corresponding to the parameter t-th should be taken into account to describe the different importance of different criteria. In this section, we focus on how to deal with this situation.

In literature, researchers discussed the notion of weight for soft sets [38], fuzzy soft sets [15, 53], intuitionistic fuzzy soft sets [22 , 50], and interval-valued fuzzy soft sets [48, 61]. Here, we recall the concept of weighted IFSS asbelow.

Definition 5.1. Let X = {x₁, ⋯ , x_m} be the set of objects and E = {e₁, ⋯ , e_n} be the set of parameters. A weighted IFSS is a triple (f, E, ω) where (f, E) is an IFSS over X and ω : E → [0, 1] is a weight function specifying the weight ω_t = ω (e_t) for each attribute e_t ∈ E.

According to Definition 5.1, any typical IFSS can be considered as a weighted IFSS such that the weights of all parameters are supposed equal.

In [10], the authors defined the concept of weighted overall score value for weighted FSSs. Regarding Definition 9 of [10], we develop the concept of weighted overall score value ${\bar{S}}_{i}$ for weighted IFSS as below.

Definition 5.2. Let ω = (ω₁, ⋯ , ω_n) ^T, such that $\sum_{t = 1}^{n} ω_{t} = 1$ , be the weight vector corresponding to each parameter e_t, for t = 1, ⋯ , n. The weighted overall score value ${\bar{S}}_{i}$ is defined as follows: ${\bar{S}}_{i} = \sum_{t = 1}^{n} ω_{t} . r_{i} (e_{t}; α, β)$ (8)

An adjustable approach to MAGDM based on weighted IFSSs is given in Algorithm 2 as an extension of Algorithm 1.

Algorithm 2

Obviously, Algorithm 2 is an extension of the Algorithm 1. We first consider the tabular form (or matrix form) of intuitionistic fuzzy soft sets with respect to the weights of parameters and then, take the weights of parameters into consideration for computing the weighted overall score value ${\bar{S}}_{i}$ instead of overall score value S_i. Similarly as Algorithm 1, in the last step of Algorithm 2 one can back to the second step and change the former threshold to select the better final optimal decision.

To illustrate the above idea, let us consider the below example.

Example 5.1. (Example 4.1 continued) Suppose that decision makers in Example 4.1, imposed the weight vector ω = (0.1, 0.3, 0.2, 0.4) ^T for parameters e₁, e₂, e₃, and e₄. Thus, we have weight mapping ω : E → [0, 1] that ω (e₁) =0.1, ω (e₂) =0.3, ω (e₃) =0.2, and ω (e₄) =0.4.

Three IFSSs (f₁, E) , (f₂, E) , and (f₃, E) in Example 4.1 are changed into weighted IFSSs (f₁, E, ω), (f₂, E, ω), and (f₃, E, ω) with their tabular representations as in Table 10–12 (Step 1 and Step 2).

Table 10

Tabular representation of weighted IFSS (f₁, E, ω)

A	e₁, ω₁ = 0.1	e₂, ω₂ = 0.3	e₃, ω₃ = 0.2	e₄, ω₄ = 0.4
a ₁	(0.6,0.2)	(0.7,0.1)	(0.3,0.3)	(0.7,0.2)
a ₂	(0.5,0.4)	(0.6,0.4)	(0.8,0.1)	(0.5,0.2)
a ₃	(0.7,0.2)	(0.2,0.4)	(0.5,0.2)	(0.8,0.2)
a ₄	(0.7,0.1)	(0.7,0.2)	(0.4,0.5)	(0.7,0.1)
a ₅	(0.5,0.4)	(0.8,0.1)	(0.5,0.3)	(0.5,0.2)

Table 11

Tabular representation of weighted IFSS (f₂, E, ω)

A	e₁, ω₁ = 0.1	e₂, ω₂ = 0.3	e₃, ω₃ = 0.2	e₄, ω₄ = 0.4
a ₁	(0.6,0.1)	(0.5,0.2)	(0.3,0.2)	(0.7,0.1)
a ₂	(0.5,0.2)	(0.6,0.3)	(0.7,0.3)	(0.6,0.3)
a ₃	(0.5,0.4)	(0.3,0.3)	(0.5,0.3)	(0.7,0.2)
a ₄	(0.6,0.3)	(0.7,0.2)	(0.5,0.4)	(0.6,0.3)
a ₅	(0.6,0.3)	(0.6,0.4)	(0.3,0.6)	(0.8,0.1)

Table 12

Tabular representation of weighted IFSS (f₃, E, ω)

A	e₁, ω₁ = 0.1	e₂, ω₂ = 0.3	e₃, ω₃ = 0.2	e₄, ω₄ = 0.4
a ₁	(0.7,0.2)	(0.5,0.3)	(0.3,0.4)	(0.8,0.2)
a ₂	(0.6,0.3)	(0.4,0.4)	(0.6,0.2)	(0.4,0.2)
a ₃	(0.5,0.1)	(0.2,0.5)	(0.4,0.6)	(0.6,0.2)
a ₄	(0.8,0.2)	(0.7,0.2)	(0.5,0.3)	(0.5,0.3)
a ₅	(0.8,0.2)	(0.6,0.2)	(0.4,0.3)	(0.7,0.1)

Step 3. Let decision makers use (α, β) = (0.6, 0.3) as the threshold value pair.

Steps 4 and 5. Then, we have the matrices ${R_{1}}_{0.6}^{0.3} (e_{t})$ , ${R_{2}}_{0.6}^{0.3} (e_{t})$ , and ${R_{3}}_{0.6}^{0.3} (e_{t})$ ; and the comparison matrices $G_{0.6}^{0.3} (e_{t})$ and ${EQ}_{0.6}^{0.3} (e_{t})$ for t = 1, 2, 3, 4 that were computed in Example 4.1 (see Steps 3 and 4 in Example 4.1).

Step 6. By formula in (8), the weighted overall score values ${\bar{S}}_{i}$ , 1 ≤ i ≤ 5, can be obtained as below: $\begin{matrix} {\bar{S}}_{1} & = & (3 \times 0.1) + (1 \times 0.3) \\ + (0 \times 0.2) + (3 \times 0.4) = 1.8, \\ {\bar{S}}_{2} & = & (0 \times 0.1) + (1 \times 0.3) \\ + (4 \times 0.2) + (0 \times 0.4) = 1.1, \\ {\bar{S}}_{3} & = & (0 \times 0.1) + (0 \times 0.3) \\ + (0 \times 0.2) + (3 \times 0.4) = 1.2, \\ {\bar{S}}_{4} & = & (3 \times 0.1) + (4 \times 0.3) \\ + (0 \times 0.2) + (1 \times 0.4) = 1.9, \\ {\bar{S}}_{5} & = & (1 \times 0.1) + (2 \times 0.3) \\ + (0 \times 0.2) + (1 \times 0.4) = 1.1 . \end{matrix}$

Step 7. So, we have the following order a₄ ≽ _Ea₁ ≽ _Ea₃ ≽ _Ea₂, a₅ over the universal set A.

Step 8. It follows that the maximum weighted overall score value is 1.9 and so, the optimal decision is to select a₄, while a₂ and a₅ should not be selected after specifying weights for different experts and parameters.

Therefore, the acceptance region is {a₄}, the rejection region is {a₂, a₅} and applicants {a₁, a₃} cannot be judged (boundary region), although a₁ may be considered as sub-optimal choice.

We compare the effectiveness of proposed method here with some existing methods by the following example.

Example 5.2. (Example 5.1 continued) Let us reconsider Example 5.1 with method in [60].

Based on method [60], we first compute the (0.6, 0.3)-level soft sets of f₁_E, f₂_E and f₃_E (see Table 13, and then find the collective IFSS ${f_{1}}_{0.6}^{0.3} \tilde{\lor} {f_{2}}_{0.6}^{0.3} \tilde{\lor} {f_{3}}_{0.6}^{0.3}$ which is given in Table 14.

Table 13

Tabular representation of ${f_{i}}_{0.6}^{0.3}$ , i = 1, 2, 3

A	e₁, ω₁ = 0.1	e₂, ω₂ = 0.3	e₃, ω₃ = 0.2	e₄, ω₄ = 0.4
a ₁	1	1	0	1
a ₂	0	0	1	0
a ₃	1	0	0	1
a ₄	1	1	0	1
a ₅	0	1	0	0
a ₁	1	0	0	1
a ₂	0	1	1	1
a ₃	0	0	0	1
a ₄	1	1	0	1
a ₅	1	0	0	1
a ₁	1	0	0	1
a ₂	1	0	1	0
a ₃	0	0	0	1
a ₄	1	1	0	0
a ₅	1	1	0	1

Table 14

Table of ${f_{1}}_{0.6}^{0.3} \tilde{\lor} {f_{2}}_{0.6}^{0.3} \tilde{\lor} {f_{3}}_{0.6}^{0.3}$

A	e ₁	e ₂	e ₃	e ₄	Weighted
					Choice
					Value ${\bar{c}}_{i}$
a ₁	1	1	0	1	0.8
a ₂	1	1	1	1	1
a ₃	1	0	0	1	0.5
a ₄	1	1	0	1	0.8
a ₅	1	1	0	1	0.8

Thus, using method [60], the following ranking of applicants is obtained as a₂ ≻ a₁ ∼ a₄_sima5 ≻ a₃.

However, based on method [60], a₂ is the best candidate, the weak rational preference relationship ≽_E suggests applicant a₄. Moreover, applicants a₂ and a₅ in our approach can be seen as incomparable alternatives, while all alternatives are comparable on the basis of method [60]. This issue is shown in Figs. 3 and 4.

Fig.3

Non-linear ordering system.

Fig.4

Linear ordering system.

6 Discussion

Until now, various intuitionistic fuzzy soft-based techniques have been proposed to solve decision-making problems. Some of them have made effort to apply a threshold value pair in order to convert IFSS into the crisp soft set and solve the problem of decision-making in a precise environment [22 , 60], while, in [25, 40] the concept of similarity measure plays the most important role.

In [22 , 60], the concept of choice value, computed based on the positive parameters, plays the key role. But, real-world decision situations are much more complicated due to dealing with both benefit and cost parameters. Our proposed methods, so-called Algorithm 1 and Algorithm 2, overcome this issue by considering the negation of negative parameters as positive ones.

On the other hand, ordering system proposed in [22 , 60] is built based on a total ordered structure. But, in reality there are situations in which some objects may be incomparable. To cope this difficulty, a rational weak preference relationship, which represents the decision makers’ preference of alternatives at some certain levels of membership degrees and non-membership degrees, is proposed to rank objects. Due to the nature of preference relationship as a non-linear ordered structure, there may exist some objects which are incomparable.

Moreover, existing algorithms [22 , 60] only consider the optimum as their output. The binary decision model consists of acceptance region involves optimal objects and rejection region as the complement set of the acceptance region is followed by methods [22 , 60]. However, in real-world problems under imprecise environment, three-way decision including acceptance region, boundary region, and rejection region are more popular (see [55 –57]). The proposed method follows three-way decision instead of two-way decision. So, the outputs from Algorithm 1 and Algorithm 2 consist of optimal objects (acceptance set), objects should not be selected (rejection set), and the complement of union of acceptance set and rejection set, known as boundary set.

From the above comparison results and discussion, the advantages of our proposed method can be summarized as Table 15.

Table 15
Comparison of existing methods and proposed methods

Methods Output Comparison Advantages Limitations

Jiang et al.’s method [49] Optimal object, Less calculation steps, Not applicable for group decision-making,

All objects are comparable Applicable at different utility level Not applicable for negative/cost parameters

Zhang’s method [60] Optimal object, Less calculation steps, Not applicable for negative/cost parameters

All objects are comparable Applicable at different utility level,

Suitable for group decision-making

Mao at al.’s method [22] Optimal object, Applicable at different utility level, Not applicable for negative/cost parameters

All objects are comparable Suitable for group decision-making

New proposed algorithm Optimal object, Applicable at different utility level, More calculation steps

Sub-optimal object, Suitable for group decision-making,

Worst object, Applicable for negative/cost

Some objects may be parameters

incomparable

Methods	Output Comparison	Advantages	Limitations
Jiang et al.’s method [49]	Optimal object,	Less calculation steps,	Not applicable for group decision-making,
	All objects are comparable	Applicable at different utility level	Not applicable for negative/cost parameters
Zhang’s method [60]	Optimal object,	Less calculation steps,	Not applicable for negative/cost parameters
	All objects are comparable	Applicable at different utility level,
		Suitable for group decision-making
Mao at al.’s method [22]	Optimal object,	Applicable at different utility level,	Not applicable for negative/cost parameters
	All objects are comparable	Suitable for group decision-making
New proposed algorithm	Optimal object,	Applicable at different utility level,	More calculation steps
	Sub-optimal object,	Suitable for group decision-making,
	Worst object,	Applicable for negative/cost
	Some objects may be	parameters
	incomparable

7 Conclusion

In this paper, we discussed the main difficulty of some well-known existing intuitionistic fuzzy soft-based approaches for solving decision-making and focused to overcome this problem by developing a new methodology for ranking data based on a non-linear ordering structure. We developed the proposed method for solving MAGDM in [10] to IFSSs. First, a preference relationship, which provides the decision makers’ preferences at different alternatives, was constructed. Then, objects were ranked based on a new overall score value which is adopted by this preference relationship. Moreover, we extended our method into the weighted intuitionistic fuzzy soft sets where the parameters have their own importance degrees which are not necessarily equal. Accordingly, two algorithms are designed to solve decision making problems based on intuitionistic fuzzy soft sets. The first algorithm is applied for solving group decision-making based on intuitionistic fuzzy soft sets, while the second one is utilized to solve group decision-making by weighted intuitionistic fuzzy soft sets.

Conflicts of interests

The authors declare that they have no conflicts of interests regarding publication of this article and they do not have direct financial relation that might lead to conflict of interest for any of the author.

Footnotes

Acknowledgments

This work is partially supported by the Institute for Mathematical Research, Universiti Putra Malaysia Grant No. 5527179.

References

Kharal

and Ahmad

, Mappings on fuzzy soft classes, Advances in Fuzzy Systems (2009), 3308–3314. Article ID 407890.

Kharal

and Ahmad

, Mappings on soft classes, New Mathematics and Natural Computation 7(3) (2011), 471–481.

Khalil

A.M.

, Commentary on “Generalized intuitionistic fuzzy soft sets with applications in decision-making”, Applied Soft Computing 37 (2015), 519–520.

Pawlak

, Rough Sets, International Journal of Computer and Information Sciences 11(5) (1965), 341–356.

Roy

A.R.

and Maji

P.K.

, A fuzzy soft set theoretic approach to decision making problems, Journal of Computational and Applied Mathematics 203 (2007), 412–418.

Sezgin

and Atagun

A.O.

, On operations of soft sets, Computers and Mathematics with Applications 61 (2011), 1457–1467.

Zahedi Khameneh

, Kilicman

and Salleh

A.R.

, Fuzzy soft product topology, Annals of Fuzzy Mathematics and Information 7(6) (2014), 935–947.

Zahedi Khameneh

, Kilicman

and Salleh

A.R.

, Fuzzy soft boundary, Annals of Fuzzy Mathematics and Information 8(5) (2014), 687–703.

Zahedi Khameneh

, Kilicman

and Salleh

A.R.

, An adjustable method for data ranking based on fuzzy soft sets, Indian Journal of Science and Technology. DOI: 10.17485/ijst/2015/v8i21-72587

10.

Zahedi Khameneh

, Kilicman

and Salleh

A.R.

, An adjustable approach to multi-criteria group decision-making based on a preference relationship under fuzzy soft information, International Journal of Fuzzy Systems. DOI: 10.1007/s40815-016-0280-z

11.

Tanay

and Kandemir

M.B.

, Topological structure of fuzzy soft sets, Computers and Mathematics with Applications 61 (2011), 2952–2957.

12.

Chen

, Tsang

E.C.C.

, Yeung

D.S.

and Wang

, The parameterization reduction of soft sets and its applications, Computers and Mathematics with Applications 49 (2005), 757–763.

13.

Molodtsov

, Soft set theory-first results, Computers and Mathematics with Applications 37 (1999), 19–31.

14.

Feng

, Li

, Davvaz

and Irfan Ali

, Soft sets combined with fuzzy sets and rough sets: A tentative approach, Soft Computing 14 (2010), 899–911.

15.

Feng

, Jun

Y.B.

, Liu

and Li

, An adjustable approach to fuzzy soft set based decision making, Journal of Computational and Applied Mathematics 234 (2010), 10–20.

16.

Feng

, Li

and Leoreanu-Fotea

, Application of level soft sets in decision making based on interval-valued fuzzy soft sets, Computers and Mathematics with Applications 60 (2010), 1756–1767.

17.

Feng

, Li

, Çağman

and Enginoğlü

, Generalized uniint decision making schemes based on choice value soft sets, European Journal of Operation Research 220 (2012), 162–170.

18.

Aktaş

and Çağman

, Soft sets and soft groups, Information Sciences 177 (2007), 2726–3332.

19.

Aktaş

and Çağman

, Soft decision making methods based on fuzzy sets and soft sets, Journal of Intelligent and Fuzzy Systems 30(5) (2016), 2797–2803.

20.

Deli

and Çağman

, Intuitionistic fuzzy parameterized soft set theory and its decision making, Applied Soft Computing 28 (2015), 109–113.

21.

Deli

and Karataş

, Interval valued intuitionistic fuzzy parameterized soft set theory and its decision making, Journal of Intelligent and Fuzzy Systems 30(4) (2016), 2073–2082.

22.

Mao

, Yao

and Wang

, Group decision making methods on intuitionistic fuzzy soft matrices, Applied Mathematical Modelling 37 (2013), 6425–6436.

23.

Atanassov

K.T.

, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986), 87–96.

24.

Zadeh

L.A.

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

25.

Agarwal

, Biswas

K.K.

and Hanmandlu

, Generalized intuitionistic fuzzy soft sets with applications in decision-making, Applied Soft Computing 13 (2013), 3552–3566.

26.

Ali

M.I.

, Feng

, liu

and Min

W.K.

, On some new operations in soft set theory, Computers and Mathematics with Applications 57(9) (2009), 1547–1553.

27.

Gorzalczany

M.B.

, A method of inference in approximate reasoning based on interval-valued fuzzy sets, Fuzzy Sets and Systems 21 (1987), 1–17.

28.

Kule

and Dost

, A textural view of semi-separation axioms in soft fuzzy topological spaces, Journal of Intelligent and Fuzzy Systems 30(4) (2016), 2037–2053.

29.

Shabir

and Naz

, On soft topological spaces, Computers and Mathematics with Applications 61 (2011), 1786–1799.

30.

Çağman

and Enginoğlü

, Soft set theory and uni-int decision making, European Journal of Operation Research 207 (2010), 848–855.

31.

Çağman

and Enginoğlü

, Soft matrix theory and its decision making, Computers and Mathematics with Applications 59(10) (2010), 3308–3314.

32.

Çağman

, Karataş

and Enginoğlü

, Soft topology, Computers and Mathematics with Applications 62 (2011), 351–358.

33.

Çağman

, Karataş

and Enginoğlü

, Fuzzy soft matrix theory and its application in decision making, Iranian Journal of Fuzzy Systems 9(1) (2012), 109–119.

34.

Çağman

and Karataş

, Intuitionistic fuzzy soft set theory and its decision making, Journal of Intelligent and Fuzzy Systems 24(4) (2013), 829–836.

35.

Maji

P.K.

, Biswas

and Roy

A.R.

, Soft set theory, Computers and Mathematics with Applications 45 (2003), 555–562.

36.

Maji

P.K.

, Biswas

and Roy

A.R.

, Fuzzy soft set, Journal of Fuzzy Mathematics 9(3) (2001), 589–602.

37.

Maji

P.K.

, Biswas

and Roy

A.R.

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

38.

Maji

P.K.

and Roy

A.R.

, An application of soft sets in a decision making problem, Computers and Mathematics with Applications 44 (2002), 1077–1083.

39.

Maji

P.K.

, Roy

A.R.

and Biswas

, On intuitionistic fuzzy soft sets, Journal of Fuzzy Mathematics 12(3) (2004), 669–683.

40.

Muthukumar

and Sai Sundara Krishnan

, A similarity measure of intuitionistic fuzzy soft sets and its application in medical diagnosis, Applied Soft Computing 41 (2016), 148–156.

41.

Borzooei

R.A.

, Mobini

and Ebrahimi

M.M.

, The category of soft sets, Journal of Intelligent and Fuzzy Systems 28(1) (2015), 157–167.

42.

Chatterjee

, Majumdar

and Samanta

S.K.

, Type-2 soft sets, Journal of Intelligent and Fuzzy Systems 29(2) (2015), 885–898.

43.

Das

and Kar

, Group decision making in medical system: An intuitionistic fuzzy soft set approach, Applied Soft Computing 24 (2014), 196–211.

44.

Hussain

and Ahmad

, Some properties of soft topological spaces, Computers and Mathematics with Applications 62 (2011), 4058–4067.

45.

Roy

and Samanta

T.K.

, A note on fuzzy soft topological spaces, Annals of Fuzzy Mathematics and Information 3(2) (2012), 305–311.

46.

Min

W.K.

, A note on soft topological spaces, Computers and Mathematics with Applications 62 (2011), 3524–3528.

47.

Yang

, Lin

T.Y.

, Yang

, Li

and Yu

, Combination of interval-valued fuzzy set and soft set, Computers and Mathematics with Applications 58(3) (2009), 521–527.

48.

Jiang

, Tanga

, Chen

, Liu

and Tanga

, Intervalvalued intuitionistic fuzzy soft sets and their properties, Computers and Mathematics with Applications 60 (2010), 906–918.

49.

Jiang

, Tang

and Chen

, An adjustable application to intuitonistic fuzzy soft sets based decision making, Applied Mathematical Modelling 35 (2011), 824–836.

50.

Jiang

, Tang

, Liu

and Chen

Zh.

, Entropy on intuitionistic fuzzy soft sets and on interval-valued fuzzy soft sets, Information Sciences 240 (2013), 95–114.

51.

Jun

Y.B.

, Soft BCK/BCI-algebras, Computers and Mathematics with Applications 56 (2008), 1408–1413.

52.

Jun

Y.B.

, Lee

K.J.

and Zhan

, Soft p-ideals of soft BCIalgebras, Computers and Mathematics with Applications 58 (2009), 2060–2068.

53.

Yang

, Tan

and Meng

, The multi-fuzzy soft set and its application in decision making, Applied Mathematical Modelling 37 (2013), 4915–4923.

54.

Yang

, Wang

, Zhang

and Zhang

, Commentary on Generalized intuitionistic fuzzy soft sets with applications in decision-making, Applied Soft Computing 40 (2016), 427–428.

55.

Yao

Y.Y.

, Wong

S.K.M.

and Lingras

, A decision-theoretic rough set model, Methodologies for Intelligent Systems 5 (1990), 17–24.

56.

Yao

Y.Y.

and Zhao

, Attribute reduction in decision-theoretic rough set models, Information Sciences 178 (2008), 3356–3373.

57.

Yao

Y.Y.

, Three-way decisions with probabilistic rough sets, Information Sciences 180 (2010), 341–353.

58.

Kong

, Gao

and Wang

, Comment on “A fuzzy soft set theoretic approach to decision making problems”, Journal of Computational and Applied Mathematics 223(2) (2009), 540–542.

59.

Kong

, Gao

, Wang

and Li

, The normal parameter reduction of soft sets and its algorithm, Computers and Mathematics with Applications 56 (2008), 3029–3037.

60.

Zhang

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

61.

Zhang

and Zhang

, A novel approach to multi attribute group decision making based on trapezoidal interval type-2 fuzzy soft sets, Applied Mathematical Modelling 37 (2013), 4948–4971.