Another view on generalized interval valued intuitionistic fuzzy soft set and its applications in decision support system

Abstract

As a combination of an interval-valued intuitionistic fuzzy soft set (IVIFSS) and interval-valued intuitionistic fuzzy set (IVIFS), the existing notion of the generalized interval-valued intuitionistic fuzzy soft set (GIVIFSS) is clarified and reformulated. We define two types of containment in GIVIFSS. With this new perspective, the g-union, g-intersection, g^∗-union, g^∗-intersection, OR, AND, g-necessity and g-possibility operations are defined for GIVIFSSs. The properties and relations between operations of GIVIFSSs are investigated. An algorithm is proposed for solving MADM problems using GIVIFSS. A descriptive example is presented to see the applicability of the proposed method. Results indicate that the proposed technique is more effective and generalize over the previous models of interval-valued fuzzy sets.

Keywords

Interval-valued intuitionistic fuzzy set interval-valued intuitionistic fuzzy soft set generalized interval-valued intuitionistic fuzzy soft set multi-attribute decision making

1 Introduction

The real world is full of imprecision, vagueness and uncertainty. In our daily life, we deal mostly with unclear concepts rather than exact ones. Dealing with imprecision is a big problem in many areas such as economics, medical science, social science, environmental science and engineering. In recent years, model vagueness has become interested in many authors. Many classical theories such as fuzzy set theory [1], probability theory, vague set theory [2], rough set theory [3], intuitionistic fuzzy set [4] and interval mathematics [5] are well known and effectively model uncertainty. These approaches show their inherent difficulties as pointed out by Molodtsov [6], because of intensive quantity and type of uncertainties. In Reference [6], Molodtsov defines the soft set which is a new logical instrument for dealing uncertainties.

Soft set theory attracts many authors because it has a vast range of applications in many areas like the smoothness of functions, decision making, probability theory, data analysis, measurement theory, forecasting and operations research [6 –10]. Nowadays, many authors work to hybridize the different models with soft set and achieved results in many applicable theories. Maji defines the fuzzy soft set and intuitionistic fuzzy soft set [11, 12]. Then the further extensions of soft sets like the generalized fuzzy soft set [13], the interval-valued fuzzy soft set [14], the soft rough set [15], the vague soft set [16], the trapezoidal fuzzy soft set [17], the neutrosophic soft set [18], the intuitionistic neutrosophic soft set [19], the multi-fuzzy soft set [20] and the hesitant fuzzy soft set [21] are introduced. Agarwal defines the generalized intuitionistic fuzzy soft set (GIFSS) [22] which has some problems pointed out by Feng [23] and redefined GIFSS.

In Reference [24], Coung defines the picture fuzzy set which is an extension of the fuzzy set and intuitionistic fuzzy set. Yang [25], introduced the picture fuzzy soft set and applied them to decision-making problems. In [26], Khan defines the generalized picture fuzzy soft set and applied them to decision making problems. For study more about decision making, we refer to [27 –35].

The soft matrix and its operations in a soft set were introduced by Cagman [36]. The notion of soft discernibility matrix $(SDM)$ is given by Feng and Zhou, which not only provide the best choice but also an order relation among all alternatives [37]. In [38], Feng introduced an adjustable approach for the fuzzy soft set and an adjustable approach for the intuitionistic fuzzy soft set is defined by Jiang [39]. In [25], an adjustable $SDM$ is define for picture fuzzy soft sets.

1.1 Related work and motivation

In 1987 [40], Gorzalczany introduced the interval-valued fuzzy set (IVFS) which is an extension of the fuzzy set. In 1989 [41], Atanassov defines the interval valued intuitionistic fuzzy set (IVIFS) which is an extension of an intuitionistic fuzzy set. In 2008 [42], Yang combined the soft set and IVFS and studied their properties. In 2016 [43], Zhenhua defines the generalized interval valued intuitionistic fuzzy set. In [44], Jiang defines the IVIFSS and discuss its properties. 1n [45], Li introduces the extension operator and algebraic properties for IVIFS. In [46], Zhang introduces a novel technique in IVIFSS. In [47], Xu introduces different matrices in interval-valued fuzzy numbers and used it for decision making. In [48], Wen defines the definite integrals for the aggregating continuous interval-valued intuitionistic fuzzy information. In [49], Ramalingam uses fuzzy interval-valued decision criteria for ranking features in multi-modal 3D face recognition. Peng [50] proposed an algorithms for interval-valued fuzzy soft sets in stochastic multi-criteria decision making based on regret theory and prospect theory with combined weight. Peng [51] proposed an algorithms for interval-valued fuzzy soft sets in emergency decision making based on WDBA and CODAS with new information measure.

The idea of generalized interval-valued intuitionistic fuzzy soft set defines by Wu [52] is very encouraging in decision-making since it considers how to capitalize an additional interval-valued intuitionistic fuzzy input from the director to minimize any possible perversion in the data provided by evaluating specialists. Since the director is responsible for the department, so he reviews and scrutinizes the general quality of evaluation made by experts groups. But the idea was not presented clearly and has some deficiencies and problems which are point outed in Section 3.

The interval fuzzy environment is better than the intuitionistic and picture fuzzy environment because instead of giving a crisp value, the decision-maker has an interval (range) to put his evaluation. Also, the proposed model is the generalization of all existing models of interval valued fuzzy models because it is the consist of IVIFS and IVIFSS.

Following the above line of exploration, the purpose of this paper is to clarify and reformulated the concept of GIVIFSS. This is done by a combination of IVIFS and IVIFSS. Using the present definition of GIVIFSS, we have introduced the two types of containment for GIVIFSS. We clarify and redefine the null GIVIFSS and absolute GIVIFSS. We define the g-union, g-intersection, g^∗-union, g^∗-intersection, OR operation, AND operation, g-necessity operations and g-possibility operations for GIVIFSS. The properties of these operations are investigated like De Morgans laws and many more presented in the text. An algorithm is proposed for solving MADM problems using GIVIFSS. We give a criterion to find a weight vector by using g-novel expectation score function. A descriptive example is presented to see the applicability of the proposed method. Results indicate that the proposed technique is more effective and generalize then previous models of interval-valued fuzzy sets.

The rest of the paper is organized as follows. Section 2 consists of the basic definitions related to fuzzy set and interval-valued fuzzy set. Section 3 highlights the problems in the concept define in [52] and reformulated with two types of containment in GIVIFSS. Section 4 also point out some deficiencies in [52] concept and based on new concept different operations are define and their properties are investigated. In Section 5, an algorithm is proposed for solving decision making problems using GIVIFSS and for obtaining weight vector the criteria is presented. In Section 6, a case study of construction is discussed to strengthen the method we proposed. Finally, Sections 7 and 8 consist of comparison of the proposed methods with the methods already presented in the literature and conclusion, respectively.

2 Preliminaries

In this section, we present the basic definitions of fuzzy set, intuitionistic fuzzy set (IFS), soft set, interval-valued fuzzy set (IVFS), IVIFS and IVIFSS.

A fuzzy set is defined by Zadeh [1], which handles uncertainty based on the view of gradualness effectively.

Definition 1. [1] A membership function $u_{\hat{A}} : \hat{Y} \to [0, 1]$ defines the fuzzy set $\hat{A}$ over the $\hat{Y}$ , where $u_{\hat{A}} (ℓ)$ particularized the membership of an element $ℓ \in \hat{Y}$ in fuzzy set $\hat{A}$ .

Like a membership degree on an element in a fuzzy set, human intuition suggests that there is a non-membership degree of an element in a set. In [4], an IFS defined by Atanassov to sketch the imprecision of human beings when needed the judgments over the elements.

Definition 2. [4] An IFS $\hat{A}$ over the universe $\hat{Y}$ is defined as $\hat{A} = {(ℓ, u_{\hat{A}}, v_{\hat{A}}) | ℓ \in \hat{Y}},$ where $u_{\hat{A}} : \hat{Y} \to [0, 1]$ and $v_{\hat{A}} : \hat{Y} \to [0, 1]$ are the degree of positive membership and degree of negative membership, respectively. Furthermore, it is required that $0 \leq u_{\hat{A}} (ℓ) + v_{\hat{A}} (ℓ) \leq 1$ .

A soft set is introduced by Molodtsov [6], which provides an effective framework to dealings with imprecision with the parametric point of view, i.e. each element is judged by some criteria of attributes.

Definition 3. [6] Let $\hat{Y}$ be a universal set, $\hat{P}$ a parameter space, $\hat{A} \subseteq \hat{P}$ and $P (\hat{Y})$ the power set of $\hat{Y}$ . A pair $(\hat{F}, \hat{A})$ is called a soft set over $\hat{Y}$ , where $\hat{F}$ is a set valued mapping given by $\hat{F} : \hat{A} \to P (\hat{Y})$ .

In [40], Gorzalczany defines the IVFS in 1987. An IVFS is a straightforward extension of the fuzzy set, where we get a membership interval instead of single value.

Definition 4. An IVFS A in Y is defined as $\hat{A} = {(ℓ, u (ℓ)) | ℓ \in \hat{Y}},$ where u (ℓ) is an interval and $u : \hat{Y} \to I$ , where I contains the subinterval of [0,1].

An extension of IVFS, namely, IVIFS is defined by Atanassov [41] in 1989 as follows.

Definition 5. An IVIFS $\hat{A}$ in $\hat{Y}$ is defined as $\hat{A} = {(ℓ, u (ℓ), v (ℓ)) | ℓ \in \hat{Y}},$ where u (ℓ) and v (ℓ) are the intervals, i.e. $u : \hat{Y} \to I$ and $v : \hat{Y} \to I$ , where I contains the subinterval of [0,1] with the condition sup u (ℓ) + sup v (ℓ) ⩽1 . If we denote $u = [u, \bar{u}]$ and $v = [v, \bar{v}]$ then union, intersection and complement of IVIFSs $\hat{A}$ and $\hat{B}$ are defined as follows $\begin{matrix} \hat{A} \cup \hat{B} & = {(ℓ, [\sup (u_{\hat{A}} (ℓ), u_{\hat{B}} (ℓ)), \sup ({\bar{u}}_{\hat{A}} (ℓ), {\bar{u}}_{\hat{B}} (ℓ))], \\ [\inf (v_{\hat{A}} (ℓ), v_{\hat{B}} (ℓ)), \inf ({\bar{v}}_{\hat{A}} (ℓ), {\bar{v}}_{\hat{B}} (ℓ))] | ℓ \in \hat{Y}}; \\ \hat{A} \cap \hat{B} & = {(ℓ, [\inf (u_{\hat{A}} (ℓ), u_{\hat{B}} (ℓ)), \inf ({\bar{u}}_{\hat{A}} (ℓ), {\bar{u}}_{\hat{B}} (ℓ))], \\ [\sup (v_{\hat{A}} (ℓ), v_{\hat{B}} (ℓ)), \sup ({\bar{v}}_{\hat{A}} (ℓ), {\bar{v}}_{\hat{B}} (ℓ))] | ℓ \in \hat{Y}}; \\ {\hat{A}}^{c} & = {(ℓ, v_{\hat{A}} (ℓ), u_{\hat{A}} (ℓ)) | ℓ \in \hat{Y}} . \end{matrix}$ We denote the interval valued intuitionistic fuzzy value (IVIFV) $p = ([u, \bar{u}], [v, \bar{v}])$ with the condition that $\bar{u} + \bar{v} ⩽ 1 .$

In [44], Jiang introduces the IVIFSS, which is an extension of IVIFS.

Definition 6. Let $\hat{Y}$ be a universal set, $\hat{P}$ a parameter space, $\hat{A} \subseteq \hat{P}$ and $IVIFS (\hat{Y})$ the set of all IVIFS over $\hat{Y}$ . A pair $(\hat{F}, \hat{A})$ is called an IVIFSS over $\hat{Y}$ , where $\hat{F}$ is a set valued mapping given by $\hat{F} : \hat{A} \to IVIFS (\hat{Y})$ .

In [55], Xu defines some operators of IVIFSs, one of them is interval valued intuitionistic fuzzy weighted averaging (IVIFWA) operator which defines as follows.

Definition 7. [55] Let $p_{i} = ([u, \bar{u}], [v, \bar{v}])$ (i = 1, 2, 3, . . . , n) be an IVIFVs. Then IVIFWA operator is a function defined pⁿ → p such that ${IVIFWA}_{\hat{ω}} (p_{1}, p_{2}, . . ., p_{n}) = \oplus_{i = 1}^{n} {\hat{ω}}_{i} p_{i}$ $\begin{matrix} = & ([1 - \prod_{i = 1}^{n} (1 - u_{i})^{{\hat{ω}}_{i}}, 1 - \prod_{i = 1}^{n} (1 - {\bar{u}}_{i})^{{\hat{ω}}_{i}}], \\ [\prod_{i = 1}^{n} (v_{i})^{{\hat{ω}}_{i}}, \prod_{i = 1}^{n} ({\bar{v}}_{i})^{{\hat{ω}}_{i}}]), \end{matrix}$ where $\hat{ω} = ({\hat{ω}}_{1}, {\hat{ω}}_{2}, . . ., {\hat{ω}}_{n})^{T}$ is the weight vector with each ${\hat{ω}}_{i} > 0$ and $\sum_{i = 1}^{n} {\hat{ω}}_{i} = 1$ .

In [53], Yager introduces the golden rule, i.e., a sophisticated representative value of Atanassov type intuitionistic membership grades to compare different solutions.

Definition 8. Let $u = [u, \bar{u}]$ be an interval valued fuzzy value (IVFV). Then golden rule for representative value of u is defined as $Rep (u) = M + (1 / 2 - M) \times R,$ where $M = (u + \bar{u}) / 2$ and $R = \bar{u} - u$ are the mean and range of u, respectively.

In [54], Nayagam defines the novel accuracy function Ł for IVIFV as follows.

Definition 9. Let $p = ([u, \bar{u}], [v, \bar{v}])$ be an IVIFV. Then novel accuracy function L of an IVIFV of p is defined as $\begin{matrix} L (p) = & \frac{u + \bar{u} - \bar{v} (1 - \bar{u}) - v (1 - u)}{2}, \\ L (p) \in [- 1, 1] . \end{matrix}$

3 Generalized interval valued intuitionisticfuzzy soft sets

GIVIFSS and its basic notions are discussed in this section. First we mention the definition of GIVIFSS unabridged from [52] as follows.

Definition 10. Let $\hat{Y} = {ℓ_{1}, ℓ_{2}, . . ., ℓ_{n}}$ denote a universal set of elements and the collection of all interval-valued intuitionistic fuzzy subsets of $\hat{Y}$ is denoted by $IVIF (\hat{Y})$ . Let $\hat{P} = {ℏ_{1}, ℏ_{2}, . . ., ℏ_{m}}$ be a set of parameters and $\hat{ρ}$ be an interval-valued intuitionistic fuzzy subset of $\hat{P}$ . The pair $(\hat{Y}, \hat{P})$ is called as a soft universe. Let $\hat{A} \subseteq \hat{P}$ and $\hat{F}$ be a mapping given by $\hat{F} : \hat{A} \to IVIF (\hat{Y})$ and ${\hat{F}}_{\hat{ρ}}$ by ${\hat{F}}_{\hat{ρ}} : \hat{A} \to IVIF (\hat{Y}) \times IVIF$ . A GIVIFSS ${\hat{F}}_{\hat{ρ}}$ over the soft universe $(\hat{Y}, \hat{P})$ is defined as ${\hat{F}}_{\hat{ρ}} (ℏ) = (\hat{F} (ℏ), \hat{ρ} (ℏ)),$ where $\hat{F} (ℏ) \in IVIF (\hat{Y})$ and $\hat{ρ} (ℏ) \in IVIF$ . $\hat{F} (ℏ)$ represents the elements of $\hat{Y}$ in the IVIFSS, and $\hat{ρ} (ℏ)$ indicates a senior expert’s assessment on the elements of $\hat{Y}$ in $\hat{F} (ℏ)$ .

Remark 1. Above mentioned definition has some difficulties which we would like to point out first.

From above mentioned definition 10, IVIF is an interval-valued intuitionistic fuzzy set. Thus the representation $IVIF (\hat{Y}) \times IVIF$ does not meaningful representation that’s why not well defined.

Similarly, in definition 10, $\hat{ρ} (ℏ) \in IVIF$ but $\hat{ρ} (ℏ)$ is not an interval-valued intuitionistic fuzzy set.

To overcome the issues in above mentioned definition 10, an improved definition of GIVIFSS is presented as follows.

Definition 11. Let $\hat{Y}$ be a universal set and $\hat{A} \subseteq \hat{P}$ . By a GIVIFSS we mean a triple $(\hat{F}, \hat{A}, \hat{ρ})$ , where $(\hat{F}, \hat{A})$ is an IVIFSS over $\hat{Y}$ and $\hat{ρ} : \hat{A} \to L$ is an IVIFS in $\hat{A}$ .

Keeping the idea of decision making in mind, we called $(\hat{F}, \hat{A})$ the basic interval valued intuitionistic fuzzy soft sets (BIVIFSS) and $\hat{ρ}$ is called the parametric interval-valued intuitionistic fuzzy sets (PIVIFS) of the $(\hat{F}, \hat{A}, \hat{ρ})$ . We denote the collection of all GIVIFSS over $\hat{Y}$ is ${GIVIFSS}^{\hat{P}} (\hat{Y})$ , where $\hat{P}$ is a parametric space and ${GIVIFSS}_{\hat{A}} (\hat{Y})$ for the fixed parametric space $\hat{A} \subseteq \hat{P}$ .

Example 1. Consider a GIVIFSS $(\hat{F}, \hat{A}, \hat{ρ})$ over $\hat{Y}$ and $\hat{A} \subseteq \hat{P}$ , where $\hat{Y}$ consists of the five laptops under consideration of a decision makers to purchase $\hat{Y} = {ℓ_{1}, ℓ_{2}, ℓ_{3}, ℓ_{4}, ℓ_{5}}$ . The parameter space consist of attributes $\hat{P} = {ℏ_{1}, ℏ_{2}, ℏ_{3}, ℏ_{4}, ℏ_{5}, ℏ_{6}}$ , where each ℏ_i stands for “keyboard/touch pad", “portability", “hard drive/RAM", “battery life" and “cheap", respectively. Let $\hat{A} = {ℏ_{1}, ℏ_{4}, ℏ_{5}} \subset \hat{P}$ chosen by a decision maker. In the view of criteria “keyboard/touch pad", “battery life" and “cheap" are the most useful characteristics for evaluation. Decision maker made an evaluation and respective results are described by the GIVIFSS $(\hat{F}, \hat{A}, \hat{ρ})$ presented in Table 1.

Table 1
The GIVIFSS

ℏ_i ℏ₁ ℏ₄ ℏ₅

ℓ₁ ([0.50, 0.70] , [0.20, 0.25]) ([0.60, 0.70] , [0.16, 0.21]) ([0.73, 0.79] , [0.10, 0.16])

ℓ₂ ([0.70, 0.80] , [0.10, 0.15]) ([0.60, 0.70] , [0.20, 0.30]) ([0.40, 0.60] , [0.20, 0.30])

ℓ₃ ([0.73, 0.84] , [0.10, 0.14]) ([0.55, 0.60] , [0.20, 0.40]) ([0.60, 0.90] , [0.04, 0.10])

ℓ₄ ([0.68, 0.78] , [0.10, 0.20]) ([0.63, 0.73] , [0.20, 0.26]) ([0.56, 0.70] , [0.18, 0.28])

ℓ₅ ([0.48, 0.58] , [0.30, 0.40]) ([0.40, 0.60] , [0.20, 0.30]) ([0.45, 0.65] , [0.25, 0.35])

τ ([0.40, 0.70] , [0.20, 0.30]) ([0.58, 0.65] , [0.30, 0.35]) ([0.30, 0.40] , [0.40, 0.50])

ℏ_i	ℏ₁	ℏ₄	ℏ₅
ℓ₁	([0.50, 0.70] , [0.20, 0.25])	([0.60, 0.70] , [0.16, 0.21])	([0.73, 0.79] , [0.10, 0.16])
ℓ₂	([0.70, 0.80] , [0.10, 0.15])	([0.60, 0.70] , [0.20, 0.30])	([0.40, 0.60] , [0.20, 0.30])
ℓ₃	([0.73, 0.84] , [0.10, 0.14])	([0.55, 0.60] , [0.20, 0.40])	([0.60, 0.90] , [0.04, 0.10])
ℓ₄	([0.68, 0.78] , [0.10, 0.20])	([0.63, 0.73] , [0.20, 0.26])	([0.56, 0.70] , [0.18, 0.28])
ℓ₅	([0.48, 0.58] , [0.30, 0.40])	([0.40, 0.60] , [0.20, 0.30])	([0.45, 0.65] , [0.25, 0.35])
τ	([0.40, 0.70] , [0.20, 0.30])	([0.58, 0.65] , [0.30, 0.35])	([0.30, 0.40] , [0.40, 0.50])

Next we mention the containment in GIVIFSS which was initiated in [52] as follows.

Definition 12. Let ${\hat{F}}_{\hat{ρ}} = (\hat{F} (ℏ), \hat{ρ} (ℏ))$ and ${\hat{G}}_{\hat{σ}} = (\hat{G} (ℏ), \hat{σ} (ℏ))$ be two GIVIFSSs over the universe $(\hat{Y}, \hat{P})$ . ${\hat{F}}_{\hat{ρ}}$ is the subset of ${\hat{G}}_{\hat{σ}}$ , if

$\hat{ρ} \subseteq \hat{σ}$ , and

$\hat{F} (ℏ)$ is an IVIF subset of $\hat{G} (ℏ)$ , $\forall ℏ \in \hat{P}$ .

Remark 2. This definition does not established the concept of containment in GIVIFSS due to following deficiencies.

It is possible that ${\hat{F}}_{\hat{ρ}}$ and ${\hat{G}}_{\hat{σ}}$ are defined on two different parametric sets $\hat{A}$ and $\hat{B}$ which are the subsets of parametric space $\hat{P}$ . But this situation is simply ignored in the given definition.

Suppose that the parametric set $\hat{A}$ and $\hat{B}$ are disjoint then IVIFSs $\hat{ρ}$ and $\hat{σ}$ are in different universe and can not be discussed. Also the second condition can not be discussed in this case.

To overcome the deficiencies in definition 12, we defined containment based on our new definition of GIVIFSS.

Definition 13. Let $Γ_{1} = (\hat{F}, \hat{A}, \hat{ρ})$ and $Γ_{2} = (\hat{G}, \hat{B}, \hat{σ})$ be two GIVIFSSs over $\hat{Y}$ and $\hat{A}, \hat{B} \subseteq \hat{P}$ . Then Γ₁ is called the GIVIFS F-subset of Γ₂ denoted as Γ₁ ⊆ _FΓ₂ if $\hat{A} \subseteq \hat{B}$ and

$\hat{F} (ℏ) \subseteq \hat{G} (ℏ)$ for all $ℏ \in \hat{A}$ ;

$u_{\hat{ρ}} (ℏ) ⩽ u_{\hat{σ}} (ℏ)$ , ${\bar{u}}_{\hat{ρ}} (ℏ) ⩽ {\bar{u}}_{\hat{σ}} (ℏ)$ , $v_{\hat{ρ}} (ℏ) ⩾ v_{\hat{σ}} (ℏ)$ and ${\bar{v}}_{\hat{ρ}} (ℏ) ⩾ {\bar{v}}_{\hat{σ}} (ℏ)$ for all $ℏ \in \hat{A}$ .

Definition 14. Let $Γ_{1} = (\hat{F}, \hat{A}, \hat{ρ})$ and $Γ_{2} = (\hat{G}, \hat{B}, \hat{σ})$ be two GIVIFSSs over $\hat{Y}$ and $\hat{A}, \hat{B} \subseteq \hat{P}$ . Then Γ₁ is called the GIVIFS M-subset of Γ₂ denoted as Γ₁ ⊆ _MΓ₂ if $\hat{A} \subseteq \hat{B}$ and

$\hat{F} (ℏ) = \hat{G} (ℏ)$ for all $ℏ \in \hat{A}$ ;

Definition 15. Let $Γ_{1} = (\hat{F}, \hat{A}, \hat{ρ})$ and $Γ_{2} = (\hat{G}, \hat{B}, \hat{σ})$ be two GIVIFSSs over $\hat{Y}$ and $\hat{A}, \hat{B} \subseteq \hat{P}$ . Then Γ₁ is called the GIVIFS equal to Γ₂ denoted as Γ₁ = Γ₂ if $\hat{A} = \hat{B}$ , $\hat{F} (ℏ) = \hat{G} (ℏ)$ for all $ℏ \in \hat{A}$ and $\hat{ρ} = \hat{σ}$ .

4 Basic operations of GIVIFSS

In this section, we point out some problems in already defines operations for GIVIFSSs then remove and redefine the operations for GIVIFSSs. We define different operations, namely, g-union, g-intersection, g^∗-union, g^∗-intersection, AND-operation, OR-operation, g-necessity and g-possibility operation. Also some properties and relations within these operations are discussed in this section.

The operations of union and intersection of GIVIFSSs was introduced by Wu [52] as follows.

Definition 16. The union of ${\hat{F}}_{\hat{ρ}}$ and ${\hat{G}}_{\hat{σ}}$ , denoted by ${\hat{F}}_{\hat{ρ}} \cup {\hat{G}}_{\hat{σ}}$ , is given as follows: ${\hat{F}}_{\hat{ρ}} \cup {\hat{G}}_{\hat{σ}} = (\hat{F} (ℏ) \circ \hat{G} (ℏ), \hat{ρ} (ℏ) \circ \hat{σ} (ℏ)), \forall ℏ \in \hat{P},$ where $\hat{F} (ℏ) \circ \hat{G} (ℏ) \in IVIF (\hat{Y})$ , $\hat{ρ} (ℏ) \circ \hat{σ} (ℏ) \in IVIF$ , ∘ is any t-conorm.

Definition 17. The intersection of ${\hat{F}}_{\hat{ρ}}$ and ${\hat{G}}_{\hat{σ}}$ , denoted by ${\hat{F}}_{\hat{ρ}} \cap {\hat{G}}_{\hat{σ}}$ , is given as follows: ${\hat{F}}_{\hat{ρ}} \cap {\hat{G}}_{\hat{σ}} = (\hat{F} (ℏ) * \hat{G} (ℏ), \hat{ρ} (ℏ) * \hat{σ} (ℏ)), \forall ℏ \in \hat{P},$ where $\hat{F} (ℏ) * \hat{G} (ℏ) \in IVIF (\hat{Y})$ , $\hat{ρ} (ℏ) * \hat{σ} (ℏ) \in IVIF$ , ∗ is any t-norm.

Remark 3. We can easily trace the difficulties in above mentioned definitions. Here is a brief analysis present.

Let ${\hat{F}}_{\hat{ρ}}$ and ${\hat{G}}_{\hat{σ}}$ have parametric set $\hat{A}$ and $\hat{B}$ , respectively. It is possible that $\hat{A}$ and $\hat{B}$ are different even disjoint. But in above definition parameters set are ignored and consider only space $\hat{P}$ .

Since $\hat{F} (ℏ)$ and $\hat{G} (ℏ)$ are IVIFS, but we know t-norm and t-conorm are binary functions on interval [0,1]. Thus representations $\hat{F} (ℏ) \circ \hat{G} (ℏ)$ and $\hat{F} (ℏ) * \hat{G} (ℏ)$ are not used appropriately.

Also from definition 10, IVIF is an interval-valued intuitionistic fuzzy set. Thus representations like $\hat{ρ} (ℏ) \circ \hat{σ} (ℏ)$ and $\hat{ρ} (ℏ) * \hat{σ} (ℏ)$ are improperly used.

To overcome the deficiencies raised above, we defined union and intersection of GIVIFSS as follows.

Definition 18. GIVIFSS Let $Γ_{1} = (\hat{F}, \hat{A}, \hat{ρ})$ and $Γ_{2} = (\hat{G}, \hat{B}, \hat{σ})$ be two GIVIFSSs over $\hat{Y}$ . Then g-union is denoted by $(\hat{H}, \hat{C}, \hat{τ}) = (\hat{F}, \hat{A}, \hat{ρ}) ⊔_{g} (\hat{G}, \hat{B}, \hat{σ})$ and defined as, for all $ℏ \in \hat{C} = \hat{A} \cup \hat{B}$ , $\begin{matrix} \hat{H} (ℏ) = {\begin{matrix} \hat{F} (ℏ), & if ℏ \in \hat{A} \ \hat{B}, \\ \hat{G} (ℏ), & if ℏ \in \hat{B} \ \hat{A}, \\ \hat{F} (ℏ) \cup \hat{G} (h), & if ℏ \in \hat{A} \cap \hat{B}; \end{matrix} \\ u_{\hat{τ}} (ℏ) = {\begin{matrix} u_{\hat{ρ}} (ℏ), & if ℏ \in \hat{A} \ \hat{B}, \\ u_{\hat{σ}} (h), & if ℏ \in \hat{B} \ \hat{A}, \\ w_{1}, & if ℏ \in \hat{A} \cap \hat{B}; \end{matrix} \\ v_{\hat{τ}} (ℏ) = {\begin{matrix} v_{\hat{ρ}} (ℏ), & if ℏ \in \hat{A} \ \hat{B}, \\ v_{\hat{σ}} (ℏ), & if ℏ \in \hat{B} \ \hat{A}, \\ w_{2}, & if ℏ \in \hat{A} \cap \hat{B} . \end{matrix} \end{matrix}$ where w₁ = [sup $(u_{\hat{ρ}} (ℏ), u_{\hat{σ}} (ℏ)),$ sup $({\bar{u}}_{\hat{ρ}} (ℏ), {\bar{u}}_{\hat{σ}} (ℏ))]$ and w₂ = [inf $(v_{\hat{ρ}} (ℏ), v_{\hat{σ}} (ℏ)),$ inf $({\bar{v}}_{\hat{ρ}} (ℏ), {\bar{v}}_{\hat{σ}} (ℏ))]$ .

Definition 19. GIVIFSS Let $Γ_{1} = (\hat{F}, \hat{A}, \hat{ρ})$ and $Γ_{2} = (\hat{G}, \hat{B}, \hat{σ})$ be two GIVIFSSs over $\hat{Y}$ . Then g-intersection is denoted by $(\hat{H}, \hat{C}, \hat{τ}) = (\hat{F}, \hat{A}, \hat{ρ}) ⊓_{g} (\hat{G}, \hat{B}, \hat{σ})$ and defined as, for all $ℏ \in \hat{C} = \hat{A} \cup \hat{B}$ , $\begin{matrix} \hat{H} (ℏ) = & {\begin{matrix} \hat{F} (ℏ), & if ℏ \in \hat{A} \ \hat{B}, \\ \hat{G} (ℏ), & if ℏ \in \hat{B} \ \hat{A}, \\ \hat{F} (ℏ) \cap \hat{G} (h), & if ℏ \in \hat{A} \cap \hat{B}; \end{matrix} \\ u_{\hat{τ}} (ℏ) = & {\begin{matrix} u_{\hat{ρ}} (ℏ), & if ℏ \in \hat{A} \ \hat{B}, \\ u_{\hat{σ}} (h), & if ℏ \in \hat{B} \ \hat{A}, \\ w_{1}, & if ℏ \in \hat{A} \cap \hat{B}; \end{matrix} \\ v_{\hat{τ}} (ℏ) = & {\begin{matrix} v_{\hat{ρ}} (ℏ), & if ℏ \in \hat{A} \ \hat{B}, \\ v_{\hat{σ}} (ℏ), & if ℏ \in \hat{B} \ \hat{A}, \\ w_{2}, & if ℏ \in \hat{A} \cap \hat{B} . \end{matrix} \end{matrix}$ where w₁ = [inf $(u_{\hat{ρ}} (ℏ), u_{\hat{σ}} (ℏ)),$ inf $({\bar{u}}_{\hat{ρ}} (ℏ), {\bar{u}}_{\hat{σ}} (ℏ))]$ and w₂ = [sup $(v_{\hat{ρ}} (ℏ), v_{\hat{σ}} (ℏ)),$ sup $({\bar{v}}_{\hat{ρ}} (ℏ), {\bar{v}}_{\hat{σ}} (ℏ))]$ .

Definition 20. Let $Γ_{1} = (\hat{F}, \hat{A}, \hat{ρ})$ and $Γ_{2} = (\hat{G}, \hat{B}, \hat{σ})$ be two GIVIFSSs over $\hat{Y}$ and $\hat{C} = \hat{A} \cap \hat{B} \neq \emptyset$ . Then g^∗-union is denoted by $(\hat{H}, \hat{C}, \hat{τ}) = (\hat{F}, \hat{A}, \hat{ρ}) ⊔_{g^{*}} (\hat{G}, \hat{B}, \hat{σ})$ and defined as for all $ℏ \in \hat{C} = \hat{A} \cap \hat{B}$ , we have

$\hat{H} (ℏ) = \hat{F} (ℏ) \cup \hat{G} (h)$ ;

$u_{\hat{τ}} (ℏ) = [$ sup $(u_{\hat{ρ}} (ℏ), u_{\hat{σ}} (ℏ)),$ sup $({\bar{u}}_{\hat{ρ}} (ℏ), {\bar{u}}_{\hat{σ}} (ℏ))]$ ;

$v_{\hat{τ}} (ℏ) = [$ inf $(v_{\hat{ρ}} (ℏ), v_{\hat{σ}} (ℏ)),$ inf $({\bar{v}}_{\hat{ρ}} (ℏ), {\bar{v}}_{\hat{σ}} (ℏ))]$ .

Definition 21. Let $Γ_{1} = (\hat{F}, \hat{A}, \hat{ρ})$ and $Γ_{2} = (\hat{G}, \hat{B}, \hat{σ})$ be two GIVIFSSs over $\hat{Y}$ and $\hat{C} = \hat{A} \cap \hat{B} \neq \emptyset$ . Then g^∗-intersection is denoted by $(\hat{H}, \hat{C}, \hat{τ}) = (\hat{F}, \hat{A}, \hat{ρ}) ⊓_{g^{*}} (\hat{G}, \hat{B}, \hat{σ})$ and defined as for all $ℏ \in \hat{C} = \hat{A} \cap \hat{B}$ , we have

$ℏ \in \hat{C} = \hat{A} \cap \hat{B}$ , $\hat{H} (ℏ) = \hat{F} (ℏ) \cap \hat{G} (h)$ ;

$u_{\hat{τ}} (ℏ) = [$ inf $(u_{\hat{ρ}} (ℏ), u_{\hat{σ}} (ℏ)),$ inf $({\bar{u}}_{\hat{ρ}} (ℏ), {\bar{u}}_{\hat{σ}} (ℏ))]$ ;

$v_{\hat{τ}} (ℏ) = [$ sup $(v_{\hat{ρ}} (ℏ), v_{\hat{σ}} (ℏ)),$ sup $({\bar{v}}_{\hat{ρ}} (ℏ), {\bar{v}}_{\hat{σ}} (ℏ))]$ .

Remark 4. The notion of g-union and g-intersection become identical with g^∗-union and g^∗-intersection, respectively, when we have same set of parameters for two or more GIVIFSSs.

Now we mention the definitions of null and absolute GIVIFSS, proposed by Wu [52].

Definition 22. A GIVIFSS $Θ_{\hat{ρ}} (ℏ) = (\hat{F} (ℏ), \hat{ρ} (ℏ))$ is called a null GIVIFSS, if $\hat{F} (ℏ) = ([0, 0], [1, 1])$ , $\hat{ρ} (ℏ) = ([0, 0], [1, 1])$ , for all $ℏ \in \hat{P}$ .

Definition 23. A GIVIFSS $Ω_{\hat{ρ}} (ℏ) = (\hat{F} (ℏ), \hat{ρ} (ℏ))$ is called an absolute GIVIFSS, if $\hat{F} (ℏ) = ([1, 1], [0, 0])$ , $\hat{ρ} (ℏ) = ([1, 1], [0, 0])$ , for all $ℏ \in \hat{P}$ .

Remark 5. Deficiencies in above mentioned definitions are easily traced. For example, it is incorrect to write $\hat{F} (ℏ) = ([0, 0], [1, 1])$ and $\hat{F} (ℏ) = ([1, 1], [0, 0])$ because $\hat{F} (ℏ)$ is an IVIFS.

To fixed the problem in above definitions we redefined null and absolute GIVIFSS as follows.

Definition 24. A GIVIFSS $Θ_{\hat{ρ}} = (\hat{F}, \hat{A}, \hat{ρ})$ is called a null GIVIFSS, if

for all $ℏ \in \hat{A}$ , $u_{\hat{F} (ℏ)} = [0, 0]$ and $v_{\hat{F} (ℏ)} = [1, 1]$ ;

for all $ℏ \in \hat{A}$ , $u_{\hat{ρ} (ℏ)} = [0, 0]$ and $v_{\hat{ρ} (ℏ)} = [1, 1]$ .

Definition 25. A GIVIFSS $Ω_{\hat{ρ}} = (\hat{F}, \hat{A}, \hat{ρ})$ is called an absolute GIVIFSS, if

for all $ℏ \in \hat{A}$ , $u_{\hat{F} (ℏ)} = [1, 1]$ and $v_{\hat{F} (ℏ)} = [0, 0]$ ;

for all $ℏ \in \hat{A}$ , $u_{\hat{ρ} (ℏ)} = [1, 1]$ and $v_{\hat{ρ} (ℏ)} = [0, 0]$ .

Definition 26. Suppose $Γ = (\hat{F}, \hat{A}, \hat{ρ})$ be a GIVIFSS over $\hat{Y}$ . The complement of Γ is defined as the GIVIFSS $(\hat{F}, \hat{A}, \hat{ρ})^{c} = (\hat{G}, \hat{A}, \hat{σ})$ where

$(\hat{G}, \hat{A})$ is the complement of BIVIFSS i.e., ${\hat{F}}^{c} (ℏ) = \hat{G} (ℏ) = {(ℓ, v_{\hat{F} (ℏ)} (ℓ), u_{\hat{F} (ℏ)} (ℓ)) | \forall ℓ \in \hat{Y} and ℏ \in \hat{A}}$ ;

$\hat{σ}$ is the complement of PIVIFS i.e., ${\hat{ρ}}^{c} = \hat{σ} = {(ℏ, v_{\hat{ρ}} (ℏ), u_{\hat{ρ}} (ℏ)) | \forall ℏ \in \hat{A}}$ .

Now we mention some basic properties of GIVIFSSs.

Theorem 1. Let $Γ = (\hat{F}, \hat{A}, \hat{ρ})$ be a GIVIFSS over $\hat{Y}$ . If $Θ = ({\hat{F}}_{Θ}, \hat{A}, {\hat{ρ}}_{Θ})$ and $Ω = ({\hat{F}}_{Ω}, \hat{A}, {\hat{ρ}}_{Ω})$ are the null and absolute GIVIFSSs, respectively, then we have

Γ ⊔ _gΓ = Γ ⊔ _g
^∗Γ = Γ;

Γ ⊓ _gΓ = Γ ⊓ _g
^∗Γ = Γ;

Γ ⊔ _gΘ = Γ ⊔ _g
^∗Θ = Γ;

Γ ⊓ _gΘ = Γ ⊓ _g
^∗Θ = Θ;

Γ ⊔ _gΩ = Γ ⊔ _g
^∗Ω = Ω;

Γ ⊓ _gΩ = Γ ⊓ _g
^∗Ω = Γ.

Proof. These assertions can easily obtained from definitions 18, 19, 20, 21, 24 and 25. □ Theorem 2. Let $Γ_{1} = (\hat{F}, \hat{A}, \hat{ρ})$ and $Γ_{2} = (\hat{G}, \hat{B}, \hat{σ})$ be two GIVIFSSs over $\hat{Y}$ . Then we have

$(Γ_{1} ⊔_{g} Γ_{2})^{c} = Γ_{1}^{c} ⊓_{g} Γ_{2}^{c}$ ;

$(Γ_{1} ⊓_{g} Γ_{2})^{c} = Γ_{1}^{c} ⊔_{g} Γ_{2}^{c}$ ;

$(Γ_{1} ⊔_{g^{*}} Γ_{2})^{c} = Γ_{1}^{c} ⊓_{g^{*}} Γ_{2}^{c}$ ;

$(Γ_{1} ⊓_{g^{*}} Γ_{2})^{c} = Γ_{1}^{c} ⊔_{g^{*}} Γ_{2}^{c}$ .

Proof. The proof of these assertions can easily be obtained from definitions 18, 19, 20, 21 and 26. □ Example 2. Consider a GIVIFSS $Γ_{1} = (\hat{F}, \hat{A}, \hat{ρ})$ over $\hat{Y}$ in example 1. Let $\hat{B} = {ℏ_{2}, ℏ_{3}, ℏ_{5}, ℏ_{6}} \subset \hat{P}$ and $Γ_{2} = (\hat{G}, \hat{B}, \hat{σ})$ be a GIVIFSS written in Table 2.

Table 2
The GIVIFSS

ℏ_i ℏ₂ ℏ₃ ℏ₅ ℏ₆

ℓ₁ ([0.60, 0.80] , [0.13, 0.18]) ([0.61, 0.72] , [0.13, 0.21]) ([0.70, 0.79] , [0.15, 0.21]) ([0.76, 0.86] , [0.10, 0.14])

ℓ₂ ([0.50, 0.60] , [0.21, 0.31]) ([0.62, 0.74] , [0.12, 0.23]) ([0.43, 0.63] , [0.23, 0.33]) ([0.34, 0.56] , [0.22, 0.34])

ℓ₃ ([0.73, 0.80] , [0.10, 0.18]) ([0.45, 0.60] , [0.22, 0.34]) ([0.64, 0.89] , [0.04, 0.11]) ([0.61, 0.91] , [0.04, 0.09])

ℓ₄ ([0.61, 0.71] , [0.15, 0.25]) ([0.36, 0.53] , [0.22, 0.32]) ([0.54, 0.70] , [0.19, 0.28]) ([0.66, 0.77] , [0.13, 0.23])

ℓ₅ ([0.58, 0.68] , [0.23, 0.30]) ([0.47, 0.67] , [0.21, 0.30]) ([0.45, 0.65] , [0.22, 0.35]) ([0.35, 0.50] , [0.27, 0.37])

τ ([0.43, 0.73] , [0.13, 0.23]) ([0.51, 0.65] , [0.31, 0.35]) ([0.33, 0.43] , [0.41, 0.51]) ([0.43, 0.53] , [0.23, 0.43])

ℏ_i	ℏ₂	ℏ₃	ℏ₅	ℏ₆
ℓ₁	([0.60, 0.80] , [0.13, 0.18])	([0.61, 0.72] , [0.13, 0.21])	([0.70, 0.79] , [0.15, 0.21])	([0.76, 0.86] , [0.10, 0.14])
ℓ₂	([0.50, 0.60] , [0.21, 0.31])	([0.62, 0.74] , [0.12, 0.23])	([0.43, 0.63] , [0.23, 0.33])	([0.34, 0.56] , [0.22, 0.34])
ℓ₃	([0.73, 0.80] , [0.10, 0.18])	([0.45, 0.60] , [0.22, 0.34])	([0.64, 0.89] , [0.04, 0.11])	([0.61, 0.91] , [0.04, 0.09])
ℓ₄	([0.61, 0.71] , [0.15, 0.25])	([0.36, 0.53] , [0.22, 0.32])	([0.54, 0.70] , [0.19, 0.28])	([0.66, 0.77] , [0.13, 0.23])
ℓ₅	([0.58, 0.68] , [0.23, 0.30])	([0.47, 0.67] , [0.21, 0.30])	([0.45, 0.65] , [0.22, 0.35])	([0.35, 0.50] , [0.27, 0.37])
τ	([0.43, 0.73] , [0.13, 0.23])	([0.51, 0.65] , [0.31, 0.35])	([0.33, 0.43] , [0.41, 0.51])	([0.43, 0.53] , [0.23, 0.43])

Since $ℏ_{5} \in \hat{A} \cap \hat{B}$ , therefore g^∗-union and g^∗-intersection of Γ₁ and Γ₂ is given as $\begin{matrix} (\hat{F}, \hat{A}) ⊓_{g^{*}} (\hat{G}, \hat{B}) & = \hat{F} (ℏ_{5}) ⊓ \hat{G} (ℏ_{5}) = {([0.70, 0.79], [0.15, 0.21]) / ℓ_{1}, ([0.40, 0.60], [0.23, 0.33]) / ℓ_{2}, \\ ([0.60, 0.89], [0.04, 0.11]) / ℓ_{3}, ([0.54, 0.70], [0.19, 0.28]) / ℓ_{4}, ([0.45, 0.65], [0.25, 0.35]) / ℓ_{5}}, \\ \hat{ρ} ⊓_{g^{*}} \hat{σ} & = {([0.30, 0.40], [0.41, 0.51]) / ℏ_{5}}, \\ (\hat{F}, \hat{A}) ⊔_{g^{*}} (\hat{G}, \hat{B}) & = \hat{F} (ℏ_{5}) ⊔ \hat{G} (ℏ_{5}) = {([0.73, 0.79], [0.10, 0.16]) / ℓ_{1}, ([0.43, 0.63], [0.20, 0.30]) / ℓ_{2}, \\ ([0.64, 0.90], [0.04, 0.10]) / ℓ_{3}, ([0.56, 0.70], [0.18, 0.28]) / ℓ_{4}, ([0.45, 0.65], [0.22, 0.35]) / ℓ_{5}}, \\ \hat{ρ} ⊔_{g^{*}} \hat{σ} & = {([0.33, 0.43], [0.40, 0.50])) / ℏ_{5}} . \end{matrix}$

We have find the g-union and g-intersection of Γ₁ and Γ₂ which are given in Tables [3,4].

Table 3

The g-union Γ₁ ⊔ _gΓ₂

ℏ_i	ℏ₁	ℏ₂	ℏ₃
ℓ₁	([0.50, 0.70] , [0.20, 0.25])	([0.60, 0.80] , [0.13, 0.18])	([0.61, 0.72] , [0.13, 0.21])
ℓ₂	([0.70, 0.80] , [0.10, 0.15])	([0.50, 0.60] , [0.21, 0.31])	([0.62, 0.74] , [0.12, 0.23])
ℓ₃	([0.73, 0.84] , [0.10, 0.14])	([0.73, 0.80] , [0.10, 0.18])	([0.45, 0.60] , [0.22, 0.34])
ℓ₄	([0.68, 0.78] , [0.10, 0.20])	([0.61, 0.71] , [0.15, 0.25])	([0.36, 0.53] , [0.22, 0.32])
ℓ₅	([0.48, 0.58] , [0.30, 0.40])	([0.58, 0.68] , [0.23, 0.30])	([0.47, 0.67] , [0.21, 0.30])
τ	([0.40, 0.70] , [0.20, 0.30])	([0.43, 0.73] , [0.13, 0.23])	([0.51, 0.65] , [0.31, 0.35])
ℏ_i	ℏ₄	ℏ₅	ℏ₆
ℓ₁	([0.60, 0.70] , [0.16, 0.21])	([0.73, 0.79] , [0.10, 0.16])	([0.76, 0.86] , [0.10, 0.14])
ℓ₂	([0.60, 0.70] , [0.20, 0.30])	([0.43, 0.63] , [0.20, 0.30])	([0.34, 0.56] , [0.22, 0.34])
ℓ₃	([0.55, 0.60] , [0.20, 0.40])	([0.64, 0.90] , [0.04, 0.10])	([0.61, 0.91] , [0.04, 0.09])
ℓ₄	([0.63, 0.73] , [0.20, 0.26])	([0.56, 0.70] , [0.18, 0.28])	([0.66, 0.77] , [0.13, 0.23])
ℓ₅	([0.40, 0.60] , [0.20, 0.30])	([0.45, 0.65] , [0.22, 0.35])	([0.35, 0.50] , [0.27, 0.37])
τ	([0.58, 0.65] , [0.30, 0.35])	([0.33, 0.43] , [0.40, 0.50])	([0.43, 0.53] , [0.23, 0.43])

Table 4

The g-intersection Γ₁ ⊓ _gΓ₂

ℏ_i	ℏ₁	ℏ₂	ℏ₃
ℓ₁	([0.50, 0.70] , [0.20, 0.25])	([0.60, 0.80] , [0.13, 0.18])	([0.61, 0.72] , [0.13, 0.21])
ℓ₂	([0.70, 0.80] , [0.10, 0.15])	([0.50, 0.60] , [0.21, 0.31])	([0.62, 0.74] , [0.12, 0.23])
ℓ₃	([0.73, 0.84] , [0.10, 0.14])	([0.73, 0.80] , [0.10, 0.18])	([0.45, 0.60] , [0.22, 0.34])
ℓ₄	([0.68, 0.78] , [0.10, 0.20])	([0.61, 0.71] , [0.15, 0.25])	([0.36, 0.53] , [0.22, 0.32])
ℓ₅	([0.48, 0.58] , [0.30, 0.40])	([0.58, 0.68] , [0.23, 0.30])	([0.47, 0.67] , [0.21, 0.30])
τ	([0.40, 0.70] , [0.20, 0.30])	([0.43, 0.73] , [0.13, 0.23])	([0.51, 0.65] , [0.31, 0.35])
ℏ_i	ℏ₄	ℏ₅	ℏ₆
ℓ₁	([0.60, 0.70] , [0.16, 0.21])	([0.70, 0.79] , [0.15, 0.21])	([0.76, 0.86] , [0.10, 0.14])
ℓ₂	([0.60, 0.70] , [0.20, 0.30])	([0.40, 0.60] , [0.23, 0.33])	([0.34, 0.56] , [0.22, 0.34])
ℓ₃	([0.55, 0.60] , [0.20, 0.40])	([0.60, 0.89] , [0.04, 0.11])	([0.61, 0.91] , [0.04, 0.09])
ℓ₄	([0.63, 0.73] , [0.20, 0.26])	([0.54, 0.70] , [0.19, 0.28])	([0.66, 0.77] , [0.13, 0.23])
ℓ₅	([0.40, 0.60] , [0.20, 0.30])	([0.45, 0.65] , [0.25, 0.35])	([0.35, 0.50] , [0.27, 0.37])
τ	([0.58, 0.65] , [0.30, 0.35])	([0.30, 0.40] , [0.41, 0.51])	([0.43, 0.53] , [0.23, 0.43])

Definition 27. Let $Γ_{1} = (\hat{F}, \hat{A}, \hat{ρ})$ and $Γ_{2} = (\hat{G}, \hat{B}, \hat{σ})$ be two GIVIFSSs over $\hat{Y}$ . Then the “AND" operation of Γ₁ and Γ₂ is denoted as Γ₁ ∇ _gΓ₂ and defined as $(\hat{F}, \hat{A}, \hat{ρ}) \nabla_{g} (\hat{G}, \hat{B}, \hat{σ}) = (\hat{H}, \hat{C}, \hat{τ})$ , where 1) $\hat{H} (a, b) = \hat{F} (a) \cap \hat{G} (b)$ , for all $(a, b) \in \hat{C} = \hat{A} \times \hat{B}$ , we have $\hat{H} (a, b) = {(ℓ, w_{1}, w_{2}) | forall ℓ \in \hat{Y}},$ where $w_{1} = [\inf (u_{\hat{F} (a)} (ℓ), u_{\hat{G} (b)} (ℓ)), \inf ({\bar{u}}_{\hat{F} (a)} (ℓ), {\bar{u}}_{\hat{G} (b)} (ℓ))]$ , $w_{2} = [\sup (v_{\hat{F} (a)} (ℓ), v_{\hat{G} (b)} (ℓ)), \sup ({\bar{v}}_{\hat{F} (a)} (ℓ), {\bar{v}}_{\hat{G} (b)} (ℓ))]$ . 2) $\hat{τ} (a, b) = \hat{ρ} (a) \cap \hat{σ} (b)$ and for all $(a, b) \in \hat{C} = \hat{A} \times \hat{B}$ , we have $\begin{matrix} \hat{τ} (a, b) = ([\inf (u_{\hat{ρ}} (a), u_{\hat{σ}} (b)), \inf ({\bar{u}}_{\hat{ρ}} (a), {\bar{u}}_{\hat{σ}} (b))], \\ [\sup (v_{\hat{ρ}} (a), v_{\hat{σ}} (b)), \sup ({\bar{v}}_{\hat{ρ}} (a), {\bar{v}}_{\hat{σ}} (b))]) . \end{matrix}$

Definition 28. Let $Γ_{1} = (\hat{F}, \hat{A}, \hat{ρ})$ and $Γ_{2} = (\hat{G}, \hat{B}, \hat{σ})$ be two GIVIFSSs over $\hat{Y}$ . Then the “OR" operation of Γ₁ and Γ₂ is denoted as Γ₁ ▵ _gΓ₂ and defined as $(\hat{F}, \hat{A}, \hat{ρ}) ▵_{g} (\hat{G}, \hat{B}, \hat{σ}) = (\hat{H}, \hat{C}, \hat{τ})$ , where

1) $\hat{H} (a, b) = \hat{F} (a) \cup \hat{G} (b)$ and for all $(a, b) \in \hat{C} = \hat{A} \times \hat{B}$ , we have $\hat{H} (a, b) = (ℓ, w_{1}, w_{2}) | forall ℓ \in \hat{Y}},$ where $w_{1} = [\sup (u_{\hat{F} (a)} (ℓ), u_{\hat{G} (b)} (ℓ)), \sup ({\bar{u}}_{\hat{F} (a)} (ℓ), {\bar{u}}_{\hat{G} (b)} (ℓ))]$ , $w_{2} = [\inf (v_{\hat{F} (a)} (ℓ), v_{\hat{G} (b)} (ℓ)), \inf ({\bar{v}}_{\hat{F} (a)} (ℓ), {\bar{v}}_{\hat{G} (b)} (ℓ))]$ .

2) $\hat{τ} (a, b) = \hat{ρ} (a) \cup \hat{σ} (b)$ and for all $(a, b) \in \hat{C} = \hat{A} \times \hat{B}$ , we have $\hat{τ} (a, b) = (w_{1}, w_{2}),$ where $w_{1} = [\sup (u_{\hat{ρ}} (a), u_{\hat{σ}} (b)), \sup ({\bar{u}}_{\hat{ρ}} (a), {\bar{u}}_{\hat{σ}} (b))]$ , $w_{2} = [\inf (v_{\hat{ρ}} (a), v_{\hat{σ}} (b)), \inf ({\bar{v}}_{\hat{ρ}} (a), {\bar{v}}_{\hat{σ}} (b))]$ .

Example 3. Consider the two GIVIFSSs $Γ_{1} = (\hat{F}, \hat{A}, \hat{ρ})$ and $Γ_{2} = (\hat{G}, \hat{B}, \hat{σ})$ from the Examples 1 and 2. We can calculate the AND and OR operations of Γ₁ and Γ₂ as follows.

For $(ℏ_{1}, ℏ_{2}) \in \hat{A} \times \hat{B}$ , we have $\begin{matrix} \hat{F} (ℏ_{1}) \cap \hat{G} (ℏ_{2}) = {([0.50, 0.70], [0.20, 0.25]) / ℓ_{1}, \\ ([0.50, 0.60], [0.21, 0.31]) / ℓ_{2}, ([0.73, 0.80], \\ [0.10, 0.18]) / ℓ_{3}, ([0.61, 0.71], [0.15, 0.25]) / ℓ_{4}, \\ ([0.48, 0.58], [0.30, 0.40]) / ℓ_{5}}, \\ \hat{ρ} (ℏ_{1}) \cap \hat{σ} (ℏ_{2}) = ([0.40, 0.70], [0.20, 0.30]) . \end{matrix}$

Similarly, we can make all the calculations for Table 5.

Table 5

$(\hat{F}, \hat{A}, \hat{ρ}) \nabla_{g} (\hat{G}, \hat{B}, \hat{σ}) = (\hat{H}, \hat{C}, \hat{τ})$

$\hat{A} \times \hat{B}$	ℏ₂	ℏ₃	ℏ₅	ℏ₆
ℏ₁	$\hat{F} (ℏ_{1}) \cap \hat{G} (ℏ_{2})$	$\hat{F} (ℏ_{1}) \cap \hat{G} (ℏ_{3})$	$\hat{F} (ℏ_{1}) \cap \hat{G} (ℏ_{5})$	$\hat{F} (ℏ_{1}) \cap \hat{G} (ℏ_{6})$
ℏ₄	$\hat{F} (ℏ_{4}) \cap \hat{G} (ℏ_{2})$	$\hat{F} (ℏ_{4}) \cap \hat{G} (ℏ_{3})$	$\hat{F} (ℏ_{4}) \cap \hat{G} (ℏ_{5})$	$\hat{F} (ℏ_{4}) \cap \hat{G} (ℏ_{6})$
ℏ₅	$\hat{F} (ℏ_{5}) \cap \hat{G} (ℏ_{2})$	$\hat{F} (ℏ_{5}) \cap \hat{G} (ℏ_{3})$	$\hat{F} (ℏ_{5}) \cap \hat{G} (ℏ_{5})$	$\hat{F} (ℏ_{5}) \cap \hat{G} (ℏ_{6})$
$\hat{A} \times \hat{B}$	ℏ₂	ℏ₃	ℏ₅	ℏ₆
[.5ex] ℏ₁	$\hat{ρ} (ℏ_{1}) \cap \hat{σ} (ℏ_{2})$	$\hat{ρ} (ℏ_{1}) \cap \hat{σ} (ℏ_{3})$	$\hat{ρ} (ℏ_{1}) \cap \hat{σ} (ℏ_{5})$	$\hat{ρ} (ℏ_{1}) \cap \hat{σ} (ℏ_{6})$
ℏ₄	$\hat{ρ} (ℏ_{4}) \cap \hat{σ} (ℏ_{2})$	$\hat{ρ} (ℏ_{4}) \cap \hat{σ} (ℏ_{3})$	$\hat{ρ} (ℏ_{4}) \cap \hat{σ} (ℏ_{5})$	$\hat{ρ} (ℏ_{4}) \cap \hat{σ} (ℏ_{6})$
ℏ₅	$\hat{ρ} (ℏ_{5}) \cap \hat{σ} (ℏ_{2})$	$\hat{ρ} (ℏ_{5}) \cap \hat{σ} (ℏ_{3})$	$\hat{ρ} (ℏ_{5}) \cap \hat{σ} (ℏ_{5})$	$\hat{ρ} (ℏ_{5}) \cap \hat{σ} (ℏ_{6})$

Table 6

$(\hat{F}, \hat{A}, \hat{ρ}) ▵_{g} (\hat{G}, \hat{B}, \hat{σ}) = (\hat{H}, \hat{C}, \hat{τ})$

$\hat{A} \times \hat{B}$	ℏ₂	ℏ₃	ℏ₅	ℏ₆
ℏ₁	$\hat{F} (ℏ_{1}) \cup \hat{G} (ℏ_{2})$	$\hat{F} (ℏ_{1}) \cup \hat{G} (ℏ_{3})$	$\hat{F} (ℏ_{1}) \cup \hat{G} (ℏ_{5})$	$\hat{F} (ℏ_{1}) \cup \hat{G} (ℏ_{6})$
ℏ₄	$\hat{F} (ℏ_{4}) \cup \hat{G} (ℏ_{2})$	$\hat{F} (ℏ_{4}) \cup \hat{G} (ℏ_{3})$	$\hat{F} (ℏ_{4}) \cup \hat{G} (ℏ_{5})$	$\hat{F} (ℏ_{4}) \cup \hat{G} (ℏ_{6})$
ℏ₅	$\hat{F} (ℏ_{5}) \cup \hat{G} (ℏ_{2})$	$\hat{F} (ℏ_{5}) \cup \hat{G} (ℏ_{3})$	$\hat{F} (ℏ_{5}) \cup \hat{G} (ℏ_{5})$	$\hat{F} (ℏ_{5}) \cup \hat{G} (ℏ_{6})$
$\hat{A} \times \hat{B}$	ℏ₂	ℏ₃	ℏ₅	ℏ₆
[.5ex] ℏ₁	$\hat{ρ} (ℏ_{1}) \cup \hat{σ} (ℏ_{2})$	$\hat{ρ} (ℏ_{1}) \cup \hat{σ} (ℏ_{3})$	$\hat{ρ} (ℏ_{1}) \cup \hat{σ} (ℏ_{5})$	$\hat{ρ} (ℏ_{1}) \cup \hat{σ} (ℏ_{6})$
ℏ₄	$\hat{ρ} (ℏ_{4}) \cup \hat{σ} (ℏ_{2})$	$\hat{ρ} (ℏ_{4}) \cup \hat{σ} (ℏ_{3})$	$\hat{ρ} (ℏ_{4}) \cup \hat{σ} (ℏ_{5})$	$\hat{ρ} (ℏ_{4}) \cup \hat{σ} (ℏ_{6})$
ℏ₅	$\hat{ρ} (ℏ_{5}) \cup \hat{σ} (ℏ_{2})$	$\hat{ρ} (ℏ_{5}) \cup \hat{σ} (ℏ_{3})$	$\hat{ρ} (ℏ_{5}) \cup \hat{σ} (ℏ_{5})$	$\hat{ρ} (ℏ_{5}) \cup \hat{σ} (ℏ_{6})$

$\begin{matrix} \hat{F} (ℏ_{1}) \cup \hat{G} (ℏ_{2}) = {([0.60, 0.80], [0.13, 0.18]) / ℓ_{1}, \\ ([0.70, 0.80], [0.10, 0.15]) / ℓ_{2}, ([0.73, 0.84], \\ [0.10, 0.14]) / ℓ_{3}, ([0.68, 0.78], [0.10, 0.20]) / ℓ_{4}, \\ ([0.58, 0.68], [0.23, 0.30]) / ℓ_{5}}, \\ \hat{ρ} (ℏ_{1}) \cap \hat{σ} (ℏ_{2}) = ([0.43, 0.73], [0.13, 0.23]) . \end{matrix}$ Similarly, we can make all the calculations for Table 6.

Theorem 3. Let $Γ_{1} = (\hat{F}, \hat{A}, \hat{ρ})$ and $Γ_{2} = (\hat{G}, \hat{B}, \hat{σ})$ be two GIVIFSSs over $\hat{Y}$ . Then we have

$(Γ_{1} \nabla_{g} Γ_{2})^{c} = Γ_{1}^{c} ▵_{g} Γ_{2}^{c}$ ;

$(Γ_{1} ▵_{g} Γ_{2})^{c} = Γ_{1}^{c} \nabla_{g} Γ_{2}^{c}$ .

Proof. The proof can be easily obtained from Definitions 26, 27 and 28. □ Definition 29. Let $(\hat{F}, \hat{A}, \hat{ρ})$ be a GIVIFSS over $\hat{Y}$ . Then the “g-necessity" operation of $(\hat{F}, \hat{A}, \hat{ρ})$ is denoted as $□_{g} (\hat{F}, \hat{A}, \hat{ρ})$ and defined as $□_{g} (\hat{F}, \hat{A}, \hat{ρ}) = (\hat{G}, \hat{A}, \hat{τ})$ where,

if $\hat{F} (ℏ) = {(ℓ, [u_{\hat{F} (ℏ)} (ℓ), {\bar{u}}_{\hat{F} (ℏ)} (ℓ)], [v_{\hat{F} (ℏ)} (ℓ), {\bar{v}}_{\hat{F} (ℏ)} (ℓ)]) | \forall ℓ \in \hat{Y} and ℏ \in \hat{A}}$ , then $\hat{G} (ℏ) = {(ℓ, [u_{\hat{F} (ℏ)} (ℓ), {\bar{u}}_{\hat{F} (ℏ)} (ℓ)], [1 - {\bar{u}}_{\hat{F} (ℏ)} (ℓ), 1 - u_{\hat{F} (ℏ)} (ℓ)]) | \forall ℓ \in \hat{Y} and ℏ \in \hat{A}}$ ,

if $\hat{ρ} = {(ℏ, [u_{\hat{ρ}} (ℏ), {\bar{u}}_{\hat{ρ}} (ℏ)], [v_{\hat{ρ}} (ℏ), {\bar{v}}_{\hat{ρ}} (ℏ)]) | \forall ℏ \in \hat{A}}$ , then $\hat{σ} = {(ℏ, [u_{\hat{ρ}} (ℏ), {\bar{u}}_{\hat{ρ}} (ℏ)], [1 - {\bar{u}}_{\hat{ρ}} (ℏ), 1 - u_{\hat{ρ}} (ℏ)]) | \forall ℏ \in \hat{A}}$ .

Definition 30. Let $(\hat{F}, \hat{A}, \hat{ρ})$ be a GIVIFSS over $\hat{Y}$ . Then the “g-possibility" operation of $(\hat{F}, \hat{A}, \hat{ρ})$ is denoted as $◊_{g} (\hat{F}, \hat{A}, \hat{ρ})$ and defined as $◊_{g} (\hat{F}, \hat{A}, \hat{ρ}) = (\hat{G}, \hat{A}, \hat{τ})$ where,

if $\hat{F} (ℏ) = {(ℓ, [u_{\hat{F} (ℏ)} (ℓ), {\bar{u}}_{\hat{F} (ℏ)} (ℓ)], [v_{\hat{F} (ℏ)} (ℓ), {\bar{v}}_{\hat{F} (ℏ)} (ℓ)]) | \forall ℓ \in \hat{Y} and ℏ \in \hat{A}}$ , then $\hat{G} (ℏ) = {(ℓ, [1 - {\bar{v}}_{\hat{F} (ℏ)} (ℓ), 1 - v_{\hat{F} (ℏ)} (ℓ)], [v_{\hat{F} (ℏ)} (ℓ), {\bar{v}}_{\hat{F} (ℏ)} (ℓ)]) | \forall ℓ \in \hat{Y} and ℏ \in \hat{A}}$ ,

if $\hat{ρ} = {(ℏ, [u_{\hat{ρ}} (ℏ), {\bar{u}}_{\hat{ρ}} (ℏ)], [v_{\hat{ρ}} (ℏ), {\bar{v}}_{\hat{ρ}} (ℏ)]) | \forall ℏ \in \hat{A}}$ , then $\hat{σ} = {(ℏ, [1 - {\bar{v}}_{\hat{ρ}} (ℏ), 1 - v_{\hat{ρ}} (ℏ)], [v_{\hat{ρ}} (ℏ), {\bar{v}}_{\hat{ρ}} (ℏ)]) | \forall ℏ \in \hat{A}}$ .

Theorem 4. Let $(\hat{F}, \hat{A}, \hat{ρ})$ and $(\hat{G}, \hat{B}, \hat{σ})$ be two GIVIFSSs over $\hat{Y}$ . Then we have

$□_{g} [(\hat{F}, \hat{A}, \hat{ρ}) ⊔_{g} (\hat{G}, \hat{B}, \hat{σ})] = [□_{g} (\hat{F}, \hat{A}, \hat{ρ})] ⊔_{g} [□_{g} (\hat{G}, \hat{B}, \hat{σ})]$ ;

$□_{g} [(\hat{F}, \hat{A}, \hat{ρ}) ⊓_{g} (\hat{G}, \hat{B}, \hat{σ})] = [□_{g} (\hat{F}, \hat{A}, \hat{ρ})] ⊓_{g} [□_{g} (\hat{G}, \hat{B}, \hat{σ})]$ ;

$□_{g} [(\hat{F}, \hat{A}, \hat{ρ}) ⊔_{g^{*}} (\hat{G}, \hat{B}, \hat{σ})] = [□_{g} (\hat{F}, \hat{A}, \hat{ρ})] ⊔_{g^{*}} [□_{g} (\hat{G}, \hat{B}, \hat{σ})]$ ;

$□_{g} [(\hat{F}, \hat{A}, \hat{ρ}) ⊓_{g^{*}} (\hat{G}, \hat{B}, \hat{σ})] = [□_{g} (\hat{F}, \hat{A}, \hat{ρ})] ⊓_{g^{*}} [□_{g} (\hat{G}, \hat{B}, \hat{σ})]$ ;

$□_{g} [□_{g} (\hat{F}, \hat{A}, \hat{ρ})] = □_{g} (\hat{F}, \hat{A}, \hat{ρ})$ .

Proof. The proof of these assertions can easily be obtained from definitions 29 and 30. □ Theorem 5. Let $(\hat{F}, \hat{A}, \hat{ρ})$ and $(\hat{G}, \hat{B}, \hat{σ})$ be two GIVIFSSs over $\hat{Y}$ . Then we have

$◊_{g} [(\hat{F}, \hat{A}, \hat{ρ}) ⊔_{g} (\hat{G}, \hat{B}, \hat{σ})] = [◊_{g} (\hat{F}, \hat{A}, \hat{ρ})] ⊔_{g} [◊_{g} (\hat{G}, \hat{B}, \hat{σ})]$ ;

$◊_{g} [(\hat{F}, \hat{A}, \hat{ρ}) ⊓_{g} (\hat{G}, \hat{B}, \hat{σ})] = [◊_{g} (\hat{F}, \hat{A}, \hat{ρ})] ⊓_{g} [◊_{g} (\hat{G}, \hat{B}, \hat{σ})]$ ;

$◊_{g} [(\hat{F}, \hat{A}, \hat{ρ}) ⊔_{g^{*}} (\hat{G}, \hat{B}, \hat{σ})] = [◊_{g} (\hat{F}, \hat{A}, \hat{ρ})] ⊔_{g^{*}} [◊_{g} (\hat{G}, \hat{B}, \hat{σ})]$ ;

$◊_{g} [(\hat{F}, \hat{A}, \hat{ρ}) ⊓_{g^{*}} (\hat{G}, \hat{B}, \hat{σ})] = [◊_{g} (\hat{F}, \hat{A}, \hat{ρ})] ⊓_{g^{*}} [◊_{g} (\hat{G}, \hat{B}, \hat{σ})]$ ;

$◊_{g} [◊_{g} (\hat{F}, \hat{A}, \hat{ρ})] = ◊_{g} (\hat{F}, \hat{A}, \hat{ρ})$ .

Proof. The proof of these assertions can easily be obtained from definitions 29 and 30. □ Theorem 6. Let $(\hat{F}, \hat{A}, \hat{ρ})$ be a GIVIFSS over $\hat{Y}$ . Then we have

$□_{g} (\hat{F}, \hat{A}, \hat{ρ}) \subseteq_{F} (\hat{F}, \hat{A}, \hat{ρ}) \subseteq_{F} ◊_{g} (\hat{F}, \hat{A}, \hat{ρ})$ ;

$◊_{g} □_{g} (\hat{F}, \hat{A}, \hat{ρ}) = □_{g} (\hat{F}, \hat{A}, \hat{ρ})$ ;

$□_{g} ◊_{g} (\hat{F}, \hat{A}, \hat{ρ}) = ◊_{g} (\hat{F}, \hat{A}, \hat{ρ})$ .

Proof. The proof of these assertions can easily be obtained from definitions 27, 28, 29 and 30. □ Theorem 7. Let $Γ_{1} = (\hat{F}, \hat{A}, \hat{ρ})$ and $Γ_{2} = (\hat{G}, \hat{B}, \hat{σ})$ be two GIVIFSSs over $\hat{Y}$ . Then we have

□_g (Γ₁ ∇ _gΓ₂) = (□ _gΓ₁) ∇ _g (□ _gΓ₂);

□_g (Γ₁ ▵ _gΓ₂) = (□ _gΓ₁) ▵ _g (□ _gΓ₂);

◊_g (Γ₁ ∇ _gΓ₂) = (◊ _gΓ₁) ∇ _g (□ _gΓ₂);

◊_g (Γ₁ ▵ _gΓ₂) = (◊ _gΓ₁) ▵ _g (◊ _gΓ₂).

5 A generalized interval-valued intuitionisticfuzzy soft set based MADM process

In this section, we have proposed an algorithm based on GIVIFSS and some related notions for solving MADM problems. Also, we have proposed a criterion for obtaining weight vector from PIVIFS using g-novel expectation score function and define representative score function for comparing IVIFVs.

Definition 31. Let $p = ([u, \bar{u}], [v, \bar{v}])$ be an IVIFV. The g-novel expectation score function φ of p is defined as $\begin{matrix} φ (p) = \frac{u_{p} + {\bar{u}}_{p} - {\bar{v}}_{p} (1 - {\bar{u}}_{p}) - v_{p} (1 - u_{p}) + 2}{4}, \\ φ (p) \in [0, 1] . \end{matrix}$

Definition 32. Let $(\hat{F}, \hat{A}, \hat{ρ})$ be a GIVIFSS over $\hat{Y}$ . Then an aggregated interval-valued intuitionistic fuzzy decision value (AIVIFDV) of the alternative $ℓ \in \hat{Y}$ is defined as $X_{Γ} (ℓ) = \oplus_{ℏ \in \hat{A}} \frac{φ_{\hat{ρ} (ℏ)}}{d} \hat{F} (ℏ) (ℓ),$ for all $ℓ \in \hat{Y}$ . Where the operator is defined in definition 7 and d is defined by using g-novel expectation score function as follows $\sum_{h \in \hat{A}} φ_{\hat{ρ} (ℏ)} = d < + \infty .$

Definition 33. Let $p = ([u, v]) = ([u, \bar{u}], [v, \bar{v}])$ be an IVIFV. Then score representative function λ of p is defined as $\begin{matrix} λ (p) = & (M_{u} + (1 / 2 - M_{u}) \times R_{u}) \\ - (M_{v} + (1 / 2 - M_{v}) \times R_{v}), \end{matrix}$ λ (p) ∈ [-1, 1],

where $M_{u} = (u + \bar{u}) / 2$ , $M_{v} = (v + \bar{v}) / 2$ , $R_{u} = \bar{u} - u$ and $R_{v} = \bar{v} - v$ are the mean and range of u and v, respectively.

The idea of the generalized interval-valued intuitionistic fuzzy soft set is very encouraging in decision-making since it considers how to capitalize an additional interval-valued intuitionistic fuzzy input from the director to minimize any possible perversion in the data provided by evaluating specialists.

5.1 Algorithm

Input: GIVIFSSs.Output: Optimal alternative.

Let $\hat{Y} = {ℓ_{1}, ℓ_{2}, . . ., ℓ_{n}}$ , and $\hat{P} = \hat{A} \cup \hat{B} = {ℏ_{1}, ℏ_{2}, . . ., ℏ_{m}}$ . Two expert groups input the two BIVIFSSs $(\hat{F}, \hat{A})$ and $(\hat{G}, \hat{B})$ over $\hat{Y}$ separately. The director examines the general quality of evaluation made by experts groups and input two PIVIFSs $\hat{ρ}$ and $\hat{σ}$ , which completes the formulation of two GIVIFSSs $Γ_{1} = (\hat{F}, \hat{A}, \hat{ρ})$ and $Γ_{2} = (\hat{G}, \hat{B}, \hat{σ})$ .

The g-union of Γ₁ and Γ₁ is calculated by using definition 18, $Γ = Γ_{1} ⊔_{g} Γ_{2} = (\hat{H}, \hat{C}, \hat{τ})$ .

By using IVIFS $\hat{τ}$ and definition 31, calculate the weight vector as follows ${\hat{ω}}_{i} = \frac{φ_{\hat{τ}} (ℏ_{i})}{d}, \forall i \in {1, 2, . . ., m},$ where $d = \sum_{h \in \hat{A}} φ_{\hat{τ} (ℏ)}$ such that each ${\hat{ω}}_{i} ⩾ 0$ and $\sum_{i = 1}^{m} {\hat{ω}}_{i} = 1$ .

By using definition 32, calculate the aggregated interval-valued intuitionistic fuzzy decision value (AIVIFDV) as follows, $X_{Γ} (ℓ) = \oplus_{ℏ \in \hat{A}} {\hat{ω}}_{i} \hat{F} (ℏ) (ℓ) .$

Calculate score representative value of each AIVIFV using definition 33.

On the basis of score representative value calculate in step 5, rank the alternatives and alternative with the highest value is an optimal choice.

Remark 6. In this remark, we try to understand the proposed algorithm that how we apply it to real life problems. The above-mentioned algorithm is designed to solve multi attribute decision making problems. Here we take two groups of experts but we can take any finite number of groups. In this algorithm, experts used their proficiencies and give BIVIFSSs $(\hat{F}, \hat{A})$ and $(\hat{G}, \hat{B})$ . The director reviews and scrutinizes the general quality of evaluation made by experts groups instead of evaluating all the alternatives with respect to every characteristic and gives two PIVIFSs $\hat{ρ}$ and $\hat{σ}$ .

After that, we use g-union to integrate the information from two GIVIFSSs. Other operations are also used to integrate information for some other models. In next step, we calculate AIVIFVs of all alternatives by aggregating the information extracted from $(\hat{H}, \hat{C}, \hat{τ})$ . After that, for comparing AIVIFVs we use score representative function. We also use the novel accuracy function 2 for comparing AIVIFVs.

In the sequence, a case study of construction of office for Honda company in Bangkok is discussed to show the supremacy of the newly proposed technique and its related notions.

6 Case study

The Honda company need to build his office in Bangkok. Since it is a very big project, it involves a very complicated evaluation and decision-making. Different parameters like “rich portfolios”, “strong risk management” used as a criterion for different construction companies. Specialists are consulted by the director for their professional opinions to select the felicitous alternative.

Suppose $\hat{Y} = {ℓ_{1}, ℓ_{2}, ℓ_{3}, ℓ_{4}, ℓ_{5}, ℓ_{6}}$ , be the construction companies under consideration. A committee which contains the experts from different departments like construction, architecture, planning departments, management, finance management and engineering is designed under the director. Let $\hat{P} = {ℏ_{1}, ℏ_{2}, ℏ_{3}, ℏ_{4}, ℏ_{5}, ℏ_{6}}$ is the criteria on the basis committee make evaluation, where each ℏ_j stands for “rich portfolios", “strong risk management”, “credentials”, “time to completion”, “modern equipment and technology” and “a skilled team”, respectively. For better evaluation the director divides the committee into two groups. The set of attributes $\hat{A} = {ℏ_{1}, ℏ_{4}, ℏ_{5}, ℏ_{6}}$ be the set of attributes which is assigned to the first group and $\hat{B} = {ℏ_{2}, ℏ_{3}, ℏ_{5}, ℏ_{6}}$ is assigned to the second group. The evaluation of alternatives is made by these two groups and results presented in the form of BIVIFSSs $(\hat{F}, \hat{A})$ and $(\hat{G}, \hat{B})$ accordingly. The two PIVIFSs $\hat{ρ}$ and $\hat{σ}$ is given by the director after scrutinizes the work done by two expert groups generally that complete the formulation of two GIVIFSSs $(\hat{F}, \hat{A}, \hat{ρ})$ and $(\hat{G}, \hat{B}, \hat{σ})$ , $\begin{matrix} \hat{F} (ℏ_{1}) = {([ & 0.50, 0.70], [0.20, 0.25]) / ℓ_{1}, ([0.70, 0.80], [0.10, 0.15]) / ℓ_{2}, ([0.73, 0.84], [0.10, 0.14]) / ℓ_{3}, \\ ([0.68, 0.78], [0.10, 0.20]) / ℓ_{4}, ([0.48, 0.58], [0.30, 0.40]) / ℓ_{5}, ([0.42, 0.52], [0.33, 0.43]) / ℓ_{6}}, \\ \hat{F} (ℏ_{4}) = {([ & 0.60, 0.70], [0.16, 0.21]) / ℓ_{1}, ([0.60, 0.70], [0.20, 0.30]) / ℓ_{2}, ([0.55, 0.60], [0.20, 0.40]) / ℓ_{3}, \\ ([0.63, 0.73], [0.20, 0.26]) / ℓ_{4}, ([0.40, 0.60], [0.20, 0.30]) / ℓ_{5}, ([0.48, 0.60], [0.30, 0.38]) / ℓ_{6}}, \\ \hat{F} (ℏ_{5}) = {( & [0.73, 0.79], [0.10, 0.16]) / ℓ_{1}, ([0.40, 0.60], [0.20, 0.30]) / ℓ_{2}, ([0.60, 0.90], [0.04, 0.10]) / ℓ_{3}, \\ ([0.56, 0.70], [0.18, 0.28]) / ℓ_{4}, ([0.45, 0.65], [0.25, 0.35]) / ℓ_{5}, ([0.60, 0.70], [0.15, 0.25]) / ℓ_{6}}, \\ \hat{F} (ℏ_{6}) = {([ & 0.60, 0.79], [0.11, 0.21]) / ℓ_{1}, ([0.50, 0.60], [0.22, 0.32]) / ℓ_{2}, ([0.63, 0.89], [0.04, 0.10]) / ℓ_{3}, \\ ([0.59, 0.69], [0.16, 0.28]) / ℓ_{4}, ([0.54, 0.65], [0.20, 0.35]) / ℓ_{5}, ([0.41, 0.51], [0.32, 0.42]) / ℓ_{6}} \\ \hat{ρ} = {([ & 0.40, 0.70], [0.20, 0.30]) / ℏ_{1}, ([0.58, 0.65], [0.30, 0.35]) / ℏ_{4}, ([0.30, 0.40], [0.40, 0.50]) / ℏ_{5}, \\ ([0.47, 0.57], [0.31, 0.41]) / ℏ_{6}}, \end{matrix}$ $\begin{matrix} \hat{G} (ℏ_{2}) = {([0.60, 0.80], [0.13, 0.18]) / ℓ_{1}, ([0.50, 0.60], [0.21, 0.31]) / ℓ_{2}, ([0.73, 0.80], [0.10, 0.18]) / ℓ_{3}, \\ ([0.61, 0.71], [0.15, 0.25]) / ℓ_{4}, ([0.58, 0.68], [0.23, 0.30]) / ℓ_{5}, ([0.53, 0.63], [0.27, 0.37]) / ℓ_{6}}, \\ \hat{G} (ℏ_{3}) = {([0.61, 0.72], [0.13, 0.21]) / ℓ_{1}, ([0.62, 0.74], [0.12, 0.23]) / ℓ_{2}, ([0.45, 0.60], [0.22, 0.34]) / ℓ_{3}, \\ ([0.36, 0.53], [0.22, 0.32]) / ℓ_{4}, ([0.47, 0.67], [0.21, 0.30]) / ℓ_{5}, ([0.58, 0.68], [0.20, 0.28]) / ℓ_{6}}, \\ \hat{G} (ℏ_{5}) = {([0.70, 0.79], [0.15, 0.21]) / ℓ_{1}, ([0.43, 0.63], [0.23, 0.33]) / ℓ_{2}, ([0.64, 0.89], [0.04, 0.11]) / ℓ_{3}, \\ ([0.54, 0.70], [0.19, 0.28]) / ℓ_{4}, ([0.45, 0.65], [0.22, 0.35]) / ℓ_{5}, ([0.78, 0.88], [0.02, 0.12]) / ℓ_{6}}, \\ \hat{G} (ℏ_{6}) = {([0.76, 0.86], [0.10, 0.14]) / ℓ_{1}, ([0.34, 0.56], [0.22, 0.34]) / ℓ_{2}, ([0.61, 0.91], [0.04, 0.09]) / ℓ_{3}, \\ ([0.66, 0.77], [0.13, 0.23]) / ℓ_{4}, ([0.35, 0.50], [0.27, 0.37]) / ℓ_{5}, ([0.65, 0.75], [0.13, 0.23]) / ℓ_{6}} \\ \hat{σ} = {([0.43, 0.73], [0.13, 0.23]) / ℏ_{2}, ([0.51, 0.65], [0.31, 0.35]) / ℏ_{3}, ([0.33, 0.43], [0.41, 0.51]) / ℏ_{5}, \\ ([0.43, 0.53], [0.23, 0.43]) / ℏ_{6}} . \end{matrix}$

In second step, we calculate the g-union of $(\hat{F}, \hat{A}, \hat{ρ})$ and $(\hat{G}, \hat{B}, \hat{σ})$ , i.e., $(\hat{F}, \hat{A}, \hat{ρ}) ⊔_{g} (\hat{G}, \hat{B}, \hat{σ}) = (\hat{H}, \hat{C}, \hat{τ})$ and the result is presented in Table 7. After that by using the PIVIFS $\hat{τ}$ , we get the weight vector by using definition 31 as follows ${\hat{ω}}_{j} = \frac{φ (ℏ_{j})}{d}$ for all j ∈ {1, 2, . . . , 6} and the following weight vector is obtained $\begin{matrix} \hat{ω} = (0.172747, 0.180744, 0.172484, 0.178216, \\ 0.131921, 0.163888)^{T}, \end{matrix}$ to be used to calculate the AIVIFDVs $X_{Γ} (ℓ_{i})$ (for all i ∈ {1, 2, . . . , 6}). Table 8 has the details regarding the calculation of weight vector. Next using weight vector, AIVIFDVs $X_{Γ} (ℓ_{i})$ (for all i ∈ {1, 2, . . . , 6}) is calculated as $\begin{matrix} X_{Γ} (ℓ_{i}) & = {IVIFWA}_{\hat{ω}} (\hat{H} (ℏ_{1}) (ℓ_{i}), \hat{H} (ℏ_{2}) (ℓ_{i}), \\ \hat{H} (ℏ_{3} (ℓ_{i}), \hat{H} (ℏ_{4}) (ℓ_{i}), \hat{H} (ℏ_{5}) (ℓ_{i}), \hat{H} (ℏ_{6}) (ℓ_{i})) . \end{matrix}$

Table 7
The g-union $Γ_{1} ⊔_{g} Γ_{2} = (\hat{H}, \hat{C}, \hat{τ})$

ℏ_i ℏ₁ ℏ₂ ℏ₃

ℓ₁ ([0.50, 0.70] , [0.20, 0.25]) ([0.60, 0.80] , [0.13, 0.18]) ([0.61, 0.72] , [0.13, 0.21])

ℓ₂ ([0.70, 0.80] , [0.10, 0.15]) ([0.50, 0.60] , [0.21, 0.31]) ([0.62, 0.74] , [0.12, 0.23])

ℓ₃ ([0.73, 0.84] , [0.10, 0.14]) ([0.73, 0.80] , [0.10, 0.18]) ([0.45, 0.60] , [0.22, 0.34])

ℓ₄ ([0.68, 0.78] , [0.10, 0.20]) ([0.61, 0.71] , [0.15, 0.25]) ([0.36, 0.53] , [0.22, 0.32])

ℓ₅ ([0.48, 0.58] , [0.30, 0.40]) ([0.58, 0.68] , [0.23, 0.30]) ([0.47, 0.67] , [0.21, 0.30])

ℓ₆ ([0.42, 0.52] , [0.33, 0.43]) ([0.53, 0.63] , [0.27, 0.37]) ([0.58, 0.68] , [0.20, 0.28])

τ ([0.40, 0.70] , [0.20, 0.30]) ([0.43, 0.73] , [0.13, 0.23]) ([0.51, 0.65] , [0.31, 0.35])

ℏ_i ℏ₄ ℏ₅ ℏ₆

[.5ex] ℓ₁ ([0.60, 0.70] , [0.16, 0.21]) ([0.73, 0.79] , [0.10, 0.16]) ([0.76, 0.86] , [0.10, 0.14])

ℓ₂ ([0.60, 0.70] , [0.20, 0.30]) ([0.43, 0.63] , [0.20, 0.30]) ([0.50, 0.60] , [0.22, 0.32])

ℓ₃ ([0.55, 0.60] , [0.20, 0.40]) ([0.64, 0.90] , [0.04, 0.10]) ([0.63, 0.91] , [0.04, 0.09])

ℓ₄ ([0.63, 0.73] , [0.20, 0.26]) ([0.56, 0.70] , [0.18, 0.28]) ([0.66, 0.77] , [0.13, 0.23])

ℓ₅ ([0.40, 0.60] , [0.20, 0.30]) ([0.45, 0.65] , [0.22, 0.35]) ([0.54, 0.65] , [0.20, 0.35])

ℓ₆ ([0.48, 0.60] , [0.30, 0.38]) ([0.78, 0.88] , [0.02, 0.12]) ([0.65, 0.75] , [0.13, 0.23])

τ ([0.58, 0.65] , [0.30, 0.35]) ([0.33, 0.43] , [0.40, 0.50]) ([0.47, 0.57] , [0.23, 0.41])

ℏ_i	ℏ₁	ℏ₂	ℏ₃
ℓ₁	([0.50, 0.70] , [0.20, 0.25])	([0.60, 0.80] , [0.13, 0.18])	([0.61, 0.72] , [0.13, 0.21])
ℓ₂	([0.70, 0.80] , [0.10, 0.15])	([0.50, 0.60] , [0.21, 0.31])	([0.62, 0.74] , [0.12, 0.23])
ℓ₃	([0.73, 0.84] , [0.10, 0.14])	([0.73, 0.80] , [0.10, 0.18])	([0.45, 0.60] , [0.22, 0.34])
ℓ₄	([0.68, 0.78] , [0.10, 0.20])	([0.61, 0.71] , [0.15, 0.25])	([0.36, 0.53] , [0.22, 0.32])
ℓ₅	([0.48, 0.58] , [0.30, 0.40])	([0.58, 0.68] , [0.23, 0.30])	([0.47, 0.67] , [0.21, 0.30])
ℓ₆	([0.42, 0.52] , [0.33, 0.43])	([0.53, 0.63] , [0.27, 0.37])	([0.58, 0.68] , [0.20, 0.28])
τ	([0.40, 0.70] , [0.20, 0.30])	([0.43, 0.73] , [0.13, 0.23])	([0.51, 0.65] , [0.31, 0.35])
ℏ_i	ℏ₄	ℏ₅	ℏ₆
[.5ex] ℓ₁	([0.60, 0.70] , [0.16, 0.21])	([0.73, 0.79] , [0.10, 0.16])	([0.76, 0.86] , [0.10, 0.14])
ℓ₂	([0.60, 0.70] , [0.20, 0.30])	([0.43, 0.63] , [0.20, 0.30])	([0.50, 0.60] , [0.22, 0.32])
ℓ₃	([0.55, 0.60] , [0.20, 0.40])	([0.64, 0.90] , [0.04, 0.10])	([0.63, 0.91] , [0.04, 0.09])
ℓ₄	([0.63, 0.73] , [0.20, 0.26])	([0.56, 0.70] , [0.18, 0.28])	([0.66, 0.77] , [0.13, 0.23])
ℓ₅	([0.40, 0.60] , [0.20, 0.30])	([0.45, 0.65] , [0.22, 0.35])	([0.54, 0.65] , [0.20, 0.35])
ℓ₆	([0.48, 0.60] , [0.30, 0.38])	([0.78, 0.88] , [0.02, 0.12])	([0.65, 0.75] , [0.13, 0.23])
τ	([0.58, 0.65] , [0.30, 0.35])	([0.33, 0.43] , [0.40, 0.50])	([0.47, 0.57] , [0.23, 0.41])

Table 8

Weights calculated from the IVIFS $\hat{τ}$

$\hat{X}$	$\hat{τ}$	$φ_{\hat{τ}} (ℏ_{j})$	${\hat{ω}}_{j}$
ℏ₁	([0.40, 0.70] , [0.20, 0.30])	0.7225	0.172747
[.5ex] ℏ₂	([0.43, 0.73] , [0.13, 0.23])	0.7559	0.180744
[.5ex] ℏ₃	([0.51, 0.65] , [0.31, 0.35])	0.7214	0.172484
[.5ex] ℏ₄	([0.58, 0.65] , [0.30, 0.35])	0.7454	0.178216
[.5ex] ℏ₅	([0.33, 0.43] , [0.40, 0.50])	0.5518	0.131921
[.5ex] ℏ₆	([0.47, 0.57] , [0.23, 0.41])	0.6855	0.163888
[.5ex]

For instance, the AIVIFDVs $X_{Γ} (ℓ_{1})$ can be obtained as follows: $\begin{matrix} X_{Γ} (ℓ_{1}) & = \oplus_{j = 1}^{6} \hat{ω_{j}} (\hat{H} (ℏ_{j}) (ℓ_{i}) \\ = ([0.63857, 0.76802], [0.13447, 0.19000]) . \end{matrix}$ The remaining AIVIFDVs are given as

$\begin{matrix} X_{Γ} (ℓ_{2}) = & ([0.573100, 0.685340], [0.166486, 0.258455]), \\ X_{Γ} (ℓ_{3}) = & ([0.634315, 0.803521], [0.098855, 0.183169]), \\ X_{Γ} (ℓ_{4}) = & ([0.595983, 0.713083], [0.157359, 0.253097]), \\ X_{Γ} (ℓ_{5}) = & ([0.491572, 0.639719], [0.224666, 0.329996]), \\ X_{Γ} (ℓ_{6}) = & ([0.580433, 0.690634], [0.170299, 0.289943]) . \end{matrix}$

Now, we calculate the score representative value of each AIVIFDV by using definition 33 and the values are $\begin{matrix} λ (X_{Γ} (ℓ_{1})) = & 0.586131, λ (X_{Γ} (ℓ_{2})) = 0.457697, \\ λ (X_{Γ} (ℓ_{3})) = & 0.645216, λ (X_{Γ} (ℓ_{4})) = 0.495622, \\ λ (X_{Γ} (ℓ_{5})) = & 0.321494, λ (X_{Γ} (ℓ_{6})) = 0.452720 . \end{matrix}$ Hence from score expectation values, it is easy to see that the alternative ℓ₃ is optimal and an order relation of all alternatives is $ℓ_{3} ≻ ℓ_{1} ≻ ℓ_{4} ≻ ℓ_{2} ≻ ℓ_{6} ≻ ℓ_{5} .$

7 Comparative analysis

In this section, our proposed method is compared with other methods to demonstrate its preferences.

First, we compare our idea with the idea proposed in [52], we have seen that the definition of GIVIFSS is not clear and have some deficiencies and problems pointed out in Sections 3 and 4. Many related notions like containment, union, intersection, null GIVIFSS and absolute GIVIFSS have some difficulties which are removed by redefining the GIVIFSS by combining IVIFSS and IVIFS. If we compare our method with the method proposed by Feng [23] and Khan [26], then they are working in intuitionistic and picture fuzzy environment, respectively, while we are working in an interval-valued intuitionistic fuzzy environment. This environment is better than the intuitionistic and picture fuzzy environment because instead of giving a crisp value, the decision-maker has an interval (range) to put his evaluation. When we compare our method with the method proposed in [46], a novel approach for IVIFSS is used to solve the decision making problem. But in many cases, it fails to generate an order relation among alternatives and give the same choice values. In weighted technique, no proper criteria to obtain weighs of parameters are mentioned in [46].

But in our method, we are working in a more general environment because in the sense of decision making, an extra IVIFS reduce the possible distortion in previous evaluation. We have proposed a proper method to obtain a weight vector by using IVIFS. By scanning the proposed algorithm we not only get optimal alternative but an order relation among all alternatives.

8 Conclusion

The idea of generalized interval-valued intuitionistic fuzzy soft set defined by Wu [52] is very encouraging in decision-making since it considers how to capitalize an additional interval-valued intuitionistic fuzzy input from the director to minimize any possible perversion in the data provided by evaluating specialists. Since the director is responsible for the department, so he reviews and scrutinizes the general quality of evaluation made by experts groups instead of evaluating all the alternatives with respect to every characteristic. But the Wu idea was not presented clearly and has some deficiencies and problems. We clarified and redefine the GIVIFSS and its related notions. We have introduced the two types of containment for GIVIFSS. We clarify and redefine the null GIVIFSS and absolute GIVIFSS. We define the g-union, g-intersection, g^∗-union, g^∗-intersection, OR operation, AND operation, g-necessity operations and g-possibility operations for GIVIFSS. The properties of these operations are investigated like De Morgans laws and many more presented in the text. An algorithm is proposed for solving MADM problems using GIVIFSS. We have given a criterion to find a weight vector by using g-novel expectation score function. A descriptive example is presented to describe the applicability of the proposed method. Results indicate that the proposed technique is more effective and generalize then previous models of interval-valued fuzzy sets because it includes the IVIFS and IVIFSS. Also, all the problems related to the GIVIFSS are fixed that’s why the model become more effective and reliable for decision-making problems.

As future directions, it would be interesting to consider how to apply GIVIFSSs and related notions to other applications such as forecasting, multi attribute classifications, sorting problems and data analysis. We consider it to find the best design concept for new product development and by using soft likelihood functions more reliable methods for forensic crime investigation. Also, we consider it for pattern recognition and medical diagnosis by defining similarity measures and entropy on GIVIFSS. We will define some preference relation for GIVIFSSs. We will consider new styles of decision making technology in interval valued intuitionistic fuzzy environment for filling up the blanks, such as interactive or dynamic multiple attribute (group) decision making.

Footnotes

Acknowledgments

The authors acknowledge the financial support provided by the Center of Excellence in Theoretical and Computational Science (TaCS-CoE), KMUTT. Moreover, Wiyada Kumam was financially supported by the Rajamangala University of Technology Thanyaburi (RMUTTT) (Grant No.NSF62D0604).

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