Abstract

We introduce the concept of generalised interval-valued fuzzy soft set and its operations and study some of their properties. We give applications of this theory in solving a decision making problem. We also introduce a similarity measure of two generalised interval-valued fuzzy soft sets and discuss its application in a medical diagnosis problem: fuzzy set; soft set; fuzzy soft set; generalised fuzzy soft set; generalised interval-valued fuzzy soft set; interval-valued fuzzy set; interval-valued fuzzy soft set.

1. Introduction

Molodtsov [1] initiated the theory of soft set as a new mathematical tool for dealing with uncertainties which traditional mathematical tools cannot handle. He has shown several applications of this theory in solving many practical problems in economics, engineering, social science, medical science, and so forth. Presently, work on the soft set theory is progressing rapidly. Maji et al. [2, 3], Roy and Maji [4] have further studied the theory of soft set and used this theory to solve some decision making problems. Maji et al. [5] have also introduced the concept of fuzzy soft set, a more general concept, which is a combination of fuzzy set and soft set and studied its properties. Zou and Xiao [6] introduced soft set and fuzzy soft set into the incomplete environment, respectively. Alkhazaleh et al. [7] introduced the concept of soft multiset as a generalisation of soft set. They also defined the concepts of fuzzy parameterized interval-valued fuzzy soft set [8] and possibility fuzzy soft set [9] and gave their applications in decision making and medical diagnosis. Alkhazaleh and Salleh [10] introduced the concept of a soft expert set, where the user can know the opinion of all experts in one model without any operations. Even after any operation, the user can know the opinion of all experts. In 2011, Salleh [11] gave a brief survey from soft set to intuitionistic fuzzy soft set. Majumdar and Samanta [12] introduced and studied generalised fuzzy soft set where the degree is attached with the parameterization of fuzzy sets while defining a fuzzy soft set. Yang et al. [13] presented the concept of interval-valued fuzzy soft set by combining the interval-valued fuzzy set [14, 15] and soft set models. In this paper, we generalise the concept of fuzzy soft set as introduced by Maji et al. [5] to generalised interval-valued fuzzy soft set. In our generalisation of fuzzy soft set, a degree is attached with the parameterization of fuzzy sets while defining an interval-valued fuzzy soft set. Also, we give some applications of generalised interval-valued fuzzy soft set in decision making problem and medical diagnosis.

2. Preliminary

In this section, we recall some definitions and properties regarding fuzzy soft set and generalised fuzzy soft set required in this paper.

Definition 2.1 (see [15]). An interval-valued fuzzy set 𝑋 on a universe π‘ˆ is a mapping such that ([]π‘‹βˆΆπ‘ˆβŸΆInt0,1),(2.1) where Int([0,1]) stands for the set of all closed subintervals of [0,1], the set of all interval-valued fuzzy sets on π‘ˆ is denoted by 𝑃(π‘ˆ).

Suppose that ξ‚ξ‚π‘‹βˆˆπ‘ƒ(π‘ˆ), for all π‘₯βˆˆπ‘ˆ, πœ‡π‘₯(π‘₯)=[πœ‡βˆ’π‘₯(π‘₯),πœ‡+π‘₯(π‘₯)] is called the degree of membership of an element π‘₯ to 𝑋. πœ‡βˆ’π‘₯(π‘₯) and πœ‡+π‘₯(π‘₯) are referred to as the lower and upper degrees of membership of π‘₯ to 𝑋 where 0β©½πœ‡βˆ’π‘₯(π‘₯)β©½πœ‡+π‘₯(π‘₯)β©½1.

Definition 2.2 (see [14]). The subset, complement, intersection, and union of the interval-valued fuzzy sets are defined as follows. Let 𝑋,π‘Œβˆˆπ‘ƒ(π‘ˆ), then(a)the complement of 𝑋 is denoted by 𝑋𝑐 where πœ‡ξ‚π‘‹π‘(π‘₯)=1βˆ’πœ‡ξ‚π‘‹ξ‚ƒ(π‘₯)=1βˆ’πœ‡+𝑋(π‘₯),1βˆ’πœ‡βˆ’ξ‚π‘‹ξ‚„(π‘₯),(2.2)(b)the intersection of 𝑋 and ξ‚π‘Œ is denoted by ξ‚ξ‚π‘Œπ‘‹βˆ© where πœ‡ξ‚π‘‹βˆ©ξ‚π‘Œξ€Ίπœ‡(π‘₯)=inf𝑋(π‘₯),πœ‡ξ‚π‘Œξ€»=ξ‚ƒξ‚€πœ‡(π‘₯)infβˆ’ξ‚π‘‹(π‘₯),πœ‡βˆ’ξ‚π‘Œξ‚ξ‚€πœ‡(π‘₯),inf+𝑋(π‘₯),πœ‡+ξ‚π‘Œ,(π‘₯)(2.3)(c)the union of 𝑋 and ξ‚π‘Œ is denoted by ξ‚ξ‚π‘Œπ‘‹βˆͺ where πœ‡ξ‚π‘‹βˆͺξ‚π‘Œξ€Ίπœ‡(π‘₯)=sup𝑋(π‘₯),πœ‡ξ‚π‘Œξ€»=ξ‚ƒξ‚€πœ‡(π‘₯)supβˆ’ξ‚π‘‹(π‘₯),πœ‡βˆ’ξ‚π‘Œξ‚ξ‚€πœ‡(π‘₯),sup+𝑋(π‘₯),πœ‡+ξ‚π‘Œ;(π‘₯)(2.4)(d)𝑋 is a subset of π‘Œdenoted by π‘‹βŠ†π‘Œ if πœ‡βˆ’π‘‹(π‘₯)β‰€πœ‡βˆ’π‘Œ(π‘₯) and πœ‡+𝑋(π‘₯)β‰€πœ‡+π‘Œ(π‘₯).

Definition 2.3 (see [14]). The compatibility measure πœ‘(𝐴,𝐡) of an interval-valued fuzzy set 𝐴 with an interval-valued fuzzy set 𝐡 (𝐴 is a reference set) is given by πœ‘ξ€Ίπœ‘(𝐴,𝐡)=βˆ’(𝐴,𝐡),πœ‘+ξ€»(𝐴,𝐡),(2.5) such that πœ‘βˆ’ξ€Ίπœ‘(𝐴,𝐡)=min1(𝐴,𝐡),πœ‘2ξ€»,πœ‘(𝐴,𝐡)+(ξ€Ίπœ‘π΄,𝐡)=max1(𝐴,𝐡),πœ‘2(ξ€»,𝐴,𝐡)(2.6) where πœ‘1(𝐴,𝐡)=maxπ‘₯βˆˆπ‘‹ξ€½ξ€Ίπœ‡minβˆ’π΄(π‘₯),πœ‡βˆ’π΅(π‘₯)ξ€»ξ€Ύmaxπ‘₯βˆˆπ‘‹ξ€Ίπœ‡βˆ’π΄(ξ€»,πœ‘π‘₯)2(𝐴,𝐡)=maxπ‘₯βˆˆπ‘‹ξ€½ξ€Ίπœ‡min+𝐴(π‘₯),πœ‡+𝐡(π‘₯)ξ€»ξ€Ύmaxπ‘₯βˆˆπ‘‹ξ€Ίπœ‡+𝐴.(π‘₯)(2.7)

Theorem 2.4 (see [14]). Consider arbitrary, nonempty interval-valued fuzzy sets 𝐴, 𝐡, and 𝐢 from the family of 𝑖𝑣𝑓(𝑋) and a compatibility measure in the sense of Definition 2.3. Then,(a)π΄πœ‘(𝐴,𝐴)=[1,1]={1}, (b)πœ‘(𝐴,𝐡)=[0,0]={0}β‡”π΄βˆ©π΅=βˆ…, (c)in general πœ‘(𝐴,𝐡)β‰ πœ‘(𝐡,𝐴).

Let π‘ˆ be a universal set and 𝐸 a set of parameters. Let 𝑃(π‘ˆ) denote the power set of π‘ˆ and π΄βŠ†πΈ. Molodtsov [1] defined soft set as follows.

Definition 2.5. A pair (𝐹,𝐸) is called a soft set over π‘ˆ, where 𝐹 is a mapping given by πΉβˆΆπΈβ†’π‘ƒ(π‘ˆ). In other words, a soft set over π‘ˆ is a parameterized family of subsets of the universe π‘ˆ.

Definition 2.6 (see [5]). Let π‘ˆ be a universal set, and let 𝐸 be a set of parameters. Let πΌπ‘ˆ denote the power set of all fuzzy subsets of π‘ˆ. Let π΄βŠ†πΈ. A pair (𝐹,𝐸) is called a fuzzy soft set over π‘ˆ where 𝐹 is a mapping given by πΉβˆΆπ΄βŸΆπΌπ‘ˆ.(2.8)

Definition 2.7 (see [12]). Let π‘ˆ={π‘₯1,π‘₯2,…,π‘₯𝑛} be the universal set of elements and 𝐸={𝑒1,𝑒2,…,π‘’π‘š} be the universal set of parameters. The pair (π‘ˆ,𝐸) will be called a soft universe. Let πΉβˆΆπΈβ†’πΌπ‘ˆ, where πΌπ‘ˆ is the collection of all fuzzy subsets of π‘ˆ, and let πœ‡ be a fuzzy subset of 𝐸. Let πΉπœ‡βˆΆπΈβ†’πΌπ‘ˆΓ—πΌ be a function defined as follows: πΉπœ‡(𝑒)=(𝐹(𝑒),πœ‡(𝑒)).(2.9) Then, πΉπœ‡ is called a generalised fuzzy soft set (GFSS in short) over the soft set (π‘ˆ,𝐸). Here, for each parameter 𝑒𝑖, πΉπœ‡(𝑒𝑖)=(𝐹(𝑒𝑖),πœ‡(𝑒𝑖)) indicates not only the degree of belongingness of the elements of π‘ˆ in 𝐹(𝑒𝑖) but also the degree of possibility of such belongingness which is represented by πœ‡(𝑒𝑖). So we can write πΉπœ‡(𝑒𝑖) as follows: πΉπœ‡ξ€·π‘’π‘–ξ€Έ=π‘₯1𝐹𝑒𝑖π‘₯ξ€Έξ€·1ξ€Έ,π‘₯2𝐹𝑒𝑖π‘₯ξ€Έξ€·2ξ€Έπ‘₯,…,𝑛𝐹𝑒𝑖π‘₯𝑛𝑒,πœ‡π‘–ξ€Έξƒͺ,(2.10) where 𝐹(𝑒𝑖)(π‘₯1),𝐹(𝑒𝑖)(π‘₯2),…,𝐹(𝑒𝑖)(π‘₯𝑛) are the degree of belongingness and πœ‡(𝑒𝑖) is the degree of possibility of such belongingness.

Definition 2.8 (see [13]). Let π‘ˆ be an initial universe and 𝐸 a set of parameters. 𝑃(π‘ˆ) denotes the set of all interval-valued fuzzy sets of π‘ˆ. Let π΄βŠ†πΈ. A pair (𝐹,𝐴) is an interval-valued fuzzy soft set over π‘ˆ, where 𝐹 is a mapping given by ξ‚ξ‚πΉβˆΆπ΄β†’π‘ƒ(π‘ˆ).

3. Generalised Interval-Valued Fuzzy Soft Set

In this section, we generalise the concept of interval-valued fuzzy soft sets as introduced in [13]. In our generalisation of interval-valued fuzzy soft set, a degree is attached with the parameterization of fuzzy sets while defining an interval-valued fuzzy soft set.

Definition 3.1. Let π‘ˆ={π‘₯1,π‘₯2,…,π‘₯𝑛} be the universal set of elements and 𝐸={𝑒1,𝑒2,…,π‘’π‘š} the universal set of parameters. The pair (π‘ˆ,𝐸) will be called a soft universe. Let ξ‚ξ‚πΉβˆΆπΈβ†’π‘ƒ(π‘ˆ) and πœ‡ be a fuzzy set of 𝐸, that is,πœ‡βˆΆπΈβ†’πΌ=[0,1], where 𝑃(π‘ˆ) is the set of all interval-valued fuzzy subsets on π‘ˆ. Let ξ‚πΉπœ‡ξ‚βˆΆπΈβ†’π‘ƒ(π‘ˆ)×𝐼 be a function defined as follows: ξ‚πΉπœ‡ξ‚€ξ‚ξ‚(𝑒)=𝐹(𝑒),πœ‡(𝑒).(3.1) Then, ξ‚πΉπœ‡ is called a generalised interval-valued fuzzy soft set (GIVFSS in short) over the soft universe (π‘ˆ,𝐸). For each parameter 𝑒𝑖, ξ‚πΉπœ‡(𝑒𝑖)=(𝐹(𝑒𝑖)(π‘₯),πœ‡(𝑒𝑖)) indicates not only the degree of belongingness of the elements of π‘ˆ in 𝐹(𝑒𝑖) but also the degree of possibility of such belongingness which is represented by πœ‡(𝑒𝑖). So we can write ξ‚πΉπœ‡(𝑒𝑖) as follows: ξ‚πΉπœ‡ξ€·π‘’π‘–ξ€Έ=π‘₯1𝐹𝑒𝑖π‘₯ξ€Έξ€·1ξ€Έ,π‘₯2𝐹𝑒𝑖π‘₯ξ€Έξ€·2ξ€Έπ‘₯,…,𝑛𝐹𝑒𝑖π‘₯𝑛𝑒,πœ‡π‘–ξ€Έξƒͺ.(3.2)

Example 3.2. Let π‘ˆ={π‘₯1,π‘₯2,π‘₯3} be a set of universe, 𝐸={𝑒1,𝑒2,𝑒3} a set of parameters, and let πœ‡βˆΆπΈβ†’πΌ. Define a function ξ‚πΉπœ‡ξ‚βˆΆπΈβ†’π‘ƒ(π‘ˆ)×𝐼 as follows: ξ‚πΉπœ‡ξ€·π‘’1ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0.3,0.62[],π‘₯0.7,0.83[]ξ‚Όξ‚Ά,𝐹0.5,0.8,0.6πœ‡ξ€·π‘’2ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0.1,0.42[],π‘₯0,0.33[]ξ‚Όξ‚Ά,𝐹0.1,0.5,0.5πœ‡ξ€·π‘’3ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0.7,0.82[],π‘₯0.1,0.23[]ξ‚Όξ‚Ά.0,0.4,0.3(3.3) Then, ξ‚πΉπœ‡ is a GIVFSS over (π‘ˆ,𝐸).
In matrix notation, we writeξ‚πΉπœ‡=βŽ›βŽœβŽœβŽœβŽœβŽ[][][]⎞⎟⎟⎟⎟⎠0.3,0.6][0.7,0.8][0.5,0.80.60.1,0.4][0,0.3][0.1,0.50.50.7,0.8][0.1,0.2][0,0.40.3.(3.4)

Definition 3.3. Let ξ‚πΉπœ‡ and 𝐺𝛿 be two GIVFSSs over (π‘ˆ,𝐸). ξ‚πΉπœ‡ is called a generalised interval-valued fuzzy soft subset of 𝐺𝛿, and we write ξ‚πΉπœ‡βŠ†ξ‚πΊπ›Ώ if(a)πœ‡(𝑒) is a fuzzy subset of 𝛿(𝑒) for all π‘’βˆˆπΈ,(b)𝐹(𝑒) is an interval-valued fuzzy subset of 𝐺(𝑒) for all π‘’βˆˆπΈ.

Example 3.4. Let π‘ˆ={π‘₯1,π‘₯2,π‘₯3} be a set of three cars, and let 𝐸={𝑒1,𝑒2,𝑒3} be a set of parameters where 𝑒1 = cheap, 𝑒2 = expensive, 𝑒3 = red. Let ξ‚πΉπœ‡ be a GIVFSS over (π‘ˆ,𝐸) defined as follows: ξ‚πΉπœ‡ξ€·π‘’1ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0.1,0.32[],π‘₯0.5,0.73[]ξ‚Όξ‚Ά,𝐹0.3,0.5,0.4πœ‡ξ€·π‘’2ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0,0.32[],π‘₯0,0.23[]ξ‚Όξ‚Ά,𝐹0.1,0.3,0.4πœ‡ξ€·π‘’3ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0.5,0.62[],π‘₯0.1,0.13[]ξ‚Όξ‚Ά.0.1,0.3,0.1(3.5) Let 𝐺𝛿 be another GIVFSS over (π‘ˆ,𝐸) defined as follows: 𝐺𝛿𝑒1ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0.3,0.62[],π‘₯0.7,0.83[]ξ‚Όξ‚Ά,𝐺0.5,0.8,0.6𝛿𝑒2ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0.2,0.42[],π‘₯0.2,0.33[]ξ‚Όξ‚Ά,𝐺0.3,0.5,0.5𝛿𝑒3ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0.7,0.82[],π‘₯0.2,0.43[]ξ‚Όξ‚Ά.0.2,0.5,0.3(3.6) It is clear that ξ‚πΉπœ‡ is a GIVFS subset of 𝐺𝛿.

Definition 3.5. Two GIVFSSs ξ‚πΉπœ‡ and 𝐺𝛿 over (π‘ˆ,𝐸) are said to be equal, and we write ξ‚πΉπœ‡=𝐺𝛿 if ξ‚πΉπœ‡ is a GIVFS subset of 𝐺𝛿 and 𝐺𝛿 is a GIVFS subset of ξ‚πΉπœ‡. In other words, ξ‚πΉπœ‡=𝐺𝛿 if the following conditions are satisfied: (a)πœ‡(𝑒) is equal to 𝛿(𝑒) for all π‘’βˆˆπΈ,(b)𝐹(𝑒) is equal to 𝐺(𝑒) for all π‘’βˆˆπΈ.

Definition 3.6. A GIVFSS is called a generalised null interval-valued fuzzy soft set, denoted by ξ‚βˆ…πœ‡ if ξ‚βˆ…πœ‡ξ‚βˆΆπΈβ†’π‘ƒ(π‘ˆ)×𝐼 such that ξ‚βˆ…πœ‡ξ‚€ξ‚ξ‚(𝑒)=𝐹(𝑒)(π‘₯),πœ‡(𝑒),(3.7) where 𝐹(𝑒)=[0,0]=[0] and πœ‡(𝑒)=0 for all π‘’βˆˆπΈ.

Definition 3.7. A GIVFSS is called a generalised absolute interval-valued fuzzy soft set, denoted by ξ‚π΄πœ‡ if ξ‚π΄πœ‡ξ‚βˆΆπΈβ†’π‘ƒ(π‘ˆ)×𝐼 such that ξ‚π΄πœ‡ξ‚€ξ‚ξ‚(𝑒)=𝐹(𝑒)(π‘₯),πœ‡(𝑒),(3.8) where 𝐹(𝑒)=[1,1]=[1], and πœ‡(𝑒)=1 for all π‘’βˆˆπΈ.

4. Basic Operations on GIVFSS

In this section, we introduce some basic operations on GIVFSS, namely, complement, union and intersection and we give some properties related to these operations.

Definition 4.1. Let ξ‚πΉπœ‡ be a GIVFSS over (π‘ˆ,𝐸). Then, the complement of ξ‚πΉπœ‡, denoted by ξ‚πΉπ‘πœ‡ and is defined by ξ‚πΉπ‘πœ‡=𝐺𝛿, such that 𝛿(𝑒)=𝑐(πœ‡(𝑒)) and 𝐺(𝑒)=̃𝑐(𝐹(𝑒)) for all π‘’βˆˆπΈ, where 𝑐 is a fuzzy complement and ̃𝑐 is an interval-valued fuzzy complement.

Example 4.2. Consider a GIVFSS ξ‚πΉπœ‡ over (π‘ˆ,𝐸) as in Example 3.2: ξ‚πΉπœ‡=βŽ›βŽœβŽœβŽœβŽœβŽ[][][]⎞⎟⎟⎟⎟⎠0.3,0.6][0.7,0.8][0.5,0.80.60.1,0.4][0,0.3][0.1,0.50.50.7,0.8][0.1,0.2][0,0.40.3.(4.1) By using the basic fuzzy complement for πœ‡(𝑒) and interval-valued fuzzy complement for 𝐹(𝑒), we have ξ‚πΉπ‘πœ‡=𝐺𝛿 where 𝐺𝛿=βŽ›βŽœβŽœβŽœβŽœβŽ[][][]⎞⎟⎟⎟⎟⎠0.4,0.7][0.2,0.3][0.2,0.50.40.6,0.9][0.7,1][0.5,0.90.50.2,0.3][0.8,0.9][0.6,10.7.(4.2)

Proposition 4.3. Let ξ‚πΉπœ‡ be a GIVFSS over (π‘ˆ,𝐸). Then, the following holds: ξ‚€ξ‚πΉπ‘πœ‡ξ‚π‘=ξ‚πΉπœ‡.(4.3)

Proof. Since ξ‚πΉπ‘πœ‡=𝐺𝛿, then ξ‚€ξ‚πΉπ‘πœ‡ξ‚π‘=𝐺𝑐𝛿(4.4) but, from Definition 4.1,  𝐺𝛿=(̃𝑐(𝐹(𝑒)),𝑐(πœ‡(𝑒))), then 𝐺𝑐𝛿===𝐹̃𝑐̃𝑐𝐹(𝑒),𝑐(𝑐(πœ‡(𝑒)))𝐹(𝑒),πœ‡(𝑒)πœ‡.(4.5)

Definition 4.4. Union of two GIVFSSs (ξ‚πΉπœ‡,𝐴) and (𝐺𝛿,𝐡), denoted by ξ‚πΉπœ‡βˆͺ𝐺𝛿, is a GIVFSS (ξ‚π»πœˆ,𝐢) where 𝐢=𝐴βˆͺ𝐡 and ξ‚π»πœˆξ‚βˆΆπΈβ†’π‘ƒ(π‘ˆ)×𝐼 is defined by ξ‚π»πœˆξ‚€ξ‚ξ‚(𝑒)=𝐻(𝑒),𝜈(𝑒)(4.6) such that 𝐻(𝑒)=𝐹(𝑒)βˆͺ𝐺(𝑒) and 𝜈(𝑒)=𝑠(πœ‡(𝑒),𝛿(𝑒)), where 𝑠 is an 𝑠-norm and 𝐻(𝑒)=[sup(πœ‡βˆ’ξ‚πΉ(𝑒),πœ‡βˆ’ξ‚πΊ(𝑒)),sup(πœ‡+𝐹(𝑒),πœ‡+𝐺(𝑒))].

Example 4.5. Consider GIVFSS ξ‚πΉπœ‡ and 𝐺𝛿 as in Example 3.4. By using the interval-valued fuzzy union and basic fuzzy union, we have ξ‚πΉπœ‡βˆͺ𝐺𝛿=𝐻𝑣, where 𝐻𝑣𝑒1ξ€Έ=π‘₯1ξ€Ίξ€»,π‘₯sup(0.1,0.3),sup(0.3,0.6)2ξ€Ίξ€»,π‘₯sup(0.5,0.7),sup(0.7,0.8)3ξ€Ίξ€»ξƒ°ξƒͺ=π‘₯sup(0.3,0.5),sup(0.5,0.8),max(0.4,0.6)ξ‚΅ξ‚»1[],π‘₯0.3,0.62[],π‘₯0.7,0.83[]ξ‚Όξ‚Ά.0.5,0.8,0.6(4.7) Similarly, we get 𝐻𝑣𝑒2ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0.2,0.42[],π‘₯0.2,0.33[]ξ‚Όξ‚Ά,𝐻0.3,0.5,0.5𝑣𝑒3ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0.7,0.82[],π‘₯0.2,0.43[]ξ‚Όξ‚Ά.0.2,0.5,0.3(4.8) In matrix notation, we write ξ‚π»π‘£βŽ›βŽœβŽœβŽœβŽœβŽ[][][]⎞⎟⎟⎟⎟⎠(𝑒)=0.3,0.6][0.7,0.8][0.5,0.80.60.2,0.4][0.2,0.3][0.3,0.50.50.7,0.8][0.2,0.4][0.2,0.50.3.(4.9)

Proposition 4.6. Let ξ‚πΉπœ‡, 𝐺𝛿, and 𝐻𝑣 be any three GIVFSSs. Then, the following results hold.(a)ξ‚πΉπœ‡βˆͺ𝐺𝛿=𝐺𝛿βˆͺξ‚πΉπœ‡.(b)ξ‚πΉπœ‡ξ‚πΊβˆͺ(𝛿βˆͺ𝐻𝑣𝐹)=(πœ‡βˆͺ𝐺𝛿𝐻)βˆͺ𝑣. (c)ξ‚πΉπœ‡βˆͺξ‚πΉπœ‡βŠ†ξ‚πΉπœ‡. (d)ξ‚πΉπœ‡βˆͺξ‚π΄πœ‡=ξ‚π΄πœ‡. (e)ξ‚πΉπœ‡βˆͺξ‚βˆ…πœ‡=ξ‚πΉπœ‡.

Proof. (a) ξ‚πΉπœ‡βˆͺ𝐺𝛿=ξ‚π»πœˆ.
From Definition 4.4, we have ξ‚π»πœˆξ‚(𝑒)=(𝐻(𝑒)(π‘₯),𝜈(𝑒)) such that 𝐻(𝑒)=𝐹(𝑒)βˆͺ𝐺(𝑒) and 𝜈(𝑒)=𝑠(πœ‡(𝑒),𝛿(𝑒)).
But 𝐻(𝑒)=𝐹(𝑒)βˆͺ𝐺(𝑒)=𝐺(𝑒)βˆͺ𝐹(𝑒) (since union of interval-valued fuzzy sets is commutative) and 𝜈(𝑒)=𝑠(πœ‡(𝑒),𝛿(𝑒))=𝑠(𝛿(𝑒),πœ‡(𝑒)) (since 𝑠-norm is commutative), then 𝐺𝛿βˆͺξ‚πΉπœ‡=ξ‚π»πœˆ.
(b) The proof is straightforward from Definition 4.4.
(c) The proof is straightforward from Definition 4.4.
(d) The proof is straightforward from Definition 4.4.
(e) The proof is straightforward from Definition 4.4.

Definition 4.7. Intersection of two GIVFSSs (ξ‚πΉπœ‡,𝐴) and (𝐺𝛿,𝐡), denoted by ξ‚πΉπœ‡βˆ©ξ‚πΊπ›Ώ, is a GIVFSS (ξ‚π»πœˆ,𝐢) where 𝐢=𝐴βˆͺ𝐡 and ξ‚π»πœˆξ‚βˆΆπΈβ†’π‘ƒ(π‘ˆ)×𝐼 is defined by ξ‚π»πœˆξ‚€ξ‚ξ‚(𝑒)=𝐻(𝑒),𝜈(𝑒)(4.10) such that 𝐻(𝑒)=𝐹(𝑒)∩𝐺(𝑒) and 𝜈(𝑒)=𝑑(πœ‡(𝑒),𝛿(𝑒)), where 𝑑 is a 𝑑-norm and 𝐻(𝑒)=[inf(πœ‡βˆ’ξ‚πΉ(𝑒),πœ‡βˆ’ξ‚πΊ(𝑒)),inf(πœ‡+𝐹(𝑒),πœ‡+𝐺(𝑒))].

Example 4.8. Consider GIVFSS ξ‚πΉπœ‡ and 𝐺𝛿 as in Example 4.5. By using the interval-valued fuzzy intersection and basic fuzzy intersection, we have ξ‚πΉπœ‡βˆ©ξ‚πΊπ›Ώ=𝐻𝑣, where 𝐻𝑣𝑒1ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯inf(0.1,0.3),inf(0.3,0.6)2[],π‘₯inf(0.5,0.7),inf(0.7,0.8)3[]ξ‚Όξ‚Ά=π‘₯inf(0.3,0.5),inf(0.5,0.8)min(0.4,0.6)ξ‚΅ξ‚»1[],π‘₯0.1,0.32[],π‘₯0.5,0.73[]ξ‚Όξ‚Ά.0.3,0.5,0.4(4.11) Similarly, we get 𝐻𝑣𝑒2ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0,0.32[],π‘₯0,0.23[]ξ‚Όξ‚Ά,𝐻0.1,0.3,0.4𝑣𝑒3ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0.5,0.62[],π‘₯0.1,0.13[]ξ‚Όξ‚Ά.0.1,0.3,0.1(4.12) In matrix notation, we write ξ‚π»π‘£βŽ›βŽœβŽœβŽœβŽœβŽ[][][]⎞⎟⎟⎟⎟⎠(𝑒)=0.1,0.3][0.5,0.7][0.3,0.50.40,0.3][0,0.2][0.1,0.30.40.5,0.6][0.1,0.1][0.1,0.30.1.(4.13)

Proposition 4.9. Let ξ‚πΉπœ‡, 𝐺𝛿, and 𝐻𝑣 be any three GIVFSSs. Then, the following results hold.(a)ξ‚πΉπœ‡βˆ©ξ‚πΊπ›Ώ=ξ‚πΊπ›Ώβˆ©ξ‚πΉπœ‡.(b)ξ‚πΉπœ‡ξ‚πΊβˆ©(π›Ώβˆ©ξ‚π»π‘£ξ‚πΉ)=(πœ‡βˆ©ξ‚πΊπ›Ώξ‚π»)βˆ©π‘£. (c)ξ‚πΉπœ‡βˆ©ξ‚πΉπœ‡βŠ†ξ‚πΉπœ‡. (d)ξ‚πΉπœ‡βˆ©ξ‚π΄πœ‡=ξ‚πΉπœ‡. (e)ξ‚πΉπœ‡βˆ©ξ‚βˆ…πœ‡=ξ‚βˆ…πœ‡.

Proof. (a) ξ‚πΉπœ‡βˆ©ξ‚πΊπ›Ώ=ξ‚π»πœˆ.
From Definition 4.7, we have ξ‚π»πœˆξ‚(𝑒)=(𝐻(𝑒)(π‘₯),𝜈(𝑒)) such that 𝐻(𝑒)=𝐹(𝑒)∩𝐺(𝑒) and 𝜈(𝑒)=𝑑(πœ‡(𝑒),𝛿(𝑒)).
But 𝐻(𝑒)=𝐹(𝑒)∩𝐺(𝑒)=𝐺(𝑒)∩𝐹(𝑒) (since intersection of interval-valued fuzzy sets is commutative) and 𝜈(𝑒)=𝑑(πœ‡(𝑒),𝛿(𝑒))=𝑑(𝛿(𝑒),πœ‡(𝑒)) (since 𝑑-norm is commutative), then ξ‚πΊπ›Ώβˆ©ξ‚πΉπœ‡=ξ‚π»πœˆ.
(b) The proof is straightforward from Definition 4.7.
(c) The proof is straightforward from Definition 4.7.
(d) The proof is straightforward from Definition 4.7.
(e) The proof is straightforward from Definition 4.7.

Proposition 4.10. Let ξ‚πΉπœ‡ and 𝐺𝛿 be any two GIVFSSs. Then the DeMorgan’s Laws hold:(a)(ξ‚πΉπœ‡βˆͺ𝐺𝛿)𝑐=ξ‚πΊπ‘π›Ώβˆ©ξ‚πΉπ‘πœ‡.(b)(ξ‚πΉπœ‡βˆ©ξ‚πΊπ›Ώ)𝑐=𝐺𝑐𝛿βˆͺξ‚πΉπ‘πœ‡.

Proof. (a) Consider ξ‚πΉπ‘πœ‡βˆ©ξ‚πΊπ‘π›Ώ=ξ‚€=((̃𝑐(𝐹(𝑒)),𝑐(πœ‡(𝑒)))∩(̃𝑐(𝐺(𝑒)),𝑐(𝛿(𝑒))))=((̃𝑐(𝐹(𝑒))βˆ©Μƒπ‘(𝐺(𝑒))),(𝑐(πœ‡(𝑒))βˆ©π‘(𝛿(𝑒))))((𝐹(𝑒))βˆͺ(𝐺(𝑒)))̃𝑐,(πœ‡(𝑒)βˆͺ𝛿(𝑒))𝑐=ξ‚€ξ‚πΉπœ‡βˆͺ𝐺𝛿𝑐.(4.14)
(b) The proof is similar to the above.

Proposition 4.11. Let ξ‚πΉπœ‡, 𝐺𝛿, and 𝐻𝑣 be any three GIVFSSs. Then, the following results hold. (a)ξ‚πΉπœ‡ξ‚πΊβˆͺ(π›Ώβˆ©ξ‚π»π‘£ξ‚πΉ)=(πœ‡βˆͺ𝐺𝛿𝐹)∩(πœ‡βˆͺ𝐻𝑣). (b)ξ‚πΉπœ‡ξ‚πΊβˆ©(𝛿βˆͺ𝐻𝑣𝐹)=(πœ‡βˆ©ξ‚πΊπ›Ώξ‚πΉ)βˆͺ(πœ‡βˆ©ξ‚π»π‘£).

Proof. (a) For all π‘₯∈𝐸, πœ†ξ‚πΉ(π‘₯)βˆͺ(𝐺(π‘₯)βˆ©ξ‚π»(π‘₯))ξ‚Έξ‚΅πœ†(π‘₯)=supβˆ’ξ‚πΉ(π‘₯)(π‘₯),πœ†βˆ’ξ‚πΊ(π‘₯)βˆ©ξ‚π»(π‘₯)ξ‚Άξ‚΅πœ†(π‘₯),sup+𝐹(π‘₯)(π‘₯),πœ†+𝐺(π‘₯)βˆ©ξ‚π»(π‘₯)=ξ‚ƒξ‚€πœ†(π‘₯)ξ‚Άξ‚Ήsupβˆ’ξ‚πΉ(π‘₯)ξ‚€πœ†(π‘₯),infβˆ’ξ‚πΊ(π‘₯)(π‘₯),πœ†βˆ’ξ‚π»(π‘₯),ξ‚€πœ†(π‘₯)sup+𝐹(π‘₯)ξ‚€πœ†(π‘₯),inf+𝐺(π‘₯)(π‘₯),πœ†+𝐻(π‘₯)=ξ‚ƒξ‚€ξ‚€πœ†(π‘₯)infsupβˆ’ξ‚πΉ(π‘₯)(π‘₯),πœ†βˆ’ξ‚πΊ(π‘₯)(ξ‚ξ‚€πœ†π‘₯),supβˆ’ξ‚πΉ(π‘₯)(π‘₯),πœ†βˆ’ξ‚π»(π‘₯)(,ξ‚€ξ‚€πœ†π‘₯)infsup+𝐹(π‘₯)(π‘₯),πœ†+𝐺(π‘₯)ξ‚ξ‚€πœ†(π‘₯),sup+𝐹(π‘₯)(π‘₯),πœ†+𝐻(π‘₯)(π‘₯)=πœ†(𝐹(π‘₯)βˆͺ𝐺(π‘₯))∩(𝐹(π‘₯)βˆͺ𝐻(π‘₯))(𝛾π‘₯),πœ‡(π‘₯)βˆͺ(𝛿(π‘₯)∩𝜈(π‘₯))𝛾(π‘₯)=maxπœ‡(π‘₯)(π‘₯),𝛾𝛿(π‘₯)∩𝜈(π‘₯)𝛾(π‘₯)=maxπœ‡(π‘₯)𝛾(π‘₯),min𝛿(π‘₯)(π‘₯),π›Ύπœˆ(π‘₯)𝛾(π‘₯)ξ€Έξ€Ύ=minmaxπœ‡(π‘₯)(π‘₯),𝛾𝛿(π‘₯)𝛾(π‘₯),maxπœ‡(π‘₯)(π‘₯),π›Ύπœˆ(π‘₯)𝛾(π‘₯)ξ€Έξ€Ύ=minπœ‡(π‘₯)βˆͺ𝛿(π‘₯)(π‘₯),π›Ύπœ‡(π‘₯)βˆͺ𝜈(π‘₯)(ξ€Ύπ‘₯)=𝛾(πœ‡(π‘₯)βˆͺ𝛿(π‘₯))ξ‚βˆ©(πœ‡(π‘₯)βˆͺ𝜈(π‘₯))(π‘₯).(4.15)
(b) Similar to the proof of (a).

5. AND and OR Operations on GIVFSS with Application

In this section, we give the definitions of AND and OR operations on GIVFSS. An application of this operations in decision making problem has been shown.

Definition 5.1. If (ξ‚πΉπœ‡,𝐴) and (𝐺𝛿,𝐡) are two GIVFSSs, then β€œ(ξ‚πΉπœ‡,𝐴) AND (𝐺𝛿,𝐡)” denoted by (ξ‚πΉπœ‡ξ‚πΊ,𝐴)∧(𝛿,𝐡) is defined by ξ‚€ξ‚πΉπœ‡ξ‚βˆ§ξ‚€ξ‚πΊ,𝐴𝛿=𝐻,π΅πœ†ξ‚,𝐴×𝐡,(5.1) where ξ‚π»πœ†(𝛼,𝛽)=(𝐻(𝛼,𝛽),πœ†(𝛼,𝛽)) for all (𝛼,𝛽)βˆˆπ΄Γ—π΅, such that 𝐻(𝛼,𝛽)=𝐹(𝛼)∩𝐺(𝛽) and πœ†(𝛼,𝛽)=𝑑(πœ‡(𝛼),𝛿(𝛽)), for all (𝛼,𝛽)βˆˆπ΄Γ—π΅, where 𝑑 is a 𝑑-norm.

Example 5.2. Suppose the universe consists of three machines π‘₯1,π‘₯2,π‘₯3, that is,π‘ˆ={π‘₯1,π‘₯2,π‘₯3}, and consider the set of parameters 𝐸={𝑒1,𝑒2,𝑒3} which describe their performances according to certain specific task. Suppose a firm wants to buy one such machine depending on any two of the parameters only. Let there be two observations ξ‚πΉπœ‡ and 𝐺𝛿 by two experts 𝐴 and 𝐡, respectively, defined as follows: ξ‚πΉπœ‡ξ€·π‘’1ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0.1,0.32[],π‘₯0.5,0.73[]ξ‚Όξ‚Ά,𝐹0.3,0.5,0.4πœ‡ξ€·π‘’2ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0,0.32[],π‘₯0,0.23[]ξ‚Όξ‚Ά,𝐹0.1,0.3,0.4πœ‡ξ€·π‘’3ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0.5,0.62[],π‘₯0.1,0.13[]ξ‚Όξ‚Ά,𝐺0.1,0.3,0.1πœ‡ξ€·π‘’1ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0.3,0.52[],π‘₯0.2,0.63[]ξ‚Όξ‚Ά,𝐺0.4,0.5,0.3πœ‡ξ€·π‘’2ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0.3,0.52[],π‘₯0.4,0.63[]ξ‚Όξ‚Ά,𝐺0,0.3,0.1πœ‡ξ€·π‘’3ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0.1,0.62[],π‘₯0.4,0.73[]ξ‚Όξ‚Ά.0.2,0.3,0.2(5.2) To find the AND between the two GIVFSSs, we have (ξ‚πΉπœ‡,𝐴) AND (𝐺𝛿𝐻,𝐡)=(πœ†,𝐴×𝐡), where ξ‚π»πœ†ξ€·π‘’1,𝑒1ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0.1,0.32[],π‘₯0.2,0.63[]ξ‚Όξ‚Ά,𝐻0.3,0.5,0.3πœ†ξ€·π‘’1,𝑒2ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0.1,0.32[],π‘₯0.4,0.63[]ξ‚Όξ‚Ά,𝐻0,0.3,0.1πœ†ξ€·π‘’1,𝑒3ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0.1,0.32[],π‘₯0.4,0.73[]ξ‚Όξ‚Ά,𝐻0.2,0.3,0.2πœ†ξ€·π‘’2,𝑒1ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0,0.32[],π‘₯0,0.23[]ξ‚Όξ‚Ά,𝐻0.1,0.3,0.3πœ†ξ€·π‘’2,𝑒2ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0,0.32[],π‘₯0,0.23[]ξ‚Όξ‚Ά,𝐻0,0.3,0.1πœ†ξ€·π‘’2,𝑒3ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0,0.32[],π‘₯0,0.23[]ξ‚Όξ‚Ά,𝐻0.1,0.3,0.2πœ†ξ€·π‘’3,𝑒1ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0.3,0.52[],π‘₯0.1,0.13[]ξ‚Όξ‚Ά,𝐻0.1,0.3,0.1πœ†ξ€·π‘’3,𝑒2ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0.3,0.52[],π‘₯0.1,0.13[]ξ‚Όξ‚Ά,𝐻0,0.3,0.1πœ†ξ€·π‘’3,𝑒3ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0.1,0.62[],π‘₯0.1,0.13[]ξ‚Όξ‚Ά.0.1,0.3,0.1(5.3) Now, to determine the best machine, we first compute the numerical grade π‘Ÿπ‘βˆˆπ‘ƒ(π‘₯𝑖) for each π‘βˆˆπ‘ƒ such that π‘Ÿπ‘βˆˆπ‘ƒξ€·π‘₯𝑖=π‘₯βˆˆπ‘ˆπ‘ξ‚€ξ‚€βˆ’π‘–βˆ’πœ‡βˆ’π»ξ€·π‘π‘–ξ€Έξ‚+𝑐(π‘₯)+π‘–βˆ’πœ‡+𝐻𝑝𝑖(π‘₯).(5.4) The result is shown in Tables 1 and 2. Let 𝑃={𝑝1=(𝑒1,𝑒1),𝑝2=(𝑒1,𝑒2),…,𝑝9=(𝑒3,𝑒3)}. Now, we mark the highest numerical grade (indicated in parenthesis) in each row excluding the last row which is the grade of such belongingness of a machine against each pair of parameters (see Table 3). Now, the score of each such machine is calculated by taking the sum of the products of these numerical grades with the corresponding value of πœ‡. The machine with the highest score is the desired machine. We do not consider the numerical grades of the machine against the pairs (𝑒𝑖,𝑒𝑖), 𝑖=1,2,3, as both the parameters are the same:Score (π‘₯1)=(1βˆ—0.1)+(1.1βˆ—0.1)=0.21, Score (π‘₯2)=(1.3βˆ—0.1)+(1.3βˆ—0.2)=0.39,Score (π‘₯3)=(0.3βˆ—0.3)+(0.3βˆ—0.2)=0.15.The firm will select the machine with the highest score. Hence, they will buy machine π‘₯2.

Definition 5.3. If (ξ‚πΉπœ‡,𝐴) and (𝐺𝛿,𝐡) are two GIVFSSs, then β€œ(ξ‚πΉπœ‡,𝐴) OR (𝐺𝛿,𝐡)” denoted by (ξ‚πΉπœ‡ξ‚πΊ,𝐴)∨(𝛿,𝐡) is defined by ξ‚€ξ‚πΉπœ‡ξ‚βˆ¨ξ‚€ξ‚πΊ,𝐴𝛿=𝐻,π΅πœ†ξ‚,𝐴×𝐡,(5.5) where ξ‚π»πœ†(𝛼,𝛽)=(𝐻(𝛼,𝛽),πœ†(𝛼,𝛽)) for all (𝛼,𝛽)βˆˆπ΄Γ—π΅, such that 𝐻(𝛼,𝛽)=𝐹(𝛼)∩𝐺(𝛽) and πœ†(𝛼,𝛽)=𝑠(πœ‡(𝛼),𝛿(𝛽)), for all (𝛼,𝛽)βˆˆπ΄Γ—π΅, where 𝑠 is an 𝑠-norm.

Example 5.4. Consider Example 5.2. To find the OR between the two GIVFSSs, we have (ξ‚πΉπœ‡,𝐴) OR (𝐺𝛿𝐻,𝐡)=(πœ†,𝐴×𝐡), where ξ‚π»πœ†ξ€·π‘’1,𝑒1ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0.3,0.52[],π‘₯0.5,0.73[]ξ‚Όξ‚Ά,𝐻0.4,0.5,0.4πœ†ξ€·π‘’1,𝑒2ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0.3,0.52[],π‘₯0.5,0.73[]ξ‚Όξ‚Ά,𝐻0.4,0.5,0.4πœ†ξ€·π‘’1,𝑒3ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0.1,0.62[],π‘₯0.5,0.73[]ξ‚Όξ‚Ά,𝐻0.3,0.5,0.4πœ†ξ€·π‘’2,𝑒1ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0.3,0.52[],π‘₯0.,0.63[]ξ‚Όξ‚Ά,𝐻0.4,0.5,0.4πœ†ξ€·π‘’2,𝑒2ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0.3,0.52[],π‘₯0.4,0.63[]ξ‚Όξ‚Ά,𝐻0.1,0.3,0.4πœ†ξ€·π‘’2,𝑒3ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0.1,0.62[],π‘₯0.4,0.73[]ξ‚Όξ‚Ά,𝐻0.2,0.3,0.4πœ†ξ€·π‘’3,𝑒1ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0.5,0.62[],π‘₯0.2,0.63[]ξ‚Όξ‚Ά,𝐻0.4,0.5,0.3πœ†ξ€·π‘’3,𝑒2ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0.5,0.62[],π‘₯0.4,0.63[]ξ‚Όξ‚Ά,𝐻0.1,0.3,0.1πœ†ξ€·π‘’3,𝑒3ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0.5,0.62[],π‘₯0.4,0.73[]ξ‚Όξ‚Ά.0.2,0.3,0.2(5.6)

Remark 5.5. We use the same method in Example 5.2 for the OR operation if the firm wants to buy one such machine depending on any one of the parameters only.

Proposition 5.6. Let (ξ‚πΉπœ‡,𝐴) and (𝐺𝛿,𝐡) be any two GIVFSSs. Then, the following results hold:(a)𝐹((πœ‡ξ‚πΊ,𝐴)∧(𝛿,𝐡))𝑐𝐹=(πœ‡,𝐴)π‘ξ‚πΊβˆ¨(𝛿,𝐡)𝑐, (b)𝐹((πœ‡ξ‚πΊ,𝐴)∨(𝛿,𝐡))𝑐𝐹=(πœ‡,𝐴)π‘ξ‚πΊβˆ§(𝛿,𝐡)𝑐.

Proof. Straightforward from Definitions 4.1, 5.1, and 5.3.

Proposition 5.7. Let (ξ‚πΉπœ‡,𝐴), (𝐺𝛿,𝐡), and (ξ‚π»πœ†,𝐢) be any three GIVFSSs. Then, the following results hold:(a)(ξ‚πΉπœ‡ξ‚πΊ,𝐴)∧((𝛿𝐻,𝐡)∧(πœ†ξ‚πΉ,𝐢))=((πœ‡ξ‚πΊ,𝐴)∧(𝛿𝐻,𝐡))∧(πœ†,𝐢), (b)(ξ‚πΉπœ‡ξ‚πΊ,𝐴)∨((𝛿𝐻,𝐡)∨(πœ†ξ‚πΉ,𝐢))=((πœ‡ξ‚πΊ,𝐴)∨(𝛿𝐻,𝐡))∨(πœ†,𝐢), (c)(ξ‚πΉπœ‡ξ‚πΊ,𝐴)∨((𝛿𝐻,𝐡)∧(πœ†ξ‚πΉ,𝐢))=((πœ‡ξ‚πΊ,𝐴)∨(𝛿𝐹,𝐡))∧((πœ‡ξ‚π»,𝐴)∨(πœ†,𝐢)), (d)(ξ‚πΉπœ‡ξ‚πΊ,𝐴)∧((𝛿𝐻,𝐡)∨(πœ†ξ‚πΉ,𝐢))=((πœ‡ξ‚πΊ,𝐴)∧(𝛿𝐹,𝐡))∨((πœ‡ξ‚π»,𝐴)∧(πœ†,𝐢)).

Proof. Straightforward from Definitions 5.1 and 5.3.

Remark 5.8. The commutativity property does not hold for AND and OR operations since 𝐴×𝐡≠𝐡×𝐴.

6. Similarity between Two GIVFSS

In this section, we give a measure of similarity between two GIVFSSs. We are taking the set theoretic approach because it is easier to calculate on and is a very popular method too.

Definition 6.1. Similarity between two GIVFSSs ξ‚πΉπœ‡ and 𝐺𝛿, denoted by 𝐹𝑆(πœ‡,𝐺𝛿), is defined by π‘†ξ‚€ξ‚πΉπœ‡,𝐺𝛿=ξ‚ƒπœ‘βˆ’ξ‚€ξ‚ξ‚πΊξ‚πΉ,β‹…π‘š(πœ‡,𝛿),πœ‘+ξ‚€ξ‚ξ‚πΊξ‚ξ‚„πœ‘πΉ,β‹…π‘š(πœ‡,𝛿),suchthatβˆ’ξ‚€ξ‚ξ‚πΊξ‚ξ‚€πœ‘πΉ,=min1𝐺𝐹,,πœ‘2𝐺,πœ‘πΉ,+ξ‚€ξ‚ξ‚πΊξ‚ξ‚€πœ‘πΉ,=max1𝐺𝐹,,πœ‘2𝐺,𝐹,(6.1) where πœ‘1𝐺=⎧βŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺ⎩𝐹,0,ifπœ‡βˆ’ξ‚πΉπ‘–βˆ‘(π‘₯)=0,βˆ€π‘–,𝑛𝑖maxπ‘₯βˆˆπ‘‹ξ‚»ξ‚΅πœ‡minβˆ’ξ‚πΉπ‘–(π‘₯),πœ‡βˆ’ξ‚πΊπ‘–(π‘₯)ξ‚Άξ‚Όβˆ‘π‘›π‘–maxπ‘₯βˆˆπ‘‹ξ‚Έπœ‡βˆ’ξ‚πΉπ‘–ξ‚Ήπœ‘(π‘₯),otherwise,2𝐺=βˆ‘πΉ,𝑛𝑖maxπ‘₯βˆˆπ‘‹ξ‚»ξ‚΅πœ‡min+𝐹𝑖(π‘₯),πœ‡+𝐺𝑖(π‘₯)ξ‚Άξ‚Όβˆ‘π‘›π‘–maxπ‘₯βˆˆπ‘‹ξ‚΅πœ‡+𝐹𝑖,π‘šβˆ‘||||(π‘₯)(πœ‡(𝑒),𝛿(𝑒))=1βˆ’πœ‡(𝑒)βˆ’π›Ώ(𝑒)βˆ‘||||.πœ‡(𝑒)+𝛿(𝑒)(6.2)

Definition 6.2. Let ξ‚πΉπœ‡ and 𝐺𝛿 be two GIVFSSs over the same universe (π‘ˆ,𝐸). We say that two GIVFSS are significantly similar if πœ‘βˆ’(𝐹,𝐺)β‹…π‘š(πœ‡,𝛿)β©Ύ1/2.

Theorem 6.3. Let ξ‚πΉπœ‡, 𝐺𝛿, and ξ‚π»πœ† be any three GIVFSSs over (π‘ˆ,𝐸). Then, the following hold: (a)in general 𝐹𝑆(πœ‡,𝐺𝛿𝐺)≠𝑆(𝛿,ξ‚πΉπœ‡), (b)πœ‘βˆ’(𝐹,𝐺)β‰₯0 and πœ‘+(𝐹,𝐺)≀1, (c)ξ‚πΉπœ‡=𝐺𝛿𝐹⇒𝑆(πœ‡,𝐺𝛿)=1, (d)ξ‚πΉπœ‡βŠ†ξ‚πΊπ›ΏβŠ†ξ‚π»πœ†ξ‚πΉβ‡’π‘†(πœ‡,ξ‚π»πœ†ξ‚πΊ)⩽𝑆(𝛿,ξ‚π»πœ†), (e)ξ‚πΉπœ‡ξ‚βˆ©ξ‚πΊπ›Ώξ‚πΉ=βˆ…β‡”π‘†(πœ‡,𝐺𝛿)=0.

Proof. (a) The proof is straightforward and follows from Definition 6.1.
(b) From Definition 6.1, we haveπœ‘1𝐺=⎧βŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺ⎩𝐹,0,ifπœ‡βˆ’ξ‚πΉπ‘–βˆ‘(π‘₯)=0,βˆ€π‘–,𝑛𝑖maxπ‘₯βˆˆπ‘‹ξ‚»ξ‚΅πœ‡minβˆ’ξ‚πΉπ‘–(π‘₯),πœ‡βˆ’ξ‚πΊπ‘–(π‘₯)ξ‚Άξ‚Όβˆ‘π‘›π‘–maxπ‘₯βˆˆπ‘‹ξ‚΅πœ‡βˆ’ξ‚πΉπ‘–ξ‚Ά(π‘₯),otherwise.(6.3)
If πœ‡βˆ’ξ‚πΉπ‘–(π‘₯)=0, for all 𝑖, then πœ‘βˆ’(𝐹,𝐺)=0, and, if πœ‡βˆ’ξ‚πΉπ‘–(π‘₯)β‰ 0, for some 𝑖, then it is clear that πœ‘βˆ’(𝐹,𝐺)β‰₯0.
Also since πœ‘+(𝐹,𝐺)=max(πœ‘1(𝐹,𝐺),πœ‘2(𝐹,𝐺)), suppose that πœ‘1(𝐹,𝐺)=1 and πœ‘2(𝐹,𝐺)=1, then πœ‘+(𝐹,𝐺)=1, that means, if πœ‘1(𝐹,𝐺)<1 and πœ‘2(𝐹,𝐺)<1, then πœ‘+(𝐹,𝐺)≀1.
(c) The proof is straightforward and follows from Definition 6.1.
(d) The proof is straightforward and follows from Definition 6.1.
(e) The proof is straightforward and follows from Definition 6.1.

Example 6.4. Let ξ‚πΉπœ‡ be GIVFSS over (π‘ˆ,𝐸) defined as follows: ξ‚πΉπœ‡ξ€·π‘’1ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0.3,0.72[],π‘₯0.4,0.83[]ξ‚Όξ‚Ά,𝐹0.1,0.3,0.4πœ‡ξ€·π‘’2ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0.5,0.62[],π‘₯0.1,0.33[]ξ‚Όξ‚Ά,𝐹0,0.4,0.6πœ‡ξ€·π‘’3ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0.7,0.92[],π‘₯0.1,0.53[]ξ‚Όξ‚Ά.0.8,1,0.8(6.4) Let 𝐺𝛿 be another GIVFSS over (π‘ˆ,𝐸) defined as follows: 𝐺𝛿𝑒1ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0.1,0.42[],π‘₯0.5,0.73[]ξ‚Όξ‚Ά,𝐺0.2,0.3,0.3𝛿𝑒2ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0.6,0.82[],π‘₯0.5,0.63[]ξ‚Όξ‚Ά,𝐺0.4,0.8,0.7𝛿𝑒3ξ€Έ=π‘₯ξ‚΅ξ‚»1[],π‘₯0.4,0.72[],π‘₯0.3,0.53[]ξ‚Όξ‚Ά.0.5,0.7,0.6(6.5) Here, βˆ‘||||π‘š(πœ‡(𝑒),𝛿(𝑒))=1βˆ’πœ‡(𝑒)βˆ’π›Ώ(𝑒)βˆ‘||||||||+||||+||||πœ‡(𝑒)+𝛿(𝑒)=1βˆ’(0.4βˆ’0.3)(0.6βˆ’0.7)(0.8βˆ’0.6)||||+||||+||||πœ‘(0.4+0.3)(0.6+0.7)(0.80+0.6)β‰…0.8824,1𝐺=(𝐹,max{min(0.3,0.1),min(0.4,0.5),min(0.1,0.2)}+max(0.3,0.4,0.1)+max(0.5,0.1,0)+max(0.7,0.1,0.8)max{min(0.5,0.6),min(0.1,0.5),min(0,0.4)}+max(0.3,0.4,0.1)+max(0.5,0.1,0)+max(0.7,0.1,0.8)max{min(0.7,0.4),min(0.1,0.3),min(0.8,0.5)})=max(0.3,0.4,0.1)+max(0.5,0.1,0)+max(0.7,0.1,0.8)max{0.1,0.4,0.1}+max{0.5,0.1,0}+max{0.4,0.1,0.5}=max(0.3,0.4,0.1)+max(0.5,0.1,0)+max(0.7,0.1,0.8)0.4+0.5+0.5πœ‘0.4+0.5+0.8=0.824,2𝐺=𝐹,(max{min(0.7,0.4),min(0.8,0.7),min(0.3,0.3)}+max(0.7,0.8,0.3)+max(0.6,0.3,0.4)+max(0.9,0.5,1)max{min(0.6,0.8),min(0.3,0.6),min(0.4,0.8)}+max(0.7,0.8,0.3)+max(0.6,0.3,0.4)+max(0.9,0.5,1)max{min(0.9,0.7),min(0.5,0.5),min(1,0.7)})=max(0.7,0.8,0.3)+max(0.6,0.3,0.4)+max(0.9,0.5,1)max{0.4,0.7,0.3}+max{0.6,0.3,0.4}+max{0.7,0.5,0.7}=max(0.7,0.8,0.3)+max(0.6,0.3,0.4)+max(0.9,0.5,1)0.7+0.6+0.70.8+0.6+1=0.833.(6.6) Then, πœ‘βˆ’ξ‚€ξ‚ξ‚πΊξ‚ξ‚€πœ‘πΉ,=min1𝐺𝐹,,πœ‘2ξ‚€ξ‚ξ‚πΊπœ‘πΉ,=min(0.824,0.833)=0.824,+ξ‚€ξ‚ξ‚πΊξ‚ξ‚€πœ‘πΉ,=max1𝐺𝐹,,πœ‘2𝐺𝐹,=max(0.824,0.833)=0.833.(6.7) Hence, the similarity between the two GIVFSSs ξ‚πΉπœ‡ and 𝐺𝛿 will be π‘†ξ‚€ξ‚πΉπœ‡,𝐺𝛿=ξ‚ƒπœ‘βˆ’ξ‚€ξ‚ξ‚πΊξ‚πΉ,β‹…π‘š(πœ‡,𝛿),πœ‘+𝐺=[]=[].𝐹,β‹…π‘š(πœ‡,𝛿)(0.824)β‹…(0.8824),(0.833)β‹…(0.8824)0.727,0.735(6.8) Therefore, ξ‚πΉπœ‡ and 𝐺𝛿 are significantly similar.

7. Application of Similarity Measure in Medical Diagnosis

In this section, we will try to estimate the possibility that a sick person having certain visible symptoms is suffering from dengue fever. For this, we first construct a GIVFSS model for dengue fever and the GIVFSS of symptoms for the sick person. Next, we find the similarity measure of these two sets. If they are significantly similar, then we conclude that the person is possibly suffering from dengue fever. Let our universal set contain only two elements β€œyes” and β€œno,” that is, π‘ˆ={𝑦,𝑛}. Here, the set of parameters 𝐸 is the set of certain visible symptoms. Let 𝐸={𝑒1,𝑒2,𝑒3,𝑒4,𝑒5,𝑒6,𝑒7}, where 𝑒1 = body temperature, 𝑒2 = cough with chest congestion, 𝑒3 = loose motion, 𝑒4 = chills, 𝑒5 = headache, 𝑒6 = low heart rate (bradycardia), and 𝑒7 = pain upon moving the eyes. Our model GIVFSS for dengue fever π‘€πœ‡ is given in Table 4, and this can be prepared with the help of a physician.

Now, after talking to the sick person, we can construct his GIVFSS 𝐺𝛿 as in Table 5.

Now, we find the similarity measure of these two sets by using the same method as in Example 6.4, where, after the calculation, we get πœ‘βˆ’(𝑀,𝐺)π‘š(πœ‡,𝛿)β‰…0.22<1/2. Hence the two GIVFSSs are not significantly similar. Therefore, we conclude that the person is not suffering from dengue fever.

8. Conclusion

In this paper, we have introduced the concept of generalised interval-valued fuzzy soft set and studied some of its properties. The complement, union, intersection, β€œAND,” and β€œOR” operations have been defined on the interval-valued fuzzy soft sets. An application of this theory is given in solving a decision making problem. Similarity measure of two generalised interval-valued fuzzy soft sets is discussed, and its application to medical diagnosis has been shown.

Acknowledgment

The authors would like to acknowledge the financial support received from Universiti Kebangsaan Malaysia under the Research Grants UKM-ST-06-FRGS0104-2009 and UKM-DLP-2011-038.