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 on a universe U is a mapping such that

()

where Int([0,1]) stands for the set of all closed subintervals of [0,1], the set of all interval-valued fuzzy sets on U is denoted by

Suppose that , for all x ∈ U, is called the degree of membership of an element x to X. and are referred to as the lower and upper degrees of membership of x to X where .

Definition 2.2 (see [14].)The subset, complement, intersection, and union of the interval-valued fuzzy sets are defined as follows. Let , then

(a)
the complement of is denoted by where
()
(b)
the intersection of and is denoted by where
()
(c)
the union of and is denoted by where
()
(d)
X is a subset of Ydenoted by X⊆Y if and .

Definition 2.3 (see [14].)The compatibility measure of an interval-valued fuzzy set A with an interval-valued fuzzy set B (A is a reference set) is given by

()

such that

()

where

()

Theorem 2.4 (see [14].)Consider arbitrary, nonempty interval-valued fuzzy sets A, B, and C from the family of ivf(X) and a compatibility measure in the sense of Definition 2.3. Then,

(a)
,
(b)
,
(c)
in general .

Let U be a universal set and E a set of parameters. Let P(U) denote the power set of U and A⊆E. Molodtsov [1] defined soft set as follows.

Definition 2.5. A pair (F, E) is called a soft set over U, where F is a mapping given by F : E → P(U). In other words, a soft set over U is a parameterized family of subsets of the universe U.

Definition 2.6 (see [5].)Let U be a universal set, and let E be a set of parameters. Let I^U denote the power set of all fuzzy subsets of U. Let A⊆E. A pair (F, E) is called a fuzzy soft set over U where F is a mapping given by

()

Definition 2.7 (see [12].)Let U = {x₁, x₂, …, x_n} be the universal set of elements and E = {e₁, e₂, …, e_m} be the universal set of parameters. The pair (U, E) will be called a soft universe. Let F : E → I^U, where I^U is the collection of all fuzzy subsets of U, and let μ be a fuzzy subset of E. Let F_μ : E → I^U × I be a function defined as follows:

()

Then, F_μ is called a generalised fuzzy soft set (GFSS in short) over the soft set (U, E). Here, for each parameter e_i, F_μ(e_i) = (F(e_i), μ(e_i)) indicates not only the degree of belongingness of the elements of U in F(e_i) but also the degree of possibility of such belongingness which is represented by μ(e_i). So we can write F_μ(e_i) as follows:

()

where F(e_i)(x₁), F(e_i)(x₂), …, F(e_i)(x_n) are the degree of belongingness and μ(e_i) is the degree of possibility of such belongingness.

Definition 2.8 (see [13].)Let U be an initial universe and E a set of parameters. denotes the set of all interval-valued fuzzy sets of U. Let A⊆E. A pair is an interval-valued fuzzy soft set over U, where is a mapping given by .

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 U = {x₁, x₂, …, x_n} be the universal set of elements and E = {e₁, e₂, …, e_m} the universal set of parameters. The pair (U, E) will be called a soft universe. Let and μ be a fuzzy set of E, that is, μ : E → I = [0,1], where is the set of all interval-valued fuzzy subsets on U. Let be a function defined as follows:

()

Then,

is called a generalised interval-valued fuzzy soft set (GIVFSS in short) over the soft universe (U, E). For each parameter e_i,

indicates not only the degree of belongingness of the elements of U in

but also the degree of possibility of such belongingness which is represented by μ(e_i). So we can write

as follows:

()

Example 3.2. Let U = {x₁, x₂, x₃} be a set of universe, E = {e₁, e₂, e₃} a set of parameters, and let μ : E → I. Define a function as follows:

()

Then,

is a GIVFSS over (U, E).

In matrix notation, we write

()

Definition 3.3. Let and be two GIVFSSs over (U, E). is called a generalised interval-valued fuzzy soft subset of , and we write if

(a)
μ(e) is a fuzzy subset of δ(e) for all e ∈ E,
(b)
is an interval-valued fuzzy subset of for all e ∈ E.

Example 3.4. Let U = {x₁, x₂, x₃} 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 be a GIVFSS over (U, E) defined as follows:

()

Let

be another GIVFSS over (U, E) defined as follows:

()

It is clear that

is a GIVFS subset of

Definition 3.5. Two GIVFSSs and over (U, E) are said to be equal, and we write if is a GIVFS subset of and is a GIVFS subset of . In other words, if the following conditions are satisfied:

(a)
μ(e) is equal to δ(e) for all e ∈ E,
(b)
is equal to for all e ∈ E.

Definition 3.6. A GIVFSS is called a generalised null interval-valued fuzzy soft set, denoted by if such that

()

where

and μ(e) = 0 for all e ∈ E.

Definition 3.7. A GIVFSS is called a generalised absolute interval-valued fuzzy soft set, denoted by if such that

()

where

, and μ(e) = 1 for all e ∈ E.

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 be a GIVFSS over (U, E). Then, the complement of , denoted by and is defined by , such that δ(e) = c(μ(e)) and for all e ∈ E, where c is a fuzzy complement and is an interval-valued fuzzy complement.

Example 4.2. Consider a GIVFSS over (U, E) as in Example 3.2:

()

By using the basic fuzzy complement for μ(e) and interval-valued fuzzy complement for

, we have

where

()

Proposition 4.3. Let be a GIVFSS over (U, E). Then, the following holds:

()

Proof. Since , then

()

but, from Definition 4.1,

, then

()

Definition 4.4. Union of two GIVFSSs and , denoted by , is a GIVFSS where C = A ∪ B and is defined by

()

such that

and ν(e) = s(μ(e), δ(e)), where s is an s-norm and

Example 4.5. Consider GIVFSS and as in Example 3.4. By using the interval-valued fuzzy union and basic fuzzy union, we have , where

()

Similarly, we get

()

In matrix notation, we write

()

Proposition 4.6. Let , , and be any three GIVFSSs. Then, the following results hold.

(a)
.
(b)
.
(c)
.
(d)
.
(e)
.

Proof. (a) .

From Definition 4.4, we have such that and ν(e) = s(μ(e), δ(e)).

But (since union of interval-valued fuzzy sets is commutative) and ν(e) = s(μ(e), δ(e)) = s(δ(e), μ(e)) (since s-norm is commutative), then .

(b) 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 and , denoted by , is a GIVFSS where C = A ∪ B and is defined by

()

such that

and ν(e) = t(μ(e), δ(e)), where t is a t-norm and

Example 4.8. Consider GIVFSS and as in Example 4.5. By using the interval-valued fuzzy intersection and basic fuzzy intersection, we have , where

()

Similarly, we get

()

In matrix notation, we write

()

Proposition 4.9. Let , , and be any three GIVFSSs. Then, the following results hold.

(a)
.
(b)
.
(c)
.
(d)
.
(e)
.

Proof. (a) .

From Definition 4.7, we have such that and ν(e) = t(μ(e), δ(e)).

But (since intersection of interval-valued fuzzy sets is commutative) and ν(e) = t(μ(e), δ(e)) = t(δ(e), μ(e)) (since t-norm is commutative), then .

(b) 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 and be any two GIVFSSs. Then the DeMorgan’s Laws hold:

(a)
.
(b)
.

Proof. (a) Consider

()

(b) The proof is similar to the above.

Proposition 4.11. Let , , and be any three GIVFSSs. Then, the following results hold.

(a)
.
(b)
.

Proof. (a) For all x ∈ E,

()

(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 and are two GIVFSSs, then “() AND ” denoted by is defined by

()

where

for all (α, β) ∈ A × B, such that

and λ(α, β) = t(μ(α), δ(β)), for all (α, β) ∈ A × B, where t is a t-norm.

Example 5.2. Suppose the universe consists of three machines x₁, x₂, x₃, that is, U = {x₁, x₂, x₃}, 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 and by two experts A and B, respectively, defined as follows:

()

To find the AND between the two GIVFSSs, we have

AND

, where

()

Now, to determine the best machine, we first compute the numerical grade r_p∈P(x_i) for each p ∈ P such that

()

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 μ. The machine with the highest score is the desired machine. We do not consider the numerical grades of the machine against the pairs (e_i, e_i), i = 1,2, 3, as both the parameters are the same:

(x₁) = (1*0.1) + (1.1*0.1) = 0.21,
(x₂) = (1.3*0.1) + (1.3*0.2) = 0.39,
(x₃) = (0.3*0.3) + (0.3*0.2) = 0.15.

The firm will select the machine with the highest score. Hence, they will buy machine x₂.

Table 1. (

	x₁	x₂	x₃	μ
(e₁, e₁)	[0.1,0.3]	[0.2,0.6]	[0.3,0.5]	0.3
(e₁, e₂)	[0.1,0.3]	[0.4,0.6]	[0,0.3]	0.1
(e₁, e₃)	[0.1,0.3]	[0.4,0.7]	[0.2,0.3]	0.2
(e₂, e₁)	[0,0.3]	[0,0.2]	[0.1,0.3]	0.3
(e₂, e₂)	[0,0.3]	[0,0.2]	[0,0.3]	0.1
(e₂, e₃)	[0,0.3]	[0,0.2]	[0.1,0.3]	0.2
(e₃, e₁)	[0.3,0.5]	[0.1,0.1]	[0.1,0.3]	0.1
(e₃, e₂)	[0.3,0.5]	[0.1,0.1]	[0,0.3]	0.1
(e₃, e₃)	[0.1,0.6]	[0.1,0.1]	[0.1,0.3]	0.1

Table 2. Numerical grade r_p∈P(x_i).

	p₁	p₂	p₃	p₄	p₅	p₆	p₇	p₈	p₉
x₁	−0.8	−0.5	−0.8	0	0	0	(1)	(1.1)	(0.8)
x₂	(0.4)	(1.3)	(1.3)	−0.3	−0.2	−0.3	−0.8	−0.7	−0.7
x₃	(0.4)	−0.8	−0.5	(0.3)	(0.1)	(0.3)	−0.2	−0.4	−0.1
μ	0.3	0.1	0.2	0.3	0.1	0.2	0.1	0.1	0.1

Table 3. Grade table.

	p₁	p₂	p₃	p₄	p₅	p₆	p₇	p₈	p₉
x_i	x₂, x₃	x₂	x₂	x₃	x₃	x₃	x₁	x₁	x₁
Highest grade	—	1.3	1.3	0.3	—	0.3	1	1.1	—
μ	0.3	0.1	0.2	0.3	0.1	0.2	0.1	0.1	0.1

Definition 5.3. If and are two GIVFSSs, then “() OR ” denoted by is defined by

()

where

for all (α, β) ∈ A × B, such that

and λ(α, β) = s(μ(α), δ(β)), for all (α, β) ∈ A × B, where s is an s-norm.

Example 5.4. Consider Example 5.2. To find the OR between the two GIVFSSs, we have OR , where

()

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 and be any two GIVFSSs. Then, the following results hold:

(a)
,
(b)
.

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

Proposition 5.7. Let , , and be any three GIVFSSs. Then, the following results hold:

(a)
,
(b)
,
(c)
,
(d)
.

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 × B ≠ B × 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 and , denoted by , is defined by

()

where

()

Definition 6.2. Let and be two GIVFSSs over the same universe (U, E). We say that two GIVFSS are significantly similar if .

Theorem 6.3. Let , , and be any three GIVFSSs over (U, E). Then, the following hold:

(a)
in general ,
(b)
and ,
(c)
,
(d)
,
(e)
.

Proof. (a) The proof is straightforward and follows from Definition 6.1.

(b) From Definition 6.1, we have

()

If , for all i, then , and, if , for some i, then it is clear that .

Also since , suppose that and , then , that means, if and , then .

(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 be GIVFSS over (U, E) defined as follows:

()

Let

be another GIVFSS over (U, E) defined as follows:

()

Here,

()

Then,

()

Hence, the similarity between the two GIVFSSs

and

will be

()

Therefore,

and

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 “yes” and “no,” that is, U = {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 M_μ is given in Table 4, and this can be prepared with the help of a physician.

Table 4. Model GIVFSS for dengue fever.

M_μ	e₁	e₂	e₃	e₄	e₅	e₆	e₇
y	1	0	0	1	1	1	1
n	0	1	1	0	0	0	0
μ	1	1	1	1	1	1	1

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

Table 5. GIVFSS for the sick person.

F_α	e₁	e₂	e₃	e₄	e₅	e₆	e₇
y	[0.3,0.4]	[0.2,0.5]	[0,0.2]	1	[0.4,0.6]	0	[0.3,0.4]
n	[0.6,0.9]	[0.5,0.7]	[0.6,0.8]	[0.3,0.5]	1	[0.4,0.6]	[0.3,0.5]
δ	0.3	0.5	0.4	0.6	0.1	0.5	0.2

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 . 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, “AND,” and “OR” 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.

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Generalised Interval-Valued Fuzzy Soft Set

Abstract

1. Introduction

2. Preliminary

3. Generalised Interval-Valued Fuzzy Soft Set

4. Basic Operations on GIVFSS

5. AND and OR Operations on GIVFSS with Application

6. Similarity between Two GIVFSS

7. Application of Similarity Measure in Medical Diagnosis

8. Conclusion

Acknowledgment

References

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Generalised Interval-Valued Fuzzy Soft Set

Abstract

1. Introduction

2. Preliminary

3. Generalised Interval-Valued Fuzzy Soft Set

4. Basic Operations on GIVFSS

5. AND and OR Operations on GIVFSS with Application

6. Similarity between Two GIVFSS

7. Application of Similarity Measure in Medical Diagnosis

8. Conclusion

Acknowledgment

References

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