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In 1999, Molodtsov introduced the concept of soft set theory as a general mathematical tool for dealing with uncertainty. By combining the multi-fuzzy set and soft set models, Y. Yang, X. Tan and C. Meng introduced the concept of multi-fuzzy soft sets and studied some of its operations, such as complement, “AND”, “OR”, Union and Intersection. They also gave an algorithm to analyze a decision problem using multi-fuzzy soft set. In this paper, we introduce the concept of multi-interval-valued fuzzy soft set (M-IVFSS). We also define its basic operations, namely complement, union, intersection, AND and OR. Finally, we give an application of this concept in decision-making problem.

Most of the problems in engineering, medical science, economics, environments etc. have various uncertainties. Molodtsov [

The concept of soft fuzzy set and some properties of soft fuzzy set are discussed in 2008 by Yao et al. [

Chaudhuri and K. De in 2009 [

Also in 2010 Xiao et al. [

Alkhazaleh et al. [

In this section we recall some definitions and properties required in this paper.

Definition 1 [

where

Suppose that

x to X.

Definition 2 [

as follows: Let

1) The complement of

2) The intersection of

3) The union of

4) X is a subset of Y denoted by

Molodtsov defined soft set in the following way. Let U be a universe and E be a set of parameters. Let

Definition 3 [

In other words, a soft set over U is a parameterized family of subsets of the universe U. For

Definition 4 [

Definition 5 [

Definition 6 [

where

tion of multi-fuzzy set

Remark 7 [

Remark 8 [

fuzzy set. If

changed into a normalized multi-fuzzy set.

Definition 9 [

set of dimension k, denoted by

dimension k, denoted by

Definition 10 [

ing relations and operations:

1)

2)

3)

4)

5)

Definition 11 [

A multi-fuzzy soft set is a mapping from parameters to

Example 12 [

Definition 13 [

1)

2)

In this case, We write

Definition 14 [

where

Definition 15 [

AND

Definition 16 [

OR

Definition 17 [

Definition 18 [

In this section we introduce the concept of multi-interval-valued-fuzzy soft sets as generalisation of definition given by [

Before we define the concept of multi-interval-valued-fuzzy soft sets, we define the concept of multi-interval- valued-fuzzy sets as follows:

Definition 19 Let k be a positive integer, a multi-intrval-valued fuzzy set

where

Definition 20 A pair

A multi-interval-valued-fuzzy soft set is a mapping from parameters to

family of multi-interval-valued-fuzzy subsets of U. For

ximate elements of the multi-interval-valued fuzzy soft set

Example 21 Suppose that

Definition 22 Let

1)

2)

In this case, We write

Example 23 Consider Example 21 where

and

It is clear that

Definition 24 The complement of a multi-interval-valued-fuzzy soft set

denoted by

where

Example 25 Consider Example 21 where

By using interval-valued fuzzy complement for

Proposition 26 Let

Proof.

Let

But from Definition 24

Definition 27 Union of two multi-interval-valued-fuzzy soft sets

is the multi-interval-valued-fuzzy soft set

Example 28 Consider Example 21 where

and

By using the interval-valued fuzzy union we have

Proposition 29 Let

1)

2)

3)

Proof.

1)

From Definition 27 and by consider the case when

2) The proof is straightforward from Definition 27.

3) The proof is straightforward from Definition 27.

Definition 30 Intersection of two multi-interval-valued-fuzzy soft sets

Example 31 Consider Example 28. By using the interval-valued fuzzy intersection we have

Proposition 32 Let

1)

2)

3)

Proof.

1)

From Definition 30 and by consider the case when

2) The proof is straightforward from Definition 30.

3) The proof is straightforward from Definition 30.

Proposition 33 Let

1)

2)

Proof.

1)

2) The proof is similar to the above progress.

Proposition 34 Let

1)

2)

Proof. a) For all

b) Similar to the proof of a.

Definition 35 If

“

Example 36 Consider Example 21. By using the interval-valued fuzzy union we have

Definition 37 If

Example 38 Consider Example 21. By using the interval-valued fuzzy intersection we have

Proposition 39 Let

1)

2)

Proof.

a) Suppose that

Therefore,

where

Now, take

Therefore,

Then

b) Similar to the proof of a.

In this section, we define an aggregate interval-valued fuzzy set of an MIVFS-set. We also define MIVFS- aggregation operator that produces an aggregate interval-valued fuzzy set from an MIVFS-set and its parameter set. Also we give an application of this operator in decision making problem.

Definition 40 Let

where

Is an interval-valued fuzzy set over U. The value

where

In the following example, we present an application of MIVFS-aggregation operator to solve a decision making problem.

Example 41

Step 1 Let the constructed MIVFS-set,

Step 2 The aggregate interval-valued fuzzy set can be found as

Step 3

Thus, we have

Step 4 The decision is any one of the elements in S where

As a generalisation of multi-fuzzy soft set and by combining this concept and interval-value fuzzy set, the concept of the multi-interval-valued fuzzy soft set is introduced and some of its properties studied. The com- plement, union and intersection, operations have been defined on the multi-interval-valued fuzzy soft set. An application of this theory is given in solving a decision making problem. We hope that our work would help enhancing this study on multi-fuzzy soft sets for the researchers.

We thank the Editor and the referee for their comments. Research of S. Alkhazaleh is funded by Shaqra University, Saudi Arabia. This support is greatly appreciated.

ShawkatAlkhazaleh, (2015) The Multi-Interval-Valued Fuzzy Soft Set with Application in Decision Making. Applied Mathematics,06,1250-1262. doi: 10.4236/am.2015.68118