Most of the problems in engineering, medical science, economics, environments etc. have various uncertainties. Molodtsov [1] initiated the concept of soft set theory as a mathematical tool for dealing with uncertainties. Chen et al. [2] and Maji et al. in [3] and [4] studied some different operations and application of soft sets. Furthermore, Maji et al. [5] presented the definition of fuzzy soft set and Roy et al. [6] presented the applications of this notion to decision making problems. By using fuzzy sets, Ahmad and Kharal in 2009 [7] studied this theory and defined arbitrary fuzzy soft union and fuzzy soft intersection and proved DeMorgan’s Inclusions and DeMorgan’s Laws in fuzzy soft set theory. In 2010, Feng et al. [8] gave deeper insights into decision making based on fuzzy soft sets. They discussed the validity of the Roy-Maji method and showed its true limitations. By means of level soft sets, they presented an adjustable approach to fuzzy soft set based decision making and gave some illustra- tive examples. Moreover, the weighted fuzzy soft set is introduced and its application to decision making is also investigated.

The concept of soft fuzzy set and some properties of soft fuzzy set are discussed in 2008 by Yao et al. [9] and the relations of fuzzy soft sets and soft fuzzy sets are compared by instances. Kharal and Ahmad in 2009 [10] defined the concept of a mapping on classes of fuzzy soft sets and studied the properties of fuzzy soft images and fuzzy soft inverse images of fuzzy soft sets, and supported them with examples and counterexamples.

Chaudhuri and K. De in 2009 [11] defined the concepts of soft relation and fuzzy soft relation and applied them to solve a number of decision making problems. The advantages of fuzzy soft relation compared to other paradigms are discussed. Jiang et al. [12] presented in 2011 an extended fuzzy soft set theory by using the concepts of fuzzy description logics to act as the parameters of fuzzy soft sets. They also defined some opera- tions for the extended fuzzy soft sets. Moreover, they proved that certain DeMorgan’s laws hold in the extended fuzzy soft set theory with respect to these operations. In 2010, Majumdar and Samanta [13] defined generalised fuzzy soft sets and studied some of their properties. They also gave applications of generalised fuzzy soft sets in decision making problem and medical diagnosis problem.

Also in 2010 Xiao et al. [14] proposed a combined forecasting approach based on fuzzy soft sets (CFFSS) by using an export dataset of Chongqing Municipality China from 1993 to 2006 and compares a CFFSS with the combined forecasting approach based on the rough sets theory (CFRS). This approach constructed the fuzzy membership function and the tabular form of the fuzzy soft sets model. Çaman et al. in 2011 [15] introduced fuzzy parameterized (FP)-soft sets and their related properties and proposed a decision making method based on FP-soft set theory. Çaman et al. in 2010 [16] defined fuzzy parameterized fuzzy soft sets (fpfs-sets) and their operations. They then presented the decision making method on the fpfs-set theory and provided an example that demonstrated that this method can work successfully. It can be applied to problems on many fields that contain uncertainty.

Alkhazaleh et al. [17] generalized the concept of fuzzy soft set to possibility fuzzy soft set and they gave some applications of this concept in decision making and medical diagnosis. They also introduced the concept of fuzzy parameterized interval-valued fuzzy soft set [18] , where the mapping is defined from the fuzzy set parameters to the interval-valued fuzzy subsets of the universal set, and gave an application of this concept in decision making. Alkhazaleh and Salleh [19] introduced the concept of soft expert sets where the user can know the opinion of all experts in one model and also gave an application of this concept in decision-making problem. Alkhazaleh and Salleh [20] generalized the concept of a soft expert set to fuzzy soft expert set, which is a more effective and useful. They also defined its basic operations, namely complement, union, intersection, AND and OR, and gave an application of this concept in decision-making problem. They also studied a mapping on fuzzy soft expert classes and its properties. As a generalization of Molodtsov’s soft set, Alkhazaleh et al. [21] presented the definition of a soft multiset and its basic operations such as complement, union and intersection. In 2012 Alkhazaleh and Salleh [22] introduced the concept of fuzzy soft multiset as a combination of soft multiset and fuzzy set and studied its properties and operations. They presented the applications of this concept to decision- making problems. In 2012 Salleh et al. [23] introduced the notion of multiparameterized soft set and studied its properties. Yang et al. [24] presented the concept of interval-valued fuzzy soft set by combining the interval- valued fuzzy set [25] [26] and soft set models. In 2011 Sebastian and Ramakrishnan [27] proposed the concept of the multi-fuzzy set which is a more general fuzzy set using ordinary fuzzy sets as building blocks, its mem- bership function is an ordered sequence of ordinary fuzzy membership functions. The notion of multi-fuzzy sets provides a new method to represent some problems which are difficult to explain in other extensions of fuzzy set theory, such as the color of pixels. Yang et al. [28] in 2012 introduced the concept of multi-fuzzy soft set which is a combination of multi-fuzzy set and soft set and studied and its basic operations such as complement, union and intersection. They also introduced the application of this concept in decision making. For more information on soft set and fuzzy soft set, see Abdul Razak Salleh 2011 [29] . 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.

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

Definition 1 [26] An interval-valued fuzzy set on a universe U is a mapping such that

where stands for the set of all closed subintervals of, the set of all interval-valued fuzzy sets on U is denoted by.

Suppose that 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 [25] The subset, complement, intersection and union of the interval-valued fuzzy sets are defined

as follows: Let then

1) The complement of is denoted by where

2) The intersection of and is denoted by where

3) The union of and is denoted by where

4) X is a subset of Y denoted by if and

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

Definition 3 [1] A pair is called a soft set over U, where F is a mapping

In other words, a soft set over U is a parameterized family of subsets of the universe U. For may be considered as the set of -approximate elements of the soft set.

Definition 4 [5] Let U be an initial universal set and let E be a set of parameters. Let denote the power set of all fuzzy subsets of U. Let A pair is called a fuzzy soft set over U where F is a mapping given by

Definition 5 [24] Let U be an initial universe and E be a set of parameters. denotes the set of all interval-valued fuzzy sets of U. Let. A pair is an interval-valued fuzzy soft set over U, where

is a mapping given by.

Definition 6 [27] Let k be a positive integer, a multi-fuzzy set in U is a set of ordered sequences

where. The function is called the multi-membership func-

tion of multi-fuzzy set; k is called the dimension of. The set of all multi-fuzzy sets of dimension k in U is denoted by.

Remark 7 [27] Clearly, a multi-fuzzy set of dimension 1 is a Zadeh fuzzy set, and a multi-fuzzy set of dimension 2 with is an Atanassov intuitionistic fuzzy set.

Remark 8 [27] If, then the multi-fuzzy set of dimension k is called a normalized multi-

fuzzy set. If for some, we redefine the multi-membership degree

as then the non-normalized multi-fuzzy set can be

changed into a normalized multi-fuzzy set.

Definition 9 [27] Let. If, then is called the null multi-fuzzy

set of dimension k, denoted by. If, then is called the absolute multi-fuzzy set of

dimension k, denoted by.

Definition 10 [27] Let and

be two multi-fuzzy sets of dimension k in U. We define the follow-

ing relations and operations:

1) iff and.

2) iff and.

3)

4)

5)

Definition 11 [28] A pair is called a multi-fuzzy soft set of dimension k over U, where is a mapping given by

A multi-fuzzy soft set is a mapping from parameters to. It is a parameterized family of multi- fuzzy subsets of U. For may be considered as the set of e-approximate elements of the multi- fuzzy soft set.

Example 12 [28] Suppose that is the set of color cloths under consideration, is the set of parameters, where stands for the parameter? color? which consists of red, green and blue, stands for the parameter? ingredient? which is made from wool, cotton and acrylic, and stands for the parameter? price? which can be various: high, medium and low. We define a multi-fuzzy soft set of dimension 3 as follows:

;

;

.

Definition 13 [28] Let. Let and be two multi-fuzzy soft sets of dimension k over U. is said to be a multi-fuzzy soft subset of if

1);

2)

In this case, We write.

Definition 14 [28] The complement of a multi-fuzzy soft set of dimension k over U is denoted by and is defined by

where is a mapping given by.

Definition 15 [28] If and are two multi-fuzzy soft sets of dimension k over U the “

AND”, denoted by, where .

Definition 16 [28] If and are two multi-fuzzy soft sets of dimension k over U the “

OR”, denoted by, where,.

Definition 17 [28] Union of two multi-fuzzy soft sets and of dimension k over U, is the multi-fuzzy soft set where and

Definition 18 [28] Intersection of two multi-fuzzy soft sets and of dimension k over U, is the multi-fuzzy soft set where and

In this section we introduce the concept of multi-interval-valued-fuzzy soft sets as generalisation of definition given by [28] . We also give basic properties of this concept.

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 in U is a set of ordered sequences

where. The function is called the multi-membership func- tion of multi-interval-valued-fuzzy set; k is called the dimension of. The set of all multi-interval- valued-fuzzy sets of dimension k in U is denoted by.

Definition 20 A pair is called a multi-interval-valued-fuzzy soft set of dimension k over U, where is a mapping given by

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

family of multi-interval-valued-fuzzy subsets of U. For may be considered as the set of e-appro-

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

Example 21 Suppose that is the set of color cloths under consideration, is the set of parameters, where stands for the parameter “color” which consists of red, green and blue, stands for the parameter “ingredient” which is made from wool, cotton and acrylic, and stands for the parameter “price” which can be various: high, medium and low. We define a multi-interval-valued-fuzzy soft set of dimension 3 as follows:

;

;

.

Definition 22 Let. Let and be two multi-interval-valued-fuzzy soft sets of dimen- sion k over U. is said to be a multi-interval-valued-fuzzy soft subset of if

1);

2)

In this case, We write.

Example 23 Consider Example 21 where and. Clearly,. Suppose and are two multi-interval-valued-fuzzy soft sets of dimension 3 defined as follows

;

;

and

;

;

.

It is clear that is multi-interval-valued-fuzzy soft subset of.

Definition 24 The complement of a multi-interval-valued-fuzzy soft set of dimension k over U is

denoted by and is defined by

where is a mapping given by.

Example 25 Consider Example 21 where

;

;

.

By using interval-valued fuzzy complement for we have

;

;

.

Proposition 26 Let be a MIVFSS of dimension k over U. Then the following holds:

Proof.

Let then

But from Definition 24 then

Definition 27 Union of two multi-interval-valued-fuzzy soft sets and of dimension k over U,

is the multi-interval-valued-fuzzy soft set where and

Example 28 Consider Example 21 where and. Suppose and are two multi-interval-valued-fuzzy soft sets of dimension 3 defined as follows

;

;

;

and

;

;

;

By using the interval-valued fuzzy union we have, where

;

;

.

Proposition 29 Let and be any three MIVFSSs of dimension k over U. Then the following results hold:

1).

2)

3)

Proof.

1)

From Definition 27 and by consider the case when as the other cases are trivial, we have (since the union of interval-valued fuzzy sets is commutative).

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 and of dimension k over U, is the multi-interval-valued-fuzzy soft set where and

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

, where

;

;

.

Proposition 32 Let and be any three MIVFSSs of dimension k over U. Then the following results hold:

1).

2)

3)

Proof.

1)

From Definition 30 and by consider the case when as the other cases are trivial, we have

(since intersection of interval-valued fuzzy sets is commutative).

2) The proof is straightforward from Definition 30.

3) The proof is straightforward from Definition 30.

Proposition 33 Let and be any two MIVFSSs of dimension k over U. Then the DeMorgan’s Laws hold:

1).

2).

Proof.

1).

2) The proof is similar to the above progress.

Proposition 34 Let and be any three MIVFSSs of dimension k over U. Then the following results hold:

1)

2)

Proof. a) For all,

b) Similar to the proof of a.

Definition 35 If and are two multi-interval-valued-fuzzy soft sets of dimension k over U the

“AND”, denoted by, where,

.

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

;

;

;

;

;

;

;

;

.

Definition 37 If and are two multi-fuzzy soft sets of dimension k over U the “OR

”, denoted by, where.

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

;

;

;

;

;

;

;

;

.

Proposition 39 Let and are two MIVFSSs of dimension k over U. Then the following results hold:

1),

2),

Proof.

a) Suppose that

Therefore, Now,

where

Now, take

Therefore,

Then and are the same. Hence, proved.

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. Then a MIVFS-aggregation operator, denoted by, is defined by

where

Is an interval-valued fuzzy set over U. The value is called an aggregate interval-valued fuzzy set of

. Here, the membership degree of u is defined as follows:

where is the cardinality of A.

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, , be given as follows:

;

;

.

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

Step 3, compute the score of such that

Thus, we have

Step 4 The decision is any one of the elements in S where. In our example, is the best choice because.

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