In 1999 Molodtsov introduced the concept of soft set theory as a general mathematical tool for dealing with uncertainty. Alkhazaleh et al. in 2011 introduced the definition of a soft multiset as a generalization of Molodtsov's soft set. In this paper we give the definition of fuzzy soft multiset as a combination of soft multiset and fuzzy set and study its properties and operations. We give examples for these concepts. Basic properties of the operations are also given. An application of this theory in decision-making problems is shown.

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

In this section, we recall some basic notions in soft set theory, soft multiset theory, and fuzzy soft set. Molodtsov defined soft set in the following way. Let

A pair

Let

All the following definitions are due to Alkhazaleh and Salleh [

Let

In other words, a soft multiset over

For any soft multiset

For two soft multisets

This relationship is denoted by

Two soft multisets

Let

The complement of a soft multiset

A soft multiset

A soft multiset

A soft multiset

A soft multiset

The union of two soft multisets

The intersection of two soft multisets

In this section, we introduce the definition of a fuzzy soft multiset, and its basic operations such as complement, union, and intersection. We give examples for these concepts. Basic properties of the operations are also given.

Let

In other words, a fuzzy soft multiset over

Suppose that there are three universes

Let

We can logically explain the above example as follows: we know that

If

For any fuzzy soft multiset

Consider Example

For two fuzzy soft multisets

This relationship is denoted by

Two fuzzy soft multisets

Consider Example

The complement of a fuzzy soft multiset

Consider Example

A fuzzy soft multiset

Consider Example

A fuzzy soft multiset

Consider Example

A fuzzy soft multiset

Consider Example

A fuzzy soft multiset

Consider Example

If

The proof is straightforward.

In this section we define the operation of union and intersection and give some examples by using the basic fuzzy union and intersection.

The union of two fuzzy soft multisets

Consider Example

If

The proof is straightforward.

The intersection of two fuzzy soft multisets

Consider Example

If

The proof is straightforward.

We begin this section with a novel algorithm designed for solving fuzzy soft set-based decision-making problems, which was presented in [

Roy and Maji [

Input the fuzzy soft sets

Input the parameter set

Compute the corresponding resultant fuzzy soft set

Construct the comparison table of the fuzzy soft set

Compute the score of

The decision is

If

In this section we suggest the following algorithm to solve fuzzy soft multisets-based decision-making problem, which is a generalization of the algorithm given by Salleh and Alkhazaleh in [

Input the fuzzy soft multiset

Apply RMA to the first fuzzy soft multiset part in

Redefine the fuzzy soft multiset

Apply RMA to the second fuzzy soft multiset part in

Redefine the fuzzy soft multiset

Apply RMA to the third fuzzy soft multiset part in

The decision is

Let

Let

By using the basic fuzzy union we have

Tabular representation:

0.3 | 0.8 | 1 | 0.8 | 0.4 | 0.9 | 1 | 0.8 | |

0.4 | 0.9 | 0.8 | 0.6 | 0.6 | 0.6 | 0.9 | 0.7 | |

0.9 | 0.3 | 0.7 | 0.1 | 0.7 | 0.7 | 0.8 | 1 | |

0.7 | 0.8 | 0 | 0.5 | 0.7 | 0.5 | 0.4 | 0.9 |

The comparison table for the first resultant fuzzy soft multiset part will be as in Table

Comparison table:

8 | 5 | 5 | 5 | |

3 | 8 | 4 | 5 | |

3 | 4 | 8 | 6 | |

3 | 3 | 3 | 8 |

Next we compute the row-sum, column-sum, and the score for each

Score table:

Row-sum | Column-sum | Score | |
---|---|---|---|

23 | 17 | 6 | |

20 | 20 | 0 | |

21 | 20 | 1 | |

17 | 24 | −7 |

From Table

Now we redefine the fuzzy soft multiset

Now we apply RMA to the second fuzzy soft multiset part in

Tabular representation:

0 | 0 | 0.8 | 0.5 | 0 | 0.8 | 0.8 | 0 | |

0 | 0 | 0.6 | 0.3 | 0 | 0.8 | 0.8 | 0 | |

0 | 0 | 0.3 | 0.1 | 0 | 0.5 | 0.5 | 0 |

The comparison table for the second resultant fuzzy soft multiset part of

Comparison table:

8 | 8 | 8 | |

6 | 8 | 8 | |

4 | 4 | 8 |

Next we compute the row-sum, column-sum, and the score for each

Score table:

Row-sum | Column-sum | Score | |
---|---|---|---|

24 | 18 | 6 | |

22 | 20 | 2 | |

16 | 24 | −8 |

From Table

Now we redefine the fuzzy soft multiset

Now we apply RMA to the third fuzzy soft multiset part in

Tabular representation:

0 | 0 | 0.5 | 0.5 | 0 | 0.8 | 0.5 | 0 | |

0 | 0 | 0.5 | 0.3 | 0 | 0.8 | 0.6 | 0 | |

0 | 0 | 0.7 | 0.4 | 0 | 1 | 0.8 | 0 |

The comparison table for the second resultant fuzzy soft multiset part of

Comparison table:

8 | 7 | 5 | |

7 | 8 | 4 | |

7 | 8 | 8 |

Next we compute the row-sum, column-sum, and the score for each

Score table:

Row-sum | Column-sum | Score | |
---|---|---|---|

20 | 22 | −2 | |

19 | 23 | −2 | |

23 | 17 | 6 |

From Table

Then from the above results the decision for Mr. X is

In this paper we have introduced the concept of fuzzy soft multiset and studied some of its properties. The operations complement, union, and intersection have been defined on the fuzzy soft multisets. An application of this theory is given in solving a decision-making problem.

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.