Abstract
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.
1. Introduction
Most of the problems in engineering, medical science, economics, environments, and so forth have various uncertainties. Molodtsov [1] initiated the concept of soft set theory as a mathematical tool for dealing with uncertainties. After Molodtsov’s work, some different operations and application of soft sets were studied by Chen et al. [2] and Maji et al. [3, 4]. Furthermore, Maji et al. [5] presented the definition of fuzzy soft set and Roy and Maji [6] presented the applications of this notion to decision-making problems. Alkhazaleh et al. [7] 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 [8], 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. In 2012 Alkhazaleh and Salleh [9] introduced the concept of generalised interval-valued fuzzy soft set and studied its properties and application. Alkhazaleh and Salleh [10] introduced the concept of soft expert sets where the user can know the opinion of all experts in one model and gave an application of this concept in decision-making problem. Salleh et al. in 2012 [11] introduced and studied the concept of multiparameterized soft set as a generalization of Molodtsov’s soft set. Alkhazaleh et al. [12] as a generalization of Molodtsov’s soft set, presented the definition of a soft multiset and its basic operations such as complement, union, and intersection. Salleh and Alkhazaleh [13] studied the application of soft multiset in decision making problem. In 2011 Salleh gave a brief survey from soft sets to intuitionistic fuzzy soft sets [14]. In this paper we give the definition of a fuzzy soft multiset, a more general concept, which is a combination of fuzzy set and soft multiset and studied its properties. We also introduce its basic operations, namely, complement, union, and intersection, and their properties. An application of this theory in a decision-making problem is given.
2. Preliminaries
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 be a universe and a set of parameters. Let denote the power set of and .
Definition 2.1 (see [1]). A pair is called a soft set over , where is a mapping In other words, a soft set over is a parameterized family of subsets of the universe . For may be considered as the set of -approximate elements of the soft set .
Definition 2.2 (see [5]). Let be an initial universal set and let be a set of parameters. Let denote the power set of all fuzzy subsets of . Let . A pair is called a fuzzy soft set over , where is a mapping given by
All the following definitions are due to Alkhazaleh and Salleh [12].
Definition 2.3. Let be a collection of universes such that and let be a collection of sets of parameters. Let where denotes the power set of , and . A pair is called a soft multiset over , where is a mapping given by .
In other words, a soft multiset over is a parameterized family of subsets of . For may be considered as the set of -approximate elements of the soft multiset . Based on the above definition, any change in the order of universes will produce a different soft multiset.
Definition 2.4. For any soft multiset , a pair is called a -soft multiset part and is an approximate value set, where and.
Definition 2.5. For two soft multisets and over , is called a soft multisubset of if(1)and(2), where , and .
This relationship is denoted by . In this case is called a soft multisuperset of .
Definition 2.6. Two soft multisets and over are said to be equal if is a soft multisubset of and is a soft multisubset of .
Definition 2.7. Let , where is a set of parameters. The NOT set of denoted by is defined by where .
Definition 2.8. The complement of a soft multiset is denoted by and is defined by = , where is a mapping given by , .
Definition 2.9. A soft multiset over is called a seminull soft multiset, denoted by , if at least one of the soft multiset parts of equals .
Definition 2.10. A soft multiset over is called a null soft multiset, denoted by , if all the soft multiset parts of equal .
Definition 2.11. A soft multiset over is called a semiabsolute soft multiset, denoted by if for at least one , ,, and.
Definition 2.12. A soft multiset over is called an absolute soft multiset, denoted by , if , .
Definition 2.13. The union of two soft multisets and over , denoted by , is the soft multiset where , and ,
Definition 2.14. The intersection of two soft multisets and over , denoted by , is the soft multiset where , and ,
3. Fuzzy Soft Multiset
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.
Definition 3.1. Let be a collection of universes such that and let be a collection of sets of parameters. Let where denotes the set of all fuzzy subsets of , and . A pair is called a fuzzy soft multiset over , where is a mapping given by .
In other words, a fuzzy soft multiset over is a parameterized family of fuzzy subsets of . For may be considered as the set of -approximate elements of the fuzzy soft multiset . Based on the above definition, any change in the order of universes will produce a different fuzzy soft multiset.
Example 3.2. Suppose that there are three universes , , and . Suppose that Mr. X has a budget to buy a house, a car and rent a venue to hold a wedding celebration. Let us consider a fuzzy soft multiset which describes “houses,” “cars,” and “hotels” that Mr. X is considering for accommodation purchase, transportation purchase, and a venue to hold a wedding celebration, respectively. Let = , = and = .
Let be a collection of sets of decision parameters related to the above universes, where
Let and , such that
Suppose that
Then we can view the fuzzy soft multiset as consisting of the following collection of approximations:
Each approximation has two parts: a predicate and an approximate value set.
We can logically explain the above example as follows: we know that where expensive house, expensive car, and expensive venue. Then, We can see that the membership value for house is 0.2, so this house is not expensive for Mr. X; also we can see that the membership value for house is 0.8, this means that the house is expensive, and since the membership value for house is 0, then this house is absolutely not expensive. Now, since the first set is concerning expensive houses, then we can explain the second set as follows: the membership value for car is 0.8, so this car is expensive (this car maybe not expensive if the first set is concerning cheap houses), also we can see that the membership value for car is 0.4, this means that this car is not so expensive for him, and since the membership value for car is 0.6, then this car is quite expensive. Now, since the first set is concerning expensive houses and the second set is concerning expensive cars, then we can also explain the third set as follows: since the membership value for venue is 0.8, so this venue is expensive (this venue maybe not expensive if the first set is concerning cheap houses or/and the second set is concerning cheap cars), also we can see that the membership value for venue and is 0.7, this means that this venue is almost expensive. So depending on the previous explanation we can say the following.
If is the fuzzy set of expensive houses, then the fuzzy set of relatively expensive cars is , and if is the fuzzy set of expensive houses and is the fuzzy set of relatively expensive cars, then the fuzzy set of relatively expensive hotels is . It is clear that the relation in fuzzy soft multiset is a conditional relation.
Definition 3.3. For any fuzzy soft multiset , a pair is called a -fuzzy soft multiset part and is a fuzzy approximate value set, where , = , , and.
Example 3.4. Consider Example 3.2. Then is a -fuzzy soft multiset part of .
Definition 3.5. For two fuzzy soft multisets and over , is called a fuzzy soft multisubset of if (a) and(b) is a fuzzy subset of ,where and.
This relationship is denoted by . In this case is called a fuzzy soft multisuperset of .
Definition 3.6. Two fuzzy soft multisets and over are said to be equal if is a fuzzy soft multisubset of and is a fuzzy soft multisubset of .
Example 3.7. Consider Example 3.2. Let Clearly . Let and be two fuzzy soft multisets over the same such that Therefore, .
Definition 3.8. The complement of a fuzzy soft multiset is denoted by and is defined by = , where is a mapping given by where is any fuzzy complement.
Example 3.9. Consider Example 3.2. By using the basic fuzzy complement which is , we have
Definition 3.10. A fuzzy soft multiset over is called a seminull fuzzy soft multiset, denoted by , if at least one of a fuzzy soft multiset parts of equals .
Example 3.11. Consider Example 3.2. Let us consider a fuzzy soft multiset which describes “stone houses,” “cars,” and “hotels” with Then a seminull fuzzy soft multiset is given as
Definition 3.12. A fuzzy soft multiset over is called a null fuzzy soft multiset, denoted by , if all the fuzzy soft multiset parts of equal .
Example 3.13. Consider Example 3.2. Let us consider a fuzzy soft multiset which describes “stone houses,” “very cheap classic cars,” and “hotels in Kajang” with Then a null fuzzy soft multiset is given as
Definition 3.14. A fuzzy soft multiset over is called a semi-absolute fuzzy soft multiset, denoted by , if for at least one , and.
Example 3.15. Consider Example 3.2. Let us consider a fuzzy soft multiset which describes “wooden houses,” “cars,” and “hotels” with Then a semi-absolute fuzzy soft multiset is given as
Definition 3.16. A fuzzy soft multiset over is called an absolute fuzzy soft multiset, denoted by , if , .
Example 3.17. Consider Example 3.2. Let us consider a fuzzy soft multiset which describes “wooden houses,” “expensive classic cars,” and “hotels in Kuala Lumpur” with Then an absolute fuzzy soft multiset is given as
Proposition 3.18. If is a fuzzy soft multiset over , then (a), (b), (c), (d), (e).
Proof. The proof is straightforward.
4. Union and Intersection
In this section we define the operation of union and intersection and give some examples by using the basic fuzzy union and intersection.
Definition 4.1. The union of two fuzzy soft multisets and over , denoted by , is the fuzzy soft multiset , where , and , where with s as an s-norm.
Example 4.2. Consider Example 3.2. Let Suppose and are two fuzzy soft multisets over the same such that By using the basic fuzzy union (maximum) we have where
Proposition 4.3. If , , and are three fuzzy soft multisets over , then (a), (b), (c), where is defined by (4.1)(d), (e), where and is defined by (4.1)(f) where ,(g), (h), (i) where ,(j) where .
Proof. The proof is straightforward.
Definition 4.4. The intersection of two fuzzy soft multisets and over , denoted by , is the fuzzy soft multiset , where , and , where , with as a -norm.
Example 4.5. Consider Example 4.2. By using the basic fuzzy intersection (minimum) we have where
Proposition 4.6. If , , and are three fuzzy soft multisets over , then (a), (b), (c), where is defined by (4.6)(d), where is defined by (4.6)(e) where ,(f) where (g), where and is defined by (4.6)(h), (i), where and is defined by (4.6)(j) where and is defined by 4.6.
Proof. The proof is straightforward.
5. Fuzzy Soft Set-Based Decision Making
We begin this section with a novel algorithm designed for solving fuzzy soft set-based decision-making problems, which was presented in [6].
5.1. Roy and Maji’s Original Algorithm Using Scores
Roy and Maji [6] used the following algorithm to solve a decision-making problem. (a)Input the fuzzy soft sets , and .(b)Input the parameter set as observed by the observer.(c)Compute the corresponding resultant fuzzy soft set from the fuzzy soft sets and and place it in tabular form.(d)Construct the comparison table of the fuzzy soft set and compute and for .(e)Compute the score of .(f)The decision is if .(g)If has more than one value, then any one of may be chosen.
5.2. A Fuzzy Soft Multiset Theoretic Approach to Decision-Making Problem
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 [13]. We note here that we will use the abbreviation RMA for Roy and Maji’s a Algorithm.(a)Input the fuzzy soft multiset which is introduced by making any operations between and.(b)Apply RMA to the first fuzzy soft multiset part in to get the decision .(c) Redefine the fuzzy soft multiset by keeping all values in each row where is maximum and replacing the values in the other rows by zero, to get .(d)Apply RMA to the second fuzzy soft multiset part in to get the decision .(e)Redefine the fuzzy soft multiset by keeping the first and second parts and apply the method in step (c) to the third part.(f)Apply RMA to the third fuzzy soft multiset part in to get the decision .(g)The decision is .
5.3. Application in a Decision-Making Problem
Let , , and be the sets of “houses,” “cars,” and “hotels”, respectively. Let be a collection of sets of decision parameters related to the above universes, where
Let Suppose Mr. X wants to choose objects from the sets of given objects with respect to the sets of choice parameters. Let there be two observations and by two experts and , respectively. Let
By using the basic fuzzy union we have Now we apply RMA to the first fuzzy soft multiset part in to take the decision from the availability set . The tabular representation of the first resultant fuzzy soft multiset part will be as in Table 1.
The comparison table for the first resultant fuzzy soft multiset part will be as in Table 2.
Next we compute the row-sum, column-sum, and the score for each as shown in Table 3.
From Table 3, it is clear that the maximum score is 6, scored by .
Now we redefine the fuzzy soft multiset by keeping all values in each row where is maximum and replacing the values in the other rows by zero:
Now we apply RMA to the second fuzzy soft multiset part in to take the decision from the availability set . The tabular representation of the second resultant fuzzy soft multiset part of will be as in Table 4.
The comparison table for the second resultant fuzzy soft multiset part of is as in Table 5.
Next we compute the row-sum, column-sum, and the score for each is shown in Table 6.
From Table 6, it is clear that the maximum score is 6, scored by .
Now we redefine the fuzzy soft multiset by keeping all values in each row where is maximum and replacing the values in the other rows by zero:
Now we apply RMA to the third fuzzy soft multiset part in to take the decision from the availability set . The tabular representation of the third resultant fuzzy soft multiset part of is as in Table 7.
The comparison table for the second resultant fuzzy soft multiset part of is as in Table 8.
Next we compute the row-sum, column-sum, and the score for each as shown in Table 9.
From Table 9, it is clear that the maximum score is 6, scored by .
Then from the above results the decision for Mr. X is .
6. Conclusion
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.
Acknowledgments
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.