Advances in Decision SciencesVolume 2011 (2011), Article ID 479756, 18 pagesdoi:10.1155/2011/479756

Research Article

Possibility Fuzzy Soft Set

Shawkat Alkhazaleh, Abdul Razak Salleh, and Nasruddin Hassan

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor Darul Ehsan, Malaysia

Received 4 January 2011; Revised 14 May 2011; Accepted 30 May 2011

Academic Editor: C. D. Lai

Copyright © 2011 Shawkat Alkhazaleh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We introduce the concept of possibility fuzzy soft set and its operation and study some of its properties. We give applications of this theory in solving a decision-making problem. We also introduce a similarity measure of two possibility fuzzy soft sets and discuss their application in a medical diagnosis problem.

1. Introduction

Fuzzy set was introduced by Zadeh in [1] as a mathematical way to represent and deal with vagueness in everyday life. After that many authors have studied the applications of fuzzy sets in different areas (see Klir and Yuan [2]). Molodtsov [3] initiated the theory of soft sets 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. Maji et al. [4, 5] have further studied the theory of soft sets and used this theory to solve some decision-making problems. They 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 [6], and also Roy and Maji used this theory to solve some decision-making problems [7]. Alkhazaleh et al. [8] introduced soft multiset as a generalization of Molodtsov’s soft set. They also introduced in [9] the concept of fuzzy parameterized interval-valued fuzzy soft set and gave its application in decision making. Zhu and Wen in [10] incorporated Molodtsov’s soft set theory with the probability theory and proposed the notion of probabilistic soft sets. In [11] Chaudhuri et al. defined the concepts of soft relation and fuzzy soft relation and then applied them to solve a number of decision-making problems. Majumdar and Samanta [12] defined and studied the generalised fuzzy soft sets where the degree is attached with the parameterization of fuzzy sets while defining a fuzzy soft set. In this paper, we generalise the concept of fuzzy soft sets as introduced by Maji et al. [6] to the possibility fuzzy soft set. In our generalisation of fuzzy soft set, a possibility of each element in the universe is attached with the parameterization of fuzzy sets while defining a fuzzy soft set. Also we give some applications of the possibility fuzzy soft set in decision-making problem and medical diagnosis.

2. Preliminaries

In this section, we recall some definitions and properties regarding fuzzy soft set and generalised fuzzy soft set required in this paper.

Let be a universe set, and let be a set of parameters. Let denote the power set of and .

Definition 2.1 (see [3]). pair is called a soft set over , where is a mapping given by . In other words, a soft set over is a parameterized family of subsets of the universe .

Definition 2.2 (see [6]). Let be an initial universal set, and let be a set of parameters. Let denote the power set of all fuzzy subsets of . Let pair is called a fuzzy soft set over where is a mapping given by .

The following definitions and propositions are due to Majumdar and Samanta [12].

Definition 2.3. Let be the universal set of elements, and let be the universal set of parameters. The pair will be called a soft universe. Let and be a fuzzy subset of , that is, , where is the collection of all fuzzy subsets of . Let be a function defined as follows: Then is called a generalized fuzzy soft set (GFSS in short) over the soft universe . Here for each parameter indicates not only the degree of belongingness of the elements of in but also the degree of possibility of such belongingness which is represented by . So we can write as follows: where are the degrees of belongingness and is the degree of possibility of such belongingness.

Definition 2.4. Let and be two GFSSs over . is said to be a generalised fuzzy soft subset of if (i) is a fuzzy subset of ;(ii) is also a fuzzy subset of , forall .In this case, we write .

Definition 2.5. Union of two GFSSs and , denoted by , is a GFSS , defined as such that where , , and is any norm.

Definition 2.6. Intersection of two GFSSs and , denoted by , is a GFSS , defined as such that where , , and is any norm.

Definition 2.7. GFSS is said to be a generalised null fuzzy soft set, denoted by , if such that , where , forall and forall .

Definition 2.8. GFSS is said to be a generalised absolute fuzzy soft set, denoted by , if such that , where , forall and forall .

Proposition 2.9. Let be a GFSS over . Then the following holds: (i),(ii),(iii),(iv),(v),(vi).

Proposition 2.10. Let , and be any three GFSSs over . Then the following holds: (i), (ii),(iii), (iv).

Definition 2.11. Similarity between the two GFSSs and , denoted by , is defined as follows: such that , where

Definition 2.12. Let and be two GFSSs over the same universe . We call the two GFSSs to be significantly similar if .

Proposition 2.13. Let and be any two GFSSs over . Then the following holds: (i),(ii),(iii),(iv),(v).

3. Possibility Fuzzy Soft Sets

In this section, we generalise the concept of fuzzy soft sets as introduced by Maji et al. [6]. In our generalisation of fuzzy soft set, a possibility of each element in the universe is attached with the parameterization of fuzzy sets while defining a fuzzy soft set.

Definition 3.1. Let be the universal set of elements and let be the universal set of parameters. The pair will be called a soft universe. Let and be a fuzzy subset of , that is, , where is the collection of all fuzzy subsets of . Let be a function defined as follows: Then is called a possibility fuzzy soft set (PFSS in short) over the soft universe . For each parameter indicates not only the degree of belongingness of the elements of in but also the degree of possibility of belongingness of the elements of in , which is represented by . So we can write as follows: Sometime we write as . If , we can also have a PFSS .

Example 3.2. Let be a set of three blouses. Let be a set of qualities where = bright, = cheap, and = colourful, and let . We define a function as follows: Then is a PFSS over . In matrix notation, we write

Definition 3.3. Let and be two PFSSs over . is said to be a possibility fuzzy soft subset (PFS subset) of , and one writes if(i) is a fuzzy subset of , forall,(ii) is a fuzzy subset of .

Example 3.4. Let be a set of three cars, and let be a set of parameters where = cheap, = expensive, and = red. Let be a PFSS over defined as follows: Let be another PFSS over defined as follows: It is clear that is a PFS subset of .

Definition 3.5. Let and be two PFSSs over . Then and are said to be equal, and one writes if is a PFS subset of and is a PFS subset of .In other words, if the following conditions are satisfied:(i) is equal to ,(ii)is equal to .

Definition 3.6. PFSS is said to be a possibility null fuzzy soft set, denoted by , if such that where , and .

Definition 3.7. PFSS is said to be a possibility absolute fuzzy soft set, denoted by , if such that where and .

Example 3.8. Let be a set of three blouses. Let be a set of qualities where = bright, = cheap, and = colorful, and let . We define a function which is a PFSS over defined as follows: Then is a possibility null fuzzy soft set.Let , and we define the function which is a PFSS over as follows: Then is a possibility absolute fuzzy soft set.

Definition 3.9. Let be a PFSS over (. Then the complement of , denoted by , is defined by such that and , where is a fuzzy complement.

Example 3.10. Consider the matrix notation in Example 3.2: By using the basic fuzzy complement, we have where

4. Union and Intersection of PFSS

In this section, we introduce the definitions of union and intersection of PFSS, derive some properties, and give some examples.

Definition 4.1. Union of two PFSSs and , denoted by , is a PFSS defined by such that and where is an norm.

Example 4.2. Let and . Let be a PFSS defined as follows: Let be another PFSS over defined as follows: By using the basic fuzzy union, we have , where Similarly we get In matrix notation, we write

Definition 4.3. Intersection of two PFSSs and , denoted by , is a PFSS defined by such that and where is a fuzzy norm.

Example 4.4. Consider the Example 4.2 where and are PFSSs defined as follows: By using the basic fuzzy intersection, we have , where Similarly we get In matrix notation, we write

Proposition 4.5. Let , and be any three PFSSs over . Then the following results hold: (i),(ii),(iii), (iv).

Proof. The proof is straightforward by using the fact that fuzzy sets are commutative and associative.

Proposition 4.6. Let be a PFSS over . Then the following results hold: (i),(ii),(iii), (iv), (v), (vi).

Proof. The proof is straightforward by using the definitions of union and intersection.

Proposition 4.7. Let , and be any three PFSSs over . Then the following results hold: (i), (ii).

Proof. For all , We can use the same method in (i).

5. AND and OR Operations on PFSS with Applications in Decision Making

In this section, we introduce the definitions of AND and OR operations on possibility fuzzy soft sets. Applications of possibility fuzzy soft sets in decision-making problem are given.

Definition 5.1. If and are two PFSSs then “ AND ”, denoted by is defined by where , such that and .

Example 5.2. Suppose the universe consists of three machines , and there are three parameters 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 defined as follows: Then where Similarly we get In matrix notation, we have Now to determine the best machine, we first mark the highest numerical grade (values with underline mark) in each row. Now the score of each of such machines is calculated by taking the sum of the products of these numerical grades with the corresponding possibility . The machine with the highest score is the desired machine. We do not consider the numerical grades of the machine against the pairs , as both the parameters are the same. Then the firm will select the machine with the highest score. Hence, they will buy machine (see Table 1).

Table 1: Grade table.

H

×
×

0.6

0.6
0.4

0.8
0.4

×
×

0.4

0.6
0.4

0.9
0.2

×
×

Score.Score.Score.

Definition 5.3. If and are two PFSSs then “ OR ”, denoted by , is defined by where , such that and .

Example 5.4. Let ; consider and as in Example 5.2. suppose now the firm wants to buy a machine depending on any one of two parameters. Then we have where Similarly we get In matrix notation, we have Now to determine the best machine, we first mark the highest numerical grade (value with underline mark) in each row. Now the score of each of such machines is calculated by taking the sum of the products of these numerical grades with the corresponding possibility . The machine with the highest score is the desired machine. We do not consider the numerical grades of the machine against the pairs , as both the parameters are the same. Then the firm will select the machine with the highest score. Hence, they will buy the machine (see Table 2).

Table 2: Grade table.

H

×
×

0.9
1

0.7
0.4

1
0.6

×
×

0.8
0.6

1
0.6

0.9
1

×
×

Score.Score.Score.

6. Similarity between Two Possibility Fuzzy Soft Sets

Similarity measures have extensive application in several areas such as pattern recognition, image processing, region extraction, coding theory, and so forth. We are often interested to know whether two patterns or images are identical or approximately identical or at least to what degree they are identical.

Several researchers have studied the problem of similarity measurement between fuzzy sets, fuzzy numbers, and vague sets. Majumdar and Samanta [12–14] have studied the similarity measure of soft sets, fuzzy soft sets, and generalised fuzzy soft sets.

In this section, we introduce a measure of similarity between two PFSSs. The set theoretic approach has been taken in this regard because it is easier for calculation and is a very popular method too.

Definition 6.1. Similarity between two PFSSs and , denoted by , is defined as follows: such that where

Definition 6.2. Let and be two PFSSs over . We say that and are significantly similar if .

Proposition 6.3. Let and be any two PFSSs over . Then the following holds: (i), (ii), (iii), (iv),(v).

Proof. The proof is straightforward and follows from Definition 6.1.

Example 6.4. Consider Example 4.2 where and are defined as follows: Here Similarly we get and . Then Similarly we get and . Then Hence, the similarity between the two PFSSs and is given by

7. Application of This Similarity Measure in Medical Diagnosis

In the following example 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 model possibility fuzzy soft set for dengue fever and the possibility fuzzy soft set 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, . Here the set of parameters is the set of certain visible symptoms. Let , where is body temperature, cough with chest congestion, loose motion, chills, headache, low heart rate (bradycardia), pain upon moving the eyes, breathing trouble, a flushing or pale pink rash comes over the face, low blood pressure (hypotension), and Loss of appetite.

Our model possibility fuzzy soft set for dengue fever is given in Table 3, and this can be prepared with the help of a physician.

Table 3: Model PFSS for dengue fever.

1
0
0
1
1
1
1
0
1
1
0

1
1
1
1
1
1
1
1
1
1
1

0
1
1
0
0
0
0
1
0
0
1

1
1
1
1
1
1
1
1
1
1
1

After talking to the sick person, we can construct his PFSS as in Table 4. Now we find the similarity measure of these two sets (as in Example 6.4), here . Hence the two PFSSs are not significantly similar. Therefore, we conclude that the person is not suffering from dengue fever.

Table 4: PFSS for the sick person.

0.3
0.7
0.5
0.3
0.4
0.1
0
0.7
0
0.4
0.2

0.6
0.2
0.1
0.5
0.5
0.8
0.7
0.2
1
0.4
0.5

0.6
0.1
0.4
0.5
0.4
0.6
0.7
0.1
0.8
0.5
0.4

0.5
0.6
0.4
0.5
0.3
0.6
0.4
0.1
0.5
0.6
0.7

8. Conclusion

In this paper, we have introduced the concept of possibility fuzzy soft set and studied some of its properties. Applications of this theory has been given to solve a decision-making problem. Similarity measure of two possibility fuzzy soft sets is discussed and an application of this to medical diagnosis has been shown.

Acknowledgment

The authors would like to acknowledge the financial support received from Universiti Kebangsaan Malaysia under the research grant UKM-ST-06-FRGS0104-2009. The authors also wish to gratefully acknowledge all those who have generously given their time to referee our paper.

References

L. A. Zadeh, “Fuzzy sets,” Information and Computation, vol. 8, pp. 338–353, 1965.

G. J. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic: Theory and Applications, Prentice-Hall, India, 6th edition, 2002.

D. Molodtsov, “Soft set theory-first results,” Computers & Mathematics with Applications, vol. 37, no. 4-5, pp. 19–31, 1999.

P. K. Maji, A. R. Roy, and R. Biswas, “Soft set theory,” Computers & Mathematics with Applications, vol. 45, no. 4-5, pp. 555–562, 2003.

P. K. Maji, A. R. Roy, and R. Biswas, “An application of soft sets in a decision making problem,” Computers & Mathematics with Applications, vol. 44, no. 8-9, pp. 1077–1083, 2002.

P. K. Maji, A. R. Roy, and R. Biswas, “Fuzzy soft sets,” Journal of Fuzzy Mathematics, vol. 9, no. 3, pp. 589–602, 2001.

A. R. Roy and P. K. Maji, “A fuzzy soft set theoretic approach to decision making problems,” Journal of Computational and Applied Mathematics, vol. 203, no. 2, pp. 412–418, 2007.

S. Alkhazaleh, A. R. Salleh, and N. Hassan, “Soft multisets theory,” Applied Mathematical Sciences, vol. 5, no. 72, pp. 3561–3573, 2011.

S. Alkhazaleh, A. R. Salleh, and N. Hassan, “Fuzzy parameterized interval-valued fuzzy soft set,” Applied Mathematical Sciences, vol. 5, no. 67, pp. 3335–3346, 2011.

P. Zhu and Q. Wen, “Probabilistic soft sets,” in Proceedingsof the IEEE International Conference on Granular Computing (GrC '10), pp. 635–638, 2010.

A. Chaudhuri, K. De, and D. Chatterjee, “Solution of the decision making problems using fuzzy soft relations,” International Journal of Information Technology, vol. 15, no. 1, pp. 78–106, 2009.

P. Majumdar and S. K. Samanta, “Generalised fuzzy soft sets,” Computers & Mathematics with Applications, vol. 59, no. 4, pp. 1425–1432, 2010.

P. Majumdar and S. K. Samanta, “On similarity measure of fuzzy soft sets,” in Proceedings of the International Conference on Soft Computing and Intelligent Systems (ICSCIS '07), pp. 40–44, Jabalpur, India, December 2007.

P. Majumdar and S. K. Samanta, “Similarity measure of soft sets,” New Mathematics and Natural Computation, vol. 4, no. 1, pp. 1–12, 2008.