Advances in Decision SciencesVolume 2011 (2011), Article ID 757868, 12 pagesdoi:10.1155/2011/757868

Research Article

Soft Expert Sets

Shawkat Alkhazaleh and Abdul Razak Salleh

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Selangor DE, 43600 Bangi, Malaysia

Received 15 June 2011; Revised 12 September 2011; Accepted 19 September 2011

Academic Editor: C. D. Lai

Copyright © 2011 Shawkat Alkhazaleh and Abdul Razak Salleh. 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.

In 1999, Molodtsov introduced the concept of soft set theory as a general mathematical tool for dealing with uncertainty. Many researchers have studied this theory, and they created some models to solve problems in decision making and medical diagnosis, but most of these models deal only with one expert. This causes a problem with the user, especially with those who use questionnaires in their work and studies. In our model, the user can know the opinion of all experts in one model. So, in this paper, we introduce the concept of a soft expert set, which will more effective and useful. We also define its basic operations, namely, complement, union intersection AND, and OR. Finally, we show an application of this concept in decision-making problem.

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 operations and application of soft sets were studied by Chen et al. [2] and Maji et al. [3, 4]. Alkhazaleh et al. [5] introduced the concept of soft multisets as a generalization of soft set. They also defined in [6, 7] the concepts of possibility fuzzy soft set and fuzzy parameterized interval-valued fuzzy soft set and gave their applications in decision making and medical diagnosis. Many researchers have studied this theory, and they created some models to solve problems in decision making and medical diagnosis, but most of these models deal only with one expert, and if we want to take the opinion of more than one expert, we need to do some operations such as union, intersection, and so forth. This causes a problem with the user, especially with those who use questionnaires in their work and studies. In our model the user can know the opinion of all experts in one model without any operations. Even after any operation on our model the user can know the opinion of all experts. So in this paper we introduce the concept of a soft expert set, which will be more effective and useful. We also define its basic operations, namely, complement, union intersection AND and OR and study their properties. Finally, we give an application of this concept in a decision-making problem.

2. Preliminaries

In this section, we recall some basic notions in soft set theory. Molodtsov [1] defined soft set in the following way. Let be a universe and be 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 .

The following definitions are due to Maji et al. [3].

Definition 2.2. For two soft sets and over , is called a soft subset of if(i), (ii)for all are identical approximations.This relationship is denoted by . In this case, is called a soft superset of .

Definition 2.3. Two soft sets and over a common universe are said to be soft equal if is a soft subset of and is a soft subset of .

Definition 2.4. Let be a set of parameters. The NOT set of denoted by is defined by where not .

Definition 2.5. The complement of a soft set is denoted by and is defined by where is a mapping given by .

Definition 2.6. A soft set over is said to be a NULL soft set denoted by , (null-set).

Definition 2.7. A soft set over is said to be an absolute soft set, denoted by .

Definition 2.8. If and are two soft sets then AND denoted by , is defined by where .

Definition 2.9. If and are two soft sets, then OR denoted by , is defined by where .

Definition 2.10. The union of two soft sets and over a common universe is the soft set where , and ,

The following definition is due to Ali et al. [8] since they discovered that Maji et al.’s definition of intersection in [3] is not correct.

Definition 2.11. The extended intersection of two soft sets and over a common universe is the soft set where , and ,

3. Soft Expert Set

In this section, we introduce the concept of a soft expert set, and give definitions of its basic operations, namely, complement, union, intersection, AND, and OR. We give examples for these concepts. Basic properties of the operations are also given.

Let be a universe, a set of parameters, and a set of experts (agents). Let be a set of opinions, and .

Definition 3.1. A pair is called a soft expert set over , where is a mapping given by where denotes the power set of .

Note 3.2. For simplicity we assume in this paper, two-valued opinions only in set , that is, , but multivalued opinions may be assumed as well.

Example 3.3. Suppose that a company produced new types of its products and wishes to take the opinion of some experts about concerning these products. Let be a set of products, a set of decision parameters where denotes the decision “easy to use,” “quality,” and “cheap,” respectively, and let be a set of experts.Suppose that the company has distributed a questionnaire to three experts to make decisions on the company's products, and we get the following: Then we can view the soft expert set as consisting of the following collection of approximations: Notice that in this example the first expert, , “agrees” that the “easy to use” products are , and . The second expert, , “agrees” that the “easy to use” products are and , and the third expert, , “agrees” that the “easy to use” products are and . Notice also that all of them “agree” that product is “easy to use.”

Definition 3.4. For two soft expert sets and over is called a soft expert subset of if(i), (ii). This relationship is denoted by . In this case is called a soft expert superset of .

Definition 3.5. Two soft expert sets and over are said to be equal if is a soft expert subset of and is a soft expert subset of .

Example 3.6. Consider Example 3.3. Suppose that the company took the opinion of the experts once again after the products have been in the market for a month.Suppose Clearly . Let and be defined as follows: Therefore .

Definition 3.7. Let be a set of parameters and a set of experts. The NOT set of denoted by , is defined by where is not .

Definition 3.8. The complement of a soft expert set is denoted by and is defined by where is a mapping given by .

Example 3.9. Consider Example 3.3. Then

Definition 3.10. An agree-soft expert set over is a soft expert subset of defined as follows:

Example 3.11. Consider Example 3.3. Then the agree-soft expert set over is

Definition 3.12. A disagree-soft expert set over is a soft expert subset of defined as follows:

Example 3.13. Consider Example 3.3. Then the disagree-soft expert set over is

Proposition 3.14. If is a soft expert set over , then (i), (ii), (iii).

Proof. The proof is straightforward.

Definition 3.15. The union of two soft expert sets and over denoted by , is the soft expert set where , and ,

Example 3.16. Consider Example 3.3. Let Suppose and are two soft expert sets over such that Therefore

Proposition 3.17. If , , and are three soft expert sets over , then (i), (ii).

Proof. The proof is straightforward.

Definition 3.18. The intersection of two soft expert sets and over denoted by is the soft expert set where , and

Example 3.19. Consider Example 3.16. Then

Proposition 3.20. If , , and are three soft expert sets over , then (i), (ii).

Proof. The proof is straightforward.

Proposition 3.21. If , , and are three soft expert sets over , then (i), (ii).

Proof. The proof is straightforward.  

Definition 3.22. If and are two soft expert sets over then AND denoted by , is defined by where .

Example 3.23. Consider Example 3.3. Let Suppose and are two soft expert sets over such that Therefore

Definition 3.24. If and are two soft expert sets then OR denoted by , is defined by where .

Example 3.25. Consider Example 3.23. Then

Proposition 3.26. If and are two soft expert sets over , then (i), (ii).

Proof. See Maji et al. [3].

Proposition 3.27. If , , and are three soft expert sets over , then (i), (ii), (iii), (iv).

Proof. Straightforward from Definitions 3.22 and 3.24.

4. An Application of Soft Expert Set

Maji et al. [4] applied the theory of soft sets to solve a decision-making problem using rough mathematics. In this section, we present an application of soft expert set theory in a decision-making problem. The problem we consider is as below.

Assume that a company wants to fill a position. There are eight candidates who form the universe . The hiring committee considers a set of parameters, where the parameters stand for “experience,” “computer knowledge,” “young age,” “good speaking,” and “friendly,” respectively. Let be a set of experts (committee members). Suppose

In Tables 1 and 2 we present the agree-soft expert set and disagree-soft expert set, respectively, such that if then otherwise , and if then otherwise where are the entries in Tables 1 and 2.

Table 1: Agree-soft expert set.

1
1
0
1
0
0
1
1

0
0
1
0
1
0
0
1

0
0
1
1
1
0
1
0

1
0
0
0
0
0
1
1

1
1
1
0
1
0
0
1

1
0
0
1
1
0
0
1

1
0
1
1
1
1
0
1

0
0
1
1
1
0
1
0

1
0
0
1
1
0
0
1

1
0
0
1
1
0
0
1

1
0
1
1
0
1
1
1

1
1
0
1
0
0
1
1

1
0
0
0
0
0
1
1

1
0
0
0
0
1
1
1

1
0
1
0
1
0
1
1

Table 2: Disagree-soft expert set.

0
0
1
0
0
1
0
0

1
1
0
1
0
1
1
0

1
1
0
0
0
1
0
1

0
1
1
1
1
1
0
0

0
0
0
1
0
1
1
0

0
1
1
0
0
1
1
0

0
1
0
0
0
0
1
0

1
1
0
0
0
1
0
1

0
1
1
0
0
1
1
0

0
1
1
0
0
1
1
0

0
1
0
0
1
0
0
0

0
0
1
0
1
1
0
0

0
1
1
1
1
1
0
0

0
1
1
1
1
0
0
0

0
1
0
1
0
1
0
0

The following algorithm may be followed by the company to fill the position.

Algorithm 4.1.     (1)input the soft expert set ,(2)find an agree-soft expert set and a disagree-soft expert set,(3)find for agree-soft expert set,(4)find for disagree-soft expert set,(5)find ,(6)find , for which .

Then is the optimal choice object. If has more than one value, then any one of them could be chosen by the company using its option.

Now we use this algorithm to find the best choices for the company to fill the position. From Tables 1 and 2 we have Table 3.

Table 3

Then , so the committee will choose candidate 8 for the job.

Acknowledgments

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 the referees for their constructive comments.

References

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D. Chen, E. C. C. Tsang, D. S. Yeung, and X. Wang, “The parameterization reduction of soft sets and its applications,” Computers & Mathematics with Applications, vol. 49, no. 5-6, pp. 757–763, 2005.

P. K. Maji, R. Biswas, and A. R. Roy, “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.

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, “Possibility fuzzy soft set,” Advances in Decision Sciences, vol. 2011, Article ID 479756, 18 pages, 2011.

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M. I. Ali, F. Feng, X. Liu, W. K. Min, and M. Shabir, “On some new operations in soft set theory,” Computers & Mathematics with Applications, vol. 57, no. 9, pp. 1547–1553, 2009.