Research Article  Open Access
IntervalValued Vague Soft Sets and Its Application
Abstract
Molodtsov has introduced the concept of soft sets and the application of soft sets in decision making and medical diagnosis problems. The basic properties of vague soft sets are presented. In this paper, we introduce the concept of intervalvalued vague soft sets which are an extension of the soft set and its operations such as equality, subset, intersection, union, AND operation, OR operation, complement, and null while further studying some properties. We give examples for these concepts, and we give a number of applications on intervalvalued vague soft sets.
1. Introduction
Uncertain or imprecise data are inherent and pervasive in many important applications in areas such as economics, engineering, environmental sciences, social science, medical science, and business management. There have been a number of researches and applications in the literature dealing with uncertainties such as Molodtsov’s [1] who initiated the concept of soft set theory as a mathematical tool for dealing with uncertainties which is free, fuzzy set theory by Zadeh [2], rough set theory by Pawlak [3], vague set theory by Gau and Buehrer [4], intuitionistic fuzzy set theory by Atanassov [5], and interval mathematics by Atanassov [6]. Alkhazaleh et al. [7] also introduced the concept of fuzzy parameterized intervalvalued fuzzy soft sets and established its application in decision making. Alkhazaleh et al. [8] further introduced soft multisets as a generalization of Molodtsov's soft set. Bustince and Burillo [9] studied the difference between vague soft sets and intuitionistic fuzzy soft sets. Furthermore, soft sets, soft groups, and new operation in soft set theory were studied by Aktas and Cagman [10], Ali et al. [11], Maji et al. [12–14], Jiang et al. [15], and Alhazaymeh et al. [16].
The purpose of this paper is to further extend the concept of vague set theory by introducing the notion of a vague soft set and deriving its basic properties. The paper is organized as follows. Section 1 is the introduction followed by Section 2 which is preliminaries of vague set and vague soft set. Section 3 presents the basic concepts and definitions for a vague soft set and a vague set and then redefines the concept of the intersection of two soft sets. Section 4 introduces the notion of an intervalvalued vague soft set and discusses its properties. Concluding remarks and open questions for further investigation are provided in Section 5.
2. Preliminaries
A soft set is a mapping from a set of parameters to the power set of a universe set. However, the notion of a soft set, as given in its definition, cannot be used to represent the vagueness of the associated parameters. In this section, we provide the concept of a vague soft set based on soft set theory and vague set theory and the basic properties.
Let be a universe, a set of parameters, the power set of the vague sets on , and .
Definition 1 (see [1]). A 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 subset of the universe .
Definition 2 (see [17]). A pair is called a vague soft set over , where is a mapping given by .
In other words, a vague soft set over is a parameterized family of the universe . For , is regarded as the set of approximate of the vague soft set .
Definition 3 (see [17]). For two vague soft sets and over universe , we say that is the vague subset of , if , and are identical approximations. This relationship is denoted by .
Definition 4 (see [17]). Two vague soft sets and over universe are said to be vague soft equal if is a vague soft subset of and is a vague soft subset of .
Definition 5 (see [17]). The complement of vague soft set is denoted by and is defined by , where is a mapping given by , , .
Definition 6 (see [17]). A vague soft set over is said to be a null vague soft set denoted by , , and , .
Definition 7 (see [17]). A vague soft set over is said to be an absolute vague soft set denoted by , , and , .
Definition 8 (see [17]). If and are two vague soft sets over . “ and ” denoted by “” which is defined by , where , , .
Definition 9 (see [17]). If and are two vague soft sets over . “ or ” denoted by “” is defined by , where , and , .
Definition 10 (see [18]). An intervalvalued fuzzy set is a mapping such that where stands for the set of all closed subintervals of , the set of all intervalvalued fuzzy sets on is denoted by .
The complement, intersection, and union of the intervalvalued fuzzy sets are defined in [19] 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
We used these definitions to introduce the concept of intervalvalued vague soft set. Also, we extend these definitions to provide some basic operation on intervalvalued vague soft set, such as equality, subset, intersection, union, AND operation, OR operation, complement, and null.
3. IntervalValued Vague Soft Set
In this section, we introduce the state of intervalvalued vague soft set and some operations. These are equality, subset, intersection, union, AND operation, OR operation, complement, and null.
Let be an initial universe, a set of parameters, the power set of intervalvalued vague sets on , and . The concept of an intervalvalued vague soft set is given by the following proposed definition.
Definition 11. A pair is called an intervalvalued vague soft set over , where is a mapping given by .
In other words, an intervalvalued vague soft set over is a parameterized family of an intervalvalued vague set of the universe .
Example 12. Consider an intervalvalued vague soft set , where is the set of three cars under the consideration of the decision maker for purchase, which is denoted by , and are the parameters set, where sporty, family, utility}. The intervalvalued vague soft set describes the “attractiveness of the cars" to this decision maker.
Suppose that
The intervalvalued vague soft set is a parameterized family of intervalvalued vague soft set on , and sporty cars , , , family cars , , , utility cars , , .
Definition 13. For two intervalvalued vague soft sets and over universe , we say that is an intervalvalued vague soft subset of , if and , and are identical approximations. This relationship is denoted by . Similarly, is said to be intervalvalued vague soft superset of and is an intervalvalued vague soft subset of as follows denoted by . Consider the definition of vague subsets. Let and be two vague sets of the universe . If , and , then the vague set are included by , denoted by , where .
Definition 14. Two intervalvalued vague soft sets and over universe are said to be intervalvalued vague soft equal if is an intervalvalued vague soft subset of and is an intervalvalued vague soft subset of .
Definition 15. Let be a parameter set. The not set of denoted by is defined by , where = not .
Definition 16. The complement of an intervalvalued vague soft set is denoted by and is defined by , where is a mapping given by , , and , .
Example 17. We used the descriptions from Example 12 to illustrate the complement of intervalvalued vague soft set as not sporty cars , , , not family cars = , , , not utility cars , , .
Definition 18. An intervalvalued vague soft set over is said to be a null intervalvalued vague soft set denoted by , if , , , and , .
Definition 19. An intervalvalued vague soft set over is said to be an absolute intervalvalued vague soft set denoted by , if , , , and , .
Definition 20. If and are two intervalvalued vague soft sets over , “ and ” is an intervalvalued vague soft set denoted by “” which is defined by , where , , that is .
Definition 21. If and are two intervalvalued vague soft sets over , “” is an intervalvalued vague soft set denoted by “” which is defined by , where , , that is , , .
Definition 22. The union of two intervalvalued vague soft sets and over a universe is an intervalvalued vague soft set , where and ,Hence, .
Definition 23. The intersection of two intervalvalued vague soft sets and over a universe is an intervalvalued vague soft set , where and , Hence, .
Theorem 24. If and are two intervalvalued vague soft sets over , then one has the following properties:(i);(ii).
Proof. Assume that , where and Since , then we have , where for all and . Hence,Since and , then we have .
Suppose that , where and we take Therefore, and are the same operators. Thus, ;
(ii) the proof is similar to that of (i).
4. An Application on IntervalValued Vague Soft Set
In this section, we provide an application of intervalvalued vague soft set.
Let be the set of apartments having different furnishings and rental, with the parameters set, fully furnished, partially furnished, empty, monthly, yearly, weekly}. Let and denote two subsets of the set of parameters . Also let represent the furnished fully furnished, partially furnished, empty} and represent the rental = {monthly, yearly, weekly}.
Assuming that an intervalvalued vague soft set describes the “apartments having furnished,” an intervalvalued vague soft set describes the “apartments having rental.” These intervalvalued vague soft sets may be computed as below.
An intervalvalued vague soft set is defined as apartments being fully furnished = , , , partially furnished = , , , empty = , , .
An intervalvalued vague soft set is defined as apartments having monthly rental = , , , apartments having yearly rental = , , , apartments having weekly rental = , , .
Let and be a two intervalvalued vague soft sets over the common universe . After performing some operations (such as AND and OR) on an intervalvalued vague soft set for some particular parameters of and , we obtain another intervalvalued vague soft set. The newly obtained intervalvalued vague soft set is termed as a resultant intervalvalued vague soft set of and .
Suppose that Mr. X is interested to rent an apartment on the basis of his choice parameters, which constitute the subset fully furnished, monthly, partially furnished} of the set , and = {partially furnished, security, yearly}, where both and . This means that out of available apartments in , he is to select an apartment which qualify all parameters of an intervalvalue vague soft sets and . The problem is to select the apartment which is most suitable with the choice parameters of Mr. X.
To solve this problem, we require some concepts in the soft set theory of Molodtsov [1], which are presented below.
Consider the above two intervalvalued vague soft sets and as in Tables 1 and 2, respectively. If we perform “ AND ,” then we will have parameters of the form , . If we require the intervalvalued vague soft set for all the parameters , then the resultant intervalvalued vague soft set for the intervalvalued vague soft sets and will be .
As such, after performing the “” for some parameters, the tabular representation of truth membership of the intervalvalued vague soft set will likely take a form as in Table 3.
Representation of the falsemembership function of and are shown in Tables 4 and 5.
After performing the “ AND ” for parameters tabular representation of falsemembership of the intervalvalued vague soft set will take a form as in Table 6.
Representations of the comparison for truthmembership function and falsemembership function are shown in Tables 7 and 8, respectively.


Tables 9 and 10 show the calculated truthmembership score and falsemembership score, respectively, while Table 11 shows the final score.



Clearly the maximum score is , which is the score of the apartment . Mr. X's best option is to rent apartment , while his second best choice will be .
5. Conclusion
In this paper, the basic concept of a soft set is reviewed. We introduce the notion of an intervalvalued vague soft set as an extension to the vague soft set. The basic properties of intervalvalued vague soft sets are also presented. These are complement, null, union, intersection, quality, subsets, “AND” and “OR” operators, and the application with respect to the intervalvalued vague soft set is illustrated.
It is desirable to further explore the applications of using the intervalvalued vague soft set approach to problems such as decision making, forecasting, and data analysis.
Acknowledgments
The authors are indebted to Universiti Kebangsaan Malaysia for funding this research under the Grant of UKMGUP2011159.
References
 D. Molodtsov, “Soft set theory—first results,” Computers and Mathematics with Applications, vol. 37, no. 45, pp. 19–31, 1999. View at: Google Scholar
 L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, no. 3, pp. 338–353, 1965. View at: Google Scholar
 Z. Pawlak, “Rough sets,” International Journal of Computer & Information Sciences, vol. 11, no. 5, pp. 341–356, 1982. View at: Publisher Site  Google Scholar
 W. L. Gau and D. J. Buehrer, “Vague sets,” IEEE Transactions on Systems, Man and Cybernetics, vol. 23, no. 2, pp. 610–614, 1993. View at: Publisher Site  Google Scholar
 K. T. Atanassov, “Intuitionistic fuzzy sets,” Fuzzy Sets and Systems, vol. 20, no. 1, pp. 87–96, 1986. View at: Google Scholar
 K. T. Atanassov, “Operators over interval valued intuitionistic fuzzy sets,” Fuzzy Sets and Systems, vol. 64, no. 2, pp. 159–174, 1994. View at: Google Scholar
 S. H. Alkhazaleh, A. Razak Salleh, and N. Hassan, “Possibility fuzzy soft set,” Advances in Decision Sciences, vol. 2011, Article ID 479756, 18 pages, 2011. View at: Publisher Site  Google Scholar
 S. H. Alkhazaleh, A. Razak Salleh, and N. Hassan, “Soft multisets theory,” Applied Mathematical Sciences, vol. 72, pp. 3561–3573, 2011. View at: Google Scholar
 H. Bustince and P. Burillo, “Vague sets are intuitionistic fuzzy sets,” Fuzzy Sets and Systems, vol. 79, no. 3, pp. 403–405, 1996. View at: Google Scholar
 H. Aktas and N. Cagman, “Soft sets and soft groups,” Information Sciences, vol. 177, no. 13, pp. 2726–2735, 2007. View at: Publisher Site  Google Scholar
 M. I. Ali, F. Feng, L. Xiaoyan, K. M. Won, and M. Shabir, “On some new operations in soft set theory,” Computers and Mathematics with Applications, vol. 57, no. 9, pp. 1547–1553, 2009. View at: Publisher Site  Google Scholar
 P. K. Maji, R. Biswas, and A. R. Roy, “Fuzzy soft set theory,” The Journal of Fuzzy Mathematics, vol. 3, pp. 589–602, 2001. View at: Google Scholar
 P. K. Maji, A. R. Roy, and R. Biswas, “An application of soft sets in a decision making problem,” Computers and Mathematics with Applications, vol. 44, no. 89, pp. 1077–1083, 2002. View at: Publisher Site  Google Scholar
 P. K. Maji, R. Biswas, and A. R. Roy, “Soft set theory,” Computers and Mathematics with Applications, vol. 45, no. 45, pp. 555–562, 2003. View at: Publisher Site  Google Scholar
 Y. Jiang, Y. Tang, Q. Chen, H. Liu, and J. Tang, “Intervalvalued intuitionistic fuzzy soft sets and their properties,” Computers and Mathematics with Applications, vol. 60, no. 3, pp. 906–918, 2010. View at: Publisher Site  Google Scholar
 K. Alhazaymeh, S. Abdul Halim, A. R. Salleh, and N. Hassan, “Soft intuitionistic fuzzy sets,” Applied Mathematical Sciences, vol. 6, no. 54, pp. 2669–2680, 2012. View at: Google Scholar
 W. Xu, J. Ma, S. Wang, and G. Hao, “Vague soft sets and their properties,” Computers and Mathematics with Applications, vol. 59, no. 2, pp. 787–794, 2010. View at: Publisher Site  Google Scholar
 S. H. Alkhazaleh, A. Razak Salleh, and N. Hassan, “Fuzzy parameterized intervalvalued fuzzy soft set,” Applied Mathematical Sciences, vol. 67, pp. 3335–3346, 2011. View at: Google Scholar
 M. B. Gorzalczany, “A method of inference in approximate reasoning based on intervalvalued fuzzy sets,” Fuzzy Sets and Systems, vol. 21, no. 1, pp. 1–17, 1987. View at: Google Scholar
Copyright
Copyright © 2012 Khaleed Alhazaymeh and Nasruddin Hassan. 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.