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 interval-valued 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 interval-valued vague soft sets.
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 [
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
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
A pair
In other words, a soft set over
A pair
In other words, a vague soft set over
For two vague soft sets
Two vague soft sets
The complement of vague soft set
A vague soft set
A vague soft set
If
If
An interval-valued fuzzy set
The complement, intersection, and union of the interval-valued fuzzy sets are defined in [ the complement of the intersection of the union of
We used these definitions to introduce the concept of interval-valued vague soft set. Also, we extend these definitions to provide some basic operation on interval-valued vague soft set, such as equality, subset, intersection, union, AND operation, OR operation, complement, and null.
In this section, we introduce the state of interval-valued vague soft set and some operations. These are equality, subset, intersection, union, AND operation, OR operation, complement, and null.
Let
A pair
In other words, an interval-valued vague soft set over
Consider an interval-valued vague soft set
Suppose that
The interval-valued vague soft set
For two interval-valued vague soft sets
Two interval-valued vague soft sets
Let
The complement of an interval-valued vague soft set
We used the descriptions from Example
An interval-valued vague soft set
An interval-valued vague soft set
If
If
The union of two interval-valued vague soft sets
The intersection of two interval-valued vague soft sets
If
Suppose that
(ii) the proof is similar to that of (i).
In this section, we provide an application of interval-valued vague soft set.
Let
Assuming that an interval-valued vague soft set
An interval-valued vague soft set
An interval-valued vague soft set
Let
Suppose that Mr. X is interested to rent an apartment on the basis of his choice parameters, which constitute the subset
To solve this problem, we require some concepts in the soft set theory of Molodtsov [
Consider the above two interval-valued vague soft sets
Representation of truth-membership function of
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Representation of truth-membership function of
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As such, after performing the “
Representation of truth membership.
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Representation of the false-membership function of
Representation of false-membership function of
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Representation of false-membership function of
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After performing the “
Representation of false membership.
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Representations of the comparison for truth-membership function and false-membership function are shown in Tables
Comparison table for truth-membership function.
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6 | 2 | 3 |
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4 | 6 | 3 |
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2 | 0 | 6 |
Comparison table for false-membership function.
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6 | 2 | 4 |
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5 | 6 | 5 |
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3 | 0 | 6 |
Tables
Truth-membership score.
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Row sum |
Column sum |
Truth-membership score = |
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11 | 12 | −1 |
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13 | 8 | 5 |
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8 | 12 | −4 |
False-membership score.
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Row sum |
Column sum |
Truth-membership score = |
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12 | 14 | −2 |
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16 | 8 | 8 |
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9 | 15 | −6 |
Final score table.
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Truth-membership score = |
False-membership score = |
Final score = |
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−1 | −2 | 1 |
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5 | 8 | −3 |
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−4 | −6 | 2 |
Clearly the maximum score is
In this paper, the basic concept of a soft set is reviewed. We introduce the notion of an interval-valued vague soft set as an extension to the vague soft set. The basic properties of interval-valued 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 interval-valued vague soft set is illustrated.
It is desirable to further explore the applications of using the interval-valued vague soft set approach to problems such as decision making, forecasting, and data analysis.
The authors are indebted to Universiti Kebangsaan Malaysia for funding this research under the Grant of UKM-GUP-2011-159.