Fuzzy assessment for sampling survey defuzzification by signed distance method
Introduction
Statistical analysis via sampling survey is a powerful market research tool to acquire the useful information. Traditionally, we compute statistics with sample data by questionnaires according to the thinking of binary logic. There usually exists two different methods, both multiple-item and single-item choices, while using linguistic variables as rating item. We use mark or unmark to determine the choice for each item, i.e., the marked item is represented by 1, while the other unmark item is represented by 0. Generally speaking, the linguistic variable possesses the vague nature (Lin and Lee, 2008, Lin and Lee, 2009, Sun and Wu, 2006). This kind of result may lead to an unreasonable bias since the human thinking is full with fuzzy and uncertain. Fuzzy sets theory was introduced by Zadeh (1965) to deal with problem in which vagueness is present, linguistic value can be used for approximate reasoning within the framework of fuzzy set theory (Zadeh, 1975a, Zadeh, 1975b, Zadeh, 1976) to effectively handle the ambiguity involved in the data evaluation and the vague property of linguistic expression, and normal triangular fuzzy numbers are used to characterize the fuzzy values of quantitative data and linguistic terms used in approximate reasoning.
With regard to fuzzy decision-making problem, Lee (1996) applied fuzzy set theory to evaluate the aggregative risk in software development under fuzzy circumstances. Lin and Lee (2008) presented facility site selection model using fuzzy set theory. Lin and Lee (2009) presented the fuzzy assessment on sampling survey analysis. Lin and Lee, 2008, Lin and Lee, 2009 applied a value m which belongs to the closed interval [0, 1] to represent the reliability or membership grade in the fuzzy sense of marking item, and presented the method to treat the aggregated assessment of the main item Bj, and the integrated assessment. Lin and Lee (2009) presented method to treat the crisp or fuzzy multiple/single choices.
As the result that if the membership function of the triangular fuzzy number is not an isosceles triangle, then, based on the maximum membership grade principle, to defuzzify the triangular fuzzy number by the signed distance is better than by the centroid method. In this study, we propose a model to do assessment analysis for sampling survey with the linear order character of fuzzy linguistics by signed distance method (Yao & Wu, 2000). The proposed fuzzy assessment method on sampling survey analysis is easily to assess the sampling survey and do the aggregative evaluation. The paper is organized as follows. Section 1 is introduction. Section 2 is the preliminaries. Aggregative assessment method for sample survey with fuzzy linguistics by signed distance method is in Section 3. Section 4 is the numerical example. The conclusion of this study is given in Section 5.
Section snippets
Preliminaries
For the proposed algorithm, all pertinent definitions of fuzzy sets are given below (Yao and Wu, 2000, Zadeh, 1965, Zimmermann, 1991). Definition 2.1 Fuzzy set: If X is a collection of objects denoted generically by x then a fuzzy set in X is a set of ordered pairs: is called the membership function of x in which maps X to the closed interval [0, 1] that characterizes the degree of membership of x in . Definition 2.2 Fuzzy point: Let be a fuzzy set on R = (−∞, ∞). It is called a fuzzy point if
Fuzzy linear order linguistics
We consider that the fuzzy linguistics L1, L2, … , Ln with the corresponding series of fuzzy numbers , , … , , wherefor q = 1, 2, … , n. Then, we have . Therefore, we can make a conclusion that these fuzzy linguistics L1, L2, … , Ln with the corresponding series of fuzzy numbers are linear order respective to the signed distance method.
Aggregative assessment for sampling survey by signed distance method
In most cases, questionnaire of sampling survey exists many topics and
Example implementation
We use the contents of main items and sub-items shown in Lin and Lee (2008) as an example to implement the proposed algorithms. Example Assume that we have the following main items, weights, sub-items and the answer numbers as shown in Table 3.
By algorithm (I) and (II), we can have the following computing results:
- (A)
By algorithm (I)
- (10)
The integrated assessment of the main items B1, B2, B3, B4, B5 are:respectively.
- (20)
The integrated
- (10)
Conclusion
As the result that if the membership function of the triangular fuzzy number is not an isosceles triangle, then, based on the maximum membership grade principle, to defuzzify triangular fuzzy number by the signed distance is better than by the centroid method. However, if the membership function of the triangular fuzzy number is an isosceles triangle, then to defuzzify the triangular fuzzy number by the signed distance is equal to by the centroid method based on the maximum membership grade
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