Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 495351, 14 pages

http://dx.doi.org/10.1155/2013/495351

## Evaluating the Efficiency of School Preceptors by Fuzzy Risk Analysis

^{1}Department of Engineering Technology, Texas Southern University, Houston, TX 77004, USA^{2}Department of Mathematics, Urmia Branch, Islamic Azad University, Urmia, Iran

Received 26 January 2013; Accepted 13 March 2013

Academic Editor: Tofigh Allahviranloo

Copyright © 2013 Rasoul Saneifard and Rahim Saneifard. 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

This paper presents a new method for evaluating the efficiency of school preceptors based on fuzzy number arithmetic operations. It uses fuzzy numbers to represent fuzzy grades. The fuzzy weights of criteria are automatically generated from the opinions of evaluators. The simplified fuzzy number arithmetic operations are used for calculating the average of fuzzy numbers. It can evaluate the efficiency of school preceptors in a more flexible and more intelligent manner.

#### 1. Introduction

The ranking of fuzzy numbers has been a concern in fuzzy Multiple Attribute Decision Making since its inception. More than 20 fuzzy ranking indices have been proposed since 1976. Various techniques are applied to compare the fuzzy numbers. Some of these ranking methods have been compared and reviewed by Bortolan and Degani [1]. Chen and Hwang [2] thoroughly reviewed the existing approaches and pointed out some illogical conditions that arise among them. Among the existing ranking methods, centroid index methods are extensively studied and applied to many decision making problems. Recently, S. J. Chen and S. M. Chen [3] point outed the drawback of the existing centroid index ranking methods and proposed a new centroid index method for ranking fuzzy numbers based on the Center of Gravity (COG) point. They applied the COG based ranking method to a human selection problem based on Fuzzy Number Induced Order Weighted Averaging operator (2003). However, the COG based ranking method presented by S. J. Chen and S. M. Chen [3] still has some drawbacks, that is, it cannot correctly calculate the ranking order of the generalized fuzzy numbers in some situations. Also, Chang et al. [4] established the criteria for high school preceptor appraisal. They used fuzzy linguistic questionnaires to evaluate the efficiency of high school preceptors and they used fuzzy linguistic integrating techniques to calculate preceptor’s fuzzy grades, and then used Lee and Li’s method to rank preceptors’ fuzzy grades. Because Chang et al.’s methods have drawbacks in preceptors’ appraisal, it is necessary to develop a new method to overcome these drawbacks.

In this paper, we propose a new centroid index method for ranking fuzzy numbers. First we briefly introduce some existing centroid index ranking methods of fuzzy numbers. Then we propose a new method to evaluate the efficiency of school preceptors based on fuzzy number arithmetic operations. The proposed method can overcome the drawbacks of the other methods.

#### 2. The Centroid Formulae for Fuzzy Numbers

A fuzzy number is a convex fuzzy subset of the real line and is completely defined by its membership function. Let be a fuzzy number, whose membership function can generally be defined as [5–9], where is a constant, and are two strictly monotonically and continuous mappings from to closed interval . When , then is a normal fuzzy number; otherwise, it is said to be a nonnormal fuzzy number. If the membership function is piecewise linear, then is referred to as a trapezoidal fuzzy number and is usually denoted by, . In particular, if , the trapezoidal fuzzy number is reduced to a triangular fuzzy number. Since and are both strictly monotonically and continuous functions, their inverse functions exist and should also be continuous and strictly monotonically. Let and be the inverse functions of and , respectively. Then and should be integrable on the closed interval . In other words, both and should exist. In the case of trapezoidal fuzzy number, the inverse function and can be analytically expressed as In order to determine the centroid point of a fuzzy number , Wang et al. [10] provided the following centroid formulae: The ranking value of the fuzzy number is defined as follows [11]: The larger the value , the better the ranking of .

In [12], the authors presented a centroid-index ranking method for ordering fuzzy numbers. The centroid point of a fuzzy number is , where and are the same as formula and in [12]. The ranking value of the fuzzy number is defined as follows: The larger the value , the better the ranking of . In [3], S. J. Chen and S. M. Chen proposed a simple method to obtain COG point of fuzzy numbers. If is a generalized fuzzy number, where , then the COG point of is as follows: After obtaining the COG point of fuzzy number , where , the ranking value can be calculated as where The larger the value , the better the ranking . However, this method has a drawback in that it cannot correctly rank generalized fuzzy numbers in some situations. The example is used to show the drawback Chen’s method.

*Example 1. *Two generalized fuzzy number and are shown as follows (Figure 1):
It can be easily to obtain the COG points of fuzzy numbers and , respectively, as follows, and . By applying Chen method, we have and . The ranking result shows that ranking order is . However, it can be easily seen that the correct order is .

To compensate for these shortcomings, a new index of ranking fuzzy numbers is constructed in this article.

#### 3. A New Ranking Method for Generalized Fuzzy Numbers

In this section, the authors propose a novel technique for ranking of fuzzy numbers based on the distance method. The new method not only considers the centroid point of fuzzy numbers, but also considers the min crisp value of a fuzzy number.

Assume that there are generalized fuzzy numbers , where , , , , is any real value. The value denotes the maximum membership value of the generalized trapezoidal fuzzy number , where . The proposed method for ranking fuzzy numbers is now presented as follows.

*Step 1. *For each generalized trapezoidal fuzzy number , where , if the fuzzy number is not a standardized generalized trapezoidal fuzzy number, where the universe of discourse of the generalized trapezoidal fuzzy number is , then translate the generalized trapezoidal fuzzy number into a standardized generalized trapezoidal fuzzy number shown as follows:
where , , or and .

*Step 2. *Use formula (6) to calculate the centroid point of each standardized generalized trapezoidal fuzzy number , where .

*Step 3. *Calculate the standard deviation of each standardized generalized trapezoidal fuzzy number from (8). The standard deviation indicates the degree of dispersion of the standardized generalized trapezoidal fuzzy number , where .

*Step 4. *Use the standard deviation and of the centroid point to derive a new value shown as follows:
where , to obtain a new point .

*Step 5. *Use the new point to calculate the ranking value of the fuzzy numbers as follows:
where denotes the minimum value of the and , and . From formula (12), we can see that can be considered as the Euclidean distance between the point and the point . We can say that the larger the value of , the better the ranking of , where .

*Example 2. *As shown above, two generalized fuzzy numbers are shown as follows:
First, the COG points of the fuzzy numbers and can be obtained as follows, respectively.
Then, translate the COG points of fuzzy numbers and into Index point. The results can be shown as follows: ,
,
The ranking value of fuzzy numbers and can be calculated, respectively, as follows:
The ranking order of fuzzy numbers and is . It can be seen that the proposed method can overcome the drawback of the one presented in [12].

#### 4. A Method for High School Preceptors Evaluation

In this section, we present a new method for high school preceptors appraisal, where the criteria and the fuzzy linguistic questionnaires used for preceptor appraisal in [3] are adopted as shown in Tables 1 and 2, respectively. In the fuzzy linguistic questionnaire shown in Table 3, Saneifard et al. used five linguistic levels to evaluate teachers’ performance regarding the subcriterion , that is, very low (VL), low (L), middle (M), high (H) and very high (VH). They used the triangular fuzzy numbers , , , , and corresponding to the linguistic satisfaction levels VL, L, M, H, and VH, respectively, where , , , , and . The fuzzy grade of the subcriterion is calculated as follows: where denotes the index of the criterion, denotes the index of the subcriterion, denotes the index of linguistic levels, denotes the th linguistic satisfaction level of the criterion , , , denotes the degree that the teacher satisfies the th satisfaction level of the criterion , and . Saneifard et al. also used fuzzy linguistic questionnaires to determine the weight of a criterion. A fuzzy linguistic questionnaire for determining the weights of a criterion is shown in Table 3. The fuzzy weight of the criterion is calculated as follows: where denotes the index of the criterion, denotes the index of linguistic levels, denotes the th importance level of the criterion , , , denotes the degree of percentage that the criterion satisfies the th importance level, and . The proposed method for high school preceptors appraisal is now presented as follows.

*Step 1. *Determine the fuzzy weight of each criterion. Let each evaluator use the fuzzy linguistic questionnaire shown in Table 3 to evaluate the importance of each criterion. Then, use (19) to calculate the fuzzy weight of each criterion evaluated by each evaluator, where a fuzzy weight represented by a triangular fuzzy number should satisfy the rules.

If , then let ; if , then let . For example, assume that there are five evaluators , , , , and nine preceptors , , , , , , , , being evaluated. After the five evaluators fill out the fuzzy linguistic questionnaires, we get the result shown in Table 4. After applying (19) to calculate the fuzzy weight of each criterion evaluated by each evaluator, we get the calculating results as shown in Table 5. Based on (12), we rank the fuzzy weights with respect to each criterion evaluated by the evaluators, where , we drop the fuzzy weights with the smallest ranking value and the largest ranking value. Then, we calculate the average of the remaining fuzzy weights using the addition operations and the division operations of fuzzy numbers. Finally, we can get the fuzzy weighted vector , where , denotes the average of the remaining fuzzy weights with respect to the th criterion, and .

For example, let us consider the first row of Table 6. Based on (12), we can see that . Because the fuzzy weights and have the largest ranking value and smallest ranking value, respectively, we drop the fuzzy weights and , and then we can get the fuzzy weight of criterion by calculating the average of the remaining fuzzy weights, shown as follows:
Therefore, the fuzzy weight of criterion is , where . Now we consider the second row of Table 6. Based on (12), we can see that > . Because the fuzzy weights and have the largest ranking value and smallest ranking value, respectively, we drop the fuzzy weights and , and then we can get the fuzzy weight of criterion by calculating the average of the remaining fuzzyweights, shown as follows:
In the same way, we can get the weights , , and of the criteria , , and , respectively, where , , and . In this situation, the fuzzy weighted vector is as follows:

*Step 2. *Establish the fuzzy grade matrix. Each evaluator uses the fuzzy linguistic questionnaire shown in Table 3 to evaluate the performance of high school preceptors. Then, use (18) to calculate the fuzzy grade of each subcriterion of each preceptor evaluated by each evaluator, where the fuzzy grade represented by the fuzzy number should satisfy the rules: if , then let ; if , then let . For example, assume that the fuzzy grades evaluated by the five evaluators are as shown in Tables 6, 7, 8, 9, and 10, respectively. Based on (7), we rank the fuzzy grades of a teacher with respect to a sub-criterion evaluated by the evaluators, drop the fuzzy grades with the smallest ranking value and the largest ranking value, and then calculate the average of the remaining fuzzy grades by using the addition operations and the division operations of fuzzy numbers to get the averaged fuzzy grade of the sub-criterion. In the same way we can get the averaged fuzzy grade of each sub-criterion of each teacher. For example, based on Tables 6, 7, 8, 9, and 10 and equation (12), we can rank the fuzzy grades of with respect to criterion evaluated by each evaluator, where the result is as follows: > . Because the fuzzy weights and have the largest ranking value and smallest ranking value, respectively, we drop the fuzzy weights and , and then we can get the fuzzy grade of preceptor with respect to sub-criterion by calculating the average of the remaining fuzzy weights, shown as follows:
Therefore, the fuzzy grade of preceptor with respect to sub-criterion is , where . Then, based on (7), we can rank the fuzzy grades of preceptor with respect to criterion evaluated by each evaluator. The result is shown as follows: > . Because the fuzzy weights and have the largest ranking value and smallest ranking value, respectively, we drop the fuzzy weights and , and then we can get the fuzzy grade of preceptor with respect to sub-criterion by calculating the average of the remaining fuzzy weights shown as follows:
In the same way, we can get the fuzzy grade of each teacher with respect to each sub-criterion to derive the averaged fuzzygrade table shown in Table 11. Then, we apply the simplified fuzzy numbers addition operations and division operations to the averaged fuzzy grade table shown in Table 11 to get the fuzzy grade of each teacher with respect to each criterion. Finally, we can get the fuzzy grade matrix defined as follows:
where denotes the fuzzy grade of the th preceptor with respect to the th criterion , where , and denotes the number of preceptors, and denotes the number of criteria. For example, based on Table 11, we can calculate as follows:In the same way, we can get the fuzzy grade matrix as follows:

*Step 3. *Calculate the total fuzzy grade vector , where
where denotes the total fuzzy grade of the th preceptor and . For example, based on the fuzzy weighted vector obtained in Step 1, the fuzzy grade matrix obtained in Step 2 and (28), we can calculate the total fuzzy grade of teacher as follows:
In the same way, we can get the total fuzzy grade vector shown as follows:

*Step 4. *Based on (12), calculate the ranking value of the fuzzy grade , where . The larger the value of , the better the ranking of and the better the choice of preceptor , where and denotes the number of teachers. For example, after applying (12) to rank the fuzzy numbers , , , , , , , , and of the fuzzy grade vector obtained in Step 3, we can get the ranking results as shown in Table 12. We can see that . Because has the largest ranking value, preceptor is the best choice.

#### 5. Conclusions

In this paper, we proposed a new centroid index method for ranking fuzzy numbers. First we briefly introduce some existing centroid index ranking methods of fuzzy numbers. Then we propose a new method to evaluate the efficiency of school preceptors based on fuzzy number arithmetic operations. The proposed method can overcome the drawbacks of the other methods.

#### Acknowledgment

The authors would like to express their sincere appreciation to the Department of Mathematics, Islamic Azad University, Urmia Branch in Iran for their cooperation.

#### References

- G. Bortolan and R. Degani, “A review of some methods for ranking fuzzy subsets,”
*Fuzzy Sets and Systems*, vol. 15, no. 1, pp. 1–19, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. J. Chen and C. L. Hwang,
*Fuzzy Multiple Attribute Decision Making*, Springer, Berlin, Germany, 1993. - S. J. Chen and S. M. Chen, “A new method for handling multicriteria fuzzy decision-making problems using FN-IOWA operators,”
*Cybernetics and Systems*, vol. 34, no. 2, pp. 109–137, 2003. View at Publisher · View at Google Scholar · View at Scopus - J. R. Chang, C. H. Cheng, and C. Y. Kuo, “Conceptual procedure for ranking fuzzy numbers based on adaptive two dimensions dominance,”
*Soft Computing*, vol. 10, no. 3, pp. 74–103, 2006. View at Google Scholar - D. Dubois and H. Prade, “Operations on fuzzy numbers,”
*International Journal of Systems Science*, vol. 9, no. 6, pp. 613–626, 1978. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Saneifard, T. Allahviranloo, F. Hosseinzadeh, and N. Mikaeilvand, “Euclidean ranking DMUs with fuzzy data in DEA,”
*Applied Mathematical Sciences*, vol. 60, pp. 2989–2998, 2007. View at Google Scholar - R. Saneifard and R. Ezzati, “Defuzzification through a bi-symmetrical weighted function,”
*Australian Journal of Basic and Applied Sciences*, vol. 4, no. 10, pp. 4976–4984, 2010. View at Google Scholar · View at Scopus - R. Saneifard, “Ranking L-R fuzzy numbers with weighted averaging based on levels,”
*International Journal of Industrial Mathematics*, vol. 2, pp. 163–1173, 2009. View at Google Scholar - R. Ezzati and R. Saneifard, “A new approach for ranking of fuzzy numbers with continuous weighted quasi-arithmetic means,”
*Mathematical Sciences*, vol. 2, pp. 143–158, 2010. View at Google Scholar - Y.-M. Wang, J.-B. Yang, D.-L. Xu, and K.-S. Chin, “On the centroids of fuzzy numbers,”
*Fuzzy Sets and Systems*, vol. 157, no. 7, pp. 919–926, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C.-H. Cheng, “A new approach for ranking fuzzy numbers by distance method,”
*Fuzzy Sets and Systems*, vol. 95, no. 3, pp. 307–317, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T.-C. Chu and C.-T. Tsao, “Ranking fuzzy numbers with an area between the centroid point and original point,”
*Computers & Mathematics with Applications*, vol. 43, no. 1-2, pp. 111–117, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet