Evaluating the Efficiency of School Preceptors by Fuzzy Risk Analysis

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.


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.

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][6][7][8][9], when  2 ≤  ≤  3 ,    () when  3 ≤  ≤  4 , 0, otherwise, where 0 ≤  ≤ 1 is a constant,    : [ In order to determine the centroid point ( 0 ,  0 ) 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 (2.2) and (2.3) 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.
After obtaining the COG point of fuzzy number , where  = ( 1 ,  2 ,  3 ,  4 ;   ), the ranking value Rank () can be calculated as where The larger the value Rank (), 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.
To compensate for these shortcomings, a new index of ranking fuzzy numbers is constructed in this article.

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.
Step 1.For each generalized trapezoidal fuzzy number   , where 1 ≤  ≤ , if the fuzzy number   is not a standardized generalized trapezoidal fuzzy number, where the universe of discourse of the generalized trapezoidal fuzzy number   is [0, ], then translate the generalized trapezoidal fuzzy number   = ( 1 ,  2 ,  3 ,  4;    ) into a standardized generalized trapezoidal fuzzy number  *  shown as follows: where Step 2. Use formula (6) where Step 5. Use the new point ( where  min denotes the minimum value of the  1 ,  2 ,  3 and  4 , and 1 ≤  ≤ .From formula (12), we can see that ( *  ) can be considered as the Euclidean distance between the point ( ) and the point ( min , 0).We can say that the larger the value of ( *  ), the better the ranking of  *  , where 1 ≤  ≤ .(13) 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].

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  To be able to search data on the web to be used in teaching 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   , 1 ≤  ≤ 5,   ∈ { 1, 2, 3, 4, 5}, (  ) denotes the degree that the teacher satisfies the th satisfaction level of the criterion   , and ∑ 5 =1 (  ) = 1.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   , 1 ≤  ≤ 5,   ∈ { 1, 2, 3, 4, 5}, (  ) denotes the degree of percentage that the criterion   satisfies the th importance level, and ∑ 5 =1 (  ) = 1.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  < 1, then let  = 1; if  > 5, then let  = 5.For example, assume that there are five evaluators  1 ,  2 ,  3 ,  4 ,  5 and nine preceptors  1 ,  2 ,  3 ,  4 ,  5 ,  6 ,  7 ,  8 ,  9 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 1 ≤  ≤ 5, 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  = [ 1  2  3  4  5 ]  ,   denotes the average of the remaining fuzzy weights with respect to the th criterion, and 1 ≤  ≤ 5.
For example, let us consider the first row of Table 6.Based on (12), we can see that (20) Therefore, the fuzzy weight of criterion  1 is  1 , where  1 = (3.07,4.07, 5).Now we consider the second row of Table 6   In the same way, we can get the weights  3 ,  4 , and  5 of the criteria  3 ,  4 , and  5 , respectively, where  3 = (2.87,3.87, 4.87),  4 = (2.98,3.98, 4.97), and  5 = (2.9, 3.9, 4.9).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  < 1, then let  = 1; if  > 5, then let  = 5.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      (2.9, 3.9, 4.9) (2.8, 3.8, 4.8) (2.9, 3.9, 4.9) (3.3, 4.3, 5) (2.9, 3.9, 4.9)

VL L M H VH VL L M H VH VL L M H VH VL L M H VH VL L M H VH
and smallest ranking value, respectively, we drop the fuzzy weights 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 1 ≤  ≤ , 1 ≤  ≤  and  denotes the number of preceptors, and  denotes the number of criteria.For example, based on (26) 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 1 ≤  ≤ .For example, based on the fuzzy weighted vector  obtained in Step

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
1 ,  2 ] → [0,] and    : [ 3 ,  4 ] → [0,] are two strictly monotonically and continuous mappings from  to closed interval [0, ].When  = 1, 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,  = ( 1 ,  2 ,  3 ,  4 ; ).In particular, if  2 =  3 , 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    : [0, ] → [ 1 ,  2 ] and    : [0, ] → [ 3 ,  4 ] be the inverse functions of    and    , respectively.Then    () and    () should be integrable on the closed interval [0, ].In other words, both ∫ Step 4. Use the standard deviation   *  and  * to calculate the centroid point ( *  *  ,  *  *  ) of each standardized generalized trapezoidal fuzzy number  *  , where 1 ≤  ≤ .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 1 ≤  ≤ . ) to derive a new value  * ŝ  *  shown as follows:

Table 1 :
The criteria for evaluating the efficiency of high school preceptors.

Table 2 :
A fuzzy linguistic questionnaire for evaluating the efficiency of a preceptor.

Table 3 :
A fuzzy linguistic questionnaire for determining the weight of the criterion   .
Because the fuzzy weights (3.75, 4.75, 5) and (2.8, 3.8, 4.8) have the largest ranking value and smallest ranking value, respectively, we drop the fuzzy weights (3.75, 4.75, 5) and (2.8, 3.8, 4.8), and then we can get the fuzzy weight  2 of criterion  2 by calculating the average of the remaining fuzzyweights, shown as follows:

Table 4 :
The result after five evaluators till out the fuzzy linguistic questionnaires.

Table 5 :
The fuzzy weight of each criterion evaluated by each evaluator.
(7)nd then we can get the fuzzy grade  11,1 of preceptor  1 with respect to sub-criterion  11 by calculating the average of the remaining fuzzy weights, shown as follows:Therefore, the fuzzy grade of preceptor  1 with respect to sub-criterion  11 is  11,1 , where  11,1 = (2.67,3.67,4.67).Then, based on(7), we can rank the fuzzy grades of preceptor  2 with respect to criterion  12 evaluated by each evaluator.Because the fuzzy weights (3.7, 4.7, 5) and (2.3, 3.3, 4.3) have the largest ranking value and smallest ranking value, respectively, we drop the fuzzy weights (3.7, 4.7, 5) and (2.3, 3.3, 4.3), and then we can get the fuzzy grade  12,1 of preceptor  1 with respect to sub-criterion  12 by calculating the average of the remaining fuzzy weights shown as follows: = (2.8,3.8, 4.73) .

Table 11 ,
we can calculate  11 as follows:

Table 6 :
Fuzzy grades evaluated by the evaluator

Table 7 :
Fuzzy grades evaluated by the evaluator

Table 10 :
Fuzzy grades evaluated by the evaluator (12)he fuzzy grade matrix  obtained in Step 2 and (28), we can calculate the total fuzzy grade  1 of teacher  1 as follows:Step 4. Based on(12), calculate the ranking value (  ) of the fuzzy grade   , where 1 ≤  ≤ .The larger the value of (  ), the better the ranking of   and the better the choice of preceptor   , where 1 ≤  ≤  and  denotes the number of teachers.For example, after applying(12)to rank the fuzzy numbers  1 ,  2 ,  3 ,  4 ,  5 ,  6 ,  7 ,  8 , and  9 of the fuzzy grade vector  obtained in Step 3, we can get the ranking results as shown in Table12.We can see that  5 >  2 >  4 >  1 >  3 >  6 >  9 >  7 >  8 .Because  5 has the largest ranking value, preceptor  5 is the best choice.

Table 12 :
Ranking values of the total fuzzy grades.