Evaluating the Efficiency of School Preceptors by Fuzzy Risk Analysis

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 𝑅 and is completely defined by its membership function. Let 𝐴 be a fuzzy number, whose membership function 𝑓_𝐴(𝑥) can generally be defined as [5–9], 𝑓_𝐴(𝑥)=𝑓^𝐿_𝐴(𝑥)when𝑎₁≤𝑥≤𝑎₂,𝜔when𝑎₂≤𝑥≤𝑎₃,𝑓^𝑅_𝐴(𝑥)when𝑎₃≤𝑥≤𝑎₄,0,otherwise, (1) where 0≤𝜔≤1 is a constant, 𝑓^𝐿_𝐴∶[𝑎₁,𝑎₂]→[0,𝜔] and 𝑓^𝑅_𝐴∶[𝑎₃,𝑎₄]→[0,𝜔] are two strictly monotonically and continuous mappings from 𝑅 to closed interval [0,𝜔]. When 𝜔=1, then 𝐴 is a normal fuzzy number; otherwise, it is said to be a nonnormal fuzzy number. If the membership function 𝑓_𝐴(𝑥) is piecewise linear, then 𝐴 is referred to as a trapezoidal fuzzy number and is usually denoted by, 𝐴=(𝑎₁,𝑎₂,𝑎₃,𝑎₄;𝜔). In particular, if 𝑎₂=𝑎₃, the trapezoidal fuzzy number is reduced to a triangular fuzzy number. Since 𝑓^𝐿_𝐴(𝑥) and 𝑓^𝑅_𝐴(𝑥) are both strictly monotonically and continuous functions, their inverse functions exist and should also be continuous and strictly monotonically. Let 𝑔^𝐿_𝐴∶[0,𝜔]→[𝑎₁,𝑎₂] and 𝑔^𝑅_𝐴∶[0,𝜔]→[𝑎₃,𝑎₄] be the inverse functions of 𝑓^𝐿_𝐴 and 𝑓^𝑅_𝐴, respectively. Then 𝑔^𝐿_𝐴(𝑦) and 𝑔^𝑅_𝐴(𝑦) should be integrable on the closed interval [0,𝜔]. In other words, both ∫^𝜔₀𝑔^𝐿_𝐴(𝑦)𝑑𝑦 and ∫^𝜔₀𝑔^𝑅_𝐴(𝑦)𝑑𝑦 should exist. In the case of trapezoidal fuzzy number, the inverse function 𝑔^𝐿_𝐴(𝑦) and 𝑔^𝑅_𝐴(𝑦) can be analytically expressed as 𝑔^𝐿_𝐴𝑦=𝑎₁+𝑎₂−𝑎₁𝑦𝜔𝑔,0≤𝑦≤𝜔,_𝑅𝐴𝑦=𝑎₄−𝑎₄−𝑎₃𝑦𝜔,0≤𝑦≤𝜔. (2) In order to determine the centroid point (𝑥₀,𝑦₀) of a fuzzy number 𝐴, Wang et al. [10] provided the following centroid formulae: 𝑥₀(𝐴)=∫^𝑎2_𝑎1𝑥𝑓^𝐿_𝐴(𝑥)𝑑𝑥+∫^𝑎3_𝑎2(𝑥𝜔)𝑑𝑥+∫^𝑎4_𝑎3𝑥𝑓^𝑅_𝐴(𝑥)𝑑𝑥∫^𝑎2_𝑎1𝑓^𝐿_𝐴(𝑥)𝑑𝑥+∫^𝑎3_𝑎2(𝜔)𝑑𝑥+∫^𝑎4_𝑎3𝑓^𝑅_𝐴(𝑥),𝑑𝑥𝑦₀(𝐴)=∫^𝜔₀𝑦𝑔^𝑅_𝐴𝑦−𝑔^𝐿_𝐴𝑦𝑑𝑦∫^𝜔₀𝑔^𝑅_𝐴𝑦−𝑔^𝐿_𝐴𝑦.𝑑𝑦 (3) The ranking value 𝑅(𝐴) of the fuzzy number 𝐴 is defined as follows [11]: 𝑅(𝐴)=𝑥²₀(𝐴)+𝑦²₀(𝐴). (4) The larger the value 𝑅(𝐴), the better the ranking of 𝐴.

In [12], the authors presented a centroid-index ranking method for ordering fuzzy numbers. The centroid point of a fuzzy number 𝐴 is (𝑥_𝐴,𝑦_𝐴), where 𝑥_𝐴 and 𝑦_𝐴 are the same as formula (2.2) and (2.3) in [12]. The ranking value 𝑆(𝐴) of the fuzzy number 𝐴 is defined as follows: 𝑆(𝐴)=𝑥_𝐴×𝑦_𝐴. (5) The larger the value 𝑆(𝐴), the better the ranking of 𝐴. In [3], S. J. Chen and S. M. Chen proposed a simple method to obtain COG point of fuzzy numbers. If 𝐴 is a generalized fuzzy number, where 𝐴=(𝑎₁,𝑎₂,𝑎₃,𝑎₄;𝑤), then the COG point (𝑥^∗_𝐴,𝑦^∗_𝐴) of 𝐴 is as follows: 𝑥^∗_𝐴=𝑦^∗_𝐴𝑎₂+𝑎₃+𝑎₁+𝑎₄1−𝑦^∗_𝐴2,𝑦^∗_𝐴=𝜔𝑎₃−𝑎₂/𝑎₄−𝑎₁+26,𝑎₁≠𝑎₄,12,𝑎₁=𝑎₄. (6) After obtaining the COG point of fuzzy number 𝐴, where 𝐴=(𝑎₁,𝑎₂,𝑎₃,𝑎₄;𝑤_𝐴), the ranking value Rank(𝐴) can be calculated as Rank(𝐴)=𝑥^∗_𝐴+𝑤_𝐴−𝑦^∗_𝐴𝑠_𝐴̂×𝑦^∗_𝐴+0.5^1−𝑤𝐴, (7) where 𝑠_𝐴̂=∑⁴_𝑖=1𝑎_𝑖−𝑎²3,𝑎𝑎=¹+𝑎₂+𝑎₃+𝑎₄4. (8) The larger the value Rank(𝐴), the better the ranking 𝐴. 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 𝐴 and 𝐵 are shown as follows (Figure 1): (),().𝐴=−0.01,−0.01,−0.01,−0.01;1𝐵=0.01,0.01,0.01,0.01;0.8 (9) It can be easily to obtain the COG points of fuzzy numbers 𝐴 and 𝐵, respectively, as follows, (𝑥^∗_𝐴,𝑦^∗_𝐴)=(−0.01,0.5) and (𝑥^∗_𝐵,𝑦^∗_𝐵)=(0.01,0.4). By applying Chen method, we have 𝑅(𝐴)=0.99 and 𝑅(𝐵)=0.989. The ranking result shows that ranking order is 𝐴≻𝐵. However, it can be easily seen that the correct order is 𝐴≺𝐵.

Figure 1 

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 𝑛 generalized fuzzy numbers 𝐴₁,𝐴₂,…,𝐴_𝑛, where 𝐴_𝑗=(𝑎_1𝑗,𝑎_2𝑗,𝑎_3𝑗,𝑎_4𝑗;𝑤_𝐴𝑗), 1≤𝑗≤𝑛, 0<𝑤_𝐴𝑗≤1, 0≤𝑎_1𝑗≤𝑎_2𝑗≤𝑎_3𝑗≤𝑎_4𝑗≤𝑘, 𝑘 is any real value. The value 𝑤_𝐴𝑗 denotes the maximum membership value of the generalized trapezoidal fuzzy number 𝐴_𝑗, where 1≤𝑗≤𝑛. The proposed method for ranking fuzzy numbers 𝐴₁,𝐴₂,…,𝐴_𝑛 is now presented as follows.

Step 1. For each generalized trapezoidal fuzzy number 𝐴_𝑗, where 1≤𝑗≤𝑛, if the fuzzy number 𝐴_𝑗 is not a standardized generalized trapezoidal fuzzy number, where the universe of discourse of the generalized trapezoidal fuzzy number 𝐴_𝑗 is [0,𝑘], then translate the generalized trapezoidal fuzzy number 𝐴_𝑗=(𝑎_1𝑗,𝑎_2𝑗,𝑎_3𝑗,𝑎_4𝑗;𝑤_𝐴𝑗) into a standardized generalized trapezoidal fuzzy number 𝐴^∗_𝑗 shown as follows: 𝐴^∗_𝑗=𝑎_1𝑗𝑘,𝑎_2𝑗𝑘,𝑎_3𝑗𝑘,𝑎_4𝑗𝑘;𝑤_𝐴^∗_𝑗=𝑎^∗_1𝑗,𝑎^∗_2𝑗,𝑎^∗_3𝑗,𝑎^∗_4𝑗;𝑤_𝐴^∗_𝑗, (10) where 𝑤_𝐴𝑗=𝑤_𝐴^∗_𝑗, 0<𝑤_𝐴^∗_𝑗≤1, 0≤𝑎^∗_1𝑗≤𝑎^∗_2𝑗≤𝑎^∗_3𝑗≤𝑎^∗_4𝑗≤1 or −1≤𝑎^∗_1𝑗≤𝑎^∗_2𝑗≤𝑎^∗_3𝑗≤𝑎^∗_4𝑗≤1 and 1≤𝑗≤𝑛.

Step 2. Use formula (6) to calculate the centroid point (𝑥^∗_𝐴^∗_𝑗,𝑦^∗_𝐴∗𝑗) of each standardized generalized trapezoidal fuzzy number 𝐴^∗_𝑗, where 1≤𝑗≤𝑛.

Step 3. Calculate the standard deviation 𝑠_𝐴^∗_𝑗 of each standardized generalized trapezoidal fuzzy number 𝐴^∗_𝑗 from (8). The standard deviation 𝑠_𝐴^∗_𝑗 indicates the degree of dispersion of the standardized generalized trapezoidal fuzzy number 𝐴^∗_𝑗, where 1≤𝑗≤𝑛.

Step 4. Use the standard deviation 𝑠_𝐴^∗_𝑗 and 𝑦^∗_𝐴∗𝑗 of the centroid point (𝑥^∗_𝐴^∗_𝑗,𝑦^∗_𝐴∗𝑗) to derive a new value 𝑦^∗𝑠_𝐴∗𝑗̂ shown as follows: 𝑦^∗𝑠_𝐴∗𝑗̂=𝑤_𝐴^∗_𝑗−𝑦^∗_𝐴∗𝑗𝑠̂_𝐴^∗_𝑗×𝑦^∗_𝐴∗𝑗+0.5^{1−𝑤𝐴∗𝑗}, (11) where 1≤𝑗≤𝑛, to obtain a new point (𝑥^∗_𝐴^∗_𝑗,𝑦^∗𝑠_𝐴∗𝑗̂).

Step 5. Use the new point (𝑥^∗_𝐴^∗_𝑗,𝑦_𝐴^∗𝑠_∗𝑗) to calculate the ranking value 𝑆(𝐴^∗_𝑗) of the fuzzy numbers 𝐴^∗_𝑗 as follows: 𝑆𝐴^∗_𝑗=𝑥^∗_𝐴^∗_𝑗−𝑎_min²+𝑦^∗𝑠_𝐴∗𝑗̂−0^21/2=𝑥^∗_𝐴^∗_𝑗−𝑎_min²+𝑦^∗𝑠̂_𝐴∗𝑗^21/2, (12) where 𝑎_min denotes the minimum value of the 𝑎_1𝑗,𝑎_2𝑗,𝑎_3𝑗 and 𝑎_4𝑗, and 1≤𝑗≤𝑛. From formula (12), we can see that 𝑆(𝐴^∗_𝑗) can be considered as the Euclidean distance between the point (𝑥^∗_𝐴^∗_𝑗,𝑦^∗𝑠_𝐴∗𝑗̂) and the point (𝑎_min,0). We can say that the larger the value of 𝑆(𝐴^∗_𝑗), the better the ranking of 𝐴^∗_𝑗, where 1≤𝑗≤𝑛.

Example 2. As shown above, two generalized fuzzy numbers are shown as follows: (),().𝐴=−0.01,−0.01,−0.01,−0.01;1𝐵=0.01,0.01,0.01,0.01;0.8 (13) First, the COG points of the fuzzy numbers 𝐴 and 𝐵 can be obtained as follows, respectively. 𝑥^∗_𝐴^∗,𝑦^∗_𝐴∗=(),𝑥−0.01,0.5_∗𝐵∗,𝑦^∗_𝐵∗=(0.01,0.4). (14) Then, translate the COG points of fuzzy numbers 𝐴 and 𝐵 into Index point. The results can be shown as follows:   𝑥^∗_𝐴^∗=−0.01, 𝑦_𝐴^∗𝑠_∗=(𝑤_𝐴^∗−𝑦^∗_𝐴∗)^𝑠𝐴∗×(𝑦^∗_𝐴∗+0.5)^{1−𝑤𝐴∗}=1, (15)  𝑥^∗_𝐵^∗=0.01, 𝑦_𝐵^∗𝑠_∗=(𝑤_𝐵^∗−𝑦^∗_𝐵∗)^𝑠𝐵∗×(𝑦^∗_𝐵∗+0.5)^{1−𝑤𝐵∗}=1. (16)The ranking value of fuzzy numbers 𝐴 and 𝐵 can be calculated, respectively, as follows: 𝑆(𝐴)=1,𝑆(𝐵)=1.0002. (17) The ranking order of fuzzy numbers 𝐴 and 𝐵 is 𝐵≻𝐴. 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 𝑋_𝑖𝑗, that is, very low (VL), low (L), middle (M), high (H) and very high (VH). They used the triangular fuzzy numbers 1, 2, 3, 4, and 5 corresponding to the linguistic satisfaction levels VL, L, M, H, and VH, respectively, where 1=(0,1,2), 2=(1,2,3), 3=(2,3,4), 4=(3,4,5), and 5=(4,5,6). The fuzzy grade 𝐺(𝑥_𝑖𝑗) of the subcriterion 𝑋_𝑖𝑗 is calculated as follows: 𝐺𝑥_𝑖𝑗=∑⁵_𝑘=1𝑥_𝑖𝑗𝑘𝑓𝑥_𝑖𝑗𝑘∑⁵_𝑘=1𝑓𝑥_𝑖𝑗𝑘, (18) where 𝑖 denotes the index of the criterion, 𝑗 denotes the index of the subcriterion, 𝑘 denotes the index of linguistic levels, 𝑥_𝑖𝑗𝑘 denotes the 𝑘th linguistic satisfaction level of the criterion 𝑋_𝑖𝑗, 1≤𝑘≤5, 𝑥_𝑖𝑗𝑘∈{1,2,3,4,5}, 𝑓(𝑥_𝑖𝑗𝑘) denotes the degree that the teacher satisfies the 𝑘th satisfaction level of the criterion 𝑋_𝑖𝑗, and ∑⁵_𝑘=1𝑓(𝑥_𝑖𝑗𝑘)=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 𝑊(𝑥_𝑖) of the criterion 𝑋_𝑖 is calculated as follows: 𝑊𝑥_𝑗=∑⁵_𝑘=1𝑥_𝑖𝑘𝑓𝑥_𝑖𝑘∑⁵_𝑘=1𝑓𝑥_𝑖𝑘, (19) where 𝑖 denotes the index of the criterion, 𝑘 denotes the index of linguistic levels, 𝑥_𝑖𝑘 denotes the 𝑘th importance level of the criterion 𝑋_𝑖, 1≤𝑘≤5, 𝑥_𝑖𝑘∈{1,2,3,4,5}, 𝑓(𝑥_𝑖𝑘) denotes the degree of percentage that the criterion 𝑋_𝑖 satisfies the 𝑘th importance level, and ∑⁵_𝑘=1𝑓(𝑥_𝑖𝑗𝑘)=1. The proposed method for high school preceptors appraisal is now presented 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 𝑋_𝑖.

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 (𝑘,𝑚,𝑛) should satisfy the rules.
If 𝑘<1, then let 𝑘=1; if 𝑛>5, then let 𝑛=5. For example, assume that there are five evaluators 𝐸₁, 𝐸₂, 𝐸₃, 𝐸₄, 𝐸₅ and nine preceptors 𝐴₁, 𝐴₂, 𝐴₃, 𝐴₄, 𝐴₅, 𝐴₆, 𝐴₇, 𝐴₈, 𝐴₉ 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 𝑋_𝑖 evaluated by the evaluators, where 1≤𝑖≤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 𝑊, where 𝑊=[𝑤₁𝑤₂𝑤₃𝑤₄𝑤₅]^𝑡, 𝑤_𝑖 denotes the average of the remaining fuzzy weights with respect to the 𝑖th criterion, and 1≤𝑖≤5.
For example, let us consider the first row of Table 6. Based on (12), we can see that (3.3,4.3,5)>(3.2,4.2,5)>(3,4,5)=(3,4,5)>(2.3,3.3,4.3). Because the fuzzy weights (3.3,4.3,5) and (2.3,3.3,4.3) have the largest ranking value and smallest ranking value, respectively, we drop the fuzzy weights (3.3,4.3,5) and (2.3,3.3,4.3), and then we can get the fuzzy weight 𝑤₁ of criterion 𝑋₁ by calculating the average of the remaining fuzzy weights, shown as follows: 𝑤₁=)⊕()⊕(=)⊕()⊕(⊘()=().[(3.2,4.2,53,4,53,4,5)]⊘3[(3.2,4.2,53,4,53,4,5)]3,3,33.07,4.07,5 (20) Therefore, the fuzzy weight of criterion 𝑋₁ is 𝑤₁, where 𝑤₁=(3.07,4.07,5). Now we consider the second row of Table 6. Based on (12), we can see that (3.75,4.75,5)>(3.6,4.6,5)>(3.1,4.1,5) > (2.9,3.9,4.9)>(2.8,3.8,4.8). 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 𝑤₂ of criterion 𝑋₂ by calculating the average of the remaining fuzzyweights, shown as follows: 𝑤₂=)⊕()⊕()⊕()⊕(⊘()=().[(3.6,4.6,53.1,4.1,52.9,3.9,4.9)]⊘3[(3.6,4.6,53.1,4.1,52.9,3.9,4.9)]3,3,33.2,4.2,4.97 (21) In the same way, we can get the weights 𝑤₃, 𝑤₄, and 𝑤₅ of the criteria 𝑋₃, 𝑋₄, and 𝑋₅, respectively, where 𝑤₃=(2.87,3.87,4.87), 𝑤₄=(2.98,3.98,4.97), and 𝑤₅=(2.9,3.9,4.9). In this situation, the fuzzy weighted vector 𝑊 is as follows: 𝑤𝑊=¹𝑤₂𝑤₃𝑤₄𝑤₅=()()()()().3.07,4.07,5.003.20,4.20,4.972.87,3.87,4.872.98,3.98,4.972.90,3.90,4.90 (22)

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.

Table 6 

Fuzzy grades evaluated by the evaluator 𝐸₁.

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 (𝑘,𝑚,𝑛) should satisfy the rules: if 𝑘<1, then let 𝑘=1; if 𝑛>5, then let 𝑛=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 𝐴₁ with respect to criterion 𝑋₁₁ evaluated by each evaluator, where the result is as follows: (3.7,4.7,5)>(2.8,3.8,4.8)>(2.7,3.7,4.7) > (2.5,3.5,4.5)=(2.5,3.5,4.5). Because the fuzzy weights (3.7,4.7,5) and (2.5,3.5,4.5) have the largest ranking value and smallest ranking value, respectively, we drop the fuzzy weights (3.7,4.7,5) and (2.5,3.5,4.5), and then we can get the fuzzy grade 𝑔_11,1 of preceptor 𝐴₁ with respect to sub-criterion 𝑋₁₁ by calculating the average of the remaining fuzzy weights, shown as follows: 𝑔_11,1=)⊕()⊕(=)⊕()⊕(⊘()=().[(2.8,3.8,4.82.7,3.7,4.72.5,3.5,4.5)]⊘3[(2.8,3.8,4.82.7,3.7,4.72.5,3.5,4.5)]3,3,32.67,3.67,4.67 (23) Therefore, the fuzzy grade of preceptor 𝐴₁ with respect to sub-criterion 𝑋₁₁ is 𝑔_11,1, where 𝑔_11,1=(2.67,3.67,4.67). Then, based on (7), we can rank the fuzzy grades of preceptor 𝐴₂ with respect to criterion 𝑋₁₂ evaluated by each evaluator. The result is shown as follows: (3.7,4.7,5)>(3.2,4.2,5) > (2.6,3.6,4.6)=(2.6,3.6,4.6)>(2.3,3.3,4.3). 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 𝑔_12,1 of preceptor 𝐴₁ with respect to sub-criterion 𝑋₁₂ by calculating the average of the remaining fuzzy weights shown as follows: 𝑔_12,1=)⊕()⊕(=)⊕(⊘()=().[(3.2,4.2,52.6,3.6,4.62.6,3.6,4.6)]⊘3[(3.2,4.2,52.6,3.6,4.6)(2.6,3.6,4.6)]3,3,32.8,3.8,4.73 (24) 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 𝐺 defined as follows: 𝐴𝐺=𝑋𝑋₁𝑋₂⋯𝑋_𝑘𝐴₁𝑔₁₁𝑔₁₂⋯𝑔_1𝑘𝐴₂𝑔₂₁𝑔₂₂⋯𝑔_2𝑘𝐴⋮⋮⋱⋯⋮_𝑛𝑔_𝑛1𝑔_𝑛2⋯𝑔_𝑛𝑘, (25) where 𝑔_𝑖𝑗 denotes the fuzzy grade of the 𝑖th preceptor 𝐴_𝑖 with respect to the 𝑗th criterion 𝑋_𝑗, where 1≤𝑖≤𝑛, 1≤𝑗≤𝑘 and 𝑛 denotes the number of preceptors, and 𝑘 denotes the number of criteria. For example, based on Table 11, we can calculate 𝑔₁₁ as follows:𝑔₁₁=(2.67+2.8+2.97+2.33+2.72+2.38+2.83+1.85+2.93)9,(3.67+3.8+3.97+3.33+3.72+3.38+3.83+2.85+3.93)9,(4.67+4.73+4.92+4.33+4.5+4.38+4.75+3.85+4.83)9=(2.61,3.61,4.55). (26)In the same way, we can get the fuzzy grade matrix 𝐺 as follows:𝐴𝐺=¹𝐴₂𝐴₃𝐴₄𝐴₅𝐴₆𝐴₇𝐴₈𝐴₉()()()()()()()()().2.61,3.61,4.55)(2.41,3.41,4.40)(2.44,3.44,4.38)(2.63,3.63,4.52)(2.28,3.28,4.282.58,3.58,4.54)(2.54,3.54,4.53)(2.49,3.49,4.47)(2.45,3.45,4.45)(2.49,3.49,4.422.57,3.54,4.52)(2.77,3.77,4.65)(2.58,3.58,4.58)(2.32,3.32,4.32)(2.10,3.10,4.102.70,3.70,4.65)(2.56,3.56,4.54)(2.46,3.46,4.46)(2.49,3.49,4.48)(2.25,3.25,4.252.36,3.36,4.33)(2.44,3.44,4.43)(2.79,3.79,4.78)(2.42,3.42,4.36)(2.67,3.67,4.642.33,3.33,4.20)(2.25,3.25,4.23)(2.29,3.29,4.23)(2.20,3.20,4.15)(1.91,2.91,3.912.40,3.40,4.38)(1.96,2.96,3.96)(2.57,3.57,4.46)(2.22,3.22,4.22)(1.55,2.41,3.412.01,2.98,3.97)(2.24,3.24,4.24)(2.09,3.09,4.09)(2.31,3.31,4.27)(2.00,3.00,3.962.61,3.61,4.55)(2.41,3.41,4.40)(2.44,3.44,4.38)(2.63,3.63,4.52)(2.28,3.28,4.28 (27)

Table 7 

Fuzzy grades evaluated by the evaluator 𝐸₂.

Table 8 

Fuzzy grades evaluated by the evaluator 𝐸₃.

Table 9 

Fuzzy grades evaluated by the evaluator 𝐸₄.

Table 10 

Fuzzy grades evaluated by the evaluator 𝐸₅.

Table 11 

The averaged fuzzy grades table.

Step 3. Calculate the total fuzzy grade vector 𝑅, where 𝑔𝑅=𝐺⊗𝑊=¹¹𝑔₁₂⋯𝑔_1𝑘𝑔₂₁𝑔₂₂⋯𝑔_2𝑘𝑔⋮⋮⋱⋮_𝑛1𝑔_𝑛2⋯𝑔_𝑛𝑘⊗𝑤₁𝑤₂⋮𝑤_𝑘=𝑔₁₁⊗𝑤₁⊕𝑔₁₂⊗𝑤₂⊕⋯⊕𝑔_1𝑘⊗𝑤_𝑘𝑔₂₁⊗𝑤₁⊕𝑔₂₂⊗𝑤₂⊕⋯⊕𝑔_2𝑘⊗𝑤_𝑘⋮𝑔_𝑛1⊗𝑤₁⊕𝑔_𝑛2⊗𝑤₂⊕⋯⊕𝑔_𝑛𝑘⊗𝑤_𝑘=𝑅₁𝑅₂𝑅₃𝑅₄, (28) where 𝑅_𝑖 denotes the total fuzzy grade of the 𝑖th preceptor 𝐴_𝑖 and 1≤𝑖≤𝑛. For example, based on the fuzzy weighted vector 𝑊 obtained in Step 1, the fuzzy grade matrix 𝐺 obtained in Step 2 and (28), we can calculate the total fuzzy grade 𝑅₁ of teacher 𝐴₁ as follows: 𝑅₁=()⊗()⊕()⊗()⊕()⊗()⊕()⊗()⊕()=().2.61,3.61,4.553.07,4.07,52.61,3.61,4.553.2,4.2,4.972.61,3.61,4.552.87,3.87,4.872.98,3.98,4.972.98,3.98,4.972.61,3.61,4.55)(2.9,3.9,4.937.18,69.57,109.39 (29) In the same way, we can get the total fuzzy grade vector 𝑅 shown as follows: 𝑅𝑅=¹𝑅₂𝑅₃𝑅₄𝑅₅𝑅₆𝑅₇𝑅₈𝑅₉=()()()()()()()()(),𝑅37.18,69.57,109.3937.72,70.29,110.7637.16,69.4,109.5837.49,69.57,110.6238.02,70.72,111.3533.02,64.02,102.4132.13,62.30,100.9832.02,62.57,101.4732.58,62.84,101.55_∗1=37.18,12069.57,120109.39,𝑅120_∗2=37.72,12070.29,120110.76,𝑅120_∗3=37.16,12069.40,120109.58,𝑅120_∗4=37.49,12069.57,120110.62,𝑅120_∗5=38.02,12070.72,120111.35,𝑅120_∗6=33.02,12064.02,120102.41,𝑅120_∗7=32.13,12062.30,120100.98,𝑅120_∗8=32.02,12062.57,120101.47,𝑅120_∗9=32.58,12062.84,120101.55.120 (30)

Step 4. Based on (12), calculate the ranking value 𝑆(𝑅_𝑖) of the fuzzy grade 𝑅_𝑖, where 1≤𝑖≤𝑛. The larger the value of 𝑆(𝑅_𝑖), the better the ranking of 𝑅_𝑖 and the better the choice of preceptor 𝐴_𝑖, where 1≤𝑖≤𝑛 and 𝑛 denotes the number of teachers. For example, after applying (12) to rank the fuzzy numbers 𝑅₁, 𝑅₂, 𝑅₃, 𝑅₄, 𝑅₅, 𝑅₆, 𝑅₇, 𝑅₈, and 𝑅₉ of the fuzzy grade vector 𝑅 obtained in Step 3, we can get the ranking results as shown in Table 12. We can see that 𝑅₅>𝑅₂>𝑅₄>𝑅₁>𝑅₃>𝑅₆>𝑅₉>𝑅₇>𝑅₈. Because 𝑅₅ has the largest ranking value, preceptor 𝐴₅ is the best choice.

Table 12 

Ranking values of the total fuzzy grades.

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.