# Evaluating the Efficiency of School Preceptors by Fuzzy Risk Analysis

Authors: Rasoul Saneifard; Rahim Saneifard
Journal: Mathematical Problems in Engineering
 (2013)

---

## 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.


---

## Body

## 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 lineR and is completely defined by its membership function. Let A be a fuzzy number, whose membership function fA(x) can generally be defined as [5–9],
(1)fA(x)={fAL(x)whena1≤x≤a2,ωwhena2≤x≤a3,fAR(x)whena3≤x≤a4,0,otherwise,
where 0≤ω≤1 is a constant, fAL:[a1,a2]→[0,ω] and fAR:[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 fA(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 fAL(x) and fAR(x) are both strictly monotonically and continuous functions, their inverse functions exist and should also be continuous and strictly monotonically. Let gAL:[0,ω]→[a1,a2] and gAR:[0,ω]→[a3,a4] be the inverse functions of fAL and fAR, respectively. Then gAL(y) and gAR(y) should be integrable on the closed interval [0,ω]. In other words, both ∫0ωgAL(y)dy and ∫0ωgAR(y)dy should exist. In the case of trapezoidal fuzzy number, the inverse function gAL(y) and gAR(y) can be analytically expressed as
(2)gAL(y)=a1+(a2-a1)yω,0≤y≤ω,gAR(y)=a4-(a4-a3)yω,0≤y≤ω.
In order to determine the centroid point (x¯0,y¯0) of a fuzzy number A, Wang et al. [10] provided the following centroid formulae:
(3)x¯0(A)=∫a1a2xfAL(x)dx+∫a2a3(xω)dx+∫a3a4xfAR(x)dx∫a1a2fAL(x)dx+∫a2a3(ω)dx+∫a3a4fAR(x)dx,y¯0(A)=∫0ωy(gAR(y)-gAL(y))dy∫0ω(gAR(y)-gAL(y))dy.
The ranking value R(A) of the fuzzy number A is defined as follows [11]:
(4)R(A)=x¯02(A)+y¯02(A).
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:
(5)S(A)=x¯A×y¯A.
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 (xA*,yA*) of A is as follows:
(6)xA*=yA*(a2+a3)+(a1+a4)(1-yA*)2,yA*={ω((a3-a2)/(a4-a1)+2)6,a1≠a4,12,a1=a4.
After obtaining the COG point of fuzzy number A, where A=(a1,a2,a3,a4;wA), the ranking value Rank(A) can be calculated as
(7)Rank(A)=xA*+(wA-yA*)s^A×(yA*+0.5)1-wA,
where
(8)s^A=∑i=14(ai-a¯)23,a¯=a1+a2+a3+a44.
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 numberA and B are shown as follows (Figure 1):
(9)A=(-0.01,-0.01,-0.01,-0.01;1),B=(0.01,0.01,0.01,0.01;0.8).
It can be easily to obtain the COG points of fuzzy numbers A and B, respectively, as follows, (xA*,yA*)=(-0.01,0.5) and (xB*,yB*)=(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 1To 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 aren generalized fuzzy numbers A1,A2,…,An, where Aj=(a1j,a2j,a3j,a4j;wAj), 1≤j≤n, 0<wAj≤1, 0≤a1j≤a2j≤a3j≤a4j≤k, k is any real value. The value wAj denotes the maximum membership value of the generalized trapezoidal fuzzy number Aj, where 1≤j≤n. The proposed method for ranking fuzzy numbers A1,A2,…,An is now presented as follows.Step 1.
For each generalized trapezoidal fuzzy numberAj, where 1≤j≤n, if the fuzzy number Aj is not a standardized generalized trapezoidal fuzzy number, where  the universe of discourse of the generalized trapezoidal fuzzy number Aj is [0,k], then translate the generalized trapezoidal fuzzy number Aj=(a1j,a2j,a3j,a4j;wAj) into a standardized generalized trapezoidal fuzzy number Aj* shown as follows:
(10)Aj*=(a1jk,a2jk,a3jk,a4jk;wAj*)=(a1j*,a2j*,a3j*,a4j*;wAj*),
where wAj=wAj*, 0<wAj*≤1, 0≤a1j*≤a2j*≤a3j*≤a4j*≤1 or -1≤a1j*≤a2j*≤a3j*≤a4j*≤1 and 1≤j≤n.Step 2.
Use formula (6) to calculate the centroid point (xAj**,yAj**) of each standardized generalized trapezoidal fuzzy number Aj*, where 1≤j≤n.Step 3.
Calculate the standard deviationsAj* of each standardized generalized trapezoidal fuzzy number Aj* from (8). The standard deviation sAj* indicates the degree of dispersion of the standardized generalized trapezoidal fuzzy number Aj*, where 1≤j≤n.Step 4.
Use the standard deviationsAj* and  yAj** of the centroid point (xAj**,yAj**) to derive a new value yAj**s^ shown as follows:
(11)yAj**s^=(wAj*-yAj**)s^Aj*×(yAj**+0.5)1-wAj*,
where 1≤j≤n, to obtain a new point (xAj**,yAj**s^).Step 5.
Use the new point(xAj**,yAj**s) to calculate the ranking value S(Aj*) of the fuzzy numbers Aj* as follows:
(12)S(Aj*)=[(xAj**-amin)2+(yAj**s^-0)2]1/2=[(xAj**-amin)2+(yAj**s^)2]1/2,
where amin denotes the minimum value of the a1j,a2j,a3j and a4j, and 1≤j≤n. From formula (12), we can see that S(Aj*) can be considered as the Euclidean distance between the point (xAj**,yAj**s^) and the point (amin,0). We can say that the larger the value of S(Aj*), the better the ranking of Aj*, where 1≤j≤n.Example 2.
As shown above, two generalized fuzzy numbers are shown as follows:(13)A=(-0.01,-0.01,-0.01,-0.01;1),B=(0.01,0.01,0.01,0.01;0.8).
First, the COG points of the fuzzy numbers A and B can be obtained as follows, respectively.
(14)(xA**,yA**)=(-0.01,0.5),(xB**,yB**)=(0.01,0.4).
Then, translate the COG points of fuzzy numbers A and B into Index point. The results can be shown as follows: xA**=-0.01,
(15)yA**s=(wA*-yA**)sA*×(yA**+0.5)1-wA*=1,xB**=0.01,
(16)yB**s=(wB*-yB**)sB*×(yB**+0.5)1-wB*=1.The ranking value of fuzzy numbers A and B can be calculated, respectively, as follows:
(17)S(A)=1,S(B)=1.0002.
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 Xij, 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(xij) of the subcriterion Xij is calculated as follows:
(18)G(xij)=∑k=15xijkf(xijk)∑k=15f(xijk),
where i denotes the index of the criterion, j denotes the index of the subcriterion, k denotes the index of linguistic levels, xijk denotes the kth linguistic satisfaction level of the criterion Xij, 1≤k≤5, xijk∈{1~,2~,3~,4~,5~}, f(xijk) denotes the degree that the teacher satisfies the kth satisfaction level of the criterion Xij, and ∑k=15f(xijk)=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(xi) of the criterion Xi is calculated as follows:
(19)W(xj)=∑k=15xikf(xik)∑k=15f(xik),
where i denotes the index of the criterion, k denotes the index of linguistic levels, xik denotes the kth importance level of the criterion Xi, 1≤k≤5, xik∈{1~,2~,3~,4~,5~}, f(xik) denotes the degree of percentage that the criterion Xi satisfies the kth importance level, and ∑k=15f(xijk)=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.
Criteria
Subcriteria
Contents
X
1 (teaching)
X
11
According to the standard of the course and the needs of the students to achieve the teaching goals
X
12
To make teaching plans and to prepare teaching materials and tools properly
X
13
To present teaching materials clearly and correctly with proper methods
X
14
To bring the motivation of learning and testing and to increase students’ attention on lessons
X
15
To express clearly in speaking (helpful for comprehension and learning)
X
16
To express clearly in writing (helpful for comprehension and learning)
X
17
To make good use of various teaching methods and teaching media
X
18
To make good use of time while teaching
X
19
Giving proper feedback to students while teaching
X
2 (class managing)
X
21
Counseling students to obey the rules of life and develop good habits
X
22
To direct students to keep personal and public sanitation
X
23
Management of classroom
X
24
Establishing an appropriate rule of class
X
25
To praise and punish students properly
X
3 (professional attitude)
X
31
To take care of students actively
X
32
Working hard
X
33
Researching and pursuing further education
X
34
Being enthusiastic with working
X
4 (cooperation in administration)
X
41
To build various kinds of students’ files
X
42
To join activities of school actively
X
43
To keep good relationship with coworkers in school
X
44
To keep good relationship and interactions with students’ parents
X
45
Treating students fairly
X
46
To keep good relationship with students
X
5 (ability to use computers)
X
51
To be able to send and receive email
X
52
To be able to design a web page interacting with students
X
53
To be able to use word edit software to edit handouts and test papers
X
54
To be able to search data on the web to be used in teachingTable 2
A fuzzy linguistic questionnaire for evaluating the efficiency of a preceptor.
Fuzzy linguistic questionnaire—satisfaction levels of the criterionXij
Low
High
←
→
VL
L
M
H
VH
0%
0%
0%
60%
40%Table 3
A fuzzy linguistic questionnaire for determining the weight of  the criterionXi.
Fuzzy linguistic questionnaire—Importance levels of the criterionXi
Low
High
←
→
VL
L
M
H
VH
0%
0%
0%
60%
40%Step 1.
Determine the fuzzy weight of each criterion. Let each evaluator use the fuzzy linguistic questionnaire shown in Table3 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.
Ifk<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 Xi 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, wi 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 Table6. 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 w~1 of criterion X1 by calculating the average of the remaining fuzzy weights, shown as follows:
(20)w1=[(3.2,4.2,5)⊕(3,4,5)⊕(3,4,5)]⊘3=[(3.2,4.2,5)⊕(3,4,5)⊕(3,4,5)]⊘(3,3,3)=(3.07,4.07,5).
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:
(21)w2=[(3.6,4.6,5)⊕(3.1,4.1,5)⊕(2.9,3.9,4.9)]⊘3[(3.6,4.6,5)⊕(3.1,4.1,5)⊕(2.9,3.9,4.9)]⊘(3,3,3)=(3.2,4.2,4.97).
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:
(22)W=[w1w2w3w4w5]=[(3.07,4.07,5.00)(3.20,4.20,4.97)(2.87,3.87,4.87)(2.98,3.98,4.97)(2.90,3.90,4.90)].Table 4
The result after five evaluators till out the fuzzy linguistic questionnaires.
Criterion
EvaluatorE1
EvaluatorE2
EvaluatorE3
EvaluatorE4
EvaluatorE5
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
X
1
0
0
0
80
20
0
0
10
80
10
0
0
20
60
20
0
0
0
70
30
0
0
80
10
10
X
2
0
0
10
70
20
0
0
0
40
60
0
0
30
60
10
0
0
20
70
10
0
0
5
15
80
X
3
0
20
80
0
0
0
0
20
70
10
0
0
0
70
30
0
0
20
70
10
0
0
20
80
0
X
4
0
0
20
60
20
0
0
5
85
10
0
0
20
70
10
0
0
10
75
15
0
0
45
55
0
X
5
0
0
30
50
20
0
0
40
40
20
0
0
20
70
10
0
0
20
30
50
0
0
20
70
10Table 5
The fuzzy weight of each criterion evaluated by each evaluator.
Criterion
Evaluator
E
1
E
2
E
3
E
4
E
5
X
1
(
3.2
,
4.2
,
5
)
(
3
,
4
,
5
)
(
3
,
4
,
5
)
(
3.3
,
4.3
,
5
)
(
2.3
,
3.3
,
4.3
)
X
2
(
3.1
,
4.1
,
5
)
(
3.6
,
4.6
,
5
)
(
2.8
,
3.8
,
4.8
)
(
2.9
,
3.9
,
4.9
)
(
3.75
,
4.75
,
5
)
X
3
(
1.8
,
2.8
,
3.8
)
(
2.9
,
3.9
,
4.9
)
(
3.3
,
4.3
,
5
)
(
2.9
,
3.9
,
9
)
(
2.8
,
3.8
,
4.8
)
X
4
(
3
,
4
,
5
)
(
3.05
,
4.05
,
5
)
(
2.9
,
3.9
,
4.9
)
(
3.05
,
4.05
,
5
)
(
2.55
,
3.55
,
4.55
)
X
5
(
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
)Table 6
Fuzzy grades evaluated by the evaluatorE1.
Criteria
Subcriteria
PreceptorA1
PreceptorA2
PreceptorA3
PreceptorA4
PreceptorA5
PreceptorA6
PreceptorA7
PreceptorA8
PreceptorA9
X
1
X
11
(
2.5,3.5,4.5
)
(
2,3
,
4
)
(
1.6,2.6
,
3.6
)
(
1.5,2.5,3.5
)
(
2.8,3.8,4.8
)
(
1.6,2.6,3.6
)
(
2.9,3.9,4.9
)
(
2.5,3.5,4.5
)
(
2.3,3.3,4.3
)
X
12
(
3.2,4.2,5
)
(
1.4,2.4,3.4
)
(
2.9,3.9,4.9
)
(
2.8,3.8,4.8
)
(
1.5,2.5,3.5
)
(
4,5
,
5
)
(
1.8,2.8,3.8
)
(
1.5,2.5,3.5
)
(
1.8,2.8,3.8
)
X
13
(
3.7,4.7,5
)
(
2.9,3.9,4.9
)
(
3.7,4.7,5
)
(
3.5,4.5,5
)
(
1,1.2,2.2
)
(
3.6,4.6,5
)
(
3.6,4.6,5
)
(
2.3,3.3,4.3
)
(
1.5,2.5,3.5
)
X
14
(
1.8,2.8,3.8
)
(
1.2,2.2,3.2
)
(
2.9,3.9,4.9
)
(
3.3,4.3,5
)
(
1.5,2.5,3.5
)
(
1.6,2.6,3.6
)
(
3.3,4.3,5
)
(
1.7,2.7,3.7
)
(
2.3,3.3,4.3
)
X
15
(
1,1.6,2.6
)
(
3.3,4.3,5
)
(
1.1,2
,
2.2
)
(
2.3,3.3,4.3
)
(
2.5,3.5,4.5
)
(
1,1.2,2.2
)
(
1.8,2.8,3.8
)
(
3.35,4.35,5
)
(
2.9,3.9,4.9
)
X
16
(
1.5,2.5,3.5
)
(
3.7,4.7,5
)
(
2.4,3.4,4.4
)
(
1.7,2.7,3.7
)
(
1.4,2.4,3.4
)
(
2,3
,
4
)
(
3.5,4.5,5
)
(
1.6,2.6,3.6
)
(
3.3,4.3,5
)
X
17
(
1.4,2.4,3.4
)
(
2.7,3.7,4.7
)
(
2.8,3.8,4.8
)
(
1,1.2,2.2
)
(
3.7,4.7,5
)
(
1.7,2.7,3.7
)
(
2.1,3.1,4.1
)
(
1,1.2,2.2
)
(
2.1,3.1,4.1
)
X
18
(
1,1.1,2.1
)
(
1.6,2.6,3.6
)
(
2.9,3.9,4.9
)
(
4,5
,
5
)
(
1.2,2.2,3.2
)
(
1.7,2.7,3.7
)
(
1.9,2.9,3.9
)
(
2,3
,
4
)
(
1,1.2,2.2
)
X
19
(
2.5,3.5,4.5
)
(
3.2,4.2,5
)
(
2.7,3.7,4.7
)
(
2.9,3.9,4.9
)
(
1.4,2.4,3.4
)
(
1.2
.
2.2,3.2
)
(
3.7,4.7,5
)
(
1.2,2.2,3.2
)
(
1.7,2.7,3.7
)
X
2
X
21
(
2.1,3.1,4.1
)
(
2.1,3.1,4.1
)
(
2.3,3.3,4.3
)
(
2.6,3.6,4.6
)
(
2.9,3.9,4.9
)
(
2.7,3.7,4.7
)
(
2.6,3.6,4.6
)
(
1.5,2.5,3.5
)
(
2.5,3.5,4.5
)
X
22
(
2.7,3.7,4.7
)
(
3.2,4.2,5
)
(
2.6,3.6,4.6
)
(
1.8,2.8,3.8
)
(
2.1,3.1,4.1
)
(
2.1,3.1,4.1
)
(
1.4,2.4,3.4
)
(
2.3,3.3,4.3
)
(
2.9,3.9,4.9
)
X
23
(
2.6,3.6,4.6
)
(
1.5,2.5,3.5
)
(
3.7,4.7,5
)
(
1.9,2.9,3.9
)
(
1.6,2.6,3.6
)
(
3.2,4.2,5
)
(
1.6,2.6,3.6
)
(
1.4,2.4,3.4
)
(
3.3,4.3,5
)
X
24
(
2.5,3.5,4.5
)
(
2.6,3.6,4.6
)
(
2.5,3.5,4.5
)
(
2.6,3.6,4.6
)
(
1,1.2,2.2
)
(
2.6,3.6,4.6
)
(
1.5,2.5,3.5
)
(
1.2,2.2,3.2
)
(
2.9,3.9,4.9
)
X
25
(
3.2,4.2,5
)
(
2.5,3.5,4.5
)
(
1.5,2.5,3.5
)
(
2.5,3.5,4.5
)
(
2.9,3.9,4.9
)
(
2,3
,
4
)
(
1.8,2.8,3.8
)
(
2.7,3.7,4.7
)
(
2.1,3.1,4.1
)
X
3
X
31
(
2.2,3.2,4.2
)
(
2.6,3.6,4.6
)
(
2.1,3.1,4.1
)
(
2.1,3.1,4.1
)
(
2.6,3.6,4.6
)
(
1.6,2.6,3.6
)
(
1.8,2.8,3.8
)
(
2.9,3.9,4.9
)
(
1.7,2.7,3.7
)
X
32
(
1.7,2.7,3.7
)
(
1.8,2.8,3.8
)
(
2.6,3.6,4.6
)
(
3.7,4.7,5
)
(
3.7,4.7,5
)
(
1,1.1,2.1
)
(
2.5,3.5,4.5
)
(
2,3
,
4
)
(
1.4,2.4,3.4
)
X
33
(
2.7,3.7,4.7
)
(
1.2,2.2,3.2
)
(
2.7,3.7,4.7
)
(
2.3,3.3,4.3
)
(
1,1.2,2.2
)
(
2.5,3.5,4.5
)
(
3.7,4.7,5
)
(
1.7,2.7,3.7
)
(
2.5,3.5,4.5
)
X
34
(
2.6,3.6,4.6
)
(
2.1,3.1,4.1
)
(
1.7,2.7,3.7
)
(
2.8,3.8,4.8
)
(
1.4,2.4,3.4
)
(
3.7,4.7,5
)
(
3.5,4.5,5
)
(
1.6,2.6,3.6
)
(
1.5,2.5,3.5
)
X
4
X
41
(
2.6,3.6,4.6
)
(
2.7,3.7,4.7
)
(
2.2,3.2,4.2
)
(
2.7,3.7,4.7
)
(
2.3,3.3,4.3
)
(
3.6,4.6,5
)
(
2.1,3.1,4.1
)
(
3.7,4.7,5
)
(
1.7,2.7,3.7
)
X
42
(3.8,4.8,5)
(2.3,3.3,4.3)
(2.2,3.2,4.2)
(1.7,2.7,3.7)
(2.5,3.5,4.5)
(2.6,3.6,4.6)
(1.9,2.9,3.9)
(1.8,2.8,3.8)
(3.7,4.7,5)
X
43
(2.6,3.6,4.6)
(2.5,3.5,4.5)
(1.4,2.4,3.4)
(1.4,2.4,3.4)
(1.8,2.8,3.8)
(1.2,2.2,3.2)
(1,1.2,2.2)
(2.9,3.9,4.9)
(1.6,2.6,3.6)
X
44
(1.7,2.7,3.7)
(2.5,3.5,4.5)
(1.2,2.2,3.2)
(2.7,3.7,4.7)
(1.4,2.4,3.4)
(1.4,2.4,3.4)
(2.6,3,6,4.6)
(1,1.2,2.2)
(3.3,4.3,5)
X
45
(2.6,3.6,4.6)
(2.1,3.1,4.1)
(2.6,3.6,4.6)
(2.3,3.3,4.3)
(2.6,3.6,4.6)
(2.6,3.6,4.6)
(2.9,3.9,4.9)
(2.3,3.3,4.3)
(1,1.2,2.2)
X
46
(1.8,2.8,3.8)
(2.3,3.3,4.3)
(2.8,3.8,4.8)
(2.1,3.1,4.1)
(1.7,2.7,3.7)
(1.7,2.7,3.7)
(1.7,2.7,3.7)
(2.5,3.5,4.5)
(2.9,3.9,4.9)
X
5
X
51
(2.2,3.2,4.2)
(2.5,3.5,4.5)
(2.6,3.6,4.6)
(2.9,3.9,4.9)
(2.8,3.8,4.8)
(2.8,3.8,4.8)
(1.6,2.6,3.6)
(3.5,4.5,5)
(3.3,4.3,5)
X
52
(1.4,2.4,3.4)
(2.7,3.7,4.7)
(2.1,3.1,4.1)
(2.3,3.3,4.3)
(2.9,3.9,4.9)
(2.9,3.9,4.9)
(1,1.2,2.2)
(1.5,2.5,3.5)
(1.4,2.4,3.4)
X
53
(2.1,3.1,4.1)
(2.6,3.6,4.6)
(1.5,2.5,3.5)
(1.2,2.2,3.2)
(1.2,2.2,3.2)
(1.2,2.2,3.2)
(1.8,2.8,3.8)
(3.7,4.7,5)
(1,1.2,2.2)
X
54
(2.3,3.3,4.3)
(1.6,2.6,3.6)
(2.8,3.8,4.8)
(1.8,2.8,3.8)
(2.9,3.9,4.9)
(2.9,3.9,4.9)
(1,1.2,2.2)
(2,3,4)
(1.2,2.2,3.2)Step 2.
Establish the fuzzy grade matrix. Each evaluator uses the fuzzy linguistic questionnaire shown in Table3 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 g11,1 of preceptor A1 with respect to sub-criterion X11 by calculating the average of the remaining fuzzy weights, shown as follows:
(23)g11,1=[(2.8,3.8,4.8)⊕(2.7,3.7,4.7)⊕(2.5,3.5,4.5)]⊘3=[(2.8,3.8,4.8)⊕(2.7,3.7,4.7)⊕(2.5,3.5,4.5)]⊘(3,3,3)=(2.67,3.67,4.67).
Therefore, the fuzzy grade of preceptor A1 with respect to sub-criterion X11 is g11,1, where g11,1=(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 g12,1 of preceptor A1 with respect to sub-criterion X12 by calculating the average of the remaining fuzzy weights shown as follows:
(24)g12,1=[(3.2,4.2,5)⊕(2.6,3.6,4.6)⊕(2.6,3.6,4.6)]⊘3=[(3.2,4.2,5)⊕(2.6,3.6,4.6)(2.6,3.6,4.6)]⊘(3,3,3)=(2.8,3.8,4.73).
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:
(25)G=[AXX1X2⋯XkA1g11g12⋯g1kA2g21g22⋯g2k⋮⋮⋱⋯⋮Angn1gn2⋯gnk],
where gij denotes the fuzzy grade of the ith preceptor Ai with respect to the jth criterion Xj, 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:(26)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).In the same way, we can get the fuzzy grade matrix G as follows:(27)G=A1A2A3A4A5A6A7A8A9[(2.61,3.61,4.55)(2.41,3.41,4.40)(2.44,3.44,4.38)(2.63,3.63,4.52)(2.28,3.28,4.28)(2.58,3.58,4.54)(2.54,3.54,4.53)(2.49,3.49,4.47)(2.45,3.45,4.45)(2.49,3.49,4.42)(2.57,3.54,4.52)(2.77,3.77,4.65)(2.58,3.58,4.58)(2.32,3.32,4.32)(2.10,3.10,4.10)(2.70,3.70,4.65)(2.56,3.56,4.54)(2.46,3.46,4.46)(2.49,3.49,4.48)(2.25,3.25,4.25)(2.36,3.36,4.33)(2.44,3.44,4.43)(2.79,3.79,4.78)(2.42,3.42,4.36)(2.67,3.67,4.64)(2.33,3.33,4.20)(2.25,3.25,4.23)(2.29,3.29,4.23)(2.20,3.20,4.15)(1.91,2.91,3.91)(2.40,3.40,4.38)(1.96,2.96,3.96)(2.57,3.57,4.46)(2.22,3.22,4.22)(1.55,2.41,3.41)(2.01,2.98,3.97)(2.24,3.24,4.24)(2.09,3.09,4.09)(2.31,3.31,4.27)(2.00,3.00,3.96)(2.61,3.61,4.55)(2.41,3.41,4.40)(2.44,3.44,4.38)(2.63,3.63,4.52)(2.28,3.28,4.28)].Table 7
Fuzzy grades evaluated by the evaluatorE2.
Criteria
Subcriteria
PreceptorA1
PreceptorA2
PreceptorA3
PreceptorA4
PreceptorA5
PreceptorA6
PreceptorA7
PreceptorA8
PreceptorA9
X
1
X
11
(
2.8,3.8,4.8
)
(
3.1,4.1,5
)
(
2.6,3.6
,
4.6
)
(
2.8,3.8,4.8
)
(
1.4,2.4,3.4
)
(
1.6,2.6,3.6
)
(
2.7,3.7,4.7
)
(
2.9,3.9,4.9
)
(
2.8,3.8,4.8
)
X
12
(
2.6,3.6,4.6
)
(
1.8,2.8,3.8
)
(
3.7,4.7,5
)
(
2.8,3.8,4.8
)
(
2.3,3.3,4.3
)
(
2.6,3.6,4.6
)
(
2.7,3.7,4.7
)
(
1.8,2.8,3.8
)
(
2.3,3.3,4.3
)
X
13
(
3.1,4.1,5
)
(
2.9,3.9,4.9
)
(
1.6,2.6,3.6
)
(
1.7,2.7,3.7
)
(
2.6,3.6,4.6
)
(
1,2
,
3
)
(
3.3,4.3,5
)
(
2.3,3.3,4.3
)
(
1,2
,
3
)
X
14
(
2.5,3.5,4.5
)
(
2.7,3.7,4.7
)
(
1.8,2.8,3.8
)
(
3.6,4.6,5
)
(
3.7,4.7,5
)
(
1.8,2.8,3.8
)
(
2.7,3.7,4.7
)
(
3.1,4.1,5
)
(
2.7,3.7,4.7
)
X
15
(
3.7,4.7,5
)
(
2.5,3.5,4.5
)
(
2.7,3.7,4.7
)
(
1.2,2.2,3.2
)
(
1.7,2.7,3.7
)
(
2.3,3.3,4.3
)
(
1.2,2.2,3.2
)
(
2.7,3.7,4.7
)
(
2.1,3.1,4.1
)
X
16
(
2.6,3.6,4.6
)
(
2.8,3.8,4.8
)
(
2.5,3.5,4.5
)
(
2.6,3.6,4.6
)
(
1,1.2,2.2
)
(
3.1,4.1,5
)
(
2.1,3.1,4.1
)
(
1.9,2.9,3.9
)
(
2.9,3.9,4.9
)
X
17
(
3.1,4.1,5
)
(
1.2,2.2,3.2
)
(
1.8,2.8,3.8
)
(
2,3
,
4
)
(
1.2,2.2,3.2
)
(
1.4,2.4,3.4
)
(
2.3,3.3,4.4
)
(
1.2,2.2,3.2
)
(
2.3,3.3,4.3
)
X
18
(
1.2,2.2,3.2
)
(
3.1,4.1,5
)
(
1.5,2.5,3.5
)
(
2.7,3.7,4.7
)
(
3.7,4.7,5
)
(
2.6,3.6,4.6
)
(
1,1.1,2.1
)
(
2.1,3.1,4.1
)
(
1.4,2.4,3.4
)
X
19
(
1.2,2.2,3.2
)
(
2.1,3.1,4.1
)
(
2.6,3.6,4.6
)
(
2.9,3.9,4.9
)
(
3.2,4.2,5
)
(
3.7,4.7,5
)
(
1.4,2.4,3.4
)
(
2.3,3.3,4.3
)
(
1.7,2.7,3.7
)
X
2
X
21
(
2.6,3.6,4.6
)
(
2.7,3.7,4.7
)
(
3.1,4.1,5
)
(
2.6,3.6,4.6
)
(
2.3,3.3,4.3
)
(
2.6,3.6,4.6
)
(
2.1,3.1,4.1
)
(
2.6,3.6,4.6
)
(
2.5,3.5,4.5
)
X
22
(
2.7,3.7,4.7
)
(
2.6,3.6,4.6
)
(
1.5,2.5,3.5
)
(
1.2,2.2,3.2
)
(
3.7,4.7,5
)
(
2,3
,
4
)
(
3.7,4.7,5
)
(
3.3,4.3,5
)
(
2.9,3.9,4.9
)
X
23
(
3.7,4.7,5
)
(
2.3,3.3,4.3
)
(
2.6,3.6,4.6
)
(
2.7,3.7,4.7
)
(
2.8,3.8,4.8
)
(
1,1.2,2.2
)
(
1.7,2.7,3.7
)
(
4,5
,
5
)
(
3.3,4.3,5
)
X
24
(
1.2,2.2,3.2
)
(
3.1,4.1,5
)
(
1.8,2.8,3.8
)
(
2.1,3.1,4.1
)
(
1.5,2.5,3.5
)
(
3.5,4.5,5
)
(
1,1.2,2.2
)
(
2.3,3.3,4.3
)
(
2.9,3.9,4.9
)
X
25
(
3.1,4.1,5
)
(
2.7,3.7,4.7
)
(
3.7,4.7,5
)
(
3.7,4.7,5
)
(
2.3,3.3,4.3
)
(
3.2,4.2,5
)
(
2.3,3.3,4.3
)
(
2.7,3.7,4.7
)
(
1.7,2.7,3.7
)
X
3
X
31
(
3.7,4.7,5
)
(
2.6,3.6,4.6
)
(
3.1,4.1,5
)
(
2.8,3.8,4.8
)
(
2.6,3.6,4.6
)
(
2.3,3.3,4.3
)
(
2.8,3.8,4.8
)
(
2.7,3.7,4.7
)
(
2.5,3.5,4.5
)
X
32
(
2.1,3.1,4.1
)
(
2.8,3.8,4.8
)
(
2.5,3.5,4.5
)
(
2.3,3.3,4.3
)
(
1.2,2.2,3.2
)
(
2.3,3.3,4.3
)
(
2.7,3.7,4.7
)
(
2.9,3.9,4.9
)
(
3.3,4.3,5
)
X
33
(
3.7,4.7,5
)
(
3.7,4.7,5
)
(
2.8,3.8,4.8
)
(
2.6,3.6,4.6
)
(
2.1,3.1,4.1
)
(
2.3,3.3,4.3
)
(
2.5,3.5,4.5
)
(
2.3,3.3,4.3
)
(
1.6,2.6,3.6
)
X
34
(
2.3,3.3,4.3
)
(
2.5,3.5,4.5
)
(
2.6,3.6,4.6
)
(
1.8,2.8,3.8
)
(
3.1,4.1,5
)
(
3.6,4.6,5
)
(
3.1,4.1,5
)
(
1.5,2.5,3.5
)
(
2.5,3.5,4.5
)
X
4
X
41
(
2.5,3.5,4.5
)
(
1.5,2.5,3.5
)
(
1.8,2.8,3.8
)
(
2.7,3.7,4.7
)
(
1.8,2.8,3.8
)
(
2.9,3.9,4.9
)
(
1.8,2.8,3.8
)
(
2.9,3.9,4.9
)
(
1.4,2.4,3.4
)
X
42
(
2.6,3.6,4.6
)
(
2.7,3.7,4.7
)
(
2.7,3.7,4.7
)
(
2.7,3.7,4.7
)
(
1.5,2.5,3.5
)
(
2.3,3.3,4.3
)
(
1.7,2.7,3.7
)
(
1,1.2,2.2
)
(
2.9,3.9,4.9
)
X
43
(
3.7,4.7,5
)
(
1.2,2.2,3.2
)
(
1.4,2.4,3.4
)
(
2.5,3.5,4.5
)
(
3.7,4.7,5
)
(
1,1.2,2.2
)
(
2.5,3.5,4.5
)
(
2.5,3.5,4.5
)
(
3.3,4.3,5
)
X
44
(
1.5,2.5,3.5
)
(
2.3,3.3,4.3
)
(
3.7,4.7,5
)
(
1.8,2.8,3.8
)
(
3.2,4.2,5
)
(
1.7,2.7,3.7
)
(
2.1,3
,
1,4.1
)
(
1.6,2.6,3.6
)
(
3.2,4.2,5
)
X
45
(
2.7,3.7,4.7
)
(
1.5,2.5,3.5
)
(
3.2,4.2,5
)
(
2.3,3.3,4.3
)
(
2.9,3.9,4.9
)
(
1.2,2.2,3.2
)
(
1.4,2.4,3.4
)
(
3.7,4.7,5
)
(
1,1.1,2.1
)
X
46
(
2.3,3.3,4.3
)
(
2.7,3.7,4.7
)
(
2.9,3.9,4.9
)
(
1.8,2.8,3.8
)
(
2.5,3.5,4.5
)
(
1.4,2.4,3.4
)
(
2.5,3.5,4.5
)
(
2.1,3.1,4.1
)
(
1.7,2.7,3.7
)
X
5
X
51
(
2.8,3.8,4.8
)
(
2.1,3.1,4.1
)
(
2.7,3.7,4.7
)
(
2.5,3.5,4.5
)
(
2.8,3.8,4.8
)
(
2,3
,
4
)
(
1,1.2,2.2
)
(
1.9,2.9,3.9
)
(
1.7,2.7,3.7
)
X
52
(
1.8,2.8,3.8
)
(
1.2,2.2,3.2
)
(
2.9,3.9,4.9
)
(
1.4,2.4,3.4
)
(
2,3
,
4
)
(
1,1.2,2.2
)
(
1.8,2.8,3.8
)
(
2,3
,
4
)
(
3.3,4.3,5
)
X
53
(
2.1,3.1,4.1
)
(
3.1,4.1,5
)
(
1.2,2.2,3.2
)
(
1.8,2.8,3.8
)
(
2.9,3.9,4.9
)
(
2.1,3.1,4.1
)
(
1.2,2.2,3.3
)
(
3.2,4.2,5
)
(
2,3
,
4
)
X
54
(
2.8,3.8,4.8
)
(
2.5,3.5,4.5
)
(
2.1,3.1,4.1
)
(
2.8,3.8,4.8
)
(
2.5,3.5,4.5
)
(
1.7,2.7,3.7
)
(
1.5,2.5,3.5
)
(
1.4,2.4,3.4
)
(
1.2,2.2,3.2
)Table 8
Fuzzy grades evaluated by the evaluatorE3.
Criteria
Subcriteria
PreceptorA1
PreceptorA2
PreceptorA3
PreceptorA4
PreceptorA5
PreceptorA6
PreceptorA7
PreceptorA8
PreceptorA9
X
1
X
11
(
2.8,3.8,4.8
)
(
3.1,4.1,5
)
(
2.6,3.6
,
4.6
)
(
2.8,3.8,4.8
)
(
1.4,2.4,3.4
)
(
1.6,2.6,3.6
)
(
2.7,3.7,4.7
)
(
2.9,3.9,4.9
)
(
2.8,3.8,4.8
)
X
12
(
2.6,3.6,4.6
)
(
1.8,2.8,3.8
)
(
3.7,4.7,5
)
(
2.8,3.8,4.8
)
(
2.3,3.3,4.3
)
(
2.6,3.6,4.6
)
(
2.7,3.7,4.7
)
(
1.8,2.8,3.8
)
(
2.3,3.3,4.3
)
X
13
(
3.1,4.1,5
)
(
2.9,3.9,4.9
)
(
1.6,2.6,3.6
)
(
1.7,2.7,3.7
)
(
2.6,3.6,4.6
)
(
1,2
,
3
)
(
3.3,4.3,5
)
(
2.3,3.3,4.3
)
(
1,2
,
3
)
X
14
(
2.5,3.5,4.5
)
(
2.7,3.7,4.7
)
(
1.8,2.8,3.8
)
(
3.6,4.6,5
)
(
3.7,4.7,5
)
(
1.8,2.8,3.8
)
(
2.7,3.7,4.7
)
(
3.1,4.1,5
)
(
2.7,3.7,4.7
)
X
15
(
3.7,4.7,5
)
(
2.5,3.5,4.5
)
(
2.7,3.7,4.7
)
(
1.2,2.2,3.2
)
(
1.7,2.7,3.7
)
(
2.3,3.3,4.3
)
(
1.2,2.2,3.2
)
(
2.7,3.7,4.7
)
(
2.1,3.1,4.1
)
X
16
(
2.6,3.6,4.6
)
(
2.8,3.8,4.8
)
(
2.5,3.5,4.5
)
(
2.6,3.6,4.6
)
(
1,1.2,2.2
)
(
3.1,4.1,5
)
(
2.1,3.1,4.1
)
(
1.9,2.9,3.9
)
(
2.9,3.9,4.9
)
X
17
(
3.1,4.1,5
)
(
1.2,2.2,3.2
)
(
1.8,2.8,3.8
)
(
2,3
,
4
)
(
1.2,2.2,3.2
)
(
1.4,2.4,3.4
)
(
2.3,3.3,4.4
)
(
1.2,2.2,3.2
)
(
2.3,3.3,4.3
)
X
18
(
1.2,2.2,3.2
)
(
3.1,4.1,5
)
(
1.5,2.5,3.5
)
(
2.7,3.7,4.7
)
(
3.7,4.7,5
)
(
2.6,3.6,4.6
)
(
1,1.1,2.1
)
(
2.1,3.1,4.1
)
(
1.4,2.4,3.4
)
X
19
(
1.2,2.2,3.2
)
(
2.1,3.1,4.1
)
(
2.6,3.6,4.6
)
(
2.9,3.9,4.9
)
(
3.2,4.2,5
)
(
3.7,4.7,5
)
(
1.4,2.4,3.4
)
(
2.3,3.3,4.3
)
(
1.7,2.7,3.7
)
X
2
X
21
(
2.6,3.6,4.6
)
(
2.7,3.7,4.7
)
(
3.1,4.1,5
)
(
2.6,3.6,4.6
)
(
2.3,3.3,4.3
)
(
2.6,3.6,4.6
)
(
2.1,3.1,4.1
)
(
2.6,3.6,4.6
)
(
2.5,3.5,4.5
)
X
22
(
2.7,3.7,4.7
)
(
2.6,3.6,4.6
)
(
1.5,2.5,3.5
)
(
1.2,2.2,3.2
)
(
3.7,4.7,5
)
(
2,3
,
4
)
(
3.7,4.7,5
)
(
3.3,4.3,5
)
(
2.9,3.9,4.9
)
X
23
(
3.7,4.7,5
)
(
2.3,3.3,4.3
)
(
2.6,3.6,4.6
)
(
2.7,3.7,4.7
)
(
2.8,3.8,4.8
)
(
1,1.2,2.2
)
(
1.7,2.7,3.7
)
(
4,5
,
5
)
(
3.3,4.3,5
)
X
24
(
1.2,2.2,3.2
)
(
3.1,4.1,5
)
(
1.8,2.8,3.8
)
(
2.1,3.1,4.1
)
(
1.5,2.5,3.5
)
(
3.5,4.5,5
)
(
1,1.2,2.2
)
(
2.3,3.3,4.3
)
(
2.9,3.9,4.9
)
X
25
(
3.1,4.1,5
)
(
2.7,3.7,4.7
)
(
3.7,4.7,5
)
(
3.7,4.7,5
)
(
2.3,3.3,4.3
)
(
3.2,4.2,5
)
(
2.3,3.3,4.3
)
(
2.7,3.7,4.7
)
(
1.7,2.7,3.7
)
X
3
X
31
(
3.7,4.7,5
)
(
2.6,3.6,4.6
)
(
3.1,4.1,5
)
(
2.8,3.8,4.8
)
(
2.6,3.6,4.6
)
(
2.3,3.3,4.3
)
(
2.8,3.8,4.8
)
(
2.7,3.7,4.7
)
(
2.5,3.5,4.5
)
X
32
(
2.1,3.1,4.1
)
(
2.8,3.8,4.8
)
(
2.5,3.5,4.5
)
(
2.3,3.3,4.3
)
(
1.2,2.2,3.2
)
(
2.3,3.3,4.3
)
(
2.7,3.7,4.7
)
(
2.9,3.9,4.9
)
(
3.3,4.3,5
)
X
33
(
3.7,4.7,5
)
(
3.7,4.7,5
)
(
2.8,3.8,4.8
)
(
2.6,3.6,4.6
)
(
2.1,3.1,4.1
)
(
2.3,3.3,4.3
)
(
2.5,3.5,4.5
)
(
2.3,3.3,4.3
)
(
1.6,2.6,3.6
)
X
34
(
2.3,3.3,4.3
)
(
2.5,3.5,4.5
)
(
2.6,3.6,4.6
)
(
1.8,2.8,3.8
)
(
3.1,4.1,5
)
(
3.6,4.6,5
)
(
3.1,4.1,5
)
(
1.5,2.5,3.5
)
(
2.5,3.5,4.5
)
X
4
X
41
(
2.5,3.5,4.5
)
(
1.5,2.5,3.5
)
(
1.8,2.8,3.8
)
(
2.7,3.7,4.7
)
(
1.8,2.8,3.8
)
(
2.9,3.9,4.9
)
(
1.8,2.8,3.8
)
(
2.9,3.9,4.9
)
(
1.4,2.4,3.4
)
X
42
(
2.6,3.6,4.6
)
(
2.7,3.7,4.7
)
(
2.7,3.7,4.7
)
(
2.7,3.7,4.7
)
(
1.5,2.5,3.5
)
(
2.3,3.3,4.3
)
(
1.7,2.7,3.7
)
(
1,1.2,2.2
)
(
2.9,3.9,4.9
)
X
43
(
3.7,4.7,5
)
(
1.2,2.2,3.2
)
(
1.4,2.4,3.4
)
(
2.5,3.5,4.5
)
(
3.7,4.7,5
)
(
1,1.2,2.2
)
(
2.5,3.5,4.5
)
(
2.5,3.5,4.5
)
(
3.3,4.3,5
)
X
44
(
1.5,2.5,3.5
)
(
2.3,3.3,4.3
)
(
3.7,4.7,5
)
(
1.8,2.8,3.8
)
(
3.2,4.2,5
)
(
1.7,2.7,3.7
)
(
2.1,3
,
1,4.1
)
(
1.6,2.6,3.6
)
(
3.2,4.2,5
)
X
45
(
2.7,3.7,4.7
)
(
1.5,2.5,3.5
)
(
3.2,4.2,5
)
(
2.3,3.3,4.3
)
(
2.9,3.9,4.9
)
(
1.2,2.2,3.2
)
(
1.4,2.4,3.4
)
(
3.7,4.7,5
)
(
1,1.1,2.1
)
X
46
(
2.3,3.3,4.3
)
(
2.7,3.7,4.7
)
(
2.9,3.9,4.9
)
(
1.8,2.8,3.8
)
(
2.5,3.5,4.5
)
(
1.4,2.4,3.4
)
(
2.5,3.5,4.5
)
(
2.1,3.1,4.1
)
(
1.7,2.7,3.7
)
X
5
X
51
(
2.8,3.8,4.8
)
(
2.1,3.1,4.1
)
(
2.7,3.7,4.7
)
(
2.5,3.5,4.5
)
(
2.8,3.8,4.8
)
(
2,3
,
4
)
(
1,1.2,2.2
)
(
1.9,2.9,3.9
)
(
1.7,2.7,3.7
)
X
52
(
1.8,2.8,3.8
)
(
1.2,2.2,3.2
)
(
2.9,3.9,4.9
)
(
1.4,2.4,3.4
)
(
2,3
,
4
)
(
1,1.2,2.2
)
(
1.8,2.8,3.8
)
(
2,3
,
4
)
(
3.3,4.3,5
)
X
53
(
2.1,3.1,4.1
)
(
3.1,4.1,5
)
(
1.2,2.2,3.2
)
(
1.8,2.8,3.8
)
(
2.9,3.9,4.9
)
(
2.1,3.1,4.1
)
(
1.2,2.2,3.3
)
(
3.2,4.2,5
)
(
2,3
,
4
)
X
54
(
2.8,3.8,4.8
)
(
2.5,3.5,4.5
)
(
2.1,3.1,4.1
)
(
2.8,3.8,4.8
)
(
2.5,3.5,4.5
)
(
1.7,2.7,3.7
)
(
1.5,2.5,3.5
)
(
1.4,2.4,3.4
)
(
1.2,2.2,3.2
)Table 9
Fuzzy grades evaluated by the evaluatorE4.
Criteria
Subcriteria
PreceptorA1
PreceptorA2
PreceptorA3
PreceptorA4
PreceptorA5
PreceptorA6
PreceptorA7
PreceptorA8
PreceptorA9
X
1
X
11
(
2.7,3.7,4.7
)
(
2.9,3.9,4.9
)
(
2.3,3.3,4.3
)
(
3.1,4.1,5
)
(
3.1,4.1,5
)
(
1.4,2.4,3.4
)
(
2.1,3.1,4.1
)
(
1.4,2.4,3.3
)
(
1.8,2.8,3.8
)
X
12
(
2.3,3.3,4.3
)
(
2.7,3.7,4.7
)
(
2.3,3.3,4.3
)
(
2.8,3.8,4.8
)
(
1.4,2.4,3.4
)
(
2.7,3.7,4.7
)
(
2.3,3.3,4.3
)
(
1.9,2.9,3.9
)
(
2.8,3.8,4.8
)
X
13
(
2.8,3.8,4.8
)
(
2.7,3.7,4.7
)
(
2.6,3.6,4.6
)
(
2.2,3.2,4.2
)
(
3.1,4.1,5
)
(
2.2,3.2,4.2
)
(
1,1.2,2.2
)
(
1.8,2.8,3.8
)
(
1.7,2.7,3.7
)
X
14
(
2.7,3.7,4.7
)
(
2.3,3.3,4.3
)
(
2.1,3.1,4.1
)
(
2.7,3.7,4.7
)
(
2.7,3.7,4.7
)
(
1.8,2.8,3.8
)
(
2.2,3.2,4.2
)
(
2.7,3.7,4.7
)
(
1.7,2.7,3.7
)
X
15
(
2.3,3.3,4.3
)
(
2.2,3.2,4.2
)
(
1,1.2,2.2
)
(
3.6,4.6,5
)
(
1.5,2.5,3.5
)
(
2.3,3.3,4.3
)
(
2.7,3.7,4.7
)
(
1.4,2.4,3.4
)
(
1.2,2.2,3.2
)
X
16
(
3.6,4.6,5
)
(
3.2,4.2,5
)
(
2.7,3.7,4.7
)
(
1,1.2,2.2
)
(
2.5,3.5,4.5
)
(
1.4,2.4,3.4
)
(
3.1,4.1,5
)
(
1.8,2.8,3.8
)
(
2.8,3.8,4.8
)
X
17
(
3.1,4.1,5
)
(
3.7,4.7,5
)
(
2.6,3.6,4.6
)
(
4,5
,
6
)
(
1.4,2.4,3.4
)
(
2.3,3.3,4.3
)
(
2.5,3.5,4.5
)
(
3.3,4.3,5
)
(
2.2,3.2,4.2
)
X
18
(
3.7,4.7,5
)
(
2.7,3.7,4.7
)
(
3.1,4.1,5
)
(
2.2,3.2,4.2
)
(
1.8,2.8,3.8
)
(
1.6,2.6,3.6
)
(
2.3,3.3,4.3
)
(
1.4,2.4,3.4
)
(
3.3,4.3,5
)
X
19
(
3.2,4.2,5
)
(
2.8,3.8,4.8
)
(
2.6,3.6,4.6
)
(
2.9,3.9,4.9
)
(
2.7,3.7,4.7
)
(
1.8,2.8,3.8
)
(
2.7,3.7,4.7
)
(
2.3,3.3,4.3
)
(
2.9,3.9,4.9
)
X
2
X
21
(
3.6,4.6,5
)
(
3.1,4.1,5
)
(
2.2,3.2,4.2
)
(
2.7,3.7,4.7
)
(
2,3
,
4
)
(
1.4,2.4,3.4
)
(
1.8,2.8,3.8
)
(
1,2
,
3
)
(
1.8,2.8,3.8
)
X
22
(
2.6,3.6,4.6
)
(
2.6,3.6,4.6
)
(
2.3,3.3,4.3
)
(
2.7,3.7,4.7
)
(
1.8,2.8,3.8
)
(
2.2,3.2,4.2
)
(
2.3,3.3,4.3
)
(
1.8,2.8,3.8
)
(
2.2,3.2,4.2
)
X
23
(
1.4,2.4,3.4
)
(
1.4,2.4,3.4
)
(
2.7,3.7,4.7
)
(
1,2
,
3
)
(
3.7,4.7,5
)
(
2.3,3.3,4.3
)
(
2.3,3.3,4.3
)
(
1.2,2.2,3.2
)
(
2.7,3.7,4.7
)
X
24
(
2.7,3.7,4.7
)
(
1.8,2.8,3.8
)
(
3.2,4.2,5
)
(
3.5,4.5,5
)
(
1.8,2.8,3.8
)
(
2.2,3.2,4.2
)
(
2.2,3.2,4.2
)
(
2.3,3.3,4.3
)
(
1.4,2.4,3.4
)
X
25
(
2.3,3.3,4.3
)
(
2.7,3.7,4.7
)
(
2.1,3.1,4.1
)
(
2.8,3.8,4.8
)
(
2.6,3.6,4.6
)
(
2.7,3.7,4.7
)
(
1.4,2.4,3.4
)
(
2.9,3.9,4.9
)
(
2.3,3.3,4.3
)
X
3
X
31
(
2.7,3.7,4.7
)
(
2.8,3.8,4.8
)
(
2.8,3.8,4.8
)
(
2.7,3.7,4.7
)
(
2.3,3.3,4.3
)
(
2.7,3.7,4.7
)
(
1.8,2.8,3.8
)
(
2.7,3.7,4.7
)
(
2.3,3.3,4.3
)
X
32
(
3.7,4.7,5
)
(
2.2,3.2,4.2
)
(
2.7,3.7,4.7
)
(
1,1.2,2.2
)
(
2.7,3.7,4.7
)
(
1.7,2.7,3.7
)
(
2.2,3.2,4.2
)
(
1.8,2.8,3.8
)
(
1.6,2.6,3.8
)
X
33
(
2.6,3.6,4.6
)
(
2.3,3.3,4.3
)
(
2.5,3.5,4.5
)
(
2.7,3.7,4.7
)
(
2.8,3.8,4.8
)
(
1.5,2.5,3.5
)
(
2.7,3.7,4.7
)
(
1.4,2.4,3.4
)
(
2.8,3.8,4.8
)
X
34
(
1.4,2.4,3.4
)
(
1.8,2.8,3.8
)
(
2.6,3.6,4.6
)
(
2.7,3.7,4.7
)
(
1.2,2.2,3.2
)
(
1.2,2.2,3.2
)
(
1.5,2.5,3.5
)
(
2,3
,
4
)
(
2.8,3.8,4.8
)
X
4
X
41
(
2.2,3.2,4.2
)
(
3.1,4.1,5
)
(
2.2,3.2,4.2
)
(
2.7,3.7,4.7
)
(
1.8,2.8,3.8
)
(
3.1,4.1,5
)
(
2.9,3.9,4.9
)
(
2.8,3.8,4.8
)
(
1.4,2.4,3.4
)
X
42
(
2.3,3.3,4.3
)
(
2.7,3.7,4.7
)
(
1.8,2.8,3.8
)
(
2.7,3.7,4.7
)
(
3.1,4.1,5
)
(
2.7,3.7,4.7
)
(
3.1,4.1,5
)
(
2.2,3.2,4.2
)
(
1.4,2.4,3.4
)
X
43
(
2.2,3.2,4.2
)
(
2.7,3.7,4.7
)
(
3.1,4.1,5
)
(
3.1,4.1,5
)
(
1.7,2.7,3.7
)
(
1,1.1,2.1
)
(
2.5,3.5,4.5
)
(
1.6,2.6,3.6
)
(
3.3,4.3,5
)
X
44
(
2.7,3.7,4.7
)
(
2.5,3.5,4.5
)
(
2.7,3.7,4.7
)
(
2.2,3.2,4.2
)
(
2.2,3.2,4.2
)
(
3.6,4.6,5
)
(
3.1,4.1
,
5
)
(
1.4,2.4,3.4
)
(
1.8,2.8,3.8
)
X
45
(
2.5,3.5,4.5
)
(
2.1,3.1,4.1
)
(
1.4,2.4,3.4
)
(
3.1,4.1,5
)
(
3.7,4.7,5
)
(
3.1,4.1,5
)
(
1.7,2.7,3.7
)
(
2,3
,
4
)
(
2.1,3.1,4.1
)
X
46
(
2.3,3.3,4.3
)
(
2.7,3.7,4.7
)
(
2.3,3.3,4.3
)
(
2.2,3.2,4.2
)
(
2.7,3.7,4.7
)
(
1.8,2.8,3.8
)
(
1.4,2.4,3.4
)
(
2.2,3.2,4.2
)
(
1.4,2.4,3.4
)
X
5
X
51
(
1.8,2.8,3.8
)
(
2.3,3.3,4.3
)
(
1,1.2,2.2
)
(
1.8,2.8,3.8
)
(
3.1,4.1,5
)
(
1,2
,
3
)
(
1,1.2,2.2
)
(
1.8,2.8,3.8
)
(
1.7,2.7,3.7
)
X
52
(
2.8,3.8,4.8
)
(
2.7,3.7,4.7
)
(
1.5,2.5,3.5
)
(
2.7,3.7,4.7
)
(
2.7,3.7,4.7
)
(
1.8,2.8,3.8
)
(
1.4,2.4,3.4
)
(
2.9,3.9,4.9
)
(
1.5,2.5,3.5
)
X
53
(
2.3,3.3,4.3
)
(
2.8,3.8,4.8
)
(
2.8,3.8,4.8
)
(
3.2,4.2,5
)
(
1.5,2.5,3.5
)
(
1.4,2.4,3.4
)
(
1.4,2.4,3.4
)
(
1.4,2.4,3.4
)
(
2,3
,
4
)
X
54
(
3.6,4.6,5
)
(
2.2,3.2,4.2
)
(
2.7,3.7,4.7
)
(
2.2,3.2,4.2
)
(
2.1,3.1,4.1
)
(
1,1.2,2.2
)
(
3.7,4.7,5
)
(
2,3
,
4
)
(
3.3,4.3,5
)Table 10
Fuzzy grades evaluated by the evaluatorE5.
Criteria
Subcriteria
PreceptorA1
PreceptorA2
PreceptorA3
PreceptorA4
PreceptorA5
PreceptorA6
PreceptorA7
PreceptorA8
PreceptorA9
X
1
X
11
(
2.5,3.5,4.5
)
(
2.2,3.2,4.2
)
(
1.6,2.6,3.6
)
(
3.2,4.2,5
)
(
2.9,3.9,4.9
)
(
3.2,4.2,5
)
(
2.7,3.7,4.7
)
(
1.6,2.6,3.6
)
(
2.9,3.9,4.9
)
X
12
(
3.7,4.7,5
)
(
1.4,2.4,3.4
)
(
3.1,4.1,5
)
(
3.6,4.6,5
)
(
2.7,3.7,4.7
)
(
1.4,2.4,3.4
)
(
1.6,2.6,3.6
)
(
3.9,4.9,5
)
(
2.2,3.2,4.2
)
X
13
(
2.8,3.8,4.8
)
(
2.7,3.7,4.7
)
(
2.7,3.7,4.7
)
(
2.7,3.7,4.7
)
(
3.1,4.1,5
)
(
2.7,3.7,4.7
)
(
1.4,2.4,3.4
)
(
3.1,4.1,5
)
(
2.9,3.9,4.9
)
X
14
(
3.7,4.7,5
)
(
2.2,3.2,4.2
)
(
1.4,2.4,3.4
)
(
2.2,3.2,4.2
)
(
2,3
,
4
)
(
1.5,2.5,3.5
)
(
2.5,3.5,4.5
)
(
1.4,2.4,3.4
)
(
1.6,2.6,3.6
)
X
15
(
3.6,4.6,5
)
(
2.7,3.7,4.7
)
(
2.9,3.9,4.9
)
(
1.8,2.8,3.8
)
(
1.4,2.4,3.4
)
(
1.9,2.9,3.9
)
(
2.9,3.9,4.9
)
(
1.5,2.5,3.5
)
(
2.8,3.8,4.8
)
X
16
(
2.2,3.2,4.2
)
(
2.9,3.9,4.9
)
(
2.7,3.7,4.7
)
(
3.2,4.2,5
)
(
2,3
,
4
)
(
3.3,4.3,5
)
(
2.8,3.8,4.8
)
(
1.9,2.9,3.9
)
(
2.9,3.9,4.9
)
X
17
(
2.2,3.2,4.2
)
(
2.9,3.9,4.9
)
(
2.5,3.5,4.5
)
(
2.7,3.7,4.7
)
(
1.6,2.6,3.6
)
(
2.9,3.9,4.9
)
(
1,1.2,2.2
)
(
1,1.2,2.2
)
(
1,1.2,2.2
)
X
18
(
1.4,2.4,3.4
)
(
1.7,2.7,3.7
)
(
3.1,4.1,5
)
(
1.9,2.9,3.9
)
(
2.7,3.7,4.7
)
(
3.2,4.2,5
)
(
1.8,2.8,3.8
)
(
1.4,2.4,3.4
)
(
3.3,4.3,5
)
X
19
(
3.1,4.1,5
)
(
2.8,3.8,4.8
)
(
3.4,4.4,5
)
(
1.8,2.8,3.8
)
(
2.8,3.8,4.8
)
(
2.7,3.7,4.7
)
(
2.1,3.1,4.1
)
(
2.9,3.9,4.9
)
(
2.8,3.8,4.8
)
X
2
X
21
(
1.8,2.8,3.8
)
(
2.7,3.7,4.7
)
(
3.7,4.7,5
)
(
2.7,3.7,4.7
)
(
2.2,3.2,4.2
)
(
0.2,1.2,2.2
)
(
2.7,3.7,4.7
)
(
1.6,2.6,3.6
)
(
1.4,2.4,3.4
)
X
22
(
1.5,2.5,3.5
)
(
1.6,2.6,3.6
)
(
3.1,4.1,5
)
(
3.6,4.6,5
)
(
2.5,3.5,4.5
)
(
1.4,2.4,3.4
)
(
1.8,2.8,3.8
)
(
2.8,3.8,4.8
)
(
1.4,2.4,3.4
)
X
23
(
1.4,2.4,3.4
)
(
2.8,3.8,4.8
)
(
3.6,4.6,5
)
(
3.3,4.3,5
)
(
2.7,3.7,4.7
)
(
2.2,3.2,4.2
)
(
1.5,2.5,3.5
)
(
2.1,3.1,4.1
)
(
1,1.2,2.2
)
X
24
(
1.8,2.8,3.8
)
(
1.4,2.4,3.4
)
(
3.2,4.2,5
)
(
2.8,3.8,4.8
)
(
2.7,3.7,4.7
)
(
2.8,3.8,4.8
)
(
1.2,2.2,3.2
)
(
1.5,2.5,3.5
)
(
1.7,2.7,3.7
)
X
25
(
1.4,2.4,3.4
)
(
2.7,3.7,4.7
)
(
2.3,3.3,4.3
)
(
2.8,3.8,4.8
)
(
3.7,4.7,5
)
(
1.2,2.2,3.2
)
(
1.8,2.8,3.8
)
(
2.7,3.7,4.7
)
(
2.1,3.1,4.1
)
X
3
X
31
(
2.7,3.7,4.7
)
(
2.2,3.2,4.2
)
(
2.7,3.7,4.7
)
(
2.9,3.9,4.9
)
(
1.4,2.4,3.4
)
(
3.1,4.1,5
)
(
1.8,2.8,3.8
)
(
2.2,3.2,4.2
)
(
1.6,2.6,3.6
)
X
32
(
1.4,2.4,3.4
)
(
1.4,2.4,3.4
)
(
1.4,2.4,3.4
)
(
2,3
,
4
)
(
4,5
,
5
)
(
3.1,4.1,5
)
(
2.8,3.8,4.8
)
(
1.7,2.7,3.7
)
(
2.2,3.2,4.2
)
X
33
(
2.7,3.7,4.7
)
(
2.8,3.8,4.8
)
(
1.8,2.8,3.8
)
(
1.6,2.6,3.6
)
(
3.7,4.7,5
)
(
1.9,2.9,3.9
)
(
3.8,4.8,5
)
(
2.5,3.5,4.5
)
(
1.2,2.2,3.2
)
X
34
(
1.8,2.8,3.8
)
(
2.7,3.7,4.7
)
(
3.7,4.7,5
)
(
2.7,3.7,4.7
)
(
3.3,4.3,5
)
(
1.8,2.8,3.8
)
(
3.7,4.7,5
)
(
2.8,3.8,4.8
)
(
2.1,3.1,4.1
)
X
4
X
41
(
2.7,3.7,4.7
)
(
3.1,4.1,5
)
(
1.4,2.4,3.4
)
(
3.1,4.1,5
)
(
1.5,2.5,3.5
)
(
1.4,2.4,3.4
)
(
1.4,2.4,3.4
)
(
1.4,2.4,3.4
)
(
2.8,3.8,4.8
)
X
42
(
3.7,4.7,5
)
(
1.4,2.4,3.4
)
(
2.9,3.9,4.9
)
(
1.4,2.4,3.4
)
(
2.2,3.2,4.2
)
(
3.3,4.3,5
)
(
2.2,3.2,4.2
)
(
2.2,3.2,4.2
)
(
1.7,2.7,3.7
)
X
43
(
3.6,4.6,5
)
(
3,4
,
5
)
(
2.7,3.7,4.7
)
(
2.7,3.7,4.7
)
(
3.2,4.2,5
)
(
1.6,2.6,3.6
)
(
3.3,4.3,5
)
(
1.4,2.4,3.4
)
(
2.8,3.8,4.8
)
X
44
(
1.4,2.4,3.4
)
(
3.1,4.1,5
)
(
2.5,3.5,4.5
)
(
2.9,3.9,4.9
)
(
1.8,2.8,3.8
)
(
2.8,3.8,4.8
)
(
2.7,3.7,4.7
)
(
3.2,4.2,5
)
(
1.6,2.6,3.6
)
X
45
(
1.4,2.4,3.4
)
(
2.7,3.7,4.7
)
(
1.6,2.6,3.6
)
(
1.9,2.9,3.9
)
(
3.7,4.7,5
)
(
1.4,2.4,3.4
)
(
3.6,4.6,5
)
(
2.8,3.8,4.8
)
(
2.1,3.1,4.1
)
X
46
(
2.1,3.1,4.1
)
(
1.8,2.8,3.8
)
(
2.7,3.7,4.7
)
(
2.1,3.1,4.1
)
(
2.8,3.8,4.8
)
(
3.1,4.1,5
)
(
3.5,4.5,5
)
(
3.7,4.7,5
)
(
3.2,4.2,5
)
X
5
X
51
(
1.5,2.5,3.5
)
(
2.7,3.7,4.7
)
(
3.7,4.7,5
)
(
3.7,4.7,5
)
(
3.3,4.3,5
)
(
3.1,4.1,5
)
(
1,1.2,2.2
)
(
1.8,2.8,3.8
)
(
2.9,3.9,4.9
)
X
52
(
2.8,3.8,4.8
)
(
3.1,4.1,5
)
(
1.4,2.4,3.4
)
(
2.5,3.5,4.5
)
(
3.9,4.9,5
)
(
1.6,2.6,3.6
)
(
2,3
,
4
)
(
1.9,2.9,3.9
)
(
1,1.2,2.2
)
X
53
(
1.9,2.9,3.9
)
(
3.7,4.7,5
)
(
1.4,2.4,3.4
)
(
2.8,3.8,4.8
)
(
3.7,4.7,5
)
(
1,1.2,2.2
)
(
1,1.2,2.2
)
(
3,4
,
5
)
(
1.7,2.7,3.7
)
X
54
(
1.4,2.4,3.4
)
(
1.6,2.6,3.6
)
(
2.1,3.1,4.1
)
(
1,1.1,2.1
)
(
3.3,4.3,5
)
(
2.7,3.7,4.7
)
(
2,3
,
4
)
(
1,2
,
3
)
(
1,1.2,2.2
)Table 11
The averaged fuzzy grades table.
Criteria
Subcriteria
PreceptorA1
PreceptorA2
PreceptorA3
PreceptorA4
PreceptorA5
PreceptorA6
PreceptorA7
PreceptorA8
PreceptorA9
X
1
X
11
(
2.6,3.6,4.6
)
(
2.7,3.7,4.7
)
(
2,3
,
4
)
(
3,4
,
4.9
)
(
2.9,3.9,4.9
)
(
2,3
,
4
)
(
2.7,3.7,4.7
)
(
2.3,3.3,4.3
)
(
2.6,3.6,4.6
)
X
12
(
2.8,3.8,4.7
)
(
1.8,2.8,3.8
)
(
3,4
,
5
)
(
3,4
,
4.9
)
(
2,3
,
4
)
(
2.7,3.7,4.7
)
(
2,3
,
4
)
(
2.3,3.3,4.2
)
(
2.4,3.4,4.4
)
X
13
(
2.9,3.9,4.9
)
(
2.8,3.8,4.8
)
(
2.7,3.7,4.7
)
(
2.5,3.5,4.5
)
(
2.8,3.8,4.7
)
(
2.1,3.1,4.1
)
(
2.4,3.4,4.3
)
(
2.1,3.1,4.1
)
(
1.9,2.9,3.9
)
X
14
(
2.3,3.3,4.3
)
(
2.1,3.1,4.1
)
(
2.2,3.2,4.2
)
(
2.7,3.7,4.6
)
(
2.8,3.8,4.6
)
(
1.6,2.6,3.6
)
(
2.5,3.5,4.5
)
(
2.2,3.2,4.2
)
(
2,3
,
4
)
X
15
(
2.7,3.7,4.5
)
(
2.6,3.6,4.6
)
(
2.2,3.2,4.2
)
(
2.3,3.3,4.3
)
(
1.8,2.8,3.8
)
(
2.1,3.1,4.1
)
(
2.4,3.4,4.4
)
(
2.1,3.1,4.1
)
(
2.2,3.2,4.2
)
X
16
(
2.3,3.3,4.3
)
(
3.2,4.2,4.9
)
(
2.7,3.7,4.7
)
(
2.4,3.4,4.4
)
(
1.9,2.9,3.9
)
(
2.8,3.8,4.6
)
(
2.9,3.9,4.9
)
(
1.9,2.9,3.9
)
(
2.9,3.9,4.9
)
X
17
(
2.8,3.8,4.7
)
(
2.8,3.8,4.8
)
(
2.3,3.3,4.3
)
(
2.6,3.6,4.5
)
(
1.8,2.8,3.8
)
(
2.2,3.2,4.2
)
(
1.8,2.8,3.8
)
(
1.2,1.9,2.9
)
(
1.9,2.9,3.9
)
X
18
(
1.8,2.8,3.8
)
(
2,3
,
4
)
(
3,4
,
4.9
)
(
2.6,3.6,4.6
)
(
2.5,3.5,4.5
)
(
2,3
,
4
)
(
2,3
,
4
)
(
1.6,2.6,3.6
)
(
2.5,3.5,4.4
)
X
19
(
2.9,3.9,4.8
)
(
2.9,3.9,4.9
)
(
2.7,3.7,4.7
)
(
2.9,3.9,4.9
)
(
2.3,3.3,4.3
)
(
2.1,3.1,4.1
)
(
2.5,3.5,4.5
)
(
2,3
,
4
)
(
2.5,3.5,4.5
)
X
2
X
21
(
2.6,3.6,4.5
)
(
2.8,3.8,4.8
)
(
2.6,3.6,4.6
)
(
2.7,3.7,4.7
)
(
2.5,3.5,4.5
)
(
1.9,2.9,3.9
)
(
2.3,3.3,4.3
)
(
1.9,2.9,3.9
)
(
1.5,2.5,3.5
)
X
22
(
2.2,3.2,4.2
)
(
2.6,3.6,4.6
)
(
2.4,3.4,4.4
)
(
1.6,2.6,3.6
)
(
2.4,3.4,4.3
)
(
2.1,3.1,4.1
)
(
1.9,2.9,3.9
)
(
2.3,3.3,4.3
)
(
2.3,3.3,4.3
)
X
23
(
2.2,3.2,4.2
)
(
1.9,2.9,3.9
)
(
3.1,4.1,4.9
)
(
2.8,3.8,4.8
)
(
2.5,3.5,4.5
)
(
2,3
,
4
)
(
1.8,2.8,3.8
)
(
2.3,3.3,4.1
)
(
2.9,3.9,4.8
)
X
24
(
2.2,3.2,4.2
)
(
2.5,3.5,4.4
)
(
3,4
,
4.8
)
(
2.6,3.6,4.6
)
(
2,3
,
4
)
(
2.5,3.5,4.5
)
(
1.6,2.6,3.6
)
(
1.7,2.7,3.7
)
(
2.5,3.5,4.5
)
X
25
(
2.7,3.7,4.7
)
(
2.7,3.7,4.7
)
(
2.5,3.5,4.4
)
(
2.9,3.9,4.8
)
(
2.7,3.7,4.7
)
(
2.6,3.6,4.5
)
(
1.9,2.9,3.9
)
(
2.8,3.8,4.8
)
(
1.9,2.9,3.9
)
X
3
X
31
(
3.0,4.0,4.8
)
(
2.4,3.4,4.4
)
(
2.5,3.5,4.5
)
(
2.8,3.8,4.8
)
(
2.5,3.5,4.5
)
(
2.4,3.4,4.4
)
(
1.8,2.8,3.8
)
(
2.5,3.5,4.5
)
(
2.1,3.1,4.1
)
X
32
(
1.7,2.7,3.7
)
(
2.2,3.2,4.2
)
(
2.7,3.7,4.7
)
(
1.7,2.7,3.7
)
(
3.7,4.7,5.7
)
(
2.1,3.1,4.1
)
(
2.6,3.6,4.6
)
(
1.8,2.8,3.8
)
(
1.8,2.8,3.8
)
X
33
(
2.7,3.7,4.7
)
(
2.7,3.7,4.7
)
(
2.4,3.4,4.4
)
(
2.4,3.4,4.4
)
(
2.5,3.5,4.5
)
(
2.2,3.2,4.2
)
(
2.9,3.9,4.7
)
(
1.8,2.8,3.8
)
(
2.3,3.3,4.3
)
X
34
(
2.2,3.2,4.2
)
(
2.4,3.4,4.4
)
(
2.5,3.5,4.5
)
(
2.7,3.7,4.7
)
(
2.3,3.3,4.2
)
(
2.3,3.3,4
)
(
2.8,3.8,4.6
)
(
2.1,3.1,4.1
)
(
2,3
,
4
)
X
4
X
41
(
2.6,3.6,4.6
)
(
2.9,3.9,4.9
)
(
2.3,3.3,4.3
)
(
2.7,3.7,4.7
)
(
1.9,2.9,3.9
)
(
2.9,3.9,4.9
)
(
1.8,2.8,3.8
)
(
2.9,3.9,4.9
)
(
1.9,2.9,3.9
)
X
42
(
3.0,4.0,4.7
)
(
2.5,3.5,4.5
)
(
1.9,2.9,3.9
)
(
2.4,3.4,4.4
)
(
2.1,3.1,4.1
)
(
2.7,3.7,4.7
)
(
1.9,2.9,3.9
)
(
1.9,2.9,3.9
)
(
2.2,3.2,4.2
)
X
43
(
3.3,4.3,4.8
)
(
2.3,3.3,4.3
)
(
2.7,3.7,4.7
)
(
2.7,3.7,4.7
)
(
2.7,3.7,4.6
)
(
1.3,2
,
3
)
(
2.6,3.6,4.6
)
(
1.9,2.9,3.9
)
(
2.5,3.5,4.4
)
X
44
(
1.9,2.9,3.9
)
(
2.4,3.4,4.4
)
(
2.8,3.8,4.8
)
(
2.4,3.4,4.4
)
(
1.9,2.9,3.9
)
(
2.4,3.4,4.4
)
(
2.6,3.6,4.6
)
(
1.5,2.5,3.5
)
(
2.7,3.7,4.6
)
X
45
(
2.6,3.6,4.6
)
(
2,3
,
4
)
(
1.9,2.9,3.9
)
(
2.6,3.6,4.5
)
(
3,4
,
4.8
)
(
2.1,3.1,4.1
)
(
2.3,3.3,4.3
)
(
2.9,3.9,4.7
)
(
1.6,2.3,3.3
)
X
46
(
2.2,3.2,4.2
)
(
2.3,3.3,4.3
)
(
2.1,3.1,4.1
)
(
2,3
,
4
)
(
2.6,3.6,4.6
)
(
1.5,2.5,3.5
)
(
1.9,2.9,3.9
)
(
2.6,3.6,4.5
)
(
2.5,3.5,4.5
)
X
5
X
51
(
2.1,3.1,4.1
)
(
2.3,3.3,4.3
)
(
2.7,3.7,4.7
)
(
2.5,3.5,4.5
)
(
3.1,4.1,4.9
)
(
2.4,3.4,4.4
)
(
1.2,1.6,2.6
)
(
1.8,2.8,3.8
)
(
2.1,3.1,4.1
)
X
52
(
2.1,3.1,4.1
)
(
2.5,3.5,4.5
)
(
1.7,2.7,3.7
)
(
2.4,3.4,4.4
)
(
2.5,3.5,4.5
)
(
1.7,2.7,3.7
)
(
1.5,2.5,3.5
)
(
1.8,2.8,3.8
)
(
1.3,2
,
3
)
X
53
(
2.2,3.2,4.2
)
(
3.2,4.2,4.9
)
(
1.6,2.6,3.6
)
(
2,3
,
4
)
(
2.1,3.1,4.1
)
(
1.7,2.7,3.7
)
(
1.5,2.5,3.5
)
(
2.6,3.6,4.5
)
(
1.7,2.7,3.7
)
X
54
(
2.6,3.6,4.6
)
(
1.8,2.8,3.8
)
(
2.3,3.3,4.3
)
(
1.9,2.9,3.9
)
(
2.8,3.8,4.8
)
(
1.7,2.7,3.7
)
(
1.8,2.8,3.8
)
(
1.6,2.6,3.6
)
(
1.9,2.9,3.9
)Step 3.
Calculate the total fuzzy grade vectorR, where
(28)R=G⊗W=[g11g12⋯g1kg21g22⋯g2k⋮⋮⋱⋮gn1gn2⋯gnk]⊗[w1w2⋮wk]=[g11⊗w1⊕g12⊗w2⊕⋯⊕g1k⊗wkg21⊗w1⊕g22⊗w2⊕⋯⊕g2k⊗wk⋮gn1⊗w1⊕gn2⊗w2⊕⋯⊕gnk⊗wk]=[R1R2R3R4],
where Ri denotes the total fuzzy grade of the ith preceptor Ai 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:
(29)R1=(2.61,3.61,4.55)⊗(3.07,4.07,5)⊕(2.61,3.61,4.55)⊗(3.2,4.2,4.97)⊕(2.61,3.61,4.55)⊗(2.87,3.87,4.87)⊕(2.98,3.98,4.97)⊗(2.98,3.98,4.97)⊕(2.61,3.61,4.55)(2.9,3.9,4.9)=(37.18,69.57,109.39).
In the same way, we can get the total fuzzy grade vector R shown as follows:
(30)R=[R1R2R3R4R5R6R7R8R9]=[(37.18,69.57,109.39)(37.72,70.29,110.76)(37.16,69.4,109.58)(37.49,69.57,110.62)(38.02,70.72,111.35)(33.02,64.02,102.41)(32.13,62.30,100.98)(32.02,62.57,101.47)(32.58,62.84,101.55)],R1*=(37.18120,69.57120,109.39120),R2*=(37.72120,70.29120,110.76120),R3*=(37.16120,69.40120,109.58120),R4*=(37.49120,69.57120,110.62120),R5*=(38.02120,70.72120,111.35120),R6*=(33.02120,64.02120,102.41120),R7*=(32.13120,62.30120,100.98120),R8*=(32.02120,62.57120,101.47120),R9*=(32.58120,62.84120,101.55120).Step 4.
Based on (12), calculate the ranking value S(Ri) of the fuzzy grade Ri, where 1≤i≤n. The larger the value of S(Ri), the better the ranking of Ri and the better the choice of preceptor Ai, 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.
x
R
i
*
y
R
i
*
s
s
R
i
*
Ranking values(Ri*)
R
1
*
0.6003
0.9049
0.2462
0.9644
R
2
*
0.6076
0.9038
0.2492
0.9660
R
3
*
0.6003
0.9046
0.2471
0.9641
R
4
*
0.6046
0.9037
0.2497
0.9647
R
5
*
0.6113
0.9035
0.2502
0.9669
R
6
*
0.5540
0.9084
0.2367
0.9527
R
7
*
0.5427
0.9090
0.2351
0.9500
R
8
*
0.5445
0.9083
0.2371
0.9498
R
9
*
0.5471
0.9089
0.2355
0.9511


## 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.



---

*Source: 495351-2013-04-23.xml*