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 CHEMICAL ENGINEERING TRANSACTIONS  
 

VOL. 46, 2015 

A publication of 

 
The Italian Association 

of Chemical Engineering 
Online at www.aidic.it/cet 

Guest Editors: Peiyu Ren, Yancang Li, Huiping Song 
Copyright © 2015, AIDIC Servizi S.r.l., 
ISBN 978-88-95608-37-2; ISSN 2283-9216 

Performance Management and Evaluation Research to 
University Students 

Shaoying Yu 

Department of Music, Guangdong University of Education, Guangzhou, 510303, China. 
yushaoying@gdei.edu.cn 

For the past decades, conspicuous progress is demonstrated in all fields of China, such as economy, science 
and technology. Meanwhile, the comprehensive strength of China has been enhanced greatly. However, the 
quality of higher education should face some difficulties. It is an important Step that how to further realize and 
enhance the performance of higher education. An assessment framework is essential to be constructed to 
evaluate the performance of students in the higher school. Because of the difficulty of expressing the experts' 
opinions exactly, hesitant fuzzy sets are introduced to form a multiple attribute decision making model. A new 
distance measure is proposed to solve this evaluation problem under the framework of MAGDM. Finally, a 
case study is demonstrated to verify the reliability and applicability of the proposed method. 

1. Introduction  

From the perspective of all over the world, the competition of economy and comprehensive national strength 
between each country is indeed that of the capability of modern technology. In modern society, with the rapid 
development of knowledge economy, higher education is always a key point related to the modern technology 
of a country whether it is a developed country (Tarvid (2015)). Owing to the rapid growth of the number of 
students in higher education, the management of students is a more complex problem than ever (Yue (2015)). 
Up to 2014, there are 24,681,000 students in higher school related to China, which is from National Bureau of 
Statistics. The satisfaction for graduates to find a good job is very small. The most important issue of these 
phenomena is lack of self-perception for graduates when they are in college. The effective way to promote 
self-perception is to accept assessment from others or themselves (Wei and Peng (2015); Sheng (2014)). 
Some higher education institute introduces Higher Education Information System to explore the possibility for 
predicting the success rate of students in higher school by using data mining technology (Natel and Zwilling 
(2014)). However, it is not comprehensive to realize the performance of students by higher education 
information system. Different mining technology can result different solution, which may miss some 
information. Then, a new assessment framework is essential to construct to help us realize the performance of 
students comprehensively. This assessment problem can be considered as a multiple attribute group decision 
making (MAGDM) problem in order to introduce some MAGDM methods (Liu, Chan and Ran (2013); Feng 
and Lai (2014)).  
The multiple attribute group decision making problem is a type of multi-objective and multi-expert decision 
problem, which has applied widely in many fields such as economics (Merigo, Xu and Zeng (2013)), 
management (Liang, Pedrycz, Liu and Hu (2015)) and so on (Chen, Zhang and Dong (2015); Sassenberg, 
Landkammer and Jacoby (2014)). The essence of the MADGM problem is assembling decision-making 
information, sorting and selecting the outcome through definite means with multiple experts. Most of the 
proposals for solving MAGDM found in literature focus on cases where the information provided by experts 
represented in precise and accurate assessment values. However, in real-life, the experts have different 
cultural and educational backgrounds, experience and knowledge. Then, there are subjective and hesitant 
assessments provided by the experts, uncertain and imprecise are the basic characteristics of their 
preferences. For instance, for a selection problem of a new computer with different brand such as Microsoft, 
Lenovo, Thinkpad and others, different experts have different opinions which have the disagreements, which 
lead the difficulty for the decision maker to select appropriate brand. 

                               
 
 

 

 
   

                                                  
DOI: 10.3303/CET1546106

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Please cite this article as: Yu S.Y., 2015, Performance management and evaluation research to university students, Chemical Engineering 
Transactions, 46, 631-636  DOI:10.3303/CET1546106  

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In other words, it is difficult to express the experts' preferences accurately. Because of this, fuzzy set is 
introduced in MAFDM to solve this problem. Zadeh defined the basic concept of fuzzy sets based on the 
theory of fuzzy mathematics whose main characteristic is that: the membership function assigns to each 
element x in a universe of discourse X a membership in interval [0,1] and the non-membership degree equals 
one minus the membership degree. Since fuzzy set was proposed, it has been a popular method to solve 
uncertain and imprecise of experts' preferences (Zadeh (1965); Zimmermann (1985)). Up to 2009, Torra and 
Narukawa asserted it was often difficult to give the membership or non-membership into one value and which 
may be exist a doubt among a set of different values. Thus, they defined hesitant fuzzy sets to deal with this 
problem, which allows the membership into a set demonstrated as several possible values between 0 and 1 
(Torra and Narukawa (2009)). Moreover, in 2010, Torra proposed some simple arithmetic and geometric 
aggregation operators of hesitant fuzzy sets to aggregate the information of different experts or attributes]. 
Different operation may result different solution, so distance measure is introduced in this paper to aggregate 
the he information of different experts or attributes. 
The main contributions of this paper include the following: (1) the construction of the assessment framework of 
student management; (2) the design of expressing expert's preference by hesitant fuzzy sets; (3) the 
introduction of Hausdorff distance measure to aggregate experts' preference; (4) the application of 
assessment framework based on the proposed method. The rest of this paper is organized as follows. In 
Section 2, we construct some assessment framework of student management. Section 3 introduces some 
concepts of hesitant fuzzy set and proposes Hausdorff distance measure to evaluate management 
performance. A case study is demonstrated in Section 4. Finally, Section 5 concludes this paper. 

2. Proposed method 

In this section, we introduce hesitant fuzzy multiple group decision making methods to evaluate management 
performance of students in higher school. Based on the basic concepts of hesitant fuzzy sets, we define 
Hausdorff distance measure to aggregate information in decision matrix. 

2.1 Hesitant fuzzy sets 
Torra proposed the concept of hesitant fuzzy set which is in terms of a function when applied to a fixed set 
returns a sunset of [0, 1]. Then, in order to easily understood, Xia and Xu expressed the hesitant fuzzy set by 
mathematical symbol.  
 
Definition 1(Torra and Narukawa (2009)). Let X be a universe of discourse, then a hesitant fuzzy set H over X 
is defined as  

{ , ( ) | }= < > ∈HH x h x x X ,   (1) 

where hH(x) is a set of some values in [0, 1], symbolizing the possible membership degrees of the element x to 
H. For convenience, we call h = hH(x) a hesitant fuzzy element and H the set of all hesitant fuzzy elements. 

2.2 Distance measures 
1. Distance measure is fundamentally important in a variety of scientific fields such as decision making, 
pattern recognition, machine learning and marker prediction, lots of studies have been done on this issue.  
2. By taking into account the discrete element of hesitant fuzzy sets , and following the basic lines of 
reasoning on which the definition of distances between intuitionistic fuzzy sets are based, we define the 
Hausdorff distance measure between the hesitant fuzzy sets. 
 
Definition 2 Let M = {x, hM(x)∣x ∈ X}and N = {x, hN(x)∣x ∈ X} be two hesitant fuzzy sets in X={x1, x2, ..., xn}, we 
define the generalized weighted Hausdorff distance measure between M and N as follows: 

dg(M, N) = 
( ) ( ) ( ) ( ) 1

1

{ max }
=

 −  
n

j j
i M i N i

j
i

w h x h x
λσ σ λ

   (2) 

where λ > 0, which can be obtained according to the decision maker's risk attitude. 
 
Definition 3 Let M = {x, hM(x)∣x ∈ X}and N = {x, hN(x)∣x ∈ X}be twol hesitant fuzzy sets in X={x1, x2, ..., xn}, we 
define the weighted Euclidean- Hausdorff distance between M and N as follows： 

de(M, N) = 
( ) ( ) ( ) ( )

2
1 2

1
{ max }

=

 −  
n

j j
i M i N i

j
i

w h x h x
σ σ

   (3) 

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Definition 4 Let M = {x, hM(x)∣x ∈ X}and N = {x, hN(x)∣x ∈ X}be twol hesitant fuzzy sets in X={x1, x2, ..., 
xn}, we define the weighted Hamming- Hausdorff distance between M and N as follows：  

dh(M, N) = 
( ) ( ) ( ) ( )

1

{ max }
=

 −  
n

j j
i M i N i

j
i

w h x h x
σ σ

   (4) 

Obviously, when λ=1, dg reduces to dh; when λ=2, dg reduces to de. 
 
Definition 5. Let M and N be two hesitant fuzzy sets on X = {x1, x2, ..., xn}, then the distance measure 
between M and N is defined as d(M, N), which satisfies the following properties: 

1. (1) 0 ≤ d(M, N) ≤ 1; 
2. (2) d(M, N) = 0, if and only if M = N; 
3. (3) d(M, N) = d(N, M). 

2.3 Decision method 
When Hwang and Yoon proposed TOPSIS(technique for order preference by similarity to an ideal ), whose 
basic principle is to choose the alternative with the shortest distance from the positive ideal solution(PIS) and 
the farthest distance from the negative ideal solution(NIS), it has been acquired great attention in MAGDM 
problems. In this paper, based on the characteristic of hesitant fuzzy sets, we use the ideal of TOPSIS to 
select the alternative with the shortest distance from the positive ideal solution (PIS). 
Under hesitant fuzzy environment, the hesitant fuzzy PIS, denoted by A+ can be defined as follows: 

{ }, max 1, 2, ,+ = =kj ij
i

A x h j n    (5) 

In order to simplify computation, the Eq. (6) is applied instead of the Eq. (5) as follows. 

{ }1, 0 1, 2, ,+ = =jA j n    (6) 
4. The relative closeness coefficient of an alternative Ai with respect to the hesitant fuzzy PIS A

+ and A- is 
expressed as follows: 

i iCC d
+=

,   (7) 

where 0 1, 1, 2,...≤ ≤ =iCC i m . 
Obviously, when an alternative Ai is closer to the hesitant fuzzy PIS, CCi will be closer to 1. Thus, based on 
the closeness coefficient CCi, the ranking of all alternatives can be determined and the best alternative can be 
selected 

2.4 Procedure of assessment model 
Based on the framework demonstrated in section 2, we can propose a procedure to solve this problem, where 
attribute values take the form of the hesitant fuzzy values 
Step 1. For a assessment problem, we firstly construct a decision matrix [ ]ij m nD h ×= , where all the arguments 

ijh (i = 1, 2, ..., m; j = 1, 2, ..., n) are the hesitant fuzzy numbers, provided by the experts. As for every 

alternative iA (i = 1, 2, ..., m), the experts is invited to express evaluation or preference according to each 

attribute jC (j = 1, 2, ..., n) by a hesitant fuzzy number ijh (i = 1, 2, ..., m; j = 1, 2, ..., n) and gives the relative 

weights of the n attributes denoted as w = (w1, w2, …, wn)
T with 0 ≤ wj ≤ 1 (j = 1, 2, ..., n) and 1

1
=

=
n

ji
w .  

Step 2. According to preference of the decision maker, the decision making matrix is obtained as follow: 

11 12 1

21 22 2

1 2

×

 
 
 =
 
 
 




   


n

n
m n

m m mn

h h h

h h h
D

h h h

    (8) 

Step 3: Utilize Eq.(6) to determine the corresponding hesitant fuzzy PISA+ and the dual hesitant fuzzy NISA-. 
Step 4: According to Eq. (7), the closeness coefficient of each alternative is acquired. 

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Step 5: Rank all alternatives Ai based on the closeness coefficient CCi, where the greater the value CCi, the 
better the alternative Ai. 

3. Case study 

In this section, we design a case study to demonstrate the applicability of the proposed method. 4 professors 
from different major are invited as the experts. A vice-president is invited as the decision maker chooses any 
five students as the alternatives denoted A1, A2, A3, A4, A5 and four attributes (examination performance, 
campus activities, Evaluation and Social activity). 
Firstly, the decision maker with the experts give the weight vector of these four attributes denoted as w = (0.3, 
0.2, 0.2, 0.3)T. Secondly, they give their preferences of every student on each attribute, respectively. Thus, the 

decision maker combines the opinions of these experts to provide a hesitant fuzzy decision matrix 5 4[ ]ijD h ×=  
demonstrated in Table1. 

Table 1: Transposition of original hesitant fuzzy decision matrix 

 A1 A2 A3 A4 A5 

C1 {0.1,0.3} {0.5,0.6, 0.8} {0.1,0.4, 0.5} {0.3,0.5} {0.5,0.7} 

C2 {0.5,0.6,0.7} {0.2,0.4,0.7} {0.1,0.4} {0.3,0.4,0.6} {0.4,0.5} 

C3 {0.4,0.6,0.8} {0.1,0.3} {0.2,0.3 } {0.2,0.5,0.6} {0.3,0.6,0.8} 

C4 {0.4,0.5,0.8} {0.3, 0.6} {0.1,0.3,0.4} {0.5} {0.3,0.6} 

 
Based on Eqs. (6) and (7), it can be obtained distance of each alternative and the hesitant fuzzy PISA+ in 
Figures 1.  

Table 2: Distance measure between each alternative and the hesitant fuzzy PISA+ 

 
id
+  

1λ=
 

2λ= 6λ= 8λ=
 

α1 0.67 0.6422 0.6101 0.5824 

α2 0.7 0.6915 0.6823 0.6778 

α3 0.88 0.8511 0.8369 0.8127 

α4 0.66 0.6337 0.6285 0.617 

α5 0.62 0.605 0.582 0.573 

 

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Figure 1: Distance of each alternative to positive ideal solution 

In order to demonstrate the relationship of parameter λ with the rank-order, we select λ∈ [1, 8] to demonstrate 
the rule. It is obvious that different parameter λ has different rank. So, we interview these experts to realize 
their preference. Based on the most experts' preference, we select 	= 2. 
Therefore, according to Eqs. 6-7 and parameter = 2, the closeness coefficient and rank-order can be 
computed as (0.6422, 0.6915, 0.8511, 0.6337, 0.605). 
As mentioned above, the rank-order is demonstrated as A3   A2  A1  A4   A5. It is obviously to select A3 
is the optimal student. This result considering the professors' risk attitude can help the students further realize 
themselves and understand the gap between them with the optimal student. After this, the student can acquire 
more motivation to enhance ability of not only examination but also social skill.  

4. Conclusions and further study 

Recently, with the fast development of higher school, student management has become an important issue for 
our country. As mentioned in section 1, to construct a new assessment framework is useful way to help us 
promote student management. The key problem of it is the actual expression of the experts' preference. Thus, 
according to existing studies, we introduce hesitant fuzzy set with multiple attribute group decision making 
method including distance measure of TOPSIS to handle this assessment problem. Based on the case study, 
it can further show the contributions and innovations of the assessment framework including the following 
aspects: (1) the construction of the assessment framework of student management; (2) the demonstration of 
expressing expert's preference by hesitant fuzzy sets; (3) the introduction of Hausdorff distance measure to 
aggregate experts' preference; (4) the application of assessment framework based on the proposed method. 
Although the method is useful to evaluate management preference of students in higher school, it cannot 
solve the problems with more complex problems such as hesitant fuzzy linguistic information. In the future, we 
will extend the method to solve more complex problems. 
 

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