CHEMICAL ENGINEERINGTRANSACTIONS 
 

VOL. 51, 2016 

A publication of 

 
The Italian Association 

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

Guest Editors:Tichun Wang, Hongyang Zhang, Lei Tian 
Copyright © 2016, AIDIC Servizi S.r.l., 

ISBN978-88-95608-43-3; ISSN 2283-9216 

A Novel Water Quality Evaluation Method Based on Efficacy 
Coefficient Method 

Laijun Luo*a, Xuehui Pengb 
aSchool of Software ,Jiangxi University of Science and Technology, Nanchang,330013, China  
bChangsha Vocational & Technical College, Changsha, 410010, China  
chinaluolj@163.com 

In order to evaluate quality of the water environment, a novel water quality evaluation method is proposed. 
Water quality evaluation problem includes several influencing indices, which lead it become a multi-attribute 
decision making (MADM) problem. Then the new evaluation method is developed from the concept of a well-
known MADM method named efficacy coefficient method. In the evaluation process, coefficient of variation 
(CV) method is used to determine the weights of indices. Finally, a practical case study proved that the 
proposed water quality evaluation method is efficient and feasible. 

1. Introduction 

The rapid developments of social economy and human activities have a great impact on water resources. 
Water quality is an important factor for water resources. In order to improve water quality and protect the 
environment, it is necessary to carry out effective water quality monitoring and evaluation of water resources. 
Pollution coming from chemical, physical and biological contaminants by anthropogenic activities received 
great environmental attention all over the world (May et al., 2006; Noori et al., 2010; Li et al., 2014). There are 
various solutes in natural water, such as dissolved oxygen (DO), biochemical oxygen demand (BOD), 
chemical oxygen demand (COD) and so on, which influence water quality (Liu and Zou, 2012). Then a water 
quality evaluation problem can be regarded as a multi-attribute decision making (MADM) problem. In recent 
years, various MADM methods are developed and applied in many fields, such as supplier selection, material 
selection, information fusion (Liu and Meng, 2009; Jahan and Edwards, 2013; Ren and Yang, 2013).There are 
also many MADM methods are developed for solving water quality evaluation problems. These methods are 
fuzzy comprehensive evaluation method (Icaga, 2007), the matter element method (Kou, 2013; Wong and Hu, 
2014), comprehensive index method (Chen et al., 2010), attribute recognition method (Yu et al, 2013), set pair 
analysis (Du et al.,2014), Variable Weight Method (Ren and Zhou, 2015) and grey relation analysis (GRA) 
method (Liu, 2015). 
Efficacy coefficient method is a well known MADM method, and it defined the mathematical formula for the 
efficacy coefficient to express the contribution of variables to a system in progress process (Yang and Gao, 
2006). Efficacy coefficient method has been applied in many fields, such as sustainable development capacity 
of logistics industry (Yu, 2013), evaluation of gas explosion disaster risk (Li et al., 2013). 
For the water quality evaluation problem, this paper will develop a new evaluation method based on efficacy 
coefficient method combining with the coefficient of variation (CV) method, which is an objective method for 
determining the weights of evaluation indices.  

2. Efficacy Coefficient Method for Water quality evaluation 

This section will develop a new water quality evaluation method based on the concept of efficacy coefficient 
method. In the following discussion, we first establish a water quality evaluation model, and then propose the 
new evaluation method. Suppose that there are m water samples 

1 2
, ,...,

m
A A A  waited to be evaluated for 

their water quality grades, and each object belongs to one grade of water quality standards, which are 

                               
 
 

 

 
   

                                                  
DOI: 10.3303/CET1651010

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Please cite this article as: Luo L.J., Peng X.H., 2016, A novel water quality evaluation method based on efficacy coefficient method, Chemical 
Engineering Transactions, 51, 55-60  DOI:10.3303/CET1651010   

55



denoted by 
1 2
, ,...,

K
C C C . There are n indicators (indices) 

1 2
, ,...,

n
o o o . 

ij
x  is the measurement value of 

i
A  on the indicator 

j
o . Thus the water sample 

i
A  can be written as 

1 2
( , ,..., )

i i i in
A x x x , 1, 2,...,i m . 

Then the sample space matrix can be expressed by the following matrix: 
 

1 2

11 12 11

2 21 22 2

1 2

( )

n

n

ij m n n

m m m mn

o o o

x x xA

X x A x x x

A x x x



 
 

 
 
 
 
 

.  

 
Suppose F  is some attribute space, and 

1 2
, ,...,

K
C C C  is an ordered series of grades in the attribute space 

F .  The series satisfies the condition 
1 2 K

C C C   . Such a space can be established for the 

standard grades of every evaluation indicator.  The standard grade matrix can then be expressed with 
 

1 2

1 11 12 1

2 21 22 2

1 2

( )

K

K

ij n K K

n n n nK

C C C

o a a a

A a o a a a

o a a a



 
 

 
 
 
 
 

 

 
where 

ij
a satisfies 

1 2i i iK
a a a   or 

1 2i i iK
a a a   . 

Then for establishing the water quality evaluation model, the attribute measures are firstly to be transformed 
as follows (Liu, 2015): 
Let ( )

ijk ij k
x C    be the attribute measure of indicator value 

ij
x , which takes the attribute levels from 

the set 
k

C . The calculation formulas of  
ijk

  are given as follows:  

Suppose that 
ij

a  ( 1, 2,...,i m ) satisfies 
1 2i i iK

a a a   , then 

i) If 
1ij i

x a ,  

then 
1 2

1, 0
ij ij ijK

      ; 

ii) If 
ij iK

x a , 

then  

1 1
1, 0

ijK ij ijK
  


    ,  

iii) If 
( 1)il ij i l

a x a


  , 

then  

( 1)

( 1)

( 1) ( 1)

| | | |
,

| | | |

ij i l ij il

ijl ij l

il i l il i l

x a x a

a a a a
 





 

 
 

 
  

and  
0,

ijk
k l   or 1k l  . 

 
Then the water quality evaluation problem can be modeled as a MADM model with the following attribute 
recognition matrix  

56



1 2

11 12 11

2 21 22 2

1 2

( )

K

i i i K

i ijk n K i i i K

n in in inK

C C C

o

H o

o

  

   

  



 
 

 
 
 
 
 

. 

 

In water quality evaluation, there are many factors, such as oxygen dissolved (DO), oxygen demand 
biochemical (BOD), chemical oxygen demand (COD). The relative importance of between these factors are 
different because of the nature of the indices (factors), such as physical properties, chemical properties, 
biological properties, etc. And the corresponding pollution characteristics leads to different nature of pollution 
to some water demand of the target water body damage. Coefficient of variation (CV) method is an objective 
method for determining index weights. The steps of the CV method is given as follows: 

(i) Normalize the water sample measurement matrix ( )
ij m n

X x


  into ( )
ij m n

R r


 . 

In general, indexes can be classified into two types: the-larger-the-better type and the-smaller-the-better type. 
The normalization method is to preserve the property that the range of a normalized number 

ij
r  belongs to the 

closed interval [0,1]. Hence, the water sample measurement matrix ( )
ij m n

X x


 is transformed into the 

normalized fuzzy decision matrix  ( )
ij m n

R r


  , where 
ij

r  obtained by the following rule (Xu, 2004): 

If the jth-indicator is the-larger-the-better type, , such as dissolved oxygen (DO), then  

     
ij j

ij

j j

x x
r

x x



 





; 

If the jth attribute is the-smaller-the-better type,  such as NH3-N, then  

j ij

ij

j j

x x
r

x x



 





 

Where 
1
max{ }

j ij
i m

x x


 
  and 

1
max{ }

j ij
i m

x x


 
 . 

Obviously, [0,1]
ij

r   is the data of the ith evaluating object on the indicator 
j

o . 

(ii) Let 
1

1 m

j ij

i

x x
m 

  , 
2

1

1
( )

m

j ij j

i

s x x
m 

  , and /j j jCV s x , then the indicators’ weights can be calculated as 

follows: 

 

1

,
j

j n

j

j

CV
w

CV





1, 2,...,j n .  

We can easily to show that the weights satisfy 0,
j

w   
1

1, 1, 2,...,
n

j

j

w j n


  . 

Further the evaluation steps of the new proposed water quality evaluation method based on efficacy 
coefficient method are given as follows (Yang and Gao, 2006):  
Step 1. Establish the MADM decision matrix of water quality evaluation problem as above-mentioned. 
Step 2. Determine weights of indicators using CV method. 
Step 3. Compute efficacy coefficient of each index as follows: 

40 60
ijk ijk

d     

Where 
ijk

  is the attribute measure of indicator value 
ij

x , 1, 2...,i m  and 1, 2,...,j n  belong to the class 

k
C . 
 

57



Step 4. Compute comprehensive efficacy coefficient 
i

D  ( 1, 2,..., )i m  of each water sample: 

 
1

, 1, 2,...,j
n

w

ik ijk

j

D d i m


   

 

Step 5. Evaluate the water samples according to the following water quality recognition rule: 
If  

0
1

arg max{ }
ik

k K
k D

 
 , 

Then the sample 
i

A  belonging to the grade 
0

k . 

3. Application Example 

To illustrate the practicability and feasibility of the new water quality evaluation method, an example discussed 
in Xu et al. (2014) is adopted. 
Tongli Town, one of the six ancient towns, located in the east of the ancient canal in Suzhou City, China. It 
was built in the Song Dynasty, which has been 1000 years of history, and it is 18 kilometers away from 
Suzhou City, 80 kilometers away from Shanghai. Tongli area of 33 hectares, surrounded by five lakes, from 
net rivers will be divided into seven island townships. The ancient town is beautiful town surrounded by water. 
In 1982, Tongli Town has become the first and only to the town as a town of cultural relics protection units of 
Jiangsu province. In 1995, it was listed as the first batch of historical and cultural towns in Jiangsu province. In 
1998, the town was inscribed as the world heritage list on preparedness. With the increase in the number of 
tourists and the development of business, the local water environment has caused some pollution. This paper 
intends to evaluate the water quality. According to Chinese Surface Water Environment Quality Standards 
(GB3838-2002), this article selects the following evaluation indicators (indices): DO (

1
o ), COD (

2
o ), CODMn 

(
3

o ), BOD5 ( 4o ),TN ( 5o ), NH3N ( 6o ) and TP ( 7o ). Five water quality grades  I (Good), II (Fine), III (Ordinary), 
IV (Poor) and V (Poor) and the corresponding to standards of water quality are reported in Table 1. Indicator 
measure values of water samples are reported in Table 2. 

Table 1: National quality standards of suface waters (GB3838-2002) of China (units of mg/L) 

Grade 1o  2o  3o  4o  5o  6o  7o  

I 7.5 15 2 3 0.2 0.15 0.02 
II 6 15 4 3 0.5 0.5 0.1 
III 5 20 6 4 1 1 0.2 
IV 3 30 10 6 1.5 1.5 0.3 
V 2 40 15 10 2 2 0.4 

 

Table 2: Water monitoring data of Tongli Town  

 1o  2o  3o  4o  5o  6o  7o  

1
A  3.60 28.17 7.30 5.33 6.09 3.83 0.36 

2
A  1.65 31.00 8.07 5.45 6.35 3.35 0.34 

3
A  2.28 26.67 7.40 5.55 5.79 2.77 0.30 

4
A  4.12 25.00 6.97 4.90 5.73 2.32 0.29 

5
A  2.52 28.00 6.50 4.22 5.30 2.34 0.21 

 
The steps of the proposed method are given as follows: 
Step 1. According to Table 2, the sample space matrix 

8 7
( )

ij
X x


  is obtained as follows: 

58



3.60 28.17 7.30 5.33 6.09 3.83 0.36

1.65 31.00 8.07 5.45 6.35 3.35 0.34

2.28 26.67 7.40 5.55 5.79 2.77 0.30

4.12 25.00 6.97 4.90 5.73 2.32 0.29

2.52 28.00 6.50 4.22 5.30 2.34 0.21

2.52 25.00 7.47 5.85 5.28 2.31 0.23

1.52 34.17 9.20 6.65 7.11 3.7

X 

2 0.51

2.67 22.83 6.43 4.75 5.19 2.09 0.22

 
 
 
 
 
 
 
 
 
 
 
 
 

 

 
Take monitoring point (sample) 

1
A  as the example, the steps of the proposed method are given as follows.  

Step 2. The attribute recognition decision matrix 
1 1 7 5

( )
jk

H 


  are obtained as follows: 

1

2

3

1

4

5

6

7

0 0 0.3000 0.7 0

0 0 0.1830 0.8170 0

0 0 0.6750 0.3250 0

0 0 0.3350 0.6650 0

0 0 0 0 1

0 0 0 0 1

0 0 0 0.4 0.6

I II III IV V

o

o

o
H

o

o

o

o

 
 
 
 

  
 
 
 
 
 
 

 

 

Step 3. By CV method, the weights can be calculated as follows:  

1 2 3 4
0.1439, 0.1469, 0.1423, 0.1154,w w w w      

5 6 7
0.1321, 0.2011, 0.1183w w w    

Step 4. The comprehensive efficacy coefficient 
1i

D  of each alternative are calculated as 

11
60D  , 

21
60D  , 

31
69.5820D  , 

41
D  72.4340, 

51
D  74.0213. 

Step 5. Due to the maximum 
51

D  74.0213., so according to the water quality recognition rule, monitoring 

point (sample) 
1

A belongs to the grade V standard, and briefly denote 
1

VA  .  
Similarly, other monitoring points’ water quality results can be obtained 
as

2 3 4 5
V, V, IV, VA A A A    .  This result is in agreement with Xu et al. (2014). From the analyzed 

result, the water quality of Tongli Town is poor, which requires the local government to develop policies and 
provide financial support to improve the local water environment. 

4. Conclusion 

For the water quality evaluation problem, we develop a new water quality evaluation method based on efficacy 
coefficient method combining with coefficient of variation method. Coefficient of variation method is an 
objective weighting method which can overcome subjective randomness. This new evaluation method is easy 
to perform by using computer softwares and can be easily accepted by engineers. A case study is used to 
validate the application of the proposed method, and the example shows that the proposed method is effective 
and feasible. The proposed method can also be applied to other fields, such as multi-sensor target recognition, 
air quality evaluation, etc. 

Acknowledgment 

This work is partially supported by the National Natural Science Foundation of China (No. 11461029), Science 
and Technology Research Project of Jiangxi Provincial Education Department (No. GJJ14449 and No. 
GJJ150696) and Natural Science Foundation of Jiangxi University of Science and Technology (JXUST) (No. 
NSFJ2014-G38). 

59



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