Abstract: Given

that results of traditional assessing methods presently used are extreme values

and of poor resolution, a set pair analysis based on triangular fuzzy number is

established and a new method of calculating weight is constructed by using the weighted

average method. The case study shows that the method presents the evaluating

grade as a confidence interval which makes it more feasible and suitable to

practical conditions.

Keywords: set pair analysis;

triangular fuzzy number; super-standard multiple weight method; confidence

interval; lake eutrophication.

1

Introduction

Sustainable

management of water resources requires a more detailed understanding of

spatiotemporal trends and differences in hydrological parameters; even though

large improvements in water quality have been achieved as a result of tieless

efforts to reduce eutrophication, nutrient enrichment continue to be the main

cause of water pollution for many freshwater ecosystems in the developing world

[1]. Considering that the demand for freshwater resources is expected to

increase, protecting diminishing water resources has become one of the most important

water management and pollution control issues [2].Water quality assessment is

the back-bone of lake water management and pollution control, it provides water

resource managers with much needed information to understand the state of the

water bodies in order to make sound decisions for proper management of this

vital resource.

Many

credible water quality assessment methods have been designed and used,

including; single factor evaluation method, comprehensive water quality

evaluation method, fuzzy evaluation method, grey evaluation method, artificial

neural network method, parameter method, biological evaluation method and

nutritional status evaluation method, set pair analysis method etc [3]. All of

these methods were widely used in water quality evaluations of individual water

bodies, and of watersheds. These existing methods were used to provide

information that formed an important foundation for promoting the rational

development, planned use, and protection of water resources [4]. In recent years the comprehensive evaluation

methods are extensively used in China because the methods allows a more

thorough investigation of water quality and a more continuous depiction of the

eutrophication processes, however,

several problems are inherent in these methods. These methods include

the fuzzy comprehensive evaluation which has been proved effective in solving

problems of fuzzy boundaries and controlling the effect of monitoring errors on

assessment results [5]. However, there are still some limits when applying fuzzy

comprehensive evaluation method to water quality assessment; For example, when

the method emphasizes extreme value action, more information is lost and the

scientific character of weight value is not clear etc [6]. Set pair analysis

model has higher objectivity and reliability compared with the traditional

fuzzy comprehensive evaluation model [7]. Nevertheless; it does not deal with

the weight of the evaluation index [8].

Weight plays a key role in the comprehensive evaluation mathematical

model, reflecting the position and role of each index in the procedure of

comprehensive decision making, and directly influencing the result of the

comprehensive evaluation. For the same measured data, the low content and high

standard allowable concentration largely affect the pollution comparatively

[9]. It is difficult to establish a

unified evaluation model, which promotes the further study on the water quality

assessment methods [10].

This

paper presents a new method based on confidence interval in evaluation of lake eutrophication

that uses the resolving power of the set pair analysis model (SPA) and the

triangular fuzzy number (TFN) to deal with certain-uncertainty information in

lake assessment. It clearly defines the scientific character of weight value

using the weighted average method and furthermore gives the final evaluating

grade as a confidence interval which has a commonly held interpretation in the

scientific community and more widely has advantages of high resolution and

information utilization.

2

Methodology

The improved set pair analysis model based on triangular fuzzy

number (ISPA (TFN)) was

established as follows

2.1 Determination

of Connection Degree Formula

Set

pair analysis is based on the three-element

connection number as a-

identity, b- discrepancy, c-contrary connection degree as shown below.

(1)

In its practical

application, there are some problems that it is rough only to divide

state-space of research object into three [11] which cannot describe the

problem explicitly. Therefore, the basic expression of the degree of couplet

should be extended at different levels and at the same level. A multi-element connection

number is formed; whose expression is as shown below [12]

(2)

In order to set up a

five-element connection number in relation to I-V grades evaluation standard

commonly used in lake assessment the b-

discrepancy was expanded into

– partial

identity in discrepancy,

– entirety

discrepancy,

– partial

contrary in discrepancy,

– contrary.

The partial identity in

discrepancy is divided into identity, the partial identity in discrepancy and

the entirety discrepancy according to a certain proportion. The entirety

discrepancy is divided into the partial identity in discrepancy, the entirety

discrepancy and the partial contrary in discrepancy according to a certain

proportion. The partial contrary in discrepancy is divided into the entirety

discrepancy, the partial contrary in discrepancy and contrary. Using the above

method can deal with the difficult problem of measure scale too wide [13] as follows.

(3)

Where:

is the coefficient of partial identity in

discrepancy,

– is the

coefficient of entirety discrepancy,

– is the coefficient of partial contrary in

discrepancy,

– is the

coefficient of contrary.

2.2 Determination

of coefficient of the discrepancy degree

based on triangular

fuzzy number

Adopting analysis technique determines

the coefficients of discrepancy degree i.e

= 0.5,

= 0,

= -0.5.

Thereby, the coefficients of connection form a group of sequence: 1, 0.5, 0,

-0.5, -1 which are continuous in [-1, 1].

So the coefficients of connection have

the minimum value

the optimum value

and the maximum

value

,The coefficient of discrepancy degree basing on the

triangular fuzzy intervals under ?-cut can be constructed as follows triangular

fuzzy number TFN

.[14]. The

triangular fuzzy numbers of the coefficient of partial dissimilarity, the

coefficient of the middle difference degree, the coefficient of partial

contrast coefficient and the coefficient of the coefficient of partial

opposition are determined [15] the TFN are constructed as follows the partial

difference coefficient with triangular fuzzy number’s Ã (

) = (0, 0.5, 1), Triangular fuzzy number

difference degree coefficient in Ã (

) = (-0.5, 0, 0.5) and

Triangular fuzzy number coefficient partial contrast degree of Ã (

)= (-1, -0.5, 0).

If the confidence level ? is

given, we can obtain the confidence interval under the confidence level ? as

follows [16]:

Ã a(

) = [0.5a, -0.5a+1]

Ã a(

) = [0.5

– 0.5, – (0.5

-0.5)] (4)

Ã a(

) = [0.5

– 1, – 0.5a]

If

a=1,

then Ã a(

) =

0.5, Ã a(

) =

0, Ã a(

) = – 0.5

which is identical with the results of analysis technique

2.3 Determination of the connection degree formula based on triangular fuzzy intervals

Adopting

the triangle membership function to calculate connection components, and then

to set up a connection degree formula [17] The hierarchical connection number expression based on

the confidence interval is obtained by using the coefficients of the triangular

fuzzy numbers in the hierarchical relation as follows [16].

(5)

Where,

is the m-th evaluation index,

;

is the k-th evaluation sample,

2, …, K;

is the k-th evaluation

sample value;

,

,

,

and

.

2.4

Determination

of connection degree matrix

The connection degree value [

,

] is used to establish a connection degree matrix R as

follows.

(6)

2.5

Determination

of every evaluation index weight.

In the weighted average method, categories of

environmental factors are assigned a weight which determines how much that environmental

factor counts towards your final grade by taking the averages of a collection

of factors, some factors are assigned a greater “weight”, or importance than

others. The weighted average method considers two parts equally, one of which

is the super-standard extent, and the other is the difference among water

quality levels. In the former part, the super-standard of every index at each

evaluating object is calculated; the larger the amount of pollution, the

greater the weight; in the latter part, differences among levels of water

quality standard are taken into consideration, the worse the level, the greater

the weight [18].

The computational equation for the

index weight is:

Where,

represents the measured value of index i and

is the arithmetic mean of index i in each grading representative value;

is the typical value of index i in each grading standard. In order to

make the compositional operation, the weight of each single factor should be

normalized as follows [9]:

(8)

Where,

is the normalized weight of the evaluation

index i.

2.6

Determination of a

comprehensive assessment analysis

Adopting the comprehensive method of the addition weighting carry out

the system comprehensive estimation, as follows [19]:

Where,

(9)

The result of comprehensive estimation

is also

a confidence interval number, and

2.7

Calculation of the

grade of lake eutrophication Status

According to the relation frame of identity,

discrepancy and contrary, the projection function of the grade

and results

of comprehensive estimation are established as follows [20]:

(10)

By using the projection function equation 10,

the grade of comprehensive estimation can be determined. Let the confidence

level of interval number be a and

3

Case study

Using Qinghai lake as an example, a

case study is done to examine the feasibility and effectiveness of

the proposed method. Table 1 shows the sample

values of the evaluation index set A=

the sample indexes

values of fifteen lakes in China and Table

2 shows the evaluation standard set B=

.

Where each evaluated index threshold value of “oligotropic”,

“oligo-meso”, “mesotrophic”, “meso-eutro”, and “eutrophic”, respectively.

Similarly, for

, the graded connection degree can be defined [10] as

follows.a??????? ] 1 ,0[Îa? ) 1(mS ? ) 2(mS ? ) 3(mS ? ) 4(mS ? ) 5(mS ????????????????x???????????m??m??????k ??k ?????? M

Table 1 Surveyed data of evaluation water quality in China lakes

Lakes

TP

(mg/m3)

COD Mn

(mg/L)

SD

(m)

TN

(mg/m3)

BIO

(104/L)

Qinghai lake

20

1.40

4.50

0.22

14.60

Chao lake

30

8.26

0.25

1.67

25.30

Waihaizhong lake

40

3.53

0.92

0.87

916.00

Waihaibei lake

56

3.37

0.87

1.08

945.00

Wuhan east lake

105

10.70

0.40

2.00

1913.70

Hangzhou west lake

130

10.30

0.35

2.76

6920.00

Tianchi lake

23

4.05

1.20

0.62

51.63

Erhai lake

34

2.11

0.30

0.49

22.36

Hulun lake

80

8.29

0.50

0.13

11.60

Fuxian lake

20

1.61

7.03

0.21

19.00

Hongze lake

100

5.50

0.30

0.46

11.50

Tai lake

20

2.83

0.50

0.90

100.00

Dianchi lake

20

10.13

0.50

0.23

189.20

West lake

33

3.70

1.60

0.50

8.01

Ping lake

177

51.00

0.41

2.86

7060.00

Table 2 Standards of the Eutrophication of

the lakes

TP

(mg/m3)

TN

(mg/m3)

CODMn

(mg/L)

BIO

(104/L)

SD

(m)

Trophic state

660

>4.60

>27.10

>1000

>0.17

Eutrophic (grade V)

Table 3 Connection degree and weight vector for indexes of Qinghai Lake

Evaluation index

L

R

weight vector

-0.2961

-0.0461

0.1267

-0.2361

0.0139

0.1941

-0.0833

0.1667

0.4364

-0.1950

0.0550

0.1797

0.1455

0.5261

0.0631

Given that set pair is defined as a pair that consists of two

interrelated sets, putting together set A and B to compose set pair H with

respect to the problem W [21]. The

connection degree formula of the evaluation index can be obtained using

Equation (3) using confidence level ? as 0.75; we can obtain the confidence

interval of the coefficient of discrepancy degree and contrary degree based on

the triangular fuzzy intervals. Therefore, the connection degree value of the

evaluation index sample values

,

can be obtained using equation

(5), then using equation (6) the connection degree matrix R is constructed by

,

as shown in table 3. The weight vector

of evaluation can be obtained

using weighted average equations (7 and 8 ) as shown in table 3 and using the equation (9 and

10) the grade of lake eutrophication status assessment results are obtained and

the comparison shown in table 4 below .

Table 4 Comparison of ISPA (TFN)

with ISPA method

Lakes

Lower

limit

Upper

limit

Grades ISPA (TFN)

Grades

ISPA

Qinghai lake

2.7854

3.2911

III

III

Chao lake

3.9767

4.4141

IV

IV

Waihaizhong lake

3.8204

5.0000

IV

IV

Waihaibei lake

3.8309

5.0000

IV

IV

Wuhan east lake

4.3612

5.0000

IV

IV

Hangzhou west lake

4.7128

5.0000

V

V

Tianchi lake

3.2902

3.7254

III

III

Erhai lake

3.2126

3.8190

III

III

Hulun lake

2.0186

4.8377

III

III

Fuxian lake

2.6296

3.1623

II

II

Hongze lake

3.8488

4.2981

IV

IV

Tai lake

3.7382

4.3756

IV

IV

Dianchi lake

3.9974

4.9133

IV

IV

West lake

3.5171

3.9438

III

III

Ping lake

4.8894

5.0000

V

V

An

improved SPA evaluation model of lake eutrophication considers both single and

multiple criteria. It can be seen from Table 4 that evaluation results of the

improved SPA are coherent to the ISPA of Reference [10]. Given the fact that both the difference among

water quality levels and the proportion of the pollutant over standard are

taken into consideration

shows that ISPA (TFN) is more feasible and informative,

compared to traditional methods the proposed method is easier to interpret,

understand and provides a strong scientific basis for decision making in

Sustainable management of lakes.

4

Conclusion

(1)

The

improved set pair analysis based on triangular fuzzy numbers enriches the

traditional set pair analysis method, which is relatively simple and accurate,

and has a wide application in the evaluation of lake eutrophication. The traditional evaluation method can

only get a rough assessment of the evaluation grade, and can’t further

distinguish between the same grades. The set pair analysis based on triangular

fuzzy number cannot only give the specific evaluation grade,

but also obtain a confidence interval.

(2)

The

improved set pair analysis based on triangular fuzzy intervals the confidence interval

results are different when the confidence level a takes different values

between 0 and 1. The results are not a confidence interval but a precise value

while a=1.

(3)

The

super-standard multiple weight method calculates the weight coefficient by the

standard value of quality level of monitor, the results can reflect the degree

of contribution to subordinate grade of the monitor data of water environment

factors accurately. It is closer to the actual result of the weight than the

traditional method. This method takes into account the portions of water

environment factors that may have uneven representation, and account for them

by making the final assessment result reflect a more balanced and equal

interpretation of the monitor data accurately.