### 1. Introduction

### 2. Experimental Design

### 2.1. Background of the Nansi Lake Basin

^{3}km

^{2}. The total surface area of the lake is 1..2×10

^{3}km

^{2}with 126 km in length from north to south and 5–25 km in width from east to west, and the average depth is only 1.46 m [23].

^{4}square kilometers in the Nansi Lake Basin, mainly including Zaozhuang, Jining, and Heze sity. in order to evaluate the water quality of the Nansi Lake Basin, the five monitoring sections (Fig. 1) and the fourteen monitoring indexes were ensured with hydrological and hydraulic, basin environment, pollution sources and land use, and the indexes were estimated from the earlier studies on the point source pollution and water quality variation trends in the Nansi Lake Basin from 2002 to 2012 [24]. The Models were improved for scientific simulation at the same time.

*U*,

*U*={TN, Oils, TP, BOD, NH

_{3}-N, V-phenol, COD

_{Cr}, Cr(VI), Hg, Pb, Cd, As, CN

^{−}, Cu}. An assessment criteria set V was also established according to the National Surface Water Environmental Quality Standards of China (Chinese Environmental Protection Agency, GB3838-2002; Table 1).

### 2.2. The Design of AHP-FCE Model

### 2.3. Model Description

#### 2.2.1. Fuzzy comprehensive evaluation model

*U*based on the actual local situation. This is expressed as

##### (1)

$$U=\{\begin{array}{lllll}{u}_{\mathit{1}},\hfill & {u}_{\mathit{2}},\hfill & {u}_{\mathit{3}},\hfill & \dots ,\hfill & {u}_{n}\hfill \end{array}\}$$*n*is the number of selected assessment parameters. The assessment criteria set

*V*is established from National Surface Water Environmental Quality Standards of China. This is expressed as

##### (2)

$$V=\{\begin{array}{lllll}{v}_{\mathit{1}},\hfill & {v}_{\mathit{2}},\hfill & {v}_{\mathit{3}},\hfill & {v}_{\mathit{4}},\hfill & {v}_{m}\hfill \end{array}\}$$*m*is the number of assessment criteria categories.

*A*is established with different degree of importance. This is expressed as

##### (3)

$$A=(\begin{array}{lllll}{a}_{\mathit{1}},\hfill & {a}_{\mathit{2}},\hfill & {a}_{\mathit{3}},\hfill & \dots ,\hfill & {a}_{n}\hfill \end{array})$$##### (4)

$$\sum _{\text{i}=1}^{\text{n}}{\text{a}}_{\text{i}}=1,{\text{a}}_{\text{i}}\ge 0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\text{i}=1,2,\dots ,n$$*V*

*based on the*

_{j}*i*th factor

*u*

*. The single-factor fuzzy evaluation set*

_{i}*R*

*is expressed as*

_{i}##### (5)

$${R}_{i}=(\begin{array}{llll}{r}_{i\mathit{1}},\hfill & {r}_{i\mathit{2}},\hfill & \dots ,\hfill & {r}_{im}\hfill \end{array})$$##### (6)

$$\begin{array}{l}{R}_{\mathit{1}i}=(\begin{array}{llll}{r}_{\mathit{11}},\hfill & {r}_{\mathit{12}},\hfill & \dots ,\hfill & {r}_{\mathit{1}m}\hfill \end{array})\\ {R}_{\mathit{2}i}=(\begin{array}{llll}{r}_{\mathit{21}},\hfill & {r}_{\mathit{22}},\hfill & \dots ,\hfill & {r}_{\mathit{2}m}\hfill \end{array})\\ {R}_{ni}=(\begin{array}{llll}{r}_{n\mathit{1}},\hfill & {r}_{n\mathit{2}},\hfill & \dots ,\hfill & {r}_{nm}\hfill \end{array})\end{array}$$*R*is expressed as

##### (7)

$$R=\left[\begin{array}{cccc}{\text{r}}_{11},& {\text{r}}_{11},& \cdots & {\text{r}}_{1m}\\ {\text{r}}_{21},& {\text{r}}_{22},& \cdots & {\text{r}}_{2m}\\ \cdots & \cdots & \cdots & \cdots \\ {\text{r}}_{n1},& {\text{r}}_{n2},& \cdots & {\text{r}}_{nm}\end{array}\right]$$*r*

*(*

_{ij}*i*= 1, 2, …,

*n*;

*j*= 1, 2, …,

*m*) is the membership degree of the

*i*th assessment parameter at the

*j*th level.

*B*is expressed as

##### (8)

$$B=A\u2022R=\left({a}_{1},{a}_{2},\cdots ,{a}_{n}\right)\left[\begin{array}{cccc}{\text{r}}_{11},& {\text{r}}_{11},& \cdots & {\text{r}}_{1m}\\ {\text{r}}_{21},& {\text{r}}_{22},& \cdots & {\text{r}}_{2m}\\ \cdots & \cdots & \cdots & \cdots \\ {\text{r}}_{n1},& {\text{r}}_{n2},& \cdots & {\text{r}}_{nm}\end{array}\right]=\left({b}_{1},{b}_{2},\cdots ,{b}_{n}\right)$$*i*th row is the influence extent between

*i*th factor and all evaluation factors, and

*j*th column is the influence extent between

*j*th factor and all evaluation factors. The results can reasonably represent the comprehensive influence of all factors.

#### 2.2.2. Principal component analysis model

*X*: The original data matrix

*X*is expressed as

##### (9)

$$X=\left[\begin{array}{cccc}{\text{x}}_{11}& {\text{x}}_{12}& \cdots & {\text{x}}_{1\text{n}}\\ {\text{x}}_{21}& {\text{x}}_{22}& \cdots & {\text{x}}_{2\text{n}}\\ \cdots & \cdots & \cdots & \cdots \\ {\text{x}}_{\text{m}1}& {\text{x}}_{\text{m}2}& \cdots & {\text{x}}_{\text{mn}}\end{array}\right]$$*m*is the sample number, and

*n*is factor number. To eliminate the influence of dimension and order of magnitudes, the original data matrix

*X*was normalized via the Z-Score Method. The normalization is expressed as

*j*th samples, and

*S*

*is the standard deviation of*

_{j}*j*th samples. The formulas of $\overline{{x}_{j}}$ and

*S*

*are expressed as*

_{j}##### (11)

$$\overline{{\text{x}}_{\text{j}}}=\frac{1}{\text{m}}\sum _{\text{i}=1}^{\text{m}}{\text{x}}_{\text{ij}}$$##### (12)

$${S}_{j}=\sqrt{\frac{1}{\text{m}-1}\sum _{\text{i}=1}^{\text{m}}{({\text{x}}_{\text{ij}}-\overline{{\text{x}}_{\text{j}}})}^{2}}$$*R*

*is the correlation coefficient between*

_{ij}*X*

*and*

_{i}*X*

*, and it is expressed as*

_{j}##### (13)

$${\text{r}}_{\text{ij}}=\frac{{\displaystyle \sum _{\text{k}=1}^{\text{n}}}({\text{x}}_{\text{ij}}-\overline{{\text{x}}_{\text{i}}})\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}({\text{x}}_{\text{kj}}-\overline{{\text{x}}_{\text{j}}})}{\sqrt{{\displaystyle \sum _{\text{k}=1}^{\text{n}}}{({\text{x}}_{\text{ki}}-\overline{{\text{x}}_{\text{i}}})}^{2}{\displaystyle \sum _{\text{k}=1}^{\text{n}}}{({\text{x}}_{\text{kj}}-\overline{{\text{x}}_{\text{j}}})}^{2}}}$$##### (14)

$$R=\left[\begin{array}{cccc}{\text{r}}_{11}& {\text{r}}_{12}& \cdots & {\text{r}}_{1\text{n}}\\ {\text{r}}_{21}& {\text{r}}_{22}& \cdots & {\text{r}}_{2\text{n}}\\ \cdots & \cdots & \cdots & \cdots \\ {\text{r}}_{\text{m}1}& {\text{r}}_{\text{m}2}& \cdots & {\text{r}}_{\text{mn}}\end{array}\right]$$*λI*–

*R*| = 0 is solved by using the Jacobi Method, and eigenvalue λ is sorted according to the size as

*λ*

_{1}≥

*λ*

_{2}≥···≥

*λ*

*≥ 0.*

_{p}##### (15)

$$\text{CR}=\frac{{\lambda}_{\text{i}}}{{\displaystyle \sum _{\text{k}=1}^{\text{p}}}{\lambda}_{\text{k}}}(\text{i}=1,2,\mathrm{..}p)$$##### (16)

$$\text{ACR}=\frac{{\displaystyle \sum _{\text{k}=1}^{\text{i}}}{\lambda}_{\text{k}}}{{\displaystyle \sum _{\text{k}=1}^{\text{p}}}{\lambda}_{\text{k}}}(\text{i}=1,2,\mathrm{..}p)$$^{st}, 2

^{nd}, 3

^{rd}, ..., m

^{th}(m≥4) principal components.

##### (17)

$${I}_{\text{ij}}=\sqrt{{\lambda}_{\text{i}}}{\text{e}}_{\text{ij}}(\text{i},\mathrm{\hspace{0.17em}\u200a\u200a}\text{j}=1,2,\cdots \text{p})$$##### (18)

$$Z=\left[\begin{array}{cccc}{\text{z}}_{11}& {\text{z}}_{12}& \cdots & {\text{z}}_{1\text{n}}\\ {\text{z}}_{21}& {\text{z}}_{22}& \cdots & {\text{z}}_{2\text{n}}\\ \cdots & \cdots & \cdots & \cdots \\ {\text{z}}_{\text{m}1}& {\text{z}}_{\text{m}2}& \cdots & {\text{z}}_{\text{mn}}\end{array}\right]$$##### (19)

$$\{\begin{array}{c}{F}_{1}={1}_{11}{Z}_{1}+{1}_{21}{Z}_{2}+\cdots +{1}_{\text{n}1}{Z}_{\text{n}}\\ {F}_{2}={1}_{12}{Z}_{1}+{1}_{22}{Z}_{2}+\cdots +{1}_{\text{n2}}{Z}_{\text{n}}\\ \cdots \cdots \\ {F}_{\text{m}}={1}_{1\text{m}}{Z}_{1}+{1}_{2\text{m}}{Z}_{2}+\cdots +{1}_{\text{nm}}{Z}_{\text{n}}\end{array}$$##### (20)

$$F=\frac{{\lambda}_{1}}{{\lambda}_{1}+{\lambda}_{2}+\cdots +{\lambda}_{\text{p}}}{F}_{1}+\frac{{\lambda}_{2}}{{\lambda}_{1}+{\lambda}_{2}+\cdots +{\lambda}_{\text{p}}}{F}_{2}+\cdots \cdots +\frac{{\lambda}_{\text{p}}}{{\lambda}_{1}+{\lambda}_{2}+\cdots +{\lambda}_{\text{p}}}{F}_{\text{p}}$$### 3. Results and Discussion

### 3.1. Model Simulations

##### (21)

$${R}_{1}=\{\begin{array}{ccc}0& ,& (X>{X}_{2})\\ \text{sin}\frac{\pi}{2}\left(\frac{{X}_{2}-X}{{X}_{2}-{X}_{1}}\right)& ,& ({X}_{1}\le X\le {X}_{2})\\ 1& ,& (X<{X}_{1})\end{array}$$##### (22)

$${R}_{2}=\{\begin{array}{ccc}0& ,& (X>{X}_{3};X<{X}_{1})\\ \text{sin}\frac{\pi}{2}\left(\frac{X-{X}_{1}}{{X}_{2}-{X}_{1}}\right)& ,& ({X}_{1}\le X\le {X}_{2})\\ \text{sin}\frac{\pi}{2}\left(\frac{{X}_{3}-X}{{X}_{3}-{X}_{2}}\right)& ,& ({X}_{2}\le X\le {X}_{3})\end{array}$$##### (23)

$${R}_{3}=\{\begin{array}{ccc}0& ,& (X>{X}_{4};X<{X}_{2})\\ \text{sin}\frac{\pi}{2}\left(\frac{X-{X}_{2}}{{X}_{3}-{X}_{2}}\right)& ,& ({X}_{2}\le X\le {X}_{3})\\ \text{sin}\frac{\pi}{2}\left(\frac{{X}_{4}-X}{{X}_{4}-{X}_{3}}\right)& ,& ({X}_{3}\le X\le {X}_{4})\end{array}$$##### (24)

$${R}_{3}=\{\begin{array}{ccc}0& ,& (X>{X}_{4};X<{X}_{2})\\ \text{sin}\frac{\pi}{2}\left(\frac{X-{X}_{2}}{{X}_{3}-{X}_{2}}\right)& ,& ({X}_{2}\le X\le {X}_{3})\\ \text{sin}\frac{\pi}{2}\left(\frac{{X}_{4}-X}{{X}_{4}-{X}_{3}}\right)& ,& ({X}_{3}\le X\le {X}_{4})\end{array}$$##### (25)

$${R}_{5}=\{\begin{array}{ccc}0& ,& (X<{X}_{4})\\ \text{sin}\frac{\pi}{2}\left(\frac{X-{X}_{4}}{{X}_{5}-{X}_{4}}\right)& ,& ({X}_{4}\le X\le {X}_{5})\\ 1& ,& (X>{X}_{5})\end{array}$$*X*is the actual monitoring data for the

*i*th assessment parameter, and

*X*

*is the criteria value of the*

_{j}*i*th assessment parameter at the

*j*th level (i = 1, 2, …, n; j = 1, 2, …, m).

*X*is the real concentration of

*it*h pollutant parameter,

*S*is the standard concentration of

*i*th pollutant parameter,

*X*

*is the standard values of*

_{j}*j*th water quality, and

*n*is the water quality class. The normalization of

*W*

*is expressed as*

_{i}##### (28)

$${W}^{\prime}=\frac{{W}_{\text{i}}}{{\displaystyle \sum _{\text{i}=1}^{\text{n}}}{W}_{\text{i}}}$$### 3.2. Calculation Results via the FCE-PCA Model

^{st}principal component in spring was 5.640, and the contribution ratio was 40.289%. The eigenroot of the 1

^{st}principal component in summer was 4.226, and the contribution ratio was 30.184%. The eigenroot of the 1

^{st}principal component in autumn was 3.698, and the contribution ratio was 26.416%. The eigenroot of the 1

^{st}principal component in winter was 5.927, and the contribution ratio was 42.336%. The eigenroot of the 1

^{st}principal component in the annual mean was 5.477, and the contribution ratio was 39.118%.

^{st}principal component were As (0.933), Hg (0.931), Cd (0.929), Cr(VI) (0.926), Pb (0.925), and Cu (0.534) in the annual mean, and the 2

^{nd}principal component were TP(0.762), NH

_{3}-N(0.743), CN

^{−}(0.717), V-phenol(0.716), Oils(0.593), COD

_{Cr}(0.567), BOD(0.516), and the 3

^{rd}principal component were TN(0.770), and the 4

^{th}principal component were BOD(0.535). The related indexes of 1

^{st}principal component were Cu (0.830), NH

_{3}-N (0.767), Oils (0.760), Pb (−0.740), COD

_{Cr}(0.736), Hg (0.680), and CN

^{−}(0.655) in spring. The related indexes of the 1

^{st}principal component were COD

_{Cr}(0.732), CN

^{−}(0.731), NH

_{3}-N (0.687), Hg (0.676), Oils (0.612), and V-phenol (0.587) in summer. The related indexes of the 1

^{st}principal component were Cd (−0.902), Pb (−0.864), TN (0.732), Cr(VI) (0.675), and V-phenol (0.568) in autumn. The related indexes of the 1

^{st}principal component were Cr (VI) (0.991), Cd (0.990), Hg (0.989), Pb (0.989), As (0.989), Oils (0.727), and Cu (0.579) in winter.

### 3.3. Water Quality Assessment and Principal Pollutant Evaluation

^{−}, V-phenol, Oils, COD

_{Cr}and BOD. Furthermore, the water eutrophication should be given attention, especially towards TN, NH

_{3}-N and TP.

### 3.4. Environment Implications

### 4. Conclusions

^{st}principal components were heavy metals (e.g. As, Hg, Cd, Cr (VI), Pb etc.), the 2

^{nd}principal components were organic and toxic pollutants (e.g., V-phenol, Oils, COD

_{Cr}, CN

^{−}etc.), the 3

^{rd}and 4

^{th}principal components were TN and BOD.

_{Cr}, V-phenol, TN, TP, CN

^{−}, Hg, As, Cd, Cr, and Pb. The water quality assessment should be chosen from more comprehensive indexes and a more reasonable weight calculation method like the methods found in the FCE-PCA model. It is our hope that the combined FCE-PCA model will explain the discrepancies, enhance the efficiency and goodness-of-fit, and predict power and robustness.