# Interval Multi-Attribute Decision of Watershed Ecological Compensation Schemes Based on Projection Pursuit Cluster

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Randomize Decision-Making Matrix

#### 2.2. Dimensionless Data Treatment

#### 2.3. Projection Pursuit Cluster

_{1}, a

_{2}, …, a

_{m}) be the m-dimensional unit projection vector, then z

_{i}, the one dimensional projection eigenvalue of ${x}_{ij}$, can be shown as follows:

_{1}, z

_{2}, …, z

_{k×n}) is the projection eigenvalue set. According to the projection vector, the weight of each attribute can be calculated by Equation (4).

_{z}is the standard deviation of z(i) series, D

_{z}is the local density of z(i), and H

_{w}is the entropy of attribute weights. Those formulas are shown as:

_{z}is the mean of z(i) series, R = 0.1S

_{z}denotes the windows radius of local density, r

_{ij}= |z(i) − z(j)|, and u(t) is the unit step function, which is equal to 1 when t ≥ 0, and 0 when t < 0.

#### 2.4. Compensation Scheme Decision Making

#### 2.5. Study Area

_{1}, compensation mode, C

_{2}, compensation amount, and C

_{3}, compensation time.

## 3. Results

#### 3.1. Random Schemes and Dimensionless Treatment

_{1}–C

_{3}is carried out using Equation (2). The calculation results of the mean and standard deviation of the random generation schemes under each scheme are listed in Table 4.

#### 3.2. Weights of Each Index by PPC Method

_{1}, C

_{2}, and C

_{3}, respectively; this is consistent with the subjective weight, which showed that C

_{2}is the most important attribute. C

_{1}is also a matter of concern for the government.

#### 3.3. Decision-Making of Each Scheme

#### 3.3.1. Scatter Plot of Each Scheme

_{2}are the largest, followed by S

_{3}, S

_{4}, and S

_{1}. It can also be seen from Figure 2 that S

_{2}, S

_{3}, and S

_{4}have a lot of mixed data points. In this case, if the decision solution is made using a small number of decision data, the optimal decision results are likely to occur in S

_{4}> S

_{3}> S

_{2}.

#### 3.3.2. Statistical Analysis of Projection Eigenvalues

_{2}is better than S

_{3}, that S

_{3}is better than S

_{4}, and that S

_{4}is better than S

_{1}under the significance level of α = 0.05. When the number of random simulations is 50, 100, 500, and 1000, the decision results under the 0.01 significance level is S

_{2}> S

_{3}> S

_{4}> S

_{1}, so that abcd symbols are not marked in Table 6. However, under the significance level of 0.01 and 10 simulations, S

_{2}is less significant than S

_{3}.

_{2}.

#### 3.4. Analysis of Decision-Making Results

#### 3.4.1. Reasonableness of Decision-Making Results

_{4}and S

_{1}, the C

_{1}and C

_{3}attribute values of S

_{4}are significantly greater than those of S

_{1}, and the C

_{2}attribute of S

_{4}is slightly less than that of S

_{1}. Accordingly, the result that S

_{4}is superior to S

_{1}is reasonable.

_{2}and S

_{3}, the C

_{1}attribute of S

_{2}is equal to that of S

_{3}, the C

_{2}attribute of S

_{2}is slightly larger than that of S

_{3}and the C

_{3}attribute of S

_{2}is slightly less than S

_{3}. Considering that the importance of the C

_{3}attribute is significantly less than that of the C

_{2}attribute, S

_{2}can also be reasonably considered to be better than S

_{3}.

_{1}and C

_{3}attribute of S

_{4}are greater than that of S

_{2}, as compared with S

_{2}and S

_{4}, the C

_{2}attribute of S

_{4}is much less than that of S

_{2}. Based on the expert opinion, S

_{2}can also be reasonably considered to be better than S

_{4}. S

_{3}and S

_{4}are similar.

_{2}> S

_{3}> S

_{4}> S

_{1}is reasonable. In order to further illustrate the accuracy of the PPC results, these results are compared with other methods.

#### 3.4.2. Results of Comparison with Other Methods

_{1}, which was only 300 million Yuan in compensation, and the compensation money was paid 30 days after the pollution loss occurred. However, the compensation money is so close to the economic loss limit for ecological pollution of 232 million Yuan that the downstream Jiaxing city may not accept that scheme. From the PPC results, the scheme S

_{1}suggested in [12] has the worst performance, which further explained that the scheme S

_{1}is not suitable for selection as an implementation scheme.

_{2}scheme of this paper has the best performance; each expert is inclined to choose the combination compensation, which is combined with the DDP and MC mode. The suggested compensation amount of scheme S

_{2}is 200 million Yuan, close to the economic loss limit for ecological pollution of 232 million Yuan, indicating that the compensation amount of scheme S

_{2}is almost in line with the economic loss that is associated with ecological pollution. Meanwhile, the S

_{2}scheme also provides compensation for DDP with a step-by-step payment of 300 million Yuan, which is beneficial to both Suzhou and Jiaxing cities. When the upstream Suzhou city implements the DDP compensation mode, it would set up an economic development zone in the downstream Jiaxing city and provide continuous financial support (in S

_{2}, the total amount for DDP is 300 million Yuan), which could not only make the upstream Suzhou city gain economic benefits, but also drive the development of the downstream Jiaxing city. Then, the DDP mode in S

_{2}is a kind of economic compensation with growth and sustainability, so as to achieve a “win-win” situation for economic development and ecological and environmental protection.

## 4. Conclusions

- (1)
- The attribute values of the decision schemes are uncertain, and the standard deviation of the decision results is larger. This problem can be solved by simulating a large number of solutions through stochastic simulation techniques. According to the results of various numbers of random simulations, the decision of all the schemes can be analyzed significantly to improve the reliability of the decision results.
- (2)
- Using the PPC method can improve the accuracy of the decision result, especially the multi-attribute decision problem of the matrix element for entire interval number or a part. With the increase of the number of simulations, the scatter plot of each scheme demonstrates the superiority of the S
_{2}scheme. - (3)
- The weight of the C
_{2}attribute is the largest, followed by the compensation mode C_{1}and the compensation time C_{3}, which are consistent with subjective weight. When the number of simulations was 50, 100, 500, and 1000, S_{2}was significantly better than the other schemes at 0.01 significance levels. It is clear that the combination scheme of money compensation and development in different place is better than that of others, and it is worth using in the decision of ecological compensation of watershed water pollution.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Scatter plot of projection eigenvalues of random samples for each scheme. (

**a**) 10 random times results; (

**b**) 100 random times results; and, (

**c**) 1000 random times results.

**Table 1.**Schemes of alternative ecological compensation [12].

Schemes | Compensation Mode | Compensation Capital | Compensation Time |
---|---|---|---|

S_{1} | Money compensation (MC) | 300 million Yuan | 30 day after pollution occurs |

S_{2} | Development in different place (DDP) and MC | 200 million Yuan for MC, and 300 million Yuan for DDP | 15 day after pollution occurs, and 60 day after making DDP compensation scheme |

S_{3} | Aid project (AP) and MC | 250 million Yuan for MC, and 250 million Yuan for AP | 15 day after pollution occurs, and 60 day after making AP compensation scheme |

S_{4} | AP and DDP | 500 million Yuan | 60 day after making DDP and AP compensation scheme |

**Table 2.**Interval decision-making matrix and range of index weights [12].

Scheme | C_{1} | C_{2} | C_{3} | |||
---|---|---|---|---|---|---|

Lower | Upper | Lower | Upper | Lower | Upper | |

S_{1} | 3 | 5 | 5 | 7 | 3 | 5 |

S_{2} | 7 | 9 | 8 | 10 | 2 | 4 |

S_{3} | 7 | 9 | 7 | 9 | 3 | 5 |

S_{4} | 8 | 10 | 2 | 4 | 7 | 9 |

weight | 0.2 | 0.5 | 0.6 | 0.8 | 0.1 | 0.2 |

Scheme | No. | Original Random Samples | Dimensionless Samples | ||||
---|---|---|---|---|---|---|---|

C_{1} | C_{2} | C_{3} | C_{1} | C_{2} | C_{3} | ||

S_{1} | 1 | 3.103 | 6.496 | 4.318 | 0.015 | 0.562 | 0.331 |

2 | 3.490 | 5.569 | 4.980 | 0.070 | 0.446 | 0.426 | |

3 | 4.247 | 6.077 | 4.379 | 0.178 | 0.510 | 0.340 | |

4 | 4.982 | 6.708 | 3.414 | 0.283 | 0.589 | 0.202 | |

5 | 4.424 | 6.678 | 3.798 | 0.203 | 0.585 | 0.257 | |

6 | 3.781 | 6.053 | 4.034 | 0.112 | 0.507 | 0.291 | |

7 | 4.772 | 5.132 | 3.660 | 0.253 | 0.392 | 0.237 | |

8 | 3.394 | 6.093 | 4.814 | 0.056 | 0.512 | 0.402 | |

9 | 3.886 | 6.447 | 4.455 | 0.127 | 0.556 | 0.351 | |

10 | 3.305 | 5.496 | 3.974 | 0.044 | 0.437 | 0.282 | |

S_{2} | 11 | 8.539 | 8.028 | 3.988 | 0.791 | 0.753 | 0.284 |

12 | 7.583 | 9.017 | 2.413 | 0.655 | 0.877 | 0.059 | |

13 | 8.895 | 8.593 | 2.956 | 0.842 | 0.824 | 0.137 | |

14 | 8.488 | 8.913 | 3.189 | 0.784 | 0.864 | 0.170 | |

15 | 7.325 | 8.553 | 3.022 | 0.618 | 0.819 | 0.146 | |

16 | 8.219 | 8.452 | 2.642 | 0.746 | 0.806 | 0.092 | |

17 | 8.488 | 8.545 | 2.608 | 0.784 | 0.818 | 0.087 | |

18 | 7.161 | 9.251 | 3.920 | 0.594 | 0.906 | 0.274 | |

19 | 8.467 | 9.230 | 2.750 | 0.781 | 0.904 | 0.107 | |

20 | 7.286 | 9.947 | 2.038 | 0.612 | 0.993 | 0.005 | |

S_{3} | 21 | 8.847 | 8.275 | 2.616 | 0.835 | 0.784 | 0.088 |

22 | 8.736 | 7.933 | 3.318 | 0.819 | 0.742 | 0.188 | |

23 | 7.971 | 8.283 | 2.080 | 0.710 | 0.785 | 0.011 | |

24 | 7.027 | 8.071 | 2.424 | 0.575 | 0.759 | 0.061 | |

25 | 8.446 | 8.694 | 3.541 | 0.778 | 0.837 | 0.220 | |

26 | 7.016 | 7.505 | 3.150 | 0.574 | 0.688 | 0.164 | |

27 | 8.740 | 7.716 | 2.938 | 0.820 | 0.715 | 0.134 | |

28 | 7.421 | 7.682 | 2.676 | 0.632 | 0.710 | 0.097 | |

29 | 8.499 | 8.551 | 2.125 | 0.786 | 0.819 | 0.018 | |

30 | 8.359 | 8.145 | 2.658 | 0.766 | 0.768 | 0.094 | |

S_{4} | 31 | 8.315 | 3.527 | 8.064 | 0.759 | 0.191 | 0.866 |

32 | 9.716 | 2.629 | 7.334 | 0.959 | 0.079 | 0.762 | |

33 | 8.521 | 3.911 | 7.885 | 0.789 | 0.239 | 0.841 | |

34 | 8.222 | 2.794 | 7.536 | 0.746 | 0.099 | 0.791 | |

35 | 8.607 | 2.202 | 8.576 | 0.801 | 0.025 | 0.939 | |

36 | 8.798 | 3.723 | 7.960 | 0.828 | 0.215 | 0.851 | |

37 | 9.825 | 2.481 | 7.063 | 0.975 | 0.060 | 0.723 | |

38 | 9.622 | 3.164 | 8.462 | 0.946 | 0.146 | 0.923 | |

39 | 8.808 | 3.626 | 8.296 | 0.830 | 0.203 | 0.899 | |

40 | 8.581 | 3.433 | 7.148 | 0.797 | 0.179 | 0.735 |

Random Times | Mean | Standard Deviation | ||||
---|---|---|---|---|---|---|

C_{1} | C_{2} | C_{3} | C_{1} | C_{2} | C_{3} | |

10 | 7.248 | 6.540 | 4.430 | 2.061 | 2.288 | 2.129 |

50 | 7.214 | 6.515 | 4.461 | 2.066 | 2.358 | 2.150 |

100 | 7.232 | 6.501 | 4.495 | 1.973 | 2.325 | 2.130 |

500 | 7.255 | 6.478 | 4.490 | 2.005 | 2.372 | 2.133 |

1000 | 7.259 | 6.503 | 4.487 | 2.005 | 2.367 | 2.134 |

Random Times | Projection Vector a | Attribute Weights | H(w) | ||||
---|---|---|---|---|---|---|---|

C_{1} | C_{2} | C_{3} | C_{1} | C_{2} | C_{3} | ||

10 | 0.542 | 0.776 | 0.323 | 0.294 | 0.602 | 0.104 | 0.820 |

50 | 0.540 | 0.775 | 0.328 | 0.292 | 0.601 | 0.107 | 0.824 |

100 | 0.543 | 0.775 | 0.323 | 0.295 | 0.601 | 0.104 | 0.821 |

500 | 0.544 | 0.775 | 0.321 | 0.296 | 0.601 | 0.103 | 0.819 |

1000 | 0.544 | 0.775 | 0.321 | 0.296 | 0.601 | 0.103 | 0.819 |

Random Times | S_{1} | S_{2} | S_{3} | S_{4} |
---|---|---|---|---|

10 | 0.569 ± 0.066 dC | 1.099 ± 0.045 aA | 1.02 ± 0.082 bA | 0.838 ± 0.062 cB |

50 | 0.536 ± 0.083 D | 1.113 ± 0.068 A | 1.013 ± 0.073 B | 0.849 ± 0.078 C |

100 | 0.550 ± 0.079 D | 1.110 ± 0.077 A | 1.015 ± 0.076 B | 0.842 ± 0.076 C |

500 | 0.557 ± 0.075 D | 1.114 ± 0.074 A | 1.010 ± 0.074 B | 0.834 ± 0.077 C |

1000 | 0.557 ± 0.076 D | 1.113 ± 0.077 A | 1.018 ± 0.077 B | 0.838 ± 0.077 C |

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## Share and Cite

**MDPI and ACS Style**

Zhang, M.; Zhou, J.; Zhou, R.
Interval Multi-Attribute Decision of Watershed Ecological Compensation Schemes Based on Projection Pursuit Cluster. *Water* **2018**, *10*, 1280.
https://doi.org/10.3390/w10091280

**AMA Style**

Zhang M, Zhou J, Zhou R.
Interval Multi-Attribute Decision of Watershed Ecological Compensation Schemes Based on Projection Pursuit Cluster. *Water*. 2018; 10(9):1280.
https://doi.org/10.3390/w10091280

**Chicago/Turabian Style**

Zhang, Ming, Jinghong Zhou, and Runjuan Zhou.
2018. "Interval Multi-Attribute Decision of Watershed Ecological Compensation Schemes Based on Projection Pursuit Cluster" *Water* 10, no. 9: 1280.
https://doi.org/10.3390/w10091280