*2.2. Index System Design*

According to the characteristics of initial provincial water rights allocation, combined with the principle for "three red lines" control and absorbing the idea of coordinated and sharable development, considering the representativeness and availability of related data, we designed the index system of initial provincial water rights allocation which is shown in Table 1.

In fact, because of the specificities of water resources, initial provincial water rights allocation proves to be a complicated problem including the aspects of social development, economic growth, and the construction of ecological civilization. Consequently, the index system in Table 1 attempts to achieve the following basic aims:



**Table 1.** The index system of initial provincial water rights allocation.

The meaning of indexes can be illustrated as follows: (1) Current water use (P1) reflects the current water use situation and also expresses the respect for the provincial different history of water use; (2) Water yield of province (P2) is decided by annual average runoff. It represents the respect of priority for local water sources. Specifically, those with more water yields should be given more initial water rights; (3) Population of province (P3) reflects people who lives in the same river basin should own the same initial water rights; (4) Provincial area (P4) shows province should be given more initial water rights if it has more area; (5) Water use per capita (P5) is the ratio between provincial current water use and its population; (6) The volume of water used for irrigation per unit area (P6) is the ratio between provincial current water use and its irrigation area; (7) Ten thousand Yuan GDP water use (P7) reflects water resources use efficiency. A smaller indicator value means higher water resources use efficiency; (8) Per capita GDP (P8) is the ratio between provincial GDP and its population which can reflect the general economic development level; (9) The amount of pollutant discharge (P9) represents the idea that more provincial pollutant discharge should result in less initial water rights; (10) Discharge standard-meeting rate of wastewater (P10) means the water function area with low water quality standard should result in less initial water rights; (11) Contribution rate of the flood control for provinces (P11) reflects the ratio between provincial flood discharge and river basin flood discharge. The higher ratio means the higher contribution and they should be assigned more initial water rights, consequently; (12) Protection level of disadvantaged groups (P12) reflects the disadvantaged provinces with bad geographical location or a backward economic development level and worsening ecological environment should be effectively protected for the assignment of initial water rights. The protection level of disadvantaged groups P12 can be represented by grades scaling 1–10 [13]. The higher grades of the protection level of the disadvantaged groups mean that more initial water rights should be assigned.

#### **3. Dynamic Projection Pursuit Allocation Model of Initial Provincial Water Rights**

Initial provincial water rights allocation based on water use control is a three-dimensional dynamic decision making problem for time, index, and allocation plans. Taking full account of the primitive values of indexes which reflect developing situation of allocation system and annual

increment values of indexes which present its dynamic increment evolving situation, we will adopt dynamic projection pursuit (DPP) technology to acquire the weights of the times and indexes. Furthermore, by calculating the best projection value, we can finally obtain the optimal allocation ratios and plans for the initial provincial water rights. During this process, we choose the dynamic projection pursuit method, because it is different from the existing initial water rights allocation methods. It can reflect the dynamics of data and overcome the difficulties of determining the weights of times and indexes.

### *3.1. Basic Data Processing*

(1) *Sk* denotes the province, *k* = 1, 2, ..., *q*; *Ti* denotes the year, *i* = 1, 2, ..., *n*; *Pj* denotes the index, *j* = 1, 2, ..., *m*.

(2) Let *akij* denote the primitive value of index *Pj* for province *Sk* at year *Ti*. The analysis matrix of year *Ti* can be described by Equation (1).

$$\left(A\_{i} = \left(a\_{kij}\right)\_{q \times m}\right) \tag{1}$$

where *k* = 1, 2, ..., *q*; *i* = 1, 2, ..., *n*; *j* = 1, 2, ..., *m*.

Let *akij* denote the absolute increment value of indexes *Pj* for province *Sk* at year *Ti*, which can be written as follows:

$$a\_{ki}{}^{\prime} = a\_{ki}{}^{\prime} - a\_{k(i-1)j} \tag{2}$$

where *i* ≥ 2, where *k* = 1, 2, ..., *q*; *i* = 2, ..., *n*; *j* = 1, 2, ..., *m*. Thus, we can ge<sup>t</sup> the absolute increment value matrix for the initial provincial water rights allocation at year *Ti*, which can be presented by Equation (3).

$$A\_i{}^{\prime} = \left(a\_{ki\dot{j}}{}^{\prime}\right)\_{q \times m} \tag{3}$$

where *k* = 1, 2, ..., *q*; *i* = 2, ..., *n*; *j* = 1, 2, ..., *m*.

(3) Adopting the improved efficiency coefficient method to remove dimensions of indexes, then matrix *Ai* and *Ai* are normalized.

Let *bkij* denote the non-dimensional value of *akij*. When *akij* represents a profit type, *bkij* can be obtained by Equation (4).

$$b\_{k\bar{i}\bar{j}} = \left\{ (a\_{k\bar{i}\bar{j}} - \min\_{k} \min\_{\bar{i}} a\_{k\bar{i}\bar{j}}) / \left( \max\_{k} \max\_{\bar{i}} a\_{k\bar{i}\bar{j}} - \min\_{k} \min\_{\bar{i}} a\_{k\bar{i}\bar{j}} \right) \right\} \times 40 + 60 \tag{4}$$

Moreover, when *akij* represents cost type, *bkij* can be obtained by Equation (5).

$$b\_{\rm kj} = 100 - \left\{ (a\_{\rm kj} - \min\_{k} \min\_{i} a\_{\rm kj}) / \left( \max\_{k} \max\_{i} a\_{\rm kj} - \min\_{k} \min\_{i} a\_{\rm kj} \right) \right\} \times 40 \tag{5}$$

where *k* = 1, 2, ..., *q*, *i* = 1, 2, ..., *n*, *j* = 1, 2, ..., *m*. *bkij* ∈ [60, <sup>100</sup>].

Let *ckij* denote the non-dimensional value of *akij*. When *akij* is a profit type, *ckij* can be obtained by Equation (6).

$$c\_{kij} = \left\{ (a\_{kij}^{\prime} - \min\_{k} \min\_{i} a\_{kij}^{\prime}) / \left( \max\_{k} \max\_{i} a\_{kij}^{\prime} - \min\_{k} \min\_{i} a\_{kij}^{\prime} \right) \right\} \times 40 + 60 \tag{6}$$

Moreover, when *akij* is cost type, *ckij* can be obtained by Equation (7).

$$c\_{k\bar{i}\bar{j}} = 100 - \left\{ (a\_{k\bar{i}}\,^{\prime} - \min\_{k} \min\_{\bar{i}} a\_{k\bar{i}\bar{j}}\,^{\prime}) / (\max\_{\bar{k}} \max\_{\bar{i}} a\_{k\bar{i}\bar{j}}\,^{\prime} - \min\_{\bar{k}} \min\_{\bar{i}} a\_{k\bar{i}\bar{j}}\,^{\prime}) \right\} \times 40 \tag{7}$$

where *k* = 1, 2, ..., *q*, *i* = 2, ..., *n*, *j* = 1, 2, ..., *m*. *ckij* ∈ [60, <sup>100</sup>].

(4) The normalized matrix *Ai* can be expressed by Equation (8).

$$B\_i = \left( b\_{k\bar{i}\bar{j}} \right)\_{q \times m} \tag{8}$$

where *k* = 1, 2, ..., *q*; *i* = 1, 2, ..., *n*; *j* = 1, 2, ..., *m*.

> Moreover, the normalized matrix *Ai* can be estimated by Equation (9).

$$\mathbf{C}\_{i} = \left(\mathbf{c}\_{kij}\right)\_{q \times m} \tag{9}$$

where *k* = 1, 2, ..., *q*; *i* = 2, ..., *n*; *j* = 1, 2, ..., *m*.

(5) Comprehensively considering *bkij* and *ckij*, the comprehensive index analysis matrix can be expressed by Equation (10).

$$E\_i = \left(\mathfrak{e}\_{kij}\right)\_{q \times m} \tag{10}$$

where *i* = 2, ..., *n*, *j* = 1, 2, ..., *m*. The element of matrix *Ei* can be measured by Equation (11).

$$
\omega\_{kij} = \alpha b\_{kij} + \beta \mathfrak{c}\_{kij} \tag{11}
$$

Here, *ekij* means the weighted sum of *bkij* and *ckij*, *k* = 1, 2, ..., *q*, *i* = 2, ..., *n*, *j* = 1, 2, ..., *m*. Where, *α* and *β* respectively represent the weights of matrix *Bi* and matrix *Ci*. Moreover, 0 ≤ *α* ≤ 1, 0 ≤ *β* ≤ 1, *α* + *β* = 1.

When *<sup>α</sup>*<sup>&</sup>gt;*β*, it reflects that we give more consideration to the developing situation of indexes themselves; when *<sup>α</sup>*<sup>&</sup>lt;*β*, it means the increment or growth situation of indexes should be given more consideration; when *α* = *β* = 0.5, it illustrates the situation of the indexes themselves and increment (or growth) situation of indexes are equally important during the allocation for initial water rights.

(6) Transfer *Ei* into the provincial index analysis matrix of initial water rights allocation:

$$E\_k = \left(\mathfrak{e}\_{kij}\right)\_{n \times m} \tag{12}$$

where *k* = 1, 2, ..., *q*. Then positive ideal matrix *E*<sup>+</sup> and negative ideal matrix *E*− can be written as follows:

$$\left| \boldsymbol{e}\_{ij}^{+} = \max \{ \boldsymbol{e}\_{kij} \Big| k = 1...q \} \right. \tag{13}$$

$$e\_{ij}^- = \min\{e\_{ki}\Big|k=1...q\}\tag{14}$$

### *3.2. Descriptions of Decision Variables*

After basic data processing, the decision variables of our model can be described as follows: Let *d*(*k*) denote the one-dimensional projection value; Let *θ* = (*<sup>ω</sup>*1, ...*ωm*, *λ*2, ...*λn*) denotes projection direction, where *<sup>ω</sup>j* is the weight of index *Pj*, *λi* is the weight of year *Ti*; Let *d*∗(*k*) denote the best projection value; Let *<sup>ω</sup>Sk* denote the ratio of initial provincial water rights allocation for the province *Sk*; Let *wSk*denote quantity of the initial water rights of the province *Sk*.

### *3.3. Construct Objective Function of Projection*

The purpose of constructing projection objective function is to decrease high dimensional data into one-dimensional projection data. Each step is presented in the following:

Step 1: Synthesize matrix *Ek* into the one-dimensional projection value *d*(*k*) with projection direction *θ* = (*<sup>ω</sup>*1, ...*ωm*, *λ*2, ...*λn*).

$$d(k) = [
\sum\_{l=2}^{n} \lambda\_l (\sum\_{j=1}^{m} \omega\_j (\epsilon\_{kij} - \epsilon\_{i\bar{j}}^{-})^2)]^{0.5} / \left\{ [
\sum\_{l=2}^{n} \lambda\_l (\sum\_{j=1}^{m} \omega\_j (\epsilon\_{kij} - \epsilon\_{i\bar{j}}^{+})^2)]^{0.5} + [
\sum\_{l=2}^{n} \lambda\_l (\sum\_{j=1}^{m} \omega\_j (\epsilon\_{kij} - \epsilon\_{i\bar{j}}^{-})^2)]^{0.5} \right\} \tag{15}$$

Furthermore, *d*(*k*) is the proximity between the positive and negative ideal solution for province *Sk*.

Step 2: Based on the principle of dispersing the projection value *d*(*k*) as far as possible, we can establish projection index function *f*(*θ*) which can be shown in Equation (16).

$$f(\theta) = s\_d = \left[ \sum\_{k=1}^{q} \left( d(k) - \overline{d(k)} \right)^2 / (q - 1) \right]^{0.5} \tag{16}$$

where *d*(*k*) is the mean value of *d*(*k*), *k* = 1, 2, ..., *q*.

Step 3: For getting the best projection value *d*(*k*), the maximum value of projection indexes function should be found which can be calculated by Equation (17).

$$\begin{cases} \max f(\theta) = s\_d \\ \text{s.T.} \begin{cases} c\_1(\theta) = \sum\_{j=1}^m \omega\_j - 1 = 0 \\ c\_2(\theta) = \sum\_{i=2}^n \lambda\_i - 1 = 0, \omega\_j > 0, \lambda\_i > 0 \end{cases} \end{cases} \tag{17}$$

By solving Equation (17), the best projection value *d*∗(*k*) of the province *Sk* can be measured. The bigger *d*∗(*k*) means more advantages can be obtained by province *Sk* during the initial water rights allocation.

### *3.4. Solutions of Initial Provincial Water Rights Allocation*

Normalize the best projection value *d*∗(*k*), then the ratio of initial provincial water rights allocation *<sup>ω</sup>Sk*which can be presented by Equation (18).

$$
\omega\_{S\_k} = d^\*(k) / \sum\_{k=1}^q d^\*(k) \tag{18}
$$

Set *W* as the total amount of initial water rights allocated by the river basin. The quantity of initial water rights of the province *Sk* can be calculated by Equation (19).

$$w\_{S\_k} = \mathcal{W} \times \omega\_{S\_k} = \mathcal{W} \times \left( d^\*(k) / \sum\_{k=1}^q d^\*(k) \right) \tag{19}$$

Here, Equation (17) is non-linear programming (NP). In fact, many approaches can solve this kind of nonlinear optimization problem including chaotic optimization algorithm (COA), genetic algorithm (GA), and simulated annealing algorithm [14]. We will choose self-adoptive chaotic optimization algorithm [15] to solve Equation (17).

### **4. A Case Study of Taihu Basin**
