3.2.2. Constraints

Economic benefit constraint (maximize the economic benefit):

$$\sum\_{t=1}^{4} \sum\_{i=1}^{4} \sum\_{j=1}^{16} \sum\_{k=1}^{4} e\_{tjk}^{\pm} a\_{tijk} x\_{tijk}^{\pm} \ge f\_1^- + \lambda^{\pm} (f\_1^+ - f\_1^-) \tag{5b}$$

where *t* denotes stage of the planning year (*t* = 1, 2, 3, 4), *i* is water source (*i* = 1, 2, 3, 4, representing surface water, underground water, diverted water and recycled water), *j* means region (*j* = 1, 2, 3, . . . , 16, representing Urban, Wuan, Jize, Qiu, Quzhou, Guantao, She, Guangping, Chengan, Wei, Ci, Linzhang, Daming, Fengfeng, Yongnian and Feixiang), *k* stands for water user (*k* = 1, 2, 3, 4, representing agricultural, industrial, domestic and ecological), *e* ± *tijk* is net efficiency coefficient of water used by user *k* in region *j* in the *t* stage of planning year (yuan/m<sup>3</sup> ), *atijk* denotes water relationship provided by water source *i* to user *k* in region *j* in stage *t* of planning year (water distribution is 1, unmatched water is 0), *x* ± *tijk* means water allocation from water source *i* to user *k* in region *j* in the stage *t* of planning year (m<sup>3</sup> ).

Social benefit constraint (maximize the overall satisfaction of water users):

Considering the principle of fairness and justice, the weight coefficient *α* is introduced to balance the water satisfaction among water users and reduce the contradictions between water users and water supply departments.

$$\sum\_{t=1}^{4} \sum\_{j=1}^{16} \sum\_{k=1}^{4} \frac{a\_{tijk} \boldsymbol{\omega}\_{tijk}^{\pm}}{\mathbf{G}\_{tjk}^{\pm}} \boldsymbol{a}\_{tjk}^{\pm} \ge f\_2^- + \lambda^{\pm} (f\_2^+ - f\_2^-) \tag{5c}$$

where *G* ± *tjk* is the ideal water demand of user *<sup>k</sup>* in region *<sup>j</sup>* in stage *<sup>t</sup>* (m<sup>3</sup> ); *α* ± *tjk* means the weight coefficient of user *k* in the region *j* of the *t* stage.

Environmental constraint (minimize the chemical oxygen demand (COD) discharge of major pollutants in the region):

While achieving the economic development, the pollution in the water utilization process should be comprehensively considered. The objective function should be established to measure the COD of the main pollutants in the region, so as to realize the balanced development of environment and economy.

$$\sum\_{t=1}^{4} \sum\_{i=1}^{4} \sum\_{j=1}^{16} \sum\_{k=1}^{4} d\_{tjk}^{\pm} \mathfrak{x}\_{tijk}^{\pm} \le f\_3^+ - \lambda^{\pm} (f\_3^+ - f\_3^-) \tag{5d}$$

where *d* ± *tjk* denotes the unit oxygen consumption generated by user *k* per unit water consumption in region *j* in stage *t* (kg/m<sup>3</sup> ).

Water supply constraint:

In the *t*th stage, the sum of water supply from water source *i* to all water users is less than the maximum water supply of water source *i*.

$$\sum\_{j=1}^{16} \sum\_{k=1}^{4} \mathfrak{x}\_{tijk}^{\pm} \le \mathfrak{S}\_{ti}^{\pm} \tag{5e}$$

where *S* ± *ti* stands for the maximum available water supply of water source *<sup>i</sup>* in stage *<sup>t</sup>* (m<sup>3</sup> ). Water demand constraint:

The amount of water supplied to water users should be greater than or equal to the minimum guaranteed water demand of the user and less than or equal to the ideal water storage capacity of the user.

$$D\_{tjk}^{\\\pm} \le \sum\_{i=1}^{4} a\_{tijk} x\_{tijk}^{\\\\\pm} \le G\_{tjk}^{\\\\\pm} \tag{5f}$$

where *D* ± *tjk* means the minimum water demand of user *<sup>k</sup>* in region *<sup>j</sup>* in stage *<sup>t</sup>* (m<sup>3</sup> ).

State transition equation:

The maximum available water supply from different water sources in each stage is taken as the state variable, and the dynamic configuration of the model is realized through the water balance equation.

$$\mathcal{S}\_{ti}^{\pm} = \mathcal{S}\_{(t-1)i}^{\pm} + \mathcal{C}\_{ti}^{\pm} - \sum\_{j=1}^{16} \sum\_{k=1}^{4} a\_{(t-1)ijk} x\_{(t-1)ijk}^{\pm} \tag{5g}$$

where *C* ± *ti* is the inflow of water source *<sup>i</sup>* in stage *<sup>t</sup>* (m<sup>3</sup> ).

Water transporting capacity constraint:

The total amount of water used in each region would be limited by the water transporting capacity in the region.

$$\sum\_{k=1}^{4} \mathfrak{x}\_{tijk}^{\pm} \le \mathcal{Q}\_{tij}^{\pm} \tag{5h}$$

where *Q* ± *tij* denotes the maximum capacity of water source *i* transporting to the region *j* in stage *t* (m<sup>3</sup> ).

The COD emission constraint:

Due to serious damages of human activities to the ecological environment in recent years, more and more managers begin to pay attention to the impact of ecological environment with the development of economy. Accordingly, each region has formulated the discharge capacity of pollutant COD to control environmental pollution. Therefore, the optimal allocation of water resources should meet this requirement.

$$\sum\_{t=1}^{4} \sum\_{i=1}^{4} \sum\_{k=1}^{4} d\_{tjk}^{\pm} \mathfrak{x}\_{tijk}^{\pm} \le F\_j^{\pm} \tag{5i}$$

where *F* ± *j* is the rated of COD emission in region *j* (kg).

Nonnegative constrains:

$$
\mathfrak{x}\_{\text{tijk}}^{\pm} \ge 0 \tag{5}
$$

$$S\_{0i}^{\\\pm} = \mathbf{0} \tag{5k}$$

$$1 \ge \lambda^{\pm} \ge 0 \tag{5l}$$
