Optimal allocation of regional water resources under water saving management contract

Abstract

Water Saving Management Contract (WSMC) is a novel market-oriented business model, which can effectually allay the pressure of the regional water crisis. However, the imbalance of water demand and supply affecting the water-saving effect of WSMC will arise with the improvement of relevant water standards. Consequently, reallocating the regional water resources under WSMC is one of the crucial issues that should be solved imminently. An uncertain multi-objective programming (UMOP) model is formulated to rationally allocate the regional water resources under WSMC, taking into account the trade-offs among bene?ts of water consumption while satisfying the double control actions and water environment capacity constraints. In the model, the bene?ts of water consumption, the cost of water supply, and other parameters are considered as uncertain variables for the case of lacking sufficient data. Then, the model is transformed into the deterministic multi-objective model via inverse uncertainty distribution, which is worked out with the non-dominated sorting genetic algorithm-II (NSGA-II). Finally, the central area of Handan City in China is taken as a case study to con?rm the feasibility of the model. The results showed that the condition of water shortage has improved dramatically and which suggests the potential application in similar regions for water resources management.

Appendices

Appendix : A

Proof Proof of Theorem 1.

First, The equivalence of the objective functions are proved as follows:

• (1) Economic objective

  Since b_ij and c_ij are independent uncertain variables with regular uncertainty distributions ${{\varPhi }}_{b_{ij}}(x)$ and ${{\varPhi }}_{c_{ij}}(x)$, respectively. It follows from Theorem 2.16 in Liu (2010) that

  $$ \sum\limits_{i=1}^{K}\sum\limits_{j=1}^{J}(b_{ij}-c_{ij})a_{j}x_{ij} $$

  has an inverse uncertainty distribution

  $$ {{\varPhi}}_{1}^{-1}\left( \alpha \right) =\sum\limits_{i=1}^{K}{\sum\limits_{j=1}^{J}{\left[ {{\varPhi}}_{b_{ij}}^{-1}\left( \alpha \right) -{{\varPhi}}_{c_{ij}}^{-1}\left( 1-\alpha \right) \right]}}a_{j}x_{ij}. $$

  Then, by Theorem 6 in Liu and Ha (2010) that

  $$ \begin{aligned} E\left[ \sum\limits_{i=1}^{K}{\sum\limits_{j=1}^{J}{\left( b_{ij}-c_{ij} \right)}}a_{j}x_{ij} \right] &={{\int}_{0}^{1}}{\sum\limits_{i=1}^{K}{\sum\limits_{j=1}^{J}{\left[ {{\varPhi}}_{b_{ij}}^{-1}\left( \alpha \right) -{{\varPhi}}_{c_{ij}}^{-1}\left( 1-\alpha \right) \right]}}}a_{j}x_{ij}d\alpha\\ &=\sum\limits_{i=1}^{K}{\sum\limits_{j=1}^{J}{a_{j}}}x_{ij}{{\int}_{0}^{1}}{\left[ {{\varPhi}}_{b_{ij}}^{-1}\left( \alpha \right) -{{\varPhi}}_{c_{ij}}^{-1}\left( 1-\alpha \right) \right]}d\alpha . \end{aligned} $$

• (2) Social objective

  Since u_j is an uncertain variable with regular uncertainty distributions ${{\varPhi }}_{u_{j}}(x)$. It follows from Theorem 2.16 in Liu (2010) that

  $$ \sum\limits_{j=1}^{J}{\left( m_{j}u_{j}-a_{j}\sum\limits_{i=1}^{K}{x_{ij}} \right)} $$

  has an inverse uncertainty distribution

  $$ {{\varPhi}}_{2}^{-1}\left( \alpha \right) =\sum\limits_{j=1}^{J}{\left[ m_{j}{{\varPhi}}_{u_{j}}^{-1}\left( \alpha \right) -a_{j}\sum\limits_{i=1}^{K}{x_{ij}} \right]}. $$

  Then, by Theorem 6 in Liu and Ha (2010) that

  $$ \begin{aligned} E\left[ \sum\limits_{j=1}^{J}{\left( m_{j}u_{j}-a_{j}\sum\limits_{i=1}^{K}{x_{ij}} \right)} \right] &={{\int}_{0}^{1}}{\sum\limits_{j=1}^{J}{\left[ m_{j}{{\varPhi}}_{u_{j}}^{-1}\left( \alpha \right) -a_{j}\sum\limits_{i=1}^{K}{x_{ij}} \right]}}d\alpha\\ &=\sum\limits_{j=1}^{J}{\left[ m_{j}{{\int}_{0}^{1}}{{{\varPhi}}_{u_{j}}^{-1}}\left( \alpha \right) d\alpha -a_{j}\sum\limits_{i=1}^{K}{x_{ij}} \right]}. \end{aligned} $$

• (3) Ecological objective

  Since d_j is an variable with regular uncertainty distributions ${{\varPhi }}_{d_{j}}(x)$. It follows from Theorem 2.16 in Liu (2010) that

  $$ \sum\limits_{i=1}^{K}{\sum\limits_{j=1}^{J}{a_{j}}}x_{ij}p_{j}d_{j} $$

  has an inverse uncertainty distribution

  $$ {{\varPhi}}_{3}^{-1}\left( \alpha \right) ={{\int}_{0}^{1}}{\sum\limits_{i=1}^{K}{\sum\limits_{j=1}^{J}{a_{j}}}}x_{ij}p_{j}{{\varPhi}}_{d_{j}}^{-1}\left( \alpha \right) d\alpha . $$

  Then, by Theorem 6 in Liu and Ha (2010) that

  $$ \begin{aligned} E\left[ \sum\limits_{i=1}^{K}{\sum\limits_{j=1}^{J}{a_{j}}}x_{ij}p_{j}d_{j} \right] &={{\int}_{0}^{1}}{\sum\limits_{i=1}^{K}{\sum\limits_{j=1}^{J}{a_{j}}}}x_{ij}p_{j}{{\varPhi}}_{d_{j}}^{-1}\left( \alpha \right) d\alpha\\ &=\sum\limits_{i=1}^{K}{\sum\limits_{j=1}^{J}{a_{j}}}x_{ij}p_{j}{{\int}_{0}^{1}}{{{\varPhi}}_{d_{j}}^{-1}}\left( \alpha \right) d\alpha . \end{aligned} $$

  Secondly, the equivalence of the constraint conditions are proved as follows:

• (4) Water demand constraint.

  Since u_j is an variable with regular uncertainty distributions ${{\varPhi }}_{u_{j}}(x)$. By Theorem 1.12 in Liu (2010) we have

  $$ \begin{array}{c} \mathrm{M}\left\{ \left( a_{j}{\sum}_{i=1}^{K}{\{}x_{ij}/m_{j} \right) \leqslant u_{j}\} \right\} \geqslant \beta_{j}\\ \Longleftrightarrow \mathrm{M}\left\{ u_{j}\leqslant a_{j}{\sum}_{i=1}^{K}{x_{ij}/m_{j}} \right\} \leqslant 1-\beta_{j}\\ \Longleftrightarrow {{\varPhi}}_{u_{j}}\left( a_{j}{\sum}_{i=1}^{K}{x_{ij}}/m_{j} \right) \leqslant 1-\beta_{j}\\ \Longleftrightarrow {\sum}_{i=1}^{K}{a_{j}x_{ij}}\leqslant m_{j}{{\varPhi}}_{u_{j}}^{-1}\left( 1-\beta_{j} \right) . \end{array} $$

• (5) Pollutant emission constraint

  It follows the proof of ecological objective, thus the constraint of pollutant emission is equal to:

  $$ \sum\limits_{i=1}^{K}{a_{j}}x_{ij}p_{j}{{\int}_{0}^{1}}{{{\varPhi}}_{d_{j}}^{-1}}\left( \alpha \right) d\alpha \le Q_{j}. $$

Then the theorem is proved. □

Appendix : B

Zigzag uncertain variable.

Table 12 The unit income obtained from water consumption. (Unit:RMByuan/m³ )

Table 13 The unit cost of tsupplying water. (Unit:RMByuan/m³)

Table 14 The standard of water consumption per unit per year for water users

Table 15 The pollutant concentration in wastewater discharged from water users

Normal uncertain variable.

Table 16 The unit income obtained from water consumption. (Unit:RMByuan/m³ )

Table 17 The unit cost of supplying water. (Unit:RMByuan/m³)

Table 18 The standard of water consumption per unit per year for water users

Table 19 The pollutant concentration in wastewater discharged from water users. (Unit:g/m³)

Appendix : C

The deterministic model.

$$ \left\{\begin{array}{l} \displaystyle\max{\sum}_{i=1}^{4}{\sum}_{j=1}^{4}(b_{ij}-c_{ij})a_{j}x_{ij}\\ \displaystyle\min {\sum\limits_{j = 1}^{4} {\left( { {m_{j}}{u_{j}}-{a_{j}}\sum\limits_{i = 1}^{4} {{x_{ij}}} } \right)} } \\ \displaystyle\min {\sum}_{i=1}^{4}{\sum}_{j=1}^{4}a_{j}x_{ij}p_{j}d_{j}\\ \textup{subject to:} \\ \quad\displaystyle {\sum}_{j=1}^{4}x_{ij}\leq w_{i} \\ \quad\displaystyle {l_{j}}\leq \sum\limits_{i = 1}^{4} {{a_{j}}{x_{ij}}} \leq{m_{j}}{u_{j}} \\ \quad\displaystyle{\sum}_{i=1}^{4}a_{j}x_{ij}p_{j}d_{j} \leq Q_{j} \\ \quad\displaystyle x_{ij}\geq 0,i=1,2,\cdots,4. \quad j=1,2,\cdots,4. \end{array}\right. $$

(4)

Table 20 The unit income obtained from water consumption. (Unit:RMByuan/m³ )

Table 21 The unit cost of supplying water. (Unit:RMByuan/m³)

Table 22 The standard of water consumption per unit per year for water users

Table 23 The pollutant concentration in wastewater discharged from water users