# A Regional Water Optimal Allocation Model Based on the Cobb-Douglas Production Function under Multiple Uncertainties

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Cobb-Douglas Production Function (CD Function) for Water Demand Prediction

#### 2.2. Fuzzy Credibility-Constrained Interval Two-Stage Stochastic Programming (FCITSP) for Regional Water Allocation

_{i}represents the decision variables; c

_{i}denotes the coefficients; ${\tilde{a}}_{i}$ and $\tilde{b}$ are fuzzy coefficients of constraints; $\lambda $ denotes the credibility level; and i denotes the number of decision variables.

- (1)
- available water constraint$$\sum _{n=1}^{N}\left({W}_{n}^{\pm}-{{W}^{\prime}}_{nm}^{\pm}\right)}\le {Q}_{m}^{\pm},\text{\hspace{1em}}\forall m$$
- (2)
- regional allowable chemical oxygen demand emission constraint$$Cr\left[{\displaystyle \sum _{n=1}^{N}{d}_{n}{\delta}_{n}\left({W}_{n}^{\pm}-{{W}^{\prime}}_{nm}^{\pm}\right)\le \tilde{O}}\right]\ge \lambda ,\text{\hspace{1em}}\forall m$$
- (3)
- minimum water demand constraint$${W}_{n}^{\pm}-{{W}^{\prime}}_{nm}^{\pm}\ge W{D}_{n}^{\pm},\text{\hspace{1em}}\forall n,m$$
- (4)
- Non-negative constraint$${W}_{n}^{\pm}\ge {{W}^{\prime}}_{nm}^{\pm}\ge 0,\text{\hspace{1em}}\forall n,m$$

## 3. Application

#### 3.1. Study Area

^{2}and a population of 273,600 by the end of 2015. Minqin County presents a typical region with arid continental inland climate, characterized by low and irregular rainfall, high evaporation and eminent drought periods [8]. The dominant industry in Minqin is agriculture, which accounts for 70–80% of the total water consumption in region.

#### 3.2. Data Collection

## 4. Results and Discussion

#### 4.1. Results of Water Demand Prediction

^{2}) and the qualified rate (QR), were used in this study. R

^{2}can measure the degree of agreement between the predicted and actual data sets and QR can check the magnitude of average accuracy among all the predictions [8]. The indicators could be calculated as follows:

^{2}and QR in Table 4 demonstrated that, the precision grade of the simulation results is higher than the first grade standard for runoff forecasting accuracy and is thus relatively reliable [37]. This proved that the CD function could well simulate the relationship between water consumption and total benefits. A(t), α and β represent the technique level, asset investment, and water input contribution to total benefits, respectively. Positive exponent parameter values suggest the corresponding input elements have positive effects on the increase of benefits. Furthermore, the values of α and β can reflect the sensitivity degree of the input-element value to total economic benefits as the larger the value of these parameters, the higher the return on the corresponding investment element. The opposite results can be obtained when the exponent parameter is negative. According to the results calculated by CD function in this study (Table 4), technique factor A(t) was found to be one of the most significant factors for increasing industry benefits in Minqin during the past ten years, especially for PI. The value of A(t) is related to the development level of technical standards in the local industries, which differs greatly in developed and developing countries. Moreover, asset investment played a pivotal role in increasing total benefits for SI and TI. Due to currency devaluation and the rapid development of technique in the past ten years, higher economic benefits were obtained with less water consumption in PI and TI.

^{8}CNY, respectively. All the data were embedded within the prepared CD function to reach the minimum water demand values of sector n ($W{D}_{n}^{\pm}$) in 2016. Therefore, the $W{D}_{n}^{\pm}$ of the three industries were obtained as [22278.97, 23183.90], [926.94, 683.25], and [863.22, 1270.19] × 10

^{4}m

^{3}, respectively, which would be inputs for the optimization model.

#### 4.2. Water Resources Optimal Allocation Results

^{3}kg. All the parameters were input to the FCITSP model. The optimization results under different credibility levels $\lambda $ are shown in Figure 4. It can be found in Figure 4e that, when the credibility level is closer to 1, more water would be allocated due to higher tolerance limits for total COD discharge. When the credibility level increases, the upper bound for the SI sector would increase under low flow level. This indicates that the water target of SI would be preferentially satisfied when there is plenty room for COD emission. It is noteworthy that PI is the most sensitive sector to the changes in available water amount, which means that water distribution to PI should be first reduced firstly when the water shortage occurs. As shown in Figure 4d, the total cost would rise with the reduction of credibility level due to the more rigorous requirements of COD discharges.

#### 4.3. Discussions

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Zhao, J.; Li, M.; Guo, P.; Zhang, C.; Tan, Q. Agricultural water productivity oriented water resources allocation based on the coordination of multiple factors. Water
**2017**, 9, 490. [Google Scholar] [CrossRef] - Ji, L.; Sun, P.; Ma, Q.; Jiang, N.; Huang, G.; Xie, Y. Inexact Two-Stage stochastic programming for water resources allocation under considering demand uncertainties and response—A case study of Tianjin, China. Water
**2017**, 9, 414. [Google Scholar] [CrossRef] - Xiang, H.X.; Xiao, F.X.; Lei, X. Study on improved PCA-SVM model for water demand prediction. Adv. Mater. Res.
**2012**, 591–593, 1320–1324. [Google Scholar] - Zeng, X.T.; Li, Y.P.; Huang, W.; Chen, X.; Bao, A.M. Two-stage credibility-constrained programming with Hurwicz criterion (TCP-CH) for planning water resources management. Eng. Appl. Artif. Intell.
**2014**, 35, 164–175. [Google Scholar] [CrossRef] - Jain, A.; Varshney, A.K.; Joshi, U.C. Short-Term water demand forecast modelling at IIT kanpur using artificial neural networks. Water Resour. Manag.
**2001**, 15, 299–321. [Google Scholar] [CrossRef] - House-Peters, L.A.; Chang, H. Urban water demand modeling: Review of concepts, methods, and organizing principles. Water Resour. Res.
**2011**, 47, 1837–1840. [Google Scholar] [CrossRef] - Haque, M.M.; Rahman, A.; Hagare, D.; Kibria, G. Principal component regression analysis in water demand forecasting: An application to the Blue Mountains, NSW, Australia. Vet. Pathol.
**2016**, 45, 842–848. [Google Scholar] - Gui, Z.; Li, M.; Guo, P. Simulation-Based inexact fuzzy Semi-Infinite programming method for agricultural cultivated area planning in the shiyang river basin. J. Irrig. Drain. Eng.
**2016**, 143. [Google Scholar] [CrossRef] - Choudhury, B.U.; Sood, A.; Ray, S.S.; Sharma, P.K.; Panigrahy, S. Agricultural area diversification and crop water demand analysis: A remote sensing and GIS approach. J. Indian Soc. Remote
**2013**, 41, 71–82. [Google Scholar] [CrossRef] - Ahmed, A.A.; Fogg, G.E.; Gameh, M.A. Water use at Luxor, Egypt: Consumption analysis and future demand forecasting. Environ. Earth Sci.
**2014**, 72, 1–13. [Google Scholar] [CrossRef] - Zhu, L.Y.; Lei, X.Y.; Wen, J. Forecast analysis of Aksu city’s water demand based on quantitative quota method. J. Water Resour. Water Eng.
**2012**, 23, 16–18. (In Chinese) [Google Scholar] - Zhang, Z.G.; Shao, Y.S.; Xu, Z.X. Prediction of urban water demand based on Engel Index and Hoffmann Coefficient. Shuili Xuebao
**2010**, 41, 1304–1309. (In Chinese) [Google Scholar] - Jia, S.; Long, Q.; Wang, R.Y.; Yan, J.; Kang, D. On the inapplicability of the Cobb-Douglas production function for estimating the benefit of water use and the value of water resources. Water Resour. Manag.
**2016**, 30, 3645–3650. [Google Scholar] [CrossRef] - Zhang, D.; Guo, P. Integrated agriculture water management optimization model for water saving potential analysis. Agric. Water Manag.
**2016**, 170, 5–19. [Google Scholar] [CrossRef] - Zhang, Q.; Diao, Y.; Dong, J. Regional water demand prediction and analysis based on Cobb-Douglas model. Water Resour. Manag.
**2013**, 27, 3103–3113. [Google Scholar] [CrossRef] - Huang, G.H.; Loucks, D.P. An inexact two-stage stochastic programming model for water resources management under uncertainty. Civ. Eng. Environ. Syst.
**2000**, 17, 95–118. [Google Scholar] [CrossRef] - Li, Y.P.; Huang, G.H.; Nie, S.L.; Nie, X.H.; Maqsood, I. An interval-parameter two-stage stochastic integer programming model for environmental systems planning under uncertainty. Eng. Optim.
**2006**, 38, 461–483. [Google Scholar] [CrossRef] - Guo, P.; Huang, G.H.; He, L.; Zhu, H. Interval-parameter two-stage stochastic semi-infinite programming: Application to water resources management under uncertainty. Water Resour. Manag.
**2009**, 23, 1001–1023. [Google Scholar] [CrossRef] - Ren, C.F.; Li, R.H.; Zhang, L.D.; Guo, P. Multiobjective stochastic fractional goal programming model for water resources optimal allocation among industries. J. Water Resour. Plan. Manag.
**2016**, 142. [Google Scholar] [CrossRef] - Zhang, F.; Guo, P.; Li, M. Planting structure optimization of main crops in the middle reaches of Heihe River basin based on dual interval two stage stochastic programming. J. China Agric. Univ.
**2016**, 21, 109–116. (In Chinese) [Google Scholar] - Tan, Q.; Huang, G.H.; Cai, Y.P. Radial interval chance-constrained programming for agricultural non-point source water pollution control under uncertainty. Agric. Water Manag.
**2011**, 98, 1595–1606. [Google Scholar] [CrossRef] - Wang, Y.Y.; Huang, G.H.; Wang, S.; Li, W. A stochastic programming with imprecise probabilities model for planning water resources systems under multiple uncertainties. Stoch. Environ. Res. Risk Assess.
**2016**, 30, 2169–2178. [Google Scholar] [CrossRef] - Zeng, X.; Kang, S.; Li, F.; Zhang, L.; Guo, P. Fuzzy multi-objective linear programming applying to crop area planning. Agric. Water Manag.
**2010**, 98, 134–142. [Google Scholar] [CrossRef] - Zhang, Y.; Hang, G. Fuzzy robust credibility-constrained programming for environmental management and planning. J. Air Waste Manag.
**2010**, 60, 711–721. [Google Scholar] [CrossRef] - Li, X.M.; Lu, H.W.; Li, J.; Du, P.; Xu, M.; He, L. A modified fuzzy credibility constrained programming approach for agricultural water resources management—A case study in Urumqi, China. Agric. Water Manag.
**2015**, 156, 79–89. [Google Scholar] [CrossRef] - Zhang, C.; Guo, P. A generalized fuzzy credibility-constrained linear fractional programming approach for optimal irrigation water allocation under uncertainty. J. Hydrol.
**2017**, 553, 735–749. [Google Scholar] [CrossRef] - Lu, H.; Du, P.; Chen, Y.; He, L. A credibility-based chance-constrained optimization model for integrated agricultural and water resources management: A case study in South Central China. J. Hydrol.
**2016**, 537, 408–418. [Google Scholar] [CrossRef] - Tan, Q.; Huang, G.H.; Cai, Y.P. Multi-Source multi-sector sustainable water supply under multiple uncertainties: An inexact Fuzzy-Stochastic quadratic programming approach. Water Resour. Manag.
**2013**, 27, 451–473. [Google Scholar] [CrossRef] - Zhang, C.; Guo, P. An inexact CVaR two-stage mixed-integer linear programming approach for agricultural water management under uncertainty considering ecological water requirement. Ecol. Indic.
**2017**. [Google Scholar] [CrossRef] - Cobb, C.W.; Douglas, P.H. A theory of production. Am. Econ. Rev.
**1928**, 18, 139–165. [Google Scholar] - Aghion, P.; Howitt, P. A model of growth through creative destruction. Econometrica
**1992**, 60, 323–351. [Google Scholar] [CrossRef] - Tinbergen, J. Professor douglas’ production function. Revue de Linstitut International de Statistique
**1942**, 10, 37–48. [Google Scholar] [CrossRef] - Xie, S.L.; Zhong-Yi, K.E.; Ding, X.T. Prediction of water shortage quantity of china based on Cobb-Douglas production function. Water Sav. Irrig.
**2014**. [Google Scholar] - Li, X.; He, L.; Lu, H. Research on plan model of water resources of uncertainty agriculture based on fuzzy and credibility constrained. J. Water Resour. Water Eng.
**2014**, 108–114. (In Chinese) [Google Scholar] - Tan, Q.; Zhang, S.; Li, R. Optimal use of agricultural water and land resources through reconfiguring crop planting structure under socioeconomic and ecological objectives. Water
**2017**, 9, 488. [Google Scholar] [CrossRef] - Ren, C.; Guo, P.; Yang, G.; Li, R.; Liu, L. Spatial and temporal analyses of water resources use efficiency based on data envelope analysis and malmquist index: Case study in Gansu province, China. J. Water Resour. Plan. Manag.
**2016**, 142, 4016066. [Google Scholar] [CrossRef] - AQSIQ (General Administration of Quality Supervision, Inspection and Quarantine of the People’s Republic of China); SAC (Standardization Administration of the People’s Republic of China). Standard for Hydrological Information and Hydrological Forecasting; AQSIQ and SAC: Beijing, China, 2009. GB/T 22482-2008. (In Chinese) [Google Scholar]

**Figure 4.**Water distribution optimization results of PI (

**a**), SI (

**b**), TI (

**c**), the total cost (

**d**) and total COD emission (

**e**) respectively under different credibility.

Parameters and Variables | Meanings and Descriptions |
---|---|

$\pm $ | an interval with lower and upper bounds, “+” and “−” are the upper and lower bounds of the corresponding parameters, respectively. |

${f}^{\pm}$ | total cost (CNY). |

n | index of water sectors, n = 1,2,3, where n = 1 represents the primary industry (PI), n = 2 denotes the secondary industry (SI), n = 3 delegates the tertiary industry (TI). |

m | different flow levels of available water, m = 1, 2, 3, where m = 1 denotes low flow level, m = 2 represents medium flow level, m = 3 means high flow level. |

${C}_{n}^{\pm}$ | water use cost in sector i per m^{3} (CNY/m^{3}). |

${W}_{n}^{\pm}$ | first-stage decision variable, which denotes the allocation target for water that is promised to sector n (m^{3}). |

$E{C}_{n}^{\pm}$ | additional cost to sector n per m^{3} of water not delivered (CNY/m^{3}). |

${{W}^{\prime}}_{nm}^{\pm}$ | second-stage decision variable, which is the shortage of water to sector n when the flow is ${Q}_{m}^{\pm}$ with probability ${P}_{m}$ (m^{3}). |

$\tilde{O}$ | allowable regional total chemical oxygen demand (COD) emission $\tilde{O}=\left(\underset{\_}{O},O,\overline{O}\right)$, which is a triangular fuzzy number. |

${d}_{n}$ | primary pollutant content per unit wastewater discharge of sector n. |

${\delta}_{n}$ | sewage discharge coefficient (SDC) of sector n. |

$W{D}_{n}^{\pm}$ | minimum water demand of sector n (m^{3}), which can be simulated by CD function. |

$\lambda $ | credibility level (greater than 0.5). |

z_{n} | coefficients between 0 and 1 transferred from the first stage decision variables. |

$W{A}_{n\text{\hspace{0.17em}}opt}^{\pm}$ | optimal water allocation results of every industry. |

Industry Type | SDC ${\mathit{\alpha}}_{\mathit{n}}$ | COD Concentration (g/m^{3}) ${\mathit{d}}_{\mathit{n}}$ | Water Distribution Target (10^{4} m^{3}) ${\mathit{W}}_{\mathit{n}}^{\pm}$ | Water Use Price (CNY/m^{3}) ${\mathit{C}}_{\mathit{n}}^{\pm}$ | Excess Water Price (CNY/m^{3}) $\mathit{E}{\mathit{C}}_{\mathit{n}}^{\pm}$ |
---|---|---|---|---|---|

PI (n = 1) | 0.1 | 60 | [26,890, 29,900] | [2.5, 3.0] | [2.6, 3.1] |

SI (n = 2) | 0.5 | 100 | [1066, 1380] | [3.2, 4.5] | [3.9, 4.8] |

TI (n = 3) | 0.7 | 230 | [1290, 2139] | [2.9, 3.8] | [3.1, 4.2] |

Flow Level | Available Water ${\mathit{Q}}_{\mathit{m}}$ (10^{8} m^{3}) | Probability ${\mathit{P}}_{\mathit{m}}$ |
---|---|---|

Low flow level (L) (m = 1) | [2.39, 2.58] | 20% |

Medium flow level (M) (m = 2) | [2.54, 2.75] | 60% |

High flow level (H) (m = 3) | [2.73, 2.92] | 20% |

Item | ln A(t) | α | β | R^{2} | QR |
---|---|---|---|---|---|

PI | 15.20954 | −0.122110 | −1.16840 | 0.92 | 90% |

SI | 0.273011 | 0.389791 | 0.15251 | 0.95 | 100% |

TI | 2.398414 | 0.416841 | −0.12044 | 0.90 | 100% |

**Table 5.**The key parameters comparison between the water allocation scheme in 2016 and optimization results.

Item | 2016 | Optimization Results | |
---|---|---|---|

Water allocation schemes (10^{4} m^{3}) | PI | 24,890.00 | [23,202.87, 24,522.02] |

SI | 658.00 | 926.94 | |

TI | 1103.00 | 1270.19 | |

Total water consumption (10^{4} m^{3}) | 26,651.00 | [25,400.00, 26,719.15] | |

Total benefit (10^{8} CNY) | 71.36 | [72.56, 74.11] | |

Total cost (10^{8} CNY) | [6.75, 8.18] | [6.47, 8.26] | |

Benefit per unit water (CNY/m^{3}) | 26.78 | [27.16, 29.18] |

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## Share and Cite

**MDPI and ACS Style**

Zhang, F.; Tan, Q.; Zhang, C.; Guo, S.; Guo, P.
A Regional Water Optimal Allocation Model Based on the Cobb-Douglas Production Function under Multiple Uncertainties. *Water* **2017**, *9*, 923.
https://doi.org/10.3390/w9120923

**AMA Style**

Zhang F, Tan Q, Zhang C, Guo S, Guo P.
A Regional Water Optimal Allocation Model Based on the Cobb-Douglas Production Function under Multiple Uncertainties. *Water*. 2017; 9(12):923.
https://doi.org/10.3390/w9120923

**Chicago/Turabian Style**

Zhang, Fan, Qian Tan, Chenglong Zhang, Shanshan Guo, and Ping Guo.
2017. "A Regional Water Optimal Allocation Model Based on the Cobb-Douglas Production Function under Multiple Uncertainties" *Water* 9, no. 12: 923.
https://doi.org/10.3390/w9120923