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Inventory Theory-Based Stochastic Optimization for Reservoir Water Allocation

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Abstract

This study aims to develop an effective model for reservoir water allocation under conditions of uncertainty. To identify a practical method that increases the benefits by optimizing the water allocation policies while reducing the costs by optimizing the water transfer scheme, several stochastic programming models (EOQ-TSP models) were developed by integrating economic order quantity (EOQ) models into a two-stage stochastic programming (TSP) framework. The EOQ-TSP models are advantageous for analyzing the effects of the water inventory scheme on the reservoir water allocation benefits and better at optimizing water allocation policies while also considering uncertainties regarding different flow levels and different water inventory conditions in a water supply-inventory-demand system. Finally, the feasibility of the developed EOQ-TSP models was demonstrated by applying the models to a real-world case study. The results show that the benefits of the optimal water allocation policy will be further increased by optimizing the water transfer scheme, and these proposed models will be helpful for systematizing reservoir water management and identifying optimal reservoir water allocation plans in uncertain environments.

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References

  • Ahmed S, Tawarmalani M, Sahinidis NV (2004) A finite branch-and-bound algorithm for two-stage stochastic integer programs. Mathematical Programming, 100 (2), 355–377

  • Birge JR, Louveaux FV (1997) Introduction to stochastic programming. New York, NY: Springer

  • Dantzig GB (1983) Reminiscences about the origins of linear programming. Memoirs of the American Mathematical Society, 48 (298), 1–11

  • Duan Y, Li G, Tien JM, Huo J (2012) Inventory models for perishable items with inventory level dependent demand rate. Appl Math Model 36(10):5015–5028

    Article  Google Scholar 

  • El Saadany AMA, Jaber MY (2010) A production/remanufacturing inventory model with price and quality dependant return rate. Comput Ind Eng 58(3):352–362

    Article  Google Scholar 

  • Fu Q, Li LQ, Li M, Li TX, Liu D, Zhou ZQ (2018) An interval parameter conditional value-at-risk two-stage stochastic programming model for sustainable regional water allocation under different representative concentration pathways scenarios. J Hydrol 564:115–124

    Article  Google Scholar 

  • Hillier L (2001) Introduction to operations research, vol 938

    Google Scholar 

  • Huang GH (1998) A hybrid inexact-stochastic water management model. Eur J Oper Res 107(1):137–158

    Article  Google Scholar 

  • Huang GH, Loucks DP (2000) Aa inexact two-stage stochastic programming model for water resource management under uncertainty. Civ Eng Syst 17(2):95–118

    Article  Google Scholar 

  • Jakeman AJ, Letcher RA, Norton JP (2006) Ten iterative steps in development and evaluation of environmental models. Environ Model Softw 21(5):602–614

    Article  Google Scholar 

  • Kelman J, Stedinger JR, Cooper LA et al (1990) Sampling stochastic dynamic programming applied to reservoir operation. Water Resour Res 26(3):447–454

    Article  Google Scholar 

  • Kovacs L, Boros E, Inotay F (2007) A two-stage approach for large-scale sewer systems design with application to the Lake Balaton resort area. Eur J Oper Res 23(2):169–178

    Article  Google Scholar 

  • Li M, Fu Q, Guo VP, Singh ZCL, Yang GQ (2019) Stochastic multi-objective decision making for sustainable irrigation in a changing environment. J Clean Prod 223:928–945

    Article  Google Scholar 

  • Li M, Fu Q, Singh VP, Ji Y, Liu D, Zhang CL, Li TX (2019) An optimal modelling approach for managing agricultural water-energy-food nexus under uncertainty. Sci Total Environ 651:1416–1434

    Article  Google Scholar 

  • Li M, Fu Q, Singh VP, Liu D, Li TX (2019) Stochastic multi-objective modeling for optimization of water-food-energy nexus of irrigated agriculture. Adv Water Resour 127:209–224

    Article  Google Scholar 

  • Li YP, Huang GH, F Y, Zhou HD (2009) A multistage fuzzy-stochastic programming model for supporting sustainable water-resources allocation and management. Environ Model Softw 24:786–797

    Article  Google Scholar 

  • Li YP, Huang GH, Nie SL (2006) An interval-parameter multi-stage stochastic programming model for water resources management under uncertainty. Adv Water Resour 29(5):776–789

    Article  Google Scholar 

  • Li YP, Huang GH, Nie SL (2011) Optimization of regional economic and environmental systems under fuzzy and random uncertainties. J Environ Manag 92(8):2010–2020

    Article  Google Scholar 

  • Li W, Li YP, Li CH, Huang GH (2010a) An inexact two-stage water management model for planning agricultural irrigation under uncertainty. Agric Water Manag 97(11):1905–1914

    Article  Google Scholar 

  • Li W, Li YP, Li CH, Huang GH (2010b) An inexact two-stage water management model for planning agricultural irrigation under uncertain. Agric Water Manag 97:1905–1914

    Article  Google Scholar 

  • Loukas A, Mylopoulos N, Vasiliades L (2007) A modeling system for the evaluation of water resources management strategies in Thessaly, Greece. Water Resour Manag 21:1637–1702

    Article  Google Scholar 

  • Mobasheri F, Harboe RC (1970) A two-stage optimization model for design of a multipurpose reservoir. Water Resour Res 6(1):22–31

    Article  Google Scholar 

  • Schmitta AJ, Shen ZJM (2010) Inventory systems with stochastic demand and supply: properties and approximations. Eur J Oper Res 206(2):313–328

    Article  Google Scholar 

  • Schultz R, Stougie L, Vlerk MHVD (1996) Two-stage stochastic integer programming: a survey. Statistica Neerlandica 50(3):404–416

    Article  Google Scholar 

  • Suo MQ, Li YP, Huang GH (2011) An inventory-theory-based interval-parameter two-stage stochastic programming model for water resources management. Eng Optim 43(9):999–1018

  • Suo MQ, Li YP, Huang GH, Fan YR, Li Z (2013) An inventory-theory-based inexact multistage stochastic programming model for water resources management. Math Probl Eng 2013(4):707–724

    Google Scholar 

  • Trezos T, Yeh WW (1986) Use of stochastic dynamic programming for reservoir management. Water Resour Res 23(6):983–996

  • Zhang L, Li CY (2014) An inexact two-stage water resources allocation model for sustainable development and management under uncertainty. Water Resour Manag 28(10):3161–3178

    Article  Google Scholar 

  • Zhang ZL, Li YP, Huang GH (2014) An inventory-theory-based interval stochastic programming method and its application to beijing’s electric-power system planning. Int J Electr Power Energy Syst 62:429–440

    Article  Google Scholar 

  • Zhang L, Yin X, Xu Z et al (2016) Crop Planting Structure Optimization for Water Scarcity Alleviation in China. J Ind Ecol 20(3):435–445

    Article  Google Scholar 

  • Zhou X, Shi L, Huang B (2017) Integrated inventory model with stochastic lead time and controllable variability for milk runs. Journal of Industrial & Management Optimization 8(3):657–672

    Google Scholar 

Download references

Acknowledgements

The authors would like to acknowledge the National Science Fund for Distinguished Young Scholars (51825901), the Postdoctoral Science Foundation of Heilongjiang Province of China (LBH-Z17031), the National Key & Program of China (2018YFC0407303), the Humanity and Social Science general project of the Ministry of Education of China (18YJAZH147), and the National Natural Science Foundation of China (51809040).

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Authors and Affiliations

  1. School of Water Conservancy & Civil Engineering, Northeast Agricultural University, Changjiang Street No. 600, Harbin, Heilongjiang, 150030, People’s Republic of China

    Yaowen Xu, Qiang Fu, Yan Zhou, Mo Li, Yi Ji & Tianxiao Li

  2. Key Laboratory of Efficient Utilization of Agricultural Water Resources of Agriculture Ministry, Northeast Agricultural University, Harbin, 150030, Heilongjiang, China

    Yaowen Xu, Qiang Fu, Yan Zhou, Mo Li, Yi Ji & Tianxiao Li

Authors
  1. Yaowen Xu
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  2. Qiang Fu
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  3. Yan Zhou
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  4. Mo Li
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  5. Yi Ji
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  6. Tianxiao Li
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Correspondence to Qiang Fu or Yan Zhou.

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Appendices

Appendix 1

The applications of the EOQ-P-T model are described as follows. The EOQ-P-T model is suitable for analyzing the water allocation and transfer problems of a reservoir when the reservoir’s insufficient water is fully replenished by transferring water from the Datong River, and when assuming the replenishment has a limited speed (runoff of the Datong River). Based on model (8), the EOQ-P-T model can be depicted as follows.

$$ \operatorname{Max}\ Z=\sum \limits_{i=1}^I\sum \limits_{t=1}^T{N}_{it}\cdot {\mathrm{w}}_{it}-\sum \limits_{i=1}^I\sum \limits_{t=1}^T\sum \limits_{l=1}^L{p}_{tl}\cdot \left(\sqrt{2{Ch}_t\cdot {Cs}_t\cdot {X}_{it l}\cdot \frac{g_{tl}-{X}_{it l}}{g_{tl}}}+{Cp}_t\cdot {X}_{it l}\right) $$
(12a)
$$ s.t.\left\{\begin{array}{l}\sum \limits_{i=1}^I{w}_{it}\le {q}_{tl}+\sum \limits_{i=1}^I{x}_{it l}+{Q}_{\left(\mathrm{t}-1\right)\mathrm{l}},\forall t;l=1,2\dots l\\ {}\sum \limits_{i=1}^I{w}_{it\begin{array}{c}\end{array}\max}\ge {q}_{tl}+\sum \limits_{i=1}^I{x}_{it l}+{Q}_{\left(\mathrm{t}-1\right)\mathrm{l}}\begin{array}{c},\end{array}\forall t;l=1,2\dots l\\ {}{q}_{tl}+\sum \limits_{i=1}^I{x}_{it l}+{Q}_{\left(\mathrm{t}-1\right)\mathrm{l}}\hbox{-} \sum \limits_{i=1}^I{w}_{it}\le CR\begin{array}{c},\end{array}\forall t;l=1,2\dots l\\ {}{Q}_{\mathrm{t}\mathrm{l}}={q}_{tl}+\sum \limits_{i=1}^I{x}_{it l}+{Q}_{\left(\mathrm{t}-1\right)\mathrm{l}}\hbox{-} \sum \limits_{i=1}^I{w}_{it}\begin{array}{c},\end{array}\forall t;l=1,2\dots {l}_{\mathrm{t}};{Q}_{0l}=0\\ {}{T_{tl}}^{\ast }=\sqrt{\frac{2{Cs}_t\cdot {g}_{tl}}{Ch_t\cdot \sum \limits_{i=1}^I{X}_{it l}\cdot \left({\mathrm{g}}_{tl}-\sum \limits_{i=1}^I{X}_{it l}\right)}}\begin{array}{c},\end{array}\forall t;l=1,2\dots l\\ {}{D_{tl}}^{\ast }=\sqrt{\frac{2{Cs}_t\cdot \sum \limits_{i=1}^I{X}_{it l}\cdot {g}_{tl}}{Ch_t\cdot \left({\mathrm{g}}_{tl}-\sum \limits_{i=1}^I{X}_{it l}\right)}}\begin{array}{c},\end{array}\forall t;l=1,2\dots l\\ {}{T^{\ast}}_{jtl}={D}^{\ast }/g\\ {}{X}_{it l}\ge 0\begin{array}{c},\end{array}\forall t;l=1,2\dots l\end{array}\right.\kern0.75em $$
(12b)

where gtl is the stream flow of the Datong River when the flow level is l during period t (m3); Tjtl is the water replenishing period when the flow of the Huang River is qtl (5-year period);

Appendix 2

The applications of the EOQ-S-T model are described as follows. The EOQ-S-T model is suitable for analyzing the water allocation and transfer problems of the H.Q. reservoir when the insufficient water in the H.Q. reservoir is replenished using planned shortages. Based on model (9), the EOQ-S-T model can be depicted as follows.

$$ \operatorname{Max}\ Z=\sum \limits_{i=1}^I\sum \limits_{t=1}^T{N}_{it}\cdot {\mathrm{w}}_{it}-\sum \limits_{i=1}^I\sum \limits_{t=1}^T\sum \limits_{l=1}^L{p}_{tl}\cdot \left(\sqrt{\frac{2{Ch}_t\cdot {Cq}_t\cdot {Cs}_t\cdot {X}_{it l}}{Ch_t+{Cq}_t}}+{Cp}_t\cdot {X}_{it l}\right) $$
(13a)
$$ s.t.\left\{\begin{array}{l}\sum \limits_{i=1}^I{w}_{it}\le {q}_{tl}+\sum \limits_{i=1}^I{x}_{it l}+{Q}_{\left(\mathrm{t}-1\right)\mathrm{l}},\forall t;l=1,2\dots l\\ {}\sum \limits_{i=1}^I{w}_{it\begin{array}{c}\end{array}\max}\ge {q}_{tl}+\sum \limits_{i=1}^I{x}_{it l}+{Q}_{\left(\mathrm{t}-1\right)\mathrm{l}}\begin{array}{c},\end{array}\forall t;l=1,2\dots l\\ {}{q}_{tl}+\sum \limits_{i=1}^I{x}_{it l}+{Q}_{\left(\mathrm{t}-1\right)\mathrm{l}}\hbox{-} \sum \limits_{i=1}^I{w}_{it}\le CR\begin{array}{c},\end{array}\forall t;l=1,2\dots l\\ {}{Q}_{\mathrm{t}\mathrm{l}}={q}_{tl}+\sum \limits_{i=1}^I{x}_{it l}+{Q}_{\left(\mathrm{t}-1\right)\mathrm{l}}\hbox{-} \sum \limits_{i=1}^I{w}_{it}\begin{array}{c},\end{array}\forall t;l=1,2\dots {l}_{\mathrm{t}};{Q}_{0l}=0\\ {}{T_{tl}}^{\ast }=\sqrt{\frac{2{Cs}_t\left({Ch}_t+{Cq}_t\right)}{Ch_t\cdot {Cq}_t\cdot \sum \limits_{i=1}^I{X}_{it l}}}\begin{array}{c},\end{array}\forall t;l=1,2\dots l\\ {}{D_{tl}}^{\ast }=\sqrt{\frac{2{Cs}_t\cdot \sum \limits_{i=1}^I{X}_{it l}\left({Ch}_t+{Cq}_t\right)}{Ch_t\cdot {Cq}_t}}\begin{array}{c},\end{array}\forall t;l=1,2\dots l\\ {}{B_{tl}}^{\ast }=\sqrt{\frac{2{Ch}_t\cdot {Cs}_t\cdot \sum \limits_{i=1}^I{X}_{it l}}{Cq_t\cdot \left({Ch}_t+{Cq}_t\right)}}\begin{array}{c},\end{array}\forall t;l=1,2\dots l\\ {}{{Tq_{\begin{array}{c}\end{array}}}_{tl}}^{\ast }=\sqrt{\frac{2{Ch}_t\cdot {Cs}_t}{Cq_t\cdot \left({Ch}_t+{Cq}_t\right)\cdot \sum \limits_{i=1}^I{X}_{it l}}}\begin{array}{c},\end{array}\forall t;l=1,2\dots l\\ {}{X}_{it l}\ge 0\begin{array}{c},\end{array}\forall t;l=1,2\dots l\end{array}\right. $$
(13b)

where Cqt is the shortage cost (penalty) per unit water per unit time during period t (Yuan/ m3); Btl is the maximum water shortage quantity when the flow of the Huang River is qtl (m3); Bitl is the water shortage quantity to water user i when the flow of the Huang River is qtl; Tqtl is the water shortage period when the flow of the Huang River is qtl (5-year period).

Appendix 3

The applications of the EOQ-SP-T model are described as follows. The EOQ-SP-T model is suitable for analyzing the water allocation and transfer problems of a reservoir when the insufficient water is replenished using planned shortages and assuming the replenishment is limited by the speed of runoff in the Datong River. Based on model (10), the EOQ-SP-T model can be depicted as follows.

$$ {\displaystyle \begin{array}{l}\mathrm{M}\ \mathrm{a}\ \mathrm{x}\kern0.5em Z\\ {}=\sum \limits_{i=1}^I\sum \limits_{t=1}^T{N}_{it}\cdot {\mathrm{w}}_{it}-\sum \limits_{i=1}^I\sum \limits_{t=1}^T\sum \limits_{l=1}^L{p}_{tl}\cdot \left(\sqrt{2{Ch}_t\cdot {Cs}_t\cdot {X}_{it l}}\cdot \sqrt{\frac{Cq_t}{\left({Ch}_t+{Cq}_t\right)}}\cdot \sqrt{\frac{g_{tl}-{X}_{it l}}{g_{tl}}}+{Cp}_t\cdot {X}_{it l}\right)\end{array}} $$
(14a)
$$ s.t.\left\{\begin{array}{l}\sum \limits_{i=1}^I{w}_{it}\le {q}_{tl}+\sum \limits_{i=1}^I{x}_{it l}+{Q}_{\left(\mathrm{t}-1\right)\mathrm{l}},\forall t;l=1,2\dots l\\ {}\sum \limits_{i=1}^I{w}_{it\begin{array}{c}\end{array}\max}\ge {q}_{tl}+\sum \limits_{i=1}^I{x}_{it l}+{Q}_{\left(\mathrm{t}-1\right)\mathrm{l}}\begin{array}{c},\end{array}\forall t;l=1,2\dots l\\ {}{q}_{tl}+\sum \limits_{i=1}^I{x}_{it l}+{Q}_{\left(\mathrm{t}-1\right)\mathrm{l}}\hbox{-} \sum \limits_{i=1}^I{w}_{it}\le CR\begin{array}{c},\end{array}\forall t;l=1,2\dots l\\ {}{Q}_{\mathrm{t}\mathrm{l}}={q}_{tl}+\sum \limits_{i=1}^I{x}_{it l}+{Q}_{\left(\mathrm{t}-1\right)\mathrm{l}}\hbox{-} \sum \limits_{i=1}^I{w}_{it}\begin{array}{c},\end{array}\forall t;l=1,2\dots {l}_{\mathrm{t}};{Q}_{0l}=0\\ {}{T_{tl}}^{\ast }=\sqrt{\frac{2{Cs}_t}{Ch_t\cdot \sum \limits_{i=1}^I{X}_{it l}}}\cdot \sqrt{\frac{\left({Ch}_t+{Cq}_t\right)}{Cq_t}}\cdot \sqrt{\frac{g_{tl}}{g_{tl}-\sum \limits_{i=1}^I{X}_{it l}}}\begin{array}{c},\end{array}\forall t;l=1,2\dots l\\ {}{D_{tl}}^{\ast }=\sqrt{\frac{2{Cs}_t\cdot \sum \limits_{i=1}^I{X}_{it l}}{Ch_t}}\cdot \sqrt{\frac{\left({Ch}_t+{Cq}_t\right)}{Cq_t}}\cdot \sqrt{\frac{g_{tl}}{g_{tl}-\sum \limits_{i=1}^I{X}_{it l}}}\begin{array}{c},\end{array}\forall t;l=1,2\dots l\\ {}{B_{tl}}^{\ast }=\sqrt{\frac{2{Cs}_t\cdot {Ch}_t\cdot \sum \limits_{i=1}^I{X}_{it l}}{Cq_t\cdot \left({Ch}_t+{Cq}_t\right)}}\cdot \sqrt{\frac{g_{tl}-\sum \limits_{i=1}^I{X}_{it l}}{g_{tl}}}\begin{array}{c},\end{array}\forall t;l=1,2\dots l\\ {}{Tq_{tl}}^{\ast }=\sqrt{\frac{2{Cs}_t\cdot {Ch}_t}{Cq_t\left({Ch}_t+{Cq}_t\right)\sum \limits_{i=1}^I{X}_{it l}}}\cdot \sqrt{\frac{g_{tl}}{g_{tl}-\sum \limits_{i=1}^I{X}_{it l}}}\begin{array}{c},\end{array}\forall t;l=1,2\dots l\\ {}{T^{\ast}}_{jtl}={D}^{\ast }/g\\ {}{X}_{it l}\ge 0\begin{array}{c},\end{array}\forall t;l=1,2\dots l\end{array}\right. $$
(14b)

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Xu, Y., Fu, Q., Zhou, Y. et al. Inventory Theory-Based Stochastic Optimization for Reservoir Water Allocation. Water Resour Manage 33, 3873–3898 (2019). https://doi.org/10.1007/s11269-019-02332-6

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