Water Supply and Demand Sensitivities of Linear Programming Solutions to a Water Allocation Problem

Author(s)
Konstantine P. Georgakakos

ABSTRACT

This work formulates and implements a mathematical optimization program to assist water managers with water allocation and banking decisions to meet demands. Linear programming is used to formulate the constraints and objective function of the problem and tests of the developed program are performed with data from the Castaic Lake Water Agency (CLWA) in Southern California. The problem is formulated as a deterministic programming problem over a five year planning horizon with annual resolution. The program accepts annual water allocations from the State Water Project (SWP) in California. It then determines the least-cost feasible allocation of this water toward meeting annual demands in the five-year planning horizon. Local water sources, including water recycling, and water banking programs with their constraints and costs are considered to determine the optimal water allocation policy within the planning horizon. Although there is not enough information to fully account for the uncertainty in future allocations and demands as part of the decision problem solution for CLWA, uncertainty in the SWP allocation is considered in the tests, and sensitivity analyses is performed with respect to demand increases to derive inferences regarding the behavior of the median minimum-cost solutions and of the risk of failure to meet demand.

This work formulates and implements a mathematical optimization program to assist water managers with water allocation and banking decisions to meet demands. Linear programming is used to formulate the constraints and objective function of the problem and tests of the developed program are performed with data from the Castaic Lake Water Agency (CLWA) in Southern California. The problem is formulated as a deterministic programming problem over a five year planning horizon with annual resolution. The program accepts annual water allocations from the State Water Project (SWP) in California. It then determines the least-cost feasible allocation of this water toward meeting annual demands in the five-year planning horizon. Local water sources, including water recycling, and water banking programs with their constraints and costs are considered to determine the optimal water allocation policy within the planning horizon. Although there is not enough information to fully account for the uncertainty in future allocations and demands as part of the decision problem solution for CLWA, uncertainty in the SWP allocation is considered in the tests, and sensitivity analyses is performed with respect to demand increases to derive inferences regarding the behavior of the median minimum-cost solutions and of the risk of failure to meet demand.

Cite this paper

K. Georgakakos, "Water Supply and Demand Sensitivities of Linear Programming Solutions to a Water Allocation Problem,"*Applied Mathematics*, Vol. 3 No. 10, 2012, pp. 1285-1297. doi: 10.4236/am.2012.330185.

K. Georgakakos, "Water Supply and Demand Sensitivities of Linear Programming Solutions to a Water Allocation Problem,"

References

[1] D. A. Pierre, “Optimization Theory with Applications. Chapter 5,” Dover Publications, Inc., New York, 1986, pp. 193-258.

[2] G. B. Dantzig, “Maximization of a Linear Function of Variables Subject to Linear Inequalities,” In: T. C. Koopmans, Ed., Activity Analysis of Production and Allocation, Cowles Commission Monograph, No. 13, Wiley, New York, 1951.

[3] G. B. Dantzig, A. Orden and P. Wolfe, “The Generalized Simplex Method for Minimizing a Linear form under Linear Inequality Constraints,” Pacific Journal of Mathematics, Vol. 5, No. 2, 1955, pp. 183-195.

[4] G. B. Dantzig, “Linear Programming and Extensions,” Princeton University Press, Princeton, 1963.

[5] S. Mehrorta, “On the Implementation of a Primal-Dual Interior Point Method,” SIAM Journal on Optimization, Vol. 2, No. 4, 1992, pp. 575-601. doi:10.1137/0802028

[6] Y. Zhang, “Solving Large-Scale Linear Programs by Interior-Point Methods under the MATLAB Environment,” Technical Report TR96-01, Department of Mathematics and Statistics, University of Maryland, Baltimore, 1995.

[7] D. M. Simmons, “Nonlinear Programming for Operations Research. Chapter 2,” Prentice-Hall, Inc., Englewood Cliffs, 1975, pp. 26-70.

[8] D. P. Loucks, J. R. Stedinger and D. A. Haith, “Water Resource Systems Planning and Analysis,” Prentice-Hall, Inc., Englewood Cliffs, 1981.

[9] D. P. Ahlfeld and G. Baro-Montes, “Solving Unconfined Groundwater Flow Management Problems with Successive Linear Programming,” Journal of Water Resources Planning and Management, Vol. 134, No. 5, 2008, pp. 404-412. doi:10.1061/(ASCE)0733-9496(2008)134:5(404)

[10] I. Ioslovich and P.-O. Gutman, “Optimal Monitoring and Management of a Water Storage,” Environmental Monitoring and Assessment, Vol. 138, No. 1-2, 2008, pp. 93-100. doi:10.1007/s10661-007-9745-8

[11] S. M. Kasterakis, G. P. Karatzas, I. K. Nikolos and M. P. Papadopoulou, “Application of Linear Programming and Differential Evolutionary Optimization Methodologies for the Solution of Coastal Subsurface Water Management Problems Subject to Environmental Criteria,” Journal of Hydrology, Vol. 342, No. 3-4, 2007, pp. 270-282.

[12] M. Pulido-Velazquez, M. W. Jenkins and J. R. Lund “Economic Values for Conjunctive Use and Water Banking in Southern California,” Water Resources Research, Vol. 40, 2004, pp. 1-15. doi:10.1029/2003WR002626

[13] K. P. Georgakakos and N. E. Graham, “Potential Benefits of Seasonal Inflow Prediction Uncertainty for Reservoir Release Decisions,” Journal of Applied Meteorology and Climatology, Vol. 47, No. 5, 2008, pp. 1297-1321. doi:10.1175/2007JAMC1671.1

[14] A. J. Draper, A. Munévar, S. K. Arora, E. Reyes, N. L. Parker, F. L. Chung and L. E. Peterson, “CalSim: Generalized Model for Reservoir System Analysis,” Journal of Water Resources Planning and Management, Vol. 130, No. 6, 2004, pp. 480-489. doi:10.1061/(ASCE)0733-9496(2004)130:6(480)

[15] S. M. Mesbah, R. Kerachian and M. R. Nikoo, “Developing Real Time Operating Rules for Trading Discharge Permits in Rivers: Application of Bayesian Networks,” Environmental Modeling and Software, Vol. 24, No. 2, 2009, pp. 238-246. doi:10.1016/j.envsoft.2008.06.007

[16] N. E. Graham, K. P. Georgakakos, C. Vargas and M. Echevers, “Simulating the Value of El Nino Forecasts for the Panama Canal,” Advances in Water Resources, Vol. 29, No. 11, 2006, pp. 1667-1677. doi:10.1016/j.advwatres.2005.12.005

[17] R. E. Howitt, “Positive Mathematical Programming,” American Journal of Agricultural Economics, Vol. 77, No. 2, 1995, pp. 329-342. doi:10.2307/1243543

[18] R. Bellman, “Dynamic Programming,” Dover Edition 2003, Dover Publications Inc., Mineola, New York, 1972.

[19] H. Yao and A. Georgakakos, “Assessment of Folsom Lake Response to Historical and Future Climate Scenarios: 2. Reservoir Management,” Journal of Hydrology, Vol. 249, No. 1-4, 2001, pp. 176-196. doi:10.1016/S0022-1694(01)00418-8

[1] D. A. Pierre, “Optimization Theory with Applications. Chapter 5,” Dover Publications, Inc., New York, 1986, pp. 193-258.

[2] G. B. Dantzig, “Maximization of a Linear Function of Variables Subject to Linear Inequalities,” In: T. C. Koopmans, Ed., Activity Analysis of Production and Allocation, Cowles Commission Monograph, No. 13, Wiley, New York, 1951.

[3] G. B. Dantzig, A. Orden and P. Wolfe, “The Generalized Simplex Method for Minimizing a Linear form under Linear Inequality Constraints,” Pacific Journal of Mathematics, Vol. 5, No. 2, 1955, pp. 183-195.

[4] G. B. Dantzig, “Linear Programming and Extensions,” Princeton University Press, Princeton, 1963.

[5] S. Mehrorta, “On the Implementation of a Primal-Dual Interior Point Method,” SIAM Journal on Optimization, Vol. 2, No. 4, 1992, pp. 575-601. doi:10.1137/0802028

[6] Y. Zhang, “Solving Large-Scale Linear Programs by Interior-Point Methods under the MATLAB Environment,” Technical Report TR96-01, Department of Mathematics and Statistics, University of Maryland, Baltimore, 1995.

[7] D. M. Simmons, “Nonlinear Programming for Operations Research. Chapter 2,” Prentice-Hall, Inc., Englewood Cliffs, 1975, pp. 26-70.

[8] D. P. Loucks, J. R. Stedinger and D. A. Haith, “Water Resource Systems Planning and Analysis,” Prentice-Hall, Inc., Englewood Cliffs, 1981.

[9] D. P. Ahlfeld and G. Baro-Montes, “Solving Unconfined Groundwater Flow Management Problems with Successive Linear Programming,” Journal of Water Resources Planning and Management, Vol. 134, No. 5, 2008, pp. 404-412. doi:10.1061/(ASCE)0733-9496(2008)134:5(404)

[10] I. Ioslovich and P.-O. Gutman, “Optimal Monitoring and Management of a Water Storage,” Environmental Monitoring and Assessment, Vol. 138, No. 1-2, 2008, pp. 93-100. doi:10.1007/s10661-007-9745-8

[11] S. M. Kasterakis, G. P. Karatzas, I. K. Nikolos and M. P. Papadopoulou, “Application of Linear Programming and Differential Evolutionary Optimization Methodologies for the Solution of Coastal Subsurface Water Management Problems Subject to Environmental Criteria,” Journal of Hydrology, Vol. 342, No. 3-4, 2007, pp. 270-282.

[12] M. Pulido-Velazquez, M. W. Jenkins and J. R. Lund “Economic Values for Conjunctive Use and Water Banking in Southern California,” Water Resources Research, Vol. 40, 2004, pp. 1-15. doi:10.1029/2003WR002626

[13] K. P. Georgakakos and N. E. Graham, “Potential Benefits of Seasonal Inflow Prediction Uncertainty for Reservoir Release Decisions,” Journal of Applied Meteorology and Climatology, Vol. 47, No. 5, 2008, pp. 1297-1321. doi:10.1175/2007JAMC1671.1

[14] A. J. Draper, A. Munévar, S. K. Arora, E. Reyes, N. L. Parker, F. L. Chung and L. E. Peterson, “CalSim: Generalized Model for Reservoir System Analysis,” Journal of Water Resources Planning and Management, Vol. 130, No. 6, 2004, pp. 480-489. doi:10.1061/(ASCE)0733-9496(2004)130:6(480)

[15] S. M. Mesbah, R. Kerachian and M. R. Nikoo, “Developing Real Time Operating Rules for Trading Discharge Permits in Rivers: Application of Bayesian Networks,” Environmental Modeling and Software, Vol. 24, No. 2, 2009, pp. 238-246. doi:10.1016/j.envsoft.2008.06.007

[16] N. E. Graham, K. P. Georgakakos, C. Vargas and M. Echevers, “Simulating the Value of El Nino Forecasts for the Panama Canal,” Advances in Water Resources, Vol. 29, No. 11, 2006, pp. 1667-1677. doi:10.1016/j.advwatres.2005.12.005

[17] R. E. Howitt, “Positive Mathematical Programming,” American Journal of Agricultural Economics, Vol. 77, No. 2, 1995, pp. 329-342. doi:10.2307/1243543

[18] R. Bellman, “Dynamic Programming,” Dover Edition 2003, Dover Publications Inc., Mineola, New York, 1972.

[19] H. Yao and A. Georgakakos, “Assessment of Folsom Lake Response to Historical and Future Climate Scenarios: 2. Reservoir Management,” Journal of Hydrology, Vol. 249, No. 1-4, 2001, pp. 176-196. doi:10.1016/S0022-1694(01)00418-8