A Linear Dynamic Programming Approach to
Irrigation System Management with Depleting Groundwater

The Southern High Plains area of the United States has little rain fall. Farmers in these areas such as Texas rely most on irrigation systems. The main source of their water comes from the Ogallala Aquifer. With little rain fall and no one refilling the aquifer, it poses a threat of running dry. Many theoretical models have been created to solve this problem but are not realistic. An applied model called the recursive linear programming (RLP) technique great but does not provide information as to the optimal temporal pattern of investment and the use of stock resources. A planning method such as the linear dynamic programming formulation will help solve many of the issues the RLP technique could not. Some of these issues include: availability of underground water, the area developed for irrigation, design of the system, water requirements, irrigation capacity, and the expected annual water level decline. The linear programming formulation is used to figure out how much money goes into irrigation systems. It includes the amount of water used and how much money is spent on irrigation. This formula is used to maximize resource plans and to help profit. This is done by looking at the type of irrigation system, the pumping capacity (how much ground water is being taken out), and the surrounding land. The linear dynamic programming formulation is a combination of parametric linear programming and discrete dynamic programming. This formulation method is based on several different studies. The studies have shown that dynamic programming can be used in decomposition problems as an alternative to linear programming. The linear dynamic programming formulation is based off of the polyperiod linear programming model too which has underwent various studies. This is what forms independent systems. These systems form State or Control variables. These form the linear dynamic programming method which handles the agricultural problems with the Ogallala Aquifer. The procedure of this method contains two phases. The results base the decisions which are used year to year about how to conserve the water in the aquifer, different ways to irrigate, and drilling and pumping methods. Phase one of this method creates one year plans for farms in order to get the water out of and back into the aquifer. It examines the profits and uses it as a framework for the multiple year dynamic programs. Phase two evaluates water usage over time by computing the results of how much water is used by irrigation resources and the return to the aquifer. Phase one of the linear dynamic programming formulation examines the area, wells, activities performed on the land with the water, resources, time, and how much water. The variables include the production level with the land because that interferes with how much water is used, and the return rate to the aquifer. The well supply is a big factor because there is a need for an adequate supply in order to meet the demand due to timing and types of activities. The other big factor is annual pumpage. This is because there cannot be more consumption from irrigation systems then what is available. Lastly, phase one accounts for Irrigation Distribution Capacity which is the seasonal demand of water for irrigation of certain seasonal crops. The outcomes of these equations provide a one-year plan. Phase two examines what will maximize the net value and profit. It evaluates what can maximize future returns. The decisions are made at the beginning of each stage and it is assumed that the outcome is known. Profit is based on saturation, thickness, and current type and size of the irrigation system. Other variables include the current state of the aquifer, quantities of groundwater, and capacity of the irrigation system. These all factor water distribution and return to the aquifer in order to make a dynamic program and a multiperiod plan. The application of this method is fairly simple. Factors of each crop are laid out such as net revenue, land used, and water used. It is examined by season. This evaluation of investments helps with plans for future irrigation and water use from the aquifer. In conclusion, the linear dynamic programming formula is essential to farmers using the aquifer. Its analysis allows farmers to see if more drills and wells need to be made and if water usage is too much or too little. This method allows people to adequately use the Ogallala Aquifer with little waste and to analyze supply, number of wells, and size and types of irrigation systems.