The mathematical model of dynamic programming consists of system state transition equation, objective function and constraint conditions.

(1) state transition equation. System state transition equation is a mathematical expression describing the relationship between i+ 1 stage state variables and I stage state variables and decision variables. For example, the state transition equation of one-dimensional multi-stage decision-making process is

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(2) objective function. The optimal extreme value of the objective function depends on the nature of the management problem. If the goal is expense or cost, the minimum value should be taken, and if the goal is revenue, the maximum value should be taken.

For example, taking the maximum economic benefit as the goal of water resources management, the objective function can be expressed as

Optimal management of drainage, water supply and environmental protection in North China coalfield

According to the optimization principle, the recursive equation can be written as.

Optimal management of drainage, water supply and environmental protection in North China coalfield

(3) Constraints. Constraints on state variables and decision variables can be limited according to the state space and decision space of management problems, which are recorded as

Optimal management of drainage, water supply and environmental protection in North China coalfield