Suppose that the extreme value problem of a system can be reduced to the following nonlinear programming problem with random variable t.
Stochastic simulation and management of groundwater system
If the decision-maker must make a decision X before observing the realization of the random variable T, then it is likely that the constraint condition in (3.3) cannot be established for a certain T. Therefore, in some cases, the constraint can be broken to some extent, that is, the probability of satisfying the constraint is not less than a given constant. For each constraint condition in formula (3.3), there is a specified number αi, i= 1, 2, …, m, 1≥αi≥0, so:
Stochastic simulation and management of groundwater system
At this point, Equation (3.3) will become a probabilistic constrained programming problem:
Stochastic simulation and management of groundwater system
For example, if the decision-maker wants to design a water supply source, it is required that the groundwater level will drop the least, the impact on the environment will be the least, the water supply capacity will be the largest and the cost will be the lowest. The given constraints usually include: in a given place and time period, the groundwater level is not lower than a certain value, and the influence of groundwater level on the funnel does not exceed a certain design boundary. However, due to many factors (such as atmospheric precipitation, nearby river flow, industrial and agricultural production water, etc. It is random to influence and control the funnel of groundwater level, so it is obviously unreasonable and difficult to require the water source to meet its given restrictions under any circumstances. Therefore, the more reasonable requirement is that the designer of water source can meet the constraints with greater probability. Adopting probabilistic constrained stochastic programming model is an effective means to solve this kind of problem.
Probability constrained stochastic linear programming model can generally be expressed as:
Stochastic simulation and management of groundwater system
Where: X-N dimension decision vector;
ξ-random vector;
F(x, ξ)-objective function;
Gi(x, ξ)-random constraint function.
Equation (3.5) is a typical mathematical programming model with random parameters. Because of the existence of random variable ξ, the mathematical programming model has no clear definition. A meaningful mathematical programming problem with random parameter variables should be the following probabilistic constrained programming model:
Stochastic simulation and management of groundwater system
Where: p {} represents the probability that the event in {} holds;
α and β-given constraints and confidence level of the objective function according to the actual situation.
The condition that point X is a feasible solution is if and only if the probability measure of the event, …, p} is not less than α, that is, the probability of violating the constraint condition is less than (1-α).
No matter what random parameter ξ and function form F, for each given decision X, f(X, ξ) is a random variable, and its probability density can be expressed by function φ f (x, ξ)(f). In this way, there may be multiple -fs that make P{f(X, ξ)≥-f}≥β hold. The maximum target value -f should be the maximum value obtained by the objective function f(X, ξ) when the confidence is at least β. Namely:
Stochastic simulation and management of groundwater system
As shown in Figure 3. 1:
Figure 3. 1 Probability density function φ f (x, ξ)(f) and target value -f
Sometimes, according to the nature and stage of the problem studied, the given confidence level of constraints can be different for different constraints. That is, the constraint in (3.6) can be expressed as:
Stochastic simulation and management of groundwater system
Similarly, for the minimization problem, there are:
Stochastic simulation and management of groundwater system
As a generalization of the single-objective problem, the multi-objective probabilistic constrained programming problem can be expressed as:
Stochastic simulation and management of groundwater system