Stochastic objective function constraints

Compensating slack variables accounting for shortfall and/or surplus in production are introduced in the stochastic constraints with the following results (i) inequality constraints are replaced with equality constraints (ii) numerical feasibility of the stochastic constraints can be ensured for all events and (iii) penalties for feasibility violations can be added to the objective function. Since a probability can be assigned to each realization of the stochastic parameter vector (i.e., to each scenario), the probability of feasible operation can be measured. In this... 

Some well-known stochastic methods for solving SOO problems are simulated annealing (SA), GAs,DE and particle swarm optimization (PSO). These were initially proposed and developed for optimization problems with bounds only [that is, unconstrained problems without Equations (4.7) and (4.8)]. Subsequently, they were extended to constrained problems by incorporating a strategy for handling constraints. One relatively simple and popular sdategy is the penalty function, which involves modifying the objective function (Equation 4.5) by the addition (in the case of minimization) of a term which depends on constraint violation. Eor example, see Equation (4.9),... 

The formulation of the batch plant retrofit problem generally involves either deterministic methods based on mathematical programming such as LP (Linear Programming), NLP (Non-Linear Programming), MILP or MINLP or stochastic ones such as evolutionary algorithms. The non-linearity aspect may result from the nature of the constraints and/or the objective function. The mixed feature means that continuous and integer variables are involved in the formulation. 

A basic assumption of stochastic programming is that the probability distribution of the random variable is known. The target then is to find an optimal solution that makes the expected value of the system to be minimum (or maximum). According to the type of the objective function and constraints, the stochastic programming problem can be divided into stochastic linear programming problems and stochastic nonlinear programming problems. 

Conduct crossover operation and mutation operation of chromosome Execute the stochastic simulation algorithm of Sect. 4.3.2. Use stochastic simulation to test on feasibility of the chromosomes according to the constraints and calculate its objective function value for each valid chromosome ... 

Our case-study corresponds to a stochastic version of the boiler/turbo generator system problem presented by Edgar et al. (2001). The system may be modeled as a set of linear constraints and a linear objective function. The demand on the resources are considered as uncertain variables in the problem. The distributions used for the demands are shown... 

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