On the parametric optimization of mathematical programs with binary variables and its applications in the chemical engineering process synthesis.

摘要

In order to substantiate an optimal design decision it is imperative that the solution is checked for the stability of its performance at least in its close neighborhood. Tools for the performance of stability or sensitivity analysis of mathematical programs suitable for chemical process synthesis problems are currently unavailable. The two major stumbling blocks for their development are the nonlinearity on one hand and the presence of discrete variables on the other. This work addresses these two problems for scalar variations.; In the LP case the existence of explicit necessary and sufficient conditions makes the sensitivity analysis a trivial algorithmic procedure. In the MILP and the MINLP case however the optimal solution is reached by an implicit exhaustive search and the optimality conditions are scattered throughout the solution procedure. To reclaim these conditions and process them in order to check the feasibility or optimality of the current optimal in a neighbor point is tantamount to resolving the initial problem, and even more without managing to deduce the limits of the feasibility or optimality of the current solution.; First the relevant literature is reviewed in some detail. Extending on the existing literature, the sensitivity analysis problem in the linear case is formulated as a single point MILP that identifies exactly the optimality limits of the current optimal solution and the next optimal integer solution. The approach is based on the inclusion in the original formulation of the optimal value function which is in turn obtained from the parameterization, with respect to the continuous variables, of the optimal solution at the initial instance.; For the nonlinear case two algorithms are proposed one extending on an algorithm proposed for the linear problem and one extending on an algorithm used to solve a single point MINLP, that of the Outer Approximation.; Parametric results in two illustrative examples one of which addresses the synthesis of a chemical complex from the dual objective of optimizing the economic performance while also accommodating the toxicity hazards of the plant, and the other the planning of the production of a chemical under uncertain product demand, provide a robust decision making environment and potentially even suggest, in a comprehensive manner, the most preferrer solution.