A Lagrangean Relaxation and A Heuristic for the Pooling Problem

The pooling problem is one of the fundamental optimization problems encountered in the petroleum industry. In the pooling problem, final products are produced using two stages of blending operations. In the first stage, raw materials are mixed together to produce intermediate products. In the second stage, intermediate products and some of the raw materials are blended together according to product demand and quality requirements. Generally, the pooling problem is a nonlinear problem because the output stream qualities, which are unknown, depend on the volume, which is also unknown, and on the quality of the input streams. Specifically, nonlinearity and nonconvexity are due to the use of bilinear terms either in the quality constraints or in the objective function. Nonlinearity and nonconvexity result in several local optima, making the process of solving large-scale pooling problems to global optimality very challenging. Therefore, developing efficient heuristics for large-scale pooling problems is very desirable. Moreover, devising tight bounds on the global solutions is essential to assess the quality of the proposed heuristics. In this thesis, we use a Lagrangean relaxation approach where feasible solutions and lower bounds are generated for the pooling problem. The procedure targets all nonlinear constraints and penalizes their violation in the objective function. The resulting Lagrangean subproblem has a nonlinear objective function and linear constraints. The Lagrangean subproblem is reformulated as a mixed integer programming problem where the nonlinearities in the objective function are eliminated at the expense of using binary variables. The obtained Lagrangean lower bounds are strengthened using valid cuts that are based on the relaxed bilinear terms. In addition, at every iteration of the Lagrangean algorithm, the subproblem solutions are used to generate feasible solutions to the pooling problem. The procedure is applied to fifteen pooling problems collected from the literature. Some of these problems have a single quality and others have multiple qualities. Numerical results show that for eight solved cases, the obtained Lagrangean lower bounds are equal to the global optima, whereas for seven cases the obtained Lagrangean lower bound is on average 8.2% from the global optimum. Numerical results indicate the efficiency of the heuristic. For nine cases, the heuristic gives the global solution, and for the other cases the heuristic solutions are within 1.8% of the global optimum

Une approche exacte de résolution de problèmes de pooling appliquée à la fabrication d'aliments

Cette thèse intitulée Une approche exacte de résolution de problèmes de pooling appliquée à la fabrication d aliments , porte sur la résolution (par des méthodes exactes d optimisation) de problèmes industriels liés à la fabrication d aliments. Ces problèmes industriels traitent de l aide à la décision pour la fabrication d aliments pour des animaux et se rapprochent de problèmes biens connus de la littérature scienti que, à savoir les problèmes de pooling. La méthode présentée dans cet exposé permet de résoudre les problèmes d optimisation bilinéaires issus de cette problématique industrielle. Elle est basée un branch-and-bound résolvant des linéarisations. Une approche lagrangienne a aussi été explorée et testée pour calculer des bornes inférieures.A global approach to solve pooling problem applied to feed mix industry deals with the resolution of non linear non convex optimization problem which can occur in the feed mix industry. Feed mix industry problems are close to pooling problem, well-known in the literature. They are aimed to help decision maker in formulating feed, ie. To decide how to blend raw material to make a product satisfying nutrient and production constraints. The brand-and-bound algorithm presented in this these is aimed to solved large-scaled bilinear problems with bilinear constraints. A lagrangian approach has also been developed to obtain valid lower bound.SAVOIE-SCD - Bib.électronique (730659901) / SudocGRENOBLE1/INP-Bib.électronique (384210012) / SudocGRENOBLE2/3-Bib.électronique (384219901) / SudocSudocFranceF

Reachability analysis and deterministic global optimization of differential-algebraic systems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2012.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student-submitted PDF version of thesis.Includes bibliographical references (p. 447-460).Systems of differential-algebraic equations (DAEs) are used to model an incredible variety of dynamic phenomena. In the chemical process industry in particular, the numerical simulation of detailed DAE models has become a cornerstone of many core activities including, process development, economic optimization, control system design and safety analysis. In such applications, one is primarily interested in the behavior of the model solution with respect variations in the model inputs or uncertainties in the model itself. This thesis addresses two computational problems of general interest in this regard. In the first, we are interested in computing a guaranteed enclosure of all solutions of a given DAE model subject to a specified set of inputs. This analysis has natural applications in uncertainty quantification and process safety verification, and is used for many important tasks in process control. However, for nonlinear dynamic systems, this task is very difficult. Existing methods apply only to ordinary differential equation (ODE) models, and either provide very conservative enclosures or require excessive computational effort. Here, we present new methods for computing interval bounds on the solutions of ODEs and DAEs. For ODEs, the focus is on efficient methods for using physical information that is often available in applications to greatly reduce the conservatism of existing methods. These methods are then extended for the first time to the class of semi-explicit index-one DAEs. The latter portion of the thesis concerns the global solution of optimization problems constrained by DAEs. Such problems arise in optimal control of batch processes, determination of optimal start-up and shut-down procedures, and parameter estimation for dynamic models. In nearly all conceivable applications, there is significant economic and/or intellectual impetus to locate a globally optimal solution. Yet again, this problem has proven to be extremely difficult for nonlinear dynamic models. A small number of practical algorithms have been proposed, all of which are limited to ODE models and require significant computational effort. Here, we present improved lower-bounding procedures for ODE constrained problems and develop a complete deterministic algorithm for problems constrained by semi-explicit index-one DAEs for the first time.by Joseph Kirk Scott.Ph.D