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A decomposition technique is applied to address the simultaneous scheduling and optimal control problem of multigrade polymerization reactors. The simultaneous scheduling and control (SSC) problem is reformulated using Lagrangean decomposition as presented by Guignard and Kim. The resulting model is decomposed into scheduling and control subproblems, and solved using a heuristic approach used before by Van den Heever et al. in a different kind of problem. The methodology is tested using a methyl methacrylate (MMA) polymerization system, and the high impact polystyrene (HIPS) polymerization system, with one continuous stirred‐tank reactor (CSTR), and with a complete HIPS polymerization plant composed of a train of seven CSTRs. In these case studies, different polymer grades are produced using the same equipment in a cyclic schedule. The results of the heuristic decomposition technique are compared against those obtained by solving the problem without decomposition, whenever both solutions were available. The presence of a duality gap for the decomposed solution is observed as expected when integer variables and other nonconvexities are present. Computational times in the first two examples were lower for the decomposition heuristic than for the direct solution in full space, and the optimal solutions found were slightly better. The example related to the full scale HIPS plant was only solvable using the decomposition heuristic. © 2007 American Institute of Chemical Engineers AIChE J, 2008

This work presents a Mixed-Integer Dynamic Optimization (MIDO) formulation for the simultaneous process design, cyclic scheduling, and optimal control of a Methyl Methacrylate (MMA) continuous stirred-tank reactor (CSTR). Different polymer grades are defined in terms of their molecular weight distributions, so that state variables values during steady states are kept as degrees of freedom. The corresponding mathematical formulation includes the differential equations that describe the dynamic behavior of the system, resulting in a MIDO problem. The differential equations are discretized using the simultaneous approach based on orthogonal collocation on finite elements, rendering a Mixed Integer Non-Linear programming (MINLP) problem where a profit function is to be maximized. The objective function includes product sales, some capital and operational costs, inventory costs, and transition costs. The optimal solution to this problem involves design decisions: flow rates, feeding temperatures and concentrations, equipment sizing, variables values at steady state; scheduling decisions: grade productions sequence, cycle duration, production quantities, inventory levels; and optimal control results: transition profiles, durations, and transition costs. The problem was formulated and solved in two ways: as a deterministic model and as a two-stage programming problem with hourly product demands as uncertain parameter described by discrete distributions.2

This paper addresses the solution of simultaneous scheduling and planning problems in a production-distribution network of continuous multiproduct plants that involves different temporal and spatial scales. Production planning results in medium and long-term decisions, whereas production scheduling determines the timing and sequence of operations in the short-term. The production-distribution network is made up of several production sites distributing to different markets.The planning and scheduling model has to include spatial scales that go from a single production unit within a site, to a geographically distributed network. We propose to use two decomposition methods to solve this type of problems. One method corresponds to the extension of bi-level decomposition of Erdirik-Dogan and Grossmann (2008) to a multi-site, multi-market network. A second method is a novel hybrid decomposition method that combines bi-level and spatial Lagrangean decomposition methods. We present four case studies to observe the performance of the full space planning and scheduling model, the bi-level decomposition, and the bi-level Lagrangean method, in profit maximization problems. Numerical results indicate that in large-scale problems, decomposition methods outperform the full space solution, and that as problem size grows the hybrid decomposition method becomes faster than the bi-level decomposition alone.

We address in this paper the optimization of a multi-site, multi-period, and multi-product planning problem with sequence-dependent changeovers, which is modeled as a mixedinteger linear programming (MILP) problem. Industrial instances of this problem require the planning of a number of production and distribution sites over a time span of several months. Temporal and spatial Lagrangean decomposition schemes can be useful for solving these types of large-scale production planning problems. In this paper we present a theoretical result on the relative size of the duality gap of the two decomposition alternatives. We also propose a methodology for exploiting the economic interpretation of the Lagrange multipliers to speed the convergence of numerical algorithms for solving the temporal and spatial Lagrangean duals. The proposed methods are applied to the multi-site multi-period planning problem in order to illustrate their computational effectiveness.