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.
Since plants that form the network are subject to fluctuations in product demand or random mechanical failures, design decisions such as adding redundant units and increasing storage between units can increase the flexibility and reliability of an integrated site. In this paper, we develop a bi-criterion optimization model that captures the trade-off between capital investment and process robustness in the design of an integrated site. Design decisions considered are increases in process capacity, introduction of parallel units, and addition of intermediate storage. The Mixed-integer
This paper presents a novel mixed-integer linear programming (MILP) formulation for the Tank Farm Operation Problem (TFOP), which involves simultaneous scheduling of continuous multi-product processing lines and the assignment of dedicated storage tanks to finished products. These products are not allowed to mix in storage tanks. Therefore, once an assignment is made, it has to be maintained until the end of the operating horizon. Since all products processed by finishing lines have to go into the tank farm before being shipped, there is the potential to run out of storage, ultimately impacting the throughput of the finishing lines, a condition known as blocking. The objective of the problem is to minimize blocking of the finished lines by obtaining an optimal schedule and an optimal allocation of storage resources. The scheduling part of the model is based on the Multi-operation Sequencing (MOS) model by Mouret et al., (2011). The formulation is tested in three examples of different size and complexity. The possibility of incorporating the MILP model into a decision support system in combination a Discrete Event Simulation (DES) model of a tank farm is also discussed.
Integrated sites are tightly interconnected networks of large-scale chemical processes.Given the large-scale network structure of these sites, disruptions in any of its nodes, or individual chemical processes, can propagate and disrupt the operation of the whole network. Random process failures that reduce or shut down production capacity are among the most common disruptions. The impact of such disruptive events can be mitigated by adding parallel units and/or intermediate storage. In this paper, we address the design of large-scale, integrated sites considering random process failures. In a previous work (Terrazas-Moreno et al., 2010), we proposed a novel mixed integer linear programming (MILP) model to maximize the average production capacity of an integrated site while minimizing the required capital investment. The present work deals with the solution of large-scale problem instances for which a strategy is proposed that consists of two elements. On one hand, we use Benders decomposition to overcome the combinatorial complexity of the MILP model. On the other hand, we exploit discrete-rate simulation tools to obtain a relevant reduced sample of failure scenarios or states. We first illustrate this strategy in a small example. Next, we address an industrial case study where we use a detailed simulation model to assess the quality of the design obtained from the MILP model.
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