(582c) Simultaneous Planning and Scheduling for Multiproduct Continuous Plants

Planning and scheduling of process systems are closely linked activities. Both planning and scheduling deal with the allocation of available resources over time to perform a collection of tasks required to manufacture one or several products (Shah, 1998). However, in planning, the aim is to determine high level decisions such as production levels and product inventories for given marketing forecasts and demands over a long time horizon (e.g. months to years). Scheduling, on the other hand, is defined over a short time horizon (e.g. days to weeks) and involves lower level decisions such as the sequence and detailed timing in which various products should be processed at each equipment in order to meet the production goals set by the planning problem. The simplest alternative for solving planning and scheduling problems is to formulate a single simultaneous model that spans the entire planning horizon. The limitation of this approach is that, the size of this detailed model becomes intractable due to the potential exponential increase in the computation. Therefore, the traditional strategy for solving planning and scheduling problems is to follow a hierarchical approach in which the planning problem is solved first to define the production targets. The scheduling problem is solved next to meet these targets (Shapiro, 2001). The problem of this approach, however, is that a solution determined at the planning level does not necessarily lead to feasible schedules. Hence, there is a need to develop methods and approaches that can more effectively integrate planning and scheduling.

We propose in this paper a novel bi-level decomposition scheme that allows rigorous integration and optimization of planning and scheduling of a multiproduct continuous plant consisting of a single processing stage. We present a multiperiod MILP optimization model for simultaneous planning and scheduling that is based on a continuous time representation. Demands are assumed to be given as lower bounds and due dates are defined at the end of each week. Also, sequence dependent changeovers are taken into account. The problem is to determine the products to be produced in each week, the sequencing of products, length of production times, amounts of products to be produced and inventory levels for each product. The objective is to maximize the total profit in terms of sales revenues, operating costs, inventory costs and transition costs. In order to avoid nonlinearities in the objective that are due to the inventory cost, we develop an overestimation that can be expressed in linear form. Since the MILP model becomes computationally very expensive to solve as the length of the planning horizon increases, we propose a rigorous bi-level decomposition algorithm that reduces the computational cost of the problem. The original simultaneous model is decomposed into an aggregated upper level MILP planning problem and a lower level MILP planning and scheduling problem. The upper level problem determines the products to be produced in each week, production levels and product inventories. The aggregated upper level is a relaxation of the original simultaneous problem and thus it yields an upper bound to the profit. The lower level, which provides a lower bound on the profit, is solved in the reduced space of binary variables and determines production levels, product inventories as well as the detailed sequence of products and their corresponding processing times. The procedure iterates until the difference between the upper and the lower bounds is less than a specified tolerance. To expedite the search, novel subset, superset and capacity cuts are proposed. These cuts are used to reduce the feasible search space for the binary variables and to tighten the gap between the solutions of the two levels. The proposed integration scheme ensures consistency and optimality within a specified tolerance. Numerical examples are presented to illustrate the performance of the algorithm and to compare it with a full space solution. The results show that the proposed method is able to solve to optimality the problems at this size while the single simultaneous MILP model proved to be intractable.