Mixed-Integer Programming Models for the Scheduling and Operational Optimization of Multiproduct Continuous Plants A. Alle and J. M. Pinto* Department of Chemical Engineering, University of Sao Paulo, Avenida Professor Luciano Gualberto, Travessa 3,380, Sa ˜ o Paulo, SP, 05508-900 Brazil This paper addresses the problem of the simultaneous scheduling and optimization of the operating conditions of continuous multistage multiproduct plants with intermediate storage. First, an MINLP model, called TSPFLOW, that is based on the TSP formulation for product sequencing is proposed for the scheduling of the operation of such plants. TSPFLOW leads to a significant reduction in the number of constraints and continuous and binary variables from a formulation previously reported (Pinto, J. M.; Grossmann, I. E. Comput. Chem. Eng. 1994, 18, 797) and, consequently, a one-order-of-magnitude reduction in CPU time. Moreover, TSPFLOW solves instances two times as large as those formerly reported. Second, production rates and yields are introduced as additional optimization variables to state the simultaneous problem of scheduling with operational optimization. A linearization approach that employs the discreti- zation of nonlinear variables is presented and compared to the direct solution of the original MINLP. Results show that nonlinear restrictions are more effective than linear discrete restrictions from the points of view of both optimality and computational effort. The tradeoffs involved are very complex, and the development of a straightforward method (general rule) for optimal scheduling of the operation seems to be less effective than the proposed approach. 1. Introduction Continuous plants have traditionally been associated with single product lines. However, with the increasing need for more flexible processing systems, continuous plants that are able to process various products have become more common. In such continuous multiproduct plants, the field for scheduling is opened as questions such as when, where, and how much of each product should be produced and what the order of production should be must be answered. With respect to topology, a continuous plant can be composed of a single line or several parallel lines. Each line can consist of only one stage or several stages. A stage is a section of the plant where a unit operation (UO) occurs, such as reaction, distillation, drying, mix- ing, etc. Each stage can be composed of a single unit or several units. A unit is the piece of equipment where the UO takes place, such as a reactor, a distillation column, a dryer, a mixer, etc. Intermediate storage is necessary whenever consecu- tive stages do not operate at the same rate. Products accumulate between faster and slower stages and should be stored in tanks to avoid overflow of the slowest equipment. In plants where the unit design or adjust- ment permits the syncronous operation of all stages, intermediate storage might not be necessary. In a continuous plant, startups and shutdowns must be avoided. The plant should be scheduled to have uninterrupted operation during a cycle. A transition cost rather than a setup cost is incurred when changing from one product to another because, unlike batch plants, continuous units are not completely turned off or discharged. A transition cost and a transition time arise whenever a new product starts to be processed. These factors often refer to the cost and time associated with out-of-specification products generated while the unit is adjusted to a new set of operating conditions. The scheduling of continuous multiproduct plants involves tradeoffs between storage levels, quantities produced, cycle durations, and transition costs (see section 2). For instance, the less frequent the transitions, the smaller the associated costs, but the higher the inventory levels, and vice versa. Additionally, in a production line that has the flexibility of adjusting the production rates of its stages, better results could be achieved by joining the scheduling with the adjustment of the production rates. A combined optimization approach would intro- duce synergic effects for more effective plant operation. This comprehensive simultaneous problem would pro- vide additional degrees of freedom and thus better capture the complex tradeoffs among cycle times, inven- tory levels, and setup times. Despite the increasing interest in the scheduling of continuous multiproduct plants 1-4 and the extensive research into the optimization of the operating condi- tions of continuous processes, much less work has been reported on the simultaneous scheduling and operation of such plants. Operating considerations have been incorporated at the process development 5 and the plant design 6,7 levels. To our knowledge, the idea of combining scheduling and operational optimization is innovative in terms of continuous plants. However, batch plants have been considered under this approach. 8 That work incorpo- rated dynamic processing conditions for products in a multiproduct batch plant in the context of scheduling and equipment design. The objective was a general function of fixed design costs, operating costs, inventory costs, production revenues, etc. The scheduling problem was addressed in the scope of flowshop plants for * To whom correspondence should be addressed. E-mail: jompinto@usp.br. Phone: (5511) 3091-2216. Fax: (5511) 3813- 2380. 2689 Ind. Eng. Chem. Res. 2002, 41, 2689-2704 10.1021/ie0106480 CCC: $22.00 © 2002 American Chemical Society Published on Web 04/30/2002