Optimal retrofit design of multi-product batch plants

The problem of retrofit design of multiproduct batch plants is considered in which the optimal addition of equipment to an existing plant must be determined in view of changes in the product demands. In order to circumvent the combinatorial problem of having to analyze many alternatives the problem is formulated as a mixed-integer nonlinear program (MINLP) and solved with the outer-approximation algorithm of Duran and Grossmann. By using suitable variable transformations and approximations, the global optimum solution is guaranteed. The proposed MINLP model is also extended to the case when time-varying forecasts for product demands are given. Numerical examples are presented. University Libraries Carnegie Meflon University Pittsburgh, Pennsylvania 15213 Introduction This paper will address the problem of optimal retrofit design of multiproduct batch plants. In this problem the sizes and types of equipment of an existing multiproduct batch plant are given. Due to the changing market conditions, it is assumed that new production targets and selling prices are specified for a given set of products. The problem then consists in finding those design modifications that involve purchase of new equipment for the existing plant to maximize the profit. The production targets that are given for the retrofit problem could be fixed or be given as upper limits. In this work the production levels are treated as upper limits to account for the following possibility. If the cost of the new equipment to operate at these new production levels is more than the revenue from the increased production, then either no new equipment should be purchased or else limited additions of equipment should be made at lower production levels. Therefore, the production levels must be optimized as part of the retrofit design problem. Other assumptions that will be used in the retrofit design problem of this paper correspond to the ones that are commonly used in the optimal design of multiproduct batch plants (e.g. see Sparrow et al, 1975, Grossmann and Sargent, 1978). These assumptions include the following: The recipes for all the products are given, while processing times are specified for each of the products in each type of equipment. The products are manufactured sequentially using an overlapping production schedule. Also, it is assumed that material can be held in its processing unit until the next stage is ready. That is, the processing vessels can act as their own storage tanks. In addition, a continuous range of equipment sizes is assumed to be available, and the number of batches is permitted to be non-integer since this is usually a large number. Finally no semi-continuous equipment is considered for the plant design, although in principle this aspect could be included in the problem formulation (see Knopf et al, 1982). As will be shown in this paper the optimal retrofit design problem for multiproduct batch plants can be formulated as a mixed-integer nonlinear programming (MINLP) problem in which two possiblities for adding new equipment in each batch stage are included. The added equipment can be used to decrease cycle times or to increase the batch sizes of the different products. By using exponential transformations and piecewise linear approximations, it is shown that the outer-approximation method of Duran and Grossmann(1983) will converge to the global optimum solution of the MINLP problem. The extension to the multiperiod case where production forecasts are given for several time periods is also considered. Three examples are presented to show that the combinatorial problem in the retrofit design can be handled effectively. Options for New Equipment The design modifications that are considered in the retrofit design of a multiproduct batch plant will involve the addition of new equipment to the existing plant. Any new equipment can be utilized jn two ways: (1) to ease bottleneck stages by operating in parallel but sequentially (option C), or (2) to increase the size of the present batches by operating in parallel and in phase with the current equipment (option B). Option C increases production by decreasing the cycle time of a product, the time needed to make one batch of a product. The new equipment used in this way operates out of phase with the existing equipment. The Gantt charts of Figure 1 demonstrate how production is increased with this design alternative. As can be seen option C decreases the idle time of a unit, thus allowing for more efficient utilization of the equipment. Option B on the other hand increases production by augmenting the batch size of a product. New equipment utilized in this fashion operates in parallel and in phase with the existing equipment, as shown in Figure 2. This option takes advantage of excess volume of a unit, allowing for better utilization of the capacity of the unit. Since the two options cited above can be applied to each of the batch stages, all the alternatives for equipment addition in the retrofit of a multi-product batch plant can be embedded within a super-structure as shown in Figure 3. Although just one potential new unit per stage is shown in this figure, it is clearly possible to specify multiple units for each option at each stage. By using this superstructure representation, the retrofit problem can be formulated as a mixed-integer nonlinear program to determine the optimal design modification without having to examine all the possible alternatives. Formulation The goal of the retrofit design problem is to maximize the profit of the batch processing plant given new product demand and prices. Profit is defined here as the net income from selling the products minus the annualized investment cost. The expected net profit per unit of product P will be denoted as p.. The cost of the equipment will be approximated by a fixed-charge cost model, where K is the annualized fixed charge of equipment type j, which includes the costs of piping, instrumentation, and some installation expenses, and c. is the annualized proportionality constant of equipment type j, which accounts for the linear increase of cost with the size of the vessel. The objective function for the retrofit problem can then be formulated as: ISI M Z B N o l d Z C max Y p n B Y f 3^ ^T c (V ) T c. V ) n)