Znd. E n g . C h e m . Res. 1994,33, 2688-2701
2688
Design of Batch Chemical Plants under Market Uncertainty Sriram Subrahmanyam,?Joseph F. Pekny,'**and Gintaras V. Reklaitiso School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907-1283
Design of batch plants under demand uncertainty is studied through a mathematical programming approach. The problem is posed as a mixed integer linear program (MILP). Scheduling needs to be considered, since it forms a n integral part of batch plant design. Instead of embedding the scheduling constraints in the design problem, leading to a n extremely large problem, scheduling is considered a s a subsequent stage solved after the Design SuperProblem. This decomposition has the advantage of including scheduling considerations of complex design problems without complicating the design problem itself. Infeasible scheduling problems are used to construct inequalities to eliminate the infeasibility-causing design solutions. Demand uncertainty is considered a t the Design level a s a set of scenarios, each of which is associated with an occurrence of uncertain parameters. Probabilities are assumed to be known a priori. The model is modified to address retrofit problems. Constraints are proposed for flexibility, and a regret analysis is investigated.
Introduction There has been an increased interest in the design of batch processes in recent years due to the growth of specialty chemicals, food products, and pharmaceutical industries. Batch production is carried out typically in a facility consisting of general purpose equipment with some degree of flexibility in their arrangement and use. Sharing of equipment is an important economic opportunity in many batch operations. Though this is not a necessary feature, it is an important issue that has to be addressed in most instances. The flexibility in the equipment assignment implies that there are a number of decisions to be made with respect to the operation of the facility. The design of the plant greatly depends on the operating policy. In the past, the operating policy has been defined by the campaign lengths and the degree of similarity of the recipe structure of the various products. The limiting cases of short campaigns with high and low degrees of similarity correspond to the classical flow shop and job shop, respectively. The short campaign strategy is useful for products with low product life span, small clean-out costs and high inventory costs. This paper deals with the design of batch plants operating under long campaigns, for which inventory costs, clean-out costs, and changeover costs are significant and product life spans are long. Further classification yields three modes of operation. High similarity recipe information leads to multiproduct batch plants. In this type of operation, a single fured equipment network is used to process all products only be one in a serial fashion. It should be obvious that this operating policy can be regarded conceptually as the simplest long campaign policy. Low degree of similarity of recipe structure corresponds t o the multipurpose plant operation. Production is achieved through multiple production lines, each of which is dedicated to the production of one product. The true multipurpose plant, however, will share the equipment items between production lines to optimize the cost. To make the operation more efficient, the plant may be reconfigured after a certain period. This policy E-mail:
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finds extensive use in the pharmaceutical industry, since the changeover costs and clean-out costs are extremely high. Intermediate degree of similarity of product recipe structure leads to multiplant type of operation, where production may occur via permanent and nonidentical production lines. The design of batch plants is complicated by the combinatorial nature of assigning equipment items to tasks. There are a large number of options that are open to the designer to choose from, increasing the complexity of the problem greatly. Another source of complexity that has not been adequately addressed in past research is the uncertain nature of key parameters in the design. There are various levels of uncertainty t o be dealt with. One classification from past work is in terms of short and long term uncertainty. Short term uncertainty includes variations in flow rates, temperature, etc. to which the plant responds within a short period of time. Long term variations, such as in product demand, occur over an extended period of time. The classification can be further broken up into specific classes pertinent to the design of batch plants. 1. Sales uncertainty involves the unpredictable changes in prices and level of demand of products over the horizon. 2. Material purchase uncertainty involves the unpredictable changes in prices and level of availability of raw materials over the horizon. 3, Equipment purchase uncertainty includes the difficulties in predicting the cost and availability of equipment items. 4. Equipment reliability is the uncertainty associated with the availability of an equipment item for normal operation. 5 , Manufacturing uncertainty accounts for the variations in the processing parameters, such as yields and processing times. The motivation behind this work is to study the effect of market uncertainty on the optimal design of batch plants. Work in this area has been difficult for various reasons. One important limiting factor is that the number of parameters that affect the demand is extremely large and, therefore, predictions made on the basis of loose approximations may not be very reliable. Hence, the assumption of a continuous distribution is
0 1994 American Chemical Society
Ind. Eng. Chem. Res., Vol. 33, No. 11, 1994 2689 often not valid. Even if a continuous distribution is assumed, the model is computationally demanding and the solution of even a reasonably small problem is difficult. The aim, therefore, is to keep the complexity to a minimum but attempt t o include all the essential features of batch processing, such as scheduling. Scheduling is intimately associated with the design of batch plants, because of sharing of equipment between tasks involved in the production of different products. Scheduling, in the context of design, has been considered before by Birewar and Grossmann (1989). Previous research in this area has focused on multipurpose plant design. However, realistic scheduling considerations were not incorporated into the design. Most of the earlier work is based on the following key assumptions: (1)there is a prespecified assignment of equipment items to tasks and (2) all units of a given type can be used for the production of one product at a time. Grossmann and Sargent (1978) expressed uncertain parameters as bounded variables. The design is such that the plant specifications are met for any feasible values of uncertain parameters while optimizing over a weighted cost function which reflects the costs over the expected range of operation. Wellons and Reklaitis (1989) proposed a model for the design of multiproduct plants under uncertainty which accommodates both batch and continuous units and batch size dependent processing times. This model employs the conceptual, multistage formulation proposed by Johns et al. (1978) modified to include an infeasibility term. Constraints are divided into hard and soft categories, with uncertain parameters appearing in the hard constraints a t their upper bounds. Production bounds become soft constraints and a cost of violation is added to the objective function. Soft constraints are the set of constraints that are allowed to be violated, at the cost of a penalty, while the hard constraints cannot be violated. Swaney and Grossmann (1985) presented a two-stage approach based on a scalar index of flexibility which attempts to provide a measure of the proportion of the uncertain parameter space within which feasible operation is guaranteed. The problem then reduces to finding the maximum flexibility index over all the design and uncertain parameter realizations. Reinhart and Rippin (1987) investigated the multiproduct problem with single batch units per stage and constant processing times under variations in the product demands. They used the idea of scenarios, a set of realization of uncertain parameters. They added a constraint for every realization of the uncertain parameters, and solved the problem under the joint constraint set. Shah and Pantelides (1992) proposed a formulation for multipurpose plants design based on the State Task Network. The approach uses the idea of scenarios and adds constraints for each scenario, as proposed by Reinhart and Rippin. No intermediate storage is considered. Papageorgaki and Reklaitis (1990) proposed a formulation for the deterministic design of multipurpose plants which split the horizon into campaigns. At the end of each campaign, the equipment configuration is disassembled and reorganized for the next campaign. This model allows flexible unit-to-task assignments and nonidentical parallel units. Most of the early work (Suhami and Mah, 1982; Imai and Nishida, 1984; Vaselenak et al., 1987) on multipurpose plant design was restricted to unique unit-to-task assignments.
As will be seen in the subsequent sections, the approach followed here will be to design batch plants with the uncertainty modeled using discrete distributions. A hierarchical approach to solve the design problem is proposed, which includes scheduling a t a lower level to ensure that the resulting operation is feasible for the proposed design. The design is considered in two stages: the first solves the Design SuperProblem, which incorporates the design and aggregate scheduling constraints, and the second is the Scheduling Substage, which ensures feasibility of schedules by solving the detailed scheduling model. The advantage of our approach is that the design problem can be solved over the horizon, without neglecting scheduling or assuming cyclic schedules. The combined model for design and scheduling is extremely large and cannot be effectively solved using existing methods. The proposed model does not base its formulation on an assumed mode of operation. The mode greatly depends on the recipe structure and the degree of similarity of the recipes. The classification of the operation into one of the modes discussed may not be valid. The schedules obtained may not appear to be cyclic and the reason is that a mode of operation is not assumed, and an optimal mixed scheduling policy may arise if this is advantageous. Scheduling is required to estimate the plant capacity since the capacity obtained from the Design SuperProblem is an overestimation. The Design SuperF’roblem solution decides the amounts of the various products that may be produced considering only the aggregate scheduling constraints. Scheduling considers the dependence of tasks and precedence order and determines the plant capacity. The usual approach of grouping equipment items and identifying campaigns within which there is no sharing allowed is not used. In the past, the restrictions placed on equipment-task allocations have been such that equipment items are used for no more than one product at a time. This is clearly not optimal from the batch processing perspective as it does not allow sharing of equipment as extensively as is considered in this work. There are no restrictions placed on the assignment of equipment items to tasks. The advantage of batch processing is realized in this approach since there is no assumption about the allocation of equipment items t o single tasks. The equipment items are free t o process several tasks in sequence with the restriction that they are feasible to process the tasks and the time involved is within the time period under consideration. The aggregate scheduling constraints included in the Design Superproblem ensure that all the tasks, required for the production of the amount of products, take place within each time period. Detailed scheduling over a design horizon extending over a number of years leads to extremely large problems, the solution of which is limited by computational resources a t the present time. Our approach divides the design time scale into a number of (potentially) nonuniform planning time periods (see Figure 1). The number of time periods used depends on the available computational resources, the demand projections, and the acceptable frequency of installation of the equipment. For example, if an equipment type can only be installed every 3-months, and demands are projected on a 6-month basis, the discretization chosen will be 3 months for the Design SuperProblem. The solution of the SuperProblem yields the planned capacity of the plant along with the equipment purchase over time.
2690 Ind. Eng. Chem. Res., Vol. 33, N o . 11, 1994 year 1997
Design time scale 2017
2007
How much product? How many tasks? How much equipment?
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Probability Scheduling time scale eg. 4 hour periods
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Figure 1. Time decomposition and discretization.
Each time period of 3 months may now be used to generate detailed scheduling problems, with the details provided in the order of the processing times. Note that the scheduling problems are not solved to determine detailed operation over an entire planning horizon; rather they simply verify that the aggregate model solution can be implemented.
The Problem The design problem for a pure batch chemical plant can be defined as follows: Given (1) a set of products, the different possible scenarios describing their demand levels in each time period, their selling price, and the available production time horizon, (2) a set of available equipment items along with a set of tasks that can be processed in each, the size and cost of each of these items, (3) recipe information is available, including the processing times and stoichiometric ratios, (4)the set of feasible equipment for each task, (5) maximum allowed storage limit, which is zero for the case of unstable intermediates, (6) resource utilization levels or rates and changeover time between products along with their associated costs, (7) inventory costs, and (8)a suitable performance function involving capital and operating costs, sales revenue, and inventory costs. Determine (1)a feasible equipment configuration for manufacturing the set of products, (2) the amounts of products to be produced during each time period, subject to the demand scenarios, and (3) the number of equipment items to be bought in each time period and the size and number of storage vessels required. Some Terms and Definitions. A few definitions that are required before proceeding to the next section are as follows. 1. Recipe: The recipe is comprised of material balances which define how material flows and is transformed by activities carried out in the process. The conventional means of depicting a product recipe is through the use of a directed graph with nodes representing both tasks and materials, where each task node transforms a set of input materials into a set of output materials. Such diagrams have been used in operations research for many years for assembly problems (Carroll, 1965; Maxwell and Mehra 1968; Bryant, 1994), and more recently in chemical engineering for chemical recipes by Kondili et al. (1988). In popular terminology, the recipe shows the dependencies necessary for MRP (Manufacturing Resource Planning; Turbide (1993)).
Figure 2. Discrete representation of a continuous distribution.
2. Scenario is a particular occurrence or realization of the uncertain parameters. For example, one scenario is to produce 30 tons of product A and 45 tons of product B . Another scenario is to produce 40 tons of product A and 55 tons ofproduct B . It is obvious to note that there is greater realism in having more scenarios, but the computational demands will go up, as mentioned earlier. The number of scenarios can be a function of time. 3. Horizon refers to the entire planned time of production and operation of the plant. 4. Time period: The horizon is split into discrete intervals so as t o account for different instances of the demand scenarios over time and to make scheduling problems more amenable t o using efficient solution strategies. 5 . Equipment type refers to an equipment item of a particular capacity. This capacity may be task dependent. In this model, an equipment type is one with a fixed size. For example, a jacketed reactor of 100-gal capacity is not the same type as a jacketed reactor of 200-gal capacity. Therefore, each individual equipment item is considered as an individual type. Note here that only discrete sizes of equipment items are assumed t o be available. 6. Resource unit is any available usable entity such as the processing time, investment, etc. The resource limitations ensure that the process does not use more of each resource than the total available. For example, consider the processing time as a resource. The assumption that there are only a discrete number of sizes available for the equipment types is reasonable, since most standard process equipment is available in discrete sizes. A discrete distribution for the demand uncertainty has been chosen since, in many instances in the industry, no sufficiently detailed forecast is available to adequately construct a continuous distribution. From a solution perspective, this is an attractive feature since it eliminates the cumbersome handling of the nonlinear terms introduced by continuous distributions. If a continuous distribution is available, it could in all cases be approximated by a set of scenarios (see Figure 2). The continuous distribution may be discretized into a number of demands with the associated probabilities given by the corresponding area under the probability distribution function. Note here that there is a tradeoff between model accuracy and computation efficiency
Ind. Eng. Chem. Res., Vol. 33, No. 11, 1994 2691 Demand of Product A
Demand of Product B
Probability
1
Probability
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0.3~0.2=0.06
45 45
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0.3~0.6=0.18
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0.5~0.6=0.30
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Probabilities sum to 1 Figure 3. Scenario generation from a discrete distribution.
in selecting the number of scenarios used to represent the demand uncertainty. Scenarios are generated by considering all the demand probability distributions at the same time. For instance, consider the case of two products A and B (see Figure 3). The demand for each is described by three discrete points with point probabilities associated with them. A unique combination of two such points, one from each distribution, constitutes a scenario with the associated probability of occurrence of both points given by the product of the individual probabilities (assuming independence). Therefore, the number of scenarios that can be generated will be equal to the number of combinationsthat are possible by considering every pair (in this case, nine scenarios). In general, the number of scenarios grows exponentially with the number of products and this presents a potential problem. In this case, it would be useful to consider the independent demand probability distributions and generate scenarios through Monte Carlo type sampling. A finite number of scenarios thus generated (which will be small in number when compared to the total number of scenarios) will then be included in the design analysis. The selection of demand scenarios will be weighted by their probabilities; therefore, the higher probability demand scenarios are more likely t o be included. A third important reason to consider the scenario approach to uncertainty is that the designer can readily use intuitive forecasts in the model without dealing with continuous distributions. A realization of a scenario at any point of time may be easily imposed on the model,
which may otherwise prove cumbersome using a continuous model.
Approach The basic approach we propose is different from previous work in that nearly all the scheduling constraints that complicate the problem are excluded from the design stage. The problem is split into two stages-the design and the scheduling stages. The Design SuperProblem solution yields the equipment sizing and assignments (to tasks), the amount of each resource to be stored at the end of each time period, and also the amount of each resource to be produced, taking into account the resource limitations. The decomposition (see Figure 4) into the design and scheduling stages facilitates the development of efficient solution strategies to solve the problem. The direct simultaneous solution of the design and scheduling stages for problems of realistic scope is not possible using existing solution methods. For example, consider a design problem for a plant with a plant life of 5 years. A simultaneous design-scheduling problem formulation would lead to a schedule spanning 5 years, but it is obvious that such a formulation would involve an enormous number of variables, if the schedule is developed in terms of days. The structure of the problem can be exploited, however, to yield a smaller design problem and a series of scheduling problems. From a practical point of view, such detailed scheduling over an extended time horizon is unnecessary. The horizon is split into a specified number of time periods which will be decided on the basis of the demand
2692 Ind. Eng. Chem. Res., Vol. 33, No. 11,1994
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