A General Treatment of Uncertainties in Batch Process Planning

Ind. Eng. Chem. Res. 2001, 40, 1507-1515

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A General Treatment of Uncertainties in Batch Process Planning Yang Gul Lee* and Michael F. Malone† Department of Chemical Engineering, University of Massachusetts, Amherst, Massachusetts 01003

A general strategy for treating uncertainties in batch process scheduling was described by Lee and Malone (Int. J. Prod. Res. 2001, in print). The strategy is based on hybridization of the Monte Carlo simulation and simulated annealing techniques. In this paper, the same strategy was used to develop a flexible planning algorithm that handles the open-shop model of batch plant operation and the discrete demand pattern. Using the flexible planning algorithm, we can estimate the maximum capacity of a batch plant and make a capacity plan in the face of uncertainties in product demands and due dates. After the capacity planning, we obtain a flexible plan that maximizes the expected profit and has some free time as a source of future flexibility. An example solved with this flexible planning algorithm shows that it is possible to use only half of the maximum capacity of a batch process because of uncertainty. This means that, in order to reduce the future inventory costs, we should not plan to run a batch process at its maximum production capacity when uncertainty is involved. The lower apparent utilization of the flexible plan will reduce the impact of uncertainties in the future. 1. Introduction In process design, planning, and scheduling of batch chemical processes, a major gap between theory and practice arises on account of uncertainties in a batch process model or market parameters such as demands and due dates. Because of the strong interactions between the design and planning of batch process, many design studies have included planning models in their problem formulations, so far in a deterministic manner.1-3 However, at the time of batch process design, we can expect many instances in which we do not have enough information on uncertain parameters to carry out a deterministic design. For example, market parameters are especially uncertain. One traditional approach is to introduce an empirical oversizing factor in a nominal design.4,5 There is also a variety of programming approaches that adopt some objective function to take capital and operating costs into account.6,7 These approaches commonly introduce some degree of flexibility into the design. It has been known that more flexibility means greater capital costs, with the benefit of reduced operating costs or penalties against the violation of design limits. For production and capacity planning problems involving uncertain parameters, Ierapetritou and Pistikopoulos8 introduced a two-stage stochastic programming method, in which Gaussian quadrature formula was used to approximate the multiple integral for the expected profit evaluation. Lee9 described an uncertainty analysis for batch process scheduling, which showed that the effects of uncertainty can be minimized by avoiding overutilizing a batch process, which can be realized by introducing some amount of free time into a batch schedule. Lee and Malone10 introduced flexibility into a batch schedule by inserting free time into * To whom all correspondence should be addressed. Present address: ExxonMobil Research & Engineering, 3225 Gallows Road, Fairfax, VA 22037. E-mail: [email protected]. † E-mail: [email protected].

a batch schedule in the form of dummy batches that take some period of time but produce nothing. Because the dummy batches can be used to handle future changes in parameter values with little or no penalties, they are the source of flexibility in a batch schedule. Flexibility in the design or production plan is insurance against the effects of uncertainties in the future at the expense of capital costs in the present. Here, there arises a question that has not been answered by previous studies: how should the flexibility be used to minimize the impact of parameter variations in the future? To answer this question, Lee and Malone10 also developed an algorithm for reactive schedule adaptation, in which the cost of uncertainties, or the impact of changed parameters during batch production, is minimized. Lee and Malone11 introduced a deterministic approach to batch process planning using a simulated annealing technique in which the objective was to maximize a profit penalized by due date (lateness) penalties and inventory (earliness) costs and including sequencedependent setup costs. This approach is used in this study as a starting point to develop a flexible planning algorithm, which plays a complementary role to flexible scheduling in a general strategy to treat uncertainties in batch processing.10 2. Background To formulate a design, planning, or scheduling problem suitable for a real situation, we need to describe uncertainties in batch processes. A real obstacle is that uncertainties are time-varying: from the design stage to the actual production, the degree of uncertainties is changing incessantly. In this section, we will briefly review and contrast the ways in which such uncertainties in the demand have been described in the operations research and chemical engineering literature. Demand uncertainty has been the major target of previous work, because of its obvious impact on profitability, but it is not the only uncertain parameter in the scheduling and planning of batch processing plants.

10.1021/ie9907122 CCC: $20.00 © 2001 American Chemical Society Published on Web 02/16/2001

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Ind. Eng. Chem. Res., Vol. 40, No. 6, 2001

Figure 1. Comparison of a closed shop and an open shop.

Figure 2. Demand patterns for a closed shop and an open shop.

In the operations research field, a majority of the studies of demand uncertainty have been done for closed-shop situations, where all of the customer requests are fulfilled by inventory and production tasks are generally the result of inventory replenishment decisions.12 In contrast, for the open-shop case, all production orders are by customer request, and no inventory is stocked. Hence, production scheduling is a sequencing problem in which open orders are sequenced at each processing unit (Figure 1). Several features characterize the planning and scheduling problems of a closed shop: inventory control, continuous demand pattern, multiperiod model, and lead time. Because the production schedule for a closed shop is decided by inventory replenishment decisions, the backorder time and quantity of replenishment are the control variables to minimize the impact of demand uncertainty. In other words, the main question is how to decide the back-order point or quantity, together with minimizing the stockout (or shortage) penalty. In the description of the uncertainty, time-varying demand patterns have been used. For the case of a probabilistic demand, a common situation is that the mean and forecast demands stay constant or change slowly with time but there are typically forecast errors present.13 Another example is a time-varying demand pattern in which the mean or forecast demand changes appreciably with time, e.g., a seasonal demand. When these continuous demand patterns have been treated within a production planning and scheduling problem, a multiperiod model has been widely used since the two early studies of Manne14 and Wagner and Whitin.15 In the multiperiod model, the demand of the next period is forecast, and production planning and scheduling is done by solving the inventory replenishment problem based on the forecasted demand. In the chemical engineering literature, demand uncertainty has also been taken into account in batch process design by incorporating a multiperiod planning model into the design problem.16,17 These studies described demand uncertainty with several scenarios to simplify a normal probability distribution for the demand. Another key factor differentiating a closed shop and an open shop is the lead time, which is the difference between the request and fulfillment times of an order. In an open shop, the due date time is the key factor corresponding to the lead time because all production orders are made directly by customer request and no inventory is stocked.12 That is, meeting the due date time of every customer request is the goal of production

planning and scheduling of an open shop. So far, most of the studies of batch chemical process planning and scheduling have dealt with closed-shop cases, but only a few studies have focused on the importance of due date times, e.g., Musier and Evans18 and Ku and Karimi.19 Even though these are deterministic scheduling studies, they are the first to consider a discrete demand pattern with due date times and demand quantities at each due date time in the chemical engineering literature. Figure 2 shows two typical demand patterns. In the first type (Figure 2a), the production horizon is divided into several periods, and within each period, the demand has some anticipated probability distribution with a high degree of uncertainty. This type of description is suitable for long-range or medium-range planning, e.g., for deciding a company’s strategic plan for capacity expansion through several years or for allocating a variety of resources to each period of one horizon spanning several months or a year. However, in a demand pattern like that shown in Figure 2b, demands are made at any time in the production horizon, and the orders have lower degree of uncertainty in quantity and due date time. This demand pattern should be used to execute short-range (operational) planning, which assigns a campaign to production lines, does lot-sizing, and sequences the campaigns within a production line. Most of the previous studies dealing with multiproduct or multipurpose batch chemical plants divided the production horizon into several fixed-length periods (“multiperiod model”), and in each period, the product demand was assumed to be constant or to have some variation range, that is, batch processes operated in a closed-shop mode were considered. However, we are interested in batch plants operated in the open-shop mode in which all production orders are made directly to the batch plant by customer request and the meeting of due dates is crucial. The major benefit of an open shop is a much lower inventory than in a closed shop. In this work, we study the multiproduct batch process structure running in an open-shop mode in which due date uncertainties must be considered together with demand uncertainties. Because customer orders for an open shop are assumed to be random, we use the discrete demand pattern in Figure 2b that represents sporadic and discrete orders, each having an uncertain product demand and due date time. With this discrete description of uncertainty, we develop a flexible planning algorithm that is a major part of the general strategy (Figure 3) of treating uncertainties in batch processing such as product demands, due dates, processing times, product prices, and raw material costs.

Ind. Eng. Chem. Res., Vol. 40, No. 6, 2001 1509

Figure 3. General strategy for treating uncertainties in batch processing.

In the next section, we will explain the flexible planning algorithm. Capacity adjustment using the flexible planning algorithm will be discussed in section 4. Then, an example of a flexible plan obtained after capacity adjustment will be illustrated in the next section. In section 6, we will show the effect of key parameters (e.g., number of due dates and penalties for lateness and earliness) on the open-shop capacity under uncertainties in demands and due dates.

3. Flexible Planning We extend the flexible scheduling algorithm introduced in Lee and Malone10 in order to develop the flexible planning algorithm, which is based on a hybridization of Monte Carlo simulation and simulated annealing. The goal of this algorithm is to find a flexible schedule that has an optimal degree of flexibility and can be adapted to future changes in various uncertain parameters. 3.1. Probability Density Function. To formulate the optimization problem for flexible planning, an uncertain parameter is expressed as a random variable with a probability density function. Conceptually simple functional forms, such as the uniform distribution or the normal (Gaussian) distribution, have been used in previous stochastic optimization procedures. These functions can sometimes be chosen from accumulated historical data but are often assigned on the basis of the subjective judgment of experienced marketing personnel.20 In practice, the precise identification of a probability density function for product demand is very difficult or impossible. Therefore, a simple form, namely, the uniform distribution, is used in this study. However, any form of the probability density function for a random parameter can be used in the flexible planning algorithm. In stochastic optimization, the probability distribution function of an objective should be determined from the probability distribution functions of individual random parameters. Analytical methods can typically be used for problems having a few random parameters. However, for problems involving many random parameters or complicated probability distribution functions, a Monte Carlo simulation technique has been used as a typical numerical method. One set of parameter values is randomly sampled from the specified range of each random parameter. This set of sampled values is then used to calculate a single value of the objective function. After this procedure is repeated many times, the probability density function for the objective is obtained from the random samples. This probability distribution func-

tion approximates the exact form of the probability distribution. 3.2. Optimization Objective Function. The objective function for flexible planning is maximization of the expected profit

∫-∞∞[profit] f(p) dp}

max{E[profit]} ) max{

(1)

where E[profit] is the expected value of profit and f(p) is the probability density function of profit, which is obtained by the hybrid algorithm. The profit function in the objection function is the net present value (NPV) of revenue minus changeover costs, due date penalties, and inventory costs, which is defined as follows.11 For production line l, the net present value (NPVl) and the net penalized present value (NPPVl) are Ncl Nbi

NPVl )

∑ ∑{VpBp,l exp[-r max(tf ,td )] i)1 j)1 j

p

PRMpBp,l exp[-rtsj]} (2) Ncl

NPPVl ) NPVl -

{CS + CV + CD } ∑ i)1 i

i

i

(3)

where Ncl and Nbi are the numbers of campaigns in production line l and the number of batches in campaign i, respectively. The subscript p represents the product produced in campaign i, and Bp,l is the line batch size of product p on production line l. Note that Bp,l is decided by the smallest stage batch size of product p among all of the stages on production line l. The stage batch size (bp,k) is equal to the unit size (uk,l) on stage k divided by the size factor (Sp,k). Vp is the product value, and PRMp is the manufacturing cost, which is the sum of the raw material, processing, and labor costs for product p produced in campaign i (these costs can also be treated separately, e.g., according to differential labor rates for different shifts, etc., although we do not include this possibility here). The completion time and the starting time for batch j are tfj and tsj, respectively; tdp is the due date of product p. The discount rate for continuous compounding is r. CSi is the setup cost for campaign i and is imposed only when the product produced in the previous campaign, i - 1, is different from that of the present campaign. (If the data are available, it is simple to make these setup costs sequence-dependent, but this is not our focus here.) CVi and CDi are the inventory cost and the due date penalty charged during campaign i, respectively.

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Figure 4. Hybrid flexible planning algorithm.

The profit function is the sum of NPPVl over production lines. Nl

profit )

NPPVl ∑ l)1

(4)

3.3. Flexible Planning Algorithm. To estimate the expected profit, we need to obtain the probability density distribution of the profit function. This job can be done by a sequential (“two-stage”) algorithm, which involves the use of a Monte Carlo simulation and a simulated annealing technique. In the outer loop of the sequential algorithm, a Monte Carlo simulation algorithm samples a set of uncertain parameters, and in the inner loop, a simulated annealing algorithm searches the configuration space of a planning problem that is defined by the set of sampled parameter values and finds the production plan that gives the best profit function value. After a number of iterations of Monte Carlo sampling on the parameter space, the expected profit, E[profit], is estimated by the best profit values, as well as their probability density distribution. This sequential algorithm is conceptually simple but computationally very expensive. Therefore, the following hybrid algorithm is preferred because it can reduce the calculation time by an order of magnitude just by combining the simulated annealing and Monte Carlo simulation to estimate the expected profit. The key idea of this algorithm is that simulated annealing itself is a kind of Monte Carlo simulation, which adopts the importance-sampling technique originally developed by Metropolis et al.21 instead of random sampling.22 This

means that the deterministic optimization and Monte Carlo simulation can be hybridized, which can greatly reduce the calculation time. In the hybrid algorithm, both the generation of a random move in the original simulated annealing and the random selection of uncertain parameter values are executed in one step, as shown schematically in Figure 4. Each parallelogram in Figure 4 represents a configuration space that is defined by a set of sampled parameter values. This configuration space is searched by simulated annealing separately from Monte Carlo sampling in the sequential algorithm. In the hybrid algorithm, however, when the simulated annealing algorithm generates a random move, it randomly samples the given set of uncertain parameters and also generates a new configuration (i.e., production plan). Hence, the sequence of moves generated by simulated annealing (solid line in Figure 4) actually covers the whole parameter space. Then, the importance sampling is carried out by Metropolis’s rule, while the iteration of simulated annealing algorithm is repeated until the termination condition is satisfied. After termination of the simulated annealing algorithm, we prepare a probability density distribution from the profit function values of all of the samples (circles in Figure 4) that have been importance-sampled over the parameter space. The expected profit is estimated from this probability distribution, and a production plan having its profit value close to the expected profit is saved as an expected profit plan. After a fixed number of runs of the hybrid algorithm, the production plan having the maximum expected profit is selected as the flexible plan from among the expected plans.


Note that the optimal amount of free time (as a form of dummy campaign) can be found in the flexible plan. These dummy campaigns can be used to handle future changes in product demands or due dates with little or no penalties. The amount of free time incorporated into the flexible plan is a tradeoff between the expected NPV and the expected penalties (i.e., inventory cost and due date penalty). When there is no free time in the flexible plan, the expected penalties for earliness and lateness will be greater in the end; the expected NPV will be smaller with a large amount of free time in the flexible plan.

Table 1. Costs, Production Requirements, and Uncertainty Data for the Flexible Planning Examplea

product

product price ($/kg)

prm cost ($/kg)

due date penalty coeff. (%/kg/week)

inventory cost coeff. ($/kg/week)

A B C D

13.006 14.237 11.384 13.834

3.148 4.158 5.389 4.026

0.500 0.500 0.500 0.500

0.500 0.500 0.500 0.500

product

due date time (h)

due date uncertainty (h)

product demand (1000 kg)

A A B B C C D D

884 1787 657 1957 2572 1351 2396 1465

-400 to 400 -400 to 400 -400 to 400 -400 to 400 -400 to 400 -400 to 400 -400 to 400 -400 to 400

100.2 51.9 50.7 76.5 40.8 61.2 83.1 34.2

4. Capacity Planning In most of studies of batch chemical processes, the purpose of planning is to find the best way to use resources to meet deterministic production requirements within the limitations imposed by locations, facilities, processes, job specifications, and capacities. In reality, however, there are always uncertainties in demands and due dates. In this situation, we need to carry out capacity planning in which we first assess the capacity of our batch processes and then set up a capacity-use strategy.23 This capacity planning is intimately related to demand resource planning and location planning, but these two types of planning are beyond the scope of this paper. The flexible planning algorithm is useful for capacity planning because we can assess the maximum capacity of a batch process and obtain a capacity-use strategy while considering the uncertainties in demands, due dates, and processing times. 4.1. Criterion for Overutilization. When we have deterministic demands and due dates, it is relatively simple to determine whether our process will be overutilized or underutilized. When product demands and due dates are uncertain, however, the method for determining overutilization of a batch process is not obvious. We propose the use of the following flexible plan in determining the overutilization: If there are production campaigns after the given horizon in the flexible plan, then the batch process is overutilized. The plan obtained with the objective max{max[profit]} (called “maximum-profit plan”) from the flexible planning algorithm will have the maximum productivity or time utilization. Therefore, if we use the objective max{max[profit]} to check overutilization and start a production horizon, we risk a large inventory because of canceled orders in the future. Because the probability that we will get the maximum demand for all products simultaneously is very small, a large portion of the production amounts will turn out to be inventory when we follow the maximum-profit plan. In the more conservative plan of max{min[profit]} (called “minimumprofit plan”), the productivity of our process will be very low, and consequently, we risk significant shortages. Therefore, when there are significant uncertainties in demand, the proper way to determine whether a batch process is overutilized under uncertainties in demands is to use the objective max{E[profit]}. In other words, if there are any campaigns ending after the given horizon in the flexible plan, the batch process is overutilized. 4.2. An Example of Overutilized Case. In this example, we consider four products (A, B, C, and D) and two due dates and corresponding demands for each

a

demand uncertainty (%) 50-150 50-150 50-150 50-150 50-150 50-150 50-150 50-150

Annual discount rate (r) ) 15%. Setup cost ) $100/changeover.

Table 2. Time Data and Size Factors for the Flexible Planning Example

product

A

B

C

D

task

processing time (h)

setup time (h)

transfer time (h)

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

22.9 20.8 11.9 9.6 10.7 21.9 8.4 14.6 28.3 28.6 26.3 24.4 23.0 27.5 16.5 9.9 15.6 22.9 22.9 22.0

4.37 1.29 2.42 3.02 2.65 1.24 1.74 3.09 4.97 0.33 3.70 0.17 0.47 3.97 1.89 0.21 4.47 0.99 1.38 1.30

1.132 1.369 1.325 1.158 1.383 1.288 1.220 1.367 1.058 1.122 1.545 1.157 1.036 1.163 1.142 1.269 1.268 1.047 1.292 1.334

size factor (m3/1000

kg)

3.137 4.727 3.124 4.334 3.137 3.93 4.99 4.19 4.69 4.57 2.97 2.96 2.29 3.16 4.09 4.94 3.12 2.80 1.63 2.00

product. In the multiproduct plant used in this study, there are three production lines, each with five processing stages. In each stage, one processing unit is assigned to perform one task, and the horizon time is 3500 h. We assume that each due date has a uniform distribution between -400 and 400 h of the nominal due date. The demand for each due date has a uniform distribution between 50 and 150% of the nominal amount. The sum of nominal demands in this example is 500.4 × 103 kg (Table 1). Other parameters are assumed to have no uncertainties (Table 2). The flexible plan in Figure 5a shows that the process is significantly overutilized because a large number of campaigns start after the horizon time. This plan produces only 64% of the nominal demand within the given horizon. Therefore, we expect that it will be necessary to use additional production lines to meet all demands. The maximum-profit plan (Figure 5b) for this example produces 149% of the nominal demand within the production horizon, and this is the maximum capacity of the process. These two cases show that, because of the uncertainties in demands, we can use only less than half of the maximum capacity of our process.

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Figure 6. Example of the flexible plan.

An example of the flexible plan will be shown in the next section. 5. Example of Flexible Plan

Figure 5. Example of (a) the overutilized plan and (b) its maximum-profit plan. Each horizontal bar represents a single product campaign, and the single character on each bar is a product produced in that campaign. The empty slot between two campaigns is a dummy campaign, which produces nothing but keeps free time within the horizon.

Note that the maximum-profit plan does not always end up with the maximum capacity plan as in this case. As the inventory cost and due date penalty increase, the capacity of the maximum-profit plan can decrease, which will be discussed later in this paper. 4.3. Capacity Adjustment. If the process turns out to be overutilized, we need to adjust the production amounts until the overutilization criterion is satisfied. There are several ways to adjust the capacity, including assigning the shortage amount to processes at other locations, constructing more production lines in the present plant, and increasing the capacities of the present production lines. In the first case, there are many policies to decide which product and how much will be assigned to other processes. For instance, if there are priorities among all of the products, then remove the low-priority products or reduce their production amounts. Considering the other options, we can see the interaction between capacity planning and the design of batch processes. Ultimately, we need to solve the planning and design problem in one problem formulation. In the following example, the capacity adjustment is done by uniformly reducing the nominal amount of each product demand until the criterion of overutilization is satisfied. Then, we redo the flexible planning with the adjusted production amounts. This flexible plan will be our the best capacityuse policy under the given uncertainties in demands.

For this example, we reduced the sum of nominal demands to 216.8 × 103 kg by capacity adjustment. Each product demand has its variation between 50 and 150% of its nominal value. The flexible plan found produces 105% (227.5 × 103 kg) of the sum of nominal demands within the given horizon (Figure 6). Using this flexible plan, our strategy for dealing with uncertainties is illustrated in Figure 3. After the flexible plan is obtained at the beginning of the production horizon, the batch process starts and follows the flexible plan. If there have been some changes in uncertain parameters before one of the production schedules is executed (for example, period 2 in Figure 3), we find another flexible schedule by using the flexible scheduling algorithm with the updated parameter information. Note that the flexible schedule involves short-term batch-wise scheduling and still has some amount (but much less than the flexible planning) of uncertainties in demands and due dates. During each production period, if there are lastminute changes in an uncertain parameter, we apply the reactive schedule adaptation algorithm to obtain a new optimal solution under the new value(s) of the uncertain parameter(s). The reactive schedule adaptation is deterministic batch sequencing using the lastminute changes. When we consider situations in which the degree of uncertainty is changing before and during the production horizon, there is no explicit way to incorporate the time-varying uncertainty into the profit function. Therefore, we use two different objective functions to deal with the situation. At the beginning of a horizon and at the beginning of each scheduling period, we use the objective max{E[profit]} because there is a large degree of uncertainty. However, during the real production, we expect that most of the uncertainty is resolved, and we use the objective, max{profit} for the reactive schedule adaptation algorithm. In the rest of this paper, we will use the hybrid algorithm in order to investigate the effects of key parameters on the capacity of batch process plants under uncertainty. 6. Application of Flexible Planning Algorithm The inner loop of the hybrid algorithm generates a distribution of importance-sampled data. Using this


Figure 7. Illustration of the effect of the number of due dates. (a) Two products A and B have one due date each, and a long single-product campaign for each product is planned. (b) Two additional due dates (dotted vertical line) for the two products are introduced. (c) To reduce the due date penalties and inventory costs, the long campaigns shown in part a must be divided into smaller campaigns around the four due dates.

distribution, we can obtain the plan corresponding to E[profit]. Using this expected profit plan, we can study the effects of various parameters on the capacity of batch process plants. 6.1. Effect of Number of Due Dates. If we use a discrete demand pattern like the one in Figure 2b, the number of due dates is a key variable characterizing the two operation modes (closed-shop and open-shop). The open shop can have much more sporadic and random due dates from customers than the closed shop. The closed shop has no explicit due dates, but in practice, the end of each period in the multiperiod model can be regarded as a due date. We can question how the production capacity of a batch process changes as the number of due dates increases, i.e., from the extremes of a closed shop to an open shop. Two ways in which the number of due dates affects the capacity of a batch process can be considered. As illustrated in Figure 7, when the number of due dates increases, a long campaign must be divided into smaller fragments in order to reduce the inventory costs and due date penalties. If the number of shorter campaigns increases, two effects reduce the capacity of a batch process. One is that the total time used for setting up each campaign increases, and consequently, the “time efficiency”, a ratio between the total processing time and the operating time, deteriorates. As the total number of campaigns is increased, each campaign cannot be an exact multiple of the maximum batch size. In that case, we need to spend an entire batch cycle to produce less than the maximum batch size (a “partial batch”). These partial batches reduce the overall capacity of a batch process. We investigated the effect of the number of due dates on the capacity (i.e., the overall production amount) of a batch process when the demands are uncertain. Four different products were considered, with two due dates per product. The processing times, setup times, transfer times, and size factors for all of the tasks for each product are shown in Table 2, and the overall production amount within the horizon was estimated in five different cases with 1, 2, 4, 6, and 8 due dates per product.

Figure 8. Capacity change of the flexible plan according to the number of due dates for two different sets of setup times. A batch process with more due dates has less capacity under demand uncertainties. The longer setup times also result in lower capacity.

The demand at each due date is assumed to be uncertain between 50 and 150% of the nominal demand in Table 1. As the number of due dates per product increases, the overall production amount decreases (solid circles in Figure 8). The decrease in the overall production amount is due to both partial batches and lower time efficiencies. Because the relative ratio of setup time to processing time influences the time efficiency, we studied another set of setup times by multiplying the setup times in Table 2 by 10. With this second set of setup times, we checked the overall production amount (open circles in Figure 8). The capacity in all cases is less than that for the shorter setup times mainly because of the lower time efficiencies. 6.2. Effect of Penalties. The penalties for earliness (inventory cost) and lateness (due date penalty) are also a major factor determining the capacity of an open shop. If the inventory cost of a product is small or a product can be late without paying large penalties, it might be still profitable to produce the product even though we pay some penalties. However, if a product has high penalties, i.e., comparable to the revenue of that product, then we should actually produce less. Thus, greater penalties result in a smaller capacity for an open shop. This penalty effect is also the reason that there are free times in the flexible plan of Figure 6. These free times provide an open shop with the flexibility to handle future changes in demands and due dates. This effect is more serious when a batch process is overutilized. An example in shown Figure 9 for cases in which the penalties for an overutilized process were varied systematically. The capacity for the maximumprofit plan within the production horizon decreases significantly as the penalties increase, as explained above. However, the capacity for the flexible plan is not significantly influenced by the penalties because the capacity of the flexible plan is adjusted not to be overutilized. 6.3. Optimal Number of Due Dates. Figure 8 shows that the capacity of a batch process decreases as the number of due dates increases. By examining this result, we can see that there is a tradeoff between productivity and inventory. An open shop having a large number of due dates will have a very low productivity that the large setup times and fragmentation effects. On the other hand, a closed shop must undertake a large inventory.

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Figure 9. Capacity decrease demonstrating the effect of penalties. The maximum-profit plan is much more sensitive to the penalties than the flexible plan.

In reality, the number of due dates is given by customer orders. However, we can probe the best operation mode of a batch process under given product demands and given due dates by finding the optimal number of due dates. One way to find the optimal number is to introduce a penalty term for the due date into a profit function. However, it is not easy to estimate how much we penalize an additional due date. Another way is to use the flexible planning algorithm with various numbers of due dates and to find the number of due dates that gives us the best E[profit]. In the profit function, the net present value (NPV) depends on the productivity, whereas the penalty terms are directly related to the inventory. For a small number of due dates, a large inventory cost will reduce the profit, but in contrast, the NPV will be greater than it is for a large number of due dates. Thus, there should be an optimum in the expected profit. The expected profits for the two sets of setup times are shown in Figure 10a where the coefficients for due date penalties and inventory costs for all of the products are set to 0.5 $/kg/week. The coefficients are the cost to keep 1 kg of a product for 1 week in inventory and the penalty for 1 kg of product being late for 1 week. With the setup times in Table 2, the optimal number of due dates is two per product (rectangle in Figure 10a). The larger setup times (solid circles) do not change the optimal number but do reduce the expected profit because the time efficiency is lower. The components of the profit function illustrate the tradeoff between productivity (NPV) and inventory, as shown in Figure 10b. When the inventory cost is relatively low, the best approach is to run a batch process in the closed-shop mode and to maintain a large inventory in order to be on time at every due date. In contrast, when the inventory cost is large (we used inventory costs 25 times higher than due date penalties.), the optimal number of due dates increases (Figure 11). In this situation, we should reduce inventory costs by distributing total production into many smaller campaigns around the greater number of due dates. This means that the openshop mode is preferred. Obviously, an extreme open shop cannot be the best option, because although we reduce inventory costs, we also lose productivity. Hence, the best operation mode for batch processing plants lies between the open shop and the closed shop.

Figure 10. (a) Optimal number of due dates decided by the expected profit obtained from the flexible planning algorithm for two different sets of setup times. (b) Tradeoff between the inventory cost and the NPV for the case of smaller setup times.

Figure 11. Optimal number of due dates per product changes with different inventory cost coefficients from two due dates (0.5 $/kg/week) to six due dates (5.0 $/kg/week). To reduce the total inventory cost, we should distribute the production into many smaller campaigns around the greater number of due dates.

It is preferred to run an open shop with the optimal number of common due dates and to give benefits to a customer who has orders delivered at the common due dates. 7. Conclusion By developing a flexible planning algorithm, we completed a general strategy of treating uncertainties


in the scheduling and planning of batch processes. Using the flexible planning and scheduling and reactive schedule adaptation algorithms, we can deal with a variety of uncertainties involved in batch processing such as changes in the product demands, the due dates, the processing times, etc. An example solved with this flexible planning approach shows that it is possible to use only half of the maximum capacity of a batch process running in an open-shop mode because of uncertainties. This means that, to reduce the inventory costs, we should not plan to run a batch process at its maximum production capacity when uncertainty is involved. By using the flexible planning algorithm, we incorporate some amount of free time into the flexible plan in order to reduce the impact of uncertainties. We also showed that there is a tradeoff between productivity and inventory in a batch process. If there are uncertainties in product demands, neither the open shop nor the closed shop is the best mode of operation. In a closed shop, we have to maintain a large inventory, and in an open shop, we will lose the productivity of our processes in order to reduce the impact of the uncertainties involved in a number of due dates. The best operation mode for batch processing plants lies between the open shop and the closed shop. In other words, running an open shop with the optimal number of common due dates is preferred. By giving benefits to a customer who has orders delivered at the common due dates, the company can partially control the due dates of products. Literature Cited (1) ] Shah, N.; Pantelides, C. C. Optimal Long-Term Campaign Planning and Design of Batch Operations. Ind. Eng. Chem. Res. 1991, 30, 2308. (2) ] Papageorgaki, S.; Reklaitis, G. V. Optimal Design of Multipurpose Batch Plants. 1. Problem Formulation. Ind. Eng. Chem. Res. 1990, 29, 2054. (3) ] Voudouris, V. T.; Grossmann, I. E. Optimal Synthesis of Multiproduct Batch Plants with Cyclic Scheduling and Inventory Consideration. Ind. Eng. Chem. Res. 1993, 32, 1962. (4) ] Avriel, M.; Wield, D. J. Engineering Design Under Uncertainty. Ind. Eng. Chem. Process Des. Dev. 1969, 8, 125. (5) ] Johns, W. R.; Marketos, G.; Rippin, D. W. T. The Optimal Design of Chemical Plants to Meet Time-Varying Demands in thePresence of Technical and Commercial Uncertainty. Trans. Inst. Chem. Eng. 1978, 56, 249.

(6) ] Grossmann, I. E.; Sargent, W. H. Optimum Design of Multipurpose Chemical Plants. Ind. Eng. Chem. Process Des. Dev. 1979, 18, 343. (7) ] Halemann, K. P.; Grossmann, I. E. Optimal Process Design under Uncertainty. AIChE J. 1983, 29, 425. (8) ] Ierapetritou, M. G.; Pistikopoulos, E. N. Novel Optimization Approach of Stochastic Planning Models. Ind. Eng. Chem. Res. 1994, 33, 1930. (9) ] Lee, Y. G. A New Approach to Batch Chemical Process Planning. Ph.D. Thesis, University of Massachusetts, Amherst, MA, 1997. (10) ] Lee, Y. G.; Malone, M. F. Batch Process Schedule Optimization under Parameter Volatility. Int. J. Prod. Res. 2001, in print. (11) ] Lee, Y. G.; Malone, M. F. Flexible Batch Process Planning. Ind. Eng. Chem. Res. 2000, 39, 2045. (12) ] Graves, S. C. A Review of Production Scheduling. Oper. Res. 1981, 29, 646. (13) ] Silver, E. A.; Peterson, R. Decision Systems for Inventory Management and Production Planning; John Wiley & Son: New York, 1979. (14) ] Manne, A. S. Programming of Economic Lot Sizes. Manage. Sci. 1958, 4, 115. (15) ] Wagner, H. M.; Whitin, T. M. Dynamic Version of the Economic Lot Size Model. Manage. Sci. 1958, 5, 89. (16) ] Subrahmanyam, S.; Pekny, J. F.; Reklaitis, G. V. Design of Batch Chemical Plants under Market Uncertainty. Ind. Eng. Chem. Res. 1994, 33, 2688. (17) ] Shah, N.; Pantelides, C. C. Design of Multipurpose Batch Plants with Uncertain Production Requirements. Ind. Eng. Chem. Res. 1992, 31, 1325. (18) ] Musier, R. F. H.; Evans, L. B. An Approximate Method for the Production Scheduling of Industrial Batch Processes with Parallel Units. Comput. Chem. Eng. 1989, 13, 229. (19) ] Ku, H.; Karimi, I. A. Scheduling in Serial Multiproduct Batch Processes with Due Date Penalties. Ind. Eng. Chem. Res. 1990, 29, 580. (20) ] Norton, J. H. The Role of Subjective Probability in Evaluating New Product Ventures. Chem. Eng. Prog. Symp. Ser. 1963, 59, 49-54. (21) ] Metropolis, N.; Rosenbluth A. W.; Rosenbluth, M. N.; Teller, A. H.; Teller, E. Equation of State Calculations by Fast Computing Machines. J. Chem. Phys. 1953, 21, 1087. (22) ] Binder, K.; Heermann, D. W. Monte Carlo Simulation in Statistical Physics; Springer-Verlag: Berlin, Germany, 1988. (23) ] Barndt, S. E.; Carvey, D. W. Essentials of Operations Management; Prentice Hall: Englewood Cliffs, NJ, 1982.

Received for review September 27, 1999 Revised manuscript received April 18, 2000 Accepted January 8, 2001 IE9907122

A General Treatment of Uncertainties in Batch Process Planning

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