Robust Short-Term Scheduling of Multiproduct Batch Plants under

Ind. Eng. Chem. Res. 2001, 40, 4543-4554

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Robust Short-Term Scheduling of Multiproduct Batch Plants under Demand Uncertainty Jeetmanyu P. Vin and Marianthi G. Ierapetritou* Department of Chemical and Biochemical Engineering, Rutgers University, 98 Brett Road, Piscataway, New Jersey 08854

In this paper we are developing a strategy to quantify scheduling robustness in the face of uncertainty, to increase scheduling flexibility, and to improve system performance when unexpected events occur during scheduling execution. Several robustness metrics that are commonly used in optimization literature are adopted, and new metrics are proposed that take into account the scheduling characteristics under demand uncertainty. The deterministic model for short-term scheduling proposed by Ierapetritou and Floudas (Ind. Eng. Chem. Res. 1998, 37, 4341) that is based on a continuous-time representation is used throughout this work. To improve the schedule performance, uncertainty was considered at the scheduling stage through the multiperiod programming model. It is found that the schedule from the multiperiod formulation has a much higher robustness compared to the one determined based on the most expected demand values. Moreover, this schedule is feasible over the entire expected range of uncertainty provided the vertexes of the uncertainty range are considered as scenarios, whereas the schedule corresponding to the nominal values is most commonly infeasible over some part of the uncertain range. Further flexibility analysis studies confirmed the increased feasibility of the multiperiod schedule and showed that this schedule had a much higher capacity in terms of the total production of all products as compared to the single-period schedule. Further studies are performed in order to examine the extent of correlation of improved schedule robustness (with respect to demand uncertainty) and performance under rush order arrival. Finally, to improve the schedule ability to meet rush orders at the time of order arrival, a new methodology has been proposed that generates the deterministic schedule using an iterative procedure considering the rush orders that may arrive at different times within the time horizon. 1. Introduction Scheduling of batch processes for mutiproduct/multiperiod batch chemical plants has been a focus of a lot of research in the last 2 decades. Most of the work, though, is limited to deterministic approaches, where it is assumed that all scheduling parameters such as product demands, processing times, etc., have constant known values. However, parameter variability within the scheduling time horizon is a common situation in practice. Upon realization of parameter uncertainty, a schedule built on a deterministic approach will be suboptimal or in some cases even infeasible. Thus, it is very important to estimate the performance of a deterministic schedule under parameter uncertainty and also to take into account the uncertainty information a priori at the scheduling stage to improve schedule feasibility. In this paper, we are using the basic concepts of multiperiod programming proposed by Grossmann and Sargent2 where the basic objective is to design a flexible plant for which the design remains feasible within a given range of uncertain design parameters. To evaluate the schedule performance under uncertainty, we are using the ideas of flexibility analysis as proposed by Swaney and Grossmann,3 where the goal is to evaluate the range of feasibility of a design under a given range of parameters. Also the ideas of robustness metrics as presented by Samsatli et al.4 are utilized to capture different robustness objectives under parameter uncer* To whom correspondence should be addressed. Tel: (732)445-2971.Fax: (732)445-2421.E-mail: [email protected].

tainty. Although a significant amount of work has been done to address the issue of robustness in design, these concepts have not been extended to scheduling of batch plants. Operations research literature reports work that is concerned with robust scheduling under uncertainty. Daniels and Carillo5 have proposed a methodology for β-robust scheduling for single-machine systems with uncertain processing times. Kouvelis et al.6 consider robust scheduling of a two-machine flow shop with uncertain processing times. However, this work is applicable to sequencing in a manufacturing environment but does not consider the material balances that are of great importance in chemical plants. A common approach to alleviating uncertainty (especially demand uncertainty) is the maintenance of safety levels of products, thus enabling the absorption of disturbances. However, this results in increased costs due to inventory handling and a higher capital cost to increased plant capacity to maintain higher inventory. Research that addresses the issue of parameter uncertainty in scheduling mainly deals with estimating the performance of a deterministic schedule under parameter uncertainty and to improve its use of heuristics. Honkomp et al.7 have developed a framework which links an optimizer and a simulator for testing the quality of schedules that are built with simplified deterministic information. Rotstein et al.8 used deterministic and stochastic flexibility analysis to quantify the flexibility of a batch production schedule. A stochastic reliability metric was also used to combine the parametric uncertainty with unit availability. In their work short-term scheduling

10.1021/ie0007724 CCC: $20.00 © 2001 American Chemical Society Published on Web 09/19/2001

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Ind. Eng. Chem. Res., Vol. 40, No. 21, 2001

is modeled using a discrete time formulation that results in large MILP models. Moreover, the authors did not consider ways of improving scheduling robustness, which is one of the targets of the work presented in this paper. Most of the other research in scheduling under uncertainty has mostly focused on rescheduling algorithms which are implemented when uncertainty is actually realized and do not consider uncertainty information prior to scheduling (see Vin and Ierapetritou9 for a review of this subject). In this paper, extensive studies have been performed to estimate, quantify, and improve schedule robustness under parameter uncertainty for multipurpose/multiproduct batch chemical plants. The deterministic model for short-term scheduling proposed by Ierapetritou and Floudas1 is used throughout this work. To quantify schedule robustness, we have used wellknown robustness metrics as well as proposed new measures that take into account the scheduling characteristics under demand uncertainty. To improve schedule robustness, parameter uncertainty is taken into account at the scheduling stage through a multiperiod programming formulation. The robustness of the resulting schedule has been tested and compared to the single-period model. Flexibility analysis studies have also been carried out for both of the schedules, and their performance is tested under different rush order arrival scenarios. This has been done using the reactive scheduling methodology proposed by Vin and Ierapetritou,9 in order to investigate the extent of correlation between schedule robustness and reactive scheduling performance for rush order arrival. Finally, to improve the schedule ability to meet rush orders as soon as possible after their time of arrival, a new methodology is proposed that generates the deterministic schedule using an iterative procedure considering the various rush orders that may arrive at different times within the time horizon. The main difference with the existing approaches that deal with the problem of scheduling considering intermediate due dates (Ierapetritou et al.10 and Karimi and McDonald11) is that in the proposed methodology we also consider the expected rush order arrivals and not only a deterministic scenario with constant due dates. Thus, it is expected that the schedule obtained would exhibit a better performance on the average in terms of rush order arrival uncertainty. The effectiveness of this approach has been demonstrated with the use of an illustrating scheduling example. 2. Deterministic Schedule Robustness: SP Formulation The process described in example 1 (described in detail in Vin and Ierapetritou9) is considered here [the state task network (STN) representation is shown in Figure 1]. The deterministic problem was solved to meet a nominal demand of 55 units of P1 and 70 units of P2. The objective was makespan minimization, and the schedule obtained is shown in the form of a Gantt chart in Figure 2. Demand uncertainty is considered by assuming that the demands of products P1 and P2 vary by 40% about their nominal values following a uniform probability distribution. Thus, the demand is assumed to vary uniformly between 55 ( 22 and 70 ( 28 for P1 and P2,

Figure 1. STN representation of example 1.

Figure 2. Gantt chart for the deterministic schedule for SP formulation.

respectively. A normal distribution is also considered, and the results are discussed in section 4.3. The performance of the deterministic schedule to meet various demands within this range is evaluated in terms of the makespan required to satisfy random demands if the sequence of tasks of the deterministic schedule (i.e., allocation of tasks to units) is followed. To achieve this, a makespan minimization problem is solved in which all binary variables are fixed to the corresponding values from the deterministic schedule but the batch sizes and starting and finishing times of the tasks are allowed to vary. In particular, the following steps are considered: Step 1. Solve the deterministic problem with the objective of minimizing the makespan using the nominal values of the demand, i.e., using the following demand constraint:

∑n d(s,n) g rnom(s),

∀s∈S

The solution yields the minimum makespan corresponding to this schedule (Hdet) and the set of binary variables corresponding to allocation of tasks to units [wvdet(i,n), yvdet(j,n)]. Step 2. Consider a random demand within the expected range of uncertainty: rp(s). Fix the binary variables for this problem to those of the deterministic schedule.

wv(i,n) ) wvdet(i,n) yv(j,n) ) yvdet(j,n) Solve the problem with the objective of minimizing the makespan using the following demand constraint:

∑n d(s,n) g rp(s),

∀s∈S

to get the makespan for this problem (Hp).

Ind. Eng. Chem. Res., Vol. 40, No. 21, 2001 4545

The most commonly used metric to quantify robustness is the standard deviation

SDavg )

x∑

ptot (H

p)1

p

- Havg)2

(ptot - 1)

(1)

where Hp corresponds to the makespan of scenario p and Havg is the average makespan over all scenarios determined as ptot

∑ Hp

p)1

Havg )

Figure 3. Histogram showing the percentage of scenarios considered in robustness analysis.

Step 3. Repeat step 2 for another random demand scenario rp′(s). For the problem considered here, the random points were generated in the form of a uniform square grid of 625 points (25 points × 25 points) between 33 and 77 units for P1 and 42 and 98 units for P2. Step 2 was repeated for each point on this grid. A histogram of the resulting makespans is shown in Figure 3. It was found that out of the 625 scenarios considered the deterministic scheduling problem was infeasible, in terms of meeting the demand for 125 scenarios (20% of the scenarios). Specifically, for all of these scenarios, the demand on P2 is not met completely. It should be noted that, because of this fact, the histogram is not symmetric about the deterministic makespan even though the demands of both products vary symmetrically about their nominal values. A makespan of 8.58 is observed in most of the scenarios considered. This makespan corresponds to those scenarios where the demand on P2 is not met completely.

3. Robustness Metrics In the context of scheduling, robustness can be defined as a measure of resilience of the scheduling objective to change in the face of parameter uncertainty and disruptive events. Scheduling is performed to satisfy a variety of different objectives such as makespan minimization or maximization of profit or production. Although the analysis performed in this paper may be generalized for different scheduling objectives, makespan minimization is used throughout this paper as the primary objective. The deterministic problem is solved using the nominal (most expected) values of the parameters to get the deterministic makespan. The performance of the deterministic schedule is tested by fixing the sequence of tasks (allocation of tasks to units) through the binary variables and solving a number of makespan minimization problems to meet demand rp(s), for various scenarios p {p ∈ P}, where rp(s) ∈ {R′(s)} and represents a randomly generated set of demands for all of the products produced in the plant.

ptot

(2)

The standard deviation SDavg represents the average deviation of the various makespans Hi about the mean Havg. Note that, because of the minimum capacity constraints and existence of constant processing times, there is a significant difference between the average makespan (Havg) and the makespan determined for the most expected demand values (Hdet; Figure 3). Because the deterministic schedule is determined to satisfy the most expected values of the product demand and is thus the most likely schedule to be followed in practice, it is proposed to evaluate the deviations of Hi using Hdet instead of Havg. A metric of robustness is thus proposed which is the standard deviation about the deterministic makespan, SDdet.

SDdet )

x∑

ptot (H

p)1

p

- Hdet)2

(ptot - 1)

(3)

Note that in both cases of standard deviation evaluation, it is assumed that every demand scenario occurs with equal probability because they are sampled using a uniform distribution about the nominal values. Using SDdet enables us to represent the variability of the schedule objective as Hdet ( SDdet, with Hdet being a measure of the schedule performance if the most expected value of demand is realized and SDdet a measure of the schedule resilience to makespan change under different demand values. Another metric of robustness considered in this paper is the extent of violation, which is a one-sided robustness metric that measures deviations that are either superior or inferior to the deterministic objective. On the basis of the ideas of Samsatli et al.,4 we define (for the case of makespan minimization) the extent of violation to represent the difference of the makespan of scenario p with respect to the deterministic makespan of only those scenarios whose makespans are inferior (higher) to the deterministic one:

Eviol )

x∑ n′

(Hp - Hdet)2

p′)1

(ptot - 1)

(4)

where p′ ∈ P′ (P′ ⊂ P) is the set of all scenarios such that Hp′ > Hdet, n′ is the total number of such scenarios, and ptot is the total number of scenarios.

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3.1. Robustness Metrics under Infeasibility. As observed in section 2, the deterministic schedule might be infeasible for many demand scenarios. This is because the deterministic schedule corresponds to a fixed number of batches of each process (sufficient to meet the nominal demand). Because each batch is limited by a maximum capacity Vmax(i,j), the sequence of tasks given by the deterministic schedule has a maximum level of production associated with it and, consequently, the scenarios with larger demand values become infeasible. Note that both SDavg and SDdet assume that all scenarios {p ∈ P} correspond to feasible scenarios. To account for scenarios that might be infeasible, a new metric in the form of a corrected standard deviation (SDcorr) is proposed. In the case of infeasibility, the problem is solved to meet the maximum demand possible. This is done by incorporating slack variables in the demand constraints and minimizing the slacks in the objective function. This makespan determined in this way is denoted as Hmax. The inventory of all raw materials and intermediates is taken at the end of this schedule, and a new scheduling problem is solved that assumes that this final inventory is available as an initial inventory for the new problem. The minimum makespan is found to meet the unsatisfied demand, Hinf. The overall makespan to meet the full demand is then

Hcorr ) Hmax + Hinf

(5)

The corrected standard deviation is defined as

SDcorr )

x∑

ptot (H

p)1

act

- Hdet)2

(ptot - 1)

(6)

4. Multiperiod Programming Formulation To improve the robustness of the deterministic schedule, information about demand uncertainty is considered at the scheduling stage. This is achieved by formulating a multiperiod programming problem involving different demand scenarios within the expected range of demand variability. The objective is to find a single sequence of tasks (i.e., a common set of binary variables among all scenarios) that minimizes the average makespan over all scenarios. The mathematical model of the multiperiod programming model has the following form:

∑p H(p)

Allocation Constraints ∀ j ∈ Ji, n ∈ N

(9)

Material Balances ST(s,n,p) ) ST(s,n-1,p) - d(s,n,p) + Fpsi B(i,j,n-1,p) - Fcsi B(i,j,n,p)

∑ ∑ i∈I j∈J s

∑ ∑ i∈I j∈J

i

s

i

∀ s ∈ S, n ∈ N, p ∈ P (10)

Demand Constraints

∑ d(s,n,p) g r(s,p)

∀ s ∈ S, p ∈ P

(11)

n∈N

Duration Constraints T f(i,j,n,p) ) T s(i,j,n,p) + Rijwv(i,n) + βijB(i,j,n,p) ∀ i ∈ I, j ∈ Ji, n ∈ N, p ∈ P (12) Sequence Constraints Same Task in the Same Unit T s(i,j,n+1,p) g T f(i,j,n,p) - Hbig(2 - wv(i,n) yv(j,n)) ∀ i ∈ I, j ∈ Ji, p ∈ P, n ∈ N, n * N (13) T s(i,j,n+1,p) g T s(i,j,n,p) ∀ i ∈ I, j ∈ Ji, p ∈ P, n ∈ N, n * N (14) T f(i,j,n+1,p) g T f(i,j,n,p) ∀ i ∈ I, j ∈ Ji, n ∈ N, p ∈ P, n * N (15) T s(i,j,n+1,p) g T f(i′,j,n,p) - Hbig(2 - wv(i′,n) yv(j,n)) ∀ j ∈ J, i ∈ Ij, i′ ∈ Ij, p ∈ P, i * i′, n ∈ N, n * N (16) Different Tasks in Different Units T s(i,j,n+1,p) g T f(i′,j′,n,p) - Hbig(2 - wv(i′,n) yv(j′,n)) ∀ j, j′ ∈ J, i ∈ Ij, i′ ∈ Ij, p ∈ P, i * i′, n ∈ N, n * N (17) Completion of Previous Tasks T s(i,j,n+1,p) g

(T f(i′,j,n′,p) - T s(i′,j,n,p′)) ∑ ∑ n′∈N,n′eni′∈I j

∀ i ∈ I, j ∈ Ji, n ∈ N, p ∈ P, n * N (18) Time Horizon Constraints T f(i,j,n,p) e H(p) ∀ i ∈ I, j ∈ Ji, p ∈ P, n ∈ N (19) T s(i,j,n,p) e H(p) ∀ i ∈ I, j ∈ Ji, p ∈ P, n ∈ N (20)

subject to wv(i,n) ) yv(j,n) ∑ i∈I

ST(s,n,p) e STmax(s) ∀ s ∈ S, n ∈ N, p ∈ P

Different Tasks in the Same Unit

where Hact ) Hp if scenario p is feasible and Hact ) Hcorr if scenario p is infeasible.

minimize

Storage Constraints

(7)

j

Capacity Constraints Vijminwv(i,n) e B(i,j,n,p) e Vijmaxwv(i,n) ∀ i ∈ I, j ∈ Ji, n ∈ N, p ∈ P (8)

This model involves the same type of constraints as the deterministic formulation proposed by Ierapetritou and Floudas.1 The differences are that there is a set of constraints for every scenario and that all of the continuous variables are functions of the scenario p. The binary variables are independent of the scenarios and represent a common set of allocation of tasks to units between all of the scenarios. Note that the objective function considered here corresponds to the actual summation over the different scenarios. An alternative objective would be the expected time horizon calculated by taking into account the probability of each scenario.

Ind. Eng. Chem. Res., Vol. 40, No. 21, 2001 4547 Table 1. Scenarios Considered in the Multiperiod Formulation demand of

demand of

scenarioa

P1

P2

scenarioa

P1

P2

1 2 3

33 33 55

42 98 70

4 5

77 77

42 98

a Scenarios 1, 2, 4, and 5 correspond to the four vertexes; scenario 3 is the nominal point.

Figure 5. Makespan as a function of demand for SP and MP schedules. Figure 4. Gantt chart for MP schedule formulation.

Because the multiperiod formulation considers the demand uncertainty simultaneously with the scheduling decisions, it is expected that the resulting schedule will be more robust compared to the deterministic schedule, which is determined considering the nominal values of demand only. Note that the multiperiod formulation considers demand scenarios uniformly sampled within the expected range of variability. For two uncertain product demands, this region corresponds to a rectangle with the four vertexes corresponding to the four extremities in the demand. It can be shown that any schedule that is feasible for these four vertexes will also be feasible over the entire uncertain range (for proof refer to Appendix I). Based on this proposition, the multiperiod formulation would always result in a feasible schedule for the entire range of uncertain demands if the four vertexes are included in the scenarios. In addition, the nominal point that corresponds to the most expected values of demand is considered. In the next section, we investigate the effect of the number of scenarios on schedule robustness. 4.1. Illustrating Example. Example 1 is considered here using the multiperiod formulation. A nominal demand of 55 units of P1 and 70 units of P2 is considered. The demand is assumed to vary uniformly within [33, 77] for P1 and [42, 98] for P2. Five scenarios are considered involving the four vertexes and the most expected value of demand to guarantee feasibility and good average performance (Table 1). The multiperiod programming problem presented in the previous section was solved with the objective of minimizing the total makespan over the five scenarios. The solution yields the optimal sequence of tasks (which are common to all scenarios) and the optimal values of the batch sizes and starting and finishing times of tasks, which are different for different scenarios. To obtain the final schedule, the values that can be considered for these variables are the ones corresponding to the nominal scenario or the average over all scenarios. The schedule using the values corresponding to the nominal scenario is shown in Figure 4. The robustness of the schedule resulting from the multiperiod problem was tested following a procedure similar to the one described in section 2 using a uniform grid of 625 points (25 points × 25 points) distributed

Table 2. Comparison of the Results for Single-Period and Multiperiod Formulations property deterministic objective to meet nominal demand mean objective over all scenarios SDdet (corrected using Hact) SDavg (corrected using Hact) extent of violation (Eviol) actual production for nominal scenario

single period

multiperiod

7.5057

7.7621

8.5230

8.1011

2.3353 2.1017 1.26 P1 ) 55, P2 ) 70

0.9776 0.9168 0.58 P1) 55, P2 ) 70

within the expected range of uncertainty ([33, 77] for P1 and [42, 98] for P2). To compare the performance of the multiperiod problem against the single-period problem, the single-period problem (as described earlier in section 2) was tested for robustness using the same grid of demand scenarios. In this case, for the infeasible scenarios, the actual makespan under demand infeasibility was determined using the procedure described in section 3.1 to get the corrected makespan (Hcorr). The results for the two studies are given in Table 2. Note that both SDdet and SDavg of the multiperiod (MP) are significantly lower than that of the single period (SP), suggesting that the MP schedule is more robust compared to the SP schedule for the given range of demand uncertainty. Also, although the makespan of the SP schedule is marginally better compared to the MP schedule, the mean of the makespans over all scenarios is better for the MP than for the SP. The MP schedule thus performs much better than the SP schedule under demand uncertainty. It was also found that, as expected, the MP schedule was feasible for all of the points on the grid, although the SP schedule that satisfies the nominal demand was infeasible for 20% of the scenarios. In these cases, the infeasibilities cause another schedule to be solved to meet the unsatisfied demand, which causes the makespan to increase significantly. This trend can be observed in Figure 5, which shows the makespan as a function of the demand of P1 and P2 for the MP and SP schedules. The higher robustness of the MP schedule is achieved at the cost of a higher makespan required to meet the nominal demand. Comparison of Figures 2 and 4 shows that the MP schedule requires more batches as com-

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Table 3. Results of Different Numbers of Scenarios multiperiod property deterministic objective (nominal demand) mean objective over all scenarios SDdet (corrected using Hact) SDavg (corrected using Hact) extent of violation (Eviol)

single period (nom)

2 scenarios (max, nom)

3 scenarios (max, min, nom)

5 scenarios (vertexes, nom)

12.797

13.829

13.829

13.829

14.800

14.429

14.254

14.361

1.825 1.724 1.08

1.696 1.642 0.93

1.736 1.652 1.00

3.0789 2.337 2.37

pared to the SP to meet the nominal demand. The extra batches are necessary to maintain feasibility over the higher demand (because all scenarios have a common sequence of tasks or batches). However, the extra batches must be operated at a certain minimum operating capacity (Vmin). This may lead to overproduction; i.e., even though the nominal demand is only P1 ) 55 and P2 ) 70, the minimum amount that will be produced while operating at the minimum capacity will be greater than the required demand. Although this might happen even for a SP schedule, overproduction is more likely to happen for the MP schedule because it has more production batches. Overproduction is accompanied by a corresponding increase in the makespan to produce the extra product over the required nominal demand. Even if the production is exactly equal to the required nominal demand, the makespan for the MP can only be greater than or equal to that of the SP (because a better or lower makespan for the MP suggests that the solution for the SP is suboptimal). Hence, even though the MP schedule produces exactly the same amount of P1 and P2 as the SP schedule (55 units of P1 and 70 units of P2), this is done by splitting, for example, one batch for reaction 2 that operates in reactor 1 for the SP schedule into two batches of reaction 2 which operate in reactor 1 in the MP schedule (refer to Figures 2 and 4). Splitting of batches is not highly efficient because, for every batch to operate, a “setup” time [the constant processing time R(i,j)] is added which may increase the makespan. Therefore, even though the performance of the MP may be better under demand uncertainty, this is achieved at the cost of an increase in the makespan to meet the nominal demand. 4.2. Minimum Number of Scenarios. As shown in the previous section, the multiperiod formulation leads to improvement in schedule robustness and increased schedule feasibility as long as the vertexes are included within the set of scenarios. The only disadvantage of the multiperiod approach is the increased computational requirement given the complexity of the scheduling problem. Thus, it is of interest to find the minimum set of scenarios that have to be considered in order to improve schedule performance in the face of demand uncertainty. It is found that two scenarios (the ones that correspond to the maximum demand values and the nominal point) are sufficient to increase schedule robustness. This comes as a result of the increasing schedule flexibility because, as has been proved (see Appendix I), the consideration of the extreme points of the demand range results in a schedule with full feasibility over the entire range of uncertainty. Table 3 illustrates the results for example 1, but with increased demand (110 for product 1 and 140 for product 2). Note that there are minor differences between the results considering three and five scenarios but a significant increase of schedule robustness between the single

Figure 6. STN representation of example 2. Table 4. Results Considering Normal Probability Distribution property deterministic objective to meet nominal demand mean objective over all scenarios SDdet (corrected using Hact) SDavg (corrected using Hact) extent of violation (Eviol) actual production for nominal scenario

single period

multiperiod

7.5057

7.7621

7.6051

7.8297

0.5671 0.5583 0.19 P1 ) 55, P2 ) 70

0.3916 0.3857 0.22 P1 ) 55, P2 ) 70

period and multiperiod involving the maximum and nominal demand points. 4.3. Normal Probability Distribution. The effects of considering different probability distribution functions for demand uncertainty is investigated here in the evaluation of schedule robustness. A Monte Carlo sampling method using normal distribution is used to select the 625 grid points considered in the robustness calculation. The results for the same SP and MP schedules are presented in Table 4. It should be noted that the MP schedule is also superior here although the difference from the SP schedule is smaller compared to the difference when uniform distribution is assumed. The reason is that the points selected using normal probability distribution are more concentrated around the nominal value (most expected demand values), which is the point used for SD schedule determination. Also very few infeasibilities are encountered. 4.4. Example 2. In this example four products are produced through eight tasks from three feeds. There are nine intermediates in the system. In all, six different units are required for the whole process. The STN representation for this process is shown in Figure 6, and the required data are given in work by Vin and Ierapetritou.9 The deterministic problem was solved to meet a nominal demand of 520, 1215, 290, and 1350 units of products 1-4, respectively. The objective was makespan

Ind. Eng. Chem. Res., Vol. 40, No. 21, 2001 4549 Table 5. Results of Different Numbers of Scenarios for Example 2 multiperiod property

single period (nom)

2 scenarios (max, nom)

81 scenarios

deterministic objective to meet nominal demand mean objective over all scenarios SDdet (corrected using Hact) SDavg (corrected using Hact) extent of violation (Eviol) CPU time (to determine schedule)

4.591 4.724 0.241 0.2 0.17 0.78 s (76 nodes)

4.692 4.744 0.138 0.128 0.08 2.17 s (54 nodes)

5.591 4.663 0.2319 0.2 0.13 2171 s (165 nodes)

Figure 7. Schedule for SP for nominal demand values for example 2.

Figure 9. Schedule for MP considering 81 demand scenarios for example 2.

5. Flexibility Analysis Flexibility analysis studies were carried out on both the MP and SP schedules to determine the extent of demand that the two schedules could satisfy. The mathematical formulation for the flexibility analysis as proposed by Swaney and Grossmann3 using the vertex enumeration solution strategy has been used:

maximize δ Figure 8. Schedule for MP with nominal and max values of demand for example 2.

minimization, and the schedule obtained is shown in the form of a Gantt chart in Figure 7. Demand uncertainty is considered by assuming that the demands of all products vary by 40% about their nominal values following a uniform probability distribution. Thus, the demand is assumed to vary uniformly between 550 ( 208, 1215 ( 486, 290 ( 116, and 1350 ( 540 for products 1-4, respectively. The same analysis presented for example 1 in sections 4.1 and 4.2 was performed for this example considering two multiperiod cases with 81 scenarios (grid of 3 points in each product direction) and only 2 scenarios, one that corresponds to the nominal point and one that represents the maximum production deviation for all products. The schedules obtained are shown in Figures 8 and 9, and the results are summarized in Table 5, together with the CPU time required for the runs. Note that the MP schedule obtained with consideration of only two scenarios has much better robustness than the nominal schedule. Also note that the consideration of a large number of scenarios (81) does not increase schedule robustness, as was also observed for example 1. Thus, it can be concluded that the consideration of only two scenarios with one that corresponds to the maximum values of product demands is sufficient to improve schedule robustness.

subject to Scheduling Constraints {Constraints (7)-(20)}

∑ d(P1,n) + st(P1,nfinal) g rnom(P1) ( δ∆R(P1)

(21)

d(P2,n) + st(P2,nfinal) g rnom(P2) ( δ∆R(P2) ∑ n∈N

(22)

n∈N

where δ is the flexibility index, ∆R(P1) and ∆R(P2) are the expected demand variabilities, and nfinal is the final event point in the schedule. Thus, the left-hand sides of eqs 21 and 22 correspond to the total production in the plant (total demand met during the schedule plus final inventory) for P1 and P2, respectively. The constraints given by eqs 21 and 22 correspond to four different problems, each of them corresponding to one of the vertexes of the expected range of uncertainty. These four problems were solved by fixing the binary variables to the corresponding values of the schedule for both the MP and SP schedules, and the maximum values of δ were determined. The total production of P1 and P2 obtained from these four problems is shown in Figure 10. To explore the maximum production capability for each schedule, the following production maximization (minimization) problems are solved.

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Figure 10. Extent of production of P1 and P2 obtained using flexibility analysis and modified objective functions. Table 6. Results for Maximum and Minimum Production of P1 and P2 objective

SP

MP

max (P1 + P2)

P1 ) 86.67 P2 ) 87.75 P1) 31.73 P2 ) 36.00

P1 ) 86.67 P2 ) 117.00 P1 ) 31.73 P2 ) 36.00

min (P1 + P2)

(i) Maximize the total production of P1 and P2, i.e.

max

∑n d(P1,n) + st(P1,nfinal) + ∑n d(P2,n) + st(P2,nfinal)

subject to satisfying the nominal demand of P1 ) 55 and P2 ) 70 and scheduling constraints. (ii) Minimize the total production of P1 and P2, i.e.

min

∑n d(P1,n) + st(P1,nfinal) + ∑n d(P2,n) +

Table 7. Makespan under Reactive Scheduling for MP and SP Schedules scenario extra order of P1 (units)

extra order of P2 (units)

time of arrival Trush (h)

40 0 20 40 30 40 0 20 40 30 40 0 20 40 30 mean

0 40 20 40 30 0 40 20 40 30 0 40 20 40 30

5 5 5 5 5 6 6 6 6 6 7 7 7 7 7

SP

MP

10.4330 10.4724 9.8974 11.9829 10.6838 10.4330 10.4724 9.8974 11.9829 10.6838 11.2752 11.7509 11.7064 12.8821 12.1983 11.1168

10.6895 10.4476 10.1251 11.7936 10.8909 10.6895 10.7288 10.1538 12.2393 10.9402 11.3245 11.6954 11.7860 12.8288 12.2510 12.2389

st(P2,nfinal) subject to satisfying a demand of P1 ) 0 and P2 ) 0 and scheduling constraints. Both the MP and SP schedules were tested; the results reported in Table 6 were obtained. Note that as expected the MP schedule has a much larger range of feasibility compared to the SP schedule. As expected, the above formulations resulted in higher overall production compared to flexibility results because there are no restrictions posed to the production proportionality between the two products. It can be shown that any demand (P1, P2) that has P1 ∈(P1min, P1max) and P2 ∈ (P2min, P2max) will be a feasible demand; i.e., the demand can be met by the corresponding MP or SP schedule within the given sequence of tasks (for proof refer to Appendix II). Note that if exact satisfaction of demand is required [constraints (21) and (22) are considered as equalities], the maximum demands of P1 and P2 that can be met are much lower (see Figure 10). This is due to the fact that overproduction is not allowed in this case. 6. Reactive Scheduling Performance Testing The demand uncertainty considered in the previous sections is realized during the course of a schedule in

the form of rush order arrivals (or cancellations) that may arrive at different times within the schedule. The rush order has two characteristics: (i) the extent of the order and (ii) the time of arrival. Although the robustness metric can be used to estimate the feasibility and performance of a given sequence with respect to demand variability, it does not take into consideration the time at which the extra order might arrive. In this section, the MP and SP schedules obtained in sections 2 and 4 were evaluated in terms of their performance with respect to reactive scheduling for different rush orders. A total of 15 different cases of rush order arrivals are considered, which differ both in the extent of the order and also in the time of arrival. The methodology proposed in work by Vin and Ierapetritou9 is followed here to perform reactive scheduling after the rush order arrives. Table 7 shows the results of the minimum makespan required to meet the new demand after rush order arrival for all of these cases, using both of the MP and SP schedules as the deterministic schedules. Note that the SP schedule performs better than the MP schedule in terms of the average makespan required to meet the nominal plus the rush order demand.


Figure 11. Schedule for SP for scenario 1 under reactive scheduling.

Figure 12. Schedule for MP for scenario 1 under reactive scheduling.

sequence of tasks are allowed to change in the reactive scheduling. However, when robustness is evaluated, the sequence of tasks remains fixed. As discussed in section 4, the advantage of the MP schedule over the SP schedule is that it takes into account the demand uncertainty at the scheduling stage itself, which results in increased feasibility. This advantage disappears in reactive scheduling because the rescheduling allows the addition of extra batches to meet the new order. However, it must be noted here that, by allowing the sequence of tasks to change completely, the new schedule will be different from the original schedule, thus disrupting plant operation. As discussed by Vin and Ierapetritou,9 it is desirable to keep the plant operation as close as possible to the original schedule. In the limiting case, to have a perfectly smooth plant operation after the rush order arrival (i.e., no change in sequence of tasks), we would like to operate at the original schedule itself, provided it is feasible for the new demand after rush order arrival. As observed in section 5, the MP schedule is feasible over a much larger range of demand as compared to the SP schedule. Thus, in the case of a rush order arrival, the MP has a much higher probability of being feasible to meet the rush order compared to the SP schedule and thus would have a much better performance in the reactive scheduling testing as compared to the SP schedule. 7. Scheduling under Immediate Delivery of Extra Order

Figure 13. Schedule for SP for scenario 3 under reactive scheduling.

Figure 14. Schedule for MP for scenario 3 under reactive scheduling.

However, the results of the reactive scheduling performance are generally mixed. In some cases, the MP schedule performs better than the SP schedule and vice versa. For example, Figures 11 and 12 show the schedules obtained for the SP and MP for scenario 1 (a rush order of 40 units of P1 that arrives at Trush ) 5 h). In this case, the makespan for the MP is better than that for the SP. Figures 13 and 14 show the schedules for scenario 3 (a rush order of 20 units of P1 and 20 units of P2 that arrives at Trush ) 5 h). In this case the makespan for the SP is better than that for the MP. Comparison of these results to the results of the robustness testing (see Table 2) shows that, although the MP schedule has a much better robustness than the SP schedule, the SP performs better than the MP in the actual reactive scheduling. These results support the idea that, by improving schedule robustness, the reactive scheduling performance is not necessarily improved. The reason for this is that, for generating the best reactive schedule after a rush order arrives at Trush, the

For all of the cases of reactive scheduling considered so far, it has been assumed that the new demand after rush order arrival has to be met only at the end of the time horizon and not at any intermediate time within the schedule. In some cases it might be necessary in the case of rush order arrival to deliver the order as soon as possible after it arrives. In such a case, it is desired to have produced and met part of the demand by the time the rush order arrives. In the reactive testing done so far, this objective has not been addressed. In this section, we propose a methodology that determines a schedule that satisfies totally or at least partially the additional demand, after the rush order arrival, at a time as close as possible to the order time. This is achieved by the use of an iterative procedure to arrive at a deterministic schedule that successively considers the various rush orders in a successive manner following the order of their arrival time. At each iteration, the objective is to maximize the average partial fulfillment of the total demand for the various rush orders considered, within a time horizon equal to the rush order arrival time. The schedule obtained in this way is then fixed by fixing the binary variables and batch sizes and the next iteration is performed considering a bigger time horizon corresponding to the next set of rush order arrivals. Note that the objective is to generate the schedule that is able to cover unpredictable rush order arrivals, and thus early production of all products is desirable. Although recalculation of the whole schedule might result in a better schedule overall, this is not allowed in the proposed approach to reflect the fact that when a rush order arrives, part of the schedule has been executed and cannot be revisited. This procedure is followed until all of the rush order arrivals are taken into consideration. As the iterations proceed, the binary variables and batch sizes for the

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schedule get fixed for a larger time horizon. Once all of the rush orders have been considered, the deterministic schedule is obtained by solving a makespan minimization problem to satisfy the nominal demand. Specifically, the following steps are considered in the proposed methodology: Step 1. All expected rush orders are grouped into q groups, where each group has p(q) different rush orders that differ in their demand on P1 and P2 but all arrive at the same fixed time Trush(q). The grouping is done chronologically in the order of rush order arrival time, i.e., Trush(q) < Trush(q+1). Step 2. For the first group of rush orders, a multiperiod programming problem is generated that considers p(q)1) different scenarios. The mathematical model of the multiperiod programming problem is similar to the one described in section 4. The demand constraint (11) for a product s is modified to correspond to the satisfaction of the new demand under rush order arrival for product s:

∑ d(s,n,p) + st(s,nfinal) g r

nom

n∈N

(s,p) + r

rush

(s,p) -

slack(s,p) ∀ s ∈ S, p ∈ P(q)

where the slack variables have been added to maintain feasibility because in some scenarios the total demand cannot be achieved (in general, because Trush(q) is smaller than the deterministic makespan, the total demand will not be met within Trush(q)). To achieve production before the time of rush order arrival, the problem is considered for a time horizon corresponding to Trush(q). Thus, the time horizon constraint (19) is modified to force all tasks to at least start before Trush(q)

T s(i,j,n,p) e Trush(q) ∀ i ∈ I, j ∈ Ji, p ∈ P(q), n ∈ N Step 3. The multiperiod programming problem is solved with the following objective function:

min

∑p ∑s frac(s,p) × slack(s,p)

where

frac(s,p) )

rrush(s,p) + rnom(s)

∑s (rrush(s,p) + rnom(s))

represents the fractional demand of product s (after rush order arrival) for scenario p. Note that the objective is to minimize the total unsatisfied demand rather than to maximize the total production because the later objective function may lead to maximization of the production of one product without satisfying the demand of the other product. Minimizing the slack in production will ensure that the demand of both products will be met completely first, before the production of any one of them is maximized above the required demand. The minimization of the slack has been done by normalizing it using the parameters frac(s,p) in order to drive the production to achieve an equal fulfillment of the demand on each product. Step 4. The binary variables corresponding to the allocation of tasks to units and the batch sizes of the corresponding tasks that are obtained from the solution

Table 8. Groups of Rush Order Scenarios Considered for the Proposed Methodology rush order arrival time (h)

rush order scenarios

5

P1 ) 40, P2 ) 0 P1 ) 0, P2 ) 40 P1 ) 20, P2 ) 20 P1 ) 40, P2 ) 0 P1 ) 0, P2 ) 40 P1 ) 20, P2 ) 20 P1 ) 40, P2 ) 0 P1 ) 0, P2 ) 40 P1 ) 20, P2 ) 20

6 7

Table 9. Production of P1 and P2 for Intermediate Times time

SP

MP

proposed

5

P1 ) 41.48 P2 ) 0 P1 ) 41.48 P2 ) 0 P1 ) 41.48 P2 ) 0

P1 ) 35.72 P2 ) 0 P1 ) 35.72 P2 ) 0 P1 ) 35.72 P2 ) 0

P1 ) 21.38 P2 ) 36.08 P1 ) 21.38 P2 ) 36.08 P1 ) 49.56 P2 ) 36.08

6 7

of the multiperiod programming problem at step 3 are fixed for subsequent iterations. Step 5. The next group of rush orders (q ) 2) are taken, and steps 2-4 are repeated by considering a multiperiod problem that involves all of the rush orders that arrive at Trush(q)2). The procedure is repeated for all values of q, i.e., by successively considering the different groups of rush orders in increasing order of their arrival time. Step 6. Once all rush orders are considered, the deterministic schedule is obtained by solving a makespan minimization problem to satisfy the nominal demand. 7.1. Illustrating Example. The procedure described above was applied to an example considered before (example 1) with a nominal demand of 55 units of P1 and 70 units of P2. A total of 9 rush order scenarios were considered. These were grouped into three groups based on their time of arrival, as shown in Table 8. Because the rush orders have been grouped into three possible arrival times, three iterations are involved to fix the schedule to meet the rush orders before the deterministic schedule can be obtained. The schedule obtained after the first iteration that considers the first group of rush orders (Trush ) 5 h) is shown in Figure 15. This schedule is fixed for the next iteration which considers the rush orders arriving at Trush ) 6 h and yields the schedule shown in Figure 16. Similarly, the third iteration yields the schedule shown in Figure 17. Finally, the deterministic schedule is obtained by solving a makespan minimization problem to satisfy a nominal demand of 55 units of P1 and 70 units of P2 (Figure 18). Note that the schedule obtained at the second iteration is the same as the schedule at the first iteration because, by fixing the first iteration variables, no time is left to perform even the minimum batch (20% of the maximum capacity) on the reactors that would result in production of product P1 or P2. For example, considering (reactor1, reaction 1) with the minimum production of 10 units, it requires 2/3 + (10/50) × 2 ) 1.067 h, which would result in exceeding the time horizon of 6 h. Table 9 reports the actual production of P1 and P2 at specific intermediate times following SP and MP schedules (Figures 2 and 4) and the deterministic schedule obtained using the proposed methodology (Figure 18).


Figure 15. Schedule after the first iteration for Trush ) 5 h.

improve schedule performance, uncertainty was considered at the scheduling stage through the formulation of a multiperiod programming problem. Flexibility analysis studies confirmed the increased feasibility of the multiperiod schedule and showed that this schedule had a much bigger capacity for the total production of all products as compared to the single-period schedule. Finally, to improve schedule ability to meet rush orders as soon as possible after the arrival time, a new methodology has been proposed that generates the deterministic schedule using an iterative procedure considering the rush orders that may arrive at different times within the time horizon. Appendix I: Schedule Feasibility

Figure 16. Schedule after the second iteration for Trush ) 6 h.

Proposition. For a given sequence of tasks (i.e., fix allocation of tasks to units), if feasibility in terms of demand satisfaction is guaranteed at the vertexes of the expected uncertainty range, then this sequence is feasible for all of the points inside the range. Proof. Assume the negative. This means that there exists a point (R1*, R2*) within the expected demand ranges

R1L e R1* e R1U R2L e R2* e R2U

(AI.1)

that is infeasible, which means that either Figure 17. Schedule after the third iteration for Trush ) 7 h.

R1* > P1

(AI.2)

R2* > P2

(AI.3)

or

where P1 and P2 are the total productions of products P1 and P2, respectively, obtained from the solution of scheduling problem following the given sequence of tasks. For ease in the presentation, let us assume that Figure 18. Deterministic schedule for the proposed methodology.

Note that the deterministic schedule obtained from the proposed methodology produces both P1 and P2 at all of the intermediate times considered. Although the MP and SP schedules produce some units of P1, they do not produce any units of P2 until the end of the time horizon. This means that, for example, if a rush order of 20 units of P2 arrives at Trush ) 5 h, the deterministic schedule from the proposed methodology will be able to deliver it immediately, but the MP and SP schedules will not be able to do so. 8. Summary In this paper extensive studies have been performed to estimate, quantify, and improve schedule robustness under demand uncertainty for multiproduct/multipurpose batch plants. The deterministic model proposed by Ierapetritou and Floudas1 has been used throughout this work which is based on a continuous-time representation and the concept of event points to reduce the number of binary variables required. To quantify schedule robustness, several robustness metrics that are commonly used in optimization literature have been applied to batch plant scheduling. To

R1* > P1

(AI.4)

Because the vertexes are feasible points, there exists P1′ such that P1′ g R1U and also R1* e R1U, which results in P1′ g R1*, that contradicts our initial assumption (AI.2). Appendix II: Maximum Production Feasibility Problem Proposition. The solution of the production maximization problem described in section 5 results in the maximum values of the demand that the schedule can satisfy (R1max, R2max) given that the sequence is fixed. This proposition states that, for every point (R1′, R2′) such that R1′ e R1max and R2′ e R2max, there is always a solution to the scheduling problem for the given sequence. Proof. Assume the negation. This means that there exists a point (R1*, R2*)

R1* e R1max

(AII.1)

R2* e R2max

(AII.2)

and

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I ) set of all tasks J ) set of all units Ij ) set of all tasks that can be performed in unit j Is ) set of tasks that produce or consume process state s Ji ) set of units that are suitable for performing task i N ) set of event points within the time horizon S ) set of all involved states P ) set of all periods involved in multiperiod model

Hbig ) arbitrary large constant used in the multiperiod model Hdet ) makespan for deterministic schedule Hp ) makespan for scenario p Havg ) mean of makespan over all scenarios Hinf ) makespan to meet unsatisfied demand for an infeasible scenario Hmax ) makespan that meets maximum possible demand for an infeasible scenario Hcorr ) corrected makespan to meet total demand for infeasible scenario Hact ) actual makespan for scenario ptot ) total number of scenarios SDdet ) standard deviation with respect to deterministic makespan SDavg ) standard deviation with respect to mean of makespans SDcorr ) standard deviation with respect to corrected makespan Eviol ) extent of violation nfinal ) final event point of schedule δ ) flexibility index ∆R(s) ) parameter proportional to demand uncertainty for state s Trush(q) ) rush order arrival time for all rush orders that belong to group q slack(s,p) ) slack variable to relax demand constraint for state s for scenario p rrush(s) ) demand from rush order for state s frac(s,p) ) fractional demand for state s in period p, after rush order arrival

Parameters

Literature Cited

such that the solution to the scheduling problem for this demand (P1, P2) is infeasible, which means that

R1* > P1

(AII.3)

R2* > P2

(AII.4)

and/or

Because assuming demand (R1max, R2max) there is a solution to the scheduling problem (P1*, P2*), this means that P1* g R1max and also R1* e R1max, which results in P1* g R1*, that contradicts (AII.3). Nomenclature Indices i ) tasks j ) units n ) event points representing the beginning of a task s ) states p ) periods Sets

Vijmin ) minimum amount of material processed by task i required to start operating unit j Vijmax ) maximum capacity of the specific unit j when processing task i STmax(s) ) maximum available storage capacity for state s r(s) ) market requirement for state s at the end of the time horizon Fsip, Fsic ) proportion of state s produced and consumed from task i, respectively Rij ) constant term of processing time of task i in unit j βij ) variable term of processing time of task i at unit j H ) time horizon price(s) ) price of state s Variables wv(i,n), wvdet(i,n) ) binary variables that assign the beginning of task i at event point n for all scenarios and deterministic schedule, respectively yv(j,n), yvdet(j,n) ) binary variables that assign the utilization of unit j at event point n for all scenarios and deterministic schedule, respectively B(i,j,n,p) ) amount of material undertaking task i in unit j at event point n for scenario p d(s,n,p) ) amount of state s being delivered to the market at event point n for scenario p ST(s,n,p) ) amount of state s at event point n for scenario p T s(i,j,n,p) ) time that task i starts in unit j at event point n for scenario p T f(i,j,n,p) ) time that task i finishes in unit j while it starts at event point n for scenario p

(1) Ierapetritou, M. G.; Floudas, C. A. Effective continuous-time formulation for short-term scheduling: I. Multipurpose batch processes. Ind. Eng. Chem. Res. 1998, 37, 4341. (2) Grossmann, I. E.; Sargent, R. W. H. Optimal Design of Chemical Plants with Uncertain Parameters. AIChE J. 1978, 24, 1021. (3) Swaney, R. E.; Grossmann, I. E. An Index for Operational Flexibility in Chemical Process Design. Part I: Formulations and Theory. AIChE J. 1985, 31, 621. (4) Samsatli, N. J.; Papageorgiou, L. G.; Shah, N. Robustness Metrics for Dynamic Optimization Models under Parameter Uncertainty. AIChE J. 1998, 44, 1993. (5) Daniels, R. L.; Carrillo, J. E. β-Robust scheduling for singlemachine systems with uncertain processing times. IEE Trans. 1997, 29, 977. (6) Kouvelis, P.; Daniels, R. L.; Vairaktarakis, G. Robust scheduling of a two machine flow shop with uncertain processing times. IEE Trans. 2000, 32, 421. (7) Honkomp, S. J.; Mockus, L.; Reklaitis, G. V. A framework for schedule evaluation with processing uncertainty. Comput. Chem. Eng. 1999, 23, 595. (8) Rotstein, G. E.; Lavie, R.; Lewin, D. R. Syntheis of Flexible and Reliable Short-term Batch Production Plans. Comput. Chem. Eng. 1996, 20, 201. (9) Vin, J.; Ierapetritou, M. G. A New Approach for Efficient Rescheduling of Multiproduct Batch Plants. Ind. Eng. Chem. Res. 2000, 39, 4228. (10) Ierapetritou, M. G.; Hene, T. S.; Floudas, C. A. Effective continuous-time formulation for short-term scheduling. Multiple intermediate due dates. Ind. Eng. Chem. Res. 1999, 38, 3446. (11) Karimi, I. A.; McDonald, C. M. Planning and Scheduling of Parallel Semicontinuous Processes. 2. Short-term Scheduling. Ind. Eng. Chem. Res. 1997, 36, 2701.

Other Symbols rnom(s) ) nominal demand (most expected value) for state s rp(s) ) demand for state s in scenario p

Received for review August 21, 2000 Accepted August 1, 2001 IE0007724

Robust Short-Term Scheduling of Multiproduct Batch Plants under

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