Addressing Robustness in Scheduling Batch Processes with

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Ind. Eng. Chem. Res. 2005, 44, 1524-1534

Addressing Robustness in Scheduling Batch Processes with Uncertain Operation Times Anna Bonfill, Antonio Espun ˜ a, and Luis Puigjaner* Department of Chemical EngineeringsETSEIB, Universitat Polite` cnica de Catalunya, Avinguda Diagonal, 647, E-08028 Barcelona, Spain

The uncertainty present in any process environment and related not only to variable market demand but also to operational disturbances is usually unavoidable, and therefore, poor performance may be attained with the execution of deterministic optimal schedules. In this work, the short-term scheduling problem in chemical batch processes with variable processing times is addressed with the aim to identify robust schedules able to face the major effects driving the operation of batch processes with uncertain times, i.e., idle and wait times. The problem is modeled using a two-stage stochastic approach accounting for the minimization of a weighted combination of the expected makespan and the expected wait times. The formulation is extended to explicitly manage risk by optimizing three different robustness criteria. The application of the proposed formulation to academic and industrially based examples shows the efficiency of the proposed approach and highlights the importance of considering the uncertainty in the shortterm scheduling level. Introduction In a realistic manufacturing environment, stochastic events not only associated with variable market demands but also with operational disturbances, such as equipment breakdowns and variable operation times, are usually unavoidable. Despite the uncertainty in the environment, the scheduler has to make some sort of decisions to start production. The execution of schedules based on nominal parameter values and the implementation of rescheduling strategies to tackle the problem once the uncertainty is revealed can be cumbersome and may lead to inefficient or even infeasible production situations without previous consideration of the uncertainty. When the presence of uncertainty is known at the time of scheduling, it should be advantageous to take possible future events into consideration before they happen in order to minimize their eventual impact. The problem of contemplating the uncertainty a priori has been extensively examined in the areas of design and planning of batch plants.1-4 Within the area of production scheduling, some works have been reported related to product demand uncertainty,5-7 whereas relatively little attention has been given to the problem of processing time uncertainty in the plant operation level. Different sources of processing time variability can be identified such as the nonuniform quality of raw materials, deviations of process parameters, equipment downtime, resources outages, poor performance of control systems, operator availability, and often also combinations of various indeterminate reasons, which make it difficult to predict the exact production rates in industrial processes. The degree of variability is a function of the batch process itself, but deviations from 5% upward of the processing time appear to be typical.8 Processing time variability has two major consequences in production scheduling (see Figure 1). On one * To whom correspondence should be addressed. E-mail: [email protected]. Telephone: +34 934 016 678. Fax: +34 934 010 979.

hand, if the actual processing time of a task is longer than the scheduled one, the time spent by batches waiting for the next unit to be available increases. On the other hand, if the actual processing time is shorter than the scheduled one, idle times appear and subsequent equipment underutilization occurs. Scheduling the operations with time estimates longer than the nominal ones could eliminate the batch wait times, but at the expense of poorer plant utilization and larger cycle times. On the other hand, scheduling the process using time estimates shorter than the nominal ones should keep the plant utilization high, at the expense of increased batch wait times and unexpected delays, thus leading to eventual quality problems. Therefore, from a system performance point of view, there is a tradeoff between low batch wait times and high plant resources utilization. Different approaches have been used so far to minimize the effects of processing time uncertainty. The traditional approach consists of introducing intermediate storage devices before the bottlenecking processing units and maintaining reserve material for downstream processing. This allows decoupling the operation of the two units, allowing the execution of the schedule without modifications. However, the production of reserve material is often expensive, inefficient, or technically difficult to maintain, and separate storage units could be required for each product or intermediate with an additional cost. Furthermore, if materials leaving a processing unit are unstable and therefore consecutive operations must be performed under a zero wait transfer policy, intermediate storage is not a viable solution. Another approach consists of the modification of the process operating conditions to adjust the processing times to return to the original schedule. The major drawback of this procedure is that there may be little flexibility for the modification of these conditions to guarantee the quality of the products, an acceptable economic performance, or other production requirements such us environmental or safety.

10.1021/ie049732g CCC: $30.25 © 2005 American Chemical Society Published on Web 01/28/2005

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Figure 1. Effects of processing time variability.

Both approaches use rough estimates or simply averages of the processing times observed in previous runs. Some other proposals have been reported in the literature to deal with processing time variability, which mainly address the problem from a rescheduling point of view, that is, once the uncertainty is disclosed. For example, Cott and Macchietto8 proposed an algorithm for online schedule modification of the processing times that detects the deviations between the target and the actual schedule at short intervals and shifts the operations without sequence modifications. Relatively few other works incorporate the uncertain information into the decision level to determine robust schedules. Daniels and Kouvelis9 defined the concept of schedule robustness as the determination of the schedule with the best worst-case performance compared with the corresponding optimal solution over all potential realizations of processing times. They presented exact and heuristic solution procedures to solve the robust scheduling problem and obtain the optimal solution; the incorporation of information on processing time variability rather than focusing only on point estimates was emphasized; however, this approach was only applicable to single-stage production plants. Orç un et al.10 developed a model for scheduling batch processes with uncertain processing times to maximize the expected profit of the plant and used the chance constraint approach to define the probability of accomplishment of the timing constraints. Sanmartı´ et al.11 presented a strategy based on the generation of a robust schedule by optimizing a reliability index and the subsequent application of rescheduling strategies to adjust the schedule to the new scenario. Furthermore, Lawrence and Sewell12 compared heuristic and optimal static and dynamic solution methods to job shop scheduling problems with processing time uncertainty; it was concluded that uncertainty would deteriorate the quality of algorithmic solutions compared to simple dispatch heuristics; however, when applying static methodologies, they only evaluated the causes of the perturbations but did not explicitly incorporate uncertainty into the design of the scheduling algorithm. Basset et al.13 presented a framework based on Monte Carlo sampling to deal with

the uncertainty on processing times and equipment availability; schedules for different generated problem instances were obtained and distributions were determined and statistically analyzed to define operating policies related to production lead times, possible maintenance protocols, and inventory profile predictions. Honkomp et al.14 also proposed a framework for shortterm batch processing with uncertain processing times and equipment breakdowns based on the use of a deterministic MILP optimizer coupled with a Monte Carlo simulator to assess the quality of base schedules and rescheduling strategies over multiple replicates of a simulation. Herrmann15 presented a two-space genetic algorithm as a general technique for solving minimax and robust discrete optimization problems; the algorithm was applied to identify a schedule with the minimum worst-case makespan for a parallel machine scheduling plant with uncertain processing times. Recently, Balasubramanian and Grossmann16 proposed a branch and bound procedure with an aggregated probability model to select the sequence of jobs with minimum expected makespan for flow shop plants with uncertain processing times modeled using discrete probability distributions. Lin et al.17 developed an optimization framework to obtain reliable schedules in the presence of uncertainty in processing times, market demands, or prices; a deterministic model was derived and the inequalities with uncertain parameters were rewritten considering their worst-case values; a certain degree of relaxation was introduced to allow violations of the constraint. The approaches proposed so far that incorporate the uncertain information into the decision level try to identify a schedule with optimal expected performance or a schedule that guarantees a minimum performance measure with a certain probability. Simple production models are usually assumed (e.g., flow shop, single stage) or the main features managing the uncertainty are not included into the objective function. Therefore, critical situations that can arise during the execution of the schedule, due to the actual operation times differing from those estimated, are not explicitly addressed and not even analyzed. For example, with the

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presence of considerable wait times, the quality of sensitive or unstable materials can decrease and become even unacceptable, thus forcing the rejection of batches with the consequent increase of operational costs. Furthermore, completion times larger than those expected can lead to delays in the promised delivery dates and hence to customer dissatisfaction. All these common shortcomings have motivated the development of this work. The short-term scheduling problem in chemical batch processes with variable processing times is addressed with the aim of identifying a robust schedule that minimizes the effects of processing time variability throughout its implementation. Therefore, robustness is defined in this work as the ability to face the uncertainty when no rescheduling strategy is to be considered and with the aim of minimizing possible wait times and idle times that can occur during the execution of a schedule. The scheduling problem is addressed for a multipurpose multistage batch plant. Material balances as well as features such as batch mixing and splitting can also be contemplated using the proposed approach but have been excluded from the scope of this paper in order to focus on the problem of the uncertainty and to avoid additional computational complexities arising from the representation of time with discrete or continuous-time models. This paper is organized as follows: The proposed problem is outlined first, along with the assumptions considered. A multiobjective stochastic formulation is developed in section 3, and in section 4, it is extended to explicitly include the tradeoff among different performance criteria (risk management). The effectiveness of the proposed approach as a decision-making tool is shown and discussed through its application to academic and industrially based examples in section 5. Finally, concluding remarks are given in section 6. Problem Statement The problem comprises a multipurpose multistage batch plant with a set of orders to be fulfilled, the set of processing stages required by each order, a set of units where they can be processed, the operations required by each stage, and the processing time of each operation represented by a probability distribution. This information is modeled using the process-stage-operation hierarchical approach defined by the standard ISAS88,18,19 which provides a standard terminology and hierarchical structured models for batch processes. Following this standard, each given order has associated with it a production process, i.e., a set of activities or stages required to transform the input materials into products. Furthermore, each stage involves an ordered set of operations that must be executed one immediately after another and assigned to the same equipment unit. Three different links are considered to describe temporal constraints between operations in different stages within a process: simultaneity, instantly, and sequential. Simultaneity accounts for those operations of different stages that have to start and end at the same time. Instantly requirements are defined for those operations that have to be produced one immediately after the other. Sequential links establish a relationship between the end time of an operation and the start time of another operation; i.e., they are defined for those operations that have to be performed consecutively without immediacy requirements.

Due to the presence of uncertainty in the operation times, there is no sense in determining a detailed schedule but only the minimum information required by the plant to start the production is needed, i.e., the sequence, the assignment of units to stages, and the initial processing time of each process (batch). To account for the eventual effects arising due to the uncertain processing times, a weighted combination of the expected makespan and the expected wait time resulting from the execution of the schedule is minimized. The tradeoff between the need for low-wait times and reasonable plant utilization is implicitly considered since the minimization of the makespan accounts for the reduction of idle times. The following assumptions are made: (1) From the schedule, the control level requires only information related to the sequence, the assignment of units to stages, and the process start times. Then production proceeds according to the control recipe without rescheduling considerations. (2) The nonintermediate storage policy (NIS) between stages is assumed; that is, an intermediate product will remain in the processing unit after its production until the unit assigned for the next stage is available. (3) Within a stage, all the operations must be executed without interruption. (4) When at the end of a processing stage or before a transfer operation the next unit is not available, a wait time is introduced. On the other hand, if a unit is available before the time it is required by the next stage, idle time appears (Figure 1). (5) To account for the wait times, sequential links are defined between the last operation of a stage (if it is not a transfer operation) and the first operation of the following stage, and between a transfer operation and the previous one in the same stage. (6) A distinction is made between wait times between stages (wt) due to the blockage or unavailability of a unit and process start wait times (st) due to delays on the determined process start times (Figure 1). Uncertainty associated with operation times can be represented indistinctly by discrete or continuous probability distributions. The scenario-based representation of the uncertainty is then adopted and Monte Carlo sampling over all the probability space is performed to approximate the expectation of the objective function. Scheduling Model A multiobjective two-stage stochastic programming approach is proposed in this work to model the processing times uncertainty within the decision-making process of the short-term scheduling of multipurpose multistage batch plants. This optimization approach is appropriate since decisions must be taken to start production before the actual values of operation times are revealed, but possible scenarios are anticipated at the time of scheduling to take into account the different probable outcomes when deciding the schedule to be implemented (for an overview on stochastic techniques, refer to Birge and Louveaux20). An MILP formulation is derived based on the concept of precedence relationship between stages introduced by Meńdez et al.21,22 Decision variables related to the production sequence, the assignment of units to stages, and the process start times are considered first-stage decisions to be taken “here and now”, independently of the realization of the uncertainty, since it is assumed


that these decisions are those required by the control level of the plant to start the production. In the second stage, and once the uncertainty is revealed, the timing of the actual schedule with the sequence, the assignment, and the process start times given by the firststage decisions can be computed for each scenario anticipated. As defined in the previous section, the determined process start times act as lower bounds on the actual schedules; i.e., the start times of each order in each scenario are constrained to be at least the start times of the proposed schedule. Therefore, a detailed schedule with particular makespan and wait times is assessed for each probable scenario anticipated, i.e., for each realization of processing times. The proposed model (S1) minimizes a weighted combination of the expected values of the makespan and wait time distributions, as described next. Model S1

min

∑s [ωs(F1Mks + F2(∑i j∑ ∑ wtois + ∑i stis))] ∈J o ∈O i

(6)

Equation 2 is a first-stage constraint that establishes the assignment of one of the alternative equipment units u to each processing stage j for every order i. Other variables related to the decisions to be taken independently of the final unveiled scenario (sequence and initial batch processing times) are deduced from eqs 3-12. These constraints (referred to as second-stage constraints) are defined over all the scenarios to determine a detailed schedule for each instance. Constraints 3-5 are sequencing constraints that express the completion time of the last operation of a stage j from order i as a lower bound for the start time of the first operation of any later stage j′ from order i′ assigned to the same unit u. The first-stage binary variable Xiji′j′ is used to define the production sequence. It takes the value of 1 if stage j of order i is processed before stage j′ of order i′ in some unit u, and 0 otherwise. Equation 5 is imposed only for those stages j and j′ from the same order i that are processed in the same unit u. Since their sequence is already established, the sequencing variable will always take the value of 1. These sequencing constraints become redundant whenever the production stages j and j′ are not allocated to the same unit u. The process start times Tini are determined with constraint 6, which establishes the wait times due to delays in the orders’ start times as the difference between the current start times of the orders for each scenario and the predicted ones coming from the first stage. Equation 7 relates the start and end times of the operations o of each order i for each scenario s through the actual operation time. Simultaneity requirements between operation o in stage j and operation o′ in stage j′ of order i are defined through constraints 8 and 9. Equation 10 establishes the instantly links between those operations o and o′ in the same stage j of order i that must be performed one immediately after the other. For those operations o and o′ of order i that have to be processed sequentially without immediacy requirements, constraint 11 is provided. Through this constraint, wait times are computed. Finally, the makespan for each scenario is defined in eq 12.

(7)

Robustness Criteria

(1)

j

subject to

∑ Yiju ) 1 ∀i,

u ∈Uij

j ∈ Ji

(2)

Tinro′i′s g Tfnrois + wtois - M(1 - Xiji′j′) - M(2 Yiju - Yi′j′u) ∀s, i, i′, j ∈ Ji, o′ ∈ Oj′f ,

o ∈ Olj,

j′ ∈ Ji′,

u ∈ (Uij ∩ Ui′j′),

i < i′ (3)

Tinrois g Tfnro′i′s + wto′i′s - MXiji′j′ - M(2 - Yiju Yi′j′u) ∀s, i, i′,

j ∈ Ji,

j′ ∈ Ji′, o ∈ Ofj, u ∈ (Uij ∩ Ui′j′),

o′ ∈ Oj′l , i < i′ (4)

Tinro′i′s g Tfnrois + wtois - M(2 - Yiju Yi′j′u) ∀s, i, i′,

j ∈ Ji, j′ ∈ Ji′, o ∈ Olj, u ∈ (Uij ∩ Ui′j′), i ) i′,

o′ ∈ Oj′f , j < j′ (5)

Tinrois ) Tini + stis ∀s, i, j ∈ Jfi, o ∈ Ofj Tfnrois ) Tinrois + Topois ∀s, i,

j ∈ Ji,

o ∈ Oj

Tinrois ) Tinro′is ∀s, i,

j′ ∈ Ji,

o ∈ Oj,

j ∈ Ji,

o′ ∈ Oj′, Tfnrois ) Tfnro′is ∀s, i,

j ∈ Ji,

sim

(o, o′) ∈ O

j′ ∈ Ji,

(8)

o ∈ Oj,

o′ ∈ Oj′, (o, o′) ∈ Osim (9) Tfnrois ) Tinro′is∀s, i,

j ∈ Ji,

o ∈ Oj,

o′ ∈ Oj,

(o, o′) ∈ Ozw (10) Tfnrois + wtois ) Tinro′is∀s, i, o ∈ Oj,

j ∈ Ji,

j′ ∈ Ji,

o′ ∈ Oj′, (o, o′) ∈ Oseq (11)

Mks g Tfnrois ∀s, i,

j ∈ Jli,

o ∈ Olj

(12)

Hence, the model identifies the most appropriate initial scheduling decisions (production sequence, assignment, and process start times) that balance the needed low-wait times and reasonable plant utilization (eq 1).

The stochastic formulation presented in the previous section (model S1) optimizes an expected objective function over all potential scenarios by assuming that the decision maker is risk averse or indifferent to the variability of outcomes. Therefore, there is no guarantee that the schedule will perform at a certain level over all the uncertain space, and some scenarios can yield to highly suboptimal performances. In this sense, this formulation can be easily extended to explicitly manage the risk of poor schedule performances by optimizing different robustness criteria based on the worst-case scenario and defined by Kouvelis and Yu23 as absolute robustness, robust deviation, and relative robustness criteria. The idea is to obtain alternative robust schedules that guarantee some expected performance, while minimizing over all the scenarios the worst performance defined according to one of the robustness measures. The absolute robustness criterion (ZAR) is a minimax criterion that attempts to determine the schedule that optimizes the worst performance over all the scenarios.

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For the proposed formulation, and for a given schedule, the absolute robustness is defined according to eq 13.

∑i j∑ ∑ wtois + ∑i stis)} ∈J o ∈O

ZAR ) max{Mks + ( s

i

(13)

j

The robust deviation (ZDR) and relative robustness (ZRR) criteria are concerned with how the actual system performance compares with the optimal performance that could have been achieved if perfect information on the scenario realization had been available at the scheduling time. These criteria are known as minimax regret criteria, where regret is defined as the difference or the ratio, respectively, between the actual schedule performance and the performance of the optimal schedule that would have been executed if the scenario had been known at the decision time. Therefore, these criteria allow, respectively, the identification of the schedule with the best worst-case deviation or the best worst-case percentage deviation from optimality over all the scenarios. For a given schedule, the robust deviation and the relative robustness are defined as stated in eqs 14 and 15, respectively.

∑i j∑ ∑ wtois + ∑i stis) ∈ J o ∈O

ZDR ) max{Mks + ( s

i

j

OFOPT } (14) s ZRR ) max{(Mks + ( s

wtois + ∑stis))/OFOPT } ∑i j∑ ∑ s ∈J o ∈O i i

j

is extended (model S3) with the incorporation of two additional constraints: the robustness definition (eq 13, 14, or 15) and the corresponding constraint 16, 17 or min min 18 (Zmin AR , ZDR , ZRR are the minimum robustness values that may be eventually obtained from models S2a, S2b, and S2c, respectively).

Model S3 min (eq 1) subject to eqs 2-12, eq 13, 14, or 15, eq 16, 17, or 18 ZAR e Zmin AR

(16)

ZDR e Zmin DR

(17)

ZRR e Zmin RR

(18)

The criteria to be applied are up to the decisionmaker’s concern about risk. As stated by Kouvelis and Yu,23 the absolute robustness criterion tends to lead to conservative decisions, since they attempt to hedge against the worst possible outcome. On the other hand, robust deviation and relative robustness criteria tend to be less conservative when making the decision and also look at uncertainty as an opportunity to be exploited rather than just as a risk to be hedged against. The deviation from optimality can be used as an indicator of how much the organizational performance could be improved, if part or all of the uncertainty could be resolved.

(15)

Note that both criteria require the computation of the optimal performance for each probable scenario sampled (OFOPT ), and hence, a deterministic problem for each s particular realization of processing times has to be solved. This deterministic model derives simply from the stochastic model proposed above (model S1) considering only one scenario with the nominal operation times and excluding constraint 6. It is worth noting that when the operation times are known at the scheduling time, no delays in the process start times occur during the execution of the schedule. Therefore, only the wait times between stages arising from the application of the NIS policy are considered in the wait times term of the objective function. The minimum absolute robustness, robust deviation and relative robustness are evaluated by solving models S2a, S2b, or S2c, respectively, described below. For modeling environments such as GAMS that do not support minimax functions, the definition of the robustness criteria, eqs 13-15, is handled by inequality constraints.

Models S2a-b-c min Robustness (defined by either eq 13, 14, or 15) subject to eqs 2-12 It is important to notice that the associated scenario probabilities are not used within this formulation. Besides, while the robustness minimization will ensure that the maximum weighted combination of makespan and wait times is minimized, the scenarios with lower maximum performance may have flexibility when fixing the times. To obtain the best schedules with minimum robustness, the proposed stochastic model (model S1)

Case Studies The proposed methodology to handle the uncertainty in the operation times is applied to an academic study and an industrially based case study. The models were implemented in GAMS24 and solved using the MIP solver of CPLEX(7.0) on a AMD Athlon 2000 computer. The deterministic problem was solved first. As previously indicated, the deterministic model was obtained directly from the stochastic one excluding constraint 4 and considering only one scenario with the nominal operation times. The decisions thus obtained were evaluated in front of the different scenarios, i.e., the production sequence, the assignment, and the process start times determined were fixed, and the makespan and wait time values that would be obtained for the realization of each particular scenario were computed. Next, the multiobjective two-stage stochastic problem (model S1) was solved and the Pareto curves were obtained by parametrically varying the weight values of both criteria (makespan and wait times) in the objective function. With fixed weight values, the problem was extended to consider explicitly the robustness criteria (models S2 and S3). In both examples, a set of 100 scenarios was considered and derived from the given probability distributions of processing times using Monte Carlo sampling. Computational times of ∼150 s of CPU time for the first example and ∼2000 s of CPU time for the second case study were required (note that the main purpose of this work is to suggest a strategy for managing uncertainty at the operation level rather than to develop the most efficient solution algorithm). Example 1: Motivating Case Study. The first case study is a simple motivating example that consists of a five-product, three-stage flow shop plant studied first


Figure 2. Scheme of the flow shop plant of example 1. Table 1. Problem Data of Example 1: Orders A-E stages

units

operations

ID operation

j1 j2 j3

U1 U2 U3

reaction centrifuge drying

o1 o2 o3

Table 2. Processing Time Distributions for Example 1 k1

k2

k3

order

operation

ta

ω

t

ω

t

ω

A A A B B B C C C D D D E E E

o1 o2 o3 o1 o2 o3 o1 o2 o3 o1 o2 o3 o1 o2 o3

2 6 12 3 4 8 5 6 8 12 16 11 15 2 8

0.15 0.50 0.125 0.30 0.50 0.40 0.10 0.20 0.80 0.15 0.50 0.125 0.30 0.25 0.40

4 10 14 5 15 15 7 8 10 24 20 17 18 4 14

0.25 0.15 0.775 0.30 0.25 0.20 0.40 0.50 0.10 0.25 0.15 0.775 0.40 0.50 0.20

8 12 16 8 25 25 12 10 14 28 25 23 19 5 19

0.60 0.35 0.10 0.40 0.25 0.40 0.50 0.30 0.10 0.60 0.35 0.10 0.30 0.25 0.40

a

Arbitrary time units.

Figure 3. Pareto curve between the expected wait time and the expected makespan values for example 1.

by Balasubramanian and Grossmann.16 A scheme of this plant is shown in Figure 2. Five orders are to be produced, namely, A, B, C, D, and E. Each order involves three production stages with one operation each. Only one unit is available to process each stage, and all the orders correspond to different products (Table 1). Uncertain operation times are represented by discrete probability distributions, with three possible time realizations for each product operation in each stage. Problem data related to the processing times and their respective probabilities for each operation is given in Table 2. Figure 3 depicts the Pareto curve between the expected wait time and makespan values. Each Pareto point identifies a scheduling policy according to different preferences for wait times and plant utilization. Solu-

tions obtained by managing robustness are also illustrated along with the results obtained from the evaluation of the deterministic schedule and the schedule with minimum expected makespan obtained by Balasubramanian and Grossmann16 (see “stochastic solution” in Figure 3). If the minimum makespan is pursued, the schedule identified by the Pareto point Pa would be implemented. Otherwise, if wait times have to be avoided, the schedule identified by the Pareto point Pb would be executed, which guarantees null expected wait times over the set of selected scenarios but at the expense of poorer plant utilization. The stochastic solution obtained with weight values for the makespan and the wait time criteria fixed at 1 (Pareto point Ps) balances both objectives. This solution is detailed in Figure 4, where the exact schedule to be implemented is represented for the nominal scenario and for one of the randomly generated scenarios (data of the operation times for this scenario is reported in Table 3). Note that decisions related to the production sequence, the assignment of units to stages (fixed in this case), and the order start times are provided in a first stage, and therefore, they are fixed and independent of the scenario unveiled. Table 4 reports and compares the results obtained related to the expected makespan, the expected wait time, the order start times, the absolute robustness, the robust deviation, and the relative robustness values for alternative schedules (deterministic, stochastic with different weight values for the objectives, and the schedules obtained under the different robustness criteria with weights fixed at 1). The makespan and wait time values of the schedule that would be executed according to each alternative policy for the nominal scenario are also included (Mknom and WTnom in the table). It is important to note from Table 4 that the scheduling decisions obtained with the deterministic formulation using nominal processing times poorly face the uncertainty and overestimate the system performance. Although the makespan and wait time values of the deterministic schedule are optimal in the nominal scenario, when these decisions are used to face the uncertainty, the expected makespan raises nearly 5% and significant wait times are expected. On the other hand, the stochastic modeling with weight values fixed at 1 for both criteria in the objective function (STPs) allows the identification of an initial schedule with expected wait times reduced nearly 37% and with acceptable expected makespan. By explicitly incorporating the robustness criteria, alternative initial schedules are identified with reduced risk of poor performances while still maintaining improved expected performance with respect to the deterministic approach. The absolute robustness criterion, for example, identifies a schedule with a worst-case performance reduced by 14% and with an expected wait time value ∼56% lower with respect to the deterministic schedule. The reduction of expected wait times is even higher for the decisions identified under the relative robustness criterion (nearly 67%) despite the increase of the expected makespan and the poor performance for the nominal scenario. The consideration of the expected makespan as a single criterion was also analyzed. With the proposed stochastic modeling, considering only the makespan term in the objective function, a schedule with a minimum makespan value of 99 is obtained at the

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Figure 4. Stochastic solution for example 1. Table 3. Operation Times of a Particular Random Scenario for Case Study 1 stage

order A

order B

order C

order D

order E

j1 j2 j3

2 6 16

5 4 15

7 10 8

28 25 17

15 4 8

Table 4. Results Obtained for the Different Approaches: Deterministic (DET), Stochastic (ST), Absolute Robustness (AR), Robust Deviation (DR), and Relative Robustness (RR) FO Mknom WTnom E(Mk) E(WT) TinA TinB TinC TinD TinE ZAR ZDR ZRR

DET

STPsa

AR

DR

RR

120.4 101.0 0.0 105.7 14.7 0.0 10.0 75.0 16.0 57.0 152.0 45.0 0.62

116.2 107.0 4.0 106.9 9.3 0.0 39.0 8.0 44.0 20.0 135.0 38.0 0.43

119.3 111.3 1.0 112.8 6.5 0.0 30.2 85.3 35.2 8.8 131.6 43.5 0.59

119.2 107.0 6.5 107.8 11.5 0.0 30.0 75.5 35.0 8.0 142.5 32.5 0.45

120.5 115.8 0.76 115.7 4.8 0.0 47.8 28.7 53.0 9.3 138.1 35.1 0.35

STPb STPb

STb

99.0 0.0 123.4 99.0 153.0 105.0 67.0 0.0 0.0 99.0 151.4 110.2 73.4 0.0 13.2 0.0 0.0 68.0 2.0 119.0 0.0 29.0 92.0 79.0 8.0 34.0 6.0 41.0 15.0 47.0 c c 165.0 c c 60.0 c c 0.57

a P , F ) 1,F ) 1; P , F ) 1, F ) 0; P , F ) 0,F ) 1. b Results s 1 2 a 1 2 b 1 2 obtained from the evaluation of the schedule with optimum expected makespan identified by Balasubramanian and Grossmann16 (different assumptions were taken, see text). c Values not given for being not comparable since different weight values are used.

expense of an important increase in the expected wait times (STPa results in Table 4). This schedule differs from that identified by Balasubramanian and Grossmann16 as the one with optimum expected makespan. It is important to mention that they do not use the NIS policy but the zero-wait one, thus leading to a higher expected makespan value (106.1). Moreover, the process start times are not taken into account when evaluating the different scenarios; i.e., they are not fixed but adjusted to follow the zero-wait policy for each scenario,

which will not always be feasible in a real procedure. Actually, an increase on the operation times cannot be detected prior to the actual execution. Therefore the uncertainty in the processing times of one batch cannot be realized before the scheduled start time of the subsequent order, so the zero-wait policy will not be followed at all and unavoidable wait times will occur. Even though both approaches cannot be directly compared, it is interesting to see the usefulness of considering the predicted start times in reducing the presence of unexpected wait times. With this purpose, the schedule identified by Balasubramanian and Grossmann16 was fixed (production sequence and process start times) and evaluated in front of the different scenarios. The results are appended in the last column of Table 4. As can be observed, the schedule thus identified does not take into account the wait and idle times that can arise during its execution. Considerable expected wait times appear (13.2) and the expected makespan increases ∼4% with respect to the optimum expected makespan value reported (110.2 vs 106.1). Note that both the expected wait times and makespan values are about 42% and 3 higher, respectively, than those obtained with the multiobjective stochastic approach proposed in this work. Moreover, the variability of outcomes is also noteworthy as can be observed from the absolute robustness value, which is 25% higher than the minimum one. Finally, to highlight the influence of the uncertainty in the decision making, the schedules to be executed following different decision criteria if the scenario defined in Table 3 finally occurred are represented in Figure 5. Note that each schedule obtained under the alternative criterion has its own production sequence and minimum process start times as indicated in Table 4. Other results in terms of production efficiency would be obtained in other scenarios. The final decision is up to the decision-maker preferences or organization’s policy.

Ind. Eng. Chem. Res., Vol. 44, No. 5, 2005 1531 Table 5. Problem Data for the Industrial Case Study units

operations

ID operation

topa (min - max)

filtration

F1 F2

mixing

V

dosification spraying I spraying II washing

P1 P2 P2 P3 P4

charge solid formation discharge cleaning filling water washing cakes water discharge drying discharge charge mixing bleaching discharge pumping spraying I spraying II washing I washing II

o1 o2 o3 o4 o5 o6 o7 o8 o9 o10 o11 o12 o13 o14 o15 o16 o17 o18

10.6-22.4 30.0-34.0 8.3-16.6 2.0-3.6 8.0-8.0 50.0-70.0 10.0-12.0 6.6-11.0 5.0-5.6 simultan with o9 10.0-10.0 30.0-30.0 5.0-5.0 simultan with o1 simultan with o4 simultan with o9 simultan with o5 simultan with o6

stages

a

Figure 5. Detailed schedules that would be implemented according to different scheduling policies for example 1. (a) Deterministic; (b) stochastic; (c) absolute robustness; (d) decisions fixed from the schedule with optimum expected makespan identified in ref 16.

Figure 6. Washing subprocess scheme.

Example 2: Industrial Case Study. The second example tested consists of the scheduling of a washing subprocess of a more complex single-product production process. A scheme of the subprocess is shown in Figure 6. The process environment is essentially of a batch nature and involves 6 production stages with 18 different activities either batch operations in filters or semicontinuous auxiliary operations. The importance of addressing the operation times’ uncertainty comes from the desire to achieve high and uniform product quality. Four orders were considered for the scheduling problem. Data related to the different stages, operations of each stage, available equipment units, and processing

Arbitrary time units.

Figure 7. Pareto curve between the expected wait time and the expected makespan values for the industrial case study.

time information is given in Table 5. The uncertain processing times were represented by uniform probability distributions between the minimum and maximum values given. Figure 7 shows the tradeoff between the expected wait time and makespan values. The stochastic solution that balances both criteria with weight values fixed at 1 is represented in Figure 8. The Gantt charts detail the schedules to be executed for the nominal scenario and for one of the randomly generated scenarios (Table 6). Results of the expected makespan, the expected wait time, the order start times, the absolute robustness, the robust deviation, and the relative robustness values for the alternative schedules are reported in Table 7. The makespan and wait time values that would be attained under the alternative policies for the nominal scenario were also included. It is worthwhile to note that, for this particular case study, and except the case when only the makespan term is considered (STPa), the schedule that would be executed if the nominal scenario was finally the one realized would proceed without wait times for any of the different scheduling decisions resulting from the alternate models proposed. As was also observed through the previous motivating example, the stochastic solutions perform better over the uncertain space than the deterministic one. With the multiobjective stochastic approach with weight values fixed at 1 for both criteria in the objective

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Figure 8. Stochastic solution for the industrial case study. Table 6. Operation Times of a Particular Random Scenario for the Industrial Case Study operation

order 1

order 2

order 3

order 4

o1 o2 o3 o4 o5 o6 o7 o8 o9 o11 o12 o13

17.7 32.9 16.4 2.7 8 53.8 10.7 10.2 5.6 10 30 5

18.6 31.8 12.9 2.3 8 56.8 10.9 6.8 5.3 10 30 5

12.0 33.2 14.3 2.3 8 69.6 10.4 6.9 5.5 10 30 5

13.0 31.2 9.4 2.7 8 57.1 10.3 10.5 5.2 10 30 5

Table 7. Results Obtained for the Different Approaches: Deterministic (DET), Stochastic (ST), Absolute Robustness (AR), Robust Deviation (DR), and Relative Robustness (RR)

FO Mknom WTnom E(Mk) E(WT) Tin1 Tin2 Tin3 Tin4 ZAR ZDR ZRR

DET

STPsa

AR

DR

RR

STPa

STPb

431.7 409.3 0.0 417.7 14.0 157.0 0.0 50.3 207.3 475.9 76.3 0.19

428.4 418.9 0.0 421.7 6.7 52.9 0.0 158.9 216.9 486.4 72.7 0.18

433.0 426.1 0.0 429.3 3.7 58.6 170.5 224.1 0.0 449.1 57.0 0.15

430.7 422.1 0.0 424.9 5.8 164.9 220.1 59.4 0.0 460.2 43.9 0.11

430.7 422.1 0.0 424.9 5.8 165.0 220.1 59.4 0.0 460.0 44.1 0.11

394.0 409.3 35.2 412.9 36.1 197.6 0.0 34.8 147.0 b b b

27.0 441.3 0.0 441.6 0.54 239.3 174.0 65.0 0.0 b b b

a P , F ) 1, F ) 1; P , F ) 0.95, F ) 0.05; P , F ) 0.06, F ) s 1 2 a 1 2 b 1 2 0.94. b Values not given for being not comparable since different weight values are used.

function, the scheduling decisions identified lead to expected wait times reduced 52% and an acceptable expected makespan (1% increase) compared with the schedule based on deterministic operation times. Furthermore, an initial schedule with a relatively small increase on the expected makespan and an expected

wait time reduced by 74% was identified with the absolute robustness criterion (see Table 7). Finally, detailed schedules obtained under the alternative criteria for the particular scenario defined in Table 6 are represented in Figure 9. Note again the different production sequences, assignments of units to tasks, and process start times of each schedule. The suitability in terms of production efficiencies will depend on the final revealed scenario. Therefore, the first-stage decisions concerning the information to be sent to the control system imply a tradeoff to be solved by the decision maker according to the risk acceptability policy. Conclusions An approach to account for the processing time uncertainty when deciding the scheduling policy in multipurpose multistage batch plants is proposed in this work. A multiobjective two-stage stochastic optimization model is presented with the aim to identify not a detailed schedule but the minimum information required by the batch process control system to start the production, prior to the realization of the uncertainty. The model provides information related to the production sequence, the assignment of units to stages, and the initial batch times, which balances the expected wait times and idle times that can arise during the execution of a schedule. Risk of poor performances is also addressed by incorporating the absolute robust deviation and relative robustness criteria. The effectiveness of the proposed approach and the alternative criteria has been illustrated and compared through their application to an academic and an industrially based case studies. With the stochastic modeling developed, an improved expected performance over all the anticipated scenarios of processing times was attained, and hence, scheduling decisions could be identified that lead to significantly reduced expected wait times and acceptable line occupation. From the analysis of the scheduling policy derived from the minimization of the expected makespan as a single criterion neglecting the fixed process start


Figure 9. Detailed schedules that would be implemented according to different scheduling policies for example 2. (a) Deterministic; (b) stochastic; (c) robust deviation.

times it was shown that the eventual effects arising during the execution of the schedule when facing operation time uncertainty are ignored, thus leading to a significant increase in expected wait times or plant underutilization. Furthermore, the consideration of different robustness criteria provides the decision maker with alternative robust schedules to be implemented according to the policy followed. In summary, an attempt has been made to properly define the scheduling problem by addressing explicitly and prior to the execution of a schedule the major effects present in the operation of batch processes with uncertain times. Despite excluding material balances, the approach derived can be applied not only to simple production models but also to more general multipurpose batch plants and includes the main features of the uncertainty into the objective function. Besides, it can be seen as a valuable way to take advantage of probability information on future events to support or facilitate rescheduling strategies implemented to adjust deviations from the predicted schedule once the uncertainty is disclosed. The results obtained highlight the importance of considering the uncertainty in the decision-making. The scenario-based representation of the uncertainty has been used in this work, but further research is required to improve the performance of stochastic approaches for applications of industrial size and complexity.

Acknowledgment Financial support received from the Spanish Ministerio de Educacioń, Cultura y Deporte (FPU research grant to A.B.), and Ministerio de Ciencia y Tecnologıá (project DPI2002-00856); from the Generalitat de Catalunya (project I0353); and from the European Community (Contracts GIRD-CT-2000-00318 and MRTNCT-2004-512233) is fully appreciated. Nomenclature Indices i ) orders or processes j ) stages o ) operations s ) scenarios u ) units Sets Ji ) stages j required for the production of order i Jfi ) first stage j of order i Jli ) last stage j of order i Oj ) operations o in stage j Ofj ) first operation o in stage j Olj ) last operation o in stage j O′seq oo ) operations o and o′ in stages j and j′ to be performed sequentially ) operations o and o′ in stages j and j′ to be O′sim oo performed simultaneously

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O′zw oo ) operations o and o′ in stage j to be performed one immediately after the other Uij ) units u that can process stage j of order i Parameters M ) large number Topois ) processing time of operation o of order i in scenario s F1 ) weight value for the makespan criterion in the objective function F2 ) weight value for the wait time criterion in the objective function ωs ) probability of occurrence of scenario s Variables Mks ) makespan for scenario s OFOPT ) optimum objective function value for scenario s s stis ) delay for the start time of order i in scenario s Tfnrois ) completion time of operation o of order i in scenario s Tini ) processing start time of order i Tinrois ) processing start time of operation o of order i in scenario s wtois ) wait time after operation o of order i in scenario s Xijij′ ) binary variable denoting that stage j of order i is processed before stage j′ of order i′ Yiju ) binary variable denoting the assignment of stage j of order i to unit u ZAR ) absolute robustness criterion ZDR ) robust deviation criterion ZRR ) relative robustness criterion

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(8) Cott, B. J.; Macchietto, S. Minimizing the Effects of Batch Process Variability Using Online Schedule Modification. Comput. Chem. Eng. 1989, 13, 105-113. (9) Daniels, R.; Kouvelis, P. Robust Scheduling to Hedge Against Processing Time Uncertainty in Single Stage Production. Manage. Sci. 1995, 41, 363-376. (10) Orç un, S.; Altinel, K.; Hortaç su, O. Scheduling of Batch Processes with Operational Uncertainties. Comput. Chem. Eng. 1996, 20, S1191-S1196. (11) Sanmartı, E.; Huercio, A.; Espunã, A.; Puigjaner, L. A Combined Scheduling/Reactive Scheduling Strategy to Minimize the Effect of Process Operations Uncertainty in Batch Plants. Comput. Chem. Eng. 1996, 20, S1263-S1268. (12) Lawrence, S. R.; Sewell, E. C. Heuristic, Optimal, Static, and Dynamic Schedules when Processing Times are Uncertain. J. Oper. Manage. 1997, 15, 71-82. (13) Basset, M. H.; Pekny, J. F.; Reklaitis, G. V. Obtaining Realistic Production Plans for a Batch Production Facility. Comput. Chem. Eng. 1997, 21, S1203-S1208. (14) Honkomp, S. J.; Mockus, L.; Reklaitis, G. V. A Framework for Schedule Evaluation with Processing Uncertainty. Comput. Chem. Eng. 1999, 23, 595-609. (15) Herrmann, J. W. A Genetic Algorithm for Minimax Optimization Problems. In Congress on Evolutionary Computation; IEEE: Piscataway, NJ, 1999. (16) Balasubramanian, J.; Grossmann, I. E. A Novel Branch and Bound Algorithm for Scheduling Flowshop Plants with Uncertain Processing Times. Comput. Chem. Eng. 2002, 26, 4157. (17) Lin, X.; Janak, S. L.; Floudas, C. A. A New Robust Optimization Approach for Scheduling under Uncertainty: I. Bounded Uncertainty. Comput. Chem. Eng. 2004, 28, 1069-1085. (18) International Society for Measurement and Control. ANSI/ ISA-S88.01-1995, Batch Control, Part I: Models and Terminology, 1995. (19) International Soc. for Measurement and Control. ANSI/ ISA-S88.01-1995, Batch Control, Part II: Data Structures and Guidelines for Languages. Draft 14, 1999. (20) Birge, J. R.; Louveaux, F. Introduction to Stochastic Programming; Springer: New York, 1997. (21) Meńdez, C. A.; Henning, G. P.; Cerda´, J. An MILP Continuous-Time Approach to Short-Term Scheduling of ResourceConstrained Multistage Flowshop Batch Facilities. Comput. Chem. Eng. 2001, 25, 701-711. (22) Meńdez, C. A.; Cerda´, J. An MILP Continuous-Time Framework for Short-Term Scheduling of Multipurpose Batch Processes Under Different Operation Strategies. Optimization Eng. 2003, 4, 7-22. (23) Kouvelis, P.; Yu, G. Robust Discrete Optimization and its Applications; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1997. (24) Brooke, A.; Kendrick, D.; Meeraus, A.; Raman, R. GAMS-A User’s Guide; GAMS Development Corp.: Washington, DC, 1988.

Received for review April 5, 2004 Revised manuscript received November 18, 2004 Accepted November 23, 2004 IE049732G

Addressing Robustness in Scheduling Batch Processes with

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