RESEARCH NOTES

Scheduling Batch Production Using a Stepwise Approach

Ka-Fai Chan and Chi-Wai Hui*

Chemical Engineering Department, Hong Kong University of Science and Technology,Clear Water Bay, Hong Kong SAR, China

This paper presents a stepwise algorithm for scheduling a single-stage, multiproduct batchproduction. Instead of solving a full-scale problem, a stepwise method is proposed that addsnew production orders sequentially into the existing schedule. A continuous-time mixed-integerlinear model with some simple heuristics is used for optimizing the schedule at each schedulingstep. By doing so, solution quality and time are significantly improved when compared withtraditional approaches that tackle the full-scale scheduling.

1. Introduction

A good production scheduling method for batch pro-ductions is essential to cope with changes in marketdemands. For most batch production plants, productionorders arrive irregularly, requesting a variety of prod-ucts in different quantities with a range of due dates.Traditional scheduling methods apply simple heuristicsto scheduling the production by either full-scale re-scheduling (rescheduling everything at one time) orinsertion of new orders into the existing schedulesequentially. Although the heuristic approach does notguarantee optimality, it is commonly used because ofits reliability in producing feasible schedules as well asa reasonable solution time. The mathematical modelingapproach is another alternative for scheduling batchproduction, which has been frequently discussed in theliterature in the past decade.1 The advantage of usingthe mathematical modeling approach lies in the pos-sibility of improving solution quality. In theory, opti-mality is guaranteed for most scheduling problems ifthe solution time is not considered.

In a multiproduct batch production plant, the produc-tion schedule is made to satisfy production orders.Although production orders arrive irregularly, most ofthe reported scheduling methods tackle them all to-gether as full-scale scheduling problems. Only a fewrescheduling algorithms were proposed to avoid full-scale scheduling. Satake et al.2 proposed a simulatedannealing approach to minimize the makespan of thegeneral order shop, in which a better schedule is foundby changing the operation sequence by viewing theGantt chart of a nonoptimal schedule. To overcome theunexpected deviations in process time and unit avail-abilities, Kanakamedala et al.3 proposed a reactiveschedule modification algorithm using a search treeanalysis, which alternatively reassigns tasks to equip-ment. The algorithm selects the best alternative forcausing minimum impact on the rest of the schedule tostrike a balance between achieving a good solutionwithin a reasonable computational effort. Huercio et al.4

also presented a reactive scheduling algorithm, whichadapts the current schedule to real-time disturbances,integrated to batch production control. The advantageof this algorithm is that the production schedule canbe easily managed in real-time systems. Ko and Moon5

suggested DSMM (dynamic shift modification method),PUOM (parallel unit operation method), and UVVM(unit validity verification method), which minimize theeffects of unexpected events in the scheduling process.

In this paper, a stepwise algorithm combining simpleheuristics into a continuous-time mixed-integer linearmodel is proposed. Instead of a full-scale scheduling, theproposed method adds new production orders sequen-tially into the existing schedule.

2. Problem Definition

This paper deals with the problem of scheduling asingle-stage, multiproduct batch production plant, inwhich a certain number of production orders are already

* To whom correspondence should be addressed. E-mail:kehui@ust.hk.

Table 1. BL(j,u), Batch Length of Order j at u

u1 u2 u3 u4 u1 u2 u3 u4

j1 1.7 j11 0.5j2 0.9 j12 0.85 0.7j3 1.3 1.1 j13 1.8 1j4 1.7 j14 1.6j5 1.4 0.85 j15 1.48 1.39j6 2.4 1.8 j16 1.3j7 1.05 1.65 j17 0.7 1.05j8 2.1 j18 1.2 1.45j9 1.6 j19 1.85 0.95j10 2.6 1.9 j20 1.2 1.3

Table 2. BS(j,u), Maximum Batch Size of j at u

u1 u2 u3 u4 u1 u2 u3 u4

j1 100 j11 190j2 210 j12 140 150j3 140 170 j13 120 155j4 120 j14 115j5 90 130 j15 130 145j6 280 210 j16 185j7 390 290 j17 110 165j8 120 j18 155 160j9 200 j19 205 170j10 250 270 j20 120 145

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10.1021/ie020802y CCC: $25.00 © 2003 American Chemical SocietyPublished on Web 06/07/2003

scheduled and with some orders that are to be added tothe existing schedule. In the production plant, a fixednumber of production units (U) are available to processall production orders (O). Each order involves a singleproduct, requiring a single processing step, has apredetermined due date, and can only be processed ina subset of the units available. The production capacityof a unit depends on the order processed. The size of anorder may be larger than the size of a batch, requiringseveral batch jobs to satisfy an order. Batch jobs of thesame order are processed consecutively in the same unit.A production unit processes only one batch job at a time.The batch time of an order is fixed and is productionunit dependent. During the transition of productionorders, time is required for setting up the unit for thechangeover. The setup time is job-sequence-dependent.The objective of the scheduling is to minimize the totaltardiness by the assignment of orders to units whilesatisfying all of the above constraints.

3. Mathematical Model

The model used in this study was proposed by Huiand Gupta6 for scheduling single-stage, multiproduct

batch plants and was extended for multistage batchproduction scheduling (Hui et al.7). The model usesthree sets of bi-index decision variables, Xij, Wiu, andSiu, to represent respectively order i transferred to orderj, order i assigned to unit u, and order i being the firstorder of unit u resulting in less integer variables thantraditional formulations using tri-index binary vari-ables, resulting in shorter solution time.

4. An Example

An example of 20 orders is used to illustrate thefeatures of the proposed scheduling methods. Exampledata for batch length, BL(j,u), batch size, BS(j), unitsetup time, C(i,j), order size, JS(j), due date, Dd(j), andorder release time, RT(j), are shown in Tables 1-6,respectively. Unit release times, RTU(u), are all zeros.

In this example, we assume that a production sched-ule for the first 10 orders has already been set. TheGantt chart for this schedule is shown in Figure 1.Additional orders (J11-J20) are then to be added to theexisting schedule, requiring rescheduling of the produc-tion. The simplest way of doing this is to reschedule thewhole problem with all of the 20 orders using the pro-posed model. This provides the maximum freedom forthe scheduling, so that a global optimum schedule mightbe generated. However, because of the increased sizeof the problem, solving it as a single problem requiresenormous computing time. The model cannot generatea feasible solution within 100 000 iterations. To reachan integer solution after 500 000 iteration requires 694CPU s. The Gantt chart for this solution is shown inFigure 2, for which the overall tardiness is 36.61.

Table 3. C(i,j), Units’ Setup Time

j1 j2 j3 j4 j5 j6 j7 j8 j9 j10 j11 j12 j13 j14 j15 j16 j17 j18 j19 j20

j1 0.65 0.85 0.4 0.35 0.65j2 0.3 0.25 0.7 0.25 0.3j3 1 0.15 0.3 1.6 0.2 0.5 0.75 0.9 0.6j4 0.05 0.5 0.7 0.45 0.5 0.75j5 0.3 0.7 0.9 0.6 0.9 0.8 0.7 1.3j6 1.4 3 0.7 1.2 1.2 0.55 0.2 0.35 0.9 0.8j7 1.8 0.85 0.45 1 1.1 0.8 0.5 0.8 0.25j8 1.65 1.05 1.1 0.3j9 2.1 1.25 0.8 0.65 0.85 0.15 1.2j10 1.5 0.6 0.75 0.5 0.7 1.15 1.3 0.95 0.4 1 1.25j11 0.95 0.15 0.15 0.35 0.2j12 0.8 0.4 0.1 0.2 0.6 1.3 1 0.8 0.95j13 0.3 0.55 1.3 1.3 1.55 0.25 1.15 1.4 0.4 0.5 0.25 0.35j14 1.45 0.8 0.5 0.35 0.75 0.55 0.5 0.65j15 0.2 0.4 1.2 0.3 0.8 0.3 1.05 0.6 0.3j16 0.25 1.05 0.85 0.2 0.15j17 0.8 0.3 0.9 1.1 0.5 0.75 0.45 0.2 0.15 0.3j18 0.4 0.5 0.45 0.35 0.6 0.65 0.55 0.3 0.6 0.45j19 0.7 0.65 0.85 0.8 0.7 0.9 0.5 1.05 0.75 0.45j20 0.15 0.55 0.45 0.4 0.4 0.4

Table 4. Js(j), Order Size

order size order size order size order size

j1 550 j6 1050 j11 1000 j16 1050j2 850 j7 950 j12 850 j17 950j3 700 j8 850 j13 700 j18 850j4 900 j9 450 j14 900 j19 450j5 500 j10 650 j15 500 j20 650

Table 5. Dd(j), Due Date

orderduedate order

duedate order

duedate order

duedate

j1 10 j6 30 j11 30 j16 30j2 22 j7 37 j12 40 j17 45j3 25 j8 23 j13 30 j18 23j4 30 j9 30 j14 35 j19 40j5 38 J10 30 j15 47 j20 30

Table 6. RT(j), Order Release Time

orderrelease

time orderrelease

time orderrelease

time orderrelease

time

j1 0 j6 2 j11 3 j16 2j2 5 j7 3 j12 5 j17 3j3 0 j8 0 j13 0 j18 5j4 6 j9 2 j14 6 j19 2j5 0 j10 6 j15 5 j20 6

Figure 1. Gantt chart: initial schedule with 10 orders.

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5. Applying Stepwise Heuristics

Instead of scheduling the whole problem with 20orders, new orders are added sequentially into theexisting scheduling with the following heuristics:

Heuristic 1. When a new order is added into theexisting schedule and the order is only allowed to beprocessed in a subset of process units, the currentschedule on the other units (units that cannot processorder i) will be fixed and will not be involved in thisscheduling step. This can be simply done by removingthe orders and units that are not involved in thisscheduling step from the sets of processing orders (I)and feasible units (U). The model presented7 remainsexactly the same. For instance, when order 11 isinserted into the existing schedule, only unit 3 and theorders that are scheduled on this unit are included forthe rescheduling.

The models are formulated in GAMS (Brooke et al.8)and solved by OSL (IBM, 1991) on a 400 MHz PentiumPC. For reducing the solution time, the relative opti-mality criterion (OPTCR) is set to be 0.1, which allowsthe solver to be terminated at an integer solution within10% of the best possible integer solutions.

The resulting schedule of applying heuristic 1 isshown in Figure 3. Computing times and the resultingtardiness at all intermediate steps are shown in Table7. The procedure started with a solution of the first 10orders and step-by-step insertion of orders 11-20 intothe schedule. By application of heuristic 1, the overalltardiness of the 20-orders problem is reduced to 3.65(from 36.61). The solution time is shortened to 51.42CPU s (from 694 CPU s).

Heuristic 2. In this heuristic, units’ online times (theoverall processing time of a unit) are compared when anew order is added into the current schedule. The unit

that has the lowest online time and is allowed to processthe new order becomes the only feasible unit for process-ing the new order. By application of this heuristic, thefreedom of the scheduling is highly restricted and,therefore, occasionally results in infeasible solutions.

Figure 2. Gantt chart: rescheduled with all 20 orders simulta-neously.

Figure 3. Gantt chart: rescheduling using heuristic 1.

Table 7. Solutions: Heuristic 1

new job iteration solution time (s) total tardiness

J1-J10 436 0.89 0.2J11 22 0.16 0.2J12 201 0.39 0.2J13 94 0.33 0.2J14 14 0.22 0.2J15 148 0.38 0.2J16 72 0.33 0.2J17 433 0.93 0.2J18 6173 7.75 3.65J19 3733 5.49 3.65J20 28023 34.55 3.65total 9716 51.42 3.65

Figure 4. Solution procedure.

Figure 5. Gantt chart: applying the solution procedure.

Table 8. Solutions: Heuristic 2

new job iteration solution time (s) total tardiness

J1-J10 436 0.89 0.2J11 22 0.17 0.2J12 31 0.18 0.2J13 79 0.22 0.2J14 14 0.18 0.2J15 1 0.15 0.2J16 377 0.55 0.2J17 23 0.21 0.2J18 364 0.67 3.8J19 26 0.23 3.8J20 infeasible 0.11J20 8343 0.95 3.8total 9716 23.06 3.8

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To guarantee a feasible solution, a simple solutionprocedure is applied that is shown in Figure 4. Thesolution procedure first inserts a new order into thecurrent schedule applying heuristic 2. If infeasible so-lution occurs, heuristic 1 will be applied. In caseheuristic 1 still cannot find a feasible solution, a full-scale problem will be solved.

In the example, the procedure successfully foundfeasible solutions with very little computing effort whenapplying heuristic 2 to orders 11-19. The solver failedto generate a feasible solution when order 20 wasinserted. Therefore, a second solution is required usingheuristic 1. This step generated a feasible solution withan overall tardiness of 3.8. The overall CPU time isfurther reduced to 23.06. The results are shown inFigure 5 and Table 8.

6. Conclusions

A stepwise scheduling approach was proposed forscheduling a single-stage, multiproduct batch produc-tion by combining a continuous-time mixed-integermodel and simple heuristics. Instead of solving a full-scale problem, production orders are added into theproduction schedule one by one. The heuristic imposednew restrictions to the scheduling problem, shorteningsignificantly the solution time. To avoid infeasibilitycaused by the additional restrictions, a scheduling

procedure is proposed that relaxes the restrictionsgradually, until a feasible solution is found. The pro-posed approach enhanced the flexibility of handlingproblems with different characteristics and sizes. Incomparison with the full-scale scheduling approach, thesolution time and quality are both significantly im-proved using the stepwise approach. The comparison issummarized in Table 9.

Literature Cited(1) Reklaitis, G. V. Overview of planning and scheduling

technologies. Latin Am. Appl. Res. 2000, 30 (4), 285-293.(2) Satake, T. K.; Morikawa, N.; Nakamura, Simulated anneal-

ing approach for minimizing makespan of general job-shop. Int.J. Prod. Econ. 1999, 60-61, 515-522.

(3) Kanakamedala, K. B.; Reklaitis, G. V.; Venkatasubrama-nian, V. Reactive schedule modification in multipurpose batchchemical plants. Ind. Eng. Chem. Res. 1994, 33, 77-90.

(4) Huercio, A.; Espuna, L.; Puigjaner, Incorporating onlinescheduling strategies in integrated batch-production control. Com-put. Chem. Eng. 1995, 19, S609-S614.

(5) Ko, D.; Moon, I. Development of a rescheduling system forthe opimal operation of pipeless plants. Comput. Chem. Eng. 1999,23, S523-S526.

(6) Hui, C. W.; Gupta, A. A Bi-Index continuous time MILPmodel for Short-term Scheduling of Single-stage Multi-productBatch Plants with Parallel Line. Ind. Eng. Chem. Res. 2001, 40(25), P5960-5967.

(7) Hui, C.-W.; Gupta, A.; Meulen, H. An Novel MILP formula-tion for ShortOLINIT-Term Scheduling of Multistage Multi-Products Batch Plants. Comput. Chem. Eng. 2000, 24, 1611-1617.

(8) Brooke, A.; Kendrick, D.; Meeraus, A. GAMSsA User’sGuide (Release 2.25); The Scientific Press: San Francisco, CA,1992.

Received for review October 11, 2002Revised manuscript received May 19, 2003

Accepted May 23, 2003

IE020802Y

Table 9. Performance Comparison

solutiontime (s)

totaltardiness

full-scale rescheduling 694 36.61stepwise approach with heuristic 1 51.42 3.65stepwise approach with the proposed

solution procedure23.06 3.8

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