atul rolling heuristic

8/14/2019 Atul Rolling Heuristic

1/31

This article was downloaded by:[University of Waterloo]On: 10 October 2007Access Details: [subscription number 769429802]Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Engineering OptimizationPublication details, including instructions for authors and subscription information:http://www.informaworld.com/smpp/title~content=t713641621

A ROLLING HORIZON HEURISTIC FOR REACTIVESCHEDULING OF BATCH PROCESS OPERATIONSAli Elkamel a; Atul Mohindra ba Department of Chemical Engineering, University of Kuwait, Safat, Kuwaitb The Foxboro Company, Foxboro, MA, USA

Online Publication Date: 01 August 1999To cite this Article: Elkamel, Ali and Mohindra, Atul (1999) 'A ROLLING HORIZONHEURISTIC FOR REACTIVE SCHEDULING OF BATCH PROCESSOPERATIONS', Engineering Optimization, 31:6, 763 - 792

To link to this article: DOI: 10.1080/03052159908941396URL: http://dx.doi.org/10.1080/03052159908941396

PLEASE SCROLL DOWN FOR ARTICLE

Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf

This article maybe used for research, teaching and private study purposes. Any substantial or systematic reproduction,re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expresslyforbidden.

The publisher does not give any warranty express or implied or make any representation that the contents will becomplete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should beindependently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings,demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with orarising out of the use of this material.
http://www.informaworld.com/smpp/title~content=t713641621http://dx.doi.org/10.1080/03052159908941396http://www.informaworld.com/terms-and-conditions-of-access.pdfhttp://www.informaworld.com/terms-and-conditions-of-access.pdfhttp://dx.doi.org/10.1080/03052159908941396http://www.informaworld.com/smpp/title~content=t713641621


2/31

Eng. Opt.. 1999. Vol 31. pp. 763-792 0 1999 OPA ( O m w a r Publishers Association) N.V.Reprints available dir ml y from !he publisher Publirhcd by liccnw undcrPhotocopying pcrm ill ~d y licrnw only the Gordon and Breach Scicncc

Publishers impri nt.Printed in Malaysia.

A ROLLING H ORIZON HEURISTICFOR REACTIVE SCHED ULINGOF BATCH PROCESS OPER ATIONSALI ELKAMEL" a n d A T U L M O H I N D R A

" Departmenr of Chemical Engineering. University of Kuwait,P. 0 . Box 5969. 13060 Safat, Kuwait;The Foxboro Com pany, 33 Commercial Sir eel, C4 1-2G .Foxboro, MA 02035, USA(Received 24 February 1998)

Batch chemical plants a re dynamic processing facilities where static produc tionschedules can rarely be adhered t o due to market and operating uncertainties. O n-lineschedule modification of a prior; timing assignments and resource allocations in responseto unantipicated disruptions is done through a decomposition heuristic that uses arolling horizon implementation policy. An attempt is made to minimize the impact of thedisruptions on the original schedule near the point o f each deviation while exploiting th ecombinatorial flexibility of task and resource reassignments in future scheduling timewindows. The problem is addressed as a multiobjective optimization problem involvingcompletion time criteria, relative custom er importance, and production cost considera-tions.A rigorous analysis of problem sensitive parameters, including penalty weights andsubhorizon length, is conducted. A model plant case study is performed. Variations onstorage availability and task flexibility are investigated in an attempt to characterizedom ina nt effects of the weighting paramete rs. Results indicate that user preference canserve as a strong guide for obtaining near optimal reactive scheduling solutions. It isshow n that the combin atories can be controlled and that costly and inefficient full scalerescheduling of multipurpose production facilities can be avoided.Keywords: Chemical plant; scheduling; batch processing

INTRODUCTIONBatch production on a large scale has long been of interest to thechemical process industry. Since most products are usually developedin batches a t the bench level, direct scale-up can prove to be an


3/31

764 A. ELKAMEL AN D A. MOHINDRA

enticing option. This is especially true for pharmaceuticals and high-value-added speciality chemicals that provide high rates of return perunit sold.

The most commonly found facil i t ies for batch manufacturing aremultiproduct an d m ultipurpose plants. Mu ltiproduct facil it ies are cap-able of producing a number and variety of products in noninteractingproduction lines. Processing units and storage vessels are typicallydedicated to the production of a single product or several closelyrelated products, only one being produced durin g a given campaign o na given l ine. Multipurpose plants are also capable of producing anumber and variety of products but with the additional flexibility ofhaving nondedicated processing and storage units. Production linesinteract through the use of vessels to process and store different .materials, but cleaning cost considerations are significant.

Plant operations are governed by production planning andscheduling. Planning is a long term activity that is usually combinedwith c orpor ate goals and eco nomic forecasting. Scheduling is more ofa short-term activity in that it dictates event times and sequences onthe plant floor for day to day op eratio ns. Reactive scheduling involvesthe short term adjustment of the master schedule in response tounanticipated deviations in plant operating parameters. Typically, thescheduling of process operations has been viewed as a priori timeassignment of tasks to units and materials to storag e vessels subject tothe assunlption of deterministic plant parameters. However, in a realmanufacturing facility, there is a limit to the certainty of variousparameters such as market demand, processing t ime, and resourceavailability. This uncertainty, which is characteristic of dynamicproduction environments, is the motivating force for developing arigorous framework for reactive scheduling. Full scale rescheduling ofan entire plant a t the point of every disturbance would be disruptive,costly, and highly inefficient in terms of preserving sm ooth opera tion.

Reactive scheduling represents an extension of the general schedul-ing problem. In addit ion to making decisions about the allocationsof resources in the t ime an d space domains, addit iona l constraints asto which allocations are preferable are imposed. A complex set ofsecondary decisions is introduced into an already involved problem.Th e heirarchy of decisions that are involved in scheduling and reactive


4/31

REACTIVE BATCH SCHEDULING 765

scheduling are i l lustrated in Figure I . At the highest level, anassessment of what to produce and how to produce i t is made. Theseissues are settled before master schedule generation. Decisions mustthen be m ade abo ut the num ber and types of orders to cons ider . Theseare based on condi t ions such as market demands , p lan t capabi l i -t ies and capaci ty, and raw material avai labi l i ty. Once this is done,decisions must be ma de abo ut when produ cts can be completed. Next ,the p riori t izat ion of different orders is considered. It m ust be decided

M&r ScluduUw S & v R e d w S c b & & # S I q r Over@ Re&"FIGURE I Decision levels in reactive scheduling.


5/31

766 A. ELKAMEL AN D A . MOHINDRA

which customers should be given priority when setting due dates inthe master schedule. The allocation of resources available in the plantmust then be decided. Finally, in the purely reactive stages ofscheduling, decisions involving unit replacements and time shifting oftasks that could not be executed at predesignated times must bemade. Simultaneously, decisions about resource reallocation andpossible due date modification need to be considered. The overlapregion in the scheduling an d reactive scheduling framework, indicatedby the shading in Figure 1, is clearly of significance. Decisions madea t the higher scheduling level ab ou t due-dates, priorities, an d resourceutilization have an impact o n future decis ions that must be m ade in areactive context. Thro ugh the ap prop riate modelling of uncertaintiesin meeting due-dates, resource availabilities, and customer prefer-ences at the master scheduling level, it is possible that a reasonableam oun t of flexibility could b e left in the schedule so as to reduce thedisruptive nature of making these difficult choices during reactivescheduling.

The scheduling problem of process operations has been addressedby num erou s researchers over th e years (see for instance Refs. [ I -81).A robust and comprehensive approach has been the State Task Net-work (S TN ) developed by Kondili et a l. [9] . The major drawback ofthis approach is in the uniform discritization of time which lead tomodels with a large number of discrete variables. Elkamel et al . [ lo]suggested a decomposition strategy to remedy this situation. Later,Elkamel and Al-Enezi [ I I] proposed valid inequalities to the KondiliE I al . model [9] that strengthen the relaxation solutions to the modeland reduce the compu tational effort.

Even though reactive scheduling has been the subject of muchresearch by the operations research community [12- 141, only littlework has been reported in the chemical process industry. Cott andMacchietto [I51 proposed a n Earliest Finishing Unit (E FU ) heuris ticas part of an integrated framework for process monitoring, diagnosis ,and control. EFU operates by making use of a number of rules thatenable it to shift task starting times on affected processing units so asto complete production as quickly as possible without reassigningbatches to other processing units . The authors demonstrate theapplicability of the heuristic to some small and medium sizedproblems, and conclude its general validity. No measure of optimality


6/31

REACTIVE BATCH SCHEDUL ING 767

was taken by this approach and the impact of change on the oldschedule was n ot considered.

Kanakamedala et a/ . [16] present an approach that goes beyondEFU in terms of finding good feasible solutions to reactive schedulingproblems. They present a Least Impact (LI) heurist ic that at tempts t ominimize the number of deviations from the or iginal schedule. Thisheuristic is based on a beam search and gave a significant improve-ment over the EFU heuristic for a relatively large plant structure. Thisheurist ic , however , is not readily adaptable to handle other plantstructures, and the authors call for fur ther work in mathematicalprogramming that may result in a more r igorous reactive schedulingf ramework.

The heuristic techniques described previously for solving thereactive process scheduling problem tend to be either too broad-based, such as that developed by Fox er al. [13], resulting in poorsolution quali ty, or too myopic, such as that developed byKanakamedala et a/. [16], resulting in a good, but problem specificsolution. In this paper , a mathematical programming based heurist icfor reactive scheduling is illustrated. This heuristic focuses on thegeneration of quali ty solutions for a broad range of problems. I tcombines heuristic rules with a mathematical scheduling techniqueknown to guarantee optimal solutions through an exact algorithm,namely, branch and bound. At the t ime of a disturbance, or set ofdisturbances, to the or iginal schedule, the remainder of thescheduling horizon is divided up into subhorizons of nonuniformlength. Each scheduling subhorizon is represented by an M l L Pformulation of the sequencing constraints and order due dates thatgovern that t ime interval with an objective of keeping productioncosts at a minimum.

Th e disturbances are absorbed into a schedule by being sequentiallyforward shifted in time until they can be fully accommodated. Thisleads to a multiobjective trade-off cost problem: preserve the scheduleas much as possible, subject to cost considerations, yet also allowenough changes so that due dates are not missed and the associatedpenalty costs are not incurred. This rolling horizon strategy makes i tpossible t o minimize the cost of changes. This is do ne by applying alinear penalty function to the objective function of the MlLPsubp roblem s near the disturbances. When weighted properly, the


7/31

768 A . E L K A M EL A N D A . M O H l N D R A

penalty function can be used to generate a least impact schedulefor the subhorizon near any given disturbance. Further , in mor e dis-tant future time periods, where schedules are not as rigidly fixed as inthe present, the subsequent M IL P subproblems have diminished pen-alty terms allowing tasks to be represented into a minimum costconfiguration.

Th e next section describes the reactive scheduling model in the formof a mixed integer linear programming formulation and details thepcnalty function that is applied to the objective function in order tocontrol reassignments. The following sections describe the rules formaking subhorizon cuts and the selection cr i ter ia for the penaltyterms. Later, an example case study is considered and the rollinghorizon heuristic is illustrated. A sensitivity analysis on the variouspenalty coefficients is carried out. Finally, conclusions about theperformance of the heuristic are given along with future improvementconsiderations.

M A T H E M A T I C A L M O D E L F O R R E A CT IV ES C H E D U L I N G R E P R E S E N T A T I O NTh e development of the reactive scheduling model is based o n the statetask network (STN) methodology of Kondili er ul . [9] as revised andamplified by Elkamel e / al. [17]. This form ulatio n, in its general fo rm,can be used to schedule the general chemical plant since constraintsinvolving continuous units are included in addit ion to those governingbatch ope rations. T he key assu mpt ion in this model is the uniform dis-critization of time. The model defines the following variables withrespect to the type of unit within the processing facility.

Batch Units

I if task ista r ts on batch un it ja t the beginning of t ime period I ;w, = 0 otherwiseBiil = Am ount of material that task i star ts processing in batch unit j

at the star t of t ime period t.


8/31

R E A C T IV E B A T C H S C H E D U L l N G 769

These variables serve the purpose of indicating when a batch recipetask is assigned, to which un it it is assigned, and how m uch mate rial isto be processed during the assignment.Continuous Units

1 if task i is being processed on continuous unit jyg~= { a t the star t of t ime period t;

0 otherwiseQg,= Amount of material finished being processed by task i in

continuous unit j at the end of t ime period I .These variables are the continuous unit analogs of the ab ove variablesfor batch processing units. Q,, denotes material tha t ha s completed aprocessing step since continuous processing is rate dependent.Storage Vessels

1 if storage unit j is used to sto re sta te s during t ime interval r ;z,3j,= 0 otherwiseF,,, = Amo unt o f s ta te s hat is being stored in storage vessel j during

time period I .In the non-reactive model, constraints were writ ten for the

allocation of units to tasks and for the nonpreemption of tasks.Various types of units were considered including purely batch units,continuous units, purely storage units, and batch units with allowedstorage. The main constraints in the model are the overall materialbalance constraints. These constraints included terms for the pur-chasing of raw materials and the delivery of finished goods. Variousother constraints were also employed in the model. The capacitylimitation constraints s tate that the batch sizes and the processing ratesmay range between some minimum production level and some upperphysical bound. The resource constraints employed two types of re-sources: renewable and non-renewable resources. Th e equipme ntcleaning constraints were designed to handle the cleaning of equipmen titems.


9/31

770 A. ELKAMEL A N D A . MOHlNDRA

Th e aim of solving the non-reactive scheduling problem was takento be to minimize the cost of production. T he major costs consideredwere the costs of feedstock, inventory costs, deadline violation costs,and machine operating costs. Additionally, there are rewards (profits)for satisfying orders. There are additional goals which arise in thereactive scheduling problem, and these involve penalty functions forattaining a least-impact schedule that still meets as many order duedates as possible. Flexibility in selecting the weights also enables theintroduction of o ther objectives such as o rder prioritization and pre-ferential task-unit-time reassignments. The reminder of this sectiondefines the compon ents of the linear penalty function that are added tothe objective function of the non-reactive scheduling model. In addi-tion, the modifications to the constraint set for establishing controlover lot size changes are defined.

In the approach of Kanakamedala er 01 . [I61 least impact reactivescheduling is merely defined as an attempt to minimize the numberof time shifts and unit replacem ents in the reactive solution from w hatwas originally scheduled. Along these lines, a simple linear penaltyfunction can be devised which only considers the least impact schedulemodification.objective. For batch units, the function takes the form:

and can be added to the objective function of the non-reactivescheduling model. Penalty terms for other units can also be written ina similar fashion. The goal of this function is simply to attempt topreserve all the batch task-unit-time assignments in a given schedule.By assignment is meant the configuration specified in the originalschedule. Criteria for selecting the penalty weight, au,can be as simpleas

< 0 if Wi,,= I in the original schedule. = { > 0 if Wv l= 0 in the original schedule (2 )

Clearly the effect of these penalty te rms would be t o force the solver tokeep assignments as much as possible the same as originally planned.However, such a simple function would oversimplify the reactive


10/31

REACTIVE BATCH SCHEDULING 77 1

schedule modification problem. The lumping of penalties for makingvarious potential readjustments into on e term, a*, inherently assumesthat all schedule changes are of equal impact. In practice however, itmay, for exam ple, be far less costly to m ake a time shifting adjustm entthan a unit replacement. Another shortcoming of this function is thefact that, although positive values of aij,could represent real costs ofschedule modifications, negative values do not have any physicalsignificance. There is no cash payback in reality for not making aschedule modification. Thus, selection of negative coefficients wouldbe purely heuristic and could easily lead to solutions without anymeasure of the trade-offs associated with keeping that task-unit-timeassignment as opposed to making a n alternate assignment.In order to introduce full flexibility in reactive scheduling, a measureof the trade-off costs associated with choos ing penalty weights to pre-serve schedules should be determined according to an appropriateweighting of the following objectives1. Simple time shifting of a task-unit assignment only.2. Unit replacement for a task-time assignment only.3. Batch size preservation for a particular task-unit-time assignment.4. Resource purch ase modifications and reallocations over the schedul-

ing subhorizon.

Assignment PenaltiesThese are the penalties on binary task-unit-time allocation variables.A high pe nalty co st on a parti cula r variable will have th e effect of drivingit to zero. Physically, this is the equivalent of saying "Avoid makin g thistask-unit-tim e assignment because a high cos t will be incurred".

(i) Task Time Shifting Pcnaltj~Batch Units


11/31

772 A. ELKAMEL AND A . MOHINDRA

Continuous Units

wheret,, = Start time of event disruptive to the original schedule./I1 Subhorizo n length of desired penalized interval.(,: = Weighted penalty cost of s tarting task i o n batch un it ja t t ime t .,JI;Weighted penalty cost of s tarting task i o n continuous unit j a t

time I.The constants oT and y: equal zero for the original assignmenttimes. In oth er words (17= 0 a t those times when Wv,= 1, an d yT = 0when Yii ,= I in the original formulation solution. F or those schedulesthat are not generated using the same scheduling model, the originalassignment times are assigned form ulation variable values correspond-ing to a model solution. This assignment procedure is independent ofthe algorithm o r heuristic rules used t o generate th e original schedule.

(ii) Processing Unit Replacement Penalty

Batch Units

Continuous Units

whcreB, = Set of allowed replacement units for each b atch task i at each

time I.C = Set of allowed replacement units for each continuous task i ateach time t .


12/31


a" = Weighted penalty cost of performing task i on batch unit j atI time t .$ = Weighted penalty cost of performing task i o n continuous unit jat t ime 1 .

As is the case with a: an d "r,a,? = 0 and $ = 0 for the originaltask unit time assignments. Clearly, a: a nd ay, as well as the con-tinuous unit variations, are independent parameters that enable mo recontro l over task-unit-time reassignment. Often, in industrial settings,the cost associated with finding a replacement unit is much higher thanthat of maintaining modest holding times for displaced materials. Thesegregation of the associated penalty costs is meant to reflect this fact.

(i ii) Storage Time Shifr PenaltyStorage vessels are subject to the same types of penalty terms asprocessing units. Both time shifting and unit replacement considera-tions can be imposed by similarly weighted terms.

wherezT = Weighted penalty cost of storing state s in vessel j at t ime 1.I

(iv) Storage Unit Replacement Penalty

whereS V , = Set of allowed storage vessels for each sta te s at each t ime t .

7" = Weighted penalty cost of storing state s in unit j a t t ime 1.-I


13/31

774 A . E L K A M E L AND A. MOHINDRA

Batch Size Preservation PenaltiesIn order to consider the issue of batch size preservation in the newschedule, introdu ction o f a pe nalty term is slightly. mor e involved.Previously, representation of a linear penalty for attemp ting t o pre-serve batch size was not clarified. The standard approach would be tominimize

whereB z w = Continuous variable representing batch size in the newschedule.B$ * = Continuou s value of theori ginal batch size in the unperturbed

schedule.Equation (9) is nonlinear and would force the problem to becom e a

M IN LP , which is usually mo re difficult to solve. In order to avoid thenonlinearity, a construction is applied for linearizing absolute valuesin the objective functions of optimization problems , as detailed byNemhauser and Wolsey [IS] . Tw o new continuous variables ar e addedto the objective function and one addit ional constraint is includedin the problem specification. Specifically, to overcome the difficultyof preserving batch size, in addition to task-unit assignments, thefollowing addit ions to the previously described penalty function ca n bemade

wherecry = Weighted penalty coefficient representing th e cos t of increasingV f the size of a specific batch from its a priori assigned size.a;: = Weighted penalty coefficient representing the cost o f decreasing

the size of a specific batch f rom its a priori assigned size.Positive continuous variable representing positive deviationfrom original batch size.


14/31

REACTIVE BATCH SCHEDULING 775

itij1= Positive continuous variable representing negative deviationfrom original batch size.

In addition, the following constraints are imposed:

Resource Reallocation and Inventory ControlThe parameters introduced here are meant to allow more control overscheduling conditions than merely penalizing assignment changes.Though resource profiles are often thought of as secondary toassignments, they govern the feasible range of scheduling solutions.Proper adjustment of resource profiles during reactive schedulingcan play a crucial role in meeting desired reschedu ling goals and satisfy-ing ord er delivery profiles.

(i) Storage LevelsT o the objective function, add

whereST,T= Set of stable states with allowed storage capacity in the plant.zf;' = Weighted penalty coefficient representing the cost of storing an

am ou nt of a specific state greater than a user specified amo unt.2 = Weighted penalty coefficient representing the cost of storing an

am ou nt of a specific state less than a user specified am oun t.p: = Positive cont inuo us variable representing positive deviation from

desired stora ge level.n = Positive continuous variable representing negative deviationfrom desired storage level.


15/31

776 A. ELKAMEL A N D A . MOHINDRA

And to the constraint set adddesiredC C F ~ 7: = F,,rEST, 1

WhereF:$" = Con tinuous variable representing amount of s ta te s stored at

time t in the new schedule.~ d c s i r e d=

51 Con stant specifying preferred sto rage level for state s at t imeI , possibly that of the original schedule but not necessarilySO.

( i i ) Purclrasing o Raw MaterialsThe objective function must contain

whereS T / = Set of states that can be purchased as feedstocks.o$ = Weighted penalty coefficient representing the additional expenseincurred for purchasing an am oun t of state s greater than som e

user specified amount at time r .cr;; = Weighted p enalty coefficient representing the a ddition al expense

incurred for purchasing an amo unt of state s greater than someuser specified amount at time t .

p:, = Posit ivecontinuousvariable representingpositivedeviation fromdesired purchasing level.

n f l = Positive continuous variable representing negative deviationfrom desired purchasing level.

The constraint set must again be expanded t o includeH'

desiredC C p::" +d l = P,,TEST, I


16/31

R E A C T I V E B A T C H S C H E D U L I N G 777

whereP TW = Continuous variable representing amount of state s pur-

chased at t ime t in the new schedule.p desired = p

.TI urchasing level specified fo r st at es at time t (possibly tha t ofthe original schedule but not necessarily).

SUBHORIZON LENGTH DETERMINATION

At the t ime of a disturbance, or set of disturbances the schedulinghorizon is divided up into different subhorizons. The reason fordividing up the schedule is threefold. First, doing so provides a meanof preserving the old schedule as much a s possible near the time of thedisturbance. Since sudden changes can be highly costly and potentiallydisruptive to smooth operations, segregating the costs of making re-assignments at different future times is important. Another reason fordecom position o f the schedule is that most realistic problems based o nactual scheduling data usually involve thousands of binary variablesan d are intractable by present solution method s [lo]. Thirdly, in anyproduction scenario, consideration must be given to custom ers.Deciding which task assignments and resource allocations are mostflexible with respect to reassignment is likely a direct function of rela-t ive customer imp ortance.

Selection of the subhorizon for reactive scheduling is a sensitiveissue. Where the "cuts" are ma de can radically effect the solution ou t-com e because of the sequen tial nature of pr oduc t recipes. Since recipesare connected networks, any form of discretization which attempts topartition th e network risks a loss of inform ation . This loss is attributeddirectly to the fact that when a cut is made, information a bout futureevents are not incorporated into the smaller time window. The rulespresented in this section attem pt to minimize the effects of informationloss between one scheduling subproblem and the next.

Th e first issue to be decided before making a cu t in the time horizonis purely logical. Given a priori knowledge of a scheduling algorithm,such as branch and bound for MIL P optimization, and the computingtools available to execute the algorithm, a decision must be mad e o nthe largest problems of practical size that can be solved. If the


17/31

778 A. ELKAMEL AN D A . MOHINDRA

available algorithm and tools can optimally schedule the entirehorizon, with reasonable computational effort , then there may notbe a need to make a ny cuts and r isk informa tion loss [ lo] .

Once the cut is made, backtrack the f low of material f rom order du edates and see where th at material is supposed to be, according to theoriginal schedule, at the time of the cut [lo]. Set demands for the in-termediates that should be met at the end of the subhorizon and a t thestart of the next time window. If unstable intermediates are present atthe cut, formulation devices must be employed if the cut is to beretained. The final storage level (target inventory) for the unstablematerial , Fq,, , , should be set to the desired value and an artificialdemand for the material should also be employed. The intermediaryinstability can still be denoted by setting c F,, = 0 . Th is i s doneonly to prevent a loss of inform ation th at would otherwise result in theunstable material not being made.

IMPLEMENTATION STRATEGY

A key aspect to the effective implementation of any heuristic isautomation. This is especially important in an iterative scheme such asthat presented by the rolling horizon heuristic. Solution repre sentationis also important since an inadequate display of results will lead to adilution of the value of the generation technique. The framework forimplementing the heuristic in this work appears in Figure 2. It is theaim of the reactive scheduling system to be highly interactive. Theopera tor should, with ease, be able to update information a bou t whatis occurr ing on the plant f loor . The means of doing so, as detailedbelow, is a Scheduling language interface using the language RCSPecdeveloped by Zentner et al. [19]. Followed by the information gathe-ring stage, the decision making framework that will set subhorizonlengths, make penalty assignments, and interact with an M l L P solverto get solution information. The final stage is also interactive in thatgraphical output, depicting the new schedule and inventory profiles, isreturned to the operator .

The RCSPec language for representing process scheduling problemsis a natural language means for representing problems without theneed for knowledge ab ou t algorith ms and heuristics. Details of the


18/31

REACTIVE BATCH SCHEDULING

IMPLEMENTATION

ScbcduliigLanguage%'

FIGURE 2 Implementation strategy.

language and its inception can be found in the original man uscrip t. Itsuser friendliness is the primary reason for incorporating the languageinto the reactive scheduling interface. The main features of thelanguage that m ake its application practical a re the ability to a dd newkeywords, the flexibility and ease of modifying problem parameters,and its suitability to a decomposition strategy. Adding keywords thathave the effect of changing the objective function and constraint setare essential to the M I L P modification scheme of the preceding sec-tions. Th e ease of modifying the scheduling param eters is the key tothe success of a rigorous sensitivity analysis. It facilitates modificationof penalty weights, as well as enabling storage, connectivity, and inter-mediate stability specifications to be altered to force a given plant struc-ture to have dominant features attr ibutable to the problem classes


19/31

780 A . E L K A M E L A N D A . M O H I N D R A

defined. Changing the scheduling horizon and setting demand levelswithin the language context is also trivial. This makes the languageideally suitable to a decomposition strategy since the length of timewindows is easily modifiable. T he reactive scheduling problem can beviewed in two distinct stages: a pre processing stage an d a main tenanc estage. During preprocessing, information ab ou t the original schedule isdetermined followed by collection of informa tion regarding the priori-tization of orders , and calculations to make n priori reassignmentpenalty cost assessments. Dur ing the m aintenan ce stage the schedule isregularly update d acco rding to need.

Once solution generation has been completed, a cost of assessmentis made by checking to see how many penalty costs have beenincurred. This is done by comparing the a priori determined M I L Pvariable values for the original schedule with the resulting variablevalues of the reactive solution. Based on this information, a table isgenerated containing the number of time shifts, number of batch sizechanges, and number of unit replacements.

ILLUSTRATIVE CASE STUD YAN D SENSITIVITY ANALYSISThe purpose of this section is to demonstrate the ability of the rollinghorizon heuristic to solve reactive scheduling problems involving vari-ous, at times simultaneous, conflicts. A model plant s tructu re con-taining features not previously considered in the reactive schedulingliterature serves as the basis of the study.

Test ProblemThe state task network representation for this plant appears in Fig-ure 3, and details of the plant tasks and equip ment s tructure are outlinesin Table I. A large horizon length is chosen in ord er to demo nstra te therolling horizon methodology. Note f rom Tab le I that the processing timeis a function o f the unit on which th e task is performed. Each of the threemixing tasks, three reaction task s, and two d rying tasks are allowed t ohave two dimerent processing times. This feature adds to thecombinatorial complexity of the scheduling problem.


20/31


FIGURE 3 State task network for test problem.

As the S TN shows, there a re 8 processing tasks, 3 cleaning tasks, 3feeds, 2 intermediates, one of which is unstable, and 3 shared secon-dar y resources. Th e second ary resources a re specified to have cyclicalavailability profiles. Fro m the STN it can also be noted that there aretwo products produced jointly in one interacting production line. Therelevant information about resources appears in Table 11, and storagevessel data is presented in Table 111. Table IV lists the dem and profileused in this study. A Ga ntt c hart of the original optimal schedule forthis demand pattern is illustrated in Figure 4.

In order to generate disturbances for test ing, i t was necessary todevise a generator that could randomly select from a large set ofpossibilities. The generator, using a formula that mathematicallycomputes random numbers based on a random init ial seed, wasrequired to make the following determinations: time of disturbance,type of disturbance, time of notification, and, in the case of unitunavailabilities and processing time deviations, length of disturbance.


21/31

782 A . E L K A M E L A N D A . M O H I N D R ATA BL E I Test plant task and equipment informatio n

T ~ s k o~ ue Allowed unit Unit size Processina timeMixingl M I6 1250 2( m l ) MI^ 750 4M I8 1000 4Mixing2 M 16 1250 2(m2) M I7 750 4MI8 1000 4Mixing3 M I6 1250 2(m3) MI^ 750 4M I8 1000 4React l RI 2000 6(R xl) R2 2500 6

R3 1500 4React2 RI 2000 6(Rx2) R2 2500 6R3 1500 4Rencl3 RI 2000 6(Rx3) R2 2500 6R3 1500 4Dryingl D25 750 3( d l ) D26 1000 2Drying2 D25 750 3(d2) D26 1000 2

TA BL E I1 Resource information for test caseResource Type Stability Initial Purchasable Cycl e rime . ValueC N R R Unlimited 10000 N o - 2.5D NR R Unlimited 10000 Yes - 0. 5E N R R Unlimited 10000 N o - 10ABC NR R Unlimited 0 N o - 2X NR R Zero Wai t 0 No - 0Y N RR Unlimited 0 No - 586Z NR R Unlimited 0 No - 92 5Elec RR Unlimited 1000 Yes 15 0.145Stenm RR Unlimited 1000 Yes 15 3.5Wa ter RR Unlimited 1000 Yes 20 0.08N R K - Nonrenewable Resource.R R - Renewable Resource.The possible disturbances considered in this study were unit un-availabilities, processing time deviations, new orders, order cancella-tions, order priority changes, and deadline advances. Table V list therandomly generated disturbances that were introduced to the schedule.


22/31

TABLE 1 1 1 Storage information for test caseVessel State Capac ityM I 6 C 1250D 1250EABCCD 750E 750ABC 750M I 8 C 1000D 1000E 1000ABC 1000D2 5 Y 75 0Z 75 0D26 Y 1000z 1000Vessel C C 10000Vessel D D 10000Vessel E E 10000Warehouse Y Y 10000Warehouse Z Z 10000

TA BL E IV Original demand profile for test caseProducl Due dale Amounr demanded


23/31


F I G U R E 4 Original schedule for test caseTA BLE V Disturbance details

3 3 Orde r Cancellation Third order for Orde r size: 300product Z16 New order 700 of product Y Due dole 1633 Due Date Advance Thir teenth order New due dale 33for product Z29 29 Processing time Drying task on Duration of 5 extradeviation unit D25 time periods29 29 Unit Unavnilabilitv Unit R3 From time 29 to 32

Note tha t dis turbances that d o not have the same notif ication time arenot considered simultaneously.

At t ime 3, the first two disturbances are reported. The reactivescheduling algorithm goes into effect and generates a new schedule toaccommodate the disruptions. The subhorizon length selection yieldeda time window of 23 time periods, which was set to be the flexibleregion, subject to reassignment penalty costs. The remainder of theschedule was solved without penalty costs in a time window of length22 . The reactive scheduling solution then becomes the operatingschedule for all times after time period 3. At time period 9, the nextdistu rbanc e is noticed a nd the heuristic is used again with the solution


24/31


from the first disturbances as its basis. The reactive scheduling for thisinstance was done with two time windows of length 23 , and 17, re-spectively. Once again, the solution that is found becomes the plantschedule and a final iteration of the procedure is simulated for asimultaneous conflict of processing time deviation and unit unavail-ability a t time 29, which was solvable in one time window of length 22 .

Figures 5 through 7 show the scheduling solutions obtained by theheuristic procedure. Storage vessels are not represented in the pre-ceding Gantt charts. Storage in processing vessels is illustrated byshaded regions with the name of the stored state. Plain shaded regionsrepresent cleaning tasks. The ta gs on the processing tasks are the nam eof the task that takes place and the corresponding batch size.

Note from Figure 5 that the new order disturbance is essentiallyaccom mod ated by executing some additional processing tasks after theend of the original horizon. This can be attributed to the fact that themodel plant was already operating at bottleneck capacity and couldnot acconlmodate a new order by increasing some intermediary lotsizes. The second disturbance essentially introduces an urgency con-dition on the production of a particular order. This immediately hasthe effect of causing th e solver t o execute processing tasks as quickly aspossible to meet the new deadline. Note the increased use of storage

FIGURE 5 Solution after disturbance I .


25/31

786 A . E L K A ME L A N D A . M O H l N D R A

for intermediate prod uct ABC in Figure 6 . This is an indication th atmixing tasks are being don e faster than the bottlenecked reactors canaccommodate .

Figure 7 dem onstra tes the resolution of tw o simu ltane ous conflicts.Reactor R3 becomes unavailable from time 29 to 32 and the pre-empted task is subsequently restarted. The dar k shading indicates theunavailability. Sim ultaneously, the half shaded box on u nit D25's time-line indicates a processing time deviation. Observe that since thesechanges happened so close to the end of the preplanned horizon, rela-tively larger lot sizes and timing adjustments were made in order tominimize late deliveries. Tables VI through Vl l I list the schedulechanges tha t occurred a t each reactive scheduling point. In these tablesa reassignment is defined a s a task-unit-time assignm ent in the newschedule. Similarly, a new assignment is taken to be on e that ca nno t bematched to any assignment in the original schedule. The net penaltycost is the sum cos t of schedule modifications plus the cost of du e dat aviolations. U nit replac ements did not take place in these cases becauseof unit replacement penalties th at were larger than time shift penalties.This serves as an indica tor o f the types o f trade-off issues that arise inreactive scheduling, and of the sensitivity of the penalty function modelto parameter selection.


26/31

REACTIVE BATCH SCHEDUL ING

F I G U R E 7 Solution after disturbance 3TA BL E VI Schedule modifications made after the first disturbance set (Net PenaltyCost = 2 .9899e+06)Task Unir Start rime Amouni Tim e shift Barcll size chanpe

ReassignedReassignedReassignedNew taskNew taskNew taskNew taskNew taskNew task


27/31

TABLE Vll Schedule modifications m ade after the second disturbance set (NetPenalty Cost = 3.10459e+09)Tnsk Unir Slnrt r i m Anrounr Tim e shilr Barc l~ ize chaneeR xl 'Rx2'm l 'm 3Rx3'd l 'm l '1112d ld2Rx lRx 3m2d lRx 3d2m3 'm3Rx 2d ld lrnlRx ld 2d 2d 2

-400ReassignedReassignedReassignedReassignedReassignedNew taskNew taskNew taskNew taskNew task

TABLE Vlll Schedule modifications made after the third disturbance set (Ne t PenaltyCost = 2.5133e+09)T d Unir Srart rin ~c Amount Tim e slrfi Barch size cl~nn ge

-d l D25 36 200 2dl ' D25 39 750 8d2' D26 31 900 - 45 0Rx I' R 1 33 1875 I 87 5Rx3' R3 32 1000 5 500Rx2' R2 33 1875 - -625m3' M I8 34 1000 I 625m2' MI7 38 750 2Rx3' R3 38 I500 6Rx l' RI 41 2000 2 125Rx3 R3 43 750 6 -750d 2 D 2 6 36 300 - Re;~ssignedd2 D26 39 500 - Reassignedd2 D26 4 1 300 - Reassignedd2 D26 45 375 - Reassigned111 M I6 31 1250 - New taskd l D25 42 300 - New taskd l D 2 5 47 550 - New task


28/31

R E A C T IV E B A T CH S C H E D U L I N G

Sensitivity StudyThe purpose of this section is to discuss the effect of various modelparameters on the solut ions obtained by the suggested rol ling horizontechnique. The parameters that requi re t es t ing are the subhor izonlength and the pe nal ty param eters . In order to ch eck the sensi tivity ofsolut ions to the select ion of each sub horizo n cut , tests were performedfor the first disturbance set with the minimu m c ut taken to be the oneused t o obtain the actual solut ion. Several runs were done, each t imeincreasing the subh orizon length of the fi rs t subpro blem . It was fou ndthat the length of the t ime window selected affects the outcome of thereact ive schedule modificat ion. Th e main t rend is that as the sub-horizon length increases, the numb er and magnitude o f task-t ime-unitshifts decreases. I t is also of no te tha t com plete task-unit-t ime reassign-men ts d o not take place as often in the penalized region. Rathe r, thesolver opts to make larger modificat ions in the future t ime windowswith the lowest penalty costs.

The sensi t ivi ty with respect to penal ty parameters is checked bydefiningschedulingproblem classes in orde r to distinguish those sche dul-ing problems that will have certain "dominat ing" features that shouldmake them insensi t ive to certain penal ty parameter variat ions duringreactive scheduling. These are: the storage, the batch size, the resourcecons t ra ined the ass ignment cons t ra ined , and due-date dominatedclasses. The storag e dom inated class represents schedules where storagecapaci ty is avai lable and nearly al l materials in the p lant a re s table overnearly the ent ire scheduling horizon. The batch size dominated classrefers to right schedules where ba tch sizes are close to unit o r bottleneckcapaci ties and /or fixed du e to constra ints on the m erging and spl i t t ing ofbatches. Th e resource constrained class represents schedules whereaddit ional outside raw material purchasing is not al lowed onceexecution of the plan is underway. The assignment constrained classinvolvesschedulesin which unit replacementsare not al lowed and /or areextremely cost ly. Final ly, the due-date d omin ated class corresponds toscheduling problems where the costs of missing order deadlines areextremely high comp ared to oth er costs in the object ive funct ion.

Th e port ion of the schedule after the third disturbance w as selectedfor sensitivity analysis. This portion is different from other points inthe schedule because it lies at the edge of the flexible region and the


29/31


beginning of the free region. The scheduling subproblem at this pointwas solved repeatedly while varying each of the penalty parametersindependently. Plant parameters were also modified and the solutionruns repeated for cases of each problem class defined earlier. Th e mainresults are summarized below. For each problem class, solutions werefound first with mixed penalties. Then each individual parameter wastested at a high value and a low value corresponding to the prioritybounds on the given parameter.

From the numerical trials for the assignment constrained class, nosensitivity was exhibited to the penalty terms for batch size preser-vation. Storage control parameters also had no impact o n the problemclass. The only param eter tha t exhibited noticeable sensitivity was a:.Increasing the unit replacement penalties only had the effect ofincreasing th e objective cost.

Results for the batch size dominated class exhibited greatest sensi-tivity to timeshifting penalties. Unitrep lacem ent an d batchsize penaltiesdid not show any im pact upon adjustment. Fixed changeover costs thatwere not part of the penalty terms may have been the reason why unitreplacements did n ot occ ur. Increased storage penalties only resulted inincreased computation time but not in a different solution outcome.

Time shifting, batch size, and storage penalty terms all exhibitedsignificant sensitivity for the due data dominated class. Unit replace-ment penalties again were not a cause of sensitivity. This stronglysensitive class confirms tha t trade-off consid erations are significant inthe objective function. Delivery deadlines represented by objectiveweighted terms have a s trong impact on the amount of reactivescheduling that m ay tak e place.

For the resource constrained class, only unit replacement terms didnot demonstrate a great deal of sensitivity. All other terms had cleareffects. This is of note because it correlates with the hypothesis thatthere is a str ong relation between resource specifications and problemdifficulty. When such a high degree of sensitivity exists, even slightperturbations to the objective function coefficient could lead toexponential increases in problem difficulty.

As was expected, all parameters for the s torage dominated classexhibited a great deal of sensitivity with regard to solution control.Because of the enhanc ed assignment flexibility created by the additio nof more storage space in the plant, the number of feasible assignments


30/31

REA CTI V E BA TCH SCH E D U L I N G 7 91

grows dram atically and the solver is able to search m any reassignmentcombinations.

CONCLUSION

In this paper, a m athem atical pro gramm ing based heuristic which uses arolling horizon planning strategy to solve reactive scheduling problem shas been presented. T he consideratio n of user preference and relativecustomer importance has led to a penalty function approach. Weightson certa in assignment variables are incorporated into the objective func-tion of an M I L P formulation with the aim of both preserving andmodifying the original schedule in a near optimal fashion. The mainshortcom ing of this approach has been the larger and varied numb erand types of user specified, subjective parameters, in addition to anumb er of model induced parameters , such as subhorizon length. Inorder to characterize the sensitivity of the model to these parameters,initial studies of some of the parameters have been do ne on a test case.Visible response to subh orizon length varia tion has been observed an d itappears tha t by loo king at a longer t ime horizon, an improved quality ofreactive modifications can be made , especially when faced with multiple,simultaneous disruptions. Indications from the results are thatparam eter sensitivity in the model is a function o f plant characteristics.DiKerent penalty weights appear to have a stronger impact when theplant has certain dominant characteristics.References

[I] Reklaitis, G. V. (1992). Overview of scheduling and planning of batch processoperations. NATO AS/, Borclr Processing Systems Engineering, Anlalya, Turkey.[2] Zentner, M . G . and Reklaitis, G. V. (1992). An interval-based mathematical modelfor the scheduling of resource-constrained batch chemical processes. NATO AS/,Barck Processing Sysrem s Engineering, Antnlya, Turkey.[3] Zentner, M . G., Pekny, J. F., Raklai tis , G. V. and Gupta . J. N. D. (1994). Practicalconsiderations is using model based optimization for the scheduling and planningof batch/semicontinuous processes. Process C o~~ rro l .(4), 259-280.[4 ] Pekny, J . F. and Miller, D. L. (1991). Exact solution of the no-wait flowshopscheduling problem with a comparison to heuristic methods. Compurers orrdClremicul Engineering, lS(1 I), 741 -748.151 Kudva, G., Elkamel, A., Pekny, J . F. and Reklaitis, G. V. (1994). A heuristicalgorithm for scheduling batch and semi-continuous plants with productiondeadlines, intermediate storage limitations, and equipment changeover costs.C o m pr . C / ~ o ? r .ng., 18(9), 859-875.


31/31

792 A . E L K A M EL A N D A . M O H I N D R A

[6] Musier. R. Fitt a nd E vans, L. B. (1989). An ap proxim ate method ior theproduction scheduling of industrial batch processes with parallel units. Conipt.Chmrl. Big. , 13, 229-238.171 Pekny, J . F., Miller, D. L. and M cRae, G. 5. (1990). An exact parallel algorithm forscheduling when production costs depend on consecutive system states. Compr.C l ~ e r r ~ .rig., 14, 1009- 1023.[8] Wellons, M. C. and R eklaitis, G . V. (1989). Optimal schedulegeneration fo ra single-product production, I-problem formulation. Compr. Chern. Dig ., 13,201 -212.[9] Kondili, E., Pantelides, C. C. and Sargent, R. W. H. (1988). A general algorithmfor scheduling batch operations. Proceedings, Third Inreniutionul Syriipo.~ilrn~r!Proccss Sj~sre~iisrigirieering, Sydney. Australia, p p. 62-75.[lo] Elkamel, A., Zentner, M., Pekny, J. F. and Reklaitis, G. V. (1997). A de-composition heuristic for scheduling the general batch chemical plant, Erig. Opr.,28, 299-330.[I I] Elkmnel. A, and Al-Enezi, G. (1998). Structured valid inequalities and separationin optimal scheduling of the resource-constrained batch chemical plant. Murh.Engrifi I d . . h(4 ), 29 1-3 18.[I21 Nof, S. Y., Rajan, V. N. and Frederick, S. W. (1990) Knowledge-based dynamicreal-time scheduling and rescheduling: A review and some annotated references.Researcli Me~irororrrl~rrr~o . 8 9 - 1 6 , School of Industrial Engineering, PurdueUniversity, West Lahyelte, IN.1131 Fox, M. S. an d Sm ith, S. F. (1984). ISIS-A knowledge-based system for Factoryscheduling. E.vperr Sysrnris, I ( ] ) , 25-49.[I41 Ow, P. S., Smith. S. F . and Thiriez, A. (1988). Reactive plan revision. Proc. Se~wi rliNat7 Cortf. A / , M IT Press, Cambridge, Mass., pp. 77-82.[I51 Cott. B. J . and Macchietto, S. (1989). Strategies for operating batch plants subjectto variability - 21 performance assessment. C li oi ~. rig. Res. Des., 67(6), 593-599.1161 Kanakamedala, K. B., Reklaitis, G. V. and Venkata S ubr am ani an, V. (1994).Reactive scheduling modification in multipurpose batch chemical plants. lr~rlusrrialar~d r~girleerhlgClremistry Research, 33(1), 77-90.[I71 Elkamel, A, , Zentner, M. G., Pekny, J . F. and Reklaitis, G . V. (1992). An enhanceduniform discritimtion model for the batchlsemi-continuous che n~ical plantscheduling problem. ClPAC Report . School of Chemical Engineering, PurdueUniversity, West Lafayette, IN 47907. USA.[I X] Nemhnuser , G . L. and W olsey, L. A. (1988). Integer a r r d C o m h i n a r o r i ~ ~ l O p ~ i n ~ i z a t i o ~ ~ .Wiley, New Yor k.[I91 Zentner. M. G . , Elkamel, A,, Pekny, J . F. and Reklaitis. G . V. (1997). A languagefor describing process scheduling problems. Conip . Chenl. E I I ~ . ,2(1-2), 125- 145.

atul rolling heuristic

Documents