both R1 and R2 instead of just R1 to Reation 3, the output will not be higher, if the supply

fromthe previousreations doesnot reahaertainlevel. Iftwo dierentases

c

^and

c ^′

^has

the same maximal revenue, but

c

^{uses only a (not neessarilyreal) subset of units for eah}

3

Intheoriginal example,10%of theoutputof separationis

IntAB

^{that isreyled. This isnegleted,}

asthepresentedalgorithmisnotapableofaddressingproblemswithloopsintheirreipe.

Case Reation 1 Reation 2 Reation 3 Max revenue

1

R1 R1 R1

^86.00

2

R1 R1 R2

^71.67

3

R1 R1 R1&R2

^86.00

4

R1 R2 R1

^53.75

5

R1 R2 R2

^53.75

6

R1 R2 R1&R2

^53.75

7

R1 R1&R2 R1

^114.67

8

R1 R1&R2 R2

^71.67

9

R1 R1&R2 R1&R2

^139.75

10

R2 R1 R1

^86.00

11

R2 R1 R2

^71.67

12

R2 R1 R1&R2

^86.00

13

R2 R2 R1

^53.75

14

R2 R2 R2

^53.75

15

R2 R2 R1&R2

^53.75

16

R2 R1&R2 R1

^89.58

17

R2 R1&R2 R2

^71.67

18

R2 R1&R2 R1&R2

^89.58

19

R1&R2 R1 R1

^86.00

20

R1&R2 R1 R2

^71.67

21

R1&R2 R1 R1&R2

^86.00

22

R1&R2 R2 R1

^53.75

23

R1&R2 R2 R2

^53.75

24

R1&R2 R2 R1&R2

^53.75

25

R1&R2 R1&R2 R1

^114.67

26

R1&R2 R1&R2 R2

^71.67

27

R1&R2 R1&R2 R1&R2

^139.75

Table 4.1: 27dierent"xed reipes" for the exampleby Kondiliet al.[66℄

reation of those used by

c ^′

^{, than}

c

^dominates

c ^′

^{. As an example, ase 9 dominates ase}

27, and ase 24 is dominated by all of the ases 4,5,6,13,14,15,22,23. Obviously, if a ase

is dominated by an other, it an be exluded from the investigation without using loosing

the guarantee for the global optimality. For the example above, the ases, whih are not

dominated by an other are shown in Table 4.2ordered by the maximal revenue.

Case Reation 1 Reation 2 Reation 3 Max revenue

4

R1 R2 R1

^53.75

Table 4.2: Non-dominatedases for the example by Kondiliet al.[66℄

After this redution, still, 11 dierent ases should be given as input to the S-graph

algorithm. Inordertofurther reduethis number, several asesanbemergedtogether. As

an example ases 4 and 5 are idential exept for the seletionfor the third Reation,thus

these two asesan be mergedtogertherby the assignments

R1

^,

R2

^,

R1 ∨ R2

^forReations

1,2, and 3,respetively. Applying the same idea, the 6 ases shown in Table 4.3remain.

Case Reation1 Reation2 Reation 3 Max revenue

4,5,13,14

R1 ∨ R2 R2 R1 ∨ R2

^53.75

Table 4.3: Merged non-dominated ases for the example by Kondili et al.[66℄

Toallof these mergedases, the reipe graphan begenerated, asshown inFigure4.5.

4.4 Empirial tests

In this setion,the results of the implemented algorithmare presented via three examples.

For eah example, the problem has been saled and solved with dierent time horizons.

At eah ase altogether 18 variants of the throughput maximization algorithm have been

Figure 4.5: The orresponding reipe graphs forthe 6ases in Table 4.3

tested and ompared. These variants add up by the possible ombinations of 3 dierent

subproblem seletionrules, 3 dierent update and 2 dierent feasibility subroutines.

Eah of the subproblem seletionstrategies rst explores how many bathes of a single

produt an be produed within the given time horizon, then they explore the bounded

region

LEX in alexiographial order,

BFS viabreath-rst searh,

DFS via depth-rst searh.

The update subroutines:

E empty, noonguration isremoved via the bound of a feasible solution

F only the best axial onguration is used for removing some of the ongurations

after the initializationphase

U the updatefuntion removesallthe ongurations thathas lowerrevenue thanthe

urrently best feasible solution

Last, themakespanminimization(MM)andthedesribed feasibilitytester(FT)isused

for the evaluation of the ongurations. At eah run, the time limitof 1 hour was used for

the solveralgorithm.

The subsetionspresentonlysomehighlightsoftheresultsduetospaelimitations. The

tableofalloftheresultsanbefoundinSetionC.1. Whenthealgorithmshavenotreahed

the 1 hour time limit,they delivered the same globallyoptimal solution.

A generalexperiene is that the type of the feasibility tester have amajoreet onthe

CPUtimes,asexpeted. Usually,mostoftheCPUtimeistaken forthetestingofinfeasible

ongurations, wherethe FTdoesnotgainanyadvantageofstoppingaftertherst feasible

solution. The initialbound of the FT subroutine however, ause a signiantredution on

the searh spae.

4

Thus, in the following subsetions the variants having MM subroutine

are not investigated.

Forallof the examples,NIS storage poliy were onsidered for allintermediates.

4.4.1 Pharmaeutial ase study

This example istaken froma multinationalpharmaeutial ompany. There are 5 hair and

skinareprodutsprodued, allofthemviatwoonseutivesteps: mixingandpaking. For

the paking,there are 3 idential pakinglines available,and the pakingtime is uniformly

12 hours for all of the produts. There are 4 mixingvessels available, however, they dier

in appliability and proessing times due to the dierent stirrer designs. The detailed

proessing times are given intable 4.4

4

Someadditionaltestswererunusingthemakespanminimizationsubroutinewiththeinitialbound,and

theresultswerelosetothatoftheFTapproah.

Produt Revenue (u)

Proessingtime (h)

Mixingvessels

Paking lines

V1 V2 V3 V4

Cream1 2 10 5 5 12

Cream2 3 12 10 7 12

Conditioner 1 12 12

Shampoo 3.5 8 13 12

Lotion 1.5 10 6 9 12

Table 4.4: Proessingtimes and revenue data for the pharmaeutial ase study

A summary of thetest results are shown onalogarithmi sale inFigure4.6, wherethe

olors orrespond to onguration seletion rules, and the shapes for the dierent update

funtions.

Figure4.6: Summary of results forthe pharmaeutialase study

Based onthese results the following onlusionsan bedrawn for this example:

•

^{For most of the ases, the} alternative solutionapproahes ould not outperform eah other more than amagnitude.

•

^{For small problems (with time horizon less than 29) the CPU times do not inrease}

drastially.

•

^{After 29hours of time horizon, the CPUtimes have adrastial inrease.}

•

^{Forlarger problems the} alternativeswith the E update funtiondominate the others, and there is only a negligibledierene between them for the dierent onguration

seletionstrategies.

4.4.2 Agrohemial example

In this exampleherbiide isprodued through three reations, aseparation, and an

evapo-ration. The owsheet of the problemis given in Figure4.7.

Figure4.7: Flowsheet of the agrohemialproess for herbiide prodution

The proessing times, apaities of the units, output ratios, and other details of the

probleman befound in the paperby Majozi and Friedler[82℄. Although thereis onlyone

produt, the bath sizes are not xed, and two units may work parallel onthe same bath.

Thus the algorithm of Setion 4.3 need to be applied to reate xed reipes. After this

preproessing steps, there are two dierent xed reipes as shown in Figure 4.8, where the

revenues of

F 1

^and

F 2

^{are 3.7 and 4.5ost units,} respetively.

Figure4.8: S-graphof the two xed reipesfor the agrohemialproess

The resultsof the testsare shown inTable 4.5, andthe best CPUtimes areindiated in

eah row.

Time CPUtime (s)

horizon LEX BFS DFS

(h) U F E U F E U F E

13 3.06 3.07 3.05 3.08 3.04 3.05 3.03 3.08 3.07

14 339 340 205 405 405 414 405 402 269

15 453 451 453 583 582 591 589 583 581

16 3600 3600 3600 3600 3600 3600 3600 3600 3600

Table 4.5: Test results forthe agrohemialexample

It is easy to see, that in ase of this problem, the inrease in the time horizon has a

drastieet onthe putimes. Eahadditionalhourinthetime horizonresulted inatleast

one magnitudegrowth inthe CPUtime. Basedonthis fewresults,itan beobserved, that

the Eupdatefuntiondominatesheretheothertwoaswell. Moreover, ingeneral,theLEX

onguration seletionstrategy proved tobe the most eient.

4.4.3 Literature example

For the last omparisons, the example from Setion 4.3 is taken. As desribed in that

setion,there are 6xed reipesfor this problemwith two produts. The resultsofthe test

are shown inTable 4.6, and again,the best CPU times are indiated in eah row.

Time CPUtime (s)

horizon LEX BFS DFS

(h) U F E U F E U F E

14 16.88 15.84 15.21 14.95 18.52 21.07 22.63 22.53 21.01

15 179 134 113 208 144 133 437 279 225

16 1260 1159 984 964 1496 1512 1594 1629 1403

17 3600 3600 3600 3600 3600 3600 3600 3600 3600

Table 4.6: Test resultsfor the exampleof Setion 4.3

As in the ase of the previous examples, the LEX-E alternative provides good results.

However, interestingly, the BFS-U strategy oftenhave good results.

Summary and onluding remarks

In this hapter, the S-graph framework has been extended to throughput orrevenue

maxi-mizationproblems. Thealgorithmis basedontheenumerationand maximumsearhofthe

feasible set of ongurations, i.e., bath numbers, whih are produable in the given time

horizon. The possibilities for dierent variations of the general algorithm were presented

and thouroughly ompared after implementation. The results showed that the approah is

apabletosolvethe onsidered set of problems,and provide theoptimalsolution eiently

when the orret variationis applied.

Related publiation

•

^{Holzinger, T., T. Majozi, M. Hegyhati, F. Friedler, An automated algorithm for}

throughput maximizationunder xed time horizon in multipurpose bath plants:

S-Graphapproah,17thEuropean Symposium onComputerAidedProess Engineering

Elsevier, 24,649 -654 (2007).

Related onferene presentations

•

^{Majozi, T., F. Friedler, T. Holzinger, M. Hegyhati, S-graph based} ontinuous-time approah for throughput maximization in multipurpose bath plants, presented at:

IFORS 2008,Johannesburg, SouthAfria, July 13-18, 2008

•

^{Hegyhati M.,T. Holzinger,T. Majozi,F. Friedler,T}ransformingSTNbased shedul-ingproblemstoS-graphrepresentation,presentedat: XIXPolishConfereneof

Chem-ialand Proess Engineering,Rzeszów, Poland, September3-7, 2007.

•

^{Holzinger,T.,T.Majozi,M.Hegyhati,F.Friedler,Throughput}Maximizationin Mul-tipurpose Bath Plants: S-graphvs Time Point BasedMethods, presented at: PRES

2007, Ishia Island, Italy,June 24-27, 2007.

•

^{Holzinger, T.,T. Majozi, M. Hegyhati, F. Friedler, An Automated Algorithm for}

ThroughputMaximizationUnder Fixed Time HorizoninMultipurpose BathPlants:

S-GraphApproah,presentedat: ESCAPE17,Buharest,Romania,May27-30,2007.

•

^{Holzinger, T.,T. Majozi, M. Hegyhati, F. Friedler, Using S-graph for Throughput}

Maximization in Multipurpose Bath Plants, presented at: VOCAL 2006, Veszprem,

Hungary, Deember12-15, 2006.

Limited- and Zero-wait storage poliies

in the S-graph framework

TheS-graphframeworkwasoriginallydeveloped forNIS-UWand UIS-UWstoragepoliies,

and the only published storage poliy extension of the S-graph framework[109℄ foused on

CIS-UW poliy. However, in many industrial appliation LW or ZW poliies are required,

as ertainintermediates lose some physialor hemialpropertiesover time, that would be

importantfor theupomingtask. Inthis hapter,several new approahesare introduedto

taklethesestoragepoliies. AsithasalreadybeenmentionedinSetion1.2, inase ofZW

poliy,the restritiononthe infrastrutureis irrelevant. However, LW itselfdoesnot dene

the storagepoliy. In this hapter LW will referto NIS-LW poliy,and the approahes are

desribed forthat,althoughthey ouldeasilybemodiedtoaddressUIS-LWpoliyaswell.

Moreover, if an approah is presented for LW poliy, it an be automatially applied for

ZW ases, as itis a speial ase of LW. Finally,it is not assumed anywhere, that allof the

intermediates share the same restrition for waiting time, i.e., some of them are ZW, LW,

or even UW.

Note that UW is a relaxation of ZW/LW, thus, all of the shedules with ZW or LW

poliies onsome intermediates remainfeasible if allof the LWintermediatesare set toUW

poliy. Later on, the terminology UW-relaxation will be used if the LW restritions of a

problemare disregardedthis way.

Most of the onstraints in sheduling has a "greater or equal" nature,e.g., a task must

startlater thanthenishingoftheprevioustaskinthereipe,ortheprevioustaskassigned

to the same unit. LW onstraints, however, dene a "not later then" thus "smaller or

equal" typeof onstraint,whihannotdiretly beexpressed by the ordinary S-graphars.

There are, however, several ways toaddress this poliy, whih are detailedin the following

subsetions.

In setion 5.1a hybridapproah isintrodued,where the S-graphbased branhing

pro-edure is extended with an LP based boundingfuntion that also inludes the "smaller or

equal" type of LW onstraints.

The approah introdued in setion 5.2 relies on the fat that a typial "less or equal"

type of LW onstrains, suh as

T ₁ ^start ≤ T ₂ ^start + t ^proc

^{an easily be onverted to a} "greater or equal form":

T ₂ ^start ≥ T ₁ ^start + (−t ^proc )

^{, whih an be modeled by regular S-graph ars.}

Theweightoftheedge,however, beomesnegative(non-positivetobepreise), whihneeds

some additionalare.

In setion5.3twoapproahes are introduedfor problems withonlyUW and ZW

inter-mediateswithoutintroduingnegativeweighted ars inthe S-graph. Addressing LW stages

is possible through a modelingonversion.

Last, setion 5.4 ompares the performane of these approahes through several

exam-ples.

5.1 Auxilary LP model

Thesmaller-or-equaltypeofonstraintsofLWpoliyanbeaddressedbyanLPmodelthat

an be formulated for eah subproblem. The mathematial model an simply built based

on the S-graph

(N, A 1 , A 2 )

^:

•

^A non-negative ontinuous variable is assigned to eah node, whih represents its startingtime:

S i

^{, for all}

i ∈ N

^.

•

^{Shedulearsrepresentsimpleorderingintime,thusaonstraintintheformof}

S i ^′ ≥ S i

is addedfor all

(i, i ^′ ) ∈ A ₂

^.

•

^{For all reipe ar, a similar onstraint is added:}

S i ^′ ≥ S i + w i,i ^′

^{for all reipe ar}

(i, i ^′ ) ∈ A ₁

^{, where}

w i,i ^′

^{isthe weight of it.}

•

^{Last, for eah task}

i

^{with LW poliy, the onstraint}

S i ^′ ≤ S i + w i,i ^′ + maxwait i

^is

added, where

i ^′

^{is an upomingtask of}

i

^{, i.e.,}

(i, i ^′ ) ∈ A ₁

^,and

maxwait i

^{the maximal}

allowed waitingtime.

Thesolutionofthismathematialmodelwillnotprovidesharperboundsthanthelongest

path algorithm. However, it willdetet if a ertain subproblem is not feasible due to wait

restritions. Insuhaase,the orrespondingbranhofthe searhtree ispruned obviously.

Note that in ase of a partialshedule that is not even feasible for the UW-relaxation,the

S-graphapproahwilldetettheinfeasibilityintheusualmannerbyndingadiretedyle,

thus the solutionof the LPmodelisnot neessary.

Advaned LP approah

The most seriousdrawbak of the previouslymentioned approahis thatthe mathematial

modelneeds to be formulated ateah subproblem,taking alot of CPU time, whilethe LP

itselfdoesnot providea sharperboundthan the longestpath approah. There is,however,

amoresophistiatedway ofusinganLPforLWand ZWpoliies,assuggestedinsubsetion

2.3.3.

Instead of formulating a separate LP model for eah subproblem, the LP relaxation of

the wholepreedene basedmodelan bebuiltforthe rootsubproblemimmediately. Then,

ateahdeision,thismodelisopied,andsomeofthe binaryvariablesarexed byhanging

their upperor lower bounds.

Though, these models are muhbigger than the formerly introduedones, their bitwise

opying and modifying some variable bounds takes less time than building up the same

model from srath. Moreover, as they have the same numberof variablesand onstraints,

the solution of the parent LP model an be used as a starting basis for the dual simplex

algorithm. Finally, the solution of these models provide a tighter bound than the longest

path algorithmitself.

Note thatsimilarlytotheoriginalapproah,there isnoneed toopy the LPrelaxation,

modify and solveitif the shedulegraph ontains ayle or thelongestpath ishigher than

the urrentupper bound.

Moreover, as the models at the subproblems are the same exept for the bounds on

variables, it may be more beneial not toopy the model, but use it as a global variable,

and adjust the bounds ateah subproblem.

5.2 Combinatorial approah with negative weighted ars

As ithas alreadybeen mentioned,the less-or-equaltypeonstraints inthe form of

T ₁ ^start ≤ T ₂ ^start + t ^proc

^{aneasilybeonvertedtoa}greater-or-equalform:

T ₂ ^start ≥ T ₁ ^start + (−t ^pt )

^{. This}

transformation,however, introduesnon-positiveweightsonthears. This ideahas already

been used inforthe Alternativegraphmodeltomodelsimilarsituations[88,98,24,25,22,

21℄

The limited waiting times an easily be modeled by negative weighted ars. If there is

a task

i

^{with proessing time}

t ^pt _i

^{, and maximal waiting time}

t ^maxwait _i

^{, then two reipe ars}

should be inserted into the S-graphas illustrated inFigure 5.1:

•

^{An ar with weight}

t ^pt _i

^from

i

^{toits subsequent task(s).}

•

^{An ar with weight}

−t ^pt _i − t ^maxwait _i

^{fromthe subsequent task(s) of}

i

^to

i

^.

It isobvious tosee that these ars express exatlythe desiredonstraints. In the gure

the intervalinwhihtask

i ^′

^{anstart (from}

T _i ^start + t ^pt _i

^to

T _i ^start + t ^pt _i + t ^maxwait _i

^{)is indiated}

by blue olor onthe time axis.

There are, however, some aspets of the algorithm that must be taken are of with the

introdution of these ars.

•

^{The longest path algorithmmust be adjusted aordingly. As the urrent}

implemen-tation maintainsa longest pathmatrix throughout the algorithm,there is noneed to

hange anything with this.

Figure5.1: Modeling LW poliywith negativeweighted ars

•

^{Eventhereipegraphwill}immediatelyontainseveralyles. Theirweightisnegative, or in the ase of ZW ars, 0. Naturally, negative weighted yles are not of great

interest. They an simplybe disregarded. However, the 0weighted yles need extra

are, assomeof themonlyrepresenta ZWonnetion,while theothers modela ross

transfer (See setion 3.2). The algorithm must be modied in order to report only

those zero-weighted yles that donot have reipe ars in them.

5.3 Combinatorial approah without negative weights

In this setion two approahes are desribed to takle ZW poliy. LW stages are not

on-sidered, but they an be addressed via a modeling transformation desribed in the last

subsetion.

Asbrieydisussed before, asheduleanbelongtoone ofthethreegroupslistedbelow:

UW infeasible These shedules would be infeasible for the UW relaxation of the

problem aswell.

ZW infeasible, UW feasible These shedules are feasible for the UW relaxation

but they violate ZW onstraints

ZW feasible These shedules are feasible for the problem.

The original S-graph algorithm fails to dierentiate the shedules in the seond group

from the ones in the third. It does, however, eliminateall the shedules in the rst group.

In order to identify shedules in the seond group, the approah desribed in the previous

setion introdued additional, negative weighted ars to the problem. The shedules inthe

seond group would result in a non-negative weighted yle by using that approah, whih

onsists of two type ofalternating parts:

•

"Forward", positive ars belongingto eitherUW or ZWintermediates

•

"Bakward", negative weighted ars of ZW stages

AnexampleisgiveninFigure5.2withboththeshedulegraphandtheorrespondingGantt

hart. The same shedule is infeasibleif the outputsof tasks

i1, i2

^{, and}

i3

^{have ZWpoliy,}

as shown inFigure 5.3. The path indiated by thik ars has a weight of 6,whihis longer

than the ZW path from

i2

^to

i4

^{, thus itresults ina positiveyle.}

5.3.1 Reursive searh

Even if the negative weighted ars are not inserted to the S-graph, the positive weighted

yles an beidentiedbya searh[50℄. Suppose that apartialsheduleis ZWfeasible, and

a new shedule ar is just inserted to the graph. Obviously, if the partial shedule is now

ZW infeasible,the newly addedar must bepart of the non-negative yle.

Figure 5.2: FeasibleUW shedule

Figure5.3: Infeasible ZW shedule

Even without the negative ars, this yle ould be found by a simple approah: from

the endpoint of the new ar, it has to be heked reursively whether the starting point

of the shedule ar is reahable via "forward" and "bakward" steps with a non-negative

weight ornot.

1

The algorithmforthat searh isgiven in blok 5.1.

This approah basially heks the existene of a non-negative yle instead of

main-tainingthe extendedlongest-pathmatrix ofthe previousapproah. Thealgorithmlooksfor

suh nodes inthe S-graphina reursive way, whosestarting timean bebounded withthe

suh nodes inthe S-graphina reursive way, whosestarting timean bebounded withthe

In document Szakaszos üzemű rendszerek ütemezése: Kiterjesztések az S-gráf módszertanhoz (Pldal 77-93)