t pr later than the start of the exeution of task i . Similarly,

(i

c, i

w)

^is ^a ^ZW ^reipe ^ar ^with ^the ^weight ^of

t ^w

^, ^that ^is, ^the ^maximal ^waiting

time possibleafter the ompletion of the intermediate.

(i

c, i ^′ ), (i ^′ , i

w)

^are zero-weighted shedule ars that ensure that

i ^′

^must ^start

be-tween the ompletion and wearing of the intermediate.

(i, i ^′ )

^is ^a ^UW ^reipe ^ar ^with ^weight ^of

t ^pr

^. ^This ^ar ^is ^needed ^so ^that ^the ^shedule

ar towards the task that follows

i

ⁱⁿ ^the ^same ^unit ^ould ^start ^from

i ^′

^not ^only

from

i

c

Figure5.7: Examplefor modeltransformation of LW stages

5.4 Comparison of approahes

The eieny of the algorithms from the previous setions is illustrated on a literature

example. Theexample features 5dierentsequential produtsand 6 unitsto besheduled.

The reipe graphfor the example is shown inFigure 5.8.

Figure5.8: Example for the omparison of LW/ZWapproahes

For the sake of the omparison, eah intermediate is assumed to have ZW storage

pol-iy. The investigated algorithms were ompared on 14 dierent ongurations with bath

numbers.

These algorithmsare:

sLP is the simple LP approah

aLP is the advaned LPapproah that opies the modelfor eah subproblem

aLP' is the advaned LP approah that uses onlyone modelasa global variable

Neg isthe approah with negativeweighted ars

Re isthe approahrelying on the reursive searh

Re+ is the reursive approahwith the extended positive ar additions

Eah test run were set within a 1000 s time limit. For the 11 smaller ongurations,

the approahes have not reahed this limitexept forouple of ases, and were able tond

the optimal solution. The data for the CPU times of the approahes are given in detail in

Setion C.2 , and illustrated inFigure 5.9

Figure5.9: CPUtime of ZW/LW approahes for the smallerases

The LP basedapproahes were usually 1or2magnitudesslowerthan the ombinatorial

approahes, among whihthe negative ar based proved tobethe most eient.

Forthe 3 larger ongurations all of the approahes have reahed the 1000 seond time

limit, and thus stopped. For these ases the omparison of the quality of the best found

(probably suboptimal) solution is important, and shown in Figure 5.10. The gure also

inludesthe2largestongurations fromthepreviousones, wherethe LPbasedapproahes

have reahed the limit.

Figure 5.10: Quality of reportedsolutions of ZW/LWapproahes forthe largerases

The aLP' approah ould not even report a feasible shedule for the larger ases. The

best solutions were reported by the negative ar based approah. The extended reursive

approah had very loseresults. The other approahes utuated.

In general, itan bestated that the most favorable approahseems to bethe one using

negative weighted ars.

Summary and onluding remarks

TheoriginalalgorithmsoftheS-graphframeworkwasdeveloped forUnlimitedWaitstorage

poliies. Inthishapter,dierentoptionsofextendingtheframeworkforLimited-and

Zero-Waitstoragepoliieshasbeenpresented andinvestigated. Theseoptionswere implemented

andomparedonawidesetofexamplesintermsofomputationalneeds. Theresultsshowed

thatanextended modelwithnegative-weightedars, andslightlymodiedalgorithmsisthe

most eient option.

Related publiation

•

^Hegyhati, ^M., ^T. ^Holzinger, ^A. ^Szoldatis, ^F. ^F^riedler, Combinatorial Approah to AddressBathShedulingProblemswithLimitedStorageTime,ChemialEngineering

Transations, 25,495-500 (2011)

•

^Hegyhati,^M.,^T.^Holzinger,^A.^Szoldatis,^F.^F^riedler,CombinatorialApproahto Ad-dress BathShedulingProblemswithLimitedStorageTime,presented at: PRES'11,

Florene, Italy,May 8-11, 2011.

•

^Hegyhati, ^M., ^T. ^Majozi, ^F. ^Friedler, ^Throughput maximization in a multipurpose bathplant, presented at: PRES 2008, Prague,Czeh Republi,August 24-28,2008

Maximizing expeted prot in a

stohasti environment

InChapter 4thegeneralalgorithmfor throughputmaximizationwasintrodued. Formany

problems, however, several parameters of these types of problems are not deterministi.

Pistikopoulosetal.[104℄ gave alassiationofstohasti shedulingproblemsbased onthe

soureoftheunertaintyintheproess. Inahangingmarketenvironment,forexample,the

prie and marked demands an usually be onsidered stohasti. Both of these values an

have impat on the overall prot, as in ase of a overprodution, for example, the storage

of the surplus may result in additionalexpenses.

Obviously, in suh unertain irumstanes, it beomes ambiguous whih solution an

be onsideredasoptimal: the mostrobust one,orthe one withhighestexpetedprot,et.

Li and Ierapetritou[76℄ gave anextensivereview of the approahes that dealwith dierent

typesofunertaintiesinbathproesssheduling. Withoutattemptingtobeomprehensive,

the three main diretionof researh is illustrated inFigure 6.1.

Figure6.1: Classiationof approahes dealingwith unertainty

Preventive sheduling In this diretion of researh[72, 14℄, the shedule must be

given a-priori,and annotbemodied afterthe system starts,or the realization

of unertainevents. If thereisnoinformationabout theprobabilitydistribution

of theunertainparameters, areasonable objetive ouldbeaprodution

shed-ule, that exeeds a ertain prot, and have the highest level of robustness. On

the other hand, an estimation about the probability distribution gives room for

optimization towards the highest expeted prot.

Reative sheduling Reative approahes[92℄ assume an existing shedule, whih

gets disturbed by some unertain event. The objetive is to modify the rest

(unexeuted part) of the shedule ina way, that leads to the best available

per-formane. Unlike inthe previous ase, the optimizationisarried out, whenthe

system is already running, thus usually there is a very limited amount of time

to deliver the solution. As a result, heuristis are often favored for this

pur-pose. Onthe brightside, theapproah doesnot have todealwith any unertain

parameters, as they are already realized.

Two- and multistage approahes In a two-stage optimization approah[59℄, it is

assumed thattherearesomedeisionsthatmust bedonepriortothestartofthe

system, however, some deisionsan bealteredlater. Asanexample,a shedule

may has to be deided in advane, but the load of the units an be adjusted

after the unertain parameters our. In a multi-stageapproah,this onept is

generalized, and itis not assumed, that allof the unertain events realizeatthe

same time.

Naturally, there are a lot of developments, whih do not t into these three ategories,

some approahes, e.g., ombine preventive and reative sheduling. Sensitivity analysis in

itself is abroad area to researh,but itan alsopartiipate asa subroutine for preemptive

sheduling, espeially inase of heuristiapproahes.

ThefollowingsetionsintrodueanS-graphbasedapproahfortheshedulingof

through-putmaximizationproblemsonsideringunertainostanddemandparameterswithdisrete

probability distribution funtions. Setion 6.1 gives anexat denition of the problems to

be solved, and Setion 6.2 and 6.3 presents the S-graph approahes to address this set of

problems. In Setion6.4 anextensionis shown toontinuous probability distribution

fun-tions. Last, Setion 6.5illustratessome of the algorithmsvia an example, and draws some

onlusions.

6.1 Problem denition

The problemstobesolved are given by similarparameterstothoseof ageneralthroughput

maximization problem, i.e., eah produt is given with its reipe, along with the set of

equipmentunits andthe timehorizon. Atthis pointitisassumedthat thereisaone toone

relation between produts and reipes, i.e., there isno reipeproduing multiple produts,

and there are no two dierentreipesproduing the same produt.

There is, however, a set of unertain parameters for eah produt, whose probability

distributionis disretizedintojoint senarios. Thus, for eahdisretesenariothefollowing

parameters are given:

The objetive is to make deision about the number of bathes and provide a feasible

shedule in way to ahieve maximal expeted prot. Note that it is assumed that if the

atual produtionis higherthan the demand, the surplus isnot sold.

Basedonthepossibledeisionsrelatedtothesizesofeahbath,threedierentproblems

are identied:

Preventive problem with xed bath sizes whenthebathsizeforeahprodut

is given, and the only preventive deisionto be madeis todeide the number of

bathes for them.

Preventive problem with variable bath sizes is a more exible version of the

previousproblem,wherenot onlythenumberofbathes,but alsotheirsizesan

be altered inadvane before the unertain events realize.

Two stage problem where the bath numbers have to be deided in advane, but

the bath sizes an be alteredaordingly afterthe unertain events realized.

Setion 6.2 introdues the approahes that an solve these three dierent problems. In

Setion6.3,theseapproahesareextendedtotakleases,whenareipeanprodueseveral

produts, and the same produts an be produed by several reipes.

6.2 S-gaph based approahes

The three approahes to solve the problems presented inthe previous setion are all based

on the throughput maximization algorithm presented in Setion 4.1. The main algorithm

that examines dierent bath size ongurations and the feasibility tester an remain the

same, as inthe ase of the general throughput maximizationproblems. The subroutines to

seletaongurationandtoupdatetheset ofopenongurations anbealteredinorderto

Taking theopposite assumption would not alterthe struture of the algorithm, thesame approahes

wouldbeappliablewithmodiedinputparameters.

Theapproahforthepreventiveversionofthis asehasbeenpublishedin theliterature[72℄

ahieve higher eieny, however, any of the previously mentioned implementationswould

sue.

Thekeydierenebetween thedierentapproahesistherevenuefuntion,thatshould

provide the expeted prot, orthe highest expeted prot for a onguration, asit will be

disussed inthe followingsubsetions.

In order to simplify these desriptions, the following additional notations will be used

throughout this hapter next to the ones used previously:

In document Szakaszos üzemű rendszerek ütemezése: Kiterjesztések az S-gráf módszertanhoz (Pldal 97-106)

t pr later than the start of the exeution of task i . Similarly,

(i

c, i

w)

t w

(i

c, i ′ ), (i ′ , i

w)

i ′

(i, i ′ )

t pr

i

i ′

i

c

•

•

•

t ^w

c, i ^′ ), (i ^′ , i

i ^′

(i, i ^′ )

t ^pr

i ^′