a thesis
submitted to the department of industrial engineering
and the institute of engineering and science
of b_
ilkent university
in partial fulfillmentof the requirements
for the degree of
master of science
By
Rabia Koylu Kayan
Asso c. Prof. SelimAkt urk (Sup ervisor)
I certify that I have read this thesis and that in my
opinionit isfullyadequate, inscop e and inquality, as a
dissertationfor the degreeof Master of Science.
Asso c. Prof. Osman O~guz
I certify that I have read this thesis and that in my
opinionit isfullyadequate, inscop e and inquality, as a
dissertationfor the degreeof Master of Science.
Assist. Prof. Oya Ekin Karasan
Approvedfor the Institute of Engineering andScience:
Prof. MehmetBaray,
SCHEDULING WITH TOOL CHANGES TO MINIMIZETOTAL
COMPLETION TIME UNDERCONTROLLABLE MACHINING
CONDITIONS
Rabia Koyl u Kayan
M. S.in Industrial Engineering
Sup ervisor: Asso c. Prof. SelimAkt urk
Septemb er2001
Inthe literature,schedulingmo delsignorethe unavailabilityof thecuttingto ols.
To ol management literature considers to ol loading problem when to ols change
dueto part mix. Inpractice,to ols are changed moreoftendueto to olwear. The
studies on to ol management issues consider machining conditions as constant
values. Infact,itisp ossible to changethe pro cessing timeandto ol usagerate of
ajobbychangingthe machiningconditions. However,the machiningconditions,
such as cutting sp eed and feed rate e ect the pro cessing timeand usage rate of
the to ol inopp osite directions. Increasing the usage rates ofjobs will lead to an
increaseinnumb erofto olswitches. Pro cessingtimesandnumb erofto olswitches
aretwocomp onentsofourobjectivefunction. Thistwo-sidee ectcreatesa
trade-o b etweenpro cessing timeand to olusage rate. Therefore machiningconditions
shouldb e selectedappropriately inorder to minimizethe total completiontime.
Weprop osedasetofsingle-passdispatchingrulesandalo calsearchalgorithm
todeterminethemachineconditionsforeachjobandtoschedulethemonasingle
CNCmachinesimultaneouslyto minimizethe total completiontime.
Keywords: Scheduling,Total CompletionTime,To olManagement,
DE ~ G _ ISKEN _
IMALATKOSULLARIALTINDA KES _ IC _ I UC DE ~ G _ IS _ IM _ I DURUMUNDA TOPLAM _ IS B _ IT _ IMZAMANINIENAZLAMA
Rabia Koyl u Kayan
End ustri M uhendisli~gi Y uksek Lisans
Tez Yoneticisi: Asso c. Prof. M. SelimAkt urk
Eyl ul2001
Literat urde cizelgeleme mo delleri kesici uc mevcudiyetsizligini d us unmemistir.
Kesici uc isletim sistemi literat ur u de uc degisimini parca srasna ba~gl olarak
kesiciucy uklemeproblemiad altndaayrcaelealr. Aslndauretim kosullarnda
kesici uclar daha cok, asnmaya ba~gl olarak degistirilir. Kesici uc isletim
sistemi uzerine onerilen calsmalarda, imalat kosullar (kesme hz, b esleme
oran) sabit girdi olarak ele alnmstr. Aslnda imalat kosullarn de~gistirerek
islemezamann ve kesiciucun omr un u degistirmekm umk und ur. Ancak imalat
kosullarnnuretim zaman ve uc kullanmoran uzerindeki etkisi ters yondedir.
Uckullanmlarnartrmakdahacokucde~gisimineseb epolur.
Uretimzamanlar
ve uc de~gisim says amac fonksiyonunun iki ogesidir ve birini azaltan imalat
kosullar di~gerini arttrmaktadr. Bu y uzden toplam is bitim zamann en
azlayacakimalatkosullar secilmelidir.
Bucalsmadaherisicinimalatkosullarnnsaptanmasveislerincizelgelenmesi
problemlerini birlikte cozecek baz hzl sezgisel algoritmalar ve yerel tarama
algoritmalargelistirilmisvebu algoritmalarnp erformanslarkarslastrlmstr.
Anahtar sozcukler: Cizelgeleme, _
Is Bitim Zaman, Kesici Uc _
Isletim
Sistemi, _
ImalatKosullar,Degisken _
Abstract i Ozet ii Contents iii List of Figures v List of Tables vi 1 Introduction 1 2 Literature Review 4 2.1 To olManagement : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.1.1 Machining Conditions : : : : : : : : : : : : : : : : : : : : 6 2.1.2 To ol Replacement: : : : : : : : : : : : : : : : : : : : : : : 10 2.2 Scheduling : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 12
2.2.1 Controllable Pro cessingTimes : : : : : : : : : : : : : : : : 12
2.2.2 Machine Availability : : : : : : : : : : : : : : : : : : : : : 14
2.3 Conclusion : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 16
3 Problem Statement and Modeling 18
3.1 Problem De nition : : : : : : : : : : : : : : : : : : : : : : : : : : 18
3.2 Assumptions: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 18
3.3 Mo del Building : : : : : : : : : : : : : : : : : : : : : : : : : : : : 20
3.8 Conclusion : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 34
4 Proposed Heuristic Algorithms 35
4.1 Characteristics of the Problem : : : : : : : : : : : : : : : : : : : : 35
4.2 Single-pass Heuristic Algorithms : : : : : : : : : : : : : : : : : : : 39
4.2.1 Stage1: Setting Assignment : : : : : : : : : : : : : : : : : 40
4.2.2 Stage2: Dispatching rule: : : : : : : : : : : : : : : : : : : 42
4.2.3 Stage3: Improvements : : : : : : : : : : : : : : : : : : : : 45
4.3 The ProblemSpace GeneticAlgorithm (PSGA) : : : : : : : : : : 51
5 ExperimentalDesign 56
5.1 Exp erimentalSetting : : : : : : : : : : : : : : : : : : : : : : : : : 56
5.2 Exp erimentalResults of Single-passHeuristics : : : : : : : : : : : 59
5.3 Lo cal SearchParametersand Results : : : : : : : : : : : : : : : : 63
6 Conclusion 75
6.1 Contributions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 75
6.2 Future ResearchDirections : : : : : : : : : : : : : : : : : : : : : : 77
APPENDIX 85
A Computational Results for Single-pass Heuristics 86
B Computational Results for PSGA 93
3.1 Feasible region of machinesettings : : : : : : : : : : : : : : : : : 23
4.1 Representationof a schedule as blo cks of jobs : : : : : : : : : : : 36
4.2 Timeversus cutting sp eed : : : : : : : : : : : : : : : : : : : : : : 36
4.3 Alternativesetting pairs : : : : : : : : : : : : : : : : : : : : : : : 37
4.4 Three stages of the heuristics : : : : : : : : : : : : : : : : : : : : 40
5.1 Summaryresults of heuristicsfor 100 jobs : : : : : : : : : : : : : 61
5.1 Exp erimentaldesign factors : : : : : : : : : : : : : : : : : : : : : 57
5.2 Technical co ecientsand parameters : : : : : : : : : : : : : : : : 58
5.3 Summaryresults of heuristicsfor 100 jobs : : : : : : : : : : : : : 59
5.4 Summaryresults of heuristicsfor 200 jobs : : : : : : : : : : : : : 60
5.5 De nitions and levelsof PSGA parameters : : : : : : : : : : : : : 63
5.6 Di erentparameter combinationsfor PSGA[MFFD(dif,1by1)] : : 64
5.7 Paired samplesstatistics for PSGAparameter sets : : : : : : : : : 67
5.8 Paired samplestest resultsfor PSGA parameter sets : : : : : : : 68
5.9 Comparison of twobase heuristics of PSGA : : : : : : : : : : : : 70
5.10 Comparison of PSGA with optimalfor 30 jobs : : : : : : : : : : : 71
5.11 Paired samplesstatistics for di erentcomparisons : : : : : : : : : 71
5.12 Paired samplestest resultsfor di erentcomparisons : : : : : : : : 72
5.13 Comparison of PSGA with along run PSGA : : : : : : : : : : : : 73
5.14 Paired samplesstatistics for PSGAand a long-run PSGA : : : : : 73
5.15 Paired samplestest resultsfor PSGA and a long-run PSGA : : : 73
A.1 For 100 jobs, results of the heuristics using the six dispatching
rules with(min,knap) alternatives : : : : : : : : : : : : : : : : : : 87
A.2 For 200 jobs, results of the heuristics using the six dispatching
rules with(min,knap) alternatives : : : : : : : : : : : : : : : : : : 88
A.3 For 100 jobs, results ofthe heuristicsusing FFD : : : : : : : : : : 89
A.4 For 200 jobs, results ofthe heuristicsusing FFD : : : : : : : : : : 90
A.5 For 100 jobs, results ofthe heuristicsusing MFFD : : : : : : : : : 91
A.6 For 200 jobs, results ofthe heuristicsusing MFFD : : : : : : : : : 92
B.5 Results of PSGA for parameter set5 : : : : : : : : : : : : : : : : 98
B.6 Results of PSGA for parameter set6 : : : : : : : : : : : : : : : : 99
B.7 Results of PSGA for parameter set7 : : : : : : : : : : : : : : : : 100
B.8 Results of PSGA for parameter set8 : : : : : : : : : : : : : : : : 101
B.9 Results of PSGA for parameter set9 : : : : : : : : : : : : : : : : 102
B.10Results of PSGA for parameter set10 : : : : : : : : : : : : : : : : 103
B.11Results of PSGA for parameter set11 : : : : : : : : : : : : : : : : 104
B.12Results of PSGA for parameter set12 : : : : : : : : : : : : : : : : 105
B.13Results of PSGA for parameter set13 : : : : : : : : : : : : : : : : 106
B.14Results of PSGA for parameter set14 : : : : : : : : : : : : : : : : 107
B.15Results of PSGA for parameter set15 : : : : : : : : : : : : : : : : 108
B.16Results of PSGA for parameter set16 : : : : : : : : : : : : : : : : 109
B.17Results of PSGA for parameter set17 : : : : : : : : : : : : : : : : 110
B.18Results of PSGA for parameter set18 : : : : : : : : : : : : : : : : 111
B.19Results of PSGA for parameter set19 : : : : : : : : : : : : : : : : 112
B.20Results of PSGA for parameter set20 : : : : : : : : : : : : : : : : 113
B.21Results of PSGA for parameter set21 : : : : : : : : : : : : : : : : 114
B.22Results of PSGA for parameter set22 : : : : : : : : : : : : : : : : 115
B.23Results of PSGA for parameter set23 : : : : : : : : : : : : : : : : 116
B.24Results of PSGA for parameter set24 : : : : : : : : : : : : : : : : 117
Introduction
The scheduling of manufacturing systems has b een the subject of extensive
researchsincetheearly1950s. Themainfo cusisontheecientallo cationofone
or moreresourcesto activitiesovertime. Weadoptthe followingterminologyfor
convenience: we refer to a job which consists of one op eration, and a machine
whichis the resourcethat can p erformat mostone op eration at a time.
We restrict our attention to deterministic machine scheduling where it is
assumedthatthe data that de nea probleminstanceis known with certaintyin
advance. Weassumeindep endentjobswithsingle op erations whichare available
at time zero and do not need any setup time. Preemption is not allowed when
pro cessing the op erations of the jobs. Only interruptions are due to the change
of cuttingto ols which are subject to wear. In industry, cuttingto ols are subject
to wearb ecauseof theusagerateof jobs. Sinceto olchangesdue toto olwearare
frequentandto olchangetimesaresigni cantcomparedto cuttingtime,e ective
scheduling cannot b e done unless taking into account the cutting to ol change
instances.
In a recent study, Akturk et al. [4] fo cus on the scheduling problem with
to ol changes due to wear, but they consider the pro cessing timeof the jobs and
cuttingto ollivesasconstantvalues. However,bychangingthecuttingsp eedand
feed rate of the machine,these two values can b e controlled. Cutting sp eed and
feed rate are the machining parameters which constitute the machine settings.
jobinpro cess willuse the to olmore. To ol usage rateof a jobissimplythe ratio
of machining timeto the to ol life. Each job has di erentusage rates dep ending
onitsdepthofcut,diameter,lengthandsurface nishrequirements. Thecutting
to ol b ecomes worn when the aggregation of usage rates of jobsop erated by this
to olexceeds1,inotherwordswhen the totalmachiningtimeof the jobsexceeds
to ollife. However,to ollifeisnot constantinour problem. Increaseinusagerate
ofthe jobswillleadto morefrequentto ol changes and theto olchangetimeswill
shift the completion timesof the succeeding jobs. On the other hand, it is easy
to see that the increase in pro cessing times will increase the total completion
time. Usage rate and machining time change in opp osite directions. When the
usage rate of ajob is increasing, i.e. machinesettings increasing, the machining
time decreases, i.e. the jobs are pro cessed morerapidly. Hence, the machining
conditions, cutting sp eed and feed rate, have to b e adjusted prop erly for each
job in order to minimizethe total completion time. Considering the pro cessing
timesand usage rates of the jobs as a consequenceof the decision of machining
conditions, rather than b eing constant, the integration of the to ol management
and schedulingproblems isimproved.
Due to high investment and to oling cost of a CNC machining center,
machining and non-machining times should b e optimized by considering to ol
changes and machining conditions. Moreover, to ol change times are generally
signi cantwhencomparedtothepro cessingtimes,andto ollivesareshortrelative
to the planning horizon. Therefore, itis imp ortantto schedulethe jobs fortime
related scheduling objectives. We fo cus on the completion time and select our
objectiveas minimizing the total completiontimeof the jobs.
In thisstudy, wepresentsolution strategies to theproblemof schedulingjobs
withpro cessingtimesandusageratescontrolledbythemachiningconditionsthat
arecuttingsp eedandfeedrate. Thereisasinglepro ductionunit,aCNCmachine
with one typ e of to ol which is subject to wear. Our objectiveis minimizing the
totalcompletiontime. Theexistingstudiesintheliteratureignoretheinteraction
b etween the scheduling decisions and the to ol change requirements due to to ol
aimto show the validity of this problemand try to nd solution metho ds to ll
inthis gap inthe literature.
We rst formulate a mixed integer program to nd the optimal machining
conditions for each job and schedule of the jobs. Then we prop ose some
single-pass heuristicalgorithmsand test the p erformanceof themon a setof randomly
generatedproblems. Moreover,weprop ose aproblemspacegeneticalgorithmto
improvethesolutionquality. Inproblemspacegeneticalgorithmsabaseheuristic
hasto b e de nedwhichiscalledmany timeswithinthealgorithm. We testsome
of our single-pass heuristics as base heuristics. We select the ones which have
high p erformance in low CPU times. Finally, inserting some of the single-pass
heuristicsgivinghighp erformanceinlowCPUtimestothelo calsearchalgorithm
as base heuristics,weimprovethe solution quality.
Inthe nextchapter,aliteraturereviewon machiningconditions optimization
and to ol replacement issues in to ol management, and controllable pro cessing
times and machine availability concerns in scheduling literature are presented.
InChapter 3,aproblem de nitionis givento de nethe scop e of this study,and
mathematical formulationof the mo del is presented. Consequently, in Chapter
4, the prop osed heuristic approaches are intro duced. Exp erimental design and
resultsaregiveninChapter5,and nallytheconclusionofthisstudyispresented
Literature Review
In literature, to ol management issues and scheduling problems are studied
separately. In b oth elds, extensive research has b een done for mo deling
the systems, and for developing a variety of solution metho ds. However the
interaction b etweenthesetwolevelsof manufacturing decisionpro cesses has not
b eenaddressed by the researchers.
In order to give the related literature in an organized manner, we will start
withthe to olmanagementissuesin the following section. Then, wewillgivethe
literatureon schedulingesp eciallywithcontrollablepro cessingtimes. Finally,we
willconcludebymentioningthe drawbacksofthecurrentliteraturethatmotivate
usfor this study.
2.1 Tool Management
Flexibility is a key requirementin manufacturing systemsto cop e with mo dern
marketenvironmentwhichischaracterizedbydiversepro ducts,high qualityand
short lead time. Crama and Klundert [10] de ne the most vital comp onent of
exibility as \the ability of machines to p erform various op erations on various
pro ductsor parts". Theterm\ exible"isgenerallyused to describ etwoasp ects
ofthesystem[40]: (1)theabilitytousealternativeroutingsthroughthemachines
top erformagivensetofop erations,and(2)theabilitytosimultaneouslymachine
arecapableofcarryingmultipleto ols. Also,theversatilityofan FMSisachieved
by equipping each machine with a to ol magazine. This magazinecan hold a set
of to ols which the machine can use to p erform a succession of op erations while
incurring low setup costs when switching from one to ol to another. In reality,
FMSsareonlycapableofpro cessinga nitefamilyofpartsatanygiventime. The
exibilityor randomnessislimitedbythe allo cationof supp ortingresources such
as pallets, xture,and to ols. AsFMSs expand into thelowvolume,high variety
pro ductionenvironment,thenumb erofpallets, xture,andto ols andtheamount
ofhandling oftheseresources areincreased. Themanagementoftheseresources,
esp eciallythe to olingwhichaccountsfor ahighp ercentageoftheop erating costs
ofan automatedmanufacturingenvironment,isan absolute must. Thereforethe
mo delsincludingto olmanagementimprovesthe pro ductivity for an FMS.
Due to its direct impact on systemp erformance, itsdynamicnature and the
largeamountofinformationinvolved,theto olingproblemhas b eenconsideredas
one of the most imp ortantand complicatedissues inautomated manufacturing.
Prop er to ol management ensures that the correct to ols are on the appropriate
machines at the right time so that the desired quantities of workpieces are
manufactured and the machine utilizations are maintained. To ol inventory,
maintenance and distribution issues determine the quantity of work pro duced
and systemutilizations.
To olmanagementisan imp ortantareaofresearchwhichhasb eenextensively
studiedfornearlyahundredyears, sinceTaylor[46] rstrecognized in1907that
the machining conditions should b e optimized to minimizethe machining cost.
Malako oti and Deviprasas [32] list vital contributions on parameter selection in
metalcutting from 1907up to 1985 in their pap er.
It isstated by Stecke[41] andGray et al. [18]that approximately 50 p ercent
of U.S. annual exp enditures on manufacturing is in the metalworking industry,
and two thirds of metalworking is metal cutting. Besides b eing a critical issue
in factory integration, to ol management has direct cost implications. Kouvelis
[26] rep orts in his study that to oling accounts for 25 p ercent to 30 p ercent of
environment. The reason for such a high contribution of the to oling to the
total manufacturing cost is related to the high material removal rate in metal
cuttingpro cesses, and the consequentincreasedto ol consumptionrates and to ol
replacementfrequencies.
Kaighobadi and Venkatesh [23] state that the lack of attention to cutting
to ol related issues is a main reason for making an FMS in exible in practice.
Gray et al. [18] and Veeramani et al. [49] give extensive surveys on the to ol
management issues in automated manufacturing systems, and emphasize that
thelackofto olmanagementconsiderations has resultedinthep o or p erformance
of these systems.
2.1.1 Machining Conditions
The optimization of the machining conditions for a single op eration is a well
known problem, where the decision variables are usually the cutting sp eed and
the feedrate. Theseconditions are the keyto economicalmachiningop erations.
Knowledge of optimal cutting parameters for machining op erations is required
for pro cess planning of metal cutting op erations. Numerous mo dels have b een
develop ed with the objectiveof determiningoptimalmachiningconditions.
Malako oti and Deviprasas [32] formulate a metal cutting op eration, sp
ecif-ically for a turning op eration, as a discrete multiple objective problem. The
objectivesaretominimizecostp erpart,pro ductiontimep erpart,androughness
of the work surface, simultaneously. They discuss a heuristic gradient-based
multiple criteria decision making approach which they apply to parameter
selection in metal cutting. For the metal cutting problem, they show how
ecient alternatives can b e generated by a discrete variable approach and how
the gradient-based multiple objective approach can b e implemented to obtain
the most preferred alternative. They also discuss their software package for
micro computers as a decision supp ort system for parameter selection. They
compare their computer aided machine parameter selection (CAMPS) package
optimization algorithms with di erent machining mo dels. Their approach is
limitedb ecause oftheuse ofgradientbasedmetho dswhichare notidealfor
non-convex problems. They conclude that the generalized reduced gradient metho d
isthe most suitablefor solving machiningoptimization mo dels.
Petrop oulos [36] has used geometric programming for optimization of
machining parameters. Multi-pass turning optimization has b een addressed by
Ermerand Kromo dihardjo [15]. Theyuse acombinationof linearand geometric
programming.
Iwata et al. [22] use a sto chastic approach to solve for optimal machining
parameters. Eskicioglu and Eskicioglu [16] demonstrate the use of non-linear
programming for machining parameter optimization. Hati and Rao [19] use
sequential unconstrained minimization technique (SUMT)to solve a multi-pass
turningop eration.
Khanetal. [25]studymachiningconditionoptimizationbygeneticalgorithms
and simulatedannealing. Althoughnonlinear and non-convexmachiningmo dels
develop ed with the objective of determining optimal cutting conditions are
traditionally solved using gradient based algorithms, they study three non
gradient based sto chastic optimization algorithms and test their eciency in
solving severalb enchmarkmachiningmo delswhichare complexb ecauseof
non-linearitiesand non-convexity.
Stori et al. [42] integrate pro cess simulation in machining parameter
optimizationandprop oseametho dologyforincorp oratingsimulationfeedbackto
ne-tuneanalyticmo delsduringoptimizationpro cess. Theypresentanon-linear
programming (NLP) optimization technique used to select pro cess parameters
based on closed-form analytical constraint equations relating to critical design
requirementsand execute simulation using these pro cess parameters, providing
predictions of the critical state variables. Then, they dynamically adapt
constraint equation parameters using the feedback provided by the simulation
predictions. Theyrep eatthissequenceuntillo calconvergenceb etweensimulation
and constraintequation predictions has b een achieved.
to ol parameters to control the required surface quality. Surface nish is an
imp ortant requirement for many turned work pieces in machining op eration.
The authors dealt with the interactions b etween the cutting parameters and
surfaceroughness. Theyinvestigatedthe e ectsofto olvibrationon theresulting
surfaceroughness in the dryturningop eration of carb onsteel. Theychosea full
factorial designthat allowedto consider the three-levelinteractions b etweenthe
cuttingparameters (cuttingsp eed,feed rate, to olnose radius,depth of cut,to ol
length,and workpiece length)on the twomeasureddep endent variables(surface
roughness and to ol vibration). Their results show that the factors having the
greatestin uenceonsurface roughnessarethe secondorderinteractionsb etween
cutting sp eed and to ol nose radius, along with third-order interaction b etween
feed rate, cutting sp eed and depth of cut. They had the b est surface nish at a
lowfeedrate, a largeto ol nose radiusand ahigh cuttingsp eed. Theyconcluded
thatfeedrateandto olnoseradiuspro ducedthemostimp ortante ectsonsurface
roughness, followed by cuttingsp eed.
Kyoungetal. [27]emphasizedtheimp ortanceofselectingto olsize,to olpath,
cuttingwidth at each to olpath prop erly and calculatingthe machiningtimefor
optimal pro cess planning. Since other factors dep end on the to ol size, it is the
most imp ortant factor in their problem. They presented a metho d for selecting
optimal to ols for p o cket machiningfor the comp onentsof injection mold. They
appliedthebranchandb ound metho dtoselecttheoptimalto olswhichminimize
the machining time by using the range of feasible to ols and the breadth- rst
search.
These mo dels consider only the contribution of machining time and to oling
cost to the total cost of op eration, and they usually ignore the contribution
of the non-machining time comp onents to the op erating cost, which could b e
very signi cant for the multiple op eration case. All of the time consuming
events except the actual cutting op eration are denoted as non-machining
time comp onents. Basic setup, to ol interchanging, to ol replacing, workpiece
loading-unloading, to ol tuning, to ol approach and stabilization etc., are the
determinants of these non-machining time comp onents. These studies also
excludethe to oling issues such as the to ol availability and the to ol life capacity
limitations. Therefore, their results might lead to infeasibilities due to to ol
contentionamong op erations for alimitednumb er ofto ol typ es [34].
Akturk and Avci [5] prop osed a solution pro cedure to make to ol allo cation
andmachiningconditions selectiondecisionssimultaneously. Theyalsotakeinto
account the related to oling considerations of to ol wear, to ol availability, and
to ol replacing and loading times,since they a ect b oth the machining and
non-machining time comp onents, hence the total cost of manufacturing. In their
study, they extend single machining op eration problem (SMOP) formulation
by adding a new to ol life constraint which enables them to include to oling
issues like to ol wear and to ol availability. Furthermore, they prop ose a new
cost measure to exploit the interaction b etween the numb er of to ols required
with the machining, to ol replacing and loading times, and to ol waste cost in
conjunction with the optimum machining conditions for alternative op
eration-to olpairs. Consequently,theypreventanyinfeasibilitiesthatmighto ccurforthe
to olallo cationproblematthe systemleveldueto to olcontentionamongto ollife
restrictions through a feedback mechanism.
Akturk and Onen [6] prop osed a new algorithm to solve lot sizing, to ol
allo cation and machining conditions optimization problems simultaneously to
minimize the total pro duction cost in a CNC environment. They integrated
the system, machine and to ol level decisions for pro duction of multiple parts
consisting of multiple op erations. By this way, they avoid any infeasibility that
may o ccur due to to oland machinehour availability limitations.
In a recent study, Akturk [3] develop ed an exact approach to determine the
optimum machining conditions and to ol allo cation decisions simultaneously to
minimizethetotal pro duction cost on aCNCturningmachinewherealternative
to ols canb e usedforeachop eration. Heemphasizedtheto olmanagementissues
at the to ol level such as the optimum machining conditions and to ol
selection-allo cation decisions considering the to ol life, machining op erations and to ol
an ecientsolution pro cedureto determineconcurrentlythe optimal machining
conditions ofcutting sp eed and feedrate, the optimalop eration-to ol assignment
and optimalallo cation of to ols.
2.1.2 Tool Replacement
Acompleteto olreplacementstrategysp eci esato olchangeschedulebasedup on
theeconomicservicelivesofto olsandacontrolp olicyregardingunscheduledto ol
changes following breakage. In cases where to ol life is not deterministicand all
to ols in the magazine do not require reconditioning at the same time, the to ol
replacement problem gets more complicated. The to ol replacement p olicies are
concernedwiththecomplexdecisionsofwhentoreplaceaparticularto olandhow
manyotherto olstoreplacealongwiththisparticularto ol. Thedistributednature
ofto ollivesunderactualmachiningparametersand theoptionto changeseveral
to ols when one fails, rather than considering only exp ectedlives and single to ol
replacementsare consideredinmostrealisticreplacementstrategies. Ignoringthe
relationship b etween the pro cessing rates and the to ol replacement p olicy, and
overlo oking the impact of to olsharing on setup timesresult in de cientmo dels.
Most of the studies assume constant pro cessing times and to ol lives though
the to ol wear can have a signi cant impact on the to ol replacement frequency.
Op erational problems concerning to ol magazine arrangements and op erations
sequencingdecisionsare consideredat the systemlevelinan aggregated manner.
Possibility of to ol sharing and loading duplicate to ols due to to ol contention
among the op erations for a limited numb er of to ol typ es as a result of the
to ol availability and to ol life limitations is ignored. However, such op erational
problemsshouldb etakenintoaccountforareliablemo delingofFMSs,otherwise
the absence of such crucial constraints may lead to infeasible results. Suri and
Whitney[43]emphasizedthataninclusionoftheseissuesinthepro cessplanning
will provide an e ective decision making to ol for the short term op erational
decisionsof FMSs.
by allowing moreaccurate p ortrayal of the op eration of CNCmachineswith an
inclusion of to ol contention, to ol life, precedence and to ol magazine capacity
restrictions.
Scheduling jobs with to oling constraints can b e studied on two topics, one
assuming that each job has di erent to ol requirements and after nishing one
job,the to ols necessary forop eration ofthe secondjob isloaded to the machine.
By using the common to ols used and sequence dep endent setups, an ecient
schedule of to ols can b e determined. The second research areain job scheduling
withto olingconstraintsisbasedon theassumptionthatthereisasingleto oltyp e
and the to ol issubjectto wear. Since to olchangetimesare generally signi cant
when comparedto the pro cessing times,and to ol lives are short relative to the
planning horizon, it is imp ortant to schedule the jobs for p ossible scheduling
objectives suchas owtime,tardiness, etc.
The rst approach is not studied much in literature, esp ecially with cost
terms related to scheduling decisions. Tang and Denardo [44] study the single
machine case with given to ol requirements where to ol changes are required due
to part mix. Their objective is to minimize the numb er of to ol switches and
they provide heuristicalgorithms for job schedulingin this environmentand an
optimal pro cedure, KTNS rule. In a companion pap er [45], they also study the
case of parallel to ol switchings with the objective of minimizing the numb er of
switching instants.
Sincethe to olsare usuallychangedmoreoftenb ecauseofto olwearthanpart
mix,thesecondapproachseemsmorerealistic. Usingthisapproach,Akturketal.
[4] intro duced a schedulingproblem that considers the to ol change requirement
duetoto olwear. Intheirproblem,asingleCNCmachineandnindep endentjobs
that are ready for pro cessing at time zero are given. The job pro cessing times
are assumed to b e known, and onlyone typ e of to olwith a known, constant life
andunlimitedavailabilityisrequired. Theto olchangetimeisalsoassumedtob e
constant. They tried to nd a schedule that will minimizethe total completion
timeof the jobs. They havesummarizedand discussedthe basic characteristics,
and havetested the relativep erformanceof variousalgorithms.
2.2 Scheduling
Scheduling is concerned with determiningthe sequence in which available work
should b epro cessed to optimize systemp erformance.
Standard formulations of the schedulingproblemassume that job pro cessing
times are xed and known in advance of scheduling. In practice, pro cessing
times are often a function of the amount and mix of resource inputs allo cated
to a job. These resources can vary dep ending on the system. For instance, in a
pro duction facilitycomp osed of CNCmachines, machine cuttingsp eed and feed
rateare e ectiveparameters changing the pro cessing timesandto ol usagerates.
In a relatively lab or-intensive systems, pro cessing timetypically dep end on the
numb erand typ e of the workersallo cated to the system(Daniels et al. [11]).
2.2.1 Controllable Processing Times
Pro cessing time control and its impact on sequencing decisions and op erational
p erformance has received limited attention in the scheduling literature. Some
mo delsforsingle-pro cessor systemshaveb eendevelop edand studiedconcerning
controllable pro cessing times. Extensions to parallel-machine environments are
alsoaddressed byresearchers. A survey oftheliteratureup to 1990can b efound
inNowicki and Zdrzalka[33].
Danielsand Sarin [12] consider the problemof joint sequencing and resource
allo cation when the scheduling criterion of interest is the numb er of tardy jobs
and derivetheoretical results that aid in developing the trade-o curveb etween
the numb erof tardy jobs and the total amountof allo cated resource.
PanwalkerandRajagopalan[35]considerthestaticsinglemachinesequencing
problem with a common due date for all jobs in which job pro cessing times are
controllablewith linearcosts. Theydevelop ametho dto nd optimalpro cessing
timesand an optimal sequence to minimizea cost function containing earliness
Adiri and Yehudai [2] study the problem of scheduling identical parallel
pro cessors whose service rates can change b etween jobs. Trick [48] fo cuses
on assigning single-op eration jobs to identical machines while simultaneously
controllingthe pro cessing sp eed of each machine.
Zdrzalka [52] deals with the problem of scheduling jobs on a single machine
in which each job has a release date, a delivery time and a controllable
pro cessing time,having its own asso ciated linearlyvarying cost and prop ose an
approximation algorithmfor minimizingthe overall schedule cost.
Ishiietal. [21]considerthe problemwithparalleluniform machinesinwhich
thesp eed of amachineisacontinuousnonnegativevariable andthe compression
cost is afunction of the sp eed of the machine.
Cheng et al. [9] consider a parallel machine scheduling problem with
controllablepro cessing times,where the jobpro cessing timescan b ecompressed
through incurring an additional cost, which is a convex function of the amount
of compression. They formulate two problems, one to minimize the total
compression cost plus the total ow time, and the other to minimize the total
compression cost plus the sum of earliness and tardiness costs for the common
due date scheduleproblem.
Danielsetal. [11]investigatetheimprovementsinmanufacturingp erformance
that can b e realized by broadening the scop e of the pro duction scheduling
functionto include b oth jobsequencingand pro cessing-timecontrol through the
deploymentofa exibleresource. Theyconsideranenvironmentinwhichasetof
jobsmustb e scheduledoverasetof parallelmanufacturingcells,eachconsisting
ofasinglemachine,wherethepro cessingtimeofeachjobdep endsontheamount
of resourceallo cated to the asso ciated cell.
Karabati and Kouvelis [24] solve the simultaneous scheduling and optimal
pro cessing-times selection problem in a ow line op erated under a cyclic
scheduling p olicy. They address the simultaneous scheduling and
optimal-pro cessing-times selection problem in a multi-pro duct deterministic ow line
op erated under a cyclic scheduling approach. They provide a mo deling
selectionconsiderations. After presenting a linear program solving the
optimal-pro cessing-timesselectionproblemfora givencyclicsequence,they demonstrate
forlarge problems,how the use of a rowgeneration schemeallows themto solve
it more eciently than standard linear programming co des. For the solution
of the simultaneous scheduling and optimal-pro cessing-times selection problem,
they prop ose a simple pro cedure that iteratively solves cyclic scheduling and
optimal-pro cessing-timesselectionsubproblems forgivensequences.
The concept of controllablepro cessing timescan also b e observed inproject
management with controllable activity durations. In 1980, Vickson treats the
problem of minimizing the total weighted ow cost plus job pro cessing cost in
a single machinesequencing problem for jobs having pro cessing costs which are
linear functions of pro cessing times in his rst study [50]. In his second study
[51], he extends his initial study and presents simple metho ds for solving two
single machine sequencing problems when job pro cessing times are themselves
decisionvariableshavingtheir ownlinearlyvaryingcosts. Theobjectivesstudied
are minimizingthe total pro cessing cost plus either the average ow cost or the
maximumtardiness cost. He treats only the problemswith zero ready timeand
no precedenceconstraints.
2.2.2 Machine Availability
As discussed in Lee et al. [31] and Pinedo [37], most theoretical mo dels do
not take into account the unavailability of resources. It is usually assumed that
the machine is available at all times. However, machines are not continuously
available in the real world. Certainly, this observation is valid for the machine
to ols, and the unavailability of to ols isa morecommon situation since the to ols
actually have short lives with resp ect to the planning horizon, as rep orted by
Gray et al. [18].
Inthe literature,thereare nostudiesconsideringthe to ollifeandto olchange
time requirement due to to ol wear, and incorp orating them with scheduling
similarcharacteristicswith the schedulingwith to ol changes problem.
In literaturethree cases are discussedforthis problem. When the job cannot
b e nishedb efore thenext down p erio dof a machineand the jobhas to restart,
then the job is called non-resumable. If the job has to partially restart afterthe
machine has b ecome available, then it is called semi-resumable. If the job can
continue to b e pro cessed on the same machine after the machine has b ecome
available, then the job is calledresumable.
The researchers on schedulingwith availability constraint mostly fo cused on
machine breakdowns and maintenanceintervals. The most common objective is
minimizing the total owtime. Adiri et al. [1] considered owtime scheduling
problem when machine faces breakdowns at sto chastic time ep o chs, and repair
timeis also sto chastic. The pro cessing timesare assumedconstant and the jobs
are assumed non-resumable. They have provided the NP completeness result
of the problem, and showed that SPT minimizesexp ected total owtimewhen
timestobreakdownareexp onential. Inthecaseofsinglebreakdownandconcave
distribution function of the time to breakdown, they have again showed the
sto chastic optimality of SPT. They have also analyzed the single deterministic
breakdown case, and found a worst case p erformance b ound for SPT heuristic,
whichwas 5/4.
Lee and Liman [28] have also studied the same problem considering only
deterministicsingle scheduled maintenancecase. They not only give a simpler
pro of of NP completeness but also found a b etter b ound for SPT, b eing 9/7.
Moreovertheyhave shown that this b ound is tight.
Lee [30] studies the single machine problem for di erent p erformance
measures. He shows that the makespan for a single machine problem with
resumable availability constraint is minimized by an arbitrary sequence. The
minimization of ow time with resumable availability constraint on a single
machine problem is solved optimally by Shortest Pro cessing Time (SPT)
algorithm. In SPT, the jobs are scheduled in nondecreasing order of pro cessing
times of jobs. Minimization of maximumtardiness can b e solved optimally by
order of due dates of the jobs.
Therearealso somestudieson owshopandparallelmachineschedulingwith
anavailabilityconstraint. LeeandLiman[29]consideredtwomachinesinparallel
schedulingproblem of minimizingthe total completiontimewhere one machine
isavailableallthetimeandthe othermachineisavailablefromtimezeroup to a
xedp ointintime. Afterprovingthatthe problemisNP-complete,theyprovide
a pseudo-p olynomial dynamic programming algorithm. They also prop ose a
heuristic which is based on a slight mo di cation of SPT rule considering the
capacity of the machine with availability constraint. This heuristic is shown to
havean errorb ound of 0.50.
However,allthesestudiesassumeasinglebreakdownormaintenanceinterval.
But,intheschedulingproblemwithto olchangesthisisnotarealisticassumption
andwecanhaveseveralto olchangesinagiventimep erio dduetorelativelyshort
to ollives.
2.3 Conclusion
In the literature, pro cessing times and to ol lives are taken as constant, either
deterministic or probabilistic. However, they are closely related with the
machining conditions. Hence, the pro cessing times and to ol usage rates of the
jobs are controllable. Inthe literatureof schedulingwith controllable pro cessing
times,most of the studies assumethat the pro cessing timescan b e crashed in a
range with linear compression cost. But, for our case, the pro cessing times are
closelyrelated with to oland op eration parameters.
Another common drawback observedin scheduling literatureis that they do
nottakeaccountofto olchanges dueto to olwear althoughthe to olchangetimes
are signi cant compared to pro cessing times and to ols are changed frequently
due to wear. There are few studies considering the resource unavailability, but
the resources in scheduling theory are mostly considered as machines, without
referringto the to oling level.
considered to b e due to part mix, that is, due to di erent to oling requirements
of the parts. However,to ols have limitedlives and they are subjectto wear out
inpractice.
Asaresult,schedulingjobswhichhavecontrollablepro cessingtimesandusage
ratesdep ending on machiningconditionson aCNCmachinehavingto olchanges
due to to ol wear is an untouched topic in the literature. Bard [8] indicatesthat
\Although the single machine scheduling problem has b een studied extensively,
the added complication of to ol loading undermines the usefulness of much of
the current results". The objective of the research rep orted in this thesis is
to show how closely to ol replacement, machining conditions optimization and
scheduling of the jobs in a CNC machine are related. These topics have b een
studiedseparatelybymanyresearches,howeverthereisno study that integrates
allof these and investigatesthe interactions among them.
In this chapter, we intro duce a short review of the literature on to ol
management and scheduling issues which is related with our problem in some
asp ects, and state the similarities and diversities of our problem b etween the
problemsstudied inliterature.
In the next chapter, we give the de nition and underlying assumptions
of the problem, present the mathematical programming formulations of two
Problem Statement and
Modeling
3.1 Problem De nition
We are given N jobs with a sp eci ed depth of cut, length and diameter of the
generated surface along with maximum allowable surface roughness attributes.
The problem is scheduling these jobs on a CNC machine in order to minimize
the total completiontime. There isa single to oltyp e whichhas a constant to ol
changing time. Whenthe to ollifeisover,the to olhas tob echanged. Sincethere
isato olchangetime,sometimessigni cant,andita ects thecompletiontime,it
isimp ortantto consideritintheschedule. ThemachiningconditionsoftheCNC
machine can b echanged, and for eachjobit can b e adjustedto di erent cutting
sp eed and feed rate pair. Howeverthere are some constraintsfor these settings.
The sp eed and feed rate have to satisfy the machine p ower, surface nish and
to ol life constraints. After detecting the feasible region of sp eed and feed rate,
we have to maketwo decisions,a feasiblesetting for each job, and the sequence
of the jobs.
3.2 Assumptions
Weaimtosolvetheschedulingproblemofjobswithcontrollablepro cessingtimes
environmentto minimizethe total completiontime. The assumptions ab out the
op erating p olicy and the characteristics of the system considered in this study
are as follows:
There is a single machine which is continuously available except the to ol
changes.
Thereare N jobswith no precedencerelation,all ready at timezero.
Depth of cut, length and diameter of the surface and maximumallowable
surface roughness values for each jobare givena priori.
The pro cessing of a job can b e accomplished by a single to ol, i.e. usage
rates are smaller than one.
The parts to b e pro cessed are comp osed of asingle op eration.
Thereis one typ e of cutting to ol with aknown to ol life, i.e. total usage of
the to ol cannot exceed1.
Thereare unlimitedamount of to ols availablefor replacement.
Whenthe usagerate of to olends, i.eto olis wornout, to ol has to b etaken
o the machine, and a new one has to b e placed. The timesp ent for this
pro cess, i.e. to olchange time,is constant.
The cutting sp eed and feed rate of the machine constitute the machining
conditions and they can easily b e adjusted to new settings. However,
cutting sp eed cannot b e lower than 100 fpm and b oth cutting sp eed and
feed rate are subject to some constraints related with the p ower of the
machine, surface nishof the parts and lifeof the to ol.
Pro cessing time and to ol usage rate of the job are determined via cutting
sp eed and feed rate.
Under these assumptions, we wish to determine the optimum machining
conditions and nd a schedule that minimizesthe total completion time of the
jobs.
The notation used throughout the thesis isas follows:
; ; : sp eed,feed, depth of cut exp onentsfor the to ol
C
m
;b;c;e : sp eci c co ecientand exp onents of the machine p ower constraint
C
s
;g;h;l : sp eci c co ecientand exp onents of the surface roughness constraint
C : Taylor's to ol lifeexpression parameter
C
o
: op erating cost of the CNC machine($/min)
C
t
: cost of the to ol ($)
d
i
: depth of cut for job i (in)
D
i
: diameterof the generatedsurface forjob i (in)
L
i
: length of the generated surface for job i(in)
H : maximumavailablemachine p ower (hp)
S
i
: maximumallowable surface roughness for job i (in)
v
ij
: cutting sp eed for settingj of jobi (fpm)
f
ij
: feed rate for settingj of jobi (ipr)
U
ij
: usage rate of job i using settingj
P
ij
: machining timeofjob i using setting j (min)
T
c
: to ol change time(min)
N : numb erof the jobs
S : numb erof di erentsettings generated for eachjob
3.3 Model Building
The pro cessing times and to ol usage rates of jobs determinedby the machining
parameters, v and f, by some well known formulas as discussed in Akturk and
Avci[5]. P = D L 12 v 1 f 1
U = DLd 12Cv (1 ) f (1 )
Ifaschedule isviewedas asequenceofblo cksofjobs,whichare sep eratedby
to ol changes, the problem is deciding on the optimummachining conditions for
each job and partitioning the jobs into blo cks. Since the machining conditions
determine the machining time directly and to ol change instances indirectly,
partitioningthe jobs into minimumnumb erof blo cks do es not implyoptimality.
Increaseinthe sp eed of the machinecan cause onemoreto olto b eused b ecause
of the increase in usage rate and so one more to ol change time which will shift
the completion time of the succeeding jobs. On the other side, decrease in the
machining time of the jobs will also decrease the comletion time of those jobs.
Thistrade-o mayresult inadecreaseinthe totalcomletiontime. Thereforewe
cannot say that the solution with minimumnumb er of blo cks is optimal. This
two-sidee ectofthe machinesettings,v and f, onthe objectivefunction canb e
b etter seen b elow:
MIN N X i=1 S X j=1 N X k =1 (N k+1)P ij X ijk +T c N 1 X k =1 (N k)R k where X ijk = 8 < :
1 if job i under condition j isp ositioned at k
0 otherwise R k = 8 < :
1 if to olis replacedafter p osition k
0 otherwise
T
c
= to olchange time
The rst part of the objective function is the total completion time of the
jobs ignoring the shifts of to ol changes. The second part gives the total shift
on the completiontimedue to to ol changes. As the to ol change timedecreases,
P
ij
values dominate T
c
for the scheduling decision and numb er of to ol changes
done mayb ecome less signi cant. Thus the rst part gains imp ortanceand the
problem converges to a classical schedulingproblem. On the other hand, when
T
c
value dominates P
ij
values,the secondpart of the objectivefunctionb ecomes
more imp ortant and the problem b ecomes similar to the bin-packing problem,
but certainly not equivalent.
Two questionshave to b e answered ab out this problem. These are:
1. What is the optimalsetting pair, v and f foreach job?
2. What is the optimalsequence of these jobs?
With their settings determined,we willhave usage rate and machiningtime
data of the jobs on hand. Using this informationin addition to the sequence of
the jobs, we can get the whole schedule showing b oth the to ol change and job
pro cessing instances. In order to ease the answer of the rst question, we take
discrete setting pairs from the feasibleregion of these two machiningconditions
v and f to b e alternatives for the jobs to b e selected. In the next section, we
will explain the pro cedure of generating discrete p oints from the feasible region
as alternativesettings.
3.4 Generate Settings
InCNCmachines,wecancontrolthe machiningtimesandusagerates ofjobsby
changing the sp eed and feed rateof the machine. While changing the machining
conditions, we have constraints such as machine p ower, surface roughness and
C 0 t v ( 1) ij f ( 1) ij
1 (To ol lifeconstraint)
C 0 m v b ij f c ij
1 (Machinep ower constraint)
C 0 s v g ij f h ij
1 (Surface roughness constraint)
v ij ;f ij >0 where C 0 t = D i L i d i 12C ,C 0 m = C m d e i H , C 0 s = C s d l i S i
Werelaxtheto ollifeconstraintnow,andwillcheckitsfeasibilitylater. Ifthe
optimalsolutionofthe relaxedproblemsatis estheto ollifeconstraint,thenitis
optimalfor the overall problem. Ifitdo es not satisfy the constraint,then a new
optimalsolution should b e found. The relationship b etweenmachine p ower and
surface roughness constraintscan b e seen inFigure 3.1. The to ollife constraint
may b e in one of 4 situations. It can b e redundant, crossing machine p ower
constraint,crossingintersectionp ointofthesetwoconstraintsorcrossing surface
roughnes constraint.
machine power
surface
roughness
cutting speed
feed rate
FEASIBLE REGION
Tool life 2
Tool life 1
Tool life 3
Tool life 4
Figure 3.1: Feasible region of machinesettings
Akturk and Avci [5] prove that at least one of the surface roughness and
machinep owerconstraintsisbindingat optimalityforSMOP.Thus,anyinterior
p oint of Figure 3.1 willgive a higher machiningtime value than the ones lying
Since it will b e hard to nd the optimum settings for each job from this
continuousp olyline,somediscretep ointsare chosenas alternativesandeachjob
is assigned to one setting among these alternatives. Instead of cho osing random
p oints, we try to nd meaningful p oints on this p olyline which minimizesome
objectives. The following three objectives are used to nd the strategic p oints.
The formulations and notations are takenfromthe thesis ofOzkan [34].
machiningtime: t
m
pro cessing time: t
p =t m +T c U
total manufacturing cost : TMC =C
o t m +C o T c U +C t U Minimizing t m
means maximizing v and f, however minimizing U means
minimizing v and f (rememb er the machining time and usage rate formulas).
Therefore the rst objective will have the highest (v;f) values as optimal and
optimal(v;f)values for the second objective willnot b e less than the third one
since the weightof U in the third objectiveis higher. The characteristics of the
optimalp oints under these objectivesare stated b elow.
1. t m = D i L i 12 v 1 ij f 1 ij
Let the p oint minimizing the machining time b e (v
a ;f
a
). According to the
theorem proved by Akturk and Avci [5], at least one of the surface roughness
or machinep owerconstraints must b e tight at the optimal solution.
In case machiningp ower constraint is tight,
f a =(C 0 m ) 1=c v b=c a
plugging it into the objective,
Min machiningtime=Min v (b c)=c
a
Ifb>c>0or b<c<0,the objectivemeans minimizing v
a .
Resp ectively, b and c are the exp onents of the cutting sp eed and feed rate in
feed rate always increases the machine p ower. With this information on hand,
we can reduce the inequalityconditions only to b>c.
In case surface roughness constraint istight,
f a =(C 0 s ) 1=h v g =h a
plugging it into the objective,
Min machiningtime=Min v (g h)=h
a
Ifh >0;h >g or h<0;h<g, the objectivemeans maximizingv
a .
Resp ectively,g and hare exp onentsof thecutting sp eedand feedrate insurface
roughness constraint. g is always negative since cutting sp eed and surface
roughness are inversely related. However, increasing the feed rate increases the
surface roughness, therefore h is a nonnegative co ecient. Consequently, the
ab oveinequality conditions are always satis ed.
As a result, if b> c, (v
a ;f
a
) is always at the intersection. If (v
a ;f
a
) satis es
the to ol life constraint, then it is optimal. If it do es not satisfy, i.e. to ol life
constraint is in situation 4 as in gure 3.1, then the intersection of to ol life
and surface roughness constraintsis takenas (v
a ;f
a
). By the help of this study,
weprovedanimp ortanttheoreminadditiontotheoneAkturkandAvci[5]found.
Theorem: Thesurfaceroughnessconstraintmustb ebindingatoptimalityunder
the condition that b> c,i.e. machine p ower is moresensitive to the changes in
cuttingsp eed than feedrate.
2. t p = D i L i 12 v 1 ij f 1 ij + D i L i d i Tc 12C v 1 ij f 1 ij
Let the p oint minimizing the pro cessing time b e (v
b ;f b ). Since (v a ;f a ) is at the intersection, (v b ;f b
) cannot b e b eyond that p oint, therefore it is in the feasible
region where surface roughness constraint is tight. With this information, we
reduce the variables to 1,and simplytaking the derivativeof the objective, nd
the p oint which minimizes it. If the p oint we found at the end is feasible, i.e.
not,thentheintersectionp ointistheoptimalone. Therefore,(v a ;f a )and(v b ;f b ) coincides. 3. TMC =C 1 v 1 ij f 1 ij +C 2 v ( 1) ij f ( 1) ij where C 1 = D i L i C o 12 , C 2 = DiLid i (Ct+CoTc) 12C Thep oint(v c ;f c
),whichminimizesthetotalcostisalsoundertheintersection
p oint, therefore the same pro cedure is applied as (v
b ;f b ). However, (v c ;f c ) is
never exp ected to b e b eyond the p oint (v
b ;f
b
) since the weight of usage rate in
this objectiveismorethan the other two.
Theremayo ccur threedi erentsituationsafterdetectingthesep oints. These
are; 1. (v a ;f a ),(v b ;f b ) and (v c ;f c
)can coincideat the intersectionp oint.
2. (v a ;f a )and (v b ;f b
)can coincideat the intersectionp oint.
3. Noneof themcoincides.
Asaresult,wehaveatleast1,atmost3di erentp ointsofsettings. Randomly
generating other p oints b etween these p oints in the feasible region, we can get
as many settingsas requested. A pro cedure is develop ed to select S settings for
eachjob. Firstly,S values of cutting sp eed are generated and the corresp onding
feed rate, usage rate and machining time are calculated by using the formulas
b elow. This pro cedureis rep eatedfor each job.
f =( C s d l S v g ) 1 h U = D Ld 12Cv (1 ) f (1 ) t m = D L 12 v 1 f 1
The pro decure of selecting S values of cutting sp eed for three cases is as
follows.
Case I
We have a p oint at the intersection which minimizes the machining time,
pro cessing time and total manufacturing cost, and we have a lower b ound v
l
for the machine sp eed rate. Let v a
b e this intersection p oint and r is the
intervalb etweentwo sp eedvalues generated. riscalculated as anintegerpartof
(v a
v
l
)=(S 1) and isalso used inpro cedures of othercases. For this case, the
algorithmhas only one step to calculate S cutting sp eed values.
STEP 1. For everyj value from0 to (S 1),calculatesp eed as v
j =v
a
(jr )
Case II
Wehavetwodi erentp oints. The rstone, whichisat theintersectionp oint,
minimizesthemachiningtimeand pro cessingtimeandthe secondoneminimizes
the total manufacturing cost. Let v a
and v c
b e these two p oints. The pro cedure
of ndingS cuttingsp eed values is:
STEP 1. Calculates
1
as 1 plus integer part of (v a v c )=r . s 1
is the numb erof settingsgenerated b etweenp oints v a and v c . STEP 2. Calculater 1 as integer part of (v a v c )=s 1 . r 1
isthe intervalof sp eed used b etweenp oints v a
and v c
.
STEP3. Foreveryj valuefrom0to (s
1 1),calculatesp eed as v j =v a (jt 1 )
STEP 4. For everyj value froms1 to (S 1),
calculatesp eed as v j =v c [(j s 1 )r ] Case III
We have three di erent p oints. The rst one, which is at the intersection
p oint, minimizes the machining time, the second one minimizes the pro cessing
time, and the third one minimizes the total manufacturing cost. Let v a ;v b and v c
b e these p oints. The pro cedureof nding S sp eed values is:
STEP 1. Calculates
1
as 1 plus integer part of (v a v b )=r . s 1
is the numb erof settingsgenerated b etweenp oints v a and v b . STEP 2. Calculater 1 as integer part of (v a v b )=s 1 . r 1
isthe intervalof sp eed used b etweenp oints v a
and v b
.
STEP 3. For everyj valuefrom0to (s
1 1) calculatesp eedas v j =v a (jr 1 ) STEP 4. Calculates 2
as 1 plus integer part of (v b v c )=r . s 2
is the numb erof settingsgenerated b etweenv b and v c . STEP 5. Calculater 2 as integer part of (v b v c )=s 2 . r 2
isthe intervalof sp eed used b etweenp oints v b
and v c
.
STEP 6. For everyj value froms
1 to (s 1 +s 2 1), calculatesp eed as v j =v b [(j s 1 )r 2 ]
STEP 7. For everyj value from(s
1 +s 2 ) to (S 1), calculatesp eed as v j =v c [(j s 1 s 2 )r ]
S numb er of setting data for every N job is generated. After detecting the
feasibleregion of sp eed and feed rate, we have to make two decisions,a feasible
settingforeachjob,and the sequenceofthe jobs. As mentionedin theliterature
review chapter, these two questions are studied in the literaturesep erately and
b efore dealing with the original problem, we will intro duce two sub-problems
related with these questions. The rst sub-problem is nding a setting for each
jobgiventhe sequenceof jobs,and thesecondone is ndingthe sequenceofjobs
3.5 Find the Optimal Settings Given the
Sequence
As we mentioned in the literature review chapter, there are several studies on
machining conditions optimization. Cutting sp eed and feed rate are taken as
decision varibles in most of these studies. However, the machining conditions
are optimized for a manufacturing pro cess related objective function without
considering their impact on the scheduling problem. The problem we present
here di ers fromour originalprobleminthe way that sequenceis xed. Givena
sequenceofjobswithattributes(D ;L;d;S),we ndtheoptimumsetting(sp eed,
feedrate)foreachjobthatminimizesthe totalcompletiontime. Aftergenerating
alternative settings for eachjob as explained ab ove, a mixedinteger program is
solved to nd the optimal settings, and to ol change instances. Picking discrete
settings from the feasible region will ease cho osing a setting among alternative
settings. Since S numb er of setting pairs (sp eed, feed rate) are chosen from the
feasibleregion for eachjob, the problem reducesto assigning one setting among
S for eachjob inorder to minimizetotalcompletiontime.
Inputs are D ;L;d;S of each job, , , , C, C
t , T c of the to ol, H, C o of the machine and b;c;e;C m ;g;h;l ;C s
co ecients. Outputs are settings of the jobs,
and the instantsto ol change isdone. This data givesthe schedule ofthe jobs.
We have N jobs with a predetermined sequence. We have already found S
di erentsp eedand feedratepairsfor eachjob byconsideringthe machinep ower
and surface roughness constraints. We also have usage rate and machiningtime
dataofthe(job,setting)pair. Ouraimistoselecttheoptimalsettingamongthe
alternatives for each job which minimizesthe total completiontime on a single
CNC machine. The following mixed integer programming (MIP) mo del can b e
MIN N X i=1 S X j=1 (N i+1)P ij X ij +T c N 1 X j=1 (N j)R j ST S X j=1 X ij =1 i=1;:::;N S X j=1 U ij X ij +d i 1 d i 0 i=1;:::;N d i S X j=1 U ij X ij d i 1 +R i 0 i=1;:::;N S X j=1 U i+1j X i+1j +d i 1 i=1;:::;N d 0 =0 X ij 2f0;1g i=1;:::;N j =1;:::;S R i 2f0;1g i=1;:::;N 1 d i 0 i=1;:::;N 1 where X ij = 8 < :
1 if job i ispro cessed by using settingj,(v
ij ,f ij ) 0 otherwise R i = 8 < :
1 if to ol isreplaced afterjob i
0 otherwise
U
ij
=usage rate of job iunder setting j
P
ij
=machiningtimeof job iunder setting j
T
c
= to olchange time
The objective is to minimize total completiontime. The rst constraint set
guaranteesthatonlyonesetting,i.e. sp eed andfeedratepair,isselectedforeach
job. Thesecondandthird constraintsets maked
i
equalto totalusageofthe to ol
if thereis noto ol change,i.e. R
i
=0,or equalto 0if thereis ato olchange. The
3.6 Find the Optimal Schedule Given the
Settings
This second problem di ers from our original problem in the way that settings
of the jobs are xed, and the problem reduces to a single machine scheduling
withto olchangesto minimizethe totalcompletiontime. Thisproblemisexactly
the one Akturk et al. [4] studied. They show that this problem is NP-hard in
the strongsense. Theypresent adynamicprogrammingformulationto solvethe
problem optimally. Here,we prop ose a mixedinteger programmingformulation
forthe sameproblem. Theyshowed somesolution prop erties whichare not only
valid for this sub-problem but also for our original problem. We will deal with
these prop ertiesin detail later.
We have N jobs with predetermined machining timesand usage rates. The
problemis nding an optimalsequence, i.e. schedule, whichminimizesthe total
completion time of the jobs. The following mixed integer programming (MIP)
mo del is used to solve the problem of scheduling these jobs considering to ol
MIN N X i=1 N X j=1 (N i+1)P i X ij +T c N 1 X j=1 (N j)R j ST N X j=1 X ij =1 i =1;:::;N N X i=1 X ij =1 j =1;:::;N N X i=1 U i X ij +d j 1 d j 0 j =1;:::;N d j N X i=1 U i X ij d j 1 +R j 0 j =1;:::;N N X i=1 U i X ij+1 +d j 1 j =1;:::;N d 0 =0 X ij 2f0;1g i =1;:::;N j =1;:::;N R j 2f0;1g j =1;:::;N 1 d j 0 j =1;:::;N 1 where X ij = 8 < :
1 if job i isscheduledat p osition j
0 otherwise R j = 8 < :
1 if to olis replacedafter p osition j
0 otherwise
P
i
= machiningtimeof job i
U
i
= usage rate of job i
T
c
= to olchange time
The rstand secondconstraintsets guaranteethat one jobisassigned to one
p ositionandonep ositionisassigned toeachjob. Thethirdand fourthconstraint
setsmaked
i
equaltototalusageoftheto olifthereisnoto olchangeafterp osition
i,i.e. R
i
3.7 MIP of the Original Problem
Inthis section,we prop osea detailedmathematicalmo delfor theop eration of a
CNCmachiningcenterwhichwillincludethesystemcharacterization,thecutting
conditions and to ol liferelationship, and related constraints.
MIN N X i=1 S X j=1 N X k =1 (N k+1)P ij X ijk +T c N 1 X k =1 (N k)R k ST S X j=1 X ijk =1 i =1;:::;N k =1;:::;N N X k =1 X ijk =1 i =1;:::;N j =1;:::;S N X i=1 X ijk =1 k =1;:::;N j =1;:::;S N X i=1 S X j=1 U ij X ijk +d k 1 d k 0 k =1;:::;N d k N X i=1 S X j=1 U ij X ijk d k 1 +R k 0 k =1;:::;N N X i=1 S X j=1 U ij X ijk +1 +d k 1 k =1;:::;N d 0 =0 X ijk 2f0;1g i =1;:::;N j =1;:::;S k =1;:::;N R k 2f0;1g k =1;:::;N 1 d k 0 k =1;:::;N 1 where X ijk = 8 < :
1 if job i under settingj is scheduledat p osition k
0 otherwise R k = 8 < :
1 if to olis replacedafter p osition k
0 otherwise
P
ij
U
ij
=usage rate of job iunder setting j
T
c
= to olchange time
As in the sub-problems, the objective is to minimize the total completion
time. The rst constraint set guarantees that only one setting is selected for
each job. The second and third constraint sets guarantee that one p osition is
assigned to each job and one job is assigned to each p osition. The fourth and
fth constraint sets make d
k
equal to total usage of the to ol if there is no to ol
change after p osition k, i.e. R
k
= 0, or equal to 0 if there is a to ol change.
Finally,the sixth constraint preventsthe total usage of the to olexceeding1.
3.8 Conclusion
In this chapter,we havegiven the de nition and the underlying assumptions of
the jointschedulingand to olmanagementproblem. We presented mathematical
formulations of two sub-problems of our original problem which are studied in
literature separately. Then, we built a mathematical mo del in order to nd
the optimal machine settings for each job and schedule of these jobs giving the
minimumtotal completiontime.
In chapter4 we will concentrate on the solution of the problemusing
Proposed Heuristic Algorithms
In the previous chapter, the problem is de ned and the assumptions are listed.
Moreover,the mathematicalprogrammingformulations aregivenforthe original
problem and two sub-problems of the original one which are studied in the
literatureseparately. Akturk et al. [4] provedthe NP-hardness of their problem
which is a sub-problem of our original one. Therefore, no algorithmis likely to
b e prop osed for solving the problem optimally in p olynomial time. Hence, it is
justi ableto tryheuristic metho ds to solveour problem.
In this chapter, after giving the characteristics of the problem which willb e
usefulinsolutionpro cedures,wepresentthreestagesingle-passheuristicmetho ds
using simpledispatching rules either created by us or existing in the literature.
Furthermore,we intro ducethe problemspacegeneticalgorithmas alo calsearch
algorithminwhich single-pass heuristicsare used as base heuristics.
4.1 Characteristics of the Problem
In the problem of scheduling with to ol changes, the jobs sharing the same to ol
can b e considered as a blo ck, and a schedule can b e viewed as blo cks of jobs
separated by to ol changes. Akturk et al. [4] represent this situation as in gure
4.1.
In our problem, length of the blo cks, i.e. life of the to ols, are not constant.
Tc
Tc
Tc
block 1
block 2
block 3
block 4
Figure 4.1: Representationof a scheduleas blo cks of jobs
usagerate ofthejobs. Whenthe remainingusagerate ofto olislessthanthe job
tried b eing to b eplaced, we are facedwith two choices:
1. Either replacethe to ol with a new one, thussp end timeT
c ,
2. or, change the machining conditions to t the usage rate of the job, thus
increasemachining timeof the job.
As a result, we can stretch the blo ck to t more jobs with higher machining
times,andby this waywewillgain from the to ol change time. On the contrary,
we can constrict the blo ck in order to have low machining times. The relation
b etweenthese timecomp onentsand cutting sp eed can b eseen in gure 4.2.
cutting speed
times
machining time
non-machining time
Figure 4.2: Timeversus cuttingsp eed
