Variety-oriented design of rotary production systems

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Variety-oriented

design

of

rotary

production

systems

Olga

Battaı¨a

a,

*

,

Daniel

Brissaud

(1)

b

,

Alexandre

Dolgui

a

,

Nikolai

Guschinsky

c

a_{E´coleNationaleSupe´rieuredesMines,CNRSUMR6158LIMOS,F-42023Saint-E´tienne,France}

b_{Univ.GrenobleAlpes,LaboratoireG-SCOP,38000Grenoble,France}

c

UnitedInstituteofInformaticsProblems,NationalAcademyofSciencesofBelarus,Minsk,Belarus

1. Introduction

Managingvarietyisagreatchallengefacingindustrytoday.In

ordertoreducetheexpenses relatedtotheinitialsetupof the

systemcapacitiesrequiredbyrelatedbutdifferentitems,suchas

variants of a product family, variety-oriented planning of

capabilitiesandcapacitiesisneededonthemanufacturing-side.

Therefore,themanufacturingsystemshouldbedesignedtofitthe

productvarietytobeproduced[1].Severalapproachesfor

variety-orienteddesignofmanufacturingsystemswereproposedinthe

literature,mostlyforassemblysystems[2–5].

Thispaperstudiesvariety-orienteddesignofrotaryproduction

systemsusedformachiningparts.Insuchaproductionsystem,parts

aresequentiallymachinedonm(1,2,...,m)workingpositions.An

exampleofsuchapositionisprovidedinFig.1.Acirculartransferis

realizedfromthezeropositionwherethebilletisloadedthrough

allworkingpositions.Eachfinishedpartis unloadedatthe zero

positionbeforetheloadingofthenextbillettobeprocessed.

Ateachworkingposition,severalmachiningmodules(spindle

heads)canbeinstalledtoprocesstheoperationsassignedtothis

position. The machining modules can work sequentially or

simultaneouslyonthesamepart.Sequentialmachiningisrealized

by the use of turrets. Simultaneous machining is possible if

machiningmodulesappliedtodifferentsidesofthepartworkin

parallel.Suchproductionsystemscanusehorizontalandvertical

spindleheadsandturretstoaccesstodifferentsidesofpartsat

aworkingposition.

Such production systems are modular and can be adapted

accordingtothepartstobeproduced[6],i.e.thefixturesofparts

are changedandsomespindles aremountedor dismountedif

necessary.However, fewstudiespublishedintheliteratureon

thedesignofrotaryproductionsystemsweremostlydedicated

to the mass production case [7–12]. In difference to that

previouswork,thispaperconsidersthecaseoftheproductionof

differentvariantsofaproductfamily.Therefore,theproduction

system has to be adapted for producing different product

models. Thedesignobjectiveistochoosetheequipmenttobe

usedbytherotaryproductionsystemsuchas–turrets(aturret

has several machining modules) and spindle heads – to be

installed at all workingpositions. The goal is tominimize the

costoftheequipmentrequiredforproducingallgivenproduct

variants.Thefollowingdecisionsmustbealsomade:thechoice

of orientations of parts, the partitioning of the given set of

operations into positions and assignments them to the

equipment, and the choice of cutting modes foreach spindle

head andturret.

Thedevelopeddesignapproachoffersa mathematicalmodel

for the description of the part parameters and operations,

constraints between operations and machining modules and

technologicalconstraintsforrotaryproductionssystems(Section

2).Withtheuseofthismodel,thedesignproblemisformulatedas

acombinatorialoptimizationproblem.Amixedinteger

program-ming(MIP) approach is used tofind theoptimalsolutions. An

industrialexampleispresentedinSection3.

Keywords:

Productionplanning

Machining Reconfiguration

ABSTRACT

Thevarietyorienteddesignproblemforrotaryproductionsystemsisconsidered.Giventhemultiple partstobeproduced,theproblemistodeterminethefeasibleconfigurationsofthemachiningsystem withminimumcost.Thisproblemismodelledasacombinatorialoptimizationproblem.Constraints relatedtothedesignofmachiningunitsaswellastotheprecedenceandcompatibilityofoperationsare takenintoaccount.TheoptimizationmethodsdevelopedtosolvetheproblemarebasedonitsMIP formulation.Anindustrialexampleispresented.

Fig.1.Arotarymachiningsystem.

* Correspondingauthor.

E-mailaddress:battaia@emse.fr(O.Battaı¨a).

2. Problemstatement

In this section, themathematical modelfor variety-oriented

designproblemforrotaryproductionsystemsispresented.Letus

considerthecasewhered0productvariantshavetobeproduced

withrequiredoutputOd,d=1,2,...,d0.

LetNdbethesetofoperationsneededformachiningdthpart,

d=1,2,...,d0,withndsidestobemachined;Nds,s=1,2,...,nd,isa

subsetofoperationstoberealizedonsthsideofpartd.

The part d can be located at zero position in different

orientationsH(d)whichdefinetheinitialpartexposure.

Thesetofalldifferentoperationstobeperformedisdefinedas

N¼Sd0

d¼1N

d

. All operations p2N are characterized by the

followingparameters:

 Thelength

l

(p)oftheworkingstrokeforoperationp2N,i.e.the

distancetoberunbythetoolinordertocompleteoperationp;

 Range [

g

1(p),

g

2(p)] of feasible values of feed rate which

characterizesthemachiningspeed;

 SetH(p,j)offeasibleorientationsofthepartforoperationp2Nif

operationpcanbeassignedtospindleheadorturretoftypej

(j=1 forverticalandj=2 forhorizontal).

ItshouldbenotedthatnosolutionexistsifT_p2Nd

sHðp;jÞ¼

? foreachj2{1,2}andsomed2{1,2,...,d0},s2{1,2,...,nd}.

Let subsetNk,k=1, ...,mcontaintheoperationsfromsetN

assignedtothekthworkingposition.

Let sets Nk1 and Nk2 be the sets of operations assigned to

workingpositionkthatareconcernedbyverticalandhorizontal

machining,respectively.

Finally,letbkjbethenumberofmachiningmodules(notmore

than b0) of type j installed at the kth working position and

respectivelysubsetsNkjl,l=1,...,bkjcontaintheoperationsfromset

Nkjassignedtothesamemachiningmodule.Intheconsideredcase,

onlyoneverticalturretorspindle headcanbeusedtoperform

vertical operations at any working position. In addition, each

positioncanbeequippedwithonehorizontalturretorspindlehead.

The assignment of operations tomachining modules has to

respect the technological constraints that emanate from the

machiningprocess. First, precedence constraints specified by a

directed graph GOR_{=(N, D}OR_{) have to be respected. An arc (p,}

q)2DOR _{if and only if operation p has to be executed before}

operationq. Itshouldbenotedthatifsuchoperationspandq

belongtodifferentsidesofthepart,thentheycannotbeassigned

tothesameworkingposition.

Tolerance constraints require to perform some pairs of

operationsfrom N at the same workingposition, by thesame

turret,bythesamespindleheadorevenbythesamespindle(for

differentparts and for each pair of operations). Such inclusion

constraints are modelled by undirected graphs GSP_{=(N, E}SP_),

GST_=(N,EST_{), G}SM_=(N,ESM_)andGSS_=(N,ESS_{)wheretheedge(p,} q)2ESS_((p,q)

2ESM_,(p,q)

2EST_,(p,q)

2ESP_{)ifandonlyifoperations}

p and q must be executed by the same spindle, at the same

machiningposition(orturret).

Becauseofunfeasibletoollocationortechnological

incompati-bility,someoperationscannotbeperformedbythesamespindle

head, turret, etc. These exclusion constraints are modelled by

undirectedgraphsGDM=(N,EDM),GDT=(N,EDT),andGDP=(N,EDP)

wheretheedge(p,q)2EDM_((p,q)

2EDT_,(p,q)

2EDP_)ifandonlyif

operationspandqcannot beexecutedbythesame machining

module,turret orat thesameposition.Itshouldbenotedthat

someoperationscan beassignedtothesame spindleheadbut

nottothesameturret.

LetP=hP1,...,Pk,...,PmiisadesigndecisionwithPk=(P1k11, P2k11, ..., Pd0k11,..., P1k1bk1, P2k1bk1, ..., Pd0k1bk1, P1k21,P2k21, ..., Pd₀k21, ..., P1k2b_k1, P2k2b_k1, ..., Pd₀k2b_k1), Pdkjl=(Ndkjl, ’_dkjl), and Nj¼S d0 d¼1 Sm k¼1 Sb_kj l¼1Ndkjl,j=1,2.

The execution time tb_(P

dkjl)=L(Ndkjl)/’dkjl+

t

a of operations fromNdkjlwhere’_dkjl2[max{

g

₁(p)jp2N_dkjl},min{

g

₂(p)jp2N_dkjl}]

and L(Ndkjl)=max{

l

(p) jp2Ndkjl},

t

a is an additional time for

advanceanddisengagementoftools.

We assumethat only timeneeded forrotation oftheturret

between nonempty sets Ndkjl is taken into account and the

executiontimeisequalto:

th ðPdkjÞ¼

t

gðldmaxðPdkjÞldminðPdkjÞÞþ Xbkj l¼1 tb ðPdkjlÞ; j¼1;2; where

t

g

is an additional time for one rotation of turret,

ld maxðPdkjÞ¼maxfl¼1;2;...;bkjNdkjl6¼? andld minðPdkjÞ¼minfl¼1;2;...;bkjNdkjl6¼?   ,respectively.

Theexecutiontimeataworkingpositiontp_(P

dk)isdefinedas

tp_(P

dk)=

t

r+max{th(Pdkj)jj=1,2},where

t

risanadditionaltimefor

tablerotation.

Thenthetimetdformachiningalltheelementsofdthpartis

equaltotd_(P)=max{tp_(P

dk)jk=1,...,m0}.

Weassumethatthegivenproductivityisprovided,ifthetotal

timeT(P)formachiningOdpartsdoesnotexceedtheavailabletime

T0,i.e.TðPÞ¼ Xd0 d¼1

td_ðPÞOd T0:

Theconstraintontheproductivityisrespectedifandonlyifitis

satisfiedfor ’_dkjl=min{

g

₂(p)jp2N_dkjl},d=1, ...,d₀, k=1, ...,m, j=1,2,l=1,...,bkj.

LetC1,C2,C3,andC4betherelativecostsforoneposition,one

turret,onemachiningmoduleofaturret,andonespindlehead,

respectively. Since the vertical spindle head (if installed) is

common for several positions, its size (and thereforethe cost)

dependsonthenumberofpositionstobecovered.Letkh

minand

kh

max be theminimal and the maximal position numbers for a

commonverticalspindlehead.Thenitscostcanbeestimatedas

C4þðkhmaxkhminÞC5,whereC5istherelativecostforcoveringone

additionalpositionbyaverticalspindlehead.Iftheverticalspindle

turret isinstalled, itscost canbeestimatedby C2+C3bk1. In a

similarway,thecostC(bk2)forperformingsetofoperationsNk2

byassociatedbk2machiningmodulescanbeassessedasfollows:

Cðbk2Þ¼ 0ifbk2¼0; C4ifbk2¼1; C2þC3bk2ifbk2>1: 8 < :

ThemachinecostQ(P)iscalculatedasthetotalcostofallpieces

ofequipmentused,i.e.

Q ðPÞ¼C1mþC4signðjN1jÞ 1 Xm k¼1 signðjNk12jÞ ! þX m k¼1

signðjNk12jÞðC2þC3bk1ÞþC5ðkhmaxkhminÞ

þX

m

k¼1

Cðbk2Þ!min (1)

wheresign(a)=1ifa>0,andsign(a)=0ifa0.

Ifaverticalturretisinstalled,thenthesecondandtheforthsum

elementsareequalto0 sinceNk126¼1forsomek2{1,...,m}and

kh

max¼khmin¼0. If avertical spindleheadisinstalled, thenthe

secondsumelementisequaltoC4andthethirdsumelementis

equalto0,sincesignðjN1jÞ¼1andPmk¼1signðjNk12jÞ¼0. Ifthereis

noverticalmachininginthedesigndecision,thenthesecond,third

andfourthsummandsareequalto0,sinceN1=1,Nk12=1,k=1,

...,m,andkh

max¼khmin¼0.

Thus,theproblemistodetermine:

(1) Thenumberofpositionsm;

(2) TheorientationsofpartsH(d);

(3) Thenumberbkjofmachiningmodulesoftypej(j=1forvertical

andj=2 forhorizontal)installedatthekthposition,k=1,...,

m;

(4) Subsets Ndkjl of operations from Nd assigned to the lth

machiningmoduleoftype jatthekthposition,d=1,2, ...,

(5) Thefeedperminute’_dkjlforeachsubsetN_dkjl,d=1,2,_...,d₀, k=1,...,m,j=1,2,l=1,...,bkj.

Thegoalistominimizethemachinecostwhilerespectingall

constraints. TðPÞT0; (2) [m k¼1 [2 j¼1 [bkj l¼1 Nkjl¼N (3) Nk0_j0_l0\N k0 0_j0 0_l0 0¼? ; k0;k00¼1;...;m;j0;j00¼1;2;l0;l00¼1;...;bkj;l06¼l00 (4) HðdÞ¼\ 2 j¼1 \ p2Nj\Nd Hðp;jÞ2HðdÞ; d¼1;...;d0 (5) N1\ðNsd0[Nds00Þ2f? ;Nds0;Nds00g; d¼1;...;d0;s0;s00¼1;...;nd;s06¼s00 (6) Nj\Nds2f? ;Ndsg; j¼1;2;d¼1;...;d0;s¼1;...;nd (7) p2[ k1 k0_¼1 [2 j0_¼1 [ b_{k0 j0} l0_¼1 Nk0_j0_l0;ðp;qÞ2DOR;q2N_kjl;p2N_3j;q2N_j; k¼1;...;m; j¼1;2;l¼1;...;bkj (8) p2[ k1 k0_¼1 [2 j0_¼1 [ b_{k0 j0} l0_¼1 Nk0_j0_l0 [ [l1 l0_¼1 Nkjl0; ðp;qÞ2DOR;q2N_kjl; p;q2N_j; k¼1;...;m; j¼1;2;l¼1;...;bkj (9) [2 j¼1 [bkj l¼1 Nkjl\fp;qg            6¼1;ðp;qÞ2E SP_{;k¼1;...;m; (10)} [bkj l¼1 Nkjl\fp;qg        6¼1;ðp;qÞ2E ST_{;k¼1;...;m;j¼1;2 (11)} Nkjl\fp;qg    6¼1;ðp;qÞ2ESB_{;k¼1;...;m;j¼1;2;l¼1;...;b} kj (12) Nkjl\fp;qg    6¼1;ðp;qÞ2ESS_;k ¼1;...;m;j¼1;2;l¼1;...;bkj (13) [2 j¼1 [bkj l¼1 Nkjl\fp;qg            6¼2;ðp;qÞ2E DP_;k ¼1;...;m (14) [bkj l¼1 Nkjl\fp;qg            6¼2orbkj¼1;ðp;qÞ2E DT_;k ¼1;...;m;j¼1;2 (15) Nkjl\fp;qg6¼2;ðp;qÞ2EDB;k¼1;...;m;j¼1;2;l¼1;...;bkj signðjNk11jÞþ Xm k0_¼1;k0_6¼k signN_k0₁₂ 1; (16) signðjNk12jÞþsignðjNk21jÞ1;k¼1;...;m (17) Gdkjl2 G1Ndkjl   ;G2 Ndkjl     ;d¼1;...;d0; k¼1;...;m;j¼1;2;l¼1;...;bkj (18) bkjb0 (19) mm0: (20)

where’₁(N)=max{

g

₁(p)jp2N}and’₂(N)=min{

g

₂(p)jp2N}.

Constraint(2)introducestheproductivityrequirement.

Con-straints(3)–(4)ensurethateachoperationfromNisassignedto

onemachiningmoduleexactly.Constraint(5) obligestochoose

feasibleorientationsofparts.Constraints(6)prohibitassignments

ofoperationsformachiningelementslocatedattwodifferentsides

oftheparttoa verticalspindlehead(orturret).Constraints(7)

ensurethatalloperationsformachiningelementslocatedatthe

samesideofthepartwillbeassignedtothesametypeofspindle

head or turret. Constraints (8)–(9) provide the precedence

relationsfortheoperationsthatrequireeitherthesametypeof

machining module (vertical or horizontal) or different ones,

respectively.Inclusionconstraintsforworkingpositions,turrets,

machiningmodulesandspindleheadsareexpressedby(10),(11),

(12) and (13), respectively. Exclusion constraints for working

positions,turrets,andmachiningmodulesareintroducedby(14),

(15) and(16),respectively.Constraint(17)ensuresthatatmost

oneverticalturretwillbechosenforthemachineandifthis is

thecase,nohorizontalmachiningunitsareinstalledatthesame

working position. Constraints (18) bound the feasible values

ofthefeedperminuteforeachmachiningmodule.Thenumber

of machining modules per turret is limited by constraint (19).

Thenumberofworkingpositionsonthemachineisboundedby

(20).

Thedevelopedmodelcanbeimplementedusingmixedinteger

programming(MIP)approach.

3. Industrialexample

Thefollowing6 partsaretobemachinedonarotarytransfer

machine(Fig. 2).TheavailableproductiontimeT0=360min.The

required outputsof thepartsare(24, 24, 24,24, 48, 48)units,

respectively. Other parameters are:

t

a₌

_t

g₌

_t

r_{=0.1min. The}

possible orientations of the parts are: H(1)=H(3)={(H4–

H9),(H18–H21)}, H(5)={(H4–H9), (–)}, H(2)=H(4)=H(6)=

{(H10–H15),(H16)}. Here orientation(H4–H9)means that holes

H4–H9 aretobeassignedtoverticalmachiningmodulesand(–)

meansthatthereisnoverticalmachining.Eachoperationpcanbe

assignedeithertoverticalorhorizontalmachiningmodules.The

parametersoftheoperationsaregiveninTable 1.

Table1 Parametersofoperations. p Hole Part l(p), mm g1(p), mm/min g2(p), mm/min p Hole Part l(p), mm g1(p), mm/min g2(p), mm/min 1 H4 1 48 39.2 62.9 46 H9 3 75 44 86.5 2 H4 1 34 27.2 248 47 H18 3 29 22.3 87.6 3 H5 1 48 39.2 62.9 48 H18 3 10 28.3 106.3 4 H5 1 34 27.2 248 49 H18 3 26 59 102.9 5 H6 1 107 22.8 81.3 50 H19 3 29 22.3 87.6 6 H6 1 105 44 86.5 51 H19 3 10 28.3 106.3 7 H7 1 107 22.8 81.3 52 H19 3 26 59 102.9 8 H7 1 105 44 86.5 53 H20 3 29 22.3 87.6 9 H8 1 107 22.8 81.3 54 H20 3 10 28.3 106.3 10 H8 1 105 44 86.5 55 H20 3 26 59 102.9 11 H9 1 91 22.8 81.3 56 H21 3 29 22.3 87.6 12 H9 1 89 44 86.5 57 H21 3 10 28.3 106.3 13 H18 1 29 22.3 87.6 58 H21 3 26 59 102.9 14 H18 1 10 28.3 106.3 59 H16 4 30 54.6 68.9 15 H18 1 26 59 102.9 60 H16 4 19 31.9 197.1 16 H19 1 29 22.3 87.6 61 H16 4 19 26.9 161.6 17 H19 1 10 28.3 106.3 62 H16 4 18 26.7 160.2 18 H19 1 26 59 102.9 63 H10 4 7 35.2 105.6 19 H20 1 29 22.3 87.6 64 H11 4 7 35.2 105.6 20 H20 1 10 28.3 106.3 65 H12 4 7 35.2 105.6 21 H20 1 26 59 102.9 66 H13 4 7 35.2 105.6 22 H21 1 29 22.3 87.6 67 H14 4 7 35.2 105.6 23 H21 1 10 28.3 106.3 68 H15 4 6 35.2 105.6 24 H21 1 26 59 102.9 69 H4 5 53 39.2 62.9 25 H16 2 30 54.6 68.9 70 H4 5 34 27.2 248 26 H16 2 19 31.9 197.1 71 H5 5 53 39.2 62.9 27 H16 2 19 26.9 161.6 72 H5 5 34 27.2 248 28 H16 2 18 26.7 160.2 73 H6 5 94 22.8 81.3 29 H10 2 6 35.2 105.6 74 H6 5 92 44 86.5 30 H11 2 7 35.2 105.6 75 H7 5 94 22.8 81.3 31 H12 2 7 35.2 105.6 76 H7 5 92 44 86.5 32 H13 2 7 35.2 105.6 77 H8 5 39 22.8 81.3 33 H14 2 6 35.2 105.6 78 H8 5 37 44 86.5 34 H15 2 6 35.2 105.6 79 H9 5 94 22.8 81.3 35 H4 3 103 39.2 62.9 80 H9 5 92 44 86.5 36 H4 3 18 27.2 248 81 H16 6 30 54.6 68.9 37 H5 3 48 39.2 62.9 82 H16 6 19 31.9 197.1 38 H5 3 34 27.2 248 83 H16 6 19 26.9 161.6 39 H6 3 92 22.8 81.3 84 H16 6 18 26.7 160.2 40 H6 3 90 44 86.5 85 H10 6 6 35.2 105.6 41 H7 3 92 22.8 81.3 86 H11 6 6 35.2 105.6 42 H7 3 90 44 86.5 87 H12 6 6 35.2 105.6 43 H8 3 77 22.8 81.3 88 H13 6 6 35.2 105.6 44 H8 3 75 44 86.5 89 H14 6 6 35.2 105.6 45 H9 3 77 22.8 81.3 90 H15 6 6 35.2 105.6

Precedence and compatibility constraintsare numerous and

canbe provided on demandby the corresponding author. The

optimizationproblemwassolvedform0=5usingCPLEX12.2.The

obtainedoptimalsolutionispresentedinTable2.After

preproces-sing,themodelcontained1488variablesand4295 constraints.

Thetotalsolutiontime was21.6s. Thissolutionwasvalidated

andimplementedbyourindustrialpartner.

4. Conclusions

A problem of variety-oriented design of rotary production

systems has been studied. A mathematical model for this

optimization problem has been developed where constraints

between operations and machining modules and technological

constraints for rotaryproduction systems were integrated. The

configurationofsuchsystemsis optimizedusingmixedinteger

programming (MIP) techniques. The configuration module has

beenimplementedinadecisionsupportsystem.Thissystemcan

detect the conflicts in the constraints and guide the designer

throughtheoptimizationprocess.Themodelandthemodulehave

beenvalidatedinpractice.Thefutureresearchworkwillconcern

thereconfigurationtechniquesforrotaryproductionsystemsto

beadaptedtonewmanufacturingconditions.

Acknowledgment

ThisworkwassupportedbyPICSFrance-Belarusgrant,CNRS.

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Fig.2.Thesixpartstobemachined.

Table 2

Characteristicsoftheoptimalsolution.

Set Ndkjl Operations ofNdkjl L(Ndkjl) ’dkjl tb(Pdkjl) Ndkjl Operations ofNdkjl L(Ndkjl) ’dkjl tb(Pdkjl) N1111 13161922 29 87.6 0.43 N121114172023 10 87.6 0.22 N3111 47505356 29 87.6 0.43 N2211293031323334 7 87.6 0.18 N2121 25 30 62.9 0.58 N321148515457 10 87.6 0.22 N4121 59 30 62.9 0.58 N4211636465666768 7 87.6 0.18 N6121 81 30 62.9 0.58 N6211858687888990 6 87.6 0.17 N2122 26 19 62.9 0.4 N131115182124 26 87.6 0.4 N4122 60 19 62.9 0.4 N331149525558 26 87.6 0.4 N6122 82 19 62.9 0.4 N132124681012 105 86.5 1.32 N1123 1357911 107 62.9 1.8 N232128 18 86.5 0.31 N2123 27 19 62.9 0.4 N3321363840424446 90 86.5 1.14 N3123 353739414345 103 62.9 1.74 N432162 18 86.5 0.31 N4123 61 19 62.9 0.4 N5321707274767880 92 86.5 1.17 N5123 697173757779 94 62.9 1.59 N632184 18 86.5 0.31 N6123 83 19 62.9 0.4