Reactive scheduling in make-to-order manufacturing by mixed integer programming

Z E S Z Y T Y N A U K O W E P O L IT E C H N IK I Ś L Ą S K IE J Seria: A U T O M A T Y K A z. 144

2006 N r kol. 1727

T adeusz SAW IK

A kadem ia G ó rn ic zo -H u tn icz a

R E A C T I V E S C H E D U L I N G I N M A K E - T O - O R D E R

M A N U F A C T U R I N G B Y M I X E D I N T E G E R P R O G R A M M I N G S u m m a r y . New a lg o rith m s b ase d on m ixed in teg er p ro g ram m in g m odels are p ro p o se d for re a c tiv e schedu ling in a d y n am ic, m ak e-to -o rd er m anu fac­

tu rin g en v iro n m e n t. T h e p ro b lem o b jectiv e is to u p d a te p ro d u c tio n sche­

d u le s u b je c t to service level a n d in v en to ry c o n stra in ts, w hen ever cu sto m er o rd e rs are m odified. N u m eric al exam ples m o d eled a fte r a real-w o rld p ro ­ d u c tio n sc h e d u lin g /re sc h e d u lin g in th e electro n ics in d u s try a re p re sen ted an d som e re su lts of c o m p u ta tio n a l e x p e rim en ts a re re p o rte d .

R E A K T Y W N E H A R M O N O G R A M O W A N I E P R O D U K C J I N A Z A M Ó W I E N I E M E T O D Ą P R O G R A M O W A N I A

C A Ł K O W I T O L I C Z B O W E G O

S tr e s z c z e n i e . W p ra c y p rz ed staw io n o nowe alg o ry tm y re ak ty w n ego liar- m onog ram o w an ia p ro d u k c ji zam aw ian ej, o p a rte n a m od elach pro gram ow a­

n ia całkow itoliczbow ego. Z am ów ienia m ogą b yć m odyfikow ane p rzez o d ­ biorców w c a ły m h o ryzoncie planow ania. C elem h arm o n o g ram o w an ia je s t m in im a liz a c ja liczby spóźnionych zam ów ień oraz łączny ch zapasów m a te ­ riałó w i gotow ych w yrobów . Z astosow anie propon ow any ch algo ry tm ów ilu ­ s tr u ją p rz y k ła d y liczbow a za cz e rp n ię te z p rz em y słu elektro niczneg o oraz w yniki ek sp ery m en tó w obliczeniow ych.

1. I n t r o d u c t i o n

In m ak e-to -o rd er m a n u fa c tu rin g th e p e rfo rm an c e of p ro d u c tio n p lan n in g a n d sched u lin g is e v a lu a te d by c u sto m e r sa tisfa c tio n a n d p ro d u c tio n costs. A ty p ic a l m easu re of th e c u sto m e r sa tisfa c tio n is c u sto m e r service level, i.e., th e fra c tio n of c u sto m e r o rd e rs filled o n or before th e ir d u e d ates, e.g. [4, 7]. O n th e o th e r h a n d to achieve low' u n it p ro d u c tio n co st, b o th th e in p u t inv en to ry of p u rc h a se d m a te ria ls w a itin g for p ro cessing in th e sy stem a n d th e o u tp u t in v en to ry of finished p ro d u c ts w a itin g for delivery to th e c u sto m e rs sh ou ld b e m inim ized.

To re d u ce th e re q u ired in p u t in v en to ry of p u rc h a se d m a te ria ls, th e m a te ria ls

80 T . Sawik

sh ould b e delivered as la te as possible, i.e., th e o rd e r earlin ess sh o u ld b e as sm all as possible. O n th e o th e r h a n d th e sm aller th e earlin ess of c u sto m e r orders, th e sm aller is th e o u tp u t in v en to ry of finished p ro d u c ts co m p leted b efo re cu sto m er re q u ire d sh ip p in g d a te s a n d w aitin g for delivery to th e cu sto m ers. However, if for som e cu sto m er o rd e rs th e earliness is sm aller th a n th e m in im u m eaxliness, i.e., re a d y p erio d s a n d d u e d a te s are closer each o th e r, th e n re a llo c a tio n of orders to th e e a rlier p e rio d s w ith su rp lu s of c a p a c ity is re s tr ic te d d u e to la te r m a te ria l availability. A s a re su lt, th e n u m b er of ta r d y o rd ers m a y in crease o r even som e o rd ers m ay re m a in u n scheduled d u rin g th e p la n n in g horizon.

T h e p u rp o se of th is p a p e r is to p re se n t new a lg o rith m s, b ase d on in te ­ ger p ro g ram m in g form ulations, for reactiv e scheduling ([8, 9, 10]) in a d y n am ic, m ak e-to -o rd er m an u fa c tu rin g , w here cu sto m ers m ay m o d ify or can cel th e ir orders o r place new o rd ers d u rin g th e p lan n in g horizon. T h e p ro b lem o b jectiv e is to d y n am ica lly a ssig n /re a ssig n cu sto m er ord ers w ith v ario u s d ue d a te s to p lan n in g p e rio d s w ith lim ite d ca p acities, to m inim ize th e n u m b e r of ta r d y o rd ers a n d th e in p u t a n d o u tp u t in v en to ry over a p lan n in g horizon.

In th e lite ra tu re on p ro d u c tio n p lan n in g a n d schedu ling th e in teg er p ro ­ g ram m in g m odels have b ee n w idely used, e.g. [3, 7]. In in d u s tria l p ra c tic e , howe­

ver, th e a p p lic a tio n of in teg er p ro g ram m in g for p ro d u c tio n sched uling is lim ited , in p a rtic u la r in m ak e-to -o rd er m an u fa c tu rin g , e .g .[l, 2, 4, 5, 6]. F or exam ple, a scheduling to o l w ith rescheduling ca p ab ilities for th e se m ic o n d u c to r in d u stry , b a ­ sed on in teg er p ro g ram m in g form ulatio n is p re se n te d in [2]. H ow ever, th e m odel is solved b y an a p p ro x im a te tech n iq u e an d o p tim a l so lu tio n was n o t a tte m p te d .

T h e p a p e r is o rg anized as follows. In th e n e x t sec tio n t h e d e sc rip tio n of m ak e-to -o rd er p ro d u c tio n scheduling in a flexible flow shop is pro v id ed . T h e b a ­ sic in teg er p ro g ram m in g fo rm u latio n s for p ro d u c tio n s c h e d u lin g /re sc h e d u lin g are p re se n te d in S ection 3. R escheduling a lg o rith m s b ase d on th e p ro p o se d m ixed in teg er p ro g ram m in g m odels are describ ed in S ectio n 4 an d so m e form ulae for th e c a lc u latio n of in p u t a n d o u tp u t inv en to ry are d eriv e d in S ectio n 5. N u m eri­

cal exam ples m o deled a fte r a real-w orld, m ak e-to -o rd er electro n ics m an u fa c tu rin g a n d som e c o m p u ta tio n a l re su lts are pro v id ed in S ectio n 6. C onclusions are m ad e in th e la st section.

2. P r o b l e m D e s c r i p t i o n

T h e p ro d u c tio n sy stem u n d e r s tu d y is a flexible flowshop t h a t co nsists of m p rocessing stag es in series a n d each sta g e i E I = { 1 , . . . ,m } is m ad e u p of in, > 1 id en tical, p arallel m achines. In th e sy ste m v ario u s ty p e s of p ro d u c ts are p ro d u c e d in a m ak e-to -o rd er en v iro n m en t re sp o n d in g d ire c tly to c u sto m e r o rders.

L et J b e th e se t cu sto m er o rd ers t h a t are know n a h e a d of a p la n n in g horizon.

E ac h o rd e r j E J is d escrib ed by a trip le (c ij.d j, s3), w h e re a j is th e o rd e r arriv al d a te (e.g. th e earliest p e rio d of m a te ria l av a ilab ility ), d j is th e c u sto m e r d u e d a te (e.g. cu sto m er re q u ired sh ip p in g d a te ), an d Sj is th e size of o rd e r (th e q u a n tity

R eactiv e scheduling in m ak e-to -o rd er m a n u fa c tu rin g 81

of o rd e red p ro d u c ts of specified ty p e ). E ac h o rd e r req u ires p ro cessin g in various processing stag es, however som e o rd e rs m ay b y p ass som e stag es. L e t p jj > 0 b e th e processing tim e in stag e i of each p ro d u c t in o rd e r j € J . T h e o rd e rs are processed a n d tra n sfe rre d am ong th e sta g e s in lo ts of various size t h a t d ep e n d s on th e o rd e red p ro d u c t ty p e a n d le t bj b e th e size of p ro d u c tio n lo t for o rd e r j .

T h e p lan n in g horizon consists of h p la n n in g p e rio d s (e.g. w orkin g d ay s). L et T = { 1 , . . . h } b e th e set of p la n n in g p e rio d s a n d th e pro cessin g tim e available in p e rio d t on each m achine in s ta g e i.

T h e following tw o ty p e s of th e c u sto m e r o rd ers are considered:

1. Sm all size (single-period) o rd ers, w here each o rd e r can b e fully p ro cessed in a single tim e perio d , e.g. d u rin g one day. T h e sing le-period o rd e rs a re re ferred to as indivisible orders.

2. L arge size (m u lti-p erio d ) o rd ers, w h ere each o rd e r c a n n o t b e c o m p le te d in one p erio d a n d m u st b e sp lit a n d p ro cessed in m ore th a n one tim e p erio d . T h e m u lti-p e rio d orders are re fe rre d to as divisible orders.

In p ra ctice, tw o ty p e s of c u sto m e r o rd ers are sim u ltan e o u sly scheduled.

D enote by J 1 C J , a n d J 2 C J , re sp ectiv ely th e su b set of indivisible, a n d div isible orders, w here J l \ J J 2 = J , a n d J 1 f | <72 = 0.

It is assu m ed t h a t each c u sto m e r o rd e r j E J 1 m u st b e fully c o m p le te d in ex actly one p lan n in g p erio d , a n d each c u sto m e r o rd e r j € .72 m u st b e co m p leted in tw o consecutive p lan n in g p e rio d s, how ever, th e la tte r a ssu m p tio n ca n b e easily relax ed [5] to allow for co m p letin g of larg e o rd ers in m ore th a n tw o con secutiv e periods.

A d y nam ic, m ak e-to -o rd er m a n u fa c tu rin g en v iro n m e n t is co n sid ered w ith a d y n am ic p lan n in g horizon used to successively u p d a te p ro d u c tio n sch ed u le w hen old, y et u n co m p leted cu sto m er o rd e rs are m odified or new c u sto m e r o rd e rs arriv e d u rin g th e horizon. T h e m o d ificatio n s o f c u sto m e r o rd ers m ay in clu d e chang es of o rd er size, e.g. increase, d ecrease or ca n cellatio n , a n d /o r changes of d u e d a te s , e.g.

p o stp o n em en t of delivery d a te , o cc u rrin g d u rin g th e horizon. T h e h o rizo n ca n be progressively sh ifted to ta k e in to ac co u n t th e o rd e rs m odifications.

T h e o b jectiv e of th e lo n g -te rm re a c tiv e scheduling is to a ssig n /re a ss ig n cu­

sto m er o rders to p lan n in g p e rio d s over a p la n n in g horizo n to m ax im ize th e cu sto ­ m er service level by m inim izing th e n u m b e r of ta r d y ord ers, w ith lim ite d m ax im u m earliness of o rders a n d by th is th e lim ite d to ta l in p u t an d o u tp u t inventory.

3. M i x e d I n t e g e r P r o g r a m s f o r R e a c t i v e S c h e d u l i n g

In th is section m ixed in teg er p ro g ra m m in g form u latio n s are p ro p o se d for cu sto m er o rd ers a ssig n m e n t/re a ssig n m e n t over a lo n g -term p la n n in g ho rizo n , to m axim ize service level, im p licitly s u b je c t to th e inv en to ry c o n stra in ts . B asic no­

ta tio n is p re sen ted in T able 1.

82 T . Sawik

3.1. B asic m odel

T h e b asic m odel used to u p d a te th e c u rre n t p ro d u c tio n schedule, w hene­

ver som e cu sto m er orders are m odified is p re se n te d below. T h e e x te n t of re q u ired changes in th e c u rre n t schedule depen ds on th e ap p lied policy (see, S ection 4) an d th e changes in size an d due d a te s of th e m odified cu sto m er orders. T h e u p d a te d schedule tak es in to acco u n t th e c u rre n t in p u t in v en to ry t h a t is im p licitly consi­

d ered in th e m odel by th e u p p e r b o u n d m e on th e m ax im u m earlin ess E max of cu sto m er orders.

T ab le 1 N o tatio n : In itia l S cheduling

i —

I n d i c e s

processing stage, i € I — { 1, . . . , m } j — cu sto m er order, j G J = { 1 , . . . , n } k = p ro d u c t type, k € K = { 1 , . . . , r } t = p lan n in g p erio d , t € T = { 1, . . . , h } CLj 5 dj « S j _

I n p u t P a r a m e t e r s arrival d a te , d u e d ate, size of o rd e r j bj = p ro d u c tio n lo t for order j

C-it = processing tim e available in p e rio d t on each m ach in e in stag e

m i —

z

nu m ber of identical, p a ra lle l m ach ines in stag e i n = n um ber of cu sto m er orders to b e scheduled

Pij = processing tim e in stag e i of each p ro d u c t in o rd e r j J l C J = su b set of sm all (single-period) cu sto m er o rders J 2 C J — subset of large (m ulti-period) cu sto m er orders J k C J = su b set of cu sto m er orders for p ro d u c t ty p e k E = u p p e r lim it on m axim um earliness

U = u p p e r lim it on nu m b er of ta r d y orders

Uj —

D e c is io n v a r i a b l e s

1, if order j is co m pleted a fte r d u e d a te ; o th erw ise uj — 0 (u n it

Xjt —

p e n a lty for ta rd y orders)

1, if order j is perform ed in p e rio d f; o th erw ise Xj t = 0 (o rd er

§ \V o _ assignm ent variable)

fractio n of custo m er o rd e r j to b e processed in p e rio d t (o rd er

Emax _ a llo catio n variable)

m axim um earliness of o rd ers Usum = n u m b er of ta r d y orders

M o d e l M a x S L (E ): C u sto m er orders a ssig n m en t to M a x im ize S ervice Level subject to M a x im u m E a rlin ess constraints

R eactiv e sched uling in m ak e-to -o rd er m an u fa c tu rin g 83

M axim ize

1 ^{— U sum/n (}1⁾

or M inim ize

Usum ~ E uj (2)

j e J su b je c t to

1. O rder a ssig n m en t constraints

- each ind ivisible cu sto m er o rd e r is assigned to ex a ctly one p la n n in g perio d ,

E

x j t = 1; j G J 1 (3)

teT:

- each div isible cu sto m er o rd e r is assigned to a t m ost tw o co nsecutiv e p la n ­ ning p erio d s,

Xjt -I- Xjt+ i < 2; j G «72, t G T : a j < t < h — 1 (4) Xjt + X jj ^ 1 j j £ J 2 , t £ T , t & T ü j < it K h — 2, i ^ i - l - 2 (5) S. O rder allocation constraints

- each o rd e r m u st b e com p leted ,

E ^Vjt = 1; j € J (6)

teT: t^a.j

- each ind ivisible o rd e r is co m p leted in a single p erio d ,

Xjt = Vjt] j e J l , t e T : t > aj (7) - each div isible o rd e r is a llo c a te d am ong all th e p e rio d s t h a t are selec ted for its assig n m en t,

Xjt > Vj t ; j €

J2,

t G T : t > aj (8) - th e m in im u m p o rtio n of a divisible o rd e r allo ted to one p e rio d is n o t less th a n th e b a tc h size,

yj t ^ bj x j t / s j , j G .72, ¿ G 7 . ¿ ^ (ij (9) 3. T ardy orders constraints

- a n ind iv isib le ta r d y o rd e r is co m p leted a fte r its d u e d a te ,

U j

E

x jt\ j € J l , (10)

te T : t> dj

- a div isible ta r d y o rd e r is p a r tly assigned a fte r its d u e d a te ,

uj

> E

^{Vjt\ 3}

6

^{J 2 ( n )}

te T : t> dj

Uj < E ^{Xjt] j} € ^J2 (12)

t e T : t> dj

84 T . Sawik

4- C apacity constraints

- in every p e rio d th e d em an d on c a p a c ity a t each pro cessin g sta g e c a n n o t b e g re a te r th a n th e m ax im u m available c a p a c ity in th is p erio d ,

P ijSjV jt < c a m ; i £ l , t £ T (13)

jeJi

5. M a x im u m earliness constraints

- for each ea rly o rd e r j assigned to p e rio d t < d j, its earliness {dj — t ) c a n n o t exceed t h e m ax im u m earliness E max,

{dj - t ) x j t < Emax] j € J , t e T : t > a j (14)

Emax ^ E (15)

6. Variable n o n n eg a tivity and integrality constraints

Uj £ { 0 ,1 } ; j £ J (16)

X j t £ {0 ,1 }; j € J, t € T : t > aj (17)

0 < y j t < 1; j G J, t € T : t > a j (18)

Emax > 1, in te g e r (19)

T h e o b jectiv e fu n c tio n re p re se n ts c u sto m e r service level, i.e., th e fra c tio n of n o n d e la jred cu sto m er o rd ers to b e m ax im ized (1) or eq uiv alen tly th e n u m b er of t a r d y o rd e rs to b e m inim ized (2). T h e so lu tio n to M a x S L ( E ) d e te rm in e s th e a ssig n m e n t of indivisible c u sto m e r o rd ers to single p lan n in g p e rio d s a n d th e al­

lo c a tio n of divisible o rd ers am ong th e p a irs of consecutive p la n n in g p e rio d s such t h a t th e c u sto m e r service level is m axim ized su b je c t to lim ited m ax im u m earlin ess of o rd e rs a n d by th is th e lim ited to ta l in p u t an d o u tp u t in v en to ry level.

M odel M a x S L ( E ) ca n b e brieffy re w ritte n as follows

M a x S L { E ) = m a x { ( l) : (2) — (19)} (20) 3.2. M o d els for in itial scheduling

T h e b eg in n in g p ro d u c tio n schedule for th e o riginal cu sto m er o rd ers k now n a h e a d o f th e p la n n in g h orizon is d e te rm in e d b y solving th e follow ing sequence of tw o m ix e d in teg er p ro g ram s

1. M o d e l M axSL: C u sto m e r orders a ssig n m en t to M a x im ize Service Level

M a x S L = m a x { ( l) : (2) — (13), (16) — (18)} (21) w h e re all m a te ria ls a re assu m ed to b e available a t th e b eg in ning , i.e. aj = 1 for each o rd e r j £ J .

R eac tiv e scheduling in m ak e-to -o rd er m a n u fa c tu rin g 85

2. M o d e l M in M E (U ): C u sto m e r orders a ssig n m en t to M in im ize M a x im u m E a rlin ess subject to service level constraints

M i n M E ( U ) = m in { E max ■ Usum < U, (2) - (14), (16) - (19)} (22) w h ere 1 — U / n is th e so lu tio n value of (21)

T h e o b jectiv e fu n c tio n of (22) im p licitly lim its th e m ax im u m level of th e to ta l in p u t an d o u tp u t in v en to ry over th e p la n n in g horizon.

In th e above seq u e n tia l decision m ak in g fram ew ork, first th e so lu tio n to M a xS L d e te rm in e s th e m a x im u m service level. T h en , th e m in im u m value of th e m ax im u m earliness is found using m odel M in M E fU ) to im p lic itly lim it to ta l inven­

tory, s u b je c t to service level c o n stra in ts. T h e so lu tio n to M in M E (U ) d e te rm in e s th e o p tim a l a llo ca tio n {x* t , y *L} of cu sto m er o rd e rs am o n g p la n n in g perio ds.

4 . R e s c h e d u l i n g A l g o r i t h m s

In th is sec tio n different resch ed u lin g alg o rith m s b a se d on th e m ix ed in teg er p ro g ram m in g m odels are proposed.

L et t moci b e th e first p la n n in g p e rio d im m e d ia te ly a fte r th e o rd ers m o d i­

fication. I t is assu m ed t h a t th e c u sto m e r o rd ers co m p leted b efore t mod or w ith due d a te s sm aller th a n t mod c a n n o t b e m odified. In p ra c tic e different resched ulin g policies c a n b e app lied, from a to ta l reschedule of all re m a in in g c u sto m e r ord ers, i.e. re sch ed u le of all u n m odified o rd ers t h a t have b ee n assig ned to p e rio d s n o t less th a n t mod (a lg o rith m R E A L L ) to a n on-reschedule of all such o rd e rs (a lg o rith m R E N O N ). In a d d itio n to th e above tw o ex tre m e resch ed u lin g policies a m ed iu m re stric tiv e alg o rith m R E M A T is p ro p o se d for re sch ed u lin g of th e re m a in in g cu­

sto m e rs o rd e rs w a itin g for m a te ria l supplies, i.e. re sch ed u lin g of all u nm odified o rd ers assig ned to p erio d s g re a te r th a n t mod + E rnax.

In all th ese a lg o rith m s th e p la n n in g horizon is p ro gressively sh ifted to ta k e in to ac co u n t m o dificatio n s of th e c u sto m e r o rd ers (ch ang es of o rd e r size a n d / o r d u e d a te ) o cc u rrin g d u rin g th e horizon. T ab le 2 p re se n ts th e n o ta tio n u sed in th e rescheduling alg o rith m s. In all a lg o rith m s first th e set J 0i(i of o rd e rs re m a in in g for co m p letio n is sp lit in to tw o d isjo in t sub sets: J^[d of sch ed u lab le o rd ers an d Jgld of fixed, n o n -schedulable o rd e rs for w hich th e assig n m en t to p la n n in g p e rio d s c a n n o t b e changed. In p a rtic u la r, in a lg o rith m R E M A T t h a t acco u n ts for th e in p u t in v en to ry of p ro d u c t specific m a te ria ls supp lied b efo re t mod an d available Emax p e rio d s a h e a d of th e o rd e rs d u e d a te s a t th e la te s t, th e su b se t of non- sch ed u lab le o rd e rs c o n ta in s o rd e rs in J 0id re m a in in g for co m p letio n , such t h a t have b ee n assigned to p e rio d s in th e su b se t T ^ d = { t mod, . . . , t mod + E max} of re m a in in g p erio d s in T0id = { t mod,

In th e sequel, d e n o te b y a p o stro p h e ( ’) th e u p d a te d values of som e p a ­ ra m e te rs a n d decision v ariab les a fte r each m o d ificatio n of orders. For ex a m p le

86 T . Saw ik

T ab le 2 N o tatio n : R escheduling

T old {tm o d r ■ ■ • j ^ ’}

'T'Âr

^ old

Tnew — { h + 1 , , / ) ' }

I n p u t P a r a m e t e r s new p la n n in g horizon

th e p lan n in g p e rio d im m e d ia te ly following m odificatio n o f o rd ers

set of m odified o rd ers

su b set of orders in J re m a in in g for co m p letio n s u b se t of o rd ers in J 0y , re sp ectiv ely non- schedulable, sched ulable

set of new p la n n in g p erio d s

su b set o f re m a in in g p la n n in g p e rio d s in T su b se t of p e rio d s in T 0id w ith fixed assig n m en t of o rd ers in J 0y

u p d a te d set of o rders

u p d a te d set o f p la n n in g p erio d s

ap o stro p h e ( ’) d en o tes u p d a te d p a ra m e te rs a fte r m o d ificatio n of o rd ers

s'j d en o tes th e m odified size of cu sto m er o rd e r j ^€ J ' ⁼ J 0idU Jmod> w here

Sj j ^£ Jold 3 i l d Sj S j^, j ^G Jmod-

A l g o r i t h m R E A L L

S t e p 0. S p lit th e set J Qid of o rd e rs re m a in in g for co m p letio n in to tw o d isjo in t subsets: J^ld of sch edulable o rd ers a n d J $ d of fixed, n on -sch ed u lab le orders.

S t e p 1. D e te rm in e new p lan n in g horizon h! for th e u p d a te d set J ' of cu sto m er orders.

(23) (24)

h \ = m in { /il : r (25)

(26)

If m a x { h i, /12} < h, th e n set h! ⁼ h.

O therw ise se t h! = m a x { /ii,/i2}, Tnew = { h + 1 , . . . , h!} a n d T ' = T0i,{U T new.

R eactiv e scheduling in m ak e-to -o rd er m a n u fa c tu rin g 87

Step 2. Do n o t change th e assig n m en t in p e rio d t mo({ of p a rtia lly co m pleted, tw o -p erio d o rd ers in J2, i.e.,

yjtmod ~ yjJmod’ 3 € J% '■ x j,tmod- 1 = 1 (27) Step 3. Solve M a x S L (E), (20) for E = E max a n d s u b je c t to fixed assig n m en t

c o n stra in ts from S tep 2, to find a new schedule for th e u p d a te d set J ' of cu sto m er orders, u p d a te d set of p la n n in g p erio d s T ' a n d u p d a te d m a te ria l availab ility p erio d s

/ J m ax{ 1. dj E max} if j 6 Jold (oqx

7 = 1

m a x { t mod, d j

^{n ,1.} Z?

E m ax\

^{\ it A}

if 3 G Jmod-

⁷ . (28)

In th e a lg o rith m s R E M A T an d R E N O N p re se n te d below , S te p 1 an d S tep 3 are id en tical w ith th e co rresp o n d in g ste p s of R E A L L .

Algorithm REM AT

Step 0. S p lit th e se t ,J0id of o rd e rs re m a in in g for co m pletion in to tw o d isjo in t subsets: J ^ d of sch ed u lab le o rd ers a n d J ^ d of fixed, n o n-sched ulable o rders.

Jold = {j e ^{jOld ■} E ^{Xjt =} 1} (29)

Jold = Jold \ Jold (30)

Set T old = { t modi • • ■ i tmod T E max}.

Step 2. Do n o t change th e assig n m en t of n on-sched ulable o rd ers j € ./¿}d, i.e.,

Vjt — Vjti 3 € ^J

^m

^{, t} S ^Toui (31)

j dm odd' ddm ax+1 yjdmcxt ^max + ^{I' 3} ^ ^Jold 0 ^J~ * J d moddddoiax ^ (3— )

Algorithm R ENO N

Step 0. S p lit th e se t

J0id

of o rd e rs re m a in in g for com p letio n in to tw o disjo in t subsets: Jgtd of sch ed u lab le o rd ers a n d J $ d of fixed, n o n -sch edu lable orders.

Jold = Jold (33)

Jold = 0 (34)

Step 2. Do n o t change th e assig n m en t of all o rd ers in

J0id

, i.e.,

Vjt = V jt> j € Jold> t € T 0id (35)

88 T . Sawik

5. Input and O utput Inventory

In th is sec tio n som e form ulae a re derived for c a lc u latio n th e in p u t inven­

to ry of ra w m a te ria ls a n d th e o u tp u t inv en to ry of finished p ro d u c ts . T h e in p u t in v en to ry of p ro d u c t-sp ecific raw m a te ria ls only is co nsidered w ith no com m on m a te ria ls for d ifferen t p ro d u c t ty p e s ta k e n in to ac co u n t. F u rth e rm o re , to m ake th e ca lc u latio n s m o re tr a n s p a re n t it is assu m ed t h a t each p ro d u c t re q u ires one u n it of th e c o rresp o n d in g p ro d u ct-sp ecific m a te ria l (e.g. one p rin te d w iring b o a rd of a specific d esign p e r one electronic device of th e corresp o n d in g ty p e ). As a re su lt, for each o rd e r j th e re q u ired q u a n tity of p ro d u ct-sp ecific m a te ria l equals th e q u a n tity of th e o rd e re d p ro d u c ts s j.

T h e o rig in al a m o u n t of p ro d u c t specific m a te ria ls re q u ire d for cu sto m er o rd ers j £ J mod su ch t h a t dj — E max < t mod < d j an d su p p lied before t mod differs from th e m odified a m o u n t of th o se m a te ria ls re q u ired a fte r th e o rders m odification. A s a re su lt, th e a c tu a l in p u t in v en to ry level in p e rio d t mod m ay b e h ig h er or low er th a n th e re q u ire d level. For each p ro d u c t ty p e k £ K , th e sh o rta g e ( A I N P ^ < 0) or su rp lu s (A I N P ^ > 0) of prod u ct-sp ecific m a te ria l in v en to ry in p e rio d t mod — 1 w ith re sp e c t to th e a m o u n t re q u ired for th e m odified o rd ers j £ J mod is

I t is assu m ed t h a t th e sh o rta g e or th e s u rp lu s of p ro d u c t specific m a te ria ls is b ala n c e d w ith h ig h er or lower supplies in p erio d t mod, respectively.

T h e in p u t in v en to ry I N P { t ) of p ro d u c t-sp ecific m a te ria ls ca n b e ca lc u la te d as below.

w h ere I N P ( t moc{ — 1) is th e in p u t in v en to ry re m a in in g in p e rio d t mod — 1

In (37), th e in p u t in v en to ry I A TP ( t ) in each p e rio d t is ca lc u la te d as th e difference b etw e en th e a m o u n t of p ro d u c t-sp ecific m a te ria ls su p p lied b y p e rio d t a n d th e a m o u n t of th e se m a te ria ls p ro cessed in to finished p ro d u c ts by th is perio d . T h e first s u m m a tio n te rm w ith n eg a tiv e sign in th e rig h t h a n d side of (37) b alan ces in p e rio d t mod th e s h o rta g e or th e su rp lu s of p ro d u ct-sp ecific m a te ria ls su p p lied b y p e rio d t mod - 1.

S im ilarly, t h e o u tp u t inv en to ry O U P ( t) of finished p ro d u c ts can b e ex pres­

sed as below.

A I N P k = E ( 4 - si ) ; k e K (36)

J € J f c P i J m o d ' d j E m a x ^ t m o d ^ d j

oup(t) = E ouPd(tmod — i) +

R eac tiv e sch ed u lin g in m ak e-to -o rd er m an u fa c tu rin g 89

w here O U P ci{ tmod — 1) is th e o u tp u t in v en to ry of finished p ro d u c ts re m a in in g in p e rio d t mod - 1, d u e in p e rio d d > t mo(j.

In (38), th e o u tp u t in v en to ry O U P ( t ) in each p e rio d t is c a lc u lated as th e a m o u n t of finished p ro d u c ts p rocessed by p e rio d t befo re th e cu sto m er re q u ired sh ip p in g d ates.

T h e to ta l in v en to ry T O T ( t ) = I N P { t ) + O U P ( t) in each p erio d t can b e found b y su m m in g th e co rresp o n d in g rig h t h a n d sides of (37) a n d (38). In p a rtic u la r, th e t o t a l in p u t a n d o u tp u t in v en to ry T O T ( t mod — 1) in th e la st p erio d of th e p revious sch ed u le ca n b e exp ressed as below.

T O T ( t mod- 1 ) = £ S j { l - £ yjT ) + £ s /3 9 )

T h e first s u m m a tio n te rm in (39) is th e in v en to ry of p ro d u ct-sp ecific m a te ria ls for c u sto m e r o rd e rs d u e by p e rio d t mod — 1, a n d th e second te rm is th e inv en to ry of p ro d u c t-sp ecific m a te ria ls a n d finished p ro d u c ts of cu sto m er o rd ers d u e a fte r p e rio d t mod — 1, re sp ectiv ely w a itin g for p rocessing in th e sy stem a n d for ship p in g to custo m ers.

T h e first te r m re p re se n ts th e in p u t in v en to ry in p e rio d t mo(j — 1 of p ro d u c t- specific m a te ria ls for ta r d y o rd e rs an d is g re a te r th a n zero only if som e cu sto m er o rd ers are ta rd y , o th erw ise th is te rm is eq u a l to zero. T h e secon d te rm increases w ith th e m ax im u m earliness E max. G iven th e t a r d y ord ers, th e t o ta l in v en to ry in t mo(i — 1 in cre ases w ith E max, i.e. b o th th e in p u t inv en to ry of p ro du ct-sp ecific m a te ria ls a n d t h e o u tp u t in v en to ry of finished p ro d u c ts ca n b e re d u ced w hen re a d y p e rio d s a n d d u e d a te s of c u sto m e r o rd ers are closer.

6. C om putational Experim ents

In th is se c tio n n u m erical exam ples a n d som e c o m p u ta tio n a l re su lts are p re ­ se n te d to illu s tra te p o ssible ap p lica tio n s of th e p ro p o se d a lg o rith m s for re activ e scheduling, b ase d on th e m ixed in teg er p ro g ram m in g fo rm u lation s. T h e ex am p les a re m o d eled a fte r a re al w orld d istrib u tio n ce n te r for h ig h -tec h p ro d u c ts , w here finished p ro d u c ts a re assem bled for sh ip p in g to cu sto m ers.

T h e d is trib u tio n c e n te r is a flexible flowshop m ad e u p of six processing sta g e s w ith p a ra lle l m achines. In th e d is trib u tio n ce n te r 10 p ro d u c t ty p e s of th re e p ro d u c t g ro u p s a re assem bled. T h e p rocessin g sta g e s are th e following: m a te ria l p re p a ra tio n sta g e , w here all m a te ria ls re q u ire d for assem bly of each p ro d u c t are p re p a re d , p o stp o n e m e n t stag e, w here p ro d u c ts for som e o rd ers are cu stom ized, th re e flash in g /flex in g sta g e s in p arallel, one for each g rou p of p ro d u c ts, w h ere re q u ired softw are is dow nloaded, a n d a p ack in g stage, w h ere p ro d u c ts an d re q u ired accessories a re p ac k ed for shipping.

C u sto m e r o rd e rs re q u ire p rocessing in a t m ost four stages: m a te ria l p re p a ­ ra tio n stag e, p o stp o n e m e n t stag e, one flash in g /flex in g stag e, a n d packing stage.

However, som e o rd e rs do n o t n eed p o stp o n e m e n t.

90 T . Sawik

C u sto m e r o rd ers are s p lit in to p ro d u c tio n lo ts of fixed sizes, each to b e processed as a s e p a ra te jo b . E a c h larg e size (m u lti-p erio d ) c u sto m e r o rd e r m u st b e co m p leted in a t m o st tw o p la n n in g p e rio d s (tw o d ays).

A b rie f d esc rip tio n of th e p ro d u c tio n sy stem , p ro d u c tio n process, p ro d u c ts a n d th e beg in n in g cu sto m er o rd e rs is given below.

1. P ro d u c tio n sy stem

• six processing stages: 10 p a ra lle l m achines in each sta g e i — 1, 2; 20 p arallel m achines in each s ta g e i = 3 , 4 , 5 ; a n d 10 p a ra lle l m ach in es in stag e i = 6.

2. P ro d u c ts

• 10 p ro d u c t ty p e s of th r e e p ro d u c t groups, each to b e processed on a se p a ra te g ro u p of fla sh in g /fle x in g m achines,

• th e b eg in n in g d e m a n d is m a d e u p of 100 c u sto m e r orders, each co nsistin g of several su b o rd e rs (c u sto m e r re q u ired sh ip p in g volum es). T h e to ta l n u m b er of su b o rd e rs is 816, a n d th e b eg in n in g to ta l d e m a n d for all p ro d u c ts is 537760.

• p ro d u c tio n (a n d tra n s fe r) lo t sizes: 200. 200, 300, 100, 100, 100, 200, 200, 300, 100, re sp ectiv ely for p ro d u c t ty p e 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

3. P ro cessin g tim e s (in seconds) for p ro d u c t ty p es:

p r o d u c t t y p e / s t a g e 1 2 3 4 5 6

1 20 0 120 0 0 15

2 20 0 140 0 0 15

3 10 0 160 0 0 10

4 15 5 0 120 0 15

5 15 10 0 140 0 15

6 10 5 0 160 0 10

7 15 10 0 180 0 15

8 20 5 0 0 120 15

9 15 0 0 0 140 10

10 15 0 0 0 160 10

4. P la n n in g horizon: h — 30 days, each o f le n g th L = 2 x 9 hours.

N otice t h a t th e s u b o rd e rs in th e c o m p u ta tio n a l ex am ples p lay th e role of o rd e rs in th e m a th e m a tic a l fo rm u la tio n . Now, th e p ro b lem o b jectiv e is to as­

s ig n /re a ssig n cu sto m er su b o rd e rs over t h e p la n n in g h o rizo n to m in im ize n u m b er o f ta r d y su b o rd e rs as a m easu re of th e c u sto m e r serv ice level s u b je c t to m ax im u m ea rlin es c o n stra in ts to lim it t h e to t a l in v en to ry level. In th e c o m p u ta tio n a l ex pe­

rim e n ts th e following th r e e m o d ificatio n s of c u sto m e r o rd ers d u rin g th e p la n n in g h o rizo n a re considered:

R eactiv e scheduling in m ak e-to -o rd er m a n u fa c tu rin g 91

60000 50000 40000 30000 2 0 000 10000

0

i i Beginning Demand

—• — Beginning Production Schedule

V ' 0 & J

r f f

y *■* rr? v v y w r n t r -T-T-r-?

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

I 1 I Remaining Demand i :--3 Updated Demand

—• — Updated P roduction Schedule

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31

I I Remaining Demand i 1 i Updated Demand

—• — U pdated Production Schedule

3 5 7 9 11 13 15 17 19 21 23 25 27 29 31

60000 50000 40000 -- 30000 2 0 0 0 0 - -

10000

0

tmod = 24 » * \

I ! I" l

Remaining Demand EH.Z3 Updated Demand

—• — Updated Production Schedule

-t—r—(

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33

Fig. 1. D istrib u tio n of d em an d a n d a g g reg ate p ro d u c tio n for alg o rith m R E A L L

92 T . Sawik

T ab le 3 C o m p u ta tio n a l re su lts

Model / tjyiod Var. Bin. Cons. Nonz. Solution values^ CPU*

INITIAL SCHEDULING FOR. BEGINNING DEMAND

M axSL 29310 14656 18198 133276 u;um = o 3.60

M inME(O) 31507 15753 33057 148946 Eïnax = 6 387.88

R.EALL

M a x S L (6 )/tmod = 6 22522 11565 19779 316229 Usum = 2, h' = 31 20.61 M a x S L (6 )/tnw<i = 14 15558 8013 13116 195823 u;um = 3, h' = 32 28.24

M a x S L (6 )/tmod = 24 4059 2112 3928 40811 U;nm = 5, h' = 33 1.22

REM AT

M a x S L (6 )/tmai = 6 15817 8145 11237 186527 u;um = 3, h! = 31 9.07

M ax.SL(6)/tmod = 14 7656 3959 5610 77634 U s u m = 6, h' = 32 1.69

M a x S L (6 )/tmod = 24 222 105 373 1898 U;um = 8 ,h ' = 33 0.02

REN ON

M a x S L (6 )/tmo<¡ = 6 371 152 1281 6121 U s u m = 8, ti = 35 0.04

M a x S L (6 )/tmod = 14 537 248 1625 8479 u ;um = 10, h' = 35 0.09

M a x S L (6 )/tmcxi = 14 382 184 759 ⁴³⁴⁴ u;um = 11, h' = 36 0.09

* Hjum ' num ber of ta rd y orders, E*nax - m axim um earliness, h' - planning horizon

1 C PU seconds for proving optim ality on a P C P entium IV, 1.8GHz, RAM 1GB /C P L E X v.9.1

Fig. 2. T o tal in p u t a n d o u tp u t in v en to ry for various resch edu ling a lg o rith m s

• 13 cu sto m er orders due in p e rio d s 8-30 are m odified in p e rio d t mo(i = 6. T h e re su ltin g increase of d em an d is 70200 p ro d u c ts.

• 13 cu sto m er orders due in p erio d s 15-30 are m odified in p e rio d t moj = 14.

T h e re su ltin g increase of d e m a n d is 15950 p ro d u c ts.

• 8 cu sto m er orders d u e in p e rio d s 26-30 are m odified in p e rio d t moj = 24.

T h e re su ltin g in crease of d e m a n d is 14960 p ro d u c ts.

R eac tiv e sched uling in m ak e-to -o rd er m an u fa c tu rin g 93

T h e re su ltin g t o ta l in cre ase of d e m a n d is 101110 p ro d u c ts.

T h e c h a ra c te ristic s of in te g e r p ro g ram s M a x S L , M in M E (U ) (w ith (7 = 0) for th e in itia l scheduling a n d M a xS L ( E ) (w ith E = 6) for th e th re e rescheduling a lg o rith m s R E A L L , R E M A T a n d R E N O N , an d th e so lu tio n re su lts a re su m m ari­

zed in T able 3. T h e size of each in te g e r p ro g ra m is re p re se n te d by th e to ta l n u m b e r of variables, V ar., n u m b e r of b in a ry variables, B in., n u m b er of co n stra in ts, C ons., a n d n u m b er of nonzero elem en ts in th e c o n s tra in t m a trix , N onz. T h e la st tw o colum ns of th e ta b le p re sen t th e o p tim a l so lu tio n values of Usum for M a x S L , E max for M in M E fU ), Usum , h ' for M a x S L (E ), an d C P U tim e in seconds re q u ired to find th e proven o p tim a l so lu tio n . T h e c o m p u ta tio n a l e x p e rim en ts w ere p erfo rm ed using A M P L p ro g ram m in g lan g u ag e a n d th e C P L E X v.9.1 solver on a la p to p w ith P e n tiu m IV a t 1.8G H z a n d 1GB R A M .

T ab le 3 in d ic a te s t h a t th e b e s t re su lts (th e m in im u m n u m b e r of ta r d y o r­

d ers over th e p la n n in g horizo n a n d th e sm allest h o rizo n len g th ) are o b ta in e d for a lg o rith m R E A L L , w h ere t o ta l resch ed u le of all re m a in in g cu sto m er o rd ers is ap ­ p lied each tim e som e o rd ers are m odified. In c o n tra s t, a lg o rith m R E N O N , w here th e assig n m en t of all re m a in in g o rd e rs is n o t changed, p ro d u c es th e w orst re su lts.

O n th e o th e r h a n d R E N O N re q u ires th e least, a n d R E A L L th e g re a te st C P U tim e to find proven o p tim a l schedules.

T h e d is trib u tio n of in itia l d e m a n d ah e ad of a m o n th ly horizon, d em an d re m a in in g a n d u p d a te d a fte r each m o d ificatio n of ord ers, an d th e co rresp o n d in g p ro d u c tio n schedules o b ta in e d using sch e d u lin g /re sch ed u lin g alg o rith m R E A L L are show n in fig .l. F or a com p ariso n , fig.2 show s how th e to ta l in v en to ry of p ro d u c t specific m a te ria ls a n d finished p ro d u c ts varies over th e ho rizon for each resche­

d u lin g alg o rith m . T h e low est m ax im u m in v en to ry level is achieved for R E A L L w hereas R E N O N leads to th e h ig h est level.

7. Conclusion

In th is p a p e r various re a c tiv e sched uling policies b ase d on th e m ixed integer p ro g ram m in g m odels a re p ro p o se d for a d y n am ic, m ak e-to -o rd er m a n u fa c tu rin g e n v iro n m en t. T h e c o m p u ta tio n a l re su lts have in d ic a te d t h a t th e m od els ca n b e a p p lied for re a c tiv e sch eduling to ite ra tiv e ly u p d a te p ro d u c tio n schedule over a d y n am ic p la n n in g horizon. T h e resch ed u lin g a lg o rith m s are ca p a b le of finding p roven o p tim a l schedules in a re a so n a b le C P U tim e for large size p ro b lem s t h a t can b e e n c o u n te re d in th e in d u s tria l p ra ctice.

R E F E R E N C E S

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94 T . Saw ik

2. L iao D .Y ., C h a n g S .C ., P ei K .W ., C h an g C .M .: D aily sch ed u lin g for R& D sem ico n d u c to r fa b ricatio n . IE E E T ra n sa c tio n s on S em ico n d u cto r M an ufac­

tu rin g , 1996, 9, p. 550-560.

3. N e m h au ser G .L ., W olsey L.A.: In teg e r a n d C o m b in a to ria l O p tim iz a tio n . 1998, (W iley: N ew Y ork).

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Omówienie

W p ra c y p rz ed staw io n o nowe a lg o ry tm y reak ty w n eg o h a rm o n o g ram o w an ia p ro d u k cji zam aw ian ej, o p a rte n a m o delach p ro g ram o w an ia całkow itoliczbow ego.

Z am ów ienia m o g ą być m odyfikow ane p rzez odbiorców w c a ły m h o ryzon cie p la­

now ania. C elem h arm o n o g ram o w an ia je s t m in im a liz a c ja liczby spó źn io n y ch za­

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