A Brief Survey of Just-In-Time Sequencing for Mixed-Model Systems

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1 Iteratoa Joura of Operatos Research Iteratoa Joura of Operatos Research Vo., No., (005) A Bref Survey of Just-I-Tme Sequecg for Mxed-Mode Systems Taka Nath hamaa * ad Wesaw Kubak Facuty of Busess Admstrato, Memora Uversty of Newfoudad AB 3X5, St. Joh s, Newfoudad, Caada Abstract The cocept of peazg obs both for beg tardy ad for beg eary has prove oe of most mportat ad ferte research topcs Operatos Research. I ths survey, we cosder ust--tme mxed-mode, mut-eve suppy chas. Obtag a optma sequece a mut-eve cha s a chaegg oear teger programmg probem. Probems for two or more eves are strogy NP-hard. The probem of mmzg devatos betwee actua ad desred producto for sge-eve ca be soved effcety. Aso the mut-eve probems wth peggg assumpto are sovabe by reducg them to the sge-eve. Cycc schedues are optma for sge-eve probem. We preset varous ways of deag wth these probems such as the eegat cocept of baaced words ad dfferet optmzato techques. We provde a short revew of dfferet mathematca modes, dscuss ther compexty ad compare them. The research resuts obtaed past severa years are preseted aog wth ope probems ad possbe extesos. Keywords Noear teger programmg, Schedug theory, Just--tme systems, Baaced (eve) schedues, Baaced words, Computatoa compexty, Poyoma agorthms. INTROUCTION The cetra goa of mxed-mode or fexbe assemby processes s to crease proft by reducg costs. The ust--tme (JIT) producto systems, whch requre producg oy the ecessary product the ecessary quattes at the ecessary tme, have bee used for cotrog such fexbe assemby systems. The teto of these methods s to satsfy the customer demads for a varety of modes wthout hodg arge vetores or currg arge shortages of the products. We assume a fow e maufacturg, caed fexbe trasfer es, where eggbe swtch-over costs from oe mode to aother aow for dversfed sma-ot producto avodg producto of each mode arge-ots. The most mportat optmzato probem that has to be soved for the mxed-modes, ust--tme systems s to determe the sequece whch dfferet modes are produced. Ths souto mpacts the etre suppy cha. There has bee growg terest JIT systems research sce Mode (983), Mteburg (989), cosders the probem of determg the sequece for producg dfferet products o the e that keeps a costat rate of usage of every part used by the e. I other words, the quatty of each part used by the mxed-mode assemby e per ut of tme shoud be kept as costat as possbe. Ths aows very tte varabty the usage of each part from oe tme horzo to the other. Mode (983), states ths as the most mportat goa of a JIT producto system mpemeted by the Toyota compay. Toyota s so-caed Goa Chasg Method, a oca search heurstc, has bee most popuar for sovg the probem. The sequeces referred to as eve, baaced or far sequeces aways keep the actua producto eve ad the desred producto eve as cose to each other as possbe a the tmes. The other producto ssues studed are cyce tmes, ead tmes, work--process ad oadg (Kubak, 993; Mteburg, 989; Mteburg ad Godste, 99; Mode, 983; Okamura ad Yamasha, 979; Kbrdge ad Wester, 963). Mut-eve producto systems, where compoets requred for dfferet modes may or may ot be dstct, make the probem more chaegg tha the sge-eve producto systems where dfferet modes requre the same umber ad mx of compoets. Because of the pu ature of the JIT systems, the producto sequeces at a other ower eves are aso herety fxed as soo as the fa eve producto sequece s fxed. That s why the determato of the sequece of dfferet products at fa assemby eve s cruca. Mteburg (989), provdes a oear teger programmg formuato for the mmzato of tota devato for mxed-mode JIT producto systems uder the assumpto that the products requre approxmatey the same umber ad mx of parts. As optma sequece at the fa assemby eve woud smutaeousy acheve a eve rate of parts usage at the feeder producto eves, ths formuato ca be cosdered as a sge-eve probem. A exact expoeta tme agorthm ad two heurstcs are aso preseted Mteburg (989), Mteburg ad Godste (99), ad Mteburg ad Samo (989), exted the formuato to mut-eve assemby systems. Most of these optmzato probems woud requre eumeratve or expoeta agorthms. Mteburg et a. (990), Yeomas (997), ad Kubak et a. (997), preset dyamc programmg approaches to the mut-eve probems. We refer the reader to Groef et a. (989), Ima ad Buf (99), g ad Cheg (993), Sumchrastet et a. (99), Sumchrast ad Russe (990), Mteburg ad Godste (99), ad Mteburg ad Samo (989) for severa * Correspodg author s e-ma: dhamaa@yahoo.com 83-73X Copyrght 005 ORSTW

2 hamaa ad Kubak: A Bref Survey of Just-I-Tme Sequecg for Mxed-Mode Systems 39 heurstcs for the probem. Kubak ad Seth (99, 994), reduce the mmzato of tota devato JIT probem to a assgmet probem ad thereby preset a effcet optmzato agorthm for ths probem. The agorthm works for more geera sum obectve fuctos cosstg of oegatve covex fuctos of devatos betwee cumuatve average demad ad cumuatve producto of varous modes over tme. Steer ad Yeomas (993), foowg the optmzato agorthm for the tota devato gve Kubak ad Seth (99, 994) gve a graph theoretc optmzato agorthm for mmzg maxmum devato JIT sge-eve sequecg probem. They aso gve a agorthm for mmzg mut-eve maxmum devato JIT assemby systems uder the peggg assumpto (Steer ad Yeomas, 996). If outputs at producto eves whch feed the fa assemby eve are dedcated to the fa product to whch they w be assembed, the the probem wth peggg s equvaet to a weghted sge-eve of probem whch ca the be mmzed by modfed agorthm for u-weghted sge-eve probem. For both maxmum ad tota devatos, there are aways cycc schedues whch are optma, see Steer ad Yeomas (996), ad Kubak (003), whch sgfcaty reduces the computatoa requremets. Brauer ad Crama (004), prove that the mmzato of maxmum devato or botteeck for a sge-eve s Co-NP, but geera, the compexty of the sge eve probems remas ope for the bary ecodg. The mut-eve probem for two or more producto eves s strogy NP-hard, Kubak (993). Brauer ad Crama (004), preset a agebrac approach to the resuts of Steer ad Yeomas (993), ad formuate the sma devato coecture. Kubak (003), presets a geometrc proof of the coecture ad ater Brauer, Jost ad Kubak (004), expot the cocept of baaced words to gve aother proof of the coecture. Kubak (004, 005), presets propertes of JIT sequeces obtaed through mathematcay eegat cocept of baaced words. We refer the terested readers to Vuo (003), for a survey ad the refereces about baaced words. Bautstaet et a. (997), Kubak (993), ad Pa (00), preset effcet agorthms for maxmum devato probem based o the reducto to the botteeck assgmet probem. Bautsta et a. (996) estabsh a terestg k betwee the JIT sequecg ad the apportomet probem. A apportomet probem deas wth the aocato of seats of a egsature amog the states or provces of a ato. Bask ad Shahd (998), cosder the JIT sequecg probem as the quota method of apportomet. Coromas ad Moreo (003), vestgate reatoshps betwee the souto spaces of dfferet obectve fuctos. The pa of the paper s as foows. I Secto, we revew optmzato modes of JIT sequeces. I Sectos 3 ad 4 we survey the effcet agorthms for tota-devato ad maxmum-devato obectve fuctos, respectvey. Secto 5 summarzes the baace propertes of m-max sequeces. The cycc schedues are dscussed Secto 6. Secto 7 reates the optmaty codtos betwee dfferet obectve fuctos. Secto 8 s devoted to the study of computatoa compexty of the probems, heurstc soutos ad a dyamc programmg approach. The fa Secto 9 cudes cocusos wth possbe drectos ad ope questos for further research.. THE MATHEMATICAL PROGRAMMING FORMULATION. Mut-eve formuato A mxed-mode mut-eve suppy cha cossts of a herarchy of severa dstct producto eves (for exampe, products subassembes compoets raw-materas). I these suppy chas, the mutpe copes of dfferet modes are produced at the fa assemby eve. The assemby system aso cotas severa other ower producto eves where subassembes, compoet parts ad raw materas are ether fabrcated or purchased for use the products. Let there be L dfferet producto eves, where,,..., L wth the hghest eve, the product eve. We deote the umber of dfferet part types of eve by ad the demad for part, where,,...,, of eve by d. eotg by t p the umber of tota uts of output at eve requred to produce oe ut of product p, we have d t p d p, the depedet demad for p part of eve determed by the fa product demads d p, p,,...,. Note that t p f p ad 0 otherwse. Let d be the tota output demad d of eve. The demad rato for part of eve s r ad r at each eve,,...,l. Uder the assumpto of o-preemptve schedue, a schedue s competey defed by the sequece of product copes of the product eve. A copy s sad to be stage k, k,,...,, f k uts of product have bee produced at eve. The tota horzo w be of tme uts ad there w be k compete uts of the varous products p at eve durg the frst k stages. ue to the pu ature of the JIT system aog wth the fact that the ower eve outputs are draw as eeded by the fa product eve, the partcuar combato of the products produced at the product eve durg the frst k stages determes the ecessary cumuatve part producto at every other eve. Let x be the ecessary cumuatve producto of k output at eve durg stages through k ad y k x k be the tota output of eve durg stages through k. Ceary, the cumuatve producto of eve

3 hamaa ad Kubak: A Bref Survey of Just-I-Tme Sequecg for Mxed-Mode Systems 40 through the frst k stages s y k x k. The requred cumuatve producto for part at eve, where, through k stages w be xk t p p x pk. Fay, we coud mpose a weght w because of reatve mportace of baacg the schedue for part at eve. The feasbe souto rego s deoted by χ M { XX ( x pk) } where the varabes satsfy the foowg costrats: k p pk,,...,,,...,,,...,, p x t x L k () y x,,..., L, k,..., k k k pk p y x, k,...,, (3) () L QS( ) ( k k ) () k G X x y r Wth ths otato, the mut-eve JIT sequecg probem s equvaet to m{ GX ( ) X χ }, M where G { GAM, GQM, GAS, G QS }. By Kubak et a. (997), w x y r γ p x pk, p k k where γ p w δ p ad δ p t p r t hp. We defe a matrx Γ [ γ ] h p wth L, where γ p represetg the ( + ) m m th row ad pth coum eemet. Let X k ( x,..., ) T k x k be a vector represetg the cumuatve producto at eve through the frst k stages. The pk,,...,, (4) p k x k x x, p,...,, k,...,, (5) pk p( k ) x d, x 0, p,...,, (6) p p p0 x N,,...,,,..., L, k,...,. (7) k I ths paper, N deotes the set of a oegatve tegers. Costrat () esures that the ecessary cumuatve producto of output of eve by the ed of stage k s determed expcty by the quatty of products produced at producto eve. Costrats () ad (3) cacuate the tota cumuatve producto of eve ad, respectvey, through stages to k. Costrat (5) s to esure that the tota producto of every product over k stages s a o-decreasg fucto of k. Costrat (6) guaratees that the producto requremets for each product are met exacty. Costrats (4), (5) ad (7) esure that exacty oe product s schedued for fa assemby durg each stage. The the mxed-mode, mut-eve schedue probem s to seect X ( x pk) that mmzes oe of the foowg m-max/m-sum obectve fucto (s) G ( X) max x y r (8) AM k k k,, QM( ) max( k k ) (9) k,, G X x y r L G ( X) x y r (0) AS k k k GAM( X ) max ΓXk, k where Γ X k max { w xk ykr }. Here, ΓX k, represets the maxmum devato at stage k over a ad. Notce from the matrx represetato that at ay partcuar stage, the devato of ay part of ay eve s determed by the eve sequece. Lkewse, sum devato obectve G ( X ) QS ( X k ) k wth devato matrx Ω Ω[ w δ ] ad the Eucdea orm p m a a of a vector a (a, a,..., a m ). The sequecg probems, maxmum-devato JIT ad tota-devato JIT, are deoted by MJIT ad SJIT probems, respectvey. The probem s oe of the most fudameta probems fexbe ust--tme mxed-mode producto systems, referred to as JIT sequeces. I these formuatos, the m-sum ad m-max obectves are smar to Mteburg ad Samo (989), Steer ad Yeomas (996), ad Kubak et a. (997), respectvey. Note that the m-max obectves seek to mmze the devatos for each output at each stage, whereas the m-sum obectves are cocered for fdg the owest possbe tota devato whch may resut reatvey arge devato for a certa product. The effects of weghts sge-eve as we as other mut-eve probems are cosdered Yeomas (997), Kubak et a. (997), Steer ad Yeomas (996), Mteburg et a. (990), Mteburg ad Godste (99), Mode (983).. Sge-Leve formuato For,...,, gve products (modes), postve

4 hamaa ad Kubak: A Bref Survey of Just-I-Tme Sequecg for Mxed-Mode Systems 4 tegers (demads) d ad covex symmetrc fuctos f of a sge varabe, caed devato, a assumg mmum 0 at 0. The foowg optmzato probems have bee cosdered to mode sge-eve system. Fd a sequece s s s...s wth tota demad d of products,,,...,, where product occurs exacty d tmes that mmzes the foowg obectve fucto (s) F ( s) max f ( x kr ) () M k k, F ( s) f ( x kr ) (3) S k k where x k represets the umber of product occurreces d (copes) the prefx s s...s k, k,...,, ad r,,...,. The foowg two measures of devatos have bee studed the terature. x k kr the absoute - devato obectve, f( xk rk ) ( x k kr ) the squared - devato obectve. The whoe feasbe souto rego x { X X ( x k ) } sge-eve probem s costraed as x k k,...,, k xk, x k, +,...,, k,...,, x, d, x 0 0,...,, x N,...,, k,...,. k, Ths probem s referred to as the Product Rate Varato Probem (PRV) the terature Kubak (993). A souto of ths probem aways keeps the actua producto eve x k ad the desred producto eve r k as cose to each other as possbe a the tmes. A souto s s s...s of the sge-eve MJIT probem for modes s caed B-feasbe (or B-bouded) f max,k f (x k r k ) B hods for the matrx varabes X ( x k ). We deote the sets of a sge-eve B-feasbe soutos by x B. Note that the above formuato gves the foowg umber-theoretc terpretato of JIT sequecg probem: gve ratoa umbers r,,,...,, wth commo deomator, the probem s to fd tegers x k whch optmay approxmate the sequece (kr ) uder the cardaty ad mootocty restrctos defed above (see aso Brauer ad Crama (004) for the refereces). A mut-eve, m-max probem uder the peggg assumpto has bee reduced to a weghted sge-eve probem (Godste ad Mteburg, 988), (see aso Steer ad Yeomas, 996). Smary, the m-sum, mut-eve probem wth peggg ca be reduced to a weghted sge eve probem cosdered by Yeomas (997). Godste ad Mteburg (988), were the frst to provde mathematca formuato of peggg JIT systems (see aso Steer ad Yeomas, 996). Uder the peggg assumpto, parts of output at eve are dedcated to the partcuar product at eve to whch they w be assembed. Ths assumpto decomposes the ower eve parts that w be assembed to dfferet eve products to dsot sets. Wth ths assumpto, the mut-eve AMJIT sequecg probem subect to the costrat set χ wth p,..., M,,...,,,..., L ad k,..., ca be formuated as m max { w x kr, w t x kr }. pk,,, p pk p p pk p Sce t p f p ad 0 otherwse, the above probem s equvaet to m max { x kr } pk,,, L υ max { w t }, υ p pk p, where p p, By droppg the superfuous subscrpt, we obta the foowg weghted sge-eve AMJIT probem m max { υ x kr : X χ}, k, k,,, k,,. (4) 3. EFFICIENTLY SOLVABLE SJIT SEQUENCING I ths secto, we study the sge-eve, m-sum probems wth the obectve defed (3). Uess otherwse specfed, sge eve, m-sum probems w be deoted by SJIT. These resuts are vad for covex, symmetrc, oegatve fuctos whch take vaue 0 at 0. Let Y {(,, k) :,..., ;,..., d ; k,...,}. efe cost C 0 for (,, k) Y wth respect to the dea posto as foows Z r Z ψ < f k Z, k C 0 f k Z, k ψ f >, Z k Z for the -th copy of mode where Z uquey soves f ( kr ) f ( kr ) ad f (x) x, ψ f( r) f( r) f < Z, f( r) f( r) f Z. A subset Y of Y s caed feasbe f t satsfes the

5 hamaa ad Kubak: A Bref Survey of Just-I-Tme Sequecg for Mxed-Mode Systems 4 foowg costrats C. For each k,...,, there s exacty oe (, ),,..., ;,..., d s.t. (,, k) Y,.e., exacty oe copy product at each tme. C. For each (, ),,..., ;,..., d, there s exacty oe k, k,..., s.t. (,, k) Y,.e., each copy s produced exacty oce. C 3. If (,, k), (,, k ) Y ad k < k, the <,.e., ower dces copes are produced earer. Cosder ay set S of trpes (,, k) satsfyg C, C, C 3 ad defe the sequece s s s...s wth s k, f (,, k) S for some,...d correspodg to the set S. The the sequece s s feasbe for ay gve stace (d, d,..., d ) ad foowg resuts hod, Kubak ad Seth (994). Theorem. Let c(s) C (,, k ) for ay S Y. The, a. For ay feasbe S, t hods F ( s) c( S). + f ( ) f k kr b. If S satsfes C adc, the S* satsfyg C, C ad C 3 wth c(s) c(s*) ca be determed O() steps. Moreover, each product copes preserve the order the sequece s*as t does the sequece s. As the term f f( kr ) s k depedet of the set S, that s costat, a optma souto to SJIT woud be a mmedate cosequece f a optma set S s foud. But a optma set caot be obtaed by smpy sovg the Assgmet Probem 5 wth costrats C ad C ad the costs C wth (,, k) S, as the costrat C 3 s ot of the assgmet type. Notce that the atter costrat s esseta as t tes up the copy of a product wth the -th dea posto for the product. The ma dea of the proof s to show that there exsts at east oe optma sequece for the assgmet probem such that copy (, + ) of product shoud appear after the copy (, ). The proof s doe by mathematca ducto. Wth these costs the correspodg assgmet probem has bee formuated as foows, Kubak ad Seth (994): ( d, ) m [ Fs ( ) C x ] (5) (, ) k subect to the costrats k S x, for,..., ;,..., d, ( d, ) x, for k,...,, (, ) where x, f (, ) s assged at posto k, 0, otherwse. Observe that a obvous optma souto coud be obtaed f sequecg a copes ther dea postos were possbe wthout competto for these postos. As ths s ot the case geera, we eed to resove competto to mmze the gve obectve. Ths s doe effcety by sovg the assgmet probem, (Kubak, 993, Kubak ad Seth, 994). Reca that the assgmet probem wth m odes ca be soved O(m 3 ) tme (see Kubak, 993; Kubak ad Seth, 994, for the refereces). The approach proposed by (Kubak ad Seth 99, 994) for the tota devato product rate varato probem s appcabe to ay p orm wth Fs p, ad partcuar to -orm. I the atter case the approach mmzes maxmum devato obectve. Cosequety, souto to mut-eve m-sum probem wth peggg assumpto coud be obtaed as Kubak ad Seth (99). Steer ad Yeomas (994), ook at the m-sum probem as a weghted matchg probem a compete bpartte graph G (V,E), where weghts of the edges equa peaty costs C. The the probem s to fd a perfect matchg wth the mmum sum of the weghts. A compete bpartte graph s defed by troducg the earest ad atest competo tmes possbe for a copy (, ) of product (see Secto 4 for the defto). Moreover, a -bouded souto that s optma (f such souto exsts) coud be obtaed O( og ) tme, sce for B mpes E ( + ). A Pareto optma souto ca be foud O( og ) tme. But the exstece of -bouded soutos optma for m-sum probems s ot aways the case (see Secto 7). The foowg questo remas ope. What s mmum B such that optma souto for m-sum probem s B-bouded? It s kow that a upper boud o the optma m-sum-absoute ad m-sum-squared obectves s though the boud s ot tght (Steer ad Yeomas, 994). For the sake of competeess, we meto that severa heurstcs for sge-eve probem have aready bee vestgated g ad Cheg (993a, 993b), Godste ad Mteburg (988), Ima ad Buf (99), Mteburg ad Godste (99), Mteburg et a. (990), Mteburg ad Samo (989), Sumchrast et a. (99), Sumchrast ad Russe (990). 4. EFFICIENTLY SOLVABLE MJIT SEQUENCING I ths ad Secto 5 we study the m-max probems that are sge-eve wth the obectve defed (). Uess otherwse specfed, sge eve absoute devato m-max probems w be deoted by AMJIT. Steer ad Yeomas (993), study AMJIT probem reducg t to a sge mache schedug decso probem wth reease tmes ad due dates. They represet the probem as a perfect matchg probem a V -covex

6 hamaa ad Kubak: A Bref Survey of Just-I-Tme Sequecg for Mxed-Mode Systems 43 bpartte graph G (V V, E) where the set V {,...,} represets postos ad the set V {(, ),..., ;,..., d } represets the copes of the products. Here, for,..., ad,..., d, the otato (, ) deotes the -th copy of product mode. There exsts a edge {k, (, )} E f ad oy f k es the permssbe terva [E(, ), L(, )] V of reease tme ad due date for the -th copy of the product. They prove the foowg resut (see aso Brauer ad Crama, (004)). Lemma. Let d,..., d be ay stace of AMJIT probem. A sequece s s s...s s B-feasbe f ad oy f for a,..., ad,...d, ths sequece assgs the copy (, ) to the terva [E(, ), L(, )], where B E (, ) r ad + B L (, ) + r deote the reease date ad the due date of the copy (, ) for gve upper boud B. A terestg questo woud be to show smar cosed form formua for other measure of devato, for stace squared devato. Amogst varous versos of the earest due date agorthms for schedug ut tme obs wth reease tmes ad due dates o a sge mache (see Steer ad Yeomas (993) for the refereces), they appy a modfed verso of Gover s (967), O( E ) earest due date (E) agorthm for fdg a maxmum matchg a V -covex bpartte graph G (V V, E) such that each ascedg k V s matched to the umatched copy (, ) wth smaest due date vaue of L(, ) as defed Lemma. They cocude the foowg. Theorem. The AMJIT sequece s s -feasbe f ad oy f the V -covex bpartte graph G wth boud B has a perfect matchg. Moreover, a optma souto ca be determed by a exact pseudo-poyoma agorthm wth compexty O ( og ). Steer ad Yeomas (996), cosder weghted AMJIT probem ad show that a bary search fds a optma souto for the weghted AMJIT O( og (φ G max )) tme, where φ s a postve teger costat depedg upo probem data. The maxmum weght G max max G gves a upper boud ad LB W m G ( r ) gves a ower boud for the optma obectve vaue of the cosdered probem. Theorem 3. A optma souto to the peggg mut-eve AMJIT ca be determed by a exact pseudo-poyoma agorthm O( og (φ G max )) tme. Let B* be the optma vaue of the AMJIT probem. The for ay stace d,,..., of the AMJIT * probem, t hods that B for,..., where, Brauer ad Crama (004). gcd( d, ) A stroger upper boud has bee obtaed by Tdema (980), B. Thus we have ( ) Theorem 4. For ay stace d,,..., ( > ) of the AMJIT probem, the optma vaue B* satsfes the equaty * max, B ( ). As < ( ) whe d for a wth >, ad ( ) most practca cases, both possbtes have to be take to accout. Obvousy, B* 0 for. A stace of the AMJIT sequecg probem s defed as stadard f gcd(d,..., d ). We ca the correspodg sequece stadard. The sma devato coecture states that for 3, a stadard stace (d,..., d ) of the AMJIT probem has B* < / f ad oy f d for,..., Brauer ad Crama (004). Brauer ad Crama (004), prove the coecture for 6 ad coectured t true for a postve. Kubak (003), presets a geometrc proof that the coecture hods true for ay >. Hs proof expots a atura symmetry of reguar poygos scrbed a crce of crcumferece. Subsequety, Brauer, Jost ad Kubak (004), expot the cocept of baaced words to gve aother proof of the coecture (see Secto 5). Thus, we ca state the foowg theorem. Theorem 5. For 3, a stadard stace (d, d,..., d ) of the AMJIT probem has optma vaue B * < / f ad oy f d * for,...,, ad B. Ths resut ca be restated as foows. For gve ratoa umbers r r,..., r wth 3, t hods [ kr ] k for ay teger k f ad oy f r for,...,. The statemet observes that x, kr < / mpes [ kr ] xk, where [x] deotes the roudg of x to the cosest teger. The structure of staces wth B / becomes more compex as x k may the be equa ether to [ kr] kr / or to [ kr] kr +/ for haf-teger kr (Brauer ad Crama, 004). 5. BALANCE WORS AN AMJIT SEQUENCES Brauer ad Crama (004), Brauer, Jost ad Kubak (004), Jost (003), Kubak (003, 005), study the AMJIT sequeces as baaced words. Oe of the ma probems of baaced words practce s to costruct a k

7 hamaa ad Kubak: A Bref Survey of Just-I-Tme Sequecg for Mxed-Mode Systems 44 fte perodc sequece over a fte set of etters where each etter s dstrbuted as evey throughout the sequece as possbe ad each etter occurs wth a gve rate. Ufortuatey, the exstece of baaced sequeces for most rates s ukey. We wrte a fte word as w a a...such that a A { a, a,..., a } for a {,..., }. A factor of egth f 0 of w s word such that f aa+... a+ f. We say that the dex s the posto of the etter a the word w. The rate r of the etter a fte word w s defed as the fracto r w w where w deotes the umber of occurreces of the dex the word w. A fte sequece w w w...for whch u v δ for a wth u v s caed δ-baaced. We deote the fte repetto of a fte word w by w* ww.... A fte word s s caed perodc f s w* for some fte word w. A fte word w s caed symmetrc f w w R where w R s a mrror refecto of w. A fte baaced word s s caed symmetrc ad perodc f s w* for some fte symmetrc word w. Oe way of budg a fte word o fte etter aphabet A usg the umbers ( ) + s r d d descrbed Kubak (005). It buds a fte word as ( ) foows. Labe the pots +, N by the d d ( ) etter, cosder, + N d d ad the correspodg sequece of abes. Break the te by choosg over whe < gvg hgher prorty to a ower dex wheever a cofct eeds to be resoved. Thus a word wth age vector α,,..., d d d ad the startg pot,,..., β, referred d d d to as a hyperboc bard word Vuo (003), s obtaed. Let w be a fte word assocated wth -dmesoa hypercubc bards of age α ad startg pot β. The w s d -baaced o each etter. Moreover, the boud for the baace s aways reached Vuo (003). Jost (003), proves that for ay fte sequece of tota demad d wth maxmum devato B for product rates d, ay fte perodc word w of perod s s -baaced, -baaced or 3-baaced o each product, f B < /, B <3/4ad B <, respectvey. Ay sequece wth d for a,...,, s a -baaced word though ts maxmum devato B > for 3, Kubak (005). However, the maxmum devato B s greater tha 3/4 for the -baaced word a a a a...a a wth d for each,...,. Lkewse, the maxmum devato B s greater tha for the 3-baaced word a a a a a a...a a a wth d 3 for each,..., wth 3. Thus we have Theorem 6. Let s be a fte sequece of egth d / 3/4 wth maxmum devato B for rates r, ad et S, S ad S be the sets of sequeces wth B </, B < 3/4 or B <, / 3/4 respectvey. The S, S ad S are propery cotaed the sets of -baace, -baace ad 3-baace words, respectvey. The resut of Vuo (003) shows that the prorty based cofct resouto apped wheever there s a competto for a dea posto yeds d beg amost the same sze of the aphabet, that s. Theorem 6 shows that the cofct resouto provded by ay agorthm mmzg maxmum devato eads to d beg costat. Thus, t s cear that the cofct resouto provded by ay agorthm mmzg maxmum devato yeds a better baace tha the prorty based cofct resouto apped wheever there s a competto for a dea posto wth mode 3. Theorem 6 combed wth Theorem 4 guaratees the exstece of a optma souto the set of a 3-baaced words. However, t s a ope questo whether there aways exsts a -baaced word that optmzes AMJIT. For 3, the stadard stace satsfes the property of -baaced words Kubak (003), Brauer, Jost ad Kubak (004). Kubak (003), proves that there exsts a perodc, symmetrc ad -baaced word o 3 etters wth destes r r... r, f ad oy f the destes satsfy r (see Theorem 5). It s easy to costruct symmetrc, perodc, -baaced word wth destes gve such a sequece wth 3 etters, oe fxes a ew etter ad serts t betwee every cosecutve etters of s as we as at the begg ad ed of s to obta a sequece for +, etters wth requred propertes. The umber of staces wth -baaced property s fte case of as Brauer ad Crama (004), Kubak (003), prove that the optma vaue of the AMJIT probem s ess tha / f ad oy f oe of the demads s eve ad the other s odd. 6. THE CYCLIC MJIT AN SJIT SEQUENCES I ths secto, we dscuss the exstece of cycc sequece that are optma. As a exstg agorthms have tme compextes depedg o the magtude of the demads d,..., d ad hece o, the exstece of cycc schedue reduces computatoa tme. Therefore, the questo whether the cocateato s m of m copes of a optma sequece s for d, d,..., d s optma for md, md,...,md s mportat for JIT sequecg. Mteburg (989), Mteburg ad Samo (989), observe the exstece of cycc schedues for sum of squared devatos sge-eve. The m-sum probem have such a cycc optma souto f f f for a, where f

8 hamaa ad Kubak: A Bref Survey of Just-I-Tme Sequecg for Mxed-Mode Systems 45 s covex ad symmetrc fucto wth f (0) 0, Bautsta, Compays ad Coromas (997). Kubak ad Kovayov (998), prove that f f (x) f (x) for a wth x (0,) for symmetrc ad covex fucto f, the the cycc schedue for m-sum probem s optma. Moreover, they gve a couterexampe to show that the aswer s egatve f at east oe f s asymmetrc. A the affrmatve aswers have bee based o the foowg two observatos. The frst observato s that f w uv where u ad v are sequeces for the staces βd,..., βd ad γd,..., γd, respectvey, where β, γ are postve tegers, the F S (w) F S (u) + F S (v), Mteburg (989). The secod observato s that eve f oe reaxes the costrats x(w) d,,,...,, the there st exsts a optma sequece w* such that x(w*) d,,,...,, Bautsta, Compays ad Coromas (997). The atter cocuso does ot hod f f are dfferet though covex ad symmetrc f havg the vaues zeros at 0, Kubak ad Kovayov (998). Kubak (003), proves that the set of a optma sequeces for m-sum sge-eve probem cudes cycc sequeces for symmetrc, covex ad oegatve fuctos. I hs proof a dfferet exchage method s used. Theorem 7. Gve d,...,d et s be a optma sequece for the sge-eve m-sum probem SJIT wth covex, symmetrc ad oegatve f,,...,, a assumg mmum 0 at 0. The s m, m, s optma sequece to SJIT for md, md,..., md. A smar resut for sge-eve m-max probem MJIT coud be proved for -orm. Steer ad Yeomas (996), show that the set of optma sequeces for both weghted as we as u-weghted sge-eve m-max probems for absoute devatos cude cycc sequeces. We coecture that cycc JIT sequeces mut-eve probem are optma. 7. RELATIONS BETWEEN IFFERENT OBJECTIVES Coromas ad Moreo (003), prove the foowg. Theorem 8. Let s be ay sequece for sge-eve JIT sequecg probem. The F AS (s) F QS (s) H 0 H(s) where the costat H 0 0 depeds oy o the probem stace, ad H(s) 0 f s s a -bouded souto ad postve otherwse. Furthermore, the m-sum probems for absoute ad squared devatos have the same set of optma soutos o χ, where χ s the set of a -bouded soutos, Coromas ad Moreo (003). Moreover, ay -bouded souto optma for m-sum absoute devato probem (f exsts) s aso optma for m-sum squared devato probem, ad hece, a optma soutos for the atter probem are -bouded Coromas ad Moreo (003). If oe of the m-sum optma souto for squared devato s -bouded, the the probem for absoute devato aso does ot have - bouded souto. A optma souto to the m-sum probem wth absoute devato whch s ot -bouded may ot be optma for the m-sum probem wth squared devato Coromas ad Moreo (003). There may exst a -bouded optma souto to the atter probem eve though oe of the optma souto to the former probem s -bouded. Moreover, ether of these probems may have -bouded optma soutos Coromas ad Moreo (003). Uke the absoute devato ad squared devato obectves for m-sum probems, the sets of -bouded optma soutos wth other covex, symmetrc ad oegatve fuctos are ot the same, Coromas ad Moreo (003). The emprca resuts of Kovayov, Kubak ad Yeomas (00) refute umber of coectures about the reatoshps betwee optma soutos for dfferet obectve fuctos. 8. COMPLEXITY AN YNAMIC PROGRAMMING The questo of the exact compexty of sge-eve JIT sequecg probem remas ope Kubak (993). As the put sze of ay stace (d,, d ) s O( og d ) O ( og ), a agorthm whch s poyoma ad s oy pseudo-poyoma but ot poyoma the put sze. The probem MJIT s Co-NP but t s st ope f the probem s Co-NP-Compete or poyomay sovabe Brauer ad Crama (004). Kubak (993), proves that a verso of mut-eve m-sum probem, referred to as Output Rate Varato Probem, s NP-hard. The mut-eve m-max probem wth absoute devato obectve s strogy NP-hard, Kubak, Steer ad Yeomas (997). However, Kubak, Steer ad Yeomas (997), preset foowg dyamc programmg approach for mut-eve, m-max ad m-sum probems. Let d ( d,..., d ) ( d,..., d ) be the demad vector at eve ad et e be a ut vector of dmeso wth uty the th row. Redefe states a schedue by X ( x,..., x ),where x deotes the cumuatve producto of the product wth x d ad the cardaty of a state X as X x.the mmum vaue of the maxmum devato for a products ad parts over a parta schedues whch ead to state X s defed by ψ (X). The maxmum orm ΓX represets the maxmum devato of actua producto from desred oe over a products ad parts state X at stage k X (see Secto. for the defto of Γ). Foowg dyamc programmg P AM recurso hods for ψ (X) (Yeomas, 997): ψ (Ø) ψ (X : x 0,,,..., ) 0, ψ (X) m{max{ ψ( X e ), Γ X }: x,,,...,

9 hamaa ad Kubak: A Bref Survey of Just-I-Tme Sequecg for Mxed-Mode Systems 46 }. The space ad tme compextes of P AM are O( ( d )) + ad O(m ( d )) +, respectvey. Kubak, Steer ad Yeomas (997), gve extesve expermets to probems of practca sze. They cosder Toyota s schedug appcato descrbed Mode (983), whch requres the producto of 500 products for oe 8-hour producto shft. Two fterg heurstcs were troduced to reduce potetay vast state space to be examed dyamc programmg. They tested four eve radomy geerated probems wth tota product demads 500 ad ut weghts. For a probem wth 6, L 4 ad 400, tme requred to mpemet the agorthm s mutes, for stace. The rato of heurstc souto to the optma souto s.03. Moreover, they cocude that the souto tme of a probem strogy depeds o the umber of dfferet products but oy sghty ot o the rage of part requremets. The dyamc programmg for mut-eve m-max probem s modfed for mut-eve m-sum probem (Yeomas, 997; Kubak et a., 997). The mmum tota squared devato for a products ad parts over a parta schedues of X s defed by φ(x). For the amout of product produced X, et ( Ω X ) deoted by θ (X) be the squared sum of the devatos of actua producto from the desred oe for a products ad parts (see Secto. for the defto of Ω ). The the foowg dyamc programmg P QS recurso hods for φ (X) (Yeomas. 997, Kubak et a 997): φ(ø) φ (X : x 0,,,..., ) 0, { } φ (X) m φ( e ) + θ( ) 9. CONCLUING REMARKS X X : x,,,, }. I ths paper we revewed some research JIT sequecg that has bee carred out t ow. A umber of outstadg ad terestg questos have bee expored whch are st ope ad chaegg. The sge-eve m-sum probems wth ay covex, symmetrc, oegatve fuctos whch take the vaue zero oy at zero devato are sovabe by reducto to the assgmet probem. Ths approach appes to m-max probems as we. A pseudo-poyoma bary search for a feasbe B-bouded sequece obtaed through perfect matchg bpartte graph soves the sge-eve m-max absoute-devato probem. Ths approach ca be apped to other covex, symmetrc, oegatve fuctos. Regardess of the methods, obtag commo soutos to dfferet obectve fuctos woud sgfcaty save the compexty cost. However, the -bouded soutos obtaed va compete bpartte graphs does ot guaratee a optma souto for m-sum probems. The questo, what s mmum B such that optma souto for m-sum probem s B-bouded?, remas ope. Athough most of the sge-eve JIT probems had bee effcety soved by pseudo-poyoma agorthms depedg o the put sze of the demads, ther compexty status s ot yet cear. Eve the basc m-max absoute-devato probem s Co-NP but t s st ope whether the probem s Co-NP-Compete or poyomay sovabe. The mut-eve probems for two or more eves are strogy NP-hard. However, they are effcety sovabe f ether the products requre approxmatey the same umber ad mx of parts or the peggg assumptos are mposed. Therefore, searchg for speca propertes ths cass of probems for whch effcet agorthms exst or ookg for good approxmato agorthms woud be a terestg drecto of research ths area. The exstece of optma schedues that are cycc cosderaby reduces the computatoa requremets for ay type of JIT optmzato probem. Ths probem has bee resoved for sge-eve probems. We coecture that cycc Just--Tme sequeces mut-eve are optma as we. Oe way to dea wth JIT probems s the eegat cocept of baaced words. However, -baaced words caot be obtaed for some rates. The set of a 3-baaced words aways cotas a optma sequece for AMJIT. It s a ope questo whether there aways exsts a -baaced word that s optma for ay gve stace of AMJIT. Characterzatos of baace words to m-max squared-devato ad m-sum probems woud be a terestg probem for further research. ACKNOWLEGEMENT Ths research has bee supported by the Natura Sceces ad Egeerg Couc of Caada research grat OGP REFERENCES. Bask, M. ad Shahd, N. (998). A Smpe Approach to the Product Rate Varato Probem va Axomatcs. Operatos Research Letters, : Bask, M. ad Youg, H.P. (98). Far Represetato: Meetg the Idea of Oe Ma, Oe Vote (Yae Uversty Press, New Have). 3. Bautsta, J., Compays, R., ad Coromas, A. (997a). Modeg ad Sovg the Producto Rate Varato Probem. TOP 5: Bautsta, J., Compays, R., ad Coromas, A. (997b). Resouto of the PRV Probem. Workg Paper.I.T. 97/5, Barceoa. 5. Bautsta, J., Compays, R, ad Coromas, A. (996). A ote o the reato betwee the product rate varato (PRV) ad the apportomet probem. Joura of the Operatos Research

10 hamaa ad Kubak: A Bref Survey of Just-I-Tme Sequecg for Mxed-Mode Systems 47 Socety, 47: Brauer, N. ad Crama, Y. (004). Facts ad questos about the maxmum devato Just--Tme schedug probem. screte Apped Mathematcs, 34: Brauer, N., Jost, V., ad Kubak, W. (004). O Symmetrc Fraeke s ad Sma evatos Coecture. Les cahers du Laboratore Lebz-IMAG, 54, Greobe, Frace. 8. Coromas, A. ad Moreo, N. (003). O the reatos betwee optma soutos for dfferet types of m-sum baaced JIT optmzato probems. INFOR, 4: g, F.Y. ad Cheg, L. (993a). A smpe sequecg agorthm for mxed-mode assemby es Just--Tme producto systems. OR Letters, 3: g, F.Y. ad Cheg, L. (993b). A effectve mxed-mode assemby e sequecg heurstc for Just--Tme producto systems. Joura of Operatos Maagemet, : Gover, F. (967). Maxmum matchg a covex bpartte graph. Nava Research Logstcs Quarterey, 4: Godste, T. ad Mteburg, J. (988). The Effects of Peggg the Schedug of Just--Tme Producto Systems. Workg Paper No. 94, Facuty of Busess, McMaster Uversty, Hamto, Ot. 3. Jost, V. (003). eux probemes d approxmato ophate: Le patage proportoe e ombres etres et Les pavages equbres de Z (EA ROCO, Laboratore Lebz-IMAG). 4. Groef, H. Luss, H., Rosewe, M. ad Wahs, E. (989). Fa assemby sequecg for Just--Tme maufacturg. It. Joura of Producto Research, 7: Ha, R. (983). Zero-Ivetores. ow Joes-Irw, Homewood, IL. 6. Ima, R.R. ad Buf, R.L. (99). Sequecg JIT mxed-mode assemby es. Maagemet Scece, 37: Kbrdge, M.. ad Wester, L. (963). The Le Mode-Mx Sequecg Probem. Proceedgs of the Thrd Iteratoa Coferece o Operatos Research (Oso Egsh Uversty Press). 8. Kovayov, M.Y., Kubak, W. ad Yeomas, J.S. (00). A computatoa study of baaced JIT optmzato agorthms. Iformato Processg ad Operatoa Research, 39: Kubak, W. (005). Baacg Mxed-Mode Suppy Chas, Groupe d tudes et de Recherché e Aayse des ecsos (GERA), Motrea. 0. Kubak, W. (004). Far Sequeces (Hadbook of Schedug, Chapma & Ha/CRC Computer & Iformato Scece Seres).. Kubak, W. (003a). Cycc Just--Tme sequeces are optma. Joura of Goba Optmzato, 7: Kubak, W. (003b). O sma devatos coecture. Buet of the Posh Academy of Sceces, 5: Kubak, W. ad Kovayov, M. (998). Product Rate Varato Probem ad Greatest Commo vsor Property. Workg-Paper: 98-5, Facuty of Busess Admstrato, MUN. 4. Kubak, W., Steer, G., ad Yeomas, J.S. (997). Optma eve schedues for mxed-mode, mut-eve, Just--Tme assemby systems. Aas of Operatos Research, 69: Kubak, W. ad Seth, S.P. (994). Optma Just--Tme schedues for fexbe trasfer es. The Iteratoa Joura of Fexbe Maufacturg Systems, 6: Kubak, W. (993), Mmzg varato of producto rates Just--Tme systems: a survey. Europea Joura of Operatos Research, 66: Kubak, W. ad Seth, S.P. (99). A ote o eve schedues for mxed-mode assemby es ust- tme producto systems, Maagemet Scece, 37: Mteburg, J. ad Godste, T. (99). eveopg producto schedues whch baace part usage ad smooth producto oads for Just--Tme producto systems. Nava Research Logstcs, 38: Mteburg, J., Steer, G. ad Yeomas, S. (990). A dyamc programmg agorthm for schedug mxed-mode, Just--Tme producto systems. Mathematca ad Computer Modeg, 3: Mteburg, J. (989). Leve schedues for mxed-mode assemby es Just--Tme producto systems. Maagemet Scece, 35: Mteburg, J. ad Samo, G. (989). Schedug mxed-mode, muteve assemby es Just--Tme producto systems. Iteratoa Joura of Producto Research, 7: Mode, Y. (983). Toyota Producto Systems (Idustra Egeerg ad Maagemet Press, Norcross, GA). 33. Okamura, K. ad Yamasha, H. (979). A heurstc agorthm for the assemby e mode- mx sequecg probem to mmze the rsk of stoppg the coveyor. Iteratoa J. of Producto Research, 7: Pa, N.M. (00). Sovg the Product Rate Varato Probem (PRVP) of Large mesos as a Assgmet Probem (Ph Thess UPC, Barceoa). 35. Steer, G. ad Yeomas, S. (996). Optma eve schedues mxed-mode mut-eve JIT assemby systems wth peggg. Europea Joura of Operatoa Research, 95: Steer, G. ad Yeomas, S. (994). A bcrtero obectve for eveg the schedue of a mxed-mode, JIT assemby processes, Mathematca & Computer Modeg 0: Steer, G. ad Yeomas, S. (993). Leve schedues for ust tme producto processes, Maagemet Scece 39: St, J.W. (979). A cass of ew methods for cogressoa apportomet, SIAM Joura o Apped Mathematcs 37, Sumchrast, R., Russe, R. ad Tayor, B. (99). A comparatve aayss of sequecg procedures for mxed-mode assemby es Just--Tme producto system. Iteratoa Joura of Producto Research, 30: Sumchrast, R. ad Russe, R. (990). Evauatg mxed-mode assemby e dequecg heurstcs for Just--Tme producto systems. Joura of Operatoa Maagemet, 9: Tdema, R. (980). The charma assgmet probem. screte Mathematcs, 3: Vuo, L. (003). Baaced Words, Rapports de Recherche -006, LIAFA CNRS, Uversté Pars Yeomas, S. (997) Optma Leve Schedues for Mxed-Mode Just--Tme Assemby Systems (Ph.. Thess, McMaster Uversty, Hamto, Otaro).