A Mathematical Model for Converting Conveyor Assembly Line to Cellular Manufacturing

Transcription

1 IEMS Vol. 7, No., pp , Septeber 008. A Matheatcal Model for Convertng Conveyor Assebly Lne to Cellular Manufacturng Ikou Kaku Departent of Manageent Scence and Engneerng, Akta Prefectural Unversty, Yulhonjo, , JAPAN Tel: , Fax: , E-al: kou_kaku@akta-pu.ac.jp Jun Gong Insttute of Systes Engneerng, Key Laboratory of Integrated Autoaton of Process Industry of MOE, Northeastern Unversty, Shenyang 0004, P.R.CHINA Tel: , , E-al: gongjun@se.neu.edu.cn Jafu Tang Insttute of Systes Engneerng, Key Laboratory of Integrated Autoaton of Process Industry of MOE, Northeastern Unversty, Shenyang 0004, P.R.CHINA Tel: , , E-al: jftang@al.neu.edu.cn Yong Yn Departent of Econocs and Busness Manageent, Yaagata Unversty, Yaagata, JAPAN Tel: , E-al: yn@huan.kj.yaagata-u.ac.jp Selected paper fro APIEM 006 Abstract. Ths paper proposes a atheatcal odel for convertng conveyor assebly lne to cellular anufacturng n coplex producton envronents. Coplex producton envronents refer to the stuatons wth ult-products, varant deand, dfferent batch szes and the worker abltes varyng wth work statons and products respectvely. The odel proposed n ths paper as to deterne () how any cells should be foratted; () how any workers should be assgned n each cell; (3) and how any workers should be rested n shortened conveyor lne when a conveyor assebly lne should be converted, n order to optze syste perforances whch are defned as the total throughput te and total labor power. e refer the odel to a new producton syste. Such odel can be used as an evaluaton tool n the cases of () when a copany wants to change ts producton syste (usually a belt conveyor lne) to a new one (ncludng cell anufacturng); () when a copany wants to evaluate the perforance of ts converted syste. Sulaton experents based on the data collected fro the prevous docuents are used to estate the argnal pact that each factor change has had on the estated perforance proveent resultng fro the converson. Keywords: Lne-cell Converson, Cellular Manufacturng, Conveyer Assebly Lne, Matheatcal odel. INTRODUCTION Cellular anufacturng (CM) s a anufacturng syste n whch one (or ultple) worker carres out all of the operatons of a job, usually n a U-shaped layout. Snce t sees to be able to prove syste perforance n a changng envronent, any Japanese copanes have ntroduced CM nto ther factores to convert exstent conveyor assebly lne (CAL). An early docuent about such lne-cell converson was reported by Tsuru (998), whch s based on a questonnare of 3 factores and one consultng copany. These anonyous : Correspondng Author

2 A Matheatcal Model for Convertng Conveyor Assebly Lne to Cellular Manufacturng 6 factores converged n electronc and autooble ndustres. The an standpont of the docuent claed that CM can be recognzed as a for of the knowledge of Toyota Producton Syste whch has been hstorcally transferred to other ndustres. In recent years, after any Japanese copanes shfted ther producton organzatons to Chna, those anufacturers left behnd n Japan have been changng ther producton ways rearkably. Several anufacturng ethods have been developed for strengthenng ther copettve power of the doestc copanes. In addton, nstead conveyor ass producton, only the products what suted the needs of custoers (the knds of products are changng dynacally) should be anufactured flexbly when they were needed (the producton quanttes are also varables). For exaple, Tanaka (005) reported that there are seven anufacturng ethods have been used to correspond so-called new anufacturng n RICOH UNI- TECHNO Inc., whch s a ddle scale Japanese copany to anufacture facsles/copy achnes/prnters. Those ethods are as follows: () One worker-one achne ethod (the product wll be assebled by only one worker, he should do all of the assebly operatons); () Two workers-one achne ethod (there are too operatonal works to asseble for a larger achne that can not coplete by one worker, n such case two workers should be assgned to do ths assebly operaton); (3) Cart pullng ethod (nstead conveyor lne a cart s used as transport tool, whch s pulled aong several workers to coplete the assebly operatons); (4) Relay ethod (the for of assebly lne s exsted but the workers assgned n the lne do not only those operatons for theselves but also the operatons not assgned for the, by ther operaton ablty); (5) Conveyor assebly lne (tradtonal assebly ethod s also reaned for those large lot sze products); (6) One worker CM (only one worker does all of the assebly operatons of products usually n a U-shaped layout. The dfference wth ethod () s that the worker n the CM can do all of assebly operatons of several products, that eans he has a hgher operaton ablty.); (7) Dvson CM (several workers are assgned n one cell, they ay do the assebly operatons usng the ethods of (3), (4) or (6)). Those ethods and ther cobnatons are used to correspond flexbly dfferent knds (over 400 knds of products) and dfferent quanttes (70% of products are under 00 unts/onth) of products, and successful perforances were ganed. It should be ponted that all of these nnovatons n Japan ndustres are based on the reflecton of ass conveyor anufacturng and are for searchng ore effectve producton systes. Convertng old conveyor assebly lne to new anufacturng systes are not the goal but only the ways and eans to ncrease the productvty of copanes. A treendous acheveent of such converson s brought fro CANON Inc., a faous Japanese electronc copany. Takahash, Taya and Tahoku (003) reported that by ntroducng CM nto ther factores n CANON, snce 995 there are over 0,000 eters of belt conveyor have been wthdrawn and 70,000 square eters of workng space fro 54 related factores were epted. The total cost rate was decreased fro 6% to 50% durng past eght years. Snce then convertng CAL n Japanese anufacturers s cong nto fashon. Yn, Kaku and Murase (006) ponted out the econoc background of convertng CAL to CM n Japan based on a survey of last Japanese lterature. Suarly, Japanese anufacturers were faced wth a decreased arket deands and ncreased product varatons. To survve n such an extreely tough busness envronent, the tradtonal hgh-volue conveyor assebly lnes were no longer fulflled. Speedy adjustents were needed to handle transtons n product odels and deands. A copany s copettveness was becong dependent on whether or not t can respond to these transtons. In such an envronent, there was a trend n Japanese ndustres toward convertng conveyor assebly lnes to ore flexble anufacturng cells. Bascally, the ter of CM s not a new concept. Over the prevous decade, a seres of research artcles have nvestgated CM systes and copared the perforance of tradtonal functon layout and CM systes. For exaple, eerlöv and Hyer (989), eerlöv and Johnson (997), Johnson (005) claed that CM represents a ajor technologcal nnovaton to any anufacturng systes tradtonally based on functonal or assebly specalzaton. As a result, any anufacturng organzatons wth tradtonal functon layout anufacturng systes have ether already adopted CM or are consderng ther adopton. Sakazue (005) nvestgated a survey of Japanese lterature that ncluded total 07 docuents ( acadecs, 8 techncal reports and 77 newspaper artcles). Through coparng ther advantages and dsadvantages, he tred to explan that the so-called Aercan Cellular Manufacturng (fro tradtonal functon layout to cells) and Japanese Cell Manufacturng (fro belt conveyor lne to cells) are copletely dfferent n ter of pleentaton changes and echanss, even through there are soe slartes n ter of cell features and pleentaton. Johnson (005) used a prevous theory to explan why the assebly cells are expected to outperfor the current assebly lne. He nvestgated sulaton odels to estate the argnal pact that the operatonal factor change had on the estated perforance proveent resultng fro the converson. Kaku, Murase and Yn (008) proposed a theoretcal odel of the converson nvolved huan factors. Because the perforance proveent resultng fro the converson s dependent on those operatng factors that can prove the syste perforance and overcoe any task te ncreases caused by the loss of worker specalzaton, the cross-tranng of workers should be consdered to be a key ssue n the converson. They nvestgated a theoretcal odel to analyze the cross-tranng of workers quanttatvely by usng huan eory ablty and to suggest that nfor-

3 6 Ikou Kaku ᆞ Jun Gong ᆞ Jafu Tang ᆞ Yong Yn aton support syste can prove the cross-tranng effect. However, t should be attended that convertng CAL to CM s a new concept. hen an assebly lne s converted to cells, each cell perfors the tasks forerly assgned to nuerous statons on the lne. Dependng on the job desgn, each worker assebles ether a porton or all of the subassebly or product produced n the cell, ay also be responsble for dynacally balancng the flow of work as product x or deand levels change. On the other hand, the ncreased nuber of assebly tasks perfored by each cell worker ay ncrease the te requred for each task, whch serves to hnder the perforance proveent caused by cell converson. Moreover, because all of workers n assebly lne were not able to have sae skll perforng those assebly works, soe workers are not approprate dong those tasks assgned n cells. Those cause resulted soe copanes dd not lead to perforance proveents wth convertng ther assebly lnes to cells. Therefore, how to convert ther CAL to CM s a very coplcate decson proble for those copanes who wanted to do such converson. There are several researches reported the advantages and dsadvantages of convertng CAL to CM (see Tsuru 998, Isa and Tsuru 999, Sakazue 005, Myake 006) but those proposed suggestons are not approprate to support such decson. Our objectve s to buld up a atheatcal odel to descrbe the lne-cell converson proble and to analyze the syste perforance of such converson n a coplex producton envronent. In ths paper, the coplex producton envronent s consdered as a stuaton wth ult-products, varant deand, dfferent batch szes and varable worker abltes wth work statons and products respectvely. The reander of ths paper s organzed n the followng way. e gve a bref descrpton of the converson proble and then buld the atheatcal odel n next secton. The sulaton experents are desgned n the thrd secton. The result analyss and dscussons are gven n the fourth secton. Concludng rearks are gven n the fnal secton.. PROBLEM DESCRIPTION AND MATHEMATICAL MODEL e consder followng producton proble: there exst a tradtonal belt conveyor lne wth ultple assebly statons. orkers were assgned at each staton accordng to a tradtonal job desgn ethod but they have had ablty to do ore tasks than that were assgned to the. e assue that the worker s abltes are dfferent wth statons and products. Multple products wll be anufactured n the conveyor lne, each product s able to have dfferent batch szes but wth a known dstrbuton of deand. Products should be anufactured by a gven schedulng rule lke as Frst Coe Frst Servce (FCFS) but wth a full batch (.e., the batch splttng s not pertted). hen the products are assebled n the conveyor lne, the statons and workers used to coplete the assebly jobs are actve. Because workers have dfferent abltes to do those jobs (whch belong to statons and products) when the batch wll be fnshed s dependent on the worker wth slowest speed to do the jobs. That eans the abltes of the other workers were not useful suffcently, whch ay lead to decreasng the otvaton of workers. On the other hand, all of the products should be anufactured at the sae conveyor lne wth a fxed order; there ay be soe watng tes n the anufacturng so that we can not response flexbly to the custoer s varant deand. In ths paper, we propose KAIZEN ethods to prove the syste perforance of such conveyor lne. Assue that the workers wll do all of jobs that they can do even that are not assgned for the, there are several KAIZEN ethods to pleent the conveyor lne. For exaple, workers who have hgher abltes should help other workers n the conveyor lne; or convertng the conveyor lne to soe assebly cells; or convertng part of lne to cells for workers who have hgher abltes and rean the part of conveyor lne for workers who have lower abltes. In ths paper, we consder three types of producton systes ncludng pure CM, pure CAL and a hybrd type of CM+CAL. It does not nfluence the syste perforance ether CM s set to front or behnd of CAL (Van der Zee and Gaalan 006). For splcty and wthout lose of generalty, we assue CAL s foratted behnd CM n the hybrd producton syste as shown n Fgure. e propose a two step approach to desgn the producton syste fro Fgure. Frst step s a cell foraton approach: f there were only cells foratted n the syste (pure CM), we assgn all of workers to cells accordng to ther abltes whch are dfferent wth products and statons; f there were part of CAL be converted to cells, we assgn the workers wth hgher abltes to cells and rean the workers wth lower abltes to CAL. The case of workers can help each other n the conveyor lne just should be consdered lke as a pure CM n whch all workers n CAL are assgned n cells. Fnally, the pure CAL s the tradtonal belt conveyor lne. The second step s a schedulng approach: a frst coe frst servce (FCFS) rule s used to assgn product batches to cells or CAL. In the case of pure CAL the product batches are just scheduled accordng to the order of ther cong; n the case of pure CM the product batches are scheduled accordng to not only the order of ther cong but also the ablty of workers (that eans that product should be assgned pror to the worker (cell) who has hgher ablty to do the job). In the case of hybrd syste of CAL+CM, the product batches are frstly assgned to cells wth the FCFS rule, then assgned to CAL wth the order calculated by the earlest fnsh te rule.

4 A Matheatcal Model for Convertng Conveyor Assebly Lne to Cellular Manufacturng 63 3 Cell M Cell Cell J M CAL orker Product batch orker assgn Batch assgn Fgure. A hybrd producton syste of CM + CAL. No. of Cell No. of Cell CELL Te CAL Te Product batch Fgure. A case of schedulng n the hybrd producton syste Fgure shows an exaple of the case of CAL + CM wth four batches and three cells, where the length of rectangle chart n Fgure states the flow te of a product batch. For evaluatng the syste perforance two crtera are consdered. Frstly we defne total throughput te to represent the syste productvty that s the te of all of product batches had been fnshed. That s to say, for gven product x the new producton syste should have a shorter total throughput te. Secondly we defne total labor power (hours) to represent the work effcency that s the cuulatve workng te of all of workers assgned n the syste. Therefore, our proble s to deterne the nuber of cells and nuber of workers n each cell to nze the total throughput te and total labor power.. Proble features and assupton Followng assuptons are consdered n ths paper to construct the odel: ) Multple products are planed to assebly wth a pro-uct x. ) Products are assebled wth dfferent batches and dferent batch szes. 3) Types and batches of products are known and constant. 4) Nuber of tasks s the sae to all of product types. (If the nuber of tasks were dfferent wth products then assue the task te to do the dfferent tasks was zero so that we can treat the products wth dfferent assebly tasks). 5) If the producton syste s CAL, just one CAL s consdered. 6) Nuber of workers s the sae wth the nuber of tasks on CAL. 7) A worker only does a task n CAL. 8) Nuber of workers n each cell ay be dfferent but lted. 9) Nuber of tasks assgned to each cell s the sae. 0) Nuber of tasks assgned to each cell s at least greater than a constant. ) A worker assgned n a cell can operate all the tasks assgned n the cell. ) A product batch s just processed n a cell. 3) Setup te s consdered when two dfferent types of

5 64 Ikou Kaku ᆞ Jun Gong ᆞ Jafu Tang ᆞ Yong Yn products have been assgned nto a cell, but the setup te between two batches wth the sae product type s zero.. Notatons e defne the followng ters: Indces : Index set of workers ( =,,, ). j : Index set of cells ( j =,,, ). n : Index set of product types ( n =,,, ). : Index set of product batches ( =,,, ). Paraeters ax : Maxu nuber of workers n one cell. S n : Mnu nuber of statons n one cell. TB n : A 0- bnary varable where TB n =, f product batch s for product type n ; otherwse 0. B : Sze of product batch. T n : Standard assebly te to each task of product type n at each staton. LS n : Setup te of product type n on CAL. CS n : Setup te of product type n n CM. ε : Coeffcent of nfluencng level of skll to ultple statons for worker. η : Upper bound on the nuber of tasks for worker n one cell, f the nuber of tasks assgned to workers s over than t, the task te wll becoe longer than ever. β n : Level of skll to for worker for one task for product type n. Decson varables X =, f worker s assgned to cell j, otherwse 0. Y =, f worker s assgned to CAL, otherwse 0. P j =, f product batch s assgned to cell j, otherwse 0. L r =, f product batch s assebled by order r on CAL, otherwse 0. Z=, f CAL exsts n the syste, otherwse 0. Varables C : Coeffcent of varaton of assebly task te of worker n each cell accountng for the effect of ultple statons. CT : Actual cycle te of product batch n CM. FC : Flow te of product batch n CM. FCB : Begns te of product batch n CM. LT : Actual cycle te of product batch on CAL. FL : Flow te of product batch on CAL. FLB : Begn te of product batch on CAL..3 Proble forulaton Here we consder the producton plannng proble whch s based on a fxed product x wth M product batches and N product types. workers are assgned to the syste whch ay be pure CM or pure CAL or a hybrd type of CM+CAL syste. Gven the upper bound ax on the nuber of workers and the lower bound Sn on the nuber of statons (tasks) n one cell, the objectve s to deterne the nuber of cells and workers n each cell to nze the total throughput te and the total labor hours. The coprehensve atheatcal odel s gven n Equatons ()-(7) as below. { [ ]} M J ( j ) Z = Mn Max ( Z)( FCB + FC ) + Z( FLB + FL ) () Z = Mn P FC X + FL Y () J j = = = j= X + Y (3) Y S n (4) = X ax j (5) = X X l j > l,( l =,,, J ) (6) = = Y = Z = 0 Y = 0 (7) = here, equaton () states the objectve to nze the total throughput te of the total product batches assgnents. The total throughput te s the due te of the last copleted product batch. The frst part s the throughput te n CM. The second part s the throughput te n CAL. Equaton () states the objectve to nze the total labor power (hours) of the product batches assgnents. The total labor power s the te of all of workers anufactured all of the product batches. The frst part s the labor hours n CM. The second part s the labor hours on CAL. Because the objectve functons are too long to wrte, the detal defnton of the objectve functons are represented n the followng subsectons. Equaton (3) s the rule of worker assgnent ensures that each worker should be at ost assgned to one cell or CAL. The sgn of nequalty eans that the worker who has the worse ablty s dscarded possbly. Equaton (4) s a nu nuber of tasks n each cell whch eans f there s no task n cells, the producton syste wll becoe tradtonal CAL. Equaton (5) s a cell sze constrant because the space of a cell s lted. The value of the axu nuber of workers n one cell wll be a functon of plant sze, desgn and process technology. Equaton (6) s the rule of cell foraton ensures that the nuber of workers n pror cell s greater than that n next cell. Equaton (7) s a flag varable shown whether the CAL exsts n the syste. Ths rule can lead a saller search space of feasble solutons but guarantee the optalty of solutons.

6 A Matheatcal Model for Convertng Conveyor Assebly Lne to Cellular Manufacturng Schedulng of batch producton n CM For defnng the objectve functon n CM, the producton plan wll be scheduled wth a gven schedulng rule under the worker assgnents to CM. Frstly, a worker s level of skll s able to vary wth the nuber of tasks. If the nuber of tasks s over an upper bound η, the task te wll becoe longer. Ths can be represented as below: = + ε η = C ax(( Y ),0) (8) Secondly, the asseble task tes of a product s also able to vary wth workers. Consequently, the task te of a product s calculated by ean task te of all workers n the sae cell. Actually, the task te of product batch s represented va followng equaton: CT J N T β TB P C X n n n j = j = n = = J Pj X = j = (9) Then, usng the FCFS rule, the flow te FC and begn te FCB of product batch s represented as below. pror product batches whch are n the sae cell. Equaton () s the assgnent rule n whch a product batch s just only assgned to a cell. Equaton (3) s the FCFS schedulng rule whch eans the pror product batch ust be assebled before the next product batch. Equaton (4) are the rules of assgnng constrants, that eans a product ust be assgned to a cell n whch a worker s assgned at least..3. Schedulng of batches producton n CAL For defnng the total throughput te of the product batch assgnents n CAL, the producton plan wll be scheduled wth a gven schedulng rule under the worker assgnents to CAL. Of course, f all workers are assgned to CAL, that s the tradtonal producton syste, otherwse, that s CM+CAL hybrd producton syste. Here, the task te s calculated by the longest task te aong the workers on CAL. Actually, the task te of product batch s represented va the followng equaton: = N LT ax( T β TB Y ) (5) n n n n = Then, usng the FCFS rule, the flow te FL and begn te FLB of product batch s presented as below. N J ( ) B CT + ( ( ) ( ) ) > Y CSn TB TBnP Pj J = n j= Pj X = j= FC = N J ( ) B CT Y + CSn TBnPj = J = n j= Pj X = j= M = sj j s = j = J j = j = (0) FCB FC P P () P () FCB FCB (3) M + C = 0 X = 0, j (4) j = = here, equaton (0) states the flow te of product batch. The frst part s the process te and the B second part s the setup te, where J Pj X = j= presents the upper nteger nuber of products for each worker n the sae cell. Equaton () states the begn te of each product batch. There s no wat te between two product batches so that the begn te of one product batch s aggregaton of flow te of all of the N N Y Tnβ ntb n+ LT ( BS ) + LSn( TB( ) ntbn) > = n= n= FL = N N Y Tnβ ntbn+ LT( BS ) + LSnTBn = = n= n= (6) Lr =, LK ( r ) = r> FLB = ax( FCB+ FC, FLB K + FL K) K (7) L kr = r = M = M r = L r = (8) L = (9) r L FLB L ( FLB + TL ), M (0) r ( r ) here, equaton (6) states the flow te of product batch, the pror two parts are the flow te of product batch, the thrd part s the setup te of product batch. Equaton (7) states the begn te of each product batch. If the producton syste s the hybrd CM+CAL odel, the watng te between two product batches wll be consdered, otherwse no consderaton for watng te. In the hybrd CM+CAL odel, the begn te of product batch s the axu value between the end te of the pror product batch on CAL and the end te of product batch n CM. In the CAL odel, the begn te of product s the end te of the pror product batch whch s ordered by the FCFS rule. Equaton (8) ensures that a product batch s ust assgned to an order. Equaton (9) ensures that a order s ust assgned wth a product

7 66 Ikou Kaku ᆞ Jun Gong ᆞ Jafu Tang ᆞ Yong Yn batch. Equaton (0) ensures that the begn te of a product batch ust be late the end te of the pror product batch. 3. NUMERICAL SIMULATIONS By usng forula ()-(0), the lne-cell converson proble can be descrbed copletely that whether the conveyer assebly lne should be converted to cell(s) and how to do such converson. In the hybrd odel, for a gven nuber of workers (X + Y), the objectve functons are not lnear but bounded. Hence, we ust conduct an exhaustve search over X + Y. Snce there are (X + Y) ajor loops for cell foraton and J nor loops for schedulng, for practcal values of X + Y and J t s not coputatonally ntensve. The purpose of ths paper s to copare the perforances of CM and CAL under coplex producton envronents. Therefore we do nuercal experents to sulate the effects of each of factors nfluenced on the perforance of producton syste based on the atheatcal odel proposed above. For coparson of the perforance between new producton syste and CAL, the percentage changes are defned as below. For sple representaton, we shorten the total throughput te as TTPH and the total labor power (hours) as TLH as followng. TTPH of CM TTPH of CAL TTPH = ( )*00 TTPH of CAL TLH TLH of CM TLH of CAL = ( ) *00 TLH of CAL Clearly, the percentage changes of TTPH and TLH of pure assebly lne are zero. 3. Paraeter desgn Tables and show the paraeters used n the experents. As shown n Table, the paraeters n frst and second row are factors nfluencng the level of skll to ultple tasks for worker. η s the upper bound on the nuber of tasks for worker n a cell, whch s assued to be 0. If the nuber of tasks assgned to a worker s over than t, the task te wll becoe longer than ever. ε s used to control the varablty of level of worker s skll, whch s based on a noral dstrbuton wth a ean value 0. and a standard devaton Sply, the nter-arrval te s neglected here. The paraeters n thrd and fourth row are the setup tes n CM and CAL whch shows that the setup te n CM s saller than that n CAL. The values of the are separately nute and. nutes. The setup te n the jont odel CM+CAL s consdered as the sae wth CAL. The paraeter n ffth row s the standard task te, whch s assued to be.8 nutes. The paraeters n sxth and seventh row are the axu nuber of workers n a cell and nu nuber of tasks needed n a cell. The values of 5 and are assued. Fnally, the paraeter n eghth row s the nuber of cells whch s gven by 0, because the bggest nuber of tasks n ths experent s 0 and tasks ust be assgned to each cell at least, the value over 0 s eanngless. As shown n Table, the worker s level of skll can be consdered to be able to vary wth product types. In our sulaton experents the varaton s assued to be a noral dstrbuton wth the ean value and the range of standard devaton fro 0. to 0.3. The ore the product types wll be anufactured, the larger the varaton of worker s skll. 3. Factors desgn For coparng the perforances aong the dfferent producton syste, four knds of hybrd syste were consdered, whch ncluded Best CM (optal soluton), workers CM (two workers were forced to be n a cell), 3 workers CM (three workers were forced to be n a cell) and 6 workers CM (sx workers were forced to be n a cell). There are a lot of our sde and nsde factors whch can nfluence the syste perforance. A bref overvew of four factors used n the experents s gven n Table Product type Snce CAL s desgned for sngle product at frst, t can be consdered that the perforances of CAL wll becoe worse when the product type changes to be ultple. e set fve cases, n whch the nuber of product type ranged fro to 5, to confr ths supposton. In ths way, the hgher varablty of worker s level of skll wll happen wth ore product types n the producton syste accordng to the paraeters set above. 3.. Product batch Even CAL s consdered be sutable to sngle (less) product type, product batch s stll nfluencng the syste perforances. It can be consdered that CAL s better n a less product batch envronent, and becoe worse when the product batches ncrease. Fro ths vew pont, fve cases of dfferent nuber of product batches are arranged for nvestgatng the effects of product batches, whch are assued be fro 3 to Batch sze Batch sze can be consdered as a factor that nfluences the syste perforances ostly. The perforances of CAL wll becoe worse when the batch sze of the product changes to be saller. e set fve cases n whch the batch sze were set to be fro 0 to 00. In ths way, the effects on the syste perforances wth dfferent volue of batch szes can be nvestgated.

8 A Matheatcal Model for Convertng Conveyor Assebly Lne to Cellular Manufacturng Task sze Dstngush the outsde factors above, cross-tranng of workers s a sgnfcant nsde factor whch can nfluence the syste perforances of hybrd systes. Through cross-tranng ethodology, the worker s level of skll to deal wth ultple tasks should becoe hgher and hgher. Table. Experental paraeters. 4. Product type Fgure 3 and 4 showed the percentage changes of the two syste perforances easures wth the ncreasng of product types. Fgure 3 shows the percentage changes of total throughput te (TTPH) and Fgure 4 shows the percentage changes of the total labor hours (TLH). Paraeter Value η 0 ε N(0., 0.05) CAL( LS ) n. nutes CM ( CS ) nute n ( T n ).8 nutes ax 5 S n J 0 Note) N(0., 0.05): Noral dstrbuton( μ = 0., σ = 0.05). Table. orker s level of skll ( β n ) wth product types. Fgure 3. TTPH wth dfferent product types. Product Type N(, 0.) N(, 0.5) N(, 0.) N(, 0.5) N(, 0.3) Note) N(, 0.): Noral dstrbuton( μ =, σ = 0. ). Accordng to ths consderaton, the dfferent values of task sze are assued fro 4 to 0, whle the lower bound on the nuber of ultple tasks s fxed to be 0. If the task sze s greater than 0, the standard task te wll becoe longer. The effect of worker s ablty to deal wth ultple tasks on the perforances of the hybrd systes wll be nvestgated n the experents. 4. RESULTS AND ANALYSIS Several sulaton results are obtaned fro our experents. Here we show soe an observatons fro the standpont of nzaton of total throughput te wth the constrant of the total labor hours. Fgure 4. TLH wth dfferent product types. Fro the vewpont of TTPH Fgure 3, generally the percentage changes of all the CM systes are decreasng when the product types are ncreasng. That eans ncreasng of product types leads to better perforance n CM but worse perforance n CAL. However, except Best CM, the perforances of workers CM, 3 workers CM, 6 workers CM are not able to ex- Table 3. Experental factors Factor Product Type Product Batch Batch Sze Task Sze Product Type, -, -3, -4, Product Batch 3, 4, 5, 6, 7 N(50,5) Batch Sze 6 N(0, 5), N(30, 5), N(50, 5), N(00, 5), N(00, 5) Task Sze 5 N(50, 5) 5, 0,, 5, 0

9 68 Ikou Kaku ᆞ Jun Gong ᆞ Jafu Tang ᆞ Yong Yn ceed that of CAL n the experental stuatons. The soluton of Best CM s varyng wth product types, especally when the product type s 5 the Best CM s constructed wth two 5 workers CM + workers CAL and acheve axu 5% proveent n total throughput te. It should be notced that all of the workers n CAL also be assgned n the Best CM. But n the case of Best CM, several workers do ther jobs each other accordng to ther abltes and only the workers who have not had cross tranng were reaned at rested CAL. Consderng the constrant of TLH Fgure 4, ncreasng of product types also leads to better perforance n CM than that n CAL. hen product types are over 4, all of CM systes can prove the perforance of total labor hours n the experental stuatons. 4. Product batch Fgure 5 and 6 showed the percentage changes of the two syste perforances easures wth the sae product type and the ncreasng of product batch. Fgure 5 shows the percentage changes of total throughput te (TTPH) and Fgure 6 shows the percentage changes of the total labor hours (TLH). Fgure 5. TTPH wth dfferent product batches. Fro the vewpont of TTPH Fgure 5, because CAL s orgnally desgned for sngle product type rather than CM, the perforances of CM systes are varyng sgnfcantly but can not exceed that of CAL. Especally t can be observed fro Fgure 5 when the product batch s ore than 5 the Best CM can acheve perforance proveent n total throughput te. The soluton of Best CM s constructed wth two cells n whch 9 workers were assgned and workers were reaned n CAL. Consderng the constrant of TLH Fgure 6, ncreasng product batch s not able to lead to better perforance n alost of CM systes except the Best CM. hen product batch s ore than 4 the Best CM can acheve about 5% perforance proveent n total labor hours n the experental stuatons. 4.3 Batch sze Fgure 7 and 8 showed the percentage changes of the two syste perforances easures wth the ncreasng of batch sze. Fgure 7 shows the percentage changes of total throughput te (TTPH) and Fgure 8 shows the percentage changes of the total labor hours (TLH). Fro the vewpont of TTPH Fgure 7, when the batch sze s sall (<30), alost of CM systes had better perforance than that of CAL. However, when the batch sze s becong larger (over 50), the perforance of CM systes s becong worse and worse, ncludng the Best CM. The reason s that the bgger batch sze wll reduce the nfluence of setup te and watng te. It can be observed that the bgger the batch sze, the saller the rato of setup and watng te to the throughput te. The soluton of Best CM s constructed wth two 5 workers CM + workers CAL. hen the batch sze s under 50, there s a nus percentage change that eans we can acheve axu 5% proveent n total throughput te. hen the batch sze s over 50, even the Best CM also can not acheve any proveent n total throughput te. Consderng the constrant of TLH Fgure 8, when the batch sze s sall (<30), all of CM systes had better perforance than that of CAL. However, when the batch sze s becong larger (over 50), the perforance of CM systes s becong worse and worse. Observably, even the batch sze ncreased to 00, the Best CM stll can acheve about 3% perforance proveent n total labor hours n the experental stuatons. 4.4 Task sze Fgure 6. TLH wth dfferent product batches. Fgure 9 and 0 showed the two syste perforances easures changes (note here not use the percentage changes) wth the ncreasng of task sze. Here we just copare the perforance between the Best CM and CAL because such coparson can clarfy the essence of the converson. The soluton of Best CM s constructed

10 A Matheatcal Model for Convertng Conveyor Assebly Lne to Cellular Manufacturng 69 wth three cells n whch 9 workers were assgned and workers were reaned n CAL. Fro the vewpont of TTPH Fgure 9, ncreasng task sze ndcates that there exsts a break pont at whch the CM syste turns to be better perforance tll worse than CAL. The an reason s that the worker s level of skll becoes lower when he ust deal wth the task sze over hs ablty so that the task te wll becoe longer. The break pont s dependent on the worker s level of skll bounded by the task nuberη. Consderng the constrant of TLH Fgure. 0, there also exsts a break pont at whch the CM syste turns better perforance to worse than CAL. The dfference s that the break pont n TLH appears later (task sze equals 7) than that n TTPH (task sze equals 5) n the experental stuatons. strateges. However, such converson should be perfored very carefully accordng to ther own producton envronents. Generally the hybrd CM+CAL odel s approprate for the frst converson, then how any cells should be desgned and how any workers should be assgned n each cell are becong a very portant decson akng proble. Too workers n one cell (bg cell) or too cells n the producton syste s not the best layout. Fgure 9. TTPH wth dfferent task szes. Fgure 7. TTPH wth dfferent batch szes. Fgure 0. TLH wth dfferent task szes. Fgure 8. TLH wth dfferent batch szes. In suary, for the copanes who have had belt conveyor assebly lne to anufacture ther products, when the producton envronent s changng to a new one that has varyng product types, saller batch sze, varyng task sze, and ther workers have ore ablty to do the jobs assgned to the before and ncreasng the otvaton of workers s becong ore portant as sae as ncreasng the productvty. Convertng ther CAL to CM s a consderable choose n ther producton 5. CONCLUSIONS Convertng CAL to CM s a new knd of techncal nnovaton. The ost portant contrbuton of ths paper s buldng up a atheatcal odel to construct the lne-cell converson. Usng ths odel, we can not only desgn a new CM but also evaluate the perforance of the converted syste. Sulaton experents have shown clear nsghts of varous factors whch nfluence the syste perforance. There are stll several research works to be nvestgated n the future. The optal calculatng algorths should be developed for solvng practcal sze probles.

11 70 Ikou Kaku ᆞ Jun Gong ᆞ Jafu Tang ᆞ Yong Yn Because the proble s NP hard, Meta algorths (e.g., GA) also should be developed to overcoe the lt of optzaton ethods. The ult objectve optzaton approach also should be carred out for the odel whereas they were consdered separately n ths paper. Moreover, the schedulng rule used n ths paper s FCFS and t s a very portant factor n the odel. So that other dspatch rules (e.g., SPT (shortest producton te)) also should be consdered to be used and evaluated n the odel. The ost portant thng s to apply the odel to a real factory case to prove the correctness and utlty of the odel, even our sulaton experents are based on the data collected fro the prevous docuents. AKNOLEDGEMENTS Ths paper s fnancally supported by the foundaton of Akta Prefectural Unversty and partly by the Natural Scence Foundaton of Chna (NSFC) funded projects (706500, 70700). REFERENCES Isa, K. and Tsuru, T. (999), Cell producton and workplace nnovaton n Japan: Toward a new odel for Japanese anufacturng?, Industral Relatons, 4(), Johnson, D. J. (005), Convertng assebly lnes to assebly cells at sheet etal products: Insghts on perforance proveents, Internatonal Journal of Producton Research, 43(7), Kaku, I., Murase, Y. and Yn, Y. (008), A Study on Huan Tasks Related Perforances of Convertng Con-veyor Assebly Lne to Cellular Manufacturng, European Journal of Industral Engneerng, (), Myake, D. I. (006), The shft fro belt conveyor lne to work-cell based assebly syste to cope wth ncreasng deand varaton and fluctuaton n the Japanese electroncs ndustres, Report paper of CIRJE-F-397. Sakazue, Y. (005), Is Japanese cell anufacturng a new syste?: A coparatve study between Japanese cell anufacturng and cellular anufacturng, Journal of Japan Industral Manageent Assocaton,, Takahash, S., Taya, H., and Tahoku, H. (003), A racle CANON producton syste: It s not only Toyota, Specal report of eekly Toyo Econocs, Noveber 7-January 3, 3-36, (n Japanese). Tanaka, T. (005), Brth of a cart pullng style producton syste, IE Revew, 46(4), 3-37, (n Japanese). Tsuru, T. (998), Cell anufacturng and nnovaton of producton syste, Report of Econoc Research Insttute, Japan Socety for the Prooton of Machne Industry, H9-9, Japan, (In Japanese). Van Der Zee, D. J. and Gaalan, G. J. C. (006), Routng flexblty by sequencng flexblty - explotng product structure for flexble process plans, The Proceedngs of the Thrd Internatonal Conference on Group Technology/Cellular Manufacturng, eerlöv, U. and Hyer, N. L. (989), Cellular anufacturng n the U.S. ndustry; A survey of users, Internatonal Journal of Producton Research, 7 (9), eerlöv, U. and Johnson, D. J. (997), Cellular anufacturng at 46 user plants: pleentaton experences and perforance proveents, Internatonal Journal of Producton Research, 35(), Yn, Y., Kaku, I. and Murase, Y. (006), Convert flow shops to anufacturng cells: Lessons fro Japan, The proceedngs of Logstcs and Supply Chan Manageent (KSCM006), HONGKONG.