Design of flexible manufacturing cell considering uncertain product mix requirement

Transcription

1 Int. J. Agle Systems and Management, Vol. 3, Nos. 1/2, Desgn of flexble manufacturng cell consderng uncertan product mx requrement Amt Ra Dxt* Department of Mechancal Engneerng and Mnng Machnery Engneerng, Indan School of Mnes Unversty, Dhanbad, Inda E-mal: *Correspondng author P.K. Mshra Department of Mechancal Engneerng, M.N.N.I.T., Inda E-mal: Abstract: In the real-world manufacturng, the requrement of number and the attrbute of product may not be known exactly at the tme of desgnng the manufacturng cell. The desgn of manufacturng cell requres to dentfy machne groups and part famles n Cellular Manufacturng (CM) system. The success of a CA system s senstve to fluctuatons n the demand for products and the product mx. A large number of papers on manufacturng cell desgn have been publshed so far, but very few of them have consdered random product mx constrant at the desgn stage. Lttle work has been reported whch ncorporated real-lfe producton parameters lke operaton sequence, producton volume, batch sze, materal handlng capacty, processng tme, setup tme, machne capacty and cost factors. Consderatons of these mportant parameters make the cell formaton problem more complex, but realstc. Ths paper presents a new formulaton of the part famly/machne cell formaton problem that addresses the dynamc nature of the producton envronment by consderng a multperod forecast of product mx and demand durng the formaton of part famles and machne cells. The computatonal procedure of the algorthm has been llustrated by an example. Numercal results ndcate that the proposed methodology s flexble, effcent and may be effectve even for ndustral problems. Keywords: cellular manufacturng systems; random product mx; ntercell movement; operaton sequence. Reference to ths paper should be made as follows: Dxt, A.R. and Mshra, P.K. (2008) Desgn of flexble manufacturng cell consderng uncertan product mx requrement, Int. J. Agle Systems and Management, Vol. 3, Nos. 1/2, pp Bographcal notes: Amt Ra Dxt s the Research Fellow n the Department of Mechancal Engneerng at Motlal Nehru Natonal Insttute of Technology, Inda. He s also a Senor Lecturer n the Department of Mechancal Engneerng and Mnng Machnery Engneerng at Indan School of Mnes Unversty, Inda. He receved hs BTech n Mechancal Engneerng from BIET, Inda and hs MTech n Producton Engneerng from MNNIT, Inda. Copyrght 2008 Inderscence Enterprses Ltd.

2 38 A.R. Dxt and P.K. Mshra Hs research nterests nclude group technology, advanced manufacturng technology, operatons research and supply chan management. He s a Member of ASME and IAENG. P.K. Mshra s a Professor n the Department of Mechancal Engneerng at MNNIT, Inda. He receved PhD from Unversty of Roorkee, Inda. He receved hs BTech n Mechancal Engneerng from MNREC, Allahabad and hs MTech n Producton Engneerng from MNREC, Allahabad. Hs areas of research nclude FMS, CAPP, CAM and Genetc Algorthm. He s a Member of IE (Inda). 1 Introducton Cellular Manufacturng (CM) has been proved as an attractve compromse between flow lne producton and ob shop producton as t ncorporates the attrbutes of both of them. CM s bascally based on the phlosophy of Group Technology. Group Technology (GT) s an approach to manufacturng and engneerng management that helps to manage dversty by captalsng on underlyng smlartes n products and actvtes. In the manufacturng context, GT has been defned as a manufacturng phlosophy dentfyng smlar parts and groupng them together nto famles to take advantage of ther smlartes n manufacturng and desgn. Groupng the producton equpment nto machne cells, where each cell specalses n the producton of part famles, s called as CM. So, CM s the applcaton of the GT phlosophy n manufacturng. CM s concerned wth the creaton and operaton of manufacturng cells whch are dedcated to the producton of a set of part famles. In order to ntroduce CM, t s necessary to dentfy parts and machne types to be used n the cellular confguraton. The frst problem faced n mplementng CM s cell formaton. Cell formaton deals wth the dentfcaton of the famly of parts and the group of machnes to process these parts. The problem of cell formaton s defned as: If the number, types, and capactes of producton machnes, the number and types of parts to be manufactured, and the routng plans and machne standards for each part are known, whch machnes and ther assocated parts should be grouped together to form cells? (Wu and Salvendey, 1993). In some cells, the defnton of cell formaton s expanded to allow choce of processng operatons to acheve specfc features. Snce the last three decades, a consderable amount of researches have been drected to ease ths type of problem. Burbdge (1971) developed an ntutve method, namely Producton Flow Analyss (PFA) whch s relatvely easy to mplement. PFA may be sutable for the small sze problem, but t would defntely have dffcultes copng wth real lfe cell formaton problems when the machne-part ncdence matrx becomes more complex because of problem sze. Many approaches have been developed to deal wth the dffcultes of ntutve method. These approaches are usually classfed nto part-orented approaches (based on part characterstcs) and process-orented approaches (based on producton methods). The part-orented technques usually employ some classfcaton and codng system, and analyse parts for ther smlartes n desgn features and functonaltes. However, these do not nfluence drectly the confguraton of manufacturng cells (Choobneh, 1988). The process-orented approaches to the cell formaton are based on manufacturng

3 Desgn of flexble manufacturng cell 39 data such as producton methods, part routng nformaton and process plans. The process-orented approach s classfed nto four groups namely: Descrptve methods, Array-based methods, smlarty coeffcent methods and other analytcal methods (Yasuda and Yn, 2001). Most of the suggested algorthms/models consder bnary machne-part ncdence matrx A = [a ], wth a 1 f part type requres machne type = 0 otherwse The bnary part-machne matrx only represents the occurrence of an operaton on a machne, but not the actual ntercell movements of parts. Most of the cell formaton methods based on bnary part-machne matrx do not consder the followng: Operaton sequence of parts: the sequence of operaton s an mportant manufacturng attrbute n the desgn of CM systems. The operaton sequence may be defned as an orderng of the machnes on whch the part s sequentally processed. The sequence of operaton has an mpact on the flow of materal n the system. An ntermedate operaton of a component to be performed outsde ts cell requres two ntercell transfers whle the frst or last operaton requres only one such transfer (Choobneh, 1988). Vakhara and Wemmerlov (1990), Logendran (1991), Wu and Salvendy (1993), Yn and Yasuda (2002), Defersha and Chen (2006) and Dxt and Mshra (2007) consdered operaton sequence matrx n the desgn of manufacturng cell. Sarker and Xu (1998) presented a bref revew of cell formaton based on the operaton sequences. Random product mx: product mx refers to a set of part types to be manufactured. In practce, the product requrement s uncertan at the desgn stage of the manufacturng system. Producton volume: the producton volume s the total number of dfferent parts to be manufactured n the gven perod. The merts of ncorporatng producton volume were depcted by Gupta and Sefoddn (1990). Number of ntercell movements: an deal clusterng conssts of mutually exclusve attrbutes between two clusters of data. However, t s rare to acheve such perfect clusterng results n realty. Clusterng results often contan exceptonal elements n the case of bnary data, an exceptonal element ndcates a clusterng dscrepancy between data and attrbutes n terms of unty elements. A 1 outsde the dagonal block s called an exceptonal element and t ndcates only one ntercell movement. Parts are generally processed n batches wth unequal volumes. In fact, the volume of ntercell movement depends upon the batch sze and capacty of materal handlng devce. If the batch sze s large and the capacty of materal handlng system s lmted, then the volume of ntercell movement wll be large. Machne capacty: the basc requrement, n the desgn of a CM system, s to have the adequate capacty to process all the parts. The machne capacty has an mpact on the workload nduced by the part. In bnary part-machne matrx, the 0s nsde the dagonal block are referred as VOIDS. A vod ndcates that a machne assgned to a cell s not requred for the processng of a part n the cell. The correspondng machne s sad to be underutlsed. But the actual utlsaton of machne depends upon the machne capacty and the workload mposed by the parts on t.

4 40 A.R. Dxt and P.K. Mshra Processng and setup tmes: the processng and setup tme requred by the part on a machne s another mportant parameter. It nfluences workload and machne utlsaton. Materal handlng capacty: the materal handlng capacty s consdered as the number of parts transported to the machnes/cells. So, t depends upon the load carryng capacty and the pallet sze. In the recent years, some of the researchers consdered dfferent producton parameters n cell formaton problem (Adl et al., 1996; Askn and Subramanam, 1987; Choobneh, 1988; Dxt and Mshra, 2004; Harhalaks et al., 1990a, b; Kumar and Vannell, 1987; Kusak and Chow, 1987; Wu and Salvendy, 1993). But, lttle work has been reported n the desgn of CM systems under uncertan product mx envronment. However, t has ganed nterest from researchers n recent years. 2 Past revew Sankaran and Kaslngam (1993) developed a mxed nteger programmng model for dfferent sze of cells wthn a sngle layout. In ths model, the ntracell transfer costs of parts produced n a cell are based on the sze of the cell that s defned n terms of the number of machnes n the cell. The model only consders a sngle perod. Ths model was tested on a small problem wth very lmted set of data consstng of three cost scenaros. Each of the three scenaros requred over 6500 sec of CPU tme on a manframe computer to fnd the model s optmal cost and a soluton. For more large szed problems, the computatonal tme would be prohbtve, so, a heurstc procedure s proposed. The heurstc procedure starts out wth all parts assgned to a sngle cell. The procedure then attempts to create cells for ndvdual parts so that ntracell transfer costs are reduced. Ths procedure s tested on the example problem and very quckly generates the optmal soluton for each of the three cost scenaros. Harhalaks et al. (1994) proposed a two-stage soluton methodology for the robust cellular manufacturng desgn. They focused on product demand varatons over a system desgn horzon whch was dvded nto elementary tme perods. The obectve was to obtan a cellular desgn wth the mnmum expected ntercell materal handlng cost over the entre desgn horzon. In the frst stage, a producton volume for each product was determned and correspondng cell confguraton was obtaned by usng a heurstc method n the second stage. The frst stage began wth mappng the forecast of product demand to a set of feasble producton volumes wth gven resource capacty constrants. If suffcent capacty exsted to produce all products at ther demand level, product demand gave feasble producton volumes. Otherwse, the proected demands were used n a lnear program that gave a set of feasble producton volumes such that proft was maxmsed. Gven several product mxes, a procedure for calculatng the ont probabltes for every feasble producton mx was presented. The ont probabltes were used to evaluate the mean producton volume for each product. The heurstc method proposed by Harhalaks et al. (1990a) was used to obtan a near-optmal cell formaton n the second stage. Ths paper assumed that the product mx for each perod was the same; no new products were ntroduced nor old products dscontnued.

5 Desgn of flexble manufacturng cell 41 Product demand n each perod was assumed to be normally dstrbuted, where the mean and standard devaton were tme-nvarant. Addtonal machnes of same knd were not consdered. Wcks (1995) addressed the dynamc nature of producton envronment by consderng a multperod forecast of the product mx and resource avalablty durng the formaton of part famles and machne cells. The obectves consdered were the mnmsaton of ntercell materal handlng cost, the mnmsaton of nvestment n addtonal machnes and the mnmsaton of the cost of system reconfguraton over the plannng horzon. A mxed-nteger formulaton of the multperod Part Famles and Machne Cells (PF/MC) formaton problem was developed. The multperod PF/MC formaton procedure belongs to machne-groupng soluton strategy where machne cells are formed frst, followed by the assgnment of parts to the machne cells. The assgnment of machnes to cells over the plannng horzon s made va a genetc algorthm. A heurstc for assgnng parts to cells s also embedded n the algorthm. Song and Htom (1996) developed a methodology to ntegrate producton plannng and cellular layout va a long-run plannng horzon. The problem was formulated as a mxed-nteger problem. It contans two types of nteger programmng problems: determnng the producton quantty for each product and the tmng of adustng the cellular layout n a fnte plannng horzon perod wth dynamc demand. The obectve of the model was to mnmse the sum of nventory-holdng cost, group-setup cost, materal handlng cost and layout-adustng cost. The Benders decomposton method was employed to solve the problem. The perodc fluctuatng demand was absorbed by adustng both layout confguraton and nventory level. The demand for each part famly n each perod was assumed to be known and constant. Chen (1998) developed a mxed nteger programmng model to mnmse ntercell materal handlng and machne nvestment costs as well as cell reconfguraton cost n a dynamc CM envronment wth antcpated changes of demand or producton process for multple tme perods. The problem was decomposed nto smpler cell formaton subproblems by removng the system reconfguraton cost from the obectve functon and the correspondng couplng constrants from the model. Thus, the decomposed subproblems correspond to dfferent tme perods. The commercal optmsaton software packages were used to solve bnary-nteger programmng problems optmally. The model only examnes reconfguraton of cells n terms of the machne types ncluded n each cell and does not nclude capacty constrants to determne how many unts of each machne type are needed n each cell durng each perod. Wcks and Reasor (1999) developed an nteger model for multperod cell formaton. Ther model consdered ntercell transfer cost, machne equpment cost and machne relocaton cost. The model requres a mnmum number of machnes and parts to be assgned to each cell (the number of cells must also be specfed). Producton costs based on the sze of the cell are not consdered. The authors also propose a genetc algorthm to generate solutons to the model. Mungwattana (2000) developed a soluton approach for desgnng CM systems that addresses the dynamc and stochastc producton requrements. A smulated annealng-based heurstc was developed to obtan good solutons wthn reasonable amounts of tme. The developed heurstc was evaluated n two ways. Frst, dfferent CM desgn problems were generated and solved usng the heurstc. Then, solutons obtaned from the heurstc were compared wth lower bounds of solutons obtaned from the optmal soluton procedure (CPLEX software). The lower bounds were used nstead of

6 42 A.R. Dxt and P.K. Mshra optmal solutons because of the computatonal tme requred to obtan optmal solutons. The obectves consdered were the mnmsaton of ntercell materal handlng cost, machne nvestment cost, operatng cost and the cost of system reconfguraton over the plannng horzon. Schaller (2007) proposes an nteger model that consders part reallocaton or equpment reallocaton between cells as alternatve for the desgn of a CM system to handle long-term demand change. The obectve of the model was to mnmse the sum of amortsed machne cost, cost of reallocatng equpment and the cost of producng parts. A problem specfc heurstc (called CB procedure) and metaheurstc (tabu search) was employed to obtan the acceptable soluton. The proposed model does not consder mportant parameters lke amount of ntercell movement of parts, operatonal sequence of parts, batch sze, etc. Ths paper descrbes a soluton methodology for the problem of manufacturng cell formaton n uncertan product mx envronment. The goal of the multperod formulaton s to obtan a cellular desgn that contnues to perform well wth respect to the desgn obectves as the product mx/volume changes wth tme. Ths paper s organsed as follows: notatons and defntons are explaned n Secton 3. The problem formulaton s presented n Secton 4. The soluton methodology s presented n Secton 5. Computatonal analyss and results are presented n Secton 6. Concluson s presented n Secton 7. 3 Notatons and defntons k n Part type ndex Machne type ndex Cell type ndex Operaton type ndex m Number of machnes M = (m 1, m 2,,m,,m m ) p Number of parts P = (p 1, p 2,,p,,p p ) c Number of cells C = (c 1, c 2,,c k,,c c ) k h r pv h bs h Number of dentcal machnes of machne type for the product mx h Maxmum number of operatons for components Total producton volume of part type for the product mx h Batch sze of part type for the product mx h MHC Capacty of materal handlng devce for part t Processng tme of part type on machne type

7 Desgn of flexble manufacturng cell 43 st T O P IT Setup tme of part type on machne type Capacty of a machne type Operaton cost per hour of machne type Procurement cost of machne type Intercell movement cost per batch of part type UMC Non-utlsaton penalty cost of machne type h K UB m(k) w wc k Number of machne type whch are not utlsed for the product mx h Upper bound on cell sze (maxmum number of machnes n a cell) Number of machnes n cell type k proportonate workload nduced by part type on machne type w = ( t pvh + st [ pv h /bsh ]) T Workload on machne type n cell type k (1) wc p k = k k = 1, k X Y w (2) X Y k k 1 f machne type s n cell type k = 0 otherwse 1 f part type s assgned n cell type k = 0 otherwse OP a a r s st rs, =, Machne type ' ' s requred for part type ' ' for operaton O( p) Total number of operatons requred by part type ' p ' χ ( ab, ) th 1 f k and ( k+ 1) operaton s performed on machne type ' a' and ' b'; = 0 otherwse Ψ ab total movement between two dstnct cells c a and c b. P O( p) 1 ab A p OPp OPp p= 1 = 1 Ψ = (3a) ab, m ( ) χ (,,, + 1)

8 44 A.R. Dxt and P.K. Mshra 4 Problem formulaton Ths paper addresses two problems, frstly, the machne/part groupng problem consderng varous producton parameters and secondly, the dentfcaton of sutable cell confguraton under dynamc product mx scenaros. 4.1 Machne/part groupng problem The most fundamental obectve of cell formaton s to acheve cell ndependency whch n turn sgnfcantly smplfes shopfloor control. Further, the potental to ncrease the accountablty, responsblty and autonomy of the workers s enhanced. It also reduces materal handlng that results n less damage to work-n-process (Shafer and Rogers, 1993). To acheve cell ndependency, ntercell movements must be mnmsed. In past few years, some researchers have ncorporated operaton sequence n the formulaton of cell formaton problem. But very few of them have consdered the non-consecutve operatons on the same machnes. In practcal stuatons, same machne s used more than once n a part routng, and f such a part has to move outsde ts assgned cell, the mplcaton on materal handlng are sgnfcant (Harahalaks et al., 1990a). But Harahalaks et al. (1990a) dd not ncorporate the non-consecutve operaton consderaton due to dffcultes encountered n proposed matrx formulaton. The non-consecutve operatons on the same machne are ncorporated n the proposed formulaton. The mathematcal formulaton for the desgn of manufacturng cell s developed such that ntally machne cells are formed and then the parts are assgned to the approprate cells. However, t could be more complcated to model and would result n a large mathematcal model, whch requres a substantal amount of tme to solve. The mathematcal formulaton s gven below: m ψ = 1 m ab Mnmse F1 (3) a= 1 b= a+ 1 ma ( ) + mb ( ) Subect to constrant: mk ( ) < UB (4) c k = 1 m = 1 p = 1 X = 1 for = 1,2,3,..., m (5) k X 1 for k = 1,2,3..., c (6) k Y 1 for k = 1,2,3..., c (7) k The obectve functon (3) mnmses the total ntercell movement of parts n the system. Constrant (4) ensures that the mergng cells/groups satsfy cell sze. Constrants (5) ensures that each part can only be assgned to one cell. Constrant (6) and (7) ensures that each cell must contan at least one machne and one part.

9 Desgn of flexble manufacturng cell Performance measure Group Technology Effcency has been used to measure the performance of the proposed algorthm to group machnes nto cells and parts nto ther respectve famles. η GT,Group Technology Effcency (Harhalaks et al. 1990a), s defned as the rato of the dfference between the maxmum number of ntercell movements possble and the number of ntercell movements actually requred by the system to the maxmum number of ntercell movements possble. Mathematcally η GT can be defned as: η GT = ( I U)/ I (8) p = 1 ( ) I = pv r 1 (9) r p = 1 U pv λ (10) = 1 r= 1 r where 0 f operatons rr, +1 are performed λr = n the same cell 1 otherwse 4.2 Economc ustfcaton of cell confguraton The cell confguraton s desgned for each product mx and the effect of the uncertanty due to occurrence of the other product mx s analysed by the cost functon Z ( hs, ). It conssts of operatng cost of machnes, nvestment cost of machnes, ntercell movement cost of parts and non-utlsaton penalty cost of machne. If the CMS s desgned accordng to a specfc product mx scenaro and some other product mx demand arses, then the followng possbltes may arse 1 All machne types are beng utlsed. 2 Only few machne types are beng utlsed and the rest reman dle for the gven tme perod. 3 All machne types are utlsed, but the number of same type of machnes requred may be less. Therefore, few dentcal machnes reman dle for the gven perod of tme. 4 All machne types are utlsed, but the number of same type of machnes requred may be more due to hgh producton requrement. Therefore, more dentcal machnes have to be procured to meet the producton requrement wthn producton schedule. If the machnes reman dle n a partcular producton perod, t ndcates the poor cell confguraton plannng. Hence, a penalty cost s ntroduced whch s assumed to be 0.25 tmes the machne nvestment cost.

10 46 A.R. Dxt and P.K. Mshra Z ( hs, ) = Cost of cell confguraton, desgned for sth product mx, due to the occurrence of hth product mx. st pv Z( ) = t + O + K P hs, p m m h pv h h h = 1 = 1 bsh = 1 ( r 1) m p h pv h + UMC K + IT λr h, s = 1 = 1 bsh r= 1 The cost functon (11) represents the total sum of operatng cost of machnes, procurement cost of machnes, non-utlsaton penalty cost for machne type and cost of ntercell movement of parts (n batches) for a gven product mx h. For a gven probablty ( Ρ s ) of the occurrence of the product mx(s), the total expected cost of cell confguraton s gven as: S S ( γ ) = s s (h,s) s= 1 h= 1 (11) E PZ (12) The cell confguraton wth mnmum E ( γ s ) s selected for the desgn of the CM system. 5 Soluton methodology The desgn of CM s combnatoral complex. The number of ways n whch m machnes may be assgned to exactly k cells s gven by the Strlng number of the second knd (Venugopal and Narendran, 1992a). ( k s ) = k = 1 ( 1) k! k K m So, there wll be dstnct parttons of 10 machnes nto 4 cells, but ths number ncreases to 1,12,59,66,000 approxmately; f 19 machnes are to be parttoned nto 4 cells. However, the number of cells s usually not known n advance. But the maxmum and mnmum lmt of the number of cell can be 1 and m, respectvely. The total number of ways n whch machne-cell assgnments may be made explodes to k k ( 1) m m ( k ) = 1 s = k= 1 k= 1 k! K m Therefore, ths class of problem s NP-complete. Many approaches have been proposed by dfferent researchers to solve the problem. Heurstc approaches are also used to obtan good solutons wthn acceptable amount of tme. Numerous papers can be found n the lterature for cell formaton usng heurstcs (Adenso-Daz et al., 2005; Baykasoglu

11 Desgn of flexble manufacturng cell 47 et al., 2001; Burke and Kamal, 1992; Cao and Chen, 2004; Carpenter and Grossberg, 1987; Chen et al., 1995; Dxt and Mshra, 2004; Dobando et al., 2002; Gupta et al., 1995; Harhalaks et al., 1990a, b; Lozano et al., 2001; Muruganandam et al., 2005; Onwubolu and Mutng, 2001; Peker and Kara, 2004; Sofanopoulou, 1997; Solmanpur et al., 2004; Su and Hsu, 1998; Venugopal and Narenderan, 1992a,b; Xambre and Vlarnho, 2003; Yasuda et al., 2005; Zolfagar and Lang, 2003). In real-world manufacturng, producton requrements may not be known exactly at the tme of desgnng CM Systems. It s lkely that a set of possble producton requrements (scenaros) wth certan probabltes may be gven at the tme of desgn. Therefore, uncertanty n producton requrements needs to be ncorporated at the desgn stage of the CM System. Dealng wth uncertan producton requrements n the desgn of CM System has not been extensvely nvestgated. Sefoddn (1990) consdered the uncertanty when desgnng CM System. The algorthm proposed by Sefoddn chooses a cell confguraton from a set of cell confguratons generated from dfferent product mxes. The chosen confguraton s the one that has the lowest expected ntercell materal handlng cost. The prmary drawback n hs algorthm s that only the optmal desgns of each product mx are consdered. It s possble that a system desgn exsts wth a lower expected ntercell materal handlng cost over all product mxes, although, t s not optmal wth respect to any ndvdual product mx. Other mportant parameters lke machne requrement, utlsaton level of machnes (under utlsed/over utlsed) etc have not been consdered. In ths Paper, a procedure to obtan CM desgn solutons for a sngle-perod plannng consderng uncertanty s presented. Such a procedure can be extended for multperod plannng. The followng assumptons have been consdered: 1 there exsts a fnte number of possble product mxes (scenaros) whch can occur 2 each product mx s represented by a unque set of part types and ther demands 3 each product mx has a known probablty of occurrence. We have also appled the heurstc based approach for manufacturng cell desgn under uncertan producton requrement 5.1 Phase-I (cell confguraton for gven product mx) In phase-i, ntal basc feasble soluton s obtaned. The machne cells and part famles are dentfed at ths stage. Intally, a symmetrc matrx wth [m m] entres s constructed usng equaton (3a). An entry Ψ ab n the matrx ndcates the number of mmedate movements between dstnct machnes m a and m b. Then, the machne-par havng maxmum normalsed ntercell flow s grouped nto a machne cell and the matrx s agan revsed. The procedure of teraton s contnued untl the upper bound condton on cell sze s not volated. Once the machne cell s constructed, the parts are assgned optmally to the cells. Later, a local refnement procedure s appled to fnd the better soluton. The general procedure of the proposed heurstc algorthm s presented as follows.

12 48 A.R. Dxt and P.K. Mshra Machne cell formaton algorthm: Step 1 Intally assgn each machne type to a cell. Thus, at the begnnng of the algorthm, the number of cells s the same as the number of machne types. (Number of cells = Number of machnes). Step 2 The total mmedate movement of parts (Ψ ab ) s calculated between all pars of cells between the cells, c a and c : b P O( p) 1 ab A p OPp, OPp, 1 p= 1 = 1 Ψ = χ ab, m ( ) (, + ) A symmetrc matrx of sze (m m) s constructed from the above formulaton. Step 3 The normalsed ntercell flow T ab, s calculated between all par of cells, c a and c b. ψab Τab = ma ( ) + mb ( ) Where m(a) and m(b) are the number of machnes n cell c a and c b, respectvely. Step 4 The par of cells that corresponds to the maxmum normalsed ntercell flow s dentfed. If Te occurred (more than one cell-par has same value of normalsed ntercell flow), then the cell-par havng less number of machnes s selected. If the selected cell pars satsfy the cell sze constrant ( mk ( ) < UB), both cells are merged to form a sngle cell. At the end of ths step, the total ntercell flow s reduced by the ntercell flow value between the two cells beng merged. It s emphaszed that ths s the maxmum possble ntercell flow value that could have been reduced by mergng any par of cells at ths teraton. Hence, the values of normalsed ntercell flow have to be revsed after teraton. The number of cells has been reduced by one unt and the number of machnes n the newly consttuted cell has been ncreased. Step 5 If the maxmum normalsed ntercell flow value s greater than zero, contnue mergng cells through step 4 untl 1 the ntercell flow between all the pars of cells becomes zero, 2 t s mpossble to further merge any cells wthout volatng the cell sze constrant Part allocaton algorthm Step 1 The parts are assgned to the cell havng MAXIMUM number of machnes requred by the partcular part. If TIE occurred (more then one cell has equal number of machnes requred by the partcular part).

13 Desgn of flexble manufacturng cell 49 If operatons are n sequence n TIE cells, then the part wll be assgned to the cell havng mnmum number of total machnes. If tes persst (total number of machnes s also equal n TIE cells), part wll be assgned to the cell havng MINIMUM operaton sequence. If operatons are not n a sequence n one of the TIE cells, then the part wll be assgned to the cell havng operatons n sequence. If operatons are not n sequence n all the TIE cells, the part wll be assgned to the cell havng mnmum number of machnes. If te perssts (number of machnes s same), part wll be assgned to the cell havng ntermedate operaton number. Step 2 A refnement step s performed after all the parts have been assgned to dfferent cells. Snce the cell formaton heurstc s a greedy algorthm, that s once machnes have been merged together nto cell, they cannot be removed from ths cell, a refnement operaton s performed to mprove the soluton. Ths s performed by dentfyng the exceptonal elements (bottleneck machnes, bottleneck parts) and ther respectve cells. Step 3 Identfy the bottleneck machne whch s more nvolved for processng bottleneck parts as compared to ther regular operatons for the part famles wthn the cell. Step 4 If these bottleneck parts belong to a sngle cell, shft the bottleneck machne from ts parent cell to the cell havng bottleneck parts. If TIES occurred Number of exceptonal elements are equal to the number of operatons wthn the parent cell of the machne and f (number of parts n parent cell > number of parts n cell havng bottleneck parts), Shft the machne to the cell havng bottleneck parts. Repeat the step for all the bottleneck machnes. Step 5 Apply only step 1 once. Step 6 Stop. 5.2 Phase-II (selecton of cell confguraton) Phase-II s appled to dentfy the overloaded machnes, so that the requrement of the number of dentcal machnes can be found to select the most economcal cell confguraton. The procedure s as follows: Step 1 Evaluate the load nduced by parts on machnes w. Also, calculate the cumulatve load on machnes. = 1w P Step 2 Identfy the overloaded machnes and calculate the number of dentcal machnes requred to balance the load.

14 50 A.R. Dxt and P.K. Mshra P Step 3 If 1< = 1w < 2 two dentcal machnes of machne type are requred ( K = 2 ). Smlarly f the above lmt s 2 3, then three dentcal machnes of machne type are requred ( K = 3 ). Step 4 Determne the cell confguraton for each product mx (by applyng Phase-I). Step 5 Evaluate the cost of the cell confguraton for each product mx. Step 6 For each cell confguraton, assgn parts of other product mx to the machne cells. Step 7 Evaluate the cost of cell confguratons for all such assgnment of dfferent possble product mxes. Step 8 Determne the total expected cost of cell confguratons. Step 9 Identfy the mnmum total expected cost of cell confguraton. Step 10 Select the cell confguraton/s wth mnmum total expected cost. Step 11 Stop. 6 Computatonal analyss and results The purpose of ths secton s to provde some computatonal analyss and results that can be wdely used to benchmark the effectveness of heurstc approaches, whch may be proposed n future research n ths feld, rather than addressng the algorthmc aspect. In order to valdate the proposed heurstc, ntal part-machne sequence matrx s taken from the publshed research papers. Due to non-avalablty of few producton data, some of the nput parameters were randomly generated. The values n the surveyed publcatons are used as a gudelne for determnng the range of the values of parameters. Table 1 summarses the value for parameters. The algorthm has been mplemented n scrpt programmng n MATLAB 7.0 and the experments have been run on a Pentum IV, wth 2.40 GHz and 512 MB RAM. The example data set contans 20 machnes and 20 parts (Harhalaks et al., 1990a). The predcted demand of the parts, batch sze, machne capacty and materal handlng devce capacty has been generated as per the gudelne of Table 1. The operaton sequence has been the same as suggested by Harhalaks et al. (1990a). Table 2 shows the attrbutes of machne type, whch ncludes procurement cost of the machne and the operatng cost of machne (per hour). Machne capacty s consdered to be fxed to 2000 hr. It s assumed that machne operates 8 hr/day, 5 days/week for 50 weeks (nearly one year). Table 3 shows the attrbutes of part type such as operaton sequence, product mx demand, batch sze, pallet sze and ntercell movement cost per batch. A zero demand of a part ndcates that the part s not manufactured n the gven perod. Fve product mxes have been consdered. Each product mx s assumed to have equal probablty of occurrence.

15 Desgn of flexble manufacturng cell 51 Table1 Values of parameters Parameters Values Remark Machne part sequence matrx Taken from publshed papers In case of 0 1 matrx, the entry 1 s replaced by non-repettve dscrete unform dstrbuted number Part demand U( ) Unformly dstrbuted random no. Processng tme U(0.5 2) mn Dscrete unform dstrbuton Setup tme U( ) mn Dscrete unform dstrbuton Machne capacty 2000 hr/perod Fxed (8 hr/day, 5 days/week for 50 weeks.e. 1 year) Materal handlng U( ) Dscrete unform dstrbuton devce capacty Batch sze U(50 100) Dscrete unform dstrbuton Investment cost U(10,000 50,000) Dscrete unform dstrbuton Operatng cost/unt N(mean 50, SD 20) Normal dstrbuton machnng tme Intercell cost U(20, 50)/batch Dscrete unform dstrbuton Non-utlsaton penalty cost of machne 0.25 nvestment cost Dscrete unform dstrbuton Table 2 Machne type attrbutes (Input) Machne type Investments cost Operatng cost 1 11, , , , , , , , , , , , , , , ,

16 52 A.R. Dxt and P.K. Mshra Table 3 Part type attrbutes (Input) Part type Operaton sequence Product demand PM 1 PM 2 PM 3 PM 4 PM 5 Batch sze Intercell movement cost 1 M2-M3-M1-M4-M M3-M2-M M1-M3-M M3-M1-M4-M M1-M3-M4-M M5-M1-M2-M3-M M1-M2-M M5-M3-M4-M2-M M4-M2-M3-M5-M M3-M1-M M3-M1-M M5-M3-M1-M4-M M1-M2-M3-M M3-M4-M1-M M1-M2-M3-M M3-M2-M1-M M2-M1-M M1-M4-M2-M M2-M1-M4-M M3-M2-M4-M Note: PM = Product Mx. In the frst product mx, thrteen part types, 1, 2, 4, 5, 6, 8, 10, 11, 12, 13, 16, 17 and 19, are produced. Twenty machne types are needed n ths perod. Two unts of machne type 1, 3, 5, 6, 7, 9, 11, 12, 14 and 15 and one unt of machne type 2, 4, 8, 10, 13, 16, 17, 18, 19 and 20 are used to meet the producton requrement. Cell 1 conssts of machne types 1, 5, 9, 12 and 18, and part types 1, 12 and 17 are produced n ths cell. Cell 2 conssts of machne types 2, 3, 10 and 11, and part types 2, 4, 11 and 19 are produced n ths cell.. Cell 3 conssts of machne types 4, 6, 7, 13 and 15, and part types 5, 8, 13 and 16 are produced n ths cell. Cell 4 conssts of machne types 14, 16 and 17, and part type 6 s produced n ths cell. Cell 5 conssts of machne types 8, 19 and 20, and part type 10 s produced n ths cell. The total cost of the cell confguraton n ths perod ncludes: 1 The procurement cost of $7,36, The operatng cost of $21,37, The ntercell movement cost of $24,285.

17 Desgn of flexble manufacturng cell 53 The value of group technology effcency s The total cost of cell confguraton s $28,97,545. Table 4 shows the machne/part groupng for other product mxes. The number of machnes of each type requred to meet the product demand for dfferent product mx scenaros are gven n Table 5. Table 6 shows the total cost and group technology effcency of cell confguraton, desgn by the proposed algorthm. Table 4 Cell confguraton for each product mx scenaros Product Mx Cell confguraton Number of cells Machne groupng Part famly PM1 5 PM2 5 PM3 5 PM4 5 PM5 5 1, 5, 9, 12, 18 1, 12, 17 2, 3, 10, 11 2, 4, 11, 19 4, 6, 7, 13, 15 5, 8, 13, 16 14, 16, , 19, , 2, 10 4, 14, 20 3, 9, 11, 12, 18 2, 9, 12 4, 6, 7, 15 5, 13, 16 8, 19, 20 3, 10, 18 5, 13, 14, 16, 17 7, 15 1, 9, 10, 12, 18 1, 9, 14, 17, 20 2, 3, 11 4, 11, 19 4, 6, 7, 15 5, 8, 13 5, 13, 14, 16, 17 7, 15 8, 19, 20 3, 18 1, 9, 12, 18 1, 12, 17, 20 2, 3, 11 2, 4, 19 4, 6, 7, 15 5, 13, 16 5, 13, 14, 16, 17 6, 7, 15 8, 10, 19, 20 3, 10, 18 1, 9, 10, 12, 18 1, 9, 12, 14, 17, 20 2, 3, 11 2, 4, 11, 19 4, 6, 7, 15 5, 8, 13, 16 5, 13, 14, 16, 17 6, 7, 15 8, 19, 20 3, 10, 18

18 54 A.R. Dxt and P.K. Mshra Table 5 Number of machnes requred to meet dfferent product mx demands

19 Desgn of flexble manufacturng cell 55 Table 6 Cell confguraton cost for dfferent Product Mx Product Mx Cell confguraton Investment cost Operatng cost Intercell movement cost Total cost Group technology effcency PM1 CC1 7,36,260 21,37,000 24,285 28,97, PM2 CC2 6,90,180 17,21,800 33,462 24,45, PM3 CC3 8,21,790 21,51,100 28,108 30,00, PM4 CC4 8,54,310 25,30,400 28,217 34,12, PM5 CC5 9,95,310 27,95,500 38,391 38,29, If the second product mx (PM2) demand occurs, then the parts wll be assgned to the machne groups, arranged for frst product mx (PM1). Three unts of machne type 1, two unts of machne type 8, 10, 12, 19 and 20 and one unt of machne type 2, 3, 4, 5, 6, 7, 9, 11, 13, 14, 15, 16, 17 and 18 are used to meet the producton requrement. One more unt of machne type 1, 8, 10, 19 and 20 s requred to be procured n order to meet the producton schedule. One unt of machne type 3, 5, 6, 7, 9, 11 and 15 wll reman dle for the gven product mx scenaro. Cell 1 conssts of machne types 1, 5, 9, 12 and 18, and part types 9, 12 and 20 are produced n ths cell. Cell 2 conssts of machne types 2, 3, 10 and 11, and part types 2, 4 and 14 are produced n ths cell.. Cell 3 conssts of machne types 4, 6, 7, 13 and 15, and part types 5, 13 and 16 are produced n ths cell. Cell 4 conssts of machne types 14, 16 and 17 and part types 7 and 15 are produced n ths cell. Cell 5 conssts of machne types 8, 19 and 20, and part types 3, 10 and 18 are produced n ths cell. The total cost of the cell confguraton n ths perod ncludes: 1 Machne nvestment cost of $8,77, Operatng cost of $17,21, Intercell movement cost of $32, Non-productve penalty cost of $46,842. The total cost of cell confguraton s $26,78,301. Smlarly, for other product mxes, the total cost of cell confguraton (desgned for partcular product mx scenaro) s gven n Table 7. The mnmum total expected cost s $26,60,568 for the cell confguraton desgned for the product mx scenaro 2(PM2). Hence, ths confguraton must be selected for the desgn of CM system. The executon tme of the algorthm to obtan the result s found to be sec. The algorthm has also been appled to the problems reported n dfferent research papers and executon tme of the proposed algorthm has been shown n Table 8. The executon tme ranges from to sec.

20 56 A.R. Dxt and P.K. Mshra Table 7 Payoff Matrx for dfferent cell confguraton cost

21 Desgn of flexble manufacturng cell 57 Table 8 Executon tme of proposed algorthm for other publshed problems No. Test Instances Source Sze Number of product mxes Executon tme (seconds) 1 Lee and Chen (1997) Jayakrshana and Narender (1998) Harhalaks et al. (1994) Venugopal and Narendran (1992a) Venugopal and Narendran (1992b) Venugopal and Narendran (1992a) Harhalaks et al. (1990a) Chandrasekharan and Raagopalan (1989) Concluson In ths paper, an effcent algorthm for a probablstc machne cell formaton model to deal wth the uncertanty of the product mx for a sngle perod has been proposed. Whle most of the exstng methods of groupng are solely based on bnary machne-part ncdence matrx and a few have used ether the operatons sequence or combnaton of processng tme, setup tme, part demand, machne capacty, materal handlng capacty and lot sze, smultaneous consderaton of all these factors makes the cell formaton problem complex, but more realstc. Ths paper also addressed the machne-part groupng problem consderng all forementoned producton parameters smultaneously. It s evdent from the llustratve example that n uncertan product mx envronment, the cell confguraton wth lowest total expected cost should be mplemented. The algorthm was appled to the numercal problems reported n dfferent research papers and computatonal experence has been reported. The results obtaned suggested that the algorthm s effcent and provdes better solutons. The proposed heurstc approach s also capable of solvng the ndustral problems. Ths approach ncorporates many of the real-lfe producton parameters. Hence, t s bound to provde amcable solutons to the desgners. References Adenso-Daz, B., Lozano, S. and Egua, I. (2005) Part-machne groupng usng weghted smlarty coeffcents, Computers and Industral Engneerng, Vol. 48, pp Adl, G., Raaman, D. and Strong, D. (1996) Cell formaton consderng alternate routngs. Internatonal Journal of Producton Research, Vol. 34, No. 5, pp Askn, R.G. and Subramanan, P.S. (1987) Cost-based heurstc for group technology confguraton Internatonal Journal of Producton Research, Vol. 25, pp Baykasoglu, A., Gndy, N.N.Z. and Cobb, R.C. (2001) Capablty based formulaton and soluton of multple obectve cell formaton problems usng smulated annealng, Integrated Manufacturng Systems, Vol. 12, pp Burbdge, J.L. (1971) Producton flow analyss, Producton Engneer, Vol. 50, Nos. 4/5, p.139.

22 58 A.R. Dxt and P.K. Mshra Burke, L.I. and Kamal, S. (1992) Fuzzy art and cellular manufacturng, Artfcal Neural Networks n Engneerng, pp Cao, D. and Chen, M. (2004) Usng penalty functon and Tabu search to solve cell formaton problems wth fxed cell cost, Computers and Operatons Research, Vol. 31, pp Carpenter, G.A. and Grossberg, S. (1987) A massve archtecture for a self organzng neural pattern recognton, Computer Vson, Graphcs Image Processng, pp Chandrasekharan, M.P. and Raagopalan, R. (1989) Groupablty: an analyss of the propertes of bnary data machnes for group technology, Internatonal Journal of Producton Research, Vol. 27, pp Chen C.L., Contruvo N.A. and Baek W. (1995) A smulated annealng soluton to the cell formaton problem Internatonal Journal of Producton Research, Vol. 33, No. 9, p Chen, C.L., Contruvo, N.A. and Baek, W. (1995) A smulated annealng soluton to the cell formaton problem, Internatonal Journal of Producton Research, Vol. 33, No. 9, pp Chen, M. (1998) A mathematcal programmng model for system reconfguraton n a dynamc cellular manufacturng envronment, Annals of Operatons Research, Vol. 77, pp Choobneh, F. (1988) A framework for the desgn of cellular manufacturng systems, Internatonal Journal of Producton Research, Vol. 26, pp Defersha, F.M. and Chen, M. (2006) A comprehensve mathematcal model for the desgn of cellular manufacturng system, Internatonal Journal of Producton Economcs, Vol. 103, pp Dxt, A.R. and Mshra, P.K. (2004) Machne-cell formaton usng genetc algorthm, Natonal Conference on Advance Manufacturng and Robotcs, Central Mechancal Engneerng Research Insttute, West Bengal, Inda, pp Dxt, A.R. and Mshra, P.K. (2007) Heurstc based approach of cell formaton consderng operaton sequences, Internatonal Conference of Manufacturng Engneerng and Engneerng Management, London, ICMEEM_44 The World Congress on Engneerng 2007 Proceedngs book Volume-II ( ), pp Dobando, D., Lozano, S., Bueno, J.M. and Larraneta, J. (2002) Cell formaton usng a Fuzzy Mn-Max neural network, Internatonal Journal of Producton Research, Vol. 40, No. 1, pp Gupta, T. and Sefoddn, H. (1990) Producton data based smlarty coeffcent for machne-component groupng decson n the desgn of a cellular manufacturng system, Internatonal Journal of Producton Research, Vol. 28, No. 7, pp Gupta, Y.P., Gupta, M.C., Kumar, A. and Sundaram, C. (1995) Mnmsng total ntercell and Intracell moves cellular manufacturng system: a genetc algorthm approach, Internatonal Journal of Computer Integrated Manufacturng, Vol. 8, pp Harhalaks, G., Nag, R. and Proth, J.M. (1990a) An effcent heurstc n manufacturng cell formaton for group technology applcatons, Internatonal Journal of Producton Research, Vol. 26, pp Harhalaks, G., Nag, R. and Xe, X.L. (1990b) Manufacturng cell desgn usng smulated annealng: an ndustral applcaton, Journal of Intellgent Manufacturng, pp.1 8. Harhalaks, G., Nag, R. and Proth, J.M. (1994) Manufacturng cell formaton under random product demand, Internatonal Journal of Producton Research, Vol. 32, pp Jayakrshnan, N.G. and Narender, T.T. (1998) CASE: a clusterng algorthm for cell formaton wth sequence data, Internatonal Journal of Producton Research, Vol. 36, pp Kumar, K.R. and Vannell A. (1987) Strategc subcontractng for effcent ds-aggregated manufacturng Internatonal Journal of Producton Research, Vol. 25, No. 4, pp Kusak, A and Chow, W.S. (1987) Effcent solvng of the group technology Journal of Manufacturng systems, Vol 6, No 2, pp

23 Desgn of flexble manufacturng cell 59 Lee, S.D. and Chen, Y.L. (1997) A weghted approach for cellular manufacturng desgn: mnmsng ntercell movements and balancng workload among duplcated machnes, Internatonal Journal of Producton Research, Vol. 35, pp Logendran, R. (1991) Impact of sequence of operatons and layout of cells n cellular manufacturng, Internatonal Journal of Producton Research, Vol. 29, pp Lozano, S., Canca, D, Gurrreo, F. and Garca, J.M. (2001) Machne groupng usng sequence-based smlarty coeffcents and neural network, Robotcs and Computer Integrated Manufacturng, Vol. 17, pp Mungwattana, A. (2000) Desgnng cellular manufacturng systems for dynamc and uncertan producton requrement wth presence of routng flexblty, PhD Thess, Vrgna Polytechnc Insttute and state Unversty. Muruganandam, A., Prabharan, G., Asokan, P. and Baskaran, V. (2005) A memetc algorthm approach to the cell formaton problem, Internatonal Journal of Advance Manufacturng Technology, Vol. 25, pp Onwubolu, G.C. and Mutng, M. (2001) A genetc algorthm approach to cellular manufacturng systems, Computers and Industral Engneerng, Vol. 39, pp Peker, A. and Kara, Y. (2004) Parameter settng of the Fuzzy ART neural network to part-machne cell formaton problem, Internatonal Journal of Producton Research, Vol. 42, No. 6, pp Sankaran, S. and Kaslngam, R. G. (1993) On cell sze and machne requrements plannng n group technology systems, European Journal of Operatonal Research, Vol. 69, pp Sarker, B.R. and Xu, Y. (1998) Operaton sequences-based formaton methods: a crtcal survey, Producton Plannng and Control, Vol. 9, pp Schaller, J. (2007) Desgnng and redesgnng cellular manufacturng systems to handle demand changes, Computers and Industral Engneerng, do /.ce Sefoddn, H. (1990) A probablstc model for machne cell formaton, Journal of Manufacturng Systems, Vol. 9, No. 1, pp Shafer, S.M. and Rogers, D.F. (1993) Smlarty and dstance measures for cellular manufacturng. Part II. An extenson and comparson, Internatonal Journal of Producton Research, Vol. 31, pp Sofanopoulou, S. (1997) Applcaton of smulated annealng to a lnear model for the formulaton of machne cells n group technology, Internatonal Journal of Producton Research, Vol. 35, No. 2, pp Solmanpur, M., Vrat, P. and Shankar, R. (2004) A mult-obectve genetc algorthm approach to the desgn of cellular manufacturng systems, Internatonal Journal of Producton Research, Vol. 42, pp Song, S and Htom, K. (1996) Integratng the producton plannng and cellular layout for flexble cellular manufacturng, Internatonal Journal of Producton, Plannng and control, Vol. 7, pp Su, C-T. and Hsu, C-M. (1998) Mult-obectve machne-part cell formaton through parallel smulated annealng, Internatonal Journal of Producton Research, Vol. 36, pp Vakhara, A.J. and Wemmerlov, U. (1990) Desgnng a cellular manufacturng system: a materals flow approach based on operaton sequences, IIE Transactons, Vol. 22, pp Venugopal, V. and Narendran, T.T. (1992a) A genetc algorthm approach to the machne-component groupng problem wth multple obectves, Computers and Industral Engneerng, Vol. 22, pp Venugopal, V. and Narendran, T.T. (1992b) Cell formaton n manufacturng systems through a smulated annealng: an expermental evoluton, European Journal of Operatonal Research, Vol. 63, pp