SKP-2 ALGORITHM: ON FORMING PART AND MACHINE CLUSTERS SEPARATELY

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1 Poeedings of the 1998 Paifi Confeene on Manufatuing, August 18-20, 1998, Bisbane, Queensland, Austalia SKP-2 ALGORITHM: ON FORMING PART AND MACHINE CLUSTERS SEPARATELY Susanto,S., Kennedy,R.D. and Pie, J.W.H Depatment of Mehanial Engineeing, Monash Univesity PO Box 197, Caulfield East, Vi. 3145, Austalia A new ell fomation algoithm, alled SKP-2 (Susanto-Kennedy-Pie Algoithm 2) is intodued and outlined. Pat and Mahine-lustes ae fomed sepaately based on the fuzzy lusteing tehnique. Manufatuing ells ae fomed based on the assignment tehnique. Computational esults show that this new algoithm esults in less exeptional elements, espeially fo medium to high density mahine-pat inidene maties. Keywods: Cellula Manufatuing, Manufatuing Cell, Cell Fomation, Pat-luste, Mahine-luste. 1. INTRODUCTION In ellula manufatuing(cm), pats that equie simila mahines ae gouped into the same patluste, while mahines needed by a pat-luste ae gouped to fom a mahine-luste. A manufatuing ell is pat of a manufatuing system whee a mahine-luste is plaed to poess a pat-luste. The fist poblem to be addessed in a CM system is that of ell fomation(cf). Data fo CF is oganized into a mahine-pat inidene matix (Susanto et.al, 1997). An exeptional pat is a pat that equies poessing in anothe mahine-luste. An exeptional mahine is a mahine that poesses pats fom a diffeent pat-luste. Both exeptional pats and exeptional mahines ause inteellula movement of pats. Ideally a pat-luste is poessed in a single mahine ell fo its entie opeations, howeve, it is a ae ase (King, 1980). Numeous eseah papes have appeaed in the liteatue fo CF. Howeve, this ealy eseah assumes that eah pat an belong to only one pat luste, yet pats may exist whose membeship is muh less evident (Chu, 1995). In othe wods, only a few eseahes have addessed the issues of fuzziness in ell fomation. Chu and Hayya (1991) intodued the appliation of fuzzy lusteing in to the CF poblem and was followed by Ponnambalam and Aavindan (1994) with a mino modifiation in the initialization. The SKP-2 algoithm intodued by Susanto et.al (1997), is a signifiant modifiation to Chu and Hayya s appoah. It intodues the fifth ategoy of fomation logi, ie. foming patlustes and mahine-lustes sepaately. In the next step, a manufatuing ell is fomed by applying the assignment algoithm, whih will esult in a dense blok-diagonal matix. This pape disusses anothe vesion of the basi algoithm that also foms pat-lustes and mahine-lustes sepaately. It an be viewed as anothe modifiation to the Chu and Hayya appoah. 2. A BRIEF REVIEW The pimay objetive of lusteing is to patition a given set of objets into so-alled homogeneous lustes (goups) (Bezdek and Pal, 1992). Globally, the lusteing poblem an be divided into two main ategoies ie. had lusteing, in whih an objet belongs only to one luste, and fuzzy lusteing, in whih evey objet belongs to all lustes with diffeent degees of membeship. The fuzzy lusteing appliation in foming pat lustes and mahine-lustes involves finding maties p U M f = U V p u i [,] 01 i,; u i = 1 ; 0< u i < p i i= 1 = 1 (1) and v R p that minimize the funtion 122

2 Poeedings of the 1998 Paifi Confeene on Manufatuing, August 18-20, 1998, Bisbane, Queensland, Austalia p f 2 Jf ( Uv, ) = ( ui) ( di) = 1i = 1 whee u i : the degee of membeship of pat to belong to pat-luste i; di = x vi is the distane between two vetos x and v i ;. is any inne podut on R m ; x =( x1, x2,..., xm) T is the attibute veto of T pat ; v = ( v v ) R u = u 1..., u ; f is the degee of fuzziness, f (, 1 ). i i mi T m 1,..., is the luste ente of i ( i pi) (2) 3. THE SKP-2(Susanto-Kennedy-Pie 2) ALGORITHM The SKP-2 algoithm, as mentioned in Setion 1, does not belong to any existing ategoy o lassifiation of CF piblem, sine it foms pat-lustes and mahine-lustes sepaately. This algoithm is aimed at takling the two poblems that may aise fom Chu and Hayya s (1991) algoithm (Susanto et.al, 1997) and at maximizing the total numbe of elements in the blok diagonal of the final aangement of the inputted mahine-pat inidene matix. Let - X= ( x q ) mxp be the given mahine-pat inidene matix of m types of mahine and p type of pats, to be gouped into ( 2 < min{ m, p}) mahine and pat-lustes espetively. T - x=( x1,..., xm) and yq=( xq1,..., xqp) ae the attibute veto of the pat type ( = 1,...,p) and the attibute veto of the mahine type q (q = 1,...,m), espetively 3. 1 The Fist Step of SKP-2 (Pat-lustes Fomation) The p attibute vetos of eah pat type seve as the only known vaiable in the funtional to be minimized J U, v). In this poblem, we define a suessful pat-luste fomation solution as one that: f ( 1) onveges to non empty pat-lustes; 2) esults in unimodal solution fo } (k = 1,..., ). max{ u k k The details of the fist step in SKP-2 ae as follows: Step 1: Fix ( 2 < min{ m, p }); hoose any inne podut nom on R m, fix f (, 1 ) ; initialize ( ) U 0 ; let suess=0; l=0 and the maximum numbe of iteations be max_iteations. M f (l) Step 2: Calulate the fuzzy luste entes { v () i } whee Step 3 : IF {,..., p}, THEN if A = 12 v () l qi p ( ui ) xq = 1 = ; i=,..., ; q=,..., m p () l f ( u ) = 1 () l f 1 A = ( ) () l αl u = { u α i } ( ) max p U {,,..., } = 1 l i i= 12,,..., 1 1 (3) is singleton, then if l < max_iteations then 1) suess suess + 1 2) at the l-th iteation and at the suess-th suessful pat assignment, assign pat q to pat-luste α () l else 1) suess suess + 1 2) at the l-th iteation and at the suess-th suessful pat assignment, assign pat to pat-luste α () l, goto Step 6; else if l = max_iteations then goto Step 6 ELSE if l = max_iteations then goto Step 6 Step 4 : Detemine I and I ( = 1, 2,..., p) as follows 123

3 Poeedings of the 1998 Paifi Confeene on Manufatuing, August 18-20, 1998, Bisbane, Queensland, Austalia Step 5 : l l + 1; alulate { u () l i } Step 6 : IF suess 1 THEN PC { 1 0} I = i i ; d = x v = (4) i i I = {, 12,..., } I (5) u whee () l i 1 k = 1 = 0, 1, I suess = PC s s = 1 di ( d ) k 2 f 1, if I = k I if I k I if I (6) U, whee PC { PC i s } s= is = 12 = 12,,..., ;,,...,suess and PC is is the patluste i obtained fom the s-th suessful pat assignment, ELSE this algoithm fails to onvege to a suessful pat-luste fomation solution. 3.2 Seond Step of SKP-2(Mahine-lustes Fomation) Basially the seond step of SKP-2 is onguent to that of the fist step. While the fist step of SKP-2 exploits the pat-attibute vetos x ( = 1, 2,..., p) (whih ae olumn-vetos) to fom the patlustes, the seond step exploits the mahine-attibute veto y q (q=1,..., m) (whih ae ow-vetos), to fom the mahine-lustes. We define a suessful mahine-luste fomation solution as one that: 1) onveges to non-empty mahine-lustes and 2) esults in a unimodal solution fo max{ } (k= 1,..., ). k u qk If, afte the maximum numbe of iteations ahieved, this seond step onveges and esults in SUCCESS ( 1) suessful pat-luste fomation solutions then let us define: SUCCESS MC = U (7) whee MC S S = 1 { MCkS k 12 S 12 SUCCESS} MCS= =,,..., ; =,,..., (8) and MC ks is the mahine-luste k obtained fom the S-th suessful mahine-luste assignment. 3.3 The Thid Step of SKP-2 (The Assignment Poblem) To apply this step, it is neessay that both the fist and the seond step of SKP-2 onvege to suess suessful pat-luste solutions and SUCCESS suessful mahine-luste solutions. Let MC PC = x q (9) is ks PCkS q MCis be the degee of onfomane of pat luste k obtained fom the S-th suessful pat-luste fomation and mahine luste i obtained fom the s-th suessful mahine-luste fomation. The final assignment poblem is the solution of the following optimization poblem: subjet to maximize z ss = max MC is PC ks MC ispcks i = 1k = 1 Ω (10) Ω = 1, i = 1,..., ; s = 1,...,suess;S=1,..., SUCCESS MC (11) ispcks k = 1 Ω = 1, k = 1,..., ; s = 1,...,suess;S=1,..., SUCCESS MC (12) ispcks i = 1 124

4 Poeedings of the 1998 Paifi Confeene on Manufatuing, August 18-20, 1998, Bisbane, Queensland, Austalia Ω MC ispcks = 0o 1 (13) Note: Ω = MC 1, if the k-th pat-luste (obtained fom the S-th pat-luste fomation solution) is assigned to ispcks the i-th mahine-luste (obtained fom the s-th mahine-luste fomation solution), = 0, othewise. This assignment poblem an be solved using The Hungaian Method (Taha, 1992) o Mak s Method (Bunday, 1984). 4. COMPUTATIONAL RESULTS In this setion the pefomanes of SKP-2 and Chu and Hayya s algoithm ae ompaed, based on the obsevations applied to 16 maties with the dimension of 40x50, with density anges fom 20 to 80%. The esults ae pesented in Table Example of Computational Results The mahine-pat inidene matix XVI is displayed in Figue 1. The esults of using this matix using the Chu and Hayya s (1991) algoithm and the SKP-2 algoithm ae pesented in Setion and espetively Results fom Chu and Hayya s(1991) algoithm This algoithm was applied to matix XVI with the desied numbe of lustes =6, the degee of fuzziness m=2, ξ=0001. and Eulidean nom as the inne podut. The mahine-lustes and pat-lustes ae as follows. The Mahine-lustes : MC-1 = {4,7,15,17,35,37}; MC-2 = {6,19,20,21,23,24,30,31,38,40}; MC-3 = {8,10,36}; MC-4 = {3,22}; MC-5 = {1,5,9, 11,12,14,25,26, 27, 32,33,39} and MC-6 = {2,13,16,18,28,29,34}. The Pat-lustes : PC-1 = {1,7,18,19,20,25,33,37,43,49}; PC-2 ={14,32,38, 50}; PC-3 = {5,9,21,31,39,40,45}; PC- 4={4,10,22,28,44, 46, 47}; PC - 5 = {2,11,15, 17,23,24,26,27,29,34,35,41} and PC - 6 = {3,6,8,12,13,16,30,36,42,48}. The esulting Manufatuing Cells ae : Manufatuing Cell-i will onsist of MC-i that poesses PC-i (i = 1,2,...,6). Thee will be 333 of 1 (one) enties on the blok diagonal of the final eaangement of the matix XVI whih ontains 1609 (ie x40x50) 1 enties. Thus, Chu and Hayya s (1991) algoithm esults in 20.7% (ie. 333/1609 x 100%) of non exeptional elements. The final eaangement by this algoithm is displayed in the Figue Result fom SKP-2 algoithm The SKP-2 algoithm was also applied to matix XVI, with the desied numbe of lustes =6, the degee of fuzziness m=2, the maximum numbe of iteations max_iteations=200, and the Eulidean nom as the inne podut. Eight suessful mahine-luste fomation solutions and seven suessful pat-luste fomation solutions ae esulted fom the teatment. The desied mahine-lustes and pat-lustes ae obtained by paiing eah suessful mahineluste and pat-luste fomation solution, and applying the Assignment Algoithm to eah pai. In the ase of mahine-pat inidene matix XVI, the desied mahine-lustes ae those obtained fom the sixth suessful mahine-luste fomation solution and the desied pat-lustes ae those obtained fom 125

5 Poeedings of the 1998 Paifi Confeene on Manufatuing, August 18-20, 1998, Bisbane, Queensland, Austalia 126 the seventh suessful pat-luste fomation solution. The assignment of pat-luste to mahine-luste is outlined in Figue 3. Thee will be 442 of 1 (one) enties on the blok diagonal of the final eaangement of the matix XVI out of 1609 enties. Thus, this algoithm esults in 27.7% of non exeptional elements. The final eaangement by this algoithm is displayed in the Figue 3. Table-1. Compaison of peentage of non exeptional elements esulting fom SKP-2 and Chu and Hayya s(1991) algoithm MATRIX Density of Matix *) SKP -2 CHU & HAYYA (1991) Comm ent on SKP % I wose II bette III wose IV wose V wose VI wose 40-60% VII bette VIII bette IX bette X bette XI bette 60-80% XII bette XIII bette XIV bette XIII bette XV wose XVI bette *) Density of a matix as defined in Miltenbug and Zhang (1991) Figue 1. The Mahine-Pat Inidene Matix XVI Figue 2. The Final Reaangement of Mahine-Pat Inidene Matix XVI by Chu and Hayya s Figue 3. The Final Reaangement of Mahine-Pat Inidene Matix XVI by SKP-2 Algoithm Pat-type Numbe M a h i n e t y p e N u m b e Pat-type Numbe M a h i n e t y p e N u m b e Pat-type Numbe M a h i n e t y p e N u m b e

6 Poeedings of the 1998 Paifi Confeene on Manufatuing, August 18-20, 1998, Bisbane, Queensland, Austalia 5. CONCLUSIONS SKP-2 algoithm foms the mahine and pat luste sepaately, eah based on fuzzy lusteing onept. The assignment of pat-luste into mahine-luste to fom a manufatuing ell is done using the assignment tehnique. Computational expeiene shows that this algoithm - espeially fo medium to high density mahine-pat inidene matix - esults in less exeptional elements than Chu and Hayya s (1991) method, whih foms the mahine and pat-luste simultaneously. REFERENCES Bezdek, J.C., and Pal, S.K. (1992). Fuzzy Models fo Patten Reognition, IEEE Pess, Pisataway, USA. Bown, D.E., and Huntley, C.L. (1992). A patial appliation of simulated annealing to lusteing. Patten Reognition, Vol. 25, No. 4, p Bunday, B.D.(1994). Basi Linea Pogamming, Edwad Anold, London; Baltimoe; Md, USA, Chu, C.H., and Hayya, J.C.(1991). A fuzzy lusteing appoah to manufatuing ell fomation. Intenational Jounal of Podution Reseah, Vol. 29, p Chu, C.H. (1995). Reent advanes in mathematial pogamming fo ell fomation. In: Planning, Design, and Analysis of Cellula Manufatuing Systems (Ed. Kamani, A et.al.), p Elsevie Siene B.B., Amstedam. King, J.R. (1980). Mahine-omponent gouping in podution flow analysis: An appoah using ank ode lusteing algoithm. Intenational Jounal of Podution Reseah, Vol. 18, p MComik,W.T, Shweitze, R.J, and White,T.W.(1972). Poblem deomposition and data eoganization by lusteing tehniques. Opeations Reseah, Vol. 20, p Miltenbug, J., and Zhang, W. (1991). A ompaative evaluation of nine well-known algoithms fo solving the ell fomation poblem in goup tehnology. Jounal of Opeations Management, Vol. 10, No. 1, p Ponnambalam, S.G., and Aavindan, P. (1994). Design of ellula manufatuing systems using objetive funtional lusteing algoithms. Intenational Jounal of Advaned Manufatuing Tehnology, Vol. 9, p Susanto, S., Kennedy, R.D., and Pie, J.W.H.(1997). A Peliminay study of a fuzzy lusteing and assignment poblem-based ell fomation algoithm. In: Poeedings of The ICMA 97 (Eds. Tan, S.T., Wong, T.N., Gibson, I), pp , The Univesity of Hong Kong, Hong Kong. Taha, H. (1992). Opeations Reseah: an intodution, 5th. ed., Mamillan Publishing Company, Don Mills, Ontaio. Wemmelöv, U., and Hye, N.L.(1987). Reseah issues in ellula manufatuing. Intenational Jounal of Podution Reseah, Vol. 25,