HEURISTIC BASED APPROACH OF CELL FORMATION CONSIDERING OPERATION SEQUENCE

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1 HEUISTIC BASED APPOACH OF CELL FOATIO COSIDEIG OPEATIO SEQUECE Amit ai Dixit, embe, IAEG. P. K. isha Abstact This pape pesents, a two-stage heuistic based pocedue fo geneating pat family and machine cell fomation in Cellula anufactuing System. It deceases exceptional elements and voids which in tuns deceases intecell flow of pats and inceases utilization of machines in the cells. In the fist phase; the poblem is solved as a bottom-up aggegation pocedue fo machine gouping. Aggegation is based on the minimization of intecell flow. Late the pats ae assigned to the cells accoding to the poposed heuistic. Uppe bound on the cell size is imposed in the fist stage which is elaxed gadually in second phase. It ensues the natual cell fomation. The solution obtained at the end of fist stage is efined in the second phase. umeical examples wee tested fo gouping efficiency, gouping efficacy, global efficiency and have been compaed with esults epoted by othe eseaches. The computational esults ae encouaging Index Tems cellula manufactuing systems, pat gouping, inte-cell movement, opeation sequence, void, exceptional element I. ITODUCTIO Goup Technology (GT) is an appoach to manufactuing and engineeing management that helps to manage divesity by capitalizing on undelying similaities in poducts and activities. In the manufactuing context, GT has been defined as a manufactuing philosophy identifying simila pats and gouping them togethe into families to take advantage of thei similaities in manufactuing and design. Gouping the poduction equipment into machine cells, whee each cell specializes in the poduction of pat families, is called as cellula manufactuing. So cellula manufactuing is the application of the GT philosophy in manufactuing. Cellula anufactuing is concened with the ceation and opeation of manufactuing cells which ae dedicated to the poduction of a set of pat families. In ode to intoduce cellula manufactuing, it is necessay to identify pats and machine types to be used in the cellula configuation. anuscipt eceived ach 22, Amit ai Dixit is with the Institute of Engineeing & ual Technology, Allahabad, India (phone: ; fax: ; amitaidixit@ gmail.com). P. K. isha is with otilal ehu ational Institute of Technology, Allahabad, India. ( pkm@mnnit.ac.in). The fist poblem faced in implementing Cellula anufactuing is cell fomation. Cell fomation deals with the identification of the family of pats and the goup of machines to pocess these pats. The poblem of cell fomation is defined as: "If the numbe, types, and capacities of poduction machines, the numbe and types of pats to be manufactued, and the outing plans and machine standads fo each pat ae known, which machines and thei associated pats should be gouped togethe to fom cells?" [33]. In some cells the definition of cell fomation is expanded to allow choice of pocessing opeations to achieve specific featues. Since last thee decades, a consideable amount of eseaches have been diected to ease this type of poblem. Bubidge [3] developed an intuitive method, namely Poduction Flow Analysis (PFA) which is elatively easy to implement. PFA may be suitable fo the small size poblem, but it would definitely have difficulties coping with eal life cell fomation poblems when the machine-pat incidence matix becomes moe complex because of poblem size. A lage numbe of appoaches have been developed to deal with the difficulties of intuitive method. These appoaches ae usually classified into Pat-oiented appoaches (based on pat chaacteistics) and Pocess-oiented appoaches (based on poduction methods). The pat-oiented techniques usually employ some classification and coding system, and analyze pats fo thei similaities in design featues and functionalities. Howeve, these do not influence diectly the configuation of manufactuing cells [0]. The pocess-oiented appoaches to the cell fomation ae based on manufactuing data such as poduction methods, pat outing infomation and pocess plans. The pocess-oiented appoach is classified into fou goups namely: - Desciptive methods, Aay-based methods, Similaity coefficient methods and othe analytical methods [ 35]. ost of the suggested algoithms/models conside binay machine-pat incidence matix A, with a ij if pat i equies machine j, othewise 0. The binay pat-machine matix is incapable of pesenting the actual intecell movements of pats. Opeation sequences of pats in one of the most impotant manufactuing factos in the

2 design of cellula manufactuing systems. The opeation sequence is defined as an odeing of the machines on which the pat is sequentially pocessed. The sequence of opeation has an impact on the flow of mateial in the system. An intemediate opeation of a component to be pefomed outside its cell equies two inte-cell tansfes while the fist o last opeation equies only one such tansfe [0], [9]. Theefoe opeation sequence matix has been used in place of binay machine-pat matix. Hahalaskis [9] also consideed the same scheme, but thee wee cetain dawbacks in the pocedue [4]. ) It equies an a pioi specification of the uppe bound on the numbe of machines within a cell and the numbe of cells. This contadicts the fundamental philosophy of gouping of machines natually and the task of the analyst is to identify them if they exist [3], [7], [0]. At the design stage, the numbe of cells should be an outcome of the solution pocedue and not an input paamete. 2) Othe dawback is the ievesibility of the hieachal clusteing algoithm, i.e. once two machines (o cells) ae gouped togethe at some stage thee is no way to etace the steps even if it leads to suboptimal clusteing at the end [7], [9]. Also in the case of ties, selection is made abitaily. This pecludes fomation of bette goups at late stage. In this pape, uppe bound on the cell size is imposed initially in the fist stage to obtain basic feasible solution. The condition is elaxed in subsequent phase (efinement stage) and the cells ae fomed natually. In the case of ties, decision is taken fo pope selection based on heuistic. The pape is oganized as follows: otations and definitions ae explained in section 2. The mathematical model is pesented in section 3. The poposed algoithm is pesented in section 4. The evaluation citeia ae given in section 5. Computational esults ae pesented in section 6 to illustate the poposed algoithm. Conclusion is pesented in section 7. II. OTATIOS AD DEFIITIOS i pat type j machine type k cell type n opeation type m numbe of machines (m, m 2,,m m ). p numbe of pats P ( p, p 2,.,p p ). c numbe of cells C ( c, c 2,.,c c ). X jk if machine j is in cell k and 0 othewise Y ik e d e o if pat i is assigned in cell k and 0 othewise numbe of machines in cell numbe of pats in cell numbe of in-cell opeations, numbe of out-of-cell opeations, Θ k δ k total numbe of opeation in the k th cell total numbe of non-opeation (voids) in the k th cell ξ Compactness UB uppe bound on cell size (maximum numbe of machines in a cell) m (k) umbe of machines in cell type k. mpim machine-pat incidence matix epesenting the opeation sequence. (mpim) ij n, if nth opeation of pat i is pefomed on machine j, 0 othewise. i ψ ab the numbe of times that pat i moves fom a to b, and b is the immediate successo of a. Whee ψ i ab p a m ( mpimaε ) ( mpimab ) b a+ ε m, and value of ε incemented by in each iteation. III. ATHEATICAL ODEL The most fundamental objectives fo cell fomation ae minimization of intecell flows and maximizing machine utilization. It helps to decease the intecell movement cost. In eseach, effots ae made to minimise intecell flows and maximize machine utilization with the consideation of opeation sequence of pats. The mathematical model is given below: omalized Intecell flows: in Z w a w b ψ ma () + ab mb ( ) Subject to constaint: m( a) + mb ( ) UB (2) c k c k m j p i x jk y ik x jk y ik fo j,2,., m. (3) fo i,2,., p. (4) fo k,2,., c. (5) fo k,2,., c. (6) Equation () shows the calculation of omalized intecell flow. Constaint (2) ensues that the meging cells/goups satisfy cell size. Constaint (3) and (4) ensues that each machine and pat can only be assigned to one cell. Constaint (5) and (6) ensues that each cell must contain at least one machine and one pat. ()

3 IV. HEUISTIC SOLUTIO APPOACH The design of cellula manufactuing is combinatoially complex. Thee ae numbe of appoaches which wee poposed by diffeent eseache. Heuistic appoaches ae used to obtain good solutions within acceptable amount of time. umeous papes can be found in the liteatue fo cell fomation using heuistics[], [2], [4]-[6], [9], [2], [3], [7]-[9], [23]-[28], [30]-[32], [34], [36], [39].We have applied the two stage heuistic based appoach consideing opeation sequence to solve cell fomation. Phase I. (Initial Cell-fomation) A. achine-cell Fomation Algoithm Step : Assign each machine to a cell (umbe of cells umbe of machines). Step 2: Detemine Ψab between the cells fom the opeation sequence matix (mpim). Step 3 : Detemine the omalized Intecell flow between the cells. Step 4 : Select the minimum nomalized Intecell flow value fo the given cell-pai satisfying the limit of cell size. If tie occus (moe than one cell-pai has same value) Decision: Select cell-pai having maximum Intecell movements. (Intecell movement will be minimized) If TIE still pevails Decision: Select cell-pai having less numbe of machines. (Intacell movement will be minimized). Step 5 : ege the cell-pai to fom new cell. Step 6 0: epeat step (2-4) till uppe bound condition on cell size is not violated. Step 7: Stop. B. Pat Allocation Algoithm Step : Pat will be assigned to the cell having AXIU numbe of machines equied by the paticula pat. If the tie occus: () If opeations ae in same sequence in TIE cells: (a) Pat will be assigned to the cell having minimum numbe of machines (void will be minimum) (b) If numbes of machines ae equal then pat will be assigned to the cell having IIU opeation sequence. (Inte-cell movement will be minimum) (2) If opeations ae not in a sequence in one of the TIE cells: (a) Pat will be assigned to the cell having opeations in sequence. (Inte-cell movement will be minimum) (3) If opeations ae not in sequence in all the TIE cells: (a) Pat will be assigned to the cell having minimum numbe of machines (voids Step 2: Stop will be minimum) (b) If numbe of machines ae same then pat will be assigned to the cell having IIU opeation numbe (inte-cell movement will be minimum) Phase II. (Impovement of esult Obtained Fom Phase I) Step : Identify the exceptional elements (EE), bottleneck machines, bottleneck pats and thei espective cells fom the initial solution obtained fom Phase-I. Step 2: Identify the bottleneck machine which is moe involved fo EE as compaed to thei egula opeations fo the pat families within the cell. Step 3: If these EE ae fom the same cell (having bottleneck pats) Shift the machine to the new cell (EE elements will be educed and machine utilization will be inceased) Step 4: if TIES occued: (umbes of EE ae equal to the numbes of opeations within the paent cell of the machine.) if numbe of pats in Paent cell > numbe of pats in cell having bottleneck pats, Shift the machine to the new cell (Voids will be educed, EE will emain same and within-cell compactness will be inceased) Step 5: epeat the step (2-4) fo all the bottleneck machines. Step 6 : Apply the pat allocation algoithm. Step 7: Stop. V. EVALUATIO CITEIO Fou pefomance measues have been used to evaluate the esult of poposed algoithm with othes. These measues of pefomance ae defined as below: Gouping efficiency [7] was the fist evaluation citeia fo final esult obtained by diffeent algoithms. ( ) 2 η Gouping efficiency g ϖη + ϖ η (7) Whee ϖ is the weighting facto anging between 0 and, η is the measue of the density of s in the diagonal clustes η of the block diagonal matix and 2 measues the density of 0 s outside the diagonal cluste., that ae defined as: ϖ η e d (8) (9)

4 η 2 Gouping efficacy ed Γ e o + e o (0) () Global Efficiency [9] is the atio of the numbe of opeations that ae pefomed within cells to the total numbe of opeations in the systems. Global Efficiency ed ηo ed + eo (2) VI. COPUTATIOAL ESULTS The algoithm has been implemented in scipt pogamming in ATLAB 7.0 and the expeiment has been un on a Pentium Table. achine-pat Incidence atix (Example ) IV, with.8 GHz and 256 B A. In ode to validate the poposed heuistic, a set of poblems have been selected fom eseach papes. A. Example Conside the example of 20 machines, 20 pats. Table shows the incidence matix. The esults afte phase-i ae same as epoted by [9]. The numbe of exceptional elements was 5. In the epoted solution, machine 4 has been assigned to cell 2. achine 4 was engaged fo pefoming 2 nd opeation on pat. So this machine was pefoming only one opeation in the assigned cell. est of the time it was engaged with bottleneck pats (6 and 5 of cell 4). This fact is taken into consideation in Phase-II and the machine 4 is shifted to cell 4. As a esult numbe of exceptional elements has been educed to 4. The numbe of voids due to machine 4 in cell 2 was 3. Afte eallocation of machine 4 in cell 4, the numbe of voids due to machine 4 in cell 4 has educed to. The impoved solution is shown in Table 2. The values of the Gouping efficiency, Gouping Efficacy, and Global Efficiency of the final solution ae 0.925, , and 0.80 espectively. These values ae bette as compaed to the epoted solution [9]. components achines Table 2. Impoved pat-machine cell matix afte phase-ii-final Solution (Example ) components achines

5 B. Example 2 A lage matix used is consideed in this example. Table 3 shows the incidence matix. The impoved solution afte phase-ii is shown in Table 4. The numbe of exceptional elements in the epoted solution is 35. The numbe of exceptional elements by the poposed solution methodology has been educed to 32 in the final solution. This solution is Components bette as compaed to the solution epoted in efeence pape [4]. The values of the Gouping efficiency, Gouping Efficacy, and Global Efficiency of the final solution ae , and espectively. These values ae bette as compaed to the epoted solution [4]. Table 3. achine-pat incidence matix (Example 2) achines I. COCLUSIO A heuistic algoithm fo geneating machine cell and pat family has been developed fo cellula manufactuing system. This algoithm geneates a feasible solution by taking opeation sequence of pats into account. The algoithm compises of two phases. The fist phase foms a configuation of independent cells using bottom-up aggegation pocedue and pats ae allocated accoding to poposed heuistic. The second phase, named as impovement stage, addesses fo minimization of voids and exceptional elements. The unde-utilized machines ae identified and an attempt is made to eallocate the machines in othe cells so that the voids and exceptional elements ae emoved fom the solution matix. The heuistic algoithm was applied to the numeical poblems epoted in diffeent eseach papes and computational expeience has been epoted. The esults obtained suggested that the algoithm is efficient and povides bette solutions. The algoithm was also tested fo lage poblems and the quality solution was obtained. The poposed heuistic appoach is capable of solving the industial poblems.

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