206 3 rd Iteratioal Coferece o Mechaical, Idustrial, ad Maufacturig Egieerig (MIME 206) ISBN: 978--60595-33-7 Applicatio of Improved Geetic Algorithm to Two-side Assembly Lie Balacig Ximi Zhag, Qia Wag, Huizhi Re Sheyag Uiversity of Techology, Sheyag, 0870, Chia ABSTRACT: Takig a automotive iterior assembly lie as the research object, a mathematic model is established which iclude the two-side effect as costraits. A improved geetic algorithm is preseted to solve the two-side assembly lie balacig problem. The ecodig ad decodig method is desiged i the algorithm. The fitess fuctio is costructed, the iitial populatio selectio method ad the choice mechaism is determied to fit the algorithm. The crossover ad mutatio method of the populatio is preseted. Furthermore, the umerical result shows that the reductio of the umber of workstatio ad a better balace rate ca be obtaied by proposed improved geetic algorithm which is superior to the ormal oe. INTRODUCTION Two-side assembly lie balacig is uder the requiremets of process coditios, all the operatio elemets which esure to meet the operatio orietatio requiremet assiged to each statio, meawhile the sum of each statio s operatio time ad delay time ca fulfill the requiremets of takt time that try to balace the workstatio s operatio load. I order to solve this problem, the domestic ad foreig scholars already have some studies. Bartholdi firstly put forward two-side assembly lie balacig problem, usig FFR(First Fit Rule) heuristic rules to help productio plaig. The tabu search algorithm was itroduced ito the first category assembly lie balace optimizatio by Ozca, but by limitatio of algorithm itself, both accuracy are ot high. Kim used geetic algorithms to solve assembly lie balacig. Aass proposed a coloy algorithm for solvig two-side assembly, but test solvig eeds a log time. This paper improved geetic algorithm which the search process is oly i feasible solutio space, ad the apply it i the automotive iterior assembly lie balacig problem. 2 GETTING THE ESTABLISHMENT OF THE FIRST CATEGORY TWO-SIDE ASSEMBLY MATHEMATICAL MODEL For the first category of two-side assembly lie balacig problem, it is sought to miimize the 0 umber of workstatios withi give tact time. Assumig that I D tasks set which assiged by operatig orietatio, I={, 2,, N},ad D=L,R,E(which meas a task may be allocated to left, right or either); I j tasks set assiged i workstatio j; T j, R j ad T j ', R j ' are the operatio time ad delay time i workstatio j ad j'; P(i) tasks set before task I;Ns umbers of opeig statios; umbers of opeig positios; X ipk if task I assiged i statio p, operatio orietatio is k, the the value is,if ot is 0. Objective fuctio: mi Subject to the costraits: The task arragemets meet costraits of priority relatioship If, x ipk =,the x hgk =0 () The task must be assiged oly i oe statio p, amog g, p=, 2, 3, ad g>p; i I p= x ipk { L, R} =, k (2) Operatio time ad delay time of each workstatio must meet the takt time, i.e.: Tj = xipk ti i I (3) Here, k is L, j=2p-; T = x t j i I ipk (4) Here, k is R, j'=2p; i
For ay positio p, T j +R j C, ad T j'+ R j' C (5) Task must meet the costrait of assembly operatio orietatio, i.e.: k ( i) (6) = 2 d = L d = R (,2 ) d = E 3 DESIGN OF TWO-SIDE ASSEMBLY IMPROVED GENETIC ALGORITHM 3. Desig of codig I compariso with the oe-side assembly lie, twoside assembly lie balacig eeds to be cosidered ot oly priority work relatioship, but also the operatig orietatio costraits, with the additio that compute the statio time eeds cosider delay time. These all icrease the difficulty of its geetic algorithm desig. So startig from the work priority relatioships diagram, this paper desigs a ew codig method. The work elemets are arraged i a lie accordig to the order of assiged to workstatio. Each work elemet correspods to a gee, chromosome legth is the umber of tasks. z d, z d2, z d z d Chromosome form: { } 2 i i, I this formula the chromosome cosists of gee, z i di is No. i gee ad cosists of two elemets, z i meas No.z i work elemet, d i meas its operatig orietatio. Amog them, the left ad right of assembly lie has two values(l,r). If work elemet ca be allocated to ay side, value of d i will be selected radomly from L or R. 3.2 Iitialize the populatio The iitial populatio is geerated by usig the radom method accordig to the priority of the work elemets. Choose the free elemet which has o work tasks before or have work tasks before but already assiged to the statio ad radomly assiged them, the refresh priority matrix util the assigmet of all the work elemets is fiish. 3.3 Decodig ad calculatig fitess fuctio Accordig to the characteristics of codig, make the chromosomes decodig from left to right, as followig steps: () Judge value of d i which determie the operatig orietatio of elemet z i ad assig z i to the correspodig statio. (2) It eeds to calculate the statio time whe every process is assiged. If the sum of task completio time curret locatio ad the ext 02 distributio process time less tha or equal to the takt time, the the ext distributio process cotiue to assiged to the statio. Otherwise this statio must closed, at the same time ope the ext statio, the task will be assiged to the ext correspodig positio. After ope oe statio, we eed to judge whether it is the last statio, if so, The rest of the process with the same orietatio will be assiged to the statio. (3) Calculate the curret takt time, takt time=max(statio time).compared with origial takt time, if it is greater tha the origial, the calculatio returs the secod step. (4) Circulatio. The step 2ad 3 will be cycled util takt time equal to lower limit takt, it meas decodig is fiished. I the program of fitess fuctio, iput decodig results, ad calculate the fitess fuctio f. 3.4 Selectio (7) This paper uses the classic roulette wheel selectio method. It is a proportioal model, a kid of playback type radom sample calculatio method, through calculatig the fitess of each idividual accouts for the proportio to determie if the remai of its descedat is kept. The fitess of chromosome p is F (p), its selectio probability is: S 3.5 Cross (8) For hadlig chromosomal crossover operatio, the method of two-poit crossover is used. () Read a paret chromosomes F ad F 2 with gee poits from the iitial populatio. (2) Radomly geerated a positive iteger k i [l, ], before k- gees still remai the iformatio of F ad F 2. (3) The gees betwee k ad i make crossig, the last -k+ gees of F will be recodig accordig to the order of F 2, i a similar way, the last -k+ gees of F 2 will be recoded accordig to the order of F. (4) Cross ad geerate the chromosomes C ad C 2 which has the both similarities ad differeces with F ad F 2. 3.6 Variatio Mi f = p = F pop size ( p) / F( q) q= Usig the method of positio variatio, select the chromosome for variatio accordig to the mutatio probability. () Select iitial populatio chromosome F. (2) Radomly geerated a positive iteger k i [l,i].
(3) Before the iformatio of k- gees still remai i F. (4) The last -k+ gees of F will be recodig accordig to the codig method, fially the variatio of the chromosomes C is geerated. 4 APPLICATION EXAMPLES AND ANALYSIS The research object is a iterior trim lie i H50 assembly lie. The curret mothly yield is about 2000 uits, five days with by two shifts produced a week. Accordig to the actual productio situatio, we foud that the productio lie has problems as follows: () The operatio time delay caused by operatig orietatio costraits. (2) Waste of waitig time betwee the processes. (3) Balace efficiecy is i a lower level. Because H50 workshop has bee used for more tha te years, ow the plat eeds to use the old workshop layout process to establish a ew H55 workshop. The target is improvig H50 productio lie ad solvig the problems above, oe of the improvemets is shorte the legth of the lies. First, we divide the work elemets ad cofirm 92 elemets. MTM(method time measuremet) is used to cofirm the work time of these work elemets ad its work time is show i table. Table. Work elemet partitio. No. Time No. Time No. Time No. Time 23 24 5 47 8 70 20 2 5 25 5 48 8 7 20 3 5 26 20 49 2 72 35 4 5 27 35 50 2 73 35 5 5 28 35 5 43 74 35 6 5 29 8 52 32 75 35 7 5 30 8 53 40 76 45 8 20 3 22 54 38 77 87 9 20 32 22 55 29 78 0 0 66 33 30 56 29 79 8 45 34 55 57 25 80 8 2 3 35 5 58 25 8 75 3 64 36 0 59 36 82 75 4 43 37 4 60 53 83 70 5 6 38 6 6 53 84 70 6 54 39 2 62 57 85 89 7 42 40 8 63 0 86 8 8 42 4 6 64 6 87 64 9 74 42 57 65 26 88 64 20 30 43 34 66 90 89 32 2 24 44 75 67 5 90 37 22 26 45 6 68 20 9 20 23 68 46 6 69 20 92 45 I order to realize the populatio iitializatio, it is eeded to establish the correspodig assembly priority sequece chart which is show i figure 2. The circled umber deotes task umber. The umber ad the letter i the bracket above the circle deote operatio time ad operatio orietatio separately. Based o the first category of two-side lie balacig mathematic model, the improved geetic algorithm is used to solve the car iterior trim lie balacig problem. The populatio size is selected as 40 ad the evolutio algebra as 50. Through the programmig software to implemet the algorithm, fially we got the work elemet assembly sequece ad distributio as show i table 2. 03
Pos itio Table 2. Work elemet distributio. The elemet i left statio Left statio time Figure 2. Operatio precedece relatio diagram. The elemet i right statio Right statio time 44 2 4 5 8 0 9 3 6 7 9 29 2 48 20 2 22 23 28 26 33 34 35 44 3 44 24 29 5 2 37 28 25 30 6 36 4 4 4 27 3 7 4 8 32 3 39 40 48 5 43 42 4 57 58 43 44 34 6 35 54 45 55 52 49 47 56 53 5 46 50 48 48 7 46 59 63 67 5 8 47 62 66 76 77 32 9 28 60 70 68 74 7 75 69 6 64 44 0 33 65 72 79 87 73 78 80 85 42 45 8 83 82 84 45 2 34 89 90 9 92 86 88 45 Table 2 shows the assembly sequece of these tasks which are ultimately meet the work task precedece relatio ad orietatio costraits. After optimizatio, the umber of assembly work statios reduced from 5 to 2, the balace efficiecy reached to 93.3%.The effect before ad after optimizatio is show i table 3. Table 3. Before ad after optimizatio. Compariso Before After Effect No. of workstatios 5 2 Decrease 20% Balace efficiecy 74.64% 93.30% Icrease 8.66% Improved GA(IGA) is compared with basic GA. The results of the two optimizatio algorithms are show i figure 3. The IGA reaches almost the same level to the optimal results of basic GA with less umber of iteratios. This is because IGA is improved based o basic GA. 04 Figure 3. Compariso results of two optimizatio algorithms. 5 CONCLUSION Accordig to the characteristics of the first category of two-side assembly lie balacig problem, a correspodig mathematical model is established to describe such a balace problem. At the same time, a improved geetic algorithm is preseted ad applied to solve the automobile iterior trim assembly lie balacig problem. The improved geetic algorithm limits the search process oly withi feasible solutio space ad has less umber of iteratios tha ormal GA. The umerical results for a car iterior trim lie balacig problem show that the umber of work statios is reduced by 3 ad balacig efficiecy is icreased by 8.66%. The results also show that the improved geetic algorithm is feasible to the improvemet of the twoside assembly balace problem ad has high practical applicatio value. It ca be exteded to similar applicatio of two-side assembly areas.
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