Heuristic for Combined Line Balancing and Worker Allocation in High Variability Production Lines

Transcription

1 Heuristic for Cobied Lie Balacig ad Worker Allocatio i High Variability Productio Lies Krisha K. Krisha* [email protected] Wichita State Uiversity, Wichita,, Kasas Shokoufeh Mirzaei Califoria State Polytechic Uiversity, Pooa, Califoria Sathya Madha Solaiuthu Pachaiuthu Wichita State Uiversity, Wichita, Kasas The aufacturig systes are geerally categorized ito achie-itesive ad labor-itesive aufacturig systes. Worker allocatio plays a iportat role i deteriig the efficiecy of a labor itesive aufacturig syste. All the efforts i literature are to the address the worker allocatio proble i a deteriistic coditio. However, i ay real labor-itesive aufacturig systes such as aircraft copaies, there is a high degree of variability i workers processig tie ad quality of perforace. Thus, i this paper a ethod for lie balacig with variability i processig tie is studied ad copared to the raked positioal-weight ethod. The, the odel is exteded further for a siultaeous lie balacig ad worker allocatio i which workers differet processig tie ad quality level coe to play i a ultiple task per statio productio lie. The ai of this paper is to fill the gap betwee the two probles of lie balacig ad worker allocatio i a ucertai eviroet by balacig the risk aog the statios. Case studies were siulated usig QUEST software ad the result idicates that risk based allocatio has icreased throughput ad efficiecy of the aufacturig lie copared to deteriistic worker allocatio. I. INTRODUCTION * Correspodig Author. E-ail address: [email protected] Productio lie is a flow lie aufacturig syste i which the resources i a factory are efficietly orgaized to trasfor raw aterials ito fiished products. I a productio lie, product oves through a set of sequetial value added activities called processes. A set of processes ca for a work statio. To achieve a high efficiet productio lie, all the associated processes should be efficiet. I geeral, aufacturig systes could be classified ito achie-itesive aufacturig ad laboritesive aufacturig (Suer, 1996). I achie itesive aufacturig, productivity of the syste is priarily depeds o the total uber of achies available i the syste. I a achie-itesive aufacturig syste, ivolveet of workers is ofte liited to tasks such as loadig/uloadig parts fro the achies, trasferrig products fro oe statio to other, etc. O the cotrary, i a labor itesive aufacturig syste, perforace of a syste typically is based o worker ivolveet. Workers are equipped with sall, iexpesive equipet s perforig the processes o products. Ofte, productio lies are eployed i a high volue, low variety aufacturig eviroet which are labor-itesive. Jewelry, apparel, leather, ad sport goods aufacturig idustries are good exaples of labor-itesive 47

2 aufacturig syste (Das ad Kalita, 2009). The worker allocatio plays a iportat role i deteriig the efficiecy of a labor-itesive aufacturig syste. Hece, it is ecessary to ipose a greater iportace o worker allocatio pla i these aufacturig systes to obtai a strategic copetitive advatage (Jordo, 1977). Worker allocatio proble is defied as the allocatio of best worker to the process i a aufacturig syste, thereby icreasig perforace of the syste. Traditioally, there are several criteria which worker allocatio plas for based o the. Soe of these criteria are: workers experiece, productivity, seiority, ad ofte soe arbitrary easures (Nebhard, 2001). I a copetitive eviroet, productio efficiecy plays a iportat role i deteriig profitability of a fir. For a syste to achieve the high level of productivity ad efficiecy, workers should be allotted to the processes based o their productivity easures. I the cotext of worker allocatio the doiat productivity easures are processig tie ad quality level of worker associated with the processes. Thus, a worker with the lowest processig tie ad highest quality is ore likely to be allotted to a process. There are three differet scearios for worker allocatio: Sigle worker-sigle process, Sigle worker-multi process, ad Multi worker- Multi process. I Sigle worker-sigle process, workers possess oly oe skill. I Sigle worker-multi process, workers have ultiple skills. The perso selected as the best worker oves betwee ultiple processes ad perfors the operatios. It is ore coo i U-Shaped productio lies. I Multi worker- Multi process, ultiple workers have ultiple skills such that skills overlap withi workers. Thus, worker with highest level of productivity for a particular task is ore likely to be selected. Worker profile could be defied as the operatioal characteristics of idividual workers associated with processes. Based o the previous literature, worker allocatio could be classified ito: 1) Sole profile allocatio 2) Multi profile allocatio I sole profile allocatio, all workers with siilar skills are assued to have sae productivity easures. The differeces i productivity, as a result of iheret variability associated with workers, are ot cosidered. Several forulatios were developed to solve sole profile worker allocatio odel ad are show i Table1. Wittrock (1992) odeled a sole profile worker assiget proble as a etwork flow proble. The lexicographic objective fuctio developed by Wittrock is to axiize the capacity i a achie itesive aufacturig syste. Table 1. Forulatios - Sole Profile Allocatio FORMULATIONS AUTHORS Mixed iteger prograig Kuo ad Yag (2005), Suer ad Bera (1998), Davis ad Mabert (2000), Mi ad Shi (1993) ad Suer (1996) Heuristic Vebu ad Sriivasa (1997), Bhaskar ad Sriivasa (1997), ad Nakade ad Oho (1999) Network flow proble Wittrock, 1992 Data evelopet aalysis Ertay ad Rua, 2005 No-liear prograig Davis ad Mabert, 2000 Vebu ad Sriivasa (1997) developed a heuristic approach for worker allocatio ad product sequecig i productio lies. The objective fuctio itroduced by Vebu ad Srivasa is to iiize the akespa. Bhaskar ad Sriivasa (1997) proposed a heuristic algorith to solve the static ad dyaic worker allocatio proble. Suer (1996) preseted a two- 48

3 stage hierarchical ethod which siultaeously solves the probles of worker allocatio ad cell loadig i a labor itesive aufacturig syste. Suer ad Bera (1998) developed a odel for the worker allocatio proble whe lotsplittig betwee cells are allowed. They also cotributed the setup ties i the odel. The odel preseted by Suer ad Bera is a extesio of previous work by Suer (1996). Kuo ad Yag (2005) ipleeted ixed iteger forulatio developed by Suer ad Bera (1998) for operator staffig level decisios i a TFT- LCD ispectio ad packagig (I/P) process. Davis ad Mabert (2000) developed two atheatical odels for akig order dispatchig ad worker assiget decisios i a liked cellular aufacturig syste. Ertay ad Rua (2005) preseted a approach for fidig the optial uber of worker allocatio i cellular aufacturig by applyig the data evelopet aalysis. Nakade ad Oho (1999) proposed a heuristic for eetig dead i a U- shaped productio lie by optially selectig the iiu uber of workers which iiizes overall cycle tie. I all the research papers discussed above, it is assued that workers possess equal productivity. However, i practice, variability i worker productivity is predoiat. Therefore, i a ulti profile allocatio proble, worker differeces i ters of productivity are cosidered. The worker profile differece based o idividual workers is ore realistic assuptio which is also cosidered i this paper. Worker profile differeces are odeled based o ultiple skill levels of idividual workers. I literature several forulatios for ulti profile worker allocatio probles were forulated which soe are show i Table 2. TABLE 2. FORMULATIONS- MULTIPLE ALLOCATION FORMULATIONS AUTHORS Mixed iteger prograig Aski ad Huag (2001), Chaves, Isa, ad Lorea (2007), Miralles et al., (2008), Suer ad Tualuri (2008), Fitzpatrick ad Aski (2005), ad Nora et al., (2002) Heuristics Fowler, Wirojaagud, ad Gel (2008) ad Nebhard (2001) Noliear iteger prograig Aryaezhad, Deljoo, ad Al-e-Hashee (2009) ad Heierl ad Kolish (2009) Particle swa optiizatio techique Yaakob ad Watada (2009) Aski ad Huag (2001) developed a ixed iteger, goal prograig odel to for worker teas based o psychological, orgaizatioal ad techical factors. Worker teas were assiged to cells ad further workers were assiged to tasks i the cells. Traiig schedule for the workers was also a output of the odel developed. Mixed iteger odel developed was solved i various solutio ethods like greedy search heuristic, filtered bea search heuristic ad siulated aealig techique. Filtered bea search heuristic was capable of producig ear-optial solutio with a reasoable coputatioal tie. Fitzpatrick ad Aski (2005) fored ultiple worker teas with ultifuctioal skill requireets for cellular aufacturig. The techical ad iter persoel skills of workers were cosidered to for worker teas. The objective was to iiize the deviatio fro optial kolbe idices across all the worker teas. A heuristic was also proposed to solve ad evaluate the sae proble. Chaves, Isa, ad Lorea (2007) odeled a iteger prograig forulatio for assebly lie worker assiget ad balacig proble (ALWABP) i sheltered work ceters. The odel assued that idividual workers have 49

4 differet deteriistic processig tie values due to iheret variability betwee workers. Objective of the Iteger Prograig (IP) odel was to iiize the cycle tie by assigig task to statio ad workers to task siultaeously. Clusterig search approach was used to solve the IP odel. Miralles, Garcia, Adres, ad Cardos (2008) exteded the previous work by Chaves, Isa, ad Lorea (2007) by providig differet solutio ethodology usig brach ad boud algorith. Suer ad Tualuri (2008) developed a three stage operator allocatio procedure for worker allocatio i labor itesive aufacturig. The first stage was to geerate alterative worker staffig level cofiguratios for all the products. The secod stage was to siultaeously fid optial uber of workers i cells ad cell loadig by assigig parts to cells. Two kids of heuristics were proposed i the third step, aely, Max ad Max Mi to assig workers to task. They had assued that the processig tie of idividual workers were differet deteriistic variables depedig o the skill levels they possess. Operator skill levels were subject to chage based o learig ad forgettig ature which was deteried by the uber of ties a operator was allotted to a particular operatio. No-Liear Iteger prograig was also a extesive techique used to solve ulti profile operator allocatio proble. Aryaezhad, Deljoo, ad Al-e-hashee (2009) forulated a oliear iteger progra for Siultaeous Dyaic Cell foratio ad Worker assiget Proble (SDCWP). Workers ability was classified as differet skill levels. The odel siultaeously decided the uber of achies to be purchased, relocated (or) to be reoved i each cell, uber of workers to be hired, fired, traied, ad allocatio of workers to achies for each period. Heierl ad Kolish (2009) odeled a o-liear iteger progra for assigig ultiskilled workers to tasks cosiderig the worker learig, forgettig ad copay skill level target. Proble was solved usig COIN-OR s Ipopt. McDoald, Ellis, Ake, ad Koellig (2009) proposed a atheatical odel to assig cross traied workers to tasks i a lea aufacturig cell. The objective of the odel was to iiize the et preset cost. It was assued that the processig tie ad quality level of workers were differet deteriistic values based o curret skill depth level of workers. The output of odel also idetified traiig requireets ad job rotatio for workers. Yaakob ad Watada (2009) developed a ethodology for worker assiget i cellular aufacturig usig particle swar optiizatio techique. The research cosidered ultifuctio workers ad evaluated relatioship betwee workers ad workers to task. Evaluatio betwee workers was perfored cosiderig social, perforace ad etal factors. Heuristic procedures were developed to reduce the coplexity i atheatical odels whe the proble size icreased. Nebhard (2001) developed a heuristic approach for worker allocatio based o idividual worker learig profiles. Worker task allocatio was exaied for log ad short productio rus ad it was observed that the learig profile based allocatio icreases productivity of the syste. Siulatio was doe obtaiig idividual worker profile epirical data of learig/forgettig fro the idustry. Fowler, Wirojaagud, ad Gel (2008) developed a ixed iteger progra for akig decisios o hirig, traiig ad firig workers. Idividual workers were assued to be differet based o their geeral cogitive ability. The objective was to iiize the total cost due to hirig, traiig, firig ad issed productio costs over ultiple periods. A geetic algorith ad two LP based heuristics were preseted to solve the proble. Worker allocatio is priarily based o the processig tie ad quality level of workers. All the researches i previous literature assue 50

5 deteriistic processig tie ad quality level of workers. A chage i processig tie or quality level of ay worker will result i a odified relatioship betwee the worker ad the process, thus affectig the optial worker allocatio. Therefore, i real world scearios, ucertaity is predoiat i worker processig tie ad quality level ad a optial worker allocatio ethod should take ito cosideratio the associated ucertaities. Whe there is a ucertaity associated with worker allocatio process, it is essetial to assess ad quatify the risk associated with the workers. Accordig to Modarres (2006), risk is defied as a easure of the potetial loss occurrig due to atural or hua activities. I the cotext of worker allocatio, risk ca be defied as the potetial loss due to delay i process or due to bad quality of the product. Whe variability i processig tie ad quality level icreases, risk due to delay i process ad bad quality product also icreases. I order to reduce risk i worker allocatio process, advaced techiques have to be developed which captures the risk ad iiizes its ipact. Aog all the pre-existig literature there is o itegrated forulatio which balaces the risks aog the work statios by a optial worker allocatio pla. Hece, the ai of this paper is to fill the gap betwee the two probles of lie balacig ad worker allocatio i a ucertai eviroet by balacig the risk aog the statios. The risk is i fact a result of differece aog the workers skill i doig the sae process. The differece causes the varyig processig ties ad qualities of tasks perfored. Cosequetly, there will be a risk of havig a work statio total process tie or quality ore tha the other statios. I the forulatios provided i this paper the objectives fuctio is to iiize the probability of such icideces as well as iiizig the differece betwee the total processig ties ad qualities by a optial worker allocatio sceario. I sectio 2 of this paper a risk based ethodology for lie balacig for ultiple tasks per statio sceario is preseted. I the forulatio preseted i paper 2 the risk balacig sceario happes by cosiderig the ucertai processig ties ad with a pre-kow uber of statios. However, with a kow uber of statios the proble ight fall i a ifeasible area. To avoid the ifeasibility a iterative algorith for fidig the optial uber of statios is provided. Validatio of the proble by coparig the result with a siulatio is ivestigated. I sectio 3 ucertai qualities as well as the ucertai processig ties coes to play for the purpose of risk balacig by worker allocatio. A atheatical forulatio ad a siultaeous approach for the risk based lie balacig ad worker allocatio is proposed i this sectio. A algorith for fidig the optial uber of statio is preseted. Sectio 4 provides the results ad coclusios. II. RISK BASED LINE BALANCING I Krisha et al. (2011) risk based worker allocatio ethod for sigle task per statio balaced/ubalaced productio lie scearios were preseted. The risk based worker allocatio ethod ais to allocate the workers to the processes by iiizig the overall risk. I a productio lie with ultiple tasks perfored at each statio, the products flow i a sequetial order through the workstatios. Each workstatio has a uber of tasks allocated to it. The perforace of a ultiple task productio lie priarily depeds upo balacig the workload betwee statios ad the quality of workers allotted to the workstatios. Thus, i this sectio, a risk based lie balacig proble is cosidered. Further, the risk based lie balacig is exteded ito siultaeous risk based lie balacig ad worker allocatio proble i sectio 3 for ultiple tasks per statio sceario. The otatios used for paraeters i the risk based lie balacig approach are as follows: P j Mea processig tie of task j 51

6 P σj Stadard deviatio i processig tie of task j STP j Stadard processig tie of task j T Tie iterval S i Miiu uber of statios S ax Maxiu uber of statios D Dead durig tie iterval T RBP Risk balacig pealty LBP Lie balacig pealty Pr fg Biary precedece variable that task f has to be doe before task g Alos, i the forulatio L jk are biary decisio variables which deterie whether task j allocated to statio k (Biary variable) or ot. The assuptios for risk based lie balacig are listed below: Processig tie of each task is kow ad follows oral distributio Stadard processig tie (target processig tie) for each task is kow A sigle product dedicated ultiple task productio lie is cosidered Dead is deteriistic ad is kow for each period I the above assuptios, the processig tie is the tie take by a labor to perfor a task, whereas the stadard processig tie is the target processig tie. The proposed Risk Based Lie Balacig (RLB) odel required the followig iput paraeters for testig the approach. 1) Mea processig ties of all tasks 2) Stadard deviatio i processig tie for all tasks 3) Expected processig tie for each task 4) Precedece costraits for all tasks 2.1 Matheatical Model A No Liear Iteger Prograig (NLIP) odel is developed for solvig risk based lie balacig proble. The objective of the proposed odel is to balace the workload ad processig tie risk betwee statios. A set of Equatios i 1-4 is defied to siplify the objective fuctio. 1 S SP k k RP 1 1 erf k K k 2 SP 2 k SP P L k K k j jk j k j jk j 1 (1) (2) SP P L k K (3) S STP L k K k j ijk j 1 (4) Equatio 1 calculates the probability that the total process tie of the task assiged to statio k is ore tha the stadard process tie of the statio. Equatio 3 is the suatio of processig tie eas for the task assiged to statio k. Equatio 3 is the suatio of processig tie variaces for the tasks assiged to statio k. Equatio 4 represets the stadard processig tie of statio k. Usig these equatios the objective fuctio ad costraits for the ultiple tasks per statio sceario are forulated as follows: Objective Fuctio z z Mi RP RP RBP SP SP LBP k l k l k 1l k 1 Subject to: 52 z k 1 j 1 L jk L jk 1 j J L 1 k K jk z L hk gh (1 P rg ) M k K (5) (6) (7) (8) The objective fuctio (Equatio 5) iiizes the differece i risk ad workload betwee statios by allocatig task j to statio k. The first ter i objective fuctio iiizes the differece i processig tie risk betwee statios. The secod ter i the objective fuctio balaces the work load betwee statios by iiizig the differece i ea workload

7 betwee statios. The costraits for the risk based assebly lie balacig proble are show i 6, 7, ad 8. Costrait 6 esures that a task ca oly be allotted to oe statio. Costrait 7 esures that a statio will have a iiu of oe task. Costrait 8 esures that the precedece relatioships withi tasks are et. The biary precedece variable Pr fg restricts the task assiget to workstatio. Pr fg =1 represets that task f has to be allocated before task g Case Study 9-Task-3-Statio A ultiple task per statio productio lie with 9 tasks ad 3 statios obtaied fro a aerospace aufacturig copay is used as a saple case study to illustrate the proposed RLB odel. The expected values of processig tie for 9-task, 3 statio case study is show i Table 3. TABLE 3. EXPECTED PROCESSING TIME (MINUTES) FOR 9-TASK-3-STATION CASE STUDY Task Expected Processig Tie Actual operatio ties of all the tasks i the 9 task - 3 statio case study is show i Table 4. The actual task processig tie values are obtaied fro previous historical tie study. TABLE 4. PROCESSING TIME (MINUTES) CHART FOR 9-TASK 3-STATION CASE STUDY Task Processig Tie 13± ± ± ± 0.94 The sequece of tasks are show usig a precedece atrix. The biary precedece 14.4± ± ± ± ± 0.96 variable Pr fg, for 9 task 3 statio case study is show below: Prfg The tie iterval (T) is set to 480 hours. Miiu uber of statios (Si) is set as 3 ad the dead is assued as 700 products. Risk balacig pealty ad lie balacig pealties are $1000 for the 9 task, 3 statio case study. Oce the iput variables are obtaied, NLIP odel is solved ad the results obtaied are show i Table 5. TABLE5. RESULTS 9-TASK 3-STATION-CASE STUDYY-RISK BASED ALLOCATION Statio Statio 1 Statio 2 Statio 3 Task 4 Task 3 Task 1 3 Statio Task 7 Task 5 Task 2 Task 9 Task 8 Task 6 53

8 2.3 Validatio I order to validate the risk based lie balacig approach, a deteriistic lie balacig is perfored usig rak positioal weight ethod for the case study preseted of sectio 2.2 ad results are show i Table 6. TABLE 6. RESUTLS 9-TASK 3-STATION-CASE STUDY-DETERMINISTIC ALLOCATION STATION Statio 1 Statio 2 Statio 3 3 STATION Task 8 Task 2 Task 7 Task 9 Task 1 Task 5 Task 4 Task 6 Task 3 Siulatio is selected as the tool to copare deteriistic ad risk based allocatio. Siulatio is coducted usig Delia QUEST V5 R18 software. Siulatio is ru for 480 hours. Sice the processig tie tasks are assued to follow oral distributio, a sigle ru ay ot be sufficiet eough to eliiate radoess i output. Hece, the odel is replicated several ties. I siulatio, a certai uber of replicatios ca lead to a ore reliable result. Hece, to obtai the proper uber of replicatios for the proble Equatio9 is used. N r t 2 / 2, 1 2 h 2 (9) The optial results obtaied fro the LINGO optiizer is copared with the outputs fro the siulatio ad it is show i Table 5. TABLE 7. COMPARISON BETWEEN DETERMINISTIC AND RISK BASED METHOD Descriptio Deteriistic Method Risk based Method Percet Iproveet Throughput Value Added Tie No-Value Added Tie Figure 1 shows a copariso of each statio cycle tie betwee deteriistic ad risk based lie balacig ethodology. Thus, fro the above table ad chart, it is evidet that the risk based lie balacig produces a better balace betwee the work statios i ters of tie take to coplete the tasks assiged to the statios. This, results i a icreased throughput of 5.54% ad siultaeously value added tie is also iproved. I additio, fro Table 7 it is clear that the throughput caot eet custoers dead of 700. I this exaple, it is assued that the uber of statios is fixed. However, i the followig sectio this assuptio is relaxed to icrease the throughput to eet the custoers dead. 2.4 Optiu Nuber of Statio Deteriatio The throughput of a ultiple tasks per statio productio lie depeds o the cycle tie of the statios. Whe the uber of statios i productio lie icreases/decreases, the cycle tie of the statios also icreases/decreases respectively. Thus, it is also vital to ake decisios o uber of statios whe balacig a ultiple tasks per statio productio lie. A algorith to ake decisios o uber of statios whe balacig a productio lie is preseted below. 54

9 Cycle Tie Krisha, Krisha K., Mirzaei, Shokoufeh ad Pachaiuthu, Sathya Madha Solaiuthu FIGURE 1. CYCLE TIME COMPARISON CHART Cycle Tie Statio 1 Statio 2 Statio 3 Deteriistic Risk Algorith for optial uber of statio deteriatio: STEP 1: Start STEP 2: Set Z = iiu uber of statios ( ) STEP 3: Ru LINGO odel to obtai optial result STEP 4: Deterie throughput by siulatio STEP 5: If Throughput Dead go to STEP 7 STEP 6: Set Z= Z+1 go to STEP 3 STEP 7: Stop A flowchart for optial uber of statio deteriatio i a assebly lie is show i Figure 2. S i Case Study 9 Task To test the proposed ethodology for optial uber of statio deteriatio, sae 9 statio case study is selected. Dead is assued as 700 products ad iiu uber of statios (S i ) is set as 3. The RLB LINGO odel is solved for 3 statios ad results are show i Table 8. Siulatio is perfored with the obtaied result to fid the average throughput. The siulatio is replicated 6 ties for the 3 statio case. Tie iterval for siulatio is assued to be 480 hours. The average throughput obtaied for 3 statios-9 task case study is Sice the dead of 700 products could ot be et by 3 statios, uber of statios is icreased by oe ad RLB odel is solved for optiality. Oce the iput variables were obtaied, NLIP odel was solved ad the results are show i Table 9. Siulatio is perfored with the obtaied allocatios of workers ad tasks to fid the average throughput. The siulatio is replicated 6 ties for the 4 statio case. Tie iterval for siulatio is assued to be 480 hours. The average throughput obtaied for 4 statios-9 task case study is Thus dead of 700 products could be et by 4 statios. Hece, the optial uber of statios for the case study is 4. III. SMIMULTANIOUES RISK-BASED LINE BALANCING AND WORKER ALLOCATION I sectio 2, risk based lie balacig approach was preseted. The risk based lie balacig approach ais to allocate tasks to workstatios by balacig the risk ad ea processig tie betwee statios. The perforace of a ultiple tasks per statio productio lie priarily depeds upo 55

10 FIGURE 2. FLOW-CHART FOR OPTIMAL NUMBER OF STATION DETERMINATION TABLE 8. RESUTLS 9-TASK 3-STATION-CASE STUDY STATION Statio 1 Statio 2 Statio 3 Avg. Throughput 3 STATION Task 4 Task 7 Task 9 Task 3 Task 5 Task 8 Task 1 Task 2 Task TABLE 9. RESULTS-TASK 4-STATION-CASE STUDY Statio Statio 1 Statio 2 Statio 3 Statio 4 4 Statios Task 7 Task 9 Task 4 Task 8 balacig the workload betwee statios ad the quality of workers allotted to the workstatios.thus, a approach for siultaeously balacig the workstatio ad allocatig best worker to the workstatio is preseted i this sectio. A sigle odel u- Task 1 Task 2 Task 3 Task 5 Task 6 Avg. Throughput paced asychroous productio lie is cosidered i this sectio. The otatios used for iput paraeters i the siultaeous risk based lie balacig ad worker allocatio is as follows: 56

11 P ij Processig tie of worker i for task j P σij Stadard deviatio i processig tie of labor i for task j Qij Quality level of worker i for task j Q σij Stadard deviatio i quality level of worker i for task j SP j Stadard processig tie of task j SQ j Stadard quality level of task j DP k Delay pealty for statio k QP k Quality pealty for statio k C ij Cost of worker i for task j T Tie iterval S i Miiu uber of statios S ax Maxiu uber of statios D Dead durig tie iterval T RBP Risk balacig pealty LBP Lie balacig pealty Pr fg Biary precedece variable that task f has to be doe before task g V ik Biary variable equals 1 if labor i is allotted to statio k, 0 otherwise. Also, L ijk are biary decisio variables which defie whether Worker i is allocated to task j i statio k or ot. The assuptios for siultaeous risk based lie balacig ad worker allocatio are listed below: Processig tie of each worker is kow ad follows oral distributio Quality level of each worker is kow ad follows oral distributio Stadard processig tie of each task is kow Stadard quality level of each task is kow A worker ca oly be allocated to a sigle statio A statio has oly oe worker I the above assuptios, the processig tie is the tie take by a labor to perfor a task, whereas the stadard processig tie is the target processig tie. The proposed Siultaeous Risk Based Lie Balacig ad Worker Allocatio (SRLW) odel requires the followig iput paraeters for testig the approach: 1) Processig ties of all workers to their respective task 2) Quality level of all workers to their respective task 3) Cost of workers for each task 4) Expected processig tie for each task 5) Expected quality level for each task 6) Delay pealty associated with each statio 7) Quality pealty associated with each statio 8) Precedece costraits for all tasks 3.1 Matheatical Model A No Liear Iteger Prograig (NLIP) odel is developed for solvig SRLW proble. The proposed odel balaces the workload, risk betwee statios, ad allocates the best worker to the statio siultaeously. Equatios 10 through 18 are defied to siplify the objective fuctio. 1 S SP k k RP 1 1 erf k K k 2 SP 2 k 1 Q SQ 1 k k RQ erf k K k 2 SQ 2 k SP P L k K k ij ijk j1 i1 2 2 SP P L k K k ij ijk j1 i1 SQ Q L k K k ij ijk j1 i1 2 2 SQ Q L k K k ij ijk j1 i1 k j ijk j1 i1 (10) (11) (12) (13) (14) (15) S STP L k K (16) 57

12 Q STQ L k K k j ijk j1 i1 T CL k C L k K ij ijk j1 i1 60 (17) (18) Equatio 10 calculates the probability that the total process tie of the task assiged to statio k is ore tha the stadard process tie of the statio. Equatio 11 shows the probability that the total quality of the task perfored i statio k is less tha the stadard quality. Equatio 12 is the suatio of processig tie eas for the task assiged to statio k. Equatio 13 is the suatio of processig tie variaces for the tasks assiged to statio k. Equatio 14 is the suatio of workers ea quality i perforig the tasks assiged to statio k. Equatio 15 is the suatio of workers variaces i quality while perforig the tasks assiged to statio k. Equatio 16 represets the stadard processig tie of statio k. Equatio 17 represets the stadard quality level of statio k. usig these equatios the objective fuctio ad costraits are forulated as follows: Objective Fuctio z z Mi RP DP RQ QP CL k k k k k RP RP RBP RQ RQ RBP SP SP LBP k l k l k l k 1 l k 1 Subject to: ijk ijk ik j 1 i1 j 1 L L V i I, k K (19) (20) V 1 k K (21) ik i1 z V 1 i I (22) ik k 1 z i1 k1 i1 j1 L 1 j J ijk (23) L 1 k K ijk (24) i1 L igk k i1 h1 L ifh (1 pr gh ) M k K The objective fuctio (Equatio 19) iiizes the risk due to worker allocatio, the differece i workload betwee statios, ad the differece i risk betwee statios by allocatig worker i to task j ad task j to statio k. The first ter i the objective fuctio iiizes the risk due to statio processig tie, which is the product of probability i exceedig the expected processig tie ad statio delay pealty. The secod ter iiizes the risk due to statio quality level, which is the product of probability i quality level below expected quality level ad statio quality pealty. The third ter iiizes the cost of workers. The fourth ter iiizes the differece i processig tie risk betwee statios. The fifth ter iiizes the differece i quality risk betwee statios. The sixth ter balaces the work load betwee statios by iiizig the differece i ea workload betwee statios. Costraits 20 ad 21 esure that oly oe worker ca be allocated to a statio. Costrait 22 esures that a worker ca oly be allotted to a axiu of oe statio. Costrait 23 esures that a task ca oly be allotted to oe worker ad oe statio. Costrait 24 esures that at least oe task is allocated to a statio. Costrait 25 esures that the precedece relatioships withi tasks are ot violated. Statio delay pealty (DP k ) is the pealty icurred if the processig tie of a certai statio exceeds the expected statio processig tie. Statio delay pealty will chage as the criticality of the statio chages. Statio quality pealty (QP k ) is the pealty icurred if the quality level of a certai statio is below the expected statio quality level. The biary precedece variable Pr fg restricts the task assiget to workstatio. Pr fg =1 represets that task f has to be allocated before task g. (25) 58

13 3.2 Case Study 9 Task 6 Worker A ultiple tasks per statio productio lie with 9 task ad 6 workers is used to illustrate the proposed odel. The data for this case study is obtaied for a aerospace aufacturig copay. The expected values of processig tie ad quality level for all the tasks i the case study are show i Table 10. Actual operatio ties of all the workers i the case study are show i Table 11. The worker processig tie values are obtaied fro previous historical tie study. TABLE 10. EXPECTED PROCESSING TIME AND QUALITY LEVEL CHART FOR 9- TASK 6-WORKER CASE STUDY Task Processig Quality Tie (iute) TABLE 11. OPERATION TIME (MIN) CHART FOR 3-STATION-9-TASK 6-WORKER CASE STUDY * Task 1 Task 2 Task 3 Task 4 Task 5 Task 6 Task 7 Task 8 Task ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±0.50 *Each row is associated with a worker Actual quality level of all the workers i the case study is show i Table 12. The worker quality level values are obtaied fro previous historical tie study. The cost for all workers associated with their respective processes i case study is give i Table 13. TABLE 12. QUALITY LEVEL CHART FOR 3-STATION-9-TASK 6-WORKER CASE STUDY * Task 1 Task 2 Task 3 Task 4 Task 5 Task 6 Task 7 Task 8 Task ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±0.22 * Each row is associated with a worker 59

14 TABLE 13. WORKER COST/HOUR ($) CHART FOR 3-STATION-9-TASK 6-WORKER CASE STUDY Task /Worker Task 1 Task 2 Task 3 Task 4 Task 5 Task 6 Task 7 Task 8 Task 9 Worker /hr 10.38/hr 10.10/hr 11.73/hr Worker /hr 10.41/hr 13.41/hr 13.95/hr Worker /hr 13.67/hr 11.22/hr 10.71/hr Worker /hr 11.63/hr 12.81/hr 11.80/hr Worker /hr 12.50/hr 13.93/hr 8.73/hr Worker /hr 12.66/hr 10.91/hr 13.36/hr Delay ad quality pealty costs of statios for the case study are preseted i table Prfg The biary precedece variable Pr fg is show below: TABLE 14. DELAY AND QUALITY PENALTY COSTS OF STATION FOR 9-TASK-6 WORKER CASE STUDY statio (k) Delay pealty (DP k ) $450 $550 $500 Quality pealty (QP k ) $375 $350 $400 The tie iterval (T) is set to 2880 iutes. Miiu uber of statios (S i ) is set at 3 ad the dead is assued to be 70 products. Risk balacig pealty ad lie balacig pealties are $1000 for the 9-task-6 worker case study. The output fro the LINGO Optiizer 12.0 is obtaied ad siulatio is coducted usig Delia QUEST V5 R18 software. Sice the processig tie ad quality level of workers are assued to follow oral distributio, a sigle ru ay ot be sufficiet to eliiate radoess i output. Hece, the odel is replicated several ties. I siulatio, a certai uber of replicatios ca lead to a ore reliable result. Hece, to obtai the proper uber of replicatios for the proble Equatio 26 is used. 2 2 N t (26) r / 2, 1 2 h Table 15 shows the result obtaied fro the LINGO optiizer. Fro Ligo the throughput for 2880 iutes is calculated to be products. 60

15 TABLE 15. RESULTS 3-STATION-9-TASK 6-WORKER CASE STUDY STATION Statio 1 Statio 2 Statio 3 Throughput Task 2 Task 4 Task 1 3 STATION TASKS Task 7 Task 5 Task 3 Task 8 Task 9 Task WORKER Worker 6 Worker 3 Worker 5 Sice the average throughput obtaied for 3 statios is ot eough to eet the dead of 70 products, the uber of statios is icreased by oe ad the sae case study is solved i LINGO optiizer for 4 statios. The LINGO odel for 4 statio case is show below. The odel for the 4 statio case is solved usig LINGO Optiizer 12.0 ad the optial result for 4 statio case was obtaied. The uber of replicatios is calculated usig the replicatio size forula etioed i Equatio 26. The 4 statio case was siulated usig Delia QUEST software for 11 replicatios ad the average throughput for 2880 iutes is products (Table 16). TABLE 16. RESULTS 4-STATION-9-TASK 6-WORKER CASE STUDY Statio Statio 1 Statio 2 Statio 3 Statio 4 Throughput Task 1 Task 5 Task 3 Task 6 TASKS Task 2 4 Statio Task 7 Task 4 Task Task 9 WORKER Worker 2 Worker 6 Worker 1 Worker 3 Sice the average throughput obtaied for 4 statios is ot eough to eet the dead of 70 products, the uber of statios is icreased by oe ad the sae case study is solved i LINGO optiizer ad the optial result for 5 statio case is obtaied. The uber of replicatios is calculated usig the replicatio size forula etioed i Equatio 26. The 4 statio case is siulated usig Delia QUEST software for 25 replicatios ad the average throughput for 2880 iutes is products (Table 17). Sice the 5 statio cofiguratio s throughput is greater tha the dead of 70 products, the 5 statio is the optial uber of statios for the 9-tasks-6-worker case study. TABLE 17. RESULTS 5-STATION-9-TASK 6-WORKER CASE STUDY STATION Throughput 5 Task 7 Task 2 Task 3 Task 1 Tasks Task 8 STATION Task 9 Task 5 Task 6 Task Worker Worker 6 Worker 3 Worker 2 Worker 4 Worker 5 IV. CONCLUSION I sectio 2, the ecessity of risk based lie balacig i ultiple tasks per statio sceario was preseted. A MINLP optiizatio ethodology was proposed which allocates tasks to the statios by balacig the risk ad the ea processig ties betwee statios. The proposed approach is validated agaist a deteriistic approach (Rak Positioal Weight Method) ad a 5.6 percet iproveet i throughput was observed. A ethodology to deterie the optial uber of statios i a assebly lie was also preseted. Efficiecy of a ultiple tasks per statio productio lie depeds o lie balacig ad worker allocatio. Hece i sectio 3, the proposed risk based lie balacig approach was exteded to a siultaeous risk 61

16 based lie balacig ad worker allocatio proble. Moreover, after explaiig the ecessity for a siultaeous approach for balacig assebly lie ad worker allocatio was preseted. The two kids of risks icludig the processig tie risk ad the quality level risk were itroduced. Processig tie risk was the icrease i delay pealty cost due to ucertaity i worker processig tie. Quality risk was the icrease i quality pealty cost due to ucertaity associated with worker quality level. The proposed approach siultaeously iiizes the risk due to worker, the differece i risks betwee statios ad the differece i ea workload of statios. A ethodology to deterie the optial uber of statios i a assebly lie was also preseted. No Liear Iteger Prograig (NLIP) odel, which siultaeously balaces ad allocates the best worker to the workstatio i a ultiple tasks per statio productio lie has bee developed. The proble cosidered for this research is a sigle product proble. The ethod for balacig product lies ad assigig workers whe ultiple products are ivolved are tedious ad curretly there are o ethods that ca address this issue. The curret research could be used a startig poit to developig ulti-product lie-balacig ad worker allocatio issues. The ethod used i this research is to develop optiizatio approaches to the worker allocatio proble. The largest case study that was coducted i this research is with 5 processes ad 15 workers. Sice, the forulatio is a NLIP, as the proble size icreases, the forulatio ad coputatioal tie icreases expoetially. Hece, ew heuristics that ca adapt to large size probles have to be developed. Soe of the heuristics that ca be applied for larger size probles are geetic algoriths, tab search, ANT Coloy algoriths etc. The efficiecy of each of these heuristics ad the ipleetatio should be ivestigated. V. REFERENCES Aryaezhad, M, B., Deljoo, V., ad Al-ehashe, S, M, J, M., (2009), Dyaic cell foratio ad worker assiget proble: a ew odel, Iteratioal Joural of Advaced Maufacturig Techology, Vol. 41, No. 3-4, pp Aski, R, G., ad Huag, Y., (2001), Forig effective worker teas for cellular aufacturig, Iteratioal Joural of Productio Research, Vol.39, No. 11, pp Bhaskar, K., ad Sriivasa, G., (1997), Static ad dyaic operator allocatio probles i cellular aufacturig systes, Iteratioal Joural of Productio Research, Vol. 35, No. 12, pp Chaves, A, A., Isa, C, M., ad Lorea, L, A., (2007), Clusterig search approach for the assebly lie worker assiget ad balacig proble, Proceedigs of the 37 th Iteratioal Coferece o Coputers ad Idustrial Egieerig,-Alexadria, Egypt., pp Davis, D. J., ad Mabert, V. A., (2000), Order dispatchig ad labor assiget i cellular aufacturig systes, Decisio Scieces, Vol. 31, No. 4, pp Ertay, T., ad Rua, D., (2004), Data evelopet aalysis based decisio odel for optial operator allocatio i CMS, Europea Joural of Operatioal Research, Vol.164, No. 3, pp Fitzpatrick, E, L., ad Aski, R, G., (2005), Forig effective worker teas with ultifuctioal skill requireets, Coputers ad Idustrial Egieerig, Vol. 48, No. 3, pp Fowler, J, W., Wirojaagud, P., ad Gel, E., (2008), Heuristics for workforce plaig with worker differeces, Europea Joural of Operatioal Research, Vol. 190, No. 3, pp Heierl, C., ad Kolisch, R., (2009), Work assiget to ad qualificatio of ulti- 62

17 skilled hua resources uder kowledge depreciatio ad copay skill level targets, Iteratioal Joural of Productio Research, Vol. 48, No. 13, pp Kuo, Y., ad Yag, T., (2006), A case study o the operator allocatio decisio for TFT-LCD ispectio ad packagig process, Joural of Maufacturig Techology Maageet, Vol. 17, No. 3, pp Mcdoald, T., Ellis, K, P., Ake, E, M, V., ad Koellig, C, P., (2009), Developet ad applicatio of a worker assiget odel to evaluate a lea aufacturig cell, Iteratioal Joural of Productio Research, Vol.47, No. 9, pp Mi, H., ad Shi, D., (1993), Siultaeous foratio of achie ad hua cells i group techology: a ultiple objective approach, Iteratioal Joural of Productio Research, Vol. 31, No. 10, pp Miralles, C., Garcia, J, P., Adres, C., ad Cardos, M., (2008), Brach ad boud procedures for solvig the assebly lie worker assiget ad balacig proble. Applicatio to sheltered work ceters for disabled, Discrete Applied Matheatics, Vol. 156, No. 3, pp Nakade, K., ad Oho, K., (1999), A optial worker allocatio proble for a U-shaped productio lie, Iteratioal Joural of Productio Ecooics, Vol , pp Nebhard, D, A., (2001), Heuristic approach for assigig worker to tasks based o idividual learig rates, Iteratioal Joural of Productio Research, Vol. 39, No. 9, pp Nora, B, A., Tharaphorphilas, W., Needy, K, L., Bidada, B., ad Warer, R, C., (2002), Worker assiget i cellular aufacturig cosiderig techical ad hua skills, Iteratioal Joural of Productio Research, Vol. 40, No. 6, pp Suer, G, A., ad Tualuri, R, R., (2008), Multi-period operator assiget cosiderigskills, learig ad forgettig i labour itesive cells, Iteratioal Joural of Productio Research, Vol. 46, No. 2, pp Suer, G. A., (1996), Optial operator assiget ad cell loadig i laboritesive aufacturig cells, Coputers ad Idustrial Egieerig, Vol. 31, No. ½, pp Suer, G. A., ad Bera, I. S., (1998), Optial operator assiget ad cell loadig whe lot-splittig is allowed, Coputers ad Idustrial Egieerig, Vol. 35, No. 3-4, pp Vebu, S., ad Sriivasa, G., (1997), Heuristics for operator allocatio ad sequecig i product-lie-cells with aually operated achies, Coputers ad Idustrial Egieerig, Vol. 32, No. 2, pp Wittrock, R. J., (1992), Operator assiget ad the paraetric preflow algorith, Maageet Sciece, Vol. 38, No. 9, pp Yakoob, S, B., ad Watada, J., (2009), Particle swar optiizatio for ulti-fuctio worker assiget proble, Kowledge- Based ad Itelliget Iforatio ad Egieerig Systes, Vol. 5712/2009, pp