Proceeding of the 200 Sytem and Information Engineering Deign Sympoium Matthew H. Jone, Stephen D. Pate, and Barbara E. Tawney, ed. HEURISTIC APPROACHES TO SOLVE THE U-SHAPED LINE BALANCING PROBLEM AUGMENTED BY GENETIC ALGORITHMS Ulie Martinez William S. Duff Mechanical Engineering Department Colorado State Univerity Fort Collin CO, College Ave. 802 USA ABSTRACT U-haped production line can be decribed a a pecial type of cellular manufacturing ued in jut-in-time (JIT) and Lean Manufacturing. The U-line arrange machine around a U-haped line in the order in which production operation are performed. Operator wor inide the U- line. Thi paper addree the Type I U-LBP uing heuritic rule adapted from the imple LBP. Then thee heuritic approache are compared with the optimal olution obtained from the previou publihed reearch wor. Finally the heuritic rule are ued a gene to find optimal or near optimal olution uing a genetic algorithm adapted from the imple LBP genetic algorithm propoed by Ponnambalam, Aravindan, and Mogilewar, (2000). The reult howed that ome very imple heuritic rule produced optimal or near optimal olution. Then with the imple implementation of a genetic algorithm it i poible to ome time obtain optimal olution in the firt iteration. INTRODUCTION Nowaday companie around the world are producing high quality product to ell them at the lowet price poible. Thi i not becaue they don t want to earn more money through the ale of product. It i becaue they are facing the neceity of increaing their participation in the maret becaue competitor alo are elling product with high quality at the lowet price poible. There are everal technique to continuouly improve quality and reduce operation cot. One of thee technique i called Line Balancing. The line balancing problem conit of aigning approximately the ame amount of worload to each wortation (worer) in an aembly line. An aembly line exit when we aemble or handle any device or product in a planned, equential manner with two or more operator performing ta of repetitive wor at etablihed wortation. When the product ha many operation and the demand i high, the proce of balancing the line become more and more difficult. According to Ajenblit (998) there are two type of optimization problem for the linebalancing problem (LBP). In the Type I problem, the cycle-time (maximum amount of time unit that can be pent at each wor tation) i fixed and the objective i to minimize the required number of wor tation. The Type II attempt to minimize the maximum cycle-time given a fixed number of wor tation." The line balancing problem (LBP) aign ta to wortation atifying two requirement. Firt time required to complete the ta aigned to a wor tation mut not exceed the cycle time (rate at which the part mut be produced) and the precedence relationhip mut be atified. Then the information needed to apply the LBP i: cycle time, precedence diagram and ta time (Figure ). 6 2 2 6 2 6 8 0 Ta 3 7 Proceing time 3 7 9 CT=0 Figure. Typical Precedence Diagram with Ta Time and Cycle Time (taen from Jacon, 962) U-haped production line can be decribed a a pecial type of cellular manufacturing ued in jut-in-time (JIT) and Lean Manufacturing. The U-line arrange machine around a U-haped line in the order in which production operation are performed (Figure 2). Operator wor inide the U-line. 287
approache for the U-haped LBP uing heuritic. (See for example: Urban 998, Guerriero and Miltengurg 2003) 3 HEURISTIC RULES TO SOLVE THE U- SHAPED LBP Figure2. Straight Line Production Sytem VS U-haped Production Sytem The U-haped production ytem allow for more poibilitie on how to aign ta to wor tation leading to a fewer number of wortation than the traight line approach. A ey idea in balancing an aembly line i aigning the operation to wor tation in ome feaible equence in which they can be performed. There are an enormou number of uch feaible equence, but only ome of which may reult in high line efficiencie. 2 PREVIOUS WORK Several approache can be found in the literature to olve the imple LBP and the U-haped LBP. Some uing heuritic and ome uing exact olution procedure. Many heuritic approache can be found in the literature to olve the imple LBP. Some of the mot popular technique are dicued by Talbot, Patteron and Gehrlein (980). There are good example of exact olution to imple LBP. In 98 Bowman preented a linear programming approach to olve the LBP. In 962 Held and Karp developed an algorithm to olve the imple LBP by uing dynamic programming. In 962, Klein preented an aignment problem application to olve the imple LBP. After thee reearch wor we can find many different approache to olve the imple LBP, ome uing heuritic and ome uing optimization tool (ee Ignall 96, Dar-El 97). All of thee paper addre the problem of balancing a production ytem arranged in line. Miltenburg and Wijngaard (99) produced the firt publication that formally addreed U-haped LBP. They howed that thi ind of arrangement of the production ytem produce a more complex problem than the one we deal with when production ytem are arranged in a line. After thi reearch wor, we can find many reearch publication about the U-haped LBP. Thee paper find optimal olution for thi problem uing dynamic programming, integer programming, branch and bound technique and genetic algorithm (GA). However there are not many Ten heuritic rule are ued in thi reearch to find olution to the Type I U-haped LBP. All thee heuritic rule were previouly ued to olve the imple LBP. However, to allow them to wor for the U-haped LBP ome modification were made. The difference between the original verion and the modified verion i that ta are available for aignment to a wor tation by having all ucceor or all predeceor previouly aigned to a wor tation, and when olving for the imple LBP, ta are available for aignment by having all ucceor previouly aigned only. The firt heuritic rule i the Modified Raned Poitional Weight procedure poted by Miltenburg and Wijngaard (99).The other nine heuritic which are introduced in thi reearch for olving the U-haped LBP are: 2. Maximum Total Number of Follower Ta or Precedence Ta, 3. Minimum Total Number of Follower Ta or Precedence Ta,. Maximum Ta Time,. Minimum Ta Time, 6. Maximum Number of Immediate Follower or Immediate Precedence Ta, 7. Minimum Number of Immediate Follower or Immediate precedence Ta, 8. Minimum U-line Upper Bound, 9. Minimum U- line Lower Bound, 0. U-line Minimum Slac. Thee heuritic rule are decribed below. Let µ be the et of ta which mut precede ta, p µ be the et of ta which mut ucceed ta. Then at any time the et o aignable ta, V={ all i µ p or all µ have already been aigned} j. Maximum Raned Poitional Weight The priority function p(), called the U-line Maximum Raned Poitional Weight, i defined a: p()=max t( ) + + i), t( ) j) p i µ i µ 2. Maximum Total Number of follower Ta or Precedence Ta The priority function p maxnf (), called the U-line Maximum Total Number of Follower or Precedence Ta p maxnf ( )=max p { number of ta µ, number of ta µ } 288
3. Minimum Total Number of follower Ta or Precedence Ta The priority function p minnf (), called the U-line Minimum Total Number of Follower or Precedence Ta p minnf ( )=min p { number of ta µ, number of ta µ }. Maximum Ta Time The priority function p Mt (), called the U-line Maximum Ta Time P Mt ( )= t (). Minimum Ta Time The priority function p mt (), called the U-line Maximum Ta Time P mt ( )= t () 6. Maximum Number of Immediate Follower or Immediate precedence Ta Let µ be the et of ta which mut immediate precede ip i ta, µ be the et of ta which mut immediate ucceed ta. The priority function p maxipi (), called the U-line Maximum Number of Immediate Follower or Immediate precedence Ta. p maxipi ()=max( µ, µ } ip i 7. Minimum Number of Immediate Follower or Immediate precedence Ta The priority function p minipi (), called the U-line Maximum Number of Immediate Follower or Immediate precedence Ta. p minipi ()=max( µ, µ } ip i 8. Minimum U-line Upper Bound Let (x) + be the leat integer>x and c the cycle time. + P lb () =min ( t( ) + + i))/ c, ( t( ) i µ 0. U-line Minimum Slac j µ + t( j))/ c p The priority function P lac (), called the U-line Minimum lac P lac ()= p ub ()- p lb () ASSIGNING TASKS TO WORK STATIONS The tep by tep procedure i given below:. Read the data: ta time, ta number, cycle time and precedence relation. 2. Compute the weight for each ta uing the deired heuritic rule. 3. Ran the ta baed on the weight computed in tep 2, and give the ame ran for the ta whoe weight are equal.. Determine the et of aignable ta V and aign the ta with the bet ran calculated in tep 3. If a tie occur, brea it uing the maximum ta time. If it doe not reolve the tie, ue the maximum ta number.. Ha the cycle time at the tation been completely filled? Or no one ta time from V i le than or equal to the remaining time in the tation? If ye go to tep 6. If no, go to tep. 6. I V=Ø? If ye top. If no, open a new tation and go to tep. To illutrate thi aignment proce let u ue the heuritic rule #2 Maximum Total Number of Follower Ta or Precedence Ta and the Jacon problem (Figure ). The aignable ta for the firt tation are V={,}. Since p maxnf ( )=0 p maxnf ( )=0 we brea the tie uing the maximum ta time, max(t(),t())=max(6,)=6, we aign ta, then V={,2,,3,}a ta ha the highet priority and ufficient cycle time remaining, it i alo aigned to wortation. The remaining aignment proce i decribed on Table. The priority function p ub (), called the U-line Upper Bound p ub () + + =min N + ( t( ) + + + i)) / c, N ( t( ) j)) / c p i µ j µ 9. Minimum U-line Lower Bound The priority function p lb (), called the U-line Lower Bound 289
Table. Ta Aignment Proce for the Jacon Problem Uing Heuritic Rule 2 Maximum Total Number of Follower Ta or Precedence Ta Station V Will fit tation Ran Aigned Time left in tation 2 3 2 3 0 2 3 2 3 9 0 2 3 9 0 3 2 3 9 2 3 7 0 2 3 7 0 3 3 3 0 0 2 3 7 8 2 3 7 8 3 3 7 7 2 3 8 2 3 8 3 2 3 6 8 3 6 3 0 6 8 6 8 3 6 8 6 6 8 0 8 8 8 COMPUTATIONAL RESULTS USING THE HEURISTIC RULES Eight popular et of LBP were taen from the literature. Each problem conit of a precedence diagram, ta time and cycle time. Since each problem olved with different heuritic rule can be conidered omewhat different problem, thi data et conit of 80 problem. For each of the 8 problem balance were obtained uing the 0 heuritic rule decribed above (ee table 2 through ). Table 2. Reult Obtained for Aae and Bowman Problem Heuritic Rule Problem Aae Bowman Ta=9 Ta=8 CT=0 CT=20-30 -9 2-3 -8 3-20 -9 - -8 - -7 6-3 -8 7-20 -9 8-0 2 * -8 9-20 -9 0-0* - -30 denote tation plu 30 time unit on a ixth tation. 2-0 Denote that the time remaining in tation i zero. * Denote that the minimum number of tation (um of all ta time/cycle time) wa found. Table 3. Reult Obtained for Dar-El and Jacon Problem Heuritic Rule Problem Dar-El Ta= CT=8 Jacon Ta= CT=0 3-2* -9* 2 3-* -8* 3-8 -9* 3-* -8* - -7* 6 3-* -8* 7-8 -9* 8 3-2* -8* 9 3-* -9* 0 3-7* -* Table. Reult Obtained for Johnon and Ponnambalam Aravindan and Naidu Problem Heuritic Rule Problem Johnon Ta= CT= -0* - 2-0* -6 3-0* -9 3-0* - -0* 6-7 6-0* -7 7-0* - 8-0* - 9-0* - 0-0* - Ponnambalam Aravindan and Naidu Ta=2 CT=0 Table. Reult Obtained for Scholl and Klein and Tonge Problem Heuritic Rule Problem Scholl and Klein Ta=2 CT=0 Tonge Ta=2 CT=20 6-6 -7* 2 6-0* -7* 3 6-8 -2* 6-7 -* 8-7 -3* 6 6-6 -* 7 6-7 -* 8 6-0* -7* 9 6-8 -2* 0 6-7 -0* 290
The number of time in which a heuritic rule produced reult achieving the minimum number of wor tation are ummarized on Table 6. From thee reult we can ee that heuritic rule 8 produced reult achieving the minimum number of wor tation 6 time. Heuritic rule 2, and 0 produced reult achieving the minimum number of wor tation time. Heuritic rule,, 6 and 9 produced reult achieving the minimum number of wor tation time. Finally, heuritic rule 3, and 7 produced reult achieving the minimum number of wor tation 3 time. Table 6. Summary of Hearttring Rule Producing Balance Achieving the Minimum Number of Wor Station Rule Time achieving minimum number of wor tation 2 3 3 3 6 7 3 8 6 9 0 6 GENETIC ALGORITHMS The heuritic rule decribed earlier in thi paper can produce good olution for the U-haped LBP, however there i nothing that guarantee that will be the cae. One way to get improved olution uing thee heuritic rule can be accomplihed by uing a number of rule imultaneouly to brea tie during the ta aignment proce. It can be done with the implementation of a genetic algorithm. Genetic algorithm are earch algorithm baed on the mechanic of natural election and natural genetic. There are three baic operator found in every genetic algorithm: reproduction, croover and mutation. For thi genetic algorithm, I ue the following tep (baed on the method propoed by Ponnambalam, Aravindan and Mogilewar, (2000) for the imple LBP adapted to the U-haped LBP). Jacon problem (Figure ) i ued to illutrate the procedure.. Initialize the population randomly. Each gene in a chromoome repreent one heuritic rule; here the chromoome length i 0 becaue we are uing 0 heuritic rule (Table 9).. Aign the ta to wortation uing the heuritic rule number repreented by the gene, tep by tep. If a tie occur, follow with the next gene until the tie i broen. Continue until the 0 gene are exhauted. If at any time of the ta aignment proce the tie can not be broen after 0 gene are exhauted, aign low fitne value to that chromoome o it will not be conidered for the next generation. Similarly, aign the ta to wor tation uing the remaining chromoome (table 0). 6. Calculate the value of the objective function (number of wor tation). 7. If an optimal olution or deired reult i not obtained in the firt interaction, continue with the reproduction croover and mutation to obtain the new generation and go to tep. (for more detail on GA ee Goldenberg (989)) Table 7. Weight for Jacon Problem (Figure ) Operation Heuritic Rule 2 3 6 7 8 9 0. U-line Maximum Raned Poitional Weight 6 9 7 9 3 22 6 27 2 6 2. U-line Maximum Total Number of Follower Ta or Precedence Ta 0 3 3 3 3 3 0 3. U-lineMinimum Total Number of Follower Ta or Precedence Ta 0 2 2 2 0. Maximum Ta Time 6 2 7 2 3 6. Minimum Ta Time 6 2 7 2 3 6 6. Maximum Number of Immediate Follower or Immediate Precedence Ta 3 3 2 7. U-line Minimum Number of Immediate Follower or Immediate Precedence Ta 0 0 8. Minimum U-line Upper Bound 7 0 0 0 0 0 9 0 9 9 7 9. Minimum U-line Lower Bound 2 2 2 0. U-line Minimum Slac 6 9 8 8 9 9 7 9 8 8 6. Get the data needed for the LBP: Number of ta, ta number, cycle time and precedence relation. 2. Compute the weight for each ta uing the 0 heuritic rule decribed above ( See Table 7). 3. Ran the ta baed on the weight computed in tep 2. Give the ame ran for the ta whoe weight are the ame. (Table 8) 29
Table 8. Ta Raning Operation Heuritic Rule 2 3 6 7 8 9 0. U-line Maximum Raned Poitional Weight 6 9 8 3 7 2 2. U-line Maximum Total Number of Follower Ta or Precedence Ta 3 3 2 3 3. U-lineMinimum Total Number of Follower Ta or Precedence Ta 2 2 2 2 3 3 3 2 2. Maximum Ta Time 2 6 3 7 6 2 3 3. Minimum Ta Time 6 2 7 2 3 6 6. Maximum Number of Immediate Follower or Immediate Precedence Ta 3 3 3 3 3 3 3 3 2 7. U-line Minimum Number of Immediate Follower or Immediate Precedence Ta 2 2 2 2 2 2 2 2 2 8. Minimum U-line Upper Bound 3 3 3 3 3 2 3 2 2 9. Minimum U-line Lower Bound 2 2 2 0. U-line Minimum Slac 3 3 2 3 3 Table 9. Initial Population Chromoome 0 6 3 3 Gene 9 3 6 3 2 8 2 9 8 0 3 0 0 8 9 2 3 7 2 0 6 7 2 9 2 7 3 7 7 9 8 8 8 8 6 0 7 7 0 6 9 3 7 7 7 8 7 0 8 6 3 0 6 6 3 0 8 6 9 2 3 8 8 9 0 8 9 6 8 2 2 6 3 0 7 2 0 3 8 3 7 2 8 6 0 6 8 2 7 3 7 7 6 3 3 8 7 0 9 8 2 2 8 2 7 2 7 3 9 6 3 3 8 0 9 7 6 3 6 9 2 7 7 2 2 3 7 6 3 9 8 8 3 7 6 8 6 2 9 7 2 0 3 8 2 20 6 0 7 2 3 2 9 With the implementation of thi imple Genetic Algorithm, the minimum number of wor tation were achieved for all the example problem in the firt iteration. However if an optimum i not achieved, additional iteration can be computed uing reproduction croover and mutation to obtain an improved population with the bet gene (heuritic rule) which apply for the problem in quetion. Table 0.Aignment Proce Uing Chromoome #3, V,9,0 2 2,3,,,9,0 Will fit tation,,9,0 2,3,,,9,0 Ran,, 6,,3,3 3,3,3,3,2,2 6,3,,7,3,3 6,7 2 Rule ued 7 2 0 6 7 Station Aign ed tie 2 Time left in tation 2,3,,9,0 2, 3,,6,9,0 3 3,6,9,0 3,6,9,0,2,, 6 8 3,8,9,0 3,6,9,0 6,7,2, 9 3 3,8,7,0 7 2 7 7 0 3,8,0 3,8,0,,3 2 0 3,8 3 3 0 8 8 0 8 Note: from the aignable ta only thoe with time le than or equal to the time remaining in the tation can be aigned. 7 CONCLUSIONS The reult howed that ome heuritic rule can produce good olution for the U-haped LBP. It i clear that there i nothing that guarantee that a choen heuritic rule will produce an optimal olution. It wa hown that the addition of a genetic algorithm can improve the current olution. Thi tudy ha taen a tep in the direction of finding good heuritic rule to olve the U-haped LBP. It i poible that different heuritic rule with different problem may produce different reult. Becaue there are a large variety of U-Shaped LBP problem, it would be beneficial to replicate thi tudy with larger and more divere problem. REFERNCES Ajenblit D. A., Applying Genetic Algorithm to the U- haped Aembly Line Balancing Problem, Proceeding of the IEEE Conference on Evolutionary Computation, (992), pp. 96-0. Aae Gerald R., algorithm for Increaing Labor Productivity in U-haped Aembly Sytem, PhD Diertation, Indiana Univerity July 999. Bowman E. H. "Aembly Line Balancing by Linear Programming", Operation Reearch, Vol. 8, (960), pp. 38-389. Dar-El, E. M., Solving Large Single-model Aembly Line Balancing Problem A comparative Study, AIIE Tranaction, Vol. 7, No 3, (97), pp. 302-306. Guerriero and Miltenbur J., The Stochatic U-Line Balancing Problem, Naval Reearch Logitic, Vol. 0, No., February 2003, pp. 3-7 Goldberg, D.E., Genetic Algorithm in Search, Optimization, and Machine Learning, Addion-Weley, USA, 989. 6 0 0 3 0 292
Held M., R.M. Karp, and R. Sharehian, Aembly Line- Balancing Dynamic Programming with Precedence Contraint, Operation reearch, Vol., No. 3, (963), pp. 2-60. Ignall, E. J., Review of Aembly Line Balancing Journal of Indutrial Engineering, Vol., No (96), pp. 2-2. Jacon J.R., A Computing Procedure for a Line Balancing Problem, Management Sci., v 2, n 3, (96), pp 26-272. Johnon R.V., Optimally Balancing Large Aembly Line with FABLE, Management Sci., v 3, n 2, (988), pp 20-23. Klein M., "On Aembly Line Balancing", Operation Re earch, Vol., (963), pp. 27-28. Miltenburg G. J., Wijngaard J., The U-Line Balancing Problem, Management Science, Vol. 0, No. 0, (99), pp. 378-388. Ponnambalam, S.G. Aravindan, P. and Naidu, G. M. (2000), "Multi-objective genetic algorithm for olving aembly line balancing problem", International Journal of Advanced Manufacturing Technology, v 6, n, 2000, p 3-32 Scholl A. and Klein R., UNLINO: Optimally Balancing U-haped JIT Line, Int. J. Production Reearch, v 37, n, (999), pp 72-736. Talbot F.B., Patteron J.H. and Gehrlein, A Comparative Evaluation of Heuritic Line Balancing Technique, Management Science, v 32, n, pp 30-. Tonge, Fred M, Summary of a Heuritic Line Balancing Procedure, Management Science,, v 7, n (960), pp2-2. Urban T. L., 'Optimal balancing of U-haped aembly line", Management Science, Vol., No., (998), pp. 738-7. Indutrial Engineering, Stanford Univerity. Hi reearch interet are: High Performance Solar Collector, Solar Air Conditioning, Solar Hot Water, Proce Heat, Electric, Mathematical Optimization, Statitical Method, Manufacturing, Simulation and Quality Engineering. AUTHOR BIOGRAPHIES ULISES MARTINEZ got hi B.S. in indutrial engineering in 992 from the Durango Mexico Intitute of Technology. M.S. in indutrial engineering from the Juarez Mexico Intitute of Technology. Hi pecial area of interet are imulation of manufacturing ytem and operation reearch. He ha taught imulation and operation reearch at the Juárez Intitute of Technology and the Monterrey Intitute of Technology at Juárez Mexico ince 99. Currently he i enrolled in the Ph. D. program in indutrial engineering at Colorado State Univerity. WILLIAM S. DUFF i Aociate Profeor of Colorado State Univerity at the Mechanical Engineering Department. B. Of Mechanical Engineering, Cornell Univerity, MBA (with honor) Univerity of Pennylvania Wharton School, M.S. in Statitic, Stanford Univerity, Ph.D. in 293
29 Martinez and Duff