Single-Pass-Based Heurisic Algorihms for Group Flexible Flow-shop Scheduling Problems PEI-YING HUANG, TZUNG-PEI HONG 2 and CHENG-YAN KAO, 3 Deparmen of Compuer Science and Informaion Engineering Naional Taiwan Universiy, Taipei, 06, TAIWAN 2 Deparmen of Compuer Science and Informaion Engineering Naional Universiy of Kaohsiung, Kaohsiung, 8, TAIWAN 3 Insiue for Informaion Indusry, Taipei, 06, TAIWAN Absrac: - In his paper, four algorihms based on some single-pass heurisics are proposed o solve he flexible flow-shop group scheduling problems wih more han wo machine ceners, which have he same number of parallel machines. Single-pass heurisics need less compuaion ime han muliple-pass ones. The single-pass heurisics adoped in he paper includes he PT, he Palmer and he Gupa algorihms. Experimens are also made for a comparison of he proposed four heurisic algorihms. Experimenal resuls show ha combining LPT and PT for he firs level, followed by he Palmer algorihm for he second level, has a superior performance among all he four proposed heurisic algorihms. Key-Words: - group scheduling, flexible flow shop, PT algorihm, Palmer Algorihm, Gupa Algorihm. Inroducion In simple flow-shop problems, each machine cener has us one machine [2][5][6][7][8][9]. If a leas one machine cener has more han one machine, he problem is called a flexible flow-shop problem. Flexible flow shops are hus generalizaion of simple flow shops [3]. Scheduling obs in flexible flow shops is considered an NP-hard problem [6][22]. Recenly, group scheduling has also been proposed and discussed. In group scheduling, each ob belongs o a specific group and all he obs are processed group by group. The group scheduling problem is also NP-hard [5].The issues in group scheduling are usually solved a wo levels. A he firs level, a sequence of he obs in each group is deermined; while a he second level, a sequence of he groups is deermined [7][3][2]. This paper specifically focuses on minimizing he final compleion ime of he enire group scheduling. Four heurisic algorihms are hus developed o solve his problem. Experimens are also made o compare he performance of he proposed four algorihms. 2 Relaed Scheduling Algorihms In he pas, Johnson firs proposed an efficien algorihm which guaraneed opimaliy in a wo-machine flow-shop problem [0]. Campbell, Dudek and Smih (CDS) hen proposed a heurisic algorihm o solve he flow-shop problems of more han wo machines [2]. Palmer, Perov and Gupa also respecive heir algorihms for achieving he same purpose [6][8][9]. The laer hree scheduling algorihms can process he ob daa in only one pass. Sriskandaraah and Sehi hen presened a heurisic algorihm based on he Johnson algorihm for solving flexible flow-shop problems of wo machine ceners wih he same number of machines [22]. As o group scheduling, Yang and Chern considered he wo-machine flow shop group scheduling problems wih group removal ime and ob ransporaion ime [23]. Dannenbring proposed a heurisic algorihm which combined he advanages of he Palmer and he CDS algorihms [4]. Allison compared he performances of single-pass and muliple-pass heurisics for solving group scheduling problems []. Schaller developed a new lower bound o evaluae parial sequences in a branch-and -bound procedure for he flow-shop group scheduling problem [20]. Logendran e al. sudied he performance of differen combinaions of LN algorihm, CDS algorihm and PT algorihm o solve group flexible flow-shop problems for minimizing makesapans [2]. Hong e al. considered he flexible flow shop group scheduling problems wih wo machine ceners of he same number of parallel machines [8]. Logendran e al. invesigaed he group flexible flow-shop problems for minimizing makesapans []. Logendran e al. presened a wo-machine group scheduling problem wih sequence-dependen se-up ime [4]. Yoshida and
Hiomi developed an opimal algorihm for minimizing he oal compleion ime in a wo machine group scheduling problem wih sequence-independen se-up ime [24]. Hong e al. proposed a heurisic algorihm for solving he flexible flow shop group scheduling problems wih more han wo machine ceners [9]. Many researches are sill in progress. The hree single-passed heurisics for scheduling ob sequencing for a flow shop wih more han wo machines are inroduced below. 2. The PT Scheduling Algorihm The PT algorihm [9] was proposed by Perov o schedule ob sequencing for a flow shop wih more han wo machines. Given a se of n independen obs, each having m (m>2) asks (T, T 2,, T m, T 2, T 22,, T (m-)n, T mn ) ha mus be execued in he same sequence on m machines (P, P 2,, P m ), he PT scheduling algorihm seeks a nearly minimum compleion ime of he las ob. I ransforms he flow shop problems wih more han wo asks ino he ones wih exacly wo asks and uses he Johnson algorihm o solve hem. This algorihm is saed as follows. The PT scheduling algorihm: Inpu: A se of n obs, each having m (m > 2) asks execued respecively on he given m machines. Oupu: A schedule wih a nearly minimum compleion ime of he las ob. Sep : For each ob, calculae he following erms for even m: m / 2 C = k and = k = m D k ( m / 2) + calculae he following erms for odd m: ( m+ ) / 2 C = k and D = k m ; k = ( m+ ) / 2 C D Sep 2: Schedule and by he wo-machine Johnson algorihm, and save he corresponding sequence and makespan f. 2.2 The Gupa Scheduling Algorihm The Gupa algorihm [6] was proposed by Gupa o schedule ob sequencing also for a flow shop wih more han wo machines. This algorihm is saed as follows. The Gupa scheduling algorihm: Inpu: A se of n obs, each having m (m > 2) asks execued respecively on he given m machines.. Oupu: A schedule wih a nearly minimum compleion ime of he las ob. Sep : Form he group of obs U ha ake less ime on he firs machine han on he las, such ha U = { < m }. Sep 2: Form he group of obs V ha ake less (or equal) ime on he las machine han on he firs, such ha V = { m }. Sep 3: For each ob J i in U, find he minimum of ( ki + (k+)i ) for k = o m-; resaed, se: ( m ) π i = min ( ki + ( k + ) i). Sep 4: For each ob J in V, find he minimum of ( k + (k+) ) for k = o m-; resaed, se: ( m ) π = min ( k + ( k+ ) ). Sep 5: Sor he obs in U in ascending order of π i 's; if wo or more obs have he same value of π i, sor hem in an arbirary order. Sep 6: Sor he obs in V in descending order of π 's; if wo or more obs have he same value of π, sor hem in an arbirary order. Sep 7: Schedule he obs on he machines in he sored order of U, hen in he sored order of V. 2.3 The Palmer Scheduling Algorihm The Palmer algorihm [8] was proposed by Palmer o schedule ob sequencing also for a flow shop wih more han wo machines. This algorihm is saed as follows. The Palmer scheduling algorihm: Inpu: A se of n obs, each having m (m > 2) asks execued respecively on he given m machines. Oupu: A schedule wih a nearly minimum compleion ime of he las ob. Sep : Find he value π for each ob J as follows: m / 2 i= π = ( m 2i + ) + ( m 2i + ), i ( m+ i) where i represens he execuion ime of he i-h ask T i in ob J. Sep 2: Sor he obs in descending order of π 's; if wo or more obs have he same value of π, sor hem in an arbirary order. Sep 3: Schedule he obs on he machines in he sored order. 3 Four Single-Pass-Based Heurisic Algorihms The issues in group scheduling are usually solved a
wo levels. A he firs level, a sequence of he obs in each group is deermined; while a he second level, a sequence of he groups is deermined. As he group scheduling problems sill follow flow-line arrangemen, i is possible o exend some heurisics of solving convenional flow shop scheduling problems o solving group scheduling problems. Four single-pass-based heurisic algorihms are hen developed in his session o solve he flexible flow-shop group scheduling problems wih more han wo machine ceners, which have he same number of parallel machines. Four differen combinaions are considered, each for a heurisic algorihm. The firs one (H) combines LPT and PT for he firs level o deermine he sequence of obs in each group, and uses Palmer for he second level o deermine he sequence of groups. The second one (H2) combined LPT wih Palmer for he firs level and used PT for he second level. The hird one (H3) combines LPT and PT for he firs level, and uses Gupa for he second level. The fourh one (H4) combines LPT and Gupa for he firs level, and uses PT for he second level. Below, he firs proposed heurisic algorihm is described in deail o illusrae he idea. The oher hree heurisic algorihms can be similarly derived. 4 The Heurisic Algorihm H The firs heurisic algorihm firs combines LPT and PT for he firs level o deermine he sequence of obs in each group, and hen uses Palmer for he second level o deermine he sequence of groups. The proposed algorihm is saed as follows. The LPT-PT-Palmer heurisic algorihm: Inpu: l groups of obs, each wih m (m > 2) asks, o be execued respecively on each of m machine ceners wih p parallel machines. Oupu: A schedule wih a subopimal compleion ime. Level : Deermining he ob sequence in each group: Sep : Se he variable k o one, where k is used o represen he number of he curren group o be processed. Sep 2: Repea Seps 3 o 5 unil k > l. Par : Forming he machine groups: Sep 3: Form p machine groups, each of which conains one machine from each machine cener. Each machine group can be hough of as a simple flow shop F, F 2,, F p. Sep 4: Iniialize he compleion ime f, f 2,, f p of each flow shop F, F 2,, F p o zero. Par 2: Assigning obs o machine groups: Sep 5: For each ob J k, find is oal execuion ime: k = k + 2k + + mk, = o n, k = o l. Sep 6: Sor he obs in descending order of processing ime k ; if any wo obs have he same k values, sor hem in an arbirary order. Sep 7: Find he flow shop F i wih he minimum processing ime f i among all he flow shops; if wo flowshops have he same minimum f i value, choose one arbirarily. Sep 8: Assign he firs ob J k in he sored lis o he chosen flow shop F i, which has he minimum compleion ime f i among all p flow shops. Sep 9: Add he oal ime k of ob J k o he needed oal ime of he chosen flow shop, F i ; ha is: f i = f i + k. Sep 0: Remove ob J k from he ob lis. Sep : Repea Seps 7 o 0 unil he ob lis is empy. Afer Sep, obs are clusered ino p groups and are allocaed o he p-machine flow shops. Par 3: Dealing wih he ob sequencing in each flow shop: Sep 2: For each flow shop F i, se he iniial compleion ime of he machines f i ( = o m, i = o p) o zero. Sep 3: Find he compleion ime of each flow shop f i by he PT algorihm saed in Secion 2. Sep 4: Save he corresponding ob sequence. Sep 5: Se k = k +. Afer Sep 5, he individual ob sequence for each group has been found. Level 2: Deermining he group sequence in he whole schedule: Sep 6: Se he processing ime mc k needed for n obs in group k a machine cener, ( = o m, k = o l) as: n mc k = i= ik i = o n, = o m, k = o l.,
Sep 7: Find he group sequence by he Palmer algorihm saed in Secion 2 according o mc k, = o m, k = o l. Sep 8: Schedule he groups based on he group sequence and schedule he ob sequence in each flow-shop of each group o find he final compleion ime. 5 An Example for he Firs Heurisic Algorihm An example is given in his session o demonsrae he applicaion of he firs heurisic algorihm (H). Assume here are hree groups and each of hem has five obs, J i o J 5i (i = o 3). Also assume each ob has hree asks o be scheduled via hree operaions. Each operaion is execued by a machine a he corresponding machine cener. Each machine cener includes wo parallel machines. Assume he execuion imes of hese obs are lised in Table. The algorihm proceeds as follows. Table. Processing imes for he hree groups of obs G G 2 G 3 J J 2 J 3 J 4 J 5 J 2 J 22 J 32 J 42 J 52 J 3 J 23 J 33 J 43 J 53 k 9 4 7 6 9 6 4 7 2 5 8 2 8 5 9 2k 7 8 3 2 9 3 2 6 7 3 7 5 2 7 3 3k 8 7 3 4 6 3 2 5 9 4 3 2 6 3 3 The proposed algorihm firs deermines he appropriae ob sequence in each of he hree groups. Each group of obs can be scheduled independenly. The processing seps are decomposed ino hree pars. Par firs forms wo machine groups, F, F 2, each of which is hough of as a wo-machine flow-shop. Par 2 hen assigns he obs in each group o he machine groups. Resuls for he example are shown in Table 2. Table 2. The obs allocaed o each flow shop for each ob group F J 5, J 3, J 4 J 42, J 52 J 33, J 43, J 23 F 2 J, J 2 J 32, J 2, J 22 J 3, J 53 Par 3 hen uses he PT algorihm o deal wih ob sequencing in each flow shop for each group. The resuls are shown in Table 3. Table 3. The ob sequence in each flow shop in each group G G 2 G 3 F J 5, J 3, J 4 J 42, J 52 J 43, J 33, J Job sequence 23 F 2 J 2, J J 32, J 2, J 22 J 3, J 53 The seps a level 2 are hen execued o deermine he group sequence in he whole schedule. The processing ime of each group of obs a each machine cener is firs calculaed and shown in Table 4. Table 4. The processing ime of each group of obs a each machine cener Machine Cener G G 2 G 3 Processing Time Machine Cener 35 24 32 Machine Cener 2 29 2 24 Machine Cener 3 28 23 7 In Table 4, he processing ime for processing he firs asks of all he obs in Group a machine cener is 35, for processing he second asks a machine cener 2 is 29, and for processing he hird asks a machine cener 3 is 28. Similarly, he processing ime evaluaed for Group 2 is 24, 2, and 23, respecively, and for Group 3 is 32, 24, and 7, respecively. The Palmer procedure is hen used o schedule he hree groups according o he processing ime a each machine cener. The obained group sequence for his example is G 2, G, G 3. All he groups of obs are hen scheduled according o he above group sequence ogeher wih is bes ob sequence in each flow shop. The final scheduling resuls are shown in Figure. The final compleion ime is 62. Flowshop i G G 2 G 3 Jobs allocaed
m m 2 42 52 32 2 22 5 3 2 4 43 33 23 3 53 m 2 m 22 242 252 232 22 222 25 23 22 2 24 243 233 223 23 253 m 3 m 23 342 352 332 32 322 35 33 32 3 34 343 333 323 33 353 Figure : The final scheduling resul in he example 6 Experimens This secion repors on experimens made o show he performance of he proposed scheduling algorihms. The experimenal design was similar o he one conduced by Logendran e al. [2]. Our experimens were implemened by Visual C++ on an AMD Ahlon(m) XP.8 GHz CPU, wih 52RAM and Windows XP 2002 operaing sysem. Four parameers were considered o define he problem srucure. They were number of machine ceners, number of parallel machines in each machine cener, number of groups, and number of obs in each group. Three differen values for each of he four parameers were considered: 3, 7, and 0 for he number of machine ceners; 3, 5, and 7 for he number of parallel machines in each machine cener; 7, 0, and 5 for he number of groups; and 0, 20, and 30 for he number of obs in each group. These selecions could idenify a wide variey of problem sizes. The execuion ime of each ask was randomly generaed from a uniform disribuion of inegers beween 5 and 50. Tweny differen runs were execued for each se of problems, and in each run he makespan and he compuaion ime were measured. All he proposed four heurisic algorihms ran very fas. They ook no longer han second, even for he larges problem in he experimens. Since each parameer had hree differen values and weny runs were execued, a oal of 620 (3 4 * 20) problems were solved. Three performance measures were considered o compare he performance of he proposed four heurisic algorihms. They were average makespan, average rank, and percenage of he bes soluions. The average makespan was evaluaed as he average of he makespans obained from all he 620 problems. The average rank was evaluaed as he average of he ranks obained from all he problems. For each problem, he heurisic algorihm wih he minimum makespan would be given he rank of 4, while he heurisic one wih he maximum makespan would be given. If several heurisic algorihms had he same makespan, hey were given he same average rank. The percenage of he bes soluions was evaluaed as he percenage of he problems in which a heurisic algorihm go he minimum makepan among all he four ones. In each run, he heurisic algorihm wih he minimum makepan would be added one o is value of bes soluions. The oal percenage of he bes soluions for all he four heurisic algorihms migh possibly exceed 00 percens since he algorihms wih he same minimum makespan would all add one o heir own values of bes soluions. The resuling performance measures for he proposed four heurisic algorihms are shown in Table 5. From Table 5, i is easily seen ha he firs proposed algorihm (H), ouperformed he oher hree heurisic algorihms for all he hree measures. Table 5. The resuling performance measures for he proposed four heurisic algorihms Algorihm H H2 H3 H4 Average makespan 904.72 92.08 97.87 92.0 Average rank 2.8 2.4 2.26 2.52 Percenage of bes soluions 45.74 23.58 23.2 26.85 Alhough Table 5 shows H is wih he minimum average makepan, i doesn mean H is always superior o he oher hree heurisics in he saisical sense. 7 Conclusion The flexible flow-shop group scheduling problems wih more han wo machine ceners, which have he same number of parallel machines, have been examined in his paper. They have been solved a wo levels. A he firs level, a sequence of he obs in
each group is deermined; while a he second level, a sequence of he groups is deermined. Four heurisic algorihms, each involving a single-pass heurisic respecively a each level, have hus been developed o solve he problems. Four differen combinaions have been considered, each for a heurisic algorihm. The firs one (H) combines LPT and PT for he firs level o deermine he sequence of obs in each group, and uses Palmer for he second level o deermine he sequence of groups. The second one (H2) combines LPT wih Palmer for he firs level and uses PT for he second level. The hird one (H3) combines LPT and PT for he firs level, and uses Gupa for he second level. The fourh one (H4) combines LPT and Gupa for he firs level, and uses PT for he second level. Experimens have also been made for a comparison of he proposed four heurisic algorihms. Four parameers, including number of machine ceners, number of parallel machines in each machine cener, number of groups, and number of obs in each group, have been considered o define differen es problems. The experimenal resuls show ha H has ouperformed he ohers for all parameer combinaions. Thus, combining LPT and PT for he firs level, followed by he Palmer algorihm for he second level, has a superior performance among all he four proposed heurisic algorihms. References: [] J. D. Allison, Combining Perov's heurisic and he CDS heurisic in group scheduling problems, Proceedings of he 2h annual conference on Compuers and indusrial engineering, Vol. 9, No. -4, 990, pp. 457-46. [2] H. G. Campbell, R. A. Dudek and M. L. Smih, A heurisic algorihm for he n ob, m machine sequencing problem, Managemen Science, Vol. 6, 970, pp. B630-B637. [3] S. C. Chung and D. Y. Liao, Scheduling flexible flow shops wih no seup effecs, The 992 IEEE Inernaional Conference on Roboics and Auomaion, 992, pp. 79-84. [4] D. G. Dannenbring, An evaluaion of flowshop sequencing heurisics, Managemen Science, Vol. 23, No., 977, pp. 74-82. [5] R. A. Dudek, S. S. Panwalkar and M. L. Smih, The lessons of flowshop scheduling research, Operaions Research, Vol. 40, 992, pp. 7-3. [6] J. N. D. Gupa, A funcional heurisic algorihm for he flowshop scheduling problem, Operaions Research, Vol. 40, 97, pp. 7-3. [7] T. P. Hong, G. B. Horng, P. Y. Huang and C. L. Wang, Solving flexible flow-shop problems by LPT and LN scheduling algorihms, The Inernaional Conference on Informaics, Cyberneics, and Sysems, 2003, pp. 849-853. [8] T. P. Hong, P. Y. Huang and G. Horng, Two group flexible flow-shop scheduling algorihms for wo machine ceners, The 33rd Inernaional Conference on Compuers and Indusrial Engineering, 2004. [9] T. P. Hong, P. Y. Huang, G. Horng, and C. Y. Kao, Solving flexible flow-shop group scheduling problems by he PT algorihm, WSEAS Transacions on Mahemaics, Issue 4, Vol. 4, 2005, pp. 506-53. [0] S. M. Johnson, Opimal wo- and hree-sage producion schedules wih se-up imes included, Naval Research Logisics Quarerly, Vol., 954, pp. 6-68. [] R. Logendran, S. Carson and E. Hanson, Group scheduling in flexible flow shops, Inernaional Journal of Producion Economics, Vol. 96, No. 2, 2005, pp. 43-55. [2] R. Logendran, L. Mai and D. Talkingon, Combined heurisics for bi-level group scheduling problems, Inernaional Journal of Producion Economics, Vol. 38, 995, pp. 33-45. [3] R. Logendran and N. Nudasomboon, Minimizing he makespan of a group scheduling problem: a new heurisic, Inernaional Journal of Producion Economics, Vol. 22, 99, pp. 27-230. [4] R. Logendran, N. Salmasi and C. Sriskandaraah, Two-machine group scheduling problems in discree pars manufacuring wih sequence-dependen seups, Compuers and Operaions Research, Vol. 33, No., 2006, pp. 58-80. [5] R. Logendran and C. Sriskandaraah, Two-machine group scheduling problem wih blocking and anicipaory seups, European Journal of Operaional Research, Vol. 69, No. 3, 993, pp. 467-48. [6] T. E. Moron and D. W. Penico, Heurisic Scheduling Sysems wih Applicaions o Producion Sysems and Proec Managemen, John Wiley & Sons Inc., New York, 993. [7] M. Nawaz, J. E. E. Enscore and I. Ham, A heurisic algorihm for he m-machine, n-ob flow-shop sequencing problem, Omega, Vol., No., 983, pp. 9-95. [8] D. S. Palmer, Sequencing obs hrough a muli-sage process in he minimum oal ime - a quick mehod of obaining a near opimum,
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