Three Algorithms for Flexible Flow-shop Scheduling

Transcription

1 Aercan Journal of Appled Scences 4 (): ISSN Scence Publcatons Three Algorths for Flexble Flow-shop Schedulng Tzung-Pe Hong, 2 Pe-Yng Huang, 3 Gwoboa Horng and 3 Chan-Lon Wang Departent of Coputer Scence and Inforaton Engneerng, Natonal Unversty of Kaohsung, Kaohsung 8, Tawan, R.O.C. 2 Departent of Coputer Scence and Inforaton Engneerng Natonal Tawan Unversty, Tape 06, Tawan, R.O.C. 3 Departent of Coputer Scence Natonal Chung-Hsng Unversty, Tachung 402, Tawan, R.O.C. Abstract: Schedulng s an portant process wdely used n anufacturng, producton, anageent, coputer scence, and so on. Approprate schedulng can reduce ateral handlng costs and te. Fndng good schedules for gven sets of obs can thus help factory supervsors effectvely control ob flows and provde solutons for ob sequencng. In sple flow shop probles, each achne operaton center ncludes ust one achne. If at least one achne center ncludes ore than one achne, the schedulng proble becoes a flexble flow-shop proble. Flexble flow shops are thus generalzaton of sple flow shops. In ths paper, we propose three algorths to solve flexble flow-shop probles of ore than two achne centers. The frst one extends Srskandaraah and Seth s ethod by cobnng both the LPT and the search-and-prune approaches to get a nearly optal akespan. It s sutable for a edu-szed nuber of obs. The second one s an optal algorth, entrely usng the search-and-prune technque. It can work only when the ob nuber s sall. The thrd one s slar to the frst one, except that t uses Petrov s approach (PT) to deal wth ob sequencng nstead of searchand-prune. It can get a polynoal te coplexty, thus beng ore sutable for real applcatons than the other two. Experents are also ade to copare the three proposed algorths. A trade-off can thus be acheved between accuracy and te coplexty. Key words: schedulng, flexble flow shop, LPT schedulng, search, PT schedulng INTRODUCTION Schedulng s an portant process wdely used n anufacturng, producton, anageent, coputer scence, and so on. In sple flow-shop probles, each achne center has ust one achne [,3,4,9-]. If at least one achne center has ore than one achne, the proble s called a flexble flow-shop proble. Flexble flow shops are thus generalzaton of sple flow shops [2]. Schedulng obs n flexble flow shops s consdered an NP-hard proble [8,2]. The proble addressed n the paper s a specal case of the flexble flow shop proble. We assue each achne center has the sae nuber of parallel achnes whch to the best of authors knowledge s the frst of ts knd. Ths paper specfcally focuses on nzng the total copleton te of flexble flow shop. Three algorths have been developed to solve flexble flow-shop schedulng probles wth ore than two achne centers. The frst one extends Srskandaraah and Seth s ethod by cobnng both the LPT [5] and the search-and-prune approaches to get a nearly optal akespan. The LPT approach s frst used to assgn obs to each achne group (flow shop). The search-and-prune approach s then used to deal wth ob sequencng. The second one s an optal algorth, entrely usng the search-and-prune technque. The thrd one s slar to the frst one except that t uses Petrov s approach (PT) [] to deal wth ob sequencng nstead of search-and-prune. Experental results show that the thrd proposed algorth can save uch coputatonal te when copared to the other two although ts akespans ay be a lttle larger. Partcularly, the thrd one has the polynoal te coplexty, avodng the ntractable probles occurrng n the other two algorths. In addton, the te coplextes and akespans by the frst algorth le between those by the other two. A trade-off for these three algorths can thus be acheved between accuracy and te coplexty. In the past, Johnson frst proposed an effcent algorth whch guaranteed optalty n a two- Correspondng Author: Tzung-Pe Hong, Departent of Coputer Scence and Inforaton Engneerng, Natonal Unversty of Kaohsung, Kaohsung 8, Tawan, R.O.C. 887

2 achne flow-shop proble [6]. Paler, Petrov and Gupta then respectvely proposed ther algorths for solvng the flow-shop probles of ore than two achnes [4,0,]. The three schedulng algorths could process the ob data n only one pass. Capbell, Dudek and Sth (CDS) then proposed a heurstc algorth for achevng the sae purpose []. It, however, needed to process the ob data n ultple passes. Logendran and Nudtasoboon also proposed a ult-pass algorth to solve t [7]. Srskandaraah and Seth then presented a heurstc algorth based on the Johnson algorth for solvng flexble flow-shop probles of two achne centers wth the sae nuber of achnes [2]. Many researches n ths feld are stll n progress. As entoned above, flexble flow-shop probles are NP-hard. No algorths can fnd the optal solutons n polynoal te. In the past, Srskandaraah and Seth proposed a heurstc algorth to solve the proble of two achne centers, and the copleton te of the derved schedules was close to the optu. In ths paper, we generalze t and propose three algorths to solve the flexble flow-shop probles of ore than two achne centers. Soe related schedulng algorths are frst ntroduced as follows. The dscovery of schedulng algorths for a set of ndependent tasks wth arbtrary executon te and an arbtrary nuber of processors s a classc sequencng proble of wde nterest and applcaton. Aong the proposed schedulng algorths, the LPT (Longest- Processng-Te-frst) schedulng algorth s the splest one and s wdely used n any real-world stuatons. Gven a set of n ndependent tasks (T to T n ), each wth arbtrary executon te (t to t n ), and a set of parallel processors or achnes (P to P ), the LPT schedulng algorth assgns the task wth the longest executon te (aong those not yet assgned) to a free processor whenever ths processor becoes free. For cases when there s a te, an arbtrary te-breakng rule can be assued. The fnshng te by the LPT schedulng algorth s n general not nal. The coputatonal te spent by the LPT schedulng algorth s, however, uch lower than that by an optal schedulng algorth. A. J. Appled Sc., 4 (): The PT algorth [] was proposed by Petrov to schedule ob sequencng for a flow shop wth ore than two achnes. Gven a set of n flow-shop obs, each havng (>2) tasks (T, T 2,, T, T 2, T 22,, T (- )n, T n ) that ust be executed n the sae sequence on 888 achnes (P, P 2,, P ), the PT schedulng algorth seeks a nearly nu copleton te of the last ob. It transfors the flow shop probles wth ore than two tasks nto the ones wth exactly two tasks and uses the Johnson algorth to solve the. Srskandaraah and Seth [2] proposed a heurstc algorth for solvng the flexble flow-shop proble of two achne centers and the copleton te of the derved schedules was close to the optu. Srskandaraah and Seth decoposed the proble nto the followng three subprobles and solved each heurstcally: Part : For the achne groups, each of whch contans a achne fro each center. Part 2: Use the LPT ethod to assgn obs to each achne group (flow shop). Part 3: Deal wth ob sequencng and tng usng the Johnson algorth. In ths paper, we wll extend above approaches to solve the flexble flow-shop probles of ore than two achne centers. Assuptons and Notaton: Assuptons and notaton used n ths paper are descrbed n ths secton. Assuptons:. Jobs are not preeptve 2. Each ob has ( > 2) tasks wth processng tes, executed respectvely on each of achne centers. 3. All achne centers have the sae nuber of parallel achnes. The Search-And-Prune Schedulng Procedure For Job Sequencng: The search-and-prune procedure proposed n ths paper s used to schedule obs sequencng for a flow shop wth ore than two achnes. An upper bound s used to ncrease the perforance of the procedure. The procedure wll act as the thrd part n the frst two algorths proposed later. Gven a set of n flow-shop obs, each havng (>2) tasks (T, T 2,, T, T 2, T 22,, T (-)n, T n ) that ust be executed n the sae sequence on achnes (P, P 2,, P ), schedulng seeks the nu copleton te of the last ob. The procedure s stated as follows. The search-and-prune procedure wth an upper bound for ob sequencng: Input: A set of n obs, each havng ( > 2) tasks executed respectvely on each of the gven achnes

3 A. J. Appled Sc., 4 (): Output: A schedule wth a nu copleton te of the last ob. Step : Set the ntal upper bound v of the fnal copleton te as. Step 2: For each possble perutaton of task sequence, do the followng steps. Step 3: Set the ntal copleton te d of the achne M ( = to, s the nuber of tasks n a ob) to zero. Step 4: Assgn the frst ob J n ts schedule sequence generated n Step 2 to the achnes such that J s frst task T s assgned to M, T 2 s assgned to M 2,, T s assgned to M. Step 5: Add the processng te t to the copleton te d of the frst achne M ; that s: d = d + t. Step 6: If d s larger than v, go to Step 2 for tryng another perutaton. Step 7: Set d k + = ( d k, d k + ) + t( k + ), for k = to (-). Step 8: If d k + s larger than v, go to Step 2 for tryng another perutaton; otherwse, do the next step. Step 9: Reove ob J fro the sequence. Step 0: Repeat Step 4 to 9 untl the ob sequence s epty. Step : Set the copleton te d as the copleton te d of ts -th achne. Step 2: If d s saller than v, then set v = d. Step 3: Repeat Step 2 to Step 2 untl all possble perutatons have been tested. Step 4: Set v as the fnal copleton te of the ob schedulng and save the schedule that gves the nu total copleton te. After Step 4, schedulng s fnshed and an optal copleton te for a flow shop has been found. group (flow shop). The thrd part deals wth ob sequencng and tng usng the search-and-prune procedure for a flow shop. The proposed algorth s stated below. The proposed LPT_ Search-and-prune flexble flowshop algorth: Input: A set of n obs, each havng ( > 2) tasks, to be executed respectvely on each of achne centers wth p parallel achnes. Output: A schedule wth a suboptal copleton te. Part : Step : Step 2: Part 2: Step 3: Step 4: Step 5: Step 6: Step 7: Forng the achne groups For p achne groups, each of whch contans one achne fro each achne center. Each achne group can be thought of as a sple flow shop F, F 2,, F p. Intalze the copleton te f, f 2,, f p of each flow shop F, F 2,, F p to zero. Assgnng obs to achne groups For each ob J, n, fnd ts total executon te tt = t + t t. Sort the obs n descendng order of processng te tt ; f any two obs have the sae tt values, sort the n an arbtrary order. Fnd the flow shop F wth the nu processng te f aong all the flow shops; f two flowshops have the sae nu f value, choose one arbtrarly. Assgn the frst ob J n the sorted lst to the chosen flow shop F, whch has the nu copleton te f, aong all p flow shops. Add the total te tt of ob J to the needed total te of the chosen flow shop, F ; that s: f = f + tt. The frst algorth for schedulng on a flexble flow shop wth ore than two achne centers: A heurstc algorth for solvng flexble flow-shop probles of two achne centers s proposed by Srskandaraah and Seth n 989 [2]. In ths paper, we generalze t to solve flexble flow-shop probles of ore than two achne centers. The proposed flexble flow-shop algorth s based on the LPT and the proposed search-and-prune approaches to anage ob schedulng. The algorth s decoposed nto three parts as Srskandaraah and Seth s ethod was. The frst part fors the achne groups, each of whch contans a achne fro each center. The second part uses the LPT ethod to assgn obs to each achne 889 Step 8: Step 9: Reove ob J fro the ob lst. Repeat Steps 5 to 8 untl the ob lst s epty. After Step 9, obs are clustered nto p groups and are allocated to the p achne flow shops. Part 3: Dealng wth ob sequence n each flow shop Step 0: For each flow shop F, set the ntal copleton te of the achnes f ( = to, = to p) to zero. Step : Fnd the copleton te of each flow shop f by the proposed search-and-prune procedure n Secton 4.

4 A. J. Appled Sc., 4 (): Step 2: Fnd the fnal copleton te p ff = ( f ) aong the copleton te = of all the flow shops. After Step 2, schedulng s fnshed and a total copleton te ff has been found. An Exaple For The Proposed Heurstc Algorth: Assue fve obs, J to J 5, each havng three tasks (t, t 2, t 3 ), are to be scheduled va three operatons. Each operaton s executed by a achne at the correspondng achne center. Each achne center ncludes two parallel achnes. Assue the executon tes of these obs are lsted n Table. The algorth proceeds as follows. Table : Processng tes for the fve obs Executon te t t 2 t 3 Job Part : Step : Step 2: Part 2: Step 3: J J J J J Forng the achne groups: For two achne groups, F and F 2, each of whch s thought of as a three-achne flowshop. Wthout lose of generalty, we ay assue the flowshops are constructed as follows: F { }, F 2 { }, where s the -th achne n the -th center. Intalze f = f 2 = 0, where f s the ntal copleton te of F. Assgnng obs to achne groups: For each ob J, = to 5, fnd ts total executon te tt = t + t 2 + t 3. For exaple, the total processng te of ob s calculated as: tt = t + t2 + t3 = = 4. The total processng tes of the other obs can be slarly found and the results are lsted n Table 2. Table 2: The total processng tes of the fve obs Step 4: Job total processng te tt J 4 J 2 8 J 3 J 4 0 J 5 6 Sort the obs J to J 5 n a descendng order of the total processng te ( tt ). The followng results are obtaned: Job lst = {J 5, J, J 3, J 4, J 2 }. Step 5: Fnd the nu f between the two Step 6: Step 7: flowshops F and F 2. Snce both the total processng tes of the two flowshops are equal to zero, any arbtrary flowshop can be chosen. Wthout lose of generalty, assue F s chosen. Assgn the frst ob J 5 n the sorted lst to the chosen flowshop F. Add the total processng te tt 5 of ob J 5 to the needed total te of the chosen flowshop F. Thus: f = f + tt 5 = = 6. After Step 7, the results of allocatng J 5 to the flowshop F are shown n Table 3. Step 8: Reove the ob J 5 fro the ob lst. After J 5 s reoved, the ob lst s then as follows: Job lst = {J, J 3, J 4, J 2 }. Step 9: Repeat Steps 5 to 8 untl the ob lst s epty. After Step 9, obs are clustered nto two groups and are respectvely allocated to the two flowshops. Results are shown n Table 4. Table 3: The flowshops wth allocated obs and total processng te Flowshop allocated obs total processng te F J 5 6 F 2 None 0 Table 4: The obs n each flow shop Flowshop Jobs allocated F J 5, J 4 F 2 J, J 3, J 2 890

5 A. J. Appled Sc., 4 (): Part 3: Dealng wth ob sequencng n each flow shop: Step 0: In each flow shop F, set the ntal copleton te of the achnes f = 0 ( = to 3, = to 2). Step : Fnd the copleton te of each flow shop f by the proposed search-and-prune procedure n Secton 4. The results are found as follows: f = 20, f 2 = 8. Step 2: Fnd the al fnal copleton te ff between the copleton tes of both the flow shops. We can thus get: ff = 20. ff has been found ff s then output as the fnal total copleton te. The schedule obtaned by the above steps s shown n Fg Machne Center t 4 t t 5 Machne Center 2 t 24 t 3 t 2 Machne Center 3 t 34 t 2 t 25 t 23 t 3 t 35 t Fg. : The fnal schedulng result n the exaple The Second Algorth: In the frst algorth, the LPT ethod s used to assgn obs to achne groups. The ob sequencng and tng n each group s then done by the search-and-prune procedure. The tasks n a set of clustered obs are executed n the sae achne group. The akespans obtaned n the above way do not guarantee to be optal. For gettng an optal schedule, the tasks n a set of obs ay be executed n dfferent achne groups. In ths secton, we thus propose another schedulng algorth based on the search-and-prune technque to get the optal solutons, whch can also be used to easure the perforance of the frst algorth. The proposed optal algorth s stated below. t 33 t 32 The proposed optal flexble flow-shop algorth: Input: A set of n obs, each havng ( > 2) tasks, to be executed respectvely on each of achne centers wth p parallel achnes. Output: A schedule wth an optal copleton te. Step : Set the ntal upper bound v of the fnal copleton te as. Step 2: For each possble cobnaton of task allocaton and perutaton of task sequence, do the followng steps. Step 3: In each achne center, set the ntal copleton te of each achne to zero. Step 4: Set the varable g to one, where g represents the nuber of the current achne center to be processed. Step 5: Schedule the frst tasks of all obs n the achnes of the frst achne center. That s, for each task T of the -th ob allocated to the -th achne D n the frst achne center, do the followng substeps accordng to the schedulng order n the perutaton and cobnaton generated: (a)add the processng te t to the copleton te d of the achne D. That s: d = d + t, and c = d. (b)if d s larger than v, neglect all the perutatons and cobnatons wth ths sequence n the frst achne center and go to Step 2 for tryng another perutaton and cobnaton. Step 6: Set g = g +. Step 7: Schedule the g-th tasks of all obs n the achnes of the g-th achne centers accordng to the perutaton and cobnaton generated. For each task T g of the -th ob allocated to the -th achne D g n the g-th achne center, do the followng substeps n the scheduled order: (a)fnd the copleton te d g of the achne D g as: d g = (d g, c (g-) )+ t g, and c g = d g. (b)if d g s larger than v, neglect all the perutatons and cobnatons wth ths sequence n the frst g achne centers and go to Step 2 for tryng another perutaton and cobnaton. Step 8: Repeat Steps 6 and 7 untl g >. Step 9: Set the copleton te d of the current 89

6 p schedule = ( d ) = n the -th achne center. A. J. Appled Sc., 4 (): aong the p achnes Step 0: If d s saller than v, then set v = d. Step : Repeat Steps 2 to 0 untl all the possble perutatons and cobnatons have been tested. Step 2: Set the optal fnal copleton te of the ob schedulng ff = v. After Step 2, a globally optal copleton te ff has been found. In the above two algorths, the perutatons and cobnatons of task sequences or achne centers ust be tested, causng the executon te s ntractable n the worst case. Below, we propose another heurstc algorth to reduce the coputaton te. The Thrd Algorth: The thrd algorth s based on the PT approach to anage ob schedulng. The algorth s decoposed nto three parts as the frst algorth. The frst part fors the achne groups, each of whch contans a achne fro each center. The second part uses the LPT ethod to assgn obs to each achne group (flow shop). The thrd part deals wth ob sequencng and tng usng the PT procedure for a flow shop. The proposed algorth s stated below. The proposed LPT_PT flexble flow-shop algorth: Input: A set of n obs, each havng ( > 2) tasks, to be executed respectvely on each of achne centers wth p parallel achnes. Output: A schedule wth a nearly copleton te. Part : Forng the achne groups: The sae as n the frst algorth. Part 2: Assgnng obs to achne groups: The sae as n the frst algorth. Part 3: Dealng wth ob sequencng n each flow shop: Step 0: For each flow shop F, set the ntal copleton te of the achnes f ( = to, = to p) to zero. Step : For each ob, calculate t D = t k and k = ( / 2) + t D = t k k = ( + ) / 2 / 2 C t = t k and k = ( ) / 2 C t k for even, and + t = k= for odd. Step 2: Schedule the obs n each flow shop f accordng to the t and t values by the C Johnson algorth. Denote the schedule n F as Q F. D 892 Step 3: For each flowshop F, assgn the frst ob J n Q F to the achnes such that J s assgned to F, J 2 s assgned to F 2,, and J s assgned to F. Step 4: Add the processng te t to the copleton te of the frst achne f ; that s: f = f + t. Step 5: Set f (k+) =(f k, f (k+) )+ t (k+), for k = to (-). Step 6: Reove ob J fro Q F. Step 7: Repeat Steps 3 to 6 untl Q F s epty. Step 8: Set the fnal copleton te of each flowshop f = the copleton te of the -th achne f. Step 9: Fnd the u fnal copleton p te ff = ( f ) aong the copleton te = of all the flowshops. After Step 9, schedulng s fnshed and a total copleton te ff has been found. Experents: Ths secton reports on experents ade to show the perforance of the proposed schedulng algorths. They were respectvely pleented by Vsual C++ at an AMD Athlon(t) XP 800+ PC. In the frst part of the experents, fve sets of probles were tested, respectvely for 3 to 7 obs. Each ob has three tasks and each achne center has two parallel achnes. The executon te of each task was randoly generated n the range of 5 to 50. Each set of probles was executed for 20 tests and the akespans and coputaton tes were easured. The proposed optal approach dd not work for ore than seven obs n lted te of 0 hours n our envronents due to the large aount of coputaton te. The optal approach consdered all possble cobnatons and used a prunng technque to ncrease ts effcency. The akespans obtaned n ths way were optal. The akespans for probles of three to seven obs by the three proposed ethods are shown n Fg. 2 to 4. Makespan The Optal Algorth The Frst Algorth The Thrd Algorth Test nuber Fg. 2: Makespans of 20 tests for three obs

7 A. J. Appled Sc., 4 (): The Optal Algorth The Frst Algorth The Thrd Algorth The Optal Algorth The Frst Algorth The Thrd Algorth 55 Makespan Makespan Average akespan Test nuber Fg. 3: Makespans of 20 tests for fve obs The Optal Algorth The Frst Algorth The Thrd Algorth Test nuber Fg. 4: Makespans of 20 tests for seven obs The Optal Algorth The Frst Algorth The Thrd Algorth Job nuber Fg. 5: Average akespans obtaned by the three proposed algorths Fro Fgs. 2 to 4, t s easly seen that the akespans by the proposed three algorths have the followng relaton: Algorth 3 > Algorth > Algorth 2. It s totally consstent wth our expectaton. The devaton percentages for the frst and the thrd algorths fro the optal algorth for processng dfferent nubers of obs are shown n Table 5. The average devaton percentage for the frst and the thrd proposed heurstc algorth fro the optal algorth s respectvely 5.05% and 6.9%. Note that the devaton rate for the second algorth s 0% snce t s an optal approach. CPU t e ( n) Job nuber Fg. 6: The average CPU tes for processng dfferent nubers of obs Table 5: The dstrbuton of devaton rates for dfferent nubers of obs and the run nuber s 20 Proble Sze n Run nuber No. Optals The frst algorth Largest Devato n (%) Average Devato n (%) No. Optals The thrd algorth Largest Devato n (%) Average Devato n (%) Total In the second part of the experents, we extend the ob nuber to 25. The average akespan for probles wth three to twenty-fve obs are shown n Fg. 5 for coparson. Note that the optal approach can process no ore than seven obs n ths envronent. The average CPU tes for probles of three to twenty-fve obs are shown n Fg. 6. The optal algorth proposed cannot run over seven obs n ten hours due to ts hgh te coplexty. Fro Fgs. 5 and 6, t s easly seen that the frst and the thrd algorths got a lttle larger akespans than the second one dd. The coputatonal te needed by the second algorth was, however, uch larger than that needed by the other two approaches, especally when the ob nuber was large. Actually, 893

8 A. J. Appled Sc., 4 (): snce the flexble flow-shop proble s an NP-hard proble, the second approach can work only for a sall nuber of obs. As to the frst and the thrd algorths, the latter got a lttle larger akespan but used less coputatonal te than the forer one. The forer can be appled for solvng a edu-szed proble. At the last part, experents for large ob nubers rangng fro 3 to 8000 were executed for verfyng the effcency of the thrd approach. The average CPU tes for dfferent obs are shown n Fg. 7. It can be observed that all the executon tes are less than 0.8 seconds. Hence, the thrd approach s feasble and effcent even for a large nuber of obs. It s thus ore sutable than the other two proposed approaches for real applcatons. CPU te (sec) The Thrd Algorth Job nuber Fg. 7: The average CPU tes for processng 3 to 8000 obs by the thrd approach CONCLUSION Approprate schedulng cannot only reduce anufacturng costs but also reduce the ateral handlng cost and te. Fndng good schedules for gven sets of obs can thus help factory supervsors control ob flows and provde for good ob sequencng. Schedulng obs n flexble flow shops has long been known an NP-hard proble. In ths paper, we propose three algorths to solve flexble flow-shop probles of ore than two achne centers. The frst one extends Srskandaraah and Seth s ethod by cobnng both the LPT and the search-and-prune approaches to get a nearly optal akespan. It s sutable for a eduszed nuber of obs. The second one s an optal algorth, entrely usng the search-and-prune technque. It can work only when the ob nuber s sall. The thrd one s slar to the frst one, except that t uses Petrov s approach (PT) to deal wth ob sequencng nstead of search-and-prune. It can get a polynoal te coplexty, thus beng ore sutable for real applcatons than the other two. Experental results show that the coputatonal tes by the proposed three algorths have the followng relaton: Algorth 3 < Algorth < Algorth 2, and the akespans have the followng relaton: Algorth 3 > Algorth > Algorth 2. It s totally consstent wth our expectaton. A trade-off can thus be acheved between accuracy and te coplexty. The choce aong the three proposed approaches to solve a flexble flow-shop proble thus depends on the proble sze, the allowed executon te and the allowed error. In the future, we wll consder other task constrants, such as setup tes, due dates and prortes. REFERENCES. Capbell, H. G., R. A. Dudek and M. L. Sth, 970. A heurstc algorth for the n ob, achne sequencng proble. Manageent Scence., 6: B630-B Chung, S. C. and D. Y. Lao, 992. Schedulng flexble flow shops wth no setup effects. The 992 IEEE Internatonal Conference on Robotcs and Autoaton., pp: Dudek, R. A., S. S. Panwalkar and M. L. Sth, 992. The lessons of flowshop schedulng research. Operatons Research., 40: Gupta, J. N. D., 97. A functonal heurstc algorth for the flowshop schedulng proble. Operatons Research., 40: Hong, T. P., C. M Huang and K. M. Yu, 998. LPT schedulng for fuzzy tasks. Fuzzy Sets and Systes., 97: Johnson, S. M., 954. Optal two- and three-stage producton schedules wth set-up tes ncluded. Naval Research Logstcs Quarterly., : Logendran, R. and N. Nudtasoboon, 99. Mnzng the akespan of a group schedulng proble: a new heurstc. Internatonal Journal of Producton Econocs., 22: Morton, T. E. and D. W. Pentco, 993. Heurstc Schedulng Systes wth Applcatons to Producton Systes and Proect Manageent. John Wley & Sons Inc., New York. 894

9 A. J. Appled Sc., 4 (): Nawaz, M., J. E. E. Enscore and I. Ha, 983. A heurstc algorth for the -achne, n-ob flowshop sequencng proble. Oega., (): Paler, D. S., 965. Sequencng obs through a ult-stage process n the nu total te- a quck ethod of obtanng a near optu. Operatonal Research Quarterly., 6(): Petrov, V. A., 966. Flow Lne Group Producton Plannng. Busness Publcatons, London. 2. Srskandaraah, C. and S. P. Seth, 989. Schedulng algorths for flexble flow shops: worst and average case perforance. European Journal of Operatonal Research., 43: