Genetic Algorithm-based Hybrid Methods for a Flexible Single-operation Serial-batch Scheduling Problem with Mold Constraints

Transcription

1 Sensors & Transducers, Vol. 55, Issue 8, August 0, pp. -4 Sensors & Transducers 0 by IFSA Genetc Algorthm-based Hybrd Methods for a Flexble Sngle-operaton Seral-batch Schedulng roblem wth Mold Constrants Hegen Xong, Hual Fan, Gongfa L School of chne Automaton of Wuhan Unversty of Scence Technology, No. 947, Hepng Road, Qngshan Dstrct, Wuhan Cty, 4008,Chna Tel.: , fax: E-mal: xonghegen@.com Receved: 4 July 0 /Accepted: August 0 /ublshed: 0 August 0 Abstract: Schedulng n a manufacturng system s currently a research hotspot. Ths paper proposes a flexble, sngle-operaton, seral-batch schedulng problem wth mold constrants that arse from njecton moldng producton. In such a schedulng problem, three decson problems exst: product batch splttng, sequencng, resource selecton. A dvde--conquer framework s ntroduced to solve the three decson problems. A hybrd algorthm combnng the genetc algorthm for sequencng wth the heurstc algorthm for resource selecton s developed. We desgn an expermental case to whch the correspondng smulaton schedulng process s appled. We present a detaled analyss of the smulaton results, whch ndcate that the soluton framework the hybrd algorthm are effectve. Copyrght 0 IFSA. Keywords: Seral-batch schedulng, Mold constrants, Injecton moldng producton, Genetc algorthm, Dspatchng rule.. Introducton The schedulng problem n a manufacturng system has been extensvely studed over the past decades. Based on the processng method, schedulng problems can be classfed nto two types: unprocessng batch processng schedulng problems. In batch processng schedulng, jobs are processed n batches. Ths process ndcates that ether two or more jobs are processed smultaneously n a machne or jobs n the same famly are processed contnuously serally n a machne. The frst case s generally referred to as parallel batch schedulng, whereas the latter s called seral-batch schedulng (SBS). Batch processng schedulng has attracted consderable research nterest for ts practcal engneerng applcatons. In 984, Santos studed sngle-machne schedulng problems wth batchng sequencng decsons wth lead-tme consderatons. Thereafter, over 00 studes on batch schedulng have been conducted. A great deal of research has been performed by Kovalyov, Brucker, otts, who studed molds, algorthms, as well as the complexty of SBS for sngle [, ] for parallel machnes [, 4]. The models presented were characterzed by the avalablty of all jobs at tme zero, set-up tme consderaton, weghted delay-assocated objectve functons. The algorthms that were commonly adopted n prevous studes nclude the dynamc programmng, branch bound, smulated annealng, taboo search, genetc algorthms [-], wth dynamc programmng beng the most wdely appled. Moreover, for sngle-machne mult- Artcle number _SI_09

2 Sensors & Transducers, Vol. 55, Issue 8, August 0, pp. -4 machne SBS, Ghosh developed a backward dynamc programmng model [7], whch s an mprovement over dynamc programmng [8], as well as a pseudopolynomal dynamc programmng method [9]. Webster et al. studed the complexty of SBS for parallel machnes [0]. A sngle-machne SBS for flow tme mnmzaton was explored by son et al. []. Cheng also conducted a great deal of research on batch schedulng, partcularly on snglemachne SBS problems [,, ]. Cheng proposed several knds of dynamc programmng methods that consdered pseudo-polynomal-tme. In ths paper, we address a new, flexble, sngleoperaton SBS problem wth mold constrants. In ths problem, multple products each have a batch wth a predetermned sze, a release tme, a due date, an mportance measurement (.e., weght). The jobs for all products have only one operaton (e.g., njecton processng) that has to be processed on a machne by usng a mold based on the product famly. The number of avalable molds for a famly job may be more than one. A job can be processed by a machne selected from a specfed machne set. However, the matchng degree n terms of technology, processng economcs, processng velocty for a machne wthn the same job-specfed machne set may dffer. Moreover, dfferences exst n terms of the colors of jobs belongng to dfferent product famles. When a machne processes a deep-color job before a lghtcolor job, ths machne has to be cleaned. For the schedulng problem, the objectve s to mnmze the total weghted tardness (TWT) of all jobs. The remer of ths paper s organzed as follows: Secton states the proposed schedulng problem. Secton presents a detaled soluton to such problem. A case study on the schedulng problem s descrbed n detal n Secton 4. Secton 5 presents the schedulng results correspondng analyss. Fnally, the concluson s presented n Secton.. roblem Descrpton The problem addressed n ths paper can be defned as follows: n batches of products {,,,,, n } are to be processed. The famles due dates of dfferent batches are not the same. In a shop, ma machnes,,, } mo sets of molds { ma { Mo, Mo,, Momo Mo } can be used. For product, we denote n as ts batch sze w, co, p, r, d as ts weght, color, job processng tme, release tme, due date, respectvely. When a job s to be processed, a correspondng mold must be equpped to the correspondng machne. For a job of, the mold can be selected from Mo { Mo, Mo,, Moj,, Moq }, where q denotes the number of avalable molds for. The processng machne wll be one among those n a machne set expressed as the followng matrx: k m k m k m l l kl ml () In the machne matrx, we dvde the machnes nto m grades accordng to ther technologcal economc matchng degree wth. All machnes n,,,, } have the best { l matchng degree, whereas all machnes n m { m, m,, ml, } have the worst matchng degree. To solve ths schedulng problem, we ntally splt all product batches nto several lots, after whch we schedule all the lots. The schedulng process nvolves the sequencng of lots the selecton of the optmal machne the optmal mold, such that the TWT of all products s mnmzed [4]. Let s denote the processng startng tme of ; let c t be the processng fnshng tme the weghted tardness, respectvely. The TWT T of all products can thus be derved as follows: T t,,, n,,, n w max(0, c. roblem Solvng Methods.. Solvng Framework d ) () To solve the gven schedulng problem, three decsons have to be made: () splttng of all product batches nto lots; () optmally sequencng all lots; () selectng the optmal processng machne mold. A dvde--conquer framework s ntroduced to address the aforementoned decsons (Fg. ). Frst, all batches of products were splt based on a splttng rule. We then sequenced all product lots by usng the genetc algorthm, where a chromosome corresponds to a lot sequence. Fnally, a chromosome s decoded accordng to the decson rule resource selecton, whle smultaneously calculatng the processng tme ntervals calculatng. The splttng result, the lot sequence, the schedule comprse the soluton to the schedulng problem.

3 Sensors & Transducers, Vol. 55, Issue 8, August 0, pp. -4 Fg.. Framework for solvng the schedulng problem... roduct Batch Splttng As prevously mentoned, when a job s to be processed on a machne, a mold must be equpped beforeh. Thus, ths procedure requres a relatvely long setup tme. When a machne processes a job belongng to one product famly that s then mmedately preceded by a job from another product famly, the dsassembly of the prevous mold the assembly of the subsequent one would cost a great deal of tme. To reduce setup tmes whle mprovng processng effcency by fully utlzng mold resources, we splt the product batches based on the followng rules: For, f only one mold s avalable,.e., q =, then the batch of product need not be splt, the products should consecutvely be processed on a machne wth the same mold. For, f more than one mold s avalable,.e., q, the batch of product can possbly be splt nto several lots for concurrent processng on dfferent machnes wth dfferent molds. Thus, we splt the batch accordng to the equpartton prncple x nto q lots that are denoted as, x x,,, q. Let b x denote the sze of. Then, b x n / q x,,, q, () n ( q ) n / q x q where x s the floor functon about x. Notably, gven that splttng precedes resource selecton, we splt a product batch nto small lots, such that for product, the number of lots equals the statc avalable quantty of mold q. Thus, the decson space for subsequent schedulng wll be enlarged. However, consderng dynamc factors such as mold mantenance breakdown when schedulng, the dynamc avalable quantty of a mold may be only one. Thus, all lots may be assgned to the same mold the same machne for consecutve processng.. Genetc Algorthm for Lot Sequencng To sequence all product lots, we adopt the genetc algorthm based on symbol encodng, where every chromosome represents a sequencng scheme. By schedulng the sequencng scheme, we can make the processng resource decson as well as calculate the processng tmes TWT.... Chromosome Encodng By splttng all product batches, we can derve the number of product lots x, where,,, n, x,,, q. Let N sub denote the number of lots of all product batches. We thus obtan N sub q,,, n (4) Every legal chromosome s an orderly symbol strng comprsng x (,,, n; x,, q ) wth length N sub. For example, three product batches exst:,,, each of whch are splt nto three lots:,, ;,, ;,,, respectvely. Fg. llustrates a case of a legal chromosome. Fg.. Case of a legal chromosome.... Selecton Operator In ths paper, we use tournament selecton as the selecton operator. Commonly used methods nclude bnary ternary tournament selecton. Evdently, a greater number of chromosomes partcpatng n a tournament results n the hgher homoplasy hgher average ftness of such chromosomes. Thus, 4

4 Sensors & Transducers, Vol. 55, Issue 8, August 0, pp. -4 the evolutonary procedure wll facltate rapd convergence. However, ths procedure may cause populaton dversty to worsen catastrophe to occur. To acheve a tradeoff between convergence rate populaton dversty, we found that an approprate number of ndvduals to be consdered n tournament selecton may be derved accordng to the populaton sze as follows: 4 f : popu 0 f : popu 40, (5) else where popu denotes the populaton sze. For example, let popu 0, let bnary tournament selecton be adopted. For each nstance that two chromosomes are drawn through to smple rom samplng wth replacement, the ftness of each chromosome s evaluated, the ndvdual wth the hgher ftness s retaned for the subsequent genetc operaton. Ths process s repeated untl 0 chromosomes are selected.... Crossover Operator In the proposed genetc algorthm, a chromosome s a symbol strng comprsng the representaton of all product lots. Thus, a legal chromosome must satsfy the completeness exclusveness crtera. art mappng crossover s adopted to ensure that the offsprng are legal. We take a sample that ncludes three product batches:,,, each of whch are splt nto three lots. Fg. (a) represents two parent ndvduals. After applyng part mappng crossover, we derve the offsprng ndvduals as shown n Fg. (b). parent parent offsprng offsprng (a) parent chromosomes. (b) offsprng chromosomes. Fg.. Example of part mappng crossover...4. Mutaton Operator Mutaton s mplemented n the manner smlar to a basc swap operaton,.e., a chromosome s romly selected whle two loc are romly generated. Genes located n the two loc are then mutually exchanged. Takng the offsprng chromosome n Fg. as an example, let the loc be. The resultng mutaton s then shown n Fg. 4. Fg. 4. An example of mutaton..4. Resource Decson In Subsecton., we dscussed that sequencng only provdes the dspatchng order for all product lots. Hence, we should schedule all lots so that the most sutable resources,.e., processng machne mold, could be assgned processng tme ntervals could be calculated. Consderng the technologcal economc matchng degree of machnes wth respect to the product famly TWT objectve, we ntroduce a two-step combnaton rule: Earlest Tardness Tme (ETT) + Best tchng Degree (BMD), whch operates as follows: when a product lot s to be scheduled, a processng resource combnaton (a machne a mold) wth the least weghted tardness wll be assgned for the lot. If more than one processng resource combnaton wth the same least weghted tardness exsts, then the one wth the hghest matchng degree wll be selected. On the other h, f more than one processng resource combnaton wth the same least weghted tardness the same hghest matchng degree exsts, any one can be selected at rom. When decdng on resources, we must calculate the weghted tardness of the lot. Thus, the setup, startng, fnshng tmes are ntroduced n the followng subsectons..4.. Setup Tme Molds are ndspensable technologcal equpment for njecton processng. Before a job s processed, a mold must be set up on a machne. Setup tme, whch s usually exceeds the processng tme of a sngle job, s ths requred. Gven that a mold can be assembled before a job arrves that setup tmes are ndependent of the number of jobs processed on the mold, setup tme should not be ntegrated wth processng tme. The composton of setup tmes can be classfed nto four cases: a machne processes a product lot wthout any forerunner, because of whch assemblng tme occurs; a machne successvely processes two lots of products belongng to the same famly, because of whch no setup tme occurs; a machne successvely processes two lots of products belongng to dfferent famles the color of the former product s the same as or lghter than that of the latter, because of whch the dsassemblng tme of the mold for the former product lot the assemblng tme of the mold for the subsequent product occur; fnally, the color of the former product s darker than that of the latter, because of whch the setup tme wll nclude the dsassemblng, 5

5 Sensors & Transducers, Vol. 55, Issue 8, August 0, pp. -4 assemblng, washng tmes. Let the product lot to be scheduled now be. The selected machne s ( ); the selected mold s Mo ( Mo Mo ); the prevous product lot ts mold are Mo, respectvely. We then provde the followng conventons: If two molds are the same, Mo Mo ; If the color of product s lghter than that of product, co co ; f the color of product s the same as or darker than that of product, co co. For product, we denote the assemblng, dsassemblng, washng tmes as t asm, t dasm,, respectvely. The setup tme, denoted as t setup t wash, can then be presented as the number of cavtes. Let denote the number of cavtes of Mo, such that dentcal jobs can be processed n an njecton technologcal cycle. The processng tme for these jobs s equal to that for processng a sngle job,.e., p. As prevously mentoned, the processng velocty for a machne wthn the same job-specfed machne set may be dfferent. Thus, we adopt the velocty factor v for to measure processng velocty. For the product lot, the sze of whch s b, the processng tme bp s calculated by bp b p / v where x s the celng functon about x., (8) t _ aux t 0 t t t asm wash asm asm t t dasm dasm t wash f : empty f : Mo Mo f : Mo Mo f : Mo Mo f : Mo Mo co co co co co co co co ().4.. Startng Tme In the proposed schedulng problem, for each product lot to be processed, the followng condtons must be satsfed: () the product lot reaches the correspondng machne; () at least one correspondng mold s avalable at each moment; () at least one correspondng machne s avalable at each moment; (4) the mold s assembled on the machne. Wth respect to product lot, the avalable tme of the mold s denoted as t Thus, the startng tme, denoted as computed by s max{ r, t max( t asm, c t t Mo aval wash dasm whch s llustrated n Fg. 5.. Mo aval s, can be )} (7) Fg. 5. Illustraton of startng tme computaton Fnshng Tme Weghted Tardness For a product lot, the fnshng tme c can be calculated based on the startng processng tmes by usng the followng formula: c s max{ r, t max( t bp asm b / p v, c t t Mo aval wash dasm the weghted tardness t w max(0, c )} t can be computed by d ) (9) (0).4.. rocessng Tmes of a roduct Lot In njecton moldng producton, a mold usually has a mult-cavty feature, such that t can smultaneously process more than one job accordng.4.5. Resource Selecton After generatng a processng sequence for the product lots, the most sutable resources should be assgned for each lot. Accordng to the resource

6 Sensors & Transducers, Vol. 55, Issue 8, August 0, pp. -4 decson combnaton rule EET+BMD, we can derve the resource selecton algorthm as procedure. rocedure [rocedure for resource selecton] t benchmark : benchmark of weghted tardness selected : selected machne selected Mo : selected mold Begn t benchmark For k= to k=m For l= to l= y k : kl For j= to q End f End for End for End for End : Mo j Mo t w max(0, c d ) If ( t tbenchmark ) benchmark : t t selected : selected Mo Mo : Referrng to the defnton of the technologcal economc matchng degree of machnes, we can easly determne that matchng degree can be measured by usng the loop varable k. A smaller value of k denotes a hgher matchng degree. In the resource selecton procedure, loop varable k progressvely ncreases as the loop runs, such that resources wth a hgher matchng degree wll be selected automatcally. 4. Case Study 4.. Case Informaton In ths secton, we dscuss a case to llustrate the schedulng problem the proposed solvng methods. In the proposed case, 5 batches of products are to be processed n a shop wth eght sets of machnes sets of molds. The nformaton on product batches s presented n Table. The detals on the machnes molds are presented n Table Table, respectvely. The washng tmes necessary for a machne to process two dfferent colors of products successvely are gven n Table 4. Sometmes, a color taboo exsts for when two products are processed successvely n the same machne. For example, f the color of the prevously processed product s black, the mmedate successor should not be a whte product. To mplement such a taboo, we set an approprate duraton for the washng tmes, e.g., 000 unts of tme, as shown n Table 4. Table. Informaton on product batches. Basc nformaton Resource nformaton rocessng ID Color Batch sze Weght Release tme Due date chne set Mold set tme whte { Mo, } Mo 4 yellow { Mo 5 } blue 4 { Mo } red { Mo 7 8 5} black { Mo } whte { Mo, 5 7 Mo8 7 8 whte { Mo 4 9} red { Mo 0} 5 { Mo 4 blue , Mo 5 yellow { Mo } } 7

7 Sensors & Transducers, Vol. 55, Issue 8, August 0, pp. -4 Table. Informaton on machnes. ID Intal avalable tme Velocty factor ID Table. Informaton on molds. Number of cavtes Intal avalable tme Assemblng tmes Dsassemblng tmes Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Table 4. Washng tme for color swtchng. Color of Color of subsequent product prevous product whte yellow red blue black whte yellow red blue black Smulaton Schedulng the Analyss of the Results 5.. Smulaton Schedulng the Results To analyze evaluate the proposed schedulng methods, we develop a smulaton schedulng system usng Java programmng language. We conduct 0 runs of smulaton schedulng for the above case usng the system. In our smulaton schedulng runs, the operaton parameters of the genetc algorthm are set as follows: populaton sze s 0, crossover probablty s 0.8, mutaton probablty s 0., the maxmum teratons are 00. The eltst selecton strategy s adopted n the genetc algorthm. Results of the 0 runs of smulaton schedulng are shown n Table 5. In these results, the optmal ftness,.e., the mnmal TWT, s 97., the correspondng soluton s llustrated n Fg. 7. Table 5. Results of 0 runs of smulaton schedulng based on the genetc algorthm. Results Number of Generaton th Best Run runs for the best ftness tmes(s) ftness roduct Batch Splttng Chromosome Encodng Accordng to the proposed rules for the splttng of product batches,,, 0 4 can each be splt nto two product lots, such that all batches would ntally have two sets of avalable molds. However, other product batches cannot be splt, gven that each of them has only one set of mold s avalable. Thus, we have 9 product lots to be scheduled. A legal chromosome wth 9 genes s llustrated n Fg.. Fg. 7. Gantt chart of the best schedulng soluton based on the genetc algorthm. Fg.. Example of a legal chromosome. In our smulaton schedulng, dspatchng rules, wth whch we am at evaluatng the proposed genetc algorthm, are also adopted n product lot sequencng. Based on the features of the proposed schedulng problem, we propose two dspatchng rules: Shortest 8

8 Sensors & Transducers, Vol. 55, Issue 8, August 0, pp. -4 Estmate of Batch rocessng Tme (SEBT) Least Estmate of Batch Slack (LEBSL). For a x product lot, the estmates of batch processng tme of batch slack, respectvely denoted by x p x bˆ bˆ s, are defned as follows. Estmate of batch processng tme: For any of a product lot, multple processng machnes are avalable, such that the processng tmes are machnedependent. We can only derve an estmate of the batch processng tmes before the resource decson s made. By gnorng the velocty factor, bˆ p x can be derved as b pˆ p b () Estmate of batch slack: Gven that the startng processng tmes of a product lot are dependent on the selected machne the selected mold, slack cannot be calculated accurately before the resource decson s made. We thus provde the estmate of batch slack as follows: bsˆ x x d ( r bpˆ ) () Wth respect to SEBT, the prorty of a product x x lot s gven as Z bp ˆ, wth respect to x x LEBSL, the prorty s gven as Z bs ˆ.The product lot wth the least Z x s chosen for loadng. Table presents the detaled results of the rulebased smulaton schedulng. The Table shows that the TWTs for SEBT LEBSL are , respectvely. roduct lots chne Mold Table. Results of rule-based smulaton schedulng. Results based on SBT (TWT=85.0) Startng Fnshng chne tme tme Mo Results based on LEBSL (TWT=44.8) Startng Fnshng Mold tme tme Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo

9 Sensors & Transducers, Vol. 55, Issue 8, August 0, pp Analyss of Results 5... Comparatve Analyss of the Results of Genetc Algorthm-based the Rule-based Schedulng The mnmal TWT for genetc algorthm-based schedulng s 97., those for SEBT LEBSL are , respectvely. Genetc algorthm-based schedulng evdently outperforms rule-based schedulng. An effectve dspatchng rule should generally reflect the effects of the man factors on schedulng. However, n the proposed schedulng problem, several factors affect the schedulng results ncludng product batch nformaton, multple machnes wth dfferent technologcal economc matchng degree, multple molds, sequence-dependent washng tmes. Thus, developng a rule wth sngle-valued prorty ncludng all nfluencng factors s dffcult. Thus, achevng a satsfactory schedulng result by usng rule-based sequencng for the schedulng problem proposed n ths paper s dffcult Analyss of the Results of Genetc Algorthm-based Schedulng Table 5 shows that when populaton sze s 0 the maxmum teratons are 00, eght out of 0 runs of smulaton schedulng acheve the best ftness,.e., 97. two runs acheve a ftness of 5.. We conduct another schedulng test wth populaton sze 50 maxmum teratons of 00, n whch all 0 runs acheved the best ftness at 97.. The results ndcate that the genetc algorthm-based solvng methods proposed n ths paper have good stablty. Fg. 7 llustrates the best schedulng soluton through a Gantt chart. Based on Fg. 7, the detaled analyses are gven as follows: In the case wth mold assemblng tmes, the startng tme for the frst product lot on a machne lags behnd the ntal avalable tme of the machne. For example, the ntal avalable tme of s 0. However, the assemblng tme of Mo, whch s assgned to the frst product lot, s 50. Thus, the startng tme of s 0 (0 plus 50). On 5, the red product lot s processed mmedately before the whte product lot 0. Ths order s unreasonable because red s darker than whte, thus requrng washng tmes. However, we note that no tardness occurred wth the two product lots n ths processng sequence. However, f we swtch the sequence, wll be delayed. Furthermore, gven that the weght of s relatvely large ( w =), a large weghted tardness wll be ncurred for. The same condton also appears on ( 4 precedes 0 ). As mentoned n Subsecton., splttng a product batch nto small lots ams at enlargng the decson space for schedulng. If an dle mold s avalable for a product lot at ts schedulng moment, parallel processng of the same famly of product lots may occur. Otherwse, dfferent lots belongng to the same product batch wll be scheduled to be processed consecutvely on the same machne to mnmze setup tmes. For example, product batch s splt nto two lots,. However, these lots are ) processed assgned to the same machne ( consecutvely. The same condton also occurs for 4. Wth respect to, such results not only reduce setup tmes but also ensure ther due date (.e., tardness s zero). For 4, tardness occurs for product lot 4. However, to prevent the delay of other product lots, the tardness of 4 wll be larger f we assgn t to another machne. Fg. 7 shows that parallel processng occurs for product batch 0, whch s splt nto two lots assgned to dfferent machnes ( 5 ) to processed usng dfferent equpped molds ( Mo 7 Mo 8 ). Such an arrangement ensures 0 s due date. Otherwse, f 0 s not splt, tardness wll be nevtable. s arranged for processng on machne. Ths arrangement seems unreasonable because the matchng degree of s lower than those of, whch also belong to the processng machne set of. However, has the largest velocty factor (.e.,.5), such that the due date of can be ensured. If s assgned to or, the respectve velocty factors of whch are 0.8.0, tardness wll nevtably occur for.. Conclusons Ths paper addressed a flexble, sngle-operaton SBS problem wth mold constrants arsng from njecton moldng producton. To solve ths problem, we propose solvng methods ncludng genetc algorthm-based sequencng rule-based resource decson. A case wth 5 product batches s studed by conductng smulaton schedulng. In the case study, we also propose two rules, SEBT 40

10 Sensors & Transducers, Vol. 55, Issue 8, August 0, pp. -4 LEBSL, whch are adopted durng product lot sequencng. The results the analyss show that the genetc algorthm-based sequencng method has good stablty evdently outperforms the rulebased sequencng method. Acknowledgements Ths research was support by Research Foundaton of Educaton Bureau of Hube rovnce, Chna (Grant No. D00). The authors are grateful to the anonymous revewers for ther valuable suggestons comments. References []. S. Albers,. Brucker, The complexty of onemachne batchng problem, Dscrete Appled thematcs, Vol. 47, Issue, 99, pp [].. Brucker, M. Y. Kovalyov, Sngle machne batch schedulng to mnmze the weghted number of late jobs, thematcal Methods of Operatons Research, Vol. 4, Issue, 995, pp. -8. []. M. Y. Kovalyov, Y. M. Shafransky, Batch schedulng wth deadlnes on parallel machnes: An N-hard case, Informaton rocessng Letters, Vol. 4, Issue, 997, pp [4].. Brucker, M. Y. Kovalyov, Y. M. Shafransky, F. Werner, Batch schedulng wth deadlnes on parallel machnes, Annals of Operatons Research, Vol. 8, Issue 0, 995, pp. 40. [5]. C. N. otts, V. A. Strusevch, T. Tautenhahn, Schedulng batches wth smultaneous job processng for two-machne shop problems, Journal of Schedulng, Vol. 4, Issue, 00, pp []. T. C. E. Cheng, M. Y. Kovalyov, Sngle machne batch schedulng wth sequental job processng, IIE Transactons, Vol., Issue 5, 00, pp [7]. J. B. Ghosh, Batch schedulng to mnmze total completon tme, Operatons Research Letters, Vol., Issue 5, 994, pp [8]. J. B. Ghosh, Gupta J. N. D., Batch schedulng to mnmze maxmum lateness, Operatons Research Letters, 997,, pp [9]. E. Erel, J. B. Ghosh, Batch schedulng to mnmze the weghted number of tardy jobs, Computers & Industral Engneerng, Vol. 5, Issue, 007, pp [0]. S. T. Webster, The complexty of schedulng job famles about a common due date, Operatons Research Letters, Vol. 0, Issue, 997, pp []. A. J. son, E. J. Anderson, Mnmzng flow tme on a sngle machne wth job classes setup tmes, Naval Research Logstcs, Vol. 8, Issue, 99, pp. 50. []. T. C. E. Cheng, Z. L. Chen, C. Oguz, One-machne batchng sequencng of multple-type tems, Computers & Operatons Research, Vol., Issue 7, 994, pp []. L. L. Lu, C. T. Ng, T. C. E. Cheng, Schedulng jobs wth agreeable processng tmes due dates on a sngle batch processng machne, Theoretcal Computer Scence, Vol. 74, Issue -, 007, pp [4]. H. A. J. Crauwels, C. N. otts, L. N. V. Wassenhove, Local search heurstcs for sngle-machne schedulng wth batchng to mnmze the number of late jobs, European Journal of Operatonal Research, Vol. 90, Issue, 99, pp Copyrght, Internatonal Frequency Sensor Assocaton (IFSA). All rghts reserved. ( 4