A NEW HYBRID RESCHEDULING POLICY BASED ON CUMULATIVE DELAY

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1 A NEW HYBRID RESCHEDULING POLICY BASED ON CUULATIVE DELAY Haruhko Suwa, Toshhsa Fujwara Department of Industral and Systems Engneerng, Setsunan Unversty 17-8, Ikedanaka-mach, Neyagawa, Osaka , JAPAN Abstract The present study consders a when-to-schedule polcy n onlne producton schedulng, whch can be applcable to the tmng of reschedulng under a dynamc envronment wth unforeseeable events. Ths paper extends our prevous work and proposes a new hybrd polcy for reschedulng ntroducng reschedulng drven by a cumulatve delay and a forced reschedulng to mantan the qualty of a current schedule. We examne, through some computatonal experments, some propertes of the proposed polcy by applyng t to sngle-machne dynamc schedulng problems wth dsturbance where we mnmze total tardness as well as total frequency of reschedulng. It s also demonstrated that the proposed polcy can outperform a typcal perodc reschedulng polcy wth less frequency of reschedulng under parallel machne dynamc schedulng. Keywords: when-to-schedule polcy, cumulatve task delay, schedulng frequency, sngle machne dynamc schedulng, parallel machne dynamc schedulng 1 INTRODUCTION In the real crcumstances, manufacturng actvtes are faced wth a vast range of uncertantes. A scheduler s always charged wth accommodatng changes n the system state. In practce, t s often performed to reschedule, revse the exstng schedule, or let the system gradually absorb the delay, e.g., by rght-shft operatons [1] at some ponts n tme where the system shows changes of ts state. These actvtes are genercally called onlne schedulng or dynamc schedulng. One of the mportant factors n buldng decsonmakng models of onlne schedulng s how well they can manage uncertantes n the manufacturng envronment. A number of studes addressng ths knd of problems have taken one of three approaches: One s to utlze job prortes every tme a faclty becomes dle to select the next job to be processed. Another s to carry out reschedulng perodcally. The other s to revse a current schedule whenever an unexpected event alterng system status occurs. The frst approach, whch s referred to as complete reactve schedulng, s known as dspatchng rules. Varous knds of dspatchng rules have been proposed over the last four decades [2]. The framework of smulaton based schedulng s also wdely used to fnd out approprate dspatchng rules among several canddates. The second approach, whch s referred to as rollng schedules or perodc reschedulng, solves a mult-perod schedulng problem at prevously planned tme ponts and mplements only the frst perod s schedule [3]. The last approach, whch s referred to as predctve-reactve schedulng, wll be effectve snce t ams to cope wth unexpected events by revsng an affected perod of the exstng schedule. Predctve-reactve schedulng s called smply reactve schedulng n the area of producton schedulng and artfcal ntellgence. ost studes addressng reactve schedulng have addressed a how-toschedule polcy, whch dscuss a method of schedule revson, by means of knowledge-based technologes [4, 5, 6, 7] on the bass of an event-drven approach. The termnology event s used here to mean a factor of uncertantes whch may affect the progress of a whole schedule, such as urgent job arrvals, job cancellatons, tools breakage, procurement delays, under- or overestmaton of processng tme or release tme of jobs. In ths approach, substantal events alterng the system status over tme are supposed to be classfed nto () crtcal ones requrng prompt reacton and () those whch can be gnored because every event need not requre an mmedate schedule change [8]. There s, however, no generc model for such event classfcaton. Ths mples that an typcal event-drven approach reactng drectly to contngences has the dsadvantage that t depends on shop floor crcumstances. A few past approaches have addressed buldng a generalzed decson model focusng on a when-to-schedule polcy whch determnes schedule decson ponts n tme[9]. From the above pont of vew, Suwa and Sandoh[10] have proposed a when-to-schedule polcy for job shop schedulng based on cumulatve task delays, whch wll be capable of buldng a generalzed model to determne the tme to schedule. Under ths polcy, schedule nspecton s performed to detect ts delays at planned tmes and make a judgment whether or not reschedule (or schedule revson) should be conducted at each ndvdual planned nspecton tme on the bass of the cumulatve sze of delays. Suwa [11] has extended the cumulatve delay based polcy and appled t to sngle machne dynamc schedulng wth urgent jobs. It s clamed that usage of task delay to trgger schedulng has the followng potental capablty for practcal schedulng decson-makng: (1) to reduce frequent schedulng whch does not necessarly mprove the stuaton wth much expense for ts operaton, (2) to avod overreactng to dsturbances ncurred by the classfcaton problem of events whch strongly depends on an ndvdual shop floor crcumstance, and (3) to smplfy the montorng of the schedule status n schedulng systems. Ths paper extends our prevous work and proposes a new hybrd polcy for reschedulng whch ntroduces cumulatve delay based reschedulng and a forced reschedulng to mantan the qualty of a current schedule. The former reschedulng s trggered when a cumulatve task delay exceeds crtcal cumulatve delay D, and the latter reschedul-

2 ng s performed after prescrbed T unt tme has passed snce reschedulng was conducted most recently before the current tme pont. We frst examne, through some computatonal experments, some propertes of the proposed polcy by applyng t to sngle-machne dynamc schedulng problems wth urgent jobs where we mnmze total tardness as well as total frequency of reschedulng. Then the proposed polcy s compared wth the conventonal perodc and event-drven reschedulng polcy [11] Fnally, It s demonstrated that the proposed polcy outperforms a typcal perodc polcy by applyng them to parallel machne dynamc schedulng wth due dates and sequence-dependent setups where urgent jobs arrve randomly. 2 CLASSIFICATION OF TIING POLICIES Let S 0 and H denote a predctve schedule startng at tme zero and a plannng horzon. We consder nspectons to the exstng schedule s carred out n order to detect schedule delays at planned tme τ = τ ( = 1,,, τ > 0) over perod (0, H] wth τ H where τ 0 denotes the startng pont of schedule executon. Here nspecton ndcates a montorng actvty of the current schedule status and s executed perodcally at ntervals τ. In each nspecton tme τ, the reschedulng decson-makng wll be done as requred. In ths case, jobs assgned after τ are rescheduled (resequenced). The schedulng problem of these jobs can be solved n the framework of statc schedulng. The nspecton tme where reschedulng was actually carred out s called as a reschedulng pont. Focusng on the above two actvtes, nspecton and reschedulng, we can classfy when-to-schedule polces nto the followng three types: perodc reschedulng polcy The faclty s rescheduled perodcally and the newly released schedule s mplemented on a rollng horzon bass[9]. Reschedulng s carred out at τ = H/ where τ = H/, and s the number of reschedules over the perod (0, H]. even-drven reschedulng polcy Perodc nspectons are executed at planned ntervals τ whch should be set to consderably a smaller value. When an unexpected event occurs durng (τ 1, τ ] ( > 0), we make a judgement whether or not reschedulng should be conducted at τ. If t s determned to rearrange the exstng jobs sequence, reschedulng for unprocessed jobs s performed. However, n typcal event-drven reschedulng polces as seen n reactve schedulng systems [5, 6], prompt reschedulng tends to be carred out n reacton to dsturbance, whch mples that, n fact, no judgement of reschedulng s made. perodc and event-drven reschedulng polcy Ths polcy takes a combnaton scheme lyng between two extremes, the perodc approach and the eventdrven approach. In ths polcy, the exstng schedule s updated perodcally at T = (lτ), 2T,, lt (lt H, (l + 1)T > H), and event-drven reschedulng s conducted for the occurrence of an unexpected event. Fgure 1 summarzes the above three types of when-to-schedule polces. The marks and n Fg. 1 respectvely ndcate nspecton tme ponts and reschedulng tme ponts. Early studes on perodc reschedulng polces provde a generc framework for analyses of rollng schedules [3, 12]. H 2H 3H H (a) Perodc reschedulng polcy H τ 0 τ 1 τ 2 τ 3 τ 4 τ 5 τ 6 τ τ (b) Event-drven reschedulng polcy τ 0 τ 1 τ 2 τ 3 τ 4 τ 5 τ 6 τ τ H 2H 3H H (c) Perodc and event-drven reschedulng polcy Fgure 1: When-to-schedule polces. The dsadvantage of the perodc approaches s that compromse of system performance wll be occurred n the face of sgnfcant changes n system state. In the event-drven approach, substantal events alterng the system status over tme are supposed to be classfed nto () crtcal ones requrng prompt reacton and () those whch can be gnored because every event does not requre an mmedate schedule change [8]. There s, however, no generc model for such event classfcaton. Ths mples that an typcal event-drven approach reactng drectly to contngences has the dsadvantage that t depends on shop floor crcumstances. From the vewpont of reducng overreacton, Suwa and Sandoh [10] have proposed an event-drven reschedulng polcy for job shop schedulng wth random machne breakdowns based on cumulatve task delays, whch wll be capable of buldng a generalzed model to determne the tme to schedule. Under ths polcy, schedule nspecton s performed to detect ts delays at planned tmes and make a judgment whether or not schedule revson should be conducted at each ndvdual planned nspecton tme on the bass of the cumulatve sze of delays. In ths study, they showed through computatonal experments that a cumulatve task delay tends to ncrease remarkably and rapdly wth elapsed tme of the schedule. It was also shown that when we emphasze the maxmum tardness of the schedule as well as the schedule revson frequency, the proposed polcy tends to outperform the event-drven reschedulng polcy (.e. reschedulng s conducted every tme a machne breakdown occurs ). The applcablty of the perodc and even-drven approach has been ponted out by a number of lteratures [8, 12]. Suwa [11] has also proposed the combned approach on the cumulatve delay bass. In ths study, the perodc and event-drven polcy was appled to sngle machne dynamc schedulng wth urgent jobs and sequence dependent setup tmes where total setups as well as frequency of reschedulng s mnmzed. A seres of computatonal experments reveal that the polcy has a good performance n mnmzng total setup tmes wth less frequency of reschedulng. However, n some specfc crcumstances, the proposed ap-

3 proach the reschedulng frequency tends to be excess snce two approaches are smply combned. In ths study, we extend our prevous work[11] to a new hybrd, complementary approach, n whch the perodc approach and the event-drven approach are effectvely combned. 3 HYBRID RESCHEDULING POLICY 3.1 Cumulatve Delay For a planned nspecton tme τ, let us denote, by τ l (l 1), the nspecton tme where reschedulng was actually carred out most recently before τ. Further, let J A denote a set of fnshed jobs at τ and J P expresses a set of unfnshed jobs at τ. The realzed schedule, expressed by S A, over perod perod (0, τ ] mght nclude some delays, and let S P denote the predctve schedule after τ. Note that the predctve schedule S P results from reschedulng at τ l. Consder a job j on the schedule S P gven by l j J P l = { J P l C j (S P l ) τ, C j (S A ) > τ } 1 where C j ( ) denotes the completon tme of job j on a schedule *. A set J P represents a set of jobs on S P whch are supposed to fnsh before τ and have note fnshed at τ 1. A l cumulatve task delay D at τ s defned by D = D 1 + δ j (1) where δ expresses a delay of job j ( J P ( φ)) over perod j (τ 1, τ ] and s gven by: δ j = j J P l 0, f C j (S P l ) = C j(s A ) τ (= τ τ 1 ), f C j (S P l ) τ 1 and C j (S A ) > τ τ C j (S P l ), f C j (S P l ) > τ 1 and C j (S A ) > τ C j (S A ) τ 1, f C j (S P l ) τ 1 and C j (S A ) τ C j (S A ) C j(s P l ), f C j (S P l ) > τ 1 and C j (S A ) τ It should be noted that C j (S A ) > τ holds even when job j has not fnshed at τ. 3.2 Procedure At each nspecton τ, the followng tmng of reschedulng s consdered: (1) If τ = τ l + T (T > 0), reschedulng for J P s performed, or (2) If τ < τ l + T, calculate the cumulatve task delay D on S A. If D s found to exceed a prescrbed threshold called a crtcal cumulatve delay D, reschedulng for J P s performed. Otherwse, no reschedulng s carred out but the unfnshed jobs are shfted to the rght as necessary as possble n order to keep feasblty of the exstng schedule. Note that the values of the parameters τ, T and D should be gven beforehand and that the cumulatve task delay D s set to zero after generatng a new schedule S P at reschedulng pont τ. In the followng, reschedulng nvoked by T s reffered to as T-drven reschedulng, and cumulatve task delay based reschedulng as D -drven reschedulng. In the hybrd reschedulng polcy, f the value of T s consderably large, e.g. T > H, the polcy wll perform as an (2) τ 0 τ 1 τ 2 τ 3 τ 4 τ 5 τ 6 τ τ Cumulatve task delay T Fgure 2: Hybrd reschedulng polcy Elapsed tme Fgure 3: Behavor of cumulatve task delay event-drven reschedulng polcy on the cumulatve delay bass. Whle the value of D s consderably large, we can utlze the perodc reschedulng polcy. Ths property ndcates that the hybrd reschedulng polcy can be vewed as a generalzed model of delay based when-to-schedule polcy. Fgure 3 depcts an llustratve landscape of a cumulatve task delay when τ = 50, D = 5000 and T = 500. In Fg. 3, D -drven reschedulng was conducted for fve tmes over perod [1000, 2300], and T-drven reschedulng was performed at 2800 unt tme snce a cumulatve task delay at ths pont n tme dd not exceed the value of D. Compared to the prevous when-to-schedulng polcy (Fg.1(c)), the hybrd approach s expected to reduce reschedulng frequency. In the next secton, ths property s examned by computatonal schedulng smulatons. 3.3 Comparson wth conventonal reschedulng polcy Smulaton schemes We here consder a sngle-machne envronment where jobs of type y (y = 1, 2,, N I ) ncludng urgent jobs arrve randomly durng the schedule executon. It s assumed that a setup tme s yy (> 0, y y ) s requred when a job of type y starts mmedately after a job of type y fnshed. Suppose that the setup tme before processng a job of the same type as a job fnshed prevously s zero. The objectve of reschedulng s to fnd a process order of unfnshed jobs so that total tardness s mnmzed. As a how-to-schedule polcy, we utlzed a smple schedulng algorthm [11]. Ths algorthm groups jobs of the same type and reschedules the jobs from the last schedule whch has not been completed, along wth newly released jobs up D*

4 Table 1: Smulaton Schemes (1) Sngle machne model (a) problem generaton job types N I = 5 total jobs N J = 1000 processng tme Un f orm(1, 10) setup tme Un f orm(5, 10) arrval nterval Exp(1/λ y ), y = 1, 2, 3, 4, 5 arrval rate λ y = 0.01y, y = 1, 2, 3, 4, 5 nstances 5 (b) dsruptons urgent jobs job types Un f orm(1, 5) arrval Exp(1/λ) λ = , 0.005, 0.02 to the reschedulng pont. The procedure of re-sequence for unfnshed jobs s summarzed as follows: Step 1 Jobs of the same type are grouped. Step 2 The jobs are sequenced by FIFO wthn each group. Step 3 The group of the jobs of the same type as the last type processed on the schedule S A starts the new schedule S P, and Step 4 The remanng groups are sequenced by FIFO of the frst job n each group. It s consdered that urgent jobs have a tght deadlne and they are processed pror to the exstng job lst (a set of unfnshed jobs). Table1 shows schemes for schedulng smulaton. Table1(a) summarzes the scheme for generaton of fve problem nstances. Fve job types are consdered and each job of type y arrves followng an exponental dstrbuton wth mean 1/λ y. Table1(b) shows a scheme used to generate urgent jobs. It s postulated that the tme between arrval of two consecutve urgent jobs s ndependent and dentcally dstrbuted, and that t follows an exponental dstrbuton wth mean 50, 200 or 400. From Table1, 15 scenaros were randomly generated. Conventonal reschedulng polcy To observe some propertes of the proposed hybrd polcy, t s compared to our conventonal reschedulng polcy [11],.e., delay-drven and perodc reschedulng polcy mentoned n Secton 2. In ths polcy, the followng tmng of reschedulng s consdered. (1) f the cumulatve task delay D at τ s found to exceed a prescrbed threshold D, reschedulng s performed, and (2) the exstng schedule s always modfed perodcally at planned tmes T(= lτ), 2T,, LT, (LT T, (L+1)T > T ) where l s a postve nteger. Ths ndcates that perodc reschedulng wll be conducted at nspecton tmes τ l, τ 2l,, τ Ll. The conventonal polcy takes a combnaton scheme lyng between two extremes, the perodc approach and the event drven approach, whch mght ncur redundant reschedulng actons. Computatonal results In both reschedulng polcy, the value of D was set to 6000 and T = 800. Note that the conventonal polcy wth D = 6000 provdes good schedules wth less frequency [11]. The nterval of nspecton τ was set to 1. The proposed polcy (Hy) and the perodc and event-drven reschedulng polcy (PE) were appled to each of 15 scenaros for 10 tmes. The followng evaluaton ndces were calculated: %R S = %R F = S Hy S PE 100 S PE (3) F Hy F PE 100 F PE (4) where S α and F α express respectvely the average of total setups and the average of reschedulng frequency by whento-schedule polcy α whch corresponds to Hy or PE. Table 2 summarzes results of smulaton n whch the values n brackets ndcate the standard devaton of %R S and %R F of 10 smulaton runs for each scenaro. The frequency of reschedulng by the proposed polcy tends to be smaller than that by PE especally n the cases of λ = 50 or 200. From the vewpont of robustness, the standard devaton of the frequency by Hy s smaller than that by PE n 14 scenaros. These results ndcate that the proposed polcy s effectve n terms of reducton of reschedulng frequency. The results of total setups reveal that there s no dfference n reschedulng qualty between Hy and PE. It s far to say that the hybrd reschedulng polcy can outperform the perodc and event-drven polcy n the crcumstances nvestgated here. 4 APPLICATION TO DYNAIC SCHEDULING WITH A- CHINES IN PARALLEL In ths secton, the proposed polcy s appled to parallel machne dynamc schedulng summarzed n Table3. The prevous secton dealt wth total setup tmes as a schedule evaluaton, whle n ths secton we consder mnmzaton of total tardness at reschedulng ponts. The due date of each job s gven by ts arrval tme r, the predctve earlest completon tme C and a parameter α whch determnes tghtness of the due date. As a reschedulng decson-makng, we utlze lst schedulng. A lst at τ conssts of all jobs n J P and, at a reschedulng pont, jobs n the exstng lst are resequenced based on the how-to-schedule polcy descrbed n secton 3.3 before dspatchng jobs to machnes. oreover, n Table 3(a), the sze of job group s set to fve and we suppose that jobs are sequenced by EDD frst wthn each group. It s also postulated that the tme between arrval of two consecutve urgent jobs s ndependent and dentcally dstrbuted, and that t follows an exponental dstrbuton wth mean 100, 200 or 400. From Table1, 40 nstances were randomly generated. The values of three parameters D, T and τ n the proposed polcy were set to: D = 3000, 5000, 7000, T = 100, 200, 300,, 1000 (10 patterns), τ = 50,

5 Table 2: Comparson of Proposed Polcy (Hy) wth Perodc/Event-Drven Polcy (PE) Instances 1/λ F Hy F PE %R F S Hy S PE %R S (1.3) 21.0 (2.0) (125.6) (198.0) (2.0) 12.4 (1.3) (55.2) (49.1) (0.5) 9.2 (1.2) (35.4) (32.1) (1.4) 21.3 (2.2) (139.5) (111.8) (0.3) 12.1 (1.3) (50.0) (46.2) (0.5) 9.7 (0.9) (37.6) (29.3) (0.6) 20.6 (1.2) (89.6) (115.4) (0.5) 12.5 (1.1) (25.7) (31.6) (0.0) 10.2 (0.8) (24.6) (31.1) (1.4) 20.8 (1.8) (128.0) (104.2) (0.7) 14.0 (1.9) (28.7) (40.6) (0.4) 11.0 (12.4) (42.2) (42.6) (1.2) 22.1 (2.2) (98.7) (133.0) (0.8) 14.0 (1.6) (34.7) (38.1) (0.3) 10.3 (0.9) (41.3) (57.3) -1.4 overall overall 0.3 whch yelds 30 sub cases of the proposed polcy. The proposed polcy s compared wth a perodc reschedulng polcy n order to examne the performance of the proposed polcy. Under the perodc reschedulng polcy, the exstng schedule (the exstng lst, n fact) s updated at kt (k = 1, 2, ) where T s an nterval of consecutve reschedulng pont. The value of T was set to: T = 100, 200, 300,, Each of sub cases and the perodc reschedulng polcy were appled to generated 40 nstances. Each schedulng smulaton lasted untl the number of fnshed jobs ncludng urgent jobs reaches Fgure 4 shows the performances of the proposed polcy and the perodc reschedulng polcy from the vewpont of Pareto optmalty. The horzontal axs of Fg. 4 expresses the mean number of reschedules, whle the vertcal axs sgnfes the average of total tardness over 20 nstances. The proposed polcy wth D = 3000 sgnfcantly outperforms the perodc reschedulng polcy rrespectve of densty of arrved urgent jobs. However, some sub cases wth larger value of D provdes worse solutons than those by the perodc approach. As a whole, these result ndcate that there s possblty for the proposed polcy to outperform the perodc reschedulng polcy when we focused on total tardness of the whole schedule as well as the number of reschedules The results also mply that t should be needed to buld a model to determne a sutable value of D, whch should be ncluded n our future work. 5 CONCLUDING REARKS Ths paper focused on a new when-to-schedule polcy n onlne schedulng and proposed a hybrd reschedulng polcy on the cumulatve delay bass. The cumulatve task delay can be vewed as aggregated nformaton of dsturbances derved from the dfferences between the predctve schedule and the realzed schedule. Frst, the proposed polcy has been appled to sngle machne dynamc schedulng wth urgent jobs n order to nvestgate ts property. Through some computatonal experments, t reveals that the hybrd reschedulng polcy can outperform the perodc and event-drven reschedulng polcy. Then we appled the proposed polcy to parallel machne dynamc schedulng n order to examne the performance of the polcy. A seres of computatonal experments demonstrated that there s possblty for the proposed polcy to outperform a typcal perodc reschedulng polcy n the envronments nvestgated here. There s a need for further research to observe the behavour of a cumulatve task delay n multple machne envronments and to compare wth an event-drven reschedulng polcy. Future work also ncludes the applcaton to other shop models. 6 ACKNOWLEDGENTS Ths work s partly supported by the Grants-n-Ad for Scentfc Research of the Japan Socety for the Promoton of Scence (JSPS), No REFERENCES [1] Wu, S. D., Storer, R. H. and Chang, P. C., 1993, One- achne Reschedulng Heurstcs wth Effcency and Stablty as Crtera, Computers and Operatons Research, 20, 1, [2] Blackstone, J. H., Phllps, D. T. and Hogg, G. L., 1982, A state-of-the-arts Survey of Dspatchng Rules for anufacturng Job-Shop Schedulng, Internatonal Journal of Producton Research, 20,

6 Table 3: Smulaton Schemes (2) Parallel machne model (a) problem generaton machnes m = 2 job types N I = 5 total jobs N J = 1000 processng tme p y of type y y + 4, y = 1, 2, 3, 4, 5 due date r + (1 + α)(c r), α = 0.1 setup tme Un f orm(1, 5) arrval nterval Exp(1/λ y ), y = 1, 2, 3, 4, 5 arrval rate λ y = 0.01y, y = 1, 2, 3, 4, 5 group (lot) sze 5 nstances 20 (b) dsruptons urgent jobs job types Un f orm(1, 5) processng tme U(5, 10) arrval Exp(1/λ) λ = , 0.01 [3] Baker, K. R. and Peterson, D. W., 1979, An Analytc Framework for Evaluatng Rollng Schedules, anagement Scence, 25, 4, [4] Noronha, S. J. and Sarma, V. V. S., 1991, Knowledge- Based Approaches for Schedulng Problems: A Survey, IEEE Transactons on Knowledge and Data Engneerng, 3, 2, [5] Kerr, R. K. and Szelke, E., 1995, Artfcal Intellgence n Reactve Schedulng, Chapman & Hall. [6] Smth, S.F., 1995, Reactve Schedulng Systems, In: Intellgent Schedulng Systems (Brown, D. E., and Scherer, W. T. (Ed.)), Kluwer Academc Publshers, [7] Suh,.S., Lee, A., Lee, Y. J. and Ko, Y. K., 1998, Evaluaton of Orderng Strateges for Constrant Satsfacton Reactve Schedulng, Decson Support Systems, 22, [8] Church L. K. and Uzsoy R., 1992, Analyss of Perodc and Event-Drven Reschedulng Polces n Dynamc Shops, Internatonal Journal of Computer Integrated anufacturng, 5, 3, [9] Vera, G. E., Herrmann, J. W. and Ln, E., 2000, Analytcal odels to Predct the Performance of a Sngle-achne System under Perodc and Event- Drven Reschedulng Strateges, Internatonal Journal of Producton Research, 38, 8, [10] Suwa H. and Sandoh H., 2003, Cumulatve Delays Drected Reactve Schedulng for Job Shop Problems, Proceedngs of 17th Internatonal Conference on Producton Research, Total tardness Total tardness Number of reschedules (a) λ = : D* =3000 :D* =5000 : D*=7000 : Perodc reschedulng polcy : D* =3000 :D * =5000 : D*=7000 : Perodc reschedulng polcy Number of reschedules (b) λ = 0.01 Fgure 4: Proposed polcy v.s. perodc reschedulng polcy [11] Suwa H., 2007, A New When-to-Schedule Polcy n Onlne Schedulng Based on Cumulatve Task Delays, Internatonal Journal of Producton Economcs ( n press ). [12] uhlemann A. P., Lockett A. G. and Farn C. K., 1982, Job Shop Schedulng Heurstcs and Frequences of Schedulng, Internatonal Journal of Producton Research, 20,