Research Article Heuristic for Stochastic Online Flowshop Problem with Preemption Penalties

Transcription

1 Discrete Dyamics i Nature ad Society Volume 2013, Article ID , 10 pages Research Article Heuristic for Stochastic Olie Flowshop Problem with Preemptio Pealties Mohammad Bayat, Mehdi Heydari, ad Mohammad Mahdavi Mazdeh Departmet of Idustrial Egieerig, Ira Uiversity of Sciece ad Techology, Tehra , Ira Correspodece should be addressed to Mohammad Bayat; mohammadbayat@iust.ac.ir Received 20 May 2013; Accepted 26 August 2013 Academic Editor: Oswaldo Luiz do Valle Costa Copyright 2013 Mohammad Bayat et al. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. The determiistic flowshop model is oe of the most widely studied problems; whereas its stochastic equivalet has remaied a challege. Furthermore, the preemptive olie stochastic flowshop problem has received much less attetio, ad most of the previous researches have cosidered a opreemptive versio. Moreover, little attetio has bee devoted to the problems where a certai time pealty is icurred whe preemptio is allowed. This paper examies the preemptive stochastic olie flowshop with the objective of miimizig the expected makespa. All the jobs arrive overtime, which meas that the existece ad the parameters of each job are ukow util its release date. The processig time of the jobs is stochastic ad actual processig time is ukow util completio of the job. A heuristic procedure for this problem is preseted, which is applicable wheever the job processig times are characterized by their meas ad stadard deviatio. The performace of the proposed heuristic method is explored usig some umerical examples. 1. Itroductio The stochastic flowshop problems have ot peetrated very far ad remai challegig. The stochastic two-machie flowshop problem is iheretly more complex tha its determiistic couterpart. This complexity is much more whe the preemptive olie versio of the problem is cosidered. I other words, the jobs are allowed to be preempted ad restarted. We assume that preemptio ca occurr oly o machie1itheflowshopproblem.moreover,allthejobs arrive overtime, which idicates that the existece ad the parameters of each job are ukow util its release date. I the previous stochastic studies, preemptios are usually assumedtobe free whichmeasthateachjobcabepreempted at ay poit of time ad resumed later without icurrig a pealty. However, this is ot usually the case i practice. I may cases, such as meltig furaces, the time that has bee spet before the preemptio o a job is lost ad cosidered as the preemptio pealty. Schedulig has become a well-studied problem ad there are literally tremedous efforts o providig solutio strategiesforvariouskidsofmodeligformulatiossuchasjob shop ad flowshop. There are literally may applicatios for flowshop problem (e.g., Defersha ad Che [1]; Mahavi et al. [2]; Braglia et al. [3]). Two-machie flowshop problem with makespa objective fuctio ad determiistic processig timeca be optimallysolved by Johso s rule[4]. This problem would be NP-hard if it cosists of three or more machies [5]. Therefore, umerous heuristic algorithms have bee preseted for solvig such problems i various studies. Framia et al. have reviewed some of these articles [6]. Ruiz ad Maroto referred to 53 articles o the heuristics preseted for miimizig makespa i permutatio flowshop problem [7]. I the stochastic flowshop problem, the processig time is a radom variable. This simple differece leads to may complexities i stochastic problems. Makio developed a sequecig rule to fid the schedule that miimizes the expected makespa i a flowshop problem with two jobs ad expoetially distributed processig time [8]. Frostig ad Adiri ivestigated three-machie flowshop stochastic schedulig with a objective of miimizig distributio of schedule legth [9]. Sethi et al. offered feedback productio plaig i a stochastic two-machie flowshop based o asymptotic aalysis ad computatioal results [10]. Elmaghraby ad Thoey

2 2 Discrete Dyamics i Nature ad Society studied the two-machie stochastic flowshop problem with arbitrary processig time distributios [11]. I the stochastic flowshop problem with two machies ad expoetial processig times, the expected makespa value would be miimized i the case that the jobs are sorted oicreasigly i terms of parameter (1/μ i1 1/μ i2 ).Thismethodwas proposed by Talwar ad is kow as Talwar s Rule [12]. Later, Cuigham ad Dutta proved its optimality [13]. Ku ad Niu obtaied a sufficiet coditio for stochastic domiace adshowedthattalwar sruleyieldsastochasticallymiimal makespa [14]. Soroush ad Allahverdi preseted a stochastic two-machie flowshop schedulig problem with total completio time criterio [15]. Laha ad Chakraborty preseted a efficiet stochastic hybrid heuristic for flowshop schedulig [16]. Portougal ad Trietsch applied Johso s Rule to stochastic problems [17]. They utilized, mea of the processig time of each job as its processig time i Johso s Rule. Moreover, Kalczyski ad Kamburowski applied Talwar s Rule for Weibull distributio [18]. Baker ad Altheimer used three heuristic methods for the flowshop problem with m machies, supposig geeral distributios for processig times. They ivestigated the performace of these methods i a set of problems usig simulatio ad oticed that these methods had ear-optimal performace [19]. Baker ad Trietsch also explored three heuristic methods for the twomachie stochastic model with a geeral distributio fuctio. They compared Johso s Heuristic method ad Talwar s Heuristic method (applyig mea of the processig time istead of job processig time i these two methods) ad also the heuristic method of chagig eighborig pairs (two eighborig jobs are separately cosidered ad are displaced if their order ca be optimized) ad figured out that oe of these methods domiate the other [20]. Heydari et al. proposed a heuristic method for miimizatio of expected valueofthetotalweightedcompletiotimeisiglemachie problem with preemptio pealties [21]. I this paper, the heuristic proposed by Baker ad Trietsch has bee applied for sequecigthejobswhicharepresetitheshop. I spite of the various studies i the last decades, a cosiderable research has ot bee performed o preemptive olie flowshopproblemswithstochasticprocessigtimes.ithis paper, a heuristic method to this problem is preseted. The logic applied i this research is emphasizig o miimizig idlig times i the secod machie. I this method, decisio about the preemptio of oe job at arrival of a ew job is made based o the idlig time produced i the secod machie. I this research, we assume that the processig time is a radom variable with ormal distributio, ad processig times of differet jobs are idepedet from each other. The remaider of this paper is orgaized as follows: research assumptios ad defiitios are preseted i Sectio 2. ISectios3 ad 4, opreemptiveadpreemptive stochastic models are reviewed. The, the proposed heuristic method is preseted i Sectio 5. I Sectio 6, a umerical example has bee solved with the proposed algorithm. The performace aalysis of the proposed heuristic method is preseted i Sectio 7. Fially, the coclusios are discussed i the last sectio. 2. Research Assumptios ad Defiitios I the stochastic case, t ip is used to deote (radom) processig time of job i o machie p, butweretaie(t ij ) to represet its expected value. Ad, r i is the release time of job i, which ca oly be kow right o or after r i.theprocessig time of each job is a radom variable with ormal distributio t ip N(μ ip,σ 2 ip ) ad jobs are idepedet of each other. Our objective is to fid a schedule to miimize the expected makespa. The applied symbols are as follows: t ip : processig time of job i o machie p. μ ip : mea of processig time of job i o machie p. σ 2 ip : variace of processig time of job i o machie p. I i : idle time of the secod machie from completig the job i 1util startig the job i. The cosidered assumptios i this article are as follows: (i) Machies have costat speed that caot be varied. (ii) The order of processig the jobs is the same o the first ad secod machies. (iii) Machies are ready to be utilized at zero time. (iv) Every machie ca operate at most oe job at a time. (v) Iitiatio of ay job o the secod machie would be after completio of the job o the first machie. (vi) Preemptio of the jobs ca be occurred oly o machie 1. We cosider the preemptio-repeat model i which the job curretly beig processed may be preempted at ay poit of time. However, by preemptig a job, all the ogoig progress is cosidered to be lost. Therefore, if the job is restarted at some later momet i time, the it has to be processed from the begiig. 3. Nopreemptive Stochastic Flowshop Problem For geeral distributios without ay special coditios o processig times, oly oe thorough solutio is kow for the stochastic flowshop problem, proposed by Makio [8]. Theorem 1. I the two-machie stochastic flowshop problem with two jobs, job i precedes job j i a optimal sequece if E(mi {t i1,t j2 }) E (mi {t i2,t j1 }). (1) Based o Theorem 1, Baker ad Trietsch proposed a method that uses the properties of a adjacet pair wise iterchage (API) [20]. They assume that the processig times are radom variables with ormal distributio. I this method, the coditio for job i to precede job j takes the followig form: μ i1 σ i1j2 [φ (z i1j2 )+z i1j2 Φ(z i1j2 )] μ i2 σ i2j1 [φ (z i2j1 )+z i1j2 Φ(z i2j1 )]. (2)

3 Discrete Dyamics i Nature ad Society 3 This form of the API procedure uses the properties of ormal distributios ad is called the API Heuristic. I this method, the expected miimum of two variables with ormal distributio is give by E [mi {X, Y}] =μ X σ XY [φ (z XY ) +z XY Φ(z XY )], (3) where φ ad Φ deote the desity fuctio ad the cdf of the stadard ormal. I additio, μ XY =μ X μ Y ad σ 2 XY =σ2 X + σ 2 Y ad z XY =μ XY /σ XY [22]. I this paper, for sequecig the jobs preset at the shop, this heuristic method has bee utilized. 4. Preemptive Stochastic Flowshop Problem Sice the objective fuctio of this study is expected makespa miimizatio, ad cosiderig the fact that makespa icrease is due to icrease i idlig time i the secod machie, this paper focused o miimizig the sum of expected idlig times of the secod machie. As show i Figure 1 the value of makespa is as follows: C max = I i + t i2 E [C max ]=E[ I i ]+E[ t i2 ], where I i is the idlig time of machie 2 before the startig of job i. Sciece the value of E[ t i2] is costat; miimizig E[ I i] leads to miimum value of makespa. Idlig time of machie 2 is as follows: I 1 =t 11, I 2 = max {0, t 11 +t 21 t 12 I 1 }, I 2 = max {0, t 11 +t 21 +t 31 t 12 t 22 I 1 I 2 }, (5). k I k = max {0, t i1 k 1 t i2 k 1 I i }. Thus,thetotalidletimeofthesecodmachiewillbecomputed as follows: I 1 +I 2 = max {t 11,t 11 +t 21 t 12 }, I 1 +I 2 +I 3 = max {t 11,t 11 +t 21 t 12, t 11 +t 21 +t 31 t 12 t 22 },. I i = max [t 11, max { k t i1 k 1 t i2 }]. 2 k Assume that U k = k k 1 t i1 (4) (6) t i2. (7) Therefore, the sum of idlig times of machie 2 is give by I i = max (U k). (8) 1 k S 1 S 2 M 1 : M 2 : M 1 : M 2 : M 1 : M 2 : t 11 t 21 t 31 t 41 t 12 t 22 t 32 I 1 I 3 I 4 Figure 1: Two machie flowshop problem. Set A Set A t i1 t i1 r j r j t i1 t j1 t i1 I i I j t j2 t j1 t i2 t i1 I i t j2 t i2 t 42 Set B Set B Figure 2: Schedules S 1 ad S 2 at arrival of job j i preemptio-repeat mode. I the two-machie stochastic olie flowshop problem, if i schedule 1 (S 1 ) job i precedes job j adischedule2(s 2 ), these two jobs are replaced, the E(C S 1 max ) E(CS 2 max ) if E[max 1 k {U k (S 1 )}] E[max 1 k {U k (S 2 )}], (9) where E[max 1 k {U k (S 1 )}] ad E[max 1 k {U k (S 2 )}] are the expectedvaluesofthesumofidligtimeomachie2i schedule 1 ad 2. As metioed before, miimizig the expected value of total idlig time i machie 2 leads to expected makespa miimizatio. Sice the expected value of makespa i S 1 is less tha its value i S 2, the idle time of machie 2 i S 1 will be less tha its value i S 2. I preemptive olie problem, assume that job i with processig time t i1 is beig processed o machie 1. Let t i1 be the remaiig processig time of job i ad the time spet for processig of job i is t i1. Therefore, we have t i1 =t i1 +t i1. (10) Suppose that the ew job j with the processig time t jp ad the release time r j arrives at the shop (Figure 2). The ew job j becomes kow at its release time. Let A be the subset of completed jobs o machie 1 ad B the subset of ucompleted jobs. First, the priority of job j is determied with the API heuristic method. If the priority of job j is more tha the job i the oe of the two S 1 ad S 2 schedules would occur as show i Figure 2;otherwise,itwill be added to list of ucompleted jobs. I S 1 job i is preempted because of the higher priority for job j, butis 2 job j will be processed after completio of job i.asmetioedbefore,

4 4 Discrete Dyamics i Nature ad Society totalidletimeofmachie2utilstartofjobi ad j i S 1 ad S 2 schedulesisasfollows: U j (S 1 )= t h1 +t i1 +t j1 t h2, U i (S 1 )= h1 +t t i1 t h2, U i (S 1 )= t h1 +t i1 +t j1 +t i1 t j2 t h2, U i (S 2 )= t h1 +t i1 t h2, U j (S 2 )= t h1 +t i1 +t j1 t i2 t h2. (11) Therefore, the expected value of makespa i S 1 is less tha its value i S 2 ad preemptio will occur if where β is a costat value. Hece, U j (S 1 ) ad U i (S 1 ) are radom variables with ormal distributio ad the followig meas ad variaces: U j (S 1 ) N(λ β+t i1 μ h2 +μ j1,σ 2 j1 + σ 2 h2 ), U i (S 1 )=N(λ β+t i1 μ h2 +μ j1 +μ i1 μ j2,σ 2 j1 +σ2 i1 +σ2 j2 + I additio, for U j (S 2 ) ad U i (S 2 ) we have σ 2 h2 ). (16) U i (S 2 )=N(λ β μ h2 +μ i1,σ 2 i1 + σ 2 h2 ), E[max {U i (S 1 ),U j (S 1 ),U i (S 1 )}] <E[max {U i (S 2 ),U j (S 2 )}] U j (S 2 )=N(λ β μ h2 +μ i1 +μ j1 μ i2, (17) U j(s 1)>U i (S 1) E[max {U j (S 1 ),U i (S 1 )}] <E[max {U i (S 2 ),U j (S 2 )}] E [mi { U j (S 1 ), U i (S 1 )}] >E[mi { U j (S 2 ), U i (S 2 )}]. (12) The processig of the jobs i the subset A has bee completed o machie 1, ad their total processig time is a costat value. Thus, we assume that t h1 =λ, (13) where λ is a costat value. Some of the jobs i the subset A have bee completed ad some of them have ot bee completed util time r j.leta be the subset of completed jobs o machie 2, ad let A be the subset of ucompleted jobs o this machie. Thus, A A =A, t h2 = t h2 + t h2. (14) σ 2 i1 +σ2 j1 +σ2 i2 + σ 2 h2 ). I order for decisio makig about preemptio, the followig iequality should be explored: Assume that E[mi { U j (S 1 ), U i (S 1 )}] >E[mi { U j (S 2 ), U i (S 2 )}]. X= U j (S 1 ), Y= U i (S 1 ), X = U i (S 2 ), Y = U j (S 2 ). (18) (19) The expected miimum of two variables with ormal distributio is calculated as discussed before. Thus, for computatio of E[mi{X, Y}] we have μ X = λ+β t i1 + μ h2 μ j1, μ XY =μ i1 μ j2, Therefore, t h2 is costat value t h2 is a radom variable with ormal distributio σ 2 XY =2σ2 j1 +σ2 i1 +σ2 j2 +2 σ 2 h2, (20) t h2 =β, t h2 N( μ h2, σ 2 h2 ), (15) z XY = μ XY σ XY E [mi {X, Y}] =μ X σ XY [φ (z XY )+z XY Φ(z XY )],

5 Discrete Dyamics i Nature ad Society 5 Start (i 1), (k total cout of jobs i the shop) Complete job i ad the: i i+1 Prioritize ucompleted jobs from i to k accordig to API heuristic method. Start the processig of job i No Is i the last job? Yes Complete job i No Has a ew job arrived to the shop before completio of job i? Yes Ed No Is the priority of ew job more the job i accordig to API? (j ew job) Yes No E[mi{X, Y}] > E[mi{X, Y }] Yes Preemptio refused (k k+1) Preempt job i ad start the processig of job j (i job j), (k k+1) Figure 3: Proposed heuristic algorithm. ad E[mi{X,Y }] is as follows: μ X = λ+β+ μ h2 μ i1, μ X Y =μ j1 μ i2, σ 2 X Y =2σ2 i1 +σ2 j1 +σ2 i2 +2 σ 2 h2, E[mi {X,Y }] z X Y = μ X Y σ X Y, =μ X σ X Y [φ (z X Y )+z X Y Φ(z X Y )]. (21) job j (assumig that the job j priority is greater tha the job i basedoapi)ifadolyif E [mi {X, Y}] >E[mi {X,Y }]. (22) Proof. As metioed before, by preemptio of job i at arrival of job j the expected value of total idlig time i machie 2 decreases if E[mi{X, Y}] > E[mi{X,Y }]. 5. Proposed Algorithm I this sectio, a heuristic algorithm is proposed to miimize the expected value of makespa for the preemptive stochastic olie flowshop problem with two machies. I additio, a schematicillustratioofthealgorithmisprovidedifigure3. Therefore, the preemptio coditio will be as preseted i Lemma 2. Lemma 2. I the preemptio-repeat stochastic olie flowshop problem with two machies, job i will be preempted at arrival of Step 0 (parameter defiitio). The parameter k deotes the total umber of jobs i the shop ad i is the job couter parameterwhittheprimaryvalueof1.processigtimeof each job is a radom variable with ormal distributio.

6 6 Discrete Dyamics i Nature ad Society Table 1: Specificatios of jobs. i μ i1 σ i1 μ i2 σ i2 r i Table 2: Differeces betwee E[mi{t i1,t j2 }] ad E[mi{t i2,t j1 }]. j i r 5 =15 Arrival time of job i is deoted by r i. Parameter j deotes the ewjobthatarrivesattheshop. Step 1. Prioritize all ucompleted jobs with API heuristic method, from i to k. Step 2. Processthejobi utilitiscompleted,oraewjob arrives at the shop. If a ew job arrives at the shop the assig idex j to its characteristics ad go to Step 4, otherwise go to ext step. Step 3.Aftercompletioofjobi,ifi is the last job the fiish the algorithm; otherwise, icremet the couter i by oe ad go to Step 1. i i+1. (23) Step 4. If the priority of the ew job accordig to API heuristic method is more tha job i the follow the algorithm, otherwisegotostep7. Step 5.IfE[mi{X, Y}] > E[mi{X,Y }] the go to ext step, otherwisegotostep7. Step 6.Thepriorityofjobj is more tha job i.thus,jobi will be preempted ad job j will be preferred. Ad sice the, the idex i will be used for job j. Icremet the couter k by oe, adthegotostep2. k k+1, i job j. (24) Step 7.Preemptioisotallowedadtheprocessigofjobi will be cotiued. Icremet the couter k by oe, ad the go to Step Numerical Example k k+1. (25) I this sectio, the performace of the proposed algorithm is evaluated through a umerical example. Suppose a stochastic two-machieflowshopproblemwith6jobsadspecificatios give i Table 1. I Table 1, r i is arrival time of job i. Moreover, we assume that the actual processig time of a job o each machie is equal to its mea processig time o that machie. Accordig the proposed algorithm, first, the jobs preset at the shop at 4.1: 4.2: 4.3: t 41 t 41 t 21 t 21 Preemptio pealty t 41 r 6 =24 t 31 t 31 t 11 t 11 t 51 t 21 t 61 t 31 t 11 t 51 Figure 4: Schedulig a sample problem with proposed algorithm. time zero should be scheduled with API heuristic method. We ru the API algorithm by startig with the sequece Table 2 shows the differeces betwee E[mi{t i1,t j2 }] ad E[mi{t i2,t j1 }] for these jobs, with job i correspodig to a row ad j to a colum. For istace, the calculatio of E[mi{t 11,t 22 }] is as follows: E[mi {t 11,t 22 }] =μ 11 σ (11)(22) [φ (z (11)(22) )+z (11)(22) Φ(z (11)(22) )], μ 11 = 17, μ (11)(22) =μ 11 μ 12 =17 13=4, σ 2 (11)(22) =σ2 11 +σ2 22 = = , E[mi {t 11,t 22 }] z (11)(22) = μ (11)(22) σ (11)(22) = 1.55, = [ ] = (26) Similarly, E[mi{t 12,t 21 }] = , adthefirstetryi Table 2 is the differece = I Table 2,accordigtoAPImethod,egativevaluesidicate stable sequeces. For istace, 1-2 is ot a stable sequece because the value of row 1 ad colum 2 is positive. Therefore, these two jobs should be substituted. Hece, API heuristic with altered order of the jobs yields the stable sequece as show i Figure 4 (4.1). The machies will start to process the jobs accordig to the sequece Job 5 with release time r 5 =15arrives at the shop as show i Figure 4 (4.1). Job 2 is beig processed atthistime.accordigtoapi,thepriorityofjob5islesstha the priority of job 2 because if we schedule jobs 2, 5 with this

7 Discrete Dyamics i Nature ad Society 7 method,thethejob5willhaslesspriority.thus,thepreemptio is ot allowed. Processig of the jobs is cotiued ad durig the processig of job 3, job 6 with release time r 6 =24 arrives to the shop as show i Figure 4 (4.2). Accordig to API,Thepriorityofjob6ismorethathepriorityofjob3at this time. Thus, the parameters at this time are as follows: i=3, i job 3, j job 6, t 31 =3. (27) Accordig to the proposed algorithm, satisfactio of the iequality E[mi{X, Y}] > E[mi{X,Y }] should be explored i order for decisio makig about the preemptio: Table 3: Completio time of jobs o machies 1 ad 2. i R i1 C i1 R i2 C i2 r i to a lower boud of the optimum makespa that is defied as follows: C OPT max max [r i +E(t i1 )+E(t i2 )], i {1,2,3,...,} C OPT max mi [r i +E(t i1 )] + E(t i2 ), i {1,2,3,...,} μ X = λ+β t i1 + μ h2 μ j1 C l max = max { max [r i +E(t i1 )+E(t i2 )], i {1,2,3,...,} (29) = = 6, μ XY =μ i1 μ j2 =16 14=2, σ 2 XY =2σ2 j1 +σ2 i1 +σ2 j2 +2 σ 2 h2 = = , z XY = μ XY σ XY = E [mi {X, Y}] =μ X σ XY [φ (z XY )+z XY Φ(z XY )] = ( ) = (28) Similarly, E[mi{X,Y }] = Sciece E[mi{X, Y}] > E[mi{X,Y }], preemptio is allowed. Job 3 will be preempted ad job 6 will start. Cosequetly, the schedulig of jobs o machie 1 will be as show i Figure 4 (4.3). The computatio of completio time of the jobs o machie1ad2areprovideditable3,wherec ij is the completio time of job i o machie. As metioed before, it is assumed that the actual processig time of the job is equal to the mea of processig time. Therefore, the expected value of makespa is equal to 85 for this example. 7. Performace Aalysis I order to evaluate the performace of the proposed algorithm, it was applied o a variety of problems with differet sizes. Thus, 20,000 problems i 20 categories with differet quatities for the umber of jobs ad the processig time specificatios have bee produced. The results are compared C OPT max Cl max. mi [r i +E(t i1 )] + E(t i2 )} i {1,2,3,...,} Parameter ρ is the performace guaratee of the proposed method (MB) ad called approximatio factor if C MB max ρ Cl MB max C max ρ COPT max, (30) where C MB max ad COPT max are the expected values of makespa that the MB method ad a optimal method, respectively, achieve o each istace. Other assumptios are as follows: (1) Release dates are geerated usig uiform distributio withi the iterval [0, 1000]. (2) All jobs are preemptio-restart. (3) The actual processig time is assumed to be equal to themeaofprocessigtime. (4) We have produced 1000 problems for each category ad based o the compariso betwee results of each method, the maximum value of factor ρ is calculated for these problems (worse case). To compare the performace of the proposed method (MB) agaist OPT; first, the performace is aalyzed with respect to the umber of jobs which is varies from 10 to 100 ad 10 categories are produced. The processig time follows a ormal distributio. We assume that the mea of processig time ad stadard deviatio of processig time are radom values withi the iterval [10, 20] ad [1, 3], respectively.sicethe mea of processig time follows uiform distributio, the expected value of the processig time is E(μ ij )=15.Table4 ad Figure 5 summarize the details of our implemetatios. Secod ad third colums of Table 4 are the mea of makespa for 1000 produced problems i each category. Whereas

8 8 Discrete Dyamics i Nature ad Society Table 4: Makespa value of the proposed algorithm versus lower boud of optimum. No. of jobs C MB max 10 3 C OPT max 10 3 ρ max MB/OPT μ [10 20], σ [13], r i [0 1000] Number of jobs Figure 5: ρ versus the umber of jobs. the forth colum is the maximum value of factor ρ amog 1000 problems i each category. As ca be observed, the proposed method (MB) performs up to times worse tha the other oe whe umber of jobs approaches 65. Note that whe exteds beyod 65, the desity of jobs will icrease illogically ad we have E(μ ij ) > 1000 while r i < Whe the desity of jobs icreases illogically, the waitig time of jobs icreases ad the idle time of machie 2 decreases. Thus, the performace of proposed algorithm almost improves. The performace of the proposed method has also bee evaluated for differet values of the mea of processig time. Tecategorieshavebeeproducedadtheumberofjobs adthevalueofstadarddeviatiohavebeefixedatcostat values ( =25, σ=3). The results are show i Table 5 ad Figure 6. Whe the mea of processig time approaches to the iterval [10, 70], the performace of the algorithm is i the worst coditio because E(μ ij ) ad max{r i } are almost equal to For the ext itervals, the desity of jobs icrease illogically ad the idle time of machie 2 decrease. Hece, the performace of MB algorithm almost improves. Moreover, the performace of the proposed method has bee evaluated with respect to distributio of process times as Table 5: The performace of the proposed algorithm (MB) agaist OPTwheretheumberofjobsis25. Mea of processig time C MB max 10 3 C OPT max 10 3 ρ max [10 20] [10 30] [10 40] [10 50] [10 60] [10 70] [10 80] [10 90] [10 100] [10 110] MB/OPT [10 20] [10 30] [10 40] =25, σ=3, r i [0 1000] [10 50] [10 60] [10 70] [10 80] Mea of processig time [10 90] Figure 6: ρ versus mea of processig time. the uiform distributio [1, 100] with the coefficiet of variatio beig 0.05, 0.1, ad 0.2. The umber of jobs varies from 10 to 100, ad te categories have bee produced. This mode of fixig the process-time distributio ca yield better isight about the evaluatio of computatioal performace of the algorithm. The processig times are geerated usig uiform distributio withi the iterval [1, 100]. It is assumed that the releasedatesareradomvalueswithitheiterval[0, 3000]. Takig ito cosideratio the fact that the mea of processig time is uiformly distributed, the expected value of the processig time is E(μ ij ) = The details of implemetatios have bee summarized i Table 6 ad Figure 7.Whe takes a value more tha 60, a illogical icrease will occur i the desity of jobs ad we have E(μ ij ) > 3000 while r i < Hece, a icrease is observed i the waitig time of jobs adtheidletimeofthesecodmachiedecreases.therefore, the performace of the proposed algorithm improves. 8. Cocludig Remarks I this paper, preemptive stochastic olie schedulig problem was ivestigated for two-machie flowshop. The objective fuctio was miimizig the expected value of makespa. Ispired by Johso s heuristic method for stochastic flowshop problem with a emphasis o miimal idle times of secod machie, a heuristic method was preseted. [10 100] [10 110]

9 Discrete Dyamics i Nature ad Society 9 Table 6: The performace of the proposed algorithm versus lower boud of optimum. Coefficiet of variatio No. of jobs C MB max 10 3 C OPT max 10 3 ρ max C MB max 10 3 C OPT max 10 3 ρ max C MB max 10 3 C OPT max 10 3 ρ max MB/OPT Number of jobs Coefficiet of variatio: Figure 7: ρ versus the umber of jobs ad the coefficiet of variatio. The implemetatio of the proposed method was demostrated usig a umerical example. The performace evaluatio of the proposed method has bee doe with comparig to a lower boud of optimum. Primary results idicated that theproposedmethodhadtheapproximatiofactorlesstha 2 for a wide rage of problems. The proposed method utilized the properties of the ormal distributios, ad this method ca be used as a heuristic method for other distributios, as log as their meas ad variaces are available. Schedulig with preemptio pealties is a ovel research area i schedulig field especially for olie stochastic problems ad we strogly believe that this research could be well exteded for problems with stochastic setup times, other distributio fuctio for processig time, ad flowshop problem with more tha 2 machies. Refereces [1] F. M. Defersha ad M. Che, Mathematical model ad parallel geetic algorithm for hybrid flexible flowshop lot streamig problem, Iteratioal Advaced Maufacturig Techology,vol.62,o.1 4,pp ,2012. [2] M. Mahavi Mazdeh, F. Zaerpour, ad F. Firouzi Jahatigh, A fuzzy modelig for sigle machie schedulig problem with deterioratig jobs, Iteratioal Idustrial Egieerig Computatios,vol.1,o.2,pp ,2010. [3] M. Braglia, M. Frosolii, R. Gabbrielli, ad F. Zammori, CONWIP card settig i a flow-shop system with a batch productio machie, Iteratioal Idustrial Egieerig Computatios,vol.2,o.1,pp.1 18,2011. [4] S. M. Johso, Optimal two- ad three-stage productio schedules with setup times icluded, Naval Research Logistics Quarterly,vol.1,o.1,pp.61 68,1954. [5]M.R.Garey,D.S.Johso,adR.Sethi, Thecomplexityof flowshop ad jobshop schedulig, Mathematics of Operatios Research,vol.1,o.2,pp ,1976. [6]J.M.Framia,J.N.D.Gupta,adR.Leiste, Areview ad classificatio of heuristics for permutatio flow-shop schedulig with makespa objective, the Operatioal Research Society, vol. 55, o. 12, pp , [7] R. Ruiz ad C. Maroto, A comprehesive review ad evaluatio of permutatio flowshop heuristics, Europea Operatioal Research,vol.165,o.2,pp ,2005. [8] T. Makio, O a schedulig problem, the Operatios Research Society Japa,vol.8,pp.32 44,1965. [9] E. Frostig ad I. Adiri, Three-machie flowshop stochastic schedulig to miimize distributio of schedule legth, Naval Research Logistics Quarterly,vol.32,o.1,pp ,1985. [10] S. Sethi, H. Ya, Q. Zhag, ad X. Y. Zhou, Feedback productio plaig i a stochastic two-machie flowshop: asymptotic aalysis ad computatioal results, Iteratioal Productio Ecoomics,vol.30-31,pp.79 93,1993. [11] S. E. Elmaghraby ad K. A. Thoey, The two-machie stochastic flowshop problem with arbitrary processig time distributios, IIE Trasactios,vol.31,o.5,pp ,1999. [12] P. P. Talwar, A ote o sequecig problems with ucertai job times, JouraloftheOperatiosResearchSocietyofJapa,vol.9, pp , [13] A. A. Cuigham ad S. K. Dutta, Schedulig jobs, with expoetially distributed processig times, o two machies of aflowshop, Naval Research Logistics Quarterly, vol. 20, pp , [14] P. S. Ku ad S. C. Niu, O Johso s two-machie flow shop with radom processig times, Operatios Research,vol.34,o. 1,pp ,1986.

10 10 Discrete Dyamics i Nature ad Society [15] H. M. Soroush ad A. Allahverdi, Stochastic two-machie flowshop schedulig problem with total completio time criterio, Iteratioal Idustrial Egieeri,vol.12,o. 2, pp , [16] D. Laha ad U. K. Chakraborty, A efficiet stochastic hybrid heuristic for flowshop schedulig, Egieerig Applicatios of Artificial Itelligece, vol. 20, o. 6, pp , [17] V. Portougal ad D. Trietsch, Johso s problem with stochastic processig times ad optimal service level, Europea Operatioal Research,vol.169,o.3,pp ,2006. [18] P. J. Kalczyski ad J. Kamburowski, A heuristic for miimizig the expected makespa i two-machie flow shops with cosistet coefficiets of variatio, Europea Operatioal Research,vol.169,o.3,pp ,2006. [19] K. R. Baker ad D. Altheimer, Heuristic solutio methods for thestochasticflowshopproblem, Europea Operatioal Research,vol.216,o.1,pp ,2012. [20] K. R. Baker ad D. Trietsch, Three heuristic procedures for the stochastic two-machie flowshop problem, Schedulig,vol.14,o.5,pp ,2011. [21] M. Heydari, M. Mahavi Mazdeh, ad M. Bayat, Stochastic olie schedulig with preemptio pealties, Mathematics ad Computer Sciece, vol. 6, pp , [22] K. R. Baker ad D. Trietsch, Priciples of Sequecig ad Schedulig, Joh Wiley & Sos, Hoboke, NJ, USA, 2009.

11 Advaces i Operatios Research Advaces i Decisio Scieces Applied Mathematics Algebra Probability ad Statistics The Scietific World Joural Iteratioal Differetial Equatios Submit your mauscripts at Iteratioal Advaces i Combiatorics Mathematical Physics Complex Aalysis Iteratioal Mathematics ad Mathematical Scieces Mathematical Problems i Egieerig Mathematics Discrete Mathematics Discrete Dyamics i Nature ad Society Fuctio Spaces Abstract ad Applied Aalysis Iteratioal Stochastic Aalysis Optimizatio