under Flexible Problem Periodic machine flexible and periodic availability availability complexity error of 2% on1 .iust.ac.ir/

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1 Iteratioal Joural of Idustrial Egieerig & Productio Research (08) March 08, Volume 9, Number pp DOI:..068/ijiepr Miimizig the Number of Tardy Jobs i the Sigle Machie Schedulig Problem uder Bimodal Flexible ad Periodic Availability Costraits Omolbai Mashai & Ghasem Moslehi Omolbai Mashai, Departmet of Idustrial ad Systems Egieerig, Isfaha Uiversity of Techology, Ghasem Moslehi, Departmet of Idustrial ad Systems Egieerig, Isfaha Uiversity of Techology, KEYWORDS sigle machie, flexible periodic costraits, bimodal, tardy jobs, brach-ad-boud. ABSTRACT I sigle machie schedulig problems with costraits, machies are ot available for oe or more periods of. I this paper, a sigle machie schedulig problem with flexible ad periodic costraitss is ivestigated. I this problem, the maximum cotiuous worig for each machie ca icrease i a stepwise maer ad ca have two differet values. I additio, the duratio of u for each period depeds o the maximum cotiuous worig of the machie i that same period, agai with two differet values. The objective is to miimize the umber of tardy jobs. I the first stage, the complexity of the problem is ivestigated; the, a biary iteger programmig model, a heuristic algorithm, ad a brach-ad- boud algorithm are proposed i the secod stage. Computatioal results of solvig 680 sample problems idicate that the brach- ad-boudd algorithm is capable of ot oly solvig problems up to 0 jobs, but also of optimally solvig 94.76% of the total umber of problems. Based o the computatioal results, a mea average error of % is obtaied for the heuristic algorithm. 08 IUST Publicatio, IJIEPR. Vol. 9, No., All Rights Reserved. Itroductio o []). Depedig o how the begiig of Whe dealig with schedulig problems with costraits, at least oe machie is uavailable for oe or more periods of. This may result from a umber of causes icludig u period is determied, schedulig problems with costraits are divided ito two groups: fixed ad flexible. Moreover, if fixed (flexible) u periods occur at breadows, maiteace, tool chage, or predetermied itervals durig the plaig overlapped plaig horizos. May researchers horizo, the such costraits are called periodic have ivestigated the effects of fixed (flexible) costraits. I costraitss o schedulig problems i various areas such as productio plaig, cost aalysis, ad other idustrial applicatios (for examplee [], problems with fixed costraits, the startt, fiish, ad duratio of the u periods are all predetermied. I problems with flexible costraits, the * duratios of the u periods are Correspodig author: Ghasem Moslehi predetermied. However, the start s of these moslehi@cc.iut.ac.ir periods are decisio variables. I the latter August 06; revised 4 December 07; accepted category, i some problems, u occurs Received 6 Jauary 08 Iteratioal Joural of Idustrial Egieerig & Productio Research, March 08, Vol. 9, No.

2 6 Omolbai Mashai & Ghasem Moslehi at the predetermiedd itervals, while, i some others, the maximum cotiuous worig of the machie is costat, ad the start of the u periods may be equal to or less tha this maximum value. O the other had, if the duratio of u periods is a fuctio of the coditios of the machie or the operator, the the costrait is called variable u. Refereces [3-6] may be cosulted for more details o schedulig problems with fixed costraits. A umber of studies have bee reported o flexible costraits. Yag et al. [7] were the first to itroduce the sigle machie schedulig problem with oe flexible u period withi a give iterval. I this study, maespa miimizatio with o preemptio, symbolized by r fa C, was studied. While the max problem was proved to be NP-hard, a heuristic algorithm was proposed for solvig the problem. Che [8] studied the same problem reported i [7] cosiderig flow. He proposed two biary iteger programmig models to solve problems up to jobs. I a later study, Che [9] dealt with the same problem with the objective of total tardiess ad developed two biary iteger programmig models capable of solvig problems up to 8 jobs. More recetly, Che [] studied the sigle machie schedulig problem with flexible periodic costraitss to miimize the maespa, which is deoted by r fpa C. He proposed max oe heuristic algorithm ad oe mixed iteger liear programmig model that was capable of solvig problems up to 0 jobs. To addresss the sigle machie schedulig problem with a flexible costraitt ad with the objective of miimizig total completio, Yag et al. [] proposed oe heuristic algorithm, oe dyamic programmig algorithm, ad oe brach-ad- problems up boud algorithm capable of hadlig to 400 jobs. Gaji et al. [] ivestigated the sigle machie schedulig problem with a flexible u period to miimize the maximum earliess. To fid the optimal solutio to each of these problems, they proposed a heuristic algorithm ad a brach-ad-boud algorithm capable of dealig with 4000 jobs. Schedulig problems with flexible costraits, where maximum cotiuous worig of the machie has a predetermied value, was first itroduced by Qi et al. [3]. They assumed predetermied values for both maximum cotiuous worig ad duratio of u costraits. To miimize the total Miimizig the Number of Tardy Jobs i the Sigle Machie Schedulig Problem... completio, they proposed threee heuristic algorithms ad oe brach-ad-boud algorithm capable of solvig problems up to 0 jobs. Graves ad Lee [4] cosideredd the same problem reported i [3] i which jobs are semi-resumablare maximumm lateess ad the objective fuctios ad weighted sum of jobs completio. For solvig these problems, they proposed two dyamic programmig algorithms. Moshiove ad Sarig [5] dealt with the sigle machie schedulig problem with oe flexible costrait with the objective of total weighted completio. They developed a dyamic programmig algorithm ad a heuristic algorithm for solvig the problem. Low et al. [6] itroduced five heuristic algorithms for solvig the sigle machie schedulig problem with flexible ad periodic costraits ad the objective of miimizig the maespa. Sbihi ad Varier [7] aalyzed the sigle machie schedulig problem with flexible ad periodic costraits ad the objective of maximum tardiess. They proposed oe heuristic ad oe brach-ad-boud algorithm capable of hadlig 5 jobs. I the studies performed i the field of schedulig problems with variable costraits, duratio of the u periods is a liear or expoetial fuctio of cotiuous worig of the machie, while the start of u costraits ad their duratios are determied by such factors as deterioratio durig processig jobs or tool agig. These costraits have bee ivestigated by [8-0]. Mashai ad Moslehi [] discussed a ew sigle machie schedulig problem uder bimodal flexible periodic costraits. The objective fuctio was miimizig the total completio. They assumed that, i each period, maximumm cotiuous worig of the machie ca have two fixed ad predetermied values, ad the u start is a decisio variable depedig o the maximum cotiuous worig of the machie. They supposed that the duratio of u costrait i each period ca have two differet values. Hece, i each period, if ay icrease i the cotiuous worig of machie is required to improve the objective fuctio, the duratio of u at the ed of that period will icrease to a costat value. I additio, preemptio is ot allowed. It should be oted that, i real world, it is possible to maitai ad service a machie (such as the millig machie) after a determied cotiuous Iteratioal Joural of Idustrial Egieerig & Productio Research, March 08, Vol. 9, No.

3 Miimizig the Number of Tardy Jobs i the Sigle Machie Schedulig Problem... worig. If maiteace activities caot be performed for whatever reaso, the the cotiuous worig of the machie will icrease. Hece, it is atural that the machie eeds more for service, oilig, scrubbig, etc., ad, thereby, ehacig the u of the machie []. Mashai ad Moslehi [] preseted a heuristic ad a brach-ad-boud algorithm to solve the problem. Their proposed brach-ad-boud algorithm was able to cope with problems up to jobs. They were the first oes who ivestigated the sigle machie schedulig problem with flexible periodic costraits ad two values for the cotiuous worig of the machie i each period. I this paper, the objective fuctio is miimizig the umber of tardy jobs uder the same circumstaces, supposed by Mashai ad Moslehi []. I Sectio, problem defiitio, symbolss ad otatios are preseted; i additio, the problem complexity is ivestigated. Sectio 3 presets a developedd biary iteger programmig model, while Sectio 4 presets relevat theorems ad lemmas. I Sectio 5, a heuristic algorithm for obtaiig the ear optimal solutio is proposed; i Sectio 6, details of the proposed brach-ad- boud algorithm are discussed. I Sectio 7, a geeralized form of the problem with several ids of u periods is itroduced. Sectios 8 ad 9 are devoted to computatioal results ad cocludig remars, respectively. Omolbai Mashai & Ghasem 7 Moslehi. Problem Defiitio ad Its Complexity A set of idepedet jobs { J, J,, J } is available at zero to be processed o a sigle machie. Setup s are idepedet of sequeces of jobs; they, however, are part of the processig for each job. I each period, the maximum cotiuous worig of the machie ca have either of two values T ad T (T > T ), ad the duratio of each u period, which depeds o cotiuous worig of the machie, ca have either of the two values W ad W (W > W ), respectively. Sice maximum cotiuous worig of the machie ad duratio of the u period ca have two differet values i each period, such costraits are called the bimodal flexible ad periodic costraits as depicted i Fig.. Jobs scheduled betwee each of two u periods are called a batch. I Fig., values q to q 4 represet the total processig of the scheduled jobs for batches to 4, respectively. This problem is symbolized as r- fpa,bm i Ui. I this paper, the followig otatios are used: : Number of jobs. p i : Processig of job J i for i=,,,. d i : Due date of job J i for i=,,,. C i : Completio of job Ji for i=,,,. K : Number of batches eeded for schedulig all jobs. K * : Number of batches i the optimal schedule. B : the th batch for =,,, K. : Number of scheduled jobs i the th batch such that K for,,, K q T T T T q T q3 T T q T T 4 J J W J J 3 J 0 W W W Fig.. Sigle machie schedulig problem uder bimodal flexible ad periodic costraits the maximum cotiuous worig of the J i : Job J i which is scheduled i the th batch for machie for p=,. i=,,, ad =,,, K. C i : Completio of job J i which is scheduled : Sequece of scheduled jobs. i the thh batch for i=,,, ad =,,, : Sequece of uscheduled jobs. K. C() : Completio for partial sequece. q : Total processig of the jobs scheduled i T i : Tardiess of job J i calculated through Eq. (). the th batch for =,,, K. T = max{0,c T p : Maximum cotiuous worig of i i -d i } i =,,, ( ) machie for p=,. U i : A biary variable. It is if job J i is tardy; W p : Duratio of u correspodig to otherwise, it is zero for i=,,,. 3 J 4 J 4 J Iteratioal Joural of Idustrial Egieerig & Productio Research, March 08, Vol. 9, No.

4 8 i U i S : Startig of the th batch for =,,, K. Qi et al. [3] showed that the sigle machie schedulig problem with flexible ad periodic costraits ad with the objective fuctio of miimizig maximum completio of jobs, where the maximum cotiuous worig of the machie i each period has a predetermied value, is NP-hard. I the followig, usig Theorem, it is show that problem r- fpa,bm i Ui is also NP-hard. Theorem : The problem r -fpa,bm NP-hard. i= Ui is Proof: I problem r- fpa, bm, assume that T equals T ad W equals W. The, this problem trasforms to the problem r- fpa i Ui i which maximum cotiuous worig of the machie has a predetermied value. Therefore, the complexity of the problem r- fpa,bm Ui is at least as equal as the problem i r- fpa i Ui. r- fpa,bm i U NP-hard problem. 3. Mathematical Modelig I this Sectio, a mathematical model is developed for the problem r- fpa,bm i U i. To do so, the followig variables are defied: xi : A biary variable. If job J i is scheduled i batch, its value will be ; otherwise, it will be zero for i=,,, ad = =,,, K. yp : A biary variable equal to if the maximum cotiuous worig of the machie i the th batch equals T p ; otherwise, it is zero for p=, ad =,,, K. h ij : This is a parameter. If job J i is located before job J i i the EDD sequecee (arragig the jobs i the o-decreasig order of their ormal due dates), the its value will be ; otherwise, it will be zero for i,j=,,, ad j i. Mathematical model: miimize i= Ui () Subject to: K x i Omolbai Mashai & Ghasem Moslehi : Number of tardy jobs. NP-hard [3], the problem r- fpa i U i is also NP-hard. As a result, the problem Ui has at least the complexity of a. Sice the problem r is i,,, i Ui i pi. xi p Tp. yp,,, K fpa C max (3) (4) Miimizig the Number of Tardy Jobs i the Sigle Machie Schedulig Problem... S 0 S r i r pi. r xi r p Wp. yp, 3,, K Ci S pi j i pi. hij. xi i,,, ;, 3,, K (7) Ci M( xi ) C i i,,, ;, 3,, K (8) Ci di M. Ui p yp xi, yp, Ui {0,} i,,,,,, K (5) (6) (9) () p, ; i,,, ;,,,K () Equatio () represets the objective fuctio for the umber of tardy jobs. Accordig to Eq. (3), each job ca be scheduled just i oe batch. Equatio (4) determies that the total processig s of the scheduled jobs i the th batch are less tha or equal to the maximum cotiuous worig of the machie. Start for the first batch is zero, as i Eq. (5). Eq. (6) calculates the start of the th batch for =, 3,, K. I Eq. (7), the completio of job J i i the th batch is represeted ad the completio of job J i is restricted i Eq. (8). Equatio (9) determies tardiess of job J i. I Eqs. (8) ad (9), M is a large positive umber. Accordig to Eq. (), the duratio of u after the cotiuous worig of the machie i each batch ca have oly values W ad W. Equatio () deals with the biary characteristics of the decisio variables. I this model, the umber of variable x i is K, the umber of variable y p is K, ad the umber of variable U i is ; therefore, the total umber of variables i the model will be K+ K+. Equatio (3) of the model should be feasible for variables ad Eq. (4) ad ( ) should also be held for K batches. Evetually, sice Eqs. (5), (6), (7), (8), ad (9) ca be merged, the K equatios will be feasible by implemetig these costraits. Hece, the umber of costraits i the model will be + K+ K. Note thatt parameter h ij should be determied by EDD arragemet of jobs. The mathematical model of r- fpa,bm i Ui problem is solved by CPLEX of the GAMS software ad has bee capable of solvig problems up to jobs i less tha 3600 secods. 4. Lemmas ad Theorems I this Sectio, a umber of lemmas ad theorems are itroduced ad proved based o specificatios of the problem r- fpa,bm i Ui. First, the relevat theorems to fid a lower boud for the Iteratioal Joural of Idustrial Egieerig & Productio Research, March 08, Vol. 9, No.

5 Miimizig the Number of Tardy Jobs i the Sigle Machie Schedulig Problem... problem r- fpa,bm i Ui are aalyzed. Leee [] showed that the optimal solutio to the problem ra i U i ca be calculated by Moore ad Hudgso algorithm. I his method, jobs are iitially scheduled usig Moore ad Hudgso algorithm. The, the duratio of the u period is addedd to the completio of those jobs, which are scheduled after the u period. Accordigly, i Theorem, a lower boud for the problem r- fpa,bm i U i is proposed. Theorem : The optimal solutio to the problem i= Ui is a lower boud for the problem r -fpa,bm U i i= Proof: Sice the solutio to = problem is a lower boud for problem r a i=ui as show i [], the umber of tardy jobs will defiitely ot decrease whe u periods are added. Therefore, it ca be cocluded that the optimal solutio to problem i= Ui is a lower boud for problem r- fpa,bm i Ui. Corollary : I problem r- fpa,bm i U i, it is assumed that is the partial sequece of scheduled jobs ad is the set of uscheduled jobs. Usig Theorem, if the jobs i set are scheduled by Moore ad Hudgso algorithm without costraits ad the umber of tardy jobs has cosequetly becomee equal to i Ui, the i Ui i U i will be a lower boud for the partial sequece. To fid aother lower boud for problem r -fpa,bm U i i= sigle machie schedulig problem with flexible ad periodic costraits is proposed where the maximum cotiuous worig of the machie i each period is T ad duratio of u is W. Whe preemptio is allowed, this problem is symbolized by r fpa i U i ad whe preemptio is ot allowed, the problem is deoted by r fpa iui. To solve problem r fpa, algorithm H ca be used. i Ui., first, a i= Ui lower boud for the Algorithm H Step 0. Arrage the jobs o the basis of EDD rule ad ame them i the same order. Set,, ad { J, J,, J }. Set parameters i,, Ci ad Omolbai Moslehi Mashai & Ghasem q equal to zero. Set = ad go to Step. Step. Set i= i+. If i=+ +, the go to Step 4; otherwise, calculate q q p i. The, omit job J i from set ad put it i set. If q T, the set Ci Ci pi ad go to Step ; otherwise, set T q,, C i Ci pi W, ad q pi ; the, go to Step. Step. Calculate the tardiess of job J i i. If it is ot tardy, the go to Step ; otherwise, go to Step 3. Step 3. If job J i is the first tardy job i the sequece, the choose a job with the largest processig from set.. Omit this job from set ad put it i set π. Go to Step. Step 4. Schedule the jobs i set π at the ed of the sequece i a arbitrary order. Algorithm H operates similarly to Moore ad Hudgso algorithm, yet oly with the followig slight differece. If the total cotiuous worig of the machie i each period exceeds a predetermied value (which is T ), the the operatio is cut ad a u period is implemeted before further processig cotiues. As a result, the complexity of algorithm H will be O ( log ). Sice the optimal solutio to problem ra iui is obtaied from Moore ad Hudgso algorithm [], i Theorem 3, it is show that algorithm H provides the optimal solutio to problem r fpa iui. Note that, i problem r fpa i Ui, the duratio of each u period is W ad the distace betwee two cosecutive u periods is exactly T. Theorem 3: The solutio yielded by algorithm H is the optimal solutio to problem r - fpa i= Ui. Proof: Suppose that, i sequece S, job J i is scheduled i batch B ad all jobs before job J i are ot tardy. I this sequece which is displayed i Fig., the set of jobs φ is scheduled from the first batch to batch B. If the jobs i set φ are scheduled without u ad if the legths of the etire u periods are summed up ad scheduled at the ed of this set, the sequece S is obtaied as show i Fig. 3. I sequece S, ulie i sequece S, the jobs i set φ will remai u-tardy whose completio has ot icreased. Meawhile, sice the processig ad legths of u periods are fixed, the completio of the jobs i both sequeces S ad S, placed before job J i i batch 9 Iteratioal Joural of Idustrial Egieerig & Productio Research, March 08, Vol. 9, No.

6 0 Omolbai Mashai & Ghasem Moslehi B, will ot chage; as a result, they will remai u-tardyproblem ra Hece, our problem trasforms to i which the startt of the u period equals the total processig of the jobs i set φ ad the legth of their u period is W. Sice the optimal solutio to problem r obtaied from Moore ad Hudgso algorithm [], it may be cocluded that the optimal solutio to problem r fpa ca also be obtaied from the same algorithm. Note that, i the optimal solutios to both problems ra ad r fpa i Ui, of the sequecee with a arbitrary order. I Theorem 4, a lower boud is proposed for problem r fpa bm. Theorem 4: I problem r- fpa i= Ui, it is assumed that the maximum cotiuous worig of machie is T ad u duratio is W for each period. Hece, the solutio to this problem by H algorithm is a lower boud for the problem r - fpa,bm i= Ui. Proof: Accordig to Theorem 3, the solutio to the problem rfpa is obtaied from algorithm H. This value is deoted by F. Suppose that F * is the optimal solutio to the schedulig problem with bimodal flexible ad periodic costraits where preemptio is allowed. This problem is deoted by r fpa, bm i Ui. Sice the maximum cotiuous worig of the machie i problem r fpa is always larger tha or equal to its correspodig value for problem r fpa bm Ui ad as the u period, i i U i i Ui tardy jobs are scheduledd at the ed, i Ui i Ui i Ui ( ) a i Ui is is always shorter tha or equal to its correspodig value i problem r fpa, bm, the relatio * i Ui i U i Miimizig the Number of Tardy Jobs i the Sigle Machie Schedulig Problem... * * F F is always true. O the other had, suppose * thatt F is the objective fuctio for the problem r fpa, bm i U * i. It is apparet that F is always larger tha or equal to the objective fuctio of the same problem whe preemptio is * * * * allowed, meaig F F ; cosequetly y, F F. Corollary : I problem r fpa, bm i U i, it is assumed that is the partial sequece of scheduled jobs ad is the set of uscheduled jobs. Assume that preemptio is allowed: the duratio of cotiuous worig of the machie i each period beig T ad the legth of each u period beig W. So, if the jobs i set are scheduled usig H algorithm, the umber of tardy jobs is i Ui. As a result, the value of i Ui i Ui is a lower boud for partial sequecee. Corollary 3: I problem i U i, there is o costrait ad if ay costrait is added to the problem, the the completio s will ot decrease. As a result, the umber of tardy jobs i the optimal solutio to problem i U i is always smaller tha or equal to its correspodig value i the optimal solutio of problem r fpa i Ui. Theorem 5: I the optimal solutio of problem r- fpa,bm i= = Ui, the order of jobs i each batch follows the procedure of Moore ad Hudgso algorithm. Proof: Schedulig of jobs i each batch is similar to the solutio of problem i U i where there are o costraits ad the optimal solutio ca be obtaied from Moore ad Hudgso algorithm. T W T W W T W J J f J g J h Ji 0 B B B - Fig.. Sequece S i Theorem 3. B (-).W J J f J g J h J i 0 B Fig. 3. Sequece S i Theorem 3. B Iteratioal Joural of Idustrial Egieerig & Productio Research, March 08, Vol. 9, No.

7 Miimizig the Number of Tardy Jobs i the Sigle Machie Schedulig Problem... Usig the result of Theorem 5, the followig three coclusios may be draw: Corollary 4: I problem r fpa, bm i Ui, the domiat sets cosist of those sequeces i which the order of jobs i a batch will follow the EDD order if there are o tardy jobs therei. Corollary 5: I problem r fpa, bm i Ui, the domiat sets are those sequecess i which if a job i oe of the batches becomes tardy, the the remaiig jobs i that batch will also become tardy. Corollary 6: I problem r fpa, bm i Ui, the domiat sets iclude those sequeces i whichh if job J i is scheduled i batch B ad if, further, it has become tardy, the there are o jobs i that batch before J i whose processig s are larger tha that of Ji. Lemma : I problem r- fpa,bm i=ui, there is oe optimal solutio i which Eq. () holds for job J i, where job J i is the first job i batch B +. Tp q pi Ji B,,, ; p {,} () Proof: It is clear that if job J i ca be scheduledd i batch B, the, by schedulig it i batch B, its completio will defiitely be shorter tha that i the alterative situatio where it is scheduled i batch B +. Therefore, either its tardiess will icrease, or the completio s of other jobs will chage. Lemma : I problem r- fpa,bm i= Ui, the domiat sets cosist of sequeces i which all tardy jobs are located at the ed of the sequece. Proof: Cosider sequece S i which job Ji is tardy ad there is at least oe job without tardiess followig it. If job J i is trasferred to the ed of sequece S, the sequece S is obtaied. I cotrastt to S, completio of the etire jobs except J i will either remai fixed or decreasee i S, while the total umber of tardy jobs will ot icrease. Corollary 7: If job J i from set is added to the ed of the partial sequece ad it becomes tardy, the the cardiality of set, i.e.,, ca be calculated ad compared to the umber of tardy jobs i a feasible solutio, i.e., i U i. If Eq. (3) is true, the schedulig the jobs of i the partial sequece may be igored. i U i (3) Lemma 3: Whe job J i is scheduled i set σ at Omolbai Mashai & Ghasem Moslehi the ed of partial sequece σ, if J i becomes tardy ad Eq. (4) holds, the by addig set σ to the ed of the partial sequece σ, a complete solutio with at most σ tardy jobs will be obtaied regardless of the job orders ad legth of machies i batches. i U i (4) Proof: Accordig to Lemma, if job J i is tardy, the the whole jobs i set will also become tardy. Therefore, arragemet of these jobs has o impact o improvig the objective fuctio. I additio, sice their amout of tardiess is of o sigificace, the they ca be scheduled by cosiderig arbitrary maximum. I this way, if the umber of jobs i set is smaller tha i Ui, the a complete solutio is obtaied which has less tardy jobs tha the previous solutio. Lemma 4: I partial sequece ( σ,w,ji ), if job J i is scheduled i batch B ad is tardy, the this partial sequece will be domiated by the partial sequece (σ,ji ) where q - - +pi T. Proof: Cosider C( ) to be the completio for partial sequece. The, tardiess of job J i i the partial sequece ( σ,w, Ji ) is obtaied by Eq. (5). Ti C( ) W pi di (5) I additio, the tardiess of job J i i the partial sequece ( σ,j i ) ca be calculated from Eq. (6). ' Ti max{0, C( ) pi di} (6) Compariso of these two equatios reveals that it is possible for the tardiess of J i i ( σ,ji ) to become zero ad, cosequetly, the partial sequece ( σ,w, Ji ) to be domiated by partial sequece ( σ,ji ). Lemma 5: Suppose that, i the partial sequece ( σ,j i ), job J i is scheduled i batch B ad is tardy. If theree is a job J j i set σ such that it is ot tardy i the partial sequece ( σ,jj ) ad that maximum of the machie i B does ot chage, the the partial sequece ( σ,j i ) will be domiatedd by ( σ,j j ). Proof: Accordig to lemma, if J i is tardy, the all the jobs i set will become tardy. Thus, J i will be defiitely tardy i the partial sequece ( σ,ji ). Sice job J j i the partial sequece (σ,jj ) is ot tardy, the the umber of tardy jobs i (σ, J j ) will be less tha that i ( σ,ji ). Therefore, the Iteratioal Joural of Idustrial Egieerig & Productio Research, March 08, Vol. 9, No.

8 Omolbai Mashai & Ghasem Moslehi Miimizig the Number of Tardy Jobs i the Sigle Machie Schedulig Problem... partial sequece (σ,ji ) will be domiated by belog to B. I sequece S, as preseted i Fig. ( σ,jj ). 6, the completio of partial sequece is C( Lemma 6: If i the optimal solutio to ), ad the total processig of the jobs i problem i= U i, the start of the first is cosidered to be P. Exchagig jobs J i ad J tardy job is Q ad the maximumm earliess of j i sequece S leads to sequece S (, Jj,, jobs is E max, the there is a optimal solutio i ) show i Fig. 7. Cosequetly, to problem r- fpa,bm Notatio ca be cocluded for these two i= Ui i which those sequeces. jobs that begi after Q+Emax will be defiitely Notatio : If Eqs. ( 7) ad (8) are true, the tardy. sequece S will be domiated by S. Proof: Assume that the jobs i sequece S, as show i Fig. 4 are scheduled usig Moore ad pj dj p i di (7) Hudgso algorithm without costraits. It ca be see that job J j belogs to dj di (8) the set of o-tardy jobs with a earliess of E j. If at least oe u period is added to 5. The Heuristic Algorithm H this problem ad if job J j is the first to begi after I this sectio, a heuristic algorithm called H is Q+Emax, the sequece S will be obtaied as proposed to obtai a ear-optimal solutio to the show i Fig. 5. I sequece S, relatio problem r fpa, bm i U i. Q+Emax pj Ej is true; thus, job J j will be I algorithm H, jobs are iitially arraged by the defiitely tardy. Therefore, i problem EDD rule. The, a iitial order of jobs willl be r fpa, bm i Ui, those jobs which begi after obtaied by algorithm H. I the followig, it is assumed that just oe u period exists Q+Emax will be tardy. for schedulig the remaiig jobs; therefore, two It should be oted that Che [3] studied the sigle machie schedulig problem with fixed periodic costraits ad with the objective fuctio of the umber of tardy jobs as batches will exist for schedulig the jobs. Oe of these batches is located before ad the other is located after the u period. I orderr to schedule the jobs i the first batch, the best batch show by r pa iui. To solve this problem, legth ad, thereby, the best legth of Che proposed five poits which also hold true u period should be chose such that for problem r fpa, bm the umber of tardy jobs becomes miimized. i Ui. I this paper, The, this batch ad its u period are attempt is made to beefit from these poits. cosideredd fixed ad the jobs withi it will be However, our ivestigatios show that oly oe of these poits, itroduced below as Notatio, ept fixed as well. I the secod step, the same procedure is adopted ca be used i the problem r fpa, bm i Ui for the remaiig jobs ad the process cotiues ad will be itroduced i the followig as util the etire jobs are scheduled. Based o what Notatio. wet above i each batch, the impacts of selectig I order to itroduce Notatio, it is required to the maximum machie of legth T or T is calculated, ad this will be used to choose cosider sequece S (, Ji,, Jj ) i which the duratio of the u period. ad are partial sequeces ad { Ji,, J j } Early Jobs i i U i tarly Jobs i i U i 0 J J j J r- Q J r Jy J y+ Q+max E i i Fig. 4. Sequece S i Lemma 6 J 0 J J g Q J h Q+ max E i i J j J Fig. 5. Sequece S i Lemma 6 Iteratioal Joural of Idustrial Egieerig & Productio Research, March 08, Vol. 9, No.

9 Miimizig the Number of Tardy Jobs i the Sigle Machie Schedulig Problem... Omolbai Moslehi Mashai & Ghasem 3 0 J C(σ ) J i Ji+ Fig. 6. Sequece S i Notatio J j- J j 0 J J j C(σ ) Ji+ Fig. 7. Sequece S i Notatio 3; otherwise, schedulee job J i i the locatio of j. Suppose that some jobs i partial sequece are Update the jobs i as { Ji } ad go to scheduled i batches to B -. Two scearios ca Step. be imagied for the maximum machie Step 3: Suppose that the maximumm of i batch B. I the first sequece, the machie for the th batch is T. Schedule job represeted by sequece S i Fig. 8, this J i i batch B + ad use algorithm H withi the maximum value is T ad r jobs are scheduledd i maximum cotiuous worig of machie this batch. Followig this batch, a u T ad the u duratio W to schedule period of the legth W is set. The, the the remaiig jobs. Put the umber of tardy jobs remaiig jobs are scheduled usig the order resultig from this sequece ito N( T ). Go to obtaied from algorithm H ad without costrait. Step 4. I the secod sceario, represeted by S i Fig. Step 4: Set q pi. Put the set of jobs ito 9, it is assumed that the maximumm of the machie is T ad y jobs ( y>r) ca be processed i batch B. After this batch, oe a set φ ad create a list. Step 4-: Select the first job from the set of jobs φ ad ame it J g. u period with legth W is scheduled Step 4-: Set pii. ad the remaiig jobs are scheduled usig the Step 4-3: If order obtaied from algorithm H. he put job J g i list ad If the umber of tardy jobs i sequece S is less update set as { Jg } ad set. If tha that i sequecee S, the sequece S better ad the maximum of is the, the go to Step 4-5; otherwise, go to Step 4-. machie i the curret batch should be cosidered Step 4-4: If as T ; otherwise, T should be adopted. I go to Step 4-5. Step 4-5: Suppose that the maximum additio to the symbols itroduced earlier, the of the machie i the th batch is T followig symbols are also used i algorithm H : ad schedule job J i: Job umber. i accompaied with the jobs i list M i this set. Schedule the jobs i M usig j: Locatio i the schedule. algorithm H withi the maximum cotiuous N( T ) : Number of tardy jobs resultig from worig of the machie T ad the schedulig jobs with maximum of T. u duratio W ad put the resultig N( T ) : Number of tardy jobs resultig from umber of tardy jobs ito N( T ). Go to Step 5. schedulig jobs with maximum of T. Step 5: If N( T ) N ( T ), the go to Step 6. Algorithm H cosists of the followig steps: Otherwise, go to Step 7. Step 0: Set { J, J,, J } ad parameters i, j, Step 6: Schedule oe u period with, q ad y equal to zero ad =. the legth of W i locatio j ad schedule job J i Step : By uig algorithm H withi the maximum cotiuous worig of machie i locatio j+. Set =+, j=j+ +, ad q pi, T ad u duratio W, obtai a iitial ad the go to Step. arragemet of jobs ad ame them based o this Step 7: I locatios to j+y, schedule the jobs i order. list M. Schedule oe u period with Step : Select the jobs based o the order the legth of W i locatio j+y+ +. Set i=i-+y, assiged i Step. Set i=i+ ad j=j+. If =+, j= =j+y+, ad q =0. Update set as i=+, the go to Step 8; otherwise, calculate M ad go to Step. q q pi. If q T the set y=i ad go to Step Step 8: Calculate the objective fuctio. J j- J i Iteratioal Joural of Idustrial Egieerig & Productio Research, March 08, Vol. 9, No.

10 4 Omolbai Mashai & Ghasem Moslehi Step of algorithm H has a complexity of O( log ). Steps 3 ad 4 are of complexity O ( ). Therefore, algorithm H has a Miimizig the Number of Tardy Jobs i the Sigle Machie Schedulig Problem... complexity of O ( log ) ). T J J J J r + J + J K+ B W Fig. 8. Sequece S i algorithm H T J J J J r J y + J + J K+ B W Fig. 9. Sequece S i the algorithm H H 6. The Brach-ad-Boudd Algorithm I order to solve problem r- fpa,bm i= U i, a depth first search brach-ad-boud algorithm has bee used. I this algorithm, jobs are iitially arraged based o the EDDD order ad reumbered i that same order. This arragemet is tae as the iput to the brach-ad-boud algorithm. I this algorithm, the solutio of algorithm H is used as the upper boud, Theorem 4 as the lower boud, ad Theorem 5, Lemma, Lemma 3, ad Notatio as domiace rules. After schedulig each job J i by the brach-ad- braches i the tree, idicatig thatt there are two choices to schedule after schedulig job J i : job J i+ or a u period. Cosequetly, the boud algorithm, there will be two possible umber of braches will be at most!. As show i Fig., whe schedulig job J i+ i B, after job J i, there will be two braches. I first brach, oe costrait with value W (or W ) ca be scheduled after J i. I the secod oe, J i+ to will be scheduledd after J i till J q T or T. The, a costrait with value W (or W) ca be cosidered after J. I each brach, if ay of lemmas or theorems is violated, the brach will be fathomed. I additio, if both of braches caot be fathomed, the the brach with smaller umber of tardy jobs should be cotiued ad other oe fathomed. W (or r W ) J i+ + J i i J i+ J W (or W ) Fig.. The brach-ad-boudd tree 7. Geeralizatio of r- fpa,bm i= U i Some of the lemmas ad theorems proposed for the problem r-fpa,b bm i= U i ca be geeralized to a more geeral form represeted by r- fpa,mmm i= U i with multiple maximum cotiuous worig of the machie. I the multi-modal problem, the maximum cotiuous worig of the machie i each period ca adopt P differet values cosistig of T, T,, TP ad the legth of each u period ca have P differet values cosistig of W, W,, WP depedig o the maximum cotiuous worig Iteratioal Joural of Idustrial Egieerig & Productio Research, March 08, Vol. 9, No.

11 Miimizig the Number of Tardy Jobs i the Sigle Machie Schedulig Problem... of the machie. The mathematical model proposed i Sectio ca be easily geeralized to problem r- fpa,mm i= = Ui. To do so, it will suffice to chage the boudaries of p i Eqs. (4), (6), ad () from to P. Moreover, Theorems, 4, ad 6, Lemmas to 6, ad Notatio will ot deped o the legth of u period or o the legth of batches; thus, they are etirely true for problem r- fpa,mm i= U i. Note thatt a lower boud for problem r- fpa,mm i=ui ca be obtaied usig Theorem 4. To do so, it will suffice to calculate the optimal solutio to problem r fpa i= U i usig H algorithm with two coditios: maximum duratio of the cotiuous worig of the machie is max p P Tp ad the legth of the u period is mi p P Wp. O the other had, Lemma 4 is geeralized to Lemma 7 for problem r- fpa,mm i=ui. Lemma 7: Suppose that job J i i the partial sequece ( σ,w p,ji ) is scheduled i batch B ad is tardy. The, this partial sequece will be domiated by the partial sequece (σ, J i ) where q - +pi Tp ad p p>. d Omolbai Mashai & Ghasem 5 Moslehi umber i the set {8,,, 4, 6, 8, 0}. Also, T has bee selected from {.4 T,.8 T } ad W.6 W. Ay possible permutatio of C, Q, T, T, W ad W is called a series, yieldig 4 (i.e., 3 ) series. I each series, for each, problems are geerated (p=) to create a total umber of 680 (4 7 ) problems. Computatioal results of solvig the sample problems are preseted i Table, where specificatios of the series, umber of problems solved by the proposed brach-ad-boud method, ad the mea average error of the heuristic algorithm are reported. Sice the umber of tardy jobs i a sample problem may become zero, a appropriate measure has bee utilized to calculate the error of algorithm H i order to evaluate its performace. Moslehi ad Jafari [4] used opt H as a measure for comparig 8. Computatioal Results To evaluate the performaces of heuristic algorithm H ad the proposed brach-ad-boud algorithm, they were coded i the C# programmig laguage ad executed o a Itel (R) Core Due 3.6 GHz with GB RAM i WINDOWS 7 eviromet. If executio of the problem exceeded 3600 secods, the brach- termiated. I order to geeratee sample problems, the approach proposed i Che [3] was used. Accordig to this approach, processig s are radomly geerated from a uiform discrete distributio over the iterval [, ]. Due dates are also radomly geerated usig the followig uiform discrete distributio. Q ad-boud algorithm would be automatically i C. Q, i p i C. i pi (9) I Eq. (9), C is the tardiess parameter chose from the set {0., 0.6}; Q is the parameter of due dates chose from the set {0., 0. 6}; T has the values {, 5, 0}, ad W =6. The umber of jobs, i.e.,, is assumed to be ay the results obtaied from both algorithm H ad the brach-ad-boud algorithm. I this measure, opt H is the optimal value ad is the objective fuctio of algorithm H both of which are opt calculated from i ( Ui ). Therefore, H is always largerr tha or equal to uity, ad the deomiator of the measure will ever be zero. opt The closer H gets to, the higher the efficiecy of the algorithm H will be. The algorithm obtais the optimal value whe this measure becomes equal to. Uder the colum opt H, the mea percetage errors of algorithm H are reported, ad this value is calculated from optimal solutios which are obtaied withi the 3600 secod boudary. I additio, the average percetages of the odes fathomed to the total umber of crossed odes as well as the average percetages of odes fathomed based o the fathomig reaso are reported. It is worth metioig that some of the lemmas ad theorems have ot bee used i the brach-ad- iefficiecy. The last colum i the Table boud algorithm due to their computatioal presets the average executio s i secods for problems that were optimally solved usig algorithms H ad the brach-ad-boud method. It is also clear from Table that sice the total umber of states i problem r- fpa,bm i= U i is at most!, ay icreasee i problem dimesios i each series results i a correspodig icrease i the solutio of the brach-ad-boud Iteratioal Joural of Idustrial Egieerig & Productio Research, March 08, Vol. 9, No.

12 6 Omolbai Mashai & Ghasem Moslehi algorithm. Fig. presets the umber of problems whose optimal solutios were obtaied usig either algorithms H or the brach-ad-boud icludes all of the 4 i less tha 3600 secods. This series based o chages i parameters C ad Q. Clearly, it ca be cocluded from the compariso of the umber of problems solved i each series that by decreasig C ad Q values, i more problems, the optimal solutios were obtaied i less tha Fig. shows that whe Q decreases, the umber of problems whose optimal solutios are obtaied from algorithm H will decrease. Geerally, lowerig the rages of due dates leads to reduced optimal solutios obtaied i less tha 3600 secods. O the other had, the solutio for series to 4 reduces whe T icreases. This is because the umber of batches eeded to schedule the jobs decreases ad, cosequetly, more optimal solutioss are obtaied i less tha 3600 secods. Fig. 3 presets the tred i the chages i the umber of problems solved by H ad brach- ad-boud algorithms with respect to T. As observed i Fig. 3, the umber of s that H algorithm reaches optimal solutios icreases whe T icreases. The reaso may lie i the reduced umber of batches eeded for schedulig the jobs, which thereby reduces the errors resultig from iappropriate selectio of u duratios. Based o our computatioal results, the mea percetage error for algorithm H is percet, which cofirms the high capability of the algorithm. Fig. 4 displays the mea solutio s obtaied from the brach-ad-boud algorithm versus T. It ca be observed that this parameter decreases by icreasig T. It ca be see i Fig. 5 that i series with idetical values of T, solutio decreases whe T - T is larger tha T. The reaso is that the umber of batches eeded for schedulig jobs reduces i the cases where there is a batch with a maximum of T i the optimal solutio. Miimizig the Number of Tardy Jobs i the Sigle Machie Schedulig Problem... Fig.. Performaces of H ad brach-ad- boud algorithms versus chages i C ad Q Fig.. Performaces of H ad brach-ad- i boud algorithms versus chages Q Fig. 3. Performaces of the H ad brach- i T ad-boud algorithms versus chages Iteratioal Joural of Idustrial Egieerig & Productio Research, March 08, Vol. 9, No.

13 Miimizig the Number of Tardy Jobs i the Sigle Machie Schedulig Problem... Fig. 4. Mea solutio recorded for the brach-ad-boudd algorithm versus T Fig. 5. Mea solutio obtaied by brach-ad-boud algorithm i series with idetical values of T 9. Coclusios ad Suggestios for Future Studies I this paper, the sigle machie schedulig problem with bimodal flexible ad periodic costraits was ivestigated. Accordig to this defiitio, it is assumed that, i Omolbai Mashai & Ghasem 7 Moslehi each period, the maximum cotiuous worig of the machie may adopt either of two differet values, ad the duratio of each u period depeds o the maximum of the machie i that period which ca also adopt either of two differet values. Our objective was to miimize the umber of tardy jobs i a problem deotedd by r-fpa,bm i= U i. I order to obtai the optimal solutio, a biary iteger programmig model was iitially proposed. It was show that optimal solutios to problems of larger scales are ot possible due to the complexity of the model. The, several lemmas ad theorems were itroduced ad proved. Fially, a heuristic algorithm ad a efficiet brach-ad-boud algorithm were proposed to solve the problem. Domiace rules, lower bouds, ad upper bouds were implemeted i both algorithms. The proposed brach-ad-boud algorithm was foud to be capable of solvig problems up to 0 jobs. I additio, i sectio 7, the problem was scaled up to ivestigate situatios where there are multiple- modal costraits. It was demostrated that most of lemmas ad theorems, which are proposed for bimodal problem, also hold for the multiple-moda al problem. Further studies were suggested to ehace the efficiecy of the brach-ad-boudd algorithm through improvig heuristic algorithm H as well as the lower boud of the problem. I additio, it will be desirable to ivestigate the sigle machie schedulig problem with bimodal flexible ad periodic costraits i which two predetermiedd itervals are cosidered i each period for the occurrece of u ad i which the legth of each u period depeds o the iterval chages ad ca be selected i a stepwise maer. Series Series C 0. Q 0. Tab.. Computatioal results for sample problems. Numbe Average Avg. Average percetage of fathomed r of perceta solvig opt odes by Optima ge of l H etire Lower Theor Lemm Notati Lem istac fathome H B& &B boud em 5 a o ma 3 es d odes Iteratioal Joural of Idustrial Egieerig & Productio Research, March 08, Vol. 9, No.

14 8 Omolbai Mashai & Ghasem Moslehi Miimizig the Number of Tardy Jobs i the Sigle Machie Schedulig Problem... T W T.4 T W.6 W Series C Q T W T.8 T W.6 W Series C Q T W T.4 T W.6 W : No optimal solutio achieved by brach-ad-boud ad H algorithms. : Optimal solutio achieved by H algorithm ad brach-ad-boud did t use. Series Series 4 C 0. Q 0. T 5 W 6 T.8 T W.6 W Series 5 C 0. Q 0. T 0 W 6 T.4 T W.6 W Series 6 C 0. Q Number of Optimal istaces opt H Tab.. cotiued. Average percetagee Avg. ercetage of fathomed odes by solvig of etire fathomed Lower Theorem Lemma Notatio Lemma H B&B odes boud Iteratioal Joural of Idustrial Egieerig & Productio Research, March 08, Vol. 9, No.

15 Miimizig the Number of Tardy Jobs i the Sigle Omolbai Mashai & Ghasem Machie Schedulig Problem... Moslehi 9 T 0 W 6 T.8 T W.6 W Series Series 7 C 0. Q 0.6 T W T W 6 Series 8 C 0. Q 0.6 T W T W 6 Series 9 C 0. Q 0.6 T W.4 T.6 W.8 T.6 W T 5 W 6.4 T.6 W Tab.. cotiued. Average Avg. Average percetage of fathomed perceta solvig Number of opt odes by ge of Optimal H etire istaces Lower Theor Lemm Notat Lemm fathome H B&B boud em 5 a io a 3 d odes Series Series C 0. Q 0.6 T 5 W 6 T.8 T Number of Optimal istaces opt H Tab.. cotiued. Average percetagee Avg. ercetage of fathomed odes by solvig of etire fathomed Lower Theorem Lemma Notatio Lemma H B&B odes boud Iteratioal Joural of Idustrial Egieerig & Productio Research, March 08, Vol. 9, No.

16 30 Omolbai Mashai & Ghasem Moslehi W.6 W 0 Series 8 C 0. Q 0.6 T 0 W 6 T.4 T W.6 W Series C 0. Q 0.6 T 0 W 6 T.8 T W.6 W Miimizig the Number of Tardy Jobs i the Sigle Machie Schedulig Problem Series Series 3 C 0.6 Q 0. T W T W 6 Series 4 C 0.6 Q 0. T W T W 6 Series 5 C 0.6 Q 0. T W.4 T.6 W.8 T.6 W T 5 W 6.4 T.6 W Tab.. cotiued. Average Avg. Average percetage of fathomed Number perceta solvig opt odes by of ge of Optimal H etire Lower Theor Lemm Notat Lemm istaces fathome H B&B boud em 5 a io a 3 d odes Iteratioal Joural of Idustrial Egieerig & Productio Research, March 08, Vol. 9, No.

17 Miimizig the Number of Tardy Jobs i the Sigle Omolbai Mashai & Ghasem Machie Schedulig Problem... Moslehi 3 Tab.. cotiued. Series Average Number Avg. opt percetage Average percetage of fathomed odes by of solvig of etire Optimal H fathomed Lower Theorem Lemma Notatio Lemma H istaces B&B odes boud 5 3 Series 6 C 0.6 Q 0. T W T.8 T W.6 W Series 7 C 0.6 Q 0. T W T.4 T W.6 W Series 8 C 0.6 Q 0. T W T.8 T W.6 W Series Series 9 C 0.6 Q 0.6 T W 6 T.4 T W.6 W Series 0 C 0.6 Q 0.6 T Tab.. cotiued. Average Avg. Average percetage of fathomed perceta solvig Number of opt odes by ge of Optimal H etire istaces Lower Theor Lemm Notat Lemm fathome H B&B boud em 5 a io a 3 d odes Iteratioal Joural of Idustrial Egieerig & Productio Research, March 08, Vol. 9, No.

18 3 Omolbai Mashai & Ghasem Moslehi W 6 T.8 T W.6 W Series C 0.6 Q 0.6 T 5 W 6 T.4 T W.6 W Miimizig the Number of Tardy Jobs i the Sigle Machie Schedulig Problem Tab.. cotiued. Average Number Avg. opt percetage Average percetage of fathomed odes by of solvig Series of etire Optimal H fathomed Lower Theorem Lemma Notatio Lemma H istaces B&B odes boud Series C 0.6 Q T W T.8 T W.6 W Series C Q T W T.4 T W.6 W Series C Q T W T.8 T W.6 W Research, Vol. 0, No. 3, (009), pp. 7- Refereces 6. [] Jalali Naii, S..G., Aryaezhad, M.B., Jabbarzadeh, A., & Babaei H., Coditio [] Riahi, M. ad Asarifard, M., Maiteace based maiteace for two-compoet improvemet of ball bearigs for idustrial systems with reliability ad cost applicatios, Iteratioal Joural of cosideratios, Iteratioal Joural of Idustrial Egieerig & Productio Idustrial Egieerig & Productio Iteratioal Joural of Idustrial Egieerig & Productio Research, March 08, Vol. 9, No.