AN EXACT AND META-HEURISTIC APPROACH FOR TWO-AGENT SINGLE-MACHINE SCHEDULING PROBLEM

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1 Joural of Marie Sciece ad Techology, Vol. 2, No. 2, pp (203) 25 DOI: 0.69/JMST AN EXACT AND META-HEURISTIC APPROACH FOR TWO-AGENT SINGLE-MACHINE SCHEDULING PROBLEM We-Hug Wu Key words: schedulig, sigle-machie, two-aget, geetic algorithm. ABSTRACT I may real-life applicatios, it ca be ofte foud that multiple agets compete o the usage of a commo processig resource i differet applicatio eviromets ad differet methodological fields, such as artificial itelligece, decisio theory, operatios research, etc. Moreover, schedulig with multiple agets is relatively uexplored. Based o this observatio, this paper attempts to study a sigle-machie schedulig problem where the objective is to miimize the total tardiess of the first aget with the costrait that o tardy job is allowed for the secod aget. I this study, we provide a brach-ad-boud algorithm ad a geetic algorithm for the optimal ad ear-optimal solutios. We also report a computatioal experimet to evaluate the impact of the parameters ivolvig with proposed problem simulatio settigs. I. INTRODUCTION Schedulig with multiple agets has received growig attetio i recetly years. Agetis et al. [] ad Baker ad Smith [3] were idepedetly the first authors to itroduce the cocept of multi-aget ito schedulig problems. Yua et al. [30] addressed two dyamic programmig recursios i Baker ad Smith [3] ad developed a polyomial-time algorithm for the same problem. Cheg et al. [9] cosidered the feasibility model of multi-aget schedulig o a sigle machie where each aget s objective fuctio is to miimize the total weighted umber of tardy jobs. Ng et al. [23] studied a two-aget schedulig problem o a sigle machie, where the objective is to miimize the total completio time of the first aget with the restrictio that the umber of tardy jobs of the secod aget caot exceed a give umber. Agetis Paper submitted 06/08/2; revised 2//2; accepted 0/28/3. Author for correspodece: We-Hug Wu ( wu40226@kjc.edu.tw). Departmet of Busiess Admiistratio, Kag-Nig Juior College of Medical Care ad Maagemet, Taipei, Taiwa, R.O.C. et al. [2] cosidered the schedulig problems whe several agets, each owig a set of o-preemptive jobs, compete to perform their respective jobs o oe shared processig resource. Each aget wats to miimize a certai cost fuctio, which depeds o the completio times of its jobs oly. Cheg et al. [9] studied multi-aget schedulig o a sigle machie where the objective fuctios of the agets are of the max-form. Lee et al. [8] cosidered a multi-aget schedulig problem o a sigle machie i which each aget is resposible for his ow set of jobs ad wishes to miimize the total weighted completio time of his ow set of jobs. Besides, for more multiple-aget works with time-depedet, we refer readers to Liu ad Tag, Cheg et al., Wa et al., Liu et al., Wu et al., Mor ad Mosheiov, Nog et al., ad Yi et al., etc. [7, 0, 9-22, 24, 26-29]. For more recet schedulig problems faced by the maufacturig idustry, but are from the same aget, the reader ca refer to Hsu et al. [6], Shyr ad Lee [25]. Due to the importace of multiple agets competig o the usage of a commo processig resource i differet applicatio eviromets ad differet methodological fields, we studied two-aget schedulig o a sigle machie. The objective is to miimize the total tardiess of the jobs of the first aget with the restrictio that o tardy job is allowed for the secod aget. The remaider of this paper is orgaized as follows: I Sectio II, the problem statemet is give. I Sectio III, some domiace properties ad a lower boud are preseted. I Sectio IV, the details of three geetic algorithms are described. I Sectio V, the extesive computatioal experimets to assess the performace of all of the proposed algorithms are reported. The coclusio is give i the last sectio. II. PROBLEM FORMULATION The problem is described as follows. There are jobs which belogs to oe of the agets AG 0 or AG. For each job j, there is a ormal processig time p j, a due date d j, ad a aget code I j, where I j = 0 if J j AG 0 or I j = if J j AG. All the jobs are available at time zero. Uder a schedule S,

2 26 Joural of Marie Sciece ad Techology, Vol. 2, No. 2 (203) let C j (S) be the completio time of job j, T j (S) = max{0, C j (S) d j } be the tardiess of J j ad U j (S) = if T j (S) > 0, ad zero otherwise. The objective of this paper is to fid a optimal schedule to miimize j= Tj( S)( I j) = 0 subject to U ( S) I 0. j= j j = III. BRANCH-AND-BOUND ALGORITHM The classical sigle-machie total tardiess problem without agets was proved to be NP-hard. Thus, our problem is also NP-hard. Moreover, o relative computatioal results from the algorithm viewpoits for the problem have bee reported. Thus, we will attempt to use the brach-ad-boud techique ad a geetic algorithm to search for the optimal solutio ad ear optimal solutio, respectively. Below we will develop the brach-ad-boud techique icorporatig with some domiace rules to help searchig for the optimal solutio. Below are some adjacet properties.. Domiace Properties I this subsectio, some adjacet domiace rules are first derived by usig the pairwise iterchage method. Let S ad S 2 deote two give job schedules i which the differece betwee S ad S 2 is a pairwise iterchage of two adjacet jobs i ad j. That is, S = (σ, i, j, σ ) ad S 2 = (σ, j, i, σ ), where σ ad σ each deote a partial sequece. I additio, let t be the completio time of the last job i σ. Property. If jobs i, j AG 0, p i < p j, ad t > max{d i p i, d j p j }, the S domiates S 2. Proof: From t > max{d i p i, d j p j }, we have ad T ( S ) = t+ p d, () i i i T ( S ) = t+ p + p d. (2) j i j j T ( S ) = t+ p d, (3) j 2 j j T ( S ) = t+ p + p d, (4) i 2 j i i From Eqs. ()-(4), ad p i < p j, we have [ T ( S ) + T ( S )] [ T ( S ) + T ( S )] = [2 p + p ] [2 p + p ] > 0 ad j 2 i 2 i j j i i j C ( S ) = C ( S ). j i 2 Therefore, S domiates S 2. The proof is completed. Property 2. If job i AG 0, job j AG, t + p i < d i < t + p i + p j, ad t + p i + p j < d j, the S domiates S 2. Proof: From job i AG 0, job j AG, ad t + p i < d i < t + p i + p j, it imply that T i (S ) = 0 ad T i (S 2 ) = t + p i + p j d i. Meawhile, because t + p i + p j < d j, we have T j (S ) = 0. Therefore, we have [T j (S 2 ) + T i (S 2 )] > [T i (S ) + T j (S )]. Property 3. If job i AG 0, job j AG, t + p i < d i ad t + p i + p j < d j, the S domiates S 2. Proof: From job i AG 0, job j AG, ad t + p i > d i, it imply that job i is tardy i S ad S 2. T i (S ) = t + p i d i ad T i (S 2 ) = t + p i + p j d i. Meawhile, because t + p i + p j < d j, we have T j (S ) = 0 ad T j (S 2 ) = 0. Therefore, we have [T j (S 2 ) + T i (S 2 )] > [T i (S ) + T j (S )]. Next, we give a propositio to determie the feasibility of the partial schedule. Let (π, π c ) be a sequece of jobs where π is the scheduled part with k jobs ad π c is the uscheduled part with (-k) jobs. Amog the uscheduled jobs, let p () = mi{ pj} ad d() = mi { d j}. Moreover, J j π c c J j π AG let C [k] be the completio times of the last job i π. Also, let π ad π deote the uscheduled jobs i AG 0 arraged i the weighted smallest processig times (SPT) order ad the uscheduled jobs i AG arraged i the earliest due date rule (EDD) order, respectively. Property 4. If all the uscheduled jobs belog to AG 0 ad C[ k ] max c{ d j}, the schedule (π, π c ) is domiated by j π schedule (π, π ). Proof: Sice C[ k ] max c{ d j}, all the uscheduled jobs are j π from AG 0 ad tardy. So the SPT rule yields a optimal subschedule. Property 5. If all the uscheduled jobs belog to AG ad o tardy job ca be foud i schedule (π, π ), the schedule (π, π c ) is domiated by schedule (π, π ). Proof: Similar to Property 4. Property 6. If C[ k ] + p() > d(), the (π, π c ) is ot a feasible sequece. Proof: Sice C[ k ] + p() d() > 0, oe job of AG amog the uscheduled jobs must be tardy. So (π, π c ) is ot a feasible solutio. 2. A Lower Boud A simple lower boud of the partial sequece will be developed i the followig. Assume that π is a partial schedule

3 W.-H. Wu: A Exact ad Meta-Heuristic Approach for Two-Aget Sigle-Machie Schedulig Problem 27 i which the order of the first k jobs is determied ad let π c be the uscheduled part with (-k) jobs. Amog the uscheduled jobs, there are 0 jobs from aget AG 0 ad jobs from aget AG. Moreover, let C [k] deote the completio times of the kth job i π. The completio time for the (k+j)th job is j C C + p,for j [ k+ j] [ k] ( k+ i) 0 i= The a lower boud ca be obtaied as follows T ( S)( I ) L ( S)( I ) j= j j j= j j = C j j S d j S I = j ( ( ) ( ))( ) 0 0 ( )( ) j= j j= ( j) = C S d = LB IV. GENETIC ALGORITHM The brach-ad-boud becomes very time cosumig whe the job size is gettig larger. Meawhile, a heuristic algorithm ca supply time-savig approximate solutio with small margi of error. Thus, we adopted three geetic algorithms (GAs) for ear-optimal solutio. Geetic algorithms (GAs) are itelliget radom search strategies which have bee successfully applied to fid earoptimal solutios of may complex problems [5-6, 6]. A geetic algorithm starts with a set of feasible solutios (populatio) ad iteratively replaces the curret populatio by a ew populatio. It requires a suitable ecodig for the problem ad a fitess fuctio that represets a measure of the quality of each ecoded solutio (chromosome or idividual). The reproductio mechaism selects the parets ad recombies them usig a crossover operator to geerate offsprigs that are submitted to a mutatio operator i order to alter them locally [2]. The procedures of the GA applied to solve the proposed problem were summarized i the followig. Represetatio of structure- I this study we adopt the method proposed by Etiler et al. [3] that a structure ca be described as a sequece of the jobs i the problem. Iitial populatio- We radomly geerate the iitial populatio based o Bea [4]. I order to arrive at the fial solutio more quickly, three improvemet techiques are applied i iitial sequeces. There are icludig pairwise iterchage, backward-shifted reisertio, ad forward-shifted reisertio []. I GA, iitial sequeces are improved by pairwise iterchage. While i GA 2, iitial sequeces are improved by forward-shifted reisertio. I GA 3, iitial sequeces are adopted by backward-shifted reisertio. Populatio size- The populatio size plays a importat role i the computatioal process of GA. I a prelimiary trial, the populatio size N is set at 40 i our computatioal experimet. Fitess fuctio- Followig Iyer ad Saxea [7], the fitess fuctio assigs to each member of the populatio a value reflectig their relative superiority or iferiority. Our objective is to miimize the total tardiess. The fitess fuctio of the strigs ca be calculated as follows: f( Si( v)) = max l N Tj( Sl( v)) Tj( Si( v)) j= j=, where S i (v) is the ith strig chromosome i the v-th geera tio, Tj( Si( v)) is the total tardiess of S i (v), ad f(s i (v)) j= is the fitess fuctio of S i (v). Therefore, the probability, P(S i (v)), of selectio for a schedule is to esure that the probability of selectio for a sequece with lower value of the objective fuctio is higher. Here P(S i (v)) ca be calculated as follows: N PS ( ( v)) = f( S( v))/ f( S( v)). i i l l= This is also the criterio used for the selectio of parets for the reproductio of childre. Crossover- This study adopts liear order crossover (LOX) method which is developed by Falkeauer ad Bouffouix [4]. I a pilot study, i order to protect the best schedule which has the miimum total tardiess at each geeratio, we trasfer this schedule to the ext populatio with o chage. This operatio eables us to choose the higher crossover with the crossover rate P c = 00%. Mutatio- I this study, the mutatio rates (P m ) are set at 0.3 based o our prelimiary experimet. Selectio- It is a procedure to select offsprig from parets to the ext geeratio. I our study, the populatio sizes are fixed at 40 from geeratio to geeratio. Excludig the best 0% schedule which has the miimum total tardiess, the rest 90% of the offsprigs are geerated from the paret chromosomes by the roulette wheel method. Termiatio- The proposed GA s are termiated after 500 geeratios or the objective with zero i our prelimiary experimet. V. COMPUTATIONAL EXPERIMENT A computatioal experimet was coducted to test the brach-ad-boud algorithm ad proposed geetic algorithms.

4 28 Joural of Marie Sciece ad Techology, Vol. 2, No. 2 (203) Table. Performace of the brach-ad-boud ad GA algorithms ( = 0, 2, 4). brach-ad-boud algorithm GA GA 2 GA 3 GA * τ R valid CPU time umber of odes error percetages sample size mea std mea std mea std mea std mea std mea std Average Average Average The algorithms were coded i Fortra ad ru o Compaq Visual Fortra versio 6.6 o a Itel(R) Core(TM)2 Quad CPU 2.66 GHz with 4 GB RAM o Widows XP. The experimetal desig follows Fisher s [6] framework. The job processig times were geerated from a uiform distributio over the itegers betwee ad 00. The due dates were geerated from a uiform distributio over the rage of itegers T( τ R/2) to T( τ + R/2), where τ is the tardiess factor, R is the due date rage, ad T is the sum of the processig times of all the jobs, i.e., T = p. The combiatio of (τ, R) took the values (0.25, 0.25), (0.25, 0.5), (0.25, 0.75), (0.5, 0.25), (0.5, 0.5), ad (0.5, 0.75). For the brach-ad-boud algorithm, the average ad stadard deviatio umbers of odes as well as the average ad stadard deviatio executio times (i secods) were recorded. For the three geetic algorithms, the mea ad stadard deviatio error percetages were recorded, where the error percetage was calculated as ( GA OP) / OP *00%, i where GA i is the total tardiess obtaied from the geetic algorithm ad OP is the total tardiess of the optimal schedule. The computatioal times of the heuristic algorithms were ot recorded sice they were fiished withi a secod. The computatioal experimet cosisted of small job um- i= i bers ad big job umbers. I the first part of the experimet with small job umbers, three job sizes ( = 0, 2 ad 4) were examied i the brach-ad-boud algorithm. The same sets of istaces were used to test the performace of the brach-ad-boud ad the geetic heuristic algorithms. As a cosequece, 8 experimetal situatios were tested. A set of 50 istaces were radomly tested for each case. Moreover, the algorithms were set to skip to the ext set of data if the umber of odes exceeded 0 9. The istaces with umber of odes less tha 0 9 were deoted as solvable istaces (valid sample size). The results are preseted i Table. As show i Fig. ad Table, it idicated that the umber of odes i the istaces is gettig larger as the umber of jobs icreases. The istaces with a bigger value of τ (τ = 0.5) is easily to solve tha those with a smaller value of τ (τ = 0.25). The performace of R also has the same situatio. For example, the istaces with a bigger value of R (R = 0.75) is easily to solve tha those with a smaller value of R (R = 0.25, 0.5). It ca be observed i Table that fixed = 4, the most difficult case occurs at (τ, R) = (0.25, 0.25) where o istace ca be solved out. As to the performace of the proposed GA algorithms, out of the 8 cases, the performaces of proposed geetic heuristics were ot affected as the values of τ or R varied. Most of the mea error percetages of GA, GA 2, ad GA 3 were less tha 2% or below, except oe case at (τ, R) = (0.25, 0.75) i GA has a bigger mea error percetage. However, the situatio was disappeared whe we further combied three

5 W.-H. Wu: A Exact ad Meta-Heuristic Approach for Two-Aget Sigle-Machie Schedulig Problem 29 Table 2. RDP of heuristic algorithms ( = 60, 80, 00). GA GA 2 GA 3 τ R CPU time RDP CPU time RDP CPU time RDP mea std mea std mea std mea std mea std mea std Average Average Average Average umber of odes 70,000,000 60,000,000 50,000,000 40,000,000 30,000,000 20,000,000 0,000,000 τ = 0.25 τ = 0.50 Mea error percetages 7.00% 6.00% 5.00% 4.00% 3.00% 2.00%.00% GA, τ = 0.25 GA, τ = 0.50 GA 2, τ = 0.25 GA 2, τ = 0.50 GA 3, τ = 0.25 GA 3, τ = R Fig.. Performace of the brach-ad-boud algorithms ( = 2). 0.00% R Fig. 2. Performace of the GA ~3 algorithms ( = 2). proposed three GAs ito GA * i which (GA * = mi{ga i, i =, 2, 3}. Table further idicated that the mea error percetages of GA * were reduced to 0.2% or below o matter that the values of τ or R varied. I the secod part of the experimet for large job-sized problems, the proposed heuristic algorithms were tested with three differet umbers of jobs at = 60, 80, ad 00. The mea executio time ad the mea relative deviace percetage were recorded for each heuristic. The relative deviace percetage (RDP) was give by * * ( GAi GA )/ GA 00%, where GA i is the value of the objective fuctio geerated by the ith heuristic, ad GA * = mi{ga i, i = 2, 3} is the smallest value of the objective fuctio obtaied from the heuristics. The results are summarized i Table 2. As show i Fig. 3 ad Table 2, it was observed that the mea RDP of GA is lower tha those of GA 2 ad GA 3. The overall mea RDP of GA was less tha 2%. However, there is o absolutely domiace betwee the performaces

6 220 Joural of Marie Sciece ad Techology, Vol. 2, No. 2 (203) Average of RPD 8.00% 7.00% 6.00% 5.00% 4.00% 3.00% 2.00%.00% 0.00% R 0.75 = 60, τ = 0.25 = 60, τ = 0.50 = 80, τ = 0.25 = 80, τ = 0.50 = 00, τ = 0.25 = 00, τ = 0.50 Fig. 3. Performace of the GA ~3 algorithms ( = 60, 80, 00). of the first three geetic algorithms. Thus, it is recommeded to use the GA * algorithm sice it has both accuracy ad the smallest RDP. VI. CONCLUSIONS This paper studied a sigle-machie two-aget schedulig problem here the objective is to miimize the total tardiess of the first aget with the costrait that o tardy job is allowed for the secod aget. The cotributios of this paper were; Firstly, a brach-ad-boud algorithm icorporatig with several domiaces ad a lower boud was proposed to derive a optimal solutio, ad the three geetic algorithms were provided for ear-optimal solutio. Fially, the impacts of the relative parameters about proposed problem were tested ad reported. The computatioal results also showed that with the help of the proposed heuristic iitial solutio, the brach-ad-boud algorithm ca solve the istaces up to = 4. Moreover, the computatioal experimets also showed that the proposed GA * algorithm performed quite well i terms of accuracy ad the smallest RDP. ACKNOWLEDGMENTS We are grateful to the Editor ad two aoymous referees for their costructive commets o the previous versio of our paper. REFERENCES. Agetis, A., Mirchadai, P. B., Pacciarelli, D., ad Pacifici, A., Schedulig problems with two competig agets, Operatios Research, Vol. 52, pp (2004). 2. Agetis, A., Pacciarelli, D., ad Pacifici, A., Multi-aget sigle machie schedulig, Aals of Operatios Research, Vol. 50, pp. 3-5 (2007). 3. Baker, K. R. ad Smith, J. C., A multiple-criterio model for machie schedulig, Joural of Schedulig, Vol. 6, pp. 7-6 (2003). 4. Bea, J. C., Geetic algorithms ad radom keys for sequecig ad optimizatio, ORSA Joural of Computig, Vol. 6, pp (994). 5. Beasley, D., Bull, D., ad Marti, R. R., A overview of geetic algorithms, part : fudametals, Joural of Uiversity Computig, Vol. 5, pp (993). 6. Che, J. S., Pa, J. C. H., ad Li, C. M., A hybrid geetic algorithm for the reetrat flowshop schedulig problem, Expert Systems with Applicatios, Vol. 34, pp (2008). 7. Cheg, T. C. E., Cheg, S.-R., Wu, W.-H., Hsu, P.-H., ad Wu, C.-C., A two-aget sigle-machie schedulig problem with trucated sum-ofprocessig-times-based learig cosideratios, Computers ad Idustrial Egieerig, Vol. 60, pp (20). 8. Cheg, T. C. E., Ng, C. T., ad Yua, J. J., Multi-aget schedulig o a sigle machie to miimize total weighted umber of tardy jobs, Theoretical Computer Sciece, Vol. 362, pp (2006). 9. Cheg, T. C. E., Ng, C. T., ad Yua, J. J., Multi-aget schedulig o a sigle machie with max-form criteria, Europea Joural of Operatioal Research, Vol. 88, pp (2008). 0. Cheg, T. C. E., Wu, W. H., Cheg, S. R., ad Wu, C. C., Two-aget schedulig with positio-based deterioratig jobs ad learig effects, Applied Mathematics ad Computatio, Vol. 27, pp (20).. Della Croce, F., Naraya, V., ad Tadei, R., The two-machie total completio time flow shop problem, Europea Joural of Operatioal Research, Vol. 90, pp (996). 2. Essafi, I., Matib, Y., ad Dauzere-Peres, S., A geetic local search algorithm for miimizig total weighted tardiess i the job-shop schedulig problem, Computers ad Operatios Research, Vol. 35, pp (2008). 3. Etiler, O., Toklu, B., Atak, M., ad Wilso, J., A geeric algorithm for flow shop schedulig problems, Joural of Operatios Research Society, Vol. 55, No. 8, pp (2004). 4. Falkeauer, E. ad Bouffoix, S., A geetic algorithm for job shop, Proceedigs of the 99 IEEE Iteratioal Coferece o Robotics ad Automatio, Vol., pp (99). 5. Fisher, M. L., A dual algorithm for the oe-machie schedulig problem, Mathematical Program, Vol., pp (97). 6. Hsu, C.-J., Yag, Y.-J., ad Yag, D.-L., Due-date assigmet ad optimal maiteace activity schedulig problem with liear deterioratig jobs, Joural of Marie Sciece ad Techology, Vol. 9, No., pp (20). 7. Iyer, S. K. ad Saxea, B. S., Improved geetic algorithm for the permutatio flowshop schedulig problem, Computers ad Operatios Research, Vol. 3, pp (2004). 8. Lee, K. B., Choi, B. C., Leug, J. Y. T., ad Piedo, M. L., Approximatio algorithms for multi-aget schedulig to miimize total weighted completio time, Iformatio Processig Letters, Vol. 09, pp (2009). 9. Liu, P. ad Tag, L., Two-aget schedulig with liear deterioratig jobs o a sigle machie, Lecture Notes i Computer Sciece, Vol. 5092, pp (2008). 20. Liu, P., Yi, N., ad Zhou, X., Two-aget sigle-machie schedulig problems uder icreasig liear deterioratio, Applied Mathematical Modellig, Vol. 35, No. 5, pp (20). 2. Mor, B. ad Mosheiov, G., Schedulig problems with two competig agets to miimize mimax ad misum earliess measures, Europea Joural of Operatioal Research, Vol. 206, pp (200). 22. Mor, B. ad Mosheiov, G., Sigle machie batch schedulig with two competig agets to miimize total flowtime, Europea Joural of Operatioal Research, Vol. 25, pp (20). 23. Ng, C. T., Cheg, T. C. E., ad Yua, J. J., A ote o the complexity of the problem of two-aget schedulig o a sigle machie, Joural of Combiatorial Optimizatio, Vol. 2, pp (2006). 24. Nog, Q. Q., Cheg, T. C. E., ad Ng, C. T., Two-aget schedulig to miimize the total cost, Europea Joural of Operatioal Research, Vol. 25, pp (20). 25. Shyr, O. F. F. ad Lee, Y.-L., Modelig pricig ad schedulig strategies for air cargo carriers as o-cooperative games, Joural of Marie Sciece ad Techology, Vol. 20, No. 2, pp (202). 26. Wa, G., Vakati, S. R., Leug, J. Y. T., ad Piedo, M., Schedulig two agets with cotrollable processig times, Europea Joural of Opera-

7 W.-H. Wu: A Exact ad Meta-Heuristic Approach for Two-Aget Sigle-Machie Schedulig Problem 22 tioal Research, Vol. 205, pp (200). 27. Wu, C. C., Huag, S. K., ad Lee, W. C., Two-aget schedulig with learig cosideratio, Computers & Idustrial Egieerig, Vol. 6, No. 4, pp (20). 28. Yi, Y., Cheg, S. R., Cheg, T. C. E., Wu, W. H., ad Wu, C. C., Two-aget sigle-machie schedulig with release times ad deadlies, Iteratioal Joural of Shippig ad Trasport Logistics, Vol. 5, No., pp (203). 29. Yi, Y., Wu, W. H., Cheg, S. R., ad Wu, C. C., A ivestigatio o a two-aget sigle-machie schedulig problem with uequal release dates, Computers & Operatios Research, Vol. 39, pp (202). 30. Yua, J. J., Shag, W. P., ad Feg, Q., A ote o the schedulig with two families of jobs, Joural of Schedulig, Vol. 8, pp (2005).