Minimizing Multiple Objective Function for Scheduling Machine. Problems

Transcription

1 Miimizig Multiple Objective Fuctio for Schedulig Machie Problems Dr. Saad Talib Hasso (Prof.) Deputy dea, Al-Musaib Egieerig College Uiversity of Babylo, Iraq. Shahbaa Mohammed Yousif Math. Dept, College of Educatio, Uiversity of Al-Mustasiriya, Baghdad, Iraq. Abstract The multi-criteria problem of schedulig jobs o a sigle machie was cosidered i this paper. The criteria belog to miimize total completio times, total tardiess ad total late work ad miimize total completio times, total tardiess ad maximum late work by usig some exact ad local search methods. The proposed methods for solvig these miimizatio problems were seemed to be helpful i fidig the set of all efficiet solutios. This set of all efficiet solutios is ot easy to fid, therefore, it could be preferable to have a approximatio to that set i a reasoable amout of time. Therefore brach ad boud (BAB) techique was proposed as a exact approach, while the Geetic algorithm (GA) ad Particle Swarm Optimizatio (PSO) methods were also proposed as local search methods. Keywords: Multiple objective Schedulig, Brach ad Boud, Pareto Optimal Solutios, Geetic Algorithm, Particle Swarm Optimizatio. 1. Itroductio Schedulig problems are brach of a large field of combiatorial optimizatio (CO). It was defied as a decisio makig process that is used o a regular basis i may maufacturig ad services idustries. Schedulig problems deals with the allocatio of resources to tasks over give time periods. Its goal is to miimize oe or more objectives (Piedo 2008). Schedulig theory has bee of cocer i productio problems, maufacturig models, computer sciece, idustrial maagemet, trasportatio, agriculture, hospitals ad may other applicatios (Agi 1996). Resources ad tasks were called machies ad jobs respectively. A machie ca perform at most oe job at a time. The multi-criteria schedulig problem ca be stated as follows: There are jobs to be processed o a sigle machie, each job i has processig time p i ad due date d i at which ideally should be completed. Pealties are icurred wheever a job i is completed earlier or later tha its due date d i. Multi-criteria optimizatio with coflictig objective fuctios provides a set of Pareto optimal solutios, rather tha oe optimal solutio. This set of solutios icludes the solutio that o other solutio is better tha with respect to all objective fuctios (Abdul_Razaq). Although the importace of multi-criteria schedulig has bee fouded for may years (Frech 1982); (Nelso et al. 1986); (George ad Paul 2007); (Mahmood 2014), little attetio has bee give i the literature to this topic. The brach ad boud (BAB) methods have bee first applied to schedulig by Lomicki 1965 ad Igall ad Schrage (1965), these methods accurately solve machie schedulig problems (MSP). Geetic Algorithm (GA) ca be cosidered as a class of optimizatio algorithms. GA attempts to solve problems through modelig a simplified versio of geetic process. There are may problems for which a GA approach is useful. It is, however, utraditioal if assigmet is such a problem (Sabah 2004). The Particle Swarm Optimizatio (PSO) cosidered a ew family of algorithms that may be used to fid optimal, ear optimal or approximated solutios to optimizatio problems. PSO is a extremely simple algorithm 49

2 that seems to be effective for optimizig a very wide rage of applicatios (Shi 2004). The orgaizatio of this paper is as follows: i sectio2 we preset multiple objective problems. Sectio3 ad 4 presets the mathematical model ad discusses the BAB, PSO ad GA methods. Implemetatio, experimetal results, aalysis ad coclusios are give i the last sectios. 1.1 Problem Statemet The problem of this work is to propose differet models to fid the efficiet solutio (Pareto optimal solutios) for 1//( C i, T i, V i ) ad 1//( C i, T i,v max ) problems. Complete Eumeratio Method (CEM) ad Brach Ad Boud (BAB) method ca be used. GA ad PSO as two local search methods (LSM) may also applied for multi-criteria schedulig problem. The followig otatio will be used i this paper: : umber of jobs. N : Set of jobs. p j : processig time of job j. d j : due date of job j. C j : completio time of job j. T j : the tardiess of job j. V j : the late work of job j. V max : The maximum late work The followig basic cocepts sequece ad rules were used i this paper: Defiitio (1): Shortest Processig Time (SPT) rule; Jobs are sequecig i o-decreasig order of p i, this rule was used to miimize C i for 1/ / C i problem (Smith 1956). Defiitio (2): Earliest Due Date (EDD) rule: Jobs are sequecig i o-decreasig order of d i, this rule was used to miimize T max for 1/ /T max (Joui 2000). Defiitio (3): The term "Optimize" i a multi-criteria decisio makig problem refers to a solutio aroud which there is o way to improve ay objective without worseig the other objective (Joui 2000). Defiitio (4): A feasible schedule is a Pareto optimal (PO), or o-domiated (efficiet) with respect to the performace criteria f ad g if there is o feasible schedule such that both f( ) f( ) ad g( ) g( ), where at least oe of the iequalities is strict (Hoogevee 2005). Lemma (1): (Al-Magraby's lemma) If d j p i, the there exists a optimal sequece i which job j sequecig last, for the 1/ / T i problem (Che et al. 1997). Theorem (1): (Emmo's theorem)if p i p j ad d i d j the there exists a optimal sequecig i which job i sequecig before job j, for the 1/ / T i problem (Che et al. 1997). 2. Mathematical Formulatio with Aalysis The MCP deals with schedulig the set N={1,2,,} of jobs which are processed o a sigle machie to miimize the multi-criteria. Each job i N has is to be processed o a sigle machie which ca hadle oly oe job at a time. The job i has a processig time p i ad due date d i, all jobs are available for processig at a time zero. If a schedule = (1,2,,) is give, the the earliest completio time C computed ad cosequetly the tardiess of job it i = max{c i -d i,0} ad i i p j j 1 for each job i ca be 50

3 0 V Ci di pi if if if C d, d C d p i i i i d C, i p, i 1,2,..., i 1,2,...,. i i i i i i 2,3,..., 2.1 Mai Cocepts Theorem (2): If the composite objective fuctio F is liear, the there exists a extreme schedule that miimizes F (Hoogevee 2005). Defiitio (5): The fuctio F(f,g) is said to be o-decreasig i both argumets if for ay pair of outcome values (x,y) of the fuctios f ad g, we have F(x,y) F(x+a, y+b), for each pair of o-egative values a ad b (Hoogevee 2005). Theorem (3): If the composite objective fuctio F(f,g) is o-decreasig i both argumets, the there exists a Pareto optimal schedule that miimizes F (Hoogevee 2005). I this work, the followig algorithms were required: Lawler's algorithm (LA) which solves the 1//V max problem [1], to fid miimum V max. Lawler's algorithm (LA) (Mahmood 2014) was described by the followig steps: Step (1): Let N={1,2,..,}, F is the set of all jobs with o successors ad =. Step (2): Let j* be a job such that V ( p ) mi{v ( p )}. j* i N i j F j i N Step (3): Set N=N-{j*} ad sequece job j* i last positio of, i.e. =(j*, ). Step (4): Modify F with respect to the ew set of schedulable jobs. Step (5): If = stop, otherwise go to step (2). i I this subsectio we shall try to fid efficiet (Pareto optimal) solutios for multi-criteria simultaeous problems. Defiitio (5) ca be exteded for three objectives as: Defiitio (6): The fuctio F (f 1, f 2, f 3 ) is said to be o-decreasig i its argumets if for ay outcome value (x,y,z) of the fuctios f 1, f 2 ad f 3, we have F(x,y,z) F(x+a, y+b, z+c) for each of o-egative value a, b ad c. Theorem (4): Cosider the composite objective fuctio F(f 1, f 2,...,f k ) where F is o-decreasig i all performace criteria f i, i=1, 2,, k, the there is a Pareto optimal schedule with respect to f 1,f 2,,f k that miimizes F (Mahmood 2014). Let us first cosider the formulatios of the multi-criteria simultaeous problems say 1//(f 1,f 2,f 3 ). The formulatio was as follows: Mi Multi-criteria f f f S.t. Costraits for the problem of object f 1 Costraits for the problem of object f 2 Costraits for the problem of object f 3 I multi-criteria schedulig problems, the optimal solutios were geerally called Pareto optimal solutios. 51

4 This meas that i this type of optimizatio, we geerate all efficiet (o-domiated) solutios, the allow the decisio maker to make explicit trade-offs amog these solutios. For the problem 1//(f1,f2,f3) a schedule (solutio) 1 domiates aother schedule if ad oly if 2 f 1 ( 1 ) f 1 ( 2 ), f 2 ( 1 ) f 2 ( 2 ) ad f 3 ( 1 ) f 3 ( 2 ), with at least oe strict iequality. Suppose that f 1 = C j, f 2 = T j ad f 3 { V j,v max }. Our objective is to fid a schedule S (where S is the set of all feasible schedule) that miimizes the multi-criteria problems: 1//( C j, T j, V j ) ad 1//( C j, T j,v max ), these problems belog to simultaeous optimizatio. Problem (P 1 ) ca be formulated as follows: Mi { C i, T i, V i } Subject to C i p i, i=1,2,,. T i C i -d i, i=1,2,,. (P 1 ) T i,v i 0, i=1,2,,. The V j j 1 1 // problem is NP-hard sice its o-preemptive total late work problem is NP-hard (Joui 2000). While problem (P 2 ) ca be formulated as follows: Mi { C i, T i, V max } Subject to C i p i, i=1,2,,. T i C i -d i, i=1,2,,. (P 2 ) T i,v i 0, i=1,2,,. The oly chace to miimize T i is to use the special cases for the jobs with the same processig times (see sectio 2.2). From problems (P 1 ) ad (P 2 ) we ca derive two subproblems, say (SP 1 ) ad (SP 2 ) respectively. Problem (SP 1 ) is 1/ / C i + T i + V i which ca be writte as: Mi { C i + T i + V i } Subject to C i p i, i=1,2,,. T i C i -d i, i=1,2,,. (SP 1 ) T i,v i 0, i=1,2,,. While problem (SP 2 ) is 1/ / C i + T i + V max which ca be writte as: Mi { C i + T i + V max } Subject to C i p i, i=1,2,,. T i C i -d i, i=1,2,,. (SP 2 ) T i,v i 0, i=1,2,,. The aim for the (P 1 ) ad (P 2 ) problems is to fid a processig order of the jobs o a sigle machie to miimize the sum of total completio times ad the total tardiess ad the total of late work, or maximum late 52

5 work, which are a sigle object ad ca be miimized by BAB method. 2.2 Special Cases for the Problems (P 1 ) ad (P 2 ) For the multi-criteria, if the objectives ca be optimized idividually, the oe ca deduce that the set of efficiet solutios have o more elemets oly oe with extreme values of the idividul objective fuctios. The above fact ca be see i the followig special cases: Case (1): A schedule obtaied by orderig the jobs i a o-decreasig order of thier processig times (SPT-rule) is a effeceit solutio for both problems (P 1 ) ad (P 2 ) if d (i) +p (i) C (i+1) for all i = 1,2,,-1. Case (2): From Emmo's theorem, if the SPT ad EDD rules are idetical the there exist a effeceit solutio for both problems (P 1 ) ad (P 2 ). Case (3): If p i = p, i, p is positive iteger ad a schedule obtaied by orderig the jobs i a o-decreasig order of due dates (EDD-rule) is a effeceit solutio for both problems (P 1 ) ad (P 2 ). Case (4): If d i = d, i, d is positive iteger ad a schedule obtaied by orderig the jobs i a o-decreasig order of processig times (SPT-rule) is a effeceit solutio for both problems (P 1 ) ad (P 2 ). Note that case (3) ad case (4) are special case of case (2). Case (5): From Al-Magrapy lemma, if d j i 1 p, ad p max{p }, ad this also satisfies for each job k N-{j}, i j the there exists a efficiet solutio for (P 1 )ad (P 2 ). Case (6): If satisifes Lawler's algorithm (LA),the there exist a efficiet solutio for problem (P 2 ). Case (7): If for ay schedule, C (j) d (j), j, j=1,2,,, ad satisfies SPT-rule, the the schedule gives V i =0 ad V max =0 the there exists a efficiet solutio for both (P 1 ) ad (P 2 ). 3. MSP Exact Solvig Methods 3.1 Complete Eumeratio Method (Joui 2000). Complete eumeratio methods geerate oe by oe, all feasible schedules ad the pick the best oe. For a sigle machie problem of jobs there are! differet sequeces. Hece for the correspodig m machies problem, there are (!) m differet sequeces. This method may take cosiderable time as the umber (!) m is very large eve for relatively small values of ad m. i N i 3.2 Brach ad Boud Method for (P 1 ) ad (P 2 ) This method depeds o the techiques of brach ad boud (BAB) algorithm with some modificatios. The BAB method is characterized by its brachig procedure, upper ad lower boudig procedures ad search strategy. We preset a costructive BAB algorithm to fid all or some of the efficiet solutios (Pareto optimal poits (POP)) whe the criteria C i, T i ad V i (V max ) are of simultaeous iterest i problem P 1 (or P 2 ). The mai idea of this BAB algorithm is depedig o properties of BAB algorithm ad some modificatios such as usig the defiitio of efficiet solutios ad without reset the upper boud (UB) at the last level of BAB method. Let LB1=max( C i - d i,t max (EDD)) The mai steps of the BAB algorithm are as follows: Step(1): Fid the proposed UB by SPT rule, that is sequecig the job i o-decreasig order of their processig time p i, i=1,2,...,, for this order calculate C i ( ), T i ( ) ad V i ( ) (V max ( )) ad set 53

6 UB=( C i ( ), T i ( ), V i ( )) (or UB=( C i ( ), T i ( ),V max ( ))) at the paret ode of the search tree. UB is efficiet by propositio (1) ad add this efficiet solutio to the set of POP. If T max (EDD)=0, the there exists a efficiet sequece obtaied by propositio (2), ad also add this efficiet solutio to the set of POP. Step(2):For each partial sequece of jobs (i.e., for each ode i the search tree), compute the lower boud LB( ) as follows: LB P1 ( )=(SPT( ),LB1,T max (EDD)) for (P 1 ) ad LB P2 ( )=(SPT( ),T max (EDD),Lawler( ))for (P 2 ). Step(3):Brach from each ode with LB( ) UB. Step(4):At each ode of the last level of the BAB method, if ( C i, T i, V i ) (or ( C i, T i,v max )) deote the outcome, the add this outcome to the set of POP, uless it is domiated by the previously obtaied POP. Step(5): Stop. 3.3 Brach ad Boud Method for (SP 1 ) ad (SP 2 ) I this sectio we will apply the above BAB steps for the (SP 1 ) ad (SP 2 ) subproblems with differet lower bouds. We still use the sequece satisfies SPT-rule as a UB for both subproblems. While, the lower boud for the same subproblems is as follows: LB SP1 ( )=(SPT( )+LB1+T max (EDD)) for (SP 1 ) ad LB SP2 ( )=(SPT( )+LB1+ Lawler( )) for (SP 2 ). 3.4 Experimetal Results of Applyig CEM ad BAB for (P 1 ) ad (P 2 ) For the problems (P 1 ) ad (P 2 ), ad for the subproblems (SP 1 ) ad (SP 2 ), a simulatio has bee costructed usig MATLAB10.0 i order to apply CEM ad BAB. The followig otatios were used: EV: Efficiet Value(s). NE: Number of Efficiet Values. T/s: Time i secods. Table (1) ad (2) shows the CPU time results after applyig BAB method compared with results obtaied from CEM i order to get a set of efficiet solutios, o samples of differet jobs. The results of CPU time, which geerate all solutios for 10. I table (3) we will show the performace of BAB method for the problems (P 1 ) ad (P 2 ) for =11,,25. I table (4) we will show the results accuracy whe applyig BAB compared with CEM for the subproblem(sp 1 ) ad (SP 2 )for =3,,10. Table (5) describes the results of applyig BAB method for the subproblem (SP 1 ) ad (SP 2 ) for =11,, MSP Local Search Solvig Methods There are may methods that ca be used to solve multi-criteria schedulig problems, which are to fid the set of all (some) efficiet solutios or at least approximatio to it. It is kow that the set of all efficiet solutios is difficult to fid especially for large ( 25). Therefore, it could be acceptable to fid a approximatio to the Pareto set i a reasoable time. I this paper, we will itroduce two local search methods to solve multi-criteria schedulig for both problem (P 1 ) ad (P 2 ) (ad subproblem (SP 1 ) ad (SP 2 )) to fid the set of efficiet solutios. Before we discuss each of the proposed methods, we have to talk about the commo basics betwee the two methods, these basics are: 1. Problem Defiitio ad Represetatio The most importat problem of the set of combiatorial optimizatio problems is udoubtedly the Machie Schedulig Problem (MSP). I order to fid the set of POP, we solve the problems (P 1 ) ad (P 2 ) of miimizig ( C i, T i, V i ) ad ( C i, T i,v max ) respectively. Obviously, this schedulig problem is example of 54

7 NP-complete, the work area to be explored grows expoetially accordig with umber of jobs, ad so does. The geeral complexity is!, such that jobs were must be arraged i a sigle machie. The solutio represetatio should be a iteger vector. I this particular approach we accept schedule represetatio which is described as a sequece of jobs. 2. Iitial Populatio For the iitializatio process we ca either use some heuristics startig from differet jobs, or we ca iitialize the populatio by a radom sample of permutatio of N={1,2,,}. 4.1 Geetic Algorithms Geetic Algorithms (GA s) are search algorithms based o the mechaics of atural selectio ad atural geetics. GA is a iterative procedure, which maitais a costat size populatio of cadidate solutio. Durig each iteratio step (Geeratio) the structures i the curret populatio are evaluated, ad, o the basic of those evaluatios, a ew populatio of cadidate solutios formed (Mitchell 1998). Now we will discuss the use of GA first, sice it has bee used before i MSP for may times. 1. Geetic Operators Selectio Operator The selectio method is the roulette wheel. Copyig strig accordig to their fitess value meas strigs with higher value have higher probability of cotributig oe or more offsprig i the ext geeratio (Mitchell 1998). Crossover Operator Order Crossover (OX) (Davis 1985) builds offsprig by choosig a subsequece of a tour from oe paret ad preservig the relative order of cities from the other paret. For example, two parets (with two cut poits marked by ' ') Idividual ad Idividual Would produce the, offsprig i the followig way; First, the segmets betwee cut poits are copied ito offsprig: Offsprig1 x x x x x ad Offsprig2 x x x x x This sequece is placed i the first offsprig (startig from the secod cut poit), the the Offsprig1 ad Offsprig2 are: Offsprig Offsprig Mutatio Operator After the ew geeratio has bee determied, the chromosomes are subjected to a low rate mutatio process. For our problems we apply the two poit mutatio operator to itroduce geetic diversity ito the evolvig populatio of permutatio, which radomly selects two elemets i the chromosome ad swap them (Abdil_Razaq 2013). 2. Geetic Parameters For MSP, from our experiece, the followig parameters are preferred to be used: populatio size (pop size =20), probability of crossover (Pc = 0.7), probability of mutatio Pm =0.1 ad some hudreds umber of 55

8 geeratios. 4.2 Particle Swarm Optimizatio PSO is a extremely very simple cocept, which it ca be implemeted without complex data structure. No complex or costly mathematical fuctios are used, ad it does t require a great amout of memory (Ribeiro 2003). The PSO algorithm depeds i its implemetatio i the followig two relatios: v id = w * v id + c 1 * r 1 * (p id -x id ) + c 2 * r 2 * (p gd -x id ) (a) x id = x id + v id (b) where w is the iertia weight, c 1 ad c 2 are positive costats, r 1 ad r 2 are radom fuctio i the rage [0,1], x i =(x i1,x i2,,x id ) represets the i th particle; p i =(p i1,p i2,,p id ) represets the (pbest) best previous positio (the positio givig the best fitess value) of the i th particle; the symbol g represets the idex of the best particle amog all the particles i the populatio, v i =(v i1,v i2,,v id ) represets the rate of the positio chage (velocity) for particle i (Ribeiro 2003). The origial procedure for implemetig PSO is as follows: 1. Iitialize a populatio of particles with radom positios ad velocities o d-dimesios i the problem space. 2. PSO operatio icludes: a. For each particle, evaluate the desired optimizatio fitess fuctio i d-variables. b. Compare particle's fitess evaluatio with its pbest. If curret value is better tha pbest, the set pbest equal to the curret value, ad pa i equals to the curret locatio x i. c. Idetify the particle i the eighborhood with the best success so far, ad assig it idex to the variable g. d. Chage the velocity ad positio of the particle accordig to equatio (a) ad (b). 3. Loop to step (2) util a criterio is met. Like the other evolutioary algorithms, a PSO algorithm is a populatio based o search algorithm with radom iitializatio, ad there is a iteractio amog populatio members. Ulike the other evolutioary algorithms, i PSO, each particle flies through the solutio space, ad has the ability to remember its previous best positio, survives from geeratio to aother (Abdul_Razaq 2013). A umber of factors will affect the performace of the PSO. These factors are called PSO parameters, these parameters are (Keedy 1995): 1. Number of particles i the swarm affects the ru-time sigificatly, thus a balace betwee variety (more particles) ad speed (less particles) must be sought. 2. Maximum velocity (v max ) parameter. This parameter limits the maximum jump that a particle ca make i oe step. 3. The role of the iertia weight w, i equatio (3a), is cosidered critical for the PSO s covergece behavior. The iertia weight is employed to cotrol the impact of the previous history of velocities o the curret oe. 4. The parameters c 1 ad c 2, i equatio (3a), are ot critical for PSO s covergece. However, proper fie-tuig may result i faster covergece ad alleviatio of local miima, c 1 tha a social parameter c 2 but with c 1 + c 2 = The parameters r 1 ad r 2 are used to maitai the diversity of the populatio, ad they are uiformly distributed i the rage [0,1]. 56

9 5. Experimetal Results of Applyig GA ad PSO for (P 1 ) ad (P 2 ) For the problems (P 1 ) ad (P 2 ), ad for the subproblems (SP 1 ) ad (SP 2 ), a simulatio has bee costructed usig Delph10.0 i order to apply GA ad PSO for =3,,1000 jobs. We use the followig otatios: MBV: Mea of Best Value(s). MEV: Mea of Efficiet Value(s). MAE: Mea Absolute Error betwee the MBV ad MEV. AAE: Average of Absolute Error comparig MBV with EV obtaied from SPT rule. 5.1 Experimetal Results for Problems (P 1 ) ad (P 2 ) For the problems (P 1 ) ad (P 2 ), a simulatio has bee costructed usig Delph10.0 i order to apply GA ad PSO. Table (6) ad (7) show the experimetal results of applyig the local search methods; GA ad PSO for problems (P 1 ) ad (P 2 ) respectively compared with CEM for =3,,10. Table (8) ad (9) show the experimetal results of applyig the local search methods; GA ad PSO for problems (P 1 ) ad (P 2 ) respectively compared with BAB for =11,,25. Figure (1) ad (2) shows the MAE behavior for both PSO ad GA applied for (P 1 ) ad (P 2 ) respectively, with =11,,25. Table (10) ad (11) show the experimetal results of applyig the local search methods; GA ad PSO (compared with other) for problems (P 1 ) ad (P 2 ) respectively for =(30,(10),90) ad (100,(100),1000). Figure (3) ad (4) show the AAE behavior for both PSO ad GA applied for (P 1 ) ad (P 2 ) respectively, while figure (5) shows the time compariso for the two problems with =30,,100,, Experimetal Results for Subproblems (SP 1 ) ad (SP 2 ) Table (12) ad (13) show the experimetal results of applyig the local search methods; GA ad PSO for subproblems (SP 1 ) ad (SP 2 ) respectively compared with CEM for =3,,10. We use the followig otatios: BV: Best Value obtaied from GA ad PSO. OV: Optimal Value obtaied from CEM ad BAB. AE: Absolute Error betwee the BV ad OV. AES: Absolute Error betwee the BV, of GA ad PSO, ad BV obtaied from SPT. NI: Number of Iteratios. Table (14) ad (15) show the experimetal results of applyig the local search methods; GA ad PSO for subproblems (SP 1 ) ad (SP 2 ) respectively compared with BAB for =11,,30. Table (16) ad (17) show the experimetal results of applyig the local search methods; GA ad PSO (compared with other) for subproblems (SP 1 ) ad (SP 2 ) respectively for =(40,(10),90), (100,(100),1000) ad Figure (6) ad (7) show the AES behavior for both PSO ad GA applied for (SP 1 ) ad (SP 2 ) respectively, while figure (8) shows the time compariso for the two subproblems with =40,,100,,1000, Aalysis of the Experimetal Results 1. For this paper, a differet umber of jobs () are used for sigle machie, startig from =3(1)10, =30(10)100, =200(100)1000 ad =2000, with umber of iteratios which is suitable with to solve the problems (P 1 ) ad (P 2 ). 2. The criteria of testig the efficiecy of local search method (GA ad PSO) are calculated, these criteria represeted by, the value of the triple-objective fuctio of exact efficiet solutios for the problems (P 1 ) 57

10 ad (P 2 ) of this paper (which are calculated from CEM for 10 ad from BAB for 11 25), the approximated local search best efficiet solutios (BV) ad their average (MBV), the MAE ad AAE, while for subproblems (SP 1 ) ad (SP 2 ) we calculate the BV, AE ad AES, ad the time which complete the sigle experimet, lastly, the iteratio which foud the correspodig BV. 3. For problems (P 1 ) ad (P 2 ): from tables (6) ad (7) for =3,,10, GA ad PSO gives approximated results from EV ad from each other ad that is clear from MAE values. from tables (8) ad (9) for =11,,25, GA ad PSO gives approximated results from EV but GA serves better from PSO ad that is clear from MAE values (see figures (1) ad (2)). from tables (10) ad (11) for =30,,1000, GA gives better results from PSO ad that is clear from AAE values (see figures (3) ad (4)). from figure (5), for =30,,1000, PSO gives better performace i time from GA. 4. For subproblems (SP 1 ) ad (SP 2 ): from tables (12) ad (13) for =3,,10, GA ad PSO gives approximated results from OV ad from each other ad that is clear from AE values. from tables (14) ad (15) for =11,,30, GA ad PSO gives approximated results from OV ad from each other ad that is clear from AE values. from tables (16) ad (17) for =40,,2000, PSO gives better results from GA ad that is clear from AE values (see figures (5) ad (6)). from figure (8), for =40,,2000, PSO gives better performace i time from GA. 7. Coclusios ad future Works 1. I this paper, we applied exact (CEM ad BAB) ad local search methods (GA ad PSO) to solve the problems (P 1 ) ad (P 2 ) ad the subproblems (SP 1 ) ad (SP 2 ). 2. It's ot easy to fid a efficiet solutio to problem of multi-objective fuctios, especially for more tha double-objective fuctios. 3. From this paper, we see, i geeral, that the GA gives accuracy results better tha PSO, while PSO gives better performace time tha GA. 4. As special case, its ot hard to solve hierarchical subproblems derived from (P 1 ) ad (P 2 ). 5. A hybrid ca be doe to improve the performace of local search methods, e.g. usig simulated aealig or tabu search. Refereces Abdul-Razaq T. S. ad H. Ali F. H., (2013). Algorithms for Schedulig a Sigle Machie to Miimize Total Completio Time ad Total Tardiess. The 2 d Iteratioal Coferece o Mathematical Scieces-ICMA, Oct. Agi N., (1996). Optimal Seekig with Brach ad Boud. Mgmt. Sci. 13, Che T, Qi X. ad Tu F., (1997). Sigle Machie Schedulig to Miimize Weight Earliess Subject to Maximum Tardiess. Computer of Operatio Research, 24, Davis, L., (1985) Applyig Adaptive Algorithms to Epistatic Domais, Proceedigs of the Iteratioal Joit Coferece o Artificial Itelligece pp Frech S., (1982). Sequecig ad Schedulig: A Itroductio to the Mathematics of the Job-Shop. Joh Wiley & Sos, New York. George S., Paul S., (2007). Pareto optima for total weighted completio time ad maximum lateess o a 58

11 sigle machie. Discrete Applied Mathematics 155, Hoogevee J. A., (2005). Ivited Review Multicriteria Schedulig. Europea Joural of Operatio Research 167, Igall I. ad Schrage L., (1965). Applicatios of the brach ad boud techique to some flow shop schedulig problems. Oper. Res. 13, Joui L.,(2000). Multi-Objective Noliear Pareto-Optimizatio. Lappeerata Uiversity of Techology, Keedy J. ad Eberhart R. C.(1995). Particle Swarm Optimizatio. Proceedigs of IEEE Iteratioal Coferece o NN, Piscataway, pp Lomicki Z., (1965). A brach ad boud algorithm for the exact solutio of the three machie schedulig problem. Oper. Res. 16, Mahmood A. A., (2014). Multicriteria Schedulig: Mathematical Models, Exact ad Approximatio Algorithms", Ph.D. Thesis, Uiversity of Al-Mustasiriya, College of Sciece, Dept. of Mathematics. Mitchell M., (1998). A Itroductio of Geetic Algorithms.Abroad Book. Nelso R.T., Sari R.K., ad Daiels R.L., (1986). Schedulig with multiple performace measures: The oe-machie case. Maagemet Sciece 32, Piedo M. L., (2008). Schedulig: Theory, Algorithms ad Systems. New York: Spriger. Ribeiro P. F. ad Kyle W. S., (2003). A Hybrid Particle Swarm ad Neural Network Approach for Reactive Power Cotrol. Member. /egir302/reactivepower-pso-wks.pdf. Sabah M. S., (2004). A Comparative Study betwee Traditioal Geetic Algorithms ad Breeder Geetic Algorithms. M. Sc. Thesis, AL-Nahrai Uiversity. Shi Y., (2004).Particle Swarm Optimizatio, Electroic Data Systems, Ic. Kokomo, IN 46902, USA Feature Article, IEEE Neural Networks Society, February. Smith W. E., (1956).Various Optimizer for Sigle-Stage Productio. Naval Res. Logist. Quart.3, Table (1) Applyig CEM ad BAB methods o (P 1 ) for =3,..,10. Values of (P 1 ) CEM BAB NE EV T/s NE EV T/s 3 2 (16,1,1),(17,0,0) (16, 1,1),(17,0,0) (32,13,9),(33,12,8) (32,13,9),(33,12,8) (37,9,7),(38,8,7) (37,9,7),(38,8,7) (61,20,15),(64,23,13) (61,20,15),(64,23,13) (83,35,19),(86,34, 21) (83,35,19),(91,43,17) (117,61,26),(120,60,28 ) (117,61,26),(127,71,24) (134,73,30),(135,71,28) (134,73,30),(135,71,28) (170,101,35),(171,99,33) (170,101,35),(171,99,33)

12 Table (2) Applyig CEM ad BAB methods o (P 2 ) for =3,..,10. Values of (P 2 ) CEM BAB NE EV T/s NE EV T/s 3 2 (16,1,1),(17,0,0) (16,1,1),(17,0,0) (32,13,5),(33,12,5) (32,13,5),(33,12,5) (37,9,4),(38,8,4) (37,9,4),(38,8,4) (61,20,9),(64,23,7) (61,20,9),(64,23,7) (83,35,9),(86,34,9) (83,35,9),(91,43,7) (117,61,9),(120,60,9) (117,61,9),(127,71,7) (134,73,9),(135,71,9) (134,73,9),(135,71,9) (170,101,9),(171,99,9) (170,101,9),(171,99,9) Table (3) Applyig BAB method for (P 1 ) ad (P 2 ) for =11,,25. Values of (P 1 ) Values of (P 2 ) BAB BAB NE EV T/s NE EV T/s (159,59,25),(160,56,25),(161,53, 25) 1 16 (159,59,9),(160,56,9),(161,53, 9) (198,85,30),(199,82,30),(200,79,30) 1 16 (198,85,9),(199,82,9),(200,79,9) (211,91,31),(211,94,30),(212,89,32) 4 15 (211,91,9),(212,89,9),(213,86,9) (266,126,40),(266,129,39), (267,124, 41) 6 7 (266,126,9),(267,124,9),(268,121,9) (281,133,41),(281,135,39),(282,132, 43) 3 6 ( 281,133, 9),(282,132,9),( 283, 130,9) (318,154,40),(319,151,43),(319,153, 39 ) 4 7 (318,154, 9),(319, 151,9),(320,150,9) (421,291,59),(422,289,57), (423,288,56) 2 5 (421,291, 9),(422,289, 9),(423, 288, 9) (439,234,52),(440,231,55),(440,233,51) (439,234,10),(440,231,10),(441,230,10) (711,527,93),(712,528,92),(713,529,90) 16 1 ( 711,527,10) (895,664,95),(896,663,97),(898,662,101) (895,664,10),(896,663,10),(898,662,10) (992,746,101),(993,745,103),(995,744,107) (992,746,10),(993,745,10),(995,744,10) (1106,848,108),(1107,847,110),(1109,846,114) (1106,848,10),(1107,847,10),(1109,846,10) (1151,880,112),(1152,883,110 ),(1159,879,113) (1151,880,10),(1159,879,10),(1160,877,10) (840,568,86),(841,567,85),(842,565,83) (840,568,10),(841,567,10),(842,565,10) (934,648,92),(935,647,91),(936,645,89) (934,648,10),(935,647,10),(936,645,10)

13 Table (4) Applyig CEM ad BAB methods for subproblem (SP 1 ) ad (SP 2 ) for =3,...,10. SP 1 SP 2 CEM T/s BAB T/s CEM T/s BAB T/s = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Table (5) Applyig BAB methods for the subproblem(sp 1 ) ad (SP 2 ) for =11,..,30. SP 1 SP 2 BAB T/s BAB T/s = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

14 Table (6) Applyig GA & PSO compared with CEM for(p 1 ) for =3,,10. GA PSO CEM MAE MBV T/s MBV T/s MEV PSO GA 3 (16.5,0.5,0.5) 1,1 (16.5,0.5,0.5) 1,1 (16.5,0.5,0.5) (0,0,0) (0,0,0) 4 (32.5,12.5,8.5) 0,2 (32.5,12.5,8,5) 0,2 (32.5,12.5,8.5) (0,0,0) (0,0,0) 5 (38.6,8,5.4) 0,5 (38.6,8,5.4) 0,5 (38.6,8,5.4) (0,0,0) (0,0,0) 6 (62.5,21.5,14) 0,2 (62.5,21.5,14) 1,2 (62.5,21.5,14) (0,0,0) (0,0,0) 7 (87,35,20.5) 1,2 (87,37.3,19) 1,3 (86.7,37.3,19) (0.004,0,0) (0.004,0.06,0.08) 8 (96.4,45.4,19.6) 0,5 (94.7,44.3,19.67) 0,3 (96.4,45.4,19.6) (0.02,0.02,0.004) (0,0,0) 9 (137.5,70.5,29) 0,2 (137.4,7,29.6) 1,5 (139.8,75,28.3) (0.02,0.03,0.05) (0.02,0.06,0.03) 10 (176,101,34.5) 1,2 (175.3,103,34.75) 0,4 (176.8,104,33.3) (0.008,0.01,0.05) (0.004,0.03,0.04) Table (7) Applyig GA & PSO compared with CEM for(p 2 ) for =3,,10. GA PSO CEM MAE MBV T/s MBV T/s MEV PSO GA 3 (16.5,0.5,0.5) (16.5,0.5,0.5) (16.5,0.5,0.5) (0,0,0) (0,0,0) 4 (32.5,12.5,9.5) (32.5,12.5,9.5) (32.5,12.5,8.5) (0,0,0.1) (0,0,0.1) 5 (38.6,8,9.5) (38.6,8,9.5) 1 (38.6,8,5.4) (0,0,0.8) (0,0,0.8) 6 (62.5,21.5,9.5) 1 (62.5,21.5,9.5) (62.5,21.5,14) (0,0,0.3) (0,0,0.3) 7 (86.67,37.33,9.5) 1 (87.67,37.33,9.5) (86.67,37.33,19) (0.1,0,0.5) (0,0,0.5) 8 (96.4,45.4,9.5) 1 (96.4,45.4,19.6) 1 (96.4,45.4,19.6) (0,0,0) (0,0,0.5) 9 (139.75,75,9.5) 1 (139.71,75.29,9.5) 1 (139.75,75,28.25) (0.0003,0.004,0.7) (0,0,0.7) 10 (176.83,104.67,9.5) 1 (174,101.75,9.5) 1 (176.75,104,33.25) (0.02,0.02,0.7) (0.0005,0.006,0.7) Where 0< <1. Table (8) Applyig GA & PSO compared with BAB for(p 1 ) for =11,,25. N GA PSO BAB MAE MBV T/s MBV T/s MEV PSO GA 11 (163.1,55.1,23.9) 1 (171.7,59.7,21.7) 1 (166,56,22) (0.034,0.065,0.015) (0.017,0.015,0.085) 12 (202.6,83.6,28.4) 1 (214,95.7,26.7) 1 (207,84,27) (0.034,0.861,0.012) (0.021,0.005,0.052) 13 (214.8,87.2,30.8) 1 (238.8,112.8,26.5) 1 (221.6,92.9,28.8) (0.078,0.213,0.079) (0.031,0.061,0.07) 14 (275,133.8,38.3) 1 (293.7,146.3,37) 1 (276.6,127.9,37.8) (0.062,0.144,0.021) (0.006,0.046,0.012) 15 (285.5,131,42.3) 1 (316.5,166.3,38) 1 (291,134.2,40.7) (0.088,0.239,0.066) (0.019,0.024,0.039) 16 (322,149,44.5) 1 (382.2,219.6,39.4) 2 (325.2,152.2,42.5) (0.175,0.443,0.072) (0.01,0.021,0.048) 17 (425.3,292.7,57) 1 (444,311.5,57.5) 1 (424.8,290.2,55.7) (0.045,0.073,0.033) (0.001,0.008,0.024) 18 (444.3,234.5,55) 2 (490,278.3,54.7) 1 (447.6,232.1,54.8) (0.095,0.199,0.001) (0.007,0.01,0.005) 19 (727.6,543.8,91.8) 2 (811.3,628.2,89.7) 2 (734,552.3,86.6) (0.105,0.137,0.036) (0.009,0.016,0.06) 20 (916.8,682.6,99.6) 1 (942.5,706.5,97.5) 2 (908.4,664.2,95.8) (0.038,0.064,0.018) (0.009,0.028,0.04) 21 (1008.5,756.8,105) 2 (1105.3,851.5,102.3) 2 (998.8,740.3,102.5) (0.107,0.15,0.002) (0.01,0.022,0.024) 22 (1116.8,854.4,112.2) 2 (1223.8,952.5,110.5) 2 (1112.8,842.3,109.5) (0.1,0.131,0.009) (0.004,0.014,0.025) 23 (1153.8,879.8,115) 2 (1235,959.5,113) 2 (1163.5,880.6,111.5) (0.061,0.09,0.013) (0.008,0.001,0.031) 24 (842,569.3,86.3) 2 (1004.8,724.6,86) 2 (848.2,570.8,83.5) (0.185,0.269,0.03) (0.007,0.003,0.033) 25 (941,651.7,92.3) 2 (1077.7,800.7,90) 2 (942.7,651.3,89.5) (0.142,0.229,0.006) (0.002,0.0005,0.032) 62

15 Table (9) Applyig GA & PSO compared with BAB for(p 2 ) for =11,,25. GA PSO BAB MAE MBV T/s MBV T/s MEV PSO GA 11 (166,58,9.5) 2 (173.7,61,9.5) 1 (177.3,67.4,7.4) (0.021,0.095,0.277) (0.063,0.139,0.277) 12 (203.8,86.6,9.5) 1 (218,98.6,9.5) 2 (219.3,96.4,7.4) (0.006,0.023,0.277) (0.071,0.101,0.277) 13 (219.3,90.7,9.5) 2 (231.8,108.5,9.5) 2 (177.3,67.4,7.4) (0.307,0.61,0.277) (0.237,0.346,0.277) 14 (271.1,130.1,9.5) 1 (292.5,143.5,9.5) 1 (271.3,120.1,9) (0.078,0.194,0.056) (0.0006,0.083,0.056) 15 (283,132,9.5) 1 (318,173.8,9.5) 1 (285,129.3,9) (0.116,0.343,0.056) (0.007,0.021,0.056) 16 (324,149.17,9.5) 1 (378.6,211.2,9.5) 2 (322.3,148.3,9) (0.175,0.424,0.056) (0.005,0.006,0.056) 17 (426.8,295.4,9.5) 2 (458.5,325.5,9.5) 2 (423.6,288.2,9) (0.082,0.129,0.056) (0.008,0.025,0.056) 18 (445,233,9.5) 1 (493.3,289,9.5) 1 (462.7,243.5,9.5) (0.066,0.187,0) (0.038,0.043,0) 19 (742,560.1,9.5) 1 (796.2,609.8,9.5) 1 (711,527,10) (0.12,0.157,0.05) (0.044,0.063,0.05) 20 (911.2,659.8,9.5) 2 (953,722.7,9.5) 2 (902,657.9,10) (0.057,0.098,0.05) (0.01,0.003,0.05) 21 (1020.8,762,9.5) 1 (1075.8,817,9.5) 2 (999,739.9,10) (0.077, 0.104,0.05) (0.022,0.03,0.05) 22 (1117.8,855.4,9.5) 1 (1216.3,940.3,9.5) 2 (1113,841.9,10) (0.092,0.117,0.05) (0.004,0.016,0.05) 23 (1153,883.7,9.5) 2 (1257.3,978.3,9.5) 2 (1159.3,876,10) (0.085,0.117,0.05) (0.005,0.009,0.05) 24 (842.5,568,9.5) 2 (999.8,729,9.5) 2 (865.6,588.8,9.6) (0.155,0.238,0.006) (0.027,0.035,0.006) 25 (935.7,648.3,9.5) 2 (1028,749.5,9.5) 2 (961.3,670.6,9.6) (0.069,0.118,0.006) (0.027,0.033,0.006) Figure (1): MAE behavior for PSO ad GA applied for (P 1 ) with =11,,25. 63

16 Figure (2): MAE behavior for PSO ad GA applied for (P 2 ) with =11,,25. Table (10) Applyig GA & PSO for(p 1 ) for =(30,(10),90) ad (100,(100),1000). GA MBV MAE T/s MBV AAE T/s PSO 30 (1457.3,1130.5,126.5) (0.0058,0.0094,0.0417) 2 (1658.5,1334.3,125.5) (0.1446,0.1914,0.0492) 2 40 (3545.5,3114,208.5) (0.0016,0.0013,0.0024) 2 (3540,3110,209) (0.0000,0.0000,0.0000) 2 50 (4287.6,3746.2,239.3) (0.0086,0.0100,0.0191) 2 (5124,4578.3,235) (0.2054,0.2344,0.0369) 3 60 (6907.8,6242.5,305.8) (0.0048,0.0052,0.0105) 3 (7974.5,7297.5,304.8) (0.1599,0.1751,0.0138) 4 70 (9391.6,8628.8,365.4) (0.0134,0.0149,0.0124) 4 ( ,10222,363) (0.1852,0.2023,0.0189) 4 80 ( , ,387.8) (0.0309,0.0349,0.0206) 4 ( , ,389.7) (0.1929,0.2118,0.0160) 4 90 ( , ,519.4) (0.0061,0.0064,0.0088) 4 (19846, ,519.67) (0.1602,0.1700,0.0083) ( , ,521.3) (0.0160,0.0172,0.0126) 4 (22697, ,515.6) (0.2344,0.2503,0.0235) ( , ,1059.8) (0.0066,0.0069,0.0067) 11 (90736,88447,1059.7) (0.2235,0.2307,0.0069) ( , ,1673.2) (0.0102,0.0105,0.0041) 17 ( , ,1670) (0.2358,0.2407,0.0060) ( , ,2199.3) (0.0160,0.0163,0.0031) 27 ( , ,2191.3) ,0.2808,0.0067) ( , ,2743.4) (0.0043,0.0043,0.0024) 40 (605573, ,2739.5) (0.2738,0.2774,0.0038) ( , ,3264.4) (0.0590,0.0596,0.0038) 55 (808538, ,3266.5) (0.1676,0.1693,0.0032) ( , ,3830.6) (0.0422,0.0426,0.0022) 88 ( , ,3829.5) (0.1877,0.1894,0.0025) ( , ,4505.1) (0.0093,0.0094,0.0013) 111 ( , ,4496.7) (0.2478,0.2498,0.0032) ( , ,4934.3) (0.0164,0.0165,0.0014) 121 ( , ,4926.6) (0.2971,0.2992,0.0029) ( , ,5489) (0.0252,0.0254,0.0015) 164 ( , ,5488) (0.1850,0.1862,0.0016) 52 64

17 Table (11) Applyig GA & PSO for(p 2 ) for =(30,(10),90) ad (100,(100),1000). GA PSO MBV MAE T/s MBV AAE T/s 30 (1452.4,1124.2,9.5) (0.0023,0.0038,0.0610) 2 (1613.3,1287.3,9.5) (0.1134,0.1494,0.0063) 2 40 (3540,3110,9.5) (0.0000,0.0000,0.0442) 2 (3540,3110,9.5) (0.0000,0.0000,0.0152) 2 50 (4348,3806.8,9.5) (0.0228,0.0264,0.0721) 2 (4716,4176.5,9.5) (0.1094,0.1260,0.0816) 3 60 (6908.6,6240,9.5) (0.0049,0.0048,0.0691) 4 (8050.4,7380,9.5) (0.1710,0.1884,0.0302) 4 70 (9337,8579,9.5) (0.0076,0.0091,0.0231) 3 ( , ,9.5) (0.2107,0.2305,0.0656) 4 80 (10728,977,9.5) (0.0073,0.0085,0.0984) 5 ( ,11149,9.5) (0.1358,0.1502,0.0759) 5 90 ( ,16075,9.5) (0.0014,0.0012,0.0695) 4 ( , ,9.5) (0.2163,0.2294,0.0765) ( , ,9.5) (0.0162,0.0175,0.0693) 5 ( , ,9.5) (0.2015,0.2151,0.0553) ( , ,9.5) (0.0053,0.0056,0.0712) 11 (93489, ,9.5) (0.2606,0.2691,0.0813) ( , ,9.5) (0.0038,0.0039,0.0587) 17 ( , ,9.5) (0.2081,0.2123,0.0355) ( , ,9.5) (0.0037,0.0038,0.0003) 26 (382810, ,9.5) (0.2374,0.2412,0.0472) ( , ,9.5) (0.0133,0.0135,0.0894) 36 ( , ,9.5) (0.2752,0.2788,0.0779) ( , ,9.5) (0.0265,0.0268,0.0315) 50 ( ,853033,9.5) (0.2420,0.2445,0.0545) ( , ,9.5) (0.0440,0.0444,0.0824) 78 ( , ,9.5) (0.2516,0.2539,0.0493) ( , ,9.5) (0.0987,0.0995,0.0153) 92 ( , ,9.5) (0.2469,0.2488,0.0884) ( , ,9.5) (0.0339,0.0341,0.0050) 116 ( , ,9.5) (0.2458,0.2476,0.0204) ( , ,9.5) (0.0005,0.0005,0.0887) 146 ( , ,9.5) (0.3039,0.3058,0.0651) 50 Figure (3): AAE behavior for PSO & GA applied for (P 1 ) with =30,,100,,

18 Figure (4): AAE behavior for PSO & GA applied for (P 2 ) with =30,,100,,1000. Figure (5): Time compariso for GA & PSO for (P 1 ) ad (P 2 ) with =30,,100,,

19 Table (12) Applyig GA & PSO compared with CEM for(sp 1 ) for =3,,10. GA PSO CEM AE BV NI T/s BV NI T/s OV GA PSO = = = = = = = = = = = = = = = = = = = = = = = = Table (13) Applyig GA & PSO compared with CEM for(sp 2 ) for =3,,10. GA PSO CEM AE BV NI T/s. BV NI T/s. OV GA PSO = = = = = = = = = = = = = = = = = = = = = = = = Table (14) Applyig GA & PSO compared with BAB for(sp 1 ) for =11,,30. GA PSO BAB AE BV NI T/s BV NI T/s OV GA PSO = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

20 = = = = = = = = = = = = = = = Table (15) Applyig GA & PSO compared with BAB for(sp 2 ) for =11,,30. GA PSO BAB AE BV NI T/s BV NI T/s OV GA PSO = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Table (16) Applyig GA & PSO for(sp 1 ) for =(40,(10),90), (100,(100),1000), GA PSO BV AES T/s BV AES T/s = = = = = = = = = = = = = = = = = =

21 = = = = = = = = = = = = = = = = Table (17) Applyig GA & PSO for(sp 2 ) for =(40,(10),90), (100,(100),1000), GA PSO BV AES T/s BV AES T/s = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

22 Figure (6): AES behavior for PSO & GA applied for (SP 1 ) with =40,,100,,1000. Figure (7): AES behavior for PSO & GA applied for (SP 2 ) with =40,,100,,

23 Figure (8): Time compariso for GA & PSO for (SP 1 ) ad (SP 2 ) with =40,,100,,

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