Development of a Quality Control Programme for steel production: A case study

Transcription

1 IOSR Journal of Mechancal and Cvl Engneerng (IOSR-JMCE) e-iss: ,p-ISS: X, Volume 11, Issue 5 Ver. I (Sep- Oct. 2014), PP Development of a Qualty Control Programme for steel producton: A case study O.E. Isaac and A.K. Le-ol Department of Mechancal Engneerng, Faculty of Engneerng, Rvers Unversty of Scence and Technology, Port Harcourt, Rvers State gera Abstract: In the steel mang process, effectve schedulng s needed for mprovement of productvty. Ths paper study a dynamc process wth a release tme, where the process tmes of job may change durng producton process due to uncertantes, the objectve s to ensure contnuty of the producton process and just n tme delvery of fnal products. A soluton methodology s developed whch combne a model predctve control (MPC) strategy based approach and lagrangan relaxaton algorthm. The MPC approach tacle the parallel process schedulng problem, and the rollng horzon approach allows applyng lagrangan relaxaton algorthm to solve the model of the schedulng problem n a rollng fashon. Computatonal experments are carred out comparng the proposed. Method wth the passve adjustment method often adopted by some qualty control engneers. The result shows that the proposed method yelds sgnfcantly better results. Keywords: model predctve control, lagrangan relaxaton algorthm. OMECLATURE I = Set of all jobs, I = {1, 2 }, s the total number of jobs M =Total number of dentcal parallel machnes p = Processng tme of job b = Startng tme of job r = Release tme of job M = umber of machnes avalable at tme K = Total number of tme unts to be consdered R = Total number of rollng wndows durng the whole tme horzon U (t) = Set of jobs that have not been processed untl the decson pont t W = Weght of job Sp = Steel mae respond Rf = refnng furnace Cf = Converter furnace HFS = Hybrd flow shop C = completon tme of job n a rollng wndow, I, t s a tme pont (C = means that the job completes at the end of tme unt ) O C I = Completon tme of job n the ntal schedule, I =1 job s processed at tme unt, I, = 1, 2.., 0 I. Introducton The metal-formng ndustry s an mportant ln n the manufacturng chan, supplyng extrusons, tubes, plates and sheets to many major manufacturng enterprses, ncludng the automoble, arcraft, housng and food servces, and beverage ndustres (Balarshnan and Brown 1996). Iron and steel producton ncludes several process phases (ron-mang, steel-mang contnuous castng and steel rollng), and s very extensve n nvestment and energy consumpton. It s also characterzed by hgh-temperature hgh-weght materal flow wth complcated technologcal processes. To accommodate customer requrements for dfferent types of fnshed products wth fluctuatng demands, dfferent rollng mlls n the steel-rollng phase are desgned wth suffcent producton capacty. Snce the steel-mang process (SP) phase needs expensve and energy-extensve equpment and runs n a contnuous mode, ts capacty s usually below the total capacty of the rollng stage. Effectve schedulng of SP resources s therefore vtal, especally n today s hghly compettve global steel maret. The steel-mang process conssts of three stages: steel-mang, refnng and contnuous castng. Each stage further ncludes parallel machnes, as shown n fgure.1 The followng s a bref descrpton of the producton process. 73 Page

2 In the steel-mang stage, carbon, sulphur, slcon, and other mpurty contents of molten ron are reduced to desrable levels by burnng wth oxygen n a converter (CF) or electrc are furnace. The output from ths stage s molten steel wth the man alloy elements. The basc unt of steel-mang producton s a charge, whch s defned as a job n SP schedulng. It refers to the concurrent smeltng n the same converter. The setup and mantenance tmes at the steel-mang stage are comparatvely short. The steel n one charge many be cast nto dfferent slabs that are used to produce fnshed steel products for dfferent customer orders. Charges must be so desgned that (1) the orders n the same charge have dentcal steel grade and gauge; (2) the slab wdths for these orders should be wthn certan lmts; (3) delvery dates of the orders are close to each other; and (4) total slab weght ranges from 95% to 100% of the furnace capacty. The molten steel from the steel-mang stage s poured nto ladles that are transported by a crane to a refnng furnace (RF) for refnng. The operaton at ths stage further refnes the chemcals and elmnates mpurtes n the molten steel or adds the requred alloy ngredents. If no RF s avalable when a new charge arrves, the charge has to wat untl one of the RFs becomes avalable. The watng tme of a charge causes the charge temperature to drop, and reheatng s needed, Energy consumpton thus grows as the watng tme ncreases Lxn (2002). The duraton of the operaton at the refnng stage s usually smlar to that of the operaton at the steel-mang stage. After refnng, molten steel s poured nto a tundsh for castng. In the castng stage, molten steel flows down from a hole at the bottom of the tundsh nto the crystallzer, the nput unt, of a contnuous caster. The molten steel contnuously soldfes nto slabs at the bottom of the caster. A sequence of charges that are consecutvely cast on the same ntermedate ladle and on the same contnuous caster usng the same crystallzer s called a cast, whch s defned as a job group n SP schedulng. Charges n the same cast need to satsfy the followng technologcal constrants. (1).Releases-tme related events: (a) Arrval delay of raw materals at upstream operatons. (B).Machne breadown at upstream operaton(c)qualty flaw of lqud steel at upstream operaton (d) Change of processng speed at upstream operaton (2).Processng tme related events:(a). Change of castng speed (b).onlne castng wdth adjustng (c). Caster stream breadown (e). Unsatsfactory qualty of castng slabs. The charges n each cast and ther sequence are fxed. The dagrams below show the followng producton process. SP schedulng s then to decde the schedule of these jobs on the process. Plannng and schedulng problems n ron and steel producton have not drawn as much attenton of the producton and operatons management researchers as many other ndustres, such as metal cuttng and electroncs manufacturng Tang and others. (2001). Because steel-mang schedulng wth practcal constrants s extremely complex, most approaches to SP schedulng treat the problem at three levels (umoao and Morshta 1991): (1) sub-schedulng, whch fulfls the schedulng of ndvdual charge sets; (2) rough schedulng, whch merges sub-schedules; and (3) optmal schedulng, whch elmnates machne conflcts. (Tang and others 2000), the sub-schedulng and rough schedulng were realzed through human-computer nter-acton, a nonlnear mathematcal model was bult for the machne conflct problem, and the model was then converted nto a lnear programmng model for easy soluton. Kalagnanam and others. (2000) addressed the problem of satsfyng customer orders n an order boo wth surplus slab nventory before schedulng the producton for the remanng orders. The problem was formulated as a bcrtera multple napsac problem wth addtonal constrants and was then solved by usng a networ-based heurstc. Cowlng and Rezng (2000) developed a mxed nteger programmng model and a heurstc for ntegrated producton plannng of a steel contnuous caster and hot strp mlls. Change and others. (2000) studed the lot plannng problem to group charges nto casts for a contnuous slab caster n a ntegrated steel mll. An nteger-programmng model and an effcent heurstc were developed the column generaton approach combned wth a smple round-off scheme. ths paper, we develop an ntegrated model for the SP 74 Page

3 schedulng problem ncludng all four SP stages, and present a Lagrangan relaxaton method to solve t. The steel-mang plant n gera s taen as the research bacground, and the goal s to generate daly schedules for steelmang process and help the gera steel company planner to mae better short term(for a shft) schedulng decson and also produce a qualty steel for ts customer. Fgure 1.2 Steel Mang Process Flow Chart: Scrap Processng to Ladle Treatment 75 Page

4 Fgure 1.3: Process Flow Chart for Contnuous Castng II. Mathematcal Formulaton Of The Problem The problem characterstcs and modelng requrements After the charges and casts are defned by lot plannng, the tas of SP schedulng s to determne when and where (on whch devce) each charge should be processed at each producton stage. The followng general assumptons usually made n HFS schedulng are also vable for ths problem. a. All charges follow the same process route: Steel-mang, refnng, and then contnuous castng. At each stage, a charge can be processed on any one of the machnes at that stage, and the parallel machnes at that stage are dentcal. b. A machne can process at most one job at a tme. c. A job can be processed on at most one machne at any tme. d. Job processng s non-pre-emptve. However, t s clear from the ntroducton that the problem has some specal features as compared wth the schedulng n a general HFS. These features are summarzed as follows. 1. The number of charges to be scheduled wthn a shft s not large. However, each charge contans more than 100 tons of molten steel at hgh temperature. Transfer tmes between stages needs to be consdered. 2. Some specfed charges (a cast) must be processed as a group on the same caster and there are precedence constrants among the charges wthn a group at the castng stage. 3. Set-up and removal tmes are requred, respectvely, before and after a whole cast s processed on a caster. These tmes are separated from the processng tmes. 4. Idle tme on the caster between charges wthn a cast (cast brea) s undesrable and nvolves cost. 5. Watng tme of charges between the processng at dfferent stages causes a temperature drop and results n cost for heatng. 6. Both earlness and tardness on job completon lead to cost, e.g. for nventory or compensaton to customers. SOLUTIO FRAMEWORK Model predctve control (MPC) s an advanced control technque, whch has been wdely used n the process ndustres. The usefulness of MPC s that t can enable the system to mae the approprate changes to counteract foreseen dsturbances effect n advance by ntroducng a feed forward effect based on a model and forecast of future behavor. Due to ths feature, predctve control can reduce the nfluence of parameters of the 76 Page

5 schedulng object and the uncertanty of crcumstance effectvely. As a tactcal decson polcy, MPC can offer a good decson framewor that best fts the dynamc problem wth dsturbances. The logc of the MPC wll be appled to model and tacle the dynamc schedulng wth dsturbances n ths paper. The appeal of the proposed method over tradtonal approaches to the dynamc schedulng problem n the ever-changng envronment ncludes the followng: (1) Under an MPC based framewor, a rollng horzon approach can handle the dsturbances n the dynamc envronment. (2) The applcaton of Lagrangan relaxaton embedded nto the rollng horzon strategy tacles schedulng problems. The problem under study s frst solved based on avalable nformaton at the begnnng to generate an ntal schedule (past output). The counteracton of new outputs and reference trajectory s selected to be the nput of the controller n or eep the process as close as possble to the reference trajectory. The establshed model s optmzed by the classcal methodology (Lagrangan relaxaton) to obtan the future nputs or control sgnals based on the past and current values. Once the control decsons are obtaned, the frst control sgnal s sent to the system whle all the next control sgnals are rejected. Then, the horzon advances one step to the next rollng wndow and the soluton procedure s terated untl the last rollng wndow s reached. ROLLIG HORIZO IMPLEMETATIO Rollng horzon procedures have been developed for several dfferent problems. Snce t optmzes the system n a lmted tme horzon nstead of globally optmzng the system, t s very sutable for the dynamc schedulng problems n varyng envronment. Wth ths strategy, the complcated dynamc schedulng problem can be solved n a rollng horzon fashon. The rollng horzon used here s a lttle dfferent from the common rollng horzon approach. Suppose that the orgnal schedulng horzon (o, ) s dvded nto R tme perods wth equal length T. For the practcal steel factory, a natural day or shft s generally taen as the schedulng horzon and an ntal schedule s made for the whole horzon. Before the start of each tme perod, job nformaton may change due to dsturbances n the producton and thus reschedulng s needed usng the orgnal schedule as a reference. In the commonly used rollng horzon approach, at each decson pont the reschedulng problem would be solved consderng a rollng horzon wth fxed number of tme perods. To tae nto account all avalable nformaton, each reschedulng model n the rollng horzon method used here adjusts the schedule for all the remanng tme perods n the same day or shft. We call the tme nterval from the decson pont to the end of the schedulng horzon the rollng wndow. As tme progress, the wndow s shortened by one tme perod wth the startng pont of t movng forward whle ts end s fxed to the end of the (orgnal) schedulng horzon. Model for a Rollng Wndow. The followng notaton s used to present the mathematcal model. Terms are defned n the omenclature secton. : 2 1C1 b1 r PO mn mze 1 w (1) 1 2 p 1: mn mze wc ( b r) ( C C 0 ) 2 (2) cu cu U s.t r b I U (t) (3) M at, K (4) U t p, U( t) C, U( t); at, K (6) C - P + 1 K (1-1 ), U (t); (at, K) (7) 0,1, U( t); at, K (8) In ths paper, parallel process schedulng wth release tmes n the dynamc envronment s consdered. A mxed nteger nonlnear programmng model s presented for the schedulng (reschedulng) problem n the rollng horzon. Intal schedule based on estmated parameters for the whole schedulng horzon s stll useful to provde the nternal planners and external customers wth an expected completon tme of ther orders. At the begnnng tme, the problem wth objectve 1 s frst solved based on avalable nformaton and then an ntal feasble soluton s obtaned ( C O I, I ). A new schedule s generated perodcally wth objectve 2 subjected to the newly obtaned producton nformaton durng each rollng wndow [at, K]. Each schedulng (5) 77 Page

6 model corresponds to a fxed value of a. Objectve 2 conssts of mnmzng followng three terms: (a) total weghted completon tmes of jobs, whch s the goal n the determnstc envronment; (b) energy consumpton of jobs caused by watng to be processed after they are released (Due to hgh temperature of the products durng the producton process, a watng penalty s ntroduced to the objectve n order to reduce energy consumpton caused by watng for processng. Here, the energy consumpton of a job s regarded as a quadratc functon the job s watng tme); (c) total devaton of actual job completon tmes from those n the ntal schedule. The devaton chosen may reduce the effect of the system arsng from dynamc dsturbances n the process of schedulng as much as possble. Constrants 3 reflect that a job can be processed only after ts release tmes. Constrants 4 are the machne capacty constrants. They ndcate that all these machne requrements are satsfed wth the number of avalable machnes at that tme. Machne capacty cannot be exceeded at each tme unt. Constrants 5 ensure that the tme ntervals for whch a job occupes one of the machnes must to equal to the requred amount of tme for processng. Constrants 6 ensure that a job occupes on of the machnes untl the processng s fnshed. Constrants 7 ensure that a job starts occupyng a machne from ts begnnng tme. Constrants. 5-7 together represent the tme ntervals for whch a job s processed on a machne. And fnally, Based on the MPC framewor, the model establshed for schedulng problem correspondng to each rollng wndow s dfferent from the most exstng tradtonal schedulng models n three aspects. (1) The watng penalty s ntroduced to the objectve for savng energy. (2) The model can tacle the dynamc schedulng wth dsturbances. (3) The schedulng devaton s taen as the objectve to reduce the nconvenence cost caused by uncertantes. LAGRAGIA RELAXATIO ALGORITHM Lagrangan relaxaton s an effcent optmzaton method. The major advantage s that t can provde a lower bound for problems.(fsher1981).it has been appled to a wde range of schedulng problems such as the parallel machne schedulng problem.(luh and others 1990) or the job shop schedulng problem. Lagrangan relaxaton s presented to solve the model correspondng to each rollng wndow n ths wor. The couplng constrants are relaxed and embedded as a penalty term multpled by Lagrange multplers n the objectve functon, therefore leadng to a relaxed verson of the prmal model, whch can be decomposed nto several ndependent job-level sub problems to reduce the problem sze. It s suffcent to obtan a near-optmal schedule wth quantfable performance wthn a reasonable computaton tme. The feasble solutons to Lagrangan relaxaton at the begnnng s an ntal soluton usng objectve 1. Durng the future rollng wndow, the schedulng problem s rescheduled agan based on the update nformaton n the wndow and schedulng result s adjusted. The job-level decomposton of the model s ntroduced by relaxng constrants 4 wth Lagrangan multplers (u). The frst solvng LR problem s denoted by LO, and the LR problem solved at each rollng wndow s denoted by L1, respectvely. 2 (L0) mn C I wc b r u M mn 2 = wc b r u um c mn 2 = wc C P 1 r u um (9) c mn 2 (L1) wc b r c U w c u M U t U mn 2 wc b r C C c U mn 2 wc ) C p 1 r c 0 2 = u = U t t um 78 Page

7 o 2 c c u t Development of a Qualty Control Programme for steel producton: A case study t um The job level subproblems are solved optmally by enumeraton. The lower bound together wth the upper bound and the devaton n the lnng constrants s used to update the Lagrange multplers accordng to a subgradent updatng procedure. In our algorthm, the subgradent optmzaton method s used to update the n n n n multplers (u) at each teraton gven byu 1 u gu, where u s a vector of multplers (u), n s n n the teraton ndex, and g (u) the subgradent of L equal to M. And s the step sze at the nth n n n 2 teraton defned as = ( L L )/ // gu // 0, < < 2, where L * s estmated by the best feasble objectve value found so far and L n s the dual value at the n th teraton. III. Computatonal Experment To test the performance of the method and to study the characterstcs of the solutons, a computatonal experment has been carred out on randomly generated problem nstances, whch were desgned to reflect practcal stuatons n ron and steel ndustres. The method was mplemented by usng vsual basc.net 2008 and the experment was carred out on a HP pavlon dv5 noteboo 2.27GHZ PC. LAGRAGIA ALGORITHM The Algorthm Procedure: The followng are the step to step algorthm process of evaluatng the objectve functon of Lagrangan relaxaton from steel producton schedulng Step1: Declaraton of varables used n the program Step2: Input the values of, whch s the array sze. Step3: Input the values of W C, U and M to compute the values of b, r, (b -r ) 2 W C ( M ) Step 4: Calculate the values of array L o and P o usng the formula: L o = mn wc + 2 K b 1 r + u IK M K C 1 I 1 K I I P o = Mnmze w C b r Step 5: Clc transfer to send the values of L o and P o to the phase II of the algorthm s as follows: Step 1: Declaraton of all varable used n phase II Step 2: Input the value of whch s the sze of the array Step 3: Enter the number of jobs machnes and number of rollng wndow whch maes up the problem structure Step 4: Enter the values of the runnng tme n seconds Step 5: Intalze of sum lo = o, sum Po = o, sum lmp = o, sum rt = o Step 6: Calculate the mprovement (%) usng the formular: mprovement = (L o - P o )/ P o x 100% Step 7: Compute the sum of L o, P o Improvement and runnng tme. Step 8: Calculate the averages of L o, P o, Improvement and runnng tme. Step 9: Plot the graph of mprovement aganst number of machnes used n processng Step 10: End program IV. Generaton Of Problem Instances To generate representatve problem nstances, we examned the actual producton data from a Steel Producton Frm n gera, whch s the largest and most advanced ron and steel enterprse n gera, ts annual producton s over 18 mllon tons of steel, and ts auto sheet accounts for more than 6% of the domestc maret share. In the steel-mang plant n gera every worday s dvded nto three shfts. Planners need to carry out SP schedulng n every shft for the next shft. The SP schedule generally ncludes about 5-7 casts for each worday, and a cast conssts of 3-5 charges subject to technologcal constrant. The maxmum number of charges to be scheduled n each worday s about 35 (about 12 per shft). Based on the above, the number of (10) 79 Page

8 charges to be scheduled s set to be 12 for each nstance. Snce the mnmum schedulng tme unt n the ron and steel plant s n an exact number of mnutes, the mnute s taen as the basc tme unt. The plannng horzon s set to be 480 mnutes, as ths study ntends to solve the SP schedulng problem for an eght-hour shft. Two other parameters are chosen to represent the problem structure as descrbed below: 1. The number of casts s set to vary at three levels: 3, 4, and The number of machnes at each stage s set to vary at three levels: 3, 4, and 5. In order to reduce expermental cases, t s assumed that every stage has the same number of machnes. However, our method presented here can deal wth practcal problems ncludng dfferent numbers of machnes for dfferent stages. The combnaton of parameter levels gves nne problem scenaros, and for each scenaro, ten dfferent problem nstances were randomly generated. Thus, a total of 90 problem nstances were used n the experment. Accordng to practcal data of the steel-mang plant of Delta steel. The proposed method wth the PAM. The PAM s an often used method by human schedulers for schedulng n systems wth uncertantes, and t consders dsturbances of the jobs durng each rollng wndow. At each rollng wndow, the newest data and nformaton s frst updated. The jobs after beng dsturbed are compulsorly moved to be processed at the avalable tme of the machnes. And, the subsequent jobs whch follow the dsturbed jobs n the ntal schedule need to be adjusted respectvely. Ths procedure stops untl all jobs are scheduled n the last rollng wndow. Fgure 4.1 Effect of the number of rollng wndows on algorthm performance (Dsturbance on release tme of jobs). Problem structures LO PO Improvement (%) Runnng tme (s) Problem o. (Jobs x machnes x o. of Rollng wndows) 1 60 x 3 x x 3 x x 3 x x 3 x x 3 x x 3 x x 3 x x 3 x x 3 x x 3 x Average Fgure 4.1 Effect of the number of rollng wndows on algorthm performance (dsturbance on release tme of jobs). For the experments, the objectve value by usng PAM s denoted as PO. And the objectve value by usng the proposed approach s denoted as LO. The column mprovement s calculated by (PO - LO)/LO x 100%.To guarantee optmal soluton In ths secton, extensve experments are conducted based on dfferent parameter confguratons to analyze the performance of the propose approach. For the tests, the number of jobs ranges from 20 to 100. The number of machnes vares at three levels (3, 4, and 5). The number of rollng wndows vares at four levels (2, 4, 6, and 8). The data for a job are randomly generated as follows. An nteger processng tme P s generated 80 Page

9 from the unform dstrbuton [5, 15], and an nteger release tme r are generated from the unform dstrbuton [1, ½ P ]. An nteger weght w s generated from the unform dstrbuton [10, 15]. Dsturbances are represented n a stochastc fashon. The entre tme horzon s dvded nto a number of rollng wndows wth dynamc events on unscheduled jobs. As done n ths paper, we study two nds of dynamc events occurrng durng each rollng wndow based on the practcal producton of the steel complex, ncludng varatons of the process tmes and the release tmes on jobs. The dsturbance scope for the processng tme of each job s generated from the unform dstrbuton [0, max (P(r t)/10, 20)]. The dsturbance scope for the release tme of each job s generated from the unform dstrbuton [0, max (r t, 20)), where t s the decson pont of the current rollng wndow. The combnaton of the above parameter levels generates 108 dfferent problem scenaros, for each of whch 10 dfferent problem nstances have been tested. A perodc rollng horzon schedulng strategy s adopted n ths paper where a natural day or shft s taen as the whole tme horzon. In order to llustrate the performance of the proposed method wth respect to the dfferent number of jobs, we carry out the computatonal experments and the results are presented n the number of rollng wndows s set to dfferent values to test ts mpact on soluton qualty. Illustrate the soluton performance aganst dfferent rollng wndow szes and dfferent numbers of jobs for dsturbance on processng tmes of jobs and release tmes of jobs, respectvely. There s a clear mpact of the number of rollng wndows on soluton qualty. As the number of rollng wndows ncreases, mprovement becomes larger gradually whch means that the soluton qualty of the proposed approach becomes better steadly, compared to the soluton of the PAM. As the number of rollng wndow ncreases, the dsturbances n the dynamc schedulng envronment can be measured and handled n a tmely manner, whch results n mprovement on the qualty of the solutons, although computaton tme ncreases. The experments are carred out based on one nds of stuatons ncludng the dsturbance of the jobs processng tmes and the dsturbances of the jobs release tmes. Upon the bass of results presented n Table 4.1, we can mae the followng observatons about our methods. V. Concluson When the number of jobs and the number of rollng wndows are fxed, the number of machnes ncrease n most cases. Ths s consstent wth the ntuton that for a fxed number of jobs and rollng wndows, when the number of machnes ncreases, there s more choce of machnes for each job and the problem becomes easer to solve. 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