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1 Advanced Materals Research Vols (2011) pp (2011) Trans Tech Publcatons, Swtzerland do: / Solvng the Two-Obectve Shop Schedulng Problem n MTO Manufacturng Systems by a Novel Genetc Algorthm Ll Yao 1,2, a, Habo Sh 2, b, hang Lu 2, c, Zhonghua Han 1,2, d 1 Graduate Unversty of the hnese Academy of Scences, Beng, hna; 2 Key Laboratory of Industral Informatcs, Shenyang Insttute of Automaton, hnese Academy of Scences Shenyang, Shenyang, hna. a b c d Keywords: Make-To-Order; two-obectve shop schedulng; Genetc Algorthm. Abstract. In ths paper, a novel genetc algorthm (GA) s proposed to solve the two-obectve shop schedulng problem n make-to-order (MTO) manufacturng systems. Ths algorthm can ensure that all obs meet ther deadlnes; smultaneously, t can satsfy another performance goal whch the enterprse pursues. Referrng to the prncple of populaton updatng wth survval of the fttest n tradtonal genetc algorthm and takng advantage of the dea of two sub-modules, the novel algorthm s controlled by the two nested closed-loops, and the strategy that feasble solutons are preferred whle nfeasble solutons are remade s employed to make the search forward. Fnally the novel algorthm and the tradtonal algorthm are used to solve the two-obectve hybrd flow-shop schedulng problem (HFSP) n MTO manufacturng systems. The result shows that the novel algorthm has an obvous advantage and good feasblty compared wth the tradtonal algorthm. Introducton As customer requrements become varous and ndvdual, make-to-order (MTO) producton s accepted by a lot of manufactores. In a MTO enterprse, the planners organze producton accordng to customer orders and sale contracts, and how to fulfll customer orders on tme s crucal [1, 2]. In MTO manufacturng systems, the producton data should be accurate, and reasonable plans such as balancng the producton capacty, solvng the bottleneck problem of constrans, mantanng equpments and nstruments wth reasonable arrangements, optmzng the producton process, controllng the work shop obs and so on are very mportant. But the m mportant thng s to ensure that all obs meet ther deadlnes. Schedulng s the key process n the computer ntegrated producton system, whch s the lnk between management and control. It determnes the specfc processng paths, work tmes, machnes and operatons for each processng obect. Excellent producton schedulng plays an mportant role n ncreasng economc effcency and mprovng the producton system. ertanly excellent producton schedulng can fulfll customer orders on tme, at the same tme reasonable schedulng also can optmze some other performance goals, such as c mnmzaton and punshment mnmzaton. In MTO enterprses, schedulng software usually adopt the method of rule-based reverse schedulng. The method takes the order tme as start tme and arranges each process from back to front; fnally we can get the latest start-tme of each ob. Ths method can better guarantee the due date. In addton to the delvery performance, MTO enterprse decson-makers usually hope to reduce the c of producton or mprove resource utlzaton through reasonable schedulng, but reverse schedulng method can t solve the mult-obectve problem. Tradtonal mult-obectve optmzaton methods maybe can optmze these obectves [3, 4, 5, 6, 7], but can t ensure the performance that all obs complete on tme. In ths paper, a novel genetc algorthm s proposed, whch can ensure that all obs meet ther deadlnes and can optmze another performance goal whch the enterprse pursues. All rghts reserved. No part of contents of ths paper may be reproduced or transmtted n any form or by any means wthout the wrtten permsson of TTP, (ID: /08/11,02:42:34)

2 1316 Advanced Manufacturng Technology Features and Schedulng Performance Goals of MTO Enterprse Features of MTO Enterprse. The features of MTO enterprse nclude multple varety and small batch products, large fluctuatons n demand, orders changng frequently, complcated processng technques and so on. Automoble manufacturng ndustres mly are MTO manufacturng systems. Snce these features n MTO enterprse, t s dffcult to deal wth the schedulng on manual. Many enterprses whch arrange the schedulng on manual have experenced the followng problems: Incomplete plan results materal storage and work broken frequently, and the producton effcency and the proft are engulfed serously; dfferent processng pace results that the slack at the begnnng and speed up towards the end phenomenon arses n the assembly department. So mplementaton of automatc schedulng has a great sgnfcance n MTO enterprse. Performance Goals of MTO Manufacturng Systems. The m mportant thng for the MTO enterprse s to ensure that all obs meet ther deadlnes. It can be descrbed as Eq.1: U = 0, n U = 0 U :{ = 1 U = 1, f f m D. (1) > D m Where s the ob number; n s the total number of obs; m s the last stage; ob s fnshed at stage m; D s the due date of the ob. t shows that the ob meets the deadlne, = 0 U s a varable, when m s the tme when the m s earler than D, U =. U ; otherwse the ob delays the completon, 1 In addton to the delvery performance, m MTO enterprses try ther best to reduce the producton c. Eq.2 s the mathematcal descrpton. For shop schedulng, the producton c s always named as drect producton c ncludng three parts: machne workng c, machne watng c and ob storage c. They are descrbed as Eq.3-Eq.5. work wat storage mn{ + + }. (2) Y F P m n M work v = k k k = 1 = 1 k= 1. (3) work s the machne workng c, and t s manly consttuted by deprecaton expenses of fxed assets, loss c of coolng flud and machnng tools, the wages of producton unt managers and so v on. In Eq.3, M s the total number of machnes at the stage ; F s processng rate of the machne k at stage ; Yk s a varable, when the ob s processed on machne k at stage, Y k = 1, otherwse Y k = 0 ; P k s the processng tme when the ob s processed on machne k at stage. Eq.3 shows that the machne workng c s equal to the product sum of machne processng rate and machne processng tme. k T F m M wat S s = k k = 1 k= 1. (4) wat s the machne watng c, and t manly contans machne dlng deprecaton and machne s s mantenance c. In Eq. 4, T s the watng tme before machne k starts to work at stage ; F s k the watng rate of the machne k at stage. The machne watng c s equal to the product sum of machne watng rate and watng tme. = S F n m w (, 1). (5) storage = 1 = 2 k

3 Advanced Materals Research Vols storage s the ob storage c. For many enterprses, there are some quench tmes between two consecutve processes. Some enterprses put obs nto storehouse durng these quench tmes; others have the onlne storage management, and they put obs near the current machne or the next machne whch wll start to process them. Anyway, storage wll result the mantenance c n order to ensure the product qualty, and ths c s named as storage c n shop schedulng. In Eq.5, S s the tme when the ob starts to be processed at stage ; w F s the storage rate of the ob ;, 1 s the tme when the ob s fnshed at stage -1.The storage c s equal to the product sum of quench tme and storage rate. There are some other performance goals n MTO enterprses such as the mprovement of resource utlzaton; however they do not have to be mentoned here. Two-Obectve Genetc Algorthm based on Sub-Module For MTO enterprses, t looks lke tradtonal two-obectve optmzaton problem to acheve the order deadlne goal and optmze another enterprse own performance goal. Actually t s dfferent from the tradtonal optmzaton problem. The goal that must ensure all productons meet ther deadlnes s an absolute goal n MTO enterprse but not an optmzaton goal. Therefore the tradtonal mult-obectve optmzaton algorthm can t solve the two-obectve optmzaton n MTO enterprse. In ths paper, a novel genetc algorthm s proposed to solve the two-obectve shop schedulng problem n MTO manufacturng systems. The Idea of the New Algorthm. The two sub-module dea s appled nto the search process n the new algorthm. It s controlled by two nested closed-loops, the nner of whch s controlled by order delvery ftness and the outer one s controlled by optmzaton goal ftness. In the search process, the nner loop works wth the manner that populatons change by comparson. There are two solutons n the nner loop: feasble soluton and nfeasble soluton. The search wll ump out of the nner loop when the feasble soluton arses; otherwse the unfeasble soluton wll be changed by crossover and mutaton operatons untl the feasble soluton arses. The outer loop s controlled by the tradtonal GA search strategy. The algorthm s controlled by two nested closed-loops and goes forward. The stoppng crteron of the outer loop s as same as the tradtonal GA s. It s ether the number of evoluton generatons or threshold value. Smlarly, the termnaton condton of the nner loop should be gven n order to prevent the order deadlne from beng unreasonable. Operatons of the New Algorthm. The new algorthm s descrbed as follows: Step1 Intalze mutaton factor P m, crossover factor P c, the populaton sze N p, the number of the nner loop evoluton generatons, the number of the outer loop evoluton generatons. Let K=0, and randomly ntalze populaton P(0). Step2 Determne whether the outer loop stoppng crteron s met, f the outer loop stoppng crteron s met, ump out of the search and output the best soluton; otherwse go to step3. Step3 Let the ndvdual number m=0. Step4 Let the nner loop current evoluton generaton =0. Step5 Evaluate whether the ndvdual m meet the order deadlne, f the ndvdual soluton meet the deadlne, ump to Step8; otherwse go to Step6 Step6 Let =+1, and determne whether the nner loop stoppng crtera s met, f the nner loop stoppng crtera s met, ump out of the search and output the warnng nformaton; otherwse go to step 7. Step7 Randomly select another ndvdual to perform crossover and mutaton operatons wth the old ndvdual, and produce a new ndvdual. Then return to Step5. Step8 Let m=m+1, and evaluate whether m s greater than N p, f m s greater than N p, go to Step9; otherwse return to Step4. Step9 Evaluate the optmzaton obectve values of all ndvduals, and pck up the best ndvdual to put asde.

4 1318 Advanced Manufacturng Technology Step10 Perform crossover and mutaton operatons for the orgnal populaton to produce a new populaton. Step11 Let K=K+1, and return to Step3. Smulaton and omparson The new two-obectve algorthm based on sub-module s smulated by vsual studo 2008 software, and s compared wth the tradtonal two-obectve optmzaton algorthm based on weght-set. They are used to solve the two-obectve hybrd flow-shop schedulng problem (HFSP) n MTO manufacturng systems, The HFSP can be descrbed as follows [8, 9, 10] : there are n obs to be processed, and each ob must experences m stages n the same drecton; there s at least one machne at each stage and not less than one stage exstng multple machnes; each work pece should complete one process at each stage; each process can work on any machne at the same stage. The HSFP model s shown n Fg. 1. m M 1 m M 2 m M m Fg. 1 Hybrd flow-shop schedulng problem detaled descrpton In order to verfy the feasblty and effectveness of the new algorthm, we select 4 sets of dfferent schedulng schemes to make comparson and analyss. Table 1 lsts some parameters of the 4 sets, some parameters are produced randomly, such as the number of machnes n each process, the processng tme, order due date, machne workng c rate, machne watng c rate and ob storage rate, and others are computed by Eq.1-Eq.5. The two obectves are delvery performance and c optmzaton obectve. Table 1 Some parameter sets of schedulng schemes Parameter sets The number of obs The number of stages Gen (outer loop) N p P c P m (nner gen loop) The range of machne number The weght of c obectve The weght of delvery obectve random random random random After smulated, the results are shown n Table 2 and Fg.2 Fg.5. Table 2 shows the two obectve values of the two algorthms. The four fgures show the compared teratve curves of the two algorthms. Table 2 Obectve values Obectve values The new algorthm The old algorthm The number of the tardy obs Producton c The number of the tardy obs Producton c

5 Advanced Materals Research Vols Fg. 2 ompared teratve curves(1) Fg. 3 ompared teratve curves(3) Fg. 4 ompared teratve curves(2) Fg. 5 ompared teratve curves(4) In Fg. 2, Fg. 3, Fg.4, Fg. 5, red curve ndcates the teratve process of the new algorthm based on sub-module, and the green curve ndcates the teratve process of the tradtonal algorthm based on weght-set. These four fgures show that the new algorthm has the faster convergent speed n search process of optmzaton obectve. Table 1 shows that the new algorthm can ensure all obs meet ther deadlne, but the tradtonal algorthm can t do t. Therefore t can say that the novel algorthm has an obvous advantage and good feasblty compared wth tradtonal algorthm. onclusons For the schedulng of MTO manufacturng systems, the m mportant thng s to ensure all obs meet ther deadlnes; at the same tme, many MTO enterprses hope to make ther own nterests maxmze through producton schedulng. Both of the rule-based reverse schedulng method and the tradtonal mult-obectve optmzaton algorthm can t solve the two-obectve problem well. In ths paper, a novel genetc algorthm s proposed whch can ensure that all obs are completed on tme. In addton, t also can better optmze another performance whch the enterprse pursues. In a word, the novel algorthm has the better feasblty, practcablty and maneuverablty to solve two-obectve shop schedulng problem n MTO manufacturng systems. Acknowledgements Ths work s fnancally supported by the Natonal Hgh Technology Research and Development Program 863 -Program Foundaton of hna (2011ZX , 2007AA ), the Natural Scence Foundaton of hna ( ).

6 1320 Advanced Manufacturng Technology References [1] Pnguan Huang, Hu L and Ldong Han, n: Order Schedulng Problems n Make-To-Order Manufacturng Systems, IEEE Internatonal onference. vol.4 (2005), p.2179 [2] R. onterno and Y.. Ho, n: Order Schedulng Problem n Manufacturng systems, Robotcs and Automaton, A: 1987 IEEE Internatonal onference, vol. 4(1987), p.124 [3] Y. Betul, M.Y. Mehmet: Expert System wth Applcatons. Vol.37 (2010), p.1361 [4] K. Deb, A. Pratap, S. Agarwal: IEEE Transactons on Evolutonary omputaton. Vol.6 (2002), p.182 [5] Mn Lu, heng Wu: Robotcs and omputer-integrated Manufacturng. Vol.20 (2004), p.225 [6] P..hang, J..Hseh, S.G.Ln: Int. J. Producton Economcs. Vol.79 (2002), p.171 [7] F.Ballestn, R.Blanco: omputers & operatons research. Vol.38 (2001), p.51 [8] Ruz, Ruben: European Journal of Operatonal Research. Vol.205 (2010), p.1 [9] E.Tallard: European Journal of Operatonal Research. Vol.64 (1993), p.278 [10] Lng Wang, n: Shop Schedulng wth Genetc Algorthms, edted by Tsnghua Unversty Press, In hnese, Beng, hna, (2003)

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