Color Coated Coil Scheduling in Cold Rolling PlantUsing Artificial Neural Network

Yanyan Zhang Lixin TangThe Logistics Institute, Northeastern University, Shenyang, China

Abstract-This paper investigates the problem ofgenerating satisfactory solution algorithm is to determine a tour of selected cities, from aproduction sequence of steel coils for color coating process in cold rolling larger set of available cities, while considering that: a cost is incurredplant The characteristic of scheduling involves two parts of work. for travel between cities; a penalty incurred for cities left unvisited; aOne is to optimally choose steel coils from available ones, the other is to prize is collected for visiting each city.generate a satisfactory processing plan so as to minimize production cost In this paper we try to solve the PCTSP by discrete HopfieldTo solve the problem, this paper transfers all the production networks, which has long been the focus of applying artificial neuralrequirements into the distance between any pair of coils, reduces the network to combinatorial optimization problem [4, 5]. The originalscheduling problem to Prize-Collecting Traveling Salesman Problem application of Hopfield network to the well-known TSP is done by(PCTSP) and develops pertinent Hopfield network algorithm, taking full Hopfield and Tank [6]. And most ofthe subsequent research is basedadvantage of the enlarged solving space of PCTSP. The performance of on it or focuses on the traditional TSP [7, 8, 9] or TSP like problemsthe proposed approach is verified by randomly generated problem [10]. By now no work has been reported on Hopfield network forinstances. Computational results show that the proposed approach PCTSP that we are facing. For this reason, we develop a 3-optoutperforms the existing heuristics with respect to solution quality. heuristics combined Hopfield network method to approach the color

I. INTRODUCTION coating scheduling.Iron and steel industry is the support industry ofnational economy III. PROBLEMDESCRIPTION

and the important measurement of a country's overall strength. A. Color coatingprocess introductionAround the world, with the fast development of industrialization and The color coated coil uses cold rolled coils and galvanized coils associal consumption, the demand for steel sheets including color the basic materials. After the surface treatment (includes degreasing,coated coil and galvanized coil is increasing continuously. In China, rinsing, chemical transforming), by applying and dressing all kinds ofmarket research shows there is 3000 of supply shortage that has to be paints to the surface in continuous roller applying method throughmet by importation. Such situation implies the promising prospect of primary coating and finish coating, then baking and cooling and colorcolor coated coil products. To optimally organize production can be coated products are manufactured. Having the advantage of excellentone of the ways to reduce the production cost and improve the processibilty, beautiful appearance, corrosion-resistant and highproductivity of color coating. This paper addresses the problem of added values, color coated coils have been broadly used in manycoils scheduling in color coating production and aims at generating an fields of industry. Figure 1 gives the structure of color coated coil.optimal coils sequence with minimal production cost. The 2# color During the color coating operation, the paint material container needscoating production line of cold rolling mill in Baosteel Complex is clearing when contracts with different colors are processed.taken as the background. Frequent paint replacement will on one hand increase production cost

II. LITERATuRE REVIEW and work intensity, and decrease the efficient work rate andFor similar kind of scheduling problem, Okano et. al. [1] addressed production quality on the other hand.

the finishing line scheduling in the steel industry and stated that mostof sequencing constraints can be represented as a distance function fiihl r(f cheil transformation

between each pair of coils. They developed efficient heuristics _ij1I1I1jalgorithm and obtained satisfactory results. Lopez et al. [2]investigated the Hot Strip Mill Production Scheduling Problem(HISMPSP) in the steel industry. They modeled the HSMPSP as a

generalization of the PCTSP with multiple and conflicting objectivesand constraint and proposed a Tabu Search based heuristic method todetermnine good approximate solutions. finish eolar (bL

The PCTSP and other generalizations of Traveling Salesman primary_ o or

Problem (TSP) are based on the same "prize-collecting" idea. The ch.mic~i rI~ tiooPCTSP was introduced by Balas and Martin [3] as a model forscheduling the daily operation of a steel rolling mill. The PCTSP Fig. 1 The structure ofcolor coated coil

1-4244-0020-1/05/$20.OO a2005 IEEE 1

B. Production features and constraints description of the wti the weight ofcoil i.schedulingproblem k,k, the processing position, k, k1 = 1,2, ... n.

In Baosteel complex, current color coated plan is completed by Vik the output ofa neuron.manual work. The generated schedules differ greatly from each Uik the state of a neuron.other because of the difference of men's experience. And oneschedule generation needs a few hours' work. the threshold vleioft nteurn.

Production scheduling pursues two goals: the feasibility and Sk the conecti wedigtbetween tw osneuons.the optimality. The former goal is gained by satisfying the hard Sn theimmedia esueedingkprocessng iconstraints in the process ofproduction, a schedule is called a on'the 'saemachine, Ske{k±1ik±2.n}feasible one when all the hard constraints are satisfied. In our dproblem, there are two types of hard constraints. The first one immediately after coilj ,which is calculated by transforming the

is that in one plan the descending sort of width must be width, the thickness, the due date and the paint changing cost

observed, not any width reverse will be allowed. The second between adjacent coil i and coilj. When no color switch is needed,one is that in order to guarantee production continuity, no gap the distance is the specification difference between adjacent coils.

between adjacent coils belonging to the same plan will be When color switch is needed, the distance is expressed as the sum

permitted. The latter goal is affected by soft constraints, which of the above mentioned items and the production cost caused bywill affect the performance of the schedule. In our problem the the switch. The importance level of each requirement can befirst soft constraint is the holding timrne: Longer holding time realized by adjusting and balancing the corresponding weight to a

than the longest allowable restriction will result in unnecessary proper degree. Given the tanformation, a parallel is madepretreatment. The second is the satisfying degree of the due between the color coated coil scheduling and the PCTSP. The

date. The third is the upper and lower bound restrictions of the salesman is considered the machine, the cities are the steel coils

production line capacity. Therefore, the problem tries to and the distances are the specification difference between adjacentminimize the total switch time, material loss and the penalty of coils. We get the networks formulation.due date, taking into all the above mentioned restrictions in the Objective function

n n n n n

color coating production. minyE Edjvikvj Sk +Epi(1-vIk )2 (6)IV. MATHEMATICAL FORMULATION i=1 ,=l1j.i k=1 i=l k=1

A. Basic assumptions Subject toOur formulation is constructed under the following n n n (7)* S~~~~~~~~~~~~~~~~~~~~~v v = 0(7

assumptions: ,, dk 1k 1kik1) The number ofavailable coils and their features are known.2) All coils to be coated go through primary coating and finish = o (8)

coating process in turn, which can not be treated at the same time. k=1 i=1 j=1,j#i3) The color coating at each process can not be interrupted. 2

B. The decisions andthe objective ofthe schedule (YYvik -numi ==0 (9)Our goal is to work out daily plan ofcolor coating scheduling in i=1 k=1)

cold rolling plant. There are two important activities associatedwith the color coating scheduling problem. The first one is E E wt . WT (10)concerned with the coils we should choose to schedule i=1 k-1

(selection), and the second one with the position in the The energy function is:sequence we should assign to the selected coils (sequencing).These two tasks have to be considered simultaneously. This A1 E1 didVIkVs++ 2 (PI(1-vik)2)paper tries to construct accurate mathematic model that can 2 i=1 j=lji k=l i=l k=1comprehensively express the practical features of color coatingproduction and design corresponding intelligent algorimth that can + A3 E E Vik Vik + 4 E E E Vik Vjksuccessively obtain satisfactory scheduling results. 2 i=k1 k1=k1i,kk 2 k=1 i=1 j=

C Notations andnetworkformulationn thenumberofall available coils to be processed. + A5 ( v, -num) + A6 F(WT-VVwtvl (11)i,j- the coil identifier,i=1,2 ...,.......... n. 2 y~ikY ) 2 ii.k)lWT the color coated coil production line capacity.Heewinrdcthfoligfuto:Pi the penalty paid for unselected coil i .

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Fu x.O (12) each unselected coil, and the distance between all pair of coils.

| x2 x < 1 To demonstrate the performance ofour approach across-the-board,The objective is to minimize the distances between all pairs of we design three sets ofpenalty which has different range ofvalues.

adjacent coils and the total penalties of unscheduled coils. We choose the heuristics proposed by Laporte and Martello [11]Constraints (7) ensure that there are at most one 1 in each row. for the PCTSP to evaluate the performance of our approach. To

Constraints (8) guarantee that there are at most one 1 in each make it convenient for comparing (the original heuristics pursues

column. Constraints (9) require that the total number of 1 im the maximal sum ofprize), we do a little modification to it, and the

permutation matrix should be num. The determination of modified heuristics algorithm is as follows:parameter num is as such: bei

T:={vl };num = n - (,f1 wti - WT) / aver, aver = E VVt, / n find the minimal ratio (d1±+d,1Yc) such thatj#1;

Where aver is the average weight of the available coils, form the tour T:=(vl, vj,vl)parameter num is mtroduced to restrict the number of coils in improvemrue;current sequence. Constraints (10) mean that total weights in the repeatcoated sequence should not exceed the production line capacity find v, T and (v,vS) consecutive vertices in T such thatrestriction. A1, A2, A3,A4, A5, A6 are positive coefficients. Zci5C and [(d,±+dc,-d,r)-p1]/cj is minimal;

From the energy function we can deduce the connection ifno suchv, and (v,vS) exist then improve :falseweights and the threshold value in the following form: else insert vy in T between v, and v,Wik jkl Al (d, (1--j )5k,8 k )A2ppiP A385j (1 -8k,k I until improve = falsely '1K ') kl,Sk 211 1 ]\k,k1/end

- A4(\1-bk,k I -A5 -A G(WT - WtV)nnwt (13) The sequence obtained by the above heuristics is further

-4(- kI-156-1E tjVkl tj optimized using 2-opt heuristics. Table 1 shows the perfonnancel k1t1 demonstrations of the proposed approach compared with the

ik = A2 pi - A5num - A6WTG(WT - E E WtjVjkl )Wtj (14) above heuristics corresponding to different satisfying levelj=1 kl=1 (denoted bya ). For each problem size, the average objective value

Where < tI i c ( ) fo x . 0 (15) ofIc runs is given, and the column identified with "volume"1 0 i .jJ xI x < O indicates the satisfying degree ofthe production line capacity. With

V SOLVING STRATEGYAND COMPUTATIONAL ExPERIMENTs respective to different penalty range, different satisfying degree ofA. Solving strateg the line capacity, the performance comparison between the existing

In traditional TSP, there must be one and only one 1 in each row heuristics and our approach is given in Fig. 2 and Fig. 3.and each column in the permutation matrix. While it is known to VI. CONCLUSIONSall that the traditional Hopfield neural network often stops in local In this paper we have examined the color coating schedulingminima or generates infeasible solution. Under such situation, the problem and formulated it as a Hopfield network formulation to

solution is infeasible no matter how small the objective may be. minimize the producing costs and the total penalties ofHowever, in our problem, the feasibility of the generated solution unscheduled coils. An efficient approach which combines Discreteis defined as follows. A certain row or colunm is allowed to have Hopfield neural network and 3-opt heuristics is proposed. The

none 1, implying that ever infeasible solution may be accepted if performance comparison results demonstrate that our method is

other constraints are satisfied, where not all the coils are required to better than the existing heuristics for PCTSP concening thebe selected. Therefore the solution space is larger than the solution quality and that the network can outstanding realize thetraditional TSP. We just take full advantage of such possibility goal ofminimizing the total penalties of all unscheduled coils. Weoffered by the networks to obtain better solution. can conclude that our approach exhibits in the following aspects.

Considering the ability of Hopfield network for sequencing, we First, it generates large enough size of the color coatingthen adopt 3-opt heuristics to optimally arrange the selected coils to sequence, keeping the size of each plan results in possibly largeminimize total distances ofall adjacent coils. productivity within the production line capacity restriction. Second,B. Computational experiments it generates schedules that include a much lower penalty values,On the PentiumlV 2.4G HZ computer, the proposed algorithm that is, coils that have larger penalty values, smaller weights and

is implemented by C-- language and the simulation experiments larger switch cost are of lower priority when being choosing.are conucted.The ranomly gnerate data st incldes.th Third, the "distances" caused by paint switch have been reduced

weight of each coil, the prize for each selected coil, the penalty for dramnatically, which means that the continuity ofproduction can be

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Table 1. Performancedemonstafonoftheproposedappfoach

a size heuristics NN + 3-opt 1000

obj-value volume obj-value volume 00020 1.0000 0.5767 0.9819 0.894930 1.0000 0.3600 0.7157 0.8184 06000

0.2 50 1.0000 0.2058 0.5307 0.9095 0500080 1.0000 0.4711 0.4849 0.8959 04000100 1.0000 0.4920 0.4412 0.8664 03000

0~200020 1.0000 0.5767 0.9819 0.8949 0100030 1.0000 0.5446 0.7849 0.8184 OW

P1 0.5 50 1.0000 0.5832 0.6607 0.909580 1.0000 0.5031 0.5057 0.8959 m o o mi

100 1.0000 0.5988 0.5280 0.8664 .. .200 1.0000 0.9818 0.9428 0.8649 Fig. 2 Comparison ofobjective function values between the two methods.20 1.0000 0.8189 0.9642 0.8949 f30 1.0000 0.8062 0.7198 0.8274 Satisfying Degree of the production line capacity

0.8 50 1.0000 0.8458 0.7486 0.9095

80 1.0000 0.8183 0.6232 0.8959100 1.0000 0.8346 0.7048 0.8664 1.0000

0.900020 1.0000 0.6078 0.6563 0.9231 0.8000° [30 1.0000 0.9985 0.7739 0.7269 0.7000

0.2 50 1.0000 0.9700 0.7370 0.9204 0.50000.4000

80 1.0000 0.9011 0.6592 0.9076 0.3000100 1.0000 0.9598 0.8081 0.9309 0.2000

0. 100020 1.0000 0.6078 0.6563 0.9231 0.000030 1.0000 0.9985 0.7742 0.7772

P2 0.5 50 1.0000 0.9700 0.7370 0.9204 HeuHO o .ristic. Our Alg.riYoahmM80 1.0000 0.9011 0.6592 0.9076 Fig3 Compar in in degree Inductionalcapaci rh.100 1.0000 0.9598 0.8081 0.9309 Com paio of satifyin d e tp in e t20 1.0000 0.9444 1.0000 0.9412 REFERENCEs30 1.0000 0.9985 0.7794 0.8900

0.8 50 1.0000 0.9700 0.7370 0.9204 [1] H. Okano, A. J. Davenpor M. Trumbo, C. Reddy, K. Yoda and M. Amano.80 1.0000 0.9011 0.6592 0.9076 Finshing Line Scheduling in the Steel Industry. Journal of Research &100 1.0000 0.9598 0.8081 0.9309 Development 2004,48(5/6), 811-830.20 1.0000 0.8367 0.8590 0.894730 1.0000 0.9985 0.7739 0.7269 [2] L. Lopez, M. W. Carter and M. Gendreau. The hot strip fll production

0.2 50 1.0000 0.9974 0.8994 0.9378 scheduling problem: A tabu search approach. European Jouial of Operational80 1.0000 0.9923 0.8889 0.9437 Research. 1998,106,317-335.100 1.0000 0.9936 0.9389 0.949320 1.0000 0.8367 0.8590 0.8947 [] E. Balas and G. Maitin. ROLL-A-ROUND: Software package for scheduling the30 1.0000 0.9985 0.7742 0.7772 rounds of a rolling fill. Copyright Balas andAmartin Associates, 104 Maple

P3 0.5 50 1.0000 0.9974 0.8994 0.9378 Heights Road, Pittsburgh, PA. 1985.

30 1.0000 0.9985 0.7794 0.8900 [5] KA. Smith. Hopfield neural networks of timetabling gforulations, methods0.8 50 1.0000 0.9974 0.8994 0.9378 and comparative Results . Ner al 1994,r7,681-690.80 1.0000 0.9923 0.8889 0.9437 adcmaaiersls optr nufilEgneig 03 4

100 1.0000 0.9936 0.9389 0.9493 283-284.

guaranteed, there are only a few times needed to change color, [6] J. J. Hopfield and D. W. TanL Neural computation of decisions in optimizationnaturlly the time for changing paint and cleansing the rollers is problems. Biological Cybernetics. 1985, 52,141-152.reduced, and the paint loss is decreased as well. In addition, our [7] L. I. Burke. Adaptive Neural Networks for the Traveling Salesman Problem:

approach can generate color coating schedule much faster than the Insights from0Operations Research. NeuralNetworks. 1994,7,681-690.manual work, which shortens the scheduling time, makes the [8] C. K. Looi. Neural Network Methods in Combinatorial Optimization, Computersoperation stuff disengage from the bothersome work, have more and OperationsResearch. 1992,19,191-208.tifme to im:prove, evaluate and optimize the schedule. [9] J. Y. Potvin. The Traveling Salesman Problem: A Neural Network Perspective.

ACKNOWLEDGMENT ORSA Journal on Computing 1993, 5, 328-348.

This research is partly supported by National Natural Science [10] R. Andresol, M. Gendreau, and J.-Y. Potvin. A Hopfield-Tank Neural NetworkFoundation for Distinguished Young Scholars ofChina (Grant No. Model for the Generalized Traveling Salesman Problem, in Meta-Heuristics:70425003), National Natural Science Foundation of China (Grant Advances and Trends in Local Search Paradigms for Optimization. S. Voss et al.No. 60274049) and (Grant No. 70171030), Fok Ying Tung (eds.), KiuwerAcademic Publishers, Boston. 1997,393-402.Education Foundation and the Excellent Young Faculty Program [11] G. Laporte and S. Martello. The Selective Traveling Salesman Problem.and the Ministry ofEducation, China. DiscreteAppliedMathematics. 1990,26, 193-207.

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