A novel hybrid algorithm for scheduling steel-making continuous casting production

Computers & Operations Research 36 (2009) 2450 -- 2461

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Computers &Operations Research

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Anovel hybrid algorithm for scheduling steel-making continuous casting production

Arezoo Atighehchian∗, Mehdi Bijari, Hamed TarkeshDepartment of Industrial Engineering, Isfahan University of Technology, Isfahan, Iran

A R T I C L E I N F O A B S T R A C T

Available online 30 October 2008

Keywords:Steel-making–continuous casting schedulingAnt colony optimizationNon-linear optimization

In this paper, steel-making continuous casting (SCC) scheduling problem (SCCSP) is investigated. Thisproblem is a specific case of hybrid flow shop scheduling problem accompanied by technological con-straints of steel-making. Since classic optimization methods fail to obtain an optimal solution for thisproblem over a suitable time, a novel iterative algorithm is developed. The proposed algorithm, namedHANO, is based on a combination of ant colony optimization (ACO) and non-linear optimization methods.The solution construction in HANO is broken up into two phases. The first phase determines the discretevariables (corresponding to job-machine assignment and sequencing), while the second phase determinesthe continuous ones (corresponding to timing of the jobs on their assigned machines) through a non-linear optimization method.The efficiency of HANO is compared with a heuristic algorithm as a real case used at Mobarakeh SteelCompany (MSC), the biggest steel factory in the Middle East. In addition, the proposed algorithm is com-pared with Genetic Algorithm, as a search method for both discrete and continuous variables, throughsolving several instances.Numerical results reveal the higher efficiency of the proposed approach compared with the heuristic oneused at MSC. Furthermore, the efficiency of HANO is compared with GA to show that HANO enjoys abetter performance in more than 95% of the cases while in the remaining 5%, its performance efficiencyshows no difference.

© 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Iron and steel industry is the cornerstone of an industrializedeconomy. Since it is capital and energy intensive, companies haveconstantly laid great emphasis on technological advances to be em-ployed in the production process in order both to increase productiv-ity and to save energy. On the other hand, production scheduling isa key tool which can improve machine productivity, reduce materialand energy consumption, and cut down production costs. Produc-tion scheduling aims to determine starting and ending times of jobson machines so that a certain measure of performance is optimized.

Steel-making–continuous casting (SCC) production schedulingproblems (SCCSP) should determine in what sequence, at whattime, and on which device, the molten steel should be arranged atvarious production stages from steel-making to continuous casting.Unlike general production scheduling in machinery industry, SCCSPhas to meet special requirements of the steel production process.

∗ Corresponding author. Tel.: +983113915515; fax: +983113872858.E-mail addresses: [email protected] (A. Atighehchian),

[email protected] (M. Bijari), [email protected] (H. Tarkesh).

0305-0548/$ - see front matter © 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.cor.2008.10.010

In the SCC process, the products being processed are handledat high temperatures and converted from liquid (molten steel) intosolid (slabs). There are extremely strict requirements on materialcontinuity and flow time.

The whole scheduling process can be divided into two steps:

(1) Cast sequencing which determines job sequence in each castingmachine.

(2) Sequencing and scheduling of steel-making process and timingof the jobs on continuous casting machines.

This paper will concentrate on the second scheduling step bygetting the cast sequencing from the higher level planning. Thesecond step needs to consider resource constraints and techno-logical requirements to ensure practical feasibility of the resultingschedule. Also the synchronization of different production stagesmust be taken into consideration in order to reach productioncontinuity, to increase productivity, and to cut down the total pro-duction costs including casting interruptions, waiting-time betweenoperations, transportation cost, and so on. We maintain that theproblem defined here with its particular assumptions (Section 3) isnot reported in the literature and that it finds real applications inindustry.

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A. Atighehchian et al. / Computers & Operations Research 36 (2009) 2450 -- 2461 2451

Fig. 1.

As can be seen from Fig. 1, the structure of the production systemfor the SCC is similar to a hybrid flow shop (HFS). The SCC problemis a more complicated form of HFS because of its additional practi-cal constraints and its more complex scheduling criteria. Gupta [1]demonstrated that the two-stage HFS scheduling problem with theobjective of minimizing the make span is NP-complete [1].

In this paper a novel hybrid algorithm is proposed based on thecombination of ant colony optimization (ACO) algorithm and classicoptimization methods. The solution construction is broken up intotwo phases and uses the concept of the ACO algorithm. The proposedalgorithm is not limited to steel-making problems but can be appliedto other job-machine scheduling problems as well.

The simultaneous of use ACO and non-linear optimization meth-ods, the employment of efficient heuristic information to guide theACO search, and the development of an initial solution for solvingthe non-linear model of the second phase form the main character-istics of the proposed approach.

The steel-making plant at MSC (Mobarakeh Steel Company) in Is-fahan is taken as the research background and the goal is to generatedaily schedules for SCC. The efficiency of HANO is evaluated againsta heuristic algorithm used at MSC. In addition, the proposed ap-proach is compared with the Genetic Algorithm as a meta-heuristicalgorithm to search both discrete and continuous variables simulta-neously through solving several instances. The experimental resultsshow that the proposed approach is an effective method for opti-mizing production continuity and minimizing the total cost.

In the following sections, a brief review of related previous workwill first be presented (Section 2). Section 3 will deal with the SCCproduction process and the scheduling problem. Section 4 will de-scribe the proposed algorithm. In Section 5, a Genetic Algorithm willbe designed for solving this problem to compare its results withHANO algorithm's results. Numerical results will be reported in Sec-tion 6. Conclusions will make Section 7.

2. Literature review

Solving production planning problems in SCC production is animportant research topic and has been widely explored. Various pro-duction planning and scheduling techniques for SCC production havebeen reported in the literature.

All the methods used for SCC production scheduling can be clas-sified into four categories:

(1) Operations research methods.(2) AI methods: Expert systems, intelligent search methods (meta-

heuristic approaches), constraint satisfaction methods.

(3) Human–machine coordination methods.(4) A-teams methods (multi-agents methods).

A review of previous research on integrated steel production plan-ning and scheduling can be found in Tang et al. [2]. Because steel-making scheduling with practical constraints is extremely complex,most previous methods used to simplified assumptions in solvingthe problem. Some approaches to SCC scheduling treat the problemat three levels [3]: sub-scheduling, which fulfills the scheduling ofindividual job sets; rough scheduling, which merges sub-schedules;and optimal scheduling, which eliminates machine conflicts.

Tang et al. [4] developed an integer programming formulationwith a “separable” structure. Their solution methodology is based oncombining Lagrangian relaxation, dynamic programming and heuris-tics. Harjunkoski and Grossmann [5] developed a decompositionstrategy for solving large scheduling problems using mathematicalprogramming methods. Lee et al. [6] solved a scheduling problemfor operating the continuous caster by using the concept of a spe-cial class of graphs known as interval graphs. Ouelhadj et al. [7]described a new model for robust predictive/reactive scheduling ofSCC based on the use of multi-agents, tabu search and heuristic ap-proaches. Ferretti et al. [8] presented the algorithmic solution basedon ant system metaheuristic, for an industrial production inventoryproblem in a steel continuous casting plant. The model takes intoaccount the relevant parameters of the finite-capacity productionsystem. The study focuses on the optimization of the production se-quence of the billets.

Bellabdaoui and Teghem [9] first reviewed the recent works incontinuous casting planning and then focused on a model inspiredfrom an application of SCC by Arcelor Group in Liege, Belgium. Theypresented a formulation with mixed integer linear programming.

Attempts have been made to apply different methods to steelproduction planning and scheduling problems. However, no system-atic research has been carried out yet on the general structure, themodel, or, the algorithm that can be applied to the case of a steel-making plant [10].

The advantages and disadvantages of different methods used forSCC production scheduling are discussed in Tang et al. [2]. Consid-ering the advantages and disadvantages of these methods, a com-bination of operations research, expert systems, and visualizationtechniques may provide great potentials for effectively solving inte-grated production scheduling problems [2].

3. Problem definition

In this section, the overall assumptions of this problem are de-fined following a description of the SCC production process and itsrelated scheduling problem.

3.1. SCC production process and scheduling problem

The steel-making process consists of three stages: steel-making,refining and continuous casting. Each stage further includes parallelmachines, as shown in Fig. 1. The following is a brief description ofthe production process of MSC.

In the steel-making stage the impurities of molten iron are re-duced to desirable levels by burning with oxygen in an electric arcfurnace (EAF). The basic unit of steel-making production is a charge,which is defined as a “job” in SCC scheduling. It refers to the concur-rent smelting in the same furnace. The steel in one charge may becast into different slabs that are used to produce finished steel prod-ucts for different customer orders. Each charge is defined throughits related gauge and Grade. Grade is a product quality descriptionincluding both chemical and physical properties of the charge. Themolten steel from the steel-making stage is poured into ladles thatare transported by a crane to a refining furnace or ladle furnace (LF)

2452 A. Atighehchian et al. / Computers & Operations Research 36 (2009) 2450 -- 2461

for refining. If no LF is available when a new charge arrives, thecharge has to wait until one of the LFs becomes available. The wait-ing time of a charge causes the charge temperature to drop, andthen reheating is needed. Energy consumption thereby grows as thewaiting time increases. After refining, molten steel is poured into aTundish, the input unit of a continuous caster, for casting.

Depending on the grade, the charge should not spend more thana certain time, the hold-time (wait time), between the LF tap and thebeginning of the casting in order to keep the charge temperatureswithin a required range. Also there is a maximum limit on the totalwaiting time during processing; otherwise, the production of a grademay fail and the product may not meet its quality requirementswithout additional costs.

A sequence of charges that are consecutively cast on the samecontinuous caster is called a cast, which is defined as a “job group” inSCC scheduling. Charges in the same cast need to satisfy the followingtechnological constraints: (1) steel grades for adjacent charges haveto be identical or similar; (2) the slab gauge of different charges hasto be identical; (3) the differences in slab widths of the charges inthe same cast must be within a certain limit and the width jumpbetween adjacent charges cannot exceed a given maximum value;(4) the total number of charges in a cast must be between a givenlower bound and an upper bound that is determined by the life ofthe Tundish; and (5) delivery dates of different charges in the samecast should be as close as possible.

No set-up time is required on the caster between adjacent chargesin the same cast. However, a relatively long set-up time is requiredbetween two casts in the same caster.

When a continuous steel flow is broken, the caster needs mainte-nance, which involves costs and a delay in production. This happens,for instance, when the product type is changed. If the two subse-quent products are very similar, it may be possible to mix them andgo on without stopping. Otherwise, the caster needs to be stoppedfor service. Moreover, the caster can only be run continuously for alimited number of products due to the extreme conditions. So con-tinuous casting can be considered as one of the main challenges insteel production planning.

As mentioned earlier, charges are “jobs” and casts are “job groups”in SCC scheduling. The sequence of the charges on the casting ma-chines are defined at the planning level by considering the ordersof rolling stage, the limitations of casting stage, the delivery times,etc. After planning, the orders in each charge and their sequence arefixed. The charges in each cast and their sequence are also fixed. SCCscheduling is then to decide the sequence and schedule of these jobson the steel-making machines (EAFs and LFs) and also timing thejobs on casting machines.

Our literature review showed that the problem at hand, as specif-ically defined here with the particular assumptions made, has neverbeen investigated before. Unlike many previous papers, the objectiveof minimizing the tardiness and earliness is not taken into accountin this paper because the casting sequence is received from a highlevel planning and remains unchanged.

3.2. Assumptions

N different products are planned on continuous casting machinesto be produced. After the charges and their sequence on castersare defined by planning, the task of SCC scheduling is to determinewhen and where (on which device) each charge should be processedat each production stage. The following general assumptions areconsidered for this problem:

(1) All charges follow the same process route: steel-making, refin-ing, and then continuous casting. At each stage, a charge canbe processed on any one of the machines at that stage, and the

parallel machines at that stage are identical whereas the pro-cessing times of parallel machines may be different.

(2) A machine can at most process one job at a time.(3) A job can at most be processed on one machine at any time.(4) The sequence of the charges on casting machines remains un-

changed.(5) For the two consecutive operations for the same charge, only

when the preceding operation has been finished, can the imme-diate next one be started.

(6) For two consecutive charges processed on the same machine,only when the preceding charge has been finished, can the im-mediate next one be started.

(7) Setup time for different machines is independent of sequenceand charge properties.

(8) The processing times are independent of charge properties.(9) The transport resources are not considered in the model as-

suming that they are available but transportation time and therelated cost are considered.

The objective is to ensure continuity of the production processthroughminimizing a cost function consisting of the following terms:

• The production quantity value. Since the objective function shouldbe minimized, the production value is considered negative. Itmeans that more production leads to less objective value.

• Total casting interruption cost.• Total molten steel temperature drop cost. It is considered as a

linear function of the total waiting time of the heats.• Total cost of poor quality. It is considered as a linear function of the

total violation from the maximum limit of waiting time betweenprocessing.

• Transportation cost.

The formulas for numerically evaluating the terms of the objectivefunction are given in Section 4.3.

4. The proposed algorithm

In the proposed algorithm, ACO and non-linear optimizationmethods are hybridized efficiently. The first ACO meta-heuristic,called ant system Dorigo et al. [11–13], was inspired by the studyof the behavior of ants. The works of Colorni et al. [14], Dorigo et al.[13,15], Drigo and Gambardella [16], Dorigo and Di Caro [17] offerdetailed information on the workings of the algorithm.

In the proposed algorithm, each solution is constructed throughtwo phases. The first phase is accomplished via the concepts of ACOalgorithm. The output of this phase is a feasible sequence of the jobs.Based on the sequencing obtained, the job scheduling is determinedin the second phase through a non-linear optimization method.

For constructing a solution, each ant finds a sequencing of thecharges on different machines of production stages (first phase:primary planning) and then applies the mathematical optimizationmodel to schedule the given primary planning (second phase). Af-ter evaluating each solution, the pheromone matrix is updated andthe algorithm continues until it reaches its stopping criteria. Accord-ing to considered stopping criteria the algorithm is stopped when itcannot reach to better solution in 100 sequential iterations. MATLABsoftware is used to implement the proposed algorithm. Fig. 2 showsthe overall structure of the whole algorithm.

The inputs and the outputs of each phase are:

• First phase inputs:◦ the sequence of the charges on casting machines;◦ the machines conditions and the parameters of the problem(processing time, setup time, . . . ).


Fig. 2.

• First phase outputs:◦ the machine assigned to each job and the sequence of thecharges on related machines in current iteration of the algo-rithm;

◦ the primary scheduling of the jobs.• Second phase input:

◦ sequencing obtained from the first phase in current iteration.• Second phase output:

◦ the job scheduling which is determined through a non-linearoptimization method.

4.1. First phase: primary planning

In this phase, a primary planning is constructed using the para-metric probabilistic model. This planning determines the machineassigned to each job and the sequence of the charges on relatedmachines. Also it determines the primary scheduling of the jobs inwhich each job is scheduled in earliest possible time using the SGS(serial schedule generation scheme) method as a standard heuristicmethod for project scheduling problem. Let us first define the re-quired notations below:

• Activity-mode: the modes of each activity show the differentroutes of resources through which the activity can be processed.

• Candidate charges: the charges whose proceeds have been sched-uled.

• Candidate set: this set contains the candidate Activity-modes orthe different routes of processing candidate charges.

• Pijk(t): probability of assigning product i to jth EAF and kth LF intth iteration.

• �ijk(t): the amount of pheromone related to assigning a product ito jth EAF and kth LF in tth iteration.

• �ijk(t): The heuristic information related to assigning a product ito jth EAF and kth LF in tth iteration.

• sw: The solution constructed by an ant to select for updatingpheromone trail in the current iteration.

• sgb: The best solution of the algorithm from the beginning of thetrail.

4.1.1. First phase algorithmStartUntil all the jobs are scheduled:• Select the first available machine from EAFs.• For each element of candidate set:

◦ Calculate the value of heuristic information related to selec-tion of this candidate

◦ Calculate this probability:

Pijk(t) =aijk∑i∈Daijk

(1)

where:

aijk(t) = (�ijk(t))� × (�ijk(t))

� (2)

• Select the winner among the candidates based on the aboveprobability

• Schedule the winner candidate in earliest possible time• Updating the problem environmentEnd;This phase is done in the simulated environment and all of the

jobs are scheduled primarily using SGS method, so all jobs are sched-uled without conflicts.

4.1.2. The particular features of the first phase designIn this section, we describe the adaptations to the ACO meta-

heuristic that we have developed in order to treat themulti-objectiveproblem addressed in this paper.

• The heuristic rules: the lower are the casting interruption cost,the waiting time cost and the transportation cost of the alterna-tive, the higher will be the probability of selecting the alternative.Analyzing the trend of ACO algorithm with using heuristic rulesand without them shows that the selected heuristic rules have agreat effect on searching trend of ACO for a same problem.The values of the heuristic rules are calculated based on the pri-mary scheduling of the jobs. With this primary scheduling, it canbe calculated whether the job can be cast continuously with itsprevious job or not. Based on the primary timing if there is a cast-ing interruption the related cost is one of the elements of heuris-tic rules. The transportation cost and the waiting time cost arealso can be calculated based on the time needed for transport be-tween different machines and the needed waiting time betweenprocesses for each job in the primary scheduling.

• To start with a good solution, the first ant of the first iteration con-structs the solution with selecting the candidate that has greaterheuristic information in each step.

• Pheromone trail updating: In order to exploit the best solutionsfound during the current iteration or during the run of the algo-rithm, the ant finding the best solution in the current iteration andthe one finding the best solution from the beginning of the trailadd pheromone after each iteration. Pheromone updating rule is


developed as follows:

�ijk(t + 1) ={(1 − �)�ijk(t) + ��ijk(t + 1) if i − j − k ∈ sw

(1 − �)�ijk(t) otherwise(3)

��ijk(t + 1) = L × (�MAX(t) − �ijk(t) + 1) ×∣∣∣∣∣ f (s

w)

f (sgb)

∣∣∣∣∣ (4)

where L is the step parameter, �MAX is the maximum amount ofpheromone in pheromone matrix in tth iteration and f(s) is theobjective function related to solution “s”.

|f (sw)/f (sgb)| is the quality function of solution “sw”; note thatthe objective function value considered in this problem is negative.The greater the value of |f(sw)| in proportion to |f(sgb)|, the higherthe quality of solution “sw”.

This mechanism can increase the performance of the algorithmbecause:

1. When the algorithm has converged or is very close to conver-gence, it will increase the pheromone trails proportionate to theirdifferences by up to maximum pheromone trail limit. The ba-sic idea is to facilitate the exploration by increasing the prob-ability of selecting solution components with low pheromonetrails.

2. The greater the quality of the solution, the more the pheromonethat is added to its components.

4.1.3. Parameter tuningIn order to tune the parameters of the ACO algorithm designed

(L, �, �, � and the number of ants (A)) each parameter is consideredin three levels.

The influence of three main parameters (�,�,�) and their inter-actions on the ACO algorithm is analyzed using the 33 full factorialdesign with six replications. The data is transformed into normaldistribution using Box–Cox transformation [18]. The results of ana-lyzing by Matlab software are given in Appendix A.

Duncan's multiple range test [19] is used for comparing all pairsof means for each factor and tuning these parameters. The results ofthis test are given in Appendix A.

For tuning the other parameters of the ACO algorithm designed(L, the number of ants (A) and the initial value of pheromone matrix(�0)), each of them is varied in the proper range while the otherparameters are kept unchanged; the algorithm is, then, run 10 times.Taking account of the average, the standard deviations of the resultsand also the average run time of the algorithm, the proper valuesfor these parameters are obtained.

The selected values for the parameters are given in Table 1.

4.2. Second phase: scheduling the primary planning

Based on the sequencing obtained from the first phase, in thesecond phase the job scheduling is determined through a non-linearoptimization method. The following symbols are used in definingthe problem parameters and variables.

Table 1ACO parameters

L � � � A �0

0.25 1 2 0.05 5 20

4.2.1. Notations

Cmax: the finish time of all planned chargesInputs:

N: the total number of production chargesEAFno(i): EAF number of charge “i”LFno(i): LF number of charge “i”CCno(i): CCM number of charge “i”Dt1(i,j): transportation time from EAF number “i” to LF number “j”Dt2(i,j): transportation time from LF number “i” to CCM number “j”Prev(i): previous charge of charge “i” on related machine.T(j,k): processing time of charge on jth machine of kth stage of pro-cessMaxWCCM: max limit for waiting time before casting stageBuffermax: the maximum limit of total waiting time between pro-cessing.Tundishlife = m: the Tundish life is equal to “m” continuous chargesSetupTime: Setup time of casting machine after casting interruptionPL: scheduling interval lengthW(i): related gauge of products from charge i (i ∈ �)f(i′,i): determines the gauge change between two adjacent charges

f (i′, i) ={1 if W(i′)�W(i)0 otherwise

i, i′ ∈ � (5)

Grade(i): related grade of products from charge iSeq(i′,i): determines the grade change on adjacent products

Seq(i′, i) ={1 if Grade(i′)�Grade(i)0 otherwise

i, i′ ∈ � (6)

Setup(k): setup time of kth stage of machines (k = 1,2)C1: cost of a casting interruptionC2: waiting time cost of charges in ladle for each time unitC3: cost of poor quality due to violation from the maximumwait-ing time limit for each time unitC4: each product valueC5: time unit value. It is calculated based on the maximum pos-sible production per unit of time multiplied by C4FirstSeq: set of all first unscheduled charges of CCMs

For charge i (i ∈ FirstSeq) the information about the status of thelast scheduled charge of related CCM is the input of the schedulingmodel.

Decision variables:

ftEAF(i): finish time of charge “i” on related EAFstLF(i): start time of charge “i” on related LFftLF(i): finish time of charge “i” on related LFstCC(i): start time of charge “i” on related CCMftCC(i): finish time of charge “i” on related CCMBL: a slack variable for waiting-time violationsSetupTundish(i): setup time of CCM before charge “i”

4.2.2. Scheduling model formulation

Min Z = C1 ∗N∑i=1

(dp(i) × 10) + C2 ∗N∑i=1

(stCC(i) − ftEAF(i))

+ C3 ∗N∑i=1

BL(i) + C5 ∗ Cmax

ftEAF(i)� ftEAF(Prev(i)) + Setup(1) + T(EAFno(i), 1) ∀i (C1)

stLF(i)� ftEAF(i) + Dt1(EAFno(i), LFno(i)) ∀i (C2)

stLF(i)� ftLF(Prev(i)) + Setup(2) ∀i (C3)


ftLF(i) = stLF(i) + T(LFno(i), 2) ∀i (C4)

stCC(i) − ftLF(i) − Dt2(LFno(i),CCno(i))

�MaxWCCM ∀i (C5)

stLF(i) − ftEAF(i) − Dt1(EAFno(i), LFno(i))

+ stCC(i) − ftLF(i) − Dt2(LFno(i),CCno(i))

�buffer max+BL(i) ∀i (C6)

stCC(i)� ftLF(i) + Dt2(LFno(i),CCno(i)) ∀i (C7)

stCC(i)� ftCC(Prev(i)) + SetupTundish(i) ∀i (C8)

stCC(i)� ftCC(Prev(i)) + SetupTime × f (i, Prev(i)) ∀i (C9)

stCC(i)� ftCC(Prev(i)) + SetupTime

× Seq(i, Prev(i)) ∀i (C10)

ftCC(i) = stCC(i) + T(CCno(i), 3) ∀i (C11)

stCC(i) − dp(i) − dp2(i) = ftCC(Prev(i)) ∀i (C12)

ftCC(i)�C max ∀i (C13)

SetupTundish(i) = SetupTime ∗ dp(i) ∗ 10 ∀i (C14)

SetupTundish(i) +i+m−1∑j=i+1

SetupTundish(j)�SetupTime ∀i (C15)

dp2(i) = dp(i) ∗ 10 ∗ dp2(i) ∀i (C16)

BL(i)�0, SetupTundish(i)�0, ftEAF(i)�0, stLF(i)�0

ftLF(i)�0, stCC(i)�0, ftCC(i)�0

0�dp(i)�0.1, dp2(i)�0

∀i ∈ {1, 2, . . . ,N}The objective is to schedule the products while minimizing a cost

function that consists of these terms: casting interruption cost whichis the linear function of the number of casting interruptions, moltensteel temperature drop cost which is the linear function of totalwaiting time of the charges, cost of poor quality which is the linearfunction of the violation from the maximum waiting time limit, andthe Cmax cost. The production constraints between adjacent prod-ucts are determined from Eqs. (C1) to (C3) and from Eqs. (C8) to(C10) for casting machines. The production constraint enforcing theorder of operations is given in Eqs. (C2) and (C7). Eqs. (C4) and (C11)define the finish time of the products. Eq. (C5) sets an upper limitfor the waiting time before casting operation. Eq. (C6) defines theviolation of the waiting time from its upper bound. Eq. (C12) com-putes the difference between the time of two adjacent charges onCCM. This is used to determine the number of casting interruptions.Eq. (C13) is used to define Cmax. Eq. (C14) defines the relation-ship between Setup Tundish and Setup Time. Eq. (C15) defines theTundish life limitation. Defining Eq. (C16) when casting interruptionoccurs (dp2(i) > 0), the dp variable gets the value equal to 0.1; so∑N

i=1(dp(i)i × 10) determines the number of casting interruptions.In order to achieve suitable solutions for this model using clas-

sical optimization methods, it is necessary to prepare a good initialsolution.

The strategy presented in this paper for solving this NLP probleminvolves relaxing the model to an LP model. The optimal solutionobtaining from the LP model, which is not necessarily the feasiblesolution of the main problem, is used as an initial solution of the NLPmodel. Experiments show that this strategy will yield local optimalsolutions for complex NLP models.

By adding dp2COST ∗ (∑N

i=1dp2(i)) to the objective function andby omitting the NLP equation (Eq. (C16)) from the NLP model, anLP model is constructed, where dp2COST is the coefficient of penalty

cost for variable dp2. Adding this element to the objective functioncauses the variable dp to get to its maximum value (0.1) when cast-ing interruption occurs. Solving this model prepares a good initialsolution for the NLP model. Conopt solver is used for solving bothNLP and LP models. This solver is based on the GRG algorithm firstsuggested by Abadie and Carpentier [20]. Details on the algorithmcan be found in [21,22].

4.3. Evaluation of the solution

The different terms of the objective function can be calculatedfrom the results of the first and second phase of the algorithm. Theproduction value of the scheduling interval is calculated based on thetotal scheduled jobs during the scheduling interval (PL) multipliedby C4. The total scheduled jobs during PL is calculated based on thefollowing formula:

total production =N∑i=1

DPR(i) (7)

where

DPR(i) ={1 if ftCC(i)�PL0 otherwise

(8)

Casting interruption cost is calculated based on the number ofcasting interruptions multiplied by C1. The number of casting inter-ruptions is one of the outputs of the algorithm second phase whichis determined by

∑Ni=1(dp(i) × 10) as described in Section 4.2.2.

Molten steel temperature drop cost is calculated based on thetotal waiting time of the heats multiplied by C2. The total waitingtime of the heats (TW) is calculated based on Eqs. (9) and (10).

TW =N∑i=1

Wait(i) (9)

where

Wait(i) = stLF(i) − ftEAF(i) − Dt1(EAFno(i), LFno(i))

+ stCC(i) − ftLF(i) − Dt2(LFno(i),CCno(i)) (10)

Cost of poor quality is calculated based on the violation from themaximum waiting time limit multiplied by C3. The total violationtime from the maximum limit of waiting time between processing isthe output of second phase of the algorithm. This value is calculatedby

∑Ni=1BL(i).

The last terms of the objective function is the transportation costwhich is derived from the first phase of the algorithm. The objectivefunction value is sum of the values of the above terms.

Note that as the objective function should be minimized, theproduction value will be negative and so will the objective function.It means that more production value leads to less objective value.

The higher the negative objective function value of the solution,the more will be its utility. Fig. 3 shows the effect of applying thesecond phase of the algorithm to the primary planning and also thetrend of HANO algorithm.

5. Designing the genetic algorithm

The proposed algorithm is compared with a Genetic Algorithm asa search method for both discrete and continuous variables. For thisproblem, each chromosome of GA is designed in five main parts: thefirst part represents the selected heat to be scheduled on selectedEAF. Each element in this part determines the priority of the relatedheat for scheduling, the second part determines the selected LF forthis heat, and the third, forth and fifth parts of the chromosomesshow the suitable waiting time before starting EAF, LF and CCM


Fig. 3.

Table 2Comparison of results

Algorithm Result Deviation from lower bound � tavg

Best Average Best (%) Average (%)

HANO −69037 −68282 5 6 389.04 302.05GA −66308 −64879 9 11 959.91 95.128Heuristic algorithm −52863 27 1.78

Fig. 4.

process of each heat. Then by applying the timing algorithm basedon SGS method that considers all the limitations, the start and endtimes of the heats are adjusted.

A fitness value of each individual is computed to represent thequality of the solution. GA operators (selection, crossover, and mu-tation) and their parameters are adjusted through solving several

instances. The stopping criterion of GA is the same as HANO. Thealgorithm is stopped when it cannot reach to better solution in 100sequential iterations.

6. Numerical results

We did not come across any problem as defined in this paper.Although there are different works cited in the literature, the differ-ences exist between the objective function and the limitations.

The efficiency of HANO is evaluated against a heuristic algorithmconventionally used at MSC to solve a real-life problem. In addi-tion, the proposed algorithm is compared with the designed GeneticAlgorithm through solving several instances.

For a real-life problem from MSC with eight EAFs, four LFs andfour CCMs, GA and HANO are run 10 times and the best result, theaverage and the standard deviations (�) of the results, the averagetime to reach the best solution (tavg) as well as the deviation fromestimated lower bound are shown in Table 2. The last row of thetable shows the result of the heuristic algorithm conventionally usedat MSC. The results are obtained using a Pentium 4 and CPU 1.33GHzPC.

The table shows that the average and the best result of HANO isbetter but it requires more average computational time in compari-son with the GA and the Heuristic algorithm.

The lower bound is provided for this real-life problem from MSC.For this problem the first stage (EAF) is the bottleneck of the shop


floor. So themaximumnumber of production can be calculated basedon this stage. By considering the minimum required transportationtime between stages and by assuming the number of casting inter-ruptions and waiting time to be zero the lower bound for this prob-lem can be obtained. The deviation of these three algorithms fromthe lower bound is shown in Table 2. The small deviation of HANOfrom the lower bound, not only shows the superiority of HANO thanGA and heuristic algorithm, but also confirms the efficiency of theproposed algorithm.

Also HANO is compared with the Genetic Algorithm through solv-ing several randomly generated instances. Comparison was basedon 200 test problems with 6 to 16 machines and 44 to 108 jobsthrough a bottleneck of different process stages. To study the per-formance of the algorithm, the ratio of the best solution's objectivefound through HANO to one found via GA (HANOobj/GAobj) is eval-uated. Considering that the objective function value is negative, if(HANOobj/GAobj) >1, then the performance of HANO is higher thanthat of GA.

Experimental results show that the efficiency of the proposedapproach is independent of the process bottleneck. In the 200 testproblems, the average of HANOobj/GAobjwas 1.59 and their standarddeviation was 1.02. In these instances, HANOobj/GAobj varied in therange 0.96 and 8.7 while this value was lower than 1 in only 3.5%of the instances. Fig. 4 shows the frequency of the results in therange. This figure verified the efficiency of the proposed approach.The abstracted results are given in Appendix B.

7. Conclusion

In this paper, the Steel-making continuous casting (SCC) schedul-ing problem is investigated. Since classic optimization methods failto obtain optimal solutions for this problem over a proper time, thispaper proposes a novel hybrid algorithm to find sufficiently good so-lutions. The proposed approach, Hybrid Ant Colony and Classic Op-timization methods (HANO), uses the advantages of meta-heuristicalgorithms and non-linear optimization methods simultaneously.Concurrent use of ACO and non-linear optimization methods, use ofefficient heuristic information to guide the ACO search, and prepa-ration of an initial solution for solving the non-linear model of thesecond phase are the main characteristics of the HANO.

The efficiency of HANO was evaluated against a heuristic algo-rithm used at MSC as a real-life problem. In addition, the proposedapproach was compared with a Genetic Algorithm through solvingseveral instances. Numerical results indicated the efficiency of theproposed approach compared with the mentioned heuristic algo-rithm. Furthermore, the efficiency of HANO as compared to that ofGA is shown to be independent of the process bottleneck. HANO hada better performance in more than 95% of the instances and had asimilar performance in the remaining 5%.

Table A1The average result for different levels of (�,�,�)

� = 0.002 � = 0.005 � = 0.1

� = 1 � = 2 � = 4 � = 1 � = 2 � = 4 � = 1 � = 2 � = 4

� = 1 −55542.2 −58436.3 −58378.2 −56650.2 −57933 −58117.8 −55189.5 −57943.7 −57088.7� = 2 −57112.8 −58149.7 −58137.8 −56896.2 −57325 −57240 −55938.2 −57707.2 −56430.7� = 3 −56764.2 −57644 −57210.3 −56506.8 −56916.3 −56650.3 −53987.3 −56615.2 −56289.8

Table A2Results of tuning L

L Best Average � iavg tavg tmean

0.125 −58569 −57914 310.52 73.87 75.6 181.730.25 −59952 −58344 805.29 119.8 122.58 224.30.5 −58121 −56863 1198.6 66.6 68.145 170.7

Acknowledgments

The authors would like to thank Dr. Gh. Moslehi and Dr. A.Z.Hamedani from Isfahan University of Technology, for their scientificadvice for this paper. The authors would also like to thank review-ers for their constructive comments, which helped to improve thequality of this paper.

Appendix A. Parameter tuning

Three main parameters (�,�,�).Three levels of three main parameters (�,�,�) of the ACO algo-

rithm and the average of six replications for each combination arereported in Table A1.

Analyzing results using 33 full factorial design byMatlab softwareshows that all three factors (�,�,�) are statistically significant. Alsotwo-factor interactions between (�,�) and (�,�) are significant. Two-factor interaction between (�,�) and also three-factor interaction arenot significant.

Duncan's multiple range test for (�,�,�):The Duncan's test shows that for factor (�) levels 1 and 2 (� = 1,

2) are the same and are significantly different from level 3 (� = 4).So level 1 is chosen for the algorithm. For factor (�) Levels 2 and3 are the same and they are significantly different from level 1. Solevel 2 (� = 2) is chosen for the algorithm. For factor (�) levels 1 and2 are the same and are significantly different from level 3. Level 2(� = 0.05) is chosen for the algorithm.

Step parameter (L): This parameter is varied in the proper rangewhile the other parameters are kept unchanged. For each value of L,the algorithm is run 10 times (each time the algorithm is continueduntil reaching to stopping criteria). The results of considering threedifferent values for L are given in Table A2.

NI = 10Best: the best solution of NI times running of the algorithmAverage: Average of the best solutions in NI times running of thealgorithm�: the standard deviation of the best solutions in NI times runningof the algorithmiavg: average of required iterations for reaching to the best solutiontavg: average of required time for reaching to the best solution (s)tmean: average of required time to reach to stopping criteria (s)

Table A3The result of tuning “A”

A Best Average � iavg tavg tmean

3 −59625 −57789 869.5 120.6 83.24 177.65 −59952 −58344 805.3 119.8 122.5 224.310 −60648 −58983 933.2 122 228.5 467.8


As it is shown in Table A2, the algorithm reach to better solutionsunder L = 0.25. So L = 0.25 is chosen for the algorithm.

A (ants number): The results of different values for A are given inTable A3.

Appendix B.

Prob. ID # EAF # LF # CCM Processing time (min) Result T-best (s) # Jobs

EAF LF CCM HANO GA Heuristicalgorithm

HANO GA

1 2 4 2 138 48 203 −14318 −9761 −8855 157.18 56.84 442 2 2 2 157 197 47 −11870 −10354 −9174 31.32 123.54 443 3 4 2 176 55 175 −16295 −10750 −9829 148.48 84.1 444 2 4 2 166 56 168 −15952 −12026 −10767 98.02 162.4 445 2 3 2 119 40 136 −17724 −14548 −12930 30.74 118.9 456 3 3 2 176 58 129 −17207 −13222 −11598 164.72 121.22 457 2 2 2 90 128 44 −15365 −11454 −10263 114.26 136.88 458 4 3 2 172 46 109 −19757 −13995 −12576 167.62 173.42 459 4 4 2 153 50 96 −21538 −15123 −13879 3.48 135.72 4610 2 4 2 90 191 56 −18640 −16096 −14416 88.74 150.8 4611 4 2 2 160 93 42 −20283 −20141 −17891 162.4 173.42 4612 5 2 2 151 54 87 −24914 −23842 −21493 85.84 98.6 4613 3 3 2 108 128 56 −23640 −21366 −18813 1.74 153.7 4714 7 4 2 290 59 58 −28808 −25341 −21171 120.64 127.6 4715 4 2 2 165 59 55 −23084 −15915 −12825 150.8 139.2 4716 3 4 2 111 45 80 −25181 −23012 −19848 128.76 101.5 4717 4 3 2 119 119 46 −20683 −12760 −11450 145 171.1 4818 5 2 2 157 77 55 −20210 −17293 −15783 157.76 168.78 4819 8 4 2 158 152 56 −22984 −14433 −13603 40.02 114.84 4820 8 2 2 177 76 58 −20860 −9853 −9223 169.36 122.96 4821 5 3 2 127 45 76 −29954 −30433 −27170 86.42 167.04 4922 2 4 2 75 43 51 −26545 −17005 −13451 13.92 132.24 4923 5 2 2 170 56 74 −23498 −17248 −15089 121.22 131.08 4924 3 2 2 108 42 54 −32858 −24386 −19930 135.72 140.94 4925 6 3 2 108 107 55 −20810 −2525 −2365 164.14 100.92 5026 6 2 2 214 42 49 −33188 −22894 −18866 151.96 149.06 5027 4 4 2 132 55 71 −26845 −24726 −21979 134.56 145.58 5028 7 2 2 171 47 71 −25737 −20268 −18552 100.92 168.78 5029 4 4 2 91 140 49 −26360 −22410 −19981 1.16 165.3 5130 2 2 2 70 52 56 −26926 −19682 −17028 46.98 143.84 5131 5 4 2 117 139 53 −24736 −19035 −17162 167.04 169.94 5132 6 4 2 165 136 40 −25194 −21841 −19840 127.6 135.72 5133 7 4 2 120 134 55 −24419 −14185 −13319 75.4 118.32 5234 6 4 2 200 58 57 −36564 −34332 −29134 170.52 100.34 5235 2 3 2 66 55 51 −32504 −23673 −20046 96.86 130.5 5236 4 2 2 84 49 66 −22407 −2646 −2377 106.14 121.22 5237 6 2 2 154 57 66 −26109 −20200 −18351 85.84 174 5338 5 2 2 145 64 53 −19530 −8671 −7447 13.34 96.28 5339 6 2 2 147 59 64 −26197 −24578 −22360 81.2 170.52 5340 5 2 2 83 63 57 −28176 −15740 −14548 4.06 144.42 5341 6 4 2 127 126 52 −25990 −17814 −15972 55.68 153.7 5442 6 2 2 82 43 63 −25644 −14146 −13380 157.18 163.56 5443 8 3 2 163 92 47 −28470 −20065 −18300 174 171.1 5444 5 4 2 115 122 41 −30811 −26816 −23332 1.74 145.58 5445 4 2 2 101 61 50 −24450 −17873 −16112 1.74 99.18 5446 5 4 2 151 46 46 −39078 −28130 −22241 94.54 171.68 5547 5 3 2 104 51 59 −34961 −39678 −35931 132.24 145.58 5548 4 4 2 116 56 40 −33239 −24677 −20320 80.62 107.3 5549 4 4 2 116 52 55 −28762 −14719 −12733 169.36 167.62 5550 4 4 2 116 57 55 −36608 −29124 −24522 148.48 147.9 5651 5 2 2 144 44 46 −33497 −26862 −21368 49.88 164.14 5652 4 4 2 115 46 49 −28329 −14792 −11920 1.16 158.92 5653 8 2 2 91 53 57 −35350 −29689 −28006 104.4 169.94 56

The algorithm reach to better solutions when A = 10 but becauseof the effect of this parameter on run time of the algorithm, A = 5 isselected.




HANO GA

54 8 2 2 224 46 51 −35756 −25494 −20735 174.58 171.68 5755 7 4 2 136 58 56 −35828 −37926 −34583 153.7 167.04 5756 8 3 2 148 54 56 −27123 −11107 −9759 168.2 142.1 5757 6 3 2 159 56 56 −34630 −37419 −31240 121.22 137.46 5758 3 4 2 83 53 41 −31633 −21773 −18276 57.42 116.58 5859 6 2 2 90 43 55 −30270 −21838 −20126 15.66 139.2 5860 6 2 2 164 45 49 −38684 −29192 −24535 1.16 171.1 5861 7 2 2 143 47 54 −35016 −28263 −24804 81.78 169.94 5862 4 4 2 107 41 52 −34864 −22830 −18674 34.22 158.34 5963 7 2 2 90 53 49 −25810 −10532 −9532 84.68 172.26 5964 6 4 2 158 58 44 −37430 −40024 −31792 153.7 146.74 5965 5 3 2 127 78 49 −24020 −7183 −6071 66.12 169.36 5966 6 4 2 155 52 50 −40718 −30985 −26558 130.5 169.94 6067 6 3 2 133 56 51 −25353 −20991 −18617 81.78 165.88 6068 2 4 2 50 53 43 −42174 −37918 −30194 112.52 58.58 6069 2 3 2 50 60 41 −35282 −35052 −29289 1.74 104.4 6070 6 2 2 96 43 50 −33177 −19931 −17883 143.26 143.84 6171 5 4 2 100 43 48 −27352 −12932 −11254 1.16 165.3 6172 8 3 2 95 57 47 −34163 −23222 −21595 77.72 174.58 6173 6 3 2 98 41 46 −31679 −15762 −13727 174.58 127.02 6174 5 4 2 105 58 44 −32464 −24964 −21012 118.9 158.92 6275 6 3 2 95 43 42 −37396 −28781 −25059 52.2 137.46 6276 6 3 2 102 41 39 −28117 −16024 −14154 139.2 171.1 6277 8 4 2 109 57 32 −36153 −30632 −25880 69.6 168.78 6378 7 4 2 93 59 31 −35742 −22225 −19186 48.14 167.62 6379 2 4 3 166 43 372 −12976 −9458 −8054 53.36 29 7180 2 2 3 167 238 55 −9330 −4519 −3985 51.04 163.56 7181 2 4 3 154 47 345 −12962 −5164 −4626 51.62 147.32 7182 2 2 3 176 41 329 −13002 −8567 −7153 172.84 172.26 7283 2 2 3 157 51 311 −13494 −8423 −7179 102.08 59.74 7284 2 2 3 171 51 305 −13665 −9728 −7961 1.16 96.86 7285 2 4 3 130 51 282 −15629 −9431 −8304 18.56 93.38 7386 2 3 3 113 45 249 −16549 −10528 −8998 71.34 62.64 7387 2 3 3 107 45 239 −16585 −10218 −8841 86.42 128.18 7388 2 3 3 104 223 53 −13579 −10750 −9436 104.98 87 7489 2 4 3 120 293 46 −11138 −10990 −9102 5.8 170.52 7490 2 2 3 141 57 219 −15898 −10703 −8722 174 141.52 7491 3 3 3 139 47 194 −18978 −15172 −12675 106.72 98.02 7592 4 2 3 176 43 192 −16929 −13703 −11648 56.26 153.12 7593 4 4 3 176 250 53 −14120 −12459 −10910 64.38 130.5 7594 2 4 3 96 45 182 −19093 −14858 −12790 80.62 56.26 7695 3 3 3 137 58 180 −19500 −14983 −12507 147.9 142.68 7696 2 2 3 88 116 54 −16447 −12140 −10384 53.94 78.3 7697 4 3 3 167 51 165 −21201 −18561 −15467 108.46 108.46 7798 3 2 3 105 103 47 −18721 −12706 −10894 157.76 160.66 7799 3 4 3 119 56 146 −23258 −24722 −20351 127.6 140.36 77100 4 3 3 166 56 126 −24202 −20439 −16930 146.16 71.34 78101 4 2 3 154 49 125 −23075 −14866 −11759 6.38 104.4 78102 4 4 3 147 164 44 −21610 −18374 −14827 144.42 165.88 78103 8 2 3 126 59 118 −22140 −12034 −11012 128.76 167.62 79104 8 2 3 172 56 116 −24486 −14740 −13058 160.08 140.36 79105 4 4 3 122 47 116 −25478 −20780 −17798 2.9 158.34 79106 3 2 3 111 55 47 −30915 −29461 −21530 88.74 153.12 80107 8 2 3 163 49 110 −24709 −13807 −11872 83.52 150.8 80108 7 2 3 121 52 105 −25630 −12454 −10971 45.24 168.78 80109 6 3 3 149 104 50 −26974 −16565 −13821 74.82 104.98 81110 4 4 3 106 44 99 −30904 −25706 −20756 125.28 145.58 81111 3 2 3 96 44 49 −34704 −24172 −17964 66.12 152.54 81112 6 4 3 158 126 48 −24648 −18467 −14958 72.5 157.76 82113 6 2 3 137 58 94 −22892 −19748 −17231 109.04 118.9 82114 2 2 3 59 43 49 −38607 −36160 −26636 172.84 67.86 82




HANO GA

115 7 2 3 158 58 57 −34895 −23388 −19404 161.82 134.56 83116 6 2 3 169 42 86 −29906 −26280 −20546 34.8 162.98 83117 6 4 3 154 48 86 −34892 −23579 −19074 132.24 166.46 83118 6 3 3 82 59 82 −31652 −19908 −17406 115.42 172.84 84119 6 3 3 158 56 80 −32759 −25097 −20866 31.32 164.72 84120 8 2 3 97 48 78 −26259 −4043 −3658 84.68 60.9 84121 7 2 3 141 49 75 −26315 −16540 −13773 148.48 159.5 85122 5 3 3 123 58 49 −32913 −28636 −20505 135.14 168.2 85123 5 3 3 112 57 46 −31130 −22722 −17560 6.38 138.04 85124 3 3 3 66 52 53 −40445 −30064 −22430 142.68 160.66 86125 8 4 3 110 87 54 −31330 −8214 −7084 123.54 174.58 86126 7 4 3 137 53 62 −44432 −35104 −28798 1.74 157.18 86127 7 3 3 144 48 51 −35764 −11355 −8348 156.6 172.26 87128 4 4 3 82 56 48 −43058 −36955 −29042 141.52 166.46 87129 5 3 3 102 57 59 −44996 −38326 −31399 172.26 172.26 87130 5 3 3 84 45 61 −38129 −25397 −20630 80.62 172.26 88131 7 4 3 119 80 48 −34139 −13670 −10884 133.98 166.46 88132 7 3 3 91 53 60 −32557 −4060 −3350 104.98 153.7 89133 8 3 3 137 51 44 −33691 −16259 −12236 72.5 171.1 89134 2 3 4 140 303 50 −9573 −8766 −7507 2.32 66.12 90135 2 2 4 167 46 392 −14757 −7477 −5923 6.38 57.42 90136 2 4 4 168 52 340 −14674 −11690 −9114 131.66 162.98 90137 2 3 4 117 253 59 −11721 −9409 −7737 26.1 146.74 91138 2 4 4 152 54 324 −16511 −13745 −10812 168.78 63.8 91139 3 4 4 171 55 300 −16700 −15133 −12406 6.38 64.96 91140 2 4 4 102 45 296 −17049 −12343 −10545 129.92 148.48 92141 3 2 4 166 41 284 −16866 −7233 −5891 151.96 140.94 92142 2 2 4 111 49 235 −18064 −13624 −10332 55.68 124.12 92143 3 3 4 120 49 208 −19963 −13305 −10540 139.78 136.3 93144 2 3 4 101 53 204 −19328 −16907 −12639 172.26 121.8 93145 2 4 4 82 58 202 −20473 −10371 −8241 95.12 80.04 93146 2 3 4 90 60 201 −20445 −16097 −12300 91.64 163.56 94147 2 3 4 88 55 192 −20194 −22544 −17698 164.72 110.78 94148 4 4 4 132 191 58 −17017 −12231 −9878 130.5 116.58 94149 4 3 4 132 44 187 −22025 −21729 −18157 12.18 133.98 95150 7 2 4 160 58 174 −21091 −8536 −7215 59.16 161.82 95151 4 2 4 143 41 172 −20532 −5518 −4344 145 168.78 95152 7 2 4 286 55 57 −26747 −28027 −20782 73.66 154.28 96153 5 3 4 154 119 56 −22998 −16369 −13432 154.86 128.76 96154 7 2 4 125 53 155 −22283 −6814 −5918 116 163.56 96155 4 3 4 146 53 152 −25331 −20752 −16595 107.88 136.88 97156 5 3 4 170 52 149 −28896 −22293 −17697 135.14 137.46 97157 2 2 4 72 54 49 −33184 −27417 −18598 136.3 120.64 97158 4 4 4 94 138 47 −24414 −20027 −16522 37.7 172.26 98159 6 2 4 144 60 137 −23730 v11329 −9050 114.84 154.86 98160 6 2 4 140 41 135 −26428 −20919 −17695 41.76 164.14 98161 7 4 4 173 128 59 −27002 −17418 −14448 22.04 170.52 99162 2 2 4 63 43 44 −38305 −26623 −18223 158.34 111.94 99163 6 2 4 104 50 126 −24680 −10716 −9240 90.48 161.82 99164 7 2 4 219 59 58 −34442 −36531 −26803 69.6 136.88 100165 5 4 4 121 125 44 −26244 −20047 −16336 64.38 172.26 100166 8 2 4 246 49 46 −33448 −30377 −20803 57.42 172.84 100167 5 2 4 85 45 123 −26160 −10851 −9110 46.4 153.7 101168 5 4 4 137 122 42 −26698 −27043 −21739 12.76 147.9 101169 5 3 4 141 91 54 −27991 −26014 −19352 69.02 171.68 101170 7 3 4 126 60 120 −35059 −22489 −18827 42.92 174 102171 5 3 4 122 51 117 −31974 −17273 −13823 116.58 160.66 102172 7 3 4 198 60 56 −34049 −24906 −16674 96.86 174 102173 6 4 4 112 112 59 −30631 −23811 −19851 146.16 172.26 102174 7 2 4 196 48 46 −32370 −26335 −20356 84.68 173.42 103175 3 2 4 83 46 53 −37699 −32468 −23813 24.36 174 103




HANO GA

176 5 2 4 97 55 57 −37050 −30133 −24132 3.48 160.08 103177 8 3 4 97 55 109 −34233 −15496 −13248 142.1 172.84 104178 7 4 4 171 107 55 −31765 −31100 −23067 69.02 156.6 104179 8 4 4 172 50 107 −42165 −28215 −22293 1.74 165.88 104180 3 3 4 80 53 45 −38394 −34153 −23947 9.28 143.84 105181 6 2 4 132 45 105 −32375 −15725 −12941 106.72 174 105182 8 3 4 148 78 51 −30394 −22392 −18096 24.36 172.84 105183 8 3 4 148 44 104 −37294 −30686 −24798 152.54 169.94 105184 8 3 4 202 52 52 −44120 −45982 −30846 173.42 174 106185 6 4 4 133 59 100 −40102 −29133 −22043 2.32 161.82 106186 8 2 4 112 41 89 −34802 −16300 −13449 13.92 155.44 106187 5 4 4 111 49 59 −52964 −40996 −25865 161.24 151.96 106188 2 3 4 44 46 58 −44478 −32329 −21795 172.84 110.2 106189 7 4 4 89 59 88 −45725 −36758 −30721 2.32 174 107190 8 4 4 104 82 58 −36154 −22313 −18609 146.16 172.84 107191 7 3 4 83 46 81 −39327 −30234 −25540 131.08 171.68 107192 8 4 4 116 49 80 −46257 −37661 −29321 164.14 174 107193 2 4 4 39 59 50 −40171 −29059 −21052 30.74 168.2 107194 3 4 4 57 51 47 −48021 −39775 −27761 3.48 168.2 108195 3 4 4 54 59 54 −44540 −40537 −30659 11.6 158.92 108196 7 4 4 123 50 50 −67475 −67423 −46649 167.04 130.5 108197 6 4 4 101 55 49 −53191 −44385 −30558 16.82 168.78 108198 5 4 4 83 50 54 −64801 −51262 −37774 3.48 171.68 108199 2 4 4 27 52 44 −45604 −44221 −31956 150.22 173.42 108200 8 4 4 89 49 42 −50912 −38932 −28634 87 174.58 108

# EAF: number of EAFs; # LF: number of LFs; # CCM: number of CCMs; processing time: the processing time of each machine (min); T-best:the required time to reach to best result (s); # jobs: number of jobs.

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A novel hybrid algorithm for scheduling steel-making continuous casting production

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