LR-Based Integrated Model for Scheduling Steelmaking-Continuous Casting-Hot Rolling*

ELSEVIER

Copyright © IFAC New Technologies for Automation of Metallurgical Industry, Shanghai, P.R. China, 2003

IFAC PUBUCATIONS www.clsevier.comllocatclifac

LR-BASED INTEGRATED MODEL FOR SCHEDULING STEELMAKING-CONTINUOUS CASTING-HOT ROLLING· Baolin Zbu1,l,

Haibin Yu'

I.Shenyang Institute 0/Automation, Chinese Academy o/Sciences, 110016 PR. China 2. College 0/Business Administration, Northeast University. 110006 P R. China Telephone: (86- 24) 23970100 - 3338 E-mail.·[email protected]

Abstract: With the development of production technologies in iron and steel enterprises, integrated production technology comes into use for steelmaking and continuous casting as well as hot rolling. In the paper, the limitation of the traditional existing planning and scheduling methods which separately consider the process of steelmaking-continuous casting or hot rolling is broken through. A scheduling model is presented to integrate the three working procedures, namely, steelmaking, continuous casting and continuous rolling. Then lagrangian relaxation algorithm is proposed to get the optimal solution of the scheduling model. Finally, numerical examples are given to demonstrate the effectiveness of the integrated model and optimization methods. Copyright©2003 IFAC Keywords: Lagrangian Relaxation, Steelmaking, Continuous Casting, Continuous Rolling

steel scheduling problems have been studied through typical mathematical mode ling, such as travel salesmen problem (TSP), Bin-packing problem, Shortest-path, Knapsack problem, have been proved to be NP-hard. Presently, many researches are being focused on the modeling of steelmaking -continuous casting (SCC) and hot rolling (HR). Solving operational management problems for SCC and HR production have been widely reported. Mentioned especially, Lixin Tang et al.developed a mathematical programming model, orientated on TIT idea, for solving machines conflicts in SCC production scheduling in the CIMS environment, and gave the mathematical representation of the model. Xiong Chen et al. put forward a mathematical model to HR scheduling problem of vehicle routing with time window to describe the rolling batch planning, and solved it using genetic algorithm (GA). Lixin Tang et al. proposed a mUltiple traveling salesmen problem (MTSP) for rolling scheduling using the parallel

1. Introduction For energy saving, newly emerged production technologies, such as hot-charging rolling, directcharge rolling, direct-rolling, etc. make it is possible for the three working procedures (steelmaking, continuous casting, continuous rolling) to be closely linked in primary steelmaking, and turn the iron and steel production into an integrated system . Unlike general production scheduling in machine industry, iron and steel production scheduling problems have to meet special requirements of steel production process. There are extremely strict require -ments on material continuity and flow time, e.g. processing time on various devices, transportation and waiting time between operations. Many iron and

-This paper is supported by NSF (No. 69974039, 5999(470)

\07

strategy, and solved it with GA too. To the best of our knowledge,up to now, the researches on mathematical programming model for SCR in literature have been rarely reported. This paper breaks through the limit of ~he fonner planning and scheduling which only consider SCC or HR procedure separately. For the first time, an integrated production scheduling model for SCR is presented here based on batch planning, and then apply lagrangian relaxation (LR) to SCR model to get the optimal scheduling solution. Finally, numerical examples are given to demonstrate the effectiveness of the integrated SCR scheduling model and the optimization algorithm.

established model. After that, synthesized the batch planning of charges, continuous casting slabs, rolling units, and made the continuous casting slabs to be a batch planning with respect to charges and rolling units, so that made the working procedures closely connected, and eventually formed integrated production scheduling of SCR. 3. SCR Integrated Model

The Steelmaking, continuous casting, hot rolling procedure in iron and steel enterprises are well known as the three big working procedures among which exits not long the balances of material flow and resource, but also that of time. Under conditions of market economy, for solving the problem of integrated production scheduling, it should start from the customers' contracts, and establish the manufacturing plan which satisfy the customers' demand and the constraints in the working procedures. In traditional iron and steel production, three big working procedures are independent to each other, the interaction among them is very slight, which outcomes that production plans are established and conducted individually, moreover, there are a lot of semi-manufactured goods in the working procedures used as middle inventory. From the above statements, it can be seen that traditional production management is only applicable to a certain working procedure, and less considered to the up and/or down working procedures, so that only achieve local or partial optimization for a single segment or working procedure.

The remainder of this paper is organized as follows. Section 2 introduces the relevant technologies about SCR. In section 3, based on the SCC, HR and the relation of Charge -Slabs-Rolling unit Batch Planning models, a SCR integrated production scheduling model is established, then LR are explored to compute the SCR model. The computational results and analyses are given in section 4. Finally, conclusions are drawn in section 5. 2.Background The continuous casting-continuous rolling (CCR) is a novel production manner developed from the integration of newest technologies of SCR. The development of CCR production dramatically promotes the revolution in technology and management, at the same time, raises higher level demand to the production management technologies, especially to planning and scheduling. The CCR production can be categorized into the following kinds: CC-HCR (Continuous casting-Hot Charge Rolling), CC-DHCR (Continuous casting-Direct Hot Charge Rolling), CC-DR (Continuous casting- Direct Rolling), CC-CCR(Continuous casting-Cold Charge Rolling).

3.1 SCC Model Description Generally speaking, the connection between SCC mainly concerns how to raise the amount of charges in the casts. Most modem iron and steel enterprises put steelmaking procedure and continuous casting procedure together into the same production area. This action is lucrative whether to process control or to the production scheduling. The continuous casting procedure is strictly concerned with ingredient, temperature and arrive time of molten steel. How to coordinate the procedures to minimize the conflicts of equipments and the temperature loss of molten steel between steelmaking and continuous is the main goal in SCC scheduling.

In the production of CCR, especially in CC-DHCR and CC-DR, direct connection of three big working procedures, steelmaking, casting, rolling, are realized. Facing this highly integrated continuous productive system, traditional management schemes are no long practicable, novel management, namely computer based integrated production management, must be taken. With regard to CC-DHCR or CC-DR, this paper established an integrated scheduling model, analyzed the manufacturing technologies of SCR, considered the problems exist in manufacturing technologies and working procedures, such as continuous production, just-in-time (nT) delivery and so on, and introduces JIT to objective function. In the establishment of scheduling, we build up an integrated production scheduling model based on batch planning of charge, cast, rolling unit, and then, ordered and scheduled the rolling units on the rolling machine respectively. Next, eliminated the machine collision by optimizing the

The

SCC

production

scheduling

is time-table

planning of charges based on batch planning. The model is developed under the following assumptions, which are made based on practical requirements in steelmaking-continuous casting production scheduling. 1) Batch plans (charge, cast) have been established.

2)The sequence of sub-schedules remains unchanged. 3) At most one charge can be processed on a machine

108

at a time.

possible.

Using the above assumptions, the model for the optimal scheduling problem is formulated as follows:

(b) Molten steel temperature drop cost in terms of

M

L

M in ~ = N

L L

(3) (4)

3.2 HR Model Description

Jer,J1'(iJ'fer.

XS1(iJ)J

-xi} ~Ty

Xi ,SP(i , J) -

(i

(1)

~(Xm..iJ) -Xi} -I; -tj J1'(iJ))

+

j.4

The following constraints are considered in the model to guarantee that there will be no machine conflicts in the schedule generated. Constraint (2) for the two contiguous operations, for the same charge, only when the preceding operation has been completed, the immediately next one can be started. Constraint (3) for two contiguous charge processed on the same machine, only when the preceding charge has been completed, the immediately next one can be started. Constraint (4) setup time and interval are required from cast to cast on the same caster.

dtC~(iJ)j -Xi} -I;)

~~Ja;~(iJ~

Ell',

xi} ~ Ti}

j e

XSi(i .j).J - Xij

(ie""j e1r;,SI(i,})e",)

1t i ' SP

(2)

+ t j ,SP(i , j) (i, j)

E 1t

J

~ Ty + SSl(i.j~j + f.J

(i E 'If.,j E 1rJ. . ~,SI(i,j) E 'If p) k,p = 1,2, .. M,k Xi} ~O

(i e ""j e tr i )

waiting time from operation to operation.

* p)

Generally, Hot Rolling working procedure concerns how to increase the number of slabs in the rolling units. Hot rolling puts forward strict demand on the temperature, width, gauge and hardness and arrival time of continuous casting slabs. How to coordinate the procedures to minimize the conflicts of equipments and the temperature loss of slabs between continuous casters and hot rollers is the main goal in hot rolling scheduling. The HR production scheduling is time-table planning of slabs based on batch planning.

(5)

'" : the set of all charges. '" = { 1, 2, ... N} . N is the total number of production charges. '"k: the kth rolling set of charges, M is the total of caster,

ke{l,2,~; j:# k , j,k e {1, .. M},

"'j (\ "'k = tP

and '" 1 U '" 2 ... U '" M = "'; tr: the set of all machines.

1ti

:

the set of all machines used for the ith The model is developed under the following assumptions, which are made based on practical requirements in hot rolling production scheduling .

charge. ~: the set of all continuous casters, ~ C 1t;

SI (i, j) : the immediate successor charge i processed on machine j. SP(i,j): the immediate

4) Batch plans (rolling unit) have been established.

success or machine of machine j for processing charge i.

5)The sequence of sub-schedules remains unchanged. 6) At most one slab can be processed on a machine at a time.

elle: coefficient of continuous casting break loss penalty for continuous casting 1. cri}: coefficient of

Using the above assumptions, the model for the optimal scheduling problem is formulated as follows:

penalty cost for the waiting time of charge i after finishing process on machine j.

'F;j:

processing

time of charge i on machine j . t jm: transportation time from machine j to machine m. f.J : interval time from caster to caster. S Sl(i,j),j : setup time of caster k

L

N{

+

on machine j. xi}: starting time of charge i on

L j~.
machinej.

+~(~ +P,1;) j9r,

The objective function (1) for this model is to ensure continuity of the production process. This is achieved through minimizing a cost function consisting of the following terms: (a) Casting break loss penalties exerted to ensure that charges in the same caster are casting continuously as

109

a:(~;'<;(iJ) -~ -Ty -~.
(6)

b> Xij_

° (.

b) IE'I/ b· ,JE1[;

'l/b : the set of all slabs. 'l/b

= {I, 2, ...N b}.

(10)

The objective function (6) for this model is to ensure

Nb is

continuity of the production process and JIT delivery of fmal

the total number of production slabs. 0/: the Ith rolling set of slabs, L is the total of rolling unit,

lE {I, ...L}. j"# 1; j, 1 E {1 , ... L} , 0j n 0, et>

, and 01 U 02 •• •

°=

the set of all machines used for the

ith slab.

W:

the set of all continuous rollers,

W C 1[,

Iwl = D . D;:

is

achieved

through

(c) Rolling break loss penalties exerted to ensure that

L 'l/b . 1[b: the set of all

1[; b :

This

following terms:

=

machines.

products.

minimizing a cost function consisting of the

slabs in the same rolling unit are rolling unit as continuously as possible.

Due date of slab i. C; :

(d) Slabs temperature drop cost in terms of waiting

completion time of slab i. SIb (i, j) : the immediate

time from operation to operation.

successor slab of slab i processed on machine j . Spb(i,j): the immediate successor machine of

(e) Earliness/tardiness penalty used to ensure that each slab is delivered as punctually as possible.

machine j for processing slab i.

The following constraints are considered in the model to guarantee that there will be no machine conflicts in the schedule generated. Constraint (7) for the two contiguous operations for the same slab, only when the preceding operation has been completed, the immediately next one can be started. Constraint (8) for two contiguous slabs processed on the same machine, only when the preceding slab has been completed, the immediately next one can be started. Constraint (9) setup time and interval are required from rolling unit to unit on the same rolling unit.

rl/ : coefficient of rolling break loss penalty for rolling 1.

er; : coefficient

of penalty cost for the

waiting time of slab i after finishing process on machine j . E;: Earliness of slab i, defined as:

nir(o,xij +Ty -.Q) and j

E 1[; ( l W

.1'; :

tardiness

of slab i , defined as: max(O,xij +Ty -.Q) and

j

E 1[; ( l W.

of slab i. j.

a;, p; : weight

T; :

processing

(value or important)

tJ,SP(;,j):

transportation time from machine j to

machine m.

,i: interval time · from rolling unit to

rolling unit.

S~( I,J . .)b ,J. : setup time of or rolling 1 on

machine j.

3.3 SCR Integrated Production Scheduling Model

time of slab i on machine

In SCC scheduling model, charge is the minimal job. However, in the HR scheduling model, slab is the minimal job. The relation of rolling unit, charge and slab is presented in Fig.I.

X!: starting time of slab i on machine j .

.

------------------, Slab 1

SCC

,,

HR Slab 2

-".'

Scheduling Model

Scheduling Model

(Slab)

(Charge)

I

SlabN

Fig.I . The relation of rolling unit, charge and slab

The SCR integrated production scheduling is a special multi-jobs, multi-machines, multi -procedures job shop scheduling based on batch plarming. Based on SCC, HR and the relation of Charge- Slabs -Rolling unit Batch Planning, the model for the optimal SCR is formulated as follows: Min(Z) (11)

s.t : (2), 0), (4), (5), (7), (8), (9), Where Z

(IQ)

= Z\ + Z3 4. Lagrangian Relaxation for SCR

Recently, scheduling methodologies based on Iagrangian relaxation have been proven to be

110

computationally efficient and have provided near optimal solutions to scheduling problems.

two continuous casters (CC) and two continuous rollers (CR). Consider eight charges, four rolling wits and forty slabs. Basic model parameters are given in Table I. The processing time of slabs are given at random.

4.1 Lagrangian Relaxation

The constraint (4), (9) can be relaxed by using the nonnegative Lagrangian Relaxation multipliers 1f/CP'

A,g:

CF

cc

RF

(12)

MinZ={Z+

L

7

5

1Zi,.(lt +T, +Ss(iJli +,u -l":v(iJli )

CR

2

i<¥\p,,", S(iJ'w,Ap4.2. ..,Mpp

8

CF

subject to

CR

(2), (3), (5), (7), (S), (IO) Fig.2. SCR Process Routes

The Lagrangian Relaxation dual to problem (12)

Max {Min (Z ')}

tr"" ,.1.11 ~o

subjectto

(13)

x.

'" ={1,2 .. . S}, '1'1 ={ 1,2}, "'2 ={3,4},

{7,S} . 1f = {1,2,3,4,5,6}, 1f1 = {1,3,5}, 1f2 =

(2), (3), (5), (7), (S), (IO)

{2,4,5},1f3 = {1,3,6},1f4 = {2,4,6}, 1fs = {l,3,5},1f6

The minimization operation in (13) has been brought inside the summation because the minimum of the sum is the sum of the minima when the cast (or rolling unit) are independent. This results in a minimization sub-problem for each cast, rolling unit and HT for charge and slab.

= {2,4,5},1f7 = {1,3,6},1fs = {2,4,6}; ~ =(5,6); N = S,

",b ={1,2 .. .40},

Z·.

The difference between

(J)

= {7,S};

(ie~jen:,

=

SSJ(;,j)

= P

=5

SI(i,j) e '1', SP(i,j) e 1f;).

° b

(j~(;,j)

=

Sb SIb (; ,j)

=

P

b

= 3

sf(i,])er/, S?(i,j)e1f: ),

Nb =40, L=4. Table. I The 'relation of rolling unit, charge and slab Rolling unit

slab

charge

Due date

I

1-5

I

130

6-10

2

120

11-15

3

75

16-20

4

60

21-25

5

90

26-30

6

100

31-35

7

lOO

36-40

8

95

2

3

Z·· and 4

Z· is known as the duality gap, an upper bound of

.'

(jsp(; ,j)

1 ={1,2 ... 10}, 02 ={11,12 ... 20}, 0 3 ={21,22 ... 30}, 04 ={31,32 .. .40}; 1fb = {7,S};

The value of the dual function Z·' is a lower

bound on

M = 4.

(i e ""j e 1f;,

The sub-problems can be solved by common linear programming and dynamic programming etc. We can obtain a near optimal schedule with quantifiable performance and within reasonable computation time. To solve the dual problem, the sub-gradient method is commonly used to solve the problem. Because of the stopping criterion used. In general, the solution to the dual problem is associated with an infeasible schedule, because some constraints can be violated. To construct a feasible schedule, the list-scheduling is then applied. In this list-scheduling procedure, a list is created by arranging slabs in certain order. If the constraint is violated, a greedy heuristic algorithm determines which slab should begin and which should be delayed. Once a feasible schedule is obtained, the corresponding value of the objective function Z is an upper bound on the optimal objective

Z·,

"'3 ={5,6}, "'4 =

.'

the duality gap is provided by ( Z - Z ) / Z , which is a measure of the sub-optimality of the feasible schedule.

First, the SCC and HR production model were solved using common programming methods (linear and dynamical programming) respectively. ZI =307,

4.2 An Example

Z2 =261, Z =56S. Fig.2 presents an example of the SCR integrated production scheduling. In this example there are two converter furnaces (CF), two refining furnaces (RF),

The SCR integrated production model for eliminating

1ll

machine conflicts in this example was built using the formulation presented in the last sections. The simulation is carried out on Pill800, visual C++6.0 by using LR. All the multipliers were initialized to zero. The lower bound is obtained 440, and the feasible schedule has a cost Z =499 with a relative duality gap of 13.2%. 5. Conclusion This paper studied the SCR scheduling problem. Based on the SCC and HR scheduling models, an integrated production scheduling model was established which analyzed the production technologies of working procedures. Complying with practical cases, the integrated model considered the problems exist in production technology and working procedures, such as continuous manufacturing, just-in-time delivery of orders, etc. To solve the integrated scheduling model, the LR algorithm is proposed to get the optimal solution of the scheduling model. Numerical examples were given to demonstrate the effectiveness of the integrated model and optimization methods.

RERERENCES Lixin Tang, Jiyin Liu, Zihou Yang, A mathematical programming model for scheduling steelmaking continuous casting production (2000). European Journal of Operational Research. Xiong Chen and Xinhe Xu, Modeling rolling batch planning as vehicle routing problem with time windows( 1998). Computer Ops Res. Lixin Tang, Jiyin Liu, Aiying Rong, Zihou Yang. A multiple traveling salesman problem model for hot rolling scheduling in Shanghai Baogang Iron & Steel Complex (2000). European Journal of

Operational Research M. L. Fisher, Optimal Solution of Scheduling Problems Using Lagrange Multipliers, Part I (1973), Operation Res. M. L. Fisher, Lagrangian Relaxation method for solving integer programming problems (1981). Management Science. Hoitomp DJ, Peter, B.Luh., A practical approach to job-shop scheduling problems (1993). IEEE Trans. Robotics and Automation. Peter,B.Luh,

Dong

Chen

,and

Lakshman. An

effective approach for job-shop scheduling with uncertain processing requirements (1999). IEEE Trans.Robotics and Automation.

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