철강공정의 일정계획: I. 수학적 모델링
문성득, 이동엽, 박선원*
*한국과학기술원 화학공학과
Scheduling of Primary Steelmaking Processes
Part I: Mathematical Modeling
Sungdeuk Moon, Dong-Yup Lee and Sunwon Park*
*Dept. of Chem. Eng., KAIST
Introduction
The overall steelmaking process can be classified into three stages: ironmaking, primary
steelmaking and finishing. Since many literatures have described detail operations for each
steelmaking process, attention will be focused on primary steelmaking for the purpose of this
paper. Scheduling in the primary phase of steelmaking is concerned with the following issues
(Lee et al., 1996): utilization of manufacturing units, allocation of production among parallel
manufacturing units, specification of batches, specification of rolling groups, sequencing of
batches/rolling groups for manufacturing units, coordination of schedules between production
stages, and rescheduling.
From the scheduling point of view, the primary steelmaking process is a sequential
multipurpose process with parallel equipment (Voudouris and Grossmann, 1996; Moon and
Hrymak, 1999). Moon and Hrymak (1999) proposed a mathematical model for scheduling
the special case of multipurpose batch plants in which the production paths of all products
have the same processing direction, namely sequential multipurpose batch plants. In
particular, the scheduling problems of the multipurpose batch plants under a mixed product
campaign (MPC) were addressed. The idea of this model is based on the definition of an
event slot, which is used to determine a production sequence. In this paper, a novel
continuous time formulation for scheduling of the primary steelmaking process is proposed.
The model is tailored to accommodate batch and continuous operation modes where the
scheduling problems in steelmaking industry have been focused. The mathematical model
was based on the work of Moon and Hrymak where a new continuous time formulation was
presented to effectively address the problem of short-term scheduling in sequential
multipurpose batch plants with MPC.
Motivating Example
Figure 1 shows a motivating example that a primary steelmaking process consists of two
BOFs, two LMFs (RH and AP types), two CCs, a HPR, and a HSM. In this example, hot
iron and scrap are fed into a BOF to generate an amount of molten steel. In this case we
BOF1
LMF1 (RH)
CC1
20%
RF
HPR
80%
Plate
Scrap
Molten
Steel
Hot iron
BOF2
20%
Refined
Steel
LMF2 (AP)
CC2
Slab
RF
HSM
80%
Hot rolled coil
Fig 1. Motivating example for a primary steelmaking process.
Table 1. Processing data for motivating example
Equipment
Prod.
BOF1
(unit 1)
BOF2
(unit 2)
LMF1
(unit 3)
LMF2
(unit 4)
CC1
(unit 5)
A
8.0
8.0
10.0
-
10.0
10.0
32.0
18.0
B
8.0
8.0
-
11.0
10.0
10.0
32.0
18.0
A
-
-
-
-
-
-
250.0
400.0
B
-
-
-
-
-
-
250.0
400.0
Processing
Time
(sec/ton)
Demand of
Product
(ton)
CC2
HPR
HSM
(unit 6) (unit 7) (unit 8)
assumed that the molten steel consists of 80 % hot iron and 20 % scrap. The molten steel is
moved to an LMF by a ladle. After a metallurgical treatment at LMF, the ladle moves to a CC
to make solid products, slabs. Finally, plates and coils are manufactured at proper rolling
mills, HPR and HSM, by consuming slabs. We assumed in this paper all slabs except the
once to be sold should move through reheat furnace before entering each rolling mill, as
shown in Figure 1. Products A and B, two different metallurgical grades, are manufactured
by treatment through LMF1 (RH type) and LMF2 (AP type), respectively. In addition, three
different shapes, slab, plate, and hot rolled coil, for each product are produced. Table 1 lists
process data for the motivating example. Slabs can be considered as semifinal products since
some of those produced are used to manufacture the final products, plates and coils, in the
primary steelmaking process and the rest is directly delivered to customers. Hence slabs have
no market requirement but a specific price in the market although plates and coils have
market demands, as shown in this table.
Mathematical Model
In the case when product i is assigned to event slot k, a production path r corresponding to
product i should be simultaneously allocated to the same slot. In order to assign product i
and production path r to time slot k, integer variables X ikr is defined as
1, if product i is operated with production path r in event slot k
X ikr  
0, otherwise

By using this concept, a mathematical model is formulated as the following objective
function and constraints:
Objective: Minimization of Makespan, MS
MS  Tekj
(1)
 k  {K}, j  {CC1, CC2, HPR, HSM}
Constraints:
X
ikr
 ni
 iI
(2)
X
ikr
1
 kK
(3)
 iI
 k K , j  J
(4)
 iI , k K , j  J
(6)
k  K r  Ri
i I r  R i
ni  N i
Tekj  H
Va
min
X
rRi
Vaikj 
ikr
Vy
s S i
Vaikj  Va
isk
max
X
rRi
ikr
 iI , k K , j  {CC1, CC2 }
(5)
(7)




 Va max 1   X ikr   Vaikj ' Vaikj  Va max 1   X ikr 
 r R  R 
 r R  R 
ij
ij '
ij
ij '




 iI , k K , j, j ' J , j '  j


(8)
Tekj  Tskj  Vaikj PTij
 k K , j  J  {HPR, HSM}
(10)
Tekj  Tskj  Vyisk PTij
 k  K , j  {HPR, HSM }, s  S j
(11)
 iI , s Si
(12)
 k K , j, j ' J , j '  j
(13)
 k K , j  {CC1, CC2} , j '  {HPR, HSM}
(14)
iI
Vy
k K
isk
iI
 REQis
Tskj'  Tekj   U (1 
X
iI rRij  Rij '
ikr
)
Tskj'  Tskj    X ikr PTij  j '
iI rRij
Ts k 1, j  Tekj  0
Ts k 1, j  Tekj  0
Vysmin  Vyisk  Vysmax
 k K , j  J  CC1, CC2 
 k K , j  CC1, CC2 
 iI , k K , s  {platei , coili }
(15)
(16)
(17)
Results for Scheduling of the Motivating Example
For the motivating example, an MILP solver, CPLEX 6.5 through GAMS (Brooke et al.,
1998) is used to solve the scheduling problems for the primary steelmaking process. In order
to verify the scheduling solution for the primary steelmaking process, we assumed that the
optimal solution is a solution within the integrality gap of 10E-6.
Figure 2 shows the optimal sequence with minimum makespan for the motivating example
with 6 batches to be processed. The optimal solution corresponds to the value of X ikr
variables as illustrated in Figure 2(a). At first event slot (k = 1), for example, Product A is
assigned and three units, BOF1, LMF1, and CC1 (defined as Production path 1 of Product A
in this paper; r = 1) are used to manufacture one batch of Product A. In order to satisfy the
production requirement, 3 batches of Product A and 3 of B are produced in this problem.
Figure 2(b) shows a Gantt chart for the optimal solution. As shown in this figure, CC1 was
used just one time at the first slot. On the other hand, CC2 was used for manufacturing. This
is because of the operating restriction of CC, which there is no idle time between the
consecutive operations of a given CC.
Conclusions
In this paper, a novel continuous time formulation for scheduling of the primary
steelmaking process is proposed. The model is tailored to accommodate batch and
continuous operation modes where the scheduling problems in steelmaking industry have
been focused. The mathematical model was presented to effectively address the problem of
short-term scheduling in sequential multipurpose batch plants with mixed production
campaign (MPC).
Acknowledgement
This work was supported in part by the Korea Science and Engineering Foundation
(KOSEF) through the Automation Research Center at Pohang University of Science and
Technology (POSTECH)
References
Brooke, A.; Kendrich, D.; Meeraus, A.; Raman R. GAMS-A user’s guide; GAMS
Development Corporation: Washington, NY, 1998.
Lee, H. S.; Murthy, S. S.; Haider, S. W.; Morse, D. V. Primary Production Scheduling at
Steelmaking Industries. IBM J. Res. Develop. 1996, 40, 231.
Moon, S.; Hrymak, A. N. A Novel MILP Model for Short Term Scheduling of a Special
Class of Multipurpose Batch Plants. Ind. Eng. Chem. Res. 1999, 38, 2144.
Voudouris, V. T.; Grossmann, I. E. MILP Model for Scheduling and Design of a Special
Class of Multipurpose Batch Plants. Computers Chem. Engng. 1996, 20, 1335.
Time Slot
Unit
slot 1
0.0
slot 2
slot 3
0.67
0.67
slot 5
slot 4
1.40
1.40
2.07
2.07
2.73
slot 6
2.73
3.40
BOF1
0.0
0.67
BOF2
LMF1
0.67
1.50
1.50
0.67
1.58
1.58
2.42
2.42
2.73
2.07
3.57
2.98
3.40
4.32
5.00
5.83
LMF2
CC1
1.50
2.33
2.42
3.33
3.33
4.17
4.17
5.00
CC2
1.58
2.37
1.61
2.33
2.37
3.43
3.43
4.61
4.61
4.87
4.87
5.14
5.14
6.03
HPR
HSM
2.33
3.08
3.08
3.91
Prod. A
(a)
4.33
5.08
5.08
5.53
5.53
6.03
Prod. B
Unit
0.67
BOF1
k1
1.40
k3
2.07
k4
2.73
3.40
k5
Prod. A
k6
Prod. B
0.67
BOF2
k2
0.67
1.50
k1
LMF1
0.67
1.58
LMF2
2.42
3.57
k5
2.07
k2
2.98
3.40
k4
1.50
4.32
k6
2.33
k1
CC1
1.58
CC2
2.42
k2
1.58
HPR
3.33
1
2
3.91
k3
3
5.83
k6
4.61 4.87
k3
3.08
k2
5.00
k5
3.43
k2
2.33
k1
0
k4
2.37
1.61
4.17
k3
k1
HSM
(b)
2.73
k3
4.33
5.53
k5
5
6.03
k6
5.08
k4
4
5.14
k4 k5
6.03
k6
6
Time (hour)
Fig 2. Optimal solution corresponding to the minimum makespan for the motivating
example, (a) event slots representation and (b) Gantt chart of the sequence..