This paper addresses the optimal scheduling of material in a steel plant. The genetic algorithm is adapted to handle various constraints in the processing mills

Ackerman, J.

Ali, M.M.

Kaelo, P.

English

Mathematics in Industry Information Service Eprints Archive

SCHEDULING OF MATERIAL THROUGH A
STEEL PLANT
M. M. Ali∗, P. Kaelo∗ and J. Ackerman †
Abstract
This paper addresses the optimal scheduling of material in a steel plant.
The genetic algorithm is adapted to handle various constraints in the
processing mills.
1 Introduction
Scheduling problems arise frequently in industry. This paper is concerned
with the scheduling of steel material in a steel plant. The problem arose
at Columbus Stainless Steel, a well known steel company in South Africa
(Ackerman and Waldeck, 2004). The problem can be stated as follows.
Given the weekly demand for the steel products, what should be the daily
scheduling of steel mills in the plant in order to satisfy the demand and
constraints in the mills?
Coils of various types are processed through steel mills to produce steel
of various outgauges. A stockpile of coils needs to be processed through
a number of available mills per day. Mills are constrained by the types of
coils in that certain coils can not be processed by certain mills. Given the
constraints, a scheduling algorithm has to be developed that will balance
the workloads of all mills while maximizing the output material in tons.
We have used a modified genetic algorithm (Ali and To¨rn, 2004) for this
purpose.
∗School of Computational and Applied Mathematics, University of Witwatersrand, Pri-
vate Bag 3, Wits 2050, South Africa. e-mail: mali@cam.wits.ac.za, pkaelo@cam.wits.ac.za
†Columbus Stainless Steel, Middlesburg, South Africa. e-mail: ackerman.johan
@columbus.co.za
103
104 M.M. Ali, P. Kaelo and J.Ackerman
2 The problem
In this section we present the scheduling problem. Figure 1 shows the layout
of the problem. The coils to be processed are stored (see the buffer on the
left hand side in Figure 1). They have different widths and incoming gauges.
They also belong to two processing groups, namely Groups 1 and 2. Group 1
consists of coils of widths 1100, 1300 and 1500 and Group 2 consists of coils
of widths 1100 and 1300 only. The processing of the coils is done by three
mills: mill 1, mill 2 and mill 3. Mill 1 processes coils of widths 1100, 1300
and 1500 while the other two mills process coils of widths 1100 and 1300
only. However, mill 1 does not process coils of processing Group 2.
BUFFERBUFFER
AP2
VBA
Mill 1
Mill 2
Mill 3
Figure 1: Problem layout.
The output from the three mills is then stored in the second buffer.
The output consists of material of different outgauges. For coils of width
1500, the mill 1 produces outgauges of at least 1mm and for coils of widths
1100 and 1300 it produces a minimum outgauge of 0.9mm. On the other
hand, mill 2 produces a minimum outgauge of 0.4mm while mill 3 produces a
minimum outgauge of 0.25mm for all coils that they process. The maximum
outgauge is 6mm. The output in the second buffer is then distributed to
either Annealing and Pickling 2 (AP2) or Vertical Bright Annealing (VBA).
VBA processes the output materials that have final outgauge of 0.3mm and
materials from processing Group 2.
Each mill has a different processing time (or average tons per hour)
for each coil type it processes. The problem is complex in the sense that
the processing time of a coil by a mill is normally distributed with a given
Scheduling of Material through a Steel Plant 105
mean and standard deviation for each coil. Given the constraints of the
mills and data for coil types, the total processing time has to be minimized.
Alternatively, the production rates of the mills (mill 1, mill 2 and mill 3)
have to be maximized. We have adapted the genetic algorithm to suit the
above optimization problem.
3 Genetic Algorithms (GA)
The genetic algorithm (GA) is a global optimization technique based on nat-
ural selection and the genetic reproduction mechanism (Michalewicz, 1996).
GA maintains a set S, called the population set, of candidate solutions,
where each solution is known as a chromosome. Central to GA is the natu-
ral evolution of the set S. At each generation of GA a new S evolves from
the old S, i.e. each generation updates the set S. As the generation proceeds
the set of solutions in S converges to the global minimum. In the basic GA,
three steps are involved in the evolution from one generation to the next.
These are as follows:
- evaluation of f at each new member of the current set S,
- stochastic selection of solutions (parents) from the current set S with
a bias in the selection towards better solutions,
- reproduction of new points (children) from the selected points (par-
ents) using the two genetic operations: crossover and mutation. The
crossover operation is achieved by taking two selected solutions from S
and exchanging their part(s). Mutation is achieved by simply flipping
the element of chromosomes with some probability.
This cycle of evaluation, selection and reproduction, terminates after a de-
sired number of generations.
The GA Algorithm
Step 1 Determine the initial set S. Generate N random solutions. Set
k = 0.
Step 2 Generate solutions to replace solutions in S.
• Selection : select m ≤ N solutions from S as parents, with prob-
ability of each point (string) being selected proportional to its
fitness.
106 M.M. Ali, P. Kaelo and J.Ackerman
• Crossover : pair the solutions (parents) and generate m new so-
lutions (offspring), which replaces the m worst solutions (least
fittest chromosomes) in S.
• Mutation : mutate an element of each solution (chromosome)
with probability pµ, say pµ = 0.001.
Step 3 Update S. Replace m bad parents in S with m new children, set
k := k + 1 and go to Step 2.
4 Results
The standard genetic algorithm described in the previous section has been
adapted to the scheduling problem considered in this paper. The adapted
version differs from the standard version in that the crossover operation uses
only one parent to create two children. We have also used pµ = 0. These
changes have to be considered due to the special structure of the problem.
The initial setting of S requires N random solutions; each random solution
consists of three parts. The first part corresponds to the coils or material
that are assigned to mill 1, the second part is assigned to mill 2 and the
third part for mill 3. For instance a random solution (chromosome) in S has
the following form:
x = (x1
1
, x2
1
, · · · , xn1
1
, x1
2
, x2
2
, · · · , xn2
2
, x1
3
, x2
3
, · · · , xn3
3
), (1)
where xji are random selections from the first buffer and n1+n2+n3 =M (M
is the total number of coils to be processed). Note that the first n1 variables
represent the material that is processed by mill 1, the second batch of n2
variables represent material processed by mill 2 and the last n3 are processed
by mill 3.
In each generation, GA calculates 2m solutions (children) using the ex-
isting solutions in S. It then replaces 2m bad parents in S with 2m new
children. Two new solutions (children) are created by crossover using one
parent. For example, if
x = (x1
1
, x2
1
, · · · , xn1
1
, x1
2
, x2
2
, · · · , xn2
2
, x1
3
, x2
3
, · · · , xn3
3
) (2)
is a parent selected, then two children are produced either by crossing ele-
ments of part 1 of x (coils in mill 1) with the elements of part 2 or part 3
(coils in mill 2 or mill 3), or by crossing elements of part 2 (coils in mill 2)
with elements of part 3 of x (coils in mill 3). However, when doing the
Scheduling of Material through a Steel Plant 107
crossover, care must be taken to avoid having material that is processed by
only one mill falling into the mill(s) that do not process them, so that the
solutions are always feasible. Thus we do the crossover between the elements
of say two mills if they can be processed by both mills. We used tournament
selection (Ali and To¨rn, 2004) to select m parents from S, for crossover. We
ran GA 10 times with different values for N and m. However, the results
presented in Table 1 were obtained using N = 50 and m = 3.
In principle, n1, n2 and n3 can be taken as variables. However, consider-
ing the availability of the mills per day as well as the capacity of each mill,
we considered n1 = n2 = n3 for the results presented here. We have consid-
ered three instances of the scheduling problem, namely the buffer consisting
of 30 coils, 45 coils and 99 coils respectively. The full details of each group of
coils can be found in Ackerman and Waldeck (2004). We only describe the
optimal scheduling for the problem using 30 coils. For 150 generations, the
optimal scheduling obtained by GA is presented in Table 1. The first ten
rows of Table 1 are the coils that are processed by mill 1, the next ten rows
are processed by mill 2 and the last ten rows are processed by mill 3. The
first five columns of Table 1 show the properties of the coils, i.e. the types
of the coils, group to which the coils belong, width of the coils, incoming
gauges and the desired outcoming gauges respectively. Columns six to eight
show the average tons per hour of each coil by mill 1, mill 2 and mill 3
respectively.
108 M.M. Ali, P. Kaelo and J.Ackerman
Type Group Width Ingauge Outgauge mill 1 tph mill 2 tph mill 3 tph
1 Type 1 1 1500 6 2.8 16.44
2 Type 3 1 1500 6 6 34.91
3 Type 2 1 1500 6 4.2 21.39
4 Type 9 1 1500 3.2 1 18.59
5 Type 1 1 1300 1.5 1 23.54
6 Type 3 1 1300 1.5 1 24.95
7 Type 2 1 1300 1.5 1 22.13
8 Type 1 1 1100 4.5 1.8 16.26
9 Type 1 1 1100 4.5 2.3 25.05
10 Type 1 1 1100 5.5 2.8 25.26
11 Type 1 2 1300 3.2 1 8.13
12 Type 1 2 1300 3.2 1.8 18.66
13 Type 1 2 1100 5.5 1 7.04
14 Type 1 2 1100 4.5 1 8.03
15 Type 2 1 1300 2.3 1 8.56
16 Type 1 1 1100 3.2 2.3 20.14
17 Type 5 2 1300 3.2 1 5.67
18 Type 5 2 1300 4.5 1 9.11
19 Type 5 2 1100 3.2 1 4.38
20 Type 8 2 1100 1.5 1 7.32
21 Type 2 2 1300 1.5 1 27.82
22 Type 1 2 1100 3.2 1 8.27
23 Type 3 2 1300 3.2 1 11.16
24 Type 6 2 1300 3.2 1 6.9
25 Type 1 2 1100 5.5 1.8 14.78
26 Type 7 2 1300 3.2 1 10.5
27 Type 3 1 1100 3.2 2.3 26
28 Type 1 1 1300 2.3 1 11.61
29 Type 4 1 1300 3.2 1 22.95
30 Type 1 1 1100 3.2 1.8 24.92
Table 1: Optimal solution found for 30 coils.
5 Conclusions
We adapted an algorithm for the scheduling problem based on real data. We
have highlighted the complexity of the problem and modified the standard
genetic algorithm for the optimal scheduling. The daily production capacity
has been maximized (or daily scheduling time is minimized). The optimal
scheduling is certainly more efficient than the current practice of manual
scheduling. The quality of scheduling will result in savings in Columbus
Stainless Steel. Further research is underway to consider large scale schedul-
ing that can take a longer plaining horizon.
Scheduling of Material through a Steel Plant 109
Acknowledgement
M.M. Ali acknowledges support of this work under the National Research
Foundation of South Africa grant number 2053240.
References
J. Ackerman and G. Waldeck (2004). Columbus Stainless Steel, Middles-
burg, South Africa, Private Communication.
M. M. Ali and A. To¨rn (2004). Population set based global optimization
algorithms : some modifications and numerical studies, Computers and Op-
erations Research, 31, 1703-1725.
Z. Michalewicz (1996). Genetic Algorithms + Data Structures = Evolution
Programs, Springer-Verlag, Berlin.


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