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Linear Programming-
Electric-Arc Furnace Steelmaking

Data Processing' Application
Introduction . . . . . . . 1
Problem Profile. . . 2
Problem Economics 3
Single- Furnace Model Formulation 3
Input Data Requirements. 4
Example Problem . . . . . . . . . 4
Cost Constraint. . . . . . . . . 8
Charge Material Supply Equations . 9
Specification and Control Constraints 10
Multifurnace Model Formulation . 19
Summary . . . . . . . . . . . 21
Output Reports. . . . . . . . 21
Basis Variables Report 22
Slacks Report. . . . 24
DO. D/J Report . . . . . . . . . 26
Cost Range Report . 27
Summary . . 28
Bibliography. . . . . . . 29

Copies of this arid other IBM publications can be obtained through IBM branch
offices. Address comments concerning the contents of this publication to
IBM, Technical Publications Department, 112 East Post Road, White Plains, N. Y. 10601

The introduction of linear programming (LP) has produced remarkable
and diverse benefits in a number of industries. Recent applications of
LP techniques by metal producers -- notably to control costs and quality
in alloy blending -- suggest a variety of new applications. The purpose
of this manual is to demonstrate the application of LP in the production of
steel in electric-arc furnaces -- a process which, because it involves
complex blending and quality control, is particularly responsive to LP
techniques. The immediate and more obvious LP results enable the steel
producer to:

1. Minimize the cost of both initial and supplemental furnace charges

2. Minimize and possibly eliminate off-compositions

3. Maintain accurate scrap inventory records

4. Purchase and sell most economically

5. Evaluate plant operating changes

6. Interpret historical charge data in terms of operating relationships
to develop more efficient operation

Contrary to popular belief, little mathematical knowledge or skill is
required to formulate an LP model. Nor does the operation of the com-
puter and the analysis of computer results require any advanced technical
skill. Linear programming requires nothing more than the expression of
all the elements in the process -- plant operating practices, charge
materials, specifications, etc. -- in the form of simple linear equations.
A general explanation of basic linear programming appears in the IBM
data processing application manual An Introduction to Linear Pro-
gramming (E20-8171), which should be read in conjunction with this

To demonstrate the methods and advantages of LP in steel production, we
shall present a typical production problem as a basis for the development
of an LP model which can be solved by the IBM 1620/1311 Linear Pro-
gramming System. With minor modifications the model can be run on
any of IBM's LP systems.


The basic process consists of the following phases:

1. An initial charge of sc rap and alloying material is melted in a
furnace by electrical energy supplied through carbon or graphite

2. Oxygen, supplied through lances, is blown through the molten bath to
burn off impurities. As a consequence, a slag forms which contains,
in addition to the oxidized impurities, a significant quantity of iron
oxide and oxides of expensive alloying metals (such as chromium).

3. In alloy steelmaking much of the metallic oxide in the slag is reduced
by the addition of silicon -- for example, in the form of high-silicon,
low-impurity chrome silicides.

4. In medium-Iow-, and low-carbon steelmaking, the initial slag is
raked and poured off, and a second slag is either formed or placed
on the bath. This slag serves to eliminate remaining contaminants
and protect the metal bath from contamination by reaction with the
furnace atmosphere.

5. When the metal bath is brought to end specifications and temperature,
the steel is poured out into ingots, molds, etc.

Because the electric furnace allows close control of both composition and
temperature, it is in widespread use in medium -low-carbon steel produc-
tion and has become the primary producer in stainless and alloy st~el

The fundamental problem is to produce a specified steel at the lowest
possible cost. In order to achieve least-cost production, the producer
must consider a complex variety of fa.ctors which, immediately or
ultimately, contribute to the costs of production. The more obvious
variable factors include price, grade, and availability of initial charge
scraps,price and quantity of required additives, and heat time (that is,
price and quantity of required energy). Less obvious factors that
markedly affect costs include refractory erosion, oxygen rate and lance
position, and quality control. The least tangible, and possibly the most
important, factor that contributes to the formulation of consistently
accurate bids (especially for steel orders) is an accurate log of heat
histories -- to serve as the basis for predicting operating efficiency and
revising operating practices.


In most cases a wide variety of scraps, differing in composition, physical
condition, and price, are available for the initial charge in the electric-
arc furnace. Further, the available quantity of each scrap, as well as
its price, fluctuates. The primary economic problem, then, is to deter-
mine the composition of an initial charge that will produce the specified
steel at least cost. The nature of the initial charge will affect the cost of
furnace operation (since different scraps will require different optimum
furnace temperature and blow time). Further, the nature of the initial
charge, in conjunction with the furnace operation during the melt and
decarburization of the charge, affects the cost in terms of relatively
expensive reducing and finishing additives.

The crucial interrelation among the several phases of steel production
makes it exceedingly difficult to determine the least-cost initial charge,
optimum furnace operation, and least-co.st supplemental charge. This
difficulty is vastly compounded by commo.n fluctuations in the availability
of specific scraps, since the alteration of anyone component in the initial
charge will alter all the relationships required for least-cost production.
Heretofore, steel producers employing manual calculation to determine
initial furnace charge often used expensive scrap that came close to
matching the alloy specification requirements together with expensive
pure metals and additives. An increasing number of steelmakers, how-
ever, are profiting from the application of linear programming, which
enables the producer to examine all possible combinations and quickly
determine the most economical furnace charge. Further, by serving to
"force" overstocked scrap types in least-cost charges, LP can contribute
to the achievement and maintenance of ideal inventory procedures.


A linear programming model for steel production is a mathematical
representation, in the form of linear equations, of all lmown and esti-
mated factors relevant to the production of the specified steel. To
demonstrate the method for formulating such a model, we postulate a .
specific problem and relatively ideal conditions -- the production of
20,000 lbs. of low-carbon stainless steel from four initially available
charge materials. In actual practice a larger number of materials are
available to the furnace operator; regardless of their variety and compo-
sition (the factors that complicate manual calculation), they can easily
be included in the LP model, increasing the model's size'but not its

Input Data Requirements

The following basic data is required to formulate the LP model:

1. Specifications of alloy to be produced

2. Pounds of alloy required

3. Composition analysis of all raw materials

4. Per-pound cost of all raw materials

5. Inventory levels of all raw materials (scrap and reducing and
finishing additives)

6. Special raw-material restrictions (for example, ingot weights)

7. Current operating practices (for example, basicity levels)

8. Furnace characteristics (for example, maximum permissible

Most of this information is available from purchasing, cost accounting,
inventory accounting, or other sources and is probably used in existing
systems for computing furnace charges. Where exact information cannot
be readily obtained, estimates should be made, since it is an easy matter
to change the input data and re-solve the problem once an optimal solu-
tion has been obtained .. Indeed, the rapid calculation of the effect of
changes in the input is a prime advantage of the LP approach. Moreover,
the accumulation of a log of heat histories will result in increasingly
precise estimates.

Example Problem

We wish to produce 20,000 lbs. of steel with the specifications shown in
Figure 1. The four initial charge materials available are steel scrap,
430 grade steel scrap, high-carbon ferrochrome, and low-carbon ferro-
chrome. They may be priced and analyzed as shown in Figure 2.

Since market variations frequently influence the choice of initial charge
materials, our model must be responsive to the fifth element in the list
of input data requirements: inventory levels. Hence we will assume that
the availability of 430 grade scrap and high- and low-carbon ferrochrome
is limited to 2000 lbs. each. We can invoke similar limitations, depend-
ing on market conditions, to vary the quantities of any of the charge
elements at any phase of the process.

We will not postulate here any special raw-material restrictions, though
forcing the use of ingot weights may be an important production problem.
(This aspect of the problem will be discussed in the section on output
basis variables.)

Chromium minimum 16.0%
Silicon maximum 1. 0%
Manganese maximum 1. 0%
Carbon maximum 0.05%

Figure 1. Problem specifications

Steel 430 Grade High-Carbon Low-Carbon
Scrap Scrap Ferrochrome Ferrochrome

Cost per lb. $0.02 $0.075 $0.27 $0.40

Chromium 0 16.0% 55.6% 65.0%
Manganese 1.0% 1. 0% 0 0
Silicon 0.2% 0.95% 2.0% 1. 0%
Carbon 0.6% 0.12% 8.0% 0.09%
Iron 98.2% 81. 43% 34.4% 33.91%

Figure 2. Analysis of materials

The complex thermochemistry and tight controls required in the production
of the specified steel introduce problems best handled by an adaptive
rather than a static model, especially when the scrap analysis is uncertain.

1. The composition of the initial charge and the amount and variety of
reducing and finishing additives are established by a linear program,
based on final metal specifications, cost and composition of available
charge materials, and plant capacity.

2. Based on carbometer analysis and spectograph analysis of the melt,
a new linear program is formulated to determine accurately the
quantities of reducing and finishing additives required to achieve the
specified steel at least-cost.

For our purposes we need develop only the first of these programs. In
practice, the second model can be developed quite easily from the first.

The schematic of the LP model matrix (Figure 3) graphically illustrates
the steelmaking process. The detailed model matrix is shown in Figure 4.

Every source of the various elements which make up the final alloy
appears at the head of a matrix column, which is called a problem activity.
Cost, maximum and minimum specifications, and symbolic designations
for the processes which alter the element quantities provided by the
sources appear at the ends of the matrix rows, called problem constraints.
Consider the first four columns of the blending section of the matrix in

Column Activity Names
/ r__________________________~I~--------------------------~,

;Cost C C
1 2 RHS
Blending 1


Right-Hand Side
Decarburization or Specifications

..JI ..JI
:;:. :;:.C\1

,'-____________________________, -______________ ~ ____________J/
Single Variable Bounds

Figure 3. Schematic of LP matrix model -- single-furnace

Figure 4. We have, in effect, transferred the data given in Figure 2 to
our matrix. Each of the four sources for the initial charge heads a
column and is assigned a symbolic name (mnemonic). Similarly, each of
the rows is symbolically named .. Figure 5 defines these column and row
mnemonics for the blending section of the matrix.

Column Names

5 5 H L R R F
T P C C C M 5 C F T I C 5 C I L C
5 4 F F R N I E E I 5 R 4 F 5 I F 5
C 3 C C I I I I I C C 5 3 C F M C I
P 0 R R T T T T T W R I 0 R E E R 5
Name No.

Value 1 .02 .075 .27 .40 0 0 0 0 0 0 0 .27 .075 .40 0 .01 .40 0 Cost

CR 2 0 .16 .556 .65 -1 =0

MN 3 .01 .01 0 0 -1 =0

51 4 .002 .0095 .02 .01 -1 =0

C 5 .006 .0012 .08 .0009 -1 =0

FE 6 .982 .8143 .334 .3391 -1 =0

TOTCHG 7 1 1 1 1 1 -1 =0

CR5LAG 8 1 1 -.074 -1 =0

TOTCRS 9 .074 .95 .39 .17 .65 .65 1 =3400

MN/CR 10 .. .98 .01 1 =200

FEsLAG 11 .075 -1 -1 =0

ENDFE 12 1 .18 .8143 .3391 -.05 .3391 ~16,2oo

C5PEC 13 -5 5 -.25 12 9 -.25 9 ~loo,ooo

BA5E 14 2.14 2.7606 -2 =0

515PEC 15 -.395 .43 .0095 .01 -.238 .01 ~2oo

TOTAL 16 -1 -1 1 -.05 .57 1 1 -.05 1 ~ 20,000

TOTRS4 17 1 1 ~2000

TOTRCF 18 1 1 1 ~2000


Figure 4. LP matrix model -- single-furnace problem
Item Name Mnemonic

Steel scrap STSCP

430 grade scrap SP43 0

High-carbon ferrochrome HCFCR

Low-carbon ferrochrome LCFCR

Chromium initially charged CRIT

Manganese initially charged MNIT

Silicon initially charged SIlT

Carbon initially charged CElT

Iron initially charged FElT

Total initial charge weight TICW

Element or Control Name Mnemonic

Price per pound (of initial charge materials) VALUE

Chromium CR

Manganese MN

Silicon SI

Carbon C

Iron FE

Total elements charged TOTCHG

Figure 5. Mnemonic table -- blending section of matrix

COST CONSTRAINT (Objective Function)

The first problem constraint row (1) incorporates the price per pound of
each scrap and additive. Hence the cost of the specified. steel may be
expressed by the linear equation:

0.02 STSCP + 0.075 SP430 + 0.27 HCFCR ... =COST,
where in each term the coefficient is the price per pound, and the mne-
monic is the quantity in pounds, of the material to be charged. The
"solution will give the quantity of each material required to produce the
specified steel at minimum cost.


Rows 2 through 6 establish the quantities of each element in the initial
charge material. The second row, for instance, establishes the quantity
of chromium in the initial charge. It is a linear summation of the pounds
of chromium per pound of material in each of the scraps to be blended.
Since steel scrap contains no chromium, a zero appears as the coefficient
for STSCP in the CR equation. As Figure 2 indicates, 430 grade scrap,
high-carbon ferrochrome, and low-carbon ferrochrome contain 16%",
55.6%, and 65% chromium, respectively. We can, consequently, express
the chromium in the initial charge as follows:

O. STSCP + 0.16 SP430 + 0.556 HCFCR + 0.65 LCFCR = CR,

where the mnemonics are the variable quantities of raw material to be
computed. We then provide a problem activity column for the total
chromium in the initial charge (CRIT). Thus,

O. STSCP + 0.16 SP430 + 0.556 HCFCR + 0.65 LCFCR - 1. CRIT = O.
Similarly, the quantities of manganese, silicon, carbon, and iron in the
initially available materials are indicated in matrix rows 3, 4, 5, and 6,
and a total-element-in-initial-charge column is formed for each. The
final factor in the first section of the matrix is a problem activity column
for the total initial charge weight (TICW) and a constraint row (7) in-
dicating that the total weight of all the separate elements charged minus
the total weight of the initial charge equals zero:

CRIT + MNIT + SIlT + CElT + FElT - TICW = o.
The limitation of 430 grade steel and low-carbon ferrochrome to 2000 lbs.
each appears in the last two constraint rows of the matrix (17 and 18);
but the limitation of high-carbon ferrochrome, because it is employed at
only one phase in the process, is introduced in the column devoted to that
material. The number of rows directly affects calculation time and,
further, is the determinant of machine capacity. Therefore, this fea-
ture -- the ability to bound any single activity without employing a row --
makes an important contribution to the computer's speed and problem
capacity, and becomes particularly useful when solving multifurnace

In order to blend an initial charge properly we must consider not only
the final specifications of the desired steel but also changes in the total
weight of each of the initially charged elements resulting from the de-
carburiz at ion , reducing, and refining processes.


Recent research in the thermochemistry of steel production makes it
possible to predict and allow for reactions and losses that occur during
decarburization and reducing. Such research emphatically demonstrates
that special factors -- hearth material, heat size, variations in initial
charge metal percentages, etc. -- contribute to the empirical result.
Yet a sufficiently reasonable correlation between calculated estimates
and end metal composition has been observed to justify the use of such
estimating techniques in production. Consequently, we have employed
a number of relationships which hold in the production of low-carbon
steel in order to