A Survey of
Mathematical Programming Applications
in Integrated Steel Plants
Goutam Dutta
Indian Institute of Management
Vastrapur, Ahmedabad 380015, India
Robert Fourer
Department of Industrial Engineering and Management
Sciences
Northwestern University
Evanston, Illinois 602083119, U.S.A.
ABSTRACT
Mathematical programming techniques were used in
the steel industry as early as 1958, and many applications of optimization
in steel production have been reported since then. In this survey,
we summarize published applications in the largest steel plants by type,
including national steel planning, product mix optimization, blending,
scheduling, set covering, and cutting stock.
Copyright © 19982001
Goutam Dutta and Robert Fourer. Draft: Please do not quote
without the authors' permission. We gratefully acknowledge financial
support from the American Iron and Steel Institute and another American
steel company.
Table of Contents
1. Introduction
An integrated steel plant is a complex industrial
system in which numerous products are routed through different series of
production units. The sales, cost, and net profit of each product
are functions of many variables. If the operating manager makes decisions
that result in suboptimal operations, a significant savings or income
opportunity can be lost. In this paper, we survey mathematical programming
applications to the following classes of problems in integrated steel plants:

National steel planning

Productmix optimization

Blending in blast furnaces, coke ovens or steel foundries

Scheduling, inventory and distribution

Set covering

Cutting stock optimization
Applications in fifteen different countries in four
continents have been reported from 1958. Prior to our current paper, there
have been four surveys. Mihailor (1961), which surveys
34 papers, is an elementary aid for engineers and metallurgists. This survey
also gives an overview of how linear programming models can be applied
in a steel plant. Gercuk (1961) is a nonmathematical
survey devoted to the subject of linear programming and some of its applications,
mainly in composition of charges, loading of equipment and transportation
of equipment. The work by McCulloch and Bandopadhay (1972)
gives a broad overview of operations research models, a significant proportion
of which are in the areas of mathematical programming and largescale optimization.
A study by Rao
et al. (1993) gives a classificatory
review of OR applications in strategic planning, operational planning and
tactical planning.
The paper is written for two audiences. The first
is the management science practitioner in industry who is looking for possible
areas of applications of optimization techniques in an integrated steel
plant. The second is the academic researcher who is looking for potential
research areas in integrated steel plants. An
elementary knowledge of integrated steelmaking operations is desirable,
but not essential. The reader interested in acquiring a detailed knowledge
of iron and steel production is referred to AISE Steel
Foundation (1998).
In this paper, we consider all of the front end
of integrated steel making operations, from ironmaking to finished steel
production, but have not considered applications in mines and quarries.
Emphasis has been placed on the real world implementation of the models.
A brief description of an integrated steel plant is given in Section 2,
prior to the survey in sections 39.
2. An Overview of an Integrated Steel Plant
Figure 1 describes an iron and steel making plant
having four stages: iron making, steel making, primary rolling and finishing
rolling. The output of each stage becomes the input to the following stage.
In the iron making stage, the blast furnaces are used to convert iron ore,
sinter and other raw materials into molten iron called hot metal. Hot metal
is supplied to the steel melting shops where the process of steel making
is either BOF (Basic Oxygen Furnace), OHF (Open Hearth Furnace) or EOF
(Energy Optimizing Furnace). The molten steel from BOF is either
sent to the continuous caster or poured into various ingot molds. The molten
steel from other shops is cast into ingots.
Figure 1. Diagram of flows
through an integrated steel plant.
In the primary rolling stage, ingots are shipped
to the soaking pits where they are heated by a mixture of gases to a uniform
temperature, before being rolled into blooms and slabs in the Blooming
Mill. The blooms are further rolled in the Sheet Bar and Billet Mill into
either sheet bars or billets. In the finishing rolling operation, the slabs,
sheet bars, strip bars and billets are the input materials to various finishing
mills. The slabs are rolled in the Plate Mill into high tensile and wear
resistant plates or ordinary mild steel plates. The sheet bars are further
rolled in the Sheet Mills into high silicon, LPG (Liquid Petroleum gas)
and galvanized sheets. The Strip Mill converts the strip bars into cold
rolled or ordinary strips which go either to the market or to the Tube Making
Plant.
Billets from the Sheet Bar and Billet Mill go
either to the conversion agents or to the Merchant Mill where they are
rolled into twisted bars, angles, octagons. The blooms are further rolled
into seamless gothics (for seamless tubemaking) or into structurals in
the Medium and Light Structural Mill.
3. National Steel Planning Models
Before describing applications developed for specific
integrated steel plants, we mention in this section several steel planning
models for national economies, using linear programming techniques.
National Steel Planning Model
in the United States
Tsao and Day (1971) develop
a process analysis model of production in the US on a national level. A
technology matrix, which represents the technology structure, is estimated
using engineering and metallurgical information. This matrix together with
the detailed cost, sales and revenue figures is then used in a linear programming
model of short run allocations of the steel industry as a whole. The linear
programming model's solution is obtained and compared with available industry
statistics for each year from 19551968. Although Tsao and Day claim to
have a fairly good results, a later study by Nelson (1971)
reported that the model had an error in the treatment of coking coal production.
Nelson attempted to correct this deficiency and presented a correlated
matrix for this stage of production.
Mexican Steel Model
This study by Kendrick, Meeraus
and Alatorre (1984) develops two static models for production planning
and one dynamic model for investment analysis. The two static models, formulated
as linear programming models, are mixed production and transportation problems.
Inputs are prices of raw materials, operations and shipments, demands,
facility capacities and input and output coefficients for each productive
unit. Outputs are optimal product distributions. The dynamic model, formulated
as a mixed integer program, incorporates time factors and deals with the
investment issues in five time periods of three years each. The inputs
are similar to those in the static models but the output also includes investment decisions.
Stochastic Programming Model
for Investment Planning in India
Anandalingam (1987) discusses
a stochastic programming model for investment planning in environments
where demand projections and technological coefficients are not known with
certainty. The model has been used primarily for strategic planning rather
than operational planning. The usual programming formulation of an industrial
process is extended to incorporate parameters and demand uncertainties
by modeling it as a stochastic linear program with simple recourse (SLPR).
The SLPR is solved using the less restrictive assumption that only the
means and variances of the stochastic entities (but not their distributions)
are known. The methodology is applied to the study of the steel industry
in India with a novel way of modeling investment and economies of scale.
4. ProductMix Optimization Models
In an integrated steel plant, the problem of determining
the optimum production level at various stages is of great practical importance.
This is so because the profit is sensitive to the product mix and not merely
to the total volume of production. Because of the complexity, suboptimal
workable solutions are generally obtained by experience. Although these
solutions when implemented achieve good plant utilization, profits/revenue
from these solutions are considerably less than the potential profit/revenue
that could have been accrued using the optimum product mix. The optimum
product mix changes from month to month and with the mill, furnace availability,
and demand for the product in the market. Pioneering work in this area
by Fabian (1958) was undertaken at Kaiser Steel Company,
and since then a number of applications in this area have been reported.
ProductMix Model at Kaiser
Steel Company
An integrated steel plant has a choice of the use
of various materials and production processes. The economical usage rate
of all materials is a function of a number of variables. Some of the most
important variables are the market price of some materials, notably various
grades of steel scrap. This scrap price fluctuates, and therefore requires
the periodic determination of economical usage rate. The work of Fabian
(1958, 1967) is a cost minimization linear programming model that has
four submodels: one for iron making, one for steel making, and one each
for shop loading for rolling operations and finishing operations. The models
of various stages of production are connected to form a "Master Model"
of an integrated steel plant. The detailed formulation at each stage and
the principles of integration are also discussed in these papers. The model
considers all the technoeconomical constraints like the capacity balance,
material balance, productdependent yield and thermal energy balance (in
the form of enthalpy balance). However, the oxygen balance and electrical
energy balance are not discussed.
Large Scale Database Model
for American Iron & Steel Institute
Fourer (1997) presents a model
which grew out of a project to design an optimization package for steel
mill planning. Because this project was supported by the American Iron
and Steel Institute (AISI) and not any particular steel company, it was
based on a generic model. Any steel plant could customize the model to
its own operation, simply by supplying its own data. Users of this model
would be concerned mainly with entering and maintaining their data and
with reporting the optimal production levels. The model is generic in nature
and can be transported to other similar industries like coal mining and
oil refineries.
This work has been used in a number of steel plants
such as LTV, Dofasco, and Armco. Dofasco has used this database optimization
software to generate models in excess of 1000 variables and Armco has developed
an equivalent of this software in a spreadsheet (Excel) using the same
solver but with a variety of reports and diagrams customized to the company's
requirements. In the LTV steel plant, it was suggested to use this model
in two plant production and distribution problems.
On the basis of the above model, the importance
of inventories and the linkage between the time periods was investigated
by Hung (1991). See the later discussion of the
AISI Inentory Model for details.
Models for Production Planning
in the United Kingdom
Lawrence and Flowerdew (1963)
develop an economic model of steel production that focuses on on the
application to the individual processes. A single cost model is constructed
containing input and output variables, cost of variables and operations,
relationships between and restrictions on the variables, technical relationships,
and flow restrictions. A simplex type tableau is then constructed for a
simplified model, and the optimal solution is then computed.
Bandyopadhay (1969) proposes
a linear programming model that allocates different capacities between
two processes for production planning, namely the Basic Oxygen Furnace
and the Open Hearth Furnace. The model is a cost minimization model with
all the technological and financial constraints. The model can also predict
the required operation level of blast furnaces and lime burning plants
at different levels of total steel production.
German Model at Hoesch Siederlandwerke
Bielfield, Walter and Wartman
(1986) at Hoesch Siegerland Werke AG (HSW) in Germany have developed
a set of accounting matrices for budgets for planning. The company had
a revenue of one billion Deutsche Marks, and its main products were cold
rolled, hotdip galvanized, electrogalvanized, and organic coated sheet
steel. The complexity of the steel company's structure and operation and
rapid environmental changes forced the HSW management to replace a manual
system with a computerbased strategic planning system having the objective
of improving efficiency and performing mass calculations and cost accounting
more efficiently. This is a linear programming model with the multiple
objectives. These objectives may be maximizing revenue, minimizing total
cost or cost per ton of steel produced. The model has about 2500 constraints
and 3000 structural variables.
ProductMix Optimization
Models in Indian Steel Plants
In India, the prices of half of all steel products
were controlled by the Government from the fifties until 1991. In this
environment, two interesting applications of planning have been reported.
During the past fifteen years, India has been
affected by an energy shortage. The crisis is most significant in the eastern
part of India where the gap between supply and demand is greatest. The
poor capacity utilization of some power plants (which supply power to the
steel plant) makes the operation of energy consuming plants extremely difficult.
In the operation of a steel plant, some of the energy consuming processors
(called essential loads) require a fixed amount of power and cannot be
switched off, even in the event of power crisis. In this environment, the
operating manager of a plant has no other option but to switch off those
processors that are not essential loads. Optimal allocation of electrical
energy is thus a very important decision for the management of the steel
plant.
Dutta, Sinha and Roy (1990),
Dutta
et
al. (1994) and Sinha
et al. (1995)
deal with the development and implementation of a mathematical model for
optimal allocation of electrical energy in a plant of Tata Steel. The
guiding principle of the model is that in the case of a power shortage,
power is allocated to those nonessential loads which have a higher
profitability (based on a mixed integer linear programming model).
Although a number of studies (Hunneault and Galiana,
1991; McCutcheon,
1988) have reported the optimal use of
power plants, such studies have addressed the issue with a cost
minimization modeling approach for power generating and distributing
plants. Others have studied the most profitable use of an integrated
steel plant (Fabian, 1958; Bielfield, Walter and Wartman, 1986; Baker et al. 1987) where the problem has
been addressed as a cost minimization or profit maximization linear
programming model.
In the Tata Steel application, the steel plant
has been modeled with a (contribution to) profit maximization
objective, with energy as a limiting constraint. This is the pioneering
attempt in India where the mathematical programming model has been
implemented not only for long term strategic planning decisions, but
also for short term operating decisions. This use is not only in an
integrated steel plant, but also in an integrated steel plant
vertically integrated with a tube manufacturing plant which requires
higher complexity. The model considers all the technical and economical
and environmental constraints such as the balance of capacity,
materials, energy and oxygen. It is an optimization model of an
integrated steel plant with blast furnaces, steel melting shops and
primary and finishing mills in a global energy crisis environment or
hot metal shortage situation. The model has different objectives:
maximizing profit contribution, minimizing cost or maximizing
production; it has about 1000 constraints and 1000 variables. Its
outputs are converted to a priority list of the facilities to be
switched off during the energy crisis. The roundtheclock
implementation of the model has improved the profitability of the steel
plant significantly from 1986.
The Steel Authority of India Limited (SAIL), the
largest steel company in India, is a multiproduct company producing a
wide range of products from its five integrated steel plants at Bhilai,
Bokaro, Durgapur, Burnpur and Rourkella. The salable outputs from these
plants can be divided into pig iron, semifinished steel, and finished steel.
Another interesting option among these five steel plants is that of interplant
transfers. This arises because of the imbalances at various stages of production
across SAIL steel plants. Sharma and Sinha (1991)
describe an optimization model for determining the optimal product mix
for the integrated steel plants of SAIL. The paper begins with a discussion
of various issues relevant to the choice of an optimum product mix in a
steelmaking operation. Some planned applications of the model are also
discussed.
Models of Production Planning
in Zambia
Sashidhar and Achray (1991a)
deal with the problem of production planning in a steel mill with
the objective of maximizing capacity utilization. The model is formulated
as a maximum flow problem in a multiple activity network. The production
is usually planned against customer orders and different customers are
assigned different priorities. The model takes into account the priorities
assigned to the customers and the order balance position. An algorithm
is presented for solving the multiple activity network formulation for
production planning with the customer priorities in a steel mill.
In another paper, Sashidhar
and Achray (1991b) discuss the problem of allocating the major components
of process costs to various quantities of products produced in a melting
shop of an alloy and special steel manufacturing unit. Quadratic programming
techniques are used to estimate the consumption pattern of important operational
materials. These consumption patterns cannot be directly allocated to each
quality of steel. Use of quadratic programming helps to arrive at more
realistic and accurate routewise and qualitywise costing at the melting
shop.
Model of Production Planning
in Algeria
Sarma (1995) describes an application
of lexicographical goal programming at Societe Nationale de Siderurgie,
Algeria. This is the only steel manufacturing plant at Algeria which caters
to the domestic needs for steel production in several other industries
such as railways, building, and bridge construction. Initially, an optimal
solution is found which gives an indication of the optimal aspiration level
of the management. The lexicographical approach has helped the management
to spell out aspiration levels of several principal objectives such as
profitability, capacity utilization of some key plants, and production
quantity of some key products.
5. Blending Models
Generally, these problems are formulated as cost
minimizing linear programming models. The thermochemical metallurgical
processes in blast furnaces, coke ovens and iron and steel foundries are
expressed as a set of constraints in a linear programming problem. The
solution indicates a minimum cost selection of input materials in a production
planning context. In addition to the plant or facility availability constraint,
it considers the limitations of input and output materials. These limitations
are given in the form of composition balance equations (such as carbon
or sulfur balance) or as constraints on the basicity ratio (the ratio of
lime to the silica plus alumina).
Blast Furnace/Cupola Blending
Models in the United States
The blending of different ores or input charge materials
in the blast furnace of a steel plant is known as a "blast furnace burdening
problem." The results obtained from Fabian (1967)
enable a producer to determine:
1. Least cost raw materials blending
2. Optimal furnace scheduling
3. Long range production planning
4. Optimal raw materials inventory levels
5. Optimal purchasing policies
6. Optimal maintenance planning
The cost minimizing output gives the solutions to
the LP problem, the total cost of the burden, metallurgical analysis, heat
balance report, burdening sheet, the marginal values of each resource,
the reduced cost coefficients, parametric analysis in ranges, and availability
of the facilities.
Metzger and Schwarzreck
(1961) describe an application of linear programming for the determination
of least cost cupola charging in an iron foundry. Their paper gives a numerical
example with actual data, describes the evolution of the solution, discusses
the difficulties overcome in developing the final version of the model,
and summarizes cost savings.
Blending Model in the United
Kingdom
Beale, Coen and Flowerdew (1965)
propose a model in which the variables are usages, in a given time period,
of ore and other materials, output of pig iron, and levels of certain factors
that depend on the of mix of materials. In the real world, some of these
models are nonlinear and a separable programming approach is useful. Representing
each nonlinear function of single variable as a piecewiselinear approximation
based on a finite number of points, the problem can be solved by a slightly
modified linear programming procedure. The same approach is repeated for
nonlinear functions of more than one variable.
Blending Model in Belgium
This objective of this study, Hernandez
and Proth (1982), was to save valuable metals whose supplies are uncertain
and/or have to be imported. The problem of selecting the charge materials
from available stocks in order to produce alloys as cheaply as possible
is extremely important to foundries producing microcomponent alloys, such
as bronze and special steel. The production of alloys at the lowest price
from a number of stocks of scrap alloys of various composition and from
unalloyed metals is achieved through the use of a new algorithm. The method
differs from normal linear programming and avoids the shortcomings of known
algorithms. The algorithm gives either an optimal solution or a "good"
solution close to optimal. The system has been implemented to give an improvement
in profit. In addition, the paper addresses the practical aspects of introducing
this software.
Blending Model in Sweden
This work by Westerberg, Bjorklund
and Hultman (1977) was done at Fagersta AB, Sweden and the Contact
Research Group for Applied Mathematics, Royal Institute of Technology in
Stockholm. The problem was modeled as a traditional blending model with
the additional constraint that some of the variables should be integer
valued. The Company produced stainless steel in HF (High Frequency) furnaces
and used up to 15 different types of scrap and alloys which are melted
together. The linear programming model is a cost minimization model
with constraints given by weight restrictions and metallurgical
composition restrictions. The implementation of the model has
decreased the cost of raw material by 5 percent which is equivalent to
$200, 000 per year.
Blending Models in East European
Countries
Muteanu and Rado (1960) solve
a blending problem in a Rumanian steel plant that deals with the raw material
loading of an ironsmelting furnace in such a way as to obtain an optimal
production plan at minimum net cost of pig iron, taking into account definite
prescribed production.
Another blending model by Taraber
(1963) has been reported in Yugoslavia and this model has an objective
of profit maximization. It provides an elementary example of the use of
the linear programming and in deciding the composition of furnace charge
for blast furnace.
6. Scheduling, Inventory and Distribution Models
In this section we discuss scheduling problems for
continuous casters and hot strip mills, as well as problems of distribution,
inventory, and supplychain design.
Scheduling Model at LTV Steel
In 1983, LTV Steel Company started up a twin strand
continuous slab caster to convert molten steel to solid steel slabs. Located
at LTV's Cleveland Works, the caster was scheduled by a computerbased
system that included a heuristic algorithm developed by Box
and Herbe (1988).
A casting sequence is required to meet all the
operating and metallurgical constraints of sequencing slabs for production.
The casting sequence also defines a sequence of heats  batches of molten
steel  in which each 250ton increment of the cast slab is of the same
metallurgical grade. The problem of sequencing slabs from the requisitions
on a single strand of a caster is similar to a knapsack problem, where
the most important orders from the order book are given the greatest value.
The complexity of the problem increases for a
twin strand caster, which produces two simultaneous and independent production
streams from one source of molten steel. The problem becomes like a routing
problem for two knapsack constrained traveling salesmen, traveling on two
interdependent itineraries. The "pool" of cities is available to both salesmen,
but their paths are mutually exclusive because a slab for a requisite order
can be produced only once. Further the two salesmen must arrive at certain
cities at the same time because of constraints imposed by successive heats.
Both production streams begin with the same heat, and the sequence ends
when the last heat is consumed. Thus the sequence must end on both strands
at roughly the same time.
The caster scheduling model determines the requisitions
that are to be filled in a sequence of heats, the order of slabs produced
in the sequence and the nature of heats needed to produce the specified
slabs in the specified sequence. A heuristic is used since the combined
problem (synchronizing, sequencing and assignment) is very complex and
some of the constraints are difficult to state mathematically in a form
suitable for inclusion in mathematical programming formulations. The objective
function is pseudocost per ton for producing a given cast sequence. It
is not the total cost, but rather the relative savings of continuous casting
compared to teeming (that is, casting by pouring molten steel into molds).
This system annually saves over $1.95 million by reducing personnel and
increasing production. Also, using the schedules determined, the design
capacity of the caster has been surpassed by 50 percent.
Scheduling Models at Bethlehem
Steel
In the late seventies, Bethlehem steel needed 40006000
cast iron and steel rolls every year to manufacture product of various
shapes in its 100 mills located throughout the U.S.A.. The rolls were first
cast at foundries and then machined in a large generalized machine shop
with 35 machines. In this context, Jain, Stott and
Vasold (1978) developed and implemented an order book balancing procedure
with a combination of linear programming and heuristics for improvement
in order book balancing when demand exceeds supply. The objective function
of the linear program is to maximize the total tonnage of rolls produced
subject to machine availability and supply and demand constraints. The
implementation of the model has improved efficiency and customer service,
reduced workinprocess inventories and machine setup time, and improved
due date performance.
Stott and Douglas (1981)
describe a scheduling system for oceangoing vessels that are employed
in shipping raw materials from around the world to Bethlehem's plants.
There are four subsystems encompassing a range of time scales: Voyage Estimation,
Preferential Employment, Single Vessel Scheduling, and Multiple Vessel
Scheduling. At the time of publication, this system had been running for
more than 4 years and had resulted in several tangible and intangible benefits
and had led to a number of spinoff projects.
A significant portion of scheduling and sequencing
problems in the steel industry can be formulated as zeroone integer programming
problems. Typically these applications cannot be solved using an exact
branch and bound approach. Vasko et al. (1993a)
discuss an intuitive user controlled variable tolerance approach to depthfirst
branchandbound algorithms. Several scenarios of a specific realworld
example problem illustrate how the parameters in the variable tolerance
approach have an impact on the solution quality and execution time.
The optimal design of production through a hot
strip mill is characterized by multiple and conflicting objectives. Jacobs,
Wright and Cobb (1988) propose an optimization model for this situation.
Considering the hot strip mill as an isolated facility, a "just in time"
delivery scenario is modeled as a goal program. A case study of the Burns
Harbor Plant is reported.
Newhart, Stott and Vasko (1993)
approach the optimal design of the supply chain in two phases, using a
mathematical programming formulation and a spreadsheet model. First the
mathematical programming and heuristic solution approach are used to minimize
the distinct number of product types held at different points in the supply
chain. Then a spreadsheet model is used to estimate the safety stock needed
to absorb the random fluctuations in both demand and the lead time throughout
the system. The implementation of this twophase approach allowed management
of Bethlehem Steel to quantify the effect of inventory required for locating
parts of the supply chain in different geographical areas. This study was
a critical factor used by top management to clarify a final decisionmaking
process.
Optimal assignment of structural steel shapes
to rail cars is an important logistics problem in the steel industry. Vasko
et
al. (1994) discuss an application that incorporates weight, dimension
and customer loading constraints. The formulation is a generalized binpacking
problem which is solved by modifying and extending previous algorithms.
It has been used extensively for one of the Bethlehem's high tonnage customers,
providing very good practical and implementable results that achieve the
desired goals.
Vonderembse and Haessler (1982)
present an effective algorithm for combining customer order sizes so as
to economically schedule the longitudinal ripping of cast slabs. This solution
process can assist decision makers in selecting master slab widths and
in designing width limitations for future casters. It entails more than
the minimization of trim loss, because other costs are relevant. This procedure
has been successfully used by the production control department.
Inventory Model for American Iron & Steel Institute
On the basis of the steel productmix optimization
model discussed by Fourer (1997), the importance of
inventories and the linkage between the time periods was investigated by
Hung
(1991). Data for the plate mill and the batch annealing process of
Bethlehem, Armco and LTV were used in an empirical study, sponsored by
the American Iron and Steel Institute. Relations between the inventory
level for plate mills and the batch annealing process were determined by
least squares and least absolute deviation regressions.
A twostep procedure for production scheduling
was also proposed. It first assigns slabs to each plate order and then
sequences the rolling jobs. The slab assignment was formulated as a linear
programming model with the objective of either maximizing yield, maximizing
revenue or maximizing profit. Both the optimal slab assignment and the
slab inventory mix are determined by the slab assignment model. The job
sequencing problem then finds a job sequence that fulfills the operational
constraints and also maximizes plate quality.
Dynamic Scheduling at Ensidesa
Steel in Spain
After building a new steel plant, Empresa National
Siderurgica implemented automatic control in various production sections,
giving the process computers continuous and complete information throughout
the production process. Making use of this information, Diaz
et
al. (1991) developed an automatic coordinating system for each
facility in the plant. In this system, the operator selects a set of heats
to produce and makes a predetermined production scheme from various preplanned
strategies. The system then arranges the heats accordingly and simulates
the delay and the idle times that could occur if the operator chooses that
scheme. Unlike some American steel plants (where the sequences last for
dozens of heats) the Spanish steel plants have short sequences (six or
seven sequences per day). As the sequences are short, the objective is
to maximize the time the casters are producing slabs.
Scheduling Model at a Canadian
Steel Plant
This work by Boukas, Haurie and
Soumis (1990) is an optimization model of productivity in a steel
plant subject to global energy constraints. The plant has four arc furnaces
and three continuous casting machines. In electric arc furnaces, the allocation
of energy, the fusion phase of the total production cycle, is of critical
importance. The problem is to define the start time and the duration of
a production cycle in combination with a power schedule which meets the
energy requirements of the different furnaces and a global power supply
limit for the whole plant. The problem is formulated as a combination of
an optimization problem and an optimal control problem. The authors have
proposed a twolevel algorithm which shows nine percent improvement in
productivity on some test data.
7. Set Covering Applications
In this section, we discuss applications of the set
covering approach in the area of assignment of slabs to orders, metallurgical
grade assignment, and selecting optimal ingot sizes. All studies in this
section have been reported at facilities of Bethlehem Steel.
Optimal Ingot Size Determination
After installation of a new ingot mold striping facility
in 1984, Bethlehem Steel developed a twophase procedure for selecting
optimal ingot dimensions, as reported in a series of publications
(Vasko, 1984; Vasko and Wilson,
1984a; Vasko and Wilson, 1984b; Vasko and Wilson, 1986; Vasko,
Wolf and Stott, 1987; Vasko and Wolf, 1988;
Vasko et al., 1989a). Previously,
Bethlehem had been using
about a dozen ingot mold sizes. Based on experience it was established
that any increase in the number of distinct mold sizes would result in
a significant increase in inventory and material handling cost.
The twophase procedure is used for selecting
the optimal ingot dimensions and internal mold dimensions. This procedure
also incorporates research in yield improvement and a variety of metallurgical
and operational constraints. Only marginal improvement would have been
possible if the old mold sizes had been retained. Phase I of the procedure
generates feasible ingot mold dimensions consistent with the constraints;
Phase II then uses a set covering approach to select, from the feasible
sizes generated, the ingot dimensions and ingot mold dimensions that minimize
the number of distinct mold sizes required to produce the finished products.
On the basis of the results of this model and trial mill tests, full production
use of new mold sizes influenced the entire plant operation and resulted
in annual savings of over $8 million.
Metallurgical Grade Assignment
Another application of the Phase II method
mentioned above is a metallurgical grade assignment model by Vasko et al. (1989b). The installation of
a continuous caster required an accompanying production planning and
control system. This module, responsible for assigning metallurgical
grades to customer orders, uses a minimum cardinality set covering
approach that not only minimizes the number of metallurgical grades
(required to satisfy a given collection of customer orders), but also
incorporates a preference for priority orders. The algorithm is used in
a twopass mode to quickly generate very good solutions to these large
scale (up to 1000 zeroone variables and 2500 constraints) optimization
problems.
Later papers (Woodyatt et.
al., 1992; Woodyatt et al., 1993)
have discussed the limitations of the above method and have suggested a
combination of set covering and fuzzy set methods. In order to use this
approach to assign metallurgical grades to a collection of customer orders,
metallurgists must first specify the set of all grades that satisfy the
requirements and specifications of those orders. However, the set of all
metallurgical grades that meet a customer's requirements is not well defined.
In their paper, the authors have discussed a methodology where each customer
order defines a fuzzy subset of the set of all metallurgical grades. They
have also defined a membership function that is based on the likelihood
of the grade meeting the customer specifications. The methodology addresses
the tradeoff between minimizing the number of grades used to produce a
collection of customer orders versus maximizing the likelihood that customer
specifications will be met.
Assigning Slabs to Orders
Another important problem in the steel industry is
the assignment of semifinished slabs to orders. Instances may be too large
(12000 to 16000 zeroone integer variables) to be solved in a reasonable
amount of computer time. Vasko et al. (1994)
have described a transportation formulation of the problem that can be
solved using a network optimization code. Then, using rounding heuristics,
the result can be used to provide a practical solution. The methodology,
formulations and algorithms are generic and can be used to solve a large
variety of set covering applications in steel and other industries.
8. Cutting Stock Problems
As reported by Tokuyama and Nomuyuki
(1981) of Sumitomo Metal Industries, Japan, the characteristics of
the cutting stock problems in the iron and steel industries are as follows:

There are a variety of criteria such as maximizing
yield and increasing efficiency.

Cutting problems are usually accompanied by inventory
stocking problems.
Practical algorithms that give near optimal
solutions in the real world have been developed. In their paper, Tokuyama
and Nomuyuki discuss applications to one dimensional cutting of large sections
and two dimensional cutting of plates. The following other applications
have also been reported.
Cutting Stock Optimization in
American Steel Plants
In a continuous caster, master slabs are produced
that are wider than the rolling mill can process. Haessler
and Vonderembse (1979) describe the master slab cutting stock problem
and present a linear programming based procedure for solving it. The primary
objective is to fill as many orders as possible
without generating any loss. This is realistic as the cut slab can be spread
and squeezed at the known limits at the rolling mills to obtain the desired
coil length. An example is presented and solved.
In a plate mill, surplus rectangular plates (flat
pieces of steel used in production of railroad cars, ships, and boilers)
of nonstandard dimensions are generated as byproducts of the batch steel
making process. An important implementation of the two dimensional cutting
stock problem is the application of customer plate orders directly to the
surplus steel plates. Although high yield cutting patterns for surplus
plates are very desirable, the following other considerations are also
important:
1. Cutting few orders from each surplus
plate (productivity reasons).
2. Cutting most of the high priority orders from
the plates (customer service considerations)
3. Cutting orders from a plate for as few distinct
customers as possible (logistical concerns).
Vasko, Wolf and Stott (1989)
and Vasko (1989) present a formulation in a fuzzy
environment that addresses these concerns. A solution procedure is outlined
and practical implementation at Bethlehem Steel's Sparrows Point Plant
is described in Vasko, Wolf and Pflugrad (1991).
The plant can produce narrow width customerplate orders (typically 10
to 24 inches) efficiently when its 60 inch plate mill is not operating.
The heuristic procedure is used to map these orders into mother plates
for production in the 160 inch plate mill. This procedure was implemented
as a module in the plant's production planning and control system and has
been used daily to generate mother plate dimensions and cutting patterns.
In another application, Vasko
et
al. (1992) discuss a method that combines set covering and cutting
stock applications for improving Bethlehem Steel's customer service. Some
of the customer orders are slit from master coils into a number of narrower
and smaller coils to fit specific manufacturing needs. To serve these customers,
Bethlehem has developed a mathematical model that generates optimal coil
widths and slitting patterns. The model has the following objectives:
1. Minimize the number of slitter setups
2. Maximize the material utilization
3. Generate minimum excess inventory
4. Generate minimum shortfall against forecast
demand
The linear program also generates coil widths that
optimally utilize the company's facilities. This system is viewed by the
customers as a value added service provided by Bethlehem Steel.
Vasko and Wolf (1994) address
the problem of determining what rectangular sizes should be stocked in
order to satisfy a bill of materials composed of smaller rectangles. They
first generate a large number of stock sizes ideally suited to the bill
of materials; then they use an uncapacitated facility location algorithm
to consolidate the stock sizes down to an acceptable number. Once the solution
of finding rectangular stock sizes is known, a second program is used to
map the bill of materials onto plates of the chosen sizes. The practicality
of the approach is demonstrated by generating a cutting plan for a real
world bill of materials having 392 distinct order sizes and over 7700 order
pieces.
In a mill finishing a structural shape such as
an Ibeam, once the final product is produced, it is cut according to the
customer's order length. The actual length may not be known precisely until
just before cutting. Also if the production rate of the mill is higher
than the cutting rate of the bars, then trying to generate cutting patterns
with the number of cuts per bar close to the average number of cuts per
bar will maximize primary saw (hotsaw) cutting and reduce the number of
cuts that have to be made at the secondary saw (coldsaw). Vasko
et
al. (1993b) discuss a branchandbound algorithm that generates
high yield, balanced cutting in real time based on the precise length of
the bar leaving the mill and arriving at the saw.
Cutting Stock Applications in
a German Steel Plant
Pohl and Kaiser (1982) develop
a cut length optimization program for the computer controlled Siege GeisWeid
AG rolling mill. They describe a procedure for cutting the rolling strand
lengths into marketable lengths. The total rolling strand length is computed
by comparison of volume and speed of billets, merchant bars (after the
first rolling block), and finished products. The speeds and lengths are
determined by measuring rollers in the front part and without contact at
the rear end of the mill. The cooling bed lengths are divided according
to the optimization computation and are conveyed under computer control
to two cuttingoff machines, which cut into marketable finished lengths.
9. Other Applications
The continuous casting machine can be used to eliminate
a number of processing steps associated with the traditional ingot/bloom
based production sequence. However, a given continuous caster can produce
only a small number of bloom thicknesses. This creates a problem for selecting
those continuouscaster configurations that would maximize utilization.
Vasko
and Friedel (1982) present a dynamic programming formulation that maximizes
the cast bloom tonnage that can be processed through one of the Bethlehem
Steel's finishing mills. Without the aid of such a model, selecting the
highest productivity would have two conflicting considerations. The first
factor is that as the number of casterproduced bloom thicknesses increases,
the caster setup time and the configuration complexity increases. The second
factor is that as the number of thicknesses decrease, the cast tonnage
processed through the finishing mill is reduced, owing to reheating furnace
and cooling bed limitations. The model results were transmitted to the
plant management and were used in conjunction with other information to
determine the most economic caster configurations.
The ElectroSlag Remelting (ESR) process was developed
for melting special alloys that were difficult to produce in conventional
electrical arc furnaces. Gower, Hahn and Tarby (1970)
describe an application of dynamic programming simulation to determine
an ESR operating policy that is predicted to maximize cumulative profit
over a number of stages.
10. Conclusion and Extensions
Although steel is a basic industry for the growth
of a nation, relatively few applications of mathematical programming have
been reported in comparison with other industries such as oil, airlines,
and semiconductors. Also, very little work has been done in the area of
inventory control and manufacturing control for steel plants. However,
it is noteworthy that four applications (Jain, Stott
and Vasold, 1978; Box and Herbe, 1988; Vasko
et
al., 1989a; and Sinha et al., 1995)
have been selected as finalists in the Management Science Achievement Award
(Edelman Competition). This gives an indication of the potential financial
benefit of applying optimization techniques to the problems of the steel
industry.
From the survey of different applications and
our personal experience in the modeling of steel plants, the following
can be considered as potential areas for future work:
1. Simultaneous optimization of productmix, inventory
and transportation problems over multiple periods. This would represent
an extension of Fabian (1958) to the multiperiod
case with inventory and transportation requirements as additional constraints.
2. Cutting stock optimization to maximize overall
yield of multistage production processes. This would go beyond most previous
work on the cutting stock problem, which has used single stage models.
3. Scheduling problems in the continuous caster.
4. Stochastic linear programming models where
not only the means and variances of the stochastic entities but also their
distributions are known.
5. Any research that increases the reliability
and validity of the data. The success of mathematical programming models
depends heavily on availability of relevant data. Often the desired
data does not exist, or must be collected from multiple sources.
Glossary
For a much more detailed glossary, see Everything You
Always Wanted to Know About Steel . . . A Glossary of Terms and
Concepts by Michelle Applebaum.
Billets: Mostly square steel shapes in
the range of 50mm x 50mm to 125mm x 125mm. They may be semifinished
or finished products depending on the customer. Blooms are rolled into
billets.
Blast Furnace: A facility that coverts
iron and other raw materials to hot metal (liquid iron at a very high
temperature). A typical blast furnace is about 30 m high and produces
500 to 10000 tons of hot metal per day.
Blooms: Steel shapes that have a
crosssection smaller than ingots but larger than billets. They are
square or slightly oblong, mostly in the range of 150mm x 150mm to
300mm x 300mm. Ingots are rolled into Blooms.
Coils / Wire Rods: The smallest round
sections of steel that can be produced by hot rolling. The sizes of
rods vary from 5.5mm to 12.7mm. Generally, rods are wound into coils of
about 760mm inside diameter that weigh from 450 to 2000 kilograms.
Continuous Caster: A facility between
the basic oxygen furnaces and the rolling and finishing mills. It casts
slabs and billets directly from the liquid metal, bypassing the ingot
stage.
Heat: A batch of liquid steel, varying
from about 50 tons to 300 tons depending upon the technology and type
of the blast furnace.
Hot Strip Mill: The rolling mill that
reheats and rolls steel slabs into hot bands, steel strips that are
typically 0.10 inches thick and 50 to 60 inches wide.
Ingot: Individual shapes cast by
pouring liquid steel into individual molds. With continuous casters
becoming more and more common in steel making, ingots are tending to
become obsolete.
Ladle: A ceramiclined open container
used to transport and hold a heat of molten steel.
Mixer: A reservoir for storing and
heating hot metal from a blast furnace before it is sent to subsequent
production steps. Its purpose is to maintain consistency in the
composition, variation, and temperature of the hot metal.
Pig Iron: The metallic product of the
blast furnac,e containing over 90 % iron.
PusherScraper: A machine used to
transport raw materials, like iron ore, from one point to another. Some
pushers are also used to take coke out of the coke ovens.
Rollers: The objects through which
ingots are passed to produce finished steel. Rollers are also used in
other production steps to reduce cross sectional area of the product.
Slab: The intermediate product from a
continuous caster or a roughing mill. It is always oblong in shape,
mostly 50 to 230mm thick and 610 to 1250 mm wide.
Slag: The fusible material formed by
the chemical reaction of a flux with gangue of an ore, with ash from a
fuel, or with impurities oxidized during the refining of a metal.
Surface Quality: The presence or
absence of flaws in the surface of steel strip or sheet. The incidence
of these flaws are extremely sensitive to the process of steel making
and/or slab casting or Ingot teeming.
Teeming: The process by which molten
steel is poured into ingot moulds.
Thermal Chamber Lateral Deformation:
Deformation of refractories due to high temperature.
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BioData of Authors
Goutam
Dutta teaches Quantitative Methods and Operations Management at the
Indian Institute of Management,
Ahmedabad. He earned his Ph.D. in 1996 in Industrial Engineering
and Management Sciences from Northwestern University. He was a faculty
member at the London School of Economics in 19961997. His research
interests are OR practice, optimization, system dynamics, decision
support systems, realworld applications of MS/OR models, and OR in
developing countries. He won (jointly) the IFORS Prize for OR in
Development in 1993 and the Franz Edelman Prize in 1994. He chaired the
IFORS OR for Developing Countries Prize Competition in 1999.
Robert
Fourer is Professor of Industrial
Engineering and Management Sciences at Northwestern University. He holds a B.S
in Mathematics from MIT, and a Ph.D. in
Operations Research from Stanford University. His research interests encompass all aspects of optimization algorithm and software design. He is codeveloper (with David Gay and Brian Kernighan of Bell Laboratories) of the AMPL modeling language, and is coauthor of AMPL: A Modeling Language for Mathematical Programming; his work in this area was recognized by the 1993 INFORMS Computing Society Award. His work on planning models for steel production has been supported in part by the American Iron and Steel Institute and several member companies.
