A Survey of 
Mathematical Programming Applications 
in Integrated Steel Plants

Robert Fourer
Department of Industrial Engineering and Management Sciences
Northwestern University
Evanston, Illinois 60208-3119, U.S.A.

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ABSTRACT

Copyright © 1998-2001 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  Draft: Please do not quote without the authors' permission.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
   
• 2. An Overview of an Integrated Steel Plant
   
• 3. National Steel Planning Models
  • National Steel Planning Model in the United States
  • Mexican Steel Model
  • Stochastic Programming Model for Investment Planning in India
     
• 4. Product-Mix Optimization Models
  • Product-Mix Model at Kaiser Steel Company
  • Large Scale Database Model for American Iron & Steel Institute
  • Models for Production Planning in the United Kingdom
  • German Model at Hoesch Siederlandwerke
  • Product-Mix Optimization Models in Indian Steel Plants
  • Models of Production Planning in Zambia
  • Model of Production Planning in Algeria
     
• 5. Blending Models
  • Blast Furnace/Cupola Blending Models in the United States
  • Blending Model in the United Kingdom
  • Blending Model in Belgium
  • Blending Model in Sweden
  • Blending Models in East European Countries
     
• 6. Scheduling, Inventory and Distribution Models
  • Scheduling Model at LTV Steel
  • Scheduling Models at Bethlehem Steel
  • Inventory Model for AISI
  • Dynamic Scheduling at Ensidesa Steel in Spain
  • Scheduling Model at a Canadian Steel Plant
     
• 7. Set Covering Applications
  • Optimal Ingot Size Determination
  • Metallurgical Grade Assignment
  • Assigning Slabs to Orders
     
• 8. Cutting Stock Problems
  • Cutting Stock Optimization in American Steel Plants
  • Cutting Stock Applications in a German Steel Plant
     
• 9. Other Applications
   
• 10. Conclusion and Extensions
   
• Glossary
   
• References
  • Bio-Data of Authors

1. Introduction

• National steel planning
• Product-mix optimization
• Blending in blast furnaces, coke ovens or steel foundries
• Scheduling, inventory and distribution
• Set covering
• Cutting stock optimization

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 iron-making 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 3-9.

2. An Overview of an Integrated Steel Plant

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 tube-making) or into structurals in the Medium and Light Structural Mill.

3. National Steel Planning Models

National Steel Planning Model in the United States

Mexican Steel Model

Stochastic Programming Model for Investment Planning in India

4. Product-Mix Optimization Models

Product-Mix Model at Kaiser Steel Company

Large Scale Database Model for American Iron & Steel Institute

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

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

Product-Mix Optimization Models in Indian Steel Plants

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 non-essential 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 round-the-clock 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 multi-product 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, semi-finished steel, and finished steel. Another interesting option among these five steel plants is that of inter-plant 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

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 route-wise and quality-wise costing at the melting shop.

Model of Production Planning in Algeria

5. Blending Models

Blast Furnace/Cupola Blending Models in the United States

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

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

Blending Model in Belgium

Blending Model in Sweden

Blending Models in East European Countries

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

Scheduling Model at LTV Steel

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 250-ton 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 pseudo-cost 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

Stott and Douglas (1981) describe a scheduling system for ocean-going 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 spin-off projects.

A significant portion of scheduling and sequencing problems in the steel industry can be formulated as zero-one 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 depth-first branch-and-bound algorithms. Several scenarios of a specific real-world 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 two-phase 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 decision-making 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 bin-packing 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

A two-step 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

Scheduling Model at a Canadian Steel Plant

7. Set Covering Applications

Optimal Ingot Size Determination

The two-phase 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

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

8. Cutting Stock Problems

• There are a variety of criteria such as maximizing yield and increasing efficiency.
• Cutting problems are usually accompanied by inventory stocking problems.

Cutting Stock Optimization in American Steel Plants

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 by-products 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).

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

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 I-beam, 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 branch-and-bound 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

9. Other Applications

The Electro-Slag 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

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 product-mix, inventory and transportation problems over multiple periods. This would represent an extension of Fabian (1958) to the multi-period case with inventory and transportation requirements as additional constraints.

2. Cutting stock optimization to maximize overall yield of multi-stage 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 semi-finished 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 cross-section 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 ceramic-lined 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.

Pusher-Scraper: 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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