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Integration of Continuous Caster and Hot Strip Mill Planning for Steel Production 9/11/99

Integration of Continuous Caster and Hot Strip Mill Planning for

Steel Production

Peter Cowling∗, Wafa Rezig†

Abstract

In this paper we present a model and a solution approach for integration of steel continuous

casters and hot strip mills to provide more responsive steel production at lower unit cost. We

describe the production environment and survey existing methods for planning continuous

casters and hot strip mills. Since these processes lie at the solid/liquid interface we use “virtual”

slabs, corresponding to possible solid forms of liquid steel, in order to integrate schedules for

these two processes. We model the planning problem as a hybrid network. Our model is solved

using a combination of mathematical programming and heuristic techniques and we show that

the solutions provided are very nearly optimal. The approach which we describe has been

implemented at several steel mills worldwide and has demonstrated significant savings.

1. Introduction

The steel industry faces increasing competitive challenges. Throughout the 1990s, world-wide

production of crude steel has remained static at about 750,000,000 tonnes per year whilst the

price of steel, as measured by the European Union’s steel price index, fell by more than 7%

between 1990 and 1996. At the same time, the European Union’s price index for all industrial

output, excluding construction, rose by 7%. Between 1987 and 1996 wages have increased by

50% in some European Union countries, while the number of people employed in the European

iron and steel industries has fallen from 420,000 to 300,000 in the same period. About 88% of

all European steel production passes through the continuous casting and hot rolling processes

which we will describe in this paper, with older ingot casting methods becoming less

important. The European average export price of hot rolled steel coils has fallen from 370 ECU

per tonne in 1989 to 263 ECU per tonne in 1996 (Eurostat 1997). Against this background of

reduced revenues and increased costs, steel industries have had to drastically cut costs whilst

improving product quality and customer focus. In particular European steel manufacturers have

∗ School of Computer Science and IT, University of Nottingham, Nottingham NG7 2RD, UK. Email:[email protected]

† A. I. Systems, J. Wybran Ave, B-1070 Brussels, Belgium. Email: [email protected]


had to use improved quality and a more responsive position towards customers in order to

maintain their market position relative to imported steel, particularly from the far east which

continues to have much lower labour costs than Europe. One of the tools in the drive to cut

costs, increase quality and improve customer satisfaction has been improved planning and

scheduling systems. This paper will describe a generally applicable decision support system for

integrating the planning and scheduling of continuous caster and hot strip mill.

We will consider the operations of a typical integrated steel plant, which uses raw materials

such as iron ore, coke, scrap steel and alloying materials to produce finished products such as

steel plates and coils. Liquid steel is cast into a solid steel band using a continuous caster. The

band is then cut into slabs. The slabs are rolled into coils in the hot strip mill. In the usual

case, which we consider in this paper, there is a slabyard where slabs may be held between

casting and rolling so as to decouple these two processes which have very different scheduling

constraints and objectives. In addition to the production pull due to customer requirements for

steel coils, there is a production push due to the necessity to cast all liquid steel into slabs, since

the liquid steel cannot be stored. These processes will be described in more detail in the next

section. As has been indicated above, the majority of steel production passes through this

processing route. Two recent surveys point to the need for improved integration between

continuous caster planning and hot strip mill planning for improved operating performance.

Dorn (1996) discusses the application of expert systems in the steel industry, pointing out that,

despite considerable improvements in scheduling systems for individual processes, a lack of

co-ordination between these scheduling systems is an important gap which needs to be filled.

Haspel (1995) discusses the need to increase the planning horizon, saying that “It is the

optimisation of the plant with respect to product quality and related process factors on a long

term basis that is the key to the ultimate level of savings arising from the application of

optimising technology.”

We have developed detailed models of the processes which govern the planning of both the

continuous caster and the hot strip mill (Baccus, Cowling, Vaessen, Van Nerom 1995,

Cowling 1995, A.I. Systems 1995, A. I. systems 1997) in order to consider how we may

arrive at an integrated plan for both of these processes for a period of two to three weeks.

Scheduling of the continuous caster is governed by the production push of liquid steel, which

must be cast as it arrives, whereas hot strip mill scheduling is governed principally by the

production pull of customer orders for coils of steel. As we will see later, the constraints

governing these two processes are entirely different. The short term scheduling problem for

each process considered in isolation is itself difficult, containing a wealth of NP-hard packing,

sequencing and scheduling problems (Garey and Johnson 1979). The complexity of each

process leads to a situation where we must consider short term scheduling constraints in order

to arrive at realistic medium term plans. Improved integration between continuous caster


scheduling, hot strip mill scheduling and medium term planning will improve reliability of the

planning process and give greater confidence that planned targets may be achieved. We realise

this integration through the use of “virtual” slabs that represent ways in which client orders

may be cast. These may be scheduled with real and planned slabs at both the continuous caster

and the hot strip mill to determine when, and to what specification, each client order should be

sent to the caster and what type of schedules will be required. The model we have developed

may be solved quickly to near optimality using heuristics and mathematical programming

techniques. Since the solution time is low, the model may be run repeatedly in order to arrive at

a compromise between increased customer satisfaction, due to fulfilment of coil orders in the

hot strip mill, and reduced production costs and quality improvements due to improved

integration of continuous caster and hot strip mill, leading to better schedules for each process.

The models and heuristics, which we describe here, have been implemented at several steel

mills and have made significant savings for these mills.

In section 2, we will describe the production environment for the primary steel making process

from the continuous caster to the hot strip mill. In section 3 we will give an overview of the

different techniques for planning both continuous caster and hot strip mill, followed by a

description of the automatic coffin selection problem central to the synchronisation of the two

processes. Section 4 outlines the mathematical formulation of the problem. Section 5 presents

the heuristic methods we have developed to handle this kind of model efficiently. Section 6

gives the main experimental results and we present conclusions and directions for further work

in section 7.

2. The Production environment

Most steel production facilities have a make-to-order production philosophy. However, large

differences between batch size, which may be 300 tonnes of liquid steel, and order size, which

may be as small as a few tonnes, in addition to a wide variety of manufacturing processes

having different technological constraints, mean that stock is inevitable. The systems that we

have developed are for this type of environment.

The processes involved in steel making are illustrated in figure 1. A flow of molten iron is

maintained by the blast furnace. Stopping this process is enormously expensive, so each blast

furnace usually produces iron continuously for more than ten years. The molten iron, or pig

iron, then passes into the primary steel making processes. The principle processes for primary

steel making are basic oxygen or electric arc furnace, ladle treatment facility, continuous caster

and hot strip mill. These processes are the subject of our planning system and we will discuss

them in detail later. After passing through the primary steel making processes the steel coils


may then pass through further finishing processes, such as pickling to remove surface oxides

and annealing to improve mechanical properties.

The primary steel making processes are concerned with the transformation of liquid iron into

semi-finished solid steel products, via a complex manufacturing process. This is an unusual

manufacturing situation and is the reason why planning and scheduling techniques which have

successfully been applied in other manufacturing industries, such as MRP and ERP systems,

have enjoyed little application to primary steel making processes. Capacity planning tools,

which are unable to take account of the complex scheduling constraints of the primary

steelmaking process, have not achieved success. Finishing processes, which involve the

transformation of solid steel via manufacturing processes, have been more successfully tackled

using traditional capacity planning techniques (Lee, Murthy, Haider and Morse 1996).

Blast Furnace

Electric Arc Furnace

Basic Oxygen Converter

Ladle treatment

Continous Casting

Hot Strip Mill

Section Mill

Tube Mill

Liquid Iron

Liquid Crude Steel

Ladles of liquid

Steel

Slabs Coils

Sectional

Steel

Tubes

Ores, coke, gas, coal

ScrapFi

ni

sh

in

g

Figure 1. Steel making environment

In the basic oxygen furnace, oxygen is passed through the molten iron to reduce the carbon

content. Scrap steel and alloying additives may also be added at this stage. The resulting raw

steel is then processed further in a ladle treatment facility which aims to produce molten steel of

the correct grade or chemistry by subjecting the steel to processes which reduce carbon content

and adding alloying additives such as nickel and manganese. A customer may be interested

only in the mechanical properties of the steel, which may be achieved by many different grades.

The customer is often indifferent to which of these grades he finally receives.

Casting is the process where liquid steel is brought into defined shapes, dimensions and

weights. Slabs are used to make flat coiled products and plates etc. They are typically 150-250


mm thick, 500-2000 mm wide and 10-20 meters long. Blooms and billets have smaller width

and thickness dimensions and are used to make long products such as pipes. During

continuous casting, a ladle of liquid steel or heat weighing up to 300 tonnes passes from the

ladle, via a tundish with an adjustable discharge device into a water-cooled copper mould. The

shape of this mould defines the shape of steel. Before casting, the bottom of the mould is

sealed with a so-called dummy bar. As soon as the bath of liquid steel reaches its intended steel

level, the mould starts to oscillate vertically in order to prevent the strand adhering to its walls.

The red-hot strand, solidified at the surface zones, is drawn from the mould, first with the aid

of a dummy bar, and later by driving rolls. Because of its liquid core, the strand must be

carefully sprayed and cooled down with water. It must also be supported by rolls on all sides

until it has completely solidified. This prevents the still thin rim zone from disintegrating. The

refractory lining of the tundish and the pipe through which the molten steel travels have a

limited life and must be replaced every few heats. Certain grades of steel result in more wear

than others. The mould often has adjustable width and the width may evolve gradually during

casting in order to cast a range of different dimensions. In general the width will decrease while

the sequence is cast due to the potential problems associated with widening out. When the steel

strand is made wider it is possible that the molten steel may break out of the thin solidified

shell, which will require a substantial and costly clean up operation. Certain grades have a

degree of compatibility and may be cast consecutively, giving rise to one or more slabs with

mixed chemistry. Other grades are incompatible and must be separated by stopping the casting

process or carrying out a tundish change, to avoid mixing. Once it has completely solidified,

the strand can be divided into slabs by making longitudinal slits and lateral cuts. Intensive

cooling leads to a homogeneous solidification microstructure with favourable technological

properties. Casting speeds depend upon dimensions and the number of strands that are

simultaneously cast. Typically, speeds about 0.6 to 3.5mm per second are possible for slabs.

Customer orders for coils will usually specify only coil dimensions and mechanical properties.

This gives rise to a situation where any given customer’s order may be cast at one of several

grades and at a range of dimensions. The flexibility of the slab dimensions depends upon the

ability of the hot strip mill to stretch and squeeze the slabs into coils of the correct dimensions.

The flexibility in grade means that orders can be batched together more easily, which is very

useful given that orders may be for several tonnes only and a heat is of a single grade and may

weigh 300 tonnes. Certain customer orders may be upgraded, i.e. they may be made at a

higher quality grade than that ordered by the customer. Whilst this may occasionally be useful

for batching, it is undesirable since it will reduce the profit on the order and increase customer

expectation. A number of production goals have to be met, particularly the minimisation of

stock production and order upgrading, the respect of orders due dates and priority, the

maximisation of the throughput through each tundish and minimisation of the downtime due to


stopping and starting the casting process. Creating stock is inevitable given the large difference

between batch size and order size. When we have finished all orders that fit into a grade-width

combination, we have to assign orders of a lower quality (upgrading) or create stock.

The hot strip mill transforms steel slabs into steel coils by subjecting the slabs first to high

temperatures in a furnace and then to high pressures through a series of rolls. Slabs may be

hot-charged directly from the continuous caster into the furnace or may be allowed to cool in a

buffer known as the slabyard. Significant energy savings may be realised by hot-charging.

Work rolls are in direct contact with the hot steel strand, and backup rolls exert pressure on the

work rolls. The work rolls become worn quite quickly and so planning takes place in shifts of a

few hours. In general the some or all of the work rolls which are in contact with the steel band

are replaced at each shift and larger backup rolls are replaced every few weeks. If the work

rolls are not replaced sufficiently often, then wear to them will mark subsequent steel coils. If

the backup rolls are not replaced sufficiently often, then wear to them will mark the work rolls

and subsequent steel coils. Rolls themselves are expensive and the time lost in production while

replacing rolls is also expensive, so it is desirable to maximise the throughput between roll

changes, while still maintaining coil quality. When there is a difference between the width,

thickness or hardness of two consecutive coils in the sequence for a given shift, the pressure

settings of some of the rolls must be changed. Changing these settings introduces a risk that

coil quality standards may not be attained. This risk increases with the magnitude of the jump in

dimensions and hardness between two coils. The marking which occurs on each work roll

occurs right across the area where the roll is in contact with the strand, which limits the overall

length of the coils milled before a roll change must occur. The marking is especially

pronounced at the edges where the band meets the roll and this wear means that there must be

an overall tendency for the coils milled in a given shift, between work roll changes, to decrease

in width.

We will call the sequence of slabs to be rolled in a particular shift a programme. Each

programme must have a particular ‘shape’ due to technical properties of the mill. Initially, we

must warm up the rolls with easy (narrow, thick, soft, high tolerance) coils. Then we roll

difficult (wide, thin, hard, low tolerance) coils, while the rolls are fresh, before decreasing

width due to marking at the edges where a coil meets each roll (see figure 2). We end up with a

particular shape, which is somewhat morbidly referred to as a coffin shape or simply coffin.

The term coffin arises since the profile of the widths in a programme tends to be in a shape of a

coffin. The coffin defines which slabs may be rolled and the shape of the width, thickness,

hardness and tolerance profiles as well as the overall coil length for the shift. Coffin shapes are

defined by a rolling expert, taking into account the production capabilities of the hot strip mill

and commercial objectives. The engineering processes which govern the milling process, and


in particular the roll wear, are not completely understood so the rolling expert must bring his

own experience to bear in deciding which coffins are possible from a production point of view

and viable from a commercial point of view. It is then clear that each steel plant will have a

different view as to which coffin types are possible. The population of possible coffin types

will also change dynamically in response to changes in plant machinery, understanding of

production processes and commercial factors.

time

Programme 1Programme 2

Coilwidth

Rollchange Roll

change

Rollchange

Figure 2. Programme Width Profile.

The scheduler and medium term planner of the hot strip mill have to make a choice of which

coffin types will be made over the current planning horizon. The choice of which coffin types

to make is affected by several considerations. Commercial considerations, particularly coil

priority and due date, must be considered. Steel availability in the slabyard buffer and from

planned continuous caster production determines the material that may be sequenced. Energy

use in the furnace must be minimised through hot charging and pairing up half-length slabs,

which would otherwise take up the same furnace area as a full sized slab. The situation of the

hot strip mill machinery must also be considered. For example, it may be impossible to roll

certain coils due to backup roll wear. Certain, difficult-to-roll coils should be rolled whenever

possible. Other coils, which may be placed early in a sequence and used to warm up the mill

machinery, must be conserved for use in later coffin sequences.

In some plants the choice of coffin is carried out on a regular basis, with only one or two shifts

being planned at a time. In others a fixed weekly sequence may be adopted. The choice of

which coffin type to make affects the range of customer orders which may be rolled in a given

coffin. Taking into account all these factors it is not easy for a human planner to select even a

single coffin type, let alone create a medium term plan. Furthermore, when planning several

programmes at one time, it is difficult to maintain approximately equal quality for each of the

programmes planned. All of the conflicting objectives and constraints must be taken into

account to arrive at a reasonable schedule. Deciding which coffins are to be milled is a difficult

process when considered in isolation and becomes even more difficult when we must also take


into account the possibility of influencing future continuous caster production and balancing the

amounts of steel sent to each of the downstream finishing processes.

3. A survey of techniques for planning continuous caster and hot strip mill

In most steel companies, the principal production planning and scheduling techniques are

essentially manual techniques with little or no computerised decision support beyond that

provided by a central mainframe database. These manual techniques are based on the know

how and the professional experience of expert people who have worked in the plant for years.

Recently there have been several attempts to introduce automation and optimisation principles to

handle the planning and scheduling problems within a steel plant because of the complexity of

the problems involved in steelmaking, on one hand, and the great possibility of saving energy

and money, on the other hand. Other benefits from computerised decision support may include

reduced planner workload, reduced company dependency on key planning staff, more uniform

plan quality and a greater ability for high level managers to influence lower level planning and

scheduling decisions. We will compare here the manual techniques in common use with recent

decision support systems from the academic literature and in the commercial domain, which

support the planning and scheduling of continuous caster and hot strip mill.

3.1. Continuous caster planning

The complex medium term planning problem for the continuous caster is simplified by the

planner in order to arrive at a problem which is tractable to a manual solution process. For

example width changes may be disallowed within a sequence, or tundish changes may be

planned infrequently or never, or grades may be grouped into artificial categories for

sequencing, with grade incompatibilities being resolved on the factory floor. These simplified

plans are often hard to precisely implement and this makes integration with upstream and

downstream processes difficult. The push of molten steel drives such a planning system, with a

“find something to do with it when it arrives” philosophy.

The potential for computerised decision support in this area is highly significant, which may

lead to better plan adherence and hence integration with upstream and downstream processes, a

reduction of stock production, less quality upgrading, longer and better casting sequences and

improved meltshop management. By increasing the range of constraints and objectives which

may be taken into account and planning over a longer time horizon, we may provide a solution

which may result in globally better plant operations, leading to lower costs, higher quality and

better customer responsiveness.


Numao and Morishita (1991), Numao (1994), Dorn and Slany (1994), Slany (1996), and Dorn

and Shams (1996) describe expert systems approaches to solving the short-term scheduling

problem for steel making and continuous casting. These approaches attempt to capture some

useful subset of the knowledge of a human scheduling expert in a computer and use simple

heuristics, while possibly interacting with a human planner, to guide schedule construction.

Lally, Biegler and Henein (1987), Vaessen and Van Nerom (1994) and Tong, Silverman and

Clausen (1994) use optimisation techniques for continuous caster planning. All these

approaches consider this problem in isolation, where there is no communication and integration

with other planning systems or reaction to real-time events. Indeed Dorn and Shams (1996)

recognise this as one of the obstacles to using their system in practice. Notman, Tullo, Neesam

and Poulter (1993) describe the Merlin system, which provides a graphical user interface to aid

manual generation of steel making and caster schedules, with a 9 day planning horizon. It aids

the scheduler when real-time events occur with a limited schedule repair capability.

3.2. Hot strip mill planning

The information that is used to make schedules for the hot mill comes from several sources.

Data concerning the slabs currently in the slabyard and client orders are usually held in the plant

computer systems. Caster planning and scheduling systems provide information as to which

slabs will arrive in the slabyard in the immediate future. However, it is not always possible for

caster schedules to be met and this information may be unreliable. Other rules and procedures

governing the technical capabilities of the mill are usually known by a small group of engineers

and schedulers. These rules change rapidly in response to conditions in the plant and external

forces. Other inputs to the scheduling process come from sales and marketing divisions and

from strategic directives emanating from higher management.

The planner must first decide on the coffin type to be rolled. This higher level decision must be

taken sufficiently in advance for the necessary mill preparation, for example the installation of

work and back up rolls, to be possible. The choice of coffin is a difficult decision. A system

for supporting this decision can result in significant savings and better plant integration.

Typically the manual planning of a coffin to be rolled over a single eight-hour shift will take the

scheduler a couple of hours. The long planning time means that it is difficult to react to

unforeseen production events, particularly on night and weekend shifts when a scheduler might

not be available. When several programme schedules are generated at the same time, we

observe a steady decline in schedule quality. Too much scheduler time is spent in the mundane

activity of coil sequencing, when this time could be better spent investigating higher level

issues of mill scheduling (Cowling 1999).


One of the most difficult tasks, which the scheduler must carry out in formulating a detailed

schedule, is to provide a sequence that minimises the changes in width, thickness and hardness

between consecutive slabs. This sequencing task is related to the travelling salesman problem

(Lawler, Lenstra, Rinooy Kan, Schmoys 1985) which is a well-studied problem of

combinatorial optimisation. There have been several papers written about decision support for

hot strip mill planning. Short-term planning systems for the hot strip mill in the literature may

be divided into two groups. Jacobs, Wright and Cobbs (1988), Balas and Martin (1991),

Sasidhar and Achary (1991), Petersen, Sørensen and Vidal (1992) and Assaf, Chen and

Katzberg (1997) all use mathematical programming techniques to solve simplified models of

the hot strip mill scheduling problem. Cowling (1995), Stauffer and Liebling (1997) and

Lopez, Carter, Gendreau (1998) propose more generally applicable models which are solved

using heuristic optimisation techniques. Once again all these models treat the hot strip mill as an

isolated problem, without consideration of real-time events or communication with external

planning systems. Using a simple model produces more nearly optimal solutions, which may,

however, be harder to implement in practice. Using a more complex model presents no

absolute guarantee of distance from optimality in the model, but gives results that are usually

easier to apply in practice. Each of these approaches has its merits and deficiencies.

3.3. Integration of Continuous Caster and Hot Strip Mill Planning

Lee, Murthy, Haider and Morse (1996) consider a prototype planning system tailored for LTV

Steel, which generates a weekly aggregated schedule for continuous casters and hot strip mills

simultaneously. The generated plan is static and cannot react to real-time information. The

problem at LTV steel is largely driven by casting constraints, with the hot strip mill as

subordinate. There is no concept of communication between the processes to arrive at a

compromise schedule. Dorn and Kerr (1994) discuss a system for co-ordination HSM and CC

schedules in the short term using fuzzy sets to model all constraints. This interesting approach

has been applied to a simple model of each process.

We have developed a system for integrated planning of the continuous caster and hot strip mill

in order to achieve a better synchronisation of steelmaking, casting and rolling. The connection

between the two production systems yields a reduction in slab inventory, more responsive and

predictable production planning, improved quality and reduced production costs through longer

and better continuous caster and hot strip mill sequences between set ups. The integration of the

two systems is achieved by selecting the orders to be fulfilled over a medium-term planning

horizon of several days. Hence we allow the production “pull” of (semi-) finished coil orders to

exert a greater influence on continuous caster planning, while still dealing with the production

push of liquid steel. To demonstrate that the selection is realisable we must choose the coffin


types that will be used in the hot strip mill to roll the selected orders. In addition to the usual hot

strip mill constraints and objectives, this coffin choice must be further subject to constraints

from the casting and finishing processes. This is performed by the automatic coffin selection

module. This module chooses the best coffins over a short time horizon (1-2 days, or about 3

coffins), the medium time horizon (3-5 days, or about 12 coffins) and especially the long time

horizon (15 days, or about 25 coffins). The coffins are selected according to the optimisation of

executed orders based on information from the production environment. We illustrate the

integration between the continuous caster and hot strip mill planning in figure 3.

Production Orders

Virtual slabyard Real slabyard Coffins Coffi n configuration Group constraints

Coff in sequencePlanned slabs

Slab generation

Automatic coffin selection

Casting constraints/objectives

HSM-planner CC-planner

Planned coils

Figure 3. Automatic coffin selection system.

The starting point is the order book with the customers’ orders in terms of physical and

chemical properties, tolerances, quantity and priority. It is also common practice to place

anticipated future orders, particularly for internal customers, in the order book. The second

level is formed by the data needed by the automatic coffin selection module to generate the best

coffin sequence over a fixed time horizon. The real slabyard consists of physical slabs in the

slabyard as well as slabs which are planned in the current caster schedule. In parallel, the coffin

selection uses a virtual slabyard, which is created from coil orders not in the current caster

schedules. Each virtual slab corresponds to a customer order. Since many customer orders can

be made from real slabs having a variety of different chemical grades and physical dimensions,

each virtual slab may have a range of possible grades and dimensions. Hence the effect of a

virtual slab is to indicate with sufficient flexibility to both caster and hot strip mill scheduling

processes the sort of material that is required. The caster scheduling process can use this

indication to produce a schedule which makes use of the flexibility of the definition of a virtual


slab to create a good caster schedule, while the hot strip mill will create schedules of improved

quality since the realisation of each virtual slab will fit well into the coffin types chosen for

sequencing. These virtual slabs are the key to effective integration of the two production

processes.

The coffin selection module also needs information about all the available coffin types as well

as the coffin selection configuration. The automatic coffin selection module divides time up into

discrete planning periods. This configuration gives for each period of the time horizon the list

of admissible coffins and the number of coffins to be made. The configuration allows coffin

selection to take account of backup roll wear, machine breakdown, planned maintenance, etc.

Finally the system allows “group” constraints to fulfil capacity requirements on the downstream

production lines after the hot strip mill, such as pickling, painting, cold rolling or simply client

destination lines and the capacity requirements of upstream steelmaking and casting processes.

The planner can quickly assign a score to each real, planned and virtual slab using the graphical

user interface, by altering the weights assigned to due dates, client priority and production

difficulty. The coffin selection module then takes into account all the weighted objectives and

constraints to generate a coffin sequence of maximised score. At this point, which is arrived at

in a matter of a couple of minutes of computer processing time, the planner may stop and use

the chosen coffin sequence to schedule the hot strip mill and the virtual slab choice to schedule

the continuous caster. This is particularly useful when fast rescheduling is required. In normal

operation, using this coffin sequence, the hot strip mill scheduling system will create detailed

hot strip mill schedules by filling all the coffins with the current, the planned and the virtual

slabs. These schedules will take into account many more detailed constraints and objectives

relating to coil quality, furnace utilisation, slabyard management and roll wear. The resulting

schedule will be used to generate a steel demand for the continuous caster scheduling system,

using the demand for virtual steel slabs to generate hard and soft constraints. The hard

constraints are concerned with the steel demand needed to warm up rolls at the start of a rolling

programme. The soft constraints for the continuous caster scheduling are related to steel

demands given by the virtual slabs that can be taken into account by appropriately adjusting

scores associated to the corresponding orders in the caster schedule. The continuous caster

scheduling system will use this constraint and score information to generate steel orders and

strand plans. This way the continuous caster schedule provides a forward push on the hot strip

mill schedule and conversely the hot strip mill schedule (and the client orders) provides a

backward pull on the continuous caster schedules. Having generated suitable continuous caster

and hot strip mill schedules, the human planner may be unhappy with certain features of either

schedule. He may then change objective weighting and constraint parameters to improve the

resulting integrated plans. In order to achieve the best performance of this integrated push-pull


system we may perform the loop several times to reach the best equilibrium between the two

processes.

4. Mathematical formulation

The model for the problem of automatic coffin selection consists of an assignment of slabs to a

set of coffins, which give rise to a maximised total score. Each coffin is divided into coffin

zones, which are characterised by slab selection criteria that specify the allowable slabs for the

zone. Each coffin and coffin zone have minimum and maximum length requirements, where

length is measured in terms of coil wear, which corresponds to the total length of the coils

rolled adjusted for metallurgical properties and coil dimensions. A simplified example is shown

in figure 4.


Coffin

Name

Min Length

(coil wear m)

Max length

(coil wear m)

Simple

coffin type

60000 75000

Zone Name Min Length

(coil wear m)

Max length

(coil wear m)

Allowable Coils

(dimensions in mm)

Start zone

Ascent zone

Difficult

dimensions

zone

Descent

zone

3000

10000

12000

40000

5000

15000

15000

50000

900 <= Coil width <= 1100

3 <= Coil thickness <= 4

Use soft coils

Use high tolerance coils

900 <= Coil width <= 1500

2.5 <= Coil thickness <= 4.5

1300 <= Coil width <= 1500


OR

1500 <= Coil width <= 1600


700 <= Coil width <= 1400


Use no hard coils

Figure 4. Coffin type.

We can represent this model by a tripartite graph. In this graph, the first level of nodes

represents the real, planned and virtual slabs or coils with their corresponding scores and coil

lengths adjusted for metallurgical properties. The second level represents the zone nodes with

their minimum and maximum zone lengths. And the third level refers to coffin nodes with

minimum and maximum length requirements (see figure 5.).


Slabs/CoilsZones Coffins

i j k

[lj’, Lj’] [lk, Lk]

Period

[nl , Nl ]

Period

Group

[cm , Cm]

Group

Group

wi , sijk

Figure 5. Network Model.

An arc connects a slab/coil node to a coffin zone node if it is permissible for the coil to be rolled

in the corresponding coffin zone. An arc connects each coffin zone node to the corresponding

coffin node. The coffin nodes are grouped into time periods, corresponding to the time buckets

in the scheduling horizon, with a specified number of coffins to be made in each period. We

may introduce an empty period to take maintenance into account. Slab/coil nodes are grouped to

reflect minimum/maximum capacity constraints for upstream and downstream processes. We

obtain the following mathematical model:

{ }{ } (8)1,0

(7),,1,0

(6)

(5)1

(4)

(3)

(2),

(1)Maximise

ky

kjix

lNyn

ix

mCxwc

kyLxwyl

ZjkyLxwyl

xs

k

ijk

l

Pk

kl

j k

ijk

m

Gi j k

ijkim

i j

kkijkikk

kk

i

jijkikj

i j k

ijkijk

l

m

∀

∀∀∀

∀≤≤

∀≤

∀≤≤

∀≤≤

∀∀′≤≤′

�

� ��

� ��

� � �


Where:

��=

��=

otherwise.0

used,iskcoffinif1

otherwise0

k coffin ofjzone toassignedisislabif1

k

ijk

y

x

sijk : score of slab i associated to zone j of coffin k.

wi : length of slab i in metres, adjusted for coil wear.

′ ′l Lj j, : minimum and maximum lengths respectively for zone j.

l Lk k, : minimum and maximum lengths respectively for coffin k.

c Cm m, : minimum and maximum lengths respectively for group m.

ll Nn , : minimum and maximum number of coffins respectively to be made in period l.

Gm = { i | slab i is in group m}

Zk = { j | zone j is in coffin k}

Pl = { k | coffin k is in period l}.

The objective function (1) consists of a maximisation of the total score of assigned slabs. We

may give a different score for the assignment of each slab to each zone and coffin and hence

each period. Together with a powerful graphical user interface, this allows us to ensure that

cold slabs in the slabyard and slabs in current caster plans are used before virtual slabs. It also

allows us to influence the structure of each coffin, for example ensuring that as many difficult,

wide coils are rolled in each coffin as possible.

Constraints (2) and (3) define the coffin structure for each coffin rolled. Constraint (2) ensures

that the total length of assigned slabs in each zone falls within the minimum and maximum

length range. Constraint (3) guarantees that the total length of coils assigned to a coffin is in the

minimum maximum length permitted by roll wear. In the same way constraint (4) ensures that

the total length of coils planned belonging to a group lies between the minimum and the

maximum length of this group. The group constraints come from both upstream and

downstream processes. Downstream processes, such as pickling to remove surface oxides,

may have capacity constraints. Upstream caster constraints may also be modelled, since for

example we might wish to limit the amount of low carbon steel cast in a given period, since this

steel must be cast slowly and the caster and steelmaking processes must remain approximately

in step. With constraint (5) we guarantee that a coil is assigned at most to one zone. And finally


constraint (6) sets that the total number of coffins made is within a minimum and maximum

range for each period.

5. Resolution method

Our mathematical model for the coffin selection problem is a mixed integer programming model

with a huge number of binary variables. In large practical problems, we must handle 2000 to

8000 slabs, 3 to 15 coffins and more or less 10 zones per coffin, with up to 40 coffins in the

planning horizon. In general, this kind of model is difficult to solve even with a small number

of variables. Indeed our problem contains the strongly NP-hard bin packing problem (Garey

and Johnson 1979). This and the sizes of the instances tackled mean that it is highly unlikely

that we can guarantee to solve our model exactly. One way to cope with this type of problem is

to relax some constraints. For example we could relax the integrality requirements on the x

variables (which become continuous variables), whilst keeping the y variables integral.

Experience of problems with such a "packing" flavour, such as the bin packing problem

discussed earlier, leads us to believe that most x variables will be integral in the resulting

relaxed solution. Knowledge of the production environment leads us to believe that for non-

integral x values we may arrive at a sensible interpretation via some kind of rounding scheme.

Hence it should be possible to implement our relaxed solution as a production plan without

significant change. This intuition has been borne out, as our results show. It is certainly not

true than non-integer y values have a reasonable interpretation or work around in the production

environment, hence these variables are not relaxed. However, even if we relax integrality

requirements on the x variables, the problem remains NP-hard and it is appropriate to find fast

heuristics for setting y variables. We have to keep in mind that, in this practical context, a good

solution rather than an optimal one has to be obtained as quickly as possible, especially in this

case where we may choose to go through the planning loop several times.

An efficient way to cope with this model is to relax the x variables and fix the y variables

iteratively, by solving efficiently the resulting subproblem. After heuristically choosing y

values to satisfy constraints (6) and (8), we can formulate the resulting subproblem as a

minimum cost flow problem. We may use the structure of the minimum cost flow problem to

provide solutions to this subproblem substantially more quickly than using standard linear

programming techniques. The corresponding network is shown in figure 6. In this network,

the sources are the slab/coil nodes with a coil i being a source providing wi resource units. The

minimum and maximum length constraints can be transformed, in the context of a flow model,

into capacity constraints on the arcs. The main difference with the previous network is that here

we add a fourth node type corresponding to the group nodes. Capacity constraints on group


nodes and arcs leading to them are modified to reflect minimum and maximum constraints on

steel, which must not be rolled, if the group constraint is to be satisfied. There is a single sink

of demand Σiwi . If a slab is to be rolled this is represented by a flow from the slab/coil node

source through the corresponding zone and coffin node(s). If a slab is not rolled, it passes

through its corresponding group node. The cost per unit flow from a slab/coil node to a group

node is S = maxi,j,k{ sijk}. The cost per unit flow from a slab/coil node to a zone node is S –

(sijk/wi) . Hence if all wi units flow from a slab coil node through a zone and a coffin arc, then

this flow will be sijk units cheaper than if they go via a group node. All other arcs have zero cost

per unit flow. Then the objective of the minimum cost flow problem will be Σ i Swi – F, where

F represents the sum of the scores of the slabs, and fractions of slabs, which have been chosen

for milling.

Slabs/Coils

(sources) Zones

Coffins

Groups

[cm , Cm]

+wi

-Σi wi

[lk, Lk]

Sink

[lj’, Lj’]cost S – (sijk /wi)

cost S

Figure 6. Minimum cost flow network for approximate evaluation of coffin selection.

This problem will only be a minimum cost flow problem if the slab groups are disjoint. In

practice this requirement on groups is reasonable and makes it simpler for the user to

understand the impact of changes in group constraints. However, a certain amount of ingenuity

is required in some production situations and more flexible groups can be modelled using linear

constraints, except that the resulting model is no longer a minimum cost flow model and hence

requires more solution time. This flexibility has not been required in practice.


We solve this problem in three stages, detailed in the algorithm given in figure 7 below. In the

first phase, an initial solution is constructed greedily by adding one by one the best coffin until

the minimum required number of coffins for each period is reached. At each iteration, the best

coffin is determined by the resolution of the corresponding minimum cost flow problem. This

is easily handled for our network by simply setting to zero the minimum and maximum arc

capacities corresponding to unused coffins. Once we obtain an initial feasible solution, we

perform a second phase by adding the best coffin at each step until no more coffins can be

added which improve our solution. We then apply a local search heuristic to improve the

solution. We investigate a local search neighbourhood where we increase by one the number of

coffins of one type, while decreasing the number of coffins of another type, within a given

period. Any improvements are implemented immediately and the search continues until no

improvement is possible. A local optimum is reached in a matter of a couple of minutes. In

order to evaluate a new solution in each of the three stages, we solve a minimum cost flow

problem to determine the score of the new solution.

Instead of the simple constraint (6), which specifies a minimum and a maximum number of

coffins per period, we may specify a minimum and maximum number for each coffin type

together with set-up costs. Then, the automatic coffin selection module will choose the number

of coffins so as to plan the minimum possible number of long coffins that will satisfy mill wide

production requirements. Hence we may achieve a balance between the set up cost of each

added coffin (beyond the minimum number required) and the value of rolled material within

this coffin.

When describing the algorithm it is useful to introduce the following notation:

S (y1,y2,…,yn) = score of the minimum cost flow problem obtained for the network resulting

from given values of the y variables.


Algorithm (Automatic coffin selection)

/* Find initial feasible solution for y greedily */y = (y1,y2,…,yn) = (0,0,…,0)For each period l

Repeat nl times /* nl is the minimum number of coffins for period l */best_score = -∞For each coffin k in period l

If S(y1,y2,…yk-1, yk+1,yk+1,yk+2,…,yn) > best_scorebest_coffin = k best_score = S(y1,y2,…yk-1, yk+1 ,yk+1,yk+2,…,yn)

End IfEnd For

End Repeaty = (y1,y2,…ybest_coffin-1, ybest_coffin+1 ,ybest_coffin+1,ybest_coffin+2 ,…,yn)

End For

/* Now add coffins greedily until no addition improves total score */improving_coffin_found = FALSEDo

best_score = S(y1,y2,…,yn)For each coffin k with Σym for m∈Pl < Nl /* Space left in this period */

If S(y1,y2,…yk-1, yk+1,yk+1,yk+2,…,yn) > best_scorebest_coffin = k best_score = S(y1,y2,…yk-1, yk+1 ,yk+1,yk+2,…,yn)improving_coffin_found = TRUE

End IfEnd ForIf improving_coffin_found

y = (y1,y2,…ybest_coffin-1, ybest_coffin+1 ,ybest_coffin+1,ybest_coffin+2 ,…,yn)End If

Until improving_coffin_found is FALSE

/* Local Search Improvement */Do

improving_swap_found = FALSEFor each period l

For each pair of coffins a,b in period l with yb>0If S(y1,y2,…, ya+1,…, yb-1,…,yn) > S(y1,y2,…,yn)

y = (y1,y2,…, ya+1,…, yb-1,…,yn)improving_swap_found = TRUE

End IfEnd For

End ForUntil improving_swap_found is FALSE

Figure 7. The heuristic solution method used.


Of course the solution obtained may not be optimal but it is a highly acceptable solution in

terms of computation time and plan quality. In particular the solution obtained has been

observed to be substantially better than existing solutions produced manually and greatly

extends the possible planning horizon above that possible using a purely manual planning

process.

6. Results

To gain an idea of the distance of our heuristically generated solution from optimality, a

comparison was done with an upper bound obtained on the basis of the linear relaxation of the

integer programming model given in section 4.

The tests are realised on different samples of data referring to different industrial environment.

For each steel company we performed several runs for each set of slab data, using different

coffin type data and number of coffins to be rolled, with each data set coming from a real

planning problem. The scores and wear values for each coil and the parameters of each coffin

were given by a group of steel engineers. The average gap over the runs between the upper

bound and the solution are given in the table below (see figure 6). We see that the solutions are

almost all very close to optimality, with the gap usually below 1%, particularly for the larger

problems. Note that the upper bound is usually strictly greater than the optimal solution,

especially for smaller problems. The heuristic performs better with a high number of coffins

which corresponds to the natural use of our system. These are precisely the problems that it

is hardest for the manual planners to solve. An enlarged search space and use of the heuristic

given allow our approach to find much better solutions.

In 96% of the above cases the best solution is obtained subsequently to the first step of the

method. The heuristic has been enhanced so that the local search improvement is only called if

the gap between the upper bound and the heuristic solution is greater than a fixed percentage

(usually 5%). This version of the solution method allowed us to decrease the execution time by

about 20 to 30%, while retaining a robust approach that can deal with the difficult problems,

which require use of the local optimisation heuristic. The system is fast enough to be reused

many times a day to take into account all the changes that can occur in this kind of industrial

environment. Execution times of a couple of minutes are typical for large problems.


Number of

Slabs

Number of

Coffin Types

Number of

Coffins to be

planned

Relative Difference

%

7 7 4.38

7 10 2.07

10 12 4.06

5267 10 15 1.81

10 20 0.05

10 25 0.05

10 30 0.02

7 7 0.10

7 10 2.68

5 12 0.01

8510 8 15 1.23

7 20 0.48

7 25 0.21

7 30 0.45

13 2 0.73

12 3 0.04

12 4 0.61

1007 12 5 0.22

12 6 0.63

12 7 0.69

12 8 0.98

Figure 8. Relative difference between heuristic solution and upper bound.

In an industrial setting, our approach must be validated in the long-term to allow comparison

with current manual planning techniques in terms of profitability and customer satisfaction.

Initial results show significant improvements in plan quality over manual systems. Several steel

engineers on observing the system agree that our system allows much better schedules to be

produced for both continuous caster and hot strip mill and that production pushes and pulls are

balanced to give better schedule integration. Production upstream and downstream of the

continuous caster and hot strip mill proceeds more smoothly due to the capacity restrictions

which are handled by the group constraints in our system.

We now perform a comparison between several manually generated plans and the plans

generated by our system, according to the score calculated by another widely used optimisation

tool, an automatic sequencer for the hot strip mill called the BetaPlanner (Cowling 1995).


The tests were performed on an industrial data with 5215 slabs (all real in this case since

manual planners only currently produce hot strip mill sequences for real slabs) and 10 different

coffin types. We performed three tests where 3, 5 and 11 coffins are to be generated. The

scores obtained using a manual approach and using our system and the relative improvement

offered by our system are given in the following table. The improved score results from

improvements in production throughput, particularly of urgent orders and improved production

quality through each coffin having a form closer to the ideal coffin specification (see figure 9).

Number of coffins BetaPlanner score BetaPlanner score Relative improvement

Manual sequence Automatic sequence %

3 428,264 503,841 15

5 1,456,764 1,531,256 5

11 2,259,091 2,689,389 16

Figure 9. BetaPlanner evaluation.

To evaluate our system’s ability to generate good long range plans, we used another sample of

industrial data, with 8416 slabs (again all real, to enable comparison with manual plans), and a

total demand during the production period of 87668 tons to be rolled using 3 different types of

coffins. Here we wish to minimise the number of coffin rounds used to roll the material,

requiring modification to constraint (6) in our model. For this sample of data, the number of

rounds, determined by the manual planner, was 40. With our system, only 37 rounds were

required. This results in a significant decrease of set up time, while maintaining a high score

and hence good coil quality and due date adherence. The steel mill for which this test was

conducted has an automatic system for changing the work rolls between coffins, giving rise to

a set up time between coffins of 45 minutes. If 3 out of every 40 set ups were to be avoided

then the whole gain would be about 50 hours per year, which is very significant in such a

capital-intensive industry. The gain could be much larger in steel plants without an automatic

roll changing system where the overall set up time can be as high as 2 hours between coffins.

Other tests were realised to study the distribution of the material over the coffins, using real,

planned and virtual slabs. No comparison is possible here with manual plans since this

represents a new way of working for the steel plant. We used industrial data with 4300

slabs/coils (10% real, 40% planned and 50% virtual), 7 types of coffins, and a requirement that

we make 20 coffins in a single period.

The rounds obtained present some significant advantages over manually generated ones, like

improved grouping of client orders, which can often only be delivered after the last coil is

rolled, and a good product mix having a lot of diverse dimensions. This mix enables us to find


hot strip mill schedules having smooth changes in dimensions between adjacent coils to give

improved coil quality.

0

20

40

60

80

100

120

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Coffins

Len

gth

Figure 10. Length distribution.

Figure 10, where each coffin types are shaded differently shows that our system generates

coffins of more or less uniform length within each coffin type category. This yields better

throughput and coil quality than most manual sequences, which tend to start with coffins which

are too long and end with coffins which are too short.

Figure 11 shows the change of the distribution of score over time, which corresponds to the

placement of urgent slabs early in the sequence. This is a highly desirable feature of our

heuristic solution procedure.

0

0.5

1

1.5

2

2.5

3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Coffins

Sco

re/L

en

gth

Figure 11. Score Distribution.

Figure 12 illustrates the distribution of material type (real, planned and virtual) within the

coffins. We observe the desirable tendency that the real material is absorbed as soon as possible

in the first coffins. However this objective is balanced against the need for good coffin shape

and we see that a small amount of real material is kept until the 12th coffin, to ensure good coil

quality for coils in that coffin. The quantity of planned material decreases with coffin index and

time, so that the virtual material can increasingly be used to guide future continuous caster


production. Early virtual material is “hot charge” material, which can be cast while still hot,

providing significant energy savings.

0

500

1000

1500

2000

2500

3000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Coffins

Weig

ht Planned Material

Virtual Material

Real Material

Figure 12. Slab distribution.

Tests have also been done on unbalanced demands with unusually large orders for certain

products. The reaction of our system was perceived to be excellent in relation to the problem.

The coffin rounds were individually under-optimised to provide a globally better solution that

absorbed the whole demand.

Capacity balancing constraints, particularly for downstream production processes are also

handled well by the system, with good trade-offs between capacity requirements and overall

plan quality.

7. Conclusion

We have carried out a detailed analysis of the medium-term planning problem for integrating the

schedules of a continuous caster and a hot strip mill, together with the upstream and

downstream processes which affect those schedules. We have developed a model which allows

us to take all of these factors into account. The model uses “virtual slabs” to guide the choice of

steel to be cast and the choice of coffin types to be rolled. Our system should work with any

continuous caster and hot strip mill scheduling systems which can use information of this type,

and has been proven to work with manual and automatic scheduling systems, where our

systems guidance makes the detailed scheduling process easier. The model developed results in

an NP-hard problem, so we have developed relaxations and heuristics to solve the model.

Since the system may need to be run several times before an acceptable plan is found, it is

critical that the solutions are found quickly, even for very large problems, and this is one of the


important design objectives which was met. We have demonstrated the quality of our results,

both qualitatively and quantitatively. The system has been tested and installed at several steel

plants worldwide, with promising initial results.

Current planning practices do not take into account all the factors which we have considered for

the steel plants which we have investigated, since the manual procedures used cannot handle

this level of complexity. There are some organisational and political issues to be resolved in

implementing the system, which are outside the scope of this paper. Most particularly, many

steel plants are driven by one or other of the continuous caster or the hot strip mill medium-term

plan, where the other steel making process simply falls into line. Our system allows for a more

“democratic” approach where the requirements of each process are considered, which will

clearly reduce the perceived influence of the management and schedulers of the “leading”

process of a plant of this type. Since directives to install a system of this type usually come

from board level, in general these working practices have been changed with the minimum of

grumbles. Production staff in the “leading” process have perceived better schedule quality,

anyway, and staff in the “following” process have observed much better schedules, making

acceptance easier.

The principal method for connectig the continuous caster process, which transforms liquid steel

into solid slabs, and the hot strip mill which processes the solid slabs is via “virtual slabs”

which allow us to make use of the flexibility which we have to cast the liquid steel into a variety

of different physical forms, so as to create good schedules upstream and downstream of the

solid/liquid interface. We imagine that these ideas could be applied to other process industries

which have intermediate products at the liquid/solid interface, for example in other metal

industries, plastics and glass.

Further work to improve the system could increase the level of detail handled for the

continuous caster and hot strip mill schedules or upstream and downstream processes.

However, we believe that through extensive access to steel engineers and schedulers, we have

produced a system and a model which achieves a successful balance between being sufficiently

complex and detailed to be useful, but not so complex that it is too difficult to use in practice.

We have demonstrated that our heuristic generates near-optimal plans most of the time, but of

course there is still some room to make the heuristic faster and more powerful. This has not

been carried out since the benefits from the small available improvements, apparent from our

tests of closeness to optimality, could not be justified commercially.


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