Delft University of Technology
FACULTY MECHANICAL, MARITIME AND MATERIALS ENGINEERING
Department Maritime and Transport Technology Mekelweg 2 2628 CD Delft the Netherlands Phone +31 (0)15-2782889 Fax +31 (0)15-2781397 www.mtt.tudelft.nl
Specialization: Transport Engineering and Logistics Report number: 2015.TEL.7927
Title: Production scheduling in a steelmaking plant
Author: Philip Schilder
Title (in Dutch) Het plannen en inroosteren van de productie in een staal fabriek.
Assignment: literature Confidential: no
Initiator (university): dr. Francesco Corman Initiator (company): none
Supervisor: dr. Francesco Corman
Delft University of Technology
FACULTY OF MECHANICAL, MARITIME AND MATERIALS ENGINEERING
Department of Marine and Transport Technology Mekelweg 2 2628 CD Delft the Netherlands Phone +31 (0)15-2782889 Fax +31 (0)15-2781397 www.mtt.tudelft.nl
Student: Philip Schilder Assignment
type: Literature
Supervisor (TUD): dr. Francesco Corman Creditpoints
(EC): 10
Supervisor
(Company) none Specialization: TEL
Report number: 2015.TEL.7927 Confidential: No
Preface
This report is the result of researching the field of production scheduling in steelmaking plants. The research is performed in context of the literature assignment, which is part of the master program Transportation Engineering and Logistics at the Delft Technical University. My interest in steel manufacturing roots back to my bachelor thesis. During my bachelor thesis I studied steel behaviour on behave of Tata Steel production facility located in
Ijmuiden in the Netherlands. Studying literature and writing this report has given me a wider scope and a more in depth understanding of the production of steel. During the time I worked on the assignment, Dr. Francesco Corman has supported me. I want to thank Dr Francesco Corman for his support and his patience.
List of abbreviations
ABC Artificial Bee Colony
ACO Ant Colony optimization
AI Artificial Intelligence
ANN Artificial Neural Network
AOD Argon Oxygen Decarburization
B&B Branch-and-Bound
BOF Basic Oxygen Furnace
CC Continuous casting
CC-CCR Continuous Casting and Cold Charge Rolling CC-HCR Continuous Casting and Hot Charge Rolling CC-DHCR Continuous Casting and Direct Hot Charge Rolling CC-HDR Continuous Casting and Hot Direct Charge Rolling CCA Colonial Competitive Algorithm
CP Constraint Programming
CS Constraint Satisfaction
CSA Conventional Subgradient Algorithm
DBD Delta-Bar-Delta
DE Differential Evolution
EAF Electric Arc Furnace
EO Extremal Optimization
FCFS First Come First Serve
FOA Fruit-fly Optimization Algorithm
GA Genetic Algorithms
HFS Hybrid Flow Shop
HSM Hot Strip Mill
JIT Just In Time
LAF Ladle Arc Furnace
LD Linz-Donawitz (converter)
LD Lagrangian Dual
LF Ladle Furnace
LR Lagrangian Relaxation
LWR Least Work Remaining
MILP Mixed Integer Linear Programming
MIP Mixed Integer Programming
NN Neural Network
NP Non-deterministic Polynomial-time
PSO Particle Swarm Optimization
RH Ruhrstahl-Haraeus refiner
SA Simulated Annealing
SCC Steelmaking Continuous Casting
SPT Shortest Processing Time
TS Tabu Search
TSP Travelling Salesman Problem
Table of Contents
Preface ... 3
List of abbreviations ... 4
Introduction ... 7
1. Basics of steel making ... 9
2. Planning and scheduling ... 12
3. Steelmaking Continuous Casting ... 16
4. Steel continuous casting solution methods ... 19
4.1 Mathematical programming SCC ... 19
4.2 Heuristics and Artificial Intelligence for the SCC ... 25
4.3 Hybrid methods ... 32
4.4 Simulation methods ... 36
5. Hot rolling scheduling ... 38
6. Integrated steel scheduling: CC-‐Hot rolling-‐Cold rolling ... 40
7. Discussion ... 48
7.1 Solution methods ... 48
7.2 Scheduling environment, objective and efficiency ... 49
7.3 Dynamic steel scheduling ... 51
8. Conclusion ... 53
8.1 Conclusion ... 53
8.2 Future research ... 55
9. References ... 56
Introduction
The iron and steel industry is an essential industry, as it serves as a driving force for any industrial economy [1]. Especially for developing countries the steel industry is of major importance. The steel industry in China has developed itself over the last decades into the worlds biggest. The magnitude of this industry provides strong support for the development of manufacturing industries, infrastructure and construction projects. An example from the 1970s and 1980s can be found in South Korea, which became one of the newly industrialized countries due to their rapid growth of its steel industry [1].
Also in the future, the iron and steel industry will continue to remain the backbone of the modern economy by providing solutions to the sustainability of the future economy. Innovative solutions like lighter vehicles, renewable energy and for instance energy conserving machinery will be provided by this vital industry. (Steel as the backbone of modern economy, Nae Hee Han, LCA and Steel Seminar, Beijing, 19july 2012, World Steel Association)
The competitive nature of the global steel market along with the push towards Just-In-Time production in today’s manufacturing environment has made production management
increasingly important for international steel makers in order to differentiate themselves from competitors [2], [3]. Optimal production order scheduling has a lot of benefits as it increases the profit and reduces the consumption of materials and energy. It also reduces lead times, improves machine productivity and improves customer satisfaction, which rely on the reliability of production order fulfilment time [2], [4], [5], [6]. However, production
scheduling in the steel industry is one of the most difficult industrial scheduling problems [7]. This holds in particular for the modern integrated process of steelmaking: Continuous
Casting and hot rolling that directly connects the steelmaking furnace, the continuous caster and the hot rolling mill with hot metal flow and makes a synchronized production.[8] Such a process has many advantages over the traditional cold charge process, but it also brings new challenges for production planning and scheduling. The Steelmaking-continuous casting (SCC) process is one of the two critical elements of the integrated steel scheduling approach. Continuous casting is characterized by several complex constraints, a combinational nature, requirements on material continuity and flow time, as well as the requirements to ensure practical feasibility.[6] It is often regarded as the bottleneck in iron and steel production.[9] [10][11] Furthermore, it is responsible for most of the steel production in the world and has largely replaced the conventional ingot casting/rolling for the production of semi-finished steel shape products.[12]
The manual approach of steel scheduling is time consuming and does not allow a thorough exploration of the solution space resulting in sub-optimal schedules [13]. Suboptimal solutions are not preferred, especially as steelmaking is a high investment and high-energy consumption industry.[14] In general, the identification of the optimum design of an industrial problem is often hard due to the size of the problem and lack of knowledge [13]. Therefore, exhaustive research has been done in order to find optimal solutions for industrial problems like the SCC problem.
This literature research reviews the problem definitions and modelling approaches for the steelmaking process. In addition, a comparison of solution approaches is given to solve the scheduling problems. The main question in this review is: what are the most important aspects of the steelmaking process and which effective methods are used to plan and schedule this process.
In the next chapters the steel planning and scheduling process will be further discussed and evaluated. First an introduction of the steel making process is given. Then the review focuses on the continuous casting-scheduling problem. After an overview and comparison of different
solution methods for the SCC, the hot rolling scheduling approaches are discussed. Although the proposed hot rolling scheduling methods show similarities with the ones used for the continuous casting problems, the problem itself has an unique character. Finally the integration of steel scheduling discusses the whole steel making process, which is a combination of continuous casting and hot rolling.
1. Basics of steel making
The basis of all steels is iron (Fe) and can be found in nature in bounded form as iron oxide (Fe2O3). The iron ore that is mined contains impurities. It must be purified first, before it can be processed. This happens in a Pellet or Sinter Plant. Here the first step consists of iron that is blended and then baked into pellets (pelletizing) or lumps (sintering). After this the pellets and lumps are exposed to heat treatments and is cooled again. This way the quality that is needed for the blast furnace can be ensured. (www.ulcos.org)
Besides the iron ore, coal is used inside a blast furnace as the fuel. Coal, consisting of
carbons and hydrocarbons, is also mined and also needs to be purified in a coke oven before it can be used in the blast furnace.
Before they are being charged in a coke oven the coals are first blended and pulverized. In the coke oven the coals are carbonized by pyrolysis and transformed into a hard porous mass called coke. The coke is then cooled and transported to the blast furnace.
Inside the blast furnace the iron and the coke are charged at the top and slowly descends till it melts at the bottom of the furnace. The carbon monoxide reduces the iron oxide to iron and carbon dioxide. A part of the waste gas can also be used to preheat the air used by the furnace.
Together with the pulverized coal the temperature reaches 1500 degrees Celsius. At this temperature the liquid iron is tapped periodically at the bottom of the furnace and
transported, often by rail, to the steel shop inside ladles. [www.ulcos.org] In the meantime raw materials continue to be charged at the top of the furnace. This continuous process goes on throughout the life of the furnace, which is on average 10 years. The continuous nature of this process is important since shutting the blast furnace down will likely result in a rebuild. Besides the cost of 70 to 100 million, the one-year rebuild time is also graceless [15]. The consumption of the continuous supply of hot molten iron is an important constraint in the whole process and has great influence on the discrete downstream processes.
Hot molten iron from the blast furnace contains a lot of carbon. This makes the material inapplicable for deformations and for instance welding. The next step is to convert the iron to steel with the use of a Basic Oxygen Furnace (BOF). Inside the BOF pure oxygen is blown into the iron at supersonic speed through a lance positioned in the mouth of the BOF. This removes the impurities, like carbon or silicon [14], and most of all reduces the level of carbon. An alternative name for the BOF is LD converter, which stands for Linz & Donawitz converter. Adding burnt lime makes sure a liquid slag is made which will absorb the
impurities so they can be separated [16].
The other method of steelmaking commonly found in the literature, regarding continuous casting, is by using an Electric Arc Furnace (EAF). The lack of iron mines prompted the development of production by EAF [17]. This type of steelmaking takes solid steel scrap, DRI, Hot Briquette Iron or pig iron and melts them to produce steel. The EAF uses electricity to melt the feedstock, although many furnaces also use oxygen, carbon and hydrocarbon fuels to accelerate the process. Although there is some removal of impurities during EAF steelmaking, there is in comparison with the BOF not the need for the removal of a lot of carbon. [16] Whether a company uses an EAF or an integrated site, with blast furnace and BOF, depends on the economics, location and history. In the past there was obviously not a lot of scrap available and even today there is not enough scrap steel to meet the world demand. Steel produced by the integrated route (BOF) or EAF route is the same apart from minor differences that are important when it comes to specific special steel grades. [16] Another challenge occurs on this stage. The continuous input of hot melted iron has to be consumed by the converters and transformed into output, which is in complete discrete heat lots. Other constraints on this stage lie in the sequence of the different grades that has to be made and the maximum number of consecutive heats [15].
The next step in the process is the refining stage or ladle metallurgy. It is the stage, between the BOF/EAF and the casting process, in which the heat is transformed into a customer demand specific steel composition. The duration of refining is usually similar to that of the steelmaking stage.[14] In the refining stage, the crude steel is poured in a ladle and alloys, like Nickel and Manganese, are added to reach a specific steel grade. Besides fine alloying the refining stage can also include desulphurization, dephosphorization and for instance degassing [18].
Different secondary steelmaking equipment is used to treat the steel, like the common Ladle Furnace (LF). During this stage, the steel is not melted but maintained a specific
temperature. An alternative way to keep the steel hot is to use oxidation of aluminium. This is called a Composition Adjustment System with Oxygen Blowing (CAS-OB). Less capital is needed than for the use of a LF, but it can only be used for steel grades with high Al contents [16].
Hydrogen, which is dissolved in the steel, can cause cracking problems in final products, so its removal is essential. The most common degasser is the Ruhrstahl-Hausen (RH) vacuum refiner. This refiner is later on recognized as the bottleneck of the steel continuous caster process in the integrated Baosteel steel facility [10]. In a RH-refiner unit the steel is made to circulate inside a vacuum chamber, which is lowered into the ladle [16].
After refining, the steel can be casted. Valves in the bottom of the ladle control the flow of hot steel. For ingots the steel is directly poured into separate moulds and for continuous casting the steel is first shed into a tundish on top of the continuous caster. This is basically the first of the three parts of the overall steelmaking process. It is also called the primary steelmaking process and is illustrated in figure 1.
Figure 1: primary steelmaking process
The primary steelmaking process is the part in which the continuous output of the blast furnace is processed further in BOF’s in the form of discrete ladles/heats/batches. In the case the steel plant model only considers an electric arc furnace, the output already is in batches.
The output of the primary steelmaking process consists of slabs, billets and blooms. Basic dimensions and applications of these semi-finished products are:
• Slabs: steel pieces with elongated rectangular cross section [4]. Typical dimensions of slabs are 150-320 mm thickness, 500-3000 mm width and a length of 10-20 m [19]. • Billets: steel pieces with square cross sections (e.g. 150 x 150mm)
• Blooms: steel pieces with square sections (e.g. 300 x 400mm), larger than billets. [4] The second part of the overall steelmaking process is the hot rolling stages, in which the output of the continuous casters is transformed into final- or intermediate products. The hot rolling process transforms the slabs, blooms and billets into practically finished products such as plates, pipes and wires. An overview of different processing techniques that can be found
Figure 2: Processing of slabs, blooms and billets
http://s274.photobucket.com/user/sethmcs/media/degarmo_18-1.jpg.html
The last part is the cold rolling and finishing line where the customer specifications, as final dimensions and surface quality, are met. [7] Which steps are taken depends on the
customer and application. For instance if the plate thickness must be reduced, the cold rolling process ensures the right thickness is met.
The key processes of the overall steelmaking process are illustrated in figure 3.
Figure 3: Integrated steel working plant
2. Planning and scheduling
Planning and scheduling are of major importance in the steel scheduling process. Results of improved schedules are an increase in performance and a stronger position in the market. In order to create a feasible schedule, decisions must be made and problems must be solved. Different phases exist in determining the plan and schedule. The first phase of the process consists of planning the quantities, mix of products and the sequence of the batches. The second phase, the actual scheduling, is in comparison with the planning phase the more complex part. The schedule represents a time-sequenced introduction of products into a production system in order to execute the plan efficiently [20]. Also, the schedule specifies the initial and completion date of each operation, on each machines, for all production orders to be scheduled [21]. All the resources and operational constraints must be taken into
account in the scheduling step to ensure practical feasibility [21], [22].
In the planning and scheduling process of the steel scheduling problem two important sets of decision-making problems are discussed. These are the batching (planning) and scheduling decisions [10].
Batching decisions include transforming primary requirements of orders into production batches. These primary requirements consist of metallurgical composition and for instance the quantity and the date of the order. An example of processing ordered quantities into batches is shown in figure 4.
Figure 4: grouping orders into batches.[23]
Two important aspects in the batching decisions are charges and casts: [10]
A charge is a basic unit of steelmaking production, which refers to a batch of molten steel loaded in a ladle. A charge is also referred to as job.
A cast is a sequence of charges that are consecutively produced via the same tundish, which is a receptacle located at the part of the continuous caster. A cast is also referred to as job-group.
Decisions must be made how to consolidate slabs into charges and charges into casts. The charge batching problem and the cast batching problem are the key planning problems for daily operational management in an iron and steel enterprise [10]. These decisions can balance the requirements of materials in downstream production lines, improve the level of costumer satisfactory and reduce the production costs.
The charge-batching problem is to make a decision that answers how to consolidate slabs into charges, so that issues of slab widths and trade-offs between the costs resulted by open ordered slabs and by upgraded slabs are considered [10]. The charge has to be between certain chemical limits to ensure the quality of the slabs [23]. These limits are the weight percentages of the elements in the steel. To combine ordered quantities the chemical
If a customer orders slabs that differ in steel-grade and largely in width, multiple charges must be used. The weight of the slabs belonging to a customer nowadays is in most cases less than that of a batch. The residue of the charge is used for open ordered slabs or for customers that ordered slabs with a lower quality. In the first case, where the slabs are not immediately linked to customer, the slabs will be stocked resulting in inventory costs. The second case will result in a lower profit as the customer pays for a lower quality-grade slab [10].
In order to schedule these batches expert knowledge is required, but even with highly skilled schedulers optimization of the batches is extremely challenging [23]. Therefore, methods such as genetic algorithms are used to ensure a feasible schedule [23].
The cast-batching problem is to make decisions that answers how to group charges into casts to satisfy all constraints, so that the utilization of the tundish is maximized while steel-grade changeover cost is minimized [10]. The tundish acts as a buffer and is located above the continuous caster. After a charge is poured form the ladle into the tundish, the hot molten steel flows through the bottom into the continuous caster. In the meantime another charge can be fed on the top of the tundish. So, the tundish makes sure that there is a continuous flow of charges into the continuous caster. The tundish and continuous caster is shown in figure 5.
[12] Figure 5. The tundish of a continuous caster.[12]
When a tundish is used and reaches its life it has to be replaced. The time before
replacement is the period in which charges can be produced on the tundish. The goal is to group as many charges into a cast so that the tundish can be used efficiently. Also the charges must be casted without interruption, otherwise the tundish deteriorates and a setup is necessary to substitute it [6]. Tundish optimization, as sub-problem of the integrated batch problem, is presented by Dong [24]. Variable neighbourhood search is used specifically on the tundish process to optimise the number of tundish, additional cost of technical
operations and balance the throughput.
Grouping charges is not an easy task, as the process should take the requirements of the orders and constraints of the process into account. First of all, charges with different steel-grade cannot be produced on the same tundish. Only steel steel-grade compatible charges can be grouped, because of their similar chemical composition (quality prescriptions) [23]. Grades in a sequence should be arranged ensuring low carbon product precede high carbon products [25].
When two charges with different metallurgical properties are sequenced adjacent, slabs with hybrid steel-grades will be produced. These slabs are considered as inferior quality products, thereby resulting in steel-grade changeover cost [10].
Besides the steel-grade compatibility also the width changes constraint should be taken into account. The determinate casting widths for two adjacent charges should change gradually and there is a maximum number of width changes [9]. Also, the charges in the same cast must be sequenced in order to ensure that their slab widths descend. Furthermore, the delivery dates of different charges in the same cast should be as close as possible [5]. Scheduling decisions include allocating, sequencing and timing charges and casts on the corresponding facilities, for instance from steelmaking to continuous casting [10]. Most of the literature considers the steel scheduling problem as a static problem in which all the information is available at the beginning of the planning horizon [26]. However, rescheduling the initial schedule is a common characteristic of the scheduling process and dynamic
scheduling is of greater value in real life scheduling than static scheduling. This is because the static model is often not feasible when unexpected time events occur. These real-time events can be classified into two categories:[26][15]
• Resource related, for instance machines breakdowns, loading limits, and unavailable materials or tools, operator illness etc.
• Job-related, for instance rush orders, due date change, cancelation or changes in processing priority, changes in order processing times etc.
Scheduling the coils on the hot strip mill is to generate a sequence of turns and a schedule of rolling coils in each turn considering the constraints of the hot strip mill [15]. If too many coils are produced on the same rolls, the rolls will wear out and affect the quality of the coils. If the rolls are not fully used this will result in more necessary rolls and more frequent set up and maintenance costs. The most important scheduling constraints for the hot rolling section consist of width and gauge changes [15]. These changes in width and gauge have to occur tranquil.
The smooth changes in the coil width is translated into scheduling the coils in such a way that the width profile of a turn is a coffin shape were the coils should be scheduled in non-increasing order of width. The smooth changes-in-gauge is a necessary constraint because the roughing and finishing mills have to make adjustments in the force applied to each slab. The scheduling is restricted by several other factors:[15]
• Product quality • Process efficiency
• Target delivery due dates
In steelmaking and other process industries, two approaches, reactive scheduling and preventive scheduling, can be used to address uncertainties in the production process.[6] Reactive scheduling is a process to revise the generated schedule from nominal parameters when a disruption occurs during the execution of the schedule. An example of reactive scheduling is a human scheduling expert solving problems by inferring knowledge from experience when disruptions occur [27]. The scheduler uses knowledge from past experience and reconsiders all the possible actions [27].
Preventive scheduling systematically seeks for methods to accommodate future uncertainty. Preventive scheduling approaches consists of two-stage stochastic programming, parametric programming, fuzzy programming, chance constraint programming, robust optimization techniques and conditional value at risk [6].
According to Roy, Hinduja and Teti [13] All the different solution methods are part of the overall classification of optimization methods and techniques. In this classification the different methods can be multi-modal, multi-objective, constrained based, dealing with real life requirements, implementing the techniques as a designer tool or a combination of these aspects [13].
Also some techniques are combinations of methods, the so-called hybridized methods, adopting the advantages (and disadvantages) each single technique has. A certain technique in a hybrid method can disguise the shortcomings of another.
The classification according to Roy, Hinduja and Teti is shown in figure 6[13].
3. Steelmaking Continuous Casting
In the Basics of steelmaking chapter, different options are discussed for the conversion and refining stages. In general, all primary steelmaking problems consist of three sections: converting, refining and continuous casting. This process is shown in figure 7.
Figure 7. primary steel making.
The primary steel making is often referred to as the bottleneck in iron and steel production, due to the different challenges, its discrete and continuous process involvement and because its production capacity is in general lower than that of hot and cold rolling [9], [10], [11]. Therefore, the first part, also known as the steelmaking-continuous casting (SCC) section, is critical for the entire steel making process and is of great essence for survey.
The output of these stages, the slabs, billets and blooms, are not only of different
dimensions but also of different steel grades depending on the order of the customer. These semi-finished products must be scheduled in a way, so they can be processed upstream to satisfy the demand of the customer.
This is not an easy task, especially with the occurrence of events creating uncertainties. Even more challenging is the already mentioned transition of continuous metal flow into discrete heat lots. Above that, each section can households multiple parallel machines en so multiple machines can be assigned to process the job. The existence of multiple parallel machines is typical for modern day steel making, just as the direction in which the hot metal moves. With these characteristics, the continuous casting process is often classified as a Hybrid Flowshop (HFS) model.
A HFS is composed of a series of production stages, each of which has several parallel machines. Jobs are processed through all stages, each stage with identical machines, in the same direction [28], [29]. The following general assumptions are usually made in HFS scheduling [14][22]:
• A machine can process at most one job at a time. • The number of processing stages m is at least 2 [30]. • At least one stage has more than one machine parallel [30]. • A job can be processed on at most one machine at a time.
• Job processing is non-pre-emptive, so that a contiguous time processing time length is needed to process the operation [28].
• All charges follow the same route. • Parallel machines are identical.
[5] Figure 8. A HFS model containing multiple parallel machines at each stage.
The HFS is a complex combinatorial problem encountered in many real world applications. The SSC process is an even more complicated form of the HFS. An example of the SCC process is shown in figure 8. The SCC process characterizes itself with additional features [14]:
• Casts must be processed as a group on the same caster and there are precedence constraints among the charges within a group [14].
• So, each machine (caster) of the last stage must process multiple jobs consecutively as a cast (batch). The operation sequence of the jobs in a cast is predefined. The processing machine of a cast at the last stage is known [29].
• A sequence-independent setup time is incurred between two adjacent casts on the same machine [6]. No setup-time is required on the caster between adjacent charges [5]. A removal time is also needed to clean the equipment after finishing of the cast [14].
• Transportation time between stages needs to be considered due to the weight of the charges. Transportation time, setup time and removal time is separated from the processing time [29], [14].
• Waiting time of charges between the processing at different stages causes
temperature drop and results in additional cost for reheating. Earliness and tardiness on job completion lead to cost, e.g. for inventory or compensation to customers [14]. • Dynamic processing times [9][26].
Between the different stages the temperature of the steel drops which result in additional costs. One of the goals of the planning and scheduling process can therefore be to minimize the waiting times between the different stages. The temperature drop is not the only
relevant aspect and might not even be the main focus of the research, as can be seen in the goals and objectives of the different solution approaches. Nevertheless, all solutions focus on efficient planning and scheduling of the SCC process. It is a crucial part for the performance of the production process and to ensure quality products, punctual delivery, increased productivity and reduced costs.
Steelmaking-continuous casting (SCC) production scheduling problems are to determine in what sequence, at what time and on which device molten steel should be arranged at various production stages from steelmaking to continuous casting [4], [22].
The classification of Roy [13] is used for all kind of optimisation techniques in different industries. Which method connects the best depends on the problem domain, requirements, design variables, constraints, objective functions and the environment in which the strategy should be applied. In the next chapter several methods are presented to deal with the
(optimal) steelmaking-continuous casting scheduling problem. Also the differences and difficulties related to these approaches are discussed.
Basically there are four categories in which the steel-scheduling problem solving approaches can be classified:
• Mathematical programming/ Operation research methods • Heuristics and Artificial Intelligence (AI)
• Hybridized methods, using a combination of the above solution strategies • Simulation based models
4. Steel continuous casting solution methods
4.1 Mathematical programming SCC
The first discussed category is the Mathematical Programming or Operation Research strategies. These methods establish optimization models for planning and production
scheduling problems and obtain optimal solutions by means of accurate algorithms [1]. The larger the problem and the more in depth the problem is described, the harder it is to find a solution in a reasonable amount of time. For instance, specific mixed integer
programming approaches may require exponential computation times and
linear-programming methods require large number of discrete variables and constraints for a valid model [25] [31]. The applicability of mathematical programming models is therefore often restricted to small problems, taking into the account NP (nondeterministic polynomial time) – Completeness of most scheduling problems [21].
It is very hard to match the model with the real life case, which is likely to be multi-dimensional and multi-objective [13]. Therefore, the model is often simplified. Simplifying reduces computational time but has a side effect that it also affects the model’s ability to describe the real problem. Another frequently used method to solve a complex problem is to decompose it into several less complex subproblems. Despite the issues regarding
mathematical programming models, they are frequently used for the steel continuous casting process and other industries to find exact optimal solutions.
A just-in-time based non-linear mathematical model for solving machine conflicts in the continuous casting problem is proposed by Tang et al. (2000)[4] The SCC process consists of three stages, each with two machines. This results in two converters, two refiners and two continuous casters. The two machines at each stage can connect with any of the two machines of the next. Like in other researches precedence constrains are used in the model to ensure charges can only begin when a precedence operation or charge has finished. Two typical JIT aspects are emphasized in the research: punctual delivery and production
operations continuity. The objective function consists of four penalty costs: cast break loss, charge waiting time, earliness and lateness. The last two terms are responsible for the nonlinearity. To solve the model with the use of standard software packages the non-linear model is converted into a linear programming model. In order to transform the model the set of machines is separated into two subsets, one for the casters and one for the converters plus refiners. The final step of the proposed model is a man-machine interactive method in which the user can directly manipulate the optimal reference schedule. The overall approach is an effective method and results show that the average waiting time was reduced
dramatically. The steelmaking-casting process routes for the proposed method are shown in figure 9.
Tang et al. (2002) later uses an integer-programming model to meet the same JIT objective requirements. The problem environment is the same as it describes three stages with parallel converters, refiners and continuous casters. The difference lies in the number of parallel machines, which is variable. They also state that their model can be modified easily for systems with more stages [14]. This improved structure is presented in figure 10.
Figure 10. Steelmaking-casting process proposed by Tang et al. [14]
The structure of both problems is similar to the Hybrid Flowshop (HFS) structure from the continuous casting problem description. The Hybrid flowshop continuous casting model of tang et al. is also characterized by additional practical constraints such as job groups and more complex practical constraints than standard HFS models, involving precedence scheduling criteria, waiting times and due dates [14].
Because most mathematical programming methods considering HFS scheduling problems are NP-hard and often describe large-scale applications, obtaining a high-quality schedule in a reasonable amount of time is extremely difficult [29]. In other words, if the size of the (HFS) problem increases so does the time required to solve it. It is therefore that most studies solve HFS problems with the use of heuristics or the deterministic branch and bound method [14]. Nonetheless, mathematical programming methods are proposed as heuristics cannot provide an easy measurement on solution quality [28].
Relaxing a mathematical programming model is a technique to solve a complex problem, like the HFS. In Lagrangian Relaxation (LR), usually a set of coupling constraints from the
problem is relaxed and introduced into the objective function while weighted by a vector of Lagrangian Multipliers [2]. The idea of LR is to formulate a relaxed problem, using a vector value, which is easier to solve than the original one [2]. In general, LR is a accurate
decomposition and coordination approach that can yield near-optimal schedules in an acceptable computing time [29]. Lagrangian Relaxation (LR) provides a solution for the NP-hardness of the HFS problem.
Tang (2002) used LR to solve the complex SCC problem. First the entire planning, in the model proposed by Tang, is divided into small time units, such that all the time parameters are of integer units [14]. In this integer model the solution approach differs from the earlier proposed method by Tang (2000) as the model is solved through Lagrangian Relaxation (LR), backward dynamic programming to solve the subproblems and a heuristic procedure. In the LR example of Tang et al. coupling constraints, including machine capacity constraints and precedence constraints for charges on the caster, are relaxed. Because the initial
solution to the relaxed problem is generally infeasible due to constraint violation, a two-phase heuristic is used to construct a feasible solution. Later optimal values of the
Lagrangian Multipliers are found by using a subgradient algorithm to solve the Lagrangian Dual (LD) [14]. Because LR does not guarantee optimal solutions, the duality gap is used as
between the upper bound, best feasible solution, and the lower bound [2]. Results show that an increase in the number of machines also results in an increase of optimality and a
decrease in computation time. This is due to the fact that each machine is less accounted for when the total of machines available to process the charges is larger. Furthermore, the duality gap and the computational times increase when the number of casts increases. This is because there are more precedence constraints on the charges.
The paper proved that the LR approach has emerged as an efficient, effective and practical approach for HFS scheduling. Also, different other relaxations or decoupling strategies have been proposed to solve these scheduling problems [29]:
Nishi et al. presented a LR and cut generation for the steel flowshop scheduling problem to minimise the total weighted tardiness [32]. Each stage contains of a single machine in this permutation flowshop model. The research considers a sequence-dependent setup time. That is, the setup time is not only needed before an operation, but also depends on the operation processed just before. This time the setup time constraints are relaxed as well as the machine capacity constraints. It is difficult to obtain a good lower bound when solving the Lagrangian Dual (LD), because of the increased search space of Lagrange multipliers. The infeasibility, caused by constraint violation, is handled by incorporating cuts.
Furthermore, the cuts improve the lower bound [32].
A new dynamic programming model has been developed to solve the subproblems. Another merit of the method is that there are no tuning parameters, except for the step size of the subgradient method used to derive near optimal Lagrange multipliers.
The method of Nishi et al. is compared to: standard LR approaches, branch and bound method, standard solver CPLEX and provided a better lower bound than these conventional methods used for comparison. Also, the proposed method is able to find near-optimal solutions with a small duality gap for small-scale instances.
Mao et al. [29] model HFS continuous casting problem as a mixed integer programming (MIP) problem. In their model two different refining steps are considered making it a four-stage hybrid flowshop model. The model of Mao et al. is shown in figure 11.
Figure 11. Steelmaking-continuous casting process Mao et al. [29].
The objective in their research is minimizing the total weighted earliness/tardiness penalties and job waiting. In general, the tardiness should be penalized more than the earliness [25]. Mao et al. use the LR approach to relax the machines capacity constraints, because these
constraints couple different types of variables or different jobs on the same machine [29]. Another method, as proposed earlier, is the relaxation of the operation precedence
constraints. This type of relaxation is also discussed later on in a model considering the integration of steelmaking continuous casting and hot rolling. However, relaxing the
operation precedence constraints in the problem of Mao et al. results in subproblems, which are also NP-hard.
After the relaxation of the machine capacity constraints, two subproblems are further solved. The first, a linear programming model, is easily solved with the use of standard software packages (e.g. CPLEX). The second subproblem contains integer and continuous variables and is harder to solve. The relaxed subproblem is decomposed into two tractable
subproblems by separating the continuous and integer variables. After that, time horizon approaches and boundedness detection are further explored to handle the unboundedness of the decomposed subproblem. Furthermore, the Lagrangian Dual (LD) is solved with the use of an improved subgradient level algorithm with global convergence. The improved subgradient algorithm does not need the optimal dual in advance, like in the conventional subgradient algorithm (CSA) [29]. In general, the solution of a dual is associated with an infeasible schedule, because the constraints are not satisfied. A simple heuristic, similar to list scheduling, is therefore used to generate a feasible schedule.
Comparison, among conventional and other algorithms, demonstrates that the proposed improved subgradient method, with three fine-adjusting strategies is the most feasible one in terms of robustness, duality gap and running time [29].
Bellabdaoui and Teghem (2006) [9]provided a MIP model for a the SCC scheduling problem to maximise productivity. They modelled (fig. 12) two parallel machines at each stage (steelmaking-refining-casting), and solved it using standard software packages [9].
Figure 12. SCC model Bellabdaoui and Teghem (2006) [9]
Bellabdaoui and Teghem used the model of Tang et al. for rough scheduling and extended the model. The difference between this paper and the paper of Tang et al. (2000) lies in the introduction of extra binary variables and corresponding constraints, ensuring a charge can only be processed at one machine and determining the position of the next charge in the sequence. Also, Tang et al. do not take into account the possibility of slowing down the charges of the sequence. Furthermore, the model of Bellabdaoui and Teghem is different from earlier proposed models as all charges follow the same process route between the last two stages.
Some contradictions between the constraints can appear in the model of Bellabdoui and Teghem making the mathematical programming model infeasible [9]:
• Unavailability of converters to the first charge assignment of a sequence
• A too strict limitation of the processing time of a charge at the last stage will make a necessary slowing down of the charge impossible
Later Sbihi, Bellabdaoui and Teghem (2014) presented a new mathematical programming model, which generalises the one described in Bellabdaoui and Teghem (2006), making the approach closer to reality. The SCC is now considered as a (complex) hybrid flowshop model with the same objective as before, which is maximizing the productivity by minimizing the total completion time of the sequences. In the renewed solution method, the mixed integer linear programming model has more than two sequences of charges that are dedicated to any of the production lines. The generalized model is able to provide solutions for any numbers of converters, refiners and casters. Also, the sequences processing order for each production line has been set previously [31]. A setup time constraint is applied in the last stage of the production process.
Small and large size problems are evaluated with different essential parameters, such as availability dates for converters, variation intervals of start time CC, processing times and maximum sojourn time [31]. The model is then solved with the use of CPLEX and the
efficiency is analysed based on the CPU time. The model showed robustness when the initial starting date is modified. However, for medium and large-scale instances and for problems with more than two continuous casters, the CPU time becomes too high.
Sbihi et al. also describe two possible improvements, which might be added in future research. The first improvement is to include unavailable converter time caused by
maintenance. The second is the uniformly distribution over a maximum number of charges in the case the process speed must be reduced to ensure continuity of the casting. Now only a single charge is affected by an implemented slowdown [31].
Finally, the case in which the number of sequences is too large to consider all the possible orders, the proposed MILP may be integrated within a meta-heuristic that can optimise the order of sequences [31]. This is also considered in the previous work of Bellabdaoui and Teghem (2006) [9].
As described in the examples above, mathematical programming methods seek for the globally best solution for a given objective, but global optimization becomes very time consuming with a large number of variables (>30) [13]. The computational time also increases with equality constraints and noisy objective functions. Furthermore, it is not always possible to identify if the solution is really a global optimum.
With gradient information and second derivatives a local optimum can be found, but in order to define the optimum, knowledge about the design space is required. Gradient-based approaches are also not suitable for real-life applications when the models describing the problems are not differentiable [13].
Because of uncertainty in steel scheduling and other real life optimisation situations, less sensitive solutions as robust optimisation are preferred. Robust solutions are areas in the design space where a significant change in design variable leads to an insignificant change in performance [13]. The discussed global, local and robust optimisation are illustrated in figure 13.
Yun Ye et al. introduced a robust optimization and stochastic programming approach to cope with uncertainty in steelmaking continuous casting operations [6]. Their hybrid flowshop model is shown in figure 14 and consists of parallel machines at four different stages: Electric Arc Furnace (EAF), Argon Oxygen Decarburization (AOD), Ladle Furnace (LF) and Continuous Caster (CC). The focus in this paper is uncertainty caused by demand, which is bounded, symmetric and bounded, following a known distribution and an unknown
probability distribution [6]. The idea is to create a model that is robust against demand uncertainty. The demand parameter is defined as the number of ladles (charges) in each cast.
Figure 14: Steelmaking continuous casting stages Yun et al.[6]
Earlier Lin et al. (2012) developed a novel and effective unit-specific event based continuous-time formulation for deterministic production scheduling of steelmaking continuous casting and extended the rolling horizon approach [6]. In the rolling horizon approach, a production planning is made for a fixed number of periods for which the demand is known [33]. The rolling horizon approach is used in the paper to decompose the entire MILP problem. Computational results show that this approach reduces the computational time and generates the same or better feasible solution than without applying the rolling horizon approach [6].
The solution from the framework of Yun Ye et al. is guaranteed to be feasible for the whole space of the uncertain demand parameters. While the robust optimization model aims at finding a single schedule that is immune to all the possible realizations of uncertainty, a two stage stochastic integer programming method provides the flexibility to implement different operational decisions after the realization of uncertainty [6]. The two-stage stochastic programming method was also studied to cope with uncertainty. In this programming formulation binary variables used to assign casts to event points and units are selected as the first stage decision variables. The second stage variables are the continuous variables describing the timing of the operations. The objective function is again minimizing the expected makespan. The uncertainty assumptions of two-stage stochastic programming method lead to a total number of 334 scenarios! To make the approach computationally tractable, a novel scenario reduction method has been implemented. The selected scenarios are incorporated into the stochastic programming model [6]. The method shows the
importance of a robust method for the steel-scheduling environment.
Harjunkoski and Grossmann developed a strategy for the four-stage (EAF-AOD-LF-CC) SCC problem with the objective of minimizing the makespan. In the first stage, only one EAF can be used for processing per time due to power constraints [25]. All the other stages consist of
a single machine. The SCC process is therefore considered a flowshop, since there is no parallel equipment and all groups follow the same route.
The solution strategy is a decomposition method, creating sub problems that are easier to solve than the original problem. However, in this paper the initial problem is split into sub problems in a natural way using the special features of steel making and avoiding the need for explicit constraints [25].
First customer orders are grouped into heats in the pre-processing phase. In this phase, heuristic ordering rules and MILP are used. Then the sequence of orders in each group is scheduled using a jobshop formulation (MILP). After that, the sequence of grouped is
scheduled as flowshop problem, again with MILP. Finally, LP and/or MILP methods determine the setup times and allocation.
While the global optimality is not guaranteed, comparison with theoretical estimates indicate that the method solution lie within 1-3 % of global optimum [25].
The research showed another decomposition strategy, disaggregating a complex model in a natural way. The method is able to provide a solution to a problem, which is due to its size virtually impossible to solve as a single scheduling problem. The problems solved by Lagrangian Relaxation also use a decomposition strategy with additional heuristics to deal with complex time consuming models. The result of such a combination is a reduced amount of discrete variables and/or and improved relaxation of the original problem (Elkamel,
Zentner, Pekny & Reklaitis, 1997; Lee, Bhaskaran & Pinedo, 1997; Roslöf, Harjunkoski, Westerlund & Isaksson, 2001)[25].
4.2 Heuristics and Artificial Intelligence for the SCC
In contrast to Mathematical programming methods, heuristics do not guarantee optimality, but their lower computational effort is an important advantage. This is the case especially in large problems where computational complexity of analytical procedures becomes excessively high [21]. In the previous chapter the solution strategies primarily consists of mathematical programming models, sometimes supported by simple heuristics. This chapter focuses on different types of heuristics as primary solution method.
This part of the review also uses the hybrid flowshop structure as the typical problem environment. The following papers summarize the heuristic and Artificial Intelligence (AI) methods to solve the Steel Continuous Casting problem. Heuristics methods are in fact one of the two most commonly employed approaches for hybrid flowshop scheduling problems [34][28]. The other one, which is branch-and-bound (B&B), is in comparison with heuristics less preferable as it is time consuming and it only obtains optimal solutions for small-scale hybrid flowshop problems [34].
With respect to heuristic methods, Pacciarelli and Pranzo (2004) [3]adopted an alternative graph formulation to describe an m-stage Hybrid Flow Shop problem and modelled the problem by means of the alternative graph. For a plant to be qualified as a hybrid flowshop, at least one stage must have more than one facility [35]. The layout of the stainless steel production, with the two ladle furnaces is presented in figure 15.
The main layout differences between this model and the earlier considered model by Harjunkoski and Grossmann is that now only one EAF is considered and that the ladles are grouped in lots. In the model of Harjunkoski and Grossmann each ladle is independent. Pacciarelli and Pranzo solved the problem by a beam search procedure, which is a heuristic search strategy first used in AI for speech recognition by Lowerre (1976) [3]. It consists of a modified branch and bound method in which only the branches that are most likely to lead to an optimal solution are explored. At each level the beam width determines how many candidate nodes are saved. The beam width also determines the quality of the solution. It is obvious that a higher value of the beam width also leads to an increase of computing time [3]. The performance of the modified technique proved to be satisfactory when looking at the solution quality, computational time, accuracy of the model and robustness to random perturbations of the processing times. The model can also be implemented in other complex scheduling problems and the model can be extended with more general constraints and objective functions. In the “standard” form of the HFS steel scheduling problem, discussed in the problem description, unlimited capacity of the buffers is a general assumption [30]. This method considers finite buffer space between the stages, as in real-life steel scheduling limited storage between two consecutive operations is available.
The beam search method is suitable as an offline and online scheduling tool due to its speed. Thus, the augmented time consumption of typical B&B methods is overcome and it can be used to quickly reschedule the production after unpredicted events such as production delays or new orders.
This problem with one caster in a plant, where the allocating problem is not relevant, can be solved within 1 to 3 % from optimality [18]. A comparison with optimal algorithms can determine the exact degree of sub optimality.
Under heuristic procedures we can distinguish constructive heuristics and
improvement heuristics. The basic approach is to apply a constructive heuristic to generate a feasible initial solution, the so-called seed, and improve this solution with an iterative
improvement heuristic [21].
Iterative improvement techniques perform local search and make use of random and
knowledge-based decisions to save computation time and escape local optima [36]. Iterative improvement heuristics include AI (artificial intelligence) methods and are often added to guide the search to a globally better solution [37].
Multi-agent systems are an emerging field of AI techniques [38]. The advantage of AI techniques is that they are effective in considering complex practical constraints and can simulate the decision process of a human scheduler [1].
Zarandi et al. propose a multi-agent base solution method for the electric arc furnace steel making process. Each process is assigned to an agent, which is able to communicate with other agents. An adaptive neuro-fuzzy inference system was used to generate knowledge for the agents. Furthermore, contract net protocol is used as negotiation protocol [38]. The authors suggest future work in which multiple agents per process are available.
The paper takes into account multiple stages of the steelmaking process: EAF, refining stages and casting, but differs form the other papers as it only describes casting with the use of ingots and not continuous casting. Nonetheless, the multi-agent based method shows a lot of potential for the steel-scheduling problem and is further discussed in the integrated steel-scheduling chapter.
AI models, can be further subdivided into: Expert systems (inference engine and knowledge based), intelligent search methods (random heuristic search methods including evolutionary computing: genetic algorithms (GA) and particle swarm optimization, simulated annealing (SA) and tabu search (TS)), meta-heuristic approaches (higher level heuristics) and constraint-satisfaction (CS) [8].
Dorn and Dorn developed a framework to compare four iterative stochastic search heuristics for the continuous casting-scheduling problem: iterative deepening, tabu search, genetic algorithms and random search [36]. A good comparison of the different techniques can be made as they use the same initial schedule set, modification operators, stopping criteria and evaluation function is used. The evaluation function consists of the optimization functions and gradual constraint satisfaction, which relaxes the constraints to cope with the over-constrained problem. Different modification operators are used and evaluated to exchange and shift jobs in the schedule. Tabu search gave the best result, outperforming the other iterative improvement techniques. This is the case, when the depth of search is limited and when the proposed heuristic to repair the main conflict (constraint violation) in the schedule is used [36]. Iterative deepening performed the same as tabu search, but the number of schedules grows exponentially with the depth of the search. This restricts the applicability of this iterative deepening technique [36].
The advantage of all the proposed algorithms is that not only predictive but also reactive scheduling can be easily implemented [36]: a new real-time event causes new constraint violations in one of the iterative improvement methods. The performance of the schedule repair in case a real-time event occurs has to be tested with the use of simulation.
Frequently used algorithms in optimization are the evolutionary algorithms. An increase of interest in solving problems based on principles of evolution has occurred [21]. Based on the Engineering Village database for all optimization problems, it is observed that the top three design optimization algorithms used in the period (1997-2006) are: Genetic algorithms, Linear and Quadratic programming and Simulated Annealing [13].
Genetic algorithms (GA) imitate the process of natural selection and are also known as population based meta-heuristics [21][36]. The population is the set of candidate solution and members of the population are chromosomes. The first population evolves in
generations after initialization and a selecting mechanism ensures survival of the fittest [36]. Optimization is achieved by selecting the individuals for crossover or mutation according to a fitness function, which is a reflection of the quality of that individual [21]. The numeric parameters for the genetic algorithms are:
• Population size
• Number of generations • Crossover probability • Mutation probability
A too small population results in a local optimum due to fast converging, while a large population results in an exhaustive computation time [21]. Also, a too small number of generations may result in a premature solution, while a too large number of generations again increases the computation time [21]. A deliberate choice of these parameters is essential in the design process of the model, so that the computation time and solution quality are balanced.
Dorn and Girsch [39] also propose a combination of a constraint repair approach and genetic algorithms to solve defects in the schedules for the Alpine Stahl plant in Linz. The steel plant is together with another Austrian steel plant, Donawitz, responsible for the name
Linz-Donawitz (LD) converter. In the model a representation of constraint violations is used and the importance of these violations were weighted with the use of fuzzy sets. The weighted violations, that can be aggregated and standardized, evaluate the different schedules [39]. Although the genetic algorithms alone generate feasible solutions, the genetic algorithms in combination with a tabu search method provide even better results [39].
Behnamian et al propose another evolutionary algorithm. Their colonial competitive
algorithm (CCA) is applicable for the HFS steel problem as they assume that the waiting time between consecutive stages is limited. The colony and relative imperialist sequence are
determined similar to crossover in Genetic Algorithms [40]. The new meta-heuristic is a very promising method due to its superior capabilities as faster convergence and better global optimal achievement in different optimization problems (Atashpaz Gargari & Lucas, 2007) [40].
Differences between the general CCA and the proposed discrete CCA are in the solution representation and assimilation. The objective is to minimize the sum of linear earliness and quadratic tardiness in hybrid flowshop problems with sequence dependent setup times and limited waiting time. This last feature is not often addressed in SCC hybrid flow shop
scheduling, but is relevant for real-world scheduling. The outcome of the model is compared with a tabu search model and a hybrid tabu-search-scatter search algorithm. The model outperforms both these methods, works as a generalised model for the steel HFS and shows good average results in medium and large-scale test problems [40].
Besides genetic algorithms, a promising method that gains popularity is Swarm intelligence. It is the discipline that deals with natural and artificial systems composed of individuals that coordinate using decentralised control and self-organization.
(https://www.youtube.com/watch?v=D58nLNLkb0I). Swarm intelligence methods show potential for the continuous casting problem because of their advantages such as scalability, fault tolerance, adaptation, speed, modularity, autonomy, and parallelism [41].
An example of Swarm Intelligence is the Ant Colony Optimization (ACO): a system based on agents, which cooperate and simulate the natural behaviour of ants to find good solutions for problems like the travelling salesman problem [42]. The Ant Colony System consists of the following steps:[42]
a) Ants derive at a decision point
b) The ants choose randomly if they take the upper or lower path
c) The speed at which an ant moves is constant. Ants that choose the lower path will reach the opposite site quicker than ants that take the upper, longer path.
d) Ants deposit pheromone will walking. At the lower path the pheromone will accumulate at a higher rate (higher density dashed lines)
Zanoni and Zavanella present two types of methods in order to find the optimal production schedule of steel billets [17]. The first is a mathematical programming approach and the second an Ant Colony optimization method. In both models, the billet cooling area is considered an integral part of the productive system (preparation-EAF-CC). Three types of costs are introduced: Holding costs, production costs and delayed order costs. The objective is maximizing the profit of the billets and therefore minimizing the costs [17].
The first method is a mixed-integer production-inventory problem. The second, Ant Colony Optimization (ACO) approach acts as an alternative to the mixed-integer method. The choice of the ACO meta-heuristic is established by its performance in industrial cases [43]. The study focuses on production optimisation of the steel billets. The paper describes the process of the model in two different phases. The first is the combination of the EAF and the
continuous casting and the second is the cooling of the billets in the warehouse, which is considered here as an important step in the production process.
The mechanism of the ant colony optimization heuristic is first explained as an application to the Travelling salesman problem (TSP). The TSP is just as the ACO an example of constraint programming. In this application the transition probability is a compromise between the visibility and intensity. The visibility aspect ensures that there is a negative relation between the probability that a city is chosen and the distance to that city. The trail intensity states that the higher level of traffic on the road is proportional to the attractiveness of that road. In the billet-scheduling problem, instead of cities, customer orders are represented in the distance matrix. The arc length represents the setup-time to switch from one production batch to the other. A summary with the comparison of the TSP with the billets Ant System is provided in figure 16.
Figure 16. Parallelism between the TSP- and billet Ant System.[43]
The algorithm focuses on finding the most profitable production schedule for the steel billets through minimizing the costs. A sub-algorithm applied in local search and in global
optimisation, finds the best billet disposition, taking into account the available space in the warehouse and cooling times of the billets. The cooling of the billets is considered in the process, because the billets are available for delivery only when the billets are completely cooled.
In this research, the Ant System is evaluated and compared to the earlier proposed MIP by Zanoni and Zavanella. The Ant System did not always find the optimal solution, but the relative error of the solutions of the Ant System lies in the range between 0.08-0.47 %. The proposed MIP model is improved in this research to match the industrial system, making it more realistic. The improvement consists of avoiding order splitting into multiple deliveries. Also, both the improved MIP and Ant System include an algorithmic control to ensure unique delivery of one billet to the customer.
The results show that when the ratio of net productive time and total time increases
(saturation), the percentage of solved problems decreases for the MIP model. Furthermore, when the saturation reaches 80% or higher, zero problems were solved. For the Ant System the percentage of problems solved is always 100%. Obviously, there is a positive correlation between the computational time and the saturation in both approaches. Although the Ant colony system always succeeds in finding a solution, the MIP is slightly faster when it finds one.
The paper of Ferretti, Zanoni and Zavanella compared the Ant Colony system with the earlier proposed MIP model. A high saturation, which can be seen as weak point of the MIP
method, is the most relevant and interesting situation to simulate for industrial context. It amplifies the complexity and hardness of the scheduling problem. This study shows that the Ant Colony model includes the real conditions of the industrial problem and is able to find good solutions in an acceptable computation time.
Li et al presented another Swarm Intelligence algorithm for the hybrid flowshop steel-scheduling problem. The proposed fruit fly optimization method is a new heuristic similar to genetic algorithms and particle swarm optimization (PSO) that mimics the food finding behaviour of the fruit fly. The objective is to minimise the average sojourn time and the earliness/tardiness penalties.
In the fruit fly optimization algorithm (FOA), each fruit fly belongs to a population of fruit flies. The fruit flies search for an optimal space through a smell-based process, a vision based process and an exploration procedure [44]. The fruit flies improve their status by smell and the vision aspect makes sure the best fruit flies guide the worst to improve the exploitation ability. A new fruit fly replaces an old one that does not improve its location. The exploration procedure makes sure that the entire population will not converge to a local optimum, and consequently avoids a premature solution. The balance between the
exploitation and exploration ability is guaranteed by the four neighbourhood structures and a self-adaptive neighbourhood strategy [44].
Li et al. consider the FOA as a feasible method for the steel-casting problem, because FOA has only a few parameters, because it is already verified and competitive in other scheduling problems and because it has a parallel search framework in which many (meta) heuristics like local search can be embedded. This makes the solution method applicable for real-life,
problem-dependent scheduling problems [44]. The proposed method exhibits perfect
convergence ability for large-scale realistic HFS scheduling problems. The solution method is designed for a static environment. Therefore, future work is the application of the method in a dynamic environment with multi-objective constraints.
The FAO is compared to four other algorithms and showed superior results in robustness, performance and computation time. The proposed methods for comparison were the Artificial Bee Colony [41] algorithm by Pan et al., tabu search [45] by Wang et al., genetic algorithm by Ruiz and Maroto [46] and the GAS algorithm by Yaurima.
Another population based method for the integrated steel scheduling plant is proposed by Tang in: “An improved Differential evolution Algorithm for Practical Dynamic Scheduling in Steelmaking-Continuous Casting production”. As the title suggests the paper is focused on the dynamic scheduling problem and uses Differential Evolution (DE), which is one of the more recent evolutionary algorithms, to solve it [26]. DE is a simple but efficient optimization approach, uses only few parameters and has good convergence (just like the FOA [44]). It is already successfully applied in many real-world applications and has shown to be an efficient approach to many numerical optimization problems. In the paper, real-time information, like unforeseen changes, is considered which affect the sequencing, assignment and timetable of the jobs. The goal is to re-optimize the schedule when these changes occur [26].
The main task of the static SCC problem is to select the processing machine at each stage for each charge and to determine the sequence and timing for processing the charges at each machine [26]. The production data provides the cast and charge information after the charge and cast batching problem. The process data provides information about the facility, including machine breakdowns, processing time and casting speed, transfer and process route information [26]. Real-time events influence the production data as well as the process data.
In the introduction planning and scheduling chapter the occurrence of real time events, resource or job related, has already been discussed. In this paper the possible real-time events and their influence on the production and process data are summarized. This is shown in figure 17.
