### Optimizing a Black Art

 This article is based on a talk given by Alan Parker of British Steel to the Mathematical Programming Study Group. Buying Coal British Steel buys 7.4 million tonnes of coking coal a year for use in blast furnaces making iron. It transforms the coal into coke at 6 coke works which are served by 3 ports: Teesside; Immingham; and Port Talbot. Each coke works uses a single port, to which it is either adjacent or connected by road or rail. 98% of the coal which British Steel uses for making coke is imported so the problem which it faces is to decide: how much of each type of coal to import for each coke works in order to minimise the cost of the coal blend + the cost of delivering it to the coke works while meeting the coke works' requirements for coal quality; remaining within the limits of the availability of each type of coal; not exceeding the capacities of the ports or the delivery mechanisms. On the surface, therefore, the problem appears to be a straightforward application of Linear Programming. But coal is not a simple commodity nor is coke-making straightforward. There are many different types of coal with different characteristics. Coke-making itself is very much a "black art" (no pun intended) and much effort has been applied over many years to understanding the process. It is only because this work has been successful that it has been possible to build a model which is genuinely useful. Producing Coke The principle of iron-making in a blast furnace is that iron ore is reduced by carbon (from coke) to form metallic iron and carbon dioxide. Coal itself is unsuitable for use directly in the blast furnace because it would not form a permeable bed of sufficient strength and porosity to support the weight of material in the blast furnace. Coal is turned into coke by crushing it and then heating it in coke ovens at a temperature of about 1250oC for about 18 hours in the absence of air. As the coal is heated towards 600oC, it softens and fuses into a coherent mass (semi-coke). As the temperature increases to about 1000oC the semi-coke mass contracts to form a hard porous coke. Typically the weight of coke produced is about 78% of the weight of coal used. The gas evolved is used, after cleaning, as a fuel for the ovens and as an energy source throughout the steelworks. By-products extracted (tar, benzole, ammonia, etc.) are either sold or incinerated. Coke-Making Characteristics Coals are ranked by the amount of volatile matter. At the simplest level you can say that mid-range prime coking coals will produce the best coke and that the further away the coal is from prime coking coal, the less suitable it is. Coke from high volatile coals will be too weak and reactive to be used in the blast furnace whilst carbonising low volatile coals can produce dangerously high pressures within the oven with consequent irreparable damage. Viewed under the microscope, coal can be seen to be composed of three main components, or macerals, analogous to the minerals found in rocks. One of these, vitrinite, softens on heating and, in conjunction with the others, liptinite and inertinite, forms the coke matrix. All three reflect light at different intensities. The reflectance of the vitrinite is a measure of the rank of the coal. It is inversely proportional to the volatile matter content. A coal blend for blast furnace coke should have a reflectance between 1.25% and 1.35%. Although the reflectance of coals blends linearly, it is not sufficient that the average reflectance of a blend should lie between 1.25% and 1.35%. Of more importance is the reflectance distribution (see Figure 1). Figure 1: Good Reflectance Distribution for Making Coke This is a series of numbers which measure the proportion of the sample whose reflectance lies in the range 0.8 - 0.9%, 0.9 - 1.0%, 1.0 - 1.1%, 1.1 - 1.2%, etc. It has been found that if the reflectance distribution is unimodal (e.g. Figure 1), the coke will be good; if it is widely dispersed (e.g. Figure 2(a)), the coke will be indifferent. If the reflectance distribution is characterised by two separate strong peaks (e.g. Figure 2(b)) the coke will not form properly at all and may damage the coke oven. Figure 2: (a) Poor and (b) Unacceptable Reflectance Distributions In addition to rank parameters, measures of coking ability are also required to assess a coal. Of these, dilatation and fluidity, which provide empirical measures of the extent of softening and fusion on heating, are the most important. For any average level of reflectance there is a desired level of inerts. The level of inerts blends linearly but dilatation and fluidity are sub-additive and not readily predictable. As a result, when a blend of coals is being considered for use, a test batch of coke will be made in a test oven to assess its quality. In addition to these quantitative characteristics there are other classifications of coals, e.g. coking coal / filler coal / petroleum coke / coke breeze; country of origin. Some coke works constrain blends of coals by limiting the number or quantity of coals of a particular class which can be used in a blend. Coal-Buying Model The decision variables and basic structure of the coal-buying model have been outlined in the first section above. It is worth remarking that the main decision variables are semi-continuous (i.e. they either take the value 0 or are greater than some lower limit) so as to avoid having very small quantities of coking coals in blends. There are also some other integer constraints which restrict the number of types of coal of a particular class which can be imported through a port or used in a blend. The coke-making process is represented in the model primarily by blend constraints on the characteristics of the coals. The precise nature of these constraints varies from coke works to coke works depending on local preferences. As an example, at one site the reflectance distribution was required to be unimodal with its peak between 1.2% and 1.3%. This was found to be very severe and has been relaxed to permit some small deviation from the ideal. One feature of the model is that stock-holding costs can be included in the objective. To do this, the objective function is modified to include the interest lost by having capital tied up in the average coal stock. Inclusion of the stockholding option tends to favour use of fewer materials and smaller consignment sizes. Practical Experience British Steel has built the coal-buying model on a PC using XPRESS-MP as the LP software and SMART spreadsheets for data-handling and reporting. The model is used with several different scopes (national; single port; single coke works) and over several different timescales. A typical national model has 1500 rows and 1000 columns including 260 binary variables and 180 semi-continuous variables and solves to global optimality within two minutes on a Pentium P100. When the model was introduced, no integer solution could be found even after 24 hours because the coke-makers constrained the blends of coals to what they considered ideal. They relaxed the constraints on the blends when it was pointed out that many of the blends which they used for making coke lay outside their notional specification. The model is being developed further, in particular to move it away from a coal-blending model and closer to being a coke-making model by incorporating more coke properties. The objective also is being enhanced so that it reflects the credits associated with the by-products and the quantities of each of these which are produced depending on the coals used. Conclusion It is often said that processes are too complicated, messy or ill-understood to be represented using Linear Programming. Coke-making certainly fell in this class. Yet work by coal researchers at British Steel and other major steel producers has overturned this. Although coke-making is still not perfectly understood, correlations have been identified which make it possible to predict how good the coke will be from a blend of coals. This in turn has enabled British Steel to build an LP model which helps to preserve its competitiveness by keeping down the cost of an essential material for making iron. Related articles include Aluminium Smelter Benefits from MP Consultancy and Mathematical Programming in the Oil Industry. To find other articles, refer to the MP in Action page.
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