Kenneth L. Stott

A set covering approach to metallurgical grade assignment

Abstract Early in 1986 Bethlehem Steel Corporation installed two continuous slab caster machines to modernize the steelmaking facilities at two of its major plants. The installation of this equipment, at a total cost of about half a billion dollars, required accompanying production planning and control systems (PPC) in order to function efficiently. The PPC module responsible for assigning metallurgical grades to customer orders uses a minimum cardinality set covering approach which not only minimizes the number of metallurgical grades required to satisfy a given collection of customer orders, but also is able to ‘show preference’ to priority orders. The algorithm, Optsol is used in a two-pass mode to quickly generate very good solutions to these large scale (up to 1000 zero-one variables and 2500 constraints) problems. When compared to the traditional method of grade assignment, this approach has the potential to significantly improve caster productivity and to reduce semi-finished inventory.

A practical solution to a fuzzy two-dimensional cutting stock problem

Abstract An important implementation of the two-dimensional cutting stock problem is the application of customer plate orders directly to surplus steel plates. In addition to the obvious desire to determine a high yield pattern, management is also interested in: (1) cutting few orders from a surplus plate (productivity reasons), (2) cutting mostly high priority orders from the plate (customer service considerations), and (3) cutting orders from a plate for as few distinct customers as possible (logistical concerns). In this paper, we present a problem formulation in a fuzzy environment which addresses the concerns listed above. Then, using α-cut sets, a sequence of crisp cutting stock problems are generated from the fuzzy formulation. A heuristic approach is developed to efficiently solve this sequence of problems. An example is solved to illustrate this approach which is used routinely at a Bethlehem Steel Corporation plant.

The cable trench problem: combining the shortest path and minimum spanning tree problems

Let G = (V, E) be a connected graph with specified vertex υ0 &isin V, length l(e) ≥ 0 for each e &isin E, and positive parameters τ and γ. The cable-trench problem (CTP) is to find a spanning tree T such that τlτ(T) + γlγ(T) is minimized where lτ(T) is the total length of the spanning tree T and lγ(T) is the total path length in T from υ0 to all other vertices of V. Since all vertices must be connected to υ0 and only edges from E are allowed, the solution will not be a Steiner tree. Consider the ratio R = τ/γ. For R large enough the solution will be a minimum spanning tree and for R small enough the solution will be a shortest path. In this paper, the CTP will be shown to be NP-complete. A mathematical formulation for the CTP will be provided for specific values of τ and γ. Also, a heuristic will be discussed that will solve the CTP for all values of R.

A hierarchical approach for one-dimensional cutting stock problems in the steel industry that maximizes yield and minimizes overgrading

Abstract In the steel industry, as hot steel products exit the producing facility, they are cut at primary saws (hotsaws) into shorter pieces. After these pieces cool, they are inspected for defects and either applied directly to customer orders or are further cut to ordered lengths at secondary saws (cold saws). In this case study, we will describe a hierarchical algorithm, DYNACUT_CS, that efficiently and effectively generates cutting patterns for material that is to be cut at cold saws. DYNACUT_CS strives to maximize yield over all the material cut and simultaneously tries to minimize overgrading (applying higher quality material than specified by the customer). An example will be given to illustrate how the algorithm works. This approach has been implemented for a variety of products at several different Bethlehem Steel Corporation facilities.

An application combining set covering and fuzzy sets to optimally assign metallurgical grades to customer orders

Abstract In order to optimize the productivity and yield of a modern high-speed continuous casting operation, it is desirable to minimize the number of metallurgical grades that have to be melted in order to satisfy a collection of customer orders. This problem can be formulated mathematically as a minimum cardinality set covering problem (MCSCP) as long as we are able to define which metallurgical grades are applicable to each customer order. The set of applicable grades is based on the metallurgists expert opinion that the customers specifications will be met by each grade selected. In this paper, we demonstrate how the expertise of the metallurgist can be used to define fuzzy subsets of the set of all metallurgical grades such that the membership function is based on the likelihood of a grade meeting the customers specifications. These membership functions are then used to define a series of MCSCPs. A comparison of the MCSCP solutions allows us to trade off minimizing the number of grades used against maximizing the likelihood that the customer specifications will be met without difficulty. These concepts are currently being integrated into an existing production planning and control system.

Assigning slabs to orders: an example of appropriate model formulation

Abstract An important problem for a steel company is the assigning of slabs, i.e. semi-finished rectangular pieces of steel, to customer orders. Due to its discrete nature, the prolem can be formulated as a zero-one integer programming problem; however, real-world problems are too large (12,000–16,000 zero-one variable) to be solved exactly in a reasonable amount of computer time. In this paper we present a transportation formulation for this problem that can be efficiently solved using a network code of Bertsekas. Then, using rounding heuristics, the transportation solution can be transformed into a practical solution. An example is used to illustrate this approach. Empirical results indicate that excellent (low-cost) practical solutions can be generated for large-scale (300 orders and 3000 slabs) problems in less than 1 min on a 386 PC (25 MHZ).

Using fuzzy sets to assign metallurgical grades to steel

To optimize the productivity and yield of a modern high-speed continuous casting operation, it is desirable to minimize the number of metallurgical grades that have to be melted to satisfy a collection of customer orders. This problem has been addressed through the development of an expert system for selecting the set of all potential grades for each order and an optimal selection algorithm for determining the actual grades that would be required to produce all orders. As a further refinement, a fuzzy formulation with a membership function based on the likelihood of a grade meeting the customer’s specifications without difficulty has been added. This enables the plate mill to trade off minimizing the number of grades used against maximizing the likelihood that the customer specifications will be met without difficulty.

Bethlehem steel combines cutting stock and set covering to enhance customer service

Some Bethlehem Steel customers order master coils of sheet steel which are slit into a number of narrower and smaller coils to fit specific manufacturing needs. To serve these customers, Bethlehem has developed a computer-based mathematical model that generates, based on customer requirements, optimum coil widths and slitting patterns. This model consists of two main programs. First, a set covering approach is used to minimize the number of distinct thicknesses that must be considered-each thickness defines a cutting stock problem. Then, based on input parameters specified by the user, a cutting stock algorithm is used that balances the following customer objectives: (1) minimize number of slitter setups, (2) maximize material utilization, (3) generate minimum excess inventory, (4) generate minimum shortfall against forecasted demand, and (5) meet all equipment limitations. The program aso generates coil widths that optimally utilize Bethlehems facilities. Examples are given to illustrate the utility of the model.

A real-time one-dimensional cutting stock algorithm for balanced cutting patterns

In the steel industry when a finished structural shape, exits the mill upon which it was produced, it is cut into customer order lenghts. The actual lenght of the finished I-beam bar may not be known precisely until immediately before cutting. Also if the mill can produce bars faster than they can be cut, then trying to generate cutting patterns with the number of cuts per bar close to the average number of cuts per bar (balanced cutting patterns) will maximize primary saw (hotsaw) cutting and reduce the number of (more costly) cuts that have to be made at the secondary saw (coldsaw). In this paper, a PC-based branch-and-bound algorithm, DYNACUT, is discussed that generates high-yield, balanced cutting patterns in real-time (maximum three seconds/bar pattern generation time on a 486 machine) based on the precise length of the bar determined as it leaves the mill and arrives at the hotsaw for primary cutting. Examples based on actual orders will illustrate both the efficiency and effectiveness of this algorithm.

Fiddler on the Roof: - balancing trim loss and setups

An application of a classic one-dimensional cutting stock problem requires that large steel coils be slit into smaller customer required widths. In determining culling patterns for this problem, both the amount of trim loss and the number of slitter setups are major economic concerns. In theory, the optimal solution to this problem is a weighted function of trim loss and number of setups. In practice, the values of the objective function cost coefficients are not readily available. In this paper, a strategy is discussed that has been used in practice to generate a set of optimal solutions for review by management.