Floyd E. Wolf

Optimal selection of ingot sizes via set covering

In 1984, Bethlehem Steel Corporation installed a new ingot mold stripping facility at its Bethlehem Plant that is capable of handling taller ingots. In order to take maximum advantage of this new facility, we developed a two-phase, computer-based procedure for selecting optimal ingot and internal ingot mold dimensions. Phase I of this procedure generates feasible ingot and internal ingot mold dimensions consistent with both the new strippers capability and with mill constraints. Phase II then uses a set covering approach to select the optimal ingot and internal ingot mold sizes from among the feasible sizes generated. After analyzing the model, we recommended six new rectangular mold sizes to replace seven existing sizes. To date, implementation of these new ingot and mold sizes is proceeding smoothly and realizing the projected cost reduction benefits.

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.

Solving large set covering problems on a personal computer

Abstract The set covering problem (SCP) was one of the first problems shown to be NP-complete. Heuristics are commonly used on mainframe computers in order to efficiently solve large-scale SCPs. In this paper, we use a new heuristic and several existing heuristics written in FORTRAN to solve 31 large (up to 2000 variables) SCPs on an IBM PC/AT. The new heuristic, SCAMP (set covering algorithm for the microprocessor), performed the best, with solution values deviating only an average 1.8% from the optimum.

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.

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.

A large-scale application of the partial coverage uncapacitated facility location problem

The traditional, uncapacitated facility location problem (UFLP) seeks to determine a set of warehouses to open such that all retail stores are serviced by a warehouse and the sum of the fixed costs of opening and operating the warehouses and the variable costs of supplying the retail stores from the opened warehouses is minimized. In this paper, we discuss the partial coverage uncapacitated facility location problem (PCUFLP) as a generalization of the uncapacitated facility location problem in which not all the retail stores must be satisfied by a warehouse. Erlenkotters dual-ascent algorithm, DUALOC, will be used to solve optimally large (1600 stores and 13 000 candidate warehouses) real-world implemented PCUFLP applications in less than two minutes on a 500 MHz PC. Furthermore, a simple analysis of the problem input data will indicate why and when efficient solutions to large PCUFLPs can be expected.

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.

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.

A matching approach for replenishing rectangular stock sizes

Consider a replenishment problem in which several different rectangular sizes of material are stocked. Customers order rectangles of the material, but the rectangles ordered have a range on specified width as well as on specified length. To satisfy the customer requirements, the stock material can be cut once longitudinally in order to satisfy two customer requirements or not cut at all, that is, an entire stock piece of material is used to satisfy one customer requirement. If an exact match is impossible in the current planning period, the unused material must be returned to stock— an expensive and undesirable situation. In this paper, a nonbipartite weighted matching problem formulation will be given for determining the replenishment requirements of rectangular stock sizes. Then, a hybrid solution approach, capable of solving real applications (typically up to 3000 nodes) efficiently, will be discussed. This solution was implemented in September 1998 and has operated successfully since then.