Francis J. Vasko

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 computational improvement to Wang's two-dimensional cutting stock algorithm

Abstract Wang developed algorithms [Opns Res. 31, 573–586 (1983)] for the constrained two-dimensional guillotine cutting stock problem. Given a bound on trim waste based on a feasible cutting pattern, her algorithms are guaranteed to generate the optimal guillotine cutting pattern. In this paper, we discuss two algorithms: (1) an algorithm, SPAM, which quickly generates solutions to the constrained two-stage two-dimensional guillotine cutting stock problem, and (2) an enhanced version of Wangs Algorithm One [1] which significantly improves its computational performance. SPAM is used to generate an initial upper bound for the minimum trim waste of the more general (non-staged) constrained guillotine cutting stock problem. Then, this bound is used in both Wangs Algorithm One and the enhanced version of it to solve 120 cutting stock problems. On average, the enhanced version was more than 25 times faster than the original algorithm, and the computational benefits of the enhancements increased as the problem complexity increased.

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.

Using a facility location algorithm to solve large set covering problems

Erlenkotter has developed an efficient exact (guarantees optimality) algorithm to solve the uncapacitated facility location problem (UFLP). In this paper, we use his algorithm to solve large instances of an important subset of the UFLP; the set covering problem (SCP). In addition, we present further empirical evidence that a heuristic algorithm developed by Vasko and Wilson for the SCP is capable of quickly generating good solutions to large SCPs.

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.

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.

Coal blending models for optimum cokemaking and blast furnace operation

An important problem at an integrated steel-producing plant is the blending of different types of coals to make coke for the blast furnace operation. Historically, linear blending models were not appropriate because coal properties important for both optimum cokemaking and blast furnace operation do not combine linearly and are not completely understood. In this paper, a solution methodology is developed that utilizes two techniques: (1) a mixed integer linear programming model for blending the candidate coals to produce coke at a minimum cost and (2) binary decision tree analyses and results that are converted into model constraints to ensure the production of high-quality coke. Subsequently, the model results are used at the pilot-scale oven for testing and for validating the new, improved blend(s) that have been recommended by the model. This is an on-going need that is dictated by changing availabilities in both coal prices and sources. These steps reduce costs by both minimizing the number of blends to be tested at the pilot-scale facility and ensuring a minimum cost coal blend that is useable for the operating facilities. Hypothetical, but realistic, data are used to illustrate how the model performs.

A performance comparison of heuristics for the total weighted tardiness problem

The single machine total weighted tardiness problem is an NP-hard problem that requires the use of heuristic solution procedures when more than 50 jobs are to be scheduled. In the literature, a well-tuned simulated annealing method and a descent heuristic with zero interchanges (DESO) both generated the best solutions for a large set of randomly generated problems. Due dates are generated by defining two parameters: the relative range of due dates (RDD) and the average tardiness factor (TF). In this paper, we define several heuristics based on dynamic programming and then use these and DESO heuristics to solve 50-job, 100-job, 200-job, and 500-job problems.

Metaheuristics for meltshop scheduling in the steel industry

The scheduling of a meltshop at an integrated steel plant is a very complex and important logistical industrial problem. This problem requires the synchronization of several steelmaking furnaces, degassing facilities, ladle treatment stations, and continuous casters. In this paper, we discuss how an efficient domain-specific heuristic is combined with metaheuristic approaches in a prototype scheduling model. Specifically, given preliminary schedules for the continuous casters, the model determines the allocation, sequencing, and scheduling of batches of steel at the basic oxygen steelmaking furnaces, the degassing facilities, and the ladle treatment stations. It also makes the appropriate schedule modifications at the continuous casters. Computational results will be discussed.