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

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 metallurgist’s expert opinion that the customer’s specifications will be met by each grade selected. In this paper, the authors 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 customer’s specifications. These membership functions are then used to define a series of MCSCP’s. 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.