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TopThe Set Covering Problem And Proposed Solution Approach
In this section, we will introduce the set covering problem which is a discrete NP-complete (Karp, 1972) combinatorial optimization problem that has many industrial applications. Two important applications from the steel industry are ingot mold selection (Vasko, Wolf and Stott, 1987) and metallurgical grade assignment (Vasko, Wolf and Stott, 1989).
The set covering problem (SCP) is the problem of covering the rows of an m-row, n-column, zero-one matrix (aij) by a subset of the columns at minimum cost. A mathematical formulation for the SCP is:Minimize (1) subject to (2)
(3) where x
j is one if column
j is in the solution and zero otherwise. Constraint set (2) ensures that each row is covered by at least one column and constraint set (3) ensures that the x
j’s take on only the values zero or one.
Since the SCP is NP-complete and industrial applications typically involve large problem instances, many heuristic solution procedures have been developed over the years. For a discussion of recent heuristic and metaheuristic approaches for solving the SCP, we suggest you consult Ren, Feng, Ke and Zhang (2010).