Three-dimensional Axial Assignment Problems with Decomposable Cost Coefficients

Abstract Given three n -element sequences a i , b i and c i of nonnegative real numbers, the aim is to find two permutations φ and Ψ such that the sum ∑ n i = 1 a i bφ ( i ) Cψ ( i ) is minimized (maximized, respectively). We show that the maximization version of this problem can be solved in polynomial time, whereas we present an NP-completeness proof for the minimization version. We identify several special cases of the minimization problem which can be solved in polynomial time, and suggest a local search heuristic for the general case.