Three-dimensional axial assignment problems with decomposable cost coefficients Original Research Article

Given three n-element sequences ai, bi and ci of nonnegative real numbers, the aim is to find two permutations φ and Ψ such that the sum ∑ni = 1 aibφ(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.