Three-Index Assignment Problem

The three-index assignment problem was introduced by W.P. Pierskalla [6] as a natural extension of the classical linear assignment problem (cf. also Assignment and matching). Given a cost array C = (c _ijk )∈ R ^{n 3} the integer 0-1 formulation of the 3AP is as follows:

where I, J, K ∈ Z ⁺ are disjoint sets. Moreover, we can assume that |I| = |J| = |K| = n, since extra elements can be added to the sets as long as the associated costs are set to a very large number. We can think of c _ijk as the cost of assigning job j to be performed by worker i in machine k. It follows that x _ijk = 1 if job j is assigned to worker i in machine k, and x _ijk = 0 otherwise. The 3AP is a special case of the multidimensional assignment problem and is related to the quadratic assignment problem (cf. Quadratic assignment problem).

The 3AP can also be formulated by using permutation functions. Clearly, there are n ³ different costs, and we wish to pick out the nsmallest without violating any of the constraints....