Multi-index Assignment Problems     In the case of 3-index assignment problems there are two main models: the axial 3-index assignment problem and the planar 3-index assignment problem.           Axial 3-index Assignment Problem     We are given an n x n x n matrix C=(cijk), and we want to select n elements of C, so that there is exactly one element in each two-dimensional face (in the three orientations), and the sum of the corresponding costs is a minimum.           Mathematical model:      min  n  ∑    i = 1  n  ∑    j = 1  n  ∑    k = 1  cijkxijk   s.t.        n  ∑    j = 1  n  ∑    k = 1  xijk = 1      (i = 1,2,...n),                n  ∑    i = 1  n  ∑    k = 1  xijk = 1      (j = 1,2,...n),                n  ∑    i = 1  n  ∑    j = 1  xijk = 1      (k = 1,2,...n),                    xijk {0,1}      (i, j, k = 1,2,...n)

Planar 3-index Assignment Problem     We are given an n x n x n matrix C=(cijk), and we want to select n2 elements of C, so that there is exactly one element in each one-dimensional line (in the three orientations), and the sum of the corresponding costs is a minimum.           Mathematical model:      min  n  ∑    i = 1  n  ∑    j = 1  n  ∑    k = 1  cijkxijk   s.t.        n  ∑    k = 1  xijk = 1      (i, j = 1,2,...n),                n  ∑    i = 1  xijk = 1      (j, k = 1,2,...n),              n  ∑    j = 1  xijk = 1      (i, k = 1,2,...n),                    xijk {0,1}      (i, j, k = 1,2,...n)              To get more, read the Introduction