Linear Assignment Problems

Publisher Summary This chapter discusses the generalized assignment problem (GAP) that consists of minimizing the total cost required to assign each job to exactly one machine without exceeding the machine's available resources. All the problems analyzed are linear. The chapter discusses the Hungarian algorithms. The Hungarian algorithms look for an augmenting path by considering only zero elements of the reduced cost matrix (c*). When no such path exists, the dual variables are updated so as to increase the number of zero elements in (c*) and the augmenting path search continues. In many practical cases the cost matrix is represented as a sparse matrix. All the primal-dual algorithms are modified so as to consider this case. As the elements of the cost matrix are always scanned only along the rows and never along the columns, the entries of row i (i ɛ N) of the sparse matrix can efficiently be represented by two sets Ji and Ci containing the corresponding column indices and costs. GAP is generally solved through branch-and-bound algorithms. The most important algorithms for the exact solution of GAP are depth-first branch-and-bound algorithms.