Optimal requirement allocation among quantity-discount quoting suppliers

Abstract

We consider a problem motivated by a central purchasing organization for a major office products distributor. This purchasing organization must source a quantity of a particular resale item from a set of capacitated suppliers. In our case each supplier offers an incremental quantity discount purchase price structure. The purchaser’s objective is to obtain a quantity of a required item at minimum cost. The resulting problem is one of allocating order quantities among an approved supply base and involves minimizing the sum of separable piecewise linear concave cost functions. We develop a branch and bound algorithm that arrives at an optimal solution by generating linear knapsack subproblems with feasible solutions to the original problem.

Appendix: Proof of Result 1

(SPLCCAP) is a concave minimization problem over a bounded polyhedron, D ∩ C. Therefore, there exists at least one optimal solution which is an extreme point of D ∩ C. The set D ∩ C consists of a knapsack constraint with bounded variables. It is a well known fact that the extreme points of D ∩ C are characterized by having at most one variable q _i such that $0< q_i<u_{iK_i}$ and all other variables either at their lower or upper bound. Therefore, it then follows that there exists at least one optimal solution to (SPLCCAP) such that there is at most one supplier j for which $0<q_j<u_{jK_j}$ and for all other suppliers (i ≠ j) q _i  = 0 or $q_i=u_{iK_i}$.