A note on duality gap in the simple plant location problem

This paper studies the duality gap in the simple plant location problem, and presents general formulas for the gap when certain complementary slackness conditions are satisfied. We show that the duality gap derived by Erlenkotter [A dual-based procedure for uncapacitated facility location, Operations Research 26 (1978) 992–1009], and which has been widely used in the literature, is a special case of the formulas presented here. A counterexample demonstrates that an underlying assumption in Erlenkotter may be violated. The results may be used to obtain improved lower bounds for branch-and-bound algorithms.