The economic production quantity with rework process in supply chain management

Cardenas-Barron [L.E. Cardenas-Barron, Economic production quantity with rework process at a single-stage manufacturing system with planned backorders, Computers and Industrial Engineering 57 (2009) 1105-1113] minimizes the annual total relevant cost TC(Q,B) to find the economic production quantity with rework process at a manufacturing system and assumes that TC(Q,B) is convex. So, the solution ([email protected]?,[email protected]?) satisfying the first-order-derivative condition for TC(Q,B) will be the optimal solution. However, this paper indicates that ([email protected]?,[email protected]?) does not necessarily exist although TC(Q,B) is convex. Consequently, the main purpose of this paper is two-fold: (A)This paper tries to develop the sufficient and necessary condition for the existence of the solution ([email protected]?,[email protected]?) satisfying the-first-derivative condition of TC(Q,B). (B)This paper tries to present a concrete solution procedure to find the optimal solution of TC(Q,B).