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)TC(Q,B) to find the economic production quantity with rework process at a manufacturing system and assumes that TC(Q,B)TC(Q,B) is convex. So, the solution (Q̄,B̄) satisfying the first-order-derivative condition for TC(Q,B)TC(Q,B) will be the optimal solution. However, this paper indicates that (Q̄,B̄) does not necessarily exist although TC(Q,B)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 (Q̄,B̄) satisfying the-first-derivative condition of TC(Q,B)TC(Q,B).(B)This paper tries to present a concrete solution procedure to find the optimal solution of TC(Q,B)TC(Q,B).