On-Line Appendix for Optimal and Heuristic Echelon (r, nQ, T ) Policies in Serial Inventory Systems with Fixed Costs

In the following proofs, we approximate demand and control variables as continuous when we prove the convexity of cost functions. In addition, unless otherwise noted, we use (f(x))′ to denote the derivative with respect to x and (g(x, y))′ to denote the partial derivative with respect to the first argument x for any continuous and differentiable functions f(x) and g(x, y). Finally, we need the following result which is due to Lemma 1 of Chen (2000). For j = 1, . . . , N ,