e-companion Reiman and Wang: Asymptotically Optimal Inventory Control for Assemble-to-Order Systems

We first prove that if y * optimizes (16), then |y * j | ≤ M, 1 ≤ j ≤ n. (EC.1) Suppose that in (16), given y and D, z * (y; D) optimizes ϕ(y; D). Then the objective function is C(y) = h · E[y − Az * (y; D)] + b · E[D − z * (y; D)]. i:a ji >0 a ji b i a ji E[D i − z * i ] ≥ ζ j A j · E[D − z * ] ≥ ζ j (A j · E[D] − y j), and C(y) ≥ h · E[y − Az * ] ≥ h j (y j − A j · E[z * ]) ≥ h j (y j − A j · E[D]). (EC.3) Because y = 0 and z(y; D) = 0 is feasible for (16) in which case C(0) = b · E[D], the above inequalities indicate that a necessary condition for C(y) ≤ b · E[D] is that y j ≥ A j · E[D] − b · E[D] ζ j and y j ≤ A j · E[D] + b · E[D] h j , 1 ≤ j ≤ n, which cannot hold if |y j | > M for some j = 1, .., n. Thus to be optimal, y * must satisfy (EC.1). Apply the same logic to (12). Since y = 0 and z(y; D) = 0 is feasible, b · E[D] ≥ C + (y o) ≥ h j (y o j − A j · E[D]), 1 ≤ j ≤ n, which implies that y o j ≤ A j · E[D] + b · E[D] h j ≤ M, 1 ≤ j ≤ n.