A On-line Appendix. Proofs Proof of Lemma 1. The supplier's objective function (3) is piecewise linear. We focus on the Summary: A On-line Appendix. Proofs Proof of Lemma 1. The supplier's objective function (3) is piecewise linear. We focus on the corner-point solutions: z = 0 or q. The result follows. Proof of Lemma 2. Omitted for brevity. The proof is similar to that of Proposition 3. Proof of Proposition 1. The proof of this proposition is parallel to that of Proposition 3, except that the feasible region is pruned by the additional constraints q1(t1, t2) = 0 or q2(t1, t2) = 0. Specifically, when determining the optimal (q1, q2)(t1, t2), instead of four corner-point solutions, there are only three feasible corner-point solutions: (q1, q2)(t1, t2) = (0, 0), (0, D) or (D, 0). Proof of Proposition 2. The procedure for deriving the optimal order quantities of the buyer is similar to the proof of Proposition 3, except that for each of the four maximization problems in (A.10), there is only one feasible corner-point solution: (q1, q2)(t1, t2) = (D, D). This is the optimal quantity pair for this model. Proof of Proposition 3. At optimality, from the supplier's optimal profit (3), we have Xn(t1, t2) + pn(t1, t2)E min { qn(t1, t2), tn n ztn n [(qn, pn)(t1, t2)] } = tn Collections: Engineering