Effect of timing on reliability improvement and ordering decisions in a decentralized assembly system

Abstract

In this study, we investigate a decentralized assembly system in which the suppliers are unreliable and have uncertain production capacities. We focus on the case in which a manufacturer deals with two suppliers that provide complementary products. We assume that the suppliers have the opportunity to improve their reliabilities through investments. The manufacturer determines her order quantities and the suppliers decide on their investment amounts in their capacities. The timing of the decisions has a substantial effect on the optimal behaviors of the players. We investigate the problem under four different settings based on the sequence of events: (1) simultaneous ordering and investment in which decisions are made concurrently, (2) ordering after observation of capacities in which the manufacturer orders after observing the suppliers’ capacities, (3) ordering before realization of capacities in which the manufacturer orders after the suppliers’ investment decisions, and (4) ordering before investments in which the suppliers invest after the manufacturer’s ordering decision. We demonstrate the existence of a Pareto optimal equilibrium in the first two scenarios. In addition, we show that in the fourth scenario, there exists a unique Nash equilibrium in the suppliers’ game. Based on the realized capacities of the suppliers, it may be beneficial for them to share their production information with the manufacturer. In addition, we indicate that using a sequential decision strategy can enhance the performances of the supply chain and the members. When the suppliers are the leaders, they implement investment inflation strategies to stimulate the manufacturer to place larger orders. When the manufacturer is the leader, she uses an order inflation strategy to increase the investment of the suppliers. Our numerical analysis revealed that in different situations, the players may prefer ordering before realization of capacities scenario or ordering before investments. Finally, we extended our results to a multiple-suppliers case in which the suppliers are identical.

Appendix

Proof of Proposition 1

Assume ${\hat{q}}_1 \ne {\hat{q}}_2$ and let ${\bar{q}} = \min \{ {\hat{q}}_1, {\hat{q}}_2 \}$. Then ${\bar{q}} \ne {\hat{q}}_1$ or ${\bar{q}} \ne {\hat{q}}_2$. So $({\bar{q}}, {\bar{q}}) \ne ({\hat{q}}_1, {\hat{q}}_2)$. We show that for any values of $r_1$ and $r_2$, $\varPi ^{\text {so}}({\bar{q}}, {\bar{q}}|r_1, r_2) \ge \varPi ^\text {so}({\hat{q}}_1, {\hat{q}}_2|r_1, r_2)$.

$$\begin{aligned} \varPi ^{\text {so}}({\hat{q}}_1, {\hat{q}}_2)&= pE \left[ \min \left\{ D, {\hat{q}}_1, {\hat{q}}_2, U_1(r_1^{\text {so}}), U_2(r_2^{\text {so}}) \right\} \right] - w_1E \left[ \min \left\{ {\hat{q}}_1, U_1(r_1^{\text {so}}) \right\} \right] \\&\ - w_2 E \left[ \min \left\{ {\hat{q}}_2, U_2(r_2^{\text {so}})\right] \right\} \\ \le&pE \left[ \min \left\{ D, {\bar{q}}, U_1(r_1^{\text {so}}), U_2(r_2^{\text {so}}) \right\} \right] - w_1E \left[ \min \left\{ {\bar{q}}, U_1(r_1^{\text {so}}) \right\} \right] \\&- w_2 E \left[ \min \left\{ {\bar{q}}, U_2(r_2^{\text {so}})\right] \right\} = \varPi ^{\text {so}}({\bar{q}}, {\bar{q}}) \end{aligned}$$

$\square$

Proof of Proposition 2

• (a) Let $M=\min \{D,U_1(r_1^{\text {so}}),U_2(r_2^{\text {so}})\}$, then:

  $$\begin{aligned} Pr\left\{ M>u \right\}&= Pr\left\{ D>u \right\} Pr\left\{ U_1(r_1^{\text {so}})>u \right\} Pr\left\{ U_2(r_2^{\text {so}})>u \right\} \\&= {\bar{F}}(u) {\bar{G}}_1(u|r_1^{\text {so}}) {\bar{G}}_2(u|r_2^{\text {so}}) \end{aligned}$$

  Therefore, $\min \{ D,q^{\text {so}},U_1(r_1^{\text {so}}),U_2(r_2^{\text {so}})\} = \min \{M, q^{\text {so}} \} \le q^{\text {so}}$, and

  $$\begin{aligned} E\left[ \min \left\{ M,q^{\text {so}} \right\} \right]&= \int _{0}^{q^{\text {so}}} Pr \left[ \min \left\{ M,q^{\text {so}} \right\}>u \right] du = \int _{0}^{q^{\text {so}}} Pr \left[ M >u \right] du \\&= \int _{0}^{q^{\text {so}}}{\bar{F}}(u) {\bar{G}}_1(u|r_1^{\text {so}}) {\bar{G}}_2(u|r_2^{\text {so}})du \end{aligned}$$

  Similarly, we have $E \left[ \min \{D,U_i(r_i^{\text {so}})\} \right] = \int _{0}^{q^{\text {so}}} {\bar{G}}_i(u|r_i^{\text {so}})du$. So::

  $$\begin{aligned} \varPi ^{\text {so}}&= p\int _{0}^{q^{\text {so}}} {\bar{F}}(u) {\bar{G}}_1(u|r_1^{\text {so}}) {\bar{G}}_2(u|r_2^{\text {so}}) du - w_1 \int _{0}^{q^{\text {so}}} {\bar{G}}_1(u|r_1^{\text {so}}) du \nonumber \\&\quad - w_2 \int _{0}^{q^{\text {so}}} {\bar{G}}_2(u|r_2^{\text {so}}) du \end{aligned}$$

  (19)

  Similar to Eq. (19), we can simplify the objective functions of the suppliers as follows:

  $$\begin{aligned} \varTheta _1^{\text {so}}&= (w_1-c_1) \int _{{\underline{l}}_1}^{q^{\text {so}}} {\bar{G}}_1(u|r_1^{\text {so}}) du - v_1(r_1^{\text {so}}) \end{aligned}$$

  (20)
  $$\begin{aligned} \varTheta _2^{\text {so}}&= (w_2-c_2) \int _{{\underline{l}}_2}^{q^{\text {so}}} {\bar{G}}_2(u|r_2^{\text {so}}) du - v_2(r_2^{\text {so}}) \end{aligned}$$

  (21)

  Based on Eq. (19), the first derivative of $\varPi ^{\text {so}}$ with respect to $q^{\text {so}}$ is:

  $$\begin{aligned} \frac{\partial \varPi ^{\text {so}}}{\partial q^{\text {so}}} = p {\bar{F}}(q^{\text {so}}) {\bar{G}}_1(q^{\text {so}}|r_1^{\text {so}}) {\bar{G}}_2(q^{\text {so}}|r_2^{\text {so}}) - w_1 {\bar{G}}_1(q^{\text {so}}|r_1^{\text {so}}) - w_2 {\bar{G}}_2(q^{\text {so}}|r_2^{\text {so}}) \end{aligned}$$

  (22)

  Consider ${\hat{q}}^{\text {so}} = \underset{q}{\arg \max } \ \varPi$. Then, based on the first-order condition, we have:

  $$\begin{aligned} p {\bar{F}}({\hat{q}}^{\text {so}}) {\bar{G}}_2({\hat{q}}^{\text {so}} | r_2^{\text {so}}) - w_1 = \frac{w_2 {\bar{G}}_2({\hat{q}}^{\text {so}}|r_2^{\text {so}})}{{\bar{G}}_1({\hat{q}}^{\text {so}}|r_1^{\text {so}})}\end{aligned}$$

  (23)
  $$\begin{aligned} p {\bar{F}}({\hat{q}}^{\text {so}}) {\bar{G}}_1({\hat{q}}^{\text {so}} | r_1^{\text {so}}) - w_2 = \frac{w_1 {\bar{G}}_1({\hat{q}}^{\text {so}}|r_1^{\text {so}})}{{\bar{G}}_2({\hat{q}}^{\text {so}}|r_2^{\text {so}})} \end{aligned}$$

  (24)

  The second derivative of $\varPi ^{\text {so}}$ with respect to $q^{\text {so}}$ is:

  $$\begin{aligned} \frac{\partial ^2 \varPi ^{\text {so}}}{\partial q^{{\text {so}}^2}}&= - p f(q^{\text {so}}) {\bar{G}}_1(q^{\text {so}}|r_1^{\text {so}}) {\bar{G}}_2(q^{\text {so}}|r_2^{\text {so}}) - p {\bar{F}}(q^{\text {so}}) g_1(q^{\text {so}}|r_1^{\text {so}}) {\bar{G}}_2(q^{\text {so}}|r_2^{\text {so}}) \nonumber \\&\quad - p {\bar{F}}(q^{\text {so}}) {\bar{G}}_1(q^{\text {so}}|r_1^{\text {so}}) g_2(q^{\text {so}}|r_2^{\text {so}}) \nonumber \\&\quad + w_1 g_1(q^{\text {so}}|r_1^{\text {so}}) + w_2 g_2(q^{\text {so}}|r_2^{\text {so}}) \end{aligned}$$

  (25)

  By replacing Eqs. (23) and (24) in Eq. (25), we have:

  $$\begin{aligned} \frac{\partial ^2 \varPi ^{\text {so}}}{\partial q^{{\text {so}}^2}} ({\hat{q}}^{\text {so}})&= - p f({\hat{q}}^{\text {so}}) {\bar{G}}_1({\hat{q}}^{\text {so}}|r_1^{\text {so}}) {\bar{G}}_2({\hat{q}}^{\text {so}}|r_2^{\text {so}}) - \frac{w_2 g_1({\hat{q}}^{\text {so}}|r_1^{\text {so}}) {\bar{G}}_2({\hat{q}}^{\text {so}}|r_2^{\text {so}})}{{\bar{G}}_1({\hat{q}}^{\text {so}}|r_1^{\text {so}})} \\&- \frac{w_1 g_2({\hat{q}}^{\text {so}}|r_2^{\text {so}}) {\bar{G}}_1({\hat{q}}^{\text {so}}|r_1^{\text {so}})}{{\bar{G}}_2({\hat{q}}^{\text {so}}|r_2^{\text {so}})} \end{aligned}$$

  Because f(.), $g_1(.)$, $g_2(.)$, ${\bar{G}}_1(.)$ and ${\bar{G}}_2(.)$ are positive, then $\frac{\partial ^2 \varPi }{\partial q^{{\text {so}}^2}} ({\hat{q}}^{\text {so}}) \le 0$, which implies $\varPi ^{\text {so}}$ is quasi-concave and, therefore, ${\tilde{q}}^{\text {so}}(r_1^{\text {so}}, r_2^{\text {so}})$ is unique. The second derivatives of the suppliers’ objective functions are:

  $$\begin{aligned} \frac{\partial ^2 \varTheta _1^{\text {so}}}{\partial r_1^{{\text {so}}^2}}&= (w_1-c_1)\int _{{\underline{l}}_1}^{q^{\text {so}}} \frac{\partial ^2 {\bar{G}}_1(u|r_1^{\text {so}})}{\partial r_1^{{\text {so}}^2}} du - \frac{\partial ^2 v_1(r_1^{\text {so}})}{\partial r_1^{{\text {so}}^2}} \\ \frac{\partial ^2 \varTheta _2^{\text {so}}}{\partial r_2^{{\text {so}}^2}}&= (w_2-c_2)\int _{{\underline{l}}_2}^{q^{\text {so}}} \frac{\partial ^2 {\bar{G}}_2(u|r_2^{\text {so}})}{\partial r_2^{{\text {so}}^2}} du - \frac{\partial ^2 v_2(r_2^{\text {so}})}{\partial r_2^{{\text {so}}^2}} \end{aligned}$$

  Because ${\bar{G}}_1(.)$ and ${\bar{G}}_2(.)$ are concave and $v_1(r_1^{\text {so}})$ and $v_2(r_2^{\text {so}})$ are convex, then $\varTheta _1^{\text {so}}$ and $\varTheta _2^{\text {so}}$ are concave and, therefore, ${\tilde{r}}_1^{\text {so}}(q^{\text {so}},r_2^{\text {so}})$ and ${\tilde{r}}_2^{\text {so}}(q^{\text {so}},r_1^{\text {so}})$ are unique.

• (b) The second derivatives of $\varPi ^{\text {so}}$ with respect to $q^{\text {so}}$ and $r_1^{\text {so}}$, and with respect to $q^{\text {so}}$ and $r_2^{\text {so}}$, are:

  $$\begin{aligned} \frac{\partial ^2 \varPi ^{\text {so}}}{\partial q^{\text {so}} \partial r_1^{\text {so}}}&= p {\bar{F}}(q^{\text {so}}) {\bar{G}}_2(q^{\text {so}}|r_2^{\text {so}}) \frac{\partial {\bar{G}}_1(q^{\text {so}}|r_1^{\text {so}})}{\partial r_1^{\text {so}}} - w_1 \frac{\partial {\bar{G}}_1(q^{\text {so}}|r_1^{\text {so}})}{\partial r_1^{\text {so}}}\end{aligned}$$

  (26)
  $$\begin{aligned} \frac{\partial ^2 \varPi ^{\text {so}}}{\partial q^{\text {so}} \partial r_1^{\text {so}}}&= p {\bar{F}}(q^{\text {so}}) {\bar{G}}_2(q^{\text {so}}|r_2^{\text {so}}) \frac{\partial {\bar{G}}_1(q^{\text {so}}|r_1^{\text {so}})}{\partial r_1^{\text {so}}} - w_2 \frac{\partial {\bar{G}}_2(q^{\text {so}}|r_2^{\text {so}})}{\partial r_2^{\text {so}}} \end{aligned}$$

  (27)

  By replacing Eqs. (23) and (24) in Eqs. (26) and (27), we have:

  $$\begin{aligned} \frac{\partial ^2 \varPi ^{\text {so}}}{\partial q^{\text {so}} \partial r_1^{\text {so}}}({\tilde{q}}^{\text {so}})&= \frac{\partial {\bar{G}}_1({\tilde{q}}^{\text {so}}|r_1^{\text {so}})}{\partial r_1^{\text {so}}} \cdot \frac{w_2 {\bar{G}}_2({\tilde{q}}^{\text {so}}|r_2^{\text {so}})}{{\bar{G}}_1({\tilde{q}}^{\text {so}}|r_1^{\text {so}})} \ge 0 \\ \frac{\partial ^2 \varPi ^{\text {so}}}{\partial q^{\text {so}} \partial r_2^{\text {so}}}({\tilde{q}}^{\text {so}})&= \frac{\partial {\bar{G}}_2({\tilde{q}}^{\text {so}}|r_2^{\text {so}})}{\partial r_2^{\text {so}}} \cdot \frac{w_1 {\bar{G}}_1({\tilde{q}}^{\text {so}}|r_1^{\text {so}})}{{\bar{G}}_2({\tilde{q}}^{\text {so}}|r_2^{\text {so}})} \ge 0 \end{aligned}$$

  The second derivatives of the $\varTheta _1^{\text {so}}$ with respect to $q^{\text {so}}$ and $r_1^{\text {so}}$, and with respect to $r_1^{\text {so}}$ and $r_2^{\text {so}}$ are:

  $$\begin{aligned} \frac{\partial ^2 \varTheta _1^{\text {so}}}{\partial r_1^{\text {so}} \partial q^{\text {so}}} = (w_1-c_1)\frac{\partial {\bar{G}}_1(q^{\text {so}}|r_1^{\text {so}})}{\partial r_1^{\text {so}}} \ge 0, \text {and,} \ \ \frac{\partial ^2 \varTheta _1^{\text {so}}}{\partial r_1^{\text {so}} \partial r_2^{\text {so}}}=0. \end{aligned}$$

  Similarly, for the second supplier, we have:

  $$\begin{aligned} \frac{\partial ^2 \varTheta _2^{\text {so}}}{\partial r_2^{\text {so}} \partial q^{\text {so}}} = (w_2-c_2)\frac{\partial {\bar{G}}_2(q^{\text {so}}|r_2^{\text {so}})}{\partial r_2^{\text {so}}} \ge 0\; \text {and,} \ \ \frac{\partial ^2 \varTheta _2^{\text {so}}}{\partial r_2^{\text {so}} \partial r_1^{\text {so}}}=0. \end{aligned}$$

  Then, according to Topkis (1998), we have $\frac{\partial {\tilde{q}}^{\text {so}}(r_1^{\text {so}},r_2^{\text {so}})}{\partial r_1^{\text {so}}}$, $\frac{\partial {\tilde{q}}^{\text {so}}(r_1^{\text {so}},r_2^{\text {so}})}{\partial r_2^{\text {so}}}$, $\frac{\partial {\tilde{r}}_1^{\text {so}}(q^{\text {so}},r_2^{\text {so}})^{\text {so}}}{\partial q^{\text {so}}}$, $\frac{\partial {\tilde{r}}_1^{\text {so}}(q^{\text {so}},r_2^{\text {so}})}{\partial r_2^{\text {so}}}$, $\frac{\partial {\tilde{r}}_2^{\text {so}}(q^{\text {so}},r_1^{\text {so}})}{\partial q^{\text {so}}}$, and $\frac{\partial {\tilde{r}}_2^{\text {so}}(q^{\text {so}},r_1^{\text {so}})}{\partial r_1^{\text {so}}} \ge 0$.

$\square$

Proof of Proposition 3

Clearly, $(q^{\text {so}}, r_1^{\text {so}}, r_2^{\text {so}}) = {\mathbf {0}}$ is a feasible strategy for the manufacturer and the suppliers. Hence, the strategy set is not empty. According to Proposition 2, Part (a), $\varPi ^{\text {so}}$, $\varTheta _1^{\text {so}}$, and $\varTheta _2^{\text {so}}$ are quasi-concave. As a result, there exists a NE for the game ( Cachon and Netessine 2006). $\square$

Proof of Lemma 1

According to Proposition 2, Part (b), the optimal strategies of the players are increasing functions with respect to other players’ strategies. As a result, we cannot have a case in which an equilibrium does not dominate another one (which contradicts with the supermodularity of the game). $\square$

Proof of Proposition 4

The first derivative of $\varPi ^{\text {so}}$ with respect to $r_1^{\text {so}}$ is:

$$\begin{aligned} \frac{\partial \varPi ^{\text {so}}}{\partial r_1^{\text {so}}}&= p \int _{0}^{q^{\text {so}}} {\bar{F}}(u) {\bar{G}}_2(u|r_2^{\text {so}}) \frac{\partial {\bar{G}}_1(u|r_1^{\text {so}})}{\partial r_1^{\text {so}}} du - w_1 \int _{0}^{q^{\text {so}}} \frac{\partial {\bar{G}}_1(u|r_1^{\text {so}})}{\partial r_1^{\text {so}}} du\\&= \int _{0}^{q^{\text {so}}} \frac{\partial {\bar{G}}_1(u|r_1^{\text {so}})}{\partial r_1^{\text {so}}} \left[ p {\bar{F}}(u) {\bar{G}}_2(u|r_2^{\text {so}}) -w_1 \right] du \end{aligned}$$

Because $\varPi ^{so}$ is quasi-concave, based on Eq. (22), for any value of $u \le {\tilde{q}}^{\text {so}}(r_1^{\text {so}}, r_2^{\text {so}})$, we have $p {\bar{F}}(u) {\bar{G}}_2(u|r_2^{\text {so}}) -w_1 \ge 0$. As a result, when $q^{\text {so}} = {\tilde{q}}^{\text {so}}(r_1^{\text {so}}, r_2^{\text {so}})$, $\varPi ^{\text {so}}$ is an increasing function of $r_1^{\text {so}}$. Similarly, we can prove that when $q^{\text {so}} = {\tilde{q}}^{\text {so}}(r_1^{\text {so}}, r_2^{\text {so}})$, $\varPi ^{\text {so}}$ is an increasing function of $r_2^{\text {so}}$. In addition, we can show that:

$$\begin{aligned} \frac{\partial \varTheta _1^{\text {so}}}{ \partial q^{\text {so}}} =(w_1-c_1) {\bar{G}}_1(q^{\text {so}}|r_1^{\text {so}}) \ge 0 \ \ , \ \ \frac{\partial \varTheta _1^{\text {so}}}{ \partial r_2^{\text {so}}}=0 \ \ , \ \ \frac{\partial \varTheta _2^{\text {so}}}{ \partial q^{\text {so}}} = (w_2-c_2){\bar{G}}_2(q^{\text {so}}|r_2^{\text {so}}) \ge 0, \end{aligned}$$

and $\ \ \frac{\partial \varTheta _2^{\text {so}}}{\partial r_1^{\text {so}}}=0.$

Therefore, the profit functions of the suppliers are increasing functions with respect to the order quantity of the manufacturer. In addition, a supplier’s profit function does not vary with respect to the other supplier’s investment. Consider $({\bar{q}}^{\text {so}^{(i)}}, {\bar{r}}_1^{\text {so}^{(i)}}, {\bar{r}}_2^{\text {so}^{(i)}})$ and $({\bar{q}}^{\text {so}^{(j)}}, {\bar{r}}_1^{\text {so}^{(j)}}, {\bar{r}}_2^{\text {so}^{(j)}})$ as two NEs of the game, where $({\bar{q}}^{\text {so}^{(i)}}, {\bar{r}}_1^{\text {so}^{(i)}}, {\bar{r}}_2^{\text {so}^{(i)}}){\ge } ({\bar{q}}^{\text {so}^{(j)}}, {\bar{r}}_1^{\text {so}^{(j)}}, {\bar{r}}_2^{\text {so}^{(j)}})$. Then, we have:

$$\begin{aligned} \varPi ^{\text {so}}({\bar{q}}^{\text {so}^{(j)}}, {\bar{r}}_1^{\text {so}^{(j)}}, {\bar{r}}_2^{\text {so}^{(j)}}) \le \varPi ^{\text {so}}({\bar{q}}^{\text {so}^{(j)}}, {\bar{r}}_1^{\text {so}^{(i)}}, {\bar{r}}_2^{\text {so}^{(i)}}) \le \varPi ^{\text {so}}({\bar{q}}^{\text {so}^{(i)}}, {\bar{r}}_1^{\text {so}^{(i)}}, {\bar{r}}_2^{\text {so}^{(i)}}) \end{aligned}$$

Similarly, we have:

$$\begin{aligned} \varTheta _1^{\text {so}}({\bar{q}}^{\text {so}^{(j)}}, {\bar{r}}_1^{\text {so}^{(j)}}, {\bar{r}}_2^{\text {so}^{(j)}}) \le \varTheta _1^{\text {so}}({\bar{q}}^{\text {so}^{(i)}}, {\bar{r}}_1^{\text {so}^{(j)}}, {\bar{r}}_2^{\text {so}^{(i)}}) \le \varTheta _1^{\text {so}}({\bar{q}}^{\text {so}^{(i)}}, {\bar{r}}_1^{\text {so}^{(i)}}, {\bar{r}}_2^{\text {so}^{(i)}}) \\ \varTheta _2^{\text {so}}({\bar{q}}^{\text {so}^{(j)}}, {\bar{r}}_1^{\text {so}^{(j)}}, {\bar{r}}_2^{\text {so}^{(j)}}) \le \varTheta _2^{\text {so}}({\bar{q}}^{\text {so}^{(i)}}, {\bar{r}}_1^{\text {so}^{(i)}}, {\bar{r}}_2^{\text {so}^{(j)}}) \le \varTheta _2^{\text {so}}({\bar{q}}^{\text {so}^{(i)}}, {\bar{r}}_1^{\text {so}^{(i)}}, {\bar{r}}_2^{\text {so}^{(i)}}) \end{aligned}$$

These inequalities can be found using the fact that each player’s objective function is an increasing function with respect to the other players’ strategies. Because $({\bar{q}}^{\text {so}^{(n)}}, {\bar{r}}_1^{\text {so}^{(n)}}, {\bar{r}}_2^{\text {so}^{(n)}}) \ge ({\bar{q}}^{\text {so}^{(i)}}, {\bar{r}}_1^{\text {so}^{(i)}}, {\bar{r}}_2^{\text {so}^{(i)}})$, $\forall i$, then $({\bar{q}}^{\text {so}^{(n)}}, {\bar{r}}_1^{\text {so}^{(n)}}, {\bar{r}}_2^{\text {so}^{(n)}})$ is the dominant equilibrium. $\square$

To prove Proposition 5, first we need to prove the following lemma.

Lemma 2

The optimal order quantity of the manufacturer, ${\tilde{q}}^{\text {oc}}$, is less than or equal to the observed capacities of the suppliers.

Proof

We need to show that ${\tilde{q}}^{\text {oc}} \le \min \left\{ u_1(r_1^{\text {oc}}), u_2(r_2^{\text {oc}}) \right\}$. We use contradiction to prove the lemma. Assume ${\tilde{q}}^{\text {oc}} > \min \left\{ u_1(r_1^{\text {oc}}), u_2(r_2^{\text {oc}}) \right\}$. Consider ${\hat{q}}^{\text {oc}} = \min \left\{ u_1(r_1^{\text {oc}}), u_2(r_2^{\text {oc}}) \right\}$. We show that $\varPi ^{\text {oc}}( {\hat{q}}^{\text {oc}}|r_1^{\text {oc}}, r_2^{\text {oc}}) \ge \varPi ^{\text {oc}}({\tilde{q}}^{\text {oc}} |r_1, r_2)$.

Case 1: Assume $u_1(r_1^{\text {oc}}) \le u_2 (r_2^{\text {oc}})$.

$$\begin{aligned} \varPi ^{\text {oc}}({\tilde{q}}^{\text {oc}}|r_1^{\text {oc}}, r_2^{\text {oc}})&= pE \left[ \min \left\{ D, {\tilde{q}}^{\text {oc}}, u_1(r_1^{\text {oc}}), u_2(r_2^{\text {oc}}) \right\} \right] - w_1\min \left\{ {\tilde{q}}^{\text {oc}}, u_1(r_1^{\text {oc}}) \right\} \\&\quad - w_2\min \left\{ {\tilde{q}}^{\text {oc}}, u_2(r_2^{\text {oc}}) \right\} \\&= pE \left[ \min \left\{ D, u_1(r_1^{\text {oc}}) \right\} \right] - w_1 u_1(r_1^{\text {oc}}) - w_2\min \left\{ {\tilde{q}}^{\text {oc}} , u_2(r_2^{\text {oc}}) \right\} \\&\le pE \left[ \min \left\{ D, u_1(r_1^{\text {oc}}) \right\} \right] - w_1 u_1(r_1^{\text {oc}}) - w_2\min \left\{ {\hat{q}}^{\text {oc}}, u_2(r_2^{\text {oc}}) \right\} \\&= pE \left[ \min \left\{ D, {\hat{q}}^{\text {oc}}, u_1(r_1^{\text {oc}}), u_2(r_2^{\text {oc}}) \right\} \right] - w_1\min \left\{ {\hat{q}}, u_1(r_1^{\text {oc}}) \right\} \\&\quad -w_2\min \left\{ {\hat{q}}, u_2(r_2^{\text {oc}}) \right\} \\&= \varPi ^{\text {oc}}({\hat{q}}^{\text {oc}}|r_1^{\text {oc}}, r_2^{\text {oc}}) \end{aligned}$$

Case 2: Assume $u_1(r_1^{\text {oc}}) \ge u_2 (r_2^{\text {oc}})$. The proof is analogous. $\square$

Proof of Proposition 5

According to Lemma 2, the problem of the manufacturer can be simplified as follows:

$$\begin{aligned} \underset{q^{\text {oc}} \ge 0}{\max } \ {\hat{\varPi }}^{\text {oc}} (q^{\text {oc}})&= p E \left[ \min \left\{ D, q^{\text {oc}} \right\} \right] - (w_1+w_2) q^{\text {oc}} \nonumber \\&s.t. \nonumber \\&q^{\text {oc}} \le \min \left\{ u_1(r_1^{\text {oc}}), u_2(r_2^{\text {oc}}) \right\} \end{aligned}$$

(28)

Because minimization preserves concavity, ${\hat{\varPi }}^{\text {oc}}(q^{\text {oc}})$ is concave with respect to $q^{\text {oc}}$. Consider ${\hat{q}}^{{\text {oc}}^*}$ as the optimum solution of ${\hat{\varPi }}^{\text {oc}}(q^{\text {oc}})$ in the absence of Constraint (28). ${\hat{\varPi }}^{\text {oc}}(q^{\text {oc}})$ is the objective function of the classic news vendor problem and its optimum solution is ${\hat{q}}^{{\text {oc}}^*} = {\bar{F}}^{-1} \left( \frac{w_1+w_2}{p} \right)$, which is the critical fractile solution. Now, by considering Constraint (28) and the concavity of ${\hat{\varPi }}^{\text {oc}}(q^{\text {oc}})$, the optimum solution of the manufacturer is as follows:

$$\begin{aligned} q^{{\text {oc}}^*}(r_1^{\text {oc}}, r_2^{\text {oc}}) = \min \left\{ {\bar{F}}^{-1} \left( \frac{w_1+w_2}{p} \right) , u_1(r_1^{\text {oc}}), u_2(r_2^{\text {oc}}) \right\} . \end{aligned}$$

(b) Let $Q= {\bar{F}}^{-1} \left( \frac{w_1+w_2}{p} \right)$. By replacing Q in Eq. (22), we have:

$$\begin{aligned} \frac{\partial \varPi ^{\text {so}}}{\partial q^{\text {so}}} (Q)&= (w_1+w_2) {\bar{G}}_1(Q|r_1^{\text {so}}) {\bar{G}}_2(Q|r_2^{\text {so}}) - w_1 {\bar{G}}_1(Q|r_1^{\text {so}}) - w_2 {\bar{G}}_2(Q|r_2^{\text {so}})\\&= - w_1 {\bar{G}}_1(Q|r_1^{\text {so}}) G_2(Q|r_2^{\text {so}}) - w_2 G_1(Q|r_1^{\text {so}}) {\bar{G}}_2(Q|r_2^{\text {so}}) \le 0 \end{aligned}$$

Because $\varPi ^\text {so}$ is unimodal and $\frac{\partial \varPi ^{\text {so}}}{\partial q^{\text {so}}} (Q) \le 0$, $\forall r_1^{\text {so}}, r_2^{\text {so}}$, then we conclude that $q^{{\text {so}}^*} \le Q$. $\square$

Proof of Proposition 6

• (a) To prove Proposition 6, first, we simplify the objective functions of the suppliers. Consider $Q = {\bar{F}}^{-1} \left( \frac{w_1+w_2}{p} \right)$, which represents the optimal order quantity of the buyer without considering the capacities of the suppliers. Then, the explicit form of the first supplier’s objective function is:

  $$\begin{aligned} \varTheta _1^{\text {oc}}&= (w_1-c_1) \int _{{\underline{l}}_1}^{Q} \int _{{\underline{l}}_2}^{u_1} u_2g_2(u_2|r_2^{\text {oc}})g_1(u_1|r_1^{\text {oc}})du_2du_1 \\&\quad + (w_1-c_1)\int _{{\underline{l}}_1}^{Q} \int _{u_1}^{{\overline{l}}_2} u_1g_2(u_2|r_2^{\text {oc}})g_1(u_1|r_1^{\text {oc}})du_2du_1 \\&\quad + (w_1-c_1)\int _{Q}^{{\overline{l}}_1} \int _{{\underline{l}}_2}^{Q} u_2g_2(u_2|r_2^{\text {oc}})g_1(u_1|r_1^{\text {oc}})du_2du_1 \\&\quad + (w_1-c_1)\int _{Q}^{{\overline{l}}_1} \int _{Q}^{{\overline{l}}_2} Q g_2(u_2|r_2^{\text {oc}})g_1(u_1|r_1^{\text {oc}})du_2du_1 - v_1(r_1^{\text {oc}})\\ \end{aligned}$$

  By using partial integration, we are able to show that:

  $$\begin{aligned} \varTheta _1^{\text {oc}} = (w_1-c_1) \int _{0}^{Q} {\bar{G}}_1(u_1|r_1^{\text {oc}}) {\bar{G}}_2(u_1|r_2^{\text {oc}}) du_1 -v_1(r_1^{\text {oc}}) \end{aligned}$$

  (29)

  Similarly, the objective function of the second supplier is:

  $$\begin{aligned} \varTheta _2^{\text {oc}} = (w_2-c_2) \int _{0}^{Q} {\bar{G}}_1(u_1|r_1^{\text {oc}}) {\bar{G}}_2(u_1|r_2^{\text {oc}}) du_1 -v_2(r_2^{\text {oc}}) \end{aligned}$$

  (30)

  Considering Eq. (29), the second derivative of $\varTheta _1^{\text {oc}}(r_1^{\text {oc}})$ with respect to $r_1^{\text {oc}}$ is:

  $$\begin{aligned} \frac{\partial ^2 \varTheta _1^{\text {oc}}}{\partial r_1^{{\text {oc}}^2}} = (w_1-c_1) \int _{0}^{Q} \frac{\partial ^2 {\bar{G}}_1(u_1|r_1^{\text {oc}})}{\partial r_1^{{\text {oc}}^2}} {\bar{G}}_2(u_1|r_2^{\text {oc}})du_1- \frac{\partial ^2 v_1(r_1^{\text {oc}})}{\partial r_1^{{\text {oc}}^2}} \end{aligned}$$

  (31)

  Because $G_1(u_1|r_1^{\text {oc}})$ is convex and ${\bar{G}}_1(u_1|r_1^{\text {oc}}) = 1 - G_1(u_1|r_1^{\text {oc}})$, then ${\bar{G}}_1(u_1|r_1^{\text {oc}})$ is concave. Additionally, it is assumed that $v_1(r_1^{\text {oc}})$ is convex. Therefore, $\frac{\partial ^2 {\bar{G}}_1(u_1|r_1^{\text {oc}})}{\partial r_1^{{\text {oc}}^2}} \le 0$ and $\frac{\partial ^2 v_1(r_1^{\text {oc}})}{\partial r_1^{{\text {oc}}^2}} \ge 0$; thus, $\frac{\partial ^2 \varTheta _1^{\text {oc}}}{\partial r_1^{{\text {oc}}^2}} \le 0$. This implies that $\varTheta _1^{\text {oc}}$ is concave with respect to $r_1^{\text {oc}}$ and ${\tilde{r}}^{\text {oc}}_1(r_2^{\text {oc}})$ is unique. The uniqueness of ${\tilde{r}}^{\text {oc}}_2(r_1^{\text {oc}})$ can be proved analogously.

• (b) We have:

  $$\begin{aligned} \frac{\partial ^2 \varTheta _1^{\text {oc}}}{\partial r_1^{\text {oc}} \partial r_2^{\text {oc}}} = (w_1-c_1) \int _{{\underline{l}}_1}^{Q} \frac{\partial {\bar{G}}_1(u_1|r_1^{\text {oc}})}{\partial r_1^{\text {oc}}}\frac{\partial {\bar{G}}_2(u_1|r_2^{\text {oc}})}{\partial r_2^{\text {oc}}} du_1 \ge 0 \end{aligned}$$

  (32)

  Inequality (32) is based on the fact that ${\bar{G}}_1(u_1|r_1^{\text {oc}})$ and ${\bar{G}}_2(u_1|r_2^{\text {oc}})$ are decreasing with respect to $r_1^{\text {oc}}$ and $r_2^{\text {oc}}$, respectively. Then, according to Topkis (1998), $\frac{\partial {\tilde{r}}_1^{\text {oc}}(r_2^{\text {oc}})}{\partial r_2^{\text {oc}}} \ge 0$. The proof for $\frac{\partial {\tilde{r}}_2^{\text {oc}}(r_1^{\text {oc}})}{\partial r_1^{\text {oc}}} \ge 0$ is analogous.

$\square$

Proof of Proposition 7

Clearly, $(r_1^{\text {oc}}, r_2^{\text {oc}}) = 0$ is a feasible strategy for the suppliers. Hence, the strategy set of the suppliers is not empty. In contrast, according to the proof of Proposition 6, $\frac{\partial ^2 \varTheta _1^{\text {oc}}}{\partial r_1^{\text {oc}} \partial r_2^{\text {oc}}} \ge 0$ and $\frac{\partial ^2 \varTheta _2^{\text {oc}}}{\partial r_1^{\text {oc}} \partial r_2^{\text {oc}}} \ge 0$. As a result, the game is supermodular, which proves the existence of an NE [see Cachon and Netessine (2006)].

According to Proposition 6, $\varTheta _1^{\text {oc}}$ and $\varTheta _2^{\text {oc}}$ are concave with respect to $r_1^{\text {oc}}$ and $r_2^{\text {oc}}$, respectively. As a result, the first-order conditions can be used to find the optimal solution. The first derivatives of $\varTheta _1^{\text {oc}}$ and $\varTheta _2^{\text {oc}}$ are:

$$\begin{aligned} \frac{\partial \varTheta _1^{\text {oc}}}{\partial r_1^{\text {oc}}} = (w_1-c_1) \int _{{\underline{l}}_1}^{Q} \frac{\partial {\bar{G}}_1(u_1|r_1^{\text {oc}})}{\partial r_1^{\text {oc}}} {\bar{G}}_2(u_1|r_2^{\text {oc}})du_1- \frac{\partial v_1(r_1^{\text {oc}})}{\partial r_1^{\text {oc}}} \\ \frac{\partial \varTheta _2^{\text {oc}}}{\partial r_2^{\text {oc}}} = (w_2-c_2) \int _{{\underline{l}}_1}^{Q} \frac{\partial {\bar{G}}_2(u_1|r_2^{\text {oc}})}{\partial r_2^{\text {oc}}} {\bar{G}}_1(u_1|r_1^{\text {oc}})du_1- \frac{\partial v_2(r_2^{\text {oc}})}{\partial r_2^{\text {oc}}} \end{aligned}$$

and that completes the proof. $\square$

To prove Proposition 8, first, we need to prove the following lemma.

Lemma 3

Assume $({\bar{r}}_1^{\text {oc}},{\bar{r}}_2^{\text {oc}})$ and $(\bar{{\bar{r}}}_1^{\text {oc}},\bar{{\bar{r}}}_2^{\text {oc}})$ are two equilibriums of the suppliers’ game. Then, either $({\bar{r}}_1^{\text {oc}},{\bar{r}}_2^{\text {oc}}) \ge (\bar{{\bar{r}}}_1^{\text {oc}},\bar{{\bar{r}}}_2^{\text {oc}})$ or $({\bar{r}}_1^{\text {oc}},{\bar{r}}_2^{\text {oc}}) \le (\bar{{\bar{r}}}_1^{\text {oc}},\bar{{\bar{r}}}_2^{\text {oc}})$.

Proof

The proof follows Part (b) of Proposition 6. $\square$

Proof of Proposition 8

Based on Eq. (29), the first derivative of $\varTheta _1^{\text {oc}}$ with respect to $r_2^{\text {oc}}$ is:

$$\begin{aligned} \frac{\partial \varTheta _1^{\text {oc}}}{\partial r_2^{\text {oc}}} = (w_1-c_1) \int _{{\underline{l}}_1}^{Q} \frac{\partial {\bar{G}}_2(u_1|r_2^{\text {oc}})}{\partial r_2^{\text {oc}}} {\bar{G}}_1(u_1|r_1^{\text {oc}})du_1 \end{aligned}$$

Because $\frac{\partial {\bar{G}}_2(u_1|r_2^{\text {oc}})}{\partial r_2^{\text {oc}}} \ge 0$, then $\frac{\partial \varTheta _1^{\text {oc}}}{\partial r_2^{\text {oc}}} \ge 0$. As result, $\varTheta _1^{\text {oc}}$ is an increasing function of $r_2^{\text {oc}}$. Consider $({\bar{r}}_1^{\text {oc}^{(i)}}, {\bar{r}}_2^{\text {oc}^{(i)}})$ and $({\bar{r}}_1^{\text {oc}^{(j)}}, {\bar{r}}_2^{\text {oc}^{(j)}})$ as two NEs of the leader stage of the dynamic game where $({\bar{r}}_1^{\text {oc}^{(i)}}, {\bar{r}}_2^{\text {oc}^{(i)}}) > ({\bar{r}}_1^{\text {oc}^{(j)}}, {\bar{r}}_2^{\text {oc}^{(j)}})$, then, we have:

$$\begin{aligned} \varTheta _1^{\text {oc}}({\bar{r}}_1^{\text {oc}^{(j)}}, {\bar{r}}_2^{\text {oc}^{(j)}}) \le \varTheta _1^{\text {oc}}({\bar{r}}_1^{\text {oc}^{(j)}}, {\bar{r}}_2^{\text {oc}^{(i)}}) \le \varTheta _1^{\text {oc}}({\bar{r}}_1^{\text {oc}^{(i)}}, {\bar{r}}_2^{\text {oc}^{(i)}}) \end{aligned}$$

This inequality can be found on the basis of the fact that $\varTheta _1^{\text {oc}}(r_1^{\text {oc}}, r_2^{\text {oc}})$ is an increasing function of $r_2^{\text {oc}}$, and ${\bar{r}}_1^{\text {oc}^{(i)}}$ maximizes $\varTheta _1^{\text {oc}}(r_1^{\text {oc}}, {\bar{r}}_2^{\text {oc}^{(i)}})$. Because $({\bar{r}}_1^{\text {oc}^{(n)}}, {\bar{r}}_2^{\text {oc}^{(n)}}) > ({\bar{r}}_1^{\text {oc}^{(i)}}, {\bar{r}}_2^{\text {oc}^{(i)}})$, $\forall i$, then $\varTheta _1^{\text {oc}}({\bar{r}}_1^{\text {oc}^{(n)}}, {\bar{r}}_2^{\text {oc}^{(n)}}) \ge \varTheta _1^{\text {oc}}({\bar{r}}_1^{\text {oc}^{(i)}}, {\bar{r}}_2^{\text {oc}^{(i)}})$, $\forall i$.

The proof for $\varTheta _2^{\text {oc}}({\bar{r}}_1^{\text {oc}^{(n)}}, {\bar{r}}_2^{\text {oc}^{(n)}}) \ge \varTheta _2^{\text {oc}}({\bar{r}}_1^{\text {oc}^{(i)}}, {\bar{r}}_2^{\text {oc}^{(i)}})$, $\forall i$ is analogous.$\square$

Proof of Proposition 9

• (a) Similar to Proposition 2, Part (a), we can show that the profit function of the manufacturer is:

  $$\begin{aligned} \varPi ^{\text {ov}}&= p \int _{0}^{q^{\text {ov}}} {\bar{F}}(u) {\bar{G}}_1(u|r_1^{\text {ov}}) {\bar{G}}_2(u|r_2^{\text {ov}}) du - w_1 \int _{0}^{q^{\text {ov}}} {\bar{G}}_1(u|r_1^{\text {ov}}) du \\&\quad - w_2 \int _{0}^{q^{\text {ov}}} {\bar{G}}_2(u|r_2^{\text {ov}}) du \end{aligned}$$

  The remainder of the proof is similar to Proposition 2, Part (a).

• (b) The proof is similar to Part (b) of Proposition 2.

$\square$

Proof of Proposition 10

• (a) The explicit form of the first supplier objective function is:

  $$\begin{aligned} \varTheta _1^{\text {ov}}&= (w_1-c_1) \int _{{\underline{l}}_1}^{{\tilde{q}}^{\text {ov}}(r_1^{\text {ov}}, r_2^{\text {ov}})} {\bar{G}}_1(u|r_1^{\text {ov}}) du - v_1(r_1^{\text {ov}}) \end{aligned}$$

  By taking the first derivative of $\varTheta _1^{\text {ov}}$ with respect to $r_1^{\text {ov}}$, we have:

  $$\begin{aligned} \frac{\partial \varTheta _1^{\text {ov}}}{r_1^{\text {ov}}}&= (w_1-c_1) \int _{{\underline{l}}_1}^{{\tilde{q}}^{\text {ov}}(r_1^{\text {ov}}, r_2^{\text {ov}})} \frac{\partial {\bar{G}}_1(u|r_1^{\text {ov}})}{r_1^{\text {ov}}} du \\&\quad + {\bar{G}}_1({\tilde{q}}^{\text {ov}}(r_1^{\text {ov}}, r_2^{\text {ov}})|r_1^{\text {ov}}) \frac{\partial {\tilde{q}}^{\text {ov}}(r_1^{\text {ov}}, r_2^{\text {ov}})}{r_1^{\text {ov}}} - \frac{\partial v_1^{\text {ov}}(r_1^{\text {ov}})}{r_1^{\text {ov}}} \end{aligned}$$

  According to Proposition 9, Part (c), $\frac{\partial {\tilde{q}}^{\text {ov}}(r_1^{\text {ov}}, r_2^{\text {ov}})}{r_1^{\text {ov}}} \ge 0$. As a result, based on Proposition 9, Part (b), we have:

  $$\begin{aligned} \frac{\partial \varTheta _1^{\text {ov}}}{r_1^{\text {ov}}}(r_1, r_2) \ge (w_1-c_1) \int _{{\underline{l}}_1}^{{\tilde{q}}^{\text {ov}}(r_1, r_2)} \frac{\partial {\bar{G}}_1(u|r_1)}{r_1} du - \frac{\partial v_1^{\text {ov}}(r_1)}{r_1} = \frac{\partial \varTheta _1^{\text {ov}}}{r_1^{\text {SO}}}(r_1, r_2) \end{aligned}$$

  (33)

  Based on Inequality (33), $r_1^{\text {ov}^*} \ge r_1^{\text {so}^*}$. The proof for $r_2^{\text {ov}^*} \ge r_2^{\text {so}^*}$ is analogous.

• (b) According to Proposition 9, Part (b), ${\tilde{q}}^{{\text {ov}}}(r_1, r_2) = q^{{\text {so}}^*}(r_1, r_2)$. Also based on Part (c) of this proposition, $\frac{\partial {\tilde{q}}^{\text {ov}}(r_1^{\text {ov}},r_2^{\text {ov}})}{\partial r_1^{\text {ov}}} \ge 0$ and $\frac{\partial {\tilde{q}}^{\text {ov}}(r_1^{\text {ov}},r_2^{\text {ov}})}{\partial r_2^{\text {ov}}} \ge 0$. As a result, ${\tilde{q}}^{\text {ov}}(.)$ is an increasing function with respect to $r_1^{\text {ov}}$ and $r_2^{\text {ov}}$. Because we have $r_1^{\text {ov}^*} \ge r_1^{\text {so}^*}$ and $r_2^{\text {ov}^*} \ge r_2^{\text {so}^*}$ from Part (a), $q^{\text {ov}^*} \ge q^{\text {so}^*}$, which completes the proof.

• (c) Note $\varPi ^{\text {ov}^*}(q, r_1, r_2) = \varPi ^{\text {so}^*}(q, r_1, r_2)$. Also, according to Proposition 4, $\frac{\varPi ^{\text {ov}^*}}{r_1^{\text {ov}}} \ge 0$ and $\frac{\varPi ^{\text {ov}^*}}{r_2^{\text {ov}}} \ge 0$. Because $(r_1^{\text {ov}^*}, r_2^{\text {ov}^*}) \ge (r_1^{\text {so}^*}, r_2^{\text {so}^*})$ from Part (a) of the proposition, we have $\varPi ^{\text {ov}^*}(q^{\text {ov}^*}, r_1^{\text {so}^*}, r_2^{\text {so}^*}) \ge \varPi ^{\text {so}^*}(q^{\text {so}^*}, r_1^{\text {so}^*}, r_2^{\text {so}^*})$

• (d) Because ${\tilde{q}}^{\text {ov}}(r_1, r_2) = q^{\text {so}^*}(r_1, r_2)$, then $\varTheta _1^{\text {ov}}(r_1, r_2) = \varTheta _1^{\text {so}}(q^{\text {so}^*}, r_1, r_2)$. Based on Eq. (33), we have:

  $$\begin{aligned} \varTheta _1^{\text {ov}^*}(r_1^{\text {ov}^*}, r_2^{\text {ov}^*}) \ge \varTheta _1^{\text {ov}^*}(r_1^{\text {so}^*}, r_2^{\text {so}^*}) = \varTheta _1^{\text {ov}^*}(q^{\text {so}^*}, r_1^{\text {ov}^*}, r_2^{\text {ov}^*})\end{aligned}$$

  which completes the proof. The proof for the second supplier is analogous.$\square$

Proof of Proposition 11

• (a) The explicit form of the first supplier’s objective function is:

  $$\begin{aligned} \varTheta _1^{\text {os}}&= (w_1-c_1) \int _{{\underline{l}}_1}^{q_1^{\text {os}}} {\bar{G}}_1(u|r_1^{\text {os}}) du - v_1(r_1^{\text {os}}) \end{aligned}$$

  The second derivative of $\varTheta _1^{\text {os}}$ with respect to $r_1^{\text {os}}$ is:

  $$\begin{aligned} \frac{\partial ^2 \varTheta _1^{\text {os}}}{\partial r_1^{\text {os}^2}} = \int _{{\underline{l}}_1}^{q_1^{\text {os}}} \frac{\partial ^2 {\bar{G}}_1(u|r_1^{\text {os}})}{\partial r_1^{\text {os}^2}}du - \frac{\partial ^2 v_1(r_1^{\text {os}})}{\partial r_1^{\text {os}^2}} \le 0 \end{aligned}$$

  As a result, ${\tilde{r}}_1^{\text {os}}(q_1^{\text {os}})$ is unique. Also, $\varTheta _1^{\text {os}}(q) = \varTheta _1^{\text {so}}(q)$, which implies that ${\tilde{r}}_1^{\text {os}}(q) = r_1^{\text {so}^*}(q)$. The proof for ${\tilde{r}}_2^{\text {os}}(q_2^{\text {os}})$ is analogous.

• (b) The second derivative of $\varTheta _1^{\text {os}}$ with respect to $r_1^{\text {os}}$ and $q_1^{\text {os}}$ is:

  $$\begin{aligned} \frac{\partial ^2 \varTheta _1^{\text {os}}}{\partial r_1^{\text {os}} \partial q_1^{\text {os}}} = \frac{\partial {\bar{G}}_1(q_1^{\text {os}}|r_1^{\text {os}})}{\partial r_1^{\text {os}}} \ge 0 \end{aligned}$$

  As a result, $\frac{\partial {\tilde{r}}_1^{\text {os}}(q_1^{\text {os}})}{\partial q_1^{\text {os}}} \ge 0$ ( Topkis 1998). The proof for $\frac{\partial {\tilde{r}}_2^{\text {os}}(q_2^{\text {os}})}{\partial q_2^{\text {os}}} \ge 0$ is analogous.

• (c) Because the suppliers’ profit functions are concave, then there exists a NE for the game. Also, $\frac{\partial \varTheta _1^{\text {os}}}{\partial r_2^{\text {os}}} = 0$ and $\frac{\partial \varTheta _2^{\text {os}}}{\partial r_1^{\text {os}}} = 0$. Consequently, the investment decisions of the suppliers have no effect on one another. Because the optimal solution of the suppliers is unique, in this situation, the NE of the game is unique and can be found using first-order conditions.

$\square$

Proof of Proposition 12

• (a) Consider the following cases: the first derivative of ${\hat{\varPi }}^{\text {os}}$ with respect to $q^{\text {os}}$ is:

  $$\begin{aligned} \frac{\partial {\hat{\varPi }}^{\text {os}}}{\partial q^{\text {os}}}&= p {\bar{F}}(q^{\text {os}}) {\bar{G}}_1(q^{\text {os}}|{\tilde{r}}_1^{\text {os}}(q^{\text {os}})) {\bar{G}}_2(q^{\text {os}}|{\tilde{r}}_2^{\text {os}}(q^{\text {os}})) - w_1 {\bar{G}}_1(q^{\text {os}}|{\tilde{r}}_1^{\text {os}}(q^{\text {os}})) \nonumber \\&\quad - w_2 {\bar{G}}_2(q^{\text {os}}|{\tilde{r}}_2^{\text {os}}(q^{\text {os}})) \nonumber \\&\quad + p \frac{\partial {\tilde{r}}_2^{\text {os}}(q^{\text {os}})}{\partial q^{\text {os}}} \int _{0}^{q^{\text {os}}} {\bar{F}}(u) {\bar{G}}_1(u|{\tilde{r}}_1^{\text {os}}(q^{\text {os}})) \frac{\partial {\bar{G}}_2(u|{\tilde{r}}_2^{\text {os}}(q^{\text {os}}))}{\partial {\tilde{r}}_2^{\text {os}}} du \nonumber \\&\quad + p \frac{\partial {\tilde{r}}_1^{\text {os}}(q^{\text {os}})}{\partial q^{\text {os}}} \int _{0}^{q^{\text {os}}} {\bar{F}}(u) {\bar{G}}_2(u|{\tilde{r}}_2^{\text {os}}(q^{\text {os}})) \frac{\partial {\bar{G}}_1(u|{\tilde{r}}_1^{\text {os}}(q^{\text {os}}))}{\partial {\tilde{r}}_1^{\text {os}}} du \nonumber \\&\quad - w_1 \frac{\partial {\tilde{r}}_1^{\text {os}}(q^{\text {os}})}{\partial q^{\text {os}}} \int _{0}^{q^{\text {os}}} \frac{\partial {\bar{G}}_1(u|{\tilde{r}}_1^{\text {os}}(q^{\text {os}}))}{\partial {\tilde{r}}_1^{\text {os}}} du \nonumber \\&\quad - w_2 \frac{\partial {\tilde{r}}_2^{\text {os}}(q^{\text {os}})}{\partial q^{\text {os}}} \int _{0}^{q^{\text {os}}} \frac{\partial {\bar{G}}_2(u|{\tilde{r}}_2^{\text {os}}(q^{\text {os}}))}{\partial {\tilde{r}}_2^{\text {os}}} du \end{aligned}$$

  (34)

  Based on Eqs. (22), (23), and (24) and the fact that ${\tilde{r}}_1^{\text {os}}(q) = r_1^{\text {so}^*}(q)$, ${\tilde{r}}_2^{\text {os}}(q) = r_2^{\text {so}^*}(q)$, ${\bar{F}}(u) {\bar{G}}_2(u|{\tilde{r}}_2^{\text {os}}(q^{\text {os}})) \ge 0$, $\forall u \le q^{\text {so}^*}$, and ${\bar{F}}(u) {\bar{G}}_1(u|{\tilde{r}}_1^{\text {os}}(q^{\text {os}})) \ge 0$, $\forall u \le q^{\text {so}^*}$, then we have $\frac{\partial {\hat{\varPi }}^{\text {os}}}{\partial q^{\text {os}}}(q^{\text {so}^*}) \ge 0$. This implies that $q^{\text {os}^*} \ge q^{\text {so}^*}$.

• (b) Because ${\tilde{r}}_1^{\text {os}}(q) = r_1^{\text {so}^*}(q)$, $q^{\text {os}^*} \ge q^{\text {so}^*}$, and $\frac{\partial {\tilde{r}}_1^{\text {os}}(q^{\text {os}})}{\partial q^{\text {os}}} \ge 0$, then $r_1^{\text {os}^*} = {\tilde{r}}_1^{\text {os}} (q^{\text {os}^*}) \ge r_1^{\text {so}^*}$. The proof for $r_2^{\text {os}^*} = {\tilde{r}}_2^{\text {os}} (q^{\text {os}^*}) \ge r_2^{\text {so}^*}$ is analogous.

• (c) The first derivative of $\varTheta _1^{\text {os}}$ with respect to $q^{\text {os}}$ is:

  $$\begin{aligned} \frac{\partial \varTheta _1^{\text {os}}}{q^{\text {os}}}&= (w_1-c_1) {\bar{G}}_1(q^{\text {os}}|r_1^{\text {os}}) du \ge 0 \end{aligned}$$

  As result, $\varTheta _1^{\text {os}}$ is an increasing function of $q^{\text {os}}$. Note that $\varTheta _1^{\text {os}} (q, r_1) = \varTheta _1^{\text {so}} (q, r_1)$. Because $q^{\text {os}^*} \ge q^{\text {so}^*}$, then $\varTheta _1^{\text {os}^*}(q^{\text {os}^*}, r_1^{\text {os}^*}) \ge \varTheta _1^{\text {so}^*}(q^{\text {so}^*}, r_1^{\text {so}^*})$. The proof for $\varTheta _2^{\text {os}^*}(q^{\text {os}^*}, r_2^{\text {os}^*}) \ge \varTheta _2^{\text {so}^*}(q^{\text {so}^*}, r_2^{\text {so}^*})$ is analogous.

• (d) Because ${\tilde{r}}_1^{\text {os}}(q) = r_1^{\text {so}^*}(q)$ and ${\tilde{r}}_2^{\text {os}}(q) = r_2^{\text {so}^*}(q)$, then $\varPi ^{\text {os}^*}(q) = \varPi ^{\text {so}^*}(q)$. Based on Eqs. (22) and (34), we have:

  $$\begin{aligned} \varPi ^{\text {os}^*}(q^{\text {os}^*}) \ge \varPi ^{\text {os}^*}(q^{\text {so}^*}) = \varPi ^{\text {so}^*}(q^{\text {so}^*}), \end{aligned}$$

  which completes the proof.

$\square$

Proof of Proposition 13

• (a) Since the suppliers are similar to each other, in the equilibrium, $r_1^{\text {so}^*}=r_2^{\text {so}^*}=\cdots =r_n^{\text {so}^*}$. Similar to Lemma 3 and Proposition 4, it is fairly easy to show that the optimal decisions of the players are supermodular with respect to each other and the objective functions of the players are increasing functions with respect to each others decisions. Therefore. there exists an equilibrium which is greater than all the other possible equilibriums that results in higher profits for all the players.

• (b) Similar to Proposition 10 Part (a), we can show that $\frac{\partial \varTheta _i^{\text {ov}}}{\partial r_i^{\text {ov}}}(r_1, \ldots , r_n) \ge \frac{\partial \varTheta _i^{\text {so}}}{\partial r_i^{\text {so}}}(r_1, \ldots , r_n)$. Therefore $r_i^{\text {ov}^*} \ge r_i^{\text {so}^*}$, $\forall i$. In addition, similar to Proposition 10 Part (b), we can show that for similar suppliers’ investments, the manufacturer’s optimal order quantities in simultaneous ordering and investment and ordering before realization of capacities scenarios are equal. Since $r_i^{\text {ov}^*} \ge r_i^{\text {so}^*}$, $\forall i$ and the problem is supermodular, then $q^{\text {ov}^*} \ge q^{\text {so}^*}$.

• (c) The proof is similar to Proposition 10 Part (c).

• (d) The proof is similar to Proposition 10 Part (d).

• (e) Similar to Proposition 11 Part (a), we can show that $\frac{\partial \varPi ^{\text {os}}}{\partial q^{\text {os}}}(q^{\text {so}^*}) \ge 0$. As a result, $q^{\text {os}^*} \ge q^{\text {so}^*}$. In addition, it is easy to show that manufacturer’s and the suppliers’ objective functions are supermodular and consequently, $r_i^{\text {os}^*} \ge r_i^{\text {so}^*}$, $\forall i$.

• (f) The proof is similar to Proposition 11 Part (c)

• (g) The proof is similar to Proposition 11 Part (d)

$\square$