The distributionally robust optimization model for a remanufacturing system under cap-and-trade policy: a newsvendor approach

Abstract

This paper considers a remanufacturing system in which the remanufacturer uses both the fresh raw material and recycled material to produce new products. Only partial information of the demand distribution, including the mean and variance, is available in this system. The operational activities of the remanufacturer are major contributors to carbon emissions, and a government agency imposes a cap-and-trade (carbon emission trading) policy on the remanufacturer. In the context of the distributionally robust newsvendor model, a maxmin approach is used to solve the optimal joint collection and production quantities. To study whether implementation of a cap-and-trade policy improves the remanufacturer’s expected profit and reduces corresponding carbon emissions, the case where the government agency does not impose cap-and-trade policy is considered and compared. Finally, numerical analysis is conducted to illustrate and complement the analytical results and to investigate the influences of several main parameters on the distributionally robust newsvendor model for the remanufacturing system under a cap-and-trade policy.

Appendices

Appendices

Proof of Lemma 2

We prove the property of $f_{s}(Q)$ shown in Lemmas 2 by considering the following two observations: (i) $f_{s}(Q)$ is an increasing function of Q, and (ii) $Q_{t_{s}}$ is a unique solution of $f_{s}(Q)=0$.

Firstly, we take the first partial derivative of f(Q) with respect to Q and have

$$\begin{aligned} \frac{\partial f_{s}(Q)}{\partial Q}= & {} \frac{c_{L}e_{v}(Q-\mu )}{\sqrt{\sigma ^{2}+(Q-\mu )^{2}}}+c_{L}(2e_{n}+e_{v})\nonumber \\= & {} \frac{c_{L}(2e_{n}+e_{v})\sqrt{\sigma ^{2}+(Q-\mu )^{2}}+c_{L}e_{v}(Q-\mu )}{\sqrt{\sigma ^{2}+(Q-\mu )^{2}}} \end{aligned}$$

(A.1)

Using $\sqrt{\sigma ^{2}+(Q-\mu )^{2}}>\mu -Q$, we have $\frac{\partial f(Q)}{\partial Q}>o$, implying that $f_{s}(Q)$ is an increasing function of Q.

Secondly, rearranging equation $f_{s}(Q)=0$, we have

$$\begin{aligned} c_{L}e_{v}\sqrt{\sigma ^{2}+(Q-\mu )^{2}}=\varOmega _{s}-c_{L}(2e_{n}+e_{v})Q \end{aligned}$$

(A.2)

Equation (A.2) yields

$$\begin{aligned} Q<\frac{\varOmega _{s}}{c_{L}(2e_{n}+e_{v})}. \end{aligned}$$

(A.3)

Simplifying Eq. (A.3), we further have

$$\begin{aligned} 4c^{2}_{L}e_{n}(e_{n}+e_{v})Q^{2}+2c_{L}Q[c_{L}\mu e^{2}_{v}-\varOmega _{s}(2e_{n}+e_{v})]+\varOmega _{s}^{2}-c^{2}_{L}e^{2}_{v}(\mu ^{2}+\sigma ^{2})=0. \end{aligned}$$

(A.4)

Solving Eq. (A.4) yields that there exist two roots $Q_{t_{s}}$ and $Q_{t_{0}}$ satisfying $f_{s}(Q)=0$, where

$$\begin{aligned} Q_{t_{s}}=\frac{(2e_{n}+e_{v})\varOmega _{s}-c_{L}\mu e^{2}_{v}-e_{v}\sqrt{[\varOmega _{s}-c_{L}\mu (2e_{n}+e_{v})]^{2} +4c^{2}_{L}e_{n}(e_{n}+e_{v})\sigma ^{2}}}{4c_{L}e_{n}(e_{n}+e_{v})}, \end{aligned}$$

(A.5)

and

$$\begin{aligned} Q_{t_{0}}=\frac{(2e_{n}+e_{v})\varOmega _{s}-c_{L}\mu e^{2}_{v}+e_{v}\sqrt{[\varOmega _{s}-c_{L}\mu (2e_{n}+e_{v})]^{2} +4c^{2}_{L}e_{n}(e_{n}+e_{v})\sigma ^{2}}}{4c_{L}e_{n}(e_{n}+e_{v})}.\nonumber \\ \end{aligned}$$

(A.6)

We prove that $Q_{t_{0}}$ is not a feasible root of $f_{s}(Q)=0$ by considering the following two equations:

$$\begin{aligned} \sqrt{[\varOmega _{s}-c_{L}\mu (2e_{n}+e_{v})]^{2}+4c^{2}_{L}e_{n}(e_{n}+e_{v}) \sigma ^{2}}>c_{L}\mu (2_{n}+e_{v})-\varOmega _{s}, \end{aligned}$$

(A.7)

and

$$\begin{aligned} (2e_{n}+e_{v})\sqrt{[\varOmega _{s}-c_{L}\mu (2e_{n}+e_{v})]^{2}+4c^{2}_{L}e_{n}(e_{n}+e_{v})\sigma ^{2}}>e_{v}[c_{L}\mu (2_{n}+e_{v})-\varOmega _{s}]. \end{aligned}$$

(A.8)

From Eq. (A.8), we have

$$\begin{aligned}&\varOmega _{s}(2e_{n}+e_{v})^{2}-c_{L}\mu (2e_{n}+e_{v}) e^{2}_{v} \nonumber \\&\qquad +\,e_{v}(2e_{n}+e_{v})\sqrt{[\varOmega _{s}-c_{L}\mu (2e_{n}+e_{v})]^{2}+4c^{2}_{L}e_{n}(e_{n}+e_{v})\sigma ^{2}}\nonumber \\&\quad >4\varOmega _{s} e_{n}(e_{n}+e_{v}), \end{aligned}$$

(A.9)

and

$$\begin{aligned} Q_{t_{0}}>\frac{\varOmega _{s}}{c_{L}(2e_{n}+e_{v})}, \end{aligned}$$

(A.10)

which contradicts with Eq. (A.3).

Similar analysis to Eq. (A.8), we easily have

$$\begin{aligned} (2e_{n}+e_{v})\sqrt{[\varOmega _{s}-c_{L}\mu (2e_{n}+e_{v})]^{2}+4c^{2}_{L}e_{n}(e_{n}+e_{v})\sigma ^{2}}>e_{v}[\varOmega _{s}-c_{L}\mu (2_{n}+e_{v})]. \end{aligned}$$

(A.11)

Rearranging Eq. (A.11), we have

$$\begin{aligned}&\varOmega _{s}(2e_{n}+e_{v})^{2}-c_{L}\mu (2e_{n}+e_{v}) e^{2}_{v} \nonumber \\&\qquad -\,e_{v}(2e_{n}+e_{v})\sqrt{[\varOmega _{s}-c_{L}\mu (2e_{n}+e_{v})]^{2}+4c^{2}_{L}e_{n}(e_{n}+e_{v})\sigma ^{2}}\nonumber \\&\quad <4\varOmega _{s} e_{n}(e_{n}+e_{v}), \end{aligned}$$

(A.12)

and

$$\begin{aligned} Q_{t_{s}}<\frac{\varOmega _{s}}{c_{L}(2e_{n}+e_{v})}. \end{aligned}$$

(A.13)

which means that $Q_{t_{s}}$ satisfies Eq. (A.3).

In the following, we prove $Q_{t_{s}}>0$. Recalling $C> \max \{{\underline{C}}_{1},{\underline{C}}_{2}\}\ge {\underline{C}}_{1}$, we have

$$\begin{aligned} C>\frac{c_{L}e_{v}(\sqrt{\sigma ^{2}+\mu ^{2}}-\mu )-\delta _{e}(\delta _{c}-c_{A}+p_{s}\delta _{e})}{2c_{L}}, \end{aligned}$$

(A.14)

and

$$\begin{aligned} \varOmega _{s}>c_{L}e_{v}\sqrt{\sigma ^{2}+\mu ^{2}}. \end{aligned}$$

(A.15)

Equation (A.15) yields

$$\begin{aligned} \varOmega _{s}^{2}-\mu ^{2}e^{2}_{v}c^{2}_{L}>c^{2}_{L}e^{2}_{v}\sigma ^{2}, \end{aligned}$$

(A.16)

and

$$\begin{aligned}{}[\varOmega _{s} (2e_{n}+e_{v})-c_{L}\mu e^{2}_{v}]^{2}>e^{2}_{v}\{[\varOmega _{s}-c_{L}\mu (2e_{n}+e_{v})]^{2}+4c^{2}_{L}e_{n}(e_{n}+e_{v})\sigma ^{2}\}.\nonumber \\ \end{aligned}$$

(A.17)

Using $\varOmega _{s} (2e_{n}+e_{v})>c_{L}\mu e^{2}_{v}$, from Eq. (A.17), we have

$$\begin{aligned} \varOmega _{s} (2e_{n}+e_{v})-c_{L}\mu e^{2}_{v}>e_{v}\sqrt{[\varOmega _{s}-c_{L}\mu (2e_{n}+e_{v})]^{2}+4c^{2}_{L}e_{n}(e_{n}+e_{v})\sigma ^{2}},\nonumber \\ \end{aligned}$$

(A.18)

which means that $Q_{t_{s}}$ shown in Eq. (A.5) is higher than zero, i.e., $Q_{t_{s}}>0$. Using Eq. (A.13), we further have that $Q_{t_{s}}$ is a unique feasible root of $f_{s}(Q)=0$. Using the monotonicity of $f_{s}(Q)$, we have that $f_{s}(Q)\le 0$ if and only if $Q\le Q_{t_{s}}$, and $f_{s}(Q)>0$ if and only if $Q>Q_{t_{s}}$.

With similar analysis, we prove that the property of $f_{b}(Q)$ shown in Lemmas 2 holds and the detailed proof is omitted.

Proof of Theorem 1

Using $Q\wedge D=D-(D-Q)^{+}$ and $(Q-D)^{+}=Q-D+(D-Q)^{+}$, and from Eqs. (1) and (2), we simplify $\varPi _{0}(\tau ,Q)$ and $J(\tau ,Q)$ as

$$\begin{aligned} \varPi _{0}(\tau ,Q)=(p-v)\mu +[v-c_{n}+(\delta _{c}-c_{A})\tau ]Q-c_{L}\tau ^{2}Q^{2}-(p+c_{s}-v)E(D-Q)^{+},\nonumber \\ \end{aligned}$$

(B.1)

and

$$\begin{aligned} J(\tau ,Q)=(e_{n}+e_{v}-\delta _{e}\tau )Q-e_{v}\mu +e_{v}E(D-Q)^{+}. \end{aligned}$$

(B.2)

For model $M_{1s}$, we substitute Eqs. (B.1) and (B.2) into (7), and have

$$\begin{aligned} \varPi _{1s}(\tau ,Q)= & {} (p-v+p_{s}e_{v})\mu +p_{s}C+[v-c_{n}-p_{s}(e_{n}+e_{v})]Q \nonumber \\&+\,(\delta _{c}-c_{A}+p_{s}\delta _{e})\tau Q-c_{L}\tau ^{2}Q^{2}\nonumber \\&-\,(p+c_{s}-v+p_{s}e_{v})E(D-Q)^{+}. \end{aligned}$$

(B.3)

Using Lemma 1 to solve Eqs. (B.2) and (B.3), we further have

$$\begin{aligned} J^{F}(\tau ,Q)=(e_{n}+e_{v}-\delta _{e}\tau )Q-e_{v}\mu +\frac{1}{2}e_{v}[\sqrt{\sigma ^{2}+(Q-\mu )^{2}}-(Q-\mu )], \end{aligned}$$

(B.4)

and

$$\begin{aligned} \varPi ^{F}_{1s}(\tau ,Q)= & {} (p-v+p_{s}e_{v})\mu +p_{s}C+[v-c_{n}-p_{s}(e_{n}+e_{v})]Q \nonumber \\&+\,(\delta _{c}-c_{A}+p_{s}\delta _{e})\tau Q-c_{L}\tau ^{2}Q^{2}\nonumber \\&-\,\frac{1}{2}(p+c_{s}-v+p_{s}e_{v})[\sqrt{\sigma ^{2}+(Q-\mu )^{2}}-(Q-\mu )]. \end{aligned}$$

(B.5)

For a fixed $Q>0$, taking the first and second partial derivatives of $\varPi ^{F}_{1s}(\tau ,Q)$ with respect to $\tau$ and from Eq. (B.5), we have

$$\begin{aligned} \frac{\partial \varPi ^{F}_{1s}(\tau ,Q)}{\partial \tau }=(\delta _{c}-c_{A}+p_{s}\delta _{e})Q-2c_{L}\tau Q^{2}, \end{aligned}$$

(B.6)

and

$$\begin{aligned} \frac{\partial ^{2}\varPi ^{F}_{1s}(\tau ,Q)}{\partial \tau ^{2}}=-2c_{L} Q^{2}<0. \end{aligned}$$

(B.7)

Equation (B.7) yields that $\varPi ^{F}_{1s}(\tau ,Q)$ is concave in $\tau$ for a fixed Q. Solving $\frac{\partial \varPi ^{F}_{1s}(\tau ,Q)}{\partial \tau }=0$ and from Eq. (B.6), we have

$$\begin{aligned} \tau ^{*}_{1s}=\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}Q}. \end{aligned}$$

(B.8)

Substituting Eqs. (B.8) into (9) yields

$$\begin{aligned} \frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L} \tau _{u}}\le Q\le \frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L} \tau _{l}}. \end{aligned}$$

(B.9)

Substituting Eqs. (B.8) into (B.4) and using the expression of $f_{s}(Q)$ yield

$$\begin{aligned} J^{F}(\tau ,Q)=\frac{1}{2c_{L}}f_{s}(Q)+C. \end{aligned}$$

(B.10)

Using $J^{F}(\tau ,Q)\le C$ and from Lemma 2, we have

$$\begin{aligned} 0<Q<Q_{t_{s}}, \end{aligned}$$

(B.11)

where $Q_{t_{s}}=\frac{(2e_{n}+e_{v})\varOmega _{s}-c_{L}\mu e^{2}_{v}-e_{v}\sqrt{[\varOmega _{s}-c_{L}\mu (2e_{n}+e_{v})]^{2}+4c^{2}_{L}e_{n}(e_{n}+e_{v})\sigma ^{2}}}{4c_{L}e_{n}(e_{n}+e_{v})}$.

Substituting Eqs. (B.8) into (B.5), we have

$$\begin{aligned} \varPi ^{F}_{1s}(\tau ^{*}_{1s},Q)= & {} \frac{1}{2}(p-v-c_{s}+p_{s}e_{v})\mu +p_{s}C+\frac{(\delta _{c}-c_{A}+p_{s}\delta _{e})^{2}}{4c_{L}} +\frac{1}{2}[p+v+c_{s}-2c_{n}\nonumber \\&-\,p_{s}(2e_{n}+e_{v})]Q-\frac{1}{2}(p+c_{s}-c+p_{s}e_{v})\sqrt{\sigma ^{2}+(Q-\mu )^{2}}. \end{aligned}$$

(B.12)

Taking the first and second partial derivatives of $\varPi ^{F}_{1s}(\tau ^{*}_{1s},Q)$ with respect to Q and from Eq. (B.12), we have

$$\begin{aligned} \frac{\partial \varPi ^{F}_{1s}(\tau ^{*}_{1s},Q)}{\partial Q}=\frac{1}{2}[p+v+c_{s}-2c_{n}-p_{s}(2e_{n}+e_{v})]-\frac{(p+c_{s}-v+p_{s}e_{v})(Q-\mu )}{2\sqrt{\sigma ^{2}+(Q-\mu )^{2}}}, \end{aligned}$$

(B.13)

and

$$\begin{aligned} \frac{\partial ^{2} \varPi ^{F}_{1s}(\tau ^{*}_{1s},Q)}{\partial Q^{2}}=-\frac{1}{2}\sigma ^{2}(p+c_{s}-v+p_{s}e_{v})[\sigma ^{2}+(Q-\mu )^{2}]^{-\frac{3}{2}}<0. \end{aligned}$$

(B.14)

Equation (B.14) yields that $\varPi ^{F}_{1s}(\tau ^{*}_{1s},Q)$ is concave in Q. Solving $\frac{\partial \varPi ^{F}_{1s}(\tau ^{*}_{1s},Q)}{\partial Q}=0$ and from Equation (B.13), we have

$$\begin{aligned} Q_{0_{s}}= & {} \mu +\frac{\sigma }{2}[\sqrt{\frac{p+c_{s}-c_{n}-p_{s}e_{n}}{p_{s}(e_{n}+e_{v})+c_{n}-v}}-\sqrt{\frac{p_{s}(e_{n}+e_{v})+c_{n}-v}{p+c_{s}-c_{n}-p_{s}e_{n}}}]\nonumber \\= & {} \mu +\frac{\sigma (A_{s}-B_{s})}{2\sqrt{A_{s}B_{s}}}. \end{aligned}$$

(B.15)

Using the concavity property of $\varPi ^{F}_{1s}(\tau ^{*}_{1s},Q)$ and from Eqs. (B.9), (B.11), and (B.15), we solve the optimal robust production quantity of the remanufacturer in problem $M_{1s}$ by considering the following three cases:

Case 1 $\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{u}}\le Q_{t_{s}}\le \frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{l}}$. From Eqs. (B.9) and (B.11), we have that the feasible region of problem $M_{1s}$ becomes

$$\begin{aligned} \frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{u}} \le Q\le Q_{t_{s}}. \end{aligned}$$

(B.16)

When $Q_{t_{s}}<Q_{0_{s}}$, we have $Q^{*}_{1s}=Q_{t_{s}}$. When $\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{u}} >Q_{0_{s}}$, we have $Q^{*}_{1s}=\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{u}}$. When $\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{u}} \le Q_{0_{s}}\le Q_{t_{s}}$, we have $Q^{*}_{1s}=Q_{0_{s}}$.

Case 2 $Q_{t_{s}}>\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{l}}$. From Eqs. (B.9) and (B.11), we have that the feasible region of problem $M_{1s}$ becomes

$$\begin{aligned} \frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{u}} \le Q\le \frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{l}}. \end{aligned}$$

(B.17)

When $Q_{0_{s}}>\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{l}}$, we have $Q^{*}_{1s}=\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{l}}$. When $Q_{0_{s}}<\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{u}}$, we have $Q^{*}_{1s}=\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{u}}$. When $\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{u}} \le Q_{0_{s}}\le \frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{l}}$, we have $Q^{*}_{1s}=Q_{0_{s}}$.

Case 3 $Q_{t_{s}}<\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{u}}$. From Eqs. (B.9) and (B.11), we have that the feasible region of problem $M_{1s}$ is empty set.

Consequently, from Cases 1 and 2, we have that when $Q_{t_{s}}\ge \frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2\tau _{u}c_{L}}$, i.e., $\tau _{u}\ge \frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}Q_{t_{s}}}$, the optimal robust production quantity $Q^{*}_{1s}$ and collection rate $\tau ^{*}_{1s}$ are $Q^{*}_{1s}=\max \{\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2\tau _{u}c_{L}}, \min \{Q_{0_{s}},\min \{\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2\tau _{l}c_{L}},Q_{t_{s}}\}\}\}$ and $\tau ^{*}_{1s}=\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}Q^{*}_{1s}}$.

Proof of Corollary 1

• (i) Using the expression of $C_{0}$ to simplify $C\ge C_{0}$ yields

  $$\begin{aligned} C\ge \frac{e_{v}}{2}\left[ \sqrt{\sigma ^{2}+(\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{u}}-\mu )}-\mu \right] +\frac{(\delta _{c}-c_{A}+p_{s}\delta _{e}) (2e_{n}-e_{v}-2\delta _{e}\tau _{u})}{4c_{L}\tau _{u}}, \end{aligned}$$

  (C.1)

  and

  $$\begin{aligned} 2c_{L}C\ge c_{L}e_{v}[\sqrt{\sigma ^{2}+(\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{u}}-\mu )}-\mu ]+\frac{(\delta _{c}-c_{A}+p_{s}\delta _{e}) (2e_{n}+e_{v}-2\delta _{e}\tau _{u})}{2\tau _{u}}. \end{aligned}$$

  (C.2)

Using the expressions of f(Q) and $\varOmega$ to rearrange Eq. (C.2), we have

$$\begin{aligned} 0\ge c_{L}e_{v}\sqrt{\sigma ^{2}+\left( \frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{u}}-\mu \right) }+\frac{c_{L}(2e_{n} +e_{v})(\delta _{c}-c_{A}+p_{s}\delta _{e})}{2c_{L}\tau _{u}}-\varOmega _{s}, \end{aligned}$$

(C.3)

and

$$\begin{aligned} f_{s}(Q_{t_{s}})\ge f(\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{u}}). \end{aligned}$$

(C.4)

Recalling that $f_{s}(Q)$ is an increasing function of Q, from Eq. (C.4), we further have

$$\begin{aligned} Q_{t_{s}}\ge \frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{u}}. \end{aligned}$$

(C.5)

Using $\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{u}}>0$, $f_{s}(0)=c_{L}e_{v}\sqrt{\sigma ^{2}+\mu ^{2}}-\varOmega$, and the monotonicity of $f_{s}(Q)$, we have

$$\begin{aligned} e_{v}\sqrt{\sigma ^{2}+(\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{u}}-\mu )^{2}} +\frac{(2e_{n}+e_{v})(\delta _{c}-c_{A}+p_{s}\delta _{e})}{2c_{L}\tau _{u}}>e_{v}\sqrt{\sigma ^{2}+\mu ^{2}}.\nonumber \\ \end{aligned}$$

(C.6)

Equation (C.6) yields $C_{0}>{\underline{C}}$. From Eq. (C.5) and Theorem 1, we have that when $C\ge C_{0}$, there exist the optimal robust collection rate and production quantity such that the expected of the remanufacturer is maximized.

Similarly, simplifying $C<C_{0}$, we have $f_{s}(Q_{t_{s}})< f(\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{u}})$ and $Q_{t_{s}}<\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{u}}$. From Theorem 1, we have that when $C<C_{0}$, there does not exist the robust collection rate and production quantity such that the expected of the remanufacturer is maximized.

• (ii) From Lemma 2 and using the monotonicity of $f_{s}(Q)$, we have that if $Q^{*}_{1s}=Q_{t_{s}}$, then $J^{F}(\tau ^{*}_{1s}, Q^{*}_{1s})=C$. Otherwise, $J^{F}(\tau ^{*}_{1s}, Q^{*}_{1s})<C$.

Proof of Theorem 2

With similar analysis to problem $M_{1s}$, using Lemma1 to solve problem $M_{1b}$, we simplify the expected profit and carbon emissions of the remanufacturer in the worst demand scenario as

$$\begin{aligned} \varPi ^{F}_{1b}(\tau ,Q)= & {} (p-v+p_{b}e_{v})\mu +p_{b}C+[v-c_{n}-p_{b}(e_{n}+e_{v})]Q\nonumber \\&+\,(\delta _{c}-c_{A}+p_{b}\delta _{e})\tau Q-c_{L}\tau ^{2}Q^{2}\nonumber \\&-\,\frac{1}{2}(p+c_{s}-v+p_{b}e_{v})[\sqrt{\sigma ^{2}+(Q-\mu )^{2}}-(Q-\mu )], \end{aligned}$$

(D.1)

and

$$\begin{aligned} J^{F}(\tau ,Q)=(e_{n}+e_{v}-\delta _{e}\tau )Q-e_{v}\mu +\frac{1}{2}e_{v}[\sqrt{\sigma ^{2}+(Q-\mu )^{2}}-(Q-\mu )]. \end{aligned}$$

(D.2)

For a fixed $Q>0$, taking the first and second partial derivatives of $\varPi ^{F}_{1b}(\tau ,Q)$ with respect to $\tau$ and from Eq. (D.1), we have

$$\begin{aligned} \frac{\partial \varPi ^{F}_{1b}(\tau ,Q)}{\partial \tau }=(\delta _{c}-c_{A}+p_{b}\delta _{e})Q-2c_{L}\tau Q^{2}, \end{aligned}$$

(D.3)

and

$$\begin{aligned} \frac{\partial ^{2}\varPi ^{F}_{1b}(\tau ,Q)}{\partial \tau ^{2}}=-2c_{L} Q^{2}<0. \end{aligned}$$

(D.4)

From Eq. (D.4), we have that $\varPi ^{F}_{1b}(\tau ,Q)$ is a concave function of $\tau$ for any fixed $Q>0$. Solving $\frac{\partial \varPi ^{F}_{1b}(\tau ,Q)}{\partial \tau }=0$ and from Eq. (D.3), we have

$$\begin{aligned} \tau ^{*}_{1b}=\frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2c_{L}Q}. \end{aligned}$$

(D.5)

Substituting Eqs. (D.5) into (13), we have

$$\begin{aligned} \frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2\tau _{u}c_{L}}\le Q\le \frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2\tau _{l}c_{L}}. \end{aligned}$$

(D.6)

Substituting Eqs. (D.5) into (D.2) and from Lemma 2, we rearrange $J^{F}(\tau ^{*}_{1b},Q)\ge C$ as

$$\begin{aligned} Q\ge Q_{t_{b}}, \end{aligned}$$

(D.7)

where $Q_{t_{b}}=\frac{(2e_{n}+e_{v})\varOmega _{b}-c_{L}\mu e^{2}_{v}-e_{v}\sqrt{[\varOmega _{b}-c_{L}\mu (2e_{n}+e_{v})]^{2}+4c^{2}_{L}e_{n}(e_{n}+e_{v})\sigma ^{2}}}{4c_{L}e_{n}(e_{n}+e_{v})}$.

Taking the first and second partial derivatives of $\varPi ^{F}_{1b}(\tau ^{*}_{1b},Q)$ with respect to Q and from Eqs. (D.1) and (D.5), we have

$$\begin{aligned} \frac{\partial \varPi ^{F}_{1b}(\tau ^{*}_{1b},Q)}{\partial Q}=\frac{1}{2}[p+v+c_{s}-2c_{n}-p_{b}(2e_{n}+e_{v})]-\frac{(p+c_{s}-v+p_{b}e_{v})(Q-\mu )}{2\sqrt{\sigma ^{2}+(Q-\mu )^{2}}}, \end{aligned}$$

(D.8)

and

$$\begin{aligned} \frac{\partial ^{2} \varPi ^{F}_{1b}(\tau ^{*}_{1b},Q)}{\partial Q^{2}}=-\frac{1}{2}\sigma ^{2}(p+c_{s}-v+p_{b}e_{v})[\sigma ^{2}+(Q-\mu )^{2}]^{-\frac{3}{2}}<0. \end{aligned}$$

(D.9)

Equation (D.9) implies that $\varPi ^{F}_{1b}(\tau ^{*}_{1b},Q)$ is concave in Q. Solving $\frac{\partial \varPi ^{F}_{1b}(\tau ^{*}_{1b},Q)}{\partial Q}=0$ and from Eq. (D.8), we have

$$\begin{aligned} Q_{0_{b}}= & {} \mu +\frac{\sigma }{2}[\sqrt{\frac{p+c_{s}-c_{n}-p_{b}e_{n}}{p_{b}(e_{n}+e_{v}) +c_{n}-v}}-\sqrt{\frac{p_{b}(e_{n}+e_{v})+c_{n}-v}{p+c_{s}-c_{n}-p_{b}e_{n}}}\nonumber \\= & {} \mu +\frac{\sigma (A_{b}-B_{b})}{2\sqrt{A_{b}B_{b}}}. \end{aligned}$$

(D.10)

Using the concavity property of $\varPi ^{F}_{1b}(\tau ^{*}_{1b},Q)$ and from Eqs. (D.6), (D.7), and (D.10), we obtain the optimal robust production quantity of the remanufacturer by considering the following three cases:

Case 1 $Q_{t_{b}}< \frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2c_{L}\tau _{u}}$. From Eqs. (D.5) and (D.6), we have that the feasible region of problem $M_{1s}$ becomes

$$\begin{aligned} \frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2c_{L}\tau _{u}} \le Q\le \frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2c_{L}\tau _{l}}. \end{aligned}$$

(D.11)

When $\frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2c_{L}\tau _{l}}<Q_{0_{b}}$, we have $Q^{*}_{1b}=\frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2c_{L}\tau _{l}}$. When $\frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2c_{L}\tau _{u}}>Q_{0_{b}}$, we have $Q^{*}_{1b}=\frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2c_{L}\tau _{u}}$. When $\frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2c_{L}\tau _{u}}\le Q_{0_{b}}\le \frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2c_{L}\tau _{l}}$, we have $Q^{*}_{1b}=Q_{0_{b}}$.

Case 2 $\frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2c_{L}\tau _{u}} \le Q_{t_{b}}\le \frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2c_{L}\tau _{l}}$. From Eqs. (D.5) and (D.6), we have that the feasible region of problem $M_{1s}$ becomes

$$\begin{aligned} Q_{0_{b}} \le Q\le \frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2c_{L}\tau _{l}}. \end{aligned}$$

(D.12)

When $\frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2c_{L}\tau _{l}}<Q_{0_{b}}$, we have $Q^{*}_{1b}=\frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2c_{L}\tau _{l}}$. When $Q_{t_{b}}>Q_{0_{b}}$, we have $Q^{*}_{1b}=Q_{t_{b}}$. When $Q_{t_{b}}\le Q_{0_{b}}\le \frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2c_{L}\tau _{l}}$, we have $Q^{*}_{1b}=Q_{0_{b}}$.

Case 3 $Q_{t_{b}}>\frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2c_{L}\tau _{l}}$. From Eqs. (D.6) and (D.7), we have that the feasible region of problem $M_{1b}$ is empty set.

Consequently, from Cases 1 and 2, we have that when $Q_{t_{b}}\le \frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2\tau _{l}c_{L}}$, i.e., $\tau _{l}\ge \frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2c_{L}Q_{t_{b}}}$, the optimal robust production quantity $Q^{*}_{1b}$ and collection rate $\tau ^{*}_{1b}$ are $Q^{*}_{1b}=\min \{\frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2\tau _{l}c_{L}}, \max \{Q_{0_{b}},\max \{\frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2\tau _{u}c_{L}},Q_{t_{b}}\}\}\}$ and $\tau ^{*}_{1b}=\frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2c_{L}Q^{*}_{1b}}$.

Proof of Theorem 4

Using $Q\wedge D=D-(D-Q)^{+}$ and $(Q-D)^{+}=Q-D+(D-Q)^{+}$ to simplify Eq. (1), and from Lemma 1, we rearrange the expected worst-case profit of the remanufacturer as

$$\begin{aligned} \varPi ^{F}_{0}(\tau ,Q)= & {} \frac{1}{2}\mu (p-v-c_{s})+\frac{1}{2}Q(p+v+c_{s}-2c_{n})+(\delta _{c}-c_{A})\tau Q-c_{L}\tau ^{2}Q^{2} \nonumber \\&-\frac{1}{2}(p+c_{s}-v)\sqrt{\sigma ^{2}+(Q-\mu )^{2}}. \end{aligned}$$

(E.1)

From Eq. (E.1), for a fixed $Q>0$, taking the first and second partial derivatives of $\varPi ^{F}_{0}(\tau ,Q)$ with respect to $\tau$ yields

$$\begin{aligned} \frac{\partial \varPi ^{F}_{0}(\tau ,Q)}{\partial \tau }=(\delta _{c}-c_{A})Q-2\tau c_{L}Q^{2}, \end{aligned}$$

(E.2)

and

$$\begin{aligned} \frac{\partial ^{2} \varPi ^{F}_{0}(\tau ,Q)}{\partial \tau ^{2}}=-2c_{L}Q^{2}<0. \end{aligned}$$

(E.3)

From Eq. (E.3), we have that $\varPi ^{F}_{0}(\tau ,Q)$ is concave in $\tau$ for a fixed $Q>0$. Solving $\frac{\partial \varPi ^{F}_{0}(\tau ,Q)}{\partial \tau }=0$ and from Eq. (E.2), we have

$$\begin{aligned} \tau ^{*}_{0}=\frac{\delta _{c}-c_{A}}{2c_{L}Q}. \end{aligned}$$

(E.4)

Substituting Eqs. (E.4) into (E.1) yields

$$\begin{aligned} \varPi ^{F}_{0}(\tau ^{*}_{0},Q)= & {} \frac{1}{2}\mu (p-v-c_{s})+\frac{1}{2}Q(p+v+c_{s}-2c_{n}) +\frac{(\delta _{c}-c_{A})^{2}}{4c_{L}}\nonumber \\&-\frac{1}{2}(p+c_{s}-v)\sqrt{\sigma ^{2}+(Q-\mu )^{2}}. \end{aligned}$$

(E.5)

Substituting Eqs. (E.4) into (19) yields

$$\begin{aligned} \frac{\delta _{c}-c_{A}}{2\tau _{u}c_{A}}\le Q\le \frac{\delta _{c}-c_{A}}{2\tau _{l}c_{A}}. \end{aligned}$$

(E.6)

Taking the first and second partial derivatives of $\varPi ^{F}_{0}(\tau ,Q)$ with respect to Q and from Eq. (E.1), we have

$$\begin{aligned} \frac{\partial \varPi ^{F}_{0}(\tau ^{*}_{0},Q)}{\partial Q}=\frac{1}{2}(p+v+c_{s}-2c_{n})-\frac{(p+c_{s}-v)(Q-\mu )}{2\sqrt{\sigma ^{2}+(Q-\mu )^{2}}}, \end{aligned}$$

(E.7)

and

$$\begin{aligned} \frac{\partial ^{2} \varPi ^{F}_{0}(\tau ^{*}_{0},Q)}{\partial Q^{2}}=-\frac{1}{2}(p+c_{s}-v)\sigma ^{2}[\sigma ^{2}+(Q-\mu )^{2}]^{-\frac{3}{2}}<0. \end{aligned}$$

(E.8)

Equation (E.8) yields that $\varPi ^{F}_{0}(\tau ^{*}_{0},Q)$ is a concave function of Q. Therefore, solving $\frac{\partial \varPi ^{F}_{0}(\tau ^{*}_{0},Q)}{\partial Q}=0$ and from Eq. (E.7), we have

$$\begin{aligned} Q_{0}=\mu +\frac{\sigma }{2}(\sqrt{\frac{p+c_{s}-c_{n}}{c_{n}-v}}-\sqrt{\frac{c_{n}-v}{p+c_{s}-c_{n}}})=\mu +\frac{\sigma (A_{0}-B_{0})}{2\sqrt{A_{0}B_{0}}}. \end{aligned}$$

(E.9)

Using a constraint condition for Q shown in Eqs. (E.6) and from (E.9), we consider the following three cases to solve the optimal robust production quantity of model $M_{0}$. (i) When $\frac{\delta _{c}-c_{A}}{2\tau _{l}c_{L}}<Q_{0}$, we have $Q^{*}_{0}=\frac{\delta _{c}-c_{A}}{2\tau _{l}c_{L}}$. (ii) When $\frac{\delta _{c}-c_{A}}{2\tau _{l}c_{L}}>Q_{0}$, we have $Q^{*}_{0}=\frac{\delta _{c}-c_{A}}{2\tau _{u}c_{L}}$. and (iii) When $\frac{\delta _{c}-c_{A}}{2\tau _{u}c_{L}}\le Q_{0}\le \frac{\delta _{c}-c_{A}}{2\tau _{l}c_{L}}$, we have $Q^{*}_{0}=Q_{0}$. Consequently, from Eq. (E.4), we have the optimal robust collection rate and production quantity as $\tau ^{*}_{0}=\frac{\delta _{c}-c_{A}}{2c_{L}Q^{*}_{0}}$ and $Q^{*}_{0}=\max \{\frac{\delta _{c}-c_{A}}{2\tau _{u}c_{A}},\min \{\frac{\delta _{c}-c_{A}}{2\tau _{l}c_{A}},Q_{0}\}\}$.

Proof of Theorem 5

• (i) Define $f(x)=\mu +\frac{\sigma }{2}(x-\frac{1}{x})$, where $x>0$. By proving $\frac{\partial f(x)}{\partial x}=\frac{\sigma }{2}(1+\frac{1}{x^{2}})>0$, we have that f(x) is an increasing function of x. Using $\frac{A_{0}}{B_{0}}=\frac{p+c_{s}-c_{n}}{c_{n}-v}>\frac{p+c_{s}-c_{n}-p_{s}e_{n}}{c_{n}-v}>\frac{p+c_{s} -c_{n}-p_{s}e_{n}}{p_{s}(e_{n}+e_{v})+c_{n}-v}=\frac{A_{s}}{B_{s}}$ and from Eqs. (16) and (21), we have $Q_{0}>Q_{0_{s}}$. Similarly, using $\frac{A_{0}}{B_{0}}=\frac{p+c_{s}-c_{n}}{c_{n}-v}>\frac{p+c_{s}-c_{n}-p_{b}e_{n}}{c_{n}-v}>\frac{p+c_{s} -c_{n}-p_{b}e_{n}}{p_{b}(e_{n}+e_{v})+c_{n}-v}=\frac{A_{b}}{B_{b}}$ and from Eqs. (17) and (21), we have $Q_{0}>Q_{0_{b}}$. Consequently, we have $Q_{0}>\max \{Q_{0_{s}},Q_{0_{b}}\}$. Moreover, from the proof of Theorem  1 and 2 , and 4 , we easily have $\tau ^{*}_{1}Q^{*}_{1}=\frac{\delta \delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}}$ or $\frac{\delta \delta _{c}-c_{A}+p_{b}\delta _{e}}{2c_{L}}$ and $\tau ^{*}_{0}Q^{*}_{0}=\frac{\delta _{c}-c_{A}}{2c_{L}}$. Using $\min \{\frac{\delta \delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}},\frac{\delta \delta _{c}-c_{A}+p_{b}\delta _{e}}{2c_{L}}\}>\frac{\delta _{c}-c_{A}}{2c_{L}}$, we have $\tau ^{*}_{1}Q^{*}_{1}>\tau ^{*}_{0}Q^{*}_{0}$.

• (ii) We consider the following three cases:

Case 1 $\tau ^{*}_{0}=\tau _{u}$. From the proof of Theorem 4, we have $Q^{*}_{0}=\frac{\delta _{c}-c_{A}}{2\tau _{u}c_{L}}$. When $\tau ^{*}_{1}=\tau _{u}$, from the proofs of Theorems 1 and 2 , we have $Q^{*}_{1}=\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2\tau _{u}c_{L}}$ or $\frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2\tau _{u}c_{L}}$. We easily have $Q^{*}_{1}>Q^{*}_{0}$. When $\tau ^{*}_{1}=\tau _{l}$, we have $Q^{*}_{1}=\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2\tau _{l}c_{L}}$ or $\frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2\tau _{l}c_{L}}$ and further have $Q^{*}_{1}>Q^{*}_{0}$ with $\tau _{l}<\tau _{u}$. When $\tau _{l}<\tau ^{*}_{1}<\tau _{u}$, from the proofs of Theorems 1 and 2 , we have $Q^{*}_{1}=Q_{t_{s}}$ or $Q_{0_{s}}$ under the costraint condition of $\min \{Q_{t_{s}},Q_{0_{s}}\}>\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2\tau _{u}c_{L}}$ and $Q^{*}_{1}=Q_{t_{b}}$ or $Q_{0_{b}}$ under the costraint condition of $\min \{Q_{t_{b}},Q_{0_{b}}\}>\frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2\tau _{u}c_{L}}$. Using $\min \{\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2\tau _{u}c_{L}}, \frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2\tau _{u}c_{L}}\}>\frac{\delta _{c}-c_{A}}{2\tau _{u}c_{L}}$ yields $Q^{*}_{1}>Q^{*}_{0}$.

Case 2 $\tau ^{*}_{0}=\tau _{l}$. From the proof of Theorem 4, we have $Q^{*}_{0}=\frac{\delta _{c}-c_{A}}{2\tau _{l}c_{L}}$. When $\tau ^{*}_{1}=\tau _{l}$, from the proofs of Theorems 1 and 2 , we have $Q^{*}_{1}=\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2\tau _{l}c_{L}}$ or $\frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2\tau _{l}c_{L}}$. Using $\min \{\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2\tau _{l}c_{L}},\frac{\delta _{c}-c_{A} +p_{b}\delta _{e}}{2\tau _{l}c_{L}}\}>\frac{\delta _{c}-c_{A}}{2\tau _{l}c_{L}}$, we have $Q^{*}_{1}>Q^{*}_{0}$.

Case 3 $\tau _{l}<\tau ^{*}_{0}<\tau _{u}$. From the proof of Theorem 4, we have $Q^{*}_{0}=\mu +\frac{\sigma }{2}(\sqrt{\frac{A_{0}}{B_{0}}} -\sqrt{\frac{B_{0}}{A_{0}}})$. When $\tau _{l}<\tau ^{*}_{1}<\tau _{u}$, from the proofs of Theorems 1 and 2 , we have $Q^{*}_{1}=Q_{0_{s}}$ or $Q_{0_{b}}$ or $Q_{t_{s}}$ satisfying $Q_{t_{s}}\le Q_{0_{s}}$, or $Q_{t_{b}}$ satisfying $Q_{t_{b}}>Q_{0_{b}}$ Using $Q_{0}>\max \{Q_{0_{s}},Q_{0_{b}}\}$ and from the condition $Q_{t_{b}}<Q_{0}$, we have $Q^{*}_{1}<Q^{*}_{0}$.

• (iii) From Eq. (B.4), we have the worst-case expected carbon emissions as

  $$\begin{aligned} J^{F}(\tau ,Q)=(e_{n}+e_{v}-\delta _{e}\tau )Q-e_{v}\mu +\frac{1}{2}e_{v}[\sqrt{\sigma ^{2}+(Q-\mu )^{2}}-(Q-\mu )],\qquad \end{aligned}$$

  (F.1)

Substituting Eqs. (E.4) into (F.1), we simplify the worst-case expected carbon emissions in model $M_{0}$ as

$$\begin{aligned} J^{F}(\tau ^{*}_{0},Q)=\frac{1}{2}(2e_{n}+e_{v})Q+\frac{1}{2}e_{v}\sqrt{\sigma ^{2}+(Q-\mu )^{2}}-\frac{1}{2}e_{v}\mu -\frac{\delta _{e}(\delta _{c}-c_{A})}{2c_{L}}.\qquad \end{aligned}$$

(F.2)

Taking the first partial derivative of $J^{F}(\tau ^{*}_{0},Q)$ with respect to Q, we have

$$\begin{aligned} \frac{\partial J^{F}(\tau ^{*}_{0},Q)}{\partial Q}=\frac{1}{2}[\sigma ^{2}+(Q-\mu )^{2}]^{-\frac{1}{2}} \{(2e_{n}+e_{v})[\sigma ^{2}+(Q-\mu )^{2}]^{\frac{1}{2}}+e_{v}(Q-\mu )\}.\nonumber \\ \end{aligned}$$

(F.3)

Using $[\sigma ^{2}+(Q-\mu )^{2}]^{\frac{1}{2}}>\mu -Q$, we have $\frac{\partial J^{F}(\tau ^{*}_{0},Q)}{\partial Q}>0$, which implies that $J^{F}(\tau ^{*}_{0},Q)$ is an increasing function of Q. Therefore, when $Q^{*}_{0}>Q^{*}_{1}$, we have

$$\begin{aligned}&J^{F}(\tau ^{*}_{0},Q^{*}_{0})>J^{F}(\tau ^{*}_{0},Q^{*}_{1})\nonumber \\&\quad =\frac{1}{2}(2e_{n}+e_{v})Q^{*}_{1} +\frac{1}{2}e_{v}\sqrt{\sigma ^{2}+(Q^{*}_{1}-\mu )^{2}}-\frac{1}{2}e_{v}\mu -\frac{\delta _{e}(\delta _{c}-c_{A})}{2c_{L}}\nonumber \\&\quad >\frac{1}{2}(2e_{n}+e_{v})Q^{*}_{1} +\frac{1}{2}e_{v}\sqrt{\sigma ^{2}+(Q^{*}_{1}-\mu )^{2}}-\frac{1}{2}e_{v}\mu -\frac{\delta _{e}(\delta _{c}-c_{A}+\max \{p_{s},p_{b}\}\delta _{e})}{2c_{L}}\nonumber \\&\quad \ge J^{F}(\tau ^{*}_{1},Q^{*}_{1}). \end{aligned}$$

(F.4)

The last inequality in Eq. (F.4) holds because of Eqs. (B.5), (B.8), and (D.5).

Moreover, from Eq. (7), we have that when $C>J^{F}(\tau ,Q)$, $\varPi ^{F}_{1}(\tau ,Q)=\varPi ^{F}_{1s}(\tau ,Q)=\varPi ^{F}_{0}(\tau ,Q)+p_{s}[C-J^{F}(\tau ,Q)]$. From Eqs. (5) and (6), we have that $(\tau ^{*}_{0},Q^{*}_{0})$ is one of the feasible solutions of model $M_{1}$. We further have $\varPi ^{F}_{1}(\tau ^{*}_{0},Q^{*}_{0})=\varPi ^{F}_{0}(\tau ^{*}_{0},Q^{*}_{0})+p_{s}[C-J^{F}(\tau ^{*}_{0},Q^{*}_{0})]> \varPi ^{F}_{0}(\tau ^{*}_{0},Q^{*}_{0})$. Using the optimal property of $(\tau ^{*}_{1},Q^{*}_{1})$, we have $\varPi ^{F}_{1}(\tau ^{*}_{1},Q^{*}_{1})\ge \varPi ^{F}_{1}(\tau ^{*}_{0},Q^{*}_{0})$. Therefore, we have $\varPi ^{F}_{1}(\tau ^{*}_{1},Q^{*}_{1})\ge \varPi ^{F}_{0}(\tau ^{*}_{0},Q^{*}_{0})$ when $C>J^{F}(\tau ^{*}_{0},Q^{*}_{0})$.