Optimizing Technology R&D Supply Chain Problem Under Technology Concern Uncertainty

Abstract

Technology research and development (R&D) plays an increasingly critical role in emerging supply chains by endowing products with desirable technical features. Due to unpredictable market acceptance, the technology concern coefficient, namely, the sensitivity of price to the technology level of technological products is uncertain in the process of R&D, and usually only its partial distribution information can be obtained by referring to similar products or resorting to experts in related fields. It reflects the subjective uncertainty of technology concern, which is different from the existing relevant literature. Motivated by this challenge, we construct a new ambiguity set of possibility distributions to characterize the distribution uncertainty of technology concern and build a novel robust fuzzy tri-level optimization model for decision-making in technology R&D supply chain (tRDsc) with government subsidy. Based on the proposed ambiguity set, we have derived the closed-form equilibrium solutions to the developed robust optimization model. At the same time, the equilibrium under manufacturer subsidy is consistent with that under consumer subsidy. Besides, manufacturer subsidy is more beneficial to the supplier, while technology supplier subsidy benefits the manufacturer more when the subsidy ratio is big enough. Finally, an application case about electric vehicles is studied to illustrate the effectiveness of our new optimization method. The case study shows that the government’s subsidy strategy may change when the uncertainty perturbation parameters change. That is, taking investment efficiency as the selection criterion, the change of perturbation set will influence subsidy policy choices.

Appendices

Appendix

The Proofs of Main Results

The Proof of Theorem 1

According to Definition 1, the possibility distribution $\mu _{\xi }(x;\lambda _l,\lambda _r)$ of trapezoidal fuzzy variable $\xi$ is as follows:

$$\begin{aligned} \mu _{\xi }(x;\lambda _l,\lambda _r)=\left\{ \begin{aligned}&\frac{(1+\lambda _l)(x-r_1)}{r_2-r_1}, \qquad \ \ r_1\le x\le \frac{1}{2}(r_1+r_2),\\&\frac{(1-\lambda _l)x+\lambda _l r_2-r_1}{r_2-r_1}, \frac{1}{2}(r_1+r_2)< x\le r_2,\\&1, \qquad \qquad \qquad \qquad \qquad \qquad r_2<x\le r_3,\\&\frac{(\lambda _r-1)x-\lambda _r r_3+r_4}{r_4-r_3}, r_3<x\le \frac{1}{2}(r_3+r_4),\\&\frac{(1+\lambda _r)(r_4-x)}{r_4-r_3}, \qquad \frac{1}{2}(r_3+r_4)<x\le r_4. \end{aligned}\right. \end{aligned}.$$

Based on the expression of $\mu _{\xi }(x;\lambda _l,\lambda _r)$, one has the credibility of the fuzzy event $\{\xi \le x\}$ as follows:

$$\begin{aligned} {{\mathrm{Cr}}}\{\xi \le x\}=\left\{ \begin{aligned}&0, \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad x<r_1,\\&\frac{(1+\lambda _l)(x-r_1)}{2(r_2-r_1)},\qquad \ r_1 \le x<\frac{1}{2}(r_1+r_2),\\&\frac{(1-\lambda _l)x+\lambda _l r_2-r_1}{2(r_2-r_1)}, \frac{1}{2}(r_1+r_2)\le x<r_2,\\&\frac{1}{2}, \qquad \qquad \qquad \qquad \quad \qquad \quad \ r_2\le x< r_3,\\&1-\frac{(\lambda _r-1)x-\lambda _r r_3+r_4}{2(r_4-r_3)}, r_3 \le x< \frac{1}{2}(r_3+r_4),\\&1-\frac{(r_4-x)(1+\lambda _r)}{2(r_4-r_3)},\ \frac{1}{2}(r_3+r_4)\le x< r_4,\\&1,\qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad x\ge r_4. \end{aligned}\right. \end{aligned}.$$

By the definition of the expected value of $\xi$, we get

$$\begin{aligned} \begin{aligned} {\mathrm {E}}_{\mu _{\xi }}[\xi ]&=\int _{(-\infty ,+\infty )}x \mathrm{{d}}{{\mathrm{Cr}}} \{\xi \le x\}\\&=\int _{[r_1,r_4]} x \mathrm{{d}}{{\mathrm{Cr}}} \{\xi \le x\}\\&=\int _{[r_1,r_1]} x \mathrm{{d}}{{\mathrm{Cr}}} \{\xi \le x\}+\int _{(r_1,\frac{r_1+r_2}{2})} x \mathrm{{d}}{{\mathrm{Cr}}} \{\xi \le x\}\\&\quad +\int _{[\frac{r_1+r_2}{2},r_2]} x \mathrm{{d}}{{\mathrm{Cr}}} \{\xi \le x\}+\int _{(r_2,r_3)} x \mathrm{{d}}{{\mathrm{Cr}}} \{\xi \le x\}\\&\quad +\int _{[r_3,\frac{r_3+r_4}{2}]} x \mathrm{{d}}{{\mathrm{Cr}}} \{\xi \le x\} +\int _{(\frac{r_3+r_4}{2},r_4)} x \mathrm{{d}}{{\mathrm{Cr}}} \{\xi \le x\}\\&\quad +\int _{[r_4,r_4]} x \mathrm{{d}}{{\mathrm{Cr}}} \{\xi \le x\}\\&=\frac{r_1+r_2+r_3+r_4}{4}+\frac{\lambda _{l}(r_1-r_2)+\lambda _{r}(r_4-r_3)}{8}. \end{aligned} \end{aligned}.$$

The Proof of Theorem 2

Based on the ambiguity set ${\mathcal {F}}_{\mu }$ in Equation (4) and the expected value of $\mu _{\xi }$ in Theorem 1, the computationally tractable formulation of the robust fuzzy bi-level tRDsc model (2) is as follows:

$$\begin{aligned} \begin{aligned}&\!\!\!\max \limits _{w,\theta ,U_s} \ \ U_s \\&\mathrm{{s.t.}} \ \ \ \pi _s(w,\theta )\ge U_s \\&\ \ \ \ \ \ \ \max \limits _{d,{\hat{U}}_m} \ \ {\hat{U}}_m \\&\ \ \ \ \ \ \ \ \mathrm{{s.t.}}\ \ L(d)\ge {\hat{U}}_m \end{aligned} \end{aligned},$$

(8)

where $\displaystyle L(d)=(\alpha +\gamma \theta -\beta d-w-c)d$, $\displaystyle \pi _{s}(w,\theta )=wd-\frac{1}{2}\phi \theta ^2$, and $\gamma =\frac{r_1+r_2+r_3+r_4}{4}+\frac{\delta _{l}(r_1-r_2)+\delta _{r}(r_3-r_4)}{8}$.

For the function L(d) in the lower level, we have its second derivative $\frac{\partial ^2 L}{\partial d^2}=-\beta <0$, then L(d) is concave which ensures the maximum point of L exists. Given w and $\theta$, from the first-order conditions, the reaction function of d is obtained as follows:

$$\begin{aligned} \frac{\partial L}{\partial d}=0\Rightarrow d=\frac{\gamma \theta +\alpha -c-w}{2\beta }. \end{aligned}$$

Based on the manufacturer’s reaction, the supplier determines optimal w and $\theta$ to maximize his profit function. The Hessian matrix associated with $\pi _s$ is as following:

$$\begin{aligned} H=\left( \begin{array}{*{20}c} \frac{\partial ^2 \pi _s}{\partial w^2} &{} \frac{\partial ^2 \pi _s}{\partial w \partial \theta } \\ \frac{\partial ^2 \pi _s}{\partial \theta \partial w} &{} \frac{\partial ^2 \pi _s}{\partial \theta ^2}\\ \end{array} \right) = \left( {\begin{array}{*{20}c} -\frac{1}{\beta } &{} \frac{\gamma }{2\beta } \\ \frac{\gamma }{2\beta } &{} -\phi \\ \end{array}} \right) \end{aligned}.$$

By calculation, $\vert H\vert =\frac{4\beta \phi -\gamma ^2}{4\beta ^2}$. Suppose that $4\beta \phi -\gamma ^2>0$, we have H is negative definite. So $\pi _s$ is jointly concave in w and $\theta$. From the first-order conditions

$$\begin{aligned} \left\{ \begin{array}{l} \frac{\partial \pi _m}{\partial w}=0\\ \frac{\partial \pi _m}{\partial \theta }=0 \end{array} \right. \end{aligned},$$

we have $\theta =\frac{(\alpha -c)\gamma }{4\beta \phi -\gamma ^2}$ and $w=\frac{2\beta \phi (\alpha -c)}{4\beta \phi -\gamma ^2}$. Substituting the values of w and $\theta$ into the reaction function d and Equation (1), we get $d=\frac{\phi (\alpha -c)}{4\beta \phi -\gamma ^2}$ and $p=\frac{\beta \phi (3\alpha +c)-c \gamma ^2}{4\beta \phi -\gamma ^2}.$

The Proof of Theorem 3

From the ambiguity set ${\mathcal {F}}_{\mu }$ in Equation (4) and the expected value of $\mu _{\xi }$ in Theorem 1, we have the computationally tractable formulation of the robust fuzzy tRDsc model (5) as follows:

$$\begin{aligned} \begin{aligned}&\max \limits _{\kappa , {\hat{U}}_g^{sp}}\ \ {\hat{U}}_g^{sp} \\&\ \mathrm{{s.t.}}\ \ \ L_1^{sp}(\kappa )\ge {\hat{U}}_g^{sp}\\&\ \ \ \ \ \ \max \limits _{w,\theta ,U_s^{sp}} \ U_s^{sp} \\&\ \ \ \ \ \ \ \ \ \mathrm{{s.t.}} \ \ \ L_2^{sp}(w, \theta )\ge U_s^{sp} \\&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \max \limits _{d,{\hat{U}}_m^{sp}} \ \ {\hat{U}}_m^{sp} \\&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathrm{{s.t.}}\ \ L_3^{sp}(d)\ge {\hat{U}}_m^{sp} \end{aligned} \end{aligned},$$

(9)

where $L_1^{sp}(\kappa )=(\alpha -c)d+\gamma \theta d-\frac{1}{2}\phi \theta ^2-\frac{1}{2}\beta d^2$, $L_2^{sp}(w,\theta )=wd-\frac{1}{2}\phi \theta ^2 (1-\kappa )$, and $L_3^{sp}(d)=(\alpha +\gamma \theta -\beta d -w-c)d$,

$\gamma =\frac{r_1+r_2+r_3+r_4}{4}+\frac{\delta _{l}(r_1-r_2)+\delta _{r}(r_3-r_4)}{8}$.

By calculation, the second derivative associated with the function in the lower level is $\frac{\partial ^2 L_3^{sp}}{\partial d^2}=-\beta <0$, then $L_3^{sp}(d)$ is concave which ensures $L_3^{sp}(d)$ has the maximum point. Given $\kappa$, w, and $\theta$, based on the first-order conditions, the reaction function of d is given by

$$\begin{aligned} \frac{\partial L_3^{sp}}{\partial d}=0\Rightarrow d^{sp}=\frac{\gamma \theta +\alpha -c-w}{2\beta }. \end{aligned}.$$

   According to the manufacturer’s reaction, the supplier maximize his profit function by determining optimal w and $\theta$. The Hessian matrix associated with $L_2^{sp}(w,\theta )$ is as follows:

$$\begin{aligned} H=\left( \begin{array}{*{20}c} \frac{\partial ^2 L_2^{sp}}{\partial w^2} &{} \frac{\partial ^2 L_2^{sp}}{\partial w \partial \theta } \\ \frac{\partial ^2 L_2^{sp}}{\partial \theta \partial w} &{} \frac{\partial ^2 L_2^{sp}}{\partial \theta ^2}\\ \end{array} \right) = \left( {\begin{array}{*{20}c} -\frac{1}{\beta } &{} \frac{\gamma }{2\beta } \\ \frac{\gamma }{2\beta } &{} -\phi (1-\kappa ) \\ \end{array}} \right) \end{aligned}.$$

It can be seen that $\vert H\vert =\frac{4\beta \phi (1-\kappa )-\gamma ^2}{4\beta ^2}>0$. Suppose that $4\beta \phi (1-\kappa )-\gamma ^2>0$, we have H is negative definite. So $L_2^{sp}(w,\theta )$ is jointly concave in w and $\theta$. From the first-order conditions

$$\begin{aligned} \left\{ \begin{array}{l} \frac{\partial L_2^{sp}(w,\theta )}{\partial w}=0\\ \frac{\partial L_2^{sp}(w,\theta )}{\partial \theta }=0 \end{array} \right. \end{aligned},$$

we have $\theta ^{sp}=\frac{(\alpha -c)\gamma }{4\beta \phi (1-\kappa )-\gamma ^2}$ and $w^{sp}=\frac{2\beta \phi (\alpha -c)(1-\kappa )}{4\beta \phi (1-\kappa )-\gamma ^2}$. Substituting the values of $w^{sp}$ and $\theta ^{sp}$ into the reaction function $d^{sp}$ and Equation (1), we get $d^{sp}=\frac{\phi (\alpha -c)(1-\kappa )}{4\beta \phi (1-\kappa )-\gamma ^2}$ and $p^{sp}=\frac{(3\alpha +c)\beta \phi (1-\kappa )-c \gamma ^2}{4\beta \phi (1-\kappa )-\gamma ^2}$.

The Proof of Proposition 5

(1) From practical problem, we can see that the price cap $\alpha$ is greater than unit production cost c. Differentiating the wholesale price in Theorem 3 with respect to the technology cost coefficient $\phi$, we get $\frac{\partial w^{sp}}{\partial \phi }=-\frac{2 \beta (\kappa -1) (c-\alpha ) \gamma ^{2}}{(4 \beta \kappa \phi +\gamma ^{2}-4 \beta \phi )^{2}}<0$;

(2) Differentiating the technology quality level in Theorem 3 with respect to technology cost coefficient $\phi$, one gets $\frac{\partial \theta ^{sp}}{\partial \phi }=-\frac{\gamma (c-\alpha ) (4 \beta \kappa -4 \beta )}{(4 \beta \kappa \phi +\gamma ^{2}-4 \beta \phi )^{2}}<0$;

(3) Differentiating the production quantity in Theorem 3 with respect to technology cost coefficient $\phi$, we have $\frac{\partial d^{sp}}{\partial \phi }=-\frac{(\kappa -1) (c-\alpha ) \gamma ^{2}}{16 [\beta \phi (\kappa -1)+\frac{\gamma ^{2}}{4}]^{2}}<0$; and

(4) Differentiating the selling price in Theorem 3 with respect to technology cost coefficient $\phi$, we get $\frac{\partial p^{sp}}{\partial \phi }=-\frac{3 \beta (\kappa -1) (c-\alpha ) \gamma ^{2}}{16 [\beta \phi (\kappa -1)+\frac{\gamma ^{2}}{4}]^{2}}<0$.

The Proof of Proposition 6

(1) Differentiating the technology supplier’s equilibrium profit in Theorem 4 with respect to technology cost coefficient $\phi$, we get

$\frac{\partial \pi _{s}^{sp}}{\partial \phi }=\frac{(\kappa -1) (c-\alpha )^{2} \gamma ^{2}}{32 [\beta \phi (\kappa -1)+\frac{\gamma ^{2}}{4}]^{2}}<0$,

(2) To ensure the results are meaningful, we assume that $\kappa <1-\frac{\gamma ^2}{4\beta \phi }$ in Theorem 3. Differentiating the manufacturer’s equilibrium profit in Theorem 4 with respect to the technology cost coefficient $\phi$, we have $\frac{\partial \pi _{m}^{sp}}{\partial \phi }=\frac{\beta \phi (\kappa -1)^{2} (c-\alpha )^{2} \gamma ^{2}}{32 [\beta \phi (\kappa -1)+\frac{\gamma ^{2}}{4}]^{3}}<0$,

(3) Differentiating the equilibrium social welfare in Theorem 4 with respect to the technology cost coefficient $\phi$ and set it equal to 0, then we have

$\frac{\partial SW^{sp}}{\partial \phi }=\frac{7 (c-\alpha )^{2} [\beta (\kappa -\frac{5}{7}) (-1+\kappa ) \phi -\frac{\gamma ^{2}}{14}] \gamma ^{2}}{64 [\beta (-1+\kappa ) \phi +\frac{\gamma ^{2}}{4}]^{3}}=0.$

The solution is $\kappa = \frac{6}{7}-\sqrt{\frac{1}{49}+\frac{\gamma ^2}{14\beta \phi }}$. Therefore,

when $0<\kappa< \frac{6}{7}-\sqrt{\frac{1}{49}+\frac{\gamma ^2}{14\beta \phi }}, \frac{\partial SW^{sp}}{\partial \phi }<0$ and

when $\frac{6}{7}-\sqrt{\frac{1}{49}+\frac{\gamma ^2}{14\beta \phi }}<\kappa <1-\frac{\gamma ^2}{4\beta \phi }, \frac{\partial SW^{sp}}{\partial \phi }>0$.

(4) Differentiating the government spending in Theorem 4 with respect to technology cost coefficient $\phi$, we have $\frac{\partial GS^{sp}}{\partial \phi }=\frac{(c-\alpha )^{2} \kappa [\beta (1-\kappa ) \phi +\frac{\gamma ^{2}}{4}] \gamma ^{2}}{32 [\beta (-1+\kappa ) \phi +\frac{\gamma ^{2}}{4}]^{3}}<0$.

The Proof of Theorem 7

According to the ambiguity set ${\mathcal {F}}_{\mu }$ in Equation (4) and the expected value of $\mu _{\xi }$ in Theorem 1, the robust fuzzy tRDsc model (6) is equivalent to

$$\begin{aligned} \begin{aligned}&\max \limits _{\rho , {\hat{U}}_g^{mp}}\ \ {\hat{U}}_g^{mp} \\&\ \mathrm{{s.t.}}\ \ \ L_1^{mp}(\rho )\ge {\hat{U}}_g^{mp} \\&\ \ \ \ \ \ \ \ \ \max \limits _{w,\theta ,U_s^{mp}} \ \ U_s^{mp} \\&\ \ \ \ \ \ \ \ \ \ \mathrm{{s.t.}} \ \ \ L_2^{mp}(w,\theta )\ge U_s^{mp} \\&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \max \limits _{d,{\hat{U}}_m^{mp}} \ \ {\hat{U}}_m^{mp} \\&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathrm{{s.t.}}\ \ L_3^{mp}(d)\ge {\hat{U}}_m^{mp} \end{aligned} \end{aligned},$$

(10)

where $L_1^{mp}(\rho )=(\alpha -c)d+\gamma \theta d-\frac{1}{2}\phi \theta ^2-\frac{1}{2}\beta d^2$, $L_2^{mp}(w,\theta )=wd-\frac{1}{2}\phi \theta ^2$, $L_3^{mp}(d)=(\alpha +\gamma \theta -\beta d -w-c+\rho )d$, and

$\gamma =\frac{r_1+r_2+r_3+r_4}{4}+\frac{\delta _{l}(r_1-r_2)+\delta _{r}(r_3-r_4)}{8}$.

By calculating the second derivative of the lower function, we have $\frac{\partial ^2 L_3^{mp}}{\partial d^2}=-\beta <0$, then $L_3^{mp}(d)$ is concave which indicates the maximum point of $L_3^{mp}(d)$ exists. Given $\rho$, w, and $\theta$, based on the first-order conditions, the reaction function of d is given by

$$\begin{aligned} \frac{\partial L_3^{mp}}{\partial d}=0\Rightarrow d^{mp}=\frac{\alpha + \gamma \theta -w-c+\rho }{2\beta }. \end{aligned}.$$

According to the manufacturer’s reaction, the supplier maximizes his profit function by determining optimal w and $\theta$. The Hessian matrix associated with $L_2^{mp}(w,\theta )$ is as follows:

$$\begin{aligned} H=\left( \begin{array}{*{20}c} \frac{\partial ^2 L_2^{mp}}{\partial w^2} &{} \frac{\partial ^2 L_2^{mp}}{\partial w \partial \theta } \\ \frac{\partial ^2 L_2^{mp}}{\partial \theta \partial w} &{} \frac{\partial ^2 L_2^{mp}}{\partial \theta ^2}\\ \end{array} \right) = \left( {\begin{array}{*{20}c} -\frac{1}{\beta } &{} \frac{\gamma }{2\beta } \\ \frac{\gamma }{2\beta } &{} -\phi \\ \end{array}} \right) \end{aligned}.$$

Therefore, $\vert H\vert =\frac{4\beta \phi -\gamma ^2}{4\beta ^2}>0$. From the assumption that $\beta \phi -\gamma ^2>0$, we have H is negative definite. So $L_2^{mp}(w,\theta )$ is jointly concave in w and $\theta$. From the first-order conditions

$$\begin{aligned} \left\{ \begin{array}{l} \frac{\partial L_2^{mp}(w,\theta )}{\partial w}=0\\ \frac{\partial L_2^{mp}(w,\theta )}{\partial \theta }=0 \end{array} \right. \end{aligned},$$

we have $\theta ^{mp}=\frac{(\alpha -c+\rho )\gamma }{4\beta \phi -\gamma ^2}$ and $w^{mp}=\frac{2\beta \phi (\alpha -c+\rho )}{4\beta \phi -\gamma ^2}$. Substituting the values of $w^{mp}$ and $\theta ^{mp}$ into the reaction function $d^{mp}$, we get $d^{mp}=\frac{\phi (\alpha -c+\rho )}{4\beta \phi -\gamma ^2}$. Getting the decisions of the technology supplier and the manufacturer, the government reduces his objective function to this form that is

$$\begin{aligned} L_1^{mp}(\rho )=\frac{[\beta \phi ^2(7\alpha -7c-\rho )-\phi \gamma ^2(\alpha -c-\rho )](\alpha -c+\rho )}{2(4\beta \phi -\gamma ^2)^2}. \end{aligned}.$$

By calculating, we get $\frac{\partial ^2 L_1^{mp}}{\partial \rho ^2}=-\frac{\phi (\beta \phi -\gamma ^2)}{(4\beta \phi -\gamma ^2)^2}$. Based on the assumption that $\beta \phi -\gamma ^2>0$, then $\frac{\partial ^2 L_1^{mp}}{\partial \rho ^2}<0$ which indicates that $L_1^{mp}(\rho )$ has the maximum points. According to the first-order conditions, we have

$$\begin{aligned} \frac{\partial L_1^{mp}}{\partial \rho }=0 \Rightarrow \rho =\frac{3\beta \phi (\alpha -c)}{\beta \phi -\gamma ^2}. \end{aligned}.$$

Further, it can be obtained that all robust equilibrium solutions which are shown in Theorem 7.

The Proof of Proposition 9

(1) Note that the price cap $\alpha$ is greater than unit production cost c. Differentiating the wholesale price in Theorem 7 with respect to the technology cost coefficient $\phi$, we get $\frac{\partial w^{mp}}{\partial \phi }=-\frac{2 \beta \gamma ^{2} (\alpha -c)}{(\beta \phi -\gamma ^{2})^{2}}<0$;

(2) Differentiating the technology quality level in Theorem 7 with respect to the technology cost coefficient $\phi$, one gets $\frac{\partial \theta ^{mp}}{\partial \phi }=-\frac{\gamma \beta (\alpha -c)}{(\beta \phi -\gamma ^{2})^{2}}<0$;

(3) Differentiating the production quantity in Theorem 7 with respect to the technology cost coefficient $\phi$, we have $\frac{\partial d^{mp}}{\partial \phi }=-\frac{(\alpha -c) \gamma ^{2}}{(\beta \phi -\gamma ^{2})^{2}}<0$; and

(4) Differentiating the selling price in Theorem 7 with respect to the technology cost coefficient $\phi$, we get $\frac{\partial p^{mp}}{\partial \phi }=0$.

The Proof of Proposition 10

(1) Differentiating the technology supplier’s equilibrium profit in Theorem 8 with respect to technology cost coefficient $\phi$, we get $\frac{\partial \pi _{s}^{mp}}{\partial \phi }=-\frac{\gamma ^{2}(\alpha -c)^{2}(7\beta \phi -\gamma ^{2})}{2(\beta \phi -\gamma ^{2})^{3}}<0$;

(2) Differentiating the manufacturer’s equilibrium profit in Theorem 8 with respect to the technology cost coefficient $\phi$, we have $\frac{\partial \pi _{m}^{mp}}{\partial \phi }=-\frac{2 \beta \phi \gamma ^2 (\alpha -c)^{2}}{(\beta \phi - \gamma ^{2})^{3}}<0$;

(3) To make sense of the result, we assume that $\beta \phi -\gamma ^2 > 0$ in Theorem 8. Differentiating the equilibrium social welfare in Theorem 8 with respect to the technology cost coefficient $\phi$, we have

$\frac{\partial SW^{mp}}{\partial \phi }=-\frac{\gamma ^{2}(\alpha -c)^{2}}{2(\beta \phi - \gamma ^{2})^{2}}<0$; and

(4) Differentiating the government spending in Theorem 8 with respect to technology cost coefficient $\phi$, we have $\frac{\partial GS^{mp}}{\partial \phi }=-\frac{6\beta \phi \gamma ^2 (\alpha -c)^{2}}{(\beta \phi - \gamma ^{2})^{3}}<0$.

The Proof of Theorem 11

Based on the ambiguity set ${\mathcal {F}}_{\mu }$ in Equation (4) and the expected value of $\mu _{\xi }$ in Theorem 1, the robust fuzzy tRDsc model (7) is equivalently written as follows:

$$\begin{aligned} \begin{aligned}&\max \limits _{\epsilon , {\hat{U}}_g^{cp}}\ \ {\hat{U}}_g^{cp} \\&\ \mathrm{{s.t.}}\ \ \ L_1^{cp}(\epsilon )\ge {\hat{U}}_g^{cp} \\&\ \ \ \ \ \ \max \limits _{w,\theta ,U_s^{cp}} \ \ U_s^{cp} \\&\ \ \ \ \ \ \ \ \ \ \mathrm{{s.t.}} \ \ \ L_2^{cp}(w,\theta )\ge U_s^{cp} \\&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \max \limits _{d,{\hat{U}}_m^{cp}} \ \ {\hat{U}}_m^{cp} \\&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathrm{{s.t.}}\ \ L_3^{cp}(d)\ge {\hat{U}}_m^{cp} \end{aligned} \end{aligned},$$

(11)

where $L_1^{cp}(\epsilon )=(\alpha -c)d+\gamma \theta d-\frac{1}{2}\phi \theta ^2-\frac{1}{2}\beta d^2$,

$L_2^{cp}(w,\theta )=wd-\frac{1}{2}\phi \theta ^2$, $L_3^{cp}(d)=(\alpha +\gamma \theta -\beta d -w-c+\epsilon )d$, and

$\gamma =\frac{r_1+r_2+r_3+r_4}{4}+\frac{\delta _{l}(r_1-r_2)+\delta _{r}(r_3-r_4)}{8}$.

By calculating the second derivative of the lower function $L_3^{cp}(d)$, we have $\frac{\partial ^2 L_3^{cp}}{\partial d^2}=-\beta <0$, then $L_3^{cp}(d)$ is concave which indicates the maximum point of $L_3^{cp}(d)$ exists. Given $\epsilon$, w, and $\theta$, based on the first-order conditions, the reaction function of d is given by

$$\begin{aligned} \frac{\partial L_3^{cp}}{\partial d}=0\Rightarrow d^{cp}=\frac{\alpha + \gamma \theta -w-c+\epsilon }{2\beta }. \end{aligned}.$$

Getting the manufacturer’s reaction, the supplier maximizes his profit function by determining optimal w and $\theta$. The Hessian matrix associated with $L_2^{cp}(w,\theta )$ is as follows:

$$\begin{aligned} H=\left( \begin{array}{*{20}c} \frac{\partial ^2 L_2^{cp}}{\partial w^2} &{} \frac{\partial ^2 L_2^{cp}}{\partial w \partial \theta } \\ \frac{\partial ^2 L_2^{cp}}{\partial \theta \partial w} &{} \frac{\partial ^2 L_2^{cp}}{\partial \theta ^2}\\ \end{array} \right) = \left( {\begin{array}{*{20}c} -\frac{1}{\beta } &{} \frac{\gamma }{2\beta } \\ \frac{\gamma }{2\beta } &{} -\phi \\ \end{array}} \right) \end{aligned}.$$

Then, we get $\vert H\vert =\frac{4\beta \phi -\gamma ^2}{4\beta ^2}$. It is known that $\beta \phi -\gamma ^2>0$, then $4\beta \phi -\gamma ^2>0$ and H is negative definite. So $L_2^{cp}(w,\theta )$ is jointly concave in w and $\theta$. From the first-order conditions

$$\begin{aligned} \left\{ \begin{array}{l} \frac{\partial L_2^{cp}(w,\theta )}{\partial w}=0\\ \frac{\partial L_2^{cp}(w,\theta )}{\partial \theta }=0 \end{array} \right. \end{aligned},$$

we have $\theta ^{cp}=\frac{(\alpha -c+\epsilon )\gamma }{4\beta \phi -\gamma ^2}$ and $w^{cp}=\frac{2\beta \phi (\alpha -c+\epsilon )}{4\beta \phi -\gamma ^2}$. Substituting the values of $w^{cp}$ and $\theta ^{cp}$ into the reaction function $d^{cp}$, we get $d^{cp}=\frac{\phi (\alpha -c+\epsilon )}{4\beta \phi -\gamma ^2}$. Getting the decisions of the technology supplier and the manufacturer, the government reduces its objective function to this form, that is,

$$\begin{aligned} L_1^{cp}(\epsilon )=\frac{[\beta \phi ^2(7\alpha -7c-\epsilon )-\phi \gamma ^2(\alpha -c+\epsilon )](\alpha -c+\epsilon )}{2(4\beta \phi -\gamma ^2)^2}. \end{aligned}.$$

The second derivative $\frac{\partial ^2 L_1^{cp}}{\partial \epsilon ^2}=-\frac{\phi (\beta \phi -\gamma ^2)}{(4\beta \phi -\gamma ^2)^2}$. Based on the assumption that $\beta \phi -\gamma ^2>0$, then $\frac{\partial ^2 L_1^{cp}}{\partial \epsilon ^2}<0$ which indicates that $L_1^{cp}(\epsilon )$ has the maximum points. Based on the first-order conditions, we have

$$\begin{aligned} \frac{\partial L_1^{cp}}{\partial \epsilon }=0 \Rightarrow \epsilon =\frac{3\beta \phi (\alpha -c)}{\beta \phi -\gamma ^2}. \end{aligned}.$$

Further, it can be obtained that all robust equilibrium solutions which are shown in Theorem 11.

The Proof of Proposition 12

The equilibrium solutions under technology supplier subsidy and those under manufacturer subsidy can be seen in Theorem 3 and 7, respectively.

(1) Technology quality: $\theta ^{sp}-\theta ^{mp}=\frac{\gamma (\alpha -c)}{4\beta \phi (1-\kappa )-\gamma ^2}-\frac{\gamma (\alpha -c)}{\beta \phi -\gamma ^2}=0$, the solution is $\kappa =\frac{3}{4}$, then we have

if $0\le \kappa <\frac{3}{4}$ and then $\theta ^{sp}<\theta ^{mp}$;

if $\frac{3}{4} \le \kappa <1-\frac{\gamma ^2}{4\beta \phi }$, then $\theta ^{sp}\ge \theta ^{mp}$.

(2) Wholesale price: $w^{sp}-w^{mp}=\frac{2\beta \phi (\alpha -c)(1-\kappa )}{4\beta \phi (1-\kappa )-\gamma ^2}-\frac{2 \beta \phi (\alpha -c)}{\beta \phi -\gamma ^2}=0$, the solution is $\kappa =\frac{3\beta \phi }{\gamma ^{2}+3\beta \phi }$, then we have

if  $0\le \kappa <\frac{3\beta \phi }{\gamma ^{2}+3\beta \phi }$, then $w^{sp} < w^{mp}$; and

if $\frac{3\beta \phi }{\gamma ^{2}+3\beta \phi } \le \kappa <1-\frac{\gamma ^2}{4\beta \phi }$, then $w^{sp} \ge w^{mp}$.

(3) Production quantity: $d^{sp} - d^{mp}=\frac{\phi (\alpha -c)(1-\kappa )}{4\beta \phi (1-\kappa )-\gamma ^2}-\frac{\phi (\alpha -c)}{ \beta \phi -\gamma ^2}=0$, the solution is $\kappa =\frac{3\beta \phi }{\gamma ^{2}+3\beta \phi }$, then we have

if $0\le \kappa <\frac{3\beta \phi }{\gamma ^{2}+3\beta \phi }$, then $d^{sp} < d^{mp}$; and

if $\frac{3\beta \phi }{\gamma ^{2}+3\beta \phi } \le \kappa <1-\frac{\gamma ^2}{4\beta \phi }$, then $d^{sp} \ge d^{mp}$.

(4) Retail price: $p^{sp}-p^{mp}=\frac{(3\alpha +c)\beta \phi (1-\kappa )-c \gamma ^2}{4\beta \phi (1-\kappa )-\gamma ^2}-c \ge 0$, thus $p^{sp}\ge p^{mp}$.

The Proof of Proposition 13

The equilibrium objectives under technology supplier subsidy and those under manufacturer subsidy can be seen in Theorems 4 and 8, respectively.

(1) The technology supplier’s profit:

$\pi _s^{sp} - \pi _s^{mp}=\frac{\phi (\alpha -c)^2 (1-\kappa )}{2[4\beta \phi (1-\kappa )-\gamma ^2]}-\frac{\phi (4\beta \phi -\gamma ^2)(\alpha -c)^2}{2(\beta \phi -\gamma ^2)^2}=0$, the solution is $\kappa = \frac{3\beta \phi (2 \gamma ^{2}-5\beta \phi )}{\gamma ^{4}+2 \gamma ^{2}\beta \phi -15\beta ^{2}\phi ^{2}}$, then we have

if $0\le \kappa <\frac{3\beta \phi (2 \gamma ^{2}-5\beta \phi )}{\gamma ^{4}+2 \gamma ^{2}\beta \phi -15\beta ^{2}\phi ^{2}}$, then $\pi _s^{sp} < \pi _s^{mp}$; and

if $\frac{3\beta \phi (2 \gamma ^{2}-5\beta \phi )}{\gamma ^{4}+2 \gamma ^{2}\beta \phi -15\beta ^{2}\phi ^{2}} \le \kappa <1-\frac{\gamma ^2}{4\beta \phi }$, then $\pi _s^{sp} \ge \pi _s^{mp}$.

(2) The manufacturer’s profit:

$\pi _m^{sp} - \pi _m^{mp}=\frac{\beta \phi ^2(\alpha -c)^2 (1-\kappa )^2}{[4\beta \phi (1-\kappa )-\gamma ^2]^2}-\frac{\beta \phi ^2(\alpha -c)^2}{(\beta \phi -\gamma ^2)^2}=0$, the solution is $\kappa =\kappa _2$, then we have, if $0\le \kappa <\kappa _2$, then $\pi _m^{sp} < \pi _m^{mp}$; and if $\kappa _2 \le \kappa <1-\frac{\gamma ^2}{4\beta \phi }$, then $\pi _m^{sp} \ge \pi _m^{mp}$,

where $\kappa _2=-\frac{\sqrt{\gamma ^{8}+12 \gamma ^{6}\beta \phi -26 \gamma ^{4}\beta ^{2}\phi ^{2}+12 \gamma ^{2}\beta ^{3}\phi ^{3}+\beta ^{4}\phi ^{4}}}{32\beta ^{2}\phi ^{2}}+\frac{-\gamma ^{4}-6 \gamma ^{2}\beta \phi +31 \beta ^{2}\phi ^{2}}{32\beta ^{2}\phi ^{2}}.$

(3) Social welfare: $SW^{sp}-SW^{mp}=\frac{\phi (\alpha -c)^2 [7\beta \phi (1-\kappa )^2-\gamma ^2]}{2[4 \beta \phi (1-\kappa )-\gamma ^2]^2}-\frac{\phi (\alpha -c)^2}{2( \beta \phi -\gamma ^2)}<0$; and

(4) Investment efficiency: Based on the results of social welfare and government spending, the investment efficiency under technology supplier subsidy and manufacturer subsidy are obtained as follows, respectively, $IE^{sp}= \frac{7\beta \phi (1-\kappa )^2-\gamma ^2}{\kappa \gamma ^2}, IE^{mp}=\frac{\beta \phi -\gamma ^2}{6\beta \phi }$, then

$$\begin{aligned} \begin{aligned} IE^{sp}-IE^{mp}&=\frac{7\beta \phi (1-\kappa )^2-\gamma ^2}{\kappa \gamma ^2}-\frac{\beta \phi -\gamma ^2}{6\beta \phi }\\&=\frac{42\beta ^2 \phi ^2 (1-\kappa )^2-6\beta \phi \gamma ^2-\kappa \beta \phi \gamma ^2+\kappa \gamma ^4}{6\kappa \beta \phi \gamma ^2}\\&\ge \frac{42\beta ^2 \phi ^2 (1-\kappa )^2-7\beta \phi \gamma ^2}{6\kappa \beta \phi \gamma ^2}\ge 0.\\ \end{aligned} \end{aligned}.$$

Therefore, $0\le \kappa \le 1-\sqrt{\frac{\gamma ^2}{6\beta \phi }}$.