Appendix A: Proof of Proposition 1
The Jacobian matrix is introduced to find the system’s stable region for supply chain members to adjust their decisions. The Jacobian matrix of function is:
$$ \left( {\begin{array}{*{20}c} {J_{11} } & {J_{12} } & {J_{13} } & {J_{14} } \\ {J_{21} } & {J_{22} } & {J_{23} } & {J_{24} } \\ {J_{31} } & {J_{32} } & {J_{33} } & 0 \\ 0 & 0 & 0 & {J_{44} } \\ \end{array} } \right) $$
(A1)
The corresponding characteristic equation is as Eq. (A1):
$$ F\left( \lambda \right) = \eta_{0} \lambda^{4} + \eta_{1} \lambda^{3} + \eta_{2} \lambda^{2} + \eta_{3} \lambda^{1} + \eta_{4} \lambda^{0} $$
(A2)
where:
$$ \left\{ {\begin{array}{*{20}l} {\eta_{0} = 1\eta_{0} = 1} \hfill \\ {\eta_{1} = - \left( {J_{11} + J_{22} + J_{33} - J_{44} } \right)} \hfill \\ {\eta_{2} = { }J_{11} J_{22} - J_{12} { }J_{21} - J_{13} { }J_{31} - J_{23} { }J_{32} + { }J_{33} \left( {J_{11} + J_{22} + J_{44} } \right) + J_{44} \left( {J_{22} + J_{11} } \right)} \hfill \\ {\eta_{3} = J_{44} \left( {J_{12} { }J_{21} - J_{11} J_{22} + J_{13} { }J_{31} + J_{23} { }J_{32} - J_{33} \left( {J_{11} + J_{22} { }} \right)} \right)} \hfill \\ {\quad \quad + { }J_{31} \left( {J_{13} { }J_{22} - J_{12} { }J_{23} } \right) + { }J_{32} \left( {J_{11} { }J_{23} - J_{13} { }J_{21} } \right) + J_{33} \left( {J_{12} { }J_{21} - J_{11} { }J_{22} } \right)} \hfill \\ {\eta_{4} = { }J_{44} \left( {J_{31} \left( {J_{12} { }J_{23} - J_{13} { }J_{22} } \right) + { }J_{32} \left( {J_{13} { }J_{21} - J_{11} { }J_{23} { }} \right) + J_{33} \left( {J_{11} J_{22} - J_{12} { }J_{21} } \right)} \right)} \hfill \\ \end{array} } \right. $$
(A3)
According to the Jury test condition, the sufficient conditions for stability of \(F\left(\lambda \right)\) are as Eq. (A4):
$$ \left[ {\begin{array}{*{20}l} {\left| {\eta_{4} } \right| < \eta_{0} } \hfill \\ {\left. {F\left( \lambda \right)} \right|_{\lambda = 1} > 0} \hfill \\ {\left. {F\left( \lambda \right)} \right|_{\lambda = - 1} > 0} \hfill \\ {\left| {b_{3} } \right| > \left| {b_{0} } \right|} \hfill \\ {\left| {c_{2} } \right| > \left| {c_{0} } \right|} \hfill \\ \end{array} } \right. $$
(A4)
Note that, based on the Jury’s stability test, when a system reaches stability, the above conditions must be met.
$$ \left[ {\begin{array}{*{20}l} {\left| {\eta_{n} } \right| < \eta_{0} } \hfill \\ {\left. {F\left( \lambda \right)} \right|_{\lambda = 1} > 0} \hfill \\ {\left. {F\left( \lambda \right)} \right|_{\lambda = - 1} > 0} \hfill \\ {\left| {b_{n - 1} } \right| > \left| {b_{0} } \right|} \hfill \\ {\left| {c_{n - 2} } \right| > \left| {c_{0} } \right|} \hfill \\ \end{array} } \right. $$
(A5)
Since,
$$ b_{k} = \left[ {\begin{array}{*{20}c} {\eta_{n} } & {\eta_{n - 1 - k} } \\ {\eta_{0} } & {\eta_{k + 1} } \\ \end{array} } \right], k = 0,1,2, \ldots ,n - 1 $$
(A6)
$$ c_{k} = \left[ {\begin{array}{*{20}c} {b_{n - 1} } & {b_{n - 2 - k} } \\ {b_{0} } & {b_{k + 1} } \\ \end{array} } \right], k = 0,1,2, \ldots ,n - 2 $$
(A7)
The values of Jacobian matrix for the Dynamic–Stochastic Non-Disruption Model:
$$ \begin{aligned} \overline{{J_{11} }} & = 1 - 2\omega_{1} P_{1} \left( t \right)\left( {\varrho_{1} + \vartheta_{1} } \right) \\ & \quad + \omega_{1} \left( {\tilde{a}_{1} } \right. \\ & \quad + \frac{1}{2}\left( {l_{1} - 4P_{1} \left( t \right)\left( {\varrho_{1} + \vartheta_{1} } \right) + 2P_{2} \left( t \right)\left( {\vartheta_{1} + \vartheta_{2} } \right) + u_{1} } \right. - 2\sigma \left( t \right)v_{1} \\ & \quad \left. {\left. { + 2\left( {\varrho_{1} + \vartheta_{1} } \right)W_{1} \left( t \right) - 2\vartheta_{2} W_{2} \left( t \right) + 2\gamma_{1} \zeta \left( t \right)} \right)} \right) \\ \end{aligned} $$
(A8)
$$ \overline{{J_{12} }} = \omega_{1} P_{1} \left( t \right)\left( {\vartheta_{1} + \vartheta_{2} } \right) $$
(A9)
$$ \overline{{J_{13} }} = - \omega_{1} \upsilon_{1} P_{1} \left( t \right) $$
(A10)
$$ \overline{{J_{14} }} = \gamma_{1} \omega_{1} P_{1} \left( t \right) $$
(A11)
$$ \overline{{J_{21} }} = \omega_{2} P_{2} \left( t \right)\left( {\vartheta_{1} + \vartheta_{2} } \right) $$
(A12)
$$ \begin{aligned} \overline{{J_{22} }} & = 1 - 2\omega_{2} P_{2} \left( t \right)\left( {\varrho_{2} + \vartheta_{2} } \right) \\ & \quad + \omega_{2} \left( {\tilde{a}_{2} } \right. \\ & \quad + \frac{1}{2}\left( {l_{2} + 2P_{1} \left( t \right)\left( {\vartheta_{1} + \vartheta_{2} } \right) + u_{2} + 2\sigma \left( t \right)v_{2} - 2\vartheta_{1} W_{1} \left( t \right)} \right. \\ & \quad \left. {\left. { + 2\left( {\varrho_{2} + \vartheta_{2} } \right)\left( {W_{2} \left( t \right) - 2P_{2} \left( t \right)} \right) - 2\omega_{2} \zeta \left( t \right)} \right)} \right) \\ \end{aligned} $$
(A13)
$$ \overline{{J_{23} }} = \omega_{2} \upsilon_{2} P_{2} \left( t \right) $$
(A14)
$$ \overline{{J_{24} }} = - \gamma_{2} \omega_{2} P_{2} \left( t \right) $$
(A15)
$$ \overline{{J_{31} }} = - \omega_{3} \upsilon_{1} \sigma \left( t \right) $$
(A16)
$$ \overline{{J_{32} }} = \omega_{3} \upsilon_{2} \sigma \left( t \right) $$
(A17)
$$ \overline{{J_{33} }} = 1 - k\omega_{3} \sigma \left( t \right) + \omega_{3} \left( { - k\sigma \left( t \right) - v_{1} \left( {P_{1} \left( t \right) - W_{1} \left( t \right)} \right) + v_{2} \left( {P_{2} \left( t \right) - W_{2} \left( t \right)} \right)} \right) $$
(A18)
$$ \overline{{J_{44} }} = 1 - h\omega_{4} \zeta \left( t \right) + \omega_{4} \left( {\gamma_{1} \left( {W_{1} \left( t \right) - C_{1} \left( t \right)} \right) - \gamma_{2} \left( {W_{2} \left( t \right) - C_{2} \left( t \right)} \right) - h\zeta \left( t \right)} \right) $$
(A19)
Appendix B: Proof of Theorem 3.1
The concavity of the total profit function in decentralized mode is examined by its corresponding Hessian matrix, \(H\).
$$ H\left( {P_{1} , P_{2} ,\sigma } \right) = \left[ {\begin{array}{*{20}c} { - 2\left( {\varrho_{1} + \vartheta_{1} } \right)} & {\vartheta_{1} + \vartheta_{2} } & { - \upsilon_{1} } \\ {\vartheta_{1} + \vartheta_{2} } & { - 2\left( {\varrho_{2} + \vartheta_{2} } \right)} & {\upsilon_{2} } \\ { - \upsilon_{1} } & {\upsilon_{2} } & { - k} \\ \end{array} } \right] $$
Owing to the positive \(\varrho_{2}\) and \(\vartheta_{2}\), it can be shown that the first minor determinant \(\big({- 2\big( {\varrho_{2} + \vartheta_{2}}\big)}\big)\) is negative. Moreover, the second minor determinant \(\big( {4\varrho_{2}\vartheta_{1} - \big( {\vartheta_{1} - \vartheta_{2}} \big)^{2} + 4\varrho_{1} \big( {\varrho_{2} + \vartheta_{2}} \big)} \big)\) is positive, in case \(\big( {\vartheta_{1} - \vartheta_{2} } \big)^{2} < 4\varrho_{2} \vartheta_{1} + 4\varrho_{1} \big( {\varrho_{2} + \vartheta_{2} } \big)\). Furthermore, the third minor determinant \(\Big( {k\big( {\big( {\vartheta_{1} - \vartheta_{2} } \big)^{2} - 4\varrho_{2} \vartheta_{1} - 4\varrho_{1} \big( {\varrho_{2} + \vartheta_{2} } \big)} \big) + 2\big( {\varrho_{2} + \vartheta_{2} } \big)v_{1}^{2} - 2\big( {\vartheta_{1} + \vartheta_{2} } \big)\upsilon_{1} \upsilon_{2} + 2\big( {\varrho_{1} + \vartheta_{1} } \big)v_{2}^{2} } \Big)\) is negative, in case \(\big( {\varrho_{2} + \vartheta_{2} } \big)v_{1}^{2} + \big( {\varrho_{1} + \vartheta_{1} } \big)v_{2}^{2} < \big( {\vartheta_{1} + \vartheta_{2} } \big)\upsilon_{1} \upsilon_{2}\).□
Appendix C: The values of substituted parameters
$$ T_{1} = - \left( {k\left( {8\varrho_{2} \vartheta_{1} - 2\left( {\vartheta_{1} - \vartheta_{2} } \right)^{2} + 8\varrho_{1} \left( {\varrho_{2} + \vartheta_{2} } \right)} \right) - 4\left( {\varrho_{2} + \vartheta_{2} } \right)\upsilon_{1}^{2} + 4\left( {\vartheta_{1} + \vartheta_{2} } \right)\upsilon_{1} \upsilon_{2} - 4\left( {\varrho_{1} + \vartheta_{1} } \right)\upsilon_{2}^{2} } \right) $$
(C1)
$$ \begin{aligned} T_{2} & = \upsilon_{1} \upsilon_{2} \left( {2a_{2} + l_{2} + u_{2} - 2W_{1} \left( {2\vartheta_{1} + \vartheta_{2} } \right) - 2W_{2} \left( {\varrho_{2} + \vartheta_{2} } \right)} \right) \\ & \quad + v_{2}^{2} \left( {2a_{1} + l_{1} + u_{1} + 2W_{1} \left( {\varrho_{1} + \vartheta_{1} } \right) + 2\vartheta_{1} W_{2} } \right) \\ & \quad - k\left( {l_{2} \vartheta_{2} + 2l_{1} \left( {\varrho_{2} + \vartheta_{2} } \right) + 2a_{2} \left( {\vartheta_{1} + \vartheta_{2} } \right) + 2\vartheta_{2} u_{1} + \vartheta_{2} u_{2} + \vartheta_{1} \left( {l_{2} + u_{2} } \right)} \right. \\ & \quad + 2W_{1} \left( {2\varrho_{1} \vartheta_{2} - \vartheta_{1}^{2} + \vartheta_{1} \vartheta_{2} } \right) + 2\varrho_{2} \left( {u_{1} + 2\left( {\varrho_{1} + \vartheta_{1} } \right)W_{1} + \left( {\vartheta_{1} - \vartheta_{2} } \right)W_{2} } \right) \\ & \quad \left. { + 2\vartheta_{2} W_{2} \left( {\vartheta_{1} - \vartheta_{2} } \right) + 2\zeta \left( {2\gamma_{1} \left( {\varrho_{2} + \vartheta_{2} } \right) - \gamma_{2} \left( {\vartheta_{1} + \vartheta_{2} } \right)} \right)} \right) \\ \end{aligned} $$
(C2)
$$ \begin{aligned} T_{3} & = \upsilon_{1} \upsilon_{2} \left( {2a_{1} + l_{1} + u_{1} - 2W_{1} \left( {\varrho_{1} + \vartheta_{1} } \right) - 2W_{2} \left( {\vartheta_{1} + 2\vartheta_{2} } \right)} \right) + 4v_{2}^{2} W_{2} \left( {\varrho_{1} + \vartheta_{1} } \right) \\ & \quad + 2\upsilon_{1} \left( {\gamma_{1} \upsilon_{2} - \gamma_{2} \upsilon_{1} } \right)\zeta \\ & \quad - k\left( {2l_{2} \left( {\varrho_{1} + \vartheta_{1} } \right) + \left( {2a_{1} + l_{1} } \right)\left( {\vartheta_{1} + \vartheta_{2} } \right) + \vartheta_{2} u_{1} + 2\varrho_{1} u_{2} } \right. \\ & \quad + 2W_{1} \left( {\varrho_{1} \vartheta_{2} - \vartheta_{1}^{2} } \right) + W_{2} \left( {4\varrho_{1} \left( {\varrho_{2} + \vartheta_{2} } \right) - 2\vartheta_{2}^{2} } \right) \\ & \quad + 2\zeta \left( {\gamma_{1} \vartheta_{2} - 2\gamma_{2} \varrho_{1} } \right) \\ & \quad + \vartheta_{1} \left( {u_{1} + } \right. \\ & \quad \left. {\left. { + 2\left( {u_{2} - \varrho_{1} W_{1} + 2\varrho_{2} W_{2} + \vartheta_{2} \left( {W_{1} + W_{2} } \right) + \gamma_{1} \zeta - 2\gamma_{2} \zeta } \right)} \right)} \right) \\ \end{aligned} $$
(C3)
$$ \begin{aligned} T_{4} & = \upsilon_{2} \left( {4a_{2} \varrho_{1} + 2l_{2} \varrho_{1} + 4a_{2} \vartheta_{1} + l_{1} \vartheta_{1} + 2l_{2} \vartheta_{1} + l_{1} \vartheta_{2} + 2a_{1} \left( {\vartheta_{1} + \vartheta_{2} } \right)} \right. \\ & \quad \left. { + \vartheta_{1} u_{1} + \vartheta_{2} u_{1} + 2\varrho_{1} u_{2} + 2\vartheta_{1} u_{2} } \right) + 4\varrho_{1} \varrho_{2} \upsilon_{1} W_{1} + 4\varrho_{2} \vartheta_{1} \upsilon_{1} W_{1} \\ & \quad + 4\varrho_{2} \vartheta_{1} \upsilon_{1} W_{1} + 4\varrho_{1} \vartheta_{2} \upsilon_{1} W_{1} + 2\vartheta_{1} \vartheta_{2} \upsilon_{1} W_{1} - 2\vartheta_{2}^{2} \upsilon_{1} W_{1} - 2\varrho_{1} \vartheta_{1} \upsilon_{2} W_{1} \\ & \quad - 2\vartheta_{1}^{2} \upsilon_{2} W_{1} + 2\varrho_{1} \vartheta_{2} \upsilon_{2} W_{1} + 2\vartheta_{1} \vartheta_{2} \upsilon_{2} W_{1} - 2\varrho_{2} \vartheta_{1} \upsilon_{1} W_{2} \\ & \quad + 2\varrho_{2} \vartheta_{2} \upsilon_{1} W_{2} - 2\vartheta_{1} \vartheta_{2} \upsilon_{1} W_{2} + 2\vartheta_{2}^{2} \upsilon_{1} W_{2} - 4\varrho_{1} \varrho_{2} \upsilon_{2} W_{2} \\ & \quad - 4\varrho_{2} \vartheta_{1} \upsilon_{2} W_{2} + 2\vartheta_{1}^{2} \upsilon_{2} W_{2} - 4\varrho_{1} \vartheta_{2} \upsilon_{2} W_{2} - 2\vartheta_{1} \vartheta_{2} \upsilon_{2} W_{2} \\ & \quad + 2\zeta \left( { - 2\gamma_{1} \left( {\varrho_{2} + \vartheta_{2} } \right)\upsilon_{1} + \gamma_{2} \left( {\vartheta_{1} + \vartheta_{2} } \right)\upsilon_{1} - 2\gamma_{2} \left( {\varrho_{1} + \vartheta_{1} } \right)\upsilon_{2} } \right. \\ & \quad \left. { + \gamma_{1} \left( {\vartheta_{1} + \vartheta_{2} } \right)\upsilon_{2} } \right) \\ \end{aligned} $$
(C4)
$$ \begin{aligned} T_{5} & = C_{2} \left( {\gamma_{2} \left( {\left( {\varrho_{2} + \vartheta_{2} } \right)\left( {2k\varrho_{1} - \upsilon_{1}^{2} } \right) + k\vartheta_{1} \left( {2\varrho_{2} - \vartheta_{1} + \vartheta_{2} } \right) + \vartheta_{1} \upsilon_{1} \upsilon_{2} } \right)} \right. \\ & \quad + \gamma_{1} \left. {\left( {k\left( {\vartheta_{1} - \vartheta_{2} } \right)\left( {\varrho_{2} + \vartheta_{2} } \right) + \upsilon_{2} \left( {\left( {\varrho_{2} + \vartheta_{2} } \right)\upsilon_{1} - \vartheta_{1} \upsilon_{2} } \right)} \right)} \right) \\ & \quad + W_{1} \left( { - \gamma_{2} k\left( {\varrho_{1} + \vartheta_{1} } \right)\left( {\vartheta_{1} - \vartheta_{2} } \right)} \right. \\ & \quad + \gamma_{1} k\left( {2\varrho_{2} \vartheta_{1} + \left( {\vartheta_{1} - \vartheta_{2} } \right)\vartheta_{2} + 2\varrho_{1} \left( {\varrho_{2} + \vartheta_{2} } \right)} \right) \\ & \quad \left. { + \gamma_{1} \upsilon_{2} \left( {\vartheta_{2} \upsilon_{1} - \left( {\varrho_{1} + \vartheta_{1} } \right)\upsilon_{2} } \right) + \gamma_{2} \upsilon_{1} \left( {\left( {\varrho_{1} + \vartheta_{1} } \right)\upsilon_{2} - \vartheta_{2} \upsilon_{1} } \right)} \right) \\ & \quad - (\gamma_{2} \left( {2k\varrho_{1} \left( {\varrho_{2} + \vartheta_{2} } \right) + k\vartheta_{1} \left( {2\varrho_{2} - \vartheta_{1} + \vartheta_{2} } \right) - \left( {\varrho_{2} + \vartheta_{2} } \right)v_{1}^{2} + \vartheta_{1} \upsilon_{1} \upsilon_{2} } \right) \\ & \quad + \gamma_{1} W_{2} \left( {k\left( {\vartheta_{1} - \vartheta_{2} } \right)\left( {\varrho_{2} + \vartheta_{2} } \right) + \upsilon_{2} \left( {\left( {\varrho_{2} + \vartheta_{2} } \right)\upsilon_{1} - \vartheta_{1} \upsilon_{2} } \right)} \right) \\ \end{aligned} $$
(C5)
$$ \begin{aligned} T_{6} & = - \upsilon_{1} \upsilon_{2} \left( {\widetilde{{\overline{a}}}_{2} - l_{2} - u_{2} } \right) - v_{2}^{2} \left( {2\widetilde{{\overline{a}}}_{1} + l_{1} + u_{1} } \right) \\ & \quad - 2\overline{W}_{1} - 2\overline{W}_{1} \left( {2v_{1}^{2} \left( {l_{1} \varrho_{2} + \vartheta_{2} } \right) + v_{2}^{2} \left( {\varrho_{1} + \vartheta_{1} } \right)} \right) \\ & \quad + \upsilon_{1} \upsilon_{2} \left( {2\overline{W}_{1} \left( {2\vartheta_{1} + \vartheta_{2} } \right) + 2\overline{W}_{2} \left( {l_{1} \varrho_{2} + \vartheta_{2} } \right)} \right) - 2\vartheta_{1} v_{2}^{2} \overline{W}_{2} \\ & \quad - 2\vartheta_{1} v_{2}^{2} \overline{W}_{2} + 2\upsilon_{2} \zeta \left( {\gamma_{2} \upsilon_{1} - \gamma_{1} \upsilon_{2} } \right) \\ & \quad + k\left( {l_{2} \vartheta_{2} + 2l_{1} \left( {\varrho_{2} + \vartheta_{2} } \right) + 2\widetilde{{\overline{a}}}_{2} \left( {\vartheta_{1} + \vartheta_{2} } \right) + 2u_{1} \left( {l_{1} \varrho_{2} + \vartheta_{2} } \right)} \right. \\ & \quad + \vartheta_{2} u_{2} + \vartheta_{1} \left( {l_{2} + u_{2} } \right) \\ & \quad - \Phi \lambda_{0} \xi T\Psi \left( {\vartheta_{2} \left( {2b_{1} \left( {l_{1} + u_{1} } \right) + b_{2} \left( {l_{2} + u_{2} } \right)} \right) + 2b_{1} l_{1} \varrho_{2} \left( {1 + u_{1} } \right)} \right. \\ & \quad + b_{2} \vartheta_{1} \left( {l_{2} + u_{2} } \right) + 2\overline{W}_{1} \left( {2\varrho_{1} \left( {l_{1} \varrho_{2} + \vartheta_{2} } \right) + \vartheta_{1} \left( {2l_{1} \varrho_{2} + \vartheta_{2} } \right)} \right) \\ & \quad + 2\overline{W}_{2} \left( {l_{1} \varrho_{2} \left( {\vartheta_{1} - \vartheta_{2} } \right) + \vartheta_{2} \left( {\vartheta_{1} - \vartheta_{2} } \right)} \right) \\ & \quad \left. { + 2\zeta \left( {2\gamma_{1} \left( {l_{1} \varrho_{2} + \vartheta_{2} } \right) - \gamma_{2} \left( {\vartheta_{1} + \vartheta_{2} } \right)} \right)} \right) \\ \end{aligned} $$
(C6)
$$ \begin{aligned} T_{7} & = - l_{2} v_{1}^{2} \left( {2\widetilde{{\overline{a}}}_{2} + l_{2} + u_{2} } \right) \\ & \quad - \upsilon_{1} \upsilon_{2} \left( {2\widetilde{{\overline{a}}}_{1} + l_{1} + u_{1} - 2\overline{W}_{1} \left( {\varrho_{1} + \vartheta_{1} } \right) + 2\overline{W}_{2} \left( {\vartheta_{1} + 2\vartheta_{2} } \right)} \right) \\ & \quad - 2\vartheta_{2} v_{1}^{2} \overline{W}_{1} - 2\overline{W}_{2} \left( {v_{1}^{2} \left( {\varrho_{2} + \vartheta_{2} } \right) + 2v_{2}^{2} \left( {\varrho_{1} + \vartheta_{1} } \right)} \right) \\ & \quad + 2\upsilon_{1} \zeta \left( {\gamma_{2} \upsilon_{1} - \gamma_{1} \upsilon_{2} } \right) \\ & \quad + k\left( {2l_{2} \left( {\varrho_{1} + \vartheta_{1} } \right) + \left( {2\widetilde{{\overline{a}}}_{1} + l_{1} } \right)\left( {\vartheta_{1} + \vartheta_{2} } \right) + \vartheta_{2} u_{1} + 2\varrho_{1} u_{2} } \right. \\ & \quad - \Phi \lambda_{0} \xi T\Psi \left( {2b_{2} \varrho_{1} \left( {l_{2} + u_{2} } \right) + b_{1} \vartheta_{2} \left( {l_{1} + u_{1} } \right)} \right) + 2\overline{W}_{1} \left( {\varrho_{1} \vartheta_{2} - \vartheta_{1}^{2} } \right) \\ & \quad + 2\overline{W}_{2} \left( {2\varrho_{1} \left( {\varrho_{2} + \vartheta_{2} } \right) - \vartheta_{1}^{2} } \right) + 2\zeta \left( {\gamma_{1} \vartheta_{2} - 2\gamma_{2} \varrho_{1} } \right) \\ & \quad + \vartheta_{1} \left( {u_{1} + 2u_{2} - \Phi \lambda_{0} \xi T\Psi \left( {b_{1} \left( {l_{1} + u_{1} } \right) + 2b_{2} \left( {l_{2} + u_{2} } \right)} \right)} \right. \\ & \quad \left. {\left. { + 2\left( { - \varrho_{1} \overline{W}_{1} + 2\varrho_{2} \overline{W}_{2} + \vartheta_{2} \left( {\overline{W}_{1} + \overline{W}_{2} } \right) + \zeta \left( {\gamma_{1} - 2\gamma_{2} } \right)} \right)} \right)} \right) \\ \end{aligned} $$
(C7)
$$ \begin{aligned} T_{8} & = 4\widetilde{{\overline{a}}}_{2} \varrho_{1} \upsilon_{2} + 2l_{2} \varrho_{1} \upsilon_{2} + 4\widetilde{{\overline{a}}}_{2} \vartheta_{1} \upsilon_{2} + \vartheta_{1} \upsilon_{2} \left( {l_{1} + 2l_{2} } \right) + l_{1} \vartheta_{2} \upsilon_{2} \\ & \quad + 2\widetilde{{\overline{a}}}_{1} \left( {\vartheta_{1} + \vartheta_{2} } \right)\upsilon_{2} + \vartheta_{1} u_{1} \upsilon_{2} + \vartheta_{2} u_{1} \upsilon_{2} + 2\varrho_{1} u_{2} \upsilon_{2} + 2\vartheta_{1} u_{2} \upsilon_{2} \\ & \quad + \Phi \lambda_{0} \xi T\Psi \left( {\upsilon_{1} \left( {2b_{1} l_{1} \varrho_{2} + b_{2} l_{2} \vartheta_{1} + \vartheta_{2} \left( {2b_{1} l_{1} + b_{2} l_{2} } \right)} \right.} \right. \\ & \quad \left. { + 2b_{1} u_{1} \left( {\varrho_{2} + \vartheta_{2} } \right) + b_{2} u_{2} \left( {\vartheta_{1} + \vartheta_{2} } \right)} \right) \\ & \quad \left. { - \upsilon_{2} \left( {b_{1} \left( {\vartheta_{1} + \vartheta_{2} } \right)\left( {l_{1} + u_{1} } \right) + 2b_{2} \left( {\varrho_{1} + \vartheta_{1} } \right)\left( {l_{2} + u_{2} } \right)} \right)} \right) \\ & \quad + \overline{W}_{1} \left( {4\varrho_{1} \varrho_{2} \upsilon_{1} + 4\varrho_{2} \vartheta_{1} \upsilon_{1} + 2\varrho_{1} \vartheta_{2} \left( {2\upsilon_{1} + \upsilon_{2} } \right) + 2\vartheta_{1} \vartheta_{2} \left( {\upsilon_{1} + \upsilon_{2} } \right)} \right. \\ & \quad \left. { - 2\vartheta_{2}^{2} \upsilon_{1} - 2\varrho_{1} \vartheta_{1} \upsilon_{2} - 2\vartheta_{1}^{2} \upsilon_{2} } \right) \\ & \quad + \overline{W}_{2} \left( {2\varrho_{2} \vartheta_{2} \upsilon_{1} - 2\varrho_{2} \vartheta_{1} \upsilon_{1} - 2\vartheta_{1} \vartheta_{2} \upsilon_{1} + 2\vartheta_{2}^{2} \upsilon_{1} - 4\varrho_{1} \varrho_{2} \upsilon_{2} } \right. \\ & \quad \left. { - 4\varrho_{2} \vartheta_{1} \upsilon_{2} + 2\vartheta_{1}^{2} \upsilon_{2} - 4\varrho_{1} \vartheta_{2} \upsilon_{2} - 2\vartheta_{1} \vartheta_{2} \upsilon_{2} } \right) \\ & \quad + 2\zeta \left( { - 2\gamma_{1} \left( {\varrho_{2} + \vartheta_{2} } \right)\upsilon_{1} + \gamma_{2} \left( {\vartheta_{1} + \vartheta_{2} } \right)\upsilon_{1} - 2\gamma_{2} \left( {\varrho_{1} + \vartheta_{1} } \right)\upsilon_{2} } \right. \\ & \quad \left. { + \gamma_{1} \left( {\vartheta_{1} + \vartheta_{2} } \right)\upsilon_{2} } \right) \\ \end{aligned} $$
(C8)
$$ \begin{aligned} T_{9} & = b_{1} \Phi \xi \lambda_{0} \Psi^{2} T\left( {l_{1} } \right. \\ & \quad \left. { + u_{1} } \right)\left( {\left( {4\gamma_{1} k\left( {\varrho_{2} + \vartheta_{2} } \right) - 2\gamma_{2} k\left( {\vartheta_{1} + \vartheta_{2} } \right) + 2\gamma_{2} \upsilon_{1} \upsilon_{2} - 2\gamma_{1} \upsilon_{2}^{2} } \right)} \right) \\ & \quad + 4\left( {\left( { - \gamma_{2} k\left( {\varrho_{1} + \vartheta_{1} } \right)\left( {\vartheta_{1} - \vartheta_{2} } \right)} \right.} \right. \\ &\quad + \gamma_{1} k\left( {2\varrho_{2} \vartheta_{1} + \left( {\vartheta_{1} - \vartheta_{2} } \right)\vartheta_{2} + 2\varrho_{1} \left( {\varrho_{2} + \vartheta_{2} } \right)} \right) + \gamma_{1} \upsilon_{2} (\vartheta_{2} \upsilon_{1} \\ & \quad \left. { - \left( {\varrho_{1} + \vartheta_{1} } \right)\upsilon_{2} } \right)\left. { + \gamma_{2} \upsilon_{1} \left( {\left( {\varrho_{1} + \vartheta_{1} } \right)\upsilon_{2} - \vartheta_{2} \upsilon_{1} } \right)} \right)\left( {\overline{C}_{1} - \overline{W}_{1} } \right) \\ & \quad - 4\left( {\left( {2\gamma_{2} k\varrho_{1} \left( {\varrho_{2} + \vartheta_{2} } \right) + \gamma_{1} k\left( {\vartheta_{1} - \vartheta_{2} } \right)\left( {\varrho_{2} + \vartheta_{2} } \right) + \gamma_{2} k\vartheta_{1} } \right)} \right.\left( {2\varrho_{2} } \right. \\ & \quad - \vartheta_{1} + \vartheta_{2} + \gamma_{1} \upsilon_{2} \left( {\left( {\varrho_{2} + \vartheta_{2} } \right)\upsilon_{1} - \vartheta_{1} \upsilon_{2} } \right) \\ & \quad \left. { + \gamma_{2} \upsilon_{1} \left( {\vartheta_{1} \upsilon_{2} - \left( {\varrho_{2} + \vartheta_{2} } \right)\upsilon_{1} } \right)} \right)\left( {\overline{C}_{2} - \overline{W}_{2} } \right) \\ \end{aligned} $$
(C9)
$$ \begin{aligned} T_{10} & = 2\gamma_{2}^{2} \left( {2k\left( {\varrho_{1} + \vartheta_{1} } \right) - \upsilon_{1}^{2} } \right) + \gamma_{1} \gamma_{2} \left( {\upsilon_{1} \upsilon_{2} - k\left( {\vartheta_{1} + \vartheta_{2} } \right)} \right) \\ & + \gamma_{1}^{2} \left( {2k\left( {\varrho_{2} + \vartheta_{2} } \right) - \upsilon_{2}^{2} } \right) \\ & + h\left( {k\left( {\left( {\vartheta_{1} - \vartheta_{2} } \right)^{2} - 4\varrho_{2} \vartheta_{1} - 4\varrho_{1} \left( {\varrho_{2} + \vartheta_{2} } \right)} \right) + 2\left( {\varrho_{2} + \vartheta_{2} } \right)\upsilon_{1}^{2} } \right. \\ & - 2\left( {\vartheta_{1} + \vartheta_{2} } \right)\upsilon_{1} \upsilon_{2} + 2\left( {\varrho_{1} + \vartheta_{1} } \right)\upsilon_{2}^{2} \\ \end{aligned} $$
(C10)
$$ \begin{aligned} T_{11} & = - \left( {\gamma_{2}^{2} k + h\left( {\upsilon_{2}^{2} - 2k\left( {\varrho_{2} + \vartheta_{2} } \right)} \right)} \right)\left( {2\overline{C}_{1}^{c} (\gamma_{1}^{2} k - hk\left( {2\widetilde{{\overline{a}}}_{1} + l_{1} + u_{1} } \right)} \right. \\ & \quad + h\left( {\upsilon_{1}^{2} - k\left( {\varrho_{1} + \vartheta_{1} } \right)} \right) - 2\overline{C}_{2}^{c} \left( {\gamma_{1} \gamma_{2} k - hk\vartheta_{2} + h\upsilon_{1} \upsilon_{2} } \right) \\ & \quad \left. { + b_{1} \Phi hk\lambda_{0} \Psi T\left( {l_{1} + u_{1} } \right)\xi } \right) \\ \end{aligned} $$
(C11)
$$ \begin{aligned} T_{12} & = \left( {2\overline{C}_{1} \left( {\gamma_{1} \gamma_{2} k - hk\vartheta_{1} + h\upsilon_{1} \upsilon_{2} } \right) - 2\overline{C}_{2}^{c} (\gamma_{2}^{2} k + h\left( {\upsilon_{2}^{2} - k\left( {\varrho_{2} + \vartheta_{2} } \right)} \right)} \right) \\ & \quad + hk\left( {2\widetilde{{\overline{a}}}_{2} + \left( {1 - b_{2} \Phi \lambda_{0} \Psi T\xi } \right)\left( {l_{2} + u_{2} } \right)} \right) \\ \end{aligned} $$
(C12)
$$ \begin{aligned} T_{13} & = \gamma_{1}^{2} kl_{2} + 4\overline{C}_{2}^{c} \gamma_{2}^{2} k\varrho_{1} - 2hkl_{2} \varrho_{1} + \overline{C}_{2}^{c} \left( {2\gamma_{1}^{2} \left( {k\varrho_{2} + k\vartheta_{2} - \upsilon_{2}^{2} } \right)} \right. \\ & \quad + 2\gamma_{2}^{2} \left( {k\vartheta_{1} - \upsilon_{1}^{2} } \right) + 2hk\left( {\vartheta_{2}^{2} - \vartheta_{1} \left( {\vartheta_{2} + 2\varrho_{2} } \right) - 2\varrho_{1} \left( {\varrho_{2} + \vartheta_{2} } \right)} \right) \\ & \quad \left. { + 2h\upsilon_{1}^{2} \left( {\varrho_{2} + \vartheta_{2} } \right) + 4h\upsilon_{2}^{2} \left( {\varrho_{1} + \vartheta_{1} } \right) + 4\gamma_{1} \gamma_{2} \left( {\upsilon_{1} \upsilon_{2} - k\vartheta_{2} } \right)} \right) \\ & \quad - hk\left( {2\widetilde{{\overline{a}}}_{1} \left( {\vartheta_{1} + \vartheta_{2} u_{1} \left( {\vartheta_{1} + \vartheta_{2} } \right)} \right) + l_{1} \left( {\vartheta_{1} + \vartheta_{2} } \right) + 2\vartheta_{1} \left( {l_{2} + u_{2} } \right)} \right. \\ & \quad \left. { + 2\varrho_{1} u_{2} } \right) + 2\overline{C}_{1}^{c} \left( {hk\left( {\left( {\varrho_{1} + \vartheta_{1} } \right)\left( {\vartheta_{1} - \vartheta_{2} } \right)} \right) + \vartheta_{2} \left( {\gamma_{1}^{2} k + h\upsilon_{1}^{2} } \right)} \right) \\ & \quad + \gamma_{1}^{2} ku_{2} + h\upsilon_{1}^{2} \left( {l_{2} + u_{2} } \right) + 2\widetilde{{\overline{a}}}_{2} \left( {\gamma_{1}^{2} k + h\left( {\upsilon_{1}^{2} - 2k\left( {\varrho_{1} + \vartheta_{1} } \right)} \right)} \right) \\ & \quad + h\upsilon_{1} \upsilon_{2} \left( {2\widetilde{{\overline{a}}}_{1} + l_{1} - 2\overline{C}_{1}^{c} \varrho_{1} - 2\overline{C}_{1}^{c} \vartheta_{1} - 2\overline{C}_{2}^{c} \vartheta_{1} - 4\overline{C}_{2}^{c} \vartheta_{2} + u_{1} } \right) \\ & \quad - \Phi \lambda_{0} \Psi T\xi \left( {b_{2} \left( {l_{2} + u_{2} } \right)\left( {\gamma_{1}^{2} k + h\left( {\upsilon_{1}^{2} - 2k\left( {\varrho_{1} + \vartheta_{1} } \right)} \right)} \right)} \right. \\ & \quad \left. { + b_{1} \left( {l_{1} + u_{1} } \right)\left( {\gamma_{1} \gamma_{2} k - hk\left( {\vartheta_{1} + \vartheta_{2} } \right) + h\upsilon_{1} \upsilon_{2} } \right)} \right) \\ \end{aligned} $$
(C13)
$$ \begin{aligned} T_{14} & = - 2\widetilde{{\overline{a}}}_{1} \gamma_{2}^{2} \upsilon_{1} - \gamma_{2}^{2} l_{1} \upsilon_{1} - \gamma_{1} \gamma_{2} l_{2} \upsilon_{1} + 2\overline{C}_{1}^{c} \gamma_{2}^{2} \varrho_{1} \upsilon_{1} + 2\overline{C}_{2}^{c} \gamma_{1} \gamma_{2} \varrho_{2} \upsilon_{1} + 4\widetilde{{\overline{a}}}_{1} h\varrho_{2} \upsilon_{1} \\ & \quad + 2hl_{1} \varrho_{2} \upsilon_{1} - 4\overline{C}_{1}^{c} h\varrho_{1} \varrho_{2} \upsilon_{1} + 2\overline{C}_{1}^{c} \gamma_{2}^{2} \vartheta_{1} \upsilon_{1} - 2\overline{C}_{2}^{c} \gamma_{2}^{2} \vartheta_{1} \upsilon_{1} + hl_{2} \vartheta_{1} \upsilon_{1} \\ & \quad - 4\overline{C}_{1}^{c} h\varrho_{2} \vartheta_{1} \upsilon_{1} + 2\overline{C}_{2}^{c} h\varrho_{2} \vartheta_{1} \upsilon_{1} - 2\overline{C}_{1}^{c} \gamma_{1} \gamma_{2} \vartheta_{2} \upsilon_{1} + 2\overline{C}_{2}^{c} \gamma_{1} \gamma_{2} \vartheta_{2} \upsilon_{1} \\ & \quad + 4\widetilde{{\overline{a}}}_{1} h\vartheta_{2} \upsilon_{1} + 2hl_{1} \vartheta_{2} \upsilon_{1} + hl_{2} \vartheta_{2} \upsilon_{1} - 4\overline{C}_{1}^{c} h\varrho_{1} \vartheta_{2} \upsilon_{1} - 2\overline{C}_{2}^{c} h\varrho_{2} \vartheta_{2} \upsilon_{1} \\ & \quad - 2\overline{C}_{1}^{c} h\vartheta_{1} \vartheta_{2} \upsilon_{1} + 2\overline{C}_{2}^{c} h\vartheta_{1} \vartheta_{2} \upsilon_{1} + 2\overline{C}_{1}^{c} h\vartheta_{2}^{2} \upsilon_{1} - 2\overline{C}_{2}^{c} h\vartheta_{2}^{2} \upsilon_{1} - \gamma_{2}^{2} u_{1} \upsilon_{1} \\ & \quad + 2h\varrho_{2} u_{1} \upsilon_{1} + 2h\vartheta_{2} u_{1} \upsilon_{1} - \gamma_{1} \gamma_{2} u_{2} \upsilon_{1} + h\vartheta_{1} u_{2} \upsilon_{1} + h\vartheta_{2} u_{2} \upsilon_{1} \\ & \quad + 2\widetilde{{\overline{a}}}_{1} \gamma_{1} \gamma_{2} \upsilon_{2} + \gamma_{1} \gamma_{2} l_{1} \upsilon_{2} + \gamma_{1}^{2} l_{2} \upsilon_{2} - 2\overline{C}_{1}^{c} \gamma_{1} \gamma_{2} \varrho_{1} \upsilon_{2} - 2hl_{2} \varrho_{1} \upsilon_{2} \\ & \quad - 2\overline{C}_{2}^{c} \gamma_{1}^{2} \varrho_{2} \upsilon_{2} + 4\overline{C}_{2}^{c} h\varrho_{1} \varrho_{2} \upsilon_{2} - 2\overline{C}_{1}^{c} \gamma_{1} \gamma_{2} \vartheta_{1} \upsilon_{2} + 2\overline{C}_{2}^{c} \gamma_{1} \gamma_{2} \vartheta_{1} \upsilon_{2} \\ & \quad - 2\widetilde{{\overline{a}}}_{1} h\vartheta_{1} \upsilon_{2} - hl_{1} \vartheta_{1} \upsilon_{2} - 2hl_{2} \vartheta_{1} \upsilon_{2} + 2\overline{C}_{1}^{c} h\varrho_{1} \vartheta_{1} \upsilon_{2} + 4\overline{C}_{2}^{c} h\varrho_{2} \vartheta_{1} \upsilon_{2} \\ & \quad + 2\overline{C}_{1}^{c} h\vartheta_{1}^{2} \upsilon_{2} - 2\overline{C}_{2}^{c} h\vartheta_{1}^{2} \upsilon_{2} + 2\overline{C}_{1}^{c} \gamma_{1}^{2} \vartheta_{2} \upsilon_{2} - 2\overline{C}_{2}^{c} \gamma_{1}^{2} \vartheta_{2} \upsilon_{2} \\ & \quad - 2\widetilde{{\overline{a}}}_{1} h\vartheta_{2} \upsilon_{2} - hl_{1} \vartheta_{2} \upsilon_{2} - 2\overline{C}_{1}^{c} h\varrho_{1} \vartheta_{2} \upsilon_{2} + 4\overline{C}_{2}^{c} h\varrho_{1} \vartheta_{2} \upsilon_{2} \\ & \quad - 2\overline{C}_{1}^{c} h\vartheta_{1} \vartheta_{2} \upsilon_{2} + 2\overline{C}_{2}^{c} h\vartheta_{1} \vartheta_{2} \upsilon_{2} + \gamma_{1} \gamma_{2} u_{1} \upsilon_{2} - h\vartheta_{1} u_{1} \upsilon_{2} - h\vartheta_{2} u_{1} \upsilon_{2} \\ & \quad + \gamma_{1}^{2} u_{2} \upsilon_{2} - 2h\varrho_{1} u_{2} \upsilon_{2} - 2h\vartheta_{1} u_{2} \upsilon_{2} \\ & \quad + \Phi \lambda_{0} \Psi T\xi \left( { - b_{2} \left( {l_{2} + u_{2} } \right)\left( { - \gamma_{1} \gamma_{2} \upsilon_{1} + h\left( {\vartheta_{1} + \vartheta_{2} } \right)\upsilon_{1} + \gamma_{1}^{2} \upsilon_{2} } \right.} \right. \\ & \quad \left. { - 2h\left( {\varrho_{1} + \vartheta_{1} } \right)\upsilon_{2} } \right) \\ & \quad \left. { + b_{1} \left( {l_{1} + u_{1} } \right)\left( {\gamma_{2}^{2} \upsilon_{1} - 2h\left( {\varrho_{2} + \vartheta_{2} } \right)\upsilon_{1} - \gamma_{1} \gamma_{2} \upsilon_{2} + h\left( {\vartheta_{1} + \vartheta_{2} } \right)\upsilon_{2} } \right)} \right) \\ \end{aligned} $$
(C14)
$$ \begin{aligned} T_{15} & = 2\gamma_{2} kl_{2} \varrho_{1} - 4\widetilde{{\overline{a}}}_{1} \gamma_{1} k\varrho_{2} - 2\gamma_{1} kl_{1} \varrho_{2} + 4\overline{C}_{1}^{c} \gamma_{1} k\varrho_{1} \varrho_{2} - 4\overline{C}_{2}^{c} \gamma_{2} k\varrho_{1} \varrho_{2} \\ & \quad + 2\widetilde{{\overline{a}}}_{1} \gamma_{2} k\vartheta_{1} + \gamma_{2} kl_{1} \vartheta_{1} - \gamma_{1} kl_{2} \vartheta_{1} + 2\gamma_{2} kl_{2} \vartheta_{1} - 2\overline{C}_{1}^{c} \gamma_{2} k\varrho_{1} \vartheta_{1} \\ & \quad + 4\overline{C}_{1}^{c} \gamma_{1} k\varrho_{2} \vartheta_{1} - 2\overline{C}_{2}^{c} \gamma_{1} k\varrho_{2} \vartheta_{1} - 4\overline{C}_{2}^{c} \gamma_{2} k\varrho_{2} \vartheta_{1} - 2\overline{C}_{1}^{c} \gamma_{2} k\vartheta_{1}^{2} \\ & \quad + 2\overline{C}_{2}^{c} \gamma_{2} k\vartheta_{1}^{2} - 4\widetilde{{\overline{a}}}_{1} \gamma_{1} k\vartheta_{2} + 2\widetilde{{\overline{a}}}_{1} \gamma_{2} k\vartheta_{2} - 2\gamma_{1} kl_{1} \vartheta_{2} + \gamma_{2} kl_{1} \vartheta_{2} \\ & \quad - \gamma_{1} kl_{2} \vartheta_{2} + 4\overline{C}_{1}^{c} \gamma_{1} k\varrho_{1} \vartheta_{2} + 2\overline{C}_{1}^{c} \gamma_{2} k\varrho_{1} \vartheta_{2} - 4\overline{C}_{2}^{c} \gamma_{2} k\varrho_{1} \vartheta_{2} \\ & \quad + 2\overline{C}_{2}^{c} \gamma_{1} k\varrho_{2} \vartheta_{2} + 2\overline{C}_{1}^{c} \gamma_{1} k\vartheta_{1} \vartheta_{2} - 2\overline{C}_{2}^{c} \gamma_{1} k\vartheta_{1} \vartheta_{2} + 2\overline{C}_{1}^{c} \gamma_{2} k\vartheta_{1} \vartheta_{2} \\ & \quad - 2\overline{C}_{2}^{c} \gamma_{2} k\vartheta_{1} \vartheta_{2} - 2\overline{C}_{1}^{c} \gamma_{1} k\vartheta_{2}^{2} + 2\overline{C}_{2}^{c} \gamma_{1} k\vartheta_{2}^{2} - 2\gamma_{1} k\varrho_{2} u_{1} + \gamma_{2} k\vartheta_{1} u_{1} \\ & \quad - 2\gamma_{1} k\vartheta_{2} u_{1} + \gamma_{2} k\vartheta_{2} u_{1} + 2\gamma_{2} k\varrho_{1} u_{2} - \gamma_{1} k\vartheta_{1} u_{2} + 2\gamma_{2} k\vartheta_{1} u_{2} \\ & \quad - \gamma_{1} k\vartheta_{2} u_{2} - \gamma_{2} l_{2} \upsilon_{1}^{2} + 2\overline{C}_{2}^{c} \gamma_{2} \varrho_{2} \upsilon_{1}^{2} - 2\overline{C}_{1}^{c} \gamma_{2} \vartheta_{2} \upsilon_{1}^{2} + 2\overline{C}_{2}^{c} \gamma_{2} \vartheta_{2} \upsilon_{1}^{2} \\ & \quad - \gamma_{2} u_{2} \upsilon_{1}^{2} - 2\widetilde{{\overline{a}}}_{1} \gamma_{2} \upsilon_{1} \upsilon_{2} - \gamma_{2} l_{1} \upsilon_{1} \upsilon_{2} + \gamma_{1} l_{2} \upsilon_{1} \upsilon_{2} + 2\overline{C}_{1}^{c} \gamma_{2} \varrho_{1} \upsilon_{1} \upsilon_{2} \\ & \quad - 2\overline{C}_{2}^{c} \gamma_{1} \varrho_{2} \upsilon_{1} \upsilon_{2} + 2\overline{C}_{1}^{c} \gamma_{2} \vartheta_{1} \upsilon_{1} \upsilon_{2} - 2\overline{C}_{2}^{c} \gamma_{2} \vartheta_{1} \upsilon_{1} \upsilon_{2} + 2\overline{C}_{1}^{c} \gamma_{1} \vartheta_{2} \upsilon_{1} \upsilon_{2} \\ & \quad - 2\overline{C}_{2}^{c} \gamma_{1} \vartheta_{2} \upsilon_{1} \upsilon_{2} - \gamma_{2} u_{1} \upsilon_{1} \upsilon_{2} + \gamma_{1} u_{2} \upsilon_{1} \upsilon_{2} + 2\widetilde{{\overline{a}}}_{1} \gamma_{1} \upsilon_{2}^{2} + \gamma_{1} l_{1} \upsilon_{2}^{2} \\ & \quad - 2\overline{C}_{1}^{c} \gamma_{1} \varrho_{1} \upsilon_{2}^{2} - 2\overline{C}_{1}^{c} \gamma_{1} \vartheta_{1} \upsilon_{2}^{2} + 2\overline{C}_{2}^{c} \gamma_{1} \vartheta_{1} \upsilon_{2}^{2} + \gamma_{1} u_{1} \upsilon_{2}^{2} \\ & \quad + \Phi \lambda_{0} \Psi T\xi \left( { - b_{2} \left( {l_{2} + u_{2} } \right)\left( {2\gamma_{2} k\left( {\varrho_{1} + \vartheta_{1} } \right) - \gamma_{1} k\left( {\vartheta_{1} + \vartheta_{2} } \right)} \right.} \right. \\ & \quad \left. { - \gamma_{2} \upsilon_{1}^{2} + \gamma_{1} \upsilon_{1} \upsilon_{2} } \right) \\ & \quad \left. { + b_{1} \left( {l_{1} + u_{1} } \right)\left( {2\gamma_{1} k\left( {\varrho_{2} + \vartheta_{2} } \right) - \gamma_{2} k\left( {\vartheta_{1} + \vartheta_{2} } \right) + \gamma_{2} \upsilon_{1} \upsilon_{2} - \gamma_{1} \upsilon_{2}^{2} } \right)} \right) \\ \end{aligned} $$
(C15)
$$ \begin{aligned} T_{16} & = \Phi \xi \lambda_{0} \Psi T\left( {b_{2} l_{2} \upsilon_{1} \upsilon_{2} \left( {l_{2} + u_{2} } \right) + \upsilon_{2}^{2} \left( {l_{1} + u_{1} } \right)} \right) \\ & \quad + 2\overline{W}_{1} \left( {\upsilon_{1} \upsilon_{2} \left( {\vartheta_{2} + 2\vartheta_{1} } \right) - 2\upsilon_{1}^{2} \left( {\varrho_{2} + \vartheta_{2} } \right) - \upsilon_{2}^{2} \left( {\varrho_{1} + \vartheta_{1} } \right)} \right) \\ & \quad + 2\upsilon_{2} \overline{W}_{2} \left( {\upsilon_{1} \left( {\varrho_{2} + \vartheta_{2} } \right) - \vartheta_{1} \upsilon_{2} } \right) + 2\upsilon_{2} \overline{\zeta }^{*} \left( {\gamma_{2} \upsilon_{1} - \gamma_{1} \upsilon_{2} } \right) \\ & \quad + k\left( {\vartheta_{2} \left( {l_{2} + u_{2} } \right) + 2\left( {\varrho_{2} + \vartheta_{2} } \right)\left( {l_{1} + u_{1} } \right) + 2\widetilde{{\overline{a}}}_{2} \left( {\vartheta_{1} + \vartheta_{2} } \right)} \right. \\ & \quad + \vartheta_{1} \left( {l_{2} + u_{2} } \right) \\ & \quad - \Phi \xi \lambda_{0} \Psi T\left( {2b_{1} \left( {\varrho_{2} + \vartheta_{2} } \right)\left( {l_{1} + u_{1} } \right) + b_{2} \left( {\vartheta_{1} + \vartheta_{2} } \right)\left( {l_{2} + u_{2} } \right)} \right) \\ & \quad + 2\overline{W}_{1} \left( {2\varrho_{1} \left( {\varrho_{2} + \vartheta_{2} } \right) + \vartheta_{1} \left( {2\varrho_{2} - \vartheta_{1} + \vartheta_{2} } \right)} \right) \\ & \quad \left. { + 2\overline{W}_{2} \left( {\left( {\vartheta_{1} - \vartheta_{2} } \right)\left( {\varrho_{2} + \vartheta_{1} } \right)} \right) + 2\overline{\zeta }^{*} \left( {2\left( {\varrho_{2} + \vartheta_{2} } \right) - \gamma_{2} \left( {\vartheta_{1} + \vartheta_{2} } \right)} \right)} \right) \\ \end{aligned} $$
(C16)
$$ \begin{aligned} T_{17} = & \;\Phi \xi \lambda_{0} \Psi T\left( {b_{2} v_{1}^{2} \left( {l_{2} + u_{2} } \right) + b_{1} \upsilon_{1} \upsilon_{2} \left( {l_{1} + u_{1} } \right)} \right) + 2\overline{W}_{1} \left( {\upsilon_{1} \upsilon_{2} \left( {\varrho_{1} + \vartheta_{1} } \right) - \vartheta_{2} v_{1}^{2} } \right) \\ & + \;\overline{W}_{2} \left( {2\upsilon_{1} \upsilon_{2} \left( {\vartheta_{1} + 2\vartheta_{2} } \right) - 2v_{1}^{2} \left( {\varrho_{2} + \vartheta_{2} } \right) - 4v_{2}^{2} \left( {\varrho_{1} + \vartheta_{1} } \right)} \right) + 2\upsilon_{1} \overline{\zeta }^{*} \left( {\gamma_{2} \upsilon_{1} - \gamma_{1} \upsilon_{2} } \right) \\ & + \;k\left( {2l_{2} \left( {\varrho_{1} + \vartheta_{1} } \right) + \left( {2\widetilde{{\overline{a}}}_{1} + l_{1} } \right)\left( {\vartheta_{1} + \vartheta_{2} } \right) + \vartheta_{2} u_{1} + 2\varrho_{1} u_{2} - \Phi \xi \lambda_{0} \Psi T\left( {2b_{2} \varrho_{1} \left( {l_{2} + u_{2} } \right)} \right.} \right. \\ & \left. { + \;b_{1} \vartheta_{2} \left( {l_{1} + u_{1} } \right)} \right) + 2\overline{W}_{1} \left( {\varrho_{1} \vartheta_{2} - \vartheta_{1}^{2} } \right) + 2\overline{W}_{2} \left( {2\varrho_{1} \left( {\varrho_{2} + \vartheta_{2} } \right) - \vartheta_{2}^{2} } \right) + 2\overline{\zeta }^{*} \left( {\gamma_{1} \vartheta_{2} - 2\gamma_{2} \varrho_{1} } \right) \\ & + \;\vartheta_{1} \left( {u_{1} + 2u_{2} - \Phi \xi \lambda_{0} \Psi T\left( {b_{1} \left( {l_{1} + u_{1} } \right) + 2b_{2} \left( {l_{2} + u_{2} } \right)} \right) + 2\left( {2\varrho_{2} \overline{W}_{2} - \varrho_{1} \overline{W}_{1} } \right.} \right. \\ & \left. {\left. {\left. { + \;\vartheta_{2} \left( {\overline{W}_{1} + \overline{W}_{2} } \right) + \overline{\zeta }^{*} \left( {\gamma_{1} - 2\gamma_{2} } \right)} \right)} \right)} \right) \\ \end{aligned} $$
(C17)
$$ \begin{aligned} T_{18} = & \; - \upsilon_{1} \left( {2u_{1} \left( {\varrho_{2} + \vartheta_{2} } \right) + u_{2} \left( {\vartheta_{1} + \vartheta_{2} } \right)} \right) + \upsilon_{2} \left( {2\left( {\varrho_{1} + \vartheta_{1} } \right)\left( {l_{2} + 2\widetilde{{\overline{a}}}_{2} } \right) + l_{1} \left( {\vartheta_{1} + \vartheta_{2} } \right)} \right. \\ & \left. { + \;2\widetilde{{\overline{a}}}_{1} \left( {\vartheta_{1} + \vartheta_{2} } \right) + u_{1} \left( {\vartheta_{1} + \vartheta_{2} } \right) + 2u_{2} \left( {\varrho_{1} + \vartheta_{1} } \right)} \right) + \Phi \xi \lambda_{0} \Psi T\left( {\upsilon_{1} } \right.\left( {2b_{1} \left( {\varrho_{2} + \vartheta_{2} } \right)\left( {l_{1} + u_{1} } \right)} \right. \\ & \left. {\left. { + \;b_{2} \left( {\vartheta_{1} + \vartheta_{2} } \right)\left( {l_{2} + u_{2} } \right)} \right) - \upsilon_{2} \left( {2b_{2} \left( {\varrho_{1} + \vartheta_{1} } \right)\left( {l_{2} + u_{2} } \right) + b_{1} \left( {\vartheta_{1} + \vartheta_{2} } \right)\left( {l_{1} + u_{1} } \right)} \right)} \right) \\ & + \;\overline{W}_{1} \left( {\upsilon_{1} \left( {4\varrho_{2} \left( {\varrho_{1} + \vartheta_{1} } \right) + 2\vartheta_{2} \left( {2\varrho_{1} + \vartheta_{1} - \vartheta_{2} } \right)} \right) + \upsilon_{2} \left( {2\vartheta_{2} \left( {\varrho_{1} + \vartheta_{1} } \right) - 2\vartheta_{1} \left( {\varrho_{1} - 2\vartheta_{1} } \right)} \right)} \right) \\ & + \;\overline{W}_{2} \left( {\upsilon_{1} \left( {2\varrho_{2} \left( {\vartheta_{2} - \vartheta_{1} } \right) - 2\vartheta_{2} \left( {\vartheta_{1} + 2\vartheta_{2} } \right)} \right) - \upsilon_{2} \left( {4\varrho_{1} \left( {\varrho_{2} + \vartheta_{2} } \right) + 2\vartheta_{1} \left( {2\varrho_{2} - \vartheta_{1} + \vartheta_{2} } \right)} \right)} \right) \\ & + \;2\overline{\zeta }^{*} \left( { - 2\gamma_{1} \upsilon_{1} \left( {\varrho_{2} + \vartheta_{2} } \right) + \left( {\vartheta_{1} + \vartheta_{2} } \right)\left( {\gamma_{2} \upsilon_{1} + \gamma_{1} \upsilon_{2} } \right) - 2\gamma_{2} \upsilon_{2} \left( {\varrho_{1} + \vartheta_{1} } \right)} \right) \\ \end{aligned} $$
(C18)
$$ \begin{aligned} T_{19} = & \;\left( {2\overline{C}_{2} \left( t \right)k\left( {\varrho_{2} + \vartheta_{2} } \right)\left( {\gamma_{1} \vartheta_{2} - 2\gamma_{2} \left( {\varrho_{1} + \vartheta_{1} } \right)} \right) - 2b_{2} \Phi \gamma_{2} kl_{2} \lambda_{0} \varrho_{1} \Psi^{2} T\xi } \right. \\ & + \;2b_{1} \Phi \gamma_{1} kl_{1} \lambda_{0} \varrho_{2} \Psi^{2} T\xi - b_{1} \Phi \gamma_{2} kl_{1} \lambda_{0} \Psi^{2} T\vartheta_{1} \xi - 2b_{2} \Phi \gamma_{2} kl_{2} \lambda_{0} \Psi^{2} T\vartheta_{1} \xi \\ & + \;2b_{1} \Phi \gamma_{1} kl_{1} \lambda_{0} \Psi^{2} T\vartheta_{2} \xi + b_{2} \Phi \gamma_{1} kl_{2} \lambda_{0} \Psi^{2} T\vartheta_{2} \xi + 2b_{1} \Phi \gamma_{1} k\lambda_{0} \varrho_{2} \Psi^{2} Tu_{1} \xi \\ & - \;b_{1} \Phi \gamma_{2} k\lambda_{0} \Psi^{2} T\vartheta_{1} u_{1} \xi + 2b_{1} \Phi \gamma_{1} k\lambda_{0} \Psi^{2} T\vartheta_{2} u_{1} \xi - 2b_{2} \Phi \gamma_{2} k\lambda_{0} \varrho_{1} \Psi^{2} Tu_{2} \xi \\ & - \;2b_{2} \Phi \gamma_{2} k\lambda_{0} \Psi^{2} T\vartheta_{1} u_{2} \xi + b_{2} \Phi \gamma_{1} k\lambda_{0} \Psi^{2} T\vartheta_{2} u_{2} \xi + b_{1} \Phi \gamma_{2} l_{1} \lambda_{0} \Psi^{2} T\upsilon_{1} \upsilon_{2} \xi \\ & + \;b_{1} \Phi \gamma_{2} \lambda_{0} \Psi^{2} Tu_{1} \upsilon_{1} \upsilon_{2} \xi - b_{1} \Phi \gamma_{1} l_{1} \lambda_{0} \Psi^{2} T\upsilon_{2}^{2} \xi - b_{1} \Phi \gamma_{1} \lambda_{0} \Psi^{2} Tu_{1} \upsilon_{2}^{2} \xi \\ & - \;4\gamma_{1} k{\Omega }\varrho_{1} \varrho_{2} \overline{W}_{1} \left( t \right) + 2\gamma_{2} k{\Omega }\varrho_{1} \vartheta_{1} \overline{W}_{1} \left( t \right) - 4\gamma_{1} k{\Omega }\varrho_{2} \vartheta_{1} \overline{W}_{1} \left( t \right) + 2\gamma_{2} k{\Omega }\vartheta_{1}^{2} \overline{W}_{1} \left( t \right) \\ & - \;4\gamma_{1} k{\Omega }\varrho_{1} \vartheta_{2} \overline{W}_{1} \left( t \right) - 4\gamma_{1} k{\Omega }\vartheta_{1} \vartheta_{2} \overline{W}_{1} \left( t \right) - 2\gamma_{2} {\Omega }\varrho_{1} \upsilon_{1} \upsilon_{2} \overline{W}_{1} \left( t \right) - 2\gamma_{2} {\Omega }\vartheta_{1} \upsilon_{1} \upsilon_{2} \overline{W}_{1} \left( t \right) \\ & \left. { + \;2\gamma_{1} {\Omega }\varrho_{1} \upsilon_{2}^{2} \overline{W}_{1} \left( t \right) + 2\gamma_{1} {\Omega }\vartheta_{1} \upsilon_{2}^{2} \overline{W}_{1} \left( t \right) + 2k\left( {\varrho_{2} + \vartheta_{2} } \right)\left( {2\gamma_{2} \left( {\varrho_{1} + \vartheta_{1} } \right) - \gamma_{1} \vartheta_{2} } \right)\overline{W}_{2} \left( t \right)} \right) \\ \end{aligned} $$
(C19)
Appendix D: Proof of Theorem 3.2
By determining the second-order derivative of the profit function, it is obvious that a positive \(h\) guarantees the concavity condition \(\left( { - h < 0} \right)\).□
Appendix E: Proof of Proposition 2
The Jacobian matrix is introduced to find the system’s stable region for supply chain members to adjust their decisions. The Jacobian matrix of function is:
$$ \left( {\begin{array}{*{20}c} {\overline{{J_{11} }} } & {\overline{{J_{12} }} } & {\overline{{J_{13} }} } & {\overline{{J_{14} }} } \\ {\overline{{J_{21} }} } & {\overline{{J_{22} }} } & {\overline{{J_{23} }} } & {\overline{{J_{24} }} } \\ {\overline{{J_{31} }} } & {\overline{{J_{32} }} } & {\overline{{J_{33} }} } & 0 \\ 0 & 0 & 0 & {\overline{{J_{44} }} } \\ \end{array} } \right) $$
The corresponding characteristic equation is as Eq. (E1):
$$ F\left( \lambda \right) = \eta_{0} \lambda^{4} + \eta_{1} \lambda^{3} + \eta_{2} \lambda^{2} + \eta_{3} \lambda^{1} + \eta_{4} \lambda^{0} $$
(E1)
where:
$$ \left\{ {\begin{array}{*{20}l} {\eta_{0} = 1} \hfill \\ {\eta_{1} = - \left( {\overline{{J_{11} }} + \overline{{J_{22} }} + \overline{{J_{33} }} - \overline{{J_{44} }} } \right)} \hfill \\ {\eta_{2} = { }\overline{{J_{11} }} { }\overline{{J_{22} }} - \overline{{J_{12} }} { }\overline{{J_{21} }} - \overline{{J_{13} }} { }\overline{{J_{31} }} - \overline{{J_{23} }} { }\overline{{J_{32} }} + { }\overline{{J_{33} }} \left( {\overline{{J_{11} }} + \overline{{J_{22} }} + \overline{{J_{44} }} } \right) + \overline{{J_{44} }} \left( {\overline{{J_{22} }} + \overline{{J_{11} }} } \right)} \hfill \\ {\eta_{3} = \overline{{J_{44} }} \left( {\overline{{J_{12} }} { }\overline{{J_{21} }} - \overline{{J_{11} }} { }\overline{{J_{22} }} + \overline{{J_{13} }} { }\overline{{J_{31} }} + \overline{{J_{23} }} { }\overline{{J_{32} }} - \overline{{J_{33} }} \left( {\overline{{J_{11} }} + \overline{{J_{22} }} { }} \right)} \right)} \hfill \\ { + { }\overline{{J_{31} }} \left( {\overline{{J_{13} }} { }\overline{{J_{22} }} - \overline{{J_{12} }} { }\overline{{J_{23} }} { }} \right) + { }\overline{{J_{32} }} \left( {\overline{{J_{11} }} { }\overline{{J_{23} }} - \overline{{J_{13} }} { }\overline{{J_{21} }} } \right) + { }\overline{{J_{33} }} \left( {\overline{{J_{12} }} { }\overline{{J_{21} }} - \overline{{J_{11} }} { }\overline{{J_{22} }} { }} \right)} \hfill \\ {\eta_{4} = { }\overline{{J_{44} }} \left( {\overline{{J_{31} }} \left( {\overline{{J_{12} }} { }\overline{{J_{23} }} - \overline{{J_{13} }} { }\overline{{J_{22} }} } \right) + \overline{{J_{32} }} \left( {\overline{{J_{13} }} { }\overline{{J_{21} }} - \overline{{J_{11} }} { }\overline{{J_{23} }} { }} \right) + \overline{{J_{33} }} \left( {\overline{{J_{11} }} { }\overline{{J_{22} }} - \overline{{J_{12} }} { }\overline{{J_{21} }} } \right)} \right)} \hfill \\ \end{array} } \right. $$
(E2)
According to the Jury test condition, the sufficient conditions for stability of \(F\left(\lambda \right)\) are as Eq. (E3):
$$ \left\{ {\begin{array}{*{20}l} {\left| {\eta_{4} } \right| < \eta_{0} } \hfill \\ {\left. {F\left( \lambda \right)} \right|_{\lambda = 1} > 0} \hfill \\ {\left. {F\left( \lambda \right)} \right|_{\lambda = - 1} > 0} \hfill \\ {\left| {b_{3} } \right| > \left| {b_{0} } \right|} \hfill \\ {\left| {c_{2} } \right| > \left| {c_{0} } \right|} \hfill \\ \end{array} } \right. $$
(E3)
By solving the above conditions, it can be considered locally stable of the model. In theory, the conditions are very complex, and the processes of solving the corresponding characteristic equation is a very complicated task.
The values of Jacobian matrix for the Stochasic-Dynamic Disruption Model:
$$ \begin{aligned} \overline{{J_{11} }} & = 1 - 2\omega_{1} \overline{P}_{1} \left( t \right)\left( {\varrho_{1} + \vartheta_{1} } \right) \\ & \quad + \frac{1}{2}\omega_{1} \left( {2\widetilde{{\overline{a}}}_{1} + l_{1} - 4\overline{P}_{1} \left( t \right)\left( {\varrho_{1} + \vartheta_{1} } \right) + 2\overline{P}_{2} \left( t \right)\left( {\vartheta_{1} + \vartheta_{2} } \right) + u_{1} } \right. \\ & \quad - 2\overline{\sigma }\left( t \right)v_{1} - b_{1} \xi \lambda_{0} \Psi T\Phi \left( {l_{1} + u_{1} } \right) + 2\left( {\varrho_{1} + \vartheta_{1} } \right)\overline{W}_{1} \left( t \right) \\ & \quad \left. { - 2\vartheta_{2} \overline{W}_{2} \left( t \right) + 2\gamma_{1} \overline{\zeta }\left( t \right)} \right) \\ \end{aligned} $$
(E4)
$$ \overline{{J_{12} }} = \omega_{1} \overline{P}_{1} \left( t \right)\left( {\vartheta_{1} + \vartheta_{2} } \right) $$
(E5)
$$ \overline{{J_{13} }} = - \omega_{1} \upsilon_{1} \overline{P}_{1} \left( t \right) $$
(E6)
$$ \overline{{J_{14} }} = \gamma_{1} \omega_{1} \overline{P}_{1} \left( t \right) $$
(E7)
$$ \overline{{J_{21} }} = \omega_{2} \overline{P}_{2} \left( t \right)\left( {\vartheta_{1} + \vartheta_{2} } \right) $$
(E8)
$$ \begin{aligned} \overline{{J_{22} }} & = 1 - 2\omega_{2} \overline{P}_{2} \left( t \right)\left( {\varrho_{2} + \vartheta_{2} } \right) \\ & \quad + \frac{1}{2}\omega_{2} \left( {2\widetilde{{\overline{a}}}_{2} + l_{2} - 4\overline{P}_{2} \left( t \right)\left( {\varrho_{2} + \vartheta_{2} } \right) + 2\overline{P}_{1} \left( t \right)\left( {\vartheta_{1} + \vartheta_{2} } \right)} \right. + u_{2} \\ & \quad + 2\overline{\sigma }\left( t \right)v_{2} - b_{2} \xi \lambda_{0} \Psi T\Phi \left( {l_{2} + u_{2} } \right) - 2\vartheta_{1} \overline{W}_{1} \left( t \right) \\ & \quad \left. { + 2\left( {\varrho_{2} + \vartheta_{2} } \right)\overline{W}_{2} \left( t \right) - 2\gamma_{2} \overline{\zeta }\left( t \right)} \right) \\ \end{aligned} $$
(E9)
$$ \overline{{J_{23} }} = \omega_{2} \upsilon_{2} \overline{P}_{2} \left( t \right) $$
(E10)
$$ \overline{{J_{24} }} = - \gamma_{2} \omega_{2} \overline{P}_{2} \left( t \right) $$
(E11)
$$ \overline{{J_{31} }} = - \omega_{3} \upsilon_{1} \overline{\sigma }\left( t \right) $$
$$ \overline{{J_{32} }} = \omega_{3} \upsilon_{2} \overline{\sigma }\left( t \right) $$
(E12)
$$ \overline{{J_{33} }} = 1 - k\omega_{3} \overline{\sigma }\left( t \right) + \omega_{3} \left( { - k\overline{\sigma }\left( t \right) - v_{1} \left( {\overline{P}_{1} \left( t \right) - \overline{W}_{1} \left( t \right)} \right) + v_{2} \left( {\overline{P}_{2} \left( t \right) - \overline{W}_{2} \left( t \right)} \right)} \right) $$
(E13)
$$ \overline{{J_{44} }} = 1 - h\omega_{4} \overline{\zeta }\left( t \right) + \omega_{4} \left( {\gamma_{1} \left( {\overline{W}_{1} \left( t \right) - \overline{C}_{1} \left( t \right)} \right) - \gamma_{2} \left( {\overline{W}_{2} \left( t \right) - \overline{C}_{2} \left( t \right)} \right) - h\overline{\zeta }\left( t \right)} \right) $$
(E14)
Appendix F: Proof of Proposition 3
The Jacobian matrix is introduced to find the system’s stable region for supply chain members to adjust their decisions. The Jacobian matrix of function is:
$$ \left( {\begin{array}{*{20}c} {\overline{{J_{11} }}^{c} } & {\overline{{J_{12} }}^{c} } & {\overline{{J_{13} }}^{c} } & {\overline{{J_{14} }}^{c} } \\ {\overline{{J_{21} }}^{c} } & {\overline{{J_{22} }}^{c} } & {\overline{{J_{23} }}^{c} } & {\overline{{J_{24} }}^{c} } \\ {\overline{{J_{31} }}^{c} } & {\overline{{J_{32} }}^{c} } & {\overline{{J_{33} }}^{c} } & 0 \\ {\overline{{J_{41} }}^{c} } & {\overline{{J_{42} }}^{c} } & 0 & {\overline{{J_{44} }}^{c} } \\ \end{array} } \right) $$
where the \(\overline{{J_{ij} }}^{c}\) is provided in Appendix D. The corresponding characteristic equation is as Eq. (F1):
$$ F\left( \lambda \right) = \eta_{0} \lambda^{4} + \eta_{1} \lambda^{3} + \eta_{2} \lambda^{2} + \eta_{3} \lambda^{1} + \eta_{4} \lambda^{0} $$
(F1)
where:
$$ \left\{ {\begin{array}{*{20}l} {\eta_{0} = 1} \hfill \\ {\eta_{1} = - \left( {\overline{{J_{11} }}^{c} + \overline{{J_{22} }}^{c} + \overline{{J_{33} }}^{c} + \overline{{J_{44} }}^{c} } \right)} \hfill \\ {\eta_{2} = \overline{{J_{44} }}^{c} \left( {\overline{{J_{11} }}^{c} + \overline{{J_{22} }}^{c} + \overline{{J_{33} }}^{c} } \right) + \overline{{J_{22} }}^{c} \left( {\overline{{J_{11} }}^{c} + \overline{{J_{33} }}^{c} } \right) - \overline{{J_{12} }}^{c} \overline{{J_{21} }}^{c} - \overline{{J_{13} }}^{c} \overline{{J_{31} }}^{c} } \hfill \\ {\quad \quad - \overline{{J_{23} }}^{c} \overline{{J_{32} }}^{c} + \overline{{J_{11} }}^{c} \overline{{J_{33} }}^{c} - \overline{{J_{14} }}^{c} \overline{{J_{41} }}^{c} - \overline{{J_{24} }}^{c} \overline{{J_{42} }}^{c} } \hfill \\ {\eta_{3} = \overline{{J_{44} }}^{c} \left( {\overline{{J_{12} }}^{c} \overline{{J_{21} }}^{c} - \overline{{J_{11} }}^{c} \overline{{J_{22} }}^{c} + \overline{{J_{13} }}^{c} \overline{{J_{31} }}^{c} + \overline{{J_{23} }}^{c} \overline{{J_{32} }}^{c} - \overline{{J_{33} }}^{c} \left( {\overline{{J_{11} }}^{c} + \overline{{J_{22} }}^{c} } \right)} \right)} \hfill \\ {\quad \quad + \overline{{J_{41} }}^{c} \left( {\overline{{J_{14} }}^{c} \overline{{J_{22} }}^{c} - \overline{{J_{12} }}^{c} \overline{{J_{24} }}^{c} + \overline{{J_{14} }}^{c} \overline{{J_{33} }}^{c} } \right)} \hfill \\ {\quad \quad + \overline{{J_{42} }}^{c} \left( {\overline{{J_{11} }}^{c} \overline{{J_{24} }}^{c} - \overline{{J_{14} }}^{c} \overline{{J_{21} }}^{c} + \overline{{J_{24} }}^{c} \overline{{J_{33} }}^{c} } \right) + \overline{{J_{31} }}^{c} \left( {\overline{{J_{13} }}^{c} \overline{{J_{22} }}^{c} - \overline{{J_{12} }}^{c} \overline{{J_{23} }}^{c} } \right)} \hfill \\ {\quad \quad + \overline{{J_{33} }}^{c} \left( {\overline{{J_{12} }}^{c} \overline{{J_{21} }}^{c} - \overline{{J_{11} }}^{c} \overline{{J_{22} }}^{c} } \right) + \overline{{J_{32} }}^{c} \left( {\overline{{J_{11} }}^{c} \overline{{J_{23} }}^{c} - \overline{{J_{13} }}^{c} \overline{{J_{21} }}^{c} } \right)} \hfill \\ {\eta_{4} = { }\overline{{J_{41} }}^{c} \left( {\overline{{J_{32} }}^{c} \left( {\overline{{J_{14} }}^{c} \overline{{J_{23} }}^{c} - \overline{{J_{13} }}^{c} \overline{{J_{24} }}^{c} } \right) + \overline{{J_{33} }}^{c} \left( {\overline{{J_{12} }}^{c} \overline{{J_{24} }}^{c} - \overline{{J_{14} }}^{c} \overline{{J_{22} }}^{c} } \right)} \right)} \hfill \\ {\quad \quad + \overline{{J_{42} }}^{c} \left( {\overline{{J_{31} }}^{c} \left( {\overline{{J_{13} }}^{c} \overline{{J_{24} }}^{c} - \overline{{J_{14} }}^{c} \overline{{J_{23} }}^{c} } \right) + \overline{{J_{33} }}^{c} \left( {\overline{{J_{14} }}^{c} \overline{{J_{21} }}^{c} - \overline{{J_{11} }}^{c} \overline{{J_{24} }}^{c} } \right)} \right)} \hfill \\ {\overline{{J_{44} }}^{c} \left( {\begin{array}{*{20}c} {\overline{{J_{31} }}^{c} \left( {\overline{{J_{12} }}^{c} \overline{{J_{23} }}^{c} - \overline{{J_{13} }}^{c} \overline{{J_{22} }}^{c} } \right) + \overline{{J_{32} }}^{c} \left( {\overline{{J_{13} }}^{c} \overline{{J_{21} }}^{c} - \overline{{J_{11} }}^{c} \overline{{J_{23} }}^{c} } \right)} \\ { + \overline{{J_{33} }}^{c} \left( {\overline{{J_{11} }}^{c} \overline{{J_{22} }}^{c} - \overline{{J_{12} }}^{c} \overline{{J_{21} }}^{c} } \right)} \\ \end{array} } \right.} \hfill \\ \end{array} } \right. $$
(F2)
According to the Jury test condition, the sufficient conditions for stability of \(F\left(\lambda \right)\) are as Eq. (F3):
$$ \left\{ {\begin{array}{*{20}l} {\left| {\eta_{4} } \right| < \eta_{0} } \hfill \\ {\left. {F\left( \lambda \right)} \right|_{\lambda = 1} > 0} \hfill \\ {\left. {F\left( \lambda \right)} \right|_{\lambda = - 1} > 0} \hfill \\ {\left| {b_{3} } \right| > \left| {b_{0} } \right|} \hfill \\ {\left| {c_{2} } \right| > \left| {c_{0} } \right|} \hfill \\ \end{array} } \right. $$
(F3)
By solving the above conditions, they can be considered locally stable of the model. In theory, the conditions are very complex, and the processes of solving the corresponding characteristic equation are very complicated tasks.
$$ \begin{aligned} \overline{{J_{11} }}^{c} & = 1 - 2\omega_{1} \overline{P}_{1} \left( t \right)^{c} \left( {\varrho_{1} + \vartheta_{1} } \right) \\ & \quad + \frac{1}{2}\omega_{1} \left( {2\widetilde{{\overline{a}}}_{1} + l_{1} + 2\overline{C}_{1} \left( t \right)^{c} \left( {\varrho_{1} + \vartheta_{1} } \right) - 4\overline{P}_{1} \left( t \right)^{c} \left( {\varrho_{1} + \vartheta_{1} } \right)} \right. \\ & \quad - 2\overline{C}_{2} \left( t \right)^{c} \vartheta_{2} + 2\overline{P}_{2} \left( t \right)^{c} \left( {\vartheta_{1} + \vartheta_{2} } \right) + u_{1} - 2\overline{\sigma }\left( t \right)^{c} \upsilon_{1} \\ & \quad \left. { - b_{1} \Phi \lambda_{0} \Psi T\xi \left( {l_{1} + u_{1} } \right) + 2\gamma_{1} \overline{\zeta }\left( t \right)^{c} } \right) \\ \end{aligned} $$
(F4)
$$ \overline{{J_{12} }}^{c} = \omega_{1} \overline{P}_{1} \left( t \right)^{c} \left( {\varrho_{1} + \vartheta_{1} } \right) $$
(F5)
$$ \overline{{J_{13} }}^{c} = - \omega_{1} \overline{P}_{1} \left( t \right)^{c} \upsilon_{1} $$
(F6)
$$ \overline{{J_{14} }}^{c} = \gamma_{1} \omega_{1} \overline{P}_{1} \left( t \right)^{c} $$
(F7)
$$ \overline{{J_{21} }}^{c} = \omega_{2} \overline{P}_{2} \left( t \right)^{c} \left( {\vartheta_{1} + \vartheta_{2} } \right) $$
(F8)
$$ \begin{aligned} \overline{{J_{22} }}^{c} & = 1 - 2\omega_{2} \overline{P}_{2} \left( t \right)^{c} \left( {\varrho_{2} + \vartheta_{2} } \right) \\ & \quad + \frac{1}{2}\omega_{2} \left( {2\widetilde{{\overline{a}}}_{2} + l_{2} - 2\overline{C}_{1} \left( t \right)^{c} \vartheta_{1} + 2\overline{C}_{2} \left( t \right)^{c} \left( {\varrho_{2} + \vartheta_{2} } \right)} \right. \\ & \quad - 4\overline{P}_{2} \left( t \right)^{c} \left( {\varrho_{2} + \vartheta_{2} } \right) + 2\overline{P}_{1} \left( t \right)^{c} \left( {\vartheta_{1} + \vartheta_{2} } \right) + u_{2} + 2\overline{\sigma }\left( t \right)^{c} \upsilon_{2} \\ & \quad \left. { - b_{2} \Phi \lambda_{0} \Psi T\xi \left( {l_{2} + u_{2} } \right) - 2\gamma_{2} \overline{\zeta }\left( t \right)^{c} } \right) \\ \end{aligned} $$
(F9)
$$ \overline{{J_{23} }}^{c} = \omega_{2} \overline{P}_{2} \left( t \right)^{c} \upsilon_{2} $$
(F10)
$$ \overline{{J_{24} }}^{c} = - \gamma_{2} \omega_{2} \overline{P}_{2} \left( t \right)^{c} $$
(F11)
$$ \overline{{J_{31} }}^{c} = - \omega_{3} \overline{\sigma }\left( t \right)^{c} \upsilon_{1} $$
(F12)
$$ \overline{{J_{32} }}^{c} = \omega_{3} \overline{\sigma }\left( t \right)^{c} \upsilon_{2} $$
(F13)
$$ \overline{{J_{33} }}^{c} = 1 - k\omega_{3} \overline{\sigma }\left( t \right)^{c} - \omega_{3} \left( {k\overline{\sigma }\left( t \right)^{c} - \overline{C}_{1} \left( t \right)^{c} \upsilon_{1} + \overline{P}_{1} \left( t \right)^{c} \upsilon_{1} + \overline{C}_{2} \left( t \right)^{c} \upsilon_{2} - \overline{P}_{2} \left( t \right)^{c} \upsilon_{2} } \right) $$
(F14)
$$ \overline{{J_{41} }}^{c} = \gamma_{1} \omega_{4} \overline{\zeta }\left( t \right)^{c} $$
(F15)
$$ \overline{{J_{42} }}^{c} = - \gamma_{2} \omega_{4} \overline{\zeta }\left( t \right)^{c} $$
(F16)
$$ \overline{{J_{44} }}^{c} = 1 - h\omega_{4} \overline{\zeta }\left( t \right)^{c} - \omega_{4} \left( {\overline{C}_{1} \left( t \right)^{c} \gamma_{1} - \overline{C}_{2} \left( t \right)^{c} \gamma_{2} - \gamma_{1} \overline{P}_{1} \left( t \right)^{c} + \gamma_{2} \overline{P}_{2} \left( t \right)^{c} + h\overline{\zeta }\left( t \right)^{c} } \right) $$
(F17)
Appendix G: Proof of Theorem 3.5
The concavity of the total profit function in the centralized mode is examined by its corresponding Hessian matrix, \(H\).
$$
H\left( \overline{P}_{1}^{c} ,{ }\overline{P}_{2}^{c} ,
\overline{\sigma }^{c} ,{ }\overline{\zeta }^{c} \right) =
\left[ \begin{array}{*{20}c} { - 2\left( {\varrho_{1} +
\vartheta_{1} } \right)} & {\vartheta_{1} + \vartheta_{2} } & { -
\upsilon_{1} } & {\gamma_{1} } \\ {\vartheta_{1} + \vartheta_{2} }
& { - 2\left( {\varrho_{2} + \vartheta_{2} } \right)} &
{\upsilon_{2} } & { - \gamma_{2} } \\ { - \upsilon_{1} } &
{\upsilon_{2} } & { - k} & 0 \\ {\gamma_{1} } & { - \gamma_{2} } &
0 & { - h} \\ \end{array} \right]
$$
All members on the main diameter are negative. Owing to the positive \({\varrho }_{1}\) and\({\vartheta }_{1}\), it can be shown that the first minor determinant \(\left(-2\left({\varrho }_{1}+{\vartheta }_{1}\right)\right)\) is negative. Moreover, in case the second minor determinant \(\left( {4\varrho_{1} \varrho_{2} + 4\varrho_{2} \vartheta_{1} - \vartheta_{1}^{2} + 4\varrho_{1} \vartheta_{2} + 2\vartheta_{1} \vartheta_{2} - \vartheta_{2}^{2} > 0} \right)\) be positive,
the third minor determinant
\( \Big(-4k{\varrho }_{1}\left({\varrho }_{2}+{\vartheta }_{2}\right)-4k{\varrho }_{2}{\vartheta }_{1}+k{\vartheta }_{1}^{2}-2k{\vartheta }_{1}{\vartheta }_{2}+k{\vartheta }_{2}^{2}+2{\varrho }_{2}{\upsilon }_{1}^{2}+2{\vartheta }_{2}{\upsilon }_{1}^{2}-2\left({\vartheta }_{1}+{\vartheta }_{2}\right){\upsilon }_{1}{\upsilon }_{2} + 2{\varrho }_{1}{\upsilon }_{2}^{2}+2{\vartheta }_{1}{\upsilon }_{2}^{2}<0\Big)\) be
negative (Considering that in this article, parameter
\(k\)
is a cost parameter and its value is large, and the rest of the
parameters are coefficients and have small values, as a result,
expressions with parameter \(k\)
determine the value of the condition, as a result,\( \left( { - 4k\varrho_{1} \left( {\varrho_{2} + \vartheta_{2} } \right) - 4k\varrho_{2} \vartheta_{1} + k\vartheta_{1}^{2} - 2k\vartheta_{1} \vartheta_{2} < 0} \right)\),
Since, because these two values \({\vartheta }_{1}{ \mathrm{and} \vartheta }_{2}\) are almost equal to
each other, so\(\left(k{\vartheta }_{1}^{2}=2k{\vartheta }_{1}{\vartheta }_{2}\right)\), and therefore
the third minor in this article is always negative), and also the
fourth minor determinant \(\Big(-2{\gamma }_{2}^{2}k{\varrho }_{1}-2{\gamma }_{1}^{2}k{\varrho }_{2}+4hk{\varrho }_{1}{\varrho }_{2}+2{\gamma }_{1}{\gamma }_{2}k{\vartheta }_{1}-2{\gamma }_{2}^{2}k{\vartheta }_{1}+4hk{\varrho }_{2}{\vartheta }_{1}-hk{\vartheta }_{1}^{2}-2{\gamma }_{1}^{2}k{\vartheta }_{2}+2{\gamma }_{1}{\gamma }_{2}k{\vartheta }_{2}+4hk{\varrho }_{1}{\vartheta }_{2}+2hk{\vartheta }_{1}{\vartheta }_{2}-hk{\vartheta }_{2}^{2}+{\gamma }_{2}^{2}{\upsilon }_{1}^{2}-2h{\varrho }_{2}{\upsilon }_{1}^{2}-2h{\vartheta }_{2}{\upsilon }_{1}^{2}-2{\gamma }_{1}{\gamma }_{2}{\upsilon }_{1}{\upsilon }_{2}+2h{\vartheta }_{1}{\upsilon }_{1}{\upsilon }_{2}+2h{\vartheta }_{2}{\upsilon }_{1}{\upsilon }_{2}+{\gamma }_{1}^{2}{\upsilon }_{2}^{2}-2h{\varrho }_{1}{\upsilon }_{2}^{2}-2h{\vartheta }_{1}{\upsilon }_{2}^{2}>0\Big)\) be
positive, (Considering that in this article, parameter
\(k\)
and \(h\)
is a cost parameter and its value is large, and the rest of the
parameters are coefficients and have small values, as a result,
expressions with parameter
\(k\) and \(h\) determine the value of the condition, so, \( \Big( 4hk\varrho_{1} \varrho_{2} + 2\gamma_{1} \gamma_{2} k\vartheta_{1} + 4hk\varrho_{2} \vartheta_{1} + 2\gamma_{1} \gamma_{2} k\vartheta_{2} + 4hk\varrho_{1} \vartheta_{2} + 2hk\vartheta_{1} \vartheta_{2} + 2h\vartheta_{1} \upsilon_{1} \upsilon_{2} + 2h\vartheta_{2} \upsilon_{1} \upsilon_{2} > 2\gamma_{2}^{2} k\varrho_{1} + 2\gamma_{1}^{2} k\varrho_{2} + 2\gamma_{2}^{2} k\vartheta_{1} + hk\vartheta_{1}^{2} + 2\gamma_{1}^{2} k\vartheta_{2} + hk\vartheta_{2}^{2} + 2h\varrho_{2} \upsilon_{1}^{2} + 2h\vartheta_{2} \upsilon_{1}^{2} + 2h\varrho_{1} \upsilon_{2}^{2} + 2h\vartheta_{1} \upsilon_{2}^{2} \Big), \) Due to the equality of
similar parameters with each other and the small values of the
parameters, almost identical values are removed from both sides of
the equation. As a result, \(\big( {4hk\varrho_{1} \varrho_{2} + 4hk\varrho_{2} \vartheta_{1} + 4hk\varrho_{1} \vartheta_{2} > 2\gamma_{1}^{2} k\varrho_{2} + hk\vartheta_{1}^{2} + 2\gamma_{1}^{2} k\vartheta_{2} + 2h\varrho_{2} \upsilon_{1}^{2} + 2h\varrho_{1} \upsilon_{2}^{2} } \big)\), according to
\(\left( h \right)\) is very big value,
this term is always satisfied, and the fourth minor in this
article is always positive) the values of decision variables will
be optimal and exclusive. Note that, \(h\)
and \(k\)
are coefficients of investment cost on greening level and sales
efforts and their values are much larger than other coefficients.
Because these two parameters are the amount of investment to
increase the levels of decision variables and are for a period and
are very large numbers. However, the magnitude of these two
parameters compared to others is part of the conditions of our
model and this is not necessarily true for other models.
Therefore, according to the conditions of the present problem, it
can be claimed that the condition of concavity is established for
the models. □
Appendix H: Proof of Theorem 3.6
By determining the second-order derivative of the profit function, it is obvious that a positive\(h\), and \(\Omega \) between 0 and 1 guarantees the concavity condition \(\left( { - h\left( {1 - {\Omega }} \right)} \right)\).□
