Appendix 1
The Hessian matrix of Eq. (10) is defined as follows:
$$H = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\frac{{\partial^{2} \Pi_{ec} }}{{\partial s^{2} }}} & {\frac{{\partial^{2} \Pi_{ec} }}{{\partial s\partial f_{cu} }}} & {\frac{{\partial^{2} \Pi_{ec} }}{{\partial s\partial f_{r} }}} \\ \end{array} } \\ {\begin{array}{*{20}c} {\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu} \partial s}}} & {\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu}^{2} }}} & {\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu} \partial f_{r} }}} \\ \end{array} } \\ {\begin{array}{*{20}c} {\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{r} \partial s}}} & {\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{r} \partial f_{cu} }}} & {\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{r}^{2} }}} \\ \end{array} } \\ \end{array} } \right],$$
(36)
where,
$$\frac{{\partial^{2} \Pi_{ec} }}{{\partial s^{2} }} = - 2\frac{D}{v\lambda },$$
(37)
$$\frac{{\partial^{2} \Pi_{ec} }}{{\partial s\partial f_{cu} }} = \frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu} \partial s}} = \frac{2D}{{\lambda v}},$$
(38)
$$\frac{{\partial^{2} \Pi_{ec} }}{{\partial s\partial f_{r} }} = \frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{r} \partial s}} = \frac{D(1 + r)}{{v\lambda }},$$
(39)
$$\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu}^{2} }} = - 2\frac{D}{(1 - \lambda )\lambda v},$$
(40)
$$\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu} \partial f_{r} }} = \frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{r} \partial f_{cu} }} = - \frac{D}{\lambda v}(1 + r),$$
(41)
$$\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{r}^{2} }} = - 2\frac{Dr}{{\lambda v}}.$$
(42)
When Eqs. (43) to (45) are established, the function is concave:
$$\left| H \right|_{1} < 0 \to \frac{{\partial^{2} \Pi_{ec} }}{{\partial s^{2} }} < 0$$
(43)
$$\left| H \right|_{2} > 0 \to \frac{{\partial^{2} \Pi_{ec} }}{{\partial s^{2} }}.\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu}^{2} }} - \frac{{\partial^{2} \Pi_{ec} }}{{\partial s\partial f_{cu} }}.\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu} \partial s}} > 0$$
(44)
$$\left| H \right|_{3} < 0 \to \left[ \begin{gathered} \frac{{\partial^{2} \Pi_{ec} }}{{\partial s^{2} }}\left( {\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu}^{2} }}.\frac{{\partial^{2} \Pi_{rc} }}{{\partial f_{r}^{2} }} - \frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu} \partial f_{r} }}.\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{r} \partial f_{cu} }}} \right) \hfill \\ - \frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu} \partial s}}\left( {\frac{{\partial^{2} \Pi_{ec} }}{{\partial s\partial f_{cu} }}.\frac{{\partial^{2} \Pi_{rc} }}{{\partial f_{r}^{2} }} - \frac{{\partial^{2} \Pi_{ec} }}{{\partial s\partial f_{r} }}.\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{r} \partial f_{cu} }}} \right) \hfill \\ + \frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{r} \partial s}}\left( {\frac{{\partial^{2} \Pi_{ec} }}{{\partial s\partial f_{cu} }}.\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu} \partial f_{r} }} - \frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu}^{2} }}.\frac{{\partial^{2} \Pi_{ec} }}{{\partial s\partial f_{r} }}} \right) \hfill \\ \end{gathered} \right] < 0$$
(45)
In order to obtain the optimum values of the decision variables, we have:
$$\frac{{\partial \Pi_{ec} }}{\partial s} = D\left[ {\left( {1 - \frac{{f_{cu} }}{(1 - \lambda )v}} \right) - (s - f_{cu} )\frac{1}{\lambda v} + \left( {\frac{{f_{cu} }}{(1 - \lambda )v} - \frac{{s - f_{r} - f_{cu} }}{\lambda v}} \right) + \frac{{f_{r} r}}{\lambda v}} \right] = 0,$$
(46)
$$\frac{{\partial \Pi_{ec} }}{{\partial f_{cu} }} = D\left[ \begin{gathered} - (s - c)\frac{1}{(1 - \lambda )v} - 1\left( {\frac{{f_{cu} }}{(1 - \lambda )v} - \frac{{s - f_{r} - f_{cu} }}{\lambda v}} \right) \hfill \\ + \left( {\frac{1}{(1 - \lambda )v} + \frac{1}{\lambda v}} \right)(s - f_{cu} ) - f_{r} r\frac{1}{\lambda v} \hfill \\ \end{gathered} \right] = 0,$$
(47)
$$\frac{{\partial \Pi_{ec} }}{{\partial f_{r} }} = D\left[ {(s - f_{cu} )\frac{1}{\lambda v} - r\left( {\left[ {1 - \frac{{s - f_{r} - f_{cu} }}{\lambda v}} \right]} \right) - \frac{{f_{r} r}}{\lambda v}} \right] = 0.$$
(48)
By solving Eq. (46) to (48), the optimum values will be equal to:
$$s^{*} = \lambda v + f_{r} ,$$
(49)
$$f_{cu}^{*} = \frac{{\lambda v + f_{r} (1 - r)}}{2},$$
(50)
$$f_{cu}^{*} = \frac{{\lambda v + f_{r} (1 - r)}}{2},$$
$$f_{r}^{*} = \frac{2\lambda v(1 - r)}{{2r - r^{2} + 1}},$$
(51)
Appendix 2
The Hessian matrix of Eq. (14) is defined as follows:
$$H = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\frac{{\partial^{2} \Pi_{ec} }}{{\partial s^{2} }}} & {\frac{{\partial^{2} \Pi_{ec} }}{{\partial s\partial f_{cu} }}} & {\frac{{\partial^{2} \Pi_{ec} }}{{\partial s\partial f_{r} }}} \\ \end{array} } \\ {\begin{array}{*{20}c} {\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu} \partial s}}} & {\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu}^{2} }}} & {\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu} \partial f_{r} }}} \\ \end{array} } \\ {\begin{array}{*{20}c} {\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{r} \partial s}}} & {\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{r} \partial f_{cu} }}} & {\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{r}^{2} }}} \\ \end{array} } \\ \end{array} } \right],$$
(52)
In other hand:
$$\frac{{\partial^{2} \Pi_{ec} }}{{\partial s^{2} }} = - 2\frac{D}{v\lambda },$$
(53)
$$\frac{{\partial^{2} \Pi_{ec} }}{{\partial s\partial f_{cu} }} = \frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu} \partial s}} = \frac{D}{\lambda v}(\theta + \psi ),$$
(54)
$$\frac{{\partial^{2} \Pi_{ec} }}{{\partial s\partial f_{r} }} = \frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{r} \partial s}} = \frac{D(1 + r)}{{v\lambda }},$$
(55)
$$\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu}^{2} }} = - 2\frac{D\theta \psi }{{(1 - \lambda )\lambda v}},$$
(56)
$$\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu} \partial f_{r} }} = \frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{r} \partial f_{cu} }} = - \frac{D}{\lambda v}(1 + \psi r),$$
(57)
$$\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{r}^{2} }} = - 2\frac{Dr}{{\lambda v}}.$$
(58)
With respect to Eq. (14), which is the condition for concavity of the function, when Eqs. (59) to (61) are established:
$$\left| H \right|_{1} < 0 \to \frac{{\partial^{2} \Pi_{ec} }}{{\partial s^{2} }} < 0,$$
(59)
$$\left| H \right|_{2} > 0 \to \frac{{\partial^{2} \Pi_{ec} }}{{\partial s^{2} }}\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu}^{2} }} - \frac{{\partial^{2} \Pi_{ec} }}{{\partial s\partial f_{cu} }}\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu} \partial s}} > 0,$$
(60)
$$\left| H \right|_{3} < 0 \to \left[ \begin{gathered} \frac{{\partial^{2} \Pi_{ec} }}{{\partial s^{2} }}\left( {\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu}^{2} }}\frac{{\partial^{2} \Pi_{rc} }}{{\partial f_{r}^{2} }} - \frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu} \partial f_{r} }}\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{r} \partial f_{cu} }}} \right) \hfill \\ - \frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu} \partial s}}\left( {\frac{{\partial^{2} \Pi_{ec} }}{{\partial s\partial f_{cu} }}\frac{{\partial^{2} \Pi_{rc} }}{{\partial f_{r}^{2} }} - \frac{{\partial^{2} \Pi_{ec} }}{{\partial s\partial f_{r} }}\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{r} \partial f_{cu} }}} \right) \hfill \\ + \frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{r} \partial s}}\left( {\frac{{\partial^{2} \Pi_{ec} }}{{\partial s\partial f_{cu} }}\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu} \partial f_{r} }} - \frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu}^{2} }}\frac{{\partial^{2} \Pi_{ec} }}{{\partial s\partial f_{r} }}} \right) \hfill \\ \end{gathered} \right] < 0.$$
(61)
In this case, to solve the second part of Eq. (14), the first derivative is used. First derivative conditions are:
$$\frac{{\partial \Pi_{ec} }}{\partial s} = D\left[ {\left( {1 - \frac{{(\psi f_{cu} }}{(1 - \lambda )v}} \right) - (s - \theta f_{cu} )\frac{1}{\lambda v} + \left( {\frac{{\psi f_{cu} }}{(1 - \lambda )v} - \frac{{s - f_{r} - \psi f_{cu} }}{\lambda v}} \right) + \frac{{f_{r} r}}{\lambda v}} \right] = 0,$$
(62)
$$\frac{{\partial \Pi_{ec} }}{{\partial f_{cu} }} = D\left[ { - \frac{(s - c)\psi }{{(1 - \lambda )v}} - \theta \left( {\frac{{\psi f_{cu} }}{(1 - \lambda )v} - \frac{{s - f_{r} - \psi f_{cu} }}{\lambda v}} \right) + \left( {\frac{\psi }{(1 - \lambda )v} + \frac{\psi }{\lambda v}} \right)(s - \theta f_{cu} ) - f_{r} r\frac{\psi }{\lambda v}} \right] = 0,$$
(63)
$$\frac{{\partial \Pi_{ec} }}{{\partial f_{r} }} = D\left[ {(s - f_{cu} )\frac{1}{\lambda v} - r\left( {\left[ {1 - \frac{{s - f_{r} - \psi f_{cu} }}{\lambda v}} \right]} \right) - \frac{{f_{r} r}}{\lambda v}} \right] = 0.$$
(64)
By solving Eqs. (62) to (64), the optimum answers will be equal to:
$$s^{*} = \lambda v + f_{r} ,$$
(65)
$$f_{cu}^{*} = \frac{{\lambda v + f_{r} (1 - r)}}{2 + \psi },$$
(66)
$$f_{r}^{*} = \frac{{v\left( {(2 + \psi )\lambda - \frac{2\theta \lambda \psi }{{(1 - \lambda )(2 + \psi )}}} \right) - c\frac{\psi \lambda }{{1 - \lambda }}}}{{1 + \frac{2\theta \psi }{{(2 + \psi )(1 - \lambda )}} + r\psi - (2 + \psi )}}.$$
(67)
Appendix 3
The Hessian matrix of Eq. (20) is defined as follows:
$$\left| H \right| < 0 \to \frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu}^{2} }} < 0 \to \left| H \right| = - 2\frac{D}{v\lambda (1 - \lambda )} < 0,$$
(68)
and
$$\frac{{\partial \Pi_{cw} }}{{\partial f_{cu} }} = D\left[ { - \left( {\frac{{f_{cu} }}{v(1 - \lambda )} - \frac{{s - f_{r} - f_{cu} }}{v\lambda }} \right) + \left( {f_{cw} - f_{cu} } \right)\left( {\frac{1}{v(1 - \lambda )} + \frac{1}{v\lambda }} \right)} \right] = 0.$$
(69)
According to Eq. (69), we have:
$$f_{cu}^{*} = \frac{{(s - f_{r} )(1 - \lambda ) + f_{cw} }}{2}$$
(70)
Appendix 4
Proof of the concavity of Eq. (23), the Hessian matrix is defined as follows:
$$H = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\frac{{\partial^{2} \Pi_{es} }}{{\partial s^{2} }}} & {\frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{cw} }}} & {\frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{r} }}} \\ \end{array} } \\ {\begin{array}{*{20}c} {\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw} \partial s}}} & {\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw}^{2} }}} & {\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw} \partial f_{r} }}} \\ \end{array} } \\ {\begin{array}{*{20}c} {\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{r} \partial s}}} & {\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{r} \partial f_{cw} }}} & {\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{r}^{2} }}} \\ \end{array} } \\ \end{array} } \right],$$
(71)
In other hand:
$$\frac{{\partial^{2} \Pi_{es} }}{{\partial s^{2} }} = - \frac{D}{v}\left( {\frac{1 + \lambda }{\lambda }} \right),$$
(72)
$$\frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{cw} }} = \frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw} \partial s}} = \frac{D}{\lambda v},$$
(73)
$$\frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{r} }} = \frac{{\partial^{2} \Pi_{es} }}{{\partial f_{r} \partial s}} = \frac{D}{v}\left( {\frac{1 + \lambda (1 + r)}{{2\lambda }}} \right),$$
(74)
$$\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw}^{2} }} = - \frac{D}{(1 - \lambda )\lambda v},$$
(75)
$$\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw} \partial f_{r} }} = \frac{{\partial^{2} \Pi_{es} }}{{\partial f_{r} \partial f_{cw} }} = - \frac{D}{2\lambda v}(1 + r),$$
(76)
$$\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{r}^{2} }} = - \frac{Dr}{v}.$$
(77)
When Eqs. (78) to (80) is established, the function is concave:
$$\left| H \right|_{1} < 0 \to \frac{{\partial^{2} \Pi_{es} }}{{\partial s^{2} }} = - \frac{D}{v}\left( {\frac{1 + \lambda }{\lambda }} \right) < 0,$$
(78)
$$\left| H \right|_{2} > 0 \to \frac{{\partial^{2} \Pi_{es} }}{{\partial s^{2} }}.\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw}^{2} }} - \frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{cw} }}.\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw} \partial s}} = \frac{{2D^{2} }}{{v^{2} \lambda }} > 0,$$
(79)
$$\begin{gathered} \left| H \right|_{3} < 0 \to \left[ \begin{gathered} \frac{{\partial^{2} \Pi_{es} }}{{\partial f_{r}^{2} }}\left( {\frac{{\partial^{2} \Pi_{es} }}{{\partial s^{2} }}.\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw}^{2} }} - \frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw} \partial s}}.\frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{cw} }}} \right) \hfill \\ - \frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cw} \partial f_{r} }}\left( {\frac{{\partial^{2} \Pi_{es} }}{{\partial s^{2} }}.\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw} \partial f_{r} }} - \frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{r} }}.\frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{cw} }}} \right) \hfill \\ + \frac{{\partial^{2} \Pi_{es} }}{{\partial f_{r} \partial s}}\left( {\frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{cw} }}.\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw} \partial f_{r} }} - \frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw}^{2} }}.\frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{r} }}} \right) \hfill \\ \end{gathered} \right] < 0. \hfill \\ \to \frac{{D^{3} }}{{\lambda^{2} v^{3} }}\left( { - (1 + \lambda )\left( {\frac{r}{1 - \lambda } - \frac{{(1 + r)^{2} }}{4\lambda }} \right) + \frac{(1 + r)(1 + \lambda (1 + r))}{{4\lambda }} - r + \frac{(1 + \lambda (1 + r))}{{2\lambda (1 - \lambda )}} - \frac{1 + r}{{2\lambda }}} \right) < 0. \hfill \\ \end{gathered}$$
(80)
In order to obtain the optimum values of the decision variables, we have:
$$\frac{{\partial \Pi_{ec} }}{\partial s} = D\left[ {\left( {1 - \frac{{s - f_{r} }}{2v}} \right) - \left( {\frac{{f_{cw} }}{2v(1 - \lambda )}} \right) - \frac{(s - c)}{{2v}} + \left( {\frac{{f_{cw} }}{2(1 - \lambda )\lambda v} - \frac{{s - f_{r} }}{2\lambda v}} \right) - \frac{{(s - f_{cw} )}}{2v\lambda } + \frac{{f_{r} r}}{2\lambda }} \right] = 0,$$
(81)
$$\frac{{\partial \Pi_{ec} }}{{\partial f_{cw} }} = D\left[ { - \frac{(s - c)}{{2(1 - \lambda )v}} - \left( {\frac{{f_{cw} }}{2\lambda (1 - \lambda )v} - \frac{{s - f_{r} }}{2\lambda v}} \right) + \left( {\frac{{(s - f_{cw} )}}{2\lambda (1 - \lambda )v}} \right) - \frac{{f_{r} r}}{2\lambda v}} \right] = 0,$$
(82)
$$\frac{{\partial \Pi_{ec} }}{{\partial f_{r} }} = D\left[ {\frac{(s - c)}{{2v}} + (s - f_{cw} )\frac{1}{2\lambda v} - \frac{{f_{r} r}}{2v} - r\left( {1 - \frac{{(s - f_{r} )r}}{2v} - \frac{{rf_{cw} }}{2\lambda v}} \right)} \right] = 0.$$
(83)
By solving Eqs. (81) to (83), the optimum answers will be equal to:
$$s^{*} = \left( {\frac{v\lambda }{{1 + \lambda }}\left[ {\frac{\lambda (\xi - 2r)}{{\lambda \xi^{2} }} + 1} \right] + \frac{c\lambda }{{1 + \lambda }}\left[ {\frac{(1 + \lambda (1 + r))\lambda r}{{\xi^{2} }} + 1} \right] - f_{cw} \left[ {2\theta \left( {1 - r} \right)} \right]} \right),$$
(84)
$$f_{cw}^{*} = \frac{{\left( {v\left( {\left[ {\frac{2(\xi - 2r)}{{\lambda \xi^{2} (1 + \lambda )}} - \frac{1}{1 + \lambda }} \right] - \left[ {\frac{{2\left( {1 + r} \right)(2r - \xi )}}{{\xi^{2} }}} \right]} \right) + c\left( \begin{gathered} \left[ {\frac{2(1 + \lambda (1 + r))\lambda r}{{(1 - \lambda )\xi^{2} }} - \frac{1}{1 - \lambda }} \right] \hfill \\ + \left[ {\frac{{\left( {1 + r} \right)(2r - \xi )}}{{\xi^{2} }}} \right] + \left[ {\frac{1}{1 - \lambda }} \right] \hfill \\ \end{gathered} \right)} \right)}}{{\left( {2\frac{1}{1 - \lambda } - \left( {1 + r} \right)\left( {\frac{2r}{{\xi^{2} }}} \right) + \frac{2}{2\lambda (1 + \lambda )}\left( {2 + \frac{(1 + \lambda (1 + r))}{{\xi^{2} }}\left( {2r} \right)} \right)} \right)}},$$
(85)
$$f^{*}_{r} = \left( {2v\lambda \left[ {\frac{\xi - 2r}{{\xi^{2} }}} \right] + \frac{{rc\lambda^{2} }}{{\xi^{2} }} + f_{cw} \frac{{\left[ {\xi 2r} \right]}}{{\xi^{2} }}} \right)$$
(86)
Appendix 5
The Hessian matrix of Eq. (29) is defined as follows:
$$\left| H \right| < 0 \to \frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu}^{2} }} < 0 \to \left| H \right| = - 2\frac{D\theta \psi }{{v\lambda (1 - \lambda )}} < 0.$$
(87)
According to Eq. (87), Eq. (29) is concave and for obtaining the optimum values of the decision variables, we have:
$$\frac{{\partial \Pi_{cw} }}{{\partial f_{cu} }} = D\left[ { - \theta \left( {\frac{{\psi f_{cu} }}{v(1 - \lambda )} - \frac{{s - f_{r} - \psi f_{cu} }}{v\lambda }} \right) + \left( {f_{cw} - \theta f_{cu} } \right)\left( {\frac{\psi }{v(1 - \lambda )} + \frac{\psi }{v\lambda }} \right)} \right] = 0.$$
(88)
By solving Eq. (88), we have:
$$f_{cu}^{*} = \frac{{\theta (s - f_{r} )(1 - \lambda ) + \psi f_{cw} }}{2\theta \psi }.$$
(89)
Appendix 6
With regard to Eq. (89) in Appendix 5, we can say:
$$H = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\frac{{\partial^{2} \Pi_{es} }}{{\partial s^{2} }}} & {\frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{cw} }}} & {\frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{r} }}} \\ \end{array} } \\ {\begin{array}{*{20}c} {\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw} \partial s}}} & {\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw}^{2} }}} & {\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw} \partial f_{r} }}} \\ \end{array} } \\ {\begin{array}{*{20}c} {\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{r} \partial s}}} & {\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{r} \partial f_{cw} }}} & {\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{r}^{2} }}} \\ \end{array} } \\ \end{array} } \right],$$
(90)
where
$$\frac{{\partial^{2} \Pi_{es} }}{{\partial s^{2} }} = - \frac{D}{v}\left( {\frac{1 + \lambda }{\lambda }} \right),$$
(91)
$$\frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{cw} }} = \frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw} \partial s}} = \frac{D}{2\lambda \theta v}(\theta + \psi ),$$
(92)
$$\frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{r} }} = \frac{{\partial^{2} \Pi_{es} }}{{\partial f_{r} \partial s}} = \frac{D}{v}\left( {\frac{1 + \lambda (1 + r)}{{2\lambda }}} \right),$$
(93)
$$\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw}^{2} }} = - \frac{D\psi }{{\theta (1 - \lambda )\lambda v}},$$
(94)
$$\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw} \partial f_{r} }} = \frac{{\partial^{2} \Pi_{es} }}{{\partial f_{r} \partial f_{cw} }} = - \frac{D}{2\lambda \theta v}(\theta + \psi r),$$
(95)
$$\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{r}^{2} }} = - \frac{Dr}{v}.$$
(96)
When Eqs. (97) to (99) exist, the function is concave:
$$\left| H \right|_{1} < 0 \to \frac{{\partial^{2} \Pi_{es} }}{{\partial s^{2} }} = - \frac{D}{v}\left( {\frac{1 + \lambda }{\lambda }} \right) < 0,$$
(97)
$$\left| H \right|_{2} > 0 \to \frac{{\partial^{2} \Pi_{es} }}{{\partial s^{2} }}\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw}^{2} }} - \frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{cw} }}\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw} \partial s}} = \left( {\frac{D}{\lambda v}} \right)^{2} \left( {\frac{\psi (1 + \lambda )}{{\theta (1 - \lambda )}} - \left( {\frac{\theta + \psi }{{2\theta }}} \right)} \right) > 0,$$
(98)
$$\begin{gathered} \left| H \right|_{3} < 0 \to \left[ \begin{gathered} \frac{{\partial^{2} \Pi_{es} }}{{\partial f_{r}^{2} }}\left( {\frac{{\partial^{2} \Pi_{es} }}{{\partial s^{2} }}\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw}^{2} }} - \frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw} \partial s}}\frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{cw} }}} \right) \hfill \\ - \frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cw} \partial f_{r} }}\left( {\frac{{\partial^{2} \Pi_{es} }}{{\partial s^{2} }}\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw} \partial f_{r} }} - \frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{r} }}\frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{cw} }}} \right) \hfill \\ + \frac{{\partial^{2} \Pi_{es} }}{{\partial f_{r} \partial s}}\left( {\frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{cw} }}\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw} \partial f_{r} }} - \frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw}^{2} }}\frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{r} }}} \right) \hfill \\ \end{gathered} \right] < 0. \hfill \\ \to \frac{{D^{3} }}{{v^{3} \lambda^{2} }}\left[ \begin{gathered} - (1 + \lambda )\left( {\frac{r\psi }{{\theta (1 - \lambda )}}} \right) - \frac{{(\theta + r\psi )^{2} }}{{4\theta^{2} \lambda }} \hfill \\ - \frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cw} \partial f_{r} }}\left( {\frac{{\partial^{2} \Pi_{es} }}{{\partial s^{2} }}\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw} \partial f_{r} }} - \frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{r} }}\frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{cw} }}} \right) \hfill \\ + \frac{{\partial^{2} \Pi_{es} }}{{\partial f_{r} \partial s}}\left( {\frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{cw} }}\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw} \partial f_{r} }} - \frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw}^{2} }}\frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{r} }}} \right) \hfill \\ \end{gathered} \right] \hfill \\ \end{gathered}$$
(99)
With respect to Eq. (100) to (102), the function is indefinite, and the maximum point, in this case obtained by:
$$\frac{{\partial \Pi_{ec} }}{\partial s} = D\left[ {\left( {1 - \frac{{s - f_{r} }}{2v}} \right) + \left( { - \frac{{\psi f_{cw} }}{2v\theta (1 - \lambda )}} \right) - \frac{(s - c)}{{2v}} + \frac{{\psi f_{cw} }}{2\theta (1 - \lambda )\lambda v} - \frac{{(2s - f_{r} - f_{cw} )}}{2\lambda v} + \frac{{f_{r} r}}{2\lambda }} \right] = 0,$$
(100)
$$\frac{{\partial \Pi_{ec} }}{{\partial f_{cw} }} = D\left[ { - \frac{{(s - c)f_{cw} }}{2\theta (1 - \lambda )v} - \left( {\frac{{f_{cw} f_{cw} }}{2\theta \lambda (1 - \lambda )v} - \frac{{s - f_{r} }}{2\lambda v}} \right) + \left( {\frac{{f_{cw} }}{2\theta \lambda (1 - \lambda )v}} \right)(s - f_{cw} ) - \frac{{f_{r} rf_{cw} }}{2\theta \lambda v}} \right] = 0,$$
(101)
$$\frac{{\partial \Pi_{ec} }}{{\partial f_{r} }} = D\left[ {\frac{(s - c)}{{2v}} + (s - f_{cw} )\frac{1}{2\lambda v} - \frac{{f_{r} r}}{2v} - r\left( {1 - \frac{{(s - f_{r} )r}}{2v} - \frac{{\psi rf_{cw} }}{2\theta \lambda v}} \right)} \right] = 0.$$
(102)
By solving Eq. (100) to (102), we have:
$$s^{*} = \left( \begin{gathered} \frac{v\lambda }{{1 + \lambda }}\left[ {\frac{\lambda (\xi - 2r)}{{\lambda \xi^{2} }} + 1} \right] + \frac{c\lambda }{{1 + \lambda }}\left[ {\frac{(1 + \lambda (1 + r))\lambda r}{{\xi^{2} }} + 1} \right] \hfill \\ - f_{cw} \left[ {\theta \left( {2 - \frac{(1 + \lambda (1 + r))}{{\xi^{2} }}\left( {\xi (1 - \theta )\varepsilon - 2\psi r} \right)} \right) - (1 - \theta )\varepsilon } \right] \hfill \\ \end{gathered} \right),$$
(103)
$$f_{cw}^{*} = \frac{{\left( \begin{gathered} v\left( {\left[ {\frac{(2 + \psi )(\xi - 2r)}{{\lambda \xi^{2} (1 + \lambda )}} - \frac{1}{1 + \lambda }} \right] - \left[ {\frac{{2\left( {\theta + r\psi } \right)(2r - \xi )}}{{\xi^{2} }}} \right]} \right) \hfill \\ + c\left( {\left[ {\frac{(2 + \psi )(1 + \lambda (1 + r))\lambda r}{{(1 - \lambda )\xi^{2} }} - \frac{1}{1 - \lambda }} \right] + \left[ {\frac{{\left( {\theta + r\psi } \right)(2r - \xi )}}{{\xi^{2} }}} \right] + \left[ {\frac{\psi }{1 - \lambda }} \right]} \right) \hfill \\ \end{gathered} \right)}}{{\left( {\frac{2\psi }{{1 - \lambda }} + \left( {\theta + r\psi } \right)\left( {\frac{\xi (1 - \theta )\varepsilon - 2r\psi }{{\xi^{2} }}} \right) + \frac{2 + \psi }{{2\lambda (1 + \lambda )\theta }}\left( {2 - \frac{(1 + \lambda (1 + r))}{{\xi^{2} }}\left( {\xi (1 - \theta )\varepsilon - 2\psi r} \right)} \right)} \right)}},$$
(104)
$$f_{r}^{*} = \left( {2v\lambda \left[ {\frac{\xi - 2r}{{\xi^{2} }}} \right] + \frac{{rc\lambda^{2} }}{{\xi^{2} }} - f_{cw} \frac{{\left[ {\xi (1 - \theta )\varepsilon - 2r\psi } \right]}}{{\xi^{2} }}} \right).$$
(105)
