Optimal pricing and replenishment decisions for non-instantaneous deteriorating items with a fixed lifetime and partial backordering under carbon regulations

Abstract

Increasing strict carbon regulations enforced by regulators are encouraging companies to seek better ways to manage inventories with a desire to reduce carbon emissions from their operations. This work illustrates an inventory model to obtain the retailer’s optimal replenishment policy for non-instantaneous deteriorating items with fixed lifetime and partial backordering under cap-and-price, cap-and-trade, and carbon tax regulations with the goal of maximizing a retailer’s profit while simultaneously reducing total carbon emissions. To investigate the work from a more general perspective, the market demand structure of perishable products is considered deterministic in nature and is reliant on both the selling price and stock level, while the shortage of items is a decreasing function of waiting time up to the next replenishment. This study examines, for the first time, the best replenishment decisions for non-instantaneous deteriorating items with an expiration date after characterizing their properties theoretically under carbon regulations. Integrating all possible cases of the optimal solutions from theoretical outcomes, different numerical examples are demonstrated, and finally, several management insights are provided by investigating the changing pattern of the optimal strategies for variation in the system parameters. The derived outcomes highlight that the cap-and-price policy performs well both economically and environmentally for the inventory decision-maker when the penalty is less than or equal to the reward for each unit of carbon emission. Furthermore, in cap-and-price regulation, the total profit is increased by $3.14\%$, $8.10\%,$ and $1.63\%$ from cap-and-trade, carbon tax, and without carbon regulation, respectively, and the total amount of emission is decreased by 5.38% from cap-and-trade and carbon tax regulation.

Appendices

Appendix 1

$$\xi = 1 + m - t,\;\psi = 1 + m - t_{d} ,\;\omega = \frac{\psi }{\chi },\;\omega_{1} = \frac{\psi }{{\chi_{1} }},\;\kappa_{1} = \left( {1 + \beta + \beta m} \right),$$

$$\kappa_{2} = \left( {6\beta m^{2} + 12\beta m - 18m + 6\beta - 18} \right),\;\kappa_{3} = \left( {9 + 3\beta + 3\beta m} \right),\;\kappa_{4} = \left( {3 - \beta - \beta m} \right),$$

$$\kappa_{5} = \left( {1 - \beta - \beta m} \right),\;\kappa_{6} = \left( {2 - \beta - \beta m} \right),\;\sigma_{1} = \left( {e^{{ - \beta t_{1} }} - e^{{ - \beta t_{d} }} } \right),\;\sigma_{2} = \left( {t_{1} e^{{ - \beta t_{1} }} - t_{d} e^{{ - \beta t_{d} }} } \right),$$

$$\sigma_{3} = \left( {t_{1}^{2} e^{{ - \beta t_{1} }} - t_{d}^{2} e^{{ - \beta t_{d} }} } \right),\;\sigma_{4} = (c_{1} + \tau .h_{e} - p\beta )\left( {1 - e^{{ - \beta t_{d} }} } \right) + \beta \left( {c_{2} e^{{ - \beta t_{d} }} + c + \tau .c_{e} } \right),$$

$$\sigma_{5} = e^{{ - \beta t_{1} }} - \beta t_{1} e^{{ - \beta t_{1} }} ,\;\sigma_{6} = 1 - \beta t_{1} - e^{{ - \beta t_{1} }} ,\;\chi = 1 + m - t_{1} ,\;\chi_{1} = 1 + m - x,$$

$$\chi_{2} = \left( {e^{ - \beta x} - e^{{ - \beta t_{d} }} } \right),\;\chi_{3} = \left( {xe^{ - \beta x} - t_{d} e^{{ - \beta t_{d} }} } \right),\;\chi_{4} = \left( {x^{2} e^{ - \beta x} - t_{d}^{2} e^{{ - \beta t_{d} }} } \right),$$

$$\chi_{5} = \left( {2xe^{ - \beta x} - \beta x^{2} e^{ - \beta x} } \right),\;\chi_{6} = \left( {e^{ - \beta x} - \beta xe^{ - \beta x} } \right),\;\chi_{7} = \left( { - \beta - \beta m + \beta x} \right),$$

$$\xi_{8} = (c_{1} + \tau .h_{e} - p\beta )\left( {\frac{{1 - e^{{ - \beta t_{d} }} }}{\beta }} \right) + c_{2} e^{{ - \beta t_{d} }} + c + \tau .c_{e} ,$$

$$\chi_{8} = - \beta \kappa_{5} e^{ - \beta x} + \beta \chi_{6} - \beta^{2} \chi_{1} \left( {1 - \beta x - e^{ - \beta x} } \right)\left( {1 + {\text{ln}}\chi_{1} } \right) + \beta^{2} \chi_{1} \left( {e^{ - \beta x} - \beta \chi_{1} - 1 + \beta x - e^{ - \beta x} \chi_{7} } \right){\text{ln}}\chi_{1} ,$$

$$\chi_{9} = \frac{{\kappa_{1} }}{{\beta^{2} \chi_{1} }}\left( {\chi_{2} \kappa_{5} + \beta \chi_{1} \left( {\beta \chi_{1} \left( {1 - \beta x} \right) + \chi_{7} e^{ - \beta x} } \right){\text{ln}}\chi_{1} + \beta \chi_{3} } \right),$$

$$\chi_{10} = \kappa_{2} + 2\kappa_{3} x - 12\beta x^{2} + 6\chi_{1} \left( {\kappa_{4} - 2\beta x} \right) + 12\beta \chi_{1}^{2} {\text{ln}}\chi_{1} + 12\chi_{1} \left( {\kappa_{4} - 2\beta x} \right){\text{ln}}\chi_{1} ,$$

$$\varrho_{1} = \left\{ {\begin{array}{*{20}c} { - (c_{1} + \tau \cdot h_{e} - \beta c_{2} - p\beta )\left\{ {\begin{array}{*{20}c} {e^{ - \beta x} \left( {1 - \beta x - \kappa_{5} } \right)\left( {\frac{{\kappa_{1} }}{{\chi_{1} \beta }} - 1} \right) + \frac{1}{{\beta^{2} \chi_{1}^{2} }}\left( {\kappa_{5} \chi_{2} + \beta \chi_{3} } \right)} \\ { - \kappa_{1} \left( {1 - \beta x - e^{ - \beta x} } \right)\left\{ {1 + \ln \chi_{1} } \right\} - \kappa_{1} \chi_{1} \beta \ln \chi_{1} \left\{ {1 + e^{ - \beta x} } \right\}} \\ \end{array} } \right\}} \\ { - \left\{ {(c_{1} + \tau \cdot h_{e} - p\beta )\left( {\frac{{1 - e^{{ - \beta t_{d} }} }}{\beta }} \right) + c_{2} e^{{ - \beta t_{d} }} + c + \tau \cdot c_{e} } \right\}\left( {\frac{{\left( {1 + \beta x} \right)}}{{\chi_{1}^{2} }} + \frac{\beta }{{\chi_{1} }}} \right)} \\ \end{array} } \right\}$$

$$\pi = \left( {t_{1} - t_{d} } \right)\kappa_{2} + \left( {t_{1}^{2} - t_{d}^{2} } \right)\kappa_{3} - 4\beta \left( {t_{1}^{3} - t_{d}^{3} } \right) - 6\chi^{2} \left( {\kappa_{4} - 2\beta t_{1} } \right){\text{ln}}\chi + 6\psi^{2} \left( {\kappa_{4} - 2\beta t_{d} } \right){\text{ln}}\psi ,$$

$$\pi_{1} = \left( {x - t_{d} } \right)\kappa_{2} + \left( {x^{2} - t_{d}^{2} } \right)\kappa_{3} - 4\beta \left( {x^{3} - t_{d}^{3} } \right) - 6\chi_{1}^{2} \left( {\kappa_{4} - 2\beta x} \right){\text{ln}}\chi_{1} + 6\psi^{2} \left( {\kappa_{4} - 2\beta t_{d} } \right){\text{ln}}\psi ,$$

$$\pi_{2} = t_{1} \kappa_{2} + t_{1}^{2} \kappa_{2} - 43t_{1}^{3} - 6\xi^{2} {\text{ln}}\chi \left( {\kappa_{4} - 2\beta t_{1} } \right) + 6\kappa_{4} \left( {1 + m} \right)^{2} \ln \left( {1 + m} \right),$$

$$\pi_{3} = \left( {e^{{ - \beta t_{1} }} - 1} \right)\kappa_{5} + \beta t_{1} e^{{ - \beta t_{1} }} ,\;\eta = \left( {\frac{{c_{3} }}{\delta } + c_{4} } \right),$$

$$\mu = \left[ {\begin{array}{*{20}c} {p + c_{2} - (c_{1} + \tau \cdot h_{e} - \beta c_{2} - p\beta )\left\{ {\frac{1}{\beta }\left( {\kappa_{5} \kappa_{6} + \beta \sigma_{2} } \right)\left( {\frac{{\kappa_{1} }}{\chi \beta } - 1} \right) + \kappa_{1} \chi \sigma_{6} \ln \chi } \right\}} \\ { - \left\{ {(c_{1} + \tau \cdot h_{e} - p\beta )\left( {\frac{{1 - e^{{ - \beta t_{d} }} }}{\beta }} \right) + c_{2} e^{{ - \beta t_{d} }} + c + \tau \cdot c_{e} } \right\}\frac{{(1 + \beta t_{1} )}}{\chi }\psi } \\ \end{array} } \right],$$

$$\mu_{1} = \left[ {\begin{array}{*{20}c} {p + c_{2} - (c_{1} + \tau \cdot h_{e} - \beta c_{2} - p\beta )\left\{ {\begin{array}{*{20}c} {\frac{1}{\beta }\left( {\kappa_{5} \chi_{2} + \beta \chi_{3} } \right)\left( {\frac{{\kappa_{1} }}{{\beta \chi_{1} }} - 1} \right)} \\ { + \kappa_{1} \chi_{1} \ln \chi_{1} \left( {1 - \beta x - e^{ - \beta x} } \right)} \\ \end{array} } \right\}} \\ {\quad - \left\{ {(c_{1} + \tau .h_{e} - p\beta )\left( {\frac{{1 - e^{{ - \beta t_{d} }} }}{\beta }} \right) + c_{2} e^{{ - \beta t_{d} }} + c + \tau \cdot c_{e} } \right\}\frac{{\left( {1 + \beta x} \right)}}{{\chi_{1} }}\psi } \\ \end{array} } \right],$$

$$\nu = \left[ {\begin{array}{*{20}c} {p + c_{2} - (c_{1} + c_{p} \cdot h_{e} - \beta c_{2} - p\beta )\left\{ {\frac{1}{\beta }\left( {\kappa_{5} \kappa_{6} + \beta \sigma_{2} } \right)\left( {\frac{{\kappa_{1} }}{\chi \beta } - 1} \right) + \kappa_{1} \chi \sigma_{6} \ln \chi } \right\}} \\ { - \left\{ {(c_{1} + c_{p} \cdot h_{e} - p\beta )\left( {\frac{{1 - e^{{ - \beta t_{d} }} }}{\beta }} \right) + c_{2} e^{{ - \beta t_{d} }} + c + c_{p} \cdot c_{e} } \right\}\frac{{(1 + \beta t_{1} )}}{\chi }\psi } \\ \end{array} } \right],$$

$$\nu_{1} = \left[ {\begin{array}{*{20}c} {p + c_{2} - (c_{1} + c_{p} .h_{e} - \beta c_{2} - p\beta )\left\{ {\begin{array}{*{20}c} {\frac{1}{\beta }\left( {\kappa_{5} \chi_{2} + \beta \chi_{3} } \right)\left( {\frac{{\kappa_{1} }}{{\beta \chi_{1} }} - 1} \right)} \\ { + \kappa_{1} \chi_{1} \ln \chi_{1} \left( {1 - \beta x - e^{ - \beta x} } \right)} \\ \end{array} } \right\}} \\ {\quad - \left\{ {(c_{1} + c_{p} .h_{e} - p\beta )\left( {\frac{{1 - e^{{ - \beta t_{d} }} }}{\beta }} \right) + c_{2} e^{{ - \beta t_{d} }} + c + c_{p} .c_{e} } \right\}\frac{{\left( {1 + \beta x} \right)}}{{\chi_{1} }}\psi } \\ \end{array} } \right],$$

$$\nu^{0} = \left[ {\begin{array}{*{20}c} {p + c_{2} - (c_{1} + c_{p} .h_{e} - \beta c_{2} - p\beta )\left[ {\kappa_{1} \chi \ln \chi \left( {1 - \beta t_{d} - e^{{ - \beta t_{d} }} } \right)} \right]} \\ { - \left[ {(c_{1} + c_{p} \cdot h_{e} - p\beta )\left( {\frac{{1 - e^{{ - \beta t_{d} }} }}{\beta }} \right) + c_{2} e^{{ - \beta t_{d} }} + c + c_{p} \cdot c_{e} } \right]\left( {1 + \beta t_{d} } \right)} \\ \end{array} } \right].$$

Appendix 2: Proof of Theorem 1

• (a) From Eq. (18), one can be written as $\mu = - \eta \left( {\frac{{\delta \left( {T - t_{1} } \right)}}{{1 + \delta \left( {T - t_{1} } \right)}}} \right) < 0$ and $\eta + \mu = \frac{\eta }{{1 + \delta \left( {T - t_{1} } \right)}} > 0.$ From our assumptions, it is clear that $T > t_{1.}$ Hence, from Eq. (21) we have $\frac{\mu }{{\delta \left( {\eta + \mu } \right)}} < 0.$ Because the numerator part, $\mu < 0;$ thus, the denominator part $\delta \left( {\eta + \mu } \right) > 0.$ From this inequality, one can find the value of $t_{1}$ as $t_{1}^{b}$ (say).

Next, from Eq. (23), $\Gamma \left( x \right)$ is defined as

$$\Gamma \left( x \right) = \left[ {\begin{array}{*{20}l} {\frac{{A + \tau \cdot A_{e} }}{D\left( p \right)} - \frac{\eta }{\delta }\ln \left[ {1 - \frac{{\mu_{1} }}{{\eta + \mu_{1} }}} \right] + \frac{\delta x - 1}{\delta }\mu_{1} - px} \hfill \\ {\quad - (p\beta - c_{1} - \tau \cdot h_{e} )\left[ {\left\{ {\psi \left\{ {\beta \left( {t_{d} - x} \right) + \kappa_{1} \ln \omega_{1} } \right\} + \frac{{e^{{\beta t_{d} }} }}{\beta }} \right\}\left( {\frac{{1 - e^{{ - \beta t_{d} }} }}{\beta }} \right) - \frac{{t_{d} }}{\beta }} \right]} \hfill \\ {\quad - (p\beta - c_{1} - \tau \cdot h_{e} + \beta c_{2} )\left[ {\frac{{\pi_{1} \kappa_{1} }}{36} + \frac{{\left[ {\kappa_{6} \left( {\chi_{2} + \beta \chi_{3} } \right) + \beta^{2} \chi_{4} } \right]}}{{\beta^{2} }} - \frac{{\kappa_{5} \chi_{2} + \beta \chi_{3} }}{{\beta^{2} }}\left[ {\beta x + \kappa_{1} \ln \chi_{1} } \right]} \right]} \hfill \\ {\quad + \left( {c + \tau \cdot c_{e} } \right)\left[ {\frac{1}{\beta }\left( {e^{{\beta t_{d} }} - 1} \right) + \psi \left[ {\beta \left( {t_{d} - x} \right) + \kappa_{1} \ln \omega_{1} } \right]} \right]} \hfill \\ {\quad - \left( {p - c - \tau \cdot c_{e} } \right)\left[ {\frac{{\ln \left\{ {1 + \delta \left( {T - x} \right)} \right\}}}{\delta } - \frac{T}{{1 + \delta \left( {T - x} \right)}}} \right]} \hfill \\ {\quad + c_{2} \left[ {\psi e^{{ - \beta t_{d} }} \left\{ {\kappa_{1} \ln \omega_{1} + \beta \left( {t_{d} - x} \right)} \right\} - \left( {x - t_{d} } \right)} \right]} \hfill \\ \end{array} } \right] = 0$$

(24)

Taking the first-order derivative of $\Gamma \left( x \right)$ with respect to $x \in \left( {t_{d} ,t_{1}^{b} } \right),$ we get

$$\frac{{{\text{d}}\Gamma \left( x \right)}}{{{\text{d}}x}} = \left[ {\begin{array}{*{20}l} {\left( {\frac{1}{\delta }\left( {1 - \delta x} \right) + \frac{\eta }{{\delta \left( {\eta + \mu_{1} } \right)}}} \right)\rangle_{1} + \mu_{1} {-}p} \hfill \\ {\quad - (p\beta - c_{1} - \tau \cdot h_{e} )\left( {\frac{{1 - e^{{ - \beta t_{d} }} }}{\beta }} \right)\left( {\frac{{\kappa_{1} }}{{\chi_{1} }} - \beta } \right)\psi } \hfill \\ {\quad - (p\beta - c_{1} - \tau \cdot h_{e} + \beta c_{2} )\left[ {\begin{array}{*{20}c} {\frac{1}{\beta }\left( {\beta \chi_{5} + \kappa_{6} \left( {\chi_{6} - e^{ - \beta x} } \right)} \right)} \\ { - \frac{1}{{\beta^{2} }}\left\{ {\left( {\beta \chi_{6} - \beta \kappa_{5} e^{ - \beta x} } \right)\left( {\beta x + \kappa_{1} \ln \chi_{1} } \right)} \right\}} \\ { + \frac{{\kappa_{1} \chi_{10} }}{36} - \frac{1}{{\beta^{2} }}\left( {\beta - \frac{{\kappa_{1} }}{{\chi_{1} }}} \right)\left( {\chi_{2} \kappa_{5} + \beta \chi_{3} } \right)} \\ \end{array} } \right]} \hfill \\ {\quad - c_{2} e^{{ - \beta t_{d} }} \psi \left( {\beta - \frac{{\kappa_{1} }}{{\chi_{1} }}} \right) - c_{2} - \frac{{\left( {p - c - \tau \cdot c_{e} } \right)\left( {\delta x - 1} \right)}}{{\left\{ {1 + \delta \left( {T - x} \right)} \right\}^{2} }}} \hfill \\ \end{array} } \right] < 0$$

(25)

So, $\Gamma \left( x \right)$ is strictly decreasing function with respect to $x \in \left[ {t_{d} ,t_{1}^{b} } \right)$. By using assumption, we get

$$\Delta_{1} = \Gamma \left( {t_{d} } \right) = \left[ {\begin{array}{*{20}l} {\frac{{A + \tau \cdot A_{e} }}{D\left( p \right)} - \frac{\eta }{\delta }\ln \left[ {1 - \frac{{\mu^{0} }}{{\eta + \mu^{0} }}} \right] + \frac{{\delta t_{d} - 1}}{\delta }\mu^{0} - pt_{d} - (p\beta - c_{1} - \tau \cdot h_{e} )\left( {\frac{{e^{{\beta t_{d} }} + \beta t_{d} - 1}}{{\beta^{2} }}} \right)} \hfill \\ {\quad - \left( {p - c - \tau \cdot c_{e} } \right)\left[ {\frac{{\ln \left\{ {1 + \delta \left( {T - t_{d} } \right)} \right\}}}{\delta } - \frac{T}{{1 + \delta \left( {T - t_{d} } \right)}}} \right] + \left( {c + \tau \cdot c_{e} } \right)\left( {\frac{{e^{{\beta t_{d} }} - 1}}{\beta }} \right)} \hfill \\ \end{array} } \right] \ge 0.$$

(26)

and it can be shown that $\mathop {\lim }\limits_{{x \to t_{1}^{b} - }} \Gamma \left( x \right) = - \infty .$ Therefore, using the intermediate value theorem, there exists a unique $t_{1}^{*} \in \left[ {t_{d} ,t_{1}^{b} } \right)$ such that $\Gamma \left( {t_{1}^{*} } \right) = 0,$ i.e., $t_{1}^{*}$ is the unique solution of Eq. (23). By using the value of $t_{1}^{*} ,$ we can find the value of $T$ (denoted by $T^{*}$) from Eq. (27) as $T^{*} = t_{1}^{*} - \frac{{\mu \left( {t_{1}^{*} } \right)}}{{\delta \left( {\eta + \mu \left( {t_{1}^{*} } \right)} \right)}}$.

• (b) Again, if $\Delta_{1} < 0,$ then from Eq. (24), we get $\Gamma \left( {t_{d} } \right) < 0.$ Since $\Gamma \left( x \right)$ is a strictly decreasing of $x \in \left[ {t_{d} ,t_{1}^{b} } \right),$ that implies $\Gamma \left( x \right) < 0$ for all $x \in \left[ {t_{d} ,t_{1}^{b} } \right)$. Thus, we cannot find any $t_{1} \in \left[ {t_{d} ,t_{1}^{b} } \right)$ such that $\Gamma \left( {t_{1} } \right) = 0;$ this completes the proof.

Appendix 3: Proof of Theorem 2

• (a) Consider ${\Pi }_{c} \left( {t_{1} ,T,p} \right) = \frac{{\Delta \left( {t_{1} ,T} \right)}}{{\Sigma \left( {t_{1} ,T} \right)}},$ where $\Sigma \left( {t_{1} ,T} \right) = T$ and

  $$\left( {t_{1} ,T} \right) = D\left( p \right)\left[ {\begin{array}{*{20}l} {pt_{1} - \frac{{A + \tau \cdot A_{e} - \tau .\varpi T}}{D\left( p \right)} + (p\beta - c_{1} - \tau \cdot h_{e} )\left[ {\left[ {\psi \left( {\beta \left( {t_{d} - t_{1} } \right) + \kappa_{1} \ln \omega } \right) + \frac{{e^{{\beta t_{d} }} }}{\beta }} \right]\left( {\frac{{1 - e^{{ - \beta t_{d} }} }}{\beta }} \right) - \frac{{t_{d} }}{\beta }} \right]} \hfill \\ {\quad + (p\beta - c_{1} - \tau \cdot h_{e} + \beta c_{2} )\left[ {\frac{{\pi \kappa_{1} }}{36} + \frac{{\left[ {\kappa_{6} \left( {\sigma_{1} + \beta \sigma_{2} } \right) + \beta^{2} \sigma_{3} } \right]}}{{\beta^{2} }} - \frac{{\kappa_{5} \sigma_{1} + \beta \sigma_{2} }}{{\beta^{2} }}\left[ {\beta t_{1} + \kappa_{1} \ln \chi } \right]} \right]} \hfill \\ {\quad - \left( {c + \tau \cdot c_{e} } \right)\left[ {\frac{1}{\beta }\left( {e^{{\beta t_{d} }} - 1} \right) + \psi \left\{ {\beta \left( {t_{d} - t_{1} } \right) + \kappa_{1} \ln \omega } \right\}} \right] + \frac{{\left( {p - c - \tau \cdot c_{e} } \right)\ln \left[ {1 + \delta \left( {T - t_{1} } \right)} \right]}}{\delta }} \hfill \\ {\quad - c_{2} \left[ {\psi e^{{ - \beta t_{d} }} \left\{ {\kappa_{1} \ln \omega + \beta \left( {t_{d} - t_{1} } \right)} \right\} - t_{1} + t_{d} } \right] - \left( {\frac{{c_{3} }}{\delta } + c_{4} } \right)\left[ {T - t_{1} - \frac{{\ln \left\{ {1 + \delta \left( {T - t_{1} } \right)} \right\}}}{\delta }} \right]} \hfill \\ \end{array} } \right]$$

  (27)

First-order derivative of $\Delta \left( {t_{1} ,T} \right)$ with respect to

$$T$$

$$\frac{{\partial \Delta \left( {t_{1} ,T} \right)}}{\partial T} = D\left( p \right)\left[ {\left( {p - c - \tau \cdot c_{e} } \right)\frac{1}{{1 + \delta \left( {T - t_{1} } \right)}} - \left( {\frac{{c_{3} }}{\delta } + c_{4} } \right)\frac{{\delta \left( {T - t_{1} } \right)}}{{1 + \delta \left( {T - t_{1} } \right)}}} \right]$$

(28)

Second-order derivative of $\Delta \left( {t_{1} ,T} \right)$ with respect to $T$

$$\frac{{\partial^{2} \Delta \left( {t_{1} ,T} \right)}}{{\partial T^{2} }} = - D\left( p \right)\left[ {\left( {p - c - \tau .c_{e} } \right)\frac{\delta }{{\left\{ {1 + \delta \left( {T - t_{1} } \right)} \right\}^{2} }} + \left( {\frac{{c_{3} }}{\delta } + c_{4} } \right)\frac{\delta }{{\left\{ {1 + \delta \left( {T - t_{1} } \right)} \right\}^{2} }}} \right] < 0,$$

(29)

$$\frac{{\partial^{2} \Delta \left( {t_{1} ,T} \right)}}{{\partial T^{2} }} = - D\left( p \right)X,$$

$${\text{where}}\quad X = \left[ {\left( {p - c - \tau .c_{e} } \right)\frac{\delta }{{\left\{ {1 + \delta \left( {T - t_{1} } \right)} \right\}^{2} }} + \left( {\frac{{c_{3} }}{\delta } + c_{4} } \right)\frac{\delta }{{\left\{ {1 + \delta \left( {T - t_{1} } \right)} \right\}^{2} }}} \right] > 0.$$

First-order derivative of $\Delta \left( {t_{1} ,T} \right)$ with respect to $t_{1}$

$$\frac{{\partial \Delta \left( {t_{1} ,T} \right)}}{{\partial t_{1} }} = D\left( p \right)\left[ {\begin{array}{*{20}l} {p + c_{2} - \left( {\frac{{c_{3} }}{\delta } + c_{4} } \right)\left( {\frac{{\delta \left( {t_{1} - T} \right)}}{{1 + \delta \left( {T - t_{1} } \right)}}} \right) - \left( {p - c - \tau \cdot c_{e} } \right)\frac{1}{{\left[ {1 + \delta \left( {T - t_{1} } \right)} \right]}}} \hfill \\ {\quad + (p\beta - c_{1} - \tau \cdot h_{e} + \beta c_{2} )\left[ {\frac{1}{\beta }\left( {\kappa_{5} \sigma_{1} + \beta \sigma_{2} } \right)\left( {\frac{{\kappa_{1} }}{\chi \beta } - 1} \right) + \kappa_{1} \chi \sigma_{6} ln\chi } \right]} \hfill \\ {\quad + \left[ {(p\beta - c_{1} - \tau \cdot h_{e} )\left( {\frac{{1 - e^{{ - \beta t_{d} }} }}{\beta }} \right) - c_{2} e^{{ - \beta t_{d} }} - c - \tau \cdot c_{e} } \right]\frac{{(1 + \beta t_{1} )}}{\chi }\psi } \hfill \\ \end{array} } \right]$$

(30)

Second-order derivative of $\Delta \left( {t_{1} ,T} \right)$ with respect to $t_{1}$

$$\frac{{\partial^{2} \Delta \left( {t_{1} ,T} \right)}}{{\partial t_{1}^{2} }} = - D\left( p \right)\left[ {\begin{array}{*{20}l} {\left( {\frac{{c_{3} }}{\delta } + c_{4} } \right)\frac{\delta }{{\left\{ {1 + \delta \left( {T - t_{1} } \right)} \right\}^{2} }} + \left( {p - c - \tau \cdot c_{e} } \right)\frac{\delta }{{\left\{ {1 + \delta \left( {T - t_{1} } \right)} \right\}^{2} }}} \hfill \\ { + \left\{ {(c_{1} + \tau \cdot h_{e} - p\beta )\left( {\frac{{1 - e^{{ - \beta t_{d} }} }}{\beta }} \right) + c_{2} e^{{ - \beta t_{d} }} + c + \tau \cdot c_{e} } \right\}\left( {\frac{\beta }{\chi } + \frac{{(1 + \beta t_{1} )}}{{\chi^{2} }}} \right)\psi } \hfill \\ { + (c_{1} + \tau \cdot h_{e} - p\beta - \beta c_{2} )\left\{ {\begin{array}{*{20}c} {e^{{ - \beta t_{1} }} \kappa_{5} - \sigma_{5} + \kappa_{1} \sigma_{6} \ln \chi } \\ { + \frac{{\kappa_{1} \left[ {\sigma_{1} \kappa_{5} + \beta \sigma_{2} } \right]}}{{\beta^{2} \chi^{2} }}} \\ { + \frac{{\kappa_{1} }}{\beta \chi }\left\{ {\begin{array}{*{20}c} {\sigma_{5} - e^{{ - \beta t_{1} }} \kappa_{5} } \\ { - \beta \chi \sigma_{6} \left( {1 + \ln \chi } \right)} \\ { - \beta \chi \ln \chi \left\{ {\begin{array}{*{20}c} {\sigma_{6} } \\ { + \beta \chi \left\{ {1 - e^{{ - \beta t_{1} }} } \right\}} \\ \end{array} } \right\}} \\ \end{array} } \right\}} \\ \end{array} } \right\}} \hfill \\ \end{array} } \right]$$

(31)

$\frac{{\partial^{2} \Delta \left( {t_{1} ,T} \right)}}{{\partial t_{1}^{2} }} = - D\left( p \right)\left[ {X + Y} \right],$ where

$$Y = \left[ {\begin{array}{*{20}l} {\left\{ {(c_{1} + \tau \cdot h_{e} - p\beta )\left( {\frac{{1 - e^{{ - \beta t_{d} }} }}{\beta }} \right) + c_{2} e^{{ - \beta t_{d} }} + c + \tau \cdot c_{e} } \right\}\left( {\frac{\beta }{\chi } + \frac{{(1 + \beta t_{1} )}}{{\chi^{2} }}} \right)\psi } \hfill \\ { + (c_{1} + \tau \cdot h_{e} - p\beta - \beta c_{2} )\left\{ {\begin{array}{*{20}l} {e^{{ - \beta t_{1} }} \kappa_{5} - \sigma_{5} + \kappa_{1} \sigma_{6} \ln \chi } \hfill \\ { + \frac{{\kappa_{1} \left[ {\sigma_{1} \kappa_{5} + \beta \sigma_{2} } \right]}}{{\beta^{2} \chi^{2} }}} \hfill \\ { + \frac{{\kappa_{1} }}{\beta \chi }\left\{ {\begin{array}{*{20}l} {\sigma_{5} - e^{{ - \beta t_{1} }} \kappa_{5} } \hfill \\ { - \beta \chi \sigma_{6} \left( {1 + \ln \chi } \right)} \hfill \\ { - \beta \chi \ln \chi \left\{ {\begin{array}{*{20}c} {\sigma_{6} } \\ { + \beta \chi \left\{ {1 - e^{{ - \beta t_{1} }} } \right\}} \\ \end{array} } \right\}} \hfill \\ \end{array} } \right\}} \hfill \\ \end{array} } \right\}} \hfill \\ \end{array} } \right]$$

$$\frac{{\partial \Delta \left( {t_{1} ,T} \right)}}{{\partial T\delta t_{1} }} = D\left( p \right)\left[ {\left( {p - c - \tau .c_{e} } \right)\frac{\delta }{{\left\{ {1 + \delta \left( {T - t_{1} } \right)} \right\}^{2} }} + \left( {\frac{{c_{3} }}{\delta } + c_{4} } \right)\frac{\delta }{{\left\{ {1 + \delta \left( {T - t_{1} } \right)} \right\}^{2} }}} \right]$$

(32)

$$\frac{{\partial \Delta \left( {t_{1} ,T} \right)}}{{\partial T\delta t_{1} }} = D\left( p \right)X$$

Therefore, the Hessian matrix for $\Delta \left( {t_{1} ,T} \right)$ is:

$$H = \left[ {\begin{array}{*{20}c} {\frac{{\partial^{2} \Delta \left( {t_{1} ,T} \right)}}{{\partial t_{1}^{2} }}} & {\frac{{\partial \Delta \left( {t_{1} ,T} \right)}}{{\delta t_{1} \partial T}}} \\ {\frac{{\partial \Delta \left( {t_{1} ,T} \right)}}{{\partial T\delta t_{1} }}} & {\frac{{\partial^{2} \Delta \left( {t_{1} ,T} \right)}}{{\partial T^{2} }}} \\ \end{array} } \right]$$

(33)

The determinant of $H$ is

$$\frac{{\partial^{2} \Delta \left( {t_{1} ,T} \right)}}{{\partial t_{1}^{2} }} \times \frac{{\partial^{2} \Delta \left( {t_{1} ,T} \right)}}{{\partial T^{2} }} - \left\{ {\frac{{\partial \Delta \left( {t_{1} ,T} \right)}}{{\partial T\delta t_{1} }}} \right\}^{2} = \left\{ {D\left( p \right)} \right\}^{2} \left[ {X + Y} \right]X - \left\{ {D\left( p \right)} \right\}^{2} X^{2} = \left\{ {D\left( p \right)} \right\}^{2} XY > 0.$$

Clearly, from the Hessian matrix $H,$ the principal minors $\frac{{\partial^{2} \Delta \left( {t_{1} ,T} \right)}}{{\partial t_{1}^{2} }} < 0$ and $\frac{{\partial^{2} \Delta \left( {t_{1} ,T} \right)}}{{\partial T^{2} }} < 0$ and ${\text{det}}\left( H \right) > 0 ;$ therefore, the profit function ${\Pi }_{c} \left( {t_{1} ,T,p} \right)$ is pseudo-concave with respect to the optimal time points $\left( {t_{1} ,T} \right)$. By our assumption $\Sigma \left( {t_{1} ,T} \right) = T > 0$ and using (Cambini and Martein (2009)) theorem, we have ${\Pi }_{c} \left( {t_{1} ,T,p} \right)$ is a pseudo-concave with respect to the optimal time point $\left( {t_{1} ,T} \right).$

• (b) If $\Delta_{1} < 0$, then clearly $\Gamma \left( x \right) < 0 \forall x \in \left[ {t_{d} ,t_{1}^{b} } \right).$

From Eq. (19) and Eq. (24), it can be written as $\frac{{\partial {\Pi }_{c} \left( {t_{1} ,T,p} \right)}}{\partial T} = \frac{D\left( p \right)}{{T^{2} }}\Gamma \left( {t_{1} } \right) < 0 \forall t_{1} \in \left[ {t_{d} ,t_{1}^{b} } \right)$ which implies ${\Pi }_{c} \left( {t_{1} ,T,p} \right)$ is strictly decreasing function of $T.$ Hence, ${\Pi }_{c} \left( {t_{1} ,T,p} \right)$ has a maximum value when $T = t_{d} - \frac{{\mu^{0} }}{{\delta \left( {\eta + \mu^{0} } \right)}}$ as $t_{1} = t_{d} .$ Therefore, ${\Pi }_{c} \left( {t_{1} ,T,p} \right)$ has a maximum value at $\left( {t_{1} ,T} \right).$

This completes the proof.

Appendix 4: Proof of Theorem 3

Second-order derivative of ${\Pi }_{c} \left( {t_{1} ,T,p} \right)$ with respect to $p$.

$$\frac{{\partial^{2} {\Pi }_{c} \left( {t_{1} ,T,p} \right)}}{{\partial p^{2} }} = \frac{1}{T}\left[ {\begin{array}{*{20}l} {\left( {2D^{\prime}\left( p \right) + pD^{\prime\prime}\left( p \right)} \right)t_{1} } \hfill \\ {\quad + \left( {2D^{\prime}\left( p \right) + \left( {p - c - \tau \cdot c_{e} } \right)D^{\prime\prime}\left( p \right)} \right)\frac{{\ln \left\{ {1 + \delta \left( {T - t_{d} } \right)} \right\}}}{\delta }} \hfill \\ {\quad + \left( {2\beta D^{\prime}\left( p \right) + (p\beta - c_{1} - \tau \cdot h_{e} )D^{\prime\prime}\left( p \right)} \right)} \hfill \\ {\quad \times \left[ {\left\{ {\psi \left( {\beta \left( {t_{d} - t_{1} } \right) + \kappa_{1} \ln \omega } \right) + \frac{{e^{{\beta t_{d} }} }}{\beta }} \right\}\left( {\frac{{1 - e^{{ - \beta t_{d} }} }}{\beta }} \right) - \frac{{t_{d} }}{\beta }} \right]} \hfill \\ {\quad + \left( {2\beta D^{\prime}\left( p \right) + (p\beta - c_{1} - \tau \cdot h_{e} + \beta c_{2} )D^{\prime\prime}\left( p \right)} \right)} \hfill \\ {\quad \times \left[ {\frac{{\pi \kappa_{1} }}{36} + \frac{1}{{\beta^{2} }}\left[ {\kappa_{6} \left( {\sigma_{1} + \beta \sigma_{2} } \right) + \beta^{2} \sigma_{3} } \right] - \frac{{\kappa_{5} \sigma_{1} + \beta \sigma_{2} }}{{\beta^{2} }}\left[ {\beta t_{1} + \kappa_{1} \ln \chi } \right]} \right]} \hfill \\ {\quad - \left( {c + \tau \cdot c_{e} } \right)D^{\prime\prime}\left( p \right)\left[ {\frac{1}{\beta }\left( {e^{{\beta t_{d} }} - 1} \right) + \psi \left\{ {\beta \left( {t_{d} - t_{1} } \right) + \kappa_{1} \ln \omega } \right\}} \right]} \hfill \\ {\quad - c_{2} D^{\prime\prime}\left( p \right)\left[ {\psi e^{{ - \beta t_{d} }} \left( {\kappa_{1} \ln \omega + \beta \left( {t_{d} - t_{1} } \right)} \right) - \left( {t_{1} - t_{d} } \right)} \right]} \hfill \\ {\quad - \left( {\frac{{c_{3} }}{\delta } + c_{4} } \right)D^{\prime\prime}\left( p \right)\left[ {\left( {T - t_{1} } \right) - \frac{{\ln \left\{ {1 + \delta \left( {T - t_{1} } \right)} \right\}}}{\delta }} \right]} \hfill \\ \end{array} } \right]$$

(34)

If the revenue $pD\left( p \right)$ is a strictly concave function of $p$, then $2D^{\prime}\left( p \right) + pD^{\prime\prime}\left( p \right) < 0$. Also, it is assumed that $D\left( p \right)$ is a decreasing function of $p$, so $D^{\prime}\left( p \right) < 0$ and $D^{^{\prime\prime}} \left( p \right) > 0$. Consequently,$\frac{{\partial^{2} {\Pi }_{c} \left( {t_{1} ,T,p} \right)}}{{\partial p^{2} }} < 0$, which means that the profit function ${\Pi }_{c} \left( {t_{1} ,T,p} \right)$, for any given value of $\left( {t_{1} ,T} \right)$, is a strictly concave function of $p,$ and hence, there exists a unique value of $p$ that maximizes ${\Pi }_{c} \left( {t_{1} ,T,p} \right).$