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An inventory model for imperfect quality items considering learning effects and partial trade credit policy

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Abstract

The incorporation of payment schemes and imperfect production has received considerable attention in the literature on inventory management; however, researchers have rarely considered these two challenges concurrently. The significance of employing both imperfect production and delay in payment is that both of these challenges affect the profit obtained directly. The total obtained profit can be mitigated by increasing the waste and reducing the number of sold perfect items and delay in payment stimulates the buyers to purchase items and settle their accounts in a specific period. This paper proposes a three-echelon inventory model that (a) Imperfect production is permissible; (b) Inspectors make errors during the inspection process and the errors can be diminished by learning from their previous performance; (c) It is practical to categorize customers into new and old types when old customers are prioritized to receive a full trade credit. As the retailer learns from the experience of screening, the probability of misclassification errors and inspection time decrease for two distinct cases of delay in payment to provide optimal replenishment and promotional decisions. To make a better connection to practical issues, a case study is elaborated. To validate the mathematical model, the impact of the inspector's learning on variables and the total profit is compared in different cases. The analysis shows that it is beneficial for the retailer to make shorter contracts, particularly with new customers to maximize the expected total profit per unit time. Eventually, a numerical experiment is conducted for each subcase to illustrate the impact of learning and determining the optimal policy for the retailer, then some implications for future contributions are outlined.

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Acknowledgements

Conceptualization, SM and MRG; Supervision, MRG; Writing-original draft, SM; Writing-review and editing, MRG and AS.

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

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Authors and Affiliations

  1. School of Industrial Engineering, Iran University of Science and Technology, Tehran, 1684613114, Iran

    Soosan Moradi, Mohammad Reza Gholamian & Arash Sepehri

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  1. Soosan Moradi
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  2. Mohammad Reza Gholamian
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  3. Arash Sepehri
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Correspondence to Mohammad Reza Gholamian.

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Appendices

Appendix 1

Expressions of \({\mathrm{X}}_{1}, {\mathrm{X}}_{2}, \dots, {\mathrm{X}}_{10}\) which are required for constructing the total profit of Subcase 1.1 are illustrated as follows.

$${\mathrm{X}}_{1}=\frac{1}{(1-\theta )}\left\{\mu (1-\theta )(1-{\mathrm{q}}_{1})+\mathrm{v}\left[\theta (1-{\mathrm{q}}_{2})+(1-\theta){\mathrm{q}}_{1}+\theta {\mathrm{q}}_{2}\right]-\upmu (1-\upgamma )(1-\theta)(1-{\mathrm{q}}_{1})(1-\beta)(1-\alpha)- \mathrm{C}-\mathrm{d}-{\mathrm{C}}_{1}(1-\theta){\mathrm{q}}_{1}- {\mathrm{C}}_{2}\theta {\mathrm{q}}_{2}+\frac{\theta {\mathrm{q}}_{2}{\mathrm{hw}}_{2}}{2}\right\}.$$
(60)
$${\mathrm{X}}_{2}=\frac{1}{\left(1-\theta \right)}\mathrm{S}\left(\left(1-\theta \right){\mathrm{q}}_{1}+\theta \left(1-{\mathrm{q}}_{2}\right)\right).$$
(61)
$${\mathrm{X}}_{3}=\frac{-1}{\left(1-\theta \right)}\left\{{\mathrm{q}}_{1}{\mathrm{w}}_{1}\left(-\mu \left(1-\theta \right)+\mathrm{v}\left(1-\theta \right)+\upmu \left(1-\gamma \right)\left(1-\theta \right)\left(1-\beta \right)\left(1-\alpha \right)-{\mathrm{C}}_{1}\left(1-\theta \right)\right)-{\mathrm{q}}_{2}{\mathrm{w}}_{2}{\mathrm{C}}_{2}\theta \right\}.$$
(62)
$${\mathrm{X}}_{4}=\frac{1}{\left(1-\theta \right)}\left\{{\mathrm{q}}_{1}{\mathrm{w}}_{1}\mathrm{S}(1-\theta )+\mathrm{S}\theta {\mathrm{q}}_{2}{\mathrm{w}}_{2}\right\}.$$
(63)
$${\mathrm{X}}_{5}=\left({\mathrm{CI}}_{\mathrm{P}}(\gamma \left(1-\alpha \right)\mathrm{S}-\alpha \left(\mathrm{N}-\mathrm{S}\right)-\left(1-\alpha \right)\left(1-\gamma \right)\left(\mathrm{N}-\mathrm{S}\right))\right).$$
(64)
$${\mathrm{X}}_{6}=\frac{1}{{\mathrm{x}}_{0}{\left(\left(1-\theta \right)\right)}^{2}}\left[\left(1-\theta \right){\mathrm{q}}_{1}{\mathrm{w}}_{1}-\theta {\mathrm{q}}_{2}{\mathrm{w}}_{2}\right].$$
(65)
$${\mathrm{X}}_{7}=\frac{1}{{\mathrm{x}}_{0}{\left(\left(1-\theta \right)\right)}^{2}}\left[\left(1-\theta \right){\mathrm{q}}_{1}+\theta -\theta {\mathrm{q}}_{2}\right].$$
(66)
$${\mathrm{X}}_{8}=\left(\frac{\mathrm{h}}{2}+\frac{{\mathrm{CI}}_{\mathrm{P}}\left(1-\alpha \right)\left(1-\gamma \right)}{2}+\frac{{\mathrm{CI}}_{\mathrm{P}}\alpha }{2}+\frac{{\mathrm{CI}}_{\mathrm{P}}\gamma \left(1-\alpha \right)}{2}\right).$$
(67)
$${\mathrm{X}}_{9}=\left(\frac{\theta {\mathrm{q}}_{2}\mathrm{h}}{2\left(1-\theta \right)}\right).$$
(68)
$${\mathrm{X}}_{10}=-\frac{1}{2}(\mu {\mathrm{I}}_{\mathrm{e}}(\gamma \left(1-\alpha \right){\mathrm{S}}^{2}+\alpha {\left(\mathrm{S}-\mathrm{N}\right)}^{2}+\left(1-\alpha \right)\left(1-\gamma \right)\beta {\left(\mathrm{S}-\mathrm{N}\right)}^{2})-{\mathrm{CI}}_{\mathrm{P}}\left(\gamma \left(1-\alpha \right){\mathrm{S}}^{2}+\left(1-\alpha \right)\left(1-\gamma \right){\left(\mathrm{N}-\mathrm{S}\right)}^{2}+\alpha {\left(\mathrm{N}-\mathrm{S}\right)}^{2}\right).$$
(69)

Expressions of \({\mathrm{X}}_{11}, {\mathrm{X}}_{12},\) and \({\mathrm{X}}_{13}\) which are required for constructing the total profit of Subcase 1.2 are illustrated as follows.

$${\mathrm{X}}_{11}=\mu {\mathrm{I}}_{\mathrm{e}}\gamma \left(1-\alpha \right)\mathrm{S}-{\mathrm{CI}}_{\mathrm{P}}\alpha \left(\mathrm{N}-\mathrm{S}\right)-{\mathrm{CI}}_{\mathrm{P}}\left(1-\alpha \right)\left(1-\gamma \right)\left(\mathrm{N}-\mathrm{S}\right).$$
(70)
$${\mathrm{X}}_{12}=\left(\frac{\mathrm{h}}{2}+\frac{\mu {\mathrm{I}}_{\mathrm{e}}\gamma \left(1-\alpha \right)}{2}+\frac{{\mathrm{CI}}_{\mathrm{P}}\left(1-\alpha \right)\left(1-\gamma \right)}{2}+\frac{{\mathrm{CI}}_{\mathrm{P}}\alpha }{2}\right).$$
(71)
$${\mathrm{X}}_{13}=-\frac{1}{2}\left(\mu {\mathrm{I}}_{\mathrm{e}}\left(\alpha {\left(\mathrm{S}-\mathrm{N}\right)}^{2}+\left(1-\alpha \right)\left(1-\gamma \right)\beta {\left(\mathrm{S}-\mathrm{N}\right)}^{2}\right)-{\mathrm{CI}}_{\mathrm{P}}\left(\left(1-\alpha \right)\left(1-\gamma \right){\left(\mathrm{N}-\mathrm{S}\right)}^{2}+\alpha {\left(\mathrm{N}-\mathrm{S}\right)}^{2}\right)\right).$$
(72)

Expressions of \({\mathrm{X}}_{14}, {\mathrm{X}}_{15}, \dots, {\mathrm{X}}_{19}\) which are required for constructing the total profit of Subcase 1.3 are illustrated as follows.

$${\mathrm{X}}_{14}=\frac{1}{\left(1-\theta \right)}\left(\mathrm{S}\theta {\mathrm{q}}_{2}+\theta {\mathrm{q}}_{2}{\mathrm{w}}_{2}\right).$$
(73)
$${\mathrm{X}}_{15}=\frac{\mathrm{S}\theta {\mathrm{q}}_{2}{\mathrm{w}}_{2}}{\left(1-\theta \right)}.$$
(74)
$${\mathrm{X}}_{16}=\left(\mu {\mathrm{I}}_{\mathrm{e}}(\alpha \left(\mathrm{S}-\mathrm{N}\right)+\left(1-\alpha \right)\gamma \mathrm{S}+\left(1-\alpha \right)\left(1-\gamma \right)\beta \left(\mathrm{S}-\mathrm{N}\right))\right).$$
(75)
$${\mathrm{X}}_{17}=\frac{{\mathrm{vI}}_{\mathrm{e}}\theta {\mathrm{q}}_{2}}{{\mathrm{x}}_{0}{\left(\left(1-\theta \right)\right)}^{2}}.$$
(76)
$${\mathrm{X}}_{18}=\frac{1}{2}\left(\mathrm{h}+\mu {\mathrm{I}}_{\mathrm{e}}(\alpha +\left(1-\gamma \right)\left(1-\alpha \right)\upbeta +\gamma (1-\alpha ))\right).$$
(77)
$${\mathrm{X}}_{19}=\frac{{\mathrm{vI}}_{\mathrm{e}}\theta {\mathrm{q}}_{2}}{(1-\theta )}.$$
(78)

Expression of \({\mathrm{X}}_{20}\) which is required for constructing the total profit of the Subcase 2.1 is illustrated as follows.

$${\mathrm{X}}_{20}=\left(\frac{{\mathrm{CI}}_{\mathrm{P}}\gamma \left(1-\alpha \right){\mathrm{S}}^{2}}{2}-\frac{\upmu {\mathrm{I}}_{\mathrm{e}}\gamma \left(1-\alpha \right){\mathrm{S}}^{2}}{2}\right).$$
(79)

Appendix 2

To satisfy the necessary conditions of concavity for Subcase 1.1, first and second-order derivatives of \({TP}_{1}\) are calculated. The second-order derivatives concerning variables should be negative.

The first derivative of \({TP}_{1}\) concerning \(e\) is obtained as follows.

$$\frac{\partial {TP}_{1}\left(e,T\right)}{\partial \mathrm{e}}=\frac{4\mathrm{b}{e}^{3}({\mathrm{X}}_{1}+{\mathrm{CI}}_{\mathrm{P}}{\mathrm{X}}_{2})}{\mathrm{F}(T)}+\frac{4\mathrm{b}{e}^{3}({\mathrm{X}}_{3}-{\mathrm{CI}}_{\mathrm{P}}{\mathrm{X}}_{4})}{T(\mathrm{F}(T))}+4\mathrm{b}{e}^{3}{\mathrm{X}}_{5}-\frac{3\mathrm{b}{e}^{2}{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\mathrm{y}\left(\mathrm{h}+{\mathrm{CI}}_{\mathrm{P}}\right)\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{T{\left(\mathrm{E}\left(T\right)\right)}^{2}{\left(\mathrm{F}\left(T\right)\right)}^{2}}+\frac{8\mathrm{b}{e}^{3}\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)\left(\mathrm{h}+{\mathrm{CI}}_{\mathrm{P}}\right)\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{\left(\mathrm{E}\left(T\right)\right){\left(\mathrm{F}\left(T\right)\right)}^{2}}-4\mathrm{b}{e}^{3}T\left({\mathrm{X}}_{8}+\frac{{\mathrm{X}}_{9}}{\mathrm{F}\left(T\right)}\right)-\frac{4{e}^{3}\left(\mathrm{G}+{\mathrm{bX}}_{10}\right)}{T}.$$
(80)

The second derivative of \({TP}_{1}\) concerning \(e\) is obtained as follows.

$$\frac{{\partial }^{2}{TP}_{1}\left(e,T\right)}{\partial {e}^{2}}=\frac{12b{e}^{2}({\mathrm{X}}_{1}+{\mathrm{CI}}_{\mathrm{P}}{\mathrm{X}}_{2})}{\mathrm{F}(T)}+\frac{12\mathrm{b}{e}^{2}({\mathrm{X}}_{3}-{\mathrm{CI}}_{\mathrm{P}}{\mathrm{X}}_{4})}{T\mathrm{F}(T)}+12\mathrm{b}{e}^{2}{\mathrm{X}}_{5}+\frac{18{\mathrm{b}}^{2}{e}^{4}{(\mathrm{a}+\mathrm{b}{e}^{4})}^{2}{\mathrm{y}}^{2}(\mathrm{h}+{\mathrm{CI}}_{\mathrm{P}})({\mathrm{X}}_{6}-T{\mathrm{X}}_{7})}{{T}^{2}{(\mathrm{E}(T))}^{3}{(\mathrm{F}(T))}^{2}}+\frac{\left(\mathrm{h}+{\mathrm{CI}}_{\mathrm{P}}\right)\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{\mathrm{E}(T)(\mathrm{F}{(T))}^{2}}\left(32{\mathrm{b}}^{2}{e}^{6}+24\mathrm{b}{e}^{2}\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)\right)-\frac{\mathrm{y}\left(\mathrm{h}+{\mathrm{CI}}_{\mathrm{P}}\right)\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{T{\left(\mathrm{E}\left(T\right)\right)}^{2}(\mathrm{F}{(T))}^{2}}\left(48{\mathrm{b}}^{2}{e}^{5}\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)+6\mathrm{b}e{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\right)-12\mathrm{b}{e}^{2}T\left({\mathrm{X}}_{8}+\frac{{\mathrm{X}}_{9}}{\mathrm{F}\left(T\right)}\right)-\frac{12{e}^{2}\left(\mathrm{G}+{\mathrm{bX}}_{10}\right)}{T}\le 0.$$
(81)

The first derivative of \({TP}_{1}\) concerning \(T\) is obtained as follows.

$$\frac{\partial {TP}_{1}\left(e,T\right)}{\partial T}=\frac{(\mathrm{a}+\mathrm{b}{e}^{4}){\mathrm{q}}_{1}{\mathrm{w}}_{1}({\mathrm{X}}_{1}+{\mathrm{CI}}_{\mathrm{P}}{\mathrm{X}}_{2})}{{T}^{2}{(\mathrm{F}(T))}^{2}}+\frac{(\mathrm{a}+\mathrm{b}{e}^{4}){\mathrm{q}}_{1}{\mathrm{w}}_{1}({\mathrm{X}}_{3}-{\mathrm{CI}}_{\mathrm{P}}{\mathrm{X}}_{4})}{{T}^{3}{(\mathrm{F}(T))}^{2}}-\frac{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)\left({\mathrm{X}}_{3}-{\mathrm{CI}}_{\mathrm{P}}{\mathrm{X}}_{4}\right)}{{T}^{2}\left(\mathrm{F}\left(T\right)\right)}-\frac{{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\left(\mathrm{h}+{\mathrm{CI}}_{\mathrm{P}}\right){\mathrm{X}}_{7}}{\left(\mathrm{E}\left(T\right)\right){\left(\mathrm{F}\left(T\right)\right)}^{2}}+\frac{2{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\left(\mathrm{h}+{\mathrm{CI}}_{\mathrm{P}}\right){\mathrm{q}}_{1}{\mathrm{w}}_{1}\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{{T}^{2}\left(\mathrm{E}\left(T\right)\right){\left(\mathrm{F}\left(T\right)\right)}^{3}}+\frac{\mathrm{b}{e}^{3}{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\mathrm{y}\left(\mathrm{h}+{\mathrm{CI}}_{\mathrm{P}}\right)\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{{\mathrm{T}}^{2}{\left(\mathrm{E}\left(T\right)\right)}^{2}{\left(\mathrm{F}\left(T\right)\right)}^{2}}-\frac{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right){\mathrm{q}}_{1}{\mathrm{w}}_{1}{\mathrm{X}}_{9}}{T{\left(\mathrm{F}\left(T\right)\right)}^{2}}-\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)\left({\mathrm{X}}_{8}+\frac{{\mathrm{X}}_{9}}{\mathrm{F}\left(T\right)}\right)+\frac{\mathrm{O}+{\mathrm{aX}}_{10}+{e}^{4}\left(\mathrm{G}+{\mathrm{bX}}_{10}\right)}{{T}^{2}}.$$
(82)

The second derivative of \({TP}_{1}\) concerning \(T\) is obtained as follows.

$$\frac{{\partial }^{2}{TP}_{1}\left(e,T\right)}{\partial {T}^{2}}=\frac{2(\mathrm{a}+\mathrm{b}{e}^{4}){\mathrm{q}}_{1}^{2}{\mathrm{w}}_{1}^{2}({\mathrm{X}}_{1}+{\mathrm{CI}}_{\mathrm{P}}{\mathrm{X}}_{2})}{{T}^{4}{(\mathrm{F}(T))}^{3}}-\frac{2\left(\mathrm{a}+\mathrm{b}{e}^{4}\right){\mathrm{q}}_{1}{\mathrm{w}}_{1}\left({\mathrm{X}}_{1}+{\mathrm{CI}}_{\mathrm{P}}{\mathrm{X}}_{2}\right)}{{T}^{3}{\left(\mathrm{F}\left(T\right)\right)}^{2}}+\frac{2\left(\mathrm{a}+\mathrm{b}{e}^{4}\right){\mathrm{q}}_{1}^{2}{\mathrm{w}}_{1}^{2}\left({\mathrm{X}}_{3}-{\mathrm{CI}}_{\mathrm{P}}{\mathrm{X}}_{4}\right)}{{T}^{5}{\left(\mathrm{F}\left(T\right)\right)}^{3}}-\frac{4\left(\mathrm{a}+\mathrm{b}{e}^{4}\right){\mathrm{q}}_{1}{\mathrm{w}}_{1}\left({\mathrm{X}}_{3}-{\mathrm{CI}}_{\mathrm{P}}{\mathrm{X}}_{4}\right)}{{T}^{4}{\left(\mathrm{F}\left(T\right)\right)}^{2}}+\frac{2\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)\left({\mathrm{X}}_{3}-{\mathrm{CI}}_{\mathrm{P}}{\mathrm{X}}_{4}\right)}{{T}^{3}\left(\mathrm{F}\left(T\right)\right)}-\frac{4{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\left(\mathrm{h}+{\mathrm{CI}}_{\mathrm{P}}\right){\mathrm{q}}_{1}{\mathrm{w}}_{1}{\mathrm{X}}_{7}}{{T}^{2}\left(\mathrm{E}\left(T\right)\right){\left(\mathrm{F}\left(T\right)\right)}^{3}}-\frac{2\mathrm{b}{e}^{3}{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\mathrm{y}\left(\mathrm{h}+{\mathrm{CI}}_{\mathrm{P}}\right){\mathrm{X}}_{7}}{{T}^{2}{\left(\mathrm{E}\left(T\right)\right)}^{2}{\left(\mathrm{F}\left(T\right)\right)}^{2}}+\frac{6{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\left(\mathrm{h}+{\mathrm{CI}}_{\mathrm{P}}\right){\mathrm{q}}_{1}^{2}{\mathrm{w}}_{1}^{2}\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{{T}^{4}\left(\mathrm{E}\left(T\right)\right){\left(\mathrm{F}\left(T\right)\right)}^{4}}+\frac{4\mathrm{b}{e}^{3}{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\mathrm{y}\left(\mathrm{h}+{\mathrm{CI}}_{\mathrm{P}}\right){\mathrm{q}}_{1}{\mathrm{w}}_{1}\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{{T}^{4}{\left(\mathrm{E}\left(T\right)\right)}^{2}{\left(\mathrm{F}\left(T\right)\right)}^{3}}-\frac{4{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\left(\mathrm{h}+{\mathrm{CI}}_{\mathrm{P}}\right){\mathrm{q}}_{1}{\mathrm{w}}_{1}\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{{T}^{3}\mathrm{E}\left(T\right){\left(\mathrm{F}\left(T\right)\right)}^{3}}+\frac{2{\mathrm{b}}^{2}{e}^{6}{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}{\mathrm{y}}^{2}\left(\mathrm{h}+{\mathrm{CI}}_{\mathrm{P}}\right)\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{{T}^{4}{\left(\mathrm{E}\left(T\right)\right)}^{3}{\left(\mathrm{F}\left(T\right)\right)}^{2}}-\frac{2\mathrm{b}{e}^{3}{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\mathrm{y}\left(\mathrm{h}+{\mathrm{CI}}_{\mathrm{P}}\right)\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{{T}^{3}{\left(\mathrm{E}\left(T\right)\right)}^{2}{\left(\mathrm{F}\left(T\right)\right)}^{2}}-\frac{2\left(\mathrm{a}+\mathrm{b}{e}^{4}\right){\mathrm{q}}_{1}^{2}{\mathrm{w}}_{1}^{2}{\mathrm{X}}_{9}}{{T}^{3}{\left(\mathrm{F}\left(T\right)\right)}^{3}}-\frac{2\left(\mathrm{O}+{\mathrm{aX}}_{10}+{e}^{4}\left(\mathrm{G}+{\mathrm{bX}}_{10}\right)\right)}{{T}^{3}}\le 0.$$
(83)

Due to the complexity of terms in second-order derivatives, the concavity of the total profit function is proved in the section of the numerical experiment.

To satisfy the sufficient condition of concavity for Subcase 1.1, the determinant of the corresponding Hessian matrix must be positive which is formulated as follows.

$$The\, determinant \,of \,H\left({TP}_{1}\left(e,T\right)\right)=\left|\begin{array}{cc}\frac{{\partial }^{2}{TP}_{1}\left(e,T\right)}{{e}^{2}}& \frac{{\partial }^{2}{TP}_{1}\left(e,T\right)}{\partial e\partial T}\\ \frac{{\partial }^{2}{TP}_{1}\left(e,T\right)}{\partial T\partial e}& \frac{{\partial }^{2}{TP}_{1}\left(e,T\right)}{\partial {T}^{2}}\end{array}\right|\ge 0.$$
(84)

Due to the complexity of terms in the Hessian matrix, it is not convenient to prove the positivity of the determinant of the Hessian matrix. Therefore, the positivity of the Hessian matrix is proved in a numerical experiment.

Appendix 3

To satisfy the necessary conditions of concavity for Subcase 1.2, first and second-order derivatives of \({TP}_{2}\) are calculated. The second-order derivatives concerning variables should be negative.

The first derivative of \({TP}_{2}\) concerning \(e\) is obtained as follows.

$$\frac{\partial {TP}_{2}\left(e,T\right)}{\partial e}=\frac{4\mathrm{b}{e}^{3}({\mathrm{X}}_{1}+{\mathrm{vI}}_{\mathrm{e}}{\mathrm{X}}_{2})}{\mathrm{F}(T)}+\frac{4\mathrm{b}{e}^{3}({\mathrm{X}}_{3}-{\mathrm{vI}}_{\mathrm{e}}{\mathrm{X}}_{4})}{T(\mathrm{F}(T))}+4\mathrm{b}{e}^{3}{\mathrm{X}}_{11}-\frac{3\mathrm{b}{e}^{2}{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\mathrm{y}\left(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}}\right)\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{T{\left(\mathrm{E}\left(T\right)\right)}^{2}{\left(\mathrm{F}\left(T\right)\right)}^{2}}+\frac{8\mathrm{b}{e}^{3}\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)\left(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}}\right)\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{\left(\mathrm{E}\left(T\right)\right){\left(\mathrm{F}\left(T\right)\right)}^{2}}-4\mathrm{b}{e}^{3}T\left({\mathrm{X}}_{12}+\frac{{\mathrm{X}}_{9}}{\mathrm{F}\left(T\right)}\right)-\frac{4{e}^{3}\left(\mathrm{G}+{\mathrm{bX}}_{13}\right)}{T}.$$
(85)

The second derivative of \({TP}_{2}\) concerning \(e\) is obtained as follows.

$$\frac{{\partial }^{2}{TP}_{2}\left(e,T\right)}{\partial {e}^{2}}=\frac{12\mathrm{b}{e}^{2}({\mathrm{X}}_{1}+{\mathrm{vI}}_{\mathrm{e}}{\mathrm{X}}_{2})}{\mathrm{F}(T)}+\frac{12\mathrm{b}{e}^{2}({\mathrm{X}}_{3}-{\mathrm{vI}}_{\mathrm{e}}{\mathrm{X}}_{4})}{T\mathrm{F}(T)}+12\mathrm{b}{e}^{2}{\mathrm{X}}_{11}+\frac{18{\mathrm{b}}^{2}{e}^{4}{(\mathrm{a}+\mathrm{b}{e}^{4})}^{2}{\mathrm{y}}^{2}(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}})({\mathrm{X}}_{6}-T{\mathrm{X}}_{7})}{{T}^{2}{(\mathrm{E}(T))}^{3}{(\mathrm{F}(T))}^{2}}+\frac{\left(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}}\right)\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{\mathrm{E}(T)(\mathrm{F}{(T))}^{2}}\left(32{\mathrm{b}}^{2}{e}^{6}+24\mathrm{b}{e}^{2}\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)\right)-\frac{\mathrm{y}\left(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}}\right)\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{T{\left(\mathrm{E}\left(T\right)\right)}^{2}(\mathrm{F}{(T))}^{2}}\left(48{\mathrm{b}}^{2}{e}^{5}\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)+6\mathrm{b}e{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\right)-12\mathrm{b}{e}^{2}T\left({\mathrm{X}}_{12 }+\frac{{\mathrm{X}}_{9}}{\mathrm{F}\left(T\right)}\right)-\frac{12{e}^{2}\left(\mathrm{G}+{\mathrm{bX}}_{13}\right)}{T}\le 0.$$
(86)

The first derivative of \({TP}_{2}\) concerning \(T\) is obtained as follows.

$$\frac{\partial {TP}_{2}\left(e,T\right)}{\partial \mathrm{T}}=\frac{(\mathrm{a}+\mathrm{b}{e}^{4}){\mathrm{q}}_{1}{\mathrm{w}}_{1}({\mathrm{X}}_{1}+{\mathrm{vI}}_{\mathrm{e}}{\mathrm{X}}_{2})}{{T}^{2}{(\mathrm{F}(T))}^{2}}+\frac{(\mathrm{a}+\mathrm{b}{e}^{4}){\mathrm{q}}_{1}{\mathrm{w}}_{1}({\mathrm{X}}_{3}-{\mathrm{vI}}_{\mathrm{e}}{\mathrm{X}}_{4})}{{T}^{3}{(\mathrm{F}(T))}^{2}}-\frac{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)\left({\mathrm{X}}_{3}-{\mathrm{vI}}_{\mathrm{e}}{\mathrm{X}}_{4}\right)}{{T}^{2}\left(\mathrm{F}\left(T\right)\right)}-\frac{{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\left(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}}\right){\mathrm{X}}_{7}}{\left(\mathrm{E}\left(T\right)\right){\left(\mathrm{F}\left(T\right)\right)}^{2}}+\frac{2{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\left(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}}\right){\mathrm{q}}_{1}{\mathrm{w}}_{1}\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{{T}^{2}\left(\mathrm{E}\left(T\right)\right){\left(\mathrm{F}\left(T\right)\right)}^{3}}+\frac{\mathrm{b}{e}^{3}{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\mathrm{y}\left(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}}\right)\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{{T}^{2}{\left(\mathrm{E}\left(T\right)\right)}^{2}{\left(\mathrm{F}\left(T\right)\right)}^{2}}-\frac{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right){\mathrm{q}}_{1}{\mathrm{w}}_{1}{\mathrm{X}}_{9}}{T{\left(\mathrm{F}\left(T\right)\right)}^{2}}-\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)\left({\mathrm{X}}_{12}+\frac{{\mathrm{X}}_{9}}{\mathrm{F}\left(T\right)}\right)+\frac{\mathrm{O}+{\mathrm{aX}}_{13}+{e}^{4}\left(\mathrm{G}+{\mathrm{bX}}_{13}\right)}{{T}^{2}}.$$
(87)

The second derivative of \({TP}_{2}\) concerning \(T\) is obtained as follows.

$$\frac{{\partial }^{2}{TP}_{2}\left(e,T\right)}{\partial {T}^{2}}=\frac{2(\mathrm{a}+\mathrm{b}{e}^{4}){\mathrm{q}}_{1}^{2}{\mathrm{w}}_{1}^{2}({\mathrm{X}}_{1}+{\mathrm{vI}}_{\mathrm{e}}{\mathrm{X}}_{2})}{{T}^{4}{(\mathrm{F}(T))}^{3}}-\frac{2\left(\mathrm{a}+\mathrm{b}{e}^{4}\right){\mathrm{q}}_{1}{\mathrm{w}}_{1}\left({\mathrm{X}}_{1}+{\mathrm{vI}}_{\mathrm{e}}{\mathrm{X}}_{2}\right)}{{T}^{3}{\left(F\left(T\right)\right)}^{2}}+\frac{2\left(\mathrm{a}+\mathrm{b}{e}^{4}\right){\mathrm{q}}_{1}^{2}{\mathrm{w}}_{1}^{2}\left({\mathrm{X}}_{3}-{\mathrm{vI}}_{\mathrm{e}}{\mathrm{X}}_{4}\right)}{{T}^{5}{\left(F\left(T\right)\right)}^{3}}-\frac{4\left(\mathrm{a}+\mathrm{b}{e}^{4}\right){\mathrm{q}}_{1}{\mathrm{w}}_{1}\left({\mathrm{X}}_{3}-{\mathrm{vI}}_{\mathrm{e}}{\mathrm{X}}_{4}\right)}{{T}^{4}{\left(F\left(T\right)\right)}^{2}}+\frac{2\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)\left({\mathrm{X}}_{3}-{\mathrm{vI}}_{\mathrm{e}}{\mathrm{X}}_{4}\right)}{{T}^{3}\left(F\left(T\right)\right)}-\frac{4{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\left(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}}\right){\mathrm{q}}_{1}{\mathrm{w}}_{1}{\mathrm{X}}_{7}}{{T}^{2}\left(E\left(T\right)\right){\left(F\left(T\right)\right)}^{3}}-\frac{2\mathrm{b}{e}^{3}{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\mathrm{y}\left(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}}\right){\mathrm{X}}_{7}}{{T}^{2}{\left(E\left(T\right)\right)}^{2}{\left(F\left(T\right)\right)}^{2}}+\frac{6{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\left(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}}\right){\mathrm{q}}_{1}^{2}{\mathrm{w}}_{1}^{2}\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{{T}^{4}\left(E\left(T\right)\right){\left(F\left(T\right)\right)}^{4}}+\frac{4\mathrm{b}{e}^{3}{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\mathrm{y}\left(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}}\right){\mathrm{q}}_{1}{\mathrm{w}}_{1}\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{{T}^{4}{\left(E\left(T\right)\right)}^{2}{\left(F\left(T\right)\right)}^{3}}-\frac{4{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\left(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}}\right){\mathrm{q}}_{1}{\mathrm{w}}_{1}\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{{T}^{3}E\left(T\right){\left(F\left(T\right)\right)}^{3}}+\frac{2{\mathrm{b}}^{2}{e}^{6}{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}{\mathrm{y}}^{2}\left(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}}\right)\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{{T}^{4}{\left(E\left(T\right)\right)}^{3}{\left(F\left(T\right)\right)}^{2}}-\frac{2\mathrm{b}{e}^{3}{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\mathrm{y}\left(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}}\right)\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{{T}^{3}{\left(E\left(T\right)\right)}^{2}{\left(F\left(T\right)\right)}^{2}}-\frac{2\left(\mathrm{a}+\mathrm{b}{e}^{4}\right){\mathrm{q}}_{1}^{2}{\mathrm{w}}_{1}^{2}{\mathrm{X}}_{9}}{{T}^{3}{\left(\mathrm{F}\left(T\right)\right)}^{3}}-\frac{2\left(\mathrm{O}+{\mathrm{aX}}_{13}+{e}^{4}\left(\mathrm{G}+{\mathrm{bX}}_{13}\right)\right)}{{T}^{3}}\le 0.$$
(88)

Due to the complexity of terms in second-order derivatives, the concavity of the total profit function is proved in the section of the numerical experiment.

To satisfy the sufficient condition of concavity for Subcase 1.2, the determinant of the corresponding Hessian matrix must be positive which is formulated as follows.

$${{The \; determinant\; of}} \,H\left({TP}_{2}\left(e,T\right)\right)=\left|\begin{array}{cc}\frac{{\partial }^{2}{TP}_{2}\left(e,T\right)}{{e}^{2}}& \frac{{\partial }^{2}{TP}_{2}\left(e,T\right)}{\partial e\partial T}\\ \frac{{\partial }^{2}{TP}_{2}\left(e,T\right)}{\partial T\partial e}& \frac{{\partial }^{2}{TP}_{2}\left(e,T\right)}{\partial {T}^{2}}\end{array}\right|\ge 0.$$
(89)

Due to the complexity of terms in the Hessian matrix, it is not convenient to prove the positivity of the determinant of the Hessian matrix. Therefore, the positivity of the Hessian matrix is proved in a numerical experiment.

Appendix 4

To satisfy the necessary conditions of concavity for Subcase 1.3, first and second-order derivatives of \({TP}_{3}\) are calculated. The second-order derivatives concerning variables should be negative.

The first derivative of \({TP}_{3}\) concerning \(e\) is obtained as follows.

$$\frac{\partial {TP}_{3}\left(e,T\right)}{\partial e}=-\frac{4{e}^{3}\mathrm{G}}{\mathrm{T}}-\frac{3\mathrm{b}{e}^{2}{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\mathrm{y}\left(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}}\right)\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{\mathrm{T}{\left(\mathrm{E}\left(\mathrm{T}\right)\right)}^{2}{\left(\mathrm{F}\left(\mathrm{T}\right)\right)}^{2}}+\frac{4\mathrm{b}{e}^{3}\left({\mathrm{X}}_{1}+{\mathrm{vI}}_{\mathrm{e}}\left({\mathrm{X}}_{2}+{\mathrm{X}}_{14}\right)\right)}{\mathrm{F}\left(T\right)}+\frac{4\mathrm{b}{e}^{3}\left({\mathrm{X}}_{3}+{\mathrm{vI}}_{\mathrm{e}}\left(-{\mathrm{X}}_{4}+{\mathrm{X}}_{15}\right)\right)}{T\mathrm{F}\left(T\right)}+4\mathrm{b}{e}^{3}{\mathrm{X}}_{16}+8\mathrm{b}{e}^{3}\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)\left(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}}\right)\left(\frac{{\mathrm{X}}_{6}-T{\mathrm{X}}_{7}}{E\left(T\right){\left(F\left(T\right)\right)}^{2}}-T{\mathrm{X}}_{17}+{\mathrm{W}}_{2}{\mathrm{X}}_{17}\right)-4\mathrm{b}{e}^{3}T\left({\mathrm{X}}_{18}+\frac{{\mathrm{X}}_{9}+{\mathrm{X}}_{19}}{\mathrm{F}\left(T\right)}\right).$$
(90)

The second derivative of \({TP}_{3}\) concerning \(e\) is obtained as follows.

$$\frac{{\partial }^{2}{TP}_{3}\left(e,T\right)}{\partial {e}^{2}}=\frac{18{\mathrm{b}}^{2}{e}^{4}{(\mathrm{a}+\mathrm{b}{e}^{4})}^{2}{\mathrm{y}}^{2}(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}})({\mathrm{X}}_{6}-T{\mathrm{X}}_{7})}{{T}^{2}{(E(T))}^{3}{(F(T))}^{2}}+\frac{12\mathrm{b}{e}^{2}({\mathrm{X}}_{1}+{\mathrm{vI}}_{\mathrm{e}}({\mathrm{X}}_{2}+{\mathrm{X}}_{14}))}{F(T)}+\frac{12\mathrm{b}{e}^{2}({\mathrm{X}}_{3}+{\mathrm{vI}}_{\mathrm{e}}(-{\mathrm{X}}_{4}+{\mathrm{X}}_{15}))}{TF(T)}+12\mathrm{b}{e}^{2}{\mathrm{X}}_{16}+32{\mathrm{b}}^{2}{e}^{6}\left(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}}\right)\left(\frac{{\mathrm{X}}_{6}-T{\mathrm{X}}_{7}}{E\left(T\right){\left(F\left(T\right)\right)}^{2}}-T{\mathrm{X}}_{17}+{\mathrm{w}}_{2}{\mathrm{X}}_{17}\right)+24\mathrm{b}{e}^{2}\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)\left(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}}\right)\left(\frac{{\mathrm{X}}_{6}-{\mathrm{TX}}_{7}}{E\left(T\right){\left(F\left(T\right)\right)}^{2}}-T{\mathrm{X}}_{17}+{\mathrm{w}}_{2}{\mathrm{X}}_{17}\right)-12\mathrm{b}{e}^{2}T\left({\mathrm{X}}_{18}+\frac{{\mathrm{X}}_{9}+{\mathrm{X}}_{19}}{F\left(T\right)}\right)-\frac{12{e}^{2}\mathrm{G}}{T}-\frac{48{\mathrm{b}}^{2}{e}^{5}\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)\mathrm{y}\left(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}}\right)\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{T{\left(E\left(T\right)\right)}^{2}{\left(F\left(T\right)\right)}^{2}}-\frac{6\mathrm{b}e{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\mathrm{y}\left(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}}\right)\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{T{\left(E\left(T\right)\right)}^{2}{\left(F\left(T\right)\right)}^{2}}\le 0.$$
(91)

The first derivative of \({TP}_{3}\) concerning \(T\) is obtained as follows.

$$\frac{\partial {TP}_{3}\left(e,T\right)}{\partial T}=\frac{{e}^{4}\mathrm{G}+\mathrm{O}}{{T}^{2}}+\frac{(\mathrm{a}+\mathrm{b}{e}^{4}){\mathrm{q}}_{1}{\mathrm{w}}_{1}({\mathrm{X}}_{1}+{\mathrm{vI}}_{\mathrm{e}}({\mathrm{X}}_{2}+{\mathrm{X}}_{14}))}{{T}^{2}{(F(T))}^{2}}+\frac{(\mathrm{a}+\mathrm{b}{e}^{4}){\mathrm{q}}_{1}{\mathrm{w}}_{1}({\mathrm{X}}_{3}+{\mathrm{vI}}_{\mathrm{e}}(-{\mathrm{X}}_{4}+{\mathrm{X}}_{15}))}{{T}^{3}{(F(T))}^{2}}-\frac{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)\left({\mathrm{X}}_{3}+{\mathrm{vI}}_{\mathrm{e}}\left(-{\mathrm{X}}_{4}+{\mathrm{X}}_{15}\right)\right)}{{T}^{2}F\left(T\right)}+{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\left(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}}\right)\left(-\frac{{\mathrm{X}}_{7}}{E\left(T\right){\left(F\left(T\right)\right)}^{2}}+\frac{2{\mathrm{q}}_{1}{\mathrm{w}}_{1}\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{{T}^{2}E\left(T\right){\left(F\left(T\right)\right)}^{3}}+\frac{\mathrm{b}{e}^{3}\mathrm{y}\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{{T}^{2}{\left(E\left(T\right)\right)}^{2}{\left(F\left(T\right)\right)}^{2}}-{\mathrm{X}}_{17}\right)-\frac{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right){\mathrm{q}}_{1}{\mathrm{w}}_{1}\left({\mathrm{X}}_{9}+{\mathrm{X}}_{19}\right)}{T{\left(F\left(T\right)\right)}^{2}}-\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)\left({\mathrm{X}}_{18}+\frac{{\mathrm{X}}_{9}+{\mathrm{X}}_{19}}{F\left(T\right)}\right).$$
(92)

The second derivative of \({TP}_{3}\) concerning \(T\) is obtained as follows.

$$\frac{{\partial }^{2}{TP}_{3}\left(e,T\right)}{\partial {T}^{2}}=-\frac{2\left({e}^{4}\mathrm{G}+\mathrm{O}\right)}{{T}^{3}}+{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\left(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}}\right)\left(-\frac{4{\mathrm{q}}_{1}{\mathrm{w}}_{1}{\mathrm{X}}_{7}}{{T}^{2}E\left(T\right){\left(F\left(T\right)\right)}^{3}}-\frac{2\mathrm{b}{e}^{3}{\mathrm{yX}}_{7}}{{T}^{2}{\left(E\left(T\right)\right)}^{2}{\left(F\left(T\right)\right)}^{2}}+\frac{6{\mathrm{q}}_{1}^{2}{\mathrm{w}}_{1}^{2}\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{{T}^{4}E\left(T\right){\left(F\left(T\right)\right)}^{4}}+\frac{4\mathrm{b}{e}^{3}{\mathrm{yq}}_{1}{\mathrm{w}}_{1}\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{{T}^{4}{\left(E\left(T\right)\right)}^{2}{\left(F\left(T\right)\right)}^{3}}-\frac{4{\mathrm{q}}_{1}{\mathrm{w}}_{1}\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{{T}^{3}E\left(T\right){\left(F\left(T\right)\right)}^{3}}+\frac{2{\mathrm{b}}^{2}{e}^{6}{\mathrm{y}}^{2}\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{{T}^{4}{\left(E\left(T\right)\right)}^{3}{\left(F\left(T\right)\right)}^{2}}-\frac{2\mathrm{b}{e}^{3}\mathrm{y}\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{{T}^{3}{\left(E\left(T\right)\right)}^{2}{\left(F\left(T\right)\right)}^{2}}\right)+\frac{2\left(\mathrm{a}+{\mathrm{be}}^{4}\right){\mathrm{q}}_{1}^{2}{\mathrm{w}}_{1}^{2}\left({\mathrm{X}}_{1}+{\mathrm{vI}}_{\mathrm{e}}\left({\mathrm{X}}_{2}+{\mathrm{X}}_{14}\right)\right)}{{\mathrm{T}}^{4}{\left(\mathrm{F}\left(\mathrm{T}\right)\right)}^{3}}-\frac{2\left(\mathrm{a}+{\mathrm{be}}^{4}\right){\mathrm{q}}_{1}{\mathrm{w}}_{1}\left({\mathrm{X}}_{1}+{\mathrm{vI}}_{\mathrm{e}}\left({\mathrm{X}}_{2}+{\mathrm{X}}_{14}\right)\right)}{{T}^{3}{\left(F\left(T\right)\right)}^{2}}+\frac{2\left(\mathrm{a}+{\mathrm{be}}^{4}\right){\mathrm{q}}_{1}^{2}{\mathrm{w}}_{1}^{2}\left({\mathrm{X}}_{3}+{\mathrm{vI}}_{\mathrm{e}}\left(-{\mathrm{X}}_{4}+{\mathrm{X}}_{15}\right)\right)}{{T}^{5}{\left(F\left(T\right)\right)}^{3}}-\frac{4\left(\mathrm{a}+\mathrm{b}{e}^{4}\right){\mathrm{q}}_{1}{\mathrm{w}}_{1}\left({\mathrm{X}}_{3}+{\mathrm{vI}}_{\mathrm{e}}\left(-{\mathrm{X}}_{4}+{\mathrm{X}}_{15}\right)\right)}{{T}^{4}{\left(F\left(T\right)\right)}^{2}}+\frac{2\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)\left({\mathrm{X}}_{3}+{\mathrm{vI}}_{\mathrm{e}}\left(-{\mathrm{X}}_{4}+{\mathrm{X}}_{15}\right)\right)}{{T}^{3}F\left(T\right)}-\frac{2\left(\mathrm{a}+\mathrm{b}{e}^{4}\right){\mathrm{q}}_{1}^{2}{\mathrm{w}}_{1}^{2}\left({\mathrm{X}}_{9}+{\mathrm{X}}_{19}\right)}{{T}^{3}{\left(F\left(T\right)\right)}^{3}}\le 0.$$
(93)

Due to the complexity of terms in second-order derivatives, the concavity of the total profit function is proved in the section of the numerical experiment.

To satisfy the sufficient condition of concavity for Subcase 1.3, the determinant of the corresponding Hessian matrix must be positive which is formulated as follows.

$${{The \;determinant \;of}}\, H\left({TP}_{3}\left(e,T\right)\right)=\left|\begin{array}{cc}\frac{{\partial }^{2}{TP}_{3}\left(e,T\right)}{{e}^{2}}& \frac{{\partial }^{2}{TP}_{3}\left(e,T\right)}{\partial e\partial T}\\ \frac{{\partial }^{2}{TP}_{3}\left(e,T\right)}{\partial T\partial e}& \frac{{\partial }^{2}{TP}_{3}\left(e,T\right)}{\partial {T}^{2}}\end{array}\right|\ge 0.$$
(94)

Due to the complexity of terms in the Hessian matrix, it is not convenient to prove the positivity of the determinant of the Hessian matrix. Therefore, the positivity of the Hessian matrix is proved in a numerical experiment.

Appendix 5

To satisfy the necessary conditions of concavity for Subcase 2.1, first and second-order derivatives of \({TP}_{4}\) are calculated. The second-order derivatives concerning variables should be negative.

The first derivative of \({TP}_{4}\) concerning \(e\) is obtained as follows.

$$\frac{\partial {TP}_{4}\left(e,T\right)}{\partial \mathrm{e}}=\frac{4\mathrm{b}{e}^{3}({\mathrm{X}}_{1}+{\mathrm{CI}}_{\mathrm{P}}{\mathrm{X}}_{2})}{F(T)}+\frac{4\mathrm{b}{e}^{3}({\mathrm{X}}_{3}-{\mathrm{CI}}_{\mathrm{P}}{\mathrm{X}}_{4})}{T(F(T))}+4\mathrm{b}{e}^{3}{\mathrm{X}}_{5}-\frac{3\mathrm{b}{e}^{2}{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\mathrm{y}\left(\mathrm{h}+{\mathrm{CI}}_{\mathrm{P}}\right)\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{T{\left(E\left(T\right)\right)}^{2}{\left(F\left(T\right)\right)}^{2}}+\frac{8\mathrm{b}{e}^{3}\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)\left(\mathrm{h}+{\mathrm{CI}}_{\mathrm{P}}\right)\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{\left(E\left(T\right)\right){\left(F\left(T\right)\right)}^{2}}-4\mathrm{b}{e}^{3}\mathrm{T}\left({\mathrm{X}}_{8}+\frac{{\mathrm{X}}_{9}}{F\left(T\right)}\right)-\frac{4{e}^{3}\left(\mathrm{G}+{\mathrm{bX}}_{20}\right)}{T}.$$
(95)

The second derivative of \({TP}_{4}\) concerning \(e\) is obtained as follows.

$$\frac{{\partial }^{2}{TP}_{4}\left(e,T\right)}{\partial {e}^{2}}=\frac{12\mathrm{b}{e}^{2}({\mathrm{X}}_{1}+{\mathrm{CI}}_{\mathrm{P}}{\mathrm{X}}_{2})}{F(T)}+\frac{12\mathrm{b}{e}^{2}({\mathrm{X}}_{3}-{\mathrm{CI}}_{\mathrm{P}}{\mathrm{X}}_{4})}{TF(T)}+12\mathrm{b}{e}^{2}{\mathrm{X}}_{5}+\frac{18{\mathrm{b}}^{2}{e}^{4}{(\mathrm{a}+\mathrm{b}{e}^{4})}^{2}{\mathrm{y}}^{2}(\mathrm{h}+{\mathrm{CI}}_{\mathrm{P}})({\mathrm{X}}_{6}-T{\mathrm{X}}_{7})}{{T}^{2}{(E(T))}^{3}{(F(T))}^{2}}+\frac{\left(\mathrm{h}+{\mathrm{CI}}_{\mathrm{P}}\right)\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{E(T)(F{(T))}^{2}}\left(32{\mathrm{b}}^{2}{e}^{6}+24\mathrm{b}{e}^{2}\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)\right)-\frac{\mathrm{y}\left(\mathrm{h}+{\mathrm{CI}}_{\mathrm{P}}\right)\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{T{\left(E\left(T\right)\right)}^{2}(F{(T))}^{2}}\left(48{\mathrm{b}}^{2}{e}^{5}\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)+6\mathrm{b}e{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\right)-12\mathrm{b}{e}^{2}T\left({\mathrm{X}}_{8}+\frac{{\mathrm{X}}_{9}}{F\left(T\right)}\right)-\frac{12{e}^{2}\left(\mathrm{G}+{\mathrm{bX}}_{20}\right)}{T}\le 0.$$
(96)

The first derivative of \({TP}_{4}\) concerning \(T\) is obtained as follows.

$$\frac{\partial {TP}_{4}\left(e,T\right)}{\partial T}=\frac{(\mathrm{a}+\mathrm{b}{e}^{4}){\mathrm{q}}_{1}{\mathrm{w}}_{1}({\mathrm{X}}_{1}+{\mathrm{CI}}_{\mathrm{P}}{\mathrm{X}}_{2})}{{T}^{2}{(\mathrm{F}(T))}^{2}}+\frac{(\mathrm{a}+\mathrm{b}{e}^{4}){\mathrm{q}}_{1}{\mathrm{w}}_{1}({\mathrm{X}}_{3}-{\mathrm{CI}}_{\mathrm{P}}{\mathrm{X}}_{4})}{{T}^{3}{(F(T))}^{2}}-\frac{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)\left({\mathrm{X}}_{3}-{\mathrm{CI}}_{\mathrm{P}}{\mathrm{X}}_{4}\right)}{{T}^{2}\left(F\left(T\right)\right)}-\frac{{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\left(\mathrm{h}+{\mathrm{CI}}_{\mathrm{P}}\right){\mathrm{X}}_{7}}{\left(E\left(T\right)\right){\left(F\left(T\right)\right)}^{2}}+\frac{2{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\left(\mathrm{h}+{\mathrm{CI}}_{\mathrm{P}}\right){\mathrm{q}}_{1}{\mathrm{w}}_{1}\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{{T}^{2}\left(E\left(T\right)\right){\left(F\left(T\right)\right)}^{3}}+\frac{{\mathrm{be}}^{3}{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\mathrm{y}\left(\mathrm{h}+{\mathrm{CI}}_{\mathrm{P}}\right)\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{{T}^{2}{\left(E\left(T\right)\right)}^{2}{\left(F\left(T\right)\right)}^{2}}-\frac{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right){\mathrm{q}}_{1}{\mathrm{w}}_{1}{\mathrm{X}}_{9}}{T{\left(F\left(T\right)\right)}^{2}}-\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)\left({\mathrm{X}}_{8}+\frac{{\mathrm{X}}_{9}}{F\left(T\right)}\right)+\frac{\mathrm{O}+{\mathrm{aX}}_{20}+{e}^{4}\left(\mathrm{G}+{\mathrm{bX}}_{20}\right)}{{T}^{2}}.$$
(97)

The second derivative of \({TP}_{4}\) concerning \(T\) is obtained as follows.

$$\frac{{\partial }^{2}{TP}_{4}\left(e,T\right)}{\partial {T}^{2}}=\frac{2(\mathrm{a}+\mathrm{b}{e}^{4}){\mathrm{q}}_{1}^{2}{\mathrm{w}}_{1}^{2}({\mathrm{X}}_{1}+{\mathrm{CI}}_{\mathrm{P}}{\mathrm{X}}_{2})}{{T}^{4}{(F(T))}^{3}}-\frac{2\left(\mathrm{a}+\mathrm{b}{e}^{4}\right){\mathrm{q}}_{1}{\mathrm{w}}_{1}\left({\mathrm{X}}_{1}+{\mathrm{CI}}_{\mathrm{P}}{\mathrm{X}}_{2}\right)}{{T}^{3}{\left(F\left(T\right)\right)}^{2}}+\frac{2\left(\mathrm{a}+\mathrm{b}{e}^{4}\right){\mathrm{q}}_{1}^{2}{\mathrm{w}}_{1}^{2}\left({\mathrm{X}}_{3}-{\mathrm{CI}}_{\mathrm{P}}{\mathrm{X}}_{4}\right)}{{T}^{5}{\left(F\left(T\right)\right)}^{3}}-\frac{4\left(\mathrm{a}+\mathrm{b}{e}^{4}\right){\mathrm{q}}_{1}{\mathrm{w}}_{1}\left({\mathrm{X}}_{3}-{\mathrm{CI}}_{\mathrm{P}}{\mathrm{X}}_{4}\right)}{{T}^{4}{\left(F\left(T\right)\right)}^{2}}+\frac{2\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)\left({\mathrm{X}}_{3}-{\mathrm{CI}}_{\mathrm{P}}{\mathrm{X}}_{4}\right)}{{T}^{3}\left(F\left(T\right)\right)}-\frac{4{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\left(\mathrm{h}+{\mathrm{CI}}_{\mathrm{P}}\right){\mathrm{q}}_{1}{\mathrm{w}}_{1}{\mathrm{X}}_{7}}{{T}^{2}\left(E\left(T\right)\right){\left(F\left(T\right)\right)}^{3}}-\frac{2\mathrm{b}{e}^{3}{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\mathrm{y}\left(\mathrm{h}+{\mathrm{CI}}_{\mathrm{P}}\right){\mathrm{X}}_{7}}{{T}^{2}{\left(E\left(T\right)\right)}^{2}{\left(F\left(T\right)\right)}^{2}}+\frac{6{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\left(\mathrm{h}+{\mathrm{CI}}_{\mathrm{P}}\right){\mathrm{q}}_{1}^{2}{\mathrm{w}}_{1}^{2}\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{{T}^{4}\left(E\left(T\right)\right){\left(F\left(T\right)\right)}^{4}}+\frac{4\mathrm{b}{e}^{3}{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\mathrm{y}\left(\mathrm{h}+{\mathrm{CI}}_{\mathrm{P}}\right){\mathrm{q}}_{1}{\mathrm{w}}_{1}\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{{T}^{4}{\left(E\left(T\right)\right)}^{2}{\left(F\left(T\right)\right)}^{3}}-\frac{4{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\left(\mathrm{h}+{\mathrm{CI}}_{\mathrm{P}}\right){\mathrm{q}}_{1}{\mathrm{w}}_{1}\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{{T}^{3}E\left(T\right){\left(F\left(T\right)\right)}^{3}}+\frac{2{\mathrm{b}}^{2}{e}^{6}{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}{\mathrm{y}}^{2}\left(\mathrm{h}+{\mathrm{CI}}_{\mathrm{P}}\right)\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{{T}^{4}{\left(E\left(T\right)\right)}^{3}{\left(F\left(T\right)\right)}^{2}}-\frac{2\mathrm{b}{e}^{3}{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\mathrm{y}\left(\mathrm{h}+{\mathrm{CI}}_{\mathrm{P}}\right)\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{{T}^{3}{\left(E\left(T\right)\right)}^{2}{\left(F\left(T\right)\right)}^{2}}-\frac{2\left(\mathrm{a}+\mathrm{b}{e}^{4}\right){\mathrm{q}}_{1}^{2}{\mathrm{w}}_{1}^{2}{\mathrm{X}}_{9}}{{T}^{3}{\left(F\left(T\right)\right)}^{3}}-\frac{2\left(\mathrm{O}+{\mathrm{aX}}_{20}+{e}^{4}\left(\mathrm{G}+{\mathrm{bX}}_{20}\right)\right)}{{T}^{3}}\le 0.$$
(98)

Due to the complexity of terms in second-order derivatives, the concavity of the total profit function is proved in the section of the numerical experiment.

To satisfy the sufficient condition of concavity for Subcase 2.1, the determinant of the corresponding Hessian matrix must be positive which is formulated as follows.

$${ {The \; determinant \;of}} H\left({TP}_{4}\left(e,T\right)\right)=\left|\begin{array}{cc}\frac{{\partial }^{2}{TP}_{4}\left(e,T\right)}{{e}^{2}}& \frac{{\partial }^{2}{TP}_{4}\left(e,T\right)}{\partial e\partial T}\\ \frac{{\partial }^{2}{TP}_{4}\left(e,T\right)}{\partial T\partial e}& \frac{{\partial }^{2}{TP}_{4}\left(e,T\right)}{\partial {T}^{2}}\end{array}\right|\ge 0.$$
(99)

Due to the complexity of terms in the Hessian matrix, it is not convenient to prove the positivity of the determinant of the Hessian matrix. Therefore, the positivity of the Hessian matrix is proved in a numerical experiment.

Appendix 6

To satisfy the necessary conditions of concavity for Subcase 2.2, first and second-order derivatives of \({TP}_{5}\) are calculated. The second-order derivatives concerning variables should be negative.

The first derivative of \({TP}_{5}\) concerning \(e\) is obtained as follows.

$$\frac{\partial {TP}_{5}\left(e,T\right)}{\partial e}=-\frac{4{e}^{3}\mathrm{G}}{T}+\frac{4\mathrm{b}{e}^{3}\left({\mathrm{X}}_{1}+{\mathrm{vI}}_{\mathrm{e}}{\mathrm{X}}_{2}\right)}{F\left(T\right)}+\frac{4\mathrm{b}{e}^{3}\left({\mathrm{X}}_{3}-{\mathrm{vI}}_{\mathrm{e}}{\mathrm{X}}_{4}\right)}{TF\left(T\right)}-\frac{3\mathrm{b}{e}^{2}{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\mathrm{y}\left(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}}\right)\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{T{\left(E\left(T\right)\right)}^{2}{\left(F\left(T\right)\right)}^{2}}+\frac{8\mathrm{b}{e}^{3}\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)\left(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}}\right)\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{E\left(T\right){\left(F\left(T\right)\right)}^{2}}+4\mathrm{b}{e}^{3}{\mathrm{X}}_{11}-4\mathrm{b}{e}^{3}\mathrm{T}\left(\frac{{\mathrm{X}}_{9}}{F\left(T\right)}+{\mathrm{X}}_{12}\right).$$
(100)

The second derivative of \({TP}_{5}\) concerning \(e\) is obtained as follows.

$$\frac{{\partial }^{2}{TP}_{5}\left(e,T\right)}{\partial {e}^{2}}=-\frac{12{e}^{2}\mathrm{G}}{T}+\frac{12\mathrm{b}{e}^{2}\left({\mathrm{X}}_{1}+{\mathrm{vI}}_{\mathrm{e}}{\mathrm{X}}_{2}\right)}{F\left(T\right)}+\frac{12\mathrm{b}{e}^{2}\left({\mathrm{X}}_{3}-{\mathrm{vI}}_{\mathrm{e}}{\mathrm{X}}_{4}\right)}{TF\left(T\right)}+\frac{18{\mathrm{b}}^{2}{e}^{4}{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}{\mathrm{y}}^{2}\left(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}}\right)\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{{T}^{2}{\left(E\left(T\right)\right)}^{3}{\left(F\left(T\right)\right)}^{2}}-\frac{48{\mathrm{b}}^{2}{e}^{5}\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)\mathrm{y}\left(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}}\right)\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{T{\left(E\left(T\right)\right)}^{2}{\left(F\left(T\right)\right)}^{2}}-\frac{6\mathrm{b}e{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\mathrm{y}\left(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}}\right)\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{T{\left(E\left(T\right)\right)}^{2}{\left(F\left(T\right)\right)}^{2}}+\frac{32{\mathrm{b}}^{2}{e}^{6}\left(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}}\right)\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{E\left(T\right){\left(F\left(T\right)\right)}^{2}}+\frac{24\mathrm{b}{e}^{2}\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)\left(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}}\right)\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{E\left(T\right){\left(F\left(T\right)\right)}^{2}}+12\mathrm{b}{e}^{2}{\mathrm{X}}_{11}-12\mathrm{b}{e}^{2}\mathrm{T}\left(\frac{{\mathrm{X}}_{9}}{F\left(T\right)}+{\mathrm{X}}_{12}\right)\le 0.$$
(101)

The first derivative of \({TP}_{5}\) concerning \(T\) is obtained as follows.

$$\frac{\partial {TP}_{5}\left(e,T\right)}{\partial T}=\frac{{e}^{4}\mathrm{G}+\mathrm{O}}{{T}^{2}}+\frac{(\mathrm{a}+\mathrm{b}{e}^{4}){\mathrm{q}}_{1}{\mathrm{w}}_{1}({\mathrm{X}}_{1}+{\mathrm{vI}}_{\mathrm{e}}{\mathrm{X}}_{2})}{{T}^{2}{(F(T))}^{2}}+\frac{(\mathrm{a}+\mathrm{b}{e}^{4}){\mathrm{q}}_{1}{\mathrm{w}}_{1}({\mathrm{X}}_{3}-{\mathrm{vI}}_{\mathrm{e}}{\mathrm{X}}_{4})}{{T}^{3}{(F(T))}^{2}}-\frac{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)\left({\mathrm{X}}_{3}-{\mathrm{vI}}_{\mathrm{e}}{\mathrm{X}}_{4}\right)}{{T}^{2}F\left(T\right)}-\frac{{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\left(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}}\right){\mathrm{X}}_{7}}{E\left(T\right){\left(F\left(T\right)\right)}^{2}}+\frac{2{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\left(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}}\right){\mathrm{q}}_{1}{\mathrm{w}}_{1}\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{{T}^{2}E\left(T\right){\left(F\left(T\right)\right)}^{3}}+\frac{\mathrm{b}{e}^{3}{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\mathrm{y}\left(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}}\right)\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{{T}^{2}{\left(E\left(T\right)\right)}^{2}{\left(F\left(T\right)\right)}^{2}}-\frac{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right){\mathrm{q}}_{1}{\mathrm{w}}_{1}{\mathrm{X}}_{9}}{T{\left(F\left(T\right)\right)}^{2}}-\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)\left(\frac{{\mathrm{X}}_{9}}{\mathrm{F}\left(\mathrm{T}\right)}+{\mathrm{X}}_{12}\right).$$
(102)

The second derivative of \({TP}_{5}\) concerning \(T\) is obtained as follows.

$$\frac{{\partial }^{2}{TP}_{5}\left(e,T\right)}{\partial {T}^{2}}=-\frac{2\left({e}^{4}\mathrm{G}+\mathrm{O}\right)}{{T}^{3}}+\frac{2\left(\mathrm{a}+\mathrm{b}{e}^{4}\right){\mathrm{q}}_{1}^{2}{\mathrm{w}}_{1}^{2}\left({\mathrm{X}}_{1}+{\mathrm{vI}}_{\mathrm{e}}{\mathrm{X}}_{2}\right)}{{T}^{4}{\left(F\left(T\right)\right)}^{3}}-\frac{2\left(\mathrm{a}+\mathrm{b}{e}^{4}\right){\mathrm{q}}_{1}{\mathrm{w}}_{1}\left({\mathrm{X}}_{1}+{\mathrm{vI}}_{\mathrm{e}}{\mathrm{X}}_{2}\right)}{{T}^{3}{\left(F\left(T\right)\right)}^{2}}+\frac{2\left(\mathrm{a}+\mathrm{b}{e}^{4}\right){\mathrm{q}}_{1}^{2}{\mathrm{w}}_{1}^{2}\left({\mathrm{X}}_{3}-{\mathrm{vI}}_{\mathrm{e}}{\mathrm{X}}_{4}\right)}{{T}^{5}{\left(F\left(T\right)\right)}^{3}}-\frac{4\left(\mathrm{a}+\mathrm{b}{e}^{4}\right){\mathrm{q}}_{1}{\mathrm{w}}_{1}\left({\mathrm{X}}_{3}-{\mathrm{vI}}_{\mathrm{e}}{\mathrm{X}}_{4}\right)}{{T}^{4}{\left(F\left(T\right)\right)}^{2}}+\frac{2\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)\left({\mathrm{X}}_{3}-{\mathrm{vI}}_{\mathrm{e}}{\mathrm{X}}_{4}\right)}{{T}^{3}F\left(T\right)}-\frac{4{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\left(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}}\right){\mathrm{q}}_{1}{\mathrm{w}}_{1}{\mathrm{X}}_{7}}{{T}^{2}E\left(T\right){\left(F\left(T\right)\right)}^{3}}-\frac{2\mathrm{b}{e}^{3}{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\mathrm{y}\left(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}}\right){\mathrm{X}}_{7}}{{T}^{2}{\left(E\left(T\right)\right)}^{2}{\left(F\left(T\right)\right)}^{2}}+\frac{6{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\left(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}}\right){\mathrm{q}}_{1}^{2}{\mathrm{w}}_{1}^{2}\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{{T}^{4}E\left(T\right){\left(F\left(T\right)\right)}^{4}}+\frac{4\mathrm{b}{e}^{3}{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\mathrm{y}\left(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}}\right){\mathrm{q}}_{1}{\mathrm{w}}_{1}\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{{T}^{4}{\left(E\left(T\right)\right)}^{2}{\left(F\left(T\right)\right)}^{3}}-\frac{4{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\left(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}}\right){\mathrm{q}}_{1}{\mathrm{w}}_{1}\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{{T}^{3}E\left(T\right){\left(F\left(T\right)\right)}^{3}}+\frac{2{\mathrm{b}}^{2}{e}^{6}{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}{\mathrm{y}}^{2}\left(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}}\right)\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{{T}^{4}{\left(E\left(T\right)\right)}^{3}{\left(F\left(T\right)\right)}^{2}}-\frac{2{\mathrm{be}}^{3}{\left(\mathrm{a}+\mathrm{b}{e}^{4}\right)}^{2}\mathrm{y}\left(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}}\right)\left({\mathrm{X}}_{6}-T{\mathrm{X}}_{7}\right)}{{T}^{3}{\left(E\left(T\right)\right)}^{2}{\left(F\left(T\right)\right)}^{2}}-\frac{2\left(\mathrm{a}+\mathrm{b}{e}^{4}\right){\mathrm{q}}_{1}^{2}{\mathrm{w}}_{1}^{2}{\mathrm{X}}_{9}}{{T}^{3}{\left(F\left(T\right)\right)}^{3}}\le 0.$$
(103)

Due to the complexity of terms in second-order derivatives, the concavity of the total profit function is proved in the section of the numerical experiment.

To satisfy the sufficient condition of concavity for Subcase 2.2, the determinant of the corresponding Hessian matrix must be positive which is formulated as follows.

$${{The \;determinant \;of}} H\left({TP}_{5}\left(e,T\right)\right)=\left|\begin{array}{cc}\frac{{\partial }^{2}{TP}_{5}\left(e,T\right)}{{e}^{2}}& \frac{{\partial }^{2}{TP}_{5}\left(e,T\right)}{\partial e\partial T}\\ \frac{{\partial }^{2}{TP}_{5}\left(e,T\right)}{\partial T\partial e}& \frac{{\partial }^{2}{TP}_{5}\left(e,T\right)}{\partial {T}^{2}}\end{array}\right|\ge 0.$$
(104)

Due to the complexity of terms in the Hessian matrix, it is not convenient to prove the positivity of the determinant of the Hessian matrix. Therefore, the positivity of the Hessian matrix is proved in a numerical experiment.

Appendix 7

Expressions of \({\mathrm{G}}_{1}, {\mathrm{G}}_{2},\) and \({\mathrm{G}}_{3}\) which are required for constructing the total profit of the base model for Subcases 1.1 and 2.1are illustrated as follows.

$${\mathrm{G}}_{1}=\left(\mathrm{O}-\mu {\mathrm{I}}_{\mathrm{e}}\frac{{\mathrm{DS}}^{2}}{2}+\frac{{\mathrm{CI}}_{\mathrm{p}}{\mathrm{DS}}^{2}}{2}\right)$$
(105)
$${\mathrm{G}}_{2}=\left(\frac{\mathrm{hD}}{2}+\frac{(\mathrm{h}{+{\mathrm{CI}}_{\mathrm{P}})\mathrm{D}}^{2}}{{\mathrm{x}}_{0}{\left(\left(1-\theta \right)\left(1-{\mathrm{q}}_{1}\right)\right)}^{2}}\left[\left(1-\theta \right){\mathrm{q}}_{1}+\theta \left(1-{\mathrm{q}}_{2}\right)\right]+\frac{\mathrm{h}\theta {\mathrm{Dq}}_{2}}{2\left(1-\theta \right)\left(1-{\mathrm{q}}_{1}\right)}+\frac{{\mathrm{CI}}_{\mathrm{p}}\mathrm{D}}{2}\right)$$
(106)
$${\mathrm{G}}_{3}=\frac{\mathrm{D}}{(1-\theta )(1-{\mathrm{q}}_{1})}\left\{\upmu \left(1-\theta \right)\left(1-{\mathrm{q}}_{1}\right)+\mathrm{v}\left[\theta \left(1-{\mathrm{q}}_{2}\right)+\left(1-\theta \right){\mathrm{q}}_{1}+\theta {\mathrm{q}}_{2}\right]-\mathrm{C}-\mathrm{d}-{\mathrm{C}}_{1}\left(1-\uptheta \right){\mathrm{q}}_{1}-{\mathrm{C}}_{2}\uptheta {\mathrm{q}}_{2}+{\mathrm{CI}}_{\mathrm{P}}\mathrm{S}\left(\left(1-\theta \right){\mathrm{q}}_{1}+\theta \left(1-{\mathrm{q}}_{2}\right)\right)\right\}+{\mathrm{CI}}_{\mathrm{p}}\mathrm{DS}$$
(107)

Appendix 8

Expressions of \({\mathrm{G}}_{4}, {\mathrm{G}}_{5},\) and \({\mathrm{G}}_{6}\) which are required for constructing the total profit of the base model for Subcases 1.2 and 2.2are illustrated as follows.

$${\mathrm{G}}_{4}=\mathrm{O}$$
(108)
$${\mathrm{G}}_{5}=\left(\frac{\mathrm{hD}}{2}+\frac{(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}}){\mathrm{D}}^{2}}{{\mathrm{x}}_{0}{\left(\left(1-\theta \right)\left(1-{\mathrm{q}}_{1}\right)\right)}^{2}}\left[\left(1-\theta \right){\mathrm{q}}_{1}+\theta \left(1-{\mathrm{q}}_{2}\right)\right]+\frac{\mathrm{h}\theta {\mathrm{Dq}}_{2}}{2\left(1-\theta \right)\left(1-{\mathrm{q}}_{1}\right)}+\frac{\mu {\mathrm{I}}_{\mathrm{e}}\mathrm{D}}{2}\right)$$
(109)
$${\mathrm{G}}_{6}=\frac{\mathrm{D}}{(1-\theta )(1-{\mathrm{q}}_{1})}\left\{\mu \left(1-\theta \right)\left(1-{\mathrm{q}}_{1}\right)+\mathrm{v}\left[\theta \left(1-{\mathrm{q}}_{2}\right)+\left(1-\theta \right){\mathrm{q}}_{1}+\theta {\mathrm{q}}_{2}\right]-\mathrm{C}-\mathrm{d}-{\mathrm{C}}_{1}\left(1-\theta \right){\mathrm{q}}_{1}-{\mathrm{C}}_{2}\theta {\mathrm{q}}_{2}+{\mathrm{vI}}_{\mathrm{e}}\mathrm{S}\left(\left(1-\theta \right){\mathrm{q}}_{1}+\theta \left(1-{\mathrm{q}}_{2}\right)\right)\right\}+\mu {\mathrm{I}}_{\mathrm{e}}\mathrm{DS}$$
(110)

Appendix 9

Expressions of \({\mathrm{G}}_{7}, {\mathrm{G}}_{8},\) and \({\mathrm{G}}_{9}\) which are required for constructing the total profit of the base model for Subcase 1.3 are illustrated as follows.

$${\mathrm{G}}_{7}=\mathrm{O}$$
(111)
$${\mathrm{G}}_{8}=\left(\frac{\mathrm{hD}}{2}+\frac{\left(\mathrm{h}+{\mathrm{vI}}_{\mathrm{e}}\right){\mathrm{D}}^{2}}{{\mathrm{x}}_{0}{\left(\left(1-\theta \right)\left(1-{\mathrm{q}}_{1}\right)\right)}^{2}}\left[\left(1-\theta \right){\mathrm{q}}_{1}+\theta \left(1-{\mathrm{q}}_{2}\right)\right]+\frac{{\mathrm{vI}}_{\mathrm{e}}{\mathrm{D}}^{2}\theta {\mathrm{q}}_{2}}{{\mathrm{x}}_{0}{\left(\left(1-\theta \right)\left(1-{\mathrm{q}}_{1}\right)\right)}^{2}}+\frac{{\mathrm{vI}}_{\mathrm{e}}\theta {\mathrm{Dq}}_{2}}{\left(1-\theta \right)\left(1-{\mathrm{q}}_{1}\right)}+\frac{\mathrm{h}\theta {\mathrm{Dq}}_{2}}{2\left(1-\theta \right)\left(1-{\mathrm{q}}_{1}\right)}+\frac{\mu {\mathrm{I}}_{\mathrm{e}}\mathrm{D}}{2}\right)$$
(112)
$${\mathrm{G}}_{9}=\frac{\mathrm{D}}{(1-\theta )(1-{\mathrm{q}}_{1})}\left\{\mu \left(1-\theta \right)\left(1-{\mathrm{q}}_{1}\right)+\mathrm{v}\left[\theta \left(1-{\mathrm{q}}_{2}\right)+\left(1-\theta \right){\mathrm{q}}_{1}+\theta {\mathrm{q}}_{2}\right]-\mathrm{C}-\mathrm{d}-{\mathrm{C}}_{1}\left(1-\theta \right){\mathrm{q}}_{1}-{\mathrm{C}}_{2}\theta {\mathrm{q}}_{2}+{\mathrm{vI}}_{\mathrm{e}}\mathrm{S}\left(\left(1-\theta \right){\mathrm{q}}_{1}+\theta \left(1-{\mathrm{q}}_{2}\right)\right)+{\mathrm{vI}}_{\mathrm{e}}\theta {\mathrm{q}}_{2}\mathrm{S}\right\}$$
(113)

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Moradi, S., Gholamian, M.R. & Sepehri, A. An inventory model for imperfect quality items considering learning effects and partial trade credit policy. OPSEARCH 60, 276–325 (2023). https://doi.org/10.1007/s12597-022-00602-3

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