Sustainable production-inventory system for perishables under dynamic fuel pricing and preservation technology investment

Abstract

In the production and inventory management of perishables, environmental considerations are gaining prominence. By reducing carbon emissions from various supply chain processes, such as production, transportation, warehousing, and waste disposal of perishable items, the present study aims to minimize the overall cost to the manufacturer through an optimized investment in green technology. Additionally, cycle time and preservation technology investment are optimized to decrease deterioration and revenue loss in order to minimize cost. The originality of the present research lies in the following considerations. Due to an increase in fuel price, the transportation cost of every subsequent order will also increase, thus resulting in an increase of average delivery cost in a production cycle. We investigate the impact of changes in fuel prices on transportation costs and production inventory model policies due to the volatile nature of fuel prices. The function of transportation cost can be used to calculate transport costs in the future. The deterioration rate is a random variable with a double triangular distribution. Precisely, the demand for any product depends on the product’s price; therefore, linear price-dependent demand is considered. Per unit production cost is a function of direct material cost, tooling cost, and manpower cost. Taking into account all the aforementioned parameters, this paper simultaneously optimizes green technology investment, preservation investment, and cycle time. To achieve the solution of the proposed sustainable production system, an optimization technique for the nonlinear function is employed. Finally, numerical experiments are conducted to validate the model. A special case of a numerical example demonstrates that the expected value of the total average cost is reduced by 10.723% when investments are made in both green and preservation technology, whereas investments in green technology alone result in a cost reduction of only 2.15%. Then, managerial implications and a discussion of findings are proposed after a sensitivity analysis that examines the model’s response to key parameter variation. The study concludes with a discussion of the limitations of current work and possible future scopes.

Appendices

Appendix 1

Proof of Lemma 1

To determine the optimal values of G, calculate the first and second-order derivatives of E [TC (T, G, ξ)] with respect to G. First-order derivatives are given below.

$$\begin{array}{l}\frac{\partial E\lbrack TC(T,G,\xi)\rbrack}{\partial G}\\=\{T-\delta(\vartheta-G\vartheta^2)\phi(T_mPE_p\\+E_h(\frac{e^{\lambda\xi}\left(P+bs-a\right)\left(T_m+\frac{e^{\lambda\xi}\left(e^{-e^{-\lambda\xi}T_mE(\theta)}-1\right)}{E(\theta)}\right)}{E(\theta)}\\+\frac{e^{\lambda\xi}\left(a-bs\right)\left(T_m-T+\frac{e^{\lambda\xi}\left(e^{e^{-\lambda\xi}\left(T-T_m\right)\theta}-1\right)}{E(\theta)}\right)}{E(\theta)})+2dlnV_e+dQwf_{ca}V_e\\+{W_dQE}_{wd})\}/T\end{array}$$

To determine the minimum cost of the production inventory system, the above-mentioned partial differentiation should satisfy the necessary condition, i.e., $\frac{\partial E [TC (T, G, \xi )] }{\partial G}=0$; from this condition, we can find the optimum value of the decision variable G.

Moreover, to satisfy optimality, i.e., convexity conditions, the sufficient condition must be satisfied, i.e., the corresponding principal minor should be positive definite.

$$Hessian\mathit\;matrix\mathit,\mathit\;H=\begin{bmatrix}\frac{\partial^2E\lbrack TC(T,G,\xi)\rbrack}{{\partial G}^2}&\frac{\partial^2E\lbrack TC(T,G,\xi)\rbrack}{\partial T\partial G}\\\frac{\partial^2E\lbrack TC(T,G,\xi)\rbrack}{\partial G\partial T}&\frac{\partial^2E\lbrack TC(T,G,\xi)\rbrack}{{\partial T}^2}\end{bmatrix}$$

If the leading minors of H are ${H}_{1}=\frac{{\partial }^{2}E [TC (T, G, \xi )] }{{\partial G}^{2}}> 0$

$$\frac{{\partial }^{2}E [TC (T, G, \xi )] }{{\partial G}^{2}}=\frac{\left\{\delta {\vartheta }^{2}\phi \left({T}_{m}P{E}_{p}+{{\varvec{E}}}_{{\varvec{h}}}\left(\frac{{e}^{\lambda \xi }\left(P+bs-a\right)\left({T}_{m}+\frac{{e}^{\lambda \xi }\left({e}^{-{e}^{-\lambda \xi }{T}_{m}E(\theta )}-1\right)}{E(\theta )}\right)}{E(\theta )}+\frac{{e}^{\lambda \xi }\left(a-bs\right)\left({T}_{m}-T+\frac{{e}^{\lambda \xi }\left({e}^{{e}^{-\lambda \xi }\left(T-{T}_{m}\right)E(\theta )}-1\right)}{E(\theta )}\right)}{E(\theta )}\right)+2dln{{\varvec{V}}}_{{\varvec{e}}}+dQw{f}_{ca}{{\varvec{V}}}_{{\varvec{e}}}+{ {W}_{d} Q E}_{wd}\right)\right\}}{T}>0$$

Appendix 2

Proof of corollary 1

For a given value of ξ and P > a- bs, and a- bs = D, from the above expression we can observe possible negative terms are ${e}^{-{e}^{-\lambda \xi }{T}_{m}E(\theta )}-1$ and $\frac{{e}^{\lambda \xi }\left({e}^{{e}^{-\lambda \xi }\left(T-{T}_{m}\right)E(\theta )}-1\right)}{E(\theta )}$

Definitely ${e}^{{e}^{-\lambda \xi }\left(T-{T}_{m}\right)E(\theta )}$> 1 and also $\frac{{e}^{\lambda \xi }\left({e}^{{e}^{-\lambda \xi }\left(T-{T}_{m}\right)E(\theta )}-1\right)}{E(\theta )}>({T}_{m}-T)$ so ${e}^{\lambda \xi }\left(a-bs\right)\left({T}_{m}-T+\frac{{e}^{\lambda \xi }\left({e}^{{e}^{-\lambda \xi }\left(T-{T}_{m}\right)E(\theta )}-1\right)}{E(\theta )}\right)$ is a very large positive value. ${e}^{-{e}^{-\lambda \xi }{T}_{m}E(\theta )}< -1$, but ${e}^{\lambda \xi }\left(a-bs\right)\left({T}_{m}-T+\frac{{e}^{\lambda \xi }\left({e}^{{e}^{-\lambda \xi }\left(T-{T}_{m}\right)E(\theta )}-1\right)}{E(\theta )}\right)$ is very large as compared to ${e}^{\lambda \xi }\left(P+bs-a\right)\left({T}_{m}+\frac{{e}^{\lambda \xi }\left({e}^{-{e}^{-\lambda \xi }{T}_{m}E(\theta )}-1\right)}{E(\theta )}\right)$ so since ${H }_{1}$> 0, corollary 1 is satisfied one can conclude $\frac{{\partial }^{2}E [TC (T, G, \xi )] }{{\partial G}^{2}}$ > 0, therefore, E [TC (T, G, ξ)] is convex in G and a unique minimum solution exists at ${G}^{*}$ hence corollary 1 is proved.

Appendix 3

Proof of Lemma 2

To determine the optimal values of T, calculate the first and second-order derivatives of E [TC (T, G, ξ)] with respect to T. First-order derivatives are given below.

$$\begin{array}{l}\frac{\partial TCE\lbrack TC(T,G,\xi)\rbrack}{\partial T}\\=\frac{G+\frac{e^{\lambda\xi}(e^{e^{-\lambda\xi}\left(T-T_m\right)E(\theta)}-1)C_h(a-bs)}{E(\theta)}+\xi+\frac{e^{\lambda\xi}(e^{e^{-\lambda\xi}\left(T-T_m\right)E(\theta)}-1)(a-bs)E_h\delta(1-(G\vartheta-\frac{G^2\vartheta^2}2)\phi)}{E(\theta)}}T\\-\frac1{T^2}(A+F+T_mP(c+\frac JP+kP)+GT+W+2dl\alpha+Q\beta_o(e^{\sigma\psi}-1)+\frac{2dl(n-1)Q\beta}{a-bs}\\+\frac{e^{\lambda\xi}C_h(P+bs-a)(T_m+\frac{e^{\lambda\xi}(e^{-e^{-\lambda\xi}T_mE(\theta)}-1)}{E(\theta)})}{E(\theta)}+\frac{e^{\lambda\xi}C_h(a-bs)(T_m-T+\frac{e^{\lambda\xi}(e^{e^{-\lambda\xi}\left(T-T_m\right)E(\theta)}-1)}{E(\theta)})}{E(\theta)}+T\xi\\+dQw\alpha f_{ca}+\delta(1-(G\vartheta-\frac{G^2\vartheta^2}2)\phi)(T_mPE_p+E_h(\frac{e^{\lambda\xi}(P+bs-a)(T_m+\frac{e^{\lambda\xi}(e^{-e^{-\lambda\xi}T_mE(\theta)}-1)}{E(\theta)})}{E(\theta)}\\+\frac{e^{\lambda\xi}(a-bs)(T_m-T+\frac{e^{\lambda\xi}(e^{e^{-\lambda\xi}\left(T-T_m\right)E(\theta)}-1)}{E(\theta)})}{E(\theta)})+2dlnV_e+dQwf_{ca}V_e+{W_dQE}_{wd}))\end{array}$$

To determine the minimum cost of the production inventory system, the above-mentioned partial differentiation should satisfy the necessary condition, i.e., $\frac{\partial E [TC (T, G, \xi )] }{\partial T}=0$; from this condition, we can find the optimum value of the decision variable T.

Moreover, to satisfy optimality, i.e., convexity conditions, the sufficient condition must be satisfied, i.e., the corresponding principal minor should be positive definite.

$$Hessian\mathit\;matrix,H=\begin{bmatrix}\frac{\partial^2E\lbrack TC(T,G,\xi)\rbrack}{{\partial G}^2}&\frac{\partial^2E\lbrack TC(T,G,\xi)\rbrack}{\partial T\partial G}\\\frac{\partial^2E\lbrack TC(T,G,\xi)\rbrack}{\partial G\partial T}&\frac{\partial^2E\lbrack TC(T,G,\xi)\rbrack}{{\partial T}^2}\end{bmatrix}$$

If the leading minors of H are ${H}_{1}=\frac{{\partial }^{2}E [TC (T, G, \xi )] }{{\partial T}^{2}}> 0$

$$\begin{array}{c}\frac{{\partial }^{2}\mathrm{E }\left[\mathrm{TC }\left(\mathrm{T},\mathrm{ G},\upxi \right)\right]}{{\partial \mathrm{T}}^{2}}= \frac{{e}^{{e}^{-\lambda \xi }\left(T-{T}_{m}\right)E\left(\theta \right)}{C}_{h}\left(a-bs\right)+{e}^{{e}^{-\lambda \xi }\left(T-{T}_{m}\right)E\left(\theta \right)}\left(a-bs\right){E}_{h}\delta \left(1-\left(G\vartheta -\frac{{G}^{2}{\vartheta }^{2}}{2}\right)\phi \right)}{T}-\\ \frac{2\left(G+\frac{{e}^{\lambda \xi }\left({e}^{{e}^{-\lambda \xi }\left(T-{T}_{m}\right)E\left(\theta \right)}-1\right){C}_{h}\left(a-bs\right)}{E\left(\theta \right)}+\xi +\frac{{e}^{\lambda \xi }\left({e}^{{e}^{-\lambda \xi }\left(T-{T}_{m}\right)E\left(\theta \right)}-1\right)\left(a-bs\right){E}_{h}\delta \left(1-\left(G\vartheta -\frac{{G}^{2}{\vartheta }^{2}}{2}\right)\phi \right)}{E\left(\theta \right)}\right)}{{T}^{2}}+\frac{1}{{T}^{3}}2\left(A+F+\right.\\ \begin{array}{c}{T}_{m}P\left(c+\frac{J}{P}+kP\right)+GT+W+2dl\alpha +Q{\beta }_{o}\left({e}^{\sigma \psi }-1\right)+\frac{2dl\left(n-1\right)Q\beta }{a-bs}+\\ \frac{{e}^{\lambda \xi }{C}_{h}\left(P+bs-a\right)\left({T}_{m}+\frac{{e}^{\lambda \xi }\left({e}^{-{e}^{-\lambda \xi }{T}_{m}E\left(\theta \right)}-1\right)}{E\left(\theta \right)}\right)}{E\left(\theta \right)}+\frac{{e}^{\lambda \xi }{C}_{h}\left(a-bs\right)\left({T}_{m}-T+\frac{{e}^{\lambda \xi }\left(-1+{e}^{{e}^{-\lambda \xi }\left(T-{T}_{m}\right)E\left(\theta \right)}\right)}{E\left(\theta \right)}\right)}{E\left(\theta \right)}+T\xi +dQw\alpha {f}_{ca}+\\ \begin{array}{c}\delta \left(1-\left(G\vartheta -\frac{{G}^{2}{\vartheta }^{2}}{2}\right)\phi \right)\left({T}_{m}P{E}_{p}+{E}_{h}\left(\frac{{e}^{\lambda \xi }\left(P+bs-a\right)\left({T}_{m}+\frac{{e}^{\lambda \xi }\left(-1+{e}^{-{e}^{-\lambda \xi }{T}_{m}E\left(\theta \right)}\right)}{E\left(\theta \right)}\right)}{E\left(\theta \right)}+\right.\right.\\ \left.\left.\left.\frac{{e}^{\lambda \xi }\left(a-bs\right)\left({T}_{m}-T+\frac{{e}^{\lambda \xi }\left(-1+{e}^{{e}^{-\lambda \xi }\left(T-{T}_{m}\right)E\left(\theta \right)}\right)}{E\left(\theta \right)}\right)}{E\left(\theta \right)}\right)+2dln{V}_{e}+dQw{f}_{ca}{V}_{e}+{ {W}_{d} Q E}_{wd}\right)\right)\ge 0\\ \end{array}\end{array}\end{array}$$

Appendix 4

Proof of Lemma 3

To determine the optimal values of T and G, calculate the first- and second-order derivative of E [TC (T, G, ξ)] with respect to T and G. First-order derivatives are given below.

$$\begin{array}{ll}\frac{\partial E [TC (T, G, \xi )] }{\partial G}\\ =\{T-\delta (\vartheta -G{\vartheta }^{2})\phi ({T}_{m}P{E}_{p}\\ \begin{array}{l}+{E}_{h}(\frac{{e}^{\lambda \xi }\left(P+bs-a\right)\left({T}_{m}+\frac{{e}^{\lambda \xi }\left({e}^{-{e}^{-\lambda \xi }{T}_{m}E(\theta )}-1\right)}{E(\theta )}\right)}{E(\theta )}\\ +\frac{{e}^{\lambda \xi }\left(a-bs\right)\left({T}_{m}-T+\frac{{e}^{\lambda \xi }\left({e}^{{e}^{-\lambda \xi }\left(T-m\right)E(\theta )}-1\right)}{E(\theta )}\right)}{E(\theta )})+2dln{V}_{e}+dQw{f}_{ca}{V}_{e}\\ \begin{array}{c}+{ {W}_{d} Q E}_{wd})\}/T\\ \frac{\partial E \left[TC \left(T, G, \xi \right)\right]}{\partial T}\\ \begin{array}{c}=\frac{G+\frac{{e}^{\lambda \xi }\left({e}^{{e}^{-\lambda \xi }\left(T-{T}_{m}\right)E(\theta )}-1\right){C}_{h}\left(a-bs\right)}{E(\theta )}+\xi +\frac{{e}^{\lambda \xi }\left({e}^{{e}^{-\lambda \xi }\left(T-{T}_{m}\right)E(\theta )}-1\right)\left(a-bs\right){E}_{h}\delta \left(1-\left(G\vartheta -\frac{{G}^{2}{\vartheta }^{2}}{2}\right)\phi \right)}{E(\theta )}}{T}\\ -\frac{1}{{T}^{2}}\left(A+F+{T}_{m}P\left(c+\frac{J}{P}+kP\right)+GT+W+2dl\alpha +Q{\beta }_{o}\left({e}^{\sigma \psi }-1\right)+\frac{2dl\left(n-1\right)Q\beta }{a-bs}\right.\\ \begin{array}{c}+\frac{{e}^{\lambda \xi }{C}_{h}\left(P+bs-a\right)\left({T}_{m}+\frac{{e}^{\lambda \xi }\left({e}^{-{e}^{-\lambda \xi }{T}_{m}E(\theta )}-1\right)}{E(\theta )}\right)}{E(\theta )}\\ +\frac{{e}^{\lambda \xi }{C}_{h}\left(a-bs\right)\left({T}_{m}-T+\frac{{e}^{\lambda \xi }\left({e}^{{e}^{-\lambda \xi }\left(T-{T}_{m}\right)E(\theta )}-1\right)}{E(\theta )}\right)}{E(\theta )}+T\xi +dQw\alpha {f}_{ca}\\ \begin{array}{l}+\delta \left(1-\left(G\vartheta -\frac{{G}^{2}{\vartheta }^{2}}{2}\right)\phi \right)\left({T}_{m}P{E}_{p}\right.\\ +{{\varvec{E}}}_{{\varvec{h}}\boldsymbol{ }}\left(\frac{{e}^{\lambda \xi }\left(P+bs-a\right)\left({T}_{m}+\frac{{e}^{\lambda \xi }\left({e}^{-{e}^{-\lambda \xi }{T}_{m}E(\theta )}-1\right)}{E(\theta )}\right)}{E(\theta )}\right.\\ \left.+\frac{{e}^{\lambda \xi }\left(a-bs\right)\left({T}_{m}-T+\frac{{e}^{\lambda \xi }\left({e}^{{e}^{-\lambda \xi }\left(T-{T}_{m}\right)E(\theta )}-1\right)}{E(\theta )}\right)}{E(\theta )}\right)\left.\left.+2dln{V}_{e}+dQw{f}_{ca}{V}_{e}+{ {W}_{d} Q E}_{wd}\right)\right)\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}$$

To determine the minimum cost of the production inventory system, the above-mentioned partial differentiation should satisfy the necessary condition, i.e., $\frac{\partial E [TC (T, G, \xi )] }{\partial G}=0$, $\frac{\partial E [TC (T, G, \xi )] }{\partial T}=0$, from this condition, we can find the optimum value of decision variables.

Moreover, to satisfy optimality, i.e., convexity conditions, the sufficient condition must be satisfied, i.e., the corresponding principal minors should be positive definite.

$$\mathrm{Hessian matrix}, H=\left[\begin{array}{cc}\frac{{\partial }^{2}E [TC (T, G, \xi )] }{{\partial G}^{2}}& \frac{{\partial }^{2}E [TC (T, G, \xi )] }{\partial T\partial G}\\ \frac{{\partial }^{2}E [TC (T, G, \xi )] }{\partial G\partial T}& \frac{{\partial }^{2}E [TC (T, G, \xi )] }{{\partial T}^{2}}\end{array}\right]$$

If the leading minors of H are ${H}_{1}=\frac{{\partial }^{2}TC\left(T, G,\xi \right)}{{\partial G}^{2}}> 0$, $\&{ H}_{2}= \left|\begin{array}{cc}\frac{{\partial }^{2}E [TC (T, G, \xi )] }{{\partial G}^{2}}& \frac{{\partial }^{2}E [TC (T, G, \xi )] }{\partial T\partial G}\\ \frac{{\partial }^{2}E [TC (T, G, \xi )] }{\partial G\partial T}& \frac{{\partial }^{2}E [TC (T, G, \xi )] }{{\partial T}^{2}}\end{array}\right|=\frac{{\partial }^{2}E [TC (T, G, \xi )] }{{\partial G}^{2}}\frac{{\partial }^{2}E [TC (T, G, \xi )]}{{\partial T}^{2}}-{\frac{{\partial }^{2}E [TC (T, G, \xi )]}{\partial G\partial T}}^{2}>0$, then minimum exists for the function $E [TC (T, G, \xi )]$ at ${T}^{*}$ & ${G}^{*}$

To calculate the principal minors, second-order derivatives are given as follows

$$\begin{array}{l}\frac{\partial^2E\lbrack TC(T,G,\xi)\rbrack}{{\partial\mathrm G}^2}\\=\{\delta\vartheta^2\phi(T_mPE_p+E_h(\frac{e^{\lambda\xi}\left(P+bs-a\right)\left({\mathrm T}_{\mathrm m}+\frac{e^{\lambda\xi}\left(e^{-e^{-\lambda\xi}{\mathrm T}_{\mathrm m}E(\theta)}-1\right)}{E(\theta)}\right)}{E(\theta)}\\\begin{array}{c}\begin{array}{c}+\frac{e^{\lambda\xi}\left(a-bs\right)\left({\mathrm T}_{\mathrm m}-T+\frac{e^{\lambda\xi}\left(e^{e^{-\lambda\xi}\left(T-m\right)E(\theta)}-1\right)}{E(\theta)}\right)}{E(\theta)})+2dlnV_e\\+dQwf_{ca}V_e+{W_dQE}_{wd})\}/T\end{array}\\\\\begin{array}{c}\frac{\partial^2TCE\lbrack TC(T,G,\xi)\rbrack}{{\partial T}^2}\\=\frac{e^{e^{-\lambda\xi}(T-T_m)E(\theta)}C_h(a-bs)+e^{e^{-\lambda\xi}(T-T_m)E(\theta)}(a-bs)E_h\delta(1-(G\vartheta-\frac{G^2\vartheta^2}2)\phi)}T\\\begin{array}{c}-\frac{2\left(G+\frac{e^{\lambda\xi}\left(e^{e^{-\lambda\xi}\left(T-T_m\right)E(\theta)}-1\right)C_h\left(a-bs\right)}{E(\theta)}+\xi+\frac{e^{\lambda\xi}\left(e^{e^{-\lambda\xi}\left(T-T_m\right)E(\theta)}-1\right)\left(a-bs\right)E_h\delta\left(1-\left(G\vartheta-\frac{G^2\vartheta^2}2\right)\phi\right)}{E(\theta)}\right)}{T^2}\\+\frac1{T^3}2(A+F+T_mP(c+\frac JP+kP)+GT+W+2dl\alpha+Q\beta_o(e^{\sigma\psi}-1)+\frac{2dl(n-1)Q\beta}{a-bs}\\\begin{array}{c}+\frac{e^{\lambda\xi}C_h(P+bs-a)(T_m+\frac{e^{\lambda\xi}(e^{-e^{-\lambda\xi}T_mE(\theta)}-1)}{E(\theta)})}{E(\theta)}+\frac{e^{\lambda\xi}C_h(a-bs)(T_m-T+\frac{e^{\lambda\xi}(e^{e^{-\lambda\xi}\left(T-T_m\right)E(\theta)}-1)}{E(\theta)})}{E(\theta)}\\+T\xi+dQw\alpha f_{ca}+\delta(1-(G\vartheta-\frac{G^2\vartheta^2}2)\phi)(T_mPE_p+E_h(\frac{e^{\lambda\xi}(P+bs-a)(T_m+\frac{e^{\lambda\xi}(e^{-e^{-\lambda\xi}T_mE(\theta)}-1)}{E(\theta)})}{E(\theta)}\\\begin{array}{c}+\frac{e^{\lambda\xi}(a-bs)(T_m-T+\frac{e^{\lambda\xi}(e^{e^{-\lambda\xi}\left(T-T_m\right)E(\theta)}-1)}{E(\theta)})}{E(\theta)})+2dlnV_e+dQwf_{ca}V_e+{W_dQE}_{wd}))\\\frac{\partial^2E\lbrack TC(T,G,\xi)\rbrack}{\partial T\partial G}=\frac{\partial^2E\lbrack TC(T,G,\xi)\rbrack}{\partial G\partial T}\\\begin{array}{c}=\frac{1-\delta(\frac{e^{\lambda\xi}(e^{e^{-\lambda\xi}\left(T-T_m\right)E(\theta)}-1)(a-bs)z}{E(\theta)})(\vartheta-G\vartheta^2)\phi}T-\frac1{T^2}(T-\delta(\vartheta-G\vartheta^2)\phi(T_mPE_p\\\begin{array}{c}{+E}_h(\frac{e^{\lambda\xi}(P+bs-a)(T_m+\frac{e^{\lambda\xi}(e^{-e^{-\lambda\xi}T_mE(\theta)}-1)}{E(\theta)})}{E(\theta)}\\+\frac{e^{\lambda\xi}(a-bs)(T_m-T+\frac{e^{\lambda\xi}(e^{e^{-\lambda\xi}\left(T-T_m\right)E(\theta)}-1)}{E(\theta)})}{E(\theta)})+2dlnV_e+dQwf_{ca}V_e\end{array}\\\begin{array}{c}{{+W}_dQE}_{wd}))\\\;\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}$$

Sufficient condition for $E [TC (T, G, \xi )]$ to be convex for a fixed value of $\xi$ is principal minors should be positive definite at the optimal values of (T, G) is shown below

$$\begin{array}{l}\begin{array}{c}{H}_{1}=\frac{{\partial }^{2}TC\left(T, G,\xi \right)}{{\partial G}^{2}}\\ =\{\delta {\vartheta }^{2}\phi ({T}_{m}P{E}_{p}+{E}_{h}(\frac{{e}^{\lambda \xi }\left(P+bs-a\right)\left({T}_{m}+\frac{{e}^{\lambda \xi }\left({e}^{-{e}^{-\lambda \xi }{T}_{m}E(\theta )}-1\right)}{E(\theta )}\right)}{E(\theta )}\\ +\frac{{e}^{\lambda \xi }\left(a-bs\right)\left({T}_{m}-T+\frac{{e}^{\lambda \xi }\left({e}^{{e}^{-\lambda \xi }\left(T-{T}_{m}\right)E(\theta )}-1\right)}{E(\theta )}\right)}{E(\theta )})+2dln{V}_{e}+dQw{f}_{ca}{V}_{e}\end{array}\\ \begin{array}{c}+{ {W}_{d} Q E}_{wd})\}/T\ge 0\\ { H}_{2}= \left|\begin{array}{cc}\frac{{\partial }^{2}E [TC (T, G, \xi )] }{{\partial G}^{2}}& \frac{{\partial }^{2}E [TC (T, G, \xi )] }{\partial T\partial G}\\ \frac{{\partial }^{2}E [TC (T, G, \xi )] }{\partial G\partial T}& \frac{{\partial }^{2}E [TC (T, G, \xi )] }{{\partial T}^{2}}\end{array}\right|\\ =\frac{{\partial }^{2}E [TC (T, G, \xi )] }{{\partial G}^{2}}\frac{{\partial }^{2}E [TC (T, G, \xi )]}{{\partial T}^{2}}-{\frac{{\partial }^{2}E [TC (T, G, \xi )]}{\partial G\partial T}}^{2}>0\end{array}\end{array}$$

Hence, we ensure that the total average cost function $E [TC (T, G, \xi )]$ is convex for a fixed value of ξ.