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Credit financing in economic ordering policies for non-instantaneous deteriorating items with price dependent demand and two storage facilities

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Abstract

The formulation of classical deteriorating inventory models is done with the common unrealistic assumption that all the items start deteriorating as soon as they arrive in the warehouse. On the contrary, in a realistic environment, it has been observed that there are several items that do not deteriorate immediately. Items such as dry fruits, potatoes, yams and even some fruits and vegetables have a shelf life and deteriorate only after a time lag. Apart from this, in today’s competitive business world, the supplier usually offers a trade credit period to his retailers to attract more sales and the retailer considers it as an alternative to price discount. The present research proposes a two warehouse inventory model for non-instantaneous deteriorating items under trade credit based on the above phenomena, where the demand rate is assumed to be a function of the selling price. Further, shortages are completely backlogged and the interest on shortages at the beginning of the cycle has also been considered. The objective of the study is to determine the retailer’s optimal replenishment policies that maximize the average profit per unit time. Conclusively, a numerical example is presented to illustrate the applicability of the proposed model. Sensitivity analysis on key parameters is provided to reveal the managerial insights.

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Acknowledgments

The authors would like to thank the Editor Endre Boros and anonymous referee for their valuable and constructive comments on earlier versions of our paper, which have led to a significant improvement in the manuscript. The first author acknowledges the support of the University Grants Commission through University of Delhi (Research Grant No. RC/2015/9677). The second author would like to thank University Grant Commission (UGC) for providing the Non-NET fellowship to accomplish this research.

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Correspondence to Chandra K. Jaggi.

Appendices

Appendix 1

For Case 1, the necessary conditions for maximizing the total average profit is given by

$$\begin{aligned} \frac{\partial \textit{TP}_{1.1} }{\partial t_r }= & {} \frac{1}{T}\left[ \frac{D\left( p \right) }{\beta }\left( {cI_p +F} \right) \left( {1-e^{\beta \left( {t_r -t_d } \right) }} \right) \right. \nonumber \\&\left. +\,\left( {cI_p -H} \right) \left\{ {We^{-\alpha \left( {t_r -t_d } \right) }+\frac{D\left( p \right) }{\alpha }\left\{ {\left( {1-e^{\alpha \left( {t_w -t_r } \right) }} \right) \left( {1-X_1 } \right) } \right\} } \right\} \right. \nonumber \\&\left. -\,FD\left( p \right) t_d e^{\beta \left( {t_r -t_d } \right) }+sD\left( p \right) \left( {T-t_w } \right) X_1 -cD\left( p \right) \left( {e^{\beta \left( {t_r -t_d } \right) }+X_1 } \right) \right. \nonumber \\&\left. -\,pI_e D\left( p \right) MX_1 -cI_p D\left( p \right) e^{\beta \left( {t_r -t_d } \right) }\left( {t_d -M} \right) \right] =0 \nonumber \\ \frac{\partial \textit{TP}_{1.1} }{\partial T}= & {} -\frac{1}{T^{2}}\left[ D\left( p \right) pT\right. \nonumber \\&\left. -\,A-F\left\{ {Zt_d -\frac{D\left( p \right) t_d^2 }{2}-Wt_d -\frac{D\left( p \right) }{\beta }\left( {t_r -t_d +\frac{1}{\beta }-\frac{e^{\beta \left( {t_r -t_d } \right) }}{\beta }} \right) } \right\} \right. \nonumber \\&-\,H\left\{ {Wt_d +\frac{W}{\alpha }\left( {1-e^{\alpha \left( {t_d -t_r } \right) }} \right) +\frac{D\left( p \right) }{\alpha }\left( {t_r -t_w -\frac{1}{\alpha }+\frac{e^{\alpha \left( {t_w -t_r } \right) }}{\alpha }} \right) } \right\} \nonumber \\&-\,sD\left( p \right) \left( {T-t_w } \right) ^{2}/2 \nonumber \\&-\,c\left\{ {W+D\left( p \right) t_d +\frac{D\left( p \right) }{\beta }\left( {e^{\beta \left( {t_r -t_d } \right) }-1} \right) -D\left( p \right) \left( {T-t_w } \right) } \right\} \nonumber \\&+\,\left\{ {\left( {pI_e D\left( p \right) M^{2}} \right) /2+pI_e D\left( p \right) \left( {T-t_w } \right) M} \right\} \nonumber \\&-\,cI_p \left\{ Z\left( {t_d -M} \right) +\frac{W}{\alpha }\left( {1-e^{\alpha \left( {t_d -t_r } \right) }} \right) +\frac{D\left( p \right) }{\alpha ^{2}}\left( {e^{\alpha \left( {t_w -t_r } \right) }-1} \right) \right. \nonumber \\&\left. -\,\frac{D\left( p \right) }{\alpha }\left( {t_w -t_r } \right) -\frac{D\left( p \right) }{2}\left( {t_d^2 -M^{2}} \right) \right. \nonumber \\&\left. {\left. { +\,\frac{D\left( p \right) }{\beta ^{2}}\left( {e^{\beta \left( {t_r -t_d } \right) }-1} \right) -\frac{D\left( p \right) }{\beta }\left( {t_r -t_d } \right) } \right\} } \right] +\frac{1}{T}\left\{ D\left( p \right) p-sD\left( p \right) \left( {T-t_w } \right) \right. \nonumber \\&\left. +\,2cD\left( p \right) +pI_e D\left( p \right) M \right\} =0 \nonumber \end{aligned}$$
$$\begin{aligned} \frac{\partial \textit{TP}_{1.1} }{\partial p}= & {} \frac{1}{T}\left[ {\left( {1-e} \right) D\left( p \right) T+F\left\{ {\frac{t_d^2 }{2}+\frac{1}{\beta }\left( {t_r -t_d +\frac{1}{\beta }\left( {1-e^{\beta \left( {t_r -t_d } \right) }} \right) } \right) } \right\} } \right. X_2 \nonumber \\&-\,\frac{H}{\alpha }\left\{ {t_r -t_w -\frac{1}{\alpha }\left( {1-e^{\alpha \left( {t_w -t_r } \right) }} \right) } \right\} X_2 \nonumber \\&-\,\frac{HD\left( p \right) }{\alpha }\left( {e^{\alpha \left( {t_w -t_r } \right) }-1} \right) X_3 +s\left( {T-t_w } \right) \left\{ {D\left( p \right) X_3 -\frac{\left( {T-t_w } \right) }{2}X_2 } \right\} \nonumber \\&-\,c\left\{ {t_d +\frac{1}{\beta }\left( {e^{\beta \left( {t_r -t_d } \right) }-1} \right) -\left( {T-t_w } \right) } \right\} X_2 \nonumber \\&-\,cD\left( p \right) X_3 +\left\{ {\frac{pI_e M^{2}}{2}+pI_e \left( {T-t_w } \right) M} \right\} X_2 -pI_e D\left( p \right) MX_3 +\frac{D\left( p \right) I_e M^{2}}{2}\nonumber \\&+\,D\left( p \right) I_e \left( {T-t_w } \right) M \nonumber \\&-\,cI_p \left\{ {\frac{1}{\alpha ^{2}}\left( {e^{\alpha \left( {t_w -t_r } \right) }-1} \right) } \right. \left. -\frac{1}{\alpha }\left( {t_w -t_r } \right) -\frac{1}{2}\left( {t_d^2 -M^{2}} \right) +\frac{1}{\beta ^{2}}\left( {e^{\beta \left( {t_r -t_d } \right) }-1} \right) \right. \nonumber \\&\left. -\,\frac{1}{\beta }\left( {t_r -t_d } \right) \right\} X_2 -\frac{cI_p D\left( p \right) }{\alpha }\left( {e^{\alpha \left( {t_w -t_r } \right) }-1} \right) X_3 \nonumber \\&\left. { -\,\left\{ {cI_p \left( {t_d -M} \right) +Ft_d } \right\} \left( {t_d +\frac{1}{\beta }\left( {e^{\beta \left( {t_r -t_d } \right) }-1} \right) } \right) X_2 } \right] =0 \end{aligned}$$
(36)
$$\begin{aligned} \frac{\partial \textit{TP}_{1.2} }{\partial t_r }= & {} \frac{1}{T}\left[ \frac{FD\left( p \right) }{\beta }\left( {1-e^{\beta \left( {t_r -t_d } \right) }} \right) \right. \\&\left. -\,\left( {cI_p +H} \right) \left\{ {We^{-\alpha \left( {t_r -t_d } \right) }+\frac{D\left( p \right) }{\alpha }\left\{ {\left( {1-e^{\alpha \left( {t_w -t_r } \right) }} \right) \left( {1-X_1 } \right) } \right\} } \right\} \right. \\&\left. -\,FD\left( p \right) t_d e^{\beta \left( {t_r -t_d } \right) }+sD\left( p \right) \left( {T-t_w } \right) X_1 -cD\left( p \right) \left( {e^{\beta \left( {t_r -t_d } \right) }+X_1 } \right) \right. \\&\left. -\,pI_e D\left( p \right) MX_1 -\frac{cI_p D\left( p \right) }{\beta }\left( {e^{\beta \left( {t_r -M} \right) }-1} \right) \right] =0 \\ \frac{\partial \textit{TP}_{1.2} }{\partial T}= & {} -\frac{1}{T^{2}}\left[ {D\left( p \right) pT-A-F\left\{ {Zt_d -\frac{D\left( p \right) t_d^2 }{2}-Wt_d -\frac{D\left( p \right) }{\beta }\left( {t_r -t_d +\frac{1}{\beta }-\frac{e^{\beta \left( {t_r -t_d } \right) }}{\beta }} \right) } \right\} } \right. \\&-\,H\left\{ {Wt_d +\frac{W}{\alpha }\left( {1-e^{\alpha \left( {t_d -t_r } \right) }} \right) +\frac{D\left( p \right) }{\alpha }\left( {t_r -t_w -\frac{1}{\alpha }+\frac{e^{\alpha \left( {t_w -t_r } \right) }}{\alpha }} \right) } \right\} -sD\left( p \right) \left( {T-t_w } \right) ^{2}/2 \\&-\,c\left\{ {W+D\left( p \right) t_d +\frac{D\left( p \right) }{\beta }\left( {e^{\beta \left( {t_r -t_d } \right) }-1} \right) -D\left( p \right) \left( {T-t_w } \right) } \right\} \\&+\,\left\{ {\left( {pI_e D\left( p \right) M^{2}} \right) /2+pI_e D\left( p \right) \left( {T-t_w } \right) M} \right\} \\&\left. -\,cI_p \left\{ \frac{W}{\alpha }\left( {e^{\alpha \left( {t_d -M} \right) }-e^{\alpha \left( {t_d -t_r } \right) }} \right) \right. \right. \\&\left. \left. +\frac{D\left( p \right) }{\alpha ^{2}}\left( {e^{\alpha \left( {t_w -t_r } \right) }-1} \right) -\frac{D\left( p \right) }{\alpha }\left( {t_w -t_r } \right) +\frac{D\left( p \right) }{\beta }\left\{ {\frac{1}{\beta }\left( {e^{\beta \left( {t_r -M} \right) }-1} \right) -\left( {t_r -M} \right) } \right\} \right\} \right] \\&+\,\frac{1}{T}\left\{ {D\left( p \right) p-sD\left( p \right) \left( {T-t_w } \right) +2cD\left( p \right) +pI_e D\left( p \right) M} \right\} =0 \end{aligned}$$
$$\begin{aligned} \frac{\partial \textit{TP}_{1.2} }{\partial p}= & {} \frac{1}{T}\left[ {\left( {1-e} \right) D\left( p \right) T+F\left\{ {\frac{t_d^2 }{2}+\frac{1}{\beta }\left( {t_r -t_d +\frac{1}{\beta }\left( {1-e^{\beta \left( {t_r -t_d } \right) }} \right) } \right) } \right\} } \right. X_2 \nonumber \\&-\,\frac{H}{\alpha }\left\{ {t_r -t_w -\frac{1}{\alpha }\left( {1-e^{\alpha \left( {t_w -t_r } \right) }} \right) } \right\} X_2 \nonumber \\&-\,\frac{HD\left( p \right) }{\alpha }\left( {e^{\alpha \left( {t_w -t_r } \right) }-1} \right) X_3 +s\left( {T-t_w } \right) \left\{ {D\left( p \right) X_3 -\frac{\left( {T-t_w } \right) }{2}X_2 } \right\} \nonumber \\&-\,c\left\{ {t_d +\frac{1}{\beta }\left( {e^{\beta \left( {t_r -t_d } \right) }-1} \right) -\left( {T-t_w } \right) } \right\} X_2 \nonumber \\&-\,cD\left( p \right) X_3 -Ft_d \left\{ {t_d +\frac{1}{\beta }\left( {e^{\beta \left( {t_r -t_d } \right) }-1} \right) } \right\} X_2 +\left\{ {\frac{pI_e M^{2}}{2}+pI_e \left( {T-t_w } \right) M} \right\} X_2\nonumber \\&-\,pI_e D\left( p \right) MX_3 +\frac{D\left( p \right) I_e M^{2}}{2} \nonumber \\&+\,D\left( p \right) I_e \left( {T-t_w } \right) M-cI_p \left\{ {\frac{1}{\alpha ^{2}}\left( {e^{\alpha \left( {t_w -t_r } \right) }-1} \right) -\frac{1}{\alpha }\left( {t_w -t_r } \right) } \right. \left. \left. +\frac{1}{\beta ^{2}}\left( {e^{\beta \left( {t_r -M} \right) }-1} \right) \right. \right. \nonumber \\&\left. \left. -\frac{1}{\beta }\left( {t_r -M} \right) \right\} X_2 -\frac{cI_p D\left( p \right) }{\alpha }\left( {e^{\alpha \left( {t_w -t_r } \right) }-1} \right) X_3 \right] =0 \end{aligned}$$
(37)
$$\begin{aligned} \frac{\partial \textit{TP}_{1.3} }{\partial t_r }= & {} \frac{1}{T}\left[ {\frac{FD\left( p \right) }{\beta }\left( {1-e^{\beta \left( {t_r -t_d } \right) }} \right) -H\left\{ {We^{-\alpha \left( {t_r -t_d } \right) }+\frac{D\left( p \right) }{\alpha }\left\{ {\left( {1-e^{\alpha \left( {t_w -t_r } \right) }} \right) \left( {1-X_1 } \right) } \right\} } \right\} } \right. \\&\left. -\,FDt_d e^{\beta \left( {t_r -t_d } \right) }+sD\left( p \right) \left( {T-t_w } \right) X_1 -cD\left( p \right) \left( {e^{\beta \left( {t_r -t_d } \right) }+X_1 }\right) \right. \\&\left. -\,pI_e D\left( p \right) MX_1 -\frac{cI_p D\left( p \right) }{\alpha }\left\{ {\left( {e^{\alpha \left( {t_w -t_r } \right) }-1} \right) \left( {X_1 -1} \right) } \right\} \right] =0 \\ \frac{\partial \textit{TP}_{1.3} }{\partial T}= & {} -\frac{1}{T^{2}}\left[ {D\left( p \right) pT-A-F\left\{ {Zt_d -\frac{D\left( p \right) t_d^2 }{2}-Wt_d -\frac{D\left( p \right) }{\beta }\left( {t_r -t_d +\frac{1}{\beta }-\frac{e^{\beta \left( {t_r -t_d } \right) }}{\beta }} \right) } \right\} } \right. \\&-\,H\left\{ {Wt_d +\frac{W}{\alpha }\left( {1-e^{\alpha \left( {t_d -t_r } \right) }} \right) +\frac{D\left( p \right) }{\alpha }\left( {t_r -t_w -\frac{1}{\alpha }+\frac{e^{\alpha \left( {t_w -t_r } \right) }}{\alpha }} \right) } \right\} -sD\left( p \right) \left( {T-t_w } \right) ^{2}/2 \\&-\,c\left\{ {W+D\left( p \right) t_d +\frac{D\left( p \right) }{\beta }\left( {e^{\beta \left( {t_r -t_d } \right) }-1} \right) -D\left( p \right) \left( {T-t_w } \right) } \right\} \\&+\,\left\{ {\left( {pI_e D\left( p \right) M^{2}} \right) /2+pI_e D\left( p \right) \left( {T-t_w } \right) M} \right\} \\&\left. {-\,cI_p \frac{D\left( p \right) }{\alpha }\left\{ {\frac{1}{\alpha }\left( {e^{\alpha \left( {t_w -t_r } \right) }-1} \right) -\left( {t_w -t_r } \right) } \right\} } \right] +\frac{1}{T}\left\{ D\left( p \right) p-sD\left( p \right) \left( {T-t_w } \right) \right. \\&\left. +\,2cD\left( p \right) +pI_e D\left( p \right) M \right\} =0 \end{aligned}$$
$$\begin{aligned} \frac{\partial \textit{TP}_{1.3} }{\partial p}= & {} \frac{1}{T}\left[ {\left( {1-e} \right) D\left( p \right) T+F\left\{ {\frac{t_d^2 }{2}+\frac{1}{\beta }\left( {t_r -t_d +\frac{1}{\beta }\left( {1-e^{\beta \left( {t_r -t_d } \right) }} \right) } \right) } \right\} } \right. X_2\nonumber \\&-\,\frac{H}{\alpha }\left\{ {t_r -t_w -\frac{1}{\alpha }\left( {1-e^{\alpha \left( {t_w -t_r } \right) }} \right) } \right\} X_2 \nonumber \\&-\,\frac{HD\left( p \right) }{\alpha }\left( {e^{\alpha \left( {t_w -t_r } \right) }-1} \right) X_3 +s\left( {T-t_w } \right) \left\{ {D\left( p \right) X_3 -\frac{\left( {T-t_w } \right) }{2}X_2 } \right\} \nonumber \\&-\,2c\left\{ {t_d +\frac{1}{\beta }\left( {e^{\beta \left( {t_r -t_d } \right) }-1} \right) -\left( {T-t_w } \right) } \right\} X_2\nonumber \\&-\,cD\left( p \right) X_3 -Ft_d \left\{ {t_d +\frac{1}{\beta }\left( {e^{\beta \left( {t_r -t_d } \right) }-1} \right) } \right\} X_2 +\left\{ {\frac{pI_e M^{2}}{2}+pI_e \left( {T-t_w } \right) M} \right\} X_2 \nonumber \\&-\,pI_e D\left( p \right) MX_3 +\frac{D\left( p \right) I_e M^{2}}{2} \nonumber \\&\left. +\,D\left( p \right) I_e \left( {T-t_w } \right) M-cI_p \left\{ \frac{1}{\alpha ^{2}}\left( {e^{\alpha \left( {t_w -t_r } \right) }-1} \right) -\frac{1}{\alpha }\left( {t_w -t_r } \right) \right\} X_2 \right. \nonumber \\&\left. -\,\frac{cI_p D\left( p \right) }{\alpha }\left( {e^{\alpha \left( {t_w -t_r } \right) }-1} \right) X_3 \right] =0 \end{aligned}$$
(38)
$$\begin{aligned} \frac{\partial \textit{TP}_{1.4} }{\partial t_r }= & {} \frac{1}{T}\left[ {\frac{FD\left( p \right) }{\beta }\left( {1-e^{\beta \left( {t_r -t_d } \right) }} \right) -H\left\{ {We^{-\alpha \left( {t_r -t_d } \right) }+\frac{D\left( p \right) }{\alpha }\left\{ {\left( {1-e^{\alpha \left( {t_w -t_r } \right) }} \right) \left( {1-X_1 } \right) } \right\} } \right\} } \right. \\&\left. { -\,FD\left( p \right) t_d e^{\beta \left( {t_r -t_d } \right) }+sD\left( p \right) \left( {T-t_w } \right) X_1 -cD\left( p \right) \left( {e^{\beta \left( {t_r -t_d } \right) }+X_1 } \right) +pI_e D\left( p \right) \left( {3t_w -2M} \right) X_1 } \right] =0 \\ \frac{\partial \textit{TP}_{1.4} }{\partial T}= & {} -\frac{1}{T^{2}}\left[ {D\left( p \right) pT-A-F\left\{ {Zt_d -\frac{D\left( p \right) t_d^2 }{2}-Wt_d -\frac{D\left( p \right) }{\beta }\left( {t_r -t_d +\frac{1}{\beta }-\frac{e^{\beta \left( {t_r -t_d } \right) }}{\beta }} \right) } \right\} } \right. \\&-\,H\left\{ {Wt_d +\frac{W}{\alpha }\left( {1-e^{\alpha \left( {t_d -t_r } \right) }} \right) +\frac{D\left( p \right) }{\alpha }\left( {t_r -t_w -\frac{1}{\alpha }+\frac{e^{\alpha \left( {t_w -t_r } \right) }}{\alpha }} \right) } \right\} -sD\left( p \right) \left( {T-t_w } \right) ^{2}/2 \\&-\,c\left\{ {W+D\left( p \right) t_d +\frac{D\left( p \right) }{\beta }\left( {e^{\beta \left( {t_r -t_d } \right) }-1} \right) -D\left( p \right) \left( {T-t_w } \right) } \right\} \\&\left. { +\,\left\{ {\left( {pI_e D\left( p \right) t_w^2 } \right) /2-pI_e D\left( p \right) t_w \left( {M-t_w } \right) +pI_e D\left( p \right) \left( {T-t_w } \right) M} \right\} } \right] \\&+\,\frac{1}{T}\left\{ {D\left( p \right) p-sD\left( p \right) \left( {T-t_w } \right) +2cD\left( p \right) +pI_e D\left( p \right) M} \right\} =0 \end{aligned}$$
$$\begin{aligned} \frac{\partial \textit{TP}_{1.4} }{\partial p}= & {} \frac{1}{T}\left[ {\left( {1-e} \right) D\left( p \right) T+F\left\{ {\frac{t_d^2 }{2}+\frac{1}{\beta }\left( {t_r -t_d +\frac{1}{\beta }\left( {1-e^{\beta \left( {t_r -t_d } \right) }} \right) } \right) } \right\} } \right. X_2 \nonumber \\&-\,\frac{H}{\alpha }\left\{ {t_r -t_w -\frac{1}{\alpha }\left( {1-e^{\alpha \left( {t_w -t_r } \right) }} \right) } \right\} X_2 \nonumber \\&-\,\frac{HD\left( p \right) }{\alpha }\left( {e^{\alpha \left( {t_w -t_r } \right) }-1} \right) X_3 +s\left( {T-t_w } \right) \left\{ {D\left( p \right) X_3 -\frac{\left( {T-t_w } \right) }{2}X_2 } \right\} \nonumber \\&-\,c\left\{ {t_d +\frac{1}{\beta }\left( {e^{\beta \left( {t_r -t_d } \right) }-1} \right) -\left( {T-t_w } \right) } \right\} X_2 \nonumber \\&-\,cD\left( p \right) X_3 +\left\{ {\frac{pI_e M^{2}}{2}+pI_e \left( {T-t_w } \right) M} \right\} X_2 +\left\{ \left( {I_e D\left( p \right) t_w^2 } \right) /2+pI_e D\left( p \right) t_w X_3 \right. \nonumber \\&\left. -\,I_e D\left( p \right) t_w \left( {M-t_w } \right) \right. \nonumber \\&\left. \left. -pI_e D\left( p \right) \left( {M-t_w } \right) X_3 -Ft_d \left\{ {t_d +\frac{1}{\beta }\left( {e^{\beta \left( {t_r -t_d } \right) }-1} \right) } \right\} X_2 +pI_e D\left( p \right) t_w X_3 \right. \right. \nonumber \\&\left. \left. +\,I_e D\left( p \right) \left( {T-t_w } \right) M-pI_e D\left( p \right) MX_3 \right\} \right] =0 \nonumber \\ \end{aligned}$$
(39)

Appendix 2

For Case 2, the necessary conditions for maximizing the total average profit is given by

$$\begin{aligned} \frac{\partial \textit{TP}_{2.1} }{\partial t_r }= & {} \frac{-D\left( p \right) }{T}\left[ {Ft_r +H\left\{ {\left( {t_d -t_r } \right) +\frac{1}{\alpha }\left( {1-e^{\alpha \left( {t_w -t_d } \right) }} \right) Y_1 } \right\} } \right. -s\left( {T-t_w } \right) Y_1 \\&-\,c\left( {e^{\beta \left( {t_r -t_d } \right) }+Y_1 } \right) +c\left( {1+Y_1 } \right) \\&\left. { -\,pI_e MY_1 +cI_p \left\{ {\left( {t_d -t_r } \right) +\frac{1}{\alpha }\left( {1-e^{\alpha \left( {t_w -t_d } \right) }} \right) Y_1 } \right\} +\left( {t_r -M} \right) } \right] -\frac{HW}{T}=0 \\ \frac{\partial \textit{TP}_{2.1} }{\partial T}= & {} -\frac{1}{T^{2}}\left[ {D\left( p \right) pT-A-\frac{FD\left( p \right) t_r^2 }{2}} \right. -H\left\{ Wt_r +D\left( p \right) t_r \left( {t_d -t_r } \right) -\frac{D\left( p \right) }{2}\left( {t_d^2 -t_r^2 } \right) \right. \\&\left. +\,\frac{D\left( p \right) }{\alpha }\left( {\frac{1}{\alpha }\left( {e^{\alpha \left( {t_w -t_d } \right) }-1} \right) -\left( {t_w -t_d } \right) } \right) \right\} \\&-\,sD\left( p \right) \left( {T-t_w } \right) ^{2}/2-c\left( {W+D\left( p \right) t_r -D\left( p \right) \left( {T-t_w } \right) } \right) \\&+\,\left\{ {\left( {pI_e D\left( p \right) M^{2}} \right) /2+pI_e D\left( p \right) \left( {T-t_w } \right) M } \right\} -cI_p \left\{ W\left( {t_d -M} \right) +D\left( p \right) t_r \left( {t_d -t_r } \right) \right. \\&\left. -\,\frac{D\left( p \right) }{2}\left( {t_d -t_r } \right) ^{2} \right. \\&+\left. \left. \frac{D\left( p \right) }{\alpha ^{2}}\left( {e^{\alpha \left( {t_w -t_d } \right) }-1} \right) -\frac{D\left( p \right) }{\alpha }\left( {t_w -t_d } \right) +\frac{D\left( p \right) }{2}\left( {t_r -M} \right) ^{2} \right\} \right] \\&+\,\frac{1}{T}\left\{ {D\left( p \right) p-sD\left( p \right) \left( {T-t_w } \right) +2cD\left( p \right) +pI_e D\left( p \right) M} \right\} =0 \end{aligned}$$
$$\begin{aligned} \frac{\partial \textit{TP}_{2.1} }{\partial p}= & {} \frac{1}{T}\left[ {\left( {1-e} \right) D\left( p \right) T-\frac{FY_2 t_r^2 }{2}} \right. -HY_2 \left\{ t_r \left( {t_d -t_r } \right) -\frac{1}{2}\left( {t_d^2 -t_r^2 } \right) \right. \nonumber \\&\left. +\,\frac{1}{\alpha }\left( {\frac{1}{\alpha }\left( {e^{\alpha \left( {t_w -t_d } \right) }-1} \right) -\left( {t_w -t_d } \right) } \right) \right\} -H\frac{D\left( p \right) }{\alpha }\left( {e^{\alpha \left( {t_w -t_d } \right) }-1} \right) Y_3 \nonumber \\&-\,sY_2 \left( {T-t_w } \right) ^{2}/2+sD\left( p \right) \left( {T-t_w } \right) Y_3 -cY_2 \left( {t_r -\left( {T-t_w } \right) } \right) -cD\left( p \right) Y_3 -pI_e D\left( p \right) MY_3 \nonumber \\&+\,Y_2 \left\{ {\left( {pI_e M^{2}} \right) /2+pI_e \left( {T-t_w } \right) M } \right\} -cI_p \left\{ {Y_2 t_r \left( {t_d -t_r } \right) -\frac{Y_2 }{2}\left( {t_d -t_r } \right) ^{2}} \right. +\frac{Y_2 }{\alpha ^{2}}\left( {e^{\alpha \left( {t_w -t_d } \right) }-1} \right) \nonumber \\&+\,\frac{D\left( p \right) }{\alpha }\left( {e^{\alpha \left( {t_w -t_d } \right) }-1} \right) Y_3 \nonumber \\&\qquad \left. {\left. {-\,\frac{Y_2 }{\alpha }\left( {t_w -t_d } \right) +\frac{Y_2 }{2}\left( {t_r -M} \right) ^{2}} \right\} } \right] =0 \end{aligned}$$
(40)
$$\begin{aligned} \frac{\partial \textit{TP}_{2.2} }{\partial t_r }= & {} \frac{-D\left( p \right) }{T}\left[ {Ft_r +H\left\{ {\left( {t_d -t_r } \right) +\frac{1}{\alpha }\left( {1-e^{\alpha \left( {t_w -t_d } \right) }} \right) Y_1 } \right\} } \right. -s\left( {T-t_w } \right) Y_1 \\&-\,c\left( {e^{\beta \left( {t_r -t_d } \right) }+Y_1 } \right) +c\left( {1+Y_1 } \right) \\&\left. { -\,pI_e MY_1 +cI_p \left\{ {\left( {t_d -M} \right) +\frac{1}{\alpha }\left( {1-e^{\alpha \left( {t_w -t_d } \right) }} \right) Y_1 } \right\} } \right] -\frac{HW}{T}=0 \\ \frac{\partial \textit{TP}_{2.2} }{\partial T}= & {} -\frac{1}{T^{2}}\left[ {D\left( p \right) pT-A-\frac{FD\left( p \right) t_r^2 }{2}} \right. -H\left\{ Wt_r +D\left( p \right) t_r \left( {t_d -t_r } \right) \right. \\&\left. -\,\frac{D\left( p \right) }{2}\left( {t_d^2 -t_r^2 } \right) +\frac{D\left( p \right) }{\alpha }\left( {\frac{1}{\alpha }\left( {e^{\alpha \left( {t_w -t_d } \right) }-1} \right) -\left( {t_w -t_d } \right) } \right) \right\} \\&-\,sD\left( p \right) \left( {T-t_w } \right) ^{2}/2-c\left( W+D\left( p \right) t_r \right. \\&\left. -\,D\left( p \right) \left( {T-t_w } \right) -D\left( p \right) T \right) \\&+\,\left\{ {\left( {pI_e D\left( p \right) M^{2}} \right) /2+pI_e D\left( p \right) \left( {T-t_w } \right) M } \right\} -cI_p \left\{ \left( {W+D\left( p \right) t_r } \right) \left( {t_d -M} \right) \right. \\&\left. -\,\frac{D\left( p \right) }{2}\left( {t_d^2 -M^{2}} \right) +\frac{D\left( p \right) }{\alpha ^{2}}\left( {e^{\alpha \left( {t_w -t_d } \right) }-1} \right) \right. \\&\left. {\left. {-\,\frac{D\left( p \right) }{\alpha }\left( {t_w -t_d } \right) } \right\} } \right] +\frac{1}{T}\left\{ {D\left( p \right) p-sD\left( p \right) \left( {T-t_w } \right) +2cD\left( p \right) +pI_e D\left( p \right) M} \right\} =0 \end{aligned}$$
$$\begin{aligned} \frac{\partial \textit{TP}_{2.2} }{\partial p}= & {} \frac{1}{T}\left[ {\left( {1-e} \right) D\left( p \right) T-\frac{FY_2 t_r^2 }{2}} \right. -HY_2 \left\{ t_r \left( {t_d -t_r } \right) -\frac{1}{2}\left( {t_d^2 -t_r^2 } \right) \right. \nonumber \\&\left. +\,\frac{1}{\alpha }\left( {\frac{1}{\alpha }\left( {e^{\alpha \left( {t_w -t_d } \right) }-1} \right) -\left( {t_w -t_d } \right) } \right) \right\} -H\frac{D\left( p \right) }{\alpha }\left( {e^{\alpha \left( {t_w -t_d } \right) }-1} \right) Y_3 \nonumber \\&-\,sY_2 \left( {T-t_w } \right) ^{2}/2+sD\left( p \right) Y_3 -cY_2 \left\{ {t_r -\left( {T-t_w } \right) } \right\} -cD\left( p \right) Y_3 \nonumber \\&+\,pI_e Y_2 M\left\{ {M/2+\left( {T-t_w } \right) } \right\} -pI_e D\left( p \right) MY_3 \nonumber \\&\left. -\,cI_p \left\{ \left( {W+Y_2 t_r } \right) \left( {t_d -M} \right) -\frac{Y_2 }{2}\left( {t_d^2 -M^{2}} \right) +\frac{Y_2 }{\alpha ^{2}}\left( {e^{\alpha \left( {t_w -t_d } \right) }-1} \right) -\frac{Y_2 }{\alpha }\left( {t_w -t_d } \right) \right. \right. \nonumber \\&\left. \left. +\,\frac{D\left( p \right) }{\alpha }\left( {e^{\alpha \left( {t_w -t_d } \right) }-1} \right) Y_3 \right\} \right] =0 \end{aligned}$$
(41)
$$\begin{aligned} \frac{\partial \textit{TP}_{2.3} }{\partial t_r }= & {} \frac{-D\left( p \right) }{T}\left[ {Ft_r +H\left\{ {\left( {t_d -t_r } \right) +\frac{1}{\alpha }\left( {1-e^{\alpha \left( {t_w -t_d } \right) }} \right) Y_1 } \right\} } \right. -s\left( {T-t_w } \right) Y_1\\&-\,c\left( {e^{\beta \left( {t_r -t_d } \right) }+Y_1 } \right) +c\left( {1+Y_1 } \right) \\&\left. { -\,pI_e MY_1 +\frac{cI_p }{\alpha }\left( {1-e^{\alpha \left( {t_w -t_d } \right) }} \right) Y_1 } \right] -\frac{HW}{T}=0 \\ \frac{\partial \textit{TP}_{2.3} }{\partial T}= & {} -\frac{1}{T^{2}}\left[ {D\left( p \right) pT-A-\frac{FD\left( p \right) t_r^2 }{2}} \right. -H\left\{ Wt_r +D\left( p \right) t_r \left( {t_d -t_r } \right) \right. \\&\left. -\,\frac{D\left( p \right) }{2}\left( {t_d^2 -t_r^2 } \right) \right. \\&\left. {+\,\frac{D\left( p \right) }{\alpha }\left( {\frac{1}{\alpha }\left( {e^{\alpha \left( {t_w -t_d } \right) }-1} \right) -\left( {t_w -t_d } \right) } \right) } \right\} -sD\left( p \right) \left( {T-t_w } \right) ^{2}/2\\&-\,c\left\{ {W+D\left( p \right) t_r -D\left( p \right) \left( {T-t_w } \right) } \right\} \\&\left. +\,\left\{ {\left( {pI_e D\left( p \right) M^{2}} \right) /2+pI_e D\left( p \right) \left( {T-t_w } \right) M } \right\} \right. \\&\left. -\,cI_p \frac{D\left( p \right) }{\alpha }\left\{ {\frac{1}{\alpha }\left( {e^{\alpha \left( {t_w -M} \right) }-1} \right) -\left( {t_w -M} \right) } \right\} \right] \\&+\,\frac{1}{T}\left\{ {Dp-sD\left( {T-t_w } \right) +3cD+pI_e DM} \right\} =0 \end{aligned}$$
$$\begin{aligned} \frac{\partial \textit{TP}_{2.3} }{\partial p}= & {} \frac{1}{T}\left[ {\left( {1-e} \right) D\left( p \right) T-\frac{FY_2 t_r^2 }{2}} \right. -HY_2 \left\{ {t_r \left( {t_d -t_r } \right) -\frac{1}{2}\left( {t_d^2 -t_r^2 } \right) } \right. \left. \right. \nonumber \\&\left. +\,\frac{1}{\alpha }\left( {\frac{1}{\alpha }\left( {e^{\alpha \left( {t_w -t_d } \right) }-1} \right) -\left( {t_w -t_d } \right) } \right) \right\} \nonumber \\&-\,H\frac{D\left( p \right) }{\alpha }\left( {e^{\alpha \left( {t_w -t_d } \right) }-1} \right) Y_3 -sY_2 \left( {T-t_w } \right) ^{2}/2+sD\left( p \right) \left( {T-t_w } \right) Y_3 \nonumber \\&-\,cY_2 \left\{ {t_r -\left( {T-t_w } \right) } \right\} \nonumber \\&+\,cD\left( p \right) Y_3 +pI_e Y_2 M\left\{ {M/2+\left( {T-t_w } \right) } \right\} -pI_e D\left( p \right) MY_3 \nonumber \\&\left. { -\,cI_p \frac{Y_2 }{\alpha }\left\{ {\frac{1}{\alpha }\left( {e^{\alpha \left( {t_w -M} \right) }-1} \right) -\left( {t_w -M} \right) } \right\} -cI_p \frac{D\left( p \right) }{\alpha }\left( {e^{\alpha \left( {t_w -M} \right) }-1} \right) Y_3 } \right] =0\nonumber \\ \end{aligned}$$
(42)
$$\begin{aligned} \frac{\partial \textit{TP}_{2.4} }{\partial t_r }= & {} \frac{-D\left( p \right) }{T}\left[ {Ft_r +H\left\{ {\left( {t_d -t_r } \right) +\frac{1}{\alpha }\left( {1-e^{\alpha \left( {t_w -t_d } \right) }} \right) Y_1 } \right\} } \right. -s\left( {T-t_w } \right) Y_1\\&-\,c\left( {e^{\beta \left( {t_r -t_d } \right) }+Y_1 } \right) +c\left( {1+Y_1 } \right) \\&\left. { -\,pI_e MY_1 } \right] -\frac{HW}{T}=0 \\ \frac{\partial \textit{TP}_{2.4} }{\partial T}= & {} -\frac{1}{T^{2}}\left[ {D\left( p \right) pT-A-\frac{FD\left( p \right) t_r^2 }{2}} \right. -H\left\{ Wt_r +D\left( p \right) t_r \left( {t_d -t_r } \right) \right. \\&\left. -\,\frac{D\left( p \right) }{2}\left( {t_d^2 -t_r^2 } \right) \right. \\&\left. {+\,\frac{D\left( p \right) }{\alpha }\left( {\frac{1}{\alpha }\left( {e^{\alpha \left( {t_w -t_d } \right) }-1} \right) -\left( {t_w -t_d } \right) } \right) } \right\} -sD\left( p \right) \left( {T-t_w } \right) ^{2}/2\\&-\,c\left\{ {W+D\left( p \right) t_r -D\left( p \right) \left( {T-t_w } \right) } \right\} \\&\left. {+\,\left\{ {\left( {pI_e D\left( p \right) t_w^2 } \right) /2+pI_e D\left( p \right) t_w \left( {M-t_w } \right) +pI_e D\left( p \right) \left( {T-t_w } \right) M} \right\} } \right] \\&+\,\frac{1}{T}\left\{ {Dp-sD\left( {T-t_w } \right) +2cD+pI_e DM} \right\} =0 \end{aligned}$$
$$\begin{aligned} \frac{\partial \textit{TP}_{2.4} }{\partial p}= & {} \frac{1}{T}\left[ {\left( {1-e} \right) D\left( p \right) T-\frac{FY_2 t_r^2 }{2}} \right. \nonumber \\&-\,HY_2 \left\{ {t_r \left( {t_d -t_r } \right) -\frac{1}{2}\left( {t_d^2 -t_r^2 } \right) } \right. \left. {+\frac{1}{\alpha }\left( {\frac{1}{\alpha }\left( {e^{\alpha \left( {t_w -t_d } \right) }-1} \right) -\left( {t_w -t_d } \right) } \right) } \right\} \nonumber \\&-\,H\frac{D\left( p \right) }{\alpha }\left( {e^{\alpha \left( {t_w -t_d } \right) }-1} \right) Y_3 -sY_2 \left( {T-t_w } \right) ^{2}/2+sD\left( p \right) \left( {T-t_w } \right) Y_3 \nonumber \\&-\,2cY_2\left\{ {t_r -\left( {T-t_w } \right) } \right\} -2cD\left( p \right) Y_3 \nonumber \\&-\,cY_2 T\left. +pI_e Y_2 \left\{ {\left( {pI_e D\left( p \right) t_w^2 } \right) /2+t_w \left( {M-t_w } \right) +\left( {T-t_w } \right) M} \right\} \right. \nonumber \\&\left. -\,pI_e D\left( p \right) t_w Y_3 \right] =0 \end{aligned}$$
(43)

where

$$\begin{aligned} X_1= & {} \frac{D\left( p \right) }{D\left( p \right) +\alpha We^{\alpha \left( {t_d -t_r } \right) }}, \quad X_2 = Y_2 = \frac{\partial D\left( p \right) }{\partial p}= -ekp^{-\left( {e+1} \right) }, \\ X_3= & {} \frac{-D\left( p \right) We^{\alpha \left( {t_d -t_r } \right) }}{eX_2 \left( {D\left( p \right) +\alpha We^{\alpha \left( {t_d -t_r } \right) }} \right) }\\ Y_1= & {} \frac{D\left( p \right) }{\left\{ {D\left( p \right) +\alpha \left( {W+D\left( p \right) \left( {t_r -t_d } \right) } \right) } \right\} }, \;and\; \\ Y_3= & {} \frac{-WY_2 }{D\left( p \right) \left\{ {D\left( p \right) +\alpha \left( {W+D\left( p \right) \left( {t_r -t_d } \right) } \right) } \right\} } \end{aligned}$$

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Jaggi, C.K., Tiwari, S. & Goel, S.K. Credit financing in economic ordering policies for non-instantaneous deteriorating items with price dependent demand and two storage facilities. Ann Oper Res 248, 253–280 (2017). https://doi.org/10.1007/s10479-016-2179-3

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