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An integrated inventory model for a deteriorating item with allowable shortages and credit linked wholesale price

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Abstract

This study proposes a single manufacturer, single retailer integrated inventory model that includes deterioration and shortages in the retailer’s inventory. The manufacturer’s production process is assumed to be imperfect as it produces a certain percentage of defective items. The retailer performs a 100  % screening process immediately on receiving a lot from the manufacturer and returns the detected defective items to the manufacturer in the next delivery. The manufacturer disposes the defective items and incurs a disposal cost. To increase sales, (s)he offers a trade credit to the retailer. The retailer’s wholesale price varies linearly with the credit period. The objective is to determine the optimal replenishment cycle time, the time of running out of stock, the length of the credit period and the number of lots from the manufacturer to the retailer so as to maximize the total profit of the integrated system. A solution algorithm is designed and illustrated through numerical examples. Furthermore, a sensitivity analysis is carried out to study the influence of the model-parameters on the optimal solution.

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Acknowledgments

The authors are thankful to the anonymous reviewers for their helpful comments and constructive suggestions on the earlier versions of the manuscript.

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Correspondence to B. C. Giri.

Appendix A

Appendix A

For a fixed value of \(n\), the necessary conditions for \(JTP_1\) to be maximized are given below:

$$\begin{aligned} \frac{\partial JTP_1}{\partial T}&= \frac{D}{T}\left\{ \left( \frac{c}{1-\gamma }+I_vvM\right) \left( \frac{e^{\theta T_1}-1+\theta (T-T_1)}{\theta T}-1\right) \right. \nonumber \\&+\frac{1}{(1-\gamma )\theta }(e^{\theta T_1}-1+\theta (T-T_1))\left[ \frac{h_mD(n-2)}{2(1-\gamma )R}\left( 1-\frac{e^{\theta T_1}-1+\theta (T-T_1)}{\theta T}\right) \right. \nonumber \\&+\left. \frac{s+w\gamma }{T}+\gamma h_{b2}\left( \frac{1-e^{-\theta _1 T}(1+\theta _1 T)}{\theta _1 T}\right) \right] +\frac{h_{b1}(e^{\theta T_1}-1-\theta T_1)}{\theta ^2 T}\nonumber \\&-\frac{1}{(1-\gamma )}\left[ \frac{h_m(n-1)T}{2}-\frac{h_mD}{2\theta (1-\gamma )}(e^{\theta T_1}-1+\theta (T-T_1))\frac{(n-2)}{R}+s+w\gamma \right. \nonumber \\&\left. +\frac{\gamma h_{b2}}{\theta _1}(1-e^{-\theta _1T})\right] -\frac{b(T^2-T_1^2)}{2T}+\frac{I_epM(2T_1-M)}{2T} +\frac{A+K+nF}{DnT}\nonumber \\&\left. +\frac{I_pv}{\theta ^2T}(e^{\theta (T_1-M)}-1-\theta (T_1-M))\right\} =0\end{aligned}$$
(A.1)
$$\begin{aligned} \frac{\partial JTP_1}{\partial T_1}&= \frac{D}{T}\left\{ (1-e^{\theta T_1})\left( \frac{c}{1-\gamma }+I_vvM+\frac{h_{b1}}{\theta }\right) - \frac{I_pv}{\theta }(e^{\theta (T_1-M)}-1)\right. \nonumber \\&\quad +\,b(T-T_1)-I_epM-\frac{(e^{\theta T_1}-1)}{(1-\gamma )}\left[ \frac{h_m(n-1)T}{2}+s\!+\!w\gamma \!+\!\frac{\gamma h_{b2}}{\theta _1}(1-e^{-\theta _1T})\right. \nonumber \\&\quad \left. \left. -\,\frac{Dh_m(n-2)(e^{\theta T_1}-1+\theta (T-T_1))}{R\theta (1-\gamma )}\right] \right\} =0\end{aligned}$$
(A.2)
$$\begin{aligned} \frac{\partial JTP_1}{\partial M}&= \frac{D}{\theta ^2T}\{\theta [I_ep(M+T-T_1)\theta -I_v(v_0+2Mv_1)(e^{\theta T_1}-1+\theta (T-T_1))]\nonumber \\&+I_p[v_1(1-2M\theta +\theta T_1)-v_{0}\theta +e^{\theta (T_1-M)}(v_{0}\theta +v_1(M\theta -1))]\}=0 \end{aligned}$$
(A.3)

Similarly, the necessary conditions for \(JTP_2\) and \(JTP_3\) to be maximized are

$$\begin{aligned} \frac{\partial JTP_2}{\partial T}&= \frac{\partial JTP_3}{\partial T}=\frac{D}{T}\left\{ \left( \frac{c}{1-\gamma }+I_vvM\right) \left( \frac{e^{\theta T_1}-1+\theta (T-T_1)}{\theta T}-1\right) \right. \nonumber \\&+\,\frac{1}{(1-\gamma )\theta }(e^{\theta T_1}-1+\theta (T-T_1))\left[ \frac{h_mD(n-2)}{2(1-\gamma )R}\left( 1-\frac{e^{\theta T_1}-1+\theta (T-T_1)}{\theta T}\right) \right. \nonumber \\&+\,\left. \frac{s+w\gamma }{T}+\gamma h_{b2}\left( \frac{1-e^{-\theta _1 T}(1+\theta _1 T)}{\theta _1 T}\right) \right] +\frac{h_{b1}(e^{\theta T_1}-1-\theta T_1)}{\theta ^2 T}\nonumber \\&-\,\frac{1}{(1-\gamma )}\left[ \frac{h_m(n-1)T}{2}-\frac{h_mD}{2\theta (1-\gamma )}(e^{\theta T_1}-1+\theta (T-T_1))\frac{(n-2)}{R}+s+w\gamma \right. \nonumber \\&\left. \left. +\,\frac{\gamma h_{b2}}{\theta _1}(1-e^{-\theta _1T})\right] -\frac{b(T^2-T_1^2)}{2T}+\frac{I_epT_1^2}{2T} +\frac{A+K+nF}{DnT}\right\} =0\end{aligned}$$
(A.4)
$$\begin{aligned} \frac{\partial JTP_2}{\partial T_1}&= \frac{\partial JTP_3}{\partial T_1}=\frac{D}{T}\left\{ (1-e^{\theta T_1})\left( \frac{c}{1-\gamma }+I_vvM+\frac{h_{b1}}{\theta }\right) +b(T-T_1)-I_epT_1\right. \nonumber \\&-\frac{(e^{\theta T_1}-1)}{(1-\gamma )}\left[ \frac{h_m(n-1)T}{2}+s+w\gamma +\frac{\gamma h_{b2}}{\theta _1}(1-e^{-\theta _1T})\right. \nonumber \\&\left. \left. -\frac{Dh_m(n-2)(e^{\theta T_1}-1+\theta (T-T_1))}{R\theta (1-\gamma )}\right] \right\} =0\end{aligned}$$
(A.5)
$$\begin{aligned} \frac{\partial JTP_2}{\partial M}&= \frac{\partial JTP_3}{\partial M}=\frac{D}{\theta T}[I_ep\theta T-I_v(v_0+2Mv_1)(e^{T_1\theta }-1+\theta (T-T_1))]= 0 \end{aligned}$$
(A.6)

The above conditions only maximize the joint total profit if the objective function is concave.

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Giri, B.C., Sharma, S. An integrated inventory model for a deteriorating item with allowable shortages and credit linked wholesale price . Optim Lett 9, 1149–1175 (2015). https://doi.org/10.1007/s11590-014-0810-2

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