An inventory model for two-parameter Weibull distributed ameliorating and deteriorating items with stock and advertisement frequency dependent demand under trade credit and preservation technology

Abstract

In this article, an inventory model is studied for ameliorating and deteriorating items with demand dependent on stock and advertisement frequency. The rate of amelioration and deterioration is represented by two-parameter Weibull distribution. The retailer invests in preservation technology to lower the rate of deterioration. It is considered that the supplier offers a trade credit period to the retailer to settle the account. Shortages in the model are allowed and partially backlogged. The primary objective of the present study is to determine the optimal advertisement frequency, preservation technology investment, cycle length, and the time of occurrence of the maximum positive level of stock and shortage that maximize the total profit per unit time of the system. An algorithm is provided to obtain the optimal solution from the developed model. Several numerical examples are provided under different situations to illustrate the model, while the concavity of the total profit function is represented graphically. Sensitivity analysis is performed to analyze the influence of the parameters on the optimal solution, while various observations and managerial insights obtained from the sensitivity analysis are discussed accordingly.

Appendices

Appendix 1

$TP_{1}(A,\xi ,T_{1},T)$ given in Eq. (23) can be written as

$$\begin{aligned} TP_{1}(A,\xi ,T_{1},T)&=\frac{1}{T}\left[ A^{\lambda }f_{1} (\xi ,T_{1},T)+A^{2\lambda }f_{2}(\xi ,T_{1})+A^{3\lambda }f_{3}(T_{1})\right. \\&\quad +\left. f_{4}(\xi ,T_{1},T) - C_{7}A \right] \end{aligned}$$

where,

$$\begin{aligned} f_{1}(\xi ,T_{1},T)&=p\left\{ a(1-\eta )T_{1} +\frac{a}{\rho }\log \{1+\rho (T-T_{1})\}\right\} -\frac{C_{2}\alpha a}{\beta +1}\left\{ (1-\eta ^{\beta })\right. \nonumber \\&\quad -\left. \beta \eta ^{\beta }(1-\eta )\right\} T_{1}^{\beta +1} -\frac{C_{3} \left( 1-m(\xi )\right) xa}{y+1} \left\{ (1-\eta ^{y})-y\eta ^{y}(1-\eta )\right\} T_{1}^{y+1}\nonumber \\&\quad - ha \left[ \frac{1}{2}(1-\eta )^{2}T_{1}^{2} +\frac{\{1-m(\xi )\}x}{(y+1)(y+2)} \left\{ y(1-\eta ^{y+2})\right. \right. \nonumber \\&\quad - \left. \left. (y+2)\eta (1-\eta ^{y}) \right\} T_{1}^{y+2} +\frac{\alpha }{\beta +1} \left\{ \eta (1-\eta ^{\beta }) -\frac{\beta }{\beta +2}(1-\eta ^{\beta +2}) \right\} T_{1}^{\beta +2}\right] \nonumber \\&\quad - \left( \frac{C_{4}a}{\rho ^{2}}+\frac{C_{5}a}{\rho } \right) \left[ \rho (T-T_{1})-\log \{1+\rho (T-T_{1})\} \right] \nonumber \\&\quad - \frac{C_{6}a}{\rho }\log \{1+\rho (T-T_{1})\} -C_{6}I_{c}a\left[ \frac{1}{2}(T_{1}-M)^{2}\right. \nonumber \\&\quad + \frac{\{1-m(\xi )\}x}{(y+1)(y+2)} \left\{ y(T_{1}^{y+2}-M^{y+2})- (y+2)T_{1} M(T_{1}^{y}-M^{y}) \right\} \nonumber \\&\quad + \left. \frac{\alpha }{\beta +1}\left\{ T_{1} M(T_{1}^{\beta }-M^{\beta })-\frac{\beta }{\beta +2} (T_{1}^{\beta +2}-M^{\beta +2})\right\} \right] \nonumber \\&\quad + \frac{pI_{e}a}{2}(M^{2}-\eta ^{2}T_{1}^{2}) \end{aligned}$$

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$$\begin{aligned} f_{2}(\xi ,T_{1})&= pab \left\{ \frac{1}{2}(1-\eta )^{2}T_{1}^{2} +\frac{\{1-m(\xi )\}x}{(y+1)(y+2)}\left\{ y(1-\eta ^{y+2})-\eta (y+2) (1-\eta ^{y}) \right\} T_{1}^{y+2}\right. \nonumber \\&\quad +\left. \frac{\alpha }{\beta +1}\{\eta (1-\eta ^{\beta }) -\frac{\beta }{\beta +2}(1-\eta ^{\beta +2})\}T_{1}^{\beta +2}\right\} - \frac{hab}{6}(1-\eta )^{3}T_{1}^{3} \nonumber \\&\quad - \frac{C_{6}I_{c}ab}{6}(T_{1}-M)^{3}+pI_{e}ab \left[ \frac{1}{6}\left\{ 3T_{1}(M^{2}-\eta ^{2}T_{1}^{2})\right. \right. \nonumber \\&\quad -\left. 2(M^{3}-\eta ^{3}T_{1}^{3})\right\} +\frac{\{1-m(\xi )\}x}{y+1} \left\{ \frac{1}{2}T_{1}^{y+1}(M^{2}-\eta ^{2}T_{1}^{2})\right. \nonumber \\&\quad +\left. \frac{y}{y+3}(M^{y+3}-\eta ^{y+3}T_{1}^{y+3}) -\left( \frac{y+1}{y+2}\right) T_{1}(M^{y+2}-\eta ^{y+2} T_{1}^{y+2})\right\} \nonumber \\&\quad + \frac{\alpha }{\beta +1}\left\{ \left( \frac{\beta +1}{\beta +2}\right) T_{1}(M^{\beta +2}-\eta ^{\beta +2}T_{1}^{\beta +2})\right. \nonumber \\&\quad -\left. \left. \frac{\beta }{\beta +3}(M^{\beta +3}-\eta ^{\beta +3} T_{1}^{\beta +3})-\frac{1}{2}T_{1}^{\beta +1} (M^{2}-\eta ^{2}T_{1}^{2})\right\} \right] \end{aligned}$$

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$$\begin{aligned} f_{3}(T_{1})&=\frac{pab^{2}(1-\eta )^{3}}{6}T_{1}^{3} +\frac{pI_{e}ab^{2}}{24}\{ 6T_{1}^{2}(M^{2}-\eta ^{2} T_{1}^{2})+3(M^{4}-\eta ^{4}T_{1}^{4}) \nonumber \\&\quad -8T_{1}(M^{3}-\eta ^{3}T_{1}^{3})\} \end{aligned}$$

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and

$$\begin{aligned} f_{4}(\xi ,T_{1},T)&= -C_{1}-C_{2}\alpha I_{0} \eta ^{\beta }T_{1}^{\beta }-C_{3} \left( 1-m(\xi )\right) x I_{0}\eta ^{y}T_{1}^{y} \nonumber \\&\quad -hI_{0}\left[ \eta T_{1} +\frac{\alpha }{\beta +1} \eta ^{\beta +1}T_{1}^{\beta +1}-\frac{\{1-m(\xi )\}x}{y+1} \eta ^{y+1}T_{1}^{y+1}\right] \nonumber \\&\quad - C_{6}I_{0}-\xi T \end{aligned}$$

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Appendix 2

$TP_{1}(A,\xi ,T_{1},T)$ given in Eq. (23) can be written as

$$\begin{aligned} TP_{1}(A,\xi ,T_{1},T)= \frac{Z}{T} \end{aligned}$$

where,

$$\begin{aligned} Z&=p\left[ aA^{\lambda }\left[ (1-\eta )T_{1}+bA^{\lambda } \left[ \frac{1}{2}(1-\eta )^{2}T_{1}^{2}+\frac{\{1-m(\xi )\}x}{(y+1)(y+2)}\left\{ y(1-\eta ^{y+2})\right. \right. \right. \right. \nonumber \\&\quad -\left. \eta (y+2)(1-\eta ^{y}) \right\} T_{1}^{y+2} +\frac{bA^{\lambda }}{6}(1-\eta )^{3}T_{1}^{3} +\frac{\alpha }{\beta +1}\{\eta (1-\eta ^{\beta })\nonumber \\&\quad -\left. \left. \left. \frac{\beta }{\beta +2}(1-\eta ^{\beta +2})\} T_{1}^{\beta +2} \right] \right] +\frac{aA^{\lambda }}{\rho } \log \{1+\rho (T-T_{1})\}\right] \nonumber \\&\quad -C_{1} -C_{2}\alpha \left[ I_{0}\eta ^{\beta } T_{1}^{\beta }+\frac{aA^{\lambda }}{\beta +1} \left\{ (1-\eta ^{\beta })-\beta \eta ^{\beta }(1-\eta )\right\} T_{1}^{\beta +1}\right] \nonumber \\&\quad -C_{3} \left( 1-m(\xi )\right) x\left[ I_{0}\eta ^{y} T_{1}^{y}+\frac{aA^{\lambda }}{y+1}\left\{ (1-\eta ^{y}) -y\eta ^{y}(1-\eta )\right\} T_{1}^{y+1}\right] \nonumber \\&\quad -h\left[ I_{0}\left[ \eta T_{1} +\frac{\alpha }{\beta +1} \eta ^{\beta +1}T_{1}^{\beta +1}-\frac{\{1-m(\xi )\}x}{y+1} \eta ^{y+1}T_{1}^{y+1}\right] \right. \nonumber \\&\quad + aA^{\lambda }\left[ \frac{1}{2}(1-\eta )^{2}T_{1}^{2} +\frac{\{1-m(\xi )\}x}{(y+1)(y+2)} \left\{ y(1-\eta ^{y+2})\right. \right. \nonumber \\&\quad -\left. (y+2)\eta (1-\eta ^{y}) \right\} T_{1}^{y+2} +\frac{bA^{\lambda }}{6}(1-\eta )^{3}T_{1}^{3} +\frac{\alpha }{\beta +1} \left\{ \eta (1-\eta ^{\beta })\right. \nonumber \\&\quad -\left. \left. \left. \frac{\beta }{\beta +2}(1-\eta ^{\beta +2})\right\} T_{1}^{\beta +2}\right] \right] -\left( \frac{C_{4}aA^{\lambda }}{\rho ^{2}} +\frac{C_{5}aA^{\lambda }}{\rho } \right) \left[ \rho (T-T_{1})\right. \nonumber \\&\quad -\left. \log \{1+\rho (T-T_{1})\}\right] - C_{6}\left[ I_{0} +\frac{aA^{\lambda }}{\rho }\log \{1+\rho (T-T_{1})\}\right] \nonumber \\&\quad - \xi T - C_{7}A- C_{6}I_{c}aA^{\lambda } \left[ \frac{1}{2}(T_{1}-M)^{2}\right. \nonumber \\&\quad + \frac{\{1-m(\xi )\}x}{(y+1)(y+2)} \left\{ y(T_{1}^{y+2}-M^{y+2})- (y+2)T_{1} M(T_{1}^{y}-M^{y}) \right\} \nonumber \\&\quad +\frac{bA^{\lambda }}{6}(T_{1}-M)^{3} +\frac{\alpha }{\beta +1}\left\{ T_{1} M(T_{1}^{\beta }-M^{\beta })\right. \nonumber \\&\quad -\left. \left. \frac{\beta }{\beta +2} (T_{1}^{\beta +2}-M^{\beta +2})\right\} \right] + pI_{e}A^{\lambda }\left[ \frac{a}{2}(M^{2} -\eta ^{2}T_{1}^{2})\right. \nonumber \\&\quad +baA^{\lambda }\left[ \frac{1}{6}\left\{ 3T_{1}(M^{2}-\eta ^{2} T_{1}^{2})- 2(M^{3}-\eta ^{3}T_{1}^{3})\right\} \right. \nonumber \\&\quad +\frac{\{1-m(\xi )\}x}{y+1}\left\{ \frac{1}{2}T_{1}^{y+1} (M^{2}-\eta ^{2}T_{1}^{2})+\frac{y}{y+3}(M^{y+3}-\eta ^{y+3} T_{1}^{y+3})\right. \nonumber \\&\quad -\left. \left( \frac{y+1}{y+2}\right) T_{1}(M^{y+2} -\eta ^{y+2}T_{1}^{y+2})\right\} +\frac{bA^{\lambda }}{24} \left\{ 6T_{1}^{2}(M^{2}-\eta ^{2}T_{1}^{2})\right. \nonumber \\&\quad +\left. 3(M^{4}-\eta ^{4}T_{1}^{4})-8T_{1}(M^{3} -\eta ^{3}T_{1}^{3})\right\} \nonumber \\&\quad + \frac{\alpha }{\beta +1} \left\{ \left( \frac{\beta +1}{\beta +2}\right) T_{1} (M^{\beta +2}-\eta ^{\beta +2}T_{1}^{\beta +2})\right. \nonumber \\&\quad - \left. \left. \left. \frac{\beta }{\beta +3}(M^{\beta +3}-\eta ^{\beta +3} T_{1}^{\beta +3})-\frac{1}{2}T_{1}^{\beta +1} (M^{2} -\eta ^{2}T_{1}^{2})\right\} \right] \right] \end{aligned}$$

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