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Joint Dynamic Pricing and Inventory Control for Perishable Products Taking into Account Partial Backlogging and Inflation

Abstract

A joint dynamic pricing and inventory model by considering the inflation and time value of money where the product has a limited lifetime is developed. The selling price depends on product lifetime and customer demand rate depends on the selling price and time. Also, the shortages are considered and partially backlogged. The main goal is to determine the optimal initial selling price, the economic order quantity, and the optimal replenishment period simultaneously in order to maximize the total net present value of profit. We represent that for given any number of replenishment the function of profit is concave. In order to find the optimal solution, the presented heuristic algorithm is applied. To represent the algorithm and procedure of solution, a numerical example is solved. The results show that applying the proposed dynamic pricing approach leads to a significant improvement in profit’s retailer. Finally, sensitivity analysis of key parameters model is performed and some managerial insights are presented.

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Correspondence to Mohammad Yavari.

Appendices

Appendix A

To prove that the Eq. (19) is strictly concave, we first obtain the following expressions:

$$ \begin{aligned} \frac{{\partial^{2} TP}}{{\partial T_{1}^{2} }} & = \frac{{e^{ - NRT} - 1}}{{m^{2} \times \left( {e^{ - RT} - 1} \right)}} \times \left[ {2bmp} \right._{0}^{2} \times e^{{ - T_{1} \left( {R - \lambda } \right)}} - amp_{0}^{2} \times e^{{ - T_{1} \left( {R - \lambda } \right)}} - 2T_{1} bp_{0}^{2} \\ & \quad \times e^{{ - T_{1} \left( {R - \lambda } \right)}} - 2bmp_{0}^{2} \times e^{{T_{1} \delta - T\delta - RT + T_{1} \lambda }} + amp_{0} \times e^{{T_{1} \delta - T\delta - RT + T_{1} \lambda }} + 2T_{1} bp_{0}^{2} \\ & \quad \times e^{{T_{1} \delta - T\delta - RT + T_{1} \lambda }} + RT_{1}^{2} bp_{0}^{2} \times e^{{ - T_{1} \left( {R - \lambda } \right)}} + RBm^{2} p_{0}^{2} \times e^{{ - T_{1} \left( {R - \lambda } \right)}} - T_{1}^{2} b\lambda p_{0}^{2} \\ & \quad \times e^{{ - T_{1} \left( {R - \lambda } \right)}} - b\lambda m^{2} p_{0}^{2} \times e^{{ - T_{1} \left( {R - \lambda } \right)}} - a\delta m^{2} p_{0} \times e^{{T_{1} \delta - T\delta - RT + T_{1} \lambda }} \\ & \quad - a\lambda m^{2} p_{0} \times e^{{T_{1} \delta - T\delta - RT + T_{1} \lambda }} + T_{1}^{2} b\delta p_{0}^{2} \times e^{{T_{1} \delta - T\delta - RT + T_{1} \lambda }} + T_{1}^{2} b\lambda p_{0}^{2} \\ & \quad \times e^{{T_{1} \delta - T\delta - RT + T_{1} \lambda }} + b\delta m^{2} p_{0}^{2} \times e^{{T_{1} \delta - T\delta - RT + T_{1} \lambda }} + b\lambda m^{2} p_{0}^{2} \times e^{{T_{1} \delta - T\delta - RT + T_{1} \lambda }} \\ & \quad - Ram^{2} p_{0} \times e^{{ - T_{1} \left( {R - \lambda } \right)}} + a\lambda m^{2} p_{0} \times e^{{ - T_{1} \left( {R - \lambda } \right)}} - 2RT_{1} bmp_{0}^{2} \times e^{{ - T_{1} \left( {R - \lambda } \right)}} \\ & \quad + 2T_{1} b\lambda mp_{0}^{2} \times e^{{ - T_{1} \left( {R - \lambda } \right)}} + T_{1} a\delta mp_{0} \times e^{{T_{1} \delta - T\delta - RT + T_{1} \lambda }} + T_{1} a\lambda mp_{0} \times e^{{T_{1} \delta - T\delta - RT + T_{1} \lambda }} \\ & \quad - 2T_{1} b\delta mp_{0}^{2} \times e^{{T_{1} \delta - T\delta - RT + T_{1} \lambda }} - 2T_{1} b\lambda mp_{0}^{2} \times e^{{T_{1} \delta - T\delta - RT + T_{1} \lambda }} \\ & \quad \left. { + \,RT_{1} amp_{0} \times e^{{ - T_{1} \left( {R - \lambda } \right)}} - T_{1} a\lambda mp_{0} \times e^{{ - T_{1} \left( {R - \lambda } \right)}} } \right] - \frac{{e^{ - NRT} - 1}}{{e^{ - RT} - 1}} \\ & \quad \times \left[ {\frac{{ce^{{T_{1} \lambda }} \times \left( {bp_{0} + a\lambda m - b\lambda mp_{0} + T_{1} b\lambda p_{0} } \right)}}{m}} \right. \\ & \quad \left. { - \frac{{c \times e^{{T_{1} \delta - T\delta - RT + T_{1} \lambda }} \times \left( {bp_{0} + a\delta m - a\lambda m - b\delta mp_{0} - b\lambda mp_{0} + T_{1} b\delta p_{0} + T_{1} b\lambda p_{0} } \right)}}{m}} \right] \\ & \quad - \frac{1}{{Rme^{{RT_{1} }} \times e^{T\delta } \times e^{NRT} - Rme^{{RT_{{}} }} \times e^{{RT_{1} }} \times e^{T\delta } \times e^{NRT} }} \times \left[ {bc_{2} p_{0} \times e^{{T_{1} \left( {R + \delta + \lambda } \right)}} - } \right.bc_{2} p_{0} \\ & \quad \times e^{{RT + T_{1} \delta + T_{1} \lambda }} + bc_{2} p_{0} \times e^{{RT + T_{1} \delta + T_{1} \lambda + NRT}} - bc_{2} p_{0} \times e^{{RT_{1} + T_{1} \delta + T_{1} \lambda + NRT}} - Rac_{2} \times e^{{RT + T_{1} \delta + T_{1} \lambda + NRT}} \\ & \quad + ac_{2} \delta m \times e^{{RT + T_{1} \delta + T_{1} \lambda + NRT}} - ac_{2} \delta m \times e^{{RT_{1} + T_{1} \delta + T_{1} \lambda + NRT}} + ac_{2} \lambda m \times e^{{RT + T_{1} \delta + T_{1} \lambda + NRT}} \\ & \quad - ac_{2} \lambda m \times e^{{RT_{1} + T_{1} \delta + T_{1} \lambda + NRT}} + ac_{2} \delta m \times e^{{T_{1} \left( {R + \delta + \lambda } \right)}} + ac_{2} \lambda m \times e^{{T_{1} \left( {R + \delta + \lambda } \right)}} + Rac_{2} m \\ & \quad \times e^{{RT + T_{1} \delta + T_{1} \lambda }} - ac_{2} \delta m \times e^{{RT + T_{1} \delta + T_{1} \lambda }} - ac_{2} \lambda m \times e^{{RT + T_{1} \delta + T_{1} \lambda }} - bc_{2} \delta mp_{0} \times e^{{T_{1} \left( {R + \delta + \lambda } \right)}} \\ & \quad - bc_{2} \lambda mp_{0} \times e^{{T_{1} \left( {R + \delta + \lambda } \right)}} + RT_{1} bc_{2} p_{0} \times e^{{RT + T_{1} \delta + T_{1} \lambda }} - T_{1} bc_{2} \delta p_{0} \times e^{{RT + T_{1} \delta + T_{1} \lambda }} - Rbc_{2} mp_{0} \\ & \quad \times e^{{RT + T_{1} \delta + T_{1} \lambda }} - T_{1} bc_{2} \lambda p_{0} \times e^{{RT + T_{1} \delta + T_{1} \lambda }} + bc_{2} \delta mp_{0} \times e^{{RT + T_{1} \delta + T_{1} \lambda }} + bc_{2} \lambda mp_{0} \times e^{{RT + T_{1} \delta + T_{1} \lambda }} \\ & \quad - RT_{1} bc_{2} p_{0} \times e^{{RT + T_{1} \delta + T_{1} \lambda + NRT}} + T_{1} bc_{2} \delta p_{0} \times e^{{RT + T_{1} \delta + T_{1} \lambda + NRT}} - T_{1} bc_{2} \delta p_{0} \times e^{{RT_{1} + T_{1} \delta + T_{1} \lambda + NRT}} \\ & \quad + Rbc_{2} mp_{0} \times e^{{RT + T_{1} \delta + T_{1} \lambda + NRT}} + T_{1} bc_{2} \lambda p_{0} \times e^{{RT + T_{1} \delta + T_{1} \lambda + NRT}} - T_{1} bc_{2} \lambda p_{0} \times e^{{RT_{1} + T_{1} \delta + T_{1} \lambda + NRT}} \\ & \quad - bc_{2} \delta mp_{0} \times e^{{RT + T_{1} \delta + T_{1} \lambda + NRT}} + bc_{2} \delta mp_{0} \times e^{{RT_{1} + T_{1} \delta + T_{1} \lambda + NRT}} - bc_{2} \lambda mp_{0} \times e^{{RT + T_{1} \delta + T_{1} \lambda + NRT}} \\ & \quad + bc_{2} \lambda mp_{0} \times e^{{RT_{1} + T_{1} \delta + T_{1} \lambda + NRT}} \left. { + T_{1} bc_{2} \delta p_{0} \times e^{{T_{1} \left( {R + \delta + \lambda } \right)}} + T_{1} bc_{2} \lambda p_{0} \times e^{{T_{1} \left( {R + \delta + \lambda } \right)}} } \right] \\ & \quad - \frac{{e^{ - NRT} - 1}}{{e^{ - RT} - 1}} \times o \times \,\left[ {a \times e^{{T_{1} \delta - T\delta - RT_{1} + T_{1} \lambda }} \times \left( {\delta - R + \lambda } \right) + a \times e^{{ - T_{1} \left( {R - \lambda } \right)}} \times \left( {R - \lambda } \right)} \right. \\ & \quad - bp_{0} \times e^{{T_{1} \delta - T\delta - RT_{1} + T_{1} \lambda }} \times \left( {\delta - R + \lambda } \right) - bp_{0} \times \,e^{{ - T_{1} \left( {R - \lambda } \right)}} \\ & \quad \times \left( {R - \lambda } \right) - \frac{{bp_{0} \times e^{{ - T_{1} \left( {R - \lambda } \right)}} \times \left( {T_{1} \lambda - RT_{1} + 1} \right)}}{m} \\ & \quad + \left. {\frac{{bp_{0} \times e^{{T_{1} \delta - T\delta - RT_{1} + T_{1} \lambda }} \times \left( {T_{1} \delta - RT_{1} + T_{1} \lambda + 1} \right)}}{m}} \right] - \frac{{e^{{ - T_{1} \left( {R - \lambda } \right)}} \times \left( {e^{ - NRT} - 1} \right)}}{{Rm \times \left( {e^{ - RT} - 1} \right)}} \\ & \quad \times c_{1} \times \left( {Ram - bp_{0} - a\lambda m + bp_{0} e^{{RT_{1} }} + a\lambda me^{{RT_{1} }} + b\lambda mp_{0} } \right. \\ & \quad \left. { + RT_{1} bp_{0} - Rbmp_{0} - T_{1} b\lambda p_{0} + T_{1} b\lambda p_{0} e^{{RT_{1} }} - b\lambda mp_{0} e^{{RT_{1} }} } \right) \\ \end{aligned} $$
figure a
figure b

Therefore, the determinant Hessian matrix H at the point (T1*, p0*) that calculated by Eqs. (20) and (21) is obtained as follows:

$$ \det \left( H \right) = \left( {\left. {\frac{{\partial^{2} TP}}{{\partial T_{1}^{2} }}} \right|_{{\left( {T_{1}^{*} ,p_{0}^{*} } \right)}} } \right) \times \left( {\left. {\frac{{\partial^{2} TP}}{{\partial p_{0}^{2} }}} \right|_{{\left( {T_{1}^{*} ,p_{0}^{*} } \right)}} } \right) - \left( {\left. {\frac{{\partial^{2} TP}}{{\partial T_{1}^{{}} \partial p_{0} }}} \right|_{{\left( {T_{1}^{*} ,p_{0}^{*} } \right)}} } \right)^{2} $$

By assuming λ < 0, |λ| > δ, m > T, bp0 < a < 2bp0 the following results are obtained:

$$ \begin{aligned} \left. {\frac{{\partial^{2} TP}}{{\partial T_{1}^{2} }}} \right|_{{\left( {T_{1}^{*} ,p_{0}^{*} } \right)}} & < 0,\left. {_{{}} \frac{{\partial^{2} TP}}{{\partial p_{0}^{2} }}} \right|_{{\left( {T_{1}^{*} ,p_{0}^{*} } \right)}} < 0,\left. {_{{}} \frac{{\partial^{2} TP}}{{\partial T_{1}^{2} }}} \right|_{{\left( {T_{1}^{*} ,p_{0}^{*} } \right)}} \\ & > \left. {\frac{{\partial^{2} TP}}{{\partial T_{1}^{{}} \partial p_{0} }}} \right|_{{\left( {T_{1}^{*} ,p_{0}^{*} } \right)}} ,\left. {_{{}} \frac{{\partial^{2} TP}}{{\partial p_{0}^{2} }}} \right|_{{\left( {T_{1}^{*} ,p_{0}^{*} } \right)}} > \left. {\frac{{\partial^{2} TP}}{{\partial T_{1}^{{}} \partial p_{0} }}} \right|_{{\left( {T_{1}^{*} ,p_{0}^{*} } \right)}} \\ \end{aligned} $$
$$ \det \left( H \right) = \left( {\left. {\frac{{\partial^{2} TP}}{{\partial T_{1}^{2} }}} \right|_{{\left( {T_{1}^{*} ,p_{0}^{*} } \right)}} } \right) \times \left( {\left. {\frac{{\partial^{2} TP}}{{\partial p_{0}^{2} }}} \right|_{{\left( {T_{1}^{*} ,p_{0}^{*} } \right)}} } \right) - \left( {\left. {\frac{{\partial^{2} TP}}{{\partial T_{1}^{{}} \partial p_{0} }}} \right|_{{\left( {T_{1}^{*} ,p_{0}^{*} } \right)}} } \right)^{2} > 0 $$

Therefore, the Hessian matrix H at the point (T1*, p0*) is negative definite. Consequently, the results show that the point (T1*, p0*) is a global maximum solution for the presented mathematical model.

Appendix B

For any given N, the first-order of TP(N, T1, p0) with respect to T1 and p0 is obtained as follows:

figure c
figure d
figure e

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Yavari, M., Zaker, H. & Emamzadeh, E.S.M. Joint Dynamic Pricing and Inventory Control for Perishable Products Taking into Account Partial Backlogging and Inflation. Int. J. Appl. Comput. Math 5, 1 (2019). https://doi.org/10.1007/s40819-018-0585-8

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Keywords

  • Perishable product
  • Dynamic pricing
  • Inflation
  • Partial backlogging