EOQ Model Under Discounted Partial Advance—Partial Trade Credit Policy with Price-Dependent Demand

Abstract

The aim of this article is to investigate an inventory model with discounted partial advance payment in a single supplier–single retailer supply chain in the presence of credit period when the demand rate is price sensitive. The lengths of the credit period, advance period, as well as rate of discount on advance payment, are specified by the supplier. Conditions for unique optimal values of the decision variables, namely, the retailer’s selling price and cycle length are obtained. Optimal values of the decision variables are determined iteratively. An algorithm is developed and a numerical example is presented to demonstrate the solution algorithm. Sensitivity analysis is conducted. It is observed that optimal cycle time is affected by the two interest rates. Optimal net profit is affected by the demand rate and the discount factor. Both, the optimal cycle time, as well as the optimal net profit is affected by the supplier’s selling price and the proportion of units for which the advance payment is made. Optimal retailer’s selling price is significantly affected by the discount factor, supplier’s selling price, price elasticity of the demand function as well as the proportion of units for which advance payment is made. We also observe that the retailer’s net profit does not decrease significantly on increasing the advance period.

Appendices

Appendix 1 (Sufficiency Conditions)

For Case I (T ≥ M_R), the second-order derivatives with respect to T and P_R are given by differentiating (4) and (5), respectively, i.e.,

$$ \frac{{\partial^{2} {\text{Net}}1\left( {{\text{T}},{\text{P}}_{\text{R}} } \right) }}{{\partial {\text{T}}^{2} }} = - \frac{{2{\text{A}}}}{{{\text{T}}^{3} }} < 0 $$

$$ \frac{{\partial^{2} {\text{Net}}1\left( {{\text{T}},{\text{P}}_{\text{R}} } \right)}}{{\partial {\text{P}}_{\text{R}}^{2} }} = - \frac{\upalpha}{2}{\text{P}}_{\text{R}}^{{ - \left( {\upbeta + 1} \right)}} \left[ {{\text{P}}_{\text{R}}^{ - 1} \left( {\upbeta + 1} \right)\left( {{\text{R}}1} \right) + \left( {\upbeta - 1} \right)\left( {2 + {\text{I}}_{\text{ER}} {\text{T}}} \right)} \right] $$

At P_R* since $\frac{\partial Net1}{{\partial P_{R} }} = 0$, we have R₁ = 0.

Since R1 = 0,

$$ \frac{{\partial^{2} {\text{Net}}1\left( {{\text{T}}, {\text{P}}_{\text{R}} } \right)}}{{\partial {\text{P}}_{\text{R}}^{2} }}\left\langle {0\,{\text{If}}\,\upbeta - 1} \right\rangle 0,\,{\text{i}}.{\text{e}}.,\,\upbeta > 1. $$

For Case II (T < M_R), the second-order derivatives with respect to T and P_R are given by differentiating (8) and (9), respectively, i.e.,

$$ \begin{aligned} \frac{{\partial^{2} {{\text{Net}}}2\left( {{{\text{T}}}, {{\text{P}}}_{{\text{R}}} } \right) }}{{\partial {{\text{T}}}^{{2}} }} & = - \frac{{2{{\text{A}}}}}{{{{\text{T}}}^{{3}} }} < 0 \\ \frac{{\partial^{2} {{\text{Net}}}2\left( {{{\text{T}}}, {{\text{P}}}_{{\text{R}}} } \right)}}{{\partial {{\text{P}}}_{{\text{R}}}^{{2}} }} & = - \frac{\upalpha}{2}{{\text{P}}}_{{\text{R}}}^{{ - \left( {\upbeta + 1} \right)}} [{{\text{P}}}_{{\text{R}}}^{{ - 1}} \left( {\upbeta + 1} \right)\left( {{{\text{R}}}2} \right) + \left( {\upbeta - 1} \right)\left( {2 + 2{{\text{I}}}_{{\text{ER}}} {{\text{M}}}_{{\text{R}}} - {{\text{I}}}_{{\text{ER}}} {{\text{T}}}} \right) \\ \end{aligned} $$

Since R2 = 0

$$ \frac{{\partial^{2} {\text{Net}}2\left( {{\text{T}}, {\text{P}}_{\text{R}} } \right)}}{{\partial {\text{P}}_{\text{R}}^{2} }}\left\langle {0,\,If\,\upbeta - 1} \right\rangle 0,\,{\text{i}}.{\text{e}}.,\,\upbeta > 1. $$

Appendix 2 (Determinant of the Hessian Matrix)

For Case I, T ≥ M_R, we have

$$ \frac{{\partial^{2} {\text{Net}}1\left( {{\text{T}}, {\text{P}}_{\text{R}} } \right) }}{{\partial {\text{T}}^{2} }} = - \frac{{2{\text{A}}}}{{{\text{T}}^{3} }} $$

$$ \frac{{\partial^{2} {\text{Net}}1\left( {{\text{T}},{\text{P}}_{\text{R}} } \right)}}{{\partial {\text{P}}_{\text{R}}^{2} }} = - \frac{\upalpha}{2}{\text{P}}_{\text{R}}^{{ - \left( {\upbeta + 2} \right)}} \left[ {{\text{P}}_{\text{R}}^{ - 1} \left( {\upbeta + 1} \right)\left( {{\text{R}}1} \right) + \left( {2 + {\text{I}}_{\text{ER}} {\text{T}}} \right)\left( {\upbeta - 1} \right)} \right] $$

On differentiating (5), we get

$$ \begin{aligned} \frac{{\partial^{2} {{\text{Net}}}1\left( {{{\text{T}}},{{\text{P}}}_{{\text{R}}} } \right) }}{{\partial {{\text{T}}}\partial {{\text{P}}}_{{\text{R}}} }} & = \frac{\upalpha}{2} {{\text{P}}}_{{\text{R}}}^{{ - \left( {\upbeta + 1} \right)}} \left\{ { - {{\text{P}}}_{{\text{R}}} {{\text{I}}}_{{\text{ER}}} \left( {\upbeta - 1} \right)} \right. \\ & \quad \left. { +\upbeta\left\{ {{{\text{h}}} + 2{{\text{P}}}_{{\text{S}}} \left[ {\left( {1 - {{\text{A}}}_{{1}} } \right){{\text{I}}}_{{1}} - {{\text{A}}}_{{2}} \left( {{{\text{I}}}_{{1}} - {{\text{I}}}_{{\text{PR}}} } \right) + {{\text{A}}}_{{1}}\uprho{{\text{I}}}_{{\text{PR}}} } \right]} \right\}} \right\} \\ \end{aligned} $$

The determinant of this Hessian matrix for Case I is

$$ \begin{aligned} {{\text{Hessian}}}1 & = - \frac{{2{{\text{A}}}}}{{{{\text{T}}}^{{3}} }}\left\{ { - \frac{\upalpha}{2}{{\text{P}}}_{{\text{R}}}^{{ - \left( {\upbeta + 2} \right)}} \left[ {{{\text{P}}}_{{\text{R}}}^{{ - 1}} \left( {{\upbeta + 1}} \right)\left( {{{\text{R}}}1} \right) + \left( {2 + {{\text{I}}}_{{\text{ER}}} {{\text{T}}}} \right)\left( {{\upbeta - 1}} \right)} \right]} \right\} \\ & \quad - \left[ {\frac{{\upalpha}}{{2}}\,{{\text{P}}}_{{\text{R}}}^{{ - \left( {\upbeta + 1} \right)}} \left\{ { - {{\text{P}}}_{{\text{R}}} {{\text{I}}}_{{\text{ER}}} \left( {\upbeta - 1} \right) +\upbeta\left[ {{{\text{h}}} + 2{{\text{P}}}_{{\text{S}}} \left( {\left( {1 - {{\text{A}}}_{{1}} } \right){\text{I}}_{1} - {{\text{A}}}_{{2}} \left( {{{\text{I}}}_{{1} }- {{\text{I}}}_{{\text{PR}}} } \right)} \right.} \right.} \right.} \right. \\ & \quad \left. {\left. {\left. {\left( { + {{\text{A}}}_{1}\uprho{{\text{I}}}_{{\text{PR}}} } \right)} \right]} \right\}} \right]^{2} \\ \end{aligned} $$

Since R1 = 0

$$ \begin{aligned} {{\text{Hessian}}}1 & = \frac{\text{A}}{{{{\text{T}}}^{3} }}\upalpha {{\text{P}}}_{{\text{R}}}^{{ - \left( {{\upbeta + 2}} \right)}} \left( {2 + {{\text{I}}}_{{\text{ER}}} {{\text{T}}}} \right)\left( {{\upbeta - 1}} \right) \\ & \quad - \left[ {\frac{{\upalpha}}{{2}}\,{{\text{P}}}_{{\text{R}}}^{{ - \left( {{\upbeta + 1}} \right)}} } \right.\left\{ { - {{\text{P}}}_{{\text{R}}} {{\text{I}}}_{{\text{ER}}} \left( {{\upbeta - 1}} \right) +\upbeta} \right.\left[ {{{\text{h}}} + 2{{\text{P}}}_{{\text{S}}} \left( {\left( {1 - {{\text{A}}}_{{1}} } \right){{\text{I}}}_{{1}} - {{\text{A}}}_{{2}} \left( {{{\text{I}}}_{{1}} - {{\text{I}}}_{{\text{PR}}} } \right)} \right.} \right. \\ & \quad \left. {\left. {\left. {\left. { + {{\text{A}}}_{{1}}\uprho{{\text{I}}}_{{\text{PR}}} } \right)} \right]} \right\}} \right]^{2} \\ \end{aligned} $$

i.e.,

$$ {\text{Hessian}}1 = {\text{AA}} - {\text{BB}} $$

where

$$ {\text{AA}} = \frac{\text{A}}{{{\text{T}}^{3} }}\upalpha\,{\text{P}}_{\text{R}}^{{ - \left( {\upbeta + 2} \right)}} \left( {2 + {\text{I}}_{\text{ER}} {\text{T}}} \right)\left( {\upbeta - 1} \right) > 0\,{\text{if}}\,\upbeta > 1. $$

$$ {{\text{BB}}} = \left[ {\frac{\upalpha}{2}\,{{\text{P}}}_{{\text{R}}}^{{ - \left( {\upbeta + 1} \right)}} \left\{ { - {{\text{P}}}_{{\text{R}}} {{\text{I}}}_{{\text{ER}}} \left( {\upbeta - 1} \right) +\upbeta\left[ {{{\text{h}}} + 2{{\text{P}}}_{{\text{S}}} \left( {\left( {1 - {{\text{A}}}_{1} } \right){{\text{I}}}_{1} - {{\text{A}}}_{2} \left( {{\text{I}}_{1} - {{\text{I}}}_{{\text{PR}}} } \right) + {{\text{A}}}_{1}\uprho{{\text{I}}}_{{\text{PR}}} } \right)} \right]} \right\}} \right]^{2} $$

Since $\frac{{\partial^{2} {\text{Net}}1}}{{\partial {\text{T}}^{2} }} < 0$, the condition for joint concavity of Net1 with respect to T and P_R is AA > BB.

For Case II, for T < M_R, we have

$$ \frac{{\partial^{2} {\text{Net}}2\left( {{\text{T}}, {\text{P}}_{\text{R}} } \right)}}{{\partial {\text{T}}^{2} }} = \frac{{ - 2{\text{A}}}}{{{\text{T}}^{3} }} $$

$$ \frac{{\partial^{2} {{\text{Net}}}2\left( {{\text{T}},{{\text{P}}}_{{\text{R}}} } \right)}}{{\partial {{\text{P}}}_{{\text{R}}}^{2} }} = - \frac{\alpha }{2}{{\text{P}}}_{{\text{R}}}^{{ - \left( {\beta + 1} \right)}} \left[ {{\text{P}}_{{\text{R}}}^{ - 1} \left( {\beta + 1} \right)\left( {{\text{R}}2} \right) + \left( {\beta - 1} \right)\left( {2 + 2{{\text{I}}}_{{\text{ER}}} {{\text{M}}}_{{\text{R}}} - {{\text{I}}}_{{\text{ER}}} {\text{T}}} \right)} \right] $$

On differentiating (9) with respect to T, we get

$$ \frac{{\partial^{2} {{\text{Net}}}2\left( {{\text{T}},{{\text{P}}}_{{\text{R}}} } \right)}}{{\partial {{\text{T}}}\partial {{\text{P}}}_{{\text{R}}}}} = \frac{\alpha }{2}{{\text{P}}}_{{\text{R}}}^{{ - \left( {\beta + 1} \right)}} \left[ {{\text{P}}_{{\text{R}}} {{\text{I}}}_{{\text{ER}}} \left( {\beta - 1} \right) +\upbeta\left( {{\text{h}} + 2{{\text{P}}}_{{\text{S}}} {{\text{A}}}_{1}\uprho{{\text{I}}}_{{\text{PR}}}} \right)} \right] $$

The determinant of the Hessian matrix for Case II is

$$ \begin{aligned} {{\text{Hessian}}}2 & = \frac{{ - 2{{\text{A}}}}}{{{\text{T}}^{3} }}\left\{ { - \frac{\alpha }{2}{{\text{P}}}_{{\text{R}}}^{{ - \left( {\beta + 1} \right)}} \left[ {{\text{P}}_{{\text{R}}}^{ - 1} \left( {\beta + 1} \right)\left( {{\text{R}}2} \right) + \left( {\beta - 1} \right)\left( {2 + 2{{\text{I}}}_{{\text{ER}}} {{\text{M}}}_{{\text{R}}} - {{\text{I}}}_{{\text{ER}}} {{\text{T}}}} \right)} \right]} \right\} \\ & \quad - \left\{ {\frac{\alpha }{2}{{\text{P}}}_{{\text{R}}}^{{ - \left( {\beta + 1} \right)}} \left[ {{{\text{P}}}_{{\text{R}}} {{\text{I}}}_{{\text{ER}}} \left( {\beta - 1} \right) +\upbeta\left( {{\text{h}} + 2{{\text{P}}}_{{\text{S}}} {{\text{A}}}_{1}\uprho{{\text{I}}}_{{\text{PR}}} } \right)} \right]} \right\}^{2} \\ \end{aligned} $$

Since at P_R2*, R2 = 0,

$$ \begin{aligned} {{\text{Hessian}}}\,2 & = \frac{{\text{A}}}{{{\text{T}}^{3} }}\alpha {{\text{P}}}_{{\text{R}}}^{{ - \left( {\beta + 1} \right)}} \left( {\beta - 1} \right)\left( {2 + 2{{\text{I}}}_{{\text{ER}}} {{\text{M}}}_{{\text{R}}} - {{\text{I}}}_{{\text{ER}}} {{\text{T}}}} \right) \\ & \quad - \left\{ {\frac{\alpha }{2}{{\text{P}}}_{{\text{R}}}^{{ - \left( {\beta + 1} \right)}} \left[ {{\text{P}}_{{\text{R}}} {{\text{I}}}_{{\text{ER}}} \left( {\beta - 1} \right) +\upbeta\left( {{\text{h}} + 2{\text{P}}_{{\text{S}}} {{\text{A}}}_{1}\uprho{{\text{I}}}_{{\text{PR}}} } \right)} \right]} \right\}^{2} \\ \end{aligned} $$

i.e.,

$$ {\text{Hessian}}2 = {\text{CC}} - {\text{DD}} $$

where

$$ {{\text{CC}}} = \frac{{\text{A}}}{{{\text{T}}^{3} }}\alpha {{\text{P}}}_{{\text{R}}}^{{ - \left( {\beta + 1} \right)}} \left( {\beta - 1} \right)\left( {2 + 2{{\text{I}}}_{{\text{ER}}} {{\text{M}}}_{{\text{R}}} - {{\text{I}}}_{{\text{ER}}} {{\text{T}}}} \right)$$

$$ {{\text{DD}}} = \left\{ {\frac{\alpha }{2}{{\text{P}}}_{{\text{R}}}^{{ - \left( {\beta + 1} \right)}} \left[ {{{\text{P}}}_{{\text{R}}} {{\text{I}}}_{{\text{ER}}} \left( {\beta - 1} \right) +\upbeta\left( {{\text{h}} + 2{{\text{P}}}_{{\text{S}}} {{\text{A}}}_{1}\uprho{{\text{I}}}_{{\text{PR}}}} \right)} \right]} \right\}^{2} $$

Since $\frac{{\partial^{{2}} {{\text{Net}}2}}}{{{\partial {{\text{T}}}^{{2}}}} } < 0$, the condition for concavity of Net2 is CC > DD.