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.
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The authors wish to thank the anonymous reviewers for their suggestions leading to some improvement in the presentation of this paper.
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Appendices
Appendix 1 (Sufficiency Conditions)
For Case I (T ≥ MR), the second-order derivatives with respect to T and PR are given by differentiating (4) and (5), respectively, i.e.,
At PR* since \( \frac{\partial Net1}{{\partial P_{R} }} = 0 \), we have R1 = 0.
Since R1 = 0,
For Case II (T < MR), the second-order derivatives with respect to T and PR are given by differentiating (8) and (9), respectively, i.e.,
Since R2 = 0
Appendix 2 (Determinant of the Hessian Matrix)
For Case I, T ≥ MR, we have
On differentiating (5), we get
The determinant of this Hessian matrix for Case I is
Since R1 = 0
i.e.,
where
Since \( \frac{{\partial^{2} {\text{Net}}1}}{{\partial {\text{T}}^{2} }} < 0 \), the condition for joint concavity of Net1 with respect to T and PR is AA > BB.
For Case II, for T < MR, we have
On differentiating (9) with respect to T, we get
The determinant of the Hessian matrix for Case II is
Since at PR2*, R2 = 0,
i.e.,
where
Since \( \frac{{\partial^{{2}} {{\text{Net}}2}}}{{{\partial {{\text{T}}}^{{2}}}} } < 0 \), the condition for concavity of Net2 is CC > DD.
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Agrawal, S., Gupta, R., Banerjee, S. (2020). EOQ Model Under Discounted Partial Advance—Partial Trade Credit Policy with Price-Dependent Demand. In: Shah, N., Mittal, M. (eds) Optimization and Inventory Management. Asset Analytics. Springer, Singapore. https://doi.org/10.1007/978-981-13-9698-4_13
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