Optimization of an Inventory Model with Demand Dependent on Selling Price and Stock, Nonlinear Holding Cost Along with Trade Credit Policy

Abstract

The demand for a product is influenced by a number of factors, including the selling price and the displayed stock level, among others. Considering this scenario, an EOQ inventory model is developed where demand is a function of both selling price and the inventory level which is one of the main contributions of this research work. Holding cost is assumed to be nonlinearly dependent on stock. Besides that, supplier grants a full trade credit policy to the retailer. This policy is very advantageous for both the counterpart—the supplier as well as the retailer. The supplier can attract more customers by offering a delay period whereas the latter enjoys the benefit of getting goods without instant payment. The proposed mathematical model aims to find out the optimal selling price and optimal length of the replenishment cycle so as to maximize the total profit of the retailer per unit time. Several theorems are well-established in order to reach to the optimal solution. A numerical example is also presented to demonstrate the suggested inventory model, and a sensitivity analysis is executed to highlight the findings of the inventory model and put forward valuable managerial insights. This research work can be helpful to the business communities facing nonlinear demand patterns. Businesses that want to offer trade credit policies but are dealing with nonlinear holding costs may also find it helpful. Sensitivity analysis can be useful in determining the impact of various cost parameters on the total generated profit.

Appendices

Appendix 1

$$\begin{aligned} f\left( T \right) & = \left[ {p\left[ {\alpha \left( {a - bp} \right)\left( {1 - \beta } \right)T} \right]^{{\frac{1}{1 - \beta }}} } \right. \\ & \quad + pI_{{\text{e}}} \left[ {M\left[ {\alpha \left( {a - bp} \right)\left( {1 - \beta } \right)T} \right]^{{\frac{1}{1 - \beta }}} } \right. \\ & \quad + \frac{1}{{\alpha \left( {a - bp} \right)\left( {2 - \beta } \right)}}\left\{ {\left[ {\left( {a - bp} \right)\left( {1 - \beta } \right)\left( {T - M} \right)} \right]^{{\frac{2 - \beta }{{1 - \beta }}}} } \right. \\ & \quad \left. {\left. { - \left[ {\alpha \left( {a - bp} \right)\left( {1 - \beta } \right)T} \right]^{{\frac{2 - \beta }{{1 - \beta }}}} } \right\}} \right] \\ & \quad - o - c\left[ {\alpha \left( {a - bp} \right)\left( {1 - \beta } \right)T} \right]^{{\frac{1}{1 - \beta }}} \\ & \quad - h\frac{{\left[ {\alpha \left( {a - bp} \right)\left( {1 - \beta } \right)T} \right]^{{\frac{\gamma + 1 - \beta }{{1 - \beta }}}} }}{{\alpha \left( {a - bp} \right)\left( {\gamma + 1 - \beta } \right)}} \\ & \quad \left. { - \frac{{cI_{{\text{p}}} \left[ {\left[ {\alpha \left( {a - bp} \right)\left( {1 - \beta } \right)\left( {T - M} \right)} \right]^{{\frac{2 - \beta }{{1 - \beta }}}} } \right]}}{{\alpha \left( {a - bp} \right)\left( {2 - \beta } \right)}}} \right] \\ \end{aligned}$$

$$g\left( T \right) = T$$

If $f^{\prime\prime}\left( T \right) < 0$, then by using the theoretical result in (10.14) it can be proved that ${\text{TP}}_{1} \left( {p,T} \right)$ is a strictly pseudo-concave function in $T$, which completes the proof of Part (1). The proof of Part (2) and Part (3) follows immediately from the proof of Part (1) of Theorem 1.

Appendix 2

Similar to Appendix 1.

Appendix 3

Since ${\text{TP}}_{1} \left( {p,T} \right)$ is a strictly pseudo-concave function in $T$, we know that $\frac{{\partial {\text{TP}}_{1} \left( {p,T} \right)}}{\partial T}$ is a decreasing function in $T$. If $\Delta > 0$, then $\mathop {\lim }\limits_{T \to \infty } \frac{{\partial {\text{TP}}_{1} \left( {p,T} \right)}}{\partial T} < 0$.

By applying mean value theorem, we know that there exists a unique $T_{1}^{*} \in \left( {M,\infty } \right)$ such that $\frac{{\partial {\text{TP}}_{1} \left( {p,T} \right)}}{\partial T} = 0$. By this, we complete the proof of $\Delta > 0$. Similarly, other theorems of Theorem 5 can be proved.