Optimal replenishment and credit policy in EOQ model under two-levels of trade credit policy when demand is influenced by credit period

Abstract

It has been observed that credit period has become a major concern for most of the retailers, because not only it has direct influence on inventory and finance but also on the demand of an item. Unfortunately, the impact of credit period on demand has received very little attention in the literature, whereas in reality length of the credit period offered has positive impact on the demand rate. The impact of credit period on demand may be instant or delayed. The aim of this paper is to determine the retailer’s optimal replenishment and credit policy in EOQ model under two-levels of trade credit policy when demand is influenced by credit period. These types of demand functions are observed in many consumer durables. Results have been illustrated with the help of a numerical example. Computational results provide some interesting policy implications.

Appendix: Concavity of π(T, N) with respect to N

Here, for a fixed value of T, we have also shown the concavity of profit functions π ₁(T, N), π ₂(T, N) and π ₃(T, N) with respect to N,

$$ \begin{aligned} \Updelta^{2} \pi_{1} (T,N) & = & (\lambda_{m} - \lambda_{0} )(1 - r)^{N} (1 - (N + 2)r)r^{2} \left\{ {P - C - \frac{ICT}{2} + \frac{{I_{e} P(M - N)^{2} - I_{p} C\left( {T + N - M} \right)^{2} }}{2T}} \right\} \\ \, & + (\lambda_{m} - \lambda_{0} )(1 - r)^{N + 1} (N + 2)r^{2} \left\{ {\frac{{I_{e} P\left( {1 - 2(M - N)} \right) - I_{p} C\left( {1 - 2\left( {T + N - M} \right)} \right)}}{2T}} \right\} \\ \, & + \left\{ {\lambda_{m} - (\lambda_{m} - \lambda_{0} )(1 - r)^{N + 2} (1 + rN + 2r)} \right\}\left( {\frac{{I_{e} P + I_{p} C}}{T}} \right) \\ \end{aligned} $$

(28)

$$ \begin{aligned} \Updelta^{2} \pi_{2} (T,N) & = & (\lambda_{m} - \lambda_{0} )(1 - r)^{N} (1 - r(N + 2))r^{2} \left\{ {P - C - \frac{ICT}{2} + I_{e} P(M - N - \frac{T}{2}} \right\} \\ \, & - (\lambda_{m} - \lambda_{0} )(1 - r)^{N + 1} (N + 2)r^{2} I_{e} P \\ \end{aligned} $$

(29)

and

$$ \begin{aligned} \Updelta^{2} \pi_{3} (T,N) & = & (\lambda_{m} - \lambda_{0} )(1 - r)^{N} (1 - r(N + 2))r^{2} \left\{ {P - C - \frac{ICT}{2} - I_{p} C(N - M + \frac{T}{2}} \right\} \\ \, & - 2(\lambda_{m} - \lambda_{0} )(1 - r)^{N + 1} (N + 2)r^{2} I_{p} C \\ \end{aligned} $$

(30)

For fixed T, Eqs. (28) to (30) are ≤0 which implies that π ₁(T, N), π ₂(T, N) and π ₃(T, N) are concave on N provided (i) N + 2 ≥ 1/r and (ii) CI _p  > PI _e . Moreover, it has also been verified graphically (Fig. 5).

Fig. 5

Concavity of π(T, N) with respect to N