The optimal ordering policy of the EOQ model under trade credit depending on the ordering quantity from the DCF approach

doi:10.1016/j.ejor.2008.04.018

European Journal of Operational Research

Volume 196, Issue 2, 16 July 2009, Pages 563-568

https://doi.org/10.1016/j.ejor.2008.04.018 Get rights and content

Abstract

This paper discusses the optimum order quantity of the EOQ model that is not only dependent on the inventory policy but also on firm’ credit policy. Here, the conditions of using a discounted cash-flows (DCF) approach and trade credit depending on the quantity ordered are discussed. We consider that if the order quantity is less than at which the delay in payments is permitted, the payment for the item must be made immediately. Otherwise, the fixed trade credit period is permitted.

This paper incorporates all concepts of a discounted cash-flows (DCF) approach, trade credit and the quantity ordered and develops a new inventory model to generalize Chung [Chung, K.H., 1989. Inventory control and trade credit revisited, Journal of the Operational Research Society 40, 495–498].

Introduction

In recent years, the literature has contained a great number of decision models for both credit decisions and inventory management, as indicated by Goyal, 1985, Chu et al., 1998, Jamal et al., 1997. However, almost all of these models suffer from the following defects: They ignore the timing and magnitude of cash-flows. In fact, it is well settled in corporate finance that the value of a firm is the present value of its cash-flow stream. The appeal of the discounted cash-flow approach to inventory modeling is that it frames the purchasing and stockholding decisions precisely.

Specifically, Chung (1989) presents the economic ordering policies in the presence of trade credit using a discounted cash-flows (DCF) approach. He divides the study into four cases:

(a)
Instantaneous cash-flows (the case of the basic EOQ model).
(b)
Credit only on units in stock when T ⩽ M (where T denotes the inventory cycle time and M denotes the credit period).
(c)
Credit only on units in stock when T ⩾ M.
(d)
Fixed credit.

In this regard, this paper is only concerned with cases (a)–(c).

On the other hand, the credit policy may seem as an alternative to price discounts because such policies are not thought to provoke competitors to reduce their prices and thus introduce lasting price reductions, or because such policies are traditional in the firm’s industry. Furthermore, Khouja and Mehrez (1996) investigate the effect of supplier credit policies on the optimal order quantity within the economic order quantity framework. The supplier credit policies addressed in Khouja and Mehrez (1996) fall into two categories. One is that supplier credit policies where terms are independent of the order quantity and the other is that supplier credit policies where credit terms are linked to the order quantity. In the latter case, suppliers use favorable credit terms to encourage customers to order large quantities. In other words, the favorable credit terms apply only at large order quantities and are used in place of quantity discounts. In this regard, this paper is only concerned with the latter case.

Combining Chung, 1989, Khouja and Mehrez, 1996, we can unify cases (a)–(c) in Chung (1989). Consequently, this paper generalizes Chung (1989).

Section snippets

The model

This section analyzes the trade credit problem using DCF approach to fully recognize the time value of money in determining the optimal policy where trade credit depends on the ordering quantity. The following notations and assumptions will be used throughout:

A theorem

(A) Suppose that $M > \frac{W}{D}$ .

Eqs. (2), (3), (4) are coincide with those of Eqs. (6), (12) and (16) in Chung (1989). Moreover, Chung and Huang (2000) has proved that PV₁(T), PV₂(T) and PV₃(T) are convex on (0, ∞).

Let $T_{i}^{*}$ be such that $\frac{d {PV}_{i} (T_{i}^{*})}{d T} = 0 (i = 1, 2, 3)$ . By convexity of PV_i(T) (i = 1, 2, 3), we have $\frac{d {PV}_{i} (T)}{d T} \{\begin{matrix} < 0 & if 0 < T < T_{i}^{*}, & (a) \\ = 0 & if T = T_{i}^{*}, & (b) \\ > 0 & if T_{i}^{*} < T . & (c) \end{matrix}$ Eq. (7) imply that PV_i(T) is decreasing on $(0, T_{i}^{*}]$ and increasing on $[T_{i}^{*}, \infty) (i = 1, 2, 3)$

Eqs. (2), (3), (4) yield ${PV}_{1}^{'} (\frac{W}{D}) = \frac{e^{- r (\frac{W}{D})}}{r {(1 - e^{- r (\frac{W}{D})})}^{2}} \{(r + h) CD [e^{r (\frac{W}{D})} - 1 - r (\frac{W}{D})] - r^{2} K\},$ ${PV}_{2}^{'} (\frac{W}{D}) = e$

The algorithm

In this section, we shall combine Section 3 to outline the algorithm to help us to decide the optimal cycle time and optimal order quantity.
The algorithm

Step 1: If $M ⩽ \frac{W}{D}$ , then go to Step 3. Otherwise, go to Step2.
Step 2:
- (1)
  If Δ₁ > 0, Δ₂ ⩾ 0 and Δ₃ > 0, then T^∗ is $T_{1}^{*}$ or $\frac{W}{D}$ associated with the least cost.
- (2)
  If Δ₁ > 0, Δ₂ < 0 and Δ₃ > 0, then T^∗ is $T_{2}^{*}$ .
- (3)
  If Δ₁ > 0, Δ₂ < 0 and Δ₃ ⩽ 0, then T^∗ is $T_{3}^{*}$ .
- (4)
  If Δ₁ ⩽ 0, Δ₂ < 0 and Δ₃ > 0, then T^∗ is $T_{2}^{*}$ .
- (5)
  If Δ₁ ⩽ 0, Δ₂ < 0 and Δ₃ ⩽ 0, then T^∗ is $T_{3}^{*}$ .
Step 3:
- (1)
  If Δ₁ > 0 and Δ₄ ⩾ 0, then T^∗ is $T_{1}^{*}$ or $\frac{W}{D}$

Numerical examples

To illustrate the results. Let us apply the proposed method to solve the following numerical examples. The following parameters K = $5/order, C = $1, D = 15units and h = 0.1/unit are used from Example 1, Example 2, Example 3, Example 4, Example 5, Example 6, Example 7, Example 8. In addition, r = $0.3/$ is used from Example 1, Example 2, Example 3, Example 4, Example 5.

Example 1

If M = 30 and W = 100, then $\frac{W}{D} = 6.6667 < M$ , Δ₁ = 25.8844 > 0, Δ₂ = 6.1336 > 0 and Δ₃ = 12139 > 0. Using Step 2 (1), we get $T^{*} = \frac{W}{D} = 6.6667$ and PV(T^∗) = 77.6665.

Example 2

Special cases

When $M > \frac{W}{D}$ , PV_∞(T) can be expressed as follows: ${PV}_{\infty} (T) = \{\begin{matrix} {PV}_{1} (T) & if 0 < T < \frac{W}{D}, \\ {PV}_{2} (T) & if \frac{W}{D} ⩽ T < M, \\ {PV}_{3} (T) & if M ⩽ T . \end{matrix}$ On the other hand, when $M ⩽ \frac{W}{D}$ , PV_∞(T) can be expressed as follows: ${PV}_{\infty} (T) = \{\begin{matrix} {PV}_{1} (T) & if 0 < T < \frac{W}{D}, \\ {PV}_{3} (T) & if \frac{W}{D} ⩽ T . \end{matrix}$ There are the following cases to occur:

(a)
When M = 0, then we have ${PV}_{\infty} (T) = {PV}_{1} (T) for all T > 0 .$ Then Eq. (21) is consistent with Eq. (6) in Chung (1989).
(b)
When W = ∞, then we have ${PV}_{\infty} (T) = {PV}_{1} (T) for all T > 0 .$ Then Eq. (22) is consistent with Eq. (6) in Chung (1989) as well.
(c)
When W = 0, then we have ${PV}_{\infty} (T) = \{\begin{matrix} {PV}_{2} (T) & if 0 < T < M, & (a) \\ {PV}_{3} (T) \end{matrix}$

Summary

This paper presents the discounted cash-flows (DCF) approach for the analysis of the optimal inventory policy in the presence of trade credit depending on the ordering quantity. If Q < W, the delay in payment is not permitted. Otherwise, the fixed trade period M is permitted. There are two cases (1) $M > \frac{W}{D}$ and (2) $M ⩽ \frac{W}{D}$ to be explored. Theorem 1 gives the solution procedure to find T^∗ when $M > \frac{W}{D}$ . Theorem 2 gives the solution procedure to find T^∗ when $M ⩽ \frac{W}{D}$ .

Numerical examples are given to illustrate

References (8)

P. Chu et al.
Economic order quantity of deteriorating items under permissible delay in payments
Computers and Operations Research
(1998)
M. Khouja et al.
Optimal inventory policy under different supplier credits
Journal of Manufacturing Systems
(1996)
K.H. Chung
Inventory control and trade credit revisited
Journal of the Operational Research Society
(1989)
K.J. Chung
Bounds on the optimum order quantity in a DCF analysis of the EOQ model under trade credit
Journal of Information and Optimization Sciences
(1999)

There are more references available in the full text version of this article.

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Production, Manufacturing and LogisticsThe optimal ordering policy of the EOQ model under trade credit depending on the ordering quantity from the DCF approach

Abstract

Introduction

Section snippets

The model

A theorem

The algorithm

Numerical examples

Special cases

Summary

Computers and Operations Research

Journal of Manufacturing Systems

Inventory control and trade credit revisited

Journal of the Operational Research Society

Bounds on the optimum order quantity in a DCF analysis of the EOQ model under trade credit

Journal of Information and Optimization Sciences

Production, Manufacturing and Logistics
The optimal ordering policy of the EOQ model under trade credit depending on the ordering quantity from the DCF approach