An Optimal Ordering Policy of the Retailers Under Partial Trade Credit Financing and Restricted Cycle Time in Supply Chain

Abstract

The traditional EOQ (Economic Order Quantity) model assumes that retailers’ capitals are unrestricting and the retailer must pay for items as soon as the retailer receives them from suppliers. However, this may not be true. In practice, the supplier will offer the retailer a delay period. This period is known as the trade credit period. Previously published papers assumed that the supplier would offer the retailer a delay period and the retailer could sell goods and earn interest or investment within the trade credit period. They assumed that the supplier would offer the retailer a delay period but the retailer would not offer the trade credit period to his/her customer. We extend their model and construct new ordering policy. In this paper, the retailer will also adopt the partial trade credit policy to his/her customer. We assume that the retailer’s trade credit period offered by the supplier is not shorter than his/her customer’s trade credit period offered by the retailer. In addition, they assumed the relationship between the supplier and the retailer is one-to-one. One thing we want to emphasize here is that the supplier has cooperative relations with many retailers. Furthermore, we assume that the total of the cycle time is restricted. Under these conditions, we model the retailers’ inventory system to determine the optimal cycle times for n retailers.

1 Introduction

Inventory management is to decide appropriate times and quantities to produce goods. It has a significant impact on the costs and profitability of many organizations. In general, the EOQ model is still a widely used model to guide the management of inventories in many industrial enterprises and service organizations. The EOQ captures the trade-off between inventory carrying cost and ordering cost for each item with accuracy.

Until now, the EOQ model assumes that the retailer’s capitals are unrestricting and the retailer must pay for the items as soon as the retailer receives the items from the supplier. However, this may not be completely true. In practice, the supplier will offer the retailer a delay period. This period is called the trade credit period. During this period, the retailer can sell goods and earn interest or investment (Chang et al. 2001; Chen and Chuang 1999; Kim et al. 1995; Hwang and Shinn 1997).

We extend their model and construct new ordering policy. We reconstruct an ordering policy to stimulate his/her customer demand to develop the retailers’ replenishment model. The retailer will adopt the partial trade credit policy to his/her customer (Huang and Hsu 2008). We assume that the retailer’s trade credit period M offered by the supplier is not shorter than his/her customer’s trade credit period N offered by the retailer (M ≥ N). Moreover, the problem we want to emphasize here is that the supplier has cooperative relations with n retailers. Furthermore, we assume that the total of the cycle time offered for n retailers is restricted. Under these conditions, we model an optimal ordering policy to determine the optimal cycle times for n retailers. This model can be formulated as a mathematical problem.

2 Model Formulations

2.1 Notation

The notations used in this paper are as follows: D: demand rate per year; A: ordering cost per order; c: unit purchasing price per item; h: unit stock holding cost per item per year excluding interest charges; $\alpha:$ customer’s fraction of the total amount owed payable at the time of placing an order offered by the retailer, $0 \le \alpha \le 1$, $I_{e}^{{}}$: interest earned per $ per year; $I_{k}^{{}}$: interest charged per $ in stocks per year by the supplier; M: the retailer’s trade credit period offered by supplier in years; N: the customer’s trade credit period offered by retailer in years; T: the cycle time in years; $TCV(T)$: the annual total inventory cost, which is a function of $T$; $T_{{}}^{*}$: the optimal cycle time of $TCV(T)$.

2.2 Assumptions

(1) Demand rate is known and constant; (2) shortages are not allowed; (3) time period is finite; (4) the lead time is zero; (5) $I_{k}^{{}} \ge I_{e}^{{}},\; M \ge N$; (6) when $T \ge M$, the account is settled at $T = M$ and the retailer starts paying for the interest charges on the items in stock with rate $I_{k}^{{}}$. When $T \le M$, the account is settled at $T = M$ and the retailer does not need to pay any interest charges; (7) the retailer can accumulate revenue and earn interest after his/her customer pays for the amount of purchasing cost to the retailer until the end of the trade credit period offered by the supplier. That is, the retailer can accumulate revenue and earn interest during the period $N$ to $M$ with rate $I_{e}^{{}}$ under the condition of the trade credit.

2.3 Model Formulation

From the above notation and assumptions, we construct a model. First, we must consider all the inventory costs. That is, the annual total inventory cost consists of the following four elements.

• 1. Annual ordering cost is as follow: $\frac{A}{T}$.

• 2. Annual stock holding cost (excluding interest charges) is as follow: $\frac{DTh}{2}$.

• 3. According to Assumption 6, there are two cases to occur in costs of interest charges for the items kept in stock per year. Annual interest payable is as follows

$$ \frac{{cI_{k} D(T - M)^{2} }}{2T},\;M \le T,\quad 0,\;N \le T < M,\;T \le N $$

• 4. According to Assumption 7, there are three cases to occur in interest earned per year. Annual interest earned is as follows:

  $$ \frac{{sI_{e} D[M^{2} - (1 - \alpha )N^{2} ]}}{2T},\;M \le T,\quad \frac{{sI_{e} D[2MT - (1 - \alpha )N^{2} - T^{2} )}}{2T},\;N \le T < M $$
  $$ sI_{e} D\left[ {M - (1 - \alpha )N - \frac{\alpha T}{2}} \right],\quad 0 \le T < N. $$

  From the above arguments, the annual total inventory cost for the retailer can be expressed as TCV(T) = ordering cost + stock-holding cost + interest payable cost − interest earned. That is, it is formulated as follows:

  $$ TCV(T) = \left\{ {\begin{array}{*{20}l} {TCV_{1} (T)} & {T \ge M,} \\ {TCV_{2} (T),} & {N \le T \le M,} \\ {TCV_{3} (T),} & {0 < T \le N,} \\ \end{array} } \right. $$
  $$ TCV_{1} (T) = \frac{A}{T} + \frac{DTh}{2} + \frac{{cI_{k} D(T - M)^{2} }}{2T} - \frac{{sI_{e} D[M^{2} - (1 - \alpha )N^{2} )}}{2T}, $$
  $$ TCV_{2}^{{}} (T) = \frac{A}{T} + \frac{DTh}{2} - \frac{{sI_{e}^{{}} D[2MT - (1 - \alpha )N_{{}}^{2} - T_{{}}^{2} ]}}{2T}, $$
  $$ TCV_{3} (T) = \frac{A}{T} + \frac{DTh}{2} - sI_{e} D\left[ {M - (1 - \alpha )N\frac{\alpha T}{2}} \right]. $$

• 5. According to Assumption 7, there are three cases to occur in interest earned per year. Annual interest earned is as follows:

  $$ \frac{{sI_{e} D[M^{2} - (1 - \alpha )N^{2} ]}}{2T},\;M \le T,\quad \frac{{sI_{e} D[2MT - (1 - \alpha )N^{2} - T^{2} )}}{2T},\;N \le T < M, $$
  $$ sI_{e} D\left[ {M - (1 - \alpha )N - \frac{\alpha T}{2}} \right],\quad 0 \le T < N. $$

  Since $TCV_{1} (M) = TCV_{2} (M)$ and $TCV_{2} (N) = TCV_{3} (N)$, $TCV(T)$ is continuous and well-defined. All $TCV_{1} (T),TCV_{2} (T),TCV_{3} (T)$ and $TCV(T)$ are defined on $T > 0$. Also, $TCV_{1} (T),TCV_{2} (T)$ and $TCV_{3} (T)$ are convex on $T > 0$. Furthermore, we have $TCV_{1}^{\prime } (M) = TCV_{2}^{\prime } (M)$ and $TCV_{2}^{\prime } (N) = TCV_{3}^{\prime } (N)$. Therefore, $TCV(T)$ is convex on $T > 0.$

  From the above arguments, the annual total inventory cost for the retailer can be expressed as TCV(T) = ordering cost + stock-holding cost + interest payable cost − interest earned. That is, it is formulated as follows:

  $$ TCV(T) = \left\{ {\begin{array}{*{20}l} {TCV_{1} (T),} & {T \ge M,} \\ {TCV_{2} (T),} & {N \le T \le M,} \\ {TCV_{3} (T),} & {0 < T \le N,} \\ \end{array} } \right. $$
  $$ TCV_{1} (T) = \frac{A}{T} + \frac{DTh}{2} + \frac{{cI_{k} D(T - M)^{2} }}{2T} - \frac{{sI_{e} D[M^{2} - (1 - \alpha )N^{2} )}}{2T}, $$
  $$ TCV_{2} (T) = \frac{A}{T} + \frac{DTh}{2} - \frac{{sI_{e} D\,[2MT - (1 - \alpha )N^{2} - T^{2} ]}}{2T}, $$
  $$ TCV_{3} (T) = \frac{A}{T} + \frac{DTh}{2} - sI_{e} D\left[ {M - (1 - \alpha )N\frac{\alpha T}{2}} \right]. $$

  Since $TCV_{1}^{{}} (M) = TCV_{2}^{{}} (M)$ and $TCV_{2} (N) = TCV_{3} (N),$ $TCV(T)$ is continuous and well-defined. All $TCV_{1} (T),TCV_{2} (T),TCV_{3} (T)$ and $TCV(T)$ are defined on $T > 0.$ Also, $TCV_{1} (T),TCV_{2} (T)$ and $TCV_{3} (T)$ are convex on $T > 0.$ Furthermore, we have $TCV_{1}^{\prime } (M) = TCV_{2}^{\prime } (M)$ and $TCV_{2}^{\prime } (N) = TCV_{3}^{\prime } (N).$ Therefore, $TCV(T)$ is convex on $T > 0.$

3 An Optimal Ordering Policy of the Retailers Under Partial Trade Credit Financing and Restricted Cycle Time

We consider an optimal ordering policy of the retailers for the following problem: (1) inventory problem we propose here is in the EOQ model; (2) the number of the retailers is n; (3) the summation of each cycle time $t_{i} \;(i = 1,2, \ldots ,n)$ is restricted to $T_{0}$; (4) we allocate the restricted cycle time $T_{0}$ to n retailers to minimize the summation of total inventory cost.

To solve the problem, we use the inventory model we have already discussed in previous section. First, we formulate the cost function for each retailer. Next, we lead to all retailers’ total cost function.

Let $t_{i}$ be a cycle time for retailer $i\;(i = 1,2, \ldots ,n).$ Then we obtain the following total inventory cost function for retailer $i$:

$$ TCV_{i} (t_{i} ) = \left\{ {\begin{array}{ll} {TCV_{1i} (t_{i} ),} & {t_{i} \ge M_{i} ,} \\ {TCV_{2i} (t_{i} ),} & {N_{i} \le t_{i} \le M_{i} ,} \\ {TCV_{3i} (t_{i} ),} & {0 < t_{i} \le N_{i} .} \\\end{array}} \right. $$

Here, new subscript $i$ represents the retailer’s number $i\;(i = 1,2, \ldots ,n).$ From the above expression, the summation of total inventory cost is shown as follows:

$$ \sum\limits_{i = 1}^{n} {TCV_{i} (t_{i} ),\quad i = 1,2, \ldots ,n,} $$

(1)

Therefore, an optimal ordering policy of the retailers under trade credit financing and restricted cycle time we propose here is represented as follows:

$$ Minimize\quad \sum\limits_{i = 1}^{n} {TCV_{i} (t_{i} ),} $$

(2)

$$ subject\;to\quad \sum\limits_{i = 1}^{n} {t_{i} \le T_{0} ,} \quad t_{i} > 0\quad (1 = 1,2, \ldots ,n), $$

(3)

Now, our aim is to minimize the objective function [Eq. (2)] under the constraints [Eq. (3)]. We differentiate Eq. (2) with respect to $t_{i}.$ We obtain as follows:

$$ TCV^{\prime } (t_{i} ) = \left\{{\begin{array}{lll}{TCV_{1i}^{\prime } (t_{i} ) = -\left[{\frac{{2A_{i} + c_{i} D_{i} M_{i}^{2} I_{ki} - s_{i}D_{i} I_{ei} (M_{i}^{2} - (1 - \alpha_{i} )N_{i}^{2})}}{{2t_{i}^{2} }}} \right] + D_{i} \left( {\frac{{h_{i} + c_{i}I_{ki} }}{2}} \right),} & {t_{i} \ge M_{i} ,} \\ {TCV_{2i}^{\prime } (t_{i} ) = - \left[ {\frac{{2A_{i} + s_{i}D_{i} (1 - \alpha_{i} )N_{i}^{2} I_{ei} }}{{2t_{i}^{2} }}}\right] + D_{i} \left( {\frac{{h_{i} + s_{i} I_{ei} }}{2}}\right),} & {N_{i} \le t_{i} < M_{i} ,} \\ {TCV_{3i}^{\prime }(t_{i} ) = - \frac{{A_{i} }}{{t_{i}^{2} }} + D_{i} \left({\frac{{h_{i} + s_{i} \alpha_{i} I_{ei} }}{2}} \right),} & {0\le t_{i} < N_{i} } \\\end{array}} \right. $$

(4)

Here, we define $\Updelta_{1i}$ and $\Updelta_{2i}$ respectively.

$$ \Updelta_{1i} = \frac{{ - 2A_{i} + D_{i} M_{i}^{2} (h_{i} + s_{i} I_{ei} ) - s_{i} D_{i} (1 - \alpha_{i} )N_{i}^{2} I_{ei} }}{{2M_{i}^{2} }},\;\Updelta_{2i} = \frac{{ - 2A_{i} + D_{i} N_{i}^{2} (h_{i} + s_{i} \alpha_{i} I_{ei} )}}{{2N_{i}^{2} }} $$

where $\Updelta_{1i} \ge \Updelta_{2i}$ ($i = 1,2, \ldots ,n$). To solve the problem, we use Lagrange’s multiplier $\lambda$. The optimal solution $\tilde{t}_{i}^{*}$ for our problem is obtained from Tables 1, 2, 3 using $\sum\nolimits_{i = 1}^{n} {t_{i} } \le T_{0}.$

Table 1 The retailer’s optimal cycle time $\tilde{t}_{i}^{*}$ for Lagrange’s multiplier $\lambda$ (Case 1)

Table 2 The retailer’s optimal cycle time $\tilde{t}_{i}^{*}$ for Lagrange’s multiplier $\lambda$ (Case 2)

Table 3 The retailer’s optimal cycle time $\tilde{t}_{i}^{*}$ for Lagrange’s multiplier $\lambda$ (Case 3)

For each retailer, we rearrange $\left\{ {\Updelta_{1i} |1 \le i \le n} \right\},$ $\left\{ {\Updelta_{2i} |1 \le i \le n} \right\},$ and $\left\{ 0 \right\}$ small order, $B_{1} \le B_{2} \le \cdots \le B_{n} \le \cdots \le B_{2n} \le B_{2n + 1}.$ If $B_{k} \ne B_{k + 1} ,$ the optimal Lagrange’s multiplier $\lambda^{*}$ only exists in the interval $B_{k} \le \lambda^{*} < B_{k + 1}$ ($0 \le k \le 2n + 1$) because the objective function is convex, where $B_{0} = - \infty$ and $B_{2n + 2} = \infty.$ In this interval, we allocate the restricted cycle time $T_{0}$ to n retailers. Then, the summation of each cycle time has to be equal to $T_{0}.$ That is to say, using $\sum\nolimits_{i = 1}^{n} {t_{i} = T_{0} } ,$ we can find the optimal Lagrange’s multiplier $\lambda^{*}$.

4 Numerical Example

In this section, we show the numerical examples. Here, there are 10 retailers in our model. The parameters for each retailer are shown in Table 4.

Table 4 The parameters for each retailer

4.1 The Optimal Retailers’ Ordering Policy with the Restricted Cycle

We consider the case of the given $T_{0}$ is equal or less than the total of optimal cycle time (0.942389). For example, we set $T_{0} = 0.62.$ In Table 5, the optimal cycle time $\tilde{t}_{i}^{*}$ and the total inventory cost $TCV_{i} (\tilde{t}_{i}^{*} )$ for each retailer are shown. The optimal Lagrange’s multiplier $\lambda^{*} = -19,244.2.$ The summation of the total inventory costs are 5,004.2731.

Table 5 The optimal cycle time $\tilde{t}_{i}^{*}$ for each retailer and the total inventory cost $TCV_{i} (\tilde{t}_{i}^{*} )$

5 Conclusion

This note is a modification of the assumption of the trade credit policy in previously published results to reflect realistic business situations. We assumed that the retailer also adopts the trade credit policy to stimulate his/her customer demand to develop the retailer’s replenishment model. In addition, their model assumed the relationship between the supplier and the retailer is one-to-one. In this paper, we assumed the supplier has cooperative relations with many retailers for more suiting and satisfying the real world problems and assumed that the summation of each retailer’s cycle time is restricted. Under these conditions, we have constructed the retailers’ inventory system as a cost minimizing problem to determine the n retailers’ optimal ordering policy.

A future study will further incorporate the proposed model more realistic assumptions such as probabilistic demand, allowable shortages and a finite rate of replenishment.