Multiple Criteria Decision Making Theory, Methods, and Applications in Engineering
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ChihTe Yang, LiangYuh Ouyang, ChangHsien Hsu, KuoLiang Lee, "Optimal Replenishment Decisions under TwoLevel Trade Credit with Partial Upstream Trade Credit Linked to Order Quantity and Limited Storage Capacity", Mathematical Problems in Engineering, vol. 2014, Article ID 736712, 14 pages, 2014. https://doi.org/10.1155/2014/736712
Optimal Replenishment Decisions under TwoLevel Trade Credit with Partial Upstream Trade Credit Linked to Order Quantity and Limited Storage Capacity
Abstract
This paper extends the previous economic order quantity (EOQ) models under twolevel trade credit such as Goyal (1985), Teng (2002), Huang (2003, 2007), Kreng and Tan (2010), Ouyang et al. (2013), and Teng et al. (2007) to reflect the reallife situations by incorporating the following concepts: (1) the storage capacity is limited, (2) the supplier offers the retailer a partially upstream trade credit linked to order quantity, and (3) both the dispensable assumptions that the upstream trade credit is longer than the downstream trade credit and the interest charged per dollar per year is larger than or equal to the interest earned per dollar per year are relaxed. We then study the necessary and sufficient conditions for finding the optimal solution for various cases and establish a useful algorithm to obtain the solution. Finally, numerical examples are given to illustrate the theoretical results and provide the managerial insights.
1. Introduction
Trade credit financing is a crucial issue and increasingly recognized as important means to increase profitability in a productioninventory system. In practice, the supplier usually allows the wholesaler a fixed permissible delay period for settling the account (i.e., an upstream trade credit) and the wholesaler in turn provides a similar credit period to its customers (i.e., a downstream trade credit). It is well known that the permissible delay in payments has two benefits: it invites new buyers who consider it to be a type of price reduction, and it may be useful as an alternative to price discount because it does not aggravate competitors to decrease their prices and thus introduce permanent price reductions (e.g., [1]).
In 1985, the EOQ model with upstream trade credit was first proposed by Goyal [2]. Following, prolific extensions of his model have been developed by researchers. For example, Aggarwal and Jaggi [3] extended Goyal’s [2] model for the deteriorating items. Jamal et al. [4] further generalized Aggarwal and Jaggi’s [3] model to allow for shortages. Teng [5] amended Goyal’s [2] model by considering the different between unit price and unit cost and found that it makes economic sense for a wellestablished wholesaler to order less quantity and take the benefits of payment delay more frequently. Chang et al. [6] developed an EOQ model for deteriorating items under supplier’s upstream trade credit linked to ordering quantity. Liang and Zhou [7] established a twowarehouse inventory model for deteriorating items under conditionally permissible delay in payment. There are several interesting and relevant papers related to the trade credits, for example, Huang [8], Ouyang et al. [9], Chen et al.[10] Chung and Huang [11], Hu and Liu [12], Min et al. [13] Giri et al. [14], Khanra et al. [15], Sarkar [16], and so forth. Nerveless, all inventory models described above only considered an upstream trade credit.
Huang [17] extended Goyal's [2] model to establish an EOQ model under two levels of trade credit policy with the downstream trade credit period being less than the upstream trade credit period . Later, Kreng and Tan [18] modified Huang’s [17] model by considering the upstream trade credit linked to the ordering quantity. Recently, Ouyang et al. [19] not only complemented the shortcomings in Kreng and Tan [18] on the interest earned and charged, but also relaxed those dispensable assumptions such that the downstream trade credit period is less than the upstream trade credit period. Other interesting and relevant papers related to twolevel trade credit such as Teng et al. [20], Liao [21], Goswami et al. [22], Min et al. [23], Ho [24], and others.
In addition, it is observed that the classical inventory models generally deal with single storage facility. The basic assumption in these models is that the manager owns a storage room with unlimited capacity. However, in practice, the manager may purchase a huge quantity of goods at a time for some reasons such as when suppliers provide price discounts for bulk purchases or trade credits to encourage the retailer to buy more. These huge stocks cannot be stored in the existing storage (the own warehouse) with limited capacity. Therefore, a rented warehouse (RW) is needed to store the excess units over the capacity of the own warehouse. An early discussion on the inventory model with twowarehouse was given by Hartely [25]. Further literatures in this direction include Sarma [26], Dave [27], Goswami and Chaudhuri [28], Pakkala and Achary [29], Bhunia and Maiti [30], Benkherouf [31], Yang [32], Huang [33], Lee and Hsu [34], Sett et al. [35], and others.
Consequently, to reflect the reallife situations, this paper extended the previous EOQ models with twolevel trade credit such as Goyal [2], Teng [5], Huang [8, 17], Ouyang et al. [19], and Teng et al. [20] by incorporating the following concepts: the storage capacity is limited, the supplier offers the retailer a partial upstream trade credit linked to order quantity, and both the dispensable assumptions of the upstream trade credit is longer than the downstream trade credit and the interest charged per dollar is larger than or equal to the interest earned per dollar are relaxed.
The rest of this paper is organized as follows. In Section 2, we describe the notation and assumptions adopted throughout this paper. Then, mathematical models are developed to minimize the total costs per year in Section 3 for various cases. In Section 4, we study the necessary and sufficient conditions and establish several theoretical results for finding the optimal solution under various situations. Numerical examples and sensitivity analysis with major parameters are given to illustrate the theoretical results and obtain some managerial insights in Section 5. Finally, conclusions are given in Section 6.
2. Notation and Assumptions
The notation used throughout this paper is as follows: : the demand rate per year; : the ordering cost per order; : the purchasing cost per unit; : the selling price per unit, with ; : the unit holding cost per year excluding interest charge in own warehouse (OW); : the unit holding cost per year excluding interest charge in rented warehouse (RW), with ; : the interest earned per dollar per year; : the interest charged per dollar per year; : the fraction of the delay payments permitted by the supplier if the order quantity is less than the preassign quantity, ; : the wholesaler’s trade credit period in years offered by the supplier; : the retailer’s trade credit period in years offered by the wholesaler; : the capacity in own warehouse; : the time interval in which maximum inventory in own warehouse is depleted to zero; that is, ; : the minimum order quantity at which full delay in payments is permitted; : the time interval in which the quantity is depleted to zero, that is, ; : the length of replenishment cycle in years; : the order quantity, where ; : the optimal length of replenishment cycle time in years; : the optimal order quantity.
The models proposed in this paper are based on the following assumptions.(1)Demand rate is known and constant.(2)Time horizon is infinite.(3)Replenishment is instantaneous and shortages are not allowed.(4)If the order quantity is greater than , then the wholesaler needs to rent an additional warehouse to hold inventory.(5)If the wholesaler’s order quantity is greater than or equal to , then fully delayed payment is permitted by its supplier. Otherwise, the partially delayed payment is permitted. That is, the wholesaler must take a loan to pay its supplier the partial payment of immediately when the order is received and then pay off the loan with entire revenue.(6)The wholesaler offers a credit period to every retailer.(7)During the credit period , sales revenue is deposited in an interest bearing account with the rate . At the end of the permissible delay , the wholesaler pays off all units sold, keeps the profit for use in other activities, and starts paying for the interest charges with the rate on loan.
3. Model Formulation
From assumptions, as (i.e., ), the full delay in payment is permitted. Otherwise, the partial delay in payment is permitted where the wholesaler must pay the supplier the amount immediately when the order is filled and pay the rest at the time . Furthermore, if the order quantity is greater than , then the wholesaler needs to rent an additional warehouse to hold inventory.
The annual total relevant cost consists of the following elements.(a)The ordering cost per year (say ) is (b)The holding cost per year excluding interest charges (say HC) is where (c)Interest earned and the interest charged.
As to calculate the interest earned and interest charged, there are two possible cases that should be considered: when (i.e., ), the full delay in payment is permitted; otherwise, the partial delay in payment is permitted where the wholesaler must pay the supplier the amount immediately when the order is filled and pay the rest at the time . That is, there are two cases that might arise: (i) full delay in payments and (ii) partial delay in payments .
Case 1 (full delay in payments ()). Based on the values of , , and , we have the following three alternative situations: , , and . Let us discuss them accordingly.
(1) . In this situation, the wholesaler receives the total revenue at time and is able to pay the supplier the total purchase cost at time (see Figure 1). Consequently, the interest charged per year (say ) is
and the interest earned per year (say ) is
(2) . In this situation, the wholesaler will sell the items and uses the sales revenue to earn interest at the rate in the interval (see Figure 2(a)). On the other hand, the wholesaler receives the pay after and pays off all units sold at time and starts paying for the interest charges with the rate on items sold after (see Figure 2(b)). As a result, the interest charged per year (say ) is
and the interest earned per year (say ) is
(3) . When , there is no interest earned for the wholesaler. In addition, the wholesaler must finance all items ordered at time at an interest charged per dollar per year and starts to pay off the loan after time (see Figure 3). Hence, the interest charged per year (say ) is
and the interest earned per year (say ) is
Consequently, from (5), (7), and (9), the interest charged per year for the case with full delay in payments (say ) is
Similarly, from (6), (8), and (10), we have that the interest earned per year for the case with full delay in payments (say ) is
Therefore, the total cost per year for the case with full delay in payments (denoted by ) is given by
where
Note that , .
(a)
(b)
Case 2 (partial delay in payments ()). In this case, the partial delay in payment is permitted where the wholesaler must take a loan to pay the supplier the amount immediately when the order is filled and pay the rest at the time M. From the constant sales revenue , the wholesaler will be able to pay off the loan at the time . Similar to Case 1, based on the values of , , and , we have the following three alternative situations: , , and . Let us discuss them accordingly.
(1) . In this situation, the wholesaler takes a loan to pay the supplier the amount immediately but receives the revenue after . That is, the wholesaler will pay off the loan from sales revenue at time and the interest earned starts from time to (see Figure 4). Consequently, the interest charged per year (say ) is
and the interest earned per year (say ) is
(2) . In this situation, the wholesaler takes a loan to pay the supplier the amount immediately but receives the revenue after . That is, the wholesaler will pays off the loan from sales revenue at time and the interest earned starts from time to (see Figure 5). After , the wholesaler starts paying for the interest charges with the rate on items sold. As a result, the interest charged per year (say ) is
and the interest earned per year (say ) is
(3) . In this situation, there is no interest earned. Due to , the wholesaler takes a loan to pay the supplier the amount immediately but receives the revenue after . The loan will be paid off from sales revenue at the time . Furthermore, the wholesaler starts paying for the interest charges with the rate on items sold after (see Figure 6). Hence, the interest charged per year (say ) is
and the interest earned per year (say ) is
Consequently, from (22), (24), and (26), the interest charged per year for the case with partial delay in payments (say ) is
Similarly, from (23), (25), and (27), we have that the interest earned per year for the case with partial delay in payments (say ) is
For convenient, we let where . Therefore, the total cost per year for the case with partial delay in payments (denoted by is given by
where
Note that and for , .
(a)
(b)
(a)
(b)
(a)
(b)
Remark 1. When (i.e., the supplier offers the full delay in payment regardless of the order quantity), which implies , we have , for and .
Remark 2. When (i.e., the capacity in own warehouse is unlimited), we have , for and .
Remark 3. When and , we have , for .
Remark 4. (i) When and , the model can be reduced to Ouyang et al. [19].
(ii) When , , and , the model is similar to Huang [17] and Teng et al. [20].
(iii) When , , and , the model is similar to Huang [8].
(iv) When , , , and , the model can be reduced to Teng [5].
(v) When , , , , and , the model is the same as Goyal [2].
Therefore, our model is in general framework that includes numerous previous models such as Goyal [2], Teng [5], Huang [8, 17], Ouyang et al. [19], and Teng et al. [20] as special cases.
4. Theoretical Results
Now, we will determine the optimal length of replenishment cycle time in years which minimizes the annual total relevant cost. First, from Remarks 1–3, we can see that the total relevant cost functions of can be reduced to , , and as or/and , where . Here we only discuss how to find the optimal length of replenishment cycle time in years that minimizes the annual total relevant cost , where . Following, we will develop an iterative algorithm to find the optimal solution for the whole problem.
The firstorder necessary condition for to be minimized is which leads to Note that , , and ; and hence is well defined. Furthermore, the secondorder sufficient condition is Therefore, is a convex function of , and in (39) satisfies . To ensure , we substitute (39) into the inequality and obtain that
, if and only if On the other hand, if , then we have for all which implies that is a strictly decreasing function of . Therefore, has a minimum value at the boundary point .
For notational convenience, we let
Then we have the following result.
Lemma 5. If , then has a minimum value at .
If , then has a minimum value at .
Similarly, the firstorder necessary condition for to be minimized is which leads to Furthermore, the secondorder sufficient condition is To ensure , we substitute (44) into the inequality , and obtain that where is defined as above, and It is noted that if , then we have Therefore, in (44) is well defined and , which implies that is a convex function of .
On the other hand, if , then we have for all which implies that is a strictly decreasing function of . Therefore, has a minimum value at the boundary point .
If , then we have for all which implies that is a strictly increasing function of . Therefore, has a minimum value at the boundary point .
From above arguments, we have proved the following result.
Lemma 6. If , then has a minimum value at .
If , then has a minimum value at .
If , then has a minimum value at .
By using analogous discussions, we can easily obtain the values of (say ) which minimizes is To ensure , we substitute (51) into the inequality and and obtain that where Conversely, if , then we have Therefore, is s strictly increasing function of , which implies that has a minimum value at the boundary point . From above arguments, we have proved the following result.
Lemma 7. If , then has a minimum value at .
If , then has a minimum value at .
It is obvious that and . Consequently, combining Lemmas 5, 6, and 7 and the facts that and for , we can obtain a table (Table 1) to determine the optimal length of cycle time that minimizes the annual total relevant cost (say ).

For Remarks 1–3, we can obtain the following tables (Tables 2–4) to determine the optimal lengths of cycle time that minimize the annual total relevant costs _{,}, and , respectively (say , , and ).
 
where , , and . 
 
where , , and . 
