Optimal Replenishment Decisions under Two-Level Trade Credit with Partial Upstream Trade Credit Linked to Order Quantity and Limited Storage Capacity

This paper extends the previous economic order quantity (EOQ) models under two-level 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 real-life 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 creditN < M and the interest charged per dollar per year is larger than or equal to the interest earned per dollar per year I c < I e 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.


Introduction
Trade credit financing is a crucial issue and increasingly recognized as important means to increase profitability in a production-inventory 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: (1) it invites new buyers who consider it to be a type of price reduction, and (2) 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 well-established 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 two-warehouse 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 two-level 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 two-warehouse 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 real-life situations, this paper extended the previous EOQ models with two-level 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: (1) the storage capacity is limited, (2) the supplier offers the retailer a partial upstream trade credit linked to order quantity, and (3) 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.

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  > ; ℎ: 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  > ℎ;   : 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, 0 ≤  ≤ 1; : 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.
(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 (1 − ) 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.

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 (1 − ) 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 OC) 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 (1 − ) 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 ( <   ).
(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 IC 11 ) is and the interest earned per year (say IE 11 ) 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 IC 12 ) is and the interest earned per year (say IE 12 ) 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 IC 13 ) is and the interest earned per year (say IE 13 ) is Consequently, from ( 5), (7), and ( 9), the interest charged per year for the case with full delay in payments (say IC 1 ) is  Similarly, from ( 6), (8), and (10), we have that the interest earned per year for the case with full delay in payments (say IE 1 ) is Therefore, the total cost per year for the case with full delay in payments (denoted by TC 1 ()) is given by where Outstanding balance Time Figure 4:  <   and  <  +  ≤ .
Note that TC 1-1 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 (1 − ) 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 (1)  ≥  + .In this situation, the wholesaler takes a loan to pay the supplier the amount (1 − ) immediately but receives the revenue after .That is, the wholesaler will pay off the loan from sales revenue at time (1 − )(/) +  and the interest earned starts from time (1 − )(/) +  to  (see Figure 4).Consequently, the interest charged per year (say IC 21 ) is and the interest earned per year (say IE 21 ) is (2)  +  ≥  ≥ (1 − )(/) + .In this situation, the wholesaler takes a loan to pay the supplier the amount (1 − ) immediately but receives the revenue after .That is, the wholesaler will pays off the loan from sales revenue at time (1 − )(/) +  and the interest earned starts from time (1−)(/)+ 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 IC 22 ) is and the interest earned per year (say IE 22 ) is (3) (1 − )(/) +  ≥ .In this situation, there is no interest earned.Due to  <   , the wholesaler takes a loan to pay the supplier the amount (1 − ) immediately but receives the revenue after .The loan will be paid off from sales revenue at the time (1 − )(/) + .Furthermore, the wholesaler starts paying for the interest charges with the rate   on items sold after (1 − )(/) +  (see Figure 6).Hence, the interest charged per year (say IC 23 ) is and the interest earned per year (say IE 23 ) is Consequently, from ( 22), (24), and ( 26), the interest charged per year for the case with partial delay in payments (say IC 2 ) is Similarly, from ( 23), (25), and ( 27), we have that the interest earned per year for the case with partial delay in payments (say IE 2 ) is For convenient, we let V = (1 − )(/) where 0 ≤ V < 1.

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 TC 22- () can be reduced to TC 11- (), TC 12- (), and TC 21- () as  = 1 or/and ℎ = , where  = 1, 2, 3.Here we only discuss how to find the optimal length of replenishment cycle time in years  22 that minimizes the annual total relevant cost TC 22- (), where  = 1, 2, 3. Following, we will develop an iterative algorithm to find the optimal solution  * for the whole problem.
Next, we will can establish the following algorithm to determine the optimal length of cycle time  * .Algorithm.
Step 1. Compare   with   .If   ≥   , then go to Step 2; otherwise, go to Step 3.

Numerical Example
To illustrate the previous results, we use a numerical example as follows.
From the results of Table 5, the following observations can be made.
(1) The wholesaler will determine whether to enjoy full or partial delay in payments based on the value of the permitted minimum order quantity with full delay in payments.For the low permitted minimum order quantity with full delay in payments (e.g.,   = 100), the wholesaler will take the fully permissible delay and pay at the end of .Otherwise, if the value of   is high enough (e.g.,   ≥ 200), then the wholesaler will take a loan to pay its supplier the partial payment of (1 − ) immediately when the order is received.
(2) The wholesaler will determine whether to rent an additional warehouse based on the value of the capacity in own warehouse.That is, when the capacity in own warehouse is low (e.g.,  = 100), the wholesaler will need to rent an additional warehouse to satisfy more goods in stocks.If the capacity in own warehouse is high enough (e.g.,  ≥ 200), then the wholesaler will no longer rent an additional warehouse.
(3) For the case of partial delay in payments (  ≥ 200), when the value of the fraction of the delay payments permitted by the supplier increases, all the optimal  values of  * ,  * , and TC( * ) decrease.The simple economic explanation for this is that the larger the fraction of the delay payments permitted, the lower the length of replenishment cycle, order quantity, and total relevant cost will be.That is, the wholesaler will reduce the order quantity to enjoy the benefit of delay in payments when the fraction of the delay payments permitted by the supplier increases.
Example 2. This example discusses the influences of changes in wholesaler's and retailer's trade credit periods on  * ,  * , and TC( * ) of Example 1.For convenience, the case with fixed  = 0.5,  = 100, and   = 200 is taken into account.According to algorithm in the previous section, we obtain the optimal cycle time and the optimal order quantity for different parameters of  ∈ {0.2, 0.25, 0.3} and  ∈ {0.2, 0.25, 0.3} as shown in Table 6.From the results in Table 5, the following observations can be made.
(1) The optimal total cost per year decreases when the value of  increases or the value of  decreases.That is, it is benefit for the wholesaler to lengthen the wholesaler's trade credit period in years offered by the supplier or shorten the retailer's trade credit period in years offered by the wholesaler.
(2) For the high value of  (e.g.,  = 0.3), the optimal order quantity increases as the value of N increases.
(3) For the high value of  (e.g.,  = 0.2), the optimal order quantity decreases as the value of  increases.
Example 3.Here we discuss the influences of changes in major parameters , , , ,, ℎ,   , and   on  * ,  * , and TC( * ) of Example 2. For convenience, the case with fixed  = 0.25 and  = 0.2 is taken into account.The sensitivity analysis is performed by changing each of the parameters by −20%, −10%, +10%, and +20%, taking one parameter at a time and keeping the remains unchanged.The computational results are shown in Table 7.
On the basis of the results of Table 7, the following observations can be made.
(1) The optimal length of replenishment cycle  * , the optimal order quantity  * , and the optimal total cost per year TC( * ) increase with the increase in the value of .
(2) It is obvious that all the values of  * ,  * , and TC( * ) decrease as the revenue parameter  or   increases.That is, both selling price per unit and interest earned per dollar per year have negative effects on the length of replenishment cycle, order quantity, and the annual total relevant cost.
(3) When the value of , , ℎ, or   decreases, the length of replenishment cycle and order quantity decrease but the total relevant cost increases.The simple economic explanation for this is that the larger cost parameters (purchasing cost, holding cost, and interest charged per dollar per year), the lower the length of replenishment cycle and order quantity, while the larger the annual total relevant cost will be.
(4) The value of  * decreases while the values  * and TC( * ) increase as the parameter  increases.

Conclusions
In this paper, we extended the previous economic order quantity (EOQ) models under two-level trade credit to reflect the following real-life situations: (1) the storage capacity is limited; (2) the supplier offers the retailer a partial upstream trade credit linked to order quantity; (3) the upstream trade credit may be longer than, equal to, or less than the downstream trade credit; and (4) the interest charged per dollar per year may be larger than, equal to, or less than the interest earned per dollar per year.In theoretical results, we studied the necessary and sufficient conditions for finding the optimal solution under various situations in Tables 1-4.Furthermore, we established a useful algorithm to obtain the optimal solution.Finally, we have provided numerical examples and sensitivity analysis with major parameters to illustrate the proposed model and understand managerial insights.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.It is our belief that our work will make some innovational and significant contributions for a wholesaler to determine his/her optimal lot size simultaneously when facing the real-life situations.

Table 2 :
The optimal length of replenishment cycle time  11 under various cases.

Table 3 :
The optimal length of replenishment cycle time  12 under various cases.

Table 5 :
Optimal solutions of Example 1.

Table 6 :
Optimal solutions for various  and .

Table 7 :
Effect of changes in major parameters of Example 2.