An EPQ model under two levels of trade credit and limited storage space

Article history: Received 25 October 2011 Received in revised form November, 2, 2011 Accepted January, 26 2012 Available online 28 January 2012 In this article, a production inventory model is developed to determine the retailer’s optimal replenishment decisions under two levels of trade credit and limited storage space. Two levels of trade credit refers that the supplier provides to his/her retailer a permissible delay period to settle the account and the retailer also in turn provides a delay period to his/her customer for paying of purchasing goods. The retailer’s credit period M offered by supplier is greater than customers credit period N provides by the retailer. This paper investigates retailer optimal replenishment policy under finite replenishment rate by minimizing the total annual inventory cost. Three theorems are developed to determine optimal cycle time, optimal relevant cost and optimal order quantity for retailers accurate and rapidly. A numerical example is used to analyze the validity of propose model, Sensitive analysis shows managerial decisions for retailer. © 2012 Growing Science Ltd. All rights reserved


Introduction
Over the years, numerous research articles have been published in different journals on trade credit under various conditions, in which the suppliers offers a delay period to settle the retailers account.The retailer takes the benefit of trade credit and sells the product and earns interest by putting generated revenue in an interest bearing account.The supplier does not charge any interest if the payment is made before credit period.However, if the payment is paid beyond the predefined period, a high interest rate is charged.Many researchers contribute on trade credit with variety of conditions.Goyal (1985) is believed to be the first who developed the economic order quantity (EOQ) model under the conditions of permissible delay in payments.Davis and Gaither (1985) presented EOQ model when supplier offers one time opportunity to delay the payments of order in case an order for additional units is placed.Chand and Ward (1987) investigated Goyal (1985) model under the assumptions of classical economic order quantity model.Shah et al. (1988) studied the same model incorporating shortages.Mondal and Phaujdar (1989c) extended this issue by considering the interest earned from the sales revenue.Shah (1993a, b) also developed EOQ models for perishable items where delay in payment is permissible.Aggarwal and Jaggi (1995) developed Goyal's (1985) model to the case of exponential deterioration, Shah and Sreehari (1996) presented an extended issue of EOQ model where the delay in payment is permitted and the capacity of own warehouse is limited.Liao (2000) and Sarker (2000a) deal this problem with inflation.Jamal et al. (1997) and Chang and Dye (2001) examined this issue with allowable shortage.Chang et al. (2001) investigated this topic in the case of linear trend demand.Mondal and Phaujdar (1989b), Hwang and Shinn (1997), Jamal et al. (1997) and Sarker et al. (2000b) extended this issue with deteriorating items.Teng (2002) modified Goyal's (1985) model by assuming the selling price is not necessary equal to purchase price.Other papers related to this area are Chang and Dye (2002), Chang el al. (2001), Chen and Chung (1999), Chung (1998a, b) Jamal et al. (2000), Sarker et al. (2000a), Salameh et al. (2003), Huang and Chung (2003), Arcelus et al. (2003), Abad andJaggi (2003), Shinn andHuang (2003), Chung and Huang (2003), Chung and Liao (2004), Chung et al. (2005), Chung and Liao (2006), Huang (2007 a ) and Liao and Chung (2009).
In above models, the supplier adopts a business strategy of permissible delay period in paying the purchasing cost to attract more customers.The retailer takes the benefit of trade credit and sells the product and earns the interest by putting generated revenue in an interest bearing account.They implicitly assumed that the buyer would pay the supplier as soon as he/she receives the items.As mentioned earlier, supplier only offers a trade credit to the retailer but retailer does not provide any trade credit to his/her customer which means we deal with one level of trade credit.However, in many real-world cases, this condition will not hold.Recently, Huang (2003) modified this assumption by assuming that the retailer will also adopt the trade credit policy to stimulate his/her customer demand to develop retailer replenishment model and this method is called two level of trade credit policy.Retailers take benefits of trade credit and place an order of large quantity of goods to supplier to reduce inventory cost.Since retailer's storage capacity is limited so he/she has to take a rented warehouse for excess quantities to store his goods.This paper is strongly related for small and medium size retailers whose storage capacity is limited especially the growing markets like in India.When retailer receives an order the delivery is first replenishment from rented warehouse because holding cost of rented warehouse is higher with the respect of own warehouse.Numerous researchers have focused in this area such as Sharma (1987), Pakkala and Achary (1992), Benkerouf (1997), Goswami and Chaudhuri (1992), Bunia and Maiti (1998), Huang et al. (2006) and Lee and Hsu (2009) and much more.Huang (2007 b ) extended Chung and Huang (2003) and Huang (2003) model in the light of two level of trade credit and determine retailer's replenishment decision with EPQ framework.Teng and Chang (2009) extended Huang (2007 b ) and Teng and Goyal (2007) model in more realistic situation in which the retailer obtains its revenue from to not from 0 to and .This paper extends Huang (2006) model by assuming finite replenishment.We developed a mathematical model to determine optimal retailer's replenishment policy under two levels of trade credit and limited storage space in EPQ framework.Three theorems are developed to accurate and rapidly determine optimal cycle time, optimal relevant cost and optimal order quantity for retailers under minimizing total cost of inventory system.Finally, a numerical example explore the validity of propose model.

Assumptions and Notations
For convenience, most assumptions and notations throughout in this paper used Huang (2006) model.

Assumptions
1) Demand rate is known and constant 2) Shortages are not allowed.
3) Time horizon is infinite.4) Replenishment is finite, known and constant 5) , , and .6) If the orders quantity is larger than retailer's OW storage capacity W, the excess quantity is stored in rented warehouse.Since, storage capacity of rented warehouse is unlimited.When orders received by retailer the items is first replenishment from rented warehouse.7) Before settling the account the retailer's takes the benefit of trade credit and sells the product and earn the interest by putting generated revenue in an interest bearing account.When the cycle time is greater than or equal to the retailer's credit period offered by the supplier.Then, the retailer settles the account at .The retailer pays for the product sold and keep his/her profit and start paying interest on unsold items at rate .When the cycle time is less than or equal to the retailer's credit period , the retailer settles the account at .Therefore, the retailer does not have to pay any interest charges.8) The retailer can accumulate revenue and earn interest when his/her customer pays the amount for purchasing goods to the retailer by the end of predefined trade credit period offered by supplier.The retailer earns the interest by putting generated revenue in an interest bearing account at the rate of per unit per year; i.e. the retailer can earn interest during the period , with rate under the condition of trade credit.9) The transportation cost between own warehouse and rented warehouse is negligible.

1)
Ordering cost per order 2) Purchasing price per item 3) Demand rate per year 4) Replenishment rate per year.(Decision variable) 5) 1 0 6) OW stock-holding cost per item per year 7) Interest earned per $ per year 8) Interest charged per $ per year 9) RW stock-holding cost per item per year (Decision variable) 10) The retailer's trade credit period offered by supplier in years 11) The customer's trade credit period offered by retailer in years (Decision variable) 12) Selling price per item (Decision variable)

13)
The rented warehouse time in years, = 0

14) w Retailer's OW storage capacity 15)
The annual total relevant cost which is a function of (Decision variable) 16) The optimal cycle time of (Decision variable)

The Model
The total annual relevant cost of inventory system consists of the following elements.Three situations may arise (1) , 2 3 .
Case.1 Suppose that / (1) Annual ordering cost (2) According to assumption (6) annual stock-holding cost (excluding interest charges ) can be obtain as follow: 1. ⁄ In this case the order quantity larger than retailer's OW storage capacity.So the retailer needs to rent the warehouse to storage the exceeding items.See in Fig. 1 Hence Annual stock holding cost = annual stock holding cost of rented warehouse + annual stock-holding cost of the storage capacity W

/
In this case order quantity is not larger than retailer's storage capacity.So the retailer will not necessary to rent warehouse to storage items.Hence (3) According to assumption (7) cost of interest charges for the items kept in stock per year can be obtained as follows: 1.
In this case, interest charges are paid for the items kept in stock 2.
as shown in Fig. 2 In this case, interest charges are paid for the items kept in stock 3.
No interest charges are paid for the items kept in stock.From the above argument the annual total relevant cost for the retailer can be expressed as = ordering cost + stock-holding cost + interest payable -interest earned.We show that the annual total relevant cost, is given by where (2) Case.2Suppose that .

If
. Eqs. 1a-e will be modified as

When
. the annual total relevant cost, consists of the following elements: (1) Annual ordering cost (2) In this case the order quantity is not larger than retailer's storage capacity.So the retailer will not necessary to rent warehouse to storage items.Case.3 Suppose that .

If
, Eqs. 1a-d will be modified as

When
, the annual total relevant cost, .consist of the following element: (1) Annual ordering cost (2) In this case the order quantity is not larger than retailer's storage capacity.So the retailer will not necessary to rent warehouse to storage items.

Determination of the optimal cycle time
In this section, we shall determine optimal cycle time for the above three situations under minimizing annual total relevant cost using differentiation Case.1 Suppose that .
Then we can rewrite Eqs. 21 25 imply that ∆ ∆ ∆ ∆ ∆ .From above arguments we know that we can obtain following results.

If
, we know as follow From Eqs. 8a-d.
From EQ. (10) we can derive with the help of derivative a follows: Differentiation with respect to , We get Similar to above procedures in cases 1 and cases 2, we substitute Eq. ( 11 The optimal cycle time with various value of , , and w s P k From the results of Table1, we can observe the optimal cycle time, optimal order quantity and optimal relevant cost with various parameters of , , , respectively.The following inferences can be made, 1.The optimal cycle time for the retailer will be decrease as replenishment rate increases.The retailer will purchase less quantity, since the replenishment rate is fast enough.2. The optimal cycle time increases when retailers' storage capacity increases.It means that the retailer will purchase more quantity.3. The optimal cycle time will decrease and total relevant cost increases as RW unit stockholding cost increases.Therefore, the retailer will purchase less.
4. The optimal cycle time will decrease as the unit selling price increases.It means that the retailer will purchase less to take the benefits of the trade credit more frequently.

Effect of changing the inventory model parameters
The sensitivity analyses are performed by varying different parameters and are given in Table 2, 3, 4, 5 and 6.
Sensitive analyses for various inventory model parameters are as follows and remaining parameters are same as from above example.5.A higher value of along with increased values of , , and results and is decreased but increases.The retailer purchased less quantity.

Conclusion
This paper developed to determine optimal retailer's replenishment policy in EPQ framework.We extends Huang (2006) model by considering finite replenishment rate.We setup three easy-to-use theorems to help the retailer in accurate and rapidly determine the optimal replenishment decisions under minimizing the total cost of inventory system.Numerical results are provided to illustrate the propose solution procedure and to provide managerial insights.(1) The retailer will purchase less quantity, since the replenishment rate is faster enough (2) when retailers storage capacity increasing.It means that the retailer will purchase more quantity.(3) When RW unit stock-holding cost is increasing, the retailer will purchase less quantity.(4) When unit selling price increases the optimal cycle time will decrease the retailer will purchase less quantity.Through sensitive analysis, we show managerial decision for retailer.

Fig. 1 .
Fig. 1.The holding cost of inventory in Rented W.O. and Own W.O.
Fig. 3.The total accumulation of interest earned when or It is observed that as is increased with an increase in values of , , and results and is decreased but increase.The retailer purchased less quantity.2. When holding cost increases with an increase in values of , , and results and is decreased but increases.The retailer purchased less quantity.3.As purchasing cost is increased with increased values of , , and results and is decreased but increases.The retailer purchased less quantity.4. If is increased with increased values of , , and results and is decreased but increases.The retailer purchased less quantity.