An EOQ model with allowable shortage under trade credit in different scenario
Introduction
In recent business environment trade credit is used as a tool by a supplier to encourage the retailer to procure a greater volume of goods, to earn a reasonable profit. On the other hand, trade credit offers a lower unit purchasing cost as well as representing an important source of short-term external finance for retailers. During this period no interest is being charged by the supplier, but beyond this period an interest is charged by the supplier under the terms and conditions agreed upon. Teng [14] illustrated two more benefits of trade credit period, firstly it attracts new buyers who consider it to be a type of price reduction, and secondly it may be applied as an alternative to price discount because it does not provoke competitors to reduce their prices and thereby introduces lasting price reductions. Moreover, the policy of granting credit terms adds not only an additional cost but also an additional dimension of default risk to the supplier (Teng et al. [15]).
In the literature, the extensive use of trade credit as an alternative has been addressed by Goyal [6] who first developed an economic order quantity model under the conditions of permissible delay in payments in which he calculated interest based on the purchasing cost of goods sold within the permissible delay period. Further, Aggarwal and Jaggi [1] developed the inventory model with an exponential deterioration rate under the condition of permissible delay in payments. Jamal et al. [10] further generalized the model with shortages. Chung [3] developed an alternative approach to the Goyal’s [6] problem. Teng [14] modified Goyal’s model by considering the difference between selling price and purchasing cost. Chung and Huang [4] developed an EPQ inventory model for a retailer when the supplier offers a permissible delay in payments. Huang [8] presented a model assuming that the retailer also offers a credit period to his customer which is shorter than the credit period offered by the supplier. Huang [9] extended his earlier model (Huang’s [8]) to investigate the retailer’s inventory policy under two levels of trade credit and limited storage capacity. Ouyang et al. [13] developed a general EOQ model with trade credit and partial backlogging for a retailer to determine its optimal shortage interval and replenishment cycle. Cheng et al. [16] extend the Goyal’s model to develop an Economic Order Quantity (EOQ) model in which the supplier offers the retailer the permissible delay period, and the retailer in turn provides the trade credit period to his/her customers. By assuming that (1) the retailer’s selling price per unit is necessarily higher than its unit cost, and (2) the interest rate charged by a supplier or a bank is not necessarily higher than the retailer’s investment return rate. Recently, Goyal et al. [7] established an appropriate EOQ model for a retailer where the supplier offers a progressive interest charge scheme. Cheng et al. [2] discussed an economic order quantity model with trade credit policy in different financial environment. They discussed the model under the conditions that the interest earned is higher than the interest charged and the interest earned is lower than the interest charged. There are several interesting and relevant papers related to trade credit such as Chang and Dye [5], [11], [12] and many more.
However, generally it is assumed that as soon as the retailer receives his order he first fulfills all his shortages and keeps the remaining units in stock to satisfy the regular demand. Here we have explored that the retailer can earn interest on the revenue generated from fulfilling the shortages in the beginning of the cycle and the revenue along with the interest can be utilize to pay off the amount at the end of the credit period. At this point of time there may be two scenarios, either he has enough amounts to settle the account with the supplier or delay incurring interest charges on the unpaid/overdue balance. Thus the determination of a retailer’s pay off time, after the expiring of credit period, is basically affected by his interest income and interest payments. However later phenomena has attractive the attentions of only few researchers whereas, former phenomena, to the best of my understanding, has so far not been addressed by the researcher. In the present study our objective is to formulate a inventory model with fully backlogged shortages under the permissible delay in payments by not only exploring the impact interest earned from revenue generated after fulfill the shortages at the beginning the of the cycle but also to determine the pay off time for the retailer. A cost minimization problem for different financial scenario, have been presented. The proposed model jointly optimizes the retailer replenishment cycle and stock-in period. Finally, the findings have been illustrated with the help of the numerical example and effects of certain parameters on the retailer ordering policy have been analyzed.
Section snippets
Notations and assumptions
The following notations are used in developing the model
D annual demand rate,
H unit stock holding cost per item per year excluding interest charges,
Ip interest charges per $ investment in stock per year,
Ie interest which can be earned per $ in a year, unit purchase price,
π unit shortage price,
p unit selling price in $, cost of placing one order, payoff time,
M permissible delay in settling accounts,
T inventory cycle length,
S1 maximum positive inventory level,
S2 maximum shortages level,
Q economic order
Mathematical formulation
In this section, some inventory models are developed for each possible case. Our purpose is to minimize the total cost function. The total variable cost per cycle C(t1, T) can be expressed aswhere
- (a)
- (b)
- (c)
- (d)
Interest earned
- (e)
Interest payable
Now calculate interest paid and earned in the following cases
Case 1.
Case 1.
To determine the optimal values of t1 and T, which minimize the total variable cost per unit time. The first and second order partial derivatives of C11(t1, T) and C12(t1, T) with respect to t1 and T are given below
Numerical example
The purposed model of the inventory system has been developed with the help of following numerical examples. The values of the parameters of the model, considered in these numerical examples are not elected from any real life case study, but these values have been seems to be realistic. All these examples have been solved to find optimal values of t1, T, B, Q along with the optimal cost of the system.
In this paper retailer’s perspective is to minimize the total cost. Table 2 reveals that the
Conclusion
Trade credit is widely used as a tool by the supplier to attract the retailers to order more. The present paper incorporates trade credit for an EOQ model with fully backlogged shortages. It is assumed that the retailer may earn interest on the revenue generated through the fulfillment of shortages quantity at the beginning of the cycle. We have also considered the payoff time, the time at which the retailer has to settle the remaining financed amount. So far, this type of consideration has not
Acknowledgement
The authors are grateful to the anonymous referees for their valuable suggestions and comments which helped immensely in improving the paper. The first author would like to acknowledge the support of the Research Grant No.: RC/R&D/2014/6820, provided by the University of Delhi, Delhi, India for conducting this research.
References (16)
- et al.
The retailer’s optimal ordering policy with trade credit in different financial environments
Appl. Math. Comput.
(2012) A theorem on the determination of economic order quantity under conditions of permissible delay in payments
Comput. Oper. Res.
(1998)- et al.
The optimal cycle time for EPQ inventory model under permissible delay in payments
Int. J. Prod. Econ.
(2003) - et al.
Optimal ordering policies when the supplier provides a progressive interest scheme
Eur. J. Oper. Res.
(2007) An inventory model under two-levels of trade credit and limited storage space derived without derivatives
Appl. Math. Model.
(2006)- et al.
Optimal payment time for a retailer under permitted delay of payment by the wholesaler
Int. J. Prod. Econ.
(2000) - et al.
Optimal inventory policy under different supplier credit policies
J. Manuf. Syst.
(1996) - et al.
Optimal pricing and ordering policy under permissible delay in payments
Int. J. Prod. Eco.
(2005)