Elsevier

Applied Mathematics and Computation

Volume 252, 1 February 2015, Pages 541-551
Applied Mathematics and Computation

An EOQ model with allowable shortage under trade credit in different scenario

https://doi.org/10.1016/j.amc.2014.12.040Get rights and content

Abstract

In present business culture, usually supplier offers a permissible delay in payments to retailer in order to stimulate his demand. However, while developing the inventory model with shortages under permissible delay in payments, it has been observed, in the literature, that researcher have not considered the fact that retailer can earn interest on the revenue generated after fulfilling the outstanding demand as soon as his new consignment arrives at the beginning of the cycle. Thus, the revenue along with the interest earn can be utilized to pay off the amount at the of 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 and the determination of a retailer’s pay off time, after the expiring of credit period, is largely affected by his interest income and interest payable. Owing to these facts, a retailer cost minimization inventory model has been developed for each scenario which jointly optimizes the cycle length and stock-in period. The model has been validated with the help of numerical example. Sensitivity analysis along with its economic interpretation has been also presented.

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 asC(t1,T)=(a)ordering cost+(b)stock holding cost+(c)shortage cost+(d)interest paid-(e)interest earnedwhere

  • (a)

    Ordering cost=A

  • (b)

    Stock holding cost=Dt12h2

  • (c)

    Shortage cost=πD(T-t1)22

  • (d)

    Interest earned

  • (e)

    Interest payable

Now calculate interest paid and earned in the following cases(1)0<Mt1<T,(2)0<t1<MT,(3)0<t1<T<M

Case 1. 0<Mt1

Case 1. 0<Mt1<T

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 belowC11(T,t1)T=1T2A+Dt12h2+πD(T-t1)22+12Ip(cDT-DMp1+12MI-D(T-t1)p(1+MI))2pD-12pDt1-M-cDT-DMp1+12MI-D(T-t1)p(1+MI)pD2Ie-pDt1-Mi-cDT-DMp1+12MIe-D(T-t1)p(1+MIe)pD1+12Iet1-M-cDT-DMp1+12MIe-D(T-t1)p(1+MIe)pDIe(T-t1)+1TπD(T-t1)+IpcDT-DMp1+12MIe-D(T-t1)p(1+MIe)(cD-Dp(1+MIe))pD+pDIet1-M-cDT-DMp

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.

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