Optimal retailer’s ordering policies under two-stage
partial trade credit financing in a supply chain
Chandra K. Jaggi*, K.K. Aggarwal and
Mona Verma
Department of Operational Research, Faculty of Mathematical Sciences, New Academic Block,
University of Delhi, Delhi 110007, India Fax: +91 11 27666672 E-mail: [email protected] E-mail: [email protected] *Corresponding author
Abstract: In this paper, an attempt has been made to investigate partial trade credit financing for two levels of supply chain, i.e. the retailer as well as the customer must make a partial payment initially in order to make them eligible for availing the delay period for the rest of their purchases. We have developed the retailer’s inventory system as a cost minimisation problem to determine the retailer’s optimal ordering policies. Further, we have deduced the models presented by Huang (2005, 2003) and Goyal (1985). It has been established numerically that offering partial trade credit in two-stage supply chain is beneficial as per retailer’s perspective. Comprehensive sensitive analysis along with a case study has also been presented.
Keywords: partial trade credit; EOQ; economic order quantity; inventory; supply chain.
Reference to this paper should be made as follows: Jaggi, C.K., Aggarwal, K.K. and Verma, M. (2012) ‘Optimal retailer’s ordering policies under two-stage partial trade credit financing in a supply chain’, Int. J. Industrial and Systems Engineering, Vol. 10, No, 3, pp.277–299.
K.K. Aggarwal is an Assistant Professor in the Department of Operational Research, University of Delhi, India. He obtained his PhD in Inventory Management and Masters in Operational Research from Department of Operational Research, University of Delhi. Prior to his appointment as Assistant Professor, he served as Scientist in National Informatics Centre (India). His research interests and teaching include inventory modelling, financial engineering and network analysis. He has published more than 15 research papers in Int. J. Production Economics, Indian Journal of Mathematics and Mathematical Sciences and Investigacion Operacional Journal.
Mona Verma is a Research Scholar in the Department of Operational Research, Faculty of Mathematical Sciences, University of Delhi, India. She has completed her MPhil in Inventory Management and Masters in Operational Research from the Department of Operational Research, University of Delhi. Currently, she is pursuing her PhD in Operational Research. Her research interest lies in inventory management and supply chain management.
1 Introduction
jointly, for the retailer. Another realistic phenomenon of the supply chain modelling is partial trade credit, i.e. the retailer/customer need to make some partial amount for their purchases in order to make themselves eligible for availing the credit period for the rest of the amount .Huang (2005) developed optimal retailer’s policy for one level where the supplier offers partial trade credit to the retailer. Adding to this, Huang and Hsu (2008) expanded the model where retailer gets full trade credit but offers partial trade credit to his/her customer. Thangam and Uthayakumar (2008) developed the partial trade credit financing in an EPQ model under the same environment.
Unfortunately, none of the researchers have incorporated the partial trade credit for supplier as well as retailer. In this paper, an attempt has been made to investigate partial trade credit financing for a two level of supply chain, i.e. the retailer as well as the customer must make a partial payment initially in order to make them eligible for availing the delay period for the rest of their purchases. We have developed the retailer’s inventory system as a cost minimisation problem to determine the retailer’s optimal ordering policies. Further, we have deduced the models presented by Huang (2003, 2005) and Goyal (1985). It has been established numerically that offering partial trade credit in two-stage supply chain is beneficial from the retailer’s perspective.
2 Assumptions and notations
We formulate a two-echelon supply chain model, for the general case, where the supplier as well as the retailer offers partial trade credit to their downstream mentor. The case of full credit then is just the special case of the general model. The nomenclature is as follows:
D demand rate per year A ordering cost per order c unit purchase price for retailer
p unit selling price for retailer, cdp
h unit stockholding cost per year excluding interest charges
e
I interest earned per $ per year for retailer p
I interest charged per $ in stocks per year for retailer
M retailer’s credit period offered by the supplier for settling the accounts N customer’s credit period offered by the retailer for settling the account D the percentage of permissible delay in payments for retailer, 0d dD 1 E the percentage of permissible delay in payments for customer, 0d dE 1 T cycle time in years
The following assumptions are made in the model: 1 Demand rate is known and constant.
2 Shortages are not allowed. 3 Time horizon is infinite.
4 The supplier offers a partial trade credit period M to the retailer. As the order is received, the retailer has to make a partial payment of (1D)cDT to the supplier. During this period, the retailer earns interest
I
e( , ,
d t
I
p)
on the revenue generated from sales at the selling price p!c, whereas if the account is not settled at M, then the supplier charges interest Ip on the remaining stock at the purchase price .c5 The retailer also offers a partial trade credit period N to his/her customer, i.e. his/her customer must make a partial payment EpDT to the retailer, and they must pay off the rest of the amount at the end of the trade credit period N(independent of M), which is offered to him by the retailer. Now the retailer earns interest Ie on the partial payment, which he receives from his/her customer.
3 The retailer’s problem
The retailer wants to stimulate his sales by offering partial trade credit E to his/her customer, while making himself eligible for the same by giving an initial payment on (1D) units to the supplier .The objective is to minimise annual total cost, which comprises of the following elements:
Annual purchasing cost = cD Annual ordering cost = A/T
Annual stockholding cost (excluding interest charges) = DTh/2
The annual total relevant cost for the retailer can be expressed as shown below: TRC(T) = Annual purchase cost + Annual ordering cost + Annual stockholding cost + Annual interest payable – Annual interest charged
After the credit period M, the retailer has to make payment on the items in stock at the interest Ip and he will earn interest at the rate Ie if the customer does not pay for the items at the specified credit period N offered by him. The two situations that may arise are as follows:
1 M tN
2 M N
Case 1 M tN: According to assumption 4 and 5, there are four subcases to occur for interest payable per year and interest earned per year for the retailer.
1
1 M
N M T
D
d d d
In this case, since the retailer makes a partial payment of (1D) units to the supplier, the interest paid, therefore, will be on the balance amount of purchases made by him.
2/ 2
Annual interest payable cIp DT DTM T
Annual interest earned e
DN DN DM M N
Therefore, the annual total relevant cost will be
2 2Annual interest earned pIe DN DN DM M N T
Thus, the annual total relevant cost is given by
Annual interest payable p
Hence, the annual total relevant cost will be
Annual interest payable cIp DT T
Annual interest earned e
Figure 3 Nd dT M
Figure 4 TdN
Consequently, the annual total relevant cost is given as
2
4
(1 )
TRC ( ) (1 )
2 p 2 e 2
A DTh T T
T cD cI D pI D M N
T
D E E
ª º
« »
¬ ¼ (4)
From the above arguments, the annual total relevant cost for the retailer can be expressed as
1
2
3
4
TRC ( ) if (5a)
1
TRC ( ) if (5b)
TRC( )
1
TRC ( ) if (5c)
TRC ( ) if (5d)
M
T T
M
T M T
T
T N T M
T T N
D
D
d
°
°
° d d
®
°
d d °
° t
Since TRC M1( / (1D)) TRC M2( / (1D)), TRC (2 M) TRC (3 M) and TRC ( )3 N 4
Therefore, Equation (5a)–(5d) imply that TRC( )T is convex on T > 0 if [1!0. Then we
can obtain the following results. Theorem 1:
Annual stockholding cost (excluding interest charges) = DTh/2
According to assumption 5 and assumption, there are two cases to occur in interest payable per year and interest earned per year, i.e.
1 MdT
Annual interest earned ( )
2
Figure 5 Interest earned when MdT
Figure 6 Interest earned when MtT
TRC(T) = Annual purchase cost + Annual ordering cost + annual stockholding cost + Annual interest payable – Annual interest charged
where
5
6
TRC ( ) if (14a)
TRC( )
TRC ( ) if (14b)
T M T
T
T M T
d
® t
¯
where
2
2 2
5
(1 ) ( )
TRC ( )
2 2 2 2
e p
pI D M
A DTh T T M
T cD cI D
T T T
E D
ª º
« »
and
We substitute Equation (23) into *
3 ,
T given by Equation (24) becomes the optimal cycle time *
T if and only if *
4 .
T dN
We substitute Equation (24) into *
2 2
3 2A N D h cIp(1
D
) pIeE
' (27)
Then we have ' ! ' ! '1 2 3 from Equations (25)–(27). Summarising the above arguments, the optimal cycle time *
T can be obtained as follows. substitute Equation (28) into *
5.
5 Special
cases
Equation (37a)–(37c) will be consistent with Equation (5a)–(5d), in Huang (2005), respectively. Equations (25) and (26) can be modified as
Thus, Theorem 2 reduces to:
Theorem 4 has been discussed in Theorem 2 of Huang (2005). Hence, Huang (2005) will be a special case of this paper.
5.2 Huang (2003) model
When M tN, p c, D 1,E 0 (which means supplier as well as retailer offers full trade credit to his/her customer), let
2 2Equation (44a)–(44c) will be consistent with Equation (5a)–(5d), in Huang (2003), respectively. Equations (26) and (27) can be modified as
2Theorem 5:
Theorem 5 has been discussed in Theorem 1 of Huang (2003). Hence, Huang (2003) will be a special case of this paper.
5.3 Goyal (1985) model
When N = 0, which means the supplier would offer the retailer a delay period but the retailer would not offer the delay period to his/her customer. Therefore, when p c, and
1
Equation (49a) and (49b) will be consistent with Equation (5a)–(5d), in Goyal (1985), respectively. Equation (26) can be modified as 2 2 ( ),
e
6 Computational
analysis
The purposes of this numerical analysis are: 1 to obtain optimal solutions for the two cases
2 to compare the results of present analysis with the previous research work 3 to use sensitivity analysis to highlight the influence of model parameters.
6.1 Numerical
analysis
Case 1 M tN
Example 1:To gain more insight into the above theory, we borrowed Huang and Hsu (2008) example, which is stated as:
A = 80, D = 2000, c = 10, p = 30, h = 7, I = 0.15,p I = 0.13, M = 0.1, N = 0.08, e
0.2
E and D= 0.09 in appropriate units.
The optimal value of T* and *
Q for the present study are T* 0.09454 year and * 1 89
Q units.
Further, the comparative analysis of proposed model with the previous research work has been summarised in Table 1.
It is clearly apparent from Table 1 that by implementing partial trade credit policy, the retailer is able to increase his ordering quantity, which implies that he is able to attract more customers, which otherwise was not possible, as evident from the Table 1. Hence, offering partial trade credit at two levels of the supply chain is beneficial for the retailer. Case 2 M N
Example 2: Let A = 80, D = 5000, c = 10, h =10, Ip 0.1, Ie 0.2, M = 0.05, N=0.06
and E 0.2, D 0.1 in appropriate units.
The optimal value of T*and Q* for the present model are *
0.057023
T year and
*
285 Q units.
The sensitivity analysis on different parameters has been presented in Section 6.2.
6.2 Sensitivity
analysis
In any decision-making situation, the change in the values of parameters may happen due to uncertainties. To examine the implications of these changes, the sensitivity analysis will be of great help in decision-making.
Firstly, we investigate the changing effects of parameters ,D p and N for fixed ,E when M tN in Table 2. Then Table 3 shows the effect of changes in ,E p and N for fixed D for the same case.
From Tables 2 and 3, we have the following inferences:
x For fixed N and ,D when p increases, the optimal cycle length for the retailer decreases, which in a way increases his inventory turnover. This implies that the retailer will be ordering more frequently, which eventually helps him to generate more revenue on the initial payment received from the customer.
x For fixed p and ,D as N increases, i.e. the credit period offered by the retailer to his customer increases, retailer’s cycle length increases so as his order quantity. This implies that the retailer is able to attract more and more customers, and he will be able to generate more revenue on the initial payment received from the customers. Lastly, for the case M N, Table 4 examines the changes in the parameter ,D p and N for fixed .E It is revealed from Table 4 that for fixed N and (1D ), as p increases, there is a decrease in the annual total cost.
Table 1 Comparative analysis
Research article T* Q*
Present study 0.09454 189
Huang (2005) 0, N E 0
0.0812 162
Huang (2003) ,
p c D 1,E 0
0.08648 173
Goyal (1985) ,
p c N 0,D 1,E 0
0.08648 173
Table 2 Optimal solutions when M tN, E fixed
1D N p *
T Q* TRC T( *)(in $)
0.9 0.02 10 0.091 183 21,489.47
30 0.081 164 21,204.26
50 0.074 149 20,898.01
0.05 10 0.093 186 21,513.06
30 0.085 170 21,282.71
50 0.079 159 21,039.65
0.08 10 0.095 191 21,556.12
30 0.090 182 21,421
50 0.087 175 21,282.18
0.5 0.02 10 0.096 193 21,410.46
30 0.084 170 21,134.23
50 0.076 154 20,834.35
0.05 10 0.097 195 21,432.99
30 0.088 176 21,209.92
Table 2 Optimal solutions when M tN, E fixed (continued)
1D N p *
T Q* TRC T( *)(in $)
0.08 10 0.099 200 21,474.06
30 0.094 188 21,343.3
50 0.090 181 21,207.4
0.1 0.02 10 0.098 197 21,375.43
30 0.086 173 21,103.42
50 0.077 156 20,806.48
0.05 10 0.099 199 21,397.49
30 0.089 179 21,177.89
50 0.082 166 20,942.23
0.08 10 0.10 204 21,437.71
30 0.09 191 21,309.2
50 0.09 183 21,174.65
Table 3 Optimal solutions when M tN, D fixed
E N p T* Q* TRC T( *)(in $)
0.2 0.02 10 0.098 197 21,375.43
30 0.086 173 21,103.42
50 0.077 156 20,806.48
0.05 10 0.099 199 21,397.49
30 0.089 179 21,177.89
50 0.082 166 20,942.23
0.08 10 0.102 204 21,437.71
30 0.095 191 21,309.2
50 0.091 183 21,174.65
0.4 0.02 10 0.098 197 21,374.37
30 0.086 172 21,099.8
50 0.077 155 20,799.79
0.05 10 0.099 199 21,390.96
30 0.088 177 21,156.03
50 0.081 163 20,902.7
0.08 10 0.101 202 21,421.32
30 0.093 187 21,256.3
Table 3 Optimal solutions when M tN, D fixed (continued)
E N p *
T Q* TRC T( *)(in $)
0.6 0.02 10 0.098 196 21,373.32
30 0.085 172 21,096.18
50 0.077 155 20,793.09
0.05 10 0.098 198 21,384.4
30 0.087 175 21,133.92
50 0.080 160 20,862.45
0.08 10 0.100 200 21,404.78
30 0.099 182 21,202.1
50 0.084 169 20,985.69
Table 4 Optimal solutions when M N, E fixed
1D p T* Q* TRC T( *)(in $)
0.9 10 0.054988 275 52,885.93
30 0.057023 285.1 52,772.93 50 0.04998 250 52,701.44
0.5 10 0.056389 282 52,807.3
30 0.05859 293 52,691.03 50 0.061072 305.3 52,569.56
0.1 10 0.057023 285.1 52,772.93
30 0.059303 296.5 52,655.17 50 0.06188 309.3 52,531.99
7 Conclusion
A future study can be incorporated in the present model into more realistic assumptions such as probabilistic demand, allowable shortages, deteriorating items or finite replenishment rate.
Acknowledgements
The authors really appreciate the constructive comments by the anonymous referees, which greatly helped in improving this paper further. The first author would like to acknowledge the support of Research Grant No. Dean (R)/R&D/2009/487, provided by University of Delhi, Delhi, India, for conducting this research. The third author would like to thank University Grant Commission (UGC) for providing the fellowship to accomplish the research.
References
Abad, P.L. and Jaggi, C.K. (2003) ‘A joint approach for setting unit price and the length of the credit period for seller when end demand is price sensitive’, Int. J. Production Economics, Vol. 83, pp.115–122.
Aggarwal, S.P. and Jaggi, C.K. (1995) ‘Ordering policies of deteriorating items with permissible delay in payments’, Journal of Operational Research Society, Vol. 46, pp.658–662.
Chang, H.J. and Dye, C.Y. (2001) ‘An inventory model for deteriorating items with partial backlogging and permissible delay in payments’, Int. J. System Science, Vol. 32, pp.345–352. Chung, K.J. (1998) ‘A theorem on the determination of economic order quantity under conditions
of permissible delay in payments’, Computers and Operations Research, Vol. 25, pp.49–52. Chung, K.J., Goyal, S.K. and Huang, Y.F. (2005) ‘The optimal inventory policies under
permissible delay in payments depending on the ordering quantity’, Int. J. Production Economics, Vol. 95, pp.203–213.
Chung, K.J. and Liao, J.J. (2006) ‘The optimal ordering policy in a DCF analysis for deteriorating items when trade credit depends on the order quantity’, Int. J. Production Economics, Vol. 100, pp.116–130.
Goyal, S.K. (1985) ‘EOQ under conditions of permissible delay in payments’, Journal of Operational Research Society, Vol. 36, pp.335–338.
Huang, Y.F. (2003) ‘Optimal retailer ordering policies in the EOQ model under trade credit financing’, Journal of Operational Research Society, Vol. 54, pp.1011–1015.
Huang, Y.F. (2005) ‘Retailer’s inventory policy under supplier’s partial trade credit policy’, Journal of the Operations Research Society of Japan, Vol. 48, pp.173–182.
Huang, Y.F. (2007) ‘Optimal retailer’s replenishment decisions in the EPQ model under two levels of trade credit policy’, European Journal of Operational Research, Vol. 176, pp.1577–1591. Huang, Y.F. and Hsu, K.H. (2008) ‘An EOQ model under retailer partial trade credit policy in
supply chain’, Int. J. Production Economics, Vol. 112, pp.655–664.
Hwang, H. and Shinn, S.W. (1997) ‘Retailers pricing and lot sizing policies for exponentially deteriorating products under the condition of permissible delay in payments’, Computers and Operations Research, Vol. 24, pp.539–547.
Jamal, A.M.M., Sarkar, B.R. and Wang, S. (2000) ‘Optimal payment time for a retailer under permitted delay in payment by the wholesaler’, Int. J. Production Economics, Vol. 66, pp.59–66.
Jaggi, C.K., Goyal, S.K. and Goel, S.K. (2008) ‘Retailer’s optimal replenishment decisions with credit-linked demand under permissible delay in payments’, European Journal of Operational Research, Vol. 190, pp.130–135.
Appendix
Case Illustration
A retailer purchases medicine from a pharmaceutical company at c = $100 per unit and sells it at p = $120 per unit. The company offers a partial credit (i.e. D 0.09) for the time period M = 0.1 year. However, if the payment is not made within the specified period then 15% interest (i.e. Ip 0.15) is charged on the outstanding amount. To avoid default risks, the retailer also offers a partial trade credit (i.e. E 0.02 and N = 0.05 year) to the credit risk customers. The retailer earns interest at the rate Ie 10% on the revenue generated. Also, the annual demand for the retailer is D = 2,000 units with holding cost h = $5 per unit per year and the ordering cost A= $250 per order.
Analysis
Using the above proposed model, it is apparent that if both the pharmaceutical company as well as the retailer offers partial trade credit to its subsequent downstream member, then the optimal cycle length and order quantity for the retailer is T* 0.1418 year and *
283,
Q respectively. However, when the retailer does not offer any credit period (i.e. E = 0 and N = 0) to his/her customer, then his cycle length and order quantity becomes T* 0.1409 year and Q* 281 units, respectively, which implies that the
