Subscribe Now Subscribe Today
Research Article
 

A Cash Flow Oriented EOQ Model with Deteriorating Items Under Permissible Delay in Payments



Kuo-Lung Hou and Li-Chiao Lin
 
Facebook Twitter Digg Reddit Linkedin StumbleUpon E-mail
ABSTRACT

This study develops an inventory model to determine an optimal ordering policy for deteriorating items with delayed payments permitted by the supplier under inflation and time discounting. This study applies the discounted cash flows approach for problem analysis. Mathematical models have been derived for obtaining the optimal cycle time and optimal payment time for item so that the annual total relevant cost is minimized. The present value of the annual total relevant cost in this inventory system is developed first, then an optimal number of replenishment, cycle time and order quantity are obtained by a solution procedure. Finally, a numerical example is given to illustrate the results.

Services
Related Articles in ASCI
Similar Articles in this Journal
Search in Google Scholar
View Citation
Report Citation

 
  How to cite this article:

Kuo-Lung Hou and Li-Chiao Lin, 2009. A Cash Flow Oriented EOQ Model with Deteriorating Items Under Permissible Delay in Payments. Journal of Applied Sciences, 9: 1791-1794.

DOI: 10.3923/jas.2009.1791.1794

URL: https://scialert.net/abstract/?doi=jas.2009.1791.1794
 

INTRODUCTION

The traditional EOQ model assumes that retailer must pay for the items as soon as the items are received. However, in real-life situations, the supplier may offer the retailer a delay period, that is trade credit period, in paying for the amount of purchasing cost. The retailer can sell the goods to accumulate revenue and earn interest before the end of trade credit period. But if the payment is delayed beyond that period, a higher interest will be charged. Such a convenience is likely to motivate customer to order more quantities because paying later indirectly reduces the purchase cost. Therefore, trade credit is an important source of financing for intermediate purchasers of goods and services and plays a large role in our economy. Recently, Hou and Lin (2008) considered an ordering policy with a cost minimization procedure for deterioration items under trade credit and time discounting. Other many related articles can be found by Goyal (1985), Aggarwal and Jaggi (1995), Jamal et al. (1997), Sarker et al. (2000), Chung et al. (2005), Chung and Liao (2006), Ouyang et al. (2006) and Huang (2007). On the other hand, many researchers have studied inventory models for deteriorating items such as volatile liquids, blood banks, medicines, electronic components and fashion goods. The analysis of deteriorating inventory problems began with Ghare and Schrader (1963), who developed a simple Economic Order Quantity (EOQ) model with a constant rate of deterioration. Since then, many related articles could be found by Heng et al. (1991), Sarker et al. (1997) and Balkhi and Benkherouf (2004). While determining the optimal ordering policy, the effects of inflation and time value of money cannot be ignored. The pioneer research in this direction was Buzacott (1975), who developed an EOQ model with inflation subject to different types of pricing policies. Other related articles can be found in Misra (1979) and Ray and Chaudhuri (1997), Liao et al. (2000) and Chung and Lin (2001).

This study develops a deterministic inventory model when a delay in payments is permissible. The effects of the inflation, deterioration and delay in payments are discussed. Moreover, an algorithm is developed to obtain the optimal the number of replenishment, cycle time and order quantity.

NOTATION AND MODEL

The following notation is used throughout the study:

H : Length of planning horizon
T : Replenishment cycle time
n : No. of replenishment during the planning horizon; n = H/T
Q : Order quantity, units/cycle
D : Demand rate per unit time, units/unit time
A : Ordering cost at time zero, $/order
c : Per unit cost of the item, $/unit
h : Inventory holding cost per unit per unit time excluding interest charges, $/unit/unit time
θ : Deterioration rate, a constant fraction of the on-hand inventory, units/unit time
r : Discount rate represent the time value of money
f : Inflation rate
R : The net discount rate of inflation
R : r-f
Ie : The interest earned per dollar per unit time
Ic : The interest charged per dollar in stocks per unit time by the supplier, Ic≥Ie
M : The permissible delay in settling account (i.e., the trade credit period)

Let I(t) be the inventory level during the first replenishment cycle. This inventory level is depleted by the effects of demand and deterioration. So, the variation of I(t) with respect to t is governed by the following differential equation:

Image for - A Cash Flow Oriented EOQ Model with Deteriorating Items Under Permissible Delay in Payments
(1)

with the boundary condition I(T) = 0. The solution of (1) can be represented by:

Image for - A Cash Flow Oriented EOQ Model with Deteriorating Items Under Permissible Delay in Payments
(2)

Consequently, initial inventory after replenishment becomes:

Image for - A Cash Flow Oriented EOQ Model with Deteriorating Items Under Permissible Delay in Payments
(3)

The present value of the total replenishment costs is given by:

Image for - A Cash Flow Oriented EOQ Model with Deteriorating Items Under Permissible Delay in Payments
(4)

and the present value of total purchasing costs is given by:

Image for - A Cash Flow Oriented EOQ Model with Deteriorating Items Under Permissible Delay in Payments
(5)

The present value of the total holding costs over the time horizon H is given by:

Image for - A Cash Flow Oriented EOQ Model with Deteriorating Items Under Permissible Delay in Payments
(6)

Case I: M≤T = H/n

In this case, the present value of the interest payable during the first replenishment cycle is:

Image for - A Cash Flow Oriented EOQ Model with Deteriorating Items Under Permissible Delay in Payments
(7)

hence, the present value of the total interest payable over the time horizon H is:

Image for - A Cash Flow Oriented EOQ Model with Deteriorating Items Under Permissible Delay in Payments
(8)

Next, the present value of the interest earned during the first replenishment cycle is:

Image for - A Cash Flow Oriented EOQ Model with Deteriorating Items Under Permissible Delay in Payments
(9)

hence, the present value of the total interest earned over the time horizon H is:

Image for - A Cash Flow Oriented EOQ Model with Deteriorating Items Under Permissible Delay in Payments
(10)

Therefore, the total present value of the costs over the time horizon H is:

Image for - A Cash Flow Oriented EOQ Model with Deteriorating Items Under Permissible Delay in Payments
(11)

Case II: M > T = H/n

The interest earned in the first cycle is the interest earned during the time period (0, T) plus the interest earned from the cash invested during the time period (T, M) after the inventory is exhausted at time T and it is given by:

Image for - A Cash Flow Oriented EOQ Model with Deteriorating Items Under Permissible Delay in Payments
(12)

hence, the present value of the total interest earned over the time horizon H is:

Image for - A Cash Flow Oriented EOQ Model with Deteriorating Items Under Permissible Delay in Payments
(13)

Since, the replenishment cost, purchasing cost and inventory holding cost over the time horizon H are the same as Case I, the total present value of the costs, TVC2(s, n), is given by:

Image for - A Cash Flow Oriented EOQ Model with Deteriorating Items Under Permissible Delay in Payments
(14)

At M = T = H/n, we find TVC1(n) = TVC2(n). Consequently, we have:

Image for - A Cash Flow Oriented EOQ Model with Deteriorating Items Under Permissible Delay in Payments

where, TVC1(n) and TVC2(n) as expressed in Eq. 11 and 14, respectively.

ALGORITHM

The following algorithm is developed to derive the optimal n, T, Q and TVC(n) values:

Step 1: Start by choosing a discrete variable n, where n is any integer number equal or greater than 1.

Step 2: If T = H/n≥M for different integer n values, derive TVC1(n) from (11); If T = H/n≤M for different integer n values, derive TVC2(n) from Eq. 14.

Step 3: Repeat Step 1 and 2 for all possible n values with T = H/n≥M until the minimum TVC1(n) is found from Eq. 11 and let n1* = n. For all possible n values with T = H/n≥M until the minimum TVC2(n) is found from Eq. 14 and let n2* = n. The n1*, n2*, TVC1(n*) and TVC2(n*) values constitute the optimal solution and the satisfy the following conditions:

Image for - A Cash Flow Oriented EOQ Model with Deteriorating Items Under Permissible Delay in Payments
(15)

Image for - A Cash Flow Oriented EOQ Model with Deteriorating Items Under Permissible Delay in Payments
(16)

Where:

Image for - A Cash Flow Oriented EOQ Model with Deteriorating Items Under Permissible Delay in Payments

and

Image for - A Cash Flow Oriented EOQ Model with Deteriorating Items Under Permissible Delay in Payments

Step 4: Select the optimal number of replenishment n* such that:

Image for - A Cash Flow Oriented EOQ Model with Deteriorating Items Under Permissible Delay in Payments

Hence, optimal order quantity Q* is obtained by substituting n* into (3) and optimal cycle time, T*, is T* = H/n*.

NUMERICAL RESULTS

An example is devised here to illustrate the results of the general model developed in this study with the following data.

The demand rate, D = 600 unit /year, the replenishment cost, A =$80/order, the holding cost excluding interest charges, h = $2.4/unit/year, the per unit item cost, c = $15/unit, the constant rate of deterioration, θ = 0.15, the net discount rate of inflation, R = $0.12/$/year, the interest charged per $ in stocks per year by the supplier, Ic = $0.18/$/year, the interest earned per $ per year, Ie = $0.16/$/year and the planning horizon, H, is 5 year. The permissible delay in settling account, M = 60 days = 60/360 years (assume 360 days per year). Using the solution procedure, we have the computational results shown in Table 1. We find the Case I is optimal option in credit policy. From the case, the minimum total present value of costs is found when the number of replenishment, n, is 23. With 23 replenishments, the optimal cycle time T is 0.217 year, the optimal order quantity, Q = 132.59 units and the optimal total present value of costs, TVC = $36296.70.

Table 1: The numerical results
Image for - A Cash Flow Oriented EOQ Model with Deteriorating Items Under Permissible Delay in Payments
*Optimal solution

CONCLUSION

This study develops a deterministic inventory model for deteriorating items over a finite planning horizon when the supplier provides a permissible delay in payments. The model considers the effects of deterioration, inflation and permissible delay in payments. Based on the DCF approach we permit a proper recognition of the financial implication of the opportunity cost in inventory analysis. In addition, we have presented an optimal solution procedure to find the optimal number of replenishment, cycle time and order quantity to minimize the total present value of costs. Finally, a numerical example is given to illustrate the results. Further research can be done for case with stochastic market demand when the supplier provides a permissible delay in payments and cash discount.

ACKNOWLEDGMENT

This research was partially supported by the National Science Research Council of Taiwan under Grant NSC 96-2416-H-240-002-MY2.

REFERENCES
1:  Aggarwal, S.P. and C.K. Jaggi, 1995. Ordering policies of deteriorating items under permissible delay in payments. J. Operational Res. Soci., 46: 658-662.
Direct Link  |  

2:  Balkhi, Z.T. and L. Benkherouf, 2004. On an inventory model for deteriorating items with stock dependent and time-varying demand rates. Comput. Operat. Res., 31: 223-240.
CrossRef  |  Direct Link  |  

3:  Buzacott, J.A., 1975. Economic order quantities with inflation. Operat. Res. Q., 26: 553-558.
Direct Link  |  

4:  Chung, K.J., S.K. Goyal and Y.F. Huang, 2005. The optimal inventory policies under permissible delay in payments depending on the ordering quantity. Int. J. Prod. Econ., 95: 203-213.
Direct Link  |  

5:  Chung, K.J. and C.N. Lin, 2001. Optimal inventory replenishment models for deteriorating items taking account of time discounting. Comput. Operat. Res., 28: 67-83.
Direct Link  |  

6:  Chung, K.J. and J.J. Liao, 2006. The optimal policy in a discounted cash-flows analysis for deteriorating items when trade credit depends on the order quantity. Int. J. Prod. Econ., 100: 116-130.
Direct Link  |  

7:  Ghare, P.M. and G..F. Schrader, 1963. A model for exponential decaying inventory. J. Ind. Eng., 14: 238-243.

8:  Goyal, S.K., 1985. Economic order quantity under conditions of permissible delay in payments. J. Operat. Res. Soc., 36: 335-338.
Direct Link  |  

9:  Heng, K.J., J. Labban and R.J. Linn, 1991. An order-level lot-size inventory model for deteriorating items with finite replenishment rate. Comput. Ind. Eng., 20: 187-197.
CrossRef  |  Direct Link  |  

10:  Hou, K.L. and L.C. Lin, 2008. An ordering policy with a cost minimization procedure for deterioration items under trade credit and time discounting. Accepted J. Stat. Manage. Syst.

11:  Huang, H.F., 2007. Economic order quantity under conditionally permissible delay in payments. Eur. J. Operat. Res., 176: 911-924.
Direct Link  |  

12:  Misra, R.B., 1979. A note on optimal inventory management under inflation. Naval Res. Logist., 26: 161-165.
CrossRef  |  

13:  Ray, J. and K.S. Chaudhuri, 1997. An EOQ model with stock-dependent demand, shortage, inflation and time discounting. Intl. J. Prod. Econ., 53: 171-180.
CrossRef  |  

14:  Sarker, B.R., A.M.M. Jamal and S. Wang, 2000. Supply chain models for perishable product under inflation and permissible delay in payments. Comput. Operat. Res., 27: 59-75.
CrossRef  |  

15:  Jamal, A.A.M., B.R. Sarker and S. Wang, 1997. An ordering policy for deteriorating items with allowable shortage and permissible delay in payments. J. Operational Res. Soc., 48: 826-833.
Direct Link  |  

16:  Liao, H.C., C.H. Tsai and C.T. Su, 2000. An inventory model with deteriorating items under inflation when a delay in payments is permissible. Int. J. Prod. Econ., 63: 207-214.

17:  Ouyang, L.Y., K.S. Wu and C.T. Yang, 2006. A study on an inventory model for non-instaneous deteriorating items with permissible delay in payments. Comput. Ind. Eng., 51: 637-651.
CrossRef  |  

18:  Sarker, B.R., S. Mukherjee and C.V. Balan, 1997. An order-level lot size inventory model with inventory-level dependent demand and deterioration. Int. J. Prod. Econ., 48: 227-236.
CrossRef  |  

©  2021 Science Alert. All Rights Reserved