An EOQ Model With Exponential Demand Rate Under Cash Discount And Permissible Delay In Payment

An EOQ Model With Exponential Demand Rate Under Cash Discount And Permissible Delay In Payment

Sushil Kumar* & U. S. Rajput

Department of Mathematics & Astronomy, University of Lucknow Lucknow -226007. U.P. India.

Corresponding Address*

Abstract

In the present paper we developed an EOQ model with exponential demand rate together with an optimal ordering policy under cash discount and permissible delay in payment to the customer. Sensitivity analysis is given with the effect of parameters in the optimal solution, since in the classical inventory models it is assumed that the customer pays to the supplier as soon as he received the items and in such cases the supplier offer a cash discount or a permissible delay to the customer.

Keywords Inventory, Cash discount, Payment Delay

1. Introduction

In the classical inventory models payment for the items paid by the suppliers depend on the payment paid by the customers and in such cases the supplier provides a fixed credit period to the customers during which no interest will be charged from the customers, after this credit period up to the end of a period interest charged paid by the customers. In such situations the customer starts to accumulate revenue on his sale and earn interest on his revenue. Goyal [1985] developed an EOQ model together with the condition of permissible delay in payments. Goyals model was extended by

Aggarwal and Jaggi [1995] with the consideration

EOQ model for deteriorating items under supplier credits when demand is stock dependent. In the earlier inventory problems discussed under the conditions of permissible delay in payment, the supplier provides not only a fixed credit period to settle the account but also gives the cash discount offer to the customers in the business market. In his model Goyal assume that the unit purchase cost of an item is equal to the selling price per unit. Chung- Tao- Chang extended the Goyals model under the assumption of constant demand with cash discount and the difference between unit price and unit purchase cost of the product.

In the present paper we developed an EOQ model with exponential demand rate together with an optimal ordering policy under cash discount and permissible delay in payment to the customers. Sensitivity analysis is given to study the effect of variation of parameters in the optimal solution.

2. Assumptions and Notations

We consider the following assumptions

1. The demand rate is R(t) eat , where

   0 a 1

2. h is the unit holding cost per unit time excluding interest charges.

3. p is the selling price per unit of the product.

4. c is the unit purchase cost.

   of deteriorating items. Further this model was

5. IC

   is the interest charged per $ per unit time in

   generalized by Jamal et al [1997] by allowing shortages also some interesting results on this study are given by Chung [1998], Sarkar et al [2000] and Teng [2001].Soni and Shah [2005] developed a mathematical model with constant rate of deterioration and the scenario of progressive credit periods, further this model was extended by Soni et al [2006] under the effect of inflation. Soni and Shah [2008] studied the Levin et al [1972] model. Nita H. Shah, Poonam Mishra [2010] developed an

   stock by the supplier.

6. Ie is the interest earned per $ per unit time.

7. S is the ordering cost per order.

8. Q is the order quantity or order size.

9. r is the cash discount rate.

10. M is the period of cash discount.

11. N is the period of permissible delay in setting account with N > M.

12. T is the inventory cycle length.

13. Shortages are not allowed.

14. The replenishment rate is infinite.

15. Time horizon is infinite.

16. The account is not settled during the time, generated sales revenue is deposited in an interest bearing account.

    At the end of the period the customer pays off all units sold, keep profit and starts paying for the interested charges on the items in stocks

17. I(t) be the inventory level at any time t.

18. Z ( T ) is the total variable cost per unit time.

3. Mathematical Formulations

The instantaneous inventory level at any time t is given by the differential equation

dI (t) eat ,

dt

0 t T

. (1)

Then the cash discount per unit time is k

With boundary conditions

I (0) I0 and I (T ) 0

r c Q

The solution of (1) is

I (t) 1 ea t eaT

a

.. (2)

rc

k

Ta

T

(1 Exp[aT ]) (6)

And the order quantity is

Q 1

1 eaT

The interest payable per unit time is

cI T

a

(3)

S

I C

C

C

1 T

I (t)dt

M

The ordering cost is

OC T

h T

(4)

cIC

T a2

1 a (T M )eaT

• ea M

  The holding cost is

  HC I (t) dt

  T

  T

  0

  (7) The interest earned per unit time is

  pI T

  h T a2

  eaT 1 a T 1 (5)

  I e tea t dt

  E

  E

  T

  T

  1

  0

  Due to cash discount, interest charged and interest earned we consider the following four cases

  Case 1 The payment is paid at M to get a cash

  p Ie

  T a2

  1 (a M 1) ea M

  discount together with T M

  Case 2 The customer pays in full at M to get a cash discount together with T<M

  Case 3 The payment is paid at N to get the permissible delay together with T N

  Case 4 The customer pays in full at N together with

  .. (8)

  So the total variable cost per unit time is

  Z (T ) S h eaT 1 aT 1 cIC 1 a (T M )eaT ea M

  1 T Ta2 Ta2

  p Ie 1 (aM 1)ea M rc 1 eaT

  T<N

  Now we find the total variable cost in each of the cases

  Case 1 Since in this case payment is paid at M and

  T M

  Ta2

  Ta

  .. (9)

  For the optimum value of

  Z

  T1

  T1

  2 Z

  Z1 (T ),

  1 0 gives T T1

  T

  for which 1 0 , T

  T 2

  Case 2 Since in this case the customer pays in full at M and T<M

  Then there is zero cash discount and the interest charged per unit time is

  cI T

  C

  C

  I C I (t) dt

  3 T

  So there is zero interest charge and the cash discount is same as that in case 1, then the interest earned per unit time is

  pI T T

  N

  cIC 1 a (T N )eaT eaN

  Ta2

  . (12)

  IE2

  e

  T

  tea t dt

  Teat dt

  The interest earned per unit time is

  pI N

  0

  pI

  M

  T eaT

  1

  IE3

  e t eat dt

  T

  pI T

  e (2eaT eaM )

  T M a2

  2

  a

  0

  pIe 1 (aN 1)ea N

  Ta2

  I e tea t

  E

  E

  2

  2

  T 0

  .. (10)

  pIe (2ea

  Therefore the total variable cost per unit time is

  T

  . (13)

  Therefore total variable cost per unit time is

  Z (T ) S h eaT (1 aT ) 1

  S h aT

  2 T Ta2

  Z3 (T ) T

• e

  2

  2

  Ta

  (1 aT ) 1

  pI 1

  eaT T

  aT aM rc aT cI

  e

  (2e

  e )

  (1 e )

  C 1 a (T N )eaT eaN

  T a2 a2 a Ta

  . (11)

  Ta2

  • pIe 1 (aN 1)ea N

Ta2

For the optimum value of

.. (14)

Z (T ), Z2 0 gives T T

for which

For the optimum value of

2 T 2

2 Z

Z (T ), Z3 0 gives T T

for which

2

T 2

0, T T2

3 T 3

2 Z

Case 3 Since in this case payment is paid at N and

3 0,

T 2

T T3

T N ,

Case 4 Since in this case the customer pays in full at N and T<N

Z1 S

T T 2

pI M 2

h cIC e

T 2

Z1 0, gives T 2 (cI

h) (S pI M 2 ) 0

T 1 C

e

.. (18)

2 Z

1

T 2

2 (S

T 3

pIe M

2 ) 0

T T1

And the condition T1 M

gives

2

2

[ pIe (h cIC )] M S

(19)

Z (T ) S hT pI

2 T e

2T M rc

Then there is no interest charged and the interest earned per unit time is

Z2 S

(20)

)M 2

then

T* T1

If S ( pIe h cIC

)M 2 then T* M

e

e

C

C

If SM 2 pI M 2 cI

If SM 2 pI M 2 cI

a2 p M 2 N 2then T* T

3

3

a2 p M 2 N 2then T* T

e C 3

Since T2 & T4 comes out to be imaginary so these two do not give the optimum value of T, If

Z1 (T1 ) Z3 (T3 ) then either T* T1

or T* T3

6. Numerical Results

We consider the parametric values of the parameters in appropriate units as

3, 0.1, 0.3, 0.1, 4, 20, 15, 0.1]

Table 1

Effect of ordering cost on optimal solution

As interest charged increases, the optimal cycle length decreases and total variable cost decreases (Negative sign comes because purchase cost is not taken into account)

7.Conclusion

In this paper we developed an economic order quantity model with exponential demand rate under the condition of cash discount and permissible delay in payment. We have seen that the increase in the ordering cost parameter and the interest charged parameter decreases the total cost also the increase in the holding cost parameter and interest earned parameter increases the total cost so the parameters

S and

IC have the significant impact on total cost

As ordering cost increases, the optimal cycle length and total variable cost decreases

Table 2

Effect of holding cost on optimal solution

for profit maximization.

References:

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4. Goyal, S. K., Economic order quantity under conditions of permissible delay in payments,

   Journal of Operational Research Society, Vol.36, pp.335-338, 1985.

5. Jamal, A. M., Sarkar, B. R. and Wang, S., An ordering policy for deteriorating items with allowable shortage and permissible delay in payments, Journal of Operational Research Society, Vol.48, pp.826-833, 1997.

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   Working Paper, William Paterson University, Wayne, New Jersey, U.S.A., 2001.

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