1. Int. J. of Mathematical Sciences and Applications,
Vol. 1, No. 3, September 2011
Copyright  Mind Reader Publications
www.journalshub.com
1265
An EOQ model with Time-dependent demand rate under trade credits
R.P.Tripathi, *Manoj Kumar
Department of Mathematics, Dehradun Institute of Technology, Dehradun, UK, India
*
Shivalik College of Engineering, Dehradun UK, India
email: rakeshprakash11@ yahoo.com, memanoj.319@ rediffmail.com
Abstract
This paper deals with the optimal policy for the customer to obtain its minimum cost when supplier
offers both permissible delay as well as a cash discount for non – deteriorating items. The demand rate
is time dependent. Four different cases are discussed. Case I: The payment is made at M1 to get a cash
discount and T ≥ M1. Case II: The customer pays in full at M1 but T < M1. Case III: The payment is
made at time M2 to get the permissible delay and T ≥ M2. Case IV: The customer pays in full at M2 but
T < M2. Finally numerical examples are given to illustrate the proposed model.
Key Words: Inventory; cash discount; delay in payments; non-deteriorating items; trade credit
1.Introduction
In a buyer – seller situation and inventory model consider a case in which depletion of inventory is caused by demand
rate. This paper considers a retailer’s model in which the deterioration rate is zero and demand rate is time dependent. In
today’s business transactions, it is common to find that the buyers are allowed some credit period before they settle the
account with the whole seller. This provides an advantage to the customers, due to the fact that they do not have to pay the
whole seller immediately after receiving the product, but delay their payment until the end of the allowed period. The
customer pays no interest during the permissible time for payment. The interest will be charged if the payment is delayed
beyond that period. To motivate fast payment stimulate more sells, the supplier also provides its customers a cash discount.
Our proposed research topic is an important to buyers when the supplier provides cash – discount and a permissible delay.
Ouyang. et al. [1] derived an EOQ model for deteriorating items under trade credit by considering cash – discount. Goyal
[2] derived an EOQ model under the condition of permissible delay. Aggarwal and Jaggi [3] extended Goyal’s model for
deteriorating items. Jamal et al. then further generalized the model to allow shortages. Haley and Higgins [4] introduced
the first model to consider the economic order quantity under conditions of permissible delay in payments with
deterministic demand, no shortage and zero lead time. Hwang and Shinn [5] considered the problem of determining the
retailer’s optimal price and lot size simultaneously when the supplier permits delay in payments. Dye [6] in their paper
considered the stock dependent demand for deteriorating items for partial backlogging and condition of permissible delay
in payment. They assumed initial stock – dependent demand function. Teng [7] provided an alternative conclusion from
Goyal [2] and mathematically proved that it makes economics sense for a buyer to order less quantity and take benefits of
permissible delay more frequently. Huang [8] extended one-level trade credit into two level trade credit to develop the
retailer’s replenishment model from the view point of the supply chain. He assumed that not only the supplier offers the
retailer trade credit but also the retailer offers the trade credit of his/her customers. This view point reflected more real life
situations in the supply chain model. Khouja [9] showed that for many supply chain configurations, complete
synchronization would result in some members of the chain being ‘losers’ in terms of cost. He viewed the economic
delivery and scheduling problem model and analyzed supply chains dealing with single and multiple components in
developing his model.
All the above mentioned articles do not use the discount cash- flow (DCF) approach for the analysis of the optimal
inventory policy when the supplier offers trade-credit. Teng [10] developed a model discounted cash – flow analysis on
inventory control under various suppliers trade – credits and obtain the optimal ordering policies to the problem. Trippi
and Lewin [11] and Kim and Chung [12] recognized the need to explore the inventory problems by using the present value
concept or DCF approach. Liao et.al [13] developed an inventory model with deteriorating items under inflation when a
delay in payment is permissible. Several studies have examined the inflammatory effect on an inventory policy. Buzacott

2. R.P.Tripathi & Manoj Kumar
1266
[14] developed an approach of model inflation by assuming a constant inflation rate. Misra [15] proposed an inflation
model for the EOQ, in which the time value of money and different inflation rates were considered. Mangianeli et. al [15]
not only reviewed and classified models appearing in previous literature, but also presented some examples with relaxed
assumptions. A cash flow oriented EOQ models with deteriorating items under permissible delay in payment were
discussed by Hou and Lin [16]. A cash flow oriented EOQ model was discussed by Tripathi et. al [17]. Tripathi and Kumar
[18] presented a paper on credit financing in economic ordering policy of time-dependent deteriorating items by using
discounted cash-flows (DCF) approach.
In this paper we established an EOQ model for non-deteriorating items and time dependent demand rate in which the
supplier provide a cash-discount and permissible delay to the customer we then established an optimal solution of the
problem. We provided several numerical examples for illustration of the theoretical results.
The rest of this paper organized as follows. In the forthcoming section we described the assumptions and notations used
through out the study. Next, we develop the mathematical model to minimize the total cost per year. In section 4 numerical
examples are given to illustrate the theoretical results. Finally we draw the conclusion and future research in section 6.
2. Assumptions and Notations
The following assumptions are used in this paper.
(1) The demand for the item is time varying.
(2) Shortages are not allowed.
(3) Replenishment is instantaneous.
(4) Time horizon is infinite.
(5) When the account is not settled the generated sells revenue is deposited in an interest bearing account. At the end
of credit period (M1 or M2), the account is settled as well as the buyer pays of all units sold and starts paying for the
interest charges on the items in stocks. That is, supplier provides a cash discount if the full payment is made within time
M1 otherwise the full payment is made within time M2. The account is settled when the payment is made.
(6) Lead time is zero.
The following notations are used throughout this paper:
)(tD = the time dependent demand rate i.e. btatD )(
h = the unit holding cost per year excluding interest charges
p = the selling price per unit
c = the unit purchasing cost, with c < p
cI = the interest charged per $ in stock per year by the supplier or bank
dI = the interest earned per $ per year
S = the ordering cost per order
Q = the order quantity
r = the cash discount rate; 0 < r < 1

3. An EOQ model with Time-dependent demand rate under trade credits
1267
1M = the period of cash discount
2M = the period of permissible delay in settling account with 12 MM 
T = the replenishment time interval
)(tI = the level of inventory at time t, Tt 0
)(TZ = the total relevant cost per year.
3. Mathematical Formulation
The variation of inventory with respect to time can be described by the following differential equation
Ttbta
dt
tdI
 0),(
)(
(1)
With boundary conditions 0)(,)0(  TIQI . The solution of (1) is given by
)(
2
)()( 22
tT
b
tTatI  (2)
and the order quantity is
2
2
T
b
aTQ  (3)
Total demand during one cycle is
2
2
T
b
aT 
The total relevant cost per year consists of the following elements:
(a) cost of placing order
T
S
 (4)
(b) cost of purchasing 





 T
b
aC
T
CQ
2
(5)
(c) cost of carrying inventory = 
T
dttI
T
h
0
)(
= 






32
bTa
hT (6)
Case I: Since the payment is maid at time 1M , the customer save rC per cycle due to price discount. From (2) we know
that the discount per year is given by

4. R.P.Tripathi & Manoj Kumar
1268






 T
b
arC
T
rC
2

(7)
According to the assumption 4, the customer pays off all units ordered at time 1M to obtain the cash discount.
Consequently, the items in stock have to be financed (at interest rate cI ) after time 1M and hence the interest payable per
year is
  
T
M
c MTbaMTr
T
C
dt
T
tI
IrC
1
)2(3))(1(
6
)(
)1( 1
2
1 (8)
Finally during [0, 1M ] period, the customer sells products and deposits the revenue into an account that earns interest dI
per dollar per year. Therefore, the interest earned per year is
 






1
0
3
1
2
1
32
)(
M
dd bMaM
T
pI
tdtbta
T
pI
(9)
Total relevant cost is given by
EarnedIntPaidIntCCCPCOTZ ..)(1 






 T
b
arC
T
S
TZ
2
)(1 + 






32
bTa
hT +  )2(3))(1(
6
1
2
1 MTbaMTr
T
C
 - 






32
3
1
2
1 bMaM
T
pId
(10)
Case II: 1MT  , In this case the customer s1ells TbTa )(  units in total at time T and has TbTarC ))(1(  to pay
the supplier in full at time 1M , consequently there is no interest payable, while the cash discount is the same as that in case
1. However, the interest earned per year is
   
T T
d
dtbtaTMdttbTa
T
pI
0 0
1 )()()( =












 )3(
62
11 TM
bTT
MapId (11)
As a result, the total relevant cost per year )(2 TZ is given by






 T
b
arC
T
S
TZ
2
)(2 + 






32
bTa
hT -  












 TM
bTT
MapId 11 3
62
(12)
Case III: 2MT  , Since the payment is made at time 2M , there is no cash discount. The interest payable per year is

T
M
c
dttI
T
CI
2
)( =  )2(3)(
6
2
2
2 MTbaMT
T
CIc
 (13)
The interest earned per year is
 






2
0
3
2
2
2
32
)(
M
dc bMaM
T
pI
tdtbta
T
pI
(14)

5. An EOQ model with Time-dependent demand rate under trade credits
1269
The total relevant cost is






 T
b
arC
T
S
TZ
2
)(3 + 






32
bTa
hT +  )2(3)(
6
2
2
2 MTbaMT
T
CIc
 - 






32
3
2
2
2 bMaM
T
pId
(15)
Case IV: 2MT  , In this case there is no interest charged. The interest earned per year is
   
T T
d
dtbtaTMdttbTa
T
pI
0 0
2 )()()( =












 )3(
62
22 TM
bTT
MapId (16)
The total relevant cost is )(4 TZ is given by






 T
b
arC
T
S
TZ
2
)(4 + 






32
bTa
hT -  












 TM
bTT
MapId 22 3
62
(17)
Theoretical results
Since 4,3,2,1,0
)(
2
2



i
T
TZi
The values of 4,3,2,1,*
 iTT i obtained from (11), (13), (16) and (18) are minimum.
The corresponding values of 4,3,2,1),( *
 iTZZ ii are also optimum (minimum).
4. Numerical examples:
Example 1: Given a = 1000units per year, h = $ 4/unit/year, cI = 0.09/year, dI = 0.06/year, C = $ 20/unit, p = $35/unit, r
= 0.02, 1M = 15 days and 2M = 30 days, S = 5
Case I: T ≥ M1 Optimal solution for different ordering costs.
Ordering Cost
S
Replenishment cycle T*
Economic order Quantity
Q(T*
)
Total Cost
Z(T*
)
10 0.0641447 64.0004 578.447
15 0.0813292 81.0007 601.944
20 0.0954686 95.0009 624.192
25 0.1077692 108.001 646.271
We obtain computational results in the sensitivity analysis on Id
Interest Rate
Id
Replenishment cycle T*
Economic order Quantity
Q(T*
)
Total Cost
Z(T*
)
0.05 0.0419812 41.0002 560.701
0.04 0.0437057 43.0002 561.041
0.03 0.0453648 45.0002 561.707
0.02 0.0469652 46.0002 562.148
0.01 0.0485129 48.0002 563.227
Example 2 Given h = $ 50/unit/year, cI = 0.09/year, dI = 0.06/year, C = $ 400/unit, p=$500/unit, r = 0.9, 1M = 15 days =
15/365 years and 2M = 30 days = 30/365 years, a =1000 units/year, b = 0.9

6. R.P.Tripathi & Manoj Kumar
1270
Case II: T < M1, Optimal solution for different ordering costs
Ordering Cost
S
Replenishment cycle T*
Economic order Quantity
Q(T*
)
Total Cost
Z(T*
)
1.4 0.0214307 21.4307 359696
1.5 0.0316328 33.3449 360154
1.6 0.0392492 41.6198 360387
Sensitivity analysis on Id
Interest Rate
Id
Replenishment cycle T*
Economic order Quantity
Q(T*
)
Total Cost
Z(T*
)
0.050 0.0405318 40.532 360536
0.051 0.0390492 39.0494 360471
0.052 0.0375071 37.5072 360403
0.053 0.0358978 35.8979 360331
0.054 0.0342117 34.2118 360257
Example 3: Given a = 1000units per year, h = $ 4/unit/year, cI = 0.09/year, dI = 0.06/year, C = $ 20/unit, p=$35/unit, r =
0.02, 1M = 15 days = 15/365 years and 2M = 30 days = 30/365 years, S = 5
Case III: T ≥ M2
Ordering Cost
S
Replenishment cycle T*
Economic order Quantity
Q(T*
)
Total Cost
Z(T*
)
25 0.091046 91.0468 638.324
30 0.100067 100.067 829.079
35 0.108343 108.340 874.331
40 0.116024 116.024 915.782
45 0.123232 123.230 954.244
Sensitivity analysis on Id
Interest Rate
Id
Replenishment cycle T*
Economic order Quantity
Q(T*
)
Total Cost
Z(T*
)
0.08 0.096130 96.1312 753.962
0.07 0.098704 98.705 766.876
0.06 0.100067 100.068 779.113
0.05 0.101979 101.98 791.181
0.04 0.103855 103.856 802.943
Example 4:Given a = 1000units per year, h = $ 4/unit/year, cI = 0.09/year, dI = 0.06/year, C = $ 20/unit, p=$35/unit, r =
0.02, 1M = 15 days and 2M = 30 days, S = 5
Case IV: T < M2
Ordering Cost
S
Replenishment cycle T*
Economic order Quantity
Q(T*
)
Total Cost
Z(T*
)
5 0.0404885 40.4887 431.401
10 0.0572593 57.2593 451.922
15 0.0701279 70.1279 479.833
20 0.0809766 80.9766 504.943

7. An EOQ model with Time-dependent demand rate under trade credits
1271
Sensitivity analysis on Id
Interest Rate
Id
Replenishment cycle T*
Economic order Quantity
Q(T*
)
Total Cost
Z(T*
)
0.08 0.404885 40.4887 864.903
0.07 0.0404885 40.4887 894.102
0.06 0.0404885 40.4887 923.303
0.05 0.0417025 41.7027 941.201
0.04 0.0404885 40.4887 981.699
5. Conclusion
In this paper we developed an EOQ model for deteriorating items with time dependent demand rate to determine the
optimal policy when the supplier provides cash-discount and / or a permissible delay in payment. We characterized the
effect of the values of parameters under replenishment cycle. Finally we provide two numerical examples to verify the
proposed model.
The proposed model in this paper can be extended in several ways. For instance we may extend non-deterioration rate to
two parameter weibull distribution. Also we could consider the demand as a function of selling price as well as time
dependent. Finally we could generalize the model for discount and inflation rates, shortages, discount, quantity etc.
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