Research Journal of Applied Sciences, Engineering and Technology 4(16): 2831-2838, 2012
ISSN: 2040-7467
© Maxwell Scientific Organization, 2012
Submitted: March 30, 2012
Accepted: March 23, 2012
Published: August 15, 2012
Economic Order Quantity Model with Two Levels of Delayed
Payment and Bad Debt
QIN Juanjuan
Business School in Tianjin University of Finance and Economics, Tianjin, 300222, China
Abstract: The purpose of this study is to determine the optimal retailer’s replenishment policies considering
the customers’ bad debt and delayed payment in the three-stage supply chain with the dominant retailer. The
effect of bad debt is analyzed on the interest earned and interest charged to build the models of the retailer’s
decision in two cases. By analyzing the model, the retailer’s optimal replenishment time and the optimal order
quantity are obtained. Furthermore, analyze the effect of parameters on the retailer’s optimal order policies.
Finally, the numerical analysis is presented to demonstrate the conclusions. The results show that the delayed
payment offered by the manufacturer becomes large, the retailer's optimal order cycle and the optimal order
quantity increases or remains the same; When the delayed payment time offered by the retailer decreases, the
retailer's optimal order cycle and the optimal order quantity increases or remains the same. When the fixed
ordering cost is reduced, the retailer's optimal order cycle and the optimal order quantity decreases or remains
the same. When the charged interest is greater than the earned interest, with the bad debt rate increasing, the
retailer's optimal order cycle and optimal order quantity is converged to a certain value.
Keywords: Bad debt, delayed payment, EOQ, the dominant retailer
INTRODUCTION
Nowadays, the retailers have more bargaining power
than the manufacturers in some supply chains. For
example, in China market, SUNING and GOME have
almost controlled the home appliances market; and in the
medium-sized cities of the food and commodity markets,
most of the market share is dominated by the Wal-Mart
and Carrefour. Therefore, the strong retailers often
required the suppliers to provide a longer extension of the
payment period, while the retailers usually provide an
extension of pay period to its customers for encouraging
the customers’ purchase behavior. Because the strong
position of the retailers in the supply chain, the delayed
payment period provided by the retailers is less than the
credit period offered by the manufacturer. For the retailers
facing the majority of customers for non-specific, there
exist the un-receivable accounts that customers will have
bad debt losses.
The delayed payment has been focused on by many
scholars. In the traditional Economic Order Quantity
model (EOQ), the manufacturer receives payment in full
of the purchase price immediately, without taking into the
delayed payment period. EOQ model with the delayed
payment is proposed by Goyal (1985), which is extended
by Chu et al. (1998) for considering the deteriorated
products. The situation with the exponential rate of
deterioration of the situation deteriorating items is
analyzed by Aggarwal and Jaggi (1995). The order
strategy is researched by Jamal et al. (1997) and Chang
and Dye (2001) allowing the shortage in the case of
delayed payment and deteriorating conditions. Teng
(2002) further considered the unit selling price is not the
same as the unit product costs in the model. The economic
production quantity model is analyzed by Chung and
Huang (2003) considering the manufacturer to the retailer
under the deferred payment policy. In addition, some
authors took into account cash discounts of the delayed
payment, for example, (Huang, 2003, 2005; Chang, 2002;
Ouyang et al., 2002).
As the development of research, there are some
papers discussing the two levels of delayed payment.
Huang (2003) first extended Goyal's model to analyze the
two levels of the trade credit policy, the manufacturer’s
credit period available to retailers as M and the retailers
provide the customers a credit period N, M>N. Qiu et al.
(2007) considers the manufacturer to give retailers the
delayed payment on purchases, the retailers give the
customers a fixed period of delayed payment of the threestage model of economic order quantity. Huang (2007)
analyzes the optimal inventory updating strategy under
the two levels of the delayed payment time, but the
delayed payment time which is given to the customers by
the retailers is less than the delayed payment time which
is given to the retailers by the manufacturer. Also, the
retailer requires the customers to pay the deposit as the
part of the retailing price to get the products.
However, Huang (2007) did not consider that when
the retailers provide the customers a credit period N, the
retailer does not receive the payment during the time
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Res. J. Appl. Sci. Eng. Technol., 4(16):2831-2838, 2012
C
C
Table 1: Notations
D
: The annual demand rate
c
: The purchasing cost per unit
Ie : The earned interest per year
M : The delayed payment time provided by the manufacturer
T
: The replenish cycle
I(t) : The inventory level in time t,, 0#t#T
A
: The order cost per order
h
: The unit holding cost
Ik : The charged interest per year
N
: The delayed payment time provided by the retailer
r
: The bad debt rate, 0#r#1
C
interval [0, T]; the customers purchase the products at
time point T, which will pay the purchasing price at the
time point T+N . Therefore, the retailer receives payment
during the time interval [T, T+N]. Jinn-Tsair and ChunTao (2009) amended the shortcomings of Huang (2007)
to relax the constraints Ik>Ie and M>N, discussing the
optimal product quantity in the two levels of the delayed
payment time.
The research on the delayed payment above did not
consider the bad debt rate of the customer. When facing
the non specific customers, the retailers will have the risk
that the accounts receivable can not be recovered. So the
retailers solving the economic order quantity, we must
consider the losses caused by the bad debts. But the
research above in the construction of retailer cost
function, does not take into consideration the bad debts to
the function model. Qin (2011) have done some research
on the delayed payment time with the customer’s bad
debt, but which is on the situation with the low bad debt
considering the retailing price larger than the production
cost. But in this study, we extends the research to
construct the models which is not relating to the high or
low rate of bad debt to obtain the optimal decisions of the
players; furthermore, I contrast the results to the EOQ
model without the delayed payment and the bad debt.
Therefore, our study further expands the delayed
payment research above, considering the retailer's
dominance in the supply chain. There are two levels of the
delayed payment time, considering the customers’ bad
debt on interest income and interest payments. Further
construct the cost function to obtain the retailers’ optimal
order quantity and the optimal order cycle. The study is
much closer to the actual situation and further enriches the
academic research of the delayed payment time.
Models: The inventory level I (t) is decreasing to meet the
customers demand. If the customers’ demand rate is D,
the inventory level over time is:
dI (t )
= − D, 0 ≤ t ≤ T
dt
C
C
The retailer has more power than the manufacturer in
the three echo supply chain, therefore, we have
M#N.
The annual demand rate is known and constant.
(1)
And meeting the initial conditions I (t) = 0, then we
can obtain the solution of Eq. (1):
I(t) = !Dt + DT, 0#t#T
(2)
The purchasing quantity of the retailer is:
Q = I(0) = DT
(3)
The annual cost associated with the retailer is TRC(T)
The ordering cost per year + the purchasing cost per year
+ the holding cost per year + the interest charged per
year-the earned interest per year.
In which: The ordering cost per year equals A/T; the
purchasing cost per year equals cD; the holding cost per
year (not including the interest) H is as follows:
T
H=
METHODOLOGY
Notations and assumption: The notations are used in the
study listed in the Table 1.
Additionally, the mathematical models proposed in
this study are based on the following assumptions:
The time horizon is infinite.
The retailing price is equal to the production Cost
(Goyal, 1985; Chu et al., 1998). In the extended
research, we can relax this assumption.
The shortages are not allowed because of the timebased competition in today’s market. The
manufacturer is offered by the manufacturer a trade
credit of M periods to settle the entire purchase cost.
On the other hand, the customers are offered by the
manufacturer a trade credit of N periods. However,
the customers have the credit risk, resulting in the
uncollectible account receivables. Uncollectible
accounts expense is recorded in the period from N to
T+N, in which individual accounts receivable are
determined to be worthless. The bad debt rate for the
individual accounts is r. By the time T+N, the
manufacturer will request the write-off of the
uncollectible accounts.
h
hDT
I (t ) dt =
2
T ∫0
The retailer's annual charged interest and annual
earned interest are related to its action strategy. Therefore,
we discuss the problem in the following two cases.
case 1: N#M#T+N
The interest charged per year: Assuming an
unlimited supplying rate of the product, the
inventory level is shown in Fig. 1a. Without
considering the impact of bad debts, the
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Res. J. Appl. Sci. Eng. Technol., 4(16):2831-2838, 2012
At time point M, the retailer needs financing to
pay for not un-receivable money. After time
point M, the retailer repays the financing money
with the sales revenue, hence, the interest
expense for the unit cycle Yk is:
I k S ACC ′A′ cI k D
(T + N − M )
=
2T
T
[( r + 1) T − (1 − r )( M − N )]
Yk =
(4)
If not considering bad debt, it means that r = 0.
The interest earned per year: The retailer start
to selling the products at the time point 0 and
begins to receive the receivable money from its
customers at time point N. At time point M, the
retailer pays money to the manufacturer for
covering the purchase cost. Therefore, the
retailers can invest the receiving money during
the time interval [N, M] with the earned interest
Ie.
Under the influence of bad debts, the earned
interest per year is:
(a) inventory level
Ye =
I e SC ′MN cI e D(1 − r )( M − N ) 2
=
2T
T
(5)
(b) The accumulated sales revenue with no bad debt
when not consider the bad debt rate, it is r = 0.
The total annual cost: Under the influence of
bad debt, the retailer’s total cost is:
(c) The accumulated sales revenue with bad debt
Fig. 1: The inventory level and the accumulated sales
revenue with N < M # T + N
the retailer's total sales revenue is presented in Fig.
1b. Taking into account the customer's bad debt
losses, the retailer's total sales revenue are shown in
Fig. 1c. The retailer’s receivable accounts, after
delayed payment time N providing to the customers,
exists a certain bad debt rate r.
Case 2: N < T + N#M: In case 2, assuming the
unlimited products’ supplying rate, the inventory
level is shown in Fig. 2a. Without considering
the impact of bad debts, the retailer's total sales
revenue is presented in Fig. 2b. Taking into
account the customer's bad debt losses, the
retailer's total sales revenue are shown in Fig. 2c.
The retailer’s receivable accounts, after delayed
payment time N providing to the customers,
exists a certain bad debt rate r.
By the assumptions we can see that at time T+N,
the retailer has written off the bad debt losses of
this cycle. At the time point M, the retailer pays
the amount of financing
with its own funds, its interest with zero. At time
point 0, the retailer begins to sell the products;
recover the receivable from the customers at
time point N; pay the purchase cost to the
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Res. J. Appl. Sci. Eng. Technol., 4(16):2831-2838, 2012
Ye′ =
I e S A′B ′MN
cI e D(1 − r )(2 M − 2 N − T )
=
T
2
(7)
The annual total cost is:
TRC2 (T ) =
hDT cI e D(1 − r )(2 M − 2 N − T )
A
+ cD +
−
T
2
2
(8)
MODEL ANALYSIS
Case: 1 N < M#T+N: To obtain the optimal solution,
take the first order and the second
order derivatives of the formula
(6) with respect to T:
(a) inventory level
∂TRC1 (T ) − A hD cI k D
= 2 +
+
[ (1 + r )T 2 −
∂T
2
T
2T 2
( M − N ) 2 (1 − r )] +
cI e D(1 − r )( M − N ) 2
2T 2
∂ 2 TRC1 (T )
1
= 3 [2 A + cD( M − N ) 2 (1 − r )( I k − I e )]
∂T 2
T
If 2A+cd(M!N)2(1!r)(Ik!Ie) > 0, we can obtain
∂ 2 TRC1 (T )
> 0 . It is clearly that TRC1(T) is strictly
∂T
2
convex function in T. Therefore, we can obtain the
optimal order cycle T*11 based on ∂TRC1 (T ) = 0. We can
∂T
(b) The accumulated sales revenue with no bad debt
obtain:
T11* =
2 A + cD( M − N ) 2 (1 − r )( I k − I e )
hD + cI k D(1 + r )
(9)
The corresponding optimal order quantity Q*11 is:
*
Q11
= DT11* =
2 AD + cD2 ( M − N ) 2 (1 − r )( I k − I e )
h + cI k (1 + r )
(10)
But in case 1, the retailer’s optimal order cycle
should meet the condition that M#T*11+N, therefore, we
can get the inequality as following:
∆ 1 = 2 A + cD( M − N ) 2 (1 − r )( I k − I e ) −
[hD + cI
(c) The accumulated sales revenue with bad debt
]
D(1 + r ) ( M − N ) 2
[
]
= 2 A − cD( M − N ) 2 2rI k + (1 − r ) I e −
Fig. 2: The inventory level and the accumulated sales revenue
with N<M#T+N
manufacturer at the time point M. Therefore,
theretailers can obtain the interest income at the time
interval [N, M] with the earned interest Ie. With the
bad debt, the earned interest per year is:
k
(11)
hD( M − N ) 2
≥0
If 2A+cD(M!N)2(1!r)(Ik!Ie)#0, we can get )1 < 0
based on the formula (11).
In case 1N#M#T+N, we can get:
C
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Res. J. Appl. Sci. Eng. Technol., 4(16):2831-2838, 2012
∂TRC1 (T )
=
∂T
optimal order cycle based on
− A hD cI k D
+
+
[ (1 + r )T 2 −
2
T2
2T 2
cI e D(1 − r )( M − N ) 2
( M − N ) 2 (1 − r )] +
2T 2
T2* =
⎛
( M + N ) 2 ⎞ ⎡ hD cI k D
2
⎟⎢
> ⎜⎜ 1 −
⎟ 2 + 2T 2 (1 + r )T −
T2
⎝
⎠⎣
[
( M − N )2 (1 − r )
] + cI D(1 − 2rT)( M − N ) ⎥⎥⎦ ≥ 0
e
2
Q*2 = DT2* =
T *1 = T *11 =
)2!2A!(hD + cIeD(1!r))(M!N)2#0
(13)
(14)
Also,
2 A + cD( M − N ) 2 (1 − r )( I k − I e )
hD + cI k D(1 + r )
)1 - )2 = cDIk(M!N)2(1!r)!cIkD(1+r)(M!N)2 # 0 (15)
= Q11* = Q11*
Analysis of case 1 and case 2: Based on the analysis of
case 1 and case 2, we can get the theorem 2:
If )1<0, then:
C
2 AD
h + cI e (1 − r )
And the retailer’s annual minimized cost is
TRC2(T*2).
But in case 2, the retailer’ optimal order cycle should
meet the condition T*2+N#M, so we can get the following
inequality:
If )1$0 then:
*
(12)
The retailer’s optimal order quantity is:
Theorem 1: in case 1 with N#M#T+N,
and Q1
2A
hD + cI e D(1 − r )
2⎤
Therefore, in case 1, when: 2A+cD(M!N)2(1!r)
(Ik!Ie)#0, the function TRC1(T) is increasing with T. We
can obtain that the optimal order cycle is T*12 = M!N and
the optimal order quantity is Q*12 = D(M!N).
Based on the analysis (1) and (2), we can get the
Theorem 1.
C
∂TRC2 (T )
= 0:
∂T
Theorem 2: when M# N,
T*1 = T*12-M-N
C
and
If )1 $ 0, the retailer’s optimal order cycle is T* =
T*11; the optimal order quantity is:
Q* = Q*11 = DT*11
Q*1 = Q*12 = D(M!N)
Proof: From the analysis above, it is easy to obtain the
Theorem 1. Based on which, we can get the retailer’s
annual minimized cost TRC1(T*1).
Case 2: N < T + N#M: To obtain the optimal solution,
take the first order and the second order
derivatives of the formula (8) with respect to T,
which is:
C
If )1 < 0 and )2 # 0, the retailer’s optimal order cycle
is T* = T*2; the optimal order quantity is:
Q* = Q*2 = DT*2;
C
If )1 < 0 and )2 > 0, the retailer’s optimal order cycle
is T* = M!N; the optimal order quantity is:
Q* = D(M!N)
∂TRC2 (T ) − A hD cI e D(1 − r )
= 2 +
+
∂T
2
2
T
Proof:
∂ TRC2 (T ) 2 A
= 3 >0
∂T 2
T
2
C
The function TRC2(T) has the minimized value for
∂ TRC2 (T )
> 0 . Therefore, we can obtain the retailer’
∂T 2
If )1 $ 0, from formula (15), we can get:
)2 = 2A!(hD + cleD(1!r))(M!N)2$0.
2
2835
In case 2, N#T+N#M, for )2$0, we can obtain that:
Res. J. Appl. Sci. Eng. Technol., 4(16):2831-2838, 2012
∂TRC2 (T ) − A hD cI e D(1 − r )
= 2 +
+
∂T
2
2
T
⎛
( M − N ) 2 ⎞ ⎛ hD cI e D(1 − r ) ⎞
≤ ⎜1−
+
⎟⎜
⎟
⎠
2
T2
⎝
⎠⎝ 2
≤0
Therefore, in case 2, if )2$0, the function TRC2(T) is
decreasing with T.
In case 1 N#T+N#M for )1 $0, based on formula
(11), we can obtain the optimal value of TRC1(T) is T*11.
With Case 1 and Case 2 analysis, when T is satisfied
N#T+N#M, we can get:
TRC2(T)$TRC2(M!N) = TRC1(M-N)$TRC1(T*11).
order quantity are increasing with M or unchanged; when
the value of N is gradually decreases, the retailer's optimal
order cycle and the optimal order quantity become larger
or unchanged; when the value of A is decreasing, the
retailer’ optimal order cycle and the order quantity is
decreasing or unchanged; when Ik$Ie, with the value of r
increasing, the retailer’ optimal order cycle is increasing
or decreasing to (M-N) and keep unchanged and the
optimal order quantity is increasing or decreasing to D(MN) and keep unchanged.
In the traditional EOQ model, not considering the
delayed payment and the bad debt, the retailer and the
customer both pay the money instantly and r = 0.
Therefore, the traditional EOQ model is the special
situation of case 1 with M = N = r = 0. So it means
Q*0 =
Therefore, the Theorem 2(1) has been proved.
C
If )1 < 0, in case 1 N# M # T+N, we can get that:
Theorem 3: when M$N and
∂TRC1 (T ) − A hD cI k D
= 2 +
+
[ (1 + r )T 2 −
∂T
2
T
2T 2
( M − N ) 2 (1 − r )] +
2A+cD(M!N)2(1!r)(Ik!Ie) > 0
cI e D(1 − r )( M − N ) 2
2T 2
C
C
C
C
⎛
( M + N ) 2 ⎞ ⎡ hD cI k D
⎟⎢
> ⎜1−
+
[ (1 + r ) T 2 −
T2
2T 2
⎠⎣ 2
⎝
( M − N ) 2 (1 − r )] +
cI e D(1 − r ) ( M − N ) 2 ⎤
⎥
2T 2
⎦
If Ie(1!r) < Ik, then Q*2 > Q*0
If Ie(1!r) > Ik, then Q*2 < Q*0
If Ie(1!r) = Ik, then Q*2 = Q*0
If Ie$Ik, then Q*11#Q*0
Proof:
C
≥0
Therefore, in case 1, the function TRC1(T) is
decreasing with T. In cased 2, for )2#0, from formula
(14), we can get T*2 the optimal solution of TRC2(T).
Based on the analysis of case 1 and case 2, we can
know that when T meets N#M#T+N, then:
TRC1(T)$TRC1(M!N) = TRC2(M!N)$TRC2(T*2)
When Ie(1!r) < Ik:
Q 2* = DT2* =
C
2 AD
= Q 0*
h + cI k
When 2A+cD(M!N)2(1!r)(Ik!Ie)>0, then we get
that:
Q *11 =
Theorem 1(1) and (2) shows that in case 1 with
N#M#T+N, if )1 < 0, the function TRC1(T) is
increasing with T; in case 2, N#T+N#M, if )2 > 0,
TRC2(T) is decreasing with the T.
≤
Therefore, when )1 < 0 and )2 > 0 , the retailer’s
optimal order cycle is T* = M!N and the optimal order
quantity is Q* = D(M!N).
From formula (11) and (14), we can get that with the
credit period M increasing, the values of )1 and )2 are
decreased. Based on the Theorem 1 and Theorem 2, it
shows that the retailer's optimal order cycle and optimal
2 AD
>
h + cI e (1 − r )
Similarly, we can prove the (2) and (3).
Therefore, the theorem 1(2) is proved. If)1< 0 and )2
#0 the retailer’s optimal order cycle is T* = T*2 and the
optimal order quantity is Q* = DT*2.
C
2 AD / h + cI k
2 AD + cI k D 2 ( M − N ) 2 (1 − r )( I k − I e )
h + cI k (1 + r )
2 AD + cD2 ( M − N )2 (1 − r )( I k − I e )
≤
h + cI k
2 AD
= Q*0
h + cI k
Theorem 3 shows that when the earned interest is
higher than the charged interest, the retailers will take
advantage of the upstream extension of the payment
period; the economic order quantity is less than the
traditional economic order quantity.
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Res. J. Appl. Sci. Eng. Technol., 4(16):2831-2838, 2012
NUMERICAL ANALYSIS
CONCLUSION
To illustrate the feasibility of the models and
theorems, we illustrate with the numerical analysis
following Assume the retailer makes its decision about the
order strategy of the products: A = 200 dollars per time,
D = 1500 units per year, c = 25 dollars per year, h = 15
dollars per year, Ie = 0.09, Ik = 0.05, M = 20/360 per year,
N = 15/360 per year, r = 0.01.
In the retailer dominant supply chain with two levels
of delayed payment, the optimal decisions of the retailer
are obtained considering the bad debt. The retailer's cost
function is established in two cases. By calculating the
functions, obtain the optimal order cycle and the optimal
order quantity.
The study found that when the delayed payment
period offered by the manufacturer M becomes large, the
retailer's optimal order cycle and the optimal order
quantity increases or remains the same; when the retailer
giving to the customers extended payment period N
decreasing, the retailer's optimal order cycle and the
optimal order quantity increases or remains the same.
When the fixed ordering cost A is reduced, the retailer's
optimal order cycle and the optimal order quantity
decreases or remains the same. When the charged interest
is greater than the earned interest, with the bad debt rate
increasing, the retailer's optimal order cycle and optimal
order quantity is converged, respectively, in (M-N) and D
(M-N). Therefore, the results not only have a certain
theoretical significance, but also can better guide
decision-making for the retailer.
Calculate to obtain )1 > 0 and )2 > 0
T*11 = 0.1192, T*2 = 0.1244
the retailer’ minimixed cost is 40781; the optimal order
cycle is T* = 0.1192; the optimal order quantity.
Q* = 178.8
Now, with the adjustment of model parameters M, N,
A, the retailer's optimal decisions are shown in Table 2, 3,
4 and 5. Through the numerical analysis, it is easier to
understand the influence of the parameters on the retailer's
optimal order cycle and the optimal order quantities.
Table 2: The retailer’s optimal decisions with the adjustment of
parameter M
The optimal
The optimal order Minimized
M
order cycle
quantity
cost
35/360
0.1202
180.2
40574
30/360
0.1197
179.6
40639
25/360
0.1194
179.1
40708
15/360
0.1191
178.7
40857
Table 3: The retailer’s optimal decisions with the adjustment of
parameter N
The optimal
The optimal
Minimized
N
order cycle
order quantity
cost
15/360
0.1192
178.8
40781
10/360
0.1194
179.1
40708
5/360
0.1197
179.6
40639
0
0.1202
180.2
40574
Table 4: The retailer’s optimal decisions with the adjustment of
parameter A
The optimal
The optimal
Minimized
A
order cycle
order quantity
cost
250
0.1333
199.9
41177
200
0.1192
178.8
40781
150
0.1033
154.9
40332
100
0.0843
126.5
39799
Table 5: The retailer’s optimal decisions with the adjustment of
parameter r
The optimal
The optimal
Minimized
r
order cycle
order quantity
cost
0.00
0.1193
179.0
40778
0.05
0.1187
178.1
40794
0.10
0.1181
177.2
40811
0.30
0.1159
173.8
40876
ACKNOWLEDGMENT
The study is funded by Tianjin City High School
Science and Technology Fund Planning Project
(20102127), Research Development Funding Project of
Tianjin University of Finance & Economics (Y1109) and
National Natural and Science Funding (70771073,
70902044).
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