DISTRIBUTOR’S ECONOMIC LOT SIZING MODEL FOR DECAYING ITEMS UNDER
SUPPLY CHAIN IN THE PRESENCE OF ORDER-SIZE-DEPENDENT DELAY IN
PAYMENTS
Seong Whan Shinn1) and Chae Soo Kim2)
1)
2)
Halla University, South Korea ([email protected])
Halla University, South Korea ([email protected])
Abstract
This paper deals with the problem of determining the distributor’s economic lot sizing policy for
exponentially decaying items under supply chain in the presence of trade credit. Assuming that the
supplier’s credit terms are already known and the length of delay is a function of the distributor’s order
size, we formulate the mathematical model from which algorithm is derived based on properties of a
solution. A numerical example is presented to illustrate the algorithm developed.
1. Introduction
A significant part of the manufacturing goods are usually kept in the supply chain, especially either
in a manufacturer’s inventory or in a distributor’s/retailer’s storage. Also, in many inventory systems
under distribution channel, purchasers(distributors or retailers) are permitted a certain fixed period(credit
period) on settling the account for the items supplied without paying any interest.
This provides an
advantage to the purchasers, due to the fact that they do not have to pay the supplier immediately after
receiving the item, but instead, can defer their payment until the end of the permitted period(Sarker et al.
[9]). Trade credit affects the conduct of business significantly for many reasons.
A major reason for a
supplier to offer a credit period to the purchasers is to stimulate the demand for the items that he
produces and the supplier usually expects that the increased sales volume can compensate the capital
losses incurred during the credit period.
For the purchasers who are permitted a certain period to pay
back for the items bought without paying any interest, trade credit is an effective means of reducing the
capital opportunity cost for stock holding. They also can earn interest on the sales of the items supplied
during that period.
In this regard, many research papers analyzed the inventory model when the
supplier permits delay in payments for an order of a product.
Chung[2], Goyal[5], and Kingsman[8]
evaluated the effects of trade credit on the economic lot sizing policy.
Recently, Shinn et al.[10]
examined the joint price and lot size determination problem for retailer under conditions of permissible
delay in payments and quantity discounts for freight cost.
All the research works mentioned above
implicitly assume that stock is depleted by customer’s demand alone. This assumption is quite valid for
products whose utility remains constant over time. However, in real life situations there are numerous
types of product whose utility does not remain constant over time and therefore, the distributor’s stock is
depleted by a combination of the customer’s demand and loss by decay.
In this case, the decrease in
utility or loss for stock of products subject to decay is usually a function of the total amount of stock on
hand. Thus, the loss due to decay must be taken into consideration while developing the distributor’s
inventory models; otherwise it will yield inaccurate results.
Ghare and Schrader[4], assuming
exponential decay of the inventory in the face of constant demand, derived a revised form of the
economic order quantity.
Cohen[3] analyzed the distributor’s price and lot size determination problem
for an exponentially decaying item.
Also, sufficient work has been done by many authors for
controlling the inventory of decaying items in the presence of trade credit. Jaggi and Aggarwal[7]
evaluated the effect of trade credit in determining the inventory policy of decaying items.
Hwang and
Shinn[6] introduced the retailer’s pricing and lot sizing policy for exponentially decaying products under
trade credit. Recently, Chu et al.[1] examined the economic lot sizing model for decaying items under
the condition of permissible delay in payments.
Note that a common assumption of the above
researches concerning trade credit is the availability of a certain length of delay that is set by the
supplier.
However, the length of delay is considered as supplier’s dominant strategy against the
competitive suppliers in expectation of increasing the sales volume.
In general, there are many factors
that determine the length of delay in a given line of business or for a given creditor. Among them,
credit risk, the size of the account, customers type, and market competition are known to be more
important.
For the size of the account, it is generally known that the length of delay tends to be stricter
on large shipments than on smaller ones, probably because the larger quantities are sold at less
advantageous prices to the seller. However, for the sake of better production and stock control, some
1
manufacturers in Korea prefer less frequent orders with larger order sizes to frequent orders with smaller
order sizes, if the distributor’s annual demand is equal.
the amount of purchase by distributor.
Thus, they associate the length of delay with
In this sense, some pharmaceutical companies and agricultural
machinery manufacturers in Korea offer a longer credit period for larger amount of purchase rather than
giving some discount on unit selling price to their distributors.
Their policy tends to make the
distributor’s lot size large enough to qualify a certain credit period break.
In this paper, an attempt has been made to develop an inventory model for obtaining the distributor’s
economic lot sizing policy of decaying items in the presence of trade credit depending on the amount of
purchase.
In the next section, we formulate the mathematical model.
For the model developed, the
properties of a solution are discussed and its solution algorithm is presented in section 3. A numerical
example is provided in Section 4, which is followed by concluding remarks in Section 5.
2. Development of the supply chain model under trade credit
The mathematical model in this paper is developed on the basis of the following notations and
assumptions.
Notations:
D : annual demand
H : unit stock holding cost per item per year excluding interest charges
r : interest charges per $ investment in stocks per year
i : interest rate which can be earned per $ in a year
C : unit purchase cost in $
S : cost of placing one order
Q : size of one order
T : time interval between successive orders
 : positive number representing the stock decaying rate( 0    1)
I(t) : stock level at time t
0
0
tc j : credit period for the amount purchased CQ, v j 1  CQ  v j ,
where tc j 1  tc j , j  1,2,  , m and v0  v1    vm , v0  0 , vm
Assumptions:
(i) The demand for the items is constant with time.
(ii) Time period is infinite.
0
2
0
0
0
0
= 
(iii) Shortages are not allowed.
(iv) Inventory is depleted not only by customer’s demand but also by decay with exponential
distribution.
(v) The supplier permits delay in payments for the items supplied and the length of delay is a
function of the amount purchased by the distributor.
(vi) The purchasing cost of the items sold during the credit period is deposited in an interest bearing
account with rate i. At the end of the credit period, the account is settled and the distributor
starts paying for the interest charges on items in stock with rate r( r  i ).
In the case of exponential decay, the rate at which stock decays will be propotional to on hand stock
level.
Being a continuous review system, it is logical to assume that depletion due to such decay and
depletion due to meeting demand will occur simultaneously. Figure 1 illustrates the time behavior of the
distributor’s stock level.
Demand rate, D is indicated by the slope of the dashed line. Accordingly,
the differential equation describing the time behavior of the system is:
dI (t )
  I (t )  D .
dt
(1)
It follows from the fact that this is a first order linear differential equation(as noted in Ghare and
Schrader[4] ) that the solution to equation (1) is
3
D
I (t )  I (0)e t 

(1  e t ) .
Now, we determine the stock loss due to decay.
(2)
Let I 0 (t ) be the stock level at time t where there
were no decaying. Then, the stock loss due to decay becomes
D


I 0 (t )  I (t )  I (0)  Dt    I (0)e t  (1  e t ) 



D
 I (t )(et  1)  Dt  (et  1) .

(3)
(4)
Therefore, the quantity ordered each cycle becomes


Q  I 0 (T )  I (T )  DT .
(5)
Since the cost structure to be defined includes the stock holding costs, it is clearly better off to have the
stock level reach zero just before reordering, i.e., I(T) = 0. Given that I(T) = 0, the following expression
for the quantity ordered each cycle results
Q
e

D
T

1 .
(6)
Also, by the condition of I (0)  Q , the stock level at time t is
I (t ) 
e

D
 (T t )

1 , 0  t  T .
(7)
Now, for the formulation of the distributor’s total cost with respect to T, we consider the supplier’s
0
0
j  1, 2, , m .
credit plan, v j 1  CQ  v j ,
By equation (6), if we denote
1  

0
0
0
v j  CD ln 
v j  1 , then the inequality v j 1  CQ  v j can be rewritten as
  CD

v j 1  CDT  v j
for j  1, 2,
, m.
(8)
Then, the distributor’s total variable cost per year, TC(T) consists of the following four elements.
(1) Cost of placing order =
S
.
T
4
(2) Cost of purchasing order =


CQ CD eT  1
.

T
T
H
T
(3) Cost of stock holding(excluding interest charges) =
(4) Capital opportunity cost for v j 1  CDT  v j :

T
0
I (t )dt 


HD eT  T  1
.
 2T
(i) Case 1( tc j  T ): (see figure 2 (a))
 Dtc j 
tc j
Interest earned per order = iC 
 2 
T
Interest payable per order = rC  I (t )dt
tc j
T
Capital opportunity cost per year =
=
rC  I(t)dt -
iCDtc j
2
tc j

rCD e
2
T
 (T tc j )
 2T
(ii) Case 2( tc j  T ): (see figure 2 (b))
  iCDtc
  (T  tc j )  1
2T
2
j
.
 DT 
Interest earned per order = iC 
T  iCDT (tc j  T )
 2 
DT
iC
T + iCDT tc j  T 
2
Capital opportunity cost per year = T
iCDT
 iCDtc j .
=
2
The distributor’s total variable cost per year TC(T) is given by
TC(T) = Ordering cost + Purchasing cost + Stock holding cost + Capital opportunity cost.
5
Therefore, depending on the relative size of tc j to T, TC (T ) has two different expressions as follows:
Case 1: T  tc j


 (T tc j )
2
  (T  tc j )  1 iCDtc j
S CD(eT  1) HD (eT  T  1)  rCD e
TC1, j (T )  



2

T
T
2T
 2T

T

,
Case 2: T  tc j
TC 2, j (T ) 
v j 1
CD
T 
vj
CD
,
j  1,2,, m .
S CD(eT  1) HD (eT  T  1)  iCDT




 iCDtc j 
2
T
T
 T
 2

,
v j 1
CD
T 

 (9)


(10)
vj
CD
,
j  1,2,, m .
3. Determination of the distributor’s economic replenishment policy
The purpose of this paper is to find the distributor’s economic replenishment cycle time T * which
minimizes the distributor’s total variable cost, TC n , j (T ) .
obtained by equation (6).
Also, an economic lot size Q * can be
Theoretically, T * can be obtained from the solution of the equation,
dTC n , j (T ) / dT  0 , but it is difficult to solve this equation for finding the exact value of T.
However,
the average life period of the item, 1 /  is greater than T. Thus, using a truncated Taylor series
expansion for the exponential function, the total cost function can be approximated as
2

S
DT ( H  C )  (r  i)CDtc j
rCDT
TC1, j (T )   CD 


 rCDtc j  , (11)


T
2
2T
2


TC 2, j (T ) 
S
DT ( H  C )  iCDT

 CD 

 iCDtc j  .
T
2
 2

(12)
Thus, it is clear that the Taylor series approximation for the exponential terms yields the credit model
without decay but with holding cost H  C . Therefore, for the normal condition( r  i ) as stated by
Goyal[5], TC n , j (T ) is a convex function for every n and j.
which minimizes TC n , j (T ) and they are:
6
And so, there exists a unique value Tn , j ,
T1, j 
T2, j 
2 S  (r  i )CDtc 2j
D ( H  C  rC )
2S
D( H  C  iC )
,
(13)
.
(14)
Also, Tn , j and TC n , j (T ) have the following properties.
Property 1.
T1, j  T1, j 1 holds for j = 1, 2,
..., m-1.
Property 2.
T2, j  T2, j 1 holds for j = 1, 2,
..., m-1.
Property 3.
For any T, TC n, j (T )  TC n, j 1 (T ) , n = 1, 2 and j = 1, 2, ..., m-1.
Properties 1 and 2 indicate that the value of T1, j is strictly increasing as j increases and the value of
T 2 , j is identical for every j.
And Property 3 implies that both TC1, j (T ) and TC 2, j (T ) are strictly
decreasing for any fixed value of T as j increases.
Also, from equation (13) and (14), we have the
following useful property.
Property 4.
For any j, if T1, j  tc j , then T2, j  tc j , which implies that TC 2, j (T ) is decreasing in T
for T  tc j . Also, if T2, j  tc j , then T1, j  tc j , which implies that TC1, j (T ) is increasing in T for
T  tc j .
Proof.
From equation (13), T1, j  tc j can be rewritten as
2 S  (r  i )CDtc 2j
D( H  C  rC )
 tc j .
(P1)
Squaring both sides of equation (P1) and rearranging,
2S  (r  i)CDtc j  D( H  C  rC )tc j ,
(P2)
2S
 tc j .
D( H  C  iC )
(P3)
2
2
7
Equation (P3) implies that T2, j  tc j and T2, j is a minimum point TC 2, j (T ) , which is a convex
Thus, TC 2, j (T ) is decreasing in T for T  tc j .
function.
T2, j  tc j implies that T1, j  tc j .
Also, from equations (P1) and (P3),
Likewise, T1, j is a minimum point of TC1, j (T ) and so TC1, j (T )
is increasing in T for T  tc j .
Q.E.D.
Now, from the above properties, we can make the following observations about the characteristics of
the total cost function for T, T  TI j  T v j 1 CD  T  v j CD , j  1, 2, , m . These


observations simplifies our search process such that only a finite number of candidate values of T needs
to be considered to find T * for the approximate model. Let k be the smallest index such that
T2, j  tc j .
Observation 1.
For T  TI j ,
v j 1
vj
j  k , we can consider the following three cases for T2, j ; T2, j 
v j 1
CD
,
vj
 T2, j .
CD
CD
v j 1
v j 1
(i) If T2, j 
, then T 
yields the minimum total annual cost for T  TI j .
CD
CD
v j 1
vj
(ii) If
, then T  T2, j yields the minimum total annual cost for T  TI j .
 T2, j 
CD
CD
vj
(iii) If
 T2, j , then we do not need to consider T for T  TI j to find T * .
CD
CD
 T2, j 
, and
Observation 2.
For T  I j ,
v j 1
vj
(iv) If
vj
CD
, tc j 
vj
 T1, j and
vj
v j 1
CD
,
 tc j  T1, j .
CD
CD
CD
v j 1
v j 1
(i) If T1, j 
, then T 
yields the minimum total annual cost for T  TI j .
CD
CD
v j 1
vj
 T1, j 
(ii) If
, then T  T1, j yields the minimum total annual cost for T  TI j .
CD
CD

vj
vj

 T1, j , then T 
(iii) If tc j 
,where v j  v j   and  is a very small positive number,
CD
CD
yields the minimum total annual cost for T  TI j .
CD
 T1, j 
j  k , we can consider the following four cases for T1, j ; T1, j 
 tc j  T1, j , then we do not need to consider T for T  TI j to find T * .
Observation 3. (Search Stopping Rule)
(i) If T  T1, j yields the minimum total annual cost for T  TI j , then T *  T1, j .
8
(ii) If T 
vj

CD
yields the minimum total annual cost for T  TI j , then T 
*
vj

CD
.
Based on the above observations, we develop the following solution procedure to determine T * for
the approximate model.
Solution algorithm
2S
and let k be the smallest index such that T2, j  tc j .
D( H  C  iC )
 tc j for all 1  j  m , then set k = m+1 and go to Step 3.
Step 1: Compute T2, j 
If T2, j
Otherwise, find the index l satisfying vl 1  T2, j CD  vl and go to Step 2.
Step 2: 2.1 If k > l, then go to Step 2.3. Otherwise, go to Step 2.2.
v v
v
v
2.2 Compute the total annual variable cost for T  T2,l , l , l 1 , , m2 and m1 , and go to
CD CD
CD
CD
Step 3.
v j 1
2.3 Compute the total annual variable cost for T =
, j  k , k  1,, m and go to Step 3.
CD
Step 3: 3.1 Set j = k - 1.
vj
3.2 If tc j 
, then go to Step 3.4. Otherwise, go to Step 3.3.
CD
3.3 In case
v j 1
v j 1
i) T1, j 
, compute the total annual variable cost for T 
and go to Step 3.4.
CD
CD
vj
ii) T1, j 
, compute the total annual variable cost for T  T1, j and go to Step 4.
CD

vj
vj

iii) T1, j 
, compute the total annual variable cost for T 
, v j  v j   where
CD
CD
 is a very small positive number and go to Step 4.
3.4 Reset j = j - 1 and go to Step 3.2.
Step 4: Select the one that yields the minimum total annual variable cost as T * for the approximate
model and stop.
4. Numerical example
9
Let D = 3223, S = $ 50, C = $ 3, H = $ 0.1, R = 0.15(= 15%), I = 0.10(= 10%) and   0.2 in
appropriate units. Also, the supplier's credit terms for the distributor’s amount of purchase is as
follows:
Total amount of purchase
Credit period
0  CQ  $1500
tc1  0.1
$1500  CQ  $3000
tc2  0.2
$3000  CQ
tc3  0.3
The distributor’s replenishment cycle time for the approximate model can be obtained through the
following steps.
Step 1. Since T2, j ( 0.1761)  tc2 ( 0.2) , k = 2.
Since T2, j CD  1703  [v1 , v 2 ) , l = 2 and go to Step 2.
Step 2.
2.1. Since k (= 2) = l (= 2), go to Step 2.2.
2.3. Compute TC 2, 2 (T2, 2 ) with equation (12).
v2
( 0.3010)  tc3 ( 0.3) , compute TC1,3 (v 2 CD) with equation (11) and
CD
go to Step 3.
And since
Step 3.
3.1. Set j = 2 - 1 = 1.
v1
( 0.1528) , go to Step 3.3.
CD
v
3.3 Since T1,1 ( 0.1682)  1 , compute the total annual cost by equation (11) for
CD

v
T  1 ( 0.1527) and go to Step 4.
CD
Step 4. Since TC 2,3 (0.3010) = 10030.13 = min. TC1,1 (0.1527, TC1,3 (0.0.3010), TC 2, 2 (0.1761), the
distributor’s economic replenishment cycle time for the approximate model becomes 0.3010
with its minimum total annual variable cost $10030.13 .
3.2. Since tc1 ( 0.1) 
5. Conclusions
In this paper, we evaluated the distributor’s economic lot sizing model for an exponentially decaying
item when the supplier permits delay in payments for an order of a item.
Recognizing that a major
reason for the supplier to offer trade credit is to stimulate the demand of the item, it is assumed that the
length of delay is a function of the distributor’s total amount of purchase. The availability of ordersize-dependent delay in payments can be justified by the principle of economies of scale, and tends to
make the distributor’s lot size larger by inducing him to qualify for a longer credit period in his
10
payments.
For the system presented, a mathematical model was developed. Recognizing that the
model has a very complicated structure, a truncated Taylor series expansion is utilized to find a solution
procedure. To illustrate the validity of the solution procedure, an example problem was solved and the
results are consistent with our expectation.
There are several interesting opportunities for future researches in this subject. The model can be
extended to the case of joint ordering of multiple items with different credit terms could be suggested.
From the supplier’ point of view, he might be interested in finding an equivalent delay-in-payments plan
to a given cost discount schedule.
References
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