Open Journal of Business and Management , 2019, 7, 427- 446 [606131]

Open Journal of Business and Management , 2019, 7, 427- 446
http://www.scirp.org/journal/ ojbm
ISSN Online: 2329- 3292
ISSN Print: 2329- 3284

DOI: 10.4236/ojbm.2019.72029 Mar. 7 , 2019 427 Open Journal of Business and Management

An Inventory Model for Ramp -Type Demand
with Two -Level Trade Credit Financing Linked
to Order Quantity
Hui-Ling Yang
Department of Computer Science and Information Engineering , Hungkuang University, Taiwan

Abstract
In the traditional economic order quantity (EOQ) model, it is assumed that
the demand rate is constant. Thereafter, many researchers developed inven-
tory model with time -varying demand to reflect sales in different phases of
product life cycle in the market. However, in practice, especially for fashion a-
ble and high- tech product, the demand rate during the growth stages of its
life cycle increases significantly with linear or exponential in the growth stage
and then gradually stabilizes, and remains near constant in the maturity
stage. It can be taken a ramp -type demand rate into account. Furthermore, in
today’s supply chain , a supplier usually offers a permissible delay in payment
to retailer s to encourage them to buy more products , and a retailer in turn
provides a downstream trade -credit period to its customers . Therefore, this
paper focus on 1) ramp -type demand rate and 2) the upstream and dow n-
stream trade credit financing linked to order quantity for retailer is consi-
dered. The objective is to find the optimal replenishment cycle and order
quantity to keep the total relevant cost per unit time as minimum as possible.
The study shows that in each case discussed, the optimal solution not only
exists but also is unique. N umerical examples are provided to illustrate the
proposed model. Finally, some relevant manage rial insights based on the r e-
sults are characterize d.

Keywords
Ramp -Type Demand , Two -Level , Trade Credit , Finance , Order Quantity

1. Introduction
1.1. Inventory Models with Ramp -Type Demand Rate
In present, high -tech manufacturing is the backbone to the Taiwan economy.
How to cite this paper: Yang , H.-L. (201 9)
An Inventory Model for Ramp -Type D e-
mand with Two -Level Trade Credit F i-
nancing Linked to Order Quantity . Open
Journal of Business and Management , 7,
427-446.
https://doi.org/10.4236/ojbm.2019.72029

Received: January 31 , 2019
Accepted: March 4 , 2019
Published: March 7 , 2019

Copyright © 201 9 by author(s) and
Scientific Research Publishing Inc.
This work is licensed under the Creative
Commons Attribution International
License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/

Open Access

H.-L. Yang

DOI: 10.4236/ojbm.2019.72029 428 Open Journal of Business and Management
During the growth and maturity stages of a high -tech product life cycle, the d e-
mand rate increases significantly with linear or exponential in the growth stage
and then remains near constant in the maturity stage. The term “ ramp type ” is
used to represent such a demand pattern. Wu [1] developed an EOQ inventory
model for Weibull distribution deterioration items with ramp type demand rate
and partial backlogging. Manna and Chaudhuri [2] also develope d an EOQ
model with ramp type demand rate in which time dependent deterioration rate
and shortages were considered. Agrawal et al. [3] provided an inventory model
with deteriorating items, ramp -type demand and partially backlogged shortages
for a two -warehouse system. Other related research articles on this field can be
found in Deng [4], Deng et al. [5], Panda et al. [6] [7], Skouri et al. [8] [9] and
their references.
1.2. Inventory Models with Permissible Delay in Payment
The traditional inventory economic order quantity (EOQ) model assumes that a
buyer must pay for items immediately a fter receiving them. However, to stimu-
late sales quantity a supplier often offers a retailer a permissible delay in pay-
ment. Thus, to offer a certain fixed credit period for his/her retailer is an alterna-
tive incentive policy to quantity discount. In early research work, Goyal [10] de-
veloped an EOQ model under conditions of permissible delay in payments, and
ignored the difference between the selling pric e and the purchase cost. Shah [11]
considered a stochastic inventory model when delays in payments are permissi-
ble. Aggarwal and Jaggi [12] extended Goyal’s model to consider the deteriora t-
ing items. Jamal et al. [13] further generalized Aggarwal and Jaggi’s model to al-
low for shortages. Teng [14] amended Goyal’s model by considering the diffe r-
ence between unit price and unit cost, and found that it makes economic sense
for a well -established buyer to order less quantity and take the benefits of the
permissible delay more frequently. Skouri et al . [15] proposed an inventory
model with ramp type demand rate under permissible delay in payment. Teng et
al. [16] established an economic order quantity model with trade credit financ-
ing for non -decreasing demand. Similarly, there are also many related articles
published in such field with different practical consideration.
1.3. Inventory Models with Two -Level Trade Credit
Huang [17] extended Goyal’s model to develop an EOQ model in which the
supplier offers the retailer a permissible delay period (i.e., an upstream trade
credit) , and the retai ler in turn provides a trade credit period ( i.e., a downstream
trade credit) to its customers. Teng and Goyal [18] complemented the shor t-
coming of Huang’s model and proposed a generalized formulation. Teng et al .
[19] obtained the retailer’s optimal ordering policy when the supplier offers a
progressive permissible delay in payments. Chen and Teng [20] provided a r e-
tailer’s optimal ordering policy for deteriorating items with maximum lifetime
under supplier’s trade cr edit. Cheng and Teng [21] proposed an inventory and

H.-L. Yang

DOI: 10.4236/ojbm.2019.72029 429 Open Journal of Business and Management
credit decision for time -varying deteriorating items with up -stream and down-
stream trade credit financing by discounted cash flow analysis. Shah [22] pro-
vided a three -layered integrated inventory model for deteriorating items with
quadratic demand and two -level trade credit fi nancing. Rameswari and Uthay a-
kumar [23] proposed an integrated inventory model for deteriorating items with
price -dependent demand under two -level trade credit policy. Several related a r-
ticles can be found in Goyal et al. [24], Huang and Hsu [25], Min et al. [26] and
their references.
1.4. Inventory Models with Trade Credit Linked to Order Quantity
Sometimes, to encourage more sales, supplier offer retailers a trade credit period
with conditional permission, if a retailer orders more than a predetermined qua n-
tity. Chang et al. [27] developed an EOQ model for deteriorating items under su p-
plier credits linked to ordering quantity. Chung and Liao [28] provided lot -sizing
decisions under trade credit depending on the ordering quantity. Ouyang et al. [29]
proposed a n economic order quantity for deteriorating items with partially pe r-
missible delay in payments linked to order quantity. Kreng and Tan [30] proposed
an inventory model under two levels of trade credit depending on the order qua n-
tity. Teng et al. [31] provided an inventory model for increasing demand under
two levels of trade credit linked to order quantity, Recently, Sash and Card e-
nas-Barrón [32] provided an inventory model which is a retailer’s decision for o r-
dering and credit policies with deteriorating items when a supplier offers o r-
der-linked credit or cash discount. Ting [33] provided some comments on the
EOQ model for deteriorating items with conditional trade credit linked to order
quantity. Similarly, other related research articles can be found in their references.
In contrast to the above papers mentioned, this paper is extended in the fo l-
lowing two ways: 1) a constant demand (or increasing demand) is extended to a
ramp -type demand function, in which the demand increases linearly and then
stays constant at the end, and 2) the supplier provides its retailer with a pe r-
missi ble delay link to order quantity while the retailer also offers a downstream
trade credit period to its customers. We establish several fundamental theoretical
results and obtain its optimal solution. We then provide several numerical e x-
amples to illustrat e the proposed model and present some important and rel e-
vant managerial insights.
The rest of the paper is structured as follows. Section2 introduces the notation
and assumption needed to develop the proposed inventory model. Section 3 fo r-
mulates the model . Section 4 discusses some theoretical results and provides an
algorithm to find the optimal solutions. Section 5 provides numerical examples to
illustrate the proposed model. Section 6 concludes the results and presents some
managerial insights. Further, provides some future research directions.
2. N otation and Assumptions
The mathematical model of the inventory problem here is based on the follo w-

H.-L. Yang

DOI: 10.4236/ojbm.2019.72029 430 Open Journal of Business and Management
ing notation and assumptions:
2.1. Notation
The following notation is used throughout this paper.
D(t) = the de mand rate at time t, we here assume that D(t) is deterministic at
a constant rate after a period of time µ, at initial the demand rate is increasing.
i.e.,
()()
(),
, ft t
Dt
ftµ
µµ< =≥ , where ()f t a bt= + , 0a>, 0b>.
dQ = the minimum order quantity at which the delay is permitted by the
supplier.
dT = the time interval that dQ units are depleted to zero due to demand.
M = the retailer’s upstream credit period offered by supplier in years.
N = the retailer’s downstream credit period offered to its buyers in years.
T = the length of replenishment cycle in years.
Q = the order qua ntity.
A = the replenishment cost per order.
h = the holding cost per unit per unit of time excluding interest charge.
p = the unit selling price.
c = the unit purchasing cost.
eI = the interest earned per dollar per year by the retailer.
pI = the interest paid per dollar per year by the retailer.
()It = the inventory level at time t.
() ijTC T = the total relevant cost per unit time for Case i and subcase j,
1 ,2 ,3 ,4i= and 1, 2j= or 3, which is a function of T.
*
ijT = the optimal replenishment cycle time of () ijTC T for Case i and su b-
case j, 1 ,2 ,3 ,4i= and 1, 2j= or 3. (i.e., *T).
*
ijQ = the optimal order quantity for Case i and subcase j, 1 ,2 ,3 ,4i= and
1, 2j= or 3. (i.e., *Q).
2.2. Assumptions
Next, the following assumptions are made to establish the mathematical inve n-
tory model.
1) Replenishment rate is instantaneous.
2) Shortages are not allowed to occur.
3) In today’s global competition, many retailers have no pricing power. As a
result, the selling price is hardly changed for many retailers. In addition, to avoid
lasting price competition, we may assume without loss of generality that the
selling price is co nstant in today’s global competition and low inflation envi-
ronment.
4) The objective here is to minimize the total relevant cost per unit of time
until the demand is no longer increasing.
5) When 0 MN−> , the buyer deposits sales revenue into an interest bearing

H.-L. Yang

DOI: 10.4236/ojbm.2019.72029 431 Open Journal of Business and Management
account from time N to M. At the end of the permissible delay, the buyer pays
off all units sold by time MN−, uses uncollected and unsold items as collateral
to apply for a loan, and pays the b ank a certain amount of money periodically
until the loan is paid off at time TN+.
3. Mathematical Formulation
Based on the above assumptions, the inventory system here is as follows. At the
beginning ( i.e., at time t = 0), the retailer orders and receives Q units of a single
product from the supplier. The inventory level is depleted gradually in the i n-
terval [0 , T] due to increasing demand from customers. At time t = T, the i n-
ventory level reaches zero. Hence, the inve ntory level at time t, I(t), can be d e-
scribed by the following differential equation :
()()
()()d,0d
d,dItft tt
Itf tTtµ
µµ= − ≤≤
= − ≤≤
(1)
with the boundary condition ()0 IT=. The solution to (1) is
()()()()()
()()222 ,0
,It a t b t a b T t
ab Tt tTµ µ µµ µ
µµ= −+ − + + − ≤≤
= + − ≤≤ (2)
Thus, the retailer’s order quantity per cycle is
() ()()202 Q I a b ab T µµ µ µ = = + ++ − (3)
From Equation (3), we can obtain the time interval dT by using the following
equations:
()()22ddQ a b ab Tµµ µ µ= + ++ − (4)
Next, based on whether the order quantity larger than the predetermined
quantity or not, we have the following two cases: 1) d TT< 2) d TT≥ .
3.1. dQQ< (i.e., dTT< )
In this case, the retailer’s order quantity is less than dQ. Hence, the permissible
delay in payment is not allowed ( i.e., M = 0). Meanwhile, the retailer offers a
permissible delay of N to its buyers. Consequently, the retailer mus t fiancé all
items ordered at time 0, and start to payoff the loan after time N. For details,
please see Figure 1 . Thus, the interest paid by the retailer is as follows. There are
two cases to be discussed. 1 ) 0 NTµ<<< 2) 0NTµ<<<
() ()
()() ()
()() ()0
0
00d d
0d 0d d , i f0
0 d d d , if 0N TN
pN
N TN
pN
N TN
pNcI I t I t N t
c IItIt I t N t N T
c I I t It N t It N t N Tµ
µ
µ
µµ
µ+
+
++−
+ + − <<<= + − + − <<<∫∫
∫∫∫
∫∫ ∫ (5)
Hence, the retailer’s total relevant cost per unit time is

H.-L. Yang

DOI: 10.4236/ojbm.2019.72029 432 Open Journal of Business and Management

Figure 1 . Graphical representation for d TT< .

()
() ()() () { }
()()
()()11
00
2223
2
22d 0d 0d d
23 2
, if 022T N TN
pN
pTC T
A h It t c I I t I t It N t T
ab T abAh
b abcI a a b TN T T N Tµ
µ
µµ µµ
µµµµµµ µ+= + + + +−
 +−= + ++
  ++ + ++ − + <<<  
   ∫ ∫∫∫
(6.1)
()
() ()()() { }
()()
()( ) ( )
( )12

00
2223
2
22
22 2d 0d d d
23 2
3 2226
, if 02T N TN
pN
pTC T
A h It t c I I t It N t It N t T
ab T abAh
bNcI a N a N b N N
abT NT N Tµ
µ
µµ µµ
µµµ µ µµ
µµµ+= + + +−+ −
 +−= + ++
 −+ + + ++ + −
+ + − + <<< ∫ ∫∫ ∫
(6.2)
3.2. dQQ≥ (i.e., dTT≥ )
In this case, based on the supplier’s trade credit M, and the last customer’s pay-
ment time T + N, we discuss the following three cases: 1 ) 0MN<< 2)
MN≥ and MTN≤+ 3) MN≥ and MTN>+ .
3.2.1. The Case of 0MN<<
Since 0MN<< , there is no interest earned for the retailer. In addition, the
retailer has to finance all items ordered after time M at an interest charged pI
per dollar per year, and start to pay off the loan after time N as shown in Figure 2 .
Consequently, the interest charged is given by
()()
()()
()()()
()()()0d d
0 d d , if 0
0d 0d d , i f 0
0 d d d , if 0N TN
pMN
N TN
pMN
N TN
pMN
N TN
pMNcI I t I t N t
cI I t I t N t M N T
c I ItIt I t N t M N T
c I I t It N t It N t Mµ
µ
µ
µµ
µ+
+
+
++−
+ − << <<
= + + − < <<<
+−+ − <∫∫
∫∫
∫∫∫
∫∫ ∫NTµ




 <<< (7)

H.-L. Yang

DOI: 10.4236/ojbm.2019.72029 433 Open Journal of Business and Management

Figure 2 . Graphical representation for d TT≥ and MN< .

Hence, the retailer’s total relevant cost per unit time is
()
() () () { }
()()
()()21
0
2223
2d 0d d
23 2
, if 02T N TN
pMN
pTC T
A h It t c I I t It N t T
ab T abAh
TcI a b T N M T M N Tµµ µµ
µµ+= + + +−
 +−= + ++
  + + − + << <<    ∫ ∫∫
(8.1)
()
() ()()() { }
()()()
()()()22
0
2223 2
2d 0d 0d d
23 2 2
, if 02T N TN
pMN
pTC T
A h It t c I I t I t It N t T
ab T ab bA h cI a M
Ta b TN M M T M N Tµ
µ
µµ µµ µµµ
µ µµ µ+= + + + +−
 +−  = + ++ + + −      
  ++ − + −+ < <<<  
 ∫ ∫∫∫
(8.2)
()
() ()() () { }
()()()
()( ) ( )
()( ) ( )23
0
2223 2
22
2d 0d 0d d
23 2 2
3 226
2 2 , if 02T N TN
pMN
pTC T
A h It t c I I t I t It N t T
ab T ab bA h cI a N M
Na N b NN
abT T N M N T MN Tµ
µ
µµ µµ µµ
µµ µµ
µµµ µ+= + + + +−
 +−  = + ++ + + −      
−+ ++ + −
+ + − ++ − + < < << ∫ ∫∫∫
(8.3)
3.2.2. The Case of MN≥ and MTN≤+
When MTN≤+ , the retailer cannot receive the last payment before the per-
missible delay time M. As a result, the retailer must finance all items sold after
time ( MN−) at time M, and pay off the loan until T + N at an interest rate of
pI per dollar per year as shown in Figure 3 . Therefore, the interest paid is given
by

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DOI: 10.4236/ojbm.2019.72029 434 Open Journal of Business and Management

Figure 3 . Graphical representation for d TT≥ , 0NM<≤ and MTN≤+ .

()
()
()() ( )d
d d , if
d d d , ifTN
pM
TN T
pM tM
TN T
pM tMcI I t M t
cI f v t M
c I f vv f vt Mµ
µµµ
µµ+
+
−
+
−−
≤=+< ∫
∫∫
∫∫ ∫ (9)
On the other hand, the retailer starts selling products at time 0, and receiving
the money at time N. Consequently, the retailer accumulates sales revenue in an
account that earns eI per dollar per year starting from N through M as shown
in Figure 3 . Therefore, the interest earned is given by
()
() () ( )
()() () ( )0

0

0 dd
d d d , if
d d d d , ifM tN
eN
M tN
eN
M N tN
eNNpI f v v t
p I f vv f vt N
p I fv v fv N v f v t Nµ
µ
µ
µµµ
µµ−
−
−+≤=
+ −+ < ∫∫
∫∫ ∫
∫∫ ∫ ∫ (10)
As a result, the retailer’s total relevant cost per unit time is
() () () {
() () ( )}
()()
()() ()
()( )()()310
0
2223
2
2 2
2d dd
d dd
23 2
22
, if 02T TN T
pM tM
M tN
eN
peTC T A h I t t cI f v t
p I f vv f vtT
ab T abAh
bc Ia b T MN p I a MN
MNa b MN MN T NMµ
µµ
µ
µµ µµ
µµµ
µµ µ+
−
−= ++
−+
 +−= + ++
 + + −− − + − 
  −  ++ −− − − <≤<    ∫ ∫∫
∫∫ ∫
(11.1)
() () () {
()() () ( )}
()()()()
()()()
()( )()()320

0
2223
2 2
2
2d dd
d d dd
223 2
2
2T TN T
pM tM
M N tN
eNN
p
eTC T A h I t t cI f v t
p I fv v fv N v f v t T
ab T abA h cI a b T M N
bpI a M N bN N M N
MNa b MN MNµ
µµ
µ
µµ µµµ
µµµ
µµ+
−
−= ++
− + −+
 +− =+ + + + + −−  
− + −− − −
  −  ++ −− − −   ∫ ∫∫
∫∫ ∫ ∫
, if 0T NM µ<<<
 (11.2)

H.-L. Yang

DOI: 10.4236/ojbm.2019.72029 435 Open Journal of Business and Management
() () ()() ( ) {
() () ( )}
()()
( ) ( )( )( )
()()()
()
330
0
2223
3
2 22
2d d dd
d dd
23 2
2 23
2T TN T
pM tM
M N tN
eNN
p
eT CT A hI ttc I f vv f vt
p I f vv f vtT
ab T abAh
TNM abcI M T N T N M
bpI a M N bN N M N
abµ
µ
µ
µµ
µ
µµ µµ
µµ µ
µµµ
µ+
−
−−
−= ++ +
−+
 +−= + ++
  +−  + −+ − − + + −−  
− + −− − −
++∫ ∫∫ ∫
∫∫ ∫
( )()()2
, if 02MNMN MN T NM µµ  − −− − − << <     (11.3)
3.2.3. The Case of MN≥ and MTN>+
Since the order quantity is larger than or equal to dQ (due to dTT≤), the r e-
tailer receives the permissible delay in payment. If TNM+< , then the retailer
receives all payments from its customers by the time T + N which is before the
permissible delay time M. Hence, the retailer has the money to pay the supplier at
time M, and does not have the interest charges. In the meantime, the retailer rec e-
ives the revenue and deposits into a bank to earn interest as shown in Figure 4 .
The interest e arned by the retailer is
()() ()
() ()
() () ( )()
()() () ( )()0
0
00 d 0d
dd 0d
d d d 0 d , if
d d d d 0dTN M
eN TN
TN tN M
eN TN
TN tN M
eN TN
TN N tN M
eN N TNpI I I t N t I t
pI f v v t I t
p I f vv f vt I t N M
p I fv v fv N v f v t I tµ
µ
µ
µµµ
µ+
+
+−
+
+−
+
+−
+−− +
= +
+ + <<=
+ −+ +∫∫
∫∫ ∫
∫∫ ∫ ∫
∫∫ ∫ ∫ ∫, if NMµ
 << (12)
Hence, the retailer’s total relevant cost per unit time is
() () () () ( ) {
()}
()()()
()()()4100
2223 2
2d d dd
0d
23 2 2
, if 02T TN tN
eN
M
TN
eT CT A hI ttp I f vv f vt
I tT
ab T ab bA h pI a M N
Tab T M N T N T N Mµ
µµ
µµ µµ µµ
µµ µ+−
+= +− +
+
 +−  = + ++ − + −      
  ++ − − − <<<+<  
 ∫ ∫∫ ∫
∫
(13.1)
() () ()() ( {
())()}4200d dd
d d 0dT TN N
eNN
tN M
TNT C T A h I t t p I fv v fv N v
f vt I t Tµ
µµ+
−
+= + − +−
++∫ ∫∫ ∫
∫∫

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DOI: 10.4236/ojbm.2019.72029 436 Open Journal of Business and Management

Figure 4 . Graphical representation for d TT≥ , 0NM<≤ and MTN>+ .

()()
()()
()()()2223
2
223 2
2
, if 0 .2eab T abAh
bpI a M N bN N T
Tab T M N T N T N Mµµ µµ
µµµ
µµ µ +−= + ++
− + −− −
  ++ − − − <<<+<  
  (13.2)
4. Theoretical R esults
To minimize the total relevant cost, taking the first and second order derivatives
of () ijTC T , 1 ,2 ,3 ,4i= , 1, 2j= or 3 with respect to T and let
() d d0ijTC T T =, we obtain the following results.
()()() { }11
11d0dpTCh ab Tc I ab T N T C TTµµ= + + + +− = , (14.1)
()() { }12
12d0dpTCh ab Tc I ab TT C TTµµ =+++− = , (14.2)
()()
12
1
2d
0dd0
djj
p
TC
TTChc I ab T
Tµ
== ++> , 1, 2j= . (15)
()()( ) { }2
2d0dj
pjTCh ab Tc I ab T NM T C TTµµ = + + + +− − = , 1, 2, 3j= . (16)
()()
22
2
2d
0dd0
djj
p
TC
TTChc I ab T
Tµ
== ++> , 1, 2, 3j= . (17)
()() { }3
3d0dj
pjTChc I ab TT C TTµ = + +− = , 1, 2j= . (18.1)
()( )
( ) ( )33
2 2
33d
d
2
0pTCh a b T cI a M T NT
bT N M TC Tµµ
µ = + + + −−  
+ −+− − 
= (18.2)
()()
32
3
2d
0dd0
djj
p
TC
TTChc I ab T
Tµ
== ++> , 1, 2j= . (19.1)

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DOI: 10.4236/ojbm.2019.72029 437 Open Journal of Business and Management
() ( ) ( )
332
33
2d0dd0
dp
TC
TTCh ab c I ab T NM T
Tµ
== + − + +− >. (19.2)
()()() { }41
41d0deTCh ab T p Iab MT N T C TTµµ = + − + −− − = , (20.1)
() ()()() { }42
42d0deTCh a b T pI bN N a b M T N TC TTµ µµ= + − − − + + −− − = , (20.2)
()()
42
4
2d
0dd0
djj
e
TC
TTCh pI a b T
Tµ
== ++> , 1, 2j= . (21)
Theorem 1. For each i, j, there exists a unique optimal cycle length ijT which
minimizes ijTC, 1 ,2 ,3 ,4i= , 1, 2j= or 3.
Proof: See Appendix .
It is not easily to find the closed -form of T from the equation of each first d e-
rivative which is equal to zero. However, we can use numerical method to find
the solution. From Theorem 1, we know that the solution minimizes the total
relevant cost function is also a global minimum. By ensuring the solution sati s-
fies the condition in each case, the following theoretical result is obtained.
Corollary 1. For d QQ< ,
(a) if 11 d TT< and 11 0 NTµ<<< , then *
11 TT= .
(b) if 12 d TT< and 12 0NTµ<<< , then *
12 TT=
Corollary 2. For d QQ≥ ,
(a) 0MN<< ,
(i) if 21 d TT≥ and 21 0 M NTµ<< << , then *
21 TT= .
(ii) if 22 d TT≥ and 22 0M NTµ< <<< , then *
22 TT= .
(iii) if 23 d TT≥ and 23 0MN Tµ < <<< , then *
23 TT=
(b) 0NM<≤ and MTN≤+ ,
(i) if 31 d TT≥ and 31 0 NMT Nµ<≤< ≤ + , then *
31 TT= .
(ii) if 32 d TT≥ and 32 0N MT Nµ<<< ≤ + , then *
32 TT= .
(iii) if 33 d TT≥ and 33 0NM T Nµ << <≤ + , then *
33 TT= .
(c) 0NM<≤ and MTN>+
(i) if 41 d TT≥ and 41 0 NT NMµ<<< +< , then *
41 TT= .
(ii) if 42 d TT≥ and 42 0N T NMµ<<< +< , then *
42 TT= .
Summarizing the results in Corollary 1 and 2, we propose the following alg o-
rithm to find the optimal solution.
Algorithm
Step 0. Input parameter values.
Step 0.1. By (4), calculate dT
Step 0.2. Compare the values of M and N. If MN< , then go to Step 1.
Otherwise, go to Step 4.
Step 1. By (14.1), (14.2), (16), calculate T, let it be 11T, 12T, 21T, 22T, 23T.
Step 2. Compare the values of ijT, 1, 2i= , 1, 2j= or 3, and dT.
Step 2.1. If 11 d TT< and 11 0 NTµ<<< , then *
11 TT= , and calc u-

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DOI: 10.4236/ojbm.2019.72029 438 Open Journal of Business and Management
late ()*
11TC T . Otherwise, set ()11 11TC T =∞.
Step 2.2. If 12 d TT< and 12 0NTµ<<< , then *
12 TT= , and calc u-
late ()*
12TC T . Otherwise, set ()12 12TC T =∞.
Step 2.3. If 21 d TT≥ and 21 0 M NTµ<< << , then *
21 TT= and
calculate ()*
21TC T , Otherwise, set ()21 21TC T =∞.
Step 2.4. If 22 d TT≥ and 22 0M NTµ< <<< , then *
22 TT= , and
calculate ()*
22TC T . Otherwise, set () 22 22TC T =∞.
Step 2.5. If 23 d TT≥ and 23 0MN Tµ < <<< , then *
23 TT= and
calculate ()*
23TC T . Otherwise, set ()23 23TC T =∞.
Step 3. Set () (){ }*min 1, 2, 1, 2 or 3ij ij TC T TC T i j= = = , then *
ij TT= is the
optimal solution, for a certain i, j and stop.
Step 4. By (18.1), (18.2), (20.1), (20.2), calculate T, let it be 31T, 32T, 33T, 41T,
42T.
Step 5. Compare the values of ijT, 3, 4i= , 1, 2j= or 3, and dT.
Step 5.1. If 31 d TT≥ and 31 0 NMT Nµ<≤< ≤ + , then *
31 TT=
and calculate ()*
31TC T . Otherwise, set ()31 31TC T =∞.
Step 5.2. If 32 d TT≥ and 32 0N MT Nµ<<< ≤ + , then *
32 TT=
and calculate ()*
32TC T , Otherwise, set ()32 32TC T =∞.
Step 5.3. If 33 d TT≥ and 33 0NM T Nµ << <≤ + , then *
33 TT=
and calculate ()*
33TC T , Otherwise, set ()33 33TC T =∞
Step 5.4. If 41 d TT≥ and 41 0 NT NMµ<<< +< , then
*
41 TT= .and calculate ()*
41TC T . Otherwise, set
()41 41TC T =∞.
Step 5.5. If 42 d TT≥ and 42 0N T NMµ<<< +< , then *
42 TT=
and calculate ()*
42TC T . Otherwise, set () 42 42TC T =∞.
Step 6. Set () (){ }*m in 3 ,4 , 1 ,2 o r 3ij ij TC T TC T i j= = = , then *
ij TT= is the
optimal solution, for a certain i, j and stop.
5. Numerical Examples
In this section, we provide two numerical examples to illustrate several distinct
theoretical results for M > N and M < N. Let the demand rate ()100 50 ft t= +
per year, A = $10 per order, h = $3/unit/year, c = $5/unit, p = $10/unit, Ip =
0.06/year, and Ie = 0.05/year.
5.1. M < N
Let M = 1/12 years, and N = 1/6 years.
1) Let 30dQ= units.
Example 1.1. Let 0.1µ= years, we know that Nµ<. By (4), we have
0.28810dT= years and by the above algorithm, we have
11 12 21 22 23 0.23986, 0.24614, 0.23995, 0.23993, 0.23998 TTTTT= = = = =
and
()11 11 88.36124 TC T = , ()12 12TC T =∞ , ()21 21TC T =∞ , () 22 22TC T =∞ ,

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DOI: 10.4236/ojbm.2019.72029 439 Open Journal of Business and Management
()23 23TC T =∞. Since 11 d TT< and 11 NTµ<< , by Corollary 1(a), we know
that the optimal solution is *
110.23986 TT= = years, and then
()()*
11 11 88.36124 TC T TC T= = . Furthermore, by (3), we have
*
1124.93522 QQ= = units.
Example 1.2. Let 0.2µ= years, we know that Nµ<. By (4), we have
0.28182dT= years and by the above algorithm, we have
11 12 21 22 23 0.23165, 0.24440, 0.23237, 0.23195, 0.23197 TTTTT= = = = =
and
()11 11TC T =∞, ()12 12 88.71856 TC T = , ()21 21TC T =∞, () 22 22TC T =∞,
()23 23TC T =∞. Since 12 d TT< and 12 NTµ<< , by Corollary 1(b), we know
that the optimal solution is *
120.24440 TT= = years, and then
()()*
12 12 88.71856 TC T TC T= = . By (3), we have *
1225.88441 QQ= = units.
2) Let 20dQ= units.
Example 1.3. Let 0.05µ= years , we know that MNµ<< . By (4), we have
0.19573dT= years and by the above algorithm, we have
11 12 21 22 23 0.24311, 0.24632, 0.24312, 0.24313, 0.24324 TTT T T= = = = =
and
()11 11TC T =∞, ()12 12TC T =∞, ()21 21 84.79927 TC T = , () 22 22TC T =∞,
()23 23TC T =∞. Since 21 d TT> and 21 M NTµ<<< , by Corollary 2(a)(i), we
know that the optimal solution is *
210.24312 TT= = years, and then
()()*
21 21 84.79927 TC T TC T= = . By (3), we have *
2124.85773 QQ= = units.
Example 1.4. Let 0.1µ= years, we know that MNµ<< . By (4), we have
0.19286dT= years and by the above algorithm, we have
11 12 21 22 23 0.23986, 0.24614, 0.23995, 0.23993, 0.23998 TTTTT= = = = =
and
()11 11TC T =∞, ()12 12TC T =∞, ()21 21TC T =∞, ()22 22 85.76229 TC T = ,
()23 23TC T =∞. Since 22 d TT> and 22 M NTµ<<< , by Corollary 2(a)(ii), we
know that the optimal solution is *
220.24440 TT= = years, and then
()()*
22 22 85.76229 TC T TC TC= = . By (3), we have *
2224.94312 QQ= = units.
Example 1.5. Let 0.2µ= years, we know that MNµ<< . By (4), we have
0.19091dT= years and by the above algorithm, we have
11 12 21 22 23 0.23165, 0.24440, 0.23237, 0.23195, 0.23197 TTTTT= = = = =
and
()11 11TC T =∞, ()12 12TC T =∞, ()21 21TC T =∞, () 22 22TC T =∞,
()23 23 86.95530 TC T = .. Since 23 d TT> and 23T N M <<<µ , by Corollary 2(a)
(iii), we know that the optimal solution is *
230.23197 TT= = years, and then
()()*
23 23 86.95530 TC T TC T= = . By (3), we have *
2324.51676 QQ= = units.
5.2. MN≥
In this subsection, let 20dQ= units,
1) Let M = 1/6 years, and N = 1/12 years.

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DOI: 10.4236/ojbm.2019.72029 440 Open Journal of Business and Management
Example 2.1. Let 0.05µ= year, we know that NMµ<≤ . By (4), we have
0.19573dT= years and by the above algorithm, we have
31 32 33 41 42 0.23967, 0.23960, 0.26351, 0.23611, 0.23360 TTTTT= = = = =
and
()31 31 81.06808 TC T = , ()32 32TC T =∞, ()33 33TC T =∞, ()41 41TC T =∞,
() 42 42TC T =∞. Since 31 d TT> and 31 NMT Nµ<<<+ , by Corollary 2(b)(i),
we know that the optimal solution is *
310.23967 TT= = years, and then
()()*
31 31 81.06808 TC T TC T= = . By (3), we have *
3124.50358 QQ= = units.
Example 2.2. Let 0.1µ= years, we know that NMµ<< . By (4), we have
0.19286dT= years and by the above algorithm, we have
31 32 33 41 42 0.23654, 0.23658, 0.25808, 0.23311, 0.23435 TTTTT= = = = =
and
()31 31TC T =∞, ()32 32 81.97423 TC T = , ()33 33TC T =∞, ()41 41TC T =∞,
() 42 42TC T =∞. Since 32 d TT> and 32 N MT Nµ<< < + , by Corollary 2(b)(ii),
we know that the optimal solution is *
320.23658 TT= = years, and then
()()*
32 32 81.97423 TC T TC T= = . By (3), we have *
3224.59067 QQ= = units.
Example 2.3. Let 0.2µ= years, we know that NMµ<< . By (4), we have
0.19091dT= years and by the above algorithm, we have
31 32 33 41 42 0.22922, 0.22946, 0.24603, 0.22611, 0.23447 TTTTT= = = = =
and
()31 31TC T =∞, ()32 32TC T =∞, ()33 33 82.41147 TC T = , ()41 41TC T =∞,
() 42 42TC T =∞. Since 32 d TT> and 33 NM T Nµ< << + , by Corollary 2(b)
(iii), we know that the optimal solution is *
330.24603 TT= = years, and then
()()*
33 33 82.41147 TC T TC T= = . By (3), we have *
3326.06367 QQ= = units.
2) Let M = 1/3 years, and N = 1/12 years.
Example 2.4. Let 0.05µ= years, we know that NMµ<< . By (4), we have
0.19573dT= years and by the above algorithm, we have
31 32 33 41 42 0.20977, 0.20953, 0.22573, 0.23617, 0.23366 TTTTT= = = = =
and
()31 31TC T =∞, ()32 32TC T =∞, ()33 33TC T =∞, ()41 41 71.91272 TC T = ,
() 42 42TC T =∞. Since 41 d TT> and 41 NT NMµ<< +< , by Corollary 2(c)(i),
we know that the optimal solution is *
410.23617 TT= = years, and then
()()*
41 41 71.91272 TC T TC T= = . By (3), we have *
4124.14471 QQ= = units.
Example 2.5. Let 0.1µ= years, we know that NMµ<< . By (4), we have
0.19286dT= years and by the above algorithm, we have
31 32 33 41 42 0.20641, 0.20653, 0.21635, 0.23336, 0.23459 TTTTT= = = = =
and
()31 31TC T =∞, ()32 32TC T =∞, ()33 33TC T =∞, ()41 41TC T =∞,
()42 42 73.05178 TC T = . Since 42 d TT> and 42 N T NMµ<< +< , by Corollary
2(c)(ii), we know that the optimal solution is *
420.23459 TT= = years, and

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DOI: 10.4236/ojbm.2019.72029 441 Open Journal of Business and Management
then ()()*
42 42 73.05178 TC T TC T= = . By (3), we have *
4224.38186 QQ= =
units.
From Table 1 , some managerial insights can be obtained. In the case of
MN< ,
a) The retailer’s total relevant cost ()*TC T increases as the predetermined
order quantity dQ increase, and the constant rate of a period time µ increase.
That is, if the predetermined order quantity (dQ) is larger and the period time
(µ) is longer, then the total relevant c ost ()*TC T for the retailer will be large.
b) As the order quantity is less than the predetermined quantity, (*
d TT<), the
optimal replenishment cycle *T and the optimal order quantity *Q increase
as the time point ( µ) increases, while in the case, the order quantity is larger
than the predetermined quantity (*
d TT> ), the optimal replenishment cycle *T
decreases as the time point ( µ) increases.
In this case, if the upstream trade credit period is less than the downstream
trade credit period, ( MN< ), then the retailer need to pay more interest than
earned, thus, the larger dQ, and µ will cause the larger total relevant cost.
From Table 2 , some managerial insights can be obtained. In the case of MN≥ ,
a) The retailer’s total relevant cost ()*TC T and the optimal order quantity
*Q increase as the constant rate of a period time ( µ) increases. That is, if the
period time ( µ) is longer, then it will cause the retailer’s total relevant cost and
the optimal order quantity to be larger.
b) The retailer’s total relevant cost ()*TC T decreases as the upstream trade
credit period M increases, since the retailer may earn more interest than paid.
c) The larger the difference of MN−, (i.e., the shorter downstream trade
credit period, and the longer upstream trade credit period ), the less the retailer’s
total relevant cost is. That is, it’s more profitable for the retailer.

Table 1 . Summary on optimal solutions for Examples 1.1- 1.5.
MN<
Example 1.1 1.2 1.3 1.4 1.5
M 1/12
N 1/6
dQ 30 20
µ 0.1 0.2 0.05 0.1 0.2
dT 0.28810 0.28182 0.19573 0.19286 0.19091
*T 0.23986 0.24440 0.24312 0.23993 0.23191
*Q 24.93522 25.88441 24.8573 24.94312 24.51676
()*TC T 88.36124 88.71856 84.79927 85.76229 86.95530
Case 3.1
*
d TT<
*NTµ<< 3.1
*
d TT<
*NTµ<< 3.2.1
*
d TT>
*M NTµ<<< 3.2.1
*
d TT>
*M NTµ<<< 3.2.1
*
d TT>
*M NTµ<<<
Corollary 1(a) 1(b) 2(a)(i) 2(a)(ii) 2(a)(iii)

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DOI: 10.4236/ojbm.2019.72029 442 Open Journal of Business and Management
Table 2 . Summary on optimal solutions for Examples 2.1- 2.5.
MN≥
Example 2.1 2.2 2.3 2.4 2.5
M 1/6 1/3
N 1/12
dQ 20
µ 0.05 0.1 0.2 0.05 0.1
dT 0.19573 0.19286 0.19091 0.19573 0.19286
*T 0.23967 0.23658 0.24603 0.23617 0.23459
*Q 24.50358 24.59067 26.06367 24.14471 24.38186
()*TC T 81.06808 81.97423 82.41147 71.91272 73.05178
Case 3.2.2
*
d TT>
NMµ<<
*MT N<+ 3.2.2
*
d TT>
NMµ<<
*MT N<+ 3.2.2
*
d TT>
NMµ<<
*MT N<+ 3.2.3
*
d TT>
NMµ<<
*MT N>+ 3.2.3
*
d TT>
NMµ<<
*MT N>+
Corollary 2(b)(i) 2(b)(ii) 2(b)(iii) 2(c)(i) 2(c)(ii)

In summary, the longer the upstream trade credit period M, the less the pr e-
determined order quantity dQ, and the less the constant rate of a period time
µ, will cause the less the retailer’s total relevant cost. However, the larger the
downstream trade credit period N, will cause the larger the retailer’s total rel e-
vant cost.
6. Con clusion s
In this study, we develop an i nventory model in a supply chain with ramp -type
demand and trade credit financing linked to order quantity . The supplier offers a
permissible delay linked to order quantity, while the retailer also provides a
downstream trade credit period to its customers. We have obtained some the o-
retical results to characterize the optimal solutions and presented several nu-merical examples to illustrate the proposed models. The results reveal that 1)
*
d TT< will cause more retailer’s total relevant cost than other cases, since there
is no upstream trade credit period allowed, the retailer need to pay more interest
than earned. 2) *
d TT> and NM< , *MT N>+ will cause less retailer’s
total relevant cost than the others, since the retailer can earn more interest than paid. It’s more profitable for the retailer in such case. 3) The retailer’s total rel e-
vant cost increase as any one of the parameter val ues
µ, dQ, N increases, while
decreases as M increases. Thus, if upstream trade credit period is longer, then
the retailer’s total relevant cost will be less, it’s more benefit for the retailer.
The model can be extended in several ways, for example, we may consider the
item with a constant deterioration rate. Also, we can extend the model to allow
for shortages and partial backlogging. Finally, we could add the pricing, adve r-
tising and quality strategies i nto consideration.

H.-L. Yang

DOI: 10.4236/ojbm.2019.72029 443 Open Journal of Business and Management
Conflicts of Interest
The author declare s no conflicts of interest regarding the publication of this p a-
per.
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DOI: 10.4236/ojbm.2019.72029 446 Open Journal of Business and Management
Appendix : Proof of Theorem 1
For 11TC , from (14.1), let
() ()()() { }11
11d0dpTCF T h ab Tc I ab T N T C TTµµ = = + + + +− = .
Differentiating () FT with respect to T, we have
()()()
1 12
1
2d d0 0d dd d0d d j jj
pTC TC
T TTC FThc I ab TT Tµ
= == = ++> .
This implies that () FT is an increasing function of T. Furthermore,
()()()() { } 1100 0lim lim lim 0pTT TF T h ab Tc I ab T N T T CT µµ
→→ →= + + + + − =−∞< ,
since 110lim
TTC T
→=∞. And
()()() ()()11 lim lim 2 0ppTTF T hc I ab T CT hc I ab µµ
→∞ →∞= ++ − = ++> ,
since
()()
()()
()()() { }()( )
()()11
2223
2
2 22lim
lim23 2
22
lim 2 By L Hospital s Rule
2T
T
p
pT
pTC T
ab T abAh
b abcI a a b TN T T
h ab Tc I ab T N T
hc I abµµ µµ
µµµµµµ
µµ
µ→∞
→∞
→∞ +− = + ++
  ++ + ++ − +  
   
= +++ +
= ++’ ’
Therefore, there exists an unique solution such that ()0 FT=. From (15), we
know that the solution which minimizes 11TC . The other cases can also be
proved by the similar way. Thus, the Theorem is proved.