The Stock and the Purchase Strategy on the Nonlinear Diversification of Price to
Time with Retailer Dominating
Shu-guang Han , Yi Xu, Jue-liang Hu
Department of mathematics, Zhejiang Sci-Tech University, Hang Zhou, China
(zist001@163.com)
Abstract -With the socio-economic changing, the retailer
plays a more significant role in the supply chain. In this
paper, a scenario is considered in which the retailer
dominates a two-echelon supply chain which consists of one
retailer and some manufacturers, and the price of raw
materials is a nonlinear function to time. The retailer first
determines the order quantity and the retail price, and then
places some orders to the manufacturers. Then the
manufacturers have to decide whether to accept the orders
according to the enterprise profit and production capacity .
We mainly develop the supply chain members' profit models
and derive the retailer's optimal ordering quantity and
retail price and the manufacturers' optimal orders which
they accepted to maximize their profits.
Keywords - supply chain, retailer dominate, nonlinear
function, ordering strategy
I. INTRODUCTION
Supply chain management (SCM) is the management
of a network of interconnected businesses involved in the
ultimate provision of product and service packages
required by customers[1]. Generally SCM can be divided
into manufacturer-led mode (MS) and retailer-led mode
(RS), according to the status of manufacturer and retailer
in the supply chain[2]. In the traditional industrial
economy era, the manufacturer that controls the
production of raw materials and technology can easily
become the dominant of the supply chain. The study of
supply chain ordering and pricing problem is mostly
concentrated on the manufacturer-driven supply chain.
References[3,4]considered manufacturer dominant the
supply chain, and developed profit models to investigate
the impact of price and remaining lifetimes for stock. Jian
Liu[5] assumed the price be linear function to time, studied
the laws of the prices, demands and profits.
With the socio-economic changing, the retailer plays
more and more important role in the supply chain, the
traditional mode that manufacturers as the leader is
gradually transformed into a retailer-driven supply chain
mode, large supermarket such as Wal-Mart and Carrefour
are the typical dominant retailers. Some scholars have
made research in this respect. Scott Webster, and Z.
Kevin Weng[6] consider the manufacturer-controlled
scenario and the distributor-controlled scenario.
References[7,8] constructed a two-period model to discuss
pricing and ordering problems for a dominant retailer.
Kebing Chen , and Tiaojun Xiao[9] considered the supply
chain consisting of one manufacturer, one dominant
retailer and multiple fringe retailers. Yang Li, Weixiang
Guo, and Lihai Wang [10] developed the equilibrium of the
retailer-led two class supply chain cooperate and noncooperative models under market demand uncertainty.
However, the change of manufacturer’s production cost is
seldom considered in the above references. Xianhao Xu,
and Siyue Nie[11] considered the supplier's production cost
and established the ordering decision model of retailer
dominating the supply chain of short life-cycle products
with some special production cost functions.
At present, some raw materials' prices have non-linear
variation trend that first increasing and then decreasing,
the manufacturer's production cost is also rapidly
changing. In this paper, we assume the price of the raw
material which the manufacturer needed is changing in
non-linear cycle with the retailer dominating the supply
chain. We establish the ordering and pricing models of
the two-echelon supply chain which is composed of one
retailer and some manufacturers.
II. NOTATIONS and ASSUMPTIONS
Consider such a supply chain system for a product
which consists of one retailer and some manufacturers,
and the retailer dominates the whole supply chain. They
make their decisions as follows: retailer determines the
order quantity and the retail price judging from the
market and other information, and then place the orders to
the manufacturers. Considering the enterprise profit and
the production capacity, the manufacturers decide
whether to accept the orders or not, if the manufacturers
accept the orders, then they determine the optimal order
quantity to maximize their profits. Some notations for
parameters and assumptions used in this paper are
following.
A. Notations
Error! Reference source not found.retail price per unit
product to consumers
 wholesale price per unit product that the manufacturer
charges from the retailer
retailer's
ordering quantity
Q
Error! Reference source not found.the price per unit raw
materials that the manufacturer
purchases
Q ' manufacturer's ordering quantity of the raw materials
Error! Reference source not found. raw materials'
amount of manufacturing unit product
Error! Reference source not found. the time of
manufacturing unit product
Error! Reference source not found. the cost of
manufacturing unit product except the
purchase cost
Error! Reference source not found. the time range of
raw materials' price changing
Error! Reference source not found. the unit cost of
retailer's stock
Error! Reference source not found. the profit of the
manufacturer
Error! Reference source not found. the optimal profit
of the manufacturer
Error! Reference source not found. the optimal
ordering quantity that the manufacturer
accepts
Error! Reference source not found. the profit of the
retailer
Proposition 1. When t1  T 2 and t1  t2  T  Q  t2  t1
B. Assumptions
1) The retailer's revenue is composed of purchase cost
and marginal profit, so the retailer can set retail price
depending on the wholesale price and marginal price. We
assume that:Error! Reference source not found. ,
where 01.
2) Demand with price dependence is assumed to follow
the additive demand type[12]:Error! Reference source
not found., where Error! Reference source not
found.andError! Reference source not found.are
respectively a constant intersection and slop of the price
curve function.
3) Suppose the diurnal price of raw materials is related to
the time and demand, the raw materials' price function to
time is Error! Reference source not found. , where
T  0 and c  0 .
2
let d M  0 , we get Q  a  (1   )(kt1  kTt1  n) .
0
2b  2ck 2 (1   )
dQ
Discussing the value of Q* according to the signs of
the second order derivative of  M , that are
III. MODEL BUILDING and SOLVING
A. Manufacturer’s Optimal Decision
The retailer determines the products' order quantity
and the retail price, and then place orders to the
manufacturers at time t1 . The manufacturers decide
whether to accept such orders according to the enterprise
profit and delivery date or not. If the manufacturers
accepted the orders, then they are required to delivery
products at time t2 . The manufacturers begin to product
after receiving the orders and deliver once after finished,
therefore the manufacturers' stock cost is not considered
here.
The manufacturer's total cost is the sum of the
purchase cost and the production cost, thus the profit of
the manufacturer is
 M  Q  Q'  nQ .
m
m
hold, the maximum profit of the manufacturer is
b
a
 *  (ck 2 
)Q*2  (
 kt 2  kTt  n)Q* .
1 
1 
Proof: When t1  T 2 , t2  t1  mQ and T 2  t1  t2  mQ  T 2
hold, we can deduce t1  t2  T  Q  t2  t1 . The purchase
m
m
M
1
1
cost of raw materials is the lowest, if purchasing raw
materials Q ' at the time t1 , and they can deliver in time.
Thus
 M  Q   Q '  nQ
1
(a  bQ)Q  (t12  Tt1  ckQ)kQ  nQ
1 
b
a
 (ck 2 
)Q 2  (
 kt12  kTt1  n)Q
1 
1 

The maximum profit of the manufacturer is
b
a
)Q*2  (
 kt12  kTt1  n)Q* .
1 
1 
d M
b
a
 2(ck 2 
)Q 
 kt12  kTt1  n
dQ
1 
1 
 M *  (ck 2 
So
,
and
b
b
b
 0 and ck 2 
 0 , ck 2 
0 .
1 
1 
1 
Lemma1. If ck 2  b  0 holds, then Q*  Q .
1 
Proof: When ck 2  b  0 and  M  ( a  kt12  kTt1  n)Q ,
1 
1 
ck 2 
in
order
to
make
the
model
significant,
let
a
 kt12  kTt1  n  0 , the function of profit is monotone
1 
increasing over the order quantity Q . Thus Q*  Q
and
 M *   M (Q* ) .
Lemma2. If ck 2  b  0 holds, then Q*  Q .
1 
Proof: When
2
b
ck 2 
 0 , we can get d 2M  2(ck 2  b )  0 .
1 
dQ
1 
Because of
a
 kt12  kTt1  n  0 , the function of profit
1 
is monotone increasing over the order quantity Q when
Q  0 . Thus we can get Q*  Q and  M *   M (Q* ) .
Lemma3. When ck 2  b  0 , if t1  t2  T  Q  t2  t1 and
0
1 
m
m
t  t T
t t
Q  Q0 hold, then Q*  Q0 ; if 1 2
 Q0  2 1 and
m
m
Q  Q0 hold, then Q*  Q ; if Q0  t2  t1 , then Q*  Q ;
m
if Q  t1  t2  T , then Q*  t1  t2  T .
0
m
m
Proof:
When
ck 2 
b
0
1 
,
we
can
get
d 2 M
b
 2(ck 2 
)  0 ,  M have the maximum
dQ 2
1 
value at the point Q0 , then we can deduce the lemma 3.
Proposition2. When t1  T 2 and 0  Q  t1  t2  T hold,
m
let d M  0 , we can get
dQ
Q1  
b
)Q*2
1 
a
(
 kt2 2  kTt2  n)Q*
1 
Proof:
When
,
and
t2  T 2  mQ
t1  T 2
T 2  t1  t2  mQ  T 2 hold, then 0  Q  t1  t2  T . The
m
raw materials' purchase cost is the lowest, when
purchasing raw materials Q ' at time t2  mQ , and they can
deliver in time. Thus the manufacturer's profit is
 M  Q  Q '  nQ
1
(a  bQ)Q  [(t2  mQ) 2  T (t2  mQ)  ckQ)]kQ  nQ
1 
b
a
 km2Q3  (kmT  ck 2  2kmt2 
)Q 2  (
 kt2 2  kTt2  n)Q
1 
1 

The maximum profit of the manufacturer is
b
)Q*2
1 
a
(
 kt2 2  kTt2  n)Q*
1 
d M
b
 3km 2Q 2  2(kmT  ck 2  2kmt2 
)Q .
dQ
1 
a

 kt2 2  kTt2  n
1 
6km 2
4(kmT  ck 2  2kmt2 

Q3  0
kmT  ck 2  2kmt2 
b
1 
2km 2
b 2
a
)  4km 2 (
 kt2 2  kTt2  n)
1


1



2km 2
b
kmT  ck 2  2kmt2 
1 
Q5  
2km 2
b 2
a
(kmT  ck 2  2kmt2 
)  4km 2 (
 kt2 2  kTt2  n)
1


1



2km 2
(kmT  ck 2  2kmt2 
Lemma5. When
b 2
a
)  3km 2 (
 kt2 2  kTt2  n)  0
1 
1 
b
hold,
and
also
kmT  ck 2  2kmt2 
0
1 
(kmT  ck 2  2kmt2 
and
Next we discuss the value of Q for two cases:
b 2
a
(kmT  ck  2kmt2 
)  3km 2 (
 kt2 2  kTt2  n)  0
1 
1 
2
and
b 2
a
)  3km 2 (
 kt2 2  kTt2  n)  0 .
1 
1 
Lemma4. If
(kmT  ck 2  2kmt2 
b 2
a
)  12km 2 (
 kt2 2  kTt 2  n)
1 
1 
6km 2
Let  M  0 , we can get
*
(kmT  ck 2  2kmt2 
b
)
1 
b 2
a
)  12km 2 (
 kt2 2  kTt 2  n)
1


1



6km 2
b
2(kmT  ck 2  2kmt2 
)
1 
Q2  
2
6km
Q4  
 M *  km 2Q*3  ( kmT  ck 2  2kmt 2 
2(kmT  ck 2  2kmt2 
4(kmT  ck 2  2kmt2 
the maximum profit of the manufacturer is
 M *  km 2Q*3  ( kmT  ck 2  2kmt 2 
b 2
a
)  3km 2 (
 kt2 2  kTt2  n)  0 ,
1 
1 
(kmT  ck 2  2kmt2 
b 2
a
)  3km 2 (
 kt2 2  kTt2  n)  0
1 
1 
holds, then Q*  Q .
Proof: When
*
a
 kt2 2  kTt2  n  0 holds, we have Q  Q .
1 
Proof:
When
kmT  ck 2  2kmt2 
b
0
1 
and
a
 kt2 2  kTt2  n  0 , we can get Q1  0 , Q2  0 . Thus
1 
when Q  0 , it has d M  0 , the function of profit is
dQ
monotone increasing over the order quantity
t  t  T . In this case, we have Q*  Q
Q ,with 0  Q  1 2
m
b 2
a
(kmT  ck 2  2kmt2 
)  3km 2 (
 kt2 2  kTt2  n)  0
1 
1 
holds, we can obtain that d M  0 for any Q because
and M *  M (Q* ) .
of 3km2  0 . So the function of profit is monotone
increasing over the order quantity Q if 0  Q  t1  t2  T .
and kmT  ck 2  2kmt  b  0 ,
2
1 
a
t  t T
*
2
,
 kt2  kTt2  n  0 if 1 2
 Q1 , then Q  Q ; if
dQ
m
Thus we get Q*  Q and
When
 M *   M (Q* ) .
Lemma 6. When
(kmT  ck 2  2kmt2 
b 2
a
)  3km 2 (
 kt2 2  kTt2  n)  0
1 
1 
1 
m
t1  t2  T
t1  t2  T , then Q*  Q ; if
and
Q1 
 Q6
Q1  Q 
1
m
m
t  t  T , then Q*  Q , where
t1  t2  T
 Q6 and Q6  Q  1 2
m
m
 M (Q6 )   M (Q1 ) .
From kmT  ck 2  2kmt  b  0 and
2
1 
a
2
,
we
can
get
0  Q1  Q4  Q2  Q5 .
 kt2  kTt2  n  0
1 
If 0  Q  Q1 or Q  Q2 , the profit function is monotone
Proof:
increasing over the order quantity Q ; if Q1  Q  Q2 , the
function of profit is monotone decreasing over the order
quantity Q . So we get the lemma 6.
Lemma 7. When
b 2
a
(kmT  ck  2kmt2 
)  3km 2 (
 kt2 2  kTt2  n)  0
1 
1 
and a  kt 2  kTt  n  0 hold, if t1  t2  T  Q , then the
2
2
5
m
1 
2
manufacturer refuses the order; if t1  t2  T  Q and
5
m
t  t T
Q5  Q  1 2
m
, then we have Q*  Q .
Proof: When a  kt 2  kTt  n  0 , we
2
2
1 
get Q4  Q1  0  Q2  Q5 . Thus if 0  Q  Q2 , it has d M  0 ,
dQ
the function of profit is monotone increasing over the
order quantity Q ; if Q  Q2 , it has d M  0 , the function of
dQ
profit is monotone decreasing over the order quantity Q .
When 0  Q  Q5 , we can get  M  0 because of
 M (0)  0 and  M (Q5 )  0 .
So the manufacturer's profit
less than zero.
Therefore if t1  t2  T  Q and 0  Q  t1  t2  T , it has
5
m
m
 M  0 , the manufacturer will refuse the order; if
t1  t2  T
and Q  Q  t1  t2  T
 Q5
5
m
m
, it has  M  0 , the
function of profit is monotone increasing over the order
quantity Q . In this case, Q*  Q and M *  M (Q* ) hold,
otherwise the manufacturer refuses the order.
Proposition3. When t1  T 2 and 0  Q  t2  t1 , the
m
maximum profit of the manufacturer is
 M *  km 2Q*3  (kmT  ck 2  2kmt2 
b
)Q*2
1 
a
(
 kt2 2  kTt2  n)Q*
1 
Proof: When t1  T 2 and t1  mQ  t2 , which
means 0  Q  t2  t1 . The purchase cost of raw materials is
m
lowest, if purchasing raw materials Q ' at time t2  mQ , and
they
can
deliver
in
the
delivery
period.
Thus
 M  Q  Q '  nQ
1

(a  bQ)Q  [(t2  mQ) 2  T (t2  mQ)  ckQ)kQ  nQ
1 
b
a
 km2Q3  (kmT  ck 2  2kmt2 
)Q 2  (
 kt2 2  kTt2  n)Q
1 
1 
The maximum profit of the manufacturer is
 M *  km 2Q*3  (kmT  ck 2  2kmt 2 
b
)Q*2 .
1 
a
(
 kt2 2  kTt2  n)Q*
1 
d M
b
2
2
2
We get dQ  3km Q  2(kmT  ck  2kmt2  1   )Q .
a

 kt2 2  kTt2  n
1 
Lemma8. When
(kmT  ck 2  2kmt2 
b 2
a
)  3km 2 (
 kt2 2  kTt2  n)  0
1 
1 
we have Q*  Q .
Lemma9. When
b 2
a
)  3km 2 (
 kt2 2  kTt2  n)  0 ,
1 
1 
b
a
kmT  ck 2  2kmt2 
 0 and
 kt2 2  kTt2  n  0 ,
1 
1 
if 0  Q  t2  t1 , then we get Q*  Q , and otherwise
m
t2  t1 .
*
Q 
m
(kmT  ck 2  2kmt2 
Lemma10. When
b 2
a
)  3km 2 (
 kt2 2  kTt2  n)  0
1 
1 
and kmT  ck 2  2kmt  b  0 , a  kt 2  kTt  n  0 ,
2
2
2
1 
1 
(kmT  ck 2  2kmt2 
if t2  t1  Q , then
m
1
Q*  Q ;
if Q  t2  t1  Q and
1
6
m
t2  t1 , then Q*  Q ; if t2  t1
 Q6 and
Q1  Q 
1
m
m
*
t t
Q6  Q  2 1 , then Q  Q ,where  M (Q6 )   M (Q1 ) ; if
m
Q
t2  t1 , then * t2  t1 .
Q 
m
m
Lemma11. When
b 2
a
)  3km 2 (
 kt2 2  kTt2  n)  0
1 
1 
and a  kt 2  kTt  n  0 , if t2  t1  Q , the manufacturer
2
2
5
1 
m
t

t
will refuse the order; if 2 1  Q and Q  Q  t2  t1 ,
5
5
m
m
*
t

t
t

t
*
then Q  Q ; if Q  2 1 , then Q  2 1 .
m
m
(kmT  ck 2  2kmt2 
B. Retailer’s Optimal Decision
The retailer determines the order quantity and the retail
price according to the market and some other information.
Suppose the daily market demand is a constant, so we can
get the product stock of retailer using EOQ model.
Proposition4. The maximum profit of the retailer is
R  
b 2
a h
Q (
 )Q
1 
1  2
Proof: The retailer's total cost is the sum of the purchase
cost and the stock cost, thus the profit of the retailer is
Q
h
 Q  Q
2
2
b 2
a h

Q (
 )Q
1 
1 2
 R  pQ  Q  h
Lemma12.
The retailer's optimal ordering quantity
2a  h  h .
a
h
h
Q

 , optimal retail price p 
2b 4 b 4b
4
Proof: It is easy to get d R   2 b Q   a  h .
dQ
1 
1  2
And let d R  0 , we get Q  a  h  h .
2b 4 b 4b
dQ
2
Because d  R  0 ,  R have the maximum value at the
2
dQ
point Q . At the moment the price is
2a  h  h .
p  a  bQ 
4
IV. CONCLUSION
In this paper, we study the ordering and pricing
problem of a two-echelon supply chain which consists of
one retailer and some manufacturers with the retailer
dominating the whole supply chain. We assume that the
products' retail price is sensitive to the demand and the
raw materials' price is related to time, and develop supply
chain members' profit models.
The differences of this paper from others are that the
price of raw materials is a nonlinear function to time, and
the order quantity and the wholesale price are determined
by the retail in the supply chain system. However the
models in this paper also have some limitations, such as
the daily market demand is a constant. In the future
research, we could extend the models that the product
daily demand is changeable.
ACKNOWLEDGMENT
This research is supported by National Science
Foundation under Grant No. 11071220 and Zhejiang
Province Science Foundation under Grant No.Y6110091,
Y6090554 and Y6090175.
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