Intelligent Control and Automation, 2011, 2, 418-423
doi:10.4236/ica.2011.24048 Published Online November 2011 (http://www.SciRP.org/journal/ica)
Dynamic Pricing Model for the Operation of
Closed-Loop Supply Chain System
Jiawang Xu1,2, Yunlong Zhu1
1Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang, Ch ina
2School of Economics & Management, She nyang Aero space Unive rs ity, Shenyang, China
Received August 2, 2011; revised August 29, 2011; accepted September 6, 2011
Abstract
A class of closed-loop supply chain system consisting of one manufacturer and one supplier is designed, in
which re-distribution, remanufacturing and reuse are considered synthetically. The manufacturer is in charge
of recollecting and re-disposal the used products. Demands of ultimate products and collecting quantity of
used products are described as the function of prices and reference prices. A non-linear dynamic pricing
model for this closed-loop supply chain is established. A numerical example is designed, and the results of
this example verified the model’s validity to price for the operation of closed-loop supply chain system.
Keywords: Closed-Loop Supply Chain, Manufacturing/Remanufacturing, Pricing, Dynamic Programming
1. Introduction
There are numerous researches on closed-loop supply
chain system which address many various topics from
definition to practical cases in real industry. Many ana-
lytic and quantitative approaches are also found in vari-
ous problems such as forecasting, production planning/
control, inventory control/management, and location. For
example, the impact of remanufacturing in economy was
studi ed by Ferre r and Ayr es [1], an d more fund amentally,
Sundin and Bras [2] provided arguments for why used
products should be remanufactured. A good overview on
quantitative models for recovery production planning
and inventory control was given by Fleischmann et al.
[3]. Der Laan and Salomon [4] pr oposed a hybrid manu-
facturing/remanufacturing system with stocking points
for serviceable and remanufacturable products. Jayara-
man et al. [5] proposed a general mixed-integer pro-
gramming model and solution procedure for a reverse
distribution problem focused on the strategic level.
Moreover, Kim et al. [6] dealt with remanufacturing
execution at operational level. They proposed a general
framework for remanufacturing system in reverse logis-
tics environment and a mathematical model to maxi-
mize the total cost savings by optimally deciding the
quantity of parts to be processed at each remanufactur-
ing facilities, the number of purchased parts from sub-
contractor.
In the course of closed-loop supply chain system op-
erations, it is also very important to decide the prices of
ultimate products and the collecting prices of used prod-
ucts. Recently, Ray and Boyaci [7] studied the optimal
pricing/trade-in strategies for durable, remanufacturable
products. They focused on the scenario where the re-
placement customers were only interested in trade-ins. In
this setting, they studied three pricing schemes: 1) uni-
form price for all customers, 2) age-independent price
differentiation between new and replacement customers,
and 3) age-dependent price differentiation between new
and replacement customers. Gu et al. [8] studied the
price decisions of recycled products based on the reverse
supply chain between the manufacturer and the retailer
by using game theory. Two non-cooperative game equi-
librium (Stackelberg equilibrium and Nash equilibrium)
and a cooperative game equilibrium (coordination in
price decision) were obtained. Guide [9] researched the
optimal collect decision when used product quality is
uncertain, and Samar [10] studied the optimal price and
collect policy for reverse logistics in electronic business.
Liang et al. [11] proposed a model to ev aluate the acqui-
sition price of the used products. This model links the
used product acqu isition price with the sale price of used
product but assumes other costs such as logistics and
remanufacturing to be deterministic. Different from these
researches above, a non-linear dynamic programming
model is established in this paper, we study the dynamic

J. W. XU ET AL.419
)
pricing problems on u ltimate products and used products
for closed-loop supply chain with remanufacturing from
the view of operation management.
2. Descriptions
2.1. Framework for Closed-Loop Supply Chain
System
Considering a multi-stage closed-loop supply chain sys-
tem consisting of one manufacturer and one supplier, in
which re-distribution, remanufacturing and reuse are
considered synthetically. The manufacturer is in charge
of recollecting and re-disposal the used products. Ac-
cording to the status of used products, the manufacturer
has four alternatives for re-disposing: 1) Repair. Some
used products are r epaired and sold w ith new products in
the same market (here, we assume that product comes
from used product repairing is same as new product); 2)
Disassemble. Some used products are disassembled, and
parts are brought back to “as new” conditions; 3) De-
compose. Some used product can’t be repaired or disas-
sembled are decomposed to raw materials and be re-
turned to supplier as his raw materials; 4) Discard. Other
used prod u cts can’t be reused are discarded.
At each stage, the manufacturer has two alternatives
for supplying materials: either ordering the required ma-
terials to suppliers or overhauling used products and
bringing those back to “as new” condition s. The quantity
of manufacturer’s outputs is determined by customers’
demands and the outputs of used products repairing. The
supplier product the materials which manufacturer or-
dering, his required raw materials come from the raw
materials market and the outputs of used products de-
composing. The framework for closed-loop supply chain
system is shown as Figure 1.
2.2. Notations
Subscript:
j
i ultimate product; (1,,jJ
(1
material of manufacturer ; ,,)iI
t stage (1,,)tT


(1
;
h raw material ,,)

hH
Decision variable:
it
p price of ultimate product j at stage t;
j
t
r col l e c t ing pri ce of us e d p roduct j at stage t;
j
t
v sales quantity of ultimate product j at stage t;
j
t
z outputs of ultimate product j at stage t;
L
j
t
z inventories of ultimate product j at stage t;
L
it
y
inventories of material i at stage t;
it
l
b orders of material i at stage t;
it supplier’s delivery quantities of material i at stage
t;
it
x
supplier’s production quantities of material i at
stage t;
L
it
x
supplier’s inventories of material i at stage t.
Parameters:
j
t
R co llecting quantity of used product j at stage t;
j
t
q
d dem a nd o f ul t imate product j at stage t;
it price of material i at stage t;
z
j
c unit variable manufacturing cost of ultimate
product j ;
z
j
hy
i
h unit inventory cost of ultimate product j; unit inventory cost of material i;
k
j

capacity consuming rate of ultimate product j;
max
K
manufacture’s maximum production capacity;
'
0
L
j
z initial inventory of ultimate product j;
z
j
o occupied inventory of unit ultimate product j;
xma
L
z total inventory level of ultimate products;
y
ij
s
BOM coefficient of ultimate product j to material
i; '
0
L
i
y
initial inventory of material i;
y
oimax
L
y occupied inventory of unit material i; total inventory level of materials;
ht
m price of raw material h at stage t;
x
i
r
c unit variable cost of material i;
hi
s
BOM coefficient of material i to raw material h;
x
i
h unit inventory cost of material i;
g
i

capacity consuming rate of material i;
ma
Gx supplier’s maximum production capacity avail-
able; '
0
L
i
x
initial inventory of material i;
x
i
o occupied inventory of unit material i;
Figure 1. Framework for closed-loop supply chain system.
Order
Deliver
Decompose Disassemble
Demands
Used products
Repair
Raw materials
market Supplier Customers Manufacturer
Discard

J. W. XU ET AL. 420
max
L
x
total inventory level of materials;
ht
s
supply quantity of raw material h at stage t;
j
t average price of raw materials decomposed from
used product j at stage t;
w
1
j
t

, 2
j
t

and 3
j
t

are repairing rate, disassembling
rate and decomposing rate of used product j at stage t
respectively, and;
123
1  
jt jt jt
1

j
t, 2
j
t

and 3
j
t

are unit repairing cost, disassem-
bling cost and decomposing cost of used product j at
stage t respectively, and1
j
t

<2
j
t

<3
j
t

;
j
d demand of ultimate product j neglecting the im-
pacts of price and vendit i on ef fo rt ;
j
e collecting quantity of used prod uct j neglecting the
impacts of collecting price and collecting efforts;
j

demand sensitive coefficient of ultimate product j
to the price;
j

demand sensitive coefficient of ultimate product j
to the variation of price;
j

collecting quantity sensitive coefficient of used
product j to collecting price;
j

collecting quantity sensitive coefficient of used
product j to the variation of collecting price.
2.3. Hypotheses
1) The process of product manufacturing and used prod-
uct remanufacturing are synchronously, and the outputs’
quality from remanufacturing is the same as that of
manufacturing, that is, the selling prices of the outputs
from manufacturing and from remanufacturing are uni-
form;
2) Demand of ultimate product decreases along with
the raising of selling price as well as the increment of
selling price. For research convenient, we set
,1
()
jtjjjtjjtj t
dd ppp


 ;
3) Collecting quantity of used product increases along
with the raising of collecting price as well as the incre-
ment of collecting price. We set
,1
()
jtjjtjtj t
Re rrr


 .
3. Pricing Model
In the closed-loop supply chain system shown as Figure
1, we consider three operation objectives: 1) At each
stage, operation of the closed-loop supply chain system
should realize coordination of participants, namely the
supplier’s delivery quantity is nicely equal to the manu-
facturer’s order quantity. 2) The Manufacturer pursues
profits maximization. 3)The supplier pursues profits
maximization. These objectives can be described as fol-
lows:
,
it it
bl it
yL
y
(1)
11 1
3
11 11
112233
11
max(( )
()
()
TJ I
PzzL
jtjtjjtjjtit itiit
tj i
TJHI yr
jtjtjkij hi
tj hi
TJ
jtjt jtjt jtjt jtjt
tj
Cpvczhzqbh
wR ss
Rr

  
 
 











 

 
 

(2)
11 1
3
11 11
max([()] )
()
TI H
SxLx
it itiitihthiit
ti h
TJHIyr
jt jt jtijhi
tj hi
Cqlhxcms
wR ss 
 
 
r
x











 
 
(3)
where,
P
C is Manufacturer’s profit, and is sup-
plier’s profit.
S
C
Transform 3 expressions above to objective program-
ming form, the operation model for the operation of
closed-loop supply chain system can be rewritten as fol-
lows.
Objective function:


11
objective 1:minTI
it it
ti
dd

 
 ;
objective 2:min
P
d

;
objec tive 3:m indS

;
where, it d

and it d

are supplier’s delivery quantity of
material i at stage t when it’s deficient and superfluous
respectively,
P
d

is deficient quantity of manufacturer’s
objective profit, S d

is deficient quantity of supplier’s
objective profit.
Objective restrictions:
0
it ititit
bld di
 ,t

  (4)
P
P
P
CddM


P
)
0
yL
y
(5)
11 1
3
11 11
1122 33
11
((
()
()
TJ I
PzzL
jtjtjjtjjtit itiit
tj i
TJHIyr
jtjtjkij hi
tj hi
TJ
jtjtjtjtjt jtjt jt
tj
Cpvczhzqbh
wR ss
Rr

 
 
 











 

 
 

(6)
SSS
CddM


S
])
r
x
(7)
11 1
3
11 11
([()
()0
TI H
SxLx
it itiitihthiit
ti h
TJHIyr
jtjtjtij hi
tj hi
Cqlhxcms
wR ss 
 
 








 


 
 
(8)
where,
P
M
is a maximum objective profit the manu-
facturer expected and it’s a constant, S
M
is a maximum

J. W. XU ET AL.421
objective profit the supplier expected and it’s also a con-
stant.
Absolute restrictions:
Supplier’s production capacity satisfies
max
1
Ig
iit
i
x
G 

t
(9)
Supplier’s inv entory of product satisfies
,1 ,
LL
iti titit
x
xxli

t (10)
'
00
LL
ii
x
xi

(11)
max
1
IxLL
iit
ioxxt 
 (12)
The restricted condition of supplier’s material supply-
ing


3
11 ,
IJ
ry
hiitijjt jtht
ij
s
xsRs 




 ht

t
t
j

t
(13)
Manufacturer’s capacity restriction at each stage
max
1
Jk
jjt
jzK


 (14)
Manufacturer’s inventory of ultimate product at each
stage
1
,1 ,
LL
jtj tjtjtjtjt
zzzvRj 

  (15)
'
00
LL
jj
zz
max
(16)
1
JzLL
jjt
joz zt 
 (17)
Manufacturer’s invent ory of material satisfies

2
,1 1,
J
LL y
it ititijjt jtjt
j
yy bszRi 


 (18)
'
00
LL
ii
y
yi

t
)
(19)
max
1
IyL L
iit
ioy yt 
 (20)
Manufacturer’s actual sales satisfies
,
jt jt
vd j (21)
Relationship between demand of ultimate product and
its price
,1
(
jtjjjtjjtj t
dd ppp



  (22)
Relationship between collecting quantity of used
product and its collecting price
,1
(
jtjjtjtj t
Re rrr)



  (23)
Nonnegative conditions:
,,,,,,,,,,,
,,,,,,0 ,,
LL L
itjt jt itititjtjt ititP
PSs
PSS jtit
bz zylx prddd
dddCCvx ijt

  (24)
4. Simulations
Set J = 1(one ultimate product), I = 1(one material), H =
1(one raw material) and T = 4 (four sta ges).
Demand of ultimate product neglecting the impacts of
sale price and vendition effort is d = 450, collecting
quantity of used product neglecting the collecting price
and collecting efforts is e = 100, demand sensitive coef-
ficients of product to the sale’s price and its variation are
0.5


and 5


respectively. Collecting quantity
Sensitive coefficients of used product to collecting price
and its variation are 2


and 20

 respectively.
Expected profits of manufacturer and supplier are all
, that is,
8
110
P
M
= and S
8
110
M
= . The
supply quantity of raw material at each stage are all 400,
initial price of ultimate product is 0, the prices
of raw material at each stage are all 40.
8
110
030p
Repairing rate, disassembling rate and decomposing
rate of used product are, and
t respectively. Initial collecting
price of used product is 0, unit cost for repairing,
disassembling and decomposing used product are
120%
t
 
)20
r
230%
t
 
330%(1,2,3,4t
 
13
t


,
28
t


and respectively. The raw material
from decomposing used product can be sold to supplier,
and its price are all
320
t
 
30
t
w

(t). Other pa-
rameters are set as below. 1, 2,3, 4
max max
400, 400KG
max max max
50, 50, 50


LLL
zyx
'''
000
0, 0, 0


LLL
zyx
10, 15, 75


xz
ccq
1, 2, 3


xy z
hh h
1, 1, 1, 1



gkry
ss
1, 1, 1


xyz
ooo
We used LINGO9.0 for solving our multi-objective
dynamic programming model. The results are shown in
Tables 1 and 2.
From results in Table 1, we can see that supplier’s de-
livery quantity of material equals to manufacturer’s order
quantity at each stage. The amount of sales quantity and
inventory is bigger than the output of ultimate product at
each stage, and the difference of these is increasing along
with operation stage. The reasons for this phenomenon is
that, as time increases, price of ultimate product and col-
lecting price of used product become gradually rational-
ize, demand of the ultimate product increases gradually,
and the collecting quantity of used product is also in-
creasing.
As results in Table 2 shown, when initial price of ul-
timate product is set 300, except stage 1, the price of
ultimate product decreases as time goes on, but the de-
mand of ultimate product increase. When initial collect-
ing price of used product is set 20, except stage 1 and
stage 2, collecting price of usd product increases gradu- e

J. W. XU ET AL. 422
Table 1. Optimal operation strategy for the closed-loop supply chain system.
Manufacturer (Profits: 297061.1) Supplier (Profits: 2 5 3 41 .0)
Stage outputs of
ultimate
product
Sales quantity
of ultimate
product
Inventories of
ultimate
product
Inventory of
materials Orders of materials Materials deliveries Materials production Materials inventory
1 187.5 190.4 0 0 183.0 183.0 183.0 0
2 400.0 363.4 50 0 379.9 379.9 379.9 0
3 400.0 433.7 50 0 349.4 349.4 349.4 0
4 400.0 513.2 0 0 305.2 305.2 305.2 0
Table 2. Demands of ultimate product and collecting quantities of used product.
Stage Prices of ultima te produ ctDem and s of ultimate product Collecting prices of used
product Collecting quantities of used product
1 319.9 190.4 9.6 14.8
2 306.6 363.4 5.2 67.0
3 281.7 433.7 10.1 168.6
4 244.6 513.2 26.4 316.0
ally, and so does collecting quantity of used product. The
varying tendency for the price of ultimate product and
the collecting price of used product are shown as Figures
2 and 3 respectively.
0
5
10
15
20
25
30
1 3 5 7 9111315171921
According to parameters above, whatever initial price
of ultimate product and collecting price of used product
are, we can obtain the same conclusions from the model:
in the long run, through the regulations of price itself and
price reference effect, the price of ultimate product is
fixed to 157.2707 and the collecting price of used prod-
uct is fixed to 27.38714.
5. Conclusions Figure 3. The varying tendency for collecting price of used
product.
From the view of operation researches, we consider a
class of closed-loop supply chain system with product
remanufacturing. The system is consists of one manu-
facturer and one supplier. The manufacturer is in charge
of recollecting and re-disposal the used products. De-
mands of ultimate products and collecting quantity of
used products are described by using prices and refer-
ence prices. A non-linear dynamic pricing model for
system operation is established. The model is constructed
as a dynamic programming problem and satisfying sev-
eral conflict objectives, such as the operating coordina-
tion of system, making the maximum profit of all par-
ticipants as much as possible. The model’s validity to
dynamic pricing for closed-loop supply chain system is
verified by a numerical example. The results of the nu-
merical example shows that, in the long run, through the
regulations of price itself and price reference effect, the
price of ultimate product and the collecting price of used
product fix to a certain value without reference to the
initial price of ultimate product and the collecting price
of used product.
0
50
100
150
200
250
300
350
13 5 79111315171921
6. Acknowledgements
This work was supported by Humanities & Social Sci-
ences Planning Fundation of Ministry of Education Grant
#11YJA630165 to J. W. Xu, and China Postdoctoral
Science Foundation Grant #2008044170 to J. W. Xu.
Figure 2. The varying tendency for the price of ultimate
product.

J. W. XU ET AL.423
7. References
[1] G. Ferrer and R. U. Ayres, “The Impact of Remanufac-
turing in the Economy,” Ecological Economics, Vol. 32,
No. 3, 2000, pp. 413-429.
doi:10.1016/S0921-8009(99)00110-X
[2] E. Sundin and B. Bras, “Making Functional Sales Envi-
ronmentally and Economically Beneficial through Prod-
uct Remanufacturing,” Journal of Cleaner Production,
Vol. 13, No. 9, 2005, pp. 913-925.
doi:10.1016/j.jclepro.2004.04.006
[3] M. Fleischmann, J. M. Bloemhof-Ruwarrd, R. Dekker, et
al., “Quantitative Models for Reverse Logistics: A Re-
view,” European Journal of Operational Research, Vol.
103, No. 1, 1997, pp. 1-17.
doi:10.1016/S0377-2217(97)00230-0
[4] E. Der Laan and M. Salomon, “Production Planning and
Inventory Control with Remanufacturing and Disposal,”
European Journal of Operational Research, Vol. 102, No.
2, 1997, pp. 264-278.
doi:10.1016/S0377-2217(97)00108-2
[5] V. Jayaraman, R. A. Patterson and E. Rolland, “The De-
sign of Reverse Distribution Networks: Models and Solu-
tion Procedures,” European Journal of Operational Re-
search, Vol. 150, No. 1, 2003, pp. 128-149.
doi:10.1016/S0377-2217(02)00497-6
[6] K. Kim, I. Song, J. Kim and B. Jeong, “Supply Planning
Model for Remanufacturing System in Reverse Logistics
Environment,” Computers & Industrial Engineering, Vol.
51, No. 2, 2006, pp. 279-287.
doi:10.1016/j.cie.2006.02.008
[7] S. Ray and T. Boyaci, “Optimal Prices and Trade-in Re-
bates for durable, Remanufacturable Products,” Manu-
facturing & Service Operations Management, Vol. 7, No.
3, 2005, pp. 208-228. doi:10.1287/msom.1050.0080
[8] Q. Gu, T. Gao and L. Shi, “Price Decision Analysis for
Reverse Supply Chain Based on Game Theory,” Systems
Engineering-Theory & Practice, Vol. 20, No. 3, 2005, pp.
20-25.
[9] T. Guide, “Matching Demand and Supply to Maximize
Profits from Remanufacturing,” Manufacturing & Service
Operations Management, Vol. 5, No. 4, 2003, pp. 303-
316. doi:10.1287/msom.5.4.303.24883
[10] K. M. Samar, “Reverse Logistics in e-Business,” Interna-
tional Journal of Physical Distribution & Logistics Ma-
nagement, Vol. 34, No. 2, 2004, pp. 70-88.
[11] Y. Liang, S. Pokharel, G. H. Lim, “Pricing Used Products
for Remanufacturing,” European Journal of Operational
Research, Vol. 193, No. 2, 2009, pp. 390-395.
doi:10.1016/j.ejor.2007.11.029