J. Serv. Sci. & Management, 2008, 1: 123-127
Published Online August 2008 in SciRes (www.SRPublishing.org/journal/jssm)
Copyright © 2008 SciRes JSSM
BBImpact of Supply Chain Coordination for Deteriorating
Goods with Stock-Dependent Demand Rate
Chuanxu Wang
PSchool of Economy and Management
Shanghai Maritime University Shanghai
China
Email: cxwang@shmtu.edu.cn
P
ABSTRACT
To analyze effects of supply chain coordination for deteriorating goods with stock-dependent demand rate, this paper
presents decision models for order quantity and ordering cycle under two scenarios( decentralized supply chain, cen-
tralized supply chain). Numerical study is carried out to demonstrate the effectiveness of the proposed models, and to
analyze the impact of supply chain coordination on supply chain profit. Sensitivity analysis is performed to study the
impact of different parameters a ssociated with the model, su ch as the rate o f deterioration, the reta iler’s purcha se cost,
the manufacturer’s production cost, the retailer’s and manufacturer’s holding cost on the supply chain profit increase
percentages generated by t he sup pl y c h a i n c oordinati on.
Keywords: supply chain coordination, deteriorating goods, ordering cycle, stock-dependent demand
1. Introduction
In real life, many inventory goods, such as agricultural
products, fashion goods, drugs and high-tech products,
are subject to depletion through spoilage, shrinkage, de-
cay and obsolescene [1]. The deterioration is quite preva-
lent and should not be disregarded. Inventory manage-
ment for deterioration goods has received many attentions
from researchers and practitioners. Most of the existing
researches focus on the EOQ-based inventory decision
models. Ghare and Schrader [2] presented the EOQ
model by considering the combined effects of demand,
usage and linear decay. Covert and Philip [3]used the
variable deterioration rate of the two-parameter Weibull
distribution, to formulate the inventory decision model
under the assumptions of a constant demand rate, with no
shortages allowed. Philip [4] modified this model by us-
ing the deterioration rate of the three-parameter Weibull
distribution. Tadikamalla [5] adopted gamma distributed
deterioration under constant demand over time, without
shortages. Moon and Lee [6] presented the EOQ model
with a normally distributed deterioration rate. Other dete-
rioration inventory models have extended prior research
by considering a time-varying demand function, with or
without shortages. Dave and Patel [7] proposed an EOQ
model under time-proportional demand, with no shortages
allowed. Sachan [8] extended their model by considering
shortages. Bahari-Kashani [9] generalized the problem by
permitting variations in both replenishment cycle length
and order quantity. Bose et al [10]developed an EOQ
model for deterioration items incorporating the effects of
inflation, time value of money, a linearly time-dependent
demand rate and shortages. Replenishment decision mod-
els under time-proportional demand and exponentially
decaying deterioration rate was developed in [1].
It is observed that large quantities of consumer goods
displayed in a supermarket generate higher demands. Sil-
ver and Peterson [11] noted that the sales at the retail
level tend to be proportional to the inventory displayed.
Gupta and Vrat [12], Mandal and Phaujdar [13], Baker
and Urban [14], Datta and Pal [15], etc developed the
EOQ models with stock-dependent demand rate. Mandal
and Phaujdar [16], Pal et al. [17] developed the inventory
models for deteriorating items with stock-dependent de-
mand rate. In this paper, we have extended these works,
on deteriorating inventory research, by considering dete-
riorating goods with stock-dependent demand in a
two-echelon supply chain consisting one manufacturer
and one retailer, the objective is to investigate the effects
of supply chain coordination on profit increase in the
supply ch ai n , an d study the impact of different parameters
associated with the model, such as the rate of deteriora-
tion, the retailer’s purchase cost, the manufacturer’s pro-
duction cost, the retailer’s and manufacturer’s holding
cost on the supply chain profit increase percentages gen-
erated by the supply chain coordination.
2. Assumptions and Notations
2.1. Assumptions
(1) The retailer replenishes the stocks from the exclusive
source on an EOQ basis. Replenishments are instantane-
ous.
(2) Lead time is assumed to be zero for the sake of sim-
plicity.
124 Chuanx u Wang
Copyright © 2008 SciRes JSSM
(3) No backorders are allowed.
(4) Demand rate is dependent on the instantaneous in-
ventory level. The demand rate ()dI of the item, when
the inventory is
I
, is considered in the form ()dI I
β
α
=,
where 0
α
> and 01
β
<< are scale and shape pa-
rameters ( Baker an d Urban 1988).
(5) The manufacturer’s production rate is greater than or
equal to the demand rate facing the retailer.
(6)The manufacturer is a make-to-order producer; it has a
lot-for-lot production policy in response to the retailer’s
demand. In this particular case, the length of the manu-
facturer’s production cycle is equal to the length of the
retailer’s replenishment cycle.
2.2. Notations
p the sale price for the retailer;
S the order cost per order for the retailer;
M
the setup cost p er lot for the manufacturer;
()
rm
cc procurement (manufacturing ) cost per unit for
retailer (manufacturer);
d the demand rate of the item in the marketplace;
q the production rate;
()
rm
hh the inventory holding cost as a fraction of the
inventory cost for the retailer(manufacturer);
()t
θ
the deterioration rate facing both the retailer and
manufacturer01
θ
≤≤
(), ()
rm
tI t inventory level at timet for the retailer
( manufacturer)
Q the order quantity for the retailer;
T the replenishment cycle ( or production cycle ) for the
retailer( manufacturer);
3. Basic Model
In this section, we first derive the profit model for the
decentralized supply chain. Later, we present the profit
model derived by considering the centralized supply
chain.
3.1. The Decentralized Supply Chain
In the decentralized supply chain, each entity within the
supply chain aims to maximize its own profit functions,
with no consideration given to its counterpart’s reaction
or profit. The retailer makes a replenishment decision
based on an EOQ policy that includes inventory holding
cost and ordering cost.
During the replenishment cycle, the change in retailer’s
inventory level depends on demand and deterioration and
is given by [1]:
() ()[ ()]
rrr
dI t
I
tIt
dt
β
θα
+=− 0tT≤≤ , (1)
As shown in Pal et al. (1993) [17], equation (1) can be
rewritten as
dttdI
tI
tI
tI
r
r
r
r−=
+
)(
)]([
)]([
)(
11 1
2
β
β
αθ
α
θ
(2)
By integrating, we get
1
)1(
1
1lnln
)]([
)]([
ln Ce
tI
tI t
r
r+=
+
ϑβ
β
β
αθ
, (3)
where 1
C is integration constant.
It can be rearranged as
t
reC
tI
θβ
β
θα
)1(
1
1
)]([
1
=
+ (4)
By using the boundary condition on inventory
(0)
r
I
Q
=
we can get
β
θα
+
=1
11
Q
C (5)
Substituting (5) in (4), the retailer’s inventory level at
time t(0tT
) can be expressed as
1(1)1/(1) 1/(1)
() [()]/
rt
ItQ e
β
βθβ β
αθα θ
−−−− −
=+ − (6)
The inventory holding cost in a cycle for the retailer
is
0()
T
rrr r
H
Chc Itdt= (7)
The retailer’s total number of deteriorated goods in a
cycle is given by
,
0
()[ ()]
T
Dr rr
QQITItdt
β
α
=−−
1(1)1/(1) 1/(1)
[()] /
T
QQe
β
βθβ β
αθα θ
−−−− −
=− +−
1 (1)/(1)
0/(1)
[( )]
Tt
Qe dt
ββθββ
ββ
αα θα
θ
−−− −
+−
(8)
The profit per unit time for the retailer can be ex-
pressed as
1 (1)/(1)/(1)
10
1(1)1/(1)1/(1),
0
()[()] /
[()]/
T
rrt
T
rr trDr
S
pcQ edtTT
hcQ edtcQ
TT
ββθββ ββ
ββθββ
αα θαθ
αα θαθ
−−− −−
−−−− −
=− +−−
+−
−−
(9)
The profit function (9) is highly nonlinear and cannot
be solved by analytical methods. We solve it by using
optimization technology on computer and get the follow-
ing optimal values:
Since the manufacturer is a make –to-order producer, it
has a lot-for-lot production policy in response to the re-
tailer’s demand. During the production cycle, the change
in manufacturer’s inventory level is due to the combined
effect of production and deterior ation:
Impact of Supply Chain Coordination for Deteriorating Goods 125
with Stock-Dependent Demand Rate
Copyright © 2008 SciRes JSSM
() ()
mm
dI tqIt
dt
θ
=− m
ttT≤≤ , (10)
where m
t and Tare the starting and stopping produc-
tion times, respectively. With a make-to-order policy, the
production quantity of the manufacturer is equal to the
demand quantity of the retailer. Therefore, we get:
0() ()
m
TT
rm
t
I
tdtItdt=
∫∫
(11)
Solving the equation (6) by using the method proposed
by Spiegel (1960)[18] , we can get:
tm eC
q
tI
θ
θ
+= 2
)( , (12)
where 2
C is constant. By using the boundary condition
on inventory () 0
mm
It=we can obtain:
m
t
e
q
C
θ
θ
−=
2
Therefore, the manufacturer’s inventory level at time
t(m
ttT≤≤ ) can be expressed as
()
()()
() m
mm
m
tt
t
tt tt
m
t
qqe
It eeqdt
θ
θθ
θθ
−−
−− −
==−
(13)
The inventory holding cost in a cycle for the manufac-
turer is
()
()
2
()
[1( )]
m
mm
m
TT
tt
mmm mmm
tt
Tt
mm m
qq
H
Chc Itdthcedt
q
hceT t
θ
θ
θθ
θ
θ
−−
−−
⎛⎞
==−
⎜⎟
⎝⎠
=−++−
∫∫
(14)
The manufacturer’s total number of deteriorated goods
in a cycle is given by
,()()
Dm m
m
QqTtITQ=−− −
(15)
The profit per unit time for the manufacturer is
,
1
()
rm mmDm
mccQ
MHCcQ
TTTT
=−−−
()
2
[1( )]
m
Tt
mm
rm
hcqeT t
cQ M
TT T
θ
θ
θ
−−
−++ −
=−−
()
() m
mTt
mm
m
cqT tcqcqe
TT T
θ
θθ
−−
−+− (16)
Based on the optimal order quantity and ordering cycle
as well as equation (11), we can obtain optimal profit for
the manufacturer usi n g (16).
3.2. The Centralized Supply Chain
In the centralized supply chain, the order quantity and
replenishment cycle are determined by considering the
total profit incurred by both the retailer and the manufac-
turer, so that the overall profit is maximized. Th e central-
ized supply chain requires information sharing between
manufacture and retailer. Sequential or concurrent engi-
neering will be beneficial to the information sharing.
In this case, the total profit per unit time for the supply
chain is the sum of Equation (9) and (16).
211
scr m
=
+=
∏∏∏
1 (1)/(1)
0/(1 )
[( )]
Tt
pQe dt
S
TT
ββθββ
ββ
αα θα
θ
−−−−
+−
1(1) 1/(1)
01/(1 )
[( )]
T
rr t
hcQ edt
T
ββθβ
β
αα θα
θ
−−− −
+−
1(1) 1/(1)
1/(1)
[( )]
rT
cQe
T
β
βθ β
β
αθ α
θ
−−− −
+−
+
()
2
[1( )]
m
Tt
mm m
hcqeT t
M
TT
θ
θ
θ
−−
−++ −
−−
()
() m
mTt
mm
m
cqT tcq cqe
TT T
θ
θθ
−−
−+− (17)
Similarly, since supply chain profit function (17) is
highly nonlinear, we solve it by using optimization tech-
nology on computer and get the optimal values.
4. Numerical Study
To illustrate the effect of the models, we give the follow-
ing numerical example: 0.5
α
=
units per time period,
0.4
β
=
, 0. 1
θ
=
, $10S
=
, $20M=, $20p
=
per
unit, 200q
=
units, 0.35
r
h=, 0.25
m
h=, 3.5
r
c=,
2.0
m
c=.
The models were implemented on a personal computer
using Mathematica version 5.2 . The results are outlined in
Table 1, revealing that the profit for centralized supply
chain is greater than that for decentralized supply chain.
Based on the numerical example considered above, we
now perform the sensitivity analysis on the effects of
changes in the model parameters such as the rate of dete-
rioration, the retailer’s purchase cost, the manufacturer’s
production cost, the retailer’s and manufacturer’s holding
cost on the optimal order quantity for the retailer, the op-
timal replenishment cycle ( or production cycle ) for the
retailer( manufacturer), the optimal supply chain profit,
and supply chain profit increase percentages generated by
the centralized policy. The effects of changes in the pa
rameters values are shown in Table 2. The sensitivity
analysis is performed by changing each of the parameters
by –50%, -20%, +30% and +50% and keeping the other
parameters unchanged. The results are demonstrated in
Table 1. The Sollution Results
Supply chain *
Q *
T r
Π m
Πsc
Π
Decentralized 12.15 2.36 5.03 3.428.45
Centralized 20.63 2.75 4.76 4.569.32
126 Chuanx u Wang
Copyright © 2008 SciRes JSSM
Table 2. Effect of Changes in the Model Parameters
Results of Optimization Procedure
Decentralized Supply Chain Centralized Supply Chain
Parameters Change (%)
*
Q *
T *
1
SC
*
Q *
T *
2
SC
Profit Increase (%)
50 10.434 2.342 6.345 18.675 2.735 7.243 14.153
20 11.876 2.351 7.421 19.538 2.747 8.256 11.252
20 13.216 2.375 9.236 21.562 2.769 9.768 5.760
θ
50 13.987 2.387 10.167 24.356 2.778 10.245 0.767
50 11.579 1.987 17.948 17.917 2.294 18.831 4.920
20 11.682 2.173 14.656 19.267 2.473 15.857 8.195
20 15.876 2.563 7.063 30.013 2.931 7.771 10.024
r
c
50 16.786 3.126 4.754 38.681 3.477 5.335 12.221
50 10.342 2.031 7.346 18.625 2.334 7.658 4.247
20 11.769 2.215 8.085 19.890 2.513 8.779 8.584
20 16.987 2.685 11.886 31.325 2.973 13.256 11.526
m
c
50 17.769 3.264 15.538 39.706 3.517 17.543 12.904
50 10.876 2.046 7.258 17.018 2.305 7.589 4.560
20 11.986 2.263 7.982 18.845 2.433 8.648 8.344
20 16.765 2.765 12.189 32.098 3.384 13.767 12.946
r
h
50 17.875 3.078 15.876 40.022 3.441 18.089 13.939
50 11.054 2.178 7.458 17.357 2.338 7.915 6.128
20 12.035 2.268 8.212 18.930 2.485 8.971 9.243
20 16.497 2.589 11.687 32.021 3.264 12.893 10.319
m
h
50 17.568 2.987 14.789 38.357 3.377 16.606 12.286
table 2. The following observations can be made from
(1) Whether the supply chain is centralized or decen-
tralized, the optimal order quantity, the optimal replen-
ishment cycle (or production cycle) , the optimal supply
chain profit are decreasing in the deterioration rate.
Meanwhile the percentages of supply chain profit in-
crease realized by employing the centralized policy are
increasing in the deterioration rates.
(2) Whether the supply chain is centralized or decen-
tralized, the optimal order quantity and replenishment
cycle (or production cycle) decrease, while the optimal
supply chain profit increases in retailer’s unit procure-
ment cost.
(3) Whether the supply chain is centralized or decen-
tralized, the optimal order quantity, the optimal replen-
ishment cycle (or production cycle) and supply chain
profit are decreasing in manufacturer’s unit manufactur-
ing cost.
(4) Whether the supply chain is centralized or decen-
tralized, the optimal order quantity, the optimal replen-
ishment cycle (or production cycle) and supply chain
profit are decreasing in retailer’s and manufacturer’s in-
ventory holding cost rate.
(5) The percentages of supply chain profit increase re-
alized by employing the centralized policy are increasing
in the deterioration rates, but are decreasing in the re-
tailer’s purchase cost, the manufacturer’s production cost,
the retailer’s and manufacturer’s holding cost.
5. Conclusions
This paper has investigated the effect of supply chain co-
ordination for deteriorating goods with stock-dependent
demand rate. Two profit models are developed with some
assumptions based on exponentially decaying deteriora-
tion rates. The numerical study is conducted to demon-
strate the effectiveness of the proposed models, and to
analyze the impact of supply chain coordination on supply
chain profit. Sensitiv ity analysis is performed to stud y the
impact of different parameters associated with the model,
such as the rate of deterioration, the retailer’s purchase
cost, the manufacturer’s production cost, the retailer’s and
manufacturer’s h olding cost on the optimal ord er quantity
for the retailer, the optimal replenishment cycle ( or pro-
Impact of Supply Chain Coordination for Deteriorating Goods 127
with Stock-Dependent Demand Rate
Copyright © 2008 SciRes JSSM
duction cycle ) for the retailer( manufacturer), the optimal
supply chain profit, and supply chain profit increase per-
centages generated by the supply chain coordination.
The following observations can be obtained from nu-
merical analysis:
(1) A centralized policy is found to be always superior
to a decentralized policy in terms of profit increase, espe-
cially when the deterioration rates are high.
(2)Whether the supply chain is centralized or decen-
tralized, the optimal order quantity , the optimal replen-
ishment cycle (or production cycle) , the optimal supply
chain profit are decreasing in the deterioration rate, re-
tailer’s unit procurement cost, manufacturer’s unit manu-
facturing cost, as well as retailer’s and manufacturer’s
inventory holding cost rate. The optimal supply chain
profit are decreasing in the deterioration rate, manufac-
turer’s unit manufacturing cost, as well as retailer’s and
manufacturer’s inventory holding cost rate. Meanwhile it
is increasing in retailer’s unit procurement cost.
(3)The percentages of supply chain profit increase re-
alized by employing the centralized policy are increasing
in the deterioration rates.
The proposed models can be used to analyze gricultural
products, fashion goods, drugs and high-tech products
supply chain. It is observed that large quantities of these
deteriorationg goods displayed in a supermarket tend to
generate higher demands. However, the models consid-
ered in this paper are somewhat idealized. In reality,
when supply chain is coordinated, there are likely to be
some costs incurred from information sharing scheme.
The information sharing could also affect production,
inventory, and other operations. The future research will
further consider these factors. These models can be ex-
tended in the future research to consider more general
deterioration rates. Another extension possibility would
be to use other replenishment policies. In addition to
profit models, the cost models can also be applied to
evaluate the extent of coordination.
6. Acknowledgement
This paper is partially supported by Shanghai Shuguang
Program under grant no. 06SG48.
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