Economic order quantity model with innov
280
Int. J. Applied Decision Sciences, Vol. 4, No. 3, 2011
Economic order quantity model with innovation
diffusion criterion having dynamic potential market
size
K.K. Aggarwal*, Chandra K. Jaggi and
Alok Kumar
Department of Operational Research, Faculty of Mathematical
Sciences, New Academic Block, University of Delhi,
Delhi 110007, India
Fax: +91-11-27666672
E-mail: kkaggarwal@gmail.com
E-mail: ckjaggi@yahoo.com
E-mail: alok_20_or@yahoo.co.in
*Corresponding author
Abstract: Introduction of a new product in the market forces the inventory
managers to consider the effects of marketing policies especially for innovation
effects at the earlier stage of the product life cycle to make the economic order
quantity (EOQ) model more realistic. Traditional EOQ models generally do not
consider the effect of marketing parameters. In this paper, a time dependent
innovation driven demand model has been introduced in the basic EOQ model
to calculate the different optimal policies. This model assumes that potential
market size is dynamic over time. The proposed model acknowledges
relationship between the innovation coefficient and the optimal policies. The
effectiveness of this model is illustrated with a numerical example and
sensitivity analysis of the optimal solution with respect to different parameters
of the system is performed.
Keywords: economic order quantity; EOQ; innovation driven demand;
dynamic potential adopters; diffusion theory.
Reference to this paper should be made as follows: Aggarwal, K.K., Jaggi, C.K
and Kumar, A. (2011) ‘Economic order quantity model with innovation
diffusion criterion having dynamic potential market size’, Int. J. Applied
Decision Sciences, Vol. 4, No. 3, pp.280–303.
Biographical notes: K.K. Aggarwal is an Assistant Professor in the
Department of Operational Research at University of Delhi, India. He obtained
his MSc in Operational Research and PhD in Inventory Management from
University of Delhi. His research interests and teaching include inventory
modelling, financial engineering and network analysis. He has published more
than 15 research papers in many scholarly journals.
Chandra K. Jaggi is an Associate Professor in the Department of Operational
Research, Faculty of Mathematical Sciences at University of Delhi, India. He
obtained his PhD, MPhil and Masters in Operational Research from the
Department of Operational Research at University of Delhi. His research
interest lies on analysis of inventory system. He has more than 31 publications
in Int. J. Production Economics, Journal of Operational Research Society,
European Journal of Operational Research, Int. J. Systems Sciences, Canadian
Copyright © 2011 Inderscience Enterprises Ltd.
Economic order quantity model with innovation diffusion criterion
281
Journal of Pure and Applied Sciences, OPSEARCH, Investigation Operational
Journal, Advanced Modelling and Optimisation, Journal of Information
and Optimisation Sciences, Int. J. Mathematical Sciences, Indian Journal of
Mathematics and Mathematical Sciences and Indian Journal of Management
and Systems.
Alok Kumar is a Research Scholar in the Department of Operational Research,
Faculty of Mathematical Sciences, University of Delhi, India. He received his
MA in Operational Research from University of Delhi. Currently, he is
pursuing his PhD in Operational Research from the Department of Operational
Research, Faculty of Mathematical Sciences, University of Delhi, India. His
area of research interest include inventory management and mathematical
modelling.
1
Introduction
The economic ordering policies often depend on demand information and demand is
mainly influenced by different marketing strategies. Therefore, the appropriate modelling
of demand is of great importance. Particularly, it is more significant for the new products,
because demands of such products are highly dynamic in nature and have less predictable
growth behaviour. In inventory system, normally demand of product is considered to be
constant, time dependent, stock dependent or probabilistic in nature. Unfortunately,
inventory modelling generally do not take into account the demand pattern of new
products. Changing needs of the society, technological breakthroughs the impact of
globalisation forces manufacturers to introduce new products in the market. When new
products are introduced in the market, their inventory management becomes significant.
The economic order quantity (EOQ) models have been explored to a great extent, in
particular with their interaction to different functional areas of business management like
production and finance. Though some models are there that used the Bass diffusion
demand model to formulate the optimal EOQ model (Chern et al., 2001) but not much of
work has been reported on the interaction of EOQ with marketing strategies. The
integration of EOQ policies with marketing policies is one of the key factors of
successful business operation. Therefore, the coordination between marketing disciplines
and inventory management is yet to be explored fully.
The proposed model includes the integration of EOQ with marketing disciplines in
which demand follows innovation diffusion process with dynamic market potential size.
Diffusion modelling of new product adoption has become an active area of marketing
research with more emphasis on inventory management since the pioneering work of
Bass (1969). This model is concerned with representing the dynamic nature of the
adoption of a new product. Since diffusion models are normative in nature and models of
this type include managerial decision variables in order to study their effect on the
diffusion process, therefore, here, role of inventory manager becomes indispensable
while managing the inventory and hence the importance of mathematical modelling in
inventory for such kind of situation is highly desired.
The diffusion of innovation within marketing has assumed greater significance
because theory of innovation diffusion relates how a new idea, a new product or a service
is accepted into a social system over time. The diffusion process in marketing is related to
282
K.K. Aggarwal et al.
penetration of a market by a new product. One of the most widely held theories of
communication in marketing is diffusion theory. Diffusion theory is actually a theory of
communication regarding how information is disseminated within a social system over
time. An innovation is an idea, object or a practice that is perceived as novel by the
members of the social system. Individuals who constitute the market may differ by the
manner in which the information about the innovation reaches to them and the manner in
which they respond to such received information. Diffusion is defined as the process by
which an innovation is communicated through certain channels over time among
members of a social system (Rogers, 1983, 2003).
Various diffusion processes have been studied in the literature. Fourt and Woodlock
(1960) gave pure innovative model, whereas Fisher and Pry (1971) are concerned with
pure imitative model. The Bass (1969) model captures both the innovative and imitative
aspects of product adoption. The Bass (1969) model, introduced over three decades ago
has been widely used in marketing by Mahajan and Muller (1979), Mahajan and Wind
(1986), Mahajan et al. (1990a, 1990b, 1995, 2000), Parker (1994), and Rogers (1995).
Bardhan and Chanda (2009) and Chanda and Bardhan (2008) have used the demand of
some products may decline due to the introduction of more advance products that
may influence the consumer’s preference. Alfares et al. (2005) have developed a
production-inventory model by integrating quality and maintenance decision. Also, a
number of researchers such as Blackman et al. (1973), Cheng and Sethi (1999), Golder
and Tellis (1998), Hamblin et al. (1973), Jaber et al. (2009), Dodson and Muller (1978),
Kalish (1985), Lilien et al. (1999), Mahajan and Robert (1978), Mahajan et al. (1990a,
1990b), Masao (1973), Roberts and Urban (1988), Rogers (1962), and Robert (1973)
have discussed innovation diffusion criterion in various forms. A theoretical analysis of
the diffusion of innovation in the marketing literature has also been well explained, which
are as follows. Sharif and Ramanathan (1981) presents some modified binomial
innovation diffusion model that incorporate dynamic potential adopter populations.
Sultan et al. (1990) considered a meta-analysis of applications of diffusion models.
Kurawarwala and Matsuo (1996) provided an integrated framework for forecasting and
inventory management of short life cycle products. Van den Bulte (2000) proposed a
model of measurement and analysis of new product diffusion acceleration. Roberts
(2000) examines the nature of emerging markets and identifies the features that make
them difficult to predict. Henard and Szymanski (2001) have studied to conduct and
present insights from a meta-analysis of the evidence on the determinants of new product
performance. Danaher et al. (2001) developed a model of marketing-mix variables and
the diffusion of successive generations of a technological innovation. Van den Bulte and
Lilien (2001) explained the underscores the importance of controlling for potential
confounds when studying the role of social contagion in innovation diffusion. Agarwal
and Bayus (2002) considered a model titled the market evolution and sales takeoff of
product innovations. Talukdar et al. (2002) have investigated the diffusion of new
product across products and countries. Furukawa and Kato (2002) developed a
conceptual model for adoption and diffusion process of a new product. Linton (2002) has
proposed a model which builds on existing knowledge of diffusion forecasting and
integrates it with the disruptive and discontinuous innovation literature. Pae and Lehmann
(2003) focuses on the diffusion patterns of the adjacent generations of technology and its
relation to the time that elapses between them. Tellis et al. (2003) developed a new
product model which comprises of the role of economics, culture and country
innovativeness. Everdingen et al. (2005) have introduced a cross-population adaptive
Economic order quantity model with innovation diffusion criterion
283
diffusion model that can be used to forecast the diffusion of an innovation at early stages
of the diffusion curve. Niu (2006) developed a piecewise-diffusion model of new-product
demands. Puumalainen et al. (2007) developed a model which proposes a set of
uncertainty sources of telecommunications industry that have notable effect on the
diffusion of innovations in the field. Chandrasekaran and Tellis (2007) have studied a
critical review of marketing research on diffusion of new products. Stremersch et al.
(2010) described the importance of time which accelerates early growth of new products.
Shinohara and Okuda (2010) proposed a dynamic innovation diffusion modelling.
Peres et al. (2010) have studied a critical review of innovation diffusion and new product
growth models.
Increments of new-buyer penetration
Cumulative percentage of buyers
Figure 1
Time
Source: Fourt and Woodlock (1960, pp.33–34)
The Fourt and Woodlock model explains the diffusion process in terms of number of
customers who have bought the product by time t by a modified exponential curve
(Figure 1). The discrete form of the model can be given as:
Qt = rQ(1 − r )t −1
(1)
where
Qt
sales at time t which is a fraction of the potential sales
Q
total potential sales as fraction of all buyers
r
rate of penetration of untapped potential
t
time period.
The model assumes that the level of adoption capability Qt can be expressed as a function
of time t and generate an exponential declining curve of new-adopter sales over time. The
values for the parameter Q and r, can be estimated based on available historical data
points in such a way that the resulting curve fits the data well. On the basis of this curve,
future adoption can be forecasted.
284
K.K. Aggarwal et al.
The Fourt and Woodlock model and many other researchers in the marketing
disciplines usually assume that the potential market size remains constant over time, but
in reality, it is not always possible. Different promotional activities, increase in
purchasing power due to increase in income, increase in population and several other
factors lead to increase in potential market size over time. Therefore, it becomes
important to include the potential market as dynamic instead of constant as far as research
regarding innovation diffusion is concerned. Sharif and Ramanathan (1982) have
considered the application of dynamic potential adopter diffusion model through their
study of diffusion of oral contraceptives in Thailand.
In this paper, an inventory model is developed where it is assumed that demand of the
product follows innovation diffusion behaviour as considered by Fourt and Woodlock
model and potential market size is dynamic. A comprehensive sensitivity analysis is
performed to highlight the impact of diffusion of innovation on the economic ordering
policies. The calculated results provide significant impetus while taking inventory
decisions for new products. This article is divided into model development, special cases
and observations and concludes with a discussion on the application.
2
Mathematical models
2.1 Assumptions and notations
Inventory model has been developed under the following assumptions:
1
the replenishment rate is instantaneous
2
lead time is zero
3
shortages are not allowed
4
demand rate is known and is governed by innovation process
5
the size of the potential market of total number of adopters is dynamic in nature
6
the coefficient of innovation is constant in nature
7
the innovation’s sales are confined to a single geographical area.
Notations used in the modelling framework are as follows:
A
Ordering cost.
C
Unit cost.
p(t)
Innovation effect at time t. Here, p(t) = p we assume that the innovation
factor that influences consumer adoption decision will remain constant
during the entire product life cycle.
I
Inventory carrying charge.
IC
Inventory carrying cost.
T
Length of the replenishment cycle.
Q
Number of items received at the beginning of the period.
Economic order quantity model with innovation diffusion criterion
K(T)
The total cost of the system per unit time.
I(t)
On hand inventory at any time t.
N0
Initial potential market size.
N(t)
Cumulative number of adopters at time t.
285
n(t) = λ(t) = S(t) The number of adoptions at time t, i.e., demand at time t.
The basic demand model used in this paper is based on the following assumptions:
1
Adoptions take place due to innovation-diffusion process and it is influenced by the
innovation-effect (mass-media) only.
2
The innovation effect is constant throughout the cycle time.
3
The potential market size is time dependent (dynamic). In this way, two form of time
dependent potential market size, i.e., linear and exponential have been considered by
taking two different cases.
The demand model used in this paper is based on the hypothesis that there exists potential
market whose size is time dependent (dynamic). With the passage of time prospective
buyers adopt the product. Using (1) and the demand assumptions, the mathematical form
of rate adoption at any time t can be given as:
φ (t ) =
f (t )
= p(t ) = p
1 − F (t )
(2)
The physical interpretation of the above model is the probability that a potential adopter
will purchase a product at time t given that no purchase has occurred till time t.
where
ϕ(t)
is the hazard rate that gives the conditional probability of a purchase in
a small time interval (t, t + ∆t), if the purchase has not occurred till time
t
f(t)
F (t ) =
∫ f (t )dt
is the likelihood of purchase at time t
t
is the cumulative likelihood of purchasing the product at time t.
0
Equation (2) can be rewritten as:
f (t ) = p [1 − F (t )] where F (t ) =
N (t )
N (t )
The number of adoptions (S(t)) at time t can be derived by multiplying f(t) with the
market size N (t ). Thus,
n(t ) = S (t ) = λ (t ) =
dN (t )
= p ⎡⎣ N (t ) − N (t ) ⎤⎦
dt
In this paper, we consider following two functional form of N (t )
(3)
286
1
2
K.K. Aggarwal et al.
when potential market size increases linearly
taking, N (t ) = N 0 (1 + gt )
g >0
when potential market size increases exponentially
taking, N (t ) = N 0 e gt
g >0
Case 1 when potential market size increases linearly
Taking, N (t ) = N 0 (1 + gt )
Therefore,
λ (t ) = n(t ) = p ⎡⎣ N 0 (1 + gt ) − N (t ) ⎤⎦
(4)
dI (t )
= −λ (t ), 0 ≤ t ≤ T
dt
(5)
The demand usage λ(t) which is a function of time plays pivotal role to shrink the
inventory size over a period of time. If in the time interval (t, t + dt) the inventory size is
dipping at the rate λ(t)dt, then the total reduction in the inventory size during the time
interval dt can be given by –dI(t) = λ(t)dt. Thus, differential equation describing the
instantaneous state of the inventory level I(t) in the interval (0, T) is given by:
Using equations (4) and (5) we get
dI (t )
= − p ⎡⎣ N 0 (1 + gt ) − N (t ) ⎤⎦
dt
(6)
(
)
The solution of the differential equation (6) is:
⎛ g⎞
⎛ g⎞
I (t ) = I (0) − N 0 pt + N 0 ⎜1 − ⎟ pt + e− pt − N 0 ⎜1 − ⎟
p
p⎠
⎝
⎠
⎝
(7)
where t = 0 ⇒ I(t) and N(t) = 0.
Since replenishment is instantaneous and shortages are not allowed.
Therefore,
∫
I (0) = Q = λ (t )dt
T
(
)
⎛ g⎞
⇒ I (0) = N 0 pT − N 0 ⎜1 − ⎟ pT + e − pT − 1
p⎠
⎝
0
(
Using the equations (7) and (8) we get
)
⎛
g⎞
I (t ) = N 0 pT − N 0 ⎜ 1 − ⎟ pT + e − pT − 1 −
p⎠
⎝
(
)
⎛
⎛ g⎞
g⎞
N 0 pt + N 0 ⎜ 1 − ⎟ pt + e− pt − N 0 ⎜1 − ⎟
p⎠
p⎠
⎝
⎝
(8)
(9)
287
Economic order quantity model with innovation diffusion criterion
Now,
∫ I (t )dt = N
T
0 pT
2
(
)
⎛
g⎞
− TN 0 ⎜ 1 − ⎟ pT + e − pT − 1 −
p⎠
⎝
(
⎡
1 − e − pT
⎛
⎛ g ⎞ T2
g⎞
T2
TN 0 ⎜1 − ⎟ − N 0 p
+ N 0 ⎜1 − ⎟ ⎢ p
+
p⎠
p ⎠⎢ 2
p
2
⎝
⎝
⎣⎢
0
) ⎤⎥
⎥
⎦⎥
The total cost per unit time, K1(T) comprises of the following elements
a
ordering cost per unit time
(OC1 ) =
b
A
T
(10)
inventory holding cost per unit time
∫
IC
I (t )dt
( IHC1 ) =
T
T
(
)
⎛
g⎞
⇒ IHC1 = N 0 pICT − ICN 0 ⎜1 − ⎟ pT + e − pT − 1 −
p⎠
⎝
0
(
− pT
⎡
⎛
⎛
g⎞
T
g ⎞ IC ⎢ T 2 1 − e
+
ICN 0 ⎜ 1 − ⎟ − N 0 pIC + N 0 ⎜1 − ⎟
p
2
p⎠
p⎠ T ⎢ 2
p
⎝
⎝
⎢⎣
c
) ⎤⎥
(11)
⎥
⎥⎦
material cost per unit time
( MC1 ) =
⎛
QC
C
g⎞
⇒ MC1 = N 0 pC − N 0 ⎜1 − ⎟ ( pT + e − pT − 1)
T
T
p⎠
⎝
(12)
Therefore, K1 (T) = OC1 + IHC1 + MC1
Using the equations (10), (11) and (12) we get
K1 (T ) =
⎛
⎛
A
g⎞
g⎞
+ N 0 pICT − ICN 0 ⎜ 1 − ⎟ ( pT + e − pT − 1) − ICN 0 ⎜1 − ⎟ −
T
p
p⎠
⎝
⎠
⎝
N 0 pIC
⎛
T
g ⎞ IC ⎡ T 2 (1 − e − pT ) ⎤
+ N 0 ⎜1 − ⎟
+
⎢p
⎥+
p ⎠ T ⎢⎣ 2
p
2
⎝
⎦⎥
N 0 pC −
⎛
C
g⎞
N 0 ⎜1 − ⎟ ( pT + e − pT − 1)
T
p⎠
⎝
The necessary condition for K1(T) to be minimum is
dK1 (T )
=0
dT
(13)
288
K.K. Aggarwal et al.
(
Therefore,
)
⎛
ICpN 0
dK1 (T )
g⎞
−A
= 0 ⇒ 2 + pICN 0 − ICN 0 ⎜1 − ⎟ p − pe − pT −
−
dT
p
2
T
⎝
⎠
⎛
⎛
g⎞
g⎞
N 0 IC ⎜1 − ⎟
N 0 IC ⎜1 − ⎟
− pT ) ⎞
2
⎛
(
p
p⎠
−
pT
e
1
⎝
⎠⎜
⎝
( pT + e− pT ) +
⎟+
+
2
⎜
⎟
p
T
2
T
⎝
⎠
⎛
⎛
g⎞
g⎞
CN 0 ⎜1 − ⎟
pCN 0 ⎜1 − ⎟
p
p⎠
⎝
⎠ ( pT + e − pT − 1) −
⎝
(1 − e− pT ) = 0,
2
T
T
Now, for K1(T) to be convex
Therefore,
d 2 K1 (T )
dT 2
d 2 K1 (T )
dT 2
(14)
≥0
⎛
g⎞
2 ICN 0 ⎜ 1 − ⎟
p ⎠ ⎡ pT 2 (1 − e− pT ) ⎤
⎝
≥0⇒ 3 +
+
⎢
⎥+
p
T
T3
⎣⎢ 2
⎦⎥
2A
⎛
⎛ g⎞
g⎞
2CN 0 ⎜1 − ⎟
ICN 0 ⎜1 − ⎟
p
p⎠
⎝
⎠ ( p − pe− pT ) +
⎝
( p − pe− pT ) ≥
2
T
T
⎛
g⎞
2 ICN 0 ⎜ 1 − ⎟
p⎠
⎛
1⎞
g ⎞⎛
⎝
( pT + e− pT ) +
p 2 e − pT CN 0 ⎜1 − ⎟ ⎜ I + ⎟ +
2
p
T
T
⎠
⎝
⎠⎝
(15)
⎛
g⎞
2CN 0 ⎜1 − ⎟
p⎠
⎝
( pT + e− pT − 1)
3
T
The solution of the equation
dK1 (T )
= 0 gives the optimum value of T provided it
dT
> 0. Since the above cost equation (13) is highly
dT 2
non-linear, the problem has been solved numerically for given parameter values. The
solution gives the optimum value T* of the replenishment cycle time T. Once T* is known
the value of optimum order quantity Q* and the optimum cost K1(T*) can easily be
obtained from the equations (8) and (13) respectively. The numerical solution for the
given base value has obtained by using software packages LINGO and Excel-Solver.
satisfies the condition
d 2 K1 (T )
Case 2 When potential market size increases exponentially
Taking, N (t ) = N 0 e gt
Therefore,
Economic order quantity model with innovation diffusion criterion
289
λ (t ) = n(t ) = p ⎡⎣ N 0 e gt − N (t ) ⎤⎦ , g > 0
(16)
dI (t )
= −λ (t ), 0 ≤ t ≤ T
dt
(17)
Let I(t) be the inventory level at time t(0 ≤ t ≤ T). The differential equation describing the
instantaneous state of the inventory level I(t) in the interval (0, T) is given by:
Using the equations (16) and (17) we get
dI (t )
= − p ⎡⎣ N 0 e gt − N (t ) ⎤⎦
dt
(18)
The solution of the differential equation (18) is:
I (t ) = I (0) − N 0 p
e gt
p 2 ⎛ e gt e − pt ⎞
+ N0
+
⎜
⎟
( g + p) ⎜⎝ g
g
p ⎟⎠
(19)
where t = 0 ⇒ I(t) = I(0) and N(t) = 0.
Since replenishment is instantaneous and shortages are not allowed.
Therefore,
∫
I (0) = Q = λ (t )dt
T
0
⇒ Q = I (0) = N 0 p
( e gT − 1)
g
p 2 ⎡ ( e gT − 1) ( e− pT − 1) ⎤
− N0
+
⎢
⎥
( g + p) ⎢⎣
g
p
⎥⎦
(20)
Using the equations (19) and (20) we get
I (t ) = N 0 p
( e gT − 1)
g
− N0
p 2 ⎡ ( e gT − 1) ( e− pT − 1) ⎤
+
⎢
⎥−
g
p
( g + p ) ⎣⎢
⎦⎥
e
p 2 ⎛ e gt e− pt ⎞
+ N0
+
N0 p
⎜
⎟
( g + p) ⎝ g
g
p ⎠
gt
Now,
∫
T
I (t )dt = N 0 pT
0
N0 p
( e gT − 1)
g
( e gT − 1)
g2
− N0
+ N0
Tp 2 ⎡ ( e gT − 1) ( e− pT − 1) ⎤
+
⎢
⎥−
( g + p ) ⎣⎢
g
p
⎦⎥
p 2 ⎡ ( e gT − 1) ( e− pT − 1) ⎤
−
⎢
⎥
( g + p ) ⎢⎣ g 2
p2
⎥⎦
The total cost per unit time, K2(T) comprises of the following elements
a
ordering cost per unit time
(21)
290
K.K. Aggarwal et al.
( OC2 ) =
b
A
T
(22)
inventory holding cost per unit time
( e gT − 1)
IC
−
I (t )dt ⇒ IHC2 = ICN 0 p
( IHC2 ) =
T
g
∫
T
0
ICN 0
p 2 ⎡ ( e gT − 1) ( e− pT − 1) ⎤
+
⎢
⎥−
( g + p) ⎣⎢
g
p
⎦⎥
ICN 0 p
c
( e gT − 1)
Tg
2
+ N0
ICp 2 ⎡ ( e gT − 1) ( e− pT − 1) ⎤
−
⎢
⎥
( g + p ) T ⎣⎢ g 2
p2
⎦⎥
material cost per unit time
( MC2 ) =
( e gT − 1)
QC
Cp 2
⇒ MC2 = N 0 Cp
− N0
( g + p)T
T
gT
(23)
⎡ ( e gT − 1) ( e− pT − 1) ⎤
+
⎢
⎥ (24)
g
p
⎢⎣
⎥⎦
Therefore, K2 (T) = OC2 + IHC2 + MC2
Using the equations (22), (23) and (24) we get
K 2 (T ) =
( e gT − 1)
A
p 2 ⎡ ( e gT − 1) ( e − pT − 1) ⎤
+ ICN 0 p
− ICN 0
+
⎢
⎥−
T
g
g
p
( g + p ) ⎣⎢
⎦⎥
ICN 0 p
N 0Cp
( e gT − 1)
Tg 2
( e gT − 1)
gT
+ N0
− N0
ICp 2 ⎡ ( e gT − 1) ( e− pT − 1) ⎤
−
⎢
⎥+
( g + p )T ⎣⎢ g 2
p2
⎦⎥
Cp 2
( g + p )T
⎡ ( e gT − 1) ( e− pT − 1) ⎤
+
⎢
⎥
g
p
⎢⎣
⎥⎦
(25)
The necessary condition for K2(T) to be minimum is:
dK 2 (T )
=0
dT
Therefore,
⎛
I⎞
CpN 0 ⎜1 − ⎟
CpN 0 e gT
g
dK 2 (T )
−A
⎝
⎠ gT
=0⇒ 2 −
( e − 1) +
dT
T
gT 2
Cp N 0 ⎛ e gT − 1 e− pT − 1 ⎞
+
⎜
⎟−
p ⎠
( g + p )T 2 ⎝ g
2
⎛
I I⎞
⎜1 + − ⎟
T
g⎠
⎝
+
T
1⎞
⎛
Cp 2 N 0 ⎜ I + ⎟
T ⎠ ( gT
⎝
e − e− pT ) −
( g + p)
ICp 2 N 0 ⎛ e gT − 1 e− pT − 1 ⎞ ICp 2 N 0 ⎛ e gT e− pT ⎞
−
+
⎜
⎟+
⎜
⎟
p ⎠
p 2 ⎟⎠ ( g + p )T ⎝ g
( g + p )T 2 ⎜⎝ g 2
(26)
Economic order quantity model with innovation diffusion criterion
Now, for K2(T) to be convex
d 2 K 2 (T )
dT 2
291
≥0
Therefore,
⎛
I⎞
2CpN 0 ⎜1 − ⎟
2
d K 2 (T )
2A
⎝ g ⎠ ( e gT − 1) + 2Cp N 0 ( e gT − e− pT ) +
0
≥
⇒
+
dT 2
T3
gT 3
( g + p )T 2
2
2 ICp 2 N 0 ⎛ e gT − 1 e− pT − 1 ⎞ ICp 2 N 0 gT
( e − e− pT ) ≥
−
⎜
⎟+
p 2 ⎠⎟ ( g + p )T
( g + p)T 3 ⎝⎜ g 2
⎛
⎛
I⎞
I I⎞
CpN 0 e gT ⎜ 1 − ⎟ CpN 0 e gT ⎜ 1 + − ⎟
g
T
g ⎠ ICpN 0 e gT
⎝
⎠+
⎝
+
+
T2
T2
T3
1⎞
⎛
Cp 2 N 0 ⎜ I + ⎟
2Cp 2 N 0 ⎛ e gT − 1 e− pT − 1 ⎞
T⎠
⎝
+
⎜
⎟+
( g + p)
p ⎟⎠
( g + p)T 3 ⎜⎝ g
( ge gT + pe− pT ) + 2ICp
N 0 ⎛ e gT e− pT
+
⎜
p
( g + p)T 2 ⎝⎜ g
The solution of the equation
2
(27)
⎞
⎟⎟
⎠
dK 2 (T )
= 0 gives the optimum value of T provided it
dT
> 0. Since the above cost equation (25) is highly
dT 2
non-linear, the problem has been solved numerically for given parameter values. The
solution gives the optimum value T* of the replenishment cycle time T. Once T* is known
the value of optimum order quantity Q* and the optimum cost K2(T*) can easily be
obtained from the equations (20) and (25) respectively. The numerical solution for the
given base value has obtained by using software packages LINGO and Excel-Solver.
satisfies the condition
d 2 K 2 (T )
2.2 Special case
When N (t ) becomes constant over a period of time say N then demand rate, i.e.,
λ(t) = n(t) will reduce to the classical Fourt and Woodlock model. The objective here is
to study the different managerial policies under the assumption that both innovation
factor and potential market size remain constant for the entire product life cycle. For this
case, the total cost per unit time, K(T) is given by
K (T ) =
A NC (
NIC (
1 − e − pT ) +
1 − e− pT ) − NICe− pT
+
T
T
pT
(28)
292
K.K. Aggarwal et al.
For optimum total cost, the necessary criterion is
dK (T )
− A NCpe − pT NC (1 − e − pT )
=0⇒ 2 +
−
+
dT
T
T
T2
NICe − pT NIC (1 − e − pT )
−
+ NICpe − pT = 0
2
T
pT
The solution of the equation
(29)
dK (T )
= 0 gives the optimum value of T provided it
dT
> 0. Since the above cost equation is highly non-linear,
dT 2
the problem has been solved numerically for given parameter values. The solution gives
the optimum value T* of the replenishment cycle time T. Once T* is known the value of
optimum order quantity Q* and the optimum cost K(T*) can easily be obtained by
following the same procedure as obtained in the above two cases. The numerical solution
for the given base value has obtained by using software packages LINGO and
Excel-Solver.
satisfies the condition
3
d 2 K (T )
Solution procedures
The solution procedure has been summarised in the following algorithm
Step 1
Input all parameter values such as unit cost, coefficient of innovation, potential
market size, ordering cost etc. for each case separately.
Step 2
Compute all possible values of T separately for the equations (14), (26) and (29)
as the case may be.
Step 3
Select the appropriate value of T for each case by using the equations (13), (25)
d 2 K (T )
> 0.
and (28) and by satisfying the above stated condition such as
dT 2
The above steps are used for all replenishment cycles using appropriate parameter values.
In order to obtain the appropriate values of T we need to follow the above procedure with
the help of LINGO and EXCEL-Solver software packages.
4
Numerical examples
A numerical example is presented in the following to illustrate the effectiveness of the
proposed model. A hypothetical example has the following parameter values in
appropriate units.
A = $1, 000/order, C = $1, 050/unit, I = 30%
Economic order quantity model with innovation diffusion criterion
293
Using the solution procedure described above, the optimal values of cycle length, order
quantity and total cost under different cases have been presented in Tables 1 to 5 for
different parameter values. Also, to prove the validity of the model numerically and to get
the appropriate parameter values, the references have been considered as Sharif and
Ramanathan (1981), Chandrasekaran and Tellis (2007), Sultan et al. (1990), Talukdar
et al. (2002) and Van den Bulte and Stremersch (2004).
•
The mean value of the coefficient of innovation for a new product usually lies
between 0.0007 and 0.03 (Sultan et al., 1990; Talukdar et al., 2002; Van den Bulte
and Stremersch, 2004).
•
The mean value of the coefficient of innovation for a new product is usually 0.001
for developed countries and 0.0003 for developing countries (Talukdar et al., 2002).
Case 1 When potential market size increases linearly
For this case, Sharif and Ramanathan (1981) considered the Coleman model and
explained by taking the case study of credit card banking in the USA. Here, Sharif and
Ramanathan (1981) regard this model as Model-1(b) and took the following parameter
values.
N 0 = 1459.77, α = g = 0.1292
Table 1 shows the changes in the values of T, Q and K1(T) for variations in the values of
coefficient of innovation for fixed value of g, whereas, Table 2 describes the changes
with respect to g for fixed value of coefficient of innovation.
Table 1
Changes in optimal values due to p
Sensitivity analysis on p for g = 0.13
T
K1(T*)
Q*
0.003
0.95
6,636
4.41
0.004
0.83
8,473
5.10
0.005
0.75
10,273
5.73
0.006
0.68
12,046
6.20
0.007
0.64
13,799
6.79
0.008
0.60
15,536
7.26
0.009
0.56
17,260
7.60
0.01
0.54
18,972
8.13
0.02
0.39
35,724
11.63
0.03
0.32
52,102
14.23
P
*
294
K.K. Aggarwal et al.
Table 2
Changes in optimal values due to g
Sensitivity analysis on g for p = 0.003
*
T
K1(T*)
Q*
0.13
0.95
6,636
4.41
0.14
0.94
6,662
4.38
0.15
0.93
6,688
4.35
0.16
0.91
6,713
4.27
0.17
0.90
6,738
4.23
0.18
0.89
6,762
4.20
0.19
0.88
6,786
4.17
0.20
0.87
6,810
4.13
0.21
0.86
6,834
4.10
0.22
0.85
6,857
4.06
g
Case 2 When potential market size increases exponentially
In this case, Sharif and Ramanathan (1981) took the following parameter values.
N 0 = 1928.55, α = g = 2.5863 × 102
Table 3 shows the changes in the values of T, Q and K2 (T) for variations in the values of
coefficient of innovation for fixed value of g, whereas, Table 4 describes the changes
with respect to g for fixed value of coefficient of innovation.
Table 3
Changes in optimal values due to p
Sensitivity analysis on p for g = 0.13
T
K2(T*)
Q*
0.003
0.99
8,065
5.77
0.004
0.86
10,391
6.67
0.005
0.77
12,680
7.46
0.006
0.71
14,942
8.25
0.007
0.66
17,184
8.94
0.008
0.61
19,410
9.44
0.009
0.58
21,623
10.09
0.01
0.55
23,825
10.63
0.02
0.40
45,461
15.42
0.03
0.33
66,715
19.05
P
*
295
Economic order quantity model with innovation diffusion criterion
Table 4
Changes in optimal values due to g
Sensitivity analysis on g for p = 0.003
*
T
K2(T*)
Q*
0.13
0.99
8,065
5.77
0.14
0.98
8,087
5.74
0.15
0.96
8,123
5.65
0.16
0.94
8,158
5.55
0.17
0.92
8,193
5.46
0.18
0.91
8,228
5.42
0.19
0.89
8,262
5.32
0.20
0.87
8,295
5.22
0.21
0.86
8,328
5.18
0.22
0.75
8,635
4.67
g
4.1 Special case
In this case, Sharif and Ramanathan (1981) considered the following parameter values.
N 0 = 2441.78, α = g = 0
Table 5 shows the changes in the values of T, Q and K(T) for variations in the values of
coefficient of innovation.
Table 5
Changes in optimal values due to p
Sensitivity analysis on p
T*
K(T*)
Q*
0.003
0.91
5,047
6.65
0.004
0.81
6,425
7.89
0.005
0.70
7,818
8.52
0.006
0.62
9,194
9.06
p
Following graphs show the behaviour of optimal total cost and optimal cycle length with
the changes in the coefficient of innovation for different cases.
Case 1 When potential market size increases linearly
Figure 2 presents the impact of coefficient of innovation on optimal total cost and
Figure 3 shows the impact on optimal cycle length.
296
K.K. Aggarwal et al.
Figure 2
Innovation effects vs. optimal cost (see online version for colours)
Figure 3
Innovation effect vs. optimal time (see online version for colours)
Figure 4
Innovation effect vs. optimal cost (see online version for colours)
Economic order quantity model with innovation diffusion criterion
297
Case 2 When potential market size increases exponentially
Figure 4 presents the impact of coefficient of innovation on optimal total cost and
Figure 5 shows the impact on optimal cycle length.
Figure 5
Innovation effect vs. optimal time (see online version for colours)
4.2 Special case
Figure 6 presents the impact of coefficient of innovation on optimal total cost and
Figure 7 shows the impact on optimal cycle length.
It can be seen that it is very difficult to prove the convexity of total cost curves in
equations (13), (25) and (28) analytically. The convexity of total cost curves has been
checked graphically for all the cases. Figures 8(a) to 8(c) illustrate the convexity of total
cost curves for different cases.
Figure 6
Innovation effect vs. optimal cost (see online version for colours)
298
K.K. Aggarwal et al.
Figure 7
Innovation effect vs. optimal time (see online version for colours)
Figure 8
Cost graphs showing convexity of the cost functions for (a) case 1, (b) case 2 and
(c) special case (see online version for colours)
(a)
(b)
(c)
Economic order quantity model with innovation diffusion criterion
5
299
Observations
We study the effect of changes in the system parameters on the optimal values of total
cost K(T*), the optimal cycle length T* and the optimal order quantity Q(T*). The results
obtained in the numerical exercise are very encouraging and are consistent with the
reality. From the results of numerical exercise and the graphical representation
(Figures 2, 3, 4, 5, 6 and 7) it has been observed that:
a
As the values of p increase keeping other parameters constant, the value of T* start
decreasing, the optimal order quantity Q(T*) increases slightly while the optimal cost
K(T*) increases significantly as depicted in Tables 1, 3 and 5. This is consistent with
the reality as more investment on promotion will increase the diffusion of a product
in the market resulting in shrinkage of the optimal reorder cycle time as a result
optimal cost is increased.
b
As the values of g increases keeping other parameters constant the value of T* and
the optimal order quantity Q(T*) decreases marginally while there is slight increase
in the optimal cost K(T*) as depicted in Tables 2 and 4. This kind of phenomenon
usually occurs in the real life situation as increment in g makes the potential adopter
dynamic which forces the inventory manager to keep the inventories according to the
changing pattern of the cycle time to get the total cost optimal.
6
Managerial implications
Economy is being globalised and the market opportunities are being opened up because
of cut in tariffs by the government of different countries to promote their products in the
market as a result new products are entered into the market. Also, innovative research
contributes the production of new products which are largely accepted by the many
sections of the society because of its innovativeness and usefulness. Now, when new
products are entered into the market their inventory management becomes a formidable
task for inventory managers and it becomes more crucial when potential market size is
dynamic. Because from a managerial perspective, it is important to focus on both
customer satisfaction and cost of inventory obsolescence to better manage inventory
planning and replenishment for sustained growth of the company. These circumstances
necessitate embracing the right balance between the customers need and the inventory
investment required to meet that demand. An overestimating demand may lead to buy too
much and exposes the firm to potential losses from liquidating overstocks where as
underestimating demand leads to backorders, cancellations and unsatisfied customers. In
dynamic environment, the determination of adopter categories is important for
developing market strategies for different kinds of adopters and for targeting market
prospects for a new product. Mathematical models have proved to be ideal tools to
control these effects and can be used for scheduling and managing the inventories. Thus,
it becomes necessary to formulate a procurement schedule of the inventories for any
organisation to success in a competitive environment. In this model, we have formulated
a procurement schedule and developed an economic ordering policies by taking into
account the sensitive nature of coefficient of innovation when the potential market size is
dynamic. This kind of model will always be helpful for the inventory manager of any
300
K.K. Aggarwal et al.
organisation for scheduling and managing the inventories when new products are
introduced in the dynamic potential market.
7
Conclusions
To manage and control inventory of a new product becomes a challengeable task for
almost every organisation because it requires a huge amount of investment of capital to
purchase inventory. Therefore, it becomes indispensable to develop a mathematical
model to get rid of these problems, as mathematical models are ideal tools to control and
manage the inventories of a new product in most economical way. In this paper, a time
dependent innovation driven demand model with dynamic potential market size has been
introduced in the basic EOQ model to calculate the different optimal policies. The
proposed model acknowledged relationship between the innovation coefficient and the
optimal policies in a dynamic market environment of two kinds. The results are very
encouraging and the findings are consistent with the idea that optimal EOQ policies are
highly sensitive towards the dynamics of innovation coefficient in a dynamic
environment and it is imperative to identify a trend. A numerical example followed by
sensitivity analysis on model parameters has been provided to verify the results obtained
in the real life situation. A simple solution procedure in the form of algorithm is
presented to determine the optimal cycle time and optimal order quantity of the cost
function. A special case has also been formulated which excludes the dynamic nature of
the potential market size.
A few limitations in our approach that suggests areas for future research, are as
follows. As the proposed EOQ model uses the extension of pure innovative demand
model, hence it ignores imitation coefficient (Bass, 1969) which is an important factor in
consumer adoption decision. The existence and the uniqueness of the different cost
functions have been shown numerically because of its highly non-linear nature. Thus
research on some alternative approach to get the optimal analytical solution of the
problem is important. In further research, we would like to extend the proposed model for
multiple generation situations, shortages, partial lost-sales, quantity discount etc.
Acknowledgements
The authors would like to thank the anonymous reviewers and the editor for their
constructive comments and suggestions.
References
Agarwal, R. and Bayus, B.L. (2002) ‘The market evolution and sales takeoff of product
innovations’, Management Science, Vol. 48, No. 8, pp.1024–1041.
Alfares, H.K., Khursheed, S.M. and Noman, S.M. (2005) ‘Integrating quality and maintenance
decisions in a production-inventory model for deteriorating items’, International Journal of
Production Research, Vol. 43, No. 5, pp.899–911.
Bardhan, A.K. and Chanda, U. (2009) ‘A mathematical model for diffusion of products with
multiple generations of a high technology product’, International Journal of Operational
Research, Vol. 4, No. 2, pp.214–229.
Economic order quantity model with innovation diffusion criterion
301
Bass, F.M. (1969) ‘A new product growth model for consumer durables’, Management Sci.,
Vol. 15, pp.215–227.
Blackman, W.A., Jr., Seligman, E.J. and Solgliero, G.C. (1973) ‘An innovation index based upon
factor analysis’, Technological Forecasting and Social Change, Vol. 4, pp.301–316.
Chanda, U. and Bardhan, A.K. (2008) ‘Modelling innovation and imitation sales of multiple
technological generation products’, Journal of High Technology Management Research,
Vol. 18, No. 2, pp.173–190, DOI: 10.1016/j.hitech. 2007.12.004.
Chandrasekaran, D. and Tellis, G.J. (2007) ‘A critical review of marketing research on diffusion of
new products’, Review of Marketing Research, pp.39–80.
Cheng, F. and Sethi, S. (1999) ‘A periodic review inventory model with demand. influenced by
promotion decisions’, Management Science, Vol. 45, No. 11, pp.1510–1523.
Chern, M.S., Teng, J.T. and Yang, H.L. (2001) ‘Inventory lot-size policies for the bass diffusion
demand models of new durable products’, Journal of the Chinese Institute of Engineers,
Vol. 24, No. 2, pp.237–244.
Danaher, P.J., Hardie, B.G.S. and Putsis, W.P., Jr. (2001) ‘Marketing-mix variables and the
diffusion of successive generations of a technological innovation’, Journal of Marketing
Research, Vol. 38, No. 4, pp.501–514.
Dodson, J.A., Jr. and Muller, E. (1978) ‘Models of new product diffusion through advertising and
word-of-mouth’, Journal of Marketing, Vol. 24, No. 15, pp.1–26.
Everdingen, Y.M.V., Aghina, W.B. and Fok, D. (2005) ‘Forecasting cross-population innovation
diffusion: a Bayesian approach’, International Journal of Research in Marketing, Vol. 22,
pp.293–308.
Fisher, J.C. and Pry, R.H. (1971) ‘A simple substitution model of technological change’,
Technological Forecasting and Social Change, Vol. 3, pp.75–88.
Fourt, L.A. and Woodlock, J.W. (1960) ‘Early prediction of market success for new grocery
products’, Journal of Marketing, Vol. 25, pp.31–38.
Furukawa, R. and Kato, H. (2002) ‘A conceptual model for adoption and diffusion process of a new
product’, Review of Marketing Science, Vol. 1, No. 1, pp.1–28.
Golder, P.N. and Tellis, G.J. (1998) ‘Beyond diffusion: an explanatory approach to modeling the
growth of durables’, Journal of Forecasting, Vol. 17, pp.259–280.
Hamblin, R.L., Jacobsen, R.B. and Miller, J.L. (1973) A Mathematical Theory of Social Change,
John Wiley & Sons, Inc., New York, NY.
Henard, D.H. and Szymanski, D.M. (2001) ‘Why some new products are more successful than
others’, Journal of Marketing Research, Vol. 38, pp.362–375.
Jaber, M.Y., Bonney, M. and Moualek, I. (2009) ‘An economic order quantity model for an
imperfect production process with entropy cost’, International Journal of Production
Economics, Vol. 118, pp.26–33.
Kalish, S. (1985) ‘A new product adoption model with price advertising and uncertainty’,
Management Science, Vol. 31, No. 12, pp.1569–1585.
Kurawarwala, A.A. and Matsuo, H. (1996) ‘Forecasting and inventory management of short
life-cycle products’, Operations Research, Vol. 44, No. 1, pp.131–150.
Lilien, G.L., Kotler, P. and Moorthy, K.S. (1999) Marketing Models, Prentice-Hall of India Private
Limited, New Delhi.
Linton, J.D. (2002) ‘Forecasting the market diffusion of disruptive and discontinuous innovation’,
IEEE Transactions on Engineering Management, Vol. 49, No. 4, pp.365–374.
Mahajan, V. and Muller, E. (1979) ‘Innovation diffusion and new-product growth models in
marketing’, Journal of Marketing Fall, Vol. 43, pp.55–68.
Mahajan, V. and Robert, A.P. (1978) ‘Innovation diffusion in a dynamic potential adopter
population’, Management Science, Vol. 24, No. 15.
Mahajan, V. and Wind, Y. (1986) Innovation Diffusion Models of New Product Acceptance,
Ballinger Publishing Company, Cambridge, MA.
302
K.K. Aggarwal et al.
Mahajan, V., Muller, E. and Bass, F.M. (1990a) ‘New product diffusion models in marketing:
a review and directions for research’, The Journal of Marketing, Vol. 54, No. 1, pp.1–26.
Mahajan, V., Muller, E. and Srivastava, R.K. (1990b) ‘Determination of adopter, categories using
innovation diffusion models’, Journal of Marketing, Vol. 27, No. 1, pp.37–50.
Mahajan, V., Muller, E. and Bass, F.M. (1995) ‘Diffusion of new products empirical
generalizations and managerial uses’, Marketing Science, Vol. 14, No. 3, pp.G79–G88.
Mahajan, V., Muller, E. and Wind, Y. (2000) New-Product Diffusion Models, Kluwer Academic
Publisher, Boston, MA.
Masao, N. (1973) ‘Advertising and promotion effects on consumers response to new products’,
Journal of Marketing Research, Vol. 10, pp.242–249.
Niu, S.C. (2006) ‘A piecewise-diffusion models of new-product demands’, Operations Research,
Vol. 54, No. 4, pp.678–695.
Pae, J.H. and Lehmann, D.R. (2003) ‘Multigeneration innovation diffusion: the impact of
intergeneration time’, Journal of the Academy of Marketing Science, Vol. 31, No. 1, pp.36–45.
Parker, P.M. (1994) ‘Aggregate diffusion forecasting models in marketing: a critical review’,
International Journal of Forecasting, Vol. 10, pp.353–380.
Peres, R., Muller, E. and Mahajan, V. (2010) ‘Innovation diffusion and new product growth
models: a critical review and research directions’, International Journal of Research in
Marketing, Vol. 27, pp.91–106.
Puumalainen, K., Sundqvist, S. and Huiskonen, J. (2007) ‘Managing uncertainty in forecasting the
diffusion of telecommunications innovations’, Journal of Scientific and Industrial Research,
Vol. 66, pp.299–304.
Robert, G.B. (1973) ‘A model for measuring the influence of promotion and consumer demand’,
Journal of Marketing Research, Vol. 10, No. 4, pp.380–389.
Roberts, J.H. (2000) ‘Developing new rules for new markets’, Journal of the Academy of
Marketing Science, Vol. 28, No. 1, pp.31–44.
Roberts, J.H. and Urban, G. (1988) ‘Modeling multivariate utility, risk, and belief dynamics for
new consumer durable brand choice’, Management Science, Vol. 34, No. 2, pp.167–185.
Rogers, E.M. (1962) Diffusion of Innovations, The Free Press, New York.
Rogers, E.M. (1983, 2003) Diffusion of Innovations, 3rd ed., The Free Press, New York.
Rogers, E.M. (1995) Diffusion of Innovations, 4th ed., The Free Press, New York.
Sharif, M.N. and Ramanathan, K. (1981) ‘Binomial innovation models with dynamic potential
adopter population’, Technological Forecasting and Social Change, Vol. 20, pp.63–87.
Sharif, M.N. and Ramanathan, K. (1982) ‘Polynomial innovation diffusion models’, Technological
Forecasting and Social Change, Vol. 21, pp.301–323.
Shinohara, K. and Okuda, H. (2010) ‘Dynamic innovation diffusion modeling’, Comput. Econ,
Vol. 35, pp.51–62.
Stremersch, S., Muller, E. and Peres, R. (2010) ‘Does new product growth accelerate across
technology generations’, Journal of Marketing, Vol. 21, No. 2, pp.103–120.
Sultan, F., Farley, J.U. and Lehmann, D.R. (1990) ‘A meta-analysis of diffusion models’, Vol. 27,
No. 1, pp.70–77.
Talukdar, D., Sudhir, K. and Ainslie, A. (2002) ‘Investigating new product diffusion across
products and countries’, Marketing Science, Vol. 21, No. 1, pp.97–114.
Tellis, G.J., Stremersch, S. and Yin, E. (2003) ‘The international take off of new products: the role
of economics, culture, and country innovativeness’, Marketing Science, Vol. 22, No. 2,
pp.188–208.
Economic order quantity model with innovation diffusion criterion
303
Van den Bulte, C. (2000) ‘New product diffusion acceleration: measurement and analysis’,
Marketing Science, Vol. 19, No. 4, pp.366–380.
Van den Bulte, C. and Lilien, G.L. (2001) ‘Medical innovation revisited: contagion versus
marketing effort’, The American Journal of Sociology, Vol. 106, No. 5, pp.1409–1435.
Van den Bulte, C. and Stremersch, S. (2004) ‘Social Contagion and income heterogeneity in new
product diffusion: a meta-analytic test’, Marketing Science, Vol. 23, No. 4, pp.530–544.
Int. J. Applied Decision Sciences, Vol. 4, No. 3, 2011
Economic order quantity model with innovation
diffusion criterion having dynamic potential market
size
K.K. Aggarwal*, Chandra K. Jaggi and
Alok Kumar
Department of Operational Research, Faculty of Mathematical
Sciences, New Academic Block, University of Delhi,
Delhi 110007, India
Fax: +91-11-27666672
E-mail: kkaggarwal@gmail.com
E-mail: ckjaggi@yahoo.com
E-mail: alok_20_or@yahoo.co.in
*Corresponding author
Abstract: Introduction of a new product in the market forces the inventory
managers to consider the effects of marketing policies especially for innovation
effects at the earlier stage of the product life cycle to make the economic order
quantity (EOQ) model more realistic. Traditional EOQ models generally do not
consider the effect of marketing parameters. In this paper, a time dependent
innovation driven demand model has been introduced in the basic EOQ model
to calculate the different optimal policies. This model assumes that potential
market size is dynamic over time. The proposed model acknowledges
relationship between the innovation coefficient and the optimal policies. The
effectiveness of this model is illustrated with a numerical example and
sensitivity analysis of the optimal solution with respect to different parameters
of the system is performed.
Keywords: economic order quantity; EOQ; innovation driven demand;
dynamic potential adopters; diffusion theory.
Reference to this paper should be made as follows: Aggarwal, K.K., Jaggi, C.K
and Kumar, A. (2011) ‘Economic order quantity model with innovation
diffusion criterion having dynamic potential market size’, Int. J. Applied
Decision Sciences, Vol. 4, No. 3, pp.280–303.
Biographical notes: K.K. Aggarwal is an Assistant Professor in the
Department of Operational Research at University of Delhi, India. He obtained
his MSc in Operational Research and PhD in Inventory Management from
University of Delhi. His research interests and teaching include inventory
modelling, financial engineering and network analysis. He has published more
than 15 research papers in many scholarly journals.
Chandra K. Jaggi is an Associate Professor in the Department of Operational
Research, Faculty of Mathematical Sciences at University of Delhi, India. He
obtained his PhD, MPhil and Masters in Operational Research from the
Department of Operational Research at University of Delhi. His research
interest lies on analysis of inventory system. He has more than 31 publications
in Int. J. Production Economics, Journal of Operational Research Society,
European Journal of Operational Research, Int. J. Systems Sciences, Canadian
Copyright © 2011 Inderscience Enterprises Ltd.
Economic order quantity model with innovation diffusion criterion
281
Journal of Pure and Applied Sciences, OPSEARCH, Investigation Operational
Journal, Advanced Modelling and Optimisation, Journal of Information
and Optimisation Sciences, Int. J. Mathematical Sciences, Indian Journal of
Mathematics and Mathematical Sciences and Indian Journal of Management
and Systems.
Alok Kumar is a Research Scholar in the Department of Operational Research,
Faculty of Mathematical Sciences, University of Delhi, India. He received his
MA in Operational Research from University of Delhi. Currently, he is
pursuing his PhD in Operational Research from the Department of Operational
Research, Faculty of Mathematical Sciences, University of Delhi, India. His
area of research interest include inventory management and mathematical
modelling.
1
Introduction
The economic ordering policies often depend on demand information and demand is
mainly influenced by different marketing strategies. Therefore, the appropriate modelling
of demand is of great importance. Particularly, it is more significant for the new products,
because demands of such products are highly dynamic in nature and have less predictable
growth behaviour. In inventory system, normally demand of product is considered to be
constant, time dependent, stock dependent or probabilistic in nature. Unfortunately,
inventory modelling generally do not take into account the demand pattern of new
products. Changing needs of the society, technological breakthroughs the impact of
globalisation forces manufacturers to introduce new products in the market. When new
products are introduced in the market, their inventory management becomes significant.
The economic order quantity (EOQ) models have been explored to a great extent, in
particular with their interaction to different functional areas of business management like
production and finance. Though some models are there that used the Bass diffusion
demand model to formulate the optimal EOQ model (Chern et al., 2001) but not much of
work has been reported on the interaction of EOQ with marketing strategies. The
integration of EOQ policies with marketing policies is one of the key factors of
successful business operation. Therefore, the coordination between marketing disciplines
and inventory management is yet to be explored fully.
The proposed model includes the integration of EOQ with marketing disciplines in
which demand follows innovation diffusion process with dynamic market potential size.
Diffusion modelling of new product adoption has become an active area of marketing
research with more emphasis on inventory management since the pioneering work of
Bass (1969). This model is concerned with representing the dynamic nature of the
adoption of a new product. Since diffusion models are normative in nature and models of
this type include managerial decision variables in order to study their effect on the
diffusion process, therefore, here, role of inventory manager becomes indispensable
while managing the inventory and hence the importance of mathematical modelling in
inventory for such kind of situation is highly desired.
The diffusion of innovation within marketing has assumed greater significance
because theory of innovation diffusion relates how a new idea, a new product or a service
is accepted into a social system over time. The diffusion process in marketing is related to
282
K.K. Aggarwal et al.
penetration of a market by a new product. One of the most widely held theories of
communication in marketing is diffusion theory. Diffusion theory is actually a theory of
communication regarding how information is disseminated within a social system over
time. An innovation is an idea, object or a practice that is perceived as novel by the
members of the social system. Individuals who constitute the market may differ by the
manner in which the information about the innovation reaches to them and the manner in
which they respond to such received information. Diffusion is defined as the process by
which an innovation is communicated through certain channels over time among
members of a social system (Rogers, 1983, 2003).
Various diffusion processes have been studied in the literature. Fourt and Woodlock
(1960) gave pure innovative model, whereas Fisher and Pry (1971) are concerned with
pure imitative model. The Bass (1969) model captures both the innovative and imitative
aspects of product adoption. The Bass (1969) model, introduced over three decades ago
has been widely used in marketing by Mahajan and Muller (1979), Mahajan and Wind
(1986), Mahajan et al. (1990a, 1990b, 1995, 2000), Parker (1994), and Rogers (1995).
Bardhan and Chanda (2009) and Chanda and Bardhan (2008) have used the demand of
some products may decline due to the introduction of more advance products that
may influence the consumer’s preference. Alfares et al. (2005) have developed a
production-inventory model by integrating quality and maintenance decision. Also, a
number of researchers such as Blackman et al. (1973), Cheng and Sethi (1999), Golder
and Tellis (1998), Hamblin et al. (1973), Jaber et al. (2009), Dodson and Muller (1978),
Kalish (1985), Lilien et al. (1999), Mahajan and Robert (1978), Mahajan et al. (1990a,
1990b), Masao (1973), Roberts and Urban (1988), Rogers (1962), and Robert (1973)
have discussed innovation diffusion criterion in various forms. A theoretical analysis of
the diffusion of innovation in the marketing literature has also been well explained, which
are as follows. Sharif and Ramanathan (1981) presents some modified binomial
innovation diffusion model that incorporate dynamic potential adopter populations.
Sultan et al. (1990) considered a meta-analysis of applications of diffusion models.
Kurawarwala and Matsuo (1996) provided an integrated framework for forecasting and
inventory management of short life cycle products. Van den Bulte (2000) proposed a
model of measurement and analysis of new product diffusion acceleration. Roberts
(2000) examines the nature of emerging markets and identifies the features that make
them difficult to predict. Henard and Szymanski (2001) have studied to conduct and
present insights from a meta-analysis of the evidence on the determinants of new product
performance. Danaher et al. (2001) developed a model of marketing-mix variables and
the diffusion of successive generations of a technological innovation. Van den Bulte and
Lilien (2001) explained the underscores the importance of controlling for potential
confounds when studying the role of social contagion in innovation diffusion. Agarwal
and Bayus (2002) considered a model titled the market evolution and sales takeoff of
product innovations. Talukdar et al. (2002) have investigated the diffusion of new
product across products and countries. Furukawa and Kato (2002) developed a
conceptual model for adoption and diffusion process of a new product. Linton (2002) has
proposed a model which builds on existing knowledge of diffusion forecasting and
integrates it with the disruptive and discontinuous innovation literature. Pae and Lehmann
(2003) focuses on the diffusion patterns of the adjacent generations of technology and its
relation to the time that elapses between them. Tellis et al. (2003) developed a new
product model which comprises of the role of economics, culture and country
innovativeness. Everdingen et al. (2005) have introduced a cross-population adaptive
Economic order quantity model with innovation diffusion criterion
283
diffusion model that can be used to forecast the diffusion of an innovation at early stages
of the diffusion curve. Niu (2006) developed a piecewise-diffusion model of new-product
demands. Puumalainen et al. (2007) developed a model which proposes a set of
uncertainty sources of telecommunications industry that have notable effect on the
diffusion of innovations in the field. Chandrasekaran and Tellis (2007) have studied a
critical review of marketing research on diffusion of new products. Stremersch et al.
(2010) described the importance of time which accelerates early growth of new products.
Shinohara and Okuda (2010) proposed a dynamic innovation diffusion modelling.
Peres et al. (2010) have studied a critical review of innovation diffusion and new product
growth models.
Increments of new-buyer penetration
Cumulative percentage of buyers
Figure 1
Time
Source: Fourt and Woodlock (1960, pp.33–34)
The Fourt and Woodlock model explains the diffusion process in terms of number of
customers who have bought the product by time t by a modified exponential curve
(Figure 1). The discrete form of the model can be given as:
Qt = rQ(1 − r )t −1
(1)
where
Qt
sales at time t which is a fraction of the potential sales
Q
total potential sales as fraction of all buyers
r
rate of penetration of untapped potential
t
time period.
The model assumes that the level of adoption capability Qt can be expressed as a function
of time t and generate an exponential declining curve of new-adopter sales over time. The
values for the parameter Q and r, can be estimated based on available historical data
points in such a way that the resulting curve fits the data well. On the basis of this curve,
future adoption can be forecasted.
284
K.K. Aggarwal et al.
The Fourt and Woodlock model and many other researchers in the marketing
disciplines usually assume that the potential market size remains constant over time, but
in reality, it is not always possible. Different promotional activities, increase in
purchasing power due to increase in income, increase in population and several other
factors lead to increase in potential market size over time. Therefore, it becomes
important to include the potential market as dynamic instead of constant as far as research
regarding innovation diffusion is concerned. Sharif and Ramanathan (1982) have
considered the application of dynamic potential adopter diffusion model through their
study of diffusion of oral contraceptives in Thailand.
In this paper, an inventory model is developed where it is assumed that demand of the
product follows innovation diffusion behaviour as considered by Fourt and Woodlock
model and potential market size is dynamic. A comprehensive sensitivity analysis is
performed to highlight the impact of diffusion of innovation on the economic ordering
policies. The calculated results provide significant impetus while taking inventory
decisions for new products. This article is divided into model development, special cases
and observations and concludes with a discussion on the application.
2
Mathematical models
2.1 Assumptions and notations
Inventory model has been developed under the following assumptions:
1
the replenishment rate is instantaneous
2
lead time is zero
3
shortages are not allowed
4
demand rate is known and is governed by innovation process
5
the size of the potential market of total number of adopters is dynamic in nature
6
the coefficient of innovation is constant in nature
7
the innovation’s sales are confined to a single geographical area.
Notations used in the modelling framework are as follows:
A
Ordering cost.
C
Unit cost.
p(t)
Innovation effect at time t. Here, p(t) = p we assume that the innovation
factor that influences consumer adoption decision will remain constant
during the entire product life cycle.
I
Inventory carrying charge.
IC
Inventory carrying cost.
T
Length of the replenishment cycle.
Q
Number of items received at the beginning of the period.
Economic order quantity model with innovation diffusion criterion
K(T)
The total cost of the system per unit time.
I(t)
On hand inventory at any time t.
N0
Initial potential market size.
N(t)
Cumulative number of adopters at time t.
285
n(t) = λ(t) = S(t) The number of adoptions at time t, i.e., demand at time t.
The basic demand model used in this paper is based on the following assumptions:
1
Adoptions take place due to innovation-diffusion process and it is influenced by the
innovation-effect (mass-media) only.
2
The innovation effect is constant throughout the cycle time.
3
The potential market size is time dependent (dynamic). In this way, two form of time
dependent potential market size, i.e., linear and exponential have been considered by
taking two different cases.
The demand model used in this paper is based on the hypothesis that there exists potential
market whose size is time dependent (dynamic). With the passage of time prospective
buyers adopt the product. Using (1) and the demand assumptions, the mathematical form
of rate adoption at any time t can be given as:
φ (t ) =
f (t )
= p(t ) = p
1 − F (t )
(2)
The physical interpretation of the above model is the probability that a potential adopter
will purchase a product at time t given that no purchase has occurred till time t.
where
ϕ(t)
is the hazard rate that gives the conditional probability of a purchase in
a small time interval (t, t + ∆t), if the purchase has not occurred till time
t
f(t)
F (t ) =
∫ f (t )dt
is the likelihood of purchase at time t
t
is the cumulative likelihood of purchasing the product at time t.
0
Equation (2) can be rewritten as:
f (t ) = p [1 − F (t )] where F (t ) =
N (t )
N (t )
The number of adoptions (S(t)) at time t can be derived by multiplying f(t) with the
market size N (t ). Thus,
n(t ) = S (t ) = λ (t ) =
dN (t )
= p ⎡⎣ N (t ) − N (t ) ⎤⎦
dt
In this paper, we consider following two functional form of N (t )
(3)
286
1
2
K.K. Aggarwal et al.
when potential market size increases linearly
taking, N (t ) = N 0 (1 + gt )
g >0
when potential market size increases exponentially
taking, N (t ) = N 0 e gt
g >0
Case 1 when potential market size increases linearly
Taking, N (t ) = N 0 (1 + gt )
Therefore,
λ (t ) = n(t ) = p ⎡⎣ N 0 (1 + gt ) − N (t ) ⎤⎦
(4)
dI (t )
= −λ (t ), 0 ≤ t ≤ T
dt
(5)
The demand usage λ(t) which is a function of time plays pivotal role to shrink the
inventory size over a period of time. If in the time interval (t, t + dt) the inventory size is
dipping at the rate λ(t)dt, then the total reduction in the inventory size during the time
interval dt can be given by –dI(t) = λ(t)dt. Thus, differential equation describing the
instantaneous state of the inventory level I(t) in the interval (0, T) is given by:
Using equations (4) and (5) we get
dI (t )
= − p ⎡⎣ N 0 (1 + gt ) − N (t ) ⎤⎦
dt
(6)
(
)
The solution of the differential equation (6) is:
⎛ g⎞
⎛ g⎞
I (t ) = I (0) − N 0 pt + N 0 ⎜1 − ⎟ pt + e− pt − N 0 ⎜1 − ⎟
p
p⎠
⎝
⎠
⎝
(7)
where t = 0 ⇒ I(t) and N(t) = 0.
Since replenishment is instantaneous and shortages are not allowed.
Therefore,
∫
I (0) = Q = λ (t )dt
T
(
)
⎛ g⎞
⇒ I (0) = N 0 pT − N 0 ⎜1 − ⎟ pT + e − pT − 1
p⎠
⎝
0
(
Using the equations (7) and (8) we get
)
⎛
g⎞
I (t ) = N 0 pT − N 0 ⎜ 1 − ⎟ pT + e − pT − 1 −
p⎠
⎝
(
)
⎛
⎛ g⎞
g⎞
N 0 pt + N 0 ⎜ 1 − ⎟ pt + e− pt − N 0 ⎜1 − ⎟
p⎠
p⎠
⎝
⎝
(8)
(9)
287
Economic order quantity model with innovation diffusion criterion
Now,
∫ I (t )dt = N
T
0 pT
2
(
)
⎛
g⎞
− TN 0 ⎜ 1 − ⎟ pT + e − pT − 1 −
p⎠
⎝
(
⎡
1 − e − pT
⎛
⎛ g ⎞ T2
g⎞
T2
TN 0 ⎜1 − ⎟ − N 0 p
+ N 0 ⎜1 − ⎟ ⎢ p
+
p⎠
p ⎠⎢ 2
p
2
⎝
⎝
⎣⎢
0
) ⎤⎥
⎥
⎦⎥
The total cost per unit time, K1(T) comprises of the following elements
a
ordering cost per unit time
(OC1 ) =
b
A
T
(10)
inventory holding cost per unit time
∫
IC
I (t )dt
( IHC1 ) =
T
T
(
)
⎛
g⎞
⇒ IHC1 = N 0 pICT − ICN 0 ⎜1 − ⎟ pT + e − pT − 1 −
p⎠
⎝
0
(
− pT
⎡
⎛
⎛
g⎞
T
g ⎞ IC ⎢ T 2 1 − e
+
ICN 0 ⎜ 1 − ⎟ − N 0 pIC + N 0 ⎜1 − ⎟
p
2
p⎠
p⎠ T ⎢ 2
p
⎝
⎝
⎢⎣
c
) ⎤⎥
(11)
⎥
⎥⎦
material cost per unit time
( MC1 ) =
⎛
QC
C
g⎞
⇒ MC1 = N 0 pC − N 0 ⎜1 − ⎟ ( pT + e − pT − 1)
T
T
p⎠
⎝
(12)
Therefore, K1 (T) = OC1 + IHC1 + MC1
Using the equations (10), (11) and (12) we get
K1 (T ) =
⎛
⎛
A
g⎞
g⎞
+ N 0 pICT − ICN 0 ⎜ 1 − ⎟ ( pT + e − pT − 1) − ICN 0 ⎜1 − ⎟ −
T
p
p⎠
⎝
⎠
⎝
N 0 pIC
⎛
T
g ⎞ IC ⎡ T 2 (1 − e − pT ) ⎤
+ N 0 ⎜1 − ⎟
+
⎢p
⎥+
p ⎠ T ⎢⎣ 2
p
2
⎝
⎦⎥
N 0 pC −
⎛
C
g⎞
N 0 ⎜1 − ⎟ ( pT + e − pT − 1)
T
p⎠
⎝
The necessary condition for K1(T) to be minimum is
dK1 (T )
=0
dT
(13)
288
K.K. Aggarwal et al.
(
Therefore,
)
⎛
ICpN 0
dK1 (T )
g⎞
−A
= 0 ⇒ 2 + pICN 0 − ICN 0 ⎜1 − ⎟ p − pe − pT −
−
dT
p
2
T
⎝
⎠
⎛
⎛
g⎞
g⎞
N 0 IC ⎜1 − ⎟
N 0 IC ⎜1 − ⎟
− pT ) ⎞
2
⎛
(
p
p⎠
−
pT
e
1
⎝
⎠⎜
⎝
( pT + e− pT ) +
⎟+
+
2
⎜
⎟
p
T
2
T
⎝
⎠
⎛
⎛
g⎞
g⎞
CN 0 ⎜1 − ⎟
pCN 0 ⎜1 − ⎟
p
p⎠
⎝
⎠ ( pT + e − pT − 1) −
⎝
(1 − e− pT ) = 0,
2
T
T
Now, for K1(T) to be convex
Therefore,
d 2 K1 (T )
dT 2
d 2 K1 (T )
dT 2
(14)
≥0
⎛
g⎞
2 ICN 0 ⎜ 1 − ⎟
p ⎠ ⎡ pT 2 (1 − e− pT ) ⎤
⎝
≥0⇒ 3 +
+
⎢
⎥+
p
T
T3
⎣⎢ 2
⎦⎥
2A
⎛
⎛ g⎞
g⎞
2CN 0 ⎜1 − ⎟
ICN 0 ⎜1 − ⎟
p
p⎠
⎝
⎠ ( p − pe− pT ) +
⎝
( p − pe− pT ) ≥
2
T
T
⎛
g⎞
2 ICN 0 ⎜ 1 − ⎟
p⎠
⎛
1⎞
g ⎞⎛
⎝
( pT + e− pT ) +
p 2 e − pT CN 0 ⎜1 − ⎟ ⎜ I + ⎟ +
2
p
T
T
⎠
⎝
⎠⎝
(15)
⎛
g⎞
2CN 0 ⎜1 − ⎟
p⎠
⎝
( pT + e− pT − 1)
3
T
The solution of the equation
dK1 (T )
= 0 gives the optimum value of T provided it
dT
> 0. Since the above cost equation (13) is highly
dT 2
non-linear, the problem has been solved numerically for given parameter values. The
solution gives the optimum value T* of the replenishment cycle time T. Once T* is known
the value of optimum order quantity Q* and the optimum cost K1(T*) can easily be
obtained from the equations (8) and (13) respectively. The numerical solution for the
given base value has obtained by using software packages LINGO and Excel-Solver.
satisfies the condition
d 2 K1 (T )
Case 2 When potential market size increases exponentially
Taking, N (t ) = N 0 e gt
Therefore,
Economic order quantity model with innovation diffusion criterion
289
λ (t ) = n(t ) = p ⎡⎣ N 0 e gt − N (t ) ⎤⎦ , g > 0
(16)
dI (t )
= −λ (t ), 0 ≤ t ≤ T
dt
(17)
Let I(t) be the inventory level at time t(0 ≤ t ≤ T). The differential equation describing the
instantaneous state of the inventory level I(t) in the interval (0, T) is given by:
Using the equations (16) and (17) we get
dI (t )
= − p ⎡⎣ N 0 e gt − N (t ) ⎤⎦
dt
(18)
The solution of the differential equation (18) is:
I (t ) = I (0) − N 0 p
e gt
p 2 ⎛ e gt e − pt ⎞
+ N0
+
⎜
⎟
( g + p) ⎜⎝ g
g
p ⎟⎠
(19)
where t = 0 ⇒ I(t) = I(0) and N(t) = 0.
Since replenishment is instantaneous and shortages are not allowed.
Therefore,
∫
I (0) = Q = λ (t )dt
T
0
⇒ Q = I (0) = N 0 p
( e gT − 1)
g
p 2 ⎡ ( e gT − 1) ( e− pT − 1) ⎤
− N0
+
⎢
⎥
( g + p) ⎢⎣
g
p
⎥⎦
(20)
Using the equations (19) and (20) we get
I (t ) = N 0 p
( e gT − 1)
g
− N0
p 2 ⎡ ( e gT − 1) ( e− pT − 1) ⎤
+
⎢
⎥−
g
p
( g + p ) ⎣⎢
⎦⎥
e
p 2 ⎛ e gt e− pt ⎞
+ N0
+
N0 p
⎜
⎟
( g + p) ⎝ g
g
p ⎠
gt
Now,
∫
T
I (t )dt = N 0 pT
0
N0 p
( e gT − 1)
g
( e gT − 1)
g2
− N0
+ N0
Tp 2 ⎡ ( e gT − 1) ( e− pT − 1) ⎤
+
⎢
⎥−
( g + p ) ⎣⎢
g
p
⎦⎥
p 2 ⎡ ( e gT − 1) ( e− pT − 1) ⎤
−
⎢
⎥
( g + p ) ⎢⎣ g 2
p2
⎥⎦
The total cost per unit time, K2(T) comprises of the following elements
a
ordering cost per unit time
(21)
290
K.K. Aggarwal et al.
( OC2 ) =
b
A
T
(22)
inventory holding cost per unit time
( e gT − 1)
IC
−
I (t )dt ⇒ IHC2 = ICN 0 p
( IHC2 ) =
T
g
∫
T
0
ICN 0
p 2 ⎡ ( e gT − 1) ( e− pT − 1) ⎤
+
⎢
⎥−
( g + p) ⎣⎢
g
p
⎦⎥
ICN 0 p
c
( e gT − 1)
Tg
2
+ N0
ICp 2 ⎡ ( e gT − 1) ( e− pT − 1) ⎤
−
⎢
⎥
( g + p ) T ⎣⎢ g 2
p2
⎦⎥
material cost per unit time
( MC2 ) =
( e gT − 1)
QC
Cp 2
⇒ MC2 = N 0 Cp
− N0
( g + p)T
T
gT
(23)
⎡ ( e gT − 1) ( e− pT − 1) ⎤
+
⎢
⎥ (24)
g
p
⎢⎣
⎥⎦
Therefore, K2 (T) = OC2 + IHC2 + MC2
Using the equations (22), (23) and (24) we get
K 2 (T ) =
( e gT − 1)
A
p 2 ⎡ ( e gT − 1) ( e − pT − 1) ⎤
+ ICN 0 p
− ICN 0
+
⎢
⎥−
T
g
g
p
( g + p ) ⎣⎢
⎦⎥
ICN 0 p
N 0Cp
( e gT − 1)
Tg 2
( e gT − 1)
gT
+ N0
− N0
ICp 2 ⎡ ( e gT − 1) ( e− pT − 1) ⎤
−
⎢
⎥+
( g + p )T ⎣⎢ g 2
p2
⎦⎥
Cp 2
( g + p )T
⎡ ( e gT − 1) ( e− pT − 1) ⎤
+
⎢
⎥
g
p
⎢⎣
⎥⎦
(25)
The necessary condition for K2(T) to be minimum is:
dK 2 (T )
=0
dT
Therefore,
⎛
I⎞
CpN 0 ⎜1 − ⎟
CpN 0 e gT
g
dK 2 (T )
−A
⎝
⎠ gT
=0⇒ 2 −
( e − 1) +
dT
T
gT 2
Cp N 0 ⎛ e gT − 1 e− pT − 1 ⎞
+
⎜
⎟−
p ⎠
( g + p )T 2 ⎝ g
2
⎛
I I⎞
⎜1 + − ⎟
T
g⎠
⎝
+
T
1⎞
⎛
Cp 2 N 0 ⎜ I + ⎟
T ⎠ ( gT
⎝
e − e− pT ) −
( g + p)
ICp 2 N 0 ⎛ e gT − 1 e− pT − 1 ⎞ ICp 2 N 0 ⎛ e gT e− pT ⎞
−
+
⎜
⎟+
⎜
⎟
p ⎠
p 2 ⎟⎠ ( g + p )T ⎝ g
( g + p )T 2 ⎜⎝ g 2
(26)
Economic order quantity model with innovation diffusion criterion
Now, for K2(T) to be convex
d 2 K 2 (T )
dT 2
291
≥0
Therefore,
⎛
I⎞
2CpN 0 ⎜1 − ⎟
2
d K 2 (T )
2A
⎝ g ⎠ ( e gT − 1) + 2Cp N 0 ( e gT − e− pT ) +
0
≥
⇒
+
dT 2
T3
gT 3
( g + p )T 2
2
2 ICp 2 N 0 ⎛ e gT − 1 e− pT − 1 ⎞ ICp 2 N 0 gT
( e − e− pT ) ≥
−
⎜
⎟+
p 2 ⎠⎟ ( g + p )T
( g + p)T 3 ⎝⎜ g 2
⎛
⎛
I⎞
I I⎞
CpN 0 e gT ⎜ 1 − ⎟ CpN 0 e gT ⎜ 1 + − ⎟
g
T
g ⎠ ICpN 0 e gT
⎝
⎠+
⎝
+
+
T2
T2
T3
1⎞
⎛
Cp 2 N 0 ⎜ I + ⎟
2Cp 2 N 0 ⎛ e gT − 1 e− pT − 1 ⎞
T⎠
⎝
+
⎜
⎟+
( g + p)
p ⎟⎠
( g + p)T 3 ⎜⎝ g
( ge gT + pe− pT ) + 2ICp
N 0 ⎛ e gT e− pT
+
⎜
p
( g + p)T 2 ⎝⎜ g
The solution of the equation
2
(27)
⎞
⎟⎟
⎠
dK 2 (T )
= 0 gives the optimum value of T provided it
dT
> 0. Since the above cost equation (25) is highly
dT 2
non-linear, the problem has been solved numerically for given parameter values. The
solution gives the optimum value T* of the replenishment cycle time T. Once T* is known
the value of optimum order quantity Q* and the optimum cost K2(T*) can easily be
obtained from the equations (20) and (25) respectively. The numerical solution for the
given base value has obtained by using software packages LINGO and Excel-Solver.
satisfies the condition
d 2 K 2 (T )
2.2 Special case
When N (t ) becomes constant over a period of time say N then demand rate, i.e.,
λ(t) = n(t) will reduce to the classical Fourt and Woodlock model. The objective here is
to study the different managerial policies under the assumption that both innovation
factor and potential market size remain constant for the entire product life cycle. For this
case, the total cost per unit time, K(T) is given by
K (T ) =
A NC (
NIC (
1 − e − pT ) +
1 − e− pT ) − NICe− pT
+
T
T
pT
(28)
292
K.K. Aggarwal et al.
For optimum total cost, the necessary criterion is
dK (T )
− A NCpe − pT NC (1 − e − pT )
=0⇒ 2 +
−
+
dT
T
T
T2
NICe − pT NIC (1 − e − pT )
−
+ NICpe − pT = 0
2
T
pT
The solution of the equation
(29)
dK (T )
= 0 gives the optimum value of T provided it
dT
> 0. Since the above cost equation is highly non-linear,
dT 2
the problem has been solved numerically for given parameter values. The solution gives
the optimum value T* of the replenishment cycle time T. Once T* is known the value of
optimum order quantity Q* and the optimum cost K(T*) can easily be obtained by
following the same procedure as obtained in the above two cases. The numerical solution
for the given base value has obtained by using software packages LINGO and
Excel-Solver.
satisfies the condition
3
d 2 K (T )
Solution procedures
The solution procedure has been summarised in the following algorithm
Step 1
Input all parameter values such as unit cost, coefficient of innovation, potential
market size, ordering cost etc. for each case separately.
Step 2
Compute all possible values of T separately for the equations (14), (26) and (29)
as the case may be.
Step 3
Select the appropriate value of T for each case by using the equations (13), (25)
d 2 K (T )
> 0.
and (28) and by satisfying the above stated condition such as
dT 2
The above steps are used for all replenishment cycles using appropriate parameter values.
In order to obtain the appropriate values of T we need to follow the above procedure with
the help of LINGO and EXCEL-Solver software packages.
4
Numerical examples
A numerical example is presented in the following to illustrate the effectiveness of the
proposed model. A hypothetical example has the following parameter values in
appropriate units.
A = $1, 000/order, C = $1, 050/unit, I = 30%
Economic order quantity model with innovation diffusion criterion
293
Using the solution procedure described above, the optimal values of cycle length, order
quantity and total cost under different cases have been presented in Tables 1 to 5 for
different parameter values. Also, to prove the validity of the model numerically and to get
the appropriate parameter values, the references have been considered as Sharif and
Ramanathan (1981), Chandrasekaran and Tellis (2007), Sultan et al. (1990), Talukdar
et al. (2002) and Van den Bulte and Stremersch (2004).
•
The mean value of the coefficient of innovation for a new product usually lies
between 0.0007 and 0.03 (Sultan et al., 1990; Talukdar et al., 2002; Van den Bulte
and Stremersch, 2004).
•
The mean value of the coefficient of innovation for a new product is usually 0.001
for developed countries and 0.0003 for developing countries (Talukdar et al., 2002).
Case 1 When potential market size increases linearly
For this case, Sharif and Ramanathan (1981) considered the Coleman model and
explained by taking the case study of credit card banking in the USA. Here, Sharif and
Ramanathan (1981) regard this model as Model-1(b) and took the following parameter
values.
N 0 = 1459.77, α = g = 0.1292
Table 1 shows the changes in the values of T, Q and K1(T) for variations in the values of
coefficient of innovation for fixed value of g, whereas, Table 2 describes the changes
with respect to g for fixed value of coefficient of innovation.
Table 1
Changes in optimal values due to p
Sensitivity analysis on p for g = 0.13
T
K1(T*)
Q*
0.003
0.95
6,636
4.41
0.004
0.83
8,473
5.10
0.005
0.75
10,273
5.73
0.006
0.68
12,046
6.20
0.007
0.64
13,799
6.79
0.008
0.60
15,536
7.26
0.009
0.56
17,260
7.60
0.01
0.54
18,972
8.13
0.02
0.39
35,724
11.63
0.03
0.32
52,102
14.23
P
*
294
K.K. Aggarwal et al.
Table 2
Changes in optimal values due to g
Sensitivity analysis on g for p = 0.003
*
T
K1(T*)
Q*
0.13
0.95
6,636
4.41
0.14
0.94
6,662
4.38
0.15
0.93
6,688
4.35
0.16
0.91
6,713
4.27
0.17
0.90
6,738
4.23
0.18
0.89
6,762
4.20
0.19
0.88
6,786
4.17
0.20
0.87
6,810
4.13
0.21
0.86
6,834
4.10
0.22
0.85
6,857
4.06
g
Case 2 When potential market size increases exponentially
In this case, Sharif and Ramanathan (1981) took the following parameter values.
N 0 = 1928.55, α = g = 2.5863 × 102
Table 3 shows the changes in the values of T, Q and K2 (T) for variations in the values of
coefficient of innovation for fixed value of g, whereas, Table 4 describes the changes
with respect to g for fixed value of coefficient of innovation.
Table 3
Changes in optimal values due to p
Sensitivity analysis on p for g = 0.13
T
K2(T*)
Q*
0.003
0.99
8,065
5.77
0.004
0.86
10,391
6.67
0.005
0.77
12,680
7.46
0.006
0.71
14,942
8.25
0.007
0.66
17,184
8.94
0.008
0.61
19,410
9.44
0.009
0.58
21,623
10.09
0.01
0.55
23,825
10.63
0.02
0.40
45,461
15.42
0.03
0.33
66,715
19.05
P
*
295
Economic order quantity model with innovation diffusion criterion
Table 4
Changes in optimal values due to g
Sensitivity analysis on g for p = 0.003
*
T
K2(T*)
Q*
0.13
0.99
8,065
5.77
0.14
0.98
8,087
5.74
0.15
0.96
8,123
5.65
0.16
0.94
8,158
5.55
0.17
0.92
8,193
5.46
0.18
0.91
8,228
5.42
0.19
0.89
8,262
5.32
0.20
0.87
8,295
5.22
0.21
0.86
8,328
5.18
0.22
0.75
8,635
4.67
g
4.1 Special case
In this case, Sharif and Ramanathan (1981) considered the following parameter values.
N 0 = 2441.78, α = g = 0
Table 5 shows the changes in the values of T, Q and K(T) for variations in the values of
coefficient of innovation.
Table 5
Changes in optimal values due to p
Sensitivity analysis on p
T*
K(T*)
Q*
0.003
0.91
5,047
6.65
0.004
0.81
6,425
7.89
0.005
0.70
7,818
8.52
0.006
0.62
9,194
9.06
p
Following graphs show the behaviour of optimal total cost and optimal cycle length with
the changes in the coefficient of innovation for different cases.
Case 1 When potential market size increases linearly
Figure 2 presents the impact of coefficient of innovation on optimal total cost and
Figure 3 shows the impact on optimal cycle length.
296
K.K. Aggarwal et al.
Figure 2
Innovation effects vs. optimal cost (see online version for colours)
Figure 3
Innovation effect vs. optimal time (see online version for colours)
Figure 4
Innovation effect vs. optimal cost (see online version for colours)
Economic order quantity model with innovation diffusion criterion
297
Case 2 When potential market size increases exponentially
Figure 4 presents the impact of coefficient of innovation on optimal total cost and
Figure 5 shows the impact on optimal cycle length.
Figure 5
Innovation effect vs. optimal time (see online version for colours)
4.2 Special case
Figure 6 presents the impact of coefficient of innovation on optimal total cost and
Figure 7 shows the impact on optimal cycle length.
It can be seen that it is very difficult to prove the convexity of total cost curves in
equations (13), (25) and (28) analytically. The convexity of total cost curves has been
checked graphically for all the cases. Figures 8(a) to 8(c) illustrate the convexity of total
cost curves for different cases.
Figure 6
Innovation effect vs. optimal cost (see online version for colours)
298
K.K. Aggarwal et al.
Figure 7
Innovation effect vs. optimal time (see online version for colours)
Figure 8
Cost graphs showing convexity of the cost functions for (a) case 1, (b) case 2 and
(c) special case (see online version for colours)
(a)
(b)
(c)
Economic order quantity model with innovation diffusion criterion
5
299
Observations
We study the effect of changes in the system parameters on the optimal values of total
cost K(T*), the optimal cycle length T* and the optimal order quantity Q(T*). The results
obtained in the numerical exercise are very encouraging and are consistent with the
reality. From the results of numerical exercise and the graphical representation
(Figures 2, 3, 4, 5, 6 and 7) it has been observed that:
a
As the values of p increase keeping other parameters constant, the value of T* start
decreasing, the optimal order quantity Q(T*) increases slightly while the optimal cost
K(T*) increases significantly as depicted in Tables 1, 3 and 5. This is consistent with
the reality as more investment on promotion will increase the diffusion of a product
in the market resulting in shrinkage of the optimal reorder cycle time as a result
optimal cost is increased.
b
As the values of g increases keeping other parameters constant the value of T* and
the optimal order quantity Q(T*) decreases marginally while there is slight increase
in the optimal cost K(T*) as depicted in Tables 2 and 4. This kind of phenomenon
usually occurs in the real life situation as increment in g makes the potential adopter
dynamic which forces the inventory manager to keep the inventories according to the
changing pattern of the cycle time to get the total cost optimal.
6
Managerial implications
Economy is being globalised and the market opportunities are being opened up because
of cut in tariffs by the government of different countries to promote their products in the
market as a result new products are entered into the market. Also, innovative research
contributes the production of new products which are largely accepted by the many
sections of the society because of its innovativeness and usefulness. Now, when new
products are entered into the market their inventory management becomes a formidable
task for inventory managers and it becomes more crucial when potential market size is
dynamic. Because from a managerial perspective, it is important to focus on both
customer satisfaction and cost of inventory obsolescence to better manage inventory
planning and replenishment for sustained growth of the company. These circumstances
necessitate embracing the right balance between the customers need and the inventory
investment required to meet that demand. An overestimating demand may lead to buy too
much and exposes the firm to potential losses from liquidating overstocks where as
underestimating demand leads to backorders, cancellations and unsatisfied customers. In
dynamic environment, the determination of adopter categories is important for
developing market strategies for different kinds of adopters and for targeting market
prospects for a new product. Mathematical models have proved to be ideal tools to
control these effects and can be used for scheduling and managing the inventories. Thus,
it becomes necessary to formulate a procurement schedule of the inventories for any
organisation to success in a competitive environment. In this model, we have formulated
a procurement schedule and developed an economic ordering policies by taking into
account the sensitive nature of coefficient of innovation when the potential market size is
dynamic. This kind of model will always be helpful for the inventory manager of any
300
K.K. Aggarwal et al.
organisation for scheduling and managing the inventories when new products are
introduced in the dynamic potential market.
7
Conclusions
To manage and control inventory of a new product becomes a challengeable task for
almost every organisation because it requires a huge amount of investment of capital to
purchase inventory. Therefore, it becomes indispensable to develop a mathematical
model to get rid of these problems, as mathematical models are ideal tools to control and
manage the inventories of a new product in most economical way. In this paper, a time
dependent innovation driven demand model with dynamic potential market size has been
introduced in the basic EOQ model to calculate the different optimal policies. The
proposed model acknowledged relationship between the innovation coefficient and the
optimal policies in a dynamic market environment of two kinds. The results are very
encouraging and the findings are consistent with the idea that optimal EOQ policies are
highly sensitive towards the dynamics of innovation coefficient in a dynamic
environment and it is imperative to identify a trend. A numerical example followed by
sensitivity analysis on model parameters has been provided to verify the results obtained
in the real life situation. A simple solution procedure in the form of algorithm is
presented to determine the optimal cycle time and optimal order quantity of the cost
function. A special case has also been formulated which excludes the dynamic nature of
the potential market size.
A few limitations in our approach that suggests areas for future research, are as
follows. As the proposed EOQ model uses the extension of pure innovative demand
model, hence it ignores imitation coefficient (Bass, 1969) which is an important factor in
consumer adoption decision. The existence and the uniqueness of the different cost
functions have been shown numerically because of its highly non-linear nature. Thus
research on some alternative approach to get the optimal analytical solution of the
problem is important. In further research, we would like to extend the proposed model for
multiple generation situations, shortages, partial lost-sales, quantity discount etc.
Acknowledgements
The authors would like to thank the anonymous reviewers and the editor for their
constructive comments and suggestions.
References
Agarwal, R. and Bayus, B.L. (2002) ‘The market evolution and sales takeoff of product
innovations’, Management Science, Vol. 48, No. 8, pp.1024–1041.
Alfares, H.K., Khursheed, S.M. and Noman, S.M. (2005) ‘Integrating quality and maintenance
decisions in a production-inventory model for deteriorating items’, International Journal of
Production Research, Vol. 43, No. 5, pp.899–911.
Bardhan, A.K. and Chanda, U. (2009) ‘A mathematical model for diffusion of products with
multiple generations of a high technology product’, International Journal of Operational
Research, Vol. 4, No. 2, pp.214–229.
Economic order quantity model with innovation diffusion criterion
301
Bass, F.M. (1969) ‘A new product growth model for consumer durables’, Management Sci.,
Vol. 15, pp.215–227.
Blackman, W.A., Jr., Seligman, E.J. and Solgliero, G.C. (1973) ‘An innovation index based upon
factor analysis’, Technological Forecasting and Social Change, Vol. 4, pp.301–316.
Chanda, U. and Bardhan, A.K. (2008) ‘Modelling innovation and imitation sales of multiple
technological generation products’, Journal of High Technology Management Research,
Vol. 18, No. 2, pp.173–190, DOI: 10.1016/j.hitech. 2007.12.004.
Chandrasekaran, D. and Tellis, G.J. (2007) ‘A critical review of marketing research on diffusion of
new products’, Review of Marketing Research, pp.39–80.
Cheng, F. and Sethi, S. (1999) ‘A periodic review inventory model with demand. influenced by
promotion decisions’, Management Science, Vol. 45, No. 11, pp.1510–1523.
Chern, M.S., Teng, J.T. and Yang, H.L. (2001) ‘Inventory lot-size policies for the bass diffusion
demand models of new durable products’, Journal of the Chinese Institute of Engineers,
Vol. 24, No. 2, pp.237–244.
Danaher, P.J., Hardie, B.G.S. and Putsis, W.P., Jr. (2001) ‘Marketing-mix variables and the
diffusion of successive generations of a technological innovation’, Journal of Marketing
Research, Vol. 38, No. 4, pp.501–514.
Dodson, J.A., Jr. and Muller, E. (1978) ‘Models of new product diffusion through advertising and
word-of-mouth’, Journal of Marketing, Vol. 24, No. 15, pp.1–26.
Everdingen, Y.M.V., Aghina, W.B. and Fok, D. (2005) ‘Forecasting cross-population innovation
diffusion: a Bayesian approach’, International Journal of Research in Marketing, Vol. 22,
pp.293–308.
Fisher, J.C. and Pry, R.H. (1971) ‘A simple substitution model of technological change’,
Technological Forecasting and Social Change, Vol. 3, pp.75–88.
Fourt, L.A. and Woodlock, J.W. (1960) ‘Early prediction of market success for new grocery
products’, Journal of Marketing, Vol. 25, pp.31–38.
Furukawa, R. and Kato, H. (2002) ‘A conceptual model for adoption and diffusion process of a new
product’, Review of Marketing Science, Vol. 1, No. 1, pp.1–28.
Golder, P.N. and Tellis, G.J. (1998) ‘Beyond diffusion: an explanatory approach to modeling the
growth of durables’, Journal of Forecasting, Vol. 17, pp.259–280.
Hamblin, R.L., Jacobsen, R.B. and Miller, J.L. (1973) A Mathematical Theory of Social Change,
John Wiley & Sons, Inc., New York, NY.
Henard, D.H. and Szymanski, D.M. (2001) ‘Why some new products are more successful than
others’, Journal of Marketing Research, Vol. 38, pp.362–375.
Jaber, M.Y., Bonney, M. and Moualek, I. (2009) ‘An economic order quantity model for an
imperfect production process with entropy cost’, International Journal of Production
Economics, Vol. 118, pp.26–33.
Kalish, S. (1985) ‘A new product adoption model with price advertising and uncertainty’,
Management Science, Vol. 31, No. 12, pp.1569–1585.
Kurawarwala, A.A. and Matsuo, H. (1996) ‘Forecasting and inventory management of short
life-cycle products’, Operations Research, Vol. 44, No. 1, pp.131–150.
Lilien, G.L., Kotler, P. and Moorthy, K.S. (1999) Marketing Models, Prentice-Hall of India Private
Limited, New Delhi.
Linton, J.D. (2002) ‘Forecasting the market diffusion of disruptive and discontinuous innovation’,
IEEE Transactions on Engineering Management, Vol. 49, No. 4, pp.365–374.
Mahajan, V. and Muller, E. (1979) ‘Innovation diffusion and new-product growth models in
marketing’, Journal of Marketing Fall, Vol. 43, pp.55–68.
Mahajan, V. and Robert, A.P. (1978) ‘Innovation diffusion in a dynamic potential adopter
population’, Management Science, Vol. 24, No. 15.
Mahajan, V. and Wind, Y. (1986) Innovation Diffusion Models of New Product Acceptance,
Ballinger Publishing Company, Cambridge, MA.
302
K.K. Aggarwal et al.
Mahajan, V., Muller, E. and Bass, F.M. (1990a) ‘New product diffusion models in marketing:
a review and directions for research’, The Journal of Marketing, Vol. 54, No. 1, pp.1–26.
Mahajan, V., Muller, E. and Srivastava, R.K. (1990b) ‘Determination of adopter, categories using
innovation diffusion models’, Journal of Marketing, Vol. 27, No. 1, pp.37–50.
Mahajan, V., Muller, E. and Bass, F.M. (1995) ‘Diffusion of new products empirical
generalizations and managerial uses’, Marketing Science, Vol. 14, No. 3, pp.G79–G88.
Mahajan, V., Muller, E. and Wind, Y. (2000) New-Product Diffusion Models, Kluwer Academic
Publisher, Boston, MA.
Masao, N. (1973) ‘Advertising and promotion effects on consumers response to new products’,
Journal of Marketing Research, Vol. 10, pp.242–249.
Niu, S.C. (2006) ‘A piecewise-diffusion models of new-product demands’, Operations Research,
Vol. 54, No. 4, pp.678–695.
Pae, J.H. and Lehmann, D.R. (2003) ‘Multigeneration innovation diffusion: the impact of
intergeneration time’, Journal of the Academy of Marketing Science, Vol. 31, No. 1, pp.36–45.
Parker, P.M. (1994) ‘Aggregate diffusion forecasting models in marketing: a critical review’,
International Journal of Forecasting, Vol. 10, pp.353–380.
Peres, R., Muller, E. and Mahajan, V. (2010) ‘Innovation diffusion and new product growth
models: a critical review and research directions’, International Journal of Research in
Marketing, Vol. 27, pp.91–106.
Puumalainen, K., Sundqvist, S. and Huiskonen, J. (2007) ‘Managing uncertainty in forecasting the
diffusion of telecommunications innovations’, Journal of Scientific and Industrial Research,
Vol. 66, pp.299–304.
Robert, G.B. (1973) ‘A model for measuring the influence of promotion and consumer demand’,
Journal of Marketing Research, Vol. 10, No. 4, pp.380–389.
Roberts, J.H. (2000) ‘Developing new rules for new markets’, Journal of the Academy of
Marketing Science, Vol. 28, No. 1, pp.31–44.
Roberts, J.H. and Urban, G. (1988) ‘Modeling multivariate utility, risk, and belief dynamics for
new consumer durable brand choice’, Management Science, Vol. 34, No. 2, pp.167–185.
Rogers, E.M. (1962) Diffusion of Innovations, The Free Press, New York.
Rogers, E.M. (1983, 2003) Diffusion of Innovations, 3rd ed., The Free Press, New York.
Rogers, E.M. (1995) Diffusion of Innovations, 4th ed., The Free Press, New York.
Sharif, M.N. and Ramanathan, K. (1981) ‘Binomial innovation models with dynamic potential
adopter population’, Technological Forecasting and Social Change, Vol. 20, pp.63–87.
Sharif, M.N. and Ramanathan, K. (1982) ‘Polynomial innovation diffusion models’, Technological
Forecasting and Social Change, Vol. 21, pp.301–323.
Shinohara, K. and Okuda, H. (2010) ‘Dynamic innovation diffusion modeling’, Comput. Econ,
Vol. 35, pp.51–62.
Stremersch, S., Muller, E. and Peres, R. (2010) ‘Does new product growth accelerate across
technology generations’, Journal of Marketing, Vol. 21, No. 2, pp.103–120.
Sultan, F., Farley, J.U. and Lehmann, D.R. (1990) ‘A meta-analysis of diffusion models’, Vol. 27,
No. 1, pp.70–77.
Talukdar, D., Sudhir, K. and Ainslie, A. (2002) ‘Investigating new product diffusion across
products and countries’, Marketing Science, Vol. 21, No. 1, pp.97–114.
Tellis, G.J., Stremersch, S. and Yin, E. (2003) ‘The international take off of new products: the role
of economics, culture, and country innovativeness’, Marketing Science, Vol. 22, No. 2,
pp.188–208.
Economic order quantity model with innovation diffusion criterion
303
Van den Bulte, C. (2000) ‘New product diffusion acceleration: measurement and analysis’,
Marketing Science, Vol. 19, No. 4, pp.366–380.
Van den Bulte, C. and Lilien, G.L. (2001) ‘Medical innovation revisited: contagion versus
marketing effort’, The American Journal of Sociology, Vol. 106, No. 5, pp.1409–1435.
Van den Bulte, C. and Stremersch, S. (2004) ‘Social Contagion and income heterogeneity in new
product diffusion: a meta-analytic test’, Marketing Science, Vol. 23, No. 4, pp.530–544.