Modeling New Product Diffusion under Uncertainty
Jayendran Venkateswaran and Siddhartha Paul
Industrial Engineering and Operations Research,
IIT Bombay, Powai, Mumbai 400076
Abstract
Diffusion of new, and innovative products in a potential market has been well studied in
literature, using the famous Bass Model as the primary model. Recently a few works have also
focussed on the supply planning and distribution strategies for new product introduction under
uncertainty. This paper aims to further our understanding on the effect of supply and demand
uncertainty on the overall market share.
1.0 Introduction
Bass diffusion model (Bass, 1969) is not new in marketing literature and perhaps it has been
studied extensively since decades for predicting demands of new innovative products. Most of
the literature until recently have assumed product diffusion under infinite supply (Mahajan et al.,
1990; Peres et al., 2010) while in reality, the constraints on supply affect the diffusion dynamics
significantly. There is few literature (Ho et al., 2002; Kumar & Swaminathan, 2003; Higuchi &
Troutt, 2004; Shen et al., 2011; Negahban et al., 2014; Negahban & Smith, 2016) to study Bass
diffusion model and supply chain model combined. Ho et al. (2002) have considered a more
generalized Bass model, where the total population is decomposed into four classes viz. Potential
customer, waiting in queue, adopter, and lost-sale and studied in the presence of supply
constraint. Using optimal control theory, authors established closed form expressions for the
demand and sales trajectories over the product lifecycle. They further concluded that the product
launch can be delayed for building up an initial inventory stockpile but demand fulfillment
should not be delayed when the firm is having inventory. Kumar & Swaminathan (2003) have
modified the classic Bass diffusion model in order to capture the effect of supply constraint on
future period demands. They have concluded that always selling the maximum of demand and
inventory may not be optimal but rather the firm should wait to build up the optimal initial
inventory level and then start selling in order to reduce lost sales. Higuchi & Troutt (2004) have
shown the importance of integration of supply chain model with product diffusion model, taking
a video game product of Bandai Co. as a case study. Bandai Co. had launched a video game viz.
Tamagotchi™
in 1996 which immediately gained popularity through word of mouth and
exceeded the supply capacity. Realizing such huge demand, after some time delay when Bandai
Co. increased their production capacity, people lost their interest in the product and resulted in
huge unsold inventory leftover for Bandai Co. Negahban et al. (2014) have developed an agent-
based simulation model for managing production level after launching a new product and found
effect of various factors (such as social network structure, production strategy etc.) on
production/inventory cost and lost sales. They have used a classical Bass model to forecast the
actual demand of future periods and dynamically update the parameters of forecasting Bass
model at every decision cycle, while the actual diffusion process was modelled as a variant of
Bass diffusion model with multiple customer stages. Negahban et al. (2016) have studied the
impact of demand and supply uncertainties on optimal production and sales plan for new
innovative products using Monte Carlo simulation. Authors have used a model similar to
modified Bass model, as proposed by Kumar & Swaminathan (2003), for simulating product
diffusion. They have concluded that the optimal sales and production plan for deterministic
setting may not remain optimal under uncertainty of demand and supply. However, the authors
have used a relatively simple ordering rule, where any production lead time is not considered.
The other stream of literature (Tan, 2002; Gupta & Maranas, 2003; Li et al., 2009) in supply
chain management have studied the effect of demand and supply uncertainty on customer
satisfaction, production cost etc. considering demand as exogenous to the system/model and thus
not valid for new product diffusion.
2.0 The Bass Model
The classical Bass model is as shown in Figure 1. The target population (NV) is divided into the
population of potential adopters (P) and the population of adopters (A). The total adoption rate of
the new product is then defined as the sum of the adoptions resulting from advertising (and any
external means) and from word of mouth (and any implicit positive feedback driven by the
adopters population). The underlying equations are given as follows:
AdoptionRate, AR(t)= pP(t) + qP(t)A(t)/N
Adoptions= Advertising _Adoptions~ word of ~ mouth
ie)
Where, p is the coefficient of innovation, g the coefficient of innovation. The term pP indicates
the Adoptions from advertising and qPA/N indicates the adoptions from word of mouth. The
population of adopters (4) and potential adopters (P) evolve as follows:
P(t)=f -AR(t) and Ald=f AR(t)
Potential
Adopters, P
Adopters, A
Adoption Rate
y
Adoption from
Advertising Adoption
from Word of
Mouth
Target
Population, N
Coefficient of
Imitation, q
Coefficient of
Innovation, p
Figure 1: Stock-Flow representation of the Bass Model
The behavior of the model is characterized by the two loops. The marketing effect loop, which
solves the problem of startup since the adoption from advertising is independent of the Adopter
population (Sterman, 2001). The adoptions from the advertising display an accelerated delay,
followed by an exponential decay. The word of mouth loop essentially drives the model to
exhibit the classic S-shaped growth of Adopters. The adoptions from word of mouth grows
rapidly, peaks, and then declines as the Adopters saturate the market. The steady state is reached
when the entire target population become adopters, i.e., A(t*) = N, or P(t*) = 0. The total
duration ¢* can be termed as the duration of diffusion.
The above Bass model includes several assumptions:
e The target population (N) is fixed and does not change with time. This may be reasonable
for diffusions happening at a small time scale. However, for models with diffusions
spanning larger time scale, the effect of births, deaths, migration etc may need to be
incorporated.
e The effect of word of mouth strictly results in a positive impact.
e The values of p and g remain unchanged with time.
e There is uniform mixing of the population, that is everyone can come into contact with
anyone else.
e The price of the product has no effect on the size of potential adopters nor on the
adoption rate.
e The adoption of the new product is considered to be independent of the adoption (or non-
adoption) of other products, independent of social, economic and political conditions.
e Customer decision instantaneously changes from potential adopter to an adopter. Multi-
stage models can be used to capture the evolution of customers through an ‘awareness’
stage and ‘showing interest’ stage before becoming adopters.
e There are no repeat or replacement purchases.
e Sufficient supply of new product inventory is available to meet the required adoption
rate.
e Spatial or geographical spread of diffusion is not considered.
Many works have been presented in literature where the Bass model is extended by relaxing one
or more of the above assumptions, as discussed earlier.
Interestingly, it is noted that for a given p and gq, the behavior of the Bass model is independent
of the population. That is, the fraction of adopters (A) to the target population (N) remains
independent of N. This behavior is illustrated in Figure 2(a) and 2(b). The values of p and q are
taken as 0.008 and 0.25. Fig. 2(a) shows the adoption rates when N = 4,000, 7,000 and 13,000. In
Fig. 2(b), the fractional rate of adoption (= AR(é)/ N) is plotted. It is clearly observed that the
model behavior is independent of N.
Adoption Rate fractional Rate of Adoption
MUZAN =p
o FE
5 10 15 20 25 30 35 40 45 5 w
‘Tame (Month) 51045 33 OS
Adoption Rate: TargetPop:
Adoption Rate : TargetPop=13000_§ ————_——
fractional TatgePop=4000
feactonal Rave of Adoption: TargeDop=13000
Adoption Rate TaegePep=7000
2(a) Adoption Rate 2(b) Fractional rate of adoption
Figure 2: Dynamics with different values of target population V
Further, the model can also be expected to be robust to any demand variations. Since the
populations are conserved (i.e., A(f) = N - P(t)), random perturbations in the adoption rates from
its mean (as governed by the Bass model) will not affect the total diffusion duration t*. Demand
uncertainty can be incorporated in the model by including a perturbation term, in computing the
Adoption Rate, where is a random variable with zero mean and non-zero variance, as:
AR(t)=pP(t)+qP(t)A(t)/N+e (2)
It is noted that to ensure the conservation of population (A(¢) = N - P(t) and prevent backward
flow, it is to be ensured that the adoption rate does not exceed the current population of potential
adopters, and is not negative. This is achieved by:
AR(t)=Min(Max( pP(t)+qP(t) A(t)/N+e,0), P) (3)
Figure 3(a) and 3(b) show the dynamics of diffusion under demand uncertainty. The values of p,
q and N are taken as 0.008, 0.25 and 4000. Figure 3(a) shows the adoption rates for different
settings of . In Fig. 3(b), the adopters (A(¢)) is plotted. It is clearly observed that demand
uncertainty per se affects neither the total adopters nor the diffusion duration. These behaviors,
while making the Bass model robust, also significantly overlooks the impact of supply on the
spread of diffusions. This paper explores the impact of supply uncertainty on the Bass model
dynamics.
Adoption Rate
35 30 3S 7 7 =
‘Tame (Month)
Adoption Rate : Epsion-UT-50,50}
‘Adoption Rate : DemandCertain
Figure 3(a) Dynamics of adoption rate AR(t) under uncertain demand
Adopters, A
5000
3750 =
2500
3 10 15 20 25 30 35 4045 30
Time (Month)
"Adopters, A"
"Adopters, A"
"Adopters, A"
"Adopters, A"
“Adopters, A" : DemandCertain
Figure 3(b) Dynamics of Adopters A(t) under uncertain demand
3.0 Exploring and Extending the Bass Model
3.1 Constrained Supply
The effect of supply can be modeled as a co-flow, as shown in Figure 4. A stock of inventory is
maintained, and the adoption rate can then be set to not exceed the available inventory on hand,
i(t). The underlying equations of the extended model are as follows:
Inventory (1)
SupplyRate
Potential
Adopters,
Adoption Rate
A Target
Adoption from Population, N
Advertising Adoption
a from Word of rain
Mouth
Coefficient of ™ Coefficient of
Innovation, p Imitation, q
Figure 4: SFD model of extended Bass model with explicit modeling of supply
AR(t)=Min( pP(t)+qP(t) A(t)/N,I(t))
a i ia 2
(4)
Desired - Adoption~ Rate, DAR(t)
I(t)=SupplyRate (t)- AR(t) ®)
Now, suppose the cumulative supply is more than the target population, and all potential adopters
wait as-long-as-required for the product, then the dynamics are quite intuitive. In periods when
the on-hand inventory exceeds the adoption rate, the dynamics will be comparable to the original
Bass model. In periods when the adoption rate is constrained by the inventory (/(‘) > 0), the
system will exhibit a linear growth in adopters. Eventually, the target population of adopters will
be reached, with the delay in diffusion proportional to the supply delays, i.e., the number of days
inventory was insufficient to meet desired demand. Figure 5 illustrates the behavior of the system
for varied patterns of supply (constant rate of supply and batch supply). When the supply is at a
constant rate of 100 units/ period, the adoption rate in the first 10 periods equals the desired
adoption rate. The accumulation of excess inventory in periods 1 to 5 provides to meet the
demand in period 6 to 10. In periods 11 to 35, the inventory availability determines the adoption
rate, and after 36 time periods, the adoption follows an exponential decay, reaching a saturation
of 4000 in 50 periods. The total duration of diffusion t* has increased when compared to the
scenario of non-delayed supply. Suppose the products are supplied in a batch size of 1000 at
time periods 0, 10, 20 and 30, the dynamics will be similar to the red color lines in the graphs in
Fig. 5. The cumulative adopters reach 2000 (exhausting the first two deliveries) in period 15
which results in a plateau (AR=0) until the next lot arrives. This alternate sales and periods of
idleness continue until the target is reached. It can be observed that the increase in the duration
of diffusion ¢* is approximately equal to the number of periods when JNV is less than the desired
adoption rate (in the example considered, the ¢* increases by about 11 periods). Also, of the total
50 periods of diffusion, in 10 periods (i.e. 20%) the distribution does not take place (i.e., on days
of zero AR(t)). This can have a serious impact on the motivation of future performance of the
sales workforce, which is not considered in the current model.
Adoption Rate Adopters, A
1000
Sih :
3101520353035 40S
5 10 15 2 25 30 35 40 45 50 Time (Month)
‘Time (Month)
Adoption Rate
Adoption Rate
‘Adoption Rate
Figure 5: AR(t) and A(t) under different supply scenarios
3.2 Constrained Supply with abandonment
It will be interesting to next understand the dynamics when the potential adopters are impatient.
That is, unavailability of the product (say, due to supply delay) creates a negative impression
resulting in some proportion of potential unsatisfied customers abandoning the future purchase of
the product (they might have purchased another equivalent product from the market or simply
lost interest due to delay). This aspect is captured by including another outflow to the stock of
Potential Adopters, called as Abandon Rate (AbR(t)). AbR(t) is a product of the abandonment
fraction f and the instantaneous proportion of dissatisfied customers. Also, the total target
population N is reduced by AbR in order to conserve the population. The extended SFD model
that includes abandonment is shown in Figure 6.
Inventory (I)
SupplyRate
Abandonment ;
sae i S to Coefficient of|
ion ‘
pcidoption |. ad — Imitation, q
“P ‘Mouth
Figure 6: SFD model of extended Bass model with explicit modeling of supply & abandonment
The additional underlying equations are as follows:
P(t)={ ~AR(t) AbR(0) (6)
AbR(t)=f* (DAR(t)- AR(t))=max( pP(t)+qP(t) A(t)/N(t)-I(t),0) (7)
N(t)=[ - AbR(t) (8)
Simulation results comparing the performance with and without abandonment are presented in
Figure 7. The values of p, g, f and N are taken as 0.008, 0.25, 0.25 and 4000. Two different
supply patterns are considered. When the supply is at a constant rate of 100 units/ period, it can
be seen that the total adopters saturate (reach) 3550 with 450 abandoning the purchase of the
product (see Figure 7(a)). However, from the manufacturer’s point of view the sales rate
(adoption rate) is actually quite stable until period 30 after which it decays exponentially, and
saturating at 3550 adopters. Suppose the products are supplied in a batch size of 1000 at time
periods 0, 10, 20 and 30 (see Figure 7(b)). The sales (adoption) first and the second batch of
1000 are identical under with and without abandonment. The third batch of 1000 is also sold
(adopted) but over a longer duration (5 periods instead of 4 periods). However, the last batch of
1000, when supplied, never gets fully sold. Also, interestingly, both supply patterns results in
almost the number of adopters post-abandonment. The key implication of the above results is
that the manufacturers will not be aware of the potential decline in adoptions until very late,
resulting in unspent inventory, additional costs and loss of market, since abandonment rates are
quite difficult to estimate.
Adoption Rate Adopters, A
4000
3000
2000
1000
3 10 15 20 25 30 35 40 45 St 9
Time (Month) 5 10 15 20 25 30 35 40 45
‘Time (Month)
"Adopters,
‘Adapters,
Adopters, A”
Figure 7(a): AR(t) and A(t) when supply is a constant at 100 units per period and abandonment
Adopters, A
Adoption Rate 4000
300
3000
225
2000
150
1000
6
0
0 5 10 15 20 25 30 35 40 45
5 10 152 3 35° «40045, Time (Month)
Time (Month) “Adopters, A”
: "Adopters, A" Si
"Adopters, A"
Figure 7(b): AR(t) and A(t) under batch supply of 1000 units per 10 periods and abandonment
3.3 Implications of a grand product launch
Most new products are launched in a ‘grand’ manner to help kick start the adoption of the
product. This refers to the initial quantity of products sold (adopted) by highly concentrated
efforts aimed to create awareness among the target population. However, post this, the
manufacturer may choose to significantly reduce the efforts on advertising and rely more on the
word of mouth effects generated by the initial set of adoptions. This is captured in the original
Bass model by modifying the coefficient of innovation parameter p, such that p takes a high
initial value (modeled as a PULSE() input) followed by the significantly lower value.
Simulation experiments are conducted to quantify the effects of the grand product launch
(p=0.008 in all periods except in period | when p = 0.05), under limited supply and with
abandonment. The results of these experiments are presented in Figure 8.
Figure 8(a) displays the adoption rates under grand launch with infinite supply. It is observed, as
expected, that there is a sharp increase in sales (~230 units) in the period since more products are
adopted in the early stages, which is further reinforced by the word-of-mouth effects resulting in
the overall reduction of the duration of diffusion by 3 periods. Figure 8(b) summarizes the results
under a constant supply of 100 units/ period (with an initial stock of 100). Figure 8(c)
summarizes the results when products are supplied in a batch size of 1000 at time periods 0, 10,
20 and 30. The total adopters for both supply patterns are lesser when the big launch is carried
out as compared with the case without a big launch. Also, it is seen that the batch supplied
saturated at a lower level of adopters (~3355) as compared when to constant supply (~3370).
Adoption Rate
i
0
1s
2 25 3
Time (Month)
3:
5
40
45
doption Rat
‘Adoption Rate
Figure 8(a): Result of Bass model with and without
ch
Adoption Rate
3
10
15
20 30
Time (Month)
Apo ae
Figure 8(b): Result of extended Bass model under c:
Adoption Rate
10
15
2 253
‘Time (Month)
30
35
40
45
Adopters, A
4000
3000
2000
1000
0
5101520253035 4045
Time (Month)
"Adopters, A"
"Adopters, A" jal aunch
a grand launch
Adopters, A.
3 10 Is 20 25 30 35 40 45 Si
Time (Month)
onstant supply, abandonment, grand launch
Adopters, A
4000
3000
2000
1000
o
510153035305
Time (Month)
Adopters A": Supply ate
Ad Sip
Figure 8(c): Result of extended Bass model under batch supply, abandonment and grand launch
4. Conclusions and Future Work
Simple extensions of the Bass model has been discussed in this paper to help further the
understanding of the uncertainties in demand and supply on the new product diffusion dynamics.
Batched supplies can cause significant periods of idleness (adoption rate equals 0) leading to loss
of motivation. Suppose customers decide to abandon their purchases, then the total target
population will never be reached, and more importantly, the manufacturer will not be aware of
this until the very end of the diffusion period. Also, it is observed that grand launches of
products, if not followed up with regular supply will result in an overall decrease in adoptions.
On going research work is being carried out to understand, model and analyse the diffusion of a
new product at multiple unconnected geographical regions, but which are constrained by a
common supplier of materials. Also, explicit modeling of the impact of idle periods (periods with
adoption rates) on the motivation of the workers can be investigated. Further, given the (real life)
adoption rates of products which includes periods of idleness (either due to supply uncertainty or
demand uncertainty) the estimation of the coefficient of innovation and imitations, along with the
reasons for supply delays are of interest. These investigations will help in designing a robust plan
for diffusion of new products under uncertainty.
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