4.1 Assessing sales amount
Following is the operating procedure of generating sales prediction.
When t = T1, there is only generation 1 existing in the market. By applying sales data of generation 1, managers can easily predict by applying Bass’s method. This forecast is accurate because no sales of generation 1 switch to generation 2. However, the forecast appears deviation since generation 2 has launched.
According to our previous discussion, we abandon the isolating data from generation 1 and generation 2 because the individual data from both generations actually affected by two potential adopters.
Therefore, combining both data to generate an aggregate data is a precise way to find . The finding of aggregate sales S(t) doesn’t directly help managers because what managers care is the sales and the future market position of their products. But by applying equation 20, we can produce the forecasting sales
τ
Generation2
Generation1
Figure 10 Assess sales amount
T1 T2
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curve of S1(t) ( ) and can, of course, calculate S2(t).
From equation 23, obviously, we can get S1(t) and S2(t) after estimating the constant k. By applying the sales of generation 1 and generation 2 to equation 18 as Fisher and Pry, a certain constant k is available. Finally, we obtain S1(t) and S2(t) through the aggregate sales S(t) and the estimated constant k.
4.2 Assessing density function
The same as above method, using the data during the time before the launching time (τ of generation 2, the density function f1 and the cumulative density function F1
can be estimated by the coefficients p1, q1, m1. We emphasize that f1 cannot be used to generate the accurate amount of sales when other generations appear but can help produce the potential density function f2 of next generation.
The method is simple. From figure 11, sales of generation 2 at a given time compose of switch from S1 and original sales of generation 2 S2. Thus, the gap
Original curve of Generation2 Original S2
Switch from S1
Practical curve of Generation2
Figure 11 Components of S2
(23)
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between the practical sales of generation1 and the forecast of generation1 generated by f1 must be the switching part to generation 2.
Where S12(t) is the switching part to generation 2
Since the sales of generation 2 consist of the potential adopters of generation 2 and the switch of generation 1,
Practical data of generation 2 S2(t) minus the switching part S12(t) brings S2original.
By treating S2original as the original sales data of generation 2, we are able to produce the coefficients of p2, q2, m2 and then the density function f2 can be formed. F2 can also be found by integrating f2 but the density function f2 is enough for us to calculate the variable α(t) which represents the transitional fraction of old generation.
4.3 The transitional fraction α(t)
According to equation 20 that , using numerical
quadrature rules through matlab can solve α(t). And if we let the time t be infinity, the ultimate allocation state of both generations can be estimated.
From above equation, managers will know how many adopters of generation 1 change to generation 2 and realize the whole practical markets of both generations they need to serve.
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Estimating α(t) at any time is unnecessary because both S1(t) and S2(t) are available by previous method and temporary cumulative sales Y1(t) and Y2(t) aren’t very
important to managers. However, we can still estimate them:
As the procedure of calculating α(t), both integrations of S1(t) and S2(t) need using numerical quadrature rules by computer to solve it.
4.4 The timing of sales peak
Generation 1:
By setting equation 22 that
equal to zero, we can find that S1(t) reaches its sales maximum and minimum when
and .
4.4.1 Two extreme values
Comparing t1* and t2*, which is bigger depends on the amount of three coefficients.
If t1* is earlier than t2*, it means t1* is the timing of sales peak and t2* is timing of the minimum sales. Since S1(t) goes up at early stage and is replaced step by step by generation2, the presumption that the earlier time t1* is the timing of sales peak is reasonable. Note that the timing t2* is equal to the timing of sales peak of the whole market category S(t) (Bass,1969). Thus, the timing of sales minimum of S1(t) is the timing of sales peak of S(t). This means a large percentage of S1(t) transforms to
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generation2 and generation2 probably has a quite large potential adopters. Both conditions make S1(t) reaches its maximum at t1* and minimum at t2*. In contrast, if t2* is earlier than t1*, it means the timing of sales peak of generation1 is equivalent to the whole market, standing for that there is rare percentage of S1(t) changes to
generation2 and the potential adopters of generation1 is much bigger than generation2.
This situation results in a small constant k and causes a longer time t1*.
4.4.2 Only one extreme value
If k is less than p plus q, we cannot count t1* because of the definition of napierian logarithm (ln). In other words, the timing of sales peak of generation1 is equal to the whole market. Therefore, like above discussion, there is rare percentage of S1(t) changes to generation2 and the potential adopters of generation1 is much bigger than generation2.
4.5 Sales peak
Generation 1:
If generation1 reaches its sales peak at T = t1*, we replace t1* with and S(t) with the coefficients p, q, m into equation 20 that . The sales peak in terms of the coefficients p, q, m is:
If generation1 reaches its sales peak at T = t2*, the outcome will be:
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Finally, we know when is the timing of sales peak and the amount of sales peak of generation 1 as long as estimating the aggregate coefficients p, q, m and the constant k. In the end, we have to point that we can obtain two curves S1(t) and S2(t) by generating coefficient k and aggregate sales curve. Consequently, we are able to find the sales peaks and the timings of sales peaks of both generations easily by observing the curves we produced.
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