An evaluation of the time-varying extended logistic, simple logistic, and Gompertz models for forecasting short product lifecycles

An evaluation of the time-varying extended logistic, simple logistic, and Gompertz

models for forecasting short product lifecycles

Charles V. Trappey, Hsin-Ying Wu

Management Science, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan

a r t i c l e

i n f o

Article history: Received 2 May 2008 Accepted 20 May 2008

Keywords:

Extended logistic model Technology forecasting Simple logistic model Gompertz model Short product lifecycle

a b s t r a c t

Many successful technology forecasting models have been developed but few researchers have explored a model that can best predict short product lifecycles. This research studies the forecast accuracy of long and short product lifecycle datasets using simple logistic, Gompertz, and the time-varying extended logistic models. The performance of the models was evaluated using the mean absolute deviation and the root mean square error. Time series datasets for 22 electronic products were used to evaluate and compare the performance of the three models. The results show that the time-varying extended logistic model fits short product lifecycle datasets 70% better than the simple logistic and the Gompertz models. The findings also show that the time-varying extended logistic model is better suited to predict market capacity with limited historical data as is typically the case for short lifecycle products.

Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction

With the rapid introduction of new technologies and fast design to satisfy consumer demand, electronic products and services are often replaced within a few years. The product lifecycle for elec-tronic goods, which used to be about 10 years in the 1960s, fell to about 5 years in the 1980s and is now less than two years for consumer electronic products such as cell phones and computers. As product lifecycles become shorter, less data are available for market analysis and technology forecasting. Given the current market situation, smaller datasets must be used to forecast future trends of new electronic products and services. Hasted and Ehlers

[1]define a small dataset as the dataset which covers only short

time intervals with fewer than 30 data points.

A product lifecycle is typically divided into four stages that

in-clude introduction, growth, maturity and decline[2]. During the

introduction stage, the product is new to the market with little awareness and as a result there is slow sales growth. The growth stage, on the other hand, is characterized by a period of rapid sales growth resulting from the product being widely accepted by the marketplace. As sales growth declines, the product enters the ma-ture stage, and finally, when the marketplace is saturated with the product or a substitute product is introduced, product sales de-cline. The product lifecycle is often modeled using growth curves or sigmoidal curves which have an inflection point and approaches a fixed limit[3–9].

Growth curves (the first derivative of the product lifecycle

curve) are widely used in technology forecasting [10–16] since

technology product growth is often very slow during the introduc-tion stage (e.g., a new product replacing a mature product) which is then followed by rapid exponential growth when barriers to prod-uct adoption fall. The growth then approaches a market share limit. The limit reflects the saturation of the marketplace with the prod-uct or the replacement of the prodprod-uct with another. The curve also models an inflection or break point where growth ends and decline begins.

Many growth curve models have been developed to forecast the penetration rate of technology based products with the simple lo-gistic curve and the Gompertz curve the most frequently

refer-enced [5,6,9,12]. However, when using these two models to

forecast market share, care must be taken to set the upper limit of the curve correctly or the prediction will become inaccurate

[9]. The upper limit is the maximum possible value and represents

the maximum penetration rate or sales volume. Setting the upper limit to growth can be difficult and ambiguous. If the product will likely be popular and used for decades, then the upper limit is set to 100% of the penetration rate. This means that the product will be completely replaced only after everyone in the market has pur-chased the product. However, when marketers consider new tech-nology products such as computer games or new model cell phones, the value for the upper limit to market share growth can be difficult to estimate. That is, a computer game can be quickly re-placed by another game after only reaching 10% market share.

In order to avoid the problem of estimating the market share capacity for the simple logistic and the Gompertz models, Meyer

and Ausubel[14]proposed the extended logistic model. Under this

1474-0346/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.aei.2008.05.007

* Corresponding author. Tel.: +886 3 5727686; fax: +886 3 5713796.

E-mail addresses:trappey@faculty.nctu.edu.tw(C.V. Trappey),cindywu.ms94g@ nctu.edu.tw(H.-Y. Wu).

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model, the capacity (or upper limit) of the curve is not constant but

is dynamic over time. Meyer and Ausubel[14]also proposed that

technology innovations do not occur evenly through time but in-stead appear in clusters or ‘‘innovation waves”. Thus, they formu-lated an extended logistics model which is a simple logistics model with a carrying capacity k(t) that is itself a logistics function of time. Therefore, the researchers extend the constant capacity (k) of the simple logistic model by embedding the carrying capacity in the constant. This study applies the embedded carrying capacity concept to the study of electronics products using a time-varying extended logistic model.

The emergence of short product lifecycles has been addressed in

the supply chain and inventory management literature [7,8,17]

and there is general agreement that improved prediction of these lifecycles will benefit the management of supply chains, invento-ries, and product design. However, these new technology lifecycles are a modern phenomenon and the datasets (which characteristi-cally have fewer data points and shorter time periods) challenge the assumptions and applications of traditional forecasting methods.

Traditional forecasting models, like the simple logistic and Gompertz models, require that the upper limit of the curve be estimated prior to the forecast. Since it is difficult to estimate the demand of a new product or the arrival of a substitute prod-uct with limited data, traditional approaches are considered unreliable and inaccurate. Therefore, a time-varying extended lo-gistic model with flexible capacity is proposed where the capac-ity (or upper limit) of the curve is not constant but is dynamic over time.

The proposition of this research is that the time-varying ex-tended logistic model is better than the simple logistic and the Gompertz models when forecasting both long and short product lifecycles. Six time-series datasets describing market penetration rates and 16 datasets describing cumulative sales volumes were used to evaluate model performance. The electronic consumer goods datasets consist of six sets representing long product lifecy-cles and 16 sets representing short product lifecylifecy-cles.

Section1of this paper provides an introduction and Section2

discusses the challenges of forecasting short product lifecycles.

Section 3 describes traditional and newly developed technology

forecasting models including the simple logistic model, the Gom-pertz model, and the time-varying extended logistic model. Section

4describes the analytical process of this study and Section5 pro-vides an empirical case that compares the performance of the mod-els. The last section provides a summary and conclusion as well as the limitations of the study.

2. Forecasting short product lifecycles

Short product lifecycles have become more common in high technology and fashion-based industries which need to continu-ously introduce new consumer products to remain competitive

[7,17]. New electronic products with more functions, faster speed, and finer quality are continuously being introduced and quickly re-place models which may only be one year old. Quell et al.[18] ana-lyzed 37 types of home appliance from 1922 to 1979 and demonstrated that the shortening of product lifecycles is an impor-tant issue for product designers and planners. Given the reality of this market condition, the development of new forecasting tech-niques will improve the competitive response and manufacturing strategy of companies.

In 1969, Bass proposed a diffusion model to forecast the sales

volume of new products[3]that used the adoption rates of

innova-tors and imitainnova-tors. Innovainnova-tors are buyers that are not influenced by the previous buyers when making purchase decisions while

imita-tors are those who are influenced by earlier buyers. The Bass model has been widely applied by practitioners and modified by research-ers to forecast short product lifecycles.

Kurawarwala and Matsuo [7] proposed a growth model that

forecasts the seasonal sales volume demand of short product life-cycles based on the Bass diffusion model. Thirty-eight monthly data points for five different personal computer products were used to estimate seasonal demand and to compare the fit and forecast performance for three models. The measures used for model comparison were the sum of squared error (SSE), the root mean squared error (RMSE), and the mean absolute deviation

(MAD). Zhu and Thonemann [17] used the discrete version of

the Bass diffusion model and improved on Kurawarwala and

Mat-suo[8]model to develop an adaptive forecasting algorithm. The

demand data for a PC manufacturer was used to test the

forecast-ing performance of the algorithm. Chen [19] proposed an

ex-tended logistic model, which is called the time-varying exex-tended logistic model. This research uses the model from Chen’s study of seven home appliance datasets to demonstrate that the ex-tended logistic model improved the forecast of both long and short lifecycle datasets.

Lackman[20] reported that the simple logistic and the

Gom-pertz models are suitable for forecasting high technology products.

Morrison[6] also showed that the simple logistic and the

Gom-pertz models can be used to forecast the growth of new products. However, when the author applied the models, the upper limit was

set subjectively. Bengisu and Nekhili[9]used the simple logistic

and the Gompertz models to predict emerging technologies using publications and patents from science and technology databases

and Boretos[21]used the simple logistic model to show that the

diffusion of mobile phone technology follows an S-curve.

Meade and Islam[12]compared 17 growth models based on 25

time series datasets describing the telecommunications market. Their literature review shows that the simple logistic model is the most widely used. The authors conclude that basic forecasting models using two or three parameters, such as the simple logistic and Gompertz model, offer the best forecasting performance. Their research used datasets for traditional land-line telephones to com-pare forecasting models. However, the classic telephone intro-duced in the 1960s and which remained in use through the 1980s has a long product lifecycle that lasted over 30 years. When there are sufficient data points, the trajectory of the product growth curve is clear and the point of inflection can be calculated. If the point of inflection can be estimated, then the upper limit of the simple logistic and the Gompertz models can also be esti-mated. The simple logistic model is symmetric about the point of inflection. So if the inflection point is defined, the upper limit is twice the market share that occurs at the inflection point. For the Gompertz model, the point of inflection occurs at 37.79% of the upper limit and the upper limit can also be calculated when

the inflection point is found. Bengisu and Nekhili[9]showed that

the simple logistic and the Gompertz models are quite valid if the upper limit is correctly identified. However, the data points may not be sufficient (too few) to see the point of inflection and to set the correct upper limit when forecasting short lifecycle prod-ucts. Therefore, a model with more parameters, for example, the time-varying extended logistic model, is needed to project the tra-jectory of the growth curve. The time-varying extended logistic model uses a dynamic upper limit that can be estimated from the data.

There is little published research which compares the perfor-mance of forecasting models used on short product lifecycle data-sets. Thus, this study compares the fit and forecast performance of the simple logistic, the Gompertz, and the time-varying extended logistic models.

3. Technological forecasting models

Many models have been used in forecasting. This section intro-duces the models which are used in this study and derives the underlying formulas for each.

3.1. Simple logistic curve model

Most biological growth follows an S-shape curve or logistic

curve which best models growth and decline over time[14]. Since

the adoption of technology and technology-based products is sim-ilar to biological growth, the simple logistic model is widely used for technology forecasting. Many new forecasting models were proposed based on the simple logistic model and include innova-tions such as the Bass diffusion model and extended logistic model

[12]. The most important characteristic of simple logistic model is that it is symmetric about the point of inflection. This feature indi-cates that the process which will happen after the point of inflec-tion is the mirror image of the process that happened before the point.

The model for the simple logistic curve is controlled by three coefficients, a, b, and L is expressed as

yt¼

L

1 þ aebt ð1Þ

where ytis the value of interest, L is the maximum value of yt, a de-scribes the location of the curve, and b controls the shape of the curve. To estimate the parameters for a and b, the equation of the simple logistic model is transformed into a linear function using natural logarithms. The linear model is expressed as

Yt¼ lnðyt=L  ytÞ ¼  lnðaÞ þ bt ð2Þ

where the parameter a and b are then estimated using a simple lin-ear regression. The simple logistic model (Eq.(1)) and the linear model (Eq.(2)) are quoted from Martino’s book[22]and the

deriva-tions are shown inAppendix 1.

3.2. Gompertz model

The Gompertz model was first used to calculate mortality rates in 1825 and has been widely applied to technology forecasting

[22]. Although the Gompertz curve is similar to the simple logistic curve, it is not symmetric about the inflection point which occurs at t = (ln(b)/k). The Gompertz model reaches the point of inflection early in the growth trend and is expressed as

yt¼ Leae bt

ð3Þ

where L is the upper bound which should be set before estimating the parameters a and b. Similar to the methodology of estimating the parameters of the simple logistic model, natural logarithms are used to transform the original Gompertz model to linear equation:

Yt¼ lnðlnðL=ytÞÞ ¼ lnðaÞ  bt ð4Þ

and then the parameters are estimated[22]. Eqs. (3) and (4) are

quoted from Martino’s book [22]and the derivations are shown in

Appendix 2.

Although the predictive performance of the simple logistic model and the Gompertz model has been validated by many

researchers[13], the models have definite limitations when used

to forecast short product lifecycles. The reason is that it is almost impossible to estimate the correct upper limit for a new product when it is first introduced to market place.

Fig. 1depicts the importance of setting the correct upper limit in the simple logistic and the Gompertz models. As can been seen

inFig. 1, curves A and B start at the same point but have different upper limits, L1 and L2. Since the upper limits are set at different level, the two curves are different, and the prediction results will also be different.

3.3. Time-varying extended logistic model

The simple logistic model and the Gompertz model assume that the capacity of technology adoption is fixed and there is an upper bound to growth for these models. However, the adoption of new technology is seldom constant and changes over time. Therefore, researchers have proposed a dynamic carrying capacity and the

carrying capacity can be any function[14,23]. As shown by Meyer

and Ausubel[14], the original form of simple logistic model is writ-ten as dyt dt ¼ b  L  yt 1  yt L   ð5Þ

Let

a

dyt dt ¼

a

where L is the upper limit of the logistic curve and k(t) is the time-varying capacity function similar to the logistic curve.

In Meyer and Ausubel’s study, a special k(t) was set to represent a technology which has a bio-logistic growth rate. The setting of

k(t) in our research comes from Chen’s study[19]with

kðtÞ ¼ 1  d  ect _ð7Þ

and c and d are parameters that are estimated. The value of d can be any number and the value of c larger than zero. Our research as-sumes that the penetration rate capacity will fluctuate with time and may reach 100% but may also be as low as 30% or 50%. The rea-son for this assumption is that some new products may be intro-duced to the market and substitute older products. Thus, a product may not always achieve 100% market penetration and may be replaced earlier than expected.

Finally, the time-varying extended logistic model is expressed as yt¼ kðtÞ 1 þ a  ebt¼ 1  d  ect 1 þ a  ebt ð8Þ

where k(t) is the capacity that fluctuates with time, and a, b, c, and d are the parameters computed using a nonlinear least squared esti-mation method provided by a statistic software package like SY-STAT. When this model is tested using sales volume data, the equation is changed to Nt¼ m  yt¼ m  1  d  ect 1 þ a  ebt ð9Þ

L1

Time

L2

A

B

where Ntis cumulative volume by time t, and the coefficient m rep-resents the total market sales which is estimated using nonlinear least squares method.

The time-varying extended logistic model is similar to the Bass diffusion model and can be viewed as a special case of this model. The Bass model was developed for predicting sales volume, whereas the time-varying extended logistic model can be modified to predict sales volume as well as other proxies including market penetration rates. Therefore, the time-varying extended logistic model was selected for evaluation over the Bass model.

4. Analytical process

In order to test the forecast accuracy of the simple logistic, Gompertz, and the time-varying extended logistic models, the ana-lytical process is divided into two steps.

Step 1: Model estimation

The first step is used to estimate the models. After reserving the last five data points to test forecast accuracy of the simple logistic, Gompertz, and the time-varying extended logistic models, the remaining data points were used to fit the three models. For the

simple logistic and the Gompertz model, Eqs.(2) and (4)are used

to estimate coefficients using a simple linear regression. For the time-varying extended logistic model, the coefficients of the mod-els are estimated using nonlinear least squares with SYSTAT statis-tical software. After the coefficients were computed and the models fitted, the estimated values were calculated.

Step 2: Fit and forecast performance

After the models are constructed, the fit and forecast perfor-mance between the three models is conducted. The test consists of checking residuals between actual values and estimated values

to measure model performance[7,13]. Two measurements, mean

absolute deviation (MAD) and root mean square error (RMSE) are used to calculate residuals.

For the simple logistic and the Gompertz models, the upper lim-it must be set to obtain accurate results. Setting different upper limit levels of these two models will achieve different prediction results and the fit and predict performance will also be influenced. Thus, several upper limits of the simple and the Gompertz models were set to determine which upper limit would yield the best fit performance.

For forecast performance, the derived models are used to fore-cast the last five data points of the datasets. In this study, the mean absolute deviation (MAD) and root mean square error (RMSE) are used to measure performance as recommended in the literature

[7,13,24]. The mathematical representations are shown below:

MAD ¼ PT t¼1jyt ^ytj n ð10Þ RMSE ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PT t¼1ðyt ^ytÞ2 n s ð11Þ

where ytis the actual value at time t, ^ytis the estimate at time t, and n is the number of observations. These measurements are based on the residuals, which represent the distance between real data and predictive data. Consequently, if the values of these measures are small, then the fit and prediction performance is acceptable.

5. Empirical results

Twenty-two time-series datasets describing Taiwan penetration rate and cumulative sales volume of electronic products were col-lected to test the forecast accuracy of the simple logistic model, the Gompertz model and the time-varying extended logistic model. The datasets for market penetration rates were providing by the

Directorate General of Telecommunications and Chunghwa

Tele-com Co. and the Directorate General of Budget[25,26]. The market

penetration rate datasets cover six products including color TVs, telephones, washing machines, asymmetric digital subscriber lines (ADSL), mobile Internet subscribers, and broadband networks. The cumulative sales volume datasets were provided by the Taiwan

Market Intelligence Center[27]. These datasets cover 16 products

including LCD-TV, 19 in. LCD monitors, digital cameras with charge coupled device image sensors (CCD DC), digital cameras with more than five million pixels (DC > 5 million), 802.11g wireless local area networks devices (WLAN 802.11g), cable modems, com-bo optical disk drives (comcom-bo ODD), Barecom-bone computers (Bare-bone), China personal wireless access systems (China PAS), LCD panels for TV, LCD panels for notebooks, color mobile phones with 65K pixels (color-65k mobile phone), servers, over 30-in. wide LCD-TVs (LCD-TV > 30 in.), Voice over Internet Protocol Integrated Access Devices (VoIP IAD), and Voice over Internet protocol (VoIP) routers.

As shown inTable 1, the estimated sample period, predicted

sample period, and sample sizes are presented. The data for color TVs, telephones and washing machines are yearly data points. Since the sample period for these data is greater than 30 years, these products depict a complete product lifecycle (Fig. 2). This re-search classifies the data for color TVs, telephones and washing machines as long product lifecycles with large datasets for fore-casting. The other datasets (ADSL, mobile Internet subscribers, etc.) represent products rapidly brought to market and are catego-rized as short lifecycle products with limited or small (less than 30 data points) datasets for forecasting.

Fig. 2 shows the penetration rate for the six products and

Fig. 3shows the cumulative sales volume for the 16 short lifecycle

Table 1

Estimated and predicted sample period and sample size

Proxy Product Estimated

sample period Predicted sample period Sample size From To From To Penetration rate Color TV 1974 1999 2000 2004 31 Telephone 1970 1999 2000 2004 35 Washing Machine 1974 1999 2000 2004 31 ADSL 2000Q2 2005Q2 2005Q3 2006Q3 26 Mobile Internet 2001Q4 2005Q2 2005Q3 2006Q3 20 Broadband network 2000Q2 2005Q2 2005Q3 2006Q3 26 Cumulative sales volume LCD-TV 2003Q1 2006Q1 2006Q2 2007Q2 18 19 in. LCD monitor 2003Q1 2006Q1 2006Q2 2007Q2 18 CCD DC 2003Q1 2006Q1 2006Q2 2007Q2 18 DC > 5 million 2003Q1 2006Q1 2006Q2 2007Q2 18 WLAN (802.11g) 2003Q1 2006Q1 2006Q2 2007Q2 18 Cable modem 2003Q1 2006Q1 2006Q2 2007Q2 18 Combo ODD 2003Q1 2006Q1 2006Q2 2007Q2 18 Barebones 2003Q1 2006Q1 2006Q2 2007Q2 18 China PAS 2003Q1 2006Q1 2006Q2 2007Q2 18 LCD panel for TV 2003Q1 2006Q1 2006Q2 2007Q2 18 LCD panel for notebook 2003Q1 2006Q1 2006Q2 2007Q2 18 Color-65k mobile phone 2003Q1 2006Q1 2006Q2 2007Q2 18 Server 2003Q1 2006Q1 2006Q2 2007Q2 18 LCD-TV > 30 in. 2004Q1 2006Q2 2006Q3 2007Q2 14 VoIP IAD 2004Q1 2006Q2 2006Q3 2007Q2 14 VoIP router 2004Q1 2006Q2 2006Q3 2007Q2 14 Source: Directorate General of Budget and Directorate General of Telecommunica-tions and Chunghwa Telecom Co.25,26, and Market Intelligence Center Taiwan27.

products.Fig. 2shows that color TVs, telephones, and washing ma-chines have entered the mature stage of the product lifecycle. Therefore, a clear upper limit for these products can be set. On the other hand, the curves for ADSL, mobile Internet subscribers, and other short lifecycle products are still evolving, making it dif-ficult to define the stage of product lifecycle or to predict when these products will stop growing.

For the long product lifecycle datasets, the upper limit is set at 100%. For the short product lifecycle datasets, different upper limits

are set to achieve the best estimates. The possible upper limit for the short lifecycle is set at three different levels to include optimistic, a conservative, and a pessimistic settings. An optimistic upper limit means that the product is new to the market and has potential to grow. A pessimistic setting means that the product almost reached the upper limit to market growth. Between the optimistic and pes-simistic limits is the conservative setting. The conservative setting models a product that has been in the market for a while and has reached about one-third or one-half of the upper limits to growth.

0 20 40 60 80 100 120 Satu ra tio n (%) 1964 1970 1974 1976 19781980 198219841986 1988 1990 1992 1994 1996 1998 2000.062000.122001 .06 2001 .12 2002 .06 2002 .12 2003 .06 2003 .12 2004 .06 2004 .12 2005 .06 2005 .12 2006 .06 Time

Color TV Telephone Washing machine ADSL Mobile internet Broadband network

Fig. 2. Market growth for saturation datasets.

0 20000 40000 60000 80000 100000 120000 140000 160000 2003Q1 2003Q2 2003Q3 2003Q4 2004Q1 2004Q2 2004Q3 2004Q4 2005Q1 2005Q2 2005Q3 2005Q4 2006Q1 2006Q2 2006Q3 2006Q4 2007Q1 2007Q2 Time Cumulative sa

les volume (thous

and

)

LCD-TV 19”LCD monitor CCD DC DC >5m

WLAN (802.11g) Cable Modem Combo ODD Barebone

China PAS LCD panel for TV LCD panel for notebook Mobile color-65k

Server LCD-TV >30” VoIP IAD VoIP Router

Note: the unit of wireless internet is ten thousand. Source: ASIP, Market Intelligence Center, Taiwan

The penetration rate datasets use upper limits of 100%, 50%, and 30%. However, since the current penetration rate of the mobile Internet is 40%, the pessimistic setting is change to 50% and conser-vative upper limit is changed to 60%. For the cumulative sales vol-ume datasets, the upper limit is set based on the multiple of the most recent observation as recommended by Meade and Islam

[13]. The optimistic, conservative, and pessimistic upper limits

are 5 times, 3 times, and 1.5 times the most recent observation which is based on the cumulative sales volume of the second quar-ter in 2007. In fact, 5 times the most recent observation means that the proportion of the current cumulative sales volume to maximum sales volume (upper limit) is 20%. Thus, the current

cumulative sales volume only reaches 20% of the upper limit and there is still 80% of the maximum sales volume remaining to sell. So the setting of 5 times the most recent observation is an optimis-tic setting. A pessimisoptimis-tic setting of the cumulative sales volume dataset is set at 1.5 times the most recent observation. Using the cable modem dataset as an example, the most recent cumulative sales volume is 70,106,900 units and the pessimistic upper limit is 105,160,350 units. This means that the current cumulative sales volume has already reached two-third of the upper limit and has entered the mature stage of the product lifecycle.

Tables 2 and 3provide the fit and forecast performance for the penetration dataset. The evaluation rule is that the smaller the

va-Table 2

Fitting performance measures for the time-varying extended logistic, Gompertz, and the simple logistic models – penetration rate datasets

Model Index Color TV Phone Washing machine ADSL Mobile Internet Broadband network

L = 100% L = 50% L = 30% L = 100% L = 60% L = 50%a

L = 100% L = 50% L = 30%

Extended logistic MAD 0.0053 0.0063 0.0098 0.0036 0.0057 0.0027

RMSE 0.0071 0.0083 0.0118 0.0043 0.0073 0.0032

Gompertz MAD 0.0297 0.0290 0.0192 0.0154 0.0140 0.0102 0.0134 0.0077 0.0059 0.0146 0.0114 0.0075

RMSE 0.0502 0.0440 0.0290 0.0228 0.0170 0.0119 0.0156 0.0094 0.0076 0.0176 0.0132 0.0084 Simple logistic MAD 0.0361 0.0323 0.0276 0.0329 0.0274 0.0212 0.0337 0.0256 0.0217 0.0243 0.0075 0.0153 RMSE 0.0551 0.0453 0.0384 0.0483 0.0372 0.0264 0.0439 0.0304 0.0248 0.0328 0.0084 0.0180 Note: L – upper limit.

Boldface number means the best performance among three models.

a

The current saturation rate of mobile Internet is over 30%.

Table 3

Forecasting performance measures for the time-varying extended logistic, Gompertz, and the simple logistic models – penetration rate datasets

Models Index Color TV Phone Washing machine ADSL Mobile Internet Broadband network

L = 100% L = 50% L = 30% L = 100% L = 60% L = 50%a

L = 100% L = 50% L = 30%

Extended logistic MAD 0.0025 0.0049 0.0021 0.0140 0.0117 0.0036

RMSE 0.0026 0.0057 0.0034 0.0147 0.0152 0.0041

Gompertz model MAD 0.0042 0.0125 0.0095 0.1037 0.0671 0.0345 0.0627 0.0132 0.0146 0.0842 0.0531 0.0237 RMSE 0.0043 0.0130 0.0100 0.1060 0.0688 0.0352 0.0699 0.0154 0.0166 0.0867 0.0545 0.0243 Simple logistic model MAD 0.0045 0.0171 0.0140 0.2846 0.1688 0.0833 0.2397 0.0991 0.0503 0.1895 0.1185 0.0556 RMSE 0.0046 0.0174 0.0143 0.2933 0.1716 0.0839 0.2503 0.1021 0.0518 0.1958 0.1210 0.0562 Note: L – upper limit.

Boldface number means the best performance among three models.

a

The current saturation rate of mobile Internet is over 30%.

Table 4

Fitting performance measures for the time-varying extended logistic, Gompertz, and the simple logistic models – cumulative shipment volume data Model Saturation specification Index LCD-TV 19 in. LCD monitor CCD DC DC > 5 million WLAN 802.11g Cable modem Combo ODD Barebones China PAS Panel for TV LCD-TV > 30 in. VoIP IAD Extended logistic MAD 51 167 451 161 586 207 435 165 398 147 11 82 RMSE 58 212 593 188 772 282 538 184 483 182 15 100 Gompertz 5*2007Q2 volume MAD 192 364 1314 205 3219 1022 2204 1412 12,717 430 54 199 RMSE 268 434 1542 274 4634 1293 2802 1740 14,616 579 82 222 3*2007Q2 volume MAD 141 412 998 297 2026 770 1696 1036 12,298 305 70 198 RMSE 174 495 1120 440 2637 927 2095 1208 14,041 412 104 224 1.5*2007Q2 volume MAD 134 13,488 989 582 2026 354 678 553 2762 401 99 314 RMSE 172 15,179 1226 954 2510 443 740 655 3357 575 143 386 Simple logistic 5*2007Q2 volume MAD 900 1958 3853 2461 14,679 2484 4570 3543 14,105 2445 65 660 RMSE 1767 3925 5931 5439 27,124 3656 6438 5080 16,670 4943 119 1021 3*2007Q2 volume MAD 823 1803 3487 2238 13,007 2259 4076 3170 9084 2231 54 588 RMSE 1569 3521 5160 4842 23,147 3213 5542 4390 11,961 4371 98 875 1.5*2007Q2 volume MAD 656 1456 2615 1744 9414 1705 2820 2270 6280 1773 29 423 RMSE 1148 2635 3440 3562 14,974 2187 3478 2817 7605 3182 49 551

lue for MAD and RMSE, the better the fit and prediction

perfor-mance. As shown inTables 2 and 3, the time-varying extended

lo-gistic model has the best fit and prediction for both long and short lifecycle products.Tables 4 and 5provide the fit and forecast

per-formance for the cumulative sales volume dataset.Table 4shows

that time-varying extended logistic model has the best fit

perfor-mance for all products.Table 5shows that the time-varying

ex-tended logistic model has the best forecast performance for the majority of the products.

Table 6summarizes the comparative results of the time-varying extended logistic model, the simple logistic and the Gompertz models. The fit and forecast performance are compared and ranked using the root mean square error (RMSE) which is widely used for measuring the performance[7,13,23]. As can be seen inTable 6, the time-varying extended logistic model has the best predictive per-formance for 13 products among the 18 products for which the model converged. The model has the second best predictive performance for 4 products, and the worst predictive performance for LCD-TV > 30 in. data. The Gompertz model predicts best for 4 product datasets and has the second best forecast performance for three models. The simple logistic model only predicts well for

the LCD-TV > 30 in. data. In summary, the time-varying

extended logistic model is 70% better in prediction than the other models.

In order to test whether the root mean square error of the time-varying extended logistic (RMSEei) is smaller than error of the sim-ple logistic (RMSEsi) and Gompertz (RMSEgi) models, we first

calcu-late the statistics RMSEei RMSEsi and RMSEei RMSEgi. Then

these statistics are used to test the null hypotheses that RMSEei= RMSEsi(H0a) and RMSEei= RMSEgi (H0b) using one-tail sign test. The reason why the one-tail sign test is chosen is because the dis-tribution of RMSE is unknown and the sample size is small, so a nonparametric test is used. A sign test only needs a count of the number of sample value exceeding a defined constant which is equal to zero in this case[28].

Table 7 presents the P-values for the fit and forecast perfor-mance between the time-varying extended logistic model, the

sim-ple logistic model and the Gompertz model. As shown inTable 7,

all P-values of sign test are smaller than 0.05, which means there are statistically significant differences among the three models at the 95% level. Further, the time-varying extended logistic model outperforms than the simple logistic and Gompertz models in both fit and forecast performance.

The simple logistic and the Gompertz models are limited by the shape of the growth curve. For example, the simple logistic curve is symmetric about the point of inflection, so when the datasets do not have these characteristics, the simple logistic does not predict well. The Gompertz curve is an asymmetric S-curve and the Gom-pertz model reaches the inflection point before the market

pene-tration has reached half the upper limit[13]. Thus, the Gompertz

model may be more suitable for certain types of short lifecycle

products than the simple logistic model. As shown in Tables 3

and 5, the wrong capacity will lead to an error in prediction. If industrial policy or enterprise decisions are made based on a model

Table 5

Forecasting performance measures for the time-varying extended logistic, Gompertz, and the simple logistic models – cumulative shipment volume data Model Saturation specification Index LCD-TV 19 in. LCD monitor CCD DC DC > 5 million WLAN 802.11g Cable modem Combo ODD Barebones China PAS Panel for TV LCD-TV > 30 in. VoIP IAD Extended logistic MAD 265 6072 923 10,068 12,030 2253 1878 2040 3115 5217 497 1342 RMSE 301 7758 1157 12,434 12,505 2911 2029 2153 3305 5846 734 1740 Gompertz 5*2007Q2 volume MAD 1820 2372 11,176 1813 63,530 7910 23,077 14,504 16,194 5076 583 1548 RMSE 1913 2706 12,062 2303 69,259 8238 24,689 15,519 20,024 5433 697 2034 3*2007Q2 volume MAD 280 6805 3223 5194 32,902 3396 15,549 7971 11,017 604 887 302 RMSE 360 7814 3452 5582 35,553 3419 16,541 8421 13,492 703 1077 403 1.5*2007Q2 volume MAD 2707 13,488 9608 14,066 11,484 4205 2651 3031 14,329 6885 1364 2069 RMSE 3116 15,179 10,662 15,342 12,692 5052 2764 3530 14,703 7865 1668 2426 Simple logistic 5*2007Q2 volume MAD 26,450 64,846 70,320 104,201 365,213 36,609 61,892 53,560 43,670 78,644 2433 11,727 RMSE 29,106 72,596 76,969 116,523 396,302 39,564 66,579 58,014 51,122 86,272 3142 14,568 3*2007Q2 volume MAD 17,740 44,268 50,134 68,273 241,342 26,680 44,945 38,728 78,007 51,581 1643 8250 RMSE 18,836 47,682 53,567 73,370 254,235 28,212 47,516 41,109 81,173 54,536 2060 10,020 1.5*2007Q2 volume MAD 6734 16,090 16,953 25,192 86,925 9222 16,668 13,437 29,775 19,611 295 2516 RMSE 6786 16,182 17,113 25,473 87,717 9271 17,002 13,582 30,183 19,761 356 2881

Note: Boldface number means the best performance among three models.

Table 6

Fitting and Forecasting performance ranks of the extended logistic, Gompertz, and the simple logistic models

Product Fitting Forecasting

Extended logistic Gompertz Simple logistic Extended logistic Gompertz Simple logistic Color TV 1 2 3 1 2 3 Phone 1 2 3 1 2 3 Washing machine 1 2 3 1 2 3 ADSL 1 2 3 1 2 3 Mobile Internet 1 2 3 1 2 3 Broadband network 1 3 2 1 2 3 LCD-TV 1 2 3 1 2 3 19 in. LCD monitor 1 2 3 2 1 3 CCD DC 1 2 3 1 2 3 DC > 5 million 1 2 3 2 1 3 WLAN (802.11g) 1 2 3 1 2 3 Cable modem 1 2 3 1 2 3 Combo ODD 1 2 3 1 2 3 Barebones 1 2 3 1 2 3 China PAS 1 2 3 1 2 3 LCD panel for TV 1 2 3 2 1 3 LCD-TV > 30 in. 1 3 2 3 2 1 VoIP IAD 1 2 3 2 1 3

Note: 1 means the model with the lowest RMSE and best performance. For the Gompertz and the simple logistic models, the capacity with the lowest RMSE is compared.

using the wrong upper limit, a serious forecast error can be made. Since the time-varying extended logistic model uses more param-eters to capture the trend of products, the fit and forecast perfor-mance are improved.

This research used 22 product datasets to test the performance of the simple logistic, the Gompertz and the time-varying extended logistic model. However, the datasets for LCD panel for notebooks, color-65k mobile phones, servers, and VoIP routers, would not con-verge when using the time-varying extended logistic model to esti-mate the coefficients. A similar situation was reported by Meade

and Islam[13]. Their research used 25 telecommunications market

datasets to compare the performance of 17 growth curve models. For their study, half of the datasets would not converge when esti-mating the coefficients of models. Our study showed four products would not converge among 22 products yielding a proportion less than 20%. Therefore, our convergence results are consistent with earlier research.

When using the simple logistic and Gompertz models, the upper limit (L) must be set and then a linear transformation

method is applied to calculate parameters using Eqs. (2) and

(4). Since only two parameters are estimated, it is easy for the

models to converge. However, since the upper limit of the time-varying extended logistic model is dynamic with time, more parameters are needed to capture the trace. Therefore, the linear

transformation method used in the simple logistic and Gompertz models cannot be used to estimate the parameters and a nonlin-ear estimation method must be used. For the cumulative sales volume dataset, five parameters are estimated using 14–18 data points which causes an increase in nonconvergence for the ex-tended logistics model.

The four products with data that would not converge provide some insight. These datasets are linear and the curve for the

col-or-65k mobile phone has an obvious jump (Fig. 4). Meade and

Islam’s research[12]used telephone data from Sweden to

com-pare the simple logistic, extended logistic, and the local logistic models. They concluded that the extended logistic model had the worst performance. Although the setting of the extended lo-gistic model is different with this research, Meade and Islam’s study serves as a useful example. The growth curve of the Swed-ish telephone dataset is linear. Therefore, the time-varying ex-tended logistic model should not have been used. If forecasters wish to apply the time-varying extended logistic model, then they should confirm that the data has an S-shape prior to the forecast.

6. Discussion and conclusion

This study compares the fit and prediction performance of the simple logistic, Gompertz, and the time-varying extended logistic models for 22 electronic products. Since the simple logistic and Gompertz curves require the correct upper limit settings for accu-rate market growth accu-rate predictions, these two models may not be suitable for short product lifecycles with limited data. Therefore, to solve this problem, the time-varying extended logistic model was tested. Since the time-varying extended logistic model estimates the time-varying capacity from the data, it tends to perform better for both long and short lifecycle products if the data are not linear. The results show that the time-varying extended logistic model outperforms the simple logistic and the Gompertz models in most of product datasets where the data has the beginnings of an S-shape.

When forecasting the future growth and market for products, forecasters need to study the shape and the characteristics of

0 20000 40000 60000 80000 100000 120000 140000 160000 2003Q1 2003 Q2 2003 Q3 2003 Q4 2004 Q1 2004 Q2 2004Q3 2004Q4 2005 Q1 2005Q2 2005 Q3 2005 Q4 2006 Q1 2006 Q2 2006 Q3 2006 Q4 2007Q1 2007 Q2 Time C

umulative sales volume (thous

and

)

LCD panel for notebook Mobile color-65k Server VoIP Router

Note: the unit of wireless internet is ten thousand. Source: ASIP, Market Intelligence Center, Taiwan

Fig. 4. Market growth for LCD panel for notebooks, color-65k mobile phones, servers, and VoIP routers. Table 7

The P-value of sign test

Fitting performance Forecasting performance H0a: RMSEei= RMSEsi 0*** 0.0001*** H1a: RMSEei< RMSEsi H0b: RMSEei= RMSEgi 0 *** 0.0482*** H1b: RMSEei< RMSEgi

Note: The equation for P-value of one-tail sign test can be expressed as P-valueofsigntest ¼Psþ k¼0 n0 k   ð0:5Þn_{, where S}+

= the number of RMSEei> RMSEsi

(or RMSEgi); n0= the number of (n  S0); n = sample size; S0= the number of

RMSEei= RMSEsi(or RMSEgi); RMSEei= the RMSE value of time-varying extended

logistic model; RMSEsi= the RMSE value of the simple logistic model; RMSEgi= the

RMSE value of the Gompertz model.

***

the growth curve before selecting a suitable model. Although the time-varyin g extended logistic model has better performanc e for forecasting short lifecycle products, care must be taken when using this model. The extended logistic model may only be suitable for data that grows as an S-curve and may not be suitable for linear data or for curves with many anomalous data points. A possible solution for these types of datasets may be to apply smoothing techniques or data re-interpret ation techniqu es. Smoothing and data re-interpretati on techniqu es were first proposed by Tukey [29] and are commonly used in exploratory data analysis. Further research can be conducted using Tukey smoothing and data re-interp retation to see if the extended logistic model can be forced to converge and therefore find broad-er applications for a wider range of short product lifecycle datasets. Acknowled gement This research project was partially supported by the Taiwan National Science Council. Appendix 1 y T ¼ L 1 þ a e  bt ð1 Þ y t þ y t_a e  bt ¼ L y t_a e  bt ¼ L  y t a e  bt ¼ L  y t y t 1 a e bt ¼ y t L  y t ln 1 a e bt  ¼ ln y t L  y t   ln a þ bt ¼ ln y t L  y t  ¼ Y t Y t ¼ ln ðy t =L  y t_Þ¼ ln a þ bt ð2 Þ Appendix 2 y t ¼ Le  a e  bt ð3 Þ y t L ¼ e  a e  bt ln y t L  ¼ a e  bt ln L y t ¼ a e  bt Y t ¼ ln ln L y t  ¼ ln ða e  bt Þ Y t ¼ ln ðln ðL =y t_ÞÞ ln a  bt ð4 Þ Appendix 3

Cumulative sales volume dataset Sample period Products LCD-TV 19 in. LCD monitor CCD DC DC > 5 million WLAN (802.11g) Cable modem Combo ODD

Barebone China PAS LCD panel for TV LCD panel for notebook Mobile color-65k Server LCD-TV > 30 in.

VoIP IAD VoIP router 2003Q1 26.900 49.921 1540.638 33.759 56.588 1752.000 2946.440 2300.000 3815.000 25.340 2559.158 390.000 410.000 2003Q2 63.100 389.660 3706.334 132.559 215.973 3614.000 5230.362 4720.000 9096.000 97.940 4956.234 490.000 850.000 2003Q3 164.100 763.660 6537.814 302.049 500.027 5379.000 8882.726 7229.000 14799.000 359.369 8059.782 2893.000 1295.000 2003Q4 407.500 1245.868 9196.234 602.352 958.930 7552.000 12461.777 10677.000 21275.000 795.749 11955.402 7557.000 1792.000 2004Q1 787.500 2173.868 12021.892 1159.524 1572.550 9832.000 16351.273 14115.000 30759.000 1307.749 15826.702 11353.000 2295.000 84.000 425.000 30.800 2004Q2 1194.500 3273.868 15788.372 1905.414 2244.595 12385.000 20438.203 17378.000 39869.000 2280.749 19619.702 14937.000 2809.000 146.000 1055.200 68.800 2004Q3 1603.900 4025.868 20911.238 3480.309 3154.315 15865.000 26021.378 21389.000 48429.000 3207.749 23735.702 19398.000 3334.000 225.000 1869.500 267.600 2004Q4 2178.600 5417.368 25733.319 5189.401 4499.246 19520.000 32006.914 25943.000 55984.000 4821.049 28662.202 24106.000 3900.000 338.000 2739.500 769.300 2005Q1 2993.600 7735.268 30158.223 7221.865 5928.487 22604.000 36858.493 31263.000 62744.000 6606.549 33278.502 28003.000 4487.300 509.000 3698.500 1309.300 2005Q2 4059.600 10957.768 36820.687 10907.315 7716.311 26086.000 41830.072 36523.000 69737.900 9046.649 38795.202 32758.000 5083.605 781.000 4773.500 1923.300 2005Q3 5432.200 15307.768 45611.350 15795.081 9722.111 31326.000 47856.446 42451.000 76124.574 12112.899 44874.202 37099.000 5699.927 1219.000 6337.500 2345.300 2005Q4 7101.652 20404.988 55382.907 22373.650 12482.311 35912.000 53790.890 49106.000 81922.139 17078.509 52274.002 54268.000 6353.141 1712.000 8489.500 2661.300 2006Q1 8883.652 26032.988 63007.848 28958.121 14918.811 40315.000 58765.654 55533.000 85990.326 21967.809 59708.202 69451.000 7014.901 2403.000 11166.500 2967.300 2006Q2 10896.652 32548.245 71902.554 37172.826 17202.311 45434.900 64048.872 61134.000 89875.326 26745.219 67162.002 84473.000 7681.721 3284.000 13853.950 3471.300 2006Q3 13379.652 41731.245 83420.201 48149.297 19519.511 51546.900 68619.391 67911.000 93475.326 33127.219 74899.002 101103.000 8369.621 4330.000 16776.350 3880.300 2006Q4 16097.652 52201.245 94502.554 59281.062 21876.301 56997.900 73063.269 74853.000 96525.326 40059.219 83663.352 117994.000 9076.997 5505.000 19727.650 4178.300 2007Q1 18563.652 61755.245 102643.731 67488.821 24246.615 63382.900 76889.575 81808.000 99545.326 46143.219 92258.352 132689.000 9789.320 6699.000 22328.650 4575.300 2007Q2 21794.652 71821.525 114524.165 79399.864 26800.515 70106.900 81083.108 86487.000 102795.326 54545.239 102214.352 146140.000 10506.950 8355.000 25181.450 4836.300 Unit: thousand.

Source: Market Intelligence Center Taiwan27.

C.V. Trappey, H.-Y. Wu /Advanced Engineering Informatics 22 (2008) 421–430 429

References

[1] J.W. Havstad, C.L. Ehlers, Attractor dimension of nonstationary dynamical systems from small data sets, Physical Review A 39 (1989) 845–853. [2] P. Kotlor, Marketing Management, 11th ed., Prentice Hall, New Jersey, 2003. [3] F.M. Bass, A new product growth model for consumer durables, Management

Science 15 (1969) 215–227.

[4] V. Mahaian, E. Muller, F.M. Bass, New product diffusion models in marketing: a review and directions for research, Journal of Marketing 54 (1990) 1–26. [5] J.S. Morrison, Life-cycle approach to new product forecasting, The Journal of

Business Forecasting Methods and Systems 14 (2) (1995) 3–5.

[6] J.S. Morrison, How to use diffusion models in new product forecasting, The Journal of Business Forecasting Methods and Systems 15 (2) (1996) 6–9. [7] A.A. Kurawarwala, H. Matsuo, Forecasting and inventory management of short

life-cycle products, Operations Research 44 (1) (1996) 131–150.

[8] A.A. Kurawarwala, H. Matsuo, Product growth models for medium-term forecasting of short life cycle products, Technological Forecasting and Social Change 57 (1998) 169–196.

[9] M. Bengisu, R. Nekhili, Forecasting emerging technologies with the aid of science and technology databases, Technological Forecasting and Social Change 73 (2006) 835–844.

[10] L.D. Frank, An analysis of the effect of the economic situation on modeling and forecasting the diffusion of wireless communications in Finland, Technological Forecasting and Social Change 71 (2004) 391–403.

[11] R.R. Levary, D. Han, Choosing a technological forecasting method, Industrial Management 37 (1995) 14.

[12] N. Meade, T. Islam, Forecasting with growth curves: an empirical comparison, International Journal of Forecasting 11 (1995) 199–215.

[13] N. Meade, T. Islam, Technological forecasting – model selection, model stability, and combining models, Management Science 44 (1998) 1115–1130. [14] P.S. Meyer, J.H. Ausubel, Carrying capacity: a model with logistically varying

limits, Technological Forecasting and Social Change 61 (1999) 209–214.

[15] P.S. Meyer, J.W. Yung, J.H. Ausubel, A primer on logistic growth and substitution, Technological Forecasting and Social Change 61 (1999) 247–271. [16] L.P. Rai, N. Kumar, Development and application of mathematical models for

technology substitution, PRANJANA 6 (2003) 49–60.

[17] K. Zhu, U.W. Thonemann, An adaptive forecasting algorithm and inventory policy for products with short life cycles, Naval Research Logistics 51 (2004) 633–653.

[18] W. Qualls, R.W. Olshavsky, R.E. Michaels, Shortening of the PLC – an empirical test, Journal of Marketing 45 (1981) 76–80.

[19] C.-P. Chen, The test oftechnological forecasting models, Master thesis, National Chiao Tung University, 2005.

[20] C.L. Lackman, Logit forecasting of high tech products, Industrial Management 35 (2) (1993) 20–21.

[21] G.P. Boretos, The future of the mobile phone business, Technological Forecasting and Social Change 74 (2007) 331–340.

[22] J.P. Martino, Technological Forecasting for Decision Making, third ed., McGraw-Hill, New York, 1993.

[23] J.E. Cohen, Population growth and the earth’s human carrying capacity, Science 269 (1995) 241–246.

[24] T. Islam, N. Meade, Forecasting the development of the market for business telephones in the UK, The Journal of the Operational Research Society 47 (1996) 906–918.

[25] Directorate General of Telecommunications and Chunghwa Telecom Co, Taiwan,<http://www.dgt.gov.tw/english/flash/index.shtml>, 2007. [26] Directorate General of Budget, Accounting and Statistics, Executive Yuan,

Taiwan,<http://eng.dgbas.gov.tw/mp.asp?mp=2>, 2007.

[27] Market Intelligence Center (MIC), Advisory & Intelligence Service Programs (AISP), Taiwan,<http://mic.iii.org.tw/intelligence/>, 2007.

[28] P. Sprent, N.C. Smeeton, Applied Nonparametric Statistical Methods, fourth ed., Chapman & Hall, Boca Raton, FL, 2007.