Quadratic-interval Bass model for new product sales diffusion

Expert Systems with Applications 36 (2009) 8496–8502

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Quadratic-interval Bass model for new product sales diffusion Fang-Mei Tseng a,*, Yi-Chung Hu b a b

Department of International Business, Yuan Ze University, 135 Yuan-Tung Road, Chung-Li, 32003 Taiwan, ROC Department of Business Administration, Chung Yuan Christian University, Taiwan, ROC

a r t i c l e

i n f o

Keywords: Bass model Fuzzy regression analysis Gompertz model Logistic model

a b s t r a c t An appropriate new product diffusion method is vital to a business firm for developing marketing strategies aimed at potential adopters. The Bass (1969) [Bass, F. M. (1969). A new product growth model for consumer durables. Management Science, 15, 215–227] model is an important innovation diffusion model used to forecast the growth speed and the potential market volume of innovative products and relies on statistics to explain the relationships between the dependent and independent variables. However, fuzzy relationships are more appropriate for describing the relationships between dependent and independent variables, since these relationships require less data than traditional models to generate reasonable estimates of parameters. Therefore, we have combined fuzzy regression with the Bass model to develop a quadratic-interval Bass diffusion model, and we have applied the models to three cases. When insufficient data are available, quadratic-interval Bass diffusion models are potentially useful tools. However, when there is high variability in the data, the quadratic-interval Bass model should not be used. Our practical application shows that the quadratic-interval Bass model is an appropriate tool that can reveal the best and worst possible sales volume outcomes. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction The diffusion of an innovation has traditionally been defined as the process by which that innovation is communicated through certain channels over time among the members of a social system (Roger, 1983). Communication channels constitute one key element. Diffusion theory’s main focus is on communication channels, which are the means by which information about an innovation is transmitted to or within the social system. Besides, diffusion theory provides estimates of the product’s diffusion speed and total market potential. The speed of diffusion represents the sales increase in any period, while the market potential is the product’s maximum total sales after its introduction. This is crucial for corporations when developing marketing strategies and maintaining or improving their competitive advantage. Models representing the time series diffusion of new products have a long history of application in marketing. Bass (1969) integrated the modified exponential model (Fourt & Woodlock, 1960) and the logistic model (Mansfield, 1961) to propose a new product diffusion model, which is well known and widely used in developing product life-cycle curves and for forecasting initial purchases of new products. Diffusion models typically require at least 6–10 observations to generate reasonable parameter estimates (Heeler & Hustad, 1980). This is a problem for new products, because early * Corresponding author. Tel.: +886 3 4638800x2691; fax: +886 3 435 4624. E-mail address: [email protected] (F.-M. Tseng). 0957-4174/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2008.10.078

sales data are lacking. Moreover, the Bass model explains the deviations between estimation and observations through measurement error, which is problematic because the data are precise values that do not include measurement errors. In addition, if a phenomenon under consideration does not have stochastic variability but is also uncertain in some sense, it is more natural to seek a fuzzy functional relationship for the given data, which may be either fuzzy or crisp. That is to say, a fuzzy phenomenon should be modeled by a fuzzy functional relationship. Fuzzy regression analysis was first proposed by Tanaka, Uejima, and Asai (1982), who used a fuzzy linear system as a regression model to solve a fuzzy environment problem and to avoid modeling error. Because the membership functions of fuzzy sets are often described as possibility distributions, this approach is usually referred to as a possibility regression analysis (Dubois & Prade, 1980; Tanaka, 1987; Tanaka et al., 1982). Tanaka and Lee (1998) proposed an interval regression analysis based on a quadratic programming approach. This quadratic programming approach produces more diverse spread coefficients than a linear programming approach, and integrates the property of central tendency in least squares analysis and the possibility property in fuzzy regression. Kim, Moskowitz, and Koksalan (1960) found that fuzzy regression can be a viable alternative to statistical linear regression in estimating regression parameters when the data set is insufficient. Therefore, this paper proposes a quadratic-interval Bass model that combines quadratic-interval regression with the Bass

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innovation diffusion model to solve a fuzzy relationship between explanatory and response variables and to provide forecasts of sales to decision makers. This model requires fewer observations than traditional innovation diffusion model. We have applied the models to three cases to demonstrate the performance, and find that it make good forecasts and appear to be appropriate tools. This remainder of this paper is organized as follows: in Section 2, we review the Bass model and the quadratic-interval regression model; in Section 3, the quadratic-interval Bass model is formulated and proposed; in Section 4, the quadratic-interval Bass model is applied to three cases; and conclusions are discussed in Section 5.

tions. A generalized model of fuzzy linear regression is as follows (Tanaka et al., 1982):

yj ¼ A0 þ A1 x1j þ    An xnj ¼

n X

Ai xij ¼ x0j A

ð3Þ

i¼1

where xj ¼ ð1; x1j ; . . . ; xnj Þ0 is a real input vector of independent variables, n is the number of variables, and A = (A0,    , An)’ represents a vector of the fuzzy parameters in the model. Instead of using a crisp value, the ith fuzzy parameter bi after the L-type fuzzy numbers of Dubois and Prade (1980), (ai, ci)L, the possibility distribution is

lAi ðAi Þ ¼ Lfðai  Ai Þ=ci g 2. Bass model and fuzzy regression model To explain the proposed model, the Bass innovation diffusion model and the fuzzy regression model are described in the following sections. 2.1. Bass innovation diffusion model The most famous first-purchase diffusion model in marketing research was the Bass innovation diffusion model (Bass, 1969), which combined the modified exponential function (Fourt & Woodlock, 1960) and the logistic function diffusion model. The Bass (1969) model is well known and widely applied in developing product life-cycle curves, while also being used to forecast the sales volume of initial purchases of new products. The basic assumption of the model is that the timing of a consumer’s initial purchase is related to the number of previous buyers. Bass (1969) presumes that initial purchases of the product are made by both ‘‘innovators” and by ‘‘imitators”. Innovators and imitators are distinguished by how they are influenced by buying patterns; the number of people who have already bought the product does not influence the timing of an innovator’s initial purchase, but does influence an imitator. Imitators ‘‘learn”, in some sense, from those who have already bought the product. Assuming that m is the total market potential for the new product and F(t) and f(t) represent the cumulative and noncumulative proportion of adopters, N(t) = mF(t) and n(t) = mf(t) are the cumulative and noncumulative number of adopters in time t. To determine the growth of a new durable product due to the diffusion effect, Bass (1969) suggested the following differential equation:

dNðtÞ q ¼ mf ðtÞ ¼ p½m  NðtÞ þ NðtÞ½m  NðtÞ dt m q ¼ pm þ ðq  pÞNðtÞ  N2 ðtÞ m

nðtÞ ¼

ð1Þ

where p and q are the coefficients of innovation and imitation, respectively, that represent the adopters in F(t) due to mass media  (p[m  N(t)]) or interpersonal communications mq NðtÞ½m  NðtÞ . Bass (1969) suggested a least squares method to estimate the parameters p, q, and m from discrete time series data, and used the following analogue:

nðtÞ ¼ a0 þ a1 Nt1 þ a2 N2t1 ; a0 ¼ pm;

a1 ¼ ðq  pÞ;

t ¼ 2; 3; . . . where q and a2 ¼ m

ð2Þ

2.2. Quadratic-interval regression model The basic idea of fuzzy regression theory is that the residuals between estimators and observations are produced by uncertainties in the model parameters rather than by measurement errors, and a possibility distribution is used to deal with practical observa-

ð4Þ

where L is a membership function type. Fuzzy parameters in the form of triangular fuzzy numbers are used

(

lAi ðAi Þ ¼

1  jai cAi j ; ai  ci 6 Ai 6 ai þ ci

) ð5Þ

i

0;

otherwise

where lAi ðAi Þ is the membership function of the fuzzy set which is represented by parameter Ai, ai is the center of the fuzzy number, and ci is the width or spread around the center of the fuzzy number. According to Zadeh’s extension principle (1965), the membership function of the fuzzy number yj ¼ x0j A can be defined by a membership function using pyramidal fuzzy parameter b, as follows:

8 0 0 > < 1  jyj  xj aj=c jxj j for xj –0; lY~ ðyj Þ ¼ 1 for xj ¼ 0; > : 0 for xj ¼ 0;

yj ¼ 0;

ð6Þ

yj –0:

where a and c denote vectors of model parameter values and spreads, respectively, for all of the model parameters, and j denotes the jth observation, j = 1,2,    , m. Finally, this method uses the criterion of minimizing the total vagueness and the sum of squared distances between the estimated output centers and the observed output, S, which reflects both properties of least squares and possibilistic approaches (Tanaka & Lee, 1998).

minimize S ¼ k1

m m X X ðyj  ax0j Þ2 þ k2 c 0 jxj jjxj j0 c j¼1

ð7Þ

j¼1

P 0 where m j¼1 jxj jjxj j is a (n + 1)  (n + 1) symmetric positive definite matrix and k1 and k2 are weight coefficients. A matrix is a positive definite if and only if all of the eigenvalues of the matrix are positive. The weight coefficients k1 and k2 in Eq. (7) have an important role in formulating fuzzy regression models. For example, if we use a large value of k1 compared to k2, a more central tendency is expected, i.e., the obtained central regression line would tend to be the regression line obtained by least squares regression. However, if we use a large value of k2 compared to k1, we reduce the fuzziness of the model. cAt the same time, this approach also considers that the membership degree of each observation yj is greater than an imposed threshold possibility, as h, he[0, 1]. This criterion simply expresses the fact that the fuzzy output of the model should ‘cover’ all of the data points y1, y2,    , ym to a certain level, h. The value of the h level that is chosen will influence the widths, c, of the fuzzy parameters:

lY ðyj Þ P h 8j ¼ 1; 2; . . . ; m

ð8Þ

where the index j denotes the jth observation. Finding the interval regression parameters is formulated by Tanaka and Lee as a quadratic programming problem (1998):

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minimize

S ¼ k1

m m X X ðyj  a0 xj Þ2 þ k2 c0 jxj jjxj j0 c j¼1

subject to

minimize S ¼ k1

m m X X ðnt  a0 xt Þ2 þ k2 c 0 jxt jjxt j0 c t¼1

j¼1

xj a þ ð1  hÞc 0 jxj j P yj ;

j ¼ 1; 2; . . . ; m;

xj a  ð1  hÞc 0 jxj j 6 yj ;

j ¼ 1; 2; . . . ; m;

ð9Þ

Simultaneously, this approach takes into account the condition that the membership degree of each observation nt is greater than an imposed threshold possibility as h, he[0, 1]. This criterion simply expresses the fact that the fuzzy output of the model should ‘cover’ all the data points n1, n2,    , nm to a certain h level. The selection of the h level value will influence the widths, c, of the fuzzy parameters

cP0 0

where a = (a0, a1,    , an) and c0 = (c0, c1,    , cn) are vectors of unknown variables. Kim et al. (1960) found that fuzzy linear regression is a viable alternative to statistical linear regression in estimating regression parameters when the data set is insufficient to support statistical regression analysis and/or when the regression model is inappropriate (i.e., because there are vague relationships among variables and the model is poorly specified). However, fuzzy regression should not used when the data is of poor quality (i.e., when there are outliers in the data or when the data are highly variable).

ln~ ðnt Þ P h 8t ¼ 1; 2; . . . ; m

ð14Þ

The index t refers to the number of nonfuzzy data used in constructing the model. Then, the problem of finding the fuzzy regression parameters was formulated by Tanaka and Lee (1998) as a linear programming problem

minimize

S ¼ k1

m m X X ðnt  a0 xt Þ2 þ k2 c 0 jxt jjxt j0 c t¼1

t¼1

subject to xt a þ ð1  hÞc 0 jxt j P nt ; 3. Quadratic-interval Bass model

xt a  ð1  hÞc 0 jxt j 6 nt ;

t ¼ 2; 3; . . . ; m

ð15Þ

t ¼ 1; 2; . . . ; m

where a0 = (a0, a1,    , an) and c0 = (c0, c1,    , cn) are vectors of unknown variables. The procedure of the quadratic-interval Bass model is as follows:

ð10Þ

Step 1: Fit the Bass model. This step gives the optimum solution ^0 ; a ^1 ; a ^2 Þ, which is used as of the parameter a ¼ ða0 ; a1 ; a2 Þ ¼ ða the input data set in Step 2. Step 2: Determine the minimal fuzziness using the Eq. (9) and a ¼ ða0 ; a1 ; a2 Þ.

A quadratic-interval Bass innovation diffusion model is described with a fuzzy parameter:

nt ¼ A0 þ A1 x1 þ A2 x2 ¼ ha0 ; c0 i þ ha1 ; c1 ix1 þ ha2 ; c2 ix2

t ¼ 1; 2; . . . ; m

cP0

The quadratic-interval Bass model is constructed using the Bass innovation diffusion model and Tanaka’s interval regression (Tanaka & Lee, 1998). The Bass innovation diffusion model representing a conventional simple regression was transformed as follows:

NðtÞ ¼ a0 þ a1 Nt1 þ a2 N2t1 ;

ð13Þ

t¼1

ð11Þ

N 2t1 .

where x1 ¼ Nt1 ; x2 ¼ According to Eq. (11) and using the extension principle, the membership function of the fuzzy number nt = A0 + A1x1 + A2x2 can be defined by a membership function using pyramidal fuzzy parameter A, as follows:

8 0 0 > < 1  jnt  xt aj=c jxt j for xt –0 ln~ ðnt Þ ¼ 1 for xt ¼ 0; > : 0 for xt ¼ 0;

nt ¼ 0 nt ¼ 0

4. Empirical results The performance of the quadratic-interval Bass model was compared with three popular new product sales forecasting models, namely, the Gompertz, the logistic, the quadratic-interval Gompertz, the quadratic-interval logistic and the Bass models, using three data sets, namely, the inventory of cars in the Netherlands, cellular phones in Portugal, and worldwide personal computer (PC) demand. My previous work (Tseng, 2008) used these three data sets to compare the Gompertz, the logistic, the quadraticinterval Gompertz model and the quadratic-interval logistic model and found that when the diffusion process of the time series gave rise to rapid early growth and relatively slow growth later in the series, the quadratic-interval Gompertz model appeared to be the

ð12Þ

where a and c denote vectors of model parameter values and spreads, respectively, for all model parameters; and j denotes the tth observation, j = 1, 2, . . . Finally, this method uses the criterion of minimizing the total vagueness and sum of squared distances between the estimated output centers and the observed output, S, defined as

6000

4000 3000 2000 1000

Years Fig. 1. The (smoothed) stock of cars in the Netherlands, 1965–1989.

89 19

87 19

85 19

83 19

81 19

79 19

77 19

75 19

73 19

71 19

69 19

67 19

65

0 19

Stocks (Unit: 1000)

5000

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5500

5000

4500

Stock

4000

3500 Stock (unit: 1000) 95%C.I. of GOMPERTZ

3000

95%C.I. of GOMPERTZ 99%C.I. of logistic

2500

99%C.I. of logistic Upper bound of quadratic Gompertz Lower bound of quadratic Gompertz

2000

Upper bound of quadratic logistic Lower bound of quadratic logistic

1500

19 65 19 66 19 67 19 68 19 69 19 70 19 71 19 72 19 73 19 74 19 75 19 76 19 77 19 78 19 79 19 80 19 81 19 82 19 83 19 84 19 85 19 86 19 87 19 88 19 89

1000

Year Fig. 2. The forecasts of the Gompertz and Quadratic-interval Gompertz models.

Table 1 The parameter estimates of Gompertz, logistic and Bass models. Parameter

Stock of cars Estimate **

Cellular phones

Worldwide PCs

R2

Estimate

R2

Estimate

R2

0.857

15552.02 8.4348** 0.2374

0.987

Not convergence

Gompertz

m a b

5962.00 1.500** 0.104**

Logistic

m b0 b1

5520.62** 1.274** 0.166**

0.9975

6736.448** 3.6025** 0.2563**

0.9975

290.898** 4.308** 0.1965**

0.989

Bass

a0 a1 a2

3558.04** 0.0397** 8.51E7**

0.9923

220.9839** 0.1949** 1.74E06**

0.9982

3.8788** 0.1733** 4.054E05**

0.970

Table 2 The parameter estimates of quadratic-interval Gompertz, quadratic-interval logistic and quadratic-interval Bass models. Stock of cars

Cellular phones

Worldwide PCs

Parameter

Estimate

Estimate

Estimate

Quadratic Gompertz

ha0 ; c0 i ha1 ; c1 i

h0.7841, 0.1882i h0.104, 0.0995i

h0.3369, 1.5014i h0.2374, 0i

Cannot calculate

Quadratic logistic

ha0 ; c0 i ha1 ; c1 i

h1.274, 0.0645i h0.166, 0i

h3.6025, 0.4137i h0.2563,0.001i

h4.308, 0.4137i h0.1965, 0.001i

Quadratic Bass

ha0 ; c0 i ha1 ; c1 i ha2 ; c2 i

h3558.04, 2016.43i h0.0397, 0i h8.51E–7, 0i

h220.9839, 215.465i h0.1949, 0i h1.74E–06, 0i

h3.8788, 5.566i h0.1733, 0i h4.054E–05, 1.853E05i

most suitable. By contrast, when the diffusion process resulted in slow early growth and relatively rapid later growth, the quadratic-interval logistic models performed better. This paper com-

pares my previous work with the Bass and the quadratic-interval Bass model to demonstrate their performance. The details are presented in Sections 4.1–4.3.

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Subscribers

5000 4000 3000 2000 1000

2 20

00

.Q

4 19

99

.Q

2 99 19

19

19

98

98

.Q

4 .Q

2 .Q

4 .Q 19

97

.Q 97 19

19

2

4 .Q 96

.Q 96 19

19

95

.Q

2

4

0

Times Fig. 3. Cumulative number of cellular phone subscribers in Portugal.

7000 stock 95% C.I. of logistic

6000

Number of subscribers (in thousands)

95% C.I. of logistic Upper bound of quadratic interval logistic Lower bound of quadratic interval logistic

5000

Upper bound of quadratic Bass Lower bound of quadratic Bass 95%C.I. of Bass

4000

95%C.I. of Bass

3000

2000

1000

19 95 .Q 19 4 96 .Q 19 1 96 .Q 19 2 96 .Q 19 3 96 .Q 19 4 97 .Q 1 19 97 .Q 2 19 97 .Q 3 19 97 .Q 4 19 98 .Q 19 1 98 .Q 2 19 98 .Q 3 19 98 .Q 4 19 99 .Q 1 19 99 .Q 2 19 99 .Q 3 19 99 .Q 4 20 00 .Q 1 20 00 .Q 2

0

Time Fig. 4. The forecasts of the logistic and quadratic-interval logistic model.

4.1. The inventory of cars in the Netherlands

150

This paper used the data set of Franses (1994), who proposed a Gompertz curving fitting method and used the inventory of cars in the Netherlands from 1965 to 1989 to examine the performance of the model. The smoothed series is depicted in Fig. 1. The procedures of the proposed model are described in Sections 4.1.1.

100

50

4.1.1. Building the quadratic-interval Bass model Step 1: Fit the Bass model: Using the data set provided by Franses (1994), we applied the Bass model in Eq. (1) to estimate param^1 ¼ 0:0397, and ^0 ¼ 3558:04, a eters. The estimation results are a ^2 ¼ 8:51E  7, and the R square is 0.9923; 95% confidence intera vals are shown in Fig. 2. This step gives the optimum solution of the parameter a ¼ ða0 ; a1 ; a2 Þ ¼ ð35580:4; 0:0397; 8:51E  7Þ, which is used as an input data set in Step 2. Step 2: Determine the minimal fuzziness using (15) and by setting a* = (35580.4, 0.0397, 8.51E7) and h = 0.5. The following quadratic-interval Bass model is obtained using the LINGO package software (1999) and the estimated equation is shown in Eq. (15):

99

96

93

90

87

84

81

0

Time Fig. 5. Worldwide PC shipments.

nt ¼ h3558:04; 2016:43i þ h0:0378; 0ix1t þ h8:51E  07; 0ix2t

ð16Þ

4.1.2. Comparisons The parameter estimation from the different models is shown in Tables 1 and 2 and the confidence interval and possibility interval

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PC

350 95% C.I. of Bass model 95%C. I. of Bass model upper bound of quadratic interval Bass model

300

lower bound of quadratic interval Bass model 95% confidence interval of logistic

250

95% confidence interval of logistic

PC

Upper bound of quadratic interval logistic lower bound of quadratic interval logistic

200

150

100

50

0 81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

Year Fig. 6. The forecasts of the Bass and quadratic-interval Bass models.

curve are shown in Fig. 2. The interval of the quadratic-interval Gompertz model is narrower than the 95% confidence interval of the Gompertz model and all of the actual data are located in the interval of the quadratic-interval Gompertz model; however, not all of the actual data are located in the 95% confidence interval of the Gompertz model. This means that the quadratic-interval Gompertz model has greater prediction capability. However, the interval of the quadratic-interval logistic model is larger than the 95% confidence interval of the logistic model. These forecasting intervals show that the quadratic-interval Gompertz model is the most appropriate for predicting the best and worst possible sales volumes. 4.2. Cellular phone subscribers in Portugal Botelho and Pinto (2004) used the exponential growth model, the Gompertz model, and the logistic model to examine the diffusion pattern of cellular phones in Portugal, and found the logistic model to be the best. Their time-series data of the cumulative number of subscribers runs from Quarter 4 of 1995 to Quarter 2 of 2000, and is depicted in Fig. 3. We have omitted parameter estimation procedures; the results of the parameter estimation are shown in Tables 1 and 2, and the confidence interval and possibility interval curve are shown in Fig. 4. Because of the diffusion process of this time series, growth is initially slow and then relatively rapid during the maturing phases; this results in poor forecasting performance by the Gompertz and the quadratic-interval Gompertz model. We have omitted the forecasting interval curves in Fig. 4. According to (Fig. 4), the interval of the quadratic-interval logistic model is narrower

than the 95% confidence interval of the logistic model and all of the actual data are located in the intervals. The interval of the quadratic-interval Bass model is narrower than the 95% confidence interval of the Bass model, and all of the actual data are also located in the intervals. Based on the forecasting performance, the quadratic-interval Bass model is narrower than the other models, which means that the quadratic-interval Bass model has greater predictive ability. 4.3. Worldwide PC demand Worldwide PC demand from 1981 to 1999 was used in an examination of PC demand using the Bass model (http:// www.utdallas.edu/mzjb/bass.ppt accessed 10 July, 2006) as shown in Fig. 5. We have omitted the procedures of parameter estimation; the parameter estimations are shown in Tables 1 and 2, and the confidence interval and the possibility interval curve are shown in Fig. 6. According to (Table 1), the parameter estimation of the Gompertz model is not significant; we therefore do not discuss the Gompertz and quadratic-interval Gompertz model in this data set. According to (Fig. 6), the interval of the quadratic-interval Bass model is narrower than the 95% confidence interval of the Bass model; and all of the actual data locate in the intervals. The interval of the quadratic-interval logistic model is narrower than the 95% confidence interval of the logistic model; all of the actual data are located in the intervals. The forecasting performance of the quadratic-interval Bass model is narrower than that of the other models. The predictive ability of this model is encouraging.

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4.4. Comparisons and discussion Three diffusion models, the Gompertz model, the logistic model, and the Bass model, were each used to determine the relationship between input and output data based on probability theory; the relationship between input and output is defined using a crisp function. The diffusion models are similar to statistical analyses, in which the validity of the results improves as more data are included. Thus, large amounts of input and output data are generally needed to reflect the objectivity of a phenomenon (Heeler and Hustad (1980) reported that at least 6–10 observations are required for accurate predictions). However, for operations research problems, the available input and output data are usually insufficient, so the data used should be selected by decision-makers to reflect their own preferences and experience. The quadratic-interval diffusion models that are produced using this selected input and output data reflect the concept of possibility, not probability. The possibilistic regression is formulated to obtain the smallest interval system, including all of the selected data, so that it can provide the possible interval. Therefore, if the data are not sufficient, quadratic-interval diffusion models are potentially useful tools. However, when there are outlier data or when there is high variability in the data, the quadratic-interval diffusion models should not be used. The forecasting performance comparisons of the six models in the three data sets reveal no convincing evidence that one model outperforms the others; performance depends on the specific time series pattern. In the first set of time series data, the inventory of cars in the Netherlands, the diffusion process leads to rapid growth during the early phase, and relatively slow growth when approaching the maturing phases. The quadratic-interval Gompertz model has the best forecasting performance. However, in the other two time series, shown in Figs. 3 and 5, the diffusion processes initially give rise to slow growth, then relatively rapid growth during the maturing phases. In this situation, the quadratic-interval Bass model outperforms the other models; the Gompertz model is unsuitable for these two data sets. 5. Conclusions We have combined Tanaka’s quadratic-interval regression model with the Bass models, to create the quadratic-interval Bass mod-

el. We have then used them to forecast sales performance for three sample data sets. The empirical analyses show that the quadraticinterval Bass diffusion model can be applied to new product sales forecasting using the sales histories of similar products, and can reveal the best- and worst-case sales volume outcomes. Moreover, when the diffusion process of the time series leads to slow early growth and relatively rapid later growth, the quadratic-interval Bass model performs better. In practice, we suggest that decision makers draw scatter diagrams to determine the diffusion patterns, and then choose the appropriate diffusion model. Acknowledgment This work was partially supported by funding from the National Science Council of the Republic of China (NSC 94-2416-H-155-008-) References Bass, F. M. (1969). A new product growth model for consumer durables. Management Science, 15, 215–227. Botelho & Pinto, L. C. (2004). The diffusion of cellular phones in Portugal. Telecommunications Policy, 28, 427–437. Dubois, D., & Prade, H. (1980). Theory and applications, fuzzy sets and systems. New York: Academic Press. Fourt, L. A., & Woodlock, J. W. (1960). Early prediction of market success for grocery products. Journal of Marketing, 25, 31–38. Franses, P. H. (1994). Fitting a Gompertz curve. Journal of Operational Research Society, 45, 109–113. Heeler, R., & Hustad, T. (1980). Problems in predicting new product growth for consumer durables. Management Science, 26(10), 1007–1020. Kim, K. J., Moskowitz, H., & Koksalan, M. (1960). Fuzzy versus statistical linear regression. European Journal of Operational Research, 92, 417–434. Mansfield, E. (1961). Technical change and the rate of imitation. Econometric, 29, 741–766. Roger, E. M. (1983). Diffusion of innovations (3rd ed.). New York: The Free Press. Tanaka, H. (1987). Fuzzy data analysis by possibility linear models. Fuzzy Sets and Systems, 24, 363–375. Tanaka, H., & Lee, H. (1998). Interval regression analysis by quadratic programming approach. IEEE Transactions on Fuzzy Systems, 6, 473–481. Tanaka, H., Uejima, S., & Asai, K. (1982). Linear regression analysis with fuzzy model. IEEE Transactions on Systems Man and Cybernetics, 12, 903–907. Tseng, F. M. (2008). Quadratic-interval innovation diffusion models for new product sales forecasting. Journal of Operational Research Society, 59(8), 1120–1127. Accessed 10.07.06. Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8, 338–353.