## 1. INTRODUCTION

With the increase in new technologies on the competitive market, research on innovation diffusion models and their applications is increasing (Lee and Huh, 2017). The innovations that spread through the population of potential adopters represent the diffusion (Rogers, 1983). Innovation diffusion models depict the concepts and various results applicable to technology innovation, adoption, and implementation studies. The diffusion patterns of multiple generation innovations are considered in multigeneration diffusion models (MGDMs) (Mahajan and Muller, 1996;Norton and Bass, 1987).

The field of multi-generation technology diffusion applies well-used terminology such as technology substitution, switching, leapfrogging, repeat purchase, and market mixed effects. A substitution phenomenon occurs when the new innovation has advantages over the previous generation (Norton and Bass, 1987). The switching of innovation is the shifting of suitable technology as required by the consumers (Heide and Weiss, 1995). Latecomers can shift to vintage technology, bypass large investments, and adopt the best technology based on the idea of leapfrogging (Lee and Lim, 2001). A repeat purchase is the purchase of the same technology innovation product as that made on a previous occasion. The userperceived functions with respect to repeat purchase behaviors were described by Bass and Bass (2001). The entire marketing system, including product, price, place, people, process, physical evidence, and promotion effects, is studied based on the market mixed phenomenon as studied by Danaher *et al*., 2001). The substitution, switching, leapfrogging, repeat purchase, and market mixed characteristics are considered when modeling the diffusion of multi-generation innovations.

The first multi-generation substitution study was carried out by Fisher and Pry (1971) who used mathematical models and demand relations between one generation and the products of the successive generation. The pioneer MGDM was studied by Norton and Bass (1987), estimated based on the Bass (1969) diffusion parameters. The model showed that the potential market increases by introducing new-generation products. Subsequently, many researchers contributed different approaches with differ-ent effects. For example, Jun and Park (1999) studied the choice-based model; Zsifkovits and Günther (2015) proposed a computer-simulated agent-based model (ABM); and Maier (1998) first established a new product diffusion model using a system dynamics (SD) approach.

The behavior of an agent can be simulated and viewed in an ABM, which is also known as individualbased model (Grimm and Railsback, 2005). The SD model explains the nonlinear characteristics of the complex system using the stocks, flows, feedback loops, table functions, and time delay under a certain system boundary (Forrester, 1997;Haraldsson, 2000;Sterman, 2002;Zulkepli *et al*., 2012). The simulation-based ABM and SD are based on bottom-up and top-down approaches, respectively (Ding *et al*., 2018). The mathematical equation- based model presents simplified results with quantitative data and the computer simulation-based model performs powerful serials, considering multiple systems in a short amount of time including quantitative as well as qualitative analytic solutions (Fishwick, 2001;Kogan, 2010;Maria, 1997;Schwarz *et al*., 2009;Velten, 2009).

The MGDMs have been used for demand forecasting and trend analysis. Many studies focused on the application of MGDMs. The demand forecasting of highdefinition television in the United States was studied by Bayus (1993) considering historical data of the home appliance industry. Speece and Maclachlan (1995) studied the diffusion of milk container technology using Norton and Bass’s (1987) model, which showed the improvement of forecasting with pricing and growth effects as studied by Speece and Maclachlan (1992). The MGDMs were used to study the high technology (smart technology), consumable goods, and service and system technology generations considering different factors such as pricing, timing, seasonal and diffusion effects as technology substitution, repeat purchase, and leapfrogging.

The adoption of a successive generation model (Islam and Meade, 1997) was analyzed with the diffusion data for cellular phones in European countries and IBM computer data considering environmental and technologic variables such as the network capacity, geographic and demographic proportion, and price of connection. Islam and Meade (1997) generalized the Norton and Bass (1987) model based on the principle of constancy, which means that the innovation and imitator coefficients of all generations remain unchanged. The diffusion of the mobile technology of the successive generation was studied by Johnson and Bhatia (1997) who showed that the Norton and Bass (1987) model has a better regression output and recommended the use of the model for environmental technology applications. The diffusion speed was investigated by Van den Bulte (2000) who empirically studied electrical household durables and showed that the speed changes due to the customer behavior, demographic changes, economic conditions, and product features. Similarly, the widely used Norton and Bass (1987) MGDM was examined by Lim and Kim (2017) for demand forecasting of 5^{th} Generation(5G) mobile telecommunication in Korea before being launched. The model was examined with 2G, 3G, and 4G technology which performed the accurate fit and using such diffusion trend, new parameters for 5G were estimated. The predictions of the 5G technology diffusion will be the same as 4G. The application of MGDMs are increasing. Thus, studies of different MGDMs are important.

In this study, we surveyed several MGDMs. The results show that the models are studied based on different approaches and applications are increasing because of their importance in this competitive world. Initially, pure mathematical equation-based modeling was used and nowadays, with computer-based simulation approaches are used. Figure 1 shows the cumulative studies on multigeneration diffusion models and simulations where initially modeled with pure equation-based mathematical approaches and recently with advanced computer simulation- based modeling approaches are introduced.

However, a review article that focuses on multigeneration technology diffusion theory, models, and their applications cannot be found in the literature. This paper presents an overview of MGDMs that have been reported in 44 research publications between 1971 and 2018. Different effects, such as substitution, repeat purchase, switching, leapfrogging, and marketing mix effects, are considered to study and model the innovation diffusion of high- technology, consumable goods, and service technology generations. High-technology products are information communication technology (ICT) and smart technology products; consumable goods include products such as glass, plastic bags, and paper bags for milk packaging systems; and the service technology includes innovative processing systems and products. Some of MGDMs and their suggested features are summarized in Table 1.

This research conducted with three steps a) literature search, b) selection of eligible papers, and c) summarizing. The literature search was performed with keywords, such as multi-generation diffusion, diffusion model, innovation diffusion model, and multi-generation technology diffusion, using the Google Scholar, Science Direct, Wiley Library, Web of Science, Research gate, Elsevier, and Springer websites and eligible papers were selected. Eligible papers selection and summarizing process of the MGDMs and their applications is studied based on articles published between 1971 and 2018.

The rest of this survey paper is organized as follows. Multiple studies of MGDMs are described in Section 2. The mathematical models are discussed in Section 2.1 based on different diffusion model categories with their emperical experimental data as high-technology product, consumable goods, and service and system technology diffusion models. The computer simulation models, including System Dynamics Models (SDMs) and Agent- Based Models (ABMs), are explained in Section 2.2. and the conclusions are drawn in Section 3.

## 2. MULTI-GENERATION DIFFUSION MODEL (MGDM) STUDIES

The first substitution model was studied by Fisher and Pry (1971) and multi-generation diffusion theory modulation on marketing was by Norton and Bass (1987). Many subsequent models were based on different ap-proaches, initially, pure equation-based mathematical and recently with computer simulation-based approaches.

### 2.1 Mathematical Models

Based on the technology product data used on empirical analysis of MGDMs, the mathematical models are categorized into multi-generation high-technology product, consumable product, and service and system technology diffusion models in this study. Some of the researchers also suggested that their model can be used for multicategory innovation diffusion, as shown in Table 1.

#### 2.1.1 High-Technology Product Diffusion Models

Many contributions were made to the diffusion analysis of high-technology products. The pioneer model on the diffusion theory of adoption and substation, which is based on the dynamic sales behavior of the successive generation of high-technology products, was established by Norton and Bass (1987). Their model was derived using Bass’s (1969) diffusion parameters and empirical analysis was done for product demand forecasting of semiconductor industry. The mathematical model for the new-product demands of three generations can be expressed as follows:

where *S _{i}*(

*t*) refer to the sales of

*i*generation products in time

^{th}*t*,

*m*is the market potential. ${m}_{i}={a}_{i}{M}_{i}$,

_{i}*a*is the average repeated buying rate among adopters;

_{i}*M*is the incremental number of ultimate adopters; and

_{i}*t*is the time since the launch of the

_{i}*i*generation product. The cumulative distribution function $F\left({t}_{i}\right)=\frac{1-{e}^{-\left(p+q\right){t}_{i}}}{1+\frac{q}{p}{e}^{-\left(p+q\right){t}_{i}}}$, where,

^{th}*p*and

*q*are the coefficient of innovation and imitation for the

*i*generation product, respectively. The series of technology generations is plotted in Figure 1. The figure shows that the market potential increases with increasing technology generation.

^{th}Most of the models regarding high-technology diffusion, such as the Norton and Bass (1987) model and its derivatives, are widely used. Mahajan and Muller’s (1996) timing, diffusion, and substitution study led to the multigeneration process of diffusion and process of substitution model. This model was a generalization of the Norton and Bass (1987) and Wilson and Norton (1989) models, which was used to predict sales and determine the optimal timing of introduction of new-generation technology. The empirical analysis of the model was validated using the 1955-1994 sales data for IBM mainframe computers. The rate of sales diffusion by Mahajan and Muller (1996) can be expressed as:

where *x _{i}* is the number of cumulative adopters of the

*i*generation product,

^{th}*N*is the market potential of the

_{i}*i*generation,

^{th}*a*is the coefficient of external influence,

_{i}*b*is the word of mouth coefficient, and

*x*is the sum of adopters of all generation products.

The multi-generation choice-based product diffusion model established by Jun and Park (1999) provided the best fit compared with previous models incorporating diffusion and choice effects. The model was tested using IBM mainframe (computer systems) market data (type I) and worldwide dynamic random-access memory (DRAM) market data (type II). Based on the model results, Jun and Park (1999) claimed that the Norton and Bass (1987) and Mahajan and Muller (1991) models can only be applied to type I data.

Similarly, Kim *et al*. (2001) carried out a choicebased diffusion study and proposed a model considering both initial and repeat purchase, including leapfrogging behavior, for multi-generation technology products based on the single generation diffusion models of Chatterjee and Eliashberg (1990) and Lattin and Roberts (1989). This model was tested with multi-generation IBM Personal Computer (PC) and Macintosh market survey data, which were fitted to explain the individual consumers’ real purchase statistics. This model incorporated with pricing factors.

Kim *et al*. (1999) proposed a substitution and diffusion model to forecast International Mobile Telecommunications- 2000 (IMT-2000) subscribers in Korea considering previous models and the exogenous impact of the market, which was not studied by Norton and Bass (1987). Their model was also a generalization of Norton and Bass’s (1987) MGDM.

Kim *et al*. (2000) proposed an inter-category model based on Asian Wireless Telecommunications service market data and the relation with inter-product category dynamics of the IT industry. Multi-category effects were studied by Peterson and Mahajan (1978) and Bayus (1987) investigated single-generation diffusion including related and contingent product categories. Favorable and unfavorable product categories and the competition between them, which guided new technology or process development, were analyzed and modeled by Kim *et al*. (2000). The model was derived from that of Norton and Bass (1987) considering product categories and the estimation of the market potential, similar to Jain and Rao’s (1990) study. The model was calibrated with Hong Kong Telecommunication market data and implemented in the Korean market. The model yielded a better fit compared with previous models.

The demand dynamics of fast-tech products were studied and modeled by Bass and Bass (2001). In their adoption and repeat sales model, the total sales of the product are considered to be the sum of the adoption and repeat sales. They identified the potential market for each generation, total systems in use based on the time, and systems in use each time. The empirical analysis was performed with eight DRAM generations and nine PC generators and the results were compared with those obtained by Norton and Bass (1987). The model yielded a better fit.

The diffusion of technological innovation, considering first time sales and subscriptions and the impact of the marketing mix variable for the successive generation, was studied using Danaher *et al*.’s (2001) diffusion model. Their empirical study was carried out to estimate the price of two-generation telephones in European countries. The results showed that the demand increases with decreasing price. The total number of 1^{st} and 2^{nd} generation technology subscribers in the Danaher *et al*. (2001) model can be calculated as follows:

where *γ _{t}* = leapfrogging multiplier,

*γ*= 0 for $t<{\tau}_{2},\hspace{0.17em}{\tau}_{i}$ is the time of the launch of the

_{t}*i*generation technology, and

^{th}*m*is the market potential of the

_{i}*i*generation products. The switching multiplier

^{th}*δ*and the modified adoption cumulative distributive function (CDF) can be expressed ${F}_{i}\left(t|{\tau}_{i}\right)=\frac{1-{e}^{-\left({p}_{i}+{q}_{i}\right)\varnothing \left(t|{\tau}_{i}\right)}}{1+q{e}^{-\left({p}_{i}+{q}_{i}\right)\varnothing \left(t|{\tau}_{i}\right)}},\varnothing \left(t|{\tau}_{i}\right)={\displaystyle \sum _{j}^{t}{\tau}_{i}+1}\text{exp}\left[{\beta}_{i}{x}_{i}\left(j\right)\right]={\displaystyle {\int}_{{\tau}_{i}}^{t}{\varnothing}_{i}\left(t\right)du}$ where

_{t}*β*is the mapping function. as:

Similarly, in the study of Kim *et al*.(2002), the small office/home office (SOHO) professional’s labor force adoption patterns of multi-generation IT products as Kim *et al*. (2001) were examined using the first and repeat purchase model. The study was carried out using data for four different generations of IBM PCs and three generations of Apple computers from 1980 to 1992.

A MGDM for IT-intensive products considering cannibalization and competition effects between generations was established by Shen and Altinkemer (2008). The model was tested for game console diffusion using the data for three Sony and Microsoft game consoles. The results showed that Sony’s product cannibalization was minimal and Microsoft’s had strong, which represents a challenge for Sony’s market. Their research results also showed that the first-mover effect yields a better diffusion performance. The model was based on previous models, including those of Bass (1969) and Bass and Bass (2001), and adoption timing and characteristics, as used by Rogers (1983) and Kerin *et al*. (1992).

The competition-substitution of the different generation of products and competition-complementary effects of different product categories were analyzed by Zhang *et al*. (2008) based on previous models (Kim *et al*., 2000;Norton and Bass, 1987;Speece and Maclachlan, 1992). Zhang *et al*. (2008) considered the price function similar to Speece and Maclachlan (1992) and followed Kim *et al*. (2000) to form a new multi-generation model, which was tested using Chinese Telecommunication products and yielded a better accuracy than the Bass (1969), Norton and Bass (1987), and Kim *et al*. (2000) models. The multiplier of the cumulative adoption function *f _{kn}*(

*t*) of

*n*generation product of

*k*-category at the time

*t*was:

where *P _{kn}*(

*t*) is the price of

*n*generation product of

*k*category at time

*t*, ${\overline{P}}_{kn}\left(t\right)$ is the average price of the

*k*-category product in time

*t*.

*η*is the price elasticity which was greater than zero.

_{kn}The relative changes due to the consumers’ behavior on diffusion parameters for both first time and repeat purchases of multi-generation technology products were studied using the first time purchase and repeat sales model by Chanda and Bardhan (2007). The new model was established based on the previous models of Bass (1969), Danaher *et al*. (2001), Mahajan and Muller (1991), and Norton and Bass (1987) and validated using six generations of DRAM shipment data of the semiconductor industry, a black and white and color TV production database of Television Industry (India), four generations of IBM data of the Mainframe Industry(USA), and two generations (wireless and analogue) of data of the Wireless Industry (USA). The model was established in the Bayesian framework (Lee *et al*., 2003) to provide a realistic lost-sales interpretation for the first to second generations and the parameters were estimated using a nonlinear least squares (NLS) method. Chanda and Bardhan (2007) defined the cumulative number of adopters of the fourgenerations product model on the market as follows:

a. Single-generation product on the market;

b. Two-generation products on the market;

c. Third generation products on the market;

d. Fourth generation products on the market;

where, *A _{j}* &

*R*are the potential and actual adaptors of earlier generation,

_{j}*β*(

_{n}*t*) is the potential of repeat/upgrade purchasers of the

*n*−1 generation product who switched to the generation

*n*product, ${B}_{n}\left(t\right)={R}_{n-1}\left(t\right){\beta}_{n}(t),\hspace{0.17em}\text{and}\hspace{0.17em}{\alpha}_{n}^{\text{'}}and{\alpha}_{n}^{\text{'}\text{'}}$ are the first and second generation adoption fraction when the generation product

*n*was on the market, respectively. The other parameters were defined above.

The MGDM considering the season factor (S), rate of market growth (M), price (P), repeat purchase (R), and technology substitution (T) is the SMPRT model established by Chien *et al*. (2010). Chien *et al*. (2010) integrated the Bass (1969), Bass *et al*. (1994), Kurawarwala and Matsuo (1998), and Norton and Bass (1987) models for semiconductor product demand forecasts and examined their model using four generations of the semiconductor foundry in Hsinchu Science Park (Taiwan). The demand of the SMPRT model was ${\widehat{s}}_{t}\left(t\right)\times {\alpha}_{t}\times \mathrm{exp}\left({g}_{t}\right)$, where ${\widehat{s}}_{t}\left(t\right)$ represents the estimated sales of products of generation *i* at time, *α _{t}* is the seasonal factors at time

*t*, and

*g*is the growth rate at time

_{t}*t*.

The market diffusion parameters were estimated using the NLS method and the function was revised as follows: $\text{Min}{\displaystyle {\sum}_{i=1}^{n}{\displaystyle {\sum}_{t=1}^{l}{\left[{S}_{i}\left(t\right)-{\widehat{s}}_{i}\left(t\right)\times \alpha \left(t\right)\times \mathrm{exp}\left({g}_{t}\right)\right]}^{2}}}$ where *l* is the number of periods, *n* is the number of generations *t* for generation *i*, and *S _{i}*(

*t*) represents the actual sales of the product at time

*t*based on Norton and Bass (1987). Figure 2

A generalized Norton–Bass Model including leapfrogging and switching adoption was introduced by Jiang and Jain (2012). The model was compared with other models (Mahajan and Muller, 1991;Jun and Park, 1999;Danaher *et al*., 2001); it yielded a better performance. The empirical analysis was carried out using world telecommunication and ICT indicator data. The leapfrogging multiplier of their study was: ${u}_{i+1}\left(t\right)={\widehat{y}}_{i}\left(t\right){F}_{i+1}\left(t-{\tau}_{i+1}\right),\hspace{0.17em}t\ge {\tau}_{\left(i+1\right)}$. The cumulative number of leapfrogging adoptions was; ${U}_{i+1}={\displaystyle {\int}_{{\tau}_{i*1}}^{t}{u}_{i+1}\left(\theta \right)d\theta}$, Similarly, the switching adoption from generation *i* to *i* + 1 was; ${W}_{i+1}\left(t\right)={\widehat{Y}}_{i}\left(t\right){f}_{i+1}\left(t-{\tau}_{i+1}\right),t\ge {\tau}_{i+1}$ and cumulative numbers of switching adoption was; ${W}_{i+1}={\displaystyle {\int}_{{\tau}_{i*1}}^{t}{w}_{i+1}\left(\theta \right)d\theta ,}$ where *θ* is the substitutability factor; *f _{i}*(

*t*) is the diffusion rate at time t, which was also derived from Bass (1969) and Norton and Bass (1987);${\widehat{y}}_{i}\left(t\right)$ is the noncumulative adoption rate of generation

*I*; and ${\widehat{Y}}_{i}\left(t\right)$ is the cumulative number of the adoption rate at time

*t*. The generalized Norton–Bass model (Jiang and Jain, 2012) was derived as:

The number of sales units in the modified GNB model was:

where the adoption units for the last generation were:

The discrete time population growth multi-generation model considering the demand for successive generation products was studied by Li *et al*. (2013). The empirical test of the model was carried out using Intel’s microprocessor sales data and compared with the previous Bass (1969), Jun and Park (1999), and Norton and Bass (1987) models, which showed a better fit than other models. This model was formulated by assuming that each customer purchases one product and never purchases the same generic product again; switches from a competitors’ product with the different aged population were considered in the initial forecast window. The sales volume of each generation product at time *t* in the proposed model was:

where ${x}_{j}\left(t\right){P}_{ji}$ represents the sales due to upgrade, $y\left(t\right){P}_{yi}I\left(i\u03f5j\left(t\right)\right)$ is the brand switching, $\alpha \left(t\right){s}_{i}\left(t-1\right)$ is the market expansion function, ${P}_{ji}={p}_{ij}+{q}_{ij}{x}_{j}$ is the fractional flow rate from population *i* to *j*, Bass diffusion parameters (Bass, 1969), *H* is the time period, *x* is the population of the product, *P _{yi}* is the fractional flow rate from

*x*to product

_{i}*y*,

*P*is the fractional flow rate from

_{yi}*y*to

*x*,

_{i}*I*(⋅) is an indicator function, and

*α*(

*t*) is the demand growth rate. The researchers modeled their linear matrix model as: $s=X\left(\beta \right)\beta +\epsilon ,$ where

*s*is the sales matrix,

*ε*is the error matrix,

*β*is the product strength, and

*X*(

*β*) is the sales data matrix of function

*β*(Greene, 2008).

The customer behavior, considering substituting, switching, and leapfrogging effects, was studied by Kapur *et al*. (2013) who also used the function of the Bass (1969) diffusion model. The proposed multi-generation model (Kapur *et al*., 2013) was examined using six-generation shipment DRAM data and three generations of IBM data collected from 1974-1997. The results were compared with that obtained using the Norton and Bass (1987) model. Accurate fits were obtained, except for the first generation of DRAM products. The cumulative adoption of the i^{th} generation product when *n* generation products were on the market can be expressed as follows:

where, ${F}_{i}\left(t-{\tau}_{i}\right)$ is the Bass cumulative adoption fraction at time *t* for *i ^{th}* generation product introduction at time

*τ*,

_{i}*N*is the the market potential of the

_{i}*i*generation product, substitution of the product

^{th}*i*by

*k*generation product,

^{th}

switching form products i^{th} generation to products of the j^{th} generation,

and leapfrogging the intermediate generations,

Based on the consumers’ forward-looking behavior (Song and Chintagunta 2003) and the cross-generation effect taken from the Bass (1969) diffusion model, the multi-generation product diffusion (MGPD) was modeled by Shi *et al*. (2014). The model was tested with the sales data for eight high-tech products of Nintendo, Sony, Microsoft, and Apple. The results were compared with those of previous models of Jun and Park (1999), Mahajan and Muller (1991), and Norton and Bass (1987). A better fit was yielded.

A multi-generation model considering the price of the technology product and time of diffusion using the Cobb- Douglas production function (Cobb and Douglas, 1928) was studied by Kapur *et al*. (2015). Their two-dimensional multi-generation model was validated using six-generation DRAM shipment data of the semiconductor industry. The marketing effort function in the model was: $\Psi (t,\hspace{0.17em}P)={t}^{\alpha}{P}^{1-\alpha}$, where *t* is the time, *P*is the price, and *α* is the degree of impact of the time on the adoption process. The cumulative distributive adoption function of the *i*^{th} generation product was: ${F}_{i}({\Psi}_{t-{\tau}_{0}},\hspace{0.17em}P)=\frac{1-{e}^{-\left({p}_{i}+{q}_{i}\right){\left(t-{\tau}_{0}\right)}^{\alpha}{P}^{1-\alpha}}}{1+\frac{q}{p}{e}^{-\left({p}_{i}+{q}_{i}\right){\left(t-{\tau}_{0}\right)}^{\alpha}{P}^{1-\alpha}}}$. The adoption numbers were estimated using their previous study as Kapur *et al*. (2013).

The multi-stage mixed influence model for technological generation products based on the awareness (Kalish, 1985) was analyzed using Chanda and Das (2015) mixed influence model. The model was validated with worldwide DRAM shipment data obtained from 1978 to 1992. Hung *et al*.’s (2017) established a modified multigeneration innovation diffusion and substitution model [modified after Lotka–Volterra’s predator-prey interactions (Lotka, 1922;Volterra, 1926) and Bass’s (1969) model]. Their model was validated with DRAM industry sales data. They considered the total market share of the product to be 100%. The parameters $\overline{x}\left(t\right)\hspace{0.17em}\text{and}\hspace{0.17em}\overline{y}\left(t\right)$ were the average market shares of old and new technology products, respectively. The adoption product rate of the modified model (Hung *et al*., 2017) was:

where *A*, *B*, and *C* were, diffusion parameters.

#### 2.1.2 Consumable Products Diffusion Models

Many models consider the multi-generation diffusion of consumable products. The models of Fisher and Pry (1971) and Norton and Bass (1987) and their derivatives were used to study the diffusion of consumable products. A simple substitution model was established by Fisher and Pry (1971) to study technology advancements and the demand relationship between the products of one generation and those of another generation. This probability model was used to substitute plastic floors for woods floors in houses, detergents for shopping and water-based paints and oil-based paints. If α is half of the annual fraction growth rate in the early years, *t*_{0} is the time at which half of the substitution is completed. The fraction of the substitution *f* follows an S-shaped curve:

Similarly, Speece and Maclachlan (1992) established a model for fluid milk packaging, which is a consumable product type-diffusion model, and used it to forecast three generations (glass, paper cartons, and plastic) of packaging technology considering the pricing factors. The model (Speece and Maclachlan, 1992) was an extension of the Norton and Bass (1987) model, where the CDF *F*(*t _{i}*) was multiplied with the pricing function

*G*(

_{i}*P*). The updated CDF was:

Two different forms of the function G_{i}(*P*) were investigated in the model:

$\begin{array}{l}\text{i)}\hspace{0.17em}{G}_{i}\left(P\right)={\left(\frac{{p}_{i}}{P}\right)}^{-\eta}\hspace{0.17em}\text{and}\\ \text{ii)}\hspace{0.17em}{\text{G}}_{i}\left(P\right)=\text{exp}\left[-\eta \left(\frac{{P}_{i}}{P}\right)\right]\end{array}$

where *η* is the price sensitivity coefficient (a negative sign indicates the decreasing demand with increasing price), *P _{i}* is the price of the product of the

*i*generation, and

^{th}*P*is the market price of the product, which is defined as the sales (

*S*)-weighted average price of the products of three generations:

The impacts of foresight, technology substitution, and market expansion on the introduction of secondgeneration products based on the optimal pricing policy of the monopolist market was studied by Padmanabhan and Bass (1993). The market expansion due to endogenous changes and impact of the foresight on the pricing strategy of the firm were considered as Bayus (1992) and pricing strategies on different effects as Mahajan and Muller (1991).

The discrete choice model by Kim *et al*. (2005) was used to forecast the multi-generation products demand considering discrete choice diffusion dynamics based on Moore’s law (Mooer, 1996) including the learning by doing aspect. The empirical study of Kim *et al*. (2005) was carried out using different DRAM generation products. The model yielded a better fit than the Jun and Park (1999), Mahajan and Muller (1991), Norton and Bass (1987), and Speece and Maclachlan (1992) models.

Based on the heuristic theory of adjustment and anchoring and considering the price effect and consumers behavior, Tsai (2013) established a new multi-generation model for LCD TV market diffusion based on Norton and Bass’s (1987) model. The diffusion parameters were estimated using the NLS method and sales data for LCD TVs with different screen sizes (26′′, 32′′, 42′′ and 46′′ ) from the first quarter of 2003 to the fourth quarter of 2010 were examined. The results showed that the reduction of the price of the demand increased with every generation.

Similarly, Guo and Chen (2018) established a new MGPD model for the firm profit and strategies for pricing and timing. They considered the cannibalization effects by using initial seeding. Their model was tested using iPhone sales data from 2009-2012. A better fit was yielded than with the Norton–Bass model. Using the Bass diffusion parameter (Bass, 1969), the purchase to own (PTO) and subscribe to use (STU) product diffusion was studied by Jiang and Qu (2018). A generalized multigeneration diffusion framework was established to determine the optimal market entry period of new generation products.

#### 2.1.3 Services and System Technology Diffusion Models

There are few models regarding services and systemrelated multi-generation technology diffusion. Using the Norton and Bass (1987) diffusion model, the cost–benefit ratio (CBA) of the management service system was studied by Sohn and Ahn (2003). Michalakelis *et al*. (2010) studied 2G and 3G service diffusion are presented. A multi- generation technology diffusion model, which can be applied to determine the best systems with respect to the CBA, was established by Sohn and Ahn (2003) based on a competition and substitution approach. Monte Carlo simulations were performed to determine factors affecting the CBA of a Cybernetic Building System (CBS) using the data applied in Chaphman’s research (Chapman, 2000). The Sohn and Ahn model (2003) was an extension of the Speece and Maclachlan (1995) and Tarn and Hui (1999) models. They introduced the performance per cost (*P*/*C*) for the *i ^{th}* generation, applied it to Speece and Maclachlan’s (1995) model, and studied the multigeneration new technology diffusion from an economic viewpoint:

where *ρ* is the sensitivity coefficient of *P*/*C*, $\frac{{p}_{i}\left(t\right)}{{C}_{i}\left(t\right)}$ is the performance/cost is the cost of saving/installation cost of the *i ^{th}* generation at time

*t*, and $\frac{P\left(t\right)}{C\left(t\right)}$ is the average performance cost for all generations at time

*t*.

Similarly, the technology innovation diffusion, including the generic substitution, was modeled and evaluated by Michalakelis *et al*. (2010). They assumed that the saturation level of the market is not constant throughout the diffusion process and proposed a diffusion model with a dynamic ceiling. The model was examined using the 2G and 3G data for European countries; accurate forecasts were obtained. The number of adopters of the product at each point of time *t* was expressed as:

Where $a=\frac{d\overline{N}(t)}{dM},{N}_{0}$
, *N*_{0} is the number of adopters at time *t*_{0}, ${\overline{N}}_{0}$ is the initially estimated saturation level in the absence of a next generation, *s* is the diffusion rate, *t* is the time period, *M* is the corresponding estimated saturation level, *γ* is the diffusion rate of the generation, and *M* is the market potential of the new-generation product.

### 2.2 Computer Simulation Models

A computer simulation-based model performs powerful serials, considers multiple complex systems in a short amount of time, and provides quantitative as well as qualitative analytic solutions (Hockney and Eastwood, 1983;Kogan, 2010, Roozmand *et al*., 2011). The mathematical models are quantitative models. The SD macroand agent-based micro-modeling approaches were used for simulation-based MGDMs. Examples are discussed below.

#### 2.2.1 System Dynamics Model (SDM)

Maier (1998) first established a multi-generation product substitution and diffusion model, which was based on the Bass (1969) diffusion model, considering various factors influencing the diffusion, such as the price, and product capabilities using an SD approach. This model was used to map the substitution among successive generations using the marketing mix behaviors. Maier (1998) examined his model using four generations of Intel microprocessor data and confirmed that the model yields a good fit. The SDM proposed by Maier (1998) is summarized in Figure 3.

Similarly, Kreng and Wang (2013) studied successive generation innovation diffusion using the Nike Golf company’s Nike SQ SUMO 5000 and Nike SQ SUMO golf clubs data and an SD approach. The model was also based on Bass’s innovation diffusion (Bass, 1969) and a SD approach (Sterman, 2002).

#### 2.2.2 Agent-Based Model (ABM)

Lin *et al*. (2011) established a dynamics pricing agent model and simulated the multi-generation product line with a cannibalization effect. Each product line was considered to be an agent, which was adjusted based on the sales price considering the market demand. Similarly, Kilicay-Ergin *et al*. (2015) carried out pricing strategy analysis using a multi-generation product line considering different cannibalization conditions. Zsifkovits and Günther (2015) studied different innovation diffusion resistances in a multi-generation product environment using an ABM. The model was simulated using AnyLogic and used to examine fuel cell vehicles based on different scenarios depending on the technology availability, timed communication strategy, and with and without investments. An ABM for MGPD based on a micro-level perspective considering the social network effect was established by Günther (2017). The model was verified with the computer diffusion data for the German market from 1994– 2013. Examples of ABM entities and dynamics are presented in Figure 4.

## 3. CONCLUSION

In this study, the progress of multi-generation innovation diffusion studies based on 44 research papers published from 1971 to 2018 is reviewed. Insights from different research streams on multi-generation innovation diffusion are summarized. This study shows that MGDM research are limited because of the complexity of the models. The MGDMs are used for demand forecasting and trend analysis. Many studies focused on the application of MGDMs. Initially, MGDMs were modeled mathematically. However, nowadays, computer simulationbased approaches, such as ABMs and SDs, are also used. Mathematical equation-based models are quantitative. In contrast, computer simulation-based approaches include complex diffusion dynamics and quantitative as well as qualitative analyses. The top-down macro analysis is generally performed with SD methods and bottom-up micro analysis is carried out with ABM methods applying simulation- based modeling approaches.

Norton and Bass (1987) model is the pioneer in MGDM. Subsequently, many researchers contributed models based on different approaches. Bass’s (1969) cumulative distribution functions with different influencing factors are widely adopted in MGDMs. Researchers studied multi-generation diffusion with repeat, switching, leapfrogging, and substation models including different market characteristics.

Future directions with respect to MGDMs can be summarized as follows: first, some areas have not been addressed using previous models such as coexisting manual, semi-manual, and automatic service technology generations and the competition between generations. Thus, future research should focus on multi-generation service technologies and their applications; and second, there are many econometric single-generation technology diffusion models, such as the Bass, exponential growth, and Gompertz models, and logistic models (Meade and Islam, 2006;Stoneman and Battisti, 2010). The MGDMs should be modeled and compared with the cumulative distribution function of Bass and other models in the fu-ture. Finally, mixed micro–macro analytical MGDMs representing a combination with agent-based and SD approaches have not been established, yet, but should be considered in the future.