Discrete and Dynamic Optimization Problems in Operation Management
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A Dynamical Innovation Diffusion Model with Fuzzy Coefficient and Its Application to Local Telephone Diffusion in China
Abstract
This paper studies the innovation diffusion problem with the affection of urbanization, proposing a dynamical innovation diffusion model with fuzzy coefficient, and uses the shifting rate of people from rural areas stepping into urban areas to show the process of urbanization. The numerical simulation shows the diffusion process for telephones in China with Genetic Algorithms and this model is effective to show the process of innovation diffusion with the condition of urbanization process.
1. Introduction
Technological innovation promotes the evolution of industries by altering the competitive market structure and value chains of industries [1]. The market success of an innovation is determined not only by technological performance, but also by the interaction of numerous factors. Consequently, it is understandable that many studies choose to investigate the key factors that influence the acceptance or the rejection of an innovation during the process of diffusion. Based on these key factors including price and advertising, researchers have contributed to the development of the diffusion theory by suggesting analytical models for describing and forecasting the diffusion process of an innovation in a social system [2â€“16] such researchers include Fourt, Mansfield, and Mahajan. Fourt and Mansfield discussed the diffusion pattern in external influence such as mass advertisement or internal influence such as oral communication separately [8, 12]. The sales model for forecasting TV made by Bass in 1969 has settled the base for the following research on innovation diffusion with both external factors and internal factors taken into account [3]. However, the potential adopters in these models are static or fixed at the time an innovation is introduced and remains constant over the diffusion process. Obviously, such an assumption is not tenable with regard to either theory or practice.
In response to this shortcoming, Mahajan and Peterson proposed a dynamical diffusion model where the potential is permitted to vary over time [13]. There are many relevant factors affecting the potential, including socioeconomic conditions in the social system, increases or decreases in the population of the social system, government actions, and efforts to influence the diffusion process, such as advertising and pricing. Such a proposition was followed and developed by Lackman, Dodson, and several other researchers [17â€“24]. Kalish, for example, conjectured that the qualitative structure of an optimal pricing policy will remain the same upon the inclusion of additional marketing variables such as advertising, in the diffusion model. Jain and Rao show that price affects the diffusion rate via the coefficients of external and internal influence [20].
Because different adopters have different attitudes of risk or decisive patterns, there will be great difference in adoption possibility and rate. The consumption habit, the living standard, and the environment around also make a great difference in the adopting rate. For this reason, a diffusion model is presented in a different section in the paper [25]. The author believes that the characteristics of adopters in different patches make differences in the diffusion rate. Furthermore, there will be migration between patches, which would also have an effect on innovation diffusion. With great economic development and urbanization, we should not ignore the potential adopters who have changed their economic conditions or status. They will have great consumption demands for goods which they would not have even considered before. For example, the data of telephone subscribers of China in 2002â€“2006 has got tremendous growth which cannot be explained by any diffusion model before. Furthermore, the rate of urbanization development is different in different stages. So it is more acceptable if we take the uncertain rate instead of the deterministic rate to simulate the process. We use the rate of people from rural areas stepping into urban to show the process of the urbanization development of this area. Consequently, this paper constructs a fuzzy innovation diffusion model and tries to find the principles of innovation diffusion with the effects of population increasing and urbanization. In this context, model uncertainty is portrayed through the fuzzy transition process from nonuser to the adopter of an innovation because of the uncertain rate of urbanization.
We organize this paper as follows. In Section 2, an innovation diffusion model with the population increasing and the conversion between two different colonies is constructed. In Section 3, a stability analysis is done on the population increasing and the diffusion rate of different groups. An empirical analysis is also done with the data of telephone subscribers of urban and rural areas in China compared with the Bass model in Section 4. Some concluding remarks are finally given in Section 5.
2. Modeling
There is a comparative independence and stability between colony 1 and colony 2. Each of the two colonies has its own main communication styles, broadcasting channels and ways, and consumption habits. However, there also exists migration between them for specific reasons such as improved living standards. The members moving into new community will be accepted into the new consumption system since they are affected by the new environment. In the interior of a community, for some related external factors and oral communication, nonusers adopt innovation and serve as users, or users give up the innovation for some reasons after experiencing it.
2.1. Conceptional Model
Based on the foundation above, the concrete convert relations and relations between the inside and outside of the two communities are shown in Figure 1 and Table 1, respectively. The arrowheads denote the flow directions of the members.

2.2. Mathematical Model
Hypothesis 1. The members of each colony are divided into users and nonusers. The nonusers include those who adopted the innovation and have given up using it.
Hypothesis 2. The internal increasing rate of the two colonies obeys the logistic distribution functions.
Hypothesis 3. The nonusers of each colony are still nonusers when entering another colony, which can be shown in Figure 1 in the conversion way of and , and the users of each colony can either keep on or give up using the innovation for various reasons when entering another colony.
Hypothesis 4. The number of the members in one colony shifting to another colony is proportional to the number of the total members of the colony, and the coefficient is fuzzy for the uncertainty of urbanization.
Based on the conceptional model, the hypothesis, and the conversion relations above, we build the increase equation for variables , , where represents the number of members in colony in time , . According to the empirical statistics, the birthrate and ratio to death are linear functions of the population of the colony. Suppose that is the birthrate of colony , is the birthrate of colony without a resource limit, and is the birthrate of colony with a resource limit, where and are both positive constants. The ratio of death of colony is , where is the natural death ratio without a resource limit and is the ratio of death increasing with a resource limit; that is to say, the birthrate will decrease with the increase of population and the death ratio will increase with the increase of population. Thus, the number of members increased in time in colony 1 is and according to Hypothesis 2. From Hypothesis 4 and the conversion way of and , the number of increased members shifting from colony 2 in time is , and the number of members shifting into colony 2 from conversion way of and is , as seen in Figure 1. Based on the analysis above, the increase equation of is
Similarly, the increase equation of is
Consequently the increasing model of coloniesâ€™ members is governed by
Suppose that is the number of users in colony at time , and is the number of nonusers in colony at time . Now we will build the equation of variable in colony 1. From conversion way in Figure 1, because of the internal oral communication and medium, the number of nonusers who have changed to become users is , in which is the probability for nonusers to become users caused by medium, and is the probability for nonusers to become users caused by oral communication between users and nonusers. Owing to natural deaths, the decreasing number of users is . From the conversion way in Figure 1, there are users who have to give up using the innovation for some reason, because of movement of population, there are users shifting out of colony 1 and users shifting into colony 1 from colony 2. Here is the probability for the members of colony converting into another one and is the probability for users to continue using the innovation in colony after converting into another colony. Based on the analysis above, the increase equation of variable is
Similarly, the increase equation of variable is
Consequently, we can say that the dynamical equations of innovation diffusion based on members shifting between colony 1 and colony 2 are governed by
This model focuses on the analysis of the affection on the customers by the process of urbanization. The urbanization process is expressed by the mutual conversion between various consumer groups. That is to say, with economic development and improved living standards, the lowconsumption groups constantly improve their levels of consumption and step into consumer groups of higher levels. From the right expression of the following equation: is fatally affected by natural population increase, and the policy of family planning, and little affected by economic development, while is mainly affected by economic development. Although certain regularities of economic development can be followed, the effect of this on the customers is uncertain and will vary during the diffusion process of an innovation. However, the models mentioned in the introduction, including system (6), are crisp and obviously cannot meet the actual changes in the law of development. Since these models stressed some main influencing factors, the impact of which on the consumers might be exaggerated, and the combined effects of other factors were ignored, some serious disturbance might occur. Moreover, during the process of diffusion, the impact on the diffusion rate by these factors is of great difference in different stages. For this reason, we try to define as fuzzy numbers and use fuzzy theory to express the uncertainty of urbanization of China.
3. The Stability Analysis on the Model
Since the market is attracted to the variety of innovation usersâ€™ size and the market potential, we only discuss the variables of members , and the variables of users , . Before discussing the stability of the members increasing model, we will introduce some related concepts and then make the stability analysis on the innovation diffusion.
3.1. Related Concepts
We now recall some definitions needed through the paper.
Let denote the set of upper semicontinuous normal fuzzy sets on with compact support. That is, if , then is upper semicontinuous, supp is compact, and there exists at least one for which . The set of , , is
Definition 1. If for , , there exists such that , then we say that the Hukuhara difference of and exists, call the difference of and , and denote .
The approach of Hukuhara differentiation suffers a grave disadvantage in so far as the solution has the property that diam () in nondecreasing in . Consequently, this formulation cannot really reflect any of the reach behaviors of ordinary differential equations such as periodicity, stability, bifurcation, and the like and is ill suited for modeling. However, HÃ¼llermeier suggested a different formulation of the FIVP based on a family of differential inclusions at each level, , where now , the space of nonempty convex compact subsets of . The idea is that the set of all such solution would be the level of a fuzzy set in the sense that all attainable sets , , are levels of a fuzzy set on . Considering to be the solution of the fuzzy DE , thus captures both uncertainty and the rich properties of differential inclusions in one and the same technique. It has been shown that the solution set and attainability set are fuzzy sets under fairly relaxed conditions on [26].
Theorem 2 (Staking Theorem [27]). Let be a family of compact subsets satisfying the following:(1) for all ;(2) for ;(3) for any nondecreasing sequence in .
Then there is a fuzzy set such that . In particular, if the are also convex, then . Conversely, the level sets of any satisfy these conditions, while if , then are also convex.
Property 1 (Lyapunov Stability, [26]). Let be nonempty and suppose that be such that initial value problems
have solutions for every and . So, the interval of existence of solutions is .
A set is stable for the inclusion (10) if for all and , there exists such that implies that on for every solution of (10). If is the attainability set of (10), this may be rephrased as implies that on . If is independent of and depends only on , , (10) is said to be uniformly stable for the inclusion. If as , and is (uniformly) stable, the set is said to be (uniformly) asymptotically stable.
Let , and consider the fuzzy differential equation(FDE): interpreted as a family of differential inclusions. Set and identify the FDE with the family of differential inclusion: where is an open subset of containing ,â€‰â€‰ and .
Theorem 3 (see [26]). Let , and let be an open set in containing . Suppose that is upper semicontinuous and write for all . Let the boundedness assumption with constants , , hold for all and the inclusion: Then the attainable sets of the family of inclusions are the level sets of a fuzzy set . The solution sets of (14) are the level sets of a fuzzy set defined on , where .
Property 2 (Lyapunov Stability of a Family of Differential Inclusion, [26]). If is a fuzzy set and are closed subsets of , define the distance from and Hausdorff separation, respectively, by
The significance of these definitions is that the metric space of fuzzy sets, , is the distance of from and is the analogue of in . Correspondingly, is the Hausdorff separation between with respect to the metric and is the analogue of the Hausdorff separation in .
Let 0 be the fuzzy singleton , write , and denote the open unit ball in by .
A set is stable for the FDE (12) if for all and there exists such that implies that on , where is the fuzzy attainability set defined by the family (12); that is, implies that on . If is independent of and depends only on , for the FDE (12) is said to be uniformly stable. If as and is (uniformly) stable, the set is said to be (uniformly) asymptotically stable. Most frequently, will consist of a single fuzzy set .
Theorem 4. Let be an open subset of containing the origin, let , and let be the flow of the nonlinear system (12). Suppose that and that has eigenvalues with negative real part and eigenvalues with positive real part. Then there exists a dimensional differentiable manifold tangent to the stable subspace of the linear differential inclusions: at 0 such that for all , and for all , and there exists an dimensional differentiable manifold tangent to the unstable subspace of (16) at 0 such that for all , and for all ,
Proof. If and , system (12) can be written as
where , , , , and . This in turn implies that for all , there is a such that and imply that
Furthermore, for all matrix there is an invertible matrix such that
in which the eigenvalues of matrix have negative real part and the eigenvalues of the matrix have positive real part. We can choose sufficiently small that for ,
Letting , the system (19) then has the form
where , , where and satisfies the Lipschitztype condition (20) above.
Consider the system (23). Let
where , . Then, for and , , , and
It is not difficult to see that with chosen in (22), we can choose sufficiently large and sufficiently small that
Next consider the integral equation:
If is a continuous solution of this integral equation, then it is a solution of the differential inclusion (23). We now solve this integral equation by the method of successive approximations. Let
Assume that the induction hypothesis
holds for and . It clearly holds for , provided . Then using the Lipschitztype condition (20) satisfied by the function and the above estimates on and , it follows from the induction hypothesis that for , ,
provided , that is, provided we choose . In order that the condition (20) for the function , it suffices to choose .; that is, we choose . It then follows by induction that (30) holds for all and . Thus, for and ,
This last quantity approaches zero as and therefore is a Cauchy sequence of continuous functions. Then we have
uniformly for all and . Taking the limit of both sides of (29), it follows from the uniform convergence that the continuous function satisfies the integral equation (27) and hence the differential equation (23). It follows by induction and the fact that that is a differentiable function of for and . Thus, it follows from the uniform convergence that is a differentiable function of for and . The estimate implies that
for and .
It is clear from the integral equation (27) that the last components of vector do not enter computation and hence they may be taken as 0. Thus, the components of the solution satisfy the initial conditions:
for . We define the functions
Then the initial values satisfy
according to the definition (36). These equations then define a differentiable manifold for . Furthermore, if is a solution of the differential inclusion (23) with , that is, with , then
It follows that if is a solution of (23) with , then for all and it follows from the estimate (34) that as . It can also be shown that if is a solution of (23) with not , then as . From definition (36) we have
for and ; that is, the differentiable manifold is tangent to the stable subspace of the linear system at 0.
The existence of the unstable manifold of (23) is established in exactly the same way by considering the system (23) with ; that is,
The stable manifold for this system will then be the unstable manifold for (23). Note that it is also necessary to replace the vector by the vector in order to determine the dimensional manifold by the above process. This completes the proof.
If the original point is replaced with the equilibrium point , fussify it and the level set of is called equilibrium core, denoted by . Now we prove that such as Theorem 4; that is, and has eigenvalues with negative real part and eigenvalues with positive real part, then there exists a dimensional differentiable manifold tangent to the stable subspace of the linear differential inclusions (16) at such that for all , and for all , then the equilibrium core is stable, and there exists an dimensional differentiable manifold tangent to unstable subspace of (16) at such that for all , and for all , then the equilibrium core is stable.
For all corresponding to , That is, , if , According to Property 2, set to be stable. Similarly, the equilibrium core corresponding to is unstable and can be proved in the same way.
3.2. Stability Analysis on Coloniesâ€™ Members
Here we will discuss the stability of the coloniesâ€™ members based on the the following different cases: (a) there is no shifting between members of different colonies; (b) one colonyâ€™s members shift into another while anotherâ€™s members do not; (c) the two coloniesâ€™ members are shifting into another, respectively.
In this section, we define as triangular fuzzy numbers, just as Figure 2. From Figure 2, . And in system (6) follow this definition.
Theorem 5. Suppose that , system (3) has one and only one positive equilibrium core, which is stable in .
Proof. If , system (3) has four equilibrium cores , , , , where , . It is easy to achieve that the equilibrium cores , , are unstable and equilibrium core is stable. So is stable in . This completes the proof.
This indicates that if there is no shifting between the two colonies, the members of the two colonies will trend their maximum eventually and maintain stability.
Theorem 6. Supposing , , if , system (3) has one and only one positive equilibrium core, which is stable in .
Proof. If , , system (3) is reduced to
Obviously, when , is stable for the second equation of system (45). Suppose that is a positive solution of the following equation:
then
and system (45) has unique positive equilibrium core .
Assuming , , and , , system (45) can be written as
then the coefficient matrix of the linear system of system (48) is
The feature values of it are both less than 0, so the equilibrium core is stable.
Consequently, if , the equilibrium core (, is stable in . This completes the proof.
This indicates that if there is only one colonyâ€™s member shifting into another while the otherâ€™s members are not shifting, the members of the two colonies will trend their maximum eventually and maintain stability if satisfying some certain conditions.
Theorem 7. Supposing , , system (3) has one and only one positive equilibrium core, which is stable in .
Proof. If , , we will show that system (3) has a unique positive equilibrium core. By setting the righthand side of (3) to 0, we obtain
By the equation , we have
It is easy to see that , , ; therefore, is convex. Notice that , , and that is concave. The parabolic curves have one and only one positive equilibrium core , . Now we will prove that the equilibrium core is stable. Supposing , , and , , system (3) can be written as
Then the coefficient matrix of the linear system of system (52) is
Now let us check , â€‰â€‰ and ,
and since , is the equilibrium core of system (3), , will satisfy the following equations, respectively:
Consequently, , and then .
From (55), we can learn that
Therefore,
From (56) we can conclude that and . As far as ,â€‰â€‰ clearly comes into existence. Therefore, the equilibrium core is stable. Consequently, the equilibrium core is stable in . This completes the proof.
Accordingly, the members of the two colonies will trend their maximum eventually and maintain stability if satisfying certain conditions whether there are members shifting from one colony into another or not.
3.3. Stability Analysis on Innovation Diffusion
Since the equilibrium core in system (3) is stable and we are interested in the asymptotic behavior of system (6), we only discuss the stability of system (6) on the base of system (3) in itâ€™s equilibrium core. That is to say, we only discuss the stability of innovation diffusion when the members of the two colonies are in their stability. Suppose that the global equilibrium core of system (3) is (, , and , . At the same time suppose , , then we have the following theorem.
Theorem 8. Systems have one and only one positive equilibrium core, which is stable in .
Proof. If , system (58) has a unique positive equilibrium core , where . Obviously, this equilibrium core is stable in ; that is to say, if there is no shifting between two colonies, this innovation will occupy its potential eventually and remain stable.
If , , system (58) can be reduced as
Take the second equation of system (6) into consideration. is the unique globe equilibrium core of this equation in , where
Assume that is the positive solution of the following equation:
then follow
Define , , then system (59) can be rewritten as
The coefficient matrix of the linear system of system (63) is
Since the feature values of are both less than 0, the equilibrium core (, is stable. So this equilibrium core is stable in .
If , , the demonstration process is similar to the process of Theorem 4. We can conclude that there exits a unique positive equilibrium core and the equilibrium core is stable. Consequently, the innovation will reach its maximal potential and keep stable. This completes the proof.
4. The Empirical Analysis
Owing to the household management system of planned economy in history, China has divided the population into urban population and rural population. Owing to the strict control, rare shifting, huge gaps between industry and agriculture, and the different environment and living level, there exists great difference between urban population and rural population in communication channels, information spreading, and consumption habit. However, with the reform and exposure to the outside world, the economic development converts China into urban society from village society, and the inhabitants begin to move. According to the related news, there is 1% rural population moving into town per year since the 1980s. The ratio between urban population and rural population may be 7â€‰:â€‰3 after 60 years. Thus, it is not enough to discuss their consumption systems, respectively, without taking the population moving between the cities and countries into consideration. Based on this, we will try to build the dynamical model for telephone diffusion under the condition of the existing large population moving between town and country.
To make it easier, we suppose that this model satisfies following hypotheses:(1)since the standard of living in town is superior to that in the country of China, we suppose , which indicates that there are no people in China shifting from town into country;(2)since the communication by telephone is popular and necessary, we suppose , which indicates that the telephone users in country will continue using it after shifting into town and users in town will not give up using it;(3)the population in China will reach 1.6 billion and maintain stable, which is obtained according to some related research on population in China [28].
4.1. Model Conversion
In this section, we first use the definition of subduction defined by Zadeh, which means , in which , . And then we use the difference to simulate the diffusion process of local telephone in China.
According to the hypothesis defined above, system (58) can be rewritten as
According to the definition of subduction defined by article [29], system (65) can be rewritten as
That is,
Then we used Genetic Algorithms (GA) to model the coefficient of model (67), which were shown in Table 4, where error is the permitted maximum error ratio between the data predicted by model and history data.
If we take the coefficient value as shown in Table 2 to the minimum of the target values in Table 3, the plots of potential curve of telephone in urban area or in rural area governed by model (67) are shown in Figure 3 or Figure 4, respectively.



In Figure 3, the red line is made up of history data and the green solid line and the green dot line represent the upper limit and the lower limit of the telephone consumers in urban area, respectively.
In Figure 4, the red line is made up of history data and the blue solid line and the blue dot line represent the upper limit and the lower limit of the telephone consumers in rural area, respectively.
From Figures 3 and 4, the fuzzy simulation effect is not perfect, and the fuzzy intervals of the consumer numbers are too big to predict the diffusion process of innovation. We could only find the time of inflection point and the maximal potential market from Figures 3 and 4. So we try to use the definition of Hukuhara difference, which, , comes into exist. Consequently, we have the following system: