Parameter analysis and identification of the multiple-advanced manufacturing mode diffusion model

Abstract

To better understand the development of advanced manufacturing modes in the competitive diffusion, this article analyzes and identifies the parameters of the multiple-advanced manufacturing modes diffusion model. First, considering the diffusion characteristics, the competitive multiple-advanced manufacturing modes diffusion model is described. Second, various parameters of the diffusion model are analyzed to reveal their influence on the diffusion process. Third, a parameter identification model and a further parameter identification algorithm based on an improved genetic algorithm are proposed. Finally, based on a practical application of the diffusion model, the parameter identification model is constructed, and optimized parameters are obtained for the multiple-advanced manufacturing mode diffusion model. And the resulting model output is compared and contrasted with real data. The proposed model and algorithm provide a new approach to the identification of parameters in the diffusion model, which allows us to better understand the diffusion rules, help enterprises to predict the diffusion status of advanced manufacturing modes, and provide a decision-making basis for enterprises and governments.

Introduction

With the ongoing development of information and scientific technologies, advanced manufacturing modes and advanced manufacturing technologies (AMTs) have developed rapidly. To date, a variety of advanced manufacturing modes have emerged, such as remanufacturing/green manufacturing, Just In Time (JIT), network manufacturing, supply chain management, computer-integrated manufacturing systems (CIMSs), synchronous manufacturing, virtual manufacturing, and total quality management (TQM). Most studies have focused on the theory and implementation of advanced manufacturing modes, and have advanced their deep development. However, there has been little reference to quantitative studies on the implementation status of advanced manufacturing modes.

The implementation of advanced manufacturing systems involves the acceptance, adaptation, and application (collectively, the diffusion process) of advanced manufacturing modes. In practice, the diffusion of advanced manufacturing modes is not the proliferation of a particular mode, but a competitive diffusion process among various modes. Meanwhile, the capability of an enterprise directly determines their likelihood of implementing advanced manufacturing modes. Therefore, it is necessary to consider the enterprise capability and the competition between various advanced manufacturing modes to further clarify the diffusion rules of advanced manufacturing modes.

The diffusion process of advanced manufacturing modes is a competition between multiple modes. To better reveal the diffusion rules of advanced manufacturing modes, the parameters of the multiple-advanced manufacturing mode diffusion model must be analyzed and identified. System parameter identification involves choosing those parameters that best fit the data, so as to make the mathematical model accord with the real system.

Thus, in this article, based on a competitive diffusion model of advanced manufacturing modes, the influence of parameters on the diffusion model is analyzed. Then parameter identification model of the multiple-advanced manufacturing diffusion is proposed and solved based on an improved genetic algorithm (IGA) to obtain the optimal values. Utilizing system parameter identification will allow the model to be more representative of the real system, thus enabling more accurate predictions of the future implementation of advanced manufacturing modes. This study is intended to help enterprises and governments to better understand the diffusion rules of advanced manufacturing modes, and thus allow them to more accurately predict the implementation status and make reasonable decisions.

Related work

With expanding markets and global competition, it has become an ongoing trend for enterprises to adapt to various targeted markets and enhance their core competitiveness by implementing new manufacturing modes. Studies of manufacturing modes have focused on the following aspects: (1) the concept, classification, and characteristic study of the manufacturing mode; ¹ (2) knowledge representation and operation of manufacturing modes;^2–5 (3) relative technologies involved in the various manufacturing modes; ⁶ and (4) case studies of advanced manufacturing mode applications.^7,8 As an example of (1), Sun and Cao ¹ classified the advanced manufacturing modes from the perspective of the manufacturing philosophy, system method, and detailed technologies according to the mode characteristics. In terms of (2), Yan ² proposed a formal representation approach to manufacturing mode, that is, knowledge mesh, which solved the representation and reconfiguration operations of manufacturing modes. In terms of (3), Davrajh and Bright ⁶ seek to propose a method of holistically managing product quality in a manufacturing environment with high customer input and product variety. In terms of (3), Thomas and Barton ⁷ provided details of a survey conducted into 260 manufacturing small and medium enterprises (SMEs) operating in a range of industrial sectors. The survey is a follow-up to a survey performed 3 years earlier on the same companies, and it is targeted in this case on investigating the migratory characteristics of AMT implementation in manufacturing SMEs. These studies have laid the foundations for advanced manufacturing modes, meanwhile, and it is also meaningful to understand the application status of advanced manufacturing modes.

Although the acceptance and implementation of the advanced manufacturing mode are also an innovative diffusion, it is the innovation of recognition and concept, which is very different to product innovation.^9–12 Various studies on new production and technology diffusion have been studied, which have mainly focused on the following aspects: (1) diffusion concepts; (2) diffusion models; (3) related aspects of diffusion models such as parameter evaluation methods, and so on; and (4) application of diffusion models in different fields. As an example of (1), Everett M. Rogers proposed the typical innovation diffusion theory, in which diffusion rules for the innovation in a social system and the famous “S” curve were presented. In terms of (2), different diffusion models have emerged since the first presentation of the diffusion model (Bass model) by Bass in 1969. For example, Yan and Ma ⁹ proposed the competitive diffusion model of repurchased products in knowledgeable manufacturing. In terms of (3), Wang et al. ¹¹ studied the mechanism and laws of products diffusion in multiple markets in terms of substitute products diffusion. In terms of (4), Zhang and Lu ¹² proposed a repeat purchase model to predict the development of the Chinese mobile phone market in the next period. Thus far, most research on the diffusion of advanced manufacturing modes has taken the form of qualitative studies. For example, Li et al. ¹³ studied the advanced manufacturing mode implementation in Huawei Technologies Ltd. The aim was to build an advanced manufacturing model for the global value network path, and also to put forward countermeasures and suggestions for developing a world-class advanced manufacturing mode. There is little in the literature concerning quantitative studies on the implementation of advanced manufacturing modes. Xue and Cao ¹⁰ first studied the acceptance of CIMS, but gave only a simple multi-stage diffusion model in which competition between manufacturing modes was not considered. On this basis, Xue and Cao ¹⁴ further discussed diffusion characteristics and mechanisms in advanced manufacturing modes, and built a diffusion model of an advanced manufacturing mode in which the competition between the prototype and improved mode was considered, although the capability of the enterprise was not considered. Then, considering enterprise capability and competition between manufacturing modes, Xue et al. ¹⁵ proposed the competition diffusion model of multiple-advanced manufacturing modes in cluster environment. However, as there are many parameters in the diffusion model, the influence of various parameters is not analyzed, and the parameter identification is not considered.

At present, the system parameter identification research falls into two categories: (1) system parameter identification methods and (2) applications of system parameter identification. As an example of (1), the infinite impulse response (IIR) system identification task is formulated as an optimization problem, and a recently introduced cat swarm optimization (CSO) is used to develop new population-based learning rules for the model in Panda et al. ¹⁶ In terms of (2), Kim and Lynch ¹⁷ proposed a system identification strategy for single-input multi-output (SIMO) subspace system identification based on Markov parameters. The method is specifically customized for embedment within the decentralized computational framework of a wireless sensor network. System parameter identification has been successfully used in different application environments, but the identification of diffusion model parameters has not been attempted.

Parameter analysis on the multiple-advanced manufacturing mode diffusion model

The diffusion of advanced manufacturing modes is a process with limited rationality. Similar to the diffusion model in Xue et al., ¹⁵ under the influence of current adopters, the government, and the external environment, potential enterprises (those that have not implemented the advanced manufacturing mode) may transfer to different advanced manufacturing modes. Also, if one mode has advantages over the others, those using the first mode may transfer to the more advanced one. Based on the similar assumptions and analysis in Xue et al., ¹⁵ enterprises are divided into $x_1,x_2,y_1,y_2,\dots ,y_n$ , and in a given period, the competition diffusion model is built as follows

$$\{\begin{array}{c}\hfill \frac{d{x}_1}{\mathrm{dt}}=-\beta x_1-{\displaystyle \sum _{k=1}^n}(p_k+q_{1k}\frac{\mathrm{am}-x_1}{\mathrm{am}})x_1\hfill \\ \multicolumn{1}{c}{\frac{d{x}_2}{\mathrm{dt}}=\beta x_1-{\displaystyle \sum _{k=1}^n}(p_k+g{q}_{2k}\frac{\mathrm{bm}-x_2}{\mathrm{bm}})x_2}\\ \multicolumn{1}{c}{\frac{d{y}_1}{\mathrm{dt}}=(p_1+q_{11}\frac{\mathrm{am}-x_1}{\mathrm{am}})x_1+(p_1+g{q}_{21}\frac{\mathrm{bm}-x_2}{\mathrm{bm}})x_2-{\displaystyle \sum _{j=2}^n}{\theta }_{1j}{y}_1}\\ \multicolumn{1}{c}{\begin{array}{ccc}\hfill \hfill & \hfill ⋮\hfill & \hfill \hfill \end{array}}\\ \multicolumn{1}{c}{\frac{d{y}_k}{\mathrm{dt}}=(p_k+q_{1k}\frac{\mathrm{am}-x_1}{\mathrm{am}})x_1+(p_k+g{q}_{2k}\frac{\mathrm{bm}-x_2}{\mathrm{bm}})x_2}\\ \multicolumn{1}{c}{+{\displaystyle \sum _{i=1}^{k-1}}{\theta }_{\mathrm{ik}}{y}_i-{\displaystyle \sum _{j=k+1}^n}{\theta }_{\mathrm{kj}}{y}_k}\\ \multicolumn{1}{c}{\begin{array}{ccc}\hfill \hfill & \hfill ⋮\hfill & \hfill \hfill \end{array}}\\ \multicolumn{1}{c}{\frac{d{y}_n}{\mathrm{dt}}=(p_n+q_{1n}\frac{\mathrm{am}-x_1}{\mathrm{am}})x_1+(p_n+g{q}_{2n}\frac{\mathrm{bm}-x_2}{\mathrm{bm}})x_2+{\displaystyle \sum _{i=1}^{n-1}}{\theta }_{\mathrm{in}}{y}_i}\end{array}$$

(1)

From the multiple-advanced manufacturing mode diffusion, the following analytic conclusions can be obtained.

Theorem 1

When ${\theta }_{\mathrm{ij}}=0$ , the diffusion model represented by equation (1) has the similar form of the Bass model.

Proof

Suppose the maximum number of each potential adopters is ${\overline{N}}_1(t)$ , ${\overline{N}}_2(t)$ , and the number of adopters is $N_1(t)$ , $N_2(t)$ , then $x_1=a[m-N(t)]={\overline{N}}_1(t)-N_1(t)$ , $x_2=b[m-N(t)]={\overline{N}}_2(t)-N_2(t)$ , $\overline{N}(t)=m$ , $y_1+y_2+\cdots +y_n=N(t)$

$$\stackrel{·}{N}(t)={\stackrel{·}{y}}_1+{\stackrel{·}{y}}_2+\cdots +{\stackrel{·}{y}}_n=[{\displaystyle \sum _1^n}{p}_k+(q_{11}+q_{12}+\cdots +q_{1n})\frac{N(t)}{{\overline{N}}_1(t)}]({\overline{N}}_1(t)-N_1(t))+[{\displaystyle \sum _1^n}{p}_k+g(q_{21}+q_{22}+\cdots +q_{2n})\frac{N(t)}{{\overline{N}}_2(t)}]({\overline{N}}_2(t)-N_2(t))$$

This is the similar form of the Bass model, which completes the proof of the Theorem 1.

Theorem 2

When ${\theta }_{\mathrm{ij}}=0$ , $P(0,0,m-y_2-\cdots -y_n,y_2,\dots ,y_n)$ is the balanced point and steady point of the system. Furthermore, when $t\to +\infty ,\underset{t\to +\infty }{lim}{x}_1(t)=0,\underset{t\to +\infty }{lim}{x}_2(t)=0$ , and $y_1+y_2+\cdots +y_n=m$ .

Proof

According to the definition of balanced points, the balanced point of system (2) is $P(0,0,m-y_2-\cdots -y_n,y_2,\dots ,y_n)$ .

As for $P(0,0,m-y_2-\cdots -y_n,y_2,\dots ,y_n)$ , its simplified linear matrix is

$$J=\left[\begin{array}{ccccc}\hfill -\alpha -{\displaystyle \sum _1^n}{p}_k-q_{11}-q_{12}-\cdots -q_{1n}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{\alpha }& \hfill -{\displaystyle \sum _1^n}{p}_k-g(q_{21}+q_{22}+q_{23})\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{p_1+q_{11}}& \hfill p_k+g{q}_{21}\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{⋮}& \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill \\ \multicolumn{1}{c}{p_k+q_{1k}}& \hfill p_k+g{q}_{2k}\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill 0\hfill \end{array}\right]$$

The eigenvalue of the matrix can be obtained, that is, ${\lambda }_1=-g{q}_{23}-3p-g{q}_{21}-g{q}_{22}$ , ${\lambda }_2=0$ , ${\lambda }_3=0$ , ${\lambda }_4=-\alpha -3p-q_{11}-q_{12}-q_{13}$ , and ${\lambda }_1<0$ , ${\lambda }_2=0$ , ${\lambda }_3=0$ , ${\lambda }_4<0$ . Thus, $P(0,0,m-y_2-y_3,y_2,y_3)$ is the steady point of the system.

Theorem 3

In system (1), $x_1(t)$ is strictly monotonically increasing with $p_k$ .

Proof

$$\begin{array}{cc}\multicolumn{1}{c}{\frac{d{x}_1}{\mathrm{dt}}=}& -\alpha x_1-(p_1+q_{11}\frac{\mathrm{am}-x_1}{\mathrm{am}})x_1\hfill \\ \multicolumn{1}{c}{}& -(p_2+q_{12}\frac{\mathrm{am}-x_1}{\mathrm{am}})x_1-\cdots -(p_n+q_{1n}\frac{\mathrm{am}-x_1}{\mathrm{am}})x_1\hfill \end{array}$$

$$\frac{\mathrm{am}}{x_1[{\displaystyle \sum _1^n}{q}_{1k}{x}_1-\mathrm{am}(\alpha +{\displaystyle \sum _1^n}{p}_k+{\displaystyle \sum _1^n}{q}_{1k})]}d{x}_1=\mathrm{dt}$$

thus

$$x_1=\frac{\mathrm{am}(\alpha +{\displaystyle \sum _1^n}{p}_k+{\displaystyle \sum _1^n}{q}_{1k})}{{\displaystyle \sum _1^n}{q}_{1k}-c_0{e}^{(\alpha +{\displaystyle \sum _1^n}{p}_k+{\displaystyle \sum _1^n}{q}_{1k})t}}$$

As

$$\frac{\partial x_1}{\partial p_k}=\frac{\mathrm{am}\{{\displaystyle \sum _1^n}{q}_{1k}+c_0{e}^{(\alpha +{\displaystyle \sum _1^n}{p}_k+{\displaystyle \sum _1^n}{q}_{1k})t}[t(\alpha +{\displaystyle \sum _1^n}{p}_k+{\displaystyle \sum _1^n}{q}_{1k})-1]\}}{{({\displaystyle \sum _1^n}{q}_{1k}-c_0{e}^{(\alpha +{\displaystyle \sum _1^n}{\mathrm{P}}_k+{\displaystyle \sum _1^n}{q}_{1k})t})}^2}$$

$\partial x_1/\partial P_k>0$ , which completes the proof of Theorem 4.

As the competition diffusion model is differential equations, it is difficult to obtain its analytic solutions. Thus, MATLAB 7.1 was used to simulate the diffusion model. To analyze and discuss the influence of relevant parameters on the diffusion process, we use supply chain management (labeled as mode A), green manufacturing (labeled as mode B), and TQM (labeled as mode C) in Xue et al. ¹⁵ to discuss the parameter influence of the diffusion model. For quantitative results, other parameters of the model are given as shown in Table 1.

Table 1.

Parameters of the model.

Parameter	$m$	$b$	$\alpha$	$g$	$p$
Value	271,835	0.89	0.17	1.44	0.09
Parameter	$q_{11}$	$q_{12}$	$q_{13}$	$q_{21}$	$q_{22}$
Value	0.7414	0.5459	0.3955	0.1343	0.2279
Parameter	$q_{23}$	${\theta }_{12}$	${\theta }_{13}$	${\theta }_{23}$	$a$
Value	0.2326	0.2887	0.1590	0.2975	0.11

• 1. Influence of $g$ . To study the influence of $g$ on the competitive diffusion model, the values shown in Table 2 were applied, and the influence on the diffusion of $x_1(t)$ , $x_2(t)$ , $y_1(t)$ , $y_2(t)$ , and $y_3(t)$ was observed, as shown in Figure 1.

Table 2.

Different values of $g$ .

Case	1	2
Value	1.0032	1.2

Figure 1.

Influence of $g$ on diffusion (.fig drawn by MATLAB, then it is saved as .jpg).

In Figure 1, the solid lines represent case 1, and the dotted lines represent case 2. As $g$ is increased, the diffusion of $x_1(t)$ is not influenced, whereas that of $x_2(t)$ , $y_1(t)$ , $y_2(t)$ , and $y_3(t)$ occurs sooner. The peak values of $y_1(t)$ and $y_2(t)$ also become larger, that is, the number of enterprises implementing modes A and B increases.

The discuss on the influence of $g$ shows that the government support does not work on the enterprises that have the capacity to implement the advanced manufacturing mode, but are yet to do so. However, the government support can accelerate the diffusion of advanced manufacturing modes in the enterprises that that do not have the capacity to implement the advanced manufacturing mode, enterprises that have implemented the modes. In the meantime, not only the peak value of enterprises implementing modes A and B increases, but the peak time occurs sooner. In the same way, the influence of other parameters can be analyzed as follows.

• 2. Influence of $p_k$ . With an increase in $p_1$ , the diffusion of $x_1(t)$ , $x_2(t)$ , $y_1(t)$ , $y_2(t)$ , and $y_3(t)$ occurs sooner, and the influence on $x_2(t)$ , $y_1(t)$ , $y_2(t)$ , and $y_3(t)$ is greater than on $x_1(t)$ . Meanwhile, the peak value of $y_1(t)$ becomes larger, but that of $y_2(t)$ decreases. As $p_2$ is increased, the diffusion of $x_1(t)$ , $x_2(t)$ , $y_1(t)$ , $y_2(t)$ , and $y_3(t)$ occurs sooner, and the peak value of $y_1(t)$ decreases, but that of $y_2(t)$ increases. With an increase in $p_3$ , the diffusion of all variables occurs sooner, and the peak value of both $y_1(t)$ and $y_2(t)$ decreases.

The discuss on the influence of $p$ shows that different influencing coefficients of potential adopters to implement the mode $k$ works on the diffusion process of advanced manufacturing modes, and they can accelerate the diffusion of advanced manufacturing modes in all kinds of enterprises. However, the influence degree of different coefficients is various, and the influence on the peak time and peak value is different. Thus, it is meaningful to adjust the diffusion time and amount according to the development of advanced manufacturing modes.

• 3. Influence of $q_{1k}$ , $q_{2k}$ . With an increase in $q_{11}$ , the diffusion of $x_1(t)$ , $x_2(t)$ , $y_1(t)$ , $y_2(t)$ , and $y_3(t)$ occurs sooner, and the peak value of $y_1(t)$ increases, whereas that of $y_2(t)$ decreases. With an increase in $q_{12}$ , the diffusion of $x_1(t)$ , $x_2(t)$ , $y_1(t)$ , $y_2(t)$ , and $y_3(t)$ occurs sooner, and the peak value of $y_1(t)$ decreases, whereas that of $y_2(t)$ increases. With an increase in $q_{13}$ , the diffusion of $x_1(t)$ , $x_2(t)$ , $y_1(t)$ , $y_2(t)$ , and $y_3(t)$ occurs sooner, and the peak value of both $y_1(t)$ and $y_2(t)$ decreases. With an increase in $q_{21}$ , the diffusion of $x_1(t)$ is unchanged, but that of $x_2(t)$ , $y_1(t)$ , $y_2(t)$ , and $y_3(t)$ occurs sooner. The peak value of $y_1(t)$ increases, but that of $y_2(t)$ decreases. With an increase in $q_{22}$ , the diffusion of $x_1(t)$ is unchanged, but that of $x_2(t)$ , $y_1(t)$ , $y_2(t)$ , and $y_3(t)$ occurs sooner. The peak value of $y_1(t)$ decreases, but that of $y_2(t)$ increases. With an increase in $q_{23}$ , the diffusion of $x_1(t)$ is unchanged, but that of $x_2(t)$ , $y_1(t)$ , $y_2(t)$ , and $y_3(t)$ occurs sooner. The peak values of $y_1(t)$ and $y_2(t)$ decrease.

The discussion on the influence of $q_{1k}$ and $q_{2k}$ shows that different inherent diffusion rates of $x_1$ to $y_k$ and $x_2$ to $y_k$ work on the diffusion process of advanced manufacturing modes, and they influence the number of enterprises implementing the different manufacturing modes. However, the influence degree of different coefficients is various, and the influence on the peak time and peak value is different. Thus, we can adjust different inherent diffusion rates to obtain the desirable diffusion time and amount according to the development of advanced manufacturing modes.

• 4. Influence of ${\theta }_{\mathrm{ij}}$ . With an increase in ${\theta }_{12}$ , the diffusion of $x_1(t)$ , $x_2(t)$ , and $y_3(t)$ is unchanged, but that of $y_1(t)$ and $y_2(t)$ occurs sooner. The peak value of $y_1(t)$ decreases, but that of $y_2(t)$ increases. With an increase in ${\theta }_{13}$ , the diffusion of $x_1(t)$ and $x_2(t)$ is unchanged, but that of $y_1(t)$ , $y_2(t)$ , and $y_3(t)$ occurs sooner. The peak values of $y_1(t)$ and $y_2(t)$ decrease. With an increase in ${\theta }_{23}$ , the diffusion of $x_1(t)$ , $x_2(t)$ , and $y_1(t)$ is unchanged, but the peak value of $y_2(t)$ decreases.

The discussion on the influence of ${\theta }_{\mathrm{ij}}$ shows that different competition transferring coefficient from $y_i$ to $y_j$ works on the diffusion process of advanced manufacturing modes, and they influence the number of enterprises implementing the different manufacturing modes. However, the influence degree of different competition transferring coefficient is various, and the influence on the peak time and peak value is different. Thus, we can adjust competition transferring coefficient to obtain the desirable diffusion time and amount according to the development of advanced manufacturing modes.

• 5. Sensitivity analysis. The sensitivity $S(\mathrm{xx},\mathrm{yy})$ is defined as follows: when parameter $\mathrm{xx}$ is changed so as to affect the output variable $\mathrm{yy}$ , then the analysis expression of sensitivity $S(\mathrm{xx},\mathrm{yy})$ is given by the expression

$$S(\mathrm{xx},\mathrm{yy})=\left|\frac{\Delta \mathrm{yy}}{\Delta \mathrm{xx}}\right|$$

The sensitivity of the peak values of $y_1$ and $y_2$ to the parameters $g$ , $p_1$ , $q_{11}$ , and ${\theta }_{12}$ is now analyzed. First, three of the four parameters are fixed, and the peaks of $y_1$ and $y_2$ are examined based on the increase of the fourth parameter between 1% and 10% of its value, as shown in Tables 3 and 4. The calculated sensitivity values for changes in $g$ , $p_1$ , $q_{11}$ , and ${\theta }_{12}$ are illustrated in Figures 2 and 3 for $y_1$ and $y_2$ , respectively.

Table 3.

Sensitivity of the peak of $y_1$ to $g$ , $p_1$ , $q_{11}$ , and ${\theta }_{12}$ .

Sensitivity	Variation
	1%	2%	3%	4%	5%	6%	7%	8%	9%	10%
$S(g,y_1)$	1000	800	633	575	1220	1583	1771	1875	1900	1900
$S(p_1,y_1)$	55,100	51,350	49,767	46,000	43,700	42,617	40,957	39,150	37,367	36,370
$S(q_{11},y_1)$	100	150	133	125	140	133	129	125	122	130
$S({\theta }_{12},y_1)$	15,100	14,850	14,533	14,275	14,020	13,533	12,943	12,475	12,078	11,730

Table 4.

Sensitivity of the peak of $y_2$ to $g$ , $p_1$ , $q_{11}$ , and ${\theta }_{12}$ .

Sensitivity	Variation
	1%	2%	3%	4%	5%	6%	7%	8%	9%	10%
$S(g,y_2)$	500	450	1000	1250	1320	1350	1328	1287	1233	1180
$S(p_1,y_2)$	5300	5250	6267	6200	6060	5983	5900	5800	5689	5570
$S(q_{11},y_2)$	0	0	33	25	20	17	14	13	22	20
$S({\theta }_{12},y_2)$	20,300	19,800	19,300	18,825	18,360	17,900	17,471	17,063	16,667	16,280

Figure 2.

Sensitivity of the peak of $y_1$ to $g$ , $p_1$ , $q_{11}$ , and ${\theta }_{12}$ (.mgf drawn by Matlab, then it is saved as .jpg).

Figure 3.

Sensitivity of the peak of $y_2$ to $g$ , $p_1$ , $q_{11}$ , and ${\theta }_{12}$ (.mgf drawn by Matlab, then it is saved as .jpg).

From Table 3 and Figure 2, we can see that the sensitivity of the peak of $y_1$ to $p_1$ and ${\theta }_{12}$ decreases, which means that as $p_1$ and ${\theta }_{12}$ increase, their impact on the peak of $y_1$ will decrease. However, there does not appear to be any strict correlation between the sensitivity of the peak of $y_1$ to $g$ and $q_{11}$ . Similarly, from Table 4 and Figure 3, we can see that an increase in ${\theta }_{12}$ leads to the sensitivity of the peak of $y_2$ decreasing. No strict correlation in the sensitivity of the peak of $y_2$ to $g$ , $p_1$ , or $q_{11}$ was observed.

Our analysis shows that the impact of various parameters on the diffusion of the advanced manufacturing mode is obvious. Changes in parameter values have a significant impact on the diffusion process of the advanced manufacturing mode, and the impact of different parameters on the diffusion process varies. Therefore, it is necessary to obtain the suitable parameters so as to better reveal the diffusion rules in the implementation of advanced manufacturing modes.

Parameter identification of the diffusion model

Parameter identification model

System parameter identification involves choosing those parameters that best fit the data, so as to make the mathematical model accord with the real system. System parameter identification methods include value function, least square method, and so on, and the least square method is widely used in different fields. Based on the thought of least square method, the square of the distance between the value in the diffusion model and their practical values is taken as the objective function. However, the relation in the diffusion model satisfies the differential equation, which cannot be solved by directly using the least square method. Fortunately, the parameter identification model is an optimization model which can be solved by the enumeration search methods. Therefore, to identify the parameters in the multiple-advanced manufacturing mode diffusion model, parameters are taken as decision-making variables, and the square of the distance between the estimated values by diffusion model and their real values is taken as the objective, thus, parameter identification problem is an optimization problem as follows.

• Objective function

$$\min {\displaystyle \sum _{i=1}^{m\prime }}{\displaystyle \sum _{j=1}^n}{(y_{\mathrm{ij}}-y_{\mathrm{ij}}^{*})}^2$$

where $y_{\mathrm{ij}}$ is the estimated number of enterprises implementing the jth manufacturing mode in the ith year and $y_{\mathrm{ij}}^{*}$ is the actual number of enterprises implementing the jth manufacturing mode in the ith year.

• Constraints

$$\{\begin{array}{c}\hfill \frac{d{x}_1}{\mathrm{dt}}=-\beta x_1-{\displaystyle \sum _{k=1}^n}(p_k+q_{1k}\frac{\mathrm{am}-x_1}{\mathrm{am}})x_1\hfill \\ \multicolumn{1}{c}{\frac{d{x}_2}{\mathrm{dt}}=\beta x_1-{\displaystyle \sum _{k=1}^n}(p_k+g{q}_{2k}\frac{\mathrm{bm}-x_2}{\mathrm{bm}})x_2}\\ \multicolumn{1}{c}{\frac{d{y}_1}{\mathrm{dt}}=(p_1+q_{11}\frac{\mathrm{am}-x_1}{\mathrm{am}})x_1+(p_1+g{q}_{21}\frac{\mathrm{bm}-x_2}{\mathrm{bm}})x_2-{\displaystyle \sum _{j=2}^n}{\theta }_{1j}{y}_1}\\ \multicolumn{1}{c}{\begin{array}{ccc}\hfill \hfill & \hfill ⋮\hfill & \hfill \hfill \end{array}}\\ \multicolumn{1}{c}{\frac{d{y}_k}{\mathrm{dt}}=(p_k+q_{1k}\frac{\mathrm{am}-x_1}{\mathrm{am}})x_1+(p_k+g{q}_{2k}\frac{\mathrm{bm}-x_2}{\mathrm{bm}})x_2+{\displaystyle \sum _{i=1}^{k-1}}{\theta }_{\mathrm{ik}}{y}_i-{\displaystyle \sum _{j=k+1}^n}{\theta }_{\mathrm{kj}}{y}_k}\\ \multicolumn{1}{c}{\begin{array}{ccc}\hfill \hfill & \hfill ⋮\hfill & \hfill \hfill \end{array}}\\ \multicolumn{1}{c}{\frac{d{y}_n}{\mathrm{dt}}=(p_n+q_{1n}\frac{\mathrm{am}-x_1}{\mathrm{am}})x_1+(p_n+g{q}_{2n}\frac{\mathrm{bm}-x_2}{\mathrm{bm}})x_2+{\displaystyle \sum _{i=1}^{n-1}}{\theta }_{\mathrm{in}}{y}_i}\\ \multicolumn{1}{c}{y_{i3}=m-x_{i1}-x_{i2}-y_{i1}-y_{i2}}\\ \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{{\eta }_0\le g,p_1,p_2,p_3,{\theta }_{12},{\theta }_{13},{\theta }_{23},q_{11},q_{12},q_{13},q_{21},q_{22},q_{23}\le {\eta }_1}\end{array}$$

Model solution by IGA

To optimize parameters of diffusion model, the proper optimization model solution must also be studied. As constrains of parameter identification model are differential equations, the parameter identification is not the general linear programming, nonlinear or dynamic programming problem, thus, the model cannot be solved by common linear programming methods or nonlinear programming methods (e.g. Newton method and gradient method). Fortunately, implicit enumeration search methods like genetic algorithm (GA)^18–20 can be utilized to solve the parameter identification model. As the general GA has the shortcoming of low searching speed, long search time, an IGA is proposed to solve the parameter identification problem. The detailed chromosomes and genetic operators in this article are as follows:

1. The chromosomes adopt symbolic coding, and all parameters are used to describe solutions, as shown in Figure 4. The chromosomes represent the value of each parameters to be identified, and they are arranged in the order shown in Figure 4. With the genetic operations, the chromosomes are changed, and different parameter values are obtained, which satisfies the differential equations as well as other constraints.

2. Selection. Selection operator in IGA utilizes the roulette wheel selection; parents are chosen on the basis of their fitness rate as shown in equation (2)

$$p_{\mathrm{opi}}=\frac{{\epsilon }_{\mathrm{opi}}}{{\displaystyle \sum _{\mathrm{opi}=1}^{\mathrm{Popsize}}}{\epsilon }_{\mathrm{opi}}}\mathrm{opi}=1,2,\dots ,\mathrm{popsize}$$

(2)

Figure 4.

Chromosomes in parameter identification model (.vsd drawn by Visio).

where $P_{\mathrm{opi}}$ is a chromosome’s selection probability, ${\epsilon }_{\mathrm{opi}}$ is the chromosome’s correlation degree, and $q(\mathrm{kk})={\displaystyle \sum _{i=1}^{\mathrm{kk}}}{P}_{\mathrm{opi}}$ . Then $q(\mathrm{kk})$ is compared with $r$ , which satisfies uniform distribution [0,1]. If $r>q(1)$ , then the first chromosome is selected, while if $r>q(1)$ and $q(\mathrm{kk}-1)<r\le q(\mathrm{kk})(2\le \mathrm{kk}\le \mathrm{popsize})$ , then the kkth chromosome is selected.

• Crossover. Crossover operator adopts two-point crossover, as shown in Figure 5. And self-adaptive crossover probability is introduced to avoid its dependence on the initial value, as shown in equation (3)

$$p_c=\{\begin{array}{c}\hfill p_{c1}-\frac{(p_{c1}-p_{c2})(\epsilon \prime -{\epsilon }_{\mathrm{avg}})}{{\epsilon }_{\max }-{\epsilon }_{\mathrm{avg}}}\epsilon \prime \ge {\epsilon }_{\mathrm{avg}}\hfill \\ \multicolumn{1}{c}{p_{c1}\epsilon \prime <{\epsilon }_{\mathrm{avg}}}\end{array}$$

(3)

Figure 5.

Crossover operation (.vsd drawn by Visio).

where $P_c$ is a chromosome’s crossover probability, $P_{c1},P_{c2}$ are initial given values.

• Mutation. Mutation operation is shown in Figure 6. Similar to crossover operation, self-adaptive mutation probability is adopted, as shown in equation (4)

$$p_m=\{\begin{array}{c}\hfill p_{m1}-\frac{(p_{m1}-p_{m2})({\epsilon }_{\mathrm{avg}}-\epsilon )}{{\epsilon }_{max}-{\epsilon }_{\mathrm{avg}}}\epsilon \ge {\epsilon }_{\mathrm{avg}}\hfill \\ \multicolumn{1}{c}{p_{m1}\epsilon <{\epsilon }_{\mathrm{avg}}}\end{array}$$

(4)

Figure 6.

Mutation operation (.vsd drawn by Visio).

where $P_m$ is a chromosome’s mutation probability, $\epsilon$ is the fitness value of the mutation individual, and $P_{m1},P_{m2}$ are initial given values.

Based on the above improvements, IGA is utilized to obtain the optimal parameters in the multiple-advanced manufacturing mode diffusion model.

Algorithm 1

Parameter identification algorithm based on IGA

• Step 1. Parameters of the IGA are initialized, which include initial iteration $g\_\mathrm{ctr}\leftarrow 0$ , maximum iteration $g\_\mathrm{num}=2000$ , $\mathrm{popsize}=50$ , and part parameters related with the diffusion model as shown in Table 1.

• Step 2. The initial population is generated randomly, which include the generation of $\mathrm{popsize}$ chromosomes and chromosome generation as shown in Figure 4.

• Step 3. The population is divided into $\mathrm{mm}$ sub-populations ${\mathrm{pop}}_i(i=1,2,\dots ,\mathrm{mm})$ uniformly.

• Step 4. The mmth sub-populations are conducted the following operations

• Crossover. According to equation (3), the $P_c$ of any two chromosomes in ${\mathrm{pop}}_i(i=1,2,\dots ,\mathrm{mm})$ is calculated, then the pair with maximum $P_c$ will be conducted the crossover operation, and the optimal chromosomes will be reserved.

• Mutation. According to equation (4), if the $P_m$ of each chromosome in ${\mathrm{pop}}_i$ is calculated, then the one with maximum $P_m$ will be performed in the mutation operation, and the optimal chromosomes will be reserved.

• The sub-population is updated. The best chromosomes in sub-population are compared, and the best chromosome in the population is reserved.

Step 5. The present generation is judged when each sub-population is calculated. And if it’s greater than or equal to $g\_\mathrm{num}$ , then the ending condition is satisfied and the chromosome with best fitness is obtained. Otherwise, $g\_\mathrm{ctr}\leftarrow g\_\mathrm{ctr}+1$ , and go to Step 4.

According to the IGA, the optimal parameters of the multiple-advanced manufacturing mode diffusion model are obtained.

Examples

Background

Parameter identification is related to real applications. We use supply chain management, green manufacturing, and TQM in Xue et al. ¹⁵ to discuss the simulation of the diffusion model and its parameter identification. Related data are given in Table 5.

Table 5.

Number of enterprises implementing the modes.

Mode	Time
	2005	2006	2007	2008	2009	2010
$A$	58	132	230	370	743	1061
$B$	328	335	704	1180	1340	1742
$C$	1000	1238	1771	2139	2405	2617

From the above background, the initial time of the system is set to 2005. From the data in Chinese industrial economy statistical yearbook in 2006, the number of enterprises with the potential to implement advanced manufacturing modes was found to be $m=29,774$ . The ratio of large enterprises and medium enterprises was $a=0.08$ , and the ratio of enterprises that go bankrupt each year in large and medium enterprises was $\beta =0.12$ . Based on the above data, a parameter identification model can be obtained.

Parameter identification using an IGA

The parameters in the diffusion model in section “Parameter identification of the diffusion model” include $g$ , $p$ , $q$ , $\theta$ , $a$ , $b$ , and $\beta$ , and it remains to identify $g$ , $p$ , $q$ , and $\theta$ . To this end, an objective function should first be obtained. As the diffusion model expresses the adoption status of the advanced manufacturing mode, we will use the square of the distance between the $y_{\mathrm{ij}}$ and their real values to show the error in the estimated values. Meanwhile, the constraints of the diffusion model should be satisfied. Thus, we combine the parameters of the competitive diffusion model with the data in Table 1 to form a parameter identification model.

• 1. Objective function

$$min{\displaystyle \sum _{i=1}^6}{{\displaystyle \sum _{j=1}^3}(y_{\mathrm{ij}}-y_{\mathrm{ij}}^{*})}^2$$

• 2. Constraints

$$\{\begin{array}{cc}\multicolumn{1}{c}{}& \frac{d{x}_{i1}}{\mathrm{di}}=-\beta x_{i1}-(p_1+q_{11}\frac{\mathrm{am}-x_{i1}}{\mathrm{am}})x_{i1}-(p_2+q_{12}\frac{\mathrm{am}-x_{i1}}{\mathrm{am}})x_{i1}-(p_3+q_{13}\frac{\mathrm{am}-x_{i1}}{\mathrm{am}})x_{i1}\hfill \\ \multicolumn{1}{c}{}& \frac{d{x}_{i2}}{\mathrm{di}}=\beta x_{i1}-(p_1+g{q}_{21}\frac{\mathrm{bm}-x_{i1}}{\mathrm{bm}})x_{i2}-(p_2+g{q}_{22}\frac{\mathrm{bm}-x_{i2}}{\mathrm{bm}})x_{i2}-(p_3+g{q}_{23}\frac{\mathrm{bm}-x_{i2}}{\mathrm{bm}})x_{i2}\hfill \\ \multicolumn{1}{c}{}& \frac{d{y}_{i1}}{\mathrm{di}}=(p_1+q_{11}\frac{\mathrm{am}-x_{i1}}{\mathrm{am}})x_{i1}+(p_1+g{q}_{21}\frac{\mathrm{bm}-x_{i2}}{\mathrm{bm}})x_{i2}-{\theta }_{12}{y}_{i1}-{\theta }_{13}{y}_{i1}\hfill \\ \multicolumn{1}{c}{}& \frac{d{y}_{i2}}{\mathrm{di}}=(p_2+q_{12}\frac{\mathrm{am}-x_{i1}}{\mathrm{am}})x_{i1}+(p_2+g{q}_{22}\frac{\mathrm{bm}-x_{i2}}{\mathrm{bm}})x_{i2}+{\theta }_{12}{y}_{i1}-{\theta }_{23}{y}_{i2}\hfill \\ \multicolumn{1}{c}{}& y_{i3}=m-x_{i1}-x_{i2}-y_{i1}-y_{i2}\hfill \\ \multicolumn{1}{c}{}& m=29774,a=0.08,b=0.92,\beta =0.12,\hfill \\ \multicolumn{1}{c}{}& 1\le g\le 1.01,\hfill \\ \multicolumn{1}{c}{}& 0.2\le q_{11},q_{12},q_{13},q_{21},q_{22},q_{23}\le 0.7\hfill \\ \multicolumn{1}{c}{}& 0.0001\le p_1,p_2,p_3\le 0.01\hfill \\ \multicolumn{1}{c}{}& 0\le {\theta }_{12},{\theta }_{13},{\theta }_{23}\le 0.1\hfill \end{array}$$

The initial values of $x_{i1},x_{i2},y_{i1},y_{i2},\mathrm{and}{y}_{i3}$ are 1117; 27,271; 58; 328; and 1000, respectively

and

$$y_{\mathrm{ij}}^{*}=\left[\begin{array}{ccc}\hfill 58\hfill & \hfill 328\hfill & \hfill 1000\hfill \\ \multicolumn{1}{c}{132}& \hfill 335\hfill & \hfill 1238\hfill \\ \multicolumn{1}{c}{230}& \hfill 704\hfill & \hfill 1771\hfill \\ \multicolumn{1}{c}{370}& \hfill 1180\hfill & \hfill 2139\hfill \\ \multicolumn{1}{c}{743}& \hfill 1340\hfill & \hfill 2405\hfill \\ \multicolumn{1}{c}{1061}& \hfill 1742\hfill & \hfill 2617\hfill \end{array}\right]$$

• 3. According to the above objective function and constraints, Algorithm 1 is utilized to obtain the optimal parameter values using MATLAB 7.1. The values obtained are shown in Table 6.

Table 6.

Optimal parameter values.

Parameter	$m$	$a$	$b$	$\gamma$	$g$
Value	29,774	0.08	0.92	0.12	1.0032
Parameter	$p_1,p_2,p_3$	$q_{11}$	$q_{12}$	$q_{13}$	$q_{21}$
Value	0.0001	0.2	0.4688	0.6715	0.2700
Parameter	$q_{22}$	$q_{23}$	${\theta }_{12}$	${\theta }_{13}$	${\theta }_{23}$
Value	0.3200	0.2444	0.0756	0.0556	0.0588
Parameter	$x_1$	$x_2$	$y_1$	$y_2$	$y_3$
Value	1117	27,271	58	328	1000

Based on the data in Table 6, the number of enterprises implementing the three modes can be estimated, as shown in Table 7. Table 7 shows the results of using the diffusion model to estimate the number of enterprises implementing each advanced manufacturing mode, using the parameters identified through the GA. There is some error between the estimated and real values, but this is controlled below 15% in average, which is feasible in practice. To show the effectiveness of the model, the results were subjected to a T test, and the results are shown in Table 8. From Table 8, the confidence interval of the mean difference between the actual and predicted values includes zero, which indicates that the difference between actual and predicted values is small. A $P$ value of 0.572 indicates that the actual and predicted values are close. Thus, the parameter values obtained by the IGA are reasonable.

Table 7.

Estimated and actual numbers of enterprises implementing the three modes.

Mode	Time
	2005	2006	2007	2008	2009	2010
$A$ (real)	58	132	230	370	743	1061
Estimated	58	175	246	350	578	1066
Error	0	32.58%	6.96%	5.41%	22.21%	0.47%
$B$ (real)	328	335	704	1180	1340	1742
Estimated	328	578	717	872	1210	1864
Error	0	72.54%	1.85%	26.10%	9.70%	7.00%
C (real)	1000	1238	1771	2139	2405	2617
Estimated	1000	1390	1625	1841	2192	2861
Error	0	12.28%	8.24%	13.93%	8.86%	9.32%

Table 8.

T test results for estimated and real values.

	Sample	Mean	Standard deviation	Average standard error
Real value	18	1077	797	188
Estimated value	18	1053	780	184
Error	18	24.6	161.4	38.0
95% confidence interval	(−55.7, 104.8)
$p$	0.527

Applying strategies

The competition diffusion model and its parameter identification model disclose the diffusion characteristics and rules of the multiple-advanced manufacturing modes. In reality, enterprises can combine the principles, methods, and conclusions with their own conditions, to fully understand the developing rules of advanced manufacturing modes, and to make the best decisions concerning when to implement the advanced manufacturing mode and which mode to implement. Moreover, enterprises should understand the influence of the different parameters so as to better implement advanced manufacturing modes.

As to the strategies of government regulation, government can make corresponding strategies to change relative parameters so as to regulate the diffusion of advanced manufacturing modes, for example, to accelerate the diffusion, the following strategies can be adopted:

1. Set technological transferring and applying centers of advanced manufacturing modes. Government can promote the implementation of advanced manufacturing modes by providing relative technologies and implementation guidance for enterprises.

2. Establish demonstration enterprises so as to obtain the demonstration effect on related industries.

3. Provide preferential loans for enterprises. Preferential loans can greatly promote the implementation of advanced manufacturing modes.

Conclusion

The development of advanced manufacturing modes and technologies provides a powerful conceptual method for enterprises. In order to make rational decisions, enterprises must not only understand the philosophies and technologies of the advanced manufacturing mode, but also know the applicable diffusion rules.

In this article, we proposed parameter identification methods based on a GA to obtain a more reasonable diffusion model. The model provided a better explanation of the diffusion mechanism of the advanced manufacturing modes, thus predicting their future application more accurately. In tandem with this, a parameter analysis and identification can help enterprises and governments to understand the advanced manufacturing diffusion processes. Governments can develop appropriate policies to influence the relevant model parameters, thereby influencing the diffusion of advanced manufacturing modes.