# The Role of Industry 4.0 and Circular Economy for Sustainable Operations: The Case of Bike Industry
## 3 Model Formulation
While the main supported functionality of the presented case supports the key constituents of a condition monitoring data process chain, from data acquisition and signal pre-processing, to detection, diagnosis, and prediction, the prime focus of this study has not been to develop a new approach for such data processing, but showcase that such processing can be delivered through low-cost IoT-enabled simple architecture. **Case 2: (Without I4.0 technology) \(\beta<\gamma\)**  
In this case, one way to support the design process of ETO building components is to use a coherent platform model that focuses on the reuse of engineering assets, which is more of the skills and knowledge to accomplish higher efficiency during development. Obviously, from Equations (14) and (15), \(t_{1}\) and \(t_{4}\) can be uniquely determined as functions of \(p\). Then, if we are trying to find the optimal solution of \((t_{1},t_{4},p)\), a reasonable thing is to substitute \(t_{1}\) and \(t_{4}\) given by Equations (14) and (15) into Equation (13), and then obtain:  
\[ \begin{array}{rl}L(p^{*})&=\frac{(p^{*}-\theta_{1}D)\left(h_{e}+h_{g}\right) \gamma h_{f}}{Dh_{e}k_{1}}+2k_{3}\\ &-\frac{1}{h_{e}}\left[\frac{(p^{*}-\theta_{1}D)\left(h_{e}+h_{g}\right)}{k_{1 }}\right]^{2}\left[\frac{k_{1}t_{1}}{(p^{*}-\theta_{1}D)^{2}}+\frac{k_{2}}{(p^ {*}-D)^{2}}\right].\end{array}  \tag{16}\]  
Taking the first-order derivative of \(L(p)\) with respect to \(p\):  
\[\frac{dL(p)}{dp}=\Bigg{\{}\frac{\theta_{1}\big{(}h_{e}+h_{g}\big{)}\gamma h_{ f}}{h_{e}k_{1}}-\frac{2}{h_{e}}\Bigg{[}\frac{\big{(}h_{e}+h_{g}\big{)} \gamma}{k_{1}}\Bigg{]}^{2}(k_{1}t_{1}{}^{*}+k_{2}\theta_{1})\Bigg{\}}.\]  
It is not difficult to understand that \(L(p)\) is a strictly decreasing function during \(p\in\big{[}p^{L},\infty\big{)}\). To that end, the following results were posed:  
**Theorem 1**. : _For any given \(t_{1}\geq 0\),_  
_(a) If_ \(L(p)<0\)_, then the solution_ \((t_{1}^{*},t_{4}^{*},p^{*})\) _which minimizes_ \(AC(t_{1},t_{4},p)\) _not only exists but also is unique, and_ \(p^{*}\in(0,\infty)\)_._  
_(b) If_ \(L(p)\geq 0\)_, then optimal value of_ \(p\) _is_ \(p^{*}\to 0\)_. The production system should not be opened._  
The proof of this theorem is given in Appendix A.  
Summarizing the above results, we obtain the following Algorithm 1. ```
1:STEP 1: Start with \(i=1\) and \(t_{4j}\);
2:STEP 2: Put \(t_{4j}\) into Equation (15) to obtain the corresponding value of \(t_{1}\) and then from Equation (16) to calculate \(L(p)\)
3:STEP 3: If \(L(p)<0\), put \(\tilde{t}_{1}\) into Equation (16) to obtain the corresponding value of \(p\), i.e., \(\widetilde{p}_{j,\tau+1}\) otherwise, let \(\widetilde{p}_{j}=0\);
4:STEP 4: If the difference between \(p_{j,\tau}\) and \(p_{j,\tau+1}\) is sufficiently small, set \(\widetilde{p}_{j}=p_{j,\tau+1}\). Otherwise, set \(p_{j,\tau+1}=p_{j,\tau}+\varepsilon\), where \(\varepsilon\) is any small positive number, and set \(\tau=\tau+1\); then, go back to STEP 2;
5:STEP 5: Substitute \(t_{1}=\widetilde{t}_{1j}\) and \(p_{j}=\widetilde{p}_{j}\) into Equation (10) to calculate the value of \(AC(t_{1},t_{4},p)\). The objective is to determine the optimal the number of shipments, lot size per shipment and capital expenditure that minimizes the joint total expected cost per unit time of the integrated supply chain. ## 4 Application Example

In this section, we use one case study to show how CE can be used to help in the explaining of the proposed model. By means of two numerical examples and sensitivity analysis, the study has collected rich data which enable descriptions of managerial insights of the EPQ model with CE. ### CE in the Context of a Bike Company

In the following, we introduce a well-established OEM bike company in Taiwan. [PERSON] et al. [44] proposed that the successful design of a folding bike should take into account the function, material properties, and fabrication. It is essential to include all key stakeholders in the process design.