# The Role of Industry 4.0 and Circular Economy for Sustainable Operations: The Case of Bike Industry
## 3 Model Formulation
In light of these concerns, this case study has led to the following observations: (1) with a simulation model, companies must be automated and measure the impact of technological tools on productivity and variability without upsetting daily production activities; (2) the company uses real-time to power decision making, resulting in the total cost per unit time shrinking by 2.9%. An important area for future research in the years to come will be to investigate to which degree and for which types of products, materials, and personalization is expectedly profitable. Conceptualization, Y.-F.H. and C.-Y.S. ; methodology, C.-Y.S. ; software, M.-W.W., H.-F.H. and I.-S.L. ; validation, C.-Y.S., Y.-F.H. and M.-W.W.; formal analysis, M.-W.W.; investigation, C.-Y.S. ; resources, M.-W.W.; data curation, M.-W.W. and H.-F.H. ; writing--original draft preparation, C.-Y.S., M.-W.W., I.-S.L. and Y.-F.H. ; writing--review and editing, C.-Y.S., M.-W.W. and Y.-F.H. ; visualization, Y.-F.H. ; supervision, Y.-F.H. ; project administration, Y.-F.H. All authors have read and agreed to the published version of the manuscript. This research was funded by Chaoyang University of Technology, project no. TCJ-111J940. Not applicable. Figure 7: Design (\(t_{1}\)) and sales promotions time (\(t_{4}\)) without I4.0. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** Not applicable. **Acknowledgments:** The researchers would like to thank Chaoyang University of Technology for funding publication of this project, LUNGE INDUSTRY CO., LTD. (LASCO) for technical support (e.g., materials used for experiments). **Conflicts of Interest:** The authors declare no conflict of interest. ## Appendix A

In order to gain a better understanding of CE issues through I4.0, the same notation in the production system. order sufficient conditions (SOSC) for a minimum value. Taking the second partial derivatives of \(AC(t_{1},p|t_{4})\) with respect to \(t_{1}\) and \(p\), respectively, yields. \[\mathrm{H}_{11}=\frac{\partial^{2}AC(t_{1},p|t_{4})}{\partial t_{1} ^{2}}\bigg{|}_{(t_{1},p)=(t_{1}^{*},p^{*})}=\frac{Dt_{4}}{2(T+t_{4})^{2}}\bigg{[} \big{(}h_{e}+h_{g}\big{)}\gamma+h_{e}\frac{k_{1}}{p-\theta_{1}D}Dt_{4}\bigg{]}>0,\]

\[\mathrm{H}_{22}=\left.\frac{\partial^{2}AC(t_{1},p|t_{4})}{\partial p^{2}} \right|_{(t_{1},p)=(t_{1}^{*},p^{*})}=h_{e}D^{2}t^{2}4\bigg{[}\frac{k_{1}t_{1} }{(p-\theta_{1}D)^{3}}+\frac{k_{2}}{(p-D)^{3}}\bigg{]}>0,\]

and

\[\mathrm{H}_{12}=\mathrm{H}_{21}=\left.\frac{\partial AC(t_{1},p|t_{4})}{ \partial t_{1}\partial p}\right|_{(t_{1},p)=(t_{1}^{*},p^{*})}=0.\]

Therefore, we can determine the nature of stationary point \((p^{*},{t_{1}}^{*})\) by considering value of Hessian matrix and whether Hessian is negative definite. \[\mathrm{det}(H) =\mathrm{H}_{11}\times\mathrm{H}_{22}-\mathrm{H}_{12}\times \mathrm{H}_{21}\] \[=\left[\big{(}h_{e}+h_{g}\big{)}\gamma+\frac{h_{e}k_{1}Dt_{4}}{p -\theta_{1}D}\right]\times\left[\frac{h_{e}D^{2}t_{2}^{2}k_{1}t_{1}}{(p- \theta_{1}D)^{3}}+\frac{h_{e}D^{2}t_{2}^{2}k_{2}}{(p-D)^{3}}\right]>0.\]

2. From Equation (11), for \(t_{1}\in\big{[}\hat{t}_{1},\infty\big{)}\), we obtain that \(\frac{\partial AC(t_{1},t_{2},p)}{\partial t_{1}}>0\). This implies that a large value of \(t_{1}\) causes a higher value of \(AC(t_{1},t_{4},p)\). Hence, the minimum value of \(AC(t_{1},t_{4},p)\) occurs at the point \(t_{1}^{*}=t_{1}^{L}\). It seems reasonable to conclude that the production system will not be opened. This completes the proof. \(\square\)

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