# The Role of Industry 4.0 and Circular Economy for Sustainable Operations: The Case of Bike Industry
## 3 Model Formulation
 \tag{7}\]  
* Rework costs (\(RC\));The rework costs for ETO components and general components per cycle (denoted by \(RC\)) is given by:  
\[RC=\Big{[}\Big{(}r_{f}\theta_{f}+r_{g}\theta_{1}\Big{)}p+r_{e}\theta_{2}p_{1} \Big{]}T, \tag{8}\]
* Production costs (\(PC\)); The production costs for finished products per cycle (denoted by \(PC\)) is given by: \[PC=\Big{[}p\Big{(}c_{pf}+c_{pg}\Big{)}+c_{pe}p_{1}\Big{]}T,\] (9)  
The objective function of the proposed model consists of four parts to minimize the total cost per unit time and is given by optimizing \(t_{1}\), \(t_{4}\), and \(p\). Therefore, the total cost per unit time (denoted by \(AC(t_{1},t_{4},p)\)) is given by:  
\[ \begin{array}{ll}AC(t_{1},t_{4},p)&=\frac{1}{T+t_{4}}(SC+HC+RC+PC)\\ &=\frac{1}{T+t_{4}}\Big{\{}S+\alpha\big{[}(1-\gamma)c_{me}+\gamma c_{mg}\big{]} +\frac{1}{2}h_{f}[(p-D)T(t_{2}+t_{3}+t_{4})]\\ &+\frac{1}{2}h_{e}[(p-D)t_{1}t_{2}+(p-\theta_{1}D)t_{2}(t_{2}+t_{3})+Dt_{4}(t_ {3}+t_{4})]\\ &+\frac{1}{2}h_{g}[t_{2}^{\prime}((p_{1}-\theta_{2}D)(t_{2}+t_{3})+t_{1}(p_{ 1}-D))+Dt_{4}(t_{3}+t_{4})]\\ &+\frac{1}{2}\big{[}h_{e}(p-D)+h_{g}(p_{1}-D)\big{]}Tt_{1}\\ &+\Big{[}\Big{(}r\theta_{f}+r_{g}\theta_{1}+c_{pf}+c_{pg}\Big{)}p+\big{(}r_{e} \theta_{2}+c_{pe}\big{)}p_{1}\Big{]}T\Big{\}}.\end{array}  \tag{10}\]  
In order to solve this nonlinear programming problem, we first ignore the restriction, and take the first-order derivation of \(AC(t_{1},t_{4},p)\) with respect to \(t_{1}\), \(t_{4}\), \(p\), respectively. We obtain:  
\[\frac{\partial AC(t_{1},t_{4},p)}{\partial t_{1}}=\frac{Dt_{4}}{2(T+t_{4})} \bigg{[}(h_{e}+h_{g})\gamma-h_{e}\frac{(\beta+\gamma)(\beta-\gamma)}{p-\theta _{1}D}Dt_{4}\bigg{]}=0, \tag{11}\]  
\[ \begin{array}{ll}\frac{\partial AC(t_{1},t_{4},p)}{\partial t_{4}}&=\frac{1 }{2}h_{f}(p-D)t_{1}+\frac{1}{2}h_{e}Dt_{1}\gamma\\ &-\Big{[}\frac{(\beta+\gamma)(\beta-\gamma)D}{p-\theta_{1}D}+\frac{D(1+\beta) (1-\beta)}{p-D}+1\Big{]}h_{e}Dt_{4}\\ &+\frac{1}{2}h_{g}Dt_{1}(2\gamma+\beta)-h_{g}Dt_{4}\Big{[}\frac{1}{2}(1+\beta )+1\Big{]}=0\end{array}  \tag{12}\]  
and:  
\[ \begin{array}{ll}\frac{\partial AC(t_{1},t_{4},p)}{\partial p}&=\frac{1}{T+ t_{4}}\Big{\{}\frac{1}{2}h_{f}T(T+t_{4})\\ &-\frac{1}{2}h_{e}D^{2}t^{2}\bigg{[}\frac{(\beta+\gamma)(\beta-\gamma)t_{1}}{ (p-\theta_{1}D)^{2}}+\frac{(1+\beta)(1-\beta)}{(p-D)^{2}}\bigg{]}\\ &+\bigg{[}\Big{(}r_{f}\theta_{f}+r_{g}\theta_{1}\Big{)}T\bigg{]}\\ &+\bigg{[}\Big{(}c_{pf}+c_{pg}\Big{)}T\bigg{]}\end{array}  \tag{13}\]  
where \(k_{1}=(\beta+\gamma)(\beta-\gamma)>0,k_{2}=(1+\beta)(1-\beta)>0\) and \(k_{3}=\big{(}r_{f}\theta_{f}+r_{g}\theta_{1}+c_{pf}+c_{pg}\big{)}>0\). To find the optimal solution of \((t_{1},t_{4},p)\), let \(\partial AC(t_{1},t_{4},p)/\partial t_{1}=0,\partial AC(t_{1},t_{4},p)/ \partial t_{4}=0\), and \(\partial AC(t_{1},t_{4},p)/\partial p=0\), simultaneously. By solving these equations, the feasible solution for \(t_{4}\) should be chosen from \(0\) to \(\overset{\wedge}{t_{4}}\), where:  
\[\overset{\wedge}{t_{4}}=\frac{(p-\theta_{1}D)\big{(}h_{e}+h_{g}\big{)}\gamma}{ Dh_{e}k_{1}}. \tag{14}\]  
Thus, we can obtain the following result: once we get the value \(t_{4}\in(0,\overset{\wedge}{t_{4}})\), a corresponding position \({t_{1}}^{*}\) can be uniquely determined by the following equation:  
\[{t_{1}}^{*}={t_{4}}^{*}\frac{2\Big{\{}\Big{(}\frac{k_{1}D}{p-\theta_{1}D}+ \frac{Dk_{2}}{p-D}+1\Big{)}h_{e}+h_{g}\Big{[}\frac{1}{2}(1+\beta)+1\Big{]} \Big{\}}}{\frac{h_{f}(p-D)}{D}+h_{e}\gamma+h_{g}(2\gamma+\beta)}. \tag{15}\]The below two cases show all possible situations of \(t_{1}\): (i) \(\beta>\gamma\) and (ii) \(\beta<\gamma\). **Case 1: (With I4.0 technology) \(\beta>\gamma\)**  
In this case, customers place more order products, they typically select or identify parameters to meet their requirements. ETO components are critical to have a robust and flexible production system in place to help manage that complexity. The key result is the demonstration of the concept of 14.0-enabled monitoring through low-cost devices.