# Developing an I4.0 Cyber-Physical System to Enhance Efficiency and Competitiveness in Manufacturing
## 4 Case Study--A Cold-Chain Insulated Packaging Manufacturing Process
### Material Consumption Estimation  
The CPS needs to estimate the time for any one of the raw materials, i.e., bubble wrap, tape, and branding label, to be almost depleted, and then issue a warning to the human operator at the right time.  
Figure 2 provides an overview of the estimation problem. The radius of the core \(r_{\text{core}}\) is a known constant. The radius of the roll \(r_{\text{roll}}(t)\) is the function of time \(t\) and its value at the beginning of the process, i.e., \(r_{\text{roll}}(0)\) is given. The height of a single layer of the materials \(h\) and the feed rate \(v\) of the material are assumed to be unknown constants. The estimation process involves estimating \(h\) based on \(r_{\text{roll}}(t)\) and \(v\) measured from sensors. Then, the remaining length of material on the roll \(l_{\text{remain}}\) is estimated based on \(t\), \(v\), and \(r_{\text{roll}}(t)\). The time for the material to be completely depleted is then estimated based on \(v\) and \(l_{\text{remain}}\). Details on the approximation are elaborated as follows.  
Suppose a roll of material has been fed into the machine with a constant rate of \(v\) for \(\tau\) amount of time; therefore, the radius of the roll has been changed from \(r_{\text{roll}}(t)\) to \(r_{\text{roll}}(t+\tau)\), where \(r_{\text{roll}}(t+\tau)\leq r_{\text{roll}}(t)\). The change in the cross-sectional area (CSA) of the roll, \(\Delta A_{\text{roll\_CSA}}\), is expressed as  
\[\Delta A_{\text{roll\_CSA}}=\pi[r_{\text{roll}}^{2}(t)-r_{\text{roll}}^{2}(t+ \tau)] \tag{1}\]  
To ease understanding, an illustration of \(\Delta A_{\text{roll\_CSA}}\) and its relationships to other parameters is provided in Figure 3.  
With the length of material \(l\) being fed into the machine approximated as \(v\tau\), the height of a single layer of material, \(h\), can be estimated as  
\[h\approx\frac{\Delta A_{\text{roll\_CSA}}}{v\tau}=\frac{\pi[r_{\text{roll}}^{2} (t)-r_{\text{roll}}^{2}(t+\tau)]}{v\tau} \tag{2}\]  
The estimation of \(h\) allows the system to continuously monitor the variations in material thickness throughout the process and reduce human interventions in measuring and entering that manually.  
With \(r_{\text{roll}}(t)\) and \(r_{\text{core}}\), the average length of material \(l_{\text{avg}}\) to be unwound from the roll per revolution is expressed as  
\[l_{\text{avg}}=\frac{2\pi(r_{\text{roll}}(t)+r_{\text{core}})}{2}. \tag{3}\]  
Using \(h\), \(r_{\text{roll}}(t)\), and \(r_{\text{core}}\), the number of layers on the roll can be approximated as  
\[n_{\text{layer}}\approx\frac{r_{\text{roll}}(t)-r_{\text{core}}}{h}. \tag{4}\]  
Figure 2: An overview of the material unwinding process.  
Therefore, \(l_{\text{remain}}\) can be approximated as  
\[l_{\text{remain}}\approx n_{\text{layer}}l_{\text{avg}}. \tag{5}\]  
An illustration of this approach is shown in Figure 4. Apparently, this approximation is overestimated as it assumes a constant radius for each layer, an even thickness of the material, and no air gaps between layers on the roll. However, the benefit of such an approach is its simplicity in implementation and low demand for computational power.