## 6 Conclusions  
In this article, a mathematical model was developed for Industry 4.0 based yogurt filling machine in the Laboratory of Computer Integrated Manufacturing, Industrial Engineering Department, King Saud University, Saudi Arabia. The objective of the model is to maximize the speed of the conveyor belt while satisfying the constraints of the maximum speed of the conveyor belt, maximum feed rates of yogurt and flavor valves, and the availability of orders at a given time. The model was solved optimally by the data of a set of orders from customers. The results were then considered for single-dimensional priority rules, and the average values of different performance measures were found for decision purposes. From the presented research, pertinent conclusions drawn are the following:  
1. The feed rate of the yogurt valve is the controlling and main decision variable. The speed of the conveyor belt, the feed rate of the flavor valve, the waiting and processing times, and all three performance measures depend on the feed rate of the yogurt valve. The speed of the conveyor belt is maximized by maximizing the feed rate of the yogurt valve, and hence the throughput is increased. Moreover, the feed rate should be within a safe limit to avoid splashes;
2. The change in length of the conveyor belt affects the speed of the conveyor belt, but as long as the feed rate of the yogurt valve is constant, the feed rate of the flavor valve, the waiting and processing times, and all three performance measures remain unchanged. As the distance between any two points on the conveyor belt is decreased, the speed of the conveyor belt decreases. The reason is that the mathematical model allows a cup to reach from one point to another point in a time equal to the yogurt of flavor filling time;
3. Normally, the percentage of yogurt in a cup is close to 100, and the model allows equal yogurt and flavor filling time; hence, a flavor valve of a smaller size is required in the system. The size of the flavor valve depends on the percentage of flavor in a cup;
4. The results of the model were used in single-dimension rules, and three performance measures were calculated. The choice of the management is to value the average minutes past due (keeping promises to customers) the most. In case of changing the feed rate of the yogurt valve and keeping the length of the conveyor belt constant, the EDD rule resulted in lower values of average minutes past due, while in case of changing the length of the conveyor belt and keeping the feed rate of yogurt valve constant, the SPT rule resulted in lowest value of average minutes past due. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Performance Measure** & **EDD** & **SPT** & **FCFS** \\ \hline Average flow time (minutes) & 26.19 & 24.65 & 27.82 \\ Average minutes early (minutes) & 0.29 & 1.73 & 2.81 \\ Average minutes past due (minutes) & 10.99 & 10.88 & 15.13 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: The effect of change in length of conveyor belt on performance measures. In the future, an automatic yogurt filling system with all nozz (yogurt valve and flavor valves) at a single point will be compared with the existing system to establish which one provides the best desired results. Conceptualization, B.S. and R.K.; methodology, M.R., B.S. and R.K.; software, R.K.; validation, M.R., R.A. and W.S. ; formal analysis, W.S. ; investigation, R.K.; resources, B.S. ; data curation, R.K.; writing--original draft preparation, R.K.; writing--review and editing, W.S. and R.K.; visualization, R.A.; supervision, R.A.; funding acquisition, B.S. All authors have read and agreed to the published version of the manuscript. This study received funding from King Saud University, Saudi Arabia, through researchers supporting project number (RSP-2021/145). Additionally, the APCs were funded by King Saud University, Saudi Arabia, through researchers supporting project number (RSP-2021/145). Not applicable. The data presented in this study are available on request from the corresponding authors.