## 3 Mathematical Model
### Decision Variables
\[\frac{\beta_{ij\#k}}{V_{ij\#k}}l\leq\textit{Maximum}\ S_{b}\qquad i=1,2, \ldots,I\ y=1,2,\ldots,Y\ j=1,2,\ldots,J\ f=1,2,\ldots,F\ k=1,2,\ldots,K \tag{2}\]  
\[\frac{\gamma_{ij\#k}}{v_{ij\#k}}l\leq\textit{Maximum}\ S_{b}\qquad i=1,2, \ldots,I\ y=1,2,\ldots,Y\ j=1,2,\ldots,J\ f=1,2,\ldots,F\ k=1,2,\ldots,K \tag{3}\]  
\[\beta_{ij\#k}\leq\textit{Maximum}\ \beta_{ij\#k}\qquad i=1,2,\ldots,I\ y=1,2, \ldots,Y\ j=1,2,\ldots,J\ f=1,2,\ldots,F\ k=1,2,\ldots,K \tag{4}\]  
\[\gamma_{ij\#k}\leq\textit{Maximum}\ \gamma_{ij\#k}\qquad i=1,2,\ldots,I\ y=1,2, \ldots,Y\ j=1,2,\ldots,J\ f=1,2,\ldots,F\ k=1,2,\ldots,K \tag{5}\]  
\[\frac{60W_{ij\#k}\beta_{ij\#k}}{V_{ij\#k}}-4\geq D_{ij\#k}\quad i=1,2,\ldots, I\ y=1,2,\ldots,J\ f=1,2,\ldots,J\ f=1,2,\ldots,F\ k=1,2,\ldots,K \tag{6}\]  
\[\frac{V_{ij\#k}}{\beta_{ij\#k}}=\frac{v_{ij\#k}}{\gamma_{ij\#k}}\qquad\that the number of cups processed within a given time must be greater than or equal to the total required number of cups of required volumes of yogurt and flavor. Finally, the Equation (7) satisfy that the filling times of yogurt and flavor in a cup of required volume should be equal.  
In order to avoid congestion of cups on the conveyor belt, the solution of the proposed model results in five equal times, namely the traveling time from placement point on the conveyor belt to yogurt filling point, yogurt filling time, traveling time from yogurt filling point to flavor filling time, flavor filling time, and traveling time from the flavor filling point to the leaving point on the conveyor belt.  
The objective function, as well as the constraints, is linear in the decision variables; hence, the model falls in the domain of linear programming and can be solved in an adequate amount of time. Nonlinear modeling was avoided in the construction of the model. However, by the use of approximation techniques, nonlinear models are converted into linear ones and can be solved using non-exact/metaheuristic approaches, but it requires relatively higher computational time.  
Each cup goes through different states, starting from placement on the conveyor belt as an empty cup to leaving the conveyor belt as a completely filled cup with required volumes of yogurt and flavor. The various times related to the process are arrival waiting time, transportation time, filling time, and total processing time. The arrival waiting time is the time in which an empty cup is waiting in queue for placement at the starting point on the conveyor belt to start moving towards the yogurt valve for filling. The transportation time is the time taken by an empty cup in moving from placement point on the conveyor belt to yogurt filling point, or the time taken by a yogurt filled cup in moving from yogurt filling point to the flavor filling point, or the time taken by a yogurt- and flavor-filled cup in moving from the flavor filling point to the leaving point on the conveyor belt. The yogurt and flavor filling time is the time consumed with filling a cup with yogurt and flavor of required volumes, respectively.  
The waiting time in the queue of \(n^{\text{th}}\) empty cup \(E_{n}\), the processing time of a cup \(P\), and the waiting time for \(n^{\text{th}}\) completely filled cup \(F_{n}\) can be written in the form of equations in terms of the required volume of yogurt and yogurt filling rate. Equations (8) and (9) show the waiting time for the \(n^{\text{th}}\) empty cup in the queue for placement on the starting point at the conveyor belt to start moving towards the yogurt filling point and the total processing time of a cup, respectively, while Equation (10) is used to find the waiting time for \(n^{\text{th}}\) completely filled cup.  
\[E_{n}=(n-1)\frac{V_{yj/k}}{\beta_{iyj/k}} i=1,2,\dots,I y=1,2,\dots,Y j=1,2,\dots,J f=1,2,\dots,F k=1,2,\dots,K \tag{8}\] \[P=5\frac{V_{yj/k}}{\beta_{iy/k}} i=1,2,\dots,I y=1,2,\dots,Y j=1,2,\dots,J f=1,2,\dots,F k=1,2,\dots,K\] (9) \[F_{n}=(n+4)\frac{V_{yj/k}}{\beta_{iy/k}} i=1,2,\dots,I y=1,2,\dots,Y j=1,2,\dots,J f=1,2,\dots,F k=1,2,\dots,K \tag{10}\]  
The benefits of the proposed model are: the solution of a problem takes less computational time due to linearity, it is used for throughput maximization purposes, and it is used for size selection of the flavor valve.