## 4 Solution Procedure  
The adopted solution procedure is divided into a few stages, which are shown in Figure 3. As a set of orders is received, all values of parameters are evaluated, and the different percentages of yogurt and flavors for cups of required different volumes are determined. All values of parameters are then used as columns in a linear programming model, and the problem is solved using the simplex algorithm. The required time, required volumes of yogurt and flavors, and all other measures are calculated by solving the problem optimally. The results are then shared and documented, and the filling of cups with required volumes of yogurt and flavor is started. For illustration purposes of the solution procedure, a small problem is solved concerned with the filling of one type of yogurt and three different types of flavors. The flavors are strawberry, blueberry, and mango, represented by the numbers 1, 2, and 3 in the mathematical model, respectively, while the yogurt is represented by the number 1 only. In a cup filled with the required volume of yogurt, only one flavor is then filled over the yogurt. The different flavors are also filled at a single point located on the conveyor belt. The total length of the conveyor belt is considered 90 cm and is divided into three equal parts. The three parts are length from placement point to yogurt filling point, length from yogurt filling point to flavor filling point, and the length of flavor filling point to the leaving point on the conveyor belt. Hence, each of these lengths is equal to 30 cm. The maximum speed of the conveyor belt is considered as 10 cm/sec, and the maximum feed rates of the yogurt and flavor valve are 50 mL/sec and 25 mL/sec, respectively. A set of orders is described in Table 1.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Order No. ** & **Volume (mL)** & **Yogurt (\%)** & **Flavor (\%)** & \(D_{y\text{/  
Some constraints must be considered during the filling process. Operational constraints include the imposition of minimum and maximum volumes of cups to optimize the use of the machinery available and the satisfaction of the customer specification of minimum and maximum volumes of yogurt and flavors. Technological constraints include the minimum and maximum sizes of a cup, the limited number of flavors, minimum and maximum speeds of conveyor belts, and minimum and maximum feed rates of the yogurt and flavor filling valves. In order to solve the problem optimally, a linear programming model was developed. The objective of the model is to maximize the yogurt and flavor filling rates and hence the speed of the conveyor belt for the purpose of maximizing the throughput. Equation (11) shows the objective function of throughput maximization. \[\text{Maximize}:\;Z=30\Big{(}\frac{\beta_{11111}}{279}+\frac{\gamma_{11111}}{ 21}+\frac{\beta_{21232}}{270}+\frac{\beta_{21313}}{30}+\frac{\beta_{31313}}{540 }+\frac{\gamma_{31313}}{60}+\frac{\beta_{41424}}{475}+\frac{\beta_{51325}}{855} +\frac{\beta_{51352}}{45}+\frac{\beta_{51686}}{837}+\frac{\gamma_{61686}}{63} \Big{)} \tag{11}\]  
The constraints of the model limit the maximum speed of the conveyor belt to less than or equal to 10 cm/sec. Moreover, the maximum feed rates of yogurt and flavor values are restricted to 50 mL/sec and 25 mL/sec, respectively. The set of customer orders includes demand for different volumes of yogurt and flavors and different waiting times for cups. The constraints in order 1 can be written as:  
\[30\frac{\beta_{11111}}{279}\leq 10 \tag{12}\]  
\[30\frac{\gamma_{11111}}{21}\leq 10 \tag{13}\]  
\[\beta_{11111}\leq 50 \tag{14}\]  
\[\gamma_{11111}\leq 25 \tag{15}\]  
\[\frac{\beta_{11111}}{279}=\frac{\gamma_{11111}}{21} \tag{16}\]  
\[\frac{\beta_{11111}}{279}=\frac{\gamma_{11111}}{21} \tag{17}\]  
Similar constraints are written for the remaining orders from customers, and the constraints of the model are solved simultaneously to find the optimal values of all decision variables satisfying all constraints and maximizing the throughput.