Filling Process Optimization of a Fully Flexible Machine through Computer Simulation and Advanced Mathematical Modeling

Abstract

It is possible to optimize the yogurt and flavor filling process through a fully flexible machine that can accommodate different types of yogurt and flavors, allowing for rapid adjustment of filling parameters such as volume, speed, and feed rate. Previously, researchers focused on developing a yogurt filling machine and presented their findings across varied machine configurations. The contribution of this study comprises two key elements: configuring the machine to achieve full flexibility, wherein yogurt and any flavor can be filled at any designated filling station, and devising a novel mathematical model to optimize the newly configured machine settings. A real-life problem within the context of yogurt filling has been solved using the proposed model and results have been compared with the previously published models. It has been found that the proposed model for the fully flexible machine settings outperformed the previously published models, achieving a significant margin of improvement.

1. Introduction

Optimizing the filling process of machines by customizing the machine settings allows the process to be optimized to achieve optimal results. In this approach, machine settings are adjusted to suit the specific needs of the product filling and the production line on which the machine operates. By utilizing fully flexible machine settings, manufacturers can improve the filling process’s accuracy, efficiency, and consistency, leading to significant cost savings, minimizing the processing time and improved product quality. With machines capable of filling containers of different volumes, the machine’s settings may be adjusted to ensure that each container is filled correctly and efficiently. As a result, there can be significant reductions in product waste, an increase in productivity, and an increase in profitability. Optimizing the filling process through fully flexible machine settings can enhance the efficiency and quality of the manufacturing process while reducing costs [1].

An objective of the parallel machine scheduling is to minimize the makespan or the time it takes to complete all jobs by simultaneously scheduling jobs on multiple identical machines. J. Lee et al. [2] worked on the makespan minimization of the parallel machine scheduling problem and to obtain optimal solutions, a mathematical model was presented. A NP-hard problem was considered. First, a feasible solution was obtained and the solution was then improved in a constructive way. A real problem from industry was considered to test the performance of the proposed algorithm and was found very efficient for most practical cases. To minimize two objectives simultaneously, S. Wang et al. [3] considered a scheduling problem of identical parallel machines. The two objectives considered in the problem were the minimization of the makespan and the energy. The reasons for working on these objectives were that a minimum makespan results in high utilization of the machines and energy being the major portion of the total cost of the manufacturing company. The contributions of the research include the derivation of a bi-objective scheduling problem of identical parallel machines from a real-world manufacturing company, adaption of an augmented ε-constraint method and comparison and validation of the performance of the proposed method. D. Hu et al. [4] studied a scheduling problem on parallel machines and the problem formulated as a mixed-integer linear programming model with the objective to minimize the makespan. Two genetic algorithms were developed and their performances were then evaluated. While implementing genetic algorithm, the results showed that a greedy assignment scheme works better than a random assignment scheme. S. Ozpeynirci et al. [5] worked on the integration of scheduling and tool assignment problems through mixed-integer programming approach with the objective to minimize the makespan. A tabu search algorithm was developed for finding near-optimal solution in a reasonable computational time instead of NP-hard which requires extremely higher computational time with the increase in the problem size. E. Canakoglu et al. [6] proposed a new mathematical model for resource constrained parallel machine problem with additional covering constraints and with the objective to minimize makespan. The proposed methodology efficiently resulted in a high number of jobs as compared to the number of machines. The proposed tabu search algorithm resulted in a balanced workload distribution over the employees and attained low mean absolute deviation value.

A combinatorial optimization procedure seeks to find the best solution from a finite number of potential solutions. A set of constraints is typically described by a combinatorial structure optimized according to the objective function. A combinatorial optimization problem is finding the optimal combination of elements that optimizes the objective function. A comprehensive introduction to combinatorial optimization has been provided by B. Korte et al. [7], which discusses both the theoretical foundations and practical algorithms for solving optimization problems. A new crossover operator for genetic algorithms has been proposed by Arram et al. [8]. The operator is intended to improve the performance of genetic algorithms in the context of combinatorial optimization. Using graph-based neural networks and combinatorial optimization, Gannouni et al. [9] developed a novel approach to production scheduling. Using a real-world production scheduling problem, the method’s performance was assessed. Compared with traditional scheduling methods, the proposed approach significantly reduces setup waste and generates feasible schedules that meet production requirements. Penna et al. [10] proposed modeling the timetable scheduling problem with job shop scheduling techniques. They explained how to formulate the problem as a job shop scheduling problem, where events and activities are considered jobs, and time slots and locations are considered machines. Job-shop scheduling was presented by El-Kholany et al. [11] as one of the most well-known combinatorial optimization problems, which involves the allocation of resources and sequencing of operations to minimize the makespan or time needed to complete the work. Several factors were highlighted, including the complexity of the problem and the necessity for effective problem-solving methods.

Several factors and variables must be considered to optimize the filling process, such as the type of container, product characteristics, filling equipment, packing procedures, and operational parameters that affect the process. A study by Kopanos et al. [12] highlighted the difficulty of production scheduling in food processing due to multiple product types, limited resources, processing time constraints, and interdependencies. It emphasized the need for mathematical models and algorithms. Wang et al. [13] proposed a mathematical model and a mixed-integer linear programming (MILP) formulation could be developed to achieve the shortest possible makespan about parallel-batching machines as well as non-identical job sizes. Chen et al. [14] presented an approach to optimize factor settings in a pharmaceutical filling process, identifying the most critical factors and determining their optimal settings to optimize efficiency and quality. Based on the integrated nature of the problem, Ferreira et al. [15] investigated how to simultaneously determine the production quantity for different flavors of soft drinks and schedule production operations to meet customer demands while considering various constraints. Several optimization opportunities were identified and potential strategies were proposed by H. Wang et al. [16] for enhancing the performance of the system. It was concluded that automated drug-filling systems require modifications to their hardware, software, workflow, and operational procedures to be optimized.

As part of the operations management and production planning, scheduling and sequencing are closely related concepts. To maximize resource utilization, meet deadlines, and reduce workflow overhead, these activities aim to establish the order and timing of tasks or activities. Among the essential factors of efficient scheduling are reducing production costs and improving overall operational effectiveness, as highlighted by Strohhecker et al. [17]. Heuristics have been proposed for loading and sequencing to minimize the total completion time, each with algorithmic logic and decision rules. In a complex environment such as an industrial setting, G. Da Col et al. [18] acknowledged that job shop scheduling is a complex problem that must be addressed to maximize resource utilization, minimize job completion times, and increase productivity. Baldo et al. [19] presented an optimization approach for scheduling problems. They developed a mathematical model that incorporated the various constraints and objectives of the brewery industry, including production capacity, variations in demand, and storage constraints. As demonstrated by the experimental results, the optimization approach effectively solved the brewery industry’s lot sizing and scheduling problems. Basso et al. [20] formulated and solved the bottling scheduling problem in the wine industry heuristically. It was believed that the findings contributed to improving scheduling processes, resource utilization, and operational efficiency in the wine industry. A scheduling model was proposed by Niaki et al. [21] to optimize the production process and improve resource utilization in the yogurt industry to minimize production time, maximize resource utilization, and accurately deliver products promptly.

In the mathematical modelling for process optimization, S. Rezig et al. [22] proposed a new approach and the contribution includes linear optimization mathematical model for production planning and to show the importance of the mathematical models for industrial issues. To express the effectiveness of the proposed model, a problem was solved and the results were presented. Toledo et al. [23] proposed a mixed-integer linear programming model for the description of an industrial problem relevant to soft drinks. An integrated solution approach was presented as the decisions made in one stage had consequences on another stage of the problem. Using CPLEX software (v. 20.1.0), the goal was to obtain the optimal solutions within one hour of running the problem on the computer. P. Kumar et al. [24] worked on the optimization and the performance analysis of soft drink bottle filling system using the Particle Swarm Optimization (PSO) technique. Equations were derived and different effects were studied. The performance was optimized using the PSO technique when the results were discussed with managers of the plant. For the minimization of the total weighted number of tardy jobs, S. Guo et al. [25], presented job scheduling with different weights on a single machine. The problem aimed at finding a feasible schedule. A real-life problem of lot sizing and production scheduling in the beverage industry was presented by M. Samouilidou et al. [26], where the production facility considered was identified as multistage and multiproduct. The objective was to generate an optimal production schedule which can satisfy a demand. The results showed that the optimal schedule leaded to better productivity of the plant and reduced the utilization and cost on the resources of labors.

Under the “Vision 2023” program of the Kingdom of Saudi Arabia, efforts have been started for many years for the digital transformation of all sectors of the country. To compete with global challenges, it was essential to renovate and standardize manufacturing and production industries. To upgrade the Industry 4.0 knowledge of the students, B. Salah et al. [27] worked on the learning methodology of the students at the University. The proposed methodology translated the potential of the students from theoretical to applied side. It was expected to induce a robust curriculum consisting of mechatronics and digitalized instrumentation relevant to Industry 4.0. The students were involved in different activities relevant to Industry 4.0 including 3D printing and familiarity with different components of the yogurt filling machine. It can be noted that many components were fixed in the yogurt filling machine, the filling process was performed but it was not an optimized filling process. There was a need to make a mathematical model and the filling process be optimized. B. Salah et al. [28] made changes in the settings of the yogurt filling machine and the filling of yogurt and all three flavors was performed at two different points. A mathematical model was developed with the objective of minimizing the filling time or maximizing the speed of the conveyor belt subject to constraints on the maximum allowable speed of the conveyor belt and feed rates of the filling nozzles. A real-life problem was solved and optimized results were achieved. The results were then used in the one-dimensional rules to find a rule which is better than the other ones. J. Chen et al. [29] modified the machine settings and proposed the filling of yogurt and flavors from a single point. The mathematical model was slightly modified and the simultaneous filling resulted in reduced filling time when the same earlier presented problem was solved through the proposed model. The results of the model were used as input in the one-dimensional rules. The results also showed that the model in [29] is 1.05-fold faster than the model presented in [28]. Y. Cui et al. [30] further modified the machine settings and proposed dedicated filling points for each of the three flavors. Due to dedicated filling points, it was not possible to mix more than one flavor with yogurt as customer may demand. The objective function of the model in [30] was similar to that of [29]; however, the constraints were slightly different from the earlier presented models. The same problem considered in [28,29] was solved using the modified model for dedicated filling points for each flavor, resulted in 2.55- and 2.41-fold faster than the models presented in [28,29], respectively. B. Salah et al. [31] combined several concepts of Industry 4.0 and proposed a control system for system improvement and remodeling of the yogurt filling machine. A new controller called Raspberry Pi 4 Model B (Raspberry Pi Trading Ltd., Cambridge, UK) was added in the system for controlling the NFC signal. For the purpose of minimizing the human intervention, the concepts of Industry 4.0 were implemented throughout the yogurt filling machine successfully.

The existing machine in Case I [28] was filling yogurt and flavors at two different points, and the filling times of yogurt and flavors were considered equal. Due to the filling at two distinct points, there was a possibility of filling the yogurt and flavors at a single point and reducing the processing time further. Hence, Case II [29] was presented, and yogurt and flavors were filled under a unified head nozzle. The limitations of Case II [29] include the non-simultaneous filling of flavors into different cups. As customers can order assorted flavors mixed with yogurt and further reduce their waiting time, Case III [30] was introduced, with dedicated filling points for the other flavors. Case III [30] made filling the yogurt and flavors possible simultaneously, minimizing the processing time.

In this research article, the settings of the yogurt filling machine are further modified, where the yogurt and all three flavors can be filled at any of the three filling points. Through the machine settings, it was possible to fill the yogurt and any of the three flavors from any filling point in the machine and hence the system is made fully flexible. A new mathematical model has been presented with the objective to minimize the filling time subject to constraints relevant to the speed of the conveyor belt, the yogurt and flavor filling times and the idle times of yogurt and flavor filling nozzles. A real-life problem has been solved to test the model considering the customer order dependent and independent variables. It has been found that the new model produced better results than the previously published models for the yogurt filling machine.

The different sections in this article are as follows. Section 1 presents the introduction and brief literature survey of fully flexible machines, parallel machine scheduling, combinatorial optimization, filling process optimization, sequencing and scheduling, and yogurt filling machines. In contrast, Section 2 provides the problem description. The proposed mathematical model for the fully flexible parallel machines is explained in Section 3, whereas Section 4 illustrates the solution procedure by providing an example. The results and discussion appear in Section 5; the conclusion and further recommendations are written in Section 6.

2. Problem Description

To address the limitations of the existing yogurt filling machines regarding flexibility and efficiency, the newly developed system can ideally switch between multiple flavors, production volumes, and packaging sizes while maintaining a higher level of productivity. Figure 1a illustrates the unified head containing four filling nozzles, which fill predetermined volumes of base yogurt and three different flavors into the cups. The cups, initially empty, are positioned at the entry point and subsequently moved to the filling area employing a conveyor belt. Once the filling process is completed, the cups are directed to the exit point, where a robotic arm removes them from the system. Figure 1b illustrates the rear view of the machine, highlighting a panel attached with sensors, switches, and buttons. At the same time, the diaphragm pumps are engaged to facilitate the transfer of yogurt and flavors from the tanks to the filling nozzles.

The main components used in the fully flexible yogurt filling machine are yogurt and flavor filling nozzles, conveyor belt, filling point allocation system, cup detection and orientation system, flavor and yogurt supply system, automated cleaning and sanitizing system, Human–Machine Interface (HMI), Industry 4.0 technologies, quality control system, Raspberry Pi 4 Model B Controller and NFC Signal.

The machine settings were changed, and the system was fully flexible to reduce the processing time further. The complete flexibility of the machine can be achieved by fixing three unified heads working in parallel at various positions. Figure 2 shows three machines that can fill any flavor at any time. The empty cups are loaded on the conveyor belt and moved towards the filling points. Two conveyor belts work under each unified head, with two nozzles in each. The stationary conveyor belt is represented by zero, while the belt in motion is represented by 1. The unified head relates to all tanks of assorted flavors and yogurt through small pipes where large volumes of flavors and yogurt are available to fill the cups with the required quantities. The cups then move towards the exit point, where a robotic arm is used to remove them from the system.

The limitations of the b. Salah et al. [28] model include limited flexibility, inefficient use of resources and lack of optimization, while the limitations of the j. Chen et al. [29] model consist of limited simultaneous filling, inability to handle assorted flavors and limited flexibility, whereas the limitations of the y. Cui et al. [30] model comprise dedicated filling points, inability to mix flavors, limited scalability. The current proposed model overcomes these limitations by allowing filling of yogurt and any flavor at any of the three filling points, enabling simultaneous filling of yogurt and multiple flavors, optimizing filling time and conveyor belt speed, adapting to changing customer demands and assorted flavor requests and working in parallel to fill the cups with required volumes.

The fully flexible yogurt filling machine has been designed as an innovative equipment to streamline the yogurt and flavor filling process. This system consists of three unified heads, each equipped with two nozzles, enhancing flexibility and efficiency. The machine is designed to simultaneously manage various yogurt filling requirements due to the integration of multiple heads and nozzles. As given in

Table 1. All flavors and yogurt filling combinations through three unified heads (Head-I, II, and III), each with two nozzles (N-I and N-II).

Yogurt	Blueberry						Strawberry						Mango
	Head-I		Head-II		Head-III		Head-I		Head-II		Head-III		Head-I		Head-II		Head-III
	N-I	N-II	N-I	N-II	N-I	N-II	N-I	N-II	N-I	N-II	N-I	N-II	N-I	N-II	N-I	N-II	N-I	N-II
1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	1	0	0	0	0	0	1	0	0	0	0	0	1	0	0	0	0	0
	0	1	0	0	0	0	0	1	0	0	0	0	0	1	0	0	0	0
	0	0	1	0	0	0	0	0	1	0	0	0	0	0	1	0	0	0
	0	0	0	1	0	0	0	0	0	1	0	0	0	0	0	1	0	0
	1	0	1	0	0	0	1	0	1	0	0	0	1	0	1	0	0	0
	0	1	0	1	0	0	0	1	0	1	0	0	0	1	0	1	0	0
	0	0	0	0	1	0	0	0	0	0	1	0	0	0	0	0	1	0
	0	0	0	0	0	1	0	0	0	0	0	1	0	0	0	0	0	1
	1	0	0	0	1	0	1	0	0	0	1	0	1	0	0	0	1	0
	0	1	0	0	0	1	0	1	0	0	0	1	0	1	0	0	0	1
	0	0	1	0	1	0	0	0	1	0	1	0	0	0	1	0	1	0
	0	0	0	1	0	1	0	0	0	1	0	1	0	0	0	1	0	1
	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0
	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1

, all combinations allow operators to access the information quickly and efficiently they need and determine which combinations are possible. This reduces the time they need to spend thinking and makes it easier to find the desired combination.

Several advantages are associated with fully flexible yogurt filling machines, making them a desirable possibility for mixing yogurt with assorted flavors. Some benefits of using a fully flexible yogurt filling machine include improved flexibility, product variety, customization, efficiency and productivity, cost savings, and adaptability to market trends. Automation also reduces the risk of human error and increases efficiency, leading to lower production costs.

There are few benefits of the proposed machine settings. The economic benefits include increased efficiency and improved productivity while the operational benefits include flexibility and reduced idle time. Similarly, the maintenance benefits include reduced downtime, extended equipment life, simplified maintenance and reduced energy consumption. Few other benefits are improved customer satisfaction and competitive advantage in the market.

The use of fully flexible yogurt filling machines has many advantages, but they also have certain limitations. Some limitations are that they are more expensive than traditional filling machines, are complex to operate and maintain, require appropriate training and expertise from the operator, have increased maintenance requirements, had compatibility issues, and require floor space.

3. Mathematical Modelling

Considering the primary goal of minimizing the makespan of the system while deciding the optimal allocation of all jobs to the available machines, a linear programming model has been developed. A variety of constraints and equations have been taken into consideration in the development of the model. The model formulation has several essential components, including indices, parameters, and decision variables. The linear programming model allows for the planning and allocating jobs to machines based on constraints and equations. Together, these elements form the foundation for adequate scheduling and allocation of jobs.

The mathematical modeling process is divided into two stages. A model for the filling speed of conveyor belts, filling time, idle time and job processing time is developed in the first stage. The purpose of this stage is to represent and analyze the dynamics of the filling process in as much detail as possible. In the second stage, the model is intended to address the problem of parallel machines to identify and explain the optimal combinations of filling times for different types of orders. Formulating and solving the second stage allows the model to determine the most efficient schedule and allocation of resources, ultimately improving the efficiency of the filling system.

Stage I
Indices
a	volume of yogurt	a∈A
b	yogurt type	b∈B
c	volume of flavor	c∈C
d	flavor type	d∈D
e	total volume of yogurt and flavor(s)	e∈E
f	filling machine in the system	f∈F
g	belt number in a machine	g∈G
h	different types of total volumes	h∈H
i	number of machines	i∈I
j	number of jobs	j∈J
Parameters
$L_t$	total length centimeter (cm)
$l$	half of the total length of the conveyor belt centimeter (cm)
$V_{abcde}$	yogurt volume in the total volume of a cup milliliter (mL)
$v_{abcde}$	flavor volume in the total volume of a cup milliliter (mL)
$S_{\max }$	conveyor belt maximum allowable speed centimeter per second (cm/s)
$P_j$	processing time of job j on any machine second (s)
$P_{ij}$	processing time of job j on machine i second (s)
$C_i$	completion time of a set of jobs on machine i second (s)
$C_{\max }$	maximum completion time of a set of jobs assigned to any machine second (s)
$m_{fg}$	binary number used for stationary and moving states of the conveyer belts unit less
$T_b$	yogurt nozzle idle time second (s)
$T_d$	flavor nozzle idle time second (s)
$t_b$	filling time of yogurt second (s)
$t_d$	filling time of flavor second (s)
$S_a$	actual speed of the belt centimeter per second (cm/s)
$S_c$	calculated speed of the belt centimeter per second (cm/s)
$C$	a number used in filling time calculations, where $Min{P}_j\le C\le {\displaystyle \sum _{j=1}^J{P}_j}$ second (s)
Decision and Resulting Variables
$X_{ij}$	1 if machine i is used to process job j, or else 0 unit less
${\delta }_{abcde}$	yogurt valve feed rate milliliter per second (mL/s)
${\alpha }_{abcde}$	strawberry flavor valve feed rate milliliter per second (mL/s)
${\beta }_{abcde}$	blueberry flavor valve feed rate milliliter per second (mL/s)
${\gamma }_{abcde}$	mango flavor valve feed rate milliliter per second (mL/s)

The following are few equations used in the mathematical calculations of the yogurt filling process.

$$S_a=\min \left\{\min \left(l{\displaystyle \frac{{\delta }_{abcde}}{V_{abcde}}},l{\displaystyle \frac{{\alpha }_{abcde}}{v_{abcde}}},l{\displaystyle \frac{{\beta }_{abcde}}{v_{abcde}}},l{\displaystyle \frac{{\gamma }_{abcde}}{v_{abcde}}}\right),S_{\max }\right\} a\in A,b\in B,c\in C,d\in D,e\in E$$

(1)

$$t_{\delta }={\displaystyle \frac{V_{abcde}}{{\delta }_{abcde}}} a\in A,b\in B,c\in C,d\in D,e\in E$$

(2)

$$t_{\alpha }={\displaystyle \frac{v_{abcde}}{{\alpha }_{abcde}}} a\in A,b\in B,c\in C,d\in D,e\in E$$

(3)

$$t_{\beta }={\displaystyle \frac{v_{abcde}}{{\beta }_{abcde}}} a\in A,b\in B,c\in C,d\in D,e\in E$$

(4)

$$t_{\gamma }={\displaystyle \frac{v_{abcde}}{{\gamma }_{abcde}}} a\in A,b\in B,c\in C,d\in D,e\in E$$

(5)

$$S_c={\displaystyle \frac{l}{t_{\delta }}}$$

(6)

$$T_{\delta }={\displaystyle \frac{l}{S_{\max }}}-{\displaystyle \frac{l}{S_c}} \mathrm{if} {\displaystyle \frac{l}{S_{\max }}}-{\displaystyle \frac{l}{S_c}}>0 \mathrm{otherwise} 0$$

(7)

$$T_{\alpha }=\max (t_{\delta },t_{\alpha },t_{\beta },t_{\gamma })-t_{\alpha }$$

(8)

$$T_{\beta }=\max (t_{\delta },t_{\alpha },t_{\beta },t_{\gamma })-t_{\beta }$$

(9)

$$T_{\gamma }=\max (t_{\delta },t_{\alpha },t_{\beta },t_{\gamma })-t_{\gamma }$$

(10)

$${\displaystyle \sum _{f=1}^F{m}_{fg}}=1 g\in G$$

(11)

$${\displaystyle \sum _{f=1}^F{\displaystyle \sum _{g=1}^G{m}_{fg}}}=3$$

(12)

$$P_j={\displaystyle \frac{V_{abcde}}{{\delta }_{abcde}}},{\displaystyle \frac{v_{abcde}}{{\alpha }_{abcde}}},{\displaystyle \frac{v_{abcde}}{{\beta }_{abcde}}},{\displaystyle \frac{v_{abcde}}{{\gamma }_{abcde}}} a\in A,b\in B,c\in C,d\in D,e\in E$$

(13)

Equation (1) is used to find the actual speed of the conveyor belt while Equations (2)–(5) are utilized to find the filling time of the required volumes of yogurt, flavor 1, flavor 2 and flavor 3, respectively. Equation (6) is used to find the calculated speed of the conveyer belt whereas Equations (7)–(10) calculate the idle times of the yogurt, flavor 1, flavor 2 and flavor 3 nozzles, respectively. Equation (11) states that one out of the two belts in a fully flexible machine should be in motion while Equation (12) enforces that three out of the six belts in the system must be in motion at any time during the filling process. Equation (13) finds the filling times of yogurt and flavors of any order.

In Stage II, the model is used to optimize the filling process. The solution of the model results in optimal values of the filling times for each order.

Stage II

$$\mathbf{Minimize}\mathbf{:} Z=C-{\displaystyle \sum _{a=1}^A{\displaystyle \sum _{b=1}^B{\displaystyle \sum _{c=1}^C{\displaystyle \sum _{d=1}^D{\displaystyle \sum _{e=1}^E\left(\frac{V_{abcde}}{{\delta }_{abcde}}+\frac{v_{abcde}}{{\alpha }_{abcde}}+\frac{v_{abcde}}{{\beta }_{abcde}}+\frac{v_{abcde}}{{\gamma }_{abcde}}\right)}}}}}$$

(14)

Subject to the following constraints and equations:

$${\displaystyle \sum _{j=1}^J{P}_j{D}_j{X}_{ij}\le C} i\in I$$

(15)

$${\displaystyle \sum _{i=1}^I{X}_{ij}}=1 j\in J$$

(16)

$$X_{ij}\in \{0,1\} i\in I, j\in J$$

(17)

$$C_i={\displaystyle \sum _{j=1}^J{P}_j{D}_j{X}_{ij}} i\in I$$

(18)

$$C_{\max }=\underset{i=1}{\overset{I}{\max }}{C}_i$$

(19)

$${\delta }_{abcde}\le Maximum {\delta }_{abcde} a\in A,b\in B,c\in C,d\in D,e\in E$$

(20)

$${\alpha }_{abcde}\le Maximum {\alpha }_{abcde} a\in A,b\in B,c\in C,d\in D,e\in E$$

(21)

$${\beta }_{abcde}\le Maximum {\beta }_{abcde} a\in A,b\in B,c\in C,d\in D,e\in E$$

(22)

$${\gamma }_{abcde}\le Maximum {\gamma }_{abcde} a\in A,b\in B,c\in C,d\in D,e\in E$$

(23)

$${\displaystyle \frac{l}{S_a}}-{\displaystyle \frac{V_{abcde}}{{\delta }_{abcde}}}\ge 0 a\in A,b\in B,c\in C,d\in D,e\in E$$

(24)

$${\displaystyle \frac{l}{S_a}}-{\displaystyle \frac{v_{abcde}}{{\alpha }_{abcde}}}\ge 0 a\in A,b\in B,c\in C,d\in D,e\in E$$

(25)

$${\displaystyle \frac{l}{S_a}}-{\displaystyle \frac{v_{abcde}}{{\beta }_{abcde}}}\ge 0 a\in A,b\in B,c\in C,d\in D,e\in E$$

(26)

$${\displaystyle \frac{l}{S_a}}-{\displaystyle \frac{v_{abcde}}{{\gamma }_{abcde}}}\ge 0 a\in A,b\in B,c\in C,d\in D,e\in E$$

(27)

The objective function 14 linked with all constraints and equations of the model is used to minimize the processing and filling times of the cups assigned to different filling stations of the system. Constraint 15 states that the processing time of the jobs assigned to any machine in the system should be less than or equal to the total completion time. Equation (16) states that a job must be assigned to any of the machines in the system for filling while Equation (17) is used for binary restrictions. Equation (18) states that the processing time on any machine is equal to the summation of the filling times of the jobs assigned to that machine. Equation (19) shows that the makespan is equal to the maximum value of the processing times of any of the available machines used in the filling of the yogurt and flavors. Inequalities (20)–(23) restrict the feed rates to be less than or equal to the maximum allowable values of the feed rates of yogurt, flavor 1, flavor 2 and flavor 3, respectively. Inequalities (23)–(26) states that either the filling times of yogurt, flavor 1, flavor 2 and flavor 3 are equal or less than to the time in which an empty cup reaches the filling point from the entry point or a completely filled cup moves from the filling point to the exit point, respectively.

Using a linear programming model, the machine can be programmed to optimize the filling process, allowing it to allocate resources more efficiently, reduce the total time it takes to fill the cups and make better decisions in real time based on changes in demand or other factors. Furthermore, the model can be adapted to changing conditions, allowing it to ensure optimal performance. The cups can be filled with exact quantities of yogurt and flavors using automated filling machines in a much shorter time.

4. Solution Procedure

Once the set of orders from customers is received, various quantities of yogurt and flavors are combined in a cup. Typically, the yogurt content outweighs the flavor(s) by a significant margin. In this case, a minimum of 75% of yogurt can be ordered in a cup and a maximum of 25% of any flavor or combination of flavors can be ordered in a cup. Ensuring the total volume falls within the permissible upper and lower limits is essential. The upper limit of total volume of a cup is 1500 mL while the minimum limit of total volume is 250 mL.

Table 2. Orders for yogurt mixed with flavors and the percentages of yogurt and flavors.

Order No.	Total Volume of Cup (mL)	Percentage of Yogurt and Flavor(s) in the Total Volume of Cup (%)				Number of Cups
		Yogurt	Flavor I	Flavor II	Flavor III
1	1500	75	10	10	5	5
2	1500	80	0	10	10	9
3	1500	85	10	5	0	10
4	1250	85	15	0	0	10
5	1250	90	0	10	0	5
6	1250	95	0	0	5	10
7	1000	80	10	0	10	5
8	1000	85	0	10	5	8
9	1000	90	10	0	0	10
10	750	75	10	5	10	8
11	750	80	10	0	10	8
12	750	85	10	0	5	11
13	500	85	0	5	10	7
14	500	90	5	5	0	10
15	500	95	0	5	0	12
16	250	80	10	0	10	20
17	250	85	0	10	5	17
18	250	90	5	5	0	35

illustrates the demand of customers for 18 total volumes of yogurt and flavors, including the respective yogurt and flavor percentages. Notably, previous studies only involved a single flavor mixed with the yogurt, whereas the current model allows for combining multiple flavors and thus the system is fully flexible. The last column of Table 2 shows the number of cups required in each order. Flavor I, II, and III are used to represent blueberry, strawberry, and mango flavors, respectively.

Table 2 provides the parameters influenced by customer orders. In contrast,

Table 3. Customer order independent parameters.

S. No.	Parameter	Value	Unit
1	Conveyor belt maximum allowable speed	10	cm/s
2	Conveyor belt total length	50	cm
3	Maximum volume that a customer can order in a single cup	1500	mL
4	Minimum volume that a customer can order in a single cup	250	mL
5	Volume of yogurt container	300	L
6	Volume of the containers of flavors	75	L
7	Yogurt valve maximum feed rate	100	mL/s
8	Flavor valve maximum feed rate	33.34	mL/s

displays the parameters independent of customer orders. Considering the values of these parameters resolves the problem, leading to an optimal solution.

The LP problem was solved using the LP_Solve tool (v. 5.5.2.5), employing a computer with a Core i7 processor running at 1.99 GHz. The computations were conducted within a reasonable time, yielding the desired results. LP_Solve, a solver tool for MILP, was utilized for this purpose. The speed of the conveyor belt and the feed rates of the yogurt and flavor valves are directly interconnected.

Table 4. The solution of the proposed model, the values of the decision variables, the filling times of different nozzles and the processing time of cups.

Order No.	Feed Rate (mL/s)				Filling Time of Nozzle (s)				Processing Time (s)
	${\mathit{\delta }}_{\mathit{a}\mathit{b}\mathit{c}\mathit{d}\mathit{e}}$	${\mathit{\alpha }}_{\mathit{a}\mathit{b}\mathit{c}\mathit{d}\mathit{e}}$	${\mathit{\beta }}_{\mathit{a}\mathit{b}\mathit{c}\mathit{d}\mathit{e}}$	${\mathit{\gamma }}_{\mathit{a}\mathit{b}\mathit{c}\mathit{d}\mathit{e}}$	Yogurt	Flavor I	Flavor II	Flavor III
1	100	33.34	33.34	33.34	11.25	4.50	4.50	2.25	11.25
2	100	0	33.34	33.34	12.00	0.00	4.50	4.50	12.00
3	100	33.34	33.34	0	12.75	4.50	2.25	0.00	12.75
4	100	33.34	0	0	10.63	5.62	0.00	0.00	10.63
5	100	0	33.34	0	11.25	0.00	3.75	0.00	11.25
6	100	0	0	33.34	11.88	0.00	0.00	1.87	11.88
7	100	33.34	0	33.34	8.00	3.00	0.00	3.00	8.00
8	100	0	33.34	33.34	8.50	0.00	3.00	1.50	8.50
9	100	33.34	0	0	9.00	3.00	0.00	0.00	9.00
10	100	33.34	33.34	33.34	5.63	2.25	1.12	2.25	5.63
11	100	33.34	0	33.34	6.00	2.25	0.00	2.25	6.00
12	100	33.34	0	33.34	6.38	2.25	0.00	1.12	6.38
13	100	0	33.34	33.34	4.25	0.00	0.75	1.50	4.25
14	100	33.34	33.34	0	4.50	0.75	0.75	0.00	4.50
15	100	0	33.34	0	4.75	0.00	0.75	0.00	4.75
16	100	33.34	0	33.34	2.00	0.75	0.00	0.75	2.00
17	100	0	33.34	33.34	2.13	0.00	0.75	0.37	2.13
18	100	33.34	33.34	0	2.25	0.37	0.37	0.00	2.25

shows how the model optimizes the conveyor belt speed to maximize the feed rates, resulting in the highest possible values. The speed of the belt relies on the feed rates and consistently remains below the maximum allowable limit. Consequently, the calculated conveyor belt speed is essential in filling. Typically, the flavor percentages are lower than the yogurt percentage in the total volume required, causing the flavors to be filled before the yogurt. This leads to idle time for the flavor nozzles while waiting to fill the subsequent cup. On the other hand, the yogurt valves operate at full capacity mostly without any idle time. The idle time of yogurt value occurs when a cup is filled earlier than the arrival of next cup to the filling point. This happens in case when the calculated speed exceeds the maximum allowable speed of the conveyer belt. The processing times for the entire order quantity of cups are provided in the last column of Table 4.

The speed of the conveyor belt is inversely proportional to the volume of yogurt and flavor(s) in a cup. For large volumes, the speed of the belt is small where for small volumes, the speed of the belt increases. As can be seen in

Table 5. The calculated, maximum allowable and actual speed of the conveyor belt and the idle times yogurt and flavor nozzles.

Order No.	Speed of the Conveyor Belt (cm/s)			Idle Time of Yogurt and Flavors Nozzles (s)
	Sc	Smax	Sa	Yogurt	Flavor I	Flavor II	Flavor III
1	4.44	10	4.44	0.00	6.75	6.75	9.00
2	4.17	10	4.17	0.00	12.00	7.50	7.50
3	3.92	10	3.92	0.00	8.25	10.50	12.75
4	4.71	10	4.71	0.00	5.00	10.63	10.63
5	4.44	10	4.44	0.00	11.25	7.50	11.25
6	4.21	10	4.21	0.00	11.88	11.88	10.00
7	6.25	10	6.25	0.00	5.00	8.00	5.00
8	5.88	10	5.88	0.00	8.50	5.50	7.00
9	5.56	10	5.56	0.00	6.00	9.00	9.00
10	8.89	10	8.89	0.00	3.38	4.50	3.38
11	8.33	10	8.33	0.00	3.75	6.00	3.75
12	7.84	10	7.84	0.00	4.13	6.38	5.25
13	11.76	10	10.00	0.75	4.25	3.50	2.75
14	11.11	10	10.00	0.50	3.75	3.75	4.50
15	10.53	10	10.00	0.25	4.75	4.00	4.75
16	25.00	10	10.00	3.00	1.25	2.00	1.25
17	23.53	10	10.00	2.88	2.13	1.38	1.75
18	22.22	10	10.00	2.75	1.88	1.88	2.25

, the calculated speed increases as the volume decreases. The speed of the belt can be increased till the maximum allowable limit which is 10 cm/s. The actual speed is equal to the maximum speed of the belt in case when the calculated speed is exceeding the maximum allowable speed. There is no idle time of yogurt nozzle when the speed of the belt is less than the maximum speed limit. When the speed of the belt is less than the maximum limit, an empty cup reaches in a time equal to the filling time of the cup under the nozzle. When the speed of the belt exceeds 10 cm/s, an empty cup reaches in a time more than the filling time of the cup under the nozzle and thus the yogurt nozzle remains idle till the empty cup reaches under the nozzle. The volume of flavor is very less than the yogurt volume, and the flavor is normally filled earlier than the yogurt, thus once the required volume of flavor is filled then the flavor nozzle remains idle till the next cup reaches for filling.

Since each machine is fully flexible, thus any ordered cup can be fulfilled using any of the available machines. In the parallel machine problem (P₃||C_max), it is essential to minimize customer waiting time and deliver the complete set of orders simultaneously by ensuring that the processing time combinations on the different available machines are almost similar. Figure 3 illustrates the processing of orders across the three available parallel working machines, resulting in the total processing times of machines 1, 2, and 3 as 417.000 s, 416.625 s, and 416.875 s, respectively. The total work content amounts to 1250.500 s on the three machines, with an average machine load of 416.833 s and a makespan value (C_max) of 417.000 s.

At any machine, the first empty cup when reaches to the filling point, it takes 5 s to reach. As there is no filling started, the conveyor belt moves over the length of 50 cm of the belt with maximum allowable speed (10 cm/s). Similarly, when all cups are filled, the last cup moves with maximum speed to the leave the system. Equation (28) is used for this relation and 5 s at the start and end used at each machine.

$$P_{ij}={\displaystyle \frac{l}{S_{\max }}}+{\displaystyle \sum _{j=1}^J{P}_j}+{\displaystyle \frac{l}{S_{\max }}} i\in I$$

(28)

5. Results and Discussion

In this section, two types of analysis have been performed. The first one is to change the feed rate of the yogurt valve from 100 mL/s, with an increment of 25 units, to 200 mL/s and check its impact on the yogurt filling time, the speed of the belt, the idle time of yogurt nozzle and the filling time of yogurt nozzle. In the second type of analysis, the processing times for similar orders of the proposed and previously published models are compared, and the differences among them are shown.

5.1. The Effect of Change in the Feed Rate of the Yogurt Valve on the Filling Time of Yogurt in a Cup, the Speed of the Conveyor Belt, the Idle Time of the Yogurt Nozzle and the Processing Time of an Order

For the analysis purpose, the feed rates of all nozzles of the flavors are kept at 100 mL/s, and the maximum speed of the conveyor belt is equal to 10 cm/s. The feed rates of yogurt valves are changed to find their impact on the yogurt filling time, the speed of the belt, the idle time of yogurt nozzle and the filling time of yogurt nozzle Also, the results of the proposed model are compared with those of the previously published models. Feed rates for each order are evaluated for the belt speed, which is precisely equal to its maximum permissible value and has no idle time for the yogurt valve in most cases.

The increase in the feed rates of yogurt nozzles decreases the filling time of yogurt. As depicted in Figure 4, the yogurt filling time is the largest for all orders when the feed rate of the nozzle is considered equal to 100 mL/s. The filling time decreases with the increase in the value of the yogurt feed rate and is the lowest when the feed rate of the nozzle reaches 200 mL/s. For large volumes, the change in the yogurt filling time per 25 mL/s change in yogurt feed rate is higher when compared to smaller demanded volumes of yogurt, and vice versa.

The increase in the feed rates of yogurt nozzles increases the speed of the conveyor belt. As shown in Figure 5, the speed of the conveyor belt is slowest when the feed rate of the nozzle is considered equal to 100 mL/s. The speed increases with the increase in the value of the yogurt feed rate and is the fastest for all orders when the feed rate of the nozzle reaches 200 mL/s. For large volumes, the change in the yogurt filling time per 25 mL/s change in yogurt feed rate is minor when compared to smaller demanded volumes of yogurt, and vice versa. For small required volumes, the change in the speed of the conveyor belt is higher than the large volumes in the set of orders. It can be noted that, in this analysis, the maximum allowable speed of the belt is not considered and the speed exceeds the limit.

In Figure 6, the upper limit of the speed of the belt is considered equal to 10 cm/s. The speed of the belt is lowest for feed rate equal to 100 mL/s. As the feed rate is increased by 25 units till 200 mL/s, the speed of the belt increases. For small volumes, the calculated speed may increase the upper limit, but is actually kept at 10 cm/s during the filling processing. When the calculated speed is more than 10 cm/s and the belt is actually run at 10 cm/s, the cup under filling is filled earlier at 100 mL/s than the next empty cup reaches to the filling point. During this time, the yogurt nozzle remains idle and waits for the empty cup to reach to the filling point.

Keeping the maximum speed of the conveyor belt at 10 cm/s, the required amount of yogurt is filled in the cup through the yogurt nozzle. For most of the orders, the speed of the conveyor belt is less than or equal to the maximum allowable limit. However, in many cases, the speed of the conveyor belts exceeds the allowable limit for different values of the feed rates, as shown in Figure 7.

For a 100 mL/s feed rate, the speed of the belt is less than or equal to 10 cm/s for the orders starting from 1 to 12. These are orders of higher volumes of yogurt than orders starting from 13 to 20. As the volumes of the orders 13–20 decreases, the speed of the belt exceeds 10 cm/s but is kept at 10 cm/s. Due to keeping the speed of the belt at 10 cm/s and the feed rate of yogurt nozzle at 100 mL/s, the cup is filled earlier than the arrival of the empty cup to the filling point and hence the yogurt nozzle waits for the arrival of empty cup and remains idle. It can be noted that for large volumes of yogurt, the filling time is more and thus the speed of the belt is slow and vice versa.

As the speed of the belt is equal to half of the length of belt divided by the time in which an empty cup reaches to the filling point from entry point or reaches to exit point from filling point. This time is normally equal to the filling time of the cup. Where the calculated speed exceeds 10 cm/s and the conveyor belt is run at 10 cm/s, this time is greater than the filling time of the cup. The filling time can be represented by the relationship given in Inequality (29) as follows.

$$\mathrm{Filling} \mathrm{time} \ge {\displaystyle \frac{l}{S_a}}$$

(29)

The relationship given in Inequality (29) can be used when the speed of the conveyor belt is less than or equal to the maximum allowable limit (10 cm/s). In case when the actual speed of the conveyor belt is equal to 10 cm/s, i.e., the conveyor belt is run at 10 cm/s, the relationship given in the Equation (30) are used.

$$\mathrm{Filling} \mathrm{time}+\mathrm{idle} \mathrm{time}={\displaystyle \frac{l}{S_{\max }}}$$

(30)

As can be seen in Figure 8, for orders with the calculated speed of the belt less than 10 cm/s, the filling time is more than 5 s, while for orders with a calculated speed greater than or equal to 10 cm/s and when the belt is actually run at 10 cm/s, the sum of the filling and idles times of the yogurt nozzle is exactly equal to 5 s.

It can be noted that for large volumes, the filling time is higher than that for smaller volumes of the yogurt. The filling time decreases with the increase in the feed rate of the yogurt nozzle.

5.2. Comparison of the Proposed Model with the Previously Published Models

The proposed model is compared with the previously published models, i.e., B. Salah et al. [28], J. Chen et al. [29] and Y. Cui et al. [30], as can be seen in Figure 9. During the comparison, the yogurt feed rate was kept at 100 mL/s. When processing all orders at the above value of the parameter, the B. Salah et al. [28] model takes longer than the other models. However, the current proposed model takes less time than all other models.

In the set of orders, few require more than one flavor to be mixed with the yogurt. As there are dedicated filling points for each flavor in the case of Y. Cui et al. [30], for each flavor, the cup is initially passed through the dedicated line where the filling nozzle is available to fill the required flavor. The additional time to fill the second or third flavor in the cup is added to the initial processing time. Therefore, the processing time of Y. Cui et al. [30] study is more than that of Chen et al. [29] in most cases.

The increase in the yogurt nozzle’s feed rate decreases processing time. As shown in Figure 10, the previously published models have been compared with the proposed models, and as the feed rates are increased, the average processing time for all orders decreases. The proposed model results in the lowest average processing times for all orders, while B. Salah et al. [28] results in the highest values.

Solving the problem of 18 orders on the 3 parallel working machines, with a maximum allowable feed rate of the yogurt nozzle of 100 mL/s and the maximum allowable speed of the belt of 10 cm/s, the processing times recorded by the B. Salah et al. [28], J. Chen et al. [29] and Y. Cui et al. [30] model are 97.4 s, 82.6 s and 84.2 s, respectively, while the proposed model takes 67.8 s to complete the filling process of all cups demanded by customers. This means that the proposed model is 1.43-, 1.21- and 1.24-fold faster than B. Salah et al. [28], J. Chen et al. [29] and Y. Cui et al. [30] models, respectively.

6. Conclusions

This article compares the processing times of the different machine settings of the yogurt filling machine. Previously published articles on process optimization have been reported by changing the machine settings and revising the mathematical model accordingly. In the current study, the machine’s settings are changed to fully flexible, where the yogurt and all flavors are filled in the cup at any filling point of the machine. A mathematical model has been developed for the proposed machine settings, where each filling point in the machine works in parallel to fill the required volumes of yogurt and flavors simultaneously. To compare the model with the previously published models on the machine, a real-life problem is solved, where the customer demand comprises 18 orders for different volumes of yogurt and flavors. The total processing times of machines 1, 2, and 3 are 417.000 s, 416.625 s, and 416.875 s, respectively. The total work content amounts to 1250.500 s on the three machines, with an average machine load of 416.833 s and a makespan value (C_max) of 417.000 s. After comparison with previous published models, the proposed model is 1.43-, 1.21- and 1.24-fold faster than the B. Salah et al. [28], J. Chen et al. [29] and Y. Cui et al. [30] models, respectively.

Implementing the fully flexible yogurt filling machine model in real-world settings may face challenges such as technical complexity, equipment modifications, operator training, maintenance and repair, diverse customer demand and significant investment in equipment, software, and training. Addressing these challenges is important to successfully implementing the proposed model and realizing its benefits in real-world yogurt filling machine operations.

In the future, to further automate the yogurt filling machine and reduce human intervention, some of the technologies that can be employed are Machine Learning, Internet of Things, Advanced Conveyor Belt Systems and Industry 4.0 technologies.