Data Collection

According to ^{Hernández et al. (2014}), data collection involves conducting a detailed plan of techniques that lead to unifying data for a specific purpose. Thus, the data collected are the results of a series of observations made over several days, perfected to obtain the expected results. For ^{Behar (2008}), data collection refers to the use of a wide variety of techniques and tools to develop information systems. Table 1 shows the time noted down for each workstation in the production process, following the sampling.

The number of observations required was determined based on the 12 observations listed in Table 1. Out of the 12 cycles noted, the first 10 measurements made at each of the stations of each cycle were preliminary taken; the observed times (TO) were computed; the ranges (R) were determined subtracting the smallest from the largest; the value of S’ was found, considering D for 10 observations, n =10, D = 3.078, as a factor to determine the standard deviation; the S’/TM ratios were obtained; and, finally, it was determined that the highest value corresponded to the closing process, as can be observed in Table 2.

Subsequently, the value of s is calculated for the closing process, which is the pivotal element, since it has the highest S’/TM quotient. The corresponding value is found in the Student’s t table. For a sample with the 10 preliminary observations and to correct for bias and obtain a more accurate answer, 1 is subtracted from the sample size. Using n-1 = 9, an error of 0.025, the value of t for n = 9 is obtained from the table of Student’s t distribution values, and it is determined that t = 2.2622. Also, α = (0.025 + 0.025) = 0.05 is considered. The values of s are calculated as follows:

s = 13.80

Lastly, the number of observations is calculated as shown below:

N = (ts/TMk)² = (2.2622 * 13.80 / 184.8 * 0.05)²

N = 11.415, then N = 12

Therefore, two additional observations must be added to the 10 preliminary selected, totaling 12 as shown in Table 1.

Model Formulation

Upon defining the problem and collecting data, the next step was to plan the model to represent the essence of the production process of the Manufacturing company. ^{Shannon (1988}) states that simulation is the process of developing a model of a real system, using past experience to understand its behavior or to evaluate new strategies while observing the restrictions imposed by a specific criterion or a set of standards for its proper functioning. Hence, the model under study focuses on the well-defined departments within the production process of the company. First, we observe the elaboration of the springs that make up the structure of the mattress, then we move to the machine that assembles the innerspring structure, then to the clinching area where the structure is reinforced, this is followed by quilting, closing and, finally, packaging.

Model Building

Real system models accurately represent the real world to be modeled. Model building is all about simplification. It is conducted to gain a better understanding of an aspect of the real world, as well as to explicitly clarify the meaning of complex relationships that exist in reality. According to ^{García et al. (2006}), ProModel focuses on the manufacturing processes of one or several assembly and manufacturing products, among others. Although minimal details of reality have been omitted, the model shows reality in all its aspects, allowing us to understand the existing complex relationships defined as exogenous and endogenous variables. Consequently, the results obtained in the research are valid.

Figure 1 shows the company’s production system in the ProModel simulation program. The locations are the physical representation of the workstations where the entities that determine the production process arrive and leave. The locations included in the model are Innerspring, Clinching, Upholstery, Quilting, Closing, and Packing. The entities represent the overall product, that is, the mattress at the different manufacturing stages. For instance, we have springs, innerspring, reinforced innerspring, upholstered innerspring, quilted innerspring, closed mattress, and packed mattress. The arrivals of the entities determine their entry into the system; the simulation model starts with the arrival of the springs, as shown in Figure 1. Also, the model spans from the arrival of the spring at the raw material (RM) warehouse to the finished goods (FG) warehouse, where the finished mattresses arrive.

The processes are the set of operations that occur in the locations, channeling the times, resources, and other occurrences related to the entities. Variables are counters that helps us track the number of products being processed at a given moment in real time. For example, total number mattresses, rejected mattresses, structured mattresses, upholstered mattresses, quilted mattresses, closed mattresses, and packed mattresses. The resources are the workers that are part of the system, such as operator in charge of structuring, innerspring, upholstery, quilting, closing, and packing, as well as inspectors and technicians. According to the historical data for each location, the frequency distribution for each workstation that best represents the model to be simulated is determined, as shown in Table 3.

Simulation Results

Based on the information gathered by means of techniques used in methods engineering, the results obtained for the respective analysis are presented. Similar to the conceptualization of the current model, the locations were taken as a starting point. The movement of the entities throughout the locations of the production process was also considered a principle. Table 4 shows the results of a run of the current simulation model expressed in seconds.

A comparison of the actual data and the data from the results obtained from the current model was made following the normal operation of the model (Table 7). Figure 2 shows the similarity of the data, thus indicating that the model is within the acceptance ranges, as verified using a Student’s t-test. First, data normal behavior is determined according to the following statistical hypothesis and decision rule:

H_o: The analyzed data have a normal behavior.

H_a: The analyzed data do not have a normal behavior.

Decision rules: If p ≥ 0.05, H_o is accepted.

If p < 0.05, H_a is rejected.

According to Table 5, p > 0.05 and H_a is accepted; therefore, data follow normal behavior. Subsequently, we conducted Student’s t-test considering the following statistical hypothesis and decision rule:

H_o. There is no significant difference between the means of the real and ProModel data.

H_a. There is a significant difference between the means of the real and ProModel data.

Decision rule: If p > 0.05, H_o is accepted.

If p ≤ 0.05, H_a is accepted.

From the results in Table 6, it is concluded that p > 0.05; therefore, H_a is accepted. There is no significance difference between the means of real data and ProModel data.

The first 20 responses of the current model are taken from Table 8 for the calculation of the number of replications needed for the model as a pilot test. Based on the results to determine the sample size, 21 replications are necessary to statistically validate the model. First, it was necessary to determine this number of responses of the replicate using the classical method, according to the formula taken from ^{Tamashiro and Yacarini (2018}).

Where:

N : number of replications

n : sample size

𝒕 𝒏−𝟏,𝟏−∝/𝟐 : Critical value of Student’s t-distribution.

α : Significance level

s(n) : Standard deviation of the sample

e : Error between the population mean and the sample mean.

The following formula was used to determine the absolute error of the sample:

Based on these data, it was possible to analyze the behavior of the replications and determine the number of responses needed to validate the model. First, a sample mean of 1692.45 and a sample deviation of 4.978 were determined. This information made it possible to calculate the sampling error.

e = 2.093 * 4.978 / 20 = 2.33

Subsequently, the number of replications was calculated considering the sampling error value (2.33) at 95% confidence level.

𝑵= 𝟐.𝟎𝟗𝟑 𝟐.𝟑𝟑 ∗ 𝟒.𝟗𝟕𝟖 ² = 20.0008

Using this result, we corroborated that 21 replications are sufficient to validate the current model simulation. For a more detailed analysis, Table 8 shows 30 replications of the current model, while Figure 3 shows the stability diagram of these replications, showing a fixed value trend.

From the results, a bottleneck was detected in the innerspring area. Therefore, we decided to create a model by adding a new innerspring mattress machine to the production line, along with some other improvements shown in Table 9.

The principle of the improved model (Figure 4) was to move the entities throughout the locations. A new innerspring mattress machine was introduced to streamline the flow of the production line. All the improvements listed in Table 9 were also implemented, thus reducing the cycle times at the stations considerably. In addition, the transportation times in the current model were determined based on the distance from the workstation to the work-in-process warehouse. Distance times are already included in the cycle times of this model, where flow is continuous; however, an average time of 10 seconds is considered for picking up the product at the workstations and finishing the process. The normal distribution adapted to the proposed model was used, as shown in Table 10. The results of the improved model run are presented in Table 11.

For the calculation of the number of replicates required for the improved model as a pilot test to determine the sample size, the first 20 responses of the runs are taken form Table 12. The number of responses of the replications was determined by the classical method, using the formula of the previous model. The absolute error of the sample was also determined using the same formula. These data made it possible to determine the number of responses needed to validate the model statistically. Also, the sample deviation was determined at 7.4544, allowing us to calculate the sampling error.

e = 2.093 * 7.4544 / 20 = 3.4887

Subsequently, the number of replications was calculated considering the sampling error value (3.4887) at 95% confidence level.

𝑵= 𝟐.𝟎𝟗𝟑 𝟑.𝟒𝟖𝟖𝟕 ∗ 𝟒.𝟗𝟕𝟖 ² = 20.0000

Using this result, we corroborated that 20 replications are sufficient to validate the current model simulation. For a more detailed analysis, Table 12 shows 30 replications of the improved model, while Figure 5 shows the stability diagram of these replications, showing a fixed value trend.

The model has a rather disorganized behavior at the beginning of the replications. The expected results then present a downward trend, but as the replications continue to increase, the model manages to stabilize at a fixed value.

^{Gutiérrez (2010}) states that productivity is related to the results obtained in a process or a system. Results for calculating productivity with the current method follow the formula below.

Productivity Index = Production/Resources

Productivity Ratio = 1712 mattresses/208 hours = 8.230 mattresses/hour

Productivity with the proposed method was calculated using the same formula as above, obtaining the results shown below.

Productivity Index = 3373 mattresses/208 hours = 16.216 mattresses/hour

From the results, a 97.02% increase in productivity is obtained using the proposed method.

A statistical analysis was performed to ratify the validity of the model. Table 13 shows 30 replications before improvements, and the same number for the proposed method for testing the hypothesis.

Using the data in Table 13, the normality test is performed, considering the following statistical hypothesis and decision rule.

H_o: The analyzed data have a normal behavior

H_a: The analyzed data do not have a normal behavior.

Decision rule: If p > 0.05, H_o is accepted. If p ≤ 0.05, H_a is accepted.

From the results in Table 14, p < 0.05, and the null hypothesis (H_o) is rejected; therefore, the data do not have a normal behavior. Consequently, a non-parametric statistic test must be used for hypothesis testing; the Wilcoxon test was chosen. Statistical hypothesis and decision rule for this case are presented below.

H_o. The implementation of methods engineering does not improve the productivity of the production process of a manufacturing company.

H_a. The implementation of methods engineering does improve the productivity of the production process of a manufacturing company.

Decision rule: If p > 0.05, H_o is accepted. If p ≤ 0.05, H_a is accepted.

In Table 15, it is observed that the comparison between the mean and median of productivity before is lower than the mean and median of productivity after. However, the decision rule states that when p-value is less than or equal to 0.05, the H_o is rejected and H_a is accepted. The Wilcoxon test statistic can be used to support this result.

From the results in Table 16, it is concluded that p < 0.05; therefore, the null hypothesis (H_o) is rejected. This means that the samples correspond to different populations. In this case, the second sample corresponds to an improvement in production through the application of methods engineering. The application of method engineering improves the productivity of the production process of a manufacturing company.