4.2. Structuring for Test
To achieve the objectives of the research, which proposes the creation of a model that guides the prediction of reduced setup time, with the adoption of optimization tools, it is necessary to carry out a certain number of tests, in different process configurations, to generate a quantity of data that can provide a function capable of responding to most situations that may be encountered.
As mentioned before, it was evaluated those three factors are relevant to the operation, namely: machine group, SKU number and planning horizon. As the machine groups are already known (slasher dye and loop dye) and we know that the planning horizon will vary from one to four weeks, it remains to define the number of SKUs. For this, an analysis was carried out on the company’s data to identify the number of processed SKUs by the dyeing machine throughout 2018. According to
Table 1, we can identify each machine, the number of dyeing batches and the SKU number processed throughout the month. A single SKU can be processed more than once throughout the month, so the number of batches is greater than the number of SKUs.
From these data, it was possible to build a histogram demonstrating the frequency of the number of processed SKU per month for the two groups of machines, as shown in
Scheme 1. The importance of this parameter is due to its variability throughout the year, mainly due to market seasonality and fashion trends. Thus, a range of SKU quantities to be tested was defined, which could guarantee a good range of results.
The histograms were thought to select many SKUs for testing to obtain a range of results, which serve as parameters for constructing a linear regression equation. This can be used in other companies with the same similar process.
Therefore, for the slasher dye group, the SKU number defined was: 4, 6, 8, 9 and 12. As for the loop dye group, the SKU number was: 5, 7, 9, 11, 12, 13 and 16. This dataset should cover the possible occurrences.
Table 2 shows how the configurations of the number of SKUs per dye machine were structured.
With the definition of these configurations, it was possible to advance the measurement of setup times according to the empirical sequencing through the company’s reports. From the report, it was possible to extract, for each machine, the processing sequence throughout each month, fractionate the horizons. With the setup time matrices, it was possible to measure the setup time to go from SKU i to SKU j.
A total of 52 tests were carried out, 24 tests for the slasher dye machine group and 28 tests for the loop dye machine group.
Figure 3 shows an example of setup time measurement performed on machine 7, with 13 SKU, broken down into four horizons.
It is adapting the TSP model to the case study, starting from the need to produce a given amount of n dyeing batches for each SKU, which must be sequenced to obtain the shortest setup time, represented by in the Equation (1). There will be situations that, for the same SKU, the demand will be greater than one dyeing batch. Thus, for each batch to be processed, its code will be indexed to construct the setup time matrix. In this way, the same SKU can appear more than once in the matrix.
To ensure a better understanding, let’s take the situation illustrated in
Figure 3, where the development of actions to optimize the sequencing of machine seven of the loop group will be presented.
For the machine in question, tests were performed for three SKU situations, as shown in
Table 2. However, it is enough to describe only one situation for a good understanding, as the rationale is the same for all other cases. Therefore, the situation with 13 SKUs will be described, where the reference month was August 2018, as shown in
Table 1.
It can be noted that a SKU was incorporated into the setup time matrix that was called zero and that it has a setup time equal to zero to any other SKU. This device was used so that when completing a sequencing cycle, the return from the last processed SKU would go to SKU 0 because in the real situation, a new sequencing would hardly start from the SKU that was the origin of the previous sequencing.
For the simulations, LINGO software was used, version 17, licensed educational version, and the implementation was performed on a Dell branded equipment, i7 processor, 16 Gb, SSD.
After processing the data in the simulator, the results obtained can be viewed in
Figure 4.
Figure 5 presented by the LINGO software [
28], shows the fill-in of the matrix model composed of values in zeros and ones, varying only in the number of variables and constraints.
Considering that this is the application of the traveling salesman problem, there is a limitation of the problem considering that it is a large problem that may involve thousands of variables and restrictions. In this aspect, it is necessary to research the possibility of applying large scale decompositions, namely, Benders, Danzig–Wolfe or cross-decomposition decompositions, to the traveling salesman problem to obtain an exact solution. Other options are structured in hybrid methods [
29].
For large models structured, considering the traveling salesman problem approach becomes necessary for other methodologies [
30,
31,
32].
After measuring the setup times for all the suggested settings in four kind horizons, one week, two weeks, three weeks and four weeks, the SKU number referring to the whole horizon, we came to
Table 6.
The collected data made it possible to analyze the correlation between the setup time variation with the parameters, machine group, SKU number, and programming horizon. The information obtained was submitted to estimate the linear correlation coefficients (r) using the Minitab 17 software.
The linear correlation coefficient, sometimes called Pearson’s product moment correlation coefficient, is represented by the letter r and takes values from −1 to 1 and measures the degree of the linear relationship between the paired values x and y in a sample. When
r = 1, it represents the perfect and positive correlation between two variables, and for
r = −1, it means a perfect negative correlation between two variables. That is, while one increases, the other decreases [
33].
The Pearson’s correlation coefficient is calculated according to the following formula:
where:
n—represents the number of data pairs present;
x,
y—are the sample values you want to evaluate the correlation.
The analysis was performed separately for each machine group, and the correlation between % variation with the number of SKU and % variation with the programming horizon was evaluated. Absolute variation with SKU number and absolute variation with programming horizon.
Below in
Table 7 are presented the results obtained for the two groups of machines.
It was chosen to study the correlation of both the relative variation and the absolute variation of setup times to assess possible divergences in the behaviour of the linear correlation coefficients. As it can be seen, there is a difference in the reading of information between the two situations.
Contrary to what was expected, the correlation of the number of SKUs with variation in setup time presented practically null correlation coefficients for both machine groups. That is, there is no correlation between the number of SKUs with variation in setup time. However, the planning horizon showed a strong negative correlation with the interpretation in setup time, with the value of r for slasher dye and loop dye group for the relative variation being −0.644 and −0.679, respectively. For the absolute deviation, it was −0.928 and −0.859, respectively. That is, the absolute variation presented a more substantial relationship degree with the planning horizon. The negative correlation shows that, as the planning horizon increases, the optimization system is more efficient is in reducing setup time. In a way, this result was already expected, as not only in dyeing sequencing, as in other areas, the more you can anticipate the needs, the better the resources are used.
Based on the results presented, we can propose a model based on linear regression that can predict the variation in setup time. However, such interpretation must refer to relative deviation, despite having presented a weaker degree of relationship when compared to absolute variation. It is not advisable to propose a model that indicates the possible absolute variation in setup time. This, when replicated in another operation with a similar process, will undoubtedly face different setup times between pairs of SKUs.
Therefore, we will describe the relationship through the straight-line equation representing the relationship between the variation in setup time with the planning horizon.
Given a collection of paired sample data, the simple regression equation describes the relationship between two variables.
Further, its representation is given by Equation (8):
where:
y—dependent variable or response variable;
x—independent variable or predictor variable;
—the y-intercept;
—angular coefficient.
Applying the simple linear regression method to both machine groups, the following data are obtained, shown in
Scheme 2:
Therefore, obtained the formulation can be tested in other companies.
Loop dye:
where: h—planning horizon.