For the present analysis, the records from the years 2015, 2016 and 2017 were considered, because it can be assumed that the behavior of customers in search of the service is maintained in recent years. Giving a total of 395 962 cases analyzed.
3.2.1. Study of customer arrival
For the analysis, it was necessary to identify the strange data, those that are not due to a normal behavior corresponding to the arrival of customers to the service system, these data were excluded since they generated erroneous estimates and inconveniences in the simulation of the system.
It is common for this atypical data to be presented, this system works in real life and is subject to uncontrollable variables. In Figure 5, the black dots correspond to the atypical data, which were not taken into account in the analysis of descriptive statistics.
Figure 5.
Box and whisker diagrams for the analysis period by years.
Figure 5.
Box and whisker diagrams for the analysis period by years.
Table 1.
Descriptive statistics of daily care.
Table 1.
Descriptive statistics of daily care.
| Day |
Min |
Max |
Stocking |
Standard deviation |
| Monday |
570 |
924 |
737 |
70 |
| Tuesday |
526 |
836 |
674 |
69 |
| Wednesday |
475 |
817 |
645 |
69 |
| Thursday |
456 |
783 |
617 |
63 |
| Friday |
456 |
762 |
601 |
66 |
The demand for the care service is differentiated per day, in addition, on Mondays an average of 737 attentions are experienced, while on Fridays, the average number of people attended is 601. To standardize the arrival rate between days, it is calculated by:
The database obtained, refers to the following data collection, the system opened its doors at 8h 00min 00s and the last client arrived at 18h 2min 37s, the operating time on that day was equal to 10 h, 2 min and 37 s or, 602, 62 min. And when considering that, on the same day 684 people were attended, it is obtained that the arrival rate is . By applying this procedure it was obtained:
Table 2.
Arrival fees.
| Day |
Media |
Standard deviation |
| Monday |
0,857 |
0,100 |
| Tuesday |
0,946 |
0,192 |
| Wednesday |
1,001 |
0,335 |
| Thursday |
1,005 |
0,109 |
| Friday |
1,048 |
0,116 |
Subsequently, the distribution function of customer arrival was identified, which was applied to represent this phenomenon.
Figure 6 shows that the distributions that most closely approximate the real data generated by the arrival of customers are the gamma distribution and the normal, the exponential is totally deviated. To avoid subjectivity in choosing the distribution, the goodness-of-fit test was applied (Montgomery and Runger, 2003). By means of the tests of Kolmogorov Smirnov the following results were obtained:
Figure 6.
Distribution of arrival time – Monday.
Figure 6.
Distribution of arrival time – Monday.
Table 3.
Goodness of fit test – Monday.
Table 3.
Goodness of fit test – Monday.
| Distribution |
E(x) |
Where(x) |
p-value |
Conclusion |
| Exponential |
0,857 |
0,100 |
0,000 |
Rejected |
| Gamma |
1,005 |
0,109 |
0,486 |
Accepted |
| Normal |
1,048 |
0,116 |
0,386 |
Accepted |
The gamma and normal distribution can be used to represent the arrival rate on Mondays (it is accepted that they come from these distributions), in this case, it is preferable to use the gamma distribution. The same procedure was applied for the rest of the days. The arrival rates served by the care center follow a gamma distribution with the following parameters.
Table 4.
Goodness of fit test – Monday.
Table 4.
Goodness of fit test – Monday.
| Day |
E(x) |
Where(x) |
| Monday |
0,857 |
0,010 |
| Tuesday |
0,930 |
0,009 |
| Wednesday |
0,972 |
0,012 |
| Thursday |
1,004 |
0,011 |
| Friday |
1,047 |
0,013 |
3.2.2. Study of customer service
For the study of the distribution of customer service time, there was a need to identify the processes that are carried out, the 395 962 clients were served under the following scheme.
Figure 7.
Proportion of care by process.
Figure 7.
Proportion of care by process.
To continue, there was a need to determine how long the servers are delayed in each of the procedures performed, for this the analysis of descriptive statistics was applied, with which it was obtained.
Table 5.
Descriptive statistics by care process.
Table 5.
Descriptive statistics by care process.
| Process |
Min |
Max |
Media |
Standard deviation |
| A |
0,02 |
26,85 |
7,46 |
6,35 |
| And |
0,02 |
26,77 |
7,90 |
6,21 |
| F |
0,03 |
12,18 |
3,60 |
2,73 |
| L |
0,02 |
32,97 |
9,18 |
7,76 |
| In |
0,02 |
32,37 |
8,33 |
7,77 |
Once the procedure was applied to estimate the distributions of attention times, it was found that no distribution adjusts to the behavior in the attention of each of the processes. We considered 32 distributions with the EasyFit® software, and none were functional, therefore, to identify the behavior of the attention time for each of the processes, the control letters were applied. The processes analyzed are of the massive type, therefore the tools to be applied correspond to the control letters. With which the following control letters were obtained.
Figure 7.
Control letter for process A
Figure 7.
Control letter for process A
Figure 7.
Control letter for process A .
Figure 7.
Control letter for process A .
In the control letters it was observed that the attention process (any of them) is in imbalance, the control limits established by the processes are higher than the specification limits, in addition, it is common to observe points outside the specification limits. As for the ranges, an important distortion is seen since instead of reducing the ranges under which the attention times appear, it was possible to review that they are expanded, with this, the increase in the variability of the process was demonstrated.
For process F, no observations were obtained during the last year, because, as of 2017, the shifts that were attended with process F, are now attended in process A. To finalize the diagnosis, the process capability indices were applied. With which the following results were obtained:
Considering the index, it can be observed that the variation of the processes is too high with reference to the range of the specification limits, with an average of , the processes are not able to meet the specifications. As for the real capacity index, which takes into account the position, it was possible to see that there is a greater capacity in relation to the lower limit, and for the upper limit the capacity of this process is much lower, that is, most of the time, the observations were above the upper limit, He also shows that apart from the fact that the processes are not capable, they are off-center with a tendency to be outside the upper limit. As for the index, evaluates how centered the processes are, with respect to the interval defined by the specification limits, the observed values (> 20%), it was possible to confirm that the processes are off-center. The previous indicators, at least consider intervals for their calculation, unlike the Taguchi index, this is much more acid in terms of evaluating what the processes are capable of, and the values obtained showed a very poor performance, since the variation they have is too large to meet the value offer of the company.