From the beginning of the study of the probability of repairable component failures, until now, operational availability is considered as the capacity of a component to fulfill its function when required [1-4]. This indicator originated because operational systems cannot guarantee 100% of the continuous flow of their operation, due to interruptions that occur as a result of the occurrence of unforeseen failures or the execution of preventive maintenance activities; therefore, they cannot fulfill their function during all the required time.

For this reason, availability is one of the most important indicators for decision making within operational and maintenance management [3,5], frequently used in the planning of the operation of productive and services systems.

Depending of system operational capacity, a small variation in availability represent very important variations in the operational consequences. For example, commonly in production systems, a small decrease in availability represents a considerable reduction in production [6]. So, the calculation of availability must have a high degree of accuracy.

The problem is that the calculation of the availability of a single component that is done by dividing the real operating time for the desired operating time [1,4] is consistent; but for series and parallel systems, the current calculation method [7-9] is analogous to that of reliability [10]; even though the availability and reliability are totally different.

Reliability is a purely stochastic process, which indicates the probability of satisfactory operation [3,11], whose value starts in one after the start-up or immediately after the repair and tends to zero when time tends to infinity; while, at the time the fault occurs and while it lasts, it takes the value of zero.

On the other hand, the operational availability is eminently deterministic, whose value in conjunction with other indicators measures the performance of the component during the analyzed time period; therefore, it expresses the result of events that occurred, and that for no reason is probabilistic. For this reason, the current equations for the calculation of the availability on series and parallel systems are questionable; however, these equations are widely used both in scientific research [12-14] and in the production of books [15,16].

The objective of this research is to evaluate these equations by analyzing the quantity of products that the system stops manufacturing due to the unavailability of each of its components, in order to know its level of precision and its calculation error. In order to do so, three operational models have been designed with two components each of equal operational capacity; set up in series, active parallel and passive parallel, respectively.

The results are obtained through the iteration of 5000 samples of size 250 with the Monte Carlo method, where a range of values of the down times of each component are entered randomly, in the current equations of the availability on series and parallel systems, to later evaluate its effectiveness, comparing these results with the t statistic.

Before the statistical treatment of the data, it is important to make a comparison of the current and real equations of the series and parallel systems, this is done through the three-dimensional graphs of Fig. 6, where the dependent variable is the non-manufactured products which is represented on the z axis; while the independent variables are defined by the down times 1 and 2 corresponding to the x and y axes respectively.

Figure 4 Source: The authors.

In Fig. 6 it may be observed that in all cases there is a positive tendency that starts from the origin; however, when the down time reaches 20 hours, the non-manufactured production differs between current and real methods. In order to have a better appreciation of these differences, the non-manufactured production is tabulated in Tables 1, 2 and 3 when the down times are extreme (0 and 20 hours).

Table 1

Source: The authors.

Table 2

Source: The authors.

Table 3

Source: The authors.

As indicated in Table 1, on series system, half the required time (20 hours) the component 1 fails and the other half (20 hours) fails the component 2, giving a total of 40 hours, but the system did not operate during this time. Therefore, the non-manufactured production is equal to the nominal production (4000 units), as indicated in eq. (7). However, with the current equation, 3000 non- manufactured units were obtained, which suggests that, although the system did not operate at any time, 1000 units were manufactured. Result that is impossible to obtain in reality.

Figure 5 Source: The authors.

For the active parallel system to have a capacity of 100 units per hour (which is the initial condition of the simulation), each component has to operate with a capacity of 50 units per hour. If in the half of the required time, the component 1 fails and component 2 fails in the other half of required time, the system operates only whit one component at a time, which means that its capacity is reduced by half, generating a non-manufactured production of 2000 units, as indicated in the last row of Table 2 with eq. (11)

While with the eq. (9), the non-manufactured production is less than the real one; but the most remarkable thing occurs when only one of the two active parallel component has a down time equal zero, the non-manufactured production is not generated, reaching the nominal production; although, only one component is operated; thus, eq (9) does not fit as expected.

Figure 6 Source: The authors.

Concerning passive parallel systems, their components operate one at a time. In no case operate both at the same time, since the component 2 is a backup, coming into operation, only when the first has failed. These systems are the most reliable, generating non-manufactured production when the main component fails at the same time that the backup component is inoperative (eq. (12) of Table 3).

On the other hand, eq. (9) admits down times without any condition, enabling the generation of non-manufactured production when the two components fail at different time points. So, the eq. (9) neither fits as expected in these systems.

So far, significant differences may be observed between the real data and those obtained with the current equations; but to make a more effective comparison it is necessary to analyze a lot of cases. For this purpose, the simulation of the algorithm of Fig. 4 is used; in which, the results of the first ten and the last iteration are indicated in Table 4, where DT₁ and DT₂ are the down times of components 1 and 2 of the systems models of Fig. 2 and 3 respectively, those were obtained randomly in order to be replaced in eq. (6), (7), (9), (11) and (12) in each of the iterations.

Table 4

Source: The authors.

The relative frequency histograms of the results obtained in the 5000 iterations are indicated in Fig. 7, where it is observed that the data does not seem to have a normal distribution, for this reason an analytical normality test is carried out using the Kolmogorov Smirnov method with the modification of Lilliefors, obtaining as a result that none of the variables evaluated are normally distributed, with a confidence of 99%, since in each case the p-value of the test statistic is less than 0.01 (Table 5).

Figure 7 Source: The authors.

Table 5

Source: The authors.

Due to the large amount of data that may be obtained through the simulations, it is decided to apply the central limit theorem with 5000 samples of size 250, which is sufficient for the sample means to be distributed normally. The results of the first 10 sample means and the last one is indicated in Table 6 and their respective relative frequency histograms are shown in Fig. 8, which seem to be normally distributed.

Table 6

Source: The authors.

Figure 8 Source: The authors.

Indeed, the analysis of normality by the Kolmogorov Smirnov method with the modification of Lilliefors indicated in Table 7, shows that the sample means of the variables evaluated from the non-manufactured production on series and parallel systems are normally distributed, with a 99% confidence, since in each case the p-value of the statistic test is greater than 0.01.

Table 7

Source: The authors.

In the statistical comparison by means of the box diagrams of Fig. 9. it can be seen that there is a clear difference between the results of the non-manufactured products calculated with the current equations and the real ones on series and parallel systems. This observation must be confirmed by using analytical statistical methods.

Figure 9 Source: The authors.

Given that, the sample means of the variables have a normal distribution, it proceeds to evaluate the efficacy of the current equations by means of the parametric statistic t; where the H₀ corresponds to the equality of the sample means, which indicates that the current equations are effective. The averages of the sample means are indicated in Table 8, while the results of this test are shown in Table 9.

Table 8

Source: The authors.

Table 9

Source: The authors.

Since the p-value obtained with the t statistic is less than 0.01 for each of the systems (Table 9), the H₀ is rejected, therefore, there is sufficient evidence to demonstrate that the two samples have different measures with the 99% confidence in each case, which means that the current equations for the calculation of the availability on series and parallel systems, give different results in comparison with the real ones.

The production losses are proportional to the down time of each system component, for this reason this calculation may be used to deduce an equation that relates the non-manufactured production with the availability of each component of series and parallel system.

The non-manufactured products resulting from the simulations using the Monte Carlo method are not normally distributed in the three systems analyzed; however, the sample means when applying the central limit theorem are distributed normally. They allow to evaluate the data by parametric methods.

The t statistic test that allows to compare of sample means distributions, shows that the current equations are ineffective with a confidence of 99%.