3.1.1 Description and decomposition of electrical test table

It is a matter of clearly identifying the elements of the machines to be studied in order to analyze, for each element, the risks of dysfunction. We proceed with a general analysis followed by operational analysis. It is first of all necessary to formulate the need in the form of simple functions that the equipment must fulfill by answering three basic questions shown on the standardized tool called the horned beast diagram illustrated in Figure 1.

Operational analysis is the analysis of the decomposition of the electric test table; this analysis will be made by a simple decomposition into blocks of element presented in Figure 2.

#### 3.1.2 Determination of the transition laws

After describing the electric test table to be studied, we are confronted with the problem of determining the failure and repair function of this equipment.

In order to model the breakdown/repair process of a repairable system we have used the history of the operating time of this equipment. The failure history of the electric test table machine enabled us to analyze the data and calculate the availability of this machine along the past period. After any calculation made, the availability of the machine is found to be of the order of 62.73%.

The choice of a particular model is statistically tested to select the model best suited to the observed failure and repair times. The protocol to be used for the study of repairable equipment has been developed by Ascher and Feingold [23].

squares) and the non-parametric staircase function. The calculation is weighted

Impact of Improving Machines' Availability Using Stochastic Petri Nets on the Overall…

DOI: http://dx.doi.org/10.5772/intechopen.82445

The decision to accept or reject the null hypothesis concerns the value p. If the value p is greater than 0.05, the null hypothesis is accepted, and if the value p is less

On the basis of the history of operating times, we used the data processing software MINITAB17 [25] in order to obtain the results mentioned in Figure 3.

0.522: This corresponds to the law: normal distribution with BOX-Cox

The Box-Cox transformation is a power transformation, W = Y\*λ, in which

According to the results obtained and by comparing the different values of P, we

more extensively at the ends of the distribution.

• H0: the data come from a specific distribution.

• H1: the data does not come from a specific distribution.

The hypothesis test is defined as:

Decomposition of electric test table.

than 0.05, the hypothesis is rejected.

note that the largest values of p are:

Minitab determines the best value for λ.

transformation.

83

Figure 2.

3.1.3 Mean time between failure modeling

Using the Anderson-Darling statistic and its p-value, we make the decision that the model, the exponential, the Weibull or Log normal, is the one that best adjusts the data. The Anderson-Darling test [24] is used to check if a sample of the data comes from a population with a specific distribution.

Minitab computes the Anderson-Darling statistic using the weighted quadratic distance between the fit line of the probability diagram (based on the chosen law and using the maximum likelihood estimation method or estimates of the least

Figure 1. Horned beast for the electric test table.

Impact of Improving Machines' Availability Using Stochastic Petri Nets on the Overall… DOI: http://dx.doi.org/10.5772/intechopen.82445

Figure 2. Decomposition of electric test table.

whose role is to check the flow of the electric current and therefore it ensures the

To achieve this result, we present the electrical test table and its components. Then, availability is calculated based on the history provided by the company. After this, we find the law that will help us anticipate future failures and when these failures will occur. This law enables us to develop a high-quality preventive main-

It is a matter of clearly identifying the elements of the machines to be studied in order to analyze, for each element, the risks of dysfunction. We proceed with a general analysis followed by operational analysis. It is first of all necessary to formulate the need in the form of simple functions that the equipment must fulfill by answering three basic questions shown on the standardized tool called the

Operational analysis is the analysis of the decomposition of the electric test table;

After describing the electric test table to be studied, we are confronted with the

In order to model the breakdown/repair process of a repairable system we have used the history of the operating time of this equipment. The failure history of the electric test table machine enabled us to analyze the data and calculate the availability of this machine along the past period. After any calculation made, the

The choice of a particular model is statistically tested to select the model best suited to the observed failure and repair times. The protocol to be used for the study

Using the Anderson-Darling statistic and its p-value, we make the decision that the model, the exponential, the Weibull or Log normal, is the one that best adjusts the data. The Anderson-Darling test [24] is used to check if a sample of the data

Minitab computes the Anderson-Darling statistic using the weighted quadratic distance between the fit line of the probability diagram (based on the chosen law and using the maximum likelihood estimation method or estimates of the least

this analysis will be made by a simple decomposition into blocks of element

problem of determining the failure and repair function of this equipment.

of repairable equipment has been developed by Ascher and Feingold [23].

availability of the machine is found to be of the order of 62.73%.

comes from a population with a specific distribution.

main technical function of the wiring harnesses produced.

3.1.1 Description and decomposition of electrical test table

horned beast diagram illustrated in Figure 1.

3.1.2 Determination of the transition laws

presented in Figure 2.

Maintenance Management

Figure 1.

82

Horned beast for the electric test table.

tenance plan in order to increase the availability of the equipment.

squares) and the non-parametric staircase function. The calculation is weighted more extensively at the ends of the distribution.

The hypothesis test is defined as:


The decision to accept or reject the null hypothesis concerns the value p. If the value p is greater than 0.05, the null hypothesis is accepted, and if the value p is less than 0.05, the hypothesis is rejected.

#### 3.1.3 Mean time between failure modeling

On the basis of the history of operating times, we used the data processing software MINITAB17 [25] in order to obtain the results mentioned in Figure 3.

According to the results obtained and by comparing the different values of P, we note that the largest values of p are:

0.522: This corresponds to the law: normal distribution with BOX-Cox transformation.

The Box-Cox transformation is a power transformation, W = Y\*λ, in which Minitab determines the best value for λ.

Figure 3. Distribution laws for MTBF.

0.925: This corresponds to the law: normal distribution with Johnson transformation.

After the application of a Johnson transformation, the data closely follow a normal distribution; indeed, the value of p is high and virtually all data points lie within the confidence limits of the Henry line.

Box-Cox transformation: λ = 0

Johnson transformation: 1.15528 + 0.453754 Ln((X – 10.5919)/(13309.4 X))

### 3.1.4 Mean time to repair modeling

Based on the history of the same equipment for the same time period, and using the MINITAB17 data processing software in order to obtain the results mentioned in Figure 4.

We note that the largest values of P are:

0.55: This corresponds to the law: normal law with transformation of BOX-Cox. The Box-Cox transformation is a power transformation, W = Y λ, in which

Minitab determines the best value for λ.

0.145: This corresponds to the law: normal law with Johnson transformation.

After the application of a Johnson transformation, the data closely follow a normal distribution; indeed, the value of p is high and virtually all data points lie within the confidence limits of the Henry line.

Box-Cox transformation: λ = 1

Johnson transformation: 6.98629E16 + 0.898955 Ln((X 1.44460)/ (11.5554 X))

#### 3.1.5 Stochastic Petri Net modeling for preventive maintenance plan

After the analysis and calculations, the following stochastic Petri net was realized in Figure 5.

After modeling the maintenance function of the electric test table, it became easy to predict the next breakdown and when it will occur. So, we developed a high-

Impact of Improving Machines' Availability Using Stochastic Petri Nets on the Overall…

DOI: http://dx.doi.org/10.5772/intechopen.82445

quality preventive maintenance plan based on the modeling realized.

Figure 4.

Figure 5.

85

Stochastic petri net modeling.

Distribution laws for MTTR.

Impact of Improving Machines' Availability Using Stochastic Petri Nets on the Overall… DOI: http://dx.doi.org/10.5772/intechopen.82445

Figure 4. Distribution laws for MTTR.

0.925: This corresponds to the law: normal distribution with Johnson

within the confidence limits of the Henry line.

We note that the largest values of P are:

within the confidence limits of the Henry line.

Minitab determines the best value for λ.

Box-Cox transformation: λ = 1

Box-Cox transformation: λ = 0

3.1.4 Mean time to repair modeling

After the application of a Johnson transformation, the data closely follow a normal distribution; indeed, the value of p is high and virtually all data points lie

Johnson transformation: 1.15528 + 0.453754 Ln((X – 10.5919)/(13309.4 X))

Based on the history of the same equipment for the same time period, and using the MINITAB17 data processing software in order to obtain the results mentioned in

0.55: This corresponds to the law: normal law with transformation of BOX-Cox. The Box-Cox transformation is a power transformation, W = Y λ, in which

0.145: This corresponds to the law: normal law with Johnson transformation. After the application of a Johnson transformation, the data closely follow a normal distribution; indeed, the value of p is high and virtually all data points lie

Johnson transformation: 6.98629E16 + 0.898955 Ln((X 1.44460)/

After the analysis and calculations, the following stochastic Petri net was real-

3.1.5 Stochastic Petri Net modeling for preventive maintenance plan

transformation.

Distribution laws for MTBF.

Maintenance Management

Figure 3.

Figure 4.

(11.5554 X))

ized in Figure 5.

84

After modeling the maintenance function of the electric test table, it became easy to predict the next breakdown and when it will occur. So, we developed a highquality preventive maintenance plan based on the modeling realized.

#### Maintenance Management

The application of this preventive maintenance plan will have an important impact on the availability of the equipment which is best manifested in the progress of the availability which reached the value of 97.05%.

3.2.1.1 The tests of the laws on Minitab 17

DOI: http://dx.doi.org/10.5772/intechopen.82445

The hypothesis test is defined as:

than 0.05, we reject the hypothesis.

3.2.1.2 Adjustment tests on XL-Stat

method of estimation of maximum likelihood.

Its estimated parameters are grouped in Table 2.

• H0: the data come from a specific distribution;

• H1: The data does not come from a specific distribution.

distribution.

for TTR.

hypotheses H0.

distribution (2).

Figure 7.

87

Anderson-Darling analysis of the TTR.

Minitab proposes Anderson-Darling Statistics and its p-value to make the decision on which model the data is distributed in. The Anderson-Darling test [24] is used to test whether a sample of the data comes from a population with a specific

Impact of Improving Machines' Availability Using Stochastic Petri Nets on the Overall…

The decision to accept or reject the null hypothesis concerns the value p. If the value p is greater than 0.05, we accept the null hypothesis, and if the value p is less

From the results obtained we notice that the value of p-value of the TTR for all

Using this software, we performed a law fit with a risk of 5% and we adopted the

The different laws tested and their value of P can be summarized in Table 1.

The distribution that best fits the data for the fit test is the Weibull

On the basis of the history of operating times, we used the data processing software MINITAB17 [25] in order to obtain the results mentioned in Figure 7

distributions is greater than 0.05. This last value allows us to accept the

Thus, the strength of the stochastic petri nets was demonstrated as a tool of modeling allowing the availability of the machine studied to be improved.

## 3.2 Sieve machine (food sector)

In this second application, we present our study to improve the efficiency of a production line in a food company. We are interested in improving availability. To achieve this result, we find laws that will help us anticipate future failures and when these failures will occur. We model then and simulate the machine maintenance. The laws found enable us to develop a high-quality preventive maintenance plan in order to increase the availability of the equipment using stochastic Petri Nets.

## 3.2.1 The normality test

This test shown in Figure 6 is considered as the first step of the statistical mastery which makes it possible to analyze the normality of the data by using the probability plot (p-value\*) that is to say the probability that two samples are identical by using a test hypothesis.

The hypothesis test is defined as:


The decision to accept or reject the null hypothesis concerns the value p. If the value p is greater than 0.05, we accept the null hypothesis, and if the value p is less than 0.05, we reject the hypothesis.

For the TTR we found p-value = 0.335 > 0.05. So we will accept the hypothesis H0 and we can say that the data of the TTR follow the normal law.

Figure 6. Normality test for TTR.

Impact of Improving Machines' Availability Using Stochastic Petri Nets on the Overall… DOI: http://dx.doi.org/10.5772/intechopen.82445

## 3.2.1.1 The tests of the laws on Minitab 17

The application of this preventive maintenance plan will have an important impact on the availability of the equipment which is best manifested in the progress

Thus, the strength of the stochastic petri nets was demonstrated as a tool of

In this second application, we present our study to improve the efficiency of a production line in a food company. We are interested in improving availability. To achieve this result, we find laws that will help us anticipate future failures and when these failures will occur. We model then and simulate the machine maintenance. The laws found enable us to develop a high-quality preventive maintenance plan in order to increase the availability of the equipment using stochastic Petri Nets.

This test shown in Figure 6 is considered as the first step of the statistical mastery which makes it possible to analyze the normality of the data by using the probability plot (p-value\*) that is to say the probability that two samples are iden-

The decision to accept or reject the null hypothesis concerns the value p. If the value p is greater than 0.05, we accept the null hypothesis, and if the value p is less

For the TTR we found p-value = 0.335 > 0.05. So we will accept the hypothesis

H0 and we can say that the data of the TTR follow the normal law.

modeling allowing the availability of the machine studied to be improved.

of the availability which reached the value of 97.05%.

3.2 Sieve machine (food sector)

Maintenance Management

3.2.1 The normality test

tical by using a test hypothesis.

The hypothesis test is defined as:

than 0.05, we reject the hypothesis.

Figure 6.

86

Normality test for TTR.

• H0: the data follow the normal law;

• H1: the data do not follow the normal law.

Minitab proposes Anderson-Darling Statistics and its p-value to make the decision on which model the data is distributed in. The Anderson-Darling test [24] is used to test whether a sample of the data comes from a population with a specific distribution.

The hypothesis test is defined as:


The decision to accept or reject the null hypothesis concerns the value p. If the value p is greater than 0.05, we accept the null hypothesis, and if the value p is less than 0.05, we reject the hypothesis.

On the basis of the history of operating times, we used the data processing software MINITAB17 [25] in order to obtain the results mentioned in Figure 7 for TTR.

From the results obtained we notice that the value of p-value of the TTR for all distributions is greater than 0.05. This last value allows us to accept the hypotheses H0.

#### 3.2.1.2 Adjustment tests on XL-Stat

Using this software, we performed a law fit with a risk of 5% and we adopted the method of estimation of maximum likelihood.

The different laws tested and their value of P can be summarized in Table 1. The distribution that best fits the data for the fit test is the Weibull distribution (2).

Its estimated parameters are grouped in Table 2.

Figure 7. Anderson-Darling analysis of the TTR.


#### Table 1.

The TTR distributions and their p-value.


#### Table 2.

The estimated parameters of Weibull distribution (2).

The XL-stat software offers several tests to ensure this distribution: Kolmogorov–Smirnov test and chi-square test.

Following the same steps, after having conducted the normality test for TBF, we found p-value = 0.020 < 0.05. So we will accept the hypothesis H1 and we can say that the data of the TBF do not follow the normal law.

According to the method of maximum likelihood, we estimate the value of the

σ 1.726

Performance indicator Distribution law Parameters Value TTR Exponential λ<sup>1</sup> 0.952 TBF Normal Log μ 2.097

Distribution p-Value α (%) Weibull (1) 0.560 32.43 Exponential 0.638 10.65 Log-normal 0.774 21.40 Gamma (2) 0.901 28.72 Normal 0.929 27.37 Logistic 0.964 22.06 GEV 0.966 20.60 Weibull (2) 0.969 32.43

Impact of Improving Machines' Availability Using Stochastic Petri Nets on the Overall…

Distribution p-Value α (%) Exponential 0.382 80.4 Fisher-Tippett (2) 0.593 1.13 Normal 0.642 0.37 Logistic 0.759 0.25 Log-normal 0.800 0.5 Gamma (2) 0.882 7.31 Weibull (2) 0.906 4.40

For our case, we worked on modeling a single machine (Sieve) using stochastic Petri nets because we can use this type of model to take into account probabilistic events such as the failure of a machine moreover it allows to model tasks with non deterministic execution times and to evaluate the performances of the system. The Petri Net modeling in Figure 8 represents the behavior of a sieve.

parameter of each law (Table 5).

Parameters estimated for each law.

This network has two places:

• Pl1: Running

89

Table 3.

Table 4.

Table 5.

The TTR distributions and their risk α.

DOI: http://dx.doi.org/10.5772/intechopen.82445

The TBF distributions and their risk α.

• Pl2: Out of order

3.3 Petri net modeling for preventive maintenance

The Anderson-Darling analysis showed that, for TBF, the value of p for some distributions is less than 0.05. As a result, we can conclude that none of the distributions is the one that best fits the data. Hence, the need to make further adjustment tests on the XL-Stat software that will be most appropriate.

#### 3.2.2 Determination of transition laws

All tests that are already done do not give an exact distribution for our database. To solve this problem we compare the risk α (the risk of rejecting the null hypothesis H0 when it is true) for the distributions whose value of p-value is greater than 0.5. (H0: data comes from a specific distribution) (Tables 3 and 4).

After several adjustment tests on the two performance indicators TTR and TBF and using the comparison between the risk values α, we find that the good distribution for the TTR is the Exponential distribution and for the TBF is a normal Log distribution.

Impact of Improving Machines' Availability Using Stochastic Petri Nets on the Overall… DOI: http://dx.doi.org/10.5772/intechopen.82445


#### Table 3.

The TTR distributions and their risk α.


#### Table 4.

The TBF distributions and their risk α.


#### Table 5.

The XL-stat software offers several tests to ensure this distribution:

Following the same steps, after having conducted the normality test for TBF, we found p-value = 0.020 < 0.05. So we will accept the hypothesis H1 and we can say

Parameters Value Standard error Beta 1.919 0.659 Gamma 1.187 0.267

Distribution p-Value Normal standard 0.010 Student 0.010 Fisher-Tippett (1) 0.064 Gumbel 0.089 Gamma (1) 0.191 Erlang 0.200 Khi<sup>2</sup> 0.516 Weibull (1) 0.560 Exponential 0.638 Log-normal 0.774 Gamma (2) 0.901 Normal 0.929 Logistic 0.964 GEV 0.966 Weibull (2) 0.969

The Anderson-Darling analysis showed that, for TBF, the value of p for some distributions is less than 0.05. As a result, we can conclude that none of the distributions is the one that best fits the data. Hence, the need to make further adjust-

All tests that are already done do not give an exact distribution for our database. To solve this problem we compare the risk α (the risk of rejecting the null hypothesis H0 when it is true) for the distributions whose value of p-value is greater than

After several adjustment tests on the two performance indicators TTR and TBF and using the comparison between the risk values α, we find that the good distribution for the TTR is the Exponential distribution and for the TBF is a normal Log

Kolmogorov–Smirnov test and chi-square test.

The estimated parameters of Weibull distribution (2).

3.2.2 Determination of transition laws

distribution.

88

Table 1.

Table 2.

The TTR distributions and their p-value.

Maintenance Management

that the data of the TBF do not follow the normal law.

ment tests on the XL-Stat software that will be most appropriate.

0.5. (H0: data comes from a specific distribution) (Tables 3 and 4).

Parameters estimated for each law.

According to the method of maximum likelihood, we estimate the value of the parameter of each law (Table 5).

#### 3.3 Petri net modeling for preventive maintenance

For our case, we worked on modeling a single machine (Sieve) using stochastic Petri nets because we can use this type of model to take into account probabilistic events such as the failure of a machine moreover it allows to model tasks with non deterministic execution times and to evaluate the performances of the system.

The Petri Net modeling in Figure 8 represents the behavior of a sieve. This network has two places:


And two transitions:


The simulation results are presented in Tables 6 and 7.

From the results of the simulation, it can be seen that the residence time of the sieve in working order is equal to 24.13 times the residence time in the state of failure. For the crossing frequencies of the two transitions, they are approximately equal in view of the nature of our operating process, that is to say a loop sequence. The modeling allowed us to conclude that our machine has a good availability as

<sup>714</sup>:<sup>39</sup> <sup>þ</sup> <sup>29</sup>:<sup>612</sup> <sup>¼</sup> 96% (4)

the time of breakdowns is negligible compared to the time of stay in operation. The application of this preventive maintenance plan will have an important impact on the availability of the equipment which is best manifested in the simulation conducted. According to the calculation of the performance using this model-

Impact of Improving Machines' Availability Using Stochastic Petri Nets on the Overall…

Thus, the strength of the stochastic petri nets was demonstrated as a tool of

In these applications, we have demonstrated the robustness of stochastic petri nets in the field of maintenance in two different sectors for the improvement of

The objective of this chapter is to demonstrate the robustness of stochastic petri nets in the field of maintenance for the improvement of machine availability. We presented the modeling of the maintenance function in a production site with stochastic Petri nets by using two performance indicators: the mean time between failures and the mean time to repair to improve the equipment performance.

The determination of the distribution law is essential for each statistical study and provides a powerful and reliable model for the evaluation of the equipment performance in the automotive sector and the food sector. After determining these laws we switched to modeling Petri nets, we proposed the establishment of an effective preventive maintenance plan which aims at increasing the reliability, thus reducing the probability of failures. Consequently, we increase the machines' avail-

ability and then the overall equipment effectiveness.

Our study is based on the values of the two indicators (MTBF and MTTR) calculated within a company working in automotive sector and another one operating in food sector. The determination of the distribution law is essential for each statistical study and provides a powerful and reliable model for the evaluation of the equipment performance by incorporating preventive maintenance. After that, we developed a preventive maintenance plan that improves availability of machines. For each sector, we obtained an increase in availability rate thanks to SPNs

ing, we find an average availability of the sieve which equals:

DOI: http://dx.doi.org/10.5772/intechopen.82445

3.4 Synthesis

methodology.

4. Conclusion

91

machines' availability.

<sup>D</sup> <sup>¼</sup> <sup>714</sup>:<sup>39</sup>

modeling allowing the availability of the machine studied to be improved.

After modeling the maintenance function of the sieve, it became easy to predict the next breakdown and when it will occur. So, we developed a high-quality preventive maintenance plan based on the modeling realized.

Figure 8. Modeling with Petri nets.


Table 6.

Simulation result by GRIF-residence time of places.


Table 7.

Simulation result by GRIF-Frequency of sorting period transitions.

Impact of Improving Machines' Availability Using Stochastic Petri Nets on the Overall… DOI: http://dx.doi.org/10.5772/intechopen.82445

The modeling allowed us to conclude that our machine has a good availability as the time of breakdowns is negligible compared to the time of stay in operation.

The application of this preventive maintenance plan will have an important impact on the availability of the equipment which is best manifested in the simulation conducted. According to the calculation of the performance using this modeling, we find an average availability of the sieve which equals:

$$\text{D} = \frac{714.39}{714.39 + 29.612} = 969\% \tag{4}$$

Thus, the strength of the stochastic petri nets was demonstrated as a tool of modeling allowing the availability of the machine studied to be improved.

#### 3.4 Synthesis

And two transitions:

Maintenance Management

Figure 8.

Table 6.

Table 7.

90

Modeling with Petri nets.

Name Number Residence

time

Simulation result by GRIF-residence time of places.

Pl2:2 2 2.9612E1 7.3264 3.9801E-

Simulation result by GRIF-Frequency of sorting period transitions.

Standard deviation (σ) Number of jets

2

Name ID Frequency of sorting period transitions

Tr1: 1 1 3.07E1 Tr2: 2 2 3.06E1

Pl1:1 1 7.1439E2 7.3264 0.9602 9.8473E3 0.9 0.3162

Mean standard deviation Number of jets

9.8473E3 0.1 0.3162

Final standard deviation

• Tr1: Equipment failure

• Tr2: Equipment repaired

The simulation results are presented in Tables 6 and 7.

ventive maintenance plan based on the modeling realized.

From the results of the simulation, it can be seen that the residence time of the sieve in working order is equal to 24.13 times the residence time in the state of failure. For the crossing frequencies of the two transitions, they are approximately equal in view of the nature of our operating process, that is to say a loop sequence. After modeling the maintenance function of the sieve, it became easy to predict the next breakdown and when it will occur. So, we developed a high-quality pre-

> In these applications, we have demonstrated the robustness of stochastic petri nets in the field of maintenance in two different sectors for the improvement of machines' availability.

Our study is based on the values of the two indicators (MTBF and MTTR) calculated within a company working in automotive sector and another one operating in food sector. The determination of the distribution law is essential for each statistical study and provides a powerful and reliable model for the evaluation of the equipment performance by incorporating preventive maintenance. After that, we developed a preventive maintenance plan that improves availability of machines.

For each sector, we obtained an increase in availability rate thanks to SPNs methodology.

#### 4. Conclusion

The objective of this chapter is to demonstrate the robustness of stochastic petri nets in the field of maintenance for the improvement of machine availability. We presented the modeling of the maintenance function in a production site with stochastic Petri nets by using two performance indicators: the mean time between failures and the mean time to repair to improve the equipment performance.

The determination of the distribution law is essential for each statistical study and provides a powerful and reliable model for the evaluation of the equipment performance in the automotive sector and the food sector. After determining these laws we switched to modeling Petri nets, we proposed the establishment of an effective preventive maintenance plan which aims at increasing the reliability, thus reducing the probability of failures. Consequently, we increase the machines' availability and then the overall equipment effectiveness.

Maintenance Management
