# Predictive Maintenance Scheduling with Failure Rate Described by Truncated Normal Distribution

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Complexity of Scheduling Problems

#### 1.2. Production and Maintenance Planning Practices

#### 1.3. Goals and Approaches

- Methods of achieving the reliability parameters of the truncated normal distribution, even in the case of the absence of complementary and reliable data on historical failure-free times;
- Methods for obtaining the best maintenance and production schedules where the goal is to maximize stability and robustness.

## 2. A Model of Failures

_{i}

_{,1},…,X

_{i}

_{,Ni}in the i-th period [(i − 1)T, iT], i = 1, …, r + 1, had normal distributions with parameters m ∈ R and σ > 0, truncated to the positive half axis. The value N

_{i}represents the number of failures (being, in general, a random variable) detected in the i-th period. It is worth noting that the cumulative distribution function (CDF) F(⋅) of such a distribution has the Equation (2):

_{i}(⋅) of the arbitrary random variable X

_{i}

_{,j}, where i = 1, …, N

_{i}, had the following Equation (3):

_{i}and σ

_{i}are parameters depending on the number of period i.

_{i}

_{,k}) and the variance Var(X

_{i}

_{,k}) of the normal distribution with parameters m

_{i}and σ

_{i}truncated to the positive half axis, with the PDF defined in (3), are given, respectively, by the following Equation (4):

_{i}.

_{i}

_{,k}, as the failure occurs, a repair time Y

_{i}

_{,k}begins immediately, and so on. Repair times Y

_{i}

_{,1}, …, Y

_{i}

_{,Ni}for i = 1, …, r + 1, are supposed to be exponentially distributed with PDFs g

_{i}(⋅) as seen in Equation (6):

_{i}> 0 is known for i = 1, …, r + 1, E(Y

_{i}

_{,k})—the mean value of Y

_{i}

_{,k}, and Var(Y

_{i}

_{,k})—the variance of Y

_{i}

_{,k}.

_{i}

_{,1}; in other words, we “deleted” the residual repair time Y

_{i−}

_{1,Ni}in the i-th period [(i − 1)T, iT]. Thus, we can write:

_{i}

_{,k}, Y

_{i}

_{,k}, for i = 1, …, r + 1, and k = 1, …, N

_{i}are supposed to be totally independent. Thus, the evolution of the system can be observed on successive cycles Z

_{i}

_{,k}= X

_{i}

_{,k}+ Y

_{i}

_{,k}, i = 1, …, r + 1, k = 1, …, N

_{i}which are independent random variables with PDFs defined as follows:

#### 2.1. Maximum Likelihood Approach

_{i}

_{,k}, k = 1, …, N

_{i}for any i = 1, …, r + 1, where n

_{i}is the observed value of N

_{i}. Thus, we have the following observations:

_{i}and σ

_{i}of truncated normal distribution as data x

_{1,1}, x

_{1,2}, …, x

_{1,n1}which are independent and identically distributed random variables. We defined the likelihood function as follows [32,33]:

_{1}and σ

_{1}, we obtain the following system of normal equations:

_{1}given by Equation (18) into Equation (19) we eliminated m

_{1}numerically, using one of the approximations of the cumulative distribution function and the probability density function of the standard normal distribution. Indeed, using the Maclaurin expansion, we obtain:

#### 2.2. Empirical Moments Approach

_{1,1}, x

_{1,2}, …, x

_{1,n1}for period [0, T] as in the previous section. We introduced the sample mean and the variance as follows [32,33]:

#### 2.3. Renewal Theory Approach

_{1}, ξ

_{2},… were nonnegative and independent random variables with the same distribution function B(t); then the following stochastic process:

_{1}, ξ

_{2},… with renewal moments ${t}_{n}={\displaystyle {\sum}_{i=1}^{n}{\xi}_{i}},n=1,2,\dots $

_{1}(t) [34]; thus, we obtained:

_{1}and σ

_{1}, we used a method of the Laplace or Laplace–Stieltjes in order to invert the right sides of Equations (27) and (28) on argument s. The right sides in Equations (27) and (28) are described by R

_{1}(t, m

_{1}, σ

_{1}) and R

_{2}(t, m

_{1}, σ

_{1}), respectively.

^{2}(t), respectively:

_{1}, …, n

_{r}are numbers of failures physically observed in successive periods [0, T], [T, 2T], …, [(r − 1)T, rT] and $\overline{n}=\frac{1}{r}{\displaystyle {\sum}_{i=1}^{r}{n}_{i}.}$ From the system of equations:

_{1}and σ

_{1}.

_{2}and σ

_{2}with n

_{2}instead of n

_{1}as the first estimator in Equation (29). Using ${\widehat{m}}_{1},\dots ,{\widehat{m}}_{r}$ and ${\widehat{\sigma}}_{1},\dots ,{\widehat{\sigma}}_{r}$ we found predictions for values ${\widehat{m}}_{r+1}$ and ${\widehat{\sigma}}_{r+1}$ using the classical regression.

#### 2.4. Predictions of Reliability Characteristics

_{0}= rT, the first failure occurs after time t:

_{i}

_{,1}, …, X

_{i}

_{,Ni}had normal distributions with parameters m ∈ R and σ > 0, truncated to the positive half-axis and estimated using the described approaches (Section 2.1, Section 2.2 and Section 2.3):

_{r}

_{+1,1}] (37). The prediction of MTTF was effective if ex-post error was less than 0.05.

_{r+1,1}on the stability and robustness criteria was investigated for robust schedule u. Different values of reliability predictions achieved by the described approaches were used to generate schedules. Reliability parameters that guarantee the achievement of the most stable and robust schedules in the event of disruptions were adopted.

## 3. A Predictive Scheduling Model of Production and Maintenance

_{z,vj}is the end time of operation v

_{j}of job j, v

_{j}= 1, …, V

_{j}, j = 1, …, J, t

_{r,vj}is the start time of operation v

_{j}of task j, d

_{j}is the deadline of job j, D

_{j}is delay in completing job j, and I

_{l}is the idle time of machine l, l = 1,…,L.

_{j}of job j in predictive schedule u; $s{t}_{j,{v}_{j}}\left(u*\right)$ is the start time of operation v

_{j}of job j in reactive schedule u*;

## 4. Mean Time to Failure MTTF Prediction

_{i}in period i. Suppose the predicted value of α

_{36}was 0.5, the expected distribution value (the mean repair time) was 2 h. After the machine failure, the rescheduled operations could be performed on parallel machines.

_{1}, …, m

_{35}and σ

_{1}, …, σ

_{35}of normal distributions truncated to the positive half-axis described the failure-free times. First, the parameters were estimated using the maximum likelihood method. Parameter σ

_{1}was numerically estimated using Equation (18) and parameter m

_{1}was estimated using Equations (19)–(21). After finding successive estimators m

_{1}, …, m

_{35}and σ

_{1}, …, σ

_{35}, values of ${\widehat{m}}_{36}$ and ${\widehat{\sigma}}_{36}$ were estimated for the future period [35T, 36T] using the least squares method or Gauss–Newton method (Table 2). The estimated models were described using the significance test for coefficients, standard error, R

^{2}, value of coefficient belonging to the interval with probability 95%, and p-value in Table 3. Values of estimators m

_{1}, …, m

_{35}and σ

_{1}, …, σ

_{35}together with the fitted functions are plotted in Figure 2.

_{1}, …, m

_{35}and σ

_{1}, …, σ

_{35}were computed using the formulas achieved for the empirical moments approach (Equations (23) and (24)). The functions describing m

_{1}, …, m

_{35}and σ

_{1}, …, σ

_{35}are presented in Table 2 and Figure 2. The predicted values of ${\widehat{m}}_{36}$ and ${\widehat{\sigma}}_{36}$ were equal 74.95 and 45.92 for the horizon [35T, 36T]. Mean time to failure was EX = 88.69. The increased probability of the machine failure was indicated at the time window [23, 125 + MTTR] (Equations (39) and (40)).

_{36,1}was equal to 80, 82,… or 100 and repair-time equaled 3 h.

## 5. Computer Simulation Results and Discussion

_{m}

_{+1,1}= 70, 72, …, E, where E depends on the size of the scheduling problem and represents the machine end time. The y-axis represents values of criteria SR (48), QR (49), and MIROS (51) of reactive schedules, where the maximum value obtained for the quality robustness and solution robustness criterion equaled to 1, QR(u**) = 1, SR(u**) = 1.

#### 5.1. Estimation of Reliability Characteristics for the Automotive Industry

_{1}= 229.23, m

_{2}= 0.06, m

_{3}= 395.95, m

_{4}= 170.79, m

_{5}= 110.97, m

_{6}= 199.81, m

_{7}= 127.54, m

_{8}= 104.80 h, and σ

_{1}= 229, σ

_{1}= 140.75, σ

_{2}= 0, σ

_{3}= 394.14, σ

_{4}= 119.33, σ

_{5}= 65.32, σ

_{6}= 213.52, σ

_{7}= 138.24, σ

_{8}= 79.66 h. The values ${\widehat{m}}_{9}$ and ${\widehat{\sigma}}_{9}$ of the truncated normal distribution were predicted using the regression method. The trend function describing m

_{1}, …, m

_{8}was f(m) = 4.3469⋅9

^{2}+ 28.624⋅9 + 149.44. The trend function describing σ

_{1}, …, σ

_{8}was f(σ) = −6.3399⋅9

^{2}+ 53.103⋅9 + 66.579. The normal distribution truncated to the positive half-axis was described by parameters ${\widehat{m}}_{9}$ =54.95 and ${\widehat{\sigma}}_{9}$ = 30.97, and the expected value equaled to EX = 57.53 for December. The technical inspection of the resource was planned at time 57 h. The increased probability of the machine failure was indicated at the time window [42, 71 + MTTR].

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**The period of increased risk of failure [a, b + mean time of repair (MTTR)] for the truncated normal distribution.

**Figure 2.**The models describing parameters $\widehat{\sigma}$, $\widehat{m}$ achieved using the Gauss–Newton method and the fitted functions.

**Figure 3.**The impact of the critical machine failure time on the solution robustness (SR), quality robustness (QR) and weighted function (FFr) of SR and QR for scheduling problems for reliability characteristics obtained using the maximum likelihood method (dataset I) and empirical moments method (dataset II).

**Figure 4.**Accuracy of prediction of failure-free time obtained using the empirical moments approach.

**Figure 5.**Reactive schedules assessed using solution robustness (SR), quality robustness (QR), and weighted function of SR and QR (FFr) for reliability input data anticipated using the maximum likelihood approach (dataset I) and empirical moments method (dataset II).

**Table 1.**Failure-free times x

_{i}

_{,k}of the critical machine collected for scheduling horizons, i = 1,2,…,35.

The Number of Failures of the Critical Machine (and Failure-Free Times) in Scheduling Horizon i | ||||
---|---|---|---|---|

9(90,90,100,110,100, 110,110,130,130) | 7(100,105,105, 120,120,105,105) | 9(110,90,105,120, 120,100,110,130,90) | 6(105,120,140, 160,180,200) | 10(90,130,135,90, 75,80,125,90,80,100) |

11(100,90,95,95,90 ,90,85,85,90,90,80) | 8(100,140,150, 140,180,100,105,80) | 6(50,100,150,150, 280,270) | 8(200,100,120, 130,150,90,100,90) | 9(90,130,100,90, 130,120,130,90,100) |

9(80,90,100,110,120, 125,120,100,110) | 7(100,120,125,135, 140,150,160) | 10(110,90,95,90,80, 100,90,100,90,90) | 11(100,90,80,80, 85,80,90,90,85,70,70) | 10(90,95,100,100, 90,150,80,90,80,80) |

9(80,90,100,110,120, 125,120,120,130) | 7(100,120,125,135, 140,140,160) | 8(90,90,100,110,120, 120,140,160) | 7(100,110,120, 145,155,150,160) | 9(120,110,105,105, 110,100,100,110,120) |

9(80,85,100,110,115, 110,120,120,140) | 8(80,100,120,125,135, 140,140,150) | 9(75,80,85,90,110, 120,130,140,150) | 9(70,80,100,110, 115,120,135,130,140) | 7(100,110,120, 130,140,140,180) |

11(50,60,60,70,80,85, 100,110,115,120,125) | 10(55,60,65,65,80,90, 110,120,140,145) | 7(90,100,130,140, 150,160,170) | 10(70,75,80,85,90, 95,100,110,120,120) | 11(55,55,60,80,85,90, 90,100,120,130,130) |

11(45,60,60,65,80,90, 100,110,115,135,135) | 11(45,55,60,65,70,90, 90,110,120,120,150) | 10(40,55,60,90,95, 90,110,125,135,155) | 9(80,90,90,95,100, 110,125,135,155) | 10(55,60,80,90,90, 90,105,125,135,140) |

**Table 2.**Calculation of parameters ${\widehat{m}}_{i}$, ${\widehat{\sigma}}_{i}$ and reliability characteristic mean time to failure (MTTF).

$\mathbf{The}\mathbf{Prediction}\mathbf{of}{\widehat{\mathit{\sigma}}}_{\mathit{m}+1}$$,{\widehat{\mathit{m}}}_{\mathit{m}+1}\mathbf{and}\mathit{MTTF}\mathbf{Using}\mathbf{the}\mathbf{Gauss-Newton}\mathbf{Method}$ | ||||
---|---|---|---|---|

Maximum Likelihood Approach | Empirical Moments Approach | |||

${\widehat{m}}_{36}$ | y = 95.70 + 0.429x − 0.023x^{2} | 80.18 | y = 99.25 − 0.988x + 0.009x^{2} | 76.39 |

${\widehat{\sigma}}_{36}$ | y = 52.27 + 0.09x − 0.023x^{2} | 44.38 | y = 63.27 − 1.464x + 0.028x^{2} | 47.22 |

The Prediction of ${\widehat{\sigma}}_{m+1}$, ${\widehat{m}}_{m+1}$, and MTTF Using the Least Squares Method | ||||

Maximum Likelihood Approach | Empirical Moments Approach | |||

${\widehat{m}}_{36}$ | y = −0.0239x^{2} + 0.4291x + 95.705 | 80.17 | y = 0.0103x^{2} – 1.0867x + 99.906 | 74.13 |

${\widehat{\sigma}}_{36}$ | y = $0.001{x}^{8\cdot {10}^{-15}}$ | 44.41 | y =0.0281x^{2} – 1.4997x + 63.531 | 45.95 |

**Table 3.**Evaluation of the estimated models achieved using the Gauss–Newton method with a maximal number of iterations: 50.

$\widehat{\mathit{m}}$ Estimated Using the Maximum Likelihood Approach | |||||||
---|---|---|---|---|---|---|---|

Coefficient | SE | t Statistic | R^{2} | p-Value | 95% Confidence Interval (Left End) | 95% Confidence Interval (Right End) | |

a0 | 95.70475 | 8.430837 | 11.35175 | 0.095 | 0.000000 | 78.53170 | 112.8778 |

a1 | 0.42915 | 1.079850 | 0.39741 | 0.693702 | −1.77044 | 2.6287 | |

a2 | −0.02390 | 0.029096 | −0.82125 | 0.417581 | −0.08316 | 0.0354 | |

$\widehat{\sigma}$ Estimated using the maximum likelihood approach | |||||||

a0 | 52.27978 | 6.464445 | 8.087281 | 0.040 | 0.000000 | 39.11214 | 65.44743 |

a1 | 0.09115 | 0.827988 | 0.110085 | 0.913030 | −1.59541 | 1.77771 | |

a2 | −0.00862 | 0.022310 | −0.386394 | 0.701762 | −0.05406 | 0.03682 | |

$\widehat{m}$ Estimated using the empirical moments approach | |||||||

a0 | 99.25615 | 12.08471 | 8.213366 | 0.083 | 0.000000 | 74.64040 | 123.8719 |

a1 | −0.98821 | 1.54785 | −0.638439 | 0.527731 | −4.14108 | 2.1647 | |

a2 | 0.00981 | 0.04171 | 0.235257 | 0.815508 | −0.07514 | 0.0948 | |

$\widehat{\sigma}$ Estimated using the empirical moments approach | |||||||

a0 | 63.27331 | 11.33485 | 5.58219 | 0.06209691 | 0.000004 | 40.18497 | 86.36164 |

a1 | −1.46463 | 1.45181 | −1.00883 | 0.320622 | −4.42186 | 1.49260 | |

a2 | 0.02830 | 0.03912 | 0.72350 | 0.474628 | −0.05138 | 0.10798 |

**Table 4.**The quality of best basic schedules obtained for job shop problems using the MOIA (Multi-objective immune algorithm).

The Size of the Problem | The Job Sequence | Cmax | F | T | I | FFy |
---|---|---|---|---|---|---|

7 × 6 | {5 2 3 4 6 0 1} | 117 | 273 | 330 | 0 | 1.02 |

8 × 7 | {5 7 2 3 4 0 6 1} | 126 | 335 | 434 | 0 | 1.02 |

9 × 8 | {8 5 2 3 0 4 7 6 1} | 137 | 415 | 553 | 0 | 1.01 |

10 × 9 | {8 5 2 3 4 6 9 0 7 1} | 164 | 530 | 756 | 0 | 0.99 |

11 × 10 | {3 8 2 5 4 7 6 9 0 1 10} | 169 | 623 | 863 | 0 | 1.05 |

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**MDPI and ACS Style**

Paprocka, I.; Kempa, W.M.; Ćwikła, G. Predictive Maintenance Scheduling with Failure Rate Described by Truncated Normal Distribution. *Sensors* **2020**, *20*, 6787.
https://doi.org/10.3390/s20236787

**AMA Style**

Paprocka I, Kempa WM, Ćwikła G. Predictive Maintenance Scheduling with Failure Rate Described by Truncated Normal Distribution. *Sensors*. 2020; 20(23):6787.
https://doi.org/10.3390/s20236787

**Chicago/Turabian Style**

Paprocka, Iwona, Wojciech M. Kempa, and Grzegorz Ćwikła. 2020. "Predictive Maintenance Scheduling with Failure Rate Described by Truncated Normal Distribution" *Sensors* 20, no. 23: 6787.
https://doi.org/10.3390/s20236787