# Integrated Optimization Model for Maintenance Policies and Quality Control Parameters for Multi-Component System

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## Abstract

**:**

## 1. Introduction

## 2. Problem Description

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- Type 1 (${F}_{1}$) is related to the mechanical failure of the machines in the system.
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- Type 2 (${F}_{2}$) is quality-related and is observed when the production process goes into an out-of-control state. When such failures are observed, an immediate shutdown occurs, and all corrective actions are carried out to restore the process to its normal operation (i.e., the in-control state). However, the process may also worsen due to external causes, such as operator mistakes, bad quality parts, environmental effects, etc. In this case, the process is reset to the in-control state.

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- Each automatic machine can process only one part at a time, which imposes a single characteristic to quality (CTQ).
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- Failure modes (${F}_{1}$ and ${F}_{2}$) are independent. Failure reports from the company’s records were used to obtain these probabilities.
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- The required resources to detect, maintain and restore the process are always available, so no waiting times are considered.

## 3. Proposed Integrated Optimization Model

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- Step 1: Defining the problem. The performance of the manufacturing system is significantly impacted by the breakdown of machines with multi-component.
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- Step 2: Select the quality control chart and maintenance policy to develop the model.
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- Step 3: Select the production system.
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- Step 4: Monitor the selected machine with multi-component using a CUSUM chart.
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- Step 5: Monitor the failure and repair rate. In this step, the data related to the mean time between failure and the mean time to repair all selected components were gathered and fitted for suitable distributions.
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- Step 6: Develop the integrated model based on CM, PM intervals, and CUSUM chart parameters.
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- Step 7: Solve the developed integrated maintenance policy and quality control mathematical model.
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- Step 8: Sensitivity analyses were conducted to illustrate the robustness of the developed model due to the stochastic nature of the problem under investigation.
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- Step 9: Discuss the obtained results.

- $E\left[{C}_{f}\right]$: the expected cost of false alarms, which includes the cost of both investigating and analyzing the false alarms, and it is given as

- $E\left[{C}_{s}\right]$: the expected sampling cost per cycle, which can be calculated as follows [45]:

- $E\left[{L}_{in}\right]$: the quality loss per unit of time in the control state, which is calculated using the Taguchi loss function (TLF), and is given as [45]

- ${E\left[{L}_{out}\right]}_{MC}$: the quality loss per unit time when the process is in an out-of-control state due to machine degradation, which is also calculated using the Taguchi loss function (TLF) and is given as [45]

- ${E\left[{L}_{out}\right]}_{E}$: the quality loss per unit of time when the process is in an out-of-control state due to external factors ($E$), which is also calculated using the Taguchi loss function (TLF) and is given as [45]

- $E\left[{C}_{r}\right]$: the expected cost of detecting and repairing the process due to external causes ($E$).

- ${E\left[{C}_{CM}\right]}_{{F}_{2}}$: the expected cost of resetting and restoring the process (through CM) after a downtime of Type 2 (${F}_{2}$), which is calculated as follows:

## 4. Case Study

## 5. Results

_{E}= 1, and the shift due to machine failure is δ

_{m/c}= 0.8, which happens randomly and causes the process means to shift from μ

_{0}to (μ

_{0}+ δ). The process quality monitoring is being monitored by using a CUSUM chart analysis. At this stage, the sought-after critical to quality characteristic (CTQ) is the shortage in filling the bottle’s water volume. According to the adapted quality control policy, samples of filled water bottles were taken from the production line every hour (h), inspected by the naked eye, confirmed, and measured using a liquid measuring cup. The CUSUM chart is implemented as shown in Figure 4. It is evident that there is a shift in the process mean. This shift in the process is due to failure in the Seals and O-rings of the filling head (component number one) and valve battery (component number three). To solve this issue in the quality acceptance and to eliminate the process shifts, the PM must be scheduled and periodically replace the O-rings, change the O-rings of the valve battery, or change the card and check the inner springs of the valve battery.

#### 5.1. Design of Experiments Based on One-Factor-at-a-Time

#### 5.2. Sensitivity Analysis

#### 5.3. Preventive Actions to Reduce the Defects

## 6. Conclusions

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- Managers can control the quality of produced units and monitor the production line and its different states using the proposed solution steps.
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- The proposed methodology helps to identify the optimal preventive maintenance interval needed to improve production output and minimize downtime with enhanced product quality.
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- A maintenance plan for a multi-component system was developed based on the optimal value of preventive maintenance interval. The result showed that the total cost was reduced by approximately 50% compared with the current system.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## List of Abbreviations

Notation | Descriptions | |

Decision variables | ||

h | Sample frequency (average number of samples obtained in one second (NO/second) | |

d | Decision Interval (hours) | |

k | Coefficient of control limit | |

n | Sample size | |

${T}_{i}^{PM}$ | Preventive maintenance interval (hours) | |

Parameters | ||

CTQ | Critical to quality characteristic | |

$f\left(\u220e\right)$ | The normal density function of quality characteristic (∎) | |

${n}_{i}^{CM}$ | Number of corrective maintenance actions | |

${n}_{i}^{PM}$ | Number of preventive maintenance actions | |

$E\left[{L}_{in}\right]$ | Quality loss per unit time in the control state (USD) | |

${E\left[{L}_{out}\right]}_{MC}$ | Quality loss per unit of time due to machine degradation (USD) | |

${E\left[{L}_{out}\right]}_{E}$ | Quality loss per unit of time due to external factors ($E$) (USD) | |

$N$ | Number of components | |

${n}_{i}^{PM}$ | Number of preventive maintenance for ith component | |

P | Production rate (Carton/hour) | |

S | Expected number of samples while the process is in-control | |

${R}^{\prime}$ | The proportion of non-conforming units when the process is in-control state | |

${R}_{{\delta}_{MC}}^{\prime}$ | Probability of non-conforming items produced due to machine failure | |

${R}_{{\delta}_{E}}^{\prime}$ | Probability of non-conforming items produced due to external factors | |

δ | Magnitude of shift | |

μ_{0} | Target value (mm) | |

σ_{0} | Standard deviation | |

λ | Process failure rate | |

λ_{1} | Failure rate due to an external factor | |

λ_{2} | Failure rate due to machine degradation | |

αi | The shape parameter of Weibull distribution for ith component | |

γi | Scale parameter of Weibull distribution for ith component | |

Δ | Tolerance factor | |

Cost parameters | ||

A | Scrap or rework cost (USD) | |

${C}_{CM}$ | Corrective maintenance cost (USD) | |

${C}_{pM}$ | Preventive maintenance cost (USD) | |

${\left[CQ\right]}_{P-F}$ | Cost of process quality loss (${\left[CQ\right]}_{P-F}$) (USD) | |

$E\left[{C}_{p}\right]$ | The expected cost of process quality (USD) | |

$E\left[{C}_{f}\right]$ | The expected cost of false alarms (USD) | |

$E\left[{C}_{s}\right]$ | Expected sampling cost per cycle (USD) | |

$E\left[{C}_{r}\right]$ | The expected cost of detecting and repairing the process due to external factors ($E$) (USD) | |

${E\left[{C}_{CM}\right]}_{{F}_{2}}$ | The expected cost of resetting and restoring the process (through CM) after a downtime of Type 2 (USD) | |

C_{f} | Cost of investigating a false alarm per unit of time (USD) | |

${C}_{S}$ | Cost of resetting (USD) | |

${C}_{PM}$ | The expected cost of preventive maintenance (PM) (USD) | |

${\left[CQ\right]}_{P-F}$ | The expected total cost of quality loss due to process failure | |

${E\left[TC\right]}_{\left(M\ast Q\right)CUSUM}$ | Expected total cost per unit of time (USD/hour) | |

F | Fixed cost of the sample (USD) | |

${FC}_{i}^{CM}$ | Fixed cost for corrective maintenance (USD) | |

${FC}_{p}^{CM}$ | Fixed cost of corrective maintenance of ith component (USD) | |

${FC}_{i}^{PM}$ | Fixed cost of preventing maintenance of ith component(USD) | |

${C}_{L}$ | The labor cost (USD/hour) | |

${C}_{PL}$ | Cost of production lost (USD/hour) | |

V | The variable cost of the sample | |

Statistical properties parameters | ||

${ARL}_{0}$ | Average run length in-control state (average number of samples taken before a false alarm occurs) | |

ARL_{1} | Average run length in an out-of-control state | |

${ARL}_{1}^{E}$ | Average run length due to external factors | |

${ARL}_{1}^{MC}$ | Average run length due to machine failure | |

Time parameters | ||

${t}_{i}^{CM}$ | Repair times required to perform corrective maintenance (hours) | |

${t}_{i}^{PM}$ | Repair times required to perform preventive maintenance (hours) | |

MTF | Mean time between process failure (hours) | |

${t}_{a}$ | An estimate of the time it takes to determine if assignable causes have occurred (hours) | |

${t}_{c}$ | Cycle time (hours) | |

t_{e} | Time evaluation period (hours) | |

t_{f} | False alarm search time (hours) | |

t_{r} | Expected time to reset the process (hours) | |

${t}_{i}^{CM}$ | Time required for corrective maintenance of ith component (hours) | |

${t}_{i}^{PM}$ | Time required for preventive maintenance of ith component (hours) | |

t_{s} | Time for the sample and plot a chart (hours) |

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**Figure 5.**The impact of selected decision parameters on the expected total cost. (

**a**) illustrates the impact of sample frequency (h) on the anticipated total cost, (

**b**) the anticipated total cost decrease with the increase in decision interval (d), (

**c**) the expected cost variations with the change in the control limit coefficient (k), (

**d**) the total cost decreases with the increase in PM interval until.

(N_{i}) | Components | Number | Shape Parameter αi | Scale Parameter γi (hr) | Component Cost during Replacement (USD) | Sub-Component/Consumable Cost through Repair (USD) | TR_{CMi}(h) | TR_{PMi}(h) |
---|---|---|---|---|---|---|---|---|

1 | Filling Head | 54 | 19,440 (USD 360/pc) | - | - | - | ||

Seals and O-rings for filling heads | 108 set | 3.7761 | 1294.3 | 8700 (USD 80.55/pc) | 90 | 2 | 4 | |

2 | Flowmeter | 54 | 1.4934 | 4426.8 | 162,000 (USD 3000/pc) | n/a | 1.25 | n/a |

3 | Valve Battery | 14 | 1.7101 | 3745.1 | 14,000 (USD 1000/pc) | 350 | 1 | 9 |

4 | Fill box Flux | 27 | 1.3329 | 4768.7 | 26,490 (USD 981/pc) | n/a | 1.5 | n/a |

Parameter | Value | Parameter | Value |
---|---|---|---|

$P$ (cartons/h) | 600 | $F$ (USD/bottle) | 0.016 |

${t}_{s}$ (h) | 0.33 | $V$ (USD/bottle) | 0.036 |

${t}_{f}$ (h) | 0.5 | ${C}_{r}$ | 0.138 |

${t}_{a}$ (h) | 0.5 | ${C}_{f}$ | 9.2 |

${t}_{r}$ (h) | 0.25 | ${C}_{s}$ | 9 |

$L$(USD)/h | 10 | Δ | 3 |

${C}_{PL}$ USD/h | 8 | A USD/bottle | 0.069 |

${\delta}_{E}$ | 1 | ${\delta}_{MC}$ | 0.8 |

Variables | Optimal Value |
---|---|

n | 1 |

d | ≃10 |

h | 5 |

k | 0.10166 |

${T}_{i}^{PM}$ | 800 |

E[TC]_{(M}_{∗}_{Q}_{) CUSUM} | 81.5479 |

Decision Variables | Ranges |
---|---|

n | 1 |

d | 8–≃10 |

h | 4–5 |

k | 0.10166 |

${T}_{i}^{PM}$ | 750–850 |

**Table 5.**Results of the sensitivity analysis and influence range of the basic variables on ${E\left[TC\right]}_{\left(M\ast Q\right)CUSUM}$.

Parameter | Basic Level 1 | Level 2 (+10%) | Level 3 (+20%) | E[CT]_{(M}_{∗}_{Q}_{)CUSUM} | |||
---|---|---|---|---|---|---|---|

Basic Level | Level 2 | Level 3 | Range | ||||

ts | 0.33 | 0.363 | 0.396 | 81.5479 | 81.5479 | 81.5479 | 81.5479 |

tf | 0.5 | 0.55 | 0.6 | 81.5479 | 81.5479 | 81.5479 | 81.5479 |

ta | 0.5 | 0.55 | 0.6 | 81.5479 | 81.5479 | 81.5479 | 81.5479 |

tr | 0.25 | 0.275 | 0.3 | 81.5479 | 81.5479 | 81.5479 | 81.5479 |

A | 0.069 | 0.0759 | 0.0828 | 81.5479 | 81.5479 | 81.5479 | 81.5479 |

F | 0.016 | 0.176 | 0.0192 | 81.5479 | 81.5479 | 81.5479 | 81.5479 |

V | 0.036 | 0.0396 | 0.0432 | 81.5479 | 81.5479 | 81.5479 | 81.5479 |

Cr | 0.138 | 0.1518 | 0.1656 | 81.5479 | 81.5479 | 81.5479 | 81.5479 |

Cf | 9.2 | 10.12 | 11.04 | 81.5479 | 81.5479 | 81.5479 | 81.5479 |

Cs | 9 | 9.9 | 10.8 | 81.5479 | 81.5479 | 81.5479 | 81.5479 |

Lp | 8 | 8.8 | 9.6 | 81.5479 | 89.6698 | 97.7117 | 81.5479–97.7117 |

L | 10 | 10.10 | 12 | 81.5479 | 81.5496 | 81.5815 | 81.5479–81.5815 |

No. | Components | |
---|---|---|

1 | Seals and O-rings for filling heads | |

Problem | Leakage in water | |

Root causes | Replace the Seals and O-rings | |

Solution | Periodically replace the O-rings | |

2 | Flowmeter | |

Problem | Water not flowing or continually flows | |

Root causes | Malfunction with the internal electronic card | |

Solution | Replace the flowmeter | |

3 | Valve Battery | |

Problem | Water not flowing according to the set level | |

Root causes | The presence of an air leak from the O-rings | |

An issue with the electronic card of the valve battery | ||

An issue in the inner Spring of the valve | ||

Solution | Change the O-rings of the valve battery or change the card. | |

Change the inner springs of the valve battery | ||

4 | Fillbox Flux | |

Problem | The filling cycle is not initiated, or filling heads are not working | |

Root causes | An issue with the motherboard. | |

The problem with the data cable | ||

The problem with the control system | ||

Solution | Check the electronic card for repair or replacement if available otherwise, replace it with a new Fillbox | |

Repair the cable data or replace | ||

Reinstall the program for the Fill box to reset the system |

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## Share and Cite

**MDPI and ACS Style**

Nasr, M.M.; Naji, F.; Amrani, M.; Ghaleb, M.; Alqahtani, K.N.; Othman, A.M.; Abualsauod, E.H. Integrated Optimization Model for Maintenance Policies and Quality Control Parameters for Multi-Component System. *Machines* **2023**, *11*, 435.
https://doi.org/10.3390/machines11040435

**AMA Style**

Nasr MM, Naji F, Amrani M, Ghaleb M, Alqahtani KN, Othman AM, Abualsauod EH. Integrated Optimization Model for Maintenance Policies and Quality Control Parameters for Multi-Component System. *Machines*. 2023; 11(4):435.
https://doi.org/10.3390/machines11040435

**Chicago/Turabian Style**

Nasr, Mustafa M., Fadia Naji, Mokhtar Amrani, Mageed Ghaleb, Khaled N. Alqahtani, Asem Majed Othman, and Emad Hashiem Abualsauod. 2023. "Integrated Optimization Model for Maintenance Policies and Quality Control Parameters for Multi-Component System" *Machines* 11, no. 4: 435.
https://doi.org/10.3390/machines11040435