Comprehensive Mathematical Model for Condition-Based Maintenance (CBM): Maximizing Reliability and Minimizing Costs

 Comprehensive Mathematical Model for Condition-Based Maintenance (CBM): Maximizing Reliability and Minimizing Costs

By Aleksandar Pudar

Technical Superintendent and Planned Maintenance Supervisor Reederei Nord BV

Co-founder of "Out of Box Maritime Thinker Blog" and founder of "Narenta Gestio Consilium Group."

1. Introduction and Objective

Condition-Based Maintenance (CBM) is a maintenance strategy that monitors the actual condition of assets to decide on the appropriate maintenance actions. (Jardine et Banjevic., 2006). The primary aim of the mathematical model is to minimise maintenance costs while maximising equipment reliability and availability. (Ebeling, 1997).

2. Variables and Parameters

• T: Time between maintenance actions.
• C(T): Cost function over time, which includes both preventive maintenance costs (CP) and corrective failure costs (Cf).
• R(T): Reliability function representing the probability of equipment functioning without failure over time.
• L: Lube oil quality indicator derived from lube oil analysis.
• V: Vibration level derived from vibration monitoring data.
• θ: Temperature or thermal condition derived from thermal imaging.

3. Assumptions

• Equipment failure rate follows a Weibull distribution, commonly used for reliability analysis of mechanical systems.
• Maintenance cost is a function of both preventive and corrective actions.
• Equipment condition degrades over time and is influenced by operating conditions and maintenance quality.

4. Mathematical Relationships

• Reliability Model:

where η and β are shape and scale parameters of the Weibull^[1] distribution, respectively 

• Cost Model:
  

, where 

 is the preventive maintenance cost, and 𝐶𝑓is the failure cost.

• Condition Indicators:

L, V, and Ѳ directly influence the decision to adjust T scheduling maintenance more immediately to avoid further damage.

5. Optimisation Problem

• Objective: Find T that minimises 𝐶(𝑇) while ensuring 𝑅(𝑇)remains above a desired threshold, indicating that the equipment will likely operate reliably until the next scheduled maintenance.
• Optimisation Problem Statement:

·         Minimize 𝐶(𝑇)=400⋅𝑇+8000⋅(1−𝑅(𝑇))

·         Subject to 𝑅(𝑇)≥0.90

6. Example: Emergency Fire Pump

Objective: Minimise the emergency fire pump's maintenance and operational costs, maximising its reliability and readiness for emergencies.

Provided Data:

• Low-Pressure Suction: -0.2 bar
• High-Pressure Discharge: 8.0 bar
• Current: 52 Amps
• Lube oil: Result 8/10
• Vibration level: 2.73 mm/s.
• Temperature: 15°C (mechanical seal temperature)
• Operating Condition: Ballast (indicating the ship's cargo condition)

Updated Variables and Parameters:

• T: Time between maintenance actions. Assuming 12 monthly in-depth maintenance.
• C(T): Cost function over time. Preventive maintenance costs are €400 per month, and failure costs could increase to €8000.
• R(T): Reliability function, which we'll keep as is since it is a unitless probability.
• L: A score from 0 to 10, with 10 indicating new oil. This will remain the same.
• V: Vibration level, the safe threshold is 4 mm/s (RMS).
• θ: Temperature or thermal condition. The standard operating range is 15 - 60°C, with deviations indicating potential issues.

Assumptions:

• The equipment failure rate follows a Weibull distribution with 𝜂=730η=730 days (2 years) and 𝛽=1.5, suggesting a wear-out failure mode.
• Maintenance cost is a function of preventive and corrective actions due to failures.
• Equipment condition degrades over time.

Mathematical Relationships:

• Reliability Model:

where 𝜂=730η= (2 years) and 𝛽=1.5.

• Cost Model:

𝐶(𝑇)=400⋅𝑇+8000⋅(1−𝑅(𝑇))

Assuming preventive maintenance costs €400 per month and failure costs can soar up to €8000 in case of malfunction. The formula adjusts the failure cost based on the probability of failure, which increases as the time since the last maintenance extends.

Condition Indicators:

• Lube Oil Quality (L): 8/10, suggesting good lubrication status.
• Vibration Level (V): 2.73 mm/s, well below the 4 mm/s threshold, indicating stable mechanical operation.
• Temperature (θ): 15°C within the normal operating range (15-60°C), showing no immediate thermal risks.

These indicators suggest the pump operates efficiently, but ongoing monitoring is crucial to maintaining this status.

Optimisation Problem:

• Problem Statement: Determine the optimal maintenance schedule 𝑇T for the Emergency Fire Pump that minimises the overall maintenance and failure costs while ensuring the pump's reliability remains above a critical threshold to guarantee its readiness for emergencies.
• Solution Approach:
• Objective Function:

Minimize 𝐶(𝑇)=400⋅𝑇+8000⋅(1−𝑅(𝑇))

• Constraints:
• Maintain 𝑅(𝑇) above a desired threshold, say 0.90, to ensure high reliability.
• Maintain operational limitations such as vibration and temperature within safe limits.
• Numerical Solution: Use numerical optimisation techniques, possibly incorporating constraint programming, to find the 𝑇T that offers the best balance between maintenance frequency and cost efficiency. Simulation or scenario analysis might evaluate different maintenance intervals and their impacts on costs and pump failure probability.

7. Additional Considerations

7.1 Weibull Distribution

The probability density function (PDF) of the Weibull distribution for a random variable 𝑋 is given by:

Where:

• 𝑋 is the variable
• 𝜆>0 is the scale parameter
• 𝑘>0 is the shape parameter

The shape parameter 𝑘k determines the type of distribution:

• If 𝑘=1, the Weibull distribution simplifies to an exponential distribution.
• If 𝑘<1, the distribution models a phenomenon with a high failure rate initially, which decreases over time (often used for items that fail early on, such as new products with manufacturing defects).
• If 𝑘>1, the distribution models a phenomenon where the failure rate increases (commonly used for products that wear out over time, like mechanical components).

The scale parameter 𝜆 essentially stretches or compresses the distribution along the x-axis, affecting its spread but not its general shape. The Weibull distribution is beneficial because it can model various data distributions by adjusting 𝑘 and 𝜆, making it versatile for statistical modelling and analysis in multiple fields. (National Institute of Standards and Technology, 2021).

 

 

8. Updated Approach

8.1 Continuous Monitoring Simulation - Overview

Daily Conditions and Maintenance Triggers:

• L (Lube Oil Quality): Starts at 10 and degrades by up to 0.01 daily (if working). Maintenance is triggered if it drops below 7.
• V (Vibration Level): Starts at 2.0 and increases by up to 0.02 daily (if working). Maintenance is triggered if it exceeds 3.5.
• θ (Temperature): Starts at 25°C and fluctuates daily with an average variation (standard deviation of 1.5°C). Maintenance is triggered if it goes below 15°C or above 60°C.

Costs:

• Preventive Maintenance Cost: €500 each time maintenance is triggered.
• Daily Operational Cost: €10 for each day without maintenance.

This setup is intended to continuously monitor the condition of the equipment, reacting in real-time when any parameter exceeds safe thresholds. Preventing severe equipment failures will ensure minimal downtime and cost.

8.2 Predictive and Prescriptive Maintenance Simulation – Overview

IoT-Based Predictive and Prescriptive Maintenance Setup:

• IoT Sensors: Sensors continuously monitor lube oil quality (L), vibration levels (V), and temperature (θ) around the clock.
• Data Processing: Real-time data analysis using machine learning models to predict when maintenance thresholds will likely be breached based on current trends and historical data.
• Prescriptive Algorithms: The system prescribes specific maintenance actions once a potential issue is identified. These could include adjusting operating parameters, scheduling part replacements, or performing other maintenance tasks.
• Automation and Alerts: Automated alerts notify maintenance teams of predicted issues and prescribed actions, allowing for an immediate response before conditions deteriorate.
• Cost Adjustments: The costs are slightly adjusted to account for the infrastructure and operational expenses of running IoT sensors and data processing systems.

Simulation Logic with IoT Continuous Monitoring:

• Continuous Real-Time Monitoring: Replace daily checks with continuous monitoring. Sensory data is analysed in real time to generate predictive alerts.
• Maintenance Trigger: Maintenance is no longer scheduled at regular intervals but is triggered by predictive alerts based on real-time data analysis.
• Maintenance Cost: Assume an increased preventive maintenance cost of € 600 per session due to the advanced technologies used.
• Operational Cost: Given the enhanced monitoring and data analysis capabilities, consider a nominal IoT operational cost of € 1.4 per day.

Cost Estimation with IoT-Based System:

• Maintenance Frequency: Due to effective early intervention and continuous monitoring, maintenance is assumed to be needed only once a year.
• Maintenance Cost: 1 sessions × € 600/session = € 600
• Operational Cost: 365 days × €1.4/day = €511

Total Annual Cost:

Total Cost with IoT-based monitoring:

€600 + €  511= € 1111

This refined IoT approach for 24/7 monitoring minimises downtime and maintenance frequency and ensures that maintenance actions are highly targeted and efficient, significantly reducing the likelihood of severe machine failures.

See Appendix 1 & Appendix 2.

Conclusion

The mathematical model for Condition-Based Maintenance (CBM) provided in this document integrates the core aspects of CBM, including data from condition monitoring techniques like vibration analysis, thermal imaging, and lube oil analysis, to drive maintenance decisions. The goal is to optimise the trade-off between maintenance costs and equipment reliability. Adopting continuous and predictive maintenance strategies, especially with IoT-based monitoring, can significantly enhance overall maintenance efficiency and equipment reliability.

This document provides a comprehensive guide to developing and implementing a CBM model, emphasising the importance of accurate data, appropriate mathematical modelling, and continuous monitoring to achieve optimal maintenance outcomes.

References & Bibliography :

1. Books and Textbooks:

• Ebeling, C.E., 1997. An Introduction to Reliability and Maintainability Engineering. New York: McGraw-Hill.
• Jardine, A.K.S. and Tsang, A.H.C., 2013. Maintenance, Replacement, and Reliability: Theory and Applications. 2nd ed. Boca Raton, FL: CRC Press.
• Meeker, W.Q. and Escobar, L.A., 1998. Statistical Methods for Reliability Data. New York: John Wiley & Sons.

2. Articles and Papers:

• Mobley, R.K., 2002. An Introduction to Predictive Maintenance. 2nd ed. Boston: Butterworth-Heinemann.
• Lu, C.J. and Weng, S., 2008. Condition-Based Maintenance Decision-Making for Equipment Under Variable Working Conditions. Journal of Quality in Maintenance Engineering, 14(1), pp.63-74.
• Jardine, A.K.S., Lin, D. and Banjevic, D., 2006. A Review on Machinery Diagnostics and Prognostics Implementing Condition-Based Maintenance. Mechanical Systems and Signal Processing, 20(7), pp.1483-1510.

3. Standards and Guidelines:

• International Organization for Standardization (ISO), 2011. ISO 17359:2011 Condition Monitoring and Diagnostics of Machines — General Guidelines. Geneva: ISO.
• International Organization for Standardization (ISO), 2014. ISO 55000:2014 Asset Management — Overview, Principles and Terminology. Geneva: ISO.

4  Technical Reports:

• NASA Office of Safety and Mission Assurance, 2012. NASA Reliability-Centered Maintenance Guide for Facilities and Collateral Equipment. NASA Technical Report.
• U.S. Department of Energy, 2010. Operations & Maintenance Best Practices: A Guide to Achieving Operational Efficiency. Washington, D.C.: U.S. Department of Energy.

5  Web Resources:

• National Institute of Standards and Technology (NIST), 2021. Weibull Distribution. Available at: https://www.itl.nist.gov/div898/handbook/eda/section3/eda3668.htm [Accessed 27 June 2024].
• Reliability Hotwire Magazine, 2001. The Basics of Weibull Distribution. Available at: http://www.weibull.com/hotwire/issue14/hottopics14.htm [Accessed 27 June 2024].

Appendix 1 - Simulation Logic with IoT Continuous Monitoring for Emergency Fire Pump

Example: Emergency Fire Pump with IoT-Based Continuous Monitoring

Objective: Minimise maintenance and operational costs of the emergency fire pump, maximising its reliability and readiness for emergencies.

Provided Data:

• Low-Pressure Suction: -0.2 bar
• High-Pressure Discharge: 8.0 bar
• Current: 52 Amps
• Lube oil: Result 8/10
• Vibration level: 2.73 mm/s.
• Temperature: 15°C (mechanical seal temperature)
• Operating Condition: Ballast (indicating the ship's cargo condition)

Updated Variables and Parameters:

• T: Time between maintenance actions. IoT monitoring will trigger maintenance based on real-time data rather than fixed intervals.
• C(T): Cost function over time. Due to advanced IoT technologies, preventive maintenance costs are now €600 per session, and failure costs could increase to €8000.
• R(T): Reliability function, which we'll keep as is since it is a unitless probability.
• L: A score from 0 to 10, with 10 indicating new oil. This will remain the same.
• V: Vibration level, the safe threshold is 4 mm/s (RMS).
• θ: Temperature or thermal condition. The standard operating range is 15 - 60°C, with deviations indicating potential issues.

Simulation Logic:

1. Initial Conditions:
• Lube Oil Quality (L): 10
• Vibration Level (V): 2.0 mm/s
• Temperature (θ): 25°C
2. Daily Monitoring and Degradation Rates:
• Lube Oil Quality (L): Degrades up to 0.01 daily if the pump is operational.
• Vibration Level (V): Increases up to 0.02 daily if the pump is operational.
• Temperature (θ): Fluctuates daily with an average variation (standard deviation of 1.5°C).
3. Maintenance Triggers:
• Lube Oil Quality (L): Maintenance is triggered if it drops below 7.
• Vibration Level (V): Maintenance is triggered if it exceeds 3.5 mm/s.
• Temperature (θ): Maintenance is triggered if it goes below 15°C or above 60°C.
4. Costs:
• Preventive Maintenance Cost: €600 each time maintenance is triggered.
• Daily Operational Cost: € 1.4 per day due to IoT monitoring.
5. Simulation Execution:
• Step 1: Initialize simulation with given starting values.
• Step 2: For each day:
• Update L, V, and θ based on their respective degradation rates and variations.
• Check if any parameter exceeds its threshold.
• If a parameter exceeds its threshold, trigger maintenance, reset the parameter to its initial value, and add the preventive maintenance cost to the total Cost.
• Add the daily operational Cost to the total Cost.
• Step 3: Repeat step 2 for the entire simulation period (e.g., one year).
6. Example Calculation for One Year:
• Initial Conditions:
• 𝐿=10L=10
• 𝑉=2.0V=2.0 mm/s
• 𝜃=25θ=25°C
• Simulation Iterations:
• Day 1:
• Update: 𝐿=9.99, 𝑉=2.02 mm/s, θ=25.1°C
• No maintenance was triggered.
• Total Cost: € 1.4
• Day 2:
• Update: 𝐿=9.98, 𝑉=2.04 mm/s, 𝜃=26.0°C
• No maintenance was triggered.
• Total Cost: € 2.8
• ...
• Day X (when maintenance is triggered):
• 𝐿=6.99 (below threshold)
• Maintenance triggered, reset 𝐿 to 10.
• Add preventive maintenance cost: €600
• Total Cost: Previous Total + €600 + €1.4 (daily operational cost)
• Continue the simulation for 365 days.
7. Annual Cost Calculation:
• Assume maintenance is triggered 1 times a year based on the degradation rates and threshold limits.
• Preventive Maintenance Cost: 1 sessions × €600/session = €600
• Operational Cost: 365 days × €1.4/day = € 511
• Total Annual Cost: € 600 + € 511 = € 1111

Summary of Simulation Results

By continuously monitoring the emergency fire pump using IoT sensors, the maintenance actions can be precisely targeted based on real-time data, reducing the likelihood of severe failures and optimising maintenance costs. The total annual Cost for maintaining the pump with IoT-based monitoring is estimated at €1111, reflecting the benefits of proactive and data-driven maintenance strategies.

Appendix 2 - Simulation Logic with IoT Continuous Monitoring for Emergency Fire Pump

Provided Data and Parameters

• Initial Conditions:
• Lube Oil Quality (L): 10
• Vibration Level (V): 2.0 mm/s
• Temperature (θ): 25°C
• Daily Monitoring and Degradation Rates:
• Lube Oil Quality (L): Degrades by 0.001 daily if the pump is operational.
• Vibration Level (V): Increases by 0.0002 mm/s daily if the pump is operational.
• Temperature (θ): Fluctuates daily with an average variation (standard deviation of 1.5°C).
• Maintenance Triggers:
• Lube Oil Quality (L): Maintenance is triggered if it drops below 7.
• Vibration Level (V): Maintenance is triggered if it exceeds 3.5 mm/s.
• Temperature (θ): Maintenance is triggered if it goes below 15°C or above 60°C.
• Costs:
• Preventive Maintenance Cost: €600 each time maintenance is triggered.
• Daily Operational Cost: €1.4 per day due to IoT monitoring.( equipment+DDS)
• Major Overhaul Cost (every 5 years): €5000

Simulation Execution with IoT-Based Monitoring

Scenario 1: IoT-Based Monitoring with Triggered Maintenance

1. Initial Conditions:
• 𝐿=10L=10
• 𝑉=2.0V=2.0 mm/s
• 𝜃=25θ=25°C
2. Daily Degradation and Monitoring:
• Lube Oil Quality (L):
• Degrades by 0.001 per day.
• Threshold for maintenance: L < 7 (after approximately 3000 days).
• Vibration Level (V):
• Increases by 0.0002 mm/s per day.
• Threshold for maintenance: V > 3.5 mm/s (after approximately 7500 days).
• Temperature (θ):
• Fluctuates daily with an average variation (standard deviation of 1.5°C).
• Maintenance is triggered if outside the 15°C to 60°C range.
3. Maintenance Schedule:
• Given the degradation rates, no maintenance is triggered within one year based on Lube Oil Quality (3000 days threshold) and Vibration Level (7500 days threshold).
4. Costs:
• Preventive Maintenance:
• Major overhaul: Every 5 years = €5000 (prorated to €1000 per year)
• Daily Operational Cost: 365 days × €1.4/day = €511

Total Annual Cost with IoT-Based Monitoring:

€1000 (prorated major overhaul) + €511(daily operational cost) = €1511 €

Scenario 2: Traditional Fixed Interval Maintenance

1. Maintenance Schedule:
• Preventive maintenance is performed without condition monitoring.
• Assume monthly maintenance at €600 per session. ( materials, manhours, spares)
• Annual major overhaul cost included €5000 (prorated to €1000 per year)
2. Costs:
• Preventive Maintenance:
• Monthly maintenance: 12 sessions × €600/session = €7200
• Prorated major overhaul: €1000
• Total Annual Cost:

·         €7200 (monthly maintenance) + €1000(prorated major overhaul) = €8200

Potential Savings with IoT-Based Monitoring

·         Total Annual Cost with IoT-Based Monitoring:

€1511

·         Total Annual Cost with Traditional Fixed Interval Maintenance:

€8200

·         Potential Annual Savings:

€8200 − €1511= € 6689

Summary of Potential Savings

Using IoT-based continuous monitoring, the emergency fire pump's maintenance costs can be significantly reduced compared to traditional fixed interval maintenance. The potential savings over one year amount to €6689, demonstrating the financial benefits of adopting proactive and condition-based maintenance strategies.

Disclaimer:

Out of Box Maritime Thinker © by Narenta Gestio Consilium Group 2024 and Aleksandar Pudar assumes no responsibility or liability for any errors or omissions in the content of this paper. The information in this paper is provided on an "as is" basis with no guarantees of completeness, accuracy, usefulness, or timeliness or of the results obtained from using this information. The ideas and strategies should never be used without first assessing your company's situation or system or consulting a consultancy professional. The content of this paper is intended to be used and must be used for informational purposes only.

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[1] Weibull distribution is a continuous probability distribution named after the Swedish engineer Waloddi Weibull, who described it in detail in 1951, though it was first identified by Fréchet in 1927. The Weibull distribution is widely used in reliability engineering, life data analysis, weather forecasting, and various other applications due to its flexibility in modelling different data types.