# Optimum Preventive Maintenance Replacement Time in Clusters Using BlockSim

In certain cases, replacing a component preventively (i.e., before it fails) may make better economic sense than replacing the component correctively (i.e., when it fails). For any given component, you must first determine whether preventive replacement is appropriate; if it is, then the next task is to identify the best time to replace the component. Earlier articles in Reliability Hotwire and Reliability Edge have discussed issues surrounding optimum replacement times, such as choosing the optimum maintenance interval and optimum preventive maintenance replacement time for a single component. However, those articles focused on determining the best maintenance strategy for a single component. When a system consists of multiple components, it may be cheaper and/or more efficient to perform preventive maintenance and replacements in clusters. This article will discuss using BlockSim to determine optimum preventive maintenance replacement time in groups via a simple example.

## Preventive Replacement Applicability and Calculation of Optimum Time

Preventive replacement of a component is appropriate only if the component’s vulnerability to failure increases with time and if its preventive (planned) replacement cost (CP) is significantly less than its corrective (unplanned) replacement cost (CU). In other words, the component should have an increasing failure rate and it should be cheaper to replace it preventively than to replace it after it fails.

If both of these requirements are satisfied, the cost per unit of operating time can be plotted against the operating time (Figure 1). Corrective replacement cost increases with operating time as the component is more likely to fail, which requires more corrective actions. On the other hand, preventive replacement cost decreases as operating time increases because the longer the component operates between the preventive maintenances, the less the cost will be. The total cost is the sum of these two costs. A minimum cost is achievable at time t (the optimum preventive replacement time for the component) where the time between preventive replacements is maximized.

Figure 1: Cost per Unit of Operating Time vs. Operating Time

Assuming that the component is replaced upon failure or after a time of operation, t, the total cost can be mathematically represented as:

where:

• R(t) is the reliability of the component at time t.
• CP is the cost of planned replacement.
• CU is the cost of unplanned replacement.

The optimum replacement time interval, t, is the time that minimizes CPUT(t). Hence, the optimum replacement time can be obtained by solving for t:

These calculations and the theory behind them are explained for a single component. However, in the Synthesis version of BlockSim, the Optimum Replacement window can be used to determine the optimum replacement time either for an individual block or for multiple blocks in a diagram simultaneously. While working with multiple blocks, the calculations can be for individual blocks or for one or more groups of blocks.

## A Note About Clustering the Components

In BlockSim, a K-means clustering algorithm is used to group the components. Fundamentally, K-means clustering aims to classify the objects based on their attributes or features into K number of groups, where K is a positive integer. The sum of squares of distances between data and the corresponding cluster centroid is minimized. [2]

If the number of data points is less than the number of clusters, then each data point is recognized as the centroid of the cluster. Hence, the number of clusters will be equal to the number of data points. In BlockSim, the number of clusters, K, is a value entered by the user. If the number of groups entered is greater than or equal to the number of blocks being analyzed, then the results (optimum replacement times) will be equal to the ones calculated individually.

However, when the number of clusters entered is smaller than the number of data points, then the distance to all centroids will be calculated for each individual data point and the minimum distance will be acquired. Each data point will be assigned to the cluster that has the minimum distance from this data point. Each centroid location needs to be adjusted based on the current updated data because the location of the centroids are not precise. After determining the new location of each centroid, all the data points are assigned again to their new closest centroid. This process is repeated until the cluster locations are fixed, which means no data points are moving to another cluster anymore.

## Example

There are three options in BlockSim for calculating the optimum replacement times:

1. Calculating optimum replacement times individually.
2. Calculating a common optimum replacement time for all selected components.
3. Calculating optimum replacement times for K number of clusters.

This basic example briefly examines options 1 and 3.

### Problem Statement

A car manufacturer’s maintenance team would like to suggest to their customers an optimum replacement time for each of the car’s components that require routine maintenance. From earlier studies, we know the reliability data for each of the components, as well as the planned and unplanned replacement costs. It should be noted that the Weibull distributions of the components have shape parameters (β) greater than 1, which satisfies the requirement of having failure rates increasing with time. The second requirement is also met because all the preventive replacement costs are significantly less than the corrective replacement costs.

 Component Reliability Data (2P-Weibull) [miles] Planned Replacement Cost [\$] Unplanned Replacement Cost [\$] Oil & Oil Filter β:6 η:10,000 30 5,000 Water Pump β:2 η:80,000 700 4,000 Timing Belt β:2.5 η:80,000 800 4500 Air Filter β:3 η:15,000 25 1,500 Spark Plugs β:1.5 η:30,000 80 250 Fuel Filter β:3 η:30,000 70 800

### The Setup

Because the car’s proper functioning depends on the functioning of all the components, they are arranged in series in the reliability block diagram (RBD). Even though this will affect the reliability of the system (car), the arrangement does not have any effect on the optimum replacement times of the components, individually or in clusters.

After entering the reliability data for each component, we open the Optimum Replacement window and enter the planned and unplanned replacement costs, as shown next.

### The Results

When we click Calculate in the Optimum Replacment window, we then can specify whether the calculation should be performed for each component individually, for all components together or for clusters of components. First, we will examine the individual optimum replacement times in this example. We select the first option and click OK to get the individually calculated optimum replacement time for each component.

The calculation results are shown next.

The results suggest that, for example, replacing the oil and oil filter every 3,263 miles is the optimum to keep the cost at its minimum while not having to worry about the car breaking down.

It is clear from these results that the replacement time for the oil and oil filter is very close to the replacement time for the air filter. Likewise, the water pump, timing belt and spark plugs have replacement times in close proximity to each other. Therefore, it makes more sense to do the maintenance of those items together rather than taking the car for maintenance twice in a short period of time. The replacement time for the fuel filter, on the other hand, is not in proximity to any other component’s replacement time. Thus, clustering the components in three groups may make the most sense.

We click Calculate again and this time we choose the last option and specify to use three clusters.

The results are shown next.

As expected, the oil and oil filter replacement is grouped with the air filter replacement, while the water pump, timing belt and spark plugs replacements are also clustered together. The third group is the fuel filter replacement by itself because its optimum replacement time was not close to that of any other component in the system.

## Conclusion

In this article, we presented the theory and mathematical formulation behind the optimum replacement calculations, individually and in clusters, with a basic example. The result of the example suggests that the car manufacturer may recommend to change the oil and oil filter with the air filter at around 3,200, miles and to replace the water pump and timing belt while replacing the spark plugs at around 37,000 miles. The fuel filter needs to be replaced by itself at around 10,000 miles. This will allow the customers to avoid extra trips for maintenance while supplying the highest reliability in clusters with the minimum cost.

## References

[2] Teknomo, Kardi. K-Mean Clustering Tutorials. Available: http://people.revoledu.com/kardi/tutorial/kMean/index.html