Patent Publication Number: US-11651627-B2

Title: Sensor network for optimized maintenance schedule

Description:
FIELD 
     One embodiment is directed generally to a sensor network, and in particular to sensor network that determines an optimized maintenance schedule for assets. 
     BACKGROUND INFORMATION 
     The Internet of Things (“IoT”) is the extension of Internet connectivity into physical devices and everyday objects. Embedded with electronics, Internet connectivity, and sensors, these devices can communicate and interact with others over the Internet, and they can be remotely monitored and controlled. In some instances, an IoT device can be used to monitor the operating conditions and status of a particular “asset” such as a vehicle part 
     SUMMARY 
     Embodiments determine an optimized maintenance schedule for a maintenance program that includes multiple levels, each level including at least one asset (i.e., asset type) and at least one of the levels including a plurality of assets. Embodiments receive historical failure data for each of the assets, the historical failure data generated at least in part by a sensor network. For each asset, embodiments generate a probability density function (“PDF”) using kernel density estimation (“KDE”). For each asset, based on a reliability rate threshold, embodiments determine a cumulative density function (“CDF”) using the PDF. For each asset, embodiments determine an optimized time to failure (“TTF”) using the CDF. Embodiments then create the schedule for each level that includes a minimum TTF for the assets at each level. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG.  1    is an overview diagram of elements of an optimized maintenance schedule network/system that can implement embodiments of the invention. 
         FIG.  2    is a block diagram of optimized maintenance schedule server of  FIG.  1    in the form of a computer server/system in accordance with an embodiment of the present invention. 
         FIG.  3    is a graph illustrating an MTTF calculation for an asset. 
         FIG.  4    is a graph illustrating another MTTF calculation for an asset. 
         FIG.  5    is a graph illustrating another MTTF calculation for an asset. 
         FIG.  6    is a histogram illustrating the historical failure rate for car batteries in accordance to embodiments of the invention. 
         FIG.  7    is a graph illustrating the KDE curve for car batteries for generated from the histogram of  FIG.  6    in accordance to embodiments of the invention. 
         FIG.  8    is a graph of the PDF curve for the failure pattern for car batteries that is generated from the KDE curve of  FIG.  7    in accordance to embodiments of the invention. 
         FIG.  9    is a graph of the PDF curve for the failure pattern for car batteries in accordance to embodiments of the invention. 
         FIG.  10    is a graph of the PDF curve for the failure pattern for car batteries in accordance to embodiments of the invention. 
         FIG.  11    is a graph of the PDF curve for the failure pattern for car batteries in accordance to embodiments of the invention. 
         FIG.  12    illustrates a proposed list of maintenance tasks for assets as inputs for which a maintenance schedule needs to be determined in accordance with embodiments of the invention. 
         FIG.  13    is a functional block diagram of functionality to determine an optimized maintenance schedule for the example of  FIG.  12    in accordance to embodiments. 
         FIG.  14    illustrates additional views of graphs of  FIG.  13    in accordance to embodiments. 
         FIG.  15    illustrates additional views of graphs of  FIG.  13    in accordance to embodiments. 
         FIG.  16    is a graph of multiple PDF curves for sub-groups of a single asset (e.g., a battery or a car filter) in accordance to embodiments of the invention. 
         FIG.  17    is a flow diagram of the functionality of the optimized maintenance schedule module of  FIG.  2    for determining an optimized maintenance schedule for an asset using a network of sensors in accordance with one embodiment. 
     
    
    
     DETAILED DESCRIPTION 
     Embodiments gather sensor data for a type of asset and, using a non-parametric kernel density approach, determine a probability density function curve for a failure pattern of the asset part. Embodiments determine a cumulative probability of failure from the probability density function curve and then determine an optimized maintenance schedule with multiple levels for the type of asset or for multiple different type of assets that may be part of a maintenance program. 
       FIG.  1    is an overview diagram of elements of an optimized maintenance schedule network/system  150  that can implement embodiments of the invention. Sensor based network  150  includes multiple sensors  101  that form a sensor network  150  in combination with one or more networks  110 . Each of sensors  101  can be considered an Internet of Things (“IoT”) device with the associated processing and communication capabilities. System  150  may include a relatively large number of sensors  101  (hundreds, thousands, or even millions or billions of sensors, for example). 
     An IoT device can be any device that has a sensor attached to it and can transmit data from one object to another or to people with the help of the Internet. IoT devices include wireless sensors, software, actuators, and computer devices. They are attached to a particular object that operates through the Internet, enabling the transfer of data among objects or people automatically without human intervention. Each of sensors  101  can include a processor/controller, and a communication interface that uses protocols such as Modbus, Zigbee, or proprietary protocols, to connect to an Edge Gateway. 
     In network  150 , each sensor  101  may be coupled, directly or indirectly, to an asset in order to monitor the use of the asset and ultimately to determine an optimized maintenance schedule for the asset, as disclosed below. The type of asset can be any asset that can be monitored and typically requires a maintenance schedule. Examples of assets can include any type of vehicle part, such as a battery, a filter, a brake, etc., an industrial part such as a pump or a compressor or motor, an electronic part such as a rotating hard drive, a fan, etc. 
     Each of sensors  101  communicate, wirelessly or wired, through one or more networks  110 . Networks  110  include the Internet, but may also include private on-premise networks that ultimately interface with the Internet as well as any other type of network that allows sensors  101  to communicate. Sensors  101  can be part of the “IoT Asset Monitoring Cloud Service” from Oracle Corp. 
     An optimized maintenance schedule system/server  10  is coupled to networks  110  to send and receive data from sensors  101 . Optimized maintenance schedule server  10  provides the optimized maintenance schedule determination functionality disclosed herein. In general, optimized maintenance schedule server  10  monitors data acquired by each of sensors  101  for purposes of accumulating the data and then performing technical calculations to determine optimized maintenance schedules for each type of asset. 
     Sensors  101  can be largely dispersed geographically, such as with temperature sensors located throughout the world, or more locally dispersed. For example, a machine can be fitted with sensors  101  to monitor its operation. These sensors measure quantities such as temperature, pressure, and vibration amplitude for the different parts of the machine. If there is some malfunction or any other abnormality, some of these readings will deviate significantly from the norm. For example, it may be the case that a small part of the engine is overheated when compared to the rest of the engine, or that the entire engine is overheated when compared to the rest of the machine. 
       FIG.  2    is a block diagram of optimized maintenance schedule server  10  of  FIG.  1    in the form of a computer server/system  10  in accordance with an embodiment of the present invention. Although shown as a single system, the functionality of system  10  can be implemented as a distributed system. Further, the functionality disclosed herein can be implemented on separate servers or devices that may be coupled together over a network. Further, one or more components of system  10  may not be included. 
     System  10  includes a bus  12  or other communication mechanism for communicating information, and a processor  22  coupled to bus  12  for processing information. Processor  22  may be any type of general or specific purpose processor. System  10  further includes a memory  14  for storing information and instructions to be executed by processor  22 . Memory  14  can be comprised of any combination of random access memory (“RAM”), read only memory (“ROM”), static storage such as a magnetic or optical disk, or any other type of computer readable media. System  10  further includes a communication device  20 , such as a network interface card, to provide access to a network. Therefore, a user may interface with system  10  directly, or remotely through a network, or any other method. 
     Computer readable media may be any available media that can be accessed by processor  22  and includes both volatile and nonvolatile media, removable and non-removable media, and communication media. Communication media may include computer readable instructions, data structures, program modules, or other data in a modulated data signal such as a carrier wave or other transport mechanism, and includes any information delivery media. 
     Processor  22  is further coupled via bus  12  to a display  24 , such as a Liquid Crystal Display (“LCD”). A keyboard  26  and a cursor control device  28 , such as a computer mouse, are further coupled to bus  12  to enable a user to interface with system  10 . 
     In one embodiment, memory  14  stores software modules that provide functionality when executed by processor  22 . The modules include an operating system  15  that provides operating system functionality for system  10 . The modules further include an optimized maintenance schedule module  16  that determines an optimized maintenance schedule for an asset using a network of sensors, and all other functionality disclosed herein. System  10  can be part of a larger system. Therefore, system  10  can include one or more additional functional modules  18  to include the additional functionality, such as the “IoT Asset Monitoring Cloud Service” from Oracle Corp. or the “Oracle Maintenance Cloud.” A file storage device or database  17  is coupled to bus  12  to provide centralized storage for modules  16  and  18 , including data generated by the sensors in the form of messages or data points. In one embodiment, database  17  is a relational database management system (“RDBMS”) that can use Structured Query Language (“SQL”) to manage the stored data. 
     In one embodiment, particularly when there are a large number of distributed files at a single device, database  17  is implemented as an in-memory database (“IMDB”). An IMDB is a database management system that primarily relies on main memory for computer data storage. It is contrasted with database management systems that employ a disk storage mechanism. Main memory databases are faster than disk-optimized databases because disk access is slower than memory access, the internal optimization algorithms are simpler and execute fewer CPU instructions. Accessing data in memory eliminates seek time when querying the data, which provides faster and more predictable performance than disk. 
     In one embodiment, database  17 , when implemented as an IMDB, is implemented based on a distributed data grid. A distributed data grid is a system in which a collection of computer servers work together in one or more clusters to manage information and related operations, such as computations, within a distributed or clustered environment. A distributed data grid can be used to manage application objects and data that are shared across the servers. A distributed data grid provides low response time, high throughput, predictable scalability, continuous availability, and information reliability. In particular examples, distributed data grids, such as, e.g., the “Oracle Coherence” data grid from Oracle Corp., store information in-memory to achieve higher performance, and employ redundancy in keeping copies of that information synchronized across multiple servers, thus ensuring resiliency of the system and continued availability of the data in the event of failure of a server. 
     In one embodiment, system  10  is a computing/data processing system including an application or collection of distributed applications for enterprise organizations, and may also implement logistics, manufacturing, and inventory management functionality. The applications and computing system  10  may be configured to operate with or be implemented as a cloud-based networking system, a software-as-a-service (“SaaS”) architecture, or other type of computing solution. 
     In general, an asset can be part of an “asset group”. For example, parts of an automobile can be an asset of an asset group such as a Toyota Prius or Camry. There may be a pre-defined maintenance program with multiple levels for that asset group, particularly when the asset group is an automobile. For example, for a Toyota Camry, there may be the following suggested maintenance program levels:
         A 5,000 mile service with operation set x where x=(Oil Filter change, Brake pad change);   A 10,000 mile service with operation set y where y=(Engine check, Door check);   A 30,000 mile service with operation set z where z=(Battery change, Tire rotation).       

     In the above programs, operations of y will include x given 10,000 is a multiple of 5,000, and so on for other programs. The program may be based on the calendar (e.g., every 3 month service) or miles. However, these maintenance programs may not be optimal in terms of the failure patterns of the assets. In general, assets (e.g., machinery, electronic parts, automobile parts, etc.) do not follow a normal distribution. As a result, determining an optimal maintenance schedule is challenging which often results in either the failure of assets or the wastage of the useful lifespan of the assets. 
     Specifically, known maintenance programs are typically generated based on “Mean Time to Failure” (“MTTF”) calculations, which measure the average time or miles until a part fails. It is calculated by summing the historical time or miles to failure for each failed part and dividing by the number of parts. MTTF is typically used to set part replacement intervals: ideally, replacing parts before failure, but not too soon before failure. 
       FIG.  3    is a graph  300  illustrating a MTTF calculation for an asset (i.e., a type of asset instead of an individual asset). In the example of  FIG.  3   , and all other examples that follow unless otherwise specified, the asset is a car battery. However, embodiments are applicable to any type of asset monitored by a sensor in a sensor network, such as sensor network  150  of  FIG.  3   . 
     In the example of  FIG.  3   , the MTTF for car batteries is 25k miles as indicated at  301 . Knowing this, a maintenance manager may set the maintenance interval to replace batteries at 20k. This is well before the average battery fails, but not so soon that too much useful battery life is wasted. In this example, most of the failures occur within  1  standard deviation of the mean and the distribution is fairly normal. 
       FIG.  4    is a graph  400  illustrating another MTTF calculation for an asset. Unlike  FIG.  3   , the time to failure (“TTF”) in  FIG.  4    is not as “normal” but is more typical for assets such as car batteries. In this example, most batteries still fail at around 25k miles, but the long-lasting outliers raise the MTTF to 31k (at  401 ), well above the mileage where the majority of batteries fail. Knowing the MTTF is 31k miles, a maintenance manager may set the maintenance interval to replace batteries at 26k. However, more than half of the batteries would have failed before 26k. Therefore, for this distribution, using the MTTF to set maintenance intervals would result in a lot of battery failures. 
       FIG.  5    is a graph  500  illustrating another MTTF calculation for an asset. In this example, most batteries still fail at around 25k miles, but the fast-failing outliers lower the average to 17.5k (at  501 ), well below the mileage where the majority of batteries fail. Knowing the MTTF is 17.5k miles, a maintenance manager may set the maintenance interval to replace batteries at 12.5k miles. However, that means most batteries will be replaced before they&#39;ve even reached half of their average lifespan of 25k miles. Therefore, for this distribution, setting maintenance intervals using MTTF means replacing batteries too soon and wasting lots of money. 
     In general, using the MTTF alone as is done with most known solutions, is not useful for maintenance planning because it does not factor in deviation, density, probability, etc. Further, it does not give any indication where maintenance intervals should be set in relation to the mean. Specifically, it fails to determine whether the interval should be 1 standard deviation below the mean as a safety buffer, 10% below, 5k miles below, etc. 
     In contrast, embodiments use a non-parametric kernel density approach to estimate the probability density function (“PDF”) of the asset failures. The estimated PDF is used to derive a cumulative density function (“CDF”) for the asset failures. An input of reliability rate targets for the asset part is received. Using the reliability target rate as a threshold, respective values from the CDF is picked up as the optimum maintenance schedule for an asset part. 
       FIG.  6    is a histogram  600  illustrating the historical failure rate for car batteries in accordance to embodiments of the invention. The information is received from sensors coupled to the batteries (or any other asset that is used). Histogram  600  provides information on how many miles were driven when the car battery failed. It is assumed, for purposes of embodiments, that historical failure information is the best predictor of future failure rate. In embodiments, historical information is fed to a system looking at the unscheduled work orders at the factory from data received from other service locations. Specifically, when the battery that was not supposed to fail actually does fail, the vehicle with the battery is brought to a service location (e.g., dealership, independent vehicle maintenance facility, etc.) and the miles and other sensor readings are captured using sensors attached to the battery and other parts, or a general sensor attached to the vehicle. Using this historical data for the battery and other parts, an optimized maintenance schedule is calculated as disclosed below. 
     Embodiments apply a kernel density estimation (“KDE”) algorithm to the histogram to estimate a probability density function (“PDF”). KDE is a non-parametric way to estimate the probability density function of a random variable and is efficient to compute and maintain in a streaming environment. In one embodiment, the curve using KDE is calculated as follows: 
                     f   ^     h     ⁡     (   x   )       =         1   n     ⁢       ∑     i   =   1     n     ⁢       K   h     ⁡     (     x   -     x   i       )           =       1   nh     ⁢       ∑     i   =   1     n     ⁢     K   ⁡     (       x   -     x   i       h     )               ,         
where K is the kernel (i.e., a non-negative function) and h&gt;0 is a smoothing parameter referred to as the bandwidth. The kernel function used in embodiments is the following Gaussian function:
 
     
       
         
           
             
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       FIG.  7    is a graph  700  illustrating the KDE curve  701  for car batteries for generated from histogram  600  of  FIG.  6    in accordance to embodiments of the invention. 
       FIG.  8    is a graph  800  of the PDF curve  801  for the failure pattern for car batteries that is generated from KDE curve  701  of  FIG.  7    in accordance to embodiments of the invention. PDF curve  801  is the same as curve  701  (i.e., it is generated using KDE). The area under PDF curve  801  for a mileage range gives the probability of failure occurring within that mileage range. The total range in the example of  FIG.  8    is from 0 to 50k miles. Therefore, 100% of the area under the curve is between 0 and 50k miles, meaning that there is a 100% probability that a battery will fail somewhere between 0 and 50k miles. 
       FIG.  9    is a graph  900  of the PDF curve for the failure pattern for car batteries in accordance to embodiments of the invention. The example of  FIG.  9    illustrates a determination of probability of a battery failing between 25k and 30k miles given this PDF curve. For the range from 25k to 30k miles, the area  901  under the curve is about 20% of the total area under the curve, so the probability of failure in this range is roughly 20%. 
       FIG.  10    is a graph  1000  of the PDF curve for the failure pattern for car batteries in accordance to embodiments of the invention. The PDF curve is also used to determine the cumulative probability of failure by taking the area  1001  under the curve for the range from 0 to x miles. Cumulative probability  1001  provides an estimate of:
         1. What percentage of batteries will fail by x miles?   2. What is the chance of any individual battery failing by x miles?       

       FIG.  11    is a graph  1100  of the PDF curve for the failure pattern for car batteries in accordance to embodiments of the invention. In the example of  FIG.  11   , the area  1101  under the PDF curve from 0 miles to 20k miles represents approximately 15% of the total area under the curve. This means that by 20k miles, approximately 15% of the batteries in the population will have failed. 
     Businesses often set reliability rate targets for assets and parts. The minimum acceptable reliability rate is the complement of the maximum acceptable failure rate:
         1. Failure Rate=1−Reliability Rate; and   2. Reliability Rate=1−Failure Rate.
 
For example, if minimum acceptable reliability rate is 85%, then 1−0.85=0.15, or a maximum acceptable failure rate of 15%. For example, if a business has a minimum acceptable reliability rate of 85% for batteries, the business is meeting its target reliability goal as long as less than 15% of batteries fail. Stated differently, how many miles can batteries be used before 15% of them fail?
       

     Given a minimum reliability rate, and a PDF for an asset, embodiments calculate a threshold (i.e., an optimum failure rate of an asset) using heuristics by leveraging business driven reliability targets and the CDF of the obtained PDF for an asset. The threshold in embodiments is determined by determining the mean of the distribution and creating a window (i.e., upper and lower bound) around the mean using a span in the same units as that of the failure asset part. Embodiments then determine the mean of the window and find the CDF of the point. If the CDF of this point is approximately close to the reliability target set by the business, the mean is returned as the new optimized schedule for that part (e.g., the amount of miles where the asset is replaced). Otherwise, the upper bound or the lower bound of the window is moved depending on if the difference between the CDF of the calculated point and the reliability target set by the business and a new CDF is calculated. 
       FIG.  12    illustrates a proposed list of maintenance tasks for assets as inputs for which a maintenance schedule needs to be determined in accordance with embodiments of the invention. The input includes three different maintenance programs/levels: intermediate service  1201 , gold service  1202  and major service  1203 . Each of these programs have different types of assets (i.e., car parts) defined under them and each type of asset is intended to be replaced as part of the respective service at a specified number of miles/days that is pre-determined by embodiments. Specifically, a cabin air filter and an engine air filter are replaced under intermediate maintenance program  1201  at  1210 , brake pads, a drive belt and a fuel filter are replaced under gold maintenance program  1202  at  1211 , and a battery and spark plugs are replaced under major service  1203  at  1212 . 
     Embodiments of the invention, based on the failure pattern of each asset, determine a new optimized TTF to be implemented for each of the maintenance programs, including programs such as shown in  FIG.  12    that replace multiple different assets. In other words, embodiments determine the optimized miles or days to be assigned for each of the programs  1201 - 1203 . For the determination, input is received from a plurality of sensors coupled (directly or indirectly) to a plurality of each of the assets to acquire failure data. Some details of the acquired data is shown in the example of  FIG.  12   . For example, the cabin air filter has data from 30 assets (at  1210 ) as input and the failures range from 753-869 days, and from 22,637-27,183 miles (shown at  1220 ). Similarly, for the engine filter belonging to the same intermediate service program  1201 , for the data from 49 assets (at  1210 ), failures occur from 430-572 days and from 10,465-14,121 miles (shown at  1221 ). 
     Embodiments can determine an optimized maintenance schedule value at a program level (e.g., programs  1201 - 1203 ) or at an individual asset level. At the program level, an example is a determination of “what” is the recommended maintenance schedule for intermediate service program  1201 . At an asset level, the maintenance schedule for an individual type of asset (e.g., a cabin air filter) is determined. 
     Embodiments determine the optimized schedule using the PDF for each individual asset (e.g., the engine air filter and the cabin air filter) from the sensor data and based on a customer provided threshold (i.e., a desired reliability specified by the business) or calculated threshold. 
     The optimized schedule (e.g., number of miles or days) for a maintenance program is determined as the minimum of the Time To Failures (“TTF”) calculated on the operations (part) under the specific maintenance program. For example, for intermediate service  1201 , the minimum of the TTF of both the cabin air filter and the engine air filter is used as the optimized schedule. 
       FIG.  13    is a functional block diagram of functionality to determine an optimized maintenance schedule for the example of  FIG.  12    in accordance to embodiments. For each of programs  1201 - 1203 , a KDE curve is used to generated an individual probability density function for each operation and for each measuring unit. Specifically, the KDE for the cabin air filter is shown at  1320  for miles and at  1321  for days (in graphs  1301  and  1302 , respectively). For days (there is no miles units for gold program  1202 ), the KDE for the brake pads is shown at  1325 , the fuel filter is shown at  1326  and the drive belt is shown at  1327  (in graph  1303 ). For major service  1203 , the KDE for the battery is shown for miles at  1331  and for days at  1332 , and the KDE for the spark plugs is shown for miles at  1333  and for days at  1334  (in graphs  1304  and  1308 , respectively). 
     From the calculated PDF and using the reliability rate to determine a threshold, embodiments determine a CDF value, which is the TTF calculated for individual operation. Then, for all calculated TTFs under a single maintenance program a minimum value is selected as a calculated TTF for that program for a particular measuring unit (e.g., days, miles, etc.). If the calculated TTF for a maintenance program is less than the existing program, this is a case of “PULL IN” (i.e., reduce the maintenance schedule). If the calculated TTF for a maintenance program is greater than the existing TTF, this is a case of “PUSH OUT” (i.e., increase the maintenance schedule). 
       FIG.  14    illustrates additional views of graphs  1301  and  1302  of  FIG.  13    in accordance to embodiments.  FIG.  15    illustrates additional views of graphs  1304  and  1308  of  FIG.  13    in accordance to embodiments. 
     For example, embodiments output  1350  for intermediate service  1201  a calculated TTF value of 13,853 miles (versus a 15,000 miles existing TTF), shown at  1401  of  FIG.  14   , and a calculated TTF value of 509 days (versus a 600 days existing TTF), shown at  1402  of  FIG.  14   , and an 85% and 90% confidence level, respectively. The outputs are the minimum value when services have more than one asset (i.e., in  1301 , the TTF for the engine air filter is 13,853 miles at  1401  and the TTF for the cabin air filter is 25,551 miles at  1403 ). 
     Additionally, a “confidence level” is calculated which specifies the confidence percentage of the recommend TTF prediction. Confidence level is calculated in embodiments by splitting the data into a training and a test data set and using a k-fold cross-validation procedure. Embodiments create a PDF on the training data set and validate the training data set on the test data sets over a number of iterations using the following steps:
         1) Implement a K-fold cross validation split on the training set (5 folds in one embodiment).   2) Using trained data and the test split for all the folds, execute the following for all folds:
           a) train a KDE model on the training split and estimate scores on training set data.   b) Get min and max scores assigned to the training points by the KDE model.   c) Generate similar scores for the test set of the split.   d) Compute the percentage of test points within the score bounds of training points.   
           3) Take the average of this percentage over all folds, which is the confidence value.       

       FIG.  16    is a graph  1600  of multiple PDF curves for sub-groups of a single asset (e.g., a battery or a car filter) in accordance to embodiments of the invention. For example, curve  1601  is for cars and trucks in Nevada, Arizona and Texas, and curve  1602  is for minivans in North America. In embodiments, machine learning is used to cluster asset groups into sub-groups with similar failure patterns and characteristics. This allows more refined models to be created to improve predictive accuracy for each sub-group. Machine learning (e.g., clustering based on a unique combination of parameters) can identify which attributes (location, configuration, usage patterns, etc.) have the greatest predictive power. 
       FIG.  17    is a flow diagram of the functionality of optimized maintenance schedule module  16  of  FIG.  2    for determining an optimized maintenance schedule for an asset using a network of sensors in accordance with one embodiment. In one embodiment, the functionality of the flow diagram of  FIG.  17    is implemented by software stored in memory or other computer readable or tangible medium, and executed by a processor. In other embodiments, the functionality may be performed by hardware (e.g., through the use of an application specific integrated circuit (“ASIC”), a programmable gate array (“PGA”), a field programmable gate array (“FPGA”), etc.), or any combination of hardware and software. The functionality of  FIG.  17    is performed for a maintenance program that has multiple levels, each level which may involve the replacement of multiple assets. 
     At  1702 , historical sensor failure data from each of the assets of the program is received. A reliability rate threshold for each asset is also received. 
     At  1704 , for each asset, KDE is used to generate a PDF curve from the historical failure data. 
     At  1706 , for each of the assets, based on the reliability rate threshold, a CDF for each asset is determined. Specifically, the reliability rate threshold is an allowable percentage of that asset that would be allowed to fail, and the CDF is the area under the PDF curve that equals the reliability rate threshold and the corresponding miles/days of the x-axis. 
     At  1708 , from the CDF, an optimized time to failure in one or more different units (e.g., days or miles) for the replacement or failure for each of the assets is determined. 
     At  1710 , at each maintenance program level that has multiple assets (or for levels with only a single asset), the minimum calculated TTF at  1708  is used as schedule for that program. 
     As disclosed, embodiments determine an optimized maintenance schedule for a maintenance program with multiple levels using a PDF and CDF for each asset to determine a TTF for each asset, and then using the minimum TTF for each level. 
     Several embodiments are specifically illustrated and/or described herein. However, it will be appreciated that modifications and variations of the disclosed embodiments are covered by the above teachings and within the purview of the appended claims without departing from the spirit and intended scope of the invention.