Using Sensor Data to Improve the Management of Spare Parts

Jennifer K. Ryan, Zhi Zeng, Xi KanDepartment of Industrial & Systems Engineering

Rensselaer Polytechnic Institute

Rong LiWashington University

Condition Monitoring

• Process of collecting real-time sensor information from a functioning device to make predictions regarding health condition and lifetime of device

• Key topics in condition monitoring:– Sensors and sensor technology

– Signal identification and processing

– Condition assessment and operational decision-making

Benefits of Condition Monitoring

• Improve maintenance planning and scheduling

– Prevent unexpected failures, minimize unscheduled downtime and unnecessary maintenance

• Improve part procurement & inventory control of service parts

– Particularly useful for devices that are subject to deterioration and require periodic maintenance and/or replacement

Relevant Literature• Much of research on condition monitoring focuses on

identifying quantities which allow successful diagnosis of a device’s health

• Research on Bayesian estimation of degradation signal models that can be used to predict device failure– N. Gebraeel, M. Lawley, R. Li and J.K. Ryan, “Residual Life

Distributions from Component Degradation Signals: A Bayesian Approach,” IIE Transactions, 37 (2005), p. 543-557

• Research on inventory management when there is learning about demand– Bayesian inventory models with exogenous demand

Overview of Approach

1. Develop & estimate degradation signal model to capture device degradation

2. Use degradation signal model & observed signals to derive remaining life distribution for each individual functioning device

3. Use remaining life distributions to derive demand distribution for each machine

4. Use demand distribution to develop inventory control model & determine optimal stocking level

Degradation Signal

• Quantity computed from sensor information that captures current state of device and provides information on how condition is likely to evolve in future

– Must reflect deteriorating condition of device

– Must exhibit clear trend

– Must have a well-defined failure threshold (F)

Degradation Signal

• Vibration analysis of ball bearings:

– As bearings degrade, the vibration emitted increases

– When vibration reaches specified threshold, said to have failed

• Oil analysis for machines:

– Spectrometry to monitor concentration of wear metals in oil

– Concentration increases with run time, significant increases seen when abnormal wear due to machine or lubricant problems

– Manufacturers often specify limits for these concentrations, indicate whether machine condition is acceptable or abnormal

Degradation Signal Model

– Deterministic parameter, f: • Same across population of devices

– Stochastic parameters, q and b:• Differ across population of devices

– Random error terms, e(t): • Brownian motion noise process with volatility s

0)},(exp{)( ttttS

100 200 300 400 500 6000

0.005

0.01

0.015

0.02

0.025

Time

Signal

This figure shows a typical degradation signal for exponential model with Brownian motion error process.

Posterior Distribution of q, b

Bayesian Updating

Prior Distribution

Real-time Sensor Readings

S1, …, Sn

Bayesian Estimation of Degradation Signal Model Parameters

Degradation Signal

Historyfrom

Population

Condition Monitoring

of Individual Machine

• Special Case: q is unknown and = 1b

– If we work with logged signal, degradation model becomes

– q is called the degradation rate

– Assume Normal prior distribution for q

– Observe signal values, S1, …, Sn, at times t1, …, tn

Bayesian Estimation of Degradation Signal Model Parameters

0 ttttS ),()(

• Posterior distribution of q after nth signal observation is Normal with mean mn and standard deviation sn :

Bayesian Estimation of Degradation Signal Model Parameters

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n

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nnn

t

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• Here t0 is time between observations, F is failure threshold, and s2 is volatility of BM, dn-1 is number failures in period n-1

• Posterior distribution of q after nth signal observation is Normal with mean mn and standard deviation sn :

Bayesian Estimation of Degradation Signal Model Parameters

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• This posterior distribution of q depends on: Sn, Sn-1, dn-1, mn-1, sn-1

Predicting Remaining Life

• Device failure is defined as signal reaching pre-established failure threshold, F

• To predict time until signal reaches threshold:

– Use results for distribution of first passage time for Brownian motion process:

• T(Sn, F | q) = time to failure given most recent signal observation Sn, failure threshold F, for a given degradation rate, q

Predicting Remaining Life

• If q is known, remaining life distribution is:

• If q is not known, must take expectation over posterior distribution of q

t

tFSFS2exp

t

tFS1tF,STP

n2n

nn

Using Remaining Life Distribution

• Each time monitor device, update remaining life distribution

• Use this distribution to derive distribution of demand for replacement parts for an individual machine

• Compute distribution of aggregate demand across set of machines serviced out of same stock point

• Use aggregate demand distribution to make spare parts inventory management decisions

Predicting Demand for Spare Parts

• Deriving probability mass function of demand:

– T(S,F) = time required for signal to evolve from S to F

• Distribution of time required for k parts to each evolve from S = 0 to S = F is same as of time required for one part to evolve from S = 0 to S = k F

– PMF of demand can therefore be written as:

n1nnn1nn

nn1nn

ttF1k,STPttkF,STP

StSkt,tDP

Predicting Demand for Spare Parts

• When degradation parameter, q, is known:

– Distribution of demand depends only on most recent signal observation, Sn:

P{ D(tn,tn+1) = k| Sn }

• When degradation parameter, q, is unknown:

– Distribution of demand depends on posterior distribution of q :

P{ D(tn,tn+1) = k| Sn,Sn-1,dn-1,mn,sn }

This figure shows the probability mass function of demand for several values of the degradation parameter, q.

This figure shows the distribution of demand for several values of the volatility of the Brownian motion error process, s.

This figure shows the distribution of aggregate demand across m machines for several different values of m.

m = 1 m = 2 m = 3

m = 5m = 4

Inventory Management of Spare Parts

• Given distribution of demand for service parts, construct an inventory model to find optimal policy for managing inventory of spare parts

– Periodic review, finite horizon, linear holding and penalty costs, emergency orders to fill backorders

– Derived demand is non-stationary and exhibits correlation, depends on observed degradation signals

• A dynamic order-up-to policy is optimal

– Order-up-to level is updated each period based on the observed degradation signal values

Inventory Management of Spare Parts

• Solving the dynamic program to find optimal order-up-to levels is computationally infeasible for problems with more than one machine

– For a problem with m machines and known degradation parameters, there are m + 1 state variables

– For a problem with m machines and unknown degradation parameters, there are 5m + 1 state variables

Inventory Management of Spare Parts

• Form of optimal policy provides guidance on how to use real-time sensor information to improve inventory management

– Extend results of Lovejoy (1990) to develop a myopic critical fractile policy

• Optimal inventory level is smallest x satisfying

• Can bound the performance of this heuristic policy

– Combine DP and simulation to find optimal policy for settings with multiple machines

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Gap between expected cost-to-go for the myopic policy and optimal policy for single machine with known drift parameter

Performance of Myopic Policy

# of Machines Average Minimum Maximum

1 3.45% 2.95% 4.4%

5 0.58% 0.03% 1.2%

Performance Gap: Myopic vs. Optimal Policy

# of Machines Policy Average Minimum Maximum

1Optimal 28.2 27.8 28.7

Myopic 0.41 0.23 0.78

5Optimal 61.6 60.2 64.6

Myopic 18.8 18.1 19.2

Computational Time: Myopic Policy vs. Optimal Policy

Comparison of total cost with and without Bayesian updating, for problem with single machine with known drift parameter

Replacement Decisions

• So far, assumed device will be replaced when observed signal value first reaches an exogenously specified replacement threshold

• Consider approaches for determining optimal threshold, with goal of minimizing total cost of preventative and corrective replacements

• Assume the observed degradation signal is not a perfect predictor of device failure

Replacement Decisions

• Degradation signals are often defined based on quantities that are easily observed & measured

• Due to limitations of sensor technology or space/budget constraints, some critical aspects of degradation may be hard to measure

• Consider case in which observed signal is an imperfect indicator of degradation

Replacement Decisions

• Whitmore, et al (1998) consider a model in which failure is not deterministically related to the observed degradation signal

– Suggest a bivariate Wiener process to model both the observed degradation signal and the actual unobserved failure process

• Crack length vs. metal fatigue

– Demonstrate how parameters of bivariate Wiener process model used can be estimated using MLE

Bivariate Weiner Process

Bivariate Wiener Process:

X(t): Unobservable true failure process

Y(t): Observable degradation process

• {X(t), Y(t)} has mean vector q = (qx, qy) and covariance matrix S

• Assume {X(0), Y(0)} ={0, 0}

Replacement Policy

• Assumptions:

– Failure occurs when unobserved process X(t) first reaches a fixed failure threshold level, F

– Replacement occurs when observed process Y(t) first reaches a fixed replacement threshold, R

• Preventive replacement, with cost CP , occurs if Y(t) reaches R before X(t) reaches F

• Corrective replacement, with cost CF , occurs if X(t) reaches F before Y(t) reaches R

– Assume CF > CP

Replacement Policy

• TF = first passage time of unobserved process, X(t), given the barrier F

• TR = first passage time of observed process, Y(t), given the barrier R

– Preventative replacement occurs if TR < TF

– Corrective replacement occurs if TF < TR

Preventative Replacement

F

TF

Y

X

TR

Replacement Policy

• Find R to minimize average cost per unit time, AC(R)

• Let T = min(TF ,TR) = time between replacement

• Find R to minimize this long run average cost

– Numerical results indicate that -AC(R) is unimodal, and hence a unique optimal R* exists

TE

TTPCTTPCRAC RFFRFP

Numerical Results

sx sy qx qy CF r

R* ↓ ↓ ↓ ↑ ↓ ↑

T* ↓ ↓ ↓ ↑ ↓ ↑

P* ? ? ? ? ↓ ↓

AC* ↑ ↑ ↑ ↑ ↑ ↓

P* = probability of failure before replacement

Bayesian Estimation

• Developed a Bayesian updating approach to estimate the parameters of the bivariate Brownian motion

• Conducted a series of numerical tests to study the value of the Bayesian updating approach

– Comparison to non-Bayesian approach in which replacement threshold R chosen based on prior distribution, with no updating

Bayesian Estimation

• Bayesian approach:

– Reduces the average cost by 5-20%

– Increases percentage of cycles with preventative replacement by 2-22%

• Bayesian approach provides the most benefit when prior standard deviation is large and when cost of corrective failure is high

Numerical Results

Metric Group Non-Bayesian Bayesian % change Parameter

changed

% cycles with

prevent replacemt

Base 71.2% 84.5% +15.8%1 62.9% 81.2% +22.6% 0x increased

2 87.5% 94.2% +7.1% F increased3 84.8% 86.1% +1.5% CF increased

Average cost

Base 0.19 0.17 -12.9%1 0.27 0.21 -28.9% 0x increased

2 0.10 0.10 -5.4% F increased3 0.28 0.23 -20.2% CF increased

Conclusions

• Real time sensor data, together with a properly estimated degradation model, can be used to improve operational decision-making

• Bayesian updating approaches can be used to combine the degradation characteristics of a population of devices (prior distribution) with the characteristics (sensor data) obtained from an individual machine