Is it a Baby or a Bathtub? & How Many Fish? Is it a Baby or a - PowerPoint PPT Presentation

Is it a Baby or a Bathtub? & How Many Fish? Is it a Baby or a Bathtub? & How Many Fish? Two Studies in Applied Computing Two Studies in Applied Computing Stanford University Department of Stanford University Department of Electrical Engineering Electrical Engineering Computer Systems Colloquium (EE380) Computer Systems Colloquium (EE380) February 8, 2006 February 8, 2006 Rose Ray, Ph.D. Rose Ray, Ph.D. Principal Scientist Principal Scientist Exponent, Inc. Exponent, Inc.

Outline Outline • Case 1: Is it a Baby or a Bathtub? • Case 1: Is it a Baby or a Bathtub? – Problem Description – Problem Description – Available Data – Available Data – Statistical Failure Analysis – Statistical Failure Analysis – Hazard Functions – Hazard Functions – Calculation Methods – Calculation Methods • Parametric • Parametric Life table • Life table • – Results – Results • Case 2: How Many Fish? • Case 2: How Many Fish? – Problem Description – Problem Description – Available Data – Available Data – Statistical Method – Statistical Method – Monte Carlo Simulation – Monte Carlo Simulation – Results – Results • Questions • Questions

Case 1: Problem Description Case 1: Problem Description • A new design of battery powered small • A new design of battery powered small appliance is introduced in the fall. appliance is introduced in the fall. – Initial sales are primarily for the holiday – Initial sales are primarily for the holiday gift market gift market – In January the manufacturer begins to – In January the manufacturer begins to receive complaints of overheating receive complaints of overheating batteries batteries – Incident dates are as early as Dec 25 – Incident dates are as early as Dec 25 – Failures may pose a safety hazard – Failures may pose a safety hazard • Should the product be recalled? • Should the product be recalled?

Available Data Available Data • For Each Reported Failure • For Each Reported Failure – Age at failure – Age at failure – Description of the failure mode (based – Description of the failure mode (based upon analysis of returned product) upon analysis of returned product) – Sales and Production – Sales and Production • Product sales by Production Lot • Product sales by Production Lot – Number Sold – Number Sold – Current age of Product – Current age of Product

Available Data: Failure Data Available Data: Failure Data A ge at A ge at In cid en t_ O b s S ales In cu d en t W ith d raw al T yp e 1 2 9 7 3 1 3 0 b urn/ho t 6 0 0 2 4 5 L eak 2 3 8 0 2 1 6 0 L eak 4 2 5 6 0 5 7 5 L eak 2 9 7 0 7 9 0 H o t 5 6 1 5 1 8 2 5 1 0 5 b urn/ho t 8 2 5 5 9 1 5 3 0 b urn/ho t 5 0 0 1 0 4 5 b urn/ho t 8 9 1 5 1 9 1 4 6 0 b urn/ho t 10 2 0 0 0 1 7 5 b urn/ho t 5 0 0 2 9 0 b urn/ho t 11 12 6 0 0 3 1 0 5 b urn/ho t 13 1 0 0 0 2 9 0 b urn/ho t

Available Data: Sales Data Available Data: Sales Data (Exposure Data) (Exposure Data) C u rrent A ge (A ge at O b s S ales W ith draw al) 1 20,101 T otal . 2 5,532 P roduction 30 L ot 6 3 1,100 P roduction 45 L ot 5 2,321 P roduction 60 4 L ot 4 4,560 P roduction 75 5 L ot 3 6 4,470 P roduction 90 L ot 2 7 2,118 P roduction 105 L ot 1

Statistical Failure Analysis Statistical Failure Analysis • Probability distribution of time to failure is • Probability distribution of time to failure is key information for determining key information for determining appropriate course of action. appropriate course of action. • Typical failure time distributions • Typical failure time distributions – Weibull – Weibull – Exponential – Exponential – Log Normal – Log Normal • F(t) =probability of failure at or before • F(t) =probability of failure at or before time t time t

Hazard Function: H(t) Hazard Function: H(t) • H(t) = probability of failure at time t • H(t) = probability of failure at time t conditional on survival to time t. conditional on survival to time t. • H(t) =dF(t)/{1-F(t)} • H(t) =dF(t)/{1-F(t)} • H(t) =failure rate at time t • H(t) =failure rate at time t • Typical hazard patterns • Typical hazard patterns – Infant mortality (burn in) – Infant mortality (burn in) – Constant (memory less) – Constant (memory less) – Wear out – Wear out – Bathtub – Bathtub

Hazard Patterns Hazard Patterns 0.060 0.060 0.050 0.050 Hazard Function Hazard Function 0.040 0.040 0.030 0.030 0.020 0.020 0.010 0.010 Infant Infant Mortality Mortality 0.000 0.000 Age 0 15 30 45 60 75 Age 0 15 30 45 60 75

Hazard Patterns Hazard Patterns 0.060 0.060 0.050 0.050 Hazard Function Hazard Function 0.040 0.040 0.030 0.030 Constant 0.020 0.020 0.010 0.010 Infant Mortality 0.000 0.000 Age Age 0 15 30 45 60 75 0 15 30 45 60 75

Hazard Patterns Hazard Patterns 0.060 0.050 Hazard Function 0.040 0.030 0.020 Wear Out 0.010 Infant Mortality 0.000 Age 0 15 30 45 60 75

Hazard Patterns Hazard Patterns 0.060 0.050 Hazard Function 0.040 Bathtub 0.030 0.020 Wear Out 0.010 Infant Mortality 0.000 Age 0 15 30 45 60 75

Calculation Methods: Parametric Calculation Methods: Parametric • Example: Weibull Distribution • Example: Weibull Distribution – Family of Failure time distributions – Family of Failure time distributions – Can model both infant mortality or – Can model both infant mortality or wear out. wear out. – Single Weibull cannot model both – Single Weibull cannot model both • Hazard Function • Hazard Function – H(t)=( α/β )*(t/ β ) ( α -1) – H(t)=( α/β )*(t/ β ) ( α -1) � α <1 ► decreasing failure rate � α <1 ► decreasing failure rate � α =1 ► constant failure rate � α =1 ► constant failure rate � α >1 ► increasing failure rate � α >1 ► increasing failure rate

Estimated Weibull Parameters for Estimated Weibull Parameters for Case Example Case Example Analysis of Parameter Estimates Standard 95% Confidence Parameter DF Estimate Error Limits Weibull 1 5.54E+10 3.15E+11 793717.7 3.86E+15 Scale( β ) Weibull 1 0.3563 0.0976 0.2083 0.6095 Shape ( α ) Upper Bound <1 indicates Upper Bound <1 indicates indicates decreasing indicates decreasing shape is statistically shape is statistically failure rate failure rate significantly <1 significantly <1

Is it a Baby or a Bathtub? Is it a Baby or a Bathtub? • If truly an infant mortality failure mode, all • If truly an infant mortality failure mode, all or most all failures may have occurred by or most all failures may have occurred by the time complaints have been received. the time complaints have been received. – No recall action is needed. – No recall action is needed. • If failure rate is constant or increasing • If failure rate is constant or increasing with time, more failures are expected with time, more failures are expected – Recall may be necessary – Recall may be necessary

Is it a Baby or a Bathtub? Is it a Baby or a Bathtub? • Weibull estimate of decreasing failure rate • Weibull estimate of decreasing failure rate depends very strongly on the assumption depends very strongly on the assumption of Weibull form of the hazard function of Weibull form of the hazard function • At early age Weibull estimates may • At early age Weibull estimates may indicate a decreasing failure rate and indicate a decreasing failure rate and miss the start of late age failures miss the start of late age failures • Lifetable method can be used to verify • Lifetable method can be used to verify

Non Parametric Method: Lifetable Non Parametric Method: Lifetable • Calculate hazard function directly with no • Calculate hazard function directly with no assumption about shape of time to failure assumption about shape of time to failure distribution distribution – Partition data into short age intervals – Partition data into short age intervals – Calculate number – Calculate number • At risk at start of interval • At risk at start of interval • Failed during interval • Failed during interval • Withdrawn during interval • Withdrawn during interval – Hazard for interval = – Hazard for interval = Failures during interval/{At risk at start -.5 withdrawn} Failures during interval/{At risk at start -.5 withdrawn} • Compare estimated hazard at intervals toward • Compare estimated hazard at intervals toward the end of experience to determine whether the end of experience to determine whether infant mortality is the only failure pattern infant mortality is the only failure pattern