Calculation of Minimum Failure Cost For Bridge Elements in Pontis BMS

1. Calculation ofMinimum Failure Cost For Bridge Elements in Pontis BMS Pontis User Training Meeting Detroit, MI - Sep 28, 2001 Adel Al-Wazeer Steve Chase BMIS Lab.Turner Fairbank Highway Research Center, FHWA

2. Minimum Failure Cost Calculation • Minimum Failure Cost • Formulation • Variation with different environments & for different discount rates • Calculation Method • Results • Sensitivity to Fail Prob. • Calculation for all Pontis CoRe Elements • Conclusions • Introduction • Why Failure Cost is needed in Optimization • Why Failure Cost should exceed a minimum value • Long Term Cost • Formula • Calculation Method • Example

3. Why We Need Failure Cost in Optimization Model Long-term Cost of Actions Present Cost of Actions If Failure Cost Is Not Used in Optimization Model (Failure Cost = Zero) = Long-term Cost of Do Nothing = 0 Each Condition State Minimum Long-term Cost = Do Nothing Action Present Cost = 0 Do Nothing Is Recommended in All Condition States Do Something Actions Present Cost > 0 No Optimal Policies

4. Why Failure Cost Should Exceed a Minimum Value For a Maintenance Policy to Be Optimal Long-term Cost of Maintenance Should Be Minimum If Failure Cost of an Element Is Insufficient Recommended Action in the Last (Worst) Condition State Could Be “DoingNothing” Element Could Fail Long Term Cost Will Not Be Minimum Maintenance Policy Will Not Be Optimal FailureCost Should Exceed a Minimum Value to Make the MaintenancePolicyOptimal

5. Long Term Cost of Actions Long Term Cost = PresentCost Discounted Future Failure Cost  Discounted Future Costs Over All Possible Condition States + + = =  Previous Minimum Long Term Cost Previous Long Term FailureCost TransitionProbability DiscountFactor Fail Probability x x x = PresentFailureCost Previous Minimum Long Term Cost of First ConditionState DiscountFactor + x

6. LT Cost Calculation Method Long-term Cost Calculation for All Actions Excel Macro (Iterations) Assume Minimum Long-term Cost Values in Each Condition State TransitionProbabilities DiscountFactor Multiply by Multiply by Min Long Term Cost in Each State Long Term Cost of Actions in Each State  Discounted Future Costs over All Possible States Discounted Future Cost Present Cost +

7. Long Term Cost Example

8. Long Term Cost For Iteration 1

9. Long Term Failure Cost For Iteration 1

10. Long Term Cost – Iteration 1 Summary

11. Long Term Cost – Iteration 2 Summary

12. Long Term Cost - Last Iteration

13. Long Term Failure Cost - Last Iteration

14. Long Term Cost– Last Iteration Summary

15. Failure Cost & LTC of Do Nothing

16. Failure Cost Formulation For a Given Discount Factor & For Each Element Environment Long Term Cost of Do Nothing in Last Condition State Increases by Increasing Failure Cost Increase FRF Values When Do Nothing Exceeds Do Something in Last Condition State Long Term Cost Calculation Assume Failure Cost as a Multiplier of Replacement Cost in Last Condition State Multiplier Is Called Failure Replacement Factor (FRF) Select FRF Value

17. Failure Cost = RC x FRF Assume Failure Cost As a Multiplier of Replacement Cost Replacement Cost (RC) Failure Replacement Factor (FRF)

18. FRF of Minimum Failure Cost • The FRF factor that results in a Minimum Failure Cost is the one for which the minimum long term cost in the last condition state becomes Doing Something after being Doing Nothing for the previous FRF factor. Recommending an Action Do Nothing Last Condition State

19. Failure Cost – Environment 1 Environment 1 Action Recommended When FRF = 4

20. Failure Cost – Environment 2 Environment 2 Action Recommended When FRF = 3

21. Failure Cost – Environment 3 Environment 3 Action Recommended When FRF = 2

22. Min. Failure Cost = FRF * RC

23. Minimum Failure Cost Variation With Environments

24. Failure Cost For 0.91 Discount Rate Discount = 0.91 Action Recommended When FRF = 4

25. Failure Cost For 0.92 Discount Rate Discount = 0.92 Action Recommended When FRF = 3

26. Failure Cost For 0.95 Discount Rate Discount = 0.95 Action Recommended When FRF = 2

27. Minimum Failure Cost Calculation Method Last Condition State Y Should Be Slightly > 1.0 Find FC for Y = 1.0 Long-term Cost of Do Nothing Divided by Minimum Long-term Cost of Do Something LTC DN Should be Slightly > Min LTC DS Apply a Factor Slightly > 1.0 Ratio Y = LTC DN/ Min LTC DS An Action Should Be Recommended Minimum Failure Cost = FC (for Y=1) * (1.05)

28. Minimum Failure Cost Calculation Method Long term Cost of Do Nothing / Min Long Term Cost of Do Something LTC 0 / min LTCa = 1.0 Modified Failure Cost = Assumed Failure Cost * Min Long Term Cost of Do Something / Long Term Cost Of Do Nothing *1.05 Y0 = Long term Cost of Do Nothing / Min Long Term Cost of Do Something Repeat if necessary FC0 = Assumed FC Modified FC = Assumed FC * 1.0 / Y0 * 1.05 Failure Cost

29. Minimum Failure Cost Calculation Method Converges to the Min. Failure Cost Assume Any Positive Value For FRF Calculate Long Term Cost (LTC) Failure Cost = New FRF x Replacement Cost x 1.05 Factor Initial Failure Cost = Assumed FRF x Replacement Cost x 1.05 Calculate the Ratio (Min LTC of Do Something / LTC of Do Nothing) New FRF = Previous FRF * LTC Ratio Converges to 1

30. Minimum Failure Cost Calculation Method LTC FC = FRF*RC*1.05 LTC Do Nothing Min LTC DS / LTC DN Assume FRF Value Last Condition State RC Copy New FRF New FRF

31. Minimum Failure Cost Results • Failure Cost = Assumed FC * • Min LTC Do Something / LTC Do Nothing * f • Where f = Factor slightly larger than 1.0 (e.g. 1.05)

32. Minimum Failure Cost Results

33. Sensitivity of the Failure Cost to the Probability of Failure

34. Applying Method to All Pontis CoRe Element Number of Condition States Number of Actions in Each State Pontis Elements Different Models Calculate Long Term Cost for All Elements Long-term Cost Formula For Each Model Copy Model Formula to Elements Each Model Represents a Number of Elements

35. FC Models of CoRe Elements

36. Calculating The Failure Cost in Pontis • Assume any value greater than zero for the failure cost (e.g. Failure Cost = Cost of Replacing the element) • Find the long-term cost values for the last condition state (run the Optimization Model in Pontis) • Find the ratio of the minimum long-term cost of “Do Something” over the long-term cost of “Do Nothing” in the last condition state • Multiply that ratio by the assumed failure cost and by a factorslightly greater than one (e.g. 1.05) to guarantee the selection of a Do Something action MinimumFailure Cost = Assumed FC * Min LTC Do Something / LTC Do Nothing * 1.05 • Repeat Steps 2 to 4 to get the minimumFailureCost

37. CONCLUSIONS • Different element ENVIRONMENTS have different Minimum Failure Costs • Different DISCOUNTRATES Result in different Minimum Failure Costs • The major factor affecting the Minimum Failure Cost is PROBABILITY of FAILURE • Minimum Failure Cost is calculated as follows • Assume an initial failure cost value • Calculate the Long Term Cost • Minimum Failure Cost = Assumed FC * Minimum Long-Term Cost of Do Something / Long Term Cost of Do Nothing * 1.05 • Repeat Step 2 & 3 until convergence.