Analysis of 2-D Warranty Data

  • Wallace R. Blischke
  • M. Rezaul Karim
  • D. N. Prabhakar Murthy
Part of the Springer Series in Reliability Engineering book series (RELIABILITY)


In this chapter, we look at the analysis of 2-D warranty data. Here we encounter several new issues that are different from those of 1-D warranty data. We discuss two data structures and four alternative scenarios. The latter depend on what data relevant to the 2-D warranty policy have been collected. For this policy, the unavailability of information on censored items leads to difficulties in estimation of the life distribution of the items. We look at three approaches to modeling failuresᾢconditioning on usage rate, composite scale, and bivariate distribution function. The models are applied to forecasting warranty claims and warranty costs.


Supplementary Data Usage Rate Claim Data Nonparametric Estimate Usage Limit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Baik J, Murthy DNP (2008) Reliability assessment based on two-dimensional warranty data. Int J Reliab Saf 2:190ᾢ208CrossRefGoogle Scholar
  2. 2.
    Baik J, Murthy DNP, Jack N (2004) Two-dimensional failure modelling and minimal repair. Nav Res Logist 51:345ᾢ362MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Blischke WR, Murthy DNP (1994) Warranty cost analysis. Marcel Dekker, Inc., New YorkGoogle Scholar
  4. 4.
    Blischke WR, Murthy DNP (eds) (1996) Product warranty handbook. Marcel Dekker, Inc., New YorkGoogle Scholar
  5. 5.
    Duchesne T, Lawless JF (2000) Alternative time scales and failure time models. Lifetime Data Anal 6:157ᾢ179MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Gertsbakh IB, Kordonsky KB (1998) Parallel time scales and two-dimensional manufacturer and individual customer warranties. IIE Trans 30:1181ᾢ1189Google Scholar
  7. 7.
    Iskandar BP, Blischke WR (2003) Reliability and warranty analysis of a motorcycle based on claims data. In: Blischke WR, Murthy DNP (eds) Case studies in reliability and maintenance. Wiley, New York, pp 623ᾢ656Google Scholar
  8. 8.
    Iskandar BP, Murthy DNP (2003) Repair-replace strategies for two-dimensional warranty policies. Math Comput Model 38:1233ᾢ1241MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Jiang R, Jardine AKS (2006) Composite scale modeling in the presence of censored data. Reliab Eng Sys Saf 91:756ᾢ764CrossRefGoogle Scholar
  10. 10.
    Jung M, Bai DS (2007) Analysis of field data under two-dimensional warranty. Reliab Eng Sys Saf 92:135ᾢ143CrossRefGoogle Scholar
  11. 11.
    Kim HG, Rao BM (2000) Expected warranty cost of a two-attribute free-replacement warranties based on a bi-variate exponential distribution. Comput Ind Eng 38:425ᾢ434CrossRefGoogle Scholar
  12. 12.
    Kordonsky KB, Gertsbakh IB (1993) Choice of best time scale for reliability analysis. Euro J Ops Res 65:235ᾢ246MATHCrossRefGoogle Scholar
  13. 13.
    Kordonsky KB, Gertsbakh IB (1995a) System state monitoring and lifetime scales I. Reliab Eng Sys Saf 47:1ᾢ14CrossRefGoogle Scholar
  14. 14.
    Kordonsky KB, Gertsbakh IB (1995b) System state monitoring and lifetime scales II. Reliab Eng Sys Saf 49:145ᾢ154CrossRefGoogle Scholar
  15. 15.
    Kordonsky KB, Gertsbakh IB (1997) Multiple time scales and lifetime coefficient of variation: engineering applications. Lifetime Data Anal 2:139ᾢ156CrossRefGoogle Scholar
  16. 16.
    Lawless JF, Hu XJ, Cao J (1995) Methods for the estimation of failure distributions and rates from automobile warranty data. Lifetime Data Anal 1:227ᾢ240MATHCrossRefGoogle Scholar
  17. 17.
    Manna DK, Pal S, Sinha S (2006) Optimal determination of warranty region for 2D policy: A customers’ perspective. Comput Ind Eng 50:161ᾢ174CrossRefGoogle Scholar
  18. 18.
    Manna DK, Pal S, Sinha S (2007) A use-rate based failure model for two-dimensional warranty. Comput Ind Eng 52:229ᾢ240CrossRefGoogle Scholar
  19. 19.
    Manna DK, Pal S, Sinha S (2008) A note on calculating cost of two-dimensional warranty policy. Comput Ind Eng 54:1071ᾢ1077CrossRefGoogle Scholar
  20. 20.
    Moskowitz H, Chun YH (1994) A Poisson regression model for two-attribute warranty policies. Nav Res Logist 41:355ᾢ376MATHCrossRefGoogle Scholar
  21. 21.
    Murthy DNP, Iskandar BP, Wilson RJ (1995) Two-dimensional failure free warranties: Two-dimensional point process models. Oper Res 43:356ᾢ366MATHCrossRefGoogle Scholar
  22. 22.
    Murthy DNP, Xie M, Jiang R (2004) Weibull models. Wiley, New YorkMATHGoogle Scholar
  23. 23.
    Pal S, Murthy GSR (2003) An application of Gumbel’s bivariate exponential distribution in estimation of warranty cost of motorcycles. Int J Qual Reliab Manag 20:488ᾢ502CrossRefGoogle Scholar
  24. 24.
    Rai B, Singh N (2003) Hazard rate estimation from incomplete and unclean warranty data. Reliab Eng Sys Saf 81:79ᾢ92CrossRefGoogle Scholar
  25. 25.
    Rai B, Singh N (2005) A modeling framework for assessing the impact of new time/mileage warranty limits on the number and cost of automotive warranty claims. Reliab Eng Sys Saf 88:157ᾢ169CrossRefGoogle Scholar
  26. 26.
    Rai B, Singh N (2006) Customer-rush near warranty expiration limit and nonparametric hazard rate estimation from the known mileage accumulation rates. IEEE Trans Reliab 55:480ᾢ489CrossRefGoogle Scholar
  27. 27.
    Suzuki K, Karim MR, Wang L (2001) Statistical analysis of reliability warranty data. In: Balakrishnan N, Rao CR (eds) Handbook of statistics: Advances in reliability, vol 20. Elsevier Science, New York, pp 585ᾢ609Google Scholar
  28. 28.
    Suzuki K (1985a) Estimation of lifetime para-meters from incomplete field data. Technometrics 27:263ᾢ271MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Suzuki K (1985b) Nonparametric estimation of lifetime distribution from a record of failures and follow-ups. J Am Statist Assoc 80:68ᾢ72MATHCrossRefGoogle Scholar
  30. 30.
    Yang G, Zaghati Z (2002) Two-dimensional reliability modeling from warranty data. Proc Annu Reliab Maintainab Symp 272-278Google Scholar

Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  • Wallace R. Blischke
    • 1
  • M. Rezaul Karim
    • 2
  • D. N. Prabhakar Murthy
    • 3
  1. 1.Sherman Oaks,Los AngelesUSA
  2. 2.Department of StatisticsRajshahi UniversityRajshahiBangladesh
  3. 3.School of Mechanical and Mining EngineeringThe University of QueenslandBrisbaneAustralia

Personalised recommendations