A Study on Extended Warranty Models

1. A Study on Extended Warranty Models Prof. Dr. Alagar Rangan Vahid.H Khiabani Department of Industrial Engineering Eastern Mediterranean University North Cyprus

2. Outline • Introduction • Extended Warranty Model • Extended Warranty Model based on Total Sales Volume • Extended Warranty Model based on Total Sales Volume and Customer Satisfaction • Conclusions

3. Introduction • A warranty is a contractual obligation that assures the buyer that the product will perform its intended function at a specified level under the stated conditions for a specified period of time. If it fails to do so, the manufacturer will repair/replace the product at no cost or at a reduced cost to the buyer as the case may be depending on the warranty terms.

4. Introduction • The manufacturer uses warranty as a tool of advertisement in competition to other products.

5. A schematic diagram of various warranty policies

6. Introduction • Presently a large number of products are being sold in the market with long term or extended warranty policies. This is mainly due to fierce competition in the market share and customer demand. An extended warranty is the extension of the base warranty and is an obligation on the part of the manufacturer for further service to the consumer beyond the free replacement period W.

7. Introduction • On average 27% of the new car buyer’s purchased extended warranty. • Sears Roebuck reported a revenue generation of nearly US $1 billion from the sale of extended warranties. • The annual sale of extended warranties in UK alone is more than US $1 billion.

8. Literature Review • Blishke and Murthy (1994), Anisur and Chattopadhyay (2006), and Thomas and Rao (1999) • Nguyen and Murthy (1984 and 1986) • Mamer (1982) • Sahin and Polatoglu (1996) • AmitMonga and Ming J.Zuo (1998)

9. Literature Review • Mitra and Patnakar (1997) • Lutz and Padmanabhan (1998) • Hollis (1999) • Yeh Lam and Peggo Kwok Wai Lam (2001)

10. The Model • A repairable product is purchased at time t=0 which is subject to failures. The times between successive failures are independently and identically distributed with distribution function F (·). Failures are classified as type I and type II failures with probabilities q and p respectively.

11. The Model • Type I failures are minor in nature and can be minimally repaired (Barlow and Proschan, 1965) restoring the product to its original condition just prior to failure. If a product is minimally repaired at time t, then its failure rate after minimal repair is given by

12. The Model • Also it is well-known that if product failures are maintained by minimal repairs only, then the failures are governed by a non-homogeneous Poisson process with intensity function r (t). A type II failure is major and necessitates product replacement.

13. The Model • Major Repair • Minimal Repair :Major failure 0 t+x t 0 :Minor failure 0 t t+x

14. The Model • The customer at the time of the purchase buys the product along with FRW with an option for an extended warranty policy at the expiry of FRW for an additional cost. The manufacturer while agreeing to replace items on failure of any type during the FRW period W offers the following extended warranty policies:

15. Policy 1 (Tm) 0   w  

16. Policy 2 (Tk) 0   w   <k 0   w kth

17. Policy 3 (T) 0   w   <T 0   w W+T

18. Cost Analysis

19. Cost Analysis

20. Numerical Illustration

21. Numerical Illustration • The reason for writing LAP in the above form is to separate the cost parameters under the control of the manufacturer and the parameters associated with product lifetime. • For the specific choice of the cost parameters not shown here in our case, the long run average profits under different policies are given by:

22. Numerical Illustration

25. Optimization Based on Total Sales Volume • We wish to determine the optimal product price and warranty periods based on the total sales volume. The motivation for choosing such an objective is that sales volume is directly affected by these variables.

26. Optimization Based on Total Sales Volume • We assume the expected forecast sales volume q for the product to be a function of product price , warranty period W and the extended warranty period Yi and is assumed to follow the displaced log linear function

27. Optimization Based on Total Sales Volume

28. Optimization Based on Total Sales Volume • The manufacturer’s total long run average profit under the assumed demand function is seen to be

29. Optimization Based on Total Sales Volume • In our analysis the extended warranty period is fixed once the model parameters are known. Thus, we treat , the extended warranty period for policy i, as fixed and merge it with the amplitude factor so that the total long run average profit is

30. Optimization Based on Total Sales Volume Policy 1 Policy 3 Policy 2

31. EXT Warranty based on T.S.V and Customer Satisfaction • Each dissatisfied customer can impact future sales in several ways and this has serious cost implications for the manufacturer. • Though customer satisfaction with a purchased product is influenced by several reasons including product price and after sale service, the main factor that attracts (detracts) a customer to return to buy the product again is its performance over the warranty and extended warranty periods.

32. EXT Warranty based on T.S.V and Customer Satisfaction • Existing warranty models in the literature mainly focus on minimizing total warranty cost to optimize for warranty period and maintenance strategies; however they fail to take into consideration the penalty cost of customer’s non return in computing their objective functions as such a penalty seems to play a major rule. Lasser et al. (1998), Jack and Murthy (2004)

33. EXT Warranty based on T.S.V and Customer Satisfaction • We assume that the customer satisfaction of the product depends on the number of major failures during the warranty and extended warranty periods. It is tacitly assumed that the customer’s satisfaction is not influenced by minor failures.

34. EXT Warranty based on T.S.V and Customer Satisfaction If W is the FRW and T is the extended warranty period, the customer satisfaction depends on N2(W+T), the number major failures in W+T. We assume that the conditional probability function of customer return given N2(W+T) = k is specified as follows:

36. EXT Warranty based on T.S.V and Customer Satisfaction • Other forms for the customer dissatisfaction depending on the product could be chosen. For instance when the customer is highly sensitive to failures of the product, one can choose to be:

37. EXT Warranty based on T.S.V and Customer Satisfaction • Let be the probability of exactly k major failures during the warranty and extended warranty period for our extended warranty model. Then the probability Pr of the customer making a repurchase of the product is given by:

38. EXT Warranty based on T.S.V and Customer Satisfaction • Let Cp be the penalty cost to the manufacturer for a dissatisfied customer not returning to repurchase the product, then the manufacturer’s total profit function given in previous section can be rewritten as:

39. Policy 1 Policy 2 Policy 3

40. Conclusion • The present work develops a new extended warranty model with different options for the consumer. • Several extensions and generalizations are possible from here on, and in the following we will spell out a few of them:

41. Different types of failures • p and q functions of the time • General repair • Alternate forms of demand function • Optimal burn in period • Preventive maintenances • Other factors that influence the product sales

42. References • AmitMonga. and Ming J.Zuo. “Optimal system design considering maintenance and warranty”, Computers & Operations Research, 25(Issue9), pp. 691-705, September 1998. • Anisur Rahman. and Chattopadhyay, G. “Review of Long-Term Warranty Policies”, Asia-Pacific Journal of Operational Research, Vol.23, No. 4, pp. 453-472, 2006. • Barlow, R. and Proschan, F. “Mathematical Theory of Reliability”, John Wiley & Sons, New York, 1965. • Barlow, R.E. and Hunter, L.C. “Optimum preventive maintenance policies”, Operations Research, 8, pp. 90-100, 1960. • Blischke, W.R. and Murthy, D.N.P. “Strategic warranty management: A life cycle approach”, IEEE Trans. Eng Magn, vol. 47, pp. 40-54, 2000. • Blishke, W.R. and Murthy D.N.P. “Warranty Cost Analysis”, Marcel Dekker, New York, 1994. • Blishke, W.R. and Scheuer E.M. “Applications of renewal theory in analysis of free-replacement warranty”, Naval Research Logistics Quarterly, 28, pp. 193-205, 1981.

43. References • Blishke, WR. and Murthy, D.N.P. “Product warranty management-I: Taxonomy for warranty policies”, European Journal of Operational Research, 62, pp.127-148, 1992. • Boland, P. and Proschan, F. “Periodic replacement with increasing minimal repair costs at failure”, Operational Research, 30, pp. 1183-1189, 1982. • BoyanDimitrov, StefankChukova, ZohelKhalil, “Warranty Costs: An Age-Dependent Failure/Repair Model”, Naval Research Logistics, Vol. 51, 2004. • Brown, M. and Proschan, F. “Imperfect repair”, Journal of Applied probability, 20, pp. 851 - 859, 1983. • Fontenot, R.A. and Proschan, F. “Some Imperfect maintenance models”, In Reliability Theory and Models, ed. M.AbdelHameed, ErhanCinlar and Joseph Quinn. Academic Press, London, 1984. • Gerald, M.Smith. (5thEd), “Statistical Process Control and Quality Improvement”, Pearson Prentice Hall , page 41, 2004. • Glickman, T. S. and Berger, P. D. “ Optimal price and protection period decisions for a product under warranty”, Management Science, 22, pp. 1381-1389, 1976.

44. References • Hollis, A. “Extended warranties, adverse selection and after markets”, The Journal of Risk and Insurance, 66(3), pp. 321-343, 1999. • Jack, N. and Murthy, D.N.P. “Warranty servicing strategies to improve customer satisfaction”, IMA Journal of Management Mathematics, 15, pp. 111-124, 2004. • Jun Bai and Hoang Pham. “Repair-Limit Risk-Free Warranty Policies with Imperfect Repair”, IEEE Transactions on system, Man and Cybernetics-Part A: System and Humans, Vol.35, No.6, November 2005. • Kijima, M. “Some results for repairable systems with general repair”, Journal of Applied Probability, 26, pp. 89-102, 1989. • Kijima, M. Morimura, H. and Suzuki,Y. “ Periodical Replacement Problem Without Assuming Minimal Repair, European Journal of Operations Research, 37, pp.194-203, 1988. • Kim, C.S. Djamaludin, I. and Murthy, D.N.P. “Warranty and discrete preventive maintenance”, Reliability Engineering and System Safety, 84, pp. 301-309, 2004. • Kumar, UD. and Chattopadhyay, G. “Mathematical models for analysis of extended warranty”. In Proc of the fifth Asia-Pacific Industrial Engineering and Management Systems Conference, Gold Coast, Australia, pp. 249, 12-15 December, 2004.

45. References • Lassar, W.M. Folkes, V.S. Grewal, D. and Costley, C. “Consumer affective reactions to product problems when timing of warranty expiration varies”, J. Business Res., 42, pp. 265-270, 1998. • Lutz, N. and Padmabhaban, V. “Warranties, extended warranties and product quality”, International Journal of Industrial Organization, 16, pp. 463-493, 1998. • Mamer, J.W. “Cost Analysis of prorate and free replacement warranties”, Naval Research Logistics Quarterly, 29(2), pp. 345-356, 1982. • Matis, T.I. Jayaraman, R. and Rangan, A. “Optimal price and pro-rata decisions for combined warranty policies with different repair options”, IIE Transactions, in press, 2008. • Medhi, J. (2nd Ed), “Stochastic processes”, Wiley Eastern limited, 1994. • Mitra, A. and Patnakar, JG. “Market share and warranty cost for renewable warranty programs”, International Journal of Production Economics, 50, pp. 155-168, 1997. • Murthy, D.N.P. and Blishke, W.R. “Product warranty and management-II: An integrated framework for study”, European Journal of Operational Research, 62, pp. 261-281, 1992.

46. References • Murthy, D.N.P. Workshop on reliability and warranty, Operational Research Society of India, Chennai, India, December, 20, 2004 • Nguyen, D.G. and Murthy, D.N.P. “An Optimal Policy for servicing warranty”, Journal of the Operations Research Society, 37, pp. 1081-1098, 1986. • Nguyen, D.G. and Murthy, D.N.P. “Cost Analysis of Warranty Policies”, Naval Research Logistics Quarterly, 31, pp. 525-541, 1984. • Padmanabhan, V. “Usage heterogeneity and extended service contracts”, Journal of Economics and Management Strategy, 12, pp. 33-54, 1995. • Padmanabhan, V. and Rao, RC. “Warranty policies and extended service contract: theory and an application to automobiles”, Marketing Science, 12, pp. 230-247, 1993. • Pascual, R. and Ortega, J.H. “Optimal replacement and overhaul decisions with imperfect maintenance and warranty contracts”, Reliability Engineering and System Safety, pp.1-8, 2005. • Ross, Sheldon M. (2nd Ed), “Stochastic processes”, John Wiley & Sons, Inc, 1996. • Sahin, I. and Polatoglu, H. “Maintenance strategies following the expiration of warranty”, IEEE Transactions on Reliability, 45(2), pp. 220-228, 1996.

47. References • Stadje, W. and Zuckerman, D. “Optimal Strategies for some repair-replacement Models”, Advances in Applied Probability, 22, pp. 641-656, 1990. • Thomas, M.U. and Rao, S.S. “Warranty economic decision models: a summary and some suggested directions for future research”, Operations Research, 47, pp. 807-820, 1999. • Yeh Lam. and Peggo Kwok Wai Lam. “An extended warranty policy with options open to consumers”, European Journal of Operational Research, 131, pp. 514-529, 2001. 

48. Thank you for your time and attention