Extended warranty pricing considering the time of money. Hui-Ming Teng

Transcription

1 Extended warranty pricing considering the time of money Hui-Ming Teng Department of Business Administration Chihlee Institute of Technology Panchiao Taiwan R.O.C. Abstract Warranty usually plays an important role in providing customers with product assurance. Although the customers have to pay extra money to buy an extended warranty, they can obtain more protections from the company. Our study focuses on the extended warranty; the successive failure times of the repairable product are assumed to be drawn from a non-stationary Poisson process, taking into the consideration of time value of money, to figure out a more precise model of warranty cost. A numerical example is given to illustrate the model. Keywords : Warranty, computer industry, net present value. 1. Introduction Warranty is a guarantee in the form of a contract. Generally, it is provided by the manufacturer (seller) to customers for its products or services. The warranty should clearly define the responsibility and obligation for both sides. Two issues must be documented on this contract: (1) Detailed description of the functions for the product. (2) Any failure due to the faulty design, manufacturing, or poor quality needs to be addressed by the manufacturer or the retailer [1]. [email protected] Journal of Information & Optimization Sciences Vol. 27 (2), No. 2, pp c Taru Publications /06 $

2 402 H. M. TENG A new product will ultimately replace the old ones due to better functions. However, most customers are not confident in the new product. To induce confidence, extended warranty is provided. Thus provision of extended warranty has become a vital policy for manufacturers and retailers. The recent new trend is to provide different types of warranties to satisfy different kinds of products and customers. This also can consolidate the customers confidence and strengthen the willingness to buy the products [2, 3]. This concept is very important in marketing new product because it reflects appropriately the performance and reliability of the product and simultaneously reduces the potential concern from the customers. A failure on an item results in an immediate claim and other related claims. Failures over the warranty period are building upon (1) part level, and (2) system level. For most products the failure rate increases with time. Warranty implements according to the contract between the seller and the buyer during a predefined time period. In other words, the manufacturer (seller) has the responsibility to replace the part(s) or to restore the full function of the product provided it is under proper usage. There are two types of warranty regular warranty and extended warranty. The regular warranty cost is usually included in the sale price. However, to have the extended warranty, the customer has to pay extra money to get the service(s) beyond the regular warranty. Blischke and Murthy [4, 5] suggested that the manufacturers should keep providing better warranty after the product is sold. Several characteristics have to be taken into consideration for the manufacturer to determine its warranty policy [4, 6]: (1) one-dimensional/two-dimensional policy: consider age or usage or both of the products. (2) free-replacement/pro-rata policy. (3) renewing/non-renewing policy. The competition in the market of computer industries is remarkable. In order to enhance market share, the manufacturers have to keep providing new models with new add-on functions and better warranty plan. There are several special characteristics for the maintenance of computer:

3 EXTENDED WARRANTY PRICING 403 (1) Rarely need to replace part(s). (2) The failure is due to inappropriate usage by the user or system failure. (3) Customer may request for upgrade during warranty. Due to the above reasons, it is very common for the buyers to request for extended warranty. Since the warranty policy can directly affect the manufacturing cost, the manufacturer has to take into consideration of the factors that can influence the customers purchasing and also the reliability of the product during the policy making. Warranty service costs include one or more of the following elements: (1) management and handling costs (2) cost of renewing, replacements or repairs (3) cost of repairs labor and parts (4) compensations A lot of research on analyzing the warranty costs had been conducted. Stefanka and Hayakawa [7] evaluated the warranty costs over the warranty period under non-renewing free replacement policy over the life cycle of the product. Kim and Murthy [8] studied the expected warranty cost for products sold with free replacement warranty with varying usage intensity. Chattopadhyay and Murthy [9] developed probabilistic models to compute the expected warranty cost to the manufacturer when the items are sold with free replacement or pro-rata warranties. Lam and Kwok [10] derived the exact expressions of the total expected discounted cost, and the long-run average cost per unit time for a consumer and the manufacturer. Huang and Zhuo [11] proposed a Bayesian decision model for determining the optimal warranty policy for repairable products. Mitra and Patanlar [12] proposed a model where the buyer has the option to extend the warranty should the product not failure during the initial warranty period. However, little researches had even taken into the consideration of the time value of money. In reality, the warranty usually lasts more than six months, so the net present value (NPV) consideration is very important. This study uses the statistical analytic approach, other than previous studies, and the time value of money to deduce the warranty cost.

4 404 H. M. TENG 2. Notation and assumptions In this section, the following notations and assumption are used throughout this paper. T : interarrival time of failure, a random variable N : number of failure, a random variable c : expected cost of each repair α : discount rate TC : net present value (NPV) of expected warranty cost TC : expected warranty cost without considering the time value of money The time value of money is considered using the DCF approach. The interest is compounded continuously. 3. Modeling and analysis The expected warranty cost in this study considers the extended warranty, i.e. the expected warranty cost is the sum of all the costs of repair/maintenance during the contracted period of warranty after sale. The cost includes only the cost of labor but not the cost of the replaced parts. In computer industries, the warranty period is more than six months and starts right after the sale or installation. Customers can choose from the different packages (or different time period) of extended warranty for their own purpose. For a random inter-arrival time of failure, T, and a random number of failure, N[a, b], during the period [a, b], it is obvious that the longer use time will result in higher probability of failure. The failure functions are modelled by a non-stationary Poisson process with an intensity function λ(s) = λβ(λs) β 1, s > 0, β 1, an increasing function of s. Then ( b ) N[a, b] Po λ(s)ds. (1) a Assuming the warranty period for the product starts at the original date of installation, that means time = 0, the expected number of failures during time interval [a, v] is E(N(v a)) = v a λ(s)ds = λ β (v β a β ). (2) Two cases are considered for (1) β = 1, (2) β > 1 :

5 EXTENDED WARRANTY PRICING 405 (1) When β = 1, i.e. λ(s) = λ, s > 0, then the number of failure during time interval [a, v] is N[a, v] Po(λ(v a)). (3) Assuming the expected cost of each repair is c, the arrival time of the j th failure is a random variable T j with T j Γ where the p.d.f of Γ t > 0, j = 1, 2,.... ( ) 1 j,, j = 1, 2,..., (4) λ(v a) ( ) 1 j, λ(v a) is f (t) = [λ(v a)] j t j 1 e λ(v a)t, Γ( j) The warranty cost considering the time value of money for the j th failure, is C NPV j = ce α(a+t j). (5) Where α is the discount rate (assume c NPV 0 = 0). And the NPV of expected warranty cost, TC, is TC = E N ( N j=0 E T j (c NPV j ) ) = E N ( N j=1 E T j (ce α(a+t j) ) ) [ = e αa N E(ce αt j N=n ) n=1 j=1 [ = e αa n c e n=1 j=1 0 [ = e αa n c n=1 j=1 = e αa cλ(v a) n=1 α αa cλ(v a) = e α ] P(N = n) ] αt (λ(v a)) j t j 1 e λ(v a)t dt P(N = n) Γ( j) ( ) λ(v a) j P(N = n)] λ(v a) + α { n=1 [ 1 ( λ(v a) ) n ] (λ(v a)) n e λ(v a) λ(v a) + α n! [ ] (λ 2 (v a)) ne 2 λ(v a) (λ(v a)) n e λ(v a) n! n=1 λ(v a)+α n! }

6 406 H. M. TENG αa cλ(v a) = e α αa cλ(v a) = e α [1 e λ2 (v a) 2 λ(v a)+α e λ(v a)] [1 e λα(v a) ] λ(v a)+α. (6) Note. The expected warranty cost without consider the time value of money, is TC = ce(n(v a)) = cλ(v a). (2) When β > 1, the number of failure during time interval [a, v] is N[a, v] Po(λ β (v β a β )). (7) Because deducing the arrival time of j th failure from a is more complicate, we propose a heuristic algorithm. Assuming the warranty period for the product starts on the original date of installation. Let a j, j = 1, 2,..., be the expected arrival time of the j th failure from a, then a j satisfies a j λ(s)ds = j. (8) a i.e. a j = (a β + jλ β ) 1 β. The expected warranty cost for the j th failure considering the time value of money, is c NPV j = ce αa j, (9) and the NPV of the expected warranty cost, is TC = λ β (v β a β )+0.5 c NPV j, (10) j=1 where x denotes the greatest integer less than or equal to x, and 0.5 is to modify/round off λ β (v β a β ). Note. The expected warranty cost without considering the time value of money, TC = ce(n(v a)) = cλ β (v β a β ). Numerical example. When β = 1, c = 50, λ = 3, α = 0.02, using software MATHCAD, the results are shown in Table 1. When β = 2, c = 50, λ = 3, α = 0.02, the results are shown in Table 2. Both tables show the expected warranty cost to be in a ranges from 0 to 2.5, v ranges over 0.5 to 3, and β = 1, 2.

7 EXTENDED WARRANTY PRICING 407 Table 1 The expected warranty cost when β = 1 a β = 1, c = 50, λ = 3, α = 0.02 v (75) (150) (225) (300) (375) (450) (75) (150) (225) (300) (375) (75) (150) (225) (300) (75) (150) (225) (75) (150) NOTE: The values in bracket show the expected warranty cost without considering the time value of money Table 2 The expected warranty cost when β = 2 (75) a β = 2, c = 50, λ = 3, α = 0.02 v (112.5) (450) (1012) (1800) (2812.5) (4050) (337.5) (900) (1687.5) (2700) (3937.5) (562.5) (1350) (2362.5) (3600) (787.5) (1800) (3037.5) (1012.5) (2250) (1237.5) NOTE: The values in bracket show the expected warranty cost without considering the time value of money

8 408 H. M. TENG 4. Conclusion Owing to the fact that customers usually put quality of the products and post-sale service into consideration in their product selection, the manufacturers(sellers) have to provide a good warranty to increase market share. A good warranty policy can also provide a good image to the manufacturers(sellers). Provision of warranty has become an indispensable part of marketing strategy. Most customers are willing to pay an extra charge for the benefits of the extended warranty. When a manufacturer wants to incorporate the extended warranty into the product, it is important to estimate the warranty cost through statistical analysis and market research. Our study takes the time value of the money into consideration to derive an estimated warranty cost. This estimated value can provide manager a basis in deciding what type of warranty policy to use. References [1] D. N. P. Murthy, O. Solem and T. Roren, Product warranty logistics: issue and challenges, European Journal of Operational Research, Vol. 156 (2004), pp [2] S. Erevelles, A. Roy and S. L. Vargo, The use of price and warranty cues in product evaluation: a comparison of U.S. and Hong Kong consumers, Journal of International Consumer Marketing, Vol. 11 (3) (1999), pp [3] W. O. Bearden and T. A. Shimp, The use of extrinsic cues to facilitate product adoption, Journal of Marketing Research, Vol. 19 (5) (1982), pp [4] W. R. Blischke and D. N. P.Murthy, Product warranty management I, a taxonomy for warranty policies, European Journal of Operational Research, Vol. 62 (1991), pp [5] W. R. Blischke and D. N. P. Murthy, Warranty Cost Analysis, Marcel Dekker, New York, [6] S. Mondal, S. Pal and D. K. Manna, Cost estimation under renewing warranty policy an application, Quality Engineering, Vol. 16 (1) (2003), pp [7] C. Stefanka and H. Yu, Warranty cost analysis: non-renewing warranty with repair time, Applied Stochastic Models in Business and Industry, Vol. 20 (1) (2004), pp

9 EXTENDED WARRANTY PRICING 409 [8] C. S. Kim, I. Djamaludin and D. N. P. Murthy, Warranty cost analysis with heterogeneous usage intensity, International Transactions in Operational Research, Vol. 8 (2001), pp [9] G. N. Chattopadhyay and D. N. P. Murthy, Warranty cost analysis for second-hand products, Mathematical and Computer Modelling, Vol. 31 (10-12) (2000), pp [10] Y. Lam and W. L. P. Kwok, An extended warranty policy with options open to consumers, European Journal of Operational Research, Vol. 131 (3) (2001), pp [11] Y. S. Huang and Y. F. Zhuo, Estimation of future breakdowns to determine optimal warranty policies for products with deterioration, Reliability Engineering and System Safety, Vol. 84 (2) (2004), pp [12] A. Mitra and K. G. Patankar, Market share and warranty costs for renewable warranty programs, International Journal of Production Economics, Vol. 50 (1997), pp Received September, 2005