Expected warranty cost of two-attribute free-replacement warranties based on a bivariate exponential distribution

Computers & Industrial Engineering 38 (2000) 425±434

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Expected warranty cost of two-attribute free-replacement warranties based on a bivariate exponential distribution H.-G. Kim a,*, B.M. Rao b a

Department of Industrial Engineering, Dongeui University, Pusan 614-714, South Korea Department of Applied Statistics and Operations Research, Bowling Green State University, Bowling Green, OH 43403, USA

b

Accepted 24 August 2000

Abstract Many warranty policies offered by manufacturers are limited by two factors, typically time since purchase, and usage of the product. Such policies are referred to as two-attribute (or two-dimensional) warranty policies and can be described in terms of regions in a plane with one axis representing time and the other axis the usage. In this paper, we consider two-attribute warranty policies for non-repairable items. The item failures are described in terms of a bivariate exponential distribution. Analytical expressions are derived for the resulting two-dimensional renewal function and the cost of warranty. An illustrative numerical example is presented to study the effect of correlation between the warranty variables on the cost of warranty. q 2000 Elsevier Science Ltd. All rights reserved. Keywords: Two-attribute warranty policy; Bivariate exponential distribution; Two-dimensional renewal function; Warranty cost; Correlation

1. Introduction A warranty is a contractual obligation of a manufacturer or vendor in relation to product sales and services. The most common form of warranty is characterized by a time interval called the warranty period. Such warranties can be classi®ed as free replacement or prorata replacement based on the cost to the consumer of replacing an item that fails within the warranty period. A two-attribute warranty may be characterized as a region in two-dimensions (2D) with the horizontal axis representing time and the vertical axis the usage. A typical example of such a warranty is that offered for new automobiles. Here, * Corresponding author. Fax: 182-51-890-1619. E-mail address: [email protected] (H.-G. Kim). 0360-8352/00/$ - see front matter q 2000 Elsevier Science Ltd. All rights reserved. PII: S0 3 6 0 - 8 3 5 2 ( 0 0 ) 0 0 05 5 - 3

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the warranty is valid until either a pre-speci®ed time limit or a pre-speci®ed usage limit (in miles driven) is exceeded. While there is extensive literature (Kao & Smith, 1996; Kim, 1997; Mi, 1997; Rao, 1995) on time limited warranties, two-attribute warranties received very limited attention (Chun & Tang, 1999; Moskowitz & Chun, 1994; Murthy, Iskander, & Wilson, 1995; Singpurwalla & Wilson, 1998). For a complete review of all existing literature on warranty modeling we refer the readers to the excellent books on the topic by Blischke & Murthy (1994a,b) and a recent survey by Thomas and Rao (1999). The approaches used to study two-attribute warranties may be classi®ed as either 1D or 2D based on the nature of the point process used to study the warranty. The 1D approach, as the name suggests, considers a 1D point process by assuming a relationship between the two-warranty attributes or variables. Moskowitz and Chun (1994) assumed a linear relationship with non-negative coef®cients and modeled the failures for free-replacement warranties using a conditional Poisson process. Chun and Tang (1999) followed a similar approach to Moskowitz and Chun (1994) to study the warranty costs under free replacement and prorata warranties. They determined the effective warranty period and then estimated the present value of the total future repair cost. A variation of this approach proposed by Singpurwalla and Wilson (1998) uses an additive hazard approach to modeling failures indexed by two scales. Their approach consists of decomposing the joint distribution of time and usage by ®rst selecting a process appropriate to the type of damage (e.g. shocks, continuous wear or intermittent wear) and incorporating the effect of usage on time to failure as an additive hazard model. In order to avoid imposing a relationship between the two warranty variables, one needs to adopt the 2D approach. This approach involves the use of a bivariate probability distribution function for the two warranty variables. The only published paper using this approach is by Murthy et al. (1995). They considered four variations of two-attribute warranties and derived results for the expected warranty cost per item sold for each of the policies. Multivariate Pareto distributions of the ®rst and second kind as well as Beta Stacy distribution were considered as candidates for the joint distribution for the two warranty variables. The numerical results were obtained using a 2D renewal integral equation solver. As this study amply illustrates, this approach is analytically dif®cult due to the complexities associated with the 2D renewal functions. In this paper, we analyze two-attribute free-replacement warranty policies using the 2D approach. We consider a simple and elegant bivariate exponential (BVE) distribution to describe the relationship between the two warranty variables. Relative to the distributions considered in Murthy et al. (1995), Downton's BVE distribution requires fewer parameters and permits the study of the effect of correlation between the variables on the warranty costs. The remainder of the paper is organized as follows: we brie¯y discuss Downton's BVE distribution and the associated 2D renewal processes in Sections 2 and 3, respectively. In Section 4, we analyze twoattribute warranty policies and obtain the expected warranty cost per item. In Section 5, we give numerical examples and discuss some implications for the consumer and the manufacturer. Concluding remarks are presented in Section 6. 2. Downton's BVE distribution Barlow and Proschan (1975) and Johnson and Kotz (1972) describe several BVE distributions. Many of these do not have an explicit joint probability density function and most involve severe restrictions on

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the correlation between the two random variables. In this section we describe the analytically simple and elegant BVE distribution proposed by Downton (1970) and discuss its suitability in modeling twoattribute warranty modeling. The joint probability density function for Downton's BVE distribution is given by )  (  l1 l2 l1 x 1 l2 y 2…rl1 l2 xy†1=2 I0 exp 2 f …x; y† ˆ 12r 12r 12r where In …´† is the modi®ed Bessel function of the ®rst kind of nth order and r…0 # r , 1† denotes the correlation coef®cient between the random variables X and Y. The marginal density functions of X and Y, denoted by f1 …x† and f2 …y†; are exponential with means 1/l 1 and 1/l 2, respectively (Downton, 1970). f p …s1 ; s2 †; the bivariate Laplace transform of f …x; y†; is given by (Downton, 1970): L{f …x; y†} ˆ f p …s1 ; s2 † ˆ

l1 l2 : …l1 1 s1 †…l2 1 s2 † 2 rs1 s2

…1†

Similarly, F p …s1 ; s2 †; the bivariate Laplace transform of F…x; y†; the cumulative density function of f …x; y† is given by: L{F…x; y†} ˆ F p …s1 ; s2 † ˆ

f p …s1 ; s2 † : s1 s2

From the properties of Laplace transforms, we have f

p

…n†

…s1 ; s2 † ˆ ‰f p …s1 ; s2 †Šn

…2†

and F

p

…n†

…s1 ; s2 † ˆ

1 ‰f p …s1 ; s2 †Šn s1 s2

…3†

where the superscript (n) denotes n-fold convolution. It is also known from Downton (1970) that when rˆ0 F …n† …x; y† ˆ Pn …l1 x†Pn …l2 y†;

…4†

where Pn …x† is the incomplete gamma function de®ned as follows: Pn …x† ˆ

Zx un21 exp…2u† du : G…n† 0

…5†

A requisite property of f …x; y† is that the conditional expectations of X and Y must be increasing functions of the other variable (Murthy et al., 1995). From Downton (1970) we have, E…XuY ˆ y† ˆ and

12r l 1r 2y l1 l1

  12r 12r l2 Var…XuY ˆ y† ˆ 1 2r y : l1 l1 l1

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Clearly, E…XuY ˆ y† as well as Var…XuY ˆ y† increase linearly with y. This property coupled with the very small number of parameters and the analytical elegance makes this a good choice in two-attribute warranty model. One limitation of this distribution is that the marginal distributions of X and Y are necessarily exponential. However, when this condition is satis®ed to an acceptable degree, this property greatly simpli®es the ®tting of the distribution because the only other parameter that needs to be estimated is the correlation coef®cient r . This represents a signi®cant advantage because in real life item failure data available to the manufacturer is usually not suf®cient to estimate large number of parameters with precision. 3. 2D renewal processes A 2D renewal process is a sequence of independent and identically distributed non-negative random variables {…Xn ; Yn †; n $ 1} with common joint distribution function F…x; y†: Let N…x; y† denote the number of renewals over the rectangle ‰…0; x† £ …0; y†Š; with the origin being a renewal point. {Xn ; n $ 1} is a sequence of independent and identically distributed random variables with common distribution function F1 …x† ˆ F…x; 1† and de®nes the univariate renewal counting process N1 …x†: Similarly, the sequence {Yn ; n $ 1} is a 1D renewal process with common distribution function F2 …y† ˆ F…1; y† and de®nes N2 …y†: From Hunter (1974) we have, N…x; y† ˆ min{N1 …x†; N2 …y†} Analogous to the univariate theory, the 2D renewal function Mr …x; y† ; E‰N…x; y†Š is given by: Mr …x; y† ˆ

1 X

F …n† …x; y†

…6†

nˆ1

where the subscript r has been of the renewal function to highlight that it is a P included in the de®nition P function of r . Let S1n ˆ niˆ1 Xi ; and S2n ˆ niˆ1 Yi denote the points of the nth renewal for the sequences {Xn ; n $ 1}; and {Yn ; n $ 1}; respectively. Clearly, P…S1n # x† ˆ F …n† …x; 1† ˆ F1…n† …x†; and P…S2n # y† ˆ F …n† …1; y† ˆ F2…n† …y†: Since the marginal distributions of X and Y are exponential, the 1D renewal functions M1 …x† and M2 …y† are given by: M1 …x† ˆ l1 x; and M2 …x† ˆ l2 y:

…7†

Applying Laplace transformation to both sides of Eq. (6) and using Eq. (3), we obtain the bivariate Laplace transform of Mr …x; y† as: Mrp …s1 ; s2 †

f p …s1 ; s2 † : ˆ s1 s2 ‰1 2 f p …s1 ; s2 †Š

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Fig. 1. Warranty region for Policy A.

Substituting for f p …s1 ; s2) from Eq. (1) and simplifying, we obtain: Mrp …s1 ; s2 † ˆ

l1 l2 : s1 s2 ‰l2 s1 1 l1 s2 1 …1 2 r†s1 s2 Š

Some algebraic manipulation yields M0p ‰…1 2 r†s1 ; …1 2 r†s2 Š ˆ

1 Mrp …s1 ; s2 † …1 2 r†3

…8†

where M0p …s1 ; s2) is the Laplace transform of Mrp …s1 ; s2) when r ˆ 0: By appealing to the scale change property of 2D Laplace transforms (Hunter, 1974), it can be seen that    x y ; …9† ˆ …1 2 r†2 M0p ‰…1 2 r†s1 ; …1 2 r†s2 Š: L M0 12r 12r Using Eqs. (8) and (9), we obtain    x y 1 L M0 ; M p …s ; s †; ˆ 12r 12r …1 2 r† r 1 2 which upon inversion yields  Mr …x; y† ˆ …1 2 r†M0

 x y ; : 12r 12r

…10†

From Eqs. (4) and (6), M0 …x; y† ˆ

1 X nˆ1

Pn …l1 x†Pn …l2 y†:

…11†

Eqs. (10) and (11) provide a convenient means of evaluating Mr …x; y†: Established numerical methods can be employed to compute the 2D renewal function Mr …x; y† using Eq. (5). The authors have used the well known package Mathematica for this purpose. 4. Two-attribute free-replacement policies In this paper we study two of the four two-attribute warranty policies (labeled A and B) proposed by

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Fig. 2. Warranty region for Policy B.

Murthy et al. (1995). Let c denote the manufacturer's cost of each replacement and EC, the expected warranty cost. We use the superscripts A and B for the two types of policies. 4.1. Policy A The warranty coverage ends when either the maximum usage limit or maximum time limit are exceeded. For this policy, the warranty region is the rectangle ‰…0; W† £ …0; U†Š as shown in Fig. 1, where W and U are the warranty limits for time and usage, respectively. Let N A …W; U† denote the number of failures occurring under Policy A. Clearly, N A …W; U† ˆ N…W; U†; and the expected warranty cost is given by EC A …W; U† ˆ cM…W; U† with M…W; U† obtained from Eqs. (5), (10) and (11). 4.2. Policy B The warranty coverage ends when both the maximum usage limit and the maximum time limit are exceeded. For this policy, the warranty region is given by two in®nite strips shown shaded in Fig. 2. Let N B …W; U† denote the number of failures having occurred under Policy B. This is related to the 2D renewal process and the 1D processes as: N B …W; U† ˆ Max{N1 …W†; N2 …U†} ˆ N1 …W† 1 N2 …U† 2 N…W; U†: The expected warranty cost is given by EC B …W; U† ˆ c‰M1 …W† 1 M2 …U† 2 M…W; U†Š; where M1 …W† and M2 …U† are obtained from Eq. (7) and M…W; U† from Eqs. (5), (10) and (11). Table 1 Numerical example Usage

E…Xi † ˆ 1=l1 (years)

E…Yi †† ˆ 1=l2 …104 miles)

E…Yi †=E…Xi †

Light Medium Heavy

4 3 2

2.0 3.0 4.0

0.5 1.0 2.0

H.-G. Kim, B.M. Rao / Computers & Industrial Engineering 38 (2000) 425±434 Table 2 Expected number of failures for Policy A: r ˆ 0:5 U

Usage

W 0.5

1.0

1.5

2.0

0.5

Light Medium Heavy

0.0447 0.0412 0.0417

0.0735 0.0723 0.0816

0.0920 0.0957 0.1119

0.1039 0.1133 0.1369

1.0

Light Medium Heavy

0.0816 0.0723 0.0735

0.1369 0.1293 0.1369

0.1742 0.1741 0.1914

0.1994 0.2093 0.2381

1.5

Light Medium Heavy

0.1119 0.0957 0.0920

0.1914 0.1741 0.1742

0.2474 0.2381 0.2474

0.2867 0.2901 0.3123

2.0

Light Medium Heavy

0.1369 0.1133 0.1039

0.2381 0.2093 0.1994

0.3123 0.2901 0.2867

0.3662 0.3577 0.3662

Table 3 Expected number of failures for Policy B: r ˆ 0:5 U

Usage

W 0.0

0.5

1.0

1.5

2.0

0.0

Light Medium Heavy

0.0000 0.0000 0.0000

0.1250 0.1667 0.2500

0.2500 0.3333 0.5000

0.3750 0.5000 0.7500

0.5000 0.6667 1.0000

0.5

Light Medium Heavy

0.2500 0.1667 0.1250

0.3303 0.2921 0.3333

0.5515 0.4277 0.4184

0.7830 0.5710 0.5131

1.0211 0.7200 0.6131

1.0

Light Medium Heavy

0.5000 0.3333 0.2500

0.4184 0.4277 0.5515

0.6131 0.5374 0.6131

0.8258 0.6592 0.6836

1.0506 0.7907 0.7619

1.5

Light Medium Heavy

0.7500 0.5000 0.3750

0.5131 0.5710 0.7830

0.6836 0.6592 0.8258

0.8776 0.7619 0.8776

1.0883 0.8766 0.9377

2.0

Light Medium Heavy

1.0000 0.6667 0.5000

0.6131 0.7200 1.0211

0.7619 0.7907 1.0506

0.9377 0.8766 1.0883

1.1338 0.9756 1.1338

431

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Fig. 3. Surface and contour plots for E‰N A …W; U†Š:

5. Numerical examples We consider the example used by Murthy et al. (1995), which refers to the warranty offered for an automobile component with time and age measured in units of years and usage in units of 10 4 miles. Three usage levels namely, light, medium, and heavy have been chosen. The mean age E…Xi †; the mean usage E…Yi † and the mean usage rate between failures E…Yi †=E…Xi † for the three usage levels are given in Table 1. Tables 2 and 3 show the expected number of failures under warranty for the three usage levels, and various warranty limits when r ˆ 0:5: For Policy B, either W or U can be zero, in which case the policy

Fig. 4. Surface and contour plots for E‰N B …W; U†Š:

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433

Fig. 5. Effect of correlation on E‰N A …W; U†Š…W ˆ U ˆ 1†:

reduces to a single attribute policy. If U ˆ 0; the policy becomes the traditional free-replacement warranty with a warranty period W. When W ˆ 0; the policy corresponds to a similar warranty based on a usage limit U. The expected number of failures under warranty increases with warranty limits W and U for Policies A and B. It can be seen that E‰N B …W; U†Š . E‰N A …W; U†Š for a given W and U. As can be anticipated, Policy B offers better coverage to light and heavy users as seen by comparing the expected number of failures. For light users with W ˆ U; the ratio E‰N B …W; W†Š 3 E‰N A …W; W†Š decreases from about 7.4 when W ˆ 0:5 to 3.1 for W ˆ 2:0: For small W and U the manufacturer's obligations cease very early with very few claims under Policy A, whereas the manufacture has to service the warranty for a long time under Policy B. These results are similar for the heavy users. Policy A is the only policy that is currently offered by manufacturers. For policies A and B, Figs. 3 and 4 show the 2D surface and contour plots of the expected number of failures as a function of W and U for 0 # W; U # 2 for medium usage and r ˆ 0:5: The lowest contour corresponds to the expected number of failures equal to 0.03 and 0.075 and subsequent contours correspond to the expected number with 0.03 and 0.075 increments, respectively. The effects of correlation between two warranty variables on the expected number of failures are shown in Figs. 5 and 6. Positive correlation increases the expected number of failures for Policy A but reduces it for Policy B.

Fig. 6. Effect of correlation on E‰N B …W; U†Š …W ˆ U ˆ 1†:

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6. Concluding remarks We have studied two-attribute free-replacement policies through 2D renewal processes based on Downton's BVE. Downton's BVE distribution yields an analytical expression for the renewal function and permits the representation of correlation between the two variables. The results obtained here are useful in establishing warranty coverage regions and warranty reserves for two-attribute warranty policies. We are currently exploring the possibility of applying the results of this paper and estimating the parameters of the BVE distribution using real data. The two-attribute free-replacement policies for repairable items and two-attribute prorata warranty policies are worthy of further study. Acknowledgements The authors are very grateful to the anonymous referee for his/her meticulous and painstaking review and detailed comments which greatly improved the paper. This research has been supported by Dongeui University, Korea, while H. G. Kim was a Visiting Scholar at the Department of Applied Statistics and Operations Research, Bowling Green State University. References Barlow, R. E., & Proschan, F. (1975). Statistical theory of reliability and life testing, New York: Holt. Blischke, W. R., & Murthy, D. N. P. (1994). Warranty cost analysis, New York: Marcel Dekker. Blischke, W. R. & Murthy, D. N. P. (1994). Product warranty handbook New York: Marcel Dekker. Chun, Y. H., & Tang, K. (1999). Cost analysis of two-attribute warranty policies based on the product usage rate. IEEE Transactions in Engineering Management, 46, 201±209. Downton, F. (1970). Bivariate exponential distributions in reliability theory. Journal of Royal Statistical Society B, 32, 408±417. Hunter, J. J. (1974). Renewal theory in two dimensions: basic results. Advances in Applied Probability, 6, 376±391. Johnson, N. L., & Kotz, S. (1972). Distributions in statistics: continuous multivariate distributions, New York: Wiley. Kao, E. P. C., & Smith, M. S. (1996). Computational approximations of renewal process relating to a warranty problem: the case of phase-type lifetime. European Journal of Operational Research, 90, 156±170. Kim, H. G. (1997). The optimal warranty servicing for repairable products with phase-type lifetime distributions. Journal of Korean OR/MS Society, 22, 87±99. Mi, J. (1997). Warranty policies and burn-in. Naval Research Logistics, 44, 199±209. Moskowitz, H., & Chun, Y. H. (1994). A poisson regression model for two-attribute warranty policies. Naval Research Logistics, 41, 355±375. Murthy, D. N. P., Iskandar, B. P., & Wilson, R. J. (1995). Two-dimensional failure-free warranty policies: two-dimensional point process models. Operations Research, 43, 356±366. Rao, B. M. (1995). Algorithms for the free replacement warranty with phase-type lifetime distributions. IIE Transactions, 27, 348±357. Singpurwalla, N. D., & Wilson, S. P. (1998). Failure models indexed by two scales. Advances in Applied Probability, 30, 1058±1072. Thomas, M. U., & Rao, S. S. (1999). Warranty economic decision models: a summary and some suggested directions for future research. Operations Research, 47, 807±820.