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A new repair–replace strategy for items sold with a two-dimensional warranty
Computers & Operations Research 32 (2005) 669 – 682
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A new repair–replace strategy for items sold with a two-dimensional warranty B.P. Iskandara , D.N.P. Murthyb; c , N. Jackd;∗ a
Departemen Teknik Industri, Institut Teknologi Bandung, Jalan Ganesha 10, Bandung 40132, Indonesia b Department of Mechanical Engineering, The University of Queensland, St. Lucia 4072, Australia c Norwegian University of Science and Technology, Trondheim, Norway d School of Computing & Advanced Technologies, University of Abertay Dundee, Dundee DD1 1HG, Scotland, UK
Abstract For repairable items, the manufacturer has the option to either repair or replace a failed item that is returned under warranty. In this paper, we look at a new warranty servicing strategy for items sold with two-dimensional warranty where the failed item is replaced by a new one when it fails for the 2rst time in a speci2ed region of the warranty and all other failures are repaired minimally. The region is characterised by two parameters and we derive the optimal values for these to minimise the total expected warranty servicing cost. We compare the results with other repair–replace strategies reported in the literature. ? 2003 Elsevier Ltd. All rights reserved.
1. Introduction For a product sold with free replacement warranty, the manufacturer recti2es all failures occurring under warranty at no cost to the consumer. As a result, o8ering warranty implies an additional cost to the manufacturer and this cost depends on the servicing strategy. In the case of a repairable item, the options available to the manufacturer to rectify the failed item are (1) to repair it or, (2) to replace it with a new one. For the recti2cation of a failure, option (1) costs less than option (2) but a repaired item has a greater probability of failing during the remainder of the warranty period. Of particular interest to the manufacturer is to choose a proper repair–replace strategy, which minimises the expected cost of servicing the warranty. The literature on warranties is vast. A taxonomy for the di8erent warranty policies is given in Blischke and Murthy [3]. They can be broadly grouped into one- and two-dimensional policies. A one-dimensional warranty policy is characterised by a one-dimensional time interval called the ∗
Corresponding author. Tel.: +44-1382-308633; fax: +44-1382-308627. E-mail address: [email protected] (N. Jack).
0305-0548/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.cor.2003.08.011
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warranty period. A two-dimensional warranty policy is characterised by a region in a two-dimensional plane with one-dimension representing time and the other representing usage. The origin corresponds to the time instant of a sale. A typical example is an automobile warranted for 3 years or 60; 000 km of travel. For the one-dimensional case, the choice between repair and replacement has received some attention. Biedenweg [2], Nguyen and Murthy [12,13], Jack and Van der Duyn Schouten [8], and Jack and Murthy [7] have studied a variety of optimal repair–replace strategies for one-dimensional warranties. In contrast, the study of such strategies for the two-dimensional case has received less attention. Iskandar and Murthy [6] have studied two repair–replace strategies for items sold with a two-dimensional failure-free warranty. In this paper, we extend the one-dimensional warranty servicing strategy proposed in Jack and Murthy [7] to a two-dimensional warranty policy. The outline of the paper is as follows. In Section 2, we give the details of the model formulation. Of particular interest are the parameters of the strategy that need to be optimally selected to minimise the expected warranty servicing cost per unit sold. Section 3 deals with the analysis of the model to determine the optimal values of these parameters. In Section 4, we give an illustrative numerical example, and we then conclude with some comments and a brief discussion of topics for further research in Section 5. 2. Model formulation The product is sold with a two-dimensional free replacement warranty. As mentioned earlier, a two-dimensional warranty is characterised by a region in a two-dimensional plane. Di8erent regions de2ne di8erent warranty policies. See Murthy and Wilson [11] and Singpurwalla and Wilson [15] for the di8erent shapes for . In this paper we con2ne to the case where is a rectangle so that the policy expires when the item reaches an age K or the total usage exceeds a level L which ever occurs 2rst. The free replacement warranty policy requires the manufacturer to rectify all failures occurring under warranty. As a result, should a failure occur with age at failure less than K and usage at failure less than L, the manufacturer recti2es the failure at no cost to the buyer. We assume that the product is repairable so the recti2cation of a failed item can be achieved through either repair or replacement. In this case, a proper repair–replace strategy can reduce the cost of servicing the warranty. 2.1. Modelling item failures Two di8erent approaches have been proposed for modelling item failures. In the 2rst approach, the failure of items is modelled using a one-dimensional point process formulation, see for example, Moskowitz and Chun [9] and Iskandar [4]. The second approach models item failures in terms of a two-dimensional point process, see for example, Murthy et al. [10]. In this paper, we use the 2rst approach. Let t = 0 correspond to the time instant of a sale, and let Tc (t) and Xc (t) denote the age and usage of the item currently in use at time t. If X (t) denotes the total usage up to time t, then X (t) equals the total usage of the current item plus that of any previous items that have been replaced on failure
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during the interval [0; t). When no item failure occurs in [0; t), then Tc (t) = t and Xc (t) = X (t), and this is also true when all item failures are repaired and repair time is assumed to be negligible compared to item operating time. However, if the item is not repairable and has to be replaced on failure and there have been one or more failures in [0; t), then Tc (t) ¡ t and Xc (t) ¡ X (t). We model Xc (t) as a linear function of Tc (t) given by Xc (t) = RTc (t);
(1)
where R represents the usage rate that varies between users. For a given user, R is assumed to be constant over the warranty period and is modelled as a non-negative random variable with distribution function G(r) (or density function g(r)) where G(r) = P{R 6 r}:
(2)
We model item failures by a point process with an intensity function that is dependent on the age and usage of the item. However, since usage is modelled as a function of age, this means that the intensity function is only a function of t, the age of the item. Let (t|r)t denote the probability that the current working unit at time t will fail in the small interval [t; t + t) given that R = r. Conditional on R = r, failures occur according to a Poisson process with an intensity function (t|r), t ¿ 0. The structure for (t|r) is discussed in Section 2.1.1. Let N (t|r) denote the number of failures over [0; t) and T1|r the time to 2rst item failure, both conditional on R = r. The distribution function of T1|r is given by t (x|r) d x : (3) F(t|r) = 1 − exp − 0
If all failures under warranty are recti2ed through minimal repair [1] then, conditional on R = r, we have Tc (t) = t, Xc (t) = rt, and N (t|r) is a non-homogeneous Poisson process with intensity function (t|r). 2.1.1. Model structure (t|r) needs to be an increasing function of r, Tc (t) and Xc (t). This o8ers a wide choice of formulations, with the simplest being a polynomial of order one in which case (t|r) is given by (t|r) = 0 + 1 r + 2 Tc (t) + 3 Xc (t);
(4)
where the parameters i ; 0 6 i 6 3, are non-negative. This is used by Murthy and Wilson [11], whilst Moskowitz and Chun’s [9] model is a special case of this formulation. An example of a polynomial of order two would be (t|r) = 0 + 1 r + 2 Tc (t) + 3 Xc (t) + 4 Tc (t)2 + 5 Tc (t)Xc (t):
(5)
We can also consider many other di8erent formulations for (t|r), for example, polynomials involving log transformations, or non-polynomial forms. The choice would need to depend on the data available for modelling and we discuss this in the next sub-section. Finally, we need to specify the form to be used for G(r). Iskandar and Murthy [6] suggest the Uniform distribution and Gamma distribution as suitable models.
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2.1.2. Model selection The appropriate forms for (t|r) and G(r) for a speci2c application need to be determined from failure data but this topic has received little attention in the literature. One explanation is that manufacturers are reluctant to release any warranty data for obvious commercial reasons and as a result very little data is available to the public for modelling. However, this data problem is starting to show limited signs of improvement with the few case studies that have appeared recently, see for example, Iskandar and Blischke [5]. We consider the case where the following data is available for model selection. Let denote the number of items under observation and nk the number of failures of item k; 1 6 k 6 over the observation period. Let tk; j and xk; j denote the age and usage of item k at the jth failure, 1 6 k 6 ; 1 6 j 6 nk. The model selection procedure involves the following steps: Step 1: Compute rk = xk; nk =tk; nk , 1 6 k 6 . Step 2: Plot a histogram of rk , 1 6 k 6 . This requires grouping the data into M intervals with the mth interval given by [hm−1 ; hm ); where 0 = h0 ¡ h1 ¡ · · · ¡ hM ¡ ∞. A smooth curve 2tted to the histogram then yields g(r). Step 3: Let hMm denote the midpoint of the interval [hm−1 ; hm ) and Im the set of items for which the computed r (from Step 1) lies in this interval. The usage rates for these Im items can then be approximated by hMm ; and the failure times are used to provide an empirical estimate of (t|hMm ). For precise details, see Rigdon and Basu [14]. Step 4: A two-dimensional smooth curve 2tted to (t|hMm ) yields the conditional intensity function (t|r). A study of the above approach is currently a joint research project involving the second author and several other researchers. This research is also looking into model selection and parameter estimation for di8erent kinds of data. For the numerical example in Section 4 we assume that (t|r) is given by (t|r) = 0 + 1 r + 2 Tc (t)2 + 3 Xc (t)Tc (t);
(6)
where the parameters i ¿ 0 for 0 6 i 6 3. When failed items are always minimally repaired then we have Tc (t) = t, Xc (t) = rt and (6) then reduces to (t|r) = 0 + 1 r + (2 + 3 r)t 2 :
(7)
2.2. Repair–replace strategy Two repair–replace strategies (called Strategies 1 and 2) have been studied in Iskandar and Murthy [6]. In both strategies, the warranty region is divided into two sub-regions—1 and 2 . In Strategy 1, all failures occurring in 1 are recti2ed by replacement and any failure in 2 is recti2ed by minimal repair. Replacing all failed items occurring in 1 will result in decreasing the number of failures. Hence, this strategy is more appropriate for the case where the failure rate of an item is high in 1 (the early stages of the warranty). In Strategy 2, all failures occurring in 1 are recti2ed by minimal repair and any failure in 2 is recti2ed by replacement. Strategy 2 will give a better result for the case where the failure rate of an item decreases in 1 (early in the warranty) and then increases in 2 . As the failure rate increases in 2 , replacing all failed items in 2 will decrease the number of failures in this region.
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Fig. 1. Regions characterising the repair–replacement strategy.
It might be more economical to modify this strategy by replacing only the 2rst failure in 2 , instead of replacing all failed items in 2 since this action would be suOcient enough to decrease the failure rate of the item in this region. However, recti2cation through replacement of the 2rst failure in 2 would not be economical if the age, t (or the usage, x) at the failure is very close to K (or L), as it is then very close to the expiry of the warranty. Hence we need to split region 2 into two sub-regions. The following new strategy is therefore proposed. The warranty region is divided into three disjoint sets 1 , 2 , and 3 such that 1 ∪ 2 ∪ 3 = (see Fig. 1). Under this strategy, (1) all failures in 1 are minimally repaired, (2) the 2rst failure in 2 is recti2ed through replacement and subsequent failures in this region are recti2ed through minimal repair, and (3) all failures in 3 are always minimally repaired. Let c1 and c2 denote the cost of each replacement and minimal repair, respectively, and de2ne ! = c2 =c1 , where 0 ¡ ! ¡ 1. Let "˜ = {K1 ; K2 ; L1 ; L2 } represent the set of parameters of the repair– ˜ the expected warranty servicing cost per item sold. The objective replacement strategy, and EC(") ˜ is to obtain the optimal "˜ that minimises EC("). We restrict our analysis to the case where the regions 1 and 1 ∪ 2 are similar in shape. This implies that L2 =K2 = L1 =K1 . Let r2 = L2 =K2 = L1 =K1 , so the warranty servicing strategy is now characterised by the three-parameter set "˜ = {K1 ; K2 ; r2 }. These three parameters need to be selected optimally to minimise the expected warranty servicing cost. Thus, not only has the number of decision variables been reduced from four to three but the strategy is also easier to implement in practice. 3. Model analysis We assume the following: • Claims are exercised immediately, • All claims are valid,
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Fig. 2. Regions characterising case (1).
• The time for each repair and replacement action is small relative to the time between failures so it can be ignored and the actions can be viewed as instantaneous. ˜ denote the expected warranty ˜ by using a conditioning argument. Let ECr (") We obtain EC(") cost conditional on R = r. Note that the subscript corresponds to conditioning on a given r. If we de2ne r1 = L=K, then we need to consider the two cases: (1) r2 ¡ r1 and (2) r2 ¿ r1 , separately. For Case (1) [r2 ¡ r1 ], we then need to investigate the following three sub-cases: (1.1) r 6 r2 , (1.2) r2 ¡ r 6 r1 , and (1.3) r ¿ r1 as illustrated in Fig. 2. Case (1.1): [r 6 r2 ] In this case, the warranty ceases at time K. The expected warranty cost conditional on R = r, ˜ is the sum of the expected conditional warranty costs over the three periods—i.e., (0; K1 ], ECr ("), (K1 ; K2 ], and (K2 ; K]. The expected conditional warranty cost in (0; K1 ] is given by K1 (t|r) dt = c2 %(K1 |r); (7) c2 0
x where %(x|r) = 0 (t|r) dt. We now obtain the expected conditional warranty costs over (K1 ; K2 ], and (K2 ; K]. These costs depend on whether the 2rst failure after K1 occurs in (K1 ; K2 ] or in (K2 ; K]. Let T (K1 |r) denote the time at which the 2rst failure occurs after K1 conditional on R = r. The distribution function of M 1 |r), where F(K M 1 |r) = 1 − F(K1 |r). T (K1 |r) is given by (F(t|r) − F(K1 |r))= F(K Conditional on T (K1 |r) = t (and R = r), the expected conditional cost over (K1 ; K2 ] is given by K2 c1 + c 2 (x − t|r) d x; K1 ¡ t 6 K2 ; (8) t 0; t ¿ K2
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and the expected conditional cost over (K2 ; K] is given by K (x − t|r) d x; K1 ¡ t 6 K2 ; c2 K2
c2
K
(x|r) d x;
K2
675
(9)
t ¿ K2 :
Removing the conditioning on T gives the conditional expected costs over (K1 ; K2 ] and (K2 ; K] as K2 K2 K2 f(t|r) f(t|r) c1 + c 2 dt = dt (10) (x − t|r) d x {c1 + c2 %(K2 − t|r)} M 1 |r) M 1 |r) F(K F(K K1 t K1 and
K2
K1
c2
=
K2
K1
K
K2
(x − t|r) d x
f(t|r) dt + M F(K1 |r)
c2 {%(K − t|r) − %(K2 − t|r)}
∞
K2
c2
K
K2
(x|r) d x
f(t|r) dt M F(K1 |r)
M 2 |r) f(t|r) F(K dt + c2 {%(K|r) − %(K2 |r)} ; M 1 |r) M 1 |r) F(K F(K
(11)
respectively. ˜ = ECr (K1 ; K2 ) From (7), (10) and (11), we have the expected conditional warranty cost, ECr (") given by K2 1 f(t|r) dt ECr (K1 ; K2 ) = c2 %(K1 |r) + + %(K − t|r) M ! F(K1 |r) K1
M 2 |r) F(K + {%(K|r) − %(K2 |r)} : (12) M 1 |r) F(K Case (1.2): [r2 ¡ r 6 r1 ] De2ne '1 = L1 =r and '2 = L2 =r. Note that in this case the warranty ceases at time K. The expected conditional warranty cost in (0; '1 ] is given by (7) with '1 replacing K1 . The expected conditional warranty cost in ('1 ; '2 ] is given by (10) and the expected conditional warranty cost in ('2 ; K] is ˜ = ECr ('1 ; '2 ) given by (11) with '1 replacing K1 and '2 replacing K2 , respectively. Hence, ECr (") is given by '2 f(t|r) 1 + %(K − t|r) ECr ('1 ; '2 ) = c2 %('1 |r) + dt M 1 |r) ! F(' '1
M 2 |r) F(' + {%(K|r) − %('2 |r)} : (13) M 1 |r) F(' Case (1.3): [r ¿ r1 ] De2ne ' = L=r, '1 = L1 =r and '2 = L2 =r. Note that in this case the warranty ceases at time '. The expected conditional warranty cost in (0; '1 ] is given by (7) with '1 replacing K1 . The expected conditional warranty cost in ('1 ; '2 ] is given by (10) with '1 replacing K1 and '2 replacing K2 and the expected conditional warranty cost in ('2 ; '] is given by (11) with '1 replacing K1 , '2 replacing
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Fig. 3. Regions characterising case (2).
˜ = ECr ('1 ; '2 ) is given by K2 , and ' replacing K. Hence, ECr (") '2 f(t|r) 1 + %(' − t|r) ECr ('1 ; '2 ) = c2 %('1 |r) + dt M 1 |r) ! F(' '1
M 2 |r) F(' + {%('|r) − %('2 |r)} : M 1 |r) F('
(14)
Finally, on removing the conditioning on R, the expected warranty cost is given by r1 ∞ r2 ˜ ECr (K1 ; K2 )g(r) dr + ECr ('1 ; '2 )g(r) dr + ECr ('1 ; '2 )g(r) dr: EC(") = 0
r2
r1
(15)
For Case (2) [r2 ¿ r1 ], we need to investigate the following three sub-cases: (2.1) r 6 r1 , (2.2) r1 ¡ r 6 r2 , and (2.3) r ¿ r2 , as illustrated in Fig. 3. Case (2.1): [r 6 r1 ] Conditional on R = r, the expected conditional warranty cost in (0; K1 ] is given by (7). The expected conditional cost over (K1 ; K2 ] and that over (K2 ; K] are given by (10) and (11), respectively. Hence ECr (K1 ; K2 ) is given by (12). Case (2.2): [r1 ¡ r 6 r2 ] De2ne ' = L=r. Note that in this case the warranty ceases at time '. The expected conditional warranty cost in (0; K1 ] is given by (7). The expected conditional warranty cost in (K1 ; K2 ] is given by (10) and the expected conditional warranty cost in (K2 ; '] is given by (11) with ' replacing K. Hence K2 f(t|r) 1 + %(' − t|r) dt ECr (K1 ; K2 ) = c2 %(K1 |r) + M ! F(K1 |r) K1
M 2 |r) F(K + {%('|r) − %(K2 |r)} : (16) M 1 |r) F(K
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Case (2.3): [r ¿ r2 ] De2ne ' = L=r, '1 = L1 =r and '2 = L2 =r. Note that in this case the warranty ceases at time '. This case is the same as Case (1.3), hence ECr ('1 ; '2 ), which is the sum of the expected conditional warranty costs over the three periods—i.e., (0; '1 ], ('1 ; '2 ], and ('2 ; '], is given by (14). As a result, we have the expected warranty cost given by r1 r2 ∞ ˜ EC(") = ECr (K1 ; K2 )g(r) dr + ECr (K1 ; K2 )g(r) dr + ECr ('1 ; '2 )g(r) dr: (17) 0
r1
r2
˜ subject to the conThe objective is to 2nd the values of "˜ = {K1 ; K2 ; r2 } that minimise EC("); ˜ straints 0 6 K1 ; K2 6 K and 0 ¡ r2 ¡ ∞. EC(") is given by (15) when 0 ¡ r2 ¡ r1 , and by (17) when r1 ¡ r2 ¡ ∞. 4. Numerical example Suppose that the item under consideration is a component of an automobile sold with an FRW policy. We con2ne our attention to the warranty cost associated with the servicing of claims resulting from the component under consideration. The time (or age) is measured in units of years and the unit for usage is 10; 000 km. We assume that K = 2 and L = 2 , so r1 = 1; and we have a square warranty region with a maximum coverage for 2 years and 20; 000 km of usage. We consider the form for the conditional failure intensity given by (7), with the parameter values 0 = 0:1, 1 = 0:2, 2 = 0:7, and 3 = 0:7, and we assume that R is uniformly distributed over [rl ; ru ], with g(r) = 1=[ru − rl ], for rl 6 r 6 ru . We consider three sets of values for rl and ru , which broadly correspond to light, medium, and heavy users, respectively. These values and those for the corresponding mean usage rate (E[R]), mean age at 2rst failure (E[T ]) and mean usage at 2rst failure (E[X ]) are as follows: Usage type
rl
ru
E[R]
E[T ]
E[X ]
Light Medium Heavy
0.1 0.7 1.1
0.9 1.3 2.9
0.5 1.0 2.0
1.1118 years 0.9575 years 0.7755 years
0.5559 (104 km) 0.9575 (104 km) 1.5510 (104 km)
Without loss of generality, we assume that c1 =1 and consider a range of values of ! varying from 0.1 to 0.9. (When c1 = 1; the costs given in the following tables are the normalised costs relative to c1 .) The decision variables for the new strategy are K1 ; K2 ; and r2 = L2 =K2 = L1 =K1 . The grid search for ˜ was done with K1 and K2 incremented in steps of 0.1 over the range the minimization of EC(") 0.0 –2.0, and r2 incremented in steps of 0.2, starting at 0.2. Tables 1–3 give the new strategy results for the three types of usage intensity. Also shown are the expected warranty servicing costs for the two simpler strategies of (i) always performing minimal repairs, and (ii) always replacing the item at each failure in the warranty region. For each value of !, the smallest expected warranty servicing cost is shown in bold font and the corresponding strategy is the optimal strategy. Note that, the new strategy reduces to ‘always repair’ when K1 = K2 ; but the ‘always replace’ strategy is not a special case of the new strategy. However, when K1 = 0; K2 = K,
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Table 1 Expected warranty costs for light usage intensity !
K1∗
K2∗
r2∗
EC("˜ ∗ )
Always repair
Always replace
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 0.1 0.2 0.6 0.6 0.6 0.6 0.6 0.5
0.2 0.2 0.3 1.5 1.7 1.8 1.9 1.9 2.0
0.2 0.2 0.2 1.0 1.0 1.0 1.0 1.0 1.0
0.3281 0.6469 0.9656 1.1401 1.2258 1.2971 1.3627 1.4263 1.4875
0.3156 0.6312 0.9469 1.2800 1.6000 1.9200 2.2400 2.5600 2.8800
1.4193 1.4193 1.4193 1.4193 1.4193 1.4193 1.4193 1.4193 1.4193
Table 2 Expected warranty costs for medium usage intensity !
K1∗
K2∗
r2∗
EC("˜ ∗ )
Always repair
Always replace
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 0.9 1.0 0.7 0.6 0.6 0.6 0.6 0.5
0.2 1.0 1.2 1.6 1.7 1.8 1.9 1.9 2.0
0.4 0.2 0.8 1.0 1.0 1.0 1.0 1.0 1.0
0.3744 0.7319 1.0894 1.2420 1.3390 1.4249 1.5066 1.5875 1.6664
0.3555 0.7109 1.0664 1.4549 1.8186 2.1823 2.5460 2.9097 3.2735
1.5781 1.5781 1.5781 1.5781 1.5781 1.5781 1.5781 1.5781 1.5781
and r2 = 1, we have the alternative strategy of replacement at the 2rst failure followed by minimal repair at all other failures. These tables indicate that • When the item has a light or medium usage intensity, it is optimal to ‘always repair’ for small values of ! and ‘always replace’ for large values of !, and this is as to be expected. For intermediate values of !, the new strategy is the best choice, and as ! increases through this range the size of the region 1 remains relatively constant with that of 3 decreasing to zero. • In the case of high usage intensity, the ‘always repair’ strategy is optimal for small to intermediate values of !. The new strategy is optimal for 0:6 6 ! 6 0:8, and as ! increases through this range, the size of region 3 remains relatively constant whilst that of 1 decreases, allowing for possible earlier replacement of the item. The optimal strategy is ‘always replace’ for ! ¿ 0:9. It can also be seen from Tables 1–3 that r2∗ is close to 1 when the new strategy performs better than the ‘always repair’ and ‘always replace’ strategies. Tables 4–6 give a relative comparison of the new strategy, ‘always repair’, ‘always replace’, and Strategies 1 and 2 of Iskandar and Murthy
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Table 3 Expected warranty costs for heavy usage intensity !
K1∗
K2∗
r2∗
EC("˜ ∗ )
Always repair
Always replace
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 0.2 0.8 1.9 1.9 0.9 0.6 0.5 0.4
0.2 0.3 0.9 2.0 2.0 1.8 1.7 1.8 1.9
0.2 0.2 0.2 0.2 0.4 0.8 1.0 1.0 1.0
0.1505 0.2959 0.4412 0.5864 0.7313 0.8384 0.9022 0.9500 0.9902
0.1458 0.2916 0.4374 0.5839 0.7299 0.8758 1.0218 1.1678 1.3138
0.9802 0.9802 0.9802 0.9802 0.9802 0.9802 0.9802 0.9802 0.9802
Table 4 Expected warranty costs for light usage intensity when r2 = 1 !
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Strategy 1
Strategy 2
New Strategy
K1∗
EC("˜ ∗ )
K1∗
EC("˜ ∗ )
K1∗
K2∗
EC("˜ ∗ )
0.0 0.0 0.0 1.1 1.3 1.4 1.6 1.7 1.9
0.3156 0.6312 0.9469 1.1091 1.2236 1.3062 1.3633 1.4006 1.4221
2.0 2.0 2.0 0.9 0.6 0.5 0.3 0.3 0.3
0.3156 0.6312 0.9469 1.3342 1.3680 1.3876 1.4018 1.4088 1.4157
0.1 0.1 0.9 0.6 0.6 0.6 0.6 0.6 0.5
0.2 0.2 1.0 1.5 1.7 1.8 1.9 1.9 2.0
0.3386 0.6551 0.9662 1.1401 1.2258 1.2971 1.3627 1.4263 1.4875
Always repair
Always replace
0.3156 0.6312 0.9469 1.2800 1.6000 1.9200 2.2400 2.5600 2.8800
1.4193 1.4193 1.4193 1.4193 1.4193 1.4193 1.4193 1.4193 1.4193
[6] for the case r2 = 1. A one-dimensional search with a grid size of 0.1 was used to obtain the optimal values of K1 . Note that Strategies 1 and 2 reduce to ‘always repair’ when K1 = 0 and K1 = 2, respectively, and ‘always replace’ occurs when K1 = 2 and K1 = 0, respectively. From Tables 4–6, the following observations can be made about the new repair–replace strategy: • For small values of !, the optimal strategy is “always repair” and strategies 1 and 2 both reduce to this form. This is as to be expected. • For intermediate values of !, the new strategy is the best choice. For heavy usage intensity, as ! increases the region 2 increases in size whilst the regions 1 and 3 both decrease. For medium and light usage intensities, as ! increases the size of region 1 remains relatively constant whilst that of 3 decreases to zero. • In the limit as ! approaches 1, then ‘always replace’ is the best strategy and again both strategies 1 and 2 reduce to this. The new strategy is never optimal here since only one replacement is allowed.
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Table 5 Expected warranty costs for medium usage intensity when r2 = 1 !
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Strategy 1
Strategy 2
New strategy
K1∗
EC("˜ ∗ )
K1∗
EC("˜ ∗ )
K1∗
K2∗
EC("˜ ∗ )
0.0 0.0 0.0 1.1 1.3 1.4 1.6 1.7 1.9
0.3555 0.7109 1.0664 1.2286 1.3522 1.4403 1.5023 1.5424 1.5656
2.0 2.0 2.0 0.9 0.7 0.6 0.5 0.3 0.3
0.3555 0.7109 1.0664 1.4652 1.5095 1.5357 1.5555 1.5654 1.5748
0.1 0.1 1.0 0.7 0.6 0.6 0.6 0.6 0.5
0.2 0.2 1.1 1.6 1.7 1.8 1.9 1.9 2.0
0.3889 0.7476 1.0902 1.2420 1.3390 1.4249 1.5066 1.5875 1.6664
Always repair
Always replace
0.3555 0.7109 1.0664 1.4549 1.8186 2.1823 2.5460 2.9097 3.2735
1.5781 1.5781 1.5781 1.5781 1.5781 1.5781 1.5781 1.5781 1.5781
Always repair
Always replace
0.1458 0.2916 0.4374 0.5839 0.7299 0.8758 1.0218 1.1678 1.3138
0.9802 0.9802 0.9802 0.9802 0.9802 0.9802 0.9802 0.9802 0.9802
Table 6 Expected warranty costs for heavy usage intensity when r2 = 1 !
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Strategy 1
Strategy 2
New strategy
K1∗
EC("˜ ∗ )
K1∗
EC("˜ ∗ )
K1∗
K2∗
EC("˜ ∗ )
0.0 0.0 0.0 0.0 0.1 1.2 1.4 1.6 1.8
0.1458 0.2916 0.4374 0.5839 0.7383 0.8569 0.9164 0.9564 0.9795
2.0 2.0 2.0 2.0 2.0 2.0 0.6 0.4 0.2
0.1458 0.2916 0.4374 0.5839 0.7299 0.8758 0.9536 0.9688 0.9769
0.1 0.1 0.2 0.6 1.0 0.7 0.6 0.5 0.4
0.2 0.2 0.3 0.7 1.1 1.5 1.7 1.8 1.9
0.1687 0.3116 0.4543 0.5957 0.7323 0.8388 0.9022 0.9500 0.9902
For light and medium usage intensities, the warranty ceases mainly due to the age limit K. The value of the total usage when the warranty ceases is higher for medium usage intensity, so the corresponding servicing costs are larger than those for light usage intensity. However, in the case of heavy usage intensity, the warranty expires mainly due to the usage limit L. The age of the item when the warranty ceases is well below the age limit K, and so the heavy usage intensity servicing costs are smaller than the corresponding costs for medium usage intensity. However, this pattern will change depending on the parameter values and the structure of the conditional intensity function. The expected warranty servicing cost (for consumer durables and commercial products) varies from 0.5% to 10% of the sale price depending on the product and the manufacturer. Since the annual sales of such products runs into trillions of dollars, even a small reduction in the expected warranty servicing cost implies savings of billions of dollars annually. For intermediate values of ! the new servicing strategy o8ers considerable potential savings to manufacturers.
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5. Conclusions In this paper, we have studied a new repair–replace strategy for a repairable item sold with a two-dimensional warranty. The strategy is characterised by four parameters and these can be selected to minimise the expected warranty cost. We con2ned our attention to the case where one of the parameters is constrained so that the problem is reduced to a three-parameter optimisation. We carried out a relative comparison with two other strategies—‘always repair’ and ‘always replace’. The new strategy, which is easy to administer, did signi2cantly better than these two simple strategies for intermediate values of ! (the ratio of repair cost to replacement cost). The results of this paper can be extended in several ways, and a few of these are discussed below • The analysis for the case where all four parameters are independent can be carried out in a manner similar to that in this paper but would be much more complex. • In our formulation, the warranty is characterised by a rectangular region . It is decomposed into three-sub-regions 1 , 2 and 3 with these being de2ned in terms of two rectangles within . An extension would be to look at regions that are not necessarily characterised by rectangles. • The true optimal repair–replace strategy and hence the optimal shapes of the regions 1 and 2 within the rectangular region are still unknown. They can be determined by modifying the dynamic programming approach used by Jack and Van der Duyn Schouten [8] for the one-dimensional case. This topic is currently under investigation. • In the new strategy in this paper, it is assumed that all failed items can be repaired. However, in many real situations, some failures are not repairable (termed type 1 failures) whilst others (called type 2 failures) are repairable. As a result, a repair–replace strategy can be formulated where the action to rectify the failed item also depends on the type of failure. These are only a few of the many topics for further research. Acknowledgements The authors thank the reviewer for the constructive comments on an earlier version of the paper. References [1] Barlow RE, Proschan F. Mathematical theory of reliability. New York: Wiley; 1965. [2] Biedenweg FM. Warranty analysis: consumer value vs manufacturers cost. Unpublished PhD thesis, Stanford University, USA, 1981. [3] Blischke WR, Murthy DNP. Warranty cost analysis. New York: Marcel Dekker; 1994. [4] Iskandar BP. Modelling and analysis of two dimensional warranty policies. Unpublished PhD thesis, The University of Queensland, Brisbane, Australia, 1993. [5] Iskandar BP, Blischke WR. In: Blischke WR, Murthy DNP, editors. Case studies in reliability and maintenance. New York: Wiley; 2003. [6] Iskandar BP, Murthy DNP. Repair replace strategies for two-dimensional warranty. Proceedings of the First Western Paci2c and Third Australia–Japan Workshop on Stochastic Models in Engineering, Technology and Management, New Zealand, September 1999. p. 206 –13.
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[7] Jack N, Murthy DNP. Servicing strategies for items sold under warranty. Journal of the Operations Research Society 2001;52:1284–8. [8] Jack N, Van der Duyn Schouten F. Optimal repair–replace strategies for a warranted product. International Journal of Production Economics 2000;67:95–100. [9] Moskowitz H, Chun YH. A Poisson regression model for two-attribute warranty policy. Naval Research Logistics 1994;41:355–76. [10] Murthy DNP, Iskandar BP, Wilson RJ. Two-dimensional failure free warranties: two-dimensional point process models. Operations Research 1995;43:356–66. [11] Murthy DNP, Wilson RJ. Modelling two-dimensional failure free warranties. Proceedings of the Fifth Symposium on Applied Stochastic Models and Data Analysis, Granada, Spain, 1991. [12] Nguyen DG, Murthy DNP. An optimal policy for servicing warranty. Journal of the Operations Research Society 1986;37:1081–8. [13] Nguyen DG, Murthy DNP. Optimal replace–repair strategy for servicing warranty sold with warranty. European Journal of Operational Research 1989;39:206–12. [14] Rigdon SE, Basu AP. Statistical methods for the reliability of repairable systems. New York: Wiley; 2000. [15] Singpurwalla ND, Wilson S. The warranty problem: its statistical and game theoretic aspects. SIAM Review 1993;35:17–42.
www.elsevier.com/locate/dsw
A new repair–replace strategy for items sold with a two-dimensional warranty B.P. Iskandara , D.N.P. Murthyb; c , N. Jackd;∗ a
Departemen Teknik Industri, Institut Teknologi Bandung, Jalan Ganesha 10, Bandung 40132, Indonesia b Department of Mechanical Engineering, The University of Queensland, St. Lucia 4072, Australia c Norwegian University of Science and Technology, Trondheim, Norway d School of Computing & Advanced Technologies, University of Abertay Dundee, Dundee DD1 1HG, Scotland, UK
Abstract For repairable items, the manufacturer has the option to either repair or replace a failed item that is returned under warranty. In this paper, we look at a new warranty servicing strategy for items sold with two-dimensional warranty where the failed item is replaced by a new one when it fails for the 2rst time in a speci2ed region of the warranty and all other failures are repaired minimally. The region is characterised by two parameters and we derive the optimal values for these to minimise the total expected warranty servicing cost. We compare the results with other repair–replace strategies reported in the literature. ? 2003 Elsevier Ltd. All rights reserved.
1. Introduction For a product sold with free replacement warranty, the manufacturer recti2es all failures occurring under warranty at no cost to the consumer. As a result, o8ering warranty implies an additional cost to the manufacturer and this cost depends on the servicing strategy. In the case of a repairable item, the options available to the manufacturer to rectify the failed item are (1) to repair it or, (2) to replace it with a new one. For the recti2cation of a failure, option (1) costs less than option (2) but a repaired item has a greater probability of failing during the remainder of the warranty period. Of particular interest to the manufacturer is to choose a proper repair–replace strategy, which minimises the expected cost of servicing the warranty. The literature on warranties is vast. A taxonomy for the di8erent warranty policies is given in Blischke and Murthy [3]. They can be broadly grouped into one- and two-dimensional policies. A one-dimensional warranty policy is characterised by a one-dimensional time interval called the ∗
Corresponding author. Tel.: +44-1382-308633; fax: +44-1382-308627. E-mail address: [email protected] (N. Jack).
0305-0548/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.cor.2003.08.011
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warranty period. A two-dimensional warranty policy is characterised by a region in a two-dimensional plane with one-dimension representing time and the other representing usage. The origin corresponds to the time instant of a sale. A typical example is an automobile warranted for 3 years or 60; 000 km of travel. For the one-dimensional case, the choice between repair and replacement has received some attention. Biedenweg [2], Nguyen and Murthy [12,13], Jack and Van der Duyn Schouten [8], and Jack and Murthy [7] have studied a variety of optimal repair–replace strategies for one-dimensional warranties. In contrast, the study of such strategies for the two-dimensional case has received less attention. Iskandar and Murthy [6] have studied two repair–replace strategies for items sold with a two-dimensional failure-free warranty. In this paper, we extend the one-dimensional warranty servicing strategy proposed in Jack and Murthy [7] to a two-dimensional warranty policy. The outline of the paper is as follows. In Section 2, we give the details of the model formulation. Of particular interest are the parameters of the strategy that need to be optimally selected to minimise the expected warranty servicing cost per unit sold. Section 3 deals with the analysis of the model to determine the optimal values of these parameters. In Section 4, we give an illustrative numerical example, and we then conclude with some comments and a brief discussion of topics for further research in Section 5. 2. Model formulation The product is sold with a two-dimensional free replacement warranty. As mentioned earlier, a two-dimensional warranty is characterised by a region in a two-dimensional plane. Di8erent regions de2ne di8erent warranty policies. See Murthy and Wilson [11] and Singpurwalla and Wilson [15] for the di8erent shapes for . In this paper we con2ne to the case where is a rectangle so that the policy expires when the item reaches an age K or the total usage exceeds a level L which ever occurs 2rst. The free replacement warranty policy requires the manufacturer to rectify all failures occurring under warranty. As a result, should a failure occur with age at failure less than K and usage at failure less than L, the manufacturer recti2es the failure at no cost to the buyer. We assume that the product is repairable so the recti2cation of a failed item can be achieved through either repair or replacement. In this case, a proper repair–replace strategy can reduce the cost of servicing the warranty. 2.1. Modelling item failures Two di8erent approaches have been proposed for modelling item failures. In the 2rst approach, the failure of items is modelled using a one-dimensional point process formulation, see for example, Moskowitz and Chun [9] and Iskandar [4]. The second approach models item failures in terms of a two-dimensional point process, see for example, Murthy et al. [10]. In this paper, we use the 2rst approach. Let t = 0 correspond to the time instant of a sale, and let Tc (t) and Xc (t) denote the age and usage of the item currently in use at time t. If X (t) denotes the total usage up to time t, then X (t) equals the total usage of the current item plus that of any previous items that have been replaced on failure
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671
during the interval [0; t). When no item failure occurs in [0; t), then Tc (t) = t and Xc (t) = X (t), and this is also true when all item failures are repaired and repair time is assumed to be negligible compared to item operating time. However, if the item is not repairable and has to be replaced on failure and there have been one or more failures in [0; t), then Tc (t) ¡ t and Xc (t) ¡ X (t). We model Xc (t) as a linear function of Tc (t) given by Xc (t) = RTc (t);
(1)
where R represents the usage rate that varies between users. For a given user, R is assumed to be constant over the warranty period and is modelled as a non-negative random variable with distribution function G(r) (or density function g(r)) where G(r) = P{R 6 r}:
(2)
We model item failures by a point process with an intensity function that is dependent on the age and usage of the item. However, since usage is modelled as a function of age, this means that the intensity function is only a function of t, the age of the item. Let (t|r)t denote the probability that the current working unit at time t will fail in the small interval [t; t + t) given that R = r. Conditional on R = r, failures occur according to a Poisson process with an intensity function (t|r), t ¿ 0. The structure for (t|r) is discussed in Section 2.1.1. Let N (t|r) denote the number of failures over [0; t) and T1|r the time to 2rst item failure, both conditional on R = r. The distribution function of T1|r is given by t (x|r) d x : (3) F(t|r) = 1 − exp − 0
If all failures under warranty are recti2ed through minimal repair [1] then, conditional on R = r, we have Tc (t) = t, Xc (t) = rt, and N (t|r) is a non-homogeneous Poisson process with intensity function (t|r). 2.1.1. Model structure (t|r) needs to be an increasing function of r, Tc (t) and Xc (t). This o8ers a wide choice of formulations, with the simplest being a polynomial of order one in which case (t|r) is given by (t|r) = 0 + 1 r + 2 Tc (t) + 3 Xc (t);
(4)
where the parameters i ; 0 6 i 6 3, are non-negative. This is used by Murthy and Wilson [11], whilst Moskowitz and Chun’s [9] model is a special case of this formulation. An example of a polynomial of order two would be (t|r) = 0 + 1 r + 2 Tc (t) + 3 Xc (t) + 4 Tc (t)2 + 5 Tc (t)Xc (t):
(5)
We can also consider many other di8erent formulations for (t|r), for example, polynomials involving log transformations, or non-polynomial forms. The choice would need to depend on the data available for modelling and we discuss this in the next sub-section. Finally, we need to specify the form to be used for G(r). Iskandar and Murthy [6] suggest the Uniform distribution and Gamma distribution as suitable models.
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2.1.2. Model selection The appropriate forms for (t|r) and G(r) for a speci2c application need to be determined from failure data but this topic has received little attention in the literature. One explanation is that manufacturers are reluctant to release any warranty data for obvious commercial reasons and as a result very little data is available to the public for modelling. However, this data problem is starting to show limited signs of improvement with the few case studies that have appeared recently, see for example, Iskandar and Blischke [5]. We consider the case where the following data is available for model selection. Let denote the number of items under observation and nk the number of failures of item k; 1 6 k 6 over the observation period. Let tk; j and xk; j denote the age and usage of item k at the jth failure, 1 6 k 6 ; 1 6 j 6 nk. The model selection procedure involves the following steps: Step 1: Compute rk = xk; nk =tk; nk , 1 6 k 6 . Step 2: Plot a histogram of rk , 1 6 k 6 . This requires grouping the data into M intervals with the mth interval given by [hm−1 ; hm ); where 0 = h0 ¡ h1 ¡ · · · ¡ hM ¡ ∞. A smooth curve 2tted to the histogram then yields g(r). Step 3: Let hMm denote the midpoint of the interval [hm−1 ; hm ) and Im the set of items for which the computed r (from Step 1) lies in this interval. The usage rates for these Im items can then be approximated by hMm ; and the failure times are used to provide an empirical estimate of (t|hMm ). For precise details, see Rigdon and Basu [14]. Step 4: A two-dimensional smooth curve 2tted to (t|hMm ) yields the conditional intensity function (t|r). A study of the above approach is currently a joint research project involving the second author and several other researchers. This research is also looking into model selection and parameter estimation for di8erent kinds of data. For the numerical example in Section 4 we assume that (t|r) is given by (t|r) = 0 + 1 r + 2 Tc (t)2 + 3 Xc (t)Tc (t);
(6)
where the parameters i ¿ 0 for 0 6 i 6 3. When failed items are always minimally repaired then we have Tc (t) = t, Xc (t) = rt and (6) then reduces to (t|r) = 0 + 1 r + (2 + 3 r)t 2 :
(7)
2.2. Repair–replace strategy Two repair–replace strategies (called Strategies 1 and 2) have been studied in Iskandar and Murthy [6]. In both strategies, the warranty region is divided into two sub-regions—1 and 2 . In Strategy 1, all failures occurring in 1 are recti2ed by replacement and any failure in 2 is recti2ed by minimal repair. Replacing all failed items occurring in 1 will result in decreasing the number of failures. Hence, this strategy is more appropriate for the case where the failure rate of an item is high in 1 (the early stages of the warranty). In Strategy 2, all failures occurring in 1 are recti2ed by minimal repair and any failure in 2 is recti2ed by replacement. Strategy 2 will give a better result for the case where the failure rate of an item decreases in 1 (early in the warranty) and then increases in 2 . As the failure rate increases in 2 , replacing all failed items in 2 will decrease the number of failures in this region.
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Fig. 1. Regions characterising the repair–replacement strategy.
It might be more economical to modify this strategy by replacing only the 2rst failure in 2 , instead of replacing all failed items in 2 since this action would be suOcient enough to decrease the failure rate of the item in this region. However, recti2cation through replacement of the 2rst failure in 2 would not be economical if the age, t (or the usage, x) at the failure is very close to K (or L), as it is then very close to the expiry of the warranty. Hence we need to split region 2 into two sub-regions. The following new strategy is therefore proposed. The warranty region is divided into three disjoint sets 1 , 2 , and 3 such that 1 ∪ 2 ∪ 3 = (see Fig. 1). Under this strategy, (1) all failures in 1 are minimally repaired, (2) the 2rst failure in 2 is recti2ed through replacement and subsequent failures in this region are recti2ed through minimal repair, and (3) all failures in 3 are always minimally repaired. Let c1 and c2 denote the cost of each replacement and minimal repair, respectively, and de2ne ! = c2 =c1 , where 0 ¡ ! ¡ 1. Let "˜ = {K1 ; K2 ; L1 ; L2 } represent the set of parameters of the repair– ˜ the expected warranty servicing cost per item sold. The objective replacement strategy, and EC(") ˜ is to obtain the optimal "˜ that minimises EC("). We restrict our analysis to the case where the regions 1 and 1 ∪ 2 are similar in shape. This implies that L2 =K2 = L1 =K1 . Let r2 = L2 =K2 = L1 =K1 , so the warranty servicing strategy is now characterised by the three-parameter set "˜ = {K1 ; K2 ; r2 }. These three parameters need to be selected optimally to minimise the expected warranty servicing cost. Thus, not only has the number of decision variables been reduced from four to three but the strategy is also easier to implement in practice. 3. Model analysis We assume the following: • Claims are exercised immediately, • All claims are valid,
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Fig. 2. Regions characterising case (1).
• The time for each repair and replacement action is small relative to the time between failures so it can be ignored and the actions can be viewed as instantaneous. ˜ denote the expected warranty ˜ by using a conditioning argument. Let ECr (") We obtain EC(") cost conditional on R = r. Note that the subscript corresponds to conditioning on a given r. If we de2ne r1 = L=K, then we need to consider the two cases: (1) r2 ¡ r1 and (2) r2 ¿ r1 , separately. For Case (1) [r2 ¡ r1 ], we then need to investigate the following three sub-cases: (1.1) r 6 r2 , (1.2) r2 ¡ r 6 r1 , and (1.3) r ¿ r1 as illustrated in Fig. 2. Case (1.1): [r 6 r2 ] In this case, the warranty ceases at time K. The expected warranty cost conditional on R = r, ˜ is the sum of the expected conditional warranty costs over the three periods—i.e., (0; K1 ], ECr ("), (K1 ; K2 ], and (K2 ; K]. The expected conditional warranty cost in (0; K1 ] is given by K1 (t|r) dt = c2 %(K1 |r); (7) c2 0
x where %(x|r) = 0 (t|r) dt. We now obtain the expected conditional warranty costs over (K1 ; K2 ], and (K2 ; K]. These costs depend on whether the 2rst failure after K1 occurs in (K1 ; K2 ] or in (K2 ; K]. Let T (K1 |r) denote the time at which the 2rst failure occurs after K1 conditional on R = r. The distribution function of M 1 |r), where F(K M 1 |r) = 1 − F(K1 |r). T (K1 |r) is given by (F(t|r) − F(K1 |r))= F(K Conditional on T (K1 |r) = t (and R = r), the expected conditional cost over (K1 ; K2 ] is given by K2 c1 + c 2 (x − t|r) d x; K1 ¡ t 6 K2 ; (8) t 0; t ¿ K2
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and the expected conditional cost over (K2 ; K] is given by K (x − t|r) d x; K1 ¡ t 6 K2 ; c2 K2
c2
K
(x|r) d x;
K2
675
(9)
t ¿ K2 :
Removing the conditioning on T gives the conditional expected costs over (K1 ; K2 ] and (K2 ; K] as K2 K2 K2 f(t|r) f(t|r) c1 + c 2 dt = dt (10) (x − t|r) d x {c1 + c2 %(K2 − t|r)} M 1 |r) M 1 |r) F(K F(K K1 t K1 and
K2
K1
c2
=
K2
K1
K
K2
(x − t|r) d x
f(t|r) dt + M F(K1 |r)
c2 {%(K − t|r) − %(K2 − t|r)}
∞
K2
c2
K
K2
(x|r) d x
f(t|r) dt M F(K1 |r)
M 2 |r) f(t|r) F(K dt + c2 {%(K|r) − %(K2 |r)} ; M 1 |r) M 1 |r) F(K F(K
(11)
respectively. ˜ = ECr (K1 ; K2 ) From (7), (10) and (11), we have the expected conditional warranty cost, ECr (") given by K2 1 f(t|r) dt ECr (K1 ; K2 ) = c2 %(K1 |r) + + %(K − t|r) M ! F(K1 |r) K1
M 2 |r) F(K + {%(K|r) − %(K2 |r)} : (12) M 1 |r) F(K Case (1.2): [r2 ¡ r 6 r1 ] De2ne '1 = L1 =r and '2 = L2 =r. Note that in this case the warranty ceases at time K. The expected conditional warranty cost in (0; '1 ] is given by (7) with '1 replacing K1 . The expected conditional warranty cost in ('1 ; '2 ] is given by (10) and the expected conditional warranty cost in ('2 ; K] is ˜ = ECr ('1 ; '2 ) given by (11) with '1 replacing K1 and '2 replacing K2 , respectively. Hence, ECr (") is given by '2 f(t|r) 1 + %(K − t|r) ECr ('1 ; '2 ) = c2 %('1 |r) + dt M 1 |r) ! F(' '1
M 2 |r) F(' + {%(K|r) − %('2 |r)} : (13) M 1 |r) F(' Case (1.3): [r ¿ r1 ] De2ne ' = L=r, '1 = L1 =r and '2 = L2 =r. Note that in this case the warranty ceases at time '. The expected conditional warranty cost in (0; '1 ] is given by (7) with '1 replacing K1 . The expected conditional warranty cost in ('1 ; '2 ] is given by (10) with '1 replacing K1 and '2 replacing K2 and the expected conditional warranty cost in ('2 ; '] is given by (11) with '1 replacing K1 , '2 replacing
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Fig. 3. Regions characterising case (2).
˜ = ECr ('1 ; '2 ) is given by K2 , and ' replacing K. Hence, ECr (") '2 f(t|r) 1 + %(' − t|r) ECr ('1 ; '2 ) = c2 %('1 |r) + dt M 1 |r) ! F(' '1
M 2 |r) F(' + {%('|r) − %('2 |r)} : M 1 |r) F('
(14)
Finally, on removing the conditioning on R, the expected warranty cost is given by r1 ∞ r2 ˜ ECr (K1 ; K2 )g(r) dr + ECr ('1 ; '2 )g(r) dr + ECr ('1 ; '2 )g(r) dr: EC(") = 0
r2
r1
(15)
For Case (2) [r2 ¿ r1 ], we need to investigate the following three sub-cases: (2.1) r 6 r1 , (2.2) r1 ¡ r 6 r2 , and (2.3) r ¿ r2 , as illustrated in Fig. 3. Case (2.1): [r 6 r1 ] Conditional on R = r, the expected conditional warranty cost in (0; K1 ] is given by (7). The expected conditional cost over (K1 ; K2 ] and that over (K2 ; K] are given by (10) and (11), respectively. Hence ECr (K1 ; K2 ) is given by (12). Case (2.2): [r1 ¡ r 6 r2 ] De2ne ' = L=r. Note that in this case the warranty ceases at time '. The expected conditional warranty cost in (0; K1 ] is given by (7). The expected conditional warranty cost in (K1 ; K2 ] is given by (10) and the expected conditional warranty cost in (K2 ; '] is given by (11) with ' replacing K. Hence K2 f(t|r) 1 + %(' − t|r) dt ECr (K1 ; K2 ) = c2 %(K1 |r) + M ! F(K1 |r) K1
M 2 |r) F(K + {%('|r) − %(K2 |r)} : (16) M 1 |r) F(K
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Case (2.3): [r ¿ r2 ] De2ne ' = L=r, '1 = L1 =r and '2 = L2 =r. Note that in this case the warranty ceases at time '. This case is the same as Case (1.3), hence ECr ('1 ; '2 ), which is the sum of the expected conditional warranty costs over the three periods—i.e., (0; '1 ], ('1 ; '2 ], and ('2 ; '], is given by (14). As a result, we have the expected warranty cost given by r1 r2 ∞ ˜ EC(") = ECr (K1 ; K2 )g(r) dr + ECr (K1 ; K2 )g(r) dr + ECr ('1 ; '2 )g(r) dr: (17) 0
r1
r2
˜ subject to the conThe objective is to 2nd the values of "˜ = {K1 ; K2 ; r2 } that minimise EC("); ˜ straints 0 6 K1 ; K2 6 K and 0 ¡ r2 ¡ ∞. EC(") is given by (15) when 0 ¡ r2 ¡ r1 , and by (17) when r1 ¡ r2 ¡ ∞. 4. Numerical example Suppose that the item under consideration is a component of an automobile sold with an FRW policy. We con2ne our attention to the warranty cost associated with the servicing of claims resulting from the component under consideration. The time (or age) is measured in units of years and the unit for usage is 10; 000 km. We assume that K = 2 and L = 2 , so r1 = 1; and we have a square warranty region with a maximum coverage for 2 years and 20; 000 km of usage. We consider the form for the conditional failure intensity given by (7), with the parameter values 0 = 0:1, 1 = 0:2, 2 = 0:7, and 3 = 0:7, and we assume that R is uniformly distributed over [rl ; ru ], with g(r) = 1=[ru − rl ], for rl 6 r 6 ru . We consider three sets of values for rl and ru , which broadly correspond to light, medium, and heavy users, respectively. These values and those for the corresponding mean usage rate (E[R]), mean age at 2rst failure (E[T ]) and mean usage at 2rst failure (E[X ]) are as follows: Usage type
rl
ru
E[R]
E[T ]
E[X ]
Light Medium Heavy
0.1 0.7 1.1
0.9 1.3 2.9
0.5 1.0 2.0
1.1118 years 0.9575 years 0.7755 years
0.5559 (104 km) 0.9575 (104 km) 1.5510 (104 km)
Without loss of generality, we assume that c1 =1 and consider a range of values of ! varying from 0.1 to 0.9. (When c1 = 1; the costs given in the following tables are the normalised costs relative to c1 .) The decision variables for the new strategy are K1 ; K2 ; and r2 = L2 =K2 = L1 =K1 . The grid search for ˜ was done with K1 and K2 incremented in steps of 0.1 over the range the minimization of EC(") 0.0 –2.0, and r2 incremented in steps of 0.2, starting at 0.2. Tables 1–3 give the new strategy results for the three types of usage intensity. Also shown are the expected warranty servicing costs for the two simpler strategies of (i) always performing minimal repairs, and (ii) always replacing the item at each failure in the warranty region. For each value of !, the smallest expected warranty servicing cost is shown in bold font and the corresponding strategy is the optimal strategy. Note that, the new strategy reduces to ‘always repair’ when K1 = K2 ; but the ‘always replace’ strategy is not a special case of the new strategy. However, when K1 = 0; K2 = K,
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Table 1 Expected warranty costs for light usage intensity !
K1∗
K2∗
r2∗
EC("˜ ∗ )
Always repair
Always replace
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 0.1 0.2 0.6 0.6 0.6 0.6 0.6 0.5
0.2 0.2 0.3 1.5 1.7 1.8 1.9 1.9 2.0
0.2 0.2 0.2 1.0 1.0 1.0 1.0 1.0 1.0
0.3281 0.6469 0.9656 1.1401 1.2258 1.2971 1.3627 1.4263 1.4875
0.3156 0.6312 0.9469 1.2800 1.6000 1.9200 2.2400 2.5600 2.8800
1.4193 1.4193 1.4193 1.4193 1.4193 1.4193 1.4193 1.4193 1.4193
Table 2 Expected warranty costs for medium usage intensity !
K1∗
K2∗
r2∗
EC("˜ ∗ )
Always repair
Always replace
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 0.9 1.0 0.7 0.6 0.6 0.6 0.6 0.5
0.2 1.0 1.2 1.6 1.7 1.8 1.9 1.9 2.0
0.4 0.2 0.8 1.0 1.0 1.0 1.0 1.0 1.0
0.3744 0.7319 1.0894 1.2420 1.3390 1.4249 1.5066 1.5875 1.6664
0.3555 0.7109 1.0664 1.4549 1.8186 2.1823 2.5460 2.9097 3.2735
1.5781 1.5781 1.5781 1.5781 1.5781 1.5781 1.5781 1.5781 1.5781
and r2 = 1, we have the alternative strategy of replacement at the 2rst failure followed by minimal repair at all other failures. These tables indicate that • When the item has a light or medium usage intensity, it is optimal to ‘always repair’ for small values of ! and ‘always replace’ for large values of !, and this is as to be expected. For intermediate values of !, the new strategy is the best choice, and as ! increases through this range the size of the region 1 remains relatively constant with that of 3 decreasing to zero. • In the case of high usage intensity, the ‘always repair’ strategy is optimal for small to intermediate values of !. The new strategy is optimal for 0:6 6 ! 6 0:8, and as ! increases through this range, the size of region 3 remains relatively constant whilst that of 1 decreases, allowing for possible earlier replacement of the item. The optimal strategy is ‘always replace’ for ! ¿ 0:9. It can also be seen from Tables 1–3 that r2∗ is close to 1 when the new strategy performs better than the ‘always repair’ and ‘always replace’ strategies. Tables 4–6 give a relative comparison of the new strategy, ‘always repair’, ‘always replace’, and Strategies 1 and 2 of Iskandar and Murthy
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Table 3 Expected warranty costs for heavy usage intensity !
K1∗
K2∗
r2∗
EC("˜ ∗ )
Always repair
Always replace
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 0.2 0.8 1.9 1.9 0.9 0.6 0.5 0.4
0.2 0.3 0.9 2.0 2.0 1.8 1.7 1.8 1.9
0.2 0.2 0.2 0.2 0.4 0.8 1.0 1.0 1.0
0.1505 0.2959 0.4412 0.5864 0.7313 0.8384 0.9022 0.9500 0.9902
0.1458 0.2916 0.4374 0.5839 0.7299 0.8758 1.0218 1.1678 1.3138
0.9802 0.9802 0.9802 0.9802 0.9802 0.9802 0.9802 0.9802 0.9802
Table 4 Expected warranty costs for light usage intensity when r2 = 1 !
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Strategy 1
Strategy 2
New Strategy
K1∗
EC("˜ ∗ )
K1∗
EC("˜ ∗ )
K1∗
K2∗
EC("˜ ∗ )
0.0 0.0 0.0 1.1 1.3 1.4 1.6 1.7 1.9
0.3156 0.6312 0.9469 1.1091 1.2236 1.3062 1.3633 1.4006 1.4221
2.0 2.0 2.0 0.9 0.6 0.5 0.3 0.3 0.3
0.3156 0.6312 0.9469 1.3342 1.3680 1.3876 1.4018 1.4088 1.4157
0.1 0.1 0.9 0.6 0.6 0.6 0.6 0.6 0.5
0.2 0.2 1.0 1.5 1.7 1.8 1.9 1.9 2.0
0.3386 0.6551 0.9662 1.1401 1.2258 1.2971 1.3627 1.4263 1.4875
Always repair
Always replace
0.3156 0.6312 0.9469 1.2800 1.6000 1.9200 2.2400 2.5600 2.8800
1.4193 1.4193 1.4193 1.4193 1.4193 1.4193 1.4193 1.4193 1.4193
[6] for the case r2 = 1. A one-dimensional search with a grid size of 0.1 was used to obtain the optimal values of K1 . Note that Strategies 1 and 2 reduce to ‘always repair’ when K1 = 0 and K1 = 2, respectively, and ‘always replace’ occurs when K1 = 2 and K1 = 0, respectively. From Tables 4–6, the following observations can be made about the new repair–replace strategy: • For small values of !, the optimal strategy is “always repair” and strategies 1 and 2 both reduce to this form. This is as to be expected. • For intermediate values of !, the new strategy is the best choice. For heavy usage intensity, as ! increases the region 2 increases in size whilst the regions 1 and 3 both decrease. For medium and light usage intensities, as ! increases the size of region 1 remains relatively constant whilst that of 3 decreases to zero. • In the limit as ! approaches 1, then ‘always replace’ is the best strategy and again both strategies 1 and 2 reduce to this. The new strategy is never optimal here since only one replacement is allowed.
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Table 5 Expected warranty costs for medium usage intensity when r2 = 1 !
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Strategy 1
Strategy 2
New strategy
K1∗
EC("˜ ∗ )
K1∗
EC("˜ ∗ )
K1∗
K2∗
EC("˜ ∗ )
0.0 0.0 0.0 1.1 1.3 1.4 1.6 1.7 1.9
0.3555 0.7109 1.0664 1.2286 1.3522 1.4403 1.5023 1.5424 1.5656
2.0 2.0 2.0 0.9 0.7 0.6 0.5 0.3 0.3
0.3555 0.7109 1.0664 1.4652 1.5095 1.5357 1.5555 1.5654 1.5748
0.1 0.1 1.0 0.7 0.6 0.6 0.6 0.6 0.5
0.2 0.2 1.1 1.6 1.7 1.8 1.9 1.9 2.0
0.3889 0.7476 1.0902 1.2420 1.3390 1.4249 1.5066 1.5875 1.6664
Always repair
Always replace
0.3555 0.7109 1.0664 1.4549 1.8186 2.1823 2.5460 2.9097 3.2735
1.5781 1.5781 1.5781 1.5781 1.5781 1.5781 1.5781 1.5781 1.5781
Always repair
Always replace
0.1458 0.2916 0.4374 0.5839 0.7299 0.8758 1.0218 1.1678 1.3138
0.9802 0.9802 0.9802 0.9802 0.9802 0.9802 0.9802 0.9802 0.9802
Table 6 Expected warranty costs for heavy usage intensity when r2 = 1 !
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Strategy 1
Strategy 2
New strategy
K1∗
EC("˜ ∗ )
K1∗
EC("˜ ∗ )
K1∗
K2∗
EC("˜ ∗ )
0.0 0.0 0.0 0.0 0.1 1.2 1.4 1.6 1.8
0.1458 0.2916 0.4374 0.5839 0.7383 0.8569 0.9164 0.9564 0.9795
2.0 2.0 2.0 2.0 2.0 2.0 0.6 0.4 0.2
0.1458 0.2916 0.4374 0.5839 0.7299 0.8758 0.9536 0.9688 0.9769
0.1 0.1 0.2 0.6 1.0 0.7 0.6 0.5 0.4
0.2 0.2 0.3 0.7 1.1 1.5 1.7 1.8 1.9
0.1687 0.3116 0.4543 0.5957 0.7323 0.8388 0.9022 0.9500 0.9902
For light and medium usage intensities, the warranty ceases mainly due to the age limit K. The value of the total usage when the warranty ceases is higher for medium usage intensity, so the corresponding servicing costs are larger than those for light usage intensity. However, in the case of heavy usage intensity, the warranty expires mainly due to the usage limit L. The age of the item when the warranty ceases is well below the age limit K, and so the heavy usage intensity servicing costs are smaller than the corresponding costs for medium usage intensity. However, this pattern will change depending on the parameter values and the structure of the conditional intensity function. The expected warranty servicing cost (for consumer durables and commercial products) varies from 0.5% to 10% of the sale price depending on the product and the manufacturer. Since the annual sales of such products runs into trillions of dollars, even a small reduction in the expected warranty servicing cost implies savings of billions of dollars annually. For intermediate values of ! the new servicing strategy o8ers considerable potential savings to manufacturers.
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5. Conclusions In this paper, we have studied a new repair–replace strategy for a repairable item sold with a two-dimensional warranty. The strategy is characterised by four parameters and these can be selected to minimise the expected warranty cost. We con2ned our attention to the case where one of the parameters is constrained so that the problem is reduced to a three-parameter optimisation. We carried out a relative comparison with two other strategies—‘always repair’ and ‘always replace’. The new strategy, which is easy to administer, did signi2cantly better than these two simple strategies for intermediate values of ! (the ratio of repair cost to replacement cost). The results of this paper can be extended in several ways, and a few of these are discussed below • The analysis for the case where all four parameters are independent can be carried out in a manner similar to that in this paper but would be much more complex. • In our formulation, the warranty is characterised by a rectangular region . It is decomposed into three-sub-regions 1 , 2 and 3 with these being de2ned in terms of two rectangles within . An extension would be to look at regions that are not necessarily characterised by rectangles. • The true optimal repair–replace strategy and hence the optimal shapes of the regions 1 and 2 within the rectangular region are still unknown. They can be determined by modifying the dynamic programming approach used by Jack and Van der Duyn Schouten [8] for the one-dimensional case. This topic is currently under investigation. • In the new strategy in this paper, it is assumed that all failed items can be repaired. However, in many real situations, some failures are not repairable (termed type 1 failures) whilst others (called type 2 failures) are repairable. As a result, a repair–replace strategy can be formulated where the action to rectify the failed item also depends on the type of failure. These are only a few of the many topics for further research. Acknowledgements The authors thank the reviewer for the constructive comments on an earlier version of the paper. References [1] Barlow RE, Proschan F. Mathematical theory of reliability. New York: Wiley; 1965. [2] Biedenweg FM. Warranty analysis: consumer value vs manufacturers cost. Unpublished PhD thesis, Stanford University, USA, 1981. [3] Blischke WR, Murthy DNP. Warranty cost analysis. New York: Marcel Dekker; 1994. [4] Iskandar BP. Modelling and analysis of two dimensional warranty policies. Unpublished PhD thesis, The University of Queensland, Brisbane, Australia, 1993. [5] Iskandar BP, Blischke WR. In: Blischke WR, Murthy DNP, editors. Case studies in reliability and maintenance. New York: Wiley; 2003. [6] Iskandar BP, Murthy DNP. Repair replace strategies for two-dimensional warranty. Proceedings of the First Western Paci2c and Third Australia–Japan Workshop on Stochastic Models in Engineering, Technology and Management, New Zealand, September 1999. p. 206 –13.
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