*Corresponding author. Tel:#31-13-4662230; fax:# 31-13-4663394.
E-mail addresses: [email protected] (N. Jack), [email protected] (F. Van der Duyn Schouten).
Optimal repair}replace strategies for a warranted product
Nat Jack
!
, Frank Van der Duyn Schouten
"
,
*
!University of Abertay Dundee, Bell Street, Dundee, Scotland DD1 1HG, UK
"Center for Economic Research, Tilburg University, P.O. Box 90153, 5000 LE Tilburg, The Netherlands
Abstract
When a repairable product is sold with a free replacement warranty, the manufacturer has the option of rectifying a failure by either repairing the failed item or replacing it with a new one. In this paper, repairs are assumed to be minimal in the sense that they leave the product in exactly the same condition it was in prior to failure. We discuss the form of the optimal repair}replace strategy that minimises the expected cost of servicing the warranty over the warranty period. ( 2000 Elsevier Science B.V. All rights reserved.
Keywords: Free replacement warranty; Repair}replace strategies; Minimal repair
1. Introduction
A warranty is a contractual agreement between a manufacturer (seller) and a consumer (buyer) which requires the manufacturer to rectify all fail-ures occurring within the warranty period. Under a free replacement warranty, no charge is made to the consumer for these recti"cation actions, which can be either repairs or replacements by new prod-ucts. The choice of repair versus replacement is made by the manufacturer and this will depend on the relevant costs, the lifetimes of repaired and new products, and the distance to the end of the war-ranty period at the occurrence of a failure. It is therefore important for the manufacturer to devise a maintenance strategy which minimises the cost of servicing the warranty.
Optimal warranty servicing strategies have been studied by a number of authors. Biedenweg [1], assuming that repaired items have independent and identically distributed lifetimes di!erent from that of a new item, showed that the optimal strategy had the following simple form:Replace with a new item at any failure occurring up to a certain time measured from the initial purchase and then repair all other failures that occur during the remainder of the war-ranty period. This strategy is based on the idea that replacements close to the end of the warranty peri-od are not in the interest of the manufacturer. The idea of splitting the warranty period into distinct intervals for repair and replacement was continued by Nguyen and Murthy [2,3]. Nguyen and Murthy [2] assumed replacement of failures during the sec-ond interval of the warranty period using a stock of used items. Nguyen and Murthy [3] extended Biedenweg's [1] model by adding a third interval where failed items are either replaced or repaired and a new warranty is given at each failure. The strategy investigated by Murthy and Nguyen [4] involved estimating the cost of a minimal repair
with the decision to replace or repair depending on whether this estimated cost exceeded a certain thre-shold or not. These models are also discussed in detail in the warranty servicing chapters of Blischke and Murthy [5,6].
Assaf and Levikson [7] also discussed a "nite horizon maintenance problem but not speci"cally in a warranty context. In their model, at each item failure,ndi!erent replacement items are available di!ering in cost of purchase and lifetime distribu-tion. The optimal strategy splits the"nite warranty period into n (possibly empty) intervals such that for each interval a particular replacement item from the set should be chosen in case a failure occurs within this interval.
Also many in"nite horizon maintenance prob-lems involving minimal repair have been studied. Of particular interest are the early papers of Muth [8] and Phelps [9]. Muth [8] was the "rst to introduce the strategy where minimal repairs are performed on an item up to age¹with replacement by a new item at the"rst failure after¹. Phelps [9] used a semi-Markov decision model to prove that this is the optimal strategy to use over an in"nite horizon.
The problem of "nding the optimal repair}
replace strategy during a warranty period corres-ponds to a "nite horizon maintenance problem where repairs are assumed to be minimal. We pro-pose the form of the optimal repair}replace strat-egy for a manufacturer maintaining a warranted product. We also discuss a simpler strategy involv-ing splittinvolv-ing the warranty period into two distinct intervals where replacements and then repairs are performed. In an example we show that this strat-egy is only slightly inferior to the optimal stratstrat-egy.
2. Model formulation and analysis
The model proposed in this paper is used to investigate a manufacturer's optimal warranty ser-vicing strategy over a warranty period of "nite length. To indicate the mathematical complexity of this model we summarise the models and results from Assaf and Levikson [7] and Phelps [9].
Firstly, we consider the Assaf and Levikson [7]
"nite horizon model where, at an item failure,
a choice from a set ofndi!erent replacement items is available.
Let
<(t) :"the minimal expected total cost withttime units left in the horizon given that the item has just failed.
Then<(t) is the solution of the following optimality equation:
i denotes the purchase cost of itemi, while
f
i(x) is the density function of the lifetime
distribu-tion of itemi. Also letF
i(x) denote the cumulative distribution
function of the lifetime of item i andk
i its mean.
Assaf and Levikson [7] prove the following theorem.
Theorem 1. If the items can be ordered according to the inequalities then the optimal policy is completely specixed by n!1 switching points t
1)t2)2)tn~1. If a failure occurs at timet't
n~1,then the item which is the highest in the ordering (i.e. which has the smallest average cost per unit time)is chosen for the replacement; if a failure occurs at time t with t
n~2(t)tn~1,then the item which is second high-est in the ordering is chosen, and so on.
Note that the inequalities (2) and (3) imply that C
i/ki'Cj/kj, so item j has a smaller average
cost per unit time than itemi. Also it is worthwhile to note that some of the intervals can be empty, implying that not necessarily every item will be used.
criterion in this model is the total discounted costs over the in"nite horizon, where the continuous discount factor is denoted bya.
Let
<
a(h) :"minimal expected total discounted costgiven that an item has just failed at ageh.
Then<
a(h) satis"es the following optimality equa-tion:
Here F(x) denotes the cumulative lifetime
distribution of a new item, whileF
h(x)"(F(h#x)!
F(h))/(1!F(h)) is the conditional cumulative
distri-bution function of the residual lifetime of an item of
age h. Further C. denotes the (expected) cost of
a minimal repair andC
3 the cost of a replacement item, where it is assumed thatC
.(C3. Phelps [9] proves the following theorem.
Theorem 2. If the item's lifetime distribution is IFR then <
a(h) is non-decreasing in h and the optimal stationary strategy is to repair if the age of the item at failure is less than a critical threshold ¹ and to replace if the age is greater than¹.
Now, we consider our "nite horizon warranty model where the choice the manufacturer has when a product fails is either to replace with a new product or to do a minimal repair.
Let
<(t,h) :"minimal expected total cost withttime
units remaining in the warranty period given that the product has just failed at ageh.
Then<(t,h) satis"es the optimality equation:
<(t,h)"Min
G
CThe complexity of this model is re#ected by the fact that the value function depends both onh(the age of the failing component) as well ast(the remaining length of the warranty period). As such it combines elements that are present in both the models of Phelps [9] and Assaf and Levikson [7].
With respect to the structure of the optimal policy for this repair}replace model we make the following conjecture. Let= denote the length of the warranty period.
Conjecture 1. If F(x) is IFR then, for every
t, 0(t(=, there exists a control limit ¸(t) such
that a minimal repair is performed on failure with
t time units remaining in the warranty period i+the
actual age of the failing unit is less than or equal to
¸(t).
Although we got ample numerical evidence of the correctness of this conjecture, we have not been able to prove it. As will be shown by the numerical examples given below the functional form of ¸(t) can be rather general. In any case¸(t) is certainly not monotonic int. It is worth mentioning that¸(t) is bounded from above by =!t. When
¸(t)"=!t,∀t3(0,=), then the optimal policy
for the manufacturer is always to do a minimal repair at failure and never to replace.
From the numerical examples it follows that
¸(t)"=!tboth in the neighbourhood of 0 and
=. This makes sense because in both areas a repair
will be preferred over a replacement (either because the failing unit is too new or the distance to the end of the warranty period is too short). We conclude this section with some numerical examples.
Example 1. Let the lifetime of a new product (in years) be Weibull distributed with
F(x)"l!e~(jx)b.
For the costs we chooseC
."1 andC3"2.
Using standard techniques from dynamic pro-gramming the function¸(t) can be evaluated nu-merically from the optimality equation (5).
Fig. 1. Control limit function for="2 andb"2.
Fig. 2. Control limit function for="3 andb"2.
Fig. 3. Control limit function for="2 andb"3.5.
Table 1
Comparison of optimal static and dynamic strategies
C
3 1.01 1.5 2.0 2.5 5.0
qH 0 0.28 0.55 2 2
C(qH) 1.913 2.762 3.476 4.000 4.000
<(2, 0) 1.913 2.694 3.228 3.668 4.000
new product lifetime"0.886 years). Figs. 3 and 4 apply whenbincreases to 3.5 (mean new product lifetime"0.899 years), giving a more sharply increasing hazard rate.
The following general observations can be made. When the length of the warranty period increases the optimal control limit function exhibits a con-stant behaviour except for the boundary regions (near 0 and=). This constant value corresponds to the critical control limit of the in"nite horizon problem. When the hazard rate increases more sharply the control limit function also tends to show more#uctuation.
From Figs. 2}4 it follows that our conjecture does not have a counterpart by reversing the dimensions. In other words, it is apparently not true that for any agehof the failing unit there exists a critical threshold distance to the end of the war-ranty period, such that a replacement is done if and only if the actual distance to the end of the warranty period is below this threshold.
Finally, we investigate the e!ect of using a more simple static strategy instead of the above optimal dynamic one. The strategy we consider involves splitting the warranty period into two intervals as suggested by Nguyen and Murthy [2]. Replace-ment by a new product occurs at any failure during [0, =!q) and minimal repair takes place at any failure during (=!q,=], whereqis the decision variable.
Using renewal arguments an expression for the expected total warranty servicing cost as a function ofqis easy to derive:
whereM(x) denotes the renewal function,R(x) the cumulative hazard function and FM(x) the survival function of the distribution functionF.
Note thatC(0)"C
3M(=) (corresponding to al-ways replace) andC(=)"C
.R(=) (corresponding
to always repair). Always repair is therefore prefer-red to always replace i!
C
3/C.'R(=)/M(=). (7)
Example 2. Suppose the lifetime of a new product (in years) again has a Weibull distribution with
F(x)"1!e~x2. ChooseC
."1, and="2 years.
Table 1 gives the optimalqvalues and a compari-son of the costs of the static and optimal dynamic strategies over a range of values ofC
3.
Note that result (7) implies that in this example always repair is preferred to always replace when C3'2.11. The di!erence between the optimal dynamic policy and the optimal static policy is quite small for this example. This di!erence may increase with increasing values of the warranty period length. More sophisticated static policies could be considered, e.g., a policy with two critical values q
1andq2. Under such a policy there will be repairs when a failure occurs beforeq
1 and beyondq2.
3. Conclusions
sample numerical evidence exists about the validity of this conjecture, a formal mathematical proo is still lacking. Since the overall optimal policy can have a rather complicated structure, which makes it hard to implement in practice, we also propose policies of a more simple structure. For warranty periods of realistic length the best policy within the simple structure class performs only slightly less than the overall optimal policy.
References
[1] F.M. Biedenweg, Warranty analysis: Consumer value vs manufacturers cost, Unpublished Ph.D. Thesis, Stanford University, Stanford, CA, 1981.
[2] D.G. Nguyen, D.N.P. Murthy, An optimal policy for servic-ing warranty, Journal of the Operational Research Society 37 (1986) 1081}1088.
[3] D.G. Nguyen, D.N.P. Murthy, Optimal replace}repair strategy for servicing items sold with warranty, European Journal of Operational Research 39 (1989) 206}212. [4] D.N.P. Murthy, D.G. Nguyen, An optimal repair cost limit
policy for servicing warranty, Mathematical and Computer Modelling 11 (1988) 595}599.
[5] W.R. Blischke, D.N.P. Murthy, Warranty Cost Analysis, Marcel Dekker, New York, 1994.
[6] W.R. Blischke, D.N.P. Murthy, Product Warranty Hand-book, Marcel Dekker, New York, 1996.
[7] D. Assaf, B. Levikson, On optimal replacement policies, Management Science 28 (1982) 1304}1312.
[8] E.J. Muth, An optimal decision rule for repair vs replace-ment, IEEE Transactions on Reliability R-26 (1977) 179}181. [9] R.I. Phelps, Optimal policy for minimal repair, Journal of
