Reserves

Managing Warranties: Funding a Warranty Reserve

and

Outsourcing Prioritized Warranty Repairs

by

Peter S. Buczkowski

A dissertation submitted to the faculty of the University of North Carolina at Chapel

Hill in partial ful llment of the requirements for the degree of Doctor of Philosophy in

the Department of Statistics and Operations Research.

c 2004

Peter S. Buczkowski

ALL RIGHTS RESERVED

ii

ABSTRACT

Peter S. Buczkowski:

MANAGING WARRANTIES: FUNDING A WARRANTY

RESERVE AND OUTSOURCING PRIORITIZED WARRANTY REPAIRS

(Under the direction of Professor Vidyadhar G. Kulkarni)

We consider two problems central to the managing of warranty costs by the manufacturer.

First, we consider funding an interest-bearing warranty reserve with contributions after

each sale. The problem for the manufacturer is to determine the initial level of the

reserve fund and the amount to be put in after each sale, so as to ensure that the

reserve fund covers all the warranty liabilities with a prespeci ed probability over a xed

period of time. We assume a non-homogeneous Poisson sales process, random warranty

periods, and an exponential failure rate for items under warranty. We derive the mean

and variance of the reserve level as a function of time and provide a heuristic to aid the

manufacturer in its decision.

We also consider the problem of outsourcing warranty repairs to outside vendors

when items have priorities in service. The manufacturer has a contract with a xed

number of repair vendors. The manufacturer pays a xed fee for each repair done by

a vendor which is independent of the repair type and priority class but depends on the

vendor. There are a xed number of items under warranty, and each item belongs to

one of a xed number of priority classes. The manufacturer also pays for holding costs

incurred when the items are at the vendors, the holding cost being higher for the higher

priority items. The vendors provide a pre-emptive priority for an item over all other items

of lower priority. We focus on static allocation of the warranty repairs; that is, we assign

iii

all items to the vendors at the beginning of the warranty period. We give the known

algorithm to optimally solve the one priority class problem and solve the multi-priority

class problem by formulating it as a convex minimum cost network ow problem. Then,

we give numerical examples to illustrate the cost bene ts of a multi-priority structure.

iv

ACKNOWLEDGMENTS

I am sincerely grateful to my advisor, Dr. Vidyadhar Kulkarni, for his knowledge,

guidance, time, and support through my studies at Chapel Hill. His comments greatly

improved this dissertation from beginning to end.

I would also like to express my gratitude to my committee members: Dr. Suheil

Nassar, Dr. Jayashankar M. Swaminathan, Dr. Eylem Tekin, and Dr. Jon W. Tolle, for

their help and suggestions on my research. Special thanks to Dr. Mark Hartmann for

donating his time and providing invaluable suggestions, without which this dissertation

would not be complete. Let me also extend thanks to my classmates, especially Wei

Huang, Bala Krishnamoorthy, Michelle Opp, and Rob Pratt. Their advice and support

is evident in many areas of this thesis.

Finally, I would like to thank my wife Gretchen for her support and understanding

over the last four years.

v

CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

1 Introduction 1

1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Organization of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . 6

2 Funding a Warranty Reserve 8

2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Notation and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Probability Distribution of the Number of Items Under Warranty . . . . 11

2.3.1 Distribution of Xn(t) . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3.2 Distribution of Xo(t) . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Di
erential Equations for Moments of R(t) . . . . . . . . . . . . . . . . . 15

2.5 Solution for Moments of R(t) . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5.1 Example: Constant Warranty Period and Constant Sales Rate . . 25

2.6 Deciding the Values of c and R0 . . . . . . . . . . . . . . . . . . . . . . . 28

2.6.1 Distribution of R(t) . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.6.2 Heuristic for Deciding c and R0 . . . . . . . . . . . . . . . . . . . 30

2.7 Numerical Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

vi

3 Warranty Reserve: Extensions 36

3.1 Random contribution to the reserve after each sale . . . . . . . . . . . . 36

3.2 Multiple products using a single reserve . . . . . . . . . . . . . . . . . . . 37

3.3 Xo(t) is known during [0; T] . . . . . . . . . . . . . . . . . . . . . . . . . 38

4 Outsourcing Prioritized Warranty Repairs 42

4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2 Notation and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.3 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.4 Single Priority Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.5 Minimum Cost Network Flow Problems . . . . . . . . . . . . . . . . . . . 52

4.5.1 Convex Network Problems . . . . . . . . . . . . . . . . . . . . . . 55

4.6 Network Flow Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.6.1 Single Priority Class . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.6.2 Two Priority Classes . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.6.3 Multiple Priority Classes . . . . . . . . . . . . . . . . . . . . . . . 61

4.7 Computational Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.8 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.9 Cost Bene ts of the Multi-Priority Approach . . . . . . . . . . . . . . . . 68

4.10 Selecting the Values of Ki . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5 Conclusions and Future Work 74

Bibliography 77

vii

LIST OF TABLES

2.1 Suggested Values for q . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.1 Arc Properties for Two-priority Network . . . . . . . . . . . . . . . . . . 59

4.2 Arc Properties for m-priority Network . . . . . . . . . . . . . . . . . . . 63

4.3 Costs and Service Rates for Each Vendor . . . . . . . . . . . . . . . . . . 68

4.4 Average Holding Costs for Vendors . . . . . . . . . . . . . . . . . . . . . 69

4.5 Vendor Properties for Similar Cost Example . . . . . . . . . . . . . . . . 70

4.6 Arc Properties for the Reward Network . . . . . . . . . . . . . . . . . . . 72

4.7 Costs and Service Rates for Each Vendor . . . . . . . . . . . . . . . . . . 73

viii

LIST OF FIGURES

2.1 Example of Warranty Reserve Account . . . . . . . . . . . . . . . . . . . 9

2.2 Examples of Con dence Bands for R(t) . . . . . . . . . . . . . . . . . . . 29

2.3 Expected Reserve for Various Values of X(0) . . . . . . . . . . . . . . . . 34

4.1 Network Model of Single Priority Problem . . . . . . . . . . . . . . . . . 57

4.2 Network Model of Two-priority Problem . . . . . . . . . . . . . . . . . . 59

4.3 Network Model of m Priority Problem . . . . . . . . . . . . . . . . . . . 62

4.4 Network Model of Reward Problem . . . . . . . . . . . . . . . . . . . . . 72

ix

Chapter 1

Introduction

1.1 Overview

Since the Magnuson-Moss Warranty Act of 1975 [33], manufacturers are required to

provide a warranty for all consumer goods which cost more than $15. Warranties play

an important role in the consumer-manufacturer relationship. They o
er assurance to

the consumer that their purchase will achieve certain performance standards through at

least the warranty period. The manufacturers use warranties as a marketing tool and

they limit their liability.

When designing product warranties, the manufacturers must decide on many issues,

such as warranty policy, length of warranty period, repair policy, and quality control.

They also have to plan to cover the costs associated with the warranty. An issue of critical

importance to the manufacturers is managing the costs associated with the warranty

e
ectively. Our research investigates two key questions of planning for these costs.

The rst is of funding a warranty reserve account with contributions made after

each sale. A warranty reserve is used to accommodate all of the costs associated with the

servicing of a warranty of a product. We model a policy that is currently implemented in

industry; that of adding a fraction of each sale to the reserve fund. There are a variety

1

of goals that a manufacturer may have regarding its warranty reserve. Two general goals

are to keep the reserve above some target dollar amount B > 0 and to not have an

excessive amount of money in the reserve. The reasoning behind these goals is simple:

a shortage requires extra administrative costs and may even have legal rami cations,

while an excessive surplus locks money in the reserve that may be more useful for other

business interests. Achieving these goals requires careful planning.

We also consider the problem of outsourcing warranty repairs to outside vendors

when items have priority levels. For example, some warranty contracts specify the repair

turnaround time (e.g. 1 day, 3 days, or 7 days). With careful management, repair outsourcing

can be a major bene t to the manufacturer. A smooth operation can improve

customer satisfaction and turnaround times, while allowing the manufacturer to maintain

its focus on production. While the manufacturer may have a central repair depot,

it often is not e
ective to ship items to the depot due to time and cost constraints. Thus

it might be bene cial to choose repair vendors distributed geographically so as to be

close to the customers. The manufacturer must seek a balance between cost savings and

customer service. If not, some customers will be lost because of poor service. Repair

outsourcing is an especially important problem when considering priorities because high

priority customers will typically inict greater loss if the manufacturer does not meet

their expectations.

1.2 Literature Review

Warranty theory has been heavily studied over the past two decades. Blischke and

Murthy [5] wrote a comprehensive reference for the subject. They discuss many di
erent

types of warranty policies, including many warranty policies currently implemented

in industry. Numerous cost and optimization models are developed from both the consumer's

and the manufacturer's point of view, including life cycle and long-run average

2

cost models. We use these models to compute the expected warranty cost of a product

in our numerical examples.

Many of the early papers on warranty theory discuss the costs and other e
ects

that are associated with warranties. Glickman and Berger [13] consider the e
ect of

warranty on demand by assuming that demand increases as the warranty period increases.

Warranty costs a
ect both the buyer and the seller. Mamer [23] wrote the rst

paper to provide a comprehensive model of both the buyer's and seller's expected costs

and long-run average costs for the free replacement warranty. Our research focuses on

the manufacturers' view of warranty costs.

The concept of a warranty reserve is a topic of many research works. The initial

papers on warranty reserves discussed here consider a xed product lot size throughout

the life cycle of a product (or equivalently, a xed cumulative failure rate). Menke [25]

wrote one of the rst papers to address the warranty reserve problem. He concentrates

on calculating the expected warranty cost over a given warranty period for two types

of pro-rata warranty policies (linear rebate and lump-sum rebate) assuming a constant

product failure rate. Amato and Anderson [2] extend Menke's model by allowing the

reserve fund to accrue interest, requiring the consideration of discounted costs. A comparison

to Menke's results is made, concluding that discounting signi cantly reduces the

expected warranty reserve over longer periods of time. Both models are rather limited in

scope because they only consider pro-rata warranty policies and an exponential failure

distribution.

Balcer and Sahin [4] derive the moments of the total replacement cost for both

the free-replacement and pro-rata warranty policies during the product life cycle. They

assume that successive failure times form a renewal process.

Mamer [24] uses renewal theory to model repeated product failures over a life cycle

of the product. He incorporates discounting in his model and allows for a general failure

distribution. However, he does not consider the sales process nor compute a reserve.

3

Tapiero and Posner [32] allow for a portion of each sale to be set aside for future

warranty costs. The contributions to the reserve fund and the items sold occur at a

constant rate. The claims are generated according to a compound Poisson Process and

they use a sample path technique to compute the long-run probability distribution of the

warranty reserve.

Eliashberg, Singpurwalla, and Wilson [12] calculate the reserve for a product

whose failure rate is indexed by two scales, time and usage. They allow for a general

failure rate and assume a form of imperfect repair. The warranty reserve is computed to

minimize a loss function for the manufacturer.

Ja, et al. [18] compute the distribution of the total discounted warranty cost over

the life cycle of the product. They analyze the discounted warranty cost of a single sale

under many di
erent policies and then consider di
erent stochastic sales processes. A

single contribution to the reserve is made at the beginning of the life cycle. However, the

subtractions from the reserve due to warranty costs are tracked as a function of time.

Another application related to the warranty reserve problem is the insurance premium

problem. An insurance company must decide on the monthly premium to charge

a certain class of customer. Low premiums result in loss to the insurer, while high premiums

result in loss of business to the competition. A discussion of this can be found

in [30]. There are other related problems, including the funding of a company's pension

plan. Many of these problems are solved using actuarial models, particularly collective

risk (loss) models (see [22] and [9] for references on this subject). However, the current

models do not incorporate the number of policies insured by the company at any given

time.

The works described above illustrate many di
erent models to compute the warranty

reserve. However, they assume that the reserve is either funded at the beginning

of the product sales period or at a constant rate. We extend this research by modeling

4

contributions to the reserve after each sale and allowing the cumulative warranty claim

rate to depend on the sales process.

We now turn to the warranty repair outsourcing problem. At its most basic structure,

the static allocation model reduces to a resource allocation problem with integer

variables. Without considering priorities, the problem has a separable objective function.

This problem has been widely studied in the literature. Gross [15] rst proposed a simple

greedy algorithm to nd the optimal solution if the objective is convex.

Several authors have since expanded the problem. Ibarki and Katoh [16] provide

a comprehensive review of resource allocation problems and algorithms to solve

them. Their bibliography provides a review of the literature up to 1988. Bretthauer and

Shetty [6], [7] also give a survey of a generalization: the nonlinear knapsack problem.

They provide a proof of the greedy algorithm by the generalized Lagrange multiplier

method. Zaporozhets [34] gives an alternate proof of the greedy algorithm. Opp, et al.

[28] describes the greedy algorithm in detail for the convex separable resource allocation

problem and its application to our problem without priorities. Also discussed are

some computational issues associated with the application, mostly regarding the expected

queue length.

Once priorities are considered, the objective is no longer separable. We extend the

previous research by providing an algorithm to optimally solve the closed static allocation

problem with priorities. We have developed a new proof of the greedy algorithm when

there is only one priority class, and give a new algorithm to handle the special structure of

the objective when there are multiple priority classes. Finally, we investigate the bene ts

of a multi-priority structure for the manufacturer.

5

1.3 Organization of the Dissertation

In Chapter 2, we address the problem of funding a warranty reserve. In the rst two

sections, we provide an overview of the problem and the notations and assumptions used

throughout Chapters 2 and 3. In Section 2.3, we derive the probability distribution for

the number of items under warranty at time t. We follow that with di
erential equations

for the rst and second moment of the reserve level in Section 2.4. The general solutions

to these equations are provided in the following section along with the special case of

a constant warranty period. We provide a heuristic for determining the values of the

contribution amount after each sale and the initial reserve level in Section 2.6 and some

simulation results in Section 2.7.

We consider three extensions of the warranty reserve problem in Chapter 3:

 The reserve contribution after the jth sale is a random variable. (Section 3.1)

 The manufacturer maintains a single reserve fund for multiple products or multiple

warranties. (Section 3.2)

 The remaining lifetimes of the items sold prior to time t are known. (Section 3.3)

Next, we turn to the warranty repair outsourcing problem in Chapter 4. After

a brief problem overview, we state the notation and assumptions of the problem in

Section 4.2. In Section 4.3, we derive the cost function and state the optimization problem

for the model. We provide the known algorithm to solve the single priority problem in

Section 4.4 and give a new proof of the algorithm. Our algorithm to solve the m-priority

problem uses network concepts. We give a brief overview of minimum cost network ow

problems in Section 4.5. Then we reformulate the optimization problem as a convex

minimum cost ow problem and provide the algorithm to solve the problem. We provide

the simpli ed algorithm for the one- and two-priority case and give the general algorithm

for the m-priority case. In Section 4.7, we discuss the computational issues that arise in

6

the problem and provide an example in the following section. In Section 4.9, we illustrate

the cost bene ts of the priority structure. We provide two examples: the rst with very

di
erent holding costs between the high and low priority customers and the second with

relatively similar holding costs between the high and low class customers. We complete

the discussion of the outsourcing problem in Section 4.10 by presenting an optimization

problem for the manufacturer when the customer pays additional monies for priority in

service.

7

Chapter 2

Funding a Warranty Reserve

2.1 Overview

In this chapter, we consider the problem of funding a warranty reserve account. We

consider a manufacturer who adjusts its warranty reserve at a series of xed time points

(e.g. at times 0; T; 2T; : : :). In this dissertation, we consider a single period [0; T]. The

manufacturer must decide on the initial amount in the reserve at the beginning of the

period and the contribution amount from each sale. We derive the mean and variance of

the reserve level as a function of time and provide a heuristic to aid the manufacturer in

its decision.

2.2 Notation and Assumptions

We begin by introducing some notation and assumptions. We de ne R(t) as the amount

in the reserve at time t, where t = 0 represents the beginning of the period. The reserve

fund accrues interest at constant rate
> 0. At each sale, an amount c is contributed

to the account. The manufacturer must decide on the initial reserve level, R0, and the

contribution amount to the reserve from each sale, c, at the beginning of the period.

Let S(t) be the total number of sales up to time t. We assume that fS(t); t  0g is

8

a nonhomogeneous Poisson Process with a known rate function (
) (we call this an

NPP ((
))). Each item is under warranty for a random amount of time. The warranty

durations are independent and identically distributed with common cdf F(
) and mean w.

Also, the warranty durations are independent of any future failures. Note that this allows

for a constant warranty period. The customer always makes a warranty claim at each

product failure. We assume instantaneous repair and that the repair times of a given

item follow a Poisson Process with rate . The repair cost of the ith failure (at time Yi)

is Di, a random variable. The Di's are i.i.d. and are independent of the failure time. Let

D(t) be the total undiscounted cost of all claims up to time t; hence

D(t) =

X

i:Yit

Di:

Let X(t) denote the number of items under warranty at time t and Sj denote the

time of the jth sale. The manufacturer observes the number of items under warranty at

time 0 to aid in his determination of R0 and c. The manufacturer may or may not know

the remaining warranty lifetimes of the items under warranty at time 0; we consider both

cases. Figure 2.1 illustrates the evolution of the warranty reserve over time.

S Y 2 2

1 D

D2

D3

dollars

0

S

c

c

time

1 1 3 Y Y

R

Figure 2.1: Example of Warranty Reserve Account

9

For computational purposes, it is helpful to distinguish between the e
ects of the

items sold since time 0 from the items sold before time 0. We will break X(t) into two

parts: let Xn(t) represent the number of items under warranty at time t that were sold

after time 0, and let Xo(t) represent the number of items under warranty at time t that

were sold prior to time 0. We write

R(t) = Rn(t) + Ro(t);

where Rn(t) is the portion of the reserve related to the new items Xn(t), and Ro(t) is the

portion of the reserve related to the old items Xo(t). Thus, in Rn(t), we add contributions

from new purchases and only subtract the claims generated by new items. In Ro(t), there

are no new contributions, so we only subtract claims generated by old items. Similarly,

we de ne Dn(t) (Do(t)) as the total undiscounted claims from time 0 to t generated

by the new (old) items. It is convenient to de ne Rn(0) = 0 and Ro(0) = R0. In our

model we track both Rn(t) and Ro(t) for ease in computation, while the manufacturer

just tracks R(t).

We will calculate rst and second moments for some of the functions R(t), S(t),

X(t), D(t) and their components (Rn(t);Ro(t), etc.). We represent this by using lower

case for the rst moment and using lower case with a subscript of 2 for the second moment

(e.g. r(t) = E[R(t)] and r2(t) = E[R2(t)]). Any exception to this will be mentioned at

the appropriate place throughout the thesis. Also, we will use
h to indicate the change

in a function from t to t + h. For example,
hR(t) = R(t + h) �� R(t). Finally, we will

use the standard o(h) notation for a function g(h) when

lim

h!0

g(h)

h

= 0:

10

2.3 Probability Distribution of the Number of Items

Under Warranty

In this section we derive the distributions for Xn(t) and Xo(t).

2.3.1 Distribution of Xn(t)

First we explore the fXn(t); t  0g process. At time t, items are purchased according to

an NPP((
))). The amount of time an item is under warranty is a random variable with

cdf F(
). We assume there is no capacity on the total number of items under warranty

at any time. Therefore, we can model the fXn(t); t  0g process as an Mt=G=1 queue

with arrival rate (
) and service time distribution F(
).

The following result was established independently by Palm [29] and Khintchine

[21]. Most recently, Eick, Massey, and Whitt [11] provided a simpler proof of this result

and developed some further results for the Mt=G=1 queue.

Theorem 1 Let Q(t) be the number of items in an Mt=G=1 queue at time t with arrival

rate (
) and i.i.d. service times S with cdf F(
). At time t, there are 0 items in the queue.

Then, for each time point t  0, Q(t) has a Poisson distribution with mean

E

2

4

Zt

t��S

(u)du

3

5 =

Zt

0

 (t �� u) [1 �� F(u)] du:

Therefore, the moments of Xn(t) are

xn(t) =

Zt

0

 (t �� u) [1 �� F(u)] du; and (2.1)

xn

2 (t) = xn(t) + (xn(t))2 : (2.2)

11

2.3.2 Distribution of Xo(t)

We consider two possible cases for the items sold prior to time 0: either the manufacturer

fully knows the remaining warranty durations of all items under warranty at time 0 or that

the remaining warranty durations are i.i.d. random variables with common cdf Q(t). The

former case is rather easy to handle { the entire sample path of Xo(t) is a deterministic

function. If the remaining warranty durations are unknown, the probability that the

remaining warranty duration of an item is greater than t, given that it was under warranty

at time 0, is 1 �� Q(t). Hence,

Xo(t)  Bin (X(0); 1 �� Q(t)) ;

where X(0) is the number of items under warranty at time 0. The moments are

xo(t) = X(0)(1 �� Q(t)); and (2.3)

xo

2(t) = X(0)Q(t) (1 �� Q(t)) + (xo(t))2 : (2.4)

One choice for Q(t) is obtained from the stationary distribution of the remaining

service times in an M=G=1 queue in steady state. From Takacs [31], we have the

following lemma.

Lemma 1 (Takacs, Theorem 3.2.2) : Let X(t) be the number of items under warranty

at time t, and let Li(t) denote the remaining warranty period of item i under

warranty. The sales process is a Poisson process. If w < 1, we have

lim

t!1

P (Li(t) < xi 8 i = 1; : : : ; k jX(t) = k) =

Yk

i=1

1

w

Zxi

0

[1 �� F(s)] ds;

and the limiting distribution is independent of the initial state.

12

Therefore, under the assumption of Poisson input in steady state, we have that

the remaining warranty distributions are independent of each other and the probability

that an item is still under warranty at time t, given that it was under warranty at time 0

is

1 �� Q(t) =

1

w

Z1

t

[1 �� F(s)] ds; (2.5)

where Q(t) is determined by Lemma 1, i.e.

Q(t) =

1

w

Zt

0

[1 �� F(s)] ds: (2.6)

This result can be extended to the case of a non-homogeneous Poisson Process,

as shown in the following lemma.

Lemma 2 Let X(t) be the number of items under warranty at time t, and let Li(t)

denote the remaining warranty period of item i under warranty. Suppose that the sales

process begins at time ��A, and fS(t); t  ��Ag is a non-homogeneous Poisson with rate

function (
). We have

P (Li(t) < xi 8 i = 1; : : : ; k jX(t) = k) =

Yk

i=1

tR+A

0

[F(s + xi) �� F(s)] (t �� s)ds

tR+A

0

[1 �� F(s)] (t �� s)ds

:

Proof. Since the sales process is an NPP((
)), we know that for t  ��A,

P(X(t) = k) = exp

0

@��

Zt+A

u=0

(1 �� F(u))(t ��