# Warranty Optimization in a Dynamic Environment

Authors: Nedialko B. Dimitrov; Stefanka Chukova
Journal: Journal of Applied Mathematics and Decision Sciences
 (2009)

---

## Abstract

A product warranty is an agreement offered by a producer to a consumer
to replace or repair a faulty item, or to partially or fully reimburse
the consumer in the event of a failure. Warranties are very
widespread and serve many purposes, including protection for producer,
seller, and consumer. They are used as signals of quality and
as elements of marketing strategies. In this study we review the notion
of an online convex optimization algorithm and its variations,
and apply it in warranty context. We introduce a class of profit functions,
which are functions of warranty, and use it to formulate the
problem of maximizing the company's profit over time as an online
convex optimization problem. We use this formulation to present an
approach to setting the warranty based on an online algorithm with
low regret. Under a dynamic environment, this algorithm provides
a warranty strategy for the company that maximises its profit over
time.


---

## Body

## 1. Introduction

A product warranty is an agreement offered by a
producer to a consumer to replace or repair a faulty item, or to partially or
fully reimburse the consumer in the event of a failure. Warranties are very
widespread and serve many purposes, including protection for producer, seller,
and consumer. They are used as signals of quality and as elements of marketing
strategies. A general treatment of warranty analysis is given by Blischke and
Murthy [1, 2].From the buyer's point of view, the main role of a
warranty in any business transaction is protectional. Specifically, the
warranty assures the buyer that faulty item will either be repaired or replaced
at no cost or at a reduced cost. A second role of warranty is informational, as
it implicitly sends out a message regarding the quality of the product and
could influence buyer's purchase decision.The main role of warranty from the producer's point of
view is also protectional. Warranty terms may and often do specify the use and
conditions of use for which the product is intended and provide for limited
coverage or no coverage at all in the event of misuse of the product. A second
important purpose of warranty for the seller is promotional. As buyers often
infer a more reliable product when a long warranty is offered, this has been
used as an effective advertising tool. In addition, warranty has become an
instrument, similar to product performance and price, used in competition with
other manufacturers in the marketplace.Despite the fact that warranties are so commonly used,
the study of warranties in many situations remains an open problem. This may
seem surprising since the fulfillment of warranty claims may cost companies
large amounts of money. Underestimating true warranty costs will result in
losses for a company, overestimating them will result in uncompetitive product
prices. The data relevant to the modeling of warranty costs in a particular
industry are usually highly confidential, since they are commercially
sensitive. Much warranty analysis therefore takes place in internal research
divisions in large companies.The common warranty parameters of interest analyzed
and evaluated are the expected warranty cost and the expected warranty cost per
unit time over the warranty length for a particular item as well as the
life cycle of the product; see Chukova and Hayakawa [3, 4]. Typically, the warranty
length and the warranty policy are assumed to be known, which identifies the
failure model. Based on the adopted model of the failure process, the total
expected total warranty cost and sometimes the variance of this cost are
evaluated.The study presented here deviates from the traditional
framework in warranty analysis. For simplicity we assume that the warranty is
one-dimensional and nonrenewing, that is, the warranty is identified by its
length, and it starts at the time the item is sold or the service has began. We
consider time periods, such that the manufacturer's profit functions, as
functions of warranty, and the warranty may vary from time period to time
period. In general, we assume that the optimal warranty and the profit
functions are unknown, but the profit for the assigned warranty in any
particular time period is known. The aim of this study is to present an
approach that will assure that if the warranty varies in a particular way,
suggested by an online algorithm, under reasonable, quite general assumptions
on the profit functions, the long run average of the manufacturer's profit will
be comparable with profit if the optimal warranty was known at the time
the product was launched on the market.The outline of this paper is as follows. In Section2
we present a brief overview of the online algorithm approach. The profit model
is introduced in Section 3, and it is analysed, using an online algorithm, in
Section 4. Section 5 contains concluding remarks.


## 2. Miscellaneous Results: Online Algorithm

In this paper,
we concern ourselves with profit maximization, thus we consider the online
convex programming problem with a sequence of concave functions and a
maximization objective. In its simplest form, an online convex programming
problem(F,{p1,p2,…}) consists of a
feasible region F∈ℜn and an infinite
sequence of concave “profit” functions {p1,p2,…}, each going from F to ℜ. An algorithm for the online convex programming
problem 𝒜(F,{p1,p2,…}) is an algorithm
that produces a point wi, which is a function only of the points w1,w2,…,wi−1, each previously produced by the algorithm, and the
first (i−1) functions p1,…,pi−1. The regret of an algorithm is defined
as(2.1)R(T)=(maxw*∈F∑i=1Tpi(w*))−∑i=1Tpi(wi).Interpreting, the regret
measures the performance of the algorithm, which does not know pi before
producing wi, to pick the single best point w* in the feasible
region F given knowledge
of all the pi's in advance.Online convex optimization, introduced by Zinkevich
[5], was originally
motivated by the notion of playing repeated games. Imagine playing an
infinitely repeating game that proceeds in rounds. In round i we must pick a
strategy knowing only the strategies we have chosen in the previous rounds,
from 1 to (i−1), and the payoffs we received in those rounds. That is
the motivation of the algorithm, which produces the point wi knowing only w1,w2,…,wi−1 and the first (i−1) functions p1,…,pi−1. Each wi can be thought
of as the strategy in the ith round, and each pi can be thought
of as the payoff function in the ith round. The
payoff function may change from round to round arbitrarily, since we do not
know the strategies adopted by opponents in the game. In the repeated game
settings, the regret then measures the amount of utility lost by a player who
follows the strategy as specified by the algorithm versus picking the single
best strategy to follow in all rounds.Zinkevich exhibits an algorithm𝒜f(F,{p1,p2,…}), in full-information settings (see the Appendices),
with regret R(T)=O(T), which gives(2.2)limT→∞R(T)T≤0.Interpreting, in the limit,
following the strategies specified by the algorithm produces the same per
period profit as picking the optimal single strategy. The quantity R(T)/T is commonly
referred to as the average regret.Online convex optimization has clear industrial
applications. For example, consider a company producing a product. The
company's profit could be a concave function of the warranty offered by the
company. However, the profit does not only depend on the warranty, but it could
also depend on the types of products offered by competitors or the changing
demands of customers. The profit function of the company in periodi could be
thought of as the pi in the online
convex optimization problem, and the wi could be the
warranty offered by the company in period i. An algorithm with low regret gives a warranty
strategy for the company to follow that maximizes the company's profit over
time.One of the main hurdles to applying Zinkevich's algorithm
directly is that it requires full knowledge of the functionpi after round i. In specific, Zinkevich's algorithm uses the gradient
of the function pi. However, in realistic settings, such as the
example in the previous paragraph, a company may not know the entire function pi. Instead, all the company learns in round i is the value of pi(wi). In other words, all the company learns is the amount
of profit the company made in round i, not the entire profit function. Flaxman et al. [6] exhibit an
algorithm for online convex optimization, 𝒜b(F,{p1,p2,…}), in bandit
settings (see the Appendices), using only the value pi(wi) of the profit
function of the previous round and with regret R(T)=O(T3/4).Another concern with the direct application of online
convex optimization is that the average regret results are in the limit as the
number of rounds goes to infinity. Traditional industries, such as car
manufacturing, have warranty on the order of years. Thus, even a few periods of
the repeated profit maximization may take a human lifetime. However, warranties
come in many varieties, and today's markets can be largely autonomous. For
example, consider a competition between online brokerage firms. A firm could
offer a warranty on the amount of time required to execute a purchase or sell
an order. The warranty offered could change dynamically throughout the trading
day. The broker's customers could themselves be automated programs that
dynamically choose which brokerage firm to use to execute trades. In such a
scenario it is easy to imagine thousands of profit maximization rounds per day.
Regardless of the plausibility of using online convex optimization in a
specific application, the average regret results imply the startling conclusion
that a company can attain nearly maximum profit in a dynamically changing
environment, without knowing anything about the future.In this paper, we study online convex optimization as
applied to the warranty applications described in this section.


## 3. The Profit Model

In what follows, we propose a general form of profit
functions{p1,p2,…} to be used with the two online convex optimization
algorithms 𝒜f and 𝒜b for the
warranty optimization examples in Section 2. Firstly, similarly to Bell et al. [7], we
define the market share function m(w) as a function
of warranty w as
follows(3.1)m(w)=a+gwa+gw+c,where a is a parameter
of initial “attractiveness” or “reputation” of the company, g is the increase
of the total attractiveness (a+gw) of the company
per unit increase of warranty, and c is the total
attractiveness of the competitors of the company in the marketplace. It is easy
to see that m(w) is an
increasing function of w.This form of the profit function is appropriate in
modeling different market structures. For example, ifc=0, the company has a monopoly in the marketplace,
whereas altering the value of c will model the
arrival or departure of a competitor.To gain some intuition on the market share function,
suppose that the warrantyw is zero. We
then have (3.2)m(0)=aa+c.One can think of this equation as follows. Suppose a customer picks
which company to use randomly, but with weights proportional to the company's
attractiveness. The form of m(0) in (3.2) is the
probability the customer selects to do business with our company instead of a
competitor, given that the company assigns no warranty to its products. Another
interpretation of (3.2) is that, if the company assigns no warranty to its
products it will have m(0) share of the
market. Now, consider form (3.1) and let w→∞. We have limw→∞m(w)=1, which means that if the company offers a large
warranty it will dominate the entire market.Now, using the
market share functionm(w) given in (3.1),
we introduce the profit function p(w), again as a function of warranty. We
propose(3.3)p(w)=Pm(w)−Rm(w)F(w),where P is a constant
equal to the total market value of the considered industry, R is a constant
equal to the penalty of total recall of all sold products, and F(w) is the
cumulative distribution function of the lifetime X of the product.
The latter function represents the quality and reliability of the production
and governs the process of failures and related warranty claims. We assume a
linear relationship between P and R of the
following form:(3.4)R=γP,1.0≤γ≤2.0.In the case of γ≤1, even if all products are recalled, offering a large
warranty will guarantee that the company will end up with a profit. On the
other hand, if 2.0≤γ, in order to avoid heavy penalties, the most
appropriate strategy for the company is to sell the product with no warranty.
Therefore, in both of these cases the optimal strategy of the company is known,
and we will focus our study on the nontrivial case of 1.0≤γ≤2.0.


## 4. Modeling a Dynamic Environment

In what follows we display the performance of𝒜f and 𝒜b in several
differing models of a dynamic environment. First, we present an environment
with a quality improvement under two failure scenarios: a gradual failure
modeled with an exponential lifetime distribution and a shock failure modeled
with a Weibull lifetime distribution. Second, we present an environment with
increasing competition again under two scenarios: a gradual increase in
competition and a shock increase in competition. Finally, we present an
environment where we increase the penalty for faulty products. We show that in
all these environments, the algorithms 𝒜f, and 𝒜b perform well as
compared to algorithm “opt_fixed” which selects a single, optimal warranty
for all rounds, even though neither 𝒜f nor 𝒜b know the future
profit functions. As algorithm 𝒜b is a randomized
algorithm and its theoretical guarantees are in expectation, in each scenario
we present the expected behavior of 𝒜b over 50 independent
runs. In addition, we include in the comparison the algorithm “opt_round”
that selects the optimal warranty in each round.


### 4.1. Environment with Quality Improvement

Refer to the profit functions defined in (3.3). Our next
goal is to use these functions for decision making related to warranty, in
environment with quality improvement. We model the dynamic environment with
quality improvement by using the cumulative distribution function FX(w) of the lifetime X of the product.
We consider two cases.Case 1.
Firstly, we assume thatX~Exp(λ), that is,(4.1)FX(w)=1−e−λw,and the mean time to failure is E(X)=1/λ. Based on (4.1) we define a sequence of profit
functions as(4.2)pi(w)=Pm(w)−Rm(w)(1−e−λiw),and use them in full-information
settings, that is, with 𝒜f(F,{p1,p2,…}) as well as in
bandit settings, that is, with 𝒜b(F,{p1,p2,…}). The results are presented in Figure 1.
In this example, we model quality improvement by
additively increasing the parameter of the exponential distribution
representing the lifetime of the product. The mean of the distribution changes
linearly from4 to 8. The resulting profit functions are presented in
Figure 1(a). As you can see, in later rounds, as the quality increases, the
company can offer a larger warranty to capture a larger fraction of the market
and thus receive higher profit. Figure 1(b) shows the warranty offered by the
various algorithms. Algorithm 𝒜f starts by
offering a zero warranty, the imposed initial starting point, and follows an
upward trend as the rounds increase. Algorithm 𝒜b has an initial
starting point, dictated by the algorithm itself, around the middle of the
feasible region. In all our examples, the feasible region is {w∣0≤w≤15}, that is, the acceptable warranty is between 0 and 15. That is why, initially, the warranty of 𝒜b decreases from 7.5 and then
increases as the rounds increase. Figure 1(c) shows the profit earned in each
round by each algorithm. The figure illustrates the benefits of using 𝒜b, as it closely
follows the profit received by optimizing the warranty in each round, but it
assumes very limited information of the profit functions. Figure 1(d) shows how
the average regret of 𝒜b decreases to
zero as the rounds increase. In other words, the per period loss of 𝒜b as compared to
following the optimal fixed warranty decreases to zero as the rounds increase.
Even better results are pictured in Figure 1(d) for 𝒜f; however, it
assumes knowledge of the gradient of the profit function in each round, where
as 𝒜b only assumes
knowledge of the evaluation of the profit at a single point.In (a) are represented several profit functions,
withpi representing
the profit function in round i. In (b) is represented the warranty offered,
as a function of the round number, by various algorithms. (c) represents
the profit earned, as a function of the round number, by each algorithm.
Finally, (d) represents the average regret of the two algorithms, 𝒜f and 𝒜b. The algorithm “opt_fixed” selects a single,
optimal warranty for all rounds. The algorithm “opt_round” selects the
optimal warranty in each round.
(a)
Profit functions(b)
Warranty period(c)
Profit(d)
Average
regretCase 2.
Secondly, we assume thatX~Weibull(γ), that is,(4.3)FX(w)=1−e−wγ,and the mean time to failure is E(X)=Γ(1+1/γ) and create the
sequence of profit functions(4.4)pi(w)=Pm(w)−Rm(w)(1−e−wiγ). In this
example, we introduce quality improvement with a Weibull lifetime distribution.
In a Weibull distribution, there is a sharp threshold at which most products
fail. That is why in Figure 2(a) the profit functions fall sharply as the
warranty increases. Figure 2(c) represents the profit of the various algorithms.
Notice that the profit for “opt_fixed” begins negatively and sharply
increases as the rounds increase. This is because the single warranty chosen by
“opt_fixed” in early rounds is greater than the failure threshold of the
product, but is less than the failure threshold in later rounds. The profit
earned by 𝒜b is negative in
early rounds, since 𝒜b begins with an
initial point in the middle of the feasible region, which is much larger than
the failure threshold of the initial Weibull distributions. As the rounds
increase, 𝒜b decreases the
warranty, as pictured in Figure 2(b). Since both the failure threshold increases
and 𝒜b decreases the
warranty in later rounds, 𝒜b eventually
begins to make a profit. In late rounds, 𝒜b begins to
approach the performance of 𝒜f and
“opt_round”, which outperform “opt_fixed”, since they can increase the
warranty as the failure threshold increases. As expected, since 𝒜f outperforms
”opt_fixed”, the average regret for 𝒜f, pictured in
Figure 2(d), is negative. The average regret for 𝒜b increases in
the early rounds, while 𝒜b is making poor
profit, and quickly decreases in the later rounds.These figures represent the algorithms' performance
with a Weibull lifetime distribution. In a Weibull lifetime distribution, there
is a sharp threshold before which most products are functioning properly and
after which most products have failed. That is why in (a) we see profit
functions that fall sharply as the warranty increases.
(a)
Profit functions(b)
Warranty period(c)
Profit(d)
Average regret


### 4.2. Environment with Increasing Competition

We model the increase in competition in the profit
function through the parameterc included in the
market share function (3.1). In this example, we additively increase the
competition from 2 to 50, with the parameter a set to 1. Interpreting, this means that initially the company
has roughly 1 to 2 odds of
attracting a customer. Toward the final round, the company has only 1 to 50 odds of
attracting a customer, thus the competition has increased. Figure 3(b), through
the graph for “opt_round”, shows that the warranty that should be offered by
the company increases as competition increases. This is to capture a larger
fraction of the market as dictated by expression (3.1). The warranty of 𝒜b decreases
throughout, as it initially begins at the middle of the feasible region.
Algorithm 𝒜f, on the other
hand, begins initially with a warranty of zero and closely follows the
performance of “opt_round”. Figure 3(d) shows that 𝒜b looses less
than 15% of the total
profit at the end of the example. This percentage would decrease to zero as the
rounds go to infinity, by the results of Flaxman et al. [6].In this example, the competition increases additively
from2 to 50. As the competition increases, the company's share of
the market decreases and so does the profit as shown in (a). (d),
shows the regret in round i as a percentage
of the total profit received by “opt_fixed” from rounds 0 through i. It illustrates that the theoretical results showing
that the average regret tends to zero as the rounds tend to infinity translate
into results showing that the percent of profit missed by 𝒜b and 𝒜f tends to zero
as the round tends to infinity.
(a)
Profit
functions(b)
Warranty period(c)
Profit(d)
Regret as percentThis example shows the algorithm behavior when there
is a shock increase in competition. In round2000, the arrival of a competitor decreases the market
share of the company significantly; the value of c jumps from 2 to 50. This leads to the sudden drop pictured in Figure 4(c),
for all warranty settings, even for “opt_round”. Though the profit for all
algorithms has a sudden drop, it is interesting to see the algorithm's reaction
in changing the warranty, pictured in Figure 4(b). Again, due to the different
information settings of 𝒜f and 𝒜b,
algorithm 𝒜f is near the
optimal setting before the competition increase and needs a short time to
readjust after the increase. On the other hand, 𝒜b begins, as
usual, in the middle of the feasible region and is decreasing the warranty
toward the optimal setting in the initial rounds. After the competition
increase, 𝒜b continues to
decrease the warranty but at a slower pace. Even though the warranty offered
by 𝒜b seems far from
the warranty offered by the other algorithms, Figure 4(d) shows that its regret,
as a percentage of the total profit gained by “opt_fixed”, is once again
steadily decreasing toward zero.These graphs represent a shock increase in
competition. In round2000, the parameter c in the market
share increases from 2 to 50.
(a)
Profit functions(b)
Warranty period(c)
Profit(d)
Regret as percent


### 4.3. Environment with Changeable Penalties

In this example, we study a linear increase in the
penalty from a faulty product. In specific, we alter the ratioγ between P and R in the profit
function (3.3) from 1 to 2. A larger γ models a larger
cost to replace a failed item. As can be seen in Figure 5(a), as the penalty
for a faulty product increases, the optimal warranty goes to zero. Figure 5(b)
shows how the warranty of 𝒜f starts at zero,
increases until it passes the optimal warranty offered by “opt_round”, and
decreases back toward zero. It can also be seen that 𝒜b starts with a
warranty of 7.5 and decreases
toward zero. Figure 5(c) shows a similar performance of “opt_round”,
“opt_fixed”, and 𝒜f. In that
figure, it is clear that 𝒜b starts with a
poor performance, but in the long run approaches the performance of
“opt_fixed”. The graph in Figure 5(d) can be explained through understanding
the performance of algorithm 𝒜b, which is
outlined in the appendices.Additive increase in penalties.
(a)
Profit functions(b)
Warranty period(c)
Profit(d)
Regret as percentIn our penalty example, the optimal warranty
approaches zero quickly. So, algorithm𝒜b cannot set a
warranty close to zero because of the algorithm's projection to a subset of the
feasible region. As the algorithm's parameter α approaches
zero, 𝒜b can set a
warranty closer and closer to zero. Thus, we can expect the regret shown in
Figure 5(d) to decrease toward zero at a speed of O(n6), matching that of the parameter α.


## 4.1. Environment with Quality Improvement

Refer to the profit functions defined in (3.3). Our next
goal is to use these functions for decision making related to warranty, in
environment with quality improvement. We model the dynamic environment with
quality improvement by using the cumulative distribution function FX(w) of the lifetime X of the product.
We consider two cases.Case 1.
Firstly, we assume thatX~Exp(λ), that is,(4.1)FX(w)=1−e−λw,and the mean time to failure is E(X)=1/λ. Based on (4.1) we define a sequence of profit
functions as(4.2)pi(w)=Pm(w)−Rm(w)(1−e−λiw),and use them in full-information
settings, that is, with 𝒜f(F,{p1,p2,…}) as well as in
bandit settings, that is, with 𝒜b(F,{p1,p2,…}). The results are presented in Figure 1.
In this example, we model quality improvement by
additively increasing the parameter of the exponential distribution
representing the lifetime of the product. The mean of the distribution changes
linearly from4 to 8. The resulting profit functions are presented in
Figure 1(a). As you can see, in later rounds, as the quality increases, the
company can offer a larger warranty to capture a larger fraction of the market
and thus receive higher profit. Figure 1(b) shows the warranty offered by the
various algorithms. Algorithm 𝒜f starts by
offering a zero warranty, the imposed initial starting point, and follows an
upward trend as the rounds increase. Algorithm 𝒜b has an initial
starting point, dictated by the algorithm itself, around the middle of the
feasible region. In all our examples, the feasible region is {w∣0≤w≤15}, that is, the acceptable warranty is between 0 and 15. That is why, initially, the warranty of 𝒜b decreases from 7.5 and then
increases as the rounds increase. Figure 1(c) shows the profit earned in each
round by each algorithm. The figure illustrates the benefits of using 𝒜b, as it closely
follows the profit received by optimizing the warranty in each round, but it
assumes very limited information of the profit functions. Figure 1(d) shows how
the average regret of 𝒜b decreases to
zero as the rounds increase. In other words, the per period loss of 𝒜b as compared to
following the optimal fixed warranty decreases to zero as the rounds increase.
Even better results are pictured in Figure 1(d) for 𝒜f; however, it
assumes knowledge of the gradient of the profit function in each round, where
as 𝒜b only assumes
knowledge of the evaluation of the profit at a single point.In (a) are represented several profit functions,
withpi representing
the profit function in round i. In (b) is represented the warranty offered,
as a function of the round number, by various algorithms. (c) represents
the profit earned, as a function of the round number, by each algorithm.
Finally, (d) represents the average regret of the two algorithms, 𝒜f and 𝒜b. The algorithm “opt_fixed” selects a single,
optimal warranty for all rounds. The algorithm “opt_round” selects the
optimal warranty in each round.
(a)
Profit functions(b)
Warranty period(c)
Profit(d)
Average
regretCase 2.
Secondly, we assume thatX~Weibull(γ), that is,(4.3)FX(w)=1−e−wγ,and the mean time to failure is E(X)=Γ(1+1/γ) and create the
sequence of profit functions(4.4)pi(w)=Pm(w)−Rm(w)(1−e−wiγ). In this
example, we introduce quality improvement with a Weibull lifetime distribution.
In a Weibull distribution, there is a sharp threshold at which most products
fail. That is why in Figure 2(a) the profit functions fall sharply as the
warranty increases. Figure 2(c) represents the profit of the various algorithms.
Notice that the profit for “opt_fixed” begins negatively and sharply
increases as the rounds increase. This is because the single warranty chosen by
“opt_fixed” in early rounds is greater than the failure threshold of the
product, but is less than the failure threshold in later rounds. The profit
earned by 𝒜b is negative in
early rounds, since 𝒜b begins with an
initial point in the middle of the feasible region, which is much larger than
the failure threshold of the initial Weibull distributions. As the rounds
increase, 𝒜b decreases the
warranty, as pictured in Figure 2(b). Since both the failure threshold increases
and 𝒜b decreases the
warranty in later rounds, 𝒜b eventually
begins to make a profit. In late rounds, 𝒜b begins to
approach the performance of 𝒜f and
“opt_round”, which outperform “opt_fixed”, since they can increase the
warranty as the failure threshold increases. As expected, since 𝒜f outperforms
”opt_fixed”, the average regret for 𝒜f, pictured in
Figure 2(d), is negative. The average regret for 𝒜b increases in
the early rounds, while 𝒜b is making poor
profit, and quickly decreases in the later rounds.These figures represent the algorithms' performance
with a Weibull lifetime distribution. In a Weibull lifetime distribution, there
is a sharp threshold before which most products are functioning properly and
after which most products have failed. That is why in (a) we see profit
functions that fall sharply as the warranty increases.
(a)
Profit functions(b)
Warranty period(c)
Profit(d)
Average regret


## 4.2. Environment with Increasing Competition

We model the increase in competition in the profit
function through the parameterc included in the
market share function (3.1). In this example, we additively increase the
competition from 2 to 50, with the parameter a set to 1. Interpreting, this means that initially the company
has roughly 1 to 2 odds of
attracting a customer. Toward the final round, the company has only 1 to 50 odds of
attracting a customer, thus the competition has increased. Figure 3(b), through
the graph for “opt_round”, shows that the warranty that should be offered by
the company increases as competition increases. This is to capture a larger
fraction of the market as dictated by expression (3.1). The warranty of 𝒜b decreases
throughout, as it initially begins at the middle of the feasible region.
Algorithm 𝒜f, on the other
hand, begins initially with a warranty of zero and closely follows the
performance of “opt_round”. Figure 3(d) shows that 𝒜b looses less
than 15% of the total
profit at the end of the example. This percentage would decrease to zero as the
rounds go to infinity, by the results of Flaxman et al. [6].In this example, the competition increases additively
from2 to 50. As the competition increases, the company's share of
the market decreases and so does the profit as shown in (a). (d),
shows the regret in round i as a percentage
of the total profit received by “opt_fixed” from rounds 0 through i. It illustrates that the theoretical results showing
that the average regret tends to zero as the rounds tend to infinity translate
into results showing that the percent of profit missed by 𝒜b and 𝒜f tends to zero
as the round tends to infinity.
(a)
Profit
functions(b)
Warranty period(c)
Profit(d)
Regret as percentThis example shows the algorithm behavior when there
is a shock increase in competition. In round2000, the arrival of a competitor decreases the market
share of the company significantly; the value of c jumps from 2 to 50. This leads to the sudden drop pictured in Figure 4(c),
for all warranty settings, even for “opt_round”. Though the profit for all
algorithms has a sudden drop, it is interesting to see the algorithm's reaction
in changing the warranty, pictured in Figure 4(b). Again, due to the different
information settings of 𝒜f and 𝒜b,
algorithm 𝒜f is near the
optimal setting before the competition increase and needs a short time to
readjust after the increase. On the other hand, 𝒜b begins, as
usual, in the middle of the feasible region and is decreasing the warranty
toward the optimal setting in the initial rounds. After the competition
increase, 𝒜b continues to
decrease the warranty but at a slower pace. Even though the warranty offered
by 𝒜b seems far from
the warranty offered by the other algorithms, Figure 4(d) shows that its regret,
as a percentage of the total profit gained by “opt_fixed”, is once again
steadily decreasing toward zero.These graphs represent a shock increase in
competition. In round2000, the parameter c in the market
share increases from 2 to 50.
(a)
Profit functions(b)
Warranty period(c)
Profit(d)
Regret as percent


## 4.3. Environment with Changeable Penalties

In this example, we study a linear increase in the
penalty from a faulty product. In specific, we alter the ratioγ between P and R in the profit
function (3.3) from 1 to 2. A larger γ models a larger
cost to replace a failed item. As can be seen in Figure 5(a), as the penalty
for a faulty product increases, the optimal warranty goes to zero. Figure 5(b)
shows how the warranty of 𝒜f starts at zero,
increases until it passes the optimal warranty offered by “opt_round”, and
decreases back toward zero. It can also be seen that 𝒜b starts with a
warranty of 7.5 and decreases
toward zero. Figure 5(c) shows a similar performance of “opt_round”,
“opt_fixed”, and 𝒜f. In that
figure, it is clear that 𝒜b starts with a
poor performance, but in the long run approaches the performance of
“opt_fixed”. The graph in Figure 5(d) can be explained through understanding
the performance of algorithm 𝒜b, which is
outlined in the appendices.Additive increase in penalties.
(a)
Profit functions(b)
Warranty period(c)
Profit(d)
Regret as percentIn our penalty example, the optimal warranty
approaches zero quickly. So, algorithm𝒜b cannot set a
warranty close to zero because of the algorithm's projection to a subset of the
feasible region. As the algorithm's parameter α approaches
zero, 𝒜b can set a
warranty closer and closer to zero. Thus, we can expect the regret shown in
Figure 5(d) to decrease toward zero at a speed of O(n6), matching that of the parameter α.


## 5. Conclusions and Future Research Directions

In this paper we have presented a framework for
analysis of warranty using an online convex optimization algorithm. We have
introduced a class of profit functions that can be used to model a competitive
market with warranties. We have shown that under incomplete information
regarding the future changes in the environment, the decision maker could
choose a warranty strategy that achieves a profit similar to the profit, that
could have been generated by the unknown optimal warranty. In specific, we use
the results of Zinkevich and Flaxman et al. to exhibit strategies achieving
near optimal profits, that is, strategies with regret approaching zero in a long
term. We exhibit several settings of changing environment and show that in each
of these, the online algorithms can provide a reasonable support in warranty-related decision making.This study demonstrates that it is feasible for a
company to maximize profit through adjusting warranty in a dynamic environment,
without knowledge of the current or future market conditions. However, the
algorithms presented here do have explicit limitations that should be noted
before use in a real environment. First, as most optimization algorithms, the
algorithms presented in the paper are guaranteed to work for convex objective
functions. However, if the profit function of the company is not convex, it is
possible for the algorithm to get stuck in a local optimum. Furthermore, as
mentioned earlier, some products, such as cars, may not be appropriate for use
with these algorithms because of the real-time length of a round, which is on
the order of years. As demonstrated, specifically for the bandit algorithm, a
large number of rounds are required to approach the optimal warranty period.Furthermore, we are able to identify two possible
directions for further research. One option is to focus on reducing the
limitations of the used online algorithms. It would be interesting to see if
these algorithms can be coupled with existing algorithms for avoiding local
optima. For example, is it possible to pair the bandit algorithm with simulated
annealing? What would such a pairing do to the regret guarantees of the
original bandit algorithm? Would such a pairing deliver good performance in
avoiding local optima? Another possible direction for further research is to
try to apply our results to a real data; related to the performance of
brokerage firms. Firstly, it will be challenging to find the appropriate set of
real data. Moreover, it would be interesting to come up with a method for
estimating the parameters of the profit function from real data; parameters
such as the total market size, the failure CDF, and the market share as a
function of warranty period. Such an estimation would make it possible to
investigate the application of these algorithms in a realistic situation.



---

*Source: 414507-2009-04-15.xml*