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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.

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 [

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 [

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 Section

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

Online convex optimization, introduced by Zinkevich
[

Zinkevich exhibits an algorithm

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 period

One of the main hurdles to applying Zinkevich's algorithm
directly is that it requires full knowledge of the function

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.

In what follows, we propose a general form of profit
functions

This form of the profit function is appropriate in
modeling different market structures. For example, if

To gain some intuition on the market share function,
suppose that the warranty

Now, using the
market share function

In what follows we display the performance of

Refer to the profit functions defined in (

Firstly, we assume that

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 from

In (a) are represented several profit functions,
with

Profit functions

Warranty period

Profit

Average regret

Secondly, we assume that

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.

Profit functions

Warranty period

Profit

Average regret

We model the increase in competition in the profit
function through the parameter

In this example, the competition increases additively
from

Profit functions

Warranty period

Profit

Regret as percent

This example shows the algorithm behavior when there
is a shock increase in competition. In round

These graphs represent a shock increase in
competition. In round

Profit functions

Warranty period

Profit

Regret as percent

In this example, we study a linear increase in the
penalty from a faulty product. In specific, we alter the ratio

Additive increase in penalties.

Profit functions

Warranty period

Profit

Regret as percent

In our penalty example, the optimal warranty
approaches zero quickly. So, algorithm

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.

As mentioned earlier, we concern ourselves with profit
maximization. Thus, consider an online convex programming problem consisting of
a maximization objective, a feasible region

The feasible
region

a
closed set, that is, for any sequence

a nonempty set;

a convex set.

The profit functions are differentiable.

There exists

For all

For all

The projection
of

The regret of

A function

Assume that the
profit function

In bandit
settings, after the

The algorithm then has three main parameters that
change as the round number increases. Using the notation of Flaxman et al., the
first parameter