A Prorata
Warranty Model for NonRepairable Products
In the current
consumer market, a product’s warranty is one of the important
factors in the consumer’s decisionmaking process. From vehicles
to home appliances, when several products perform similar
functions, customers usually prefer to purchase the product that
provides the better warranty. In this article, we will discuss
several warranty policies. One example will be given for
calculating the predicted warranty cost.
A warranty is a contract or an
agreement under which the manufacturer of a product or service
must agree to repair, replace or provide service when the
product fails or the service does not meet the customer’s
requirement before a specified time (warranty length) [1].
There are three commonlyused warranty policies: the ordinary
free replacement warranty, the
unlimited free replacement warranty and the prorata
warranty.
Ordinary Free Replacement
Warranty: Under this warranty, if a product fails before the
end of the warranty period because of quality or reliability
issues, it will be replaced or repaired at no cost to the
customer. The repaired product is then covered by an ordinary
free replacement warranty. The length of the warranty for the
repaired product is equal to the remaining length of the
original warranty. The ordinary free warranty is used by many
vehicle and home appliance companies.
Unlimited Free Replacement
Warranty: Under this warranty, if a product fails before the
end of the warranty period, it will be replaced or repaired at
no cost to the customer. The repaired product is then covered by
a new unlimited free replacement warranty. The length of the new
warranty is equal to the length of the original one. The
unlimited free replacement warranty is used for small electronic
appliances that have high early failure rates. The length of the
unlimited free replacement warranty usually is short.
Prorata Warranty: Under
this warranty, if an item fails before the end of the warranty
period, it is replaced at a cost that depends on the age of the
item at the time of failure. The replacement item is then
covered by an identical new warranty. This type of warranty
sometimes is also called partial warranty, since only a part of
the initial cost is covered. It is usually used for
nonrepairable items such as tires and batteries.
If a product is covered by a
warranty, the manufacturers need to decide the length of the
warranty and predict the warranty cost. Sometimes the warranty
length is affected by the competitors on the market. For
example, no one will buy a new car with only a 1 year limited
warranty since many cars are covered by 5, 7 or even 10 year
warranty assurances. Once the warranty policy has been decided,
the amount of capital that must be allocated to cover the future
warranty cost needs to be predicted. In the following section,
we use an example to illustrate how to predict the warranty cost
for the prorata warranty policy.
Example:
A prorata warranty is applied to a product. Assume that the
unit price before adding the warranty cost is $50 and the
warranty length is five years or 1825 days. It is also known
that the failure time distribution is a Weibull distribution
with a shape parameter of 1.2 and a scale parameter of 5,600
days. What should be the unit price in order to cover the
warranty cost?
Let’s define the following
variables:
 c': unit price
before adding the warranty cost
 r: expected
warranty cost per unit
 c: unit price after adding
the warranty cost, c = c’ + r
 N(t): number of
failure at time
t
 L: product lot size
for warranty cost determination
 w: duration of the
warranty period
 C(t): prorata
customer cost at time
t, C(t)=c[1(t/w)]
 T_{c}:
total warranty cost of a lot of size
L
The probability density
function (pdf) for a Weibull distribution is [2]:
and the cumulative distribution
function (cdf) is:
The expected number of failures
for the whole lot by time t is:
.
The total number of failures in
the interval from t to t+dt is:
.
The expected total cost for the
failures from t to t+dt is:
.
So, the total expected cost for
the warranty period is:
The warranty cost per unit is:
For this example:
From the above equation, we can
solve for the warranty cost per unit r = 6.117. For the
integral, a mathematic software package such as MathCAD can be
used to get the result.
Therefore, to cover the
warranty cost, the price per unit should be:
If the lot size is 10,000, the
estimated warranty cost will be $61,170. For different warranty
lengths, the calculated unit price is given in the following
table.
Warranty Length (Year)

Unit Price ($) 
2.00 
51.99 
2.50 
52.64 
3.00 
53.27 
3.50 
53.94 
4.00 
54.65 
4.50 
55.37 
5.00 
56.12 
5.50 
56.89 
6.00 
57.67 
6.50 
58.48 
7.00 
59.30 
7.50 
60.15 
8.00 
61.01 
8.50 
61.89 
9.00 
62.78 
9.50 
63.69 
10.00 
64.62 
This is also shown in the
following figure.
Figure 1: Unit Price for Different Warranty Lengths
If customers will not buy the
product for more than $60, then the warranty length should not
be longer than 7 years.
The above example also can be
solved using simulation. You can use ReliaSoft's
RENO,
Weibull++ or
BlockSim
software to obtain the desired results. The following steps will
show how to estimate the warranty cost in BlockSim.
First, create a block with the
following settings:
Second, conduct the simulation
with the End Time = 1825.
Make sure you save the
simulation results by selecting
Save Log of Simulations on the Display/Other Settings
page, as shown next.
Click Select Results to
save the Time to the First Failure (TTFF).
The saved failure times will
look like the file shown next.
For each simulation run (the
seed column), you can calculate the prorata time using 1  t_{i}
/1825. If the item fails, t_{i} is the failure
time. Otherwise, it is the suspension time 1825.
where K is the number of
simulations and A denotes the entire equation. As the
number of simulations increases, the value of A approaches the
analytical solution:
Using the above simulation
settings, we get A = 0.115, while the analytical solution is
0.109. The warranty cost per unit is calculated by:
Solving the above equation, we
get
r = 6.497. The unit price for the product after adding
the warranty cost is $56.497, which is very close to the
analytical solution of $56.117.
Conclusion
In this article, we discussed three common kinds of warranty
policies, which are typically used for different types of
products. We used an example to calculate the expected warranty
cost for a prorata warranty policy and we provided both
analytical and and simulation results. The analytical solution
requires solving a complex integral, so it can be cumbersome for
engineers to use. Simulation is a simple and easy method for
calculating warranty costs that can provide relatively accurate
results. Note that the ordinary free replacement warranty policy
and the unlimited free replacement warranty policy are quite
simple and also can be easily modeled in BlockSim.
References
[1] Elsayed, E. A. Reliability Engineering,
Massachusetts: Addison Wesley, 1996.
[2] ReliaSoft Corporation, Life Data Analysis
Reference, Tucson, AZ: ReliaSoft Publishing, 2007.
