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A methodology for design for
warranty with focus on reliability
and warranty policies
Prashant M. Ambad and Makarand S. Kulkarni
Mechanical Engineering Department, Indian Institute of Technology Delhi,
New Delhi, India
Abstract
Purpose – The purpose of the paper is to develop a conceptual framework that integrates the
technology and commercial issues early at the design stage to minimize warranty costs in the most
effective and efficient manner and also to develop a model for optimization of warranty with specific
focus on reliability and warranty policies.
Design/methodology/approach – The critical issues in warranty are addressed which affect the
warranty cost. An optimization model to achieve multiple goals like minimization of the warranty cost
and improving the reliability of the product is developed using genetic algorithm as a solution
methodology. The model is illustrated with a real case of automobile engine.
Findings – The results of the optimization show improvement in mean time between failures (MTBF)
which results due to improvement in the product reliability and also the targeted warranty cost is
achieved.
Research limitations/implications – The model developed needs to be further extended with
inclusion of additional decision variable such as support level offered and more objectives such as
attractiveness of the warranty from the customer’s view point and spares cost to the customer.
Originality/value – The paper provides the help to the designers at the design stage to take the
decisions related to warranty in deciding the warranty parameters.
Keywords Warranty policies, Reliability, Mean time between failures, Warranty optimization,
Warranty cost, Genetic algorithms
Paper type Research paper
1. Introduction
The complexity of the products in the recent past has increased significantly to meet
the ever increasing needs and expectations of consumers. In the purchase decision of a
product, buyers typically compare characteristics of comparable models of different
brands available in the market. When competing brands are nearly identical, it is very
difficult, in many instances, to choose a particular product solely on the basis of the
product-related characteristics such as price, features, product quality, finance offered
by the manufacturer, and so on. In such situations, post-sale factors – warranty,
support level, maintenance, spare parts cost and their availability, etc., are important in
the choice of the product. Of these, warranty is a one of the most influential factors that
is known to the buyer at the time of purchase.
A failure can occur early in an item’s life due to manufacturing defects or at a later
time in its life due to degradation which is dependent on age and usage. Consumers
need assurance that the product they purchase will be able to carry out its intended
function over a period of time and protected against product failures in the early
phases of product usage. Manufacturers provide the protection by offering warranty.
A warranty is a contract between the buyer and the manufacturer associated
with the sale of a product. Warranty typically specifies the performance that is to be
The current issue and full text archive of this journal is available at
www.emeraldinsight.com/0972-7981.htm
Journal of Advances in Management
Research
Vol. 10 No. 1, 2013
pp. 139-155
rEmerald Group Publishing Limited
0972-7981
DOI 10.1108/09727981311327811
139
Design for
warranty
expected from the product, conditions of use and the rectification available to the buyer
in case if failure occurs.
A warranty of any type, since it involves an additional service associated with a
product, will lead to potential costs to the manufacturer beyond those associated with
the design, manufacture and sale of the product. The warranty servicing costs may
vary from 2 to 10 percent of sale price depending on the product and the manufacturer
( Murthy, 2007). The warranty cost strongly depends on the reliability of the product
which in turn depends upon several factors, some of which are controlled by the
manufacturer, such as the decisions made during the design and development stage.
Some of the factors are related to the consumer such as the product usage pattern and
the operating environment and maintenance.
To address the issues related to warranty, there is a need to develop a framework
which addresses the critical parameters which affect the decisions related to warranty
and integrate the issues in such a way that the warranty cost can be minimized.
In the present study, a warranty optimization model is developed considering
multiple target goals like:
(1) limiting the warranty cost to a certain proportion of the engine price; and
(2) improving the mean time between failures (MTBF) to a certain proportion as
compared to existing one in order to improve the reliability of the engine.
The decision variables considered are component alternatives, type of warranty policy
and warranty duration. The model is demonstrated using a case study of an engine
manufacturer. The optimization is carried out using genetic algorithm (GA) approach.
The outline of the paper is as follows. Section 2 reviews the literature related to
the parameters affecting warranty cost and warranty optimization, Section 3 proposes
a conceptual framework that links warranty parameters and warranty cost. In Section
4, a warranty optimization model is developed. Finally, paper concludes with remarks
in Section 5.
2. Literature review
Depending upon the type of product, an appropriate warranty policy needs to be
selected. A lot of research has been carried on for selection of an appropriate warranty
policy and warranty length by considering either the manufacturer’s or the buyer’s
point of view. Offering an attractive warranty policy is typically costly, especially for
products that deteriorate quickly.
Blischke and Murthy (1992) have proposed a taxonomy for warranty policies for
new products and grouped these policies into number of categories. Murthy and
Chattopadhyay (1999) have developed policies and taxonomy for second hand
products. Murthy and Blischke (2006) have discussed in detail about the warranty
polices and the associated warranty cost analysis. Over the last decade, manufacturers
have started offering extended warranty which provides the customer with additional
protection beyond the normal warranty at an additional cost. The popularity of
extended warranty has resulted in third parties providing these services.
Reliability of a product conveys the concept of dependability, successful operation or
performance and the absence of failures. Failures over the warranty period are closely
linked to product reliability. The reliability of a product gets determined by the
decisions made during the pre-production stages (frontend, design and development)
as well as the production stage of the product life cycle ( Blischke and Murthy, 2000).
Murthy et al. (2008, 2009) have dealt with reliability decision making during the
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front-end (or feasibility) and the design and development stages of new product
development. Murthy (2007) has given a brief overview of reliability as well as
warranty and discussed some new issues and the challenges for future research.
Doganaksoy et al. (2006) have described methods of implementing quality control
techniques for various processes in business enterprises and to improve reliability of
equipment, based on analysis of data pertaining to warranty.
Preventive maintenance (PM) over the warranty period has a greater impact on the
warranty servicing cost. It is worthwhile to carryout maintenance as it affects
the overall health of the product in future. Murthy and Jack (2003) have reviewed the
literature pertaining to warranty and maintenance and suggested areas for future
research. Park and Pham (2012) developed warranty cost models considering a
periodic PM policy with both corrective maintenance and PM and also determines
three decision variables including warranty period, repair time limit and periodic
maintenance cycles. Huang and Yen (2009) have developed a two-dimensional
warranty model in which the customer is expected to perform appropriate PM. The
warranty policy that maximizes the manufacturers’ profits is determined. Ben-Daya
and Noman (2006) have developed an integrated model that simultaneously considers
inventory production decisions, PM schedule and warranty policy for a deteriorating
system that experiences shifts leading to an out of control state. Jung and Park (2003)
have developed an approach for optimal periodic PM policies following the expiration
of warranty by minimizing the expected long-run maintenance cost per unit time.
Djamaludin et al. (2001) have developed a framework to study warranty and maintenance.
Kim et al. (2004) have proposed a model to determine discrete time instants when PM
actions are to be carried out over the warranty period.
Warranty logistics deals with various issues relating to the servicing of warranty.
Proper management of warranty logistics is needed not only to reduce the warranty
servicing cost but also to ensure customer satisfaction. Dı
´az et al. (2012) addressed the
problem of improving warranty management programs through logistic support
planning and presented a framework for the military industry in which logistic
support strategies are widely applied. Murthy et al. (2004) have linked the literature on
warranty and on logistics and then discussed the different issues in warranty logistics.
Blanchard et al. (1995) have dealt with maintenance management and Blanchard (1998)
with some related logistical issues. Loomba (1996) discussed the linkage between
product distribution and service support channels.
In the literature, the optimization of warranty is carried out in which objective
functions for optimization are the minimization of the expected warranty cost and
price, maximization of the expected profit per product and market share, improvement
of reliability ( Mitra and Patankar, 1988, 1997; Yun, 1997; Liu et al., 2006; Lu and
Chiang, 2008; Wu et al., 2009; Saidi-Mehrabad et al., 2010; Shafiee et al., 2011; Shafiee
and Zuo, 2011; Park and Pham, 2012; Faridimehr and Niaki, 2012). The most commonly
used constraints in the literature are related to warranty reserves and life cycle costs
(Kleyner and Sandborn, 2008; Mitra and Patankar, 1990; Painton and Campbell, 1995).
The decision variables typically used are type and duration of the warranty policy,
reliability of the product and selling price ( Mohan et al., 2009; Wang et al., 2010; Huang
et al., 2007; Rai and Singh, 2005; Chun and Tang, 1999; Blischke and Vij, 1996).
Goal programming has been used by a number of authors in the literature to solve
the single as well as multi objective problems ( Mahdavi et al., 2009; Javadi et al., 2008).
Different meta-heuristic methods like GA, simulated annealing are used as solution
techniques for solving the optimization problems. Monga and Zuo (1998) presented a
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study on reliability based design of a series-parallel system and used GA to obtain
optimal values of system design, burn-in period for different lengths of warranty,
PM intervals and replacement time. Deb (1999, 2001) and Hu et al. (2007) used
multi-objective GA to solve the goal programming problems.
Some of the literature also focussed on achieving customer satisfaction through
improvements in the warranty parameters by making changes in the design. Manna
et al. (2006) and Maronick (2007) focussed on maximization of customer’s utility in
terms of warranty duration for the different warranty policies. Researchers also
focussed on determination of the customer’s satisfaction in terms of warranty claims
after making improvements in the design ( Majeske and Herrin, 1998; Yang and
Zaghati, 2002; Jack and Murthy, 2004; Lassar et al., 1998). Price and Dawar (2002)
and Noll (2004) have examined warranty as a signal for quality and studied its impact
from the customer’s perspective.
As can be seen from the above review, minimization of warranty cost is a
multidimensional problem. It is studied either as a single objective or multi-objective
optimization problem. In addition to this, the complexity of the problem is very high
due to the way in which these parameters affect warranty cost.
In the next section, we present a conceptual framework for modeling the impact
of some of the most important parameters discussed in the present section on the
warranty cost and eventually on profits. Subsequently, a warranty optimization model
is also developed with specific focus on reliability and warranty polices.
3. The proposed conceptual framework
From the business perspective, there can be multiple goals such as market share, total
profits, etc. Warranties not only impact total sales in a positive manner, but also impact
warranty cost and profit margin. Figure 1 shows the link between warranty
parameters and warranty cost.
As shown in Figure 1, four decisions namely warranty policy, product reliability,
maintenance and warranty logistics are important from warranty point of view.
The warranty policy related decisions decide the type and duration of warranty.
Design
decisions
Warranty
policy
Product
reliability
Maintenance
Warranty
logistics
Revenue
Profit Need for
improvement
Customer
satisfaction
Brand
image
Warranty
cost
Warranty
claims
Sales
Price
Figure 1.
Conceptual framework
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Reliability-related decisions reflect the expected useful life of the component, which is
through the appropriate choice of a component alternative. The maintenance decisions
are mainly concerned with the type of maintenance to be carried out during the
warranty period. The warranty logistics decisions are related to deciding the service
delivery network and management of spares. The details about the warranty related
decisions at the design stage are discussed in the subsequent paragraphs.
As the failure frequency of a product is dependent on the reliability of the product, it
will also significantly affect the warranty claims. With increase in product’s reliability,
the number of warranty claims will decrease. Not having adequate reliability is costlier
as failures result not only in higher warranty costs but also reduced sales and revenue
due to the negative impact of customer satisfaction resulting from product failures.
The data related to inherent reliability of components can be obtained from component
manufacturers, from field failure records and from life tests conducted at the design
and development stage.
The warranty claims data can be used for analysis of failures and subsequently
making improvements into the design of the product. Warranty data are the
primary mode of performance feedback regarding the ability of the product to perform
its intended function in the hands of the customer. Warranty data are an
accumulation of all incidents reported during the warranty period. The warranty
period is that time and/or mileage during which the manufacturer will repair, with no
charge or minimum charge to the customer, all incidents that occur during the system/
vehicle warranty. The warranty claim data can be obtained from the dealers and
service agents. Many web-based warranty management softwares are available
in the market, which can help in effective processing of warranty data (SAS Institute
Inc, 2008).
In a competitive environment, the total demand for first purchase sales depends
on product attributes. These include the sales price and the warranty terms of the
products. Manufacturer’s reputation depends on the reputation of its products
and has a strong influence on the first purchase decisions of new customers. Product
reputation has a similar effect on repeat purchase decisions. The sales data of the
product compared with the competitor’s sales volume as well as share of the product
in the market can be used to set the benchmarking and building the reputation in
the market.
The type of maintenance will affect the number of failures and consequently the
number warranty claims. For a given warranty period, the manufacturer can minimize
the expected number of warranty claims through optimal maintenance decision
making, which reduces the likelihood of failures. Optimal preventive actions need to be
viewed from a life cycle perspective for the buyer and manufacturer.
Apart from the design-related issues, warranty logistics plays an important role in
controlling the warranty cost. The manufacturer needs a dispersed network of service
facilities that store spare parts and provide a base for field service. The service delivery
network requires a diverse collection of human and capital resources and careful
attention must be paid to both the design and the control of the service delivery system.
The data related to the requirement of spare parts can be obtained from the sales
department. Warranty logistics-related data can be acquired from the service agents
and dealers, who provide service to the customers.
The feedback for a product can be obtained through customers in the form of
feedback reports, consumer surveys and warranty claim data/failure data obtained
from the dealers and service agents. Collected data provides ability to an organization
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to track product failures and defects in order to recover warranted repair/part/labour
costs from suppliers. Warranty claim data will be useful in early warning/detection of
bad design, poor production processes, defective parts, poor materials, etc.
Manufacturers analyze field reliability data to enhance the quality and reliability of
their products and to improve customer satisfaction. A quick feedback mechanism
needs to be developed in order to make the improvements as early as possible and
reducing the number of defective units to be produced.
Warranty optimization using all the parameters and decision variables identified in
the above framework is extremely complex. However, the optimization can be
attempted by reducing the complexity to some extent. This can be done by considering
a few significant decision variables out of the entire set.
In the next section a methodology for warranty optimization is presented with
specific focus on component reliability, warranty policy type and duration.
4. Warranty optimization
In this section, a model for solving the warranty optimization problem is developed and
demonstrated using a real life case of an automobile engine. We consider a case where
the manufacturer of the engine wants to improve the reliability of the product by
choosing appropriate component alternatives and also restrict the corresponding
warranty cost to the certain proportion of the engine price by choosing appropriate
warranty-related parameters. The decision variables considered are component
alternatives, type of warranty policy and duration of warranty policy.
Currently the manufacturer offers a non-renewing free replacement warranty (FRW-
NR) with a duration of one year. In the present paper, four types of warranty policies
namely, FRW-NR, non-renewing pro-rata warranty (PRW-NR), renewing free replacement
warranty (FRW-R), renewing pro-rata warranty (PRW-R) have been considered.
The details of the methodology used are mentioned in following section.
4.1 Parameters estimation for failure distribution
As the two parameter Weibull distribution is most commonly used to model the failure
behaviour of the mechanical components, the same is used in the present study to
model the failure behaviour of components of the engine.
In the present study, the failure data were collected from the warranty database and
the field failure records available at the service centers.
Components for which sufficient data were available, the standard distribution
fitting method using the maximum likelihood Estimation procedure was used.
However, there were cases where the field failure data was not sufficient. In such
cases using only warranty data was not sufficient and an expert judgment-base
approach was used for parameter estimation. This method was first proposed by Jager
and Bertsche (2004) and a modified version was developed and used by Lad and
Kulkarni (2010).
In both these studies, the method was proven to be sufficiently accurate and robust
for parameter estimation. A brief account of the method is given below.
The idea behind the expert judgment-based parameter estimation is to determine
the distribution parameters using the knowledge and experience of the experts. This is
achieved by modeling the expert knowledge for a continuous random variable t, i.e.,
time to failure (lifetime) of any component for a given probability distribution. The use
of expert knowledge to model a two parameter Weibull distribution is discussed in the
following section.
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The probability density function of the time to failure twith Weibull distribution is
given by:
fðtÞ¼ b
Z
t
Z
b1
!
et
Z
ðÞ
b
b;Z;tX0ð1Þ
In this method, the knowledge available with the expert regarding value of tis
captured and is converted into estimation of distribution parameters.
The data related to expert judgment for the components of the engine is collected
from the experts from the engine manufacturing company, OEM and service centers
who are directly involved with the post-sales support of the engine.
Experts are asked about the time period when the components mostly fail or most
likely to fail. This value indicates the mode of the time to failure distribution and can be
estimated with differentiating the Equation (1) with respect to tand equating it to zero.
Thus mode Xcan be given by:
tmod e ¼X¼Zb1
b
1
bð2Þ
Next, the experts are asked about the longest failure free life of the component they
have observed. From the methodology proposed by Jager and Bertsche (2004), it is
assumed that experts know about how many parts they have already replaced. So it
can be assumed that sample size is known and it is represented by N. With this
information the failure probability for the longest life time Ycan be determined using
Equation (3) for the approximation of rank dependent failure probabilities (Ebeling,
2000):
Ft
z
ðÞ
z0:3
mþ0:4ð3Þ
For the longest observed life (Y), z¼m¼N:
FYðÞ
N0:3
Nþ0:4ð4Þ
Thus the failure probability at the longest observed life (Y) for a Weibull distribution is
given by:
FYðÞ1et=bðÞ
bð5Þ
Solving the Equations (2) and (5) simultaneously, the value of the Weibull distribution
parameters is obtained. In the present study, the value of the F(Y) is assumed as 0.990.
In the present paper only critical components are considered, the criticality depends
upon the cost of the component and criticality and consequences of failure. So there are
29 components considered in the engine as critical components. The component
alternatives along with their distribution parameters and cost are given in the Table I.
The component alternatives are selected based on the similar components used by
the manufacturer in other types of engine models and variants but could be used
directly or with a little modifications in the engine under study.
4.2 Warranty cost models
The warranty costs for different warranty policies considered in the present paper are
calculated using Equations (6)-(9) as follows ( Blischke and Murthy, 1994).
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Design for
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In the case of FRW-NR policy, the manufacturer agrees to repair or provide
replacements for failed items free of charge up to a warranty duration wfrom the time
of the initial purchase. The cost function is given by Equation (6):
WCFRW NR ¼Cc;0ptipw
0;otherwise
till X
m
i¼1
tipwð6Þ
where, C
c
is the component cost and wis the warranty duration t
i
is time to failure.
In case of PRW-NR policy, the manufacturer agrees to refund a fraction of
the purchase price should the item fail before warranty duration wfrom the time of the
initial purchase. The cost function is given by the following equation:
WCPRW NR ¼Cc1ti
w
;0ptipw
0;otherwise
till X
m
i¼1
tipwð7Þ
In the case of FRW-R policy, the manufacturer agrees to either repair a failed item or
provide a replacement free of charge up to warranty duration wfrom the initial
Alternative 1 Alternative 2
Sl. no. Component Cost (Rs.) ZbCost (Rs.) Zb
1 Crank Case 2,559.82 13,496.37 3.88 2,816.12 16,149.15 3.52
2 Starter Motor 2,309.13 8,747.26 2.83 2,673.18 10,410.83 3.26
3 Fuel Pump Assly 1,835.32 8,199.72 3.62 2,034.33 9,811.42 2.53
4 Crank Shaft 1,488.09 9,932.81 2.18 1,737.25 10,907.09 3.00
5 Nozzle 1,350.53 9,634.98 3.45 1,564.59 11,536.38 3.35
6 Cyl. Head 1,184.33 9,331.37 2.42 – – –
7 FMA 1,100.00 13,496.37 3.88 1,255.53 14,758.32 2.83
8 BP Kit 1,089.61 8,333.48 1.98 1,316.19 9,951.897 2.54
9 Regulator 681.34 10,602.86 4.40 – – –
10 Ex. Muffler 672.42 8,199.72 3.62 792.11 9,760.83 3.00
11 Delivery Valve 587.38 4,373.63 2.83 645.35 5,214.75 2.51
12 CAM Shaft 495.84 28,851.96 1.79 582.27 31,696.49 2.25
13 Feed Pump 323.33 47,555.60 1.39 395.35 52,173.49 1.86
14 Crank Shaft Support 284.43 38,249.60 2.51 305.20 45,657.77 2.85
15 Connecting Rod Assly. 267.88 4,665.68 2.43 310.73 5,566.89 2.52
16 Lub Oil Pump 258.09 18,083.35 1.90 262.26 19,713.56 2.31
17 CAM and Follower Set 242.68 29,065.05 1.61 261.41 31,707.47 1.53
18 Governor Gear 181.57 5,554.26 3.36 206.34 6,663.39 3.00
19 Governor Support Assly 201.26 5,301.43 4.40 – – –
20 Valve Set 154.01 10,146.41 2.07 171.96 12,167.24 2.52
21 Roller Tappet 126.44 5,301.43 4.40 151.58 6,355.22 4.00
22 Set of Rocker lever 122.13 6,420.42 2.29 139.78 7,649.58 2.81
23 Diaphragm 80.85 2,915.75 2.83 89.90 3,471.40 3.10
24 High Pressure Pipe 74.86 11,108.52 3.36 82.38 13,240.73 3.53
25 LE Bearing 48.24 12,767.51 1.79 55.82 13,917.72 2.00
26 Set of Bushes 61.60 49,520.84 1.70 67.81 54,164.85 1.83
27 Bush STD 49.44 49,520.84 1.70 56.25 54,164.85 2.56
28 Bush F.W.E. Side 45.07 49,520.84 1.70 51.97 54,164.85 2.22
29 Set of Valve Guides 37.15 14,013.14 2.79 43.11 15,348.83 3.21
Tabl e I.
Component alternatives
with cost and distribution
parameters
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purchase. Whenever there is a replacement, the failed item is replaced by a new
one with a new warranty whose terms are identical to those of the original warranty.
The cost function is given by the following equation:
WCFRWR¼Cc;0ptipw
0;otherwise
till tipwð8Þ
In the case of PRW-R policy, the manufacturer agrees to provide, at a pro-rated cost, a
replacement item for failed items up to a warranty duration wfrom the time of the
initial purchase. The replacement item is covered under warranty identical to that of
the original item purchased. The cost function is given by the following equation:
WCPRW R¼Cc1ti
w
;0ptipw
0;otherwise
till tipwð9Þ
In the present paper a simulation-based approach is used for calculation of warranty
costs for different policies. All the components are assumed as a non-repairable. The
process used for simulation is as given below:
(1) First the number of simulations are set as n.
(2) For each simulation, the following process is repeated for mnumber of
iterations. The number of iterations depend upon the fulfillment of the
condition for given warranty policy:
.Randomly generate values of reliability R(t) with uniform distribution
between 0 and 1.
.The value of t
i
, i.e. time to failure can be calculated by Equation (10) with a
given value of Zand b. This is given by:
ti¼Zln RðtiÞ½
1
bð10Þ
where i¼1, 2, 3, y,m
.Calculate the warranty cost using the cost functions for each of the
warranty policy is given in Equation (6)-(9).
(3) Calculate the cumulative warranty cost for each of the iteration till the
condition for given warranty policy gets satisfied. This gives warranty cost per
simulation.
(4) Repeat the step number 2 and 3 for the nnumber of simulations.
(5) The warranty cost for the nsimulation is added up. This cost is then divided
with number of simulations nto get the warranty cost per unit.
In the present case, the target is to restrict the warranty cost to around 2 percent of the
price of the engine, which is 35,000. So the target cost for the warranty cost in this case
is Rs. 700, which has been used as one of the goals for optimization.
Another requirement is that the MTBF obtained from the new solution should be
higher than the existing one. The improvement in MTBF is achieved through selecting
the proper alternatives available with the manufacturer. Higher reliability can be
achieved by using the components with higher reliability, but this is achieved at a
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Design for
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higher cost. Choosing a proper alternative is a difficult task as trade-off needs to be
achieved between cost and reliability. In the present case, the target for MTBF is set as
10 percent higher than the existing MTBF for five years of duration.
The MTBF for the engine under study is determined using the simulation-based
approach explained earlier, except that simulation is run for five years of duration. The
duration of five years is taken based on the typical duration before a major overhaul.
The MTBF of an engine with ncomponents is determined using the Equation (11) and
is given by (Ebeling, 2000):
MTBF ¼t
Pn
i¼1NiðtÞð11Þ
where, tis time period over which MTBF is calculated.
N
i
(t) is the expected number of failures for ith component during time period t.
An assumption made here is that all the components of the engine are independent
and failure of one component does not cause failure of other component. Using the
above methodology the MTBF considering five years of duration for the current design
was determined as 390 hr. As the target set for the MTBF is 10 percent higher than the
current MTBF, the value to be achieved is 429 hr. This has been used as one of
the goals for the optimization problem.
4.3 Objective function
The objective function is formulated using a deviation function for each of the target
goals. The deviation function is calculated using a quadratic loss function (Taguchi
et al., 1989; Ross, 2005). The warranty cost is represented by a smaller the better type of
quadratic loss function, whereas the MTBF is represented by a larger the better type
of quadratic loss function. Using above two deviation functions the objective function
is formulated as a sum of the deviation functions and can be stated as follows.
Objective function:
Minimize XwMTBF d
MTBFþwWC dþ
WC
¼0:501:10MTBFCurrent
ðÞ
2
MTBFTarget
2þ0:50
WC Target
2
0:02Engine PriceðÞ
2
ð12Þ
where, w
MTBF
and w
WC
are weights for target goals; d
MTBF
and d
wc
þ
are the negative
and positive deviation functions for MTBF and warranty cost, respectively;
MTBF
Current
and MTBF
Target
are current and target MTBF; WC
Target
is the target
warranty cost.
In the present study, the manufacturer provided same preference to both the target
goals so a weightage of 0.5 is given to each of the goals in Equation (12).
The optimization is carried out using a GA (Deb, 1999, 2001). The problem
addressed in the paper is a mixed integer nonlinear programming (MINLP)
problem and GA is a good choice for solving MINLP problems (Gantovnik et al., 2003;
Ponsich et al., 2008).
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Start
Select population size and no. of generations
Generate random population for component alternatives,
warranty type and duration
Find out fitness function value for the population
Find out cumulative function values and normalized
fitness function values for the population
Form a mating pool by selecting best chromosomes
from population. It will be new population with good
chromosomes may occur more than once
Select two parents from the mating pool and using
crossover produce two new off-springs
Crossover finished ?
Select single off-spring and using mutation operator
produce new off-spring
Repair the chromosomes
Elite selection: select the best fitness value population
among the previous population and replace it with worst of
the current population
Find out fitness values for the new population
Find out maximum fitness function value from
the population
Is best solutions
achieved ?
Mutation finished ?
Finish
Yes
No
No
No
Yes
Yes
Figure 2.
Flow chart for genetic
algorithm process for
warranty optimization
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As the GA is naturally suitable for maximization problem (Mathew, 2012), the fitness
function is converted to a maximization function and is given as:
Fitness function ¼1
Objective function ð13Þ
4.4 Solution methodology and results
The GA parameters used to solve warranty optimization problem are: population size:
200, number of generations: 200, cross-over probability: 0.9 and mutation rate: 0.01.
The flowchart in Figure 2 shows the optimization process using GA.
The results of the optimization are shown in Table II.
Sl. no. Component
Warranty duration
(in months)
Component
alternative Warranty policy
1 Crank Case 23 2 PRW-NR
2 Starter Motor 21 2 PRW-NR
3 Fuel Pump Assly. 25 2 PRW-R
4 Crank Shaft 22 2 PRW-NR
5 Nozzle 19 2 PRW-NR
6 Cylinder Head 20 2 PRW-R
7 FMA 21 2 PRW-NR
8 BP Kit 20 2 PRW-NR
9 Regulator 17 2 PRW-NR
10 Ex. Muffler 19 2 PRW-NR
11 Delivery Valve 18 2 PRW-NR
12 CAM Shaft 28 2 PRW-NR
13 Feed Pump 17 2 PRW-NR
14 Crank Shaft Support 17 2 PRW-NR
15 Connecting Rod Assly. 20 2 PRW-NR
16 Lub Oil Pump 16 1 PRW-NR
17 CAM and Follower Set 18 1 PRW-NR
18 Governor Gear 28 2 PRW-NR
19 Governor Support Assly. 21 2 PRW-NR
20 Valve Set 25 2 PRW-R
21 Roller Tappet 16 2 PRW-NR
22 Set of Rocker lever 21 2 PRW-NR
23 Diaphragm 27 2 PRW-NR
24 High Pressure Pipe 18 1 PRW-NR
25 LE Bearing 21 1 PRW-NR
26 Set of Bushes 16 1 PRW-R
27 Bush STD 26 1 FRW-R
28 Bush F.W.E. Side 20 1 PRW-R
29 Set of Valve Guides 21 2 PRW-NR
Table II.
Results of optimization
Warranty cost (Rs.) MTBF (hrs)
Goal set 2 % of engine price 1.10 MTBF
Current
Target value 700 429
Achieved value 692 472
Table III.
Comparison of achieved
goal with target goals
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Result obtained through optimization shown in Table II. It can be seen from the results
that different component alternatives have got selected and each component is offered
with a different warranty policy and duration. This could be justified based
on the fact that customer wants the warranty period to be as long as possible.
The manufacture on the other hand would want to offer a shorter warranty period to
lower the warranty cost. The solution obtained is such that the warranty period offered
is proportional to the expected life of a component. This will result into the customer
perceiving a higher value in terms of warranty offered at the same time the warranty
cost constraint set by the manufacturer are also met.
Based on the reliability of the components higher warranty period is offered for
the components based on their reliability so that higher customer satisfaction can be
achieved. The values of the goal achieved through optimization are shown in Table III.
It can be seen that warranty cost goal is achieved as per the target set by the
manufacturer. The target set for the MTBF is also achieved and the value of the MTBF
obtained is higher than the target value of the 429 hr., which will result into improved
reliability of the engine. Figure 3 shows how the fitness value converges towards the
solution. It can be seen that the solution gets stable after 130 generations. The time
required for obtaining the solution was 20 hr.
As the distribution parameters were determined using expert judgment, there are
chances of error in the values provided by the experts. To estimate the effect of error in
0 50 100 150 200
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Number of generations
Fitness value
Figure 3.
Fitness value for the
achieved solution
Case no. % of error in judgment Warranty cost (Rs.) MTBF (hr)
110% in Xand 10% in Y855.35 427.16
210% in Xand þ10% in Y989.99 452.25
3þ10% in Xand 10% in Y620.34 426.10
4þ10% in Xand þ10% in Y592.64 516.15
Tabl e IV.
Effect of error in expert
judgment on distribution
parameter estimation
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the expert judgment, the solution achieved is tested for robustness using a
combination of cases for a 710 percent error in the values of mode (X) and maximum
observed life (Y). The results for different combinations of error are shown in Table IV.
From the Table IV, it can be seen that case no. 1 gives higher values of warranty
whereas the value of the MTBF is almost near set target. In case no. 2, the target value
of the MTBF is achieved but the warranty cost is high. While in case no. 3, the
warranty cost target is achieved whereas MTBF value is slightly less than the target
value. In case no. 4, the target values are achieved for both warranty cost and MTBF.
It can be seen that even though the goals are not achieved in all the cases, the deviation
from the target values is small and the solution can be considered to be sufficiently
robust.
5. Conclusion
In this paper, a conceptual framework is proposed that integrates the technology and
commercial issues early at the design stage of a product to minimize the warranty costs
in the most effective and efficient manner. Warranty costs can be reduced by taking
warranty-related decisions at the design stage. The model for optimization of warranty
is developed with a focus on reliability and warranty policies. The target goals set for
the optimization are to restrict the warranty cost to a predetermined proportion of the
engine price and to achieve a higher value of MTBF as compared to the current value.
The model is illustrated using the case of an engine manufacture. The robustness of the
result are evaluated through calculation of warranty cost and MTBF in the case of
710 percent error in the expert judgment.
The conceptual framework proposed in this paper will further motivate researchers
to work in this area. The model developed will help the designers to take appropriate
decisions related to improvement in the reliability and selection of warranty
parameters in order to minimize the warranty cost.
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Hahn, G.J., Meeker, W.Q. and Doganaksey, N. (2006), “Improving reliability through warranty
data analysis”, Quality Progress, November, pp. 63-67.
Corresponding author
Prashant M. Ambad can be contacted at: pmambad@mech.iitd.ac.in
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