Exercises
Exercises for Chapter 1
1. List some intangible and tangible products and discuss their similarities and
dissimilarities.
2. Discuss the possible consequences of failures for a car from the owner and the
manufacturer perspectives.
3. Rank the eight quality dimensions in terms of their importance for a specific product.
4. Discuss what the quality of higher education is.
Exercises for Chapter 2
1. Discuss main issues in product design.
2. List some products for which a life cycle cost analysis should be carried out.
3. For a specific product, list some important product performances to be considered in
its design process.
4. Give some real-world examples of hot standby, cold standby and warm standby.
5. Discuss why one should pay more attentions on the upstream of PLC to ensure
product quality.
6. List some products that are suited to be remanufactured.
Exercises for Chapter 3
1. Given the failure rate of the Weibull distribution, derive the other reliability basic
functions.
2. For the Weibull distribution with   2.75 and   100 , calculate the mean life,
B10 life and tradeoff BX life using Microsoft Excel and its Solver.
3. Derive the expressions of the Skewness and kurtosis of the Weibull distribution.
4. Derive the expression of m(t ; k ) for the gamma distribution.
5. Draw the four basic functions for the gamma distribution with u  5 and v  20
using Microsoft Excel. Calculate the mean life and tradeoff BX life using Microsoft
Excel and its Solver.
6. Derive the expression of mean life with censoring for the exponential distribution.
7. Discuss the similarities and differences between the power-law model and the
Weibull cumulative hazard function.
8. Discuss the differences between the component bathtub curve and system bathtub
curve.
Exercises for Chapter 4
1. Reproduce Example 4.1 for p  0.9 .
2. Reproduce Example 4.2 for p  0.9 .
3. Derive the mean and variance of X for the zero-inflated Poisson distribution.
4. Generate a random sample of the chi-squared variable with n  5 .
5. Discuss under what conditions the failure rate function of the two-fold Weibull
competing risk model is decreasing, bathtub-shaped and increasing.
6. Assume that the component life follows the Weibull distribution. Derive the
distribution of the test duration of the sudden death testing.
7. Extend Eqs. (4.39) and (4.41) to the case of n  3 .
Exercises for Chapter 5
1. Generate a random sample of the chi-squared variable and fit the generated sample to
the gamma distribution using the method of moments, least square method and
maximum likelihood method, respectively.
2. Fit the gamma and lognormal distributions to the first bus-motor failure dataset and
evaluate their goodness-of-fits.
3. Fit the 2-fold Weibull competing risk model to all the five bus-motor failure datasets.
4. Fit the exponential-Weibull competing risk model to all the five bus-motor failure
datasets, and test the appropriateness of the fitted models.
5. Fit the 2-fold Weibull competing risk model and the exponential-Weibull competing
risk model to the data set shown in Table 5.3, respectively, and select the best model.
Exercises for Chapter 6
1. Consider the exponential-Weibull model given by R(t )  exp[t /  (i)  (t /  )  ]
with  (i)  a  (b / i)c , i  1, 2,... , and fit it to the bus-motor data shown in Table 5.2
using the MLM.
2. Fit the data in Table 6.1 to the power-law model using the LSM, and test the
normality and randomness of the residuals.
3. The numbers of exercises for all the chapters in this book are respectively: 4, 6, 8, 7,
5, 3, 4, 7, 4, 8, 7, 6, 8, 7, 5, 6, 6. Carry out the tests for sationarity, randomness and
normality of this dataset.
Exercises for Chapter 7
1. List the main cost elements of LCC of a business-to-business product from the
perspective of manufacturer and discuss why the product designer needs to evaluate
the LCC of a design alternative.
2. Discuss which performance characteristics are important for a customer good and
which performance characteristics are important for a capital good.
3. Rank the X’s in terms of their importance for a capital good.
4. Quantify the correlations betweens the X’s in the form of matrix using 0, 1 and 2 to
denote no, weak and strong correlations, respectively.
Exercises for Chapter 8
1. Reproduce Example 8.1.
2. Derive the loss function for a smaller-the-better quality characteristic, which follows
an exponential distribution.
3. Assume that the number of parameters is 3 and each parameter has 3 levels. Give the
factorial design table.
4. Assume that the number of parameters is 10, each parameter has 3 levels, and each
level is of equal probability. Give a random design table with 50 experiments.
5. For Exercise 4, generate a mixed experiment plan of Taguchi design and random
design.
6. Reproduce Example 8.2.
7. Assuming C p  1.25 , reproduce Example 8.3.
Exercises for Chapter 9
1. For Example 9.2, find the minimal value of n so that RS (t  1)  0.9 .
2. Discuss the similarities and dissimilarities between fault-tree diagram and reliability
block diagram.
3. For
Example
9.3,
assume
that
the
importance
factor
is
given
by
wi   i / max( j ,1  j  n) , and the values of  i ’s are those shown in Example 9.4.
Reallocate the failure rate and compare the results with those obtained in Example
9.4.
4. In Example 9.5, find the value of x so that the failure rate equals 0.1.
Exercises for Chapter 10
1. Suppose that the temperature stress is defined as s  e1000/(273T ) . Discuss the link
between the proportional hazard model and the Arrhenius model.
2. Discuss similarities and dissimilarities between ALT and ADT models.
3. Assuming that the lifetime follows the lognormal distribution, reanalyze Example
10.1.
4. Discuss whether or not the model given by (10.27) is appropriate for modeling the
data in Table 10.3.
5. Suppose that the prior life distribution is the Weibull distribution with the shape
parameter 2.5 and the scale parameter 10; and the stress-life relationship is
 (s)  s / s0 . The parameters of the test scheme are k  1 , s0  1 , s1  2 , t1  4
and n1  20 . Compute the performances of this test plan.
6. In Exercise 5, if the Weibull shape parameter is   3.5, compute the performance
and compare the results with those obtained for   2.5.
7. Confine ni Fi (ti )  6 , reanalyze Option 5 of Example 10.2.
8. Discuss the test design problem involving two factors and two stress levels through
incorporating the empirical approach with the Taguchi experiment design method.
Exercises for Chapter 11
1. Discuss the similarities and dissimilarities between a stage growth curve and the
overall growth curve.
2. Discuss when the interval MTBF should be used and when the instantaneous MTBF
should be used.
3. Redo Example 11.3 for the model whose S ( j ) is the reliability function of a
Poisson distribution.
4. Draw the mean cumulative function of the data in Table 11.5, and discuss whether the
power-law model is an appropriate model for fitting the data.
5. Draw the mean cumulative functions of the failure data of Modes 3 and 12 in Table
11.6 and discuss whether the power-law model is an appropriate model for modeling
the failure processes of these two failure modes.
6. Draw the mean cumulative function of all the data in Table 11.6, and discuss whether
the power-law model is an appropriate model for fitting the data.
7. Fit the data in Table 11.10 to Eq. (11.9) with G (t ) being the lognormal cdf and
inverse Weibull cdf, respectively.
Exercises for Chapter 12
1. Revise the plots of Figure 12.1 into the plot of p versus  /  .
2. Discuss the main factors that impact quality failure.
3. Assume p1  0.2 , redo Example 12.2.
4. Assume c2  1 , redo Example 12.2.
5. Discuss the similarities and dissimilarities between TQM and Six-sigma quality.
6. Discuss quality improvement of higher education by applying the eight principles of
quality management.
Exercises for Chapter 13
1. Draw the OC curves based on the binomial distribution for c  0 and n  100 , 150
and 200, respectively.
2. Draw the OC curves based on the binomial distribution for n  100 and c  0 , 1
and 2, respectively.
3. Draw the OC curves based on the hypergeometric distribution for c  0 , N  1000
and n  100 , 150 and 200, respectively.
4. In Example 13.2, if X is a larger-the-better characteristic, design the sampling
scheme.
5. In Example 13.2, if X is a smaller-the-better, design the sampling scheme.
6. Reproduce Example 13.3 for r  5 .
7. In Example 13.4, assume that the lifetime follows the lognormal distribution with
 l  0.4 . Reproduce this example.
8. Reproduce Example 13.5 for L  80 .
Exercises for Chapter 14
1. Discuss the similarities and dissimilarities between statistical process control and
statistical quality control.
2. Describe the seven graphical tools mentioned in Section 14.1.
3. Let r be a random number in (0, 1) and p be the probability of the change from
an in-control state to an out-of-control state. If r  p , the state change occurs;
otherwise, the state does not change. Generate several stochastic processes for
p  0.001 and calculate the average run length using the simulation.
4. Calculate C p for the resulting X chart in Example 14.1.
5. Calculate the fraction of nonconformance for C p  1 .
6. Calculate the process capability index C p if the fraction of nonconformance is 5%.
7. Assume that the values of Di in Example 14.3 are the total numbers of defects of
the samples. Design the u chart and the c chart.
Exercises for Chapter 15
1. Discuss the similarities and dissimilarities between burn-in and ESS.
2. In Example 15.1, if  decreases from 10 to 5, and c p increases from 1000 to
2000, optimize the screening limit and discuss whether such an improvement is
beneficial.
3. Assuming  n  1 , reanalyze Example 15.2.
4. Assuming L  150 , reanalyze Example 15.2.
5. Discuss the similarities and dissimilarities between component-level burn-in and
system-level burn-in.
Exercises for Chapter 16
1. Consider the Weibull distribution with shape parameter   2.5 and scale parameter
  10 . Compute the renewal function M (20) using Eq. (6.2) , Eq. (16.5) through
Eq. (16.8) and compare the results.
2. Reanalyze Example 16.1 for   2 .
3. Generate five failure-repair processes for Example 16.2.
4. For Example 16.3, study the influence of  on the expected number of warranty
claims.
5. Reanalyze Example 16.3 for the one-dimensional PRW policy with   1 .
6. In Example 16.4, compute the related costs under the assumption that the usage limit
is increased from U  20 to 50.
Exercises for Chapter 17
1. Compare the repair cost limit policy with repair time limit policy and discuss their
applicable situations.
2. For the special case of T   , derive relevant expressions of the failure counting
policy with a reference age.
3. Table 1 shows a set of lifetime data of vehicle batteries. Find the trade-off BX life
of the battery.
Table 1 Life data of vehicle batteries in days
687
1103
1115
1131
1149
1151
1240
1251
1723
1884
2006
2625
1173
4. Assume that the time to failure follows the Weibull distribution with shape parameter
2.5 and scale parameter 10, and cost parameters are ( c1 , c2 , c3 ) = (1, 1.5, 2). Find the
optimal inspection interval associated with the periodic inspection scheme with
perfect PM.
5. Assume that the time to failure follows the Weibull distribution with shape parameter
2.5 and scale parameter 10, and the time of each inspection is 0.01. Find the optimal
inspection interval associated with the availability objective for the periodic
inspection scheme with minimal repair.
6. The reliability parameters of five components are shown in Table 2. Assume that cost
parameters ( c p , c f , cs ) are (1, 4, 0.4). Design a two-level group replacement policy
and specify their opportunistic replacement windows.
Table 2 Reliability parameters of components
Component
1
2
3
4
5

2.9
2.1
3.5
2.8
2.7
 , hours
2428
3157
1879
4579
3125