Limitations of Acceptance Sampling

As one of the earliest methods of quality control, acceptance sampling is closely related to inspection of output of a process, or testing of a product. Acceptance sampling is defined as the inspection and classification of samples from a lot randomly and decision about disposition of the lot. At the beginning of the concept of quality conformance back in the 1930s, the acceptance sample took the whole effort of quality improvement. The most widely used plans are given by the Military Standard tables (MIL STD 105A), which were developed during World War II. The last revision (MIL STD 105E) was issued in 1989, but cancelled in 1991. The standard was adopted by the American Society for Quality as ANSI/ASQ A1.4.

Due to its less proactive nature in terms of quality improvement, acceptance sampling is less emphasized in current quality control systems. Usually, methods of lot sentencing include no inspection, 100% inspection, and acceptance sampling. Some of the problems with acceptance sampling were articulated by Dr. W. Edwards Deming [2], who pointed out that this procedure, while minimizing the inspection cost, does not minimize the total cost to the producer. To minimize the total cost to the producer, Deming indicated that inspection should be performed either 100% or not at all, which is called Deming’s “All or None Rule.” In addition, acceptance sampling has several disadvantages compared to 100%

inspection [20]:

• There are risks of accepting “bad” lots and rejecting “good” lots.

• Less information is usually generated about the product or process.

• Acceptance sampling requires planning and documentation of the acceptance sampling procedure.

2.8 Conclusions

The quality of a system is defined and evaluated by the customer [29]. A system has many qualities and we can develop a utility or customer satisfaction measure based on all of these qualities. The design process substitutes the voice of the customer with engineering or technological characteristics. Quality function deployment (QFD) plays an important role in the development of those characteristics. Quality engineering principles can be used to develop ideal values or targets for these characteristics. The methodology of robust design is an integral part of the quality process [30–32]. Quality should be an integral part of all the elements of the enterprise, which means that it is distributed throughout the enterprise and also all of these elements must be integrated together as shown inFigure 2.8.

Statistical quality control is a primary technique for monitoring the manufacturing process or any other process with key quality characteristics of interests. Before we use statistical quality control to monitor the manufacturing process, the optimal values of the mean and

Quality Control 2-21

integrated andTo distributed

Transition from

FIGURE 2.8 Integrated and distributed quality management.

standard deviation of the quality characteristic are determined by minimizing the variability of the quality characteristic through experimental design and process adjustment techniques.

Consequently, the major goal of SQC (SPC) is to monitor the manufacturing process, keep the values of mean and standard deviation stable, and finally reduce variability. When the process is in control, all the assignable causes are not present and consequently the probability to produce a nonconforming unit is very small. When the process changes to out of control, the probability of nonconformance may increase, and the process quality deteriorates significantly. As one of the primary SPC techniques, the control charts we discussed in this chapter can effectively detect the variation due to the assignable causes and reduce process variability if the identified assignable causes can be eliminated from the process.

The philosophy of Deming, Juran, and other quality gurus implies that the responsibility for quality spans the entire organization. It is critical that the management in any enterprise recognize that quality improvement must be a total, company-wide activity, and that every organizational unit must actively participate. Statistical quality control techniques are the common language of communication about quality problems that enables all organizational units to solve problems rapidly and efficiently.

References

1. Juran, J.M. and Gryna, F.M. Quality Planning and Analysis: From Product Development Through Usage, 3rd edition. McGraw-Hill, New York, 1993.

2. Deming, W.E., Quality, Productivity, and Competitive Position, Massachusetts Institute of Technology, Center for Advanced Engineering Study, Cambridge, MA, 1982.

3. American Society for Quality Control, ASQC Glossary and Tables for Statistical Quality Control, Milwaukee, 1983.

4. Shewhart, W.A., Economic Control of Quality of a Manufactured Product, D. Van Nostrand Company, New York, 1931.

5. Mizuno, S. and Akao, Y. (Eds), Quality Function Deployment: A Company-Wide Quality Approach, JUSE Press, Tokyo, 1978.

6. Akao, Y. (Ed), Quality Function Deployment: Integrating Customer Requirements into Product Design, translated by Mazur, G.H., Productivity Press, Cambridge, MA, 1990.

7. Montgomery, D.C., Design and Analysis of Experiments, 5th edition, John Wiley & Sons, New York, 2001.

8. Kapur, K.C. and Feng, Q., Integrated optimization models and strategies for the improve-ment of the Six Sigma process. International Journal of Six Sigma and Competitive Advan-tage, 1(2), 2005.

9. Montgomery, D.C., The economic design of control charts: A review and literature survey, Journal of Quality Technology, 14, 75–87, 1980.

10. Breyfogle, F.W., Implementing Six Sigma: Smarter Solutions Using Statistical Methods, 2nd edition, John Wiley & Sons, New York, 2003.

11. Pyzdek, T., The Six Sigma Handbook, Revised and Expanded: The Complete Guide for Greenbelts, Blackbelts, and Managers at All Levels, 2nd edition, McGraw-Hill, New York, 2003.

12. Yang, K. and El-Haik, B., Design for Six Sigma: A Roadmap for Product Development, McGraw-Hill, New York, 2003.

13. De Feo, J.A. and Barnard, W.W., Juran Institute’s Six Sigma Breakthrough and Beyond:

Quality Performance Breakthrough Methods, McGraw-Hill, New York, 2004.

14. Duncan, A.J., Quality Control and Industrial Statistics, 5th edition, Irwin, Homewood, IL, 1986.

15. Stoumbos, Z.G., Reynolds, M.R., Ryan, T.P., and Woodall, W.H., The state of statistical process control as we proceed into the 21st century, Journal of the American Statistical Association, 95, 992–998, 2000.

16. Box, G. and Luceno, A., Statistical Control by Monitoring and Feedback Adjustment, John Wiley & Sons, New York, 1997.

17. Hicks, C.R. and Turner, K.V., Fundamental Concepts in the Design of Experiments, 5th edition, Oxford University Press, New York, 1999.

18. ASTM Publication STP-15D, Manual on the Presentation of Data and Control Chart Analysis, 1916 Race Street, Philadelphia, PA, 1976.

19. Alwan, L.C., Statistical Process Analysis, McGraw-Hill, New York, 2000.

20. Montgomery, D.C., Introduction to Statistical Quality Control, 5th edition, John Wiley & Sons, New York, 2005.

21. Western Electric, Statistical Quality Control Handbook, Western Electric Corporation, Indianapolis, IN, 1956.

22. Chandra, M.J., Statistical Quality Control, CRC Press LLC, Boca Raton, FL, 2001.

23. Kane, V.E., Process capability indices, Journal of Quality Technology, 18, 1986.

24. Kotz, S. and Lovelace, C.R., Process Capability Indices in Theory and Practice, Arnold, London, 1998.

25. Roberts, S.W., Control Chart Tests Based on Geometric Moving Averages, Technometrics, 42(1), 97–102 1959.

26. Hotelling, H., Multivariate Quality Control, In: Techniques of Statistical Analysis, Edited by Eisenhart, C., Hastay, M.W., and Wallis, W.A., McGraw-Hill, New York, 1947.

27. Duncan, A.J., The economic design of charts used to maintain current control of a process, Journal of the American Statistical Association, 51, 228–242, 1956.

28. Woodall, W.H. and Montgomery, D.C., Research issues and ideas in statistical process control, Journal of Quality Technology, 31(4), 376–386, 1999.

29. Kapur, K.C., An integrated customer-focused approach for quality and reliability, Inter-national Journal of Reliability, Quality and Safety Engineering, 5(2), 101–113, 1998.

30. Phadke, M.S., Quality Engineering Using Robust Design, Prentice-Hall, Englewood Cliffs, NJ, 1989.

31. Taguchi, G., Introduction to Quality Engineering, Asia Productivity Organization, Tokyo, 1986.

32. Taguchi, G., System of Experimental Design, Volume I and II, UNIPUB/Krauss Interna-tional, White Plains, New York, 1987.

Quality Control 2-23

Appendix

Constants and Formulas for Constructing Control Charts

X and R Charts X and S Charts

Chart for Chart for

Averages Averages

X Chart for Ranges (R) X Chart for Standard Deviations (S)

Devisors for Divisors for

Factors for Estimate of Factors for Factors for Estimator of Factors for Subgroup Control Standard Control Limits Control Standard Control Limits

Size Limits Deviation Limits Deviation

6 0.483 2.534 – 2.004 1.287 0.9515 0.030 1.970

7 0.419 2.704 0.076 1.924 1.182 0.9594 0.118 1.882

8 0.373 2.847 0.136 1.864 1.099 0.9650 0.185 1.815

9 0.337 2.970 0.184 1.816 1.032 0.9693 0.239 1.761

10 0.308 3.078 0.223 1.777 0.975 0.9727 0.284 1.716

LCL =x − A2R

Guide for Selection of Charts for Attributes:

Nonconforming

• p chart for proportion of units nonconforming, from samples not necessarily of constant size: (Ifn varies, use n or individual ni.)

UCLpˆ=p + 3

• np chart for number of units nonconforming, from samples of constant size:

UCL =np + 3

np(1 − p) Centerline =np

LCL =np − 3

np(1 − p)

• c chart for number of nonconformities, from samples of constant size:

LCL =c − 3√ c CL =c UCL =c + 3√

c

• u chart for number of nonconformities per unit, from samples not necessarily of constant size: (If n varies, use n or individual n_i.)

LCL =u − 3

u n CL =u

UCL =u + 3

u n

3

Reliability

Lawrence M. Leemis

The College of William and Mary

3.1 Introduction. . . . 3-1 3.2 Reliability in System Design. . . . 3-3

Structure Functions^•Reliability Functions

3.3 Lifetime Distributions. . . . 3-10 Survivor Function^•Probability Density Function^•

Hazard Function^•Cumulative Hazard Function^• Expected Values and Fractiles^•System Lifetime Distributions

3.4 Parametric Models. . . . 3-17 Parameters^•Exponential Distribution^•Weibull

Distribution

3.5 Parameter Estimation in Survival Analysis. . . . 3-22 Data Sets^•Exponential Distribution^•Weibull

Distribution

3.6 Nonparametric Methods. . . . 3-32 3.7 Assessing Model Adequacy . . . . 3-36 3.8 Summary. . . . 3-40 References. . . . 3-41

3.1 Introduction

Reliability theory involves the mathematical modeling of systems, typically comprised of components, with respect to their ability to perform their intended function over time. Reli-ability theory can be used as a predictive tool, as in the case of a new product introduction;

it can also be used as a descriptive tool, as in the case of finding a weakness in an existing system design.

Reliability theory is based on probability theory. The reliability of a component or sys-tem at one particular point in time is a real number between 0 and 1 that represents the probability that the component or system is functioning at that time. Reliability theory also involves statistical methods. Estimating component and system reliability is often per-formed by analyzing a data set of lifetimes.

Recent tragedies, such as the space shuttle accidents, nuclear power plant accidents, and aircraft catastrophes, highlight the importance of reliability in design. This chapter describes probabilistic models for reliability in design and statistical techniques that can be applied to a data set of lifetimes. Although the majority of the illustrations given here come from engineering problems, the techniques described here may also be applied to problems in actuarial science and biostatistics.

Reliability engineers concern themselves primarily with lifetimes of inanimate objects, such as switches, microprocessors, or gears. They usually regard a complex system as a

3-1

collection of components when performing an analysis. These components are arranged in a structure that allows the system state to be determined as a function of the component states. Interest in reliability and quality control has been revived by a more competitive international market and increased consumer expectations. A product or service that has a reputation for high reliability will have consumer goodwill and, if appropriately priced, will gain in market share.

Literature on reliability tends to use failure, actuarial literature tends to use death, and point process literature tends to use epoch, to describe the event at the termination of a lifetime. Likewise, reliability literature tends to use a system, component, or item, actuarial literature uses an individual, and biostatistical literature tends to use an organism as the object of a study. To avoid switching terms, failure of an item will be used as much as possible throughout this chapter, as the emphasis is on reliability. The concept of failure time (or lifetime or survival time) is quite generic, and the models and statistical methods presented here apply to any nonnegative random variable (e.g., the response time at a computer terminal).

The remainder of this chapter is organized as follows. Sections 3.2 through 3.4 contain probability models for lifetimes, and the subsequent remaining sections contain methods related to data collection and inference.

Mathematical models for describing the arrangement of components in a system are introduced in Section 3.2. Two of the simplest arrangements of components are series and parallel systems. The notion of the reliability of a component and system at a particular time is also introduced in this section. As shown in Section 3.3, the concept of reliability generalizes to a survivor function when time dependence is introduced. In particular, four different representations for the distribution of the failure time of an item are considered:

the survivor, density, hazard, and cumulative hazard functions.

Several popular parametric models for the lifetime distribution of an item are investi-gated in Section 3.4. The exponential distribution is examined first due to its importance as the only continuous distribution with the memoryless property, which implies that a used item that is functioning has the same conditional failure distribution as a new item. Just as the normal distribution plays a central role in classical statistics due to the central limit theorem, the exponential distribution is central to the study of the distribution of lifetimes, as it is the only continuous distribution with a constant hazard function. The more flexible Weibull distribution is also outlined in this section.

The emphasis changes from developing probabilistic models for lifetimes to analyzing lifetime data sets in Section 3.5. One problem associated with these data sets is that of censored data. Data are censored when only a bound on the lifetime is known. This would be the case, for example, when conducting an experiment with light bulbs, and half of the light bulbs are still operating at the end of the experiment. This section surveys methods for fitting parametric distributions to data sets. Maximum likelihood parameter estimates are emphasized because they have certain desirable statistical properties. Section 3.6 reviews a nonparametric method for estimating the survivor function of an item from a censored data set: the Kaplan–Meier product-limit estimate. Once a parametric model has been chosen to represent the failure time for a particular item, the adequacy of the model should be assessed.

Section 3.7 considers the Kolmogorov–Smirnov goodness-of-fit test for assessing how well a fitted lifetime distribution models the lifetime of the item. The test uses the largest vertical distance between the fitted and empirical survivor functions as the test statistic.

We have avoided references throughout the chapter to improve readability. There are thousands of journal articles and over 100 textbooks on reliability theory and applications.

As this is not a review of the current state-of-the-art in reliability theory, we cite only a few key comprehensive texts for further reading. A classic, early reference is Barlow and

Reliability 3-3 Proschan (1981). Meeker and Escobar (1998) is a more recent comprehensive textbook on reliability. The analysis of survival data is also considered by Kalbfleisch and Prentice (2002) and Lawless (2003). This chapter assumes that the reader has a familiarity with calculus-based probability and statistical inference techniques.

In document Operations Research Applications. a. Ravi Ravindran (Page 74-81)