Engineering Statistics for Manufacturing Systems

Engineers not only provide the science behind the design of the products of today's technological revolution, they also help ensure that the production of those products -- and virtually any mass manufactured product -- remains within specification. To do these things, engineers use the tools and techniques of applied statistics. Through the application of various statistical tools and techniques, engineers can evaluate, improve, and optimize the processes and products of manufacturing systems. Statistics is used in the control of the various manufacturing processes and in quality control. In addition, engineering statistics is used in the design and analysis of experiments that are used to characterized, improve, qualify, and optimize the manufacturing processes as well as product performance and reliability. Through the use of statistics, engineers help keep processes in control and contribute to the effectiveness and efficiency of the organization.

Engineers are essential for the success of any large-scale manufacturing operation. Not only do engineers provide the science behind the design of the products of today's technological revolution, they also help ensure that the production of those products remains within specification. However, it is not only the production of high technology products that require engineering expertise. The large-scale production of any product, including clothing, paper products, and canned goods, requires the expertise of an engineer to help make sure that the product is being produced within the specifications for which it was designed. This is important not only because the consumer expects a certain level of quality and consistency in a product (for example, one would not purchase a can of peas in which the lid was not tightly sealed), but also because the rejection of products that are outside of specifications or manufacturing processes that result in excess waste cost the organization money.

Uses of Applied Statistics in Engineering

Applied statistics is one of the disciplines that engineers use in their tasks at manufacturing facilities. Through the application of various statistical tools and techniques, engineers can evaluate, improve, and optimize the processes and products of manufacturing plants. Engineering statistics are used in several areas of manufacturing systems.

Quality Control

First, statistics are used in the control of the various manufacturing processes and in quality control. This discipline comprises a set of procedures or processes that help ensure that products or services comply with predefined quality criteria or otherwise meet the requirements of the client or customer. Quality control activities include:

• The collection and statistical analysis of data to determine whether the process includes systematic (i.e., nonrandom) variation in quality.
• Monitoring and inspecting products or services vis à vis predefined specifications or quality standards.
• Determining the cause of variation.
• Developing and implementing changes to help meet target quality goals.

The tools and techniques of statistical process and quality control have numerous benefits. In particular, they help engineers monitor manufacturing processes to determine when a problem has occurred so that the process can be brought back under control. When a process is stable and assignable causes of variation have been eliminated, these tools and techniques can help engineers analyze manufacturing processes and compare their output with specified tolerances.

Experiment Design & Analysis

In addition, engineering statistics are used in the design and analysis of engineering experiments that are used to characterize, improve, qualify, and optimize the manufacturing processes as well as product performance and reliability. The proper design of experiments and application of the appropriate statistical technique allows engineers to draw valid conclusions from the quality and other data collected. Engineers also use statistics techniques in the development of mathematical models that represent the system or situation being studied so that impact of various variables on the end result can be better understood. The use of statistical tools in the analysis of data from experimental studies assists engineers in better understanding the data that are gathered from various manufacturing processes. Statistically analyzed experimental designs have a wide range of applications to manufacturing processes. An understanding of what statistics can and cannot do helps engineers to better design experiments that will yield the answers to questions about manufacturing processes. Specifically, this understanding can help in the development of better hypotheses that can be realistically tested using real world data. In addition, this understanding can help engineers interpret the results of experiments and to apply them to the real world in order to continuously improve manufacturing processes.

Solving Engineering Problems

The successful solution of engineering problems such as the improvement of manufacturing processes must be based on an understanding of variability and how to apply the principles of mathematical statistics to real world problems. Mathematical statistics is a branch of mathematics that deals with the analysis and interpretation of data. Mathematical statistics provides the theoretical underpinnings for various applied statistical disciplines, including engineering statistics, in which data are analyzed to find answers to quantifiable questions. Engineering statistics is the application of these tools and techniques to the analysis of real world problems for the purpose of business decision making. Specifically, the discipline of engineering statistics is concerned with the collection, presentation, analysis, and use of data in order to solve real world problems and make practical decisions. Statistical methods are useful in helping engineers understand the underlying variability that can be observed in systems and phenomena as are frequently observed in manufacturing processes. For example, in manufacturing, some proportion of products always has defects no matter how standardized or efficient the process. Statistics can help engineers better understand why this occurs and design processes or equipment that will help reduce the number of defective products produced.

Applications

Process & Quality Control

Statistical process control is the application of statistical techniques to measure and analyze the variations that occur in any production process and to monitor the consistency with which the processes used in manufacturing result in products that are within their design specifications. Statistical quality control is the use of statistical techniques to measure and improve the quality the processes. These processes have nothing to do with the quality of the product itself: it only monitors whether or not that product is within specification. Statistical process and quality control are not tools for improving the quality of the design, but only help monitor whether or not a product is being manufactured as designed.

Errors in Manufacturing Processes

Modern manufacturing and automation technology go a long way toward ensuring that products can be repeatedly and reliably manufactured within the specifications for which they should be designed. However, although it would be tempting to assume that production lines would repeatedly produce quality products without adjustment, this assumption flies in the face of the laws of physics and of probability and physics. Even using the most advanced manufacturing technologies and processes that minimize the impact of the human being and concomitant human error in the process, error in the form of defects and waste will always occur. One type of error commonly found in manufacturing processes is "noise," random variability that occurs naturally. For example, just as the amount or quality of ore produced from a mine varies naturally from day to day, so too can the amount or quality of products manufactured using that ore change from day to day. Changes in quality or quantity of the ore, for example, can affect the inputs into the production line (e.g., lower quality ore may result in greater breakage of the products that were made from it). A second kind of error commonly found in manufacturing processes is due to problems with the process, equipment, materials, or humans working the line. Engineers can use statistical tools and techniques to continually improve manufacturing processes in order to increase the quality of the product.

Shewhart Control Charts

One of the primary tools used in statistical process and quality control are Shewhart control charts. These tools are simple graphing procedures that help quality control engineers and managers monitor processes and determine whether or not they are in control. Control charts are based on two statistical ideas.

• The first assumption is that random noise is naturally occurring and will be evident in any manufacturing process.
• The second assumption is that within a random process, there is a certain amount of regularity. As a result of this regularity, a variable will differ from its mean by more than two standard deviations only five percent of the time (i.e., one occurrence in 20). A process is said to be within statistical control if it performs within the limits of its capability within these parameters.

X-bar Charts

Quality control charts are used to make measurements and to check for compliance. The X-bar chart (so named because the mathematical symbol for the arithmetic mean is X ) is a chart of the means of some characteristic of the product (e.g., acceptability of solder joints) of small random samples taken from the production line over time. The means are plotted over time on a chart that contains a center line (i.e., the mean for the process) and upper and lower control limits. The center line shows the location of the arithmetic mean of the means of the samples. The upper control limit is a line showing the location of three standard deviations above the center line and the lower control limit shows the location of three standard deviations below the center line. When all the points plotted on the chart fall between the upper and lower control limits, the process is considered to be in control. On the other hand, if computed sample means fall outside the control limits, the process is considered to be out of control and the process is stopped so that an assignable cause can be determined. Some assignable causes are easily explained by passing phenomena that are unlikely to occur again. Other assignable cause, however, are more serious and require corrective action (e.g., replacing a defective part or machine, retraining of employees, switching suppliers). A sample X-bar chart is shown in Figure 1. Other types of quality control charts include R charts that keep track of the range, p charts track the proportion of defective products, c charts track the number of defects, and s charts that examine sample variance.

In addition to Shewhart control charts; there are more sophisticated charting methods available. For example, multivariate charting methods are available that allow the quality control engineer to monitor several related variables simultaneously. In addition, methods are available for charting a single measurement rather than a sample (e.g., moving average charts, exponentially weighted moving average charts) and cumulative sum methods that are more sensitive than Shewhart control charts for detecting small, consistent changes.

Experimental Research & Modeling

In real world manufacturing situations, it is often important not only to monitor processes, but also to investigate ways that revealed problems can be solved. As shown in Figure 2, the engineering approach to problem solving comprises several steps.

• Before engaging in data collection, the engineer first needs to develop a clear and concise description of the problem.
• After the problem has been clearly defined, the next step in the engineering problem solving process is to identify the important factors that bound the problem or play a role in the solution. This step often results in a tentative list that is revisited and revised as new data are compiled.
• After the important factors have been tentatively identified, the next step is the development of an initial model of the situation, system, or subsystem being analyzed. The model should be based on scientific or engineering knowledge of the problem. Part of this process is the development and articulation of a conceptual model that describes the situation or system under investigation. This conceptual model may also be used in the development of a later mathematical or computer model that mathematically represents the system or situation being studied, including articulation of the assumptions used in building the model and any limits within which it applies.
• After the model is developed, an experimental research paradigm is designed and data collected to test how well the model reflects the real world situation. Based on the statistical analysis of the data collected, the model is then refined to better assist in developing a solution to the problem.
• Additional empirical research may then be conducted to confirm that the proposed solution to the problem is both effective and efficient.
• Based on the results of this experimental research, the engineer then draws conclusions and makes recommendations on the best way to proceed in order to solve the problem.

Design of Experiments & Research

The design and conduct of experiments is often a part of real world engineering problem solving for manufacturing processes. In general, the goal of research is to describe, explain, and predict observed phenomena. For example, an engineer may want to determine which of two alternative manufacturing processes produces more products within tolerance than the other process. Similarly, it might be of interest to know whether the equipment on any of three product manufacturing lines is more likely to break down than the others and whether or not this is attributable to chance. Statistically analyzed research can help engineers answer such questions.

A good research study allows one to control the situation so that the research is only measuring what it is supposed to measure and includes as many of the relevant factors as possible so that the research fairly emulates the real world experience. In the simplest research design, a stimulus (e.g., an alternative production process) is tested and a response is observed and recorded (e.g., the number of widgets produced without defects).

Experimental Variables

There are three types of variables that are important in research: independent, dependent, and extraneous variables. The variables of most concern in the design of a research study is the independent variable, which is the stimulus or experimental condition that is hypothesized to affect behavior, and the dependent variable, which is the observed effect on behavior caused by the independent variable. As shown in Figure 3, however, other variables need to be considered during the research study.

Extraneous variables are variables that can affect the outcome of the experiment (i.e., the number of defects per hour) but which have nothing to do with the independent variable itself. For example, if the person performing manufacturing process A is less skilled in operating the machine on that line than is the person performing manufacturing process B, it is likely that process B will appear to be a better alternative not because of the type of machine used in the manufacturing process (which was the dependent variable under consideration, but because of the extraneous variable of the amount of experience of the machine operator. There are many such variables in any experiment that are extraneous to the research question being asked but that still affect the outcome of the research. As much as possible, these variables need to be controlled. Although it is impossible to control for the effects of literally every possible extraneous variable, the more of these that are accounted for and controlled in the experimental design, the more meaningful the results will be.

Real World Observation

As with the engineering problem solving process, research design starts with a theory based on real world observation (Figure 4). For example, from personal experience with the two types of manufacturing processes, an engineer may have observed that process A tends to result in fewer defects than does process B. From these observations, s/he could form an empirically testable hypothesis concerning the relative effectiveness of the two processes (e.g., "Process A results in fewer defects than does process B"). To find out if the hypothesis is true, the engineer would operationally define the various terms (i.e., constructs) in the hypothesis (e.g., specify what a "defect" is). The engineer would then run the experiment, using both processes in a controlled setting where the extraneous variables were kept as constant as possible. The results of the trials would then be statistically analyzed using inferential statistics, and -- based on the statistical significance of the answer -- the engineer would determine whether it was likely to be cost effective to use process A or process B.

Conclusion

Statistical tools and techniques are invaluable to engineers working with manufacturing systems. Statistical processes and quality controls use various charting methods to help keep manufacturing processes in control and detect where they can be improved. Experiments allow engineers to better understand and interpret real world data. Statistical modeling can help maximize the information gained from process data. Through the use of statistics, engineers help keep processes in control and contribute to the effectiveness and efficiency of the organization.

Terms & Concepts

Data: (sing. datum) In statistics, data are quantifiable observations or measurements that are used as the basis of scientific research.

Dependent Variable: The outcome variable or resulting behavior that changes depending on whether the subject receives the control or experimental condition (e.g., a consumer's reaction to a new cereal).

Hypothesis: An empirically testable declaration that certain variables and their corresponding measure are related in a specific way proposed by a theory.

Independent Variable: The variable in an experiment or research study that is intentionally manipulated in order to determine its effect on the dependent variable (e.g., the independent variable of type of cereal might affect the dependent variable of the consumer's reaction to it).

Inferential Statistics: A subset of mathematical statistics used in the analysis and interpretation of data. Inferential statistics are used to make inferences such as drawing conclusions about a population from a sample and in decision making.

Mathematical Statistics: A branch of mathematics that deals with the analysis and interpretation of data. Mathematical statistics provides the theoretical underpinnings for various applied statistical disciplines, including engineering statistics, in which data are analyzed to find answers to quantifiable questions.

Mean: An arithmetically derived measure of central tendency in which the sum of the values of all the data points is divided by the number of data points.

Quality Control: A set of procedures or processes that help ensure that products or services comply with predefined quality criteria or otherwise meet the requirements of the client or customer. Quality control activities include the collection and statistical analysis of data to determine whether the process includes systematic (i.e., nonrandom) variation in quality. Quality control activities include monitoring and inspecting products or services vis à vis predefined specifications or quality standards, determining the cause of variation, and developing and implementing changes to help meet target quality goals.

Standard Deviation: A measure of variability that describes how far the typical score in a distribution is from the mean of the distribution. The standard deviation is obtained by determining the deviation of each score from the mean (i.e., subtracting the mean from the score), squaring the deviations (i.e., multiplying them by themselves), adding the squared deviations, and dividing by the total number of scores. The larger the standard deviation, the farther away it is from the midpoint of the distribution.

Statistical Significance: The degree to which an observed outcome is unlikely to have occurred due to chance.

Variable: An object in a research study that can have more than one value. Independent variables are stimuli that are manipulated in order to determine their effect on the dependent variables (response). Extraneous variables are variables that affect the response but that are not related to the question under investigation in the study.

Bibliography

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Tsai, J. C., & Kuo, C. H. (2012). A novel statistical tolerance analysis method for assembled parts. International Journal of Production Research, 50(12), 3498-3513. Retrieved December 2, 2013 from EBSCO Online Database Business Source Premier. http://search.ebscohost.com/login.aspx?direct=true&db=buh&AN=77058513

Suggested Reading

Alexander, C. (2013). Elements of an engineering-based integrity management program. Pipeline & Gas Journal, 240(3), 108-111. Retrieved December 2, 2013 from EBSCO Online Database Business Source Premier. http://search.ebscohost.com/login.aspx?direct=true&db=buh&AN=86444066

Boyd, A. H. & Radson, D. (1998). Statistical analysis of downtime severity rates. International Journal of Production Research, 36(3), 603-615. Retrieved October 5, 2007, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=6484114&site=bsi-live

Coleman, S. Y., Arunakumar, G., Foldvary, F., & Feltham, R. (2001). SPC as a tool for creating a successful business measurement framework. Journal of Applied Statistics, 28(3/4), 325-334. Retrieved August 21, 2007, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=4395729&site=bsi-live

Dean, B. R., Hand, S., & Kaye, M. (1997). A cost based strategy for assessing improvements in manufacturing processes. International Journal of Production Research, 35(4), 955-968. Retrieved October 5, 2007, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=6657700&site=bsi-live

Leitnaker, M. G. & Cooper, A. (2005). Using statistical thinking and designed experiments to understand process operation. Quality Engineering, 17(2), 279-289. Retrieved August 21, 2007, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=17003962&site=bsi-live

Moeller, C. (2012). Basics of Engineering Statistics. New Delhi: World Technologies. Retrieved December 2, 2013 from EBSCOhost eBook Collection. http://search.ebscohost.com/login.aspx?direct=true&db=nlebk&AN=397636&site=ehost-live

Montgomery, D. C. (1999). Experimental design for product and process design and development. Journal of the Royal Statistical Society, 48(2), 159-177. Retrieved October 5, 2007, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=4520053&site=bsi-live