statistical quality control

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STATISTICAL QUALITY CONTROL Objectives After reading this unit you should be able to:  Apply statistical thinking to quality improvement  Use 7 tools to troubleshoot quality problems  Determine process capability of a manufacturing process  Explain what a control chart is and how to design one  Select appropriate control charts for different applications  Understand the principles behind sampling methods  Understand the role of 6-sigma and other advanced QC methods

Structure 1. 2. 3. 4.

5.

6.

7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

What is Statistical Quality Control? Process Capability: A Discerning Measure of Process Performance The Seven Quality Improvement Tools Control Charts for Variables Data 4.1 Constructing xbar and R Charts and Establishing Statistical Control 4.2 Interpreting Abnormal Patterns in Control Charts 4.3 Routine Process Monitoring and Control 4.4 Estimating Plant Process Capability 4.5 Use of Warning Limits 4.6 Modified Control Limits 4.7 Key Points about Control Charts Special Control Charts for Variables Data 5.1 Xbar and s-Charts 5.2 Charts for Individuals Control Charts for Attributes 6.1 Fraction Nonconforming (p) Chart. 6.2 Variable Sample Size p Chart 6.3 np-Charts for Number Nonconforming 6.4 Charts for Defects Choosing the Correct SPC Chart Key Points about Control Chart Construction Implementing SPC Review Questions about Control Charts Self Assessment Questions about SPC Acceptance Sampling What are Taguchi Methods? The Six-Sigma Program Key Words Further Readings and References

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WHAT IS STATISTICAL QUALITY CONTROL?

The concept of TQM is basically very simple. Each part of the organization has customerssome external and many internal. Identifying what the customer requirements are and setting about to meet them is the core of a total quality approach. This requires a good management system, methods including statistical quality control (SQC), and teamwork. A well-operated, documented management system provides the necessary foundation for the successful application of SQC. Note, however, that SQC is not just a collection of techniques. It is a strategy for reducing variability, the root cause of many quality problems. SQC refers to the use of statistical methods to improve and enhance quality and through it customer satisfaction. However, this task is seldom trivial because real world processes are affected by numerous uncontrolled factors. For instance, within every factory, conditions fluctuate with time. Variations occur in the incoming materials, in machine conditions, in the environment and in operator performance. A steel plant, for example, may purchase good quality ore from a mine, but the physical and chemical characteristics of ore coming from different locations in the mine may vary. Thus, everything isn't always "in control." Besides ore, in steel making furnace conditions may vary from heat to heat. In welding, it is not possible to form two exactly identical joints and faulty joints may occur occasionally. In a cutting process, the size of each piece of material cut varies; even the most high-quality cutting machine has some inherent variability. In addition to such inherent variability, a large number of other factors may also influence processes (Figure 1.1). Many of these variations cannot be predicted with certainty, although sometimes it is possible to trace the unusual patterns of such variations to their root cause(s). If we have collected sufficient data from these variations, we can tell, in terms of probability, what is most likely to occur the next if no action is taken. It we know what is likely to occur the next given certain conditions, we can take suitable actions to try to maintain or improve the acceptability of the output. This is rationale of statistical quality control.

Ambient temperature, vibration, humidity, supply voltage, etc. Labour Training level

The Process Control Variables: Set points for temperature, cutting speed, raw material specs, recipe etc. Raw material quality/quantity State Variables measured here

Variation in Output Variables: Quality of Finished Product; Level of Customer Satisfaction

Figure 1.1: Input, Environmental and Output Variables of a Process


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Input factors x1, x2, x3, …

Process y = f(x)

Output responses y1, y2, y3, …

Figure 1.2: An Idealized Process Model

Another prospect in which statistical methods can help to improve product quality is the design of products and processes. It is now well-understood that over 2/3rd of all product malfunctions may be traced to their design. Indeed, the characteristics or quality of a product depend greatly on the choice of materials, settings of various parameters in the design of the product and the production process settings. In order to locate an optimal setting of the various parameters which gives the best product, we may consider using models governing the outcome and the various parameters, if such models can be established by theory or through experimental work. Such a model is diagrammatically shown in Figure 1.2. However, in many cases, a theoretical quality control model y = f(x) relating the final output responses (y1, y2, y3, …) and the input parameters (x1, x2, x3, …) is either extremely difficult to establish or mathematically intractable. The following two examples illustrate such cases. Example 1: In bakery industry, the taste, tenderness and texture of a kind of bread depends on various input parameters such as the origin of the flour used, the amounts of sugar, the amount of baking powder, the baking temperature profile and baking time, and the type of oven used, and so on. In order to improve the quality of the bread produced, the baker may use a model which relates the input parameters and the output quality of the bread. To find theoretical models quantifying the taste, tenderness and texture of the bread produced and relate these quantities to the various input parameters based our present scientific knowledge is a formidable task. However, the baker can easily use statistical methods in regression analysis to establish empirical models and use them to locate an optimal setting of the input parameters. Example 2: Sometimes there are great difficulties in solving an engineering problem using established theoretical models. The heat accumulated on a chip in an electronic circuit during normal operation will raise the temperature of the chip and shorten its life. In order to improve the quality of the circuit, the designer would like to optimize the design of the circuit so that the heat accumulated on the chip will not exceed a certain level. This heat accumulated can be expressed theoretically in terms of other parameters in the circuit using a complicated system of ten or more daunting partial differential equations which can be used to optimize the circuit design. However, it is usually not possible to solve such a system analytically, and to solve it numerically using the computer also has computational difficulties. In this situation, a statistical methodology known as design of experiments (DOE) can be used to find an optimal design of the circuit without going through the complicated method of solving partial differential equations. In other cases control may need to be exercised even on-linewhile the process is in progressbased on how the process is performing, to maintain product quality. Thus in statistical quality control problems are numerous and diverse. SQC engages the following three methodologies: 1. Acceptance Sampling This method is also called "sampling inspection." When products are required to be inspected but it is not feasible to inspect 100% of the products, samples of the product may be taken for inspection and conclusions drawn using the results of inspecting the samples. This technique specifies how to draw samples from a population and what rules to use to determine the acceptability of the product being inspected.


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2. Statistical Process Control (SPC) Even in an apparently stable production process, products produced are subject to random variations. SPC aims at controlling the variability of process output using a device called the control chart. On a control chart, a certain characteristic of the product is plotted. Under normal conditions these plotted points are expected to vary in a "usual way" on the chart. When abnormal points or patterns appear on the chart, it is a statistical indication that the process parameters or production conditions might have changed undesirably. At this point an investigation is conducted to discover unusual or abnormal conditions (e.g. tool breakdown, use of wrong raw material, temperature controller failure, etc.). Subsequently, corrective actions are taken to remove the abnormality. In addition to the use of control charts, SPC also monitors process capability, an indicator of the adequacy of the manufacturing process to meet customer requirements under routine operating conditions. In summary, SPC aims at maintaining a stable, capable and predictable process. Note, however, that since SPC requires processes to display measurable variation, it is ineffective for quality levels approaching six-sigma though it is quite effective for companies in the early stages of quality improvement efforts. 3. Design of Experiments Trial and error can be used to run experiments in the design of products and design of processes, in order to find an optimal setting of the parameters so that products of good quality will be produced. However, performing experiments by trial and error unscientifically is frequently very inefficient in the search of an optimal solution. Application of the statistical methodology of "design of experiments" (DOE) can help us in performing such experiments scientifically and systematically. Additionally, such methods greatly reduce the total effort used in product or process development experiments, increasing at the same time the accuracy of the results. DOE forms an integral part of Taguchi methodstechniques that produce high quality and robust product and process designs.

The Correct Use of Statistical Quality Control Methods The production of a product typically progresses as indicated in the simplified flow diagram shown in Figure 1.3. In order to improve the quality of the final product, design of experiments (DOE) may be used in Step 1 and Step 2, acceptance sampling may be used in Step 3 and Step 5, and statistical process control (SPC) may be used in Step 4.

Design of the Product

Design of the Process

Procurement of materials and parts

Production

Dispatch of the Product

Figure 1.3: Production from Design to Dispatch

There are several benefits that the SPC approach brings, as follows:  There are no restrictions as to the type of the process being controlled or studied, but the process tackled will be improved.  Decisions guided by SPC are based on facts not opinions. Thus a lot of 'emotion' is removed from problems by SPC.  Quality awareness of the workforce increases because they become directly involved in the improvement process.  Knowledge and experience of those who operate the process are released in a systematic way through the investigative approach. They understand their role in problem solving, which includes collecting facts, communicating facts, and making decisions.  Management and supervisors solve problems methodically, instead of by using a seat-ofthe-pants style. Prepared for the Extension Program of Indira Gandhi Open University

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Natural Variation of the process

Natural Variation of the process

Specification range

Specification range

Figure 1.4 A Capable Process: Natural variation is within spec range


Figure 1.5 A Process that is not Capable: Natural variation exceeds spec range

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PROCESS CAPABILITY: A DISCERNING MEASURE OF PROCESS PERFORMANCE

We introduce now an important concept employed in thinking statistically about real life processes. Process capability is the range over which the "natural variation" of a process occurs as determined by the system of common or random causes; that is, process capability indicates what the process can deliver under "stable" conditions when it is said to be under statistical control. The capability of a process is the fraction of output that can be routinely found to be within specifications. A capable process has 99.73% or more of its output within specifications (Figures 1.4 and 1.5). Process capability refers to how capable a process is of making parts that are within the range of engineering or customer specifications. Figure 1.4 shows the distribution of the dimension of parts for a machining process whose output follows the bell-shaped normal distribution. This process is capable because the distribution of its output is wholly within the spec range. The process shown by Figure 1.5 is not capable. Process Control on the other hand refers to maintaining the performance of a process at its current capability level. Process control involves a range of activities such as sampling the process product, charting its performance, determining causes of any excessive variation and taking corrective action. As mentioned above, the capability of a process is an expression of the comparison of product specs to the range of natural variability seen in the process. In simple terms, process capability expresses the proportion or fractional output that a process can routinely deliver within the specifications. A process when subjected to a capability study answers two key questions, "Does the process need to be improved?" and "How much does the process need to improved?" Knowing process capability allows manufacturing and quality managers to predict, quantitatively, how well a process will meet specs and to specify equipment requirements and the level of control necessary to maintain the firm's capability. For example, if a design specs require a length of metal tubing to be cut within one-tenth of an inch, a process consisting of a worker using a ruler and hacksaw will probably result in a large percentage of nonconforming product. In this case, the process, due to its high inherent or natural variability, is not capable of meeting the design specs. Management would face here three possible choices: (1) measure each piece and either re-cut or scrap nonconforming tubing, (2) develop a better process by investing in new technology, or (3) change the specifications. Such decisions are usually based on economics. Remember that under routine production, the cost to produce one unit of the product (i.e., its unit cost) whether the product ultimately ends up falling within or outside specs is the same. Rather, the firm may be forced to raise the market price of the within-spec products (those that are acceptable to customers) and thus weaken its competitive position. "Scrap and/or rework out-of-spec or defective parts" is therefore a poor business strategy since labour and materials have already been invested in the unacceptable product produced. Additionally, inspection errors will probably allow some nonconforming products to leave the production facility if the firm aims at making parts that just meet the specs. On the other hand, new technology might require substantial investment the firm cannot afford.


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Distribution of part dimensions

-3

-2

-

Mean

+

+2

+3

Natural variation of the process Lower Spec Limit

Upper Spec Limit

Figure 2.1: A Capable Process: Output is wholly within spec limits

Distribution of part dimensions Natural variation of the process -3

-2

-

Mean

Lower Spec Limit

+

+2

+3

Upper Spec Limit

Figure 2.2: A Process with Natural Variability equal to Spec Range Distribution of part dimensions

-3

-2

-

Mean

+

+2

+3

Natural variation of the process Spec Limits

Figure 2.3: A Process with Natural Variability wider than Spec Limits. This process is not capable.


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Distribution of part dimensions

-3

-2

-

Mean

+

+2

+3

Natural variation of the process Lower Spec Limit

Upper Spec Limit

Figure 2.4: An off-centered Process

Changes in design, on the other hand, may sacrifice fitness-for-use requirements and result in a lower quality product. Thus, these factors demonstrate the need to consider process capability during product design and in the acceptance of new contracts. Many firms now require process capability data from their vendors. Both ISO 9000 and QS 9000 quality management systems require a firm to determine its process capability. Process capability has three important components: (1) the design specifications, (2) the centering of the natural variation, and (3) the range, or spread, of variation. Figures 2.1 to 2.4 illustrate four possible outcomes that can arise when natural process variability is compared with product specs. In Figure 2.1, the specifications are wider than the natural variation; one would therefor expect that this process will always produce conforming products as long as it remains in control. It may even be possible to reduce costs by investing in a cheaper technology that permit a larger variation in the process output. In Figure 2.2, the natural variation and specifications are the same. A small percentage of nonconforming products might be produced; thus, the process should be closely monitored. In Figure 2.3, the range of natural variability is larger than the specification; thus, the current process would not always meet specifications even when it is in control. This situation often results from a lack of adequate communication between the design department and manufacturing, a task entrusted to manufacturing engineers. If the process is in control but cannot produce according to the design specifications, the question should be raised whether the specifications have been correctly applied or if they may be relaxed without adversely affecting the assembly or subsequent use of the product. If the specifications are realistic and firm, an effort must be made to improve the process to the point where it is capable of producing consistently within specifications. Finally, in Figure 2.4, the capability is the same as in Figure 2.2, but the process average is off-center. Usually this can be corrected by a simple adjustment of a machine setting or recalibrating the inspection equipment used to capture the measurements. If no action is taken, however, a substantial portion of output will fall outside the spec limits even though the process has the inherent capability to meet specifications. We may define the study of process capability from another perspective. A capability study is a technique for analyzing the random variability found in a production process. In every manufacturing process there is some variability. This variability may be large or small, but it is always present. It can be divided into two types:  Variability due to common (random) causes  Variability due to assignable (special) causes Prepared for the Extension Program of Indira Gandhi Open University

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The first type of variability is said to be inherent in the process and it can be expected to occur naturally within a process. It is attributed to a multitude of factors which behave like a constant system of the chances affecting the process. Called common or random causes, such factors include equipment vibration, passing traffic, atmospheric pressure or temperature changes, electrical voltage or humidity fluctuations, changes in operator's physical or emotional conditions, etc. Such are the forces that determine whether a coin when tossed will end up showing a head or tail when on the floor. Together, however, these "chances" form a unique, stable and describable distribution. The behaviour of a process operating under such conditions is predictable (Figure 2.5). Inherent variability may be reduced by changing the environment or the technology, but given a set of operating condition, this variability can never be completely eliminated from a process. Variability due to assignable causes, on the other hand, refers to the variation that can be linked to specific or special causes that disturb a process. Examples are tool failure, power supply interruption, process controller malfunction, adding wrong ingredients or wrong quantities, switching a vendor, etc. Predicted variability

Time

Figure 2.5: Common Causes of Variation Variability cannot be present, but no Assignable Causes predicted ? ? ? ? ? ?

Figure 2.6: Both Common and Assignable Causes affecting the process


?

Time ?

?

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Assignable causes are fewer in number and are usually identifiable through investigation on the shop floor or an examination of process logs. The effect (i.e., the variation in the process) caused by.an assignable factor, however, is usually large and detectable when compared with the inherent variability seen in the process. If the assignable causes are controlled properly, the total process variability associated with them can be reduced and even eliminated. Still, the effect of assignable causes cannot be described by a single distribution (Figure 2.6). A capability study measures the inherent variability or the performance potential of a process when no assignable causes are present (i.e., when the process is said to be in statistical control). Since inherent variability can be described by a unique distribution, usually a normal distribution, capability can be evaluated by utilizing the properties of this distribution. Recall that capability is the proportion of routine process output that remains within product specs. Even approximate capability calculations done using histograms enable manufacturers to take a preventive approach to defects. This approach is in contrast with the traditional two-step process: production personnel make the product while QC personnel inspect and screen out products that do not meet specifications. Such QC is wasteful and expensive since it allows plant resources including time and materials to be put into products that are not salable. It is also unreliable since even 100 percent inspection would fail to catch all defective products. SPC aims at correcting undesirable changes in the output of a process. Such changes may affect the centering (or accuracy) of the process, or its variability (spread or precision). These effects are graphically shown in Figure 2.

             

Output

Target

 

             

Distribution of output



 

Accuracy 

Accuracy 

Accuracy 

Accuracy 

Precision 

Precision 

Precision 

Precision 

Figure 2.7: Process Accuracy and Precision Lower Control Limit for xbar

Upper Control Limit for xbar

Distribution of xbar

-3

Distribution of individual measurements (x)

-2

-

Mean

+

+2

+3

Natural variation of the process {x} Lower Spec Limit for x

Upper Spec Limit for x

Figure 2.5: Distribution of Averages (xbar) compared to Distribution of Individuals (x) Prepared for the Extension Program of Indira Gandhi Open University

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Control Limits are Not an Indication of Capability Those new to SPC often have the misconception that they don't need to calculate capability indices. Some even think that they can compare their control limits to the spec limits. This is not true, because control limits look at the distribution of averages (xbar, p, np, u, etc.) while capability indices look at the distribution of individual measurements (x). The distribution of x for a process will always be more spread out than the distribution of its xbar values (Figure 2.5). Therefore, the control limits are often within the specification limits but the plus-andminus-3-sigma distribution of individual part dimensions (x) is not. The statistical theory of the "central limit theorem" says that the averages of samples or subgroups {xbar} follow more closely a normal distribution. This is why we can easily construct control charts on process data that are themselves not normally distributed. But averages cannot be used for capability calculation because capability evaluates individual parts delivered by a process. After all, parts get shipped to customers, not averages.

What Capability Studies Do for You Capability studies are most often used to quickly determine whether a process can meet specs or how many parts will exceed the specifications. However, there are numerous other practical uses:  Estimating percentage of defective parts to be expected  Evaluating new equipment purchases  Predicting whether design tolerances can be met  Assigning equipment to production  Planning process control checks  Analyzing the interrelationship of sequential processes  Making adjustments during manufacture  Setting specifications  Costing out contracts Since a capability study determines the inherent reproducibility of parts created in a process, it can even be applied to many problems outside the domain of manufacturing, such as inspection, administration, and engineering. There are instances where capability measurements are valuable even when it is not practical to determine in advance if the process is in control. Such an analysis is called a performance study. Performance studies can be useful for examining incoming lots of materials or onetime-only production runs. In the case of an incoming lot, a performance study cannot tell us that the process that produced the materials is in control, but it may tell us by the shape of the distribution what percent of the parts are out of specs or more importantly, whether the distribution was truncated by the vendor sorting out the obvious bad parts.

How to set up a Capability Study Before we set up a capability study, we must select the critical dimension or quality characteristic (must be a measurable variable) to be examined. This dimension is the one that must meet product specs. In the simplest case, the study dimension is the result of a single, direct product and measurement process. In more complicated studies, the critical dimension may be the result of several processing steps or stages. It may become necessary in these cases to perform capability studies on each process stage. Studies on early process stages frequently prove to be more valuable than elaborate capability studies done on later processes since early processes lay the foundation (i.e., constitute the input) which may affect later operations. Prepared for the Extension Program of Indira Gandhi Open University

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Once the critical dimension is selected, data measurements can be collected. This can be accomplished manually or by using automatic gaging and fixturing linked to a data collection device or computer. When measurements on a critical dimension are made, it is important we ensure that the measuring instrument is as precise as possible, preferably one order of magnitude finer than the specification. Otherwise, the measuring process itself will contribute excess variation to the dimension data as recorded. Using handheld data collectors with automatic gages may help reduce errors introduced by the process of measurement, data recording, and transcription for post processing by computer. The ideal situation for data collection is to collect as much data as possible over a defined time period. This will yield a reliable capability number since it is based upon a large sample size. In the practice of process improvement, determining process capability is Step 5: Step 1 Step 2 Step 3 Step 4 Step 5 Step 6

Gather process data Plot the data on control charts. Find the control limits. Get the process in control (in other words, identify and eliminate assignable causes). Calculate process capability. If process capability is not sufficient, improve the process (reduce its inherent variation), and go back to Step 1.

Capability Calculations Condition 1: The Process Must be in Control! Process capability formulas commonly used by industry require that the process must be in control and normally distributed before one takes samples to estimate process capability. All standard capability indices assume that the process is in control and the individual data follow a normal distribution. If the process is not in control, capability indices are not valid, even if they appear to indicate the process is capable. Three different statistical tools are used together to determine whether a process is in control and follows a normal distribution. These are  Control charts  Visual analysis of a histogram  Mathematical analysis of the distribution to test that the distribution is normal. Note that no single tool can do the job here and all three must be used together. Control charts (discussed in detail later in this Unit) are the most common method for maintaining a process in statistical control. For a process to be in control, all points plotted on the control chart must be inside the control limits with no apparent patterns (e.g., trends) be present. A histogram (described below in Section 3) allows us to quickly see (a) if any parts are outside the spec limits and (b) what the distribution's position is relative to the specification range. If the process is one that is naturally a normal distribution, then the histogram should approximate a bell-shaped curve if the process is in control. However, note that a process can be in control but not have its individuals following a normal distribution if the process is inherently non-normal.

Capability Calculations Condition 2: The Process Must be Inherently Normal Many process naturally follow a bell-shaped curve (a normal distribution) but some do not. Examples of non-normal dimensions are roundness, squareness, flatness and positional tolerances; they have a natural barrier at zero. In these cases, a perfect measurement is zero (for example, no ovality in the roundness measurement). There can never be a value less than zero. The standard capability indices are not valid for such non-normal distributions. Tests for normality are available in SPC text books that can assist you to identify whether or not a process is normal. If a process is not normal, you may have to use special capability measures that apply to non-normal distributions [1].


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Capability Index Formulas We first define some special terms. USL stands for Upper Specification Limit and LSL for Lower Specification Limit. Midpoint is the center of the spec limits. The midpoint is also frequently referred to as the nominal value or the target. Tolerance is the distance between the upper and lower spec limits (tolerance = USL - LSL). The standard deviation for the distribution of individual data, one important variable in all the capability index calculations, can be determined in either of two ways. The standard text book formula for , the population standard deviation, when the true process mean () is known and population size (N) is finite is N

 

(x i 1

i

  )2

N

The term population here means all parts produced, not just a sample. In practice, however, we usually we work with a sample of the population, a handful of items collected or sampled from the production line, since this is more practical. In this case, the formula for standard deviation (s) is as follows. n

s 

(x i 1

i

 xbar) 2

n 1

when xbar is the sample average given by

xbar 

n

x i 1

i

/n

s symbolizes sample standard deviation and n is the sample size. s is an estimator of . The standard deviation of a distribution is an indication of the dispersion (or variability) present in the population of the datathe higher is variability, the larger will be s (and ). When over 30 individual observations are taken, the above formulas for population standard deviation () and sample standard deviation (s) yield virtually the same numerical result. In the following pages we will use the symbol  to denote the term standard deviation as we continue our discussion of process capability indices. Standard deviation may also be estimated using Rbar (the average of the sample or subgroup ranges Ri) and a constant that has been developed by statisticians for this purpose. The formula for estimating sigma is:

ˆ 

Rbar d2

" hat" represents the estimated standard deviation. Rbar is the average of the sample ranges {Ri} for a sample period i when the process is in control. The constant d2 varies by sample size (n) and is listed in Table 2.1. It is important that you remember that the process data or the individual measurements {xi} must be normally distributed and in control in order to use the estimated value of  in process capability calculations. If both of these conditions are not met, the estimated standard deviation or  value will not be valid. If the process is normally distributed and in control, either method is acceptable and usually yields about the same result. Also, remember that Prepared for the Extension Program of Indira Gandhi Open University

14 neither actual nor estimated  used in calculating capability will be meaningful if the process is inherently non-normal. Separate special methods are available for finding capability measures for non-normal distributions arising from non-normal processes.

TABLE 2.1: FACTORS FOR CALCULATING CONTROL CHART LIMITS

n 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

A 2.121 1.732 1.500 1.342 1.225 1.134 1.061 1.000 0.949 0.905 0.866 0.832 0.802 0.775 0.750 0.728 0.707 0.688 0.671 0.655 0.640 0.626 0.612 0.600

xbar Chart A2 A3 1.880 2.659 1.023 1.954 0.729 1.628 0.577 1.427 0.483 1.287 0.419 1.182 0.373 1.099 0.337 1.032 0.308 0.975 0.285 0.927 0.266 0.886 0.249 0.850 0.235 0.817 0.223 0.789 0.212 0.763 0.203 0.739 0.194 0.718 0.187 0.698 0.180 0.680 0.173 0.663 0.167 0.647 0.162 0.633 0.157 0.619 0.153 0.606

c4 0.7979 0.8862 0.9213 0.9400 0.9515 0.9594 0.9650 0.969 0.9727 0.9754 0.9776 0.9774 0.9810 0.9823 0.9835 0.9845 0.9854 0.9862 09869 0.9876 0.9882 0.9887 0.9892 0.9896

B3 0 0 0 0 0.030 0.118 0.185 0.239 0.284 0.321 0.354 0.382 0.406 0.428 0.448 0.466 0.482 0.497 0.510 0.523 0.534 0.545 0.555 0.565

s Chart B4 B5 3.267 0 2.568 0 2.266 0 2.089 0 1.970 0.029 1.882 0.113 1.815 0.179 1.761 0.232 1.716 0.276 1.679 0.313 1.646 0.346 1.618 0.374 1.594 0.399 1.572 0.421 1.552 0.440 1.534 0.458 1.518 0.475 1.503 0.490 1.490 0.504 1.477 0.516 1.466 0.528 1.455 0.539 1.445 0.549 1.435 0.559

B6 2.606 2.276 2.088 1.964 1.874 1.806 1.751 1.707 1.669 1.637 1.610 1.585 1.563 1.544 1.526 1.511 1.496 1.483 1.470 1.459 1.448 1.438 1.429 1.420

d2 1.128 1.693 2.059 2.326 2.534 2.704 2.847 2.970 3.078 3.173 3.258 3.336 3.407 3.472 3.532 3.588 3.640 3.689 3.735 3.778 3.819 3.858 3.895 3.931

d3 0.853 0.888 0.880 0.864 0.848 0.833 0.820 0.808 0.797 0.787 0.778 0.770 0.763 0.756 0.750 0.744 0.739 0.734 0.729 0.724 0.720 0.716 0.712 0.708

R Chart D1 D2 0 3.686 0 4.358 0 4.698 0 4.918 0 5.078 0.204 5.204 0.388 5.306 0.547 5.393 0.687 5.469 0.811 5.535 0.922 5.594 1.025 5.647 1.118 5.696 1.203 5.741 1.282 5.782 1.356 5.820 1.424 5.856 1.487 5.891 1.549 5.921 1.605 5.951 1.659 5.979 1.710 6.006 1.759 6.031 1.806 6.056

D3 0 0 0 0 0 0.076 0.136 0.184 0.223 0.256 0.283 0.307 0.328 0.347 0.363 0.378 0.391 0.403 0.415 0.425 0.434 0.443 0.451 0.459

If you have both estimated and actual capability indices available, choose one method and stay with it. Avoid the temptation to look at both and choose the one that is better, since this will introduce variation in results.

The Cp Index The most commonly used capability indices are Cp and Cpk. Cp, a measure of the dispersion of the process, is the ratio of tolerance to 6. The formula is

Cp 

Tolerance 6

The quantity "6" in the Cp formula is derived from the fact that, in a normal distribution, 99.73% of the parts will be within a 6 apread, i.e., within (  3) when the process is being disturbed only by random or chance causes. Suppose that the tolerance and measurements on a process are as follows: USL = 5.0, LSL = 1.0, midpoint of spec = 3.0, average of sample averages ( xbar ) = 2.0,  = 0.5. Then tolerance = (5.0 - 1.0) = 4.0, xbar + 3 = 3.5 and xbar - 3 = 0.5. Then the CP for the process output data sampled is 4.0/3.0 = 1.33.


D4 3.267 2.574 2.282 2.114 2.004 1.924 1.864 1.816 1.777 1.744 1.717 1.693 1.672 1.653 1.637 1.622 1.608 1.597 1.585 1.575 1.566 1.557 1.548 1.541

15

As you can see from the Cp formula, values for Cp can range from near zero to very large positive numbers. When Cp is less than 1, tolerance is less than the 6the inherent or natural spread of the process output. When Cp is greater than 1, tolerance is greater than the 6. For a process, the greater the Cp index number, the higher its process capability is.

" Cp" alone, however, does not tell the complete story about the process! Note that Cp is only a measure of the dispersion or spread of the distribution. It is not a measure of the "centeredness" (where the mean of the process output is in relation to the spec midpoint). An off-centered process could be a problematic situation. An example is the situation shown by Figure 2.4. Both Figures 2.2 and 2.4 display processes that have a Cp = 1.0. But the process shown by Figure 2.4 has a significant fraction of its output falling outside (below) the lower spec limit. This is why Cp is never used alone as a measure of process capability. Cp only shows how good would the process be if the process could be centered with some adjustments. The alternative capability index is Cpk, described below.

The Cpk Index While Cp is only a measure of dispersion, Cpk measures both dispersion and centeredness. The Cpk foirmula takes into account both the process spread and the location of the process average in relation to the spec midpoint. The formula is as follows.

 USL  mean   mean  LSL  C pk  The _ lesser _ of  and  3 3     "The lesser of" actually determines how capable the process is on the worst side. Using the data of the previous example we obtain Cpk = The lesser of (5.0 - 2.0)/1.5 or (2.0 - 1.0)/1.5 = Min (2.0, 0.67) = 0.67 The greater the Cpk value is the higher is the fraction of the output meeting specs. Hence, the better is the process. A Cpk value greater than 1.0 means that the 6 spread of the data falls completely within the spec limits. An example is the process shown in Figure 1.4. A Cpk value of 1.0 indicates that 99.73% of the parts produced by the process would be within the spec limits. In this process only about 3 out of a thousand parts would be scrapped or rejected. In other words, such a process just meets specs. Do we need to improve the process (i.e., reduce its inherent variability) further? Improvement beyond just meeting specs may greatly improve the quality of fitness of the parts during assembly and also cut warranty costs. The different special process conditions detectable by Cpk calculations are as follows.


16

Cpk value > 1.0

Process Output Distribution All output completely within spec limits

= 1.0

One end of the 6 spread falls on a spec limit, the other may be within the other spec limit

0  Cpk  1.0

Part of the 6 spread falls outside spec limit

Negative Cpk , i.e., Cpk  0.0

Process mean is not within spec limits

Many companies with QS 9000 registration, demand their vendors to demonstrate Cpk capabilities of 1.33 or beyond. A Cpk of 1.33 has about 99.994% of products within specs. A Cpk value of 2.0 is the coveted "six sigma" quality level. To reach this stage advanced SQC methods including design-of-experiments (DOE) would be required. At this level no more than 3 or 4 parts per million products produced would fall outside the spec limits. Such small variation is not visible on xbar-R control charts in the normal operation of the process. Remember that control and capability are two very different concepts. As shown in Figure 2.6, in general, a process may be capable or not capable, or in control or out of control, independently of each other. Clearly, we would like every process to be both capable and in (statistical) control. If a process is neither capable nor in control, we must take two corrective actions to improve it. First we should get it in a state of control by removing special causes of variation, and then attack the common causes to improve its capability. If a process is capable but not in control (as the above example illustrated), we should work to get it back in control. Control state In Control Capable Capability state

Out of Control

Ideal process

Not Capable

Figure 2.6 Capability versus Control (Arrows indicate the Directions of Appropriate Management Action). Statistical Process Control (SPC) Methodology Control charts, like the other basic tools for quality improvement, are relatively simple to use. In general, control charts have two objectives, (a) help restore accuracy of the process so that the process average stays near the target, and (b) help minimize variation in the process to ensure that good precision is maintained in the output (see Figure 2.7). Control charts have three basic applications: (1) to establish a state of statistical control, (2) to monitor a process and signal when the process goes out of control, and (3) to determine process capability. The following is a summary of the steps required to develop and use control charts. Steps 1 through 4 focus on establishing a state of statistical control; in step 5, the charts are used for ongoing monitoring; and finally, in step 6, the data are used for process capability analysis. Prepared for the Extension Program of Indira Gandhi Open University

17

1.

2.

3.

4.

5.

6.

Preparation a. Choose the variable or attribute to be measured b. Determine the basis, size, and frequency of sampling. c. Set up the correct control chart. Data Collection a. Record the data b. Calculate relevant statistics: averages, ranges, proportions, and so on. c. Plot the statistics on the chart. Determination of trial control limits a. Draw the center line (process average) on the chart. b. Compute the upper and lower control limits. Analysis and interpretation a. Investigate the chart for lack of control b. Eliminate out-of-control points. c. Re-compute control limits if necessary. Use as a problem-solving tool a. Continue data collection and plotting. b. Identify out-of-control situations and take corrective action. Use the control chart data to determine process capability, if desired.

In Section 3 we review the "seven quality improvement tools"simple methods popularized by the Japanesethat can do a great deal in bringing a poorly performing process into control and then to improve it further. In Section 4 we discuss the SPC methodology in detail and the construction, interpretation, and use of the different types of process control charts. Although many different charts will be described, they will differ only in the type of measurement for which the chart is used; the same analysis and interpretation methodology described applies to each of them.


18

3

THE SEVEN QUALITY IMPROVEMENT TOOLS

In SPC, numbers and information form the basis for decisions and actions. Therefore, a thorough data recording systemmanual or otherwisewould be an essential enabler for SPC. In order to allow one to interpret fully and derive maximum use of quality-related data, over the past fifty years a set of simple statistical 'tools' have evolved. These tools offer any organization a easy means to collect, present and analyze most of such data. In this section we briefly review these tools. An extended description of them may be found in the quality management standard ISO 9004-4 (1994). In the 1950's Japanese industry began to learn and apply statistical methods in earnestness, methods that American statisticians Walter Shewhart and W Edward Deming developed in the 1930's and 1940's to help manage quality. Subsequently, progress in continuous quality improvement in Japan led to significant expansion of the many simple statistical tools on shop floor. Kaoru Ishikawa, head of the Japanese Union of Scientists and Engineers (JUSE), later formalized the use of these tools in Japanese manufacturing with the introduction of the 7 Quality Control (7 QC) tools. The seven Ishikawa tools reviewed below are now an integral part of quality control on the shop floor around the world. Many Indian industries use them routinely.

3.1

Flowchart

The flowchart lists the order of activities in a project or process and their interdependency. It expresses detailed process knowledge. To express this knowledge certain standard symbols are used. The oval symbol indicates the beginning or end of the process. The boxes indicate action items while diamonds indicate decision or check points. The flowchart can be used to identify the steps affecting quality and the potential control points. Another effective use of the flowchart would be to map the ideal process and the actual process and to identify their differences as the targets for improvements. Flowcharting is often the first step in Business Process Reengineering (BPR).

START

no MIX

CHECK

RE-MIX

ok STOP

3.2

Histogram

The histogram is a bar chart showing a distribution of variable quantities or characteristics. An example of a "live" histogram would be to line up by height a group of students enrolled in a course. Normally, one individual would be the tallest and one the shortest, with a cluster of individuals bunched around the average height. In manufacturing, the histogram can rapidly identify the nature of quality problems in a process by the shape of the distribution as well as the width of the distribution. It informally establishes process capability. It can also help compare two or more distributions.


19

Lower Spec Limit

35

Upper

 Tolerance  Spec Limit

30 25

Frequency

20 15 10 5 5.1

5.2

5.3

5.4

5.5

5.6

5.7

Weight (gms)

3.3

Pareto Chart

The Pareto chart, as shown below, indicates the distribution of effects attributable to various causes or factors arranged from the most frequent to the least frequent. This tool is named after Wilfredo Pareto, the Italian economist who determined that wealth is not evenly distributed and some of the people have most of the money. This tool is a graphical picture of the relative frequencies of different types of quality problems with the most frequent problem type obtaining clear visibility. Thus the Pareto chart identifies the vital few and the trivial many and it highlights problems that should be worked first to get the most improvement. Historically, 80% problems are caused by 20% of the factors.

35% 30% 25%

Percent defective

20% 15% 10 % 5%

P o o r d e s i g n

d i m e n s i o n

Operator errors Calibration

p a r t s

Material Misc.

Categories

3.4

Cause and Effect Diagram

The cause and effect diagram is also called the fishbone chart because of its appearance and the Ishikawa diagram after the man who popularized its use in Japan. Its most frequent use is to list the causes of some particular quality problem or defect. The lines coming of the core horizontal line are the main causes while the lines coming off those are subcauses. The cause and effect diagram identifies problem areas where data should be collected and analyzed. It is used to develop reaction plans to help investigate out-of-control points found on control charts. It is also the first step for planning design of experiments (DOE) studies and for applying Taguchi methods to improve product and process designs.


20

Man

Machine speed

training

attention tool

Defect mixing

quality

quantity inspection

Methods

3.5

Materials

Scatter Diagram

The scatter diagram shows any existing the pattern in the relationship between two variables that are thought to be related. For example, is there a relationship between outside temperature and cases of the common cold? As temperatures drop, do cases of the common cold rise in number? The closer the scatter points hug a diagonal line, the more closely there is one-to-one relationship between the variables being studied. Thus, the scatter diagram may be used to develop informal models to predict the future based on past correlations.

Negative Correlation observed

20 Cases of Common 15 Cold/100 10 persons 5 0

30 40 50 60 70 Outdoor Temperature (F)

3.6

Run Chart

The run chart shows the history and pattern of variation. It is plot of data points in time sequence, connected by a line. Its primary use is in determining trends over time. The analyst should indicate on the chart whether up is good or down is good. This tool is used at the beginning of the change process to see what the problems are. It is also used at the end (or check) part of the change process to see whether the change made has resulted in a permanent process improvement.

A Run Chart of Sample Averages

Average

0.85 0.8 0.75 0.7

Sample # Prepared for the Extension Program of Indira Gandhi Open University

29

27

25

23

21

19

17

15

13

11

9

7

5

3

1

0.65

21

3.7

Control Chart

Whereas a histogram gives a static picture of process variability, a run chart or a control chart illustrates the dynamic performance (i.e., performance over time) of the process. The control chart in particular is a powerful process quality monitoring device and it constitutes the core of statistical process control (SPC). It is a line chart marked with control limits at 3 standard deviations () above and below the average quality level. These limits are based on the statistical studies of shop data conducted in the 1030s by Dr Walter Shewhart. By comparing certain measures of the process output such as xbar, R, p, u, c etc. (see Section 4) to their control limits one can determine quality variation that is due to common or random causes and variation that is produced by the occurrence of assignable events (special causes). Failure to distinguish between common causes and special causes of variation can actually increase the variation in the output of a process. This is often due to the mistaken belief that whenever process output is off target, some adjustment must be made. However, knowing when to leave a process alone is an important step in maintaining control over a process. Equally important is knowing when to take action to prevent the production of nonconforming product. Using actual industry data Shewhart demonstrated that a sensible strategy to control quality is to first eliminate the special causes with the help of the control chart and then systematically reduce the common causes. This strategy reduces the variation in process output with a high degree of reliability while it improves the acceptability of the product. Statistical process control (SPC) is actually a methodology for monitoring a process to (a) identify the special causes of variation and (b) signal the need to take corrective action when it is appropriate. When special causes are present, the process is deemed to be out of control. If the variation in the process is due to common causes alone, the process is said to be in statistical control. A practical definition of statistical control is that both the process averages and variances are constant over time (Figure 2.5). Such a process is stable and predictable. SPC uses control charts as the basic tool to improve both quality and productivity. SPC provides a means by which a firm may demonstrate its quality capability, an activity necessary for survival in today's highly competitive markets. Also, many customers (e.g., the automotive companies) now require the evidence that their suppliers use SPC in managing their operations. Note, however, that since SPC requires processes to display measurable variation; even though it is quite effective for companies in the early stages of quality efforts, it becomes ineffective in producing improvements once quality level approaches six-sigma. Before we leave this section, we repeat again that process capability calculations make little sense if the process is not in statistical control because the data are confounded by special (or assignable) causes and thus do not represent the inherent capability of the process. The simple tools described in this section may be good enough to enable you to check this. To see this, consider the data in Table 3.1, which shows 150 measurements of a quality characteristic from a manufacturing process with specifications 0.75  0.25. Each row corresponds to a sample of size = 5 taken every 15 minutes. The average of each sample is also given in the last column. A frequency distribution and histogram of these data is shown in Figure 3.1. The data form a relatively symmetric distribution with a mean of 0.762 and standard deviation 0.0738. Using these values, we find that Cpk = 1.075 and form the impression that the process capability is at least marginally acceptable. Some key questions, however, remain to be answered. Because the data were taken over an extended period of time, we cannot determine if the process remained stable throughout that period. In a histogram the dimension of time is not considered. Thus, histograms do not allow you to distinguish between common and special causes of variation. It is unclear whether any special causes of variation are influencing the capability index. If we plot the average of each sample against the time at which the sample was taken (since the time increments between samples are equal, the sample number is an appropriate surrogate for time), we obtain the run chart shown in Figure 3.6.


22

Upper limit Frequency LSL = 0.5 0 0.55 1 0.6 1 0.65 10 0.7 14 0.75 40 0.8 31 0.85 37 0.9 14 0.95 1 USL = 1 0 More 0 Average 0.762 Std Dev 0.0738

Histogram Frequency

50 40 30 20 10

USL = 1

0.9

0.8

0.7

0.6

LSL = 0.5

0

Bin

Figure 3.1 Frequency Distribution and Histogram of a Typical Process

Table 3.1 Thirty samples of Quality Measurements. Sample # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

X1 0.682 0.787 0.780 0.591 0.693 0.749 0.791 0.744 0.769 0.718 0.787 0.622 0.657 0.806 0.660 0.816 0.826 0.828 0.805 0.802 0.876 0.855 0.762 0.703 0.737 0.748 0.826 0.728 0.803 0.774

X2 0.689 0.860 0.667 0.727 0.708 0.714 0.713 0.779 0.773 0.671 0.821 0.802 0.822 0.749 0.681 0.817 0.777 0.829 0.719 0.756 0.803 0.783 0.705 0.837 0.723 0.686 0.803 0.721 0.892 0.837

X3 0.776 0.601 0.838 0.812 0.790 0.738 0.689 0.660 0.641 0.708 0.764 0.818 0.893 0.859 0.644 0.768 0.721 0.865 0.612 0.786 0.701 0.722 0.804 0.759 0.776 0.856 0.764 0.820 0.740 0.872

X4 0.798 0.749 0.785 0.775 0.758 0.719 0.877 0.737 0.644 0.850 0.658 0.872 0.544 0.801 0.747 0.716 0.770 0.778 0.938 0.815 0.789 0.856 0.805 0.975 0.748 0.811 0.823 0.772 0.816 0.849

X5 0.714 0.779 0.723 0.730 0.671 0.606 0.603 0.822 0.725 0.712 0.708 0.727 0.750 0.701 0.728 0.649 0.809 0.872 0.807 0.801 0.672 0.751 0.809 0.732 0.732 0.838 0.886 0.639 0.770 0.818

Average 0.732 0.755 0.759 0.727 0.724 0.705 0.735 0.748 0.71 0.732 0.748 0.768 0.733 0.783 0.692 0.753 0.781 0.834 0.776 0.792 0.768 0.793 0.777 0.801 0.743 0.788 0.82 0.736 0.804 0.83

The run chart hints that the mean might have shifted up at about sample #1 In fact, the average for the first 16 samples turns out to be 0.738 while for the remaining samples it is 0.789. Therefore, although the overall average is close to the target specification (0.75), at no time was the actual process operating centered near the target. In the next section you will see why we should conclude that this process is not in statistical control and therefore we should not pay much attention to the process capability index Cpk calculated as 1.075. SPC exists because there is, and will always be, variation in the characteristics of materials, in parts, in services, and in people. With the help of the simple tools described in this section SPC can provide us the means to understand and assess such variability, and then manage it.


23

4

CONTROL CHARTS FOR VARIABLES DATA

As we mentioned in the previous section, the control chart is a powerful process quality monitoring device and it constitutes the core of statistical process control (SPC). In the SPC methodology, knowing when to leave a process alone is an important step in maintaining control over a process. Control charts enable us to do that. Equally important is knowing when to take action to prevent the production of nonconforming product. Indeed, failure to distinguish between variation produced by common causes and special causes can actually increase the variation in the output of a process. Again, control charts empower us here. When the chart crosses any of its control limits, special causes are indicated to be present. The process is now deemed to be out of control and is investigated to remove the source of disturbance. Otherwise, when variation stays within control limits, it is indicated to be due to common causes alone. Now the process is said to be in "statistical control" and it should be left alone. Statistical control is defined as the state in which both the process averages and variances are constant over time and hence the process output is stable and predictable (Figure 2.5). Control charts help us in bringing a process within such control. Most process that deliver a "product" or a "service" may be monitored by measuring their output over time and then plotting these measurements appropriately. However, processes differ in the nature of those output. Variables data are those output characteristics that are measurable along a continuous scale. Examples of variables data are length, weight, or viscosity. By contrast, some output may only be judged to be good or bad, or "acceptable" or "unacceptable", such as print quality of a photocopier or defective knots produced per meter by a weaving machine. In such cases we categorize the output as an attribute that is either acceptable or unacceptable; we cannot put it on a continuous scale as done with weight or viscosity. However, SPC methodology provides us with a variety of different types of control charts to work with such diversity. For variables data control charts most commonly used are the "xbar" chart and the "R-chart" (range chart). The xbar chart is used to monitor the centering of the process to help control its accuracy (Figure 2.4). The R-chart monitors the dispersion or precision of the process. Range R rather than standard deviation is used as a measure of variation simply to enable workers on the factory floor perform control chart calculations by hand, as done for example in the turbine blade machining shop in BHEL, Hardwar. For large samples and when data can be processed by a computer, the standard deviation is a better measure of variability.

4.1

Constructing xbar and R-Charts and Establishing Statistical Control

The first step in developing xbar and R-charts is to gather data. Usually, about 25 to 30 samples are collected. Samples between size 3 and 10 are generally used, with 5 being the most common. The number of samples is indicated by k, and n denotes to sample size. For each sample i, the mean (denoted by xbari) and the range (Ri) are computed. These values are then plotted on their respective control charts. Next, the overall mean (x_doublebar) and average range (Rbar) calculations are made. These values specify the center lines for the xbar and R-charts, respectively. The overall mean is the average of the sample means xbari. k

x  x  ...  xn xbar  1 2 n

x _ doublebar 

The average range is similarly computed, using the formulas Prepared for the Extension Program of Indira Gandhi Open University

 xbar

i

i 1

k

24

Ri  max( xi )  min( xi ) k

Rbar 

R i 1

i

k

The average range and average mean are used to compute control limits for the R-and xbar charts. Control limits are easily calculated using the following formulas: UCLR = D4 Rbar LCLR = D3 Rbar UCLxbar = x_doublebar + A2 Rbar LCLxbar = x_doublebar - A2 Rbar where the constants D3, D4 and A2 depend on sample size n and may be found in Table 2.1. Control limits represent the range between which all points are expected to fall if the process is in statistical control, i.e., operating only under the influence of random or common causes. If any points fall outside the control limits or if any unusual patterns are observed, then some special (called assignable) cause has probably affected the process. In such case the process should be studied using a "reaction plan" (Figure 4.1), process logs and other tools and devices to determine and eliminate that cause. Note, however, that if assignable causes are affecting the process, then the process data are not representative of the true state of statistical control and hence the calculations of the center line and control limits would be biased. To be effective, SPC requires the center line and the control limit calculations to be unbiased. Therefore, before control charts are set up for roitine use by the factory, any out-or-control data points should be eliminated from the data table and new values for x_doublebar, Rbar, and the control limits re-computed, as illustrated below. In order to determine whether a process is in statistical control, the R-chart is always analyzed first. Since the control limits in the xbar chart depend on the average range, special causes in the R-chart may produce unusual patterns in the xbar chart, even when the centering of the process is in control. (An example of this is given later in this unit). Once statistical control is established for the R-chart, attention may turn to the xbar chart.

Figure 4.2 A Control Chart Data Sheet Part Name: Operator: Date Time Sample1 2 3 4 5 xbar R Notes

Process Step: Machine:

Gage:


Spec Limits: Unit of Measure:

25

Figure 4.2 shows a typical data sheet used for recording. This form provides space (under "Notes") for descriptive information about the process and for recording of sample observations and computed statistics. Subsequently the control charts are drawn. The construction and analysis of control charts may be best seen by example as follows. Example 3: Control Charts for Silicon Wafer Production. The thickness of silicon wafers used in the production of semiconductors must be carefully controlled during wafer manufacture. The tolerance of one such wafer is specified as  0.0050 inches. In one production facility, three wafers were randomly selected each hour and the thickness measured carefully (Figure 4.3). Subsequently, xbar and R were calculated. For example, the average of the first sample was xbar1 = (41 + 70 + 22)/3 = 113/3 = 44.33 The range of the first sample was Ri = 70-22 = 48. (Note: In practice, xbar and R calculations may be rounded to the nearest integer for simplicity). The calculations of sample averages, range, overall mean, and the control limits are shown on the worksheet displayed in Figure 4.3. The average range is the sum of the sample ranges (676) divided by the number of samples (25); the overall mean is the sum of the sample averages (1,221) divided by the number of samples (25). Since sample size is 3 here, the factors used in computing the control limits are A2 = 1.023 and D4 = 2.574 (Table 2.1). For sample size of 6 or less, factor D3 is 0; therefore, the lower control limit (LCL) on the range chart is zero. The center line and control limits are drawn on the charts shown in Figure 4.3. Note that as a convention, out-of-control points are noted directly on the charts. On examining the R chart first we infer that the process is in control because all points lie within the control limits and no unusual patterns exist. On the xbar chart, however, the xbar value for sample 17 lies above the upper control limit. On investigation we find that some suspicious cleaning material had been used in the process at this point (an assignable cause of variation). Therefore, data from sample 17 should be eliminated from the control chart calculations and the control limits re-done. Figure 4.4 shows the revised calculations after sample 17 was removed. The revised center lines and control limits are shown. The resulting xbar and R charts both appear to be in control.

Figure 4.3 Silicon Wafer Thickness Data as observed Date Time 1 2 3 4 5

12/1 12/1 12/1 12/1 12/1 12/1 13/1 13/1 13/1 13/1 13/1 13/1 14/1 14/1 14/1 14/1 14/1 14/1 15/1 15/1 15/1 15/1 15/1 15/1 16/1 8:00 11:0 2:00 5:00 8:00 11.0 8:00 11:0 2:00 5:00 8:00 11.0 8:00 11:0 2:00 5:00 8:00 11.0 8:00 11:0 2:00 5:00 8:00 11.0 8:00

41 78 84 60 46 64 43 37 50 57 24 78 51 41 56 46 99 71 41 41 22 62 64 44 41 70 53 34 36 47 16 53 43 29 83 42 48 57 29 64 41 86 54 2 39 40 70 52 38 63 22 68 48 25 29 56 64 30 57 32 39 39 50 35 36 16 98 39 53 36 46 46 57 60 62

Calculations: Sum

133 199 166 121 122 136 160 110 136 172 105 165 158 105 156 103 283 164 96 116 108 178 173 142 166

Average 44. 66. 55. 40. 40. 45. 53. 36. 45. 533 35

= Xbar Range

=R Notes:

33 48 Gas flow adju sted

33 25

33 50

33 35

67 18

33 48

33 21

67 13

33 28

51

18

AC malf uncti on


55 39

52. 67 7 Pum p stall ed once

35

52

12

28

34. 94. 54. 33 33 67 30 13 32 New clea ning tried

32 51

38. 67 5

36 24

59. 567 433 55. 33 33 24 12 22 22

26

Control Limit Calculations Calculation basis: Rbar X_doublebar Spec Midpoint A2  Rbar UCLxbar = x_doublebar + A2  Rbar LCLxbar = x_doublebar - A2  Rbar UCLR = D4  Rbar UCLR = D3  Rbar

All Subgroups included 676/25 = 27 1221/25 = 48.8 50.0 1.02327 = 26 76.4 21.2 69.5 0.0

The Initial xbar Control Chart 1 Average 44. = Xbar 33 x_double 48. bar 8 76.4 UCL Sample #

xbar

LCLxbar

2 66. 33 48. 8 76.4

3 55. 33 48. 8 76.4

4 40. 33 48. 8 76.4

5 40. 67 48. 8 76.4

6 45. 33 48. 8 76.4

7 53. 33 48. 8 76.4

8 36. 67 48. 8 76.4

9 45. 33 48. 8 76.4

10 11 533 35

12 55

13 14 15 16 17 52. 35 52 34. 94. 67 33 33 48. 48. 48. 48. 48. 48. 48. 48. 8 8 8 8 8 8 8 8 76.4 76.4 76.4 76.4 76.4 76.4 76.4 76.4

18 19 20 21 22 54. 32 38. 36 59. 67 67 33 48. 48. 48. 48. 48. 8 8 8 8 8 76.4 76.4 76.4 76.4 76.4

23 24 25 567 433 55. 33 48. 48. 48. 8 8 8 76.4 76.4 76.4

21.2 21.2 21.2 21.2 21.2 21.2 21.2 21.2 21.2 21.2 21.2 21.2 21.2 21.2 21.2 21.2 21.2 21.2 21.2 21.2 21.2 21.2 21.2 21.2 21.2

Point beyond UCL, i.e., out of control

INITIAL Xbar CONTROL CHART

Xbar

100 75

UCL

50

Center lin x_double

25

LCL

0 1

3

5

7

9

11

13

15

17

19

21

23

25

Sample # Sample # 1 2 Range = 48 25

3 50

4 35

5 18

6 48

7 21

8 13

9 28

10 51

11 18

12 39

13 7

14 12

15 28

16 30

17 13

18 32

19 51

20 5

21 24

22 24

23 12

24 22

25 22

R

UCLR LCLR

69.5 69.5 69.5 69.5 69.5 69.5 69.5 69.5 69.5 69.5 69.5 69.5 69.5 69.5 69.5 69.5 69.5 69.5 69.5 69.5 69.5 69.5 69.5 69.5 69.5 0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

INITIAL R CHART 80 70

UCL

60

R

50 40 30 20 10 LCL

0


27

Figure 4.4 Revised Control Chart Calculations To reveal the inherent variability of the process, the control limits must be re-calculated, after removing any "out of control" points.

Calculation basis: Rbar x_doublebar Spec Midpoint A2  Rbar UCLxbar = x_doublebar + A2  Rbar LCLxbar = x_doublebar A2  Rbar UCLR = D4  Rbar UCLR = D3  Rbar

All Subgroups included 676/25 = 27 1221/25 = 48.8 50.0 1.02327 = 26 76.4

Subgroup #17 Removed 663/24 = 26 1127/24 = 40 50.0 1.02326 = 28.2 75.2

21.2

18.8

69.5 0.0

71.0 0.0

The Revised Xbar Control Limits and Chart Sample 1 2 3 4 5 6 7 8 9 10 11 # Average 44. 66. 55. 40. 40. 45. 53. 36. 45. 533 35

= Xbar

33 40

x_double bar

75. 2 LCLxbar 18. 8

UCLxbar

12

13

14

15

55

52. 67

35

52

16

17

18

19

20

21

22

23

24

25

40

40

40

40

40

40

40

40

40

40

40

40

40

40

34. Om 54. 32 38. 36 59. 567 433 55. 33 itte 67 67 33 33 d 40 40 40 40 40 40 40 40 40 40

75. 2 18. 8

75. 2 18. 8

75. 2 18. 8

75. 2 18. 8

75. 2 18. 8

75. 2 18. 8

75. 2 18. 8

75. 2 18. 8

75. 2 18. 8

75. 2 18. 8

75. 2 18. 8

75. 2 18. 8

75. 2 18. 8

75. 2 18. 8

75. 2 18. 8

33

33

33

67

33

33

67

33

75. 2 18. 8

75. 2 18. 8

75. 2 18. 8

75. 2 18. 8

75. 2 18. 8

75. 2 18. 8

75. 2 18. 8

75. 2 18. 8

75. 2 18. 8

REVISED Xbar CONTROL CHART 80

UCL

70 60 Xbar

50

x_doublebar

40 30 20

LCL

10 0 Sample #

Sample # Range

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

48

25

50

35

18

48

21

13

28

51

18

39

7

12

28

30

Om -it

32

51

5

24

24

12

22

22

=R

UCLR UCLR

71. 71. 71. 71. 71. 71. 71. 71. 71. 71. 71. 71. 71. 71. 71. 71. 71. 71. 71. 71. 71. 71. 71. 71. 71. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0


28

REVISED R CHART

R

80 70 60 50

UCL

40 30 20 10 0

LCL

Sample #

4.2

Interpreting Abnormal Patterns in Control Charts

When a process is in statistical control, the points on a control chart fluctuate randomly between the control limits with no recognizable, non-random pattern. The following checklist provides a set of general rules for examining a process to determine if it is in control: 1. No points are outside control limits. 2. The number of points above and below the center line is about the same. 3. The points seem to fall randomly above and below the center line. 4. Most points, but not all, are near the center line, and only a few are close to the control limits. The underlying assumption behind these rules is that the distribution of sample means is normal. This assumption follows from the central limit theorem of statistics, which states that the distribution of sample means approaches a normal distribution as the sample size increases regardless of the original distribution. Of course, for small sample sizes, the distribution of the original data must be reasonably normal for this assumption to hold. The upper and lower control limits are computed to be three standard deviations from the overall mean. Thus, the probability that any sample mean falls outside the control limits is very small. This probability is the origin of rule 1. Since the normal distribution is symmetric, about the same number of points fall above as below the center line. Also, since the mean of the normal distribution is the median, about half the points fall on either side of the center line. Finally, about 68 percent of a normal distribution falls within one standard deviation of the mean; thus, most but not all points should be close to the center line. These characteristics will hold provided that the mean and variance of the original data have not changed during the time the data were collected; that is, the process is stable. Several types of unusual patterns arise in control charts, which are reviewed here along with an indication of the typical causes of such patterns. One Point Outside Control Limits A single point outside the control limits (see Figure 4.5) is usually produced by a special cause. Often, the R-chart provides a similar indication. Once in a while, however, such points are normal part of the process and occur simply by chance. A common reason for a point falling outside a control limit is an error in the calculation of xbar or R for the sample. You should always check your calculations whenever this occurs, Other possible causes are a sudden power surge, a broken tool, measurement error, or an incomplete or omitted operation in the process.


28

Figure 4.5 Single Point Outside Control Limits 100

Xbar

80 60 40 20 0 Sample #

Sudden Shift in the Process Average An Unusual number of consecutive points falling on one side of the center line (see Figure 4.6) is usually an indication that the process average has suddenly shifted. Typically, this occurrence is the result of an external influence that has affected the process, which would be considered a special cause. In both the xbar and R-charts, possible causes might be a new operator, a new inspector, a new machine setting, or a change in the setup or method.

Figure 4.6 Shift in Process Average 100

Xbar

80 60 40 20 0 Sample #

If the shift is up in the R-chart, the process has become less uniform. Typical causes are carelessness of operators, poor or inadequate maintenance, or possibly a fixture in need of repair. If the shift is down in the R-chart, the uniformity of the process has improved. This might be the result of improved workmanship or better machines or materials. As mentioned, every effort should be made to determine the reason for the improvement and to maintain it. Three rules of thumb are used for early detection of process shifts. A simple rule is that if eight consecutive points fall on one side of the center line, one could conclude that the mean has shifted. Second, divide the region between the center line and each control limit into three equal parts. Then if (1) two of three consecutive points fall in the outer one-third region between the center line and one of the control limits or (2) four of five consecutive points fall within the outer two-thirds region, one would also conclude that the process has gone out of control.

Cycles Cycles are short, repeated patterns in the chart, alternating high peaks and low valleys (see Figure 4.7). These patterns are the result of causes that come and go on a regular basis. In the xbar chart, cycles may be the result of operator rotation or fatigue at the end of a shift, different gauges used by different inspectors, seasonal effects such as temperature or humidity, or differences between day and night shifts. In the R-chart, cycles can occur from maintenance schedules, rotation of fixtures or gauges, differences between shifts, or operator fatigue. Prepared for the Extension Program of Indira Gandhi Open University

29

Figure 4.7 Cycles 100

Xbar

80 60 40 20 0 Sample #

Trends A trend is the result of some cause that gradually affects the quality characteristics of the product and causes the points on a control chart to gradually move up or down from the center line. As a new group of operators gains experience on the job, for example, or as maintenance of equipment improves over time, a trend may occur. In the xbar chart, trends may be the result of improving operator skills, dirt or chip buildup in fixtures, tool wear, changes in temperature or humidity, or aging of equipment. In the R-chart, an increasing trend may be due to a gradual decline in material quality, operator fatigue, gradual loosening of a fixture or a tool, or dulling of a tool. A decreasing trend often is the result of improved operator skill or work methods, better purchased materials, or improved or more frequent maintenance.

Hugging the Center Line Hugging the center line occurs when nearly all the points fall close to the center line (see Figure 4.8). In the control chart, it appears that the control limits are too wide. A common cause of hugging the center line is that the sample includes one item systematically taken from each of several machines, spindles, operators, and so on. A simple example will served to illustrate this pattern. Suppose that one machine produces parts whose diameters average 508 with variation of only a few thousandths; a second machine produces parts whose diameters average 502, again with only a small variation. Taken together, parts from both machines would yield a range of variation that would probably be between 500 and 510, and average about 505, since one will always be high and the second will always be low. Even though a large variation will occur in the parts taken as a whole, the sample averages will not reflect this variation. In such a case, a control chart should be constructed for each machine, spindle, operator, and so on. An often overlooked cause for this pattern is miscalculation of the control limits, perhaps by using the wrong factor from the table, or misplacing the decimal point in the computations.

Figure 4.8 Hugging the Centre Line 100

Xbar

80 60 40 20 0 Sample # Prepared for the Extension Program of Indira Gandhi Open University

30

Hugging the Control Limits This pattern shows up when many points are near the control limits with very few in between. It is often called a mixture and is actually a combination of two different patterns on the same chart. A mixture can be split into two separate patterns. A mixture pattern can result when different lots of material are used in one process, or when parts are produced by different machines but fed into a common inspection group.

Instability Instability is characterized by unnatural and erratic fluctuations on both sides of the chart over a period of time (see Figure 4.9). Points will often lie outside both the upper and lower control limits without a consistent pattern. Assignable causes may be more difficult to identify in this case than when specific patterns are present. A frequent cause of instability is over adjustment of a machine, or the same reasons that cause hugging the control limits.

Figure 4.9 Out-of-Control Indicators 100

2 of 3 above two 

8 points on one side of center line

Xbar

80

UCL

60 40 20 0

LCL 4 of 5 below one 

Sample #

As suggested earlier, the R-chart should be analyzed before the xbar chart, because some outof-control conditions in the R-chart may cause out-of-control conditions in the xbar chart. Also, as the variability in the process decreases, all the sample observations will be closer to the true population mean, and therefore their average, xbar, will not vary much from sample to sample. If this reduction in the variation can be identified and controlled, then new control limits should be computed for both charts.

4.3

Routine Process Monitoring and Control

After a process is determined to be in control, the charts should be used on a daily basis to monitor production, identify any special causes that might arise, and make corrections as necessary. More important, the chart tells when to leave the process alone! Unnecessary adjustments to a process result in nonproductive labor, reduced production, and increased variability of output. It is more productive if the operators themselves take the samples and chart the data. In this way, they can react quickly to changes in the process and immediately make adjustments. To do this effectively, training of the operators is essential. Many companies conduct in-house training programs to teach operators and supervisors the elementary methods of statistical quality control. Not only does this training provide the mathematical and technical skills that are required, but it also give the shop-floor personnel increased quality-consciousness.


31

Introduction of control charts on the shop floor typically leads to improvements in conformance, particularly when the process is labor intensive. Apparently, management involvement in operators' work produces positive behavioral modifications (as first demonstrated in the famous Hawthorne studies). Under such circumstances, and as good practice, management and operators should revise the control limits periodically and determine a new process capability as improvements take place. Another important point must be noted. Control charts are designed to be used by production operators rather than by inspectors or QC personnel. Under the philosophy of statistical process control, the burden of quality rests with the operators themselves. The use of control charts allows operators to react quickly to special causes of variation. The range is used in place of the standard deviation for the very reason that it allows shop-floor personnel to easily make the necessary computations to plot points on a control chart. The experience even in Indian factories such as the turbine blade machining shop in BHEL Hardwar strongly supports this assertion. The right approach taken by management in ingraining the correct outlook among the workers appears to hold the key here.

4.4

Estimating Plant Process Capability

After a process has been brought to a state of statistical control by eliminating special causes of variation, the data may be used to find a rough estimate process capability. This approach uses the average range Rbar rather than the estimated standard deviation of the original data. Nevertheless, it is a quick and useful method, provided that the distribution of the original data is reasonably normal. Under the normality assumption, the standard deviation (x) of the original data {x}can be estimated as follows:  (= x) = Rbar/d2 where d2 is a constant that depends on the sample size and is also given in Table 2.1. Therefore, process capability may be determined by comparing the spec range to 6 x. The natural variation of individual measurements is given by x_doublebar  3 x. The following example illustrates these calculations. Example 4: Estimating Process Capability for Silicon Wafer Manufacture In this example, the capability (Cp and Cpk, see Section 2.4) calculations for the silicon wafer production data displayed in Figure 4.3 are shown. The overall distribution of the data is indicated by Figure 4.10. For a sample of size (n) = 3, d2 is 1.693. The ULx and LLx represent the upper and lower 3x limits on the data for individual observations. Thus, wafer thickness is expected to vary between -19 and 95.9. The "zero point" of acceptable wafers is the lower specification, meaning that the thickness of all wafers produced is expected, without adjustment, to vary from 0.0019 below the lower specification to 0.0959 above the lower specification. Therefore, Cp = 100/98 = 1.02 Thus, Cp for this process looks OK. However, the lower and upper capability indices are Cpl = (47 - 0)/48.9 = 0.96 Cpu = (100 - 47)/48.9 = 1.08 This gives a Cpk value equal to 0.96, which is less than 1.0. This analysis suggests that both the centering and the variation of the wafer manufacturing process must be improved. The actual fraction of the output falling within the spec range or tolerance may be calculated in a step-by-step manner as follows.


32

Step 1: Find Modified (corrected) Control limits Note: The initial xbar control chart (Figure 4.3) shows one xbar point (Sample #17) out of control (beyond UCLxbar). This point should be removed and the control limits should be recalculated. This is done as follows.

Calculation basis: Rbar x_doublebar Spec Midpoint A2  Rbar UCLxbar = x_doublebar + A2  Rbar LCLxbar = x_doublebar - A2  Rbar UCLR = D4  Rbar UCLR = D3  Rbar

All Subgroups included 676/25 = 27 1221/25 = 48.8 50.0 1.02327 = 26 76.4 21.2 69.5 0.0

Subgroup #17 Removed 663/24 = 26 1127/24 = 40 50.0 1.02326 = 28.2 75.2 18.8 71.0 0.0

Step 2: Find the revised Process Standard Deviation (x) x = Rbar/d2 = 26/1.693 = 16.30

Step 3: Compare tolerance limits (specification) with the revised 6x US (Upper Spec Limit) = 100.0 LS (Lower Spec Limit) = 0.0 US - LS = 100.0 - 0.0 = 100.0 6x = 6  16.30 = 981

Distribution of xbar

0.002

LLx LSL = 0 47 ZLSL = (0 - 47)/16.3 = z-2.88 = (100 - 47)/16.3 = 3.25

ULx USL =100 ZUSL = (100 - 47)/16.3 = 3.25

Figure 4.10 Process Capability Probability Computations


33

Step 4: Compute upper and lower ( 3x) limits of process variation under statistical control: ULx = x_doublebar + 3x = 40 + 3  16.30 = 95.9 LLx = x_doublebar - 3x = 40 - 3  16.30 = -1.9 If the individual observations are normally distributed, then the probability of being out of specification can be computed. In the example above we assumed that the data are normal. The revised mean (estimated by x_doublebar) is 47 and the standard deviation (x) is 98/6=16.3. Figure 4.10 shows the z calculations for specification limits of 0 and 100. These z values are used to find the area (= probability of finding x) between 0 and the mean (47) is 0.4980 as determined from the standard normal distribution table. Thus 0.2 percent of the output (wafer production or {x}) would be expected to fall below the lower specification. The area to the right of 100 is approximately zero. Therefore, all the output can be expected to meet the upper specification.

Control limits are not specification limits! A word of caution deserves emphasis here. Control limits are often confused with specification or "spec" limits. Spec limits, normally expressed in engineering units, indicate the range of variation in a quality characteristic that is acceptable to the customer. Specification dimensions are usually stated in relation to individual parts for "hard" goods, such as automotive hardware. However, in other applications, such as in chemical processes, specifications are stated in terms of average characteristics. Thus, control charts might mislead one into thinking that if all sample averages fall within the control limits, all output will conform to specs. This assumption is not true. Control limits relate to sample averages while specification limits relate to individual measurements. A sample average may fall within the upper and lower control limits even though some of the individual observations are out of specification. Since xbar = x/n, control limits are narrower than the natural variation in the process (Figure 2.5) and they do not represent process capability.

4.5

Use of Warning Limits

When a process is in control, the xbar and R charts may be used to make decisions about the state of the process during its operation. We use three zones on the charts to help in the routine management of the process. The zones are marked on the charts as follows. Zone 1 (it falls between the upper and lower WARNING LINES (or  2 sigma lines) . If the plotted points fall in this zone, it indicates that the process has remained stable and actions or adjustments are unnecessary. Indeed any adjustment here may increase the amount of variability. Zone 2 (it falls between Zone 1 and Zone 3). Any point found in Zone 2 suggests that there may have been an assignable change and another sample must be taken to check that out. Zone 3 (it falls beyond the UPPER or LOWER CONTROL LIMIT). Any points falling in this zone indicates that the process should be investigated and thatn if action is taken, the latest estimate of x_doublebar value and Rbar should be used to revise the control limits.


34

xbar

WARNING: Point in Zone 2  Take second sample

Possible Results of second sampling

ZONE 3

 Investigate and adjust

ZONE 2

 Investigate

ZONE 1

 Do nothing

Upper Control Limit Upper Warning Limit

ZONE 1

X_double bar

ZONE 1 ZONE 1

Lower Warning Limit Lower Control Limit

ZONE 2 ZONE 3

Sample #

Figure 4.11 ACTIONS FOR SECOND SAMPLE FOLLOWING A WARNING SIGNAL IN ZONE 2

4.6

Modified Control Limits

Modified control limits often are used when process capability is very good. For example, suppose that the process capability of a factory is only 60 percent of tolerance (Cp = 1.67) and that the process mean can be controlled by a simple machine adjustment. Management may quickly discover the impracticality of investigating every isolated point that falls outside the usual control limits because the output is probably still well within specifications. In such cases, the usual control limits may be replaced with the following modified control limits: URLx = USL - Am Rbar LRLx = LSL + Am Rbar where URLx is the upper reject level, LRLx is the lower reject level and USL and LSL are the upper and lower specifications respectively. A m values are determined by statistical principles and these are shown in Table 4.1. The modified control limits allow for more variation than the ordinary control limits and still provide high confidence that the product produced will remain within specification. It is important to note that modified limits apply only if process capability is at least 60 to 75 percent of tolerance. However, if the mean must be controlled closely, a conventional xbar-chart should be used even if the process capability is good. Also if the process standard deviation (x) is likely to shift, don't modify control limits.

Table 4.1 Factors for Control Limts and Standard Deviation Sample Size n 2 3 4 5 6

A2 1.880 1.023 0.729 0.577 0.483

D4 3.267 2.574 2.282 2.114 2.004

d2 1.128 1.693 2.059 2.326 2.534


Am 0.779 0.749 0.728 0.713 0.701

35

Example 5: Computing Modified Control Limits for the Silicon Wafer Case: Shown below is the calculations for the silicon wafer thickness example considered in this section. Since the sample size is 3, Am = 0.749. Therefore, the modified limits are URLx = US - AmR = 100-0.7(27) = 79.3 LRLx = LS + AmR = 0+0.749(26) = 20.7 Observe that if the process is centered on the nominal, the modified control limits are "looser" (wider) than the ordinary control limits. For this example, before the modified control limits are implemented, the centering of the process would first have to be corrected from its current (estimated) value of 40 to the specification midpoint of 50.0.

4.7      

 

Key Points about Control Charts Every process exhibits some variability. Variability is caused by a multitude of factors that affect a manufacturing or service process while it is operating. In SPC, a process is assumed to be disturbed by two types of factors called respectively (1) random causes and (2) assignable causes. Random causes are many, but their individual effect is small. The combined effect of all random causes is manifested as the inherent variability of the process. A performance measure called "process capability" quantifies the inherent variability of a process. Process capability measures the fraction of the total process output that falls within tolerance (or spec range) when no assignable factors are affecting the process. Assignable factors on the other hand are few in number (we can usually "put our finger on it"), but their effect is large and noticeable on an appropriately drawn "control chart." When disturbed by an assignable cause, the average value of the process output may deviate from its desired target value, hence the process may lose accuracy. Process spread or dispersion may also widen, causing a worsening of precision. Hence, the appearance of assignable causes should be detected quickly and corrective steps should be taken to remove such factors from affecting the process. Mean or sample average (xbar) indicates the extent of accuracy of process output (location of the output distribution relative to the target). Standard deviation, or range (R), indicates the precision of the output. The purpose of drawing control charts is to determine when the process should be left alone and when it should be investigated and if necessary, adjusted or rectifiedto remove any assignable cause affecting it.


36

5

SPECIAL CONTROL CHARTS FOR VARIABLES

Several alternatives to the popular xbar and R-chart for process control of variables measurements are available. This section discusses some of them.

5.1

Xbar and s-charts

An alternative to using the R-chart along with the xbar chart is to compute and plot the standard deviation s of each sample. Although the range has traditionally been used, since it involves less computational effort and is easier for shop-floor personnel to understand, using s rather than R has its advantages. The sample standard deviation is a more sensitive and better indicator of process variability, especially for larger sample sizes. Thus, when tight control of variability is required, s should be used. With the use of modern calculators and personal computers, the computational burden of computing s is reduced or eliminated, and s has thus become a viable alternative to R. The sample standard deviation is computes as n

s

 (x i 1

i

 xbar) 2

n 1

To construct an s-chart, compute the standard deviation for each sample. Next, compute the average standard deviation sbar by averaging the sample standard deviations over all samples. (Notice that this computation is analogous to computing R). Control limits for the s-chart are given by UCLs = B4 sbar LCLs = B3 sbar where B3 and B4, are constants found in Table 2.1. For the associated xbar chart, the control limits derived from the overall standard deviation are UCLxbar = x_doublebar + A3 sbar LCLxbar = x_doublebar - A3 sbar where A3 is a constant that is a function of sample size (n) may be found in Table 2.1. Observe that the formulas for the control limits are equivalent to those for xbar and R-charts except that the constants differ.

Example 6: Constructing xbar and s-Charts. To illustrate the use of the xbar- and s-charts, consider the data given below. These data represent measurements of deviations from a nominal specification for some machined part. Samples of size 10 are used; for each sample the mean and standard deviation have been computed. The average (i.e., overall) mean is computed to be x_doublebar = 0.108, and the average standard deviations is sbar = 1.791. Since the sample size is 10, B3 = 0.284, B4 = 1.716, and A3 = 0.975. Hence, the control limits for the s-chart are


37

LCLs = 0.284 (1.791) = 0.509 UCLs = 1.716 (1.791) = 3.073 For the xbar chart, the control limits are LCLxbar = 0.108 - 0.975 (1.791) = - 1.638 UCLxbar = 0.108 + 0.975 (1.791) = 1.854 The xbar and s-charts are shown in Figures 5.1 and 5.2 respectively. The charts indicate that this process is not in control, and an investigation as to the reasons for the variation, particularly in the xbar chart, is warranted.

Data and Calculations for Example 6 Number of samples = 25, sample size = 10. Sample # 1 2 3 4 5 6 7 8 9 10

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

1 8 6 9 7 9 2 7 9 7

9 4 0 3 0 0 3 4 8 3

0 8 0 0 3 1 2 0 2 3

1 1 0 2 1 1 2 0 0 1

-3 -1 0 -4 0 1 0 -2 0 -2

-6 2 0 0 2 -1 2 0 -3 0

-3 -1 0 -2 -1 -1 -3 0 -2 -2

0 -2 -3 -1 -2 1 -3 0 -3 -2

2 0 -1 -1 -3 0 1 -3 -1 0

0 0 -2 -1 -1 0 -1 -2 -2 0

-3 -2 2 -1 1 -2 -2 -1 1 1

-12 2 0 -4 -1 4 2 -3 -4 0

-6 -3 0 0 -8 -4 -6 -1 -1 -2

-3 -5 5 0 -5 1 5 -4 -1 -5

-1 -1 -1 -2 -1 0 -2 -1 0 -1

-1 -2 -2 0 -4 0 -2 -4 -1 0

-2 2 -1 0 -1 -1 2 -1 1 -2

0 4 0 0 0 3 0 0 1 0

0 3 -3 3 3 1 0 1 2 -2

1 2 1 1 -3 2 1 -2 3 0

1 2 2 1 2 2 1 1 1 2

-1 0 2 -1 2 2 -1 0 0 -1

0 0 -1 0 1 0 0 0 -1 0

1 0 0 1 1 2 0 0 -1 0

2 2 1 2 -1 2 2 1 -1 2

Xbar 6.5 3.4 1.9 0.9 -1.1 -0.4 -1.5 -1.5 -0.6 -0.9 -0.6 -1.6 -3.1 -1.2 -1 -1.6 -0.3 0.8 0.8 0.6 1.5 0.2 -0.1 0.4 1.2 X_doubl 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 ebar UCLxbar 1.85 1.85 1.85 1.85 1.85 1.85 1.85 1.85 1.85 1.85 1.85 1.85 1.85 1.85 1.85 1.85 1.85 1.85 1.85 1.85 1.85 1.85 1.85 1.85 1.85 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 LCLxbar 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

Std_dev 2.83 3.13 2.47 0.73 1.59 2.50 1.08 1.43 1.57 0.87 1.71 4.52 2.80 3.91 0.66 1.50 1.49 1.47 2.09 1.83 0.52 1.31 0.56 0.84 1.22 8 4 0 8 5 3 0 4 8 6 3 7 7 0 7 6 4 6 8 8 7 7 8 3 9 UCLs 3.07 3.07 3.07 3.07 3.07 3.07 3.07 3.07 3.07 3.07 3.07 3.07 3.07 3.07 3.07 3.07 3.07 3.07 3.07 3.07 3.07 3.07 3.07 3.07 3.07 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 LCLs 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9

xbar Chart Calculations for Example 6 Xbar 6.5 3.4 1.9 0.9 -1.1 -0.4 -1.5 -1.5 -0.6 -0.9 -0.6 -1.6 -3.1 -1.2 -1 -1.6 -0.3 0.8 0.8 0.6 1.5 0.2 -0.1 0.4 1.2 X_doubl 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 ebar UCLxbar 1.85 1.85 1.85 1.85 1.85 1.85 1.85 1.85 1.85 1.85 1.85 1.85 1.85 1.85 1.85 1.85 1.85 1.85 1.85 1.85 1.85 1.85 1.85 1.85 1.85 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 LCLxbar 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8


38

Xbar

Figure 5.1 Xbar CHART

8 6 4 2 0 -2 -4

1

3

5

7

9

11

13

15

17

19

21

23

25

Sample #

s-Chart for Example 6 Std_dev 2.83 3.13 2.47 0.73 1.59 2.50 1.08 1.43 1.57 0.87 1.71 4.52 2.80 3.91 0.66 1.50 1.49 1.47 2.09 1.83 0.52 1.31 0.56 0.84 1.22 8 4 0 8 5 3 0 4 8 6 3 7 7 0 7 6 4 6 8 8 7 7 8 3 9 UCLs 3.07 3.07 3.07 3.07 3.07 3.07 3.07 3.07 3.07 3.07 3.07 3.07 3.07 3.07 3.07 3.07 3.07 3.07 3.07 3.07 3.07 3.07 3.07 3.07 3.07 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 LCLs 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9


39

Figure 5.2 s-CHART 5 4

s

3 2 1 0 1

4

7

10

13

16

19

22

25

Sample #

5.2

Charts for Individuals

With the development of automated inspection for many processes, manufacturers can nor easily inspect and measure quality characteristics on every item produced. Hence, the sample size for process control is n = 1, and a control chart for individual measurements also called an x-chart can be used. Other examples in which x-charts are useful include accounting data such as shipments, orders, absences, and accidents; production records of temperature, humidity, voltage, or pressure; and the results of physical or chemical analyses. With individual measurements, the process standard deviation can be estimated and threesigma control limits used. As shown earlier, Rbar/d2 provides an estimate of the process standard deviation. Thus, an x-chart for individual measurements would have "three-sigma" control limits defined by UCLx = x_average + 3 Rbar / d2 LCLx = x_average - 3 Rbar / d2 Sample of size 1, however, do not furnish enough information for process variability measurement. However, process variability can be determined by using a moving average of ranges, or a moving range, or n successive observations. for example, a moving range for n = 2 is computed by finding the absolute difference between two successive observations. The number of observations used in the moving range determines the constant d 2; hence, for n = 2 from Table 2.1, d2 = 1.128. In a similar fashion, larger values of n can be used to compute moving ranges. The moving range chart has control limits defined by UCLR = D4 Rbar LCLR = D3 Rbar which is comparable to the ordinary range chart.

Example 7:

Constructing an x-Chart with Moving Ranges.

Consider a set of observations measuring the percentage of cobalt in a chemical process as given in Figure 5.3.

Figure 5.3 Data and Calculations for Example 7


40

Observation # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

X

LCLx

CLx

UCLx

3.75 3.8 3.7 3.2 3.5 3.05 3.5 3.25 3.6 3.1 4 4 3.5 3 3.8 3.4 3.6 3.1 3.55 3.65 3.45 3.3 3.75 3.5 3.4

2.56 2.56 2.56 2.56 2.56 2.56 2.56 2.56 2.56 2.56 2.56 2.56 2.56 2.56 2.56 2.56 2.56 2.56 2.56 2.56 2.56 2.56 2.56 2.56 2.56

3.498 3.498 3.498 3.498 3.498 3.498 3.498 3.498 3.498 3.498 3.498 3.498 3.498 3.498 3.498 3.498 3.498 3.498 3.498 3.498 3.498 3.498 3.498 3.498 3.498

4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43

Moving Range 0.05 0.1 0.5 0.3 0.45 0.45 0.25 0.35 0.5 0.9 0 0.5 0.5 0.8 0.4 0.2 0.5 0.45 0.1 0.2 0.15 0.45 0.25 0.1

LCLr

UCLr

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15

The moving range is computed as shown by taking absolute values moving range is the difference between the first two observations: |3.75 - 3.80| = 0.05 The second moving range is computed as |3.80 - 3.70| = 0.10 From these data we find that LCLR = 0 and UCLR = (3.267)(0.352) = 1.15 The moving range chart, shown in Figure 5.5, indicates that the process is in control. Next, the x-chart is constructed for the individual measurements: LCLx = 3.498 - 3(0.352)/1.128 = 2.56 UCLx = 3.498 + 3(0.352)/1.128 = 4.43 The two charts indicate that the process is in control.


41

Some caution is necessary when interpreting patterns on the moving range chart. Points beyond control limits are signs of assignable causes. Successive ranges, however, are correlated, and they may cause patterns or trends in the chart that are not indicative of out-ofcontrol situations. On the x-chart, individual observations are assumed to be uncorrelated; hence, patterns and trends should be investigated. Control charts for individuals have the advantage that specifications can be drawn on the chart and compared directly with the control limits. Some disadvantage also exist:  Individuals charts are less sensitive to many of the conditions that can be detected by xbar and R-charts; for example, the process must vary a lot before a shift in the mean is detected.  Also, short cycles and trends may appear on an individual's chart and not on an xbar or Rchart.  Finally, the assumption of normality of observations is more critical than for xbar and Rcharts; when the normality assumption does not hold, greater chance for error is present. To summarize, SPC for variables data progresses in three stages: 1. Examination of the state of statistical control of the process using xbar and R charts, 2. A process capability study to compare process spread and its location to specifications, and 3. Routine process control using control charts.


42

6

CONTROL CHARTS FOR ATTRIBUTES

Attributes quality data assume only two valuesgood or bad, pass or fail. Attributes usually cannot be measured, but they can be observed and counted and are useful in quality management in many practical situations. For instance, in printing packages for consumer products, color quality can be rated as acceptable or not acceptable, or a sheet of cardboard either is damaged or is not. Usually, attributes data are easy to collect, often by visual inspection. Many accounting records, such as percent scrapped, are also usually readily available. However, one drawback in using attributes data is that large samples are necessary to obtain valid statistical results.

6.1

Fraction Nonconforming (p) Chart

Several different types of control charts are fused for attribute data. One of the most common is the p-chart (introduced in this section). Other types of attributes charts are presented in the next chapter. One distinction that we must make is between the terms defects and defectives. A defect is a single nonconforming quality characteristic of an item. An item may have several defects. The term defective refers to items having one or more defects. Since certain attributes charts are used for defectives while others are used for defects, one must understand the difference. In quality control literature, the term nonconforming is often used instead of defective. A p-chart monitors the proportion of nonconforming items produced in a lot. Often it is also called a fraction nonconforming or fraction defective chart. As with variables data, a p-chart is constructed by first fathering 25 to 30 samples of the attribute being measured. The size of each sample should be large enough to have several nonconforming items. If the probability of finding a nonconforming item is small, a large sample size is usually necessary. Samples are chosen over time periods so that any special causes that are identified can be investigated. Let us suppose that k samples, each of size n, are selected. If y represents the number nonconforming items or defectives in a particular sample, the proportion nonconforming is (y/n). Let pi be the fraction nonconforming in the ith sample; the average fraction nonconforming pbar for the group of k samples then is

pbar  p 

p1  p2  ...  pk k

This statistic pbar reflects the average performance of the process. One would expect a high percentage of samples to have a fraction nonconforming within three standard deviations of p. As estimate of the standard deviation is given by

sp 

p(1  p) n

Therefore, upper and lower control limits may be given by

UCLp  p  3s p UCLp  p  3s p If LCLp is less that zero, a value of zero is used.


43

Analysis of a p-chart is similar to that of the xbar or R-chart. Points outside the control limits signify an out-of-statistical-control situation, i.e., the process has been disturbed by an assignable factor. Patterns and trends should also be sought to identify the presence of assignable factors. However, a point on a p-chart below the lower control limit or the development of a trend below the center line indicates that the process might have improved, since the ideal is zero defectives. However, caution is advised before such conclusions are drawn, because errors may have been made in computation. An example of a p-chart is presented next. Example 8: Constructing a p-Chart. The mail sorting personnel in a post office must read the PIN code on a letter and divert the letter to the proper carrier route. Over one month's time, 25 samples of 100 letters were chosen and the number of errors was recorded. This information is summarized in Figure 6.1. The average fraction nonconforming, pbar is found to be

pbar  p 

0.03  0.01  ...  0.01  0.022 25

The standard deviation is computed as

sp 

Figure 6.1: Sample # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

0.022(1  0.022)  0.01467 100 Data and Control Limit Calculations for Example 8 Number of errors 3 1 0 0 2 5 3 6 1 4 0 2 1 3 4 1 1 2 5 2 3 4 1 0 1

Sample Size 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100

p = Fraction Nonconforming 0.03 0.01 0.00 0.00 0.02 0.05 0.03 0.06 0.01 0.04 0.00 0.02 0.01 0.03 0.04 0.01 0.01 0.02 0.05 0.02 0.03 0.04 0.01 0.00 0.01


LCLp

UCLp

0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022

0.066 0.066 0.066 0.066 0.066 0.066 0.066 0.066 0.066 0.066 0.066 0.066 0.066 0.066 0.066 0.066 0.066 0.066 0.066 0.066 0.066 0.066 0.066 0.066 0.066

44

Figure 7.6.2 Attribute (p ) Chart for Example 8

p (Fraction Defective)

0.07

UCLp

0.06 0.05 0.04 0.03

pbar

0.02 0.01

LCLp

0

1

3

5

7

9

11

13

15

17

19

21

23

25

Sample #

Thus, the upper control limit, UCLp, is 0.022 + 3(0.01467) = 0.066, and the lower control limit, LCLp is 0.022 - 3(0.01467) = - 0.022. Since this later figure is negative and fraction nonconforming (pbar) can never be negative, for LCLp zero (0) is used. The control chart for this example is shown in Figure 6.2. The sorting process appears to be in statistical control. Any values found above the upper control limit or evidence of upward trend might indicate the need for re-training the personnel.

6.3

Variable Sample Size p Charts

Often 100 percent inspection is performed on process output during fixed sampling periods; however, the number of units produced in each sampling period may vary. In this case, the pchart would have a variable sample size. One way of handling this is to compute a standard deviation for each individual sample. Thus,

Std _ Dev  p(1  p) / ni p3

p(1  p) ni

where the number of observations in the Ith sample is nI,. The control limits for this sample will be given by where

p

 number _ nonconfor min g n i

Example 7: Variable sample Size. Figure 6.3 shows 20 samples with varying sample sizes. The value of pbar is computed as

p

18  20  14  ...  18 271   0.0909 137  158  92  ...  160 2980


45

Therefore, control limits for sample #1 would be

LCLp  0.0909  3 UCLp  0.0909  3

0.0909(1  0.0909)  0.017 137 0.0909(1  0.0909)  0.165 137

Note carefully that because the sample sizes vary, control limits would be different for each sample. The p-chart is shown in Figure 6.4. Points 13 and 15 are outside the control limits.

Figure 6.4 p CHART WITH VARIABLE SAMPLE SIZE Variable control limits that depend on sample size ni

p (Fraction Defective)

0.2

UCL

0.15 0.1 0.05

LCL 0 1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19 20

Sample #

Figure 6.3

Data and Calculations for Example 7

Sample # I

Value

Sample Size (nI)

p = Fraction Nonconforming

Std Dev

LCLp

UCLp

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

18 20 14 6 11 22 6 9 14 12 8 13 5 15 25 12 16 12 15 18

137 158 92 122 86 187 156 117 110 142 140 179 196 163 140 135 186 193 181 160

0.1314 0.1266 0.1522 0.0492 0.1279 0.1176 0.0385 0.0769 0.1273 0.0845 0.0571 0.0726 0.0255 0.0920 0.1786 0.0889 0.0860 0.0622 0.0829 0.1125

0.0245648 0.0228741 0.0299764 0.0260311 0.031004 0.0210258 0.230203 0.0265815 0.0274143 0.0241284 0.0243001 0.0214905 0.0205374 0.0225206 0.0243001 0.0247461 0.0210822 0.0206964 0.0213714 0.0227307

0.0172 0.0223 0.001 0.0128 0 0.0279 0.0219 0.0112 0.0087 0.0186 0.018 0.0265 0.0293 0.0234 0.018 0.0167 0.0277 0.0289 0.0268 0.0227

0.1646 0.1596 0.1809 0.169 0.184 0.154 0.16 0.1707 0.1732 0.1633 0.1638 0.1554 0.1526 0.1585 0.1638 0.1652 0.1542 0.153 0.1551 0.1591


46

6.3

np-charts for Number Nonconforming

In the p-chart, the fraction nonconforming of the ith sample is given by pi = yi/n where yi is the number found nonconforming and n is the sample size. Multiplying both sides of the equation pi = yi/n, yields yi = npi That is, the number nonconforming is equal to the sample size times the proportion nonconforming. Instead of using a chart for the fraction nonconforming, an equivalent alternativea chart for the number of nonconforming items is useful. Such a control chart is called an np-chart. The np-chart is a control chart for the number of nonconforming items in a sample. To use the np-chart, the size of each sample must be constant. Suppose that two samples of sizes 10 and 15 each have four nonconforming items. Clearly, the fraction nonconforming in each sample is difference between samples. Thus, equal sample size are necessary to have a common base for measurement. Equal sample sizes are not required for p-charts, since the fraction nonconforming is invariant to the sample size. The np-chart is a useful alternative to the p-chart because it is often easier to understand for production personnelthe number of nonconforming items is more meaningful than a fraction. Also, since it requires only a count, the computations are simpler. The control limits for the np-chart, like those for the p-chart, are based on the binomial probability distribution. The center line is the average number of nonconforming items per sample as denoted by npbar, which is calculated by taking k samples of size n, summing the number of nonconforming items yi in each sample, and dividing by k. That is

npbar 

y1  y 2  ...  y k k

An estimate of the standard deviation is

s npbar  npbar (1  pbar ) where pbar = (npbar)/n. Using three-sigma limits as before, the control limits are specified by

UCLnpbar  npbar  3 npbar (1  pbar ) LCLnpbar  npbar  3 npbar (1  pbar ) Example 8: An np-chart for a Post Office. The np data for the post office example discussed earlier is given in Figure 6.5. The average number of errors found is:

np 

3  1  ...  0  1  2.2 25


47

To find the standard deviation, we compute

p

2.2  0.022 100

Then,

sn p  2.2(1  0.022)  1.4668 The control limits are then computed as

UCLn p  2.2  3(1.4668)  6.6 LCLn p  2.2  3(1.4668)  2.20 Since the lower control limit is less than zero, a value of LCL = 0 is used. The control chart for this example is given in Figure 6.6.

Figure 6.5 Data and Calculations for Example 8

Sample # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number Nonconforming (np) 3 1 0 0 2 5 3 6 1 4 0 2 1 3 4 1 1 2 5 2 3 4 1 0 1

LCLnp 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0


UCLnp 6.6 6.6 6.6 6.6 6.6 6.6 6.6 6.6 6.6 6.6 6.6 6.6 6.6 6.6 6.6 6.6 6.6 6.6 6.6 6.6 6.6 6.6 6.6 6.6 6.6

48

Figure 6.6

7 6 5 np 4 3 2 1 0

np Chart for Example 8 UCLnp

LCLnp 1

3

5

7

9

11

13

15

17

19

21

23

25

Sample #

6.4

Charts for Defects

Recall that a defect is a single nonconforming characteristic of an item, while the term defective refers to an item that has one or more defects in it. In some situations, quality assurance personnel may be interested not only in whether an item is defective but also in how many defects it has. For example, in complex assemblies such as electronics, the number of defects is just as important as whether the product is defective. Two charts can be applied in such situations. The c-chart is used to control the total number of defects per unit when subgroup size is constant. If subgroup sizes are variable, a u-chart is used to control the average number of defects per unit. The c-chart is based on the Poisson probability distribution. To construct a c-chart, first estimate the average number of defects per unit, cbar, by taking at least 25 samples of equal size, counting the number of defects per sample, and finding the average. The standard deviation (sc) of the Poisson distribution is the square root of the mean and yields

sc  cbar Thus, three-sigma control limits for the c-chart are given by

UCLc  cbar  cbar

LCLc  cbar  cbar Example 9: Constructing a c-Chart Figure 6.7 shows the number of machine failures in a factory over a 25-day period. The total number of failures is 45; therefore, the average number of failures per day is cbar = 45/25 = 1.8 Control limits for the c-chart here are therefore given by UCLc = 5.82 LCLc = -2.22 or zero The chart is shown in Figure 6.8 and appears to be in control. Such a chart can be used for continued control or for monitoring the effectiveness of a quality improvement program.


49

Figure 6.7 Data and Calculations for Example 9 Sample # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of Defects (c) 2 3 0 1 3 5 3 1 2 2 0 1 0 2 4 1 2 0 3 2 1 4 0 0 3

LCLc 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

UCLc 5.82 5.82 5.82 5.82 5.82 5.82 5.82 5.82 5.82 5.82 5.82 5.82 5.82 5.82 5.82 5.82 5.82 5.82 5.82 5.82 5.82 5.82 5.82 5.82 5.82

UCL

LCL

As long as sample size is constant, a c-chart is appropriate. In many cases, however, the subgroup size is not constant or the nature of the production process does not yield discrete, measurable units. For example, suppose that in an auto assembly plant, several different models are produced that vary in surface area. The number of defects will not then be a valid comparison among different models. Other applications, such as the production of textiles, photographic film, or paper, have no convenient set of items to measure. In such cases, a


50

standard unit of measurement is used, such as defects per square foot or defects per square inch. The control chart used in these situations is called a u-chart. The variable u reprsents the average number of defects per unit of measurement, that is ui = ci/ni, where ni is the size of subgroup #i (such as square feet). The center line ubar for k samples each of size ni is computed as follows:

ubar 

c1  c2  ...  ck n1  n2  ...  nk

The standard deviation su of the ith sample is estimated by

su  ubar / ni The control limits, based on three standard deviations for the ith sample are then

UCLu  ubar  3 ubar / ni LCLu  ubar  3 ubar / ni Note that if the size of the subgroups varies, so will the control limits. This result is similar to the p-chart with variable sample sizes. In general, whenever the sample size n varies, the control limits will also vary.

Example 10: Constructing a u-Chart A catalog distributor ships a variety of orders each day. The packing slips often contain errors such as wrong purchase order numbers, wrong quantities, or incorrect sizes. Figure 6.10 shows the error data collected during August, 1999. Since the sample size varies each day, a u-chart is appropriate. To construct the chart, first compute the number of errors per slip as shown in column 3. The average number of errors per slip, ubar, is found by dividing the total number of errors (217) by the total number of packing slips (2,843). This is ubar = 217/2843 = 0.076 The standard deviation for a particular sample size ni is therefore

su  0.076 / ni Figure 7.6.9 u CHART

u (Defects/Unit)

0.25 0.2 0.15

Variable Control limits

0.1 0.05


29 31

25 27

23

19 21

15 17

9

11 13

7

5

3

1

0

51

The control limits (LCLu and UCLu) are shown in Figure 6.10. As with a p-chart, individual control limits will vary with the sample size, ni. The control chart is shown in Figure 6.9. One point (sample #2) appears to be out of control.

Figure 6.10 Sample # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

Data and Calculations for Example 10 No.of Defects cI 8 15 6 13 5 5 3 8 4 6 7 4 2 11 13 6 6 3 8 9 8 2 9 5 13 5 8 6 7 4 8

Sample Size ni 92 69 86 85 123 87 74 83 103 60 136 80 70 73 89 129 78 88 76 101 92 70 54 83 165 137 79 76 147 80 78

ui = ci/ni = Defects/unit 0.0870 0.2174 0.0698 0.1529 0.0407 0.0575 0.0405 0.0964 0.0388 0.1000 0.0515 0.0500 0.0286 0.1507 0.1461 0.0465 0.0769 0.0341 0.1053 0.0891 0.0870 0.0286 0.1667 0.0602 0.0788 0.0365 0.1013 0.0789 0.0476 0.0500 0.1026

Standard Deviation su 0.0288036 0.0332596 0.0297915 0.0299662 0.0249109 0.0296198 0.0321163 0.0303251 0.0272222 0.0356669 0.0236904 0.0308885 0.0330212 0.0323355 0.0292951 0.0243246 0.031282 0.029451 0.0316909 0.0274904 0.0288036 0.0330212 0.0375963 0.0303251 0.021508 0.0236038 0.0310834 0.0316909 0.0227868 0.0308885 0.031282

LCLu 0 0 0 0 0.0016 0 0 0 0 0 0.0053 0 0 0 0 0.0034 0 0 0 0 0 0 0 0 0.0118 0.0055 0 0 0.008 0 0

UCLu 0.1627 0.1761 0.1657 0.1662 0.1511 0.1652 0.1727 0.1673 0.158 0.1833 0.1474 0.169 0.1754 0.1733 0.1642 0.1493 0.1702 0.1647 0.1714 0.1588 0.1627 0.1754 0.1891 0.1673 0.1409 0.1471 0.1696 0.1714 0.1447 0.169 0.1702

One application of c-charts and u-charts is in a quality rating system. When some defects are considered to more serious than others, they can be rated, or categorized, into different classes. For instance, A - Very serious, B - serious, C - moderately serious, D - not serious Each category can be weighted using a point scale, such as 100 for A, 50 for B, 10 for C, and 1 for D. These points, or demerits, can be used as the basis for a c- or u-chart that would measure total demerits or demerits per unit, respectively. Such charts are often used for internal quality control and as a means of rating suppliers. Prepared for the Extension Program of Indira Gandhi Open University

52

7

CHOOSING THE CORRECT SPC CHART

Confusion often exists over which chart is appropriate for a specific application, since the cand u-charts apply to situations in which the quality characteristics inspected do not necessarily come from discrete units. The key issue to consider is whether the sampling unit is constant. For example, suppose that an electronics manufacturer produces circuit boards. The boards may contain various defects, such as faulty components and missing connections. Because the sampling unit - the circuit board - is constant (assuming that all boards are the same), a c-chart is appropriate. If the process produces boards of varying sizes with different numbers of components and connections, then a u-chart would apply. As another example, consider a telemarketing firm that wants to track the number of calls needed to make one sale. In this case, the firm has no physical sampling unit. However, an analogy can be made with the circuit boards. The sale corresponds to the circuit board, and the number of calls to the number of defects. In both examples, the number of occurrences in relationship to a constant entity is being measured. Thus, a c-chart is appropriate.

Figure 7.1 Control Chart Selection

Quality Characteristic

Use x- and moving range charts

n > 1?

no n  10? yes Use xbar and s charts

Use xbar and R charts

Type of Attribute

Constant sample size?

yes

Use p- or np-charts

no Use p-chart with variable sample size


Constant sampling unit?

no Use uchart

Use cchart

43

8.

KEY POINTS ABOUT CONTROL CHART CONSTRUCTION



Statistical process control (SPC) is a methodology for monitoring a process to identify special causes of variation and signal the need to take corrective action when appropriate.



Capability and control are independent concepts. Ideally, we would like a process to have both high capability and be in control. If a process is not in control, it should first be brought into control before attempting to evaluate process capability



Control charts have three basic applications: (1) Establishing a state of statistical control, (2) Monitoring a process to identify special causes, and (3) Determining process capability.



Control charts for variables data include: xbar- and R-charts; and s-charts, and individual and moving range charts. xbar- and s-charts are alternatives to xbar- and R-charts for larger sample sizes. The sample standard deviation provides a better indication of process variability than the range. Individuals charts are useful when every item can be inspected and when a long lead time exists for producing an item. Moving ranges are used to measure the variability in “individuals” or x charts.



A process is in control if no points are outside control limits; the number of points above and below the center line is about the same; the points seem to fall randomly above and below the center line; and most points (but not all) are near the center line, with only a few close to the control limits.



Typical out-of-control conditions are represented by sudden shifts in the mean value, cycles, trends, hugging of the center line, hugging of the control limits, and instability.



Modified control limits can be used when the process capability is known to be good. These wider limits reduce the amount of investigation of isolated points that would fall outside the usual control limits.



Charts for attributes include p-, np-, c- and u-charts. The np-chart is an alternative to the p-chart, and controls the number nonconforming for attributes data. Charts for defects include the c-chart and u-chart. The c-chart is used for constant sample size and the uchart is used for variable sample size.


44

9

IMPLEMENTING SPC

The original methods of SQC have been available for over 75 years now; Walter Shewhart's first book on control charts was written in 1924. However, studies show that managers still do not understand variation. Where do you find the motivation? Studies in industry show that when used properly, SQC or SPC reduce the cost of qualitythe cost of excessive inspection, scrap, returns from customers and warranty service. Good quality, as the Japanese have demonstrated, edges out competition, builds repute and along with that brings in new customers, raises morale and expands business. Good quality culture also propagates: Companies using SPC frequently require their suppliers to use them also. This generates considerable benefit. Where there is low use of SPC, the major reason often is lack of knowledge of variation and the importance of understanding it in order to improve customer satisfaction. Successful firms on the other hand repeatedly show certain characteristics:  Their top management understand variation and the importance of SPC methods to successfully manage it. They do not delegate this task to the QC department.  All people involved in the use of the technique understand what they are being asked to do and why it would help them, and  Training, followed by clear and written instructions on agreed procedures are systematically introduced and followed up through audits. The above principles form the core of the general principles of good quality management, be it through ISO 9000, QS 9000 or TQM. The bottom line is that you must find your own motivation to create and deliver quality. If you do it as a fad, you will neither get the results nor maintain credibility with your own people about initiatives that you take in your management interventions. So, to summarize,  Studies show that to succeed with SPC you must understand variation, however boring that may appear.  When SPC is used properly, quality costs go down.  Low usage of SPC is associated with lack of knowledge and training, even in senior management.  SPC needs to be systematically introduced.  A step-wise introduction of SPC would include - review of management systems, - review of requirements and design specs, - emphasis on the need for process understanding and control, - planning for education and training, - tackling one problem at a time based on customer complaints/feedback, - recording of detailed data, - measuring process capability and - making routine use of data to manage the process. The first prudent step toward implementing SPC in an organization would be "to put the house in order," which may be done by getting the firm registered by ISO 9000.


45

10 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

REVIEW QUESTIONS ABOUT CONTROL CHARTS Define statistical process control and discuss its advantages What does the term in statistical control mean? Explain the difference between capability and control. What are the disadvantages of simply using histograms to study process capability? Discuss the three primary applications of control charts. Describe the difference between variables and attributes data. What types of control charts are used for each? Briefly describe the methodology of constructing and using control charts. What does one look for in interpreting control charts? Explain the possible causes of different out-of-control indicators. How should control charts be used by shop-floor personnel? What are modified control limits? Under what conditions should they be used? How are variables control charts used to determine process capability? Describe the difference between control limits and specification limits. Shy is the s-chart sometimes used in place of the R-Chart? Describe some situations in which a chart for individual measurements would be used. Explain the concept of a moving range. Why is a moving range chart difficult to interpret? Explain the difference between defects and defectives. Briefly describe the process of constructing a p-chart. What are the key differences compared with an x-chart? Does an np-chart provide any different information than a p-chart? Why would an npchart be used? Explain the difference between a c-chart and a u-chart. Discuss how to use charts for defects in a quality rating system. Describe the rules for determining the appropriate control chart to use in any given situation.


46

11 1.

SELF ASSESSMENT QUESTIONS ABOUT SPC Thirty samples of size 3 listed in the following table were taken from a machining process over a 15-hour period. a. Compute the mean and standard deviation of the data. b. Compute the mean and range of each sample and plot them on control.

Sample 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

2.

3.55 3.61 3.61 4.13 4.06 4.48 3.25 4.25 4.35 3.62 3.09 3.38 2.85 3.59 3.60 2.69 3.07 2.86 3.68 2.90 3.57 2.82 3.82 3.14 3.97 3.77 4.12 3.92 3.50 4.23

Observations 3.64 3.42 3.36 3.50 3.28 4.32 3.58 3.38 3.64 3.61 3.28 3.15 3.44 3.61 2.83 3.57 3.18 3.69 3.59 3.41 3.63 3.55 2.91 3.83 3.34 3.60 3.38 3.60 4.08 3.62

4.37 4.07 4.34 3.61 3.07 3.71 3.51 3.00 3.20 3.43 3.12 3.09 4.06 3.34 2.84 3.28 3.11 3.05 3.93 3.37 2.72 3.56 3.80 3.80 3.65 3.81 3.37 3.54 4.09 3.00

Forty samples of size 5 listed in the following table were taken from a machining process over a 25-hour period. a. Compute the mean and standard deviation of the data. b. Compute the mean and range of each sample and plot them on control charts. Does the process appear to be in statistical control? Why or why not?

Data 1 2 3 4 5

Sample Number  1 2 9.999 10.022 9.992 9.998 10.002 10.037 10.003 9.994 10.009 10.003

3 10.001 10.006 10.002 9.993 10.011

4 10.007 10.006 10.004 10.018 10.011

5 10.011 9.979 9.991 9.996 9.994


6 10.019 10.017 10.018 10.008 10.018

7 10.015 10.015 9.978 10.006 9.997

8 9.988 9.990 10.008 10.002 9.989

9 9.980 10.001 10.013 9.998 10.015

10 10.017 10.017 9.988 10.010 9.980

47

3.

1 2 3 4 5

11 9.980 10.038 9.990 9.996 10.016

12 10.004 9.990 10.002 10.003 9.996

13 10.025 9.989 9.981 10.006 9.998

14 9.992 10.023 10.019 9.990 10.003

15 9.985 10.002 10.008 10.008 9.998

16 9.977 9.975 10.002 10.021 9.989

17 9.996 9.991 10.005 10.009 9.977

18 10.014 10.010 10.000 10.001 10.006

19 10.001 9.979 10.001 10.015 10.009

20 9.982 9.975 9.976 10.012 9.994

1 2 3 4 5

21 10.010 10.003 9.990 10.010 10.015

22 9.988 10.004 10.001 9.995 9.977

23 9.991 9.996 10.020 10.002 10.022

24 10.005 10.003 10.027 9.996 9.970

25 9.987 9.993 9.992 9.987 10.008

26 9.994 10.007 10.013 9.997 10.014

27 9.994 9.987 10.027 10.030 9.989

28 9.972 9.994 9.969 10.011 9.985

29 10.018 10.007 9.980 9.987 10.014

30 9.985 10.010 9.998 10.033 9.994

1 2 3 4 5

31 10.009 10.013 10.008 9.990 10.008

32 9.987 10.012 10.015 9.995 10.021

33 9.990 9.973 9.996 9.990 9.980

34 9.985 10.038 9.991 9.988 9.986

35 9.991 9.999 9.989 10.014 9.997

36 10.002 9.989 9.983 10.013 9.980

37 10.045 9.993 10.007 9.990 10.010

38 9.970 9.999 9.989 9.999 10.014

39 10.019 9.989 9.998 9.997 9.986

40 9.954 10.011 10.003 9.987 10.005

Suppose that the following sample means and standard deviations are observed for samples of size 5. Construct xbar and s-charts for these data. Sample # 1 2 3 4 5 6 7 8 9 10

X 2.15 2.07 2.10 2.14 2.18 2.11 2.10 2.11 2.06 2.15

S 0.14 0.10 0.11 0.12 0.12 0.12 0.14 0.10 0.09 0.08

Sample # 11 12 13 14 15 16 17 18 19 20

X 2.10 2.19 2.14 2.13 2.14 2.12 2.08 2.18 2.06 2.13

S 0.17 0.13 0.07 0.11 0.11 0.14 0.17 0.10 0.06 0.14

4.

Construct charts for individuals using both two-period and three-period moving ranges for the following observations: 9.0, 9.5, 8.4, 11.5, 10.3, 12.1, 11.4, 10.0, 11.0, 12.7, 11.3, 12, 12.6, 12.5, 13.0, 12.0, 11.2, 11.1, 11.5, 12.5, 12.1

5.

The fraction defective for an automotive piston is given below for 20 samples. Two hundred units are inspected each day. Construct a p-chart and interpret the results. Sample # 1 2 3 4 5 6 7 8 9 10

Fraction Defective 0.11 0.16 0.12 0.10 0.09 0.12 0.12 0.15 0.09 0.13

Sample # 11 12 13 14 15 16 17 18 19 20


Fraction Defective 0.16 0.23 0.15 0.12 0.11 0.11 0.14 0.16 0.10 0.13

48

12

ACCEPTANCE SAMPLING

Sampling inspection is a screening mechanism used to separate acceptable products from products of poor quality; it actually does not improve the quality of any product. The most obvious way to tell whether a product item is acceptable or not is to inspect it or to use it. If this can be done to every item, that is, a 100% inspection can be performed, there would be no need to use acceptance sampling. However, in many cases, it is not economical nor possible to do a 100% inspection. For instance, if the cost of inspection for an item is higher the value of the item, which usually is true for low-cost massed-produced products such as injectionmolded parts or flash light bulbs, a 100% inspection is not justified; if equipment cost and labor cost to inspect an item are high, only a small fraction of the product can be inspected. If the inspection is a destructive test (for example, a life test for electronic components or a car crash test), obviously a 100% inspection will destroy all the products. When inspection is necessary and 100% inspection is not possible, acceptance sampling can be employed. A sampling plan is a method for guiding the acceptance sampling process. It specifies the procedure for drawing samples to inspect from a batch of products and then the rule for deciding whether to accept or reject the whole batch based on the results of this inspection. The sample is a small number of items taken from the batch rather than the whole batch. The action of rejecting the batch means not accepting it for consumption and this may include downgrading the batch or selling it at a lower price returning it to its supplier or vendor. Suppose that a sampling plan specifies that (a) n item are drawn randomly from a batch to form a sample and (b) the batch is rejected if and only if c or more of these n items are defective or non-conforming. An operating characteristic curve, or OC-curve, of a sampling plan is defined as the plot of the probability that the batch will be accepted (Pa(p)) against the fraction p of defective products in the batch. The larger is Pa(p), the more it is likely that the batch is accepted. A higher likelihood of acceptance benefits the producer. On the other hand, the smaller is Pa(p), the harder it will be for the batch to be accepted. This would benefit and even protect the consumer who would want some assurance against receiving bad products and would prefer accepting batches with a low p (fraction defective) value.

12.1

AQL and RQL

In order to specify a sampling plan with Pa(p) characteristics as described above, the numbers n and c must be correctly specified. This specification requires us to specify two batch quality or fraction defective levels first, namely the AQL (acceptable quality level) and the RQL (rejection quality level) values. AQL and RQL (explained below) are two key quality parameters frequently used in designing a sampling plan. An ideal sampling plan is regarded to be one that accepts the batch with 100% probability if the fraction of defective items in it is less than or equal to AQL and that rejects the batch with a 100% probability if the fraction of defective items in the batch is larger than AQL.

Most customers realize that perfect quality (e.g., a batch or lot of parts containing no defective parts in it) is perhaps impossible to expect; some defectives will always be there. Therefore, the customer decides to tolerate a small fraction of defectives in his/her purchases. However, the customer certainly wants a high level of assurance that the sampling plan used to screen the incoming lots will reject lots with fraction defective levels exceeding some decidedly poor quality threshold called the rejection quality level or RQL. In reality, RQL is a defect level that causes a great deal of heartache once such a lot enters the customer's factory. The supplier of those parts, on the other hand, wants to ensure that the customer's acceptance sampling plan will not reject too many lots with defect levels that are certainly within the customer's tolerance, i.e., acceptable to the customer on the average. Generally, the customer sets a quality threshold here also, called AQL. This is actually the worst lot fraction defective that is acceptable to the customer in the shipments he/she receives on an average basis. Prepared for the Extension Program of Indira Gandhi Open University

49

12.2

The Operating Characteristic (OC) Curve of a Sampling Plan

A bit of thinking will indicate that only error-free 100% inspection of all items in a lot would accept the batch with 100% probability if the fraction of defective items in it is less than or equal to AQL and reject the lot with a 100% probability if the fraction of defective items in it is larger than AQL. Such performance cannot be realized othewise. The OC-curve of such a sampling plan is shown in Figure 12.1.

1.0 Pa(p)

0.0 0.0

AQL

p (fraction defective)

Figure 12.1 The Ideal OC Curve

In order to correctly evaluate the OC-curve of an arbitrary (not 100%-inspection) sampling plan with parameters (n, c) we need to harness certain principles of probability theory as follows. Suppose that the batch is large or the production process is continuous, so that drawing a sample item with or without replacement has about the same result. In such cases we are permitted to assume that the number of defective items x in a sample of sufficiently large size n follows a binomial probability distribution. Pa(p) is then given by the expression c n Pa( p)     p x (1  p) n  x x 0  x 

Now, if the producer is willing to sacrifice a little bit so that the batch with fraction defective AQL is accepted with a probability at least (1- ) where  is a small positive number, and the consumer is willing to sacrifice a little so that the batch with fraction defective RQL is accepted with a probability of at most , where  is a small positive number and RQL > AQL, then the following two inequalities can be established. c

 AQL (1  AQL) x

n x

1

n x



x 0

n

 RQL (1  RQL) x

x  c 1

From the above two inequalities, the numbers n and c can be solved (but the solution may not be unique). The OC-curve of such a sampling plan is shown in Figure 12.2. The number  is call the producer's risk, and the number  called the consumer's risk. As a common practice in industries, the magnitude of  and  is usually set at some value from 0.01 to 0.1. Nomograms are available for obtaining solution(s) of the inequalities given above easily. Prepared for the Extension Program of Indira Gandhi Open University

50

The above sampling plan is called a single sampling plan (Figure 12.3) because the decision to accept or reject the batch is made in a single stage after drawing a single sample from the batch being evaluated. To set up a practical single sampling procedure you would need to specify 1) AQL, 2) , the consumer's risk, 3) RQL, and 4) , the producer's risk. The plan itself may be developed from a tool called the"Larsen Nomogram" given in statistical quality control texts and handbooks. A typical plan specifying AQL = 0.01, = 0.05, RQL = 0.06 and  = 0.10 will have sample size n = 89 and the maximum number of defectives allowed (c) = 2.

 1.0 Pa(p)

0.0 0.0

 AQL

RQL

p (fraction defective)

Figure 12.2 OC Curve of a Practical Sampling Plan

Sample n items

Inspect all items in the sample: d defectives found

Yes d < c?

Accept the lot

No Reject the lot

Figure 12.3 The Single Sampling Plan


51

12.3

DOUBLE SAMPLING PLANS

A double sampling plan is one in which a sample of size n1 is drawn and inspected, the batch is accepted rejected according to whether the number d1 of defective items found in the sample is  r1 or  r2 (where r1 < r2); and if the number of defective items lies between r1 and r2, a further sample of size n2 is drawn and inspected, and the batch is accepted or rejected according to whether the total number d1 + d2 of defective items in the first and second sample is  r3 or > r3. This procedure is shown diagrammatically in Figure 12.4.

Sample n1 items

Inspect all items in this sample: d1 defectives found

d1 < r1?

Yes

Accept the lot

No Yes d1 > r2?

Reject the lot

No Sample n2 more items from the lot

Inspect all items in the second sample: d2 defectives found

d1+d2 > r3?

Yes

Accept the lot

No

Reject the lot

Figure 12.4 The Double Sampling Plan Prepared for the Extension Program of Indira Gandhi Open University

52

The average sample number (ASN) of a double sampling plan is Pa1(p) n1 + (1 - Pa1(p)) n2, where Pa1(p) is the probability that the batch will be accepted upon inspection of the first sample. If the value of p in the batch is very low or very high, a decision can usually be made within inspection of the first sample, and the ASN of a double sampling plan will be smaller than the sample size of a single sampling plan with the same producer's risk () and consumer's risk (). The idea of double sampling plan can be extended to construct sampling plans of more than two stages, namely multiple sampling plans. A sequential sampling plan is a sampling plan in which one item is drawn and inspected each time, in such a way that if a decision (of accepting or rejecting the batch) can be made upon inspection of the first item, the process is stop; if a decision cannot be made, a second item will be drawn and inspected, and if decision can be made upon inspection of the second item, the plan is stopped; otherwise, the third item is drawn and inspected, and so on, until a decision can be made. However, multiple sampling plans and sequential sampling plans are not so commonly used, because their implementation in practice is more complicated that single and double sampling plans.

12.4

AOQ AND AOQL (AVERAGE OUTGOING QUALITY LIMIT

The quantity [p Pa(p)] is called the average outgoing quality (AOQ) of a sampling plan at fraction defective p. This is the quality to be expected at the customer's end if the sampling plan is used consistently and repeatedly to accept or reject lots being received. It is clear that if p = 0, then AOQ = 0. If p = 1, that is, all the product items in the batch are defective, then Pa(p) of any sampling plan that we are using should be equal to 0 for otherwise the plan would not have any utility. Hence AOQ = 0 also when p = 1. Since AOQ can never be negative, it has a global maximum point in the range (0,1); this maximum is called the average outgoing quality limit (AOQL) of the sampling plan. The graph of AOQ against p for a typical sampling plan is shown in Figure 12.5. Although a sampling plan can be specified by setting the producer's risk () and consumer's risk () at AQL and RQL, the quantity AOQL can also be used to specify a sampling plan.

AOQ AOQL

Lot Fraction Defective p Figure 12.5 The AOQ Curve of a Sampling Plan

Another type of sampling plan which is different from the above is called continuous sampling procedure (CSP). The rationale of CSP is that, if we are not sure that the products produced from a process is of good quality, a 100% inspection will be adopted; if the quality of products is found to be good, then only a fraction of the products will be inspected. In the simplest CSP, initially 100% is performed; during 100% inspection if no defective items are found after a specified number of items are inspected (which means that the quality of product produced is perhaps good), 100% inspection is stopped and only a fraction f of the products is inspected. During fraction inspection if a defective item is found (which means that the quality of products might have deteriorated), then fraction inspection is stopped and 100% inspection is resumed. More refined CSP's have also been constructed, for example, by setting f at 1/2 at the first stage, 1/4 at the second stage, and so on. Prepared for the Extension Program of Indira Gandhi Open University

53

12.5

VARIABLES SAMPLING PLANS

All sampling plans described above are called attribute sampling plans, because the inspection procedure is based on a "go"/ "no go" basis, that is, an item is either regarded as non-defective and accepted, or it is regarded as defective and not accepted. Variable sampling plans are sampling plans in which continuous measurements (such as dimensional or weight checks) are made on each item in the sample, and the decision as to whether to accept or to reject the batch is based on the sample mean or the average of the measurements obtained from all items contained in the sample. A variable sampling plan can be used, for example, when a product item is regarded as acceptable if a certain measurement x (diameter, length, hardness, etc.) of it exceeds a pre-set lower spec limit L; otherwise the item is regarded as not acceptable (see Figure 12.5.6).

Unacceptable items

L

x

Figure 12.6 Distribution of Individual Measurements (x)

Measurements {x} of the products produced would vary from item to item, but these measurements have a population mean µ, say. When µ is much larger than L, we can expect that most items will have an x value greater than L and all such items would be acceptable; when µ is much less than L, we can expect that most items will have x values less than L and all such items would not be acceptable. A variable sampling plan can be constructed by specifying a sample size n and a lower cut-off value c for the sample mean xbarn such that if the sample of size n is drawn and all items in this sample are measured, the lot is accepted if the sample mean xbarn exceeds c. The lot is rejected otherwise. We require that when the population mean of the product produced is µ1 or larger, the lot is accepted with a probability of at least 1 - , and when the population mean is µ2 or smaller, the lot is accepted with probability only  or less, where  = producer's risk and  = consumer's risk (Figure 12.7).

Unacceptable Lots

2

Acceptable Lots

1

c



Xbarn



Figure 12.7 Producer's () and Consumer's () Risks


54 Suppose that x is the value of x such that the probability for x < x is . According to the criterion given above we can drive that

n



( x  x ) ( 1   2 )

x  n

 2  c 

x  1 n

The above system of inequalities may not have a unique (n, c) solution. From elementary statistical theory, if the form of x is known (for example, when x follows a normal distribution), from these inequalities we can determine a minimum value for the sample size n (which is an integer), and a range for the cut-off point c for the sample mean xbar. Such a sampling plan is a called single specification limit variable sampling plan. If an upper specification limit U instead of a lower specification limit L is set for x, we only need to consider the lower specification limit problem with (-x) replacing x and (-U) replacing L. When a product item is regarded as acceptable only if a certain measurement x of it lies between a lower specification limit L and an upper specification limit U, a double specification limit variable sampling plan will be used. In a double specification limit variable sampling plan, a sample size n, a lower cut-off value cL and an upper cut-off value cU for the sample mean xbar are specified. A batch is accepted if any only if the sample mean of a sample of size n from the batch lies between cL and cU . Calculations for cL,, cU, and n are more complicated than the single specification case.

MIL-STD-105E SAMPLING PLANS International standards for sampling plans are now available. Many of these are based on the work of Professors Dodge and Rohmig. The plan than originally was developed for single and multiple attribute sampling for the US army during WW II is now widely used in industries. It is called the MIL-STD-105E. An equivalent Indian standard known as IS 2500 has been published by the Bureau of Indian Standards. Many other official standards for various attribute sampling plans (such as those based on AOQ, or CSP's, and so on) and variable sampling plans (assuming the variable has a normal distribution, when the population variance is known or unknown, and so on) have been published by the US government and the British Standards Institution.

Sampling Inspection is only a Screening Tool! Before we end this section, we stress again that acceptance sampling or sampling inspection is only a screening tool for separating batches or lots of good quality products from batches of poor quality products. To some extent this screening assures the quality of incoming parts and materials. Actually, the use of sampling plans does help an industry to do this screening more effectively than the drawing samples arbitrarily. Therefore, sampling inspection can be used during purchasing, for checking the quality of incoming materials, whenever one is not sure about the conditions and QC procedures in use in the vendor's plant. Acceptance sampling can also be used for the final checking of products after production (Figure 1.3). This, to a limited degree, assures the quality of the products being readied for a customer before they are physically dispatched and even Motorola uses acceptance sampling as a temporary means of quality control until permanent corrective actions can be implemented. But note that unlike SPC, acceptance sampling does not help in the prevention of the production of poor quality products.


55

13

WHAT ARE TAGUCHI METHODS?

In this section we discuss briefly methods that belong in the domain of quality engineering, a recently formalized discipline that aims at developing products whose superior performance delights the discriminating usernot only when the package is opened, but also throughout their lifetime of use. The quality of such products is robust, i.e., it remains unaffected by the deleterious impact of environmental or other factors often beyond the users' control. Since the topic of quality engineering is of notably broad appeal, we include below a brief review of the associated rationale and methods. The term “quality engineering” (QE) was used till recently by Japanese quality experts only. One such expert is Genichi Taguchi (1986) who reasoned that even the best available manufacturing technology was by itself no assurance that the final product would actually function in the hands of its user as desired. To achieve this Taguchi suggested the designer must “engineer” quality into the product, just as he/she specifies the product’s physical dimensions to make the dimensions of the final product correct. QE requires systematic experimentation with carefully developed prototypes whose performance is tested in actual field conditions. The object is to discover the optimum setpoint values of the different design parameters, to ensure that the final product would perform as expected consistently when in actual use. A quality-engineered product has robust performance.

13.1

Taguchi's Thoughts on Quality

Taguchi, a Japanese electrical engineer by training, is credited to have made several contributions in the management and assurance of quality. Taguchi studied the methods of design of experiments (DOE) at the Indian Statistical Institute in the 1950s and later applied these methods in a very creative manner to improve product and process design. His methods now form the foundation of engineering design methodology in many leading industries around the world, including AT&T, General Motors and IBM. In the 1980s his methods were popularized in the USA by Madhav Phadke and Raghu Kacker of the Bell Laboratories. Taguchi's contributions may be classified under the following three headings:  The loss function  Robust design of products and production processes  Simplified industrial statistical experiments

Loss to Society

Taguchi's view of loss to society: "Target is best; Loss rises continuously"

Lower Spec

Target

Upper spec

Traditional view of loss to society: "Within spec is OK, outside spec is bad"

Performance Characteristic

Figure 13.1 Loss to Society increases whenever Performance deviates from the Target


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The essence of the loss function concept may be stated as follows. Whenever a product deviates from its target performance, it generates a loss to society (Figure 13.1). This loss is minimum when performance is right on target, but it grows gradually as one deviates from the target. Such a philosophy suggests that the traditional "if it is within specs, the product is good" view of judging a product's quality is not correct. If your foot size is 7, then a shoe of size different from 7 will cause you inconvenience, pain, loose fit, and even embarrassment. Under such conditions it is meaningless to seek a shoe that meets a spec given as (7  x). To state again, the loss function philosophy says that for a producer, the best strategy is to produce products as close to the target as possible, rather than aiming at "being within specifications." The other contributions of Taguchi are a methodology to minimize performance or quality problems arising due to non-ideal operating or environmental conditions, and a simplified method known as orthogonal array experiments to help conduct multi-factor experiments toward seeking the best product or process design. These ideas may be described as follows.

13.1

The Secret of Creating a Robust Design

A practice common in traditional engineering design is sensitivity analysis. For instance, in traditional electronic circuit design, as well as the development of performance design equations, sensitivity analysis of the circuit developed remains a key step that the designer must complete before his job is over. Sensitivity analysis evaluates the likely changes in the device's performance, usually due to element value tolerances or due to value changes with time and temperature. Sensitivity analysis also determines the changes to be expected in the design’s performance due to factor variations of uncontrollable character. If the design is found to be too sensitive, the designer projects the worst-case scenarioto help plan for the unexpected. However, studies indicate that worst-case projections or conservative designs are often unnecessary and that a “robust design” can greatly reduce off-target performance caused by poorly controlled manufacturing conditions, temperature or humidity shifts, wider component tolerances used during fabrication, and also field abuse that might occur due to voltage/frequency fluctuations, vibration, etc. Robust design should not be confused with rugged or conservative design, which adds to unit cost by using heavier insulation or high reliability, high tolerance components. As an engineering methodology robust design seeks to reduce the sensitivity of the product/process performance to the uncontrolled factors through a careful selection of the values of the design parameters. One straightforward way to produce robust designs is to apply the "Taguchi method". The Taguchi method may be illustrated as follows. Suppose that a European product (Swiss chocolate bars) is to be introduced “as is” in a tropical country where the ambient temperature rises to 45C. If the European product formulation is directly adopted, the result may be molten bars on store shelves in Bombay and Singapore and gooey hands and dirty dresses, due to the high temperature sensitivity of the Swiss chocolate recipe (Curve 1, Figure 13.2). The behavior of the bar’s plasticity may be experimentally explored to determine its robustness to temperature, but few product designers actually attempt this. Taguchi would suggest that we do here some special “statistical experiments” in which both the bar’s formulation (the original Swiss and perhaps an alternate prototype formulation that we would call “X”) and ambient temperature would be varied simultaneously and systematically and the consequent plasticities observed.


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Curve 1

P l a s t i c i t y

Curve 2

Temperature Figure 13.2 DEPENDENCE OF PLASTICITY ON AMBIENT TEMPERATURE

Response at the highest ambient temperature

P l a s t i c i t y

Robust behavior of “X” Unacceptable variation in Eu’s plasticity due to temperature rise

Response at the lowest ambient temperature Formulation “X”

European (Eu) formulation

Figure 13.3 INTERACTION OF THE EFFECTS OF TEMPERATURE AND CHOCOLATE FORMULATION

Taguchi was able to show that by such experiments it is often possible to discover an alternate bar design (here an appropriate chocolate bar formulation) that would be robust to temperature. The trick, he said, is to uncover any “exploitable” interaction between the effect of changing the design (e.g. from the Swiss formulation to Formulation “X”) and temperature. In the language of statistics, two factors are said to interact when the influence of one on a response is found to depend on the setting of the other factor (Montgomery, 1997). Figure 13.3 shows such an interaction, experimentally uncovered. Thus, a “robust” chocolate bar may be created for the tropical market if the original Swiss formation is changed to Formulation “X”.


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13.2

Robust Design by the "Two-step" Taguchi Method

Note that a product’s performance is “fixed” primarily by its design, i.e., by the settings selected for its various design factors. Performance may also be affected by noiseenvironmental factors, unit to unit variation in material, workmanship, methods, etc., or due to aging/deterioration (Figure 13.4). The breakthrough in product design that Taguchi achieved renders performance robust even in the presence of noise, without actually controlling the noise factors themselves. Taguchi’s special “designnoise array” experiments (Figure 13.5) discover those optimum settings. Briefly, the procedure first builds a special assortment of prototype designs (as guided by the “design array”) and then tests these prototypes for their robustness in “noisy” conditions. For this, each prototype is “shaken” by deliberately subjecting it to different levels of noise (selected from the “noise array”, which simulates noise variation in field conditions). Thus performance is studied systematically under noise in order to find eventually a design that is insensitive to the influence of noise.

Noise

Design Factor A

Variable Response

Product

Design Factor B Figure 13.4

DESIGN AND NOISE FACTORS BOTH IMPACT RESPONSE

“X”  Eu

Design Array Figure 13.5

Temp 50C Temp 40C Temp 30C Temp 20C Temp 10C Temp 0C Noise Array

=

Plasticity X_50 Plasticity Eu_50 ... Plasticity X_0 Plasticity Eu_0 (Experimental results)

DESIGNNOISE ARRAY EXPERIMENTS AND THEIR RESULTS

To guide the discovery the “optimum” design factor settings Taguchi suggested a two-step procedure. In Step 1, optimum settings for certain design factors (called “robustness seeking factors”) are sought so as to ensure that the response (for the bar, plasticity) becomes robust (i.e., the bar does not collapse into a blob at least up to 50C temperature). In Step 2, the optimum setting of some other design factor (called the “adjustment” factor) is sought to put the design’s average response at the desired target (e.g., for plasticity a level that is easily chewable). Prepared for the Extension Program of Indira Gandhi Open University

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For the chocolate bar design problem, the alternative design “factors” are two candidate formulationsone the original European, and the other that we called “X”. Thus, the design array would contain two alternatives “X” and “Eu.” “Noise” here is ambient temperature, to be experimentally varied over the range 0C to 50C, as seen in the tropics. Figure 13.5 shows the experimental outcome of hypothetical designnoise array experiments. For instance, response value Plasticity X_50 was observed when formulation X was tested at 50C. Figure 13.3 is a compact illustrative display of these experimental results. It is evident from Figure 13.3 that Formulation X’s behavior is quite robust even at tropical temperatures. Therefore, the adaptation of Formulation “X” would make the chocolate bar “robust”, i.e., its plasticity would not vary much even if ambient temperature had wide swings.

Taguchi's Orthogonal Array Experiments Rather typically, the performance of a product (or process) is affected by a multitude of factors. It is also well-known that over 2/3rd of all product malfunctions may be traced to the design of the product. To the extent basic scientific knowledge allows the designer to guide his/her design, the designer does his/her best to come up with selection of design parameter values that would ensure good performance. Frequently though, not everything can be predicted by theory and experimentation or prototyping must be resorted to and the design must be empirically optimized. The planning of such experimental multiple-factor investigations falls in the domain of statistical design of experiments (DOE). Many text book methods for conducting multi-factor experiments, however, are too elaborate and cumbersome. This has discouraged many practicing engineers from trying out this powerful methodology in real design and optimization work. Taguchi observed this and popularized a class of simpler experimental plans that can still reveal a lot about the performance of a product or process, without the burden of heavy theory. An example is shown below. A fire extinguisher is to be designed so as to effectively cover flames in case of a fire. The designer wishes to achieve this by either using higher pressure inside the CO 2 cylinder, or by altering the nozzle design. Theoretical models of such systems using computational fluid dynamics (CFD) are too complex and too cumbersome to optimize. The question to be answered is, which is more effectivehigher pressure or a wider nozzle? The Taguchi orthogonal array experimental scheme would set up the following experimental plan consisting of only four specifically designed experiments. The results are shown in the table and on the associated graph. It is clear from the factor effect plots of the results that the designer would be much better off by increasing pressure. Nozzle diameter seems to have little effect on the extent of the area covered by the extinguisher.

Exp t#

Nozzle dia

CO2 Pressure

Observed Spray Area

1

5 MM

2 BARS

0.8 M2

2

10 MM

2 BARS

0.9 M2

3

5 MM

4 BARS

1.6 M2

4

10 MM

4 BARS

1.9 M2

A r e a

5


10

Nozzle

2

4 CO2

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14

THE SIX SIGMA PRINCLPLE

The six sigma principle is Motorola's own rendering of what is known in the quality literature as the zero defects (ZD) program. Zero defects is a philosophical benchmark or standard of excellence in quality proposed by Philip Crosby. Crosby explained the mission and essence of ZD by the statement "What standard would you set on how many babies nurses are allowed to drop?" ZD is aimed at stimulating each employee to care about accuracy and completeness, to pay attention to detail, and to improve work habits. By adopting this mind-set, everyone assumes the responsibility toward reducing his or her own errors to zero. One might think that having three-sigma quality, i.e., the natural variability ( x  3x) equal to tolerance (= upper spec limit - lower spec limit, or in other words, Cp = 1.0) would mean good enough quality. After all, if distribution is normal, only 0.27% of the output would be expected to fall outside the product's specs or tolerance range. But what does this really mean? An average aircraft consists of 10,000 different parts. At 3-sigma quality, 27 of those parts in an assembled aircraft would be defective. At this performance level, there would be no electricity or water in Delhi for one full day each year. Even four-sigma quality may not be OK. At four-sigma level there would be 62.10 minutes of telephone shutdown every week. You might wish to know where performance is today. Restaurant bill errors are near 3-sigma. Payroll processing errors are near 4-sigma. Wire transfer of funds in banks is near 4-sigma and so is baggage mishandling by airlines. The average Indian manufacturing industry is near the 3-sigma level. Airline flight fatality rates are at about 6.5 sigma level (0.25 per million landings). At two-sigma level a company's cost of returns, scrap, rework and erosion of market share costs over a third of its yearly sales. For the typical Indian, a 1-hour train delay, an incorrect eye operation or drug administration, or no electricity or water half a day is no surprise; he/she routinely experiences even worse performance. Quantitatively, such performance is worse than two-sigma. Can this be called acceptable? One- or two-sigma performance is downright noncompetitive. Besides adopting TQM as the way to conduct business, many companies worldwide are now seriously looking at six-sigma benchmarks to assess where they stand. Six sigma not only reduces defects and raises customer acceptability, it has been now shown at Allied Signal Inc., Motorola, Raytheon, Bombardier Aerospace and Xerox that it can actually save money as well. Therefore, it is no surprise that Motorola aggressively set the following quality goal for itself in 1987 and then didn't want to stop till they achieved it: Improve product and services quality ten times by 1989, and at least one hundred fold by 1991. Achieve six-sigma capability by 1992. With a deep sense of urgency, spread dedication to quality to every facet of the corporation, and achieve a culture of continual improvement to assure total customer satisfaction. There is only one goal: zero defectsin everything we do.

The Steps to Six Sigma The concept of six-sigma qualityshrinking the inherent variation in a process to half of the spec range (Cp = 2.0) while allowing the mean to shift at most 1.5 sigma from the spec midpoint (the target quality)is explained by Figure 14.1. The area under the shifted curves beyond the six sigma range (the tolerance limits) is only 0.0000034, or 3.4 parts per million. If the process mean can be controlled to within  1.5 x of the target, a maximum of 3.4 defects per million pieces produced can be expected. If the process mean is held exactly on target, only 2.0 defects per billion would be expected. This is why within its organization Motorola defines six sigma as a state of the production or service unit that represents "almost perfect quality." Prepared for the Extension Program of Indira Gandhi Open University

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Process Mean  3 = Inherent variability = half of Tolerance

- 6

LSL

- 1.5

+ 1.5

Target

+ 6

USL

Process Mean is allowed to vary in this range Tolerance

Figure 14.1 The Six Sigma Process: USL = Mean + 3, LSL = Mean - 3

Motorola prescribes six steps to achieve the six-sigma state, as follows. Step 1: Identify the product you create or service you provide. Step 2: Identify the customer(s) for your product or service and determine what they consider important. Step 3: Identify your needs to provide the product or service that satisfies the customer. Step 4: Define the process for doing the work. Step 5: Mistake-proof the process and eliminate waste effort. Step 6: Ensure continuous improvement by measuring, analyzing and controlling the improved process. Many companies have adopted the Measure-Analyze-Improve-Control cycle to step into six sigma. Typically they proceed as follows:  Select critical-to-quality characteristics  Define performance standards (the targets to be achieved)  Validate measurement systems (to ensure that the data is reliable)  Establish Product Capability (how good are you now?)  Define performance objectives  Identify sources of variation (seven tools etc.)  Screen potential causes (correlation studies, etc.)  Discover relationship between variablescauses or factorsand the output (DOE)  Establish operating tolerances for input factors and output variables  Validate the measurement system  Determine process capability (Cpk) (Can you deliver? What do you need to improve?)  Implement process controls One must audit and review nonstop to ensure that one is moving along the charted path.


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The aspect common between six sigma and ZD ("zero defects") is that both concepts require the maximum participation by the entire organization. In other words, they require that unrelenting effort by management and the involvement of all employees. Companies such as General Motors have used a four-phase approach to seek six sigma: 1.

2. 3. 4.

Measure: Select critical quality characteristics though Pareto charts; determine the existing frequency of defects, define target performance standard, validate the measurements system and establish existing process capability. Analyze: Understand when, where, and why defects occur by defining performance objectives and sources of variation. Improve: Identify potential causes, discover cause-effect relationships, and establish operating tolerances. Control: Maintain improvements by validating the measurement system, determining process capability, and implementing process control systems.

It is reported that in GM a new culture has been created. An individual or team devotes all its time and energy to solving one problem at a time, designs solutions with customers' assistance, and helps to minimize bureaucracy in supporting the six-sigma initiative.

Beyond TQM The Japanese have recently evolved the "Bluebird Plan," which is a "third option" beyond SPC and TQM designed to achieve the four objectives business excellence. These objectives include establishing corporate ethics, maintaining and boosting international competitiveness, ensuring stable employment and improving national quality of life. The Bluebird Plan provides a forum for government, labor and management to discuss the actions which need to be taken. In Japan the plan set out an action program for reform for the three years 1997-1999 which was noted to be a critical time that would determine the direction of Japan's future. Striking about the plan is the employers' acceptance that the relationship between labor and management is an imperative "stabilizing force in society." Thus it reaches beyond the tenets of TQM.
