Bibliography
4.3 Statistical Methods of Process Monitoring
Statistical process control is a powerful collection of problem-solving tools useful in achieving process stability and improving process capability through the reduction of variability. The natural variability in a process is the effect of many small unavoidable causes. This natural variability is also called a “stable system of chance causes” [31]. A process is said to be in statistical control when it operates under only chance causes of variation. On the other hand, unnatural variation may be observed and assigned to a root cause. These unnatural sources of variability are referred to as assignable causes. Assignable causes can range from improperly adjusted machines to human error. A process or service operating under assignable causes is said to be out of SPC.
4.3.1 Seven Tools of SPC
Statistical process control can be applied to any process and relies on seven major tools, sometimes called the magnificent seven [31]:
1. Histogram 2. Check sheet 3. Pareto chart
4. Cause and effect diagram 5. Defect concentration diagram
TABLE 4�2 Logistics Variable Data
PM# Performance Metric Source
1 Cycle time for receipt of material Defense Logistics Agency [26]
2 Cycle time for issuance of stock Defense Logistics Agency [26]
3 Cost of nonconformance Arkansas Best Freight [30]
4 Cost of maintenance Lucent Technologies [30]
5 Transportation cost J.B. Hunt [30]
6 Inventory on hand Lucent Technologies [30]
7 Customer inquiry time Defense Logistics Agency [30]
6. Scatter diagram 7. Control chart
Histogram: A histogram represents a visual display of data in which three properties can be seen (shape, location or central tendency, and scatter or spread). The typical histogram is a type of bar chart with the vertical bars ordered horizontally by value of a variable. The vertical scale measures frequencies.
Check sheet: A check sheet is a very useful tool in the collection and interpretation of data. For example, a check sheet may capture data for a histogram. Events are tallied in categories. A check sheet should clearly specify the type of data to be collected as well as any other information useful in diagnosing the cause of poor performance.
Pareto chart: The Pareto chart is simply a frequency distribution (or histogram) of attribute data arranged by category. The Pareto chart is a very useful tool in identifying the problems or defects that occur most frequently. It does not identify the most important defects; it only identifies those that occur most frequently. Pareto charts are widely used for identifying quality-improvement opportunities.
Cause and effect diagram: The cause and effect diagram is a tool frequently used to analyze potential causes of undesirable problems or defects. Montgomery [31] suggests a list of seven steps to be followed when constructing a cause and effect diagram: (i) define the problem, (ii) form the team to perform the analysis, (iii) draw the effect box and the centerline, (iv) specify the major potential cause categories and join them as boxes connected to the centerline, (v) identify the possible causes and classify them, (vi) rank the causes to identify those that impact the problem the most, and (vii) take corrective action.
Defect concentration diagram: The defect concentration diagram is a picture of the process or product.
The different types of defects or problems are drawn on the picture, and the diagram is analyzed to determine the location of the problems or defects.
Scatter diagram: The scatter diagram is used to identify the potential relationship between two variables. Data are plotted on an x-y coordinate system. The shape of the scatter diagram indicates the possible relationship existing between the two variables.
Control chart: The control chart is a graphical display of a quality characteristic that has been measured or computed from a sample versus the sample number or time.
4.3.2 Control Charts in the Logistics Area
To separate assignable causes from the natural process variation, we make use of control charts. Control charts are the simplest procedure of on-line SPC (Fig. 4.1). These charts make possible the diagnosis and correction of many problems, and help to improve the quality of the service provided. Control charts also help in preventing frequent process adjustments that can increase variability. Through process improvements, control charts often provide assurance of better quality at a lower cost. Therefore, a control chart is a device for describing in a precise manner exactly what is meant by statistical control [27].
A control chart contains a centerline that represents the in-control average of the quality characteris-tic. It also contains two other horizontal lines called the upper control limit (UCL) and lower control limit (LCL). If a process is in control, most sample points should fall within the control limits. These limits are typically called “3-sigma (3σ) control limits.” Sigma represents the standard deviation (a mea-sure of variability, or scatter) of the statistic plotted on the chart. The width of the control limits is inversely proportional to the sample size n.
Control charts permit the early detection of a process that is unstable or out of control. However, a control chart only describes how a process is behaving, not how it should behave. A particular control chart might suggest that a process is stable, yet the process may not actually be satisfying customer requirements.
4.3.3 Error Types
In a control chart, the distance between the centerline and the limits controls decision-making based on error. There are two types of statistical error. A type I error, also known as a false alarm or producer’s risk, results from wrongly concluding that the process is out of control when in fact it is in control.
A type II error, also known as the consumer’s risk, results from concluding that the process is in control when it is not. Widening the control limits in a control chart decreases the risk of a type I error, but at the same time increases the risk of a type II error. On the other hand, if the control limits are moved closer to the centerline, the risk of having a type I error increases while decreasing the risk of a type II error.
4.3.4 AT&T Run Rules
The identification of nonrandom patterns is done using a set of rules known as run rules. The classic Western Electric (AT & T) handbook [32] suggests a set of commonly used decision tools for detecting nonrandom patterns on control charts. A process is out of control if any one of the following applies:
1. One data point plots outside the 3-sigma limits (UCL, LCL).
2. Two out of three consecutive points plot outside the 2-sigma limits.
3. Four out of five consecutive points plot outside the 1-sigma limits.
4. Eight consecutive points plot on one side of the centerline.
The rules apply to one side of the centerline at a time. For example, in the case of rule 2, the process is judged out of control when two out of three consecutive points falling beyond the 2-sigma limits are on the same side of the centerline.
4.3.5 Types of Control Charts
Quality is said to be expressed by variables when a record is made of an actual measured quality charac-teristic. The _x , R, and S control charts are examples of variables control charts. When samples are of size one, individual (I) charts are suggested for monitoring the mean, and moving range (MR) charts are suggested for monitoring the variance. On the other hand, when a record shows only the number of articles conforming or nonconforming to certain specified requirements, it is said to be a record by attributes. The p, c, and u charts are examples of control charts for attribute data. One other important control chart is the moving centerline exponentially weighted moving average (EWMA), which is very effective in monitoring data that are not independent.
Control Chart
0 1 2 3 4 5 6
Sample Number
Upper Control Limit Center Line Lower Control Limit Data
2 3 4 5 6 7 8 9 10
1
FIGuRE 4�1 Control chart.
4.3.5.1 Control Charts for Variable Type Data
When dealing with variable data, it is usually necessary to control both the mean value of the quality characteristic and its variability. To monitor the mean value of the product, the _x control chart is often used. Process variability can be monitored with a control chart for the standard deviation called the S chart, or a control chart for the range, called the R chart. The R chart is the more widely used. The _x and R (or S) charts are among the most important and useful on-line SPC techniques. When the sample size, n, is large, n > 12, or the sample size is variable, the S chart is preferred to the R chart for monitoring variability.
4.3.5.2 Control Charts for Attribute Data
It is known that many quality characteristics cannot be represented numerically. Items inspected are usually classified as conforming or nonconforming to the specifications of that quality characteristic.
This type of quality characteristic is called an attribute. Attribute charts are very useful in most indus-tries. For example, from the logistics perspective, it is often necessary to monitor the percentage of units delivered on-time, on-budget, and in compliance with specifications.
The p-chart is used to monitor the fraction nonconforming from a manufacturing process or a service. It is based on the binomial distribution (number of successes in n trials) and assumes that each sample is independent. The fraction nonconforming is defined as the ratio of nonconforming items in a population to the total number of items in that population. Each item may have a number of quality characteristics that are examined simultaneously. If any one of the items being scrutinized does not satisfy the requirements, then the item is classified as nonconforming. The fraction nonconforming is usually expressed as a decimal, although it is occasionally expressed as the percent nonconforming.
There are many practical situations in which working directly with the total number of defects or nonconformities per unit or the average number of nonconformities per unit is preferred over the frac-tion nonconforming. The c-chart assumes that the occurrence of nonconformities in samples of constant size is rare. As a result, the occurrence of nonconformity is assumed to follow the Poisson probability distribution. The inspection unit must be the same for each sample.
4.3.5.3 Control Chart for Moving Centerline Exponentially Weighted Moving Average The use of variable control charts implies the assumption of normal and independent observations.
If the assumption of normality is violated to a moderate degree, the _x control chart used to monitor the process average will work reasonably well due to the central limit theorem (law of large numbers).
However, if the assumption of independence is violated, conventional control charts do not work well.
Too many false alarms disrupt operations and produce misleading results. The moving centerline exponentially weighted moving average (EWMA) is effectively a one-step-ahead predictor to monitor processes when data are correlated. The moving centerline EWMA chart is also recommended for use in the logistics arena for a performance metric that is subject to seasonal variation.
4.3.6 Construction of Control Charts
Table 4.3 (variable type) and Table 4.4 (attribute, or fraction nonconforming type) summarize the parameters and equations for commonly used control charts applicable to logistics performance measurement.