STATISTICAL QUALITY CONTROL (SQC)


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1 Statistical Quality Control 1 SQC consists of two major areas: STATISTICAL QUALITY CONTOL (SQC)  Acceptance Sampling  Process Control or Control Charts Both of these statistical techniques may be applied to two kinds of data. 1. Attribute Data: when the quality characteristic being investigated is noted by either its presence or absence and then classified as Defective or NonDefective. Example: Conforming or nonconforming Pass or fail Good or bad 2. Variable Data: The characteristics are actually measured and can take on a value along a continuous scale. Example: Length, Weight Sometimes variable data can be transformed into attribute data. For example, the specifications required for a shaft diameter (X) is 2" plus or minus 0.01". If X falls within 1.99" and 2.01", then the shaft diameter is conforming to specifications and hence is classified as good. If X < 1.99" or X > 2.01", then the shaft diameter is not conforming to specifications and hence classified as bad. Thus, attribute data does not have information of how much good or how much bad? which the variable data would have, because it would record the exact measurements of each shaft. We will first study Acceptance Sampling.
2 2 Statistical Quality Control Acceptance Sampling: Inspection provides a means for monitoring quality. For example, inspection may be performed on incoming raw material, to decide whether to keep it or return it to the vendor if the quality level is not what was agreed on. Similarly, inspection can also be done on finished goods before deciding whether to make the shipment to the customer or not. However, performing 100% inspection is generally not economical or practical, therefore, sampling is used instead. Acceptance Sampling is therefore a method used to make a decision as to whether to accept or to reject lots based on inspection of sample(s). The objective is not to control or estimate the quality of lots, only to pass a judgment on lots. Using sampling rather than 100% inspection of the lots brings some risks both to the consumer and to the producer, which are called the consumer's and the producer's risks, respectively. We encounter making decisions on sampling in our daily affairs. Example: LOT (N) SAMPLE (n) STATISTICAL Inference is made on the quality of the lot by inspecting only the small sample drawn from the lot.
3 Statistical Quality Control 3 There are several Acceptance Sampling Plans:  Single Sampling (Inference made on the basis of only one sample)  Double Sampling (Inference made on the basis of one or two samples)  Sequential Sampling (Additional samples are drawn until an inference can be made) etc. We will do Single Sampling plans only in this course. Single Sampling Plans A Single Sampling plan is characterized by n (the sample size) which is drawn from the lot and inspected for defects. The number of defects (d) found are checked against c (the acceptance number) and the procedure works as follows (clearly, d = 0, 1, 2, n): Example: Suppose n=100 and c=3, which means that if the number of defectives in the sample (d) is equal to 0, 1, 2, or 3, then the lot will be accepted, and if d is 4 or more, then the lot will be rejected.
4 4 Statistical Quality Control As mentioned earlier, inherent in a sampling plan are producer s and consumer s risk. These risks can be depicted by the following table: Lot is Good Decision Accept No Error eject Error (Producer s isk) Bad Error (Consumer s isk) No Error Formally, these risks are written as: where α : The producer's risk, is the probability that a lot with AQL will be rejected. β : The consumer's risk, is the probability that a lot with LTPD will be accepted. Acceptable Quality Level (AQL) = The quality level acceptable to the consumer Lot Tolerance Percent Defective (LTPD) = The level of "poor' quality that the consumer is willing to tolerate only a small percentage of the time. In general, both the producer and the consumer want to minimize their risks. The choice of a well designed sampling plan can help both the producer and the consumer maintain their respective risks at acceptable levels to both. For example, α = 5% for AQL of 0.02 and β = 10% for LTPD of 0.08.
5 Statistical Quality Control 5 Keeping c constant: What is the effect on producer s risk? What is the effect on consumer s risk? Keeping n constant: What is the effect on producer s risk? What is the effect on consumer s risk?
6 6 Statistical Quality Control The Theory Behind Process Control Let s now turn our attention to the second major area of SQC, namely Process Control or Control Charts, which directly affect the quality of a production or service process. Every production process has a natural variation. For example, a process making shafts is adjusted so that the shaft diameter will be 2". However, due to the natural variation in the manufacturing process, not every shaft coming off the production line will have a diameter of exactly 2". There will be some unexplained variation around the nominal value of 2". Therefore, some tolerance is built into the design of the product to allow for this natural (random) variation. However, if the process goes out of control, the variation may become more than that allowed by the design indicating the presence of variation that can be explained(e.g., defective raw material, untrained worker, etc.). In this case some action needs to be taken, the machine can be readjusted, replaced etc. The control charts show when the variation in the process is within the limits of the natural variation and when it goes out of control. Below are pictures that show various incontrol and outofcontrol situations for a process.
7 Statistical Quality Control 7 Even when the process is in control, we need to make sure that the mean of the process is in conformance with specifications as shown below.
8 8 Statistical Quality Control Continuous improvement in the process is possible by reducing the variation around the mean as shown below.
9 Statistical Quality Control 9 Charts Used with Variable Data: Control charts are of two types corresponding to the type of data that is used, namely variable or attribute data. We will study the popular control charts of both these types. X and Charts (mean and range charts) are commonly used in dealing with variable data to monitor the quality of a manufacturing process. The reason that both the charts have to be used together is that both the mean and the variation (spread) have to be under control. ecall that the variable data consists of actual measurements (e.g., shaft lengths, weight of bags in lbs, etc.). Let us take an example of variable data that is pertinent for the acid content in a certain chemical product. The operator measured and recorded the acid content of a sample of 4 units at a time at regular intervals for at least 25 times. This variable data and the calculations performed with it are shown on the following table. Also, given are the variable control charts ( X and charts) for the data.
10 10 Statistical Quality Control The Control Limits (UCL = Upper Control Limit and LCL = Lower Control Limit with the mean of the data as the central line) for X and Charts are established as follows: X Charts X = g i g X i UCL x = X + 3σ LCL = X 3σ x x x where: X = average of subgroup averages (the central line in the chart) X i = average of the ith subgroup g = number of subgroups σ x is further estimated using the range information (i.e., 3σ x = A 2 ); as such the control limit calculations are much simplified. The simplified control limits are as follows: UCL = X + x A2 LCL = X x A2 where A 2 is a factor available in tables for different sample sizes (see table below). Charts: = g i g i UCL = + 3σ LCL = 3σ where = average of subgroup ranges (the central line in the chart) i = range if the ith subgroup g = number of subgroups Similarly, control limit calculations are much simplified and are: UCL = D4 LCL = D3 where D 3 and D 4 are factors available in tables for different sample sizes (see table below). Factors for Control Limits n A 2 D 4 D
11 Statistical Quality Control 11 Let us now calculate the control limits for the given data, starting first with the ange () chart. This is done first because the X chart requires in determining its control limits. Therefore, naturally we need to first check if the chart is under control and use that in the control limits of the X chart. Chart: n = = X = A 2 = D 3 = D 4 = UCL = D 4 LCL = D 3 = = Does the chart show that the process is under control? Yes or No and why? X Chart: UCL = X + A 2 LCL = X  A 2 = = Does the X chart show that the process is under control? Yes or No and why?
12 12 Statistical Quality Control Another Example: The St. Patrick's Hospital is starting a quality improvement project on the time to admit a patient using X and Charts. Determine the limits for the X and charts and check to see if there are any outofcontrol points. Subgroup Number OBSEVATION X 1 X 2 X 3 X Subgroup Number OBSEVATION X 1 X 2 X 3 X n = = X = A 2 = D 3 = D 4 = Chart: UCL = D 4 LCL = D 3 = = Does the chart show that the process is under control? Yes or No and why? X Chart: UCL = X + A 2 LCL = X  A 2 = = Does the X chart show that the process is under control? Yes or No and why?
13 Statistical Quality Control 13 Charts Used with Attribute Data PChart, also known as the fraction or percent defective chart, is commonly used in dealing with attribute data to monitor the quality of a manufacturing process. The mean percent defective ( p ) is the central line. The upper and lower control limits are constructed as follows: The mean proportion defective ( p ): The standard deviation of p: p = Total Number of Defectives Total Number Inspected p( 1 p) σ p = n where n = sample size. Control Limits are: UCL = p + Z σ p LCL = p Z σ p or UCL = p + Z p( 1 p) n LCL = p Z p( 1 p) n Usually the Z value is equal to 3 (as was used in the X and charts), since the variations within three standard deviations are considered as natural variations. However, the choice of the value of Z depends on the environment in which the chart is being used, and on managerial judgment.
14 14 Statistical Quality Control Example: A computer manufacturer collects data from the final test of its product starting from the end of January and all through February. Each day a sample of 2000 items are inspected and the number of items in the sample that do not conform to specifications is recorded. The data is shown below: Subgroup Number Number Percent Subgroup Number Number Percent Number Inspected Defective Defective Number Inspected Defective Defective (day) (day) n = p = σ p = UCL = LCL =
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