SPECIAL CONTROL CHARTS

INDUSTIAL ENGINEEING APPLICATIONS AND PACTICES: USES ENCYCLOPEDIA SPECIAL CONTOL CHATS A. Sermet Anagun, PhD STATEMENT OF THE POBLEM Statistical Process Control (SPC) is a powerful collection of problem-solving tools, such as histograms, Pareto diagrams, cause-effect diagrams, check sheets, stratification, scatter diagrams, and control charts, useful in achieving process stability and improving capability through the reduction of variability. A control chart, also known as Shewhart control chart and mostly preferred among the other tools, has an ability to determine whether there are variation created by causes in the process and to reduce or adjust the variation, if applicable and required, by taking the necessary corrective actions regarding to the causes that created the variation. When the control charts are properly used, they may: be applied by operators for ongoing control of a process, help the process perform consistently and predictably, allow the process to achieve higher quality, lower cost, and higher effective capacity, provide a common language for discussing the process performance, distinguish special from common causes of variation as a guide to local or management action. In order to use control charts as intended, first, a proper control chart should be selected suitable for the process characteristics, such as: manufacturing type and volume (bulk, continuous, or discrete), type of inspection and strategy (destructive or non-destructive testing, in-process or pre-process inspection), cost of inspection, inspection time, quality characteristics of the product being produced within the process (quantitative or qualitative), distribution of quality characteristic(s) lot or sample size. Second, to make the process, which is generally defined as a combination of people, machines, and other equipment, raw materials, methods, and environment that produces products as planned, stable or to keep the process in control, the causes of variations, if applicable, should be determined and interpret effectively assuming that the proper control chart has been implemented. In any process, regardless of how well the process is designed and maintained, a certain amount of inherent or natural variability, variation due to chance causes, may occur. When chance causes, which are inevitable, difficult to detect or identify, are in affect, a process is considered to be in a state of statistical control. Any attempt to adjust for this kind of variation results in over control and is likely to throw the process out of control. On the other hand, even if a process is in control, variation due to machine and operator performances and characteristics of incoming materials or other causes may occur within a stable and predictable process. If unnatural patterns are observed, special causes responsible for the condition must be determined and interpreted effectively so that these disturbances such as operator fatigue, tool wear, different incoming materials, voltage fluctuations, or systematic adjustment of the process may be eliminated from the process by taking the necessary corrective actions. The control charts are used to detect and eliminate unwanted special causes of variation occurred during a period of time where a certain amount of products have already been manufactured. It should be recalled that these special causes of variation have an adverse effect on the overall output of the process, not just individual product characteristics. In order to obtain the expected benefits of using control charts as a problem-solving tool, the following issues should be taken into consideration: Key characteristics that best indicate the control or out-of-control status of a process should be selected for control purposes. Thus, quality of many product characteristics may be improved with applying control charts on the key characteristics as the features which have the greatest influence upon the product fit, performance, service life as mutually agreed with the customer, Copyright 1999 INTENATIONAL JOUNAL OF INDUSTIAL ENGINEEING

INDUSTIAL ENGINEEING APPLICATIONS AND PACTICES: USES ENCYCLOPEDIA The control charts must be applied to control a process where the same machine(s), the same machining method, and the same materials are used. There are control charts developed which may be categorized into two groups regardless of the characteristics of the products, either measurable such as length and diameter or countable such as number of defects per unit: standards are given meaning that population parameters such as mean and standard deviation are known and there is no need to collect data to establish control limits, and standards are not given meaning that population parameters are not known and control limits must be derived from at least 0-5 subgroups of data taken from the process based on a sampling policy. The commonly known control charts may not be used when: production level is low; therefore, an extensive amount of time should be waited to collect a required number of data, at least 0-5 individuals or subgroups of data, to establish control limits so to construct a control chart. production is run in a discrete form because of unexpected delays due to lack of materials or sudden changes in production scheduling; therefore, a certain amount of production may be performed by different personnel and time intervals. besides piece-to-piece and time-to-time variation, variation within the piece is needed to be measured and observed. even though the characteristics observed or dimensions measured are similar in terms of machine, material, and method used, separate control charts should be constructed for each characteristic or dimension, instead of monitoring similar characteristics on the same chart. The process should be investigated by the person, who is responsible for statistical process control, to determine whether the circumstances given above are applicable for processes or characteristics concerned to be able to select the proper control chart for evaluating the processes and making them stable. When a control chart is selected without considering the circumstances of the process or characteristics occur, the chart being constructed may not be proper to evaluate the process because each control chart has its own properties in terms of sample size, type of characteristic observed, formulation to calculate the control limits, the coefficients used in the formulation, and the interpretation of the chart constructed. SOLUTIONS TO THE POBLEM: SPC USING SPECIAL CONTOL CHATS The number of measurements in the sample is an essential criterion for selecting a control chart. Suppose, a set of samples, for instance, each has seven measurements, are taken from a process based on a sampling policy. Since the sample size is relatively large, the standard deviation should be preferred instead of the range to represent the dispersion of the process concerned. The standard deviation is considered as an effective parameter for representing the variation since it is calculated using all of the data points. On the other hand, when the sample size is relatively small, the range yields almost as good as an estimator of the variance. However, for moderate values, the range loses efficiency rapidly, as it ignores all the information in the sample between the maximum and the minimum measurements. Nevertheless, the range may be preferred to simplify the calculation; but, for this situation, based on the sample size, the average and the standard deviation chart should be used to evaluate the process effectively. Addition, using an improper control chart may often cause a problem such that an interpretation may be done as the process is out-of-control based on the selected chart even though it is under control indeed or vice versa. These conclude that the selection of a proper control chart for the purpose of statistical process control requires an extensive amount of attention in terms of properties of the control charts. In the remaining of this chapter, the control charts, namely special control charts, will be discussed based on specific examples which are prepared to demonstrate each special control chart in regard to where and when to use, how to calculate the control limits, and how to evaluate the chart constructed and similarities between the traditional control charts will be given. Some of the charts being introduced here require a minimum number of data points to be constructed and evaluated (standards are not given). For the other charts, there is not such a constraint because the control limits are derived directly from the specification limits (standards are given) and measurements are plotted on a control chart to evaluate the process concerned. Copyright 1999 INTENATIONAL JOUNAL OF INDUSTIAL ENGINEEING

INDUSTIAL ENGINEEING APPLICATIONS AND PACTICES: USES ENCYCLOPEDIA 1. Interrupted Average and ange Chart The interrupted average and range chart, which is similar to the average and range chart, may be used for discrete or low volume production. It is also used when there are unexpected delays in process due to lack of materials and sudden changes in production scheduling. Therefore, the products may be produced by different personnel and time intervals. In other words, a machine is setup to do an operation for 10 pieces and not perform that operation again for several days or weeks. Production control considerations might also dictate that more than one operator-machine combinations may be set up to run an operation simultaneously, so that defining the process becomes a much more generic than for classical production situations. The chart is started with different short runs from the process. Generally small samples (e.g., or 3) obtained from one characteristic of a product that produced within a process are taken based on tight intervals (e.g., 15 minutes). The control chart is filled out as the parts run, filed, then filled out during the next run. This activity continues until there are not any parts to produce within that time interval or the necessary samples are taken to construct a chart. For instance, suppose that a drilling operation is being performed for a diameter of a flange with specification of 0.515±0.005, and an unreasonably long time is required to accumulate necessary number of sub-grouped data because of insufficient number of flange at the time where production is being run. In this case, the drilling operation is performed as long as the number of parts available at hand, then the machine used for drilling operation and the personnel who does that operation may be switched to operate on parts with different specifications depending upon a production scheduling. For these reasons, whereas the parts are processed lot by lot in terms of time intervals, a separate work order is assigned to each lot and measurements obtained from each lot are plotted and evaluated separately according to the control limits calculated regarding to the appropriate lot. Assume that the mentioned drilling operation has been performed and the sub-grouped data, each had three samples and coded as 0.5, have been obtained as given in Table 1. Table 1. Measurements for a Drilling Operation of a Flange Work Date Sample 1 3 Order No. 06 3/10 1 16.0 16.5 16.0 16. 0.5 17.0 16.5 16.0 16.5 1.0 3 16.0 16.0 15.5 15.8 0.5 4 17.0 15.0 15.0 15.7.0 5 17.5 16.5 17.0 17.0 1.0 6 17.0 15.6 17.5 16.7 1.9 7 16.0 16.5 16.5 16.3 0.5 17 5/07 8 14.5 14.0 14.0 14. 0.5 9 15.0 14.5 15.5 15.0 1.0 10 15.5 16.0 15.5 15.7 0.5 11 15.0 15.0 15.0 15.0 0.0 1 16.0 15.5 16.5 16.0 1.0 13 16.5 17.5 18.0 17.3 1.5 0 6/0 14 17.5 17.5 18.5 17.8 1.0 15 16.0 16.5 17.0 16.5 1.0 16 16.0 15.5 16.0 15.8 0.5 17 16.0 16.0 17.0 16.3 1.0 18 16.5 17.0 16.5 16.7 0.5 19 17.0 17.5 16.0 16.8 1.5 0 16.5 16.5 16.5 16.5 0.0 The control limits of the chart for each time interval (run) may be calculated as follows: for average chart, Copyright 1999 INTENATIONAL JOUNAL OF INDUSTIAL ENGINEEING

INDUSTIAL ENGINEEING APPLICATIONS AND PACTICES: USES ENCYCLOPEDIA for range chart, k k = = k k k k = D = D + A A 4 3 k k k k where k is the time interval in which the samples are obtained, A, D 3 and D 4 are the necessary coefficients for k the sample size of n, is the grand average of the k th time interval which is obtained from the samples k averages, is the average range of the k th time interval which is obtained based on the differences between a maximum and a minimum measurements of the samples Since the measurements obtained from different personnel and time intervals, this chart help production reach a state of control and monitor variation in process over time. Thus, a proper setup for the process monitored may be easily determined by focusing on the variation during the intervals to make the process stable and predictable. In order to determine the control limits for each time intervals, first, grand averages and average range for intervals should be calculated as follows: 7 7 I I i i I i= 1 114. I i= 1 = = = 16.31 = = 7 7 7 6 6 II II i i II i= 1 93.0 II i= 1 = = = 15.53 = = 6 6 6 7 7 III III i i III i= 1 116.4 III i= 1 = = = 16.63 = = 7 7 7 7.4 = 1.06 7 4.5 6 5.5 7 = 0.75 = 0.79 The control limits for the portions of average and range are then calculated using the equations given above and the coefficients of A = 1.03, D 4 =.574, and D 3 = 0 for the sample size of 3. The calculated control limits are given in Table. Table. Control Limits for a Flange Drilling Operation The Time Intervals Control Limits I II III 17.39 16.30 17.44 15.3 14.76 15.8.73 1.93.03 0 0 0 As in the average and range chart, subgroup averages on the average chart and subgroup ranges on the range chart are plotted and the points are connected according to the time intervals where samples are taken. The control charts for the averages and ranges are given in Figure 1 and Figure, respectively. Copyright 1999 INTENATIONAL JOUNAL OF INDUSTIAL ENGINEEING

INDUSTIAL ENGINEEING APPLICATIONS AND PACTICES: USES ENCYCLOPEDIA 19 18 17 16 15 14 13 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18 19 0 Figure 1. Average Chart for a Flange 3.5 1.5 1 0.5 0 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18 19 0 Figure. ange Chart for a Flange When the charts are examined, the following results may be obtained: 1. According to the average chart, the first setup is the best among others.. Variation between setups in average chart is extensive. 3. There may be an excess tool wear in the second time interval, and few data points are out of control limits. 4. Whereas the ranges are stable, / d may be used for an estimate of standard deviation to perform a process capability study. As a conclusion, the machine should be setup and the personnel should be advised based on the setup of the first interval. For this example, a process capability study may not be performed because the process is not in control. Therefore, first, necessary corrective actions should be taken to reduce the variation, and then a process capability index may be calculated to interpret the process. The interrupted average and range chart may be interpreted and the process capability may be studied in the same manner as the average and range chart.. Multiple Variation Chart The multiple variation chart may be used when specification limits or the population mean and standard deviation are known. Consequently, there is no need to collect data to establish control limits. In contrast to other known chart such as the average and range and the average and standard deviation charts, piece-to-piece variation, variation within the piece, and time-to-time variation may be examined with this chart. Piece-to-piece variation concerns itself with different sizes of dimensions such as thickness, diameter, and the variation within the piece concerns itself with problems such as roundness and taper, the time-to-time variation, on the other hand, concerns itself with variation between the averages of the subgroups. Since three types of variation are taken into consideration with this chart, the effects of different raw materials, sources, fixtures, operators, and spindles may be easily determined and compared to improve the process performance. Copyright 1999 INTENATIONAL JOUNAL OF INDUSTIAL ENGINEEING

INDUSTIAL ENGINEEING APPLICATIONS AND PACTICES: USES ENCYCLOPEDIA Suppose that specifications of a diameter on a shaft are 0.500±0.003 inch, roundness is considered as a key characteristic to evaluate a process regarding to three different variation. Samples, each has four pieces, are taken from the process and minimum and maximum measurements of each piece are given in Table 3. Table 3. Measurements of a Diameter of a Shaft (coded as deviations from nominal) Sample Number Measurements Deviation Below Nominal (-) 1 1-0.0005 0.0035-0.000 0.0030 3 3-0.000 0.0040 4 4-0.0005 0.0040 5 1-0.0005 0.0035 6-0.0005 0.0030 7 3-0.000 0.0030 8 4 0.0000 0.0045 9 1-0.0010 0.0030 10-0.0015 0.0030 11 3-0.0010 0.0040 1 4-0.0010 0.0040 13 1-0.0015 0.0035 14-0.0005 0.0045 15 3-0.000 0.0030 16 4-0.000 0.000 Deviation Above Nominal (+) In order to construct a multiple variation chart, maximum and minimum values of the characteristic of each sample in one particular location are measured and plotted for each sample on the same vertical line as deviation from a nominal value of the characteristic examined. After selecting the proper scale, the specification limits are drawn on the chart, and the midpoints of each plot (the point between the maximum and the minimum measurements) are connected. anges are not considered with this chart. A multiple variation chart for the data given in Table 3 is depicted in Figure 3. 0.005 0.004 0.003 USL 0.00 0.001 0 CL -0.001-0.00-0.003-0.004 LSL -0.005 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 Figure 3. A Multiple Variation Chart for the Shaft Example The multiple variation chart may be interpreted as follows: If the distance between averages of each sample in the subgroup varies, this represents piece-to-piece variation, which could be caused due to tool wear, or operator influence. The process is out of control, Copyright 1999 INTENATIONAL JOUNAL OF INDUSTIAL ENGINEEING

INDUSTIAL ENGINEEING APPLICATIONS AND PACTICES: USES ENCYCLOPEDIA whereas either the minimum or the maximum measurement is out of specification limits for the appropriate pieces. If the maximum or the minimum measurement being part of a sample is out of specification limits, the cause may be variation within the piece. The distance between the two measurements for each sample represents the degree of variation within the piece. There is sudden increase and/or decrease due to excess taper for the example. Variation between the averages of each subgroup to another represents time-to-time variation, which could be caused due to setup error or uncalibrated machine tool/equipment. Even though the process is not in control, there is not a large variation between samples averages. However, necessary corrective actions should be taken to reduce variation. 3. Print Tolerance Control Chart As the multiple variation chart, the print tolerance control chart uses specification limits directly and applied statistics to establish control limits. This chart is not only good for low volume production, but also for high volume production since the control limits may be calculated without requiring any previous data. The control limits derived from the specification limits are centered around the nominal of the specification. The control limits of a print tolerance control chart are determined as follows: 1. Calculate the L factor based on the sample size of n and the assumption of the process mean is normally distributed within ±3σ limits. d L= 6 where, d is the coefficient obtained from a table based on the sample size of n.. Calculate max, the maximum that the ranges can reach and still maintain capability. max = Total specification tolerance L 3. Calculate the control limits; for average chart, = MTD+ A = MTD A max max for range chart, = D 4 = D 3 max max where, MTD is the mid-tolerance dimension (centerline of the chart), A, D 3 and D 4 are the necessary coefficients for the sample size of n. Suppose that a shaft is being machined according to specifications of 0.500±0.010 inch. While the production is running, fifteen samples with the size of 5 are taken based on a sampling policy. The obtained data are coded as values above 0.480 inch and given in Table 4. Let us examine whether the process is in control using the print tolerance chart. Copyright 1999 INTENATIONAL JOUNAL OF INDUSTIAL ENGINEEING

INDUSTIAL ENGINEEING APPLICATIONS AND PACTICES: USES ENCYCLOPEDIA Table 4. Measurements for a Characteristic of a Shaft Sample No. 1 3 4 5 1 5 3 6 3 3.8 4 3 5 4 3 5 4.0 3 6 4 5 3 4 4.4 3 4 3 3 4 5 5 4.0 5 3 4 3 4 5 3.8 6 5 4 3 3 3.4 3 7 4 5 5 4 3 4. 8 4 4 5 4 3 4.0 9 3 4 4 5 5 4. 10 4 3 3 4 3 3.4 1 11 4 5 5 5 4 4.6 1 1 4 5 6 30 6 6. 6 13 5 4 3 4 3 3.8 14 4 5 5 3 0 3.4 5 15 3 3 1 4 1.4 3 The control limits for the measured quality characteristic may be calculated using d =.36, A =0.577, D 4 =.115, and D 3 =0 for sample size of 5: 1. Calculating L,. Calculating max, L = d 6.36 = = 0.3877 6 max = Total specification tolerance L = 0.00 0.3877= 0.0078 3. Calculating the control limits for the average and the range charts, respectively; = 0.500+ (0.577)0.0078= 0.5045 = 0.500 (0.577)0.0078= 0.4955 = (.115)0.0078= 0.0165 = (0)0.0078= 0.0 After calculating the control limits, the actual averages and ranges obtained from the process are plotted and the chart is interpreted to decide whether the process is in control. The print tolerance charts for averages and ranges are given in Figure 4 and Figure 5, respectively. 8 6 4 0 18 16 14 1 10 1 3 4 5 6 7 8 9 10 11 1 13 14 15 MTD Figure 4. The Print Tolerance Chart for Averages Copyright 1999 INTENATIONAL JOUNAL OF INDUSTIAL ENGINEEING

INDUSTIAL ENGINEEING APPLICATIONS AND PACTICES: USES ENCYCLOPEDIA 0.0 0.015 0.01 CL 0.005 0 1 3 4 5 6 7 8 9 10 11 1 13 14 15 Figure 5. The Print Tolerance Chart for anges When the print tolerance charts are examined closely, it can be say that the process is out of control because the 11 th and 1 th samples are above the in the averages chart, and the process is setup and operated approximately 0.004 inch above nominal. For the range chart, on the other hand, all of the samples are within the control limits and it may be assumed that the variation is not high enough to require corrective actions. 4. PreControl Chart PreControl (PC) chart, also known as stoplight or target control chart, was developed in the early 60 s as a means of providing the advantages of control charts on the machine without burdening the operator with knowing how to construct and interpret control charts. ather, it represents a series of easily followed rules, which lead the operator to correct adjustments, or as importantly, to leave the process alone when only random chance is affecting the output. The concept is based on knowledge of the characteristics of a normal distribution. The PC chart, applicable to short and long production runs, is easy to use and it is simple to implement in the facility. ecording data and calculation are nor required; however, plotting and interpretation according to the rules may be necessary. In order to use PC chart effectively, the process should be centered between specification limits, with ±3σ equal to or better than specification limits, or a capability index, C p, of 1.00 or more. If this condition is met, the PC chart will keep the capable process centered and detect shifts that may result in making some of the parts outside of the specification limits or decreasing the probability of making defectives. The PC chart uses specification limits to establish PC limits. The PC limits (PCL), as shown in Figure 6, are established as follows: where, MTD is the mid-tolerance dimension. Total specification tolerance PCL = MTD± 4 Copyright 1999 INTENATIONAL JOUNAL OF INDUSTIAL ENGINEEING

INDUSTIAL ENGINEEING APPLICATIONS AND PACTICES: USES ENCYCLOPEDIA ED YELLOW GEEN GEEN YELLOW ED USL UPCL MTD LPCL LSL Figure 6. Graphical Description of PC Limits The center zone between the upper and the lower PC limits is one-half the print tolerance and is called green area. The sides between the PC limits and the specification limits are one-fourth of the total tolerance and are called yellow zones. Outsides the specification limits are called the red zones. As in Figure 1, he zones may be colored to make the procedure simple, understandable, and applicable. In PC chart, if the process is capable and the process is centered, around 86% of the parts will fall between the PC limits (green zones), and around 7% of the parts will fall in each of the outer sections (yellow zones). So, under the conditions of statistical control, 7% of the points plotted would be expected to occur in each outer zone or 1 time in 14 on the average. Let us consider an example of a drilling operation for a diameter of a flange with specifications of 0.500±0.00 inch. The data, 0 individual values, obtained from this operation are given in Table 5. Table 5. Measurements obtained from a drilling operation for a diameter Sample No Sample No 1 0.5030 11 0.5000 0.5015 1 0.5008 3 0.5000 13 0.5000 4 0.5005 14 0.5005 5 0.4995 15 0.5015 6 0.5005 16 0.501 4 0.5005 17 0.500 8 0.5015 18 0.5005 9 0.5005 19 0.498 10 0.5008 0 0.5015 The PC limits, for MTD is 0.500 inch and total specification tolerance is 0.004 inch, may be calculated as follows: Total PCL = MTD± specification tolerance 4 UPCL= 0.500+ 0.001= 0.501 LPCL= 0.500 0.001= 0.499 Copyright 1999 INTENATIONAL JOUNAL OF INDUSTIAL ENGINEEING

INDUSTIAL ENGINEEING APPLICATIONS AND PACTICES: USES ENCYCLOPEDIA The parts as they produced checked and measurements are usually plotted on the chart. The PC chart for the drilling operation example is shown in Figure 7. 0.504 0.503 0.50 0.501 0.5 0.499 0.498 USL UPCL MTD LPCL LSL 0.497 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18 19 0 Figure 7. PC Chart for the Drilling Operation The interpretation of the chart may be performed based on the following rules: 1. If the part measured falls outside the specification limits (red zones), the process stopped and reset.. If the part measured is between the PC limits and the specification limits (yellow zones), a second part is tested; a. If the second part is in the same yellow zone, the process is stopped and reset. b. If the second part is in the opposite yellow zone, the process is stopped and corrective actions are taken. c. If the second part is between the PC limits (green zones), continue the process. 3. When five consecutive parts fall within the PC limits, switch to frequency of gaging. 4. When a process is reset for any reason, five consecutive parts must occur within the PC limits before switching to the frequency of gaging. Graphical description of the PC rules is depicted in Figure 8. 1 a b c 3, 4 USL UPCL MTD LPCL LSL Figure 8. Graphical Description of PC ules When the PC chart constructed for the drilling operation example is evaluated based on the PC rules given above, the following results may be obtained: 1. Whereas the first sample is out of specification limits, the process should have been stopped and reset (with rule number of 1).. When the 16 th sample is taken, the process should have been stopped and reset (with rule number of a). Copyright 1999 INTENATIONAL JOUNAL OF INDUSTIAL ENGINEEING

INDUSTIAL ENGINEEING APPLICATIONS AND PACTICES: USES ENCYCLOPEDIA 3. When the 0 th sample is taken, the corrective actions should have been taken after stopping the process (with rule number of b). 4. The process may be considered in control except for the previous comments. In addition to the advantages of the traditional control charts such as indicating shifts in process centering and increases in process spread, assuring that the number of defective parts will not exceed predetermined levels, PC chart offers some valuable advantages. These advantages are given as follows: 1. Operator control becomes more practical since no recording, calculating, or plotting of data is required. However, measurements may be plotted if the consumer or management desires to keep a PC chart as statistical evidence of process control.. By adjusting the inspection frequency to a proper level, PC chart may be applied to long production runs. 3. PC chart may be used with attributes by appropriately colored go/no-go gages to initiate PC limits or even visual characteristics by assigning visual standards to the PC limits. 4. Under the assumption of that the process capability is less than the specification and the process is centered, the control limits are directly driven from the specification limits without requiring any previous data. This makes the PC chart simple to understand and easy to implement. 5. Since it works directly with the specification limits, it does not require any statistics background to explain the chart to the people who will be using the plan. Despite the advantages, PC chart has some drawbacks. Since it is assumed that the process is capable and normally distributed within ±3σ limits, central tendency is considered the only problem. Thus, sensitivity to variation in the process is reduced. While the process is monitored, the operator evaluates the process according to the easily followed rules and tries to keep the process centered. Therefore, assessment of capability may be difficult. In addition, some over-control, adjusting the process when it does not need an adjustment, and/or under-control, not adjusting the process when it does need an adjustment, may result. Finally, the PC chart, unlike the other control charts, may not be used for problem solving, but instead, may be used for monitoring a process. For that reason, the PC chart is often used to initially implement some form of statistical process control in the facility and is soon followed by more sensitive process control charts to detect, if applicable, and control the variation within the process. 5. Two- Control Chart Most of the control charts seen in the textbooks have measured piece-to-piece and time-to-time variation. The range chart measures variation from piece-to-piece in the individuals and moving average charts and between pieces within in the subgroup in the average and range charts. The time-to-time variation is plotted on the individual or average portion of the suitable chart. However, there may be another source of variation, called within-piece variation, that need to be considered for some situations. The traditional control charts such as the average and range and the average and standard deviation charts do not allow tracking and recording this type of variation. Two- control chart, which is a combination of the individuals control chart and the average and range chart, allow plotting three different sources of variation. The two- chart is especially useful in processes where withinpiece or within-group variation is as important as between-piece or between-group variation such as measures of roundness or concentricity, flatness, surface finish, thickness, a dimensional characteristic common to many parts processed in a fixture at the same time, or even hardness of a batch of parts run through a heat treat operation. This chart consists of three separate charts: an individuals or average chart for representing variation between the individuals or averages of the samples (what is happening to the process center), a range chart for representing variation piece-to-piece variation (how uniformly the process is behaving batch-to-batch), and a range chart for representing within piece variation (how uniformly the parts are treated within each batch). For instance, if a quality characteristic examined is a roundness of a diameter of a shaft, the difference between the maximum and the minimum readings of a part is called variation within the piece ( w ), and the average of the maximum and the minimum measures determines the time-to-time variation. The difference between the previous reading and the most current reading is called the piece-to-piece variation ( p ). The two- chart is constructed after 0-5 data points available and interpreted in the same manner as the other control charts. The control limits for the chart may be calculated as follows: Copyright 1999 INTENATIONAL JOUNAL OF INDUSTIAL ENGINEEING

INDUSTIAL ENGINEEING APPLICATIONS AND PACTICES: USES ENCYCLOPEDIA Individuals chart, w chart, w w = + E = E = D = D 4 3 w w p p p chart, p p = D 4 = D 3 p p where, is the average of the maximum and the minimum readings, w is the average of ranges for withinpiece variation determined by subtracting the max reading from the min reading, p is the average of ranges for piece-to-piece variation determined by subtracting the second plot point from the first plot point, and E = 3 / d, D3, and D4 are the coefficients for the sample size of. For the process capability study, an estimated standard deviation for the process is calculated using / d for each range type as where, σ = p p / d and σ = w w / d ˆ σ = σ p + σ, Painting is considered as special process. Suppose that a part which affect the speed of an aircraft while the aircraft is making some sort of movements, such as closing to and leaving from the ground sharply, in the air is being painted trough a special process. Thickness of the paint shoot up, with specifications of 0.001-0.0040 inch, is a key characteristic. For this situation, the thickness of the paint through the surface of the part is measured and recorded as minimum and maximum values. The data, coded as 0.00, for this example are given in Table 6. Table 6. Measurements for a Painting Process Sample No. Min Max i w p 1 14 16 15.0-1 16 14.0 4 1.0 3 14 19 16.5 5.5 4 13 16 14.5 3.5 5 13 16 14.5 3 0.0 6 13 15 14.0 0.5 7 14 17 15.5 3 1.5 8 13 15 14.0 1.5 9 1 14 13.0 1.0 10 13 16 14.5 3 1.5 11 14 16 15.0 0.5 1 1 15 13.5 3 1.5 13 13 15 14.0 1.5 14 1 14 13.0 1.0 15 13 16 14.5 3 1.5 16 14 16 15.0 0.5 17 15 16 15.5 1 0.5 18 14 16 15.0 0.5 19 15 17 16.0 1.0 0 13 14 13.5 1.5 The control limits of the - chart for the painting process are calculated using E=.660, D 4 =3.67, and D 3 =0 for the sample size of : w Copyright 1999 INTENATIONAL JOUNAL OF INDUSTIAL ENGINEEING

INDUSTIAL ENGINEEING APPLICATIONS AND PACTICES: USES ENCYCLOPEDIA 90.5 49 = = 14.55, w = =.45, p 0 0 = 3 = 1.1 19 for individuals chart, = 14.55+ (.660) 1.1= 17.7436 = 14.55 (.660) 1.1= 11.3064 for w chart, for p chart, w w p p = (3.67).45= 8.004 = (0).45= 0.0 = (3.67) 1.1= 3.9541 = (0) 1.1= 0.0 The - control charts for individuals, w, and p are depicted in Figures 9-11, respectively. 0 19 18 17 16 15 14 13 1 11 10 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18 19 0 CL Figure 9. Individuals Chart 10 9 8 7 6 5 4 3 1 0 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18 19 0 CL Figure 10. w Chart Copyright 1999 INTENATIONAL JOUNAL OF INDUSTIAL ENGINEEING

INDUSTIAL ENGINEEING APPLICATIONS AND PACTICES: USES ENCYCLOPEDIA 5 4 3 1 0 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18 19 0 CL Figure 11. p Chart When both w and p control charts are evaluated it may be seen that the process is having a periodic variation which is one of the special causes and there is a variation between samples, piece-to-piece variation. Variation within-piece is small; however, most of the samples are accumulated around the centerline. For the individuals chart, a decreasing trend from the 3 rd sample, an increasing trend from the 14 th sample, and, overall, fluctuations may be observed. Hence, the process requires a close examination for the determination of the reasons cause special patterns and how to eliminate those by taking what kind of corrective actions. 6. Deviation Charts The traditional control charts may be applied to continuous and homogeneous processes. In these processes, a control chart is used for a characteristic of a part. Additional charts are needed if a part has more characteristics. On the other hand, in order to construct a control chart to be used to detect whether there is a variation, at least 0-5 samples, each may consist of more than one unit, are required to satisfy the normality assumption. In some situations, the process is often completed before the number of samples required might be obtained and thus the operators do not have time to calculate the control limits to evaluate the process. Deviation charts, also known as delta charts, are developed to eliminate these drawbacks. Two types of deviation charts may be used for different circumstances. One is called deviation from nominal or nominal equals zero chart (N=0). The second one is called deviation from target chart. These charts will be discussed in details in the following paragraphs. 6.1. Deviation from Nominal Chart This chart is used to control process, which produces different parts with similar characteristics. In other words, the N=0 chart focuses on the process and characteristics rather than dimensions and part numbers (e.g. a machine produces the same diameter on different parts at different sizes) as long as the following conditions are met: The standard deviations must be essentially identical between part numbers, The process must be homogeneous, The same material must be processed using the same method and machine (one process stream), One type of characteristic per part number must be selected, The subgroup size must be constant to be able to calculate the control limits for all characteristics or parts. If these conditions are met, data from multiple parts may be merged so that the number of samples required to construct a control chart is easily obtained. The deviation from nominal chart is identical to the average and range chart, except the centerline of the chart is the nominal of the specification, which is coded 0, and the plotting points for the characteristics concerned are the deviations from the nominal, which are determined by subtracting the nominal value of the proper characteristic from the reading, such as +0.00, above the nominal, or 0.005, below the nominal. With this arrangement, since the plotting points are deviations, a machine capability study Copyright 1999 INTENATIONAL JOUNAL OF INDUSTIAL ENGINEEING

INDUSTIAL ENGINEEING APPLICATIONS AND PACTICES: USES ENCYCLOPEDIA may be easily performed assuming the same parts produced within the process and the same characteristics measured. Besides, the same patterns will be seen on the charts as if actual sizes were being plotted. Therefore, the chart is interpreted in the same manner as the traditional control charts. In order to evaluate the process properly, a range chart should be constructed at first. In the deviation from nominal chart, the ranges are calculated and plotted identical to the range chart for averages or individuals depending on the sample size of n. The plotting point is the difference between the maximum and minimum readings for the range chart for the averages. A moving range is calculated for the individual chart by subtracting the current reading from the last reading and using the absolute value of that difference. Once the ranges are calculated, a test depending on the standard deviations of the parts measured should be performed to decide whether the characteristics of the parts might be examined on the same chart. The test is performed using the following steps: 1. Calculate a ratio of Part Total where, Part is the average range of the part suspected and Total is the average of the total of all the parts including the suspected part.. Use the following criterion to make decision about the suspected part(s) Part Total 1.3 = > 1.3 the measurements for that part may be plotted on the same chart with the other a separate chart should be used for that chart parts Assume that the parts are being manufactured on a process with specifications of.015±0.00, 1.831±0.00 and 4.6795±0.00 inches, respectively. The parts have different specifications; however, they are made of the same material, and manufactured on a process using the same method. Thus, a control chart, for instance, deviations from nominal chart, may be a candidate to control that process in which the parts are being processed. Suppose the parts are processed based on schedule and the coded measurements obtained from the process are given in Table 7. Table 7. Measurements for the Parts A, B, and C Sample No. Part Nominal Deviation 1 A 15 14.9-0.1 - A 15.1 0.1 0. 3 A 15.3 0.3 0. 4 A 14.8-0. 0.5 5 B 31 31.0 0.0-6 B 31.4 0.4 0.4 7 B 31. 0. 0. 8 B 30.9-0.1 0.3 9 B 30.8-0. 0.1 10 C 79.5 79. -0.3-11 C 79.8 0.3 0.6 1 C 78.8-0.7 1.0 13 C 78.9-0.6 0.1 14 C 79.3-0. 0.4 15 C 79.8 0.3 0.5 The plot points for the averages or individuals are simply represented by deviations from nominals and plotted as in the traditional control charts. The control limits are calculated using proper formula depending on the sample size of n; average chart for n and individuals chart for n=1. Whereas each sample has a single measurement, the control limits are determined as if a control chart being used for individuals: Copyright 1999 INTENATIONAL JOUNAL OF INDUSTIAL ENGINEEING

INDUSTIAL ENGINEEING APPLICATIONS AND PACTICES: USES ENCYCLOPEDIA The control limits using E =.660, D 4 =3.67 and D 3 =0 are given: for individuals chart, for range chart, 0.80 4.5 = = = 0.0533 and = = = 0.375 m 15 m 3 1 = 0.0533+ (.660) 0.375= 0.944 = 0.0533 (.660) 0.375= 1.051 = (3.67) 0.375= 1.5 = (0) 0.375= 0.0 The control charts are depicted in Figure 1 and Figure 13 for individuals and ranges, respectively. 1.5 1 0.5 0 CL -0.5-1 -1.5 1 3 4 5 6 7 8 9 10 11 1 13 14 15 Figure 1. A Deviations from the Nominal Chart for Averages 1.4 1. 1 0.8 0.6 0.4 0. 0 1 3 4 5 6 7 8 9 10 11 1 1 3 1 4 1 5 Figure 13. A Deviations from the Nominal Chart for anges CL There is a decreasing trend during manufacturing of parts B especially starting from the 6 th sample, and a large variation may be seen within the process when the parts C are started to manufactured. The latter result, which may also be seen in the range chart, shows evidence that the parts C would be different in terms of material, machine, and method so that the test should have been performed before starting to construct the charts. Indeed, the ratio of c / Total (0.5/0.375=1.39) is greater than 1.3, which implies that the parts C should be plotted on a separate chart. Hence, the deviations from the nominal charts should be constructed and reevaluated after Copyright 1999 INTENATIONAL JOUNAL OF INDUSTIAL ENGINEEING

INDUSTIAL ENGINEEING APPLICATIONS AND PACTICES: USES ENCYCLOPEDIA eliminating the parts C. For a capability study of the deviations from nominal chart, the steps as in the other control charts are followed, but actual limits of each different characteristic and its tolerance are used for making decision whether a process is capable. 6.. Deviation from Target Chart Many processes in manufacturing need to position their process output close enough to one of the specification limits for a purpose of economics such as extension of tool life and longer solution deterioration times. In this case, the parts are produced within the print tolerance; however, some parts such as bearings, pistons, bores, and keys are processed either above or below nominal to be on the safe side especially for assembly. A control chart named deviation from the target chart is developed for the situations where similar characteristics of a part or different parts are processed and leading a target for each characteristic instead of being around nominal is important or required. In order to use the deviation from the target chart, the properties mentioned for the deviation from the nominal chart must also be satisfied because of the similarities between them. Addition, the parts or characteristics of the parts being plotted on the same chart could have different sample averages. For this case, the chart may not be easily evaluated. Besides, a result may be obtained that the process in which the parts are being manufactured is out of control due to large variation between the sample averages while the process is statistically under control (Type I error). Therefore, a test should be performed to determine which deviation chart is appropriate for the process or the parts concerned. The steps for the test are given below: 1. Calculate standard deviations of the parts taken into consideration and F 1 value, ( ) = i s n 1 and tα / ;v F1 = n where, n is the number of samples for each part concerned, t α / ; v is the value obtained from t distribution for significance level of α and degree of freedom of v=(n-1).. Use the following criterion to determine which chart is appropriate, Nominal Target > F1 s = F1 s use Deviation from the Target Chart use Deviation from the Nominal Chart For instance, a nominal value of part A is coded as 4.5, and a target value as 5.9. Seven samples are taken and standard deviation is calculated as 1.3. To make a decision, F 1 value should be determined. The required F 1 value for α =0.05 is 0.05;6.4469 F 1 = t = = 0.9 7 7 Since F1 s= 0.9 1.3= 1. 144 and 4.5-5.9 = -1.4 = 1.4 > 1.144, deviation from the target chart should be selected. As mentioned before, the control limits for the deviation from the target charts are calculated and the charts are constructed based on either average and range charts or individuals and moving range charts depending on the sample size. Let us suppose an example to introduce this particular control chart. Parts A, B, and C have different specifications, but they are made of the same material and machined using the same method. The data, each is having three pieces, obtained from the process are coded and given in Table 8. Copyright 1999 INTENATIONAL JOUNAL OF INDUSTIAL ENGINEEING

INDUSTIAL ENGINEEING APPLICATIONS AND PACTICES: USES ENCYCLOPEDIA Table 8. Measurements for the Parts A, B, and C Sample No Part 1 3 Target 1 A 93 95 91 93 90 3 4 A 89 9 83 88 90-9 3 A 94 93 89 9 90 5 4 A 96 89 97 94 90 4 8 5 A 88 86 93 89 90-1 6 6 A 91 9 87 90 90 0 5 7 B 59 55 58 57.3 58-0.7 4 8 B 63 57 59 59.7 58 1.7 6 9 B 59 61 60 60 58 10 10 B 60 55 56 57 58-1 5 11 B 58 54 56 56 58-4 1 B 60 55 59 58 58 0 5 13 C 36 30 33 33 30 3 6 14 C 9 33 31 31 30 1 4 15 C 3 6 9 9 30-1 6 16 C 34 9 7 30 30 0 7 The control limits for the deviation from the target charts; using A =1.03, D 4 =.575, and D 3 =0 for the sample size of n=3 and the number of sample m=16, are calculated as follows: = m = 9 16 = 0.565 and = 86 = = 5.375 m 16 for average chart, for range chart, = 0.565+ (1.03) 5.375= 6.06 = 0.565 (1.03) 5.375= 4.937 = (.575) 5.375= 13.841 = (0) 5.375= 0.0 The control charts are given in Figure 14 and Figure 15, respectively. 8 6 4 0 CL - -4-6 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 Figure 14. A Deviation from the Target Chart for Averages Copyright 1999 INTENATIONAL JOUNAL OF INDUSTIAL ENGINEEING

INDUSTIAL ENGINEEING APPLICATIONS AND PACTICES: USES ENCYCLOPEDIA 16 14 1 10 8 6 4 0 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 Figure 15. A Deviation from the Target Chart for anges CL When both average and range charts are examined, it may be said that process is in control; however, variation for the parts A is larger than the others. Therefore, the variation for the parts A should be decreased. 7. Short un Chart The traditional control charts such as the average and range charts and the average and standard deviation charts only allows for one quality characteristic to plotted at a time. When two characteristics are tracked, two charts are needed to evaluate the process. Deviation chart based on either nominal or target values may be used to plot two or more characteristics on a chart when the dispersion expressed in range or standard deviation are essentially identical between characteristics. The range or standard deviation is used in the formulas for calculating the upper and lower control limits on both portions of the control charts. When the dispersion expressed in range or standard deviation calculated using the samples taken from the process is large, artificially high limits may occur so that true out of control conditions could be masked. In order to make the evaluation process as effective as it should be, short run chart, also known as Z and W for the average and range (standard deviation) charts and Z and W for the individuals and moving range charts, may be used to somehow eliminate the effect of having the range or the standard deviation in the calculation of the control limits. With the short run chart, a plot point is calculated on a control chart that is independent of the range or standard deviation using a series of inequalities. A range value determined either a difference between the maximum and the minimum readings observed as in the average and range charts, or a difference between successive readings as in the individuals and range charts, lies between the upper and the lower control limits. This may be shown in the following inequality: or < < D3 < i < D4 Now, if the last inequality is divided by, which may be a historical or a target value for the process concerned, or a current value determined based on the samples taken, the following result is obtained: D i i 3 < < D 4 The control chart will have D 3 as the lower control limit, D 4 as the upper control limit, and 1.00 for the central line because it occurs when =. Therefore, the plot point for each range, i, is i = i Copyright 1999 INTENATIONAL JOUNAL OF INDUSTIAL ENGINEEING

INDUSTIAL ENGINEEING APPLICATIONS AND PACTICES: USES ENCYCLOPEDIA where, i is the ith range calculated from the current subgroup for a characteristic of a part, and is the historical average or target value for the same part. The control limits D 3 and D 4 are independent of ; however, they are a function of sample size n, which must be constant. With the short run chart, samples obtained from two or more characteristics of a part or different parts may be plotted. If two or more characteristics of a part, two diameters with different dimensions on the same part, or more than two characteristics of different parts, two diameters with different dimensions on the different parts or two different characteristics such as length and width, on the same part, are monitored, the plot points are determined based on the historical average or target value of the processes from which the samples are taken. For instance, if there are data obtained from two parts, part A and part B, related to a characteristic, two diameters with same or different dimensions, then the plot points for part A and part B may be calculated using A and B, respectively. This enables us to plot different parts with similar characteristics generated from similar processes with different averages and dispersions. The same approach is followed to determine the plot points for the average portion of the chart. The control limits for average portion of the average and range chart also contain in the formula. An average of a subgroup,, falls between the control limits. This may be shown in the following inequality: or < < A < < + A If we subtract from the inequality and divide by, we will get: ( A ) < ( ) < ( + A ) ( ) A < < A As in the range chart, the control limits A and A are dependent of sample size n, which must be constant. The central line is equal to 0.0 because it occurs when = 0, which is the ideal situation. The plot point of each average, i, is: i = where, is the average range obtained from the ranges of the samples, is the historical or current grand average, and is the average of a subgroup. The short run chart may be applied to individuals by making the necessary modifications. The ranges are calculated considering the difference between the successive data points, also called moving range. Therefore, if there is not any data available to compute a historical range average, sum of the ranges derived from the samples is divided by m-1, where, m is the number of samples. The control limits for the range portion of the chart are calculated using D 3 and D 4 for sample size of and the plot points are determined in the same manner as in the average and range chart. However, in contrast to the average and range chart, the control limits for the individuals are calculated using E for the sample size of. The control limits independent of the and, and plot points determined based on the concept discussed are given as: The control limits; E < < + E ( E ) < < ( + E ) Copyright 1999 INTENATIONAL JOUNAL OF INDUSTIAL ENGINEEING