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ANALYSIS OF NONRANDOM PATTERNS IN CONTROL AND ITS SOFTWARE SUPPORT

Darja NOSKIEVIČOVÁ

FMMI, VŠB-TU Ostrava, 17. listopadu 15, 708 33 Ostrava – Poruba, Czech Republic, [email protected]

Abstract Identification of assignable causes of the process variability and restriction and elimination of their influence are the main goals of statistical process control (SPC). Identification of these causes is associated with so called tests for special causes or runs tests. From the time of formulation of the first set of such rules (Western Electric rules) several different sets have been created (Nelson rules, AIAG rules, Boeing AQS rules, Trietsch rules). This paper deals with the comparison analysis of these sets of rules and with the analysis of the computer support of them. At the end of this paper some recommendations for the correct use of these rules are formulated. Keywords:, Common causes, special causes, runs tests, natural pattern, unnatural pattern

1. INTRODUCTION Statistical process control (SPC) is an approach to process control that has been widely used in any industrial or non-industrial fields. It is primarily a tool for understanding process variation [1]. SPC is based on so called Shewhart´s conception of the process variability. This conception distinguishes variability caused by obviously effected common causes (process is considered to be statistically stable) from variability caused by abnormal special (assignable) causes (process is considered not to be statistically stable) using control . The main goals of SPC is an identification of abnormal variability caused by special (assignable) causes with the aim to make the process stable, minimize the process variability, improve the process performance. To meet these goals SPC must be built as the problem solving instrument and the sequence of the subprocesses “Out-of control signal revelation – Root cause identification – Action acceptance – Verification of action” must be the axis of the SPC application. In this paper the stress is put on the first subprocess applying to so called Shewhart control charts. Control chart is the main SPC instrument for the analysis of the process variation over time. It is a graphical depiction of the process variation and its natural and unnatural patterns. Control chart displays a value of the characteristic that has been measured or some sample that has been computed from measured values in sample versus the sample number or time. Central line in the control chart represents the average value of the quality characteristic corresponding to the statistically stable process (only natural pattern is presented and only common causes influence the process). are set so that when the process is statistically stable, nearly all the points in control chart fall between them (for mostly applied ±3σ control limits and assumed it is 99,73% points). As long as the points are within the control limits and they show natural random pattern, the process is considered to be statistically stable. But if some points are out of the control limits or the points inside the limits show nonrandom (unnatural) pattern it is assumed that assignable cause of the abnormal process variation is present and must be removed from the process via searching for it and corrective action or some improvement must be realized. In general the points plotted on a control chart form an irregular up-and-down pattern – natural or unnatural. An advantage of using control charts is that it enables observation of the process behaviour over time to search for occurring of any unnatural pattern [2] as a symptom of the special (assignable) cause and as a signal for the reduction of the process variation and for improvement. 23. - 25. 5. 2012, Brno, Czech Republic, EU

Natural pattern is such pattern where the points fluctuate randomly. It can be characterized as follows [3]: most points lie near the central line; only few of the points spread out and reach the control limits; very rarely some point goes out of the control limit. When some or all these attributes are missing the pattern will be classified as unnatural. Many types of unnaturalness can be recognized in control charts. But several generic patterns that can be detected for various processes were defined. In table 1 we can find mostly recommended (for instance in [3], [4], [5], [6], [7]) and applied nonrandom patterns for Shewhart control charts with their description and their typical symptom in control chart. To standardize and simplify the process of the typical unnatural patterns recognition statistical tests (rules) that quantify the length of the unnatural patterns were formulated. Table 1 Unnatural pattern description

No. Unnatural (non- Pattern Description Symptom in control chart random) pattern 1 Large shift Sudden and high change Points near and or beyond control limits (strays, freaks) 2 Smaller sustained Sustained smaller change Series of points on the same side of the central shift line 3 Trends A continuous changes in one Steadily increasing or decreasing run of points direction 4 Stratification Small differences between values in A long run of points near the central line on the a long run, absence of points near both sides the control limits 5 Mixture Saw-tooth effect, absence of points A run of consecutive points on both sides of near the central line central line, all far from the central line 6 Systematic Regular alternation of high and low A long run of consecutive points alternating up Variation values and down 7 Cycle Recurring periodic Cyclic recurring patterns of points movement

2. TESTS FOR UNNATURAL PATTERNS RECOGNITION Tests for unnatural patterns supplement the evidence of the unnatural pattern given by the position of the points relative to the control limits or central line with evidence based on the of runs [8] except the Shewhart rule (point out of the control limit). They are based on the calculations that tell us the portion of points lying near the central line, near the control limits, etc. We will discuss tests that cover unnatural patterns no. 1 - 6 (see the former table), i.e. shifts, trends, mixtures, systematic variation and stratification. Table 2 contains sets of the rules that are the most well – known and applied (Shewhart rule [9], Western Electric rules [3], Nelson tests [10], ISO 2859 tests [11], special sets formulated by large companies (Boeing [12], AIAG [13]) and the most new set by Trietsch [2]. Rules 1 - 4 were defined to quickly recognize patterns linked to the shifts in the process. Rule 5 is connected with the trends, rule 6 and 8 with patterns caused by incorrect and rule 7 with abnormal oscillation. Historically first rule for the detection of the assignable cause developed W. S. Shewhart – in table 2 it is Rule 1. Set of Western Electric Rules added to the Shewhart criterion run tests covering such unnatural patterns as smaller or sustained shifts (called zone tests), mixtures, stratification and systematic variation. To be able to apply these tests the region between control limits in control chart must be divided into 6 zones each of 1 sigma width and the location of the predefined sequences of the points in relation to these zones must be evaluated. As normal distribution of the applied sample statistics is supposed, in zones C 68,27% values, in zones B 27,18% values and in zones A 4,28% values are expected when process is statistically 23. - 25. 5. 2012, Brno, Czech Republic, EU

Table 2 Tests for unnatural patterns Tests for unnatural Shewhart Western Nelson ISO AIAG Boeing Trietsch patterns Rule Electric Tests 8258 Rules ASQ Rules Rules Rule 1 1 1 1 1 1 1 One or more points more than 3 sigma from the (beyond zone A) Rule 2 2 5 5 2 5 2 of 3 consecutive points between 2 and 3 sigma from the mean (in zone A or beyond) Rule 3 3 6 6 3 6 4 of 5 consecutive points between 1 and 3 sigma from the mean (in zone B or beyond) Rule 4 4 2 2 2 4 2 8 consecutive points on one side of the mean (in 9 points 9 points 7 points 9 points zone C or beyond) Rule 5 3 3 3 3 Six consecutive points steadily increasing or decreasing Rule 6 7 7 7 15 consecutive points both above and below central 13 points line ( in zones C) Rule 7 4 4 4 14 consecutive points alternating up and down 13 points Rule 8 8 8 8 Eight points in a row on both sides of the central 5 points line with none in zones C stable. Rules 2, 3 and 4 were designed for quicker identification of the shifts from the mean as compared to the test 1 (early warning indicators). As it can be seen in table 2 the most complex as to the rate of covering all eight defined rules are Nelson rules [10]. In addition to the Westerns Electric set Nelson defined run test for the oscillation and for the trends. In rule 4 he changed the length of the run from 8 to 9 points. As compared to the Western Electric rules the sequence of the rules had been changed by Nelson, too. Standard for Shewhart control charts ISO 8250 has wholly copied the Nelson set of rules. Also Trietsch in his publication works with the Nelson rules but in rules 6, 7, 8 he has changed the length of the runs to reach better statistical properties of these tests. 23. - 25. 5. 2012, Brno, Czech Republic, EU

Rules of the companies Boeing and AIAG have less complex sets of rules based on the first four Western Electric zone rules.

3. METHODOLOGY FOR APPLICATION OF THE RULES FOR UNNATURAL PATTERN RECOGNITION In literature (see for instance [9], [3], [14], [15], [10], [16], [17], [4]) several strategies for the application of the rules for unnatural pattern recognition from different authors can be found. They have many differences but the main idea is common: runs test must not be applied routinely; it is really impractical and simultaneous application of all known rules brings a danger of unacceptable rise of the false alarm. As it can be found in the mentioned literature some authors differ in the idea what rules to apply routinely. But predominantly they recommend rule 1 in conjunction with rule 4. For additional sensitizing of the control charts some of the authors recommend rules 2 and 3. Wheeler recommends using of these rules only when increasing sensitivity of a control chart is absolutely necessary. From some strategies it can be seen an effort to apply separately rules for the identification of patterns for shifts and patterns for subgrouping problems (for instance Wheeler intents [16] that the main use of Rule 6 is at start-up of SPC than in an on-going control). Some authors put the stress on the phases of the SPC implementation when selecting suitable rules. Montgomery that „Sensitizing rules can be helpful when the control chart is first applied and the focus is on stabilizing an out-of-control process. However, once the process is reasonably stable, the routine use of these sensitizing rules to detect small shifts or to try to react more quickly to assignable causes, should be discouraged” ([4], p. 177).

4. TESTS AND TYPES OF CONTROL CHARTS When selecting suitable rules for the selected control chart it must be considered that ([10], [3], [2], [16.]):  All rules can be applied to x- and individuals chart supposing normal distribution of the variable;  Rules 1- 4 can be applied to the dispersion charts (R, s) without any modification when sample size is 5 or more (it should lead to the symmetric control limits);  Nonparametric rules (Rule 5 and 7) work sufficiently well for any continuous distribution.  The rules can be applied without any problems to attribute control charts np and c supposing that normal approximation is valid and control limits are reasonably symmetrical.  The same pays for the control charts p and u with constant control limits.

5. ANALYSIS OF SELECTED SW PRODUCTS SUPPORTING UNNATURAL PATTERNS RECOGNITION As it was mentioned in the previous chapter routine application of runs rules can lead to reducing the effectiveness of SPC implementation. Now days there are many SW products that support this part of SPC. They enable to increase effectiveness of the problem solving in SPC but at the same time they bring danger of routine application of the runs rules with its negative effects. In this charter the analysis of SW (Statgraphics [18], [19], Minitab [20]) used for the education at the Department of quality management at the University of Ostrava, Czech Republic and really excellent statistical SW used by many well known Czech universities (Statistica [21]) will be done in connection with their support to the control chart unnatural pattern recognition and with respect to the problems discussed in the previous chapter. In general all analysed SW products cover the set of Nelson rules (the same used in ISO 8258) with some little departures. All Nelson rules (all tests for assignable causes) including Shewhart criterion 1 (Rule 1) contains only SW Minitab. The rest of the analysed SW have rule 1 incorporated into the basic control chart 23. - 25. 5. 2012, Brno, Czech Republic, EU

analysis outside the set of runs tests. Statgraphics products have slightly different definition of run in rule 5, rule 8 (Statgraphics Plus) and Rule 8 (Statgraphics Centurion). Detailed characterization of every analyzed SW can be found in table 3. As to the complexity in general it can be stated that all analysed SW cover the set of Nelson rules (the same used in ISO 8258) with some little departures. Only SW Minitab contains all Nelson rules (tests for assignable causes) including Shewhart criterion 1 (rule 1). The rest of the analysed SW have rule 1 incorporated into the basic control chart analysis outside the set of runs tests ( for that reason in these SW the term „runs tests“ is used). From the point of view of the common types of the unnatural patterns defined in this paper (large shift, smaller sustained shift, trends, stratification, mixture, systematic variation) all analysed SW cover all these types except Statgraphics Plus where systematic variation is not considered. All analyzed SW cover all standard Shewhart control charts for variables and for attributes. Statgraphics products cover in addition ARIMA residuals chart and CusCore chart. To use the tests correctly the user needs to have at least basic knowledge of the statistical base of the tests. This request is to a certain extent solved only in SW Statistica. Different default runs length definition as compared to the Nelson rules can be found in the Statgraphics SW products in rule 5, rule 8 (Statgraphics Plus) and rule 7 (Statgraphics Centurion). The analyzed SW also differ in the default rules. SW Statgraphics Plus and Statistica have preset all defined rules, covering unnatural patterns as all types shifts in the mean, systematic variation, stratification and mixture; Statgraphics Centurion only first four (according to the numbering in table 2 - rules 4, 2, 3, 5, i.e. rules covering different types of shifts).WS Minitab has only first four rules ((according to the numbering in table 2 - rules 1, 4, 5, 7 covering basic types of the mean shift and systematic variation). For less experienced user setting all the tests as a default can be confusing. Better access is setting the limited number of rules. It is not good strategy to apply all tests simultaneously – it leads to the higher false alarm and to more complicated analysis of the process. The limited number of rules as a default is a smaller evil. In addition every rule is not suitable for each type of the chart. In SW Statgraphics Centurion and Minitab the user is to some extend protected from incorrect application of runs rules as to the type of control chart by the limited default set of runs that can be applied to various control charts. But most precisely it is solved in Minitab where the system itself enables to use only suitable runs tests when it is needed – for R, S, moving , np, p, u, c charts only tests 1, 4, 5 and 7 (numbering according table 2) are allowed. All the tested SW are flexible as to the choice of the run test and the run length. We can select only test or tests we need. It offers the possibility to set the unique strategy for the analysis of the process instability related to the process behaviour. The possibility to change the length of the runs can lead to the selection of the optimal length connected with the acceptable value of false signal. Except SW Minitab all tested SW enable to change definition of zones. It offers possibility to adopt the rules to control chart that is not based on the normal distribution. It is sufficiently flexible for the user but the how to decide about suitable tests or the length of runs is not appropriately detailed. Indication of patterns in control chart is solved in all analyzed SW. SW Statistica offers in addition the following function: when the out-of-control conditions are investigated and an explanation for them is found, investigator can assign descriptive labels to those out-of-control samples and explain the causes and actions that have been taken. Having causes and actions displayed in the chart will document that the centre line and the control limits of the chart are affected by special cause variation in the process. As it was mentioned above simultaneous application of more rules leads to rise of false alarm and it can lead to worse and longer identification of really affected uncommon causes. Warning about simultaneous using more rules is included only in the adviser of SW Minitab and Statistica. The same conclusion can be done as to the warning about the necessity to have a deep knowledge about the analyzed process. But without appropriate knowledge about the process the unnatural patterns analysis can´t offer acceptable results. 23. - 25. 5. 2012, Brno, Czech Republic, EU

Table 3 Multicriteria analysis of the selected SW Statgraphics Statgraphics Minitab Statistica Properties/ SW Plus V.15 Centurion v. 15 v. 16 v. XV Term used for unusual pattern Unusual pattern Unusual Nonrandom Systematic sequence pattern pattern Term used for tests Runs chart Runs tests Tests for Runs tests special causes Complexity 6 from 8 7 form 8 8 from 8 7 form 8 Statistical base of tests no no no yes Coverage of unnatural pattern  Large shift yes yes yes yes  Smaller sustained shift yes yes yes yes  Trends yes yes yes yes  Stratification yes yes yes yes  Mixture yes yes yes yes  Systematic Variation no yes yes yes Different default runs definition as 2 2 0 0 compared to Nelson rules Default number of rules 6 all First 4 First 4 (WE) 7 all Possibility to choice the rules yes yes yes yes Possibility to change the length of runs yes yes yes yes Possibility to change definition of zones yes yes no yes Indication of patterns in control chart yes yes yes yes Record of assignable causes into no no no yes control chart Record of actions into control chart no no no yes Warning about simultaneous using more no no yes yes rules Warning about the necessity to have a no no yes yes deep knowledge about process Types of control chart covered (S- S + 0 S +0 S S Shewhart, O- other) The same default set of rules for yes yes yes yes different control charts Information to that control charts are Yes (no precise) Yes (no precise) partial partial rules applicable Defined potential general assignable no no no yes causes References yes yes yes yes Description of rules yes yes yes yes Interpretation of rules no no no yes

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Description of the set of the rules can be found in each analyzed SW. But the most detailed description including possible interpretation of the rules and definition of possible assignable causes offers only SW Statistica. The former analysis of the properties of the analyzed SW showed that the best solution in the field of runs tests application offer SW Minitab and Statistica. But they also suffer from some disadvantages that can lead little experienced user to incorrect results and to complicate procedure of the process control. It can unfortunately result in ignoring these runs tests in the analysis of the process stability or failing of the whole SPC application. For that reason in the next charter simple methodology for the correct application of tests for special casus is formulated.

6. RECOMMENDATIONS FOR THE APPLICATION OF RULES FOR UNNATURAL PATTERNS RECOGNITION  Before applying runs tests or tests for special causes check the default set of tests for selected control charts.  Never routinely apply all runs tests or tests for assignable causes even if they are preset in your SW.  Check if chosen control charts have reasonably symmetrical control limits because in such situation next conclusions are valid: - all tests (Rule 1 - Rule 8) can be applied to x-bar chart and individuals chart supposing normal distribution of the variable; - rules 1- 4 can be applied to the dispersion charts (R, s) without any modification when sample size is 5 or more; - the rules can be applied without any problems to control charts np and c supposing that normal approximation is valid and control limits are reasonably symmetrical. - The same pays for control chart p and u with constant control. - Tests 5 and 7 work sufficiently well for any continuous distribution because of their nonparametric nature.  In the situation when control charts have reasonable symmetrical control limits, select suitable test according to the following schema: - Rules 6 and 8 apply at the beginning of the SPC implementation to verify rational subgrouping. - In the phase of the statistical stability ensuring start with Rule 1 and 4. - If additional sensitizing of control chart is necessary add Rules 2 and 3 or only one of them. - When your knowledge of the process has risen during the previous SPC implementation adds some of the other standard tests or your own fitted test. - Test 5 apply singularly (its marginal increase in sensitivity to real signals is more than offset by the greater increase in false alarm ([16], p. 137).  Rules 1 – 4 apply to both halves of the control chart but separately.  When using a couple of control charts (for instance x-bar and R) start with an analysis of the dispersion control chart.  Keep in mind that where the run test is violated does not always indicate where the process change had occurred.

7. CONCLUSIONS Based on the theoretical background made in the introductory chapter this paper dealt with the analysis and comparison of the several known sets of the rules for the special causes identification. Shewhart rule, Western Electric Rules, Nelson tests, ISO 2859 tests, special sets formulated by large companies (Boeing, GE) and the most new set by Trietsch were compared. In the light of the conclusions of these analyses the next multi-criteria analysis of the selected SW product with the focus on the tests for the special causes 23. - 25. 5. 2012, Brno, Czech Republic, EU

identification were done with the aim to reveal the advantages and disadvantages of every discussed SW from the point of view of the support for meeting the main goal of SPC. Based on this analysis several recommendations for the correct and effective application of these tests were then formulated.

ACKNOWLEDGEMENT This paper was elaborated in the frame of the specific research project No. SP2012/42, which has been solved at the Faculty of Metallurgy and Materials Engineering, VŠB-TU Ostrava with the support of Ministry of Education, Youth and Sports, Czech Republic.

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