{"id":1120,"date":"2013-05-16T12:04:29","date_gmt":"2013-05-16T16:04:29","guid":{"rendered":"http:\/\/www.reliabilityanalytics.com\/blog\/?p=1120"},"modified":"2013-07-04T16:06:50","modified_gmt":"2013-07-04T20:06:50","slug":"sequential-reliability-test-calculator","status":"publish","type":"post","link":"https:\/\/reliabilityanalytics.com\/blog\/2013\/05\/16\/sequential-reliability-test-calculator\/","title":{"rendered":"Sequential Reliability Test Calculator"},"content":{"rendered":"<p>A <a title=\"Sequential Reliability Test Calculator\" href=\"http:\/\/reliabilityanalyticstoolkit.appspot.com\/sequential_reliability_testing\">Sequential Reliability Testing Calculator<\/a> was recently added to the Reliability Analytics Toolkit. Sequential testing often provides a more efficient method to verify equipment reliability achievement. Really &#8220;good&#8221; equipment will be accepted much quicker and really &#8220;bad&#8221; equipment will be rejected much sooner, often resulting in fewer test hours needed than using a <a title=\"Fixed length tests summary\" href=\"http:\/\/reliabilityanalytics.com\/reliability_engineering_library\/MIL-HDBK-781A_Reliability_Test_Methods_Plans_and_Environments_1_Apr_96\/MIL-HDBK-781A_Reliability_Test_Methods_Plans_and_Environments_1_Apr_96_pp_527.htm\">military handbook 781 fixed length reliability test<\/a>. The tool provides the ability to plan a sequential reliability demonstration test for verification of equipment mean time between failure (MTBF) if it can be assumed that the equipment follows an exponential failure distribution (i.e., constant failure rate). <!--more-->Input parameters include the following:<\/p>\n<ul>\n<li>Lower test MTBF (\u03b8<sub>1<\/sub>). The test plan will reject an item whose true MTBF is \u03b8<sub>1<\/sub>\u00a0with a probability of 1 &#8211; \u03b2.<\/li>\n<li>Upper test MTBF (\u03b8<sub>0<\/sub>). The test plan will accept an item whose true MTBF is \u03b8<sub>0<\/sub>\u00a0with a probability of 1 &#8211; \u03b1. \u00a0\u00a0\u03b8<sub>0<\/sub>\u00a0= d * \u03b8<sub>1<\/sub>.<\/li>\n<li>Discrimination ratio (d). The discrimination ratio is one of the standard test plan parameters which establish the test plan envelope. d = \u03b8<sub>0<\/sub>\/\u03b8<sub>1<\/sub>.<\/li>\n<li>Consumer\u2019s risk (\u03b2). The consumer\u2019s risk is the probability of accepting equipment with a true MTBF equal to the lower test MTBF (\u03b8<sub>1<\/sub>).<\/li>\n<li>Producer\u2019s risk (\u03b1). The producer\u2019s risk is the probability of rejecting equipment with a true MTBF equal to the upper test MTBF (\u03b8<sub>0<\/sub>).<\/li>\n<li>True MTBF. The MTBF that would be observed if an infinite number of units were tested for an infinite amount of time.<\/li>\n<\/ul>\n<p><strong>Example 1<br \/>\n<\/strong>Suppose that a system had a specified lower test MTBF of 100 hours. \u00a0A test discrimination ratio of 1.5 is chosen along with a 5% consumers and producers risk:<br \/>\n<a href=\"https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential11.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1122\" title=\"sequential1\" src=\"https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential11.png\" alt=\"\" width=\"590\" height=\"490\" srcset=\"https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential11.png 590w, https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential11-300x249.png 300w, https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential11-361x300.png 361w\" sizes=\"auto, (max-width: 590px) 100vw, 590px\" \/><\/a><\/p>\n<p>The above inputs result in the test plan summarized below. The plan requires a minimum time to accept of 883 hours if no failures occur and a maximum time to accept of 6,800 hours if the test ends in truncation. The red line in the sequential test plot is the reject line while the green line is the accept line. Any failures occurring during testing are plotted on the graph and if this plot crosses the green line, the equipment accepted will have a true MTBF of greater than 100 hours with a 0.95 probability (i.e., 1 &#8211; \u03b2).<br \/>\n<a href=\"https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential2.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1123\" title=\"sequential2\" src=\"https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential2.png\" alt=\"\" width=\"783\" height=\"690\" srcset=\"https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential2.png 783w, https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential2-300x264.png 300w, https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential2-340x300.png 340w\" sizes=\"auto, (max-width: 783px) 100vw, 783px\" \/><\/a><\/p>\n<p><strong>Simulating Testing<br \/>\n<\/strong>For planning purposes, the tool allows the user to simulate testing by assuming a true MTBF and generating simulated failure times using a random number generator. Simulated failure times are generated using the reliability function for the exponential failure distribution and the\u00a0<a title=\"Generate pseudo-random numbers\" href=\"http:\/\/docs.python.org\/2\/library\/random.html\">random number generator associated with the Python programming language<\/a>. Modifying the inputs to simulate three tests and assuming a true system MTBF is 150 hours:<a href=\"https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential3.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1124\" title=\"sequential3\" src=\"https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential3.png\" alt=\"\" width=\"596\" height=\"487\" srcset=\"https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential3.png 596w, https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential3-300x245.png 300w, https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential3-367x300.png 367w\" sizes=\"auto, (max-width: 596px) 100vw, 596px\" \/><\/a><\/p>\n<p>The resulting simulations are shown below in both plot and summary table form. \u00a0For a true MTBF of 150 hours, the average test time for the three simulations is 3,803 hours.<br \/>\n<a href=\"https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential42.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1127\" title=\"sequential4\" src=\"https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential42.png\" alt=\"\" width=\"676\" height=\"729\" srcset=\"https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential42.png 676w, https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential42-278x300.png 278w\" sizes=\"auto, (max-width: 676px) 100vw, 676px\" \/><\/a><\/p>\n<p>If the true MTBF is changed from 150 hours to 250 hours and the simulation is re-run, the average test time drops to 1,978 hours &#8211; a really good MTBF is accepted much quicker, on average: \u00a0<a href=\"https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential5.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1129\" title=\"sequential5\" src=\"https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential5.png\" alt=\"\" width=\"674\" height=\"724\" srcset=\"https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential5.png 674w, https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential5-279x300.png 279w\" sizes=\"auto, (max-width: 674px) 100vw, 674px\" \/><\/a><\/p>\n<p>If the true MTBF is much less than the 100 hour lower test MTBF, say 50 hours, the simulated tests all reach reject decisions relatively quickly. \u00a0On average, it takes 779 hours to reach a reject decision:<br \/>\n<a href=\"https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential61.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1131\" title=\"sequential6\" src=\"https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential61.png\" alt=\"\" width=\"673\" height=\"718\" srcset=\"https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential61.png 673w, https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential61-281x300.png 281w\" sizes=\"auto, (max-width: 673px) 100vw, 673px\" \/><\/a><\/p>\n<p>The tool also outputs the proposed test plan in tabular form, showing all possible accept\/reject points and allows the option is to plot specific failure times that a user has entered for input 4A.<\/p>\n<p><strong>Increase Decision Risk to Lower Test Time<br \/>\n<\/strong>In the first example above, the minimum time to accept was 883 hours if no failures occurred and a maximum time to accept was 6,800 hours if the test ended in truncation. How can the test be shortened? If higher\u00a0decision risks can be accepted then test time can be significantly shortened. \u00a0In the above example, if an accept decision was reached, there was a 0.95 probability that the true MTBF was greater than 100 hours. \u00a0What would the test characteristics be if this probability was 0.70 instead of 0.95? \u00a0Change the decision risks to 30%, as shown below:<br \/>\n<a href=\"https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential7.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1136\" title=\"sequential7\" src=\"https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential7.png\" alt=\"\" width=\"587\" height=\"488\" srcset=\"https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential7.png 587w, https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential7-300x249.png 300w, https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential7-360x300.png 360w\" sizes=\"auto, (max-width: 587px) 100vw, 587px\" \/><\/a><\/p>\n<p>Now, instead of a minimum test time of 883 hours, only 254 hours will be required if no failures occur. The maximum test time is now only 680 hours, as compared to 6,800 hours for the 5% decision risk test. \u00a0If the equipment reaches an accept decision, there is a 0.7 probability that the true MTBF is 100 hours, or better. \u00a0Conversely, there is a 0.3 probability that the true MTBF is less than 100 hours. These decision risks are highlighted in the operating characteristic curve, as shown in the second picture below. \u00a0If the true MTBF is 100 hours, there is a 0.3 probability of acceptance, as highlighted in the figure. However, if the true MTBF is equal to the upper test MTBF (\u03b8<sub>0<\/sub> = 150 hours, which is the lower test MTBF(\u03b8<sub>1<\/sub>) times the discrimination ratio), then there is a 0.7 probability of acceptance.<br \/>\n<a href=\"https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential8.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1137\" title=\"sequential8\" src=\"https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential8.png\" alt=\"\" width=\"679\" height=\"694\" srcset=\"https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential8.png 679w, https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential8-293x300.png 293w\" sizes=\"auto, (max-width: 679px) 100vw, 679px\" \/><\/a><\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1139\" style=\"color: #333333; font-style: normal;\" title=\"sequential9\" src=\"https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential91.png\" alt=\"\" width=\"634\" height=\"464\" srcset=\"https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential91.png 634w, https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential91-300x219.png 300w, https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential91-409x300.png 409w\" sizes=\"auto, (max-width: 634px) 100vw, 634px\" \/><\/p>\n<p><strong>Impact of Discrimination Ratio on Test Length<br \/>\n<\/strong>The discrimination ratio d is the ratio of the upper test MTBF to lower test MTBF, d = \u03b8<sub>0<\/sub>\/\u03b8<sub>1<\/sub>. The farther apart\u00a0\u03b8<sub>0<\/sub> is from\u00a0\u00a0\u03b8<sub>1<\/sub>\u00a0(i.e., larger d) the faster a test decision is reached. For example, all other things being equal, the figure below shows testing for the same lower test MTBF of 100 hours with the test on the left having a d=1.5 (\u03b8<sub>0<\/sub>\u00a0=150)\u00a0and the test on the right having a d=3.0 (\u03b8<sub>0<\/sub>\u00a0=300). Although not visually intuitive because the plot scales are different, the area of indecision for the test on the left is far greater than the test on the right, specifically 92,552\/6,798 = 13.6 times greater. \u00a0The conclusion is that a smaller \u00a0discrimination ratio results in a greater the area of indecision and a longer test. It takes more test time to discriminate between an upper and lower test MTBF if they are close together.<br \/>\n<a href=\"https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential102.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1159\" title=\"sequential10\" src=\"https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential102.png\" alt=\"\" width=\"965\" height=\"865\" srcset=\"https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential102.png 965w, https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential102-300x268.png 300w, https:\/\/reliabilityanalytics.com\/blog\/wp-content\/uploads\/2013\/05\/sequential102-334x300.png 334w\" sizes=\"auto, (max-width: 965px) 100vw, 965px\" \/><\/a><\/p>\n<p>&nbsp;<\/p>\n<p>References<\/p>\n<ol>\n<li>Bazovsky, Igor,\u00a0<a href=\"http:\/\/www.amazon.com\/gp\/product\/0486438678?ie=UTF8&amp;tag=reliabilityan-20&amp;linkCode=as2&amp;camp=1789&amp;creative=9325&amp;creativeASIN=0486438678\">Reliability Theory and Practice<\/a><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/www.assoc-amazon.com\/e\/ir?t=reliabilityan-20&amp;l=as2&amp;o=1&amp;a=0486438678\" alt=\"\" width=\"1\" height=\"1\" border=\"0\" \/>.<\/li>\n<li>Wald, A.,\u00a0<a href=\"http:\/\/www.amazon.com\/gp\/product\/0486615790\/ref=as_li_tf_tl?ie=UTF8&amp;camp=1789&amp;creative=9325&amp;creativeASIN=0486615790&amp;linkCode=as2&amp;tag=reliabilityan-20\">Sequential Analysis.<\/a><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/www.assoc-amazon.com\/e\/ir?t=reliabilityan-20&amp;l=as2&amp;o=1&amp;a=0486615790\" alt=\"\" width=\"1\" height=\"1\" border=\"0\" \/><\/li>\n<li>Wald, A. (1945).\u00a0<a href=\"http:\/\/www.worldcat.org\/title\/sequential-tests-of-statistical-hypotheses\/oclc\/709418926\">Sequential tests of statistical hypotheses<\/a>. Annual Mathematica Statistics. 16, 117-186.<\/li>\n<li>MIL-HDBK-781A,\u00a0<a href=\"http:\/\/reliabilityanalytics.com\/reliability_engineering_library\/MIL-HDBK-781A_Reliability_Test_Methods_Plans_and_Environments_1_Apr_96\/MIL-HDBK-781A_Reliability_Test_Methods_Plans_and_Environments_1_Apr_96_pp_1.htm\">Reliability Test Methods, Plans, and Environments for Engineering Development, Qualification, and Production<\/a>.<\/li>\n<li>Epstein, B., &amp; Sobel, M. (1955).\u00a0<a href=\"http:\/\/projecteuclid.org\/euclid.aoms\/1177728595\">Sequential Life Tests in the Exponential Case<\/a>. The Institute of Mathematical Statistics.<\/li>\n<\/ol>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A Sequential Reliability Testing Calculator was recently added to the Reliability Analytics Toolkit. Sequential testing often provides a more efficient method to verify equipment reliability achievement. Really &#8220;good&#8221; equipment will be accepted much quicker and really &#8220;bad&#8221; equipment will be &hellip; <a href=\"https:\/\/reliabilityanalytics.com\/blog\/2013\/05\/16\/sequential-reliability-test-calculator\/\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[76],"tags":[78,77,42,79],"class_list":["post-1120","post","type-post","status-publish","format-standard","hentry","category-testing","tag-sequential-testing","tag-testing-2","tag-toolkit-examples","tag-wald"],"_links":{"self":[{"href":"https:\/\/reliabilityanalytics.com\/blog\/wp-json\/wp\/v2\/posts\/1120","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/reliabilityanalytics.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/reliabilityanalytics.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/reliabilityanalytics.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/reliabilityanalytics.com\/blog\/wp-json\/wp\/v2\/comments?post=1120"}],"version-history":[{"count":24,"href":"https:\/\/reliabilityanalytics.com\/blog\/wp-json\/wp\/v2\/posts\/1120\/revisions"}],"predecessor-version":[{"id":1154,"href":"https:\/\/reliabilityanalytics.com\/blog\/wp-json\/wp\/v2\/posts\/1120\/revisions\/1154"}],"wp:attachment":[{"href":"https:\/\/reliabilityanalytics.com\/blog\/wp-json\/wp\/v2\/media?parent=1120"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/reliabilityanalytics.com\/blog\/wp-json\/wp\/v2\/categories?post=1120"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/reliabilityanalytics.com\/blog\/wp-json\/wp\/v2\/tags?post=1120"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}