1、Designation: E 2555 07Standard Practice forFactors and Procedures for Applying the MIL-STD-105 Plansin Life and Reliability Inspection1This standard is issued under the fixed designation E 2555; the number immediately following the designation indicates the year oforiginal adoption or, in the case o
2、f revision, the year of last revision. A number in parentheses indicates the year of last reapproval. Asuperscript epsilon (e) indicates an editorial change since the last revision or reapproval.1. Scope1.1 This practice presents a procedure and related tables offactors for adapting Practice E 2234
3、(equivalent to MIL-STD-105) sampling plans to acceptance sampling inspection whenthe item quality of interest is life length or reliability. Factorsare provided for three alternative criteria for lot evaluation:mean life, hazard rate, and reliable life. Inspection of thesample is by attributes with
4、testing truncated at the end of someprearranged period of time. The Weibull distribution, togetherwith the exponential distribution as a special case, is used asthe underlying statistical model.1.2 A system of units is not specified by this practice.1.3 This standard does not purport to address all
5、of thesafety concerns, if any, associated with its use. It is theresponsibility of the user of this standard to establish appro-priate safety and health practices and determine the applica-bility of regulatory limitations prior to use.2. Referenced Documents2.1 ASTM Standards:2E 456 Terminology Rela
6、ting to Quality and StatisticsE 2234 Practice for Sampling a Stream of Product byAttributes Indexed by AQL3. Terminology3.1 DefinitionsThe terminology defined in TerminologyE 456 applies to this practice unless modified herein.3.1.1 acceptance quality level (AQL), nquality limit thatis the worst tol
7、erable process average when a continuing seriesof lots is submitted for acceptance sampling. E 22343.1.1.1 DiscussionThis term is often referred to as the“acceptance quality limit.”3.1.1.2 DiscussionThis definition supersedes that given inMEL-STD-105E.3.1.1.3 DiscussionA sampling plan and an AQL are
8、chosen in accordance with the risk assumed. Use of a value ofAQL for a certain defect or group of defects indicates that thesampling plan will accept the great majority of the lots orbatches provided the process average level of percent defective(or defects per hundred units) in these lots or batche
9、s are nogreater than the designated value of AQL. Thus, the AQL is adesignated value of percent defective (or defects per hundredunits) for which lots will be accepted most of the time by thesampling procedure being used. The sampling plans providedherein are so arranged that the probability of acce
10、ptance at thedesignated AQL value depends upon the sample size, beinggenerally higher for large samples than for small ones, for agiven AQL. The AQL alone does not identify the chances ofaccepting or rejecting individual lots or batches but moredirectly relates to what might be expected from a serie
11、s of lotsor batches, provided the steps indicated in this refer to theoperating characteristic curve of the plan to determine therelative risks.3.1.2 consumers risk, nprobability that a lot havingspecified rejectable quality level will be accepted under adefined sampling plan.3.1.3 double sampling p
12、lan, na multiple sampling plan inwhich up to two samplings can be taken and evaluated toaccept or reject a lot.3.1.4 limiting quality level (LQL), nquality level having aspecified consumers risk for a given sampling plan.3.1.5 lot, na definite quantity of a product or materialaccumulated under condi
13、tions that are considered uniform forsampling purposes.3.1.5.1 DiscussionThe lot for sampling may differ from acollection of units designated as a batch for other purposes, forexample, production, shipment, and so forth.3.1.6 multiple sampling plan, na sampling plan in whichsuccessive samples from a
14、 lot are drawn and after each sampleis inspected a decision is made to accept the lot, reject the lot,or to take another sample, based on quality level of thecombined samples.3.1.6.1 DiscussionWhen the quality is much less or muchmore than the AQL, the decision can be made on the firstsample, which
15、is smaller than that of a single sampling planwith equivalent acceptance quality level. For samples that are1This practice is under the jurisdiction of ASTM Committee E11 on Quality andStatistics and is the direct responsibility of Subcommittee E11.30 on Data Analysis.Current edition approved March
16、1, 2007. Published April 2007.2For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at serviceastm.org. For Annual Book of ASTMStandards volume information, refer to the standards Document Summary page onthe ASTM website.1Copyright ASTM International,
17、100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United States.close to theAQLin quality, additional samples are required andthe total sample size will be larger than the correspondingsingle sampling plan.3.1.7 sample, ngroup of items, observations, test results,or portions of m
18、aterial taken from a large collection of items,observations, test results, or quantities of material that serves toprovide information that may be used as a basis for making adecision concerning the larger collection. E 22343.2 Definitions of Terms Specific to This Standard:3.2.1 acceptance number,
19、nthe maximum number offailed items allowed in the sample for the lot to be acceptedusing a single or multiple sampling plan.3.2.2 hazard rate, ndifferential fraction of items failing attime t among those surviving up to time t, symbolized by h(t).3.2.2.1 Discussionh(t) is also referred to as the ins
20、tanta-neous failure rate at time t. It is related to the probabilitydensity and cumulative distribution functions by h(t) = f(t)/(l F(t).3.2.3 mean life, naverage time that items in the lot orpopulation are expected to operate before failure.3.2.3.1 DiscussionThis metric is often referred to as mean
21、time to failure (MTTF) or mean time before failure (MTBF).3.2.4 rejection number, nthe minimum number of faileditems in the sample that will cause the lot to be rejected undera given sampling plan.3.2.5 reliable life (rr), nlife beyond which some specifiedproportion, r, of the items in the lot or po
22、pulation will survive.3.2.6 test truncation time (t), namount of time sampleditems are allowed to be tested.3.2.7 Weibull distribution, nprobability distribution hav-ing cumulative distribution:function Ft! 5 1 expSSt ghDbD, t .gand probability densityfunction ft! 5bhSt ghDb21expSSt ghDbD3.2.7.1 Dis
23、cussionThe Weibull distribution is widely usedfor modeling product life. It can take a wide variety of shapesand also the characteristics of other types of distributions basedon the value of its parameters. g is called the location,minimum life, or threshold parameter and defines the lowerlimit of t
24、he distribution (Fig. 1). h is called the scale orcharacteristic life parameter and is equal to the 63.2 percentileof the distribution, minus g (Fig. 2). b is the shape parameter(Fig. 3). The exponential distribution is the special case whereg = 0 and b =l.4. Significance and Use4.1 The procedure an
25、d tables presented in this practice arebased on the use of the Weibull distribution in acceptancesampling inspection. Details of this work, together with tablesof sampling plans of other forms, have been published previ-ously. See Refs (1-3).3Since the basic computations requiredhave already been ma
26、de, it has been quite easy to provide thesenew factors. No changes in method or details of applicationhave been made over those described in the publicationsreferenced above. For this reason, the text portion of this reporthas been briefly written. Readers interested in further detailsare referred t
27、o these previous publications. Other sources ofmaterial on the underlying theory and approach are alsoavailable (4-7).4.2 The procedure to be used is essentially the same as theone normally used for attribute sampling inspection. The only3The boldface numbers in parentheses refer to the list of refe
28、rences at the end ofthis standard.FIG. 1 Effect of the Parameter g on the Weibull ProbabilityDensity Function, f(t)E2555072difference is that sample items are tested for life or survivalinstead of for some other property. For single sampling, thefollowing are the required steps:4.2.1 Using the table
29、s of factors provided in Annex A1,select a suitable sampling inspection plan from those tabulatedin Practice E 2234.4.2.2 Draw at random a sample of items of the size specifiedby the selected Practice E 2234 plan.4.2.3 Place the sample of items on life test for the specifiedperiod of time, t.4.2.4 D
30、etermine the number of sample items that failedduring the test period.4.2.5 Compare the number of items that failed with thenumber allowed under the selected Practice E 2234 plan.4.2.6 If the number that failed is equal to or less than theacceptable number, accept the lot; if the number failingexcee
31、ds the acceptable number, reject the lot.4.3 Both the sample sizes and the acceptance numbers usedare those specified by Practice E 2234 plans. It will be assumedin the section on examples that single sampling plans will beused. However, the matching double sampling and multiplesampling plans provid
32、ed in MIL-STD-105 can be used ifdesired. The corresponding sample sizes and acceptance andrejection numbers are used in the usual way. The specified testtruncation time, t, must be used for all samples.4.4 The probability of acceptance for a lot under thisprocedure depends only on the probability of
33、 a sample itemfailing before the end of the test truncation time, t. For thisFIG. 2 Effect of the Parameter h on the Weibull ProbabilityDensity Function, f(t)FIG. 3 Effect of the Parameter b on the Weibull ProbabilityDensity Function, f(t)E2555073reason, the actual life at failure need not be determ
34、ined; onlythe number of items failing is of interest. Life requirements andtest time specifications need not necessarily be measured inchronological terms such as minutes or hours. For example, thelife measure may be cycles of operation, revolutions, or milesof travel.4.5 The underlying life distrib
35、ution assumed in this standardis the Weibull distribution (note that the exponential distribu-tion is a special case of the Weibull). The Weibull model hasthree parameters. One parameter is a scale or characteristic lifeparameter. For these plans and procedures, the value for thisparameter need not
36、be known; the techniques used are inde-pendent of its magnitude. A second parameter is a location or“guaranteed life” parameter. In these plans and procedures, it isassumed that this parameter has a value of zero and that thereis some risk of item failure right from the start of life. If this isnot
37、the case for some applications, a simple modification inprocedure is available. The third parameter, and the one ofimportance, is the shape parameter, b.4The magnitude of theconversion factors used in the procedures described in thisreport depends directly on the value for this parameter. For thisre
38、ason, the magnitude of the parameter shall be known throughexperience with the product or shall be estimated from pastresearch, engineering, or inspection data. Estimation proce-dures are available and are outlined in Ref (1).4.6 For the common case of random chance failures with thefailure rate con
39、stant over time, rather than failures as a result of“infant mortality” or wearout, a value of 1 for the shapeparameter shall be assumed. With this parameter value, theWeibull distribution reduces to the exponential. Tables ofconversion factors are provided in Annex A1 for 15 selectedshape parameter
40、values ranging from12 to 10, the rangecommonly encountered in industrial and technical practice.The value 1, used for the exponential case, is included. Factorsfor other required shape parameter values within this rangemay be obtained approximately by interpolation. A morecomplete discussion of the
41、relationship between failure pat-terns and the Weibull parameters can be found in Refs (1-3).4.7 One possible acceptance criterion is the mean life foritems making up the lot (). Mean life conversion factors orvalues for the dimensionless ratio 100t/ have been determinedto correspond to or replace a
42、ll the p or percent defective valuesassociated with Practice E 2234 plans. In this factor, t repre-sents the specified test truncation time and the mean item lifefor the lot. For reliability or life-length applications, thesefactors are used in place of the corresponding p valuesnormally used in the
43、 use of Practice E 2234 plans for attributeinspection of other item qualities. The use of these factors willbe demonstrated by several examples (see Sections 5, 7, and 9).4.8 Annex Table 1A lists, for each selected shape parametervalue, 100t/ ratios for each of the Practice E 2234 AQLp(%) values. Wi
44、th acceptance inspection plans selected interms of these ratios, the probability of acceptance will be highfor lots whose mean life meets the specified requirement. Theactual probability of acceptance will vary from plan to plan andmay be read from the associated operating characteristic curvessuppl
45、ied in MDL-STD-105. The curves are entered by usingthe corresponding p(%) value. Annex Table 1B lists 100t/ratios at the LQL for the quality level at which the consumersrisk is 0.10. Annex Table 1C lists corresponding 100t/ ratiosfor a consumers risk of 0.05.4.8.1 These ratios are to be used directl
46、y for the usual casefor which the value for the Weibull location or thresholdparameter (g) can be assumed as zero. If g is not zero but hassome other known value, all that shall be done is to subtract thevalue for g from t to get t0and from m to get m0. Thesetransformed values, t0and m0, are then em
47、ployed in the use ofthe tables and for all other computations.Asolution in terms ofm0and t0can then be converted back to actual or absolutevalues by adding the value for g to each.5. Examples, Mean Life Ratio5.1 A Practice E 2234 acceptance sampling inspection planis to be applied to incoming lots o
48、f product for which the meanitem life is the property of interest. An acceptable mean life of2000 h has been specified, and under the plan, used lots with amean life of this value or greater shall have a high probabilityof acceptance. A testing truncation time of t = 250 h has beenspecified. From pa
49、st experience it has been determined that theWeibull distribution can be used as a life-length model and ashape parameter value of 2.5 and a location or thresholdparameter value of 0 can be assumed. Single sampling is to beused. A sample of as many as 300 items or so can be tested atone time. An appropriate sampling inspection plan shall beselected. Also, the consumers risk under use of the selectedplan shall be determined.5.1.1 Computation of the 100t/ ratio at the AQL gives100t/ = 100 3 250/2000 = 12.5. Examination of the rat
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