STUDIES ON DESIGNING CERTAIN CONTINUOUS SAMPLING PLANS

Transcription

1 STUDIES ON DESIGNING CERTAIN CONTINUOUS SAMPLING PLANS

2 CHAPTER I INTRODUCTION

3 Certain concepts, terminologies and symbols of acceptance sampling relevant to the thesis are explained in Section l. A review of certain designing methodologies including the methodologies followed in this thesis is presented in Section 2. A review cf continuous sampling plans of Dodge type, Wald-Wolfowitz type, Beattie type, and other continuous sampling plans and the author's contribution in designing certain continuous sampling plans of Dodge type are given in Section 3. SECTION 1 Comcepts and Terminologies of Acceptance Sampling. (ANSI/ASQC Standard A2 (1987) defines acceptance sampling as the methodology that deals with procedures by which decisions to accept or not-accept, are based on the results of the inspection of samples. According to Dodge (1969), the general arecis of acceptance sampling are: 1. Ict-by-lot sampling by the method of attributes, in which each unit in a sample is inspected on a go-not-go basis for one or more characteristics; 2. lot-by-lot. sampling by the method of variables, in which each unit in a sample is measured for a single cnaracteristic, such as weight or strength; -3. continuous sampling of a flow of units by the method of 1

4 cit tributes; and 4. that we may speak of as "special purpose plans, including chain sampling, skip-lot sampling, small sample plans, etc." This thesis mainly relates to designing of certain continuous sampling plans classified under the third area listed above. Certain concepts, terminologies and symbols of acceptance sampling are explained below. The following definitions are from KIL-STD-1235 C (1988), unless mentioned otherwise. Inspection : Inspection is the process of measuring, examining, testing or otherwise comparing she unit of product with the requirements. Inspection by Attributes : Inspection by attributes is inspection whereby certain characteristics of units of products are inspected and classified simply as conforming or not conforming to the specified requirements. Moving Product : The term "moving product" refers to product, which is flowing past the inspection station. In the typical case the product moves on a conveyor belt or line; however it may be moved in tote boxes, buggies or other 2

5 conveyances which are operated manually or by mobile materia1s-handling equipment. Unit of Product : The unit of product is the thing being inspected in order to determine its classification as defective or nondefective or to count the number of defects. It may be a single article, a pair, a set, a length, an area, an operation, a volume, a component of an end product or the end product itself. The unit of product may or may not be the same as the unit of purchase, supply, production or shipment. Process Average : The process average is defined as the percent defective of product submitted by the supplier for original inspection. Original inspection is the first inspection of a particular quantity of product as distinguished from the inspection of product which has been resubmitted. The phrases "process average" and "percent defective of submitted product" are used interchangeably. Conforming Unit : A conforming unit is one which meets the acceptance criteria established for the characteristic being considered. Nonconforming Unit : A nonconforming unit is one which does not meet the acceptance criteria established for the 3

6 characteristic being considered. Defect : A defect is any nonconformance of the unit of product with specified requirements. Defective : A defective is any unit of product which contains one or more defects. The two terms "defective" and "nonconforming (unit)" are interchangeably used in this thesis. Defects Concerned : The defects being inspected for while using the sampling plan. 100% Inspection : 100% inspection means the inspection of every unit of product for the defects concerned listed for an inspection station. The two terms screening and 100% inspection are used interchangeably in this thesis. Sampling Inspection : Sampling inspection means the inspection for the defects concerned where the units selected for inspection are selected by sampling. Continuous Sampling Inspection : Continuous sampling inspection is the examination or testing of units of product as they move past an inspection station. Only those units of product found by the inspector or screening screw to be nonconforming are corrected or replaced with conforming units. The rest of the production, uninspected units as 4

7 v/ell as units found to be conforming, is allowed to continue down the production line as conforming material. Sampling Frequency (f) : The sampling frequency, f, is the desired ratio between the number of units of product randomly selected and inspected at an inspection station and the number of units passing the inspection station during periods of sampling inspection. The procedure used in selecting the sample units should give each unit of product presented during periods of sampling inspection an equal chance of being selected and inspected. Process Quality : ANSI/ASQC Standard A2 (1987) defines process quality as a statistical measure for the quality of product from a given process. The most commonly used measure of process quality is the percentage or proportion of nonconforming units in the process. Continuous Sampling Plan (CSP) : ANSI/ASQC Standard A2 (1987) defines continuous sampling plan as a plan intended for application to a continuous flow of individual units of product that (1) involves acceptance or non-acceptance on a unit-by-unit basis, and (2) uses alternate periods of 100 percent inspection and sampling depending on the quality of the observed product. 5

8 Clearance Number : ANSI/ASQC Standard A2 (1987) defines clearance number as the number of successively inspected units of product that must be found acceptable during the 100% inspection sequence before action to change the amount of inspection can be taken. % Single-Level Continuous Sampling: ANSI/ASQC Standard A2 (1987) defines single-level continuous sampling as sampling inspection of consecutively produced units in which fixed sampling rate is alternated with 100% inspection depending on the quality of the observed product. Multi-Level Continuous Sampling : ANSI/ASQC Standard A2 (1987) defines multi-level continuous sampling as. sampling inspection of consecutively produced units in which two or more sampling rates are alternated with 100% inspection, or each other, depending on the quality of observed product. Operating Characteristic (OC) Curve : ANSI/ASQC Standard A2 (1987) defines the operating characteristic curve for continuous sampling plans as a curve showing the long-run percentage of product accepted during the sampling phase(s) as a function of the quality level of the process. Acceptable Quality Level (AQL): ANSI/ASQC Standard A2 (1987) defines acceptable quality level as the maximum percentage or proportion of nonconforming units (variant units) that 6

9 can be considered satisfactory as a process average. AQL is the primary index used for designing an acceptance sampling plan. When a consumer designates some specific value of AQL for a certain characteristic or group of characteristics, the supplier is informed that the consumer's acceptance sampling plan will accept the great majority of products submitted by the supplier during periods of sampling provided that the process average is no greater than the designated value of AQL. In this thesis, the AQL is taken as the quality level at which the probability of acceptance on sampling basis is Limiting Quality Level (LQL) : Limiting quality level is defined as the percentage or proportion of nonconforming units for which the consumer wishes the probability of acceptance to be restricted to a specified low value. LQL is used as an index for consumer protection for designing an acceptance sampling plan. When some specific value of LQL is designated for a certain characteristic or group of characteristics, the consumer is guaranteed that the continuous sampling plan will not accept great majority of products during periods of sampling if the process average is at or worse than LQL. In this thesis the LQL is 7

10 taken as the quality level at which the probability of accepting the product on sampling basis is only Indifference Quality Level (IQL) : Indifference quality level is defined as the quality level in the region containing quality levels between AQL and LQL and would represent a process containing a percentage or proportion of nonconforming units not low enough that it should be limited to having only a small risk of being called not acceptable on sampling basis, nor high enough that it should be limited to having only a small risk of being accepted on sampling basis. In this thesis the indifference quality level is taken as the quality level at which the probability of accepting the product on sampling basis is 0.5. Maximum Allowable Percent Defective (MAPD) : Maximum allowable percent defective is the quality level that corresponds to the point of inflection of the OC curve. It is the quality level at which the second order derivative of the OC function Pa(p) with respect to p is zero. MAPD is used as an index for acceptance sampling plans. When some specific value for a characteristic or group of characteristics is designated, the continuous sampling plan will have a tendency to accept product during periods of 8

11 sampling if the submitted quality is upto MAPD and if the submitted quality is beyond MAPD, the sampling plan will have a tendency to subject the product to screening. Average Outgoing Quality (AOQ) : ANSI/ASQC Standard A2 (1987) defines the average outgoing quality as the expected quality of outgoing product following the use of an acceptance sampling plan for a given value of incoming product quality. AOQ is usually expressed in terms of the percentage or proportion of nonconforming units in a product stream and is of practical value for continuous sampling plans. The computational formula for AOQ depends on whether or not the nonconforming units detected during sampling or 100% inspection are replaced by acceptable units. In this thesis, the AOQ is computed under the assumption that all the nonconforming units are corrected or replaced with conforming units. Average Outgoing Quality Limit (AOQL) : Average outgoing quality limit is the maximum AOQ over all possible levels of incoming quality. AOQL is used as an index for consumer protection while designing an acceptance sampling plan which is necessarily followed by a screening inspection in case of nonacceptance. 9

12 Sampling plans indexed by AOQL provide assurance that the long run average of outgoing quality, given screening and replacement of nonconforming units, will be no worse than the indexed or desired AOQL value. Producer's Risk (a) : ANSI/ASQC Standard A2 (1987) defines producer's risk as follows :- " For a given sampling plan, the probability of not accepting product the quality of which has a designated numerical value representing a level which it is generally desired to accept on sampling basis In this thesis, a is fixed at Consumer's Risk (fl) : ANSI/ASQC Standard A2 (1987) defines consumer's risk as follows :- " For a given sampling plan, the probability of acceptance of product the quality of which has a designated numerical value representing a level which it is seldom desired to accept on sampling basis ". In this thesis, B is fixed at Average Fraction inspected (AFI): MIL-STD-1235 C (1988) defines the average fraction inspected of a continuous sampling plan as the fraction of product that will be inspected over the long run if the process average is a particular value. 10

13 Lot-by-lot Sampling Vs. Continuous Sampling : A basic requirement for the application of any lot-by-lot sampling inspection plan is the formation of production lots or similar groupings of items. Some processes do not lend themselves to the formation of discrete lots. Units often flow along a continuous conveyor belt or similar conveyance. Inspection needs to be performed at discrete locations along this flow of individual units. Sampling procedures devised for handling such situations are known as continuous sampling plans. In continuous sampling, we have an acceptance criteria that is able to distinguish between product of different quality and does not depend on forming the product into separate lots. According to Wadsworth, Stephens and Godfrey (1986), there are two other ways in which continuous sampling differs from lot-by-lot sampling and they are explained below. In continuous sampling there is no fixed sample size. The sample is expressed as a fraction or percentage of the continuous flow of product. For example, the sample may be expressed as 10% of the product, denoting an f of 1/10. This can mean different things. Atleast three ways of applying the parameter f to the drawing of sample units have been identified [Derman, Johns and Lieberman (1959]. These 11

14 are systematic sampling, block or random sampling and probability sampling. The operating characteristic curve for a continuous sampling plan does not have the same meaning as for a lot-by-lot sampling plan. In lot-by-lot sampling inspection, when rectification procedures are used, the 'probability of acceptance' refers to the probability of accepting a single lot without having to inspect 100%. In continuous sampling there are no specific lots. Consequently, a different measure of evaluation of the performance of continuous sampling plans is required. The most common measure is "the percent of total production accepted on a sampling basis" as a function of the incoming percent nonconforming. This is denoted by Pa(p). Other measures are a) the average number of units inspected in a 100% screening sequence, b) the average number of units inspected during the periods of sampling, c) the average fraction of total produced units inspected in the long run, and d) the average outgoing quality. 12

15 SECTION 2 Designing Sampling Plans According to Case and Keats (1982), the sampling plan design methodologies are classified as 1) risk-based non- Bayesian, 2) risk-based Bayesian, 3) economically-based non- Bayesian and 4)economically based Bayesian approaches. The risk based non-bayesian approach is applied by vast majority of quality control practitioners due to their wider availability of tables and ease of application. In this thesis, the risk-based non-bayesian design methodology is alone considered. According to Peach (1947), the following are some of the major types of designing sampling plans based on the OC curves : 1) The plan is specified by requiring the OC curve to pass through (or nearly through) two fixed points. Tables of Cameron (1952) were based on this type of designing. 2) The plan is specified by fixing one point only through which the OC curve is required to pass, and setting up one or more conditions not explicitly in terms of the OC curves. Stephens (1981) and Govindaraju (1989) followed this type of designing. 3} The plan is specified by imposing upon the OC curve two 13

16 or nore independent conditions none of which explicitly involves the OC curves. Dodge and Romig (1959) AOQL table is an example of this type of designing. The second type of designing listed above is followed in this thesis. The classical work of Dodge (1943) on continuous sampling plan uses the well-known concept of AOQL which Dodge and Romig (1959) have introduced in their early v*ork on acceptance sampling for lots in According to Shahani (1979), the concepts of Dodge and Romig have certainly stood the test of time and these concepts are still widely used in practice and in theoretical work. AOQL seeks to provide a measure of a guaranteed level of quality that emerges as a result of the chosen sampling plan. According to Dodge (1943), continuous sampling plans are intended primarily for use in process inspection of parts or final inspection of finished articles within a manufacturing plant, where it is desired to have assurance that the percentage of nonconforming units in accepted product will be held down to some prescribed low figure. The object of such plans is to establish a limiting value of average outgoing quality which will not be exceeded no matter what quality is submitted for inspection. Thus AOQL forms a major criterion in the selection of continuous sampling 14

17 plans. In this thesis, AOQL is treated as the primary index while designing continuous sampling plans. All the continuous sampling plans discussed in this thesis involve alternating sequences of 100% inspection and sampling inspection. Planing manpower requirements for 100% inspection assumes importance because the screening crew is normally made up of production workers. According to Govindaraju (1989), using a continuous sampling plan with a given acceptable quality level (AQL) and producer's risk of 5% means that the producer is guaranteed that if the incoming quality is maintained at or better than the AQL, the percentage of production that will be accepted during the periods of sampling is atleast 95%. Similarly, a plan with a given limiting quality level (LQL) and consumer's risk of 10% assures the consumer that if the incoming quality is at or poorer than the LQL, only 10% or less of the production will be accepted in the long run during the periods of sampling. Thus a reference to AQL or LQL helps one to plan the manpower requirements for 100% inspection depending on the level of process quality maintained and the production per shift. While describing the philosophy of matching lot-by-lot sampling plans to continuous sampling plans, Storer (1956) advocated the use of AQL index and

18 Stephens (1981) advocated the use of LQL index for consumer protection. Hence both AQL and LQL may be used as secondary indices for designing continuous sampling plans. Single sampling plans are widely used for appraising the incoming product quality. When the formation of lots for inspection is impracticable or artificial, it becomes necessary to use alternate procedures such as continuous sampling plans of Dodge (1943). In such a situation it is logical to have a comparison between the continuous sampling plans and their matching single sampling plans. Wasserman (1990) presented ci strategy to match operating performance of CSP-1 plans with single sampling plans based upon the long run proportion accepted about two target operating levels AQL and LQL. He also observed that it is not possible tc match OC curves when this strategy is used. Kamaker (1950) advocated the use of indifference quality level and the relative slope of the OC curve at (IQL, 0.5) to compare acceptance sampling plans and was of the opinion that for any acceptance sampling plan its equivalent single sampling plan can be found by matching the IQL and the relative slope of the OC curve at the point of control. The indifference quality level can be used to obtain matching single sampling plans to enable comparison of continuous sampling plans with their matching single sampling plans. 16

19 Hence the IQL is used in this thesis as one of the secondary indices while designing continuous sampling plans. Bush, Leonard and Marchant (1953) considered the point of inflection as the most representative point of the OC curve. Mandelson (1962) introduced the quality level which corresponds to the inflection point on the OC curve. Mayer (1967) interpreted it as the maximum allowable percent defective and established that MAPD < IQL. Soundararajan (1975) used MAPD as an index for designing single sampling plans. The practical performance of a sampling plan is revealed by its OC curve. The shape of the OC curve will be satisfactory if it is concave at good quality levels and convex at poor quality levels. A desirable property of MAPD is that the OC curve is concave to the origin upto the quality level represented by MAPD and convex beyond MAPD. Govindaraju and Kuralmani (1992) exhibited this desirable property of MAPD and pointed out that the requirements for a satisfactory OC curve are i) MAPD > AQL and ii) Pa (MAPD) < Pa(AQL). In this thesis, MAPD is hence considered as a secondary index while providing the designing approach to continuous sampling plans. Dodge (1943) observed the following : a) For a moderate range of f values the factor i has a 17

20 stronger influence than f in determining the discrimination that the CSP-1 procedure affords between high and low levels of incoming percent defective. b) When the normal level of incoming percent defective is well below the AOQL, the AOQL value can be assured with less inspection by choosing f small and i large. But since, for a given AOQL value, the average amount of inspection approaches a minimum as f approaches 0, factors other than the minimum amount of inspection have a more important influence on the choice of the most advantageous combination of f and i values for a given set of circumstances. c) Protection against "spotty" quality, such as may arise from temporary irregularities in workmanship or materials, should receive special consideration in connection with the choice of f. The protection against spotty quality falls off very rapidly with f and the protection, considering runs of product of 1000 consecutive units each, becomes quite poor if f is less than 2%. In this thesis, the amount of inspection represented by AFI is not considered as a criterion to index continuous sampling plans, since AOQL is considered as the primary index and as observed by Dodge (1943), for a given AOQL 18

21 value factors other than the minimum amount of inspection have a more important influence on the choice of the parameters. While selecting sampling plans 'Spotty Quality' is not considered as a quality level for the purpose of indexing, since 'spotty quality' consideration receives importance in connection with the choice of f. The higher the value of f chosen, the greater will be the protection against spotty quality. In this thesis, tables are provided to enable selection of certain popularly used continuous sampling plans, when (AQL, AOQL) or (LQL, AOQL) or (IQL, AOQL) or (MAPD, AOQL) values are specified. The followii^procedure is used for designing continuous sampling plans : 1. Assume that the values for the primary index AOQL and the secondary index AQL or LQL or IQL or MAPD are given. In case of AQL and LQL, also assume that the associated risks a and 6 are specified. 2. Establish a relationship between the sampling frequency and the clearance number and the secondary index. In case of sampling plans having more than one sampling frequency (or one clearance number), the other sampling frequencies (or clearance numbers) are expressed as a function of the first sampling frequency (or the first clearance number) 19

22 3. Obtain sampling frequencies for all clearance numbers in the interval (2, IU), where IU is the value of the clearance number which corresponds to the sampling frequency 1/ For each parameter set (clearance number, sampling frequency), find the maximum of AOQ for submitted quality ranging from 0 to 1 in steps of Select the plan for which D is nonnegative and minimum where D = (AOQL specified - AOQL calculated). List of Symbols The following is the list of symbols which are frequently used in this thesis. p = The probability that an item produced by the process is nonconforming, q i = 1-p. = The clearance number, x = The clearance number less than i. f = The sampling frequency of single level continuous sampling plans, n c = 1/f. = The acceptance number of CSP-l(c) plan. = The sampling frequency of two level or three level tightened continuous sampling plans on level one. 20

23 f2 - The sampling frequency of two level or three level tightened continuous sampling plans on level two. f3 = The sampling frequency of three level tightened continuous sampling plans on level three. k = The number of sample units to be found conforming in order that the inspection will continue to be in the sampling mode. AOQL= The average outgoing quality limit specified for the purpose of selection of plans. p^ = The maximum value of average outgoing quality obtained for a specific plan. u = The average number of units inspected during the periods of 100% inspection. v = The average number of units passed during the periods of sampling inspection. F = The average fraction inspected. Pa(P)~ The probability of acceptance during the periods of sampling when the incoming product is of quality p. a = The producer's risk. 6 = The consumer's risk. A0Q(p)=The average outgoing quality when the submitted product is of quality p. Pq = The indifference quality level. Pi = The acceptable quality level.; 21

24 h0 = The relative slope of the OC curve at p0. Sj = The jth state of the process. p^j = The probability that the process transits from the state to the state Sj in one step. N = The lot size of single sampling plan. Nm = A limiting value on the lot size of single sampling plan which is used for comparison purposes. n0 = The sample size of single sampling plan. c0 = The acceptance number of single sampling plan. Fs = The average fraction inspected of single sampling plan. D(i,n)-AOQL specified - pl. 22

25 SECTION 3 Review of Dodge Type Continuous Sampling Plans Several well organized sampling plans and procedures have been published for use in the inspection of a product that is made available in individual lots or batches. But there are situations where it is neither practical nor convenient to group product articles in collective lots or batches for the purpose of inspection. This is particularly so when the product units are manufactured by a more or less continuous process on a conveyor or other straight line system with individual units flowing one after another in a progressive assembly. Dodge (1943) innovated the concept of continuous sampling inspection and introduced the first continuous sampling plan, originally referred to as the "random-order" plan and later designated as CSP-1 plan by Dodge and Torrey (1951). Other continuous sampling plans represent extensions and variations in the basic procedure of Dodge (1943). Dodge (1947) outlined several sampling plans for continuous production. Lot-by-lot sampling plans and continuous sampling plans can be used as inspection plans for continuous production. However continuous sampling plans are applicable to situations where there is continuous flow of consecutive articles or consecutive batches of articles 23

26 and these articles are submitted for inspection in the order of production. MIL-STD-1235C (1988) is the latest US Military Standard on continuous sampling plans. This Standard provides tables and procedures for applying five different types of continuous sampling plans by attributes. In this review we focus our attention on continuous sampling plans rather than sampling plans for continuous production. The usual conditions for application of continuous sampling plans are given below : 1. There is a continuous flow of units from the production process and units are offered for inspection one by one in the order of production. 2. The process is producing or is capable of producing, material whose process quality level is stable. 3. Ample space, equipment, and work force are provided at or near the site of inspection to permit rapid 100% inspection when required. 4. The inspection is relatively easy and quick (e.g. attribute inspection by visual observation or automatic inspection). 5. The inspection is nondestructive since the procedure incorporates 100% inspection. 6. The sampling procedures can apply to individual nonconformities, classes of nonconformities or 24

27 nonconforming units. When applied to classes of nonconformities it is possible for one such class to be undergoing a clearing interval for the nonconformity that caused rejection while other classes would still be under sampling. The first continuous sampling plan of Dodge (1943) has two procedures, namely procedure A and procedure B. Procedure A is applicable to a product of consecutive articles. Procedure B is applicable to a product of consecutive sublots or batches of article. Procedure A (a) At the outset, inspect 100% of the units consecutively as produced and continue such inspection until i units in succession are found clear of defects. (b) When i units in succession are found clear of defects, discontinue 100% inspection and inspect only a fraction f of the units selecting individual sample units one at a time from the flow of the product in such a manner as to assure an unbiased sample. (c) If a sample unit is found defective, revert immediately to a 100% inspection of succeeding units and continue until again i units in succession are found clear of defects as in paragraph (a). (d) Correct or replace all defective units found with good 25

28 Flow Diagran for the Operation of CSP-1 Plan. FISURE i.i

29 units. Procedure B of the random-order plan incorporates modifications of procedure for adaptation to a product consisting of flow consecutive sublots or batches of articles. In this case f is the fraction of each sublot inspected during sampling. When a sample unit is found defective, 100% inspection is started for the remainder of the sublet and is continued even into succeeding sublots if necessary until i inspected units in succession are found clear of defects. Both procedures, under CSP-1, follow the flow diagram of Figure 1.1. Necessary conventions are to treat the "order of production" of procedure A with "order of inspection" in procedure B, since strict order of production by articles is lost in sublotting. Additionally, in procedure B when it is necessary to find i inspected units in succession clear of nonconformities, the 100% inspection must be allowed to extend to the immediately succeeding lots if i nonconforming units in succession are not found in the current lot. Dodge's theoretical frame work is based on spacing between defective units when the individual units are arrayed in the order of production. If a manufacturing process is statistically controlled so that the probability 26

30 of producing a defective unit is constant, p, then defective units will have an order spacing of a random character which is expressible in terms of certain probability laws. A terminal defect sequence of i+1 successive units following the observance of a defect consists of a succession of i nondefective units followed by a defective unit and Dodge has shown that the probabilities of such sequences are the successive terms in the infinite power series. p+pq+pq where q =l-p. For evaluation of CSP-1 plans, Dodge proposed the following measures and derived formulas for them: (1) The average number of units, u, inspected during a screening phase is given by u == ( l-q1)/(pq1). (2) The average number of units, v, passed during a sampling phase is given by v = l/(fp). (3) The average fraction of total produced units, F, inspected in the long run is given by F = (u + fv)/(u+v). (4) The average outgoing quality is given by 27

31 AOQ(P) = p(l-f). (5) The average fraction of total production, Pa(p), accepted on a sampling basis is given by Pa(P) = v/(u+v). Dodge presented curves for finding i and f for a given value of AOQL and also a pt scale at the right of the curves which provides a guide concerning the protection afforded against spotty quality in a continuous run of product. The value of pt is the percent defective in a run of 1000 consecutive product units, for which the probability of acceptance by sample is 0.10 for a sample of f. It shows the percent defective which will result in a 90% chance of reverting to 100% inspection within a run of 1000 consecutive units. According to Derman, Johns and Lieberman (1959) the phrase "selecting individual sample units one at a time from the flow of product in such a manner as to assure an unbiased sample" implies atleast three interpretations of the sampling procedure while on partial inspection. These are as follows: (1) Look at every kth item. This type of sampling is denoted as systematic and has the practical disadvantage that 28

32 the particular item to be chosen is known in advance. (2) Sample every item with probability f. This type of sampling is denoted as probability sampling and has the disadvantage that the number of uninspected items is a random variable. (3) Sample only a fraction f of the units, choosing the item to be observed at random from a segment of size 1/f. This type of sampling is denoted as random sampling. Dodge showed that the AOQL of CSP-1 plan under constant p-model is given by AOQL = p*[l-f/(f+(l-f)(1-p*)1)] where p* is the solution to the equation (i+l)p-l = (1/f-l)(l-p)i+1. Lieberman (1953) presented an analysis of CSP-1 plan under the assumption that p is not a constant for each unit. He determined that the worst situation would be the one where only good units reached the inspector during phases of 100% inspection and only defective units reached the inspector during phases of sampling inspection. The outgoing quality reflected by this worst possible situation came to be called as the unrestricted average outgoing quality limit (UAOQL). When random sampling is used during sampling phase, 29

33 Lieberman showed that the UAOQL of CSP-1 plan is given by UAOQL = (1-f)/ (i+if). Derman, Johns and Lieberman (1959) proved that the UAOQL of CSP-1 plan when probability sampling is used is the same as given by Lieberman (1953). White (1965) introduced linear programming approach to compute the UAOQL and proved that the UAOQL of CSP-1 plan is the same as derived by Lieberman (1953) using probability theoretic considerations. Lieberman (1953) and White (1965) both derived the UAOQL of CSP-1 plan under the assumption that the defective materials found are replaced with nondefective materials. Endres (1967) gave a method for computing the values of UAOQL when defective material is removed and is not replaced with good material. This method is based on the solution of linear programming models. The UAOQL of CSP-1 plan under this nonreplacement assumption is given by UAOQL = (1-f)/ [l+(i-l)f]. Brugger (1967) described a simple method for computing UAOQL under the nonreplacement assumption; much of the fundamental justification for this method came from Endres (1967). Sackrowitz (1975) found the UAOQL of CSP-1 plan when probability sampling is used, and although the process need 30

34 not be in statistical control, it is assumed that incoming process quality cannot depend on the results of the inspection procedure. Sackrowitz showed that the AOQL also depends on the type of sampling done while on partial inspection and on the interaction of the production process with the inspection process. Sackrowitz's UAOQL under the assumption stated is given by UAOQL = [ (l/f)-l ] [l-(l-f)l*]/ (L* + i) where L* is the integer which maximizes [l-(l-f)l]/(l+i). He, by considering four CSP-1 plans indicated that Sackrov/itz's UAOQL is more realistic than the UAOQL of Lieberiaan (1953) and Derman, Johns and Lieberman (1959). Dodge used the algebraic methods to determine the performance measures of continuous sampling plans. Incorporation of Markov chain methods into the mathematics of continuous sampling plans was brought about by Lieberman and Solomon (1955). Roberts (1965) defined the states for CSP-l. He further solved the resulting Markov chain of CSP-1 for equilibrium probabilities of the states and derived AOQ from these probabilities. For selection of CSP-1 plans, Dodge (1943) provided 31

35 curves Cor determining values of f and i for a given value of AOQL along with pt, a measure of spotty quality. According to Dodge (1943), protection against spotty quality should receive special consideration in connection with the choice of f. Murphy (1959a) introduced a technique for selecting CSP-1 plan. He proposed a second restriction, beside AOQL, namely a choice of the total fraction of units inspected (in both 100% inspection and sampling inspection) at a value of the process average chosen by the producer, called the producer's nominal quality level (PNQL) (usually the AQL). For a given combination of AOQL, PNQL and the total fraction of units inspected at PNQL, there is only one CSP-1 plan satisfying this combination. On the basis of a comparative study, Murphy (1959a)chose a kind of stopping rule "Rule (r)h to accompany such a plan and presented a table and a formula to assist in selecting the parameter r. He presented four stopping rules and a method for comparing these rules. He finally concluded that it is reasonable and convenient to use Rule (r) as an aid in controlling quality. Murphy (1959a) presented formulas for determining f and i. Murphy (1959b) presented charts for graphical solution to the same problem. The chart given by Murphy along with the chart of Dodge (1943) is used for the selection of the parameters f and i. White (1961) presented a new graph for CSP-1 plans using f values given in H-107 (1959) and a table 32

36 of "CSP-l Exact AOQL1s (in percents)" for selected f and i values. Resnikoff (1960) proposed a method of selecting CSP-1 plan which could be used under conditions almost complementary to those assumed by Murphy (1959a). Murphy considered the selection of CSP-1 plan by specifying the PNQL which is a fraction of the AOQL and the AFI. Resnikoff (1960) considered the selection of CSP-1 plan by specifying a value of the process average which exceeds the AOQL and by selecting the value for f which minimizes the AFI at the specified value of the process average. According to Hillier (1964), both of these methods ignore the crucial question of the amount of protection provided when the process goes out cf control. Stephens (1931) developed consumer-oriented CSP- 1 plans. For developing CSP-1 plans for consumer protection based on the LQL with 0.10 consumer risk, Stephens derived relationships between the LQL and the parameters of the plan. Utilization of the plans is aided by the presentation of a nomograph and a table of f and i values indexed by LQL. Stephens also included a table of the corresponding AOQL's. Govindaraju (1989) gave procedures and a table for the selection of CSP-1 plans for given (AQL, AOQL) and (LQL, AOQL). While Stephens procedure for the selection of CSP-1 plans assumes that sampling frequency is known, Govindaraju procedure makes no assumption about the parameters of the 33

37 plan. In Chapter II of this thesis, a new procedure and four tables are given for the selection of CSP-1 plans for a given set of conditions (AQL, AOQL), (LQL, AOQL), (IQL, AOQL) and (MAPD, AOQL). While Govindaraju tables are used for given (AQL, AOQL) and (LQL, AOQL), the tables presented in Chapter II of this thesis are used for given (AQL, AOQL/AQL), (LQL, LQL/AOQL), (IQL, IQL/AOQL) and (MAPD, MAPD/AOQL). Wasserman (1990) devised a model which can be used to relate the plan performance between single sample lot acceptance procedure and CSP-1 plan. He showed that it is not generally possible to match up performance based upon OC curve expression for the two plans. Instead the plans are matched by equating expression for the long run proportion of product accepted about two target operating levels AQL and LQL under both procedures. This has been shown equivalent to matching up properties on an outgoing quality basis. He presented three tables giving matching CSP-1 plans for the acceptance number c0=0, 1,and 2. One disadvantage of this method is that more tables are required when we consider matching plans with varying cq values. Hamaker (1950) used IQL and the relative slope of the OC curve at IQL for comparing various types of acceptance sampling plans. Using Hamaker's approach matching single 34

38 Start Flow Diagram for the Operation of CSP-2 Plan. FIGURE 1.2

39 Flew!>iuriirt for h* OptratUn of CSP-3 Plan. FI6!M 1,3

40 sampling plans are found and compared in Chapter II of this thesis. An extension of CSP-1 plan introducing an acceptance number c is considered and the resulting plan, designated as CSP-1(c), is described in Chapter VIII. Formulation of the CSP-l(c) procedure, as a Markov chain, derivation of performance measures and designing CSP-1(c) plans for c=l and c 2 are also given in Chapter VIII. Dodge and Torrey (1951) proposed two additional continuous sampling plans, CSP-2 and CSP-3, which are modifications of CSP-1 plan. CSP-1 plan requires reversion to 100% inspection whenever a defective unit is found during sampling phase. "Plans CSP-2 and CSP-3 remove this feature and instead call for such reversion to 100% inspection only when one defect falls too closely on the heels of another during sampling inspection that is when their separation is smaller than a prescribed minimum spacing, k". The additional plans permit one or more defective units during the periods of sampling and the number of defective units permitted depends on the quality of the product. These plans carry a higher risk of accepting a short run of product of abnormally poor quality. Flow diagrams for CSP-2 and CSP-3 procedures are given in Figures 1.2 and 1.3 respectively. In CSP-2 plan, once sampling inspection is 35

41 started, 100% inspection is not invoked when each defective unit is found but is invoked only if a second defective unit occurs in the next k or less sample units; otherwise sampling inspection is continued. CSP-3 plan introduces a simple and effective refinement of CSP-2 plan. It calls for the inspection of four additional consecutive units whenever an allowed defective unit is found during sampling and the immediate return to 100% inspection if one of the four is found defective. According to Dodge (1970) this is a natural added procedure, popular with inspection engineers and inspectors, that increases the efficiency of the plan in the detection of one general type of quality deterioration. CSP-3 plan provides extra protection against spotty quality, that is, surges of highly defective product. Dodge and Torrey (1951) derived performance measures u, v, F, AOQ(p), and Pa(p) for CSP-2 and CSP-3 plans. Sheesley (1975) presented a computer program to evaluate F, AOQ and Pa(p) for CSP-1, CSP-2 and CSP-3 plans. Stephens (1979) used the Markov chain model of Roberts (1965) to evaluate the characteristics of CSP-1, CSP-2 and CSP-3 plans. Stephens used the flow diagrams of Roberts showing the transitions from the states for CSP-1, CSP-2 and CSP-3 plans, solved the resulting Markov chains ofcsp-2 and CSP-3 for the 36

42 equilibrium probabilities of the states and then combined these results with Roberts results for CSP-1 plan to derive a set. oi operaring characteristics namely u, v, F, AOQ(p) and Pa(p) for CSP-1, CSP-2 and CSP-3 plans under probability sampling. 3rugger (1967) described a simple method for computing UAOQL and found UAOQL's for CSP-1 and CSP-2 plans under the assumption that the defective units found are not rep laced. Dodge and Torrey (1951) presented curves for selecting the values of the parameters f and i for a given value of AOQL. These curves are used for the selection of CSP-2 plans when k =i. In order to measure the protection against the spotty quality, (in percent) values are also shown on the right side of the curves as pt scale. "In applying CSP-3 plans, the curves of CSP-2 plans may be used as an approximation for determining values of f and i for a given value of AOQL. The k =i values are equal to those for CSP-2 when AGQL is less than 2% and are less than those for CSP-2 by more than 2 units when AOQL = 10%". Thus the effect of using Dodge's CSP-2 curves for CSP-3 plans for larger values of AOQL is to give actual AOQL values slightly smaller than the charted values. Abraham (1971) provided graphs for finding AOQL and i values for CSP-1 and CSP-2 plans when the sampling rate is 37

43 given but when the defective units removed are not replaced. CSP-l and CSP-2 plans are included in the latest US Standard on continuous sampling plans MIL-STD-1235C (1988). For the purpose of selection of CSP-l and CSP-2 plans, the Standard provides tables and graphs. But CSP-3 plan is not included in the Standard. Procedures and tables are given for finding the values of the parameters of CSP-2 and CSP-3 plans respectively in Chapter III and Chapter IV for given values of (AQL, AOQL/AQL), (LQL, LQL/AOQL), (IQL, IQL/AOQL) and (MAPD, MAPD/AOQL). A simple modification of CSP-2 plan is proposed and its operating procedure, derivation of performance measures under probability sampling and presentation of procedures and tables for the selection of the modified CSP-2 plan are given in Chapter III of this thesis. The modified CSP-2 plan is designated as Tightened CSP-2 plan in this thesis. In a similar way, a Tightened CSP-3 plan is described and designing aspects are considered in Chapter IV. Lieberman and Solomon (1955) considered an extension of CSP-l plan which (a) allows for smoother transition between sampling inspection and 100% inspection, (b) requires 100% inspection only when the quality submitted is quite inferior, and (c) allows for a minimum amount of inspection when quality 38

44 Flew Diagraw for th* Operation of HLP(k=2) Plan. FI60SE 1.4

45 is definitely good. This aim is accomplished by the introduction of multi-level sampling plan which specifically allows for any number of sampling levels subject to the provision that transitions can only occur between adjacent levels. Each multi-level inspection plan (MLP) is initiated with 100% inspection. Then k sampling levels are defined with one article to be randomly selected in every group of l/f^ articles while on the kth level. Whenever i consecutive nondefective articles are inspected, the next level in the plan is used. If a defective article is found at any sampling level, say rth level, the (r-l)st level will be used for the next article to be inspected. Thus the inspection can proceed back and forth, a level at a time, from level 0 (100% inspection) to level k according to the inspection rules. A flow diagram showing the operation of MLP (k=2) plan is given in Figure 1.4. Lieberman and Solomon (1955) used a Markov chain model (random walk model with reflecting barriers) and derived the AOQ function for MLP. They have shown that when quality is exceptionally good, the infinite sampling level plan guarantees local stability and has minimum average fraction inspected. If quality is poor, then CSP-1 plan is to be preferred. They have also shown that, for a single level plan there will always exist a 39

46 Start Flsw iagr*«for tho Operation of HLP-I <ks2) Plan. FI8URE 1,5

47 plan which will yield local stability. Derman, Littauer and Solomon (1957) presented three generalizations to the MLP of Lieberman and Solomon. These plans were labelled as tightened plans because the generalizations were accomplished by altering the manner in which transitions between sampling levels could occur and by making it more difficult to reach a very low sampling frequency. The plans were designated as MLP-rxl, MLP-T, and MLP-rxs. In MLP, when a defective article is found at the kth sampling level the transition from the sampling rate 1/f^ to the rate l/f^-1 takes place. But in MLP-rxl, the transition is from the k-th sampling level to the level (kr) or to 100% inspection if (k-r)<0. MLP-T plan requires that 100% inspection be instituted whenever a defective is found. Both MLP-rxl and MLP-T plans permit a decrease in sampling to level (k+1) whenever i consecutive nondefective units are found by sampling on level k. The MLP-rxs plan is the same as MLP except when i consecutive nondefective units are found on level k. Then a reduction in sampling to level (k+s) may occur. The MLP-T plan is the tightest of the three plans. A flow diagram showing the operation of MLP-T (k=2) plan is given in Figure 1.5. A Markov chain model was used to find the AOQ functions for these plans. A two-level Tightened plan is considered in Chapter VI of this thesis. This plan is designated as MLP-T-2. While MLP-T with two 40

48 Flow Diagraw for th» Operation of C8P-T Plan. FIGURE 1.6

49 sampling Levels (k=2) has sampling frequencies f and f, the MLP-T-2 plan has ft and f2 Thus the parameters of MLP-T-2 plan are i,f3 and f2. The operating procedure and derivations of performance measures of MLP-T-2 plan under probability sampling (following Stephens approach) are given in Chapter VI. Tables are provided for the selection of MLP- T-2 plans when sampling frequencies are f-^f and f2=f/2. A three-level Tightened plan, which is an extension of MLP-T-2 plan, is considered in Chapter VII of this thesis and is designated as MLP-T-3. Just like MLP-T-2 plan, the parameters of MLP-T-3 are i, f3, f2 and f3 (the sampling rates at three levels). Fordice (1972) proposed a tightened three level continuous sampling plan, CSP-T, which is included in MIL-STD-1235C (1988). Its operating procedure is given in figure 1.6. Brugge r (197 2b) presented a simplification of the Markov chain approach to continuous sampling plan formulation. Using this method, Fordice derived AOQ and AFI functions of CSP-T plan. He also presented AFI and AOQ curves for CSP-T and CSP-1 plans. CSP-T plan is a special case of MLP-T-3 plan when f3 = f, f2 ~ f/2 and f3 = f/4. Using Stephens approach, derivations of performance measures for MLP-T-3 plan under probability sampling are also given in Chapter VII. These measures have been derived without assuming any relationship between 41

50 sampling rates except fx > f2 > f?> and hence enhance their applicability at varying sampling rates. For the selection of plans, tables are also provided when sampling frequencies are f1-f,f2=f/2 and f3=f/4. Sackrowitz (1972) considered two alternative multi-level continuous sampling plans for which rules concerning partial inspection depend, in part, on the length of time it takes to decide that the process quality is good enough so that 100% inspection may be suspended. These plans were offered by Sackrowitz as alternative plans to MLP and MLP-T plans. The plans of Sackrowitz, P* and P**, are described below: Description of p* (A) At the outset use fq inspection (100% inspection). 1) If the first i consecutive units are found clear of defects discontinue f0 inspection and begin f2 inspection. 2) Otherwise continue fq inspection until any run of i successive units are found clear of defects and then begin inspection. (B) If on fj inspection (sampling inspection) (j "* 1,2, *, in J continue until a defective is found. When this occurs,revert immediately to fq inspection and then 1) if the first i consecutive units are found clear of defects discontinue fq inspection and begin fj* 42