THE EFFICIENCY OF SEQUENTIAL SAMPLING FOR ATTRIBUTES

Transcription

1 R 229 Philips Res. Rep. 8, , 1953 THE EFFCENCY OF SEQUENTAL SAMPLNG FOR ATTRBUTES PART 11. PRACTCAL APPLCATONS by H. C. HAl\AKER : Summary The practical consequences of the theory developed in part are considered. t is shown that the simplified equations valid under Poisson conditions can be used to construct a slide rule from which the operating characteristic can he read off at once when ho and Po are given. Furthermore graphs are constructed which permit an easy and straightforward transition from the parameters Po and ho to the more. common parameters such as AQL, AOQL, etc. Résumé Les consëquences pratiques de la thëorie dëveloppëe dans la partie sont discutées. On trouve que les équations simplifiëesvalables dans les conditions de Poisson penvent être utilisées pour construire une règle à calcul dont les caractëristiques de fonctionnement peuvent être lues Ïmmédiatement ëtant donnés les pnramètres ho et Po. De plus des abaques sont ëtablies permettant une transition facile et directe des paramêtres Po et ho nux paramètres plus courants tels que AQL, AOQL, etc. Zusammenfassung Es werden die praktischen Folgerungen aus der in Teil entwickelten Theorie betrachtet. Hierbei wird gezeigt, dab die unter Poissonschen Bedingungen geltenden vereinfachten Gleichungen zur Bildung einer gleitenden Regel verwendet werden können, von der die Arbeitskennlinie unmittelbar abgelesen werden kann, wenn ho und Po gegeben sind. Weiter werden Diagramme konstrniert, welche einen leichten und direkten Übergang von den Parametern Po und ho zu den mehr gehräuchlichen Parametern wie AQL, AOQL usw. gestatten. 6. Practical conclusions From the theoretical arguments developed in part 1 1 ), we may draw the following conclusions: (A) n principle Wald's probability-ratio sequential plans constitute a three-parametric set, with decision lines in the random-walk which may be either symmetric or asymmetrie with respect to the origin. The degree of asymmetry occurring in practical applications is, however, as a rule very small. (B) A detailed theoretical investigation reveals that a strongly asyd:- metric position of the decision lines leads to sequential sampling plans that are decidedly inefficient, at least in the region of practical interest; whereas

2 / 428 H. C. HAMAKER plans with only slightly asymmetrie decision lines are practically equivalent to plans with symmetric decision lines, if these are properly adjusted. (C) Hence for practical purposes we can restrict ourselves to the use of sequential plans with symmetric decision lines. Doing so the number of parameters reduces from three to two, and the equations specifying the decision lines, the operating characteristic, and the average sample number are considerably simplified. Adopting the point of control Po and the relative slope ho, defined by (1) and (2), as the fundamental parameters, and assuming Poisson conditions these equations become: for the decision lines (22, 23) for the operating characteristic 1 P=-- 1+ e- h '1 and or p y -==--, Po e 1-1' hoy = n (_! ), l-p (24a) (24b) where y is an auxiliary parameter; for the average sample number and -2P n(p) = ho P-Po (-po)' (25) f Poisson conditions do not hold eqs (11) to (H) should he used. The gain with respect to the formulae for sequential sampling plans as usually given is obvious. From eqs (24a) and (24b) we may tabulate hoy as a function of P, and plpo as a function of Y; by means of such tables the operating characteristic can be very rapidly computed once Po and ho have been specified. We may even construct a slide rule from which the operating characteristic can be read off at once. For by (24b) the ratio pipo and the parameter y, and consequently also n y, are coupled once and for all, independently of the parameter ho; while from (24a) we find n ho + lny = nn (_! ). -P Hence if we plot y on a logarithmic scale and construct corresponding scales for Pand plpo' as in fig. 6, a change in the parameter ho will correspond to a,shift of the y and plpo' scales with respect to the P scale. Since P = 0 5 for y = 0 this point cannot be included. t turns out that (26)

3 EFFCENCY OF SEQUENTAL SAMPLNG 429 Fig. 6. Principle of the construction of a slide rule for the operating characteristic; n y is plotted on a linear scale, and a plpo scale is fixedto this n y scaleby using eq.(24b). The P scale has been drawn for ho= 1 by means of eq.(26). A change in ho will produce a shift in the P scale with respect to the n y and plpo scales. the two branches of the operating characteristic for P>0 5 and P<0 5 must he separated, the final construction being as shown in fig. 7. f we use Po and ha as the two parameters for specifying a sampling plan, the operating characteristics for all plans, whether single, double, or sequential, are found to be closely coincident when they possess the same values of these parameters. Hence the slide rule of fig. 7 constitutes a very convenient implement for constructing operating characteristics: it is sufficiently accurate for practical purposes, and can be used as a substitute for more elaborate tables.? (}6 0 7 o a Pin %; "i' 10',, j i i.i,', iis Fig. 7. Slide rule for constructing the operating characteristic. j i i r i i is Other parameters (A) Acceptable quality level (AQL) and lot tolerance (LTPD) The relative slope ha is not a convenient parameter for practical use, but it is fortunately easy to link up ha with the parameters which are commonly used. Denoting by Pos the acceptable quality level for which P = 0,95, and by Y95 the corresponding value of y, we have from (24a) Yos = 2'944/ho' t follows from (24b) that the ratio Po/Pos is a function of ho alone. The same holds evidently for the ratio PlO/PO where PlO is the lot tolerance fraction defective, and consequently also for the ratio PlO/POS'

4 430 H. C.HAMAKER With the aid of fig. 8, where Po/Pos and PlO/POS have been plotted as functions of ho, the transition from one set of parameters to another is immediately achieved. Suppose we require AQL = Pos = 1 5% and LTPD = PlO = 5%; then PlO/POS = 3,33, and starting with this value from the ordinate axis in fig. 8 and following the arrows, we find ho = 2 2 and Po/Pos = 2,15; hence Po = 3 22%. Values of Po/Pos and PlO/POS as functions of ho are given in table r in the appendix. Pip' t - ~..Ko... ~ \ p 95 -AQL p/()=ltpd 1\ PA =AOQL ~ P9i\ 1\ 1\, > '\: ==PlYR A.. -c =~ '" i"'" rt-... i'... ~ r-, ~ i"'" -...;; ~ ~ l- -- ~, '---. r- J,, HCo -..._ '"F==::::, ", :', 'ij" lj"3o ' :ij 'ilj'so ~o Fig. 8. Graphs relating the parameters Po and ha with any set of two out of the three parameters AQL, AOQL, and LTPD. (B) Average outgoing quality limit (AOQL) and lot tolerance (LTPD) The ~verage outgoing quality limit, PA = AOQL, is given by the maximum value of the product pp when ho is constant. By multiplying equations (24a) and (24b) 'Y"esee that for given ho the product pp is a function of y alone and will reach its maximum for a specific value y = YA' Hence it follows that the ratio PO/PA is again a function of ho alone, and so will he the ratio PlO/PA' Values of these functions obtained by numeri al computation have been entered in columns 5 and 6 oftable r (appendix). The ratio PlO/PA has been plotted as a third curve in fig.8 which permits

5 EFFCENCY OF SEQUENTAL SAMPLNG 431 the derivation of Po and ho from the AOQL and LTPD in exactly the same way as from AQL and LTPD. The procedure needs no further explanation. (C) Acceptable quality level (AQL) and average outgoing quality limit (AOQL) The transition from AQL and AOQL to Po and.ho could be achieved in exactly the same manner. Actually, however, these two parameters correspond to two points on the operating characteristic lying close together in the upper part of the curve. Consequently if we use this set of parameters, small variations in the value of one of them will correspond to eonsiderable variation in the operating characteristic and in the corresponding sampling plan. Since, moreover, the choice of the value of the parameters is always to some extent arbitrary, this is not a very suitable set of parameters for practical pul'p0ses. (D) The average sample number The average number of observations is another important feature in practice. For sequential sampling this number depends on the quality of the lot sampled and cannot therefore he represented by a singlefigure. But in actual applications most of the lots submitted will as a rule be accepted; and it is reasonable to assume the average sample number n 05 at the AQL = PU5 as a suitable index for the average amount of inspection labour involved *). From eq. (25) we then have ' nosp05 = ho 1 " -PO;P05 which by our previous arguments is again a function of ho alone. Likewise the products nuspa' nuspo. and n U5 PO' w:ill be functions of ho; values of these functions are given in table of the appendix and they have been plotted in fig. 9. With the aid of this graph we may now find Po and ho when nos and one of the parameters Pos' Po' PlO and PA are prescribed. For example, if we desire nos = 100' and AOQL = PA = 2%, we have and followingthe arrows in fig. 9 we obtain and consequently Po = 3 5%. ho = 2,15; n U5 Po = 3,50, *) This choise is of course somewhat arbitrary. Other values, for example ndd and POg, might also he adopted and can be treated in a similar way.

6 432 H. C. HAMAKER Thus wc see that, although ho and Po may not he the most suitable parameters for immediate use, they can he linked up with the practical parameters in an extremely simple and straightforward manner. 3 ft. er / 1/ / / 7 V 7 V 1/VA l7 [la ~ " v / / 17 P1 7/ V ~ ' ~ 's!:f!j /- -- V / / ry / V/! / 0-5 (J.4 O / V / / i ~p~ P95=AQL l- f- L.1' 1 p/o=ltpd l- f- V '\)<;;4 1/ PA=AOQL. l- f-! n9s=av. ero ge ~o:"pler- / ' size at. O 1 1ltJ5f95, '-- ~GO! J t, 5 '' "ij" "i5"~o '30' 40'50 0, CJ ho Fig. 9. Graph relating the two parameters Po and ho with the average sample number,. nos, at the acceptable quality level, and anyone of the three parameters AQL, AOQL, and LTPD. Perhaps it will do no harm if we recall that the arguments advanced in this section only hold true under Poisson conditions. To conclude wish to e~press. my gratitude to R. van Strik, A. M. van Beek, and F. J. van Dun who assisted in the 'computations and in constructing the graphs. Eindhoven, May 1953 REFERENCE 1) H. C. Hamaker, The efficiencyof sequential sampling for attributes; Part, Theory, Philips Res. Rep. 8, 35-46, 1953.

7 EFFCENCY OF SEQUENTAL SAMPLNG 433 APPENDX TABLE Numerical data relating the parameters Po and ho to the AQL, AOQL, LTPD, and the average sample number at the AQL hl) Po/Pos PlO/PO PlO/POS PO/PA PlO/PA _ _ ho nos Pos n9spa n05po nos PlO ] , ' J. 14_ , }4, Pos = AQL, PlO = LTPD, PA = AOQL, nos = average sample number for lot quality Pw