630 BELL SYSTEM TECHNICAL JOURNAL 



piece must be considered either as defective or non-defective. Hence 

 minimum values of z (^min.) will be found for many pairs {M, k) for 

 the same value of c. From this it is evident that on an M, k plane 

 there exist zones in which the acceptance numbers are identical. To 

 find the boundary lines of these zones it was noted that for certain 

 pairs {M, k) two pairs of {c, a) exist, giving the same minimum value 

 for z. These values of c were found to differ by 1 in all such cases. 

 Designating in general two such adjacent acceptance numbers as c 

 and c + 1 and corresponding values of a which satisfy the Consumer's 

 Risk as ac and ac+i, we may obtain these boundary curves from the 

 equation, 



' m=o ml m=o ml 



In using the above equation to determine these boundary curves 

 for Fig. 2, the following steps were taken: 



(1) Assume values for c and c + 1. 



(2) Determine ac and flc+i for a given value of P assuming N to 



be infinite. 



(3) For any given value for k, solve the linear equation in M 



obtained by substituting the assumed values in the above 

 equation. 



(4) Using the value of M thus found, determine the exact values 



of ac and flc+i from equation (2') for F = .10 (Fig. 3). 



(5) Using the same value of k, again solve the linear equation in M 



substituting the values of ac and ac+i obtained from step (4). 



(6) If the values ac and Oc+i obtained in step (4) satisfy the value of 



M thus found, the values of M and k define a point on the 

 boundary curve between two adjacent acceptance numbers. 

 If these values of a do not satisfy the value of M thus de- 

 termined, steps (4) and (5) may be repeated until the limiting 

 conditions are satisfied. 



Basis of Fig. 3 Giving n for Any c 



For given values of the Consumer's Risk F and acceptance number 

 c, the sample size n may be obtained from equation (2') since a = ptfi. 

 For the case P = .10 values of a are presented in Fig. 3 for selected 

 ranges of M and c. 



Basis of Fig. 4 Giving the Minimum Average Number of 

 Pieces Inspected Per Lot 



The curves in Fig. 4 represent specific values of Zmin. on an M, k 

 plane for P = .10. Each curve was obtained by substituting given 



