Golden ratio

23cb7c47-1a8d-4890-aac7-435cf13499ba 39 4 5 decimal

&[DATE] &[TIME] - &[FILENAME]

data char size clr plotG k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 31 line for data r3 r4 r5 augment 21 line :

Hypergeom

x Γ x Gamma :

N n p x hypergeom N 1 + Γ n 1 + Γ * N n + p - 1 + Γ * p 1 + Γ * x 1 + Γ x - n + 1 + Γ * N n + 1 + Γ * x N + p - 1 + Γ * x - p + 1 + Γ * / :

N 80 :

n 10 :

p 5 :

x hypergeom N n p x hypergeom :

Continuous PDF Hypergeometric

0 x ≤ x hypergeom outside lower bound if p 1 p 0.25 - 2 2 mat 2 1 sys

p hypergeom 5.7339 106 -^*

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1. Lot Acceptance Probability LAP =========================

Poisson parameter c 2 :

Poisson probability p n PA n p * - exp n p * (i^i! / i 0 c sum * :

1. Lot Acceptance Probability

From a lot of size N, many times larger than the sample size and where

the number of defectives in a sample of size n is approximately Poisson

distributed with mean n*p . The acceptance probability PA(p) is the sum

of the probabilities of finding each possible number of defectives between

0 and c. The lot is accepted for shipment if c or fewer defective items are

found. If more than c defective items are found, the entire lot is inspected,

and all defective items are replaced before the lot is shipped.

p is the actual fraction of defectives in the output from the manufacturing

process. This fraction p is unknown, it runs from 0 ... 1

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criterium of acceptance cr 0.2 :

sample size ... experiment n 15 :

Pacc cr n PA :

Pacc 0.42319

PA ε 1012 -^: p ε 1 0.001 range : i 1 p rows range probaccept i el p i el n PA eval : for p probaccept augment 41 line :

Poisson probability of acceptance

PA cr Pacc + 15 red 1 5 mat 2 1 sys

Pacc 0.42319

2. Average Outgoing Quality AOQ =========================

n 10 :

c 2 :

p PAp n p * - exp n p * (i^i! / i 0 c sum * :

PAp ε 1012 -^: p ε 1 0.001 range : i 1 p rows range probworst i el p i el PAp p i el * eval : for p probworst augment 41 line :

Quality ε 1012 -^: p ε 1 0.001 range : i 1 p rows range quality i el 1 p i el PAp p i el * - eval : for p quality augment 41 line :

worst-case probability accept.

PAp

2. Average Outgoing Quality

The outgoing product consists of lots that were accepted, which contain a

fraction p of defectives and lots that were rejected, which after replacement

have no defectives. Thus, the average fraction of defectives in the shipped

product is PA(p)*p. This quantity which is again a fraction of p, is called

average outgoing Quality AOQ. The maximum of the AOQ curve is called

the AOQ limit, and represents the worst-case quality of the outgoing

product.

If p is very small, most lots are accepted and the quality in all lots is high.

If p is close to 1, most lots will be rejected and all defectives in these lots

are replaced, so again the the outgoing product will have few defectives.

Thus, the AOQ limit typically occurs at some intermediate value of p.

The average outgoing quality represents the average fraction of defectives

in the shipped product. The AOQ in the suite below.

The AOQ limit is given by the maximum of this curve.

]]> Golden ratio

The hyperfast Golden-Ratio search [min/max]

x f x PAp x * : a b G 32 / b * 12 / a * - 12 / 5 sqrt * a b - (* + : 21 line

Code adapted from Valery Ochov Mathcad 11 f a b test TOL Extrema y1 y2 ≥ max in range ≡ y1 y2 ≤ min in range ≡ x1 a b G eval : y1 x1 f eval : x2 b a G eval : y2 x2 f eval : 41 line iter 0 : iter 1000 < a b - abs TOL ≤ break test b x2 : x2 x1 : y2 y1 : x1 a b G eval : y1 x1 f eval : 51 line a x1 : x1 x2 : y1 y2 : x2 b a G eval : y2 x2 f eval : 51 line if iter iter 1 + : 21 line if while a b + 2 / iter 21 mat 61 line :

1. All modules/functions in "Symbolic" optimiz. 2. Once in target: shrink the range 3. respect the search y1≥y2≡"max in range" y1≤y2≡"min in range"

Ex f 0.1 0.25 y1 y2 ≥ 1012 -^Extrema 1 el :

Ex Ex : 0.227

Ex Ex f 12 mat 0.227 0.1371 12 mat

worst-case probability of acceptance

PAp Ex Ex f . 15 red 1 5 mat 2 1 sys

Quality of production vs defect rate

Quality 1 Ex 1 Ex f - . 15 red 1 5 mat 3 1 sys

3. Average Total Inspection ATI ============================

ATI => Average Total Inspection p n N ATI p PAp n * 1 p PAp - (N * + :

3. Average Total Inspection

Another quantity of interest is the average number of items

in each lot that are inspected. This average total inspection

is one measure of the cost of the sampling procedure.

A sample of size n is drawn from every lot, but if the lot is

rejected, the rest of the lot must also be inspected. Thus,

the average total inspection ranges between n and N and takes

on these extreme values when p is 0 or 1.

Assume that for a lot of size N, n components are sampled

in those lots that pass and N components must be inspected

for those lots that fail ... the average total inspection curve is

calculated below ... suite of this project.

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Define lot size N 40 :

Define sample size n 10 :

plot scale factor s 0.175 :

u f s u n N ATI u diff (* :

x c 10 x ≤ (x 1 ≤ (& cases :

Golden ratio

The hyperfast Golden-Ratio search [min/max]

x f x n N ATI x diff : a b G 32 / b * 12 / a * - 12 / 5 sqrt * a b - (* + : 21 line

Code adapted from Valery Ochov Mathcad 11 f a b test TOL Extrema y1 y2 ≥ max in range ≡ y1 y2 ≤ min in range ≡ x1 a b G eval : y1 x1 f eval : x2 b a G eval : y2 x2 f eval : 41 line iter 0 : iter 1000 < a b - abs TOL ≤ break test b x2 : x2 x1 : y2 y1 : x1 a b G eval : y1 x1 f eval : 51 line a x1 : x1 x2 : y1 y2 : x2 b a G eval : y2 x2 f eval : 51 line if iter iter 1 + : 21 line if while a b + 2 / iter 21 mat 61 line :

1. All modules/functions in "Symbolic" optimiz. 2. Once in target: shrink the range 3. respect the search y1≥y2≡"max in range" y1≤y2≡"min in range"

Ex f 0.125 0.25 y1 y2 ≥ 1012 -^Extrema 1 el :

Ex Ex : 0.2

cost Ex n N ATI : 19.7

Cost of inspection for 'n' samples from 'N' lot size inspected and re-inspected ... Cost of inspection increases with samples size 'n' increases, ATI tells by how much.

s x f * n + 0 f - ( x c * n x n N ATI x c * Ex cost o 9 red 1 5 mat Ex 50 Ex 25 - 2 2 mat 5 1 sys

s x f * n + 0 f - (x c * n x n N ATI x c * Ex cost o 9 red 15 mat Ex 50 Ex 25 - 22 mat 51 sys

For firms of mass producion of constant n/N AIT is a good indication of miscellaneous parameters governing production ... [robotics, employees incentive, raw products, working conditions, sickness ...] Day by day AIT concerns all in that firm and its competitivity. Decreasing 'n' is surely NOT an alternative !