Intrepreting Consumers and Producers Risk

1
Intrepreting Consumers and Producers Risk

• Stipulated Producers Risk
• Need to specify producers risk, AQL, and
  acceptance number.
• Stipulated Consumers Risk
• Need consumers risk, LQL, and acceptance number.
• Stipulated Producers and Consumers Risk
• Need AQL, LQL, producers and consumers risk

2
Choosing an (n,c) plan

• Stipulating Producers Risk
• Set PR desired alpha (.05 in our example) AQL
  .015 and choose c
• For c 1, Table G shows np 0.355.
• We know p ( AQL .015) so solve for n
• n 0.355/.015 23.67 24. So desired plan is
  (24, 1)
• Can duplicate this for any c.
• What is consumers risk for this plan?
• LQL .08 n 24 ? (24)(.08) 1.92 1.9
• CR Pa .434 (not too good)

3
Choosing an (n,c) plan

• Stipulating Consumers Risk
• Set CR desired beta (.10 in our example),
• LQL .08 and choose c
• For c 3, np 6.681.
• p .08, n 6.681/.08 83.51 84
• (n, c) plan (84, 3)
• What is producers risk for this plan (AQL
  .018)?
• n 84, p .018, ? 1.532 1.5
• Pa .934, Prod Risk 1 Pa .066. Pretty
  good!

4
Stipulated Producers AND Consumers Risk

• For the same parameters above (AQL .018, LQL
  .08) assume that we want PR 5 and CR 10,
  what is the least n that satisfies this?
• np2/np1 LQL/AQL .09/.018 5.00
• Look this number up on Table G.
• Doesnt appear, so try closest, which are c 2
  (6.51) and c 3 (4.89, probably better because
  closer!)
• Stipulate which (PR or CR) you want to be exact.
• See which of the four possibilities you are
  happiest with (closest fit)

5

• Set PR .05, c 2
• np 0.818 (from Table G), p .018 gt n 44.45
  45 (round up for conservatism), gt (45,2)
• What is CR? ? np 45.09 4.15 .21 (using
  interpolation)
• Choose c 3
• np 1.366, p .018 gt n 75.88 76 gt(76,3)
• CR? ? np 76.09 6.84 0.9 (better!)
• Can do the exact same procedure simply setting
  consumers risk exactly and solving for
  producers risk. Then decide which is best.
• Why are we not surprised that (76, 3) has a lower
  CR than (45, 2)?

6
Challenge Problem

• In the previous example, assume the defective (p
  6) batches occur, on average, only 30
  times/year (out of 300 plant operating days) and
  that on a given day the sample of 50 shows 4
  defects and is rejected. What is the probability
  that this batch is really ok (type II error)?

7
Selecting a Sampling Plan Approach

• Use the Military Standard or ANSI/ASQC Z1.4
  Plans
• AQL
• Level of Inspection (I,II,III,S1,S2,S3,S4)
• Inspection severity (normal, tightened, reduced)
• Single or multiple sampling
• Dodge-Romig plan based on LQL. Need to know the
• average p of nonconforming lots based on recent
  data.
• this is closest to the approach which will now be
  
• discussed.

8
Selecting a Sampling Plan Lets make a decision!

• Use Economic Model with Optimization
• Calculate total cost given p, n and c
• Using a prob. distribution of p, find expected
  cost given n and c
• Choose n and c to minimize total expected cost

9
Approach 3 Economic Model for Rectifying
Sampling

• Total Cost Inspection cost cost of
  undetected defects
• Expected total cost

10
Lets make a decision!

• Talco supplies ships lots of 1000 transistors to
  the US government. Some of the time (P-good) the
  proportion defective is 5, which is acceptable,
  given the low labor rate that Talco charges.
  Otherwise (P-bad),the defect rate .10, for
  which Talco must pay a penalty of Rs1500. It
  costs Talco Rs 1 per inspection.
• If it is equally likely that p .05 as .10,
  should Talco inspect all or not? (no sampling
  available)
• At what value of P-bad would Talco be indifferent
  between 100 inspection, and no inspection?

11
Lets make a decision (with sampling)!

• Would a sample help us? How? What does the
  sample give us?
• Suppose Talco took a sample of 20, and found 2
  defective parts, should they accept or reject the
  lot?
• Is this answer invariant of the prior probability
  of good lots vs. bad lots?
• If n were set at 20, what would be the optimal
  (n, c) plan?

12
Summary

• Sampling can be helpful in many ways.
• Its not easy to figure out what kind of sampling
  plan to use or when its helpful.
• What do we need to know important factors