# Cost Benefit Analysis of Pre-screening Widgets for Faults before they Fail

I want to build a model that determines whether to pre-screen my widgets for defects. If I do pre-screen, it costs a fixed amount per check and I resolve the problem 100% of the time. If I don't pre-screen, its cost will be the cost of fixing the problem, plus the cost of materials for expired inventory. I have a whole bunch of variables that impact whether a defect will occur in each particular widget, but if it does occur and I didn't pre-screen it, it will definitely result in a fixed repair cost and a variable expired inventory cost based on cost of materials. For each widget, I need to know if the expected cost of the defect exceeds or is less than the cost of pre-screening.

Things I know:

• A - Cost of pre-screening
• B - Model to Predict Whether it will fail - based on test data - of ones it says it won't have a defect, it is right 98.5% of the time (false negatives).
• C -Model to Predict expected cost of Raw Materials for each widget produced
• D - % of repairs that will be successful and result in no expired inventory cost
• E - Fixed Cost of Repairs

Therefore, I should screen if:

A - B*(E+C*D) > 0

I have all of these pieces, but I don't know how to build this into a model.

Anyone with R or Stata skills, I would appreciate being pointed in the direction of a good package to use for building this model.

• Please let me know if this answered your question or if you have any further concerns. – Jack Ryan Apr 30 '14 at 11:25

Considering my predilection for R, I am going to do something unlikely: I would advise you not to use R or any other program. If you agree to the principle that the production should be done correctly, I think you would agree with the principle that determining the inspection criteria and the inspection itself should be as simple as possible (but you can call it a "decision tree" to justify your title):

Question 1: Is the supply of your part stable?

A1 == NO. Considering a type of widget: if the supply of this widget is not in a state of control, strongly consider 100% inspection (Deming, 1986). DONE.

A1 == YES. OK. . .

Question 2: Does the cost of testing outweigh the cost of not testing?

If the supply of this part is in control, you can adopt an all-or-nothing testing strategy (Deming, 1986), given:

p   = [historic] average percent defective [from] incoming lots of parts
(e.g. received 100 widgets Friday, 3 defective; p = 3%).
Remember: stable process exists!
k   = cost to inspect one part
l   = cost to dismantle, repair, reassemble, and test an assembly that
fails because a defective part was put into the production line.
k/l = break even quality, or break-even point (always between 0 and 1)


A2 == NO. If p > k/l inspect 100%. You will save money with each inspection by preventing expensive mistakes further down the production line. DONE.

A2 == YES. If p < k/l inspect 0%. You will lose money with each inspection by spending valuable time inspecting parts that are most likely acceptable. DONE.

The statistical proof of this is "exceedingly simple," but involves a chart which I might attempt to reproduce later if I'm feeling ambitious. I would encourage you to find a copy of the text which I created this answer from:

Deming, W. E. (1986). Out of the Crisis (Ch 15). Cambridge, MA: MIT Center for Advanced Engineering Study.