Question Set 5 (Ch 12), Reliability (609-623), Capability (630-637), and Taguchi Loss function (604-607 and equation excel sheet under the supplementary tab).

1)  In the movie “Apollo 13” there is a scene that shows a ground controller saying that the screens must be wrong because it is impossible to have quadruple failure. What did he mean

  1. What is Reliability: the probability a unit will perform as desired over a given amount of time under given conditions.
  2. What is failure rate: lambda = number of failures/total unit operating hours
  3. What is the Reliability function (R): The probability a unit will last for a given time period. R for some T (time) = e^(-lambda*T)
  4. What is mean time until failure (MTTF) and mean time between failures (MTBF)
  5. How to calculate reliability of a unit
  6. How to calculate reliability of units working in series
  7. How to calculate reliability of units working in parallel
  8. How to use reliability in a business sense

2)  Capability

  1. Capability is no more than the distance between upper and lower tolerance divided by plus or minus 3 SD = 6 SD.
  2. UTL-LTL/6SD = Cp UTL-mean/3SD = Cpu Mean-LTL/3SD
  3. To determine a standardized method of comparing processes relative to common variation relative to tolerances/specifications
  4. Stays the same when mean drifts, thus need Cpl and Cpu and Cpk
  5. Six sigma is referring to 6 SD from the norm to the tolerance or a Cp of 2.

3)  Taguchi Loss Function

  1. What is this function; Taguchi posts that cost of quality starts as soon as the specified norm/target is deviated from thus, costs steadily increase as soon as product starts deviating from the norm/target of specifications.
  2. L(x)=k((x-T)^2) k is the constant that tells you how slow or fast the cost increases. L(x) is the loss accrued when the characteristic being measured = x. X-T is the distance from norm/target to x.
  3. Want to calculate this function because you want to know what your process is costing you relative to quality. Can and should be able to calculate average cost for some process. EL(x)=k((SD^2)+(D^2)) D^2=(xbar-T)^2. Should also be able to calculate average cost for an area under the curve that is truncated due to inspection at some point EQL=k(SD^2)(1-(2z(e^(- 1*(z^2)/2))/(Φ(-z,z)(sqrt(2π)))) where Φ(-z,z) is the area under the curve between the two truncated values, z is the number of SD from norm to truncated value, k is the constant to determine width of cost function, e =2.7183 and pie = 3.142. If the mean is not centered, k(SD^2)(1-(z2(e^(- 1*(z2^2)/2)- (z1(e^(- 1*(z1^2)/2))/(Φ(-z,z)(sqrt(2π)))) this formula is in an excel sheet set up to use. The sheet is under supplementary and then equations…

4)  Cost benefit Analysis and sensitivity analysis

  1. NPV see Supplementary under NPV
  2. Pay Back cost/monthly cash flow gives number of months before cost is paid back (ignores time value of money)
  3. IRR: discount rate that provides and NPV of Zero
  4. Sensitivity analysis what happens to a, b, and c as you change variables that affect cash flow (best way is to determine a CI for NPV as you let all variables vary
  5. Area under a curve, proportion of product beyond some measure: cost calculations based on cost parameters as some metric of a product changes throughout its range of possibilities
  6. Solver: tool in excel to solve complex problems
  7. Confidence intervals: how to get to contingency planning and how to make decisions in a stochastic world. The size of the area depicted by the probability that the population parameter will be in a given area.

At how many miles would you offer replacement of a tire given average life of 50,000 miles, standard deviation of 10,000 miles, and management has made a decision not to replace any more than 2% of tires sold. Also, how would management make the 2% decision?

If you want to be 99% confident with a 2% margin of error that the results of a poll to determine what proportion of the population would vote for Bush, how many people would you have to poll? How could the poll be compromised, or how do you as a consumer of the poll trust the results?

Find n, given that we wish to estimate mu to within 10 units, with 95% confidence, assume that sigma = 100.

Calculate mu for a stand of trees, min DBH = 2 inches, max DBH = 27 inches, 99% CL, E = .5 inch. What sample size would you want to see to ensure that the interval around the mean of the sample actually contained the stand’s true mean.

What is the probability the mean of a sample of 36 units will be 50.04 when the population mean has been 50.00 with a SD of .12? Why would you want to know such a thing?

The belief is that the average evaluation score for teachers is 3.2 (is a good average) on a scale of 4. The average score of 25 teachers for a semester was 2.8 with a standard deviation of 1. Tell me, at a 95% confidence level, if there are a bunch of teachers who need fired or is there sampling error.