1. A Case Study Jake Blanchard Spring 2010 Uncertainty Analysis for Engineers1

2. Introduction These slides contain a description of a case study of an uncertainty analysis You should use this as a model for your final projects Uncertainty Analysis for Engineers2

3. The Case We are concerned with widget production The question is how many widgets should we produce in order to maximize profit Assume you are a manufacturer of widgets, which are purchased seasonally Fixed production costs are $40,000 per year The unit cost varies between $2,000 and $2,400 above the fixed costs, depending on the year. Demand typically fluctuates from 30 to 50 units per year. The off-season sales price is $500 each for the first ten units and between 0 and $500 for the remainder. The sales price is fixed at $8,000 per unit. Uncertainty Analysis for Engineers3

4. Variables P=profit M=# manufactured D=demand S=in-season sales UP=unit price UC=unit cost TO=total off-season revenue Off=off-season price (first 10) OffExtra=price for rest of widgets I=inventory (M-S) F=fixed cost TC=total cost R=revenue Uncertainty Analysis for Engineers4

5. Algorithm P=R-TC TC=F+UC*M R=UP*S+TO I=M-D Uncertainty Analysis for Engineers5

6. Input Distributions To start, assume all distributions are uniform, with the limits defined on the previous slide Also consider the case where the distributions are normal, with the same means and variances as the uniform distributions Uncertainty Analysis for Engineers6

7. Analysis What is profit, assuming all variables are at their mean (this is first order approximation of the mean)? What is first order estimate of variance? What is sensitivity for all random inputs? Plot histogram for profit. Plot histogram for normal distributions. Uncertainty Analysis for Engineers7

8. First Order Estimate of Profit Putting in all mean values and setting M=40 gives a profit of $192,000 If we vary M, the first order estimate of the mean profit is Uncertainty Analysis for Engineers8

9. First Order Estimate of Variance For M=40-, variance is estimated to be 2.1e7 $ 2 For M=40+, variance is estimated to be 1.9e9 $ 2 Uncertainty Analysis for Engineers9

10. Sensitivity 10

11. Sensitivity Uncertainty Analysis for Engineers11

12. Results for M=40 Mean Profit from MC is $170,000, compared to $190,000 first order estimate Mean variance from MC is 6.6e8, compared to estimates of 2.1e7 below 40 and 1.9e9 above 40 Uncertainty Analysis for Engineers12

13. Profit Histograms – M=30 Uncertainty Analysis for Engineers13

14. Profit Histograms – M=40 Uncertainty Analysis for Engineers14

15. Profit Histograms – M=50 Uncertainty Analysis for Engineers15

16. Mean Profit vs. M Uncertainty Analysis for Engineers16

17. Normal Distributions Now repeat for normal distributions Uncertainty Analysis for Engineers17

18. Results for M=40 Mean Profit from MC is $175,000, compared to $190,000 first order estimate (Unif dist gave $170,000) Mean variance from MC is 6.64e8, compared to estimates of 2.1e7 below 40 and 1.9e9 above 40 (Unif Dist gave 6.6e8) Uncertainty Analysis for Engineers18

19. Profit Histograms – M=30 Uncertainty Analysis for Engineers19

20. Profit Histograms – M=40 Uncertainty Analysis for Engineers20

21. Profit Histograms – M=50 Uncertainty Analysis for Engineers21

22. Mean Profit vs. M No change in this plot Uncertainty Analysis for Engineers22

23. Decision How many widgets should we manufacture? Uncertainty Analysis for Engineers23