A Case Study

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

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 Engineers

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 Engineers

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 Engineers

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

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 Engineers

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 Engineers

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 Engineers

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 Engineers

10. Sensitivity Uncertainty Analysis for Engineers

11. Sensitivity Uncertainty Analysis for Engineers

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 Engineers

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

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

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

16. Mean Profit vs. M Uncertainty Analysis for Engineers

17. Normal Distributions • Now repeat for normal distributions Uncertainty Analysis for Engineers

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 Engineers

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

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

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

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

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