5 pages of notes allowed.
Please place the solution of each problem on a separate sheet of paper, with your name on top of each sheet.
1. Describe alpha-cuts corresponding to alpha = 0.4:
2. For x1 = Small Positive and x2 = Small Negative (as in Problem 1), find the alpha-cut for y = x1 - x2 for alpha = 0.4.
3-4. Find the range of the function y = (x1 + 1)2 - x1 * x2 when x1 is between 0 and 1, and x2 is between -2 and 1, by using two methods:
5. Use fuzzy optimization to find the price of the cheapest healthy lunch at the Union. The objective function is the price, which goes from $2 (sandwich) to $10 (chicken salad); as usual, use linear interpolation to find the membership function corresponding to "minimal". To get a membership function for "healthy", use linear interpolation; assume that the $2 lunch is absolutely un-healthy, while the $10 lunch is absolutely healthy.
6. Use the crisp clustering algorithm to cluster the following 1-D data: objects are characterized by values 0.0, 1.0, 5.0, 6.0, and 7.0, we have two clusters, and the initial representatives are 0.0 and 7.5.
7. On a line x1 + x2 = 1, find a point which is the closest to the point (3,4). Use the Lagrange multipliers method.
8. Use the fuzzy clustering algorithm to cluster the data from Problem 6. Use m = 2, and run only one iteration; it if Ok not to perform computations, but you need to write down all the formulas for each step.
9-10. Given the following number of people in each age bracket: [20, 30]: 2, [30, 40]: 1, [40, 50]: 2, compute intervals for the mean age and the variance. Lower bound for the variance is for extra credit.