Name: _____________________________________________________________

10 pages of notes allowed. Please place your solution to each problem on a separate sheet of paper, with your name on top of each sheet.

1. Describe two methods of eliciting degrees of certainty from experts: polling and marking on a scale. Give numerical examples of using both methods.

2. List requirements on "and"-operations (t-norms) and "or"-operations (t-conorms). Give two example of t-norms and two examples of t-conorms. Explain, on the example of one of your t-norms, why it is not a t-conorm: list all t-conorm requirements that are satisfied by this t-norm and those that are not satisfied.

3-4. If an iPhone overheats, it is necessary to let it cool down:

- if it overheats a little bit, we need to cool it down for a short period of time;
- if it overheats a lot, we must cool it down for a long period of time.

- For "overheating a little bit": the value is 1 for x = 0 and 0 for x = 50.
- For "overheating a lot": the value is 0 for x = 0 and 1 for x = 50 (and for all larger x).
- For "short period of time": the value is 1 for u = 0 and 0 for u = 10.
- For "long period of time": the value is 0 for u = 10 and 1 for u = 20 (and for all larger values u).

5. Three people estimated the temperature as 90, 95, and 100.
Use the Least Squares method to combine these three estimates into
a single one. Explain how the Least Squares method is used to
derive a defuzzification formula; write down the resulting
formula. *For extra credit:* derive the formula for
centroid defuzzification.

6. Let x_{1} = "long period of time" and
x_{2} = "short period of time" (as in Problem 3-4).
Find the alpha-cut for x_{1} - x_{2} for
alpha = 0.6.

7. Find the range of the function
y = (x_{1} - 1)^{2} + x_{1} *
x_{2} when x_{1} is between 0 and 1, and
x_{2} is between -2 and 1, by
using two methods:

- the calculus-based method for finding the exact range, and
- the naive interval computations method for computing the enclosure for the range.

8. Use the crisp clustering algorithm to cluster the following 1-D data: objects are characterized by values 0.0, 1.0, 2.0, 6.0, and 7.0, we have two clusters, and the initial representatives are 0.5 and 7.5. Write down the formulas explaining how to use fuzzy clustering to cluster this data. What is the advantage of fuzzy clustering as compares to the crisp one?

10. Use the Lagrange multiplier method to find the minimum of a
function x_{1}^{2} + x_{2}^{2}
under the constraint x_{1} + x_{2} = 2. What if
instead, we have a fuzzy constraint -- that the sum
x_{1} + x_{2} is approximately 2, where
"approximately 2" is described by a membership function
x/2 for x from 0 to 2 and 1 - x/2 for x from 2 to 4. For the fuzzy
case, just describe
the formulas, no need to find the actual location of the minimum.