## Homeworks for the course CS 4365/CS 5354, Summer 2010

1. (Due June 9) Use linear interpolation to describe the membership function for the term "close to 75". Assume that for this function, the value is 1 for t = 75, 0 for t = 70, and 0 for t = 80.

2. (Due June 11) In class, for the example, membership functions, and "and"- and "or"-operations described in the handout, we computed the degree to which for the difference in temperature x = 3 degree, the control of u = -2 degrees is reasonable. Perform similar computations for x = 3 and for u = -6, u = -5, u = -4, u = -3, u = -2, u = -1, u = 0, and u = 1, and plot the dependence of the degree on u.

3. (Due June 14) Use the Least Squares method to estimate the value of the resistance R based on the following three measurements:

• for I1 = 0.1, we get V1 = 0.2;
• for I1 = 0.2, we get V1 = 0.38;
• for I1 = 0.4, we get V1 = 0.81.
The values I, R, and V are related by the Ohm's law V = I * R.

4. (Due June 16) Write a program, that, given the difference in temperature x, returns the control u recommended by the fuzzy control. Use the rules, membership functions, "and"- and "or"-operations, defuzzification, and whole procedure as described in the handout. Turn in the printout of the program and of the results. Be ready to show me how your program works.

5. (Due June 16) Use Zadeh's extension principle to find the membership function for y = x1 * x1 provided that you know the membership functions for x1 and x2. Use any reasonable function you want. For example, you can assume that x1 is the number of students in a group, x2 is the number of credit hours taken by each student, and y = x1 * x1 is the total number of credit hours taken by all the students from this group.

6. (Due June 21) Describe alpha-cuts corresponding to alpha = 0.1, 0.5, 0.9, and 1, for the membership functions SP (small positive) and SN (small negative) that are described in the handout (and that we used in our program).

7. (Due June 21) Let us assume that a fuzzy number is given as a nested family of intervals, with X(0.1) = [-9,9], X(0.2) = [-8,8], X(0.3) = [-7,7], X(0.4) = [-6,6], X(0.5) = [-5,5], X(0.6) = [-4,4], X(0.7) = [-3,3], X(0.8) = [-2,2], X(0.9) = [-1,1], X(1.0) = [0,0]. Use bisection (as we did in class) to find the value of the corresponding membership for x = 3.5.

8. (Due June 22) For x1 = Small Positive and x2 = Small Negative, find the alpha-cuts for y = x1 + x2 for alpha = 0.1, 0.2, ..., 0.9, 1.0.

9. (Due June 22) For x1 = Small Positive and x2 = Small Negative, find the alpha-cuts for y = x1 - x2 for alpha = 0.1, 0.2, ..., 0.9, 1.0.

10. (Due June 22) Find the range of the function f(x1) = x1 - x12 on the interval [-1,1].

11. (Due June 23) Let us assume that x1 is Small Positive, and y = (x1 - 4)2 + x1. Find the alpha-cuts for y corresponding to alpha = 0.1, alpha = 0.5, and alpha = 1.0. For each of these values alpha, use two methods:

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

11a. (Due June 24) Let us assume that x1 is Small Negative, and y = (x1 + 1)2 - x1. Find the alpha-cuts for y corresponding to alpha = 0.1, alpha = 0.5, and alpha = 0.9. For each of these values alpha, use two methods:

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

12. (Due June 25) Use the calculus-based method to find the range of a function y = (x1 - x2)2 + x1 when x1 is in the interval [0,1], and x2 is in the interval [-1,1].

12a. (Due June 24) Use the calculus-based method to find the range of a function y = x12 - x1 * x2 + 2 * x22 + x2 when x1 is in the interval [-1,1], and x2 is in the interval [-2,2].

13. (Due June 24) Let us assume that the actual temperature outside is t0 = 100 degrees. Find the temperature setting t between 75 and 100 degrees which provides the minimal use of energy under the constraint that the resulting temperature is comfortable. To get a membership function for "comfortable", use linear interpolation; assume that 75 is absolutely comfortable, 100 is absolutely not comfortable. Assume that the energy spent of cooling is proportional to the difference in temperature t0 - t. Just like we did in class, use linear interpolation to find the membership function corresponding to "minimal".

14. (Due July 8) Write a program that implements the clustering algorithm (non-fuzzy one) that we had in class.

15. (Due July 8) On a line x1 = x2, find a point which is the closest to the point (3,4). Use the Lagrange multipliers method.

16. (Due July 8) Of all rectangles with perimeter 12, find the one which the largest area. Use the Lagrange multipliers method.

17. (Due July 9) Run a numerical example of fuzzy clustering. Feel free to finish the example that we started in class.

18. (Due July 12) Given the following number of people in each age bracket: [0, 10]: 2, [10, 40]: 5, [40, 50]: 1, [50, 60]: 3, [60, 80]: 6, and [80, 100]: 1, compute intervals for the mean age and the variance.

19. (Due July 27) Solve the following linear programming problem.