1. 10/9 More optimization Test 2 is scheduled for next Monday, October 16 5-7pm Come to this room, FB 200, to take the test. This is a room change from what was previously announced.

2. Example to show an unbounded problem can have a maximum Maximize P = -2x + y, where x,y>= 0, y >= x, and y<=x+2

3. The simplex algorithm 1Find the most negative entry in the bottom row. This determines the pivot column. Bringing this variable into solution increases the objective the quickest. Note: If there are no negative entries you are already at the maximum value. Stop. 2Calculate all the positive ratios in the pivot column. Find the smallest one. This determines the pivot row. Note: if there are no positive ratios in the column, the problem is unbounded and has no maximum. Stop. 3 Perform a pivot operation using the entry in the pivot row and pivot column. 4 Repeat steps 1 thru 3 until you reach the maximum or discover the objective has no maximum.

4. Final tableau with maximum example x y s t P rhs

5. Final tableau with no maximum example x y s t P rhs -1 -1 1 0 0 10 2 -1 0 1 0 20 -1 -2 0 0 1 0

6. What kind of tableau is this? x y s t r P rhs -1 -1 1 0 0 0 -10 1 1 0 1 0 0 40 -1 1 0 0 1 0 20 -3 -2 0 0 0 1 0

7. Asset allocation: p. 240 #36 200,000 to invest x in growth, y in balanced, z in income rate of return 12% 10% 6% risk factor.1.06.02 Wants at least 50% in income at least 25% in balanced with average risk factor not to exceed.06 Find an optimum portfolio (x,y,z)

8. Initial tableau x,y,z nonbasic (decision) vars s,t,r basic (slack) vars (1 for each constraint)