11. combinations of the framing principles Suppose we want to maximize (x, y) = 12x x2...

11. combinations of the framing principles Suppose we want to maximize ω(x, y) = 12x − x2 + 18y − 3y2 − 10, subject to x+y ≤ 8, x ≥ 0 and y ≥ 0. You should verify the solution has x = 5.25 and y = 2.75. Now consider the following. (i) Initially drop the constant of −10. (ii) Notice that if the constraint were not present, we would never set x above 6 or y above 3. Doing so lowers the objective function. Similarly, we would never set x below 6 or y below 3. A slight increase whenever the variables are below the noted targets will increase the objective function. (iii) This insight implies, with the constraint present, we would never set x below 5 (because y would never be set above 3). (iv) Together, then, we can locate the best choice of x by maximizing 12x−x2 +18(8−x)−3(8−x)2, subject to the constraint 5 ≤ x ≤ 6.

(a) Try it.

(b) Carefully document the use of the three principles of consistent framing in this exercise.