HOME WORK set (2) PROBLEM SET 2.2A 2. Identify the direction of increase in z in each of the following cases: *(a) Maximize z = x1 - , (b) Maximize z = -5x1 - 6x2. (e) Maximize z = -x + 2x2. *(d) Maximize z = -3x, + x3. 7. An individual wishes to invest SS000 over the next year in two types of investment: Invest ment A yields 5% and investment B yields 8%. Market research recommends an alloca- tion of at least 25% in A and at most 50% in B. Moreover, investment in A should be at least half the investment in B. How should the fund be allocated to the two investments? What if the word (most 50%) switched to (least 50%) (solve as new problem) What if both A and B yields similar % (discuss it in both previous cases) 8. The Continuing Education Division at the Ozark Community College offers a total of 30 courses each semester. The courses offered are usually of two types practical, such as woodworking, word processing and car maintenance, and humanistic, such as histo- ry, music, and fine arts. To satisfy the demands of the community, at least 10 courses of each type must be offered each semester. The division estimates that the revenues of offering practical and humanistic courses are approximately $1500 and $1000 per course, respectively (a) Devise an optimal course offering for the college. 19. Minimize z = 3x4 + 8x2 subject to X: + 1228 2x - 3x2 so x + 2x2 30 3x; - X220 s 10 11, 1220 20. In LP, a constraint is said to be redundant if its removal from the model leaves the reasi- ble solution space unchanged. identify the redundant constraints, then show that their removal (simply by not graphing them) does not affect the solution space Maximize z = 5x + 4x2 subject to 6x + 4x2 s 24 6x + 3x2 5 22.5 x1 + x2 5 5 * + 2x2 3 6 --X1 + x S 1 x2 s 2 X1, X220 PROBLEM SET 2.28 1. Identify the direction of decrease in z in each of the following cases: *(a) Minimize 2 - 4x, - 2x2. (b) Minimize z =-3x, + x2 (e) Minimize 2 - - - 2x2. 6. Day Trader wants to invest a sum of money that would generate an annual yield of at least $10,000. Two stock groups are available: blue chips and high tech with average an- nual yields of 10% and 25%, respectively. Though high-tech stocks provide higher yield, they are more risky, and Trader wants to limit the amount invested in these stocks to no more than 60% of the total investment. What is the minimum amount Trader should in vest in each stock group to accomplish the investment goal