1. Solve the following linear programming problem graphically:
maximize = 2X1 + 3X2
subject to X1 ≤ 8
X2 ≤ 6
X1 + 2X2 ≥ 16
X1, X2 ≥ 0
2. In problem 1, how would the optimal solution change if the restrictions imposed (i.e.,the ri's) were all cut in half?
3. Solve the following linear programming problem using the general solution method:
minimize C = 3X1 + 4X2
subject to X1 + X2 ≥ 2
2X1 + 4X2 ≥ 5
X1, X2 ≥ 0
4. Form the dual to the linear programming problem presented in problem 3; then solve it to obtain the optimal value of C. Does the minimum value of C for the primal in Problem 3 equal the maximum value of C in the dual for this problem?
5. The advertising manager at Cadillac wishes to run both television and magazine ads to promote the new Cadillac GTS in the greater Chicago area market. Each 30-second television ad will reach 30,000 viewers in the target age group of buyers 35 to 55 years old.
Running one full page ad in Cool Driver magazine will reach 10,000 readers in the 35 to 55 year-old target market. To further promote the new GTS, the manager wishes to stimulate prospective buyers to come in to Chicago area dealerships to test drive the GTS. Past experience in Chicago indicates that a television ad will generate 500 test drives, while a magazine ad will generate only 250 test drives.
In order to reach the desired level of new-model penetration in the Chicago area, the advertising manager believes it is necessary to reach at least 90,000 potential buyers in the 35 to 55 age bracket and to get at least 2,000 of these potential buyers to take a test drive. Each 30- second TV ad costs $100,000 and each magazine ad costs $40,000. In reaching these objectives, the manager wishes to minimize the total expenditure on TV and magazine ads.
a. State the linear programming problem facing this advertising manager. Be sure to formulate the objective function and inequality constraints (including appropriate non-negativity constraints).
b. Solve the linear programming problem. What is the optimal number of TV ads and magazine ads? What will be the minimum possible level of total expenditures on television and magazine ads necessary to successfully promote the GTS in Chicago?
c. Suppose the local television stations, in order to reduce set-up costs, require Cadillac to run its ad two or more times. How would this constraint alter the solution to this linear programming problem.