# 1. Solve the following linear programming problem graphically:

1. Solve the following linear programming problem graphically:
maximize 2X1 3X2
subject to X1 8
X2 6
X1 2X2 16
X1
, X2 0
Units of L per period

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

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 dri- ves, 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 nonnegativity 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?

5. 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?

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