Set up the initial simplex tableau that could be used to solve the following problem by the simplex method.
The Fancy Fashions, an independent, local boutique, has $\$ 9000$ available each month for advertising. Newspaper ads cost $\$ 500$ each, and no more than 30 can run per month. Internet banner ads cost $\$ 20$ each, and no more than 60 can run per month. TV ads cost $\$ 2000$ each, with a maximum of 15 available each month. Approximately 5000 women will see each newspaper ad, 3000 will see each Internet banner, and 11,000 will see each TV ad. How much of each type of advertising should be used if the store wants to maximize its ad exposure?
Final Answer: The store should run \(\boxed{16}\) newspaper ads, \(\boxed{60}\) internet banner ads, and \(\boxed{0}\) TV ads to maximize its ad exposure.
Step 1 :Define the variables: Let \(x_1\), \(x_2\), and \(x_3\) represent the number of newspaper ads, internet banner ads, and TV ads, respectively.
Step 2 :Set up the objective function: The objective function to maximize is \(5000x_1 + 3000x_2 + 11000x_3\), which represents the total number of women who see the ads.
Step 3 :Set up the constraints: The constraints are \(x_1 \leq 30\) (no more than 30 newspaper ads), \(x_2 \leq 60\) (no more than 60 internet banner ads), \(x_3 \leq 15\) (no more than 15 TV ads), and \(500x_1 + 20x_2 + 2000x_3 \leq 9000\) (total cost cannot exceed $9000).
Step 4 :Solve the problem using the simplex method. The optimal solution is \(x_1 = 15.6\), \(x_2 = 60\), and \(x_3 = 0\).
Step 5 :Since the number of ads must be an integer, round \(x_1\) down to 16.
Step 6 :Final Answer: The store should run \(\boxed{16}\) newspaper ads, \(\boxed{60}\) internet banner ads, and \(\boxed{0}\) TV ads to maximize its ad exposure.