Question

Island Publishing Company publishes two types of magazines on a monthly basis: a restaurant and entertainment guide and a real estate guide. The company distributes the magazines free to businesses, hotels, and stores on Hilton Head Island in South Carolina. The company's profits come exclusively from the paid advertising in the magazines. Each of the restaurant and entertainment guides distributed generates $0.50 per magazine in advertising revenue, whereas the real estate guide generates $0.75 per magazine. The real estate magazine is a more sophisticated publication that includes color photos, and accordingiy it costs $0.25 per magazine to print, compared with only $0.17 for the restaurant and entertainment guide. The publishing company has a printing budget of $4,000 per month. There is enough rack space to distribute at most 18,000 magazines each month. To entice businesses to place advertisements, Island Publishing promises to distribute at least 8,000 copies of each magazine. The company wants to determine the number of copies of each magazine it should print each month to maximize advertising revenue. Formulate a linear programming model for this problem.

Answer 1

8,000 and 10,000 respectively.

Step-by-step explanation:

Linear programming problem.

Please see the enclosed picture of the graph plotting the straight lines corresponding to the boundaries of the constraint conditions given.

Let the numbers of restaurant & entertainment guides and real estate guides be represented by A and B respectively.

Given constraints are :

A ≥ 8,000 -------(1)

B ≥ 8,000 -------(2)

A + B ≤ 18,000 ------(3)

Total cost of publishing C = 0.17 A + 0.25 B ≤ $4,000

=> 17 A + 25 B ≤ 400,000 --------- (4)

Revenue R = 0.50 A + 0.75 B $

Profit P = (R - C) = 0.33 A + 0.50 B $

We plot the straight lines corresponding to the constraints (1) to (4). The region XYZ marked on the graph is the one satisfying the constraints. To find the optimum or maximum revenue or maximum profit, we find the revenue and profit at points X, Y and Z.

Y = (8000, 8000) , X = (10000, 8000), Y = (8000, 10000).

Profits and revenues:

At point Y: R = $ (0.50 * 8000 + 0.75 * 8000) = $ 10,000.

P = $ (0.33 * 8000 + 0.50 * 8000) = $ 6,640.

At point X: R = $ (0.5 * 10000 + 0.75 * 8000) = $ 11,000.

P = $ (0.33 * 10000 + 0.50 * 8000) = $7,300.

At point Z: R = $ (0.5 * 8000 + 0.75 * 10000) = $ 11,500.

P = $ (0.33 * 8000 + 0.50 * 10000) = $ 7,640.

So the revenue and profit are both maximized at point Z.

So A = 8000. B = 10,000.