A publishing company makes 2 types of magazines on a monthly basis, a Restaurant and Entertainment guide and a Real Estate guide. It distributes the magazine free in Penang island. The company profits come from advertising. Each restaurant and entertainment guide distributed generates $0.50, whereas the real estate guide generates $0.75 per guide. The restaurant and entertainment guide costs $0.17 to make and the real estate guide costs $0.25 per magazine. The company has a printing budget of $4000 per month. There's enough rack space to distribute at most 18000 magazines each month. In order to entice businesses to place advertisements they must distribute at least 8000 magazines. The company wants to determine the number of copies each magazine it should print in order to maximize revenue
(a) formulate a linear programming model for this problem

Well, I'm not sure about the ways to formulate the linear programming model, but through calculus, that shouldn't be difficult.

Revenue from selling x number of R&E guide = (0.5-0.17)x
Revenue from selling y number of RE guide = (0.75-0.25)y

At most 18000 magazines means that x + y = 180000, if you're targeting maximum revenue, you should maximise production.

Then, the net revenue is given by:

R = (0.5-0.17)x + (0.75-0.25)y

Find the derivative of this, by substituting a one of the variables using the equation x + y = 18000.

Equate the derivative to 0 to get the number of the magazine to produce to maximise revenue. Then use that information to get the number of the other magazine.

Can you post what you get?

Hi Unknown. Long time. :)

Linear programming does not require calculus.

If one graphs the lines generated from the inequalities, one looks at the vertices of their intersections.

Basic linear programming may be encountered in a pre-calc course or even in college algebra. There is an entire vocation centered around linear programming and other math. It's called Operations Research.

Usually, profit is maximized. Profit = Revenue - Cost. Therefore,

Revenue = Profit + Cost.

By 'generate', do they mean that is the profit on each copy or what they are sold for?

I will assume it means profit.

Let x = # of entertainment guides and y = # of real estate guides.

We are maximizing revenue. This is the objective function.

If 'generate' means profit, then revenue would be cost + profit: R=.67x+y

Subject to the constraints:

The cost to make is .17x+.25y\leq 4000 because they only have 4000 to spare for printing.

They only have room for 18000, so another constraint is

x+y\leq 18000

But, they must sell at least 8000.

x+y\geq 8000

Assume x and y non negative

x, y\geq 0

Excel Solve does a fine job of solving linear programming problems.

Are you familiar with it?

Using Excel Solver, the result says they should sell 16,000 real estate guides and no entertainment guides to max revenue.

This results in $16,000 total revenue.

Perhaps I misinterpreted the problem because 'generate' is rather ambiguous. I tried the other objective function and got a negative number of entertainment guides. So, perhaps it means profit.

EDIT:

To do this by hand though, we look at the intersections of the lines and where they cross the axes. The 'feasible region'.

There are vertices at (6250,11750), (0,8000), (0,16000), (8000,0), (16000,0)

We enter these into the objective function. The set that results in the largest number is the max revenue.

Doing so, we find that (0,16000) gives the largest max revenue.

R=.67(0)+16000=16000

.67(6250)+11750=15937.5

.67(0)+8000=8000

.67(16000)+0=10720

.67(8000)+0=5360

This means 16000 restaurant guides and 0 entertainment guides should be sold to max revenue.