Linear Programming : Forumulation Product Mix A carpenter specializes in making

Question

Linear Programming : Forumulation

Product Mix

      A carpenter specializes in making bookshelves and desks. Each bookshelf earns a profit of $36, while each desk earns a profit of $40. A bookshelf requires 48 ft. of wood and takes 16 hrs. to make, while a desk requires 40 ft. of wood and takes 20 hrs. to make. The carpenter has 480 ft. of wood available, and is willing to spend up to 200 hrs. to make the bookshelves and desks. How many bookshelves and desks should be made to maximize total profit?

Please just set up the problem showing the variables and constraints etc.

Explanation / Answer

Let us assume that the carpenter needs to produce 'x' bookshelves and 'y' desks.

Each bookshelf earns a profit of $36, while each desk earns a profit of $40. So, total profit = 36x + 40y

The objective here is to maximize profit. Therefore, the objective function is:

Maximize Z = 36x + 40y

A bookshelf requires 48 ft. of wood a desk requires 40 ft. of wood. The carpenter has 480 ft. of wood available.

Therefore the constraint for wood is 48x + 40y <= 480

A bookshelf requires 16 hrs. to make, while a desk requires 20 hrs. to make. The carpenter has 200 hrs. to spend.

Therefore the constraint for time is 16x + 20y <= 200

Also, x and y must be greater than zero.

So, the LPP problem is as follows:

Maximize Z = 36x + 40y

Subject to constraints:

48x + 40y <= 480

16x + 20y <= 200

x >= 0

y >= 0

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