a bakery makes both cakes and cookies. each batch of cakes reqire 2 hours in the oven and 3 hours in the decorating room. each batch of cookies need 3/2 hours in the oven and 2/3 of an hour decorating room. the oven is available no more than 15 hours a a day. while the decorating room can be used no more than 13 hours a day. how many batches of cakes and cookies should the bakery make in order to maximize profits if cookies produce a frofit of $40 per batch and cakes produce a profit of $25 per batch?

Question

MathMate · Accepted Answer

This is a linear programming problem in which we have given resources (oven and decorating room) and production options (cakes, x and cookies y).

The objective is to maximize profit, given by Z=25x+40y.

To analyze the problem, you would need to identify the constraints, namely
Oven:
2x+3y/2 ≤ 15, or
y ≤ (2/3)(15-2x), and
Decorating room:
3x+2y/3 ≤ 13, or
y ≤ (3/2)(13-3x)

Plot these as lines (equality) and consider only integers (batches).

The two lines will intersect at the point (4,6) which represents the optimal use of the resources, but does not necessarily mean the maximum profit.

The profit function is given by:
z(x,y)=25x+40y
and has to be maximized within the feasible region.

Points outside the figure are non-feasible because the constraints are violated.

Any point within the figure enclosed by the two lines and the axes is a feasible solution, as long as the points are in the integer domain.

To find the maximum or minimum profit, we do not need to check the interior points, but we need to check all points at or close to corners of the polygon.

The one that gives the maximum value of z(x,y) is the combination of (integer) batches of each kind.

Here's a link to the graph:
http://img442.imageshack.us/img442/4765/1335065908.png

If you need further help, please post.