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November 30, 2015
Posted by **Gracy** on Saturday, April 21, 2012 at 11:38pm.

- Math Analysis -graphical method -
**MathMate**, Sunday, April 22, 2012 at 8:43amThis 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.