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Linear Programming — explain what it is, please?

Please explain in your own words what how to solve a Linear programming problem. I read the textbook, but I still don’t understand it. I don’t understand any part of it. For example, can you show me how to solve this problem and please explain each step you take. Thank you so much.

I recently answered another question like this. See:

http://answers.yahoo.com/question/;_ylc=X3oDMTBxcDMxYWk4BF9TAwRzZWMDbWFpbHRvBHNsawNzdWJqZWN0;_ylv=3?qid=20080427083842AA1W03y

Hope that link works! Anyway, linear programming is a special form of the more general “mathematical programming”. Basically this is a constrained optimization problem. You want to minimize (or maximize) a particular function of several unknowns called the *objective* function, subject to a collection of inequalities called *constraints*. For linear programming, the constraints are a linear function of the unknowns.

So, to your specific example.

The key is the following: How may of each pair should be produced per day for maximum profit?

The unknowns are the number of oxfords (call this X), and the number of loafers (call this L). The shoe manufacturer wants to maximize profit, so that will be the objective function.

Each type of shoe requires 15 minutes on the cutting machine. Total time on cutting machine is (15 minutes)*(X+L).

Oxfords require 10 minutes of sewing time, and loafers require 20 minutes of sewing time. Total sewing time is (10 minutes)*X + (20 minutes * L).

Each machine has an operator (so they run in parallel) and can be run eight hours per day. (I’m sure they mean 7 hours, since you need half an hour for lunch and a 15 minute break for every four hours. Labor laws!)

The total time on each machine cannot exceed what is available.

We know:
(8 hours) * (60 minutes / hour) = 480 minutes

We have a constraint due to the cutting machine:
(15 minutes)*(X+L) <= 480 minutes
X+L <= 32

We have a constraint due to the sewing machine:
(20 minutes)*L + (10 minutes)*X <= 480 minutes
2*L + X <= 48

Profit is $15 per pair, so the total profit (objective function) is:
($15)*(X + L)

The linear programming problem is:
Maximize: ($15)*(X+L)
Subject to:
X+L <= 32
2*L + X <= 48

The constraints determine a region called the "feasible region" where the solution must lie. For linear programming, the maximum or minimum must lie at a vertex; a "bend," or extreme point at the edge of the feasible region.

Since X and L must be positive, the feasible region lies in quadrant one. It is bounded by the other two constraints, which are linear. Let X be the x-axis, and L be the y-axis.

We can re-write the constraints as follows.

L <= -X + 32
L <= (-1/2)X + 24

I've rewritten these in the usual slope-intercept form for a line. You can graph these now. The first gives a line intersecting the y-axis at L=32 and then descending to the right (negative slope). The second gives a line intersecting the y-axis at L-24 and then descending to the right (negative slope), but not as steep.

The feasible region is everything below both lines (the L<= part).

The two lines are not parallel, and will thus intersect. This is important, since the intersection is a vertex. We can find their point of intersection.

L = -X + 32
L = (-1/2)X + 24

Solving -X+32 = (-1/2)X+24 gives X=16 and L=16, so that's the point where they intersect.

Another vertex is the lower y-intercept at X=0, L=24.

Another is the origin at X=0, Y=0.

Another is the point where L=-X+32 intersects the x-axis. We find this by setting L=0, giving:
0 = -X + 32
X = 32
So this point is X=32, L=0.

The vertices are thus:
X=16, L=16
X=0, L=24
X=32, L=0
X=0, L=0

We need to compute the profit ($15)*(L+X) and see where it is maximum. This function is maximized where L+X is maximized, and there are two points: X=16,L=16 and X=32,L=0.

The shoe maker should produce 16 of each kind, or produce just Oxfords to maximize his profit.

To see why this is so, graph the region. You'll see that for X<16 the constraint from the sewing machine applies, favoring oxfords. For X>16 the constraint from the cutting machine applies, and this favors both evenly.

32 pairs saturates the cutting machine, since 32*15 = 480. Only making Oxfords uses up 10*32 = 320 minutes of sewing, but making both uses up 10*16+20*16 = 160+320 = 480 minutes, saturating the sewing machine.

Hope this helps!

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