Jeffery H. answered • 10/20/17
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Patient and Knowledgeable Professional Programmer and Computer Expert
Greetings Francisco from Fort Hill,
Thank you for your question.
You asked the following Linear Programming question:
Maximize = 25x1 + 15x2,
subject to the following constraints:
1. 3x1 + 2x2 =< 240 and
2. 2x1 + x2 =< 140.
The general process for solving linear-programming exercises is to graph the "feasibility region" created by these "constraints" to form a walled-off area on the x1,x2-plane. If this is a compact convex polytope of the plane, we know that the maximum and minimums must occur at the vertices of this polytope.
Unfortunately, the constraints you have given do not by themselves give us a compact region of the plane, instead you would need at least three constraints, in the case of two independent variables x1 and x2. Additionally, the objective function is unbounded (i.e. does not have a max or min value instead it goes to +/- infinity) over the feasibility region created by just these two constraints.
Thank you for your question.
You asked the following Linear Programming question:
Maximize = 25x1 + 15x2,
subject to the following constraints:
1. 3x1 + 2x2 =< 240 and
2. 2x1 + x2 =< 140.
The general process for solving linear-programming exercises is to graph the "feasibility region" created by these "constraints" to form a walled-off area on the x1,x2-plane. If this is a compact convex polytope of the plane, we know that the maximum and minimums must occur at the vertices of this polytope.
Unfortunately, the constraints you have given do not by themselves give us a compact region of the plane, instead you would need at least three constraints, in the case of two independent variables x1 and x2. Additionally, the objective function is unbounded (i.e. does not have a max or min value instead it goes to +/- infinity) over the feasibility region created by just these two constraints.
So I would probably assume that there are two addition constraints:
3. x1 >= 0
4. x2 >= 0,
as this is a common standard form of Linear programming problem.
It is fairly easy to solve for the vertices, which I will leave the details as an exercise for you, since I cannot include a graph in this answer post. For reference, I got the following for the vertices of this feasibility region:
{(0, 0), (70, 0), (40, 60), (0, 120)}.
All you need to do then is compute the objective function for each of these values and you get the max and the min of this objective subject to the constraints given.
{0, (25*70), (25*40 + 15*60), (15*120)}
This gives us a minimum of 0 at (0,0) and a maximum of 1900 at (40,60).
I hope this answers your question, and have a good night!
3. x1 >= 0
4. x2 >= 0,
as this is a common standard form of Linear programming problem.
It is fairly easy to solve for the vertices, which I will leave the details as an exercise for you, since I cannot include a graph in this answer post. For reference, I got the following for the vertices of this feasibility region:
{(0, 0), (70, 0), (40, 60), (0, 120)}.
All you need to do then is compute the objective function for each of these values and you get the max and the min of this objective subject to the constraints given.
{0, (25*70), (25*40 + 15*60), (15*120)}
This gives us a minimum of 0 at (0,0) and a maximum of 1900 at (40,60).
I hope this answers your question, and have a good night!
