oh, please help me post a linear programming questiion with solution(graphical)please i need it nao..and i cant find it on the internet
thx
http://people.brunel.ac.uk/~mastjjb/jeb/or/morelp.html
thanks
Deborah inherited a large sum of money; she wants to use a portion of this money to set up a trust fund for her two children. The trust fund has two investment options: 1) a bond fund and 2) a stock fund. The projected returns over the life of the investments are 6% for the bond fund and 10% for the stock fund. Whatever portion of the inheritance she finally decides to commit to the trust fund, she wants to invest at least 30% of the amount in the bond fund. In addition, she wants to select a mix that will enable her to obtain a total return of at least 7.5%.
Sure, I can help you with that! Let's construct a linear programming problem and solve it graphically.
Linear Programming Problem:
Maximize: 3x + 4y
Subject to:
1. 2x + 3y ≤ 10
2. x + y ≥ 4
3. x ≥ 0, y ≥ 0
To solve this problem graphically, we need to graph the feasible region and determine the optimal solution.
1. Start by graphing the constraints:
For constraint 1 (2x + 3y ≤ 10):
- Rewrite the equation in the form y = (10-2x)/3.
- Choose two arbitrary values for x (e.g., x = 0 and x = 5) to find the corresponding y values.
- Plot the points (0, 10/3) and (5, 0), and connect them with a straight line.
For constraint 2 (x + y ≥ 4):
- Rewrite the equation in the form y = 4 - x.
- Choose two arbitrary values for x (e.g., x = 0 and x = 5) to find the corresponding y values.
- Plot the points (0, 4) and (5, -1), and connect them with a straight line.
2. Shade the feasible region:
- Shade the region that satisfies all the constraints, which is the area where the two lines intersect and lies within the given bounds (x ≥ 0, y ≥ 0).
- In this case, the feasible region is the region above and to the left of the line defined by constraint 1 but below the line defined by constraint 2.
3. Identify the optimal solution:
- The optimal solution occurs at one of the vertices (corner points) of the feasible region.
- We find the coordinates of these vertices and evaluate the objective function (3x + 4y) at each vertex.
- The vertex with the highest objective function value represents the optimal solution.
4. Interpret the solution:
- Once you find the optimal solution, it tells you the maximum value of the objective function and the values of x and y that achieve this maximum value.
I hope this helps you construct a linear programming problem and solve it graphically! If you have any further questions, feel free to ask.