MIN z= 5x1 +2x2
st 2x1 +5x2>or equal to 10
4x1-x2>or equal to 12
x1 + x2 > or equal to 4
x1, x2 > or equal to 0
A Solve graphically for the optimal solution.
B How does one know that although x1=5, x2=3 is a feasible solution for the constraints, it will never be the optimal solution no matter what objective function is imposed?
The process of solving a linear programming problem graphically is explained and illustrated by an example.
The solution is in a PDF file.
Graphical solution of linear programming problem.
2. Problem 8
A company produces, A and B, which have profits of $9 and $7, respectively. Each unit of product must be processed on two assembly lines, where the required production times are as follows.
Product Line 1 Line 2
A 12 4
B 4 8
Total hours 60 40
A. Formulate a linear programming model to determine the optimal product mix that will maximize profit.
B. Transform this model into standard form.
3. Problem 9 -
a) Solve problem 8 graphically. List all extreme points (x1, x2, z) and indicate the optimum solution. Identify the amount of unused resources (slack) for each of the extreme points.
b) What would be the effect on the optimal solution if the production time on line 1 was reduced to 40 hours?
c) What would be the effect on the optimal solution if the profit for product B was increased from $7 to $15? How about from $7 to $20?
4. Problem 10 -
For the linear programming model formulated in problem 8 and solved in problem 9:
a) Determine the sensitivity ranges for the objective function coefficients either manually using the graphical solution or using QM for Windows. i.e.: I want you to return the minimum and maximum coefficients (profit margins) for products A and B for which the original optimum solution still applies.
b) Using manual methods, QM for Windows, determine the shadow prices for additional hours of product time on line 1 and line 2.