Question

Z Min = 1600x+2400y

Subjected to:
4x+Y>24
2x+3y>42
X+4y>36
X<14
y<14 , x, y>0
a. Solve this LP problem using both graphic and simplex method (Show all steps).
b. Is there alternative solution? If so, Identify at least, four integer alternative solutions.
c. Solve using excel solver and write the summary of both answer and sensitivity report

Answer 1

To solve the LP problem, we can use either the graphical method or the simplex method. Using the graphical method, we graph the constraints and find the corner points of the feasible region to determine the maximum value of Z Min. Using the simplex method, we set up a tableau and apply the simplex algorithm to find the optimal solution.

Step-by-step explanation:

To solve the LP problem using the graphical method, we need to graph the feasible region defined by the given constraints and then find the maximum value of Z Min = 1600x + 2400y within that region. To do this, we will:

1. Graph each constraint as a line on a coordinate plane.

2. Shade the region that satisfies all the constraints.

3. Identify the corner points of the feasible region.

4. Substitute the coordinates of each corner point into the objective function Z Min = 1600x + 2400y to find the maximum value.

Using the simplex method, we can solve the LP problem by setting up a simplex tableau and applying the simplex algorithm to find the optimal solution. This involves:

1. Writing the constraints in standard form by introducing slack, surplus, and artificial variables.

2. Setting up the initial tableau using the coefficients of the variables and the constraints.

3. Applying the simplex algorithm to iterate through different tableaus until the optimal solution is found.

Answer 2

To solve the LP problem, you can use graphical method by graphing the feasible region and identifying the highest point that satisfies the objective function. Alternatively, you can use simplex method by converting the inequalities into equations and applying the simplex algorithm. There are alternative solutions and you can find four integer alternative solutions by substituting the corner points into the objective function. Excel solver can also be used to solve the problem and generate a summary of the optimal solution and a sensitivity report.

Step-by-step explanation:

1. Graphical Method:
   - Graph the feasible region defined by the inequalities.
   - Identify the feasible region as the set of points that satisfy all the inequalities.
   - Plot the objective function Z = 1600x + 2400y on the graph.
   - Find the highest point on the feasible region that lies on the line Z = 1600x + 2400y. This point gives the maximum value of Z.
   
   
2. Simplex Method:
   - Convert the inequalities into equations by adding slack variables.
   - Set up the initial simplex tableau with the coefficients of the objective function and the slack variables.
   - Apply the simplex method to find the optimum solution.
   
   b. Yes, there are alternative solutions. To identify four integer alternative solutions, substitute the corner points of the feasible region into the objective function and select the maximum values.
   
   c. Solve the problem using Excel solver by setting up the constraints and objective function. The summary provides the optimal solution and the sensitivity report shows the range of values for the coefficients of the objective function that keep the optimal solution unchanged.