What is linear programming?
A mathematical tool to assist managers in decision-making when constraints are present.
What is the objective function in LP?
It is what we want to maximize (e.g., contribution margin) or minimize (e.g., costs).
What assumptions does LP adopt from CVP analysis?
Linearity of prices and costs, fixed costs treated as constant and irrelevant.
When can graphical methods be used in LP?
When there are only two decision variables.
What is the simplex method?
An iterative algebraic procedure that examines corner points until the optimal solution is found.
What is a basic feasible solution?
A solution where some variables are set to zero to solve for others, representing a corner point.
How do we convert ≤ constraints into equations?
By adding slack variables to represent unused resources.
What does a slack variable represent?
Unused capacity of a resource.
How does the objective function change with slack variables?
They represent unused resources or capacity, and because they do not contribute to profit or cost, they do not directly change the objective value.
What is the initial solution in simplex?
T=0, C=0, S1=100, S2=240 (all real variables set to zero).
What are basic variables in LP?
Variables in the solution mix (basis), e.g., slack variables initially.
What are nonbasic variables in LP?
Variables set to zero. They are NOT in the solution mix.
What do substitution rates represent?
The trade-off of resources when introducing one unit of a variable into the solution.
What does Cj - Zj represent?
The net profit gained by introducing one unit of a variable into the solution.
When is the solution optimal in simplex?
When all Cj - Zj values are ≤ 0.
Step 1 of simplex method?
Choose the pivot column: the variable with the largest positive Cj - Zj.
Step 2 of simplex method?
Choose the pivot row: smallest nonnegative ratio of quantity ÷ pivot column value.
Step 3 of simplex method?
Divide every number in the pivot row by the pivot number.
Step 4 of simplex method?
Update other rows using substitution with the new pivot row.
Step 5 of simplex method?
Recalculate Zj and Cj - Zj; if all are ≤ 0, the solution is optimal.
How are ≥ constraints handled in simplex?
By introducing surplus and artificial variables.
What does slack in a solution mean?
Unused resources in a constraint.
What does a positive Cj - Zj mean?
Profits can be increased by adding that variable to the solution.
What does a negative Cj - Zj mean?
Profits would decrease if that variable entered the solution.
