Answer and Explanation:
a. The feasible solution area in linear programming refers to the region of the graph that satisfies all the constraints of the model. To obtain the optimal solution graphically, we first need to plot the constraints on a graph and then identify the feasible region.
Plot the given constraints:
1. The constraint 41 + 22 ≥ 20 can be rearranged as 41 + 22 - 20 ≥ 0, which simplifies to 63 ≥ 0. This constraint represents a line.
2. The constraint -61 + 42 ≤ 12 can be rearranged as -61 + 42 - 12 ≤ 0, which simplifies to -31 ≤ 0. This constraint represents a line.
3. The constraint 1 + 2 ≥ 6 can be rearranged as 1 + 2 - 6 ≥ 0, which simplifies to -3 ≥ 0. This constraint represents a line.
Now, let's plot these lines on a graph. The feasible region is the area where all the lines intersect or overlap. The optimal solution lies within this feasible region.
b. If the coefficient of 2 in the objective function decreases to 4, it means that the coefficient of the decision variable associated with the second constraint has changed. This change will affect the objective function's slope and the optimal solution. The new optimal solution will occur at a different point in the feasible region.
c. If the second constraint, -61 + 42 ≤ 12, is removed from the given model, it means that this constraint will no longer limit the feasible region. The feasible region will change, and the optimal solution will occur within the new feasible region. The removal of this constraint may result in a larger feasible region and potentially a different optimal solution.
d. If a new constraint, 41 + 62 ≤ 24, is added to the given model, it means that this constraint will limit the feasible region. The feasible region will become smaller, as it needs to satisfy this new constraint in addition to the existing constraints. The optimal solution will now occur within this smaller feasible region, which may result in a different optimal solution compared to the original model.