LPP Description:
The given Linear Programming Problem (LPP) is a maximization problem with two decision variables, x1 and x2. The objective function is z = 3x1 + 2x2. There are four constraints, including two inequality constraints and two non-negativity constraints. The constraints are as follows:
1. x1 ≤ 4
2. x2 ≤ 6
3. 3x1 + 2x2 ≤ 18
4. x1 ≥ 0, x2 ≥ 0

Explanation:
To determine if the LPP has a unique optimal solution, is infeasible, unbounded, or has multiple optimal solutions, we need to analyze the given problem.

Feasible Region:
The feasible region is the region that satisfies all the given constraints. In this case, the feasible region is the intersection of the half-planes defined by the constraints.

- Constraint 1: x1 ≤ 4
This constraint represents a vertical line parallel to the y-axis passing through x1 = 4.

- Constraint 2: x2 ≤ 6
This constraint represents a horizontal line parallel to the x-axis passing through x2 = 6.

- Constraint 3: 3x1 + 2x2 ≤ 18
This constraint represents a straight line with a negative slope passing through the points (0,9) and (6,0).

- Non-negativity constraints: x1 ≥ 0, x2 ≥ 0
These constraints represent the positive quadrant of the coordinate plane.

Graphical Representation:
By plotting the feasible region graphically, we can determine the nature of the problem.

- Graph the lines and regions corresponding to each constraint.
- Shade the region that satisfies all the constraints.

The resulting feasible region is a polygon bounded by the lines and the coordinate axes.

Optimal Solution:
To find the optimal solution, we need to evaluate the objective function at the extreme points of the feasible region.

- Identify the vertices of the feasible region.
- Substitute the coordinates of each vertex into the objective function to calculate the corresponding objective function value.

Analysis:
After evaluating the objective function at each vertex, we can determine the nature of the problem.

- If the objective function value is unique and the maximum value, then the LPP has a unique optimal solution.
- If the objective function value is unbounded, then the LPP is unbounded.
- If the objective function value is non-existent or negative for all vertices, then the LPP is infeasible.
- If there are multiple vertices with the same maximum objective function value, then the LPP has multiple optimal solutions.

Conclusion:
From the analysis of the feasible region and the evaluation of the objective function at the vertices, we can conclude that the given LPP has multiple optimal solutions.