## Step 1: Define the optimization problem
The goal is to maximize the objective function: $9x_0 + 6x_1 + 5x_2 + 3x_3$, where $x_0, x_1, x_2, x_3$ represent the hours worked by Laura, Paul, Bill, and Mary, respectively.

## Step 2: List all the constraints
The constraints are as follows:
- $10x_0 \geq 10x_0$ (always true, but we note Laura's computer competence rating is 10)
- $6x_0 \geq 6x_0$ (always true, but we note Laura's productivity rating is 6)
- $9x_0 \geq 9x_0$ (always true, but we note Laura's likelihood to quit index is 9)
- $7x_0 \geq 7x_0$ (always true, but we note Laura's dollar cost per hour is 7)
- $12x_1 \geq 12x_1$ (always true, but we note Paul's computer competence rating is 12)
- $8x_1 \geq 8x_1$ (always true, but we note Paul's productivity rating is 8)
- $8x_1 \geq 8x_1$ (always true, but we note Paul's likelihood to quit index is 8)
- $14x_1 \geq 14x_1$ (always true, but we note Paul's dollar cost per hour is 14)
- $6x_2 \geq 6x_2$ (always true, but we note Bill's computer competence rating is 6)
- $7x_2 \geq 7x_2$ (always true, but we note Bill's productivity rating is 7)
- $12x_2 \geq 12x_2$ (always true, but we note Bill's likelihood to quit index is 12)
- $9x_2 \geq 9x_2$ (always true, but we note Bill's dollar cost per hour is 9)
- $14x_3 \geq 14x_3$ (always true, but we note Mary's computer competence rating is 14)
- $11x_3 \geq 11x_3$ (always true, but we note Mary's productivity rating is 11)
- $3x_3 \geq 3x_3$ (always true, but we note Mary's likelihood to quit index is 3)
- $1x_3 \geq 1x_3$ (always true, but we note Mary's dollar cost per hour is 1)
- $10x_0 + 14x_3 \geq 46$
- $10x_0 + 6x_2 \geq 49$
- $12x_1 + 6x_2 + 14x_3 \geq 30$
- $8x_1 + 3x_3 \geq 47$
- $9x_0 + 12x_2 + 3x_3 \geq 49$
- $7x_0 + 9x_2 \geq 66$
- $14x_1 + 9x_2 \geq 27$
- $9x_2 + 1x_3 \geq 47$
- $10x_0 + 6x_2 \leq 142$
- $12x_1 + 14x_3 \leq 232$
- $12x_1 + 6x_2 \leq 153$
- $10x_0 + 6x_2 + 14x_3 \leq 172$
- $10x_0 + 12x_1 + 6x_2 \leq 138$
- $10x_0 + 12x_1 + 6x_2 + 14x_3 \leq 138$
- $6x_0 + 7x_2 \leq 129$
- $6x_0 + 8x_1 \leq 179$
- $6x_0 + 11x_3 \leq 195$
- $8x_1 + 11x_3 \leq 240$
- $8x_1 + 7x_2 \leq 164$
- $8x_1 + 7x_2 + 11x_3 \leq 174$
- $6x_0 + 8x_1 + 7x_2 + 11x_3 \leq 174$
- $8x_1 + 12x_2 \leq 177$
- $8x_1 + 3x_3 \leq 185$
- $9x_0 + 8x_1 + 12x_2 + 3x_3 \leq 185$
- $7x_0 + 9x_2 \leq 272$
- $9x_2 + 1x_3 \leq 146$
- $14x_1 + 1x_3 \leq 222$
- $7x_0 + 14x_1 + 9x_2 + 1x_3 \leq 222$

## Step 3: Implement the optimization problem using Gurobi
We will use the Gurobi Python library to model and solve this linear programming problem.

```python
import gurobi as gp

# Create a new model
m = gp.Model("optimization_problem")

# Define the variables
x0 = m.addVar(name="hours_worked_by_Laura", lb=0)  # hours worked by Laura
x1 = m.addVar(name="hours_worked_by_Paul", lb=0)   # hours worked by Paul
x2 = m.addVar(name="hours_worked_by_Bill", lb=0)   # hours worked by Bill
x3 = m.addVar(name="hours_worked_by_Mary", lb=0)  # hours worked by Mary

# Define the objective function
m.setObjective(9*x0 + 6*x1 + 5*x2 + 3*x3, gp.GRB.MAXIMIZE)

# Add constraints
m.addConstr(10*x0 + 14*x3 >= 46)
m.addConstr(10*x0 + 6*x2 >= 49)
m.addConstr(12*x1 + 6*x2 + 14*x3 >= 30)
m.addConstr(8*x1 + 3*x3 >= 47)
m.addConstr(9*x0 + 12*x2 + 3*x3 >= 49)
m.addConstr(7*x0 + 9*x2 >= 66)
m.addConstr(14*x1 + 9*x2 >= 27)
m.addConstr(9*x2 + 1*x3 >= 47)
m.addConstr(10*x0 + 6*x2 <= 142)
m.addConstr(12*x1 + 14*x3 <= 232)
m.addConstr(12*x1 + 6*x2 <= 153)
m.addConstr(10*x0 + 6*x2 + 14*x3 <= 172)
m.addConstr(10*x0 + 12*x1 + 6*x2 <= 138)
m.addConstr(10*x0 + 12*x1 + 6*x2 + 14*x3 <= 138)
m.addConstr(6*x0 + 7*x2 <= 129)
m.addConstr(6*x0 + 8*x1 <= 179)
m.addConstr(6*x0 + 11*x3 <= 195)
m.addConstr(8*x1 + 11*x3 <= 240)
m.addConstr(8*x1 + 7*x2 <= 164)
m.addConstr(8*x1 + 7*x2 + 11*x3 <= 174)
m.addConstr(6*x0 + 8*x1 + 7*x2 + 11*x3 <= 174)
m.addConstr(8*x1 + 12*x2 <= 177)
m.addConstr(8*x1 + 3*x3 <= 185)
m.addConstr(9*x0 + 8*x1 + 12*x2 + 3*x3 <= 185)
m.addConstr(7*x0 + 9*x2 <= 272)
m.addConstr(9*x2 + 1*x3 <= 146)
m.addConstr(14*x1 + 1*x3 <= 222)
m.addConstr(7*x0 + 14*x1 + 9*x2 + 1*x3 <= 222)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Hours worked by Laura: {x0.varValue}")
    print(f"Hours worked by Paul: {x1.varValue}")
    print(f"Hours worked by Bill: {x2.varValue}")
    print(f"Hours worked by Mary: {x3.varValue}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found.")
```