## Step 1: Define the symbolic representation of the variables
The variables are ['hours worked by Paul', 'hours worked by Peggy', 'hours worked by Jean', 'hours worked by Dale']. Let's denote them as $x_0$, $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Define the symbolic representation of the problem
The objective function to maximize is $6x_0 + 3x_1 + 2x_2 + 6x_3$.

## Step 3: List the constraints
1. $21x_0 \geq 0$ is not needed as $x_0 \geq 0$ is implied,
2. $x_0 \geq 0$,
3. $x_1 \geq 0$,
4. $x_2 \geq 0$,
5. $x_3 \geq 0$,
6. $21x_0 + 21x_2 \geq 33$,
7. $21x_0 + 16x_1 \geq 38$,
8. $16x_1 + 21x_2 \geq 33$,
9. $16x_1 + 21x_2 + 17x_3 \geq 50$,
10. $21x_0 + 21x_2 + 17x_3 \geq 50$,
11. $16x_1 + 21x_2 + 17x_3 \geq 50$,
12. $21x_0 + 21x_2 + 17x_3 \geq 50$,
13. $x_0 + x_1 + x_2 + x_3$ has no direct constraint but is bounded by others,
14. $x_1 + 14x_2 \geq 0$ is not needed as $x_1, x_2 \geq 0$,
15. $17x_2 + 17x_3 \leq 103$,
16. $16x_1 + 21x_2 \leq 119$,
17. $16x_1 + 17x_3 \leq 87$,
18. $21x_0 + 21x_2 + 17x_3 \leq 53$,
19. $21x_0 + 16x_1 + 21x_2 + 17x_3 \leq 53$,
20. $x_1 + 14x_2 \leq 243$,
21. $x_0 + 19x_3 \leq 222$,
22. $x_0 + x_1 \leq 194$,
23. $x_0 + 14x_2 \leq 193$,
24. $14x_2 + 19x_3 \leq 71$,
25. $x_1 + 14x_2 + 19x_3 \leq 94$,
26. $x_0 + x_1 + 14x_2 + 19x_3 \leq 94$.

## Step 4: Correct and Simplify Constraints
Some constraints are redundant or not directly provided. Focus on provided and meaningful constraints.

## 5: Write down the problem in a standard form
Maximize $6x_0 + 3x_1 + 2x_2 + 6x_3$

Subject to:
- $21x_0 + 21x_2 \geq 33$
- $21x_0 + 16x_1 \geq 38$
- $16x_1 + 21x_2 \geq 33$
- $16x_1 + 21x_2 + 17x_3 \geq 50$
- $21x_0 + 21x_2 + 17x_3 \geq 50$
- $17x_2 + 17x_3 \leq 103$
- $16x_1 + 21x_2 \leq 119$
- $16x_1 + 17x_3 \leq 87$
- $21x_0 + 21x_2 + 17x_3 \leq 53$
- $21x_0 + 16x_1 + 21x_2 + 17x_3 \leq 53$
- $x_1 + 14x_2 \leq 243$
- $x_0 + 19x_3 \leq 222$
- $x_0 + x_1 \leq 194$
- $x_0 + 14x_2 \leq 193$
- $14x_2 + 19x_3 \leq 71$
- $x_1 + 14x_2 + 19x_3 \leq 94$
- $x_0 + x_1 + 14x_2 + 19x_3 \leq 94$
- $x_0, x_1, x_2, x_3 \geq 0$

## 6: Provide Symbolic Representation
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Paul'), 
        ('x1', 'hours worked by Peggy'), 
        ('x2', 'hours worked by Jean'), 
        ('x3', 'hours worked by Dale')
    ], 
    'objective_function': '6*x0 + 3*x1 + 2*x2 + 6*x3', 
    'constraints': [
        '21*x0 + 21*x2 >= 33',
        '21*x0 + 16*x1 >= 38',
        '16*x1 + 21*x2 >= 33',
        '16*x1 + 21*x2 + 17*x3 >= 50',
        '21*x0 + 21*x2 + 17*x3 >= 50',
        '17*x2 + 17*x3 <= 103',
        '16*x1 + 21*x2 <= 119',
        '16*x1 + 17*x3 <= 87',
        '21*x0 + 21*x2 + 17*x3 <= 53',
        '21*x0 + 16*x1 + 21*x2 + 17*x3 <= 53',
        'x1 + 14*x2 <= 243',
        'x0 + 19*x3 <= 222',
        'x0 + x1 <= 194',
        'x0 + 14*x2 <= 193',
        '14*x2 + 19*x3 <= 71',
        'x1 + 14*x2 + 19*x3 <= 94',
        'x0 + x1 + 14*x2 + 19*x3 <= 94',
        'x0 >= 0',
        'x1 >= 0',
        'x2 >= 0',
        'x3 >= 0'
    ]
}
```

## 7: Gurobi Code
```python
import gurobi

def solve_optimization_problem():
    model = gurobi.Model()

    # Define variables
    x0 = model.addVar(name="x0", lb=0)  # hours worked by Paul
    x1 = model.addVar(name="x1", lb=0)  # hours worked by Peggy
    x2 = model.addVar(name="x2", lb=0)  # hours worked by Jean
    x3 = model.addVar(name="x3", lb=0)  # hours worked by Dale

    # Objective function
    model.setObjective(6*x0 + 3*x1 + 2*x2 + 6*x3, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(21*x0 + 21*x2 >= 33)
    model.addConstr(21*x0 + 16*x1 >= 38)
    model.addConstr(16*x1 + 21*x2 >= 33)
    model.addConstr(16*x1 + 21*x2 + 17*x3 >= 50)
    model.addConstr(21*x0 + 21*x2 + 17*x3 >= 50)
    model.addConstr(17*x2 + 17*x3 <= 103)
    model.addConstr(16*x1 + 21*x2 <= 119)
    model.addConstr(16*x1 + 17*x3 <= 87)
    model.addConstr(21*x0 + 21*x2 + 17*x3 <= 53)
    model.addConstr(21*x0 + 16*x1 + 21*x2 + 17*x3 <= 53)
    model.addConstr(x1 + 14*x2 <= 243)
    model.addConstr(x0 + 19*x3 <= 222)
    model.addConstr(x0 + x1 <= 194)
    model.addConstr(x0 + 14*x2 <= 193)
    model.addConstr(14*x2 + 19*x3 <= 71)
    model.addConstr(x1 + 14*x2 + 19*x3 <= 94)
    model.addConstr(x0 + x1 + 14*x2 + 19*x3 <= 94)

    # Solve the model
    model.optimize()

    if model.status == gurobi.GRB.Status.OPTIMAL:
        print("Objective: ", model.objVal)
        print("Hours worked by Paul: ", x0.varValue)
        print("Hours worked by Peggy: ", x1.varValue)
        print("Hours worked by Jean: ", x2.varValue)
        print("Hours worked by Dale: ", x3.varValue)
    else:
        print("No solution found")

solve_optimization_problem()
```