## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Laura', 'hours worked by Ringo', 'hours worked by Mary', and '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 $3x_0 + 1x_1 + 3x_2 + 6x_3$.

## Step 3: List the constraints
The constraints are:
- $9x_0 \geq 45$ (productivity rating for Laura and Dale combined is not directly given, but we have individual ratings)
- $13x_1 + 31x_2 \geq 98$ (productivity rating for Ringo and Mary)
- $9x_0 + 13x_1 + 22x_3 \geq 83$ (productivity rating for Laura, Ringo, and Dale)
- $10x_0 + 6x_2 \geq 43$ (dollar cost per hour for Laura and Mary)
- $6x_2 + 18x_3 \geq 68$ (dollar cost per hour for Mary and Dale)
- $6x_0 + 4x_3 \geq 24$ (work quality rating for Laura and Dale)
- $6x_0 + 8x_2 \geq 49$ (work quality rating for Laura and Mary)
- $6x_0 + 1x_1 + 4x_3 \geq 56$ (work quality rating for Laura, Ringo, and Dale)
- $6x_0 + 1x_1 + 8x_2 \geq 56$ (work quality rating for Laura, Ringo, and Mary)
- $6x_0 + 1x_1 + 4x_3 \geq 32$ (another work quality rating for Laura, Ringo, and Dale)
- $6x_0 + 1x_1 + 8x_2 \geq 32$ (another work quality rating for Laura, Ringo, and Mary)
- $13x_1 + 31x_2 \leq 424$ (productivity rating for Ringo and Mary)
- $31x_2 + 22x_3 \leq 336$ (productivity rating for Mary and Dale)
- $9x_0 + 22x_3 \leq 316$ (productivity rating for Laura and Dale)
- $13x_1 + 22x_3 \leq 262$ (productivity rating for Ringo and Dale)
- $9x_0 + 31x_2 \leq 390$ (productivity rating for Laura and Mary)
- $13x_1 + 31x_2 + 22x_3 \leq 289$ (productivity rating for Ringo, Mary, and Dale)
- $9x_0 + 13x_1 + 22x_3 \leq 126$ (productivity rating for Laura, Ringo, and Dale)
- $9x_0 + 13x_1 + 31x_2 \leq 211$ (productivity rating for Laura, Ringo, and Mary)
- $9x_0 + 13x_1 + 31x_2 + 22x_3 \leq 211$ (productivity rating for all)
- $10x_0 + 6x_2 \leq 266$ (dollar cost per hour for Laura and Mary)
- $6x_2 + 18x_3 \leq 166$ (dollar cost per hour for Mary and Dale)
- $5x_1 + 6x_2 \leq 170$ (dollar cost per hour for Ringo and Mary)
- $10x_0 + 18x_3 \leq 93$ (dollar cost per hour for Laura and Dale)
- $5x_1 + 18x_3 \leq 221$ (dollar cost per hour for Ringo and Dale)
- $10x_0 + 5x_1 + 6x_2 + 18x_3 \leq 221$ (dollar cost per hour for all)
- $1x_1 + 8x_2 \leq 60$ (work quality rating for Ringo and Mary)
- $6x_0 + 8x_2 \leq 71$ (work quality rating for Laura and Mary)
- $1x_1 + 8x_2 + 4x_3 \leq 63$ (work quality rating for Ringo, Mary, and Dale)
- $6x_0 + 1x_1 + 4x_3 \leq 182$ (work quality rating for Laura, Ringo, and Dale)
- $6x_0 + 8x_2 + 4x_3 \leq 76$ (work quality rating for Laura, Mary, and Dale)
- $6x_0 + 1x_1 + 8x_2 + 4x_3 \leq 76$ (work quality rating for all)
- $21x_0 + 12x_2 \leq 193$ (likelihood to quit index for Laura and Mary)
- $12x_2 + 31x_3 \leq 251$ (likelihood to quit index for Mary and Dale)
- $21x_0 + 30x_1 + 12x_2 + 31x_3 \leq 251$ (likelihood to quit index for all)

## 4: Convert the problem into Gurobi code
```python
import gurobi

def optimize_problem():
    # Create a new Gurobi model
    model = gurobi.Model()

    # Define the variables
    x0 = model.addVar(name="x0", lb=0)  # hours worked by Laura
    x1 = model.addVar(name="x1", lb=0)  # hours worked by Ringo
    x2 = model.addVar(name="x2", lb=0)  # hours worked by Mary
    x3 = model.addVar(name="x3", lb=0)  # hours worked by Dale

    # Define the objective function
    model.setObjective(3*x0 + x1 + 3*x2 + 6*x3, gurobi.GRB.MAXIMIZE)

    # Add constraints
    model.addConstr(9*x0 + 22*x3 >= 45)
    model.addConstr(13*x1 + 31*x2 >= 98)
    model.addConstr(9*x0 + 13*x1 + 22*x3 >= 83)
    model.addConstr(10*x0 + 6*x2 >= 43)
    model.addConstr(6*x2 + 18*x3 >= 68)
    model.addConstr(6*x0 + 4*x3 >= 24)
    model.addConstr(6*x0 + 8*x2 >= 49)
    model.addConstr(6*x0 + x1 + 4*x3 >= 56)
    model.addConstr(6*x0 + x1 + 8*x2 >= 56)
    model.addConstr(6*x0 + x1 + 4*x3 >= 32)
    model.addConstr(6*x0 + x1 + 8*x2 >= 32)
    model.addConstr(13*x1 + 31*x2 <= 424)
    model.addConstr(31*x2 + 22*x3 <= 336)
    model.addConstr(9*x0 + 22*x3 <= 316)
    model.addConstr(13*x1 + 22*x3 <= 262)
    model.addConstr(9*x0 + 31*x2 <= 390)
    model.addConstr(13*x1 + 31*x2 + 22*x3 <= 289)
    model.addConstr(9*x0 + 13*x1 + 22*x3 <= 126)
    model.addConstr(9*x0 + 13*x1 + 31*x2 <= 211)
    model.addConstr(9*x0 + 13*x1 + 31*x2 + 22*x3 <= 211)
    model.addConstr(10*x0 + 6*x2 <= 266)
    model.addConstr(6*x2 + 18*x3 <= 166)
    model.addConstr(5*x1 + 6*x2 <= 170)
    model.addConstr(10*x0 + 18*x3 <= 93)
    model.addConstr(5*x1 + 18*x3 <= 221)
    model.addConstr(10*x0 + 5*x1 + 6*x2 + 18*x3 <= 221)
    model.addConstr(x1 + 8*x2 <= 60)
    model.addConstr(6*x0 + 8*x2 <= 71)
    model.addConstr(x1 + 8*x2 + 4*x3 <= 63)
    model.addConstr(6*x0 + x1 + 4*x3 <= 182)
    model.addConstr(6*x0 + 8*x2 + 4*x3 <= 76)
    model.addConstr(6*x0 + x1 + 8*x2 + 4*x3 <= 76)
    model.addConstr(21*x0 + 12*x2 <= 193)
    model.addConstr(12*x2 + 31*x3 <= 251)
    model.addConstr(21*x0 + 30*x1 + 12*x2 + 31*x3 <= 251)

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objVal)
        print("Hours worked by Laura: ", x0.varValue)
        print("Hours worked by Ringo: ", x1.varValue)
        print("Hours worked by Mary: ", x2.varValue)
        print("Hours worked by Dale: ", x3.varValue)
    else:
        print("The model is infeasible")

optimize_problem()
```

## 5: Symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Laura'), 
        ('x1', 'hours worked by Ringo'), 
        ('x2', 'hours worked by Mary'), 
        ('x3', 'hours worked by Dale')
    ], 
    'objective_function': '3*x0 + x1 + 3*x2 + 6*x3', 
    'constraints': [
        '9*x0 + 22*x3 >= 45',
        '13*x1 + 31*x2 >= 98',
        '9*x0 + 13*x1 + 22*x3 >= 83',
        '10*x0 + 6*x2 >= 43',
        '6*x2 + 18*x3 >= 68',
        '6*x0 + 4*x3 >= 24',
        '6*x0 + 8*x2 >= 49',
        '6*x0 + x1 + 4*x3 >= 56',
        '6*x0 + x1 + 8*x2 >= 56',
        '6*x0 + x1 + 4*x3 >= 32',
        '6*x0 + x1 + 8*x2 >= 32',
        '13*x1 + 31*x2 <= 424',
        '31*x2 + 22*x3 <= 336',
        '9*x0 + 22*x3 <= 316',
        '13*x1 + 22*x3 <= 262',
        '9*x0 + 31*x2 <= 390',
        '13*x1 + 31*x2 + 22*x3 <= 289',
        '9*x0 + 13*x1 + 22*x3 <= 126',
        '9*x0 + 13*x1 + 31*x2 <= 211',
        '9*x0 + 13*x1 + 31*x2 + 22*x3 <= 211',
        '10*x0 + 6*x2 <= 266',
        '6*x2 + 18*x3 <= 166',
        '5*x1 + 6*x2 <= 170',
        '10*x0 + 18*x3 <= 93',
        '5*x1 + 18*x3 <= 221',
        '10*x0 + 5*x1 + 6*x2 + 18*x3 <= 221',
        'x1 + 8*x2 <= 60',
        '6*x0 + 8*x2 <= 71',
        'x1 + 8*x2 + 4*x3 <= 63',
        '6*x0 + x1 + 4*x3 <= 182',
        '6*x0 + 8*x2 + 4*x3 <= 76',
        '6*x0 + x1 + 8*x2 + 4*x3 <= 76',
        '21*x0 + 12*x2 <= 193',
        '12*x2 + 31*x3 <= 251',
        '21*x0 + 30*x1 + 12*x2 + 31*x3 <= 251'
    ]
}
```