## Step 1: Define the symbolic representation of the variables
The variables are ['hours worked by Dale', 'hours worked by Paul', 'hours worked by Laura', 'hours worked by Jean']. 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 $5x_0 + 2x_1 + 5x_2 + 3x_3$.

## Step 3: List the constraints
The constraints are:
- $11x_0 \geq 11x_0$ (always true, but we note Dale's work quality rating is 11)
- $4x_0 \geq 4x_0$ (always true, but we note Dale's dollar cost per hour is 4)
- $9x_0 \geq 9x_0$ (always true, but we note Dale's productivity rating is 9)
- $7x_1 \geq 7x_1$ (always true, but we note Paul's work quality rating is 7)
- $1x_1 \geq 1x_1$ (always true, but we note Paul's dollar cost per hour is 1)
- $10x_1 \geq 10x_1$ (always true, but we note Paul's productivity rating is 10)
- $7x_2 \geq 7x_2$ (always true, but we note Laura's work quality rating is 7)
- $5x_2 \geq 5x_2$ (always true, but we note Laura's dollar cost per hour is 5)
- $1x_2 \geq 1x_2$ (always true, but we note Laura's productivity rating is 1)
- $3x_3 \geq 3x_3$ (always true, but we note Jean's work quality rating is 3)
- $10x_3 \geq 10x_3$ (always true, but we note Jean's dollar cost per hour is 10)
- $7x_3 \geq 7x_3$ (always true, but we note Jean's productivity rating is 7)
- $11x_0 + 7x_1 + 3x_3 \geq 16$
- $1x_1 + 5x_2 + 10x_3 \geq 14$
- $9x_0 + 7x_3 \geq 12$
- $9x_0 + 10x_1 \geq 21$
- $10x_1 + 1x_2 \geq 13$
- $-7x_2 + 1x_3 \geq 0$
- $7x_1 + 3x_3 \leq 81$
- $11x_0 + 3x_3 \leq 43$
- $7x_1 + 7x_2 \leq 31$
- $11x_0 + 7x_1 + 7x_2 + 3x_3 \leq 90$
- $11x_0 + 7x_1 + 3x_3 \leq 23$
- $11x_0 + 7x_1 + 7x_2 + 3x_3 \leq 23$
- $1x_1 + 10x_3 \leq 38$
- $1x_1 + 5x_2 \leq 33$
- $4x_0 + 10x_3 \leq 55$
- $4x_0 + 1x_1 + 10x_3 \leq 32$
- $4x_0 + 1x_1 + 5x_2 + 10x_3 \leq 32$
- $1x_2 + 7x_3 \leq 42$
- $10x_1 + 1x_2 + 7x_3 \leq 49$
- $9x_0 + 10x_1 + 7x_3 \leq 92$
- $9x_0 + 1x_2 + 7x_3 \leq 59$
- $9x_0 + 10x_1 + 1x_2 \leq 50$
- $9x_0 + 10x_1 + 1x_2 + 7x_3 \leq 50$

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

Subject to:
- $11x_0 + 7x_1 + 3x_3 \geq 16$
- $1x_1 + 5x_2 + 10x_3 \geq 14$
- $9x_0 + 7x_3 \geq 12$
- $9x_0 + 10x_1 \geq 21$
- $10x_1 + 1x_2 \geq 13$
- $-7x_2 + 1x_3 \geq 0$
- $7x_1 + 3x_3 \leq 81$
- $11x_0 + 3x_3 \leq 43$
- $7x_1 + 7x_2 \leq 31$
- $11x_0 + 7x_1 + 7x_2 + 3x_3 \leq 90$
- $11x_0 + 7x_1 + 3x_3 \leq 23$
- $11x_0 + 7x_1 + 7x_2 + 3x_3 \leq 23$
- $1x_1 + 10x_3 \leq 38$
- $1x_1 + 5x_2 \leq 33$
- $4x_0 + 10x_3 \leq 55$
- $4x_0 + 1x_1 + 10x_3 \leq 32$
- $4x_0 + 1x_1 + 5x_2 + 10x_3 \leq 32$
- $1x_2 + 7x_3 \leq 42$
- $10x_1 + 1x_2 + 7x_3 \leq 49$
- $9x_0 + 10x_1 + 7x_3 \leq 92$
- $9x_0 + 1x_2 + 7x_3 \leq 59$
- $9x_0 + 10x_1 + 1x_2 \leq 50$
- $9x_0 + 10x_1 + 1x_2 + 7x_3 \leq 50$

## 5: Implement the problem using Gurobi
```python
import gurobi as gp

# Define the model
m = gp.Model()

# Define the variables
x0 = m.addVar(name="x0", lb=0) # hours worked by Dale
x1 = m.addVar(name="x1", lb=0, integrality=gp.GRB.INTEGER) # hours worked by Paul
x2 = m.addVar(name="x2", lb=0) # hours worked by Laura
x3 = m.addVar(name="x3", lb=0) # hours worked by Jean

# Define the objective function
m.setObjective(5 * x0 + 2 * x1 + 5 * x2 + 3 * x3, gp.GRB.MAXIMIZE)

# Add constraints
m.addConstr(11 * x0 + 7 * x1 + 3 * x3 >= 16)
m.addConstr(1 * x1 + 5 * x2 + 10 * x3 >= 14)
m.addConstr(9 * x0 + 7 * x3 >= 12)
m.addConstr(9 * x0 + 10 * x1 >= 21)
m.addConstr(10 * x1 + 1 * x2 >= 13)
m.addConstr(-7 * x2 + 1 * x3 >= 0)
m.addConstr(7 * x1 + 3 * x3 <= 81)
m.addConstr(11 * x0 + 3 * x3 <= 43)
m.addConstr(7 * x1 + 7 * x2 <= 31)
m.addConstr(11 * x0 + 7 * x1 + 7 * x2 + 3 * x3 <= 90)
m.addConstr(11 * x0 + 7 * x1 + 3 * x3 <= 23)
m.addConstr(11 * x0 + 7 * x1 + 7 * x2 + 3 * x3 <= 23)
m.addConstr(1 * x1 + 10 * x3 <= 38)
m.addConstr(1 * x1 + 5 * x2 <= 33)
m.addConstr(4 * x0 + 10 * x3 <= 55)
m.addConstr(4 * x0 + 1 * x1 + 10 * x3 <= 32)
m.addConstr(4 * x0 + 1 * x1 + 5 * x2 + 10 * x3 <= 32)
m.addConstr(1 * x2 + 7 * x3 <= 42)
m.addConstr(10 * x1 + 1 * x2 + 7 * x3 <= 49)
m.addConstr(9 * x0 + 10 * x1 + 7 * x3 <= 92)
m.addConstr(9 * x0 + 1 * x2 + 7 * x3 <= 59)
m.addConstr(9 * x0 + 10 * x1 + 1 * x2 <= 50)
m.addConstr(9 * x0 + 10 * x1 + 1 * x2 + 7 * x3 <= 50)

# Solve the model
m.optimize()

# Print the solution
print("Objective: ", m.objVal)
print("x0: ", x0.varValue)
print("x1: ", x1.varValue)
print("x2: ", x2.varValue)
print("x3: ", x3.varValue)

# Symbolic representation
print("Symbolic Representation:")
print("Variables: ")
print([('x0', 'hours worked by Dale'), ('x1', 'hours worked by Paul'), ('x2', 'hours worked by Laura'), ('x3', 'hours worked by Jean')])
print("Objective Function: 5*x0 + 2*x1 + 5*x2 + 3*x3")
print("Constraints: ")
print([
    "11*x0 + 7*x1 + 3*x3 >= 16",
    "1*x1 + 5*x2 + 10*x3 >= 14",
    "9*x0 + 7*x3 >= 12",
    "9*x0 + 10*x1 >= 21",
    "10*x1 + 1*x2 >= 13",
    "-7*x2 + 1*x3 >= 0",
    "7*x1 + 3*x3 <= 81",
    "11*x0 + 3*x3 <= 43",
    "7*x1 + 7*x2 <= 31",
    "11*x0 + 7*x1 + 7*x2 + 3*x3 <= 90",
    "11*x0 + 7*x1 + 3*x3 <= 23",
    "11*x0 + 7*x1 + 7*x2 + 3*x3 <= 23",
    "1*x1 + 10*x3 <= 38",
    "1*x1 + 5*x2 <= 33",
    "4*x0 + 10*x3 <= 55",
    "4*x0 + 1*x1 + 10*x3 <= 32",
    "4*x0 + 1*x1 + 5*x2 + 10*x3 <= 32",
    "1*x2 + 7*x3 <= 42",
    "10*x1 + 1*x2 + 7*x3 <= 49",
    "9*x0 + 10*x1 + 7*x3 <= 92",
    "9*x0 + 1*x2 + 7*x3 <= 59",
    "9*x0 + 10*x1 + 1*x2 <= 50",
    "9*x0 + 10*x1 + 1*x2 + 7*x3 <= 50"
])

```
The final solution is 
```json
{
    'sym_variables': [('x0', 'hours worked by Dale'), ('x1', 'hours worked by Paul'), ('x2', 'hours worked by Laura'), ('x3', 'hours worked by Jean')],
    'objective_function': '5*x0 + 2*x1 + 5*x2 + 3*x3',
    'constraints': [
        '11*x0 + 7*x1 + 3*x3 >= 16',
        '1*x1 + 5*x2 + 10*x3 >= 14',
        '9*x0 + 7*x3 >= 12',
        '9*x0 + 10*x1 >= 21',
        '10*x1 + 1*x2 >= 13',
        '-7*x2 + 1*x3 >= 0',
        '7*x1 + 3*x3 <= 81',
        '11*x0 + 3*x3 <= 43',
        '7*x1 + 7*x2 <= 31',
        '11*x0 + 7*x1 + 7*x2 + 3*x3 <= 90',
        '11*x0 + 7*x1 + 3*x3 <= 23',
        '11*x0 + 7*x1 + 7*x2 + 3*x3 <= 23',
        '1*x1 + 10*x3 <= 38',
        '1*x1 + 5*x2 <= 33',
        '4*x0 + 10*x3 <= 55',
        '4*x0 + 1*x1 + 10*x3 <= 32',
        '4*x0 + 1*x1 + 5*x2 + 10*x3 <= 32',
        '1*x2 + 7*x3 <= 42',
        '10*x1 + 1*x2 + 7*x3 <= 49',
        '9*x0 + 10*x1 + 7*x3 <= 92',
        '9*x0 + 1*x2 + 7*x3 <= 59',
        '9*x0 + 10*x1 + 1*x2 <= 50',
        '9*x0 + 10*x1 + 1*x2 + 7*x3 <= 50'
    ]
}
```