## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Dale', 'hours worked by Laura', and 'hours worked by Bobby'. Let's denote them as $x_0$, $x_1$, and $x_2$ respectively.

## Step 2: Define the objective function in symbolic notation
The objective function to minimize is $1 \cdot x_0 + 8 \cdot x_1 + 7 \cdot x_2$.

## Step 3: Define the constraints in symbolic notation
The constraints are:
- $3x_0 \leq 77$
- $9x_0 \leq 112$
- $7x_1 \leq 77$
- $3x_1 \leq 112$
- $6x_2 \leq 77$
- $2x_2 \leq 112$
- $3x_0 + 6x_2 \geq 8$
- $7x_1 + 6x_2 \geq 20$
- $3x_0 + 7x_1 + 6x_2 \geq 20$
- $9x_0 + 3x_1 \geq 31$
- $9x_0 + 3x_1 + 2x_2 \geq 31$
- $6x_0 - 4x_1 \geq 0$
- $-9x_1 + 4x_2 \geq 0$
- $3x_0 + 6x_2 \leq 27$
- $3x_0 + 7x_1 \leq 68$
- $7x_1 + 6x_2 \leq 45$
- $9x_0 + 3x_1 + 2x_2 \leq 77$
- $x_0 \in \mathbb{Z}$
- $x_1 \in \mathbb{Z}$
- $x_2 \in \mathbb{Z}$

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

# Create a new model
m = gurobi.Model()

# Define the variables
x0 = m.addVar(name="hours_worked_by_Dale", vtype=gurobi.GRB.INTEGER)
x1 = m.addVar(name="hours_worked_by_Laura", vtype=gurobi.GRB.INTEGER)
x2 = m.addVar(name="hours_worked_by_Bobby", vtype=gurobi.GRB.INTEGER)

# Objective function
m.setObjective(x0 + 8 * x1 + 7 * x2, gurobi.GRB.MINIMIZE)

# Constraints
m.addConstr(3 * x0 <= 77)
m.addConstr(9 * x0 <= 112)
m.addConstr(7 * x1 <= 77)
m.addConstr(3 * x1 <= 112)
m.addConstr(6 * x2 <= 77)
m.addConstr(2 * x2 <= 112)
m.addConstr(3 * x0 + 6 * x2 >= 8)
m.addConstr(7 * x1 + 6 * x2 >= 20)
m.addConstr(3 * x0 + 7 * x1 + 6 * x2 >= 20)
m.addConstr(9 * x0 + 3 * x1 >= 31)
m.addConstr(9 * x0 + 3 * x1 + 2 * x2 >= 31)
m.addConstr(6 * x0 - 4 * x1 >= 0)
m.addConstr(-9 * x1 + 4 * x2 >= 0)
m.addConstr(3 * x0 + 6 * x2 <= 27)
m.addConstr(3 * x0 + 7 * x1 <= 68)
m.addConstr(7 * x1 + 6 * x2 <= 45)
m.addConstr(9 * x0 + 3 * x1 + 2 * x2 <= 77)

# Optimize the model
m.optimize()

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

## 5: Symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Dale'), 
        ('x1', 'hours worked by Laura'), 
        ('x2', 'hours worked by Bobby')
    ], 
    'objective_function': 'x0 + 8x1 + 7x2', 
    'constraints': [
        '3x0 <= 77', 
        '9x0 <= 112', 
        '7x1 <= 77', 
        '3x1 <= 112', 
        '6x2 <= 77', 
        '2x2 <= 112', 
        '3x0 + 6x2 >= 8', 
        '7x1 + 6x2 >= 20', 
        '3x0 + 7x1 + 6x2 >= 20', 
        '9x0 + 3x1 >= 31', 
        '9x0 + 3x1 + 2x2 >= 31', 
        '6x0 - 4x1 >= 0', 
        '-9x1 + 4x2 >= 0', 
        '3x0 + 6x2 <= 27', 
        '3x0 + 7x1 <= 68', 
        '7x1 + 6x2 <= 45', 
        '9x0 + 3x1 + 2x2 <= 77'
    ]
}
```