## Step 1: Define the variables and their corresponding symbolic notation
Let's denote the hours worked by Peggy, Dale, and John as $x_0$, $x_1$, and $x_2$ respectively.

## Step 2: Convert the given problem into a symbolic representation
The objective function to minimize is $1x_0 + 5x_1 + 2x_2$.

## 3: List all the constraints in symbolic notation
- $20x_0 = 20$
- $9x_0 = 9$
- $13x_0 = 13$
- $10x_0 = 10$
- $4x_0 = 4$
- $2x_1 = 2$
- $6x_1 = 6$
- $22x_1 = 22$
- $14x_1 = 14$
- $12x_1 = 12$
- $6x_2 = 6$
- $19x_2 = 19$
- $15x_2 = 15$
- $12x_2 = 12$
- $8x_2 = 8$
- $20x_0 + 6x_2 \geq 71$
- $2x_1 + 6x_2 \geq 34$
- $20x_0 + 2x_1 + 6x_2 \geq 34$
- $9x_0 + 19x_2 \geq 105$
- $6x_1 + 19x_2 \geq 99$
- $9x_0 + 6x_1 + 19x_2 \geq 99$
- $13x_0 + 22x_1 \geq 56$
- $13x_0 + 15x_2 \geq 79$
- $22x_1 + 15x_2 \geq 90$
- $13x_0 + 22x_1 + 15x_2 \geq 90$
- $10x_0 + 14x_1 \geq 105$
- $10x_0 + 12x_2 \geq 109$
- $14x_1 + 12x_2 \geq 54$
- $10x_0 + 14x_1 + 12x_2 \geq 88$
- $4x_0 + 8x_2 \geq 30$
- $4x_0 + 12x_1 + 8x_2 \geq 30$
- $7x_0 - 10x_1 \geq 0$
- $-10x_0 + 5x_2 \geq 0$
- $9x_0 + 6x_1 \leq 273$
- $9x_0 + 19x_2 \leq 204$
- $13x_0 + 22x_1 \leq 214$
- $22x_1 + 15x_2 \leq 190$
- $13x_0 + 22x_1 + 15x_2 \leq 211$
- $12x_1 + 8x_2 \leq 73$
- $4x_0 + 8x_2 \leq 139$
- $4x_0 + 12x_1 + 8x_2 \leq 60$

## 4: Create a symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Peggy'),
        ('x1', 'hours worked by Dale'),
        ('x2', 'hours worked by John')
    ],
    'objective_function': '1*x0 + 5*x1 + 2*x2',
    'constraints': [
        '20*x0 = 20',
        '9*x0 = 9',
        '13*x0 = 13',
        '10*x0 = 10',
        '4*x0 = 4',
        '2*x1 = 2',
        '6*x1 = 6',
        '22*x1 = 22',
        '14*x1 = 14',
        '12*x1 = 12',
        '6*x2 = 6',
        '19*x2 = 19',
        '15*x2 = 15',
        '12*x2 = 12',
        '8*x2 = 8',
        '20*x0 + 6*x2 >= 71',
        '2*x1 + 6*x2 >= 34',
        '20*x0 + 2*x1 + 6*x2 >= 34',
        '9*x0 + 19*x2 >= 105',
        '6*x1 + 19*x2 >= 99',
        '9*x0 + 6*x1 + 19*x2 >= 99',
        '13*x0 + 22*x1 >= 56',
        '13*x0 + 15*x2 >= 79',
        '22*x1 + 15*x2 >= 90',
        '13*x0 + 22*x1 + 15*x2 >= 90',
        '10*x0 + 14*x1 >= 105',
        '10*x0 + 12*x2 >= 109',
        '14*x1 + 12*x2 >= 54',
        '10*x0 + 14*x1 + 12*x2 >= 88',
        '4*x0 + 8*x2 >= 30',
        '4*x0 + 12*x1 + 8*x2 >= 30',
        '7*x0 - 10*x1 >= 0',
        '-10*x0 + 5*x2 >= 0',
        '9*x0 + 6*x1 <= 273',
        '9*x0 + 19*x2 <= 204',
        '13*x0 + 22*x1 <= 214',
        '22*x1 + 15*x2 <= 190',
        '13*x0 + 22*x1 + 15*x2 <= 211',
        '12*x1 + 8*x2 <= 73',
        '4*x0 + 8*x2 <= 139',
        '4*x0 + 12*x1 + 8*x2 <= 60'
    ]
}
```

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

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

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

    # Objective function
    model.setObjective(1 * x0 + 5 * x1 + 2 * x2, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(20 * x0 == 20)
    model.addConstr(9 * x0 == 9)
    model.addConstr(13 * x0 == 13)
    model.addConstr(10 * x0 == 10)
    model.addConstr(4 * x0 == 4)
    model.addConstr(2 * x1 == 2)
    model.addConstr(6 * x1 == 6)
    model.addConstr(22 * x1 == 22)
    model.addConstr(14 * x1 == 14)
    model.addConstr(12 * x1 == 12)
    model.addConstr(6 * x2 == 6)
    model.addConstr(19 * x2 == 19)
    model.addConstr(15 * x2 == 15)
    model.addConstr(12 * x2 == 12)
    model.addConstr(8 * x2 == 8)
    model.addConstr(20 * x0 + 6 * x2 >= 71)
    model.addConstr(2 * x1 + 6 * x2 >= 34)
    model.addConstr(20 * x0 + 2 * x1 + 6 * x2 >= 34)
    model.addConstr(9 * x0 + 19 * x2 >= 105)
    model.addConstr(6 * x1 + 19 * x2 >= 99)
    model.addConstr(9 * x0 + 6 * x1 + 19 * x2 >= 99)
    model.addConstr(13 * x0 + 22 * x1 >= 56)
    model.addConstr(13 * x0 + 15 * x2 >= 79)
    model.addConstr(22 * x1 + 15 * x2 >= 90)
    model.addConstr(13 * x0 + 22 * x1 + 15 * x2 >= 90)
    model.addConstr(10 * x0 + 14 * x1 >= 105)
    model.addConstr(10 * x0 + 12 * x2 >= 109)
    model.addConstr(14 * x1 + 12 * x2 >= 54)
    model.addConstr(10 * x0 + 14 * x1 + 12 * x2 >= 88)
    model.addConstr(4 * x0 + 8 * x2 >= 30)
    model.addConstr(4 * x0 + 12 * x1 + 8 * x2 >= 30)
    model.addConstr(7 * x0 - 10 * x1 >= 0)
    model.addConstr(-10 * x0 + 5 * x2 >= 0)
    model.addConstr(9 * x0 + 6 * x1 <= 273)
    model.addConstr(9 * x0 + 19 * x2 <= 204)
    model.addConstr(13 * x0 + 22 * x1 <= 214)
    model.addConstr(22 * x1 + 15 * x2 <= 190)
    model.addConstr(13 * x0 + 22 * x1 + 15 * x2 <= 211)
    model.addConstr(12 * x1 + 8 * x2 <= 73)
    model.addConstr(4 * x0 + 8 * x2 <= 139)
    model.addConstr(4 * x0 + 12 * x1 + 8 * x2 <= 60)

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print("Hours worked by Peggy:", x0.varValue)
        print("Hours worked by Dale:", x1.varValue)
        print("Hours worked by John:", x2.varValue)
        print("Objective function value:", model.objVal)
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```