## 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 minimize is $2.91x_0 + 4.55x_1 + 2.66x_2 + 5.97x_3$.

## Step 3: List all the constraints
Constraints:
- $9x_0 \leq 291$
- $10x_0 \leq 231$
- $7x_0 \leq 394$
- $8x_1 \leq 291$
- $26x_1 \leq 231$
- $24x_1 \leq 394$
- $27x_2 \leq 291$
- $31x_2 \leq 231$
- $29x_2 \leq 394$
- $11x_3 \leq 291$
- $9x_3 \leq 231$
- $29x_3 \leq 394$
- $8x_1 + 11x_3 \geq 71$
- $9x_0 + 11x_3 \geq 39$
- $9x_0 + 8x_1 \geq 61$
- $9x_0 + 27x_2 + 11x_3 \geq 72$
- $9x_0 + 29x_2 + 11x_3 \geq 72$
- $8x_1 + 27x_2 + 11x_3 \geq 72$
- $9x_0 + 27x_2 + 11x_3 \geq 52$
- $9x_0 + 29x_2 + 11x_3 \geq 52$
- $8x_1 + 29x_2 + 11x_3 \geq 52$
- $9x_0 + 8x_1 + 27x_2 \geq 66$
- $9x_0 + 29x_2 + 11x_3 \geq 66$
- $8x_1 + 27x_2 + 11x_3 \geq 66$
- $9x_0 + 8x_1 + 27x_2 + 11x_3 \geq 66$
- $26x_1 + 31x_2 \geq 45$
- $26x_1 + 9x_3 \geq 50$
- $26x_1 + 31x_2 + 9x_3 \geq 56$
- $10x_0 + 26x_1 + 31x_2 + 9x_3 \geq 56$
- $7x_0 + 24x_1 \geq 87$
- $29x_2 + 29x_3 \geq 37$
- $7x_0 + 29x_2 \geq 34$
- $7x_0 + 29x_3 \geq 34$
- $7x_0 + 24x_1 + 29x_2 + 29x_3 \geq 34$
- $4x_0 - x_1 \geq 0$
- $9x_0 + 8x_1 \leq 209$
- $8x_1 + 11x_3 \leq 209$
- $8x_1 + 27x_2 \leq 78$
- $9x_0 + 27x_2 \leq 269$
- $9x_0 + 11x_3 \leq 176$
- $9x_0 + 8x_1 + 11x_3 \leq 109$
- $9x_0 + 27x_2 + 11x_3 \leq 108$
- $10x_0 + 31x_2 \leq 109$
- $26x_1 + 9x_3 \leq 138$
- $10x_0 + 31x_2 + 9x_3 \leq 167$
- $26x_1 + 31x_2 + 9x_3 \leq 177$
- $10x_0 + 26x_1 + 31x_2 \leq 93$

## 4: Write down the problem in JSON format
```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': '2.91*x0 + 4.55*x1 + 2.66*x2 + 5.97*x3',
    'constraints': [
        '9*x0 <= 291',
        '10*x0 <= 231',
        '7*x0 <= 394',
        '8*x1 <= 291',
        '26*x1 <= 231',
        '24*x1 <= 394',
        '27*x2 <= 291',
        '31*x2 <= 231',
        '29*x2 <= 394',
        '11*x3 <= 291',
        '9*x3 <= 231',
        '29*x3 <= 394',
        '8*x1 + 11*x3 >= 71',
        '9*x0 + 11*x3 >= 39',
        '9*x0 + 8*x1 >= 61',
        '9*x0 + 27*x2 + 11*x3 >= 72',
        '9*x0 + 29*x2 + 11*x3 >= 72',
        '8*x1 + 27*x2 + 11*x3 >= 72',
        '9*x0 + 27*x2 + 11*x3 >= 52',
        '9*x0 + 29*x2 + 11*x3 >= 52',
        '8*x1 + 29*x2 + 11*x3 >= 52',
        '9*x0 + 8*x1 + 27*x2 >= 66',
        '9*x0 + 29*x2 + 11*x3 >= 66',
        '8*x1 + 27*x2 + 11*x3 >= 66',
        '9*x0 + 8*x1 + 27*x2 + 11*x3 >= 66',
        '26*x1 + 31*x2 >= 45',
        '26*x1 + 9*x3 >= 50',
        '26*x1 + 31*x2 + 9*x3 >= 56',
        '10*x0 + 26*x1 + 31*x2 + 9*x3 >= 56',
        '7*x0 + 24*x1 >= 87',
        '29*x2 + 29*x3 >= 37',
        '7*x0 + 29*x2 >= 34',
        '7*x0 + 29*x3 >= 34',
        '7*x0 + 24*x1 + 29*x2 + 29*x3 >= 34',
        '4*x0 - x1 >= 0',
        '9*x0 + 8*x1 <= 209',
        '8*x1 + 11*x3 <= 209',
        '8*x1 + 27*x2 <= 78',
        '9*x0 + 27*x2 <= 269',
        '9*x0 + 11*x3 <= 176',
        '9*x0 + 8*x1 + 11*x3 <= 109',
        '9*x0 + 27*x2 + 11*x3 <= 108',
        '10*x0 + 31*x2 <= 109',
        '26*x1 + 9*x3 <= 138',
        '10*x0 + 31*x2 + 9*x3 <= 167',
        '26*x1 + 31*x2 + 9*x3 <= 177',
        '10*x0 + 26*x1 + 31*x2 <= 93'
    ]
}
```

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

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

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

    # Objective function
    model.setObjective(2.91 * x0 + 4.55 * x1 + 2.66 * x2 + 5.97 * x3, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(9 * x0 <= 291)
    model.addConstr(10 * x0 <= 231)
    model.addConstr(7 * x0 <= 394)
    model.addConstr(8 * x1 <= 291)
    model.addConstr(26 * x1 <= 231)
    model.addConstr(24 * x1 <= 394)
    model.addConstr(27 * x2 <= 291)
    model.addConstr(31 * x2 <= 231)
    model.addConstr(29 * x2 <= 394)
    model.addConstr(11 * x3 <= 291)
    model.addConstr(9 * x3 <= 231)
    model.addConstr(29 * x3 <= 394)
    model.addConstr(8 * x1 + 11 * x3 >= 71)
    model.addConstr(9 * x0 + 11 * x3 >= 39)
    model.addConstr(9 * x0 + 8 * x1 >= 61)
    model.addConstr(9 * x0 + 27 * x2 + 11 * x3 >= 72)
    model.addConstr(9 * x0 + 29 * x2 + 11 * x3 >= 72)
    model.addConstr(8 * x1 + 27 * x2 + 11 * x3 >= 72)
    model.addConstr(9 * x0 + 27 * x2 + 11 * x3 >= 52)
    model.addConstr(9 * x0 + 29 * x2 + 11 * x3 >= 52)
    model.addConstr(8 * x1 + 29 * x2 + 11 * x3 >= 52)
    model.addConstr(9 * x0 + 8 * x1 + 27 * x2 >= 66)
    model.addConstr(9 * x0 + 29 * x2 + 11 * x3 >= 66)
    model.addConstr(8 * x1 + 27 * x2 + 11 * x3 >= 66)
    model.addConstr(9 * x0 + 8 * x1 + 27 * x2 + 11 * x3 >= 66)
    model.addConstr(26 * x1 + 31 * x2 >= 45)
    model.addConstr(26 * x1 + 9 * x3 >= 50)
    model.addConstr(26 * x1 + 31 * x2 + 9 * x3 >= 56)
    model.addConstr(10 * x0 + 26 * x1 + 31 * x2 + 9 * x3 >= 56)
    model.addConstr(7 * x0 + 24 * x1 >= 87)
    model.addConstr(29 * x2 + 29 * x3 >= 37)
    model.addConstr(7 * x0 + 29 * x2 >= 34)
    model.addConstr(7 * x0 + 29 * x3 >= 34)
    model.addConstr(7 * x0 + 24 * x1 + 29 * x2 + 29 * x3 >= 34)
    model.addConstr(4 * x0 - x1 >= 0)
    model.addConstr(9 * x0 + 8 * x1 <= 209)
    model.addConstr(8 * x1 + 11 * x3 <= 209)
    model.addConstr(8 * x1 + 27 * x2 <= 78)
    model.addConstr(9 * x0 + 27 * x2 <= 269)
    model.addConstr(9 * x0 + 11 * x3 <= 176)
    model.addConstr(9 * x0 + 8 * x1 + 11 * x3 <= 109)
    model.addConstr(9 * x0 + 27 * x2 + 11 * x3 <= 108)
    model.addConstr(10 * x0 + 31 * x2 <= 109)
    model.addConstr(26 * x1 + 9 * x3 <= 138)
    model.addConstr(10 * x0 + 31 * x2 + 9 * x3 <= 167)
    model.addConstr(26 * x1 + 31 * x2 + 9 * x3 <= 177)
    model.addConstr(10 * x0 + 26 * x1 + 31 * x2 <= 93)

    model.optimize()

    if model.status == gurobi.GRB.Status.OPTIMAL:
        print("Optimal solution found.")
        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)
        print("Objective function value: ", model.objVal)
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```