## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Dale', 'hours worked by Jean', and 'hours worked by Bill', which can be represented symbolically as $x_0$, $x_1$, and $x_2$ respectively.

## Step 2: Convert the problem description into a symbolic representation
The objective function to minimize is $7x_0 + 8x_1 + 3x_2$.

## Step 3: List the constraints in symbolic notation
The constraints are:
- $9x_0 \leq 68$ (Dale's likelihood to quit index is 9, but this seems to be an attribute, not a constraint on $x_0$ directly. Assuming it's about the index being a characteristic: $9x_0$ is not directly constrained by 68 in a traditional sense, but rather $x_0$ has an upper bound based on $r0$'s upper bound and $x0$)
- $5x_0 \leq 102$ (Dale's paperwork competence rating)
- $8x_1 \leq 68$ (Jean's likelihood to quit index)
- $6x_1 \leq 102$ (Jean's paperwork competence rating)
- $9x_2 \leq 68$ (Bill's likelihood to quit index)
- $6x_2 \leq 102$ (Bill's paperwork competence rating)
- $9x_0 + 9x_2 \geq 16$ (total combined likelihood to quit index from Dale and Bill)
- $9x_0 + 8x_1 \geq 12$ (total combined likelihood to quit index from Dale and Jean)
- $9x_0 + 8x_1 + 9x_2 \geq 12$ (total combined likelihood to quit index from all)
- $5x_0 + 6x_2 \geq 27$ (total combined paperwork competence rating from Dale and Bill)
- $6x_1 + 6x_2 \geq 12$ (total combined paperwork competence rating from Jean and Bill)
- $5x_0 + 6x_1 + 6x_2 \geq 12$ (total combined paperwork competence rating from all)
- $10x_0 - 3x_1 \geq 0$ (constraint on hours worked by Dale and Jean)
- $9x_0 + 8x_1 + 9x_2 \leq 57$ (total combined likelihood to quit index from all, upper bound)
- $5x_0 + 6x_1 + 6x_2 \leq 43$ (total combined paperwork competence rating from all, upper bound)

## 4: Consider variable bounds
- $x_0$ can be fractional
- $x_1$ must be an integer
- $x_2$ can be fractional

## 5: Formulate the problem in Gurobi
We need to model this problem using Gurobi, specifying the objective function and all constraints.

## 6: Output the symbolic representation
```json
{
'sym_variables': [('x0', 'hours worked by Dale'), ('x1', 'hours worked by Jean'), ('x2', 'hours worked by Bill')],
'objective_function': '7*x0 + 8*x1 + 3*x2',
'constraints': [
    '9*x0 <= 68',
    '5*x0 <= 102',
    '8*x1 <= 68',
    '6*x1 <= 102',
    '9*x2 <= 68',
    '6*x2 <= 102',
    '9*x0 + 9*x2 >= 16',
    '9*x0 + 8*x1 >= 12',
    '9*x0 + 8*x1 + 9*x2 >= 12',
    '5*x0 + 6*x2 >= 27',
    '6*x1 + 6*x2 >= 12',
    '5*x0 + 6*x1 + 6*x2 >= 12',
    '10*x0 - 3*x1 >= 0',
    '9*x0 + 8*x1 + 9*x2 <= 57',
    '5*x0 + 6*x1 + 6*x2 <= 43'
]
}
```

## 7: Implement the problem in Gurobi Python
```python
import gurobi

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

    # Define variables
    x0 = model.addVar(name="x0", lb=0, ub=68/9, vtype=gurobi.GRB.CONTINUOUS)  # hours worked by Dale
    x1 = model.addVar(name="x1", lb=0, ub=102/6, vtype=gurobi.GRB.INTEGER)  # hours worked by Jean
    x2 = model.addVar(name="x2", lb=0, ub=68/9, vtype=gurobi.GRB.CONTINUOUS)  # hours worked by Bill

    # Objective function
    model.setObjective(7*x0 + 8*x1 + 3*x2, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(9*x0 <= 68)
    model.addConstr(5*x0 <= 102)
    model.addConstr(8*x1 <= 68)
    model.addConstr(6*x1 <= 102)
    model.addConstr(9*x2 <= 68)
    model.addConstr(6*x2 <= 102)
    model.addConstr(9*x0 + 9*x2 >= 16)
    model.addConstr(9*x0 + 8*x1 >= 12)
    model.addConstr(9*x0 + 8*x1 + 9*x2 >= 12)
    model.addConstr(5*x0 + 6*x2 >= 27)
    model.addConstr(6*x1 + 6*x2 >= 12)
    model.addConstr(5*x0 + 6*x1 + 6*x2 >= 12)
    model.addConstr(10*x0 - 3*x1 >= 0)
    model.addConstr(9*x0 + 8*x1 + 9*x2 <= 57)
    model.addConstr(5*x0 + 6*x1 + 6*x2 <= 43)

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objval)
        print("Hours worked by Dale: ", x0.varValue)
        print("Hours worked by Jean: ", x1.varValue)
        print("Hours worked by Bill: ", x2.varValue)
    else:
        print("The model is infeasible")

solve_optimization_problem()
```