## Step 1: Define the variables and their attributes
We have three variables: 'hours worked by Bill', 'hours worked by Dale', and 'hours worked by Mary'. Let's denote these as $x_B$, $x_D$, and $x_M$ respectively. The attributes given are:
- 'r0': productivity rating with $x_B = 5$, $x_D = 7$, $x_M = 7$, and an upper bound of 92.
- 'r1': work quality rating with $x_B = 1$, $x_D = 6$, $x_M = 3$, and an upper bound of 34.

## 2: Formulate the objective function
The objective function to maximize is: $5x_B^2 + 2x_Bx_D + 4x_Bx_M + 2x_D^2 + 3x_B + x_D$.

## 3: Define the constraints
1. Bill's productivity rating is 5: $5x_B \leq 92$ and $x_B \geq 0$.
2. Bill's work quality rating is 1: $1x_B \leq 34$ and $x_B \geq 0$.
3. Dale's productivity rating is 7: $7x_D \leq 92$ and $x_D \geq 0$.
4. Dale's work quality rating is 6: $6x_D \leq 34$ and $x_D \geq 0$.
5. Mary's productivity rating is 7: $7x_M \leq 92$ and $x_M \geq 0$.
6. Mary's work quality rating is 3: $3x_M \leq 34$ and $x_M \geq 0$.
7. Total combined productivity rating from $x_D$ and $x_M$ is 13 or more: $7x_D + 7x_M \geq 13$.
8. Total combined productivity rating from $x_B$ and $x_D$ is 28 or more: $5x_B + 7x_D \geq 28$.
9. Total combined work quality rating from $x_D$ and $x_M$ is 8 or more: $6x_D + 3x_M \geq 8$.
10. Total combined work quality rating from $x_B$ and $x_M$ is 8 or more: $1x_B + 3x_M \geq 8$.
11. Total combined productivity rating from $x_B^2$ and $x_D^2$ is 67 at maximum: $5^2x_B^2 + 7^2x_D^2 \leq 67$ or $25x_B^2 + 49x_D^2 \leq 67$.
12. Total combined productivity rating from $x_B$ and $x_M$ is 31 or less: $5x_B + 7x_M \leq 31$.
13. Total combined productivity rating from $x_B$, $x_D$, and $x_M$ is 31 or less: $5x_B + 7x_D + 7x_M \leq 31$.
14. Total combined work quality rating from $x_B$ and $x_D$ is 21 at most: $1x_B + 6x_D \leq 21$.
15. Total combined work quality rating from $x_B$, $x_D$, and $x_M$ is 21 at most: $1x_B + 6x_D + 3x_M \leq 21$.
16. $x_B$ must be an integer.
17. $x_D$ can be fractional.
18. $x_M$ can be fractional.

## 4: Implement the problem in Gurobi
We will use Gurobi's Python interface to model and solve this problem.

```python
import gurobi as gp

# Create a new model
m = gp.Model("optimization_problem")

# Define the variables
x_B = m.addVar(name="hours_worked_by_Bill", vtype=gp.GRB.INTEGER)
x_D = m.addVar(name="hours_worked_by_Dale")
x_M = m.addVar(name="hours_worked_by_Mary")

# Objective function
m.setObjective(5*x_B**2 + 2*x_B*x_D + 4*x_B*x_M + 2*x_D**2 + 3*x_B + x_D, gp.GRB.MAXIMIZE)

# Constraints
m.addConstr(5 * x_B <= 92, name="Bill_productivity")
m.addConstr(x_B >= 0, name="Bill_productivity_nonneg")
m.addConstr(1 * x_B <= 34, name="Bill_work_quality")
m.addConstr(7 * x_D <= 92, name="Dale_productivity")
m.addConstr(x_D >= 0, name="Dale_productivity_nonneg")
m.addConstr(6 * x_D <= 34, name="Dale_work_quality")
m.addConstr(7 * x_M <= 92, name="Mary_productivity")
m.addConstr(x_M >= 0, name="Mary_productivity_nonneg")
m.addConstr(3 * x_M <= 34, name="Mary_work_quality")
m.addConstr(7 * x_D + 7 * x_M >= 13, name="Dale_Mary_productivity")
m.addConstr(5 * x_B + 7 * x_D >= 28, name="Bill_Dale_productivity")
m.addConstr(6 * x_D + 3 * x_M >= 8, name="Dale_Mary_work_quality")
m.addConstr(1 * x_B + 3 * x_M >= 8, name="Bill_Mary_work_quality")
m.addConstr(25 * x_B**2 + 49 * x_D**2 <= 67, name="Bill_Dale_productivity_squared")
m.addConstr(5 * x_B + 7 * x_M <= 31, name="Bill_Mary_productivity_limit")
m.addConstr(5 * x_B + 7 * x_D + 7 * x_M <= 31, name="total_productivity_limit")
m.addConstr(1 * x_B + 6 * x_D <= 21, name="Bill_Dale_work_quality_limit")
m.addConstr(1 * x_B + 6 * x_D + 3 * x_M <= 21, name="total_work_quality_limit")

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Hours worked by Bill: ", x_B.varValue)
    print("Hours worked by Dale: ", x_D.varValue)
    print("Hours worked by Mary: ", x_M.varValue)
else:
    print("The model is infeasible")
```