To solve the given optimization problem using Gurobi, we need to define the decision variables, objective function, and constraints according to the provided description. The decision variables are `cheeseburgers`, `cantaloupes`, `protein_bars`, and `blueberry_pies`. The objective is to minimize the cost function: 8.78 times the quantity of cheeseburgers added to 9.72 times the quantity of cantaloupes plus 6.84 times the quantity of protein bars plus 3.94 times the quantity of blueberry pies.

The constraints are defined based on the resource limitations and additional requirements specified in the problem description.

Here's how we can implement this using Gurobi:

```python
from gurobipy import *

# Create a model
m = Model("Optimization_Problem")

# Define decision variables (allowing for non-integer values)
cheeseburgers = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="cheeseburgers")
cantaloupes = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="cantaloupes")
protein_bars = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="protein_bars")
blueberry_pies = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="blueberry_pies")

# Objective function: Minimize cost
m.setObjective(8.78 * cheeseburgers + 9.72 * cantaloupes + 6.84 * protein_bars + 3.94 * blueberry_pies, GRB.MINIMIZE)

# Constraints based on the problem description
# Cost constraints
m.addConstr(12 * cheeseburgers <= 249, name="max_cost_cheeseburgers")
m.addConstr(cantaloupes <= 231, name="max_cost_cantaloupes")
m.addConstr(11 * protein_bars + 12 * blueberry_pies <= 181, name="max_cost_protein_bars_and_blueberry_pies")

# Fat constraints
m.addConstr(6 * cheeseburgers + 12 * cantaloupes >= 57, name="min_fat_cheeseburgers_and_cantaloupes")
m.addConstr(8 * protein_bars + 8 * blueberry_pies >= 42, name="min_fat_protein_bars_and_blueberry_pies")
m.addConstr(6 * cheeseburgers + 8 * protein_bars >= 29, name="min_fat_cheeseburgers_and_protein_bars")
m.addConstr(6 * cheeseburgers + 12 * cantaloupes + 8 * protein_bars <= 230, name="max_fat_cheeseburgers_cantaloupes_protein_bars")

# Protein constraints
m.addConstr(3 * protein_bars + 16 * blueberry_pies >= 20, name="min_protein_protein_bars_and_blueberry_pies")
m.addConstr(2 * cantaloupes + 3 * protein_bars >= 10, name="min_protein_cantaloupes_and_protein_bars")
m.addConstr(2 * cantaloupes + 16 * blueberry_pies >= 24, name="min_protein_cantaloupes_and_blueberry_pies")
m.addConstr(2 * cheeseburgers + 2 * cantaloupes + 3 * protein_bars + 16 * blueberry_pies >= 24, name="min_total_protein")

# Additional constraints
m.addConstr(-9 * cantaloupes + 10 * blueberry_pies >= 0, name="cantaloupes_and_blueberry_pies_relation")
m.addConstr(6 * cheeseburgers - 5 * protein_bars >= 0, name="cheeseburgers_and_protein_bars_relation")
m.addConstr(-9 * cheeseburgers + 7 * blueberry_pies >= 0, name="cheeseburgers_and_blueberry_pies_relation")

# Total cost constraints
m.addConstr(12 * cheeseburgers + cantaloupes >= 55, name="min_cost_cheeseburgers_and_cantaloupes")
m.addConstr(cantaloupes + 12 * blueberry_pies >= 36, name="min_cost_cantaloupes_and_blueberry_pies")
m.addConstr(11 * protein_bars + cantaloupes + 12 * blueberry_pies <= 69, name="max_cost_protein_bars_cantaloupes_blueberry_pies")

# Optimize the model
m.optimize()

# Print results if an optimal solution is found
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    for var in m.getVars():
        print(f"{var.varName}: {var.x}")
else:
    print("No optimal solution found")
```