To solve this optimization problem, we first need to define the variables, objective function, and constraints using Gurobi's Python API.

### Problem Definition

We have five variables:
- \(x_0\): chicken drumsticks
- \(x_1\): oranges
- \(x_2\): blueberry pies
- \(x_3\): cherry pies
- \(x_4\): strips of bacon

The objective is to minimize:
\[ 5.07x_0 + 4.94x_1 + 3.54x_2 + 7.76x_3 + 2.12x_4 \]

Subject to numerous constraints on carbohydrates, umami index, fat, and protein.

### Gurobi Code

```python
import gurobi

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

    # Define variables
    x0 = model.addVar(name="chicken_drumsticks", lb=0)
    x1 = model.addVar(name="oranges", lb=0)
    x2 = model.addVar(name="blueberry_pies", lb=0)
    x3 = model.addVar(name="cherry_pies", lb=0)
    x4 = model.addVar(name="strips_of_bacon", lb=0)

    # Objective function
    model.setObjective(5.07 * x0 + 4.94 * x1 + 3.54 * x2 + 7.76 * x3 + 2.12 * x4, gurobi.GRB.MINIMIZE)

    # Constraints
    # Carbohydrates
    model.addConstr(13 * x0 + 16 * x1 + 11 * x2 + 22 * x3 + 13 * x4 <= 405, name="carbohydrates")
    model.addConstr(13 * x0 + 22 * x3 >= 42, name="carbohydrates_cd_cp")
    model.addConstr(11 * x2 + 13 * x4 >= 48, name="carbohydrates_bp_sb")
    model.addConstr(16 * x1 + 13 * x4 >= 67, name="carbohydrates_o_sb")
    model.addConstr(16 * x1 + 22 * x3 >= 46, name="carbohydrates_o_cp")
    model.addConstr(13 * x0 + 16 * x1 + 13 * x4 >= 71, name="carbohydrates_co_sb")
    model.addConstr(13 * x0 + 16 * x1 + 11 * x2 >= 71, name="carbohydrates_co_bp")
    model.addConstr(13 * x0 + 16 * x1 + 13 * x4 >= 78, name="carbohydrates_co_sb_2")
    model.addConstr(13 * x0 + 16 * x1 + 11 * x2 >= 78, name="carbohydrates_co_bp_2")
    model.addConstr(13 * x0 + 16 * x1 + 11 * x2 + 22 * x3 + 13 * x4 >= 78, name="carbohydrates_all")

    # Umami index
    model.addConstr(14 * x1 + 4 * x4 >= 49, name="umami_o_sb")
    model.addConstr(14 * x0 + 4 * x4 >= 52, name="umami_cd_sb")
    model.addConstr(27 * x2 + 23 * x3 >= 34, name="umami_bp_cp")
    model.addConstr(23 * x3 + 4 * x4 >= 62, name="umami_cp_sb")
    model.addConstr(27 * x2 + 4 * x4 >= 32, name="umami_bp_sb")
    model.addConstr(14 * x0 + 17 * x1 >= 40, name="umami_cd_o")
    model.addConstr(17 * x1 + 23 * x3 + 4 * x4 >= 33, name="umami_o_cp_sb")
    model.addConstr(14 * x0 + 17 * x1 + 4 * x4 >= 33, name="umami_cd_o_sb")
    model.addConstr(17 * x1 + 27 * x2 + 23 * x3 >= 33, name="umami_o_bp_cp")
    model.addConstr(17 * x1 + 23 * x3 + 4 * x4 >= 50, name="umami_o_cp_sb_2")
    model.addConstr(14 * x0 + 17 * x1 + 4 * x4 >= 50, name="umami_cd_o_sb_2")
    model.addConstr(17 * x1 + 27 * x2 + 23 * x3 >= 50, name="umami_o_bp_cp_2")
    model.addConstr(17 * x1 + 23 * x3 + 4 * x4 >= 48, name="umami_o_cp_sb_3")
    model.addConstr(14 * x0 + 17 * x1 + 4 * x4 >= 48, name="umami_cd_o_sb_3")
    model.addConstr(17 * x1 + 27 * x2 + 23 * x3 >= 48, name="umami_o_bp_cp_3")
    model.addConstr(14 * x0 + 17 * x1 + 27 * x2 + 23 * x3 + 4 * x4 >= 48, name="umami_all")

    # Fat
    model.addConstr(10 * x2 + 3 * x3 >= 33, name="fat_bp_cp")
    model.addConstr(3 * x3 + 4 * x4 >= 41, name="fat_cp_sb")
    model.addConstr(28 * x0 + 18 * x1 + 10 * x2 + 3 * x3 + 4 * x4 >= 41, name="fat_all")
    model.addConstr(18 * x1 + 3 * x3 <= 198, name="fat_o_cp")
    model.addConstr(28 * x0 + 4 * x4 <= 60, name="fat_cd_sb")
    model.addConstr(10 * x2 + 3 * x3 <= 85, name="fat_bp_cp_2")
    model.addConstr(28 * x0 + 18 * x1 <= 183, name="fat_cd_o")
    model.addConstr(18 * x1 + 10 * x2 <= 249, name="fat_o_bp")
    model.addConstr(28 * x0 + 10 * x2 + 3 * x3 <= 76, name="fat_cd_bp_cp")
    model.addConstr(28 * x0 + 3 * x3 + 4 * x4 <= 141, name="fat_cd_cp_sb")
    model.addConstr(28 * x0 + 18 * x1 + 3 * x3 <= 89, name="fat_cd_o_cp")
    model.addConstr(18 * x1 + 10 * x2 + 3 * x3 <= 79, name="fat_o_bp_cp")
    model.addConstr(28 * x0 + 18 * x1 + 4 * x4 <= 212, name="fat_cd_o_sb")
    model.addConstr(28 * x0 + 18 * x1 + 10 * x2 <= 95, name="fat_cd_o_bp")

    # Protein
    model.addConstr(15 * x3 + 18 * x4 >= 44, name="protein_cp_sb")
    model.addConstr(24 * x0 + 6 * x1 >= 28, name="protein_cd_o")
    model.addConstr(6 * x1 + 15 * x3 >= 20, name="protein_o_cp")
    model.addConstr(6 * x1 + 18 * x4 >= 37, name="protein_o_sb")
    model.addConstr(11 * x2 + 15 * x3 >= 24, name="protein_bp_cp")
    model.addConstr(24 * x0 + 18 * x4 >= 37, name="protein_cd_sb")
    model.addConstr(6 * x1 + 11 * x2 >= 25, name="protein_o_bp")
    model.addConstr(24 * x0 + 6 * x1 + 11 * x2 + 15 * x3 + 18 * x4 >= 25, name="protein_all")
    model.addConstr(11 * x2 + 18 * x4 <= 239, name="protein_bp_sb_2")
    model.addConstr(24 * x0 + 11 * x2 + 15 * x3 <= 237, name="protein_cd_bp_cp_2")

    # Others
    model.addConstr(-8 * x2 + 5 * x3 >= 0, name="others")
    model.addConstr(11 * x2 + 22 * x3 + 13 * x4 <= 274, name="carbohydrates_bp_cp_sb")
    model.addConstr(14 * x0 + 23 * x3 <= 109, name="umami_cd_cp")
    model.addConstr(14 * x0 + 17 * x1 <= 252, name="umami_cd_o")
    model.addConstr(14 * x0 + 4 * x4 <= 145, name="umami_cd_sb")
    model.addConstr(23 * x3 + 4 * x4 <= 135, name="umami_cp_sb")
    model.addConstr(14 * x0 + 23 * x3 + 4 * x4 <= 294, name="umami_cd_cp_sb")
    model.addConstr(14 * x0 + 17 * x1 + 23 * x3 <= 89, name="umami_cd_o_cp")
    model.addConstr(27 * x2 + 23 * x3 + 4 * x4 <= 205, name="umami_bp_cp_sb")
    model.addConstr(14 * x0 + 17 * x1 + 4 * x4 <= 162, name="umami_cd_o_sb")
    model.addConstr(17 * x1 + 23 * x3 + 4 * x4 <= 327, name="umami_o_cp_sb")

    # Optimize
    model.optimize()

    # Print solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objval)
        print("Chicken drumsticks: ", x0.varValue)
        print("Oranges: ", x1.varValue)
        print("Blueberry pies: ", x2.varValue)
        print("Cherry pies: ", x3.varValue)
        print("Strips of bacon: ", x4.varValue)
    else:
        print("No solution found")

optimize()
```