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

### Problem Definition

We have the following variables:
- $x_0$: peanutbutter sandwiches
- $x_1$: oreos
- $x_2$: kiwis
- $x_3$: fruit salads
- $x_4$: milkshakes
- $x_5$: eggs

### Objective Function

Minimize: $4.47x_0 + 7.12x_1 + 3.96x_2 + 7.26x_3 + 9.54x_4 + 7.14x_5$

### Constraints

The constraints are defined based on the problem description.

```python
import gurobi as gp

# Define the model
m = gp.Model("optimization_problem")

# Define the variables
x0 = m.addVar(name="peanutbutter_sandwiches", lb=0)  # Non-negative
x1 = m.addVar(name="oreos", lb=0)  # Non-negative
x2 = m.addVar(name="kiwis", lb=0)  # Non-negative
x3 = m.addVar(name="fruit_salads", lb=0)  # Non-negative
x4 = m.addVar(name="milkshakes", lb=0)  # Non-negative
x5 = m.addVar(name="eggs", lb=0)  # Non-negative

# Objective function
m.setObjective(4.47*x0 + 7.12*x1 + 3.96*x2 + 7.26*x3 + 9.54*x4 + 7.14*x5, gp.GRB.MINIMIZE)

# Constraints
# Umami index constraints
m.addConstr(14*x0 + 4*x3 >= 12)  # Total umami from peanutbutter sandwiches and fruit salads >= 12
m.addConstr(2*x2 + 2*x4 >= 15)  # Total umami from kiwis and milkshakes >= 15
m.addConstr(2*x2 + 4*x3 >= 9)  # Total umami from kiwis and fruit salads >= 9
m.addConstr(14*x0 + 2*x2 >= 8)  # Total umami from peanutbutter sandwiches and kiwis >= 8
m.addConstr(4*x3 + 11*x5 >= 6)  # Total umami from fruit salads and eggs >= 6
m.addConstr(14*x0 + 2*x4 >= 5)  # Total umami from peanutbutter sandwiches and milkshakes >= 5
m.addConstr(13*x1 + 4*x3 >= 5)  # Total umami from oreos and fruit salads >= 5
m.addConstr(14*x0 + 11*x5 >= 14)  # Total umami from peanutbutter sandwiches and eggs >= 14
m.addConstr(4*x3 + 2*x4 >= 6)  # Total umami from fruit salads and milkshakes >= 6
m.addConstr(13*x1 + 2*x4 + 11*x5 >= 15)  # Total umami from oreos, milkshakes, and eggs >= 15
m.addConstr(14*x0 + 13*x1 + 2*x2 + 4*x3 + 2*x4 + 11*x5 >= 15)  # Total umami from all >= 15

# Fat constraints
m.addConstr(14*x2 + 13*x4 >= 50)  # Fat from kiwis and milkshakes >= 50
m.addConstr(6*x3 + 14*x5 >= 49)  # Fat from fruit salads and eggs >= 49
m.addConstr(11*x0 + 6*x3 >= 57)  # Fat from peanutbutter sandwiches and fruit salads >= 57
m.addConstr(14*x2 + 6*x3 >= 21)  # Fat from kiwis and fruit salads >= 21
m.addConstr(13*x4 + 14*x5 >= 48)  # Fat from milkshakes and eggs >= 48
m.addConstr(5*x1 + 14*x5 >= 39)  # Fat from oreos and eggs >= 39
m.addConstr(5*x1 + 13*x4 >= 25)  # Fat from oreos and milkshakes >= 25
m.addConstr(5*x1 + 14*x2 >= 43)  # Fat from oreos and kiwis >= 43
m.addConstr(14*x2 + 14*x5 >= 32)  # Fat from kiwis and eggs >= 32
m.addConstr(11*x0 + 14*x5 >= 37)  # Fat from peanutbutter sandwiches and eggs >= 37
m.addConstr(11*x0 + 5*x1 + 14*x2 + 6*x3 + 13*x4 + 14*x5 >= 37)  # Fat from all >= 37

# Linear constraints
m.addConstr(2*x1 - 4*x3 >= 0)  # 2*oreos - 4*fruit salads >= 0

# Umami index limit constraints
m.addConstr(13*x1 + 2*x2 + 4*x3 <= 88)  # Umami from oreos, kiwis, and fruit salads <= 88
m.addConstr(2*x2 + 2*x4 + 11*x5 <= 33)  # Umami from kiwis, milkshakes, and eggs <= 33
m.addConstr(13*x1 + 4*x3 + 2*x4 <= 53)  # Umami from oreos, fruit salads, and milkshakes <= 53
m.addConstr(2*x2 + 4*x3 + 2*x4 <= 29)  # Umami from kiwis, fruit salads, and milkshakes <= 29
m.addConstr(13*x1 + 2*x4 + 11*x5 <= 79)  # Umami from oreos, milkshakes, and eggs <= 79
m.addConstr(4*x3 + 2*x4 + 11*x5 <= 97)  # Umami from fruit salads, milkshakes, and eggs <= 97
m.addConstr(14*x0 + 4*x3 + 2*x4 <= 43)  # Umami from peanutbutter sandwiches, fruit salads, and milkshakes <= 43
m.addConstr(13*x1 + 2*x2 + 2*x4 <= 25)  # Umami from oreos, kiwis, and milkshakes <= 25
m.addConstr(14*x0 + 13*x1 + 2*x2 <= 99)  # Umami from peanutbutter sandwiches, oreos, and kiwis <= 99
m.addConstr(14*x0 + 13*x1 + 2*x4 <= 51)  # Umami from peanutbutter sandwiches, oreos, and milkshakes <= 51
m.addConstr(2*x2 + 4*x3 + 11*x5 <= 44)  # Umami from kiwis, fruit salads, and eggs <= 44
m.addConstr(13*x1 + 2*x2 + 11*x5 <= 35)  # Umami from oreos, kiwis, and eggs <= 35
m.addConstr(13*x1 + 4*x3 + 11*x5 <= 82)  # Umami from oreos, fruit salads, and eggs <= 82

# Fat limit constraints
m.addConstr(5*x1 + 13*x4 <= 369)  # Fat from oreos and milkshakes <= 369
m.addConstr(11*x0 + 14*x2 + 13*x4 <= 386)  # Fat from peanutbutter sandwiches, kiwis, and milkshakes <= 386
m.addConstr(11*x0 + 13*x4 + 14*x5 <= 371)  # Fat from peanutbutter sandwiches, milkshakes, and eggs <= 371
m.addConstr(11*x0 + 5*x1 + 14*x2 <= 258)  # Fat from peanutbutter sandwiches, oreos, and kiwis <= 258
m.addConstr(14*x2 + 6*x3 + 13*x4 <= 276)  # Fat from kiwis, fruit salads, and milkshakes <= 276
m.addConstr(11*x0 + 6*x3 + 13*x4 <= 194)  # Fat from peanutbutter sandwiches, fruit salads, and milkshakes <= 194

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objval)
    print("peanutbutter sandwiches: ", x0.varValue)
    print("oreos: ", x1.varValue)
    print("kiwis: ", x2.varValue)
    print("fruit salads: ", x3.varValue)
    print("milkshakes: ", x4.varValue)
    print("eggs: ", x5.varValue)
else:
    print("The model is infeasible")
```

```python
import gurobi as gp

# Define the model
m = gp.Model("optimization_problem")

# Define the variables
x0 = m.addVar(name="peanutbutter_sandwiches", lb=0)  
x1 = m.addVar(name="oreos", lb=0)  
x2 = m.addVar(name="kiwis", lb=0)  
x3 = m.addVar(name="fruit_salads", lb=0)  
x4 = m.addVar(name="milkshakes", lb=0)  
x5 = m.addVar(name="eggs", lb=0)  

# Objective function
m.setObjective(4.47*x0 + 7.12*x1 + 3.96*x2 + 7.26*x3 + 9.54*x4 + 7.14*x5, gp.GRB.MINIMIZE)

# Constraints
# Umami index constraints
m.addConstr(14*x0 + 4*x3 >= 12)  
m.addConstr(2*x2 + 2*x4 >= 15)  
m.addConstr(2*x2 + 4*x3 >= 9)  
m.addConstr(14*x0 + 2*x2 >= 8)  
m.addConstr(4*x3 + 11*x5 >= 6)  
m.addConstr(14*x0 + 2*x4 >= 5)  
m.addConstr(13*x1 + 4*x3 >= 5)  
m.addConstr(14*x0 + 11*x5 >= 14)  
m.addConstr(4*x3 + 2*x4 >= 6)  
m.addConstr(13*x1 + 2*x4 + 11*x5 >= 15)  
m.addConstr(14*x0 + 13*x1 + 2*x2 + 4*x3 + 2*x4 + 11*x5 >= 15)  

# Fat constraints
m.addConstr(14*x2 + 13*x4 >= 50)  
m.addConstr(6*x3 + 14*x5 >= 49)  
m.addConstr(11*x0 + 6*x3 >= 57)  
m.addConstr(14*x2 + 6*x3 >= 21)  
m.addConstr(13*x4 + 14*x5 >= 48)  
m.addConstr(5*x1 + 14*x5 >= 39)  
m.addConstr(5*x1 + 13*x4 >= 25)  
m.addConstr(5*x1 + 14*x2 >= 43)  
m.addConstr(14*x2 + 14*x5 >= 32)  
m.addConstr(11*x0 + 14*x5 >= 37)  
m.addConstr(11*x0 + 5*x1 + 14*x2 + 6*x3 + 13*x4 + 14*x5 >= 37)  

# Linear constraints
m.addConstr(2*x1 - 4*x3 >= 0)  

# Umami index limit constraints
m.addConstr(13*x1 + 2*x2 + 4*x3 <= 88)  
m.addConstr(2*x2 + 2*x4 + 11*x5 <= 33)  
m.addConstr(13*x1 + 4*x3 + 2*x4 <= 53)  
m.addConstr(2*x2 + 4*x3 + 2*x4 <= 29)  
m.addConstr(13*x1 + 2*x4 + 11*x5 <= 79)  
m.addConstr(4*x3 + 2*x4 + 11*x5 <= 97)  
m.addConstr(14*x0 + 4*x3 + 2*x4 <= 43)  
m.addConstr(13*x1 + 2*x2 + 2*x4 <= 25)  
m.addConstr(14*x0 + 13*x1 + 2*x2 <= 99)  
m.addConstr(14*x0 + 13*x1 + 2*x4 <= 51)  
m.addConstr(2*x2 + 4*x3 + 11*x5 <= 44)  
m.addConstr(13*x1 + 2*x2 + 11*x5 <= 35)  
m.addConstr(13*x1 + 4*x3 + 11*x5 <= 82)  

# Fat limit constraints
m.addConstr(5*x1 + 13*x4 <= 369)  
m.addConstr(11*x0 + 14*x2 + 13*x4 <= 386)  
m.addConstr(11*x0 + 13*x4 + 14*x5 <= 371)  
m.addConstr(11*x0 + 5*x1 + 14*x2 <= 258)  
m.addConstr(14*x2 + 6*x3 + 13*x4 <= 276)  
m.addConstr(11*x0 + 6*x3 + 13*x4 <= 194)  

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objval)
    print("peanutbutter sandwiches: ", x0.varValue)
    print("oreos: ", x1.varValue)
    print("kiwis: ", x2.varValue)
    print("fruit salads: ", x3.varValue)
    print("milkshakes: ", x4.varValue)
    print("eggs: ", x5.varValue)
else:
    print("The model is infeasible")
```