To solve the given optimization problem using Gurobi, we first need to understand the objective function and the constraints. The objective is to maximize the value of:

1 * (milligrams of vitamin B1) + 9 * (milligrams of iron) + 5 * (grams of protein) + 2 * (milligrams of vitamin A)

Subject to several constraints involving cognitive performance indices.

Let's denote:
- x0: milligrams of vitamin B1
- x1: milligrams of iron
- x2: grams of protein
- x3: milligrams of vitamin A

The given constraints can be summarized as follows:
1. Cognitive performance index for each nutrient.
2. Minimum total cognitive performance index from x1 and x2 is 33.
3. Maximum total cognitive performance index from x2 and x3 is 108.
4. Maximum total cognitive performance index from x0 and x2 is 181.
5. Maximum total cognitive performance index from x0, x1, and x2 is 158.
6. Maximum total cognitive performance index from x0, x2, and x3 is 177.
7. Maximum total cognitive performance index from x1, x2, and x3 is 123.
8. Maximum total cognitive performance index from x0, x1, and x3 is 141.
9. Maximum total cognitive performance index from all nutrients is 141.
10. x0 must be an integer (whole number).
11. x1 can be any real number.
12. x2 must be an integer (whole number).
13. x3 must be an integer (whole number).

Here's how we could implement this in Gurobi using Python:

```python
from gurobipy import *

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

# Define the variables
x0 = m.addVar(vtype=GRB.INTEGER, name="milligrams_of_vitamin_B1")
x1 = m.addVar(vtype=GRB.CONTINUOUS, name="milligrams_of_iron")
x2 = m.addVar(vtype=GRB.INTEGER, name="grams_of_protein")
x3 = m.addVar(vtype=GRB.INTEGER, name="milligrams_of_vitamin_A")

# Objective function
m.setObjective(1*x0 + 9*x1 + 5*x2 + 2*x3, GRB.MAXIMIZE)

# Constraints
m.addConstr(x1 + x2 >= 33)  # Minimum total cognitive performance index from x1 and x2 is 33
m.addConstr(x2 + x3 <= 108)  # Maximum total cognitive performance index from x2 and x3 is 108
m.addConstr(x0 + x2 <= 181)  # Maximum total cognitive performance index from x0 and x2 is 181
m.addConstr(x0 + x1 + x2 <= 158)  # Maximum total cognitive performance index from x0, x1, and x2 is 158
m.addConstr(x0 + x2 + x3 <= 177)  # Maximum total cognitive performance index from x0, x2, and x3 is 177
m.addConstr(x1 + x2 + x3 <= 123)  # Maximum total cognitive performance index from x1, x2, and x3 is 123
m.addConstr(x0 + x1 + x3 <= 141)  # Maximum total cognitive performance index from x0, x1, and x3 is 141
m.addConstr(x0 + x1 + x2 + x3 <= 141)  # Maximum total cognitive performance index from all nutrients is 141

# Solve the model
m.optimize()

# Print solution
for v in m.getVars():
    print(f"{v.varName}: {v.x}")

print(f"Objective: {m.objVal}")
```