## Problem Description and Formulation

The problem is an optimization problem with three variables: 'milligrams of iron', 'milligrams of vitamin C', and 'grams of fiber'. The objective is to maximize the function:

\[ 1.3 \times \text{milligrams of iron} + 6.78 \times \text{milligrams of vitamin C} + 8.38 \times \text{grams of fiber} \]

subject to several constraints related to 'kidney support index' and 'cognitive performance index'.

## Constraints

1. The total combined kidney support index from milligrams of vitamin C and grams of fiber must be no less than 78.
2. The total combined kidney support index from milligrams of iron plus grams of fiber should be 331 at maximum.
3. The total combined kidney support index from milligrams of vitamin C plus grams of fiber must be 166 at maximum.
4. The total combined kidney support index from milligrams of iron, milligrams of vitamin C, and grams of fiber has to be 166 at maximum.
5. The total combined cognitive performance index from milligrams of vitamin C plus grams of fiber must be at maximum 87.
6. The total combined cognitive performance index from milligrams of iron plus milligrams of vitamin C must be no more than 113.
7. The total combined cognitive performance index from milligrams of iron, milligrams of vitamin C, and grams of fiber must be 113 at a maximum.

## Gurobi Code Formulation

```python
import gurobipy as gp

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

# Define variables
iron = m.addVar(name="iron", lb=0)  # milligrams of iron
vitamin_c = m.addVar(name="vitamin_c", lb=0)  # milligrams of vitamin C
fiber = m.addVar(name="fiber", lb=0)  # grams of fiber

# Objective function
m.setObjective(1.3 * iron + 6.78 * vitamin_c + 8.38 * fiber, gp.GRB.MAXIMIZE)

# Constraints
m.addConstraint(2 * vitamin_c + 6 * fiber >= 78)  # Kidney support index constraint 1
m.addConstraint(33 * iron + 6 * fiber <= 331)  # Kidney support index constraint 2
m.addConstraint(2 * vitamin_c + 6 * fiber <= 166)  # Kidney support index constraint 3
m.addConstraint(33 * iron + 2 * vitamin_c + 6 * fiber <= 166)  # Kidney support index constraint 4
m.addConstraint(4 * vitamin_c + 25 * fiber <= 87)  # Cognitive performance index constraint 1
m.addConstraint(6 * iron + 4 * vitamin_c <= 113)  # Cognitive performance index constraint 2
m.addConstraint(6 * iron + 4 * vitamin_c + 25 * fiber <= 113)  # Cognitive performance index constraint 3

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Iron: {iron.varValue} milligrams")
    print(f"Vitamin C: {vitamin_c.varValue} milligrams")
    print(f"Fiber: {fiber.varValue} grams")
    print(f"Objective: {m.objVal}")
else:
    print("No optimal solution found.")
```