## Problem Description and Formulation

The problem is an optimization problem where we need to maximize the objective function: $9 \times \text{milligrams of vitamin K} + 2 \times \text{milligrams of vitamin A}$, subject to various constraints.

## Constraints

The constraints can be categorized into variable constraints and combined constraints.

### Variable Constraints

These are already given in the problem description:
- $2 \times \text{milligrams of vitamin K} \leq 142$
- $7 \times \text{milligrams of vitamin K} \leq 156$
- $5 \times \text{milligrams of vitamin K} \leq 60$
- $2 \times \text{milligrams of vitamin K} \leq 98$
- $15 \times \text{milligrams of vitamin A} \leq 142$
- $11 \times \text{milligrams of vitamin A} \leq 156$
- $8 \times \text{milligrams of vitamin A} \leq 60$
- $8 \times \text{milligrams of vitamin A} \leq 98$

However, these individual variable constraints are implicitly considered in the problem through the resource/attribute bounds.

### Combined Constraints

- $2 \times \text{milligrams of vitamin K} + 15 \times \text{milligrams of vitamin A} \geq 62$
- $7 \times \text{milligrams of vitamin K} + 11 \times \text{milligrams of vitamin A} \geq 74$
- $5 \times \text{milligrams of vitamin K} + 8 \times \text{milligrams of vitamin A} \geq 11$
- $2 \times \text{milligrams of vitamin K} + 8 \times \text{milligrams of vitamin A} \geq 34$
- $-4 \times \text{milligrams of vitamin K} + 5 \times \text{milligrams of vitamin A} \geq 0$
- $2 \times \text{milligrams of vitamin K} + 15 \times \text{milligrams of vitamin A} \leq 82$
- $7 \times \text{milligrams of vitamin K} + 11 \times \text{milligrams of vitamin A} \leq 111$
- $5 \times \text{milligrams of vitamin K} + 8 \times \text{milligrams of vitamin A} \leq 49$
- $2 \times \text{milligrams of vitamin K} + 8 \times \text{milligrams of vitamin A} \leq 98$

## Gurobi Code

```python
import gurobipy as gp

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

# Define variables
vitamin_K = m.addVar(name="vitamin_K", lb=0)  # Assuming non-negative
vitamin_A = m.addVar(name="vitamin_A", lb=0)  # Assuming non-negative

# Objective function
m.setObjective(9 * vitamin_K + 2 * vitamin_A, gp.GRB.MAXIMIZE)

# Constraints
m.addConstr(2 * vitamin_K + 15 * vitamin_A >= 62, name="cardiovascular_support_min")
m.addConstr(7 * vitamin_K + 11 * vitamin_A >= 74, name="kidney_support_min")
m.addConstr(5 * vitamin_K + 8 * vitamin_A >= 11, name="immune_support_min")
m.addConstr(2 * vitamin_K + 8 * vitamin_A >= 34, name="cognitive_performance_min")
m.addConstr(-4 * vitamin_K + 5 * vitamin_A >= 0, name="linear_constraint")

m.addConstr(2 * vitamin_K + 15 * vitamin_A <= 82, name="cardiovascular_support_max")
m.addConstr(7 * vitamin_K + 11 * vitamin_A <= 111, name="kidney_support_max")
m.addConstr(5 * vitamin_K + 8 * vitamin_A <= 49, name="immune_support_max")
m.addConstr(2 * vitamin_K + 8 * vitamin_A <= 98, name="cognitive_performance_max")

# Solve the model
m.optimize()

# Print the results
if m.status == gp.GRB.OPTIMAL:
    print("Optimal Solution:")
    print(f"Milligrams of Vitamin K: {vitamin_K.varValue}")
    print(f"Milligrams of Vitamin A: {vitamin_A.varValue}")
    print(f"Objective: {m.objVal}")
else:
    print("No optimal solution found.")
```