## Step 1: Define the optimization problem and the objective function
The goal is to minimize the objective function: $7 \times \text{milligrams of vitamin B5} + 6 \times \text{milligrams of vitamin C} + 2 \times \text{milligrams of vitamin A} + 8 \times \text{milligrams of vitamin D}$.

## Step 2: List all the constraints
1. Digestive support index of vitamin B5: $13 \times \text{milligrams of vitamin B5} \leq 157$
2. Kidney support index of vitamin B5: $10 \times \text{milligrams of vitamin B5} \leq 143$
3. Digestive support index of vitamin C: $13 \times \text{milligrams of vitamin C} \leq 157$
4. Kidney support index of vitamin C: $1 \times \text{milligrams of vitamin C} \leq 143$
5. Digestive support index of vitamin A: $9 \times \text{milligrams of vitamin A} \leq 157$
6. Kidney support index of vitamin A: $5 \times \text{milligrams of vitamin A} \leq 143$
7. Digestive support index of vitamin D: $8 \times \text{milligrams of vitamin D} \leq 157$
8. Kidney support index of vitamin D: $4 \times \text{milligrams of vitamin D} \leq 143$
9. $13 \times \text{milligrams of vitamin B5} + 9 \times \text{milligrams of vitamin A} + 8 \times \text{milligrams of vitamin D} \geq 31$
10. $13 \times \text{milligrams of vitamin C} + 9 \times \text{milligrams of vitamin A} + 8 \times \text{milligrams of vitamin D} \geq 31$
11. $13 \times \text{milligrams of vitamin B5} + 9 \times \text{milligrams of vitamin A} + 8 \times \text{milligrams of vitamin D} \geq 30$
12. $13 \times \text{milligrams of vitamin C} + 9 \times \text{milligrams of vitamin A} + 8 \times \text{milligrams of vitamin D} \geq 30$
13. $13 \times \text{milligrams of vitamin B5} + 13 \times \text{milligrams of vitamin C} + 9 \times \text{milligrams of vitamin A} + 8 \times \text{milligrams of vitamin D} \geq 30$
14. $10 \times \text{milligrams of vitamin B5} + 5 \times \text{milligrams of vitamin A} \geq 18$
15. $10 \times \text{milligrams of vitamin B5} + 4 \times \text{milligrams of vitamin D} \geq 35$
16. $10 \times \text{milligrams of vitamin B5} + 1 \times \text{milligrams of vitamin C} \geq 27$
17. $1 \times \text{milligrams of vitamin C} + 5 \times \text{milligrams of vitamin A} \geq 12$
18. $1 \times \text{milligrams of vitamin C} + 4 \times \text{milligrams of vitamin D} \geq 24$
19. $10 \times \text{milligrams of vitamin B5} + 5 \times \text{milligrams of vitamin A} + 4 \times \text{milligrams of vitamin D} \geq 25$
20. $10 \times \text{milligrams of vitamin B5} + 1 \times \text{milligrams of vitamin C} + 5 \times \text{milligrams of vitamin A} + 4 \times \text{milligrams of vitamin D} \geq 25$
21. $-10 \times \text{milligrams of vitamin B5} + 7 \times \text{milligrams of vitamin A} \geq 0$
22. $2 \times \text{milligrams of vitamin C} - 6 \times \text{milligrams of vitamin D} \geq 0$
23. $13 \times \text{milligrams of vitamin B5} + 13 \times \text{milligrams of vitamin C} \leq 83$
24. $13 \times \text{milligrams of vitamin B5} + 8 \times \text{milligrams of vitamin D} \leq 141$
25. $13 \times \text{milligrams of vitamin B5} + 9 \times \text{milligrams of vitamin A} \leq 133$
26. $13 \times \text{milligrams of vitamin B5} + 13 \times \text{milligrams of vitamin C} + 8 \times \text{milligrams of vitamin D} \leq 91$
27. $13 \times \text{milligrams of vitamin B5} + 13 \times \text{milligrams of vitamin C} + 9 \times \text{milligrams of vitamin A} \leq 132$
28. $13 \times \text{milligrams of vitamin C} + 9 \times \text{milligrams of vitamin A} + 8 \times \text{milligrams of vitamin D} \leq 145$
29. $10 \times \text{milligrams of vitamin B5} + 5 \times \text{milligrams of vitamin A} \leq 111$
30. $5 \times \text{milligrams of vitamin A} + 4 \times \text{milligrams of vitamin D} \leq 124$
31. $10 \times \text{milligrams of vitamin B5} + 4 \times \text{milligrams of vitamin D} \leq 63$

## Step 3: Implement the optimization problem using Gurobi
```python
import gurobi

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

    # Define the variables
    vitamin_B5 = model.addVar(name="vitamin_B5", lb=0)
    vitamin_C = model.addVar(name="vitamin_C", lb=0)
    vitamin_A = model.addVar(name="vitamin_A", lb=0)
    vitamin_D = model.addVar(name="vitamin_D", lb=0)

    # Objective function
    model.setObjective(7 * vitamin_B5 + 6 * vitamin_C + 2 * vitamin_A + 8 * vitamin_D, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(13 * vitamin_B5 <= 157)
    model.addConstr(10 * vitamin_B5 <= 143)
    model.addConstr(13 * vitamin_C <= 157)
    model.addConstr(vitamin_C <= 143)
    model.addConstr(9 * vitamin_A <= 157)
    model.addConstr(5 * vitamin_A <= 143)
    model.addConstr(8 * vitamin_D <= 157)
    model.addConstr(4 * vitamin_D <= 143)

    model.addConstr(13 * vitamin_B5 + 9 * vitamin_A + 8 * vitamin_D >= 31)
    model.addConstr(13 * vitamin_C + 9 * vitamin_A + 8 * vitamin_D >= 31)
    model.addConstr(13 * vitamin_B5 + 9 * vitamin_A + 8 * vitamin_D >= 30)
    model.addConstr(13 * vitamin_C + 9 * vitamin_A + 8 * vitamin_D >= 30)
    model.addConstr(13 * vitamin_B5 + 13 * vitamin_C + 9 * vitamin_A + 8 * vitamin_D >= 30)

    model.addConstr(10 * vitamin_B5 + 5 * vitamin_A >= 18)
    model.addConstr(10 * vitamin_B5 + 4 * vitamin_D >= 35)
    model.addConstr(10 * vitamin_B5 + vitamin_C >= 27)
    model.addConstr(vitamin_C + 5 * vitamin_A >= 12)
    model.addConstr(vitamin_C + 4 * vitamin_D >= 24)
    model.addConstr(10 * vitamin_B5 + 5 * vitamin_A + 4 * vitamin_D >= 25)
    model.addConstr(10 * vitamin_B5 + vitamin_C + 5 * vitamin_A + 4 * vitamin_D >= 25)

    model.addConstr(-10 * vitamin_B5 + 7 * vitamin_A >= 0)
    model.addConstr(2 * vitamin_C - 6 * vitamin_D >= 0)

    model.addConstr(13 * vitamin_B5 + 13 * vitamin_C <= 83)
    model.addConstr(13 * vitamin_B5 + 8 * vitamin_D <= 141)
    model.addConstr(13 * vitamin_B5 + 9 * vitamin_A <= 133)
    model.addConstr(13 * vitamin_B5 + 13 * vitamin_C + 8 * vitamin_D <= 91)
    model.addConstr(13 * vitamin_B5 + 13 * vitamin_C + 9 * vitamin_A <= 132)
    model.addConstr(13 * vitamin_C + 9 * vitamin_A + 8 * vitamin_D <= 145)

    model.addConstr(10 * vitamin_B5 + 5 * vitamin_A <= 111)
    model.addConstr(5 * vitamin_A + 4 * vitamin_D <= 124)
    model.addConstr(10 * vitamin_B5 + 4 * vitamin_D <= 63)

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Vitamin B5: {vitamin_B5.varValue}")
        print(f"Vitamin C: {vitamin_C.varValue}")
        print(f"Vitamin A: {vitamin_A.varValue}")
        print(f"Vitamin D: {vitamin_D.varValue}")
        print(f"Objective: {model.objVal}")
    else:
        print("No optimal solution found.")

optimize_vitamins()
```