## Step 1: Define the symbolic representation of the variables
The variables are defined as follows: 
- $x_0$ represents 'milligrams of zinc'
- $x_1$ represents 'milligrams of vitamin A'
- $x_2$ represents 'milligrams of calcium'
- $x_3$ represents 'milligrams of magnesium'

## Step 2: Define the symbolic representation of the objective function
The objective function to minimize is $5x_0 + 7x_1 + 3x_2 + 1x_3$.

## Step 3: Define the symbolic representation of the constraints
The constraints are as follows:
- $8x_0 \geq 0$ (cardiovascular support index for $x_0$)
- $5x_0 \geq 0$ (immune support index for $x_0$)
- $19x_0 \geq 0$ (cognitive performance index for $x_0$)
- $11x_1 \geq 0$ (cardiovascular support index for $x_1$)
- $2x_1 \geq 0$ (immune support index for $x_1$)
- $14x_1 \geq 0$ (cognitive performance index for $x_1$)
- $16x_2 \geq 0$ (cardiovascular support index for $x_2$)
- $7x_2 \geq 0$ (immune support index for $x_2$)
- $9x_2 \geq 0$ (cognitive performance index for $x_2$)
- $7x_3 \geq 0$ (cardiovascular support index for $x_3$)
- $3x_3 \geq 0$ (immune support index for $x_3$)
- $12x_3 \geq 0$ (cognitive performance index for $x_3$)
- $11x_1 + 7x_3 \geq 75$ (cardiovascular support index for $x_1$ and $x_3$)
- $8x_0 + 11x_1 \geq 71$ (cardiovascular support index for $x_0$ and $x_1$)
- $11x_1 + 16x_2 \geq 97$ (cardiovascular support index for $x_1$ and $x_2$)
- $8x_0 + 11x_1 + 16x_2 \geq 101$ (cardiovascular support index for $x_0$, $x_1$, and $x_2$)
- $11x_1 + 16x_2 + 7x_3 \geq 101$ (cardiovascular support index for $x_1$, $x_2$, and $x_3$)
- $8x_0 + 11x_1 + 7x_3 \geq 101$ (cardiovascular support index for $x_0$, $x_1$, and $x_3$)
- $8x_0 + 11x_1 + 16x_2 \geq 95$ (cardiovascular support index for $x_0$, $x_1$, and $x_2$)
- $11x_1 + 16x_2 + 7x_3 \geq 95$ (cardiovascular support index for $x_1$, $x_2$, and $x_3$)
- $8x_0 + 11x_1 + 7x_3 \geq 95$ (cardiovascular support index for $x_0$, $x_1$, and $x_3$)
- $8x_0 + 11x_1 + 16x_2 \geq 78$ (cardiovascular support index for $x_0$, $x_1$, and $x_2$)
- $11x_1 + 16x_2 + 7x_3 \geq 78$ (cardiovascular support index for $x_1$, $x_2$, and $x_3$)
- $8x_0 + 11x_1 + 7x_3 \geq 78$ (cardiovascular support index for $x_0$, $x_1$, and $x_3$)
- $8x_0 + 11x_1 + 16x_2 + 7x_3 \geq 78$ (cardiovascular support index for $x_0$, $x_1$, $x_2$, and $x_3$)
- $5x_0 + 3x_3 \geq 20$ (immune support index for $x_0$ and $x_3$)
- $2x_1 + 7x_2 \geq 46$ (immune support index for $x_1$ and $x_2$)
- $5x_0 + 2x_1 + 7x_2 + 3x_3 \geq 46$ (immune support index for $x_0$, $x_1$, $x_2$, and $x_3$)
- $19x_0 + 12x_3 \geq 20$ (cognitive performance index for $x_0$ and $x_3$)
- $19x_0 + 9x_2 \geq 31$ (cognitive performance index for $x_0$ and $x_2$)
- $9x_2 + 12x_3 \geq 37$ (cognitive performance index for $x_2$ and $x_3$)
- $14x_1 + 9x_2 + 12x_3 \geq 27$ (cognitive performance index for $x_1$, $x_2$, and $x_3$)
- $19x_0 + 14x_1 + 9x_2 + 12x_3 \geq 27$ (cognitive performance index for $x_0$, $x_1$, $x_2$, and $x_3$)
- $8x_0 - 5x_2 \geq 0$
- $-x_0 + 3x_3 \geq 0$
- $11x_1 + 16x_2 \leq 347$ (cardiovascular support index for $x_1$ and $x_2$)
- $8x_0 + 11x_1 + 16x_2 \leq 205$ (cardiovascular support index for $x_0$, $x_1$, and $x_2$)
- $8x_0 + 16x_2 + 7x_3 \leq 425$ (cardiovascular support index for $x_0$, $x_2$, and $x_3$)
- $8x_0 + 11x_1 + 7x_3 \leq 418$ (cardiovascular support index for $x_0$, $x_1$, and $x_3$)
- $2x_1 + 7x_2 \leq 117$ (immune support index for $x_1$ and $x_2$)
- $5x_0 + 3x_3 \leq 71$ (immune support index for $x_0$ and $x_3$)
- $19x_0 + 9x_2 \leq 159$ (cognitive performance index for $x_0$ and $x_2$)
- $9x_2 + 12x_3 \leq 127$ (cognitive performance index for $x_2$ and $x_3$)
- $19x_0 + 14x_1 + 12x_3 \leq 165$ (cognitive performance index for $x_0$, $x_1$, and $x_3$)

## 4: Implement the optimization problem using Gurobi

```python
import gurobi as gp

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

# Define the variables
x0 = m.addVar(name="milligrams_of_zinc", lb=-gp.GRB.INFINITY)
x1 = m.addVar(name="milligrams_of_vitamin_A", lb=-gp.GRB.INFINITY)
x2 = m.addVar(name="milligrams_of_calcium", lb=-gp.GRB.INFINITY)
x3 = m.addVar(name="milligrams_of_magnesium", lb=-gp.GRB.INFINITY)

# Define the objective function
m.setObjective(5 * x0 + 7 * x1 + 3 * x2 + x3, gp.GRB.MINIMIZE)

# Add constraints
m.addConstr(8 * x0 >= 0)
m.addConstr(5 * x0 >= 0)
m.addConstr(19 * x0 >= 0)
m.addConstr(11 * x1 >= 0)
m.addConstr(2 * x1 >= 0)
m.addConstr(14 * x1 >= 0)
m.addConstr(16 * x2 >= 0)
m.addConstr(7 * x2 >= 0)
m.addConstr(9 * x2 >= 0)
m.addConstr(7 * x3 >= 0)
m.addConstr(3 * x3 >= 0)
m.addConstr(12 * x3 >= 0)
m.addConstr(11 * x1 + 7 * x3 >= 75)
m.addConstr(8 * x0 + 11 * x1 >= 71)
m.addConstr(11 * x1 + 16 * x2 >= 97)
m.addConstr(8 * x0 + 11 * x1 + 16 * x2 >= 101)
m.addConstr(11 * x1 + 16 * x2 + 7 * x3 >= 101)
m.addConstr(8 * x0 + 11 * x1 + 7 * x3 >= 101)
m.addConstr(8 * x0 + 11 * x1 + 16 * x2 >= 95)
m.addConstr(11 * x1 + 16 * x2 + 7 * x3 >= 95)
m.addConstr(8 * x0 + 11 * x1 + 7 * x3 >= 95)
m.addConstr(8 * x0 + 11 * x1 + 16 * x2 >= 78)
m.addConstr(11 * x1 + 16 * x2 + 7 * x3 >= 78)
m.addConstr(8 * x0 + 11 * x1 + 7 * x3 >= 78)
m.addConstr(8 * x0 + 11 * x1 + 16 * x2 + 7 * x3 >= 78)
m.addConstr(5 * x0 + 3 * x3 >= 20)
m.addConstr(2 * x1 + 7 * x2 >= 46)
m.addConstr(5 * x0 + 2 * x1 + 7 * x2 + 3 * x3 >= 46)
m.addConstr(19 * x0 + 12 * x3 >= 20)
m.addConstr(19 * x0 + 9 * x2 >= 31)
m.addConstr(9 * x2 + 12 * x3 >= 37)
m.addConstr(14 * x1 + 9 * x2 + 12 * x3 >= 27)
m.addConstr(19 * x0 + 14 * x1 + 9 * x2 + 12 * x3 >= 27)
m.addConstr(8 * x0 - 5 * x2 >= 0)
m.addConstr(-x0 + 3 * x3 >= 0)
m.addConstr(11 * x1 + 16 * x2 <= 347)
m.addConstr(8 * x0 + 11 * x1 + 16 * x2 <= 205)
m.addConstr(8 * x0 + 16 * x2 + 7 * x3 <= 425)
m.addConstr(8 * x0 + 11 * x1 + 7 * x3 <= 418)
m.addConstr(2 * x1 + 7 * x2 <= 117)
m.addConstr(5 * x0 + 3 * x3 <= 71)
m.addConstr(19 * x0 + 9 * x2 <= 159)
m.addConstr(9 * x2 + 12 * x3 <= 127)
m.addConstr(19 * x0 + 14 * x1 + 12 * x3 <= 165)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("milligrams of zinc: ", x0.varValue)
    print("milligrams of vitamin A: ", x1.varValue)
    print("milligrams of calcium: ", x2.varValue)
    print("milligrams of magnesium: ", x3.varValue)
else:
    print("The model is infeasible")
```

## 5: Symbolic representation of the problem

```json
{
    'sym_variables': [
        ('x0', 'milligrams of zinc'), 
        ('x1', 'milligrams of vitamin A'), 
        ('x2', 'milligrams of calcium'), 
        ('x3', 'milligrams of magnesium')
    ], 
    'objective_function': '5*x0 + 7*x1 + 3*x2 + x3', 
    'constraints': [
        '8*x0 >= 0', 
        '5*x0 >= 0', 
        '19*x0 >= 0', 
        '11*x1 >= 0', 
        '2*x1 >= 0', 
        '14*x1 >= 0', 
        '16*x2 >= 0', 
        '7*x2 >= 0', 
        '9*x2 >= 0', 
        '7*x3 >= 0', 
        '3*x3 >= 0', 
        '12*x3 >= 0', 
        '11*x1 + 7*x3 >= 75', 
        '8*x0 + 11*x1 >= 71', 
        '11*x1 + 16*x2 >= 97', 
        '8*x0 + 11*x1 + 16*x2 >= 101', 
        '11*x1 + 16*x2 + 7*x3 >= 101', 
        '8*x0 + 11*x1 + 7*x3 >= 101', 
        '8*x0 + 11*x1 + 16*x2 >= 95', 
        '11*x1 + 16*x2 + 7*x3 >= 95', 
        '8*x0 + 11*x1 + 7*x3 >= 95', 
        '8*x0 + 11*x1 + 16*x2 >= 78', 
        '11*x1 + 16*x2 + 7*x3 >= 78', 
        '8*x0 + 11*x1 + 7*x3 >= 78', 
        '8*x0 + 11*x1 + 16*x2 + 7*x3 >= 78', 
        '5*x0 + 3*x3 >= 20', 
        '2*x1 + 7*x2 >= 46', 
        '5*x0 + 2*x1 + 7*x2 + 3*x3 >= 46', 
        '19*x0 + 12*x3 >= 20', 
        '19*x0 + 9*x2 >= 31', 
        '9*x2 + 12*x3 >= 37', 
        '14*x1 + 9*x2 + 12*x3 >= 27', 
        '19*x0 + 14*x1 + 9*x2 + 12*x3 >= 27', 
        '8*x0 - 5*x2 >= 0', 
        '-x0 + 3*x3 >= 0', 
        '11*x1 + 16*x2 <= 347', 
        '8*x0 + 11*x1 + 16*x2 <= 205', 
        '8*x0 + 16*x2 + 7*x3 <= 425', 
        '8*x0 + 11*x1 + 7*x3 <= 418', 
        '2*x1 + 7*x2 <= 117', 
        '5*x0 + 3*x3 <= 71', 
        '19*x0 + 9*x2 <= 159', 
        '9*x2 + 12*x3 <= 127', 
        '19*x0 + 14*x1 + 12*x3 <= 165'
    ]
}
```