```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 + 1*x3",
  "constraints": [
    "8*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",
    "-1*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",
    "x0 >= 0",
    "x1 >= 0",
    "x2 >= 0",
    "x3 >= 0"
  ]
}
```

```python
import gurobipy as gp

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

# Create variables
zinc = m.addVar(lb=0, name="zinc")
vitamin_a = m.addVar(lb=0, name="vitamin_a")
calcium = m.addVar(lb=0, name="calcium")
magnesium = m.addVar(lb=0, name="magnesium")

# Set objective function
m.setObjective(5*zinc + 7*vitamin_a + 3*calcium + 1*magnesium, gp.GRB.MINIMIZE)

# Add constraints
m.addConstr(8*vitamin_a + 7*magnesium >= 75)
m.addConstr(8*zinc + 11*vitamin_a >= 71)
m.addConstr(11*vitamin_a + 16*calcium >= 97)
m.addConstr(8*zinc + 11*vitamin_a + 16*calcium >= 101)
m.addConstr(11*vitamin_a + 16*calcium + 7*magnesium >= 101)
m.addConstr(8*zinc + 11*vitamin_a + 7*magnesium >= 101)
m.addConstr(8*zinc + 11*vitamin_a + 16*calcium >= 95)
m.addConstr(11*vitamin_a + 16*calcium + 7*magnesium >= 95)
m.addConstr(8*zinc + 11*vitamin_a + 7*magnesium >= 95)
m.addConstr(8*zinc + 11*vitamin_a + 16*calcium >= 78)
m.addConstr(11*vitamin_a + 16*calcium + 7*magnesium >= 78)
m.addConstr(8*zinc + 11*vitamin_a + 7*magnesium >= 78)
m.addConstr(8*zinc + 11*vitamin_a + 16*calcium + 7*magnesium >= 78)
m.addConstr(5*zinc + 3*magnesium >= 20)
m.addConstr(2*vitamin_a + 7*calcium >= 46)
m.addConstr(5*zinc + 2*vitamin_a + 7*calcium + 3*magnesium >= 46)
m.addConstr(19*zinc + 12*magnesium >= 20)
m.addConstr(19*zinc + 9*calcium >= 31)
m.addConstr(9*calcium + 12*magnesium >= 37)
m.addConstr(14*vitamin_a + 9*calcium + 12*magnesium >= 27)
m.addConstr(19*zinc + 14*vitamin_a + 9*calcium + 12*magnesium >= 27)
m.addConstr(8*zinc - 5*calcium >= 0)
m.addConstr(-1*zinc + 3*magnesium >= 0)
m.addConstr(11*vitamin_a + 16*calcium <= 347)
m.addConstr(8*zinc + 11*vitamin_a + 16*calcium <= 205)
m.addConstr(8*zinc + 16*calcium + 7*magnesium <= 425)
m.addConstr(8*zinc + 11*vitamin_a + 7*magnesium <= 418)
m.addConstr(2*vitamin_a + 7*calcium <= 117)
m.addConstr(5*zinc + 3*magnesium <= 71)
m.addConstr(19*zinc + 9*calcium <= 159)
m.addConstr(9*calcium + 12*magnesium <= 127)
m.addConstr(19*zinc + 14*vitamin_a + 12*magnesium <= 165)


# Optimize model
m.optimize()

# Print results
if m.status == gp.GRB.OPTIMAL:
    print('Obj: %g' % m.objVal)
    for v in m.getVars():
        print('%s %g' % (v.varName, v.x))
elif m.status == gp.GRB.INFEASIBLE:
    print("The model is infeasible.")
else:
    print("Optimization ended with status %d" % m.status)

```
