Here's the Gurobi code to solve the optimization problem:

```python
import gurobipy as gp

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

# Create variables
vitamin_e = m.addVar(vtype=gp.GRB.INTEGER, name="vitamin_e")
calcium = m.addVar(vtype=gp.GRB.CONTINUOUS, name="calcium")
vitamin_c = m.addVar(vtype=gp.GRB.CONTINUOUS, name="vitamin_c")

# Set objective function
m.setObjective(5 * vitamin_e + 1 * calcium + 8 * vitamin_c, gp.GRB.MAXIMIZE)

# Add constraints
m.addConstr(24 * vitamin_e + 29 * calcium + 6 * vitamin_c >= 45, "c0")
m.addConstr(28 * vitamin_e + 25 * calcium <= 106, "c1")
m.addConstr(25 * calcium + 13 * vitamin_c <= 70, "c2")
m.addConstr(28 * vitamin_e + 25 * calcium + 13 * vitamin_c <= 70, "c3")
m.addConstr(29 * calcium + 18 * vitamin_c <= 148, "c4")
m.addConstr(19 * vitamin_e + 29 * calcium <= 103, "c5")
m.addConstr(19 * vitamin_e + 29 * calcium + 18 * vitamin_c <= 103, "c6")
m.addConstr(24 * vitamin_e + 29 * calcium <= 137, "c7")
m.addConstr(24 * vitamin_e + 6 * vitamin_c <= 166, "c8")
m.addConstr(29 * calcium + 6 * vitamin_c <= 192, "c9")
m.addConstr(24 * vitamin_e + 29 * calcium + 6 * vitamin_c <= 192, "c10")


# Optimize model
m.optimize()

# Print results
if m.status == gp.GRB.OPTIMAL:
    print('Obj: %g' % m.objVal)
    print('vitamin_e:', vitamin_e.x)
    print('calcium:', calcium.x)
    print('vitamin_c:', vitamin_c.x)
elif m.status == gp.GRB.INFEASIBLE:
    print('The model is infeasible.')
else:
    print('Optimization ended with status %d' % m.status)

```
