## Problem Description and Symbolic Representation

The problem involves maximizing viewership given certain constraints on the number and types of commercials an energy drink company can produce.

### Symbolic Variables
- $x_1$ : Number of commercials with famous actors
- $x_2$ : Number of commercials with regular people
- $x_3$ : Number of commercials with no people

### Objective Function
Maximize viewership: $50000x_1 + 20000x_2 + 18000x_3$

### Constraints
1. Budget constraint: $10000x_1 + 3000x_2 + 2000x_3 \leq 50000$
2. Constraint on commercials with regular actors: $x_2 \leq 3$
3. Constraint on commercials with no people: $x_3 \leq \frac{1}{3}(x_1 + x_2 + x_3)$
4. Constraint on commercials with famous actors: $x_1 \geq 0.2(x_1 + x_2 + x_3)$
5. Non-negativity constraints: $x_1, x_2, x_3 \geq 0$ and are integers.

## Conversion to Gurobi Code

To solve this linear programming problem using Gurobi in Python, we first need to ensure that our constraints are properly formulated. The constraint on commercials with no people and the constraint on commercials with famous actors need to be rewritten for clarity:

3. $3x_3 \leq x_1 + x_2 + x_3$ which simplifies to $2x_3 \leq x_1 + x_2$
4. $5x_1 \geq x_1 + x_2 + x_3$ which simplifies to $4x_1 \geq x_2 + x_3$

```python
import gurobi as gp
from gurobi import GRB

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

# Define variables
x1 = m.addVar(name="famous_actors", vtype=GRB.INTEGER)  # Commercials with famous actors
x2 = m.addVar(name="regular_people", vtype=GRB.INTEGER)  # Commercials with regular people
x3 = m.addVar(name="no_people", vtype=GRB.INTEGER)     # Commercials with no people

# Objective function: Maximize viewership
m.setObjective(50000*x1 + 20000*x2 + 18000*x3, GRB.MAXIMIZE)

# Budget constraint
m.addConstr(10000*x1 + 3000*x2 + 2000*x3 <= 50000, name="budget_constraint")

# Constraint on commercials with regular actors
m.addConstr(x2 <= 3, name="regular_people_constraint")

# Constraint on commercials with no people
m.addConstr(2*x3 <= x1 + x2, name="no_people_constraint")

# Constraint on commercials with famous actors
m.addConstr(4*x1 >= x2 + x3, name="famous_actors_constraint")

# Non-negativity constraints are inherently handled by Gurobi for integer variables

# Solve the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal Solution:")
    print(f"Commercials with famous actors: {x1.varValue}")
    print(f"Commercials with regular people: {x2.varValue}")
    print(f"Commercials with no people: {x3.varValue}")
    print(f"Max Viewership: {m.objVal}")
else:
    print("No optimal solution found")
```

## Symbolic Representation in JSON Format

```json
{
  "sym_variables": [
    ["x1", "commercials with famous actors"],
    ["x2", "commercials with regular people"],
    ["x3", "commercials with no people"]
  ],
  "objective_function": "50000x1 + 20000x2 + 18000x3",
  "constraints": [
    "10000x1 + 3000x2 + 2000x3 <= 50000",
    "x2 <= 3",
    "2x3 <= x1 + x2",
    "4x1 >= x2 + x3",
    "x1, x2, x3 >= 0"
  ]
}
```