## Step 1: Define the symbolic variables
Let's define the symbolic variables for the problem:
- $x_1$ represents the number of ads in malls
- $x_2$ represents the number of ads at bus stops
- $x_3$ represents the number of ads in theatres

## Step 2: Formulate the objective function
The objective is to maximize viewership. The audience reach for each ad is given as:
- An ad in a mall reaches 50,000 viewers
- An ad at a bus stop reaches 10,000 viewers
- An ad in a theatre reaches 20,000 viewers

The objective function to maximize viewership is: $50,000x_1 + 10,000x_2 + 20,000x_3$

## 3: Define the constraints
1. **Budget Constraint**: The total cost of ads should not exceed $30,000.
- Cost of an ad in a mall: $5,000
- Cost of an ad at a bus stop: $1,000
- Cost of an ad in a theatre: $3,000
So, $5,000x_1 + 1,000x_2 + 3,000x_3 \leq 30,000$

2. **Bus Stop Ads Limit**: The number of ads at bus stops should not exceed 20.
So, $x_2 \leq 20$

3. **Theatre Ads Limit**: At most a third of the total number of ads should be in theatres.
So, $x_3 \leq \frac{1}{3}(x_1 + x_2 + x_3)$

4. **Mall Ads Minimum**: A minimum of 20% of ads should be in malls.
So, $x_1 \geq 0.2(x_1 + x_2 + x_3)$

5. **Non-Negativity Constraints**: The number of ads cannot be negative.
So, $x_1 \geq 0, x_2 \geq 0, x_3 \geq 0$

## 4: Convert constraints into standard form
- Budget Constraint: $5,000x_1 + 1,000x_2 + 3,000x_3 \leq 30,000$
- Bus Stop Ads Limit: $x_2 \leq 20$
- Theatre Ads Limit: $3x_3 \leq x_1 + x_2 + x_3$ or $2x_3 \leq x_1 + x_2$
- Mall Ads Minimum: $0.8x_1 \geq 0.2x_2 + 0.2x_3$ or $4x_1 \geq x_2 + x_3$

## 5: Symbolic Representation
```json
{
'sym_variables': [('x1', 'ads in malls'), ('x2', 'ads at bus stops'), ('x3', 'ads in theatres')],
'objective_function': '50000*x1 + 10000*x2 + 20000*x3',
'constraints': [
    '5000*x1 + 1000*x2 + 3000*x3 <= 30000',
    'x2 <= 20',
    '2*x3 <= x1 + x2',
    '4*x1 >= x2 + x3',
    'x1 >= 0', 'x2 >= 0', 'x3 >= 0'
]
}
```

## 6: Gurobi Code
```python
import gurobipy as gp

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

# Define variables
x1 = m.addVar(name="ads_in_malls", lb=0, vtype=gp.GRB.INTEGER)  # ads in malls
x2 = m.addVar(name="ads_at_bus_stops", lb=0, vtype=gp.GRB.INTEGER)  # ads at bus stops
x3 = m.addVar(name="ads_in_theatres", lb=0, vtype=gp.GRB.INTEGER)  # ads in theatres

# Objective function: Maximize viewership
m.setObjective(50000*x1 + 10000*x2 + 20000*x3, gp.GRB.MAXIMIZE)

# Constraints
m.addConstr(5000*x1 + 1000*x2 + 3000*x3 <= 30000, name="budget_constraint")
m.addConstr(x2 <= 20, name="bus_stop_ads_limit")
m.addConstr(2*x3 <= x1 + x2, name="theatre_ads_limit")
m.addConstr(4*x1 >= x2 + x3, name="mall_ads_minimum")

# Solve the model
m.solve()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal Solution:")
    print(f"Ads in malls: {x1.varValue}")
    print(f"Ads at bus stops: {x2.varValue}")
    print(f"Ads in theatres: {x3.varValue}")
    print(f"Max Viewership: {m.objVal}")
else:
    print("No optimal solution found")
```