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Your model is built. You know that Facebook has an Average ROAS of 3.0 and TV has an Average ROAS of 1.5.

The natural instinct is to say: "Cut TV budget and move it all to Facebook!"

This is a trap. Average ROAS looks backward at what happened. It ignores Saturation. If you double your Facebook spend, you might hit the "Ceiling" (Module 5), and your efficiency will plummet. To optimize, we must look at the Marginal ROAS (mROAS).

1. Average vs. Marginal

2. The Response Curve

Because we used Hill Functions (S-Curves) in our model, we can plot the relationship between Spend (X-axis) and Revenue (Y-axis) for every channel.

The "Optimal Point" is usually where the curve starts to flatten. Mathematically, this is where the first derivative of the curve equals 1.0 (or your target profit margin).

If mROAS > 1.0, you are making profit on the next dollar. Spend more.
If mROAS < 1.0, you are losing money on the next dollar. Pull back.

3. Algorithmic Optimization (Python)

You cannot solve this in your head for 10 channels simultaneously. We use a solver like scipy.optimize. We define a "constrained optimization" problem.

• Objective: Maximize Total Revenue.
• Constraint 1: Total Spend cannot exceed $1 Million.
• Constraint 2: Don't change any channel by more than +/- 20% (for stability).

from scipy.optimize import minimize

# 1. Define the Objective Function (Negative Revenue because we want to minimize the negative)
def objective_function(budget_allocation):
    total_sales = 0
    for i, channel in enumerate(channels):
        # Apply Hill Function to calculate expected sales for this spend
        sales = hill_function(budget_allocation[i], channel_params)
        total_sales += sales
    return -total_sales

# 2. Define Constraints (Sum of budget = Total Budget)
constraints = ({'type': 'eq', 'fun': lambda x: sum(x) - TOTAL_BUDGET})

# 3. Define Bounds (Don't let any channel go to $0 or Infinity)
bounds = [(current_spend[c]*0.8, current_spend[c]*1.2) for c in channels]

# 4. Run the Solver
result = minimize(objective_function, current_spend, bounds=bounds, constraints=constraints)

4. The Result: The Optimized Mix

The output of this script gives you the "Efficient Frontier." It might say:

• Facebook: Decrease spend by $50k (It is saturated).
• TV: Increase spend by $30k (It has high headroom).
• TikTok: Keep flat.

By simply moving this money around—without increasing the total budget—you might squeeze out an extra 5-10% in Total Revenue. This is the ultimate value proposition of MMM.