The idea of using mathematics to helps us make smarter and more efficient choices, is definitely fascinating to me. Recently, I tried to play with a simple yet intriguing problem that felt both interesting and practically meaningful. The problem tries to find an answer to the following question: how many electric buses should a city deploy tomorrow to cut pollution?

At first, it sounds quite easy and straightforward to be hones. But as often happens, reality comes in and she always likes to complexify things: if the predictions we have for tomorrow’s demand are wrong, even the most “optimal” deployment plan can fail miserably… and that’s the magic (and frustration) of predict-then-optimize :)

The story: a city’s bus challenge

Imagine a city that operates both electric buses (clean but limited in number) and diesel buses (polluting but higher in number).

• Electric buses are great but the city only owns 6 of them.
• Diesel buses are polluting, so the city wants to minimize their use.
• Each bus, regardless of type, can carry 50 passengers.

We will try to find a solution to: how many passengers will actually need buses tomorrow? If we predict wrong, our “optimal” allocation won’t really be optimal.

Step 1: Predict passenger demand

We’ll use a toy example where tomorrow’s passenger demand depends on temperature (I know, that’s super simplified, but it gives you the idea). Warmer days → more people ride. A tiny linear regression does our prediction trick:

import numpy as np
from sklearn.linear_model import LinearRegression

# Dummy historical data: temperature vs passengers
X = np.array([[10], [15], [20], [25], [30]])
y = np.array([300, 350, 400, 450, 480])  # passengers observed

# Train the model
model = LinearRegression().fit(X, y)

# Tomorrow’s forecast: 22°C
predicted_passengers = int(model.predict([[22]])[0])
print(f"Predicted passengers: {predicted_passengers}")

Let’s say the model predicts 420 passengers for tomorrow.

Step 2: Optimize bus allocation with OR-Tools

Knowing this forecast, the city now wants to deploy buses. The goal: cover all predicted passengers while minimizing the number of diesel-powered buses use.

from ortools.linear_solver import pywraplp

# Create solver
solver = pywraplp.Solver.CreateSolver('SCIP')

# Decision variables
electric_buses = solver.IntVar(0, 6, 'electric_buses')  # max 6 available
diesel_buses = solver.IntVar(0, 100, 'diesel_buses')    # can use many

# Constraint: cover predicted demand
solver.Add(50*electric_buses + 50*diesel_buses >= predicted_passengers)

# Objective: minimize diesel buses
solver.Minimize(diesel_buses)

solver.Solve()

print(f"Electric buses: {electric_buses.solution_value()}")
print(f"Diesel buses: {diesel_buses.solution_value()}")

For 420 passengers, the solver finds:

• 6 electric buses (300 seats)
• 3 diesel buses (150 seats)

Total = 450 seats → enough to cover demand. Pollution minimized, resources respected. Looks “optimal” :)

Step 3: Reality bites

But here’s where things get interesting…

Case 1: Real demand is lower (350 passengers)

• We oversupplied buses. Some buses run half-empty.
• Worse, we ran 3 diesel buses unnecessarily, adding pollution that wasn’t needed.

Case 2: Real demand is higher (480 passengers)

• Suddenly, our “optimal” plan leaves 30 passengers unserved.
• Some of them may switch to cars or taxis, ironically causing more pollution.

The bigger lesson

This little bus story shows why predict-then-optimize is powerful but at the same time tricky:

• Predictions drive everything → if your forecast is wrong, the optimization only amplifies the error.
• Optimality is fragile → “optimal” only means optimal under the assumptions.
• Robustness matters → sometimes, it’s better to plan conservatively (maybe always keep a buffer bus) rather than chase a brittle optimum.

To conclude

I love working through simple examples to explain complex ideas. It reminds me that optimization is never the full story… it’s a partner play with prediction. And when predictions fail (as they often do), the “perfect” plan can quickly collapse.

Simple examples like this make me see “optimal” less like a magic word and more like a careful balance between data, assumptions, and reality.