Plan Retail Promotions with Coupled Simulation

rng = np.random.default_rng(11)
n = 1095          # three years of daily history
h = 28            # the promotion window
t = np.arange(n + h)

# Each product runs its own price schedule; the three do not move together.
# Prices carry a lot of day-to-day movement, which is what identifies the price
# effects. Footfall below drifts slowly instead, so the two cannot be confused
# for each other.
promoted_price = (
    6.00
    + 0.60 * np.sin(2 * np.pi * t / 97)
    + 0.30 * np.sin(2 * np.pi * t / 29)
    + rng.normal(0, 0.30, n + h)
)
alternative_price = (
    9.00
    + 0.85 * np.sin(2 * np.pi * t / 73 + 1.1)
    + 0.40 * np.sin(2 * np.pi * t / 23)
    + rng.normal(0, 0.36, n + h)
)
related_price = (
    4.50
    + 0.45 * np.sin(2 * np.pi * t / 113 + 2.3)
    + 0.22 * np.sin(2 * np.pi * t / 31)
    + rng.normal(0, 0.24, n + h)
)

# Store footfall drifts through busy and quiet stretches, lifting every product
# on the same day. It is never given to the model, so it survives as demand that
# moves together across the products.
shock = rng.normal(0, 1, n + h)
footfall = np.zeros(n + h)
for i in range(1, n + h):
    footfall[i] = 0.92 * footfall[i - 1] + shock[i]
footfall = 1 + 0.059 * footfall
weekly = np.sin(2 * np.pi * t / 7)

# The product relationships are planted here.
promoted_demand = (
    40 * footfall
    - 5.3 * (promoted_price - 6.0)     # own-price effect
    + 2.0 * (alternative_price - 9.0)
    + 3.0 * weekly
    + rng.normal(0, 1.6, n + h)
)
alternative_demand = (
    55 * footfall
    + 7.0 * (promoted_price - 6.0)     # substitute: cheaper promoted, fewer sales
    - 4.0 * (alternative_price - 9.0)
    + 3.0 * weekly
    + rng.normal(0, 2.0, n + h)
)
related_demand = (
    30 * footfall
    - 5.0 * (promoted_price - 6.0)     # complement: cheaper promoted, more sales
    - 3.5 * (related_price - 4.5)
    + 2.0 * weekly
    + rng.normal(0, 1.4, n + h)
)

dates = pd.date_range("2023-01-01", periods=n + h, freq="D")
prices = {
    "promoted_price": promoted_price,
    "alternative_price": alternative_price,
    "related_price": related_price,
}
demands = {
    "Promoted product": promoted_demand,
    "Alternative product": alternative_demand,
    "Related product": related_demand,
}

history, future = [], []
for label, demand in demands.items():
    history.append(
        pd.DataFrame(
            {
                "unique_id": label,
                "ds": dates[:n],
                "y": np.maximum(demand[:n], 0).round(),
                **{name: values[:n] for name, values in prices.items()},
            }
        )
    )
    future.append(
        pd.DataFrame(
            {
                "unique_id": label,
                "ds": dates[n:],
                **{name: values[n:] for name, values in prices.items()},
            }
        )
    )

df = pd.concat(history, ignore_index=True)
current_X_df = pd.concat(future, ignore_index=True)

df.head()

def daily_paths(paths):
    nonnegative = paths.assign(
        TimeGPT=paths["TimeGPT"].clip(lower=0)
    )
    return nonnegative.pivot(
        index=["sample_id", "ds"],
        columns="unique_id",
        values="TimeGPT",
    )


separate_daily = daily_paths(separate_paths)
coupled_daily = daily_paths(promotion_paths)
high_demand = separate_daily.quantile(0.90)


def simultaneous_high_demand(daily):
    two_or_more = daily.gt(high_demand).sum(axis=1).ge(2)
    return two_or_more.groupby("sample_id").any().mean()


separate_risk = simultaneous_high_demand(separate_daily)
coupled_risk = simultaneous_high_demand(coupled_daily)

separate_risk, coupled_risk

(0.424, 0.538)