5. Multi-Objective Optimization for Demand Curve

In general, we can model a demand curve where we predict quantity \(q\) from price \(p\) and other covariates.

\(q = f(\theta) = f(p, x_0, x_1, \ldots, x_n)\)

If we have the model \(f(\theta)\), then we can find \((q^*, \theta^*)\) that maximizes either the total revenue or profit. Note the following.

• \(r = qp\), and

• \(t = q(p - c)\)

where

• \(r\) is the total revenue,

• \(q\) is quantity,

• \(p\) is unit price,

• \(t\) is the total profit, and

• \(c\) is the unit cost.

It is not necessarily the case that maximizing revenue is also maximizing profit. In this notebook, we will generate fake data and illustrate how we can model the demand curve and then use this model to optimize for revenue, profit and both.

5.1. Data

The data is simulated as follows

\(Q \sim \mathcal{N}(50 - 1.5 P + 100 M_q, 1) + \epsilon\),

where

• \(Q\) is the quantity,

• \(P\) is the unit price,

• \(M_q\) is the corresponding yearly quarter for the specified month, and

• \(\epsilon \sim \mathcal{N}(10, 5)\) is the error.

import numpy as np
import pandas as pd
import random

random.seed(37)
np.random.seed(37)

def f(p, month):
    e = np.random.normal(10, 5, p.shape[0])
    quarter = np.select([month < 4, month < 7, month < 11], [1, 2, 3], 4)
    return np.random.normal(50 - 1.5 * p + 100 * quarter) + e

df = pd.DataFrame({
        'p': np.random.randint(50, 100, 1_000),
        'month': np.random.randint(1, 12, 1_000)
    }) \
    .assign(q=lambda d: f(d['p'], d['month']))

5.2. Visualize quantity over price trends

Below, we show the

• spread of quantity by price and

• also the sum of the quantity by price.

import matplotlib.pyplot as plt
import seaborn as sns

fig, ax = plt.subplots(2, 1, figsize=(15, 6))

sns.boxplot(df, x='p', y='q', ax=ax[0])
df.groupby(['p'])['q'].sum().plot(kind='line', ylabel='q', ax=ax[1])

ax[0].set_title('Box plots of quantity at different prices')
ax[1].set_title('Sum of quantities for different prices')

fig.tight_layout()

We can breakdown the data by quarters and observe the spread of quantities by price. In all quarters except the last, all quantities trend downward.

_temp = df.assign(quarter=lambda d: np.select([d.month < 4, d.month < 7, d.month < 10], [1, 2, 3], 4))

fig, axes = plt.subplots(4, 1, figsize=(20, 12))

for q, ax in zip([1, 2, 3, 4], np.ravel(axes)):
    sns.boxplot(
        _temp[_temp.quarter==q],
        x='p',
        y='q',
        ax=ax
    )

    ax.set_title(f'Quantities by price, Q{q}')

fig.tight_layout()

5.3. Demand data

The demand curve data will be grouped by price-month and quantities summed over.

demand_df = df.groupby(['p', 'month'])['q'].sum().to_frame().reset_index()
demand_df

	p	month	q
0	50	1	97.630858
1	50	2	170.132677
2	50	3	181.190757
3	50	4	367.611310
4	50	5	185.221784
...	...	...	...
461	99	6	443.360809
462	99	7	210.213176
463	99	8	818.645468
464	99	9	430.012356
465	99	10	210.007255

466 rows × 3 columns

5.4. Linear modeling

We can certainly model the demand curve using linear regression. As you can see below, the coefficient corresponding to price is -3.20. The mean absolute error MAE and weighted absolute percentage error WAPE are very big.

from sklearn.linear_model import LinearRegression

X = demand_df[['p', 'month']]
y = demand_df['q']

m = LinearRegression()
m.fit(X, y)

pd.Series([m.intercept_, m.coef_[0], m.coef_[1]], ['intercept', 'p', 'month'])

Intel(R) Extension for Scikit-learn* enabled (https://github.com/intel/scikit-learn-intelex)

intercept    221.907460
p             -3.199247
month         66.123215
dtype: float64

from sklearn.metrics import mean_absolute_error

mean_absolute_error(y, m.predict(X))

175.57064604298589

np.sum(np.abs(y - m.predict(X))) / np.sum(y)

0.4666900845671383

5.5. Non-linear modeling

In the non-linear modeling approach, we use a random forest to model the demand curve. Interestingly, month is more important than price. Also, using a non-linear model cuts the MAE and WAPE by over half of what we saw with the linear model. Let’s keep this model.

from sklearn.ensemble import RandomForestRegressor

X = demand_df[['p', 'month']]
y = demand_df['q']

m = RandomForestRegressor(n_jobs=-1, random_state=37, n_estimators=40)
m.fit(X, y)

pd.Series(m.feature_importances_, X.columns)

p        0.434937
month    0.565063
dtype: float64

mean_absolute_error(y, m.predict(X))

76.80770760690886

np.sum(np.abs(y - m.predict(X))) / np.sum(y)

0.20416508320929747

5.6. Visualizing quantities space

We can visualize the relationship between quantity (z-axis), price (x-axis) and month (y-axis). As you can tell, the surface is very complicated of how price and month interact to influence quantity.

x, y = np.meshgrid(np.arange(50, 100.1, 0.1), np.arange(1, 13, 1))
z = m.predict(pd.DataFrame([(_x, _y) for _x, _y in zip(np.ravel(x), np.ravel(y))], columns=['p', 'month'])).reshape(x.shape)

x.shape, y.shape, z.shape

((12, 501), (12, 501), (12, 501))

import plotly.graph_objects as go

fig = go.Figure(data=[go.Surface(
    x=x,
    y=y,
    z=z,
    colorscale='Viridis',
    opacity=0.5,
)])

fig.update_layout(
    title=r'$q = f(p, m_q)$',
    autosize=False,
    width=600,
    height=600,
    margin={'l': 65, 'r': 50, 'b': 65, 't': 90}
)
fig.update_xaxes(title='p')
fig.update_yaxes(title=r'$m_q$')

fig.show()