# 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.

[1]:

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']))


## 5.3. Demand data

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

[4]:

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

[4]:

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.

[5]:

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)

[5]:

intercept    221.907460
p             -3.199247
month         66.123215
dtype: float64

[6]:

from sklearn.metrics import mean_absolute_error

mean_absolute_error(y, m.predict(X))

[6]:

175.57064604298589

[7]:

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

[7]:

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.

[8]:

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)

[8]:

p        0.434937
month    0.565063
dtype: float64

[9]:

mean_absolute_error(y, m.predict(X))

[9]:

76.80770760690886

[10]:

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

[10]:

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.

[11]:

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

[11]:

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

[12]:

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()