Newsvender Problem

This repository is a Python implementation of the Newsvendor Model.

Requirements

• Python>=3.8
• scipy>=1.8.1

Install

pip install git+https://github.com/takazawa/newsvendor-model

Model

Newsvendor Model

Input (for some product)

• demand $D$ (random variable)
• retail price $p$
• purchase price $c$

Problem

The standard newsvendor model is to find the optimal quantity $q_{\text{opt}} $of the product, which maximizes the expected profit $E[F(q, D)]$, where $$ F(q, D) = p\min (q, D) - cq. $$

The optimal solution to the problem is known to be

$$ q_{\text{opt}} = F^{-1} \left( \dfrac{p-c}{c} \right), $$

where $F^{-1}$ is the inverse cumulative distribution function of $D$.

Usage

The model to be imported depends on whether the probability distribution used is discrete or continuous as follows:

from nvmodel import ContinuousNewsvendorModel, DiscreteNewsvendorModel

Using the Wikipedia example, consider the following problem:

• retail price $p = 7$
• purchase price $c = 5$
• demand $D$ follows a (continuous) uniform distribution between $D_{\min}=50$ and $D_{\max}=80$.

Then the newsvendeor model is as follows:

from nvmodel import ContinuousNewsvendorModel


def p_func(x):
    d_min, d_max = 50, 80
    if d_min <= x <= d_max:
        return 1 / (d_max - d_min)
    else:
        return 0


model = ContinuousNewsvendorModel(retail_price=7,
                                  cost_price=5,
                                  probability_function=p_func)

Now, we can calculate values

• model.critical_ratio() $=\frac{p-c}{c}$
• model.optimal_quantity() $=q_{\text{opt}}$
• model.revenue(q, D) $=F(q, D)$
• model.expected_revenue(q) $=E[F(q, D)]$

>> q_opt = model.optimal_quantity()
>> q_opt
58.57142857142857

Other Samples

1. Normal Distribution

from scipy.stats import norm

from nvmodel import ContinuousNewsvendorModel


def normal_pdf(x):
    mu, sigma = 50, 20
    return norm(mu, sigma).pdf(x)


model = ContinuousNewsvendorModel(retail_price=7, cost_price=5, probability_function=normal_pdf)
q_opt = model.optimal_quantity()

>> q_opt
39.04454355287975

In the Continuous Model, the inverse CDF is computed approximately. It can also be computed exactly by explicitly giving inverseCDF as follows.

from scipy.stats import norm

from nvmodel import ContinuousNewsvendorModel


def normal_inverse_cdf(x):
    mu, sigma = 50, 20
    return mu + sigma * norm.ppf(x)


model = ContinuousNewsvendorModel(retail_price=7,
                                  cost_price=5,
                                  inverse_cdf=normal_inverse_cdf)
q_opt = model.optimal_quantity()

>> q_opt
38.68102356134274

2. Poisson Distribution

• retail price $p = 8$
• purchase price $c = 5$
• demand $D$ follows the poisson distribution (mean=25)
• salvaged revenue $s = 4$
  • a profit per unit from unsold products

from scipy.stats import poisson

from nvmodel import DiscreteNewsvendorModel


def poisson_pmf(x):
    return poisson.pmf(x, 25)


model = DiscreteNewsvendorModel(retail_price=8,
                                cost_price=5,
                                probability_function=poisson_pmf,
                                salvaged_revenue=4)
q_opt = model.optimal_quantity()

>> q_opt
28