Newsvendor problem

Authors: Morgan McCormick (mm3237), Brittany Yesner (by286), Daniel Aronson (da523), John Bednarek (jwb389)

Introduction

The mathematical application for the Newsvendor Problem dates back to 1888, when Francis Ysidro Edgeworth used the central limit theorem to find the optimal cash reserves needed to satisfy various withdrawals from depositors.¹ The namesake for the problem comes from Morse and Kimball's book from 1951, where they used the term “newsboy” to describe this specific problem.² Also referred to as “newsboy problem”, it is named by analogy with the situation faced by a newspaper vendor who must decide how many copies of the day's paper to stock in the face of uncertain demand and knowing that unsold copies will be worthless at the end of the day.

T.M Whitin in 1955 was the first to consider not only the cost minimization portion of the problem, but also the profit maximization.³ To do so he formulated a newsvendor model with price effects, where the selling price and stocking quantity are set simultaneously. He then adapted his model to include a probability distribution for demand as a function of the selling price, therefore making the price of the product a decision variable rather than an assigned variable.

In general, this model can be used in any application with a perishable good and unknown, randomized demand.

Description

The newsvendor model is a model used to determine the optimal inventory levels in operations management and applied economic applications. The assumptions for this problem usually include fixed prices and uncertain demands for perishable products with limited availability. In this model, any unit of demand, R,  over the current inventory level, n, is identified as a lost sale.

Formulation

Overview

To formulate a standard newsvendor problem to determine profit, the function is ${\textstyle E[profit]=E[s*min(x,R)]-wx}$ . In the formulation, s represents the price a unit is sold for, x represents the number of units in inventory that the vendor ordered, R is a random variable representing a probability distribution for the demand a given day, and w is the wholesale cost for the vendor to purchase materials. The goal is to optimize the profit to be a maximum. This is achieved by maximizing the amount of inventory on hand to be able to sell while also minimizing the amount of unsold inventory that is void or considered perishable at the end of the day. The salvage cost for any unsold inventory at the end of the sales period is represented by v.

The balance of being understocked and losing potential sales with the potential loss from being overstocked can be represented by the critical fractile.  This is illustrated by the formula $n=F^{-1}({s-w \over s})$ where F^-1 is the inverse of the cumulative distribution function of R.^4,5

Detailed Solution Steps

In formulation, a newsboy could purchase a given number of newspapers x one morning for a given wholesale bulk cost, b. The selling price and salvage values are known constants s and v, respectively, and the demand is given by D. The overage cost is c^o for the cost of ordering one unit too many. The cost of ordering one unit too few is the cost of underage, c^u.

The activity variables are D(ω), the realization of random demand which is assumed to be continuous; p(ω), the probability of outcome ω; S⁰(ω), the overage which is equal to $[x-D(\omega )]^{+}$; and the underage S^u(ω) which is equal to $[D(\omega )-x]^{+}$.

To calculate the wholesale cost per newspaper, w, the formula ${\textstyle w=b/x}$ is used.

The marginal profit, or net profit for the newsvendor per unit, m is found by the formula $m=s-w$.

The marginal loss, or loss for each unsold unit, l is found using the formula $l=w-v$.

The profit, P, is calculated by $P=m*x$ if every item in inventory was sold.

The expected profit, E, taking into account a given demand probability is calculated by $E=x*D*m$ if every item in inventory is sold.

The objective function can be represented as $F(x,\omega )=c^{o}S^{o}(x,\omega )+c^{u}S^{u}(x,\omega )=c^{o}[x-d(\omega )]^{+}+c^{u}[D(\omega )-x]^{+}$$F(x)=E[F(x,\omega )]=\int (c^{o}[x-D(\omega )]^{+}+c^{u}[D(\omega )-x]^{+})p(\omega )d\omega$

where the goal is to solve for $min_{x}F(x)=E[F(x,\omega )]$.

Numerical Example

A historically relevant example of the newsvendor problem would be the working conditions that led to the newsboy strike of 1899 and subsequent labor movements.

In the late nineteenth century and before the Spanish-American War, newsboys in New York City could purchase 100 newspapers for 50 cents and sell the newspapers for 8 cents each. If a paper didn’t sell, assume the publisher would buy the newspaper back at 60% cost.⁷

Assume the newspaper sales in New York City followed the following demand schedule:

Table 1: Demand in New York City

Quantity	Probability of Demand
700	0.450
800	0.300
900	0.220
1000	0.015
1100	0.010

The cost price of the newspapers is $0.05/100 = $0.005 per newspaper

The selling price of the newspapers is $0.08 per newspaper

The salvage value of the newspapers is $0.003 per newspaper

The marginal profit is equal to $0.08 - $0.005 = $0.075 per additional newspaper sold

The marginal loss is equal to $0.005 - $0.003 = $0.002 per unsold newspaper

Tabulated Values

$c^{o}=\$0.005-\$0.005(0.6)=\$0.002$per unit

$c^{u}=\$0.08$per unit

x = purchase quantity, where $x\in (700,800,900,1000,11000)$

$S^{o}(\omega )=x-\omega ,x>\omega$

$S^{u}(\omega )=\omega -x,x<\omega$

$S^{o}(\omega )=S^{u}(\omega ),x=\omega$

$F(x,\omega )=lossfunction$

$F(x,\omega )=c^{o}S^{o}(x,\omega )+c^{u}s^{u}(x,\omega )$

$F(x,\omega )=c^{o}(x-\omega )+c^{u}(\omega -x)$

$F(x,\omega )=(0.002)(x-\omega )+(0.008)(\omega -x)$

$R(x,\omega )=0.08\omega -F(x,\omega )$

The optimal quantity to purchase is 1000 in order to minimize expected loss and maximize expected revenue.

Demand Distributions

The newsvendor problem can be solved in a multitude of ways, the one uncertainty that always exists is the number of papers needed to fully maximize the profits. This can be estimated by a variety of ways, but most commonly there are uniform, normal, or lognormal distributions.

The uniform distribution estimates the probability to not change. In the case of the newspaper problem this would mean that the demand for a newspaper does not vary from day to day. This method can pose issues as the demand for papers can vary from days like Monday or Tuesday, to days like Sunday which historically have been a day recognized as always having a paper.

The next method that can be used to estimate the demand of a paper can be done using a normal distribution. A normal distribution’s standard deviation positions the curve of demand into being one that can be used to calculate the different demands that a salesman may face amongst the sales of a paper. The normal distribution allocates variations that enable the salesman to take calculated risks based on historical norms. These norms provide contextual evidence to accurately account for the demand that the seller may see.

While a normal distribution can provide estimates into how many papers may need to be printed for the public, it does not take into account the potential profit or loss that the vendor may undertake. The logarithmic method will show at what point the salesman optimal peak profit will be. The logarithmic curve is exponential and will ultimately determine the peak profit and printing point at which the business will succeed. This solution is meant to determine the optimal solution from a profit standpoint.^4,6

Applications

Beyond the namesake example of the newsvendor problem, the newsvendor problem model can be applied to a variety of other discrete optimization problems.

Personal Investments

The tradeoff between typing funds up in a stock and holding cash reserves follows the model of the newsvendor problem because putting too much much of your money in stocks could lead to having to sell stocks undervalue to free up cash whole holding too much money in cash reserves could lead to money that is under performing. The newsvendor problem can help investors find an optimal way to allow to minimize risk while allowing enough opportunity to create a large gain. With recent trends of market volatility evaluating cash positions and market exposure has become ever more important.⁶

Emergency Resources

The amount of emergency resources to hold on hand follows the model of the newsvendor problem because holding too many emergency resources could mean throwing out expensive inventory if there is no emergency and not having enough emergency resources could be disastrous in times of peril. Emergencies have the same tendencies of an unknown market. The first responders need to have an optial amount of supplies to maximize their effectiveness. If items that are perishable are sent in mass quantities it can bog down the supply lines and lead to important resources becoming expired.⁶

Manufacturing

The amount of units of a good to manufacture follows the model of the newsvendor problem because while overproduction would always meet demand, production costs increase and storage costs are introduced for the excess inventory. Manufacturers and wholesalers often rely on razor thin margins. By understanding how to limit excess storage and money that it put out into the materials themselves the business can find an accurate way of maximizing the cash flow. Inventory is often one of the crippling factors of a business. Businesses often can save money on individual units by producing larger quantities but this ultimately eats away at having a strong cash position to address the concerns of a changing market.⁶

Real Estate

House pricing in the real estate market follows the model of the newsvendor problem because if a house is priced too high it will take too long to sell and if the house is priced too low it will sell quickly but at lower price. The housing market is another investment that is exposed to a great deal of volatility and increased market risk. Markets can change rapidly from economic situations to also the crime, schools and locations around a property. By understanding the market norms one can find the adequate pricing method for a home using the newsvendor problem. Appraisers and realtors must focus on understanding these metrics to ensure the estimates are accurate.⁶

Conclusion

The newsboy formulation is used to optimize the amount of profit while minimizing the excess materials that hold no value after a given period of time. This formulation can be adapted for different probabilities and distributions of expected sales.  Additionally, nuances such as accounting for a salvage price for unsold perishable goods can also be added to the problem for added complexity to mimic a given situation. From that, the salesperson can determine how many of a perishable product should be purchased for resale at a given time.

References

1. F. Y. Edgeworth (1888). "The Mathematical Theory of Banking". Journal of the Royal Statistical Society.
2. R. R. Chen; T.C.E. Cheng; T.M. Choi; Y. Wang (2016). "Novel Advances in Applications of the Newsvendor Model". Decision Sciences.
3. Whitin, T. M. “Inventory Control and Price Theory.” Management Science, vol. 2, no. 1, 1955, pp. 61–68.
4. “Newsvendor Model.” Wikipedia, Wikimedia Foundation, 12 Nov. 2020, en.wikipedia.org/wiki/Newsvendor_model.
5. Yan Qin, Ruoxuan Wang, Asoo J. Vakharia, Yuwen Chen, Michelle M.H. Seref, “The newsvendor problem: Review and directions for future research.” European Journal of Operational Research. Volume 213, Issue 2. 2011. Pages 361-374, ISSN 0377-2217. https://doi.org/10.1016/j.ejor.2010.11.024.
6. Powell, W. B. “Newsvendor Problem.” Castle Lab Princeton, 2013, castlelab.princeton.edu/html/Presentations/ORF411_2013/ORF%20411%2015%20Newsvendor%20problem.pdf.
7. “Labor History Lesson: The ‘Newsies’ Strike.” Labor History Lesson: The "Newsies" Strike | AFT Connecticut, 25 May 2016, aftct.org/story/labor-history-lesson-newsies-strike
8. Birge, J. and Louveaux, F. Introduction to Stochastic Programming, Springer, 2011.