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. 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 . In the formulation, s represents the price a unit is sold for, n represents the number of units in inventory, 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 where F^{-1} is the inverse of the cumulative distribution function of R.^{3,4}
Detailed Solution Steps
In formulation, a newsboy could purchase a given number of newspapers n 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.
To calculate the wholesale cost per newspaper, w, the formula is used.
To calculate the marginal profit, or net profit for the newsvendor per unit, m is found by the formula .
The marginal loss, or loss for each unsold unit, l is found using the formula .
The profit, P, is calculated by if every item in inventory was sold.
The expected profit, E, taking into account a given demand probability is calculated by if every item in inventory is sold.
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.^{3,5}
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.^{5}
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.
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.^{5}
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.^{5}
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.^{5}
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.^{6}
Assume the newspaper sales in New York City followed the following demand schedule:
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
Suppose a newsboy wants to purchase a supply of 700 newspapers at a cost of $3.50.
The probability of demand for 700 newspapers is 0.45 according to the demand schedule in Table 1.
If the newsboy sold all 700 papers, the profit would be $56.00.
The expected profit would be 700 newspapers x $0.08 profit per newspaper x 0.45 = $25.20.
If the newsboy only sold 600 papers, the profit would be $48.00 and the newsboy could earn $0.30 from selling the unsold papers back to the publisher.
To determine the optimal quantity of newspapers to purchase given the demand schedule, the following are tabulated:
Demand (Probability) | |||||||
---|---|---|---|---|---|---|---|
0.450 | 0.300 | 0.220 | 0.015 | 0.010 | Expected Profit | ||
Supply | 700 | 52.5 | 52.5 | 52.5 | 52.5 | 52.5 | 52.24 |
800 | 52.3 | 60.0 | 60.0 | 60.0 | 60.0 | 56.24 | |
900 | 52.1 | 59.8 | 67.5 | 67.5 | 67.5 | 57.92 | |
1000 | 51.9 | 59.6 | 67.3 | 75.0 | 75.0 | 57.91 | |
1100 | 51.7 | 59.4 | 67.1 | 74.8 | 82.5 | 57.79 |
Given the above demand schedule, the optimal solution for the newsboy is to purchase 900 newspapers at a cost of $4.50 because this level of supply has the highest expected profit of $57.92.^{7}
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
- F. Y. Edgeworth (1888). "The Mathematical Theory of Banking". Journal of the Royal Statistical Society.
- R. R. Chen; T.C.E. Cheng; T.M. Choi; Y. Wang (2016). "Novel Advances in Applications of the Newsvendor Model". Decision Sciences.
- “Newsvendor Model.” Wikipedia, Wikimedia Foundation, 12 Nov. 2020, en.wikipedia.org/wiki/Newsvendor_model.
- 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.
- Powell, W. B. “Newsvendor Problem.” Castle Lab Princeton, 2013, castlelab.princeton.edu/html/Presentations/ORF411_2013/ORF%20411%2015%20Newsvendor%20problem.pdf.
- “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
- Birge, J. and Louveaux, F. Introduction to Stochastic Programming, Springer, 2011.