Difference between revisions of "Newsvendor problem"

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<math>S^o (\omega) = S^u (\omega), x = \omega</math>
<math>F(x,\omega) = loss function<math>
<math>F(x,\omega) = c^o S^o (x, \omega) + c^u s^u (x, \omega)<math>
<math>F(x,\omega) = c^o (x-\omega) + c^u (\omega -x)<math>
<math>F(x,\omega) = (0.002)(x- \omega)+(0.008)(\omega-x)<math>
<math>R(x,\omega) = 0.08\omega - F(x,\omega)<math>
To determine the optimal quantity of newspapers to purchase given the demand schedule, the following are tabulated:

Revision as of 00:53, 13 December 2020




T.M Whitin in 1955 was the first to consider not only the cost minimization portion of the problem, but also the profit maximization. To do so he formulated a newsvendor model with price effects, where the selling price and stocking quantity and 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.








Detailed Solution Steps




To calculate the marginal profit, or net profit for the newsvendor per unit, m is found by the formula .




The objective function can be represented as

where the goal is to solve for .

Numerical Example




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



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.

per unit

per unit

x = purchase quantity, where







Table 2: Expected Profit for a Given Supply and Demand Relation
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

Demand Distributions







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.5

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.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. 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.5




  3. “Newsvendor Model.” Wikipedia, Wikimedia Foundation, 12 Nov. 2020, en.wikipedia.org/wiki/Newsvendor_model.
  4. 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.
  5. Powell, W. B. “Newsvendor Problem.” Castle Lab Princeton, 2013, castlelab.princeton.edu/html/Presentations/ORF411_2013/ORF%20411%2015%20Newsvendor%20problem.pdf.
  6. “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