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# lec8 - 15.053 March 6 2007 Duality 1 Pricing Warning...

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1 15.053 March 6, 2007 Duality 1: Pricing Warning: heavier than usual theory ahead.

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Quotes of the Day What is a cynic? A man who knows the price of everything and the value of nothing. Oscar Wilde It's not enough to create magic. You have to create a price for magic, too. You have to create rules. Eric Burns The price of freedom is eternal vigilance. Thomas Jefferson 2
Prices as Part of Linear Programming z Shadow Prices z Reduced costs and pricing out z Prices as part of the simplex algorithm z Prices because they are useful in practice 3

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What do prices have to do with linear programming? Tim The shadow prices can be very useful in practice. Moreover, pricing comes up a lot in linear programming, much more than you would ever guess. You will see a lot of pricing in this lecture. Cleaver Normally, we introduce duality in linear programming directly. Duality involves prices and pricing out. But in this lecture, we will not talk about duality at all. We will talk about prices, especially shadow prices and simplex multipliers. 4
Example: Transfer Prices within a Firm z From Wikipedia: Transfer pricing refers to the pricing of goods and services within a multi-divisional organization. z Example, goods from the production division may be sold to the marketing division, or goods from a parent company may be sold to a foreign subsidiary, with the choice of the transfer price affecting the division of the total profit among the parts of the company. There are a lot of applications of pricing in practice. This is obvious as we are constantly paying prices for goods. What is less obvious is that prices are often used in more subtle ways. For example, firms use transfer prices in order to price goods between divisions. In principle, the transfer prices can be used to increase efficiency within a firm. In fact, linear programs can help determine transfer costs for many firms. More likely, it will be used to take advantage of different tax codes in different countries. The way to save taxes is to have the profit of a firm show up in locations with low taxes, and to have no profit in those locations with a high tax rate. 5

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Mira and Marnie’s M&M Adventure Mira and Marnie, two MIT undergraduates known as the M&M sisters, recently received a gift from their parents of 2000 pounds of gray M&Ms and 6000 pounds of red M&Ms, the MIT colors. So, they decided to go into business selling large bags of “MIT M&Ms” for frat parties. They can sell a bag with 3 pounds of red M&Ms and 2 pounds of gray M&Ms for \$20. They can purchase bags of 3 pounds of red M&Ms and 4 pounds of gray M&Ms for \$30. How many bags should Mira & Marnie buy and sell to maximize their profit. This is the M&M adventure from previous lectures. 6
Formulation as a linear program z Let x 1 be the number of 7 pound bags purchased (in thousands) z Let x 2 be the number of 5 pound bags sold (in thousands) z Measure the profit in \$10,000s.

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lec8 - 15.053 March 6 2007 Duality 1 Pricing Warning...

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