Explanation Given: A survey shows that 23% of the shoppers in seven Midwestern states regularly buy their groceries at Safe Shop. 29% regularly shop at PayNEat, and the balance shop at any one of several smaller markets. The survey indicates that 8% of the consumers who currently shop at Safe Shop will purchase their groceries at PayNEat the next time they shop and that 5% will switch to some other store. Also, 12% of the consumers who currently shop at PayNEat will purchase their groceries at Safe Shop the next time they shop, and 2% will switch to some other store. In addition. 13% of the consumers who currently shop at neither store will purchase their groceries at SafeShop the next time they shop, and 10% will shop at PayNEat. Approach: Find the equilibrium Matrix L for more than two unknowns. 1. Find the transition matrix T and the equilibrium matrix L . 2. Simplify the system of equations from L T = L and discard any redundant equations and include the equation x + y + ... + z = 1 . 3. Solve the resulting system of equations. Calculation: From the information provided, the transition matrix T is given below. T = [ 0.87 0.08 0.05 0.12 0.86 0.02 0.13 0.10 0.77 ] There are three variables, therefore, the equilibrium matrix L is given below. L = [ x y z ] The system of equation L T = L is given below. [ x y z ] [ 0.87 0.08 0.05 0.12 0.86 0.02 0.13 0.10 0.77 ] = [ x y z ] The system in the form of linear equations is given below. { 0.87 x + 0.12 y + 0.13 z = x 0.08 x + 0.86 y + 0.10 z = y 0.05 x + 0.02 y + 0.77 z = z The system of equations can be written as shown below. { − 0.13 x + 0.12 y + 0.13 z = 0 0.08 x − 0.14 y + 0.10 z = 0 0.05 x + 0.02 y − 0.23 z = 0 Multiply each element of the system by 100. { − 13 x + 12 y + 13 z = 0 8 x − 14 y + 10 z = 0 5 x + 2 y − 23 z = 0 Discard the redundant equation 5 x + 2 y − 23 z = 0 with the equation x + y + z = 1 . { − 13 x + 12 y + 13 z = 0 8 x − 14 y + 10 z = 0 x + y + z = 1 Solve the equations − 13 x + 12 y + 13 z = 0 and x + y + z = 1

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ISBN: 9780357425831

Author: Johnson

Publisher: CENGAGE L

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Chapter 11.5, Problem 7E

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The percentage of Midwestern shoppers who will be regular Safe Shop customers and the percent that will be regular PayNEat customers as a result of the trend.

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Chapter 11.5, Problem 6E

Chapter 11.5, Problem 8E

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The percentage of Midwestern shoppers who will be regular Safe Shop customers and the percent that will be regular PayNEat customers as a result of the trend.

Solution Summary: The author explains how to find the equilibrium matrix L for more than two unknowns.

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To find: The percentage of Midwestern shoppers who will be regular Safe Shop customers and the percent that will be regular PayNEat customers as a result of the trend.

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Chapter 11 Solutions

To find:

The percentage of Midwestern shoppers who will be regular Safe Shop customers and the percent that will be regular PayNEat customers as a result of the trend.