Introduction to mathematical programming
Introduction to mathematical programming
4th Edition
ISBN: 9780534359645
Author: Jeffrey B. Goldberg
Publisher: Cengage Learning
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Chapter 2.1, Problem 2P

Explanation of Solution

Relating two vectors using matrix multiplication:

Suppose that at the beginning of each year, people change the beer they drink.

Let for i=1,2,3,....xi be the number of people who prefer beer “i” at the beginning of this month, and yi be the number of people who prefer beer “i” at the beginning of next month.

Since 30% of the people who prefer beer 1 switch to beer 2 and 20% of the people switch to beer 3, so the remaining 50% continue to be with beer 1. Also, 10% of people who prefer beer 3 switch to beer 1.

Hence, the number of people who prefer beer 1 in the beginning of next month is given below:

  y1=0.5x1+0.1x3

Now, 30% of the people who prefer beer 2 switch to beer 3 and the remaining 70% of the people continue to be with beer 2. Also, 30% of people who prefer beer 3 switch to beer 2 and 30% of the people who prefer beer 1 switch to beer 2.

Hence, the number of people who prefer beer 2 in the beginning of next month is given below:

  y2=0.3x1+0.7x2+0.3x3

Then, 30% of the people who prefer beer 3 switch to beer 2 and 10% of the people switch to beer 1, so the remaining 60% continue to be with beer 3. Also, 20% of people who prefer beer 1 switch to beer 3 and 30% of the people who prefer beer 2 switch to beer 3.

Hence, the number of people who prefer beer 3 in the beginning of next month is given below:

  y3=0.2x1+0

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