Question

Consider the isolated Swiss town of Andelfingen, inhabited by 1200 families. Each family takes a weekly shopping trip to the only grocery in town, run by Mr. and Mrs. Wipf, until the day when a new, fancier (and cheaper) chain store, Migros, opens its doors. It is not expected that everybody will immediately run to the new store, but we do anticipate that 20% of those shopping at Wipf's each week switch to Migros the following week. Some people who do switch miss the personal service (and the gossip) and switch back: We expect that 10% of those shopping at Migros each week go to Wipf's the following week. The state of this town (as far as grocery shopping is concerned) can be represented by the vector x(t)=[ w(t)

m(t)
​
], where w(t) and m(t) are the numbers of families shopping at Wipf's and Migros, respectively, t weeks after Migros opens. Suppose w(0)=1200 and m(0)=0. a. Find a matrix A such that x(t+1)=Ax(t). Verify that A is a regular transition matrix. [See Problem 14.] b. How many families will shop at each store after t weeks? Give closed formulas. c. The Wipfs expect that they must close down when they have less than 250 customers per week. When does that happen?

Answer 1

To find the matrix A, we need to determine the transition probabilities between Wipf's and Migros. The transition matrix A is [0.8 0.1],[0.2 0.9] and it is a regular transition matrix. After t weeks, the number of families shopping at each store can be found using the formula x(t) = A^t * x(0). The Wipfs will close down when the number of customers at Wipf's is less than 250 per week.

Step-by-step explanation:

To find the matrix A, we need to determine the transition probabilities between Wipf's and Migros. From the information given, we know that 20% of the families shopping at Wipf's each week switch to Migros, and 10% of those shopping at Migros each week go to Wipf's the following week. This means that 80% of the families continue to shop at Wipf's each week, and 90% continue to shop at Migros. Therefore, the transition matrix A is:

A = [0.8 0.1],[0.2 0.9]

To verify that A is a regular transition matrix, we need to check if the sum of each column is equal to 1. In this case, the sum of the first column is 0.8 + 0.2 = 1, and the sum of the second column is 0.1 + 0.9 = 1. Therefore, A is a regular transition matrix.

To find the number of families shopping at each store after t weeks, we can use the formula x(t) = A^t * x(0), where x(0) is the initial vector [w(0), m(0)].

To find when the Wipfs must close down, we need to find the number of customers shopping at Wipf's each week. This can be calculated using the formula C(t) = w(t) + m(t), where C(t) is the number of customers at Wipf's and m(t) is the number of families shopping at Migros after t weeks. The Wipfs will close down when C(t) is less than 250 customers per week.