A certain community has three grocery stores. Within this community there exists a shift of customers from one grocery store to another. A study was made on January 1, and it was found that 1/4 of the population shopped at store I, 1/3 at store II, and 5/12 at store III. Each month store I retains 90% of its customers and loses 10% of them to store II. Store II retains 5% of its customers and loses 85% of them to store I and 10% to store III. Store III retain 40% of its customers and loses 50% of them to store I and 10% to store II. Assume the population of the community remains constant. (a). Find the transition matrix T and the initial market share vector v0. (b). Find the proportion of customers will each store retain by March 1. (c). Assuming the same pattern continues, what is the long-term market share.
A certain community has three grocery stores. Within this community there exists a shift of customers from
one grocery store to another. A study was made on January 1, and it was found that 1/4
of the population shopped at store I, 1/3
at store II, and 5/12
at store III. Each month store I retains 90% of its customers and loses
10% of them to store II. Store II retains 5% of its customers and loses 85% of them to store I and 10% to
store III. Store III retain 40% of its customers and loses 50% of them to store I and 10% to store II. Assume
the population of the community remains constant.
(a). Find the transition matrix T and the initial market share vector v0.
(b). Find the proportion of customers will each store retain by March 1.
(c). Assuming the same pattern continues, what is the long-term market share.
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