An offer to buy the Westmall shopping centre was made by a competitor.
One market research project suggested that if the Westmall shopping centre were sold, each of the three centres (Westmall, Grandmall and Eastmall) would continue to have regular shoppers but would attract and lose shoppers on a weekly basis.
Let `S_n` be the state matrix that shows the expected number of shoppers at each of the three centres `n` weeks after Westmall is sold.
A matrix recurrence relation that generates values of `S_n` is
`S_(n+1) = T xx S_n`
`{:(quad qquad qquad qquad qquad qquad qquad qquad text(this week)),(qquad qquad qquad qquad qquad qquad quad \ W qquad quad G qquad quad \ E),(text(where)\ T = [(quad 0.80, 0.09, 0.10),(quad 0.12, 0.79, 0.10),(quad 0.08, 0.12, 0.80)]{:(W),(G),(E):}\ text(next week,) qquad qquad S_0 = [(250\ 000), (230\ 000), (200\ 000)]{:(W),(G),(E):}):}`
- Calculate the state matrix, `S_1`, to show the expected number of shoppers at each of the three centres one week after Westmall is sold. (1 mark)
Using values from the recurrence relation above, the graph below shows the expected number of shoppers at Westmall, Grandmall and Eastmall for each of the 10 weeks after Westmall is sold.
- What is the difference in the expected weekly number of shoppers at Westmall from the time Westmall is sold to 10 weeks after Westmall is sold?
Give your answer correct to the nearest thousand. (1 mark)
- Grandmall is expected to achieve its maximum number of shoppers sometime between the fourth and the tenth week after Westmall is sold.
Write down the week number in which this is expected to occur. (1 mark)
- In the long term, what is the expected weekly number of shoppers at Westmall?
Round your answer to the nearest whole number. (1 mark)