Assignment:
Three major motel chains compete for the lucrative economy market: Sunny 6, Cloudy 7, and Rainy 8. It has been observed that customers who stay in one of the motels usually patronize the same chain again, except if they perceive the service to be poor. In particular, in case of Sunny 6, 10 % of the time the service is perceived to be poor in which case customers change to one of the other two chains with equal probability. In case of Cloudy 7, service is perceived to be satisfactory 80 % of the time; if it is not, customers switch to one of the other two chains with equal probability. Finally, 90 % of the time service at Rainy 8 is deemed to be ok; if it is not, customers always switch to Sunny 6.
(a) Set up the transition matrix P.
(b) In the long run, what percentage of customers patronizes the three motels? What are the mean recurrence times for the three motels?
(c) Management of the Cloudy 7 chain perceives that its customer loyalty is not as good as one could wish. By using of better management techniques, they may be able to reduce the proportion of poor service to 10 % (again, in case of poor service, customers switch to the other chains with equal likelihood). What are the new steady-state proportions?
(d) Given that 1 % of business is worth $500,000, what is the maximum amount that the management of Cloudy 7 should be prepared to pay to implement the new management techniques that result in the improved service?
Provide complete and step by step solution for the question and show calculations and use formulas.