### Describe the discrete-time markov chain

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Q1. A soda producer makes and sells two products, Classic Cola and Diet Cola. If this company spends X1 dollars on promotion of Classic Cola in a particular western city, it can sell at 75 X1^ 0.5 12-packs of Classic Cola each week there. Moreover, if this company spends X2 dollars on promotion of Diet Cola in a particular western city, it can sell at 50X2^0.5 12-packs of Diet Cola each week there. Each 12-pack of Classic Cola sells for \$3.00 and costs \$0.80 to produce and ship to customers in this western city. Each 12-pack of Diet Cola sells for \$3.50 and costs \$1.00 to produce and ship to customers in this western city. A total of \$10,000 is available for promotion each week in this city. The soda producer seeks to maximize its weekly profit. Formulate and solve an appropriate optimization model to help this soda producer identify the best promotional strategies for its two products.

Q2. A faculty member can be an assistant professor, an associate professor, or a full professor. In terms of departmental standing, a full professor is above an associate professor, who is in turn, above assistant professor. At the end of each academic year, 13% of assistant professors become associate professors and 18% leave the department; 21% of associate professors become full professors, and 17% leave the department; and 9% of full professors leave the department. Demotion never occurs. It is also known that 60% of faculty members enter the department as an assistant professor, 30% as an associate professor, and 10% as a full professor. When an associate professor leaves the department, 75% of the time is on good terms. When an assistant professor leaves the department, 60% of time is on "bad terms" and 40% of the time is on "good terms". When a full professor leaves the department, 90% of the time is on "good terms".

a) Describe the discrete-time Markov chain (X(t)) and transition probability matrix

b) Given that a faculty member enters the department as an assistant professor, what is the probability that he leaves the department on "bad terms"?

c) What is the probability that a faculty member leaves the department on "bad terms"?

d) Given that a faculty member enters the department as an associate professor, how long would you expect him to stay in the department?

e) On average, how many years does a faculty member stay in the department?

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