problem 1: 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.  Furthermore, 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 resolve a suitable optimization model to help this soda producer identify the best promotional strategies for its two products. 

problem 2: 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 takes place. It is as well 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, find out the probability that he leaves the department on 'bad terms'?

c) Determine 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?