MA231-2019-exam

Summer Exam MA Operational Research Methods Suitable for all candidates Instructions to candidates This paper contains 6 questions. You may attempt as many questions as you wish, but only your best answers will count towards the nal mark. All questions carry equal numbers of marks. Answers should be justi ed by showing work. Please write your answers in dark ink (black or blue) only. Time Allowed Reading Time: None Writing Time: hours and minutes You are supplied with: No additional materials You may also use: No additional materials Calculators: Electronic Calculator (as prescribed in the examina- tion regulations) © LSE ST /MA Page of

Question (a) Suppose that in the entire confectionary market there are only two brands of chocolate bar, namely bar and bar . Given that a person last purchased bar , there is a % chance that her next purchase will be bar . Given that a person last purchased bar , there is an 8 % chance that her next purchase will be bar . (i) Each consumer’s purchases can be represented as a Markov chain. Write down the transi- tion matrix. (ii) If a person is currently a bar purchaser, what is the probability that she will purchase bar two purchases from now? If a person is currently a bar purchaser, what is the probability that she will purchase bar three purchases from now? (iii) The two brands have been established in the market for many years, and the consumers’ behaviour described above has remained stable for a long time. Compute the respective market shares of bar and bar . (iv) Suppose that each consumer purchases one chocolate bar during each week ( year= weeks). Suppose there are million chocolate bar consumers, and both companies make a net pro t of £ per chocolate bar sold. For £ million per year, an advertising rm guarantees to decrease from % to % the fraction of bar consumers who switch to bar after a purchase. Should the company that makes bar hire the advertising rm? [ marks] (b) Consider a Markov chain with n + 1 states 0 , 1 , 2 , . . . , n , where 0 is an absorbing state. From state 1 , we reach in one step any of the other n states { 0 } ∪ { 2 , . . . , n } with equal probability 1 /n . From any state i ∈ { 2 , . . . , n } , we reach in one step any of the n states 1 , . . . , n with equal probability 1 /n (in particular, we stay at i with probability 1 /n ). (i) Which states are recurrent? Which states are transient? Is there a limiting distribution? (ii) What is the expected number of steps to reach absorbing state starting from state ? What is the expected number of steps to reach absorbing state starting from any of the states 2 , . . . , n ? [8 marks] © LSE ST /MA Page of