Instructions:

1.   Sufficient details should be provided in your solutions to gain higher marks, round your answers to 3 decimal places if needed.

2.   You are allowed to use online or computer tools if needed, but details (e.g., codes, snapshots) should be provided.

3.   Sloppy submissions with unclear writing might cause mark reductions.

Question  1.  [25  marks]  XJTLU Library currently holds over 500,000 books, which are added to and updated throughout the year in response to requests from academic departments. Currently the library is planning a “buy-back” event to recycle used books from students, and it is believed that this would help reduce the cost on ordering brand new books from publishers. For the sake of simplicity, we categorize the daily business volume as “high” and “low” According to past experience, a day with high business volume is be followed by a day with high volume with probability 0.6; whereas a day with low volume will become high volume on the next day with probability 0.2. If the business volumes are high for consecutive 2 days, the library will need to spend $10,000 on new books, and $30,000 if the volumes are low for both days. If the business volume is changed from low to high, the library will spend $15,000 and $20,000 for “high-to-low” Besides the base case, the library would encourage students to participate the “buy-back” event by offering a lottery that awards 5 winners of the day. This will increase the  probability of getting two consecutive high-volume days to 0.75 and there will be a 40%  chance for a low volume day to change to a high-volume day. However, the cost will be increased by 20%.

(1) Write out the state space, the set of actions, the probability transition matrices and cost matrices.      [8 marks]

(2) Write out a backward recursive formula to calculate fn (i), the optimal expected cost of stages n, n + 1,  ⋯ , N, given state i  at the beginning of stage n.        [5 marks]

(3) Generate the optimal solution for XJTLU Library for a 3-day “buy-back” event, assuming the business volume of the day right before the event is low. Show your calculations and explain the optimal strategy.              [12 marks]

Question 2. [25 marks] Use postgraduate application as the big background to design a multi-objective optimization problem and identify the Pareto-optimum. Make sure that your question contains 4 objectives and only finite number of feasible solutions.