For both problems, Python code preambles containing the input data may be found on Canvas. Please use all these variables in your code rather than hard coding fixed numbers into your calculations. You are encouraged to use the functions in the binomialPoisson.py file available on Canvas to use these functions, just put this file in the same directory as your code and put the statement

from binomialPoisson import *

at the top of your code (already done for you in the input data code preambles).

For each problem of problems 1 and 2, upload two files to Canvas. The first file should contain the output of your Python program to solve the problem, along with the answers to the following questions:

When solving this problem by dynamic programming, what are the stages?

Within each stage, are the states?

Within each stage and state, what is the decision to be made?

Within each stage and state, what is the source of uncertainty?

What is the specific interpretation of the value function f_t(i) for this problem? Your answer should explicitly mention the role of both t and i.

The second file should be the .py source code for your program to solve the problem.

Q1: Managing Repair Operations

Fuel processing units arrive at your facility for repair at a steady rate of 3 units at the beginning of each week. At the start of each week, you decide how many units you will try to repair, up to a maximum of 8. The repair process has the following cost structure:

Number of units attempted to repair	0	1	2	3	4	5	6	7	8
Cost	$ 0	$ 100	$ 110	$ 120	$ 130	$ 190	$ 200	$ 210	$ 220

(These are total costs, not costs per unit.) Every time you attempt to repair a unit, there is an 85% chance, independent of all other units, that the repair operation succeeds. Units for which the repair operation fails, plus any which you did not attempt to repair, are carried over to the next week. The costs in the above table apply not matter how many of the attempted repairs are successful. Safety regulations allow you to carry over at most 15 units to the next week; if the number of units to be carried over exceeds this amount, any excess must be sent to an external repair vendor at a cost of $150 each. Each unit carried over to the next week incurs a direct $9 waiting cost, and your firm has an internal cost of funds of 0.1% per week. The number of units you attempt to repair in any given week cannot exceed the number that are available to repair, which is the number carried over from the previous week plus the 3 that arrive at the start of that week. Just before the start of week 1, you have 2 units awaiting repair. After 12 weeks, the facility must be temporarily shut down for its annual safety cleaning, at which time any remaining units must be shipped out to the external repair vendor at the same $150 per unit cost as mentioned above.

To minimize the expected present value of the cost of the operation, find the optimal number of units to attempt to repair each week, as a function of the number of units carried over from the previous week.