### Business Decision Analytics under Uncertainty (33:136:400)

Professor Eckstein

A Dynamic Programming Problems with Binomial Distributions -- Multiple Machine
Breakdowns

Your company anticipates a need for the following number of DNA sequencing
machines over the next 10 weeks.

**Week** |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |

**Machines **
Needed |
10 |
12 |
5 |
9 |
15 |
16 |
18 |
20 |
21 |
17 |

At the beginning of each week you decide how many new machines to order at a
cost of $20,000 each. Ordered machines arrive at the beginning of the next
week. In your experience, each machine in your possession has an
independent 5% chance of breaking down in any given week. You do not
discover which machines are going to break during a week until after you have
placed your order for new machines. For simplicity,
assume that broken-down machines are complete "write-offs" and have no
value. If, after breakdowns, you do not have enough machines to meet your
requirements for the week, you incur a $5,000 cost for each machine you are
"short" for the week (this cost is incurred because you must
subcontract out your DNA sequencing work to other companies). Any machines
still working at the end of the 10-week period are assumed to have a salvage
value of $13,000 each. You currently have 11 machines in your
possession and you have room for at most 25. Assume that your firm's internal discount rate is 1% per week.

**Determine the machine-buying policy that minimizes the expected present
value of your costs for the 10-week period, net of the present value of the
ending salvage value.**