Problem: An event manager is planning how much coffee to purchase for a research conference. The conference will last for a full day, so to keep the researchers alert and, perhaps more importantly, to keep the presenters to the point when they speak, coffee is a bare necessity.
While coffee could be bought from any local retailer before or during the conference, the manager wants to place a large order with a wholesaler to reach a more favorable price. Such an order has a lead time of up to five days, so it has to be placed well ahead of the start of the conference.
The wholesaler, whom the manager plans to purchase from, offers ground beans in 5 pound bags at $85 per bag. The local retailer offers similar bags at $200 per bag. Running out of coffee is not an option, so the manager plans to use as much wholesale coffee as possible and, if it runs out, buy local retailer coffee for the remaining demand. Running out of coffee is not an option, so the manager plans to use as much wholesale coffee as possible and, if it runs out, buy local retailer coffee for the remaining demand. Unused beans cannot be sold or stored, so if any ground beans remain after the conference they are given away to the crew for free.
The manager asks one of his assistants to investigate coffee consumption at previous conferences to provide a forecast. After some data mining, the assistant presents the following data:
| Bags of coffee |
Probability |
| 6 |
0.01 |
| 7 |
0.02 |
| 8 |
0.03 |
| 9 |
0.04 |
| 10 |
0.05 |
| 11 |
0.07 |
| 12 |
0.09 |
| 13 |
0.11 |
| 14 |
0.12 |
| 15 |
0.13 |
| 16 |
0.13 |
| 17 |
0.1 |
| 18 |
0.06 |
| 19 |
0.03 |
| 20 |
0.01 |
Question 1: How many bags of coffee beans should be ordered from the wholesaler to minimize expected cost with such a buy back option? Answer in integer bags.
Question 2: What is the expected cost? Round to closest integer dollars.