**Part 1**

You sell a product for which monthly demand is Poisson with a mean of 40. The units cost you $1,500 each, and you sell them for $2,800. You can carry inventory from month to month, and estimate your inventory holding cost as $10 per unit left in inventory at the end of a month.

Every time you order, there is a fixed cost of $600, plus the $1,500 per unit cost of the products ordered. Assume that you can order at the end of the day, and the ordered stock arrives at the beginning of the following day.

You want to simulate a 24-month period, at the outset of which you have 70 units in stock. For every unit in stock at the end of this period, you assess a "salvage" credit of $1,500.

You are considering ordering policies of the following form: if the ending
inventory for a given month is less than or equal to some "threshold"
value *R*, immediately order another *Q* units. For simplicity,
assume that these units become available immediately at the beginning of the
next month.

Your boss asks you to evaluate the following possible combinations of *R*
and *Q*. Which one seems to yield the highest expected profit over
the 24 month period?

Policy |
R |
Q |

1 | 40 | 75 |

2 | 40 | 80 |

3 | 40 | 85 |

4 | 45 | 75 |

5 | 45 | 80 |

6 | 45 | 85 |

7 | 50 | 75 |

8 | 50 | 80 |

9 | 50 | 85 |

For each policy, you also wish to estimate the probability of having a "stockout"
at some time during the 24 month period. A "stockout" means that
there is insufficient stock to meet customer demand.

**Part 2**

Now, let us consider a more realistic scenario. Instead of orders arriving instantaneously, assume that they take a random amount of time to arrive, according to the following distribution:

Number of Months Until Arrival |
Probability |

1 | 0.10 |

2 | 0.40 |

3 | 0.30 |

4 | 0.20 |

That is, an order has a 10% chance of arriving at the start of the next month, a 40% chance
of arriving the following month, and so forth. We define our "inventory
position" to be the number of units in stock plus those on order. Our
policy now becomes the following: if the inventory position at the end of
the month is at or below *R*, we order *Q* units.

Suppose that we start the 24-month period with 110 units in inventory and none currently on order. Determine which policy appears to be best from among the following:

Policy |
R |
Q |

1 | 115 | 80 |

2 | 115 | 85 |

3 | 115 | 90 |

4 | 120 | 80 |

5 | 120 | 85 |

6 | 120 | 90 |

7 | 125 | 80 |

8 | 125 | 85 |

9 | 125 | 90 |