For each problem, make an Excel model, simulate the situation using YASAI, and hand in the
standard printouts for a
simulation problem.
Q1. (30 points) You operate an aircraft that flies two flights per day: an "outbound" flight from its base to a remote destination, and then and "inbound" flight back again. The aircraft can carry a mix of passengers and freight: if configured for all passengers, the aircraft holds 216 seats. Each freight pallet you configure the aircraft to hold takes away 24 seats, and you are considering setting up the aircraft to hold either 0, 1, 2, 3, 4, 5, or 6 freight pallets. The split between seats and space for pallets must be decided once a year when the aircraft undergoes "heavy maintenance"; it cannot be changed for each flight. You make a profit of $50 for each passenger carried (in either direction), and demand for seats on the outbound flight is Poisson with a mean of 160, while demand for seats on the inbound flight is Poisson with a mean of 159 (independent of the outbound flight). Demand for freight carriage is independent of passenger demand and also independent between the inbound and outbound flights. You have collected historical data on freight demand and have obtained the following:
Outbound | Inbound | |
Pallets | Probability | Probability |
0 | 6% | 7% |
1 | 8% | 9% |
2 | 18% | 18% |
3 | 20% | 19% |
4 | 21% | 17% |
5 | 16% | 16% |
6 or more | 11% | 14% |
You make a profit $275 per pallet carried (in either direction). What
aircraft configuration maximizes the average profit per day? Simulate 1000 days for each possible configuration.
Q2. (35 points) You make a perishable, volatile chemical for which you charge $2.25 per liter. You have 75 regular customers for the chemical, each of whom has an independent 90% chance of placing an order on any given day. You also get an average of 33 orders per day from other, non-regular customers; assume the number of non-regular customers per day has a Poisson distribution. Every order is for one 20-liter container.
You produce the chemical by two batch processes, called "large" and "small". The large batch process produces 800 liters of the chemical at a cost of $1300, while the small batch process produces 500 liters at a cost of $850. Each day, you can run each of these processes any whole number of times. Because it is so unstable, any chemical left unsold at the end of the day must be recycled, at a cost of $0.35 per liter.
What is the
best number of times to run each process? Use a sample size of 1000 and
consider all six possible combinations of running the large process either 1 or
2 times, and the small process either 0, 1, or 2 times. For the
choice giving the highest average profit per day, what is the average number of
liters recycled per day?
Q3. (35 points) You operate a remote work site with five pieces of diesel-powered equipment, as follows:
Probability | Min Fuel | Max Fuel | |
Equipment | Needed | Amount | Amount |
1 | 0.43 | 5 | 15 |
2 | 0.31 | 3 | 20 |
3 | 0.41 | 2 | 20 |
4 | 0.37 | 10 | 25 |
5 | 0.23 | 5 | 20 |
For example, the first piece of equipment has a 43% chance of being used on any given day, and if it is used, it requires an amount of fuel uniformly distributed between 5 and 15 gallons. Similarly, the second piece of equipment has a 31% chance of being used on any particular day, and if it is used, requires an amount of fuel uniformly distributed between 3 and 20 gallons. The remaining rows of the table are interpreted similarly.
Before the work day begins, and thus before you know which pieces of equipment will be used and how much fuel they will consume, you can "preposition" fuel at the work site at a cost of $0.22 per gallon. If any of this fuel is not used, it must (for security reasons) be shipped back from the work site at the end of the day for the same cost of $0.22 per gallon. If you end up needing more fuel than you prepositioned, the extra fuel needed must be shipped to the worksite at a higher, "emergency" cost of $0.54 per gallon.
You are considering prepositioning either 0, 5, 10, 15, 20, 25, 30, 35, or 40 gallons of fuel at the work site. Based on a sample size of 1000, determine which choice gives you the lowest average shipping daily cost. With the lowest-cost strategy, what is the average number of gallons of fuel per day that must be "emergency" shipped to the site?
Hint:
one way to produce a 1 with probability p and a 0 with probability 1
− p is to use the formula GENBINOMIAL(1,p).