Sheletering the Homeless

Sheltering the Homeless

A large municipality has a policy of providing some form of shelter on cold nights to all homeless people who ask for shelter or who are regarded as in need of help by a social service professional. This city has set up an adequate, low cost shelter that can hold 10,000 each night. However since on some nights more than 10,000 people will need a place to stay, the city has contracted with several boarding houses for extra rooms. These rooms are relatively expensive. They cost $100 per night per person if rented one night at a time, but only $200 per person total for five consecutive nights if taken for the week (5 nights.) The daily rate is only paid if the room is used. The weekly rate must be paid in full if taken by the city whether the room is used or not. As such the city has to decide whether to take a block of rooms for the week, or just take them one at a time as needed, or some combination of both.

Staff analysts at the Department of Homeless Services estimate that during the coldest stretches of the winter the number of people needing shelter on any given night can be estimated by a normal distribution with an average of 10,400 and a standard deviation of 2,000. As such for most cold nights the City’s shelter will be insufficient. On those nights when more than 10,000 need shelter, the City will need more space.

The coming weeks are expected to be cold, so the Department must decide each week if it should reserve a block of rooms (and if so how many) or just rent rooms each day as needed. The City can rent blocks of rooms for the week in the amount of either 500, 1000, 1500, 2000 or none at all. If more rooms are needed beyond that, they will be rented one day at a time.

For example, if the City decides this week to rent a block of 500 rooms, they must pay at the beginning of the week $200 times 500 ($100,000) for those 500 rooms whether they are used or not. If on Monday 10,350 people needed housing, the City would have enough rooms in this case (10,000 plus the 500 rented) but would have paid for 150 rooms it didn’t use. If the next night 10,732 people needed rooms, all 500 rooms rented for the week would be used but the City would have to pay $100 each for another 232 rooms ($23,200) for that night. The total cost for the week is the sum of the daily charges for each of the 5 days plus the one time block rental cost. Calculate a cost for each of the block of rooms’ options (0, 500, 1000, 1500, 2000.)

First, simulate one week (5 days) of demand for a cold spell and determine the least cost strategy for the City.  Simulate 1000 weeks using 123 as the initial random number seed and using the Latin Hypercube method.

Hint: Each day will have its own random number. Evaluate the cost implications of the demand using the same random number for each of the 5 choices for renting blocks of rooms. Do that for each of five (5) days and add up the five days worth of costs associated with each choice to find the least expensive. Think “Ski Jacket” problem for 5 periods, not just one.

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