The elevator problem

(Adapted from " The GAIMME report.")

MEMO

From: Your Boss
To: You
Re: Late Arrivals

I have received numerous complaints that large numbers of our employees are reaching their offices well after 9:00 a.m. due to the inability of the present three elevators to cope with the rush at the start of the day. In the present financial situation it is impossible to consider installing any extra elevators or increasing the capacity of existing ones above the current ten persons. Please investigate and let me have some possible solutions to the problem with an indication of their various advantages and disadvantages.

How to solve the problem of employees coming in late due to inability of elevators to handle peak demand?

We want to develop a mathematical model to understand the nature of this problem, and to propose a solution.

As a start, let us brainstorm some ideas about how to approach the problem.

What are your thoughts? Discuss!

Strategy 1

Perhaps a good starting point is to try to understand how severe the current problem is, in terms of time. Let us try to estimate the total time the elevators currently take to get all the employees to their respective floors. In order to do this, we need to get some more information, and also must make some simplifying assumptions.

For example, here are some items of information we need:

• What is the capacity of each elevator? (i.e., how many people can be in it?)
• How many floors are in the building?
• How many employees work on each floor?

Suppose each of the 3 elevators can accommodate 10 peolple; and there are 6 floors in the building, including a ground floor on which no employees work. There are 300 employees total, distributed approximately equally across the 5 occupied floors. So the situation looks something like this:

Floor	G	1st	2nd	3rd	4th	5th
Employees	0	60	60	60	60	60

Exercise

Estimate the total time the elevators take to get all the employees in the building to their respective floors.
If you need more information or data, please feel free to ask. Also, feel free to make any reasonable assumptions needed.

Strategy 1 (continued)

Now that we have an estimate of how much total time is required, let's think about ways to reduce it.

One example is to consider a policy in which one elevator is reserved just for the upper floors. Suppose Elevator A only serves floors 4-5, and Elevators B and C serve floors 1, 2, 3.

Elevator A		Elevator B		Elevator C
Floors 4-5 only		Floors 1-3 only		Floors 1-3 only

Exercise

Estimate the total time the elevators take to get all the employees to their respective floors with the above policy.

Discussion

• It is not difficult to try out other policies about the floor(s) each elevator serves. Try one of your own variations.

• After our simple initial approach, it is now easier to relax or change our initial assumptions and make them more realistic.
  For example suppose there is a 20% probability that an elevator must stop on the way down. How can we account for that?

• Also, suppose the number of employees working on a particulr floor is much higher than on other floors.
  How can we modify our strategy to account for it?