︠fe2aea64-92f4-4610-9b2d-b601bb6eb8adsi︠ %hide %html

The elevator problem

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. ︡eb0fa042-21c7-4007-9973-afc5bf4f2a7f︡{"hide":"input"}︡{"html":"\n

\nThe elevator problem\n

\n\n(Adapted from \n\"\nThe GAIMME report.\")\n\n

\n\n \n \n \n
\n \n
\n\n\nMEMO\n

\nFrom: Your Boss
\nTo: You
\nRe: Late Arrivals \n

\nI have received numerous complaints that large numbers of our \n employees are reaching their offices well after 9:00 a.m. due \n to the inability of the present three elevators to cope with \n the rush at the start of the day. In the present financial \n situation it is impossible to consider installing any extra \n elevators or increasing the capacity of existing ones above \n the current ten persons. Please investigate and let me have \n some possible solutions to the problem with an indication of \n their various advantages and disadvantages.\n"}︡{"done":true}︡ ︠38c3df39-9dee-4751-99a4-0b7c9dc98788si︠ %hide %html 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! ︡afd01169-9b0e-465b-83fe-30a20052d192︡{"hide":"input"}︡{"html":"\nHow to solve the problem of employees coming in late due to \ninability of elevators to handle peak demand?\n\n

\nWe want to develop a mathematical model to understand the \nnature of this problem, and to propose a solution.

\nAs a start, \nlet us brainstorm some ideas about how to approach the problem. \n

\nWhat are your thoughts? Discuss!\n\n"}︡{"done":true}︡ ︠f0a484de-d439-4217-8095-cf49f833ae4csi︠ %hide %html 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. ︡3af4006c-d0d6-4ce9-b1f4-2dfa8e82dd99︡{"hide":"input"}︡{"html":"\nStrategy 1\n\n

\nPerhaps a good starting point is to try to understand how \nsevere the current problem is, in terms of time. \nLet us try to estimate the total time the elevators currently \ntake to get all the employees to their respective floors. \nIn order to do this, we need to get some more information, and \nalso must make some simplifying assumptions.\n

\nFor example, here are some items of information we need:\n

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

\n\nSuppose each of the 3 elevators can accommodate \n10 peolple; and there are 6 floors in the building, including a \nground floor on which no employees work. There are 300 employees \ntotal, distributed approximately equally across the 5 occupied \nfloors. So the situation looks something like this:

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

\n\n\nExercise\n\n

\nEstimate the total time the elevators take to get all \nthe employees in the building to their respective floors.
\nIf you need more information or data, please feel free \nto ask. Also, feel free to make any reasonable \nassumptions needed.\n"}︡{"done":true}︡ ︠0dcbe150-d7c6-4d5c-bbf5-31d745adea67sio︠ %hide %html

One possible solution

This solution is based on the following assumptions:

Once the employees start arriving, they come in a steady stream until everyone gets to their floor.
The elevator will need to stop at every floor on the way up.
On the way down, it will not need to stop anywhere.
The elevator doors will need to open only once on each floor.
Other information we need/get from measurements:

It takes 5 seconds for the elevator to travel from one floor to the next.
It takes 15 seconds for the elevator doors to open/close and let people off at each floor.
On the ground floor it takes 25 seconds for the elevator to fill up and close its doors.

The sketch summarizes this information. It is now straightforward to compute the time it takes to complete one full trip (from the ground floor to the top, and then back down).

︡a6db0971-e53e-42eb-84c9-92eef1644188︡{"hide":"input"}︡{"html":"\n\n \n \n\n \n \n
\n\nOne possible solution\n\n

\n\nThis solution is based on the following assumptions:\n\n
\n
Once the employees start arriving, they come in a steady \nstream until everyone gets to their floor.\n
The elevator will need to stop at every floor on the way up. \n
On the way down, it will not need to stop anywhere.\n
The elevator doors will need to open only once \non each floor.\n
\n\nOther information we need/get from measurements:

\n
\n
It takes 5 seconds for the elevator to travel \nfrom one floor to the next.\n
It takes 15 seconds for the elevator doors to \nopen/close and let people off at each floor.\n
On the ground floor it takes 25 seconds for the elevator \nto fill up and close its doors.\n
\n
\nThe sketch summarizes this information. It is now straightforward \nto compute the time it takes to complete one full trip (from the \nground floor to the top, and then back down).\n \n \n
\n"}︡{"done":true}︡ ︠777171ac-8743-4223-914c-90f0df29aa74si︠ n_elev = 3; n_emps = 300; n_per_elev = 10 t_between = 5; t_stops = 15; t_ground = 25 t_full_trip = t_ground + 5*t_between + 5*t_stops + 5*t_between print "Time per elevator roundtrip =", t_full_trip, "sec" print "Total time to get", n_emps, "employees to their offices =", t_full_trip*(n_emps/n_per_elev)/n_elev, "sec" ︡b99be295-d847-4444-b165-e3c25f225dea︡{"stdout":"Time per elevator roundtrip = 150 sec\n"}︡{"stdout":"Total time to get 300 employees to their offices = 1500 sec\n"}︡{"done":true}︡ ︠13d5daee-faea-4728-bc75-9c451264f544si︠ %hide %html 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. ︡69a47674-7d81-4f25-8670-5241f1d95909︡{"hide":"input"}︡{"html":"\nStrategy 1 (continued)\n\n

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

\nOne example is to consider a policy in which one \nelevator is reserved just for the upper floors. Suppose \nElevator A only serves floors 4-5, and Elevators B and C \nserve floors 1, 2, 3.

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

\n\n\n\n\n\nExercise\n\n

\nEstimate the total time the elevators take to get all \nthe employees to their respective floors with the above \npolicy.\n\n"}︡{"done":true}︡ ︠35072758-a71f-469e-ac2f-cc39efcc2e52si︠ %hide %html 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?

︡d84f6d13-71b0-4f24-a7df-60ea29ff220e︡{"hide":"input"}︡{"html":"\nDiscussion\n\n

It is not difficult to try out other policies about \nthe floor(s) each elevator serves. Try one of your \nown variations.
\n
After our simple initial approach, it is now easier to \nrelax or change our initial assumptions and make them \nmore realistic.
\nFor example suppose there is a 20% probability that an elevator \nmust stop on the way down. How can we account for that?
\n
Also, suppose the number of employees working on a particulr \nfloor is much higher than on other floors.
\nHow can we modify our \nstrategy to account for it?\n

"}︡{"done":true}︡

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

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

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