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# Elevators problem

11 replies to this topic

### #1 ujjagrawal

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Posted 31 July 2012 - 10:35 AM

Most of the people, I understand, must not be happy with the logic by which the elevators on their building works. This is an opportunity for you to work on a logic that should be efficient, fair and a practical one.

There is a 12 floors building (excluding ground level) with 2 elevators. Can you work out what should be the ideal position for the two elevators (in terms of floor numbers), while they are not in use i.e. idle. Assume equal probability of getting calls from all 12 floors.
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### #2 kbrdsk

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Posted 31 July 2012 - 11:19 AM

Spoiler for maybe

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Posted 31 July 2012 - 05:18 PM

What is the frequency of requests for an elevator?

If we assume that only one person rides it at a time, that gives a different figure than if the elevator is constantly filling up.
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### #4 MikeD

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Posted 31 July 2012 - 07:31 PM

Spoiler for more of a guess

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hi again, everyone. Nice to be back home.

### #5 plasmid

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Posted 01 August 2012 - 06:04 AM

Spoiler for The optimal position for an elevator would be

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### #6 ujjagrawal

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Posted 01 August 2012 - 06:28 AM

Spoiler for maybe

What is the frequency of requests for an elevator?

If we assume that only one person rides it at a time, that gives a different figure than if the elevator is constantly filling up.

Spoiler for more of a guess

Spoiler for The optimal position for an elevator would be

Spoiler for Hint

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### #7 MikeD

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Posted 01 August 2012 - 10:11 AM

Spoiler for a spoiler

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hi again, everyone. Nice to be back home.

### #8 fabpig

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Posted 01 August 2012 - 01:23 PM

Spoiler for Ifs and buts

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You must not think me necessarily foolish because I am facetious,  nor will I consider you necessarily wise because you are grave.  Sydney Smith.

### #9 bushindo

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Posted 01 August 2012 - 09:46 PM

By 'ideal distribution of elevator position', do you mean that

1) You wish to minimize expected the number of floors it would take for an elevator to reach a caller whenever he/she presses the elevator button?
2) Or do you wish to minimize the expected total distance that an elevator would travel when requested (i.e., caller A request the elevator from floor i to go to floor j; the nearest elevator travels from its idle floor to floor i, takes the caller to floor j, and then return to its idle floor)?

Also, as it is, the problem is under-constrained since it leaves out an important piece of information, which is the probability of calls from the ground floor (or floor 0). Consider the following two scenarios

A) Nobody in the building travels to the ground floor. All calls from floor 1 to 12 are equally likely, and each elevator trip travels to floor 1 to 12 with equal probability. In this case, probability of calls from the ground is 0.

B) All building residents only travel from their floor to the ground, and from the ground to the their floor. Assuming that there is no net gain or loss in the number of residents, then probability of calls from ground floor is 1/2.

Both scenarios A and B satisfy the OP, but they have different implication on the resulting `optimal' idle floor. Some clarification would be appreciated.
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### #10 ujjagrawal

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Posted 03 August 2012 - 05:42 AM

By 'ideal distribution of elevator position', do you mean that

1) You wish to minimize expected the number of floors it would take for an elevator to reach a caller whenever he/she presses the elevator button?
2) Or do you wish to minimize the expected total distance that an elevator would travel when requested (i.e., caller A request the elevator from floor i to go to floor j; the nearest elevator travels from its idle floor to floor i, takes the caller to floor j, and then return to its idle floor)?

Also, as it is, the problem is under-constrained since it leaves out an important piece of information, which is the probability of calls from the ground floor (or floor 0). Consider the following two scenarios

A) Nobody in the building travels to the ground floor. All calls from floor 1 to 12 are equally likely, and each elevator trip travels to floor 1 to 12 with equal probability. In this case, probability of calls from the ground is 0.

B) All building residents only travel from their floor to the ground, and from the ground to the their floor. Assuming that there is no net gain or loss in the number of residents, then probability of calls from ground floor is 1/2.

Both scenarios A and B satisfy the OP, but they have different implication on the resulting `optimal' idle floor. Some clarification would be appreciated.

Thanks for raising above concerns... I assume, I framed the problem in bit hurry... here are the clarification to your concerns...

AIM is to minimize average waiting time...

Further assume, it's a residential building... all residents mostly travel between their floor and ground floor, so please ignore other in-between floor travels...

Hope this problem make more sense now...
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