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# Elevators problem

## Question

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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Have on elevator positioned between floors 3 and 4, the other between floors 9 and 10. As soon as one elevator gets a call the other moves into position between 6 and 7 until the first elevator is done.

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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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I would have them rest on floors 4 and 9.

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on whichever floor has an activated motion sensor that detects people walking toward the elevator. Or in the absence of that, whichever floor has a pressed elevator button, where elevator buttons are installed in approaching hallways 15 meters away from the elevator bank itself.

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Have on elevator positioned between floors 3 and 4, the other between floors 9 and 10. As soon as one elevator gets a call the other moves into position between 6 and 7 until the first elevator is done.

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.

I would have them rest on floors 4 and 9.

on whichever floor has an activated motion sensor that detects people walking toward the elevator. Or in the absence of that, whichever floor has a pressed elevator button, where elevator buttons are installed in approaching hallways 15 meters away from the elevator bank itself.

Ignore the numbers of people traveling by lift and lift capacity.

Assuming equal probability of getting calls from all 12 floors(level 1-12), what will be the probability of getting calls from ground floor (level 0)?

Note - consider two way traffic.

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The calls on ground floor would be equal to call on all the above floors. (I first had throught floor 1 was ground floor and so had equal probability of getting calls)

Since ground floor would have half of all the calls, then it would make sense to have both elevators waiting there.

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If by "their building" you mean the apartments where they live, then it is a fair assumption that people will indeed make about 50 percent of their journeys from the ground floor (I actually thought this, but discarded it when I read the proviso that all floors would have an equal number of calls).

If, however, it's an office block, it is quite possible that a person will take the lift elevator from the ground floor to their office on the (say) 5th floor, and then make several journeys during the day from the 5th floor to one or more of the other floors.........

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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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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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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...

Thanks for the clarification. If that's the case,

Assuming no net gain or net loss of residents, the probability of call from floor 0 is 1/2, while probability of calls from each remaining floor is 1/24. The 2 elevators should idle at floor 0 and floor 8.

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Thanks for the clarification. If that's the case,

Assuming no net gain or net loss of residents, the probability of call from floor 0 is 1/2, while probability of calls from each remaining floor is 1/24. The 2 elevators should idle at floor 0 and floor 8.

Floor 0 and Floor 9. It would result in the exact same average wait time.

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