Mathematical Equation: How to Decide Whether or Not to Hold Open the Door

When you are exiting or entering through a doorway and someone else nearby is about to enter or exit through that same doorway, why do we sometimes hold the door open for them to let them exit/enter or to wait for them to reach the doorway while we hold the door open for them? Usually when someone does this, they are trying to be polite and want to make exiting/entering the doorway easier for the person they are waiting on. However, there is a point where this is more of a nuisance than a convenience for both parties.

Usually the person for whom the door is being held open for feels obligated to not keep the door-holder waiting. At what point does this obligation become an issue? How far away does the person for whom the door is being held for have to be or how long does it have to take that person to reach the door before one of the two feels any sort of discomfort from this act? Is there an equation somewhere where you could input numbers into the equation and have it tell you whether or not it is appropriate to hold the door open for someone? There just might be an equation that could help you out!

There are many factors to consider before we can determine this. The ones I will take into consideration are:

1. How far from the door is the person for whom the door is being held?
If they are 200 feet away, then it might take a while for them to reach the door and you might not want to hold the door open for that long.

2. How fast is this person walking?
If they are only 10 feet away, but are moving at a speed of 1 ft/min, you might not want to hold the door open for that long.

3. Does it appear this person will benefit from the door being held open (handicapped, elderly, baby in stroller, etc)?
Some people might not be able to easily get through the doorway on their own, it would be a good idea to be polite and hold the door open for them.

4. How heavy is the door?
If it weighs 10 ounces, opening the door probably won't be an issue for most people. If it weighs 10 lbs, it would probably be helpful for the person for whom you are holding the door open (even though it would probably be easier for you if the door weighed 10 ounces as opposed to 10 lbs).

5. How fast does the door close?
If it takes 20 minutes to fully close, then you could just open it all the way and continue on your way and the door will most likely be still open when the other person reaches it.

All of these except for #3 are measurable. Thus, I will use all but #3 in the equation (#3 can override any decision you make based on the equation). The equation is going to be made using what I consider to be reasonable estimates and numbers when it comes to putting the measurable factors into something that will help us mathematically determine what to do. It can be reasoned that #1 and #2 are more important than #4 and #5. In our equation, there will be four main variables:

Y = you (deciding whether or not to hold the door open)
P = person (for whom you will potentially hold the door open)
D = door (the actual door itself)
DO = door open (this will be the decision to hold it open or not)

In the equation, we will solve for the decision DO. The decision DO will be either 1 (hold the door open), or 0 (don't hold the door open).

With all of this established, #1 from above can be found by doing:

P - D₀ [ft]

Where P is the person, and D₀ (this is going to represent the position of the doorway relative to the person for whom we are holding the door open) is the doorway position/placement. This will represent how far from the doorway the person for whom the door is being held open for is, and can be given a unit of ft.

Next, #2 from the above can be found by doing:

(P - D₀) / (T_{[(P - D0) / Pfs ]} * [(P-D₀) / P_fs]) [ft/sec]

Where P_fs is the number of feet the person is traveling with each stride they take, (P-D₀) / P_fs is the previous distance measured from #1 divided by that value to determine how many strides it will take the person, and T_{[(P - D0) / Pfs]} is the time it takes per stride. What we end up doing is dividing the whole distance between the door and the person walking to the door by the amount of time it would take them. To do this, we will need to know the stride rate of the person. Knowing this, we can calculate how many strides it will take the person, and the total time can be determined using this stride rate and total # of strides information (this is reasonably assuming the person is either walking or running toward the door).

Next, #4 from the above can be represented very simply by this:

D_w [lbs]

The D_w is the door weight (it really is how much average force is required to open the door in units of lbs). Reasonable estimates on this can be made when you have fully opened the door.

Finally, #5 from the above can be represented simply by this:

D_s [ft/sec]

The D_s is the door speed (the speed at which the door will close in units of ft/sec where the arc made by the door edge, ¼ of a circle, from the open position to the closed position is the measurement for the distance in ft for this variable).

So all we need now is one big equation to represent all of this information in a system where the end result is a 1 or a 0 which will define our decision as to whether or not we should hold the door open for the person. Here is the equation which assumes equal importance to each of the 4 conditions in the equation using reasonable values for each condition:

DO = [|[(¼ ([|(40 - P - D₀)| + (40 - P - D₀)] / [2(40 - P - D₀)]) + ¼ ([|((P - D₀) / (T_{[(P - D0) / Pfs ]} * [(P-D₀) / P_fs]) - 1)| + ((P - D₀) / (T_{[(P - D0) / Pfs ]} * [(P-D₀) / P_fs]))] / [2(1 - (P - D₀) / (T_{[(P - D0) / Pfs ]} * [(P-D₀) / P_fs]))] - 1) + ¼ ([|(D_w - 1)| + (D_w - 1)] / [2(D_w - 1)]) + ¼ ([|(D_s - (1/6))| + (D_s - (1/6))] / [2(D_s - (1/6))])) - (9/10)]| + [(¼ ([|(40 - P - D₀)| + (40 - P - D₀)] / [2(40 - P - D₀)]) + ¼ ([|((P - D₀) / (T_{[(P - D0) / Pfs ]} * [(P-D₀) / P_fs]) - 1)| + ((P - D₀) / (T_{[(P - D0) / Pfs ]} * [(P-D₀) / P_fs]))] / [2(1 - (P - D₀) / (T_{[(P - D0) / Pfs ]} * [(P-D₀) / P_fs]))] - 1) + ¼ ([|(D_w - 1)| + (D_w - 1)] / [2(D_w - 1)]) + ¼ ([|(D_s - (1/6))| + (D_s - (1/6))] / [2(D_s - (1/6))])) - (9/10)] / [2[(¼ ([|(40 - P - D₀)| + (40 - P - D₀)] / [2(40 - P - D₀)]) + ¼ ([|((P - D₀) / (T_{[(P - D0) / Pfs ]} * [(P-D₀) / P_fs]) - 1)| + ((P - D₀) / (T_{[(P - D0) / Pfs ]} * [(P-D₀) / P_fs]))] / [2(1 - (P - D₀) / (T_{[(P - D0) / Pfs ]} * [(P-D₀) / P_fs]))] - 1) + ¼ ([|(D_w - 1)| + (D_w - 1)] / [2(D_w - 1)]) + ¼ ([|(D_s - (1/6))| + (D_s - (1/6))] / [2(D_s - (1/6))])) - (9/10)]]]

Yes, the above equation is a valid one. If you were to enter in the appropriate values for the variables shown, you will get a 1 or a 0 answer which will be the mathematically correct thing to do in a situation where you have to decide whether or not to hold the door open for someone. The main reasonable assumptions used for each condition are the following:

#1: If the person is 40 ft or more away from the door, don't bother holding the door open.
#2: If the person is walking slower than 1 ft/sec, don't bother holding the door open.
#4: If the force required to push/pull open the door is less than 1 lb, don't bother holding the door open.
#5: If the door closes slower than 1/6 ft/sec, don't bother holding the door open.

I have left out many details in this equation. It can be much more detailed and accurate to account for many more factors that could affect the decision. Additionally, many of the computations required to complete the equation such as the calculation of the speed the door is closing, the accurate estimation of the amount of force required to open the door and many more have been withheld to keep the equation simple.

The basic format of the above equation is to take an expression, evaluate it, and if the expression is true/false depending on the condition, the result will be a 1 or it will be a 0. In the end 9/10 is used to evaluate the entire equation to get a final result of 1 or 0. If interested in more details in how the equation is derived, feel free to contact me.

So if this ever happens to you, you will know why:

You are walking toward a doorway. The person that is about to walk through that door notices you and motions for you to stop. You stop and the person comes running over to where you are and takes out a notebook and pencil and motions for you to start moving. You start walking toward the doorway once again, and 1 second later the person motions for you to stop again. After jotting something on the piece of paper, the person runs back to the door while counting how many feet he travels back to the door. When he gets there, he opens the door and lets it close by itself while looking at his watch. Then he measures the floor around the door to follow the arc the door made, followed by multiple opening and closing of the door as if he is seeing how heavy it is. Finally he takes a seat and starts working furiously on something he has written down on the piece of paper with his pencil.

Then, hours later, he gets up, looks at you, and motions for you to start walking again. You start walking, he opens the door, and runs off, letting the door close behind him (or he stands there holding the door open for you).

If that ever happens to you, you will know why!