the umbrella problem in my office


Once you learn about certain models used in operations research and industrial engineering, you start seeing them everywhere. I see the umbrella problem and its variants everywhere (read about the umbrella problem in this post).

I try to keep a few pens in my laptop bag at all times. These pens will drift in and out of my bag when I work. Every now and then, the pens entirely move out of my bag and I am left without a much needed pen. This happened earlier in March when I was on a trip and had planned to review a paper on my flight.

It has been a very cold winter in Wisconsin. I learned the hard way that it’s useful to keep a spare sweater or two in my office. I often forget to wear them home, and so sweaters and scarves have been accumulating in my office.

The city of Madison tries to be pedestrian friendly. They have these pedestrian flags (see picture below) that are supposed to help you cross the street safely. Each intersection has a bunch of flags in two bins with on either side of the street. I drive by a few of these intersections on the way home, and sometimes I see an intersection where all the flags are in one of the two bins. It always makes me smile.

This isn’t exactly the umbrella problem since the same pedestrian doesn’t go from one side of the street to the other indefinitely as in the umbrella problem, but the umbrella problem only needs to be slightly modified to capture the real problem here and to provide insight into how many flags should be stocked to maintain a certain reliability level.


Look for a few posts about March Madness next week!

2 responses to “the umbrella problem in my office

  • Dan Black

    I saw a similar problem in a Stockholm guidebook (my next planned holiday). On the Stockholm archipelago there are rowing boats for getting from one small island to another. You are obliged to make sure that there is always one boat on each side of a straight. If there is only one boat on your side you have to row across, “tow” another boat back and then row back again.

  • David K Smith

    The pedestrian flags raise further complications in modelling. The balance of the numbers of people flowing across the road will vary during the day, Early morning will see flow in one direction, late afternoon, those people will return. So the system may not have an equilibrium. Also, good-hearted pedestrians may notice that the numbers are unbalanced, and may take two flags with them.
    Cities which have rental bicycles have a problem related to the umbrella problem, as do car-hire companies with “rent it here, leave it there” offers. Vehicles will tend to congregate at certain points and need to be collected and reassigned. Rental bikes need to be collected and redistributed (think of the algorithms!) and car-hire companies can charge a premium for their offer (each consumer has to decide whether or not to accept the premium, the company has to decide what premium to charge).
    And yet another variant of the umbrella problem is in long-haul trucking by independent operators. The truck which goes loaded from A to B and leaves its load at B can return empty to A, or find a load from C to D where B is “near” C and A is “near” D. The operator needs to decide when “near” is good enough.

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