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If you have strategic planning problems like hub location problems, the input data often consists of average values for shipping volumes etc.

When planning capacities, it is risky to ignore the distribution of the input data (if your hub is large enough for average capacities, you may get huge problems when volumes are higher). On the other hand, the distribution is not known.

Does it make sense to derive a "most likely" distribution by methods like entropy maximization?

LarrySnyder610
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J Fabian Meier
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3 Answers3

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I come across the problem quite a bit (am a practitioner in a utility company building offshore wind farms), and here is my take:

  • As Marco said, try to get as much data about the system as possible: past shipment values, domain experts giving their opinion (treat those with a pinch of salt though) etc. then incorporate this in the best way you see fit (fit a distribution, derive ML predictive model, robust optimization etc)
  • However, if you don’t have anything available, I would not assume anything. There are two reasons for this:

1) You are introducing “magic numbers” into the problem without any justification. However, somebody who looks at your results may not know this and suddenly you end up suggesting that you have more information than you do. This makes debugging, new development and handovers more difficult.

2) On a more fundamental level, in my opinion it is the job of OR people to provide the best answer given our current knowledge of the system. We take what is there and make the most of it. It is the clients’ responsibility to provide the data to use. You can (and should) highlight what you think would make the model more accurate and hence more valuable. However it is their responsibility, and they have to live up to that.

Richard
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  • The problem: "not assuming anything" usually is also an assumption. If you don't know the probability distribution of a parameter and you treat it as constant, this assumes a lot. – J Fabian Meier Jun 28 '19 at 07:06
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    I wouldn’t even assume a distribution: if they give you average values, use average values, that’s my point. Then you can go back and say “hey, is there really nothing we can say?” But before they tell you something, I’d just use what they provide. – Richard Jun 28 '19 at 07:07
  • So if we just have average values because either this is an academic problem where you cannot "ask the customer", or the customer does not have more detailed data, then you should just give up? Strictly speaking, I cannot sensibly plan a capacity of a hub if I only know the average shipping volume. – J Fabian Meier Jun 28 '19 at 07:48
  • I would not “give up”. Rather I would say: given the information provided, this is the best I can do. If I look at it and see that it does not make sense at all, then you can either try to get more data, or say you can’t solve it. I find the idea of just assuming stuff to make it work dangerous, because I have come across code in my work that was just put in there “to make it work” and it gave wrong results. But nobody checked so everybody trusted the results, leading to misinformed decisions. – Richard Jun 28 '19 at 07:55
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The following may or may not be useful for hub location problems, but is certainly applicable to many problems pertaining to the title "Strategic planning based on average values"

If $X$ is a random variable for which only the mean, $\bar{X}$ is known, sometimes qualitative analysis can be performed to determine the direction of the error between

1) the solution obtained by using the mean of $X$ in lieu of its actual distribution

and

2) the correct solution

In particular, suppose we are interested in $E(f(X))$, the expected value (i.e., mean) of $f(X)$. If the function $f$ is known to be convex, for instance $f(x) = x^2$, or concave, for instance, $f(x) = \sqrt{x}$ or $f(x) = \text{log}(x)$, then Jensen's Inequality can be applied.

If $f(x)$ is convex, Jensen's inequality tells us that $f\bar{X}) \le E(f(X))$ I.e., substitution of $X$ by its mean causes $E(f(X))$ to be underestimated.

If $f(x)$ is concave, Jensen's inequality tells us that $f\bar{X}) \ge E(f(X))$ I.e., substitution of $X$ by its mean causes $E(f(X))$ to be overestimated.

Because it supports such qualitative analysis, Jensen's Inequality is one of the most powerful tools in applied probability, and therefore in Operations Research. Sometimes this qualitative analysis of the error goes in the direction needed to support decisions, even with missing quantitative information.

Note: In the real world, many managers, decision makers, etc, think that the only rigorous analysis is quantitative. In many cases, qualitative analysis is more rigorous and better supports decisions than inadequate quantitative analysis. So sometimes a "sales job" is needed.

Mark L. Stone
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Fabian, I believe many situations have this tension of strategic decisions that may turn out not to suit every operational situation; I think of locating charging stations for electric vehicles, etc.

In this generality, I can only come up with general answers. Of course, robust optimization comes to mind, but these days also data-driven approaches. Don't you have past data from which a distribution could be generated? This most recent article from the Montreal people may also be relevant as it employs machine learning to try to overcome the tactical/strategic information imbalance.

Marco Lübbecke
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