The problem with the problem

Operations research is a field with many problematic words, of which ‘problem’ is an interesting example. The word is generally used to describe a difficulty, but it has found an additional meaning in mathematics as a well-defined question that one may use an algorithm to answer. That would not be bad if it was not by a problem (of the first kind): both meanings are often interchangeably used.

The unavoidable consequence is that new developments targeting a given application sometimes present a meaningless criticism against mathematical problems, which are abstract entities that exist in spite of their use. Such a concern may sound pedantic because the mathematical problem is usually only serving as a proxy to its interpretation as the solution to the problem (of the first kind) associated with a given application. However, this bothers me because it is the tip of an iceberg of misguided conceptualization. And, of course, that is not how one should treat guys like the poor traveling salesman and his problem, who did not intend to do any harm against the problems of others.

Even though this is not a common mistake among reputed researchers in optimization, it happens pretty often among those in fields that need optimization as a means but do not delve further into theoretical concerns. This is bad because many of them go to the industry and will eventually come back to hire an expert in operations research to solve their most challenging problems, which will also involve detaching their needs from fixed ideas on how to model them.

I have already written previously about the meaning of problem, model, and solution in this blog, but coming across yet another bunch of writings and sayings with similar issues made me do it again. Personally, I only realized the problem with that after talking about my work with people in other fields, to which a problem has only one meaning (the full story is on that previous post). If I ever have the opportunity to teach an introductory course on OR, I would try to make this point clear in the most funny way that I could conceive (the funny part takes longer to be forgotten). As a matter of fact, this issue reminds me of a poem of my favorite writer. It is entitled XXVIII and is from The Keeper of Flocks, a book written by the Portuguese poet Fernando Pessoa under the name Alberto Caeiro (translation from the blog Fernando Pessoa: Alberto Caeiro: Complete Poems):  


Today I read almost two pages
In a book by a mystical poet
And I laughed like someone who’d cried a lot.


Mystical poets are sick philosophers
And philosophers are crazy.


Mystical poets say flowers feel
And they say stones have a soul
And they say rivers have ecstasies in the moonlight.


But flowers wouldn’t be flowers if they felt,
They’d be people;
And if stones had a soul, they’d be living things, they wouldn’t be stones;
And if rivers had ecstasies in the moonlight,
Rivers would be sick people.


You need to not know what flowers and stones and rivers are
To talk about their feelings.

Talking about the soul of stones, of flowers, of rivers,
Is talking about yourself and your false thoughts.
Thank God stones are only stones,
And rivers are nothing but rivers,
And flowers are just flowers.

Me, I write the prose of my poems
And I’m at peace,
Because I know I comprehend Nature on the outside;
And I don’t comprehend Nature on the inside
Because Nature doesn’t have an inside;
If she did she wouldn’t be Nature.

Update (same day in the night): Made some changes in the conclusion of the fourth paragraph.

2 Replies to “The problem with the problem”

  1. I ran into the same difficulties about what the word "problem". I decided to use "model" to refer to mathematical representations of decision problems. So a model has decision variables, constraints, and objective functions, whereas a decision problem is what is expressed by a decision-maker, which has goals and has decisions to make. Its not that clear, but refrees don’t seem to like speaking of "location-routing model" instead of the "location-routing problem". Still, I have yet to see a decision maker which asks a professor for help in solving his "location-routing problem".Did you get a lot of negative feedback from reviewers about your definitions of "model" and "problem"?

  2. I had some discussions about the ambivalent meaning of problem. When it comes to mathematical representations, I am preferring to use formulation instead of model for a similar reason: models have a much broader meaning.

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