# Revisiting operations research elements – part I: problem, model and solution

People often admit that they have just discovered to know nothing about something they thought they knew. I felt that about terms like “problem”, “model” and “solution” during the last year. At a first sight, those words seem quite simple to anyone: a problem is a quest one wants to resolve, a model is an insightful description of such problem and a solution is an attempt of resolving the problem. As I want to show next, big messes are made of such simple things, especially in OR.

What is your problem?

Once, when I practiced a presentation at home saying “My problem is that I have those decision variables Xi, Yi and Zi”, I was interrupted by my mother-in-law with a “No! Your problem is that you have expensive machinery, oil wells to develop and you want to maximize the profit of the company”, as I indeed had! She might not know much about combinatorial optimization problems or other related class of optimization problems, but after two daughters, Ph.D. and Post-Doc in organic chemistry and a published book, she definitely knows better what a problem is.

I used to consider the “problem” as a class of purely abstract formulations or a domain-targeted specialization of such a class. However, I was ignoring the actual problem – something that must be fixed or enhanced –, which was occasionally mentioned as a motivation for the study. Indeed, such mistake can be the first step to detach a promising subject from its context and devalue its importance. As a matter of fact, I should have done the converse path: to consider my problem as a real matter from which disclosure is motivated by the resemblance with other problems of practical importance.

What does your model explain?

For a long time, I’ve been used to denote as a “model” what I write as input to an optimization solver. However, after reading “Good and Bad Futures for Constraint Programming (and Operations Research)” by John N. Hooker, I decided to get a step back about that. In his article, Hooker emphasizes the importance of models to leverage scientific progress according to their original meaning: an elegant description of a natural phenomenon. Under that assumption, a model is as valuable as its ability to explain how a given system works and why it works that way. For sure, that is not achievable by simply stating a bunch of equations.

The input of an optimization solver is always computer source code, even if it is at a very high level of abstraction when compared to generic programming languages. Nevertheless, what you are able to describe with such programming language – its syntax – can be considered as the “model” for some sort of class of abstract formulations, and my code was just a specialization of it. Therefore, it would be more accurate to say that I’ve spent much more time coding upon a model of the problem than modeling the problem. Metaphorically, the model is an ethereal inspiration to a substantial source code (hold that thought for the next post).

What are you solving?

The term “solution” has been so often associated with variable assignments of a given problem formulation that its use in its original assertion sounds like an invitation to confusion. I felt that when reading an article about Case-Based Reasoning (CBR) applications to constraint programming (CP), in which the authors had to manage such nomenclature conflict: in CBR, a solution is understood as a technique for solving a given problem and was forcefully redefined as a “resolution”, given that, in CP – and OR in general –, a solution is the output of a technique that solves some problem. I can’t imagine a better way of handling it than theirs: the ambiguity was acknowledged and possibly taught to some unaware readers, and the article moved forward.

Maybe it is a lost battle trying to associate the term “solution” in the OR field with what the general public would expect it to be. However, if that is not possible, one must emphasize the difference, since the benefit of succeeding on that – as well as in the other cases exhibited – is to present an original work in a more accessible language to researchers of other fields. Therefore, the reward for that is a higher probability of having your effort being recognized and used somewhere else.