Towards Optimization: Modeling Mathematically Real World Problems
Humans face different problems either in business or in daily life. The complexity of problems differ based on many factors. Hence, some problems are quite easy to solve while others are more difficult to tackle like the miniature of the vehicle routing problem (VRP) shown in the figure below. Here, we might be interested to find the best solution. For instance, a solution that provides the highest revenue or the lowest cost.
With the interest highlighted above, being a sub-component of applied mathematics, mathematical modeling takes place. Still, it is intriguing to wonder whether: any real world problem can be modeled mathematically? If yes, is it always necessary? What is the approach to follow for mathematical modeling? And what challenges can be expected?
Natural Modeling
In the real world, we use quite a lot our intuition to handle different tasks. This intuition keeps growing with us and feeds on our experiences, successes, and mistakes. For instance, in factories, there are always experienced operators that have been working there for many years. They mastered the operational level and can deal with any problem very efficiently. These operators accumulated tacit knowledge, which represent their capital. To tackle issues, they use quick thinking, sketches, checking, and rapid testing. Through this approach, they can solve many problems “naturally” without having to go through complex processes. Beyond the operational level, the same is happening on the tactical and strategic levels as well as in the daily life.
Mathematical Modeling
From the other hand, mathematical modeling is a conversion process that translates a real world problem into a mathematical formulation like the one presented below. Hence, it requires a deep understanding of the problem, its parameters, its variables, its constraints, and its objectives. A formulation is not necessarily analytic, it can be a network like the VRP example shown above. Whatever the form, we seek satisfying a specific objective P, i.e. maximizing or minimizing, while respecting the bounds on the constraints. Usually, in such formulations, it is difficult to find naturally the solution, except for some relatively easy problems. Hence, computing is required to search for the best solution.
Why — Improvement vs Optimization
Understanding the key idea behind mathematical modeling is the fascinating part in this whole story. In fact, when doing natural modeling, i.e. using intuition, we may end up finding good solutions. These solutions improve the situation and are called improvements. However, there are no clues to state that they are optimal. This is the difference between improvement and optimization. Compared to natural modeling, mathematical modeling guarantees an optimal solution or in some cases a satisfactory solution close to it. On the graph, the local maximum is an improvement, the green star is a satisfactory solution, and the blue star is the optimal solution. Hence, the significance of mathematical modeling and its importance.
How — Approach
Considering the length of the process, a wise problem solver should think deeply before starting mathematical modeling. In fact, intuition is a very powerful way to handle things, especially if the solution obtained is satisfactory for the stakeholders. In this case, there is no need to get involved in a relatively longer process to design the problem with its parameters, variables, constraints, and objectives. As a rule of thumb, it is good to compare the necessary effort to build the model, solve it, and implement it with the expected results in terms of time reduction, cost benefit or revenue increase, etc. If we foresee that the optimal solution is required, then modeling mathematically deserves as proven in many real cases.
Challenges — Theory vs Practice
The most exciting part that stimulates the applied mathematics scholars is seeing the theoretical results, i.e. optimal solution obtained through mathematical modeling, implemented on the practical level. However, this is not always the case since it is difficult to formulate exactly a real world problem. Real life incorporates uncertainties, exceptions, unexplained behaviors, stochasticity, real-time changes, etc. Consequently, assumptions are usually needed to relax the model. It is quite similar to what is shown in the figure below. Still, it is possible to ensure great results through gap shortening between theory and practice. This requires a good business acumen and a close interaction with the field stakeholders who can support in variables interpretation and results tuning.
Real World Examples
There are many real world examples where mathematical modeling was very successful. We can cite the tail assignment problem (TAP), shown below, where specific aircraft are assigned to flights while ensuring producing an operational schedule which fulfills operational constraints and minimizes a cost function. For such problems, the results obtained through mathematical modeling exceed by far what humans can handle naturally because of the complexity, which is correlated with the size of the problem. As a consequence, the impact on cost reduction or revenue increase is quite significant and makes airline companies gain more without making huge changes. Optimization plays then the role of a fictive manager that can identify hidden costs reduction and revenues increase.
Like neural nets which can approximate any function, mathematical modeling can be used to tackle any real world problem. Still, it can be done only under the known set of techniques and the current state of art in mathematics. Sometimes, given the complexity of the problem, the available knowledge is not sufficient. In such cases, other methods that mix mathematical modeling with intuition take place. They are not necessary exact and are called heuristics.
What are your insights on the topic? I am looking forward to hearing them :).
