Beyond Classical Optimization Paradigms: Robustness, Fairness & Others

When solving a mathematical problem, one generally seeks the optimal solution. However, sometimes, this solution is not good enough, especially if other factors are taken into consideration. These factors include robustness, fairness, and others.

“If you optimize everything, you will always be unhappy” — Donald Ervin Knuth

In this article, we introduce the notions of robustness and fairness as well as how they might affect the optimal solution(s) of a given mathematical problem. We will also share some applications and briefly introduce some other notions.

Robustness

A robust solution to a mathematical model is a solution that is immune to data uncertainty. Let us consider the mathematical model above. In a classical context, we usually assume that the data of this model (c, A, and b) is known. While it is the case for deterministic contexts, this assumption is no longer valid in uncertain contexts where the data fluctuates. In such contexts, the optimal solution for given data values might no longer be optimal or even feasible if data changes. Thus, we seek robust solution(s), i.e., solution(s) that remain at least feasible when the model data changes.

The Price of Robustness

Of course, when seeking robust solutions, one might accept suboptimal solutions, which are far from optimal one(s), in a conservative manner, i.e., to ensure that these solutions satisfy data fluctuations. Such a decision is costly and highlights a trade-off between optimality and robustness. Exploring this trade-off, one might think of reducing the gap between robustness and optimality as much as possible, i.e., ensuring robustness without straying too far from optimality. Here comes the notion of The Price of Robustness introduced by Bertsimas and Sim in the paper below.

Fairness

There are a series of fairness definitions in the literature, including exact, proportional, and envy-free. Let us consider an allocation context. Exact fairness happens when exchanging shares will not affect any stakeholder’s outcome. Proportional fairness is when every stakeholder prefers its allocation to an allocation from an exact division. Envy-free fairness is when every stakeholder prefers its allocation to any other stakeholder’s allocation. An interesting paper to consult is Measuring unfairness feeling in allocation problems by Hoang et al.

The Price of Fairness

Similarly to robustness, when seeking fairness, one might accept suboptimal solutions to satisfy ethical or moral considerations. There is then a trade-off between the solution’s quality and its fairness. Here comes, in a similar way to robustness, the notion of The Price of Fairness introduced by Bertsimas, Faris, and Trichakis in the paper below.

Applications

An interesting application example is the portfolio optimization problem presented in the Price of Robustness paper. The suggested robust optimization approach allows capturing the trade-off between risk and return in a linear framework. It also provides solutions that ensure deterministic and probabilistic guarantees that constraints will be satisfied as data change.

For fairness, an interesting example is ambulance location and relocation when the goal is not only to allocate ambulances to prime spots such that they can quickly reach the maximum number of people but also such that the same set of people is not always at a disadvantage with respect to access to a quick service. An interesting paper is Fairness over time in dynamic resource allocation with an application in healthcare by Lodi et al., where the authors provide efficient mathematical programming formulations that improve fairness over time.

Other Notions

Robustness and fairness are not the only notions. We also have other notions, including resilience (e.g., in supply chain management) and satisfaction (e.g., in airline planning), briefly introduced below.

Resilience is the adaptive capability to face disruptions and perturbations. In the supply chain context, it is the ability of a supply chain to prepare for unexpected events, respond to disruptions, and recover from them by maintaining the continuity of operations. Here also, we seek to find resilient solutions that are not affected a lot by disruptions.

Satisfaction also can be incorporated into an optimization process. For instance, in airline planning, crew satisfaction is a critical objective for any airline. Thus, one does not look only to optimize costs but also to ensure personnel satisfaction, which directly impacts the long-term success of the airline company.

Operations research is not used only to optimize costs and profits. It is also used to identify robust, fair, resilient, and satisfactory solutions. This is the beauty of mathematical modeling, especially when it allows us to incorporate ethical, social, and legal considerations!

What do you think about the presented insights? I am looking forward to hearing your ideas :-)