Is It Reasonable to Aim for the Optimum?

When referring to those decisions that are key to a company’s strategy or day-to-day operations, the answer to the question may seem obvious: who would not want to achieve the optimum? However, if we try to go a step beyond the obvious, the question reveals multiple facets and deserves some important reflections.

So much so that we can state at the start of this article that (spoiler alert) aspiring to the optimum may be a chimera in many real-world scenarios. However, this does not reduce the importance of adopting technologies or strategies that try to lead us towards it. The paradox is served, but we will try to shed some light on it in the following sections.

Reality is Complex

It is clear to everyone that reality is complex, and increasingly so. For example, our ancestors only had to hunt for food. We need to locate factories, organise procurement, plan production, and distribute finished products to distribution centres until, after many processes, we have the goods available on the shelves. This is true for any business activity, even if our products and services move away from the factory model to much more intangible forms.

To effectively and efficiently manage increasingly complex systems, we can build models, and algorithms to solve those models. These models will be abstract representations (like the equations we all studied at school but in their sophisticated version) capable of serving us on a plate the mathematical optimum we are looking for, and which we will use to make decisions in the real system.

To build these models and run the algorithms that reveal the optimum, we need to define the indicator we want to optimise (which we call the objective function), a set of requirements we have to respect (which give rise to what we call constraints) and a set of decision variables (whose values are the ones we are looking for, and from which we will have a solution that we can implement). For example, in a network design problem in a supply chain, the objective function may be the total cost. A constraint would be the obligation to meet the demand of our customers. Finally, a variable would reflect whether we should open a warehouse in a certain location or not.

What Keeps us off the Path to the Optimum: The Simplifications and Uncertainties of the Model

However, as useful our models may be, there are considerations to bear in mind that can lead us hopelessly off-target.

First, any model is always built on simplifications and assumptions. We can assume that the cost of transport between two locations is equal to the quantity transported times a unit cost per item, and times the distance between the two locations. However, the cost will depend on how many trucks we finally use, the type of trucks and the use made of them.

Secondly, data is never as good as we would like it to be. Sometimes it is not as accurate as we would need it to be, or it simply has an inherent uncertainty that affects the model. If we build a model for the redesign of our supply chain, the demand for the next few years is anything but certain.

Because of the above, the mathematical optimum resulting from a model will not necessarily be the actual value if we implement what the model tells us. For example, the expected cost corresponding to a network design alternative will not be equal to the actual cost of our supply chain.

Therefore, the mathematical optimum will not necessarily lead to the lowest-cost alternative in reality. However, the model has a lot of value because, all in all, if the model represents reality well, the model’s optimum will not be far from the optimum of the real system, and the decisions it proposes will be very good guidelines for managing our real system.

What Takes Us Away From the Path to the Optimum: More than One Criterion for Choosing Alternatives

The reflection goes further if we think about what optimal means in each specific case. In the reality, there is rarely only one criterion that allows us to discern whether we like one solution better than another.

Continuing with the example of network design, the cost will not be the only criterion, although it will obviously be an important one. Quality of service will be another relevant criterion, and this is always in conflict with cost. So what is the point of talking about the optimum in this case, and from what point of view will one cost-quality pair be preferable to another? We can apply our knowledge of the particular domain, but it is clear that, to the extent that there is more than one criterion, the concept of optimum becomes more blurred and it is no longer meaningful to fixate on the best possible solution. This is not to say that we should give up our goal of excellence, but that the decision we take is preferable to no other without any doubt.

The models and tools at our disposal allow us to obtain the best possible balance of the different criteria according to the decision-maker’s preference. For example, we can offer the lowest cost given a minimum admissible quality of service level, or the highest quality of service without exceeding a maximum cost.

Practical Considerations: How Long We Can Wait, and the Actual Usefulness of the Decision We Choose.

If we consider the need to implement in reality – with its urgencies, organisational structures or costs involved – those decisions that mathematics prescribes, new reflections arise. That is to say, even if we admit for a moment that there is a dominant criterion and that the model represents reality so well that the value of the objective function is very precise, the search for the optimum may not be reasonable.

Whatever decision we have to make, let us ask ourselves the question: how long are we willing to wait for the optimum? In our network design example, we may be able to be a little more patient because this is a strategic decision. However, if we are scheduling last-mile delivery routes for tomorrow, we cannot afford to wait several hours. Moreover, if we want to use the model to re-route as soon as an incident occurs, we need to have the composition of the new routes in a very short period of time.

As with money, there is a discount rate on the outcome of an optimisation algorithm that depends on the time it takes to reach a solution: information about optimal routes is worthless if that information is known when they can no longer be implemented.
It is therefore reasonable to sacrifice some quality in the solution and move away from the optimum to some extent if we can evaluate more scenarios and, of course, if we can implement the decisions we get from the algorithm in a timely manner.

The Ultimate Touchstone: Cost-Effectiveness

Finally, most importantly, and again paradoxically, the optimal may be unprofitable.

Any model or algorithm requires a development effort, which implies an associated investment. It stands to reason that in order to find algorithms that are faster and find better solutions in less time, that investment has to be higher.

The greater the impact on the system that justifies the investment, the more interesting it is to aim for the optimum. Otherwise, it could happen that the development of an algorithm that makes it possible to find the optimum (or stay within a negligible distance) requires such an investment that the improvement obtained as a result of using the algorithm does not allow the investment to be recovered. We will have an algorithm that offers the optimum but has lost us money.

Back to the Initial Question

Consequently, the question is not whether we aim for the optimum. The question is what is the most cost-effective solution to our problem? Here solution does not mean “ideal solution to a mathematical problem”, it means “realistic solution to a real problem”; that solution with which the managers of a company want to undertake some kind of improvement, and which best meets the needs in terms of time, scope, cost, usability, etc.
The good news is that there is no shortage of increasingly powerful alternatives. We have exact mathematical models, but we also have heuristics, metaheuristics, hybrid strategies (simulation optimisation, metaheuristics), etc. It is this diversity that makes it essential to call on expert travelling companions.

The best news is that there are professionals capable of guiding managers to solve all the trade-offs, and to build a profitable instrument to solve each problem (which, as we have seen, is not only a mathematical problem but for which mathematics will be of enormous help). Those colleagues who, being aware of the difficulties, will not embark us on unachievable crusades, but who will not accept to stay one centimetre beyond the “optimum” of what is reasonable.

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