Mathematical Models – Why They Are Just Theory and How to Use Them in Real Life

The Reality of Mathematical Models

Models can describe how a system works now or how it could work in the future. However, it is at best a simplified version of reality or describes a to-be state.

The model of a machine will always be more accurate than that for a process involving people. Humans are by their very nature a source of variability, though toddlers are a primary source of chaos.

Models may help you understand why things work the way they do, but creating a model doesn’t make it real – or even correct.

Forces that caused a system to change may still be in effect today – and can make your new model obsolete.

If you have to add corrective factors to a model, then the model is just plain wrong. Corrective factors to an incorrect model do not make it right.

Don’t assume that current trends will continue forever. There are always limits to a system, even if we don’t see them now.

Mathematical Models and Decision Making

A model may describe an ideal state, but everyone’s version of “perfect” is different. You can optimize a model and find the lowest cost, minimal defects, highest efficiency, least waste or minimal downtime – but you cannot achieve all of these things and the ideal outcome will differ among stakeholders.

Engineers need to be careful to design systems that are not only ideal but will also work. Too many decision points creates delay that will be intolerable to those who need something now or when the lack of a decision lets the opportunity pass you by. When consumers lack choices in your streamlined process, they’ll go elsewhere or demand work-arounds. A process that cannot handle exceptions creates delays or fails. Processes that are too complex to follow or work within will fail due to human limitations like patience or budget.

Systems can arise from chaos and self-organization. Study the existing system and review options to improve it before inventing a new one. Never assume that a new system will be better than the old one.

Plan for emergencies, and model how they will affect your system. Have a plan to handle issues like rush jobs, power outages, equipment failures or disruptions to the processes in your model. Assume the worst, so that your system will still operate when life happens.

Consensus is not the basis of sound science or mathematical models. Consensus often incorrect. It may be due to incorrect ‘common sense”, political pressure to agree with the majority or the “Abilene paradox” where people go along with the original suggestion simply because it looks like group opinion. You may have a majority vote to choose underlying assumptions, adopt one model over another or give suggestions based on the results – but this doesn’t make them right.

Be careful of those who will use mathematical modeling to justify a specific course of action. They may have their own agenda, and will suggest the same actions no matter what model is used or its outcome.

Faced with constraints, managers default to demanding more management. The solution to a resource constraint is installing gatekeepers to control access, adding more review steps or additional authorizations. Unfortunately, management itself consumes resources like time and money, generally raising the cost of operations and creating more resource constraints. Look at government bureaucracies like the British National Health Service and public schools, which solve the problem of cost over-runs by hiring more managers and end up laying off nurses and teachers to pay for the increased administrative overhead.

Solutions to Common Mistakes with Mathematical Models

Models are built upon mathematical equations based upon assumptions. Assumptions can be wrong. And the underlying conditions that led to those assumptions can change. Models should not be held up as infallible, as many global warming models have been found to be wrong due to global cooling observed since 2000. Majority consensus said the root cause was carbon dioxide, while more recent reports link both the warming and subsequent cooling to CFCs and the ozone layer.

Never discard or exclude data contrary to your theory of how a system works. It likely reveals flaws in the model, additional factors at work or errors in the assumptions used to build the model. If you find yourself excluding data points because it doesn’t fit the model, you should instead admit the model is wrong instead of denying the growing evidence that the model is wrong.

Models are not reality. They are, at best, a simplistic approximation of it. They are at worst an academic exercise that becomes a waste of time. Don’t get emotionally invested in models or spend more time modeling than you do studying the physical world upon which it is based.

The act of collecting data in and of itself can alter a process. For example, a manufacturing process may be slowed down by collecting metrics while people may work faster when observed.

There are standard methods of determining variability, but there is no standard method for measuring system complexity. And variability is not a measure of complexity.

Use trial and error to improve both your models and the processes you are modeling. Accept failures and learn from them.

1 Comment so far

  1. Rooservelt Akume

    Great article. I am a senior IE student and I find modeling to be the most challenging part of operations research. With every LP model that we solved, my instructor referred to the Certificate of Optimality as a “bonafide proof” that the model works for that particular case. Of course over time the parameters will change, and this will require an upgrade of the model.

    I learned in my Quality Control class when we studied Shewhart’s charts that there’s a type II error that should be taken into account, which is the probability that after a certain number of events/occurrences, the process will shift out of control, and when this happens, the process is reevaluated and brought back in control with a new type II error calculated. So yes it’s not perfect and that’s why we need continuous improvement because the system is always changing.

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