Relative performance analysis - in the spirit of Sadi Carnot
Sadi Carnot - who?
Well, every engineer worth his salt will remember Sadi Carnot from their thermodynamic classes. The Carnot heat engine is a fundamental concept in thermodynamics - which establishes the maximum possible efficiency that any heat engine can achieve when operating between two temperature reservoirs. This is quite an important concept - and an interesting answer to an even more interesting question.
It does not matter what setting one finds oneself - be it in engineering, business, or even personal life - the question of “How well can I do?” is always relevant. Given the constraints of my situation - what is the best possible outcome that I can achieve? This is where the concept of relative performance analysis comes into play.
Relative vs Absolute
Let’s first clarify what is meant by relative performance analysis.
We can use the first law of thermodynamics as a simple benchmark - although we are absolutely not limiting ourselves to thermodynamics. The first law states that energy is conserved, i.e. it is not created or destroyed, only converted from one form to another. For a heat engine, which we want to produce work \(W\) for us, the total work output is equal to the heat input \(Q_{in}\) minus the heat rejection \(Q_{out}\). An absolute efficiency of this system is calculated by looking at the work achieved compared to the heat input, i.e. how much did we put in and how much of value did we get out, or
\[\eta_{absolute} = \frac{W}{Q_{in}} = \frac{Q_{in} - Q_{out}}{Q_{in}} = 1 - \frac{Q_{out}}{Q_{in}}\]
The relation \(W=Q_{in}-Q_{out}\) is the first law.
Given the constraints of the 2nd law - i.e. entropy must always increase - \(\eta_{absolute}\) can never be 100% - there will always be some heat rejected to the cold reservoir, i.e. \(Q_{out}>0\).
Since we can’t achieve 100% efficiency, the value of this efficiency paramenter is questionable, since we don’t know how high it can be? So what does \(\eta_{absolute}=0.74\) mean for us if we don’t know it’s upper bound? This is where Sadi Carnot comes in - and relative performance analysis.
Sadi Carnot asked the right question
Since a heat engine’s performance is limited by the 2nd law of thermodynamics, Sadi Carnot asked the right question - what is the maximum possible efficiency that a heat engine can achieve when operating between two temperature reservoirs?
His ideal heat engine - the Carnot engine - operates on a reversible cycle between a hot reservoir at temperature \(T_{hot}\) and a cold reservoir at temperature \(T_{cold}\). The efficiency of the Carnot engine is given by: \[\eta_{Carnot} = 1 - \frac{T_{cold}}{T_{hot}}\]
So it makes no relation to the actual engine itself - as it is ideal - but is only limited by the temperatures of the heat reservoirs between which it operates. By inspection, increasing the temperature difference between the hot and cold reservoirs increases the maximum possible efficiency.
Now we have an upper bound on our efficiency - and we can make better use of the absoluted efficiency by comparing it to the Carnot efficiency. This gives us the relative efficiency: \[\eta_{relative} = \frac{\eta_{absolute}}{\eta_{Carnot}} = \frac{W/Q_{in}}{1 - T_{cold}/T_{hot}}\]
Can we build a heat engine that has the ideal Carnot efficiency? No, because it would require reversible processes, which are not achievable in practice. However, by comparing our actual engine’s efficiency to the Carnot efficiency, we can assess how well our engine performs relative to the theoretical maximum - not the absolute 1st law constraint.
Applying this to other fields
The concept of relative performance analysis can be applied to various fields beyond thermodynamics. In business, for example, companies can evaluate their performance relative to industry benchmarks or best practices. Instead of just looking at absolute metrics like revenue or profit, they can assess how well they are doing compared to the best possible outcomes given their market conditions. Similarly, in personal development, individuals can measure their progress relative to their potential or goals rather than just absolute achievements.
How do we determine these benchmarks or best practices? This is where domain expertise and analysis come into play. By understanding the constraints and opportunities within a specific field, we can establish meaningful benchmarks that reflect the best possible outcomes.
We can use the tools of mathematical modelling and optimization to help us establish the optimal benchmarks - just like Sadi Carnot did for heat engines. By identifying the key variables and constraints in a given situation, we can develop models that help us understand the limits of performance and identify strategies for improvement.
Everything we do is subject to constraints - be it physical, economic, or personal. We also have certain objectives in mind for each process, be it profit or cost, etc. By adopting a relative performance analysis approach, we can gain a deeper understanding of our capabilities and identify pathways for growth and improvement. Just like Sadi Carnot’s insights revolutionized thermodynamics, applying similar principles in other fields can lead to significant advancements and better outcomes.
Can we increase performance if we are constrained?
Of course.
Since we are always subject to constraints, we can always look for ways to improve our outcomes by investigating strategies to change these constraints.
Let’s say we have a mathematical optimzation model for our process - in whatever industry or setting. We solve that problem and optain an optimal solution. That is not the end of the story. There are ways to possibly achieve a better result - but it may need some increase in cost, additional effort, or investment.
Here is where duality theory comes into play. In simple tems, duality theory provides insights into the optimal solution to a problem by determining the change in the objective function value given a change in a constraint. In real world terms, this means that we can assess how much additional profit we can achieve if we invest more resources, or how much cost we can save if we relax certain constraints. By analyzing the dual variables associated with the constraints in our optimization model, we can identify which constraints are most critical to our performance and explore strategies for improvement.
Investingating the dual solution to an optimzation problem is a key step that I emphasized to engineering studient - because the model that we set up and solved is the the only way the look at the real-world problem.