Efficiency prediction (minimum versus average)

• I
I am trying to determine the efficiency of a light source for treating plants. When using the light source, there is an equal probability of encountering two different plant species.

The light source has a certain electro-optical efficiency ##\eta_{\text{eo}}##. And each plant type has a different receptiveness (i.e. absorbance) to the light ##\eta_{\text{a}}##. So, the total efficiency of the process will be the product of the two efficiencies:

$$\eta_{\text{total}}=\eta_{\text{eo}}.\eta_{\text{a}} \tag{1}$$

Some research papers that I have read predict the efficiency of such a system to be the minimum of the two efficiencies:

$$\eta_{\text{total}}=\min\left(\eta_{\text{eo}}\eta_{\text{a}_{1}},\ \eta_{\text{eo}}\eta_{\text{a}_{2}}\right). \tag{2}$$

where the subscripts denote the plant type. However, the authors did not provide any statistical justifications for that.

So, what could be the reasoning for this? Is it not better to use the average value instead? i.e.,

$$\eta_{\text{total}}=\eta_{\text{eo}}.\left(\frac{\eta_{\text{a}_{1}}+\eta_{\text{a}_{2}}}{2}\right).$$

Any explanations would be greatly appreciated.

Dale
Mentor
2020 Award
Some research papers that I have read predict the efficiency of such a system to be the minimum of the two efficiencies:
Can you link to the papers?

mathman