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Why do special forces need to select soldiers with the same physical strength? Why do two adults of the same physique put the heavy object in the middle when lifting something? Why is it that when an adult and a child are lifting something, the weight should be placed closer to the adult? In fact, behind these practices lies the concept of energy efficiency optimization.
For the energy efficiency optimization of humans (or organisms), we can define it as that the physical arrangement method that continues to do work over a long period of time and ultimately maximizes the total amount of work, or minimizes the total physical ability. This is called human (or organisms) energy efficiency optimization Fig. 26.1.
Let's take humans as an example to illustrate below, assuming that the situation of other living creatures is similar.
26.1 Human Energy Efficiency Curve
The human energy efficiency curve can be defined like this. The abscissa is the amount of work done per unit time W/t, labor intensity, which is similar to the output power P of a machine. We use P to represent the labor intensity of human work. The ordinate is the energy efficiencyη, which represents the ability to complete the total amount of labor within a period of longer time. The energy efficiency curve of the i-th person is shown in Fig. 26.2.
Assume that the maximum energy efficiency of the first person and the i-th person are equal, but the amount of work done per unit time is different, which is equivalent to the different power of the machine, as shown in Fig. 26.3.
If the energy efficiency curves of person 1 and person i have the following relationship
where βi is a constant, we call them similarly efficient people, and β1 = 1.
Assume that there are n people in a team to complete one task.
The total work to be done in a human team is P0 that is similar to the total energy output in a machine. Assuming P0 is a fixed value. The work completed by the i-th person is Pi, which is similar to the energy output. The physical energy input of the i-th person is Wi. The total physical energy consumed by all people in the team is Pt that is similar to the total energy input in a machine.
The expression of maximizing P0 is
The expression of minimizing Pt is
Note that the arguments of the efficiency functions in the two expressions are different and they are not the identical function.
These are two different expressions of the same problem. We only need to analyze the subsequent minimization expression.
26.2 Energy Efficiency Optimal Control in a Human Team
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1.
If the n people are identical, Pt becomes
$$ P_{t} = P_{0} \mathop \sum \limits_{i = 1}^{n} \frac{{\theta_{i} }}{{\eta \left( {\theta_{i} P_{0} } \right)}} $$(26.4)
where
Consider the minimization problem of Pt
This problem can also be written as
We consider three cases:
-
(1)
n = 2
The team has two variables and has
The objective function Pt can be expressed as
The optimization condition is
easy to draw
for the optimization point. Then we have
That is, the optimal control method is to keep
The Pt is
Since the shape of the overall efficiency curve of the team is the same as that of one person, so the second derivative of the Pt is also greater than zero
Pt is the only minimum value.
The overall energy efficiency ηt of the human team is the only maximum value.
-
(2)
n = 3
The team has three variables, based on known conditions, we have
The Pt expression becomes
Assuming that θ3 is fixed and an optimization point, only θ1 and θ2 are variables, we have
Based on the conclusion of n = 2 above, we have
is the optimal point.
Assuming that θ2 is fixed and is an optimization point, only θ1 and θ3 are variables, we have
According to the conclusion of n = 2 above, we have
is the optimal point.
Similarly, assuming that θ1 is fixed and is an optimization point, only θ2 and θ3 are variables, we have
According to the conclusion of n = 2 above, we have
is the optimal point.
So, we have the optimal point
That is, the optimal control method is to keep
The minimum Pt is
The maximum overall energy efficiency ηt is
-
(3)
n = k
The human team has k variables, the above conclusion can be extended to the case of n = k, the optimal point is
That is, the optimal method is to keep
The minimum Pt is
The maximum overall energy efficiency ηt is
-
2.
If n people are the energy efficiency similarity people, the expression of Pt becomes
$$ P_{t} = P_{0} \mathop \sum \limits_{i = 1}^{n} \frac{{\theta_{i} }}{{\eta_{i} \left( {\theta_{i} P_{0} } \right)}} $$(26.34)
where
Consider the minimization problem of total power consumption
This problem can also be written as
We consider three cases:
-
(1)
n = 2
The human team has two variables and has
The objective function Pt can be expressed as
The optimization condition is
It is easy to see that
for the optimization point. Then we have
That is, the optimal control method is to keep
The Pt is
Since the shape of the overall efficiency curve of the human team is the same as that of one person, so the second derivative of the Pt is also greater than zero
Pt is the only minimum value with
The overall energy efficiency ηt of the team is the only maximum value.
-
(2)
n = 3
The team has three variables, based on known conditions, we have
The Pt expression becomes
Assuming that θ3 is fixed and is an optimization point, only θ1 and θ2 are variables, we have
Based on the conclusion of n = 2 above, there are
is the optimal point.
Assuming that θ2 is fixed and is an optimization point, only θ1 and θ3 are variables, we have
According to the conclusion of n = 2 above, there are
is the optimal point.
Similarly, assuming that θ1 is fixed and is an optimization point, only θ2 and θ3 are variables, we have
According to the conclusion of n = 2 above, we have
to be the optimal point.
Thus, the optimal control method is to keep
The minimum Pt is
The maximum overall energy efficiency ηt is
-
(3)
n = k
The human team has k variables, the above conclusion can be extended to the case of n = k, the optimal point and the optimal method is to keep
The minimum Pt is
The maximum overall energy efficiency ηt is
26.3 Optimal Number of Working People in a Human Team
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1.
A human team has m identical people, if the following equations are satisfied
$$ \eta \left( {\frac{{P_{0} }}{n}} \right) \ge \eta \left( {\frac{{P_{0} }}{n - 1}} \right) $$(26.62)$$ \begin{array}{*{20}c} {\eta \left( {\frac{{P_{0} }}{n}} \right) \ge \eta \left( {\frac{{P_{0} }}{n + 1}} \right)} \\ {n \le m} \\ \end{array}. $$
Then the number n is the optimal number of working people.
-
2.
A human team has m energy efficiency similarity people, if the following equations are satisfied
$$ \eta_{1} \left( {\frac{{P_{0} }}{{\mathop \sum \nolimits_{l = 1}^{n} \beta_{l} }}} \right) \ge \eta_{1} \left( {\frac{{P_{0} }}{{\mathop \sum \nolimits_{l = 1}^{n1} \beta_{l} }}} \right) $$(26.63)$$ \begin{array}{*{20}c} {n \le m} \\ {n1 \le m} \\ \end{array}. $$
n1 is any combination other than the optimal combination of n working people this time, and also include other combinations of n working people. The number n is the number of people with optimal working.
26.4 Optimal Scheduling for the Number of Working People
As long as people are sent out, they have to eat whether they work or not. In this regard, people are different from machines. Therefore, when a team is sent out to perform a task, as long as the number of people is determined, there is no problem of optimizing the number of people during the process. There is a problem of optimizing the number of people before departure.
-
1.
A human team has m identical people, and the number n is the optimal number of working people. If P0 increases to P01, the following relation holds:
$$ \eta \left( {\frac{{P_{01} }}{n}} \right) = \eta \left( {\frac{{P_{01} }}{n + 1}} \right) $$(26.64)$$ n \le m $$
Then P01 is the optimal scheduling point between n people working and n + 1 people working. When the total required power P0 is greater than P01, the optimal number of working people is adjusted from n to n + 1. If P0 increases until P0/n = P1m, there isn’t the point P01, then P0/n = P1m is the adjusting point from n to n + 1.
If P0 is reduced to P02, the following relation is established
Then P02 is the optimal adjusting point between the n working people and the n-1 working people. When the total required power P0 is less than P02, the optimal number of working people is adjusted from n to n-1. If P0 reduces until P0/(n-1) = P1m, there isn’t the point P02, then P0/(n-1) = P1m is the adjusting point from n to n-1.
The analysis process is shown in Fig. 26.4.
-
2.
A human team has m energy efficiency similarity people, and the number n is the number of the optimal working people. If P0 increases to P01, the following relation holds:
$$ \begin{array}{*{20}c} {\eta_{1} \left( {\frac{{P_{01} }}{{\mathop \sum \nolimits_{l = 1}^{n} \beta_{l} }}} \right) = \eta_{1} \left( {\frac{{P_{01} }}{{\mathop \sum \nolimits_{l = 1}^{k1} \beta_{l} }}} \right)} \\ {n \le m} \\ \end{array}. $$(26.66)
Then P01 is the optimal adjusting point between the operation of n working people and the operation of k1 working people. When the total required power P0 is greater than P01, the optimal number of working people is adjusted from n to k1. If P0 increases until P0/\(\mathop \sum \limits_{l = 1}^{n} \beta_{l}\) = P1m, there isn’t the point P01, then P0/\(\mathop \sum \limits_{l = 1}^{n} \beta_{l}\) = P1m is the adjusting point from n to k1.
If P0 is reduced to P02, the following relation is established
Then P02 is the optimal switching point between the operation of n transformers and the operation of k2 transformers. When the total required power P0 is less than P02, the optimal number of transformers in operation is switched from n to k2. If P0 reduces until P0/\(\mathop \sum \limits_{l = 1}^{k2} \beta_{l}\) = P1m, there isn’t the point P02, then P0/\(\mathop \sum \limits_{l = 1}^{k2} \beta_{l}\) = P1m is the adjusting point from n to k2.
The analysis process is shown in Fig. 26.5.
k1 and k2 are any combination other than the optimal combination of n working people this time, and also include other combinations of n working people. Point \( P_{0} /\mathop \sum \limits_{l = 1}^{k1} \beta_{l}\) is the point closest to \(P_{0} /\mathop \sum \limits_{l = 1}^{n} \beta_{l} \) to the left of point \(P_{0} /\mathop \sum \limits_{l = 1}^{n} \beta_{l}\). Point \(P_{0} /\mathop \sum \limits_{l = 1}^{k2} \beta_{l}\) is the point closest to \(P_{0} /\mathop \sum \limits_{l = 1}^{n} \beta_{l}\) to the right of point \(P_{0} /\mathop \sum \limits_{l = 1}^{n} \beta_{l}\). If all people are identical, we have k1 = n + 1 and k2 = n-1.
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Yao, F., Yao, Y. (2024). Energy Efficiency Optimization of Human (or Biological) Team. In: Efficient Energy-Saving Control and Optimization for Multi-Unit Systems. Springer, Singapore. https://doi.org/10.1007/978-981-97-4492-3_26
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DOI: https://doi.org/10.1007/978-981-97-4492-3_26
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