Multi-Objective
- bm_kursawe(individual)
Kursawe multi-objective function.
- Parameters
individual (Individual) – The Individual to be evaluated.
- Returns
Fitness values of the individual.
- Return type
Equations
\(f_{1}(\mathbf{x}) = \sum_{i=1}^{N-1} -10 e^{-0.2 \sqrt{x_i^2 + x_{i+1}^2} }\)
\(f_{2}(\mathbf{x}) = \sum_{i=1}^{N} |x_i|^{0.8} + 5 \sin(x_i^3)\)
Returns \(f_{1}(\mathbf{x})\) and \(f_{2}(\mathbf{x})\).
- bm_schaffer_mo(individual)
Schaffer’s multi-objective function on a one-attribute individual.
- Parameters
individual (Individual) – The Individual to be evaluated.
- Returns
Fitness values of the individual.
- Return type
Equations
\(f_{1}(\mathbf{x}) = x_1^2\)
\(f_{2}(\mathbf{x}) = (x_1-2)^2\)
Returns \(f_{1}(\mathbf{x})\) and \(f_{2}(\mathbf{x})\).
- bm_fonseca(individual)
Fonseca and Fleming’s multiobjective function.
- Parameters
individual (Individual) – The Individual to be evaluated.
- Returns
Fitness values of the individual.
- Return type
Equations
\(f_{1}(\mathbf{x}) = 1 - e^{-\sum_{i=1}^{3}(x_i - \frac{1}{\sqrt{3}})^2}\)
\(f_{2}(\mathbf{x}) = 1 - e^{-\sum_{i=1}^{3}(x_i + \frac{1}{\sqrt{3}})^2}\)
Returns \(f_{1}(\mathbf{x})\) and \(f_{2}(\mathbf{x})\).
- bm_poloni(individual)
Poloni’s multiobjective function on a two-attribute individual.
- Parameters
individual (Individual) – The Individual to be evaluated.
- Returns
Fitness values of the individual.
- Return type
Equations
\(A_1 = 0.5 \sin (1) - 2 \cos (1) + \sin (2) - 1.5 \cos (2)\)
\(A_2 = 1.5 \sin (1) - \cos (1) + 2 \sin (2) - 0.5 \cos (2)\)
\(B_1 = 0.5 \sin (x_1) - 2 \cos (x_1) + \sin (x_2) - 1.5 \cos (x_2)\)
\(B_2 = 1.5 \sin (x_1) - cos(x_1) + 2 \sin (x_2) - 0.5 \cos (x_2)\)
\(f_{1}(\mathbf{x}) = 1 + (A_1 - B_1)^2 + (A_2 - B_2)^2\)
\(f_{2}(\mathbf{x}) = (x_1 + 3)^2 + (x_2 + 1)^2\)
Returns \(f_{1}(\mathbf{x})\) and \(f_{2}(\mathbf{x})\).
- bm_dent(individual, dent_size=0.85)
- Two-objective problem with a “dent”. The individual must havetwo attributes that take values in the range of [-1.5, 1.5].
- Parameters
individual (Individual) – The Individual to be evaluated.
dent_size (float) – The size of the dent.
- Returns
Fitness values of the individual.
- Return type
Equations
\(f_{1}(\mathbf{x}) = \text{ ?}\)
\(f_{2}(\mathbf{x}) = \text{ ?}\)
Returns \(f_{1}(\mathbf{x})\) and \(f_{2}(\mathbf{x})\).
- bm_zdt_1(individual)
ZDT1 multi-objective function.
- Parameters
individual (Individual) – The Individual to be evaluated.
- Returns
Fitness values of the individual.
- Return type
Equations
\(g(\mathbf{x}) = 1 + \frac{9}{n-1}\sum_{i=2}^n x_i\)
\(f_{1}(\mathbf{x}) = x_1\)
\(f_{2}(\mathbf{x}) = g(\mathbf{x})\left[1 - \sqrt{\frac{x_1}{g(\mathbf{x})}}\right]\)
Returns \(f_{1}(\mathbf{x})\) and \(f_{2}(\mathbf{x})\).
- bm_zdt_2(individual)
ZDT2 multi-objective function.
- Parameters
individual (Individual) – The Individual to be evaluated.
- Returns
Fitness values of the individual.
- Return type
Equations
\(g(\mathbf{x}) = 1 + \frac{9}{n-1}\sum_{i=2}^n x_i\)
\(f_{1}(\mathbf{x}) = x_1\)
\(f_{2}(\mathbf{x}) = g(\mathbf{x})\left[1 - \left(\frac{x_1}{g(\mathbf{x})}\right)^2\right]\)
Returns \(f_{1}(\mathbf{x})\) and \(f_{2}(\mathbf{x})\).
- bm_zdt_3(individual)
ZDT3 multi-objective function.
- Parameters
individual (Individual) – The Individual to be evaluated.
- Returns
Fitness values of the individual.
- Return type
Equations
\(g(\mathbf{x}) = 1 + \frac{9}{n-1}\sum_{i=2}^n x_i\)
\(f_{1}(\mathbf{x}) = x_1\)
\(f_{2}(\mathbf{x}) = g(\mathbf{x})\left[1 - \sqrt{\frac{x_1}{g(\mathbf{x})}} - \frac{x_1}{g(\mathbf{x})} \sin(10\pi x_1)\right]\)
Returns \(f_{1}(\mathbf{x})\) and \(f_{2}(\mathbf{x})\).
- bm_zdt_4(individual)
ZDT4 multi-objective function.
- Parameters
individual (Individual) – The Individual to be evaluated.
- Returns
Fitness values of the individual.
- Return type
Equations
\(g(\mathbf{x}) = 1 + 10(n-1) + \sum_{i=2}^n \left[ x_i^2 - 10\cos(4\pi x_i) \right]\)
\(f_{1}(\mathbf{x}) = x_1\)
\(f_{2}(\mathbf{x}) = g(\mathbf{x}) \left[ 1 - \sqrt{ \frac{x_1}{g(\mathbf{x})}} \right]\)
Returns \(f_{1}(\mathbf{x})\) and \(f_{2}(\mathbf{x})\).
- bm_zdt_6(individual)
ZDT6 multi-objective function.
- Parameters
individual (Individual) – The Individual to be evaluated.
- Returns
Fitness values of the individual.
- Return type
Equations
\(g(\mathbf{x}) = 1 + 9 \left[ \left(\sum_{i=2}^n x_i\right)/(n-1) \right]^{0.25}\)
\(f_{1}(\mathbf{x}) = 1 - \exp(-4x_1)\sin^6(6\pi x_1)\)
\(f_{2}(\mathbf{x}) = g(\mathbf{x}) \left[1 - \left( \frac{f_{1}(\mathbf{x})}{g(\mathbf{x})}\right)^2 \right]\)
Returns \(f_{1}(\mathbf{x})\) and \(f_{2}(\mathbf{x})\).
- bm_dtlz_1(individual, count)
- DTLZ1 multi-objective function. Returns a list of size count.The individual must have at least count number of elements.
- Parameters
individual (Individual) – The Individual to be evaluated.
count (int) – Number of objectives.
- Returns
Fitness values of the individual.
- Return type
Equations
\(g(\mathbf{x}_m) = 100\left(|\mathbf{x}_m| + \sum_{x_i \in \mathbf{x}_m}\left((x_i - 0.5)^2 - \cos(20\pi(x_i - 0.5))\right)\right)\)
\(f_{1}(\mathbf{x}) = \frac{1}{2} (1 + g(\mathbf{x}_m)) \prod_{i=1}^{m-1}x_i\)
\(f_{2}(\mathbf{x}) = \frac{1}{2} (1 + g(\mathbf{x}_m)) (1-x_{m-1}) \prod_{i=1}^{m-2}x_i\)
\(f_{m-1}(\mathbf{x}) = \frac{1}{2} (1 + g(\mathbf{x}_m)) (1 - x_2) x_1\)
\(\ldots\)
\(f_{m}(\mathbf{x}) = \frac{1}{2} (1 - x_1)(1 + g(\mathbf{x}_m))\)
Where \(m\) is the number of objectives and \(\mathbf{x}_m\) is a vector of the remaining attributes \([x_m~\ldots~x_n]\) of the individual in \(n > m\) dimensions.
- bm_dtlz_2(individual, count)
- DTLZ2 multi-objective function. Returns a list of size count.The individual must have at least count number of elements.
- Parameters
individual (Individual) – The Individual to be evaluated.
count (int) – Number of objectives.
- Returns
Fitness values of the individual.
- Return type
Equations
\(g(\mathbf{x}_m) = \sum_{x_i \in \mathbf{x}_m} (x_i - 0.5)^2\)
\(f_{1}(\mathbf{x}) = (1 + g(\mathbf{x}_m)) \prod_{i=1}^{m-1} \cos(0.5x_i\pi)\)
\(f_{2}(\mathbf{x}) = (1 + g(\mathbf{x}_m)) \sin(0.5x_{m-1}\pi ) \prod_{i=1}^{m-2} \cos(0.5x_i\pi)\)
\(\ldots\)
\(f_{m}(\mathbf{x}) = (1 + g(\mathbf{x}_m)) \sin(0.5x_{1}\pi )\)
Where \(m\) is the number of objectives and \(\mathbf{x}_m\) is a vector of the remaining attributes \([x_m~\ldots~x_n]\) of the individual in \(n > m\) dimensions.
- bm_dtlz_3(individual, count)
- DTLZ3 multi-objective function. Returns a list of size count.The individual must have at least count number of elements.
- Parameters
individual (Individual) – The Individual to be evaluated.
count (int) – Number of objectives.
- Returns
Fitness values of the individual.
- Return type
Equations
\(g(\mathbf{x}_m) = 100\left(|\mathbf{x}_m| + \sum_{x_i \in \mathbf{x}_m}\left((x_i - 0.5)^2 - \cos(20\pi(x_i - 0.5))\right)\right)\)
\(f_{1}(\mathbf{x}) = (1 + g(\mathbf{x}_m)) \prod_{i=1}^{m-1} \cos(0.5x_i\pi)\)
\(f_{2}(\mathbf{x}) = (1 + g(\mathbf{x}_m)) \sin(0.5x_{m-1}\pi ) \prod_{i=1}^{m-2} \cos(0.5x_i\pi)\)
\(\ldots\)
\(f_{m}(\mathbf{x}) = (1 + g(\mathbf{x}_m)) \sin(0.5x_{1}\pi )\)
Where \(m\) is the number of objectives and \(\mathbf{x}_m\) is a vector of the remaining attributes \([x_m~\ldots~x_n]\) of the individual in \(n > m\) dimensions.
- bm_dtlz_4(individual, count, alpha)
- DTLZ4 multi-objective function. Returns a list of size count.The individual must have at least count number of elements.
- Parameters
individual (Individual) – The Individual to be evaluated.
count (int) – Number of objectives.
alpha (float) – Fitness values exponentiation factor.
- Returns
Fitness values of the individual.
- Return type
Equations
\(g(\mathbf{x}_m) = \sum_{x_i \in \mathbf{x}_m} (x_i - 0.5)^2\)
\(f_{1}(\mathbf{x}) = (1 + g(\mathbf{x}_m)) \prod_{i=1}^{m-1} \cos(0.5x_i^\alpha\pi)\)
\(f_{2}(\mathbf{x}) = (1 + g(\mathbf{x}_m)) \sin(0.5x_{m-1}^\alpha\pi ) \prod_{i=1}^{m-2} \cos(0.5x_i^\alpha\pi)\)
\(\ldots\)
\(f_{m}(\mathbf{x}) = (1 + g(\mathbf{x}_m)) \sin(0.5x_{1}^\alpha\pi )\)
Where \(m\) is the number of objectives and \(\mathbf{x}_m\) is a vector of the remaining attributes \([x_m~\ldots~x_n]\) of the individual in \(n > m\) dimensions.
- bm_dtlz_5(individual, count)
- DTLZ5 multi-objective function. Returns a list of size count.The individual must have at least count number of elements.
- Parameters
individual (Individual) – The Individual to be evaluated.
count (int) – Number of objectives.
- Returns
Fitness values of the individual.
- Return type
Equations
\(g(\mathbf{x}_m) = \text{ ?}\)
\(f_{1}(\mathbf{x}) = \text{ ?}\)
\(f_{2}(\mathbf{x}) = \text{ ?}\)
\(\ldots\)
\(f_{m}(\mathbf{x}) = \text{ ?}\)
Where \(m\) is the number of objectives and \(\mathbf{x}_m\) is a vector of the remaining attributes \([x_m~\ldots~x_n]\) of the individual in \(n > m\) dimensions.
- bm_dtlz_6(individual, count)
- DTLZ6 multi-objective function. Returns a list of size count.The individual must have at least count number of elements.
- Parameters
individual (Individual) – The Individual to be evaluated.
count (int) – Number of objectives.
- Returns
Fitness values of the individual.
- Return type
Equations
\(g(\mathbf{x}_m) = \text{ ?}\)
\(f_{1}(\mathbf{x}) = \text{ ?}\)
\(f_{2}(\mathbf{x}) = \text{ ?}\)
\(\ldots\)
\(f_{m}(\mathbf{x}) = \text{ ?}\)
Where \(m\) is the number of objectives and \(\mathbf{x}_m\) is a vector of the remaining attributes \([x_m~\ldots~x_n]\) of the individual in \(n > m\) dimensions.
- bm_dtlz_7(individual, count)
- DTLZ7 multi-objective function. Returns a list of size count.The individual must have at least count number of elements.
- Parameters
individual (Individual) – The Individual to be evaluated.
count (int) – Number of objectives.
- Returns
Fitness values of the individual.
- Return type
Equations
\(g(\mathbf{x}_m) = \text{ ?}\)
\(f_{1}(\mathbf{x}) = \text{ ?}\)
\(f_{2}(\mathbf{x}) = \text{ ?}\)
\(\ldots\)
\(f_{m}(\mathbf{x}) = \text{ ?}\)
Where \(m\) is the number of objectives and \(\mathbf{x}_m\) is a vector of the remaining attributes \([x_m~\ldots~x_n]\) of the individual in \(n > m\) dimensions.