Table 2 Optimization Model for Partitions
From: Partition-based optimization model for generative anatomy modeling language (POM-GAML)
Arg Min: \( P{\left(\sum \limits_{l=1}^N{k}_l\left|{p}_l-{p}_{Destination}\right|\right)}_t \) | ||
Subject to: | ||
Distij − |pi − pj| = 0 | for (i,j)∈At | (5) |
\( {\mathit{\cos}}^{-1}\left(\frac{\left({p}_{io}-{p}_j\right)\bullet \left({p}_i-{p}_j\right)}{\left\Vert \left({p}_{io}-{p}_j\right)\right\Vert \times \left\Vert \left({p}_i-{p}_j\right)\right\Vert}\right)-{\theta}_{ij}<0 \) | for (i,j)∈Bt | (6) |
(pio − pj)axis − (pi − pj)axis = 0 | (6a) | |
Distij − ∆dmax − |pi − pj| < 0 | for (i,j)∈Ct | (7) |
|pi − pj| − Distij − ∆dmax < 0 | (8) | |
\( 2\operatorname{sgn}\left({p}_i{p}_j.{p}_i{v}_j\right){\times \mathit{\tan}}^{-1}\left(\frac{r}{\left\Vert {v}_j\right\Vert}\right)-{a}_{ij}<0 \) | for (i,j)∈Dt | (9) |
\( \left(\frac{\tan \left({a}_{ij}\right)}{\left\Vert {v}_j\right\Vert}\right)<0 \) | (10) | |
|pi − vj| < 0 | (11a) | |
|pi − vi| = 0 | (11b) | |
i, j ∈ J and i, j ⊆ M, kl > 0, A ∩ C =  ∅ , P(A)m ∩ P(A)n =  ∅ , vi = pi, vj = pj, v ∈ V, v ∉ A, B, C | ||