**7. Crossover operator for MSA**

Besides, a special crossover operator is designed to generate a new solution from the current solution and the best solution so far in the rough search based on the representation in section 5.2. The margin and centre of the new solution (child) inherit the margin of the current solution (father) and the reversed centre of the best solution (mother), respectively. The reason to reverse the best solution is to get a different solution even two given solutions are the same solution.

**Figure 11.** An example of two layouts before the crossover operator: the current layout (F: father) and the best layout so far (M: mother)

For the detail of crossover, two sequences *Γ+* and *Γ–* are selected randomly from (*Γ1, Γ2, Γ3*) or (*Γ1, Γ2, Γ3, Γ4*, *Γ5*) for 3D packing. Let us denote the father as (*Γ<sup>f</sup> <sup>+</sup>*, *Γ<sup>f</sup> –*) which is selected from the current solution. The mother (*Γm<sup>+</sup>*, *Γm–*) is from the best solution so far. A number *i* is an integer randomly produced between 1 and *k/2*–1. The child of sequence pair (*Γ<sup>c</sup> <sup>+</sup>*, *Γ<sup>c</sup> –*) is given by *Γ<sup>f</sup> <sup>+</sup>*[*0, i*] + *Γ'm+* + *Γ<sup>f</sup> <sup>+</sup>*[(*k*–*i*–1), *k*–1] and *Γ<sup>f</sup> –*[*0, i*] + *Γ'm–* + *Γf –*[(*k*–*i*–1), *k*–1], where *Γ'm+* and *Γ'm–* are the inverse of *Γm+* – *Γ<sup>f</sup> <sup>+</sup>*[*0, i*] – *Γ<sup>f</sup> <sup>+</sup>*[(*k*–*i*–1), *k*–1] and the inverse of *Γm–* – *Γ<sup>f</sup> –*[*0, i*] – *Γ<sup>f</sup> –*[(*k*–*i*– 1), *k*–1], respectively.

To make it clearer, let us take an example to explain the crossover operator. As shown in Figure 11, the left layout is represented by *Γ+*(*m3,m4,m2,m1,m5*) and *Γ -*(*m5,m1,m2,m3,m4)* as the father, which is the capital "F" in Figure 12. The right one is *Γ+*(*m1,m3,m4,m2,m5*) and *Γ -* (*m3,m4,m1,m5,m2*) as the mother, which is the reversed capital "M" in Figure 12. If we assume the *i* be 1, the child will be the layout *Γ+*(*m3,m2,m4,m1,m5*) and *Γ -*(*m5,m2,m1,m3,m4*) as the right layout of Figure 12*,* where *Γ+*(*m*3*, ..., m5*) and *Γ -*(*m5, ..., m*4) are from the father as the margin of left picture of Figure 12, and *Γ+*(*...,m2,m4,m1,...*) and *Γ -*(*...,m2,m1,m3,...*) are from mother with an inverse order as the centre of left picture of Figure 12.

**Figure 12.** The layout (child) after the crossover operator between the current layout (F: father) and the best layout so far (M: mother) with an inverse order
