# An Iterated Hybrid Local Search Algorithm for Pick-and-Place Sequence Optimization

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## Abstract

**:**

## 1. Introduction

## 2. The Description of Pick-and-Place Sequence Optimization Problem

#### 2.1. Machine Structure and PCB Assembly Process

- (1)
- The PCB is transmitted to the workbench by PCB conveyor.
- (2)
- The PCB is fixed on the workbench, and the coordinates of points on PCB (see Figure 1 PART III) are loaded in the work coordinate system.
- (3)
- The mounting head moves to the corresponding feeder slots according to a sequence. There are several nozzles on the mounting head, and each nozzle can only grip one component. Thus, the maximum number of components to be carried by the head is the nozzle number (see Figure 1 PART II).
- (4)
- In order to identify the type of components carried by each nozzle, the mounting head should be scanned by camera before moving to workbench, so as to find the corresponding position on PCB.
- (5)
- All the components carried by the mounting head are mounted onto PCB.
- (6)
- The mounting head after mounted moves to the corresponding feeders which are assigned feeder slots, then the head repeat step (3) to step (5) until all the points on PCB are finished. Then, PCB conveyor starts loading the next PCB.

#### 2.2. The Problem Simplification and Notifications

- (1)
- Each feeder only contains one component type and can only be assigned to one feeder slot.
- (2)
- Each point on PCB needs one component and is only allowed to mount once.
- (3)
- The mounting head needs to pick up components among feeders first, and then the head moves to corresponding points on PCB without scanning procedure by camera.

**Parameters:**

${\mathit{P}}_{\mathit{X}}={\left(p{x}_{1},\dots ,p{x}_{i},\dots ,p{x}_{N}\right)}^{T}$ | The total points’ position in x direction on PCB |

${\mathit{P}}_{\mathit{Y}}={\left(p{y}_{1},\dots ,p{y}_{j},\dots ,p{y}_{N}\right)}^{T}$ | The total points’ position in y direction on PCB |

${\mathit{P}}_{\mathit{T}}={\left(p{t}_{1},\dots ,p{t}_{i},\dots ,p{t}_{N}\right)}^{T}$ | The total points need component type on PCB |

$N$ | The total number of points on PCB |

$H$ | The number of nozzles |

$S$ | The total number of feeder slots |

$t$ | The total number of component types |

$n$ | The total number of pick-and-place cycles, $n=\u23a1N/H\u23a4$ |

$sx$ | The first feeder slot position in x direction |

$sy$ | The first feeder slot position in y direction |

$gap$ | The fixed distance between adjacent feeder slots, because all the feeder slots have the same coordinate in y direction, the distance is the difference between two adjacent coordinates in x direction actually |

$pD\left(i,j\right)$ | The Euclidean distance between point $i$ and point $j$ on PCB |

$psD\left(i,p\right)$ | The Euclidean distance between point $i$ and feeder slot $p$ |

${Q}_{k}$ | The number of points to be mounted in kth pick-and-place cycle, ${Q}_{k}=H$, where $k<n$; ${Q}_{k}=N-\left(n-1\right)H$, where $k=n$ |

**Variables:**

${X}^{k}={\left({x}_{1}^{k},\dots ,{x}_{i}^{k},\dots ,{x}_{{Q}_{k}}^{k}\right)}^{T}$ | ${X}^{k}$ is the pick-and-place sequence in kth pick-and-place cycle, ${x}_{i}^{k}\in \left\{1,\dots ,N\right\},1\le i\le {Q}_{k},1\le k\le n$ |

${a}_{gq}\in \left\{0,1\right\}$ | Whether the component type $g$ is assigned to the feeder slot $q$, if $g$ is assigned to slot $q$, ${a}_{gq}=1$; otherwise, ${a}_{gq}=0$. $1\le g\le t$, $1\le q\le S$. |

#### 2.3. Mathematical Model of the Problem

## 3. Iterated Hybrid Local Search Algorithm

${f}_{1}\left(x\right)$ | The solution of pick-and-place distance in each interaction. |

${{f}^{\prime}}_{1}\left(x\right)$ | The best solution of pick-and-place total distance in iterations. |

${{X}^{\prime}}^{k}\left(1\le k\le n\right)$ | The best solution of pick-and-place sequence in iterations. |

${\mathrm{G}}^{\prime}\left(\mathrm{a}\right)$ | The best object function value of feeder assignment model in iterations. |

${a}_{gq}^{\prime}\left(1\le g\le t,1\le q\le S\right)$ | The best solution of feeder assignment in iterations. |

$\omega \left(i\right)\left(1\le i\le H\right)$ | The distance weight which is proposed based on splitting unit circle. $\omega \left(i\right)$ is used to generate initial pick-and-place sequences and it can be obtained by formula (13). |

#### 3.1. The Greedy Strategy with Distance Weight

- Step 1:
- Start calculate kth pick-and-place cycle where $k=1$.
- Step 2:
- If the first iteration involves this process, ${x}_{1}^{1}=1$ and ${x}_{1}^{k}\left(2\le k\le n\right)$ is other random encoding of points; else, ${x}_{1}^{1}=ite{r}_{x}$ (See Section 3.4) and ${x}_{1}^{k}\left(2\le k\le n\right)$ is other random encoding of points.
- Step 3:
- In kth pick-and-place cycle, as ${x}_{1}^{k}\left(2\le k\le n\right)$ is confirmed, calculate each ${\sigma}_{j,i}^{k}$ by formula (14) and let ${x}_{i}^{k}=j$ while the corresponding $j$ satisfies $\underset{1\le j\le N,j\notin {X}^{k}}{\mathrm{min}}{\sigma}_{j,i}^{k}$, from $i=2$ to $i={Q}_{k}$.
- Step 4:
- If $k=n$, output the initial sequence ${X}^{k}\left(1\le k\le n\right)$; otherwise, $k=k+1$ and repeat from step 3 to step 4.

#### 3.2. The Adjusting Strategy Based on Convex-Hull

- Step 1:
- Let $A={\left({x}_{1}^{k},{x}_{2}^{k},{x}_{3}^{k}\right)}^{\mathrm{T}}$, $U={\left({x}_{4}^{k},\dots ,{x}_{{Q}_{k}}^{k}\right)}^{\mathrm{T}}$.
- Step 2:
- Select each item ${x}_{p}^{k}$ from $U$, ${x}_{p}^{k}\in U$; Select each pair of items ${x}_{p}^{k},{x}_{p+1}^{k}$ from $A$, ${x}_{p}^{k},{x}_{p+1}^{k}\in A$. Calculate the value of $\rho \left({x}_{p}^{k},{x}_{q}^{k},{x}_{q+1}^{k}\right)$.
- Step 3:
- Find the corresponding ${x}_{p}^{k}$, where ${x}_{p}^{k}$ satisfies $\underset{{x}_{p}^{k}\in Uand{x}_{q}^{k},{x}_{q+1}^{\mathrm{k}}\in A}{\mathrm{min}}\rho \left({x}_{p}^{k},{x}_{q}^{k},{x}_{q+1}^{k}\right)$; put ${x}_{p}^{\mathrm{k}}$ into $A$ between ${x}_{q}^{k}$ and ${x}_{q+1}^{k}$, remove ${x}_{p}^{k}$ from $U$.
- Step 4:
- If $U=\varphi $, turn to step 7; otherwise, repeat from step 2 to step 3.
- Step 5:
- Update ${X}^{k}$, let ${X}^{k}=A$.

#### 3.3. Feeder Slots Selection

$lin{k}_{fg}$ | The number of links between component type $f$ and $g$, where $1\le f,g\le t$ and $f\ne g$. |

$boun{d}_{r}$ | The first or last point in each pick-and-place cycle ${X}^{k}$, where $1\le r\le 2n$. |

#### 3.4. Initialization Update and Termination Criteria

- Step 1:
- Initialize the parameters.
- Step 2:
- Initialize the pick-and-place sequence ${X}^{k}$ with $\omega \left(i\right)$ according to greedy strategy, where $1\le k\le n$.
- Step 3:
- Adjust the pick-and-place sequence ${X}^{k}$ by adjustment strategy, where $1\le k\le n$.
- Step 4:
- Calculate the fitness value ${f}_{1}\left(x\right)$.
- Step 5:
- If the best solution ${f}_{1}^{\prime}\left(x\right)>f\left(x\right)$, turn to step 6; otherwise, repeat from step 3 to step 5.
- Step 6:
- Update the record best solution ${f}_{1}^{\prime}\left(x\right)$, let ${f}_{1}^{\prime}\left(x\right)=f\left(x\right)$ and ${{X}^{\prime}}^{k}={X}^{k}$.
- Step 7:
- According to ${{X}^{\prime}}^{k}$, relevant parameters ($lin{k}_{fg}$ and $boun{d}_{r}$) are prepared and input to feeder assignment model.
- Step 8:
- Update the best solutions ${G}^{\prime}\left(a\right)$ and ${a}_{gq}^{\prime}$, let ${G}^{\prime}\left(a\right)=G\left(a\right)$ and ${a}_{gq}^{\prime}={a}_{gq}$.
- Step 9:
- If termination criteria was satisfied, turn to step 10; otherwise, repeat from step 3 to step 8.
- Step 10:
- Output the best solutions.

## 4. Experimental Results and Analysis

#### 4.1. Comparison with Exact Algorithm

#### 4.2. Comparison of the Heuristics in Different Size Instances and Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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No. | $\mathit{x}$ | $\mathit{y}$ |
---|---|---|

1 | 16 | 2 |

$2$ | 18 | 2 |

$3$ | 19 | 12 |

$4$ | 17 | 14 |

5 | 11 | 8 |

6 | 15 | 4 |

$\mathit{l}\mathit{i}\mathit{n}{\mathit{k}}_{\mathit{f}\mathit{g}}$ | $\mathit{p}{\mathit{t}}_{1}$ | $\mathit{p}{\mathit{t}}_{2}$ | $\mathit{p}{\mathit{t}}_{3}$ | $\mathit{p}{\mathit{t}}_{4}$ |
---|---|---|---|---|

$p{t}_{1}$ | 0 | 0 | 0 | 1 |

$p{t}_{2}$ | 1 | 0 | 0 | 0 |

$p{t}_{3}$ | 0 | 0 | 0 | 0 |

$p{t}_{4}$ | 1 | 1 | 0 | 0 |

(PN, TN) | No. | Object Function Value (mm) | CUP Times | |||
---|---|---|---|---|---|---|

${\mathit{D}}_{\mathit{I}\mathit{H}\mathit{L}\mathit{S}}$ | ${\mathit{D}}_{\mathit{C}\mathit{p}\mathit{l}\mathit{e}\mathit{x}}$ | $\frac{{\mathit{D}}_{\mathit{I}\mathit{H}\mathit{L}\mathit{S}}-{\mathit{D}}_{\mathit{C}\mathit{p}\mathit{l}\mathit{e}\mathit{x}}}{{\mathit{D}}_{\mathit{C}\mathit{p}\mathit{l}\mathit{e}\mathit{x}}}\times 100\%$ | $\mathit{C}{\mathit{T}}_{\mathit{I}\mathit{H}\mathit{L}\mathit{S}}$ | $\mathit{C}{\mathit{T}}_{\mathit{c}\mathit{p}\mathit{l}\mathit{e}\mathit{x}}$ | ||

(6, 5) | 1 | 280.12 | 263.37 | 6.45 | 0.35 | 0.27 |

(8, 6) | 2 | 409.41 | 383.71 | 6.78 | 0.98 | 0.83 |

(10, 6) | 3 | 598.34 | 566.02 | 5.65 | 2.21 | 2.09 |

(82, 3) | 4 | 3766 | Unsolved | - | 22.3 | - |

No. | N | t | H | n | S | (sx, sy) | $\mathbf{gap}$ | ${\mathbf{Q}}_{\mathit{k}}$ |
---|---|---|---|---|---|---|---|---|

1 | 82 | 3 | 10 | 9 | 60 | (−120, −72) | 12 | (10, 10, …, 3) |

2 | 100 | 4 | 10 | 10 | 60 | (−120, −72) | 12 | (10, 10, …, 10) |

3 | 145 | 5 | 10 | 15 | 60 | (−120, −72) | 12 | (10, 10, …, 5) |

4 | 163 | 4 | 10 | 17 | 60 | (−120, −72) | 12 | (10, 10, …, 3) |

5 | 184 | 5 | 10 | 19 | 60 | (−120, −72) | 12 | (10, 10, …, 4) |

6 | 200 | 5 | 10 | 20 | 60 | (−120, −72) | 12 | (10, 10, …, 10) |

7 | 240 | 6 | 10 | 24 | 60 | (−120, −72) | 12 | (10, 10, …, 10) |

8 | 264 | 7 | 10 | 27 | 60 | (−120, −72) | 12 | (10, 10, …, 4) |

9 | 300 | 9 | 10 | 30 | 60 | (−120, −72) | 12 | (10, 10, …, 10) |

No. | GA | IHLS | Memetic Algorithm | |||||
---|---|---|---|---|---|---|---|---|

${\mathbf{D}}_{\mathbf{best}}$ | $\mathbf{Minimum}\text{}\mathbf{CT}$ | ${\mathbf{D}}_{\mathbf{GA}}\left(\mathbf{mm}\right)$ | $\mathbf{C}{\mathbf{T}}_{\mathbf{GA}}$ | ${\mathbf{D}}_{\mathbf{IHLS}}\left(\mathbf{mm}\right)$ | $\mathbf{C}{\mathbf{T}}_{\mathbf{IHLS}}$ | ${\mathbf{D}}_{\mathbf{MA}}\left(\mathbf{mm}\right)$ | $\mathbf{C}{\mathbf{T}}_{\mathbf{MA}}$ | |

1 | 3511 | 22.3 | 3542 | 1534.0 | 3766 | 22.3 | 3511 | 1485.7 |

2 | 5509 | 29.5 | 5377 | 1512.4 | 5844 | 29.5 | 5509 | 1508.2 |

3 | 7473 | 52.4 | 7655 | 1743.5 | 7740 | 52.4 | 7473 | 1485.0 |

4 | 8166 | 65.0 | 8352 | 1744.0 | 8479 | 65.0 | 8166 | 1846.8 |

5 | 8813 | 78.4 | 10,439 | 1766.8 | 8813 | 78.4 | 9430 | 2020.5 |

6 | 9882 | 93.4 | 11,541 | 1923.4 | 9882 | 93.4 | 10,249 | 2195.0 |

7 | 11,218 | 124.2 | 14,745 | 2043.8 | 11,218 | 124.2 | 11,725 | 2596.0 |

8 | 13,591 | 174.2 | 17,610 | 2142.0 | 13,591 | 174.2 | 13,837 | 2802.0 |

9 | 14,854 | 247.0 | 20,745 | 2689.4 | 14,854 | 247.0 | 15,007 | 3310.7 |

$\mathit{k}$ | ${\mathbf{X}}^{\mathbf{k}}$ | ${\mathit{a}}_{\mathit{g}\mathit{q}}$ | Corresponding Feeder Type | Corresponding Feeder Coordinates |
---|---|---|---|---|

1 | ${X}^{1}=\left(146,219,13,27,185,224,284,259,215,233\right)$ | ${a}_{1,27}=1$ | 100004652 | (192, −72) |

2 | ${X}^{2}=\left(93,72,113,165,177,204,87,31,89,152\right)$ | ${a}_{2,23}=1$ | 100004697 | (144, −72) |

3 | ${X}^{3}=\left(108,237,153,174,101,69,210,140,126,62\right)$ | ${a}_{3,24}=1$ | 100004795 | (156, −72) |

4 | ${X}^{4}=\left(105,48,58,186,143,129,77,57,231,176\right)$ | ${a}_{4,26}=1$ | 100004883 | (180, −72) |

5 | ${X}^{5}=\left(66,104,128,268,214,184,276,280,239,238\right)$ | ${a}_{5,16}=1$ | 100005022 | (16, −72) |

6 | ${X}^{6}=\left(63,234,179,131,65,75,183,132,47,264\right)$ | ${a}_{6,3}=1$ | 100006424 | (−96, −72) |

7 | ${X}^{7}=\left(144,74,59,213,158,159,98,99,270,45\right)$ | ${a}_{7,22}=1$ | 100008025 | (132, −72) |

8 | ${X}^{8}=\left(245,29,226,187,241,\text{}251,190,188,253,260\right)$ | ${a}_{8,25}=1$ | 100030843 | (168, −72) |

9 | ${X}^{9}=\left(244,271,265,200,203,266,209,212,211,230\right)$ | ${a}_{8,15}=1$ | 100031371 | (48, −72) |

10 | ${X}^{10}=\left(293,235,269,205,206,267,279,202,\text{}236,199\right)$ | |||

11 | ${X}^{11}=\left(141,156,68,21,207,180,168,110,246,111\right)$ | |||

12 | ${X}^{12}=\left(232,181,295,257,258,229,208,248,17,262\right)$ | |||

13 | ${X}^{13}=\left(26,247,220,194,242,243,193,217,256,10\right)$ | |||

14 | ${X}^{14}=\left(223,197,282,281,196,218,263,225,164,161\right)$ | |||

15 | ${X}^{15}=\left(44,95,52,119,147,162,198,27,49,71\right)$ | |||

16 | ${X}^{16}=\left(22,7,189,123,122,83,9,216,135,138\right)$ | |||

17 | ${X}^{17}=\left(252,11,255,191,227,250,249,221,182,254\right)$ | |||

18 | ${X}^{18}=\left(195,137,134,86,170,125,80,81,192,8\right)$ | |||

19 | ${X}^{19}=\left(240,32,222,149,155,107,36,228,167,16\right)$ | |||

20 | ${X}^{20}=\left(173,92,12,201,171,150,116,54,24,261\right)$ | |||

21 | ${X}^{21}=\left(291,286,33,61,6,172,117,103,35,145\right)$ | |||

22 | ${X}^{22}=\left(287,274,37,67,28,136,114,91,120,154\right)$ | |||

23 | ${X}^{23}=\left(4,70,55,163,2,112,30,157,300,299\right)$ | |||

24 | ${X}^{24}=\left(42,73,96,151,40,94,3,133,277,273\right)$ | |||

25 | ${X}^{25}=\left(46,76,51,148,34,64,5,169,296,292\right)$ | |||

26 | ${X}^{26}=\left(19,88,50,127,90,115,43,121,290,283\right)$ | |||

27 | ${X}^{27}=\left(14,288,139,23,100,18,142,97,102,294\right)$ | |||

28 | ${X}^{28}=\left(285,289,78,85,20,124,53,106,15,130\right)$ | |||

29 | ${X}^{29}=\left(297,278,84,79,60,160,56,118,41,175\right)$ | |||

30 | ${X}^{30}=\left(298,38,82,166,1,39,178,109,25,275\right)$ |

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## Share and Cite

**MDPI and ACS Style**

Gao, J.; Zhu, X.; Liu, A.; Meng, Q.; Zhang, R.
An Iterated Hybrid Local Search Algorithm for Pick-and-Place Sequence Optimization. *Symmetry* **2018**, *10*, 633.
https://doi.org/10.3390/sym10110633

**AMA Style**

Gao J, Zhu X, Liu A, Meng Q, Zhang R.
An Iterated Hybrid Local Search Algorithm for Pick-and-Place Sequence Optimization. *Symmetry*. 2018; 10(11):633.
https://doi.org/10.3390/sym10110633

**Chicago/Turabian Style**

Gao, Jinsheng, Xiaomin Zhu, Anbang Liu, Qingyang Meng, and Runtong Zhang.
2018. "An Iterated Hybrid Local Search Algorithm for Pick-and-Place Sequence Optimization" *Symmetry* 10, no. 11: 633.
https://doi.org/10.3390/sym10110633