Saint-Gobain France為ROADEF/EURO在2018舉辦的挑戰賽所提出的切割優化問題是一個二維、三階段的切割過程，此挑戰包括一些特定的限制條件，這些限制條件使為標準切割問題開發的演算法無法直接應用於此挑戰賽。一方面，待切割的板材存在缺陷，因此它們獨一無二並且必須按照給定的順序使用。另一方面，物件按堆棧分組，每個堆棧中的物件必須按照順序切割。
過去已經有許多研究針對原料切割問題作探討，常見的方法有精確式算法、啟發式演算法以及萬用啟發式演算法。在啟發式演算法方面，如波束搜索演算法（Beam Search Algorithm）以及A*搜尋演算法（A* Search Algorithm），這些方法都可以求得不錯的近似最佳解。而在萬用啟發式演算法方面，有學者使用基因演算法（Genetic Algorithm, GA）來求解原料切割問題，不過基因演算法所求得的解品質有待提升。
本研究使用簡化群體演算法（Simplified Swarm Optimization, SSO）來求解該問題， SSO的優點是使用離散型的問題上有不錯的成果，也因為容易調整參數，故更易求得近似最佳解。本研究使用SSO的單變數更新，使更新後的解符合問題的限制條件，並且提出了使用垂直線及水平線紀錄物品擺放位置。最後，本研究利用SSO的更新機制，得到了更加優良的解品質，並且與其他演算法進行比較，如：波束搜索演算法、A*搜尋演算法。

The cutting optimization problem proposed by Saint-Gobain France for the 2018 ROADEF/EURO challenge is a two-dimensional, three-stage cutting process. The challenge includes some specific constraints that make it impossible to directly apply the algorithms developed for the standard cutting problem. On the one hand, the sheets have defects, so they are unique and must be used in order. On the other hand, the objects are grouped in stacks, and the objects in each stack must be cut in order.
There have been many studies on the raw material cutting problem, and the common approaches are exact algorithms, heuristic algorithms and generalized heuristic algorithms. In the case of heuristic algorithms, such as Beam Search Algorithm and A* Search Algorithm, these methods can find good approximate optimal solutions. As for the general-purpose heuristic algorithm, some scholars use Genetic Algorithm (GA) to solve the raw material cut problem, but the quality of the solutions obtained by GA needs to be improved.
This study uses the Simplified Swarm Optimization (SSO) algorithm to solve the problem. SSO has the advantage of good results using discrete problems and is easier to find the approximate best solution because of the ease of adjusting the parameters. This study uses single variable update of SSO to make the updated solution meet the constraints of the problem and proposes the use of vertical and horizontal lines to record the placement of items. Finally, this study uses the update mechanism of SSO to obtain better solution quality and compare with other algorithms, such as beam search algorithm and A* search algorithm.

摘要....................................................I
Abstract...............................................II
目錄..................................................III
表目錄..................................................V
圖目錄.................................................VI
第一章、 緒論............................................1
1.1 研究背景與動機...................................1
1.2 研究目的.........................................2
1.3 研究架構.........................................3
第二章、 文獻回顧.........................................5
2.1 切割和封裝問題....................................5
2.2 原料切割問題演算法................................11
2.3 簡化群體演算法（Simplified Swarm Optimization）...14
2.4 文獻回顧小節 .....................................15
第三章、 問題定義.........................................16
3.1 切割問題定義......................................16
3.2 數學模型.........................................19
第四章、 研究方法.........................................22
4.1 編解碼方式.......................................22
4.2 初始解及第一個放進箱中的物品.......................23
4.3 缺陷處理方法......................................31
4.4 適應度衡量.......................................34
4.5 SSO更新步驟......................................34
4.6 SSO求解切割問題流程圖.............................37
第五章、 實驗結果與分析....................................38
5.1 實驗資料集........................................38
5.2 SSO參數實驗設計...................................39
5.3 實驗結果比較......................................41
第六章、 結論與未來研究方向.................................50
6.1 結論..............................................50
6.2 未來研究方向......................................51
參考文獻..................................................52

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