馬克排版是一個在傳統成衣產業生產流程中常見的二維佈局問題，主要是去
排列許多不規則的圖形或模板在一張二維矩形布料上，目的在於是否能快速排出
最節省布料的佈局，通常是固定矩形布料寬度下，最小化布料長度。此二維排版問題為一種組合最佳化的問題，一般而言，有兩個重要的決策需要考量，一是圖形能夠擺放的位置；二是圖形擺放的順序。這是非常困難的問題，因此必須使用啟發式方法來處理。本研究使用了兩種現有的排列圖形方法進行模擬實驗，第一種是建設性演算法(constructive algorithm, CA)和基因演算法(genetic algorithm, GA)，由 Hifi 和 Hallah (2003)提出，是透過 CA 找出圖形可排的點位置，再利用 GA 模擬尋找圖形的排列順序。另一種方法則是利用 Kirkpatrick(1983)
所提出的模擬退火法(simulated annealing)取代原先的 GA 來做最佳路徑的搜尋。CA 主要是透過最小矩形包圍圖形的角點作為下一個欲排入圖形的參考位置
再。而 GA 和 SA 不同的地方在於，GA 是一種群集搜尋法，能透過本身運算子
找出最佳解；SA 亦是一種群集搜尋法，但會接受較差勁二維佈局解，進而避免
落入區域最佳解中。

In the traditional garment industry, marking is a common two-dimensional layout problem for arranging many irregular patterns or polygons on fabric. The objective of this problem is to minimize the use of the fabric or, commonly, minimize the fabric
length on rectangular fabric with a fixed width.
This problem is a combinatorial optimization in typology. In general, two essential decisions of the problem have to be considered. The first one is the locations where a polygon can be placed and the second one is the order in which polygons are placed. It is a very difficult problem and heuristic methods are necessary for this problem. This study tests and compares two heuristic methods. One is constructive approach and genetic algorithm (CAGA) proposed by Hifi and Hallah (2003), which includes CA for the locations of a considered polygon and GA for the order of the polygons. The other method replaces genetic algorithm with simulated annealing(SA) proposed by Kirkpatrick (1983), which also obtains the location of a polygon with CA but the order of the polygons with SA.
The purpose of CA is collecting the corner points of a minimized rectangle containing the considered polygon for reference point of next polygon. Both GA and SA are population search methods. The difference between GA and SA is that GA finds the optimal solution through bio-inspired operators, but SA would accept the worse
solution to escape local optimum.

目錄.............................................. I
圖目錄............................................ III
表目錄............................................ V
1. 緒論........................................... 1
1.1 研究背景...................................... 1
1.2 研究動機...................................... 2
1.3 問題描述及複雜度............................... 3
2. 文獻回顧....................................... 4
2.1 多邊形擺放位置................................. 4
2.1.1 CA(Constructive approach) .................. 4
2.1.2 NFP(No-fit polygon)......................... 4
2.2 多邊形擺放順序................................. 5
2.2.1 Initial sorting............................. 6
2.2.2 Local search ............................... 6
2.2.3 Simulated annealing......................... 7
2.2.4 Genetic algorithm........................... 8
3. 研究方法....................................... 10
3.1 計算圖形可擺放位置............................. 10
3.1.1 CA(Constructive approach) ................. 10
3.2 計算圖形順序編號............................... 11
3.2.1 計算圖形順序編號(GA) ........................ 11
3.2.2 計算圖形順序編號(SA)......................... 15
3.3 平行化 GA/SA.................................. 19
4. 實驗設計與結果.................................. 20
4.1 實驗問題產生................................... 20
4.2 實驗設計....................................... 20
4.3 實驗結果分析.................................... 21
5. 未來展望........................................ 25
Appendix A：部分排列結果 ........................... 26
參考文獻........................................... 35

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