A HEURISTIC APPROACH FOR SOLVING IRREGULAR PACKING PROBLEM 黃玟錫、吳泰熙,蕭育如
E-mail: 9018616@mail.dyu.edu.tw
ABSTRACT
PACKING AND CUTTING IRREGULAR OBJECTS ARE TASKS FREQUENTLY ENCOUNTERED BY THE
INDUSTRIES, SUCH AS SHOES MAKING, TEXTILE, STEEL, CLOTHING, ETC. THE GREATER SAVING IN THE WASTAGE OF MATERIALS OF THE ABOVE-MENTIONED INDUSTRIES, THE LOWER THE TOTAL
PRODUCTION COST. LOTS OF RESEARCH IN THE LITERATURE HAS BEEN DEVOTED TO DEVELOPING ALGORITHMS TO FIND THE OPTIMAL / NEAR OPTIMAL WAY OF CUTTING AND PACKING. IN THIS STUDY, WE PROPOSE A SIMULATED ANNEALING (SA) APPROACH FOR PACKING PATTERNS WITH IRREGULAR SHAPES IN A RECTANGULAR SHEET. IN ADDITION, THE SHEET ACCOMMODATING IRREGULAR SHAPES IS ALLOWED TO HAVE NON-RECTANGULAR SHAPE AND NON-UNIFORM QUALITY INSIDE. SEVERAL
PARAMETERS CONTROLLING THE MECHANISM OF SA ARE ALSO TESTED THROUGH SOME EXPERIMENTS TO ACCELERATE THE CONVERGENCE OF THE SA ALGORITHM.
Keywords : SIMULATED ANNEALING, SA, CUTTING, PACKING, IRREGULAR PATTER Table of Contents
第一章 緒論--P1 1.1 研究背景與動機--P1 1.2 研究目的--P2 1.3 研究假設--P3 1.4 研究方法與架構--P3 第二章 文獻探討--P7 2.1 物件分類與排列--P7 2.2 規則方形物件的排列問題--P9 2.3 不規則形狀物件之排列問題--P10 2.3.1 圓形物件之排列問 題--P10 2.3.2 不規則物件排列問題--P10 第三章 不規則物件排列問題之求解法--P13 3.1 問題定義--P13 3.1.1 傳統型不規則物 件排列問題--P13 3.1.2 非均質及不規則外型原片之不規則物件排列問題--P13 3.2 傳統型不規則物件之SA演算法--P14 3.2.1 預處理--P16 3.2.2 起始解--P22 3.2.3 使用率--P24 3.2.4 目標函數--P24 3.2.5 移步之種類與選取方式--P25 3.2.6 退火程序--P31 3.2.7 終止條件--P33 3.3 非均質及不規則外型原片排列問題之SA演算法--P34 3.3.1 預處理--P34 3.3.2 起始解--P35 3.3.3 使用 率--P37 3.3.4 目標函數--P37 3.3.5 移步之種類與選取方式--P39 3.3.6 退火程序--P39 3.3.7 終止條件--P39 第四章 演算結果與 分析--P40 4.1 實驗參數說明--P40 4.1.1 起始溫度、冷卻率與馬可夫鍊--P40 4.1.2 馬可夫鍊改善水準--P45 4.1.3 目標函數權重 值--P47 4.2 實例說明與驗證--P49 4.3 傳統型不規則物件排列問題之SA演算結果--P52 4.4 非均質及不規則原片排列問題之SA 演算結果--P56 第五章 結論與建議--P62 5.1 結論--P62 5.2 建議--P62 參考文獻--P64 附錄一--P77 附錄二--P78
REFERENCES
[ 1]吳泰熙、駱景堯、林東養,「多尺寸方形排列問題啟發式解法之研究」,工業工程學刊,第17卷, 75- 85, (2000) [ 2]林東養,「不規則物件排 列問題解法之研究」,大葉大學工業工程研究所碩士論文, (1998) [ 3]徐德興,「利用模擬退火演算法求解不規則物件排列及切割問題」,大 葉大學工業工程研究所碩士論 文,(2000) [ 4]ALBANO, A. AND G. SAPUPPO, "OPTIMAL ALLOCATION OF TWO- DIMENSIONAL IRREGULAR SHAPES USING HEURISTIC SEARCH METHODS", IEEE TRANSACTIONS ON SYSTEM, MAN, AND
CYBERNETICS, 10, 242- 248, (1980) [ 5]BEASLEY, J. E., "ALGORITHMS FOR UNCONSTRAINED TWO-DIMENSIONAL GUILLOTINE CUTTING", JOURNAL OF THE OPERATIONAL RESEARCH SOCIETY, 36, 297-306, (1985) [ 6]BIRO M., AND E.
BORORS, "NETWORK FLOWS AND NON-GUILLOTINE CUTTING PATTERNS", EUROPEAN JOURNAL OF OPERATIONAL RESEARCH, 16, 215-221, (1984) [ 7]BRUCE W. L., J. A. GEORGE, JENNIFER M. GEORGE, "PACKING DIFFERENT-SIZED CIRCLES INTO A RECTANGULAR CONTAINER", EUROPEAN JOURNAL OF OPERATIONAL RESEARCH, 84, 693-712, (1995) [
8]CHRISTOFIDES, N. AND C. WHITLOCK, "AN ALGORITHMS FOR TWO DIMENSIONAL CUTTING PROBLEMS" ,
OPERATIONS RESEARCH, 25, 30-44, (1977) [ 9]CHRISTOPH N. AND G. SCHEITHAUER, "TIGHTER RELAXATIONS FOR THE CUTTING STOCK PROBLEM", EUROPEAN JOURNAL OF OPERATIONAL RESEARCH,654-663,(1999) [10]DAGLI, C. H.,
"NEURAL NETWORK IN MANUFACTURING: POSSIBLE IMPACTS ON CUTTING STOCK PROB -LEMS", IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, 531-537, (1990) [11]DAGLI, C. H. AND A. HAJAKBARI, "SIMULATED ANNEALING APPROACH FOR SOLVING STOCK CUTTING PROBLEM", IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, 221-223, (1990) [12]DAGLI, C. H. AND M. Y. TATOGLU, "AN APPROACH TO TWO- DIMENSIONAL CUTTING STOCK PROBLEM
", INTERNATIONAL JOURNAL PRODUCTION RESEARCH, 25, 175-190, (1987) [13]DOWSLAND, K. A. AND W. DOWSLAND,
"PACKING PROBLEM", EUROPEAN JOURNAL OF OPERATIONAL RESEARCH, 56, 2-14, (1992) [14]FUH-HWA F. L. AND C-J.
HSIAO, "A THREE-DIMENSIONAL PALLET LOADING METHOD FOR SINGLE- SIZE BOXES" JOURNAL OF THE
OPERATIONAL RESEARCH SOCIETY, 48, 726-735, (1997) [15]G-C H., MSC AND S-J NA, DR ING, "TWO-STAGE APPROACH FOR NESTING IN TWO-DIMENSIONAL CUT TING PROBLEMS USING NEURAL NETWORK AND SIMULATED ANNEALING", PROCEEDING OF INSTITUTION OF MECHANICAL ENGINEERS, JOURNAL OF ENGINEERING MANUFACTURE, 210, 509-519, (1996) [16]GILMORE, P. C. AND R. E. GOMORY, "A LINEAR PROGRAMMING APPROACH TO THE CUTTING STOCK
PROBLEM (PART 1)", OPERATIONS RESEARCH, 9, 849-855, (1961) [17]GILMORE, P. C. AND R. E. GOMORY, "A LINEAR PROGRAMMING APPROACH TO THE CUTTING STOCK PROBLEM (PART 2)", OPERATIONS RESEARCH, 11, 863-888, (1963) [18]GILMORE, P. C. AND R. E. GOMORY, "MULTISTAGE CUTTING PROBLEMS OF TWO AND MORE DIMENSI -ONS",
OPERATIONS RESEARCH, 13, 94-120, (1965) [19]GILMORE, P. C. AND R. E. GOMORY, "THE THEORY AND COMPUTATION OF KNAPSACK FUNCTIONS", OPERATIONS RESEARCH, 14, 1045-1074, (1966) [20]GRINDE, R. B. AND T. M. CAVALIER, "A NEW ALGORITHM FOR THE MINIMAL-AREA CONVEX ENCLO -SURE PROBLEM", EUROPEAN JOURNAL OF OPERATIONAL RESEARCH, 84, 522-538, (1995) [21]GRINDE, R. B. AND T. M. CAVALIER, "CONTAINMENT OF A SINGLE POLYGON USING MATHEMATICAL PROGRAMMING", EUROPEAN JOURNAL OF OPERATIONAL RESEARCH, 92, 386-286, (1996) [22]GOULIMIS C., "OPTIMAL SOLUTION FOR THE CUTTING STOCK PROBLEM", EUROPEAN JOURNAL OF OPERATIONAL RESEARCH, 44, 197-208, (1990) [23]HENRY L. AND W. N. WAGGENSPACK, JR., "PRACTIC & EXPERIENCE NESTING OF TWO-DIMENSIONAL IRREGULAR PARTS USING A SHAPE REASONING HEURISTIC" COMPUTER-AIDED DESIGN, 29, 221-238, (1997) [24]HIFI, M.,
"THE DH/KD, "ALGORITHM: A HYBRID APPROACH FOR UNCONSTRAINED TWO -DIMENSIONAL CUTTING PROBLEMS", EUROPEAN JOURNAL OF OPERATIONAL RESEARCH, 97, 41-52, (1997) [25]HYUN-CHAN LEE, "DETERMINING IN LINEAR TIME THE MINIMUM AREA CONVEX HULL OF TWO POLYGO -NS", IIE TRANSACTIONS, 20, 338-345, (1988) [26]ISMAIL, H. S.
AND J. L. SANDERS, "TWO-DIMENSIONAL CUTTING STOCK PROBLEM RESEARCH: A RE -VIEW AND A NEW
RECTANGULAR LAYOUT ALGORITHM ", JOURNAL OF MANUFACTURING SYSTEMS, 1, 169-182, (1982) [27]ISMAIL, H. S. AND K. K. B. HON, "THE NESTING OF 2-DIMENSIONAL SHAPES USING GENETIC ALG -ORITHM", PROCEEDINGS OF THE INSTITUTION OF MECHANICAL ENGINEERS PART B-JOURNAL OF ENGINEERING MANUFACTURE, 209, 115-124, (1995) [28]J.
BLAZEWICZ, P. HAWRYLUK AND R. WALKOWIAK, "USING A TABU SEARCH APPROACH FOR SOLVING THE TWO-DIMENSIONAL IRREGULAR CUTTING PROBLEM", ANNALS OF OPERATIONS RESEARCH, 41, 313- 325, (1993)
[29]JACOBS, S., "ON GENETIC ALGORITHM FOR THE PACKING OF POLYGONS",EUROPEAN JOURNAL OF OPE -RATIONAL RESEARCH, 88,165-181, (1996) [30]JIMMY W.M., K. K. LAI, CHAN, "DEVELOPING A SIMULATED ANNEALING ALGORITHM FOR THE CUTTI -NG STOCK PROBLEM", COMPUTERS INDUSTRIAL ENGINEER, 32, 115-127, (1997) [31]KIRKPATRICK, S., C.D. GELATT. TR., AND M.P. VECCHI, "OPTIMIZATION BY SIMULATED ANNEAL -ING", SCI., 22, 671-680(1983) [32]KROGER, B., "GUILLOTINEABLE BIN PACKING: A GENETIC APPROACH", EUROPEAN JOURNAL OF OPER -ATIONAL RESEARCH, 84, 645-661, (1996) [33]LAI, K. K. AND J. XUE, "CONTAINER PACKING IN A MULTI-CUSTOMER DELIVERING OPERATION", COMPUTERS INDUSTRIAL ENGINEER, 35, 323- 326, (1998) [34]LAMOUSIN H. J., W. N. WAGGENSPACK, JR. AND G. T.
DOBSON,"NESTING OF COMPLEX 2-D PARTS WITHIN IRREGULAR BOUNDARIES", JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING, 118, 615- 622, (1996) [35]LUTFIYTA, H. AND B. MCMILLIN, "COMPOSITE STOCK CUTTING THROUGH SIMULATED ANNEALING", MATHEMATICAL COMPUTING AND MODELING, 16, 57-74, (1992) [36]MORABITO, R.
N. AND M. N. ARENALES, "AN AND-OR-GRAPH APPROACH FOR TWO-DIMENSIONAL CUTT -ING STOCK PROBLEMS", EUROPEAN JOURNAL OF OPERATIONAL RESEARCH, 58, 263-271, (1992) [37]RICHARD V., "RANDOM SEARCH IN THE ONE-DIMENSIONAL CUTTING STOCK PROBLEM", EUROPEAN JOURNAL OF OPERATIONAL RESEARCH, 95, 191-200, (1996) [38]SARKER, B. R., "AN OPTIMAL SOLUTION FOR ONE-DIMENSIONAL SLITTING PROBLEMS: A DYNAMIC PROGRAMMING APPROACH", JOURNAL OF THE OPERATIONAL RESEARCH SOCIETY, 39, 749-755, (1988) [39]SILVANO M. AND D. VIGO,
"EXACT SOLUTION OF THE TWO-DIMENSIONAL FINITE BIN PACKING PROB -LEM", MANAGEMENT SCIENCE, 44, 388-399, (1998) [40]S. K. CHENG AND K. P. RAO, "LARGE-SCALE NESTING OF IRREGULAR PATTERNS USING COMPACT
NEIGHBORHOOD ALGORITHM", MATERIALS PROCESSING TECHNOLOGY, 103, 135-140, (2000) [41]THEODORACATOS, V. E.
AND J. L. GRIMSLEY, "THE OPTIMAL PACKING OF ARBITRARILY-SHAPED POLYGONS USING SIMULATED ANNEALING AND POLYNOMIAL-TIME COOLING SCHEDULES", JOURNAL OF THE OPERATIONAL RESEARCH SOCIETY, 125, 53-70, (1995) [42]VASSILIOS E. T. AND J. L. GRIMSLEY, "THE OPTIMAL PACKING OF ARBITRARILY-SHAPED POLYGON -S USING
SIMULATED ANNEALING AND POLYNOMIAL-TIME COOLING SCHEDULES", COMPUTER METHODS IN APPLIED
MECHANICS AND ENGINEERING, 125, 53-70, (1995) [43]VICTOR P., M. SEPULVEDA, M. SOLAR AND A. GOMES, "SOLUTION FOR THE CONSTRAINED GUILLOT -INE CUTTING PROBLEM BY SIMULATED ANNEALING", COMPUTERS OPERATIONS RESEARCH, 25, 37-47 , (1998) [44]WASCHER, G., "AN LP BASED APPROACH TO CUTTING STOCK PROBLEM WITH MULTIPLE OBJECTIVES" , EUROPEAN JOURNAL OF OPERATIONAL RESEARCH, 84, 522-538, (1995) [45]YU G.S., M.V. NOVOZHILOVA AND A.V. KARTASHOV, "MATHEMATICAL MODEL AND METHOD OF SEARC -HING FOR A LOCAL EXTREME FOR THE
NON-CONVEX ORIENTED POLYGONS ALLOCATION PROBLEM", EUROPEAN JOURNAL OF OPERATIONAL RESEARCH, 92, 193-210, (1996) [46]YU G.S. AND V.N. PATSUK, "A METHOD OF OPTIMAL LATTICE PACKING OF CONGRUENT ORIENTED -POLYGONS IN THE PLANE", EUROPEAN JOURNAL OF OPERATIONAL RESEARCH, 124, 204-216, (2000) [47]Y.K.D.V. PRASAD, S. SOMASUNDARAM AND K.P. RAO, "A SLIDING ALGORITHM FOR OPTIMAL NESTI -NG OF ARBITRARILY SHAPED SHEET METAL BLANKS", INTERNATIONAL JOURNAL OF PRODUCTION RE -SEARCH, 33, 1505-1520, (1995)
