Division of Research School of Business Administration The University of Michigan CUITING AND PACKING PROBLEM: AN UPDATED LITERATURE REVIEW Working Paper #654 Paul E. Sweeney Elizabeth Ridenour Paternoster The University of Michigan FOR DISCUSSION PURPOSES ONLY None of this material is to be quoted or reproduced without the expressed permission of the Division of Research Copyright 1991 The University of Michigan School of Business Administration Ann Arbor, Michigan 48109-1234 May 1991

Cutting and Packing Problems: An Updated Literature Review by Paul E. Sweeney and Elizabeth Ridenour Paternoster Abstract More than 400 books, articles, dissertations, and working papers are included in this bibliography on cutting and packing problems. The list of published articles and dissertations is a nearly exhaustive compilation of the English language references on the subject. The list of working papers, although extensive, presents only a sample of the recent work in this field. All of the books and journal articles listed in the bibliography have been categorized according to the dimensionality of the problems studied and the solution methodologies employed. Key Words: Cutting Stock Problems Trim Loss Problems Packing Bin Packing Linear Programming Applications Heuristic Problem Solving

Cutting and Packing Problems: An Updated Literature Review by Paul E. Sweeney and Elizabeth Ridenour Paternoster Introduction Although several English language articles on cutting and packing problems appeared in the 1950's [2-12], Gilmore and Gomory's later articles on linear programming approaches to one-dimensional cutting stock problems [15, 20, 24, and 30] were the first to present techniques which could be practically applied to difficult real-world problems. Since then, articles have been published every year on the original onedimensional problem and, increasingly, on its two- and three-dimensional extensions (see Figure 1). This bibliography has been compiled in response to the increasing diversity and complexity of research on these problems. Its purposes are to: 1) provide the most exhaustive, categorized bibliography ever published on these problems, 2) provide a listing of the large number of articles published since the last similar bibliography appeared (see Dyckhoff, et.al. [251]), 3) facilitate access to publications pertaining to distinct problem types, and 4) facilitate access to publications pertaining to distinct solution methodologies. We assume the reader is familiar with cutting and packing problems, but those who require a basic introduction to the field should refer to the following:

Brown [52] Salkin and de Kluyver [80] Golden [88] Berge [119] Gilmore [124] Hinxman [147] Garey and Johnson [162] Israni and Sanders [187] Sarin [210] Coffman, Garey, and Johnson [220] Dowsland [250] Dyckhoff, Kruse, Abel, and Gal [251] Dyckhoff [347] Because this bibliography is concerned with application-oriented research, some articles of a more theoretical nature have been intentionally omitted. That is particularly true for articles on knapsack and bin packing algorithms, where a great deal of research has been done on computational complexity and worst-case analysis (see Johnson, [68]). To find other articles on those topics, refer to the literature reviews cited above (i.e., see [80], [162], or [220]). Oraanization More than 400 books, articles, dissertations, and working papers are included in this bibliography on cutting and packing problems. The list of published articles and dissertations is a nearly exhaustive compilation of the English language references on the subject. The list of working papers, although extensive, presents only a sample of the recent work in this field. While the task of categorizing such a large number of apparently diverse publications is formidable, cutting and packing problems do share a common underlying structure which can be exploited in their classification. Both types of problems focus on the division of some stock resource into smaller pieces (e.g., demanded items) in a way which minimizes the volume or quantity of the resource required. In traditional cutting stock problems, the resource is generally some type

Figure 1 Number of Publications by Year and by Dimension of Problem* 140 T 120 100 NUMBER OF CITATIONS 80 60 * ONE-DIMENSIONAL EO TWO-DIMENSIONAL I THREE-DIMENSIONAL 40 20 0 I THRU 1960 1961 - 1970 1971 - 1981 - 1980 1990 -- *Publications which cover more than one problem type are double counted.

of physical material (e.g., a large roll of paper or sheet of glass). For packing problems, the resource is an empty space (e.g., a pallet or truck). In either case, classification of the problems according to the dimensionality of the decision regarding resource division is relatively straightforward, and we have followed that convention in our categorization. In addition to physical dimensionality, we have also grouped the publications according to the solution methodologies employed. Three categories have been used: - sequential assignment heuristics - single-pattern generating procedures - multiple-pattern generating procedures Sequential heuristic techniques group demanded items into a cutting or packing pattern by using an assignment rule (e.g., first-fit decreasing in bin packing). These procedures (e.g., dynamic programming based algorithms) generate a single "optimal" pattern that can be used once or many times, but they do not consider subsequent patterns for residual demand items. Multiple pattern procedures (e.g., LP-based algorithms) consider interactions between patterns, but they require solutions to be rounded and are, therefore, also heuristic in nature. By combining these two methods of categorization, we have generated a 3 x 3 matrix to which the publications have been assigned (see Table 1). Articles which discuss more than one kind of problem may appear in more than one cell. Such will be the case, for example, with articles on multi-dimensional problems which are approached by solving one- or twodimensional sub-problems. Also, publications often cannot be clearly categorized as belonging to only one of the above three groups of solution procedures. All such articles will either be placed where they fit best or, in some cases, will appear more than once in our categorization.

The citations in Table 1 refer to the entries in the main bibliography which immediately follows it. The bibliography is arranged chronologically by year and alphabetically by author within each year. Immediately following the it are two supplementary lists. The first contains dissertations and theses. The second provides a list of recent (1986 to 1990) proceedings, presentations, and working papers. The entries in the supplementary bibliographies have not been categorized with the other books and journal articles because of the inherent difficulties in obtaining and reviewing these more informal works in process. Special Topics For those with an interest in various special topics in cutting and packing, the citations listed below refer to the main bibliography which follows Table 1: Assortment Problems: 12-13, 20, 25-27, 42, 52, 70, 78, 85, 93, 95-96, 104, 109, 138, 147, 187, 217, 222, 230, 243, 253, 259-260, 274, 297, 311, 314, 327, 332, 339, 347-348, 352 Corrugator Trim Problems: 21, 24, 36-37, 39, 52, 106, 117, 202, 253, 262, 306, 322 Cutting or Packing Items with Irregular Shapes: 49, 56-57, 67, 75, 79, 83, 97, 120, 125, 129, 132, 135, 140, 187, 210, 215, 223, 234, 239, 251, 293, 303, 310, 329, 344, 347 Cutting Material with Defects or Quality Variations: 24, 43, 47, 49, 51, 59, 98, 173, 206, 228, 233, 236, 263, 270, 316, 318, 342, 357, 359 Multi-Stage Problems: 24, 29, 43, 49, 52, 55, 59, 66, 78, 82-83, 85, 87, 99, 101, 107, 124, 130, 140, 146-147, 150, 167, 187, 194, 199-200, 206, 210, 216, 245, 250-251, 253, 279, 297, 305-306, 314, 317-318, 335, 337-339, 348-350, 355-357

Rounding: 3-4, 6, 15, 29, 41, 46, 59, 76, 89, 94, 115, 119, 124, 138, 145, 147, 160, 184, 192, 194, 231, 251, 253, 256, 269, 271, 273, 292, 305, 318, 347, 353, 356 1-1/2 Dimensional Problems: 24, 36-37, 39, 47, 51-52, 61, 101, 113, 115, 117, 128, 130, 136, 141, 148, 155, 162, 178, 181, 195-197, 220, 247, 250 -251, 262, 264, 282, 312, 321, 325-327, 336, 347 4 or More Dimensional Problems: 3, 9, 41, 52, 54, 59, 80, 103, 105, 111, 147, 158, 162, 177, 220-221, 291, 311, 347 Acknowled ament a We would like to thank Professor Harald Dyckhoff (RWTH Aachen, Germany) for his comments on an early version of this bibliography and acknowledge the influence of his ideas on our categorization approach. We would also like to bring to your attentioon the existence of a Special Interest Group on Cutting and Packing Problems (SICUP) which was formed in conjunction with the EURO IX / TIMS XXVIII Joint International Conference (Paris, July 1988). For information regarding this organization, contact either the authors or Professor Dyckhoff.

Table 1 Categorization of Articles by Problem Dimension and Solution Approach PROBLEM DIMENSION One Two Three SOLUTION APPROACH: Sequential Heuristics 45, 52, 54, 58, 62, 68-69, 73, 88, 103, 105, 111-113, 137, 142-143, 156, 162, 168, 171-172, 187, 193, 198, 208, 220 -221, 227, 230, 233, 238, 247, 264-268, 274, 284, 290, 292, 300-301, 306-307, 315, 327-328, 331, 347, 350, 356-357 Single Pattern Oriented 3, 12, 14, 26-28, 41, 44, 50, 52, 54, 60, 70-71, 77, 80, 88, 91, 93, 95-96, 100, 102, 104, 108 -110, 116, 121, 124, 131, 137, 147, 151, 157, 169, 173, 176, 187, 207, 218, 233, 236-237, 241, 251, 257, 260, 270, 274, 288, 306, 347, 352 Multiple Pattern Oriented 2-8, 10-11, 13, 15 -18, 20-23, 29, 34 -35, 38, 40, 42, 46, 52, 55, 61, 64, 76, 85-90, 94, 102, 107, 109, 115, 123-124, 126-128, 131, 137, 145-147, 160, 164 -165, 174-175, 185, 187-188, 192, i94, 201-202, 205, 213, 217-218, 231, 233, 251-253, 256, 259, 269, 271-274, 276 -278, 280, 288, 302, 304-306, 308, 312, 316, 321, 347, 353, 356-357 47, 52, 54, 84, 103, 106, 111, 120, 129, 132, 136, 139, 141, 143, 148-149, 153, 155, 162-163, 178, 181-182, 187, 189, 195-198, 208, 220 -221, 230, 247, 250, 254-255, 258, 262 -263, 268, 274, 281, 283, 289, 293, 295, 298, 313-314, 322, 324-326, 332, 335 -336, 343-347, 354 1, 19, 25, 33, 41, 43, 49, 52, 54, 57, 59, 65, 67, 72, 74 -75, 78-80, 82-83, 97-99, 114, 118-119, 122, 125, 133-135, 138, 140, 147, 154, 159, 161, 166, 177, 179, 183, 186-187, 190-191, 204, 206, 210, 212, 214-216, 219, 223-225, 228, 234, 240, 242-246, 248-251, 258, 260 -261, 274-275, 286 -287, 310, 314, 318 -319, 329-330, 335, 337-338, 340, 344, 346-348, 352 16, 24, 30, 32, 36 -38, 48, 51-53, 56, 63, 66, 81, 101, 117, 130, 147, 150, 152, 167, 170, 174, 180, 184, 187, 199-200, 203, 209-211, 222, 226, 229, 232, 235, 239, 250-251, 274, 279, 282, 285, 294, 296, 299, 303, 309, 317, 319, 323, 342, 347-349, 355, 358-359 54, 84, 103, 106, 111, 144, 162, 187, 220 -221, 250, 253, 291, 311, 320, 324, 333-334, 341, 347, 351, 354 1, 9, 41, 54, 65, 80, 92, 122, 215, 228, 236, 250, 310, 319, 329-330, 333, 347 24, 31, 158, 170, 180, 250, 297, 299, 319, 339, 347

Bibliography for Cutting and Packing Problems (Arranged chronologically by year and alphabetically therein by author) 1940 1. Brooks, R.L., C.A.B. Smith, A.H. Stone, and W.T. Tutte, "The Dissection of Rectangles into Squares," Duke Mathematical Journal, V.7 (1940), pp. 312-340. 1956 2. Paull, A.E., "Linear Programming: A Key to Optimum Newsprint Production," Pulp and Paper Magazine of Canada, V.57 No.4 (March 1956), pp. 145-150. 1957 3. Dantzig, G.B., "Discrete-Variable Extremum Problems," Operations Research, V.5 No.2 (April 1957), pp. 266-277. 4. Eisemann, K., "The Trim Problem," Management Science, V.3 No.3 (April 1957), pp. 279-284. 5. Herrmann, K. and S. Vaida, "Linear Programming in Trim Calculation," The World's Paper Trade Review, June 13, 1957, pp. 1928-1930. 1958 6. Metzger, R.W., "Stock Slitting," Elementary Mathematical Programming (Chapter 8, pp. 200-210). New York: Wiley, 1958. 7. Vajda, S., "Trim Loss Reduction," Readings in Linear Programming (Chapter 21, pp. 78-84). New York: Wiley, 1958. 8. Wolfe, H.B., "Operations Research and How It Can be Applied in the Paper Industry," Paper Trade Journal, V.142 No.38 (September 22, 1958), pp. 42-47. 1959 9. Coxeter, H.S.M., L. Few, and C.A. Rogers, "Covering Space with Equal Spheres," Mathematica, V.6 (1959), pp.147-157. 10. Goldberg, B.A., "Computer Applications in the Paper Industry," Paper Trade Journal, V.143 No.14 (April 6, 1959), pp. 48-50. 11. Johnston, F.J. and T. Pethica, "Optimisation Methods in the Paper Industry," The Paper-Maker and British Paper Trade Journal, V.137 No.3 (March 1959), pp. 68-70. 12. Sadowski, W., "A Few Remarks on the Assortment Problem," Management Science, V.6 (1959), pp. 13-24.

1960 13. Eilon, S., "Optimizing the Shearing of Steel Bars," Journal of Mechanical Engineering Science, V.2 No.2 (1960), pp. 129-142. 14. Kantorovich, L.V., "Mathematical Methods of Organizing and Planning Production," Management Science, V.6 No.4 (July 1960), pp. 366-422. 1961 15. Gilmore, P.C. and R.E. Gomory, "A Linear Programming Approach to the Cutting Stock Problem," Operations Research, V.9 No.6 (November-December 1961), pp. 849-859. 16. Morgan, J.I., "Survey of Operations Research," Paper Mill News, V.84 (March 13, 1961), pp. 10-11, 20. 17. Smith, G.W. and C. Harrell, "Linear Programming in Log Production," Forest Products Journal, V.11 (January 1961), pp. 8-11. 18. Van Duyne, R.W., "Linear Programming and the Paper Industry," TAPPI, V.44 No.5 (May 1961), pp. 189A-193A. 1962 19. Pandit, S.N.N., "The Loading Problem," Operations Research, V.10 (1962), pp. 639-646. 1963 20. Gilmore, P.C. and R.E. Gomory, "A Linear Programming Approach to the Cutting Stock Problem - Part II," Operations Research, V.11 No.6 (November-December 1963), pp. 863-888. 21. Mahoney, D. and J.L. Marley, "Can Cutting Trim Waste Cut into Your Profits? Parts 1, 2, and 3," Paperboard Packaging, October 1963 and November 1963. 22. Van Den Meerendok, H.W., J.A.G.M. Kerbosch, P. Medema, and J.H. Schouten, "Some Computational Aspects of a Trimloss Problem,"Statistica Nederlandica, V.17 No.1 (1963), pp.37-47. 1964 23. Pierce, J.F., Some Large Scale Production Scheduling Problems in the Paper Industry. Englewood Cliffs, N.J.: Prentice Hall, 1964. 1965 24. Gilmore, P.C. and R.E. Gomory, "Multistage Cutting Stock Problems of Two and More Dimensions," Operations Research, V.13 No.1 (January-February 1965), pp. 94-120. 25. Wilson, R.C., "A Packaging Problem," Management Science, V.12 No.4 (December 1965), pp. B135-B145.

26. Wolfson, M.L., "Selecting the Best Lengths to Stock," Operations Research, V.13 (1965), pp. 570-585. 1966 27. Cohen, G.D., "Comments on a Paper by M.L. Wolfson: Selecting the Best Lengths to Stock," Operations Research, V.14 (1966), p. 341. 28. Eilon, S. and R.V. Mallya, "A Note on the Optimal Division of a Container into Two Compartments," International Journal of Production Research, V.5 No.2 (1966), pp. 163-169. 29. Gl more, P.C., "The Cutting Stock Problem," IBM Proceedings on Combinatorial Problems (1966), pp.211-224. 30. ---------—. and R.E. Gomory, "The Theory and Computation of Knapsack Functions," Operations Research, V.14 No.6 (NovemberDecember 1966), pp. 1045-1074. 31. Kortanek, K. and D. Sodaro, "A Generalized Network Model for Three-Dimensional Cutting Stock Problems and New Product Analysis," Journal of Industrial Engineering, V.17 (November 1966), pp. 572-576. 32. Noel, E., "Programming of Cultting Operations for Sheet Material," U.S. Patent #3,242,573 (March 29, 1966). 1967 33. Barnett, S. and G.J. Kynch, "Exact Solution of a Simple Cutting Stock Problem," Operations Research, V.15 (1967), pp. 1051-1056. 34. Hodges, R.R., "The Trim Problem Reconsidered," Operations Research and the Design of Management Information Systems (Editor - J.F. Pierce; Chapter 18, pp. 355-359). New York: TAPPI, 1967. 35. Johns, E.C., "Heuristic Procedures for Solving the Paper Trim Problem," Operations Research and the Design of Management Information Systems (Editor - J.F. Pierce; Chapter 20, pp. 361 -369). New York: TAPPI, 1967. 36. Pegels, C.C., "Heuristic Scheduling Models for Variants of the Two-Dimensional Cutting-Stock Problem," TAPPI, V.50 No.11 (November 1967), pp. 532-535. 37. ---------—, "A Comparison of Scheduling Models for Corrugator Production," Journal of Industrial Engineering, V.18 No.8 (August 1967), pp. 466-472. 38. Pierce, J.F., "Trimming in Practice: Discussion," Operations Research and the Design of Management Information Systems (Editor - J.F. Pierce, Chapter 20). New York: TAPPI, 1967. 39. Poirier, C.C., "Automated Data Processing for the Corrugated Box Plant," Operations Research and the Design of Management Information Systems (Editor - J.F. Pierce, Chapter 24, pp. 416 -432). New York: TAPPI, 1967.

40. Struve, D.L., "Paper Machine Production Allocation by Linear Programming," Operations Research and the Design of Management Information Systems (Editor - J.F. Pierce, Chapter 16, pp. 293 -307). New York: TAPPI, 1967. 41. Weingartner, H.M. and D.N. Ness, "Methods for the Solution of the Multi-Dimensional 0/1 Knapsack Problem," Operations Research, V.15 No.1 (January-February 1967), pp. 83-103. 1968 42. Elmaghraby, S.E., "A Loading Problem in Process Type Production," Operations Research, V.16 (1968), pp. 902-914. 43. Hahn, S.G., "On the Optimal Cutting of Defective Sheets," Operations Research, V.16 No.6 (November-December 1968), pp. 1100-1114. 44. Pierce, J.F., "Application of Combinatorial Programming to a Class of All-Zero-One Integer Programming Problems," Management Science, V.15 No.3 (November 1968), pp. 191-209. 1969 45. Brown, A.R., "Selling Television Time: An Optimisation Problem," The Computer Journal, V.12 (1969), pp. 201-207. 1970 46. Beged Dov, A.G., "Some Computational Aspects of the M Paper Mills and P Printers Paper Trim Problem," Journal of Business Administration, V.1 No.2 (Winter 1969/ 1970), pp. 15-34. 47. Brichard, E., J.A. Valembois, and A. Raes, "Method for Optimally Cutting Successive Panels of Pieces from a Sheet or Strip," U.S. Patent #3,490,147 (January 20, 1970). 48. Filmer, P.J., "A Sheet Scheduling Model for a Digital Computer," Appita, V.24 No.3 (November 1970), pp. 189-195. 49. Haims, M.J. and H. Freeman, "A Multistage Solution of the Template-Layout Problem," IEEE Transactions on Systems Science and Cybernetics, SSC-6 No.2 (April 1970), pp.145-151. 50. Pierce, J.F., "Pattern Sequencing and Matching in Stock Cutting Operations," TAPPI, V.53 No.4 (April 1970), pp. 668-678. 51. Valembois, J.A. and J-M Couvreur, "Method for Obtaining Patterns for Cutting Pieces Out of Sheets or Strips," U.S. Patent #3,490,320 (January 20, 1970) 1971 52. Brown, A.R., Optimum Packing and Depletion: The Computer in Space- and Resource-Usage Problems. New York: American Elsevier Inc., 1971. 53. Eheling, C.W., "Pallet Pattern by Computer," Modern Packaging (May 1971), pp. 66-67, 70.

54. Eon, S. and N. Christofides, "The Loading Problem," Management Science, V.17 No.5 (January 1971), pp. 259-268. 55. Haessler, R.W., "A Heuristic Programming Solution to a Nonlinear Cutting Stock Problem," Management Science, V.17 No.12 (August 1971), pp. B793-B802. 56. Peleg, K. and S. Orlowski, "Optimisation of Citrus Packaging," Packaging Technology (1971), p.17. 1972 57. Adamowicz, M. and A. Albano, "A Two-Stage Solution of the Cutting Stock Problem," Information Processing 71 (Editor - C.V. Freiman; pp. 1086-1091). Amsterdam: North-Holland, 1972. 58. Greenberg, I., "Application of the Loading Algorithm to Balance Workloads," AIIE Transactions, V.4 No.4 (December 1972), pp. 337-339. 59. Herz, J.C., "Recursive Computational Procedure for TwoDimensional Stock Cutting," IBM Journal of Research and Development, V.16 (September 1972), pp. 462-469. 60. Pnevmaticos, S.M. and S.H. Mann, "Dynamic Programming in Tree Bucking," Forest Products Journal, V.22 No.2 (February 1972), pp. 26-30. 1973 61. Caruso, F.A. and J.J. Kokat, "Coil Slitting - A Shop Floor Solution," IE, V.5 (November 1973), pp. 18-23. 62. Garey, M.R., R.L. Graham, and J.D. Ullman, "An Analysis of Some Packing Algorithms," Combinatorial Algorithms (Editor - R. Rustin; pp.39-47). New York: Algorithmic Press, 1973. 63. Gribov, A.B., "Algorithm for Solving the Problem of a Plane Cutting Layout," Cybernetics, V.9 (1973), pp. 1036-1043. 64. Johnston, R.E. and S.B. Bourke, "The Development of Computer Programmes for Reel Deckle Filling," Appita, V.26 No.6 (May 1973), pp. 444-448. 1974 65. Brualdi, R.A. and T.H. Foregger, "Packing Boxes with Harmonic Bricks," Journal of Combinatorial Theory, V.B17 (1974), pp. 81 -114. 66. Dyson, R.G. and A.S. Gregory, "The Cutting Stock Problem in the Flat Glass Industry," Operational Research Quarterly, V.25 No.1 (1974), pp. 41-53. 67. Freeman, H., "Computer Processing of Line-Drawing Images," Computing Surveys, V.6 No.1 (March 1974), pp. 57-97. 68. Johnson, D.S., "Fast Algorithms for Bin Packing," Journal of Computer and System Sciences, V.8 No.3 (June 1974), pp. 272-314.

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83. ------------- and -------—, "Nesting Two-Dimensional Shapes in Rectangular Modules," Computer Aided Design, V.8 No. 1 (January 1976), pp. 27-33. 84. Akeda, Y. and M. Hori, "On Random Sequential Packing in Two and Three Dimensions," Biometrika, V.63 No.2 (1976), pp. 361-366. 85. Chambers, M.L. and R.G. Dyson, "The Cutting Stock Problem in the Flat Glass Industry - Selection of Stock Sizes," Operational Research Quarterly, V.27 No. 4 (1976), pp. 949-957. 86. Coffield, D.R. and R.M. Crisp Jr., "Extensions to the Coil Slitting Problem," International Journal of Production Research, V.14 No.5 (September 1976), pp. 625-630. 87. Coverdale, I.L. and F. Wharton, "An Improved Heuristic Procedure for a Nonlinear Cutting Stock Problem," Management Science, V.23 No.1 (September 1976), pp. 78-86. 88. Golden, B.L., "Approaches to the Cutting Stock Problem," AIIE Transactions, V.8 No.2 (June 1976), pp. 265-274. 89. Haessler, R.W., "Single-Machine Roll Trim Problems and Solution Procedures," TAPPI, V.59 No.2 (February 1976), pp. 145-149. 90. Hardley, C.J., "Optimal Cutting of Zinc-Coated Steel Strip," New Zealand Operational Research, V.4 No.2 (July 1976), pp. 92-100. 91. Nauss, R.M., "An Efficient Algorithm for the 0-1 Knapsack Problem," Management Science, V.23 No.1 (September 1976), pp. 27-31. 92. Peleg, K. and E. Peleg, "Container Dimensions for Optimal Utilization of Storage and Transportation Space," Computer Aided Design, V.8 No.3 (1976), pp. 775-781. 93. Pentico, D.W., "The Assortment Problem with Non-Linear Cost Functions," Operations Research, V.24 No.6 (November-December 1976), pp. 1129-1142. 94. Rao, M.R., "On the Cutting Stock Problem," Journal of the Computer Society of India, V.7 No.1 (December 1976), pp. 35-39. 95. Walters, J.R., "An Extension of the Assortment Problem in which Components Are Grouped to Form Assemblies," Operational Research Quarterly, V.27 No.1 (1976), pp. 169-175. 96. -----------—, "Approximate Stock Formulae for Use in Conjunction with the Assortment Problem," Operational Research Quarterly, V.27 No.4 (1976), pp. 801-804. 1977 97. Albano, A., "A Method to Improve Two-Dimensional Layout," Computer Aided Design, V.9 (1977), pp. 48-52. 98. Cheng, R.M.H. and A.W. Pila, "Maximized Utilization of Surface Areas with Defects by the Dynamic Programming Approach," ISA Transactions, V.17 No.4 (1977), pp. 61-69.

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336. Rhee, W.T. and M. Talagrand, "Optimal Bin Packing with Items of Random Sizes II," SIAM Journal on Computing, V.18 No.l (February 1989), pp. 139-151. 337. Vasko, F.J., "A Computational Improvement to Wang's TwoDimensional Cutting Stock Algoritlhm," Computers and Industrial Engineering, V.16 No.1 (1989), pp. 109-115. 338. ---------—, F.E. Wolf, and K.L. Stott, "A Practical Solution to a Fuzzy Two-Dimensional Cutting Stock Problem," Fuzzy Sets and Systems (pp. 259-275). North-Holland, 1989. 339. ---------- --- ---- and J.W. Scheirer, "Selecting Optimal Ingot Sizes for Bethlehem Steel," Interfaces, V.19 No.l (January-February 1989), pp. 68-84. 1990 340. Anderson, R., S. Kahan, and M. Schlac, "An O(nlogn) Algorithm for 1-D Tile Compaction," Lecture Notes in Computer Science (Editor - M. Nagl; pp. 287-301). Berlin: Springer-Verlag, 1990. 341. Bischoff, E.E. and M.D. Marriott, "A Comparative Evaluation of Heuristics for Container Loading," European Journal of Operational Research, V.44 No.2 (January 25, 1990), pp. 267-276. 342. Bobrowski, P.M., "Branch-and-Bound Strategies for the Log Bucking Problem," Decision Sciences, V.21 No.4 (Winter 1990), pp. 1-13. 343. Coffman, E.G., Jr. and P.W. Shor, "Average-Case Analysis of Cutting and Packing in Two Dimensions," European Journal of Operational Research, V.44 No.2 (January 25, 1990), pp. 134-144. 344. Dagli, C.H., "Knowledge-Based Systems for Cutting Stock Problems," European Journal of Operational Research, V.44 No.2 (January 25, 1990), pp. 160-166. 345. Daniels, J.J. and P. Ghandforoush, "An Improved Algorithm for the Non-Guillotine-Constrained Cutting Stock Problem," Journal of the Operational Research Society, V.41 No.2 (1990), pp. 141 -149. 346. Dowsland, K.A., "Efficient Automated Pallet Loading," European Journal of Operational Research, V.44 No.2 (January 25, 1990), pp. 232-238. 347. Dyckhoff, H., "A Typology of Cutting and Packing Problems," European Journal of Operational Research, V.44 No.2 (January 25, 1990), pp. 145-159. 348. Farley, A.A., "Selection of Stockplate Characteristics and Cutting Style for Two Dimensional Cutting Stock Situations," European Journal of Operational Research, V.44 No.2 (January 25, 1990), pp. 239-246. 349. --------—, "The Cutting Stock Problem in the Canvas Industry," European Journal of Operational Research, V.44 No.2 (January 25, 1990), pp. 247-255.

350. Ferreira, J.S., M.A. Neves, and P. Fonseca e Castro, "A TwoPhase Roll Cutting Problem," European Journal of Operational Research, V.44 No.2 (January 25, 1990), pp. 185-196. 351. Gehring, H., K. Menschner, and M. Meyer, "A Computer-Based Heuristic for Packing Pooled Shipment Containers," European Journal of Operational Research, V.44 No.2 (January 25, 1990), pp. 277-288. 352. Gemmill, D.D. and J.L. Sanders, "Approximate Solutions for the Cutting Stock 'Portfolio' Problem," European Journal of Operational Research, V.44 No.2 (January 25, 1990), pp. 167-174. 353. Golilimis, C., "Optimal Solutions for the Cutting Stock Problem," European Journal of Operational Research, V.44 No.2 (January 25, 1990), pp. 197-208. 354. Haessler, R.W. and F.B. Talbot, "Load Planning for Shipments of Low Density Products," European Journal of Operational Research, V.44 No.2 (January 25, 1990), pp. 289-299. 355. Oliveira, J.F. and J.S. Ferreira, "An Improved Version of Wang's Algorithm for Two-Dimensional Cutting Problems," European Journal of Operational Research, V.44 No.2 (January 25, 1990), pp. 256-266. 356. Stadtler, H., "A One-Dimensional Cutting Stock Problem in the Aluminum Industry and Its Solution," European Journal of Operational Research, V.44 No.2 (January 25, 1990), pp. 209-223. 357. Sweeney, P.E. and R.W. Haessler, "One-Dimensional Cutting Stock Decisions for Rolls with Multiple Quality Grades," European Journal of Operational Research, V.44 No.2 (January 25, 1990), pp. 224-231. 358. Wascher, G., "An LP-Based Approach to Cutting Stock Problems with Multiple Objectives," European Journal of Operational Research, V.44 No.2 (January 25, 1990), pp. 175-184. 359. Whitaker, D. and S. Cammell, "A Partitioned Cutting-Stock Problem Applied in the Meat Industry," Journal of the Operational Research Society, V.41 No.9 (September 1990), pp. 801-807.

Supplementary Bibliography of Dissertations and Theses Burkhalter, R., "An Investigation of Packing with Emphasis on the 2-Dimensional Pattern Cutter's Problem," Doctoral Dissertation (University of Michigan), 1964. Coverdale, I.L., "A Pattern Enumeration Approach to the Trim Loss Problem," Doctoral Dissertation (University of Hull, England), 1975. De Cani, P., "Packing Problems in Theory and Practice," Doctoral Dissertation (University of Birmingham, U.K.), 1979. Dowsland, K.A., "Two-Dimensional Rectangular Packing," Master's Thesis (University of Wales), 1982. ---------—, "Graph Theory and O.R. - An Exact Solution to the Pallet Loading Problem," Doctoral Dissertation (University of Swansea, U.K.), 1985. Dyson, R.G., "Computable Models for Planning and Controlling Warehousing Operations in the Glass Industry," Doctoral Dissertation (University of Lancaster), 1969. Eastman, W.L., "Linear Programming with Pattern Constraints," Doctoral Dissertation (Harvard University), 1958. Eng, G., "A Method for Forest Outturn Assessment and Optimal Tree Bucking," Master's Thesis (University of Canterbury), 1982. Farley, A.A., "Cutting Knife Limitations in Cutting Stock Problems," Master's Thesis (Monash University, Australia), 1979. ------, "Algorithms for Solving Cutting Stock Problems," Doctoral Dissertation (Monash University, Australia), 1986. Haessler, R.W., "An Application of Heuristic Programming to a Nonlinear Cutting-Stock Problem Occurring in the Paper Industry," Doctoral Dissertation (University of Michigan), 1968. Haims, M.JT., "On the Optimum Two-Dimensional Allocation Problem," Doctoral Dissertation (New York University), 1966. Hinxman, A.I., "The Use of Geometric Information in Heuristic Optimization," Doctoral Dissertation (University of Stirling), 1977. Johnson, D.S., "Near Optimal Bin Packing Algorithms," Doctoral Dissertation (MIT), 1973. Leung, J.Y.T., "Fast Algorithms for Packing Problems," Doctoral Dissertation (Pennsylvania State University), 1977. Mannchen, K., "Solution Methods for Two and Three Dimensional Packing Problems," Doctoral Dissertation (Universitat Karlsruhe), 1989.

Pentico, D.W., "Extensions of the Assortment Problem," Doctoral Dissertation (Carnegie Mellon University), 1971. Robinson, T.F., "Rectangular Stock Cutting Using Optimisation," Doctoral Dissertation (University of Canterbury, New Zealand), 1989. Scarborough, S.G., "The Single Dimensional Cutting Problem," Master's Thesis (Imperial College, London), 1982. Sweeney, P.E., "Alternative Manufacturing Infrastructures for a Special Class of Arborescent Production Processes," Doctoral Dissertation (University of Michigan), 1988. Van Wormer, T., "The Trimmer: A Heuristic Solution to the Trim Problem in the Corrugated Container Industry," Doctoral Dissertation (Carnegie Institute of Technology), 1963. Vonderembse, M.A., "An Investigation of Design and Operating Decisions for Continuous Casting in the Steel Industry," Doctoral Dissertation (University of Michigan), 1979. Winter, J.L., "A Class of Nunmber and Location Optimization Problems, with Applications in Inventory and Catalog Problems, Doctoral Dissertation (Arizona State University), 1979.

Supplementary Bibliography of Recent Proceedings, Presentations, and Working Papers (1986 - 1990) Alves, J.C., J.S. Ferreira, and J.F. Oliveira, "Solving TwoDimensional Cutting Problems and Comparing Well-Known Algorithms," presented at EURO IX / TIMS XXVIII Joint International Conference (Paris, July 1988). Bagchi, A. and K.V. Viswanathan, "An Exact Best-First Search Procedure for the Constrained Rectangular Guillotine Knapsack Problem," Proceedings of the American Association for Artificial Intelligence (1988). Biro, M., "Graphical Representations in DSS and Resource Constrained Scheduling," Computer and Automation Institute, Hungarian Academy of Sciences (1989). Bischoff, E.E., "Interactive Approaches to Packing Problems," presented at 10th International Conference on Production Research (Nottingham, August 14-18, 1989). ----------, "Stability Aspects of Packing Problems," presented at IFORS '90 12th Triennial Conference on Operations Research (Athens, June 26-29, 1990). ---- --—, K.A. Dowsland, and W.B. Dowsland, "Packing Problems and Solution Approaches," presented at IFORS '90 12th Triennial Conference on Operations Research (Athens, June 26-29, 1990). Blazewicz, J., M. Drozdowski, B. Soniewicki, and R. Walkowiak, "Decision Support for Cutting a Irregular Shapes-Implementation and Experimental Comparison," Technical University of Poznan, Poland (November 1990). —. —..-. ---., -.-. --- —. ---- ------------- and ----------—, "Two-Dimensional Cutting Problem-Basic Complexity Results and Algorithms for Irregular Shapes," Technical University of Poznan, Poland (November 1990). -----------—, P. Hawryluk, and R. Walkowiak, "Tabu Search Approach for Solving Two-Dimensional Irregular Cutting Problem," Technical University of Poznan, Poland (November 1990). Boros, E., "On Rectangular Subdivisions," presented at EURO IX / TIMS XXVIII Joint International Conference (Paris, July 1988). Chu, C.B., M-C Portmann. and F. Vernadat, "Flexible Method for Container Loading: First Step Towards an Expert System," presented at EURO X, The 10th European Conference on Operational Research (June 27-30, 1989). Csirik, J., J.B.G. Frenk, M. Labbe, and S. Zhang, "Fast Algorithms for Dual Bin Packing," abstract in SICUP Bulletin (April 1990). -----, ------------ ----- and ------—, "Heuristics for the 0-1 Min-Knapsack Problem," abstract in SICUP Bulletin (April 1990).

. --- ——. and E. Mate, "The Analysis of the Next-k Fit Bin Packing Heuristic," presented at EURO IX / TIMS XXVIII Joint International Conference (Paris, July 1988). Dagli, C., "Cutting Stock Problem: Combined Use of Heuristics and Optimization Methods," Ninth International Conference on Production Research (1987), pp. 843-849. -------—, "Feasibility of Integrating AI and Computer Graphics in Solving Cutting Stock Problems," presented at EURO IX / TIMS XXVIII Joint International Conference (Paris, July 1988). -------—, "An Intelligent System for Reducing Lead Time in Stock Cutting," presented at 10th International Conference on Production Research (Nottingham, August 14-18, 1989). -------—, "An Approach for a Neural Network Solution to Two Dimensional Stock Cutting Problem," presented at IFORS '90 12th Triennial Conference on Operations Research (Athens, June 26-29, 1990). de Kovel, W.F., "An Algorithm for the Loading of Containers with Block-Shaped Parcels of Arbitrary Sizes," DrNeher Laboratories of the Netherlands; Postal and Communications Services (PTT), 1990. Diegel, A., "Integer LP Solution for Large Trim Problems," presented at EURO IX / TIMS XXVIII Joint International Conference (Paris, July 1988)..-.-. --- —, "Critical Sizes in Integer Trim Problems," presented at Fourth South African Pulp and Paper Technical Conference (South Africa, 1988). -----, "Critical Aspects of Integer Trim Problems," presented at 10th International Conference on Production Research (Nottingham, August 14-18, 1989). -------—. "Integer LP Solution for Full-Reel Sheet Cutting Problems," abstract in SICUP Bulletin (May 1989). --------—, "Absolute Integer Solution for Very Large Trim Problems," presented at IFORS '90 12th Triennial Conference on Operations Research (Athens, June 26-29, 1990). Dowsland, K.A., "Some Experiments with Simulated Annealing Techniques for Packing Problems," presented at IFORS '90 12th Triennial Conference on Operations Research (Athens, June 26-29, 1990). ------------- and W.B. Dowsland, "State of the Art Survey - Packing Problems," abstract in SICUP Bulletin (November 1990). Dowsland, W.B., "Computer Aided Pack Design," Presented at Paper Industry Research Association Conference (October 1986). ------------—, "Computational Considerations in Packing Algorithm Development," presented at EURO IX / TIMS XXVIII Joint International Conference (Paris, July 1988). ---------- "Intelligent Solutions to Three-Dimensional Packing Problems," presented at 10th International Conference on Production Research (Nottingham, August 14-18, 1989).

----- ---—. "Improved Solutions to Three-Dimensional Packing Problems," presented at IFORS '90 12th Triennial Conference on Operations Research (Athens, June 26-29, 1990). Dyckhoff, H., "Approaches to Cutting and Packing Problems," presented at 10th International Conference on Production Research (Nottingham, August 14-18, 1989). ---------- "Production Theoretic Foundation of Cutting and Related Problems," abstract in SICUP Bulletin (May 1989) ----------—, "Bridges between Two Principal Model Formulations for Cutting Stock Processes," abstract in SICUP Bulletin (November 1990). ------, U. Finke, and H-J Kruse, "Standard Software for Cutting Stock Management," abstract in SICUP Bulletin (May 1989). -. --- —-. --- and H. Gehring, "Trim Loss and Inventory Planning in a Small Textile Firm," abstract in SICUP Bulletin (May 1989). -------- H-J Kruse, D. Abel, and T. Gal, "Classification of Real World Trim Loss Problems," abstract in SICUP Bulletin (May 1989). --- ---- and G. Wascher, "Bibliography of Cutting and Packing," abstract in SICUP Bulletin (December 1989). Erol, D. and I. Kara, "A Heuristic for Two Dimensional Guillotine Cutting Stock Problem," abstract in SICUP Bulletin (November 1990). Farley, A.A., "A Note on Bounding a Class of Linear Programming Problems, Including Cutting Stock Problems," abstract in SICUP Bulletin (December 1989). ----------—, "Limiting the Number of Each Piece in TwoDimensional Cutting Stock Patterns," abstract in SICUP Bulletin (December 1989). ----------—, "Planning the Cutting of Photographic Color Paper," Working Paper, Monash University (Australia, 1990). Galambos, G., "A Linear Time 5/4 Bin-Packing Algorithm," presented at EURO IX / TIMS XXVIII Joint International Conference (Paris, July 1988). Gehring, H., "A Tree-Search Algorithm for Container Packing," presented at IFORS '90 12th Triennial Conference on Operations Research (Athens, June 26-29, 1990). Gerardin, B., M. Gondran, A. Lahrichi, and J.L. Mollieres, "New Methods for the Environment Constrained Two Dimensional Bin Packing Problem," presented at EURO IX / TIMS XXVIII Joint International Conference (Paris, July 1988). Gondran, M. and A. Lahrichi, "The Compulsory Parts Theory and Its Use for Two Dimensional Bin Packing Problem," presented at EURO IX / TIMS XXVIII Joint International Conference (Paris, July 1988).

Haessler, R.W., "Cutting Stock Problems and Solution Approaches," presented at IFORS '90 12th Triennial Conference on Operations Research (Athens, June 26-29, 1990). ---- ---- and P.E. Sweeney, "Evaluating the Impact of Small Changes in Ordered Widths on Trim Yields," Working Paper No. 640, University of Michigan, School of Business Administration (July 1990). -------------- and F.B. Talbot, "Corrugator Combining," presented at EURO IX / TIMS XXVIII Joint International Conference (Paris, July 1988). ------ -- and F.B. Talbot, "A Second Generation Vehicle Planning Algorithm," presented at IFORS '90 12th Triennial Conference on Operations Research (Athens, June 26-29, 1990). Hall, N.G., "The Inventory Packing Problem," presented at the Multi-Echelon Inventory Systems Conference (Ann Arbor, July 1988). Johnson, D.S., "Average Case Analysis of One-Dimensional Packing Algorithms," presented at IFORS '90 12th Triennial Conference on Operations Research (Athens, June 26-29, 1990). Kalpic, D. and V. Mornar, "Two-Stage Two-Dimensional Cutting Stock Problems," presented at EURO X, The 10th European Conference on Operational Research (June 27-30, 1989). Kroger, B. and 0. Vornberger, "A Parallel Branch & Bound Approach for Solving a Two-Dimensional Cutting Stock Problem," presented at IFORS '90 12th Triennial Conference on Operations Research (Athens, June 26-29, 1990). Lahrichi, A., "The Compulsory Parts Theory and Its Use for Scheduling and Bin Packing Problem," presented at EURO IX / TIMS XXVIII Joint International Conference (Paris, July 1988). Lai, K.K. and.T.W.M. Chan, "A Study of the Cutting Stock Problem," abstract in SICUP Bulletin (April 1990). ---- - and -------—, "Two Dimensional Cutting Stock Algorithms: Bin Packing Approach," abstract in SICUP Bulletin (November 1990). Martello, S. and P. Toth, "An Exact Algorithm for the BinPacking Problem," presented at EURO X, The 10th European Conference on Operational Research (June 27-30, 1989). ----------- and ----—, "Knapsack Problems - Algorithms and Computer Implementations," presented at IFORS '90 12th Triennial Conference on Operations Research (Athens, June 26-29, 1990). Oliveira, J.F. and J. Ferreira, "On the Application of the Gilmore and Gomory Approach to the Two-Dimensional Cutting Stock Problem," presented at IFORS '90 12th Triennial Conference on Operations Research (Athens, June 26-29, 1990). Penders, J., Th. Snellen, and J. Look, "Packing a Roller Container - The Frame of an Algorithm," presented at IFORS '90 12th Triennial Conference on Operations Research (Athens, June 26-29, 1990).

Portmann, M-C, "An Efficient Algorithm for Container Loading," abstract in SICUP Bulletin (November 1990). Rodder, W., "Freightage-Optimization for Goods Transportation," presented at IFORS '90 12th Triennial Conference on Operations Research (Athens, June 26-29, 1990). Rode, M., "Interactive Computer-Aided Simulation of TwoDimensional Packing Problems," presented at EURO IX / TIMS XXVIII Joint International Conference (Paris, July 1988). Scheithauer, G. and J. Terno, "An Effective Branch and Bound Approach to the Cutting Stock Problem," Working Paper, Technische Universitat (Dresden, 1987). Seth, A., "Trim Loss Problems with Lexicographic Maximum Objective Function," presented at IFORS '90 12th Triennial Conference on Operations Research (Athens, June 26-29, 1990). Sweeney, P.E., "A Bibliography for Cutting Stock Problems," presented at EURO IX / TIMS XXVIII Joint International Conference (Paris, July 1988). —, ---- "Cutting Stock Decisions for Rolls with Multiple Quality Grades: Results from an Implemented Model," presented at EURO IX / TIMS XXVIII Joint International Conference (Paris, July 1988). —... —.... — and E. Ridenour, "Cutting and Packing Problems:A Categorized, Application-Oriented Research Bibliography," Working Paper No. 610, School of Business Administration, University of Michigan (1989). Tuenter, H., "On the Core-Size of the Unbounded, Uncorrelated Knapsack Problem," abstract in SICUP Bulletin (November 1990). Wascher, G., "Dealing with Multiple Objectives in Cutting Problems," presented at EURO IX / TIMS XXVIII Joint International Conference (Paris, July 1988). Woodyatt, L.R., "Computerized Grade Assignment and Order Consolidation for Improving Caster Utilization," 70th Steelmaking Conference Proceeding (March 1987). Yanassee, H.H., A. Zinober, and R.G. Harris, "Cutting Stock: Board Selection," abstract in SICUP Bulletin (November 1990).