39:
31:
113:. In that problem, the dimensions of the small rectangles are not fixed in advance. The challenge comes from the fact that the original sheet might not be rectangular - it can be any rectilinear polygon. In particular, it might contain holes (representing defects in the raw material). The optimization goal is usually to minimize the number of small rectangles, or minimize the total length of the cuts.
1206:. Each arc in this graph is colored in one of two colors: "horizontal" or "vertical". Each monochromatic directed cycle in this graph corresponds to a build. By repeatedly contracting monochromatic cycles, one can recover a recursive build-sequence that represents a cutting-pattern class. Every guillotine-graph contains between
362:. Every pattern can be represented as a recursive sequence of builds. Every recursive sequence of builds corresponds to many different patterns, which have an equivalent combinatorial structure; the set of all patterns corresponding to the same recursive-build is called a
940:) guillotine-cutting problem, the required output is a sequence of guillotine cuts producing pieces of the target dimensions, such that the total area of the produced pieces is maximized (equivalently, the waste from the raw rectangle is minimized).
1299:: it generates approximately-optimal solutions in a given amount of time, and then improves it if the user allows more time. The program was used by a specialty paper producer, and has cut the time required for sheet layout while reducing waste.
1078:
In the 2-staged variant, a further distinction is whether all strips resulting from the first stage are cut in the same locations (called "1-group") or on two different locations (called "2-group") or in possibly different locations (called
1258:. In all approaches, it is important to find good lower and upper bounds in order to trim the search-space efficiently. These bounds often come from solutions to related variants, e.g. unconstrained, staged, and non-guillotine variants.
1754:
91:
related to guillotine cutting, such as: maximize the total area of the produced pieces, or their total value; minimize the amount of waste (unused parts) of the large sheet, or the total number of sheets. They have been studied in
1043:
problem, the large rectangle has infinite height (but a fixed width), and the goal is to cut a single rectangle of each type, such that the total material used (equivalently, the total height) is minimized. It is a variant of the
881:
If, after merging, there are at least two disjoint horizontal intervals, then a vertical guillotine cut is possible; if there are at least two disjoint vertical intervals, then a horizontal cut is possible; otherwise, no cut is
3162:
M. Hifi, R. M’Hallah and T. Saadi, Approximate and exact algorithms for the double-constrained two-dimensional guillotine cutting stock problem. Computational
Optimization and Applications, Volume 42, Number 2 (2009), 303-326,
1447:
958:. The goal is then to maximize the total value of the produced pieces. The unweighted (waste-minimization) variant can be reduced to the weighted variant by letting the value of each target-dimension equal to its area (
1461:
Move the grid uniformly at random. Each object is intersected by a horizontal line with probability 1/4 and with a vertical line with probability 1/4 too, so the expected number of intersected objects is
370:
Some problems accept additional inputs, as explained below. The goal is to cut, from the raw rectangle, some smaller rectangles having the target dimensions. The following assumptions are often made:
1862:
897:
When the algorithm returns "yes", it also produces a sequence of guillotine cuts; when it returns "no", it also produces specific subsets of rectangles that cannot be separated by guillotine cuts.
1058:, there is an infinite number of stock sheets of the same dimensions, and the goal is to cut all required target rectangles using the smallest possible number of sheets. It is a variant of the
837:) can be separated by a horizontal cut. All conditions together imply that, if any set of adjacent rectangles contains more than one element, then they can be separated by some guillotine cut.
1639:
472:(there is a single rectangle of each target-dimension). The goal is to decide whether this pattern can be implemented using only guillotine cuts, and if so, find a sequence of such cuts.
1238:
review over 90 papers dealing with unstaged constrained guillotine-cutting (with quantity upper-bounds), both weighted and unweighted. There are two main approaches for exact solutions:
2681:
2271:
577:) contains the indices of all rectangles whose bottom-left corner is in the rectangle X . A cutting pattern is a guillotine pattern if and only if, for all quadruplets of indices
423:. This does not lose much generality, since the variant in which rectangles can be rotated, can be reduced to the non-rotatable variant by adding the rotated patterns explicitly.
2864:
2768:
2170:
1787:
2811:
subsets (called "levels"), and chooses the level with the largest number of rectangles. Each level can be separated by two guillotine cuts. An improved algorithm can separate
2022:
1971:
3836:
Abou Msabah, Slimane, and Ahmed Riadh Baba-Ali. "A new guillotine placement heuristic combined with an improved genetic algorithm for the orthogonal cutting-stock problem."
2809:
2590:
2366:
2074:
2316:
3056:
2900:
2710:
2619:
2494:
2462:
2103:
1582:
2534:
1552:
3053:
Gerhard Wäscher, Heike Haußner, Holger
Schumann, An improved typology of cutting and packing problems, European Journal of Operational Research 183 (2007) 1109–1130,
1641:
objects. Pick a single object from each cell, and separate it from the other objects in the same cell. The total number of objects separated in this way is at least
847:
Stop when either all subpatterns contain one rectangle (in which case the answer is "yes") or no more guillotine cuts are possible (in which case the answer is "no").
844:
At each iteration, divide a given pattern, containing at least two rectangles, into two disjoint sub-patterns using a guillotine cut, and recurse on each sub-pattern.
2197:
1920:
1893:
1644:
1516:
1488:
72:
industry. Glass sheets are scored along horizontal and vertical lines, and then broken along these lines to obtain smaller panels. It is also useful for cutting
2496:
axes-parallel rectangle, while they do not settle it, they show that, if it is correct, then it implies an O(1) approximation algorithm to the problem of
1356:
301:
refers to constructing a new rectangle by attaching two smaller rectangles. Due to the guillotine constraint, there are only two types of builds: in a
1030:
variant is a constrained variant in which each target dimension must appear at most once (i.e., the upper bound is 1). This case is associated with a
3749:"A Mathematical formulation and a lower bound for the three-dimensional multiple-bin-size bin packing problem (MBSBPP): A Tunisian industrial case"
3282:
34:
A guillotine cutting: an optimised sheet of smaller rectangles which can be divided intact through the correct series of bisecting end-to-end cuts.
1198:
propose an exact algorithm for the decision problem. Their algorithm uses a compact representation of guillotine-cutting-pattern classes, using a
3794:"The constrained two-dimensional guillotine cutting problem with defects: an ILP formulation, a Benders decomposition and a CP-based algorithm"
1319:
in the plane, and the goal is to separate them using a sequence of guillotine cuts. Obviously it may not be possible to separate all of them.
1345:
If all objects are of a similar size, then a constant fraction of them can be separated. Formally, if all objects contain a circle of radius
53:
is the process of producing small rectangular items of fixed dimensions from a given large rectangular sheet, using only guillotine-cuts. A
3857:
184:
is the number of rectangles. Often, it is allowed to have several rectangles of the same dimensions; in this case, the pair of dimensions (
1274:"A Controlled Stability Genetic Algorithm With the New BLF2G Guillotine Placement Heuristic for the Orthogonal Cutting-Stock Problem."
1226:; there is a one-to-one correspondence between such graphs and cutting-pattern classes. They then solve the optimization problem using
1109:
The special case in which there is only one type (i.e., all target rectangles are identical and in the same orientation) is called the
479:
present a stronger condition, which is both necessary and sufficient. The input rectangles are ordered from left to right, such that
809:) can be separated by a vertical cut (going between the two disjoint horizontal intervals); condition 3 implies the rectangles in E(
3617:
Abed, Fidaa; Chalermsook, Parinya; Correa, José; Karrenbauer, Andreas; Pérez-Lantero, Pablo; Soto, José A.; Wiese, Andreas (2015).
3558:
Problem presented at ACCOTA '96, Combinatorial and
Computational Aspects of Optimization Topology and Algebra, Taxco, Mexico 1996
1036:, where the goal is to decide whether it is possible to produce a single element of each target dimension using guillotine cuts.
3054:
3768:
3684:
3638:
2972:
1264:"A new guillotine placement heuristic combined with an improved genetic algorithm for the orthogonal cutting-stock problem."
17:
1794:
1075:
stages: in the first stage, only horizontal cuts are made; in the second stage, only vertical cuts are made; and so on.
3546:
Michael L. McHale, Roshan P. Shah
Cutting the Guillotine Down to Size. PC AI magazine, Volume 13, Number 1 Jan/Feb 99.
3702:"Three-dimensional guillotine cutting problems with constrained patterns: MILP formulations and a bottom-up algorithm"
1146:
for both staged and unstaged guillotine cutting. However, it was later shown that both algorithms contained errors.
1587:
901:
61:) is a straight bisecting line going from one edge of an existing rectangle to the opposite edge, similarly to a
2627:
1323:
asked whether it is possible to separate a constant fraction of them, that is, whether there exists a constant
3182:"New reduction procedures and lower bounds for the two-dimensional bin packing problem with fixed orientation"
2933:
3663:
Approximation, Randomization, and
Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)
2951:
Guillotine-cutting when the raw rectangle may have defects, but the produced rectangles must be defect-free.
2209:
1175:
497:
on the indices such that, with this permutation, the rectangles would be ordered from bottom to top, i.e.,
3270:
2814:
2718:
900:
The necessary and sufficient condition can be used to present the guillotine-strip-cutting problem as a
3852:
2123:
1759:
1120:
However, when there are two or more types, all optimization problems related to guillotine cutting are
3700:
Martin, Mateus; Oliveira, José Fernando; Silva, Elsa; Morabito, Reinaldo; Munari, Pedro (2020-11-08).
1983:
1932:
3701:
3181:
3122:
374:
All cuts have zero width. This does not lose much generality, since if each cut has a fixed width of
212:, is an arrangement of small rectangles on the stock sheet. It may be given as a sequence of points (
2773:
2554:
2325:
2034:
2920:
axes-parallel squares with weights, at least a fraction 1/80 of the total weight can be separated.
2281:
1129:
3380:
3078:
2876:
2686:
2595:
2470:
2423:
2079:
3509:
3224:"Constrained two-dimensional guillotine cutting problem: upper-bound review and categorization"
1557:
1227:
874:
Merge overlapping horizontal intervals, and merge overlapping vertical intervals. This takes O(
101:
93:
3461:
3341:
2503:
1521:
1184:
propose an algorithm for the doubly-constrained guillotine-cutting problem. It is a bottom-up
1087:
is a restricted variant of guillotine-cutting in which each cut separates a single rectangle.
3223:
2870:
2113:
1320:
1047:
921:
2206:
An analogous theorem is valid in higher dimensions too: the number of separable objects is
1749:{\displaystyle {\frac {n}{2}}/{\frac {(8R)^{2}}{\pi r^{2}}}={\frac {\pi r^{2}}{128R^{2}}}n.}
3675:
2497:
2175:
1898:
1871:
475:
An obvious necessary condition is that no two input rectangles overlap in both dimensions.
109:
88:
121:
The following terms and notations are often used in the literature on guillotine cutting.
8:
2928:
1493:
1465:
1219:
1140:
1061:
97:
3792:
Martin, Mateus; Hokama, Pedro H. D. B.; Morabito, Reinaldo; Munari, Pedro (2020-05-02).
3629:
2409:
axes-parallel squares with weights, at least 4/729 of the total weight can be separated.
3838:
2011 IEEE International
Conference on Industrial Engineering and Engineering Management
3821:
3774:
3753:
2014 International
Conference on Control, Decision and Information Technologies (CoDIT)
3729:
3599:
3251:
3036:
2024:
can be separated. They conjecture that the worst case can be attained by line segments.
1266:
2011 IEEE International
Conference on Industrial Engineering and Engineering Management
3658:
3657:
Khan, Arindam; Pittu, Madhusudhan Reddy (2020). Byrka, Jaros\law; Meka, Raghu (eds.).
3618:
3825:
3813:
3764:
3733:
3721:
3680:
3634:
3591:
3529:
3481:
3442:
3400:
3361:
3322:
3255:
3243:
3201:
3142:
3098:
3028:
1296:
1189:
929:
133:, is the raw rectangular sheet which should be cut. It is characterized by its width
3778:
3498:
O. B. G. Masden (1980), IMSOR working paper, Technical university of
Denmark, Lyngby
3040:
3805:
3756:
3713:
3670:
3624:
3603:
3581:
3521:
3473:
3434:
3392:
3353:
3318:
3314:
3235:
3193:
3164:
3134:
3090:
3020:
1185:
1071:
is a restricted variant of guillotine cutting where the cutting is made in at most
1032:
62:
3809:
3669:. Dagstuhl, Germany: Schloss Dagstuhl–Leibniz-Zentrum für Informatik: 47:1–47:22.
3302:
3060:
1442:{\displaystyle {\frac {\pi r^{2}}{128R^{2}}}n\approx {\frac {1}{40.7(R/r)^{2}}}n}
1125:
158:, are the required outputs of the cutting. They are characterized by their width
3793:
1791:
If the objects are straight line-segments, then in some instances, only at most
1276:
International
Journal of Cognitive Informatics and Natural Intelligence (IJCINI)
378:>0, then the problem can be reduced to the zero-width variant by just adding
3760:
3748:
3717:
3547:
3422:
3138:
1199:
3197:
3168:
3846:
3817:
3725:
3595:
3533:
3485:
3446:
3404:
3365:
3326:
3247:
3205:
3146:
3102:
3032:
1316:
38:
3396:
3094:
3586:
3525:
1246:(branch-and-bound). The tree-search approaches are further categorized as
3569:
3477:
3357:
3222:
Russo, Mauro; Boccia, Maurizio; Sforza, Antonio; Sterle, Claudio (2020).
3024:
1161:
925:
30:
3438:
3121:
Ben Messaoud, Said; Chu, Chengbin; Espinouse, Marie-Laure (2008-11-16).
1094:
is a 1-simple pattern such that each part is itself a 1-simple pattern.
3510:"Two Algorithms for Constrained Two-Dimensional Cutting Stock Problems"
3303:"A Polynomial Time Algorithm For The Guillotine Pallet Loading Problem"
3239:
3008:
2973:"Challenge ROADEF / EURO 2018 Cutting Optimization Problem Description"
859:
horizontal intervals, and order them from left to right; determine the
3423:"Recursive Computational Procedure for Two-dimensional Stock Cutting"
2373:
Abed, Chalermsook, Correa, Karrenbauer, Perez-Lantero, Soto and Wiese
1143:
81:
863:
vertical intervals, and order them from bottom to top. This takes O(
3180:
Carlier, Jacques; Clautiaux, François; Moukrim, Aziz (2007-08-01).
3077:
Clautiaux, François; Jouglet, Antoine; Moukrim, Aziz (2011-10-17).
1864:
of them can be separated. Particularly, for every positive integer
1222:. Sorting the vertices according to this circuit makes the graph a
3079:"A New Graph-Theoretical Model for the Guillotine-Cutting Problem"
3009:"Algorithms for Unconstrained Two-Dimensional Guillotine Cutting"
2322:
sides overall, then the separating lines can be computed in time
1121:
851:
Finding a guillotine cut for a given pattern is done as follows:
3616:
932:
problems, where the cuts are constrained to be guillotine cuts.
27:
Process of producing small rectangular items of fixed dimensions
1292:
890:-1 times. Therefore, the run-time of the entire algorithm is O(
3342:"Multistage Cutting Stock Problems of Two and More Dimensions"
2387:/2 can be separated, and there are instances in which at most
605:, at least one of the following conditions is fulfilled for E(
46:
be separated by making single bisecting cuts across the plane.
3665:. Leibniz International Proceedings in Informatics (LIPIcs).
886:
The ordering step is done once, and the merging step is done
73:
69:
1272:
Abou-Msabah, Slimane, Ahmed-Riadh Baba-Ali, and Basma Sager.
436:, there is a cutting-pattern given as a sequence of points (
3699:
3301:
Tarnowski, A. G.; Terno, J.; Scheithauer, G. (1994-11-01).
2278:
All these separable subfamilies can be constructed in time
1490:. Therefore, there exist grid-lines that intersect at most
1311:
is a related problem in which the input is a collection of
996:
on the number of pieces that can be produced of that type.
77:
3791:
3123:"Characterization and modelling of guillotine constraints"
840:
This condition can be checked by the following algorithm.
3300:
2770:
rectangles. The algorithm partitions the rectangles into
3120:
2683:
stages are allowed, then it is not possible to separate
2592:
stages are allowed, then it is not possible to separate
3221:
3179:
3076:
2944:
Some recently studied variants of the problem include:
912:
constraints, and is considered not practically useful.
3659:"On Guillotine Separability of Squares and Rectangles"
2467:
Regarding the conjecture that it is possible separate
1857:{\displaystyle O(n^{\log _{3}{2}})\approx O(n^{0.63})}
1756:
A similar argument for the case of unit squares gives
3460:
Christofides, Nicos; Whitlock, Charles (1977-02-01).
2879:
2817:
2776:
2721:
2689:
2630:
2598:
2557:
2506:
2473:
2426:
2328:
2284:
2212:
2178:
2126:
2082:
2037:
1986:
1935:
1901:
1874:
1797:
1762:
1647:
1590:
1560:
1524:
1496:
1468:
1359:
1230:
on the space of well-sorted normal guillotine graphs.
1218:
have the interesting property of containing a unique
915:
3746:
3459:
1214:-2 arcs. A special kind of guillotine graphs called
1117:
present a polynomial-time algorithm for solving it.
1098:-simple cutting patterns can be defined recursively.
3747:Baazaoui, M.; Hanafi, S.; Kamoun, H. (2014-11-01).
3462:"An Algorithm for Two-Dimensional Cutting Problems"
3307:
INFOR: Information Systems and Operational Research
2715:There is a simple 2-stage algorithm that separates
3381:"The Theory and Computation of Knapsack Functions"
3228:International Transactions in Operational Research
2894:
2858:
2803:
2762:
2704:
2675:
2613:
2584:
2528:
2488:
2456:
2360:
2310:
2265:
2191:
2164:
2097:
2068:
2016:
1965:
1914:
1887:
1856:
1781:
1748:
1633:
1576:
1546:
1510:
1482:
1441:
1150:presented a correct dynamic programming algorithm.
3283:Journal of Information Processing and Cybernetics
777:), is made up of at least two disjoint intervals.
721:), is made up of at least two disjoint intervals;
68:Guillotine cutting is particularly common in the
3844:
2105:can be separated; this conjecture is still open.
1262:Abou Msabah, Slimane, and Ahmed Riadh Baba-Ali.
407:The target dimensions cannot be rotated, i.e.,
116:
3378:
3339:
2971:Tlilane, Lydia; Viaud, Quentin (2018-05-18).
1634:{\displaystyle {\frac {(8R)^{2}}{\pi r^{2}}}}
1518:objects. Since the area of each grid cell is
781:Condition 2 implies that the rectangles in E(
468:) is the bottom-left coordinate of rectangle
244:) is the bottom-left coordinate of rectangle
147:, which are the primary inputs to the problem
3798:International Journal of Production Research
3548:http://www.amzi.com/articles/papercutter.htm
3379:Gilmore, P. C.; Gomory, R. E. (1966-12-01).
3340:Gilmore, P. C.; Gomory, R. E. (1965-02-01).
2970:
2120:times the largest contained disc), at least
3268:
3013:Journal of the Operational Research Society
1250:(starting with single rectangles and using
1164:procedures for unstaged guillotine cutting.
1124:. Due to its practical importance, various
427:
42:A non-guillotine cutting: these rectangles
3567:
2624:When the rectangles are weighted, if only
1327:such that, in any such collection of size
1104:
920:These are variants of the two-dimensional
3674:
3656:
3628:
3585:
2676:{\displaystyle o(\log {n}/\log \log {n})}
2116:(the smallest containing disc is at most
80:sheets to make furniture, and cutting of
3127:European Journal of Operational Research
1554:and the area of each object is at least
1353:, then there is a separable set of size
1349:and are contained in a circle of radius
1303:
37:
29:
3427:IBM Journal of Research and Development
3006:
2948:Guillotine-cutting in three dimensions.
2076:can be separated. They conjecture that
14:
3845:
2266:{\displaystyle n/c(R,d)(\log {n})^{d}}
668:The union of the horizontal segments (
3652:
3650:
3416:
3414:
2464:of the total weight can be separated.
2383:axes-parallel unit squares, at least
1895:disjoint intervals such that at most
337:the combined rectangle has width max(
3676:10.4230/LIPIcs.APPROX/RANDOM.2020.47
3507:
3420:
3275:-simple guillotine cutting patterns"
3217:
3215:
3158:
3156:
3116:
3114:
3112:
3072:
3070:
3068:
3002:
3000:
2998:
2996:
2500:of axes-parallel rectangles in time
724:The union of the vertical segments (
477:Ben Messaoud, Chengbin and Espinouse
3858:Optimization algorithms and methods
3630:10.4230/LIPIcs.APPROX-RANDOM.2015.1
3574:Discrete and Computational Geometry
3186:Computers & Operations Research
2318:. If the objects are polygons with
2199:is a constant that depends only on
2031:axes-parallel rectangles, at least
1453:: construct a grid with cell size 8
1224:well-sorted normal guillotine graph
1003:variant, for each target-dimension
985:variant, for each target-dimension
947:variant, for each target dimension
107:A related but different problem is
24:
3647:
3411:
2880:
2859:{\displaystyle n/\log _{3}{(n+2)}}
2763:{\displaystyle n/(1+\log _{2}{n})}
2690:
2599:
2551:axes-parallel rectangles, if only
2474:
2083:
1987:
1936:
1282:
1254:to construct the entire sheet) or
916:Finding an optimal cutting-pattern
25:
3869:
3212:
3153:
3109:
3065:
2993:
2420:-dimensional cubes with weights,
2165:{\displaystyle n/(c_{R}\log {n})}
1980:straight line segments, at least
1782:{\displaystyle {\frac {1}{27}}n.}
1021:on the number of produced pieces.
305:the combined rectangle has width
3706:Expert Systems with Applications
2939:
2909:axes-parallel squares, at least
2398:axes-parallel squares, at least
2017:{\displaystyle \Omega (n^{1/2})}
1966:{\displaystyle \Omega (n^{1/3})}
1236:Russo, Boccia, Sforza and Sterle
1115:Tarnowski, Terno and Scheithauer
3785:
3740:
3693:
3620:On Guillotine Cutting Sequences
3610:
3561:
3552:
3540:
3501:
3492:
3453:
3372:
3333:
3294:
1335:that are guillotine-separable.
665:) contains at most one element;
252:occupies a horizontal segment (
248:. In such a pattern, rectangle
3421:Herz, J. C. (September 1972).
3319:10.1080/03155986.1994.11732257
3262:
3173:
3047:
2964:
2889:
2883:
2852:
2840:
2804:{\displaystyle 1+\log _{2}{n}}
2757:
2730:
2699:
2693:
2670:
2634:
2608:
2602:
2585:{\displaystyle o(\log \log n)}
2579:
2561:
2523:
2510:
2483:
2477:
2449:
2443:
2361:{\displaystyle O(N+n\log {n})}
2355:
2332:
2305:
2288:
2254:
2239:
2236:
2224:
2159:
2135:
2092:
2086:
2063:
2046:
2011:
1990:
1960:
1939:
1851:
1838:
1829:
1801:
1676:
1666:
1604:
1594:
1535:
1525:
1424:
1409:
1196:Clautiaux, Jouglet and Moukrim
13:
1:
3810:10.1080/00207543.2019.1630773
3568:Pach, J.; Tardos, G. (2000).
3269:Scheithauer, Guntram (1993).
3007:Beasley, J. E. (1985-04-01).
2957:
2934:Hyperplane separation theorem
2069:{\displaystyle n/(2\log {n})}
1584:, each cell contains at most
3083:INFORMS Journal on Computing
2311:{\displaystyle O(n\log {n})}
2172:objects can be saved, where
1007:there is both a lower bound
902:mixed integer linear program
434:pattern verification problem
7:
1092:2-simple guillotine cutting
1085:1-simple guillotine cutting
1069:k-staged guillotine cutting
117:Terminology and assumptions
10:
3874:
3761:10.1109/CoDIT.2014.6996896
3718:10.1016/j.eswa.2020.114257
3508:Wang, P. Y. (1983-06-01).
3139:10.1016/j.ejor.2007.08.029
2895:{\displaystyle \Omega (n)}
2705:{\displaystyle \Omega (n)}
2614:{\displaystyle \Omega (n)}
2489:{\displaystyle \Omega (n)}
2457:{\displaystyle 1/2^{O(d)}}
2098:{\displaystyle \Omega (n)}
1331:there is a subset of size
1056:stock minimization problem
989:, there is an upper bound
493:. There is a permutation
273:) and a vertical segment (
3198:10.1016/j.cor.2005.08.012
3169:10.1007/s10589-007-9081-5
1929:convex objects, at least
1922:of them can be separated.
1577:{\displaystyle \pi r^{2}}
1278:13, no. 4 (2019): 91–111.
1182:Hiffi, M'Hallah and Saadi
1158:Christofides and Whitlock
1111:guillotine pallet loading
908:/4 binary variables and 2
3271:"Computation of optimal
2529:{\displaystyle O(n^{5})}
1547:{\displaystyle (8R)^{2}}
1295:program implementing an
1216:normal guillotine graphs
1130:approximation algorithms
1041:guillotine strip cutting
951:, there is also a value
428:Checking a given pattern
415:is not the same type as
364:guillotine-cutting class
1868:, there is a family of
1105:Optimization algorithms
3397:10.1287/opre.14.6.1045
3095:10.1287/ijoc.1110.0478
2896:
2860:
2805:
2764:
2706:
2677:
2615:
2586:
2530:
2490:
2458:
2362:
2312:
2267:
2193:
2166:
2099:
2070:
2018:
1967:
1916:
1889:
1858:
1783:
1750:
1635:
1578:
1548:
1512:
1484:
1443:
1228:constraint programming
102:industrial engineering
94:combinatorial geometry
47:
35:
3587:10.1007/s004540010050
3526:10.1287/opre.31.3.573
2980:Challenge ROADEF/EURO
2916:In any collection of
2913:/40 can be separated.
2905:In any collection of
2897:
2869:In any collection of
2861:
2806:
2765:
2707:
2678:
2616:
2587:
2531:
2491:
2459:
2412:In any collection of
2405:In any collection of
2402:/81 can be separated.
2394:In any collection of
2379:In any collection of
2363:
2313:
2268:
2194:
2192:{\displaystyle c_{R}}
2167:
2108:In any collection of
2100:
2071:
2027:In any collection of
2019:
1976:In any collection of
1968:
1925:In any collection of
1917:
1915:{\displaystyle 2^{k}}
1890:
1888:{\displaystyle 3^{k}}
1859:
1784:
1751:
1636:
1579:
1549:
1513:
1485:
1444:
1321:Jorge Urrutia Galicia
1309:Guillotine separation
1304:Guillotine separation
1048:Strip packing problem
521:. Given four indices
89:optimization problems
41:
33:
18:Guillotine separation
3755:. pp. 219–224.
3478:10.1287/opre.25.1.30
3358:10.1287/opre.13.1.94
3025:10.1057/jors.1985.51
2877:
2815:
2774:
2719:
2687:
2628:
2596:
2555:
2504:
2498:maximum disjoint set
2471:
2424:
2391:/2 can be separated.
2326:
2282:
2210:
2176:
2124:
2080:
2035:
1984:
1933:
1899:
1872:
1795:
1760:
1645:
1588:
1558:
1522:
1494:
1466:
1357:
1176:heuristic algorithms
208:, often called just
198:) is often called a
110:guillotine partition
3514:Operations Research
3466:Operations Research
3439:10.1147/rd.165.0462
3385:Operations Research
3346:Operations Research
2929:Geometric separator
1511:{\displaystyle n/2}
1483:{\displaystyle n/2}
1240:dynamic programming
1220:Hamiltonian circuit
1202:that they call the
1141:dynamic programming
1132:have been devised.
1062:bin-packing problem
1014:and an upper bound
904:. Their model has 3
98:operations research
76:plates, cutting of
3240:10.1111/itor.12687
3059:2016-12-20 at the
2892:
2856:
2801:
2760:
2702:
2673:
2611:
2582:
2526:
2486:
2454:
2358:
2308:
2263:
2189:
2162:
2095:
2066:
2014:
1963:
1912:
1885:
1854:
1779:
1746:
1631:
1574:
1544:
1508:
1480:
1439:
1315:pairwise-disjoint
1137:Gilmore and Gomory
1001:doubly-constrained
450:), for i in 1,...,
226:), for i in 1,...,
129:, also called the
87:There are various
51:Guillotine cutting
48:
36:
3853:Discrete geometry
3770:978-1-4799-6773-5
3686:978-3-95977-164-1
3640:978-3-939897-89-7
3623:. pp. 1–19.
2902:can be separated.
1973:can be separated.
1771:
1738:
1701:
1656:
1629:
1434:
1392:
1297:anytime algorithm
1190:best-first search
930:rectangle packing
206:A cutting-pattern
16:(Redirected from
3865:
3830:
3829:
3804:(9): 2712–2729.
3789:
3783:
3782:
3744:
3738:
3737:
3697:
3691:
3690:
3678:
3654:
3645:
3644:
3632:
3614:
3608:
3607:
3589:
3580:(2–3): 481–496.
3565:
3559:
3556:
3550:
3544:
3538:
3537:
3505:
3499:
3496:
3490:
3489:
3457:
3451:
3450:
3418:
3409:
3408:
3391:(6): 1045–1074.
3376:
3370:
3369:
3337:
3331:
3330:
3298:
3292:
3291:
3279:
3266:
3260:
3259:
3219:
3210:
3209:
3192:(8): 2223–2250.
3177:
3171:
3160:
3151:
3150:
3118:
3107:
3106:
3074:
3063:
3051:
3045:
3044:
3004:
2991:
2990:
2988:
2987:
2977:
2968:
2901:
2899:
2898:
2893:
2865:
2863:
2862:
2857:
2855:
2835:
2834:
2825:
2810:
2808:
2807:
2802:
2800:
2792:
2791:
2769:
2767:
2766:
2761:
2756:
2748:
2747:
2729:
2711:
2709:
2708:
2703:
2682:
2680:
2679:
2674:
2669:
2652:
2647:
2620:
2618:
2617:
2612:
2591:
2589:
2588:
2583:
2535:
2533:
2532:
2527:
2522:
2521:
2495:
2493:
2492:
2487:
2463:
2461:
2460:
2455:
2453:
2452:
2434:
2367:
2365:
2364:
2359:
2354:
2317:
2315:
2314:
2309:
2304:
2272:
2270:
2269:
2264:
2262:
2261:
2252:
2220:
2198:
2196:
2195:
2190:
2188:
2187:
2171:
2169:
2168:
2163:
2158:
2147:
2146:
2134:
2104:
2102:
2101:
2096:
2075:
2073:
2072:
2067:
2062:
2045:
2023:
2021:
2020:
2015:
2010:
2009:
2005:
1972:
1970:
1969:
1964:
1959:
1958:
1954:
1921:
1919:
1918:
1913:
1911:
1910:
1894:
1892:
1891:
1886:
1884:
1883:
1863:
1861:
1860:
1855:
1850:
1849:
1828:
1827:
1826:
1818:
1817:
1788:
1786:
1785:
1780:
1772:
1764:
1755:
1753:
1752:
1747:
1739:
1737:
1736:
1735:
1722:
1721:
1720:
1707:
1702:
1700:
1699:
1698:
1685:
1684:
1683:
1664:
1662:
1657:
1649:
1640:
1638:
1637:
1632:
1630:
1628:
1627:
1626:
1613:
1612:
1611:
1592:
1583:
1581:
1580:
1575:
1573:
1572:
1553:
1551:
1550:
1545:
1543:
1542:
1517:
1515:
1514:
1509:
1504:
1489:
1487:
1486:
1481:
1476:
1448:
1446:
1445:
1440:
1435:
1433:
1432:
1431:
1419:
1401:
1393:
1391:
1390:
1389:
1376:
1375:
1374:
1361:
1204:guillotine graph
1188:algorithm using
1186:branch and bound
1126:exact algorithms
1060:two-dimensional
1046:two-dimensional
1033:decision problem
303:horizontal build
152:small rectangles
63:paper guillotine
59:edge-to-edge cut
57:(also called an
21:
3873:
3872:
3868:
3867:
3866:
3864:
3863:
3862:
3843:
3842:
3834:
3833:
3790:
3786:
3771:
3745:
3741:
3698:
3694:
3687:
3655:
3648:
3641:
3615:
3611:
3570:"Cutting Glass"
3566:
3562:
3557:
3553:
3545:
3541:
3506:
3502:
3497:
3493:
3458:
3454:
3419:
3412:
3377:
3373:
3338:
3334:
3299:
3295:
3277:
3267:
3263:
3220:
3213:
3178:
3174:
3161:
3154:
3119:
3110:
3075:
3066:
3061:Wayback Machine
3052:
3048:
3005:
2994:
2985:
2983:
2975:
2969:
2965:
2960:
2942:
2878:
2875:
2874:
2839:
2830:
2826:
2821:
2816:
2813:
2812:
2796:
2787:
2783:
2775:
2772:
2771:
2752:
2743:
2739:
2725:
2720:
2717:
2716:
2688:
2685:
2684:
2665:
2648:
2643:
2629:
2626:
2625:
2597:
2594:
2593:
2556:
2553:
2552:
2517:
2513:
2505:
2502:
2501:
2472:
2469:
2468:
2439:
2435:
2430:
2425:
2422:
2421:
2350:
2327:
2324:
2323:
2300:
2283:
2280:
2279:
2257:
2253:
2248:
2216:
2211:
2208:
2207:
2183:
2179:
2177:
2174:
2173:
2154:
2142:
2138:
2130:
2125:
2122:
2121:
2081:
2078:
2077:
2058:
2041:
2036:
2033:
2032:
2001:
1997:
1993:
1985:
1982:
1981:
1950:
1946:
1942:
1934:
1931:
1930:
1906:
1902:
1900:
1897:
1896:
1879:
1875:
1873:
1870:
1869:
1845:
1841:
1822:
1813:
1809:
1808:
1804:
1796:
1793:
1792:
1763:
1761:
1758:
1757:
1731:
1727:
1723:
1716:
1712:
1708:
1706:
1694:
1690:
1686:
1679:
1675:
1665:
1663:
1658:
1648:
1646:
1643:
1642:
1622:
1618:
1614:
1607:
1603:
1593:
1591:
1589:
1586:
1585:
1568:
1564:
1559:
1556:
1555:
1538:
1534:
1523:
1520:
1519:
1500:
1495:
1492:
1491:
1472:
1467:
1464:
1463:
1427:
1423:
1415:
1405:
1400:
1385:
1381:
1377:
1370:
1366:
1362:
1360:
1358:
1355:
1354:
1339:Pach and Tardos
1306:
1289:McHale and Shah
1285:
1283:Implementations
1107:
1019:
1012:
994:
976:
970:
963:
956:
918:
836:
829:
822:
815:
808:
801:
794:
787:
776:
769:
762:
755:
743:
736:
729:
720:
713:
706:
699:
687:
680:
673:
664:
657:
650:
643:
632:
625:
618:
611:
604:
597:
590:
583:
576:
569:
562:
555:
548:
541:
534:
527:
520:
512:
506:
502:
492:
485:
466:
459:
448:
441:
430:
394:
387:
360:
353:
346:
342:
331:
324:
319:and height max(
317:
310:
292:
285:
278:
271:
264:
257:
242:
235:
224:
217:
196:
189:
170:
163:
146:
139:
127:large rectangle
119:
28:
23:
22:
15:
12:
11:
5:
3871:
3861:
3860:
3855:
3832:
3831:
3784:
3769:
3739:
3692:
3685:
3646:
3639:
3609:
3560:
3551:
3539:
3520:(3): 573–586.
3500:
3491:
3452:
3433:(5): 462–469.
3410:
3371:
3332:
3313:(4): 275–287.
3293:
3261:
3234:(2): 794–834.
3211:
3172:
3152:
3133:(1): 112–126.
3108:
3064:
3046:
3019:(4): 297–306.
2992:
2962:
2961:
2959:
2956:
2955:
2954:
2949:
2941:
2938:
2937:
2936:
2931:
2922:
2921:
2914:
2903:
2891:
2888:
2885:
2882:
2871:fat rectangles
2867:
2854:
2851:
2848:
2845:
2842:
2838:
2833:
2829:
2824:
2820:
2799:
2795:
2790:
2786:
2782:
2779:
2759:
2755:
2751:
2746:
2742:
2738:
2735:
2732:
2728:
2724:
2713:
2712:of the weight.
2701:
2698:
2695:
2692:
2672:
2668:
2664:
2661:
2658:
2655:
2651:
2646:
2642:
2639:
2636:
2633:
2622:
2610:
2607:
2604:
2601:
2581:
2578:
2575:
2572:
2569:
2566:
2563:
2560:
2541:Khan and Pittu
2538:
2537:
2525:
2520:
2516:
2512:
2509:
2485:
2482:
2479:
2476:
2465:
2451:
2448:
2445:
2442:
2438:
2433:
2429:
2416:axes-parallel
2410:
2403:
2392:
2370:
2369:
2357:
2353:
2349:
2346:
2343:
2340:
2337:
2334:
2331:
2307:
2303:
2299:
2296:
2293:
2290:
2287:
2276:
2275:
2274:
2260:
2256:
2251:
2247:
2244:
2241:
2238:
2235:
2232:
2229:
2226:
2223:
2219:
2215:
2186:
2182:
2161:
2157:
2153:
2150:
2145:
2141:
2137:
2133:
2129:
2106:
2094:
2091:
2088:
2085:
2065:
2061:
2057:
2054:
2051:
2048:
2044:
2040:
2025:
2013:
2008:
2004:
2000:
1996:
1992:
1989:
1974:
1962:
1957:
1953:
1949:
1945:
1941:
1938:
1923:
1909:
1905:
1882:
1878:
1853:
1848:
1844:
1840:
1837:
1834:
1831:
1825:
1821:
1816:
1812:
1807:
1803:
1800:
1789:
1778:
1775:
1770:
1767:
1745:
1742:
1734:
1730:
1726:
1719:
1715:
1711:
1705:
1697:
1693:
1689:
1682:
1678:
1674:
1671:
1668:
1661:
1655:
1652:
1625:
1621:
1617:
1610:
1606:
1602:
1599:
1596:
1571:
1567:
1563:
1541:
1537:
1533:
1530:
1527:
1507:
1503:
1499:
1479:
1475:
1471:
1438:
1430:
1426:
1422:
1418:
1414:
1411:
1408:
1404:
1399:
1396:
1388:
1384:
1380:
1373:
1369:
1365:
1317:convex objects
1305:
1302:
1301:
1300:
1284:
1281:
1280:
1279:
1269:
1259:
1233:
1231:
1200:directed graph
1193:
1179:
1165:
1151:
1106:
1103:
1102:
1101:
1100:
1099:
1082:
1081:
1080:
1066:
1052:
1037:
1024:
1023:
1022:
1017:
1010:
992:
979:
974:
968:
961:
954:
941:
936:In the basic (
917:
914:
884:
883:
879:
872:
855:Determine the
849:
848:
845:
834:
827:
820:
813:
806:
799:
792:
785:
779:
778:
774:
767:
760:
753:
741:
734:
727:
722:
718:
711:
704:
697:
685:
678:
671:
666:
662:
655:
648:
641:
630:
623:
616:
609:
602:
595:
588:
581:
574:
567:
560:
553:
546:
539:
532:
525:
514:
510:
504:
500:
490:
483:
464:
457:
446:
439:
429:
426:
425:
424:
405:
392:
385:
368:
367:
358:
351:
344:
340:
335:vertical build
329:
322:
315:
308:
295:
290:
283:
276:
269:
262:
255:
240:
233:
222:
215:
203:
194:
187:
168:
161:
154:, also called
148:
144:
137:
118:
115:
55:guillotine-cut
26:
9:
6:
4:
3:
2:
3870:
3859:
3856:
3854:
3851:
3850:
3848:
3841:
3840:. IEEE, 2011.
3839:
3827:
3823:
3819:
3815:
3811:
3807:
3803:
3799:
3795:
3788:
3780:
3776:
3772:
3766:
3762:
3758:
3754:
3750:
3743:
3735:
3731:
3727:
3723:
3719:
3715:
3711:
3707:
3703:
3696:
3688:
3682:
3677:
3672:
3668:
3664:
3660:
3653:
3651:
3642:
3636:
3631:
3626:
3622:
3621:
3613:
3605:
3601:
3597:
3593:
3588:
3583:
3579:
3575:
3571:
3564:
3555:
3549:
3543:
3535:
3531:
3527:
3523:
3519:
3515:
3511:
3504:
3495:
3487:
3483:
3479:
3475:
3471:
3467:
3463:
3456:
3448:
3444:
3440:
3436:
3432:
3428:
3424:
3417:
3415:
3406:
3402:
3398:
3394:
3390:
3386:
3382:
3375:
3367:
3363:
3359:
3355:
3352:(1): 94–120.
3351:
3347:
3343:
3336:
3328:
3324:
3320:
3316:
3312:
3308:
3304:
3297:
3290:(2): 115–128.
3289:
3285:
3284:
3276:
3274:
3265:
3257:
3253:
3249:
3245:
3241:
3237:
3233:
3229:
3225:
3218:
3216:
3207:
3203:
3199:
3195:
3191:
3187:
3183:
3176:
3170:
3166:
3159:
3157:
3148:
3144:
3140:
3136:
3132:
3128:
3124:
3117:
3115:
3113:
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2945:
2940:More variants
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2984:. Retrieved
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1147:
1139:presented a
1136:
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86:
84:into boxes.
67:
58:
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50:
49:
43:
2866:rectangles.
2621:rectangles.
2114:fat objects
1244:tree-search
1162:tree-search
983:constrained
926:bin packing
165:and height
140:and height
131:stock sheet
3847:Categories
3712:: 114257.
2986:2019-06-13
2958:References
2924:See also:
1174:presented
1160:presented
938:unweighted
3826:197434029
3818:0020-7543
3734:228839108
3726:0957-4174
3596:0179-5376
3534:0030-364X
3486:0030-364X
3447:0018-8646
3405:0030-364X
3366:0030-364X
3327:0315-5986
3256:195551953
3248:1475-3995
3206:0305-0548
3147:0377-2217
3103:1091-9856
3033:0160-5682
2881:Ω
2837:
2794:
2750:
2691:Ω
2663:
2657:
2641:
2600:Ω
2574:
2568:
2475:Ω
2348:
2298:
2246:
2152:
2084:Ω
2056:
1988:Ω
1937:Ω
1833:≈
1820:
1710:π
1688:π
1616:π
1562:π
1398:≈
1364:π
1248:bottom-up
1144:recursion
1113:problem.
882:possible.
454:, where (
400:in 0,...,
230:, where (
176:in 1,...,
172:and for
82:cardboard
3779:18598442
3057:Archived
3041:58059319
2982:. ROADEF
2543:proved:
2375:proved:
1341:proved:
1291:wrote a
1256:top-down
1079:"free").
945:weighted
507:≤ ... ≤
486:≤ ... ≤
333:); in a
180:, where
3604:1737527
1148:Beasley
1122:NP hard
1054:In the
1039:In the
999:In the
981:In the
943:In the
878:) time.
871:) time.
432:In the
210:pattern
3824:
3816:
3777:
3767:
3732:
3724:
3683:
3637:
3602:
3594:
3532:
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3445:
3403:
3364:
3325:
3254:
3246:
3204:
3145:
3101:
3039:
3031:
1293:Prolog
1252:builds
1168:Masden
1028:binary
44:cannot
3822:S2CID
3775:S2CID
3730:S2CID
3600:S2CID
3278:(PDF)
3252:S2CID
3037:S2CID
2976:(PDF)
2547:With
1451:Proof
1210:and 2
749:in E(
693:in E(
299:build
156:items
74:steel
70:glass
3814:ISSN
3765:ISBN
3722:ISSN
3681:ISBN
3635:ISBN
3592:ISSN
3530:ISSN
3482:ISSN
3443:ISSN
3401:ISSN
3362:ISSN
3323:ISSN
3244:ISSN
3202:ISSN
3143:ISSN
3099:ISSN
3029:ISSN
1847:0.63
1457:by 8
1407:40.7
1242:and
1172:Wang
1170:and
1156:and
1154:Herz
1128:and
1026:The
928:and
867:log
591:and
535:and
419:-by-
411:-by-
396:for
389:and
200:type
150:The
125:The
100:and
78:wood
3806:doi
3757:doi
3714:doi
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3021:doi
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1449:.
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376:d
366:.
359:j
357:h
355:+
352:i
350:h
345:j
341:i
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330:j
328:h
326:,
323:i
321:h
316:j
314:w
312:+
309:i
307:w
291:i
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287:+
284:i
282:y
277:i
275:y
270:i
268:w
266:+
263:i
261:x
256:i
254:x
250:i
246:i
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239:y
237:,
234:i
232:x
228:m
223:i
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219:,
216:i
214:x
202:.
195:i
193:h
191:,
188:i
186:w
182:m
178:m
174:i
169:i
167:h
162:i
160:w
145:0
142:H
138:0
135:W
20:)
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