CC BY
27
ISSN 1992-6502 (P ri nt)_
Vol. 17, no. 6 (59), pp. 92-94, 2013.
Vestnik UQA7U
ISSN 2225-2789 (Online) http://journal.ugatu.ac.ru
UDC 004.7
Minimizing cardinality of the set of raster points
for orthogonal packing problem
12 3
V. M. Kartak , A. V. Ripatti , M. A. Mukhacheva
1 kvmail@mail.ru, 2 ripatti@inbox.ru, 3 mukhacheva.marina@gmail.com Ufa State Aviation Technical University, Russia Submitted 2013, May 9
Abstract. We consider the N-dimensional orthogonal packing feasibility problem (OPP-N). Given a set of N-dimensional rectangular items, OOP-N is to decide whether all items can be orthogonally packed into the given rectangular container. We construct a method which finds equivalent packing for given orthogonal packing under certain criterion.
Keywords: orthogonal packing problem; raster points; linear programming.
1. INTRODUCTION
We consider following problem: m N-dimensional parallelepiped items R = {R},
i e {l,..., m} with sizes R = (r/,..., rN ) are to be
packed into a container S = (s,..., SN ) , which is
also N-dimensional orthogonal parallelepiped. The
N-dimensional orthogonal packing feasibility problem (OPP-N) asks whether all the items R can be
orthogonally packed into the container S without rotations. This problem is a main procedure of solving many discrete optimization problems [1,2,3]. Such as cutting and packing problems, resource-constrained project scheduling problem etc. All these problems are NP-hard.
The approach of F. Clautiaux [4] gives on the best computational results at this time. Author investigates the raster model of the orthogonal packing problem, which was represented as integer linear programming problem with pseudopolinomial number of variables. It is noticed, that effectively of algorithm working depends on both the number of considered items and the ratio of their sizes. For example, the solution of the problem with sizes
R1 ={(20,20), (21,21),(22,22),(23,23)} and S = (50,50) from the algorithmic point of view simpler than the solution of the problem with
R1 ={(20,20), (20,20),(20,20),(20,20)} and S = (50,50). It can be explained by the fact that on the first case has many symmetric variants in contrast to the second case.
2. EQUIVALENT SETS
Lets given set (l,L), consist of vector l e R+ and number L e R+. Define the set of binary vectors
P<(l,L) := (a: £aili < L,a e{0,1}n},
P> (l, L) := (a : £ atlt > L, a e (0,1}n }.
Using this definition, we can take the definition of the equivalent sets from [3].
Definition 1. Set (l,L) is equivalent to (l, L) if P< (l, L) = P< (l, L).
Lets known some vector w e R1. Consider the 0-1 knapsack problem, where L is the size
K(l, w, L) = max( £atWi,a e R (l, L)).
The following theorem was proved:
Theorem 1. The set (l, L) is equivalent the set (l, L) if and only if K(l, l, L) < L and K(J, l, L) < L .
Theorem 1 gives simple condition to verify the equivalence of two any sets.
In case, when L = L is equivalent, the set of all possible sets, which is equivalent to (l, L) is represented as:
n(l,L) := (ne R+ : £ nlal < L, £ 7t1a1 > L,
a e P< (l,L), a e P>(l, L)}.
The set n(l, L) is convex polytope and elements of it, may be given with using the methods of
V. M. Kartak, A. V. Ripatti, M. A. Mukhacheva • Minimizing cardinality of the set .
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linear programming generation on the base of the column generation method. Similar conversion for different type of integer programming problem is discussed in [5].
The reduced raster points set
Definition 1. The reduced raster points set for
(/, L) is
{m m i
Yfjaj ■ t/jaj < L, a e {0,1}m k
Example 1.
Let be L = 20 n = 10 l = {4, 5, 6, 7, 7, 8, 11, 13, 13, 14} then R (L,l) = {0, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} and |R(L,l)| = 18.
Using a fitness function to set n(/,L) we can generate an equivalent size with certain properties.
For example, the a fitness functions are follow:
1) k1 = min(max)(Y T ■ t e n(/, L)) is minimal (maximal) total size.
2) k = min(t1 -tn ■ t e n(/, L)) is minimal variance of lengths.
3) k3 =min(L
■ t e n(/, L),teZ" ) is minimal
length of the object L where sizes of items are integer.
Example 2.
Given: L = 20, n = 10, l = { 4, 5, 6, 7, 7, 8, 11, 13, 13, 14 }.
The reduce raster points is R (L, l) = { 0, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 } and |R (L, l)| = 18.
Using the fitness function k 3 , we have a new items size
L = 14, n = 10, l = {3, 3, 4, 5, 5, 6, 8, 9, 9, 10 }.
R (L, l) = {0, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}, |R (L, l)| = 13.
The orthogonal packing problem
Lets T] = (rj ,...,r™ ).
Defenition 2. We will say, that problem (R, S) is equivalent to (R, S), if for i = 1, N the sets (r', St) and (ri, St) is equivalent.
Theorem 2. If problems (R, S ) and (R, S)
is equivalent, then the solution of one of them may be transformed to the solution of another one.
From the theorem 2 follows, that the sets (1) may be used for the constructing of equivalent problems. It allows to choose the most successful
problem statement from the class of equivalence by using the addition of objective function to the restrictions (1). For example, the criterion of choice may have the following representation:
Computational results
For the testing was generated the set of following instances for three-dimensional case (N=3): container is represented by cube with sizes Sj = S2 = S3 = 1000. The number of items m is belong to the set {10, 15}, approximate waste value e is from the {0 %, 2 %, ..., 40 %}. Maximal ratio rmax between dimensions of each item is belong to the set {1, 3, 20}. Obviously, if rmax = 1, then we have a set of cubes. For each class (m, e, rmax) was generated ten instances. For more detail of generation method see [1].
Table 1. Minimizing the number of the raster points (k3)
Type of tests Number of inst. Num. of inst where min RRP Number of RRP before / after
Claut. 672 56 1.01
LCPS 1260 24 1.001
n = 10 900 850 1.447
n = 15 900 6 1.001
RRP is reduced raster points.
For generated test set was obtained the following results.
1. The main aim for the first criterion (the maximization of the total sum of elements in each dimension) was getting the equivalent problem with the bigger total volume of items.
The lower bound by volume for the obtained problem was increased (from 0 till 1) for 20.5 percents of tests.
2. The main aim for second criterion (the minimization of variance of items) was getting the equivalent problem with the less number of raster point. (Table 1)
4. CONCLUSION
The method, proposed in this paper, allows building the set of equivalent problem for the given one. Thereby the opportunity of getting the most successful statement of the problem from the equivalent class was obtained. Also the criterion of choice for input data was proposed.
ACKNOWLEDGMENTS
This research has been supported by grant 12-07-00631 "Information systems for solving practical multidimensional orthogonal polyhedron packing problems".
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Special issue: ITIDS + MAAO'2013
REFERENCES
1. G. Belov, V. Kartak, H. Rohling, and G. Scheithauer, "One-dimensional relaxations and LP bounds for orthogonal packing," Int. Trans. Operational Research, vol. 16, no. 6, pp. 745-766, 2009.
2. V. M Kartak, "A orthogonal packing problem: exact algorithm on based matrix model," Vestnik UGATU, vol. 9, no. 4, pp. 104-110, 2007.
3. V. M Kartak, "A new low bound for orthogonal packing problem," Vestnik UGATU, vol. 10, no. 2, pp. 154-158, 2008.
4. F. Clautiaux, A. Jouglet, J. Carlier, and A. Moukrim, "A new constraint programming approach for the orthogonal packing problem," Computers and Operations Research, vol. 35, no. 3, pp. 944-959, 2008.
5. M. Devyaterkina, A. Kolokolov, and A. Kolosov, "An uni-modular transformations for integer programming problem and an analysis of their effectiveness," in Proc. of the Institute of Mathematics and Mechanics, vol. 12, no. 2, pp. 48-62, 2010.
ABOUT AUTHORS
KARTAK, Vadim Mikhaylovich, Head of Department of Applied Informatics, BSPU, Dipl. Eng.-Programmer (UGATU, 1995). Cand. of Phys.- Math. Sci. (BSU, 1999), Dr. of Phys.-Math. Sci. (UGATU, 2012).
RIPATTI, Artem Valeryevich, Grad. Student, Dept. of Computational Mathematic and Cybernetics, Dipl. Mathematician-Programmer (UGATU, 2012).
MUKHACHEVA, Marina Andreevna, Postgrad. (PhD) Student, Dept. of Computational Mathematic and Cybernetics, Dipl. Mathematician-Programmer (UGATU, 2011).
