CC BY
36
ISSN 1992-6502 (P ri nt)______________
Vol. 17, no. 6 (59), pp. 101-102, 2013.
Vestnik UQA7U
ISSN 2225-2789 (Online) http://journal.ugatu.ac.ru
UDC 004.7
Improvements in the APLP3D for
THREE-DIMENSIONAL PACKING PROBLEM OF CYLINDERS AND PARALLELEPIPEDS
I. S. KOSHCHEEV
Ufa State Aviation Technical University, Russia Submitted 2013, July 9
Abstract. The three-dimensional packing problem of cylinders and parallelepipeds into semi-infinite container is discussed in this paper. The focus of the paper is some improvement in the algorithm which was proposed in article [1].
Keywords: APLP-3D; three-dimensional packing problem; packing of cylinders and parallelepipeds.
1. INTRODUCTION
The three-dimensional Bin packing problem has many applications in various branches of human activity. Three-dimensional Bin Packing problem is NP-hard combinatorial optimization problem. There are only two articles about the packing of cylinders and parallelepipeds [1, 2]. In the first article author proposed the exact method. This method can’t be used on the practice, because of a big amount of items it takes a lot of time. The second article is discussed a method on the base of (1+1)EA.
This paper consists of the three sections: introduction, problem statement, approach and conclusions.
2. PROBLEM STATEMENT
We formulate the statement of problem. Given: W is width of container, L is length of container, vector lc=<ri, hci> represents a collection of cylinders, vector Ip=<Wj, lj, hpj> represents a collection of parallelepipeds. Here ri is the radius of the i-th cylinder, hci is the height of i-th cylinder, Wj is the width of j-th parallelepiped, lj is the length of j-th parallelepiped, hpj is the height of j-th parallelepiped, iE[0, Nc] is number of cylinders, jE[0, Np], Np is number of parallelepipeds.
We introduce Cartesian coordinate system centered in the far bottom left corner of the container
(Fig. 1).
Solution of the problem is the set of elements <C, P>, where C=<Xci, yci, Zci> h P = <Xpj, ypj, Zpj>, (xci, yci, zci) - coordinates of the center of the lower base of i-th cylinder, iE[0, Nc], (xpj, ypj, zj - coor-
dinates of lower left far corner of the parallelepiped, jE[0, Np].
/ z
HEIGHT
| WIDTH X
/0(0,0.0)
LENGTH
Fig. 1
Collection <C, P> is called acceptable packing, if the next conditions perform:
1) Orthogonality condition for parallelepipeds and cylinders.
2) Parallelepipeds don’t overlap j, sElp (s^j):
((Xpj > Xps + Ws ) V (Xps > xpj + Wj )) V
((ypj > yps + ls) V (yps > ypj + lj)) V (1)
((zpj > zps + hps ) V (zps > zpj + hpj )).
3) Parallelepipeds don’t go beyond container (jElp):
(Xpj > 0) A (ypj > 0) A (Zpj > 0) A ((Xpj + Wj) < W) A ((ypj + lj ) < L).
4) Cylinders don’t go beyond pallets (jElc):
(Xj > rj ) A (yCj > rj ) A ((XCj + rj ) < W) A
((ycj + rj ) < L) A (Zpj > 0).
(3)
(4)
5) Cylinders and parallelepipeds don’t overlap (jElp, jElc):
(xci + r < xpj) V (xpj + Wj < xc - r) V
(ypj + lj < yci- r) V (yci + ri < ypj)
V (zci+ hci< zpj) V (zpj+ hpj< zcj).
6) Cylinders don’t overlap (i,kElc):
((xci-xc/)2 + (yci-ycf > (rci+ rck )2) V
(zci + hci < zck) V (zck + hck < zci).
(5)
Need to find acceptable packing with minimal height H:
H = max(max(zci + hci | i = 1,...,Nc),
max(zpj + hpj | j = 1,...,Np)). (6)
2. APPROACH
The approach on the base of (1+1)EA algorithm and ABLP3D procedure was proposed in[1]. Here we propose some improvements in it.
Reduce the number of potential positions. After new item is added into container some positions can became unreachable. For example if first item in container is parallelepiped there is no sense to consider positions 1 and 2 (Fig. 2 two-dimensional case is presented) and compute new item coordinates for them.
Fig. 2
There are several cases when we should eliminate some positions for the list of potential positions:
1) When added object has common faces with faces of container. We should remove one position from potential position list in this case (positions 1 in Fig. 3).
2) When we add new parallelepiped to a packed parallelepiped. We should remove two positions: one from the potential position list of already packed item and one from the potential position list of new packed item. (Positions 1 and 2 in Fig. 3).
Fig. 3
3) When we add some parallelepiped into the corner of container, we should remove two positions (described above).
faCG Fig. 3
4. CONCLUSION
Some simple improvements in the algorithm proposed in article. In some test cases, computation time was reduced sufficiently. Some complex test cases will be presented in next articles.
ACKNOWLEDGMENTS
This research has been partially supported by grants of the Russian Foundation of Basic Researches (RFBR) 12-07-00579.
REFERENCES
1. I. S. Koshcheev, "Evolutionary algorithm (1+1)EA for packing problem of cylinders and parallelepipeds into containers," in Proc. of 12th Int. Workshop on Computer Science and Information Technologies CSIT'2010, vol. 3, pp. 43-46, Ufa: USATU, 2010.
2. Yu. Stoyan and A. Chugay, "Packing cylinders and rectangular parallelepipeds with distances between them into a given region," European Journal of Operational Research, vol. 197, issue 2, pp. 446-455, 2009.
3. N. I. Yusupova, A. F. Valeeva, and R. I. Fayzrakhmanov, "A probabilistic algorithm ant colony for solving cutting of industrial materials into various geometric shapes," (in Russian), Informatsionnye Tekhnologii, no. 5, pp. 35-42, 2012.
4. A. F. Valeeva and Yu. A. Goncharova, "Practical application of population based ant colony optimization algorithm," (in Russian), Vestnik UGATU, vol. 17, no. 6 (59), pp. 75-78, 2013.
