CC BY
72
Quasi-Phi-Functions in Packing Problem
of Ellipsoids
A. Pankratov, T. Romanova, O. Khlud
Abstract - The paper considers the problem of packing a given collection of ellipsoids of revolution into a rectangular container of minimal volume. Our ellipsoids can be continuos rotated and translated. A class of radical-free quasi-phi-functions is used for an analytical description of nonoverlapping and containment constraints. We formulate the packing problem in the form of a nonlinear programming problem and propose a solution strategy, which allow us to search for local optimal packings. The actual search for a local minimum is performed by IPOPT. We provide computational results.
Index Terms - packing, ellipsoids, continuous rotations, non-overlapping, containment, quasi-phi-functions, solution algorithm, nonlinear optimization
I. Introduction
In this paper we deal with the optimal ellipsoid packing problem, which is a part of operational research and computational geometry. The problem is NP-hard [1] and has multiple applications in modern biology, mineralogy, medicine, materials science, nanotechnology, as well as in the chemical industry, power engineering etc.
our approach is based on mathematical modeling of relations between ellipsoids and thus reducing the packing problem to a nonlinear optimization problem. To this end a class of quasi-phi-functions [2] is used for analytic description of placement of ellipsoids in a rectangular container taking into account their continuous rotations and translations.
The paper is organized as follows: In Section 2 we formulate the optimal ellipsoid packing problem and give a short review of related works. In Section 3 we define quasi-phi-functions for nonoverlapping and containment constraints. In Section 4 we propose a mathematical model as a continuous nonlinear programming problem by means of quasi-phi-functions and describe a solution strategy. In Section 5 we provide our computational results. Finally we give some conclusions in Section 6.
Manuscript received March 18, 2015
Alexandr V. Pankratov is with the Institute for Problems in Machinery of National Academy of Sciences of Ukraine (Kharkov).
Tatiana E. Romanova is with the Institute for Mechanical Engineering Problems of the National Academy of Sciences of Ukraine (corresponding author to provide e-mail: [email protected]).
Olga M. Khlud is with the Kharkiv National University of Radioelectronics.
II. Problem Formulation
We consider here a packing problem in the following setting. Let Q denote a rectangular domain of length l, width w and height h. All of these dimensions may be variable, or one (two) may be fixed and the other variable. Suppose a set of ellipsoids of revolution, Ei , i є{1,2,...,n} = In , is given to be placed in Q without overlaps. Each ellipsoid E i is generated by rotation of an ellipse of semi-axes a; and b;, a; > b;, along the axis of revolution OX, therefore we assume that third semi-axe is defined as c; = b;. With each ellipsoid E; we associate its local coordinate system whose origin coincides with the center of the ellipsoid and the coordinate axes are aligned with the ellipsoid’s axes. In that system the ellipsoid is described by parametric equations x = a;cost;, y = bi sinti cosgi, z = bi sinti sing;, 0 < t;< 2n, 0 < g;< 2n. We also use a fixed coordinate system attached to the container Q. The location and orientation of each ellipsoid E i is defined by a variable vector of its placement parameters (v;, 9;). Here v; = (x;, y;, z;) is a translation vector, 9; = (91,92) is a vector of rotation parameters,
where 9;, 9; are appropriate angles from axis OX to OY, from axis OY to OZ in the local coordinate system of ellipsoid E^ The rotated by angles 91,92 and translated by vector v i ellipsoid E i is defined as E;(u) = {pєR3 :p = v; + M(9;)• p0, Vp° єEf}, where E0 denotes the non-translated and non-rotated ellipsoid E^ M(9) = M2(9 2) • M1(91) is a rotation matrix, where
M1(91)
cos 91 - sin 91 0
sin 91 cos 91 0
0 0 1
V J
(10 о ^
M2(9 2)
0 cos 9 2 V 0 sin 9 2
- sin 92 cos 92 J
Packing problem of ellipsoids. Pack the set of ellipsoids Ei, i є In, within a rectangular domain
of
Q = {(x, y, z) є R3 :0 < x < l, 0 < y < w, 0 < z < h}
minimal volume. If one of the two dimensions (l or w or h) is fixed, we need to minimize the other ones. If all are variable, it is natural to minimize the volume F = l • w • h of the container.
At present, the interest in finding effective solutions for placement problems of ellipsoids is growing rapidly (see, e.g., [4-8]). This is due to a large number of applications and an extreme complexity of methods used to handle many of them.
The remarkable method of the problem of cutting ellipses from a rectangular plate of minimal area was developed by Josef Kallrath and Steffen Rebennack, see [10]. The paper offers a good overview of related publications. For a small number of ellipses they are able to compute a globally optimal solution subject to the finite arithmetic of global solvers at hand. However, for more than 14 ellipsoids none of the nonlinear programming (NLP) solvers available in GAMS can even compute a locally optimal solution. Therefore, the authors of [10] develop polylithic approaches, in which the ellipses are added sequentially in a strip-packing fashion to the rectangle restricted in width but unrestricted in length. The rectangle’s area is minimized at each step in a greedy fashion. The sequence in which they add ellipses is random; this adds some GRASP flavor to the approach. The polylithic algorithms allow the authors to compute good solutions for up to 100 ellipses.
Paper [9] studies the problem of placing a given collection of ellipses into a rectangular container of minimal area. Radical free quasi-phi-functions are used to reduce it to a nonlinear programming problem and develop an efficient solution algorithm. The paper provides computational results with local optimal solutions for the problem (up to 120 ellipses).
The present paper proposes an approach, which is capable of handling precise ellipsoids (without
approximations) and thus finding an exact local optimal solution. The approach can be considered as some extension of quasi-phi-functions for ellipses, derived in [9], to 3D case.
III. Quasi-phi-functions for nonoverlapping and
CONTAINMENT CONSTRAINTS
Quasi-phi-functions for nonoverlapping constraints. Let Ei(ui) and Ej(uj) be two ellipsoids of revolution with
semi-axes a^b^q and a j, b j, c j.
Then, a quasi-phi-function for E^uQ and Ej(uj) may be defined as follows
ФІ (u;, u j, u ij ) = min{x(© b © J, u ІJ), X+(Ui, UJ, u ij X
X-(ui,uj,uij),X2(ui,uj,uij),x2(ui,uj,uij)}, (1)
where uij = (ti,gi,tJ,gj),
X = -( Ni,Nj) = -a ia j-Pi Pj-Y І Y j, © i = (0u
(a;, p;, yi) = M(©;)• (a;, p;, yі )T ,
cost _ sintj cosgj sintj sing:
a i =----L, P i =----i-----0 Y i =-----i----L
©j= (0j0 (a J, P J, Y j) = M(©j) • (a j, P J, Y j)T,
cos t j sin t j cos g j sin t j sin g j
a =------L, Pj =---1,Yj =- J J
j
a
bj
b
Xk = ai(x +k-xi) + Pi(y +k -Уі) + Yi(z+k-zi)-1, Xk =ai(x-k-xi) + РІ(У-k -Уі) +Yi(z-k-zi)-1,
(x +k,y +k,z+k) are coordinates of point q+k and
(xjk, yjk, zjk) are coordinates of point qjk , k = 1,2 (see Fig.1).
We derive q +k and q -k as follows:
(x +2,y +2,z +2) = vj +M(©j)M2(gj)(ajcostj,bj sin tjo/2aj)T, (x-2,У-2,z-2) = vj +M(0j)M2(gj)(aj costj,bj sintj,^2aj)T, (x +1,y+1,z +1) = vj + M(© j)M2(gj)(x+,y+,0)T,
(x -1,y-1,z -1) = vj + M(© j)M2(gj)(x-,y-,0)T,
(x +,У+) = (at, Pt) + П(-Рtj, at),
(x-,У-) = (a F Pt)-П(-Рt,a t),
(at, pt) = M1(tj)(aj,0)T, n=/2(aj)2.
Thus a nonoverlapping constraint, i.e. intEi(ui)ПintEj(uj) = 0, can be defined as
Фij(ui,uj,u'ij)> 0, where Ф'у is a quasi-phi-function of
ellipsoids E:(u:) and Ej (uj) given by (1).
Quasi-phi-functions for containment constraints. Let vertices of our rectangular container
Q = {(x, y, z) є R3 : 0 < x < l, 0 < y < w, 0 < z < h} be given as follows: {v;,i = 1,...,8} = {(0,w,0), (l,w,0),
(l, 0,0), (0,0,0), (0, w, h), (l, w, h), (l, 0, h), (0,0, h)}.
And let Ei(ui) be ellipsoid of revolution with semi-axes a;, bi and q = b;, a; >b;, i = 1,2...,n.
Then a quasi-phi-function for Ei(uQ and object Q* = R3 \int Q may be defined in the form
Ф i (u і , ui) = min{cp i1 (u), Фі2 (u), Фі3 (u)}, (2)
where (ui,ui) є r11, ui = (tix, ti2, ti3,gi^x,gi2>ЄІ3) >
0 < tik < 2n ,0 < gik < 2n ,
Ф i1(u) = min{p11(v1), ф1 1(v2), P11(v3), p11(^4), ф12 (v5), ф12 (v6), p12 CVyX p12 (v8)},
Фi2 (u) = min{p:21(v1), Ф:21(v4), Ф21(v5), p:21(v8), p22(v2X ф22(v3), p22(v6), ф22(v7)},
q22
Z
Fig. 1 Illustration to construction of a quasi-phi-function for two ellipsoids Ej (Uj) and E2 (u 2 )
Фi3(u) = min{cP31(v 1) = Ф31(v2)’ Ф31(v5)’ Ф31(v6) =
Ф32 (v3X ф32 (v4)= ф32(v7)’ ф32 (v8)},
фk1 = Aikx + Biky + Cikz + Dik -
Фk2 =-Aikx -BikY - C1kz -Dik -1,
(Aik,Bik.Cik) = M(01)(a1 cos tik,bi sint±k cos g1k,b1 sin t±k sin g*) ,
Dik =-Aikxi - BikYi - Cikzi , k = 1,2,3 .
Thus, a containment constraint, i.e.
Ei(ui) cQ» intEi(ui)ПQ* = 0, can be defined as Фl(Ui,u1) > 0, where Ф-j is a quasi-phi-function for
Ei(ui) and Q* given by (2).
IV. Mathematical Model And Solution Strategy
The vector u є Rct of all our variables can be described as follows: u = (l, w, h,u1,u2,...un,t) , where (l,w,h) denote the variable dimensions of the rectangular container Q and ui = (vi, 91) is the vector of placement parameters
for the ellipsoid Ei, i є In, where vi = (xi,yi,Zi),
91 = (91,92). The vector t denotes the vector of extra variables (for our quasi-phi-functions), defined as follows:
T=a!,g1,t1„
,...,t
2 1
V1,
'1,t'1,
2
:'1,...,t,n
3 1
Tm tm
V ’ V
'n
Tm,
”,tn,g“),
1
n
t
where tk,gk, tk
11 2
k are extra variables for the k-th
2
pair of ellipsoids, k = 1,...,m, m =
(n - 1)n 2
, and t1, g1,
1 1
t1 , g 1 , t1, g 1, are extra variables for each ellipsoid E:,
2 2 3 3
i є In . Lastly, Rct denotes the ст-dimensional Euclidean
space, where ct = 3 + 5n + 2n(n -1) + 6n = 2n + 9n + 3 is the number of the problem variables.
A mathematical model of the basic packing problem may now be stated in the following form:
min F(u) , (3)
uєW c R ct
W = {uєRct :oij >0,ф1 >0,i =1,2,...,n,j =1,2,...,n,j >i},(4)
where F(u) = l • w • h, Ф ij is a quasi-phi-function (1) defined for the pair of ellipsoids Ei and E j , (to hold nonoverlapping constraint), Фi is a quasi-phi-function (2)
defined for an ellipsoid Ei and the object Q* (to hold the
containment constraint).
our constrained optimization problem (3)-(4) is a continuous nonlinear programming problem.
We propose the following solution strategy for the problem, which involves three major stages:
1) First we generate a number of random starting points.
2) Then starting from each point obtained at Step 1 we search for a local minimum of the objective function F(u) of problem (3)-(4).
3) Lastly, we choose the best local minimum from those found at Step 2. This is our best solution of the problem (3)-
(4).
V. Computational results
Here we present a number of Instances to demonstrate the efficiency of our quasi-phi-functions. We have run our experiments on an AMD Athlon 64 X2 5200+ computer. We search for 100 local minima to each of Instances. The actual search for a local minimum is performed by IPOPT proposed in [11], which is available at an open access
noncommercial software depository (https://proiects.coin-or.org/Ipoptt .
We consider a collection of ellipsoids: {Ei,i = 1,..,12} = {(ai,bi,ci),i = 1,2,...,12} = {(5, 4, 4), (7, 5, 5), (6, 5, 5), (4, 3, 3), (5.5, 4.5, 4.5), (7.5, 5.5, 5.5), (6.5, 5.5, 5.5), (4.5, 3.5, 3.5), (5.3, 4.3, 4.3), (7.3, 5.3, 5.3), (6.3, 5.3, 5.3), (4.3, 3.3, 3.3)}.
Instance E2. Local optimal placement of ellipsoids {E;, i = 1,2} is shown in Figure 2,a. Container has volume
F* = 2192.513985 and sizes (l*,w*,h*) = (10.000000, 10.006950, 21.909912). Average time per one local minimum is 2.09 sec.
Instance E3. Local optimal placement of ellipsoids {Ei, i = 1,2,3} is shown in Figure 2,b. Container has volume
F* = 3385.008834 and sizes (l*,w*,h*) = (10.000000,
33.797139, 10.015667). Average time per one local minimum is 5.89 sec.
Instance E4. Local optimal placement of ellipsoids {Ei,i = 1,...,4}is shown in Figure 2,c. Container has
volume F* = 3539.283378 and sizes
(l*,w*,h*) = (18.273863, 10.014451, 19.340061). Average time per one local minimum is 22.76 sec.
a
b c
Fig. 2. Local optimal placement of ellipsoids in Instances: a - E2, b - E3, c - E4
Instance E5. Local optimal placement of ellipsoids {Ei,i = 1,...,5}is shown in Figure 3,a. Container has
volume F* = 4347.434370 and sizes
(l*,w*,h*) = (24.366822, 10.000252, 17.841164). Average time per one local minimum is 60.71 sec.
Instance E6. Local optimal placement of ellipsoids {Ei,i = 1,...,6}is shown in Figure 3,b. Container has
volume F* = 7687.512942 and sizes
(l*, w*, h*) = (18.960443, 19.723184, 20.557029). Average time per one local minimum is 222.36 sec.
a
b c
Fig. 3. Local optimal placement of ellipsoids in Instances: a - E5, b - E6, c - E7
Instance E8. Local optimal placement of ellipsoids {Ei,i = 1,...,8}is shown in Figure 4a. Container has
volume F* = 7998.224794 and sizes
(l*,w*,h*) = (20.223526, 19.919118, 19.854851). Average time per one local minimum is 359.88 sec.
a b
c
Fig. 4. Local optimal placement of ellipsoids in Instances: a - E8, b - E9, c - E10
Instance E9. Local optimal placement of ellipsoids {Ei,i = 1,...,9}is shown in Figure 4b. Container has
volume F = 6312.236870 and sizes
(l*,w*,h*) = (27.244026, 11.000291, 21.062399). Average time per one local minimum is 126.02 sec.
Instance E7. Local optimal placement of ellipsoids {Ei,i = 1,...,7}is shown in Figure 3,c. Container has
volume F = 8524.765214 and (l ,w ,h ) = (19.365765, 18.695366, 23.545819). Average time per one local minimum is 369.42 sec.
Instance E10. Local optimal placement of ellipsoids {Ei, i = 1, ...,10} is shown in Figure 4c. Container has
volume F* = 10263.381559 and sizes
(l*, w*, h*) = (25.780036, 18.893787, 21.071135). Average time per one local minimum is 371.23.
Instance E11. Local optimal placement of
ellipsoids {E;,i = 1, ...,11}is shown in Figure 5a. Container
has volume F* = 11860.716557 and sizes
(l*,w*,h*) = (21.945274, 27.902451, 19.369908). Average time per one local minimum is 445.95 sec.
Instance E12. Local optimal placement of ellipsoids {E;,i = 1,...,12}is shown in Figure 5b. Container has
volume F* = 11768.260385 and sizes
(l*,w*,h*) = (19.327419 19.558038 31.132438). Average time per one local minimum is 836.70 sec.
a
b
Fig. 5. Local optimal placement of ellipsoids in Instances: a - E11, b -E12
VI. Conclusions
We developed here an exact continuous NLP model of the placement problem of ellipsoids, using quasi-phi-functions. The use of quasi-phi-functions allows us to handle ellipsoids which can be continuously rotated and translated, but there is a price to pay: now the optimization has to be performed over a larger set of parameters, including the extra variables, besides placement parameters of ellipsoids. The model can be realized by the current state-of-the art local or global solvers. We are working on the improvement of our algorithms to generate feasible starting points, as well as, to reduce our problem dimension in local optimisation procedures, based on the paper [9]. We expect that efficiency of our algorithms will be increased in the future.
References
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Alexandr V. Pankratov received Doctor of Technical Sciences degree in Mathematical Modeling and Computational Methods (2013) from Institute for Problems in Machinery of National Academy of Sciences of Ukraine (Kharkov). From 2013 he is a senior researcher at the Department of Mathematical Modeling and Optimal Design, Institute for Mechanical Engineering Problems of the National Academy of Sciences of Ukraine. His current research interests include mathematical modeling, operational research, computational geometry, optimisation, packing, cutting and covering.
Tatiana E. Romanova received Doctor of Technical Sciences degree in Mathematical Modeling and Computational Methods (2003) from Institute of Cybernetics of the National Academy of Sciences of Ukraine (Kiev). From 2002 he is a senior researcher at the Department of Mathematical Modeling and Optimal Design, Institute for Mechanical Engineering Problems of the National Academy of Sciences of Ukraine. From 2005 she is a professor at the Department of Applied Mathematics, Kharkiv National University of Radioelectronics. Her current research interests include mathematical modeling, operational research, computational geometry, optimisation, packing, cutting and covering.
Olga M. Khlud received Bachelor's degree in System Analysis (2014) from Kharkiv National University of Radioelectronics. She is an undergraduate at the Kharkiv National University of Radioelectronics. Her current research interests include mathematical modeling, operational research, packing and cutting.
