=Paper=
{{Paper
|id=Vol-2300/Paper6
|storemode=property
|title=Muticriteria Model of Balanced Layout Problem of 3D-Objects
|pdfUrl=https://ceur-ws.org/Vol-2300/Paper6.pdf
|volume=Vol-2300
|authors=Inna Urniaieva,Igor Grebennik,Tatiana Romanova,Alexandr Pankratov,Anna Kovalenko
|dblpUrl=https://dblp.org/rec/conf/acit4/UrniaievaGRPK18
}}
==Muticriteria Model of Balanced Layout Problem of 3D-Objects==
https://ceur-ws.org/Vol-2300/Paper6.pdf
22
Muticriteria Model of Balanced Layout Problem of 3D-
Objects
Inna Urniaieva1, Igor Grebennik1, Tatiana Romanova2, Alexandr Pankratov2, Anna
Kovalenko1
1. Department of Computer Science, Kharkiv National University of Radioelectronics, UKRAINE, Kharkiv, 14 Nauki ave., email:
igorgrebennik@gmail.com
2. Department of Mathematical Modeling and Optimal Design, Institute for Mechanical Engineering Problems of the National Academy of
Sciences of Ukraine, UKRAINE, Kharkiv, 2/10 Pozharskogo str., email: sherom@kharkov.ua
Abstract: The paper studies the optimal layout problem we employ phi-function technique [5]. We also formalise
of 3D-objects. The problem takes into account placement behaviour constraints (equilibrium, moments of inertia, and
constraints, as well as, behaviour characteristics of the stability constraints) based on [3].
mechanical system. We construct a mathematical model The variety of forms of objective functions and
of the problem in the form of a multicriteria optimisation combinations of placement and behaviour constraints
problem and call the problem Multicriteria Balanced generates various variants of the MBLP problem.
Layout Problem (MBLP).
Keywords: Layout problem, Behaviour Constraints, II. PROBLEM FORMULATION
Placement Constraints, Multicriteria Optimisation.
=Let Ω {(x, y, z) ∈ 3 : G(x, y, z) ≥ 0} be a container
I. INTRODUCTION of given height H . We consider the following types of
3D layout optimisation problems have a wide spectrum of containers: 1) Ω ≡ C , C is a straight circular cylinder with a
practical applications. In particular, these problems arise in base of radius R,
space engineering for rocketry design. Their distinctive G(x, y, = z) min{− x 2 − y 2 + R 2 , −z + H, z} ; 2) Ω ≡ Λ , Λ
feature consists of taking into account behaviour constraints
is a paraboloid of revolution with a base of radius R = H ,
of a satellite system. Behaviour constraints specify the
requirements for system’s mechanical properties such as G(x, y, = z) min{−z − x 2 − y 2 + H, z} ; 3) Ω ≡ Ε , Ε is a
equilibrium, inertia, and stability. Many publications analyze straight circular blunted cone with lower and upper bases of
problems of the equipment layout in modules of spacecraft or radii R1 and R 2 < R1 respectively,
satellites [1, 2]. These problems are NP-hard.
In the research we consider the balance layout problem in G(x, y, = z) min{−z − H ( x 2 + y 2 − R 1 ) (R 1 − R 2 ), −z + H, z} .
the following statement: arrange 3D-objects in a container Suppose that Ω is divided by circular racks S k ,
taking into account special placement and behaviour
=
constraints so that the objective function attains its extreme k 1, 2, ..., m + 1 , into subcontainers Ω k , k = 1, 2, ..., m .
value [3]. We assume that S1 is a base of Ω . Between racks S k and
We consider here an extension of the balanced layout of S k +1 the distance t k is given.
3D-objects considered in [3,4]. The paper studies 3D
optimisation balance layout problem taking into account Family= A {A i , i ∈ I n } , I n = {1, 2, .., n} , involves the
minimal and maximal allowable distances. Classes of following shapes of objects: solid spheres of radius r ;
i i
adjusted phi-functions and adjusted quasi-phi-functions are
derived for analytical description of non-overlapping, straight circular cylinders i of radius ri and height 2h i;
containment and distance constraints. A circular cylinder, a tori i with metric characteristics (ri , h i ) , where ri is the
paraboloid, or a truncated cone are taken as a container. We distance from the center of generating circle to the axis of
consider cylinders, spheres, tores, spherecylinders and revolution, 2h is the height of , h is the radius of the
i i i
straight convex prisms as the placement objects. An exact
generating circle; spherocylinders i with metric
mathematical model of the problem in the form of NLP
problem is provided. characteristics (l i , ri , h i ) , where l i is the height of ball
The aim of this study is to develop a mathematical model segments, ri is the radius and 2h i is the height of cylinder;
of 3D layout optimisation problem taking into account
straight regular prisms and cuboids i with metric
behavior constraints in the form of multicriteria optimisation
problem. We call the problem the Multicriteria Balanced characteristics (h i , v il , ) , where 2h i is the height of i ,
Layout Problem (MBLP). v il = (x il , y il ) , l = 1, ..., s i , are vertices of the base of i
To describe placement constraints (non-overlapping of
objects, containment of objects in a container with regard for (which is a convex polygon K i ), s i is the number of
the minimal and maximal allowable distances) analytically vertices of K i .
ACIT 2018, June 1-3, 2018, Ceske Budejovice, Czech Republic
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MBLP: Pack 3D-objects A i ∈ A , i ∈ I n = {1, 2, ..., n} , J X (u)= J x + ∑ n n
(J x i cos 2 θ i + J y i sin 2 θ i ) + ∑ (y i2 + z i2 )m i − M ( y s2 + z s2 ) ,
0
inside container Ω , so that the vector function attains
= its i 1 =i 1
n n
extreme value with regard for placement and behaviour = J y 0 + ∑ (J x i sin 2 θ i + J y i cos 2 θ i ) + ∑ (x i2 + z i2 )m i − M ( x s2 + z s2 )
J Y (u)
constraints. =i 1 =i 1
The placement constraints in the MBLP problem are n n
generated by non-overlapping of objects A i , A j , i > j ∈ I n , J Z (u) = ∑ J z i + ∑ (y i2 + z i2 )m i − M ( x s2 + y s2 ) ,
=i 0=i 1
which have to be placed inside container Ω , and
J x 0 , J y 0 , J z 0 are the moments of inertia of container Ω
containment of object A i in container Ω , i ∈ I n . In
with respect to the axes of the coordinate system Oxyz ,
addition, the minimal ρ ij− and maximal ρ ij+ ≥ ρ ij− allowable
J x i , J y i , J z i , i ∈ I n , are the moments of inertia of object А i
distances between objects Ai, A j , i > j∈ In , may be
with respect to the axes of coordinate system O i x i y i z i .
specified. Also, the minimal allowable distance ρ i− between The stability constraints are defined by the following
object A i ∈ A , i ∈ I n , and the lateral surface of container Ω system of inequalities:
may be given. Without loss of generality we set ρ ij− =0 (or µ 31=
(u) min{−J XY (u) + ∆ J XY , J XY (u) + ∆ J XY } ≥ 0 ,
ρ ij+ =ϖ ) if a minimal (or a maximal) allowable distance µ 32=
(u) min{−J YZ (u) + ∆ J YZ , J YZ (u) + ∆ J YZ } ≥ 0 ,
between objects A i and A j is not given, i > j ∈ I n . Here ϖ
µ 33=
(u) min{−J XZ (u) + ∆ J XZ , J XZ (u) + ∆ J XZ } ≥ 0 ,
is a given sufficiently great number. In particular, the
condition ρ ij+ =ρ ij− provides the arrangment of objects where J XY (u), J YZ (u), J XZ (u) are the products of inertia of
system Ω A with respect to the axes of the coordinate system
A i and A j on the exact distance. We also set ρ i− =0 if a
O s XYZ , ∆ J XY , ∆ J YZ , ∆ J XZ are admissible values for
minimal allowable distance between object A i and the J XY (u), J YZ (u), J XZ (u) , respectively,
lateral surface of container Ω is not given.
1 n n
Placement constraints in the MBLP problem may be
∑ J=(J x i − J y i ) sin 2θ i + ∑ x i y i m i − Mx s y s ,
XY (u)
presented as the following: ρ ij− ≤ dist( A i , A j ) ≤=
ρ ij+ , 2 i 1 =i 1
n
i > j ∈ I n , and dist( A i , Ω * ) ≥ ρ i− , i = 1, ..., n , where =
J YZ (u) ∑ y i z i m i − My s z s ,
Ω * 3 \ int Ω .
= i =1
n
To describe the placement constraints analytically we
employ the phi-function technique [5–8].
=
J XZ (u) ∑ x i z i m i − Mx s z s .
i =1
Let us consider the constraints of mechanical
Behaviour constraints of the BLP problem we define as
characteristics of system Ω A .
the system of inequalities
The equilibrium constaints are defined by the following µ1 (u) ≥ 0, µ 2 (u) ≥ 0, µ 3 (u) ≥ 0 ,
system of inequalities:
where
µ11=(u) min{−(x s (u) − x e ) + ∆ x e , (x s (u) − x e ) + ∆ x e } ≥ 0
µ1 (u) = min{µ11 (u), µ12 (u), µ13 (u)}, (1)
µ12=
(u) min{−(y s (u) − y e ) + ∆ y e , (y s (u) − y e ) + ∆ y e } ≥ 0 ,
µ 2 (u) = min{µ 21 (u), µ 22 (u), µ 23 (u)} , (2)
µ13=
(u) min{−(z s (u) − z e ) + ∆ z e , (z s (u) − z e ) + ∆ z e } ≥ 0 ,
µ 3 (u) = min{µ 31 (u), µ 32 (u), µ 33 (u)}. (3)
where (x e , y e , z e ) is the expected position of O s ,
(∆ x e , ∆ y e , ∆ z e ) are admissible deviations from the point Here O s = (x s , y s , z s ) is the center of mass of system Ω A ,
(x e , y e , z e ) . 1 n 1 n
The constraints of moments of inertia are defined as the x s (u) = ∑
M i =1
m i x i , y s (u) = ∑ miyi ,
M i =1
following:
µ 21 (u) = −J X (u) + ∆ J X ≥ 0 , 1 n n
z s (u) = ∑
M i =1
m i z i , M = ∑ m i is the mass of system Ω A .
µ 22 (u) = −J Y (u) + ∆ J Y ≥ 0 , i =0
µ 23 (u) = −J Z (u) + ∆ J Z ≥ 0 , III. MATHEMATICAL MODEL
where J X (u), J Y (u), J Z (u) are the moments of inertia of the A mathematical model of the MBLP problem can be
system Ω A with respect to the axes of coordinate system presented in the form
O s XYZ , ∆ J X , ∆ J Y , ∆ J Z are admissible values for extrF ( p, u ) s.t. (u , p ) ∈ W (4)
J X (u), J Y (u), J Z (u) , where W {(u , p ) ∈ ξ : ϒ(u , p ) ≥ 0, µ(u , p ) ≥ 0, ζ ≥ 0} , (5)
=
ACIT 2018, June 1-3, 2018, Ceske Budejovice, Czech Republic
24
where F ( p, u ) = ( F1 ( p, u ), F2 ( p, u ), ..., Fk ( p, u )) , depending on the forms of the objective functions, shapes of
ϒ(u , p ) describes placement constraints, objects and containers, combinations of placement and
behavior constraints.
ϒ(u, p) = min{ϒ1 (u), ϒ 2 (u, p)} ,
ϒ1 (u) is responsible for non-overlapping constraints, IV. COMPUTATIONAL RESULTS
ϒ 2 (u, p) is responsible for containment constraints,
Instance 1. Let E , m 2 , H 0.6 , R1 0.5 ,
µ(u )= min{µ s (u ), s ∈ U t } is responsible for behavior
constraints, U t ∈ P(U ) , P(U ) is the power set of R3 0.3 , A { 1, 2 , 3 , 4 ,5 , 6 , 7 , 8 , 9 , 10 } ,
U = {1, 2, 3} , functions µ1 (u ), µ 2 (u ), µ 3 (u ) are given by (1)- A { 1, 3 ,5 , 7 , 9 } ,
1
A2 { 2 , 4 ,
6
, 8 , 10 } ,
(3), ζ ≥ 0 is the system of additional constraints of metric ij 0.03 , i j I 10 ,
39 0.1 ,
26 0.08 ,
characteristics of container Ω and placement parameters of
objects. If s = ∅ , i.e. behavior constraints are not involved in (xe , ye , ze ) (0, 0, 0.275) , t1 0.3 , n 10 ,
(5), then our objective function F (u ) meets mechanical { z i , i 1,...,10 }={0.19, 0.4, 0.19, 0.41, 0.24, 0.35, 0.19,
characteristics of system Ω A . 0.39, 0.18, 0.42}, { mi , i 1,...,10 }={27.8764, 20.944,
Depending on the different combinations of objective 34.5575, 16.9332, 28.4245, 22.2066, 17.2159, 19.2265, 38.4,
functions F1 ( p, u ), F2 ( p, u ), ..., Fk ( p, u ) different variants
19.9532}, r1 0.11 , r2 0.1 , r3 0.1 , h3 0.11 ,
of mathematical model (4)-(5) can be generated. The most
frequently occurring objective functions found in related r4 0.07 , h4 0.11 , r5 0.08 , h5 0.06 , r6 0.09 ,
publications are the following: 1) size of container Ω ; 2)
h6 0.05 , r7 0.08 , h7 0.05 , l7 0.06 , h8 0.06 ,
deviation of the center of mass of system Ω A from a given
point; 3) moments of inertia of system Ω A [3,9-13]. l 0.03 , s 4, h 0.12 , v (0.08, 0.1) , v =
8 9 9
91 92
Let us consider some of realisations of model (4) - (5): (0.08,-0.1), v93 =(-0.08,-0.1), v94 (0.08, 0.1) , s10 6 ,
p s.t. ( p, u ) ∈ W ⊂ ξ ,
• F ( p, u ) = h 0.12 , v
10
(0.04, 0.07) , v
(10)1 (0.08, 0) , v
(10)2 , (10)3
v(10)4 =(-0.04,-0.07),
ξ
W {( p, u ) ∈ : ϒ1 (u ) ≥ 0, ϒ 2 ( p, u ) ≥ 0,
= =(0.04,-0.07) v(10)5 (0.08, 0) ,
µ( p, u ) ≥ 0, ζ ≥ 0} v (0.04, 0.07) .
(10)6
d , ( p, u ) ∈ W ⊂ ξ ,
• F (u ) = The local-optimal solution found by NLP-solver in CAS
d = ( x s (u ) − x e ) + ( y s (u ) − y e ) + ( z s (u ) − z e ) ,
2 2 2 Math 9 (Fig. 1) is F (u * , u * ) 1.12726 106 .
W {( p, u ) ∈ ξ : ϒ1 (u ) ≥ 0, ϒ 2 ( p, u ) ≥ 0, µ 2 ( p, u ) ≥ 0,
=
µ 3 ( p, u ) ≥ 0, ζ ≥ 0} ;
• F ( p, u ) = ( F1 ( p, u ) = p, F2 ( p, u ) = d ) , ( p, u ) ∈ W ⊂ ξ ,
W {( p, u ) ∈ ξ : ϒ1 (u ) ≥ 0, ϒ 2 ( p, u ) ≥ 0, µ 2 ( p, u ) ≥ 0,
=
µ 3 ( p, u ) ≥ 0, ζ ≥ 0} ;
• F ( p, u ) = ( F1 ( p, u ) = J X ( p, u ), F2 ( p, u ) = J Y ( p, u ),
F3 ( p, u ) = J Z ( p, u )) Fig. 1 The local optimal layout of 3D-objects in Instance 1
( p, u ) ∈ W ⊂ ξ , Instance 2. Let C , m 3 , H 1 , R 0.45 ,
t2 0.35 , n 20 , A { i , i 1,..., 4, i , i 5,..., 8,
W {( p, u ) ∈ ξ : ϒ1 (u ) ≥ 0, ϒ 2 ( p, u ) ≥ 0, µ1 ( p, u ) ≥ 0,
=
i , i 9...12, i , i 13,...16, i , i 17,..., 20}
,
µ 3 ( p, u ) ≥ 0, ζ ≥ 0} .
A1 { 1, 5 , 6 ,9 , 14 , 17 } , A2 { 2 , 3 , 7 , 10 ,
III. CONCLUSIONS
15 , 18 , 20 } , A3 { 4 , 8 , 11,12
, 16 , 19 } , U t ,
In this paper we formulate the optimisation layout
problem of 3D-objects into a container taking into account ij 0.02 , i j 1,..., 20 , (xe , ye , ze ) (0, 0, 0.5) ,
placement (non-overlapping, containment, distance) and { z i , i 1,..., 20 }={0.1, 0.44, 0.46, 0.81, 0.11, 0.12, 0.46,
behaviour (equilibrium, inertia and stability) constraints. We 0.78, 0.06, 0.425, 0.76, 0.77, 0.11, 0.13, 0.46, 0.81, 012, 0.47,
call the problem as Multicriteria Balance Layout Problem
(MBLP). In order to describe placement constraints 0.82, 0.46}, { mi , i 1,.., 20 }={20.944, 15.2681, 27.8764,
analytically we employ phi-function technique. A 34.5575, 63.7115, 41.8146, 30.4106, 28.4245, 49.9649,
mathematical model of the problem in the form of 24.8714, 38.6888, 26.2637, 20.7764, 17.2159, 16.8756, 52.8,
multicriteria optimisation problem is proposed. 52.8, 52.8, 23.1489}, r1 0.1 , r2 0.09 , r3 0.11 ,
We also consider some variants of the MBLP problem
ACIT 2018, June 1-3, 2018, Ceske Budejovice, Czech Republic
25
r4 0.11 , r5 0.1 , h5 0.11 , h6 0.12 , r7 0.11 , 81, 54, 29, 94, 92, 41, 57, 77, 40, 67, 31, 47, 39, 61, 73, 83,
11, 20, 86, 72, 81, 54, 29, 94, 92, 41, 57, 77, 40, 67, 31, 47,
r8 0.11 , h8 0.08 , r9 0.08 , h9 0.07 , r10 0.09 , 39, 61, 73, 83, 11, 20, 86, 72, 81, 54, 29, 94, 92, 41, 57, 77,
40, 67, 31, 47, 39, 61, 73, 83, 11, 20, 86, 72, 81, 54, 29, 94,
h10 0.075 , r11 0.07 , h11 0.06 , r12 0.08 ,
92, 41, 57, 77, 40, 67, 31, 47, 39, 61, 73, 83, 11, 20}.
h12 0.07 , r13 0.1 , h13 0.05 , l13 0.07 , r14 0.05 , The local optimal solution found by IPOPT is
h14 0.05 , l14 0.08 , r15 0.08 , h15 0.05 , l15 0.06 , F (u * , u * ) 0.000000 (Fig. 3).
r16 0.08 , h16 0.04 , l16 0.07 , si 4 ,
v (0.11, 0.1) , v (0.11, 0.1) , v (0.11, 0.1) ,
i1 i2 i3
vi 4 (0.11, 0.1) , hi 0.12 , i 17,18,19 , s20 6 ,
v
(20)1 (0.045, 0.078) , v(20)2 =(0.09,0), v(20)3 =(0.045,-0.078),
v(20)4 (0.045, 0.078) , v(20)5 (0.09, 0) ,
v(20)6 (0.045, 0.078) , h20 0.11 . Fig. 3 The local optimal layout of 3D-objects in Instance 3
The local optimal solution found by NLP-solver in CAS
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