Balance Layout Problems: Mathematical Modeling and Nonlinear Optimization

Abstract

The paper studies the optimal layout problem of 3D-objects (solid spheres, straight circular cylinders, spherocylinders, straight regular prisms, cuboids and tori) in a container (a cylindrical, a parabolic, or a truncated conical shape) with circular racks. The problem takes into account a given minimal and maximal allowable distances between objects, as well as, behaviour constraints of the mechanical system (equilibrium, moments of inertia and stability constraints). We call the problem the Balance Layout Problem (BLP) and develop a continuous nonlinear programming model (NLP-model) of the problem, using the phi-function technique. We also consider several BLP subproblems; provide appropriate mathematical models and solution algorithms, using nonlinear programming and nonsmooth optimization methods, illustrated with computational experiments.

Appendices

Appendix 1: Phi-Functions and Quasi-Phi-Functions

1.1 Phi-Objects

Here we define a class of admissible objects for our models, called phi-objects (see, e.g., [11]). They must have interior (“main part”) and boundary (frontier). Accordingly, we require each phi-object be the closure of its interior. (In mathematical topology, closed sets that are closures of their interior are said to be canonically closed; this is what our phi-objects are.) This requirement rules out such elements as isolated points, one-dimensional curves, etc.— they do not occur in realistic applications. Figure A1a shows an invalid phi-object— it has three one-dimensional ‘whiskers’, two isolated points, and four punctured interior points (white dots). In addition, our phi-objects should not have self-intersections along their frontier, as shown in Fig. A1b because this may lead to confusion. For example, Fig. A1c shows a dark domain of which two ends touch each other like pincers; this must be prohibited. The reason is also demonstrated in the same figure: a similar object (the light grey “figure eight”) is placed so that the two objects intersect each other only in their frontiers, which is generally allowed, but in this particular case we cannot place these objects as shown because one ‘cuts’ through the other.

Fig. A1

Examples of invalid phi-objects: (a ) object with ‘whiskers’, isolated and four punctured points, (b ) object with self-intersections along its frontier, (c ) the confusion case for two objects

Mathematically, the above requirement can be stated as the following: a phi-object and its interior must have the same homotopic type (the same number of connected components, the same number of interior holes, etc.). These requirements may sound too abstract, but their practical meaning should be clear from the above example. An important property of phi-objects is that if A is a phi-object, then the closure of its complement is a phi-object, too. Figure A2 shows the examples of valid phi-objects.

Fig. A2

Examples of valid phi-objects: (a ) 2D phi-objects, (b ) 3D phi-objects

1.2 Phi-Functions

Let A and B be two phi-objects. The position of object A is defined by the vector of placement parameters (v _A, θ _A), where: v _A = (x _A, y _A) is a translation vector and θ _A is a rotation angle if $A \subset \boldsymbol{ R}^{2};v_{A} = (x_{A},y_{A},z_{A})$ is a translation vector and θ _A = (θ _z, θ _x, θ _y) are rotation angles (from axis OX to OY, from axis OY to OZ and from axis OX to OZ) if $A \subset \boldsymbol{ R}^{3}$. We denote the vector of variables for object A by u _A = (v _A, θ _A) and the vector of variables for object B by u _B = (v _B, θ _B). Object A rotated by θ _A and translated by vector v _A will be denoted by A(u _A).

In order to feasibly place two phi-objects within a containing region, we need an analytical description of the relationships between a pair of objects A and B. We employ the phi-function technique for this. Phi-functions allow us to distinguish the following three cases: A and B are intersecting so that A and B have common interior points; A and B do not intersect, i. e. A and B do not have common points; A and B are in contact, i. e. A and B have only common frontier points.

Definition A1.

Continuous and everywhere defined function $\Phi ^{AB}(u_{A},u_{B})$ is called a phi-function for objects A(u _A) and B(u _B) if

$$\displaystyle\begin{array}{rcl} & \Phi ^{AB} <0,\ \mathrm{if\ int}A(u_{A}) \cap \mathrm{ int}B(u_{B})\neq \emptyset; & {}\\ & \Phi ^{AB} = 0,\ \mathrm{if\ int}A(u_{A}) \cap \mathrm{ int}B(u_{B}) =\emptyset \ \mathrm{ and}\ frA(u_{A}) \cap frB(u_{B})\neq \emptyset;& {}\\ & \Phi ^{AB}> 0,\ \mathrm{if}\ A(u_{A}) \cap B(u_{B}) =\emptyset. & {}\\ \end{array}$$

Here frA means the boundary (frontier) and intA means the interior of object A.

Thus, $\Phi ^{AB} \geq 0 \Leftrightarrow intA(u_{A}) \cap intB(u_{B}) =\emptyset$. We employ phi-functions for the description of the containment relation A ⊆ B as the following: $\Phi _{AB^{{\ast}}} \geq 0$, where $B^{{\ast}} =\boldsymbol{ R}^{d}\setminus intB$, d = 2, 3. We emphasize that according to Definition A1, phi-function $\Phi ^{AB}$ for a pair of objects A and B can be constructed by many different formulas, and we can choose the most convenient ones for our optimization algorithms.

1.3 Quasi-Phi-Functions

In comparison with phi-functions we include auxiliary variables u ^′, which take values in some domain $U \subset \boldsymbol{ R}^{n}$ (it depends on the shapes of objects A and B), and introduce function $\Phi ^{{\prime}AB}(u_{A},u_{B},u^{{\prime}}).$ The function must be defined for all values of u _A and u _B. It must be continuous in all its variables.

Definition A2.

Continuous and everywhere defined function $\Phi ^{{\prime}AB}(u_{A},u_{B},u^{{\prime}})$ is called a quasi-phi-function for two objects A(u _A) and B(u _B) if $\mathop{\max }\limits _{u^{{\prime}}\in U}\Phi ^{{\prime}AB}(u_{A},u_{B},u^{{\prime}})$ is a phi-function for the objects.

Let us consider two convex objects A(u _A) and B(u _B) and let P(u _P) be a half-space: P(u _P) = {(x,y,z): ψ _P = α ⋅ x +β ⋅ y +γ ⋅ z +μ _P ≤ 0}, u _P = (θ _xP, θ _yP, μ _P), α = sinθ _yP, β = −sinθ _xP ⋅ cosθ _yP, γ = cosθ _xP ⋅ cosθ _yP for 3D case;

P(u _P) = {(x, y): ψ _P = α⋅ x+β⋅ y +γ _P ≤ 0}, u _P = (θ _P, γ _P), α = cosθ _P, β = sinθ _P for 2D case. A function defined by

$$\displaystyle{\Phi ^{{\prime}AB}(u_{ A},u_{B},u^{{\prime}} = u_{ P}) =\min \{\Phi ^{AP}(u_{ A},u_{P}),\Phi ^{BP^{{\ast}} }(u_{B},u_{P})\},}$$

is a quasi-phi-function for A(u _A) and B(u _B). Here $\Phi ^{AP}(u_{A},u_{P})$ is a phi-function for A(u _A) and a half-space P(u _P) and $\Phi ^{BP^{{\ast}} }(u_{B},u_{P})$ is a phi-function for B(u _B) and $P^{{\ast}}(u_{P}) =\boldsymbol{ R}^{d}\setminus intP(u_{P})$, d = 2, 3.

The latter function meets all the requirements of Definition A2. First, function $\Phi ^{{\prime}AB}$ is defined everywhere and is continuous in all its variables, since the phi-functions $\Phi ^{AP}$ and $\Phi ^{BP^{{\ast}} }$ enjoy the same properties. Based on the properties of a separated line (plane) for two convex objects the following is fulfilled:

• 1) $\mathop{\max }\limits _{u^{{\prime}}\in R^{d}}\Phi ^{{\prime}AB} <0,$ if intA(u _A) ∩ intB(u _B) ≠ ∅;

• 2) $\mathop{\max }\limits _{u^{{\prime}}\in R^{d}}\Phi ^{{\prime}AB} = 0,$ if intA(u _A) ∩ intB(u _B) = ∅ and frA(u _A) ∩ frB(u _B) ≠ ∅;

• 3) $\mathop{\max }\limits _{u^{{\prime}}\in R^{d}}\Phi ^{{\prime}AB}> 0,$ if A(u _A) ∩ B(u _B) = ∅.

It means that $\mathop{\max }\limits _{u^{{\prime}}\in R^{d}}\Phi ^{{\prime}AB}$ is a phi-function for objects A and B according to Definition A1.

1.4 Examples

Example 1.

Let v _i ¹ = (x _i ¹, y _i ¹), i = 1, …, m ₁, be the vertices of convex polygon K ₁(u ₁), and v _j ² = (x _j ², y _j ²), j = 1, …, m ₂, those of convex polygon K ₂(u ₂), and K ₁(u ₁) = {(x, y): ϕ _i ≤ 0,  i = 1, …, m ₁}, K ₂(u ₂) = {(x, y): ψ _j ≤ 0,  j = 1, …, m ₂}, ϕ _i = α _i ^′ x + β _i ^′ y + γ _i ^′, ψ _j = α _j ^{′ ′} x + β _j ^{′ ′} y + γ _j ^{′ ′}, where u ₁ = (x ₁, y ₁, θ ₁) and u ₂ = (x ₂, y ₂, θ ₂) are the placement parameters of polygons K ₁ and K ₂.

It should be noted that each point $(\widetilde{x},\widetilde{y})$ of non-translated and non-rotated convex polygon K is transformed into point (x, y):

$x =\widetilde{ x} \cdot \cos \theta _{K} +\widetilde{ y} \cdot \sin \theta _{K} + x_{K},y = -\widetilde{x} \cdot \sin \theta _{K} +\widetilde{ y} \cdot \cos \theta _{K} + y_{K},$ where (x _K,y _K) is a translation vector and θ _K is a rotation angle of K.

A phi-function for K ₁ and K ₂ can be defined in the form

$$\displaystyle{ \Phi ^{K_{1}K_{2} }\; =\;\max \{\mathop{\max }\limits _{1\leq i\leq m_{1}}\mathop{ \min }\limits _{1\leq j\leq m_{2}}\phi _{ij},\mathop{\max }\limits _{1\leq j\leq m_{2}}\mathop{ \min }\limits _{1\leq i\leq m_{1}}\psi _{ji}\}, }$$

(36)

where ϕ _ij = ϕ _i(v _j ²) = α _i ^′ x _j ² +β _i ^′ y _j ² +γ _i ^′, ψ _ji = ψ _j(v _i ¹) = α _j ^{′ ′} x _i ¹ + β _j ^{′ ′} y _i ¹ + γ _j ^{′ ′}.

Example 2.

Let us consider convex polygons K ₁ and K ₂ from Example 1.

A quasi-phi-function for K ₁ and K ₂ can be defined in the form

$$\displaystyle{ \Phi ^{{\prime}K_{1}K_{2} }(u_{1},u_{2},u_{P}) =\min \{\Phi ^{K_{1}P}(u_{ 1},u_{P}),\Phi ^{K_{2}P^{{\ast}} }(u_{2},u_{P})\}, }$$

(37)

where $\Phi ^{K_{1}P}(u_{1},u_{P}) =\mathop{ \min }\limits _{1\leq i\leq m_{ 1}}\psi _{P}(v_{i}^{1})$ is a phi-function of K ₁ and halfplane P(u _P), $\Phi ^{K_{2}P^{{\ast}} }(u_{2},u_{P}) =\mathop{ \min }\limits _{1\leq j\leq m_{2}}(-\psi _{P}(v_{j}^{2}))$ is a phi-function of K ₂ and halfplane P ^∗(u _P) = R ²∖intP(u _P).

In general, each of our phi-functions (ordinary, adjusted) is formed by operations of minimum and maximum of continuous and everywhere defined functions. The more operations of maximum take part in forming of a phi-function the more nonlinear programming subproblems we need to solve.

For example, in order to reach the global minimum for the problem of packing of two convex polygons K ₁ and K ₂ in a rectangle of minimum area, using phi-function (36), we need to solve m ₁ + m ₂ nonlinear programming subproblems optimally. See details in [18].

Alternatively, in order to reach the global minimum of the latter problem, using quasi-phi-function (37), we need to solve only one nonlinear programming problem optimally. However, in the case the problem dimension is increased by two.

We may reasonably combine phi-functions and quasi-phi-functions in our models depending on types of our objects.

Appendix 2: Moments of Inertia for Containers and Objects

• For the lateral surface of container $\Omega$ we have:

  $$\displaystyle\begin{array}{rcl} \Omega & \equiv &\mathbf{C}:\ \ J_{x_{0}} = J_{y_{0}} = \frac{1} {6}m_{0}(3R^{2} + 2H^{2}),\ \ J_{ z_{0}} = m_{0}R^{2}; {}\\ \Omega & \equiv &\boldsymbol{\Lambda }:\ \ J_{x_{0}} = J_{y_{0}} = \frac{1} {70}m_{0}H(21 + 16H),\ \ J_{z_{0}} = \frac{3} {5}m_{0}H; {}\\ \Omega & \equiv &\mathbf{E}:\ \ J_{x_{0}} = J_{z_{0}} = \frac{m_{0}} {2} \left (\frac{H^{2}(R_{1} + 3R_{2})} {3(R_{1} + R_{2})} + \frac{R_{1}^{2} + R_{2}^{2}} {2} \right ),\ \ J_{z_{0}} = \frac{m_{0}(R_{1}^{2} + R_{2}^{2})} {2}.{}\\ \end{array}$$

• For a homogeneous object A _i we have:

  $$\displaystyle\begin{array}{rcl} A_{i}& \equiv & \mathbb{S}_{i}:\ \ J_{x_{i}} = J_{y_{i}} = J_{z_{i}} = \frac{2} {5}m_{i}r_{i}^{2}; {}\\ A_{i}& \equiv & \mathbb{C}_{i}:\ \ J_{x_{i}} = J_{y_{i}} = \frac{1} {12}m_{i}(3r_{i}^{2} + 4h_{ i}^{2}),\ \ J_{ z_{i}} = \frac{1} {2}m_{i}r_{i}^{2}; {}\\ A_{i}& \equiv & \mathbb{T}_{i}:\ \ J_{x_{i}} = J_{y_{i}} = \frac{1} {8}m_{i}(4r_{i}^{2} + 5h_{ i}^{2}),\ \ J_{ z_{i}} = \frac{1} {4}m_{i}(4r_{i}^{2} + 3h_{ i}^{2}); {}\\ A_{i}& \equiv & \mathbb{S}_{\mathbb{C}i}:\ \ J_{x_{i}} = J_{y_{i}} = \frac{m_{i}} {2} \left (\frac{h_{i}(2h_{i}(l_{i}^{3} + 3l_{i}r_{i}^{2}) + (2l_{i}^{4} + 4l_{i}^{2}r_{i}^{2} + 3r_{i}^{4}))} {l_{i}^{3} + 6h_{i}r_{i}^{2} + 3l_{i}r_{i}^{2}} \right. + {}\\ & & +\!\left.\frac{7l_{i}^{5} + 5r_{i}^{2}(8h_{i}^{3} + 3l_{i}^{3} + 2l_{i}r_{i}^{2})} {10(l_{i}^{3} + 6h_{i}r_{i}^{2} + 3l_{i}r_{i}^{2})} \right ),\ \ J_{z_{i}}\,=\,\frac{m_{i}(l_{i}^{5} + 5l_{i}^{3}r_{i}^{2} + 30h_{i}r_{i}^{4} + 10l_{i}r_{i}^{4})} {10(l_{i}^{3} + 6h_{i}r_{i}^{2} + 3l_{i}r_{i}^{2})}.{}\\ \end{array}$$

For objects $A_{i} \equiv \mathbb{K}_{i}$ moments of inertia depends on a type of the cross-section polygon. For instance, we have

$J_{x_{i}} = \frac{1} {12}m_{i}(l_{i}^{2} + h_{ i}^{2}),\quad J_{ y_{i}} = \frac{1} {12}m_{i}(w_{i}^{2} + h_{ i}^{2}),\quad J_{ z_{i}} = \frac{1} {12}m_{i}(l_{i}^{2} + w_{ i}^{2})$ for cuboid, and $J_{x_{i}} = J_{y_{i}} = \frac{1} {24}m_{i}(5r_{i}^{2} + 8h_{ i}^{2})$, $J_{z_{i}} = \frac{5} {12}m_{i}r_{i}^{2}$ for straight regular prism with the regular hexagon base.

Appendix 3: Shor’s r-Algorithm

The r-algorithm is one of the Shor’s subgradient-type methods with the space transformation of variables (the space dilation) for minimisation of nonsmooth convex functions. Shor’s r-algorithms are based on two related ideas. The first idea lies in using the steepest descent method in the direction of antisubgradient of nonsmooth convex functions in the transformed space of variables. It ensures a monotonicity of a nonsmooth convex function for the minimizing the sequence which is constructed by r-algorithm. The second idea employs the operation of the space dilation in the direction of the difference of two subsequent subgradients in order to transform the space of variables; this permits to improve the properties of ravine-like functions in the transformed space. Combination of the ideas provides the accelerated convergence of r-algorithms for ravine-like functions ensuring their monotonicity (or almost monotonicity) under certain regulation of the step and the space dilation coefficients.

Let f(x) be a convex function, x be a vector of n variables. We assume that space dilation coefficients {α _k}_k = 0 ^∞ have to be greater than unity. Our r-algorithm for minimization of f(x) is an iterative procedure for finding sequence of vectors {x _k}_k = 0 ^∞ and matrices {B _k}_k = 0 ^∞ by the following rule:

$$\displaystyle{ x_{k+1} = x_{k} - h_{k}B_{k}\xi _{k},\quad B_{k+1} = B_{k}R_{\beta _{k}}(\eta _{k}),\quad k = 0,1,2,\ldots, }$$

(38)

where

$$\displaystyle{ \xi _{k} = \frac{B_{k}^{T}g_{f}(x_{k})} {\parallel B_{k}^{T}g_{f}(x_{k}) \parallel },\quad h_{k} =\arg \mathop{ \min }\limits _{h\geq 0}f(x_{k} - hB_{k}\xi _{k}), }$$

(39)

$$\displaystyle{ \eta _{k} = \frac{B_{k}^{T}r_{k}} {\parallel B_{k}^{T}r_{k} \parallel },\quad r_{k} = g_{f}(x_{k+1}) - g_{f}(x_{k}),\quad \beta _{k} = \frac{1} {\alpha _{k}} <1. }$$

(40)

Here, x ₀ is a starting point; B ₀ = I _n is a unity n × n-matrix (B ₀ is often taken to be diagonal matrix D _n with positive entries on a diagonal to make scaling of variables); h _k is a step multiplier (found from the condition of minimum of function f(x) in the direction of the normed subgradient in the transformed space of variables); α is a coefficient of the space dilation; R _β(η) = I _n + (β − 1)η η ^T is an operator of contraction of space of subgradients in the normed direction η with coefficient $\beta = \frac{1} {\alpha } <1$; g _f(x _k) and g _f(x _k+1) are subgradients of function f(x) at points x _k and x _k+1. If g _f(x _k) = 0, then x _k is a point of the minimum of function f(x), and process (38) and (40) stops.

Among r-algorithms the most efficient is r(α)-algorithm with α _k ≡ α and adaptive regulation of step h _k. The value of h _k is related to the unidimensional descent procedure in the direction of the normed antigradient in the transformed space of variables. The procedure involves parameters h ₀, q ₁, n _h, q ₂. Here h ₀ is the value of an initial step (it is used on the first iteration, and this value is sequentially refined on each iteration); q ₁ is a step decrease factor (q ₁ ≤ 1), if the descent stopping criterion is satisfied in one step; q ₂ is a step increase factor (q ₂ ≥ 1); natural number n _h specifies the number of steps in one-dimensional descent (n _h > 1)—after this number of steps the step size will be taken q ₂ times greater.

Guidance to the values of the space dilation coefficient as well as the parameters of adaptive regulation of a step is discussed in [15, pp. 104–105]. The values are aimed to improve the accuracy of finding of approximation to the minimum of the function, provided that the number of steps should not be too large (two-three per one iteration).

Stopping criteria in r(α)-algorithm is described by parameters ɛ _x and ɛ _g: calculations come to the end at point x _k+1, if $\left \|x_{k+1} - x_{k}\right \| \leq \varepsilon _{x}$ (stopping criterion by argument) or if $\left \|g_{f}(x_{k+1})\right \| \leq \varepsilon _{g}$ (stopping criterion by normed gradient, which is used for smooth functions). Abnormal program termination can happen if either function f(x) is not bounded below, or initial step h ₀ is too small and should be increased.

The following values of parameters are recommended for minimization of nonsmooth functions: α = 2 ÷ 3, h ₀ = 1. 0, q ₁ = 1. 0, q ₂ = 1. 1 ÷ 1. 2, n _h = 2 ÷ 3. If the priory bound of the distance from starting point x ₀ to the minimum point x ^∗ is given, then it is reasonable to choose initial step h ₀ to be approximately equal to ∥ x ₀ − x ^∗ ∥ .

For minimization of smooth functions the same parameters are recommended, except q ₁, that should be taken q ₁ = 0. 8 ÷ 0. 95. This can be explained in such a way: further step decreasing would provide finding a more accurate approximation to the minimum point of the function in the direction, and in the case of minimization of smooth functions this gives good rate of convergence. Under the parameters the number of descents is usually not greater than two, and after n steps the accuracy will be three-five times better. Stopping parameters ɛ _x, ɛ _g ∼ 10⁻⁶ ÷ 10⁻⁵ for minimization of a convex function (even the strongly ravine-like one) provide finding x _r ^∗ which is a fairly good approximation to the minimum point of the function.

Usually the condition $\frac{f(x_{r}^{{\ast}})-f(x^{{\ast}})} {\vert f(x^{{\ast}})\vert +1} \sim 10^{-6} \div 10^{-5}$ for nonsmooth functions (and ∼ 10⁻¹² ÷ 10⁻¹⁰ for smooth functions) is satisfied. It is confirmed by the results of numerous tests and applied calculations in linear and nonlinear programming problems, block problems with different schemes of decompositions, minimax and matrix optimization problems. It is also used for calculation of Lagrangian dual bounds in multiextremal and combinatorial optimization problems.

Shor’s r(α)-algorithm with the adaptive step regulation is realized by a number of programs. One of the simplest programs is octave-program ralgb5, which requires 5n ² arithmetical operations for each iteration [16]. The program uses octave’s function [f, g] = calcfg(x), which calculates values of function f = f(x) and its subgradient g = ∂ f(x) at point x.