Placement Problems for Irregular Objects: Mathematical Modeling, Optimization and Applications

Abstract

We describe our methodology for solving NP-hard irregular placement problems. We deal with an accurate representation of objects bounded by circular arcs and line segments and allow their free rotations within a container. We formulate a basic irregular placement problem (IRPP), which covers a wide spectrum of practical packing, cutting, nesting, clustering, and layout problems. We provide a nonlinear programming (NLP) model of the problem, employing the phi-function technique. Our model involves a large number of inequalities with nonsmooth functions. We describe a solution tree for our placement problem and evaluate the number of its terminal nodes. We reduce IRPP problem to a sequence of NLP-subproblems with smooth functions.

Our solution strategy is based on combination of discrete and continuous optimization methods. We employ two approaches to solve IRPP problem: a branching scheme algorithm and an efficient optimization algorithm, which involves a feasible starting point and local optimization procedures. To show the benefits of our methodology we present computational results for a number of new challenger and the best known benchmark instances.

Appendices

Appendix 1

We follow here the paper [3] to define placement objects. We consider phi-objects as mathematical models of our placement objects. A two-dimensional phi-object T is a canonically closed point set T ⊂ R ² (T = cl ^∗(T) = cl(int(T)) having the same homotopic type as its interior (details of definition one can find in [2] and [8]). Each composed placement object T is given by an ordered collection of frontier elements l ₁, l ₂, …, l _n , (in counterclockwise order). Each element l _i is given by tuple $(x_i ,y_i,r_i ,x_{c_i } ,y_{c_i } )$ if l _i is an arc or by tuple (x _i , y _i , 0) if l _i is a line segment, where (x _i , y _i ) and (x _i+1, y _i+1) are the end points of l _i , $(x_{c_i } ,y_{c_i } )$ is the center point of the arc generating circle C _i of radius $\left | {r_i } \right |$. We note that l _i is a “convex” arc, if r _i  > 0; l _i is a “concave” arc, if r _i  < 0 and assume (x _i+1, y _i+1) = (x ₁, y ₁) for i = n.

Given the ordered collection of line segments and circular arcs, in our program we apply the object decomposition algorithm described in [8] to obtain a set of basic objects automatically that covers our object T. The authors of the paper prove that each phi-object T, bounded by line segments and circular arcs, may be presented as a union of a finite number of basic objects of four types, including convex polygons (K), circular segments (D), hats (H), and horns (V ). Each basic object is the intersection of primitive objects (half-planes, circles, and circular holes). Illustrations of primitive and basic objects are given in Figure 25.

Fig. 25

(a) Three types of primitive objects; (b) four types of basic objects

Thus, we represent placement object T in the form

$$\displaystyle \begin{aligned} T = \bigcup_{j = 1}^n {T_j } \quad \mathrm{with} \quad T_j \in \Re, \end{aligned} $$

(25)

where ℜ is the set of basic objects.

Example A

Let us consider two objects A and B given in Figure 26a. These can be decomposed into basic objects according to formula (25), so we have A = H ∪ K, B = D ∪ V , V, D, H, K ∈ ℜ, n _A  = n _B  = 2. See Figure 26b.

Fig. 26

Object decomposition: (a) objects A and B, (b) decomposition of A and B into basic objects

We set that a circle T ≡ C is defined by its radius (r _C ) and a convex m-polygon T ≡ K is defined by its vertices (x _i , y _i , i = 1, 2, .., m).

Appendix 2

Example B1

Let a basic phi-function Φ _k in (6) be given in the form

$$\displaystyle \begin{aligned} \varphi = \min \{\max \left\{ {\varphi _1 ,\varphi _2 } \right\},\max \left\{ {\varphi _3 ,\varphi _4 } \right\}\}, \end{aligned}$$

where φ _j  ∈{f}, j = 1, 2, 3, 4, here {f} is a family of smooth functions.

This function, according to (8), can take the following equivalent form:

$$\displaystyle \begin{aligned} \varphi = \max \{\min \left\{ {\varphi _1 ,\varphi _3 } \right\},\min \left\{ {\varphi _2 ,\varphi _3 } \right\},\min \left\{ {\varphi _1 ,\varphi _4 } \right\},\min \left\{ {\varphi _2 ,\varphi _4 } \right\}\}. \end{aligned} $$

(26)

Example B2

Let us consider a region W, which is described by inequality φ ≥ 0, where φ has the form (26). We emphasize that φ ≥ 0 if $\min \left \{ {\varphi _1 ,\varphi _3 } \right \} \ge 0$ or $\min \left \{ {\varphi _2 ,\varphi _3 } \right \} \ge 0$ or $\min \left \{ {\varphi _1 ,\varphi _4 } \right \} \ge 0$ or $\min \left \{ {\varphi _2 ,\varphi _4 } \right \} \ge 0$.

Taking into account

$$\displaystyle \begin{aligned} \min \left\{ {\varphi _1 ,\varphi _3 } \right\} \ge 0 \Leftrightarrow \left\{ {\begin{array}{l} \varphi _1 \ge 0 \\ \varphi _3 \ge 0 \\ \end{array}} \right., \end{aligned} $$

(27)

$$\displaystyle \begin{aligned} \min \left\{ {\varphi _2 ,\varphi _3 } \right\} \ge 0 \Leftrightarrow \left\{ {\begin{array}{l} \varphi _2 \ge 0 \\ \varphi _3 \ge 0 \\ \end{array}} \right., \end{aligned} $$

(28)

$$\displaystyle \begin{aligned} \min \left\{ {\varphi _1 ,\varphi _4 } \right\} \ge 0 \Leftrightarrow \left\{ {\begin{array}{l} \varphi _1 \ge 0 \\ \varphi _4 \ge 0 \\ \end{array}} \right., \end{aligned} $$

(29)

$$\displaystyle \begin{aligned} \min \left\{ {\varphi _2 ,\varphi _4 } \right\} \ge 0 \Leftrightarrow \left\{ {\begin{array}{l} \varphi _2 \ge 0 \\ \varphi _4 \ge 0 \\ \end{array}} \right., \end{aligned} $$

(30)

we can conclude that the region W can be presented as a union of subregions W ₁, W ₂, W ₃ and W ₄, described by the appropriate inequality systems in (27), (28), (29) and (30).

Appendix 3

Data to Instance n4 of Section 5.3

Four types of arc objects are considered: A ₁, A ₂, A ₃, A ₄ (see Figure 27). Each object is defined by tuple l _A =(l ₁, l ₂, l ₃), where $l_i = (x_i ,y_i ,r_i ,x_{c_i } ,y_{c_i } ), i = 1,2,3$ (see Appendix 1 for details).

Fig. 27

Four types of objects in Instance n4 of Section 5.3

For the first type object: $l_{A_1 }=(-127.793885277, -31.260629710, \\ -269.683141074, -114.585285651, -300.620109728, 107.745848162, \\ -147.983732947, -855.023127662, 880.040475106, 218.929823670, \\ 25.145613652, 204.120208658, -286.311966465, -260.582463369, \\ 222.396098309)$,

For the second type object: $l_{A_2 }=(-127.793885277, -31.260629710, \\ -222.243232453, -89.597792211, -250.196951150, 107.745848162, \\ -147.983732947, -341.501046399, 419.415318598, -8.394672165, \\ 114.198987902, 144.791346024, 164.379352457, 33.240703895, \\ 1.730772263)$,

For the third type object: $l_{A_3 }=(-190.389372841, 30.002879612, \\ 263.715084806 28.896179393, 176.493468167, 45.150360598, \\ -86.720223626,227.048705406, -125.107278015, 63.491131206, \\ 51.603500338, 206.054855345, -179.259124424, -127.468038006, \\ 197.856206394)$,

For the fourth type object: $l_{A_4 }=(-190.389372841, 30.002879612, \\ 181.347924897, -17.140772642, 83.593852669, 45.150360598, -86.720223626, \\ 214.700196049, -108.608670698, 63.127480656, 51.603500338, \\ 206.054855345, 477.417362860, 197.319382676, -248.581504831)$.