Cutting and packing problems for irregular objects with continuous rotations: mathematical modelling and non-linear optimization

Abstract

We further improve our methodology for solving irregular packing and cutting problems. We deal with an accurate representation of objects bounded by circular arcs and line segments and allow their continuous rotations and translations within rectangular and circular containers. We formulate a basic irregular placement problem which covers a wide spectrum of packing and cutting problems. We provide an exact non-linear programming (NLP) model of the problem, employing ready-to-use phi-functions. We develop an efficient solution algorithm to search for local optimal solutions for the problem in a reasonable time. The algorithm reduces our problem to a sequence of NLP subproblems and employs optimization procedures to generate starting feasible points and feasible subregions. Our algorithm allows us to considerably reduce the number of inequalities in NLP subproblems. To show the benefits of our methodology we give computational results for a number of new challenger and the best known benchmark instances.

Appendices

Appendix A

We follow here the paper (Bennell et al, 2015) 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⊂R² (T=cl*(T)=cl(int(T)) having the same homothopic type as its interior (details of defenition one can find in Chernov et al (2010)). Each composed placement object T is given by an ordered collection of frontier elements l₁, l₂, …, l _n , (in counter clockwise order). Each element l _i is given by tuple 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 , is the center point of the arc generating circle C _i of radius |r _i |. 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)=(x₁, y₁) for i=n.

Given the ordered collection of line segments and circular arcs, in our programme we apply the object decomposition algorithm described by Chernov et al (2010) to obtain a set of basic objects automaticaly that covers our object T. 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 A1.

Figure A1

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

Thus, we represent placement object T in the form

where ℜ is the set of basic objects.

Example A

• Let us consider two objects A and B given in Figure A2a . These can be decomposed into basic objects according to formula (A), so we have A=H∪K, B=D∪V, V, D, H, K∈ℜ, n _A =n _B =2 (see Figure A2b).

  Figure A2

  

  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 verticies (x _i , y _i , i=1, 2, …, m).

Appendix B

Example B1

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

  

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

The transformed function takes the equivalent form according to (8)

Example B2

• Let us consider a region W described by inequality ϕ⩾0, where ϕ has the form defined in Example B1. We emphasize that ϕ⩾0 if at least min{ϕ₁, ϕ₃}⩾0 or min{ϕ₂, ϕ₃}⩾0 or min{ϕ₁, ϕ₄}⩾0 or min{ϕ₂, ϕ₄}⩾0.

Taking into account

we conclude that the region W can be presented as a union of subregions W₁, W₂, W₃ and W₄, described by the appropriate inequality systems in (B1), (B2), (B3) and (B4).

Appendix C

Data to Instance 4 of Subsection 4.2

Four types of arc objects are considered: A₁, A₂, A₃, A₄ (see Figure C1). Each object is defined by tuple l _A =(l₁, l₂, l₃), where i=1, 2, 3 (see Appendix A for details).

Figure C1

Four types of objects in Instance 4 of Subsection 4.2.

For the first type object: =(−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: =(−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: =(−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 forth type object: =(−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).