Optimization methods for cutting and packing problems Maria Teresa Alonso Martinez Universidad de Castilla-La Mancha, España Where is UCLM? Outline • Introduction and classification • Some problems – Cutting Stock Problem: – Strip Packing Problem: • Future works Column Generation Branch & Bound Cutting and packing problems? If we were capable of solving this problems… • Less production costs • Less waste of material – Less natural resources consumption – Less pollution • Waste,… • Transport – Less energy consumption http://www.dreamstime.com/stock-photo-conceptual-recycling-symbol-image3468870 Cutting & Packing Large objects ¿Patterns? Small objects Stock sheets Pieces Characteristics Dimensionality http://www.dcvamdmovers.com/moving_companies_2.JPG Characteristics Type of assignment Stock sheets Pieces All the pieces with minimum number of sheets All the sheets with the maximum value of pieces Characteristics Type of small pieces Identical pieces Weakly heterogeneous Strongly heterogeneous Characteristics Type of large objects Identical pieces Weackly heterogeneuos Strongly heterogeneous Outline • Introduction and classification • Some problems – Cutting Stock Problem: – Strip Packing Problem: • Future works Column Generation Branch & Bound Constructive algorithms : Bottom-Left Min H W 10 Constructive algorithms : Bottom-Left-Fill Constructive algorithms : Best-Fit • Burke, Kendall, Whitwell, Operations Research 2005 10 10 Squeaky Wheel Priority space Solution space Penalize New ordering Search by the geometry NEW CHALLENGES Some challenges… • Loading trucks with real constraints: weight, equilibrium,… • phases problems: make pallets and load into trucks • Design of boxes • Combination of problems: routing y loading • Irregular pieces Loading trucks with real conditions… phases problems: • Putting the boxes in pallets phases problems • Loading the pallets into trucks Boxes design • If I want to get in stock types of boxes What are the best sizes? Combination problems D Irregular pieces (Nesting) Optimization methods for cutting and packing problems Maite Alonso Universidad de Castilla-La Mancha, España [...]... HETEROGENEOUS SBSBPP FIXED DIMENSIONS DIFFERENT LARGE OBJECT OF VARIABLE DIMENSIONS MSSCSP/RCSP MBSBPP/RBPP ODP Bidimensional Strip Packing Problem n rectangular pieces Min H W Outline • Introduction and classification • Some problems – Cutting Stock Problem: – Strip Packing Problem: • Future works Column Generation Branch & Bound ... list of all possible cutting patterns p1 p2 p3 p4 p5 p6 p7 p8 Pi=times pattern i is used One-dimensional model Min 4 p1 + 4 p2 + 4.5 p3 + 4.5 p4 + 3 p5 + 3 p6 + 3.5 p7 + 4 p8 s.a + p5 p1 2p2 + p3 p1 + 2 p3 +4p4 pi ≥ 0, entero ≥ 60 +2p6 +p7 ≥ 40 +p7 +3p8 ≥ 75 Linear programming formulation Min ∑c x q∈Q s.t ∑a q∈Q q q x ≥ d i , i = 1, , m iq q xq ≥ 0, integer, ∀q ∈ Q Q = set of cutting patterns xq =... DIFFERENT LARGE OBJECT OF VARIABLE DIMENSIONS MSSCSP/RCSP MBSBPP/RBPP ODP AN EXAMPLE The two-dimensional guillotine cutting stock problem • Given a set of stock sheets, with known dimensions and costs • and a set of pieces, with known dimensions and demands The two-dimensional guillotine cutting stock problem • The problem is:how many sheets to cut? and in which way to cut them? • to satisfy the demands