Cutting packing alonso bài toán cắt vật tư

Large objects Cutting & Packing Small objects Pieces Stock sheets... Container loadingDimension Objective Type of small objects Type of large objects Shape 3D Output maximization Weakly

Trang 1

Optimization methods for cutting and packing problems

Maria Teresa Alonso Martinez

Universidad de Castilla-La Mancha, España

Trang 2

Where is UCLM?

Trang 4

Cutting and packing problems?

Trang 5

• Less production costs

• Less waste of material

– Less natural resources consumption

http://www.dreamstime.com/stock-photo-conceptual-recycling-symbol-image3468870

Trang 6

Large objects

Cutting & Packing

Small objects

Pieces Stock sheets

Trang 7

1 Dimensionality

Characteristics

http://www.dcvamdmovers.com/moving_companies_2.JPG

Trang 8

2 Type of assignment

Characteristics

All the sheets with the maximum value of pieces

All the pieces with minimum number of sheets

Trang 9

3 Type of small pieces

Characteristics

Identical pieces

Weakly heterogeneous

Strongly heterogeneous

Trang 10

4 Type of large objects

Characteristics

Identical pieces

Weackly heterogeneuos

Strongly heterogeneous

Trang 11

Irregular pieces

http://ultranest.com/UltraNest.htm

Trang 12

Classification of problems

1 Dimension (1, 2, 3)

2 Objective

• Maximize output ( fixed input)

• Minimize input (fixed output)

3 Types of small objects (pieces, boxes,….)

• Several objects (identical or different)

5 Shape of the small objects (pieces)

• Rectangular

• Irregular

Trang 13

Container loading

Dimension

Objective

Type of small objects

Type of large objects

Shape

3D Output maximization Weakly heterogeneous One object

Rectangular

Trang 14

WHAT WAS RESEARCH?

Trang 15

MAXIMIZE OUTPUT

Summary of research efforts

SMALL OBJECTS LARGE

OBJECTS

WEAKLY HETEROGENEOUS

STRONGLY HETEROGENEOUS

FIXED DIMENSIONS

IDENTICAL SSSCSP SBSBPP

DIFFERENT MSSCSP/RCSP MBSBPP/RBPP

LARGE OBJECT OF VARIABLE DIMENSIONS ODP

Trang 16

MAXIMIZE OUTPUT

OBJECTS

Trang 17

MAXIMIZE OUTPUT

OBJECTS

Trang 18

MAXIMIZE OUTPUT

OBJECTS

Trang 19

MAXIMIZE OUTPUT

OBJECTS

Trang 20

MAXIMIZE OUTPUT

OBJECTS

Trang 21

MAXIMIZE OUTPUT

OBJECTS

Trang 22

AN EXAMPLE

Trang 23

• Given a set of stock sheets , with known dimensions and costs

The two-dimensional guillotine cutting stock problem

• and a set of pieces , with known dimensions and demands

Trang 24

• The problem is:how many sheets to cut? and in which way to cut them?

The two-dimensional guillotine cutting stock problem

• to satisfy the demands of pieces completely with minimum cost of sheets

Trang 28

Linear programming formulation

Q q

x

m i

d x

a t

s

x c Min

q

i Q

q

q iq

Q q

q q

, 0

, , 1

,

.

Q q

x q ≥ 0 , integer , ∀ ∈

i

q c

i d

q i

a

q x

Q

i q i iq q

constraint of

price dual

sheet) of

(cost

pattern of

cost

piece of

demand

pattern

in appears piece

times of

number

pattern use

we times of

number

patterns cutting

of set

Trang 29

Column generation procedure

1.- Generate an initial set of m patterns Q’,

one for each type of piece

2.- Solve the linear relaxation of the problem

over the set of variables Q’

Trang 30

3.- For each stock sheet p solve the subproblem :

i

i i p

S for sheet

ng pattern

is a cutti a

a t

s

a Max

z

} , ,

{ .

∑

Column generation procedure

If, for some p, z p > c p ,

add the column to Q’ and go back to Step 2

Otherwise, the process stops

p

a a

t

s { 1 , , }

Trang 31

Subproblem of Step 3

• Given a stock sheet

• and a set of pieces

• decide how many pieces of each type to cut

• in order to maximize the total value of pieces cut

Trang 32

A column of the formulation

Trang 33

2D EXAMPLE

Trang 34

MAXIMIZE OUTPUT

OBJECTS

Trang 35

Bidimensional Strip Packing Problem

n rectangular pieces

Min H

W

Trang 36

• Exact algorithms

– Guarantee optimality

– Can be very time-consuming

Solution methods

• Heuristics & metaheuristics

– Do not guarantee optimality

– Obtain good solution in reasonable computing times

Trang 37

Preprocessing: Fixing pieces

Trang 38

Branch & Bound : branching strategy

Root node (empty solution)

Trang 40

Branch & Bound : dominance

2

Trang 41

• More efficient use of space

1

Trang 42

• Avoid studying equivalent solutions

Trang 43

Effect of dominance and symmetry criteria

5 5

5

Trang 44

Implicit enumeration: bounding

Waste = 6

If we already have a solution with total waste = 4,

we can fathom this branch

Trang 45

Constructive algorithms : Bottom-Left

10

W Min H

Trang 46

Constructive algorithms : Bottom-Left-Fill

Trang 47

Constructive algorithms : Best-Fit

• Burke, Kendall, Whitwell, Operations Research 2005

8

Trang 48

Squeaky Wheel

Penalize

New ordering Priority space Solution space

Trang 49

Search by the geometry

Trang 50

NEW CHALLENGES

Trang 52

Loading trucks with real conditions…

Trang 53

• Putting the boxes in pallets.

2 phases problems:

Trang 54

2 phases problems

• Loading the pallets into trucks.

Trang 55

Boxes design

• If I want to get in stock 3 types of boxes.

What are the best sizes?

Trang 56

Combination problems

3 2

7

Trang 57

Irregular pieces (Nesting)

Trang 58

Optimization methods for cutting and packing problems

Maite Alonso

Universidad de Castilla-La Mancha, España

Định dạng
Số trang	58
Dung lượng	2,28 MB