8/8/2019 HYBRID METHODS FOR TWO-DIMENSIONAL CUTTING STOCK AND BIN PACKING PROBLEMS

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Universos of Minho

School of Engineering

Department of Production and Systems Engineering

Engenharia para a Qualidade de Vida: MOBILIDADE E ENERGIA Semana da Escola de Engenharia -11 a 16 de Outubro de 2010

In a two-dimensional cutting stock problem (2DCSP ) i t is intended to

cut a set of rectangular items from a set of rectangular plates in sucha way that the number of used plates is minimized. The plates are

avai lab le in (a v ir tual ly ) infini te number and al l have the same

dimensions, i.e., the same width and height.

The 2DCSP and its variants arise in different industrial processes,such as the textile, the leather, the paper, thefurniture and the sheet

metal. However there arenot general and efficient methods to solve it

due to its computational complexity (NP-hard).

ELSA SILVA*

Supervisors: Filipe Alvelos, J. M. Valério de Carvalho

* elsa@dps.uminho.pt

HYBRID METHODS FOR TWO-DIMENSIONAL CUTTING STOCK AND BIN PACKING

PROBLEMS

Two sets of ³real world´ instances from furniture companies (A and

B)and two sets of instances adapted fromthe literature(C andD).

Implemented in C++ using the ILOG Cplex 11.0 callable library and

compared with other approaches from the literature.

The propose model allows to deal with some poblem

variations:  Allowing rotation of items;

 Allowing plates with multiple dimensions;

Minimizing the length of cuts;

Maximizing the value of usable leftovers;

Minimizing waste.

However, the 2DCSP is usually optimized individually and notstudied

as an integrated process in the industries. The cutting productionplanning is a repetit ive process, occurring in the beginning of each

time period, based on thecostumers demands. Search for an optimal

solution for a 2DCSP in each period of the p lanning horizon, isdifferent froma completeplanninghorizon evaluation.

Multi-Item Lot-Sizing Problem:

It is intended to determine the quantities to produce of different

products over a given number of time periods in order to satisfy theproducts¶ demand for each product in each period.

Objective: Minimize the total cost (production + storage + setup).

Multi-Item Lot-Sizing Problem:

Combined Cutting Stock and Lot-Sizing Problem:

Determine the quant ity of i tems cut in each per iod, in order tominimize the cut operation costs, the storage costs and the wastage

fromthe plates.

Can be viewed as a balance between:

-  Antecipating the production of some i tems, minimizing the wastage

 And- Not antecipating production, minimizing the storage costs.

The mixed integer programming model proposed allows solvingopt imally the integrated 2CS-LSP, by modeling the production

anticipation of some items and also by storing object remainders, in

theend of each time periodof theplanninghorizon.

Two-dimensional cutting Stock Problem

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3-st ¡ ¢ 

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Explicitly considers how thedemanded items can be obtained

from the plates.

It is based on the enumerationof all possible cuts and of all

residual plates.

Main idea: associate a decisionvariable with the number of times

the cut is performed.

Objective: minimize the number 

of cuts in plates of the initial type.

Constraints: ensure that thedemand of each item type is

satisfied and ensure that the cuts

can be made only in existing

plates.

Integer Programming Model

Cutting Patterns

Computational Results

Setof Instances

Total

number of 

instances

Number of instances optimally solved in two hours in a laptop

(1.87 GHz Intel Core Duo 2GB RAM)

2 stage

exact2 stage non exact

¦ 

stage exact

¦ 

stage

non exact

Proposed

model

Proposed

model

Lodi et

al. (2004)Proposed

model

Puchinger 

and

Raidl

(2007)

Proposed

model

 A 47 44 44 25 4¦ 

26¦ 

9

B 121 115 119 22 119 22 120

C¦ 

0¦ 

0¦ 

0 29 29 28¦ 

0

D 20 20 12 4 6¦ 

5

Total 218 209 205 80 197 79 194

Combined Cutting Stock and Lot-Sizing Problem

Combined Cutting Stock and Lot-Sizing Problem

R&D CENTRE

ALGORITMI