ALFA Project

1
ALFA Project
Universidad de Santiago de Chile Departamento de
Ingeniería Informática

• Víctor Parada
• vparada_at_dii.uchile.cl
• www.diinf.usach.cl/vparada
• www.optimos.usach.cl
• Profesor Visitante en el Departamento de
  Ingeniería Industrial Centro de Gestión de
  Operaciones CGO
• Universidad de Chile

2
Universidad de Santiago de Chile

• Image Gallery
• http//www.usach.cl/index2.php?id7196nomEstudia
  ntespag6515

3
Universidad de Santiago de Chilewww.usach.cl

• Departamento de Ingeniería Informática
• www.diinf.usach.cl
• Undergraduate Conputer Science Engineering
  (Since 1982, 700 students).
• Master in Computer Science Engineering (30
  students).
• Doctorate in Science of Computing and
  Informatic (5 students).

4

• Research Topics

5
Cutting stock problem
One-dimensional Problem
Two-dimensional Problem
6
Cutting stock problem
Two-dimensional Problem
Three-dimensional Problem
7
Cutting stock problem
a
3
Plate
b
2
3
c
e
b
1
d
b
d
5
a
e
c
a
5
f
8
Cutting Stock Problem
Example
bi xi
8 2 5 2 2 1 3 1 2 2
4
9
Cutting Stock Problem
Min Z(x) WL - S wilixi s.t. i) 0 xi bi
" iÎ R ii) xi integer " iÎ R iii) Cortes
Factibles - Guillotina - Sobreposición

• Wang, P. 1983, Opn. Res. (WA)
• Vasko, F. 1989 Comp. Ind. Eng.
• Oliveira, J. Ferreira, J. 1990, EJOR (MW)
• Viswanathan, K. Bagchi, A. 1993 Opn. Res.
• Parada, V. et al. 1995, EJOR (AAO)
• Parada, V. et al. 1995, Nissen Ed. (GAO)
• Parada, V. et al. 1998, CompOps. Res. (SA)

10
Solution Methods
Wangs algorithm
Wang (WA)

• Constructive rectangles are generated.
• Vertical and horizontal combinations .
• An internal loss is defined.
• Trim (parameter b).

Oliveira Ferreira (MW)

• Modifies Wangs Method.
• Defines external loss.

11
Solution Methods
Wangs algorithm
Algorithm - WA. Begin Choose a value for b, 0 ? b
? 1 Define L(0) F(0) p1, p2 , ... , pn k
0 While F(k) ? ? k k 1 Determine F(k), a the
set of rectangles T R1, R2 , ... , Rn
satisfying (i) T is a combination of rectangles
belonging to L(k-1), (ii) the total loss of each
Ri is less than or equal to b1HW, (iii) Ri
belonging to T do not surpass the limits bi of
each piece, Set L(k) L(k-1) ? F(k), eliminating
equivalent patterns Choose the rectangle of L(k)
with minimum total loss, End.
12
AAO
Solution Methods
And/Or Graph
13
AAO
Solution Methods
Algorithm AAO

• A solution is represented by means of And/Or
  graphs .
• A node represents a combination between pieces.
• Generalizes methods WA and MW.
• Optimal solutions can be obtained.

14
Simulated Annealing
Solution Methods
Algorithm - SA. Begin Find an initial
solution i and an initial value for T0 t
0 Repeat n 0 Repeat Generate solution
j neighbor to i d f(j) - f(i) If d ? 0
then i j Else If random(0,1) lt exp-d/T
then i j n n 1 Until n N(t) t t
1 T T(t) Until stop criterion
achieved End.
15
Solution Methods
Simulated Annealing
Algorithm Annealing (SA)
(0,0)
6
9
B1
X
(0,0)
(9,7)
G1
P2
3
P2
B2
(6,0)
(0,0)
P1
G2
(9,7)
(6,7)
7
G1
B3
(9x7)
(0,3)
(0,0)
Y
(6,3)
(6,7)
G2
P1
(0,3)
(4,3)
6x3
4x4
(6,7)
(4,7)
a)
b)
16
Genetic algorithm
Solution Methods
Genetic Algorithm (GAO)

• It is based on the evolutionary process, used
  in Genetic Algorithms.
• Syntactic binary trees representing the
  problem.
• A constructive solution is generated.
• The solution is a string.

17
Genetic algorithm
Resolution Methods
VVaaHHbbVcc
18
Numerical Results

• Analysis between WA, MW, AAO, SA and GAO.
• For WA, MW and AAO b (0, 0.08 and 1).
• For SA the Nº of iterations iter (10 and 100).
• For GAO the Size of population pop (n y 2n).
• 1000 instances of grouped problems (k 2, 39
  ).
• Platform Silicon Graphics Challenge.
• 2 processors (150 MHz).
• 256 MB RAM.
• Quality, Running time and of resolution are
  measured.

19
All Methods - of Resolution
Comparative Analysis
20
All Methods - of trim loss
Comparative Analysis
21
All Methods - Running Time
Comparative Analysis
22
Comparative Analysis
GAO and SA - Running Time
23
GAO and SA Methods - of Trim loss
Comparative Analysis
24
Characterization
Comparative Analysis
Running Time High Tresp gt 1 min. Medium
5 sec. Tresp 1 min. Low Tresp lt 5 sec.
Problem size Small k 4 medium
5 k 15 Large k ³ 16
25
Conclusions

• Several methods which resolves the Constrained
  Guillotine 2-Dimensional Cutting Problem have
  been implemented
• The theoretical results have been validated
  through 1000 instances of the CGTCP
• Problems have been organized according to their
  complexity degree
• Only GAO y SA could be considered reliable
• For small instances it is better to use exact
  algorithms

26

WWW.OPTIMOS.USACH.CL A web site to solve
on-line optimization problems


27
http//www.optimos.usach.cl
Pequeñas Empresas Problemas de optimización de
materia prima Realidad Industrial
28
http//www.optimos.usach.cl
Empresas Problemas de planificación de la
producción Problemas de pronósticos de demanda
29
http//www.optimos.usach.cl
30
Objetivo de Optimos

• Difundir conocimiento sobre el área Optimización
  de manera amena y clara a diversos tipos de
  usuarios y posibilitar la resolución de problemas
  de optimización de manera on-line.

http//www.optimos.usach.cl
31
http//www.optimos.usach.cl

• Requerimientos de empresariales
• Resolución on-line de
• Problemas de corte de piezas guillotinables.
• Problemas de corte de piezas no guillotinables.
• Problema de organización de actividades.
• Definición de horarios.
• Problema de distribución de energía eléctrica.
• Problemas de determinación de pit final...

32
http//www.optimos.usach.cl

• Requerimientos de enzeñanza
• Resolución on-line de
• Problemas de programación lineal (Simplex)
• Problemas de Programación entera (BB)
• Árboles de Cobertura de Costo mínimo.
• Flujo de Costo mínimo.
• Flujo Máximo.

33
Estructura del Sitio
Clase de Optimización
Parte Dinámica
Parte Estática
Juegos de Optimización
Métodos de Resolución de Problemas
Noticias
Eventos
Problemas
P1
P2
P3
P4
..Pn
P.M.
M.H.
PL
PNL
PE
AG
BT
SA
34

35
http//www.optimos.usach.cl

• Conclusiones
• Impacto en el sector productivo.
• Difusión de las potencialidades de la
  optimización.
• Alto costo de manutención.
• Distribución en diversas unidades académicas.