Universidad de Santiago de Chile Departamento de Ingeniería Informática ALFA Project Víctor Parada Profesor Visitante en el Departamento de Ingeniería Industrial – Centro de Gestión de Operaciones –CGO Universidad de Chile

Universidad de Santiago de Chile Image Gallery:

Universidad de Santiago de Chile Departamento de Ingeniería Informática Undergraduate Conputer Science Engineering (Since 1982, 700 students). Master in Computer Science Engineering (30 students). Doctorate in Science of Computing and Informatic (5 students).

Research Topics

One-dimensional Problem Two-dimensional Problem Cutting stock problem

Two-dimensional Problem Cutting stock problem Three-dimensional Problem

Cutting stock problem a b c d e a d e a b b c Plate f 5

b i x i 4 Cutting Stock Problem Example

Min Z(x) = WL -  w i l i x i s.t. i) 0  x i  b i  i  R ii) x i integer  i  R iii) Cortes Factibles: - Guillotina - Sobreposición Wang, P. 1983, Opn. Res. (WA) Vasko, F Comp. Ind. Eng. Oliveira, J. & Ferreira, J. 1990, EJOR (MW) Viswanathan, K. & Bagchi, A Opn. Res. Parada, V. et al. 1995, EJOR (AAO*) Parada, V. et al. 1995, Nissen Ed. (GAO) Parada, V. et al. 1998, Comp&Ops. Res. (SA) Cutting Stock Problem

Solution Methods Wang’s algorithm Wang (WA) Constructive rectangles are generated. Vertical and horizontal combinations. An internal loss is defined. Trim (parameter  ). Oliveira & Ferreira (MW) Modifies Wang’s Method. Defines external loss.

Solution Methods Wang’s algorithm Algorithm - WA. Begin Choose a value for , 0    1; Define L (0) = F (0) ={ p 1, p 2,..., p n } k = 0 While F (k)   k= k+ 1; Determine F (k), a the set of rectangles T = { R 1, R 2,..., R n } satisfying: (i)T is a combination of rectangles belonging to L (k-1), (ii)the total loss of each R i is less than or equal to  1 HW, (iii)R i belonging to T do not surpass the limits b i 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.

AAO* Solution Methods And/Or Graph P P1P2 P3

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. AAO* Algorithm AAO* Solution Methods

Simulated Annealing Algorithm - SA. Begin Find an initial solution i and an initial value for T 0 ; 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) < exp{-d/T} then i = j; n = n + 1; Until n = N(t); t = t + 1; T = T(t); Until stop criterion achieved; End. Solution Methods

Algorithm Annealing (SA) Solution Methods P1 P2 (0,0) X Y (9x7) 3 6 P2 (9,7) (6,0) G1 G2P1 (0,0) (9,7) B1 B2 B3 (0,3) (0,0) (6,3) (4,7) (4,3) (6,7) (0,3) (6,7) (0,0) (6,7) 6x3 4x4 G2 G1 a) b) 9 7 Simulated Annealing

Genetic algorithm 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. Solution Methods

Genetic algorithm Resolution Methods “VVaaHHbbVcc” V VH aaHV bccb

Analysis between WA, MW, AAO*, SA and GAO. For WA, MW and AAO*  = (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) instances of grouped problems  = [2, 39] ). Platform : Silicon Graphics Challenge. 2 processors (150 MHz). 256 MB RAM. Quality, Running time and % of resolution are measured. Numerical Results

All Methods - % of Resolution Comparative Analysis

All Methods - % of trim loss

Comparative Analysis All Methods - Running Time

Comparative Analysis GAO and SA - Running Time

Comparative Analysis GAO and SA Methods - % of Trim loss

Characterization Running Time High T resp > 1 min. Medium 5 sec.  T resp  1 min. Low T resp < 5 sec. Problem size Small   4 medium 5   15 Large  16 Comparative Analysis

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

A web site to solve on-line optimization problems

Pequeñas Empresas: Problemas de optimización de materia prima Realidad Industrial

Empresas: Problemas de planificación de la producción Problemas de pronósticos de demanda

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”.

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...

Requerimientos de enzeñanza: Resolución “on-line” de: Problemas de programación lineal (Simplex) Problemas de Programación entera (B&B) Árboles de Cobertura de Costo mínimo. Flujo de Costo mínimo. Flujo Máximo.

Estructura del Sitio NoticiasEventos Parte Dinámica P1P2P3P4..Pn Problemas PLPNLPE P.M. AGBTSA M.H. Métodos de Resolución de Problemas Parte Estática Juegos de Optimización Clase de Optimización

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