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Optimization of a Nonlinear

Workload Balancing Problem

Stefan Emet

Department of Mathematics and Statistics

University of Turku

Finland

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Outline of the talk

Introduction

Some notes on Mathematical ProgrammingMINLP methods and solversSolution principlesSome advantages and disadvantages

MINLP model for the PCB-problemObjective - Maximizing profit under cyclic operation

Some example problemsSolution results

SummaryConclusions and some comments on future research issues

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Optimization problems are usually classified as follows;

Variables Functions

continuous:

masses,volumes, flowes

prices, costs etc.

discrete:

binary {0, 1}

integer {-2,-1,0,1,2}

discrete values{0.2, 0.4, 0.6}

linear non-linear

non-convex

quasi-convex

pseudo-convex

convex

Classification of optimization problems...

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variables

continuous integer mixed

linear

nonlinear

LP ILP MILP

NLP INLP MINLP

On the classification...

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MINLP-methods..

Branch and Bound MethodsDakin R. J. (1965). Computer Journal, 8, 250-255.

Gupta O. K. and Ravindran A. (1985).Management Science, 31, 1533-1546.

Leyffer S. (2001). Computational Optimization and Applications,18, 295-309.

Cutting Plane MethodsWesterlund T. and Pettersson F. (1995). An Extended Cutting Plane Method for Solving Convex MINLPProblems. Computers Chem. Engng. Sup., 19, 131-136.

Westerlund T., Skrifvars H., Harjunkoski I. and Porn R. (1998). An Extended Cutting Plane Method for

Solving a Class of Non-Convex MINLP Problems. Computers Chem. Engng., 22, 357-365.Westerlund T. and Prn R. (2002). Solving Pseudo-Convex Mixed Integer Optimization Problems byCutting Plane Techniques. Optimization and Engineering, 3,253-280.

Decomposition MethodsGeneralized Benders Decomposition

Geoffrion A. M. (1972).Journal of Optimization Theory and Appl., 10, 237-260.

Outer Approximation

Duran M. A. and Grossmann I. E. (1986).Mathematical Programming, 36, 307-339.

Viswanathan J. and Grossmann I. E. (1990). Computers Chem. Engng, 14, 769-782.

Generalized Outer Approximation

Yuan X., Piboulenau L. and Domenech S. (1989). Chem. Eng. Process, 25, 99-116.

Linear Outer Approximation

Fletcher R. and Leyffer S. (1994).Mathematical Programming, 66, 327-349.

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NLP-subproblems:

+ relative fast convergengeif each node can be solvedfast.

- dependent of the NLPs

MINLP-methods (solvers)...

Branch&Bound

minlpbb, GAMS/SBB

Outer Approximation

DICOPT

ECP

Alpha-ECP

MILP

MILP

NLP

NLPNLP

NLP NLP

NLP

MILP and NLP-subproblems:

+ good approach if the NLPscan be solved fast, and the

problem is convex.

- non-convexities impliessevere troubles

MILP-subproblems:

+ good approach if the

nonlinear functions arecomplex, and e.g. if gradientsare approximated

- might converge slowly ifoptimum is an interior point offeasible domain.

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Workload balancing problem...

Decision variables:

yikm=1, if component i is in machine k feeder m.

zikm= # of comp. i that is assembled from machine k and feeder m.

Feeders:

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Placement-head

PCB

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Production balancing problem...

Production lines:

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Line 1:

Line 2:

Line n:

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Objective (one production line)..

where is the assembly time of the slowestmachine:

KkzttsM

m

I

i

ikmik ,...,1,..1 1

K

k

kkYc

1max

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Optimize the profits during a period :

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constraints...

(slot capacity)km

M

m

ikmik Sys 1

i

K

k

M

m

ikm dz 1 1

(component to place)

(all components set)

0

ikmiikm ydz

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PCB example problems...

Problem characteristics:

Machines 3 3 3 3 6 6 6 6

Components 10 20 40 100 100 140 160 180

Tot. # comp. 404 808 1616 4040 4040 5656 6464 7272

Variables

Binary 90 180 360 900 1800 2520 2880 3240

Integer 90 180 360 900 1800 2520 2880 3240

Constraints

Linear 172 332 652 1612 3424 4784 5464 6144

cpu [sec] 0.11 0.03 3.33 2.72 5.47 6.44 11.47 121.7

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Objective (multiline system)...

where is the assembly time of the slowestmachine:

I

i

ililLl

tz

1,,1

maxmin

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Decision variables:

zil = # of products i that are assembled on line l

til = production time of product i on line

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Summary...

Though the results are encouraging there are issues to be tackled and/or

improved in a future research (in order to enable the solving of larger problemsin a finite time);

- refinement of the models

- implementation of convexification strategies

Some references

Emet S. et al. (2010). Workload balancing in printed circuit board assembly line.Int. Journal ofAdv Manufacturing Technology, 50, 1175-1182.

Porn R. et al. (2008). Global Solution of Optimization Problems with Signomial Parts.DiscreteOptimization, 5, 108-120.

Emet S. (2004).A Comparative Study of Solving Some Nonconvex MINLP Problems, Ph.D. Thesis,bo Akademi University.

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