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An Mixed Integer Approach for

Optimizing Production Planning

Stefan Emet

Department of Mathematics

University of Turku

Finland

WSEAS Puerto de la Cruz 15-17.12.2008

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Outline of the talk

Introduction

Some notes on Mathematical ProgrammingChromatographic separation the process behind the modelMINLP model for the separation problem

Objective - Maximizing profit under cyclic operationPDA constraints

Numerical solution approachesMINLP methods and solvers

Solution principlesSome advantages and disadvantages

Some example problemsSolution results - Some different separation sequences

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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The separationproblem...

H2OC1

C2

C2C1

Column 1

A one-column-system:

dttc )(1 dttc )(2

2

2

zcD

zcu

tqF

tc kjjkjkjkj

Goal: Maximize the profitsduring a cycle, i.e.

max 1/T*(incomes-costs)

1

1 1

),(1

max iiT

i

t

t H ttydtztcy

i

i


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A two-column-system with three components:

H2O

Column 1 Column 2

waste

H2O

C1 C2 C3 C1 C2 C3

C3C2

C1

Waste

(Note 2*3 PDEs)

In general CPDEs/Column, i.e.

tot. K*Cx1i2x1i1 x2i1 x2i2

yin

1iy

in

2i

y1i1 y1i2 y1i3 y2i1 y2i2 y2i3


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K

k

T

iii

in

ki

C

j

t

t Hkjkijj ttwydtztcyp

i

i1 11

1 1),(

1max

Price of productsCycle length

Raw-material costs

ykijand ykiinare binary decision variables while tiand are continuous ones.

pjand ware price parameters. K = number of columns,T = number of timeintervals,C =number of components to be separated.

MINLP model for the SMB process...

Objective function:


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PDEs for the SMB process:

2

2

11

1z

cD

z

cu

t

cFc

t

ccFF

kj

j

kjC

l

kljlkj

kjC

l

kljlj

data)fromestimated(e.g.parametersareand,,,

,,1,,,1for

jjlj DuF

KkCj

),(),0(

)0,0(

),()()()0,(

1

1

zczc

cc

ztctxctytc

kjkj

in

jj

K

lHljlk

in

j

in

kkj

otherwise.,0

,1,,if,1)(

)()(

)()(

1

1

1

Titttt

txtx

tyty

ii

i

T

i

iliklk

T

i

i

in

ki

in

k

Logical functions:

Boundary and initialconditions:


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Integral constraints for the pure and unpure components;

)1(),(

1

kij

t

t

Hkjkij yMdtztcmi

i

Pure components:

i

i

t

t

Hkjkij dtztcm

1

),(Equality constraints:

Unpure components: )1(),(1

1

C

jl

l

kil

t

t

Hkjkij yMdtztcmi

i


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MINLP-formulation summary...

Linearconstraints

Non-linearconstraints

Boundary valueproblem

Objective


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MINLP-methods..

Branch and Bound M ethodsDakin 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.

Cutti ng Plane MethodsWesterlund T. and Pettersson F. (1995). An Extended Cutting Plane Method for Solving Convex MINLP

Problems. 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 by

Cutting 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 theproblem 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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SMB example problems...

(separation of a fructose/glucose mixture)

Problem characteristics:

Columns 1 2 3

Variables

Continuous 34 63 92Binary 14 27 71

Constraints

Linear 42 78 114

Non-linear 16 32 48

PDE:s involved 2 4 6


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Water

Mixture

Fructose

Recycle 1

Glucose

114,9 m

t=57-124.8 min t=43.5 - 57 min

t=57-116 min t= 0- 43.5 min

116-124.8 min

t=0-43.5 min


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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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Optimize the profits during a period :

Objective...

where is the assembly time of the slowestmachine:

KkzttsM

m

I

i

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

K

k

kkYc

1max


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constraints...

(slot capacity)km

M

m

ikmik Sys 1

i

K

k

M

mikm 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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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

- further development of the numerical methods

Some references

Emet S. and Westerlund T. (2007). Solving a dynamic separation problem using MINLP

techniques.Applied Numerical Matematics.

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

Westerlund T. and Prn R. (2002). Solving Pseudo-Convex Mixed Integer Optimization Problemsby Cutting Plane Techniques. Optimization and Engineering, 3, 253-280.


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