an mixed integer approach for optimizing production planning

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

Outline of the talk…

IntroductionSome notes on Mathematical ProgrammingChromatographic separation – the process behind the model MINLP model for the separation problem

Objective - Maximizing profit under cyclic operation

PDA 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


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...


variables

funct

ions

continuouscontinuous integerinteger mixedmixed

line

arli

near

nonl

inea

rno

nlin

ear

LPLP ILPILP MILPMILP

NLPNLP INLPINLP MINLPMINLP

On the classification...


The separationproblem...

H2OC1

C2

C2C1

Column 1

A one-column-system:

dttc )(1 dttc )(2

2

2

z

cD

z

cu

t

qF

t

c kjj

kjkjkj

Goal: Maximize the profits during a cycle, i.e.

max 1/T*(incomes-costs)

1

1 1

),(1

max ii

T

i

t

t H ttydtztcyi

i


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 C PDEs/Column, i.e. tot. K*C x1i2x1i1 x2i1 x2i2

yin1i yin

2i

y1i1 y1i2 y1i3 y2i1y2i2 y2i3


K

k

T

iii

inki

C

j

t

t Hkjkijj ttwydtztcypi

i1 11

1 1

),(1

max

Price of products

Cycle length

Raw-material costs

ykij and ykiin are binary decision variables while ti and τ are continuous ones.

pj and w are price parameters. K = number of columns, T = number of time

intervals, C = number of components to be separated.

MINLP model for the SMB process...

Objective function:



PDEs for the SMB process:

2

2

11

1z

cD

z

cu

t

cFc

t

ccFF kj

jkj

C

l

kljlkj

kjC

lkljlj

data) from estimated (e.g. parameters are and ,,,

,,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

iii

T

iiliklk

T

ii

inki

ink

Logical functions:

Boundary and initial conditions:



Integral constraints for the pure and unpure components;

)1(),(1

kij

t

t

Hkjkij yMdtztcmi

i

Pure components:

i

i

t

tHkjkij

dtztcm1

),(Equality constraints:

Unpure components: )1(),(1

1

C

jll

kil

t

t

Hkjkij yMdtztcmi

i


MINLP-formulation summary...

Linear Linear constraintconstraint

ss

Non-linear Non-linear constraintsconstraints

Boundary value Boundary value problemproblem

ObjectiveObjective


MINLP-methods..

Branch and Bound Methods Dakin 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 Methods Westerlund 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 Pörn R. (2002). Solving Pseudo-Convex Mixed Integer Optimization Problems by Cutting Plane Techniques. Optimization and Engineering, 3, 253-280.

Decomposition Methods Generalized 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.


NLP-subproblems:

+ relative fast convergenge if each node can be solved fast.

- 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 NLPs can be solved fast, and the problem is convex.

- non-convexities implies severe troubles

MILP-subproblems:

+ good approach if the nonlinear functions are complex, and e.g. if gradients are approximated

- might converge slowly if optimum is an interior point of feasible domain.


SMB example problems...

(separation of a fructose/glucose mixture)

Problem characteristics:

Columns 1 2 3

Variables Continuous 34 63 92 Binary 14 27 71

Constraints Linear 42 78 114 Non-linear 16 32 48

PDE:s involved 2 4 6


Feed mixture

Collect separated productsPurity requirements:

90% of product 1

90% of product 2.

Recycle

Recycle

[min]

[g/100ml] (0.03, 0.97)

c2

feed

(0.14, 0.86)

recycle

(0.9, 0.1)

c1

(0.37, 0.63)

c2

43.5 57 116124.80

4

8

12


WaterWater

MixtureMixture

FructoseFructose

Recycle 1Recycle 1

GlucoseGlucose

1114,9 m 14,9 m

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

t=57-116 mint=57-116 min t= 0- 43.5 mint= 0- 43.5 min 116-124.8 min116-124.8 min

t=0-43.5 mint=0-43.5 min


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:


Optimize the profits during a period τ:

Objective...

where τ is the assembly time of the slowest machine:

KkzttsM

m

I

iikmik ,...,1,..

1 1

K

kkkYc

1max


constraints...

(slot capacity) km

M

mikmik Sys

1

i

K

k

M

mikm dz

1 1

(component to place)

(all components set)

0 ikmiikm ydz


PCB example problems...

Problem characteristics:

Machines 3 3 3 3 6 6 6 6

Components 10 20 40 100 100 140 160 180Tot. # 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


Summary...

Though the results are encouraging there are issues to be tackled and/orimproved in a future research (in order to enable the solving of larger problems in 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, Åbo Akademi University.

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


an mixed integer approach for optimizing production planning

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