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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 Cruz15-17.12.2008. Outline of the talk…. Introduction Some notes on Mathematical Programming - PowerPoint PPT Presentation
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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
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
WSEAS Puerto de la Cruz 15-17.12.2008
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...
WSEAS Puerto de la Cruz 15-17.12.2008
variables
funct
ions
continuouscontinuous integerinteger mixedmixed
line
arli
near
nonl
inea
rno
nlin
ear
LPLP ILPILP MILPMILP
NLPNLP INLPINLP MINLPMINLP
On the classification...
WSEAS Puerto de la Cruz 15-17.12.2008
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
WSEAS Puerto de la Cruz 15-17.12.2008
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
WSEAS Puerto de la Cruz 15-17.12.2008
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:
WSEAS Puerto de la Cruz 15-17.12.2008
MINLP model for the SMB process...
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:
WSEAS Puerto de la Cruz 15-17.12.2008
MINLP model for the SMB process...
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
WSEAS Puerto de la Cruz 15-17.12.2008
MINLP-formulation summary...
Linear Linear constraintconstraint
ss
Non-linear Non-linear constraintsconstraints
Boundary value Boundary value problemproblem
ObjectiveObjective
WSEAS Puerto de la Cruz 15-17.12.2008
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.
WSEAS Puerto de la Cruz 15-17.12.2008
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.
WSEAS Puerto de la Cruz 15-17.12.2008
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
WSEAS Puerto de la Cruz 15-17.12.2008
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
WSEAS Puerto de la Cruz 15-17.12.2008
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
WSEAS Puerto de la Cruz 15-17.12.2008
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:
WSEAS Puerto de la Cruz 15-17.12.2008
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
WSEAS Puerto de la Cruz 15-17.12.2008
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
WSEAS Puerto de la Cruz 15-17.12.2008
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
WSEAS Puerto de la Cruz 15-17.12.2008
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.
WSEAS Puerto de la Cruz 15-17.12.2008