Transcript Convexification Techniques

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…
Introduction
Some notes on Mathematical Programming
Chromatographic separation – the process behind the model
MINLP model for the separation problem
Objective - Maximizing profit under cyclic operation
PDA constraints
Numerical solution approaches
MINLP methods and solvers
Solution principles
Some advantages and disadvantages
Some example problems
Solution results - Some different separation sequences
Summary
Conclusions and some comments on future research issues
WSEAS
Puerto de la Cruz
15-17.12.2008
Classification of optimization problems...
Optimization problems are usually classified as follows;
Variables
Functions
continuous:
discrete:
•masses,
volumes, flowes
•binary {0, 1}
•non-convex
•integer {-2,-1,0,1,2}
•quasi-convex
•discrete values
{0.2, 0.4, 0.6}
•pseudo-convex
•prices, costs etc.
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linear
Puerto de la Cruz
non-linear
•convex
15-17.12.2008
NLP
INLP
MINLP
LP
ILP
MILP
integer
mixed
linear
nonlinear
On the classification...
continuous
variables
WSEAS
Puerto de la Cruz
15-17.12.2008
The separationproblem...
A one-column-system:
C1
C2
H2O
 c1 (t )dt
 c2 (t)dt
Column 1
ckj
t
F
qkj
t
u
ckj
z
 Dj
Goal: Maximize the profits
during a cycle, i.e.
 2ckj
max 1/T*(incomes-costs)
z 2
C1
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C2
Puerto de la Cruz
1  T ti
max   y  c(t, zH )dt  y ti  ti1 

  i1 ti1
15-17.12.2008
A two-column-system with three components:
C2 C
3
C1
H2O
in
H2O
in
y1i
y 2i
Column 1
x1i1
y1i1
y1i2
C1
WSEAS
Column 2
y1i3
C2
C3
y 2i2
y 2i1
Waste
waste
In general C
PDEs/Column, i.e.
tot. K*C
x2i2
x2i1
x1i2
(Note 2*3 PDEs)
C1
C2
y 2i3
C3
Puerto de la Cruz
15-17.12.2008
MINLP model for the SMB process...
Objective function:
ti
 C


in
max
    p j ykij t ckj (t , z H )dt   wyki ti  ti 1 
i 1
 k 1i 1  j 1


1
K T
Price of products
Raw-material costs
Cycle length
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.
WSEAS
Puerto de la Cruz
15-17.12.2008
MINLP model for the SMB process...
PDEs for the SMB process:
C
C
ckj
 2 ckj
ckl

 ckj
 Fckj   jl
u
 Dj
1  F j  F   jl ckl 
t
z
z 2
l 1
l 1

 t
for j  1,, C , k  1,, K
F ,  j ,  jl , u and D j are parameters (e.g. estimated from data)
ckj (t ,0)  y kin (t )  c inj   xlk (t )  clj (t , z H )
l 1


in
c1 j (0,0)  c j
c (0, z )  c ( , z )
kj
kj


K
Boundary and initial
conditions:
Logical functions:
WSEAS
T
 in
in
 y k (t )   y ki   i (t )
i 1

T


 xlk (t )   xlik   i (t )
i 1


1, if t  ti 1 , ti , i  1, T
 i (t )  

0, otherwise.

Puerto de la Cruz
15-17.12.2008
MINLP model for the SMB process...
Integral constraints for the pure and unpure components;
ti
Equality constraints:
mkij   ckj (t , z H )dt
ti1
ti
Pure components:
mkij   ckj (t , z H )dt  M (1  ykij )
ti 1
Unpure components:
WSEAS
ti
C
ti 1
l 1
l j
mkij   ckj (t , z H )dt  M (1   ykil )
Puerto de la Cruz
15-17.12.2008
MINLP-formulation summary...
Objective
Linear
constraints
Boundary value
problem
Non-linear
constraints
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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
MINLP-methods (solvers)...
Branch&Bound
Outer Approximation
ECP
minlpbb, GAMS/SBB
DICOPT
Alpha-ECP
NLP
MILP
MILP
NLP
NLP
NLP
NLP
MILP-subproblems:
NLP
MILP and NLP-subproblems:
NLP-subproblems:
+ relative fast convergenge
if each node can be solved
fast.
- dependent of the NLPs
WSEAS
+ good approach if the NLPs
can be solved fast, and the
problem is convex.
- non-convexities implies
severe troubles
Puerto de la Cruz
+ 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.
15-17.12.2008
SMB example problems...
(separation of a fructose/glucose mixture)
Problem characteristics:
Columns
1
2
3
Variables
Continuous
Binary
34
14
63
27
92
71
Constraints
Linear
Non-linear
42
16
78
32
114
48
PDE:s involved
2
4
6
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Puerto de la Cruz
15-17.12.2008
Purity requirements:
Recycle
90% of product 1
Collect separated products
90% of product 2.
 c2
[g/100ml]
(0.03, 0.97)
(0.14, 0.86)
 c1
 c2
(0.9, 0.1)
(0.37, 0.63)
12
8
4
[min]
0
feed
Feed mixture
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43.5
57
recycle
116124.8
Recycle
Puerto de la Cruz
15-17.12.2008
Mixture
t=0-43.5 min
Water
Recycle 1
t=57-124.8 min
t=43.5 - 57 min
t=57-116 min
14,9 m
1
Fructose
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t= 0- 43.5 min
116-124.8 min
Puerto de la Cruz
Glucose
15-17.12.2008
Workload balancing problem...
Feeders:
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.
WSEAS
Puerto de la Cruz
15-17.12.2008
Objective...
Optimize the profits during a period τ:
 K
ck Yk



max  k 1











where τ is the assembly time of the slowest
machine:
M
I
s.t.    tik zikm
, k  1,..., K
m 1 i 1
WSEAS
Puerto de la Cruz
15-17.12.2008
constraints...
(slot capacity)
M
s
m 1
(all components set)
K
y
ik ikm
M
 z
k 1 m 1
(component to place)
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 S km
ikm
 di
zikm  di yikm  0
Puerto de la Cruz
15-17.12.2008
PCB example problems...
Problem characteristics:
Machines
3
3
3
3
6
6
6
6
Components
Tot. # comp.
10
404
20
808
40
1616
100
4040
100
4040
140
5656
160
6464
180
7272
Variables
Binary
Integer
90
90
180
180
360
360
900
900
1800
1800
2520
2520
2880
2880
3240
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/or
improved 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