Convexification Techniques

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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 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
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Classification of optimization problems...
Optimization problems are usually classified as
follows
Variables
Functions

• 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

• continuous
• masses, volumes, flowes
• prices, costs etc.

WSEAS Puerto de la Cruz 15-17.12.2008
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On the classification...
WSEAS Puerto de la Cruz 15-17.12.2008
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The separationproblem...
A one-column-system
Goal Maximize the profits during a cycle,
i.e. max 1/T(incomes-costs)
WSEAS Puerto de la Cruz 15-17.12.2008
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A two-column-system with three components
(Note 23 PDEs) In general C PDEs/Column, i.e.
tot. KC
WSEAS Puerto de la Cruz 15-17.12.2008
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MINLP model for the SMB process... Objective
function
Raw-material costs
Price of products
Cycle length
ykij and ykiin are binary decision variables
while ti and t 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
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MINLP model for the SMB process... PDEs for
the SMB process
Boundary and initial conditions
Logical functions
WSEAS Puerto de la Cruz 15-17.12.2008
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MINLP model for the SMB process... Integral
constraints for the pure and unpure components
Equality constraints
WSEAS Puerto de la Cruz 15-17.12.2008
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MINLP-formulation summary...
Objective
Linear constraints
Boundary value problem
Non-linear constraints
WSEAS Puerto de la Cruz 15-17.12.2008
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MINLP-methods..
WSEAS Puerto de la Cruz 15-17.12.2008
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MINLP-methods (solvers)...
Outer Approximation DICOPT
ECP Alpha-ECP
BranchBound minlpbb, GAMS/SBB
MILP
MILP
NLP
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.
MILP and NLP-subproblems good approach if the
NLPs can be solved fast, and the problem is
convex. - non-convexities implies severe troubles
NLP-subproblems relative fast convergenge if
each node can be solved fast. - dependent of the
NLPs
WSEAS Puerto de la Cruz 15-17.12.2008
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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 PDEs involved 2 4 6
WSEAS Puerto de la Cruz 15-17.12.2008
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Purity requirements 90 of product 1 90 of
product 2.
Collect separated products
Recycle
Feed mixture
Recycle
WSEAS Puerto de la Cruz 15-17.12.2008
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Mixture
t0-43.5 min
Water
Recycle 1
t57-124.8 min
t43.5 - 57 min
t 0- 43.5 min 116-124.8 min
t57-116 min
1
14,9 m
Fructose
Glucose
WSEAS Puerto de la Cruz 15-17.12.2008
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Workload balancing problem...
Feeders
Decision variables
yikm1, 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
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Objective...
Optimize the profits during a period t
where t is the assembly time of the slowest
machine
WSEAS Puerto de la Cruz 15-17.12.2008
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constraints...
(slot capacity)
(all components set)
(component to place)
WSEAS Puerto de la Cruz 15-17.12.2008
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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 4
784 5464 6144 cpu sec 0.11 0.03 3.33 2.7
2 5.47 6.44 11.47 121.7
WSEAS Puerto de la Cruz 15-17.12.2008
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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 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