Transcript Mathematical Programming Model

Mathematical Programming Model
Factory Automation Lab. SNU.
Jul. 30. 1999
Min, Dai ki
Contents
 Recent development in mathematical programming
modeling systems
Introduction
Modeling language extensions
Modeling interface extensions
Online optimization services
Steel industry example
New Directions in Algebraic Modeling Languages
 Paper review
Database structure for mathematical programming models
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Recent development in mathematical programming modeling systems
Introduction
 Modeler’ form - Algebraic
Conception
of Problem
Characteristics
Minimize (function of variables)
Subject to (function of variables) 
(constant)
Standard “mathematical” notation
Advantages
Familiar to everyone
Applicable to a broad variety of models
Extendible to nonlinearities, networks,
activities…
Computer
Algorithm’ s Form
of Problem
Algorithm’ s Form
of Results
Modeler’ s Form
of Results
Understanding
of Results
 Examples
 AMPL, GAMS, CPLEX, ILOG Opt Suite, OSL...
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Modeler’ s Form
of Problem
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Using an optimization model
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Recent development in mathematical programming modeling systems
Introduction
 Design objectives
Minimize disruption
Introduce few new features
Make new features follow established conventions
Impose few new rules
Maximize naturalness
Describe optimization problems in the way people think of them and
people think in many ways
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Recent development in mathematical programming modeling systems
Modeling languages extensions
 Complementarity problems
Collections of complementarity conditions
Express complementarity conditions in a natural and convenient way
 Stochastic programming
 Combinatorial optimization
Algebraic modeling languages have not been particularly successful for
modeling combinatorial optimization problems
Solver strategies
Convert to integer program
Extend branch-and-bound approach
Extend logic programming approach...
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Recent development in mathematical programming modeling systems
Modeling interface extensions
 Database features
Database links
Map optimization system’s data to relational data tables
Use ODBC
Database integration
Use relational database for a model’s data in place of the usual text files
Extending the database paradigm
Model base : Generalize the idea of a database to store whole cases models and data
Model management : Maintain a series of scenario, or library of a models
… scenarios in MathPro
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Recent development in mathematical programming modeling systems
Modeling interface extensions
 Model analysis support
Principles & practice
Access to input and output values
Access to analysis
Examples
Warm start after data change
Analysis of nonlinear functions
Interactive analysis
In analysis system’s environment : MProbe, ANALYZE
In modeling system’s environment : MIMI, AIMMS
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Recent development in mathematical programming modeling systems
Modeling interface extensions
 Application development
Model development cycle
Prototyping of model
Construction of user application
Maintenance and updating
System design approaches
Use modeling system to build application
Embed modeling system in application
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Recent development in mathematical programming modeling systems
Online optimization services
Services
URL
Services
URL
GIDEN
http://www.iems.nwu.edu/~
giden
NIMBUS
http://Nimbus.math.jyu.fi
IBM OSL
www.research.ibm.com/osl/
bench.html
Numerica
http://www.ilog.com/
other_files/Numerica/bin/nu
merica.cgi
www.rz.tu-ilmenau.de/~hqp
Decision Net
http://www.ini.cmu.edu/ema
rket
NEOS
Server
http://www.mcs.anl.gov/otc/
Server
HQP
/CODIN
MILP
/lp-solve
http://pinnacle.edrc.cmu.edu
:8080/milp.shtml
AMPL Remote Access
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http://www.ampl.com/ampl/TRYAMPL
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Recent development in mathematical programming modeling systems
Steel industry example
Optimization Approaches to Production Planning in the Steel
Industry
 Continuous deterministic & easy to solve
 Spreadsheet Tools for Planning
Reliance on rules of thumb to “maximize” profits
 Optimization Tools for Planning
Complex steel-making configurations
 Applications at Tata Steel in India
Generalized network flow linear program
4th Dimension 1.0 database software, Apple Macintosh II, XMP
linear programming library
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New Directions
in Algebraic Modeling
Languages
Recent development in mathematical programming modeling systems
 Integration
Model analysis support
Database connections
Application development tools
 Combinatorial optimization
 Online optimization services
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Database structure for
mathematical programming models
Robert Fourer
Department of Industrial Engineering and Management
Sciences, Northwestern University, IL, USA
DSS, Vol. 20, 1997, pp.317-344.
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Paper review
Contents
 Introduction
 Formulations
 Database structures
Relational structures
Hierarchical structures
 Comparisons
 Conclusions
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Paper review
Introduction
 What is the problem?
In the design and use of large-scale mathematical programming systems,
a substantial portion of the effort has no direct relation to the variable
and constraints.
 Goal of this paper
Codify some of the principles of database construction for LP
 Steel optimization model example
 This paper is based on a generic model
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Paper review
Formulations
 General formulation
1
2
3
n
Maximize  c j x j
A
j 1
n
row
i
Subject to l
l
col
j
  aij x j  uirow ,
i  1,..., m
j 1
 xj  u
col
j
j  1,..., n
,
 Specific model formulation
Generic continuous-flow production
process
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B
6
C
6
7
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Paper review
Formulations
 First formulation ;
max
c
jm
x buy

j
sell
j

x sell
-  c buy
x buy
j
j
j
jm
xijout 
( i , j )t out
xijin 
m : set of materials t : set of facilities

( j ' , j )mconv
a
in
ijk
xikact

( j , j ' )m co n v
a conv
x conv
 x sell

j
j' j
j' j
x
act
ik
x
in
ij
( i , j )t in
xijout 

c
act
ik
act
xik
( i , k )t a ct

( j , j ' )mconv
a
out
ijk
x conv
jj '
xikact
( i , j , k )Ao u t
( i , j , k )Ain
licap 
c conv
x conv
jj '
jj '
/ rikact  uicap
( i , k )t act
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Paper review
Formulations
 First formulation (con’t)
l buy
 xbuy
 u buy
j
j
j , for each j  m
l sell
 x sell
 u sell
j
j
j , for each j  m
0  x conv
,
jj '
for each ( j , j ' )  m conv
lijin  xijin  uijin ,
for each (i, j )  t in
lijout  xijout  uijout , for each (i, j )  t out
likact  xikact  uikact , for each (i, k )  t act
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Paper review
Formulations
 First formulation (con’t)
mconv  m  m conversions
t in  t  m facility inputs
t out  t  m facility outputs
t act  t  ? facility activities
Ain  t  m  ? activity inputs
Aout  t  m  ? activity outputs
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Paper review
Formulations
 Second formulation
max
c
sell
j
jm
x buy

j

x sell
-  c buy
x buy
-
j
j
j
jm
xijout 
it: jt o u t
xijin 
licap 
a
in
ijk
in
kti a ct : jAik
x
act
ik
jm j ' mco n v

j ' m: jm co n v
xikact

conv
c conv

jj ' x jj '
a conv
x conv
 x sell

j
j' j
j' j
c
it ktia ct
x
in
ij
it : jti in
xijout 


j ' m co n v
a
out
ijk
act
ik
xikact
x conv
jj '
xikact
out
kti o u t: jAik
/ rikact  uicap
kti act
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Paper review
Formulations
 Second formulation (con’t)
l buy
 xbuy
 u buy
j
j
j , for each j  m
l sell
 x sell
 u sell
j
j
j , for each j  m
0  xconv
,
jj '
for each j  m, j '  m j
lijin  xijin  uijin ,
for each i  t , j  ti
in
lijout  xijout  uijout , for each i  t,j  ti
out
likact  xikact  uikact , for each i  t,k  ti
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conv
act
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Paper review
Formulations
 Second formulation (con’t)
m conv
 m conversions from material j
j
tiin  m
inputs at facility i
tiout  m
outputs from facility i
tiact  ?
activities at facility i
Aikin  tiin
inputs to activity k at facility i
Aikout  tiout outputs from activity k at facility i
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Paper review
Database structures
- Relational structures
 General model ; α : set of constraints, β : set of variables
Maximize  c j x j
j
Subject to lirow 
 a
( i , j )
ij
for all i  
x j  uirow ,
l col
 x j  u col
j
j ,
for all j  
VARIABLES
CONSTRAINTS
row_name
row_min
row_max
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col_name
col_profit
col_min
col_optimal
col_max
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COEFFICIENTS
coeff_row -> CONSTRAINTS
coeff_col -> VARIABLES
coeff_value
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Paper review
Database structures
- Relational structures
 Principles of relational structure
Rule 1 (files)
For each set  in the model, there is a corresponding file in the database
Rule 2 (key fields)
The -file has a number of key fields equal to the dimension
 of
Rule 3 (data fields)
The -file has an additional data field for each model entity
indexed over
Rule 4 
(records)
The


-file has a record corresponding to each member of
Rule 5 (many-to-one relationships)
(..., id ,...)   id  
Foreach containment restriction of the form 
dth key in the -file has a many-to-one relationship to the
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the
-file
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Paper review
Database structures
- Relational structures
 Multi-facility production model
MATERIALS
FACILITIES
MATERIAL_CONVERSIONS
mat_name
buy_min
buy_opt
buy_cost
sell_min
sell_opt
sell_max
sell_cost
fac_name
cap_min
cap_max
from_mat -> MATERIALS
to_mat -> MATERIALS
conv_yield
conv_cost
conv_opt
FACILITY_INPUTS
in_fac -> FACILITIES
in_mat -> MATERIALS
in_min
in_opt
in_max
act_fac -> FACILITIES
act_name
act_min
act_opt
act_max
act_cost
act_cap_rate
FACILITY_OUTPUTS
ACTIVITY_INPUTS
act_in_fac -> FACILITIES
act_in_mat -> MATERIALS
act_in
act_in_rate
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ACTIVITIES
out_fac -> FACILITIES
out_mat -> MATERIALS
out_min
out_opt
out_max
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ACTIVITY_OUTPUTS
act_out_fac -> FACILITIES
act_out_mat -> MATERIALS
act_out
act_out_rate
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Paper review
Database structures
- Hierarchical structures
 General model
Maximize  c j x j
j
Subject to lirow 
 a
j
ij
x j  uirow ,
for all i  
i
l col
 x j  u col
j
j ,
for all j  
VARIABLES
col_name col_profit
col_optimal
CONSTRAINTS
row_name
row_min
row_max
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col_min
col_max
COEFFICIENTS
coeff_row -> CONSTRAINTS
coeff_value
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Paper review
Database structures
- Hierarchical structures
 Principles of hierarchical structure
Rule 1a (files)
For each unindexed set  in the model, there is a corresponding file in DB
Rule 1b (subfiles)
For each collection of sets  s indexed over s  the
corresponding to the collection  s
Rule 2 (key fields)
Rule 3 (data fields)
 -file has a subfile
An additional data field for each model entity indexed over  (or
Rule 4 (records)
Rule 5 (many-to-one relationships)
s)
j   j  
For each containment restriction of the form
,

the key record
to
 s in the -subfile has a many-to-one relationship
the -file
s
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Paper review
Database structures
- Hierarchical structures
 Multi-facility production model
MATERIALS
mat_name
buy_opt
buy_cost
sell_opt
sell_cost
buy_min
buy_max
sell_min
sell_max
CONVERSIONS
to_mat -> MATERIALS
conv_yield
conv_cost
conv_opt
FACILITEIS
fac_name
cap_min
cap_max
INPUTS
in_mat -> MATERIALS
in_min
in_opt
in_max
act_name
act_max
act_min
act_cost
act_opt
act_cap_rate
ACT_INPUTS
act_in_mat -> FACILITIES.INPUTS
act_in_rate
OUTPUTS
out_mat -> MATERIALS
out_min
out_opt
out_max
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ACTIVITES
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ACT_OUTPUTS
act_out_mat -> FACILITIES.OUTPUTS
act_out_rate
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Paper review
Comparisons
Ease of
use
Hierarchical
Relational
 Simple and straightforward
representation of the data
 Less common and minimally
standardizes
 Need a deeper understanding of
database principles
 Solid foundation
Data
storage
 All of two satisfy the normalization
 The difference in the data structure ; coeff_col & coeff_row
Data
retrieval
 Compactness vs. flexibility
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Paper review
Conclusions
 The principles of this paper shows the database structures
can be derived in a systematic way from sets and indexing
that are characteristics of mathematical programming data.
 The choice of index sets for the formulation of an
optimization problem is observed to involve certain
tradeoffs in convenience and efficiency of data access.
 Other indexing structure
Use modeling languages : MPL, AMPL, AIMMS…
 Other database type
Multidimensional database : On-Line Analytical Processing(OLAP)...
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References
 Database structures for mathematical programming models , Robert Fourer, DSS. vol.
20(1997), pp317-344.
 Recent development in mathematical programming modeling systems, Robert Fourer,
Conference of the Operational Research Society, Lancaster, September 8-10, 1998.
 AMPL new standard database features, Robert Fourer, Informs Meeting, Cincinnati,
May 2, 1999.
 Optimization Software
 http://www-fp.mcs.anl.gov/otc/Guide/SoftwareGuide/index.html
 Optimization FAQ on Linear and Nonlinear Programming.
 http://www-unix.mcs.anl.gov/otc/Guide/faq/linear-programming-faq.html
 Modeling solution ; AMPL
 http://www.modeling.com
 http://lionhrtpub.com/orms/surveys/LP/LP-survey.html
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