Past, Present, and Future

Workshop on Optimization/Modeling/Applications

Washington, DC September 19, 2003

GAMS

Alex Meeraus

GAMS Development Corporation

2

Model Structure

3

Model Data

4

Matrix Generator

5

Matrix Generator Input

6

MPS File – Column Section

7

MPS Revision File

8

MPS Output

9

WB Old Slide 1

10

WB Old Slide 2

11

WB Old Slide 3

12

WB Old Slide 4

13

WB Old Slide 5

14

WB Old Slide 6

15

WB Old Slide 7

16

Basic Principles

• Separation of model and solution methods

•Models is a data base operator and/or object

• Balanced mix of declarative and procedural approaches

• Computing platform independence

•Multiple model types, solvers and platforms

17

Computation Past

• Algorithm limits applications

• Problem representation is low priority

• Large costly projects

• Long development times

• Centralized expert groups

• High computational cost, mainframes

• Users left out

Model Present

• Modeling skill limitsapplications

• Algebraic model representation

• Smaller projects

• Rapid development

• Decentralized modeling teams

• Low computational cost, workstations

• Machine independence

• Users involved

ApplicationFuture

• Domain expertise limits application

• Off-the-shelf graphical user interfaces

• Links to other types of models

• Models embedded in business applications

• New computing environments

• Internet/web

• Users hardly aware of model

Change in Focus

18

Dinner Table Assignment

• ‘Couples” on the same table

• Mix backgrounds and interest (algorithms,applications,World Bank, GAMS staff and family)

• Allow for last minute changes

• Use a model to shed responsibility for ‘poor’ assignments

19

List of Participants

20

Problem Domains

21

New Excel Input

22

Model Definition

23

Solve Summary

24

Output Report

25

Group Balance

26

Excel Report

27

Data Connectivity

• Data Import/Export from Standard Applications– MS Office, Database, Text files, …

• Capture an Instance of a Model Failure– Reproducibility of Model/System Bugs

– Problems: Life Database/different Platforms

• Definition of Data Interface– Gams Data eXchange (GDX)

– Separation of Responsibility for Data and Model

28

Gams Data eXchange

• Gams Data eXchange (GDX):

• Complements the ASCII text data input

• Advantages:– Fast exchange of data

– Syntactical check on data before model starts

– Compile-time and Run-time Data Exchange

ApplicationGDX

GAMS

29

GDX Tools

GDX

gdxxrw(MS Office)

gams

IDE GDX Viewer

gdxdiff

gdxdump

gdxmerge

gdxsplitGDXAPI

30

Data Contract in GAMS

31

Data Contract in Excel

32

Data Contract in Excel – cont.

33

GAMS/MapInfo

Less Than 0

0 - 199

200-1000

1000-3000

Increase in Ktons Per Year

34

GAMS/MATLAB

35

36

Three Examples of GDX

• VEDA : post processing and ad-hoc information restructuring

• LOGMIP: experimental language extensions, algorithmic design studies

• Branch & Cut & Heuristic: automation of complex solver enhancements

37

The VErsatile Data Analyst (VEDA)

Amit Kanudia, KanORS Consulting Inc.

Gary A. Goldstein, IRG

38

VEDA Description• Manages GAMS Sets, Parameters and Model

Results for the purpose of dynamic viewing and re-organizing, and presentation to others.

• Allows for cross-cutting access to parameters and results according to index structure and element values.

• Supports construction of user-defined sets, definition of new tables, and manipulation of tables.

• Provides powerful filtering capabilities on both elements names and descriptions.

39

VEDA Description• Employs customizable n-dimensional Data

Cube component for displaying tables.• Full-blown graphing facilities, as well as

exporting to Excel, Word, PPT, HTML.• Web-enabled, allowing model results to be

made available over the web in user-friendly, reconfigurable tables.

• Convenient GDX2VEDA data definition file for identification of information to be ported from GAMS to VEDA.

40

VEDA Table Form – Attribute List

41

VEDA Table Form – Sets (Model&User)/Elements

42

VEDA Table Form – Search & Select Engine

43

ProcessSet/Process/Commodity/Scenario

44

VEDA Cube – Scenario/ProcessSetTotals

45

VEDA Cube – Analysis Graph

46

GDX2VEDA Specification

47

Characterization and Formulation of Disjunctions and their Relaxations

Aldo Vecchietti (*), Sangbum Lee(+) and Ignacio Grossmann(+)

(+) Chemical Engineering Department - Carnegie Mellon University(*) INGAR - Instituto de Desarrollo y Diseño

UTN - Facultad Regional Santa Fe

LOGMIP

48

Accepted Formulations

MILP and MINLP formulations are the most accepted byacademia and the industry for problems involving discretedecisions

Recent formulations

Constrained Logic Programming (CLP): general logic constraints, applied mainly for highlycombinatoric problems

Generalized Disjunctive Programming (GDP) :disjunctions and logic propositions for discretechoices

Problem Formulations

49

min Z = Σk ck + f(x) + dTysujeto a:

g(x) ≤ 0

r(x) + Dy ≤ 0Ay ≥ a

Yik

∨ hik (x) ≤ 0 k∈ SDi∈Dk ck = γik

Ω (Y) = Truex∈ Rn, y∈ 0,1q

Y∈ True, Falsem , ck ≥ 0

q y binary variables (0-1)q x y ck continuous variablesq Yik Boolean variables to establish if a

disjunction term is true or false q f(x) objective function (linear or not linear)q dTy Linear cost term q g(x) constraint (linear or not linear)

independent of discrete choicesq r(x)+Dy≤ 0 mixed integer constraint

(linear or not linear) q Ay≥ a integer constraintqΩ (Y) propositional logic relating Boolean

variables (disjunction terms)

Disjunctions

50

Big-M

Linear

[ ] ni

Ti

DiR x b x a F ∈≤∨=

∈

)y-(1M b x a iiiTi +≤

1 yi

i =∑

Mi = max aiT x – bi | xlo≤ x≤ xup

Non Linear

[ ] ni

DiR x 0 (x)h F ∈≤∨=

∈

Beaumont Surrogate

1 -N Mb

M

xa

i i

i

i i

Ti +∑≤∑ 1 -N

M)x(h

i i

i ≤∑

hi(x)≤ Mi(1-yi)

Mi = max hi(x) | xlo≤ x≤ xup

1 yi

i =∑

Big-M Relaxation

51

iupi

i

iDi

i

iiii

ni

Dii

y v v 0

D i 1, y 0 , 1 y

0 )y / v(h y

Rv x, 0 vx -

≤≤

∈≤≤∑ =

≤

∈=∑

∈

∈

y v v 0

D i 1, y 0 , 1 y

0 y b -v a

Rv x, 0 vx -

iupii

iDi

i

iiiTi

ni

Dii

≤≤

∈≤≤∑ =

≤

∈=∑

∈

∈

[ ] ni

Ti

DiR x b x a F ∈≤∨=

∈

Linear Non linear

[ ] ni

DiR x 0 (x)h F ∈≤∨=

∈

Convex Hull Relaxation

52

The x space

false,trueY

c;x,x

c

xx

Y

c

xx

Y

.t.s

c).x().x(minZ

∈

≤≤≤

=

==¬

∨

=

≤+

+−+−=

1

121

1

21

1

1

22

21

1

12

22

1

010

0

0

1

1

1111

Improper disjunction ofspecial interest in Process

Engineering

Both relaxations have the same feasible region

X1

X2

1

1

Relaxation Comparison

53

X1

X2

yConvex hull

X1

X2

y Big-M

10

0

0

1111

1

12

11

21

22

21

12

22

1

≤≤≤≤

≤≤≤+

+−+−=

y

yx

yx

yxx

.t.s

y).x().x(minZ

1010

1111

121

122

21

12

22

1

≤≤≤≤≤+

+−+−=

y;x,x

yxx

.t.s

y).x().x(minZ

X – Y Space

54

Modeling two terms disjunction

∨

2Constraint

se Fal

1 Constraint

True

IF (condition1) THENConstraints to be considered when condition1 is True

ELSEConstraints to be considered when condition1 is False

END IF

Two Term Disjunction

55

∨∨

∨

N sConstraint

N ..

2sConstraint

2

1 sConstraint

1

IF (condition1) THENConstraints to be considered when condition1 is True

ELSIF (condition2) THENConstraints to be considered when condition1 is True

ELSIF (condition3) THEN...

ELSIF (conditionN) THENConstraints to be considered when condition1 is True

END IF

N Term Disjunction

56

.3,2,1,,

0 ,5050

01

7

02

5

02

2min

21

32

321

1

3

21

3

2

2

21

1

21

=∈≥≤≤≤≤

∧¬¬⇒¬∧

=

¬∨

≤−

=≤−∨

=≤++−

++

jfalsetrueY

cx, x

)Y (Y

Y Y Y

x

Y

xx

Y

c

x

Y

c

xx

Y

x x c

j

Example

57

$ONTEXT BEGIN LOGMIP$ONTEXT BEGIN LOGMIP

IF Y('1') THENIF Y('1') THEN

EQUAT1;EQUAT1;

EQUAT2;EQUAT2;

ELSIF Y('2') THENELSIF Y('2') THEN

EQUAT3;EQUAT3;

EQUAT4;EQUAT4;

ENDIF;ENDIF;

IF Y('3') THENIF Y('3') THEN

EQUAT5;EQUAT5;

ELSEELSE

EQUAT6;EQUAT6;

ENDIF;ENDIF;

$OFFTEXT END LOGMIP$OFFTEXT END LOGMIP

OPTION MIP=LOGMIPC;OPTION MIP=LOGMIPC;

MODEL PEQUE2 /ALL/;MODEL PEQUE2 /ALL/;

SOLVE PEQUE2 USING MIP MINIMIZING Z;SOLVE PEQUE2 USING MIP MINIMIZING Z;

EQUAT1.. X('2')EQUAT1.. X('2')-- X('1') + 2 =L= 0;X('1') + 2 =L= 0;

EQUAT2.. C =E= 5;EQUAT2.. C =E= 5;

EQUAT3.. 2 EQUAT3.. 2 -- X('2') =L= 0;X('2') =L= 0;

EQUAT4.. C =E= 7;EQUAT4.. C =E= 7;

EQUAT5.. X('1')EQUAT5.. X('1')--X('2') =L= 1;X('2') =L= 1;

EQUAT6.. X('1') =E= 0;EQUAT6.. X('1') =E= 0;

INT1.. Y('1')+ Y('3') =L= 1;INT1.. Y('1')+ Y('3') =L= 1;

INT2.. Y('2')+ (1INT2.. Y('2')+ (1--Y('3')) =G= 1;Y('3')) =G= 1;

INT3.. Y('2')+ Y('3') =L= 1;INT3.. Y('2')+ Y('3') =L= 1;

FICT.. SUM(I, Y(I)) =G= 0;FICT.. SUM(I, Y(I)) =G= 0;

OBJECTIVE.. Z =E= C + 2*X('1') + OBJECTIVE.. Z =E= C + 2*X('1') + X('2');X('2');

X.UP(J)=20;X.UP(J)=20;

C.UP=7;C.UP=7;

GAMS Implementation

Branch-and-Cut & Heuristics in GAMS for MIP Problems

Hua (Edward) NiGeorge Washington University

59

Branch-and-Cut (B&C)

• B&C is an established algorithm to improve the B&B search.

• Implementation facilities:– MIP solver callback functions (CPLEX, XPRESS, …)– B&C framework (ABACUS, COIN BCP, …)

• Required Knowledge for B&C– IT knowledge (C/C++/JAVA, Solver APIs)– Mathematical programming knowledge– Application specific knowledge

• Supply GAMS users with an easy access to B&C

Design Principle

GAMS SYSTEM

GAMSSolverLink

MIP Solver(e.g. CPLEX)

B&B Callback

UserCut

Generator&

Heuristics(e.g. GAMS)

A Steiner Tree Problem

• Berlin52 – from SteinLib– 52 nodes (1 source, n’=15 sinks), 1326 edges

• Flow formulation:

• Dicut: if S⊂V and b(S)>0.

• Separation:

1,0,'0,..,min)(),()(),(),(

∈≤≤=− ∑∑∑+− ∈∈∈

yynxbxxtsyf ijijiijiij

iijji

Ajiijij

δδ

62

Cut Generator in GAMS I

63

Cut Generator in GAMS II

Computational Results

• Overhead– Time spent within the callback functions minus

MIP computation on cuts and heuristics.– 20% ~ 25%

• Performance Improvements (B&C vs. regular GAMS/CPLEX)– Steiner: 6 hours vs. 2+ days (unsolvable)– Pipeline Design: 20 minutes vs. 450 minutes

65

Convergence – Steiner Tree

626.4

835.2

1044

1252.8

0 300 600 900 1200 1500 1800

Compuation Time (seconds)

B&C&H_Dual

B&C&H_Primal

B&C_Dual

B&C_Primal

CPLEX_Dual

CPLEX_Primal

66

Convergence – Pipeline Design

1280.7

1351.85

1423

1494.15

1565.3

1636.45

0 300 600 900 1200 1500 1800

Computation Time (seconds)

B&C&H_Dual

B&C&H_Primal

CPLEX_Dual

CPLEX_Primal

B&C_Dual

B&C_Primal

67

What is a Model?

• List of Equations – Mathematical Programming (MP) Model

• Collection of several intertwined (MP) Models– Data Preparation and Calibration– “Solution” Module– Reporting Module

• Categorization of Models by answering:– Who is the User of a Model?

68

Who is the User of a Model?

• (Academic) Researcher– One time use (Research Paper)

• Domain&Model Expert– Model Results used for Consulting

• Black Box User– Model integrated in (Optimization) Application

• Each Category has its own needs– Development & Deployment

69

Research Models

• 95% of Model Source is (Equation) Algebra

• Declarative Modeling

• Set of (Benchmark) Problem Instances

• Modeling Language Differences: Few

• Taste (Syntax, Development Environment,…)

• Platform, Model Type, Solver

70

Supported Solvers

71

Available Solvers

• Growing number of MP Solvers (often out of Academic Labs)

• NEOS: >40 Solvers• Impractical to have

Interface to allModeling Languages

• “Solution”: Model Translation

72

Model Translation

73

Consulting Models

• Model is Tool for Problem Analysis

• 10% of Model Source is (Equation) Algebra

• User: Domain & Modeling Expert (not necessary the same person)

• Living Model (changes with the problem)– Lifecycle: At least 10 years

– Technology Change (Platform, Solver, …)

International CoalTrade Model

Coal MarginalCost Models

National Power Model™• Dispatch Economics• 90+ Control Areas

• Seasonal/TOD Prices, Flow & Gen.

DemandModel

GDPWeatherElectric Intensity

Transmission• Bi-Directional Simultaneous Flows• Seasonal Limits• Time-of-Day Rates

Regional Emission Limits• SO2, CO2, NOx

GenerationDatabase(all units)

GenerationCost Supply

Models

Hill and Associates, Inc.Electric Generation, Coal and Emissions Forecasting System

Utility FuelEconomics Model

• Fuel Switching• Clean-up Equip Choices

• Allowance Trading

GenerationCompliance

Models

Strategies for• SO2• NOx• Particulates

Demand• Industrial• Commercial• Residential

GenerationDatabase

(coal units)Coal Supply Curves• Cash Cost by Mine• All Regions• By Coal Type

• New Build• Gas & Oil Forecast

Coal Plant Costs

&EmissionsForecast

Plant/Area by TOD• Generation• Power Flows• Marginal Prices• Emissions

• Unit Info• Transp. Rates• Compliance Costs

Coal Plant Energy Demand

SeasonalDispatch Costs,Emission Rates

I-2

Figure I-1

75

Modeling System Requirements

• Survive in such a diverse Environment

• Compatibility – 15 year application lifecycle

• Data Connectivity/Exchange– Programs and People

• Support for Analysis/Reporting– Modeling System Tools and external Program

76

Black Box Model

• Innocent User

• Bulletproof Optimization Application– No failures: e.g. No Infeasible Models

• Model embedded in larger System:

• Optimization

• Takes Longer than one is willing to wait

• It will eventually fail

• Application

• Real Time

• Always need a Solution to Problem

77

Scheduling US Military Academy West Point

“ … each student’s daily activities are a carefully regimented balance of academic, military, and physical requirements.”

78

Modeling System Requirements

• Reliable System• Using Third Party Software (Solver)

– Keep Resource Usage of Solver in check– Solver will eventually crash

• Possibility to Implement Simple Heuristics• Platform Choice• Less important: Data Import/Export

– IT does not want Modeling System to mess with DB– Simple/Thin Interfaces (text files, XML)

79

Future Directions

• Value Added Applications

• Solution Service Providers

• Distributed System Architectures

• New Solution Approaches

• Continued Changes in the Modeling ‘Industry’