GAMS An Introduction

GAMS Development Corp. GAMS Software GmbH www.gams.com

GAMS – An Introduction

Hands-on Tutorial on Optimization

Frederik Proske & Lutz WestermannGAMS Software GmbH

Agenda

GAMS at a Glance

GAMS – Hands On Examples

Outlook

• A Model is a Model is a Model

• Parallelization on HPC Systems

• Deployment of GAMS Models

• …

GAMS at a Glance

1976 - A World Bank Slides

Vision

GAMS came into being!

The aim of this system is to provide one representation of a model which is easily understood by both humans and machines.[J. Bischopp, A. Meeraus, On the Development of a General Algebraic Modeling System in a Strategic Planning Environment. Mathematical Programming Study 20 (1982) 1–29.]

Self-documenting model. A GAMS model is a machine-executable documentation of an optimization problem.[M. Bussieck & A. Meeraus, Algebraic Modeling for IP and MIP (GAMS), Annals of Operations Research 149(1): History of Integer Programming: Distinguished Personal Notes and Reminiscences, Guest Editors: Kurt Spielberg and Monique Guignard-Spielberg, February, 2007, pp. 49-56 ]

What did this give us?

Simplified model development & maintenance

Increased productivity tremendously

Made mathematical optimization available to a broader audience (domain experts)

➢2012 INFORMS Impact Prize

➢Roots: World Bank, 1976

➢Went commercial in 1987

➢Locations➢GAMS Development Corporation (USA)➢GAMS Software GmbH (Germany)

➢Product➢The General Algebraic Modeling System

Company

Broad User Community and Network

30+ YearsGAMS Development

14,000+ licenses

Users: 50% academic, 50% commercial

GAMS used in more than 120 countries

Uniform interface to ~40 solvers

Broad Range of Application Areas

Agricultural Economics Applied General Equilibrium

Chemical Engineering Economic Development

Econometrics Energy

Environmental Economics Engineering

Finance Forestry

International Trade Logistics

Macro Economics Military

Management Science/OR Mathematics

Micro Economics Physics

Foundation of GAMS

Powerful algebraic modeling language

Open architecture and interfaces to other systems, independent layers

Declarative and Procedural Language Elements

Declarative elements

• Similar to mathematical notation

• Easy to learn - few basic language elements: sets, parameters, variables, equations, models

• Model is executable (algebraic) description of the problem

Procedural elements

• Control Flow Statements (e.g. loops, for, if,…),

• Build complex problem algorithms within GAMS

• Simplified interaction with other systems • Data exchange• APIs

Independence of Model and Operating System

Platforms supported by GAMS:

Models can be moved between platforms with ease!

Cross Platform GUI – GAMS Studio

➢ Open source Qt project (Mac/Linux/Win)➢ Published on GitHub under GPL

➢ Released in beta status➢ Group Explorer➢ Editor / Syntax coloring / Spell checks

➢ Tree view / Syntax-error navigation➢ Option Editor➢ Integrated Help➢ Model Debugging & Profiling➢ Solver selection & setup➢ Data viewer ➢ GAMS Processes Control

Independence of Model and Solver

All major commercial LP/MIP solver

Open Source Solver (COIN) Also solver for NLP, MINLP,

global, and stochastic optimization

One environment for a wide range of model types and solvers

Switching between solvers with one line of code!

Independence of Model and Data

• Declarative Modeling• ASCII: Initial model development

• GDX: Data layer (“contract”) between GAMS and applications– Platform independent– No license required– Direct GDX interfaces and

general API– …

SOLVER

Independence of Model and User Interface

API’s

• Low Level• Object Oriented: .Net, Java,

Python, C++• No modeling capability:

Model is written in GAMS• Wrapper class that

encapsulates a GAMS model

Free Model Libraries

➢ More than 1400 models!

• Experience of 30+ years • Broad user community from different areas• Lots of model templates• Strong development interface

• Consistent implementation of design principles• Simple, but powerful modeling language• Independent layers• Open architecture: Designed to interact with other

applications

• Open for new developments• Protecting investments of users

Why GAMS?

GAMS –Building blocks

• Basic elements of a model

• Syntax:set[s]

set_name ["text"] [/element [text] {,element [text]} /]{,set_name ["text"] [/element [text] {,element [text]} /] } ;

• Elements have up to 63 characters, start letter or digit or are quoted

• Elements have no value (!), that is, ‘1986’ does not have the numerical value 1986 and ‘01’ ≠ ‘1’

• Text has up to 254 characters, all in one line

• Example:

• Data in GAMS consists always of real numbers (no strings)

• Uninitialized data has the default value 0

• 3 forms to declare data:• Scalar: a single (scalar) number• Parameter: list oriented data• Table: table oriented data (at least 2 dimensions)

• Scalar Data:• Syntax:

• Example:

Data: Parameters

• Can be indexed over one or several sets

• Syntax:

• Example:

Data: Table

• Syntax:

• Example:

• No “free form”: position of elements is of importance

• tables with more than 2 dimensions are also possible

Data: Assignments

• Scalar Assignment:

• Indexed Assignment:

Data: Expressions (I)

Expression: an arbitrarily complex calculation instruction

Arithmetic operators: +, -, *, /, ** (exponentiate)

• 𝑥**𝑛 corresponds to exp(𝑛*ln(𝑥)) -> only allowed for 𝑥 > 0

• power(𝑥,𝑛) can be used instead (if 𝑛 ∊ ℤ)

Data: Expressions (II)

Indexed Operations:

Variables

Examples:

Variable Attributes

Attributes of a variable:

Examples:

Equations (I)

• Equations serve to define restrictions (constraints) and an objective function

Declaration:

• Syntax:

• Example:

Definition:

• Syntax:

[...] [...]

Equations (II)

Example:

Attributes of an equation:

Model Statements (I)

• Model = collection of equations

• Syntax:

• Example:

Model Statements (II)

• Attributes:

• Example:

Solving a Model

• Passing a model to a solver and evaluation of results

• Specification of one free variable to be minimized or maximized

• Syntax:

• the model type defines the problem class to be used for the model:

• Example:

GAMS –Hands On Examples

• What does this example show?

A Simple Transportation Problem

• It gives a first glimpse of how a problem can be formulated in GAMS

• It shows how easy it is to change model type and, consequently, solver technology

• It shows some basics of data exchange with GAMS

•Determine minimum transportation cost

•Result: city to city shipment volumes

•Discretedecisions

•E.g.: Ship at least 100 cases

•Non-linearity

•E.g.: Decreasein unit cost with growing volumes

Scenarios

•SolveLink

•Grid Facility

•GUSS

Freight: $90 case / thousand miles

Canning Plants (supply) Markets (demand)shipments

(Number of cases)

Minimize Transportation costsubject to Demand satisfaction at markets

Supply constraints

San Diego 600

Topeka

1.81.72.5

Supply(cases)

Demand(cases)

Distance(thousand miles)

New York

Chicago

Seattle

Mathematical Model Formulation

Indices: 𝑖 (Canning plants) 𝑗 (Markets) Decision variables: 𝑥𝑖𝑗 (Number of cases to ship)

Data: 𝑐𝑖𝑗 (Transport cost per case)

𝑎𝑖 (Capacity in cases) 𝑏𝑖 (Demand in cases) min 𝑐𝑖𝑗 ∙ 𝑥𝑖𝑗𝑗𝑖 (Minimize total transportation cost)

subject to 𝑥𝑖𝑗𝑗 ≤ 𝑎𝑖 ∀ 𝑖 (Shipments from each plant ≤ supply capacity)

𝑥𝑖𝑗𝑖 ≥ 𝑏𝑗 ∀ 𝑗 (Shipments to each market ≥ demand)

𝑥𝑖𝑗 ≥ 0 ∀𝑖, 𝑗 (Do not ship from market to plant)

𝑖, 𝑗 ∈ ℕ

Hands-On

GAMS Syntax (LP Model)

GAMS Syntax (Data Input)

Solution to LP model

Freight: $90 case / thousand miles Total cost: $153,675

(Number of cases)

•Non-linearity

•E.g.: Decrease in unit cost with growing volumes

Scenarios

•SolveLink

•Grid Facility

•GUSS

• Shipment volume: x (continuous variable)

• Discrete decision: ship (binary variable)

MIP Model: Minimum Shipment of 100 cases

add constraints:𝑥𝑖,𝑗 ≥ 100 ∙ 𝑠ℎ𝑖𝑝𝑖,𝑗 ∀ 𝑖, 𝑗 (if ship=1, then ship at least 100)𝑥𝑖,𝑗 ≤ 𝑏𝑖𝑔𝑀 ∙ 𝑠ℎ𝑖𝑝𝑖,𝑗 ∀ 𝑖, 𝑗 (if ship=0, then do not ship at all)

𝑠ℎ𝑖𝑝𝑖,𝑗 ∈ 0,1

ship = 0 ship = 1

Cost ($)

Not possible

x (Number of cases)

Hands-On

MIP Model: GAMS Syntax

MIP Model: Results

MIP Model: Solution

Freight: $90 case / thousand miles Total cost: $153,675

(Number of cases)

•Non-linearity

Scenarios

•SolveLink

•Grid Facility

•GUSS

MINLP: Cost Savings

ship = 0 ship = 1

Cost ($)

Not possible

The cost per case decreases with an increasing shipment volume

Replace:min 𝑖 𝑗 𝑐𝑖𝑗 ∙ 𝑥𝑖𝑗 (Minimize total transportation cost) Withmin 𝑖 𝑗 𝑐𝑖𝑗 ∙ 𝑥𝑖𝑗

𝒃𝒆𝒕𝒂 (Minimize total transportation cost)

x (Number of cases)

Hands-On

MINLP Model: GAMS Syntax

MINLP Model: Results

MINLP Model: Solution

Total cost: $115,438

(Number of cases)

•Non-linearity

Scenarios

•SolveLink

•Grid Facility

•GUSS

• Solving challenging real-world problems often involves the solution of many optimization problems

• Decomposition Methods• Scenario Analysis• Heuristics• …

• Such approaches are often chosen, if solving the problem at hand does not work with a single monolithic model, e.g.

• Due to it’s size and the required resources (e.g. memory)• Due to time restrictions (Problem should be solved in minutes

but it takes days)• …

→GAMS Grid Facility→Gather-Update-Solve-Scatter

Motivation

Controls GAMS function when linking to solver.

Model transport /all/ ;

transport.solveLink={0 %Solvelink.ChainScript%,

1 %Solvelink.CallScript%,

2 %Solvelink.CallModule%,

3 %Solvelink.AsyncGrid%,

4 %Solvelink.AsyncSimulate%,

5 %Solvelink.LoadLibrary%,

6 %solveLink.aSyncThreads%,

7 %solveLink.threadsSimulate%};

solve transport using lp minimizing z;

SolveLink Option

• ChainScript [0]: Solver process, GAMS vacates memory+ Maximum memory available to solver+ protection against solver failure (hostile link)- swap to disk

• Call{Script [1]/Module [2]}: Solver process, GAMS stays live+ protection against solver failure (hostile link)+ no swap of GAMS database- file based model communication

• LoadLibrary [5]: Solver DLL in GAMS process+ fast memory based model communication+ update of model object inside the solver (hot start)- not supported by all solvers

SolveLink Option – Sequential Solves

• Generate 100 distance scenarios with random data

• Solve these scenarios with the solveLink values…• ChainScript [0]• CallModule [2]• LoadLibrary [5]

• Compare the execution time of solving all scenarios with different solveLink settings• Hint: Look at the GAMS function jNow

SolveLink Option Sequential – Exercise

Hands-On

SolveLink Option – Sequential Solves

---- 88 PARAMETER time time for 100 scenarios

ChainScript 6.710, CallModule 2.694, LoadLibrary 0.578

• aSyncGrid [3]: GAMS starts the solution and continues in a Grid computing environment

• aSyncThreads [6]: The problem is passed to the solver in core without use of temporary files, GAMS does not wait for the solver to come back

SolveLink Option – Asynchronous Solves

• Generate 100 distance scenarios with random data

• Solve these scenarios with the solveLink values…• aSyncGrid [3]• aSyncThreads [6]

• Compare the execution time of solving all scenarios with different solveLink settings• Hint: Check the log for output about solveLink

• → Use solver CplexD instead of Cplex• Hint: Look at the following GAMS functions:

• readyCollect

• handleCollect

• handleDelete

SolveLink Option Asynchronous – Exercise

Hands-On

---- 96 PARAMETER time time for 100 scenarios

aSyncGrid 4.259, aSyncThreads 0.496

ChainScript 6.710

CallModule 2.694

LoadLibrary 0.578

aSyncGrid 4.259

aSyncThreads 0.496

• Helpful, if large ratio of solver time / GAMS time

ChainScript 29.807

CallModule 30.004

LoadLibrary 28.864

aSyncGrid 9.112

aSyncThreads 7.901

GUSS – Gather-Update-Solve-Scatteraka Scenario Solver

Generation

Solution

Update

Simple Serial Solve Loop

Scenario Solver/GUSS

Generates model once and updates the algebraic model keeping the model “hot” inside the solver.

Generation

Solution

Update

GUSS – Gather-Update-Solve-Scatteraka Scenario Solver

ChainScript 6.710

CallModule 2.694

LoadLibrary 0.578

aSyncGrid 4.259

aSyncThreads 0.496

GUSS 0.273

• trnsgrid: https://www.gams.com/latest/gamslib_ml/libhtml/gamslib_trnsgrid.html

• Simple asynchronous solves in a loop, separate collection loop

• tgridmix: https://www.gams.com/latest/gamslib_ml/libhtml/gamslib_tgridmix.html

• Asynchronous solves in combined submission & collection loop. Keep number of submitted models <= #threads

• gussgrid: https://www.gams.com/latest/gamslib_ml/libhtml/gamslib_gussgrid.html

• Asynchronous GUSS-solves in combined submission & collection loop. Keep number of submitted models <= #threads

Grid & GUSS – Examples from the model library

A Model is a Model is a Model*

* Freely adapted from the poetry of Gertrude Stein, 1874-1946, American writer

What is a Model?

• A mathematical model is a description of a system using mathematical language (from: Wikipedia)

➔Mathematical Programming (MP) Model:List of Equations

What is a Model?

• A mathematical model is a description of a system using mathematical language (from: Wikipedia)

➔Mathematical Programming (MP) Model:List of Equations

• Collection of several intertwined MP Models• Data Preparation• Data Calibration• “Solution” Module (e.g. sequential, parallel loop)• Report Module

What is a Model in GAMS?

• Defining scope of a (part of a) project/model

• IT, analysts, managers, model builders have different views

• Misunderstandings common with verbal descriptions

• Use a model to define the scope• Requirements for such a model

• Rapid prototyping (max. 1-2 man days)• Standard IO interface (Excel)• Remote execution (e.g. through Web UI)

Model as Communication Vehicle

• Optimization models do not allow for any type of vagueness• Input data requirements• Objectives and constraints• Results

• Misunderstandings result in failure of the model• Compilation/execution errors• Infeasible/unbounded MP models

• Model as a contract

Model as Analytic Framework

• Good models do not rely on contract (input data)

• Input Module (handles bad data)• Simple error checks• Analyzing and reporting complex data

problems• Good models (modeling systems) provide

access to results via independent result analyzers for non model experts

• Analytic framework help define result metric• e.g. violations of soft constraints

Model as Contract

Model as Cost Saver

• Most convincing and obvious reason for using an optimization model

• Science of better (INFORMS)

• Often exaggerated/difficult to estimate

Model Roles over Time

Communication Vehicle

Analytic Framework

Contract Cost Saver

Outlook: Parallelization on HPC SystemsFrom simple sequential to highly parallel solve statements

Off-the-shelf software

(speedup via parallelization)

How does GAMS support problem specific solution approaches that are well suited/customized for HPC?

OptimizationProblem

core core core core

memory

Multi-core shared memory

Distributed (shared) memory

core core core core

memory

core core core core

memory

core core core core

memory

Available Computing Resources

Non-zero Plot of matrix

Solve sequence of problems (e.g. in Benders Decomp.)

… …

Parallelize

…time time

……

Decompose

Use structure exploiting solver

User Annotation to define block structure

GAMS supports different levels of parallelization

GAMS provides annotation facilities and HPC solver links

OptimizationProblem

• loop body code in sequence, often with an expensive solve statement:

Sequential Solve Statements in Loops

Model generation

Solution process

Model generation

Solution process

Model generation

Solution process

• SolveLink option specifies the solver linking conventions

• Split loop in submission & collection loop:

• Model generation and loop body code in sequence

• Either file based IO or limited to shared memory

Parallel Solves – GAMS Grid Facility

mpiexec –n 5 gams myfile

Excursus 1: Message Passing Interface (MPI)

gams myfile

broadcast(db,0)

gams myfile <-- %sysenv.PMI_RANK%=0

1 2 3 4

gather(db,0)

1 2 3 4

myfile.gms – pseudo code

• Requires reorganization of the code but allows parallel solve

• Distribute/merging data easy (part of MPI)

• Network based communication• Need to make GAMS aware of

MPI → Embedded Code

Excursus 2: Embedded Code Facility

set scen 'scenario set' / scen1*scen100 /s(scen) 'dynamic secenario subset'k 'benders iterations' / k1*k1000 /;

... // preparatory workloop(k$( NOT done ),

... // setup model for master-problemsolve master min obj_master use lp;... // fix first stage variablesloop(scen,

... // setup model for sub-problemoption clear=s; s(scen) = yes;solve sub min obj_sub use lp;... // process results

);... // compute cuts for next master... // free fixed first stage variables... // set done=1 if convergence criterion is met

);... // reporting

Example: Sequential Benders Decomposition

set scen 'scenario set' / scen1*scen100 /

s(scen) 'dynamic secenario subset'

k 'benders iterations' / k1*k1000 /;

embeddedCode Python:

from mpi4py import *

comm = MPI.COMM_WORLD

pauseEmbeddedCode

... // preparatory work

$ifthen.MPI 0==%sysenv.PMI_RANK%

loop(k$( NOT done ),

... // setup model for master-problem

solve master min obj_master use lp;

... // fix first stage variables

continueEmbeddedCode:

comm.bcast([[done]] + <data for sub>, root=0)

cut = comm.gather(None, root=0)[1:]

... // gathered data → GAMS data struct.

pauseEmbeddedCode <load GAMS data struct.>

... // compute cuts

... // free fixed first stage variables

... // set done=1 if convergence criterion is met

comm.bcast([[done],<empty>], root=0)

endEmbeddedCode

... // reporting

$else.MPI

Example: Parallel Benders with mpi4py

set scen 'scenario set' / scen1*scen100 /

s(scen) 'dynamic secenario subset'

k 'benders iterations' / k1*k1000 /;

embeddedCode Python:

from mpi4py import *

comm = MPI.COMM_WORLD

pauseEmbeddedCode

... // preparatory work

$else.MPI

s(scen) = ord(scen)=%sysenv.PMI_RANK%;

while(1,

primal_solution = comm.bcast(None, root=0)

// broadcasted data → GAMS data struct.

pauseEmbeddedCode <GAMS data struct.>

abort.noerror$done 'terminating subprocess';

solve sub min obj_sub use lp;

... // process results

comm.gather(<subproblem results>), root=0 )

pauseEmbeddedCode

$endif.MPI

PMI_RANK=0 PMI_RANK>=1

• Two-stage stochastic problem emerged from energy system model• 100 scenarios• Deterministic Equivalent:

21,029,101 rows, 23,217,077 columns, 85,721,477 non-zeroes

• Benders:• Master: up to 553 rows, 177 columns, 24,911 non-zeroes• Sub: 210,282 rows 232,161 columns 696,461 non-zeroes• 19 lines of Python Code + some refactorization of GAMS code for MPI version

All runs were made with GAMS 25.1.2 on JURECA@JSC with 24 cores per node, 2.5 GHz, (Intel Xeon E5-2680 v3 Haswell), 128 GB RAM

1: single node, 16 cores, CPLEX barrier, no crossover

2: single node, 4 cores per solve statement, CPLEX barrier, advind 0

3: 17 nodes, 404 cores in total, 4 cores per solve statement, CPLEX barrier, advind 0

Computational Result(s)

TIME [sec]

Methodsub-

problemsmaster-problem total

Deterministic Equivalent1 4059.00

Seq. Benders2 2394.92 0.18 2395.10

MPI Benders3 28.35 0.16 28.51

Deployment of GAMS models

Calling GAMS from your Application

Creating Input for GAMS Model

→ Data handling using GDX API

Callout to GAMS

→GAMS option settings using Option API→Starting GAMS using GAMS API

Reading Solution from GAMS Model

→ Data handling using GDX API

ApplicationGDX

GAMS SOLVER

• Low level APIs• GDX, OPT, GAMSX, GMO, …• High performance and flexibility• Automatically generated imperative APIs for

several languages (C, C++, C#, Delphi, Java, Python, VBA, …)

• Object Oriented GAMS API• Additional layer on top of the low level APIs• Object Oriented• Written by hand to meet the specific

requirements of different Object Oriented languages

Low level APIs → Object Oriented API

• GAMS comes with several OO APIs (Python, Java, C++, C#, …) to develop applications

→ Programming required to build your applications

From GAMS Model to Visual Web User Interface

1Basic setup:

✓ Annotating GAMS model (defining the input and output data to be displayed in the WebUI)

✓ Fully functional interface by only specifying input and output data

✓ Tabular view of input (editable) and output data

✓ Graphical visualization via pivot charts

Basic Setup –GAMS Model Annotations

Basic setup:

Advanced setup: ✓ Configuration of standard graphics and UI✓ Sophisticated (custom) graphics (R API)✓ Scenario management with internal database

Advanced Setup –Configuration

✓ Configuration via JSON file✓ Access to a number of pre-

implemented tools for graphical representation

✓ Focus on configuration, not programming

Advanced Setup –Sophisticated graphics

✓ Sophisticated graphics created in R can be included as modules

✓ Access to the entire R ecosystem

✓ Easily extendable with the wide spectrum of the R programming language

Advanced setup: ✓ Configuration of standard graphics and UI✓ Sophisticated (custom) graphics (R API)✓ Scenario management with internal database

Enterprise setup:✓ User- and Application management

Initialization:

Enterprise Setup –User and Application Management

✓ Local or server-based solution✓ User authentication (e.g. LDAP,

OAuth 2.0, Google, GitHub, Facebook)

✓ Multi-Application support with docker-based technology

Yaml config file

Enhanced Model Deployment in GAMS using R/Shiny

• Application connects Web User Interface with a GAMS model

• User Interface allows ✓ Data exchange via local files or database✓ Modification of the input data ✓ Extensive visualization options✓ Comparison of different scenarios✓ Multi-user support based on Docker technology✓ User authentication

• Tool with intuitive interface for planners (configuration vs. programming)

• This “product” is currently under development. If you are interested in getting involved, please contact support@gams.com

Welcome to Jupyter @ GAMS!

GAMS Jupyter Example

Hands-On

• Runs in a browser/on a server → No local installation needed

• Allows to use notebook technology in combination with GAMS

• Notebooks allow to combine GAMS and Python• GAMS works great with well structured data and optimization

models• Python is very rich in features to retrieve, manipulate, and

visualize data that comes in all sort of ways• ➔ Combining GAMS and Python in a notebook makes is easy

to tell optimization story with text, data, graphs, math, and models

• This “product” is currently under development.Give it a try at https://jupyterhub.gams.com/hub/login

Using GAMS Jupyter Notebooks to tell “optimization stories”

Outlook: Stochastic Programming

Stochastic Programming in GAMS

EMP Information

(uncertain Par.)

ReformulatedModel

Solution

origin

al space

Deterministic Model

Translation

Solve withestablished Algorithms

EMP/SP➢ Simple interface to add

uncertainty to existing deterministic models

➢ (EMP) Keywords to describe uncertainty include: discrete and parametric random variables, stages, chance constraints, Value at Risk, ...

➢ Available solution methods:➢ Automatic generation of

Deterministic Equivalent (can be solved with any solver)

➢ Specialized commercial algorithms (DECIS, LINDO)

Transport Example – Uncertain Demand

Prob: 0.3Val: 0.90

Prob: 0.5Val: 1.00

Prob: 0.2Val: 1.10

Decisions to make➢ First-stage decision: How many units should be shipped

“here and now” (without knowing the outcome)➢ Second-stage (recourse) decision:

➢ How can the model react if we do not ship enough?➢ Penalties for “bad” first-stage decisions, e.g. buy

additional cases u(j) at the demand location:costsp .. z =e= sum((i,j), c(i,j)*x(i,j))+ sum(j,0.3*u(j));

demandsp(j) .. sum(i, x(i,j)) =g= bf*b(j) - u(j) ;

Uncertain demand factor bf

Uncertain Demand: GAMS Algebra

Uncertain Demand: Results

• The Extended Mathematical Programming (EMP) framework is used to replace parameters in the model by random variables

• Support for Multi-stage recourse problems and chance constraint models

• Easy to add uncertainty to existing deterministic models, to either use specialized algorithms or create Deterministic Equivalent (new free solver DE)

• More information:https://www.gams.com/latest/docs/UG_EMP_SP.html

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