mie376 lec 1a intro

8/9/2019 MIE376 Lec 1a Intro

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MIE376 – Mathematical Programming 1

Mathematical Programming

Within the context of

Operations Research methods


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Definition of OR

Operations Research, or simply OR is an

interdisciplinary science which deploys scientific

methods like mathematical modeling, statistics, and

algorithms to decision making in complex real-worldproblems which are concerned with coordination

and execution of the operations within an

organization. The na

ture of organization is

essentially immaterial. The eventual intention behind

using this science is to elicit a best possible solution

to a problem scientifically, which improves or

optimizes the performance of the organization.


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the name…

Started in WWII under the name of Operational Research

Also called Management Science

Traditionally associated with Industrial Engineering,

More recently as Systems Engineering

Institute for Operations Research and the Management Sciences

In 2005 INFORMS introduces its motto: The Science of Better.

In 2011 INFORMS adds a Business Analytics Section to the existingOperations Research and Management Science Sections.


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Focus on Operations

The origin and mainstay of OR

Justifies dual “homes” for OR

Industrial engineering in engineering schools,

Operations management in business schools.


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Nature Of Organization Is Immaterial

Energy,

defence,

healthcare,

transportation,

mining, forestry,

electric utilities,

retail,

manufacturing,

banking,

financial engineering,

security,

etc..


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Getting to the best solution

Each application requires a new optimization modelspecific to that situation,

BUT, there are standard model TYPES available.

At a high level can be categorized under

Descriptive versus Normative (or Prescriptive) Models

Deterministic versus Stochastic (or Probabilistic) Models


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Descriptive versus Normative Models

Descriptive Models help you understand howthe situation will change if one makes a

particular decision.

Normative Models go beyond the DescriptiveModels and explicitly advise what one ought

to do to improve the situation.


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Deterministic versus Stochastic Models

Deterministic Models do not take into account

uncertainty.

Stochastic Models take into accountuncertainties modelled as probability

distributions.


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Types of OR ModelsDeterministic Stochastic

Descriptive

Systems of Equations Linear

Non-linear

Differential

Simulation Models Discrete Event

Monte-Carlo

Queueing Theory

Markov Chains

Normative

Mathematical Programming Linear Programming (LP)

Network Models (NP)

Integer Programming (IP)

Non-linear Programming (NLP)

Quadratic Programming (QP)

Bi-Level Linear Programming

(BLLP)

Dynamic Programming (DP)

Optimal Control

Heuristic Methods

Multi-Criteria Models

Stochastic Programming Chance constrained

Recourse Models

Stochastic DP

Decision Analysis Models

Stochastic Optimal Control

Stochastic Multi-Criteria


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Mathematical Programming

Minimize or Maximize f(x1, x2, x3, …, xn)

subject to constraints

g1(x1, x2, x3,…, xn) { ≥ or ≤ or = } some constant b1

g2(x1, x2, x3,…, xn) { ≥ or ≤ or = } some constant b2…. .

gm(x1, x2, x3,…, xn) { ≥ or ≤ or = } some constant bm

For example

Minimize 3x1 + ln (x1/x2) – exp(x2 + x3)

subject to x1 + x2 + sin x3 ≥ 5


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Linear Programming

General Form

Minimize or Maximize c j x j

subject to

a1j x j { ≥ or ≤ or = }b1 a2j x j { ≥ or ≤ or = }b2.

.

.

amj

x j{ ≥ or ≤ or = }b

mx j ≥ 0

Example

Max 3x1 + 4x2

subject to2x1 + 5x2 ≤ 5

3x1 + x2 ≤ 10

7x1 + 3x2 ≤ 7

12x1 + 10x2 ≤ 8

x1 ≥ 0 and x2 ≥ 0

• all relationships are LINEAR!


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Back to the 7 Step Modeling Process

1. Formulate the Problem

2. Observe the System

3. Formulate the Model

4. Verify and Validate the Model

5. Solve the Model

6.

Recommend7. Implement


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Formulating and Verifying LP Models

Formulating your model Understand the problem formulation

LET some symbol represent some problem entity

Repeat for all problem entities

Derive linear mathematical relationships between symbols

Verifying your model Postulate an arbitrary numerical “solution” to the problem

Check its feasibility from the problem formulation

Compare with a feasibility check using your LP model Check its objective value from the problem

Compare with the objective value from your LP model


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LP Software

INFORMS regularly surveys the marketplace for LP

software systems

For the latest 2013 survey see http://www.orms-

today.org/surveys/LP/LP-survey.html

ECF labs includes the following LP software:

Excel with Solver – widely available

CPLEX, Gurobi and GAMS – all industry standard solvers OPL and AMPL - front end software work with most solvers

OPL/CPLEX and AMPL/Gurobi are popular choices

http://www.orms-today.org/surveys/LP/LP-survey.htmlhttp://www.orms-today.org/surveys/LP/LP-survey.htmlhttp://www.orms-today.org/surveys/LP/LP-survey.htmlhttp://www.orms-today.org/surveys/LP/LP-survey.html


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LP System Components

Front-end GUI/DB application

Customized software

Back-End Database System

Access, Oracle, DB2, MS-SQL

Back-End Modeling System

OPL, AMPL, GAMS

Usually separate model from data (.mod and .dat)

Back-End Solver

CPLEX, Gurobi, MINOS

Also see: http://www.gams.com/solvers/index.htm


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Typical LP System Diagram

GUI/

DB

App

Modeling

system

(.mod, .dat)

Solver

Database


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Excel Solver

GUI/

DB

App

Modeling

systemSolver

DatabaseMS Excel

Solver Add-in by Frontline


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AMPL IDE

GUI/

DB

App

Modeling

system

(.mod, .dat)

Solver

Database

Can call a variety of solvers,

most popular CPLEX and

Gurobi

New Product in Version 1

mie376 lec 1a intro

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