Agenda of Week III. LP I

LP StandardizationLP Standardization

Optimization

LP intro LP introWeek 2Week 2

1 3 4

Definition

Basic assumptions

Example

General form

Standard form

Objective : Understanding the solution of optimization problems Understanding the introduction of LP

SolvingSolving

2

How to get…

Review of Week 2

최적화 최적화

1

정의

기본구성

목적함수

제약조건

Objective : Understanding the optimization problems

Solving Optimization Problems

Theoretically

Modeling with mathematical tools Theoretically solve model by employing calculus Always optimal solutions under some conditions Impossible for complex problems LINGO or Excel: Theory Algebra

Heuristics

Confirm current status

Develop a specific logic/process improving current objective function and

repeat it

Not guarantee optimal solution

E.g.: The blind climbing

Solving Optimization Problems

LINGO

How to get…

Lecture HP: http://www.niceprof.net Lindo Co.: http://www.lindo.com

Solving Optimization Problems

LP

Optimization problem with 1st order constraints and obj. func.

General solution

Structure (Table 3-2)

Obj. func. Constraints: LHE, RHS, Equality Decision variables, Parameters Nonnegativity

LP

Basic assumptions

Proportionality Additivity Divisibility Certainty

LP

General from of LP

mnmnm

nn

nnxx

bxaxa

bxaxa

toSubject

xcxcMinMaxn

)(......

......

)(......

,

......)(

11

11111

11,...,1

LP

Decision variables n variables:

Contribution coefficients Coefficients in obj. func.:

Possible limits of resources (m resources) Right hand side constants:

Technology coefficients Coefficients in constraints:

nxx ,...,1

ncc ,...,1

mbb ,...,1

ija

Modeling Examples of LP

Example 3-2

Server problem: p.113 Lingo program

Example 3-3

P.126 Lingo program

LP

0

......

......

......

,

......

11

11111

11,...,1

i

mnmnm

nn

nnxx

xAll

bxaxa

bxaxa

toSubject

xcxcMaxn

General from of LP

Transformation

Minimization Multiply -1 to obj. func.

Non nonnegativity Decompose variable x into 2 variables Give nonnegativity to both variables

Equality constraint Decompose it into 2 constraints with ‘>=‘ and ‘<=‘ Multiply -1 to constraint with ‘>=‘

,x x