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Linear programming. Linear program: optimization problem, continuous variables, single, linear objective function, all constraints linear equalities or inequalities Applications Allocation models Operations planning models Limit load analysis in structues

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Presentation Transcript
linear programming
Linear programming
  • Linear program: optimization problem, continuous variables, single, linear objective function, all constraints linear equalities or inequalities
  • Applications
    • Allocation models
    • Operations planning models
    • Limit load analysis in structues
    • Dynamic linear programming: time-phased decision making
matrix form
Matrix form
  • Basic solution (BS): solution of AX=b, n-m redundant variables zero (nonbasic variables), n constraints active. Remaining m variables non zero (basic variables)
  • Each BS corresponds to a vertex
  • BFS, non BFS
possible solutions to a linear programming problem
Possible solutions to a linear programming problem
  • Unique solution
  • Nonunique solution
  • Unbounded solution
  • No feasible solution
simplex method
Simplex method

Idea: Start from a vertex. Move to adjacent vertex so that F decreaces. Continue until no further improvement can be made.

Facts

  • Optimum is a vertex
  • Vertex: BS
  • Moving from a vertex to adjacent vertex: swap a basic variable with a non basic variable
simplex method5
Simplex method
  • Variable with smallest negative cost coefficient will become basic
  • Select variable to leave set of basic variables so that a BFS is obtained
  • Design space convex
tableau canonical form
Tableau: canonical form

Basic variables

Nonbasic variables

example
Example

A, B, C: BS

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