Linear programming
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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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Linear programming l.jpg
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


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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


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Possible solutions to a linear programming problem

  • Unique solution

  • Nonunique solution

  • Unbounded solution

  • No feasible solution


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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


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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


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Tableau: canonical form

Basic variables

Nonbasic variables


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x2 leave

Tableau: swapping variables

Pivot element

xm+1 enter


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Example

A, B, C: BS


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