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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

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Linear programming

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  1. 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

  2. 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

  3. Possible solutions to a linear programming problem • Unique solution • Nonunique solution • Unbounded solution • No feasible solution

  4. 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

  5. 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

  6. Tableau: canonical form Basic variables Nonbasic variables

  7. x2 leave Tableau: swapping variables Pivot element xm+1 enter

  8. Example A, B, C: BS

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