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Linear programming

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- 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

- Basic solution (BS): solution of AX=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

- Unique solution
- Nonunique solution
- Unbounded solution
- No feasible solution

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

- 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

Basic variables

Nonbasic variables

x2 leave

Pivot element

xm+1 enter

A, B, C: BS