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# Linear programming PowerPoint PPT Presentation

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

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

• Dynamic linear programming: time-phased decision making

### 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

• Unique solution

• Nonunique solution

• Unbounded solution

• No feasible solution

### 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 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

Basic variables

Nonbasic variables

x2 leave

Pivot element

xm+1 enter

A, B, C: BS