An Introduction to Linear Programming

An Introduction to Linear Programming Pham Minh Tri Ph.D. Candidate and Project Officer School of Computer Engineering 18 Jan 2008 presented by

An example ,[object Object],[object Object],[object Object],[object Object],[object Object]

An example ,[object Object],[object Object],[object Object]

An example ,[object Object],[object Object],[object Object],[object Object]

An example ,[object Object],[object Object]

An example ,[object Object],[object Object],[object Object],(0,83.3) (166.7,0)

An example ,[object Object],[object Object],[object Object],[object Object],(0,83.3) (100,0) (0,150) (166.7,0)

An example ,[object Object],[object Object],[object Object],[object Object],(0,83.3) (100,0) (0,150) (166.7,0) Feasible region Feasible region is always a convex hull

An example ,[object Object],[object Object],[object Object],[object Object],(0,83.3) (100,0) (0,150) (166.7,0) Feasible Region F

An example ,[object Object],(0,83.3) (100,0) (0,150) (166.7,0) Feasible Region F Direction Of profit increase

An example ,[object Object],(0,83.3) (100,0) (0,150) (166.7,0) Feasible Region F Direction Of profit increase P = 0

An example ,[object Object],[object Object],[object Object],[object Object],(0,83.3) (100,0) (0,150) (166.7,0) Feasible Region F Direction Of profit increase P = 1192 (66,50) An optimal solution, always locates at one of the corners

2-variable Linear Programming ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Simple Algorithm: - Find the feasible region F (convex hull) - Choose the corner that has the highest profit

What about over 2 variables? ,[object Object],[object Object],[object Object],[object Object],[object Object]

Linear Programming in general ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Linear Programming, short form ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Find x to: Maximize: c T x Constraints: x >= 0 A x <= b

Linear Programming ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],y T A x <= y T b

Linear Programming ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],y T A >= c T Let’s call this feasible region G for y

Linear Programming - Duality ,[object Object],[object Object],[object Object],[object Object]

Linear Programming - Duality ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Linear Programming - Duality ,[object Object],[object Object],[object Object],Using this duality, we can convert a problem with many variables but a few constraints (hard to solve) into a problem of multiple constraints with a few variables (easy to solve).

Another example ,[object Object],[object Object],[object Object]

Another example ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Another example ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Another example ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Another example ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

An Introduction to Linear Programming

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An Introduction to Linear Programming