CE 191: Civil and Environmental Engineering Systems Analysis - Graphical LP.pdf · CE 191: Civil...
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CE 191: Civil and Environmental Engineering Systems Analysis LEC 03 : Graphical Solutions to LP Professor Scott Moura Civil & Environmental Engineering University of California, Berkeley Fall 2014 Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 1
Transcript of CE 191: Civil and Environmental Engineering Systems Analysis - Graphical LP.pdf · CE 191: Civil...
CE 191: Civil and Environmental EngineeringSystems Analysis
LEC 03 : Graphical Solutions to LP
Professor Scott MouraCivil & Environmental EngineeringUniversity of California, Berkeley
Fall 2014
Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 1
Graphical Solutions of Linear Programs
Example:
min J = 140x1 + 160x2
s. to 2x1 + 4x2 ≤ 28
5x1 + 5x2 ≤ 50
x1 ≤ 8
x2 ≤ 6
x1 ≥ 0
x2 ≥ 0
Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 2
Construction of the feasible set
Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 3
Construction of the feasible set
Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 4
Construction of the feasible set
Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 5
Construction of the feasible set
Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 6
Construction of the feasible set
Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 7
Construction of the feasible set
Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 8
Construction of the feasible set
Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 9
Feasible Set Final Result
Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 10
Isolines
Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 11
Isolines
Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 12
Isolines
Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 13
Gradient of the cost function
Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 14
Uniqueness (or not) of the cost function
Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 15
Features of the feasible set
Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 16
Features of the feasible set
Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 17
Features of the feasible set
Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 18
Features of the feasible set
Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 19
Feasible set is unbounded
Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 20
Objective function might be unbounded too
Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 21
Objective function might be bounded
Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 22
Optimum may be non-unique
Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 23
Feasible set might be empty
Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 24
Feasible set might be empty
Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 25
Feasible set might be empty
Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 26
Feasible set might be empty
Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 27
Constraint domination
Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 28
Constraint domination
Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 29
Constraint domination
dominated
Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 30
Graphical solution of LPs: A General Method
Write your LP
Successively eliminate half-‐spaces
Feasible Set Empty?
Problem Infeasible
Objec?ve Bounded?
Infinite Solu?on Solu?on
Unique?
Feasible Set Bounded?
YES NO
NO
NO YES
Corner Point Solu?on
Boundary Solu?on
YES NO
YES
Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 31
Insights from Graphical LP
Linear constraints Ax ≤ b form feasible set (possibly empty)
Feasible set is a (possibly unbounded) convex polytope
Optimal solution exists along edges (corner point or line segment)
Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 32
Danzig’s Simplex Algorithm
1 Define feasible set
2 Start at vertex. Move along vertices until obj. fcn. stops decreasing
Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 33
Example of Simplex Algorithm
Recall the LP problem:
max J = 140x1 + 160x2
s. to 2x1 + 4x2 ≤ 28
5x1 + 5x2 ≤ 50
x1 ≤ 8
x2 ≤ 6
x1 ≥ 0
x2 ≥ 0
Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 34
Feasible Set Final Result
Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 35
Start at a Vertex
Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 36
Jump to adjacent vertex
Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 37
Stop when objective stops decreasing
Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 38
Additional Reading
Revelle
Chapter 3 - A Graphical Solution Procedure and Further Examples
Simplex Algorithm
Revelle Chapter 4 - The Simplex Algorithm for Solving Linear Programs
Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 39
