CE 191: Civl and Environmental Engineering Systems...
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CE 191: Civl and Environmental Engineering Systems Analysis LEC 03 : Graphical Solutions to LP Professor Scott Moura Civl & Environmental Engineering University of California, Berkeley Fall 2013 Last Modified: September 22, 2013 Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns of LP Slide 1
Transcript of CE 191: Civl and Environmental Engineering Systems...
CE 191: Civl and Environmental Engineering Systems Analysis
LEC 03 : Graphical Solutions to LP
Professor Scott MouraCivl & Environmental EngineeringUniversity of California, Berkeley
Fall 2013
Last Modified: September 22, 2013
Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns of LP Slide 1
Graphical Solutions of Linear Programs
Example:
min J = 140x1 + 160x2
s. to 2x1 + 4x2 ≤ 285x1 + 5x2 ≤ 50
x1 ≤ 8x2 ≤ 6x1 ≥ 0x2 ≥ 0
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Construction of the feasible set
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Construction of the feasible set
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Construction of the feasible set
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Construction of the feasible set
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Construction of the feasible set
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Construction of the feasible set
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Construction of the feasible set
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Feasible Set Final Result
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Isolines
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Isolines
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Isolines
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Gradient of the cost function
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Uniqueness (or not) of the cost function
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Features of the feasible set
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Features of the feasible set
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Features of the feasible set
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Features of the feasible set
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Feasible set is unbounded
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Objective function might be unbounded too
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Objective function might be bounded
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Optimum may be non-unique
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Feasible set might be empty
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Feasible set might be empty
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Feasible set might be empty
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Feasible set might be empty
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Constraint domination
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Constraint domination
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Constraint domination
dominated
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Graphical solution of LPs: A General Method
Write your LP
Successively eliminate half spaces corresponding to constraints
Is the feasible set empty?
YES→ problem infeasible→ FINISHED
NO→ is the feasible set bounded?
NO→ is solution finite?
NO→ solution is unbounded→ FINISHED
YES→ is there a unique solution?
YES→ corner point solution→ FINISHED
NO→ boundary solution→ FINISHED
YES→ is there a unique solution?
YES→ corner point solution→ FINISHED
NO→ boundary solution→ FINISHED
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Additional Reading
RevelleChapter 3 - A Graphical Solution Procedure and Further Examples
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