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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