Linear Programming

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Linear Programming

• Optimal Solutions
• andModels Without Unique Optimal Solutions

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Models With No Solutions- Infeasibility

• Max 8X1 5X2
• s.t. 2X1 1X2 1000
• 3X1 4X2 2400
• X1 - X2 350
• X1 800
• X1, X2 0

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Solver - Infeasible

•                      

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Solver Results
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Infeasibility

• Excel When Solve is clicked
• A problem is infeasible when there are no
  solutions that satisfy all the constraints.
• Infeasibility can occur from
• Input Error
• Mis-formulation
• Simply an inconsistent set of constraints

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Models With An Unbounded Solution

• Max 8X1 5X2
• s.t. X1 - X2 350
• X1 200
• X2 200

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Solver - Unbounded

•                      

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Solver Results - Unbounded
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Unboundedness

• An unbounded solution means you left out some
  constraints you cannot make an infinite
  profit.
• Excel When Solve is clicked

Means the problem is unbounded
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Multiple Optimal Solutions

• Max 8X1 4X2
• s.t. 2X1 1X2 1000
• 3X1 4X2 2400
• X1 - X2 350
• X1, X2 0

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Multiple Optimal Solutions

• When slope of objective function line equals
  slope of binding constraint the problem can have
  multiple optimal solutions.
• The slope of a line written as aX1 bX2 d
• is
• Object function
• 8X1 4X2 Slope is -8/4 -2
• Plastic constraint
• 2X1 1X2 1000 Slope is -2/1 -2

-a/b
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Multiple Optimal Solutions

• The constraint must not be a redundant constraint
  but must be a boundary constraint.
• The objective function must move in the direction
  of the constraint
• In the previous example if the objective function
  had been MIN 8X1 4X2, then it is moved in the
  opposite direction of the constraint and (0,0)
  would be the optimal solution.
• Multiple optimal solutions allow the decision
  maker to use secondary criteria to select one of
  the optimal solutions that has another desirable
  characteristic (e.g. Max X1 or X1 3X2, etc.)

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Alternate Optimal Solution

• To find the second optimal solution
• Observe that an Allowable Increase or Allowable
  Decrease for the objective function coefficient
  of some variable Xj is 0
• Add a constraint that sets the value function
  cell to the optimal value from the first optimal
  solution.
• Change objective function to
• If the Allowable Increase 0, change objective
  to maximize Xj
• If the Allowable Decrease 0, change objective
  to minimize Xj

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Original Optimal Solution in Excel
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Multiple Optimal Solutions in Excel

• Excel Identification of multiple solutions

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Alternate Optimal Solution in Excel
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Generating the Multiple Optimal Solutions

• Any weighted average of optimal solutions is also
  optimal.
• In the previous example it can be shown that the
  two optimal extreme points are (320,360) and
  (450, 100).
• Thus .5(320,360) .5(450,100) (385,230) is
  also an optimal point that is half-way between
  these two points.
• .8(320,360) .2(450,100) (346,308) is also an
  optimal point that is 8/10 of the way up the line
  toward (320,360).

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Review

• When a linear programming model is solved it
• Has a unique optimal solution
• Has multiple optimal solutions
• Is infeasible
• Is unbounded
• Identification of each
• By Excel
• Find alternative solutions for multiple optimal
  solutions problem