LP Sensitivity Analysis

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LINEAR PROGRAMMING Introduction to Sensitivity
Analysis
Professor Ahmadi

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Chapter 3 Linear Programming Sensitivity
Analysis and Interpretation of Solution

• Introduction to Sensitivity Analysis
• Graphical Sensitivity Analysis
• Spreadsheet Solution Sensitivity Analysis

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Sensitivity Analysis

• Sensitivity analysis (or post-optimality
  analysis) is used to determine how the optimal
  solution is affected by changes, within specified
  ranges, in
• the objective function coefficients
• the right-hand side (RHS) values
• Sensitivity analysis is important to the manager
  who must operate in a dynamic environment with
  imprecise estimates of the coefficients.
• Sensitivity analysis allows the manager to ask
  certain what-if questions about the problem.

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Range of Optimality The Objective Function
Coefficients

• A range of optimality of an objective function
  coefficient is found by determining an interval
  for the coefficient in which the original
  optimal solution remains optimal while keeping
  all other data of the problem constant. (The
  value of the objective function may change in
  this range.)

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The Right Hand Sides Shadow Price (Dual Price)

• A shadow price for a right hand side value (or
  resource limit) is the amount the objective
  function will change per unit increase in the
  right hand side value of a constraint.
• The range of feasibility for a change in the
  right hand side value is the range of values for
  this coefficient in which the original shadow
  price remains constant.
• Graphically, a shadow price is determined by
  adding 1 to the right hand side value in
  question and then resolving for the optimal
  solution in terms of the same two binding
  constraints.
• The shadow price is equal to the difference in
  the values of the objective functions between the
  new and original problems.
• The shadow price for a non-binding constraint is
  0.

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Example

• Refer to the Woodworking example of chapter 2,
  where X1 Tables and X2 Chairs. The problem is
  shown below.
•  Max. Z 100X160X2
•  
• s.t. 12X14X2 lt 60 (Assembly time in hours)
• 4X18X2 lt 40 (Painting time in hours)
• The optimum solution was X14, X23, and Z580.
  Answer the following questions regarding this
  problem.

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Answer the following Questions

• 1. Compute the range of optimality for the
  contribution of X1 (Tables)
•  
•  
• 2. Compute the range of optimality for the
  contribution of X2 (Chairs)
•  
•  
• 3. Determine the dual Price (Shadow Price) for
  the assembly stage.
•  
•  
• Determine the dual Price (Shadow Price) for the
  painting stage.

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Example 2 Your Turn

• Refer to The Olympic Bike Co. (From Chapter 2)
• Solve this Problem and find the optimum solution.
• Max 10x1 15x2 (Total Weekly Profit)
• s.t. 2x1 4x2 lt 100 (Aluminum
  Available)
• 3x1 2x2 lt 80 (Steel
  Available)
• x1, x2 gt 0 (Non-negativity)

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Example 2 Olympic Bike Co.

• Range of Optimality
• Question
• Suppose the profit on deluxe frames is increased
  to 20. Is the above solution still optimal?
  What is the value of the objective function when
  this unit profit is increased to 20?
• Answer

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Example 2 Olympic Bike Co.

• Range of Optimality
• Question
• If the unit profit on deluxe frames were 6
  instead of 10 would the optimal solution change?
• Answer

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Range of Feasibility

• The range of feasibility for a change in a
  right-hand side value is the range of values for
  this parameter in which the original shadow price
  remains constant.

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Example 2 Olympic Bike Co.

• Range of Feasibility and Relevant Costs
• Question
• If aluminum were a relevant cost, what is the
  maximum amount the company should pay for 50
  extra pounds of aluminum?
• Answer

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Example 3

• Consider the following Minimization linear
  program
• Min Z 6x1 9x2 (
  cost)
• s.t. x1 2x2 lt 8
• 10x1 7.5x2
  gt 30
• 
  x2 gt 2
• x1, x2
  gt 0
• Use Excel to solve this problem.

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Example 3

• Optimal Solution
• According to the output
• x1 1.5
• x2 2.0
• Z (the objective function value) 27.00.

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Example 3

• Range of Optimality
• Question
• Suppose the unit cost of x1 is decreased to 4.
  Is the current solution still optimal? What is
  the value of the objective function when this
  unit cost is decreased to 4?
• Answer

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Example 3

• Range of Optimality
• Question
• How much can the unit cost of x2 be decreased
  without concern for the optimal solution
  changing?
• Answer

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Example 3

• Range of Feasibility
• Question
• If the right-hand side of constraint 3 is
  increased by 1, what will be the effect on the
  optimal solution?
• Answer

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A Note on Sunk Cost and Relevant Cost

• A resource cost is a relevant cost if the amount
  paid for it is dependent upon the amount of the
  resource used by the decision variables.
• Relevant costs are reflected in the objective
  function coefficients.
• A resource cost is a sunk cost if it must be paid
  regardless of the amount of the resource actually
  used by the decision variables.
• Sunk resource costs are not reflected in the
  objective function coefficients.

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Reduced Cost

• The reduced cost for a decision variable whose
  value is 0 in the optimal solution is the amount
  the variable's objective function coefficient
  would have to improve (increase for maximization
  problems, decrease for minimization problems)
  before this variable could assume a positive
  value.
• The reduced cost for a decision variable with a
  positive value is 0.