LP Sensitivity Analysis

1
LINEAR PROGRAMMING Introduction to Sensitivity
Analysis
Professor Ahmadi

2
Learning Objectives

• Understand, using graphs, impact of changes in
  objective function coefficients, right-hand-side
  values, and constraint coefficients on optimal
  solution of a linear programming problem.
• Generate answer and sensitivity reports using
  Excel's Solver.
• Interpret all parameters of reports for
  maximization and minimization problems.
• Analyze impact of simultaneous changes in input
  data values using 100 rule.

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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 1

• 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
• 4X18X2
• 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.
•  
•  
• 4. Determine the dual Price (Shadow Price) for
  the painting stage.

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Standard Computer Output

• Software packages such as Microsoft Excel and
  LINDO provide the following LP information
• Information about the objective function
• its optimal value
• coefficient ranges (ranges of optimality)
• Information about the decision variables
• their optimal values
• their reduced costs
• Information about the constraints
• the amount of slack or surplus
• the dual prices
• right-hand side ranges (ranges of feasibility)

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Sensitivity Report

• Sensitivity report has two distinct components.
• (1) Table titled Adjustable Cells
• (2) Table titled Constraints.
• Tables permit one to answer several "what-if"
  questions regarding problem solution.
• Consider a change to only a single input data
  value.
• Sensitivity information does not always apply to
  simultaneous changes in several input data values.

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Example 2 Olympic Bike Co.

• Model Formulation
• Max 10x1 15x2 (Total Weekly Profit)
• s.t. 2x1 4x2 Available)
• 3x1 2x2 Available)
• x1, x2 0 (Non-negativity)

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Example 2 Olympic Bike Co.

• Optimal Solution
• According to the output x1 (Deluxe frames)
  15,
• x2 (Professional frames) 17.5, and the
  objective function value 412.50.

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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
• The output states that the solution remains
  optimal as long as the objective function
  coefficient of x1 is between 7.5 and 22.5. Since
  20 is within this range, the optimal solution
  will not change. The optimal profit will change
  20x1 15x2 20(15) 15(17.5) 562.50

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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
• The output states that the solution remains
  optimal as long as the objective function
  coefficient of x1 is between 7.5 and 22.5. Since
  6 is outside this range, the optimal solution
  would change.

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Example 2 Olympic Bike Co.

• 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.
• Question
• What is the maximum amount the company should
  pay for 50 extra pounds of aluminum?
• Answer
• The shadow price provides the value of extra
  aluminum. The shadow price for aluminum is 3.125
  per pound and the maximum allowable increase is
  60 pounds. Since 50 is in this range, then the
  3.125 is valid. Thus, the value of 50
  additional pounds is 50(3.125) 156.25

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Example 3

• Consider the following linear program
• Min 6x1 9x2
  ( cost)
• s.t. x1 2x2
• 10x1 7.5x2
  30
• 
  x2 2
• x1, x2
  0
• Solve the above problem and perform sensitivity
  analysis.

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Range of Optimality and 100 Rule

• The 100 rule states that simultaneous changes in
  objective function coefficients will not change
  the optimal solution as long as the sum of the
  percentages of the change divided by the
  corresponding maximum allowable change in the
  range of optimality for each coefficient does not
  exceed 100.

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Range of Feasibility

• A dual price represents the improvement (increase
  or decrease) in the objective function value per
  unit increase in the right-hand side. As long as
  the right-hand side remain within the range of
  feasibility, there will be is no change in the
  shadow price. The range of feasibility is the
  range over which the dual price is applicable.

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Range of Feasibility and 100 Rule

• The 100 rule states that simultaneous changes in
  right-hand sides will not change the dual prices
  as long as the sum of the percentages of the
  changes divided by the corresponding maximum
  allowable change in the range of feasibility for
  each right-hand side does not exceed 100.

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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.

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Example 4

• In a product-mix-problem, X1, X2, X3, and X4
  indicate the units of products 1, 2, 3, and 4,
  respectively, and the linear programming model is
• MAX Z 5X17X28X36X4
•  
• S.T. 1) 3X12X24X33X4?600 Machine A hours
• 2) 4X11X22X36X4?700 Machine B hours
• 3) 2X13X21X32X4?800 Machine C hours 
• Input the data in an Excel file and save the
  file. Using Solver, solve the problem and obtain
  sensitivity results.

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Summary

• Sensitivity analysis used by management to answer
  series of what-if questions about LP model
  inputs.
• Tests sensitivity of optimal solution to
  changes
• Profit or cost coefficients, and
• Constraint RHS values.
• Explored sensitivity analysis graphically (with
  two decision variables).
• Discussed interpretation of information
• In answer and sensitivity reports generated by
  Solver.
• In reports used to analyze simultaneous changes
  in model parameter values.