Sensitivity Analysis: Shadow Price and Objective Coefficient

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Sensitivity AnalysisShadow Price and Objective
Coefficient

• Henry C. Co
• Technology and Operations Management,
• California Polytechnic and State University

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PROTRAC, Inc.

• This example was taken from p. 134, 5th Edition.
• There is another example in this PowerPoint
  presentation, taken from p. 170, 6th Edition.
  (Starting from p. 33 of this PowerPoint
  presentation.)

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• PROTRAC, Inc. produces two lines of heavy
  equipment in the same departments using the same
  machines.
• E-9 for construction applications.
• F-9 for the lumber industry.
• The marketing manager has judged that next month
  it will be possible to sell as many E-9s or F-9s
  as PROTRAC can produce.
• How many E-9s and F-9s should be produced to
  maximize next months profit contribution margin
  (revenue minus costs)?

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Decision variables E number of E-9s to be
produce, and F the number of F-9s to be
produced.
Suppose PROTRACs unit contribution margin is
5000 on each E-9 that is sold and 4000 for each
F-9.
Objective function We want to maximize the
contribution margin that we earn z(E, F)
5000E 4000F
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PROTRAC Machining Data
PROTRAC Testing Data
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The Constraints

• Availability of machining hours in Department A
  150
• 10 E 15 F ? 150
• Availability of machining hours in Department B
  160
• 20 E 10 F ? 160
• The total hours used in testing cannot fall below
  135 hours
• 30 E 10 F ? 135. (Note the sign)

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Additional Constraints

• In order to maintain the current market position,
  senior management has decreed the operating
  policy that it is necessary to make at least one
  F-9 for every three E-9s produced.
• E/F ? 3 or E 3F ? 0.
• A major dealer has ordered a total of at least
  five E-9s and F-9s (in any combination).
• E F ? 5. (Note the sign)

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The Basic Elements

Objective function z(x, y) 5000E 4000F
The Constraints
10 E 15 F ? 150 20 E 10 F ? 160 30 E 10 F ?
135. E 3F ? 0 E F ? 5.
Non-negativity Constraints
E ? 0 F ? 0
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Graphical Solution
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Corner Solutions
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Corner Solutions

• We can examine the corner solutions in three ways
• Eyeball E and F from the graph.
• Solve for each intersection (2 equations with 2
  unknowns)
• Use Solver
• Take two constraints at a time, change the
  inequality sign to equality sign.
• These two constraints will be binding (Active).

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Corner 1
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Corner 2
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Corner 3
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Corner 4
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Slacks and Surpluses

• Slack measures unused resource.
• Graphically This is the distance from the
  solution point to the constraint.
• Slack for market position balance 16.5.
• Surplus for minimum production requirement 6.5
• Surplus for minimum testing hours 70.

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The Solver Optimal Solution
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• Active or binding constraints never have slack or
  surplus
• Inactive or non-binding constraints always have
  slack or surplus
• Number of binding constraints ? number of
  decision variables

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Degenerate Solution

• Suppose we removed the 2nd constraint, and
  changed the 5th constraint
• When the Solver solution has less than m positive
  variables (decision variables, surplus and slack
  variables), where m no. of constraints, the
  solution is called degenerate, and special care
  must be taken in interpreting the sensitivity
  report.
• See example on page 172-176 (190-197 in 5th
  Edition).

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Sensitivity Analysis
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Sensitivity Analysis
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Shadow Price (E15E19)

• Value of one additional unit of resource
• Increasing Dept A resource from 150 to 151
• Objective Function increased by 50,650 - 50,500
  150 ( cell E18 of sensitivity report)!
• One extra unit of Dept A resource is worth 150!

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• Increasing Dept B resource from 160 to 161
• Objective Function increased by 50,675 - 50,500
  175 ( cell E19 of sensitivity report)!

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How far can we go?

• RHS Ranging Upper and lower boundary range over
  which shadow prices are valid. Multiple RHS
  changes are possible.

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• Increasing Dept A resource from 150 to 240
• Increasing further to 241 (Dept A no longer
  binding)

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Think about this

• What about changes in the other direction
  (decreasing a unit of resource)?

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Reduced Cost
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Objective Ranging

• Changing the objective function coefficients
  changes the slope of the objective function line.
  
• For each objective function coefficient, there is
  an upper and lower boundary range of values over
  which the optimal solution to the problem does
  not change (Column G Column H of sensitivity
  report).
• As the value of an objective coefficient changes,
  the optimal objective function value, the shadow
  prices, and the reduce costs will change, however
  the values of the optimal basic (used in
  solution) variables do not change.
• Objective ranging provides a sensitivity analysis
  of how the solution changes as we move past the
  bounds of the original optimal solution.
  However, to obtain the exact solution, the model
  must be resolved.

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• Suppose the Objective Coefficient of E is changed
  from 5,000 to 7,500.
• The optimal values for E and F remained the same
  at 4.5 and 7, respectively.

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• Suppose the Objective Coefficient of E is changed
  from 5,000 to 7,999.
• The optimal values for E and F remained the same
  at 4.5 and 7, respectively.

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• Suppose the Objective Coefficient of E is changed
  from 5,000 to 8,000.
• The optimal values for E and F have changed to
  6.857 and 2.286, respectively.

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Reduced cost

• Definition
• Both decision variables are positive at the
  optimal. Thus the reduced costs are both 0.
• The example from the 6th Edition (page 33) is
  more interesting.