Sensitivity Analysis

1
Sensitivity Analysis

• How Dependable Are Our Conclusions?

2
Sensitivity Analysis

• Key Question How sensitive is the optimal
  solution to the assumptions, estimates, or
  constraints we have made in our modeling of the
  problem?
• This applies to
• Profit or Cost Estimates
• Restrictions on Time, Capacity, Budget
• Adding or Removing Restrictions
• Including Other Variables in Obj. Function

3
How Sensitive Is Our Solution To Changes In The
Profit Coefficients?

• Range of Optimality Range of values of profit
  coefficients within which the optimal solution
  remains unchanged.
• The Wider the Range the Less Sensitive Our
  Conclusions Are to Profit Coefficient Estimates.
• The Optimal Value May Change However.

4
Time Allocation Example

• Max 5X110X2
• s.t. X1X2 lt 20 Hours
• X1 gt 10 Hours
• X2 gt 3 Hours
• Range of Optimality for 5X1
• -M to 10 Rating Points1
• Range of Optimality for 10X2
• 5 to M Rating Points1
• 1 Holding other parameters constant

5
How Much Would A Profit Coefficient Have To
Change To Take On A Value Above Its Lower bound?

• Reduced Costs
• The amount the profit coefficient of a variable
  will have to increase before the decision
  variable can increase beyond its lower bound
  (usually 0).
• The amount the optimal profit will change per
  unit increase in the decision variable
• Complementary Slackness indicates that at the
  optimal solution either the decision variable is
  at its lower bound or its reduced cost is 0.

6
Time Allocation Example Reduced Costs

• Because the optimal solution is X110 and X210
  the reduced costs for their respective
  coefficients are equal to 0 (both are above their
  lower bound of 0).
• What happens if we eliminate the constraint
  X1gt10?

7
Time Allocation Example Modified

• Max 5X110X2
• s.t. X1X2 lt 20
• X2 gt 3
• Optimal Solution
• X1 0 and X2 20
• Reduced Costs for the X1 coefficient is now -5
  points. This implies that the X1 coefficient must
  increase by 5 points before we allocate any time
  to X1 (Fun).

8
What Happens When We Make Changes To Our
Constraints?

• Any Changes We Make to Constraints that Effect
  the Feasible Region May Change the Optimal
  Solution.
• Changing the RHS (right hand side) of a Binding
  Constraint Will Change the Optimal Solution.
• Changing the RHS of a Nonbinding Constraint Only
  Effects Optimal Solution If the Changes Are
  Greater Than the Corresponding Slack or Surplus.

9
Time Allocation Example Changing RHS Constraints

• Max 5X1X2
• s.t. X1X2 lt 20 (Binding)
• X1 gt 10 (Binding)
• X2 gt 3 (Nonbinding)
• gt Changing the RHS of the first 2 constraints
  will effect the optimal solution. Increasing the
  third constraint by more than 7 (surplus 7)
  will alter optimal solution.

10
Time Allocation Example Changing RHS Constraints

• Case 1 X1X2 lt 21
• Optimal Solution X110 X211
• Case 2 X1 gt 9
• Optimal Solution X19 X211
• Case 3 X2 gt 10
• Optimal Solution X110 X210
• Case 4 X2 gt 11
• Infeasible Solution

11
How Do Changes In RHS Constraints Influence The
Obj. Function Value?

• Shadow Price The degree to which the optimal
  objective function value will change per unit
  increase in a RHS constraint.
• Interpretation
• If costs are not included in the profit
  coefficients (sunk costs) then we should be
  willing to pay up to the shadow price per unit of
  additional resource.
• If costs are included then it represents the
  incremental price we should pay above which we
  are currently paying.
• Complementary Slackness At optimal solution
  either a constraint is satisfied with equality or
  its shadow price0.

12
How Are Shadow Prices Impacted By Changes In RHS
Values?

• Range of Feasibility Set of RHS values for which
  the shadow price for a given constraint remains
  constant1.
• The Set of Binding Constraints (the constraints
  that determine the optimal solution) Remains
  Constant For All RHS Values Within the Range of
  Feasibility.
• 1 Optimal solution and obj. function value will
  change over this range.

13
Time Allocation Example Shadow Prices

• Max 5X110X2
• Shadow Price F.
  Range
• s.t. X1X2 lt 20 10 Pts. (13 to M)
• X1 gt 10 -5 Pts. (0 to 17)
• X2 gt 3 0 Pts. (-M to 10)
• InterpretationIn order- For values of RHS from
  13 to M one unit changes in RHS change the obj.
  function value by 10 points. For values of RHS
  from 0 to 17 one unit changes in RHS result in a
  -5 point change. For values of RHS from -M to 10
  unit changes in RHS result in no obj, function
  changes.