# Sensitivity Analysis of LP Solutions

```Sensitivity Analysis
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Sensitivity analysis examines how the optimal
solution will be impacted by changes in the model
coefficients due to uncertainty, error or human
intervention. In particular, it can address:
 how a change in an objective function coefficient
will affect the optimal solution
 how a change in the right hand side (RHS) of a
constraint will affect the optimal solution
 how an alternate optimal solution can be identified
Range of Optimality
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The range of optimality of an objective function
coefficient specifies an interval for the objective
function coefficient in which the original optimal
solution remains optimal, while keeping all other
coefficients in the problem constant. The value
of the objective function may change in this range.
If the coefficient is changed by more than the
Allowable Increase or Allowable Decrease, another
extreme point of the feasible region will be identified
as the optimal solution.
Alternate Optimal Solutions
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An alternate optimal solution exists if the
Allowable Increase or Allowable Decrease for
one or more variables are equal to zero and
the solution is not degenerate. (Ragsdale
pg.143)
The solution is degenerate if the RHS values
of any of the constraints have zero Allowable
Increase or Allowable Decrease .
Reduced Cost
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For each variable, the reduced cost is the per-unit
amount the product contributes to the objective less
the per-unit value of the resources it consumes.
Reduced cost represents the amount that the variable
is overpriced in minimization problems or underpriced
in maximization problems.
Reduced cost represents the amount that the
objective function will be hurt if one unit of that
variable is forced into the solution.
Alternate Optimal Solutions
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An alternate optimal solution exists if the
Final Value and the Reduced Cost entries for
a variable are both zero.
To identify the alternate optimal solution,
change Solver’s target cell to maximize the
identified zero variable value and add the
constraint that the objective must equal the
optimal value specified in the current
solution.
Binding Constraints
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The slack of a resource constraint is the amount of
the corresponding resource that will not be used if
the optimal solution is implemented.
The surplus of a minimum requirement constraint is
the amount by which the requirement will be
exceeded if the optimal solution is implemented.
A binding constraint implies that the corresponding
resource or requirement is restricting the optimal
solution and objective value. A binding constraint
has slack or surplus of zero.
A nonbinding constraint is one in which there is
positive slack or surplus when evaluated at the
optimal solution.
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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 shadow price for a nonbinding constraint is 0.
Changing the right hand side of a binding constraint
will result in a new optimal solution that will adjust to
incorporate the new requirement. This new optimal
solution must be found by resolving the model with
Solver.
When a variable is forced into the solution due to a
lower bound constraint, the reduced cost represents
shadow price of that lower bound constraint.
Graphical Interpretation
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Graphically, a shadow price is determined by
adding +1 to the right hand side value in
question, enlarging the feasible region 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.
Excel snippit to illustrate shadow prices
Range of Feasibility
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The range of feasibility for a change in
the right hand side value is the
Allowable Increase and Allowable
Decrease for the coefficient in which the