BY GEBRE S.
Sensitivity Analysis and
Duality

SensitivityAnalysis
Benefits of sensitivity analysis
Enables the decision maker to determine how a change
in one of the values of a model will impact the optimal
solution and the optimal value of the objective function
while holding all other parameters constant.
Provides the decision maker with greater insight about
the sensitivity of the optimal solution to changes in
various parameters of a problem.
Permits quick examination of changes due to improved
information relating to a problem or because of the
desire to know the potential impact of changes that are
contemplated.

Changes in Parameter Values
Categories of model parameters subject to potential
changes
◦ The value of an objective function coefficient
◦ The right-hand side (RHS) value of a constraint
◦ A coefficient of a constraint
Concerns about ranges of changes
◦ Which range pertains to a given situation?
◦ How can the range be determined?
◦ What impact on the optimal solution does a change
that is within the range have?

A Change in the RHS of a Constraint
Analysis of RHS changes begins with determination of a constraint’s
shadow price in the optimal solution.
Shadow Price indicates the impact that a one unit change in the amount of a
constraint would have on the value of the objective function.
Range of feasibility: the range of values over which the right-hand-side
(RHS) value can change without causing the shadow price to change.
Within this range of feasibility, the same decision variables will remain
optimal, although their values and the optimal value of the objective
function will change.

The final simplex table for server problem
To find the values, for each constraint, the entries
in the associated slack columns must be divided
into the values in the quantity column

The General Rule
The reverse is true for minimization
Example: The manager in the micro server problem is
contemplating a change in the level of the storage
constraint: an increase of 3 cubic feet. Determine the
revised optimal solution for the change.

Solution
For an increase in storage space of 8 cubic feet,
would be S1= 0, X1=7.5, X2=7, and Profit =800.

There are two cases:
◦ Changes for a variable that is not currently in the solution mix
◦ Changes for a variable that is currently in the solution mix.
The range over which a non-basic variable’s objective function
coefficient can change without causing that variable to enter the
solution mix is called its Range of Insignificance.
The range over which the objective function coefficient of a
variable that is in the solution can change without changing the
optimal values of the decision variable is called the Range of
optimality
A value of the objective function that falls within the range of
optimality will not change the optimal solution, although the
optimal value of the objective function will change.
AChange in an Objective Function Coefficient

If a variable is not currently in solution in a
maximization problem, its objective function
coefficient would have to increase by an amount that
exceeds the C – Z values for the variable to end up as a
basic variable in the optimal solution.
To find the range of optimality, the values in the C – Z
must be divided by the corresponding row values of the
variable in question.

Example:
Determine the range of optimality for the decision variables in the
server problem.
Solution:
For x1 we find:
The smallest positive ration is +40. Therefore, the coefficient of
X1 can be increased by Birr 40 without changing the optimal
solution. The upper is (Birr 60 + Birr 40) = Birr 100. Also, the
smallest negative ratio is -10; therefore, the lower end of the
range equal to (Birr60 - Birr10 )= Birr 50.

For x2 we find:
The range of optimality for the objective
function coefficient of x2 is Birr 30 to Birr 60.

Duality
An alternate formulation of a linear programming problem as either the
original problem or its mirror image, the dual, which can be solved to obtain
the optimal solution.
Its variables have a different economic interpretation than the original
formulation of the linear programming problem (the primal).
It can be easily used to determine if the addition of another variable to a
problem will change the optimal.

Formulation of a Dual
The number of decision variables in the primal
is equal to the number of constraints in the dual.
The number of decision variables in the dual is
equal to the number of constraints in the primal.
Since it is computationally easier to solve
problems with less constraints in comparison to
solving problems with less variables, the dual
gives us the flexibility to choose which problem
to solve.

A comparison of these two versions of the problem
will reveal why the dual might be termed the
“mirror image” of the primal.
The original objective was to minimize, whereas
the objective of the dual is to maximize.
In addition, the coefficients of the primal’s
objective function become the right-hand-side
values for the dual’s constraints, whereas the
primal’s right-hand side values become the
coefficients of the dual’s objective function.

Example

Table: Transforming the Primal into Its Dual

Exercise
Formulate the dual of the server problem

Formulating the Dual when the Primal has Mixed Constraints
In order to transform a primal problem into its dual, it is easier
if all constraints in a maximization problem are of the < variety,
and in a minimization problem, every constraint is of the >
variety.
To change the direction of a constraint, multiply both sides of
the constraints by -1.
If a constraint is an equality, it must be replaced with two
constraints, one with a < sign and the other with a > sign.
Example: Formulate the dual of this LP model.
Maximize z = 50X1 + 80X2
Subject to:
3X1 + 5X2 ≤ 45
4X1 + 2X2 ≥ 16
6X1 + 6X2 = 30
X1 , X2 ≥ 0

Final tableau of Primal solution to the server problem:
Final tableau of Dual solution to the server problem

Economic Interpretation of The Dual
Economic interpretation of dual solution results
◦ Analysis enables a manager to evaluate the potential
impact of a new product.
◦ Analysis can determine the marginal values of
resources (i.e., constraints) to determine how much
profit one unit of each resource is equivalent to.
◦ Analysis helps the manager to decide which of
several alternative uses of resources is the most
profitable.