Duality Theory and Sensitivity Analysis

DUALITY THEORY
AND SENSITIVITY
ANALYSIS

Outline
DUALITY THEORY
SENSITIVITY ANALYSIS

Importance of Sensitivity Analysis
Is the optimal solution sensitive to changes
in input parameters?
Possible reasons for asking this question:
◦ Parameter values used were only best
estimates.
◦ Dynamic environment may cause changes.
◦ “What-if” analysis may provide economical and
operational information.
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Scope of sensitivity analysis
Changes values of objective function
Changes values of constrains
Solution method apply
Graphical approach

Sensitivity Analysis of
Objective Function Coefficients.
Range of Optimality
◦ The optimal solution will remain unchanged as long
as
An objective function coefficient lies within its range of
optimality
There are no changes in any other input parameters.

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Sensitivity Analysis of
Objective Function Coefficients.
1000

X2

Range of optimality for X1 values: [4 to10]
Max, Z:10 X1 + 5X2
Max, Z: 9 X1 + 5X2
Max, Z: 8 X1 + 5X2

500

Max, Z: 7 X1 + 5X2
Max, Z: 6 X1 + 5X2
Max, Z: 5 X1 + 5X2
Max, Z: 4 X1 + 5X2

400

600

800

X1

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Sensitivity Analysis of
Right-Hand Side Values
In sensitivity analysis of right-hand sides of
constraints we are interested in the following
questions:
◦ Keeping all other factors the same, how much would the
optimal value of the objective function (for example, the
profit) change if the right-hand side of a constraint changed
by some unit?
◦ For how many additional or fewer units will this per unit
change be valid?

Any change to the right hand side of a binding
constraint will change the optimal solution.
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Modified example of “Toy Doll
Production Problem”
Refers to lecture note chapter 2: Linear Programming
Part 2.

Model summary
Max, Z = 8X1 + 5X2

(Weekly profit)

subject to;
2X1 + 1X2 ≤ 1000
3X1 + 4X2 ≤ 2400
X1 - X2 ≤ 350

(Plastic)
(Production Time)
(Mix)

Problem 1
Assume that the company have 3 plants...