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.

3

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.

5

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

6

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.

7

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...

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.

3

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.

5

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

6

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.

7

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...