Linear optimization: Sensibility analysis 1

Theory

Terminology

The german term "Sensibilitätsanalyse" was derived from the original english term "sensitivity analysis" by Müller-Merbach [Operations Research] in 1992. The term sensibility analysis, that is used in this Article, is used synonymous to that.

Use

After finding an optimal solution for a linear optimization problem by the means of the simplex algorithm, one can use the sensibility analysis to see, how much the initial data may be changed, without changing the structure of the optimal solution. In other words you can see, the robustness of your optimal programm regarding the change of the objective function or constraints.

Non basic variable change

In the following section, all relations concerning dimensions of numbers are absolute values

The smallest negative and the smallest positive quotient of the optimal solution's right hand side and the corresponding column element of the optimal solution define, when added or subtracted to the right hand Side of the initial solution, the upper and lower endpoint of the interval in which the constraints may vary.

So, you can see the robustness of the optimal solution, concerning the change of the restrictions.

Basic variable change

The smallest negative and the smallest positive quotient of the optimal solution's Objective function coefficient and the corresponding row element of the optimal solution define, when subtracted or added to the coefficients of the initial objective function, the lower and upper endpoint of the interval in which the coefficients of the objective function may vary.

By the interval of basic variables, you can see the robustness of the optimal solution, concerning the change of the objective function.

Complete detailed Example

Optimization

Text

A garden center has got a bed with a developable area of 100 m².
There is a budget of $720 available to plant roses and carrots in the bed.
However, the maximal planting area of carrots shall not exceed 60m².
The planting shall maximize profit.

The costs for the seed are as following:

• Roses: $6{\frac {\$}{m^{2}}}$      Carrots: $9{\frac {\$}{m^{2}}}$
  

The selling prices of the products are:

• Roses: $7{\frac {\$}{m^{2}}}$      Carrots: $11{\frac {\$}{m^{2}}}$
  

Formal LP

  Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): OF: x _1 + 2x_2 \xrightarrow {} max! \qquad | x_1 ; x_2 \ge 0

1)     Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): x_2+ y_1 \le 60

2)     Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): x_1 + x_2 + y_2 \le 100

3)     Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): 6x_1 + 9x_2 + y_3 \le 720

Simplex tableaus

Fehler beim Erstellen des Vorschaubildes: Die Miniaturansicht konnte nicht am vorgesehenen Ort gespeichert werden

Sensibility analysis

NBV-analysis

In Order to analyze the non basic variables we have to pick (in the final tableau)

For variable $y_{1}$

We have the absolute ratios:

• ${\frac {60}{1}}=60$

• ${\frac {10}{\frac {1}{2}}}=20$

• Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \frac {30}{-\frac{3}{2}}= -20

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): y_\text{1,min} = 60 - \min \left \{ \left | \frac{60}{1} \right |; \left |\ \frac{10}{\frac{1}{2}} \right |\ \right \} = 60-20 = 40

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): y_\text{1,max} = 60 + \min \left \{ \left | \frac {30}{-\frac{3}{2}} \right |\ \right \} = 60+20 = 80

For variable $y_{3}$

• the absolute smallest ratio of the right hand side and the coresponding negative collumn element.
  

Here it is Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \frac{10}{- \frac{1}{6}} = 60

• the absolutely smallest ratio of the right hand side and the corresponding positive collumn element.
  

Here it is ${\frac {30}{6}}=5$

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): y_\text{3,min} = 720 - \min \left \{ \left | \frac{30}{\frac{1}{6}} \right |\ \right \} = 720-180 = 540

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): y_\text{3,max} = 720 + \min \left \{ \left | \frac {10}{-\frac{1}{6}} \right |\ \right \} = 720+60 = 780

BV-analysis

In Order to analyze the basic variables we have to pick (in the final tableau)

for variable $x_{1}$

• the absolute smallest negative ratio of OF-coefficient and the corresponding row-element

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): x_\text{1,min} = -1 - \min \left \{ \left | \frac{\frac{1}{2}}{-\frac{3}{2}} \right |\ \right \} = -1-\frac{1}{3} = - \frac {4}{3}

• the absolute smallest positive ratio of OF-coefficient and the corresponding row-element

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): x_\text{1,max} = -1 + \min \left \{ \left | \frac{ \frac{1}{6} }{ \frac{1}{6} } \right |\ \right \} = -1 +1 = 0

for variable $x_{2}$

• the absolute smallest negative ratio of OF-coefficient and the corresponding row-element

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): x_\text{2,min} = -2 - \min \left \{ \left | \frac{ \frac{1}{6}} { 0 } \right |\ \right \} = -2 -\infty = -\infty

• the absolute smallest positive ratio of OF-coefficient and the corresponding row-element

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): x_\text{2,max} = -2 + \min \left \{ \left | \frac{\frac{1}{2}}{1} \right |\ \right \} = -2 +\frac{1}{2} = - \frac{3}{2}

Counterexample

To demonstrate, the funcionality of the sensibility analysis above and that it is correct we vary one of the problem's constraints.
For this example we modify $y_{3}$ to 781 instead of 720.

Fehler beim Erstellen des Vorschaubildes: Die Miniaturansicht konnte nicht am vorgesehenen Ort gespeichert werden

Now it is obvious that the basic variables in the optimal solution have changed. Thus it is clear what is meant by structural changes of the optimal solution. In this simple example the bottlenecks are now $y_{2}$ and $y_{1}$ and not $y_{3}$ and $y_{1}$ like in the first example.
The upper bound for this constraint is 780. When we choose 780 we can choose two possible pivot elements, thus it is not sure that the structure of the optimal solution will change.

Interpretation of the sensibility analysis

By the found intervals above, you can see the following:

From the interval $\textstyle \left[40;80\right]$ for variable $y_{1}$, you can see, in which the plantable Area of carrots may vary, without having an influence on the structure of the optimal solution.

For the variable $y_{3}$ the Interval $\textstyle \left[540;780\right]$ in which the budget may vary, without having an effect on the structure of the optimal solution.

For the variable $x_{1}$ an Interval from $\textstyle \left[0;{\frac {4}{3}}\right]$ in which the Cost contribution of roses may vary, without having an effect on the structure of the optimal solution.

For the variable $x_{2}$ an Interval from Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \textstyle \left [ \frac{3}{2} ; \infty \right ]

in which the Cost contribution of carrots may vary, without having an effect on the structure of the optimal solution.

Sources

OR lecturescript, OR-Course on Olat

Detailed example: Own Notes of last Year's Tutorial of Produktionswirtschaft

Authors: Arno Hornberg Philipp Stedem