View source for Scilab---FOSSEE-Optimisation-Toolbox/C2/Linear-Programming-using-linprog-function/English

'''Title of script''': '''Linear Programming using fot_linprog function'''

'''Author: Rupak Rokade, Siddharth Agarwal, Georgey John and Mankrit Singh'''

'''Keywords:Scilab console, FOSSEE Optimization Toolbox, Linear Programming, OR, Operations Research, fot_linprog, constraints, input, output, video tutorial.'''


{| border=1
! '''Visual Cue'''
! '''Narration'''
|-
|
Show Slide

'''Title Slide'''
| Hello and welcome to the spoken tutorial on “'''Linear Programming''' using '''fot underscore linprog function'''”.
|-
|
Show Slide

'''Learning Objectives'''
|
In this tutorial, we will learn how to:

*Solve '''linear programming''' problems using '''fot underscore linprog function''' in '''Scilab'''.
*Use '''fot underscore linprog function''' of '''FOSSEE Optimization Toolbox.'''
|-
|
Show Slide

'''System requirement'''
|
To record this tutorial, I am using

*'''Ubuntu 18.04'''
*'''Scilab 6.1.0'''
*'''FOSSEE Optimization Toolbox''' version 0.4.1.
|-
|
Show Slide

'''Pre-requisites'''

'''https://spoken-tutorial.org'''
|
To follow this tutorial, you should

*Install '''FOSSEE Optimization Toolbox''' version '''0.4.1''' or above
*Have basic understanding of '''Scilab''' and optimization theory
*If not, for relevant tutorials please visit this site.
|-
|
Show Slide

'''Code Files'''
|
*The files used in this tutorial have been provided in the '''Code files''' link.
*Please download and extract the files.
*Make a copy and then use them while practising.
|-
|
Show Slide

'''What is Linear Programming?'''
|'''What is Linear Programming?'''
*A '''function''' is '''linear,''' if it has a degree of one or zero.
*A '''linear program''' is a mathematical '''optimization model''' with:
**'''Linear objective function'''
**'''Linear constraints'''
|-
|
Show Slide

'''Mathematical Formulation'''
| A general form of the '''Linear Program''' is as shown.
|-
|
Show Slide

'''Example'''
|
We will now solve this example to illustrate the use of '''fot underscore linprog.'''

In this example, we will learn how to:

*Minimize the given '''function''' subject to these given constraints and bounds.

Note that the '''objective function''' and constraints are '''linear'''.
|-
|
Show Slide

'''Example'''
| I have downloaded the '''opt_linprog.sce''' file to my '''Downloads''' folder.
|-
| Cursor on the Scilab console.
| I have opened the '''Scilab console'''.
|-
| Type '''editor''' &gt;&gt; press '''Enter'''.
|
In the '''Scilab console,''' type '''editor''' and press '''Enter.'''

'''Editor''' window opens.
|-
|
Click on Open button >> locate the file '''opt_linprog.sce'''.

Click on the Ok button
|
Click on the '''Open''' button on the '''toolbar''' and locate the file '''opt_linprog.sce'''.

Then click the '''Ok''' button.

'''opt_linprog.sce''' file opens in the '''editor'''.
|-
| Show '''opt_linprog.sce''' in scilab editor.
| Now we will see the '''input arguments''' for '''fot underscore linprog'''.
|-
| Highlight the line with '''‘c’'''
| '''''c''''' is a '''vector''' of '''coefficients''' in the '''objective function'''.
|-
| Highlight the line with '''‘A’'''
|'''''A''''' is a '''matrix''' of '''coefficients''' of '''inequality''' constraints.
|-
| Highlight the line with '''‘b’'''
|'''''b''''' is a '''vector''' of the right-hand side of '''inequality''' constraints.
|-
| Highlight the line with '''‘Aeq’'''
| '''''‘Aeq’''''' is a '''matrix''' of '''coefficients''' of '''equality''' constraints.
|-
| Highlight the line with '''‘beq’'''
| '''''‘beq’''''' is a '''vector''' of the right-hand side of '''equality''' constraints.
|-
| Highlight the line with '''‘lb’'''
| '''''‘lb’''''' is a '''vector''' of lower bounds on '''''x'''''.
|-
| Highlight the line with '''‘ub’'''
| '''''‘ub’''''' is a '''vector''' of '''upper bounds''' on '''''x'''''.
|-
|

|Now we will summarize the '''output arguments'''.

'''Output arguments''' are '''''xopt, fopt, exitflag, output,''''' '''lambda'''
|-
| Highlight '''‘xopt’'''
| '''''xopt''''' is the optimal value of '''x'''.
|-
| Highlight '''‘fopt’'''
| '''''fopt''''' is the optimal '''objective function''' value.
|-
| Highlight '''‘exitflag’'''
| '''''exitflag''''' denotes the status of '''execution'''.
|-
| Highlight '''‘output’'''
| '''''Output''' is a'' '''structure''' containing detailed information about the '''optimization'''.
|-
| Highlight '''‘lambda’'''
| '''Lambda''' is a '''structure''' containing '''Lagrange multipliers''' at the optimal solution.
|-
| Highlight the line calling ‘'''fot_linprog’'''
| We will use the '''fot underscore linprog function''' to solve the example.
|-
|
Press '''CTRL + S'''

Click on the '''Execute''' button on scilab.

Select '''File with Echo''' from the drop-down.

|Save the file by pressing '''Control''' and '''‘S’''' keys simultaneously.

To '''run''' the file, click on the '''Execute''' menu.

Then click on '''File with Echo''' from the dropdown.
|-
|
Change the window to Scilab console

Highlight Optimal solution

Highlight '''‘xopt''' values'''’'''

Highlight '''‘fopt''' value'''’'''

Highlight '''‘exitflag''' value'''’'''

Highlight '''‘output''' values’

Highlight '''‘lambda''' values'''’'''
|Switch to the '''Scilab console''' to see the '''output'''.

'''Optimal solutions''' for the following are displayed on the '''Scilab console'''.

'''xopt''' values

'''fopt''' value

'''exitflag'''

'''output''' and

'''lambda.'''
|-
|
Show Slide

'''Alternate Input Arguments'''
| Now we see an alternate way of '''pass'''ing '''input arguments''' to '''fot underscore linprog.'''
|-
|
Show Slide

'''Alternate Input Arguments'''

Highlight '''‘file’'''

Highlight '''‘MPS’'''
|
'''file''' is a '''string''' stating the path of the '''mps file'''.

'''MPS (Mathematical Programming System)''' is a file format.
|-
|
Show Slide

'''Alternate Input Arguments'''

Highlight '''‘param’'''
|
It is used to present and archive:

*'''linear programming''' problems and
*'''mixed integer programming''' problems

'''param''' is a list containing '''parameters''' to be '''set'''.
|-
|
Show Slide

'''Unbounded Problems I'''
|
The problem we just saw was directly solvable using '''fot underscore linprog. '''

*There are cases when the '''optimal value''' is unbounded.
*The minimum value may go to '''negative infinity''' in the absence of suitable '''constraints'''.
|-
|
Show Slide

'''Unbounded Problems II'''
|Such problems are called '''Unbounded''' problems.
|-
|
Show Slide

'''Infeasible Problems'''
|
*There are instances where no solution exists for all the constraints.
*These problems are called '''Infeasible''' problems.

We will see an example of these constraints on '''Scilab'''.
|-
|
Switch to the scilab editor.

Replace the value.

Click on the '''Execute''' menu

Select '''File with Echo''' option
|
We will now change the previous problem to make it '''infeasible'''.

*Locate the line that defines '''ub'''
*Change the second element of '''ub''' to 0
*Click on the '''Execute''' menu,
*Click on '''File with Echo''' from the drop down

This will execute the '''Scilab code.'''
|-
|
Switch to the Scilab Console

Highlight the message '''Primal Infeasible'''
|
Switch to the '''Scilab console''' to see the '''output'''.

The '''console''' shows that the problem is '''Primal Infeasible.'''
|-
|
Show Slide

'''Summary'''
|
This brings us to the end of this tutorial.

Let us summarise.

In this tutorial, we have learnt how to:

*Use '''fot underscore linprog''' function of the '''FOSSEE Optimization Toolbox'''.
*Solve an '''LP''' example using '''fot underscore linprog''' in '''Scilab'''.
|-
|
Show Slide

'''Assignment'''

Highlight the '''constraint'''
|
As an assignment:

*Solve the same example that we '''executed''', with this additional constraint.
*The optimal value will be -0.5714 and optimal solution will be 0.2857 and 0.8571
*These are the optimal values of '''x1''' and '''x2'''
|-
|
Show Slide

'''About Spoken Tutorial Project'''
|
The video at the following link summarises the '''Spoken Tutorial''' project.

Please download and watch it.
|-
|
Show Slide

'''Spoken Tutorial Workshops'''
|
The '''Spoken Tutorial''' Project Team conducts workshops and gives certificates.

For more details, please write to us
|-
|
Show Slide

'''Answers for THIS Spoken Tutorial'''
| Please post your timed queries in this forum.
|-
| Show Slide: '''FOSSEE Forum'''
| Please post your general and technical queries on '''Scilab''' in this forum.
|-
|
Show Slide

'''Textbook Companion project'''
|
The '''FOSSEE''' team coordinates the '''Textbook Companion project'''.

We give '''Certificates''' and '''Honorarium''' to the contributors.

For more details, please visit this site..
|-
|
Show Slide

'''Lab Migration'''
|
The '''FOSSEE''' team coordinates the '''Lab Migration project'''.

For more details, please visit this site.
|-
|
Show Slide

'''Acknowledgment'''
| '''Spoken Tutorial''' and '''FOSSEE projects''' are funded by '''MoE, the Government of India'''.
|-
|
Show Slide

'''Thank you'''
|
This is Mankrit Singh, a FOSSEE intern 2021, IIT Bombay signing off

Thanks for joining.
|}