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.

Outline:

1. Define linear program

2. Linear Programming using FOT

3. General form of linear programming

4. Solve a minimizing problem where the objective function and constraints are linear and the decision variables are integers.

5. Write code to solve an integer programming problem using the FOT toolbox in scilab

6. Solve a similar minimizing problem using alternate input format using mps file

7. Define an infeasible optimization problem

8. Solving an infeasible optimization problem

9. Identify and differentiate between feasible and infeasible problems

10. Use the fot_linprog with various arguments

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 and 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 ?	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 >> 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 tool-bar 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. Video-editor: Pls put a textbox on screen."Clear console" window opens to ask are you sure you want to clear the console? - click on the yes button	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 drop down.
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 passing 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.

Contributors and Content Editors

Mankrits, Nancyvarkey