Difference between revisions of "Scilab---FOSSEE-Optimisation-Toolbox/C2/Integer-Linear-Programming-using-FOT/English"
(Created page with "'''Title of script''': '''Integer''' '''Linear Programming using fot_intlinprog function''' '''Author: Siddharth Agarwal and Mankrit Singh''' '''Keywords: FOSSEE Optimizatio...") |
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'''Title Slide''' | '''Title Slide''' | ||
− | | Welcome to the spoken tutorial on '''Integer Linear Programming''' | + | | Welcome to the spoken tutorial on '''Integer Linear Programming'''. |
|- | |- | ||
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In this tutorial, we will learn how to: | In this tutorial, we will learn how to: | ||
− | + | *Use the '''fot underscore intlinprog function''' in '''Scilab.''' | |
− | + | *Solve '''Integer linear programming''' problems using '''fot underscore intlinprog function'''. | |
− | + | ||
|- | |- | ||
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To record this tutorial, I am using | To record this tutorial, I am using | ||
− | + | *'''Ubuntu 18.04''' | |
− | + | *'''Scilab 6.1.0''' | |
− | + | *'''FOSSEE Optimization Toolbox''' version '''0.4.1''' | |
− | + | ||
− | + | ||
− | + | ||
|- | |- | ||
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To follow this tutorial, you should: | 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 | + | If not, for relevant tutorials please visit this website. |
|- | |- | ||
| | | | ||
Line 61: | Line 56: | ||
'''Code Files''' | '''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. | |
− | + | ||
|- | |- | ||
| | | | ||
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'''What is Integer Linear Programming ?''' | '''What is Integer Linear Programming ?''' | ||
− | | | + | |'''An Integer Linear Program''' is a mathematical '''optimization model''' with: |
− | '''An Integer Linear Program''' is a mathematical optimization model with: | + | |
− | + | *'''Linear objective function''' | |
− | + | *'''Linear''' constraints | |
− | + | *Some '''decision variables''' as '''integers'''. | |
− | + | ||
|- | |- | ||
| | | | ||
Line 82: | Line 74: | ||
'''Mathematical Formulation''' | '''Mathematical Formulation''' | ||
− | | A | + | | A general form of the '''Integer Linear Program''' is as shown. |
|- | |- | ||
| | | | ||
Line 93: | Line 85: | ||
In this example, we will learn how to: | In this example, we will learn how to: | ||
− | + | *Minimize the given '''function''' subjected to these given constraints and bounds. | |
− | + | *Note that the '''objective function''' and constraints are '''linear.''' | |
− | + | *The '''decision variables''' are '''integers'''. | |
− | + | ||
|- | |- | ||
| | | | ||
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Click on the OK button | Click on the OK button | ||
| | | | ||
− | Click on the ''' | + | Click on the '''Open''' button on the '''toolbar''' and locate the file '''opt_intlinprog.sce'''. |
Then click the '''OK''' button. | Then click the '''OK''' button. | ||
Line 125: | Line 116: | ||
|- | |- | ||
| Show '''opt_intlinprog.sce''' in scilab editor. | | Show '''opt_intlinprog.sce''' in scilab editor. | ||
− | | Now we will see the input arguments for '''fot underscore intlinprog'''. | + | | Now we will see the '''input arguments''' for '''fot underscore intlinprog'''. |
|- | |- | ||
| Highlight '''‘c’''' | | Highlight '''‘c’''' | ||
Line 132: | Line 123: | ||
| Highlight '''‘A’''' | | Highlight '''‘A’''' | ||
| | | | ||
− | '''A''' is a '''matrix''' of '''coefficients''' of | + | '''A''' is a '''matrix''' of '''coefficients''' of '''inequality''' constraints |
− | + | ||
− | '''inequality''' constraints | + | |
|- | |- | ||
| Highlight '''‘b’''' | | Highlight '''‘b’''' | ||
− | | '''b''' is a '''vector''' of the right-hand side of '''inequality | + | | '''b''' is a '''vector''' of the right-hand side of '''inequality''' constraints. |
|- | |- | ||
| Highlight '''‘Aeq’''' | | Highlight '''‘Aeq’''' | ||
− | | '''Aeq''' is a '''matrix''' of '''coefficients''' of '''equality | + | | '''Aeq''' is a '''matrix''' of '''coefficients''' of '''equality''' constraints. |
|- | |- | ||
| Highlight '''‘beq’''' | | Highlight '''‘beq’''' | ||
| | | | ||
− | '''beq''' is a '''vector''' of the right-hand side of '''equality | + | '''beq''' is a '''vector''' of the right-hand side of '''equality''' constraints. |
− | + | ||
− | + | ||
|- | |- | ||
| Highlight '''‘intcon’''' | | Highlight '''‘intcon’''' | ||
− | | '''intcon''' is a '''vector''' of the '''indices''' of integer variables | + | | '''intcon''' is a '''vector''' of the '''indices''' of '''integer variables'''. |
|- | |- | ||
| Highlight '''‘lb’''' | | Highlight '''‘lb’''' | ||
| | | | ||
'''lb''' is the '''lower bound''' for '''x'''. | '''lb''' is the '''lower bound''' for '''x'''. | ||
− | |||
− | |||
|- | |- | ||
| Highlight '''‘ub’''' | | Highlight '''‘ub’''' | ||
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|- | |- | ||
| Highlight '''‘options’''' | | Highlight '''‘options’''' | ||
− | | '''options''' is a | + | | '''options''' is a list containing the '''parameters''' of the '''solver''' that is to be '''set'''. |
|- | |- | ||
| Highlight Output arguments | | Highlight Output arguments | ||
− | | | + | |Now we will see the '''output arguments'''. |
− | Now we will see the output arguments. | + | |
− | Output arguments are '''xopt,''' '''fopt''', '''exitflag | + | '''Output arguments''' are '''xopt,''' '''fopt''', '''exitflag and output''' |
|- | |- | ||
| Highlight '''‘xopt’''' | | Highlight '''‘xopt’''' | ||
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|- | |- | ||
| Highlight '''‘fopt’''' | | Highlight '''‘fopt’''' | ||
− | | '''fopt''' is the optimal | + | | '''fopt''' is the optimal '''objective function''' value. |
|- | |- | ||
| Highlight '''‘exitflag’''' | | Highlight '''‘exitflag’''' | ||
− | | '''exitflag''' is the status of execution | + | | '''exitflag''' is the status of '''execution''' |
|- | |- | ||
| Highlight '''‘output’''' | | Highlight '''‘output’''' | ||
− | | '''output''' is a '''structure''' containing detailed information about the optimization. | + | | '''output''' is a '''structure''' containing detailed information about the '''optimization'''. |
|- | |- | ||
| | | | ||
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'''[xopt,fopt,exitflag,output]= fot_intlinprog(c, intcon, A, b, Aeq, beq, lb, ub,options)''' | '''[xopt,fopt,exitflag,output]= fot_intlinprog(c, intcon, A, b, Aeq, beq, lb, ub,options)''' | ||
| | | | ||
− | Here we see the ''' | + | Here we see the '''Scilab''' code to define and solve the example. |
− | We call the '''fot underscore intlinprog function''' to solve the given | + | We call the '''fot underscore intlinprog function''' to solve the given problem. |
− | + | ||
− | + | ||
|- | |- | ||
| | | | ||
− | Press '''CTRL + | + | Press '''CTRL + S''' |
Click on execute button on scilab | Click on execute button on scilab | ||
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Save the file by pressing '''Control''' and '''S''' keys simultaneously. | Save the file by pressing '''Control''' and '''S''' keys simultaneously. | ||
− | To '''run''' the file, click on the '''Execute''' | + | To '''run''' the file, click on the '''Execute menu'''. |
− | + | ||
− | + | ||
− | down. | + | Click on '''File with Echo''' from the drop-down. |
In the '''Clear Console''' window click on the '''Yes''' button. | In the '''Clear Console''' window click on the '''Yes''' button. | ||
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Highlight '''‘output''' values'''’''' | Highlight '''‘output''' values'''’''' | ||
| | | | ||
− | Switch to the Scilab '''console''' to see the output. | + | Switch to the Scilab '''console''' to see the '''output'''. |
− | We see that it prints the '''xopt''' | + | We see that it prints the |
+ | '''xopt''' values, | ||
− | '''fopt''' | + | '''fopt''' value, |
'''exitflag''' and | '''exitflag''' and | ||
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'''output''' in the '''Scilab console'''. | '''output''' in the '''Scilab console'''. | ||
− | Since this is an | + | Since this is an '''integer programming''' problem, some '''decision variables''' are '''integers'''. |
|- | |- | ||
| Show '''opt_intlinprog.sce''' in scilab editor. | | Show '''opt_intlinprog.sce''' in scilab editor. | ||
− | | Let's see an alternate way of | + | | Let's see an alternate way of '''pass'''ing '''input arguments''' to '''fot underscore intlinprog.''' |
|- | |- | ||
| | | | ||
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Highlight '''‘options’''' | Highlight '''‘options’''' | ||
| | | | ||
− | file :A string containing the path of the mps file to be read. | + | '''file''' :A '''string''' containing the path of the '''mps''' file to be read. |
− | options: A list containing the parameters of the solver that is to be set. | + | '''options''': A list containing the '''parameters''' of the '''solver''' that is to be '''set'''. |
|- | |- | ||
| | | | ||
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'''Options''' | '''Options''' | ||
| | | | ||
− | The options allow the user to set various parameters of the '''Optimization''' problem. | + | The '''options''' allow the user to set various '''parameters''' of the '''Optimization''' problem. |
Two such options are: | Two such options are: | ||
− | + | *'''MaxTime''': The maximum amount of '''CPU''' time in seconds that the '''solver''' should take. | |
− | + | *'''MaxNodes''': The maximum number of '''nodes''' that the '''solver''' should search. | |
− | + | ||
|- | |- | ||
| | | | ||
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'''Exitflags''' | '''Exitflags''' | ||
| | | | ||
− | + | *In the example we have '''executed''', you have seen the '''exitflag'''. | |
− | + | *This indicates the status of '''execution'''. | |
− | + | *The documentation explains what they mean for each '''function'''. | |
− | + | ||
|- | |- | ||
| | | | ||
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For '''fot underscore intlinprog''', they are explained briefly as follows: | For '''fot underscore intlinprog''', they are explained briefly as follows: | ||
− | 0 : Optimal Solution Found. | + | '''0 : Optimal Solution Found.''' |
|- | |- | ||
| Highlight 1 | | Highlight 1 | ||
− | | 1 : Converged to a point of primal infeasibility. | + | | '''1 : Converged to a point of primal infeasibility.''' |
|- | |- | ||
| Highlight 2 | | Highlight 2 | ||
− | | 2: Solution Limit is reached. | + | | '''2: Solution Limit is reached.''' |
|- | |- | ||
| Highlight 3 | | Highlight 3 | ||
− | | 3: Node Limit is reached. Output may not be optimal. | + | | '''3: Node Limit is reached. Output may not be optimal.''' |
|- | |- | ||
| Highlight 4 | | Highlight 4 | ||
− | | 4 : Numerical Difficulties. | + | | '''4 : Numerical Difficulties.''' |
|- | |- | ||
| Highlight 5 | | Highlight 5 | ||
− | | 5 : Maximum amount of CPU Time exceeded. | + | | '''5 : Maximum amount of CPU Time exceeded.''' |
|- | |- | ||
| Highlight 6 | | Highlight 6 | ||
− | | 6 : Continuous Solution Unbounded. | + | | '''6 : Continuous Solution Unbounded.''' |
|- | |- | ||
| Highlight 7 | | Highlight 7 | ||
− | | 7 : Converged to a point of dual infeasibility. | + | | '''7 : Converged to a point of dual infeasibility.''' |
|- | |- | ||
| | | | ||
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In this example, we will learn how to: | In this example, we will learn how to: | ||
− | + | *Use '''mps''' files for large '''optimization''' problems | |
− | + | *Use '''options'''. | |
− | + | *Interpret '''exitflags'''. | |
− | + | ||
|- | |- | ||
| | | | ||
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Show '''opt_intlinprog2.sce''' in scilab editor. | Show '''opt_intlinprog2.sce''' in scilab editor. | ||
| | | | ||
− | We will use the toolbox to solve this example. | + | We will use the '''toolbox''' to solve this example. |
Open the '''Scilab console'''. | Open the '''Scilab console'''. | ||
− | Type editor on the Scilab console and | + | Type '''editor''' on the '''Scilab console''' and press '''Enter'''. |
Open '''opt_intlinprog2.sce''' in the '''Scilab editor.''' | Open '''opt_intlinprog2.sce''' in the '''Scilab editor.''' | ||
Line 341: | Line 319: | ||
'''file = "liu.mps";''' | '''file = "liu.mps";''' | ||
| | | | ||
− | Instead of defining the integer linear program as a series of vectors and matrices, | + | Instead of defining the '''integer linear program''' as a series of '''vectors''' and '''matrices''', we can use it directly from a file. |
− | + | '''LIU''' is a well-known problem with '''1156 decision variables'''. | |
− | + | ||
− | LIU is a well-known problem with 1156 decision variables. | + | |
It would take many hours to solve if we don’t add any time constraints. | It would take many hours to solve if we don’t add any time constraints. | ||
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'''options = list('MaxNodes',[1000], 'MaxTime',[30]);''' | '''options = list('MaxNodes',[1000], 'MaxTime',[30]);''' | ||
| | | | ||
− | We define options to ensure that | + | We define options to ensure that the maximum number of '''nodes''' expanded won’t exceed 1000. |
− | + | We also ensure that the time taken by the '''solver''' doesn’t exceed 30 seconds. | |
− | + | ||
− | We also ensure that the time taken by the solver doesn’t exceed 30 seconds. | + | |
Here we call '''fot underscore intlinprog.''' | Here we call '''fot underscore intlinprog.''' | ||
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Press '''CTRL +S to''' save the file. | Press '''CTRL +S to''' save the file. | ||
− | Click on '''Execute''' button on | + | Click on '''Execute''' button on '''Scilab''' |
Click on '''File with Echo''' from the drop down. | Click on '''File with Echo''' from the drop down. | ||
− | In the Clear Console window click on the '''Yes''' button. | + | In the '''Clear Console''' window click on the '''Yes''' button. |
| | | | ||
Save the file by pressing '''Control''' and '''‘S’''' keys simultaneously. | Save the file by pressing '''Control''' and '''‘S’''' keys simultaneously. | ||
− | To '''run''' the file, click on the '''Execute''' | + | To '''run''' the file, click on the '''Execute menu'''. |
Click on '''File with Echo''' from the drop down. | Click on '''File with Echo''' from the drop down. | ||
− | In the Clear Console window click on the '''Yes''' button. | + | In the '''Clear Console''' window click on the '''Yes''' button. |
|- | |- | ||
| | | | ||
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Highlight the '''exitflag''' value | Highlight the '''exitflag''' value | ||
| | | | ||
− | Switch to Scilab console to see the output. | + | Switch to '''Scilab console''' to see the '''output'''. |
We see that '''exitflag,''' and '''output''' are displayed on the '''Scilab console'''. | We see that '''exitflag,''' and '''output''' are displayed on the '''Scilab console'''. | ||
− | The '''exitflag''' is 3, indicating that the node limit is reached and the output may not be optimal. | + | The '''exitflag''' is 3, indicating that the '''node limit''' is reached and the '''output''' may not be optimal. |
− | As the output indicates, the gap is quite huge. | + | As the '''output''' indicates, the gap is quite huge. |
|- | |- | ||
| | | | ||
Show Slide: | 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 intlinprog function''' of the '''FOSSEE Optimization Toolbox'''. | |
− | + | *Solve an '''integer linear programming''' example using '''fot underscore intlinprog''' in '''Scilab'''. | |
− | + | *Use '''options''' to exert control on the '''solver'''. | |
+ | *Read '''exitflags'''. | ||
+ | *Use '''MPS''' files as '''inputs'''. | ||
|- | |- | ||
| | | | ||
Show Slide: | Show Slide: | ||
− | ''' | + | '''Assignment''' |
| | | | ||
− | + | As an assignment: | |
− | + | ||
− | + | ||
− | + | *Solve the given example | |
− | + | *The optimal value will be 7000 and optimal solution will be '''x one''' equal to 1, '''x two''' equal to 0, '''x three''' equal to 2, and '''x four''' equal to 0. | |
− | + | ||
− | + | ||
− | + | ||
− | + | ||
|- | |- | ||
| | | | ||
Line 479: | Line 451: | ||
'''Thank you''' | '''Thank you''' | ||
| | | | ||
− | This is '''Mankrit Singh''', a '''FOSSEE''' intern 2021, '''IIT Bombay''' signing off | + | This is '''Mankrit Singh''', a '''FOSSEE''' intern 2021, '''IIT Bombay''' signing off. |
Thanks for joining. | Thanks for joining. | ||
|} | |} |
Latest revision as of 16:31, 21 June 2022
Title of script: Integer Linear Programming using fot_intlinprog function
Author: Siddharth Agarwal and Mankrit Singh
Keywords: FOSSEE Optimization Toolbox, Integer Linear Programming, OR, fot_intlinprog.
Visual Cue | Narration |
---|---|
Show Slide: Title Slide |
Welcome to the spoken tutorial on Integer Linear Programming. |
Show Slide: Learning Objectives |
In this tutorial, we will learn how to:
|
Show Slide: System requirement |
To record this tutorial, I am using
|
Show Slide: Pre-requisites |
To follow this tutorial, you should:
If not, for relevant tutorials please visit this website. |
Show Slide: Code Files |
|
Show Slide: What is Integer Linear Programming ? |
An Integer Linear Program is a mathematical optimization model with:
|
Show Slide: Mathematical Formulation |
A general form of the Integer Linear Program is as shown. |
Show Slide: Example |
We will now solve this example to illustrate the use of fot underscore intlinprog. In this example, we will learn how to:
|
Show Slide: Example |
I have downloaded the required files to my Downloads folder. |
Cursor on the Scilab console. | Now open 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_intlinprog.sce. Click on the OK button |
Click on the Open button on the toolbar and locate the file opt_intlinprog.sce. Then click the OK button. opt_intlinprog.sce file opens in the editor. |
Show opt_intlinprog.sce in scilab editor. | Now we will see the input arguments for fot underscore intlinprog. |
Highlight ‘c’ | c is a vector for the coefficients in the objective function |
Highlight ‘A’ |
A is a matrix of coefficients of inequality constraints |
Highlight ‘b’ | b is a vector of the right-hand side of inequality constraints. |
Highlight ‘Aeq’ | Aeq is a matrix of coefficients of equality constraints. |
Highlight ‘beq’ |
beq is a vector of the right-hand side of equality constraints. |
Highlight ‘intcon’ | intcon is a vector of the indices of integer variables. |
Highlight ‘lb’ |
lb is the lower bound for x. |
Highlight ‘ub’ | ub is the upper bound for x. |
Highlight ‘options’ | options is a list containing the parameters of the solver that is to be set. |
Highlight Output arguments | Now we will see the output arguments.
Output arguments are xopt, fopt, exitflag and output |
Highlight ‘xopt’ | xopt is the optimal value of x. |
Highlight ‘fopt’ | fopt is the optimal objective function value. |
Highlight ‘exitflag’ | exitflag is the status of execution |
Highlight ‘output’ | output is a structure containing detailed information about the optimization. |
Highlight [xopt,fopt,exitflag,output]= fot_intlinprog(c, intcon, A, b, Aeq, beq, lb, ub,options) |
Here we see the Scilab code to define and solve the example. We call the fot underscore intlinprog function to solve the given problem. |
Press CTRL + S Click on execute button on scilab Click on the Execute Button Click on the File with Echo button In the Clear Console window click on the Yes button. |
Save the file by pressing Control and S keys simultaneously. To run the file, click on the Execute menu. Click on File with Echo from the drop-down. In the Clear Console window click on the Yes button. |
Change the window to Scilab console Highlight ‘xopt values’ Highlight ‘fopt value’ Highlight ‘exitflag value’ Highlight ‘output values’ |
Switch to the Scilab console to see the output. We see that it prints the xopt values, fopt value, exitflag and output in the Scilab console. Since this is an integer programming problem, some decision variables are integers. |
Show opt_intlinprog.sce in scilab editor. | Let's see an alternate way of passing input arguments to fot underscore intlinprog. |
Show Slide: Alternate Input Arguments Highlight ‘file’ Highlight ‘options’ |
file :A string containing the path of the mps file to be read. options: A list containing the parameters of the solver that is to be set. |
Show Slide: Options |
The options allow the user to set various parameters of the Optimization problem. Two such options are:
|
Show Slide: Exitflags |
|
Show Slide: Exitflags Highlight 0 |
For fot underscore intlinprog, they are explained briefly as follows: 0 : Optimal Solution Found. |
Highlight 1 | 1 : Converged to a point of primal infeasibility. |
Highlight 2 | 2: Solution Limit is reached. |
Highlight 3 | 3: Node Limit is reached. Output may not be optimal. |
Highlight 4 | 4 : Numerical Difficulties. |
Highlight 5 | 5 : Maximum amount of CPU Time exceeded. |
Highlight 6 | 6 : Continuous Solution Unbounded. |
Highlight 7 | 7 : Converged to a point of dual infeasibility. |
Show Slide: Example |
We will now solve this example to illustrate the use of fot underscore intlinprog with mps files In this example, we will learn how to:
|
Show opt_intlinprog2.sce in scilab editor. Type editor >> press Enter. Show opt_intlinprog2.sce in scilab editor. |
We will use the toolbox to solve this example. Open the Scilab console. Type editor on the Scilab console and press Enter. Open opt_intlinprog2.sce in the Scilab editor. |
Highlight file = "liu.mps"; |
Instead of defining the integer linear program as a series of vectors and matrices, we can use it directly from a file. LIU is a well-known problem with 1156 decision variables. It would take many hours to solve if we don’t add any time constraints. |
Highlight options = list('MaxNodes',[1000], 'MaxTime',[30]); |
We define options to ensure that the maximum number of nodes expanded won’t exceed 1000. We also ensure that the time taken by the solver doesn’t exceed 30 seconds. Here we call fot underscore intlinprog. |
Press CTRL +S to save the file. Click on Execute button on Scilab Click on File with Echo from the drop down. In the Clear Console window click on the Yes button. |
Save the file by pressing Control and ‘S’ keys simultaneously. To run the file, click on the Execute menu. Click on File with Echo from the drop down. In the Clear Console window click on the Yes button. |
Change the window to Scilab console Highlight exitflag and output values Highlight the exitflag value |
Switch to Scilab console to see the output. We see that exitflag, and output are displayed on the Scilab console. The exitflag is 3, indicating that the node limit is reached and the output may not be optimal. As the output indicates, the gap is quite huge. |
Show Slide: Summary |
This brings us to the end of this tutorial. Let us summarise. In this tutorial, we have learnt how to:
|
Show Slide: Assignment |
As an assignment:
|
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: Acknowledgement |
Spoken Tutorial and FOSSEE projects are funded by MoE, Government of India. |
Show Slide : Thank you |
This is Mankrit Singh, a FOSSEE intern 2021, IIT Bombay signing off. Thanks for joining. |