1. El documento presenta los pasos del método simplex para resolver problemas de optimización lineal.
2. Los tres pasos son: paso inicial, prueba de optimalidad y paso iterativo.
3. Se aplica el método simplex a un ejemplo de asignación de recursos para maximizar beneficios.
2024 State of Marketing Report – by HubspotMarius Sescu
https://www.hubspot.com/state-of-marketing
· Scaling relationships and proving ROI
· Social media is the place for search, sales, and service
· Authentic influencer partnerships fuel brand growth
· The strongest connections happen via call, click, chat, and camera.
· Time saved with AI leads to more creative work
· Seeking: A single source of truth
· TLDR; Get on social, try AI, and align your systems.
· More human marketing, powered by robots
ChatGPT is a revolutionary addition to the world since its introduction in 2022. A big shift in the sector of information gathering and processing happened because of this chatbot. What is the story of ChatGPT? How is the bot responding to prompts and generating contents? Swipe through these slides prepared by Expeed Software, a web development company regarding the development and technical intricacies of ChatGPT!
Product Design Trends in 2024 | Teenage EngineeringsPixeldarts
The realm of product design is a constantly changing environment where technology and style intersect. Every year introduces fresh challenges and exciting trends that mold the future of this captivating art form. In this piece, we delve into the significant trends set to influence the look and functionality of product design in the year 2024.
How Race, Age and Gender Shape Attitudes Towards Mental HealthThinkNow
Mental health has been in the news quite a bit lately. Dozens of U.S. states are currently suing Meta for contributing to the youth mental health crisis by inserting addictive features into their products, while the U.S. Surgeon General is touring the nation to bring awareness to the growing epidemic of loneliness and isolation. The country has endured periods of low national morale, such as in the 1970s when high inflation and the energy crisis worsened public sentiment following the Vietnam War. The current mood, however, feels different. Gallup recently reported that national mental health is at an all-time low, with few bright spots to lift spirits.
To better understand how Americans are feeling and their attitudes towards mental health in general, ThinkNow conducted a nationally representative quantitative survey of 1,500 respondents and found some interesting differences among ethnic, age and gender groups.
Technology
For example, 52% agree that technology and social media have a negative impact on mental health, but when broken out by race, 61% of Whites felt technology had a negative effect, and only 48% of Hispanics thought it did.
While technology has helped us keep in touch with friends and family in faraway places, it appears to have degraded our ability to connect in person. Staying connected online is a double-edged sword since the same news feed that brings us pictures of the grandkids and fluffy kittens also feeds us news about the wars in Israel and Ukraine, the dysfunction in Washington, the latest mass shooting and the climate crisis.
Hispanics may have a built-in defense against the isolation technology breeds, owing to their large, multigenerational households, strong social support systems, and tendency to use social media to stay connected with relatives abroad.
Age and Gender
When asked how individuals rate their mental health, men rate it higher than women by 11 percentage points, and Baby Boomers rank it highest at 83%, saying it’s good or excellent vs. 57% of Gen Z saying the same.
Gen Z spends the most amount of time on social media, so the notion that social media negatively affects mental health appears to be correlated. Unfortunately, Gen Z is also the generation that’s least comfortable discussing mental health concerns with healthcare professionals. Only 40% of them state they’re comfortable discussing their issues with a professional compared to 60% of Millennials and 65% of Boomers.
Race Affects Attitudes
As seen in previous research conducted by ThinkNow, Asian Americans lag other groups when it comes to awareness of mental health issues. Twenty-four percent of Asian Americans believe that having a mental health issue is a sign of weakness compared to the 16% average for all groups. Asians are also considerably less likely to be aware of mental health services in their communities (42% vs. 55%) and most likely to seek out information on social media (51% vs. 35%).
AI Trends in Creative Operations 2024 by Artwork Flow.pdfmarketingartwork
This article is all about what AI trends will emerge in the field of creative operations in 2024. All the marketers and brand builders should be aware of these trends for their further use and save themselves some time!
2024 State of Marketing Report – by HubspotMarius Sescu
https://www.hubspot.com/state-of-marketing
· Scaling relationships and proving ROI
· Social media is the place for search, sales, and service
· Authentic influencer partnerships fuel brand growth
· The strongest connections happen via call, click, chat, and camera.
· Time saved with AI leads to more creative work
· Seeking: A single source of truth
· TLDR; Get on social, try AI, and align your systems.
· More human marketing, powered by robots
ChatGPT is a revolutionary addition to the world since its introduction in 2022. A big shift in the sector of information gathering and processing happened because of this chatbot. What is the story of ChatGPT? How is the bot responding to prompts and generating contents? Swipe through these slides prepared by Expeed Software, a web development company regarding the development and technical intricacies of ChatGPT!
Product Design Trends in 2024 | Teenage EngineeringsPixeldarts
The realm of product design is a constantly changing environment where technology and style intersect. Every year introduces fresh challenges and exciting trends that mold the future of this captivating art form. In this piece, we delve into the significant trends set to influence the look and functionality of product design in the year 2024.
How Race, Age and Gender Shape Attitudes Towards Mental HealthThinkNow
Mental health has been in the news quite a bit lately. Dozens of U.S. states are currently suing Meta for contributing to the youth mental health crisis by inserting addictive features into their products, while the U.S. Surgeon General is touring the nation to bring awareness to the growing epidemic of loneliness and isolation. The country has endured periods of low national morale, such as in the 1970s when high inflation and the energy crisis worsened public sentiment following the Vietnam War. The current mood, however, feels different. Gallup recently reported that national mental health is at an all-time low, with few bright spots to lift spirits.
To better understand how Americans are feeling and their attitudes towards mental health in general, ThinkNow conducted a nationally representative quantitative survey of 1,500 respondents and found some interesting differences among ethnic, age and gender groups.
Technology
For example, 52% agree that technology and social media have a negative impact on mental health, but when broken out by race, 61% of Whites felt technology had a negative effect, and only 48% of Hispanics thought it did.
While technology has helped us keep in touch with friends and family in faraway places, it appears to have degraded our ability to connect in person. Staying connected online is a double-edged sword since the same news feed that brings us pictures of the grandkids and fluffy kittens also feeds us news about the wars in Israel and Ukraine, the dysfunction in Washington, the latest mass shooting and the climate crisis.
Hispanics may have a built-in defense against the isolation technology breeds, owing to their large, multigenerational households, strong social support systems, and tendency to use social media to stay connected with relatives abroad.
Age and Gender
When asked how individuals rate their mental health, men rate it higher than women by 11 percentage points, and Baby Boomers rank it highest at 83%, saying it’s good or excellent vs. 57% of Gen Z saying the same.
Gen Z spends the most amount of time on social media, so the notion that social media negatively affects mental health appears to be correlated. Unfortunately, Gen Z is also the generation that’s least comfortable discussing mental health concerns with healthcare professionals. Only 40% of them state they’re comfortable discussing their issues with a professional compared to 60% of Millennials and 65% of Boomers.
Race Affects Attitudes
As seen in previous research conducted by ThinkNow, Asian Americans lag other groups when it comes to awareness of mental health issues. Twenty-four percent of Asian Americans believe that having a mental health issue is a sign of weakness compared to the 16% average for all groups. Asians are also considerably less likely to be aware of mental health services in their communities (42% vs. 55%) and most likely to seek out information on social media (51% vs. 35%).
AI Trends in Creative Operations 2024 by Artwork Flow.pdfmarketingartwork
This article is all about what AI trends will emerge in the field of creative operations in 2024. All the marketers and brand builders should be aware of these trends for their further use and save themselves some time!
A report by thenetworkone and Kurio.
The contributing experts and agencies are (in an alphabetical order): Sylwia Rytel, Social Media Supervisor, 180heartbeats + JUNG v MATT (PL), Sharlene Jenner, Vice President - Director of Engagement Strategy, Abelson Taylor (USA), Alex Casanovas, Digital Director, Atrevia (ES), Dora Beilin, Senior Social Strategist, Barrett Hoffher (USA), Min Seo, Campaign Director, Brand New Agency (KR), Deshé M. Gully, Associate Strategist, Day One Agency (USA), Francesca Trevisan, Strategist, Different (IT), Trevor Crossman, CX and Digital Transformation Director; Olivia Hussey, Strategic Planner; Simi Srinarula, Social Media Manager, The Hallway (AUS), James Hebbert, Managing Director, Hylink (CN / UK), Mundy Álvarez, Planning Director; Pedro Rojas, Social Media Manager; Pancho González, CCO, Inbrax (CH), Oana Oprea, Head of Digital Planning, Jam Session Agency (RO), Amy Bottrill, Social Account Director, Launch (UK), Gaby Arriaga, Founder, Leonardo1452 (MX), Shantesh S Row, Creative Director, Liwa (UAE), Rajesh Mehta, Chief Strategy Officer; Dhruv Gaur, Digital Planning Lead; Leonie Mergulhao, Account Supervisor - Social Media & PR, Medulla (IN), Aurelija Plioplytė, Head of Digital & Social, Not Perfect (LI), Daiana Khaidargaliyeva, Account Manager, Osaka Labs (UK / USA), Stefanie Söhnchen, Vice President Digital, PIABO Communications (DE), Elisabeth Winiartati, Managing Consultant, Head of Global Integrated Communications; Lydia Aprina, Account Manager, Integrated Marketing and Communications; Nita Prabowo, Account Manager, Integrated Marketing and Communications; Okhi, Web Developer, PNTR Group (ID), Kei Obusan, Insights Director; Daffi Ranandi, Insights Manager, Radarr (SG), Gautam Reghunath, Co-founder & CEO, Talented (IN), Donagh Humphreys, Head of Social and Digital Innovation, THINKHOUSE (IRE), Sarah Yim, Strategy Director, Zulu Alpha Kilo (CA).
Trends In Paid Search: Navigating The Digital Landscape In 2024Search Engine Journal
The search marketing landscape is evolving rapidly with new technologies, and professionals, like you, rely on innovative paid search strategies to meet changing demands.
It’s important that you’re ready to implement new strategies in 2024.
Check this out and learn the top trends in paid search advertising that are expected to gain traction, so you can drive higher ROI more efficiently in 2024.
You’ll learn:
- The latest trends in AI and automation, and what this means for an evolving paid search ecosystem.
- New developments in privacy and data regulation.
- Emerging ad formats that are expected to make an impact next year.
Watch Sreekant Lanka from iQuanti and Irina Klein from OneMain Financial as they dive into the future of paid search and explore the trends, strategies, and technologies that will shape the search marketing landscape.
If you’re looking to assess your paid search strategy and design an industry-aligned plan for 2024, then this webinar is for you.
5 Public speaking tips from TED - Visualized summarySpeakerHub
From their humble beginnings in 1984, TED has grown into the world’s most powerful amplifier for speakers and thought-leaders to share their ideas. They have over 2,400 filmed talks (not including the 30,000+ TEDx videos) freely available online, and have hosted over 17,500 events around the world.
With over one billion views in a year, it’s no wonder that so many speakers are looking to TED for ideas on how to share their message more effectively.
The article “5 Public-Speaking Tips TED Gives Its Speakers”, by Carmine Gallo for Forbes, gives speakers five practical ways to connect with their audience, and effectively share their ideas on stage.
Whether you are gearing up to get on a TED stage yourself, or just want to master the skills that so many of their speakers possess, these tips and quotes from Chris Anderson, the TED Talks Curator, will encourage you to make the most impactful impression on your audience.
See the full article and more summaries like this on SpeakerHub here: https://speakerhub.com/blog/5-presentation-tips-ted-gives-its-speakers
See the original article on Forbes here:
http://www.forbes.com/forbes/welcome/?toURL=http://www.forbes.com/sites/carminegallo/2016/05/06/5-public-speaking-tips-ted-gives-its-speakers/&refURL=&referrer=#5c07a8221d9b
ChatGPT and the Future of Work - Clark Boyd Clark Boyd
Everyone is in agreement that ChatGPT (and other generative AI tools) will shape the future of work. Yet there is little consensus on exactly how, when, and to what extent this technology will change our world.
Businesses that extract maximum value from ChatGPT will use it as a collaborative tool for everything from brainstorming to technical maintenance.
For individuals, now is the time to pinpoint the skills the future professional will need to thrive in the AI age.
Check out this presentation to understand what ChatGPT is, how it will shape the future of work, and how you can prepare to take advantage.
A brief introduction to DataScience with explaining of the concepts, algorithms, machine learning, supervised and unsupervised learning, clustering, statistics, data preprocessing, real-world applications etc.
It's part of a Data Science Corner Campaign where I will be discussing the fundamentals of DataScience, AIML, Statistics etc.
Time Management & Productivity - Best PracticesVit Horky
Here's my presentation on by proven best practices how to manage your work time effectively and how to improve your productivity. It includes practical tips and how to use tools such as Slack, Google Apps, Hubspot, Google Calendar, Gmail and others.
The six step guide to practical project managementMindGenius
The six step guide to practical project management
If you think managing projects is too difficult, think again.
We’ve stripped back project management processes to the
basics – to make it quicker and easier, without sacrificing
the vital ingredients for success.
“If you’re looking for some real-world guidance, then The Six Step Guide to Practical Project Management will help.”
Dr Andrew Makar, Tactical Project Management
Unlocking the Power of ChatGPT and AI in Testing - A Real-World Look, present...Applitools
During this webinar, Anand Bagmar demonstrates how AI tools such as ChatGPT can be applied to various stages of the software development life cycle (SDLC) using an eCommerce application case study. Find the on-demand recording and more info at https://applitools.info/b59
Key takeaways:
• Learn how to use ChatGPT to add AI power to your testing and test automation
• Understand the limitations of the technology and where human expertise is crucial
• Gain insight into different AI-based tools
• Adopt AI-based tools to stay relevant and optimize work for developers and testers
* ChatGPT and OpenAI belong to OpenAI, L.L.C.
More than Just Lines on a Map: Best Practices for U.S Bike Routes
This session highlights best practices and lessons learned for U.S. Bike Route System designation, as well as how and why these routes should be integrated into bicycle planning at the local and regional level.
Presenters:
Presenter: Kevin Luecke Toole Design Group
Co-Presenter: Virginia Sullivan Adventure Cycling Association
Ride the Storm: Navigating Through Unstable Periods / Katerina Rudko (Belka G...DevGAMM Conference
Has your project been caught in a storm of deadlines, clashing requirements, and the need to change course halfway through? If yes, then check out how the administration team navigated through all of this, relocating 160 people from 3 countries and opening 2 offices during the most turbulent time in the last 20 years. Belka Games’ Chief Administrative Officer, Katerina Rudko, will share universal approaches and life hacks that can help your project survive unstable periods when there seem to be too many tasks and a lack of time and people.
A report by thenetworkone and Kurio.
The contributing experts and agencies are (in an alphabetical order): Sylwia Rytel, Social Media Supervisor, 180heartbeats + JUNG v MATT (PL), Sharlene Jenner, Vice President - Director of Engagement Strategy, Abelson Taylor (USA), Alex Casanovas, Digital Director, Atrevia (ES), Dora Beilin, Senior Social Strategist, Barrett Hoffher (USA), Min Seo, Campaign Director, Brand New Agency (KR), Deshé M. Gully, Associate Strategist, Day One Agency (USA), Francesca Trevisan, Strategist, Different (IT), Trevor Crossman, CX and Digital Transformation Director; Olivia Hussey, Strategic Planner; Simi Srinarula, Social Media Manager, The Hallway (AUS), James Hebbert, Managing Director, Hylink (CN / UK), Mundy Álvarez, Planning Director; Pedro Rojas, Social Media Manager; Pancho González, CCO, Inbrax (CH), Oana Oprea, Head of Digital Planning, Jam Session Agency (RO), Amy Bottrill, Social Account Director, Launch (UK), Gaby Arriaga, Founder, Leonardo1452 (MX), Shantesh S Row, Creative Director, Liwa (UAE), Rajesh Mehta, Chief Strategy Officer; Dhruv Gaur, Digital Planning Lead; Leonie Mergulhao, Account Supervisor - Social Media & PR, Medulla (IN), Aurelija Plioplytė, Head of Digital & Social, Not Perfect (LI), Daiana Khaidargaliyeva, Account Manager, Osaka Labs (UK / USA), Stefanie Söhnchen, Vice President Digital, PIABO Communications (DE), Elisabeth Winiartati, Managing Consultant, Head of Global Integrated Communications; Lydia Aprina, Account Manager, Integrated Marketing and Communications; Nita Prabowo, Account Manager, Integrated Marketing and Communications; Okhi, Web Developer, PNTR Group (ID), Kei Obusan, Insights Director; Daffi Ranandi, Insights Manager, Radarr (SG), Gautam Reghunath, Co-founder & CEO, Talented (IN), Donagh Humphreys, Head of Social and Digital Innovation, THINKHOUSE (IRE), Sarah Yim, Strategy Director, Zulu Alpha Kilo (CA).
Trends In Paid Search: Navigating The Digital Landscape In 2024Search Engine Journal
The search marketing landscape is evolving rapidly with new technologies, and professionals, like you, rely on innovative paid search strategies to meet changing demands.
It’s important that you’re ready to implement new strategies in 2024.
Check this out and learn the top trends in paid search advertising that are expected to gain traction, so you can drive higher ROI more efficiently in 2024.
You’ll learn:
- The latest trends in AI and automation, and what this means for an evolving paid search ecosystem.
- New developments in privacy and data regulation.
- Emerging ad formats that are expected to make an impact next year.
Watch Sreekant Lanka from iQuanti and Irina Klein from OneMain Financial as they dive into the future of paid search and explore the trends, strategies, and technologies that will shape the search marketing landscape.
If you’re looking to assess your paid search strategy and design an industry-aligned plan for 2024, then this webinar is for you.
5 Public speaking tips from TED - Visualized summarySpeakerHub
From their humble beginnings in 1984, TED has grown into the world’s most powerful amplifier for speakers and thought-leaders to share their ideas. They have over 2,400 filmed talks (not including the 30,000+ TEDx videos) freely available online, and have hosted over 17,500 events around the world.
With over one billion views in a year, it’s no wonder that so many speakers are looking to TED for ideas on how to share their message more effectively.
The article “5 Public-Speaking Tips TED Gives Its Speakers”, by Carmine Gallo for Forbes, gives speakers five practical ways to connect with their audience, and effectively share their ideas on stage.
Whether you are gearing up to get on a TED stage yourself, or just want to master the skills that so many of their speakers possess, these tips and quotes from Chris Anderson, the TED Talks Curator, will encourage you to make the most impactful impression on your audience.
See the full article and more summaries like this on SpeakerHub here: https://speakerhub.com/blog/5-presentation-tips-ted-gives-its-speakers
See the original article on Forbes here:
http://www.forbes.com/forbes/welcome/?toURL=http://www.forbes.com/sites/carminegallo/2016/05/06/5-public-speaking-tips-ted-gives-its-speakers/&refURL=&referrer=#5c07a8221d9b
ChatGPT and the Future of Work - Clark Boyd Clark Boyd
Everyone is in agreement that ChatGPT (and other generative AI tools) will shape the future of work. Yet there is little consensus on exactly how, when, and to what extent this technology will change our world.
Businesses that extract maximum value from ChatGPT will use it as a collaborative tool for everything from brainstorming to technical maintenance.
For individuals, now is the time to pinpoint the skills the future professional will need to thrive in the AI age.
Check out this presentation to understand what ChatGPT is, how it will shape the future of work, and how you can prepare to take advantage.
A brief introduction to DataScience with explaining of the concepts, algorithms, machine learning, supervised and unsupervised learning, clustering, statistics, data preprocessing, real-world applications etc.
It's part of a Data Science Corner Campaign where I will be discussing the fundamentals of DataScience, AIML, Statistics etc.
Time Management & Productivity - Best PracticesVit Horky
Here's my presentation on by proven best practices how to manage your work time effectively and how to improve your productivity. It includes practical tips and how to use tools such as Slack, Google Apps, Hubspot, Google Calendar, Gmail and others.
The six step guide to practical project managementMindGenius
The six step guide to practical project management
If you think managing projects is too difficult, think again.
We’ve stripped back project management processes to the
basics – to make it quicker and easier, without sacrificing
the vital ingredients for success.
“If you’re looking for some real-world guidance, then The Six Step Guide to Practical Project Management will help.”
Dr Andrew Makar, Tactical Project Management
Unlocking the Power of ChatGPT and AI in Testing - A Real-World Look, present...Applitools
During this webinar, Anand Bagmar demonstrates how AI tools such as ChatGPT can be applied to various stages of the software development life cycle (SDLC) using an eCommerce application case study. Find the on-demand recording and more info at https://applitools.info/b59
Key takeaways:
• Learn how to use ChatGPT to add AI power to your testing and test automation
• Understand the limitations of the technology and where human expertise is crucial
• Gain insight into different AI-based tools
• Adopt AI-based tools to stay relevant and optimize work for developers and testers
* ChatGPT and OpenAI belong to OpenAI, L.L.C.
More than Just Lines on a Map: Best Practices for U.S Bike Routes
This session highlights best practices and lessons learned for U.S. Bike Route System designation, as well as how and why these routes should be integrated into bicycle planning at the local and regional level.
Presenters:
Presenter: Kevin Luecke Toole Design Group
Co-Presenter: Virginia Sullivan Adventure Cycling Association
Ride the Storm: Navigating Through Unstable Periods / Katerina Rudko (Belka G...DevGAMM Conference
Has your project been caught in a storm of deadlines, clashing requirements, and the need to change course halfway through? If yes, then check out how the administration team navigated through all of this, relocating 160 people from 3 countries and opening 2 offices during the most turbulent time in the last 20 years. Belka Games’ Chief Administrative Officer, Katerina Rudko, will share universal approaches and life hacks that can help your project survive unstable periods when there seem to be too many tasks and a lack of time and people.
Ride the Storm: Navigating Through Unstable Periods / Katerina Rudko (Belka G...
Metodo simplex uc
1. Pontificia Universidad Católica
Escuela de Ingeniería
Departamento de Ingeniería Industrial y de Sistemas
Clase 20 • El M´todo Simplex
e
ICS 1102 • Optimizaci´n
o
Profesor : Claudio Seebach
16 de octubre de 2006
Apuntes de Clases • Optimizaci´n • Claudio Seebach
o Programaci´n Lineal • 48
o
Pasos en el M´todo Simplex
e
• El m´todo Simplex est´ compuesto por tres pasos:
e a
Paso Inicial
Paso Iterativo
Prueba de Optimalidad
No óptima
Solución
Óptima
Fin
Apuntes de Clases • Optimizaci´n • Claudio Seebach
o Programaci´n Lineal • 49
o
2. Pasos en el M´todo Simplex
e
• Los tres pasos son:
1. Paso inicial: Determinar una soluci´n factible en un v´rtice.
o e
2. Prueba de optimalidad: La soluci´n factible en un v´rtice es ´ptima
o e o
cuando ninguna de las soluciones en v´rtices adyacentes a ella sean
e
mejores.
3. Paso iterativo: Traslado a una mejor soluci´n factible en un v´rtice
o e
adyacente (repetir las veces que sea necesario).
Apuntes de Clases • Optimizaci´n • Claudio Seebach
o Programaci´n Lineal • 50
o
Ejercicio Simplex
• Recordemos el ejemplo de las planchas de aluminio:
P ) min −800x1 − 600x2
s.a 15x1 + 5x2 ≤ 600
7x1 + 14x2 ≤ 630
0, 3x1 + 0, 3x2 ≤ 15
x 1 , x2 ≥ 0
• Introducir tres variables de holgura:
P ) min −800x1 − 600x2
s.a. 15x1 + 5x2 + x3 = 600 (minutos disponibles)
7x1 + 14x2 + x4 = 630 (m2 de aluminio)
0, 3x1 + 0, 3x2 + x5 = 15 (lts de pintura)
x 1 , x2 ≥ 0
x 3 , x4 , x 5 ≥ 0
Apuntes de Clases • Optimizaci´n • Claudio Seebach
o Programaci´n Lineal • 51
o
3. Ejercicio Simplex
1. Paso Inicial:
• Determinar una soluci´n inicial factible.
o
• Si todas las restricciones son desigualdades de menor o igual, todas
las variables son no negativas y todos los recursos disponibles son no
negativos, existe una soluci´n factible trivial:
o
– Asignar un valor cero a cada una de las variables originales.
Tipo 2
x1= x2=0
Tipo 1
Apuntes de Clases • Optimizaci´n • Claudio Seebach
o Programaci´n Lineal • 52
o
Ejercicio Simplex
• Las variables b´sicas son x3, x4, x5 y las no b´sicas son x1, x2.
a a
• En formato tableau:
x1 x2 x3 x4 x5 v.b.
15 5 1 0 0 600 x3
7 14 0 1 0 630 x4
0,3 0,3 0 0 1 15 x5
-800 -600 0 0 0 0 0
• Los coeficientes de la ultima fila se denominan costos reducidos
´
• La casilla inferior derecha del tableau indica el inverso aditivo del
valor de la funci´n objetivo.
o
Apuntes de Clases • Optimizaci´n • Claudio Seebach
o Programaci´n Lineal • 53
o
4. Ejercicio Simplex
2. Prueba de Optimalidad:
• Si el valor de Z puede mejorar (en este caso disminuir) al hacer que
una de las variables no b´sicas crezca.
a
• Esto se puede verificar observando la ultima fila del tableau (Z): si
´
todos los valores son positivos o cero, estamos en el ´ptimo, de lo
o
contrario regresamos al paso iterativo.
• Tanto la variable x1 como x2 mejoran la funci´n objetivo si aumentan
o
en una unidad ya que sus coeficientes en la fila de la funci´n objetivo
o
son estrictamente negativos, por lo que nos convendr´ que una de
a
ellas (cualquiera) entre a la base.
Apuntes de Clases • Optimizaci´n • Claudio Seebach
o Programaci´n Lineal • 54
o
Ejercicio Simplex
3. Paso Iterativo:
3.1 Parte I: Determinar la variable no b´sica que entra a la base.
a
• T´ıpicamente se escoge la variable cuyo coeficiente en la funci´n
o
objetivo sea el m´s negativo.
a
• Sin embargo podr´ escogerse cualquier variable con costo reducido
ıa
negativo y el m´todo igual converger´
e ıa.
• En nuestro ejemplo, la variable que m´s aporta a la funci´n obje-
a o
tivo es x1, por lo que ´sta ser´ la variable entrante.
e a
Apuntes de Clases • Optimizaci´n • Claudio Seebach
o Programaci´n Lineal • 55
o
5. Ejercicio Simplex
3.2 Parte II: Se determina la variable b´sica que sale de la base:
a
• Se elige la variable b´sica que primero alcanza el valor cero cuando
a
se incrementa la variable b´sica entrante.
a
• La variable saliente es x3, de min{ 600 , 630 , 0,3 } = 40.
15 7
15
3.3 Parte III: Se determina la nueva soluci´n b´sica factible.
o a
x1 x2 x3 x4 x5 v.b.
1 1
1 3 15 0 0 40 x1
• El nuevo tableau es:
0 35 −7 1 0 350
3 15 x4
• El nuevo valor ´ptimo es -32.000
o 1 −1
0 5 50 0 1 3 x5
1000 800
0 − 3 15 0 0 32.000
Apuntes de Clases • Optimizaci´n • Claudio Seebach
o Programaci´n Lineal • 56
o
Ejercicio Simplex
Tipo 2
µ2= -333.3
µ3= 53.3
x1= 40 Tipo 1
x2= 0
Apuntes de Clases • Optimizaci´n • Claudio Seebach
o Programaci´n Lineal • 57
o
6. Ejercicio Simplex
• A´n existe una variable no b´sica cuyo costo reducido es negativo: x2.
u a
• Esta variable reduce en 1000 la funci´n objetivo por cada unidad que
3 o
aumente su valor, por lo que conviene que entre a la base.
• ¿por qu´ una unidad extra del producto 2 agrega s´lo $ 1000 a la funci´n
e o 3 o
objetivo siendo que su utilidad era $600?
• Dada la combinaci´n actual de productos, y nula disponibilidad de re-
o
cursos del tipo 1 (tiempo), cada unidad extra del producto 2 exige
dejar de producir 1 de unidad del producto 1. La utilidad neta es
3
$600 − 1 · $800, esto es $ 1000 .
3 3
• La variable que sale de la base es x5:
40 350 3
min 1 , 35 , 1 = {120, 30, 15} = 15
3 3 5
Apuntes de Clases • Optimizaci´n • Claudio Seebach
o Programaci´n Lineal • 58
o
Ejercicio Simplex
• El tableau siguiente es:
x1 x2 x3 x4 x5 v.b.
1 0 101
0 −53 35 x1
0 0 107
1 −175 175
3 x4
0 1 −1
10 0 5 15 x2
5000
0 0 20 0 3 37.000
Tipo 2
x2= 15 µ3= 20
µ5= 1666.6
x1= 35 Tipo 1
Apuntes de Clases • Optimizaci´n • Claudio Seebach
o Programaci´n Lineal • 59
o
7. Ejercicio Simplex
• Prueba de optimalidad: todos los costos reducidos no b´sicos son po-
a
sitivos, por lo que estamos en una soluci´n ´ptima.
o o
• Por lo tanto, la soluci´n obtenida es la siguiente:
o
x1 = 35
x2 = 15
x3 = 0 (Restricci´n activa)
o
x4 = 175 (Holgura de aluminio)
x5 = 0 (Restricci´n activa)
o
Y el valor ´ptimo es -$37.000 (este valor calza con lo que se esperaba:
o
−32.000 − 15 ∗ 1000 ).
3
Apuntes de Clases • Optimizaci´n • Claudio Seebach
o Programaci´n Lineal • 60
o
Ejercicio Simplex
• La optimalidad de esta soluci´n es evidente si uno transforma este
o
tableau en el problema de minimizaci´n correspondiente:
o
5000
P ) min 20x3 + x5 − 37.000
3
1 5
s.a x1 + x3 − x5 = 35
10 3
7 175
x3 + x4 − x5 = 175
10 3
1
x2 − x3 + 5x5 = 15
10
x 1 , x2 ≥ 0
x 3 , x4 , x5 ≥ 0
• Dado que las variables deben ser no negativas, el valor ´ptimo no puede
o
ser inferior a -$37.000.
• Basta hacer x3 = x5 = 0 para obtener una soluci´n factible que alcanza
o
ınima para el valor ´ptimo −37.000.
la cota m´ o
Apuntes de Clases • Optimizaci´n • Claudio Seebach
o Programaci´n Lineal • 61
o
8. Ejercicio Simplex
• El tableau final entrega los multiplicadores asociados a cada uno de los
recursos necesarios para producir planchas de aluminio.
• µ1 y µ2 son los multiplicadores asociados a la no negatividad de las
variables x1 y x2, mientras que µ3, µ4 y µ5 los multiplicadores asociados
a las restricciones de minutos de trabajo, m2 de material y litros de
pintura, respectivamente.
• Los costos reducidos indican que si x3 aumenta en una unidad, la funci´n
o
objetivo empeora en $20. Es decir, µ3 = 20.
• An´logamente, µ4 = 0 y µ5 = 1666, 6 lo que es consistente con nuestros
a
resultados previos.
Apuntes de Clases • Optimizaci´n • Claudio Seebach
o Programaci´n Lineal • 62
o
Soluci´n B´sica Factible Inicial
o a
• El m´todo Simplex requiere de una soluci´n inicial factible b´sica (SIFB)
e o a
para comenzar a iterar.
• En cualquier problema de programaci´n lineal en forma est´ndar con
o a
b ≥ 0, es f´cil identificar una SIFB:
a
– Definir el conjunto de variables b´sicas como el conjunto de holguras
a
de las restricciones: xholguras = b, y las dem´s variables originales
a
(no b´sicas) iguales a cero.
a
• Los problemas que tienen restricciones de igualdad o desigualdades ”≥”,
tal que al asignar un valor cero a las variables originales del problema
no se obtiene un punto factible del dominio.
Apuntes de Clases • Optimizaci´n • Claudio Seebach
o Programaci´n Lineal • 63
o
9. Soluci´n B´sica Factible Inicial
o a
• En el ejemplo del aluminio, supongamos ahora que estamos forzados a
utilizar al menos 15 lts de pintura.
• El problema en forma est´ndar ser´ entonces:
a ıa
min −800x1 − 600x2
s.a 15x1 + 5x2 + x3 = 600
7x1 + 14x2 + x4 = 630
0, 3x1 + 0, 3x2 − x5 = 15
xi ≥ 0, i ∈ {1, 2, 3, 4, 5}
en que x3 y x4 son variables de holgura y x5 es de exceso.
Apuntes de Clases • Optimizaci´n • Claudio Seebach
o Programaci´n Lineal • 64
o
Soluci´n B´sica Factible Inicial
o a
• Como podemos ver, no el problema no posee una SIFB trivial:
Tipo 2
D
(0,0) no es factible Tipo 1
Apuntes de Clases • Optimizaci´n • Claudio Seebach
o Programaci´n Lineal • 65
o
10. Primera Fase
• Soluci´n: resolver el problema en dos fases:
o
– En la primera fase se usa el Simplex para determinar una SIFB
– En la segunda fase se usa el m´todo Simplex a partir de la SIFB
e
encontrada.
• En la primera fase se procurar´ inventar una soluci´n b´sica al problema
a o a
que sea evidente:
– Agregar´ una variable artificial yi no negativa en cada una de las
a
restricciones que no cumpla con el formato est´ndar.
a
– Esto genera inmediatamente una SBIF del nuevo problema.
• Cualquier soluci´n a este nuevo problema s´lo ser´ factible en el prob-
o o a
lema original si todas las variables artificiales son nulas.
• Reemplazar, s´lo durante la primera fase del algoritmo, la funci´n ob-
o o
jetivo del problema original por:
min yi
i
Apuntes de Clases • Optimizaci´n • Claudio Seebach
o Programaci´n Lineal • 66
o
Primera Fase
• Supongamos que el problema original es el siguiente:
min c x
s.a. A1x ≤ b1
A2x ≥ b2
A3x = b3
x ≥ 0
• El problema queda:
min yi + yi
yi ∈a2 yi ∈a3
s.a. A1x + h1 = b1
A2x − e2 + a2 = b2
A3x + a3 = b3
x, h1, e2, a2, a3 ≥ 0
• La SIFB de este problema es evidente: h1, a2 y a3 como el conjunto de
variables b´sicas.
a
Apuntes de Clases • Optimizaci´n • Claudio Seebach
o Programaci´n Lineal • 67
o
11. Primera Fase
• Es posible aplicar Simplex directamente comenzando en dicha SIBF.
• Se pueden dar dos casos:
1. En la soluci´n ´ptima a este problema todos los elementos yi tanto
o o
de a2 como de a3 son nulos, entonces basta eliminar las variables
artificiales del problema y utilizar la soluci´n ´ptima como SIFB
o o
para el problema original.
2. La soluci´n ´ptima contempla alg´n yi > 0 significa que no es posi-
o o u
ble encontrar una soluci´n en que todos los yi sean nulos.
o
Esto indica que el dominio del problema original no admite solu-
ciones factibles, es decir, es vac´ıo.
Apuntes de Clases • Optimizaci´n • Claudio Seebach
o Programaci´n Lineal • 68
o
Primera Fase
• En una primera fase, nos interesa resolver el siguiente problema:
min y3
s.a 15x1 + 5x2 + x3 = 600
7x1 + 14x2 + x4 = 630
0, 3x1 + 0, 3x2 − x5 + y3 = 15
xi ≥ 0, i ∈ {1, 2, 3, 4, 5}
y3 ≥ 0
• Su tableau asociado es el siguiente:
x1 x2 x3 x4 x5 y3 v.b.
15 5 1 0 0 0 600 x3
7 14 0 1 0 0 630 x4
0, 3 0, 3 0 0 −1 1 15 y3
0 0 0 0 0 1 0
−800 −600 0 0 0 0 0
Apuntes de Clases • Optimizaci´n • Claudio Seebach
o Programaci´n Lineal • 69
o
12. Primera Fase
• Es util incluir una fila adicional en la parte inferior del tableau con la
´
funci´n objetivo original. Esto facilita la transici´n entre la Fase 1 y la
o o
Fase 2, en la que se vuelve a la funci´n objetivo original.
o
• La funci´n objetivo de la Fase I est´ en la pen´ltima fila.
o a u
• La SBIF corresponde a x3 = 600, x4 = 630, y3 = 15. Las dem´s
a
variables (no b´sicas) son cero.
a
• Hay que ajustar la funci´n objetivo en el tableau para que contenga
o
s´lo ceros en las columnas asociadas a las variables b´sicas:
o a
x1 x2 x3 x4 x5 y3 v.b.
15 5 1 0 0 0 600 x3
7 14 0 1 0 0 630 x4
0, 3 0, 3 0 0 −1 1 15 y3
−0, 3 −0, 3 0 0 1 0 −15
−800 −600 0 0 0 0 0
Apuntes de Clases • Optimizaci´n • Claudio Seebach
o Programaci´n Lineal • 70
o
Primera Fase
• La soluci´n anterior no es ´ptima: x1 y x2 tienen costo reducido negativo
o o
• Despu´s de una iteraci´n de Simplex en el pivote destacado se alcanza
e o
el siguiente tableau:
x1x2 x3 x4 x5 y3 v.b.
25
20 1 −5 0 0 375
14 x3
1 1
21 0 14 0 0 45 x2
3 −3
200 0 140 −1 1 1, 5 y3
−3 3
200 0 140 1 0 −1, 5
600
−500 0 0 14 0 0 27000
Apuntes de Clases • Optimizaci´n • Claudio Seebach
o Programaci´n Lineal • 71
o
13. Primera Fase Gr´ficamente
a
Tipo 2
(0,45)
D
Tipo 1
Apuntes de Clases • Optimizaci´n • Claudio Seebach
o Programaci´n Lineal • 72
o
Primera Fase
• La soluci´n b´sica anterior a´n no es factible para el problema original
o a u
pues y3 a´n es positiva.
u
• La soluci´n no es ´ptima para la Fase I: el costo reducido de la variable
o o
3
x1 es negativo (− 20 ).
• Una iteraci´n adicional y obtenemos:
o
x1 x2 x3 x4 x5 y3 v.b.
−250
0 0 1 107
250
3 3 250 x3
−10
0 1 0 1 7
10
3 3 40 x2
1 0 0 −17
−20
3
20
3 10 x1
0 0 0 0 0 1 0
0 0 0 −200
7
−10000 10000
3 3 32000
• La soluci´n b´sica alcanzada es ´ptima para la Fase I, ya que todas las
o a o
variables artificiales han salido de la base.
Apuntes de Clases • Optimizaci´n • Claudio Seebach
o Programaci´n Lineal • 73
o
14. Primera Fase Gr´ficamente
a
Tipo 2
(0,45)
(10,40)
D
Tipo 1
Apuntes de Clases • Optimizaci´n • Claudio Seebach
o Programaci´n Lineal • 74
o
Fin Primera Fase
• A trav´s de operaciones fila transformamos el problema de la Fase I en
e
el siguiente:
min y3
20 250 250
s.a x 3 + x4 + x5 − y3 = 250
14 3 3
1 10 10
x2 + x4 + x5 − y3 = 40
7 3 3
1 20 20
x1 − x4 − x5 + y3 = 10
7 3 3
xi ≥ 0, i ∈ {1, 2, 3, 4, 5}
y3 ≥ 0
• La soluci´n ´ptima prescinde de la variable y3.
o o
• La soluci´n b´sica factible alcanzada es: x1 = 10, x2 = 40, x3 =
o a
250, x4 = x5 = 0 y es factible para el problema original.
• Basta eliminar la variable artificial y3 y reemplazar la funci´n objetivo
o
por la original para poder comenzar la segunda fase.
Apuntes de Clases • Optimizaci´n • Claudio Seebach
o Programaci´n Lineal • 75
o
15. Transici´n Fase I a Fase II
o
• Las v. b. x1, x2 de esta soluci´n aparecen en la funci´n objetivo original:
o o
min −800x1 − 600x2
• Es necesario realizar operaciones adicionales para que la funci´n objetivo
o
quede expresada s´lo en funci´n de las variables no b´sicas.
o o a
• Esto se obtiene gracias a mantener durante la primera fase la ultima fila
´
del tableau con la funci´n objetivo original.
o
• Basta reemplazar el valor de las v. b. en la funci´n objetivo original:
o
f (x1, x2) = −800x1 − 600x2
1 20 20 1 10 10
= −800(10 + x4 + x5 − y3 ) − 600(40 − x4 − x5 + y3)
7 3 3 7 3 3
200 10000 10000
= −32.000 − x4 − x5 + y3
7 3 3
• Esto es la expresi´n que aparece en la ultima fila del tableau.
o ´
Apuntes de Clases • Optimizaci´n • Claudio Seebach
o Programaci´n Lineal • 76
o