This document provides an overview of the topics covered in Unit V: Linear Programming. It begins with an introduction to operations research and some example problems that can be modeled as linear programs. It then discusses formulations of linear programs, including the standard and slack forms. The document outlines the simplex algorithm for solving linear programs and how to convert between standard and slack forms. It provides examples demonstrating these concepts. The key topics covered are linear programming models, formulations, and the simplex algorithm.
This presentation is trying to explain the Linear Programming in operations research. There is a software called "Gipels" available on the internet which easily solves the LPP Problems along with the transportation problems. This presentation is co-developed with Sankeerth P & Aakansha Bajpai.
By:-
Aniruddh Tiwari
Linkedin :- http://in.linkedin.com/in/aniruddhtiwari
This presentation is trying to explain the Linear Programming in operations research. There is a software called "Gipels" available on the internet which easily solves the LPP Problems along with the transportation problems. This presentation is co-developed with Sankeerth P & Aakansha Bajpai.
By:-
Aniruddh Tiwari
Linkedin :- http://in.linkedin.com/in/aniruddhtiwari
A problem is provided which is solved by using graphical and analytical method of linear programming method and then it is solved by using geometrical concept and algebraic concept of simplex method.
The Modified Distribution Method or MODI is an efficient method of checking the optimality of the initial feasible solution. MODI provides a new means of finding the unused route with the largest negative improvement index. Once the largest index is identified, we are required to trace only one closed path. This path helps determine the maximum number of units that can be shipped via the best unused route.
My talk about linear programming in NTU's APEX Club in NTU, Singapore in 2007. The club is for people who are keen on participating in ACM International Collegiate Programming Contests organized by IBM annually.
A problem is provided which is solved by using graphical and analytical method of linear programming method and then it is solved by using geometrical concept and algebraic concept of simplex method.
The Modified Distribution Method or MODI is an efficient method of checking the optimality of the initial feasible solution. MODI provides a new means of finding the unused route with the largest negative improvement index. Once the largest index is identified, we are required to trace only one closed path. This path helps determine the maximum number of units that can be shipped via the best unused route.
My talk about linear programming in NTU's APEX Club in NTU, Singapore in 2007. The club is for people who are keen on participating in ACM International Collegiate Programming Contests organized by IBM annually.
In a linear programming problem, a linear function is to be optimized subject to linear inequality constraints. The corner principle says to solve such a problem all we have to do is look at the corners of the feasibility set.
For a good business plan creative thinking is important. A business plan is very important and strategic tool for entrepreneurs. A good business plan not only helps entrepreneurs focus on specific steps necessary for them to make business ideas succeed, but it also helps them to achieve short-term and long-term objectives. As an inspiring entrepreneur who is looking towards starting a business, one of the businesses you can successfully start without much stress is book servicing café.
Importance:
Nowadays, network plays an important role in people’s life. In the process of the improvement of the people’s living standard, people’s demand of the life’s quality and efficiency is more higher, the traditional bookstore’s inconvenience gradually emerge, and the online book store has gradually be used in public. The online book store system based on the principle of providing convenience and service to people.
With the online book servicing café, college student do not need to blindly go to various places to find their own books, but only in a computer connected to the internet log on online book servicing café in the search box, type u want to find of the book information retrieval, you can efficiently know whether a site has its own books, if you can online direct purchase, if not u can change the home book store to continue to search or provide advice to the seller in order to supply. This greatly facilitates every college student saving time.
The online book servicing café’s main users are divided into two categories, one is the front user, and one is the background user. The main business model for Book Servicing Café relies on college students providing textbooks, auctions, classifieds teacher evaluations available on website. Therefore, our focus will be on the marketing strategy to increase student traffic and usage. In turn, visitor volume and transactions will maintain the inventory of products and services offered.
Online bookstore system i.e. Book Servicing Café not only can easily find the information and purchase books, and the operating conditions are simple, user-friendly, to a large extent to solve real-life problems in the purchase of the books.
When you shop in online book servicing cafe, you have the chance of accessing and going through customers who have shopped at book servicing café and review about the book you intend to buy. This will give you beforehand information about that book.
While purchasing or selling books at the book servicing café, you save money, energy and time for your favorite book online. The book servicing café will offer discount coupons which help college students save money or make money on their purchases or selling. Shopping for books online is economical too because of the low shipping price.
Book servicing café tend to work with multiple suppliers, which allows them to offer a wider variety of books than a traditional retail store without accruing a large, costly inventory which will help colle
A brief study on linear programming solving methodsMayurjyotiNeog
This small presentation includes a brief study on various linear programming solving methods. These methods (graphical & simplex) are used to solve industrial engineering related problems in practical use.
Linear ProgrammingPart 1J. M. Pogodzinskicarol.docxcroysierkathey
Linear Programming
Part 1
J. M. Pogodzinski
carol
carol
carol
carol
Agenda
• Mathematical Programming Problems
• Economic Theory and Mathematical Programming Problems
• Linear Programming Problems
• The Objective Function
• The Inequality Constraints
• The Non-Negativity Constraints (which are inequality constraints)
• Equality Constraints?
• The Feasible Set
• Does a
Solution
Exist to a Linear Programming Problem? (the existence question)
• Applications (Uses) of Linear Programming
• Solving Linear Programming Problems
• Theorems About Linear Programming
Mathematical Programming Problems
• A Mathematical Programming Problem consists of:
• An objective function
• Constraints defined somehow – equations, inequalities,…
• Little can be said about such a general problem – we need to make
assumptions about the objective function and/or about the
constraints before we can say anything about the existence of
solutions, algorithms for finding solutions (if they exist), properties of
solutions
About Objective Functions
• Very common to assume there is only one objective function
• Objective functions are either maximized or minimized – the generic term is
optimized. The specific problem determines whether maximization or
minimization is appropriate. There are deeper connections between
maximization and minimization. Maximization problems can be restated as
minimization problems. More importantly, specific maximization problems are
associated with specific minimization problems through duality.
• It is possible to consider multi-objective mathematical programming problems
(there is a legitimate topic called multi-objective linear programming)
• What do you get out of multi-objective linear programming (if there is a
solution)?
• The Pareto Frontier
• We will not consider multi-objective linear programming because it is
computationally difficult
About Objective Functions
• Example (from microeconomics): Consumers maximize utility subject
to a budget constraint
• 𝑚𝑎𝑥𝑥,𝑦 𝑈 𝑥,𝑦 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑝𝑥𝑥 + 𝑝𝑦𝑦 = 𝑀 (and 𝑥 ≥ 0 and y ≥ 0)
• We assume that 𝑈 𝑥,𝑦 is a quasi-concave continuous function
(Note: famous paper “Quasi-Concave Programming” by Kenneth J.
Arrow and Alain C. Enthoven, Econometrica, Vol. 29, No. 4 (Oct.,
1961), pp. 779-800)
• A function 𝑈 𝑥,𝑦 is quasi-concave if its upper level sets are convex
sets
Constraints
• Most common to define constraints by one or more equations or
inequalities
• Note on finite constraint sets – existence of optimum
• For example, in the consumer choice problem mentioned in the
previous slide, an equation called the budget equation defined the
constraint set - 𝑝𝑥𝑥 + 𝑝𝑦𝑦 = 𝑀 (and 𝑥 ≥ 0 and y ≥ 0)
• We might also have defined the constraint set with several
inequalities: 𝑝𝑥𝑥 + 𝑝𝑦𝑦 ≤ 𝑀 and 𝑥 ≥ 0 and y ≥ 0
• We can write the equation 𝑝𝑥𝑥 + 𝑝𝑦𝑦 = 𝑀 as two inequalities:
𝑝𝑥𝑥 + 𝑝𝑦𝑦 ≤ 𝑀 and 𝑝𝑥𝑥 + 𝑝𝑦𝑦 ≥ 𝑀
Linear ProgrammingPart 1J. M. Pogodzinskicarol.docxwashingtonrosy
Linear Programming
Part 1
J. M. Pogodzinski
carol
carol
carol
carol
Agenda
• Mathematical Programming Problems
• Economic Theory and Mathematical Programming Problems
• Linear Programming Problems
• The Objective Function
• The Inequality Constraints
• The Non-Negativity Constraints (which are inequality constraints)
• Equality Constraints?
• The Feasible Set
• Does a
Solution
Exist to a Linear Programming Problem? (the existence question)
• Applications (Uses) of Linear Programming
• Solving Linear Programming Problems
• Theorems About Linear Programming
Mathematical Programming Problems
• A Mathematical Programming Problem consists of:
• An objective function
• Constraints defined somehow – equations, inequalities,…
• Little can be said about such a general problem – we need to make
assumptions about the objective function and/or about the
constraints before we can say anything about the existence of
solutions, algorithms for finding solutions (if they exist), properties of
solutions
About Objective Functions
• Very common to assume there is only one objective function
• Objective functions are either maximized or minimized – the generic term is
optimized. The specific problem determines whether maximization or
minimization is appropriate. There are deeper connections between
maximization and minimization. Maximization problems can be restated as
minimization problems. More importantly, specific maximization problems are
associated with specific minimization problems through duality.
• It is possible to consider multi-objective mathematical programming problems
(there is a legitimate topic called multi-objective linear programming)
• What do you get out of multi-objective linear programming (if there is a
solution)?
• The Pareto Frontier
• We will not consider multi-objective linear programming because it is
computationally difficult
About Objective Functions
• Example (from microeconomics): Consumers maximize utility subject
to a budget constraint
• 𝑚𝑎𝑥𝑥,𝑦 𝑈 𝑥,𝑦 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑝𝑥𝑥 + 𝑝𝑦𝑦 = 𝑀 (and 𝑥 ≥ 0 and y ≥ 0)
• We assume that 𝑈 𝑥,𝑦 is a quasi-concave continuous function
(Note: famous paper “Quasi-Concave Programming” by Kenneth J.
Arrow and Alain C. Enthoven, Econometrica, Vol. 29, No. 4 (Oct.,
1961), pp. 779-800)
• A function 𝑈 𝑥,𝑦 is quasi-concave if its upper level sets are convex
sets
Constraints
• Most common to define constraints by one or more equations or
inequalities
• Note on finite constraint sets – existence of optimum
• For example, in the consumer choice problem mentioned in the
previous slide, an equation called the budget equation defined the
constraint set - 𝑝𝑥𝑥 + 𝑝𝑦𝑦 = 𝑀 (and 𝑥 ≥ 0 and y ≥ 0)
• We might also have defined the constraint set with several
inequalities: 𝑝𝑥𝑥 + 𝑝𝑦𝑦 ≤ 𝑀 and 𝑥 ≥ 0 and y ≥ 0
• We can write the equation 𝑝𝑥𝑥 + 𝑝𝑦𝑦 = 𝑀 as two inequalities:
𝑝𝑥𝑥 + 𝑝𝑦𝑦 ≤ 𝑀 and 𝑝𝑥𝑥 + 𝑝𝑦𝑦 ≥ 𝑀
Courier management system project report.pdfKamal Acharya
It is now-a-days very important for the people to send or receive articles like imported furniture, electronic items, gifts, business goods and the like. People depend vastly on different transport systems which mostly use the manual way of receiving and delivering the articles. There is no way to track the articles till they are received and there is no way to let the customer know what happened in transit, once he booked some articles. In such a situation, we need a system which completely computerizes the cargo activities including time to time tracking of the articles sent. This need is fulfilled by Courier Management System software which is online software for the cargo management people that enables them to receive the goods from a source and send them to a required destination and track their status from time to time.
COLLEGE BUS MANAGEMENT SYSTEM PROJECT REPORT.pdfKamal Acharya
The College Bus Management system is completely developed by Visual Basic .NET Version. The application is connect with most secured database language MS SQL Server. The application is develop by using best combination of front-end and back-end languages. The application is totally design like flat user interface. This flat user interface is more attractive user interface in 2017. The application is gives more important to the system functionality. The application is to manage the student’s details, driver’s details, bus details, bus route details, bus fees details and more. The application has only one unit for admin. The admin can manage the entire application. The admin can login into the application by using username and password of the admin. The application is develop for big and small colleges. It is more user friendly for non-computer person. Even they can easily learn how to manage the application within hours. The application is more secure by the admin. The system will give an effective output for the VB.Net and SQL Server given as input to the system. The compiled java program given as input to the system, after scanning the program will generate different reports. The application generates the report for users. The admin can view and download the report of the data. The application deliver the excel format reports. Because, excel formatted reports is very easy to understand the income and expense of the college bus. This application is mainly develop for windows operating system users. In 2017, 73% of people enterprises are using windows operating system. So the application will easily install for all the windows operating system users. The application-developed size is very low. The application consumes very low space in disk. Therefore, the user can allocate very minimum local disk space for this application.
Explore the innovative world of trenchless pipe repair with our comprehensive guide, "The Benefits and Techniques of Trenchless Pipe Repair." This document delves into the modern methods of repairing underground pipes without the need for extensive excavation, highlighting the numerous advantages and the latest techniques used in the industry.
Learn about the cost savings, reduced environmental impact, and minimal disruption associated with trenchless technology. Discover detailed explanations of popular techniques such as pipe bursting, cured-in-place pipe (CIPP) lining, and directional drilling. Understand how these methods can be applied to various types of infrastructure, from residential plumbing to large-scale municipal systems.
Ideal for homeowners, contractors, engineers, and anyone interested in modern plumbing solutions, this guide provides valuable insights into why trenchless pipe repair is becoming the preferred choice for pipe rehabilitation. Stay informed about the latest advancements and best practices in the field.
Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
Event Management System Vb Net Project Report.pdfKamal Acharya
In present era, the scopes of information technology growing with a very fast .We do not see any are untouched from this industry. The scope of information technology has become wider includes: Business and industry. Household Business, Communication, Education, Entertainment, Science, Medicine, Engineering, Distance Learning, Weather Forecasting. Carrier Searching and so on.
My project named “Event Management System” is software that store and maintained all events coordinated in college. It also helpful to print related reports. My project will help to record the events coordinated by faculties with their Name, Event subject, date & details in an efficient & effective ways.
In my system we have to make a system by which a user can record all events coordinated by a particular faculty. In our proposed system some more featured are added which differs it from the existing system such as security.
Automobile Management System Project Report.pdfKamal Acharya
The proposed project is developed to manage the automobile in the automobile dealer company. The main module in this project is login, automobile management, customer management, sales, complaints and reports. The first module is the login. The automobile showroom owner should login to the project for usage. The username and password are verified and if it is correct, next form opens. If the username and password are not correct, it shows the error message.
When a customer search for a automobile, if the automobile is available, they will be taken to a page that shows the details of the automobile including automobile name, automobile ID, quantity, price etc. “Automobile Management System” is useful for maintaining automobiles, customers effectively and hence helps for establishing good relation between customer and automobile organization. It contains various customized modules for effectively maintaining automobiles and stock information accurately and safely.
When the automobile is sold to the customer, stock will be reduced automatically. When a new purchase is made, stock will be increased automatically. While selecting automobiles for sale, the proposed software will automatically check for total number of available stock of that particular item, if the total stock of that particular item is less than 5, software will notify the user to purchase the particular item.
Also when the user tries to sale items which are not in stock, the system will prompt the user that the stock is not enough. Customers of this system can search for a automobile; can purchase a automobile easily by selecting fast. On the other hand the stock of automobiles can be maintained perfectly by the automobile shop manager overcoming the drawbacks of existing system.
NO1 Uk best vashikaran specialist in delhi vashikaran baba near me online vas...Amil Baba Dawood bangali
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Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
TECHNICAL TRAINING MANUAL GENERAL FAMILIARIZATION COURSEDuvanRamosGarzon1
AIRCRAFT GENERAL
The Single Aisle is the most advanced family aircraft in service today, with fly-by-wire flight controls.
The A318, A319, A320 and A321 are twin-engine subsonic medium range aircraft.
The family offers a choice of engines
Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
Democratizing Fuzzing at Scale by Abhishek Aryaabh.arya
Presented at NUS: Fuzzing and Software Security Summer School 2024
This keynote talks about the democratization of fuzzing at scale, highlighting the collaboration between open source communities, academia, and industry to advance the field of fuzzing. It delves into the history of fuzzing, the development of scalable fuzzing platforms, and the empowerment of community-driven research. The talk will further discuss recent advancements leveraging AI/ML and offer insights into the future evolution of the fuzzing landscape.
Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
Sachpazis:Terzaghi Bearing Capacity Estimation in simple terms with Calculati...Dr.Costas Sachpazis
Terzaghi's soil bearing capacity theory, developed by Karl Terzaghi, is a fundamental principle in geotechnical engineering used to determine the bearing capacity of shallow foundations. This theory provides a method to calculate the ultimate bearing capacity of soil, which is the maximum load per unit area that the soil can support without undergoing shear failure. The Calculation HTML Code included.
Pile Foundation by Venkatesh Taduvai (Sub Geotechnical Engineering II)-conver...
Linear programming
1. Unit V : Linear Programming
Syllabus :
•Standard and Slack forms
•Formulation of Problems as Linear Programs
•Simplex Algorithm
•Duality
•Initial Basic Feasible Solution
•Problem Formulation for
Single Source Shortest path
Maximum Flow Problem
2. Unit V : Linear Programming
Introduction :
Operations Research :
•Operations Research (OR) includes identification of a real life
problem, formulation of the problem as OR model and
providing the realistic solution to the problem. Generally these
problems are optimization problems (NP-Hard)
•OR includes the following techniques :
Linear Programming
Probability, Decision Analysis and Games
Queuing Systems
Job Sequencing – PERT / CPM
Non-Linear Programming etc.
3. Unit V : Linear Programming
Introduction :
OR Models :
Example -1: Construct a maximum area rectangle out of a
piece of wire of length L.
Example -2: Amy, Jim, John and Kelly are standing on the
east bank of a river and wish to cross to the west side using
canoe. The canoe can hold at most two people at a time. Amy
being the most athletic, can row across the river in one minute.
Jim, John and Kelly would take 2, 5, and 10 minutes
respectively to row across. If two people are their in canoe, the
slower person dictates the crossing time. The objective is for
all four people to be on the other side of the river in the
shortest time possible.
a) Identify at least two feasible plans for crossing
4. Unit V : Linear Programming
Introduction :
OR Models contd…:
b) Define the objective criterion for evaluating
the alternatives.
c) What is the shortest time for moving all four
people to the other side of the river?
Example -3: (LP Model) The Reddy Mikks company
produces both interior and exterior paints from the two raw
materials M1 and M2.The following table provides the basic
data of the problem:
Tons of Raw
Matlerial / ton
of
Daily Max.
Raw Matl
Availability in
tons
Exterior Paint Interior Paint
5. Unit V : Linear Programming
Introduction :
OR Models contd …:
Example -3 contd…: A market survey indicates that the
daily demand for interior paint cannot exceed that of exterior
paint by more than one ton. Also maximum daily demand for
interior paint is two tons.
Reddy Mikks wants to determine the optimum (best)
product mix of interior and exterior paints that maximizes the
total daily profit.
6. Unit V : Linear Programming
Introduction :
OR Models contd …:
Example -4 : Formulation of the problem as LP Model
A firm manufactures two types of products A and B,
and sells them at a profit of Rs. 2 on type A and Rs. 3 on type
B. Each product is processed on two machines G and H. Type
A requires 1 minute of processing time on machine G and 2
minutes on machine H. Type B requires 1 minute on both G
and H. The machine G is available for not more than 6 hours
and 40 minutes while machine H is available for 10 hours
during any working day. Formulate LP model to maximize
daily profit.
7. Unit V : Linear Programming
Introduction :
OR Models contd …:
Example -4 contd,,,: Formulation of the problem as LP Model
Linear Program (Mathematical Model or OR Model)
Maximize z = 2x1 + 3x2 (Type A = x1and Type B =x2)
Subject to x1 + x2 ≤ 400
2x1 + x2 ≤ 600
x1 , x2 ≥ 0
Description
Time
required in
Minutes for
products
Max.
Available
time in
Minutes per
day
Product A Product B
Processing time on G 1 1 400
Processing time on H 2 1 600
Profit per piece 2 3
8. Unit V : Linear Programming
Introduction :
OR Models contd …:
Example-5 : Political Problem :
Suppose that you are a politician trying to win an
election. Your district has three different area types – Urban,
Seburban, and Rural. These areas have registered voters
100000, 200000, and 50000 respectively. Although not all the
registered voters actually go to the polls, you decide that to
govern effectively, you would like at least half the registered
voters in each of these three regions to vote for you. You are
honorable and would never consider supporting policies in
which you do not believe. You realize, however, that certain
issues may be more effective in winning votes in certain
places. Your primary issues are building more roads, 24 hours
9. Unit V : Linear Programming
Introduction :
OR Models contd …:
Example-5 contd… : Political Problem :
Linear Program (Mathematical Model or OR Model)
Minimize x1 + x2+ x3+ x4
Subject to 2x1+ 8x2+ 0x3+10x4 ≥ 50
5x1+ 2x2+ 0x3+ 0x4 ≥ 100
3x1 - 5x2+ 10x3- 2x4 ≥ 25
x1, x1, x1, x1≥ 0
Policy
Votes in thousands
per lakh INR spent
Urban Sub-urban Rural
Build Roads -2 5 3
24 hour water supply 8 2 -5
Farm subsidies 0 0 10
LBT Tax 10 0 -2
10. Unit V : Linear Programming
Introduction :
General Linear Programs :
•In the general linear programming problem, we wish to optimize a
linear function subject to a set of linear inequalities.
•Given a set of real numbers a1, a2, … an and a set of variables
x1, x2, … , xn we define a linear function f on those variables by:
f (x1, x2, … , xn) = a1x1+ a2x2+ … + anxn = Σ ajxj
1<= j<= n
•If b is a real number and f is a linear function, then,
f (x1, x2, … , xn) = b is a linear equality and
f (x1, x2, … , xn) ≥ b or
f (x1, x2, … , xn) ≤ b are the linear inequalities
•We shall use the general term linear constraints to denote either
linear equalities or linear inequalities
•In linear programming strict inequalities are not allowed i.e.
11. Unit V : Linear Programming
Introduction :
General Linear Programs contd… :
•Formally, a linear-programming problem is the problem of either
minimizing or maximizing a linear function subject to a finite set of
linear constraints.
•If we are to minimize, then we call the linear program a
minimization linear program and if we are to maximize, then we
call the linear program a maximization linear program.
Applications of linear programming:
•Political problem (Elections)
•Optimal Product Mix in Manufacturing
•Airline Schedule of flights
•Oil company – where to drill ?
•Graph problems – Single source shortest path, vertex cover problem,
maximum flow problem
12. Unit V : Linear Programming
Introduction :
Algorithms for linear programming :
•Simplex Algorithm : Often solves general linear programs
quickly in practice in polynomial time, however it may need
exponential time for some problem instances. The algorithm moves
along the exterior of the feasible region and maintains a feasible
solution that is a vertex of the simplex at each iteration.
•Ellipsoid Algorithm : The first polynomial-time algorithm for
linear programming which runs slowly in practice.
•Interior Point Methods : These are also a polynomial-time
algorithm s which move through the interior of the feasible region,
but the final solution is a vertex. For large inputs, interior point
algorithms can run as fast as and sometimes faster than the simplex
algorithm.
•Integer linear program : If we add to a linear program the
13. Unit V : Linear Programming
Introduction :
Standard and Slack forms :
•Standard and Slack form are useful when we specify and work
with linear programs. In standard form, all the constraints are
inequalities, whereas in slack form, all constaints are equalities.
Standard Form :
In standard form, we are given n real numbers c1, c2, … , cn;
m real numbers b1, b2, … , bm and mn real numbers aij for i = 1,2,
… m and j = 1,2, … , n. We wish to find n real numbers x1, x2,
… , xn that
maximize Σ cjxj
… (1) 1<= j<= n
subject to Σ aijxj ≤ bi
… (2)
1<= i<= n
14. Unit V : Linear Programming
Introduction :
Terminology used in Linear Programs:
•Feasible solution : A setting of the variables x- that satisfies all
the constraints. Whereas A setting of the variables x- that fails to
satisfy at least one constraint is an infeasible solution.
•Objective value : A solution x- (feasible) has objective value cTx-.
•Optimal solution : A feasible solution x- whose objective value is
maximum over all feasible solutions is an optimal solution.
•Infeasible linear program: If a linear program has no feasible
solutions, then the linear program is infeasible, otherwise it is feasible.
•Unbounded linear program : If a linear program has some feasible
solutions but does not have a finite optimal value then the linear program
is unbounded. However a linear program can have finite optimal
objective value even if the feasible region is not bounded.
15. Unit V : Linear Programming
Converting linear programs into standard form:
•It is always possible to convert a linear program given as
minimizing or maximizing a linear function subject to linear
constraints, into a standard form. A linear program might not
be in standard form for any of the following four possible
reasons.
1.The objective function might be minimization rather
than maximization.
2.There might be variables without non-negativity
constraints.
3.There might be constraints with equal sign rather
than ≤ sign.
4.There might be inequality constraints but instead of
having ≤ sign, they have a ≥ sign. Example
16. Unit V : Linear Programming
Converting linear programs into standard form:
Equivalent linear programs :
•Two maximization linear programs L and L’ are equivalent if
for each feasible solution x- to L with objective value z, there
is a corresponding feasible solution x-’ to L’ with objective
value z, and for each feasible solution x-’ to L’ with objective
value z, there is a corresponding solution x- to L with objective
value z.
•A minimization linear program L and a maximization linear
program L’ are equivalent if for each feasible solution x- to L
with objective value z, there is a corresponding feasible
solution x-’ to L’ with objective value –z and for each feasible
solution x-’ to L’, with objective value z, there is a
corresponding feasible solution x- to L with objective value –z.
17. Unit V : Linear Programming
Converting linear programs into standard form:
Example
Reduce the following linear program to standard form :
minimize - 2 x1 + 3x2
subject to x1 + x2 = 7
x1 - 2x2 ≤ 4
x1 ≥ 0
Solution : Linear program in Standard form
maximize 2 x1 - 3x2’ + 3x2”
subject to x1 + x2’ - x2” ≤ 7
- x1 - x2’ + x2” ≥ -
7
x1 - 2x2’ + 2x2” ≤ 4
18. Unit V : Linear Programming
Converting linear programs into standard form:
Example
Solution contd …:
The final solution can be written as :
maximize 2 x1 - 3x2 + 3x3
subject to x1 + x2 - x3 ≤ 7
- x1 - x2 + x3 ≥ - 7
x1 - 2x2 + 2x3 ≤ 4
x1, x2 , x3 ≥ 0
19. Unit V : Linear Programming
Converting linear programs into slack form:
•To efficiently solve a linear program with the simplex
algorithm, we prefer to express it in a form in which some of
the constraints are equality constraints.
•More precisely, we shall convert it into a form in which the
non-negativity constraints are the only inequality constraints
and the remaining constraints are equalities.
•Let Σ aijxj ≤ bi … (1)
1<= i<= n
be an inequality constraint. We introduce a new variable
s and rewrite inequality as the two constraints
s = bi - Σ aijxj … (2)
1<= i<= n
s ≥ 0 … (3)
20. Unit V : Linear Programming
Converting linear programs into slack form:
•We call s as a slack variable because it measures the slack or
difference between the LHS and RHS of equation (1). It is
convenient to write slack variable on the LHS.
•We can convert each inequality constraint of a linear program
in this way to obtain an equivalent linear program in which the
only inequality constraints are the non-negativity constraints.
•While converting standard form to slack form , we shall use
xn+i (instead of s) to denote the slack variableassociated with
the ith inequality. The ith constraint is therefore,
xn+i = bi - Σ aijxj and
1<= i<= n
xn+i ≥ 0
21. Unit V : Linear Programming
Converting linear programs into slack form:
Example in standard form explained above when
converted into slack form would be,
maximize 2 x1 - 3x2 + 3x3
subject to x4 = 7 - x1 - x2 + x3
x5 = -7 + x1 + x2 - x3
x6 = 4 - x1 + 2x2 - 2x3
x1, x2 , x3, x4, x5 , x6 ≥ 0
•We call the variables on the LHS of the equalities as Basic
Variables and those on the RHS as the Non-basic Variables.
•For linear programs that satisfy these conditions, we shall omit the
words “maximize” and “subject to” as well as the explicit non-
22. Unit V : Linear Programming
Converting linear programs into slack form:
•The resulting form is called as slack form.
z = 2x1 - 3x2 + 3x3
x4 = 7 - x1 - x2 + x3
x5 = -7 + x1 + x2 - x3
x6 = 4 - x1 + 2x2 - 2x3
23. Unit V : Linear Programming
Converting linear programs into slack form:
•The slack form in general is :
z = v + Σ cjxj
1<= j<= n
xi = bi - Σ aijxj for i ε B
j ε N
where N = set of non-basic variables index
B = set of basic variables index
A = set of coefficients of non-basic variables
bi = RHS value of ith constraint
b = set of values of RHS of constraints
c = set of coefficients of objective function
v = optimal constant term which makes it
easy to
determine value of objective function.
n
24. Unit V : Linear Programming
Converting linear programs into slack form:
•Slack form tuple (N, B, A, b, c, v) for the above example is :
N = {1, 2, 3}
B = {4, 5, 6}
| 1 1 -1| | 7 |
A = |-1 -1 1| b = |-7|
| 1 - 2 2| | 4|
c = (2, -3, 3)T
v = 0
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25. Unit V : Linear Programming
Simplex Algorithm :
•The simplex algorithm is the classical method for solving
linear programs. It’s running time is not polynomial in the
worst case, but it is often remarkably fast in practice.
•Similar to Gaussian elimination used to solve simultaneous
equations, the simplex algorithm as well use Gaussian
elimination for inequalities.
An iteration of the simplex algorithm :
•Associated with each iteration there will be a “basic solution”
that we can obtain from the slack form of the linear program.
•An iteration converts one slack form into an equivalent slack
form.
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26. Unit V : Linear Programming
Simplex Algorithm :
An iteration of the simplex algorithm contd…:
•The objective value of the associated basic feasible solution
will be no less than that at the previous iteration. Objective
value of basic solution can easily be obtained by setting each
non-basic variable to 0.
•To achieve this increase in the objective value, we choose a
non-basic variable such that if we increase that variable value
from 0, then the objective value would also increase.
•The amount by which we can increase the variable is limited
by the other constraints. In particular we raise it until some
basic variable becomes 0.
•Rewrite the slack form, exchanging the roles of that basic
variable and the chosen non-basic variable.
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27. Unit V : Linear Programming
Simplex Algorithm :
An example of the simplex algorithm :
Consider the following linear program in standard
form:
maximize 3x1 + x2 + 2x3 …
(1)
subject to x1 + x2 + 3x3 ≤ 30 …
(2)
2x1 + 2x2 + 5x3 ≤ 24 …
(3)
4x1 + x2 + 2x3 ≤ 36 …
(4)
x1, x2, x3 ≥ 0 …
(5)
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28. Unit V : Linear Programming
Simplex Algorithm :
Pivoting : We can formulate the procedure for pivoting which
is used in each iteration of the simplex algorithm.
Procedure pivot (N, B, A, b, c, l, e)
// input is a linear program in slack form, entering variable and
// leaving variable.
1.Compute the coefficients of the equation for the new basic
variable i.e. for the entering variable xe, using tight constraint
equation of the leaving variable xl.
2.Compute the coefficients of the remaining constraints by
replacing the values of xe, i.e. get A^ and b^.
3.Compute the objective function by replacing value of xe &
get c^ and v^ modified.
4.Compute new sets of basic and non-basic variables
n
29. Unit V : Linear Programming
Simplex Algorithm :
The formal Simplex Algorithm :
Algorithm SIMPLEX(A, b, c)
// Input is a linear program in standard form. The procedure
// INITIALIZE-SIMPLEX checks whether input linear
program // is feasible & if so, it finds it’s slack form
& returns.
1.(N, B, A, b, c, v) = INITIALIZE-SIMPLEX(A, b, c)
2.Choose entering variable (xe ) having positive coefficient in
objective function.
3.Find the leaving variable (xl) corresponding to entering
variable using tight constraint.
4.If xl is ∞, for all the constraints then the linear program is
“unbounded” otherwise proceed.
n
30. Unit V : Linear Programming
Simplex Algorithm :
The formal Simplex Algorithm :
Algorithm SIMPLEX(A, b, c)
7. Compute & return the basic solution :
for i = 1 to n do
if i Є B
x-
i = bi
else x-
i= 0
return (x-
1, x-
2, …, x-
i)
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31. Unit V : Linear Programming
Simplex Algorithm :
Duality :
•We have seen how to solve optimization programming using
simplex algorithm. But we do not know whether it finds
actually optimal solution to a linear problem. i.e. we have not
proved that simplex algorithm finds an optimal solution to a
linear program.
•In order to do so, we shall use a powerful concept called as
“linear-programming duality”.
•Duality enables us to prove that a solution is indeed optimal.
•Given a linear program, in which the objective is to
maximize, we shall describe how to formulate a dual linear
program in which the objective is to minimize and whose
optimal value is identical to that of the original problem.
n
32. Unit V : Linear Programming
Simplex Algorithm :
Duality : Dual form of a given primal linear program :
•Given a linear program (primal) in standard form
maximize Σ cjxj
… (1) 1<= j<= n
subject to Σ aijxj ≤ bi for i = 1, 2, …,m
… (2)
1<= i<= n
xj ≥ 0 for j = 1, 2, …,n
… (3)
•We define the dual linear program as :
minimize Σ biyi
… (4) 1<= i<= m
subject to Σ aijyi ≥ cj for j = 1, 2, …,n
n
33. Unit V : Linear Programming
Simplex Algorithm :
Duality : Dual form of a given primal linear program :
•Given a linear program (primal) in standard form
maximize 3x1 + x2 + 2x3 …
(1)
subject to x1 + x2 + 3x3 ≤ 30 …
(2)
2x1 + 2x2 + 5x3 ≤ 24 …
(3)
4x1 + x2 + 2x3 ≤ 36 …
(4)
x1, x2, x3 ≥ 0 …
(5)
•We define the dual linear program as :
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34. Unit V : Linear Programming
Simplex Algorithm :
Duality :
Following are the steps to get a dual of a given primal:
1.Change maximization in primal to minimization in
dual.
2.Change the roles of coefficients on the RHS of
constraints and the coefficients in objective function.
3.Replace each inequality ≤ sign in primal by ≥ sign in
dual.
4.Each of the m constraints in primal has an associated
variable yi in dual.
5.Each of the n constraints in dual has an associated
variable xj in primal.
Example – Graphical Method
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35. Unit V : Linear Programming
Simplex Algorithm :
The initial basic feasible solution :
•The INITIALIZE-SIMPLEX first checks
whether a linear program is feasible and if
it is it produces a slack form for which the
basic solution is feasible. This ensures that
the simplex procedure always produces the
correct result.
•A linear program may be feasible, yet the
initial basic solution might not be feasible.
n
36. Unit V : Linear Programming
Simplex Algorithm :
The initial basic feasible solution contd….:
•In order to determine whether a linear program has any feasible
solutions, we will formulate an auxiliary linear program. For this
auxiliary linear program we can find a slack form for which the
basic solution is feasible. Furthermore, the solution of his auxiliary
linear program determines whether the initial linear program is
feasible and if so, it provides a feasible solution with which we can
initialize SIMPLEX.
Example : For the linear program represented by inequalities (1) to
(4). This linear program is feasible if we can find non-negative
values for x1 and x2 such that inequalities (2) and (3) are satisfied.
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37. Unit V : Linear Programming
Simplex Algorithm :
The initial basic feasible solution contd….:
Example contd….:
•The auxiliary linear program is (Laux) :
maximize - x0 … (1)
subject to 2x1 - x2 - x0 ≤ 2 … (2)
x1 - 5x2 - x0 ≤ -4 … (3)
x1, x2, x0 ≥ 0 … (4)
•Laux in slack form would be :
z = - x0
x3 = 2 - 2x1 + x2 + x0
x4 = - 4 - x1 + 5x2 + x0
In this program the basic solution would set x4 = - 4,
which is not feasible.
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38. Unit V : Linear Programming
Simplex Algorithm :
The initial basic feasible solution contd….:
Example contd….:
•We can convert this slack form into one which is feasible by
giving one call to PIVOT. Let x0 be an entering variable and
x4 be a leaving variable, then we get :
z = - 4 - x1 + 5x2 - x4
x0 = 4 + x1 - 5x2 + x4
x3 = 6 - x1 - 4x2 + x4
The associated basic solution is
(x -
0, x -
1, x -
2, x -
3, x -
4) = (4, 0, 0, 6, 0) which is
feasible.
•We now repeatedly call PIVOT until we obtain an optimal
solution to Laux .
39. Unit V : Linear Programming
Simplex Algorithm :
The initial basic feasible solution contd….:
Example contd….:
•In this case one call to PIVOT with entering variable x2 and
leaving variable x0 we get
z = - x0
x2 = 4/5 - x0 /5 + x1/5 + x4/5
x3 = 14/5 + 4 x0 /5 - 9x1/5 + x4/5
The associated basic solution is
(x -
0, x -
1, x -
2, x -
3, x -
4) = (4, 0, 0, 6, 0) which is
feasible.
This slack form is the final solution to Laux .
•If x0 is basic variable perform one (degenerate) PIVOT to
make it non-basic.
40. Unit V : Linear Programming
Simplex Algorithm :
The initial basic feasible solution contd….:
Example contd….:
•Now since x0 = 0, we can remove it from the set of
constraints. We then restore the original objective function
with appropriate substitutions made to include only non-basic
variables, we get
z = - 4/5 + 9x1/5 - x4/5
x2 = 4/5 + x1/5 + x4/5
x3 = 14/5 - 9x1/5 + x4/5
•This slack form has a feasible basic solution and we can
return it to procedure SIMPLEX.
41. Unit V : Linear Programming
Simplex Algorithm :
The initial basic feasible solution contd….:
Algorithm
Algorithm INITIAL-SIMPLEX (A, b, c)
1.Let k be the index of the minimum bi
2.If bk ≥ 0
return ({1, 2, …, n}, {n+1,n+2, …, n+m},
A, b, c, 0)
1.form Laux by adding x0 to LHS of each constraint
and set the objective function to –x0
2.let (N, B, A, b, c, v) be the resulting slack form for
Laux
42. Unit V : Linear Programming
Simplex Algorithm :
The initial basic feasible solution contd….: Algorithm
Algorithm INITIAL-SIMPLEX (A, b, c) contd…
1.if the optimal solution to Laux sets to x-
0 = 0
if x0 is basic
perform one (degenerate) PIVOT to make
x0 non-basic
form the final slack form of Laux , remove x0
from the constraints and restore the original
objective function of L, but replace each basic
variable in this objective function by the RHS of its
associated constraint
return the modified final slack form (N, B, A, b,
c, v)
43. Unit V : Linear Programming
Examples of Linear programming: Problem formulations
Now we shall examine how linear programming can be used
to solve the following problem :
1.Single Source Shortest Path Problem (A Graph
problem)
2.Maximum Flow Problem (A Network problem)
3.Vertex Cover problem (A Graph problem)
4.0/1 Knapsack problem : (A combinatorial
Problem)