The document discusses linear programming (LP), which is a mathematical optimization method that allocates resources by optimizing a linear objective function subject to linear constraints. It defines the key components of an LP problem as decision variables, an objective function, and constraints. Convex sets are also discussed as they relate to LP problems, with convex sets ensuring that the optimal solutions found by LP algorithms are globally optimal and can be efficiently obtained. Examples of convex sets are provided.
Linear programming
Application Of Linear Programming
Advantages Of L.P.
Limitation Of L.P.
Slack variables
Surplus variables
Artificial variables
Duality
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.
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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
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
Linear programming
Application Of Linear Programming
Advantages Of L.P.
Limitation Of L.P.
Slack variables
Surplus variables
Artificial variables
Duality
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
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 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 Programming Problems {Operation Research}FellowBuddy.com
FellowBuddy.com is an innovative platform that brings students together to share notes, exam papers, study guides, project reports and presentation for upcoming exams.
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# Underachievers can find peer developed notes that break down lecture and study material in a way that they can understand
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Nonlinear Programming: Theories and Algorithms of Some Unconstrained Optimiza...Dr. Amarjeet Singh
Nonlinear programming problem (NPP) had become an important branch of operations research, and it was the mathematical programming with the objective function or constraints being nonlinear functions. There were a variety of traditional methods to solve nonlinear programming problems such as bisection method, gradient projection method, the penalty function method, feasible direction method, the multiplier method. But these methods had their specific scope and limitations, the objective function and constraint conditions generally had continuous and differentiable request. The traditional optimization methods were difficult to adopt as the optimized object being more complicated. However, in this paper, mathematical programming techniques that are commonly used to extremize nonlinear functions of single and multiple (n) design variables subject to no constraints are been used to overcome the above challenge. Although most structural optimization problems involve constraints that bound the design space, study of the methods of unconstrained optimization is important for several reasons. Steepest Descent and Newton’s methods are employed in this paper to solve an optimization problem.
Operations research (OR) is an analytical method of problem-solving and decision-making that is useful in the management of organizations. In operations research, problems are broken down into basic components and then solved in defined steps by mathematical analysis.
Analytical methods used in OR include mathematical logic, simulation, network analysis, queuing theory , and game theory .The process can be broadly broken down into three steps.
1. A set of potential solutions to a problem is developed. (This set may be large.)
2. The alternatives derived in the first step are analyzed and reduced to a small set of solutions most likely to prove workable.
3. The alternatives derived in the second step are subjected to simulated implementation and, if possible, tested out in real-world situations. In this final step, psychology and management science often play important roles
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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 Programming Problems {Operation Research}FellowBuddy.com
FellowBuddy.com is an innovative platform that brings students together to share notes, exam papers, study guides, project reports and presentation for upcoming exams.
We connect Students who have an understanding of course material with Students who need help.
Benefits:-
# Students can catch up on notes they missed because of an absence.
# Underachievers can find peer developed notes that break down lecture and study material in a way that they can understand
# Students can earn better grades, save time and study effectively
Our Vision & Mission – Simplifying Students Life
Our Belief – “The great breakthrough in your life comes when you realize it, that you can learn anything you need to learn; to accomplish any goal that you have set for yourself. This means there are no limits on what you can be, have or do.”
Like Us - https://www.facebook.com/FellowBuddycom
Nonlinear Programming: Theories and Algorithms of Some Unconstrained Optimiza...Dr. Amarjeet Singh
Nonlinear programming problem (NPP) had become an important branch of operations research, and it was the mathematical programming with the objective function or constraints being nonlinear functions. There were a variety of traditional methods to solve nonlinear programming problems such as bisection method, gradient projection method, the penalty function method, feasible direction method, the multiplier method. But these methods had their specific scope and limitations, the objective function and constraint conditions generally had continuous and differentiable request. The traditional optimization methods were difficult to adopt as the optimized object being more complicated. However, in this paper, mathematical programming techniques that are commonly used to extremize nonlinear functions of single and multiple (n) design variables subject to no constraints are been used to overcome the above challenge. Although most structural optimization problems involve constraints that bound the design space, study of the methods of unconstrained optimization is important for several reasons. Steepest Descent and Newton’s methods are employed in this paper to solve an optimization problem.
Operations research (OR) is an analytical method of problem-solving and decision-making that is useful in the management of organizations. In operations research, problems are broken down into basic components and then solved in defined steps by mathematical analysis.
Analytical methods used in OR include mathematical logic, simulation, network analysis, queuing theory , and game theory .The process can be broadly broken down into three steps.
1. A set of potential solutions to a problem is developed. (This set may be large.)
2. The alternatives derived in the first step are analyzed and reduced to a small set of solutions most likely to prove workable.
3. The alternatives derived in the second step are subjected to simulated implementation and, if possible, tested out in real-world situations. In this final step, psychology and management science often play important roles
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1. OPERATION
RESEARCH
BASIC LPP AND APPLICATIONS;
VARIOUS COMPONENTS OF LP
PROBLEM FORMULATION.
CONVEX SET AND EXPLANATION
WITH EXAMPLES
ARITRA KUNDU
DEPT:CSE
ROLL NUMBER:35000120030
SEMESTER:7TH
2. 1 Introduction
2 Basic Linear Programming (LP)
3 LP Problem Formulation Components
4 Applications of LP
5 Convex Sets and LP
6 Characteristics of Convex Sets
7 Convex Set Examples
LIST OF CONTENTS
3. INTRODUCTION
Operations Research (OR) is a discipline that applies mathematical and analytical methods
to tackle complex decision-making challenges. It involves using tools like optimization,
simulation, and modeling to find the best possible solutions to problems in diverse areas
such as logistics, finance, healthcare, and engineering. OR aims to enhance efficiency,
minimize costs, and improve processes by providing data-driven insights that help
organizations make informed choices.
At its core, Operations Research uses quantitative techniques to analyze and solve real-
world problems. It's about transforming complex situations into mathematical models,
gathering and interpreting data, and using computational approaches to guide decision-
makers toward optimal outcomes. By leveraging these methods, Operations Research
assists organizations in navigating uncertainty, making strategic decisions, and improving
overall performance in a wide range of industries.
4. BASIC LINEAR PROGRAMMING
Linear Programming (LP) is a mathematical optimization method that tackles resource allocation by optimizing a linear
objective while adhering to linear constraints. It involves decision variables, an objective function, and constraints, all
expressed linearly. LP aims to maximize or minimize the objective function while satisfying constraints, finding optimal
solutions for various real-world problems like production planning, supply chain management, and financial portfolio
optimization. It excels in scenarios where decisions have linear relationships and constraints are represented as linear
equations or inequalities, making it a valuable tool for efficient decision-making in diverse fields.
Objective Function: The objective function in Linear Programming defines the quantity that needs to be optimized, whether it's
maximized (finding the highest value) or minimized (finding the lowest value). It is typically a linear mathematical expression
involving decision variables. The goal is to adjust the values of these variables to achieve the best possible value of the objective
function. For instance, in a manufacturing context, the objective might be to maximize profit or minimize costs, and the objective
function would be formulated as a linear equation involving factors like production quantities and costs.
Decision Variables: Decision variables are the unknown quantities that you're trying to determine in a Linear Programming problem.
These variables represent the choices or decisions you can make to achieve the desired outcome. They could be quantities of items to
produce, allocate, or invest in. For instance, in a production planning problem, decision variables might represent the number of units
of different products to manufacture. The objective is to find the optimal values of these variables that lead to the optimal value of the
objective function, subject to the given constraints.
Constraints: Constraints are the restrictions or limitations that define the feasible region within which the decision variables must
operate. These constraints are usually linear equations or inequalities involving the decision variables. Constraints represent real-
world limitations on resources, capacities, or other factors. For example, in a transportation problem, constraints might involve
limiting the available quantities of goods, the capacity of transportation vehicles, and the demands at various destinations. The
feasible region is the set of values for the decision variables that satisfy all constraints simultaneously.
In essence, Linear Programming seeks to find the values of decision variables that simultaneously meet all constraints while
optimizing the objective function. It's a mathematical approach to problem-solving that balances available resources and desired
outcomes, making it a powerful tool for making informed decisions in various practical scenarios.
5. LP Problem Formulation Components
Decision Variables: These are the variables that you can adjust or
control to achieve the desired outcome.
Objective Function: This is a linear equation that you want to
maximize (in case of profit) or minimize (in case of cost) based on the
decision variables.
Constraints: These are linear inequalities or equations that represent
limitations or requirements on the decision variables. They define the
feasible region, i.e., the set of solutions that satisfy all constraints.
Non-Negativity Constraints: Decision variables are typically required
to be nonnegative (i.e., greater than or equal to zero) in most LP
problems.
6.
7. Convex Sets and LP
Convex Sets and Convexity: A convex set is a mathematical concept in which, for any two points within the set, the
straight line segment connecting them also lies entirely within the set. In other words, a set is convex if it contains all the
points on the line segment connecting any two points within the set. This property is known as "convexity," and it
implies that the set doesn't have any indentations, holes, or disjointed parts. Convexity is a fundamental property in
mathematics and has significant implications in various fields, including Linear Programming (LP).
Relation to LP Solutions: Convex sets play a crucial role in Linear Programming. In LP, both the feasible region (the set of
solutions satisfying constraints) and the objective function are typically linear. The feasible region is often a convex set
because the constraints are represented by linear inequalities or equations. The optimal solution of an LP problem,
which maximizes or minimizes the objective function while staying within the feasible region, often lies at a vertex of
this convex feasible region. This vertex is a point where multiple constraints meet, and it's crucially a part of the convex
hull of the feasible region.
Intuitive Understanding: Imagine you have a convex set as a piece of flexible rubber sheet. If you place two points
anywhere on the rubber sheet, the rubber can be stretched and bent in a way that the sheet always remains between
those two points. There are no holes or creases in the sheet. Now, think of the rubber sheet as the feasible region in an LP
problem, and the two points as potential solutions that adhere to the constraints. The stretching and bending of the
rubber sheet represent all possible combinations of the decision variables within the constraints. This concept shows
how convex sets ensure that all the points on the line connecting the two solutions also lie within the feasible region,
preserving the convexity property.
In summary, convex sets ensure that the "straightforward path" between any two points within the set stays within the
set itself. In Linear Programming, the convexity of the feasible region ensures that optimization algorithms can reliably
find solutions, and the concept of convexity provides a geometric foundation for understanding the relationships
between constraints, decision variables, and optimal solutions.
8. CHARACTERISTICS OF CONVEX SET
1.Line Segment Inclusion: For any two points A and B within the set, the entire line segment connecting A and B lies within the set.
Mathematically, if λA+(1−λ)B are in the set, then λA+(1−λ)B is also in the set for 0≤λ≤1.
2.Convex Combination: Any point that can be expressed as a weighted average of two points within the set is also within the set.
Mathematically, if A and B are in the set, then λA+(1−λ)B is in the set for 0≤λ≤1.
3. Vertex Inclusion: All vertices (extreme points) of the convex set are part of the set. A vertex is a point that cannot be expressed as
a convex combination of other points within the set.
4.Affine Combination: Any point obtained by an affine combination (a linear combination with weights summing to 1) of points
within the set is also in the set.
5.No Holes or Gaps: Convex sets have no indentations, holes, or gaps in their structure.
Relevance to LP Solutions:
These characteristics of convex sets are highly relevant to Linear Programming (LP) solutions for several reasons:
1. Optimality at Vertices: In LP, the optimal solution often lies at a vertex of the convex feasible region. This is because vertices are extreme
points that cannot be improved by moving along the line segment connecting them. The property of vertex inclusion ensures that optimal
solutions are part of the feasible region.
2. Linearity of Constraints and Objectives: LP constraints are typically linear equations or inequalities, and the objective function is also linear.
The line segment inclusion property of convex sets ensures that all the points on the line segment connecting feasible solutions adhere to the
linear constraints, enabling LP solvers to efficiently explore the feasible region.
3. Convexity Ensures Global Optimality: Convexity guarantees that any local minimum or maximum within the feasible region is also a global
minimum or maximum. This property is vital for LP problems where finding the global optimum is essential.
4. Efficient Solution Search: Convexity simplifies the search for solutions by allowing LP algorithms to focus on vertices or boundary points of
the feasible region. This reduces the search space and makes optimization more efficient.
In summary, the key characteristics of convex sets ensure that LP problems have well-defined, achievable solutions that can be efficiently found
through optimization algorithms. Convexity underlies the stability, efficiency, and global optimality of solutions in Linear Programming.