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.
Linear programming
Application Of Linear Programming
Advantages Of L.P.
Limitation Of L.P.
Slack variables
Surplus variables
Artificial variables
Duality
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.
Linear programming
Application Of Linear Programming
Advantages Of L.P.
Limitation Of L.P.
Slack variables
Surplus variables
Artificial variables
Duality
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Importance:
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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.
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The Art Pastor's Guide to Sabbath | Steve ThomasonSteve Thomason
What is the purpose of the Sabbath Law in the Torah. It is interesting to compare how the context of the law shifts from Exodus to Deuteronomy. Who gets to rest, and why?
The Indian economy is classified into different sectors to simplify the analysis and understanding of economic activities. For Class 10, it's essential to grasp the sectors of the Indian economy, understand their characteristics, and recognize their importance. This guide will provide detailed notes on the Sectors of the Indian Economy Class 10, using specific long-tail keywords to enhance comprehension.
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Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
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How to Split Bills in the Odoo 17 POS ModuleCeline George
Bills have a main role in point of sale procedure. It will help to track sales, handling payments and giving receipts to customers. Bill splitting also has an important role in POS. For example, If some friends come together for dinner and if they want to divide the bill then it is possible by POS bill splitting. This slide will show how to split bills in odoo 17 POS.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
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2. Concepts of Duality
■ One of the most important discoveries in the early development of linear
programming was the concept of duality.
■ Every LPP has associated with it another LPP. The original problem is called the
“Primal” while the other is called its “Dual”.
■ The optimal solution of either problem reveals information concerning the optimal
solution of the other.
■ This fact is important because the situation can arise where the dual is easier to
solve than the primal.
3. Example … (Primal)
The Following problem can be formulated as follows:
Min z = 10x1 + 15x2, subject to constraints
5x1 + 7x2 >= 80
6x1 + 11x2 >= 100, x1 >= 0, x2 >= 0
In the formulation, we have assumed that taking more than the minimum
requirement is not harmful.This LPP will be considered as the primal problem.
4. Basis of Conversion
■ The costs associated with the objective function of one problem are just the
requirements in the other’s set of constraints.
■ The constraint coefficient matrix associated with one problem is simply the
transpose of the constraint Coefficient matrix associated with the other.
■ One of the problems is a maximization problem while the other one is a
minimization problem.
6. Symmetric Primal-Dual Problem
• Transposing the coefficient matrix.
• Interchanging the role of constant terms and the role of the objective function.
• Reverting the inequalities.
• Minimizing the objective function instead of maximizing.
7. Rules For Conversion
■ First convert the objective function to maximization form, if not already.
■ If a constraint has inequality sign “≥”, then multiply both sides by −1 and make the
inequality sign ≤.
■ If a constraint has an equality sign “=”, then it is replaced by two constraints
involving the inequalities going in opposite directions, simultaneously.
■ Every unrestricted variable is replaced by the difference of two non-negative
variables.
8. Rules of Conversion (contd…)
■ We get the standard primal form of the given LPP in which:
a) all the constraints have “≤” sign, where the objective function is of
maximization form; or
b) all the constraints have “≥” sign, where objective function is of
minimization form.
■ Finally, the dual of the given problem is obtained by:
a) transposing the rows and columns of constraint coefficients;
b) transposing the coefficients (c1, c2, ... , cn) of the objective function and
the right-side constants (b1, b2, ... , bm);
c) changing the inequalities from “≤” to “≥” sign; and
d) minimizing the objective function instead of maximizing it.
9. Duality in Linear Programming
Duality Theorem: The dual of the dual of a given primal is the primal.
■ If the kth constraint of the primal is an equality, then the dual variable wk is
unrestricted in sign. Also, if pth variable of the primal is unrestricted in sign, then
the pth constraint of the dual is an equality.
■ If the primal or the dual has a finite optimum solution, then the other problem also
possesses a finite optimum solution and the optimum value in both the cases will
be the same.
11. Dual Simplex Method
■ Such a situation is recognized by first expressing the constraints in the form “≤”
and the objective function in the maximization form.
■ After adding the slack variables and putting the problem in the tabular form, if any
of the right hand side elements are negative and if the optimality condition is
satisfied, then the problem can be solved by the dual simplex method.
■ negative element on the right hand side signifies that the corresponding slack
variable is negative.
■ In this method, we shall proceed towards feasibility maintaining optimality and at
the iteration where the basic solution becomes feasible, it becomes the optimal
basic feasible solution also.
12. Advantages of Dual Simplex
■ we do not require any artificial variables in the dual simplex method. Hence a lot of
effort is saved whenever this method is applicable.
■ The dual simplex method is similar to the standard simplex method except that the
starting initial solution in standard simplex method is feasible but not optimum
while in the dual simplex it is infeasible but optimum or better than optimum.
■ The dual simplex method works towards feasibility while simplex method works
towards optimality.