ON Fuzzy Linear Programming Technique Applicationijsrd.com
A solution approach based on fuzzy linear programming is proposed and applied to optimal machine scheduling problem. In this solution approach errors in the demand of various products during the next production period are considered to be fuzzy in nature. In conventional linear programming approach it is assumed that there is no error in the expected demand of various products. A fuzzy linear programming approach is proposed to obtain an optimal solution under fuzzy conditions. In the proposed method expected demand of products & profit are expressed by fuzzy set notations. The proposed fuzzy linear programming formulation is then transformed to an equivalent conventional linear programming problem and solutions obtained by solving this transformed linear programming problem. For illustration purpose the proposed method is applied to a profit maximization related machine scheduling problem.
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ON Fuzzy Linear Programming Technique Applicationijsrd.com
A solution approach based on fuzzy linear programming is proposed and applied to optimal machine scheduling problem. In this solution approach errors in the demand of various products during the next production period are considered to be fuzzy in nature. In conventional linear programming approach it is assumed that there is no error in the expected demand of various products. A fuzzy linear programming approach is proposed to obtain an optimal solution under fuzzy conditions. In the proposed method expected demand of products & profit are expressed by fuzzy set notations. The proposed fuzzy linear programming formulation is then transformed to an equivalent conventional linear programming problem and solutions obtained by solving this transformed linear programming problem. For illustration purpose the proposed method is applied to a profit maximization related machine scheduling problem.
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Decisions Optimization Related to the Production Within Refining and Petroche...ijtsrd
The paper underlines the use of quantitative analyses and mathematical models to optimize the decision within companies from oil and gas industry. It will be presented a case study from a refinery that use RPMS Refinery and Petrochemical Modeling System software for optimizing LPG blends. Catalin Popescu "Decisions Optimization Related to the Production Within Refining and Petrochemical Industry" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-1 , December 2018, URL: http://www.ijtsrd.com/papers/ijtsrd20189.pdf
http://www.ijtsrd.com/management/operations-management/20189/decisions-optimization-related-to-the-production-within-refining-and-petrochemical-industry/catalin-popescu
OR is defined as a scientific approach to optimal decision making through modelling of
deterministic and probabilistic systems that originate from real life.
Scientific approach: LPP, PERT/CPM, Queueing model, NLP, DP,MILP, Game
theory, heuristic programming.
Deterministic system: - a system which gives the same result for a particular set of
input, no matter how many times we recalculate it
Om0013 advanced production and operations managementsmumbahelp
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The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
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.
For more information, visit-www.vavaclasses.com
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
3. project
Within 15 days Sales target within
a month
Within a budget of
INR 25000
:
Maximum score Maximum sales
in a month Minimize the cost
TIME TIME
Pre Decided
Budget
4. Such type of problems are called
• Maximum Profit
• Minimum Cost
• Minimum use of Resources
There can be many day to day problems which need to be
solved using:
5. Linear Programming (LP) is a versatile technique for assigning a fixed amount of
Resources in such a way that some objectives are optimized and other defined
conditions are also satisfied.
The Objectives may be cost minimization or inversely profit maximization
The Technique Of Linear Programming was developed by GEORGE B.
DANTZIG while he was working with US Air force during World War 2.
primarily it was developed for solving military problems.
But now it is being used for solving wide range of problems relating to oil
refining , energy planning, pollution control education and almost in all
functional areas of Management-production,finance,marketing,personnel.
6. Linear programming and Optimization are used in various industries.
The manufacturing and service industry uses linear programming on a
regular basis. In this section, we are going to look at the various
applications of Linear programming.
1. Manufacturing industries use linear programming for analyzing
their supply chain operations. Their motive is to maximize efficiency
with minimum operation cost. As per the recommendations from
the linear programming model, the manufacturer can reconfigure
their storage layout, adjust their workforce and reduce the
bottlenecks.
2. Linear programming is also used in organized retail for shelf
space optimization. Since the number of products in the market
has increased in leaps and bounds, it is important to understand
what does the customer want. Optimization is aggressively used in
stores like Walmart, Hypercity, Reliance, Big Bazaar, etc.
7. 3. Optimization is also used for optimizing Delivery Routes. This is an
extension of the popular traveling salesman problem. The service
industry uses optimization for finding the best route for multiple
salesmen traveling to multiple cities. With the help of clustering and
greedy algorithm, the delivery routes are decided by companies like
FedEx, Amazon, etc. The objective is to minimize the operation cost
and time.
4. Optimizations are also used in Machine Learning. Supervised
Learning works on the fundamental of linear programming. A system is
trained to fit on a mathematical model of a function from the labeled
input data that can predict values from an unknown test data.
8. 1. A decision amongst alternative courses of action is required.
2. The decision is represented in the model by decision variables.
3. The problem encompasses a goal, expressed as an objective function, that the
decision maker wants to achieve.
4. Restrictions (represented by constraints) exist that limit the extent of achievement
of the objective.
5. The objective and constraints must be definable by linear mathematical
functional relationships.
1. Proportionality - The rate of change (slope) of the objective function and constraint
equations is constant.
2. Additivity - Terms in the objective function and constraint equations must be
additive.
3. Divisibility -Decision variables can take on any fractional value and are therefore
continuous as opposed to integer in nature.
4.Certainty - Values of all the model parameters are assumed to be known with
certainty (non-probabilistic).
9. ADVANTAGES OF LINEAR PROGRAMMING
Following are certain advantages of linear programming:
1.Linear programming helps in attaining the optimum use of productive
resources. It also indicates how a decision-maker can employ his productive
factors effectively by selecting and distributing (allocating) these resources.
2.Linear programming techniques improve the quality of decisions. The
decision-making approach of the user of this technique becomes more
objective and less subjective.
3.linear programming techniques provide possible and practical solutions since
there might be other constraints operating outside the problem which must be
taken into account. Just because we can produce so many units docs not
mean that they can be sold. Thus, necessary modification of its mathematical
solution is required for the sake of convenience to the decision-maker.
4.Highlighting of bottlenecks in the production processes is the most significant
advantage of this technique. For example, when a bottleneck occurs, some
machines cannot meet demand while other remains idle for some of the time.
5.Linear programming also helps in re-evaluation of a basic plan for changing
conditions. If conditions change when the plan is partly carried out, they can
be determined so as to adjust the remainder of the plan for best results.
10. 1. Linear programming model does not take into consideration the
effect of time uncertainty. Thus, the LP model should be defined in
such a way that any change due to internal as well as external
factors can be incorporated.
2. Parameters appearing in the model are assumed to be constant
but in real-life situations, they are frequently neither known nor
constant.
3. Parameters like human behaviour, weather conditions, stress of
employees, demotivated employee can’t be taken into account
which can adversely effect any organisation.
4. Only one single objective is dealt with while in real life situations,
problems come with multi-objectives.
5. Sometimes large-scale problems can be solved with linear
programming techniques even when assistance of computer is
available. For it, the main problem can be fragmented into several
small problems and solving each one separately.
11. Linear optimization models are among the most successful application of
operational research. In fact, they rank highest in economic impact. Suppose, we
consider the operations of a major integrated oil company. The key decisions in
this type of industry related to processes of:
1.Exploring for oil deposits.
2.Producing crude oil.
3.Shipping crude to various refineries.
4.Cracking the crude into several blending stock.
5.Combining the stocks into several types of petroleum
products.
6.Shipping the manufacture products from the refineries to
marketing areas.
12. understanding the given problem
Convert to a Linear Programming
problem
Two Aspect:
Formulation Part
Solution Part
13. A Company manufactures and sells 2 types of products ‘A’ and
‘B’. The cost of production of each unit of ‘A’ and ‘B’ is RS.200
and RS.150 respectively. Each unit of ‘A’ yield a profit of 20 and
each unit of ‘B‘ yields a profit of 15 on selling.
Company estimates the monthly demand of ‘A’ and ‘B’ to be
maximum value of 500 units in All. The production budget for
the month is 50000.How many units should be company
manufactures in order to earn maximum profit from its monthly
sales of ‘A’ and ‘B’?
Maximum profit ?????
Constraints Production Budget
14. products
Cost of
production per
unit
Profit per unit Total demand
A 200 20
500 units
B 150 15
Production
budget
50000 _ _
Tabular form
Now, express the problem in mathematical form.
15.
16. Step 1. Formulate the LPP problems and develop objective
function along with all the constraints function.
Step 2. Graph the feasible region and find the corner
points. The coordinates of the corner points can be
obtained by either inspection or by solving the two
equations of the lines intersecting at that point.
Step 3. Make a table listing the value of the objective
function at each corner point.
Step 4. Determine the optimal solution from the table in step
3. If the problem is of maximization (minimization) type, the
solution corresponding to the largest (smallest) value of the
objective function is the optimal solution of the LPP.
17. A furniture company produces inexpensive tables and chairs. The production process for
each is similar in that both require a certain number of hours of carpentry work and a
certain number of labour hours in the painting department. Each table takes 4 hours of
carpentry and 2 hours in the painting department. Each chair requires 3 hours of carpentry
and 1 hour in the painting department. During the current production period, 240 hours of
carpentry time are available and 100 hours in painting is available. Each table sold yields a
profit of E7; each chair produced is sold for a E5 profit. Find the best combination of tables
and chairs to manufacture in order to reach the maximum profit.
Example 2
18. The decision variables can be defined as X = number of tables to
be produced & Y = number of chairs to be produced.
Now linear programming (LP) problem can be formulated in
terms of X and Y and Profit (P).
maximize P = 7X + 5Y (Objective function) subject to 4X + 3Y ≤ 240
(hours of carpentry constraint) 2X + Y ≤ 100 (hours of painting
constraint) X ≥ 0, Y ≥ 0 (Non-negativity constraint).
Therefore the mathematical formulation of the LPP is:
Maximize: P = 7X + 5Y
Subject to: 4X + 3Y ≤ 240
2X + Y ≤ 100 X
≥ 0 , Y ≥ 0 To find the optimal solution to this LP using the
graphical method we first identify the region of feasible solutions
and the corner points of the of the feasible region.
The graph for this example is plotted in the next slide
19. In this example the corner points are (0,0),
(50,0), (30,40) and (0,80). Testing these
corner points on P = 7X + 5Y gives
Because the point (30,40) produces the
highest profit we conclude that producing 30
tables and 40 chairs will yield a maximum
profit of E410.
The graphical method is one of the easiest
way to solve a small LP problem. However
this is useful only when the decision variables
are not more than two.