The document discusses primal and dual linear programming problems. It provides examples of a primal problem about maximizing revenue from producing furniture given resource constraints, and its corresponding dual problem. The key relationships between a primal problem, its dual, and their optimal solutions are explained, including weak duality where any feasible primal solution has an objective value no greater than any feasible dual solution, and strong duality where the optimal primal and dual objectives are equal. General rules are provided for constructing the dual problem from the primal.
GAME THEORY
Terminology
Example : Game with Saddle point
Dominance Rules: (Theory-Example)
Arithmetic method – Example
Algebraic method - Example
Matrix method - Example
Graphical method - Example
GAME THEORY
Terminology
Example : Game with Saddle point
Dominance Rules: (Theory-Example)
Arithmetic method – Example
Algebraic method - Example
Matrix method - Example
Graphical method - Example
Sensitivity analysis in linear programming problem ( Muhammed Jiyad)Muhammed Jiyad
Topic includes : *Sensitivity Analysis *Objective function *Right Hand Side(RHS) *Sensitivity analysis using graph *Objective function coefficient *Reduced cost *Shadow pricing *Shadow pricing Microsoft Excel sensitivity report and solution.
The transportation problem is a special type of linear programming problem where the objective is to minimize the cost of distributing a product from a number of sources or origins to a number of destinations.
Because of its special structure, the usual simplex method is not suitable for solving transportation problems. These problems require a special method of solution.
Sensitivity analysis in linear programming problem ( Muhammed Jiyad)Muhammed Jiyad
Topic includes : *Sensitivity Analysis *Objective function *Right Hand Side(RHS) *Sensitivity analysis using graph *Objective function coefficient *Reduced cost *Shadow pricing *Shadow pricing Microsoft Excel sensitivity report and solution.
The transportation problem is a special type of linear programming problem where the objective is to minimize the cost of distributing a product from a number of sources or origins to a number of destinations.
Because of its special structure, the usual simplex method is not suitable for solving transportation problems. These problems require a special method of solution.
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.
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.
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
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
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
1. PRIMAL & DUAL
PROBLEMS,THEIR INTER
RELATIONSHIP
PREPARED BY:
130010119042 –KHAMBHAYATA MAYUR
130010119045 – KHANT VIJAY
130010119047 – LAD YASH A.
SUBMITTED TO:
PROF. S.R.PANDYA
2. CONTENT:
Duality Theory
Examples
Standard form of the Dual Problem
Definition
Primal-Dual relationship
Duality in LP
General Rules for Constructing Dual
Strong Dual
Weak Dual
3. Duality Theory
The theory of duality is a very elegant and important
concept within the field of operations research. This
theory was first developed in relation to linear
programming, but it has many applications, and
perhaps even a more natural and intuitive
interpretation, in several related areas such as
nonlinear programming, networks and game theory.
4. The notion of duality within linear programming asserts
that every linear program has associated with it a
related linear program called its dual. The original
problem in relation to its dual is termed the primal.
it is the relationship between the primal and its dual,
both on a mathematical and economic level, that is
truly the essence of duality theory.
Duality Theory (continue…)
5. Examples
There is a small company in Melbourne which has recently
become engaged in the production of office furniture. The
company manufactures tables, desks and chairs. The
production of a table requires 8 kgs of wood and 5 kgs of
metal and is sold for $80; a desk uses 6 kgs of wood and 4
kgs of metal and is sold for $60; and a chair requires 4 kgs of
both metal and wood and is sold for $50. We would like to
determine the revenue maximizing strategy for this
company, given that their resources are limited to 100 kgs
of wood and 60 kgs of metal.
6. Standard form of the Primal
Problem
a x a x a x b
a x a x a x b
a x a x a x b
x x x
n n
n n
m m mn n m
n
11 1 12 2 1 1
21 1 22 2 2 2
1 1 2 2
1 2 0
...
...
... ... ... ...
... ... ... ...
...
, ,...,
max
x
j j
j
n
Z c x
1
7. Standard form of the Dual
Problem
a y a y a y c
a y a y a y c
a y a y a y c
y y y
m m
m m
n n mn m n
m
11 1 21 2 1 1
12 1 22 2 2 2
1 1 2 2
1 2 0
...
...
... ... ... ...
... ... ... ...
...
, ,...,
min
y
i
i
m
iw b y
1
8. Definition
z Z cx
s t
Ax b
x
x
*: max
. .
0
w* : min
x
w yb
s.t.
yA c
y 0
Primal Problem Dual Problem
b is not assumed to be non-negative
9. Primal-Dual relationship
x1 0 x2 0 xn 0 w =
y1 0 a11 a12 a n1 b1
D u a l y2 0 a21 a22 a n2 b2
(m in w ) .. . . .. ... . .. . . .
ym 0 am1 am2 amn bn
Z = c1 c2 cn
10. Example
5 18 5 15
8 12 8 8
12 4 8 10
2 5 5
0
1 2 3
1 2 3
1 2 3
1 3
1 2 3
x x x
x x x
x x x
x x
x x x
, ,
max
x
Z x x x 4 10 91 2 3
12. Dual
5 8 12 2 4
18 12 4 10
5 8 5 9
0
1 2 3 4
1 2 3
1 3 4
1 2 3 4
y y y y
y y y
y y y
y y y y
, , ,
min
y
w y y y y 15 8 10 51 2 3 4
13. d x ei
i
k
i
1
d x e
d x e
i
i
k
i
i
i
k
i
1
1
d x e
d x e
i
i
k
i
i
i
k
i
1
1
Standard form!
15. Duality in LP
In LP models, scarce resources are allocated, so they
should be, valued
Whenever we solve an LP problem, we solve two
problems: the primal resource allocation problem,
and the dual resource valuation problem
If the primal problem has n variables and m constraints,
the dual problem will have m variables and n
constraints
16. Primal and Dual Algebra
Primal
j j
j
ij j i
j
j
c X
. . a X b 1,...,
X 0 1,...,
Max
s t i m
j n
Dual
i
i
ij j
i
i
Min b
s.t. a c 1,...,
Y 0 1,...,
i
i
Y
Y j n
i m
'
. .
0
Max C X
s t X b
X
'
'
. .
0
Min b Y
s t AY C
Y
17. Example
1 2
1 2
1 2
1 2
40 30 ( )
. . 120 ( )
4 2 320 ( )
, 0
Max x x profits
s t x x land
x x labor
x x
1 2
1 2 1
1 2 2
1 2
( ) ( )
120 320
. . 4 40 ( )
2 30 ( )
, 0
land labor
Min y y
s t y y x
y y x
y y
Primal
Dual
18. General Rules for Constructing Dual
1. The number of variables in the dual problem is equal to the number of
constraints in the original (primal) problem. The number of constraints in
the dual problem is equal to the number of variables in the original
problem.
2. Coefficient of the objective function in the dual problem come from the
right-hand side of the original problem.
3. If the original problem is a max model, the dual is a min model; if the original
problem is a min model, the dual problem is the max problem.
4. The coefficient of the first constraint function for the dual problem are the
coefficients of the first variable in the constraints for the original problem,
and the similarly for other constraints.
5. The right-hand sides of the dual constraints come from the objective function
coefficients in the original problem.
19. Relations between Primal and Dual
1. The dual of the dual problem is again the primal problem.
2. Either of the two problems has an optimal solution if and
only if the other does; if one problem is feasible but
unbounded, then the other is infeasible; if one is infeasible,
then the other is either infeasible or feasible/unbounded.
3. Weak Duality Theorem: The objective function value of the
primal (dual) to be maximized evaluated at any primal (dual)
feasible solution cannot exceed the dual (primal) objective
function value evaluated at a dual (primal) feasible solution.
cTx >= bTy (in the standard equality form)
20. Relations between Primal and Dual (continued)
4. Strong Duality Theorem: When there is an optimal solution, the
optimal objective value of the primal is the same as the optimal
objective value of the dual.
cTx* = bTy*
21. Weak Duality
• DLP provides upper bound (in the case of maximization) to
the solution of the PLP.
• Ex) maximum flow vs. minimum cut
• Weak duality : any feasible solution to the primal linear
program has a value no greater than that of any feasible
solution to the dual linear program.
• Lemma : Let x and y be any feasible solution to the PLP and
DLP respectively. Then cTx ≤ yTb.
22. Strong duality : if PLP is feasible and has a finite
optimum then DLP is feasible and has a finite
optimum.
Furthermore, if x* and y* are optimal solutions for
PLP and DLP then cT x* = y*Tb
Strong Duality
23. Four Possible Primal Dual Problems
Dual
Primal Finite optimum Unbounded Infeasible
Finite optimum 1 x x
Unbounded x x 2
Infeasible x 3 4