The assignment problem is a special type of linear programming problem and it is sub class of transportation problem. Assignment problems are defined with two sets of inputs i.e. set of resources and set of demands. Hungarian algorithm is able to solve assignment problems with precisely defined demands and resources.Nowadays, many organizations and competition companies consider markets of their products. They use many salespersons to improve their organizations marketing. Salespersons travel form one city to another city for their markets. There are some problems in travelling which salespeople should go which city in minimum cost. So, travelling assignment problem is a main process for many business functions. Mie Mie Aung | Yin Yin Cho | Khin Htay | Khin Soe Myint "Minimization of Assignment Problems" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-5 , August 2019, URL: https://www.ijtsrd.com/papers/ijtsrd26712.pdfPaper URL: https://www.ijtsrd.com/computer-science/other/26712/minimization-of-assignment-problems/mie-mie-aung
Enhance interval width of crime forecasting with ARIMA model-fuzzy alpha cutTELKOMNIKA JOURNAL
With qualified data or information a better decision can be made. The interval width of forecasting
is one of data values to assist in the selection decision making process in regards to crime prevention.
However, in time series forecasting, especially the use of ARIMA model, the amount of historical data
available can affect forecasting result including interval width forecasting value. This study proposes a
combination technique, in order to get get a better interval width crime forecasting value. The propose
combination technique between ARIMA model and Fuzzy Alpha Cut are presented. The use of variation
alpha values are used, they are 0.3, 0.5, and 0.7. The experimental results have shown the use of
ARIMA-FAC with alpha=0.5 is appropriate. The overall results obtained have shown the interval width
crime forecasting with ARIMA-FAC is better than interval width crime forecasting with 95% CI
ARIMA model.
The assignment problem is a special type of linear programming problem and it is sub class of transportation problem. Assignment problems are defined with two sets of inputs i.e. set of resources and set of demands. Hungarian algorithm is able to solve assignment problems with precisely defined demands and resources.Nowadays, many organizations and competition companies consider markets of their products. They use many salespersons to improve their organizations marketing. Salespersons travel form one city to another city for their markets. There are some problems in travelling which salespeople should go which city in minimum cost. So, travelling assignment problem is a main process for many business functions. Mie Mie Aung | Yin Yin Cho | Khin Htay | Khin Soe Myint "Minimization of Assignment Problems" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-5 , August 2019, URL: https://www.ijtsrd.com/papers/ijtsrd26712.pdfPaper URL: https://www.ijtsrd.com/computer-science/other/26712/minimization-of-assignment-problems/mie-mie-aung
Enhance interval width of crime forecasting with ARIMA model-fuzzy alpha cutTELKOMNIKA JOURNAL
With qualified data or information a better decision can be made. The interval width of forecasting
is one of data values to assist in the selection decision making process in regards to crime prevention.
However, in time series forecasting, especially the use of ARIMA model, the amount of historical data
available can affect forecasting result including interval width forecasting value. This study proposes a
combination technique, in order to get get a better interval width crime forecasting value. The propose
combination technique between ARIMA model and Fuzzy Alpha Cut are presented. The use of variation
alpha values are used, they are 0.3, 0.5, and 0.7. The experimental results have shown the use of
ARIMA-FAC with alpha=0.5 is appropriate. The overall results obtained have shown the interval width
crime forecasting with ARIMA-FAC is better than interval width crime forecasting with 95% CI
ARIMA model.
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A MULTIPLE – CHANNEL QUEUING MODELS ON FUZZY ENVIRONMENTIAEME Publication
In this paper, we investigated a queuing model of fuzzy environment-based a multiple channel queuing model (M/M/C) ( /FCFS) and study its performance under realistic conditions. It applies a nonagonal fuzzy number to analyse the relevant performance of a multiple channel queuing model (M/M/C) ( /FCFS). Based on the sub interval average ranking method for nonagonal fuzzy number, we convert fuzzy number to crisp one. Numerical results reveal that the efficiency of this method. Intuitively, the fuzzy environment adapts well to a multiple channel queuing models (M/M/C) ( /FCFS) are very well.
Balancing the line by using heuristic method based on cpm in salbp –a case studyeSAT Journals
Abstract
In mass production systems, line balancing plays a great role, but this is not easy even if it is a simple straight line. So, in order to
solve these problems Heuristic methods are very much desirable. It is also found that Heuristic methods play a great role in the
formation of metaheuristic methods.Therefore it is very much important to use more efficient heuristic methods. In this research
paper we presents a heuristic method that is based on critical path method for simple assembly line balancing. This research is
mainly concerned with objectives of minimizing the number of workstations, improvement of smoothness index, mean absolute
deviation (MAD) and increasing line efficiency.
Keywords-Heuristic methods,Assembly line balancing problem, Critical path method, Simple assembly line balancing.
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Airline Start up is hard issue for investors, today airline try to survive in this high competitive industry, this presenting address new tools that guide and put airline in the right way
Queuing theory: What is a Queuing system???
Waiting for service is part of our daily life….
Example:
we wait to eat in restaurants….
We queue up in grocery stores…
Jobs wait to be processed on machine…
Vehicles queue up at traffic signal….
Planes circle in a stack before given permission to land at an airport….
Unfortunately, we can not eliminate waiting time without incurring expenses…
But, we can hope to reduce the queue time to a tolerable levels… so that we can avoid adverse impact….
Why study???? What analytics can be drawn??? Analytics means ---- measures of performance such as
1. Average queue length
2. Average waiting time in the queue
3. Average facility utilization….
A Predictive Stock Data Analysis with SVM-PCA Model .......................................................................1
Divya Joseph and Vinai George Biju
HOV-kNN: A New Algorithm to Nearest Neighbor Search in Dynamic Space.......................................... 12
Mohammad Reza Abbasifard, Hassan Naderi and Mohadese Mirjalili
A Survey on Mobile Malware: A War without End................................................................................... 23
Sonal Mohite and Prof. R. S. Sonar
An Efficient Design Tool to Detect Inconsistencies in UML Design Models............................................. 36
Mythili Thirugnanam and Sumathy Subramaniam
An Integrated Procedure for Resolving Portfolio Optimization Problems using Data Envelopment
Analysis, Ant Colony Optimization and Gene Expression Programming ................................................. 45
Chih-Ming Hsu
Emerging Technologies: LTE vs. WiMAX ................................................................................................... 66
Mohammad Arifin Rahman Khan and Md. Sadiq Iqbal
Introducing E-Maintenance 2.0 ................................................................................................................. 80
Abdessamad Mouzoune and Saoudi Taibi
Detection of Clones in Digital Images........................................................................................................ 91
Minati Mishra and Flt. Lt. Dr. M. C. Adhikary
The Significance of Genetic Algorithms in Search, Evolution, Optimization and Hybridization: A Short
Review ...................................................................................................................................................... 103
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IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Sequences classification based on group technology for flexible manufacturing...eSAT Journals
Abstract Flexible cell formation is based on Group Technology. Group Technology rests on the exploitation of resemblances between products or processes, which makes the identification of products’ families and machines’ cells easier. We propose a new approach based on the language theory for product family grouping according to their manufacturing sequences. This approach uses linear sequences of the manufacturing products which are assimilated to the words of a language. We have chosen the Levenhstein distance for sequence classification. We are going to compare our method to Dice-Czekanowski and Jaccard’s methods and apply the vectorial correlation coefficient as a comparison tool between two hierarchical classifications. Keywords: manufacturing sequences, language theory, hierarchical classification, Group Technology.
International Journal of Engineering Research and Development (IJERD)IJERD Editor
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal
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.
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.
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
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.
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.
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
1. ASSIGNMENT-FALL 2013
Name:
__Narinder Kumar_________________________
Registration No:
__511225739_____________________________
Learning Center:
__ Artex Informatic ________________________
Learning Center Code:
__1688__________________________________
Course:
__ MCA _________________________________
Subject:
COMPUTER BASED OPTIMIZATION METHODS
Semester:
__IV____________________________________
Subject Code:
__MC0079_______________________________
Date of submission:
__8 December 2013________________________
Marks awarded:
________________________________________
Directorate of Distance Education
Sikkim Manipal University
II Floor, Syndicate House
Manipal – 576 104
Signature of Coordinator
Signature of Center
Signature of Evaluator
Q1:- Discuss the various application domains of Operations Research.
2. Ans:- The application of Operations research methods helps in making decisions in such
complicated situations. Evidently the main objective of Operations research is to provide a
scientific basis to the decision-makers for solving the problems involving the interaction of
various components of organization, by employing a team of scientists from different
disciplines, all working together for finding a solution which is the best in the interest of the
organization as a whole. The solution thus obtained is known as optimal solution or decision.
A few examples of applications in which operations research is currently used include:
� Designing the layout of a factory for efficient flow of materials.
� Constructing a telecommunications network at low cost while still guaranteeing QoS
(quality of service).
� Road traffic management and 'one way' street allocations i.e. allocation problems.
� Designing the layout of a computer chip to reduce manufacturing time and hence
reducing cost.
� Managing the flow of raw materials and products in a supply chain based on uncertain
demand for the finished products.
� Roboticizing human-driven operations processes.
� Globalizing operations processes in order to take advantage of cheaper materials, labor,
land or other productivity inputs.
� Scheduling:
- personnel staffing
- manufacturing steps
- Network data traffic: these are known as queueing systems.
- Sports events and their television coverage.
� Blending of raw materials in oil refineries.
� Operations research is also used extensively by various governments in Defence
Operations, Industry, Planning, Agriculture, Hospitals, Transport, Research and
Development etc.
Q2:- Explain Erlang family of distributions of service times.
Ans:- The Erlang distribution is a continuous probability distribution with wide
applicability primarily due to its relation to the exponential and Gamma distributions. The
Erlang distribution was developed by A. K. Erlang to examine the number of telephone calls
which might be made at the same time to the operators of the switching stations. This work on
telephone traffic engineering has been expanded to consider waiting times in queuing
systems in general. The distribution is now used in the fields of stochastic processes and
of biomathematics.
Probability density function
The probability density function of the Erlang distribution is
The parameter is called the shape parameter and the parameter is called the rate
parameter. An alternative, but equivalent, parameterizations (gamma distribution) uses the
scale parameter
which is the reciprocal of the rate parameter (i.e.,
):
3. When the scale parameter equals 2, the distribution simplifies to the chi-squared distribution
with 2k degrees of freedom. It can therefore be regarded as a generalized chi-squared
distribution, for even degrees of freedom.
Because of the factorial function in the denominator, the Erlang distribution is only defined
when the parameter k is a positive integer. In fact, this distribution is sometimes called the
Erlang-k distribution (e.g., an Erlang-2 distribution is an Erlang distribution with k=2). The
Gamma distribution generalizes the Erlang by allowing to be any real number, using the
gamma function instead of the factorial function.
Cumulative distribution function (CDF)
The cumulative distribution function of the Erlang distribution is:
where
is the lower incomplete gamma function. The CDF may also be
expressed as
Q3:- Explain the algorithm for solving a linear programming problem by
graphical method.
Ans:- Linear programming (LP) is a mathematical method for determining a way to achieve
the best outcome (such as maximum profit or lowest cost) in a given mathematical model for
some list of requirements represented as linear equations. More formally, linear programming
is a technique for the optimization of a linear objective function, subject to linear equality and
linear inequality constraints. Given a polytope and a real-valued affine function defined on
this polytope, a linear programming method will find a point on the polytope where this
function has the smallest (or largest) value if such point exists, by searching through the
polytope vertices.
An Algorithm for solving a linear programming problem by Graphical Method:(This
algorithm can be applied only for problems with two variables).
Step – I: Formulate the linear programming problem with two variables (if the given
problem has more than two variables, then we cannot solve it by graphical method).
Step – II: Consider a given inequality. Suppose it is in the form a1x1 + a2x2 <= b (or a1x1 +
a2x2 >= b). Then consider the relation a1x1+ a2x2= b. Find two distinct points (k, l), (c, d)
that lie on the straight line a1x1+ a2x2= b. This can be found easily: If x1= 0, then x2 = b /
a2.If x2=0, then x1 = b / a1. Therefore (k, l) = (0, b / a2) and (c, d) = (b / a1, 0) are two points
on the straight line a1x1+a2x2= b.
Step – III: Represent these two points (k, l), (c, d) on the graph which denotes X–Y-axis
plane. Join these two points and extend this line to get the straight line which represents
a1x1+ a2x2= b.
4. Step – IV: a1x1 + a2x2= b divides the whole plane into two half planes, which are a1x1+
a2x2 <= b (one side) and a1x1+ a2x2 >= b (another side). Find the half plane that is related to
the given inequality.
Step – V: Do step-II to step-IV for all the inequalities given in the problem. The intersection
of the half-planes related to all the inequalities and x1 >= 0,x2 >= 0 , is called the feasible
region (or feasible solution space). Now find this feasible region.
Step – VI: The feasible region is a multisided figure with corner points A, B,C, … (say). Find
the co-ordinates for all these corner points. These corner points are called as extreme points.
Step – VII: Find the values of the objective function at all these corner/extreme points.
Step – VIII: If the problem is a maximization (minimization) problem, then the maximum
(minimum) value of z among the values of z at the corner/extreme points of the feasible
region is the optimal value of z. If the optimal value exists at the corner/extreme point, say A
(u, v), then we say that the solution x1= u and x2= v is an optimal feasible solution.
Step – IX: Write the conclusion (that include the optimum value of z, and the co-ordinates of
the corner point at which the optimum value of z exists).
Q4:- Explain the use of finite queuing tables.
Ans:- There will be cases, where the possible number of arrivals is limited and is relatively
small. In a production shop, if the machines are considered as customers requiring service
from repair crews or operators, the population is restricted to the total number of machines in
the shop. In a hospital ward, the probability of the doctors or nurses being called for service is
governed by the number of beds in the ward. Similarly, in an aircraft the number of seats is
finite and the number of stewardesses provided by the airlines will be based on the
consideration of the maximum number of passengers who can demand service. As in the case
of a queuing system with infinite population, the efficiency of the system can be improved in
tens of reducing the average length of queues, average waiting time and time spent by the
customer in the system by increasing the number of service channels. However, such
increases mean additional cost and will have to be balanced with the benefits likely to accrue.
If the queuing system in a machine shop is under study, the cost of providing additional
maintenance crews or operators can be compared with the value of additional production
possible due to reduced downtime of the machines` In cases where it is not possible to
quantify the benefits, the management will have to base its decisions on the desired standards
for customer service
The queue discipline in a finite queuing process can be:
i)
First come-first served
ii)
Priority e.g.: Machines of high cost may be given priority for maintenance while
others may be kept waiting even if they had broken down before.
iii)
Random e.g.; in a machine shop if a single operator is attending to several
machines and several machines call for his attention at a time, he may attend first
to the one nearest to him.
The analysis of Finite Queuing Models is more complex than those with infinite population
although the approach is similar.
5. Notations: Notations used are different and are given below:
The queue discipline in a finite queuing process can be:
i)
First come-first served
ii)
Priority e.g.: Machines of high cost may be given priority for maintenance while
others may be kept waiting even if they had broken down before.
iii)
Random e.g.; in a machine shop if a single operator is attending to several
machines and several machines call for his attention at a time, he may attend first
to the one nearest to him.
The analysis of Finite Queuing Models is more complex than those with infinite population
although the approach is similar.
Notations: Notations used are different and are given below:
(i)
Find mean service timeTand mean running tirne U.
(ii)
Compute the service factor
(iii)
Select the table corresponding to the population N.
(iv)
For the given population, locate the service factor value.
(v)
Read off from tables, values of D and F for the number of service crews M.
If necessary, these values may be interpolated between relevant values of X. (vi) Calculate the
other measures L W, H, and J from the formulae given.
The overall efficiency F of the system will increase with the number of service channels (M)
provided. As mentioned earlier, addition of service crews involves cost, which should be
justified by the increase in the efficiency of the system i.e. additional running time of
machines possible. However it will be seen from the tables that as M increase the rate 0f
increase in efficiency decreases. The practical significance is that beyond a certain value of
M, it is not worthwhile increasing M as there would be no appreciable increase in the
efficiency of the system.
Q5:- Customers arrive at a small post office at the rate of 30 per hour.
Service by the clerk on duty takes an average of 1 minute per
customer
1) Calculate the mean customer time.
a. Spent waiting in line
b. Spent receiving or waiting for service.
2) Find the mean number of persons
a. in line (ii) Receiving or waiting for service
Ans:- Mean arrival rate λ = 30 customers per hour = ½ customer per minute.
Mean service rate λ = 1 per minute
Traffic intensity
P
= λ/µ
=½
a) Mean Customer Time
a. Spent waiting in line
6. E(w) = λ/( µ(µ - λ))
= 1 minute
b. Spent receiving or waiting for service
W(v) = 1/( µ - λ)
= 2 minutes
b) Find the mean number of persons
a. in line
E(m) = λ2/( µ(µ - λ))
= ½ Customer
b. Receiving or waiting for service
E(n)
= λ /( µ - λ)
= 1 Customer
Q6:- Explain the terms
I.
Saddle point
II.
Max-min and min-max principle
Ans:- I. Saddle point: For any game, if the maxi-min and the mini-max are equal, then such
games are said to have a saddle point
Steps to detect a Saddle Point
Step (1): At the right of each row, write the row minimum and ring the largest of them.
Step (2): At the bottom of each column, write the column maximum and ring the smallest
of them.
Step (3): If these two elements are same, the cell where the corresponding row and column
meet is a saddle point and the element in that cell is the value of the game.
Step (4): If the two ringed elements ate unequal, there is no saddle point, and the value of
the game lies between these two values.
Step (5): If there are more than one saddle points then there will be more than one solution,
each solution corresponding to each saddle point.
Max-min and min-max principle: Suppose player A and player B are to play a game
without knowing what the other player would do. However, player A would like to
maximize his profit and player B would like to minimize his loss. And thus, each player
would expect his opponent to be calculative.
Suppose player A plays A1. Then, his gain would be a11, a12, � a1n according as B �s
choice is B1, B2, �. Bn. Let ?1 = min {a11, a12,�a1n}. Then, ?1 is the minimum gain of
A when he plays A1. (Here, ?1 is the minimum pay-off in the first row.) Similarly, if A
plays A2, his minimum gain is ?2 which is the least pay-off in the second row. Thus
proceeding, we find that corresponding to A�s play A1, A2, �.. Am, the minimum gains
are the row minimums
. Suppose A chooses that course of which
is
maximum. This maxi-mum of the row minimum in the pay-off matrix is called maxi-min.
The maxi-min is
Similarly, by taking the minimum of the column maximums in the pay-off matrix is called
mini-max.