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
Pedagogy of Mathematics - Part II (Numbers and Sequence - Ex 2.6), Numbers and Sequence, Maths, IX std Maths, Samacheerkalvi maths, II year B.Ed., Pedagogy, Mathematics, series, Sum to n terms of an A.P.,
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Mathematics Grade 10 Learner's Module (2015). Department of Education
A Mathematical Model for the Enhancement of Stress Induced Hypoglycaemia by A...IJRES Journal
The normal distribution is a very commonly occurring continuous probability distribution. In this paper the Multivariate Normal distribution is used for finding the mgf of the curve for the enhancement of stress induced Hypoglycaemia with consideration of the variablesProlactin, ACTH, Growth Hormone, Blood Pressure, Plasma Glucose, Plasma Renin, Epinephrine, Cortisol. These variables are treated with the drugs (citalopram and tianeptine) and the joint moment generating function for the variables in Citalopram, Tianeptineand Placebo casesare found out and are given as curves in the Mathematical Results
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.
Design and verification of pipelined parallel architecture implementation in ...eSAT Publishing House
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
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.
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
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.
Pedagogy of Mathematics - Part II (Numbers and Sequence - Ex 2.6), Numbers and Sequence, Maths, IX std Maths, Samacheerkalvi maths, II year B.Ed., Pedagogy, Mathematics, series, Sum to n terms of an A.P.,
Geometric Series and Finding the Sum of Finite Geometric SequenceFree Math Powerpoints
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References:
Nivera, G. C. (2015), Grade 10 Mathematics: Pattern and Practicalities. Don Bosco Press Inc. Makati City, Philippines.
Mathematics Grade 10 Learner's Module (2015). Department of Education
A Mathematical Model for the Enhancement of Stress Induced Hypoglycaemia by A...IJRES Journal
The normal distribution is a very commonly occurring continuous probability distribution. In this paper the Multivariate Normal distribution is used for finding the mgf of the curve for the enhancement of stress induced Hypoglycaemia with consideration of the variablesProlactin, ACTH, Growth Hormone, Blood Pressure, Plasma Glucose, Plasma Renin, Epinephrine, Cortisol. These variables are treated with the drugs (citalopram and tianeptine) and the joint moment generating function for the variables in Citalopram, Tianeptineand Placebo casesare found out and are given as curves in the Mathematical Results
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.
Design and verification of pipelined parallel architecture implementation in ...eSAT Publishing House
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
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.
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
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.
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.
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.
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
Nanoparticle based charge trapping memory device applying mos technology a co...eSAT Publishing House
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.
Effect of superplasticizers compatibility on the workability, early age stren...eSAT Publishing House
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
Importance of post processing for improved binarization of text documentseSAT Publishing House
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.
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.
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.
Umts femto access point for higher data rate and better quality of service to...eSAT Publishing House
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.
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.
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.
Building extraction from remote sensing imageries by data fusion techniqueseSAT Publishing House
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.
The green aggregates for sustainable development in construction industryeSAT Publishing House
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
Partial differentiation, total differentiation, Jacobian, Taylor's expansion, stationary points,maxima & minima (Extreme values),constraint maxima & minima ( Lagrangian multiplier), differentiation of implicit functions.
Unlock a deep understanding of mathematics with our Module and Summary! Clear definitions, comprehensive discussions, relevant example problems, and step-by-step solutions will guide you through mathematical concepts effortlessly. Learn with a systematic approach and discover the magic in every step of your learning journey. Mathematics doesn't have to be complicated—let's make it simple and enjoyable!
Least Square Plane and Leastsquare Quadric Surface Approximation by Using Mod...IOSRJM
Now a days Surface fitting is applied all engineering and medical fields. Kamron Saniee ,2007 find a simple expression for multivariate LaGrange’s Interpolation. We derive a least square plane and least square quadric surface Approximation from a given N+1 tabular points when the function is unique. We used least square method technique. We can apply this method in surface fitting also.
Direct solution of sparse network equations by optimally ordered triangular f...Dimas Ruliandi
Triangular factorization method of a power network problem (in form of matrix). Direct solution can be found without calculating inverse matrix which usually considered an exhaustive method, especially in large scale network.
Comparative analysis of x^3+y^3=z^3 and x^2+y^2=z^2 in the Interconnected Sets Vladimir Godovalov
This paper introduces an innovative technique of study z^3-x^3=y^3 on the subject of its insolvability in integers. Technique starts from building the interconnected, third degree sets: A3={a_n│a_n=n^3,n∈N}, B3={b_n│b_n=a_(n+1)-a_n }, C3={c_n│c_n=b_(n+1)-b_n } and P3={6} wherefrom we get a_n and b_n expressed as figurate polynomials of third degree, a new finding in mathematics. This approach and the results allow us to investigate equation z^3-x^3=y in these interconnected sets A3 and B3, where z^3∧x^3∈A3, y∈B3. Further, in conjunction with the new Method of Ratio Comparison of Summands and Pascal’s rule, we finally prove inability of y=y^3. After we test the technique, applying the same approach to z^2-x^2=y where we get family of primitive z^2-x^2=y^2 as well as introduce conception of the basic primitiveness of z^'2-x^'2=y^2 for z^'-x^'=1 and the dependant primitiveness of z^'2-x^'2=y^2 for co-prime x,y,z and z^'-x^'>1.
ملزمة الرياضيات للصف السادس التطبيقي الفصل الاول الاعداد المركبة 2022anasKhalaf4
طبعة جديدة ومنقحة
حل تمارين الكتاب
شرح المواضيع الرياضية بالتفصيل وبأسلوب واضح ومفهوم لجميع المستويات
حلول الاسألة الوزارية
اعداد الدكتور أنس ذياب خلف
email: anasdhyiab@gmail.com
Similar to Game theory problems by an alternative simplex method (20)
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Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
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About
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Game theory problems by an alternative simplex method
1. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
_______________________________________________________________________________________
Volume: 03 Issue: 05 | May-2014, Available @ http://www.ijret.org 900
GAME THEORY PROBLEMS BY AN ALTERNATIVE SIMPLEX
METHOD
Kirtiwant P. Ghadle1
, Tanaji S. Pawar2
1
Associate Professor, Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad,
Maharashtra, India
2
Research Student, Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad,
Maharashtra, India
Abstract
In this paper, an alternative method for the solution of game problems is introduced. This method is easy to solve game problem
which does not have a saddle point. It is powerful method to reduce number of iterations and save valuable time.
Keywords: Linear programming problem, Optimal solution, Alternative simplex method, and Game problem.
--------------------------------------------------------------------***------------------------------------------------------------------
1. INTRODUCTION
Today’s life is a full of struggle and competitions. A great
variety of competitive situations is commonly seen. What
should be the bid to win a big government job in the pace of
competition from several jobs? Game must be thought of, in
abroad sense, not as a kind of sport but as competitive
situation, a kind of conflict in which somebody must win
and somebody must lose.
John Von Neumann suggestion is to solve the game theory
problems on the maximum losses. Dantzig [1] suggestion is
to choose that entering vector corresponding to which
𝑧𝑗 − 𝑐𝑗 is most negative. Khobragade et al. [2, 3, 4]
suggestion is to choose that entering vector corresponding to
which
( 𝑧 𝑗 − 𝑐 𝑗 ) 𝜃 𝑗
𝑐 𝑗
is most negative.
In this paper, an attempt has been made to solve linear
programming problem (LPP) by new method which is an
alternative for simplex method. This method is different
from Khobragade et al. Method.
2. SOLUTION OF M x N RECTANGULAR
GAME PROBLEM
By fundamental theorem of rectangular games, if mixed
strategies are allowed, there always exists a value of game.
( i.e. 𝑉 = 𝑉 = 𝑉 ).
Let the two person zero sum game be defined as follows:
Player B
Player A
𝑎11 ⋯ 𝑎1𝑛
⋮ ⋱ ⋮
𝑎 𝑚1 ⋯ 𝑎 𝑚𝑛
Let 𝑝1, 𝑝2, … , 𝑝 𝑚 and 𝑞1, 𝑞2, … , 𝑞 𝑛 be the probabilities of
two players A and B, to select their pure strategies. i.e.
𝑆𝐴 = 𝑝1, 𝑝2, … , 𝑝 𝑚 and 𝑆 𝐵 = (𝑞1, 𝑞2, … , 𝑞 𝑛 ).
Then 𝑝1 + 𝑝2 + 𝑝3 + … + 𝑝 𝑚 = 1
and 𝑞1 + 𝑞2 + 𝑞3 + … + 𝑞 𝑛 = 1,
Where 𝑝𝑖 ≥ 0 and 𝑞𝑗 ≥ 0 for all 𝑖, 𝑗.
Let the game can be defined by LPP as given below:
For player A: Minimize 𝑋 = 𝑥1 + 𝑥2 + ⋯ + 𝑥 𝑚 or =
1
𝑉
Subject to: 𝑎11 𝑥1 + 𝑎21 𝑥2 + ⋯ + 𝑎 𝑚1 𝑥 𝑚 ≥ 1
𝑎12 𝑥1 + 𝑎22 𝑥2 + ⋯ + 𝑎 𝑚2 𝑥 𝑚 ≥ 1
..................................................
..................................................
𝑎1𝑛 𝑥1 + 𝑎2𝑛 𝑥2 + ⋯ + 𝑎 𝑚𝑛 𝑥 𝑚 ≥ 1
𝑥1, 𝑥2, … , 𝑥 𝑚 ≥ 0.
For Player B: Maximize 𝑌 = 𝑦1 + 𝑦2 + ⋯ + 𝑦𝑛 or =
1
𝑉
Subject to: 𝑎11 𝑦1 + 𝑎12 𝑦2 + ⋯ + 𝑎1𝑛 𝑦 𝑚 ≤ 1
𝑎21 𝑦1 + 𝑎22 𝑦2 + ⋯ + 𝑎2𝑛 𝑦 𝑚 ≤ 1
..................................................
..................................................
𝑎 𝑚1 𝑦1 + 𝑎 𝑚2 𝑦2 + ⋯ + 𝑎 𝑚𝑛 𝑦 𝑚 ≤ 1
𝑥1, 𝑥2, … , 𝑥 𝑚 ≥ 0.
To find the optimal solution of the above LPP, it has been
observed that the player B’s problem is exactly the dual of
the player A’s problem. The optimal solution of one
problem will automatically give the optimal solution to the
other. The player B’s problem can be solved by an
alternative simplex method while player A’s problem can be
solved by an alternative dual simplex method [7].
2. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
_______________________________________________________________________________________
Volume: 03 Issue: 05 | May-2014, Available @ http://www.ijret.org 901
3. SOLVED PROBLEMS
3.1: Problem 1
Solve the following game by linear programming technique:
Player B
Player A
1 −1 −1
−1 −1 3
−1 2 −1
.
Solution: The given game has 𝑉= -1, 𝑉 = 1. It does not
possess the saddle point and the value of game -1 < V < 1.
Adding a suitable constant k = 1 to all the elements of
payoff of matrix.
Player B
Player A
2 0 0
0 0 4
0 3 0
Let the strategies of two players be:
𝑆𝐴 = 𝑝1, 𝑝2, 𝑝3 and 𝑆 𝐵 = (𝑞1, 𝑞2, 𝑞3), where 𝑝1 + 𝑝2 +
𝑝3 = 1, 𝑞1 + 𝑞2 + 𝑞3 = 1.
Then, the linear programming problem can be written as:
For player A: Minimize 𝑋 = 𝑥1 + 𝑥2 + 𝑥3 or =
1
𝑉
Subject to: 2𝑥1 ≥ 1
3𝑥3 ≥ 1
4𝑥2 ≥ 1.
𝑥1, 𝑥2, 𝑥3 ≥ 0.
For Player B:
Maximize 𝑌 = 𝑦1 + 𝑦2 + 𝑦3 or =
1
𝑉
Subject to: 2𝑦1 ≤ 1
4𝑦3 ≤ 1
3𝑦2 ≤ 1.
𝑦1, 𝑦2, 𝑦3 ≥ 0.
LPP is in standard form for player B:
Maximize 𝑌 = 𝑦1 + 𝑦2 + 𝑦3 or =
1
𝑉
Subject to: 2𝑦1 + 𝑠1 = 1
4𝑦3 + 𝑠2 = 1
3𝑦2 + 𝑠3 = 1.
𝑦1, 𝑦2, 𝑦3, 𝑠1, 𝑠2, 𝑠3 ≥ 0.
Where 𝑠1, 𝑠2, 𝑠3 are slack variables.
Simplex Table:
𝐶 𝐵 BVS 𝑋 𝐵 𝑦1 𝑦2 𝑦3 𝑠1 𝑠2 𝑠3 Ratio
0 𝑠1 1 2 0 0 1 0 0 -
0 𝑠2 1 0 0 4 0 1 0 1/4→
0 𝑠3 1 0 3 0 0 0 1 -
0 𝑠1 1 2 0 0 1 0 0 -
1 𝑦3 1/4 0 0 1 0 1/4 0 -
0 𝑠3 1 0 3 0 0 0 1 1/3→
0 𝑠1 1 2 0 0 1 0 0 1/2→
1 𝑦3 1/4 0 0 1 0 1/4 0 -
1 𝑦2 1/3 0 1 0 0 0 1/3 -
1 𝑦1 1/2 1 0 0 1/2 0 0
1 𝑦3 1/4 0 1 0 0 1/4 0
1 𝑦2 1/3 0 0 1 0 0 1/3
Since all rows and column are ignored, hence an optimum
solution has been reached. Therefore optimum solution is:
𝑦1 =
1
2
, 𝑦2 =
1
3
, 𝑦3 =
1
4
. Max. 𝑌 =
13
12
.
Thus, the optimal strategies for player B are:
𝑞1 =
𝑦1
𝑌
=
1
2
13
12
=
6
13
,
𝑞2 =
𝑦2
𝑌
=
1
3
12
13
=
4
13
,
𝑞3 =
𝑦3
𝑌
=
1/4
12/13
=
3
13
and 𝑉 =
1
𝑌
− 𝑘 =
12
13
− 1 = −
1
13
.
The optimal strategies for player A are obtained from final
table of the above problem. This is given by duality rules:
𝑋 = 𝑌 =
13
12
,
𝑥1 = 1 1 1
1/2
0
0
=
1
2
,
𝑥2 = 1 1 1
0
1/4
0
=
1
4
,
𝑥3 = 1 1 1
0
0
1/3
=
1
3
.
Hence,
𝑝1 =
𝑥1
𝑋
=
1
2
13
12
=
6
13
, 𝑝2 =
𝑥2
𝑋
=
1
4
13
12
=
3
13
,
𝑝3 =
𝑥3
𝑋
=
1/3
13/12
=
4
13
and 𝑉 =
12
13
− 1 =
−1
13
.
3. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
_______________________________________________________________________________________
Volume: 03 Issue: 05 | May-2014, Available @ http://www.ijret.org 902
3.2: Problem 2:
Two companies P and Q are competing for the same
product. Their different strategies are given in the following
payoff matrix:
Player Q
Player P
1 2 4
2 2 1
3 1 1
.
Use linear programming to determine the best strategies for
both the companies.
Solution: The given game has 𝑉= 1, 𝑉 = 2. It does not
possess the saddle point and the value of game 1 < V < 2.
Here, the payoff matrix is already non-negative.
Let the strategies of two players be:
𝑆𝐴 = 𝑝1, 𝑝2, 𝑝3 and 𝑆 𝐵 = (𝑞1, 𝑞2, 𝑞3), where 𝑝1 + 𝑝2 +
𝑝3 = 1, 𝑞1 + 𝑞2 + 𝑞3 = 1.
Then, the linear programming problem can be written as:
For player A:
Minimize 𝑋 = 𝑥1 + 𝑥2 + 𝑥3 or =
1
𝑉
Subject to: 𝑥1 + 2𝑥2 + 3𝑥3 ≥ 1
2𝑥1 + 2𝑥2 + 𝑥3 ≥ 1
4𝑥1 + 𝑥2 + 𝑥3 ≥ 1.
𝑥1, 𝑥2, 𝑥3 ≥ 0.
For Player B:
Maximize 𝑌 = 𝑦1 + 𝑦2 + 𝑦3 or =
1
𝑉
Subject to: 𝑦1 + 2𝑦2 + 4𝑦3 ≤ 1
2𝑦1 + 2𝑦2 + 𝑦3 ≤ 1
3𝑦1 + 𝑦2 + 𝑦3 ≤ 1.
𝑦1, 𝑦2, 𝑦3 ≥ 0.
LPP is in standard form for player B:
Maximize 𝑌 = 𝑦1 + 𝑦2 + 𝑦3 or =
1
𝑉
Subject to: 𝑦1 + 2𝑦2 + 4𝑦3 + 𝑠1 = 1
2𝑦1 + 2𝑦2 + 𝑦3 + 𝑠2 = 1
3𝑦1 + 𝑦2 + 𝑦3 + 𝑠3 = 1.
𝑦1, 𝑦2, 𝑦3, 𝑠1, 𝑠2, 𝑠3 ≥ 0.
Where 𝑠1, 𝑠2, 𝑠3 are slack variables.
Simplex Table:
𝐶 𝐵 BVS 𝑋 𝐵 𝑦1 𝑦2 𝑦3 𝑠1 𝑠2 𝑠3 Ratio
0 𝑠1 1 1 2 4 1 0 0 1/4→
0 𝑠2 1 2 2 1 0 1 0 1
0 𝑠3 1 3 1 1 0 0 1 1
1 𝑦3 1/4 1/4 1/2 1 1/2 0 0
0 𝑠2 3/4 7/4 3/2 0 -1/4 1 0 3/7
0 𝑠3 3/4 11/4 1/2 0 -1/4 0 1 3/11→
1 𝑦3 2/11 0 5/11 1 3/11 0 -
1/11
0 𝑠2 3/11 0 13/11 0 -
1/11
1 -
7/11
3/13→
1 𝑦1 3/11 1 2/11 0 -
1/11
0 4/11
1 𝑦3 1/13 0 0 1 4/13 -5/13 2/13
1 𝑦2 3/13 0 1 0 -
1/13
11/13 -
7/13
1 𝑦1 3/13 1 0 0 -
1/13
-2/13 6/13
Since all rows and column are ignored, hence an optimum
solution has been reached. Therefore optimum solution is:
𝑦1 =
3
13
, 𝑦2 =
3
13
, 𝑦3 =
1
13
. Max. 𝑌 =
7
13
.
Thus, the optimal strategies for player B are:
𝑞1 =
𝑦1
𝑌
=
3
13
7
13
=
3
7
,
𝑞2 =
𝑦2
𝑌
=
3
13
7
13
=
3
7
,
𝑞3 =
𝑦3
𝑌
=
1/13
7/13
=
1
7
and 𝑉 =
1
𝑌
− 𝑘 =
13
7
− 0 =
13
7
.
The optimal strategies for player A are obtained from final
table of the above problem. This is given by duality rules:
𝑋 = 𝑌 =
7
13
,
𝑥1 = 1 1 1
4/13
−1/13
−1/13
=
2
13
,
𝑥2 = 1 1 1
−5/13
11/13
−2/13
=
4
13
,
𝑥3 = 1 1 1
2/13
−7/13
6/13
=
1
13
.
Hence,
𝑝1 =
𝑥1
𝑋
=
2
13
7
13
=
2
7
, 𝑝2 =
𝑥2
𝑋
=
4
13
7
13
=
4
7
,
𝑝3 =
𝑥3
𝑋
=
1/13
7/13
=
1
7
and 𝑉 =
13
7
− 0 =
13
7
.
4. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
_______________________________________________________________________________________
Volume: 03 Issue: 05 | May-2014, Available @ http://www.ijret.org 903
3.3: Problem 3:
Solve the following game by linear programming technique:
Player B
Player A
3 −1 −3
−3 3 −1
−4 −3 3
.
Solution: The given game has 𝑉= -3, 𝑉 = 3. It does not
possess the saddle point and the value of game -3 < V < 3.
To make this payoff matrix as positive, adding a suitable
constant k = 5 to all the elements of payoff of matrix.
Player B
Player A
8 4 2
2 8 4
1 2 8
Let the strategies of two players be:
𝑆𝐴 = 𝑝1, 𝑝2, 𝑝3 and 𝑆 𝐵 = (𝑞1, 𝑞2, 𝑞3), where 𝑝1 + 𝑝2 +
𝑝3 = 1, 𝑞1 + 𝑞2 + 𝑞3 = 1.
Then, the linear programming problem can be written as:
For player A: Minimize 𝑋 = 𝑥1 + 𝑥2 + 𝑥3 or =
1
𝑉
Subject to: 8𝑥1 + 2𝑥2 + 𝑥3 ≥ 1
4𝑥1 + 8𝑥2 + 2𝑥3 ≥ 1
2𝑥1 + 4𝑥2 + 8𝑥3 ≥ 1.
𝑥1, 𝑥2, 𝑥3 ≥ 0.
For Player B:
Maximize 𝑌 = 𝑦1 + 𝑦2 + 𝑦3 or =
1
𝑉
Subject to: 8𝑦1 + 4𝑦2 + 3𝑦3 ≤ 1
2𝑦1 + 8𝑦2 + 4𝑦3 ≤ 1
𝑦1 + 2𝑦2 + 8𝑦3 ≤ 1.
𝑦1, 𝑦2, 𝑦3 ≥ 0.
LPP is in standard form for player B:
Maximize 𝑌 = 𝑦1 + 𝑦2 + 𝑦3
Subject to: 8𝑦1 + 4𝑦2 + 3𝑦3 + 𝑠1 = 1
2𝑦1 + 8𝑦2 + 4𝑦3 + 𝑠2 = 1
𝑦1 + 2𝑦2 + 8𝑦3 + 𝑠3 = 1.
𝑦1, 𝑦2, 𝑦3, 𝑠1, 𝑠2, 𝑠3 ≥ 0.
where 𝑠1, 𝑠2, 𝑠3 are slack variables.
Simplex Table:
𝐶 𝐵 B
V
S
𝑋 𝐵 𝑦1 𝑦2 𝑦3 𝑠1 𝑠2 𝑠3 Ratio
0 𝑠1 1 8 4 2 1 0 0 1/8→
0 𝑠2 1 2 8 4 0 1 0 1/2
0 𝑠3 1 1 2 8 0 0 1 1
1 𝑦1 1/8 1 1/2 1/4 1/8 0 0
0 𝑠2 3/4 0 7 7/2 -1/4 1 0 3/14
0 𝑠3 7/8 0 3/2 31/4 -1/8 0 1 7/62→
1 𝑦1 3/31 1 14/31 0 4/31 0 -1/31
0 𝑠2 11/31 0 196/31 0 -6/31 1 - 11/196
14/31 →
1 𝑦3 7/62 0 6/31 1 -1/62 0 4/31
1 𝑦1 1/14 1 0 0 1/7 -1/14 0
1 𝑦2 11/196 0 1 0 -3/98 31/196 -1/14
1 𝑦3 5/49 0 0 1 -1/98 -3/98 1/7
Since all rows and column are ignored, hence an optimum
solution has been reached. Therefore optimum solution is:
𝑦1 =
1
14
, 𝑦2 =
11
196
, 𝑦3 =
5
49
. Max. 𝑌 =
45
196
.
Thus, the optimal strategies for player B are:
𝑞1 =
𝑦1
𝑌
=
1
14
45
196
=
14
45
,
𝑞2 =
𝑦2
𝑌
=
11
196
45
196
=
11
45
,
𝑞3 =
𝑦3
𝑌
=
5/49
45/196
=
20
45
and 𝑉 =
1
𝑌
− 𝑘 =
196
45
− 5 = −
29
45
.
The optimal strategies for player A are obtained from final
table of the above problem. This is given by duality rules:
𝑋 = 𝑌 =
45
196
,
𝑥1 = 1 1 1
1/7
−3/98
−1/98
=
5
49
,
𝑥2 = 1 1 1
−1/14
31/196
−3/98
=
11
196
,
𝑥3 = 1 1 1
0
−1/14
1/7
=
1
14
.
Hence,
𝑝1 =
𝑥1
𝑋
=
5
49
45
196
=
20
45
, 𝑝2 =
𝑥2
𝑋
=
11
196
45
196
=
11
45
,
𝑝3 =
𝑥3
𝑋
=
1/14
45/196
=
14
45
and 𝑉 =
196
45
− 5 = −
29
45
.
5. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
_______________________________________________________________________________________
Volume: 03 Issue: 05 | May-2014, Available @ http://www.ijret.org 904
3.4: Problem 4:
Two companies P and Q are competing for the same
product. Their different strategies are given in the following
payoff matrix:
Player Q
Player P
−1 0 2
0 0 −1
1 −1 −1
.
Use linear programming to determine the best strategies for
both the companies.
Solution: The given game has 𝑉= -1, 𝑉 = 0. It does not
possess the saddle point and the value of game -1 < V < 0.
To make this payoff matrix as positive, adding a suitable
constant k = 2 to all the elements of payoff of matrix.
Player B
Player A
1 2 4
2 2 1
3 1 1
Let the strategies of two players be:
𝑆𝐴 = 𝑝1, 𝑝2, 𝑝3 and 𝑆 𝐵 = (𝑞1, 𝑞2, 𝑞3), where 𝑝1 + 𝑝2 +
𝑝3 = 1, 𝑞1 + 𝑞2 + 𝑞3 = 1.
Then, the linear programming problem can be written as:
For player A: Minimize 𝑋 = 𝑥1 + 𝑥2 + 𝑥3 or =
1
𝑉
Subject to: 𝑥1 + 2𝑥2 + 3𝑥3 ≥ 1
2𝑥1 + 2𝑥2 + 𝑥3 ≥ 1
4𝑥1 + 𝑥2 + 𝑥3 ≥ 1.
𝑥1, 𝑥2, 𝑥3 ≥ 0.
Min. 𝑍 = - Max. −𝑍 , −𝑍 = 𝑍∗
Max. 𝑍∗
= −𝑥1 − 𝑥2 − 𝑥3
Subject to: −𝑥1 − 2𝑥2 − 3𝑥3 ≤ −1
−2𝑥1 − 2𝑥2 − 𝑥3 ≤ −1
−4𝑥1 − 𝑥2 − 𝑥3 ≤ −1.
𝑥1, 𝑥2, 𝑥3 ≥ 0.
LPP is in standard form for player A:
Max. 𝑍∗
= 𝑥1 − 𝑥2 − 𝑥3
Subject to: −𝑥1 − 2𝑥2 − 3𝑥3 + 𝑠1 = −1
−2𝑥1 − 2𝑥2 − 𝑥3 + 𝑠2 = −1
−4𝑥1 − 𝑥2 − 𝑥3 + 𝑠3 = −1
𝑥1, 𝑥2, 𝑥3, 𝑠1, 𝑠2, 𝑠3 ≥ 0.
where 𝑠1, 𝑠2, 𝑠3 are slack variables.
Simplex Table:
𝐶 𝐵 BVS 𝑋 𝐵 𝑥1 𝑥2 𝑥3 𝑠1 𝑠2 𝑠3 Ratio
0 𝑠1 -1 -1 -2 -3 1 0 0 1
0 𝑠2 -1 -2 -2 -1 0 1 0 1/2
0 𝑠3 -1 -4 -1 -1 0 0 1 1/4→
0 𝑠1 -3/4 0 -7/4 -
11/4
1 0 -1/4
0 𝑠2 -1/2 0 -3/2 -1/2 0 0 -1/2
-1 𝑥1 1/4 1 1/4 1/4 0 1 -1/4
-1 𝑥3 3/11 0 7/11 1 -4/11 0 1/11
0 𝑠2 -4/11 0 -
13/11
0 -2/11 1 -5/11
-1 𝑥1 2/11 1 1/11 0 1/11 0 -3/11
-1 𝑥3 1/13 0 0 1 -6/13 7/13 -2/13
-1 𝑥2 4/13 0 1 0 2/13 -
11/13
5/13
-1 𝑥1 2/13 1 0 0 1/13 1/13 -4/13
Since all rows and column are ignored, hence an optimum
solution has been reached. Therefore optimum solution is:
𝑥1 =
2
13
, 𝑥2 =
4
13
, 𝑥3 =
1
13
. Min. 𝑋 =
7
13
.
Thus, the optimal strategies for player A are:
𝑝1 =
𝑥1
𝑋
=
2
13
7
13
=
2
7
, 𝑝2 =
𝑥2
𝑋
=
4
13
7
13
=
4
7
,
𝑝3 =
𝑥3
𝑋
=
1/13
7/13
=
1
7
and =
1
𝑋
− 𝑘 =
13
7
− 2 = −
1
7
.
The optimal strategies for player B are obtained from final
table of the above problem. This is given by duality rules:
𝑋 = 𝑌 =
7
13
,
𝑦1 = −1 −1 −1
−6/13
2/13
1/13
=
3
13
,
𝑦2 = −1 −1 −1
7/13
−11/13
1/13
=
3
13
,
𝑦3 = −1 −1 −1
−2/13
5/13
−4/13
=
1
13
.
Hence,
𝑞1 =
𝑦1
𝑌
=
3
13
7
13
=
3
7
, 𝑞2 =
𝑦2
𝑌
=
3
13
7
13
=
3
7
,
𝑞3 =
𝑦3
𝑌
=
1/13
7/13
=
1
7
and 𝑉 =
13
7
− 2 = −
1
7
.
6. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
_______________________________________________________________________________________
Volume: 03 Issue: 05 | May-2014, Available @ http://www.ijret.org 905
4. CONCLUSIONS
An alternative method for game theory problems to obtain
the solution of linear programming problem has been
derived. This technique is useful to apply on numerical
problems, reduces the labour work and save valuable time.
REFERENCES
[1]. G. B. Dantzig: Maximization of linear function of
variables subject to linear inequalities, In: 21-Ed. Koopman
Cowls Commission Monograph, 13, John Wiley and Sons,
Inc., New Yark (1951).
[2]. K. G. Lokhande, N. W. Khobragade, P. G. Khot:
Simplex Method: An Alternative Approach, International
Journal of Engineering and Innovative Technology, Volume
3, Issue 1, P: 426-428 (2013).
[3]. N. V. Vaidya, N. W. Khobragade: Solution of Game
Problems Using New Approach, IJEIT, Vol. 3, Issue 5,
2013.
[4]. N. W. Khobragade and P. G. Khot: Alternative
Approach to the Simplex Method-II, Acta Ciencia Indica,
Vol.xxx IM, No.3, 651, India (2005).
[5]. S. D. Sharma: Operation Research, Kedar Nath Ram
Nath, 132, R. G. Road, Meerut-250001 (U.P.), India.
[6]. S. I. Gass: Linear Programming, 3/e, McGraw-Hill
Kogakusha, Tokyo (1969).
[7]. K. P. Ghadle, T. S. Pawar, N. W. Khobragade: Solution
of Linear Programming Problem by New Approach, IJEIT,
Vol.3, Issue 6. Pp.301-307, 2013
BIOGRAPHIES
Mr. Tanaji S. Pawar, Research student,
Department of mathematics, Dr. Babasaheb
Ambedkar Marathwada University,
Aurangabad.
Dr. K. P. Ghadle for being M.Sc in Maths he
attained Ph.D. He has been teaching since
1996. At present he is working as Associate
Professor. Achieved excellent experiences in
Research for 15 years in the area of Boundary
value problems and its application. Published more than 45
research papers in reputed journals. Four students awarded
Ph.D Degree and four students working for award of Ph.D.
Degree under their guidance.