SlideShare a Scribd company logo
OPERATIONAL RESEARCH
Topic: Linear Programming Problem
Submitted To : Prof. Nilesh
Coordinators : Zeel Mathkiya (19)
               Dharmik Mehta (20)
               Sejal Mehta (21)
               Hirni Mewada (22)
               Varun Modi (23)
               Siddhi Nalawade (24)
DEFINITION OF LINEAR
PROGRAMMING
The Mathematical Definition of LP:
 “It is the analysis of problem in which a linear
  function of a number of variables is to maximised
  (minimised), when those variables are subject to a
  number of restraints in the form of linear
  inequalities”.
TERMINOLOGY OF LINEAR
PROGRAMMING
 A typical linear program has the following
  components
 An objective Function.
 Constraints or Restrictions.
 Non-negativity Restrictions.
TERMS USED TO DESCRIBE LINEAR
PROGRAMMING PROBLEMS
 Decision variables.
 Objective function.
 Constraints.
 Linear relationship.
 Equation and inequalities.
 Non-negative restriction.
FORMATION OF LPP
 Objective function
 Constraints
 Non-Negativity restrictions
 Solution
 Feasible Solution
 Optimum Feasible Solution
SOLVED EXAMPLE -1
 A Company manufactures 2 types of product H₁ & H₂. Both
  the product pass through 2 machines M₁,M₂.The time requires
  for processing each unit of product H₁,H₂.On each machine &
  the available capacity of each machine is given below:
        Product                                   Machine
                                        M₁                    M₂
      H₁                                3                       2
      H₂                                 2                       7
Available Capacity(hrs)                1800                   1400
 The availability of materials is sufficient to product 350 unit of
  H₁ & 150 of H₂.Each unit of H₁ gives a profit of Rs.25,each
  unit of H₂ gives profit of Rs.20.Formulate the above problem
  as LPP.
SOLUTION
 From manufactures point of view we need to maximise the
  profit.The profit depend upon the number of unit of product H₁
  &H₂ produced.
Let x₁= no of unit of H₁ produce
     x₂=no of unit of H₂ produce
                        x₁ ≥ 0  1
                        x₂ ≥ 0  2
                 3x₁ + 2x₂ ≤ 1800 3
                 2x₁ + 7x₂ ≤ 1400 4
                 Z= 25x₁ + 20x₂
       LPP is formed as follows:
Maximise Z= 25x₁ + 20x₂
CONTI…..
 Subject to:
                       x₁ ≥ 0
                       x₂ ≥ 0
                3x₁ + 2x₂ ≤ 1800
                2x₁ + 7x₂ ≤ 1400
CONTI…...
 A Manager of hotel dreamland plans and extancison
 not more than 50 groups attleast 5 must be executive
 single rooms the number of executive double rooms
 should be atleast 3 times the number of executive
 single rooms. He charges Rs.3000 for executive
 double rooms and Rs.1800 executive single rooms
 per day.
CONTI…..
Formulate the above problume for LPP

SOLUTION →
  The LPP is formulated as follows ;
 Let X1 = Total No. of single executive rooms
 Let X2 = Total No. of Double executive rooms
... X1 + x2 < 50
    X1 > 5
    x2 > 3 X1
 Maximise ; Z = 1800 X1 + 3000 x2
The LPP is formulated as follows
Maximise ; Z = 1800 X1 + 3000 x

Subject to ; X1 + x2 < 50
              X1 > 5
              x 2 > 3 X1
GRAPHICAL METHOD
1. Arrive at a graphical solution for the following LPP.
Maximize Z = 40x1 + 35x2
Subject to : 2x1 + 3x2 < 60
              4x1 + 3x2 < 96
              x1 , x 2 > 0
Solution : Let us consider the equation
1) 2x1 + 3x2 = 60
Put x2 = 0: 2x1 = 60
              x1 = 30
       A = (30 , 0)
Put x1 = 0 : 3x2 = 60
              x2 = 20
       B = (0 , 20)
2) 4x1 + 3x2 < 96
Put x2 = 0 : 4x1 = 96
              x1 = 24
      C = (24 , 0)
Put x1 = 0 : 3x2 = 96
              x2 = 32
      D = (0 , 32)
Y axis
40
                        Scale : Xaxis = 1 cm = 5 units
35                              Yaxis = 1 cm = 5 units

30      D

25

20 B

15

10                  p

5
                       C   A                             X axis
    O       5 10 15 20 25 30    35 40
 OBPC is the feasible region
Points       x1     x2    z
O             0      0    z=0
B             0     20    z = 40(0) + 35 (20) = 700
P            18      8    z = 40(18) + 35(8) = 1000
C            24      0    z = 40(24) + 35(0) = 960

Thus, the optimal feasible solution is x1 = 18 , x2 = 8
and z = 1000
CONTI…..
 Find the feasible solution to following LPP
Minimize Z = 6x + 5y
Subject to = x + y > 7
            x<3,y<4
            x<0,y>0
Solution : Removing Inequality in given equation
1. x + y > 7
Put y = 0 : x = 7
Put x = 0 : y = 7
The two points are : A = (7 , 0) & B = (0 , 7)
Further,
X=3,y=4
Y axis

8

7   B                               Scale : X axis = 1cm = 1 unit
                                          Y axis = 1cm = 1 unit
6

5
                    P
4

3

2

1

                                        A
        1   2   3       4   5   6   7       8                       X axis
O
CONTI…..
 As all the 3 lines intersect each other at a common
 point P( 3 , 4) it is the feasible solution to LPP

Z = 6(3) + 5(4)
  = 18 + 20
  = 38
CONCLUSION
 Linear programming is very important
 mathematical technique which enables managers
 to arrive at proper decisions regarding his area of
 work. Thus it is very important part of operations
 research.

More Related Content

What's hot

linear programming
linear programminglinear programming
linear programming
Karishma Chaudhary
 
Integer Linear Programming
Integer Linear ProgrammingInteger Linear Programming
Integer Linear Programming
SukhpalRamanand
 
Questions of basic probability for aptitude test
Questions of basic probability for aptitude test Questions of basic probability for aptitude test
Questions of basic probability for aptitude test
Dr. Trilok Kumar Jain
 
Goal Programming
Goal ProgrammingGoal Programming
Goal Programming
Evren E
 
Sequencing
SequencingSequencing
Sequencing
Rashmi Navaghane
 
Assignment Problem
Assignment ProblemAssignment Problem
Assignment Problem
Nakul Bhardwaj
 
Simplex Method
Simplex MethodSimplex Method
Simplex Method
kzoe1996
 
Profit maximization
Profit maximizationProfit maximization
Profit maximization
Abhishek Eraiah
 
Ch3
Ch3Ch3
Linear Programming
Linear ProgrammingLinear Programming
Linear Programming
Pulchowk Campus
 
Duality
DualityDuality
Duality
Sachin MK
 
Game theory and its applications
Game theory and its applicationsGame theory and its applications
Game theory and its applications
Eranga Weerasekara
 
Transportation problem ppt
Transportation problem pptTransportation problem ppt
Transportation problem ppt
Dr T.Sivakami
 
Assignment Problem
Assignment ProblemAssignment Problem
Assignment Problem
JIMS Rohini Sector 5
 
Operations research-an-introduction
Operations research-an-introductionOperations research-an-introduction
Operations research-an-introduction
Manoj Bhambu
 
Simplex method concept,
Simplex method concept,Simplex method concept,
Simplex method concept,
Dronak Sahu
 
Chapter 19 decision-making under risk
Chapter 19   decision-making under riskChapter 19   decision-making under risk
Chapter 19 decision-making under risk
Bich Lien Pham
 
Ppt on decision theory
Ppt on decision theoryPpt on decision theory
Ppt on decision theory
Bhuwanesh Rajbhandari
 
Unit.2. linear programming
Unit.2. linear programmingUnit.2. linear programming
Unit.2. linear programming
DagnaygebawGoshme
 
Simplex Method
Simplex MethodSimplex Method
Simplex Method
Sachin MK
 

What's hot (20)

linear programming
linear programminglinear programming
linear programming
 
Integer Linear Programming
Integer Linear ProgrammingInteger Linear Programming
Integer Linear Programming
 
Questions of basic probability for aptitude test
Questions of basic probability for aptitude test Questions of basic probability for aptitude test
Questions of basic probability for aptitude test
 
Goal Programming
Goal ProgrammingGoal Programming
Goal Programming
 
Sequencing
SequencingSequencing
Sequencing
 
Assignment Problem
Assignment ProblemAssignment Problem
Assignment Problem
 
Simplex Method
Simplex MethodSimplex Method
Simplex Method
 
Profit maximization
Profit maximizationProfit maximization
Profit maximization
 
Ch3
Ch3Ch3
Ch3
 
Linear Programming
Linear ProgrammingLinear Programming
Linear Programming
 
Duality
DualityDuality
Duality
 
Game theory and its applications
Game theory and its applicationsGame theory and its applications
Game theory and its applications
 
Transportation problem ppt
Transportation problem pptTransportation problem ppt
Transportation problem ppt
 
Assignment Problem
Assignment ProblemAssignment Problem
Assignment Problem
 
Operations research-an-introduction
Operations research-an-introductionOperations research-an-introduction
Operations research-an-introduction
 
Simplex method concept,
Simplex method concept,Simplex method concept,
Simplex method concept,
 
Chapter 19 decision-making under risk
Chapter 19   decision-making under riskChapter 19   decision-making under risk
Chapter 19 decision-making under risk
 
Ppt on decision theory
Ppt on decision theoryPpt on decision theory
Ppt on decision theory
 
Unit.2. linear programming
Unit.2. linear programmingUnit.2. linear programming
Unit.2. linear programming
 
Simplex Method
Simplex MethodSimplex Method
Simplex Method
 

Similar to Linear Programming Feasible Region

Lecture - Linear Programming.pdf
Lecture - Linear Programming.pdfLecture - Linear Programming.pdf
Lecture - Linear Programming.pdf
lucky141651
 
Lenier Equation
Lenier EquationLenier Equation
Lenier Equation
Khadiza Begum
 
Decision making
Decision makingDecision making
Decision making
Dronak Sahu
 
introduction to Operation Research
introduction to Operation Research introduction to Operation Research
introduction to Operation Research
amanyosama12
 
Linear Programming.ppt
Linear Programming.pptLinear Programming.ppt
Linear Programming.ppt
Abdullah Amin
 
Linear programming graphical method
Linear programming graphical methodLinear programming graphical method
Linear programming graphical method
Dr. Abdulfatah Salem
 
181_Sample-Chapter.pdf
181_Sample-Chapter.pdf181_Sample-Chapter.pdf
181_Sample-Chapter.pdf
ThanoonQasem
 
Introduction to Operations Research/ Management Science
Introduction to Operations Research/ Management Science Introduction to Operations Research/ Management Science
Introduction to Operations Research/ Management Science
um1222
 
LP special cases and Duality.pptx
LP special cases and Duality.pptxLP special cases and Duality.pptx
LP special cases and Duality.pptx
Snehal Athawale
 
Integer Programming PPt.ernxzamnbmbmspdf
Integer Programming PPt.ernxzamnbmbmspdfInteger Programming PPt.ernxzamnbmbmspdf
Integer Programming PPt.ernxzamnbmbmspdf
Raja Manyam
 
LINEAR PROGRAMING
LINEAR PROGRAMINGLINEAR PROGRAMING
LINEAR PROGRAMING
shahzadebaujiti
 
LPP, Duality and Game Theory
LPP, Duality and Game TheoryLPP, Duality and Game Theory
LPP, Duality and Game Theory
Purnima Pandit
 
Management Science
Management ScienceManagement Science
Management Science
lisa1090
 
P
PP
Integer Programming, Goal Programming, and Nonlinear Programming
Integer Programming, Goal Programming, and Nonlinear ProgrammingInteger Programming, Goal Programming, and Nonlinear Programming
Integer Programming, Goal Programming, and Nonlinear Programming
Salah A. Skaik - MBA-PMP®
 
Linear progarmming part 1
Linear progarmming   part 1Linear progarmming   part 1
Linear progarmming part 1
Divya K
 
Linear Programming Problems : Dr. Purnima Pandit
Linear Programming Problems : Dr. Purnima PanditLinear Programming Problems : Dr. Purnima Pandit
Linear Programming Problems : Dr. Purnima Pandit
Purnima Pandit
 
Linear Programming
Linear ProgrammingLinear Programming
Linear Programming
syed_shahzad786
 
Linear Programming
Linear ProgrammingLinear Programming
Linear Programming
syed_shahzad786
 
Linear Programming
Linear ProgrammingLinear Programming
Linear Programming
syed_shahzad786
 

Similar to Linear Programming Feasible Region (20)

Lecture - Linear Programming.pdf
Lecture - Linear Programming.pdfLecture - Linear Programming.pdf
Lecture - Linear Programming.pdf
 
Lenier Equation
Lenier EquationLenier Equation
Lenier Equation
 
Decision making
Decision makingDecision making
Decision making
 
introduction to Operation Research
introduction to Operation Research introduction to Operation Research
introduction to Operation Research
 
Linear Programming.ppt
Linear Programming.pptLinear Programming.ppt
Linear Programming.ppt
 
Linear programming graphical method
Linear programming graphical methodLinear programming graphical method
Linear programming graphical method
 
181_Sample-Chapter.pdf
181_Sample-Chapter.pdf181_Sample-Chapter.pdf
181_Sample-Chapter.pdf
 
Introduction to Operations Research/ Management Science
Introduction to Operations Research/ Management Science Introduction to Operations Research/ Management Science
Introduction to Operations Research/ Management Science
 
LP special cases and Duality.pptx
LP special cases and Duality.pptxLP special cases and Duality.pptx
LP special cases and Duality.pptx
 
Integer Programming PPt.ernxzamnbmbmspdf
Integer Programming PPt.ernxzamnbmbmspdfInteger Programming PPt.ernxzamnbmbmspdf
Integer Programming PPt.ernxzamnbmbmspdf
 
LINEAR PROGRAMING
LINEAR PROGRAMINGLINEAR PROGRAMING
LINEAR PROGRAMING
 
LPP, Duality and Game Theory
LPP, Duality and Game TheoryLPP, Duality and Game Theory
LPP, Duality and Game Theory
 
Management Science
Management ScienceManagement Science
Management Science
 
P
PP
P
 
Integer Programming, Goal Programming, and Nonlinear Programming
Integer Programming, Goal Programming, and Nonlinear ProgrammingInteger Programming, Goal Programming, and Nonlinear Programming
Integer Programming, Goal Programming, and Nonlinear Programming
 
Linear progarmming part 1
Linear progarmming   part 1Linear progarmming   part 1
Linear progarmming part 1
 
Linear Programming Problems : Dr. Purnima Pandit
Linear Programming Problems : Dr. Purnima PanditLinear Programming Problems : Dr. Purnima Pandit
Linear Programming Problems : Dr. Purnima Pandit
 
Linear Programming
Linear ProgrammingLinear Programming
Linear Programming
 
Linear Programming
Linear ProgrammingLinear Programming
Linear Programming
 
Linear Programming
Linear ProgrammingLinear Programming
Linear Programming
 

More from VARUN MODI

Chauvinism
ChauvinismChauvinism
Chauvinism
VARUN MODI
 
World trade organisation
World trade organisationWorld trade organisation
World trade organisation
VARUN MODI
 
Primary research on baby care product by using quantitative research method
Primary research on baby care product by using quantitative research methodPrimary research on baby care product by using quantitative research method
Primary research on baby care product by using quantitative research method
VARUN MODI
 
Amul Inventory Techniques & Other Details
Amul Inventory Techniques & Other DetailsAmul Inventory Techniques & Other Details
Amul Inventory Techniques & Other Details
VARUN MODI
 
Cost Sheet At Corporate Level
Cost Sheet At Corporate LevelCost Sheet At Corporate Level
Cost Sheet At Corporate Level
VARUN MODI
 
Export procedure
Export procedureExport procedure
Export procedure
VARUN MODI
 
Sbi mutual gold fund
Sbi mutual gold fundSbi mutual gold fund
Sbi mutual gold fund
VARUN MODI
 
Retailing & Promotion Mix
Retailing & Promotion MixRetailing & Promotion Mix
Retailing & Promotion Mix
VARUN MODI
 
Dabur
DaburDabur
Dabur
VARUN MODI
 
Can mumbai be shanghai
Can mumbai be shanghaiCan mumbai be shanghai
Can mumbai be shanghai
VARUN MODI
 
Production planning & control (ppc)
Production planning & control (ppc)Production planning & control (ppc)
Production planning & control (ppc)
VARUN MODI
 
HOW TO CONTROL YOUR ANGER
HOW TO CONTROL YOUR ANGERHOW TO CONTROL YOUR ANGER
HOW TO CONTROL YOUR ANGER
VARUN MODI
 
Skill of successful entrepreneur
Skill of successful entrepreneurSkill of successful entrepreneur
Skill of successful entrepreneur
VARUN MODI
 
Marginal costing
Marginal costing Marginal costing
Marginal costing
VARUN MODI
 

More from VARUN MODI (14)

Chauvinism
ChauvinismChauvinism
Chauvinism
 
World trade organisation
World trade organisationWorld trade organisation
World trade organisation
 
Primary research on baby care product by using quantitative research method
Primary research on baby care product by using quantitative research methodPrimary research on baby care product by using quantitative research method
Primary research on baby care product by using quantitative research method
 
Amul Inventory Techniques & Other Details
Amul Inventory Techniques & Other DetailsAmul Inventory Techniques & Other Details
Amul Inventory Techniques & Other Details
 
Cost Sheet At Corporate Level
Cost Sheet At Corporate LevelCost Sheet At Corporate Level
Cost Sheet At Corporate Level
 
Export procedure
Export procedureExport procedure
Export procedure
 
Sbi mutual gold fund
Sbi mutual gold fundSbi mutual gold fund
Sbi mutual gold fund
 
Retailing & Promotion Mix
Retailing & Promotion MixRetailing & Promotion Mix
Retailing & Promotion Mix
 
Dabur
DaburDabur
Dabur
 
Can mumbai be shanghai
Can mumbai be shanghaiCan mumbai be shanghai
Can mumbai be shanghai
 
Production planning & control (ppc)
Production planning & control (ppc)Production planning & control (ppc)
Production planning & control (ppc)
 
HOW TO CONTROL YOUR ANGER
HOW TO CONTROL YOUR ANGERHOW TO CONTROL YOUR ANGER
HOW TO CONTROL YOUR ANGER
 
Skill of successful entrepreneur
Skill of successful entrepreneurSkill of successful entrepreneur
Skill of successful entrepreneur
 
Marginal costing
Marginal costing Marginal costing
Marginal costing
 

Linear Programming Feasible Region

  • 1. OPERATIONAL RESEARCH Topic: Linear Programming Problem Submitted To : Prof. Nilesh Coordinators : Zeel Mathkiya (19) Dharmik Mehta (20) Sejal Mehta (21) Hirni Mewada (22) Varun Modi (23) Siddhi Nalawade (24)
  • 2. DEFINITION OF LINEAR PROGRAMMING The Mathematical Definition of LP: “It is the analysis of problem in which a linear function of a number of variables is to maximised (minimised), when those variables are subject to a number of restraints in the form of linear inequalities”.
  • 3. TERMINOLOGY OF LINEAR PROGRAMMING A typical linear program has the following components  An objective Function.  Constraints or Restrictions.  Non-negativity Restrictions.
  • 4. TERMS USED TO DESCRIBE LINEAR PROGRAMMING PROBLEMS  Decision variables.  Objective function.  Constraints.  Linear relationship.  Equation and inequalities.  Non-negative restriction.
  • 5. FORMATION OF LPP  Objective function  Constraints  Non-Negativity restrictions  Solution  Feasible Solution  Optimum Feasible Solution
  • 6. SOLVED EXAMPLE -1  A Company manufactures 2 types of product H₁ & H₂. Both the product pass through 2 machines M₁,M₂.The time requires for processing each unit of product H₁,H₂.On each machine & the available capacity of each machine is given below: Product Machine M₁ M₂ H₁ 3 2 H₂ 2 7 Available Capacity(hrs) 1800 1400 The availability of materials is sufficient to product 350 unit of H₁ & 150 of H₂.Each unit of H₁ gives a profit of Rs.25,each unit of H₂ gives profit of Rs.20.Formulate the above problem as LPP.
  • 7. SOLUTION  From manufactures point of view we need to maximise the profit.The profit depend upon the number of unit of product H₁ &H₂ produced. Let x₁= no of unit of H₁ produce x₂=no of unit of H₂ produce x₁ ≥ 0  1 x₂ ≥ 0  2 3x₁ + 2x₂ ≤ 1800 3 2x₁ + 7x₂ ≤ 1400 4 Z= 25x₁ + 20x₂ LPP is formed as follows: Maximise Z= 25x₁ + 20x₂
  • 8. CONTI…..  Subject to: x₁ ≥ 0 x₂ ≥ 0 3x₁ + 2x₂ ≤ 1800 2x₁ + 7x₂ ≤ 1400
  • 9. CONTI…...  A Manager of hotel dreamland plans and extancison not more than 50 groups attleast 5 must be executive single rooms the number of executive double rooms should be atleast 3 times the number of executive single rooms. He charges Rs.3000 for executive double rooms and Rs.1800 executive single rooms per day.
  • 10. CONTI….. Formulate the above problume for LPP SOLUTION → The LPP is formulated as follows ; Let X1 = Total No. of single executive rooms Let X2 = Total No. of Double executive rooms ... X1 + x2 < 50 X1 > 5 x2 > 3 X1 Maximise ; Z = 1800 X1 + 3000 x2
  • 11. The LPP is formulated as follows Maximise ; Z = 1800 X1 + 3000 x Subject to ; X1 + x2 < 50 X1 > 5 x 2 > 3 X1
  • 12. GRAPHICAL METHOD 1. Arrive at a graphical solution for the following LPP. Maximize Z = 40x1 + 35x2 Subject to : 2x1 + 3x2 < 60 4x1 + 3x2 < 96 x1 , x 2 > 0
  • 13. Solution : Let us consider the equation 1) 2x1 + 3x2 = 60 Put x2 = 0: 2x1 = 60 x1 = 30 A = (30 , 0) Put x1 = 0 : 3x2 = 60 x2 = 20 B = (0 , 20)
  • 14. 2) 4x1 + 3x2 < 96 Put x2 = 0 : 4x1 = 96 x1 = 24 C = (24 , 0) Put x1 = 0 : 3x2 = 96 x2 = 32 D = (0 , 32)
  • 15. Y axis 40 Scale : Xaxis = 1 cm = 5 units 35 Yaxis = 1 cm = 5 units 30 D 25 20 B 15 10 p 5 C A X axis O 5 10 15 20 25 30 35 40
  • 16.  OBPC is the feasible region Points x1 x2 z O 0 0 z=0 B 0 20 z = 40(0) + 35 (20) = 700 P 18 8 z = 40(18) + 35(8) = 1000 C 24 0 z = 40(24) + 35(0) = 960 Thus, the optimal feasible solution is x1 = 18 , x2 = 8 and z = 1000
  • 17. CONTI…..  Find the feasible solution to following LPP Minimize Z = 6x + 5y Subject to = x + y > 7 x<3,y<4 x<0,y>0
  • 18. Solution : Removing Inequality in given equation 1. x + y > 7 Put y = 0 : x = 7 Put x = 0 : y = 7 The two points are : A = (7 , 0) & B = (0 , 7) Further, X=3,y=4
  • 19. Y axis 8 7 B Scale : X axis = 1cm = 1 unit Y axis = 1cm = 1 unit 6 5 P 4 3 2 1 A 1 2 3 4 5 6 7 8 X axis O
  • 20. CONTI…..  As all the 3 lines intersect each other at a common point P( 3 , 4) it is the feasible solution to LPP Z = 6(3) + 5(4) = 18 + 20 = 38
  • 21. CONCLUSION  Linear programming is very important mathematical technique which enables managers to arrive at proper decisions regarding his area of work. Thus it is very important part of operations research.