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### 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.
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