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Transcript of Linear Programming Feasible Region

• 1. OPERATIONAL RESEARCHTopic: Linear Programming ProblemSubmitted To : Prof. NileshCoordinators : Zeel Mathkiya (19) Dharmik Mehta (20) Sejal Mehta (21) Hirni Mewada (22) Varun Modi (23) Siddhi Nalawade (24)

2. DEFINITION OF LINEARPROGRAMMINGThe Mathematical Definition of LP: It is the analysis of problem in which a linearfunction of a number of variables is to maximised(minimised), when those variables are subject to anumber of restraints in the form of linearinequalities. 3. TERMINOLOGY OF LINEARPROGRAMMING A typical linear program has the followingcomponents An objective Function. Constraints or Restrictions. Non-negativity Restrictions. 4. TERMS USED TO DESCRIBE LINEARPROGRAMMING 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 H & H. Boththe product pass through 2 machines M,M.The time requiresfor processing each unit of product H,H.On each machine &the available capacity of each machine is given below:Product MachineMMH3 2H 2 7Available Capacity(hrs)1800 1400 The availability of materials is sufficient to product 350 unit ofH & 150 of H.Each unit of H gives a profit of Rs.25,eachunit of H gives profit of Rs.20.Formulate the above problemas LPP. 7. SOLUTION From manufactures point of view we need to maximise theprofit.The profit depend upon the number of unit of product H&H produced.Let x= no of unit of H produce x=no of unit of H producex 0 1x 0 2 3x + 2x 1800 3 2x + 7x 1400 4 Z= 25x + 20x LPP is formed as follows:Maximise Z= 25x + 20x 8. CONTI.. Subject to: x 0 x 03x + 2x 18002x + 7x 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 LPPSOLUTION The LPP is formulated as follows ; Let X1 = Total No. of single executive rooms Let X2 = Total No. of Double executive rooms... X1 + x2 < 50X1 > 5x2 > 3 X1 Maximise ; Z = 1800 X1 + 3000 x2 11. The LPP is formulated as followsMaximise ; Z = 1800 X1 + 3000 xSubject to ; X1 + x2 < 50X1 > 5x 2 > 3 X1 12. GRAPHICAL METHOD1. Arrive at a graphical solution for the following LPP.Maximize Z = 40x1 + 35x2Subject to : 2x1 + 3x2 < 604x1 + 3x2 < 96x1 , x 2 > 0 13. Solution : Let us consider the equation1) 2x1 + 3x2 = 60Put x2 = 0: 2x1 = 60x1 = 30 A = (30 , 0)Put x1 = 0 : 3x2 = 60x2 = 20 B = (0 , 20) 14. 2) 4x1 + 3x2 < 96Put x2 = 0 : 4x1 = 96x1 = 24C = (24 , 0)Put x1 = 0 : 3x2 = 96x2 = 32D = (0 , 32) 15. Y axis40Scale : Xaxis = 1 cm = 5 units35Yaxis = 1 cm = 5 units30D2520 B1510p5 C A X axisO 5 10 15 20 25 3035 40 16. OBPC is the feasible regionPoints x1 x2zO 00z=0B 0 20z = 40(0) + 35 (20) = 700P188z = 40(18) + 35(8) = 1000C240z = 40(24) + 35(0) = 960Thus, the optimal feasible solution is x1 = 18 , x2 = 8and z = 1000 17. CONTI.. Find the feasible solution to following LPPMinimize Z = 6x + 5ySubject to = x + y > 7x