LINEAR & NON-LINEAR OPTIMIZATION.pdf - Google Drive

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*EJM011*

EJM – 011

II Semester M.E. (Power & Energy Systems) Degree Examination, January 2013 2K8 PS 211 : LINEAR & NON-LINEAR OPTIMIZATION Time : 3 Hours

Max. Marks : 100

Instruction: Answer any five full questions. 1. a) What are the characteristics of constrained optimization problems ? Briefly explain.

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b) A person has to provide 10, 12 and 12 units of chemicals A, B and C respectively to his Garden. A liquid product contains 5, 2 and 1 units of chemical A, B and C respectively per Jar and costs ` 3/- per Jar. A dry product contains 1, 2 and 4 units of chemical A, B and C respectively per packet and costs ` 2/- per packet. How many of these the person should purchase to meet the requirements at least cost ? Formulate as an LPP and solve graphically. 12 2. a) With reference to the solution of LPP by Simplex method/table when do you conclude as follows i) LPP has multiple optimum solutions ii) LPP has no limit for improvement of the objective function. iii) LPP has no feasible solution. b) What are the properties of duality ?

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c) Use K-T conditions to solve the Non-Linear Programming problems Maximize z = 7 x12 − 6 x1 + 5 x 22 Subject to x1 + 2x 2 ≤ 10 x1 – 3x2 ≤ 9 x1, x2 ≥ 0.

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3. a) Explain the procedure of solving an LPP by Dual Simplex Method.

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b) State the optimal control problem. Derive the necessary conditions for optimal control.

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c) Find the minimum of f = λ5 − 5 λ3 − 20 λ + 5 using Quadratic Interpolation Method.

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*EJM011*

EJM – 011 4. a) Use Lagrangian multiplier to solve Min Z = x12 + x 22 + x 23 . Subject to x1 + 2x2 + x3 = 4 x1 + x2 + x3 = 3. b) Define Unimodal and multimodal functions.

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c) Solve the following LPP using Revised Simplex Method : Minimize Z = 3x1 + x2 + x 3 + x4 Subject to – 2x1 + 2x2 + x3 = 4 3x1 + x2 + x4 = 6 x1 ≥ 0, x2 ≥ 0, x3 ≥ 0 and x4 ≥ 0. 5. a) Explain the following methods with the help of flow-chart. i) Powell’s method ii) Steepest Descent Method.

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b) Maximize f( x) = x 12 + 3 x 22 + 6 x 23 by univariate method starting from (2 –1 1). Obtain the value of the function after two iterations. 10 6. a) Explain Economic Dispatch by Gradient Method.

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b) Explain in brief the Genetic Algorithm Operators.

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c) How are the search directions generated in Fletcher-Reeves Method ? Explain with an example.

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7. a) Explain the optimization of Fuzzy Systems. b) Explain optimal power flow based on Newton Method.

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c) Use Hooke and Jeeve’s method to minimize the function f (x1 x 2 ) = x12 + 3x22 by taking Δx 1 = Δx 2 = 0 .5 starting from (2, –1). Obtain the value of f(x) after two iterations. ______________

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