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M. TECH. DEGREE EXAMINATION, MODEL QUESTION PAPER – I First Semester Branch: Mechanical Engineering Specialisation: Machine Design MMEMD 105-3 Engineering Optimization (Elective – I) (Regular – 2013 Admissions) Time: Three Hours Maximum: 100 Marks (Answer all questions) 1. (a) Write a note on applications of optimization. (10) (b) Design a uniform column of tubular section, with hinge joints at both ends, to carry a compressive load P = 2500 kgf for minimum cost. The column is made up of a material that has a yield stress (σy) of 500 kgf/cm2, modulus of elasticity (E) of 0.85 × 106 kgf/cm2, and weight density (ρ) of 0.0025 kgf/cm3. The length of the column is 250 cm. The stress induced in the column should be less than the buckling stress as well as the yield stress. The mean diameter of the column is restricted to lie between 2 and 14 cm, and columns with thicknesses outside the range 0.2 to 0.8 cm are not available in the market. The cost of the column includes material and construction costs and can be taken as 5W + 2d, where W is the weight in kilograms force and d is the mean diameter of the column in centimeters. (15) Or 2. (a) Write a note on Interval Halving method. (5) (b) Briefly explain quasi-Newton Method. (5) (c) Find the minimum of the function f = λ5 − 5λ3 − 20λ + 5 by the Golden section method in the interval (0, 5). (15) 3. (a) What are Kuhn-Tucker Conditions? (5) (b) A manufacturing firm producing small refrigerators has entered into a contract to supply 50 refrigerators at the end of the first month, 50 at the end of the second month, and 50 at the end of the third. The cost of producing x refrigerators in any month is given by $(x2 + 1000). The firm can produce more refrigerators in any month and carry them to a subsequent month. However, it costs $20 per unit for any refrigerator carried over from one month to the next. Assuming that there is no initial inventory, determine the number of refrigerators to be produced in each month to minimize the total cost. (20) Or 4. Minimize f (X) = 3x12 + 2x22 + 5x23 − 4x1x2 − 2x1x3 − 2x2x3 subject to 3x1 + 5x2 + 2x3 ≥ 10 3x1 + 5x3 ≤ 15 xi ≥ 0, i = 1, 2, 3 by quadratic programming. (25) 5. (a) Explain Principle of Optimality. (b) Maximize f (x1, x2) = 50x1 + 100x2 using dynamic programming method, subject to 10x1 + 5x2 ≤ 2500 4x1 + 10x2 ≤ 2000 x1 + 1.5x2 ≤ 450 x1 ≥ 0, x2 ≥ 0 (5) (20) Or 6. 7. (a) Describe the applications of integer programming. (b) Solve the following mixed integer programming problem: Minimize f = 4x1 + 5x2 subject to 10x1 + x2 ≥ 10, 5x1 + 4x2 ≥ 20 3x1 + 7x2 ≥ 21, x2 + 12x2 ≥ 12 x1 ≥ 0 and integer, x2 ≥ 0 (5) (20) Find the function x(t) that minimizes the functional with the condition that x(0) = 2. (25) Or 8. Derive the equation for the shape of a solid body of revolution for minimum drag. (25)