University of Mumbai 2007-2nd Sem M.E Mechanical Engineering Machine Design Optimization -e - Question Paper

Z e 3x_l * 2x₃ Subjected to

5x, -*-se_a >10 2x, +2x₂> 12 x, + 4x₂ >12 x x_; > 0

(t>) {i) State the necessary and sufficient condition for maximization of multi variable function f{x)

(ii) Explain the significance of Lagrange's Multiplier

2    A gear manufacturing company received an order for three specific types of gears for regular supply The 2 production planning in-charge is considering to devote the available excess capacity to one or more of the three types say S, W, B. The available capacity of the machines which might limit output and the number of machine hours required for each unit of the respective gear is also given below. The unit profit would be Rs. 10, Rs. 30 and Rs. 40 respectively for gears S, W, B.

Find how much of each gear the company should produce in order to maximize profit ?

3. (a) Discuss the procedures involved in the optimum design of gear train, minimizrg the total weight of gear train is considered as objective.

(b) Explain the application of optimization technique in Engineering applications.

4 (a) Explain greedy algorithms and its application in finding shortest path between any given vertices of a graph.

(b) Find the minimum of the function

f(x) * 10 x^#-48x⁵ + 15x + 200x³ - 120x* - 480x + 100

5. Write notes on the following

(a) Genetic algorithms

(t>) Fibonacci Method to find maxima and minima of a function

(c)    Non linear programming and constrained optimization techniques

(d)    Stochastic programming

6 (a) Explain elim nation methods and compare exhaustive search and dichotomous search (b) (i) Explain Unimodai functions

(K) F nd the minimum of f= x* i ~ in the interval (0 0,1 0) to within 10% of the exact value, i

7. (a) A tent on a square base of side 2a consists of fcur vertical sides of height b surrounded by a regular 10 pyramid of height h. If the volume enclosed by the tent is V, show that the area of canvas in the tent can

t>& BApfessodas    ^{r 4m} i/** * *

Also show that the least area of tne canvas corresponding to a given volume V, if a and h can both vary, is given by

a/5 h

3 = - _ancj_s 2b

(b) Find the dimensions of a rectangular box of volume V= 1000 cm³ for which the total length of the 12 edges 10 is a minimum using the lagrange multiplier method. Also find the change in dimensions of the box when tt e volume >s changed to 1200 cm¹ by using the value found earlier.