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Seat No.: Enrolment
GUJARAT TECHNOLOGICAL UNIVERSITY
M.E. SEMESTER II-EXAMINATION Summer 2017
Subject code: 2724712 Date:31/05/2017
Subject Name: Optimization Theory and Practice
Time: 02:30 p.m.-5:00 p.m. Total Marks: 70
Instructions:
1. Attempt all questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.
Q.1

A manufacturer produces two types of machine parts, and using lathes and milling machines. The machining times required by each part on the lathe and the milling machine and the profit per unit of each part are given below:
Machine
Machine Time Required
Profit Per Unit
Lathe
Milling Machine
5
2
200
4
4
300
If the total machining times available in a week are 500 hours on lathes and 400 hours on milling machines, determine the number of units of and to be produced per week to maximize the profit by Simplex Method.
07

i Explain the terms: Chromosome, Population, Fitness Function
ii Describe Particle Swarm Optimization algorithm.
03
04
Q.2

Use Fletcher-Reeves method to minimize the function
starting from the point
07

Minimize
subject to
by Dual Simplex Method.
07
OR

Use Graphical Method to solve the following problem:
Maximize
subject to
07
Q.3

i Find the points at which the function
has maximum and minimum values.
ii Write some engineering applications of optimization. Describe
one of them in detail.
03
04

i Consider the following optimization problem:
subject to
Use Kuhn-Tucker conditions to check whether is a
local minimum. What are the values of the Lagrange multipliers?
03
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ii Find the extreme points of the function
04
OR
Q.3

i How gradient vector and Hessian matrix for a function of
variables are useful for optimization of the function?
ii Explain the terms: Feasible Region, Unimodal Function
03
04

i Determine whether the following matrix is positive definite,
negative definite or indefinite by evaluating the signs of its
sub-matrices:
ii Maximize subject to
using Lagrange's multiplier method.
03
04
Q.4

Use Dichotomous Search Method to find the minimum of the function in the interval to within 10 of the exact value. Take
07

Find the minimum of the function
using quasi-Newton method with the starting point and the step size Use for checking the convergence.
07
OR
Q.4

Minimize the function in the interval using the Fibonacci Method with
07

Find the minimum of the function using Newton Method with the starting point Take
07
Q.5

What are the advantages of random search method? Also describe Random Jumping Method.
07

Describe the basic approach of the penalty function method.
07
OR
Q.5

i Mention the transformations used for converting a constrained
optimization problem into an unconstrained one.
ii Perform two iterations of Cauchy's steepest descent method to
minimize the function
by taking as the starting point.
03
04

Write the Sequential Linear Programming Algorithm.
07