Explain the role of optimization techniques in engineering and business.

What do you understand by deterministic and probabilistic models that are used in optimization methods?

With the help of a suitable example, explain the procedure of constrained optimization.

With the help of a suitable example, explain the procedure for solving a multi-variable optimization problem.

Consider the following function:

-24x1 2x2^2-8x2 200

By separating this function into three one-variable functions, show that the function is convex. Solve the problem by solving each one-variable function by calculus.

Solve the following LP problem by graphical method:

Maximise z 4x2

subject to

-3x1 +x2

x1 2x2 4

x2

4. Using Big M-method, solve the following LP problem:

Minimize z =12x1 20x2

subject to

6x1+ 8x2 100

7x1+ 12x2 120

x1, x2 0

Explain the concept involved in branch and bound algorithm used for solving integer programming problem.

Discuss the concept of Genetic algorithm optimization.

Using the method of Lagrangian multipliers, solve the following optimization problem:

Optimize z x1^2 l0x1 x2^2 6x2 x3^3 ­4x3

subject to

X1+x2+x3= 7

x1,x2,x3 0

Solve the following problem using the Kuhn-Tucker conditions:

Maximize z 2x1^2 -7x2^2 l2x1x2

subject to

2x1+ 5x2 98

x1, x2 0

7. Write short notes on any two of the following:

Non-traditional algorithms

Geometric programming

Optimization techniques as tools for decision-making