Explain how and why optimization techniques have been valuable in aiding executive decisions.

Discuss the various phases in solving an optimization problem.

With the help of a suitable "example, explain the role of differential calculus in solving optimization problem.

Differentiate between single and multi-variable optimization giving suitable example.

For each of the following functions determine whether it is convex, concave or neither:

=20x1 10x2

Solve the following LP problem graphically:

Maximise z 2x2
subject to: x1 ­x2
-0·5 x1 x2 2
x1 ,x2 0

4. Solve the following LP problem using simplex method:

Maximize z 3x1 2x2 5x3 subject to: x1 x2 x3
2x1 3x2 5x3 30
2x1 x2 ­x3 8
x1, x2, x3 0

Explain the concept involved in the Gomory's cutting plane method.

Explain the graphical method for solving linear goal programming problem.

6. Use the method of Lagrangian multipliers to solve the following non-linear programming problem. Does the solution maximize or minimize the objective function?

Optimize z 2x1^2 x2^2 3 x3^2 10x1 8x2 6x3 -100
subject to: x1 x2 +x3 20
x1, x2, x3 0

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

One-dimensional optimization

Dual linear programming problem

Genetic Algorithm