# Optimization in Civil Engineering July 2011

1. (a) (i) What are the engineering application of optimization methods ? Explain briefly.

(ii) A rectangular plate of size axb is to be used to prepare a box open at top by cutting four rectangular portions at each corners and folding along the edge. Find the size s of the smaller rectangular portions to be cut to obtain the maximum volume in the box.

OR

(b) (i) What are the classifications of optimization problems ? Explain any one in briefly.

(ii) A solid cone is to be moulded by using minimum material and to obtain maximum volume. Find the ratio of base diameter to height.

2 (a)What do you mean by a linear programming- problem ? Using Simplex method solve the following linear programming problem ?

(a)  Max z = xj+3 x2 -2xg Subject to

3x1 -x2 +2x3 <=7 -2 Xj +4 x2 <==12 4 X|+3 x2 +8 x3 <-7

Xj, x2, x3 >=0

OR

3 (a)What do you mean by Duality in Linear Programming ? Write the dual of the Linear Programming problem given above in and find the solution.

(b)What do you mean by a sensitivity analysis of linear programming problem ?

Using simplex method solve the following linear programming problem.

Max z = x^+3 x2 -2x3 Subject to

3X} -x2 +2x3 <=7 -2 Xi +4 x2 <=12

4             Xj+3 x2 +8 x3 <=7

xl> x2> x3 >=0

OR

(a) That do you mean by Transportation problem ?

(b) Find and solve the dual of the Linear Programming problem given above.

4 (a)What are the methods employed in solving the non­linear optimization problems. Give a brief of any one method.

(b)Find the minimum of the function f = x5 – 5 x3 -20 x + 10 using golden section method, in the interval (0.5)

OR

(a)What do you mean by direct search method employed in solving the Non-linear optimization problems. Give a brief of the method.

(b)Find the minimum of the function f=x^-5x’^ -20 x + 10 using Fibonacci method, in the interval (0,5)

5 (a) What do you mean by multi-stage decision in dynamic methods of optimization problems. Give a brief of any one method.

(b) Solve the following LP problem by dynamic programming :

Max z = 5x1 +4 x2 Subject to

OR

(a) What do you mean by dynamic programming in optimization problems. Give a brief of the method.

(b) Differentiate between an initial value problem and final value problem. How to convert an initial value problem and final value problem ?