# Operations Research

**Topics:**Optimization, Maxima and minima, Operations research

**Pages:**23 (1892 words)

**Published:**December 9, 2012

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Code: 9A05603

III B. Tech II Semester (R09) Regular Examinations, April/May 2012 OPTIMIZING TECHNIQUES

(Common to Computer Science & Engineering & Computer Science & Systems Engineering) Time: 3 hours

Max Marks: 70

1

(a)

(b)

2

Answer any FIVE questions

All questions carry equal marks

*****

State the necessary and sufficiency conditions for the minimum of the single variable function f ( x ) .

Find the minimum of the function:

f ( x ) = 10 x 6 − 48 x 5 + 15 x 4 + 200 x 3 − 120 x 2 − 480 x + 100 .

0.0

using

0.0

2

2

Minimize f ( x1 , x 2 ) = x1 − x 2 + 2 x1 + 2 x1 x 2 + x 2 starting from the point X1 =

D

L

Hooke and Jeeves’ method. Take ∆x1 = ∆x 2 = 0.8 and ε = 0.1. 3

A firm produces three products. These products are processed on three different machines. The time required to manufacture one unit of each of three products and the daily capacity of the three machines are given in the table below: Machine

M1

M2

M3

R

O

Time per unit

(minutes)

2

3

2

4

-3

2

5

--

T

N

W

U

Machine capacity

(minutes / day)

440

470

430

The profit for product 1, 2, and 3 is Rs. 4, Rs. 3 and Rs. 6 respectively. It is assumes that all the amounts produced are consumed in the market. Formulate the problem as LPP in order to determine the daily number of units to be manufactured for each product.

J

4

Describe the transportation problem. Formulate the transportation problem as a linear programming problem.

5

Using Lagrange multiplier method solve following problem:

Minimize f (X) = 1/2(x12 + x22 + x32).

subject to constraints:

g1(x) = x1 − x2 = 0.

g2(x) = x1 + x2 + x3 − 1 = 0.

6

(a)

(b)

Explain method of multipliers algorithm.

Describe briefly differences between the MOM and other transformation methods such as SUMT.

Contd. in Page 2

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Code: 9A05603

Page 2

7

(a)

(b)

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Is it correct to say that in a quadratic programming problem the objective function and the constraints both should be quadratic? If not, give your own comments. Derive the Kuhn-Tucker necessary conditions for an optimal solution to a quadratic programming problem.

A company has three media A, B and C available for advertising the product. The data collected over the past year about the relationship between the sales and frequency of advertisement in the different media as follows:

Frequency/month

Media

1

2

3

4

A

125

225

260

300

Estimated sales (units) per month

B

C

180

300

290

350

340

450

370

500

D

L

The cost of advertisement is Rs. 5000 in media A, Rs.10000 in B and Rs. 20,000 in media C. The total budget allocated for advertising the product is Rs. 40000. Determine the optimal combination of advertising media and frequency.

*****

R

O

W

U

T

N

J

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Code: 9A05603

III B. Tech II Semester (R09) Regular Examinations, April/May 2012 OPTIMIZING TECHNIQUES

(Common to Computer Science & Engineering & Computer Science & Systems Engineering) Time: 3 hours

Max Marks: 70

Answer any FIVE questions

All questions carry equal marks

*****

1

Briefly explain the applications of optimization in engineering.

2

Show that the function:

2

2

f ( x ) = 3x12 + 2 x2 + x3 − 2 x1 x2 − 2 x1 x3 + 2 x2 x3 − 6 x1 − 4 x2 − 2 x3 is convex.

3

Solve the following LP problem using graphical method and give your comment on the result:

Maximize Z = 3 X1 + 2 X2

Subject to

– 2 X1 + 3 X2 ≤

9

3 X1 – 2 X2 ≥ – 20

X1, X2 ≥ 0.

4

Explain the similarities and differences between a transportation problem and assignment problem. Can we use the transportation algorithm to solve the assignment problem. If so illustrate with an example.

5

Use the Kuhn-Tucker conditions to solve the following NLPP:

Maximize 8x12+2x2

subject to the constraints...

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