LOYOLA  COLLEGE (AUTONOMOUS), CHENNAI-600 034.

M.Sc. DEGREE  EXAMINATION  – STATISTICS

FourTh SEMESTER  – APRIL 2003

ST  4951/ S  1052   advanced operations  reSEaRCH

26.04.2003

1.00 – 4.00                                                                                             Max: 100 Marks

 

 

SECTION – A                      (10 ´ 2 = 20 Marks)

Answer ALL the questions.

1. State the Bellman’s principle of optimality.
2. Define a general non-linear programming problem.
3. What is posynomial and where it is used?
4. Define the mathematical formulation of an quadratic programming problem.
5. Define separable programming problem with an example.
6. Convert the chance constraint into equivalent deterministic constraint for the following problem.

Min  Z = 3x₁+ 4x₂

St   Pr (3x₁-2x₂ £ b₁) ³

 

x₁, x₂ ³ 0, where b₁, b₂, b₃ are independent random variables uniformly distributed

in the intervals (-2,2), (0,2) and (0,4) respectively.

7. Explain the concept of integer programming problem.
8. Explain the mathematical model of a stochastic linear program.
9. Explain a scenario where the goal programming concepts are applied.
10. State the Kuhn-Tucker conditions to solve an NLPP program.

 

SECTION – B                                   (5 ´ 8 = 40 Marks)

Answer any FIVE questions

11. Explain Wolfe’s Algorithm in solving a non linear programming problem.
12. Solve the following geometric programming problem

 

13. Explain clearly the piece wise linear approximation.
14. Solve the following Integer programming problem

Max Z = x₁+ x₂

s.t     3x₁+ 2x₂ £ 5

x₂ £ 2

x₁, x₂ ³ 0 and integers.

15. Derive the Gomary’s constraint in solving an integer programming problem
16. Solve the non linear programming problem using Lagrangian multipliers.

Maximize

s.t       x₁+ x₂ + x₃ =15

2x₁ -x₂ +2x₃ = 20

x₁,  x₂, x₃ ³ 0

 

 

 

17. Explain the concepts in solving the stochastic programming problem for

the following scenario  assuming the usual notations

(i) Aircraft Allocation problem    (ii) Two stage programming

18. Solve the following cargo loading problem assuming the usual notations when

there is no volume restriction  (i.e., Q = ¥) , W = 5 and N = 3 with the

numerical  data given below

 

item (n)	Weight (Wn)	Value (Vn)
1	2	7
2	3	10
3	1	3

 

SECTION – C                                  (2 ´ 20 = 40 Marks)

Answer any TWO questions.

19. Explain Beale’s Algorithm in solving a non linear programming problem.
20. Solve the following non linear programming problem using Beale’s Algorithm

 

s.t   x₁ +2x₂ £ 10

x₁ + x₂ £ 9

x₁ ,x₂ ³ 0

20. a) Explain Branch and Bound Technique in solving an Integer programming

problem

1. Derive the geometric -arithmetic mean inequality in solving a geometric

programming problem

21  a)   Explain the dynamic programming problem concepts in solving the Cargo

loading problem, assuming the usual notations.

1. Solve the following linear programming problem through Dynamic programming problem

Max  Z = 3x₁ + 4x₂

s.t         2x₁ + x₂ £ 40

2x₁ + 5x₂ £ 180

x₁, x₂ ³ 0

22. a) Approximate the following NLPP to LPP using separable convex

programming and piece wise linear combination concepts

 

Max f(x) = 3x₁ + 2x₂

                                                               

1. Explain the scenario of n component system in series in Reliability and

provide a solution to solve using a Dynamic programming problem.

 

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

M.Sc., DEGREE EXAMINATION – STATISTICS

  FOURTH SEMESTER – APRIL 2004

ST 4951/S 1052 – ADVANCED OPERATIONS RESEARCH

12.04.2004                                                                                                           Max:100 marks

1.00 – 4.00

SECTION – A

 

Answer ALL questions                                                                                (10 ´ 2 = 20 marks)

 

1. What is the need for an integer programming problem?
2. Define a covex function.
3. Is the following quadratic form negative definite?

j (x₁, x₂) = –

4. Is the function f(x) = x₁ separable? x = (x₁, x₂).
5. Define a quadratic programming problem.
6. Explain the Markovian property of dynamic programming.
7. When do you say the Khun-Tucker necessary conditions are also sufficient for a maximization problem?
8. Explain the need for Goal programming.
9. What is zero-one programming?
10. When do we need Geometric programming problem?

 

SECTION – B

 

Answer any FIVE questions                                                                        (5 ´ 8 = 40 marks)

 

11. Solve the following LPP by dynamic programming.

max z = 3x₁ + 4x₂

Subject to

2x₁+x₂ £ 40

2x₁+5x₂ £ 180

x₁+x₂ ≥ 0

12. State and prove the necessary condition for a function of n variables to have a minimum. Also prove the sufficient condition.
13. Derive the Gomery’s constraint for a mixed algorithm.
14. Solve by Beale’s method

max Z = 2x₁ +3x₂ –

Subject to

x₁ + 2x₂ £ 4, x₁, x₂ ≥ 0.

15. Solve by using Khun – Tucker conditions

max Z = 10x₁ + 4x₂ – 2

Subject to: 2x₁ + x₂ £ 5, x₁, x₂ ≥ 0

 

 

 

 

16. Reduce the following separable programming problem to an approximate linear programming problem.

f(x₁, x₂) = 2x₁ + 3

Subject to 4x₁ + 2£ 16, x₁, x₂ ≥ 0

17. Consider the chance constrained problem

max Z = 5x₁ + 6x₂ + 3x₃

Subject to

P_r [a₁₁ x₁ + a₁₂ x₂ + a₁₃ x₃ £ 8] ≥ .95

P_r [5x₁ + x₂ + 6x₃ £ b₂] ≥ .1, x_j ≥ 0 “j = 1,2,3

~ N

b₂ ~ N (7,9).  Reduce this problem to a deterministic model.

18. Solve

minimize f(x₁, x₂) = 3 + 2

Subject to

x₁ + x₂ = 7

x₁, x₂≥ 0

 

SECTION – C

 

Answer any TWO questions                                                                        (2 ´ 20 = 40 marks)

 

19. a) Solve the following all Integer programming problem

max  Z = x₁ + 2x₂

Subject to

x₁ + x₂ £ 7

2x₁ £ 11

2x₂ £ 7     x₁, x₂ ≥ 0, x₁, x₂ integers.

1. b) Explain branch and bound method with an example.                                            (12+8)
2. Solve by Wolfe’s method

max Z =  2x₁ + 3

Subject to

x₁ + 4 x₂  £  4

x₁ + x₂ £ 2

x­₁, x₂ ≥ 0

21. a) A student has to take examination in 3 courses A,B,C. He has 3 days available for the study.  He feels it would be best to devote a whole day to the study of the same course, so that he may study a course for one day, two days or three days or not at all.  His estimates of the grades he may get by study are as follows:-

Course  A      B    C

Days

0                  0      1     0

1                  1      1     1

2                  1      3     3

3                  3      4     3

How should he study so that he maximizes the sum of his grades?  Solve by Dynamic

Progrmming.

1. b) Solve the following using dynamic programming

min Z =

Subject to

u₁+u₂+u₃ ≥ 10,  u₁, u₂, u₃ ≥ 0                                                       (15+5)

22. a) Solve the following Geometric programming problem

f(X) = .

1. b) Explain how will you solve if there is a constraint.        (15+5)

 

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 51

FOURTH SEMESTER – APRIL 2006

                                        ST 4951 – ADVANCED OPERATIONS RESEARCH

 

 

Date & Time : 27-04-2006/9.00-12.00         Dept. No.                                                       Max. : 100 Marks

 

 

SECTION A  

Answer ALL questions.                                                                 (10 ´ 2 =20 marks)

1. Define linearly independent vectors.
2. Define a Mixed Integer Programming  Problem.
3. What is the need for Integer Programming Problems?
4. State Bellman’s principle of optimality.
5. What is meant by Separable Programming Problem?
6. Define Goal Programming Problem.
7. Write down the mathematical formulation of a Geometric Programming Problem.
8. Explain the stage and state variables in a dynamic Programming Problem.
9. Name the methods used in solving a Quadratic Programming Problem.
10. What is the need for Dynamic Programming Problem?

 

SECTION B

Answer any FIVE questions.                                                          (5 ´ 8 =40 marks)

 

1. Explain the construction of fractional cut in the Gomory’s constraint method.

 

1. State all the characteristics of a Dynamic Programming Problem.

 

1. In the network given below are different routes for reaching city B from city A passing through a number of other cities, the lengths of the individual routes are shown on the arrows. It is required to determine the shortest route from A to B. Formulate the problem as a Dynamic Programming Problem model, explicitly defining the stages, states and then find the optimal solution.

 

6

 

 

5                      3                  2                      4

 

 

 

7                   4                   2                    2

 

5

 

 

 

1. Solve the following Non-Linear Programming Problem:

Optimize Z = X ² +Y ² + W ²,

subject to X +Y + W = 1,

X, Y, W ≥ 0.

1. Derive the Kuhn-Tucker necessary conditions for solving a Generalized Non-Linear Programming Problem with one inequality constraint.

 

1. Derive the orthogonality and Normality conditions for solving the unconstrained Geometric Programming Problem.

 

1. Convert the following Stochastic Programming Problem into an equivalent deterministic model, max Z = X₁ + 2 X₂ + 5 X₃ ,subject to

P [a₁ X₁ + 3 X₂ + a₃ X₃  ≤ 10 ] ≥ 0.9,

P [ 7 X₁ + 5 X₂ + X₃  ≤ b₂ ] ≥ 0.1,

X₁, X₂, X₃  ≥ 0.

Assume that a₁, a₃ are independent normally distributed random variables with means E (a₁) = 2, E (a₃) = 5, V (a₁) = 9, V (a₃) = 16. Also assume that

b₂ ~ N (15, 25).

 

1. The manufacturing plant of an electronics firm produces two types of T.V. sets, both colour and black-and-white. According to past experiences, production of either a colour or a black-and-white set requires an average of one hour in the plant. The plant has a normal production capacity of 40 hours a week. The marketing department reports that, because of limited sales opportunity, the maximum number of colour and black-and-white sets that can be sold are 24 and 30 respectively for the week. The gross margin from the sale of a colour set is Rs. 80, whereas it is Rs. 40 from a black-and-white set.

The chairman of the company has set the following goals as arranged in the order of their importance to the organization.

1. 1. Avoid any underutilization of normal production capacity (on layoffs of production workers).
   2. Sell as many T.V. sets as possible. Since the gross margin from the sale of colour T.V. set is twice the amount from a black-and-white set, he has twice as much desire to achieve sales for colour sets as black-and-white sets.
   3. The chairman wants to minimize the overtime operation of the plant as much as possible.

Formulate this as a Goal Programming Problem.

 

SECTION C

Answer any TWO questions.                                                        (2 x 20 =40 marks)

1. Solve the following Integer Programming Problem:

Max Z = 3 X₁ +  X₂ + 3 X₃

subject to – X₁ + 2 X₂ + X₃  ≤ 4,

4 X₂ – 3 X₃  ≤ 2,

X₁ – 3 X₂ + 2 X₃  ≤ 3,

X₁, X₂, X₃  ≥ 0.

 

1. (i) Solve the following Dynamic Programming Problem (DPP):

Min Z =  subject to    = C , x _j ≥ 0 , j = 1,2, … n. C > 0.

 

(ii) Solve the following LPP by DPP technique:

Max Z = 3 X₁ + 4 X₂ ,

subject to 2 X₁ + X₂  ≤ 40,

2 X₁ + 5 X₂  ≤ 180,

X₁, X₂ ≥ 0.

 

1. Use Kuhn-Tucker necessary conditions to solve the following Generalized Non- Linear Programming Problem:

Max Z = 2 X₁ – X₁² +  X₂

subject to 2 X₁ + 3 X₂  ≤ 6,

2 X₁ + X₂  ≤ 4,

X₁, X₂ ≥ 0.

 

1. Solve the following Quadratic Programming Problem using Wolfe’s algorithm:

Max Z = 4 X₁ + 6 X₂ –  2 X₁² – 2 X₁ X₂  –  2 X₂² ,

subject to  X₁ + 2 X₂  ≤ 2,

X₁, X₂ ≥ 0.

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 54

FOURTH SEMESTER – APRIL 2007

ST 4807 – ADVANCED OPERATIONS RESEARCH

 

 

 

Date & Time: 20/04/2007 / 9:00 – 12:00          Dept. No.                                                          Max. : 100 Marks

 

 

SECTION A

(10 X 2 = 20)

 

ANSWER ALL QUESTIONS. EACH CARRIES TWO MARKS

 

1. What is a pure integer programming problem?
2. Write down the formula for mixed cut.
3. What is the basic principle used in Dynamic Programming?
4. What do you mean by problem of dimensionality?
5. When do you use e-model in stochastic programming ?
6. What are soft and hard constraints?
7. List the assumptions made in single item static model.
8. Mention how the objective function is expressed at the beginning of each iteration in Beal’s method.
9. Give two examples for setup cost.
10. What will be your conclusion when the value of the objective function at the end of Phase 1 is non zero in two-phase simplex method?

 

SECTION B

(5 X 8 = 40)

 

ANSWER ANY FIVE. EACH CARRIES EIGHT MARKS

 

1. Show that the following problem has two optimum solutions :

Maximize : subject to

 

 

 

1. Describe in detail “Fractional Algortithm”
2. Describe with an example of your choice Branch and Bound Method
3. Explain the method of solving  LPP using dynamic programming technique.
4. Explain how constrained non-linear programs are solved.
5. Describe the steps used in Beale’s Method
6. Solve the following inventory problem (Multi item static model with storage constraint).

 

 

1        10        2          0.3       1 sq. ft

2        5          4          0.1       1 sq. ft

3        15        4          0.2       1 sq. ft

 

Assume that = 25 sq. ft

 

 

1. Explain various elements of a queuing model.

 

 

SECTION C

(2 X 20 = 40)

 

ANSWER ANY TWO. EACH CARRIES TWENTY MARKS

 

 

1. Explain Capital Budgeting Model. Develop DP solution for the same and illustrate the solution for the following data :

 

Plant 1             Plant 2             Plant 3

 

Proposal          c1        R1       c2        R2       c3        R3

 

1                3          5          3          4          0          0

 

2                4          6          4          5          2          3

 

3                —          —          5          8          3          5

 

4                —          —          —          —          6          9

 

 

1. Solve the following problem by Wolf’s method.

Minimize

Subject to

,

1. Kumaravel has Rs.80000/- with him. He wants to buy shares of Lavanya & Co and Sathish & Co. The following table gives the necessary data.

 

Share Price      Annual Return                        Risk Index

 

Lavanya & Co             Rs.25/-             Rs.3/-               0.50

 

Sathish & Co               Rs.50/-             Rs.5/-               0.25

Kumaravel wants to have a minimum return of Rs.9000/- and does not like to lose

more than Rs.700/- . Formulate this as a goal programming problem and solve the

same.

 

22. In the deterministic model with instantaneous stock replenishment, no shortage, and constant demand rate, suppose that the holding cost per unit is given by for             quantities below q and for quantities above q, . Find the economic lot     size.

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

NO 52

FOURTH SEMESTER – APRIL 2008

ST 4807 – ADVANCED OPERATIONS RESEARCH

 

 

 

Date : 23/04/2008            Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

SECTION- A

Answer all the questions.                        10 x 2 = 20 Marks

 

1. When an LPP is said to have an unbounded solution?
2. Write the significance of goal programming.
3. Write a note on holding and shortage costs in an inventory system.
4. What are the behaviours of customers in queuing analysis?
5. Provide the Khun -Tucker conditions for a maximization problem.
6. How Beal’s method differs from Wolfe’s method?
7. Give an account of methods used for solving an integer programming problem.
8. Define dynamic programming problem.
9. How to find an optimal inventory policy for multiple item static model?

10.When an LPP is called stochastic?

 

                                                                        SECTION- B

Answer any five questions.                5 x 8 =  40 Marks

11. Use two-phase simplex method to

Max. Z = 5x₁ + 3x₂

Subject to

2x₁ + x₂    1

x₁  + 4x₂  6

x₁  0 and x₂  0 .

12. An item sells for $25 a unit ,but a 10 % discount is offered for lots of 150 units or more. A company uses this item at the rate of 20 units per day. The setup cost for ordering a lot is $50 and the holding cost per unit per day is $0.3. Should the company take advantage of the discount?
13. Explain multi-item EOQ model with storage limitation.
14. Derive the steady-state measures of performance for (M / M / 1):(FIFO/ / ).
15. Explain generalized Poisson queuing model.
16. Provide branch and bound algorithm for solving I P P.
17. Use dynamic programming to

Min. Z = x₁² + x₂² + x₃²

Subject to

x₁ + x₂ + x₃  15

x₁ 0 , x₂  0 and  x₃  0.

18. Explain the two- stage programming technique used in stochastic programming.

 

    …2

 

 

-2-

 

 

SECTION – C

Answer any two questions.                2 x 20 =  40 Marks

 

19.(a)  Solve the following LPP graphically:

Max. Z = 2x₁ + 5x₂

Subject to

x₁ + x₂  1  ,     x₁ – 5x₂  0  ,   5x₁ – x₂ 0  ,  x₁ – x₂ -1 ,   x₁ + x₂  6  ,

x₂  3   ,   x₁  0 and x₂  0 .

 

• Use dynamic programming to solve the following LPP:

Max.Z = 3x₁ + 5x₂

Subject to

x₁  4 ,  x₂  6 ,  3x₁ + 2x₂   18

x₁  0 and x₂  0 .

 

20. (a) Derive the probabilistic EOQ model.

(b)  Electro uses resin in its manufacturing process at the rate of 100 gallons per

month. It cost Electro $100 to place an order for a new shipment. The holding

cost per gallon per month is $2 and the shortage cost per gallon is $10.Historical

data show that the demand during lead time is uniform over the range (0, 100)

gallons. Determine  the  optimal ordering policy for Electro.

 

21. Use Wolfe’s method to solve the following QPP:

Max. Z = 2x₁ + x₂ – x₁²

Subject to

2x₁ + 3x₂   6

2x₁ + x₂     4

x₁  0 and x₂  0 .

 

22. Solve the following mixed-integer programming problem using Gomory’s cutting plane algorithm:

Max. Z = 3x₁ + x₂+ 3x₃

Subject to

-x₁  + 2x₂ + x₃    4

4x₂ – 3x₃            2

x₁  – 3x₂ + 2x₃  3

x_i   0     (i = 1,2,3) where x₁ and x₃ are integers.

 

 

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     LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

YB 49

FOURTH SEMESTER – April 2009

ST 4807 – ADVANCED OPERATIONS RESEARCH

 

 

 

Date & Time: 23/04/2009 / 9:00 – 12:00  Dept. No.                                                    Max. : 100 Marks

 

 

SECTION -A                                                                                                                                 

Answer all the questions                                                                  10 x 2 = 20 marks

 

1. When a solution to an LPP is called infeasible?
2. How dual simplex method differs from other simplex methods ?
3. Define holding and penalty costs.
4. Write basic components of a queuing model.
5. Write the significance of integer programming problem.
6. Define Dynamic Programming Problems.
7. Differentiate goal programming from other programming problems.
8. Write a note on complementary slackness condition.
9. Provide any two  applications for parallel  and sequence service systems.
10. For a single item static model if D = 100 , h = $0.02 , K = $100 and lead

time is 10 days,find the economic order quantity and re order point.

 

                                                                SECTION -B                                                                                                                                 

Answer any five questions                                                                 5 x 8 = 40 marks

 

11. Use the graphical method to solve the following LPP:

Maximize Z = 2x₁ + 3x₂

Subject to the constraints:                                                                                                           x₁ + x₂ ≤ 30 ,  x₁ – x₂ ≥ 0 , x₂ ≥ 3 , 0≤ x₁ ≤ 20 and 0 ≤ x₂ ≤ 12.

12. Write big M method algorithm.
13. Use duality to solve the following LPP:

Maximize Z = 2x₁ + x₂

Subject to the constraints:

x₁ + 2x₂ ≤ 10  ,   x₁ + x₂ ≤ 6 ,  x₁ – x₂ ≤ 2 , x₁ – 2x­₂ ≤ 1 ; x₁,x₂ ≥ 0 .

14. Write briefly about inventory management.
15. Derive the steady state measures of (M/M/1) : (GD/∞/∞) queuing model.
16. Write Beale’s algorithm to solve Quadratic Programming Problem.
17. Obtain the set of necessary and sufficient conditions  for the following NLPP.

Minimize Z = 2x₁² – 24x₁ + 2x₂² – 8x₂ + 2x₃² – 12x₃ + 200

Subject to the constraints:

x₁ + x₂ + x₃ = 11 ,  x₁,x_2, x₃ ≥ 0 .

18. Solve the following NLPP using Kuhn- Tucker conditions :

Maximize Z =  –x₁² – x₂² – x₃² + 4x₁ + 6x₂

Subject to the constraints:

x₁ + x₂  ≤ 2  ,   2x₁ + 3x₂ ≤ 12 ; x₁, x₂ ≥ 0

 

 SECTION -C                                                                                                                              

Answer any two questions                                                                    2x 20 = 40 marks

 

19.(a) Use two-phase simplex method to

Maximize Z =  5x₁ + 8x₂

Subject to the constraints:

3x₁ + 2x₂ ≥ 3 , x₁ + 4x₂ ≥ 4 , x₁ + x₂ ≤ 5 ; x₁, x₂ ≥ 0

 

• Use dynamic programming to solve:

Minimize Z = x₁² + 2x₂² + 4x₃

Subject to the constraints:

x₁ + 2x₂ + x₃ ≥ 8 ;  x₁ ,x₂ , x₃ ≥ 0.

(12 + 8 )                                                                 20(a)  Derive probabilistic EOQ model.

 

(b)  Electro uses resin in its manufacturing process at the rate of 1000 gallons

Month. It cost Electro $100 to place an order for a new shipment .The holding

Cost per gallon per month is $2 and the shortage cost per gallon is $10.Historical

data show that the demand during lead time is uniform over the  range(0, 100)

gallons. Determine the optimum ordering policy for Electro.

(10 + 10)

21.    Use Wolfe’s method to solve the following QPP:

Maximize Z = 6x₁ + 3x₂ – 4x₁x₂ – 2x₁² – 3x₂²

Subject to the constraints:

x₁ + x₂ ≤ 1  ,  2x₁ + 3x₂ ≤ 4 ; x₁, x₂ ≥ 0 .

 

22. Use cutting plane algorithm to solve the following LPP:

Maximize Z = 200x₁ + 400x₂ + 300x₃

Subject to the constraints:

30x₁ +  40x₂ + 20x₃ ≤  600

20x₁ + 10x₂ + 20x₃  ≤  400

10x₁ + 30x₂ + 20x₃  ≤  800

x₁, x₂, x₃ ≥ 0  and are integers.

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

FOURTH SEMESTER – APRIL 2011

ST 4811/4807 – ADVANCED OPERATIONS RESEARCH

 

 

Date : 09-04-2011             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

Section A

 

Answer ALL questions.                                                                                  (10 x 2 =20 marks)

1. Define General Linear Programming Problem.
2. Define Pure Integer Programming Problem.
3. What is the need for inventory control?
4. What is the behaviour of customers in a queue?
5. Define dynamic Programming Problem.
6. What do you mean by Non Linear Programming Problem?
7. Define a chance constrained model.
8. Show that Q = 2 x₁² + 2 x₂² + 3 x₃² + 2 x₁ x₂ + 2 x₂ x₃ is positive definite.
9. Write the significance of Goal Programming.
10. State the use of simulation analysis.

 

SECTION B

 

Answer any FIVE questions.                                                                    (5 x 8 =40 marks)

 

1. Apply the principle of duality to solve the following: Min Z = 2 x₁ + 2 x₂ , subject to the constraints,     2 x₁ +  4 x₂ ≥ 1,   x₁ + 2 x₂ ≥ 1,  2 x₁ + x₂ ≥ 1,      x₁ , x₂ ≥ 0.
2. Explain Generalized Poisson queuing model.
3. Explain the classical EOQ model.
4. Derive Gomory’s constraint for solving a Mixed Integer Programming Problem.
5. Use Dynamic Programming Problem to solve the following LPP; Max Z =  3 x₁ + 5 x₂ subject to the constraints,     x₁  ≤ 4, x₂  ≤ 6,    3 x₁ + 2 x₂ ≤ 18,    x₁ , x₂ ≥ 0.
6. Derive the KTNC for solving a GNLPP with one inequality constraint.

 

 

 

 

1. Find the deterministic equivalent of the following problem: Min Z =  3 x₁ + 4 x₂ subject to the constraints,     P[ 3 x₁ – 2x₂  ≤ b₁] ≥ ¾,  P[  x₁/7 + 2x₂  ≥ b_2;  x₁ + x₂ /9 ≥ b₃] = 1/4 , x₁ , x₂ ≥ 0, where b₁, b₂, and b₃ are independent random variables uniformly distributed in the intervals     (-2, 2), (0, 2), (0, 4) respectively.
2. An electronic device consists of 4 components, each of which must function for the system to function. The system reliability can be improved by installing parallel units in one or more of the components. The reliability R of a component with 1, 2 or 3 parallel units and the corresponding cost C ( in 000’s) are given in the following table. The maximum amount available for this device is  Rs. 1,00,000. Use DPP to maximize the reliability of the entire system.

	j = 1		j = 2		j = 3		j = 4
Uj	R1	C1	R2	C2	R3	C3	R4	C4
1	.7	10	.5	20	.7	10	.6	20
2	.8	20	.7	40	.9	30	.7	30
3	.9	30	.8	50	.95	40	.9	40

 

 

 

 

 

 

 

SECTION C

 

Answer any TWO questions.                                                                          (2 x 20 =40 marks)

 

1. Explain Branch and Bound algorithm for solving MIPP and hence solve the following problem:

Max z = 3 x₁+  x₂  + 3 x₃   subject to the following constraints, – x₁+  2 x₂  +  x₃ ≤ 4,

4 x₂ – 3 x₂   ≤ 2,  x₁ –  3 x₂  +  2 x₃ ≤ 3,   x₁ , x₂ , x₃  ≥ 0, x₁ , x₃  are integers.

 

20. Solve the following GNLPP using KTNC, Max Z = 2 x₁ – x₁² + x₂  subject to the constraints,         2 x₁ + 3 x₂ ≤ 6, 2 x₁ + x₂ ≤ 4,    x₁,  x₂ ≥ 0.

 

21. Max Z = 6 x₁ + 3 x₂ – 4 x₁ x₂ – 2 x₁² – 3 x₂² subject to the constraints,     x₁ +  x₂ ≤ 1,

2 x₁ + 3 x₂ ≤ 4,    x₁ , x₂ ≥ 0. Show that z is strictly concave and then solve the problem by Wolfe’s algorithm.

22. (i) Derive steady state measures of performance for (M│M│1) : (GD│∞│∞) queue system.

      (ii) Explain multi-item EOQ model with storage limitation.                                                                                          

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

FOURTH SEMESTER – APRIL 2012

ST 4811 – ADVANCED OPERATIONS RESEARCH

 

 

Date : 20-04-2012             Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

PART-A

Answer all the following:                                                                                                            (10X2=20)

 

1. When is a solution to an LPP called infeasible?
2. Define Pure Integer Programming Problem.
3. Define holding costs.
4. Write down the basic components of a queuing model.
5. Write down the significance of integer programming problem.
6. What do you mean by Non Linear Programming Problem?
7. Define Dynamic Programming Problems.
8. Define Stochastic programming.
9. Provide any two applications for parallel and sequence service systems.
10. An oil engine manufacturer purchases lubricants at the rate of Rs.42 per piece from a vendor.

The requirement of these lubricants is 1,800 per year. What should be the order quantity per

order, if the cost per placement of an order is Rs.16 and inventory carrying charge per rupee per

year is only 20 paise.

 

PART  B

Answer any FIVE of the following:                                                                                       (5 X 8 = 40)

 

11) Use the graphical method to solve the following LPP:

Minimize Z = x₁ + 0.5x₂

Subject to the constraints:

3x₁ +  2x₂ ≤ 12 , 5x₁≤ 10  x₁+ x₂ ≥ 8 , – x₁+ x₂ ≥ 4 ,  x₁≥ 0 and x₂ ≥ 0.

• Write down the simplex algorithm.
• Use duality to solve the following LPP:

Maximize Z = 2x₁ + x₂

Subject to the constraints:

x₁ + 2x₂ ≤ 10  ,   x₁ + x₂ ≤ 6 ,  x₁ – x₂ ≤ 2 , x₁ – 2x­₂ ≤ 1 ; x₁,x₂ ≥ 0 .

• Write briefly about inventory management.
• Explain Branch and Bound model for solving interger programming problem.
• Write Wolfe’s algorithm to solve Quadratic Programming Problem.
• Solve the following NLPP using lagrangian multiplier principle:

Z =  x² + y² + z²

Subject to the constraints:  x + y + z = 1 , x, y , z ≥ 0

• Explain the scope of simulation and its applications.

 

 

 

 

PART – C

 Answer any two questions:                                                                                             (2x 20 = 40)

 

19.(a) Use two-phase simplex method to

Maximize Z =  5x₁ – 4x₂ + 3x₃

Subject to the constraints:

2x₁ + x₂ -6x₃ = 20 , 6x₁ + 5x₂ + 10x₃ ≤ 76 , 8x₁ – 3x₂ +6x₃ ≤ 50 ; x₁, x₂, x₃  ≥ 0

(b) Explain the characteristics of dynamic programming problem.             (12 + 8 )

 

20) Solve the following integer programming problem using Gomory’s constraints method:

Maximise Z = 7x₁+ 9x₂

Subject to –x₁ + 3x₂ ≤ 6, 7x­₁+ x₂ ≤ 35, x₁ is a n integer and x₂ ≥ 0

(20)

21)   Use Wolfe’s method to solve the following QPP:

Maximize Z = 4x₁ + 6x₂ – 2x₁x₂ – 2x₁² – 2x₂²

Subject to the constraints:

x₁ + 2x₂ ≤ 2  ; x₁, x₂ ≥ 0 .                                                                     (20)

 

22) a) Derive the steady state differential equation for the model (M/M/1) : (GD/. (12)

 

1. b) The rate of arrival of customers at a public telephone booth follows Poisson distribution,

with an average time of 10 minutes between one customer and the next. The duration of a

phone call is assumed to follow exponential distribution, with mean time of 3 minutes.

• What is the probability that a person arriving at the booth will have to wait?
• What is the average length of the non-empty queues that form from time to time?
• The Steve Telephone Nigam Ltd. will install a second booth when it is convinced that the customer would expect waiting for atleast 3 minutes for their turn to make a call. By how much time should the flow of customers increase in order to justify a second booth?
• Estimate the fraction of a day that the phone will be in use.    (8)

 

 

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