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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