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West Bengal Institute of Technology (WBIT) 2009-4th Sem M.C.A (wbut) - exam paper

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I . .

CS/MCA/SEH-4/MM-401/09 3

ENGINEERING A MANAGEMENT EXAMINATIONS, JUNE - 2009 OPERATIONS RESEARCH AND OPTIMIZATION TECHNIQUES

SEMESTER - 4

Time : 3 Hours ] , I Full Marks : 70

Graph sheets are provided at the end of the booklet.

GROUP - A (Multiple Choice Type Question*)

1. Choose the correct alternatives for any ten of the following : 10 x 1 = 10

4 Given a system of m simultaneous equations in n unknowns { m < n ), the number of basic variables will be a) m b) n

c) m-n d) m + n. [___

ii) A two-person zero-sum game is said to be fair if

a) both the players have equal number of strategies

b) the game has a saddle point

c) the game does not have a saddle point

d) the value of the game is zero. _

iii) In an assignment problem involving four workers and three jobs, the total number of assignments possible is

a) 4 b) 3

c) 7 ............d) 21. 1

ivj In {(M/M/1): (~ /F/FO )}, average length of a non-empty queue is _* X2 n U

* H ^b) H-X

c) -- d) none of these.

(H-X)^a -

v) In a flow pattern X/_w = %f_M when the vertex A is

a) arbitrary vertex b) any vertex other than source

c) sourpe d) none of these. |

The formula for finding the minimum Inventory cost under the purchasing model without shortage is

C8/X0CA/SW-4/lO(-4Ol/O9

Vi)

V2cif

b)

d)

a} 2RCjC₃

none of these.

c)

2i?C

A simplex In two dimension Is

a) rectangle

t

c) triangle

vii)

line segment pentagon.

b)

d)

vUiJ When a positive quantity R is divided into five parts, the maximum value of their product is

a) 5k b) t k/5 >⁵

c) (k5)⁵ d) 5 (k/5 J.

The optimality condition for minimization LPP in the simplex method Is

i /

a) Zj-CjZ 0 b) Zj-Cji 0

c) Zj- Cj <0 d) none of these.

What Is the method used to solve an LPP involving artificial variables ? a) Simplex method b) Chames M method

c) - VAM d) None of these.

Consider the following game :

' Player (A)

Player (B)

	1	2	3
1	. 5	50	50
2	1	1	0-1
3 .	10	1	10

M ⁵⁰ b)

d)

50.

Then the value of the game Is * f

*:

C8/MCA/SEM-4/MM-401/09 5

xll) The total number of possible solutions for nxn assignment problem is always a) n b) n 1

c) 1 d) n I

jdifl The point of intersection of pure strategies in a game is called

a) value of the game b) saddle point

c) mixed strategy d) optimal strategy.

xiv) The system of simultaneous equations given by 2x_r + 3x2 + 4 = 0 3*! + 42 +6*0 4Xi + 5xg+ 8 m 0

is. . .

a) consistent b)

Inconsistent none of these.

% - . . c) possessing unique solution d)

GROUP - B (8hort Answer TypeQntfons)

3x5-ir

Answer any three of the following.

2. Solve the following LPP by graphical method : Minimize Z * 20 Xj + 10 Xj Subject to ; x_} + 2 40 3x_x X2& 30

4x,+ 3x> 60

3. Arrivals at a telephone booth are considered to be Poisson with an average time of ' 10 minutes between one arrival and the next. The length of the phone call is assumed

to be distributed exponentially with mean 3 minutes.

1) What Is the prolablllty that a person arriving at the booth will have to wait ?

ii) The telephone department will install a second booth when convinced that an arrival would expect waiting for at least 3 minutes for a phone call. By how much should the flow of arrivals increase in order to justify a second booth ?

ill) What Is the average length of the queue that forms time to time ?

4, Find the optimal strategies and the value of the game G whose pay-off matrix Is

	Player B
Player A	- 2	6
	5	1

There are five jobs, each of which Is to be processed through two machines M _l and M₂ In the order M, M₂ . The processing hours are the following :

Jobs	1	2	3	4	5
M,	3	8	5	7	4
m₂	4	10	6	5	8

Determine the optimal sequence of the five jobs, the minimum elapsed time and the Ideal times for the machines M_l and M₂

Find the basic solution or solutions, If there be any, of the set of equations 2x₁ + 4x₂ - 2x₃ = 1 lOXj + 3*₂ - 7x₃ = 33 Solve the assigment problem :

7.

	1	2	3
A	7	5	6
B	8	4	7
C	9	6	4

/

GROUP - C

(Long Answer Type Questions)

Answer any three questions.

3x 15 = 45

8. a) Solve the following LPP by Simplex method. ' Maximize Z = 60.x j + 50x₂

7

subject to

x_l + 2x₂ < 40 3Xj + 2x₂ < 60 x_r, x₂>0 ,

b) Solve the following linear programming problem by Chames Big M method (if possible ) :

Maximize Z = 2x_l - x₂ + 5x₃

subject to

Xj + 2x₂ + 2x₃ S 2

5 ->

2 + 3x₂ + 4x₃ = 12 4Xj + 3x₂ + 2x₃ > 24 and Xj , x₂ , x₃ S 0

8

What is the optimal value of Z?

9. a) Write down the dual of the following LPP :

Minimize Z = 3x, + x₂

subject to

2x, + x₂ 14,

Xj - x₂ > 4, Xj , x₂ > 0,

and solving the dual problem find out the optimal solution and the optimal value

of the objective function.

8

b) Find the optimal solution to the following integer programming problem : 7

Maximize Z = x₁ -x₂

subject to Xj + 2x₂ 4,

6Xj + 2x₂ < 9, Xj , x₂ S O and

Xj, x₂ are integers.

CS/MCA/SEM-4 /UU-401/09 8

10. a) Solve the following balanced Transportation Problem :

		^Di		3	Capacity
	FI	2	3	11 7	6
	F2	1	0	6 1	1
	F3	5	8	15 9	10
	Requirement	7	5	3 2	17
b)	Use dynamic programming to solve the following problems : 7
	Minimize Z	= yi + y₂²	+ y₃³ 4
	subject to y	i ⁺ y₂ ⁺ y₃	> 15
	and y	t . y₂ . y₃>.
11. a)	Solve the following two-person-zero game using the method of dominance : 8
			Player B
		5	- 10	9 0
	Player A	6	7	8 1
		8	7	15 1	*
		3	4	- 1 4
b)	Find the optimal assignments			to find the minimum cost for the assigment
	following cost matrix :				7
*			J1	J2	J3
	PI		12	24	15
**	P2		23	18	24
	P3		30	14	28
12. Use dynamic programming to solve
	Maximum	Z= Y, .	^Y2 ^y3
	Subject to constraints
		^Yl ^{+ Y}2	₊ y₃ = 5
	and Vj . Y₂ , Y₃		0.		15
4617 (12/06) \|

CS/MCA/SEM-4/MM-401/09

13. a) Write the differences between PERT and CPM.

b) The following information is given :

Activity	1 - 2	2-3	2-4	3-5	4-6	5-el 5-7		6-7
Pessimistic time (weeks)	3	9	6	8	8	0	5	8
Most likely time (weeks)	3	6	4	6	6	0	4	5
Optimistic time (weeks)	3	3	2	4	4	0	3	3

Draw the network diagram for the above. Calculate

1) Variance of each activity

U) Critical path and expected project length

US) The probability that the project will be completed in 23 weeks.

Z-value	1.9	1.91	1,92	1.93	1.94
Probability	0.9713	0.9719	0.9726	0.9732	0.9738

4 + 2 + (2+l) + 4

,4. a) Assuming that the expected times are normally distributed, find the "probability

of meeting the schedule date for the given network :

Job	i.	1 - 3	2 - 4	3-4,	4-5	3*5
V	2	9	5	2	6	8
' *m	5	12	14		6	17
h>	14	15	17	12	12	20

Schedule project completion date Is 30 days, Also find the date on which the

project manager can complete the project with a probability of 0.90.

b) A small project consists of seven activities for which the relevant data are

given :

Activity	Immediate predecessors	Time ( days)
A	-	2
B	-	1
C	A	3
D	A. B	2
E	C. D	1
F	B, D	3
G	E, F	1

I) Draw the network diagram.

10

CS/MGA/ SEM-4/MM-401 / 09

II) Indicate the critical path and calculate the total float and free float for each . activity. ' 7 15. aj The following network gives the distance in miles between pairs of entries

1, 2............ and 8. Find the shortest rout-between city 1 and 8 using

Dljkstras Algorithm. 8

b) The following table shows the jobs of a network along with their time estimates. The time estimates are in days :

Job	1 - 2	1 - 6	2 - 9	2 - 4	3 - 5	4 - 5	5 - 8	6 - 7	7 - 8
tj	3	2	6	2	5	3	1	3	4
*m	6	5	12	5	11	6	4	9	9
	15	14	30	8	17	15	7	27	28

¹ i) Find the critical path.

ii) Find the probability that the project is completed in 31 days. [ P( z -2.1667 ) * 0.0114 ].

END

4617 (12/06)

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