GTU Operations Research Model Question Paper – 2

GUJARAT TECHNOLOGICAL UNIVERSITY

ATMIYA INSTITUTE OF TECHNOLOGY AND SCIENCE, RAJKOT

M.C.A. SEM. IV Model Question Paper – 2
640003: Operations Research
MARKS: 70

INSTRUCTIONS:   

1. Attempt all the questions.

2.Figurestotherightindicate full marks

3. Assume suitable additional data,  if required.

Q.1

(a)

I) Explain close-loop in Transportation Problem and its properties.

II) Write an Economic interpretation of ui’s and vj’s.

5+2

(b)

I) Write the dual of following LPP.

Min Z = 12×1 + 8×2

Subject to        x1 + x2 >= 3

3×1 + x2 >= 7

x1, x2 >= 0

II) Determine an initial basic feasible solution to the following transportation problem by using a NWCM method. (Ans 12200)

D1 D2 D3 D4 Supply
S1 11 13 17 14 250
S2 16 18 14 10 300
S3 21 24 13 10 400
Demand 200 225 275 250

3+4

Q.2

(a)

Define Simulations with its merits and demerits. ExplainMonte CarloSimulations.

7

(b)

Find out the total maximum sales of the following Assignment problem

Sales Person

Districts                       A         B         C         D         E

1                      32        38        40        28        40

2                      40        24        28        21        36

3                      41        27        33        30        37

4                      22        38        41        36        36

5                      29        33        40        35        39

7

OR

(b)

Determine an initial basic feasible solution to the following transportation problem by using a LCM method. (Ans 180)

D1 D2 D3 D4 Supply
S1 1 2 1 4 30
S2 3 3 2 1 50
S3 4 2 5 9 20
Demand 20 40 30 10

5

Q.3

(a)

A firm makes two products X and Y, and has a total production capacity of 9 tonnes per day, X and Y requiring the same production capacity. The firm has a permanent contract to supply at least 2 tonnes of X and at least 3 tonne of Y requires 50 machine hours of production time. The daily maximum possible number of machine hours is 360. All the firm’s output can be sold, and the profit made is Rs. 80 per tonne of X and Rs. 120 per tonne of Y. It is required to determine the production schedule for maximum profit and to calculate this profit. (Use graphical method)

7

(b)

Determine an initial basic feasible solution to the following transportation problem by using a VAM method.

D1 D2 D3 D4 Supply
S1 19 30 50 10 7
S2 70 30 40 60 9
S3 40 8 70 20 18
Demand 5 8 7 14

7

OR

Q3

(a)

Formulate the given LP problem by applying simplex method.

MAX Z= 4x+3y

Subject to constraints,

2x+y<=1000

x+y<=800

x<=400

y<=700   and

x, y >=0       (140)

7

(b)

Determine an initial basic feasible solution to the following transportation problem by using a MODI method. (Ans. 149, P no 340)

D1 D2 D3 D4 Supply
S1 6 3 5 4 22
S2 5 9 2 7 15
S3 5 7 8 6 8
Demand 7 12 17 9

7

Q-4

(a)

Explain the following terms:

1)     Free float

2)     ES

3)     Looping and Dangling

4)     Merge and burst events

5)     AOA Vs AON

6)     Optimistic time

7)     Critical Path

7

(b)

A pipeline is due for repairs. It will cost Rs 10000 and lasts for three years. Alternately, new pipeline can be laid at a cost of Rs 30000 and lasts for 10 years. Assuming cost of capital to be 10 percent and ignoring salvage value, which alternative should be chosen? (p no 794)

7

 

Q-5

(a)

Use the graphical method to minimize the time needed to process the following jobs on the machines shown, i.e. each machine finds the job which should be done first. Also calculated the total elapsed time to complete both jobs.

JOB 1

JOB 2

Sequence Time(hrs)
A

B

C

D

E

3

4

2

6

2

Sequence Time(hrs)
B

C

A

D

E

5

4

3

2

6

7

(b)

Construct the network diagram with the following relationship. Find the critical path.

1.      A and B are the first activities of the project

2.      A and B precede C

3.      C precedes D

4.      B precedes E and F

5.      F and C precedes G

6.      E  precedes  H

7.      H precedes I and J

8.      D and J precedes K

9.      K precedes L

10.  G, I and L are terminal activities of the project

Activity           A         B         C         D         E          F          G         H         I           J           K         L

Duration          5          3          2          1          3          2          8          1          2          3          4          7

(weeks)

7

OR

Q-5

(a)

1) A company operating 50 weeks in a year is concerned about its stocks of copper cable. This costs rs 240 a meter and there is a demand for 8000 meters a week. Each replenishment costs rs 1050 for administration and rs 1650 for delivery, while holding coss are estimated at 25% of value held a year. Assumng no shortages are allowed what is the optimal inventory policy for the company??

How would this analysis differ if the company wanted to maximize profit rather than minimize cost?

What is the gross profit if the company sells cable for rs 360 a meter? (p no 614)

II) Explain following keywords:

  • EOQ
  • (T,S) Policy
  • Shortage cost
  • Lead time

4

3

(b)

 A construction company is considering a large building project. The mean time estimates, the variance of these time estimates, manpower requirements and precedence requirement for the various activities are given in the following table:

Activity           Immediate          Mean time estimates            Variance (days squared)

Predecessors                (days)

A                                -                       7                                  3

B                                 A                     5                                  1

C                                 B                     8                                  5

D                                 A                     20                                5

E                                  A                     11                                9

F                                  C, D                7                                  6

G                                 D                     2                                  4

H                                 D, E                 6                                  4

I                                   F,G,H              10                                9

(i)                 Draw a PERT network for the building project.

(ii)               Assuming manpower resources are unlimited, find a critical path, as well as the minimum expected completion time.

7

 

 ===== All The Best =====

Posted in Operations Research Exam Papers | Leave a comment

GTU Operations Research Model Question Paper – 1

GUJARAT TECHNOLOGICAL UNIVERSITY

ATMIYA INSTITUTE OF TECHNOLOGY AND SCIENCE, RAJKOT

M.C.A. SEM. IV Model Question Paper – 1

640003: Operations Research

MARKS: 70

INSTRUCTIONS:

1. Attempt all the questions.

2.Figurestotherightindicate full marks

3. Assume suitable additional data, if required.

Q.1

(a)

Explain the Mathematical Model of Transportation Problem.

7

(b)

Determine an initial basic feasible solution to the following transportation problem by using a NWCM method.

D1 D3 D4 Supply
S1 19 30 50 10 7
S2 70 30 40 60 9
S3 40 8 70 20 18
Demand 5 8 7 14

7

Q.2

(a)

I) Write a note on applications of Operations Research.

II) Explain Slack, surplus and artificial variables with example.

3+4

(b)

I) What is pure birth process and pure death process in queuing theory?II)Explain following terms:

a. Unit Transportation Cost

b. Feasible Solution

c. Basic Feasible Solution

d. Optimal Solution

e. Occupied cells and non- Occupied cells

f. Rim condition

2+5

OR

(b)

Determine an initial basic feasible solution to the following transportation problem by using a Penalty method. (Ans 12075)

D1 D2 D3 D4 Supply
S1 11 13 17 14 250
S2 16 18 14 10 300
S3 21 24 13 10 400
Demand 200 225 275 250

7

Q.3

(a)

Determine an initial basic feasible solution to the following transportation problem by using a u – v method or method of multipliers. (Ans. 149, P no 340)

D1 D2 D3 D4 Supply
S1 6 3 5 4 22
S2 5 9 2 7 15
S3 5 7 8 6 8
Demand 7 12 17 9

7

(b)

Use the Big M method to solve the following LP problem.

(ans: x1=4,x2=1,s1=0,s2=12, Z=23)

Min Z=5×1+3×2

Subject to constraints,

2×1+4×2<=12

2×1+2×2=10

5×1+2×2>=10

and x1, x2>=0

7

OR

Q.3

(a)

Determine an initial basic feasible solution to the following transportation problem by using a Penalty method. (Ans 12075)

D1 D2 D3 D4 Supply
S1 11 13 17 14 250
S2 16 18 14 10 300
S3 21 24 13 10 400
Demand 200 225 275 250

7

(b)

Determine an initial basic feasible solution to the following transportation problem by using a u – v method or method of multipliers. (Ans. 149, P no 340)

D1 D2 D3 D4 Supply
S1 6 3 5 4 22
S2 5 9 2 7 15
S3 5 7 8 6 8
Demand 7 12 17 9

7

Q-4

(a)

Activity Optimistic time Most likely time Pessimistic time
1-2 1 1 7
1-3 1 4 7
1-4 2 2 8
2-5 1 1 1
3-5 2 5 14
4-6 2 5 8
5-6 3 6 15

Draw network diagram and find out floats for all the activities.

Calculate the variance and standard deviation of the project length. Find out the probability of completing the project at least 4 weeks earlier than expected time. (P no – 557)

7

(b)

A company contains 10000 resistors. When any resistor fails, it is replaced. The cost of replacing a resistor individually is Rs 1 only. If all the resistors are replaced at the same time, the cost per resistor would be reduced to 35 paise. The percentage of surviving resistors say S(t) at the end of month t and the probability of failure P(t) during the month t are:

t S(t) P(t)
01

2

3

4

5

6

10097

90

70

30

15

0

-0.03

0.07

0.20

0.40

0.15

0.15

7

OR
Q-4

(a)

1) Explain Forward Pass method and backward Pass method with example.2) Explain the Dual primal relationship in brief, write the Dual of following LPP.

Min Z= x1 + x2 + x3

Subject to, x1 + 2×2 + 3×3 = 7

X1 + 3×2 <=5

2×1 + x3 >=6

x1, x2 >= 0 and x3 unrestricted.

4+3

(b)

I) Write an algorithm to store unbalanced transportation problem using lowest cost entry method.2) A bakery keeps stock of a popular brand of cake. Previous experience shows the daily demand pattern for the item with associated probabilities, as given below:

Daily demand (number) Probability
010

20

30

40

50

0.010.20

0.15

0.50

0.12

0.02

Use following random numbers to simulate the demand for next 10 days.

  • Random nos: 25 39 65 76 12 05 73 89 19 49

Also estimate the daily average demand for the cakes on the basis of simulated data.

2+5

Q-5

(a)

Find the sequence that minimizes the total elapsed time required to complete the following tasks on two machines. Also find the idle time.

Task Machine 1 Machine 2
AB

C

D

E

F

G

H

I

25

4

9

6

8

7

5

4

68

7

4

3

9

3

8

11

7

(b)

Explain EOQ model with different rates of Demand.

7

OR
Q-5

(a)

i) The production department for a company requires 3600 kg of raw material for manufacturing a particular item per year. It has been estimated that the cost of placing an order is Rs 36 and the cost of carrying inventory is 25 percent of the investment in the inventories. The price is Rs 10 per kg.The purchase manager wishes to determine an ordering policy for raw material. (p no 612)II) Explain EOQ model with constant rate of demand. 52

(b)

I) Explain value of game and saddle point with example.II) Find the range of values of p and q which will render the entry (2,2) a saddle point for the game:

Player A

Player B

B1 B2 B3
A1A2

A3

210

4

47

P

5q

6

4

3

===== All The Best =====

Posted in Operations Research Exam Papers | Leave a comment

Operations Research Pre GTU Exam Paper

GUJARAT TECHNOLOGICAL UNIVERSITY

ATMIYA INSTITUTE OF TECHNOLOGY AND SCIENCE, RAJKOT

M.C.A. SEM. IV PRE – GTU
Sub: Operations Research
MARKS: 70

INSTRUCTIONS:

1. Attempt all the questions.

2. Figures to the right indicate full marks

3. Assume suitable additional data, if required.

Q.1

(a)

1) Explain following terms:a. Basic Feasible Solution

b. Optimal Solution

c. Occupied cells and non- Occupied cells

d. Rim condition

e. Unbalanced Transportation Problem

f. Degenerate Solution

g. Non- Degenerate Solution

7

(b)

I) What is OR? Explain the role of OR in Decision Making process with its applications.II) Determine an initial basic feasible solution to the following transportation problem by using a NWCM method. (Ans 12200)

D1 D2 D3 D4 Supply
S1 11 13 17 14 250
S2 16 18 14 10 300
S3 21 24 13 10 400
Demand 200 225 275 250

4

3

Q.2

(a)

Explain the following Terms:

a. Reneging

b. Jokeying

c. Transient State

d. Line length or Queue Size

e. Pure Birth process

f. {(a/b/c) : (d/c)}

g. {(M/M/1) : (∞/SIRO)}

h. Pre-emptive Priority

7

(b)

I) Write an Economic interpretation of ui’s and vj’s.II) Five men are available to do five different jobs. Find the assignment of men to jobs that will minimize the total time taken. (Ans 13 hours)

JOBS

I II III IV V
A 2 9 2 7 1
B 6 8 7 6 1
C 4 6 5 3 1
D 4 2 7 3 1
E 5 3 9 5 1

2

5

OR

(b)

I) Determine an initial basic feasible solution to the following transportation problem by using a LCM method. (Ans 180)

D1 D2 D3 D4 Supply
S1 1 2 1 4 30
S2 3 3 2 1 50
S3 4 2 5 9 20
Demand 20 40 30 10

II) A bakery keeps stock of a popular brand of cake. Previous experience shows the daily demand pattern for the item with associated probabilities, as given below:

Daily demand (number) Probability
010

20

30

40

50

0.010.20

0.15

0.50

0.12

0.02

Use following random numbers to simulate the demand for next 10 days.

  • Random nos: 25 39 65 76 12 05 73 89 19 49

Also estimate the daily average demand for the cakes on the basis of simulated data.

3

4

Q.3

(a)

The standard weight of a special purpose brick is 5 kg and it contains two basic ingredients B1 and B2. B1 costs Rs. 5 per kg and B2 costs Rs 8 per kg. Strength considerations dictate that the brick should contain not more than 4 kg of B1 and a minimum of 2 kg of B2. Since the demand for the product is likely to be related to the price of the brick, find out graphically the minimum cost of the brick satisfying the above conditions.(ans: Z =31 A(3,2) )

7

(b)

Determine an initial basic feasible solution to the following transportation problem by using a VAM method.

D1 D2 D3 D4 Supply
S1 19 30 50 10 7
S2 70 30 40 60 9
S3 40 8 70 20 18
Demand 5 8 7 14

7

OR

Q.3

(a)

Formulate the given LP problem by applying simplex method.MAX Z= 4x+3y

Subject to constraints,

2x+y<=1000

x+y<=800

x<=400

y<=700 and

x, y >=0 (140)

7

(b)

Determine an initial basic feasible solution to the following transportation problem by using a MODI method. (Ans. 149, P no 340)

D1 D2 D3 D4 Supply
S1 6 3 5 4 22
S2 5 9 2 7 15
S3 5 7 8 6 8
Demand 7 12 17 9

7

Q-4

(a)

Activity Optimistic time Most likely time Pessimistic time
1-2 1 1 7
1-3 1 4 7
1-4 2 2 8
2-5 1 1 1
3-5 2 5 14
4-6 2 5 8
5-6 3 6 15

Draw network diagram and find out floats for all the activities.

Calculate the variance and standard deviation of the project length. Find out the probability of completing the project at least 4 weeks earlier than expected time. (P no – 557)

7

(b)

An engineering company is offered a material handling equipment A. It is priced at Rs 60000 including cost of installation and the costs for operation and maintenance are estimated to be Rs 10000 for each of the first five years, increasing every year by Rs 3000 in the sixth and subsequent years. The company expects a return of 10 percent on all its investment. What is the optimal replacement period? ( p no 799)

7

OR
Q-4:

(a)

Define Simulation. Write the steps of simulation process. Explain Monte Carlo Simulation.

7

(b)

We have five jobs each of which must be processed on the 2 machines A and B. in order AB. Processing times in hours are given in the table below: (p no 962)

JOB Machine A Machine B
12

3

4

5

51

9

3

10

26

7

8

4

7

Q-5

(a)

Find an optimal sequence for the following sequencing problems of four jobs and five machines when passing is not allowed of which processing time in hours is given below:

JOB M1 M2 M3 M4 M5
AB

C

D

76

5

8

56

4

3

24

5

3

35

6

2

910

8

6

(P NO 969) ANS: 51 hrs

7

(b)

II) Draw network diagram & find out the Critical Path by considering following activity and duration.

Activity Predecessors Duration
A - 6
B A 4
C B 7
D A 2
E D 4
F E 10
G - 2
H G 10
I J,H 6
J - 13
K A 9
L C,K 3
M I,L 5

1) In a railway marshalling yard, goods trains arrive at a rate of 30 trains per day. Assuming that the inter arrival time follows an exponential distribution and the service time distribution is also exponential with an average of 36 minutes. Calculate

a) expected queue size (line length)

b) probability that the queue size exceeds 10

if the input of trains increases to an average of 33 per day, what will be the change in a) and b) ?

4

3

OR
Q-5

(a)

1) A manufacturer has to supply his customer with 600 units of his product per year. Shortages are not allowed and the storage cost amounts to Rs 0.60 per unit per year. The set up cost per run is rs 80. find the optimum run size and the minimum average yearly cost. (p no 623)II) Explain following keywords:

  • (s,Q) Policy
  • (T,s,S) Policy
  • Ordering cost
  • Carrying cost

3

4

(b)

I) Explain MINIMAX and MAXIMIN principles and Saddle point.II) Find out the value of game and optimal strategies for each player from following payoff matrix.

Player A

Player B

B1 B2 B3
A1A2

A3

-2-5

-5

15-6

20

-2-4

-8

4

3

===== All The Best =====

Posted in Operations Research Exam Papers | Leave a comment

Simulation Assignment:

Simulation Assignment:

1)      Explain the term Simulation. Give some real life examples where simulation is used.

2)      Define types of Simulation.

3)      Write steps of Simulation Process.

4)      Give advantages and disadvantages of simulation.

5)      Explain Monte Carlo Simulation.

6)      Explain Random number generation.

Posted in Operations Research Theory | Leave a comment

Replacement and Maintenance Models Assignment

Replacement and Maintenance Models Assignment:

1)      Write a note on Types of failure.

2)      Explain the replacement policy for items whose running cost increases with time and value of money remains constant during a period.

3)      Explain the replacement policy for items whose running cost increases with time but value of money changes with constant rate during a period.

4)      Explain the replacement policy when maintenance cost increases with time and the money value decreases with constant rate.

5)      When group replacement policy is adopted?

6)      Explain the following terms:

  1. Discount rate or Depreciation value
  2. P(t) (Probability of failure of any item at age t)
  3. C(t)
  4. F(t)
  5. R(t)
  6. Individual Replacement Vs Group Replacement
  7. Mortality theorem
  8. Mortality tables
  9. Pwf (Present Worth Factor)
  10. Retrogressive Failure
  11. Progressive Failure
Posted in Operations Research Theory | Leave a comment

Queuing Theory Assignment

Queuing Theory Assignment:

1)      Explain Queuing Theory fundamentals.

2)      Explain Queuing cost vs level of services with figure.

3)      Describe the components of Queuing System Explain the structure of a Queuing System.

4)      Write a note on pattern of arrivals at the system.

5)      Explain Queue Discipline.

6)      Explain service mechanism.

7)      Explain arrangements of service facilities in series and in parallel.

8)      Explain Pure birth process and Pure death process.

9)      Write a note on performance measures of a queuing system.

10)  Explain Single Server Queuing models.

11)  Explain Model 1 {(M/M/1) : (∞/FCFS)}

12)  Explain Model 2 {(M/M/1) : (∞/SIRO)}

13)  Explain Model3 {(M/M/1) : (N/FCFS)}

 

14)  Explain the following Terms:

  1. Balking
  2. Reneging
  3. Jokeying
  4. TransientState
  5. Steady State
  6. Line length or Queue Size
  7. Queue length
  8. Pure Birth process
  9. Pure Death process
  10. j.        {(a/b/c) : (d/c)}
  11. k.      {(M/M/1) : (∞/SIRO)}
  12. l.        {(M/M/1) : (∞/FCFS)}
  13. m.    {(M/M/1) : (N/FCFS)}
  14. Pre-emptive Priority
  15. Non Pre-emptive Priority

 

Posted in Operations Research Theory | Leave a comment

PERT/CPM Assignment

PERT/CPM Assignment:

1) Explain the significance of PERT/CPM.

2) Give the basic difference between PERT and CPM.

3) Explain the Project management phases.

4) Explain PERT/CPM network components. (Events & Activities)

5) Write a note on Events.

6) Write a note on Activities.

7) Explain types of network of the activities in the project. (AOA &AON)

8) Mention the rules for AOA network construction.

9) Explain Forward Pass method.

10) What is float? Explain different types of floats.

11) Explain Backward Pass method.

12) Explain three time estimates in PERT.

 

13) Explain following terms.

  1. Event
  2. Activity (or task)
  3. Redundant (or Dummy) Activity
  4. Looping
  5. Dangling
  6. Earliest time & Latest time
  7. Ei , Li , ESij, LSij, EFij, LFij, tij
  8. Slack (or float or free) time
  9. Total float & free float & independent float
  10. Critical Path
  11. Critical Activities
  12. Non-Critical Activities
  13. E-values
  14. L-values
  15. Optimistic time (to or a)
  16. Pessimistic time (tp or b)
  17. Most likely time (tm or m)
  18. Beta distribution
  19. Activity variance and Project variance
  20. Expected Time (Te)
  21. Standard deviation of critical path

 

Posted in Operations Research Theory | Leave a comment