Download GTU (Gujarat Technological University) BE/BTech (Bachelor of Engineering / Bachelor of Technology) 2018 Winter 7th Sem New 2171901 Operation Research Previous Question Paper
1
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER ?VII (NEW) EXAMINATION ? WINTER 2018
Subject Code: 2171901 Date: 26/11/2018
Subject Name: Operation Research
Time: 10:30 AM TO 01:00 PM Total Marks: 70
Instructions:
1. Attempt all questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.
Q.1 (a) Illustrate graphically for Linear Programming Problem;
(a) No-feasible solution (b) Unbounded solution.
03
(b) A firm plans to purchase atleast 200 quintals of scrap containing high quality
metal X and low quality metal Y. It decides that the scrap to be purchased
must contain atleast 100 quintals of X-metal and not more than 35 quintals
of Y-metal. The firm can purchase the scrap from two suppliers (A and B) in
unlimited quantities. The percentage of X and Y metals in terms of weight in
the scraps supplied by A and B is given below:
Metals Suppliers A Suppliers B
X 25% 75%
Y 10% 20%
The price of supplier A?s scrap is Rs. 200 per quintal and that of supplier B?s
is Rs. 400 per quintal. Formulate this problem as LP model to minimize total
purchase cost.
04
(c) Solve by Simplex method the following L.P. problem
Minimize ?? = 500 ?? 1
+ 200 ?? 2
Subjected to the constraints,
3 ?? 1
+ 2 ?? 2
? 90
?? 1
? 10
?? 2
= 2 ?? 1
?? 1
, ?? 2
? 0
07
Q.2 (a) Discuss in brief the areas of application of Operation Research. 03
(b) Find the dual problem for the following:
Minimize ?? = 5?? 1
? 6?? 2
+ 4?? 3
Subject to the constraints,
3?? 1
+ 4?? 2
+ 6?? 3
? 9
?? 1
+ 3?? 2
+ 2?? 3
? 5
7?? 1
? 2?? 2
? ?? 3
? 10
?? 1
? 2?? 2
+ 4?? 3
? 4
2?? 1
+ 5?? 2
? 3?? 3
= 3
?? 1
, ?? 2
, ?? 3
? 0
04
(c) The data for a network is given below.
Activity
Time in days Direct cost (Rs.)
Normal Crash Normal Crash
1-2 10 4 2300 3100
1-3 12 3 1700 2600
2-4 20 10 2400 4400
3-5 10 9 1100 1500
4-5 6 6 800 800
07
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1
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER ?VII (NEW) EXAMINATION ? WINTER 2018
Subject Code: 2171901 Date: 26/11/2018
Subject Name: Operation Research
Time: 10:30 AM TO 01:00 PM Total Marks: 70
Instructions:
1. Attempt all questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.
Q.1 (a) Illustrate graphically for Linear Programming Problem;
(a) No-feasible solution (b) Unbounded solution.
03
(b) A firm plans to purchase atleast 200 quintals of scrap containing high quality
metal X and low quality metal Y. It decides that the scrap to be purchased
must contain atleast 100 quintals of X-metal and not more than 35 quintals
of Y-metal. The firm can purchase the scrap from two suppliers (A and B) in
unlimited quantities. The percentage of X and Y metals in terms of weight in
the scraps supplied by A and B is given below:
Metals Suppliers A Suppliers B
X 25% 75%
Y 10% 20%
The price of supplier A?s scrap is Rs. 200 per quintal and that of supplier B?s
is Rs. 400 per quintal. Formulate this problem as LP model to minimize total
purchase cost.
04
(c) Solve by Simplex method the following L.P. problem
Minimize ?? = 500 ?? 1
+ 200 ?? 2
Subjected to the constraints,
3 ?? 1
+ 2 ?? 2
? 90
?? 1
? 10
?? 2
= 2 ?? 1
?? 1
, ?? 2
? 0
07
Q.2 (a) Discuss in brief the areas of application of Operation Research. 03
(b) Find the dual problem for the following:
Minimize ?? = 5?? 1
? 6?? 2
+ 4?? 3
Subject to the constraints,
3?? 1
+ 4?? 2
+ 6?? 3
? 9
?? 1
+ 3?? 2
+ 2?? 3
? 5
7?? 1
? 2?? 2
? ?? 3
? 10
?? 1
? 2?? 2
+ 4?? 3
? 4
2?? 1
+ 5?? 2
? 3?? 3
= 3
?? 1
, ?? 2
, ?? 3
? 0
04
(c) The data for a network is given below.
Activity
Time in days Direct cost (Rs.)
Normal Crash Normal Crash
1-2 10 4 2300 3100
1-3 12 3 1700 2600
2-4 20 10 2400 4400
3-5 10 9 1100 1500
4-5 6 6 800 800
07
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2
Considering indirect cost as Rs. 1000 per/day.
(i) Find the normal duration of project completion and corresponding cost.
(ii) What can be minimum compression of project and corresponding total
time?
(iii) Find the optimal duration of project completion and related cost.
OR
(c) A construction company is preparing a PERT network for laying the
foundation of a new art museum. Given the following set of activities, their
predecessor requirements and three time estimates of completion time:
Activities Predecessors
Time in weeks
Optimistic Pessimistic
Most
Likely
A None 2 4 3
B None 8 8 8
C A 7 11 9
D B 6 6 6
E C 9 11 10
F C 10 18 14
G C, D 11 11 11
H F, G 6 14 10
I E 4 6 5
J I 3 5 4
K H 1 1 1
(a) Draw the PERT network.
(b) What is the expected time of the duration of the project?
(c) What is the probability that the project will be completed in 40
weeks?
Use the following data :
Z 1.0 1.22 1.57 1.87
Probability 0.8413 0.8888 0.9418 0.9693
07
Q.3 (a) Define the following terms: balking, reneging and jockeying. 03
(b) Explain in brief characteristic of queuing theory. What is traffic intensity? If
traffic intensity of a system is given to be 0.76, what percent of time the
system would be idle?
04
(c) The following is the pay-off matrix between player X and player Y. Find the
optimal strategies, their frequencies and the value of the game. Use rule of
dominance and oddment in calculations.
Player X
Player Y
A B C D
I 0.25 0.20 0.14 0.30
II 0.27 0.16 0.12 0.14
III 0.35 0.08 0.15 0.19
IV -0.02 0.08 0.13 0.00
07
OR
Q.3 (a) What do you understand by ?zero-sum? in the context of game theory?
Explain the meaning following terms used in game theory;
i. Saddle Point
ii. Pure Strategy
iii. Mixed Strategy
07
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1
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER ?VII (NEW) EXAMINATION ? WINTER 2018
Subject Code: 2171901 Date: 26/11/2018
Subject Name: Operation Research
Time: 10:30 AM TO 01:00 PM Total Marks: 70
Instructions:
1. Attempt all questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.
Q.1 (a) Illustrate graphically for Linear Programming Problem;
(a) No-feasible solution (b) Unbounded solution.
03
(b) A firm plans to purchase atleast 200 quintals of scrap containing high quality
metal X and low quality metal Y. It decides that the scrap to be purchased
must contain atleast 100 quintals of X-metal and not more than 35 quintals
of Y-metal. The firm can purchase the scrap from two suppliers (A and B) in
unlimited quantities. The percentage of X and Y metals in terms of weight in
the scraps supplied by A and B is given below:
Metals Suppliers A Suppliers B
X 25% 75%
Y 10% 20%
The price of supplier A?s scrap is Rs. 200 per quintal and that of supplier B?s
is Rs. 400 per quintal. Formulate this problem as LP model to minimize total
purchase cost.
04
(c) Solve by Simplex method the following L.P. problem
Minimize ?? = 500 ?? 1
+ 200 ?? 2
Subjected to the constraints,
3 ?? 1
+ 2 ?? 2
? 90
?? 1
? 10
?? 2
= 2 ?? 1
?? 1
, ?? 2
? 0
07
Q.2 (a) Discuss in brief the areas of application of Operation Research. 03
(b) Find the dual problem for the following:
Minimize ?? = 5?? 1
? 6?? 2
+ 4?? 3
Subject to the constraints,
3?? 1
+ 4?? 2
+ 6?? 3
? 9
?? 1
+ 3?? 2
+ 2?? 3
? 5
7?? 1
? 2?? 2
? ?? 3
? 10
?? 1
? 2?? 2
+ 4?? 3
? 4
2?? 1
+ 5?? 2
? 3?? 3
= 3
?? 1
, ?? 2
, ?? 3
? 0
04
(c) The data for a network is given below.
Activity
Time in days Direct cost (Rs.)
Normal Crash Normal Crash
1-2 10 4 2300 3100
1-3 12 3 1700 2600
2-4 20 10 2400 4400
3-5 10 9 1100 1500
4-5 6 6 800 800
07
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2
Considering indirect cost as Rs. 1000 per/day.
(i) Find the normal duration of project completion and corresponding cost.
(ii) What can be minimum compression of project and corresponding total
time?
(iii) Find the optimal duration of project completion and related cost.
OR
(c) A construction company is preparing a PERT network for laying the
foundation of a new art museum. Given the following set of activities, their
predecessor requirements and three time estimates of completion time:
Activities Predecessors
Time in weeks
Optimistic Pessimistic
Most
Likely
A None 2 4 3
B None 8 8 8
C A 7 11 9
D B 6 6 6
E C 9 11 10
F C 10 18 14
G C, D 11 11 11
H F, G 6 14 10
I E 4 6 5
J I 3 5 4
K H 1 1 1
(a) Draw the PERT network.
(b) What is the expected time of the duration of the project?
(c) What is the probability that the project will be completed in 40
weeks?
Use the following data :
Z 1.0 1.22 1.57 1.87
Probability 0.8413 0.8888 0.9418 0.9693
07
Q.3 (a) Define the following terms: balking, reneging and jockeying. 03
(b) Explain in brief characteristic of queuing theory. What is traffic intensity? If
traffic intensity of a system is given to be 0.76, what percent of time the
system would be idle?
04
(c) The following is the pay-off matrix between player X and player Y. Find the
optimal strategies, their frequencies and the value of the game. Use rule of
dominance and oddment in calculations.
Player X
Player Y
A B C D
I 0.25 0.20 0.14 0.30
II 0.27 0.16 0.12 0.14
III 0.35 0.08 0.15 0.19
IV -0.02 0.08 0.13 0.00
07
OR
Q.3 (a) What do you understand by ?zero-sum? in the context of game theory?
Explain the meaning following terms used in game theory;
i. Saddle Point
ii. Pure Strategy
iii. Mixed Strategy
07
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3
(b) Arrival rate of telephone calls at a telephone booth is according to Poisson
distribution, with an average time of 9 minutes between consecutive arrivals.
The length of telephone call is exponentially distributed with a man of 3
minutes. Find:
i. Determine the probability that a person arriving at the booth will have to
wait.
ii. Find the average queue length that forms from time to time.
iii. The telephone company will install a second booth when conveniences
that an arrival would expect to have to wait at least four minutes for the
phone. Find the increase in flow of arrivals, which will justify a second
booth.
iv. What is the probability that an arrival will have to wait for more than 10
minutes before the phone is free?
v. Find the fraction of a day that the phone will be in use.
07
Q.4 (a) Explain the meaning of following items in inventory management;
a. Re-order Level
b. Buffer Stock
03
(b) Discuss the similarity and differences between PERT and CPM. 04
(c) A stockiest of a particular commodity makes a profit of Rs. 30 on each sale
made within the same week of purchase; otherwise he incurs a loss of Rs. 30
on each item. The data on the past sales are given below:
No. of items sold
within the same week
5 6 7 8 9 10 11
Frequency 0 9 12 24 9 6 0
i. Find out the optimum number of items the stockiest should buy every
week in order to maximize the profit.
ii. Calculate the expected value of perfect information.
07
OR
Q.4 (a) Explain ABC analysis. 07
(b) The truck-owner finds from his past experience that the maintenance costs
are Rs. 200 for the first year and then increase by Rs. 2000 every year. The
cost of truck type A is Rs. 9000. Determine the best age at which to replace
the truck. If the optimum replacement is followed what will be the average
yearly cost of owning and operating the truck? Truck of type B costs Rs.
20000. Annual operating costs are Rs. 400 for the first year and then increase
by Rs. 800 every year. The truck owner has now the truck type A which is
one year old. Should it be replaced by B type truck, and if so when?
07
Q.5 (a) A hardware store procures and sells hardware items. Information on an item
is given here:
Expected annual sales = 8000 units
Ordering cost = Rs. 180 per order
Holding cost = 10% of the average inventory value
The item can be purchased according to the following schedule:
Lot size Unit price (Rs.)
1-999 22
1000-1499 20
1500-1999 19
2000 and above 18.5
Determine the best order size.
06
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1
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER ?VII (NEW) EXAMINATION ? WINTER 2018
Subject Code: 2171901 Date: 26/11/2018
Subject Name: Operation Research
Time: 10:30 AM TO 01:00 PM Total Marks: 70
Instructions:
1. Attempt all questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.
Q.1 (a) Illustrate graphically for Linear Programming Problem;
(a) No-feasible solution (b) Unbounded solution.
03
(b) A firm plans to purchase atleast 200 quintals of scrap containing high quality
metal X and low quality metal Y. It decides that the scrap to be purchased
must contain atleast 100 quintals of X-metal and not more than 35 quintals
of Y-metal. The firm can purchase the scrap from two suppliers (A and B) in
unlimited quantities. The percentage of X and Y metals in terms of weight in
the scraps supplied by A and B is given below:
Metals Suppliers A Suppliers B
X 25% 75%
Y 10% 20%
The price of supplier A?s scrap is Rs. 200 per quintal and that of supplier B?s
is Rs. 400 per quintal. Formulate this problem as LP model to minimize total
purchase cost.
04
(c) Solve by Simplex method the following L.P. problem
Minimize ?? = 500 ?? 1
+ 200 ?? 2
Subjected to the constraints,
3 ?? 1
+ 2 ?? 2
? 90
?? 1
? 10
?? 2
= 2 ?? 1
?? 1
, ?? 2
? 0
07
Q.2 (a) Discuss in brief the areas of application of Operation Research. 03
(b) Find the dual problem for the following:
Minimize ?? = 5?? 1
? 6?? 2
+ 4?? 3
Subject to the constraints,
3?? 1
+ 4?? 2
+ 6?? 3
? 9
?? 1
+ 3?? 2
+ 2?? 3
? 5
7?? 1
? 2?? 2
? ?? 3
? 10
?? 1
? 2?? 2
+ 4?? 3
? 4
2?? 1
+ 5?? 2
? 3?? 3
= 3
?? 1
, ?? 2
, ?? 3
? 0
04
(c) The data for a network is given below.
Activity
Time in days Direct cost (Rs.)
Normal Crash Normal Crash
1-2 10 4 2300 3100
1-3 12 3 1700 2600
2-4 20 10 2400 4400
3-5 10 9 1100 1500
4-5 6 6 800 800
07
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2
Considering indirect cost as Rs. 1000 per/day.
(i) Find the normal duration of project completion and corresponding cost.
(ii) What can be minimum compression of project and corresponding total
time?
(iii) Find the optimal duration of project completion and related cost.
OR
(c) A construction company is preparing a PERT network for laying the
foundation of a new art museum. Given the following set of activities, their
predecessor requirements and three time estimates of completion time:
Activities Predecessors
Time in weeks
Optimistic Pessimistic
Most
Likely
A None 2 4 3
B None 8 8 8
C A 7 11 9
D B 6 6 6
E C 9 11 10
F C 10 18 14
G C, D 11 11 11
H F, G 6 14 10
I E 4 6 5
J I 3 5 4
K H 1 1 1
(a) Draw the PERT network.
(b) What is the expected time of the duration of the project?
(c) What is the probability that the project will be completed in 40
weeks?
Use the following data :
Z 1.0 1.22 1.57 1.87
Probability 0.8413 0.8888 0.9418 0.9693
07
Q.3 (a) Define the following terms: balking, reneging and jockeying. 03
(b) Explain in brief characteristic of queuing theory. What is traffic intensity? If
traffic intensity of a system is given to be 0.76, what percent of time the
system would be idle?
04
(c) The following is the pay-off matrix between player X and player Y. Find the
optimal strategies, their frequencies and the value of the game. Use rule of
dominance and oddment in calculations.
Player X
Player Y
A B C D
I 0.25 0.20 0.14 0.30
II 0.27 0.16 0.12 0.14
III 0.35 0.08 0.15 0.19
IV -0.02 0.08 0.13 0.00
07
OR
Q.3 (a) What do you understand by ?zero-sum? in the context of game theory?
Explain the meaning following terms used in game theory;
i. Saddle Point
ii. Pure Strategy
iii. Mixed Strategy
07
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3
(b) Arrival rate of telephone calls at a telephone booth is according to Poisson
distribution, with an average time of 9 minutes between consecutive arrivals.
The length of telephone call is exponentially distributed with a man of 3
minutes. Find:
i. Determine the probability that a person arriving at the booth will have to
wait.
ii. Find the average queue length that forms from time to time.
iii. The telephone company will install a second booth when conveniences
that an arrival would expect to have to wait at least four minutes for the
phone. Find the increase in flow of arrivals, which will justify a second
booth.
iv. What is the probability that an arrival will have to wait for more than 10
minutes before the phone is free?
v. Find the fraction of a day that the phone will be in use.
07
Q.4 (a) Explain the meaning of following items in inventory management;
a. Re-order Level
b. Buffer Stock
03
(b) Discuss the similarity and differences between PERT and CPM. 04
(c) A stockiest of a particular commodity makes a profit of Rs. 30 on each sale
made within the same week of purchase; otherwise he incurs a loss of Rs. 30
on each item. The data on the past sales are given below:
No. of items sold
within the same week
5 6 7 8 9 10 11
Frequency 0 9 12 24 9 6 0
i. Find out the optimum number of items the stockiest should buy every
week in order to maximize the profit.
ii. Calculate the expected value of perfect information.
07
OR
Q.4 (a) Explain ABC analysis. 07
(b) The truck-owner finds from his past experience that the maintenance costs
are Rs. 200 for the first year and then increase by Rs. 2000 every year. The
cost of truck type A is Rs. 9000. Determine the best age at which to replace
the truck. If the optimum replacement is followed what will be the average
yearly cost of owning and operating the truck? Truck of type B costs Rs.
20000. Annual operating costs are Rs. 400 for the first year and then increase
by Rs. 800 every year. The truck owner has now the truck type A which is
one year old. Should it be replaced by B type truck, and if so when?
07
Q.5 (a) A hardware store procures and sells hardware items. Information on an item
is given here:
Expected annual sales = 8000 units
Ordering cost = Rs. 180 per order
Holding cost = 10% of the average inventory value
The item can be purchased according to the following schedule:
Lot size Unit price (Rs.)
1-999 22
1000-1499 20
1500-1999 19
2000 and above 18.5
Determine the best order size.
06
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4
(b) Solve the following transportation problem for maximum profit. Use vogel?s
approximation method to find out initial feasible solution.
Market (Per unit profit in Rs.)
A B C D
Warehouse
X 12 18 6 25
Y 8 7 10 18
Z 14 3 11 20
Availability in warehouses Demand in the markets
X = 200 units A = 180 units
Y = 500 units B = 320 units
Z = 300 units C = 100 units
D = 400 units
08
OR
Q.5 (a) What is degeneracy in transportation problems? Explain how to resolve
degeneracy in a transportation problem.
06
(b) You are given the information about the cost of performing different jobs by
different persons. The job person making ?X? indicates that the individual
involved cannot perform the particular job. Using this information, state (i)
the optimal assignment of job. (ii) the cost of such assignment.
Job
J1 J2 J3 J4 J5
Person
P1 27 18 X 20 21
P2 31 24 21 12 17
P3 20 17 20 X 16
P4 22 28 20 16 27
08
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This post was last modified on 20 February 2020