ITM341
Advanced Operations Research
Assignment No.I
Assignment Code: 2013ITM341A1 Last Date of Submission: 15th April 2013
Maximum Marks:100
Attempt all the questions. All the questions are compulsory and carry equal marks.
Section-A
Ques. 1 Explain what do you mean by Sensitivity Analysis. How is it useful in business ?
Ques. 2 Solve the following L.P. Problem using Graphical Technique
Maximize Z = 2A – 3B
Subject to the constraint:
2A + 5B ≤ 25
2A + 3B ≤ 20
A + B ≥ 4
A ≤ 10
B ≥ 3
All variables are non- negative
Ques. 3 a) Rupak who travels frequently between two cities, has two route options. Route A is a
fast four lane highway and route B is a long winding road. The highway patrol has a
limited police force. If full force is allocated to either route, Rupak is certain to receive a
Rs. 1000 challan. If the force is split 50-50 between the two routes, there is a 50%
chance that Rupak will get a challan for Rs. 1000 in Route A and only a 30% chance of
getting a challan for Rs. 1000 in Route B. Develop a strategy for Rupak and the police.
b) A Professor has three pet questions, one of which occurs on every test he gives.
The student knows his habits well. He never uses the same question twice in a row. If
he used question (one) last time, he tosses a coin and uses question ( two) if a head
comes up. If he used question ( two ) , he tosses two coins and switches to question
(three) if both coins comes up as head. If he used question ( three ), he tosses three
coins and switches to question if all three come up heads. In the long run which
question does he use most often and how frequently is it used?
Ques. 4 Solve the following Linear Programming Problem using Simplex method.
Minimize Z = 36A + 60B + 45C
Subject to constraints:
A + 2B + 2C ≥ 40
2A + B + 5C ≥ 25
A + 4B + C ≥ 50
A , B, C ≥ 0
Section-B
Fifth Avenue Industries, a nationally known manufacturer of men’s wears, produces four varieties of ties. One is an expensive all silk tie, one is an all-polyester tie, and two are blends of polyester and cotton. The following table illustrates the cost and availability (per monthly production planning period) of the three materials used in the production process:
Material Cost per yard ($) Material available per month (yards)
Silk 21 800
Poly Polyester 6 3000
Cotton 9 1600
The firm has fixed contracts with several major department store chains to supply ties. The contracts require that Fifth Avenue Industries supply a minimum quantity of each tie, but allow for a larger demand if Fifth Avenue chooses to meet that demand. Following table summarizes the contract demand for each of the four styles of ties, selling price per tie, and the fabric requirements of each variety.
Variety
of Tie Selling
Price
per tie ($) Monthly contract minimum Monthly
Demand Material required
per tie (Yards) Material Requirements
All Silk 6.70 6000 7000 0.125 100% silk
All Polyester 3.55 10000 14000 0.08 100% polyester
Ploycotton Blend I 4.31 13000 16000 0.10 50% Poly &
50% cotton
Ploycotton Blend II 4.81 6000 8500 0.10 30% Poly &
70% cotton
ITM341
Advanced Operations Research
Assignment No.II
Assignment Code: 2013ITM341A2 Last Date of Submission: 15th May 2013
Maximum Marks:100
Attempt all the questions. All the questions are compulsory and carry equal marks.
Section-A
Ques. 1 Write Short note to explain the following:
(a) Use of Sensitivity Analysis in Linear Programming. Particularly highlight the use
of “Shadow Prices” and “Reduced Cost”.
(b) Use of Goal Programming. How it differs from Linear Programming?
Ques. 2 Passengers arrive at an airport at the rate of 10 per minute and is believed to be
Poisson distributed. The check in counter can handle 12 passenger in a minute due to
computerization and the time taken by the check in counter per passenger is believed
to be exponentially distributed. Find the following
(i) What is the probability that the passenger will have to wait for check in?
(ii) On the average, how many passengers are waiting in line to check in
their baggage.
Ques. 3 Minimize: Z = 8A + 10B + 15C
Subject to the constraint:
2A + 3C ≥ 3
3A + 2B + 2C ≥ 5
5B + 4C ≥ 4
A , B, C ≥ 0
Find the solution to the above problem and also find the solution to the dual
problem.
Ques. 4 a) Discuss how a queue is classified.
b) In a repair bay machines come for repairing at the rate of 4 machine per hour and
can be assumed to be Poisson Distribution. The repair bay has a repair team
consisting of 3 members to repair these machines which are arriving for repair. At
present there is only one set of repair team consisting of 3 members. These repair
crew together can on an average take 10 minutes to repair each machine and put the
same back into operation. The down time of a machine is Rs. 200 per hour and the
crew members are to be paid Rs. 60 each per hour or part thereof. Would you advise
them to use another set of 3 member crew or stay with one crew itself.
Section-B
An airport hotel has 100 rooms. On any given night , it takes upto 105 reservations, because of the possibility of no-shows. Past records indicate that the number of daily reservations is uniformly distributed over the integer range 96 to 105. That is, each integer in this range has an equal probability of 0.1 of showing up. The no-shows are represented by the distribution given in table below: Develop a simulation model (20 iterations) to find the following measures of performance of this booking system: the expected number of rooms used per night and the percentage of nights when more than 100 rooms are claimed.
No of No-shows Probability
0 0.10
1 0.20
2 0.25
3 0.30
4 0.10
5 0.05
The set of twenty random numbers needed to solve this problem are: 2, 5, 4, 1, 5, 2, 1, 5, 3, 1, 5, 3, 4, 4, 1, 4, 5, 4, 1, 5.
Advanced Operations Research
Assignment No.I
Assignment Code: 2013ITM341A1 Last Date of Submission: 15th April 2013
Maximum Marks:100
Attempt all the questions. All the questions are compulsory and carry equal marks.
Section-A
Ques. 1 Explain what do you mean by Sensitivity Analysis. How is it useful in business ?
Ques. 2 Solve the following L.P. Problem using Graphical Technique
Maximize Z = 2A – 3B
Subject to the constraint:
2A + 5B ≤ 25
2A + 3B ≤ 20
A + B ≥ 4
A ≤ 10
B ≥ 3
All variables are non- negative
Ques. 3 a) Rupak who travels frequently between two cities, has two route options. Route A is a
fast four lane highway and route B is a long winding road. The highway patrol has a
limited police force. If full force is allocated to either route, Rupak is certain to receive a
Rs. 1000 challan. If the force is split 50-50 between the two routes, there is a 50%
chance that Rupak will get a challan for Rs. 1000 in Route A and only a 30% chance of
getting a challan for Rs. 1000 in Route B. Develop a strategy for Rupak and the police.
b) A Professor has three pet questions, one of which occurs on every test he gives.
The student knows his habits well. He never uses the same question twice in a row. If
he used question (one) last time, he tosses a coin and uses question ( two) if a head
comes up. If he used question ( two ) , he tosses two coins and switches to question
(three) if both coins comes up as head. If he used question ( three ), he tosses three
coins and switches to question if all three come up heads. In the long run which
question does he use most often and how frequently is it used?
Ques. 4 Solve the following Linear Programming Problem using Simplex method.
Minimize Z = 36A + 60B + 45C
Subject to constraints:
A + 2B + 2C ≥ 40
2A + B + 5C ≥ 25
A + 4B + C ≥ 50
A , B, C ≥ 0
Section-B
Fifth Avenue Industries, a nationally known manufacturer of men’s wears, produces four varieties of ties. One is an expensive all silk tie, one is an all-polyester tie, and two are blends of polyester and cotton. The following table illustrates the cost and availability (per monthly production planning period) of the three materials used in the production process:
Material Cost per yard ($) Material available per month (yards)
Silk 21 800
Poly Polyester 6 3000
Cotton 9 1600
The firm has fixed contracts with several major department store chains to supply ties. The contracts require that Fifth Avenue Industries supply a minimum quantity of each tie, but allow for a larger demand if Fifth Avenue chooses to meet that demand. Following table summarizes the contract demand for each of the four styles of ties, selling price per tie, and the fabric requirements of each variety.
Variety
of Tie Selling
Price
per tie ($) Monthly contract minimum Monthly
Demand Material required
per tie (Yards) Material Requirements
All Silk 6.70 6000 7000 0.125 100% silk
All Polyester 3.55 10000 14000 0.08 100% polyester
Ploycotton Blend I 4.31 13000 16000 0.10 50% Poly &
50% cotton
Ploycotton Blend II 4.81 6000 8500 0.10 30% Poly &
70% cotton
ITM341
Advanced Operations Research
Assignment No.II
Assignment Code: 2013ITM341A2 Last Date of Submission: 15th May 2013
Maximum Marks:100
Attempt all the questions. All the questions are compulsory and carry equal marks.
Section-A
Ques. 1 Write Short note to explain the following:
(a) Use of Sensitivity Analysis in Linear Programming. Particularly highlight the use
of “Shadow Prices” and “Reduced Cost”.
(b) Use of Goal Programming. How it differs from Linear Programming?
Ques. 2 Passengers arrive at an airport at the rate of 10 per minute and is believed to be
Poisson distributed. The check in counter can handle 12 passenger in a minute due to
computerization and the time taken by the check in counter per passenger is believed
to be exponentially distributed. Find the following
(i) What is the probability that the passenger will have to wait for check in?
(ii) On the average, how many passengers are waiting in line to check in
their baggage.
Ques. 3 Minimize: Z = 8A + 10B + 15C
Subject to the constraint:
2A + 3C ≥ 3
3A + 2B + 2C ≥ 5
5B + 4C ≥ 4
A , B, C ≥ 0
Find the solution to the above problem and also find the solution to the dual
problem.
Ques. 4 a) Discuss how a queue is classified.
b) In a repair bay machines come for repairing at the rate of 4 machine per hour and
can be assumed to be Poisson Distribution. The repair bay has a repair team
consisting of 3 members to repair these machines which are arriving for repair. At
present there is only one set of repair team consisting of 3 members. These repair
crew together can on an average take 10 minutes to repair each machine and put the
same back into operation. The down time of a machine is Rs. 200 per hour and the
crew members are to be paid Rs. 60 each per hour or part thereof. Would you advise
them to use another set of 3 member crew or stay with one crew itself.
Section-B
An airport hotel has 100 rooms. On any given night , it takes upto 105 reservations, because of the possibility of no-shows. Past records indicate that the number of daily reservations is uniformly distributed over the integer range 96 to 105. That is, each integer in this range has an equal probability of 0.1 of showing up. The no-shows are represented by the distribution given in table below: Develop a simulation model (20 iterations) to find the following measures of performance of this booking system: the expected number of rooms used per night and the percentage of nights when more than 100 rooms are claimed.
No of No-shows Probability
0 0.10
1 0.20
2 0.25
3 0.30
4 0.10
5 0.05
The set of twenty random numbers needed to solve this problem are: 2, 5, 4, 1, 5, 2, 1, 5, 3, 1, 5, 3, 4, 4, 1, 4, 5, 4, 1, 5.
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