aaoc+c312

BIRLA INSTITUTE OF TECHNOLOGY & SCIENCE, PILANI

HYDERABAD CAMPUS

SECOND SEMESTER - 2010-2011

TEST- I [CLOSED BOOK]

Course Name: OPERATIONS RESEARCH Course Code: AAOC C312

DATE: 02-02-2011 (WEDNESDAY) Time: 50 Minutes Max Marks: 75

Answer all questions

1. Write the steady state equation for the Birth and Death process and solve it. Using steady state probability conditions of birth and death process prove that�� . [15M]

2. A queuing system has four servers with expected service times of 20 min, 15 min, 12 min and 10 min. The

service times have an exponential distribution. Each server has been busy with a current customer for 5 minutes. Determine the expected remaining time until the next service completion. Also State and prove the property / properties that are to be used to solve the above problem. [25M]

3. Customers arrive at a one-man barber shop according to a Poisson process with a mean inter arrival time of

12 minutes. Customers spend on average of 10 minutes in the barber’s chair. (a) What is the expected number of customers in the barber shop and in the queue? (b) Calculate the percentage of time an arrival can walk straight into the barber’s chair without having to

wait. (c) How much time can a customer expect to spend in the barber’s shop? (d) Calculate the percentage of customers who have to wait prior to getting into the barber’s chair? (e) What is the probability that more than 3 customer in the system? [20M]

4. Patients arrive at a clinic according to poisson distribution at a rate of 30 patients per hour. The waiting

room does not accommodate more than 4 patients. Examination time per patient is exponential with mean rate 20 per hour. Consultancy fee per patients is Rs 500.

(a) What is his expected income per hour? (b) What is the probability that an arriving patient will not wait? (c) What is the expected waiting time until a patient is discharged from the clinic?

[15M]

BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, PILANI HYDERABAD CAMPUS

SECOND SEMESTER 2010-2011 AAOC C312: OPERATION RESEARCH

TEST II (OPEN BOOK) Date: 09-03-2011 Max Marks: 75 Day: WEDNESDAY Time: 8 AM to 8.50 AM

Note: Answer all the questions. Marks for each question are given at the end of the question.

1. Cars arrive at a service station in a Poisson Process at a mean rate of 10 per hour. The service time of a car has an exponential distribution with mean 12 minute. A repairmen has to be paid Rs 3.00 per hour. Manager of the service station estimates that the goodwill cost of a customer waiting to begin service is Rs 5.00 per hour while the goodwill cost of a customer under service is Rs 4.00 per hour. What is the optimum number of repairman (s) to minimize the mean cost per hour if there is space for only 4 cars who are either in service or waiting and if all spaces are filled up, the customer will go to another service station. (35 M)

2. Previous data suggests that the distribution of demand of pen (in dozen) during comprehensive examination period will be as follows: � 2 5 8 15 25 50 �� 0.05 0.10 0.25 0.05 0.15 0.40

Pens can be purchased from stall no “A” in the CP only at the beginning of examination at the rate of Rs 36 per dozen. The holding cost is Rs 12 per dozen calculated over the pen left over at the end of examination and penalty cost per dozen pen when it is out of stock is Rs 60. The cost of placing an order is Rs 40 and the initial inventory is 10 dozen. Find the optimal ordering policy and optimal cost. (15 M)

3. ABC Garment Company offers the following discount schedule for its good quality sweatshirts. Order Unit cost (in $) � 10 18 10 � � 50 17 50 � � 100 16 � � 100 15

A shop owner orders sweatshirts from ABC Company. The ordering cost is $45 per order. The carrying cost is $10 per unit per month and the monthly demand is 100. If the lead time is 36 days (assume that 30 days in a month) determine ��, ��∗�, �� (no shortages are allowed and the delivery is instantaneous). In minimizing cost, how many orders would be made each year? What is the time between orders? What is the reorder point? (25 M)


HYDERABAD CAMPUS


QUIZ [CLOSED BOOK]



Name: ID. Number:

Note: Each question has four possible choices. Pick the most appropriate choice and put in the box provided

as A or B or C or D. Every correct answer carries 3 marks; a WRONG answer carries -1 mark and SMALL CASE

letters / OVERWRITTEN answers carries 0 marks.

1) In a queuing system (M/M/2): (FCFS/∞/∞), λ=6 customers per hour and μ=4 customers per hour. The maintaining cost of the service station as a whole (including the salaries of both the servers etc) is Rs.40 per hour. Each customer is charged Rs.15 for service. What is the expected profit per hour of the service station?

A) Rs. 45 B) Rs.50 C) Rs.65 D) Rs.35

2) In a barber shop with only one chair, customers arrive according to Poisson distribution at an average rate of 5 customers per hour. The waiting room can accommodate at maximum 4 customers. The service time for the customer is exponential distribution with mean rate 5 per hour. What is the expected waiting time until the customer is departed from the barber shop.

A).5 B) .6 C) .7 D).8

3) In machine maintenance, a mechanic repairs four machines. The mean time between service requirements is 5 hours for each machine and forms an exponential distribution. The mean repair time is 1 hour and also follows the same distribution pattern. Also given that the average number of customers in the queue is 0.4. Find the expected number of operating machines

A) 3 B) 2 C) 1 D) 0

4) Identify the first random generate of the given size 5 from X a poisson distribution with parameters 2.5 by using the random numbers 25,95,12,34,56,67,45. (Take the random numbers in order) A) 1 B) 2 C) 3 D) 4 5) To find the Random variate for the pdf f(x) = (x-3)2/ 3, if 0≤x≤6 = 0 elsewhere The suitable scheme is A)If 0≤u<1, then X= 3+ (54u-27)1/3 B) If 0≤u≤1, then X= 3+ (54u-27)1/3 C) If 0≤u<1, then X= 3- (54u-27)1/3 D) If 0≤u≤1, then X= 3-(54u-27)1/3

Question Number

1 2 3 4 5 6 7 8 9 10

Answer B B A B A C D B D

A

6) An airline company is opening a one line reservation service. A passenger making reservation will be able to telephone the office and place his request. The operating characteristics are given below.

Inter arrival time 1 2 3 4 5 Service time 1 2 3

Probability 0.1 0.3 0.2 0.2 0.2 Probability 0.25 0.5 0.25 Random Number 21 23 45 10 20 Random Number 85 35 67

Initial customer all line is free to start. The next two events of the above simulation are: (A: Arrival D: Departure) A) A, D B) D, A C) A, A D) D, D

7) For the given hazard function Z(t) = t find the mean time of the system failure. A)√2� B)2 √� C)√π / 2 D) √π / √2 8) In a single item, static demand purchase inventory model with infinite delivery rate and no backordering, the setup cost is 100 , optimum order cycle is 10 days and the economic order quantity is 1000 units then find the holding cost of the inventory model. A) 0.01 B) 0.02 C) .04 D) 0.07 9) In a mall there is a special counter for cotton shirts and sells them during the summer. Each shirt costs Rs.400 and sells for 1000. If a shirt is not sold it is estimated that it will cost Rs.100 to keep it until the next season. Assume that there were 20 shirts left in the last year. There is no setup cost for the model. Previous data suggests that the distribution of demand will be as follows:

Demand 10 20 30 40 50 Probability 0.1 0.2 0.3 0.3 0.1

Find the optimum number of shirts to be made in this year. A) 20 B) 30 C) 40 D) 10

10) The demand of a commodity in a single period is an exponential distribution with distribution f(x) = e-x ,x>0 ; =0 elsewhere. The values of holding cost and purchase cost and shortage cost are respectively 1,3 and 5. Find R*.

A) R* =ln (3/2) B) R* =ln (2/3) C) R* = log (3/2) D) R* = log (2/3)

***********


HYDERABAD CAMPUS


QUIZ [CLOSED BOOK]



Name: ID. Number:




1) An airline company is opening a one line reservation service. A passenger making reservation will be able to telephone the office and place his request. The operating characteristics are given below.

Inter arrival time 1 2 3 4 5 Service time 1 2 3

Probability 0.1 0.3 0.2 0.2 0.2 Probability 0.25 0.5 0.25 Random Number 21 23 45 10 20 Random Number 85 35 67

Initial customer all line is free to start. The next two events of the above simulation are: (A: Arrival D: Departure) A) D, D B) D, A C) A, D D) A, A

2) For the given hazard function Z(t) = t find the mean time of the system failure. A)√2� B)2 √� C)√π / 2 D) √π / √2 3) In a single item, static demand purchase inventory model with infinite delivery rate and no backordering, the setup cost is 100 , optimum order cycle is 10 days and the economic order quantity is 1000 units then find the holding cost of the inventory model. A) 0.02 B) 0.01 C) .04 D) 0.07 4) In a mall there is a special counter for cotton shirts and sells them during the summer. Each shirt costs Rs.400 and sells for 1000. If a shirt is not sold it is estimated that it will cost Rs.100 to keep it until the next season. Assume that there were 20 shirts left in the last year. There is no setup cost for the model. Previous data suggests that the distribution of demand will be as follows:

Demand 10 20 30 40 50 Probability 0.1 0.2 0.3 0.3 0.1

Question Number

1 2 3 4 5 6 7 8 9 10

Answer D D A A B D D D C B

B

Find the optimum number of shirts to be made in this year. A) 10 B) 30 C) 40 D) 20 5) The demand of a commodity in a single period is an exponential distribution with distribution f(x) = e-x ,x>0 ; =0 elsewhere. The values of holding cost and purchase cost and shortage cost are respectively 1,3 and 5. Find R*.

A) R* =ln (2/3) B) R* =ln (3/2) C)R* = log (3/2) D) R* =log (2/3)

6) In a queuing system (M/M/2): (FCFS/∞/∞), λ=6 customers per hour and μ=4 customers per hour. The maintaining cost of the service station as a whole (including the salaries of both the servers etc) is Rs.40 per hour. Each customer is charged Rs.15 for service. What is the expected profit per hour of the service station?

A) Rs. 45 B) Rs.35 C) Rs.65 D) Rs.50

7) In a barber shop with only one chair, customers arrive according to Poisson distribution at an average rate of 5 customers per hour. The waiting room can accommodate at maximum 4 customers. The service time for the customer is exponential distribution with mean rate 5 per hour. What is the expected waiting time until the customer is departed from the barber shop.

A).5 B) .8 C) .7 D).6

8) In machine maintenance, a mechanic repairs four machines. The mean time between service requirements is 5 hours for each machine and forms an exponential distribution. The mean repair time is 1 hour and also follows the same distribution pattern. Also given that the average number of customers in the queue is 0.4. Find the expected number of operating machines

A) 2 B) 1 C) 0 D) 3

9) Identify the first random generate of the given size 5 from X a poisson distribution with parameters 2.5 by using the random numbers 25,95,12,34,56,67,45. (Take the random numbers in order) A) 4 B) 1 C) 2 D) 3 10) To find the Random variate for the pdf f(x) = (x-3)2/ 3, if 0≤x≤6 = 0 elsewhere The suitable scheme is A) If 0≤u≤1, then X= 3+ (54u-27)1/3 B) If 0≤u<1, then X= 3+ (54u-27)1/3 C) If 0≤u<1, then X= 3- (54u-27)1/3 D) If 0≤u≤1, then X= 3-(54u-27)1/3 *********


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TEST- I MAKEUP [CLOSED BOOK]


DATE: 07-02-2011 (MONDAY) Time: 50 Minutes (5.10-6.00P.M) Max Marks: 75

Answer all questions

1. A) Write the basic components of a queuing system. B) Write the assumptions of the birth and death process and derive the governing differential equation. Find its solution under the steady state conditions. C) A common phenomenon of arrivals is that it is a random process and there is a small probability of large inters arrival times and large probabilities of small inter arrival times. State and prove the theorem that is well suited to model the inter arrival times described as above. [20M]

2. Consider a birth and death process with just three attainable states (0, 1, 2) for which the steady state probabilities are P0, P1 and P2 respectively. The birth and death rates are summarized in the following table: (a) Develop the balance equations. (b) Solve for P0, P1 and P2. (c) Calculate the mean waiting time in the system and also in the queue. [15M]

3. A drive in banking service is modeled as an M/M/1 queuing system with customer arrival rate of 2 per minute. It’s desired to have fewer than 5 customers line up 99 percent of the time. How fast should the service rate be? Find the expected number of customers in the system and in queue using the above data. What is the portion of the time the server is idle. What is the expected waiting time for a customer in the system and in the queue? [20M]

4. At a railway station, only one train is handled at a time. The railway yard is sufficient only for 2 trains to wait, while the other is given signal to leave the station. Trains arrive at the station at an average rate of 6 per hour and the railway station can handle them on an average of 6 per hour. Assuming Poisson arrivals and exponential service distribution. If the handling rate is doubled, find the probabilities for the number of trains in the system. Also find the average waiting time of a new train coming into the yard. [20M] ------------------------

State Birth rate Death Rate 0 1 --- 1 1 2 2 0 2


HYDERABAD CAMPUS


TEST- II MAKEUP [OPEN BOOK]


Time: 50 Minutes Max Marks: 75

1. A company has 4 cars and the car servicing station has 2 stalls where service can be offered simultaneously. The

cars wait in such a way that when stall becomes vacant, the car at the head of the line pulls up to it. The station can

accommodate at most two cars waiting at one time. The arrival pattern is poisson with a mean of one car per

minute during the peak hours. The service time is exponential with mean 6min. [25M]

a) Find the Steady state probabilities for this system

b) Find the overall effective rate of arrival to the service station?

c) Find L, Lq,, W, Wq

2. Find the optimum order quantity for a product when the annual demand for the product is 500 units, the cost of

storage per unit per year is 10% of the unit cost and ordering cost per order is Rs. 180. The unit costs are given below.

[20M]

Quanity Unit cost

0<Q≤500

500<Q≤1500

1,500<Q≤3000

3,000<Q≤5000

Rs. 25.00

Rs. 26.00

Rs. 27.00

Rs. 28.00

3. Safety is trying to decide how many check outlines to keep open. An average of 18 customers arrives

per hour according to a poisson process and goes to the first empty check outline. If no checkout line

is empty, suppose that arriving customers form a single line to wait for the next free checkout line is

empty, suppose that arriving customers form a single line to wait for the next free checkout line. The

checkout time for each customer is exponentially distributed with mean 4 minutes. It costs $ 20 per

hour to operate a checkout line and safety estimates that it costs them $0.25 for each minute a

customer waits in the cash register area. How many registers should the store have to open? [30M]


HYDERABAD CAMPUS


QUIZ-MAKEUP [CLOSED BOOK]



Name: Sec No. ID. Number:




1) Same type of problems comes to a computing centre at a mean rate of 5 per hour in Poisson pattern. The computing centre has

three PC installed in parallel each with a exponential service rate of 5 per hour. The computer centre allows only one problem in the waiting. If more problems come, they are sent to another centre. What is the portion of idle time per PC.

A) 22/49 B)33/49 C)44/49 D)11/49

2) Consider the queuing system (M/M/1):(M/∞/∞) in which �� ! , " � 0,1,2,3, … , %� � %, " � 1,2,3, ….. Then the

Utilization factor for the defined system is

'�1 ( )*+ ,�1 ( )-*+ ��1 ( )+* .�1 ( )-+* 3) At a reservation counter with “s” reservation clerks the calls for reservation come in Poisson pattern at an average rate of 10 calls

per hour. If a caller requests information or reservation, the operator asks the caller to wait until a clerk becomes free. The service time of clerk is exponentially distributed with mean 10 minutes. Assume that the cost of one reservation clerk is Rs. 50 per day, the goodwill cost of having a customer wait as Rs.5 per hour spent waiting (before being connected to the clerk) and the expected number of customers per hour in the queue is 4. Find the expected cost per day if minimum number of servers is used?

A) 540. B) 560 C) 590 D) 580 4) Find the sum of the first two random samples (Approximately) of size five from X an exponential distribution with parameter is

2 by using the random numbers 20,23,86,09,92. (Take the random numbers in order)

A) 1.54 B) 1.64 C) 1.24 D) 1.14

Question Number

1 2 3 4 5 6 7 8 9 10

Answer B B D A B D A A D D

A

5) Generate a random sample for the following pdf f(x)= |x|, -1<x<1

A) X=(√2/ ( 1 when 0≤u<0.5, X=√1 ( 2/ 0.5≤u<1 B) X=(√1 ( 2/ when 0≤u<0.5, X=√2/ ( 1 0.5≤u<1

B) X=(√2/ ( 1 when 0≤u<0.5, X=√1 0 2/ 0.5≤u<1 D) X=(√1 0 2/ when 0≤u<0.5, X=√2/ ( 1 0.5≤u<1

6) Find the hazard rate function Z(t) for the uniform distribution defined on [0,10] at t=10 is

A) 10 B)5 C) 0 .5 D) Undefined

7) The management of the united commercial bank plans to open a one teller window. Research study has projected the following distribution for inter arrival time. Assume that the first customer arrives at zero time. Find the next two events for the following data. (A: Arrival, D: Departure)

A) A, D B) D, A C) A, A D) D, D 8) An item costs 100 Rupees and it can be produced at a rate of 50 per month and it is sold at a rate of 20 per month. The holding

cost Rs.10 per unit per month, back ordering is allowed with a cost Rs.50 per unit of back order per month and the set up cost is Rs.500 .Find the economic back order quantity up to two decimal places.

A) 6.32 B) 5.32 C) 2.32 D) 4.32.

9) The demand of an item during a single period is uniformly distributed over [0, 6]. The item can be purchased only at the beginning of the period. The purchase cost is Rs. 3 per unit. The holding cost on ending inventory is Rs 1 per unit, and shortage cost is Rs. 5 per unit shortage. The cost of placing an order is Rs. 1 and the initial inventory is Rs 1

Find the expected cost if an order is placed by using the given values. 1!23 2 ( �4∗5 6 � !7 , 823 ( 2�24∗ 6 � 957 :

A) 10 B) 20 C) 15 D) 11

10) The demand of a commodity is 100 units per day and the production rate is 200 units per day. The set up cost is Rs. 500 per production run. The cost of holding inventory is Rs.1.25 per unit/day. No shortage is allowed. The unit production cost in rupees is as follows: C0 = 10 if Q<200 =20 if 200� � 500 Where Q is the quantity produced in one run. Then EOQ is A) 200 B) 300 C) 500 D) 400

*******

Inter arrival time 5 6 7 8 Service

time 5 6 7

Probability 0.15 0.35 0.35 0.15 Probability 0.25 0.5 0.25 Random Number 25 37 91 00 Random

Number 84 01 59


HYDERABAD CAMPUS


QUIZ-MAKEUP [CLOSED BOOK]



Name: Sec No. ID. Number:




1) Generate a random sample for the following pdf f(x)= |x|, -1<x<1

A) X=(√2/ ( 1 when 0≤u<0.5, X=√1 ( 2/ 0.5≤u<1 B) X=(√1 ( 2/ when 0≤u<0.5, X=√2/ ( 1 0.5≤u<1

C) X=(√2/ ( 1 when 0≤u<0.5, X=√1 0 2/ 0.5≤u<1 D) X=(√1 0 2/ when 0≤u<0.5, X=√2/ ( 1 0.5≤u<1

2) Find the hazard rate function Z(t) for the uniform distribution defined on [0,10] at t=10 is

A) 10 B) 5 C) 0.5 D) Undefined

3) The management of the united commercial bank plans to open a one teller window. Research study has projected the following distribution for inter arrival time. Assume that the first customer arrives at zero time. Find the next two events for the following data. (A: Arrival, D: Departure)

A) A, D B) D, A C) A, A D) D, D

4) An item costs 100 Rupees and it can be produced at a rate of 50 per month and it is sold at a rate of 20 per month. The

holding cost Rs.10 per unit per month, back ordering is allowed with a cost Rs.50 per unit of back order per month and the set up cost is Rs.500 .Find the economic back order quantity up to two decimal places.

A) 6.32 B) 5.32 C) 2.32 D) 4.32.

Question Number

1 2 3 4 5 6 7 8 9 10

Answer B D A A D B B D A D

Inter arrival time 5 6 7 8 Service

time 5 6 7

Probability 0.15 0.35 0.35 0.15 Probability 0.25 0.5 0.25 Random Number 25 37 91 00 Random

Number 84 01 59

B

5) The demand of an item during a single period is uniformly distributed over [0, 6]. The item can be purchased only at the

beginning of the period. The purchase cost is Rs. 3 per unit. The holding cost on ending inventory is Rs 1 per unit, and shortage cost is Rs. 5 per unit shortage. The cost of placing an order is Rs. 1 and the initial inventory is Rs 1 .Find the

expected cost if an order is placed by using the given values. 1!23 2 ( �4∗5 6 � !7 , 823 ( 2�24∗ 6 � 957 :

A) 10 B) 20 C) 15 D) 11

6) Same type of problems comes to a computing centre at a mean rate of 5 per hour in Poisson pattern. The computing centre

has three PC installed in parallel each with a exponential service rate of 5 per hour. The computer centre allows only one problem in the waiting. If more problems come, they are sent to another centre. What is the portion of idle time per PC?

A) 22/49 B)33/49 C)44/49 D)11/49

7) Consider the queuing system (M/M/1):(M/∞/∞) in which �� ! , " � 0,1,2,3, … , %� � %, " � 1,2,3, ….. Then the

Utilization factor for the defined system is

'�1 ( )*+ ,�1 ( )-*+ ��1 ( )+* .�1 ( )-+*

8) At a reservation counter with “s” reservation clerks the calls for reservation come in Poisson pattern at an average rate of 10 calls per hour. If a caller requests information or reservation, the operator asks the caller to wait until a clerk becomes free. The service time of clerk is exponentially distributed with mean 10 minutes. Assume that the cost of one reservation clerk is Rs. 50 per day, the goodwill cost of having a customer wait as Rs.5 per hour spent waiting (before being connected to the clerk) and the expected number of customers per hour in the queue is 4. Find the expected cost per day if minimum number of servers is used?

A) 540. B) 560 C) 590 D) 580

9) Find the sum of the first two random samples (Approximately) of size five from X an exponential distribution with parameter is 2 by using the random numbers 20,23,86,09,92. (Take the random numbers in order)

A) 1.54 B) 1.64 C) 1.24 D) 1.14

10) The demand of a commodity is 100 units per day and the production rate is 200 units per day. The set up cost is Rs. 500 per production run. The cost of holding inventory is Rs.1.25 per unit/day. No shortage is allowed. The unit production cost in rupees is as follows:

C0 = 10 if Q<200 =20 if 200� � 500 Where Q is the quantity produced in one run. Then EOQ is

A) 200 B) 300 C) 500 D) 400

*******


HYDERABAD CAMPUS


COMPREHENSIVE EXAMINATION [CLOSED BOOK]

PART-A

Course Title: OPERATIONS RESEARCH Course Code: AAOC C312

DATE: 11-05-2011-AN (WEDNESDAY) Time: 60 Minutes Max Marks: 36

Name: Sec: ID. No.:

Note: 1. Write your answers only in the box given below. Each answer carries 3 marks. 2. Over writing and illegible writing will be awarded 0 marks. No negative marking. No partial marks. 3. Calculators are not allowed. Answers must be in most simplified form for fractions. Q1. At a parking lot, there are a large number of parking spaces. Cars arrive at the lot in a Poisson process at a

mean rate of 20 cars per hour. The parking time of a car is exponentially distributed with mean 15 minutes. Each car receiving the parking facility has to pay Rupees 5 (irrespective of its parking time) plus Rupees 10 per hour for the parking time. Then the expected revenue (per hour) of the parking lot is…

Data for Questions 2, 3 and 4:

A Group of engineers has two terminals available to aid in their calculations. The average computing job requires 20 minutes of terminal time, and each engineer requires some computation about once every 20 minutes i.e. the mean time between a call for service is 20 minutes. Assume these distributed according to an exponential distribution. There are four engineers in the group.

Q2. The probability that both the terminals are idle is…

Q3. The probability that more than 2 engineers are in the computing center is …

Q4. The expected number of engineers waiting to use one of the terminals is…

Q1. Q2. Q3. Q4.

Q5. Q6. Q7. Q8.

Q9. Q10. Q11. Q12.

Rs. 150 3/5 or 0.6

37.5 or 75/2 Rs. 2000

e-2/5 or e-0.4

1/20 or 0.05 9/20 or 0.45

61 3

2 5/4 or 1.25 P≤7, q ≥7

Q5. A telephone exchange has two long distance operators. During the peak hours, long distance calls arrive in a Poisson process at the mean rate of 15 per hour. The length of service on these calls is approximately exponentially distributed with mean 5 minutes. The percentage of the idle time of a server is… Q6. A gas station with only one gas pump employs the following policy: If a customer has to wait, the price is Rs 60 per gallon; if he/she does not have to wait, the price is Rs 64 per gallon. Customers arrive according to a Poisson process with a mean rate of 15 per hour. Service times at the pump have an exponential distribution with a mean of 3 minutes. Arriving customers always wait until they can eventually buy gasoline. Then the expected price of gasoline per gallon is… Q7. Using random numbers 78, 87, 32, 53, one observation of the binomial random variable � with ; � 0.52 and < � 4 is… Q8. The daily demand for a commodity produced by ABC company is approximately 100 units. The item is produced locally. The set up cost is Rs.100 per production run and the production rate is 200 units per day. The daily holding cost per unit inventory is 4 paise. It was found that the economic lot size is 1000 units for the above data. But due to some unavoidable circumstances, the set up cost has to be increased to Rs. 400, but the other costs remains the same. Then the economic lot size will be…

Q9. An integrated circuit chip has a constant failure rate of 0.02 per thousand hours. Then the probability that it will

operate satisfactorily for at least 20,000 hrs is…

Q10. The mean time to failure of a component whose failure rate >?� � @! @ where t > 0 is…

Q11.Consider a system consisting of 4 components arranged in parallel. Assume that the components function

independently and have a constant failure rate AB , then the mean life of the system in terms of the mean life of each

component is …

Q12. Consider the game whose pay-off matrix isC 245107E4;8 G. The range of values of p and q that will make the

payoff element (2, 2) a saddle point is…

********


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COMPREHENSIVE EXAMINATION [CLOSED BOOK]

PART-B

Course Title: OPERATIONS RESEARCH Course Code: AAOC C312

DATE: 11-05-2011-AN (WEDNESDAY) Time: 120 Minutes Max Marks: 84

Answer all 5 questions. Q1.Assume that the demand of a commodity in a single period is a continuous random variable uniformly distributed on (100, 400). There is an initial inventory of 150. The purchase cost is Rs. 3 per unit. The holding cost of ending inventory is Rs 1 per unit. The stock out cost per unit out of stock is Rs 5. The cost of placing an order is Rs. 100. Should the purchase be made? How many units are purchased if the purchase is to be made, in order to minimize the expected total cost? Assuming that the initial inventory 150 is not given, determine the minimum initial inventory level which will be required to prevent any fresh order for minimum expected total cost? [22] Q2. Assume that in a queuing system with one server, FCFS queue discipline, unlimited queue length, unlimited input source, the inter arrival time �� distribution is uniform over 0,2� and the service time H� distribution is given by the density function

�I� �JKLKM 12, 1 � I 214 ,2 � I 40)NO)PQ)R)

Simulate the system for 1.5 units of time. Use the random numbers 12, 32, 50, 05, 17 for IAT and 20, 60, 40 32, 71, 02 for the service times in the order they are given. Estimate S, ST, U, UT and the fraction of time the server is busy. Assume that there is one customer at time zero. Prepare your table according to the following heading in the order they are given: T, CE, CNA, CND, NS, NQ, CWTS, CWTQ, STS, CIDT, UA, IAT, NAT, US, ST, NDT, NET, NE. [22] Q3. Solve the following game by graphical method. [10]

Player B Player A

B1 B2

A1 3 -5 A2 1 -1 A3 2 -3 A4 -1 3

Q4. The time estimates of all activities of a project are as given below (in days)

(a). Construct the project network.

(b). Determine the critical path by calculating earliest occurrence of time and latest occurrence of time?

(c). What due date has 90% chance of being met? (Z.10 = 1.28) [15]

Q5. You have to load a vessel with three items. The maximum allowable weight is 10 lb. The weight per unit of different terms and their values are given below. It is required to find the loading which maximizes the values of the vessel without exceeding the weight constraint of 10 lb.?

Solve the above problem using backward recursive equation, show your calculations in tabular form. [15]

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Activity ( i-j ) 1-2 1-3 1-4 2-6 3-5 3-6 4-5 5-6 Optimistic Time 6 3 3 4 3 2 1 6 Pessimistic time 18 15 27 28 27 8 7 30 Most likely time 12 6 9 19 9 5 4 12

Item i Weight

(lb) Revenue

1 4 11 2 3 7 3 5 12

aaoc+c312

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