## St. Joseph’s College of Commerce M.Com. 2012 II Sem Operation Research Question Paper PDF Download

1. JOSEPH’S COLLEGE OF COMMERCE (AUTONOMOUS)

End Semester Examinations – MARCH / APRIL 2012

M.Com. – II Semester

OPERATION RESEARCH

Duration: 3 Hrs                                                                                                              Max. Marks: 100  Section – A

1. Answer SEVEN questions out of Ten.                   (7 x 5 = 35)

1. Discuss various steps for solving an operation research problem. Illustrate with one example from the functional area of your choice.

1. Determine the trend line and the forecast the production for the year 2013.

YEAR                          PRODUCTION

2005                            100

2006                            225

2007                            175

2008                            199

2009                            250

2010                            255

2011                            275

1. An airline organization has one reservation clerk on duty in its local branch at any given time. The clerk handles information regarding passenger reservation and flight timings. Assume that the number of customers arriving during any given period is Poisson distributed with an arrival rate of eight per hour and that the reservation clerk can service a customer in six minutes on an average, with an exponentially distributed service time:

1) what is the probability that the system is busy?

2) what is the average time a customer spends in the system?

3) what is the average length of the queue?

4) what is the number of customers in the system?

1. What is Dynamic programming? Describe the characteristics of Dynamic Programming.

1. A care hire company has one car at each of the five depots a, b, c,d, and e. A customer in each of the five towns A,B,C,D and E requires a car. The distance in miles between the depots and towns where the customers are given in the following distance matrix:
 TOWN PERSON a b c D e A 160 130 175 190 200 B 135 120 130 160 175 C 140 110 155 170 185 D 50 50 80 80 110 E 55 35 70 80 105

How the cars should be assigned to the customers so as to minimize the distance travelled.

1. A company has three factories A,B and C which supply to 4 warehouses at P,Q, R and S. The monthly production capacity (tons) A,B and C are 120, 80 and 200 resp. The monthly req ( tons) for warehouses P,Q,R and S are 60, 50, 140 and 50 resp. The transportation cost (Rs per ton) Matrix is given below:
 Warehouses Factories A B C P 4 3 7 Q 5 8 4 R 2 4 7 S 5 8 4

Use NWCR and Vogel’s Approximation Method to determine transportation distribution of product to warehouses to minimize transportation cost.

1. A company produces two types of products Type A and Type B. Product B is of superior quality and product A is of lower quality. Profits on two types of products are Rs 30 and Rs 40 respectively. The data resources required and availability of resources are given below:

Requirement

 Product A Product B Product C Raw Material 60 120 12000 Machine Hours 8 5 630 Assembly 3 4 500

How should the company manufacture the two types of products in order to get maximum overall profits?

1. Define degeneracy. Discuss with the help of examples how degeneracy can be

resolved in a transportation problem at the initial stage.

1.     Discuss the steps of forming a dual with the help of an example.

1.    Define Slack, surplus and artificial variables.

Section – B

1. Answer THREE questions  out of Five.                                      (3 x 15 = 45)

1. A small garment making unit has five tailors stitching five different types of garments. The output per day per tailor and the profit( rs ) for each type of garment are given below:
 TAILORS GARMENTS 1 2 3 4 5 A 7 9 4 8 6 B 4 9 5 7 8 C 8 5 2 9 8 D 6 5 8 10 10 E 7 8 10 9 9 PROFIT PER GARMENT 2 3 2 3 4

Which type of garment should be assigned to which tailor in order to maximize profit assuming that there are no other constraints?

1. A Company has four terminals u,v,w and x. At the start of a particular day 10,4,6 and 5 trailers respectively are available at these terminals. During the previous night 13,10, 6 and 6 trailers respectively were loaded at plants A,B,C and D. The Company dispatcher has come up with the costs between the terminals and plants as follows:
 Terminals Plants A B C D U 20 36 10 28 V 40 20 45 20 W 75 35 45 50 x 30 35 40 25

Find the allocation of loaded trailer from plants to terminals in order to minimize transportation costs.

1. A rural health centre receives a delivery of fresh blood plasma once each week from a city blood bank. The supply varies according to demand from other clinics and hospitals in the region but ranges between four and nine litres of the most widely used blood type (blood O). The number of patients per week requiring this blood varies from zero to four and each patient may need from one to four litres of blood. The delivery quantities, patient distribution and demand per patient are given below:

Table 1. Delivery Quantities:

 Litres per week Probability 4 0.15 5 0.20 6 0.25 7 0.15 8 0.15 9 0.15 total 1.00

Table 2. Patient Distribution:

 Patients per week requiring blood Probability 0 0.25 1 0.25 2 0.30 3 0.15 4 0.05 Total 1.00

Table 3. Demand per Patient

 Litres Probability 1 0.40 2 0.30 3 0.20 4 0.10

RN for Quantity Delivered: 10,31,02,53,16,40

RN for patients needing blood: 85,28,72,44,16,83

RN or quantity required: 21,06,61,96,12,67,23,65,34,82.

Determine the number of litres of blood in excess or short for a six week period using simulation technique. Assume that the blood is storable and current stock is 3 litres.

1. Analysis of data on customer arrivals at a fast food restaurant has revealed that the mean arrival rate is 45 customers per hour and is found to follow a Poisson distribution. The service time starts when a customer begins to place the order with the food server and continues until the customer has received the order.

Quantitative analysis has revealed that the probability distribution for the service time can be assumed as exponential distribution and the mean servicing rate is 60 customers per hour. Assuming that the restaurant has single channel characteristics, calculate:

1. probability that no customers are in the system
2. the average number of customers in the queue
3. average number of customers in the system
4. average time a customer spends in the waiting line
5. the average time the customer spends in the system.
6. Discuss the various techniques used under Operation research for decision making. Give examples to support your answer.

Section – C

• Compulsory Case study (No choice)                              (1 x 20 = 20)

The ABC company combines factors X and Y to form a product which must weigh 50 kgs. At least 20 kgs of X and no more than 40 kgs of Y can be used. X costs Rs 25 and Y Rs 10 per kg. Find out amount of factor X and factor Y to be used manufacturing the product. Use Simplex Method.

## St. Joseph’s College of Commerce B.Com. 2015 Operation Research Question Paper PDF Download

st. joseph’s college of commerce (autonomous)
END SEMESTER EXAMINATION – MARCH/APRIL 2015
b.com – vi semester
C1  11 602: OPERATION RESEARCH
Duration: 3 Hours                                                                                             Max. Marks: 100
SECTION – A
I) Answer ALL the questions.  Each carries 2 marks.                                        (10×2=20)
1. What is Operation Research?
2. Explain symbolic models and mention the two types of symbolic models with suitable examples.
3. Introduce suitable slack, surplus or artificial variables for the expressions:

(a)   2x + 5y = 6

(b)   450a + 500b ≥ 1500

(c)    35x + 45y +55z ≤ 175

(d)  a + b + c + d ≥ 20

4. How is an Unbalanced Transportation Problem different from an Unbalanced Assignment Problem? Explain.
5. Distinguish between CPM and PERT on any two basis.
6. State any four advantages of using  models in O.R.
7. When the solution is called degenerate and non-degenerate in case of Transportation Problem? Explain.
8. What is a Trans-shipment Model?
9. Briefly bring out any two limitations of O.R.
10. Mention the applicability of LPP in the field of Defence and Marketing.

SECTION – B

II) Answer any FOUR questions.  Each carries 5 marks.                                      (4×5=20)
11. Best Investment Analyst, is an investment firm that manages stock portfolios for a number of clients. A new client has just requested that the firm handle a Rs. 80,000 portfolio. The client would like to restrict the portfolio to a mix of the following two stocks:

 Stock Price per share Estimated annual Return per share Maximum Possible Investment A Rs. 50 Rs. 6 Rs. 50,000 B Rs. 30 Rs. 4 Rs. 45,000

If ‘x’ and ‘y’ represent the number of shares of firms A and B respectively,

(a)   Develop the objective function assuming that the client desires to maximize the total annual return.

(b)   Write the mathematical expression for each of the following three constraints:

(i)                 Total investment funds available are Rs. 80,000.

(ii)              Maximum investment in stock A is Rs. 50,000.

(iii)Maximum investment in stock B is Rs. 45,000.

(c)    Mention the Non-Negativity restriction.

12. Solve the following Transportation Problem using North West Corner Method to find the IBFS.

 From/To D E F Supply A 6 4 1 50 B 3 8 7 40 C 4 4 2 60 Demand 20 95 35 ?
13. Discuss briefly the various characteristics of Operations Research.
14. Determine the optimal assignment to minimize the total cost.

 Machine/Jobs J1 J2 J3 J4 M1 28 10 16 14 M2 4 24 12 10 M3 14 16 6 18 M4 4 8 12 20
15. A small project consists of jobs as given in the table below:

 Activity Predecessor Activity Duration (days) A — 9 B — 8 C — 15 D A 5 E B 10 F D,E 2

(i)                 Draw the network.

(ii)              Calculate the project duration and identify the critical path.

16. Briefly explain the rules followed in formulating the Dual of a given LPP.

SECTION – C
III) Answer any THREE questions.  Each carries 15 marks.                                (3×15=45)
17. Solve the following LPP using Simplex Method.

Objective function:

Maximize Z = 1000x + 4000y + 5000 z

Subject to Constraints:

3x + 3z ≤ 22

x + 2y + 3z ≤ 14

3x + 2y ≤ 14

Where x, y and z ≥0

18. (a) Briefly explain ANY SIX Techniques of Operations Research in Business with suitable examples.                                                                          (12 marks)

(b) Write short notes for the following terms:

(i) Feasible solution  (ii) Network (iii) Non-Negativity Restriction     (3 marks)

19. A company has four sales representatives who are assigned to four different sales territories. The monthly sales increase estimated for each sales representative for different sales territories (in lakhs of rupees) are shown in the following table:

 Sales Representatives/ Sales Territories I II III IV A 200 150 170 220 B 160 120 150 140 C 190 195 190 200 D 180 175 160 190

Suggest an optimal assignment and the total maximum sales increase per month. If for certain reasons sales representative ‘B’ wants only sales territory III, will the optimal assignment schedule be different? If so find that schedule and the effect on the total sales.

20. A manufacturer has distribution centres located at Agra, Allahabad and Guwahati. These centres have available 40, 20 and 40 units of his product respectively. His retail outlets require the following number of units:

Patna = 25; Varanasi = 10; Silchar = 20; Kolkata = 30 and Kanpur = 15.

The shipping cost per unit in Rs. Between each centre and outlet is given in the table below:

 Distribution Centre Retail Outlets Patna Varanasi Silchar Kolkata Kanpur Agra 55 30 40 50 50 Allahabad 35 30 100 45 60 Guwahati 40 60 95 35 30

(i)                 Determine the shipping cost using VAM (only IBFS).              (7 marks)

(ii)              If the problem is solved using NWCM and LCM will the IBFS of shipping cost vary? If yes, show the changes.                            (7 marks)

(iii)            Of all the three methods of IBFS which method is most preferred and why?                                                                                                     (1 mark)

21. Use the graphical method to solve the following LPP.

Maximize Z = 80 a + 120 b

Subject to constraints:

a + b ≤ 9

a ≥ 2

b ≥ 3

20a + 50 b ≤ 360

where, a , b ≥ 0

SECTION – D

IV) Case Study                                                                                                              (1×15=15)
22. A  project is composed of the following activities, whose time estimates are listed in the table below:

 Activity Estimated duration (weeks) Optimistic (to) Most Likely ™ Pessimistic (tp) 1-2 1 1 7 1-3 1 4 7 2-4 2 2 8 2-5 1 1 1 3-5 2 5 14 4-6 2 5 8 5-6 3 6 15

You are required to:

a.                  Draw the project network and find the expected project length and the critical path.                                                                             (6 marks)

b.                  Find the variance and standard deviation of the project on the critical activities.                                                                                        (2 marks)

c.                   Calculate earliest and latest time occurrence and total float for each activity.                                                                                             (4 marks)

d.                 What is the probability that the project will be completed within 21 weeks?                                                                                          (1.5 marks)

e.                  Suppose the manager wants 85% surety, when should he start the project?                                                                                        (1.5 marks)

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## St. Joseph’s College of Commerce VI Sem Operation Research Question Paper PDF Download

1. JOSEPH’S COLLEGE OF COMMERCE (AUTONOMOUS)

END SEMESTER EXAMINATION – MARCH /APRIL 2015

B.COM (T. t.) – VI SEMESTER
OPERATION  RESEARCH
Duration: 3 Hours                                                                                             Max. Marks: 100
SECTION – A
I) Answer ALL the questions.  Each carries 2 marks.                                        (10×2=20)
1. What is the structure of a Linear Programming Problem?
2. Mention any Four important features of Operations Research.
3. In graphical approach of solving a LPP, differentiate between feasible solution and optimum solution.
4. With the help of a rough graph plot  x – y < 0 .
5. What is a model? How are operation Research Models classified?
6. How do you treat an unbalanced transportation problem before attempting to solve?
7. What is an assignment problem? Name the method used to solve.
8. A customer arrives at a bank counter every three minutes and it requires on an average two minutes for service. Find arrival rate (λ) and service rate (µ)
9. What is (a)  Jockeying

(b) Reneging

10. Mention any 2 applications of simulation.

SECTION – B

II) Answer any FOUR questions.  Each carries 5 marks.                                      (4×5=20)
11. A factory is engaged in manufacturing  of three products A,B and C for which profits per unit are Rs. 10, Rs.6 and Rs. 4 respectively. The time taken for preparatory work, machining and packing and maximum time available is stated in the table. Determine the most profitable mix, assessing that which is produced can be sold.  Formulate the LPP. (Do not solve.)

Time Required (in hrs)

 Product Preparatory Work Machining Packing A 1 10 2 B 1 4 2 C 1 5 6 Time Available (Hrs) 100. 600. 300.

12.

Find the dual of :

Minimize  Z= 8x + 4y + 2z

Subject to ,

4x + 2y+ z  ≤ 8

3x + 2y        ≤10

x + y + z   ≥ 5

x + y         = 3

Where x, y , z ≥ 0

13. Mention the different methods used in finding an initial solution to a transportation problem. Which is the most preferable method? Give reasons.

 Warehouses Plant  Capacity Plants A B C W 12 8 18 400 X 20 10 16 350 Y 24 14 12 150 Warehouse demand 500 200 300

Find IBFS by NWCM.  Determine the cost of transportation.

14. The table below shows the possible assignments of  5 jobs to 5 machines and gives the relative cost. Find the minimum cost assignment.

 Machines → Jobs ↓ M1 M2 M3 M4 M5 J1 8 4 2 6 1 J2 0 9 5 5 4 J3 3 8 9 2 6 J4 4 3 1 0 3 J5 9 5 8 9 5

15.

A departmental store has a single cashier. During the rush hours, customers arrive at the rate of 20 customers per hour. The average number of customers that can be processed by the cashier is 24 per hour. Assume that the conditions for the single channel queuing model apply, find the following.

(a)   Probability that the cashier is busy.

(b)   Probability that you can approach the cashier directly

(c)    Probability that there are exactly 2 customers in the system.

16.

Seven Star Tours and Travels organize various trips to tourists. Number of trips organized per month varies between 5 to 10 . The number of trips organised and their respective probabilities are given in the table. Using Monte-Carlo simulation technique find the approximate number of trips organised in 6 months.

Random numbers : 67, 84, 01, 77, 90, 14

 Trips Organised 5 6 7 8 9 10 Probability 0.1 0.2 0.45 0.1 0.1 0.05
SECTION – C
III) Answer any THREE questions.  Each carries 15 marks.                                (3×15=45)
17. (a) What are the steps to be followed in the OR approach of solving problems.

(b) “ Any real life problem is an OR problem”. In this context explain five               techniques of OR.                                                                                (5+10)

18. Solve the following LPP by graphical method.

Min Z = 20x + 40y

Subject to constraints,

36x + 6y ≥ 108

3x + 12y ≥ 36

20x + 10y ≥ 100

Where x, y ≥ 0

19. Solve the following transportation problem using Vogel’s approximation method and MODI method to find minimum cost of transportation.

 Direct   Source 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

20.

Using Hungarian Method find the optimum assignment schedule and the minimum cost of assignment.

 P Q R S T A 10 5 9 18 11 B 13 9 6 12 14 C 3 2 4 4 5 D 18 9 12 17 15 E 11 6 14 19 10

21.

At The Family Doctor Clinic patients arrive at an average of 6 patients per hour. The clinic is attended by Dr. Satyanarayan. Some come for minor check up while some others require thorough inspection for the diagnosis. The doctor can check 10 patients  per hour on an average  . It can be assumed that arrivals follow Poisson Distribution and doctor’s inspection time follows an exponential distribution Determine.

(i)      The percentage of time that a patient can walk to the doctor without having to wait.

(ii)    The expected average number of patients in the system and the queue.

(iii)  The expected average waiting time (in minutes) of a patient in the system and the queue.

(iv)  The average time the doctor is free on a 7 hours working day.

SECTION – D

IV) Case Study                                                                                                              (1×15=15)
22. A flight is scheduled to leave from Delhi to Hyderabad at 10 a.m The pattern of departure of the flight is as follows:

 Delay in minutes 0 5 10 15 20 25 30 Probability 0.25 0.15 0.1 0.06 0.1 0.2 0.14

The flight normally takes 75 minutes, but actual time varies as follows:

 Actual flight time in minutes 70 75 77 78 80 Probability 0.2 0.2 0.25 0.15 0.2

If the scheduled time of arrival at Hyderabad is 11.a.m Using Monte Carlo method of simulation find out the arrival time for 8 consecutive days. How many days do flights arrive after 11.30 and how many reach before 11.10 ?

 Random no for  Delay in minutes 28 57 60 17 64 20 27 58 Random no for Actual Flight time 28 29 83 58 41 18 67 16

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## St. Joseph’s College of Commerce Operation Research Question Paper PDF Download

ST. JOSEPH’S COLLEGE OF COMMERCE (AUTONOMOUS)
END SEMESER EXAMINATION – MARCH/APRIL 2016
B.Com (T.T.) – VI SEMESTER
C2 12 602: OPERATION RESEARCH
Duration: 3 Hours                                                                                             Max. Marks: 100
SECTION – A
I) Answer ALL the questions.  Each carries 2 marks.                                        (10×2=20)
1. Give the meaning of Operation Research.
2. What do you mean by a “feasible solution” in a transportation problem?  State any 3 methods of obtaining IBFS.
3. What is an unbounded solution and a redundant solution of a LPP?
4. What is a loop in transportation problem? Draw any 2 types of loops.
5. Explain the term unbalanced and degenerancy in a Transportation Problem.
6. What is an assignment problem?
7. The Customers arrive at a booking office window every 2 minutes. It takes 1 minute for the person to serve the customer. Find Arrival rate and service rate of the queing system.
8. State the advantages of simulation.
9. Mention the steps involved in formulation of LPP.
10. List any 4 areas of LPP applications.

SECTION – B

II) Answer any FOUR questions.  Each carries 5 marks.                                      (4×5=20)
11. A firm produces two spare parts A and B using milling machine and grinding machine. The machine time required for each spare part and machine time available for each machine are given in the following table. The profit on selling each spare part is also given:

 Machines Time required per unit for Maximum time available per day Spare part A Spare part B Milling machine 10 minutes 5 minutes 2500 minutes Grinding machine 4 minutes 10 minutes 2000 minutes Profit per spare part Rs.50 Rs.100

Formulate a linear programming problem such that the number of spare parts A and B manufactured per week maximizes the profit.

12. Explain the scope of Operation Research.

13. Determine the initial basic feasible solution for the following Transportation problem by Least Cost method. The table given transportation cost from origins to destinations in ‘000s of rupees.

 Warehouses Availabilities W1 W2 W3 Factories F1 9 8 16 120 F2 15 10 17 80 F3 3 9 12 80 Requirements 150 80 50 280

14.

A company has 4 salesmen A, B, C and D. These salesmen are to be allotted to 4 districts1, 2, 3 and 4. The estimated profit per day for each salesman in each district is given in the following table. What is the optimal assignment which will yield maximum profit?

 1 2 3 4 A 16 10 14 11 B 14 11 15 15 C 15 15 13 12 D 13 12 14 15

15.

In a drive-in restaurant the arrivals follow Poisson distribution with an average of 2 cars in every 15 minutes. The restaurant  can serve the customers at the rate of 5 minutes per customer and obey exponential probability distribution. Find out  a) The probability that the service is idle.

b) Expected time that the customer has to spend in the queue.

c) Average length of the queue.

16. The daily production of mopeds in a factory varies from 146 to 154 depending upon the availability of raw material and other working conditions. The associated probabilities for various levels of production are highlighted in the following table:

 Production per day 146 147 148 149 150 151 152 153 154 Probability 0.04 0.09 0.12 0.14 0.11 0.1 0.2 0.12 0.08

The finished mopeds are transported in a specially designed lorry accommodating only 150 mopeds to the maximum. If there is any excess moped produced over and above 150, they cannot find space in the lorry and hence can be carried over to the next day’s lorry for transport. Using the following random numbers  43, 18, 26, 10, 12 simulate the production levels for 5 days and obtain the answer to the following questions:

a)      What is the average number of mopeds waiting in the factory?

b)     What will be the average number of empty spaces on the lorry?

SECTION – C

III) Answer any THREE questions.  Each carries 15 marks.                                (3×15=45)
17.  a) “Model building is the essence of OR approach”. Describe the classification  of models in detail.

b)Explain the features of Operation research

(10+5)

18. Plot a graph for the following constraints, Identify the feasible region and the Optimum solution of the LPP.

Max Z= 3x + 4y

Subject to constraints,

3x + y >= 6

x + y <=8

y<=4

x>= 2

where x, y >=0

19. Determine the initial basic feasible solution for the following transportation problem by Vogel’ Approximation Method and also find optimal solution by MODI method.

 Origins Destinations Availabilities D1 D2 D3 O1 2 8 7 100 O2 10 11 12 90 O3 5 6 9 60 O4 8 3 5 100 Requirements 80 120 150 350
20. Given the following cost matrix obtain the minimum operation time cost. Also state the optimum assignment using Hungarian method.

 Men -> Task I II III IV V A 1 3 2 3 6 B 2 4 3 1 5 C 5 6 3 4 6 D 3 1 4 2 2 E 1 5 6 5 4
21. In a market analysis, the following information was collected. Simulate for 6 trials . Workout the average profit.

RN for Selling Price         – 78, 43, 92, 87, 47, 83

RN for Unit Cost              – 23, 08, 28, 17, 73, 87

RN for sales volume        – 58, 86, 62, 06, 03, 52

 Selling Price Prob. Unit Cost Prob. Sales Volume Prob. 35 0.30 30 0.40 800 0.15 45 0.40 35 0.25 650 0.45 50 0.20 40 0.15 500 0.30 55 0.10 45 0.20 450 0.10

SECTION – D

IV) Case Study – Compulsory question.                                                                (1×15=15)
22. A departmental store has a single cashier. During the rush hours, customers arrive every 3 minutes.  The average number of customers that can be processed by the cashier is 24 per hour. Assuming that the conditions for the single channel queuing model apply, find the following

a)      Utilization rate

b)     Probability that the cashier is idle

c)      Average number of customers in the queuing system.

d)     Average time a customer spends in the system.

e)      Average number of customers in the queue.

f)       Average time a customer spends in the queue waiting for service.

g)     The number of hours the cashier is idle if he works from 9am to 6pm.

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## St. Joseph’s College of Commerce VI Sem Operation Research Question Paper PDF Download

 REG NO:

ST. JOSEPH’S COLLEGE OF COMMERCE (AUTONOMOUS)
END SEMESTER EXAMINATION – MARCH/APRIL 2016
B.Com – VI Semester
C1 11 602 :OPERATION RESEARCH
Duration: 3 Hours                                                                                             Max. Marks: 100
SECTION – A
I) Answer ALL the questions.  Each carries 2 marks.                                        (10×2=20)
1. Name any four techniques of Operation Research.
2. Give any two merits of linear programming.
3. What is a key element in Simplex Method?
4. With the help of a diagram, show what is a redundant constraint?
5. How will you solve maximization case in assignment problem?
6. Give the procedure of tracing a closed loop.
7. What is independent float?
8. How do you proceed in simplex, if there is a tie for the key column?
9. What is a dummy activity?
10. Differentiate between CPM and PERT.
SECTION – B
II) Answer any FOUR questions.  Each carries 5 marks.                                      (4×5=20)
11. A  BPO requires different numbers of employees on different days of the week. Union rules state each employee must work 5 consecutive days and then receive two days off. Find the minimum number of employees needed. Formulate the problem

Mon        Tue       Wed        Thur      Fri       Sat      Sun

Staff Needed           17            13           15            19         14         16       11

12. A construction company has four large bulldozers located at four different garages. The bulldozers are to be moved to four different construction sites. The distances in miles between the bulldozers and the construction sites are given below.

 Buldozer/site A B C D 1 90 75 75 80 2 35 85 55 65 3 125 95 90 105 4 45 110 95 115

How should the bulldozers be assigned to the construction sites in order to minimize the total distance traveled?

13. Find the initial solution to the transportation problem with the help if LCM method and NWCR

 D E F G Available A 11 13 17 14 250 B 16 18 14 10 300 C 21 24 13 10 400 Required 200 225 275 250

What is unusual about the solution that is derived?

14. Draw a network diagram for the project whose activities and their predecessor relationships are given below:

Activity:         A          B         C       D         E         F         G       H

Predecessor

Activity-        –           –          –         A         B          C       D,E     F,G

15. What do you understand by the term duality? Solve the dual given below:

Min Z= 2X1 + 3 X2 + 4X3

Subject to

2X1 + 3X2 + 5X3  ≥ 2

3X1 + X2 + 7X2  = 3

X1 + 4X2 + 6X3 < 5

X1, X2,X3 > 0

16. “Operation Research is a decision science which helps management to make better decisions.” Discuss.
SECTION – C
III) Answer any THREE questions.  Each carries 15 marks.                                (3×15=45)
17. a. What are the different types of Models used in OR?  Explain in detail.
b.  The products A, B and C are produced in three machine centres X, Y and Z. Each product involves operations involves on each of the machine centres. The time required for each operation for unit amount of each product is given below:

 Products Machine Centres X Y Z A 10 7 2 B 2 3 4 C 1 2 1

There are 100, 77 and 80 hours available at machine centres X, Y and Z respectively. The profit per unit of A,B and C is Rs 12, Rs 3 and Rs 1 respectively. Formulate the LPP .                                                               (10+5)

18. Solve by Simplex Method

Maximize    Z= 3x +4y +z

Subject to constraints,

x + 2y +3z ≤ 90

2x  +y +z ≤ 60

3x + y + 2z ≤ 80

Where x,  y,  z ≥ 0

19. Five lathes are allotted to five operators (one for each). The following table gives weekly output figures (in pieces).

 Weekly output in lathe OPERATOR L1 L2 L3 L4 L5 P 20 22 27 32 36 Q 19 23 29 34 40 R 23 28 35 39 34 S 21 24 31 37 42 T 24 29 31 36 41

Profit per piece is Rs 25. Solve the assignment problem and find the maximum profit per week.

20. Solve the following transportation problem in which cell entries represent unit costs

 D1 D2 D3 AVAILABLE Q1 2 7 4 5 Q2 3 3 1 8 Q3 5 4 7 7 Q4 1 6 2 14 REQ 7 9 18 34

Apply Modi Method to test optimality and find the optimal solution.

21. An engineering project has the following activities, whose time estimates are listed below:

 Activity Estimated duration(in months) Optimistic Most Likely Pessimistic 1-2 2 2 14 1-3 2 8 14 1-4 4 4 16 2-5 2 2 2 3-5 4 10 28 4-6 4 10 16 5-6 6 12 30

1.      Draw the project network and find critical path.

2.      Find expected duration and variance for each activity.

3.      Calculate the variance and standard deviation of the project length.

4.      What is the probability that the project will be completed at least eight months earlier than the expected time?

5.      If the project due date is 38 months, what is the probability of not meeting the due date?

SECTION – D
IV) Case Study                                                                                                              (1×15=15)
22. A farmer is engaged in breeding pigs. The pigs are fed on various products grown on the farm. Because of the need to ensure nutrient constituents, it is necessary to buy additional one or two products which we shall call A and B. The nutrient constituents (vitamins and proteins) in each of the products are given below:

 Nutrient constituents Nutrient in the product Minimum requirement of nutrient constituents A B X 36 6 108 Y 3 12 36 Z 20 10 100

Product A costs Rs 20 per unit and Product B costs Rs 40 per unit. Determine how much of products A and B must be purchased so as to provide the pigs nutrients not less than the minimum required, at the lowest possible cost. Solve graphically.

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