**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**

**Answer**__SEVEN__questions out of Ten. (7 x 5 = 35)

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

- 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

- 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?

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

- 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.

- 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.

- 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?

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

resolved in a transportation problem at the initial stage.

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

- Define Slack, surplus and artificial variables.

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__Section – B__

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

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- 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?

- 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.

- 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.

- 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:

- probability that no customers are in the system
- the average number of customers in the queue
- average number of customers in the system
- average time a customer spends in the waiting line
- the average time the customer spends in the system.
- 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)**

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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.