## Operations Research - Jun 2014

### Mechanical Engg. (Semester 7)

TOTAL MARKS: 100

TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.

(2) Attempt any **four** from the remaining questions.

(3) Assume data wherever required.

(4) Figures to the right indicate full marks.
**1 (a)** List and explain the various phases of OR problems(6 marks)
**1 (b)** What are the limitations of OR?(4 marks)
**1 (c)** A manufacturer of a line of patent medicines is preparing a production plan on medicines A and B. There are sufficient ingredients available to make 20,000 bottles of A' and 40,000
bottles of B'. But three are only 45,000 bottles into which either of the medicines can be put. Further more. it takes 3 hours to prepare enough material to till 1.000 bottles of A. it
takes 1 hour to prepare enough material to fill 1,000 bottles of 'B' and there are 66 hours available for this operation. The profit is Rs 8 per bottle for 'A' and Rs 7 per bottle for 'B'. formulate the problem as a LPP and solve by graphical method(10 marks)
**2 (a)** Define slack, surplus, and artificial variables,(6 marks)
**2 (b)** Obtain the dual of the following primal LP problem:

Minimize Z- x_{1}+x_{2}+x_{3}

Subject to x_{1}-3x_{2}+4x_{3}-5 ; 2x_{1}-2x_{2}\le3

x_{1} +2x_{3}\ge30 ; x_{1}, x_{2}\ge0(10 marks)
**3 (a)** A company has plants A, B and C which have capacity to produce 300, 200 and 500 kg respectively of a particular chemical/day. The production cost per kg in these plants are Rs0.70, Rs0.60 and Rs 0.66 respectively. Four bulk consumers have. placed orders for the products on the following books

Consumer | Kg required / day | Price offered Rs/kg |

I | 400 | 1.00 |

II | 250 | 1.00 |

III | 350 | 1.02 |

IV | 150 | 1.03 |

Shipping costs in paise/kg from plants to consumer are given below

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I | II | III | IV | |

A | 3 | 5 | 4 | 6 |

B | 8 | 11 | 9 | 12 |

C | 4 | 6 | 2 | 8 |

Workout the optimum schedule for the above situation considering all the data given(12 marks)

**3 (b)**company has a team of four salesman and there are four districts where the company wants to start its business. The company estimates that the profit/day is given below. Find the assignment of salesman to districts which gives maximum profit.

I | II | III | IV | |

A | 16 | 10 | 14 | 11 |

B | 14 | 11 | 15 | 15 |

C | 15 | 15 | 13 | 12 |

D | 13 | 12 | 14 | 15 |

**4 (a)**Explain the branch and bound method in integer programming(6 marks)

**4 (b)**Use Gomary's' fractional cutting plane method to solve the following IPP

Minimize Z=x

_{1}+4x

_{2}

5x

_{1}-4x

_{2}\le15

x

_{1},x

_{2}\ge 0 and are integers(14 marks)

**5 (a)**List the differences between PERT and CPM(5 marks)

**5 (b)**A small project consists of EIGHT activities has the following characteristics.

Activity | Preceding activity | Times estimate (weeks) | ||

t_{u} |
t_{m} |
t_{p} |
||

A | - | 2 | 4 | 12 |

B | - | 10 | 12 | 26 |

C | A | 8 | 9 | 10 |

D | A | 10 | 15 | 20 |

E | A | 7 |
7.5 |
11 |

F | B,C | 9 | 9 | 9 |

G | D | 3 | 3.5 | 7 |

H | E,F,G | 5 | 5 | 5 |

(i) Draw the PERT network for the project.

(ii) Determine the critical path and prepare the activity schedule for the project.

(iii) If a 30 week deadline is imposed. what is the probability that the project will be completed within the time limit?(15 marks)

**6 (a)**Briefly explain the queuing system and their characteristics(6 marks)

**6 (b)**A postal clerk can service a customer in 3 minutes. The service time is being exponentially distributed. The inter arrival time of customers is also experimentally distributed with an average of 12 minutes during early morning slack period and an average of 5 minutes during the afternoon peak period. Assess the average queue length and the expected waiting time in the queue during the two periods(14 marks)

**7 (a)**Explain the following

i) Pay off matrix ii) Saddle point iii) Fair game(5 marks)

**7 (b)**Explain the rule of dominance(3 marks)

**7 (c)**Use of property of dominance to solve the following game

I | II | III | IV | V | VI | |

I | 0 | 0 | 0 | 0 | 0 | 0 |

II | 4 | 2 | 0 | 2 | 1 | 1 |

III | 4 | 3 | 1 | 3 | 2 | 2 |

IV | 4 | 3 | 7 | -5 | 1 | 2 |

V | 4 | 3 | 4 | -1 | 2 | 2 |

VI | 4 | 3 | 3 | -2 | 2 | 2 |

**8 (a)**State the assumptions made while dealing with sequencing problems(4 marks)

**8 (b)**Find the sequence for the following six jobs that will minimize the total elapsed time for the three operations

Job | 1 | 2 | 3 | 4 | 5 | 6 |

Turning (A) | 3 | 12 | 5 | 2 | 9 | 11 |

Threading (B) | 8 | 6 | 4 | 6 | 3 | 1 |

Knurling (c) (Time in minute) | 13 | 14 | 9 | 12 | 8 | 13 |

**8 (c)**Use graphical method m minimi7c the time required to process the following jobs on the machines. Calculate the total elapsed time to complete the jobs. For each machine specify the job that should be done first

Machine | ||||||

Job 1 | Sequence: | A | B | C | D | E |

Time (hr) : | 6 | 8 | 4 | 12 | 4 | |

Job 2 | Sequence : | B | C | A | D | E |

Time (hr) : | 10 | 8 | 6 | 4 | 12 |