## Operations Research - Dec 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)** Define operations research. Explain the phases of operations research.(6 marks)
**1 (b)** A firm manufactures two products A and B on which the profit earned per unit are 3 and 4 respectively. Each product is processed on two machines M_{1} and M_{2}. Product A requires
one minute of processing time on MI and two minutes on M_{2}while B requires one minute on M_{1} and one minute on M_{2}. Machine M_{1} is available for not more than 7 hrs. 30 mins while machine M_{2} is available for 10 hrs during any working day. Find the number of units
of product A and B to be manufactured to get maximum profit(14 marks)
**2 (a)** Solve the following LPP using simplex method

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

Subjected to constraints

x_{2}+x_{2}≤4

x_{1}-x_{2}≤2

x_{1}, x_{2}≥0(10 marks)
**2 (b)** Solve the given problem by using Big-M method.

Maximum Z-- 2x_{1}-x_{2}

Subject to constraints:

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

4x_{1}+3x_{2},\ge 6

x_{1}-2x_{2}\le and

x_{1},x_{2}\ge0(10 marks)
**3 (a)** ABC limited has three production shops supplying a product to 5 warehouses. The cost of
production varies from shop to shop, cost of transportation from shop to shop, cost of transportation from shop to warehouses also varies. Each shop has a specific production capacity of each warehouse has certain amount of requirement. The cost of transportation
are as given below

Shop | Warehouse | Capacity | Cost of production | ||||

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

A | 6 | 4 | 4 | 7 | 5 | 100 | 14 |

B | 5 | 6 | 7 | 4 | 8 | 125 | 16 |

C | 3 | 4 | 6 | 3 | 4 | 175 | 15 |

Requirement | 60 | 80 | 85 | 105 | 70 |

Find the optimum quantity to be supplied from each shop to different warehouse at minimum cost.(12 marks)

**3 (b)**A ABC company has 5 takes and 5 Persons to perform. Determine the optimal assignment that minimizes the total cost

Jobs | Machines | ||||

A | B | C | D | E | |

P | 6 | 7 | 5 | 9 | 4 |

Q | 7 | 5 | 10 | 9 | 6 |

R | 5 | 4 | 3 | 6 | 5 |

S | 8 | 3 | 5 | 6 | 4 |

T | 4 | 7 | 5 | 6 | 6 |

**4 (a)**Explain the importance of integer programming(5 marks)

**4 (b)**Solve the following linear programming by Gomory technique

Maximum Z=x

_{1}+x

_{2}

Subject to 2x

_{1}+x

_{2}\le6

4x+5x

_{2}\le 20

x

_{2}x

_{2}\ge and intergers(15 marks)

**5 (a)**Define the following

i) Normal time

ii) Crash time

iii) Free float(6 marks)

**5 (b)**R and D activity has 7 activities for which the three time estimate are given below along with its preceding activity

<colgroup span="5" width="85"> </colgroup>

Activity | Preceding activity | Optimistic time (a) | Most likely time(m) | Pessimistic time (b) |

A | - | 4 | 6 | 8 |

B | A | 6 | 10 | 12 |

C | A | 8 | 18 | 24 |

D | B | 9 | 9 | 9 |

E | C | 10 | 14 | 18 |

F | A | 5 | 5 | 5 |

G | D,E,F | 8 | 10 | 12 |

i) Draw PERT network

ii) find EST,Lst and slack for each node

iii)Find critical path expected project duration(14 marks)

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

**6 (b)**Arrival rate of telephone call at a telephone booth are according to poisson distribution , with an average time of 9 minute between two consecutive arrivals. The length of the telephone call is assumed to be exponentially distributed, with mean 3 minute

i) Determine the probability that a person arriving at the booth will having to wait.

ii) Find the average queue length

iii) The telephone company will install a second booth when convinced that an arrival would expect to have to wait at least four minute for the phone. Find the increase in flow rate of arrival which justify a second booth.

iv) What is the probably that he will have to wait for more than 10 minute before the phone is free?(14 marks)

**7 (a)**Explain clearly the following terms

i)Pay off matrix

ii) Saddle point

iii) Fair game(6 marks)

**7 (b)**Use dominance to find the optimum strategies for the both player

$$\begin{matrix} B_{1} & B_{2} &B_{3} &B_{4} &B_{5} &B_{6} \end{matrix}\\\begin{matrix} A_{1}\\A_{2} \\A_{3} \\A_{4} \\A_{5} \end{matrix}\begin{bmatrix} 4 &2 &0 &2 &1 &1 \\4 &2 &1 &3 &2 &2 \\4 &3 &7 &-5 &1 &2 \\4 &3 &4 &-1 &2 &2 \\4 &3 &3 &-2 &2 &2 \end{bmatrix}$$(7 marks)

**7 (c)**Solve the game by graphical method

$$\begin{matrix} b_{1} &b_{2} \end{matrix}\\\begin{matrix} a_{1}\\a_{2} \\a_{3} \\a_{4} \end{matrix}\begin{bmatrix} 1 &-3 \\3 &5 \\-1 &6 \\4 &1 \end{bmatrix}$$(7 marks)

**8 (a)**Define i) Total elapsed time ii) Idle time(4 marks)

**8 (b)**List the assumption made while dealing sequencing problem(4 marks)

**8 (c)**We have five jobs which must go through the machines A,B and C in the order ABC. Determine a sequence for job taht will minimize the total elapsed time and idle time for each machine.

Job number | Processing time in hours | ||||

1 | 2 | 3 | 4 | 5 | |

Machine A | 5 | 7 | 6 | 9 | 5 |

Machine B | 2 | 1 | 4 | 5 | 3 |

Machine C | 3 | 7 | 5 | 6 | 7 |