1 (a)
Define operations research. Explain the phases of operations research.

6 M

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 M

2 (a)
Solve the following LPP using simplex method

Maximize Z-3x

Subjected to constraints

x

x

x

Maximize Z-3x

_{1}+2x_{2}Subjected to constraints

x

_{2}+x_{2}≤4x

_{1}-x_{2}≤2x

_{1}, x_{2}≥0
10 M

2 (b)
Solve the given problem by using Big-M method.

Maximum Z-- 2x

Subject to constraints:

3x

4x

x

x

Maximum Z-- 2x

_{1}-x_{2}Subject to constraints:

3x

_{1}+x_{2}-34x

_{1}+3x_{2},\ge 6x

_{1}-2x_{2}\le andx

_{1},x_{2}\ge0
10 M

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

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

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 M

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 |

8 M

4 (a)
Explain the importance of integer programming

5 M

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

Maximum Z=x

Subject to 2x

4x+5x

x

Maximum Z=x

_{1}+x_{2}Subject to 2x

_{1}+x_{2}\le64x+5x

_{2}\le 20x

_{2}x_{2}\ge and intergers
15 M

5 (a)
Define the following

i) Normal time

ii) Crash time

iii) Free float

i) Normal time

ii) Crash time

iii) Free float

6 M

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

i) Draw PERT network

ii) find EST,Lst and slack for each node

iii)Find critical path expected project duration

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 M

6 (a)
Briefly explain queuing system and its characteristics

6 M

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?

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 M

7 (a)
Explain clearly the following terms

i)Pay off matrix

ii) Saddle point

iii) Fair game

i)Pay off matrix

ii) Saddle point

iii) Fair game

6 M

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

\[\begin{matrix} B_{1} & B_{2} &B_{3} &B_{4} &B_{5} &B_{6} \end{matrix}\\\begin{matrix} A_{1}\\A_{2} \\A_{3} \\A_{4} \\A_{5} \end{matrix}\begin{bmatrix} 4 &2 &0 &2 &1 &1 \\4 &2 &1 &3 &2 &2 \\4 &3 &7 &-5 &1 &2 \\4 &3 &4 &-1 &2 &2 \\4 &3 &3 &-2 &2 &2 \end{bmatrix}\]

\[\begin{matrix} B_{1} & B_{2} &B_{3} &B_{4} &B_{5} &B_{6} \end{matrix}\\\begin{matrix} A_{1}\\A_{2} \\A_{3} \\A_{4} \\A_{5} \end{matrix}\begin{bmatrix} 4 &2 &0 &2 &1 &1 \\4 &2 &1 &3 &2 &2 \\4 &3 &7 &-5 &1 &2 \\4 &3 &4 &-1 &2 &2 \\4 &3 &3 &-2 &2 &2 \end{bmatrix}\]

7 M

7 (c)
Solve the game by graphical method

\[\begin{matrix} b_{1} &b_{2} \end{matrix}\\\begin{matrix} a_{1}\\a_{2} \\a_{3} \\a_{4} \end{matrix}\begin{bmatrix} 1 &-3 \\3 &5 \\-1 &6 \\4 &1 \end{bmatrix}\]

\[\begin{matrix} b_{1} &b_{2} \end{matrix}\\\begin{matrix} a_{1}\\a_{2} \\a_{3} \\a_{4} \end{matrix}\begin{bmatrix} 1 &-3 \\3 &5 \\-1 &6 \\4 &1 \end{bmatrix}\]

7 M

8 (a)
Define i) Total elapsed time ii) Idle time

4 M

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

4 M

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 |

12 M

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