## Operations Research - Dec 2013

### 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)** Old hens can he brought at Rs.50/- each and young hens can be brought at Rs.l00/- each. The old hens lay 3 eggs/week and young hens 5/week. Each egg cost 2 Rs. A hen cost
Rs. 5/week to feed. if a person has only Rs.2000 to spend for hem. Formulate the problem to decide how many of each kind of hens should he buy? And he cannot house more than 40 hens. formulate the problem a LPP model(10 marks)
**1 (b)** Solve the following LPP by Graphical method Z_{max} =3x_{1}+4x_{2}

subject to constrains,

5x_{1}+4x_{2}\le 200.

3x_{1}+5x_{2}\le150

5x_{1}+4x_{2}\ge 100

8x_{1}+4x_{2}\ge20(10 marks)
**2 (a)** Solve the following LPP using simplex method

Z_{ max}=10x_{1}+5x_{2}

subject to constrains,

4x_{1}+5x_{2}\le100

5x_{1}+2x_{2}\le 80

x_{2}& x_{2,/sub> \ge0}(8 marks)
**2 (b)** Solve the given problem by using Big m

Z_{min}=5x_{1}+3x_{2}

subject to constraints,

2x_{1}+4x_{1}\le12

2x_{1}+2x_{2}=10

5x_{1}+2x_{2}\ge10

x_{1}&x_{2}\ge0(12 marks)
**3 (a)** Find the optimality for given problem and initial solution by using VAM method

Destination \Origin |
W_{1} |
W_{2} |
W_{3} |
W_{4} |
Supply |

F_{1} |
2 | -2 | 2 | 1 | 3 |

F_{2} |
10 | 8 | 5 | 4 | 7 |

F_{3} |
7 | 6 | 6 | 8 | 5 |

Demand | 4 | 3 | 4 | 4 |

**3 (b)**A AML company has 5 tasks has 5 tasks and persons to perform. Determine the optimal assignment and to minimize the total cost.

Jobs | Machine | ||||

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)**A travelling salesman has to visit 4 cities. He wishes to start from a particular city, visit each city once and return to his starting point find the least cost route

$$\begin{matrix} A &B &C &D &E \end{matrix}\\\begin{matrix} A\\B \\C \\D \\E \end{matrix}\begin{bmatrix} \infty &4 &10 &14 &2 \\12 &\infty &6 &10 &4 \\16 &14 &\infty &8 &14 \\24 &8 &12 &\infty &10 \\2 &6 &4 &16 &\infty \end{bmatrix}$$(10 marks)

**4 (b)**What is integer programming? Why its is ,needed and write is needed and write the branch and bound algorithm(10 marks)

**5 (a)**The following tables gives the activities in a construction project and other related information.

Activity |
t_{0} |
t_{m} |
t_{p} |

1-2 | 20 | 30 | 46 |

1-3 | 9 | 12 | 21 |

2-3 | 3 | 5 | 7 |

2-4 | 2 | 3 | 4 |

3-4 | 1 | 2 | 3 |

4-5 | 12 | 18 | 24 |

i) Draw a pert network

ii) Calculate project duration

iii) Find the critical path

iv) Find the probability that the project will be completed within 50days.(14 marks)

**5 (b)**Define the following

i) Normal time

ii) Crash time

iii) Free float(6 marks)

**6 (a)**Define i) Fair game ii) Pure strategy iii) Mixed strategy(6 marks)

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

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

**6 (b)**Solve the game by graphical method

$$]\begin{bmatrix} 1 &-3 \\3 &5 \\-1 &6 \\4 &1 \end{bmatrix}$$(7 marks)

**7 (a)**Write the characteristics of waiting lines(5 marks)

**7 (b)**At what average rate must a clerk at super market work in order to ensure a probability of 0.9 that the customer will not have to wait longer than 12 minutes? It is assumed that there is only one counter to which customer arrive in a Poisson fashion at an average rate of 15/hr. The length of service by the clerk has an exponential distribution(7 marks)

**7 (c)**In a hair dress by saloon with one barber. the customer arrived follows Poisson distribution at an average rate of one every 45 minutes. The service time is exponentially distributed with a mean of 30 minutes. Find:

i) Average number of customer in a saloon.

ii) Average waiting time of customer before service.

iii) Average idle time of barber(8 marks)

**8 (a)**Define the following:

i) Idle time ii) Total elapsed time(4 marks)

**8 (b)**Write the assumption underlying the sequencing problem(4 marks)

**8 (c)**Find the sequence that minimizes the total elapsed time. idle time and normal time.

Machine | Jobs | ||||

A | B | C | D | E | |

M_{1} |
6 | 8 | 7 | 10 | 6 |

M_{2} |
3 | 2 | 5 | 6 | 4 |

M_{3} |
4 | 8 | 6 | 7 | 8 |