University of Mumbai 2008-2nd Sem M.E Mechanical Engineering Machine Design Optimization -e - Question Paper
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1. (a) State the various methods available for solving a multivariable optimization pr equality constraints.
(b) A uniforrr. column of rectangular cross section is to bo constructed for supporting a water tank of mass M.
It is required (i) to minimize the mass of the column for economy and (ii) to maximize the natural frequency of transverse vibration of the system for avoiding possible resonance due to wind. Formulate the problem of designing the column to avoide failure due to direct compression and buckling.
Assume the permissible compresive stress to be o.
( Hint: The natural frequency of transverse vibration of water tank (<o) by treating it as a cantilever beam with a tip mass M is given by ]
1/2
3EI
<o =
f M + m )l3 I 140 )
and buckling stress for a fixed-free column (ab) is given by
At2 bd
Water Tank on a Column
2. (a) A firm manufactures two items. It purchases castings which are then machined, bored and polished. Castings for items A and B cost Rs. 2 and Rs. 3 respectively and are sold at'Rs. 5 and Rs. 6 each respectively. Running cost of the three machines are Rs. 20, Rs. 14 and Rs. 17 50 per hour respectively.
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Capacities of the machines are :
Part A Part B
Machining Capacity 25/hr 40/hr
Boring Capacity 28/hr 35/hr
Polishing Capacity 35/hr 25/hr
Formulate the L.P. model to determine the product mix that maximizes the profit.
(b) Explair the following terms :
(v) Feasible Region
(i) Global Maxima
(ii) Relative Maxima
(iii) Global Minima (iy) Relative Minima
(vl) Non Feasible Region
(vil) Saddle Point
(viii) Constraints.
3/ fa} The manager of an oil refinery has to decide upon the optimal mix of two possible blooding 15 processes of which the inputs and outputs per production run are as follows :
|
Process |
In |
put |
Output | |
|
Crude A |
Crude B |
Gasoline X |
Gasoline Y | |
|
1 |
5 |
3 |
5 |
8 |
|
2 |
4 |
5 |
4 |
5 |
The maximum amount available o1 crude A and B is 200 units and 150 units respectively. Markel requirements show that at feast 100 units of gasoline X and 80 units of gasoline Y mus: be produced. The profits per production run from process 1 and process 2 are Rs. 3 and Rs. 4 respectively. Formulate the problem as a linear programming problem. Solve Ihe problem by the graphical method.
(b) State the necessary and sufficient condition for maximization of multivariable function 5 f(x).
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4. (a) A scaffolding system consists of three beams and six ropes as shown in figure. Each of the top ropes A and B can carry a load of W|t each of the middle ropes C and D can carry a load of W2, and each of the bottom ropes E and F can carry load of W9. If the loads acting on beams 1, 2 and 3 are x1( x2, and x3 respect vely, as shown in figure formulate the problem of finding the maximum load (x1 + x2 + x3) that can be supported by the system. Assume that the weights of beam 1, 2 and 3 are w,( w2 and w3 respectively and the weights of the ropes are negligible.
/////S/S//// / / / / / / / / / / / S Sf
6
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h
2JL -h-*
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=1
ou
6. (a) Find all the basic feasible solution of the equations :
2x,
6x,
(b) Explain Canonical and Standard Forms of linear programming problem.
- 3x, + x2 < 6
X, + 2x? < 4
x2 < - 3
(b) Explain the general linear programming problem.
5. (a) Minimize
Subjected to
(b) Explain the significance of Lagrange's Multiplier.
7. Write short notes on the following :
(a) Dynamic Programming
(b) Computer implementation of genetic algorithm
(c) The Slmplox Method
(d) Lagrange multiplier method.
5
10
10
10
10
*1 - 4*2
+ 6x2 + 2x3 + x4 = 3
4Xg + 6x4 = 2
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Attachment: |
| Earning: Approval pending. |
