JNTU
JNTU B.Tech III OPTIMIZATION TECHNIQUES Previous Question Paper (March-2024 - R16)
Note: i) Question paper consists of Part A, Part B. ii) Part A is compulsory, which carries 25 marks. In Part A, Answer all questions. iii) In Part B, Answer any one question from each unit. Each question carries 10 marks and may have a, b as sub questions.
Part A
State the optimization problem.
Answer
An optimization problem involves finding the values of decision variables that minimize or maximize an objective function subject to a set of equality or inequality constraints.
State the necessary and sufficient condition multivariable unconstrained optimization problem.
Answer
For a function f(X), the necessary condition for a local extremum at X* is that the gradient vector must be zero: ∇f(X*) = 0. The sufficient condition is that the Hessian matrix H(X*) must be positive definite for a minimum or negative definite for a maximum.
Define the fundamental theorem of LPP.
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State the procedure involved in least cost method for initial B.F.S of Transportation problem.
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State the problem of conducting last experiment in Fibonacci method and how is it resolved and with what accuracy.
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State the steps involved in steepest descent method.
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State the characteristics of constrained problem.
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State the basic approach of penalty function method.
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State the Bellman’s principle of optimality.
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Explain the concept of sub optimization.
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Part B
The ABC Electric Appliance Company produces two products: Refrigerators and ranges. Production takes place in separate departments. Refrigerators are produced in department 1 and ranges are produced in department II. The company’s two products are produced and sold on weekly basis. The weekly production cannot exceed 25 refrigerators in department I 35 ranges in department II because of the limited available facilities in these two departments. The company regularly employs a total 60 workers in the two departments. A refrigerator requires man-week of labour, while a range requires 1 week man-week of labour. A refrigerator contributes a profit of Rs 60 and a range contribute profit of Rs 40. Formulate a) Design vector b) constraints c) objective function.
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Use the Kuhn- Tucker conditions to solve the following non-linear programming problem:
St x1 + 2x2 ≤ 10; x1 - 3x2 ≤ 9; xi ≥ 0 ∀i
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Use the Charnes penality method to solve the problem Maximize Z = 4x1 + 5x2 + 2x3 subject to 2x1 + x2 + x3 ≤ 10; x1 + 3x2 + x3 ≤ 12; x1 + x2 + x3 = 6.
Answer
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Find the initial feasible solution Vogel’s method and obtain optimal solution for the following Transportation Problem.
| Factory/Warehouse | 1 | 2 | 3 | 4 | 5 | Requirement |
|---|---|---|---|---|---|---|
| A | 4 | 7 | 3 | 8 | 2 | 4 |
| B | 1 | 4 | 7 | 3 | 8 | 7 |
| C | 7 | 2 | 4 | 7 | 7 | 9 |
| D | 4 | 8 | 2 | 4 | 7 | 2 |
| Supply | 8 | 3 | 7 | 2 | 2 | - |
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Min Y = 100(x2 - x12)2 + (1 - x1)2. The starting point is (0,0) along direction S=[4, 0] by using Quadratic interpolation method. Take step size as 0.1.
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Min f(x1, x2) = x12 - 3x1x2 + 3x22 by using univariant method. Take starting point as (1, 2)T.
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Explain the procedure for finding the required feasible starting points by using the interior penalty function method.
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Explain the steps involved in the exterior penalty function method in detail.
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A company has to transport some goods from city A to city J. The cost of transportation between the different cities is given in the following network. Find the optimal route connecting cities A and J.
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Explain the steps involved in the Tabular method of solution in detail.
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