JNTU
JNTU B.Tech III OPTIMIZATION TECHNIQUES Previous Question Paper (July - 2023 - 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
Write the Kuhn -Tucker conditions for a maximization NLPP.
Answer
For a maximization problem: Maximize f(x) subject to gi(x) ≤ 0 (i=1, ..., m) and x ≥ 0, the Kuhn-Tucker conditions are:
Classify optimization problems based on (i) the number objective function(s), (ii) constraints and (iii) linearity of the objective functions and constraints.
Answer
(i) Single-objective (one function) vs Multi-objective (multiple functions) optimization.
(ii) Constrained (subject to equalities/inequalities) vs Unconstrained (no restrictions on variables).
(iii) Linear Programming (all functions are linear) vs Non-linear Programming (at least one function is non-linear).
A manufacturer has two types of machine to choose from. He must have at least three type-A of machine and one type-B of machine. The cost of the machine is 1000 for the type-A and 1200 for the type-B. The floor area taken up by the two types of machines are 4 m2 and 5 m2 respectively. The total cost must not exceed 15000 and the total available floor space is 40 m2. Suppose now that the manufacturer estimates the weekly profit from the output as 120 for each type-A machine and 100 from each type-B machine. Then, formulate a linear programming model for maximizing the profit?
Answer
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The system AX = b, X ≥ 0 is given by:
Then, justify your answers:
i) Is X = (x1, x2, x3, x4) = (4, 8, 2, 2) a feasible solution? Is it a basic feasible solution also? If not, why?
ii) Is X = (x1, x2, x3, x4) = (-10, 0, 0, 4) is basic solution? Is it feasible also? If not, why?
Answer
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Define measure of effectiveness. Why is it used for a search method?
Answer
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Write down the iteractive steps of the steepest descent method.
Answer
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At any sequence, the following penalty function is minimized: P(x, R) = f(x) + Ω(R, g(x), h(x)). Then, write down the expressions for (i) Parabolic penalty, (ii) Infinite barrier penalty, (iii) Log penalty, (iv) Inverse penalty.
Answer
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Consider the following problem: Minimize x12 + x22 subject to x1 + x2 - 1 = 0. Apply penalty function method to justify the optimal solution lies at the point (1/2, 1/2) and has objective value 1/2.
Answer
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Define Stage, State, Return function of a dynamic programming problem.
Answer
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Determine the value of u1, u2 and u3, so as to Maximize Z = u1 · u2 · u3 Subject to u1 + u2 + u3 = 10; u1, u2, u3 ≥ 0.
Answer
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Part B
Apply Lagrange’s multipliers method to solve the non-linear programming problem given below:
Optimize Z = x12 + x22 + x32
Subject to x1 + x2 + 3x3 = 2, 5x1 + 2x2 + x3 = 5, x1, x2, x3 ≥ 0.
Answer
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Apply bordered Hessian matrix concept and Kuhn – Tucker conditions to solve the following NLPP:
Optimize Z = 2x1 + 3x2 - (x12 + x22 + x32)
Subject to x1 + x2 ≤ 1; 2x1 + 3x2 ≤ 6; x1, x2 ≥ 0.
Answer
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The following table shows a simplex tableau for a maximization problem:
| C | RHS | B | x1 | x2 | x3 | x4 | x5 |
|---|---|---|---|---|---|---|---|
| 8 | 1 | x1 | 1 | 0 | 1 | β | -1 |
| 10 | 4 | x2 | 0 | 1 | 3/2 | -1/2 | 3/2 |
| Cj-Zj | γ |
Find the range values of α, β and γ for the following situations:
a) The current solution is optimal but an alternate optimum exists.
b) The current solution will yield a degenerate BFS in the next iteration.
c) The current solution is feasible but the problem has no finite optimum.
d) The current solution is feasible but the objective can be improved if x3 replaces x2 in the basis. What will be the change in the value of the objective function?
Answer
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Find the optimal Transportation cost and optimal allocations for the following problem. Is the optimal solution degenerate? Also, find an alternate solution if exits.
| D1 | D2 | D3 | Supply | |
|---|---|---|---|---|
| S1 | 16 | 18 | 22 | 95 |
| S2 | 21 | 23 | 19 | 90 |
| S3 | 26 | 29 | 18 | 85 |
| S4 | 25 | 25 | 35 | 80 |
| Demand | 100 | 200 | 50 |
Answer
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Apply Quadratic interpolation method to find the minimum of f = λ5 - 5λ3 - 20λ + 5. (Perform 2 iterations with convergence test).
Answer
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Use Powell’s method to find the optimal solution for the given problem:
Minimize f(x1, x2) = x1 - x2 + 2x12 + 2x1x2 + x22. Start with X1 = {0, 0}T. Take the probe length as ε = 0.01.
Answer
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Solve the following NLPP using the logarithmic barrier function method:
Minimize f(X) = x12 + 2x22
Subject to g(X) = 1 - x1 - x2 ≤ 0; X ∈ ℝ2.
Find the exact solution as well as perform At least 5 iterations in the iterative process.
Answer
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Solve the following NLPP using the Exterior penalty function method:
Minimize f(X) = x12 + 2x22
Subject to g(X) = 1 - x1 - x2 ≤ 0; X ∈ ℝ2.
Find the exact solution as well as perform At least 5 iterations in the iterative process.
Answer
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Use dynamic programming to discuss the general case of dividing Q into n parts u1, u2, ..., un so as to maximize ∏ ui, for ui ≥ 0.
Answer
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A man is engaged in buying and selling identical items. He operates from a warehouse that can hold 500 items. Each month he can sell any quantity that he chooses to stock at the beginning of the month. Each month, he can buy as much as he wishes for delivery at the end of the month, so long as his stock does not exceed 500 items. For the next four months he has the following error-free forecasts of cost and sale prices:
| Months | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Cost | 27 | 26 | 24 | 28 |
| Sale price | 28 | 25 | 25 | 27 |
If he currently has a stock of 200 units, what quantities should he sell and buy in the next four months? Find the solution using dynamic programming.
Answer
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