[1.41] Consider the problem: Minimize cx subject to Ax ≥b, x ≥ 0. Suppose thata new constraint is added to the problem.a. What happens to the feasible region?b. What happens to the optimal objective value z*?11 421 Congidor the problem, MiniOT whigot to0 Suanggo thot

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Answered: [1.41] Consider the problem: Minimize…

Advanced Engineering Mathematics

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Author:Erwin Kreyszig

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Chapter2: Second-order Linear Odes

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b) Using Lagrangian Multiplier Method, maximize the following utility function:U=2x+3y+ 3xySubject theto following constraint:2x+3y = 8and find out optimal values of goods 'x' and 'y', where utility of the isconsumer maximum?
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[1.41] Consider the problem: Minimize cx subject to Ax ≥b, x ≥ 0. Suppose that a new constraint is added to the problem. a. What happens to the feasible region? b. What happens to the optimal objective value z*? [1.42] Consider the problem: Minimize cx subject to Ax≥b, x ≥0. Suppose that a new variable is added to the problem. a. What happens to the feasible region? b. What happens to the optimal objective value z
28. If the objective function is to Max Z=2x+3y, what will be the optimal solution as shown below? (0.5) (0.4) 1 (6/11, 40/11) Feasible Region (2.0)
[1.43] Consider the problem: Minimize cx subject to Ax≥b, x ≥ 0. Suppose that a constraint, say constraint i, is deleted from the problem. a. What happens to the feasible region? b. What happens to the optimal objective value z*? [1.44] Consider the problem: Minimize cx subject to Ax ≥b, x ≥ 0. Suppose that a variable, say, xk, is deleted from the problem. a. What happens to the feasible region? b. What happens to the optimal objective value z*?
If the objective function is to Max Z=2x+3y, what will be the optimal solution as shown below? (o,5) (0, 4) (6/11, 40/11) Feasible Region (2.0) O D. (2,2) B. (4,0) O C. 6,0) O A. (3,1.5)
4.1-2. Consider the following problem. Z = 3x₁ + 2x₂, Maximize subject to 2x₁ + x₂ ≤ 6 x₁ + 2x₂ ≤ 6
Consider the nonlinear optimization model stated below. 2x² 18x + 2XY+Y² X + 4Y ≤ 8 (a) Find the minimum solution to this problem. at (x, y) = Min s.t. 16Y + 51 ) (b) If the right-hand side of the constraint is increased from 8 to 9, how much do you expect the objective function to change? Based on the dual value on the constraint X + 4Y ≤ 8, we expect the optimal objective function value to decrease by (c) Resolve the problem with a new right-hand side of the constraint of 9. How does the actual change compare with your estimate? , so the actual change is a decrease of If we resolve the problem with a new right-hand-side of 9 the new optimal objective function value is rather than what we expected in part (b).
Consider the following LP: Maximize z = 5x1 + 2x2 + 3x3; subject to x1 + 5x2 + 2x3 = 15; x15x2- 6x3 = 0; The starting solution consists of artificial x4 for the first constraint and slack x5 for the second constraint. Using M = 100 for the artificial variables, the optimal tableau is given below. Use the optimal inverse matrix for the LP to find the optimal solution for the dual problem. Basic Z x1 x5 xl 0 1 0 x2 23 5 -10 (y1,y2)=(3,2) (y1,y2)=(-1,3) (y1,y2)=(2,3) (y1,y2)=(5,0) (y1,y2)=(0,5) x3 7 2 -8 x4 105 1 -1 x5 0 0 1 Solution 75 15 5
7 The following questions refer to the Giapetto problem (Section 3.1). Giapetto's LP was max z = 3x, + 2x2 2x, + x2 < 100 X, + x2 < 80 < 40 (Finishing constraint) (Carpentry constraint) (Limited demand for soldiers) st. (x, = soldiers and x2 = trains). After adding slack variables S1, 82, and s3, the optimal tableau is as shown in Table 12. Use this optimal tableau to answer the following questions: a Show that as long as soldiers (x1) contribute between $2 and $4 to profit, the current basis remains optimal. If soldiers contribute $3.50 to profit, find the new optimal solution to the Giapetto problem. b Show that as long as trains (x2) contribute between $1.50 and $3.00 to profit, the current basis remains optimal. c Show that if between 80 and 120 finishing hours are available, the current basis remains optimal. Find the new optimal solution to the Giapetto problem if 90 fin- ishing hours are available. d Show that as long as the demand for soldiers is at least 20, the current basis…
What is the optimal solution to the following NLP (Use K-T conditions and the multiplier X for the constraint)? Max z = X₁ X2 s. t. 4x2 + 4x2 < 1 Select one: O a. O b. O ö O d. √2 √2 X1 = 1-4x₂ - - ₁2 - 1/2 x₁ - ²₁x₂-√²,2-₂ = X2 = = - 1 = 2 × 2 = − 1, λ = 11 X1 - x₁ = 1 / ² × ₂ = -1/² ₁2 = 1²/01 λ X1 2
2. For each of the following questions, justify your answer using complementary slack- ness and if the given solution is optimal, give an optimal solution to the dual as well. (b) Determine whether x¹ = (0,,) is an optimal solution to the following linear program. maximize subject to ₁ + x₂ + x3 2x1 + 7x₂ + x3 ≤ 8, x₁ + 3x2 + 3x3 ≤7, 2x1 + 4x3 ≤ 6, x₁ + 3x2 + x3 ≤9, I1, I2, I3 20

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[1.41] Consider the problem: Minimize cx subject to Ax ≥b, x ≥ 0. Suppose that a new constraint is added to the problem. a. What happens to the feasible region? b. What happens to the optimal objective value z*?