[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

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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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…
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
Which of the following gives an optimal value of the objective function for the LP model in this problem? a.) (0,0) b.) (0,10) c.) (4,6) d.) None of the above
CAMMIMS16 8. E.002. Consider the nonlinear optimization model stated below. Min 2x² - 18x + 2xY+Y²-16Y+53s.t. x+4Y ≤ 6 (a) Find the minimum solution to this problem. at (x, y) = (, ) (b) If the right-hand side of the constraint is increased from 6 to 7, how much do you expect the objective function to change? Based on the dual value on the constraint.x + 4Y ≤ 6, we expect the optimal objective function value to decrease by (c) Resolve the problem with a new right-hand side of the constraint of 7. How does the actual change compare with your estimate? If we resolve the problem with a new right-hand - side of 7 the new optimal objective function value is part (b). DETAILS CAMMIMS16 8.E.002. Consider the nonlinear optimization model stated below Min 2x2 18x + 2XY+Y²-16Y+ 53 s.t. X + 4y s 6 (a) Find the minimum solution to this problem. -([ at (x, y) = J G (b) If the right-hand side of the constraint is increased from 6 to 7, how much do you expect the objective function to change? Based…
Find the optimal value(s) of the objective function on the feasible set S. Z = 5x + 7y minimum maximum y (0, 6) (3, 4) S + + (0, 0) (5, 0)
Solve this LP problem: 1. Maximize z = 20x₁ + 10x₂ + x3 Subject to: 3x₁ - 3x₂ + 5x3 ≤ 50 x₁ + x₂ ≤ 10 x₁-x₂ + 4x₂ ≤ 20 X₁, X₂, X₂ ≥ 0 Requirements: a) By inspecting the constraints, determine the direction (x₁, x₂, or x²) in which the solution space is unbounded. Support your answer. b) Verify using simplex calculation.
Consider the following LP: Minimize z - 4x1 + x2 subject to 3x1 + x2 = 30 4x, + 3x2 2 60 X1 + 2x2 = 40 X1, X2 2 0 The starting solution consists of artificial x4 and xs for the first and second constraints and slack x, for the third constraint. Using M = 100 for the artificial variables, the optimal tableau is given as Basic X2 X3 X6 Solution -98.6 -100 -2 34 1 4 -2 4 X2 1 2 .6 18 X3 1 -1 1 10 Write the associated dual problem, and determine its optimal solution in two ways.
6.2.2

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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*?