Consider the following linear program.15-4A + 3B ≤ 31A 1B ≤ 5A, B 2 0(a) Graph the feasible region for the problem.B10Maxs.t.51A - 2B268 10 12 14A(c) Find the optimal solution.(A, B) =B151026810 12 14AB(d) Does an unbounded feasible region imply that the optimal solution to the linear program will be unbounded?O YesO No1510O The feasible region is unbounded because more than one optimal solution occurs without violating any of the constraints.O The feasible region is not unbounded.(b) Is the feasible region unbounded? Explain.O The feasible region is unbounded because the value of the solution may be made infinitely large without violating any of the constraints.O The feasible region is unbounded because the solution coordinates may be made infinitely large without violating any of the constraints.O The feasible region is unbounded because the value of the solution violates at least one of the constraints.246810 1214ADOB151052681012 14A

Given LP: Max Z = A - 2BSubject to-Constraint (1):-4A + 3B≤3Constraint (2): A - B≤5Non-negativity…

Answered: Consider the following linear program.…

Practical Management Science

6th Edition

ISBN:9781337406659

Author:WINSTON, Wayne L.

Publisher:WINSTON, Wayne L.

Chapter5: Network Models

Section: Chapter Questions

Problem 62P

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Consider the following LP model in standard form, with a row for the objective function Z. a) Put it into Canonical form ( or Simplex Tableau form) with basic variables X1, X2 , and X3. b) Determine the association BFS (Basic Feasible Solution) and the new formula for the objective function Z Minimize 10X1 + 4X2 Sujbject to 3X1 + 2X2 - X3 = 60 7X1 + 2X2 - X4 = 84 3X1 + 6X2 -X5 = 72 X1, X2, X3 , X4 , X5 >= 0
Consider the following primal LP problem:Maximize X1 + 2X2 – 9X3 + 8X4 – 36X5Subject to 2X2 – X3 + X4 – 3X5 ≤ 40 X1 – X2 + 2X4 – 2X5 ≤ 10 X1 ≥ 0, X2 ≤ 0, X3 ≥ 0, X4 ≥ 0, X5 ≥ 0 (a) Write the dual of the LP above, using variables Y1, Y2, etc.(b) Sketch the feasible region of the dual LP in 2 dimensions, and use the graphical method to find the dual optimalsolution (Plot an isovalue line corresponding to the feasible solution, move the line in improving direction, findthe last one touching the feasible region, and any point(s) on the intersection of the last isovalue line andfeasible region are optimal solutions)(c) Using complementary slackness conditions, - write equations which must be satisfied by the optimal primal solution X* - which primal variables must be zero?(d) Using the information in (c), determine the optimal primal solution X* (e) Compare the optimal objective values of the primal and dual solutions
Consider the following set of constraints: 48Y >= 7296; 0.25 X + 12Y >= 1824, and X + Y <= 152. Pick a suitable statement for this problem: a. Solution to this problem cannot be found without the objective function. b. The feasible region is defined by a single (unique) point. c. It is a non-linear problem - unsuitable for grphical method. d. This problem has two feasible points - one is optimal for miniization problem and other is optimal for maximization problem. e. Feasible region is represented by a line and multiple feasible points are available.
Consider the following set of constraints: -4X <= -512; -28Y <= -3584; 0.5 X + 14Y >= 1792, and 2X + 2Y <= 256. Pick a right statement for this problem: a. Feasible region is represented by a line and multiple feasible points are available. b. The feasible region is defined by a single (unique) point. c. Feasible region does not exist. d. All the options are incorrect. e. Solution to this problem cannot be found without the objective function
The optimal solution of this linear programming problem is at the intersection of constraints 1 and 2. Max 6x1 + 3x2 s.t. 4x1 + x2 ≤ 400 4x1 + 3x2 ≤ 600 x1 + 2x2 ≤ 300 x1, x2 ≥ 0 (a) Over what range can the coefficient of x1 vary before the current solution is no longer optimal? (Round your answers to two decimal places.) ------ to -------- (b) Over what range can the coefficient of x2 vary before the current solution is no longer optimal? (Round your answers to two decimal places.) ----- to -------- (c) Compute the dual value for the first constraint, second constraint & third constraint
Consider the following all-integer linear program: Max 5x1 + 8x2 s.t. 6x1 + 5x2 ≤ 28 11x1 + 5x2 ≤ 46 x1 + 2x2 ≤ 8 x1, x2 ≥ 0 and integer Find the optimal solution to the LP Relaxation. If required, round your answers to two decimal places. x1= fill in the blank 2 x2= fill in the blank 3 Optimal Solution to the LP Relaxation fill in the blank 4 Round down to find a feasible integer solution. If your answer is zero enter “0”. x1= fill in the blank 5 x2= fill in the blank 6 Feasible integer solution fill in the blank 7 Find the optimal integer solution. If your answer is zero enter “0”. x1= fill in the blank 8 x2= fill in the blank 9 Optimal Integer Solution fill in the blank 10
Consider the following all-integer linear program: Max 5x1 + 8x2 s.t. 6x1 + 5x2 ≤ 28 11x1 + 5x2 ≤ 46 x1 + 2x2 ≤ 8 x1, x2 ≥ 0 and integer Find the optimal solution to the LP Relaxation. If required, round your answers to two decimal places. x1= fill in the blank 2 x2= fill in the blank 3 Optimal Solution to the LP Relaxation fill in the blank 4 Round down to find a feasible integer solution. If your answer is zero enter “0”. x1= fill in the blank 5 x2= fill in the blank 6 Feasible integer solution fill in the blank 7 Find the optimal integer solution. If your answer is zero enter “0”. x1= fill in the blank 8 x2= fill in the blank 9 Optimal Integer Solution fill in the blank 10 Is it the same as the solution obtained in part (b) by rounding down?
Consider the following all-integer linear program. Max 5x1 + 8x2 s.t. 6x1 + 5x2 ≤ 25 10x1 + 4x2 ≤ 40 1x1 + 2x2 ≤ 8 x1, x2 ≥ 0 and integer (a) Graph the constraints for this problem. Use points to indicate all feasible integer solutions. (b) Find the optimal solution to the LP Relaxation. (Round your answers to three decimal places.) at (x1, x2) = Using this solution, round down to find a feasible integer solution. at (x1, x2) = (c) Find the optimal integer solution. at (x1, x2) = Is it the same as the solution obtained in part (b) by rounding down? YesNo
Given four decision variables A, B, C, and D, which of the following could be a linear programming problem constraint? 1A + 2B - 1C/D >=23 1A + 2A*B + 3A*B/C + 4A*B*C*D >=100 1A + 2B + 3C + 4D >=100 1A + 2B/C + 3D <=45 1A + 2B*C + 3D <=100
Consider the given LP:Maximize z = 4x1 + 6x2 + 8x3Subject to   3x1 + 2x2 + 5x3<= 30 9x1 + 2x2 + 7x3 <= 126 2x1 + 3x2 + x3<= 60 xi>= 01-Find the optimal values of Z, x1, x2, x3 by using the simplex method.3-If the RHS of the constraint 1 is 34 instead of 30, what is the new value of Z?
Consider the following all-integer linear program. Max     1x1 + 1x2 s.t. 5x1 + 7x2 ≤ 42 1x1 + 6x2 ≤ 18 2x1 + 1x2 ≤ 15 x1, x2 ≥ 0 and integer (a) Graph the constraints for this problem. Use dots to indicate all feasible integer solutions. On the coordinate plane the horizontal axis is labeled x₁ and the vertical axis is labeled x₂. A region bounded by a series of connected line segments and several points are on the graph. The line segments connect the approximate points (0, 7.5), (1, 7), (2.09, 5.48), and (3, 0). The region is above the horizontal axis, to the right of the vertical axis, and below the line segments. All ordered pairs with integer values in the region, on the series of connected line segments, but not on the horizontal nor vertical axes, are shown. On the coordinate plane the horizontal axis is labeled x₁ and the vertical axis is labeled x₂. A region bounded by a series of connected line segments and several points are…
Consider the following all-integer linear program. Max     1x1 + 1x2 s.t. 5x1 + 7x2 ≤ 42 1x1 + 5x2 ≤ 20 2x1 + 1x2 ≤ 15 x1, x2 ≥ 0 and integer (b)Solve the LP Relaxation of this problem. ( ) at (x1, x2) = ( ) (c) Find the optimal integer solution. ( ) at (x1, x2) = ( )

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