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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| Max | 1x1 | + | 1x2 | |
| s.t. | ||||
| 5x1 | + | 7x2 ≤ | 42 | |
| 1x1 | + | 5x2 ≤ | 20 | |
| 2x1 | + | 1x2 ≤ | 15 | |
| x1, x2 ≥ 0 and integer |
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- Consider the following statements about linear programming and the simplex method. Label each statement as true or false, and then justify your answer.(a) In a particular iteration of the simplex method, if there is a tie for which variable should be the leaving basic variable, then the next BF solution must have at least one basic variable equal to zero.(b) If there is no leaving basic variable at some iteration, then the problem has no feasible solutions.(c) If at least one of the basic variables has a coefficient of zero in row 0 of the final tableau, then the problem has multiple optimal solutions.(d) If the problem has multiple optimal solutions, then the problem must have a bounded feasible region.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 10Consider 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? YesNoThe linear program Max 3X1 + 2X2 is solved subject to the constraints i) X1 + X2 ≤ 10 ii) 3X1 + X2 ≤ 24 iii) X1 + 2X2 ≤ 16 and iv) non-negativity for both X1 and X2. Which of the following statements is true? A. The optimal solution occurs at the point (6, 6). B. The feasible region has five corner points. C. The optimal solution occurs at (8, 0) and the optimal value is 24. D. The optimal solution value is 41.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) = ( )
- 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…Please answer True or False: An LP can have multiple disjointed feasible regions as long as it has at least two constraints. Every LP has an optimal solution. Any point located in the interior of the feasible region (not on the boundary) can be written as a convex combination of two extreme points. Isocost and isoprofit lines are graphed while solving maximization andminimization problems, respectively. Simplex algorithm finds and evaluates the objective function value of all extreme points. A standard form LP with n variables and m constraints yields a n by m technology matrix (denoted by A). For a standard form LP with n variables and m constraints, the direction ofunboundedness is a vector with n-m elements. Row 0 coefficients, while applying the simplex algorithm to a maximizationproblem, are known as reduced price.Determine whether the problem has multiple solutions, unbounded solutions, or no feasible solutions.Maximize z = 10x1 + 11x2 + 7x3, subject to 2x1 + 3x2 − 9x3 ≤ 72 2x1 + 5x2 − 10x3 ≥ 100 x1 ≥ 0, x2 ≥ 0, x3 ≥ 0 The problem has multiple solutions. The problem has unbounded solutions. T he problem has no feasible solutions.
- Consider the following problem: Minimize Z = 5X1 + 8X2 + 3X3 + 5X4 + 12X5 Subject to X1 + 3X2 + 5X3 + 6X4 + 3X5 > 60 and all variables are > 0, except X1 which is of unrestricted sign Use the Primal-Dual relations to solve both problems (i.e., the Primal and the Dual).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