Consider the following all-integer linear program. Max 5, + By s.t. 6x₂ + 5x₂ ≤ 35 11x₂ + 4x₂ s 44 1x₂ + 2x₂ ≤ 12 x₁x₂20 and integer X₂ (a) Graph the constraints for this problem. Use points to indicate all feasible integer solutions. FELL 8 6 (c) Find the optimal integer solution. ] at (x₁, x₂) = X₂ 6 (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) = ( 2 Is it the same as the solution obtained in part (b) by rounding down? O Ves O No 8 2 4 8 2 8 X₂ @
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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 >= 0Consider 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 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
- 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 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).The 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.
- Please do not give solution in image formate thanku. Consider the following problem.Maximize Z = 5x1 + 3x2 + 4x3,Subject to2x1 + x2 + x3 ≤ 203x1 + x2 + 2x3 ≤ 30 andx1 ≥ 0, x2 ≥ 0, x3 ≥ 0You are given the information that the non-zero variables in the optimal solution are x2 and x3.Describe how you can use this information to adapt the simplex method to solve this problem in the minimum possible number of iterations (when you start from the usual initial BF solution). Do not actually perform any iterations.x1 + x2 ≤30300 ≤5x1 + 6x2x1 ≥0, x2 ≥0. From the given constraint above, provide the followinga. label the curvesb. give the point of intersection of the constraintsc. indetify the feasible regionsConsider 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?The standard form of the following linear programming model is given. Find the values of variables at the point of intersection of constraint 1 and the vertical axis (y). (Round your answers to 3 decimal places.) Maximize P = 30x + 15y + 0s1 + 0s2 subject to 6x + 12y + s1 = 19 13x + 12y + s2 = 35 and x, y, s1, s2≥ 0.We have 60 meters of fence and want to fence a triangular shaped area. Please formulate an NLP (do not try to solve) that will enable us to maximize the fenced area (Hint: The area of a triangle with sides of length a, b, and c is ( s (s – a) (s – b) (s – c))1/2, where s is half the parameter of the triangle).