Consider the following LP. Мах z%3D2х, +10х, + бх, subject to X, +2x, + x, = 3 = 4 2x, — х, Xq, X2,X, 2 0 a. Write the associated dual problem. b. Given only the information that the optimal basie variables are x, and x; , (without using simplex method) determine the associated optimal primal and dual solutions.
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- Consider the LP model and its corresponding graph below. Maximize z = 5x + 4ySubject to:3x + 2y <= 12x + 2y >= 8x >= 0y >= 0 From the given graph, what are the corner points of the feasible region? What are the values of x and y for the optimal solution? What is the maximum value of z?Over what range can the coefficient of x2 vary before the current solution is no longer optimal? Compute the dual value for the first constraint.Consider a maximization problem with the optimal tableauin the following table.z x1 x2 x3 x4 rhs1 2 1 0 0 100 3 2 1 0 30 4 3 0 1 5The optimal solution is obvious here, please determine the second best BFS to this LP.
- Find the optimal solution for the following problem. Maximize C = 4x + 12y + 3z subject to 15x + 5y + 12z ≤ 75 6x + 2y + 8z ≤ 150 and x ≥ 0, y ≥ 0. What is the optimal value of x? What is the optimal value of y? What is the optimal value of z?Use the graphical method to find the optimal solutions of the following LP Problem. Max. Z = 3x1 + 5x2 subject to 3x1 + 2x2 ≤ 18 x1 ≤ 4 x2 ≤ 6 x1, x2 ≥ 0Consider the following problem. Maximize z = 3x1 – 8x2 Construct the dual problem, and then find its optimal solution by inspection. You have to show your complete solution.
- Solve the following LP using Simplex method Minimise Z = 60x1 + 80x2 Subject to 20x1 + 30x2 ≥ 900 Resource 1 40x1 + 30x2 ≥ 1200 Resource 2 x1, x2 ≥ 0 You are required to find Optimum product mix and minimum Optimum solution so obtained is unique or has multiple optimum solution, justify your answer. Comment on the values you have got for S1 and S2 in the optimum solution, what do they represent. Check whether both the resources are fully utilized or not. If these resources are made available for one more unit what is the effect on the cost?2. Maximise 1170x1 + 1110x2Subject to: 9x1 + 5x2 ≥ 5007x1 + 9x2 ≥ 3005x1 + 3x2 ≤ 15007x1 + 9x2 ≤ 19002x1 + 4x2 ≤ 1000x1, x2 ≥ 0-Find graphically the feasible region and the optimal solution.At Long John Silver’s (LJS), Platter 1 comes with 2 pieces of fish, 3 pieces of chicken, and 5 fried shrimp, while Platter 2 comes with 3 pieces of fish, 1 piece of chicken, and 8 fried shrimp. Platter 1 sells for 8 dollars and platter 2 sells for 10 dollars. If LJS has access to 1000 pieces of fish, 1200 pieces of chicken, and 2600 fried shrimp, what is the optimal number of platters LJS should make in order to maximize its profit? Note that the constraints relate to (please use these in order) fish, chicken, and shrimp.
- At Long John Silver’s (LJS), Platter 1 comes with 2 pieces of fish, 3 pieces of chicken, and 5 fried shrimp, while Platter 2 comes with 3 pieces of fish, 1 piece of chicken, and 8 fried shrimp. Platter 1 sells for 8 dollars and platter 2 sells for 10 dollars. If LJS has access to 1000 pieces of fish, 1200 pieces of chicken, and 2600 fried shrimp, what is the optimal number of platters LJS should make in order to maximize its profit? Note that the constraints relate to (please use these in order) fish, chicken, and shrimp. Note that problem 7 had three constraints (excluding the non-negativity constraints), and setting two pairs of those constraints to equality led to two extreme points on the feasible region. What INFEASIBLE point was at the intersection of the other pair of constraints? In the optimal solution to problem 7, the optimum was found at the intersection of two of the constraints that were not non-negativity constraints. Call these “binding” constraints. One of those…Find r and h that minimize S subject to the constraint V = 54n.Consider the following all-integer linear program: Max 5x1 + 8x2 s.t. 5x1 + 6x2 ≤ 32 10x1 + 5x2 ≤ 46 x1 + 2x2 ≤ 10 x1, x2 ≥ 0 and integer Choose the correct graph which shows the constraints for this problem and uses dots to indicate all feasible integer solutions. (i) (ii) (iii) (iv) Graph (i) 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