Find the optimum solution for the following Integer Linear Programming problem using the Cutting Plane Method. Maximum Profit, z = 7x1 + 5x2 Subject to: 3x1 + 4x2 < 25 6x1 + 4x2 < 29 x1, x2 > 0
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Find the optimum solution for the following Integer Linear Programming problem using the Cutting Plane Method. Maximum Profit, z = 7x1 + 5x2 Subject to: 3x1 + 4x2 < 25 6x1 + 4x2 < 29 x1, x2 > 0 The optimal linear programming tableau is shown in Table 2. Table/Jadual 2
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3. | Solve the problem in Question 2 using the Branch and Bound Method. | |
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- (5) Consider the following linear programming model with 4 regular constraints:Maximize 3X + 5Y (a) Draw your graph in the space below:subject to: 4X + 4Y ≤ 48 (constraint #1) 4X + 3Y ≤ 50 (constraint #2) 2X + 1Y ≤ 20 (constraint #3) X ≥ 2 (constraint #4) X, Y ≥ 0 (non-negativity constraints)(a) Which of the constraints is redundant? Constraint #______.Justify by drawing a graph similar to Figure 7.14 on p.263.(b) Is point (9,3) a feasible solution? _____. Explain your answer (by analyzing each of the constraints).Constraint #1: _______________________________________________________________Constraint #2: _______________________________________________________________Constraint #3: _______________________________________________________________Constraint #4: _______________________________________________________________(c) Which of the following points yields the best solution? Underline the best solution: (7,5), (9,2), (6,6).Justify your answer (using the data from the above LP…1. Consider the following linear programming formulation: Min 5x + 2y Subject to (1) 3x + 6y ≥ 18 (2) 5x + 4y ≥ 20 (3) 8x + 2y ≥ 16 (4) 7x + 6y ≤ 42 (5) x, y ≥ 0 a. Solve the problem graphically. Specifically, show each constraint and the feasible region, draw an objective function line and identify an optimal point (the solution). When reporting the optimal solution and the corresponding objective function value, you may estimate the optimal x and y values from the graph. b. What are the optimal values of x and y, using the solver add-in? What is the corresponding value of the objective function? c. How many extreme points does the feasible region have? Enumerate them. Hint: It's from the graph. d. Change the objective function to 15x + 12y.. What is the new optimal solution(s)?2. Optimal solutionA company wants to hire 4 vendors for the sale of 4 products, they could only sell one type of product. The following table (image) indicates what each seller charges for selling each of the products. The company wants to assign each seller a product, find all possible optimal assignments by pinpointing their optimal cost. Indicate which is the optimal solution and the optimal value: a) Seller 1 is assigned product 2, seller 2 is assigned product 4, seller 3 is assigned product 3, and seller 4 is assigned product 1. For an optimal cost of 15 b) Seller 2 is assigned product 1, Seller 4 is assigned product 2, Seller 3 is assigned product 3, and Seller 1 is assigned product 4. For an optimal cost of 15 c) Seller 1 is assigned product 2, seller 2 is assigned product 4, seller 3 is assigned product 3, and seller 4 is assigned product 1. For an optimal cost of 14 d) None of the above
- 1. Given the following linear programming model: Minimize Z = 480x1 + 160x2 subject to x1 + x2 >= 40 x1 + 4x2 >= 60 3x1 + x2 >= 60 x1 >= 0, x2 >= 0 a. Solve the LP model graphically and explain the solution result. b. Develop a spreadsheet model and solve using Excel Solver. What is the optimal solution?Set up the simplex matrix used to solve the linear programming problem. Assume all variables are nonnegative.Maximize f = 5x + 9y subject to 8x + 5y ≤ 200 x + 6y ≤ 250. x y s1 s2 f first constraint second constraint objective function1. Given the following linear programming model: Minimize Z = 480x1 + 160x2 subject to x1 + x2 >= 40 x1 + 4x2 >= 60 3x1 + x2 >= 60 x1 >= 0, x2 >= 0 a. Solve the LP model graphically and explain the solution result. b. Develop a spreadsheet model and solve using Excel Solver. What is the optimal solution? 2. Provident Capital Corp. specializes in investment portfolios designed to meet the specific risk tolerances of its clients. A client contacted Provident with P2,000,000 available to invest. Provident’s investment advisor recommends a portfolio consisting of two investment funds: the Dynamic fund and the Diversified fund. The Dynamic fund has a projected annual return of 10%, and the Diversified fund has a projected annual return of 8%. The investment advisor requires that at most P1,400,000 of the client’s funds should be invested in the Dynamic fund. Provident’s services include a risk rating for each investment alternative. The Dynamic…
- 1. True/FalseA company wants to hire 4 vendors for the sale of 4 products, they could only sell one type of product. The following table (image) indicates what each seller charges for selling each of the products. The company wants to assign each seller a product, find all possible optimal assignments by pinpointing their optimal cost. answer the following:a) It is an allocation model T() F()b) It is not a transportation model T() F()c) The model is not balanced T() F()d) The problem has exactly 24 feasible points T() F()4. Consider the following linear programming problem: Maximize Z=$15x + $5y, subject to (1) 2x + y ≤ 10 and (2) 4x + 3y ≤ 24 and (3) x, y ≥ 0. Will the optimal solution change if the objective function becomes Maximize Z=$15x + $20y (constraints remain the same)? Select one: a. Can't determine given the information. b. Yes, it will change. c. No, it remains the same.Consider the following LP problem: Min 6X+ 27Y Subject to : 2 X + 9Y => 25, and X + Y <= 75. Pick a suitable statement for this problem: a. X=37.5, Y=37.5 is the only optimal solution. b. Optimal Obj. function value is 75 c. X = 0, Y = 0 is the only optimal solution. d. Optimal Obj. function value is 0
- (a) Write the dual of the following linear programming problem: Maximize Ζ = 3x1 + 5x2 + 4x3, Subject to 2x1 + 3x2 ≤ 8, 2x2 + 5x3 ≤10 3x1 + 2x2 + 4x3 ≤15 x1 , x2 , x3 ≥ 0 (b) Find the optimal solution to the given problem in (a) above, using the simplex method. (c) From the final optimal table in (b) above, find the solution to the dual problem. (d) Give the economic interpretation to the solution in (c) above.Consider the following linear programming model with 4 regular constraints:Maximize 3X + 5Y (a) Draw your graph in the space below:subject to: 4X + 4Y ≤ 48 (constraint #1) 4X + 3Y ≤ 50 (constraint #2) 2X + 1Y ≤ 20 (constraint #3) X ≥ 2 (constraint #4) X, Y ≥ 0 (non-negativity constraints)(a) Which of the constraints is redundant? Constraint #______.Justify by drawing a graph similar to Figure 7.14 on p.263.(b) Is point (9,3) a feasible solution? _____. Explain your answer (by analyzing each of the constraints).Constraint #1: _______________________________________________________________Constraint #2: _______________________________________________________________Constraint #3: _______________________________________________________________Constraint #4: ______________________________________________________________Consider the followingg linear programming problem: Max 3A + 3Bst. 2A + 4B ≤ 12 6A + 4B ≤ 24 A, B ≥ 0 The point (4.0,0.0) is: a. unbounded. b. is one of the extreme points. c. the optimal solution. d. infeasible.