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Question 3-Integer Programming (10 points). Let PB be the following binary program. P: min 2x1 x2...
Problem 1 (20 pts) Consider the mathematical program max 3x1+x2 +3x3 s.t. 2x1 +x2 + x3...
Problem 1 (20 pts) Consider the mathematical program max 3x1+x2 +3x3 s.t. 2x1 +x2 + x3 +x2 x1 + 2x2 + 3x3 +2xs 5 2x 2x2 +x3 +3x6-6 Xy X2, X3, X4, Xs, X620 Three feasible solutions ((a) through (c)) are listed below. (0.3, 0.1, 0.4, 0.9, 1.65, 1.6) (c) x Please choose one appropriate interior point from the list, and use the Karmarkar's Method at the interior point and determine the optimal solution. 25
Problem 1 (20 pts) Consider the mathematical program max 3x1+x2 +3x3 s.t. 2x1 +x2 + x3 +x2 x1 + 2x2 + 3x3 +2xs 5 2x 2x2 +x3 +3x6-6 Xy X2, X3, X4, Xs, X620 Three feasible solutions ((a) through (c...
Problem 1 (20 pts) Consider the mathematical program max 3x1+x2 +3x3 s.t. 2x1 +x2 + x3 +x2 x1 + 2x2 + 3x3 +2xs 5 2x 2x2 +x3 +3x6-6 Xy X2, X3, X4, Xs, X620 Three feasible solutions ((a) through (c)) are listed below. (0.3, 0.1, 0.4, 0.9, 1.65, 1.6) (c) x Please choose one appropriate interior point from the list, and use the Karmarkar's Method at the interior point and determine the optimal solution. 25
Problem 1 (20 pts) Consider...
6, Maximize z = 2x1 + x2 + 3x3 subject to x 3x2 5x3 s 10 2x x 20, x, 0, x320. (a) State the dua...
6, Maximize z = 2x1 + x2 + 3x3 subject to x 3x2 5x3 s 10 2x x 20, x, 0, x320. (a) State the dual problem. (b) Solve both the primal and the dual problem with any method that works. (c) Check that your optimal solutions are correct by verifying they are feasible and the primal and dual objective functions give the same value.
6, Maximize z = 2x1 + x2 + 3x3 subject to x 3x2 5x3 s...
A firm has prepared the following binary integer program to evaluate a number of potential locations...
A firm has prepared the following binary integer program to evaluate a number of potential locations for new warehouses. The firm's goal is to maximize the net present value of their decision while not spending more than their currently available capital. Max 20x1 + 30x2 + 10x3 + 15x4 s.t. 5x1 + 7x2 + 12x3 + 11x4 ≤ 21 {Constraint 1} x1 + x2 + x3 + x4 ≥ 2 {Constraint 2} x1 + x2 ≤ 1 {Constraint 3} x1...
QUESTION 13 A company has decided to use 0-1 (binary) integer programming to help to make...
QUESTION 13 A company has decided to use 0-1 (binary) integer programming to help to make some investment decisions. There are three possible investment alternatives from which to choose, but if it is decided that a particular alternative is to be selected, the entire cost of that alternative will be incurred (i.e., it is impossible to build one-half of a factory). The integer programming model is as follows: Max 5000X1+7000X2+9000X3 S.t. X1+X2+X3<=2 (only 2 may be chosen) 25000X1+32000X2+29000X3<=62000 (budget limit) 16X1+14X2+19X3<=36 (resource...
QUESTION 13 A company has decided to use 0-1 (binary) integer programming to help to make...
QUESTION 13 A company has decided to use 0-1 (binary) integer programming to help to make some investment decisions. There are three possible investment alternatives from which to choose, but if it is decided that a particular alternative is to be selected, the entire cost of that alternative will be incurred (i.e., it is impossible to build one-half of a factory). The integer programming model is as follows: Max 5000X1+7000X2+9000X3 S.t. X1+X2+X3<=2 (only 2 may be chosen) 25000X1+32000X2+29000X3<=62000 (budget limit)...
Consider the following linear program: Maximize Z-3xI+2x2-X3 Subject to:X1+X2+2 X3s 10 2x1-X2+X3 s20 3 X1+X2s15 X1, X2,...
Consider the following linear program: Maximize Z-3xI+2x2-X3 Subject to:X1+X2+2 X3s 10 2x1-X2+X3 s20 3 X1+X2s15 X1, X2, X320 (a) Convert the above constraints to equalities. (2 marks) (b) Set up the initial simplex tableau and solve. (9 marks)
Consider the following linear program: Maximize Z-3xI+2x2-X3 Subject to:X1+X2+2 X3s 10 2x1-X2+X3 s20 3 X1+X2s15 X1, X2, X320 (a) Convert the above constraints to equalities. (2 marks) (b) Set up the initial simplex tableau and solve. (9 marks)
Question 1. (30 points) You are provided with the following integer program: max := 3x +...
Question 1. (30 points) You are provided with the following integer program: max := 3x + y 8.t. x + 1.6y S8 - 5 + 6x = 15 35.5 x,y20 and integer (a) On the graph provided on the following page, use the graphical solution method to identify the feasible points on your graph. (Use the scale 1 by 1 for each small square so that you can visually detect the feasible integer solutions.) (b) Enumerate the feasible extreme points...
(45 Points) Consider the constrained optimization problem: min f(x1, x2) = 2x} + 9x2 + 9x2...
(45 Points) Consider the constrained optimization problem: min f(x1, x2) = 2x} + 9x2 + 9x2 - 6x1x2 – 18x1 X1 X2 Subject to 4x1 – 3x2 s 20 X1 + 2x2 < 10 -X1 < 0, - x2 < 0 a) Is this problem convex? Justify your answer. (5 Points) b) Form the Lagrange function. (5 Points) c) Formulate KKT conditions. (10 Points) d) Recall that one technique for finding roots of KKT condition is to check all permutations...
Question 1 - Revised Simplex Algorithm 10 marks Suppose we are solving the following linear programming problem Subject...
Question 1 - Revised Simplex Algorithm 10 marks Suppose we are solving the following linear programming problem Subject to 8x1 + 12x2 + x3 15x2 + x4 3x1 + 6x2 + X5 -120 60 = 48 x1,x2,x3, x4,x5 2 0 Assume we have a current basis of x2,xz, x5. Demonstrate your understanding of the steps of the Revised Simplex Algorithm by answering the following: a) What is the basic feasible solution at this stage? What is the value of the...
Problem 1 (20 pts) Consider the mathematical program max 3x1+x2 +3x3 s.t. 2x1 +x2 + x3 +x2 x1 + 2x2 + 3x3 +2xs 5 2x 2x2 +x3 +3x6-6 Xy X2, X3, X4, Xs, X620 Three feasible solutions ((a) through (c)) are listed below. (0.3, 0.1, 0.4, 0.9, 1.65, 1.6) (c) x Please choose one appropriate interior point from the list, and use the Karmarkar's Method at the interior point and determine the optimal solution. 25
Problem 1 (20 pts) Consider the mathematical program max 3x1+x2 +3x3 s.t. 2x1 +x2 + x3 +x2 x1 + 2x2 + 3x3 +2xs 5 2x 2x2 +x3 +3x6-6 Xy X2, X3, X4, Xs, X620 Three feasible solutions ((a) through (c)) are listed below. (0.3, 0.1, 0.4, 0.9, 1.65, 1.6) (c) x Please choose one appropriate interior point from the list, and use the Karmarkar's Method at the interior point and determine the optimal solution. 25
Problem 1 (20 pts) Consider...
6, Maximize z = 2x1 + x2 + 3x3 subject to x 3x2 5x3 s 10 2x x 20, x, 0, x320. (a) State the dual problem. (b) Solve both the primal and the dual problem with any method that works. (c) Check that your optimal solutions are correct by verifying they are feasible and the primal and dual objective functions give the same value.
6, Maximize z = 2x1 + x2 + 3x3 subject to x 3x2 5x3 s...
QUESTION 13 A company has decided to use 0-1 (binary) integer programming to help to make some investment decisions. There are three possible investment alternatives from which to choose, but if it is decided that a particular alternative is to be selected, the entire cost of that alternative will be incurred (i.e., it is impossible to build one-half of a factory). The integer programming model is as follows: Max 5000X1+7000X2+9000X3 S.t. X1+X2+X3<=2 (only 2 may be chosen) 25000X1+32000X2+29000X3<=62000 (budget limit)...
Consider the following linear program: Maximize Z-3xI+2x2-X3 Subject to:X1+X2+2 X3s 10 2x1-X2+X3 s20 3 X1+X2s15 X1, X2, X320 (a) Convert the above constraints to equalities. (2 marks) (b) Set up the initial simplex tableau and solve. (9 marks)
Consider the following linear program: Maximize Z-3xI+2x2-X3 Subject to:X1+X2+2 X3s 10 2x1-X2+X3 s20 3 X1+X2s15 X1, X2, X320 (a) Convert the above constraints to equalities. (2 marks) (b) Set up the initial simplex tableau and solve. (9 marks)
Question 1. (30 points) You are provided with the following integer program: max := 3x + y 8.t. x + 1.6y S8 - 5 + 6x = 15 35.5 x,y20 and integer (a) On the graph provided on the following page, use the graphical solution method to identify the feasible points on your graph. (Use the scale 1 by 1 for each small square so that you can visually detect the feasible integer solutions.) (b) Enumerate the feasible extreme points...
(45 Points) Consider the constrained optimization problem: min f(x1, x2) = 2x} + 9x2 + 9x2 - 6x1x2 – 18x1 X1 X2 Subject to 4x1 – 3x2 s 20 X1 + 2x2 < 10 -X1 < 0, - x2 < 0 a) Is this problem convex? Justify your answer. (5 Points) b) Form the Lagrange function. (5 Points) c) Formulate KKT conditions. (10 Points) d) Recall that one technique for finding roots of KKT condition is to check all permutations...
Question 1 - Revised Simplex Algorithm 10 marks Suppose we are solving the following linear programming problem Subject to 8x1 + 12x2 + x3 15x2 + x4 3x1 + 6x2 + X5 -120 60 = 48 x1,x2,x3, x4,x5 2 0 Assume we have a current basis of x2,xz, x5. Demonstrate your understanding of the steps of the Revised Simplex Algorithm by answering the following: a) What is the basic feasible solution at this stage? What is the value of the...
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