Consider the following IP problem:Minimize Z = 3x₁ - 2x₂subject to5x₁ - 7x₂ ≥ 3x1 ≤ 3x₂ ≤3and x₁ ≥ 0, x₂ ≥ 0 are integers.a)Use the MIP branch-and-bound algorithm to solve this model by hand. For eachsubproblem, solve its LP relaxation graphically.

Given Information:Objective function of IP is .Subject to the constraints that,To solve:The IP model…

Answered: Consider the following IP problem:…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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An LP solution always provides a lower bound to an IP minimization problem: O True O False W
Consider a set of 500 equations in 100 variables: Ax\le b, given by a_{i}^{T}x\le b_{i}. Suppose that these are not feasible, i.e., there is no solution that satisfies all of them. Show that in fact, there must be a subset of at most 101 which are not feasible. (Hint: Use duality and complementary slackness, E.g., what is the LP that finds the minimum value of slack s we have to put into Ax−1s≤b to make it feasible? The dual of such an LP could then be useful, and complementary slackness can relate variables and constraints between the two.).
Consider the following feasible region:P = {x:Ax=b, x≥0}. Now suppose that you have two objective functions: a “primary” cT x and a “secondary” dT x (think of the primary objective function as the main goal of the company, say profit, and of the secondary one as some organization-related issue, say a measure of how fairly the tasks are divided among the employees). Describe an algorithm that finds, among all vectors that maximize the primary objective function, the one that achieves the maximum value of the secondary objective function.
Consider a pair of dual LPs. none of the other four. The primal is feasible if and only if the dual is feasible. The primal is unbounded if and only if the dual is unbounded. The primal is infeasible if and only if the dual is infeasible. The primal has an optimal solution if and only if the dual has an optimal solution.
3.9. Is the reduced echelon form of a matrix unique? Justify your conclusion. Namely, suppose that by performing some row operations (not necessarily fol- lowing any algorithm) we end up with a reduced echelon matrix. Do we always end up with the same matrix, or can we get different ones? Note that we are only allowed to perform row operations, the "column operations"' are forbidden.
Question2: Zmin=3x1+4x2+5x3 2x1+x2+2x3<=120 x1+x2+x3=100 x1,x2>=0 and x3 are unconstrained Which of the following is true about the dual cycle of the given linear programming model? 1-)Dual model should have 2 dual variables (v1) and 3 dual constraints. 2-) The linear programming model given is in canonical (appropriate) form. 3-)All constraints of the dual model should be written as <=. 4-) The objective function of the dual model is Qmin. 5-) The sign of the second dual variable of the dual model must be unconstrained.
Consider the following LP: Max z = 5x1 + 3x2 + x3 subject to 2x1 + x2 + x3 S 6 X1 + 2x2 + x3 S 7 X1, X2, X3 2 0 a. Formulate and graphically solve the dual of this LP. b. Use complementary slackness to solve the max problem.
Solve the following system of equations using the Gaussian algorithm with column pivoting and backward insertion: 2x2 - 4x3 + x4 = -8x1 + 2x2 + 3x3 - x4 = 3-3x1 + 2x3 + 2x4 = -32x1 + x2 - x4 = 2 Here, column pivoting means that in the respective elimination stepsu. a row swap may be carried out in the respective elimination steps, so that the elimination elementa(m)mm is the largest element of the elimination column in terms of amount.

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Consider the following IP problem: Minimize Z = 3x1 - 2x2 subject to 5x1 - 7x2 ≥3 X1 ≤ 3 X2 ≤ 3 and x1 ≥0, x2 ≥ 0 are integers. a) Use the MIP branch-and-bound algorithm to solve this model by hand. For each subproblem, solve its LP relaxation graphically.