Consider the following partial Branch and Bound tree for a Maximization Pure Binary Integer Programming Problem where all variables are nonnegative integers. Note that the number in the circle is the optimal value of the Linear Programming Relaxation and the solution attached to the circle is the optimal solution of the Linear Programming Relaxation. 30.3 (0.40, 0.10, 1, 0.65) x1=0 x1=1 (1, 0.33, 0.88, 1) -0.4 27.2 (1, 0.40, 0.30, 0.25) x2=0 x2=1 (0, 0, 1, 0) 23.3 28.5 (1, 1, 0.04, 1) The above Branch and Bound tree seems to be incorrect. Identify ALL mistakes with this tree. For each mistake, use two sentences to explain why.
Added by Manuel G.
Close
Step 1
The optimal solution of the Linear Programming Relaxation for node 1 is incorrect. The value of 30.3 is not possible as all variables are nonnegative integers and the LP relaxation can only give fractional values. Show moreā¦
Show all steps
Your feedback will help us improve your experience
Adi S and 57 other Calculus 3 educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
Problem 12-16: Consider the single knapsack constrained binary optimization problem (with all variables binary 0/1): maximize 40x1 + 5x2 + 50x3 + 8x4 subject to 18x1 + 3x2 + 20x3 + 5x4 <= 25 which has LP relaxation optimum x* = (5/18, 0, 1, 0). 1) Perform a branch & bound using breadth-first search, evaluating left to right where the left child node has a variable constrained below the floor of its current fractional value and the right child node has that variable constrained above the ceiling of its current value. At each of your additional four nodes, you should be able to greedily solve the LP relaxation solution in your head by trying to pack the variables that give the biggest ratio of objective value per constraint cost. 2) Generate a Gomory cut from the initial basis. (This should be fairly easy since there is only one physical constraint. Thus, you need only invert a one-dimensional basis under the bounded simplex which allows relaxed binary variables non-basic at upper bound 1) Be sure to simplify the constraint in terms of the four original variables. 3) Strengthen your Gomory cut and compare it to the cover inequality x1 + x3 <= 1. Which is stronger? 4) Perform a discrete improving search that always advances to the feasible neighbor with the best objective value and uses the single-complement neighborhood permitting only one variable to be switched from 0 to 1 (or vice versa). Start from solution x = (0,0,0,0). This should lead to the optimal solution in a couple of steps!
Sri K.
Consider the following integer programming problem: max z = 7x1 + 9x2 s.t. -x1 + 3x2 ā¤ 6 7x1 + x2 ā¤ 35 x1, x2 ā Z+ And the optimal tableau (with some missing values) for its LP relaxation is: z + ax1 + bx2 + cs1 + ds2 = e x2 + 7/22s1 + 1/22s2 = 7/2 x1 - 1/22s1 + 3/22s2 = 9/2 Where s1 and s2 are two slack variables. Please complete the missing values of the tableau labeled with a, b, c, d, e. Show steps.
Recommended Textbooks
Calculus: Early Transcendentals
Thomas Calculus
Transcript
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD