Branch and bound node i may be fathomed for the following reason(s) in a minimization problem (select all that apply) O LP; is integer feasible O LP; is infeasible OLP; is feasible, but the relaxation objective is less than the incumbent objective LP; is feasible, but the relaxation objective is greater than the incumbent objective The LP; down-branch is integer feasible The LP; up-branch is infeasible
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A: here we have give the solution using geometric method to solve a linear programming problem.
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- Fill in only five of the following with a short answer 1. It can assign solution type (minimum or maximum) of control problem through 2. The function of integral control is 3. The system is in critial stable if 4. A third order system with an output equation (y = 2x₁ - x₂) its output matrix is_ 5. A system with state matrix A = [¹ 2] its overshoot value is 6. The stability of the open-loop system depandes onIf X~ N(3,9), find (a) Pr(2 0) (c) Pr(|X3|> 6).Please show steps clearly
- 4. Graph Theory and Approximation Algorithms- (a) Find the maximum flow possible from vertex s to vertex t in the given graph. Show all your steps Given an example to show why reverse flow edges should be included to get the optimal flow 12 vi V4 20 16 8 4 9. 14 V3 v2 15 (b) Given below are the distances between A, B, C, D, E, F. You have to start from A, visit all the vertices and return back to the A. Find the optimal path using the approximation algorithm discussed in class. Show all your steps. () | A BC D E F A0 4 4 3 5 5 в| 4 0 2 4 C 4 2 D 3 4 E 5 3 6 F 5 3 4 3 3 6. 4 5 4 5 4 5 5 0 inSuppose you are provided an even number of people going on an adventure, and there are only two-seated rafts. All rafts are identical, weighing 20kg each. Each person is described by two parameters, their strength and weight. The speed of a raft can be calculated as the sum of strengths of both persons sitting in it, divided by the total weight of the loaded. We would like to choose the allocation of people to rafts so as to maximize the speed at which the whole group can travel, assuming that the group travels at the speed of the slowest raft in it. Develop a python code to find the maximum speed of the group which can be achieved by optimally choosing places for each person. Sample Input: 4 50 50 50 60 70 100 100 60 Sample Output: 0.8421052590012552Problem 3: A day at the beach. A group of n people are lying on the beach. The beach is represented by the real line R and the location of the i-th person is some integer r; e Z. Your task is to prevent people from getting sunburned by covering them with umbrellas. Each umbrella corresponds to a closed interval I = [a, a + L] of length L e N, and the i-th person is covered by that umbrella if æ¡ € I. Design a greedy algorithm for covering all people with the minimum number of umbrellas. The input consists of the integers x1,..., En, and L. The output of your algorithm should be the positions of umbrellas. For example, if the input is r1 = 1, x2 = 3, r3 = 5, and L = 2, then an optimum solution is the set of two umbrellas placed at positions 2 and 5, covering intervals [1,3] and [4,6]. 1 2 3 4 5 6 Prove that your algorithm is correct and that it runs in time polynomial in n.