Consider the following flow network G = (V,E,{c(e)},s, t}. 12 4 9. 3 15 10 15 10 (b) Compute a minimum s-t cut in this network and state its capacity.
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- 1. Develop a system of linear equations for the network by writing an equation for each router (A, B, C, D, and E). Make sure to write your final answer as Ax=b where A is the 5x5 coefficient matrix, x is the 5x1 vector of unknowns, and b is a 5x1 vector of constants. 2. construct the augmented matrix [A b] and then perform row reduction using the rref() function. Write out your reduced matrix and identify the free and basic variables of the system. 3. Use MATLAB to compute the LU decomposition of A, i.e., find A = LU. For this decomposition, find the transformed set of equations Ly = b, where y = Ux. Solve the system of equations Ly = b for the unknown vector y. 4. Compute the inverse of U using the inv() function. 5. Compute the solution to the original system of equations by transforming y into x, i.e., compute x = inv(U)y 6. Check your answer for x1 using Cramer’s Rule. Use MATLAB to compute the required determinants using the det() function.Consider the minimum cost flow problem shown in the figure below (note that some arcs have negative arc costs). Modify the problem so that all arcs have nonnegative arc costs..Pipeline fluid flows are indicated below. Determine the maximum flow from Node 1 to Node 4. From To Pipeline Node Node Capacity 1 2 400 2 1 0 1 4 200 4 1 200 1 3 200 3 1 0 2 4 200 4 2 200 3 4 300 4 3 300 a. none of the answers presented apply. b. 600 c. 300 d. 700 e. 200
- Given the following nodes from A to G as shown in the diagram, determine the shortest path network from node A to all other nodes in the given cost network graph.Defend the following claim: (Maxflow-mincut theorem) A st-flow, f, should be. The three following situations are interchangeable:I. A ST-cut exists with a capacity equal to the flow f's value.A maxflow is in ii.With regard to f, there is no enhancing route.(2) Compute the reliability of a system that consists of two subsystems in series, where the first subsystem consists of components A, B in parallel and the second subsystem consists of components C, D, E in parallel. Assume that all components are independent and have reliabilities r_A, r_B, r_C, r_D and r_E. Obtain the solution in polynomial form.
- Determine whether the following statements regarding network flow holds true or false.(a) A feasible network flow from the source node to the sink node is larger than or equals the flow cross some cut;(b) A feasible network flow from the source node to the sink node is less than or equals the flow cross some cut;(c) Given a feasible flow. If we can’t find an augmented path from the source node to the sink node in the network, then the current flow is optimal;(d) A feasible flow that equals the flow cross some cut is the maximum flow.A contractor owns three excavators of different capacity. He/she wants to dispatch these excavatorsinto three different jobs. The performance of the excavators is measured as the time consumed toperform each job. The data of the time tests are shown as given in the table below. (a) Draw a network flow diagram.(b) Formulate a mathematical model to obtain the minimum time.(c) Solve this model using the Vogel’s approximation model.Consider the dual of a minimum cost network flow problem in which the costs are integer. Prove that the basic feasible solutions are integer.
- In Figure, suppose that the wellheads can be divided into two groups depending on gas pressure: a high-pressure group that includes wells 2, 3, 4, and 6 and a low-pressure group that includes wells 5, 7, 8, and 9 . Because of pressure difference, it is not possible to link the wellheads from the two groups. At the same time, both groups must be connected to the delivery point through wellhead 1. Determine the minimum pipeline network for this situation. Choices: a.) Set length of arcs 3-5, 5-3, 4-5, 5-4, 4-7, 7-4, 5-6, and 6-5 to ∞ : Total length = 53 b.) Set length of arcs 3-5, 5-3, 4-5, 5-4, 4-7, 7-4, 5-6, and 6-5: Total length = 53 c.) Set length of arcs 3-5, 5-3, 4-5, 5-4, 4-7, 7-4, 5-6, and 6-7 to ∞ : Total length = 53 d.) Set length of arcs 3-5, 5-3, 4-5, 5-4, 4-7, 7-4, and 6-5 to ∞ : Total length = 53For traffic signal control in a T junction: · Provide the allocation List · Draw the Grafcet and obtain the grafcet equation · Develop the ladder programming from the grafcet Figure below shows the T junction with three traffic signals. Each traffic light cycles through red (i.e. stop) time depends on the other two signals, red & amber (i.e. get ready to go) for 5 seconds, green (go) for 20 seconds, and amber (stop if safe to do so) for 5 seconds.Obtain a maximum flow and a minimum cut in the following network