# ps7 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY 15.053...

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M ASSACHUSETTS I NSTITUTE OF T ECHNOLOGY 15.053 – Optimization Methods in Management Science (Spring 2007) Problem Set 7 Due April 12 th , 2007 at 4:30 pm. You will need 157 points out of 185 to receive a grade of 5. Problem 1: Max Flows and Minimum Cuts (50 Points) Part A: (10 Points) Find the maximum flow from source to sink for the network in Figure 1. Part B: (10 Points) Find the maximum flow from source to sink for the network in Figure 2. Part C: (5 Points) Find a cut whose capacity equals the maximum flow for the network in Figure 1. Part D: (5 Points) Find a cut whose capacity equals the maximum flow for the network in Figure 2. Part E: (10 Points) Suppose for an S-T path we call the pitch of the path as smallest of the arc capacities on the path. Modify the shortest path algorithm to find the total pitch (sum of all the pitches) of the network in figure 2. Part F: (10 Points) Suppose for an S-T path we call the strength of the path as the amount we can send on the path is the product of the arc capacities on the path. Modify the shortest path algorithm to find the total strengths (sum of all the strengths. of the network in figure 1. Page 0 of 11

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4 t s t 4 2 3 1 4 4 4 4 2 6 9 6 1 Figure 1 s 1 3 4 2 9 3 8 6 7 6 9 9 7 Figure 2 Problem 2: Network Conversions: (32 Points; 4 Points Each) Part A: Suppose the total flow into a node of a network is restricted to be 10 or less. How can we modify a network to incorporate this restriction? Suppose a network contains a finite number of arcs and the capacity of each arc is an integer. Answer the following questions: Part B: Page 1 of 11
Explain why the Ford Fulkerson method will find the max flow in a finite number of steps. Part C: True or False: The max flow solution found by the Ford Fulkerson algorithm will be integral (Explain your answer). Part D: The network in Figure 3 has two source nodes and two sink nodes. A feasible flow must satisfy conservation of flow at nodes 1 and 2 (flow in = flow out), and we want to maximize the sum of the flows out of s 1 and s 2 . Convert this to an equivalent maximum flow problem with a single source node and single sink node. (HINT: you don’t have to delete any nodes of the network below. The transformation will involve adding nodes and arcs to the network. s 1 t 1 2 1 4 4 4 4 6 1 t 2 s 2 2 3 2 Figure 3 Remark: If the objective is to send flow from s1 to t1 and to send flow from s2 to t2, it cannot in general be modeled as a max flow problem. In this case, the problem would be a multicommodity flow problem. In parts E to H you are permitted to have a network with more than one arc from i to j, and each arc will have its own capacity. Such arcs are called parallel arcs . An example could be obtained from Figure 1 if we replaced the arc (1, 3) with two arcs both directed from node 1 to node 3 with capacities 4 and 5. In fact, two arcs with capacities 4 and 5 would be equivalent to a single arc with capacity 9 in terms of the value of the max flow. We ask you to come up with examples of networks illustrating certain properties.

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## This note was uploaded on 01/17/2012 for the course MGMT 15.053 taught by Professor Jamesorli during the Spring '07 term at MIT.

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ps7 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY 15.053...

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