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Network Flow Problems – Maximal Flow Problems

Network Flow Problems – Maximal Flow Problems. Consider the following flow network:. k 1n. k s1. 1. n. s. k 13. k 21. k 3n. 3. k s2. 2. k 23. The objective is to ship the maximum quantity of a commodity

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Network Flow Problems – Maximal Flow Problems

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  1. Network Flow Problems –Maximal Flow Problems Consider the following flow network: k1n ks1 1 n s k13 k21 k3n 3 ks2 2 k23 The objective is to ship the maximum quantity of a commodity from a source node s to some sink node n, through a series of arcs while being constrained by a capacity k on each arc.

  2. Maximal Flow Problems • Examples: • Maximize the flow through a company’s distribution network from its factories to its customers. • Maximize the flow through a company’s supply network from its vendors to its factories. • Maximize the flow of oil through a system of pipelines. • Maximize the flow of water through a system of aqueducts. • Maximize the flow of vehicles through a transportation network.

  3. Maximal Flow Problems Definitions: Flow network – consists of nodes and arcs Source node – node where flow originates Sink node – node where flow terminate Transshipment points – intermediate nodes Arc/Link – connects two nodes Directed arc – arc with direction of flow indicated Undirected arc – arc where flow can occur in either direction Capacity(kij) – maximum flow possible for arc (i,j) Flow(f ij) – flow in arc (i,j). Forward arc – arcs with flow out of some node Backward arc – arc with flow into some node Path – series of nodes and arcs between some originating and some terminating node Cycle – path whose beginning and ending nodes are the same

  4. Maximal Flow Problems – LP Formulation f 1 n f s 3 2 Objective: Maximize Flow (f) Constraints: 1) The flow on each arc, fij, is less than or equal to the capacity on each arc, kij. 2) Conservation of flow at each node. Flow in = flow out.

  5. Maximal Flow Problems – LP Formulation f 1 n f s 3 Max Z = f st s) fs1 +fs2 = f 1) f13 +f1n = fs1 +f21 2) f21 +f23 = fs2 3) f3n = f13 +f23 n) f = f3n +f1n 0 <= fij <= kij 2 • Objective: Maximize Flow (f) • Constraints: • The flow on each arc, fij, is less than or • equal to the capacity on each arc, kij. • Conservation of flow at each node. • Flow in = flow out.

  6. Maximal Flow Problems – Conversion to Standard Form What if there are multiple sources and/or multiple sinks? n1 s1 1 n2 3 s2 2

  7. Maximal Flow Problems – Conversion to Standard Form Create a “supersource” and “supersink” with arcs from the supersource to the original sources and from the original sinks to the supersink. What capacity should we assign to these new arcs? n1 f s1 n 1 f s n2 3 s2 2

  8. Maximal Flow Problems – Conversion to Standard Form What if there is an undirected arc (flow can occur in either direction)? See arc (1,2). f 1 n f s k12 3 2

  9. Maximal Flow Problems – Conversion to Standard Form Create two directed arcs with the same capacity. Upon solving the problem and obtaining flows on each arc, replace the two directed arcs with a single arc with flow | fij– fji |, in the direction of the larger of the two flows. f 1 n f s k21 k12 3 2

  10. Maximal Flow Problems – Lingo Solution

  11. Maximal Flow Problems – Excel Solution

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