Network Flow Problems Maximal Flow Problems Consider the

Network Flow Problems – Maximal Flow Problems Consider the following flow network: ks 1 k 1 n 1 s k 13 k 21 ks 2 2 3 n k 3 n k 23 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.

Maximal Flow Problems Examples: 1. Maximize the flow through a company’s distribution network from its factories to its customers. 2. Maximize the flow through a company’s supply network from its vendors to its factories. 3. Maximize the flow of oil through a system of pipelines. 4. Maximize the flow of water through a system of aqueducts. 5. Maximize the flow of vehicles through a transportation network.

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

Maximal Flow Problems – LP Formulation f 1 s n 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. f

Maximal Flow Problems – LP Formulation f 1 s n 3 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. f Max Z = f st s) fs 1 + fs 2 = f 1) f 13 + f 1 n = fs 1 + f 21 2) f 21 + f 23 = fs 2 3) f 3 n = f 13 + f 23 n) f = f 3 n + f 1 n 0 <= fij <= kij

Maximal Flow Problems – Conversion to Standard Form What if there are multiple sources and/or multiple sinks? n 1 s 1 1 3 s 2 2 n 2

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? n 1 s 1 f n 1 s 3 s 2 2 n 2 f

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 s 1 n k 12 3 2 f

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 s 1 k 21 2 k 12 n 3 f

Maximal Flow Problems – Lingo Solution

Maximal Flow Problems – Excel Solution