Transportation, Assignment and Network Models © 2007 Pearson Education

Network Flow Models Consist of a network that can be represented with nodes and arcs Transportation Model Transshipment Model Assignment Model Maximal Flow Model Shortest Path Model Minimal Spanning Tree Model

Characteristics of Network Models A node is a specific location An arc connects 2 nodes Arcs can be 1-way or 2-way

Types of Nodes Origin nodes Destination nodes Transshipment nodes Decision Variables XAB = amount of flow (or shipment) from node A to node B

Flow Balance at Each Node (total inflow) – (total outflow) = Net flow Node Type Net Flow Origin < 0 Destination > 0 Transshipment = 0

The Transportation Model Decision: How much to ship from each origin to each destination? Objective: Minimize shipping cost

The Transshipment Model Similar to a transportation model Have “Transshipment” nodes with both inflow and outflow Node Type Flow Balance Net Flow (RHS) Supply inflow < outflow Negative Demand inflow > outflow Positive Transshipment inflow = outflow Zero

Assignment Model For making one-to-one assignments Such as: People to tasks Classes to classrooms Etc.

The Maximal-Flow Model Where networks have arcs with limited capacity, such as roads or pipelines Decision: How much flow on each arc? Objective: Maximize flow through the network from an origin to a destination

Need 2 arcs for 2-way streets Road Network Example Need 2 arcs for 2-way streets

Modified Road Network

Decision Variables Xij = number of cars per hour flowing from node i to node j Dummy Arc The X61 arc was created as a “dummy” arc to measure the total flow from node 1 to node 6

Flow Balance At Each Node Node (X61 + X21) – (X12 + X13 + X14) = 0 1 Objective Function Max X61 Subject to the constraints: Flow Balance At Each Node Node (X61 + X21) – (X12 + X13 + X14) = 0 1 (X12 + X42 + X62) – (X21 + X24 + X26) = 0 2 (X13 + X43 + X53) – (X34 + X35) = 0 3 (X14+ X24 + X34 + X64)–(X42+ X43 + X46) = 0 4 (X35) – (X53 + X56) = 0 5 (X26 + X46 + X56) – (X61 + X62 + X64) = 0 6

Flow Capacity Limit On Each Arc Xij < capacity of arc ij

The Shortest Path Model For determining the shortest distance to travel through a network to go from an origin to a destination Decision: Which arcs to travel on? Objective: Minimize the distance (or time) from the origin to the destination

Ray Design Inc. Example Want to find the shortest path from the factory to the warehouse Supply of 1 at factory Demand of 1 at warehouse

Decision Variables Xij = flow from node i to node j Note: “flow” on arc ij will be 1 if arc ij is used, and 0 if not used Roads are bi-directional, so the 9 roads require 18 decision variables

Objective Function (in distance) Min 100X12 + 200X13 + 100X21 + 50X23 + 200X24 + 100X25 + 200X31 + 50X32 + 40X35 + 200X42 + 150X45 + 100X46 + 40X53 + 100X52 + 150X54 + 100X56 + 100X64 + 100X65 Subject to the constraints: (see next slide)

Flow Balance For Each Node Node (X21 + X31) – (X12 + X13) = -1 1 (X12+X32+X42+X52)–(X21+X23+X24+X25)=0 2 (X13 + X23 + X53) – (X31 + X32 + X35) = 0 3 (X24 + X54 + X64) – (X42 + X45 + X46) = 0 4 (X25+X35+X45+X65)–(X52+X53+X54+X56)=0 5 (X46 + X56) – (X64 + X65) = 1 6