1. Transportation, Assignment and Network Models
© 2007 Pearson Education

2. 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

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

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

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

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

7. 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

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

9. 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

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

11. Modified Road Network

12. 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

13. 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) =
(X12 + X42 + X62) – (X21 + X24 + X26) =
(X13 + X43 + X53) – (X34 + X35) =
(X14+ X24 + X34 + X64)–(X42+ X43 + X46) =
(X35) – (X53 + X56) =
(X26 + X46 + X56) – (X61 + X62 + X64) =

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

15. 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

16. 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

17. 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

18. Objective Function (in distance)
Min 100X X X X X X X X X X X X X X X X X X65
Subject to the constraints:
(see next slide)

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