Chapter 5:
Transportation, Assignment
and Network Models
© 2007 Pearson Education
Network Flow Models
Consist of a network that can be
represented with nodes and arcs
1. Transportation Model
2. Transshipment Model
3. Assignment Model
4. Maximal Flow Model
5. Shortest Path Model
6. 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
Origin
Destination
Transshipment
Net Flow
<0
>0
=0
The Transportation Model
Decision: How much to ship from each
origin to each destination?
Objective: Minimize shipping cost
Data
Decision Variables
Xij = number of desks shipped from factory i
to warehouse j
Objective Function: (in $ of transportation cost)
Min 5XDA + 4XDB + 3XDC + 8XEA + 4XEB +
3XEC + 9XFA + 7XFB + 5XFC
Subject to the constraints:
Flow Balance For Each Supply Node
(inflow) - (outflow) = Net flow
- (XDA + XDB + XDC) = -100
(Des Moines)
OR
XDA + XDB + XDC = 100
(Des Moines)
Other Supply Nodes
XEA + XEB + XEC = 300
XFA + XFB + XFC = 300
(Evansville)
(Fort Lauderdale)
Flow Balance For Each Demand Node
XDA + XEA + XFA = 300 (Albuquerque)
XDB + XEB + XFB = 200 (Boston)
XDC + XEC + XFC = 200 (Cleveland)
Go to File 5-1.xls
Unbalanced Transportation Model
• If (Total Supply) > (Total Demand), then for
each supply node:
(outflow) < (supply)
• If (Total Supply) < (Total Demand), then for
each demand node:
(inflow) < (demand)
Transportation Models With
Max-Min and Min-Max Objectives
• Max-Min means maximize the smallest
decision variable
• Min-Max mean to minimize the largest
decision variable
• Both reduce the variability among the Xij
values
Go to File 5-3.xls
The Transshipment Model
• Similar to a transportation model
• Have “Transshipment” nodes with both inflow
and outflow
Node Type
Supply
Demand
Transshipment
Flow Balance
inflow < outflow
inflow > outflow
inflow = outflow
Net Flow
(RHS)
Negative
Positive
Zero
Revised Transportation Cost Data
Note: Evansville is both an origin and a
destination
Objective Function: (in $ of transportation cost)
Min 5XDA + 4XDB + 3XDC + 2XDE + 3XEA +
2XEB + 1XEC + 9XFA + 7XFB + 5XFC + 2XFE
Subject to the constraints:
Supply Nodes (with outflow only)
- (XDA + XDB + XDC + XDE) = -100 (Des Moines)
- (XFA + XFB + XFC + XFE) = -300 (Ft Lauderdale)
Evansville (a supply node with inflow)
(XDE + XFE) – (XEA + XEB + XEC) = -300
Demand Nodes
XDA + XEA + XFA = 300
XDB + XEB + XFB = 200
XDC + XEC + XFC = 200
(Albuquerque)
(Boston)
(Cleveland)
Go to File 5-4.xls
Assignment Model
• For making one-to-one assignments
• Such as:
– People to tasks
– Classes to classrooms
– Etc.
Fit-it Shop Assignment Example
Have 3 workers and 3 repair projects
Decision: Which worker to assign to which
project?
Objective: Minimize cost in wages to get all
3 projects done
Estimated Wages Cost
of Possible Assignments
Can be Represented
as a Network Model
The “flow” on each arc is either 0 or 1
Decision Variables
Xij = 1 if worker i is assigned to project j
0 otherwise
Objective Function (in $ of wage cost)
Min 11XA1 + 14XA2 + 6XA3 + 8XB1 + 10XB2 +
11XB3 + 9XC1 + 12XC2 + 7XC3
Subject to the constraints:
(see next slide)
One Project Per Worker (supply nodes)
- (XA1 + XA2 + XA3) = -1
(Adams)
- (XB1 + XB2 + XB3) = -1
(Brown)
- (XC1 + XC2 + XC3) = -1
(Cooper)
One Worker Per Project (demand nodes)
XA1 + XB1 + XC1 = 1
(project 1)
XA2 + XB2 + XC2 = 1
(project 2)
XA3 + XB3 + XC3 = 1
(project 3)
Go to File 5-5.xls
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
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
Objective Function
Max X61
Subject to the constraints:
Flow Balance At Each 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)
Node
=0
=0
=0
=0
=0
=0
1
2
3
4
5
6
Flow Capacity Limit On Each Arc
Xij < capacity of arc ij
Go to File 5-6.xls
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
(X21 + X31) – (X12 + X13)
Node
= -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)
6
Go to file 5-7.xls
=1
Minimal Spanning Tree
For connecting all nodes with a minimum
total distance
Decision: Which arcs to choose to connect
all nodes?
Objective: Minimize the total distance of
the arcs chosen
Lauderdale Construction Example
Building a network of water pipes to supply water
to 8 houses (distance in hundreds of feet)
Characteristics of Minimal
Spanning Tree Problems
• Nodes are not pre-specified as origins or
destinations
• So we do not formulate as LP model
• Instead there is a solution procedure
Steps for Solving
Minimal Spanning Tree
1. Select any node
2. Connect this node to its nearest node
3. Find the nearest unconnected node and
connect it to the tree (if there is a tie,
select one arbitrarily)
4. Repeat step 3 until all nodes are
connected
Steps 1 and 2
Starting arbitrarily with node (house) 1, the
closest node is node 3
Second and Third Iterations
Fourth and Fifth Iterations
Sixth and Seventh Iterations
After all nodes (homes) are connected the total
distance is 16 or 1,600 feet of water pipe