Network Optimization ModelsChapter 10: Hillier and Lieberman
Chapter 8: Decision Tools for Agribusiness
Dr. Hurleys AGB 328 Course
Terms to Know
Nodes, Arcs, Directed Arc, Undirected Arc, Links, Directed Network, Undirected Network, Path, Directed Path, Undirected Path, Cycle, Connected, Connected Network, Tree, Spanning Tree, Arc Capacity, Supply Node, Demand Node, Transshipment Node, Sink, Source, Residual Network, Residual Capacity, Augmenting Path, Cut, Cut Value, Max-Flow Min-Cut Theorem, Feasible Solutions Property, Integer Solutions Property, Reverse Arc, Basic Arcs, NonbasicArcs
Terms to Know Cont.
Spanning Tree Solution, Feasible Spanning
Tree, Fundamental Theorem for the
Network Simplex Method, Program
Evaluation and Review Technique (PERT),
Critical Path Method (CPM), Immediate
Successor, Immediate Predecessor, Project
Network, Activity-on-Arc, Activity-on-Node,
Project Duration, Critical Path, Crashing an
Activity, Crashing the Project, Normal Point,
Crash Point, Marginal Cost Analysis
The Shortest Path Problem
A shortest path problem usually has a node known as the origin and a node known as the destination
The objective of this problem is to find the shortest path from the origin node to the destination node
The shortest path could be measured in time, distance, etc.
Since this problem is a special case of the linear programming problem, the simplex method could be used to solve it
Mathematical Model for Seervada
Shortest Path Problem
2 + 5 + 4 + 2 + 7 + 1 + 4
+ 3 + 1 + 4 + 1 + 1 + 5 + 7Subject To:
+ + = 1 = 0 + + = 0 + = 0 + + = 0 + + = 0 + = 1, , , , , , , , , , , , , (0,1)
Seervada Spreadsheet Model
Examine in Class
Minimum Spanning Tree Problems
The goal of the minimum spanning tree
problem is to connect all the nodes either
directly or indirectly at the lowest cost
The minimum spanning tree problem will
have one less link than the number of
nodes in the optimal solution
The solution to the minimum spanning
tree problem can be accomplished using
the greedy algorithm
Choose any two nodes initially and
Identify the closest unconnected node
and then connect it
Continue until all nodes have been connected to the tree
Ties can be broken arbitrarily
The Maximum Flow Problem
The purpose of the maximum flow
problem is to get as much flow through
the network based on the capacity
constraints of the network
This can be measured by the amount leaving the source or by the amount entering the sink
It has a source where supply originates
from and a sink which absorbs the supply
that makes it through the network
Augmenting Path Algorithm for
Maximum Flow Problems Identify an augmenting path that takes flow
from the source to the sink in the residual network such that every arc on this path has strictly positive residual If this path does not exist, you have the optimal
Identify the residual capacity c* by finding the minimum of the residual capacities of the arcs on the path Increase the flow in this path by c*
Decrease the residual capacities by c* for each arc on the augmenting path
Max-Flow Min-Cut Theorem
Another way of figuring out the maximum flow is by using the Max-Flow Min-Cut Theorem
The theorem states that if you have a single source and sink, then the maximum flow through the network is equal to the smallest cut value for all the cuts of the network
The cut value is found by summing up all the arcs which are directly affected by the cut of a network A cut is defined as a set of directed arcs that separate the
source from the sink
Mathematical Model for the
Seervada Max Flow Problem
= 0 + = 0
+ = 0 + + = 0 + = 0
0 5, 0 7, 0 4, 0 1,0 3, 0 2, 0 4, 0 5,0 4, 0 9, 0 1,0 6
Note: This is based on Figure 10.11 in the text.
Excel Formulation of Seervada Max
Flow Problem Examined in Class
In Class Max Flow Activity (Not
Minimum Cost Flow Problem
Requirements At least one supply node
At least one demand node
The network is directed and connected
If the node is not a supply or demand node, then it is a transshipment node
Flow through an arc is directed
There is enough arc capacity to get the total supply to the total demand
Costs are proportional to the amount of flow
The objective is to minimize cost
Minimum Cost Flow General
Mathematical Model xij = the arc representing the flow from nodes i to
cij = the cost of flow through xij uij = the arc capacity for xij bi = net flow generated at node i Supply node (bi > 0), demand node (bi < 0),
transshipment node (bi = 0)
= =1 =1
0 for each arc
Minimum Cost Flow Problem in
Excel We will examine the Distribution
Unlimited Co. in class
What is Project Management
Project management can be defined as the
coordination of activities with the potential use
of many organizations, both internal and
external to the business, in order to conduct a
large scale project from beginning to end.
There are two management science techniques
that are used for project management:
Program and Evaluation Review Technique (PERT)
Critical Path Method (CPM)
PERT was designed to examine projects from the standpoint of uncertainty.
CPM was designed to examine projects from the standpoint of costs.
PERT and CPM techniques have been combined over time.
PERT and CPM both rely heavily on the use of networks to help plan and display the coordination of all the activities for a project.
The Reliable Construction
Company Reliable has just secured a contract to
construct a new plant for a major manufacturer.
The contract is for $5.4 million to cover all
costs and any profits.
The plant must be finished in a year.
A penalty of $300,000 will be assessed if Reliable does not complete the project within 47 weeks.
A bonus of $150,000 will be paid to Reliable if the plant is completed within 40 weeks.
A distinct task that needs to be performed as part of the project.
This is a node that represents the beginning of the project.
This node represents the end of the project.
Needed Terminology Cont.
These are activities that must be completed by no later than the start time of the given activity.
Given the immediate predecessor of an activity, this activity becomes the immediate successor of each of these immediate predecessors.
If an immediate successor has multiple immediate predecessors, then all must be finished before an activity can begin.
Activity List for Reliable Construction
Activity Activity Description
A Excavate 2
B Lay the foundation A 4
C Put up the rough wall B 10
D Put up the roof C 6
E Install the exterior plumbing C 4
F Install the interior plumbing E 5
G Put up the exterior siding D 7
H Do the exterior painting E, G 9
I Do the electrical work C 7
J Put up the wallboard F, I 8
K Install the flooring J 4
L Do the interior painting J 5
M Install the exterior fixtures H 2
N Install the interior fixtures K, L 6
Questions Needed to be Answered
How can the project be displayed graphically?
How much time is required to finish the project
if no delays occur?
When is earliest start and finish times for each
activity if no delays occur?
What activities are critical bottleneck activities
where delays must be avoided to finish the
project on time?
Questions Needed to be Answered
Cont. For non bottleneck activities, how much can an
activity be delayed and yet still keep the project on time?
What is the probability of completing the project by the deadline?
What is the least amount of money needed to expedite the project to obtain the bonus?
How should costs be monitored to keep the project within budget?
A project network is a network diagram that uses nodes and arcs to represent the progression of the activities for a project from start to finish.
Three pieces of information needed:
Project Network Cont.
Two types of project networks
On this diagram, the activity is represented on an
arc, while a node is used to separate an activity