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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

min... ,,,,,,,,

,,,,,

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

Greedy Algorithm

Choose any two nodes initially and

connect them

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

max... ,,,,,,

,,,,,

+ +

= 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

Graded)

A

D

C

B

E

H

G

F

I

8

7

6

7

106 3

5

7

6

47

0

0

0

0

00

0

0

0

0

0

0

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

j

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

Subject to:

=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/CPM

PERT

PERT was designed to examine projects from the standpoint of uncertainty.

CPM

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.

Needed Terminology

Activity

A distinct task that needs to be performed as part of the project.

Start Node

This is a node that represents the beginning of the project.

Finish Node

This node represents the end of the project.

Needed Terminology Cont.

Immediate Predecessor

These are activities that must be completed by no later than the start time of the given activity.

Immediate Successor

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

Immediate

Predecessors

Estimated

Duration (Weeks)

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?

Project Network

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:

Activity information

Precedence relationship

Time information

Project Network Cont.

Two types of project networks

Activity-on-Arc (AOA)

On this diagram, the activity is represented on an

arc, while a node is used to separate an activity