Welcome to MM305 Unit 6 Seminar Larry Musolino [email protected] [email protected] Network Models

Welcome to MM305

Unit 6 Seminar

Larry Musolino

[email protected]

Network Models

mailto:[email protected]

Reminders:Reminders:

• Each Week you are responsible for:• Posting to DISCUSSION BOARD

• Must post an initial response plus at least two significant follow-up responses to fellow students

• Submit the UNIT PROJECT to DROPBOX• 2 to 3 problems where you show step by step solutions

• Complete the UNIT QUIZ• 20 multiple choice problems, one attempt, 4 hours to complete,

Open book, Open notes.

• Attend SEMINAR• Weds at 10pm ET• If you cannot attend seminar, complete the Seminar Option 2

assignment, place the completed assignment in the dropbox

More reminders:More reminders:• When submitting Unit Project, show your work, partial credit

awarded for correct method.• Review the SELF-TEST at the end of each Chapter (answers are in

the back of the text). This is good practice for the UNIT QUIZZES.• See Chap 6 Self test on page 220• Solutions to Self Test on page 309-310 (back of text)

• Review the Glossary while taking the Unit Quiz• Chap 6 Terminology explained, see page 217

• When working the problems in the UNIT PROJECT, review the examples in the TEXT. Also, there are Worked-out (SOLVED) PROBLEMS in the back of each chapter.• See Chap 6 Solved Problems on pages 217 - 219

OutlineOutline

• Three Network Models:

• Section 6.2 – Minimal Spanning Tree Technique• Minimize total distance

• Section 6.3 – Maximal Flow Technique• Maximize flow through a network

• Section 6.4 – Shortest Route Technique• Minimize the path through a network

Network ModelsNetwork Models• Minimal-Spanning Tree Technique

• [least amount of material to connect all points]• Example: connect all houses in subdivision to electrical power,

how to minimize length of cable.

• Maximal-Flow Technique • [maximum amount of material that can flow through a network]• Example: how to optimize max number of vehicles that can travel

from one location to another

• Shortest-Route Technique • [travel from one location to another while minimizing total

distance]• Example: Salesperson needs to visit 10 cities, what route will

minimize total distance traveled

Network ModelsNetwork ModelsImportant Note:• QM for Windows or Excel QM will be big timesaver for these network

model analyses.• Although simple problems can be done by hand, the analysis can be

time consuming!• For any real world problem, we typically use some technology tool

such as QM for Windows or Excel QM • We normally would not do this analysis manually !!

• Note: QM for Windows can handle all three network analysis techniques: • Minimal-Spanning Tree, Maximal Flow and Shortest-Path

• Excel QM only includes Shortest-Path and Maximal Flow but NOT Minimal-Spanning Tree Technique.

• We will use QM for Windows in the examples in this Seminar.

QM for Windows for Network AnalysisQM for Windows for Network Analysis

• After starting QM for Windows, Select Network Analysis• Then select FILE, NEW

Minimal-Spanning Tree Technique


Section 6.2• The minimal-spanning tree technique

involves connecting all the points of a network together while minimizing the distance between them.



Section 6.2 – Method (see bottom of page 206)

1. Select any node in the network2. Connect this node to the nearest node to minimize

the total distance3. Consider all the nodes that are now connected, now

connect the nearest node to minimize distance1. If there is a tie for the nearest node, select one

arbitrarily4. Repeat the 3rd step until all nodes are connected

Example – Figure 6.1 on page 207Example – Figure 6.1 on page 207• Assume we want to connect eight houses to the power

line. What is the minimum distance to connect all the eight nodes?

• The numbers on the lines between the nodes represents the distance between the houses.

Example – Figure 6.2 on page 207Example – Figure 6.2 on page 207• Step 1 - Select any node – Let’s select Node #1• Step 2 – Connect Node #1 to nearest node in order to

minimize the distance. Notice Node 3 is the smallest distance away from Node 1, distance = 2.

Example – Figure 6.3 on page 208Example – Figure 6.3 on page 208• Step 3 – Now consider that both Nodes 1 and 3 are connected. We now

connect the nearest node for nodes 1 and 3, which is not yet connected.• Notice that node 4 is distance of 2 away from node 3, so we connect this

node next. See Figure 6.3(a) on left side of below diagram.• Now we have nodes 1, 3 and 4 connected. We next look to connect the

nearest nodes and see nodes 2 and 6 are both distance of 3. We arbitrarily pick node 2. See Figure 6.3 (b) on right side of below diagram.

Example – Figure 6.4 on page 208Example – Figure 6.4 on page 208• Step 3 – Now consider that Nodes 1, 2, 3 and 4 are connected. We now

connect the nearest node, which is not yet connected.• Notice that the distance from node 2 to 5 is “3” and the distance from node 3

to 6 is “3”. We arbitrarily select to connect node 2 to 5. See Figure 6.4 (a) on left side of diagram

• Notice we do not consider to connect node 1 to 2 since these nodes are already connected to the network.

• Next, we connect nodes 3 and 6, since the distance here is “3”

Example – Figure 6.5 on page 209Example – Figure 6.5 on page 209• Step 3 – We now have nodes 1, 2, 3, 4, 5 connected. The next nearest node

is node 8 with distance of “1”. See diagram 6.5(a) on left side.• The last node to connect is node 7 at distance of 2.• The total distance to connect all eight nodes is 2+2+3+3+3+1+2 = 16

Using QM for WindowsUsing QM for Windows

• QM for Windows can be used to solve minimal spanning tree.

• Start up QM for Windows,• Select Module,

• Then Networks,• Then File, New, Minimal Spanning Tree

• Then Enter number of Branches


• For our example, we had 8 nodes and 13 branches

Using QM for WindowsUsing QM for Windows• For each branch, fill in the connecting nodes and the

“COST” which in our case means the distance between the two nodes

Distance

Using QM for WindowsUsing QM for Windows• Once all branches have been entered, click SOLVE.

Distance

Using QM for WindowsUsing QM for Windows• QM for Windows will then show which branches to

include to connect all nodes to the network, and also show the total distance.

Distance


• Solution Steps are then shown:

Worked out Solved Problem 6-1, see page 217Worked out Solved Problem 6-1, see page 217

Worked out Solved Problem 6-1, see page 217Worked out Solved Problem 6-1, see page 217• Notice there are 8 nodes and 13 branches• Enter data into QM for Windows

Problem 6-1, see page 217

Worked out Solved Problem 6-1, see page 217Worked out Solved Problem 6-1, see page 217• Click SOLVE

Problem 6-1, see page 217

Worked out Solved Problem 6-1, see page 217Worked out Solved Problem 6-1, see page 217

Problem 6-1, see page 217Total distance to minimize is 67

6.3 Maximal-Flow Technique6.3 Maximal-Flow Technique

• The maximal-flow technique allows us to determine the maximum amount of a material that can flow through a network• Example: Find the maximum number of

cars that can flow through a highway system

• Example: Find the maximum flow of water through a storm drain system

6.3 Maximal-Flow Technique6.3 Maximal-Flow TechniqueMethod for Maximal Flow Technique

• See page 2101. Pick any path from start to finish with some flow.2. Find the arc on this path with the smallest flow capacity.

Call this capacity C. (This is the maximim additional capacity that can be allocated to this route).

3. Foe each node on this path, decrease the flow capacity in the direction of the flow by the amount C. For each node on this path, increase the flow capacity in the reverse direction by amount C.

4. Repeat these steps until an increase in flow is no longer possible.

6.3 Maximal-Flow Technique6.3 Maximal-Flow TechniqueExample shown in Fig 6.6 on page 210Goal: find max number of cars that can flow from west to east.

1. Pick any path from start to finish with some flow. Note: in this diagram, the max capacity from node 1 to node 2 is “3”. The max capacity from node 2 to node 1 is “1”

6.3 Maximal-Flow Technique6.3 Maximal-Flow TechniqueExample shown in Fig 6.6 on page 210Goal: find max number of cars that can flow from west to east.

1. Pick any path from start to finish with some flow. Let’s pick the flow on the top of the diagram from Node 1 to Node 2 to Node 6.

6.3 Maximal-Flow Technique6.3 Maximal-Flow TechniqueExample shown in Fig 6.6 on page 210Step 2 - For the path from Node 1 to Node 2 to Node 6, notice the arc

with the smallest flow capacity from west to east is “2” since only 2 units from flow from node 2 to node 6. Call this capacity C (2 units). Note: Path 1 to 2 to 6 results in capacity of “2”

6.3 Maximal-Flow Technique6.3 Maximal-Flow TechniqueExample shown in Fig 6.7 on page 211Step 3 – Now we adjust the flow for this path. For each node on this

path decrease the flow from west to east by C (2 units). Then for each node on this path, increase the flow capacity in reverse direction by C (2 units).

This is the new relative capacity for this route

6.3 Maximal-Flow Technique6.3 Maximal-Flow TechniqueExample shown in Fig 6.8 on page 212Step 4 – Now let’s pick another path with some unused capacity.

Suppose we next pick Path from Node 1 to 2 to 4 to 6. The smallest flow capacity on this route is “1”. Note we now need to use the modified diagram based on changes to route node 1 to 2 to 6 from previous slide. Note: Path 1 to 2 to 4 to 6 results in capacity of “1”

6.3 Maximal-Flow Technique6.3 Maximal-Flow TechniqueExample shown in Fig 6.9 on page 213Step 5 – Now let’s pick another path with some unused capacity.

Notice there is one more path, node 1 to 3 to 5 to 6. The capacity available on this path is “2”. The resulting network diagram is shown below. Notice there are no additional paths with capacity from west to east. Note: Path 1 to 3 to 5 to 6 results in capacity of “2”

6.3 Maximal-Flow Technique6.3 Maximal-Flow TechniqueExample shown in Fig 6.9 on page 213Summary:Max capacity = 2 + 1 + 2 = 5 (this represents 500 cars)

Maximal-Flow TechniqueQM for Windows


• Start up QM for Windows• Select Module,

• Then Networks,• Then File, New, Maximal Flow




• For our example we had 6 nodes and 9 branches



• Enter Network Info: Outbound Reverse

Maximal-Flow TechniqueMaximal-Flow Technique• Click SOLVE Total Flow

Shortest-Route TechniqueShortest-Route Technique

• The shortest-route technique finds how a person or item can travel from one location to another while minimizing the total distance traveled

• It finds the shortest route to a series of destinations

Shortest-Route TechniqueShortest-Route TechniqueMethod (See Page 214)1.Find the nearest node to the starting point. Put

that distance in a box by the node.2.Find the next-nearest node to the starting point,

and put the distance in a box by the node.3.Repeat this process until you have gone

through the entire network. The last distance at the ending note is the shortest distance through the network


• We will use Example shown in Fig 6.10 on page 213.• Goal: find the shortest distance from the plant to the

warehouse through various cities (distances in miles).


• Example shown in Fig 6.11 on page 214.• We start off to note that the nearest node to the Starting

point (the plant) is Node 2 which is 100 miles away, so we connect these two nodes.

• Write the distance “100” in a box near node 2.

Shortest-Route TechniqueShortest-Route Technique• Example shown in Fig 6.12 on page 215.• Next we look for the next nearest node to the origin. We

check nodes 3, 4 and 5. • The nearest path to the origin is Path 1 to 2 to 3 at distance

of 150.• Write the distance “150” in a box near node 3.•

Shortest-Route TechniqueShortest-Route Technique• Example shown in Fig 6.13 on page 216.• Next we look for the next nearest node to the origin. We

check nodes 4 and 5.• Node 1 to 2 to 4 = 300 miles• Node 1 to 2 to 5 = 200 miles• Node 1 to 2 to 3 to 5 = 190 miles (Shortest path)

• Write the distance “190” in a box near node 5.

Shortest-Route TechniqueShortest-Route Technique• Example shown in Fig 6.14 on page 216.• Next we check nodes 4 and 6 as the last nodes.

• Node 1 to 2 to 4 = 300 miles• Node 1 to 2 to 5 to 6 = 290 miles (Shortest path)• Since node 6 is the ending node, we are done

• Write the distance “290” in a box near node 6, this is the shortest distance, and path is 1, 2, 3, 5, 6.

Using QM for Windows for Shortest RouteUsing QM for Windows for Shortest Route

• Start up QM for Windows• Select Module,

• Then Networks,• Then File, New, Shortest Route



• In our example we have 6 nodes and 9 branches


Enter data for network including distance between nodes:


Solution for shortest distance is displayed

Unit Project for Unit 6Unit Project for Unit 6

• Problem #1


• Problem #2


• Problem #3

Documents

Welcome to MM305 Unit 6 Seminar Larry Musolino [email protected] [email protected] Network Models