Welcome to MM305 Unit 6 Seminar Larry Musolino lmusolino@kaplan.edu Network Models

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: 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

Outline 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 Models Minimal-Spanning Tree Technique Maximal-Flow 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 Models Important 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 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.

Minimal-Spanning Tree Technique Section 6.2 – Method (see bottom of page 206) Select any node in the network Connect this node to the nearest node to minimize the total distance Consider all the nodes that are now connected, now connect the nearest node to minimize distance If there is a tie for the nearest node, select one arbitrarily Repeat the 3rd step until all nodes are connected

Example – 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 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 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 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 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 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

Using QM for Windows For our example, we had 8 nodes and 13 branches

Using 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 Windows Once all branches have been entered, click SOLVE. Distance

Using 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

Using QM for Windows Solution Steps are then shown:

Worked out Solved Problem 6-1, see page 217

Worked 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 217 Click SOLVE Problem 6-1, see page 217

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

6.3 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 Technique Method for Maximal Flow Technique See page 210 Pick any path from start to finish with some flow. 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). 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. Repeat these steps until an increase in flow is no longer possible.

6.3 Maximal-Flow Technique Example shown in Fig 6.6 on page 210 Goal: find max number of cars that can flow from west to east. 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 Technique Example shown in Fig 6.6 on page 210 Goal: find max number of cars that can flow from west to east. 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 Technique Example shown in Fig 6.6 on page 210 Step 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 Technique Example shown in Fig 6.7 on page 211 Step 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 Technique Example shown in Fig 6.8 on page 212 Step 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 Technique Example shown in Fig 6.9 on page 213 Step 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 Technique Example shown in Fig 6.9 on page 213 Summary: Max capacity = 2 + 1 + 2 = 5 (this represents 500 cars)

Maximal-Flow Technique QM for Windows Start up QM for Windows Select Module, Then Networks, Then File, New, Maximal Flow Then Enter number of Branches

Maximal-Flow Technique QM for Windows For our example we had 6 nodes and 9 branches

Maximal-Flow Technique QM for Windows Enter Network Info: Outbound Reverse

Maximal-Flow Technique Click SOLVE Total Flow

Shortest-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 Technique Method (See Page 214) Find the nearest node to the starting point. Put that distance in a box by the node. Find the next-nearest node to the starting point, and put the distance in a box by the node. 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

Shortest-Route Technique 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).

Shortest-Route Technique 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 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 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 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 Route Start up QM for Windows Select Module, Then Networks, Then File, New, Shortest Route Then Enter number of Branches

Using QM for Windows for Shortest Route In our example we have 6 nodes and 9 branches

Using QM for Windows for Shortest Route Enter data for network including distance between nodes:

Using QM for Windows for Shortest Route Solution for shortest distance is displayed

Unit Project for Unit 6 Problem #1

Unit Project for Unit 6 Problem #2

Unit Project for Unit 6 Problem #3