ch 9 network

1 1© 2003 Thomson© 2003 Thomson/South-Western/South-Western Slide Slide

Slides Prepared bySlides Prepared by

JOHN S. LOUCKSJOHN S. LOUCKS

St. Edward’s UniversitySt. Edward’s University


Chapter 9 Chapter 9 Network ModelsNetwork Models

Shortest-Route ProblemShortest-Route Problem Minimal Spanning Tree ProblemMinimal Spanning Tree Problem Maximal Flow ProblemMaximal Flow Problem


Computer CodesComputer Codes

Computer codesComputer codes for specific network problems for specific network problems (e.g. shortest route, minimal spanning tree, and (e.g. shortest route, minimal spanning tree, and maximal flow) are extremely efficient and can maximal flow) are extremely efficient and can solve even large problems relatively quickly. solve even large problems relatively quickly.

The Management Scientist The Management Scientist contains computer contains computer codes for each of these problems.codes for each of these problems.


Shortest-Route ProblemShortest-Route Problem

The The shortest-route problemshortest-route problem is concerned with is concerned with finding the shortest path in a network from one finding the shortest path in a network from one node (or set of nodes) to another node (or set of node (or set of nodes) to another node (or set of nodes).nodes).

If all arcs in the network have nonnegative values If all arcs in the network have nonnegative values then a labeling algorithm can be used to find the then a labeling algorithm can be used to find the shortest paths from a particular node to all other shortest paths from a particular node to all other nodes in the network.nodes in the network.

The criterion to be minimized in the shortest-route The criterion to be minimized in the shortest-route problem is not limited to distance even though the problem is not limited to distance even though the term "shortest" is used in describing the term "shortest" is used in describing the procedure. Other criteria include time and cost. procedure. Other criteria include time and cost. (Neither time nor cost are necessarily linearly (Neither time nor cost are necessarily linearly related to distance.)related to distance.)


Shortest-Route AlgorithmShortest-Route Algorithm

Note:Note: We use the notation [ ] to represent a We use the notation [ ] to represent a permanent permanent label and ( ) to represent a tentative label and ( ) to represent a tentative label.label.

Step 1:Step 1: Assign node 1 the permanent label [0,Assign node 1 the permanent label [0,SS]. ]. The first number is the distance from node 1; the The first number is the distance from node 1; the

second number is the preceding node. second number is the preceding node. Since node 1 has no preceding node, it is labeled Since node 1 has no preceding node, it is labeled SS for the starting node. for the starting node.

Step 2:Step 2: Compute tentative labels, (Compute tentative labels, (dd,,nn), for the ), for the nodes that can be reached directly from node 1. nodes that can be reached directly from node 1. dd = the = the direct distance from node 1 to the direct distance from node 1 to the node in question -- this is called the distance node in question -- this is called the distance value. value. nn indicates the preceding node on the indicates the preceding node on the route from node 1 -- this is called the preceding route from node 1 -- this is called the preceding node value. node value.



Step 3:Step 3: Identify the tentatively labeled node Identify the tentatively labeled node with the smallest distance value. Suppose it is with the smallest distance value. Suppose it is node node kk. Node . Node kk is now permanently labeled is now permanently labeled (using [ , ] brackets). If all nodes are (using [ , ] brackets). If all nodes are permanently labeled, GO TO STEP 5.permanently labeled, GO TO STEP 5.

Step 4:Step 4: Consider all nodes without permanent Consider all nodes without permanent labels that can be reached directly from the labels that can be reached directly from the node node kk identified in Step 3. For each, calculate identified in Step 3. For each, calculate the quantity the quantity t, t, wherewhere

tt = (arc distance from node = (arc distance from node kk to node to node ii) )

+ (distance value at node + (distance value at node kk).).



Step 4: (continued)Step 4: (continued)

• If the non-permanently labeled node has a If the non-permanently labeled node has a tentative label, compare tentative label, compare tt with the current with the current distance value at the tentatively labeled node distance value at the tentatively labeled node in question.in question.

If If tt < distance value of the tentatively < distance value of the tentatively labeled labeled node, replace the tentative node, replace the tentative label with (label with (tt,,kk). ).

If If tt >> distance value of the tentatively distance value of the tentatively labeled node, keep the current tentative labeled node, keep the current tentative label.label.

• If the non-permanently labeled node does not If the non-permanently labeled node does not have a tentative label, create a tentative label have a tentative label, create a tentative label of (of (tt,,kk) for the node in question.) for the node in question.

In either case, now GO TO STEP 3.In either case, now GO TO STEP 3.



Step 5:Step 5: The permanent labels identify the The permanent labels identify the shortest distance from node 1 to each node as shortest distance from node 1 to each node as well as the preceding node on the shortest well as the preceding node on the shortest route. The shortest route to a given node can route. The shortest route to a given node can be found by working backwards by starting at be found by working backwards by starting at the given node and moving to its preceding the given node and moving to its preceding node. Continuing this procedure from the node. Continuing this procedure from the preceding node will provide the shortest route preceding node will provide the shortest route from node 1 to the node in question.from node 1 to the node in question.


Example: Shortest RouteExample: Shortest Route

Find the Shortest Route From Node 1 to All Find the Shortest Route From Node 1 to All Other Nodes in the Network:Other Nodes in the Network:

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Iteration 1Iteration 1

•Step 1:Step 1: Assign Node 1 the permanent label Assign Node 1 the permanent label [0,S].[0,S].

•Step 2:Step 2: Since Nodes 2, 3, and 4 are directly Since Nodes 2, 3, and 4 are directly connected to Node 1, assign the tentative connected to Node 1, assign the tentative labels of (4,1) to Node 2; (7,1) to Node 3; labels of (4,1) to Node 2; (7,1) to Node 3; and (5,1) to Node 4. and (5,1) to Node 4.



Tentative Labels ShownTentative Labels Shown

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(4,1)(4,1)

[0,S][0,S]

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•Step 3:Step 3: Node 2 is the tentatively labeled node Node 2 is the tentatively labeled node with the smallest distance (4) , and hence with the smallest distance (4) , and hence becomes the new permanently labeled node.becomes the new permanently labeled node.



Permanent Label ShownPermanent Label Shown

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[4,1][4,1]

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•Step 4:Step 4: For each node with a tentative label For each node with a tentative label which is connected to Node 2 by just one arc, which is connected to Node 2 by just one arc, compute the sum of its arc length plus the compute the sum of its arc length plus the distance value of Node 2 (which is 4). distance value of Node 2 (which is 4).

Node 3: 3 + 4 = 7 (not smaller Node 3: 3 + 4 = 7 (not smaller than current label; do not change.)than current label; do not change.)

Node 5: 5 + 4 = 9 (assign Node 5: 5 + 4 = 9 (assign tentative label to Node 5 of (9,2) since node 5 tentative label to Node 5 of (9,2) since node 5 had no label.)had no label.)



Iteration 1, Step 4 ResultsIteration 1, Step 4 Results

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[4,1][4,1]

[0,S][0,S]

(5,1)(5,1)

(7,1)(7,1)

(9,2)(9,2)

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•Step 3:Step 3: Node 4 has the smallest tentative Node 4 has the smallest tentative label distance (5). It now becomes the new label distance (5). It now becomes the new permanently labeled node.permanently labeled node.




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[4,1][4,1]

[0,S][0,S]

[5,1][5,1]

(6,4)(6,4)

(9,2)(9,2)

33331111

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•Step 4:Step 4: For each node with a tentative label For each node with a tentative label which is connected to node 4 by just one arc, which is connected to node 4 by just one arc, compute the sum of its arc length plus the compute the sum of its arc length plus the distance value of node 4 (which is 5).distance value of node 4 (which is 5).

Node 3: 1 + 5 = 6 (replace the Node 3: 1 + 5 = 6 (replace the tentative label of node 3 by (6,4) since 6 < 7, tentative label of node 3 by (6,4) since 6 < 7, the current distance.)the current distance.)

Node 6: 8 + 5 = 13 (assign Node 6: 8 + 5 = 13 (assign tentative label to node 6 of (13,4) since node tentative label to node 6 of (13,4) since node 6 had no label.)6 had no label.)




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[4,1][4,1]

[0,S][0,S]

[5,1][5,1]

(6,4)(6,4)

(9,2)(9,2)

(13,4)(13,4)

33331111

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•Step 3:Step 3: Node 3 has the smallest tentative Node 3 has the smallest tentative distance label (6). It now becomes the new distance label (6). It now becomes the new permanently labeled node.permanently labeled node.




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[4,1][4,1]

[0,S][0,S]

[5,1][5,1]

[6,4][6,4]

(9,2)(9,2)

(13,4)(13,4)

33331111

2222 5555

7777




•Step 4:Step 4: For each node with a tentative label For each node with a tentative label which is connected to node 3 by just one arc, which is connected to node 3 by just one arc, compute the sum of its arc length plus the compute the sum of its arc length plus the distance to node 3 (which is 6).distance to node 3 (which is 6).

Node 5: 2 + 6 = 8 (replace the Node 5: 2 + 6 = 8 (replace the tentative label of node 5 with (8,3) since 8 < tentative label of node 5 with (8,3) since 8 < 9, the current distance)9, the current distance)

Node 6: 6 + 6 = 12 (replace the Node 6: 6 + 6 = 12 (replace the tentative label of node 6 with (12,3) since 12 tentative label of node 6 with (12,3) since 12 < 13, the current distance)< 13, the current distance)




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[4,1][4,1]

[0,S][0,S]

[5,1][5,1]

[6,4][6,4]

(8,3)(8,3)

(12,3)(12,3)

33331111

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[4,1][4,1]

[0,S][0,S]

[5,1][5,1]

[6,4][6,4]

[8,3][8,3]

(12,3)(12,3)

33331111

2222 5555

7777




•Step 4:Step 4: For each node with a tentative label For each node with a tentative label which is connected to node 5 by just one arc, which is connected to node 5 by just one arc, compute the sum of its arc length plus the compute the sum of its arc length plus the distance value of node 5 (which is 8).distance value of node 5 (which is 8).

Node 6: 3 + 8 = 11 (Replace the Node 6: 3 + 8 = 11 (Replace the tentative label with (11,5) since 11 < 12, the tentative label with (11,5) since 11 < 12, the current distance.)current distance.)

Node 7: 6 + 8 = 14 (AssignNode 7: 6 + 8 = 14 (Assign




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[4,1][4,1]

[0,S][0,S]

[5,1][5,1]

[6,4][6,4]

[8,3][8,3]

(11,5)(11,5)

(14,5)(14,5)33331111

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[4,1][4,1]

[0,S][0,S]

[5,1][5,1]

[6,4][6,4]

[8,3][8,3]

[11,5][11,5]

(14,5)(14,5)33331111

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•Step 4:Step 4: For each node with a tentative label For each node with a tentative label which is connected to Node 6 by just one arc, which is connected to Node 6 by just one arc, compute the sum of its arc length plus the compute the sum of its arc length plus the distance value of Node 6 (which is 11).distance value of Node 6 (which is 11).

Node 7: 2 + 11 = 13 (replace the Node 7: 2 + 11 = 13 (replace the tentative label with (13,6) since 13 < 14, the tentative label with (13,6) since 13 < 14, the current distance.)current distance.)




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[4,1][4,1]

[0,S][0,S]

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[6,4][6,4]

[8,3][8,3]

[11,5][11,5]

(13,6)(13,6)33331111

2222 5555

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•Step 3:Step 3: Node 7 becomes permanently Node 7 becomes permanently labeled, and hence all nodes are now labeled, and hence all nodes are now permanently labeled. Thus proceed to permanently labeled. Thus proceed to summarize in Step 5.summarize in Step 5.

•Step 5:Step 5: Summarize by tracing the shortest Summarize by tracing the shortest routes backwards through the permanent routes backwards through the permanent labels.labels.



Solution SummarySolution Summary

NodeNode Minimum DistanceMinimum Distance Shortest Shortest RouteRoute

2 2 4 4 1-2 1-2

3 3 6 6 1-4-3 1-4-3

4 4 5 5 1-4 1-4

5 5 8 8 1-4-3- 1-4-3-55

6 6 11 11 1-4-3-5-6 1-4-3-5-6

7 7 13 13 1-4-3-5-6-7 1-4-3-5-6-7


Minimal Spanning Tree ProblemMinimal Spanning Tree Problem

A A treetree is a set of connected arcs that does not is a set of connected arcs that does not form a cycle.form a cycle.

A A spanning treespanning tree is a tree that connects all nodes is a tree that connects all nodes of a network.of a network.

The The minimal spanning tree problemminimal spanning tree problem seeks to seeks to determine the minimum sum of arc lengths determine the minimum sum of arc lengths necessary to connect all nodes in a network. necessary to connect all nodes in a network.

The criterion to be minimized in the minimal The criterion to be minimized in the minimal spanning tree problem is not limited to distance spanning tree problem is not limited to distance even though the term "closest" is used in even though the term "closest" is used in describing the procedure. Other criteria include describing the procedure. Other criteria include time and cost. (Neither time nor cost are time and cost. (Neither time nor cost are necessarily linearly related to distance.)necessarily linearly related to distance.)


Minimal Spanning Tree AlgorithmMinimal Spanning Tree Algorithm

Step 1:Step 1: Arbitrarily begin at any node and Arbitrarily begin at any node and connect it to the closest node. The two nodes connect it to the closest node. The two nodes are referred to as connected nodes, and the are referred to as connected nodes, and the remaining nodes are referred to as unconnected remaining nodes are referred to as unconnected nodes.nodes.

Step 2:Step 2: Identify the unconnected node that is Identify the unconnected node that is closest to one of the connected nodes (break ties closest to one of the connected nodes (break ties arbitrarily). Add this new node to the set of arbitrarily). Add this new node to the set of connected nodes. Repeat this step until all connected nodes. Repeat this step until all nodes have been connected.nodes have been connected.

Note:Note: A problem with A problem with nn nodes to be connected nodes to be connected will require will require nn - 1 iterations of the above steps. - 1 iterations of the above steps.


Example: Minimal Spanning TreeExample: Minimal Spanning Tree

Find the Minimal Spanning Tree:Find the Minimal Spanning Tree:

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Iteration 1:Iteration 1: Arbitrarily selecting node 1, we see Arbitrarily selecting node 1, we see that its closest node is node 2 (distance = 30). that its closest node is node 2 (distance = 30). Therefore, initially we have: Therefore, initially we have:

Connected nodes: 1,2 Connected nodes: 1,2 Unconnected nodes: 3,4,5,6,7,8,9,10Unconnected nodes: 3,4,5,6,7,8,9,10Chosen arcs: 1-2Chosen arcs: 1-2

Iteration 2:Iteration 2: The closest unconnected node to a The closest unconnected node to a connected node is node 5 (distance = 25 to node connected node is node 5 (distance = 25 to node 2). Node 5 becomes a connected node. 2). Node 5 becomes a connected node.

Connected nodes: 1,2,5 Connected nodes: 1,2,5 Unconnected nodes: 3,4,6,7,8,9,10Unconnected nodes: 3,4,6,7,8,9,10

Chosen arcs: 1-2, 2-5Chosen arcs: 1-2, 2-5




Connected nodes: 1,2,5,7 Connected nodes: 1,2,5,7 Unconnected nodes: 3,4,6,8,9,10Unconnected nodes: 3,4,6,8,9,10Chosen arcs: 1-2, 2-5, 5-7Chosen arcs: 1-2, 2-5, 5-7

Iteration 4:Iteration 4: The closest unconnected node to a The closest unconnected node to a connected node is node 10 (distance = 20 to connected node is node 10 (distance = 20 to node 7). Node 10 becomes a connected node. node 7). Node 10 becomes a connected node.

Connected nodes: 1,2,5,7,10 Connected nodes: 1,2,5,7,10 Unconnected nodes: 3,4,6,8,9Unconnected nodes: 3,4,6,8,9

Chosen arcs: 1-2, 2-5, 5-7, 7-10Chosen arcs: 1-2, 2-5, 5-7, 7-10




Connected nodes: 1,2,5,7,10,8 Connected nodes: 1,2,5,7,10,8 Unconnected nodes: 3,4,6,9Unconnected nodes: 3,4,6,9

Chosen arcs: 1-2, 2-5, 5-7, 7-10, 10-8Chosen arcs: 1-2, 2-5, 5-7, 7-10, 10-8


Connected nodes: 1,2,5,7,10,8,6 Connected nodes: 1,2,5,7,10,8,6 Unconnected nodes: 3,4,9Unconnected nodes: 3,4,9

Chosen arcs: 1-2, 2-5, 5-7, 7-10, 10-8, 10-Chosen arcs: 1-2, 2-5, 5-7, 7-10, 10-8, 10-66



Iteration 7:Iteration 7: The closest unconnected node to a The closest unconnected node to a connected node is node 3 (distance = 20 to node 6). connected node is node 3 (distance = 20 to node 6). Node 3 becomes a connected node. Node 3 becomes a connected node.

Connected nodes: 1,2,5,7,10,8,6,3 Connected nodes: 1,2,5,7,10,8,6,3 Unconnected nodes: 4,9Unconnected nodes: 4,9

Chosen arcs: 1-2, 2-5, 5-7, 7-10, 10-8, 10-6, Chosen arcs: 1-2, 2-5, 5-7, 7-10, 10-8, 10-6, 6-36-3

Iteration 8:Iteration 8: The closest unconnected node to a The closest unconnected node to a connected node is node 9 (distance = 30 to node 6). connected node is node 9 (distance = 30 to node 6). Node 9 becomes a connected node. Node 9 becomes a connected node.

Connected nodes: 1,2,5,7,10,8,6,3,9 Connected nodes: 1,2,5,7,10,8,6,3,9 Unconnected nodes: 4Unconnected nodes: 4

Chosen arcs: 1-2, 2-5, 5-7, 7-10, 10-8, Chosen arcs: 1-2, 2-5, 5-7, 7-10, 10-8, 10-6, 6-3, 6-910-6, 6-3, 6-9



Iteration 9:Iteration 9: The only remaining unconnected The only remaining unconnected node is node 4. It is closest to connected node 6 node is node 4. It is closest to connected node 6 (distance = 45). (distance = 45).

Thus, the minimal spanning tree (displayed on Thus, the minimal spanning tree (displayed on the next slide) consists of: the next slide) consists of:

Arcs: 1-2, 2-5, 5-7, 7-10, 10-8, 10-6, 6-3, Arcs: 1-2, 2-5, 5-7, 7-10, 10-8, 10-6, 6-3, 6-9, 6-46-9, 6-4

Values: 30 + 25 + 15 + 20 + 25 + 35 + 20 Values: 30 + 25 + 15 + 20 + 25 + 35 + 20 + 30 + 45+ 30 + 45

= 245= 245



Optimal Spanning TreeOptimal Spanning Tree

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Maximal Flow ProblemMaximal Flow Problem

The The maximal flow problemmaximal flow problem is concerned with is concerned with determining the maximal volume of flow from determining the maximal volume of flow from one node (called the source) to another node one node (called the source) to another node (called the sink). (called the sink).

In the maximal flow problem, each arc has a In the maximal flow problem, each arc has a maximum maximum arc flow capacityarc flow capacity which limits the flow which limits the flow through the arc.through the arc.


Example: Maximal FlowExample: Maximal Flow

Network ModelNetwork Model

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SinkSinkSourceSource



A A capacitated transshipment modelcapacitated transshipment model (Chapter 7) (Chapter 7) can be developed for the maximal flow problem.can be developed for the maximal flow problem.

We will add an arc from node 7 back to node 1 to We will add an arc from node 7 back to node 1 to represent the total flow through the network.represent the total flow through the network.

There is no capacity on the newly added 7-1 arc.There is no capacity on the newly added 7-1 arc. We want to maximize the flow over the 7-1 arc.We want to maximize the flow over the 7-1 arc.



Modified Network ModelModified Network Model

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LP FormulationLP Formulation

(as Capacitated Transshipment Problem)(as Capacitated Transshipment Problem)

•There is a There is a variablevariable for every arc. for every arc.

•There is a There is a constraintconstraint for every node; the flow for every node; the flow out must equal the flow in.out must equal the flow in.

•There is a There is a constraintconstraint for every arc (except the for every arc (except the added sink-to-source arc); arc capacity cannot added sink-to-source arc); arc capacity cannot be exceeded.be exceeded.

•The The objectiveobjective is to is to maximize the flow over the maximize the flow over the added, sink-to-source arc.added, sink-to-source arc.




(as Capacitated Transshipment Problem)(as Capacitated Transshipment Problem)

Max Max xxkk11 ( (kk is sink node, 1 is source node) is sink node, 1 is source node)

s.t. s.t. xxijij - - xxjiji = 0 (conservation of = 0 (conservation of flow) flow) ii jj

xxijij << ccijij ( (ccijij is capacity of is capacity of ijij arc)arc)

xxijij >> 0, for all 0, for all ii and and jj (non-negativity) (non-negativity)

(x(xijij represents the flow from node represents the flow from node ii to node to node j)j)




•18 variables (for 17 original arcs and 1 added 18 variables (for 17 original arcs and 1 added arc)arc)

•24 constraints24 constraints 7 node flow-conservation constraints7 node flow-conservation constraints 17 arc capacity constraints (for original arcs)17 arc capacity constraints (for original arcs)




•Objective FunctionObjective Function

Max Max xx7171

•Node Flow-Conservation ConstraintsNode Flow-Conservation Constraints

xx7171 - - xx1212 - - xx1313 - - xx1414 = 0 (flow in & out of node 1) = 0 (flow in & out of node 1)

xx1212 + + xx4242 + + xx5252 – – xx2424 - - xx2525 = 0 (node 2) = 0 (node 2)

xx1313 + + xx4343 – – xx3434 – – xx3636 = 0 (etc.) = 0 (etc.)

xx1414 + + xx2424 + + xx3434 + + xx5454 + + xx6464 – – xx4242 - - xx4343 - - xx4545 - - xx4646 - - xx4747 = 0 = 0

xx2525 + + xx4545 – – xx5252 – – xx5454 - - xx5757 = 0 = 0

xx3636 + + xx4646 - - xx6464 - - xx6767 = 0 = 0

xx4747 + + xx5757 + + xx6767 - - xx7171 = 0 = 0



LP Formulation (continued)LP Formulation (continued)

•Arc Capacity ConstraintsArc Capacity Constraints

xx1212 << 4 4 xx1313 << 3 3 xx1414 << 4 4

xx2424 << 2 2 xx2525 << 3 3

xx3434 << 3 3 xx3636 << 6 6

xx4242 << 3 3 xx4343 << 5 5 xx4545 << 3 3 xx4646 << 1 1 xx4747 << 3 3

xx5252 << 5 5 xx5454 << 5 5 xx5757 << 5 5

xx6464 << 5 5 xx6767 << 5 5



Optimal SolutionOptimal Solution

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HomeworkHomework

Solve the following Problems:Solve the following Problems:

9- 89- 8 9- 99- 9 9- 149- 14 9- 179- 17 9- 189- 18


End of Chapter 9End of Chapter 9

ch 9 network

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