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PPT – Mathematical models of networks give us algorithms so computationally efficient that we can employ them to evaluate problems too big to be solved any other way. PowerPoint presentation | free to download - id: 6b915d-M2JjZ

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Chapter 13

- Mathematical models of networks give us

algorithms so computationally efficient that we

can employ them to evaluate problems too big to

be solved any other way. - Network Models

The Structure of Network Problems

- Networks have features of the following arcs

(e.g., roads), nodes (e.g., cities), and arc

values representing distance or flow.

Types of Networks and Applications

- Networks have been used in the following.
- There are four basic network models.
- Shortest route problems.
- Minimal spanning trees.
- Maximum flow.
- Minimum-cost maximum flow.

The Shortest Route Problem

- This problem uses the network as prop. We find

the shortest route from A to G. - The START node is evaluated first (shaded). Then

all direct links between evaluated and

unevaluated nodes are identified and the

distances back to START are computed along

connecting arcs. - The node with smallest distance, written above

it, becomes the next evaluated one and is shaded. - An arrow points from there along the connecting

arc.

The Shortest Route Problem

- Node C joins the evaluated set. An arrow is

added pointing back to A, and the cumulative

distance, 3, back to START from C is entered

above. - The process continues.
- Node B is next to join the evaluated set, at a

distance of 4 back to START.

The Shortest Route Problem

- The process continues.
- Node D is next to join the evaluated set with a

distance of 5 back to START.

The Shortest Route Problem

- The process continues.
- Node E is next to join the evaluated set with a

distance of 7 back to START.

The Shortest Route Problem

- The process continues.
- Two nodes, F and G, are next to join the

evaluated set, each with with a distance of 9

back to START.

The Shortest Route Problem

- All nodes are evaluated. The shortest route is

found by tracing back from FINISH following the

arrows. - The shortest route from A to G is
- A-B-D-E-G for a distance of C 9.

The Minimal Spanning Tree

- A tree is a set of arcs connecting nodes in such

a way that only one route involving those arcs

connects any two nodes. - Imagine an ant on a real tree. It has just one

way to walk from any leaf to another. - A spanning tree connects with all nodes.
- It is like railroad tracks connecting all cities,

but with only one routing between any two. - A minimal spanning tree has the smallest sum of

its arc distances C (tree size). - In connecting all circuit-board solder points

with gold wire, it would use the least gold. - It would have the least tracks for a railway.

Finding the Minimal Spanning Tree

- As first connected node pick any (here A). Find

all arcs directly joining connected to

unconnected. Join the shortest arc to tree. - Connected nodes are shaded. A-C joins the tree.

Finding the Minimal Spanning Tree

- As new arcs join the tree, more nodes become

connected. We consider only arcs directly

joining connected to unconnected nodes. - Arc A-B joins the tree.

Finding the Minimal Spanning Tree

- The process continues.
- Arc C-F joins the tree.

Finding the Minimal Spanning Tree

- The process continues.
- Arcs B-E, D-E, F-J, H-I, and I-J join tree.

Finding the Minimal Spanning Tree

- The process continues.
- Arc G-H joins the tree. Since all nodes are

connected, the tree has been found. The sum of

the arc lengths gives its size C.

Maximizing Flow

- Arcs in a maximum flow problem are directed and

have upper bounds. Flow moves one way. - A node is designated as the SOURCE and another as

the SINK. - Flow-augmenting paths from SOURCE to SINK are

found and flows sent over the arcs. If no path

can be found, flow is maximized. - A flow-augmenting path ordinarily involves arcs

directed away from the SOURCE toward the SINK. - But an arc can point in the opposite direction if

some of its current flow would be reduced and be

redirected to another arc. - It doesnt matter which path is used. The

possibilities shrink as more are found. - Flow into an interior node must equal the flow

out.

Maximizing Flow

- The bottleneck arc on a path has the lowest

remaining capacity. Here it is H-J.

Maximizing Flow

- Arc H-J is saturated. Flow over saturated arcs

may be decreased only. - The next paths bottleneck arc is E-I.

Maximizing Flow

- The next path has two bottleneck arcs C-F and

K-L.

Maximizing Flow

- The next path has two bottleneck arcs D-F and

I-J.

Maximizing Flow

- This path goes against the direction of J-K

flow. Some J-K flow is redirected over J-L. - The bottlenecks are B-D and I-K.

Maximizing Flow

- There are no more flow-augmenting paths. The

optimal solution has been found with maximum flow

(sum into SINK) of C 14.

Minimum-Cost Maximum Flow

- Transportation problems are special cases of

minimum-cost maximum flow problems. - The general problem has bounded arcs (routes) and

is represented as a network. - It may be solved by an elaborate procedure, the

out-of-kilter algorithm, involving shortest

routes and maximum flows. - However, it is best solved on the computer.
- QuickQuant may be used for this purpose.
- It can perform the out-of-kilter algorithm.
- The problem can also be solved as a general

linear program (with QuickQuant or Excel).

Solving with QuickQuant

- The following first iteration involves a smaller

version of the problem in the text.

Solving with QuickQuant

- The initial solution is infeasible. A series of

iterations yields the optimal solution.

NetworkTemplates

- shortest route
- maximum flow
- minimum cost maximum flow

Shortest Route for Yellow Jacket Freightways

(Figure 13-11)

This is the upper portion of Figure 13-11. The

lower portion is shown next.

1. Enter the problem name in B3.

3. If 1000 is not large enough to denote the

impossibility of going between two cities, use a

larger number.

2. Enter the distances above the diagonal in the

table B9H15. They will automatically be entered

below the diagonal.

Shortest Route for Yellow Jacket Freightways

(Figure 13-11)

This is the lower portion of Figure 13-11.

The length of the shortest route is in cell E9.

Here it is 9.

The shortest route is found from the table in

cells A20H27. Here it is A-B-D-E-G.

Shortest Route for Yellow Jacket Freightways

(Figure 13-11)

This is the lower portion of Figure 13-11.

4. Click on Tools and use Solver to find the

shortest route. The Solver Parameters dialog box

is shown on the next slide.

5. The starting point is assumed to be A and the

ending point G. If this is different adjust the

required flow in cells B31H31 accordingly.

6. For problems with more than 7 cities, expand

the distance and path tables and check to make

sure that all the formulas have the proper ranges.

Solver Parameters Dialog Box (Figure 13-12)

1. Enter the value of the objective function,

E17, in the Target Cell line, either with or

without the signs.

NOTE Normally all these entries appear in the

Solver Parameter dialog box so you only need to

click on the Solve button. However, you should

always check to make sure the entries are correct

for the problem you are solving.

2. The Target Cell is to be minimized so click

on Min in the Equal To line.

3. Enter the decision variables in the By

Changing Cells line, B21H27.

4. The constraints are entered in the Subject to

Constraints box by using the Add Constraints

dialog box shown next (obtained by clicking on

the Add button). If a constraint needs to be

changed, click on the Change button. The Change

and Add Constraint dialog box function in the

same manner.

The Add Constraint Dialog Box

Normally, all these entries already appear. You

will need to use this dialog box only if you need

to add a constraint.

3. Enter the required flow B31H31 in the

Constraint line (or B31H31).

1. Enter the net flows B30H30 (or B30H30)

in the Cell Reference line.

4. Click the OK button.

If you need to change a constraint, the Change

Constraint dialog box functions just like this

one.

2. Enter as the sign because the net flow must

be equal to the required flow, given next in Step

3.

Maximum Flow for Lulliput Telephone Company

(Figure 13-31)

This is the upper portion of Figure 13-31. The

lower portion is shown next.

1. Enter the problem name in B3.

2. (a) Enter the capacities in the table B9M20.

2. (b) A big number is entered for the upper

limit on the return flow from L to A.

Maximum Flow for Lulliput Telephone Company

(Figure 13-31)

This is the lower portion of Figure 13-31.

The maximum flow is in cell E22. Here it is 14.

The flow along each arc is found from the table

in cells A20H27. For example, cell C26 has a 1

in it. This means one unit of flow goes from A

to B.

Maximum Flow for Lulliput Telephone Company

(Figure 13-31)

This is the lower portion of Figure 13-31.

3.Click on Tools and use Solver to find the

maximum flow. The Solver Parameters dialog box is

shown on the next slide.

4. The starting point is the first node and the

ending point the last one.

5. For problems with more than 12 nodes, expand

the capacities and flows tables and check to make

sure that all the formulas have the proper ranges.

Solver Parameters Dialog Box (Figure 13-32)

1. Enter the value of the objective function,

E22, in the Target Cell line, either with or

without the signs.

NOTE Normally all these entries appear in the

Solver Parameter dialog box so you only need to

click on the Solve button. However, you should

always check to make sure the entries are correct

for the problem you are solving.

2. The Target Cell is to be maximized so click

on Max in the Equal To line.

3. Enter the decision variables in the By

Changing Cells line, B26M37.

4. The constraints are entered in the Subject to

Constraints box by using the Add Constraints

dialog box (obtained by clicking on the Add

button) as was done for the shortest route

template. If a constraint needs to be changed,

click on the Change button. The Change and Add

Constraint dialog box function in the same manner.

Minimum Cost Maximum Flow for BigCo (Figure

13-40)

This is the upper portion of Figure 13-40. The

lower portion is shown next.

1. Enter the problem name in B3.

2. Enter the costs and capacities in the table

B8G10 and the corresponding From and To names in

cells A8A9 and B7F7.

3. Enter the minimum quantities in the table

B15F16.

4. Enter the maximum quantities in the table

B21F22.

Minimum Cost Maximum Flow for BigCo (Figure

13-40)

This is the lower portion of Figure 13-40.

The minimum cost is in cell E24. Here it is

12,110.

The optimal shipping schedule is given in the

table in cells A28G29.

Minimum Cost Maximum Flow for BigCo (Figure

13-40)

This is the lower portion of Figure 13-40.

5.Click on Tools and use Solver to find the

optimal solution. The Solver Parameters dialog

box is shown on the next slide.

6. For other problems insert (or delete) the

appropriate number of rows or columns and check

to make sure that all the formulas have the

proper ranges.

Solver Parameters Dialog Box (Figure 13-41)

1. Enter the value of the objective function,

E24, in the Target Cell line, either with or

without the signs.

NOTE Normally all these entries appear in the

Solver Parameter dialog box so you only need to

click on the Solve button. However, you should

always check to make sure the entries are correct

for the problem you are solving.

2. The Target Cell is to be minimized so click

on Min in the Equal To line.

3. Enter the decision variables in the By

Changing Cells line, B27F28.

4. The constraints are entered in the Subject to

Constraints box by using the Add Constraints

dialog box (obtained by clicking on the Add

button) as was done for the shortest route

template. If a constraint needs to be changed,

click on the Change button. The Change and Add

Constraint dialog box function in the same manner.