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OR/MA 504Chapter 4Network Modeling

Introduction to Mathematical Programming

Introduction

- A large variety of linear programming applications can be represented graphically as networks.
- This chapter focuses on several such problems:
- Transshipment Problems
- Shortest Path Problems
- Maximal Flow Problems
- Transportation/Assignment Problems
- Generalized Network Flow Problems
- The Minimum Spanning Tree Problem

Network Flow Problem Characteristics

- Network flow problems can be represented as a collection of nodes connected by arcs.
- There are three types of nodes:
- Supply
- Demand
- Transshipment
- We’ll use negative numbers to represent supplies and positive numbers to represent demand.

A Transshipment Problem:The Bavarian Motor Company

+100

$30

Boston

$50

2

-200

Newark

1

Columbus

+60

$40

3

$40

$35

Richmond

$30

+80

Atlanta

4

+170

5

$25

$50

$45

$35

Mobile

+70

J\'ville

-300

6

$50

7

Defining the Decision Variables

For each arc in a network flow model

we define a decision variable as:

Xij = the amount being shipped (or flowing) from node ito node j

For example…

X12 = the # of cars shipped from node 1 (Newark) to node 2 (Boston)

X56 = the # of cars shipped from node 5 (Atlanta) to node 6 (Mobile)

Note: The number of arcs determines the number of variables!

Defining the Objective Function

Minimize total shipping costs.

MIN: 30X12 + 40X14 + 50X23 + 35X35

+40X53 + 30X54 + 35X56 + 25X65

+ 50X74 + 45X75 + 50X76

Constraints for Network Flow Problems:The Balance-of-Flow Rules

For Minimum Cost Network Apply This Balance-of-Flow

Flow Problems Where: Rule At Each Node:

Total Supply > Total Demand Inflow-Outflow >= Supply or Demand

Total Supply < Total Demand Inflow-Outflow <=Supply or Demand

Total Supply = Total Demand Inflow-Outflow = Supply or Demand

Defining the Constraints

- In the BMC problem:

Total Supply = 500 cars

Total Demand = 480 cars

- For each node we need a constraint like this:

Inflow - Outflow >= Supply or Demand

- Constraint for node 1:

–X12 – X14 >= – 200(Note: there is no inflow for node 1!)

- This is equivalent to:

+X12 + X14 <= 200

(Supply >= Demand)

Defining the Constraints

- Flow constraints

–X12 – X14 >= –200 } node 1

+X12 – X23 >= +100 } node 2

+X23 + X53 – X35 >= +60 } node 3

+ X14 + X54 + X74 >= +80 } node 4

+ X35 + X65 + X75 – X53 – X54 – X56 >= +170} node 5

+ X56 + X76 – X65 >= +70 } node 6

–X74 – X75 – X76 >= –300 } node 7

- Nonnegativity conditions

Xij >= 0 for allij

Implementing the Model

See file Fig4-1.xls

20

80

40

210

70

Optimal Solution to the BMC Problem+100

$30

Boston

$50

2

-200

Newark

1

Columbus

+60

$40

3

$40

Richmond

+80

Atlanta

4

+170

5

$45

Mobile

+70

J\'ville

-300

6

$50

7

The Shortest Path Problem

- Many decision problems boil down to determining the shortest (or least costly) route or path through a network.
- Ex. Emergency Vehicle Routing
- This is a special case of a transshipment problem where:
- There is one supply node with a supply of -1
- There is one demand node with a demand of +1
- All other nodes have supply/demand of +0

The American Car Association

+0

3.3 hrs

+1

L\'burg

5 pts

9

Va Bch

11

5.0 hrs

9 pts

2.0 hrs

4 pts

4.7 hrs

2.7 hrs

+0

9 pts

+0

4 pts

1.1 hrs

K\'ville

3 pts

G\'boro

Raleigh

5

2.0hrs

3.0 hrs

8

9 pts

10

4 pts

+0

1.7 hrs

5 pts

A\'ville

1.5 hrs

+0

6

2.3 hrs

3 pts

+0

3 pts

Chatt.

2.8 hrs

7 pts

3

Charl.

2.0 hrs

7

8 pts

+0

1.7 hrs

3.0 hrs

4 pts

4 pts

1.5 hrs

G\'ville

2 pts

4

Atlanta

+0

B\'ham

2.5 hrs

2

3 pts

1

2.5 hrs

+0

3 pts

-1

Solving the Problem

- There are two possible objectives for this problem
- Finding the quickest route (minimizing travel time)
- Finding the most scenic route (maximizing the scenic rating points)

See file Fig4-2.xls

The Equipment Replacement Problem

- The problem of determining when to replace equipment is another common business problem.
- It can also be modeled as a shortest path problem…

The Compu-Train Company

- Compu-Train provides hands-on software training.
- Computers must be replaced at least every two years.
- Two lease contracts are being considered:
- Each requires $62,000 initially
- Contract 1:
- Prices increase 6% per year
- 60% trade-in for 1 year old equipment
- 15% trade-in for 2 year old equipment
- Contract 2:
- Prices increase 2% per year
- 30% trade-in for 1 year old equipment
- 10% trade-in for 2 year old equipment

+0

$63,985

4

2

$30,231

$33,968

$28,520

$32,045

+1

5

1

3

-1

$67,824

$60,363

+0

Network for Contract 1Cost of trading after 1 year: 1.06*$62,000 - 0.6*$62,000 = $28,520

Cost of trading after 2 years: 1.062*$62,000 - 0.15*$62,000 = $60,363

Cost of trading in year 2 after trading in year 1:

1.062*$62,000 – 0.6(1.06*$62,000) = $69,663 - $39,432 = $30,231

Solving the Problem

See file Fig4-3.xls

Plants

Groves

Distances (in miles)

Supply

Capacity

21

Mt. Dora

Ocala

200,000

275,000

1

4

50

40

35

30

Eustis

Orlando

600,000

400,000

2

5

22

55

20

Clermont

Leesburg

225,000

300,000

3

6

25

Transportation & Assignment Problems- Some network flow problems don’t have trans-shipment nodes; only supply and demand nodes.

These problems are implemented more effectively using the technique described in Chapter 2.

Generalized Network Flow Problems

- In some problems, a gain or loss occurs in flows over arcs.
- Examples
- Oil or gas shipped through a leaky pipeline
- Imperfections in raw materials entering a production process
- Spoilage of food items during transit
- Theft during transit
- Interest or dividends on investments
- These problems require some modeling changes.

Coal Bank Hollow Recycling

Process 1 Process 2

Material Cost Yield Cost Yield Supply

Newspaper $13 90% $12 85% 70 tons

Mixed Paper $11 80% $13 85% 50 tons

White Office Paper $9 95% $10 90% 30 tons

Cardboard $13 75% $14 85% 40 tons

Newsprint Packaging Paper Print Stock

Pulp Source Cost Yield Cost Yield Cost Yield

Recycling Process 1 $5 95% $6 90% $8 90%

Recycling Process 2 $6 90% $8 95% $7 95%

Demand 60 tons 40 tons 50 tons

Network for Recycling ProblemPrint

-70

Newspaper

$13

1

$12

Newsprint

95%

+0

+60

90%

pulp

7

$5

80%

$11

Recycling

90%

Mixed

-50

Process 1

paper

$6

95%

5

2

$13

$8

90%

75%

Packing

+40

paper

pulp

$9

8

95%

85%

White

85%

-30

office

$6

paper

$10

Recycling

90%

3

$8

90%

Process 2

$7

6

+50

stock

85%

95%

$13

pulp

+0

9

-40

Cardboard

$14

4

Defining the Objective Function

Minimize total cost.

MIN: 13X15 + 12X16 + 11X25 + 13X26

+ 9X35+ 10X36 + 13X45 + 14X46 + 5X57 + 6X58 + 8X59 + 6X67 + 8X68 + 7X69

Defining the Constraints-I

- Raw Materials

-X15 -X16 >= -70 } node 1

-X25 -X26 >= -50 } node 2

-X35 -X36 >= -30 } node 3

-X45 -X46 >= -40 } node 4

Defining the Constraints-II

- Recycling Processes

+0.9X15+0.8X25+0.95X35+0.75X45- X57- X58-X59 >= 0 } node 5

+0.85X16+0.85X26+0.9X36+0.85X46-X67-X68-X69 >= 0 } node 6

Defining the Constraints-III

- Paper Pulp

+0.95X57 + 0.90X67 >= 60 } node 7

+0.90X57 + 0.95X67 >= 40 } node 8

+0.90X57 + 0.95X67 >= 50 } node 9

Implementing the Model

See file Fig4-4.xls

Important Modeling Point

- In generalized network flow problems, gains and/or losses associated with flows across each arc effectively increase and/or decrease the available supply.
- This can make it difficult to tell if the total supply is adequate to meet the total demand.
- When in doubt, it is best to assume the total supply is capable of satisfying the total demand and use Solver to prove (or refute) this assumption.

The Maximal Flow Problem

- In some network problems, the objective is to determine the maximum amount of flow that can occur through a network.
- The arcs in these problems have upper and lower flow limits.
- Examples
- How much water can flow through a network of pipes?
- How many cars can travel through a network of streets?

The Northwest Petroleum Company

Pumping

Pumping

Station 3

Station 1

3

4

2

6

2

6

Oil Field

1

6

Refinery

2

4

4

5

3

5

Pumping

Pumping

Station 2

Station 4

Max Flow Problem Set-Up

- Solve as transshipment problem:
- Add return arc from ending node to the starting node
- Assign demand of 0 to all nodes in network
- Maximize flow over the return arc

The Northwest Petroleum Company

Pumping

Pumping

Station 3

Station 1

+0

+0

3

4

2

6

2

6

+0

+0

Oil Field

1

6

Refinery

2

4

4

5

3

+0

5

+0

Pumping

Pumping

Station 2

Station 4

Formulation of the Max Flow Problem

MAX: X61

Subject to: +X61 - X12 - X13 = 0

+X12 - X24 - X25 = 0

+X13 - X34 - X35 = 0

+X24 + X34 - X46 = 0

+X25 + X35 - X56 = 0

+X46 + X56 - X61 = 0

with the following bounds on the decision variables:

0 <= X12 <= 6 0 <= X25 <= 2 0 <= X46 <= 6

0 <= X13 <= 4 0 <= X34 <= 2 0 <= X56 <= 4

0 <= X24 <= 3 0 <= X35 <= 5 0 <= X61 <= inf

Implementing the Model

See file Fig4-5.xls

Optimal Solution

Pumping

Pumping

Station 3

Station 1

3

3

4

2

5

6

5

2

2

6

Oil Field

1

6

Refinery

2

4

2

4

4

4

5

3

5

2

Pumping

Pumping

Station 2

Station 4

Special Modeling Considerations:Flow Aggregation

+0

$3

$5

-100

1

3

5

+75

$3

$4

$5

$4

2

4

6

+50

-100

$5

$6

+0

Suppose the total flow into nodes 3 & 4 must be at least 50 and 60, respectively. How would you model this?

Special Modeling Considerations:Flow Aggregation

+0

+0

$3

$5

-100

1

30

3

5

+75

L.B.=50

$4

$3

$4

$5

2

40

4

6

+50

-100

L.B.=60

$5

$6

+0

+0

Nodes 30 & 40 aggregate the total flow into nodes 3 & 4, respectively.

Special Modeling Considerations:Multiple Arcs Between Nodes

$8

-75

1

2

+50

$6

U.B. = 35

Two two (or more) arcs cannot share the same beginning and ending nodes. Instead, try...

+0

10

$0

$8

$6

1

2

+50

-75

U.B. = 35

Special Modeling Considerations:Capacity Restrictions on Total Supply

+75

-100

$5, UB=40

Supply exceeds demand, but the upper bounds prevent the demand from being met.

1

3

$4, UB=30

$6, UB=35

2

4

$3, UB=35

+80

-100

Special Modeling Considerations:Capacity Restrictions on Total Supply

Now demand exceeds supply. As much “real” demand as possible will be met in the least costly way.

+75

-100

$5, UB=40

1

3

$999, UB=100

+200

$4, UB=30

0

$6, UB=35

2

4

$999, UB=100

$3, UB=35

+80

-100

The Minimal Spanning Tree Problem

- For a network with n nodes, a spanning tree is a set of n-1 arcs that connects all the nodes and contains no loops.
- The minimal spanning tree problem involves determining the set of arcs that connects all the nodes at minimum cost.

Minimal Spanning Tree Example:Windstar Aerospace Company

$150

4

2

$100

$85

$75

$40

1

$80

5

$85

$90

3

$50

6

$65

Nodes represent computers in a local area network.

The Minimal Spanning Tree Algorithm

1. Select any node. Call this the current subnetwork.

2. Add to the current subnetwork the cheapest arc that connects any node within the current subnetwork to any node not in the current subnetwork. (Ties for the cheapest arc can be broken arbitrarily.) Call this the current subnetwork.

3.If all the nodes are in the subnetwork, stop; this is the optimal solution. Otherwise, return to step 2.

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