lecture 8 sep23

8/3/2019 Lecture 8 Sep23

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QMB 4701:

Managerial Operations Analysis I

NETWORK MODELING


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LAST CLASS

Network flow problems

y Three types of nodes

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

ito

nodej

y Numberof decision variables

y Numberofconstraints

Minimumcost networkflow problems

y If Total Supply > Total Demand

y Then Inflow-Outflow >= Supply or Demand

Excel:

y SUMIF(range,criteria,sum_range)2


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THE SHORTEST PATHPROBLEM

Many decision problems boil down to determining theshortest (or least costly) route or path through a

network.

y Ex. Emergency Vehicle Routing

This is a special case of a transshipment problem where:y There is one supply node with a supply of -1

y There is one demand node with a demand of +1

y All other nodes have supply/demand of +0


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THEAMERICAN CAR

ASSOCIATION

B'hamAtlanta

G'ville

Va Bch

Charl.

L'burg

K'ville

A'ville

G'boro Raliegh

Chatt.

12

3

4

6

5

7

8

9

10

11

2.5 hrs

3 pts

3.0 hrs4 pts

1.7 hrs4 pts

2.5 hrs

3 pts

1.7 hrs5 pts

2.8 hrs7 pts

2.0 hrs8 pts

1.5 hrs2 pts

2.0 hrs

9 pts

5.0 hrs9 pts

3.0 hrs4 pts

4.7 hrs

9 pts

1.5 hrs3 pts 2.3 hrs

3 pts

1.1 hrs3 pts

2.0 hrs4 pts

2.7 hrs4 pts

3.3 hrs5 pts

-1

+1

+0

+0

+0

+0

+0

+0

+0

+0

+0

4


5/31

SOLVING THEPROBLEM

There are two possible objectives for this problem

y Finding the quickest route (minimizing travel time)

y Finding the most scenic route (maximizing the scenic

rating points)

Define the decision Variable

Xij =1,chooseto leave nodei fornodej

0, dontchoosethe arc from nodei fornodej

Numberof decision variables?


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DEFINING THE OBJECTIVE FUNCTION

Min-Time objective function

Min 2.5 X12 +3 X13 +1.7 X23 +2.5X24

+1.7X35+2.8X36+2X46+1.5X47++2X56 +5X59 +3X68+4.7X69+1.5 X78

+2.3 X7,10+2X89+1.1 X8,10+3.3 X9,11

+2.7X10,11

Each flowhasits own costcoefficient.

6


7/31

DEFINING THE CONSTRAINTS

Flow constraintsX12 X13 = 1 } node 1

+X12 X23 X24 = 0 } node 2

+X13 + X23 X35 X36 = 0 } node 3

+ X24 X46 X47 = 0 } node 4

+ X35 X56 X59 = 0 } node 5

+ X36 + X46 + X56 X68 X69 = 0 } node 6

+X47 X78 X7,10 = 0 } node 7

+ X68 + X78 X89 X8,10 = 0 } node 8

+ X59 + X69 + X89X9,11 = 0 } node 9

+X7,10 + X8,10 X10,11 = 0 } node 10

+X9,11 + X10,11 = 1 } node 11

Nonnegativity conditions

Xij >= 0for all ij

7


8/31

IMPLEMENT

See file Fig5-7

8


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FORMULATE THE OBJECTIVE FUNCTION

WHEN YOU WANT TO MAXIMIZE THE SCENICSCORES

Max scenic score objective function

Max 3X12 +4X13 +4X23 +3X24

+5X35+7X36+8X46+2X47+

+9X56 +9X59 +4X68+9X69+3X78

+3X7,10+4X89+3X8,10+5X9,11+4X10,11

9


10/31

CAN YOU DRAW THE NETWORK OF

MIN-TIME PROBLEM

10B'hamAtlanta

G'ville

Va Bch

Charl.

Raliegh

12

4

7

10

11

Total driving time = 11.5


11/31

CAN YOU DRAW THE NETWORK OF

MAX-SCEN

IC-SCORE

PROBLE

M

11

Total scenic score = 35

B'hamAtlanta

Va Bch

L'burg

K'ville

A'ville

Chatt.

12

3

6

5

9

11


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PRACTICE QUESTIONS

Find the quickest route but with total 18 scenic

scores.

y Total driving time = 11.8

Find the most scenic route but with no more than13-hr total travel time.

y Total scenic score = 23

12


13/31

SUMMARY OF SHORTEST PATH PROBLEM

1. One starting point (supply, with flow=1), oneending point (demand, with flow=+1), othernodes have net flow=0.

2. Objective: It can be min total shipping cost,

min travel time, or max scenic scores, etc.3. Constraints: For each node, (inflow outflow)

has to satisfy the demand or supplyrequirement. Other constraints may apply.

4. You want to find a path optimizing your

objective.5. You should know how to draw the result

network diagram from Excel solution.

13


14/31

THE EQUIPMENT REPLACEMENT PROBLEM

THE COMPU-TRAIN COMPANY

The problem of determining when to replace

equipment is another common business problem.

Compu-Train provides hands-on software training.

Computers must be replaced at least every twoyears.

It can also be modeled as a shortest path problem

14


15/31

THE COMPU-TRAIN COMPANY

Two lease contracts are being considered:y Each equipment requires $62,000 initially

y Contract 1:

Prices increase 6% per year

60% trade-in for 1 year old equipment15% trade-in for 2 year old equipment

y Contract 2:

Prices increase 2% per year

30% trade-in for 1 year old equipment

10% trade-in for 2 year old equipment

You want to minimize total cost

Each contract can be modeled as a shortest pathproblem 15


16/31

NETWORK FOR CONTRACT 1

1 3 5

2 4

-1 +1

+0

+0 +0

$28,520

$60,363

$30,231

$63,985

$32,045

$67,824

$33,968

y Net cost = Initial investment trade-in valuey Cost of trading after 1 year: 1.06*$62,000-0.6*$62,000 = $28,520

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

y etc, etc.16


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SOLVING THE PROBLEM

See data file Fig5-12

Tips:

For each contract, you formulate a LPminimization problem in solver. Then you compare

the costs in two contracts.

17


18/31

TRANSPORTATION & ASSIGNMENT PROBLEMS

Some network flow problems dont have transshipment nodes;only supply and demand nodes.

Mt. Dora1

Eustis

2

Clermont

3

Ocala1

Orlando

2

Leesburg

3

Distances (in miles)CapacitySupply

275,000

400,000

300,000 225,000

600,000

200,000

GrovesProcessing

Plants

21

50

40

3530

22

55

25

20

18


19/31

FORMULATION

Let Xij be the amount from grove i to plant j

19

11 12 13 21 22 23 31 32 33

11 12 13

21 22 23

31 32 33

11 21 31

12 22 32

13 23

min 21 50 40 35 30 22 55 20 25

s.t. 275

400

300

200

600

x x x x x x x x x

x x x K

x x x K

x x x K

x x x K

x x x K

x x x

e

e

e

e

e

33 225

0, , 1, 2,3ij

K

x i j

e

u !

These problems are implemented more effectively in a matrix format

in Chapter 3.


20/31

TRANSPORTATION & ASSIGNMENT PROBLEMS

What if the problem is not fully interconnected?

Mt. Dora

1

Eustis

2

Clermont

3

Ocala

1

Orlando

2

Leesburg

3

Distances (in miles)CapacitySupply

275,000

400,000

300,000 225,000

600,000

200,000

GrovesProcessing

Plants

21

50

30

22

55

25

20

20


21/31

GENERALIZED NETWORKFLOW PROBLEMS

In some problems, a gain or loss occurs in flows

over arcs.

y Oil or gas shipped through a leaky pipeline

y

Imperfections in raw materials entering a productionprocess

y Spoilage of food items during transit

y Theft during transit

y Interest or dividends on investments

These problems require some modeling changes.

21


22/31

COAL BANKHOLLOW RECYCLING

A firm is doing business from transferring from

paper materials to pulp.

There are four materials, two processors, and three

types of pulps. The yields for each of processing materials and

making pulps are less than one.

The firm wants to find the best processes to satisfy

the requirement and minimize the cost.

22


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COAL BANKHOLLOW RECYCLING

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

Process 1 Process 2

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%

Newsprint Packaging Paper Print Stock

Demand 60 tons 40 tons 50 tons

23


24/31

NETWORK FOR RECYCLING PROBLEM

Newspaper

1

Mixed

paper

2

3

Cardboard

4

RecyclingProcess 1

5

6

Newsprint

pulp

7

Packingpaperpulp

8

Print

stock

pulp

9

-70

-50

-30

-40

+60

+40

+50

Whiteofficepaper

Recycling

Process 2

$13

$12

$11

$13

$9

$10

$14

$13

90%

80%

95%

75%

85%85%

90%

85%

$5

$6

$8

$6

$7

$8

95%

90%

90%

90%

95%

95%+0

+0

24


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DEFINING THE DECISIONVARIABLE AND

OBJECTIVE FUNCTION

M

inimize total cost.

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

+ 9X35+ 10X36 + 13X45 + 14X46 + 5X57

+ 6X58 + 8X59 + 6X67 + 8X68 + 7X69

25

Xij= # tons of paper shipping from node i to node j


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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

26


27/31

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

27


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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

28


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IMPLEMENTING THE MODEL

See data file Fig 5-17

Draw the result network diagram.

29


30/31

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 supplyis 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.

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PRACTICE: ASSIGNMENT PROBLEM

ASSIGNING SCHOOL BUSES TO ROUTS

Comp

any

Route

1

2 3 4 5 6 7 8

1 8200 7800 5400 3900

2 7800 8200 6300 3300 4900

3 4800 4400 5600 3600

4 8000 5000 6800 6700 4200

5 7200 6400 3900 6400 2800 3000

6 7000 5800 7500 4500 5600 6000 4200

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Objective: To use a network model to assign companies to

bus routes so that each route is covered at minimum cost

and no company is assigned to more than two routes

lecture 8 sep23

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