Tran Van Hoai 1

Network Models

Tran Van HoaiFaculty of Computer Science & Engineering

HCMC University of Technology

2010-2011

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Harmful Waste Collection at HCM city

Industrial zoneProcessing Factory

Industrial zone

Industrial zone Industrial zoneProcessing Factory2010-2011

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Typical route

Depot

2010-2011

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Solution

2010-2011

HIGHLY COMPLEX

PRACTICAL ISSUES

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Problem Constraints

• Vehicle’s capacity• Customer’s Time Window• Conflict Harmful Waste cannot transport in

the same vehicles• Maximum time for a route• …

2010-2011

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Objectives• Minimize cost travel• Minimize the number of vehicles• Balance workload among the vehicles• Minimize waiting time needed to

serve customers in their required hours

• Satisfy service requirements• …

2010-2011

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Delivery route without optimization

2010-2011

STRATEGY: GO TO THE NEAREST LOCATION FIRST

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Delivery route with optimization

2010-2011

Practical problems are much more difficult• Traffic jam (time-dependence)• Delivery time (time-window)• Carrier capacity (space-dependence)• Precedence constraint• …

Traffic jam from 6:30am to 9am

Delivery time from 9am to 10am

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Networks

• Nodes– Microchips, cities, TV stations,…

• Arcs– Wires, roads, satellite transmission,…

• Functions (defining resource)– Resource: electrical current, delivery trucks, TV program,…)

2010-2011

Network =- A set of nodes- A set of arcs (connecting nodes)- Functions defined on nodes & arcs

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Classification (1)

• Network flow models– Delivery of goods or resource from supply nodes,

thru intermediate nodes, to demand nodes– Examples: • Transportation models• Capacitated transshipment models• Assignment models• Shortest path models• Maximum flow models

2010-2011

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Classification (2)

• Network connectivity models– Link all nodes together– Examples: • Traveling salesman models• Minimal spanning tree models

2010-2011

Flow models can be modeled as LP (although they are ILP)

Connectivity models cannot modeled as LP

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Terminology (1)

2010-2011

i jFLOW Xij

CAPACITY Uij

Decisionvariable

i jDirected arc

i jUndirected arc

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Terminology (2)

2010-2011

1 3

25

4

7

6Path

Cycle

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Terminology (3)

2010-2011

1 3

25

4

7

6Tree

Spanning tree

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Transportation model

2010-2011

- m sources- Supply resource at source Si

- n destinations- Demands for resource at destination Di

- Unit shipping cost Cij between i & j

GOAL: minimize total shipping cost

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Carlton Pharmaceutical transportation network

2010-2011

5

4

7

6

Distributionwarehouses

1

3

2

Productionplants

S1=1200

S2=1000

S3=800

D1=1100

D2=400

D3=750

D4=750

3530

4032

3740

4225

4015

2028

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Assumptions (simplification)

• Constant per item shipping cost• All shipping performed simultaneously (within

fixed time frame)• Vaccine only shipped from source to

destination

2010-2011

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FormulationMIN <Total shipping cost>S.T. <Amount shipped from each

source>≤ <Supply at that source>

<Amount received at each destination>

= <Demand at that destination>

<No negative shipments>

2010-2011

• Xij: shipment from i (1,…,3) to j (4,…,7)• 12 integer variables

• Complexity increases quickly when number sources (destinations) increases

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Practical issues

• Blocked routes– Xij = 0 means no vaccine assigned to route i to j– Or ….

• Minimum/maximum shipments– Lij ≤ Xij ≤ Uij

• Production planning can be considered as transportation model

2010-2011

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Capacitated transshipment networks

2010-2011

5

4

7

6

Distributionwarehouses

1

3

2

Productionplants

S1=1200

S2=1000

S3=800

D1=1100

D2=400

D3=750

D4=750

3530

4032

3740

4225

4015

2028

Intermediate nodes(no supply, no demand)

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Capacitated transshipment

2010-2011

Constraints:- supply node: net flow out (flow out – flow in) not exceed its supply

- intermediate node: net flow out = 0- demand node: net flow out = - demand

GOAL: minimize total shipping cost(capacitated transshipment = general network model)

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Depot Max

2010-2011

Alexandria

Chevy chase

Fairfax

Gerogetown

Fall Church

Bethesda

Supply nodes

Transshipment nodes

Demandnodes

S1=10

S2=17

D5=12

D6=13$1510

$1517

$117

$7 5

$78

$1012

$6 7

$53

$206