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