1. Network Models Tran Van Hoai Faculty of Computer Science & Engineering HCMC University of Technology 2010-20111Tran Van Hoai

2. Harmful Waste Collection at HCM city Industrial zone Processing Factory Industrial zone Processing Factory 2010-20112Tran Van Hoai

3. Typical route Depot 2010-20113Tran Van Hoai

4. Solution 2010-2011Tran Van Hoai4 HIGHLY COMPLEX PRACTICAL ISSUES

5. Problem Constraints Vehicle’s capacity Customer’s Time Window Conflict Harmful Waste cannot transport in the same vehicles Maximum time for a route … 2010-20115Tran Van Hoai

6. 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 … 2010-20116Tran Van Hoai

7. Delivery route without optimization 2010-2011Tran Van Hoai7 STRATEGY: GO TO THE NEAREST LOCATION FIRST

8. Delivery route with optimization 2010-2011Tran Van Hoai8 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

9. Networks Nodes – Microchips, cities, TV stations,… Arcs – Wires, roads, satellite transmission,… Functions (defining resource) – Resource: electrical current, delivery trucks, TV program,…) 2010-2011Tran Van Hoai9 Network = - A set of nodes - A set of arcs (connecting nodes) - Functions defined on nodes & arcs Network = - A set of nodes - A set of arcs (connecting nodes) - Functions defined on nodes & arcs

10. 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 2010-2011Tran Van Hoai10

11. Classification (2) Network connectivity models – Link all nodes together – Examples: Traveling salesman models Minimal spanning tree models 2010-2011Tran Van Hoai11 Flow models can be modeled as LP (although they are ILP) Connectivity models cannot modeled as LP Flow models can be modeled as LP (although they are ILP) Connectivity models cannot modeled as LP

12. Terminology (1) 2010-2011Tran Van Hoai12 ij FLOW X ij CAPACITY U ij Decision variable ij Directed arc ij Undirected arc

13. Terminology (2) 2010-2011Tran Van Hoai13 13 2 5 4 7 6 Path Cycle

14. Terminology (3) 2010-2011Tran Van Hoai14 13 2 5 4 7 6 Tree Spanning tree

15. Transportation model 2010-2011Tran Van Hoai15 - m sources - Supply resource at source S i - n destinations -Demands for resource at destination D i - Unit shipping cost C ij between i & j - m sources - Supply resource at source S i - n destinations -Demands for resource at destination D i - Unit shipping cost C ij between i & j GOAL: minimize total shipping cost

16. Carlton Pharmaceutical transportation network 2010-2011Tran Van Hoai16 5 4 7 6 Distribution warehouses 1 3 2 Production plants S 1 =1200 S 2 =1000 S 3 =800 D 1 =1100 D 2 =400 D 3 =750 D 4 =750 35 30 40 32 37 40 42 25 40 15 20 28

17. Assumptions (simplification) Constant per item shipping cost All shipping performed simultaneously (within fixed time frame) Vaccine only shipped from source to destination 2010-2011Tran Van Hoai17

18. Formulation MIN S.T. ≤ = 2010-2011Tran Van Hoai18 X ij : shipment from i (1,…,3) to j (4,…,7) 12 integer variables Complexity increases quickly when number sources (destinations) increases

19. Practical issues Blocked routes – X ij = 0 means no vaccine assigned to route i to j – Or …. Minimum/maximum shipments – L ij ≤ Xij ≤ U ij Production planning can be considered as transportation model 2010-2011Tran Van Hoai19

20. Capacitated transshipment networks 2010-2011Tran Van Hoai20 5 4 7 6 Distribution warehouses 1 3 2 Production plants S 1 =1200 S 2 =1000 S 3 =800 D 1 =1100 D 2 =400 D 3 =750 D 4 =750 35 30 40 32 37 40 42 25 40 15 20 28 Intermediate nodes (no supply, no demand)

21. Capacitated transshipment 2010-2011Tran Van Hoai21 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 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) GOAL: minimize total shipping cost (capacitated transshipment = general network model)

22. Depot Max 2010-2011Tran Van Hoai22 Alexan dria Chevy chase Fairfax Geroge town Fall Church Bethes da Supply nodes Transshipment nodes Demand nodes S 1 =10 S 2 =17 D 5 =12 D 6 =13 $15 10 $15 17 $11 7 $7 5 8 $10 12 $6 7 $5 3 $20 6