Network Models Tran Van Hoai Faculty of Computer Science & Engineering HCMC University of Technology Tran Van Hoai

Harmful Waste Collection at HCM city Industrial zone Processing Factory Industrial zone Processing Factory Tran Van Hoai

Typical route Depot Tran Van Hoai

Solution Tran Van Hoai4 HIGHLY COMPLEX PRACTICAL ISSUES

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

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 … Tran Van Hoai

Delivery route without optimization Tran Van Hoai7 STRATEGY: GO TO THE NEAREST LOCATION FIRST

Delivery route with optimization Tran 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

Networks Nodes – Microchips, cities, TV stations,… Arcs – Wires, roads, satellite transmission,… Functions (defining resource) – Resource: electrical current, delivery trucks, TV program,…) Tran 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

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 Tran Van Hoai10

Classification (2) Network connectivity models – Link all nodes together – Examples: Traveling salesman models Minimal spanning tree models Tran 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

Terminology (1) Tran Van Hoai12 ij FLOW X ij CAPACITY U ij Decision variable ij Directed arc ij Undirected arc

Terminology (2) Tran Van Hoai Path Cycle

Terminology (3) Tran Van Hoai Tree Spanning tree

Transportation model Tran 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

Carlton Pharmaceutical transportation network Tran Van Hoai Distribution warehouses Production plants S 1 =1200 S 2 =1000 S 3 =800 D 1 =1100 D 2 =400 D 3 =750 D 4 =

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

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

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 Tran Van Hoai19

Capacitated transshipment networks Tran Van Hoai Distribution warehouses Production plants S 1 =1200 S 2 =1000 S 3 =800 D 1 =1100 D 2 =400 D 3 =750 D 4 = Intermediate nodes (no supply, no demand)

Capacitated transshipment Tran 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)

Depot Max Tran 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