1. Chapter 5: Transportation, Assignment and Network Models © 2007 Pearson Education

2. Network Flow Models Consist of a network that can be represented with nodes and arcs 1.Transportation Model 2.Transshipment Model 3.Assignment Model 4.Maximal Flow Model 5.Shortest Path Model 6.Minimal Spanning Tree Model

3. Characteristics of Network Models A node is a specific location An arc connects 2 nodes Arcs can be 1-way or 2-way

4. Types of Nodes Origin nodes Destination nodes Transshipment nodes Decision Variables X AB = amount of flow (or shipment) from node A to node B

5. Flow Balance at Each Node (total inflow) – (total outflow) = Net flow Node TypeNet Flow Origin< 0 Destination> 0 Transshipment= 0

6. The Transportation Model Decision: How much to ship from each origin to each destination? Objective: Minimize shipping cost

7. Data Decision Variables X ij = number of desks shipped from factory i to warehouse j

8. Objective Function: (in $ of transportation cost) Min 5X DA + 4X DB + 3X DC + 8X EA + 4X EB + 3X EC + 9X FA + 7X FB + 5X FC Subject to the constraints: Flow Balance For Each Supply Node (inflow) - (outflow) = Net flow - (X DA + X DB + X DC ) = -100 (Des Moines) OR X DA + X DB + X DC = 100 (Des Moines)

9. Other Supply Nodes X EA + X EB + X EC = 300 (Evansville) X FA + X FB + X FC = 300 (Fort Lauderdale) Flow Balance For Each Demand Node X DA + X EA + X FA = 300 (Albuquerque) X DB + X EB + X FB = 200 (Boston) X DC + X EC + X FC = 200 (Cleveland) Go to File 5-1.xls

10. Unbalanced Transportation Model If (Total Supply) > (Total Demand), then for each supply node: (outflow) < (supply) If (Total Supply) < (Total Demand), then for each demand node: (inflow) < (demand)

11. Transportation Models With Max-Min and Min-Max Objectives Max-Min means maximize the smallest decision variable Min-Max mean to minimize the largest decision variable Both reduce the variability among the X ij values Go to File 5-3.xls

12. The Transshipment Model Similar to a transportation model Have “Transshipment” nodes with both inflow and outflow Node TypeFlow Balance Net Flow (RHS) Supplyinflow < outflowNegative Demandinflow > outflowPositive Transshipmentinflow = outflowZero

13. Revised Transportation Cost Data Note: Evansville is both an origin and a destination

14. Objective Function: (in $ of transportation cost) Min 5X DA + 4X DB + 3X DC + 2X DE + 3X EA + 2X EB + 1X EC + 9X FA + 7X FB + 5X FC + 2X FE Subject to the constraints: Supply Nodes (with outflow only) - (X DA + X DB + X DC + X DE ) = -100 (Des Moines) - (X FA + X FB + X FC + X FE ) = -300 (Ft Lauderdale)

15. Evansville (a supply node with inflow) (X DE + X FE ) – (X EA + X EB + X EC ) = -300 Demand Nodes X DA + X EA + X FA = 300 (Albuquerque) X DB + X EB + X FB = 200 (Boston) X DC + X EC + X FC = 200 (Cleveland) Go to File 5-4.xls

16. Assignment Model For making one-to-one assignments Such as: –People to tasks –Classes to classrooms –Etc.

17. Fit-it Shop Assignment Example Have 3 workers and 3 repair projects Decision: Which worker to assign to which project? Objective: Minimize cost in wages to get all 3 projects done

18. Estimated Wages Cost of Possible Assignments

19. Can be Represented as a Network Model The “flow” on each arc is either 0 or 1

20. Decision Variables X ij = 1 if worker i is assigned to project j 0 otherwise Objective Function(in $ of wage cost) Min 11X A1 + 14X A2 + 6X A3 + 8X B1 + 10X B2 + 11X B3 + 9X C1 + 12X C2 + 7X C3 Subject to the constraints: (see next slide)

21. One Project Per Worker (supply nodes) - (X A1 + X A2 + X A3 ) = -1(Adams) - (X B1 + X B2 + X B3 ) = -1(Brown) - (X C1 + X C2 + X C3 ) = -1(Cooper) One Worker Per Project (demand nodes) X A1 + X B1 + X C1 = 1(project 1) X A2 + X B2 + X C2 = 1(project 2) X A3 + X B3 + X C3 = 1(project 3) Go to File 5-5.xls

22. The Maximal-Flow Model Where networks have arcs with limited capacity, such as roads or pipelines Decision: How much flow on each arc? Objective: Maximize flow through the network from an origin to a destination

23. Road Network Example Need 2 arcs for 2-way streets

24. Modified Road Network

25. Decision Variables X ij = number of cars per hour flowing from node i to node j Dummy Arc The X 61 arc was created as a “dummy” arc to measure the total flow from node 1 to node 6

26. Objective Function Max X 61 Subject to the constraints: Flow Balance At Each Node Node (X 61 + X 21 ) – (X 12 + X 13 + X 14 ) = 0 1 (X 12 + X 42 + X 62 ) – (X 21 + X 24 + X 26 ) = 0 2 (X 13 + X 43 + X 53 ) – (X 34 + X 35 )= 0 3 (X 14 + X 24 + X 34 + X 64 )–(X 42 + X 43 + X 46 ) = 0 4 (X 35 ) – (X 53 + X 56 ) = 0 5 (X 26 + X 46 + X 56 ) – (X 61 + X 62 + X 64 ) = 0 6

27. Flow Capacity Limit On Each Arc X ij < capacity of arc ij Go to File 5-6.xls

28. The Shortest Path Model For determining the shortest distance to travel through a network to go from an origin to a destination Decision: Which arcs to travel on? Objective: Minimize the distance (or time) from the origin to the destination

29. Ray Design Inc. Example Want to find the shortest path from the factory to the warehouse Supply of 1 at factory Demand of 1 at warehouse

30. Decision Variables X ij = flow from node i to node j Note: “flow” on arc ij will be 1 if arc ij is used, and 0 if not used Roads are bi-directional, so the 9 roads require 18 decision variables

31. Objective Function (in distance) Min 100X 12 + 200X 13 + 100X 21 + 50X 23 + 200X 24 + 100X 25 + 200X 31 + 50X 32 + 40X 35 + 200X 42 + 150X 45 + 100X 46 + 40X 53 + 100X 52 + 150X 54 + 100X 56 + 100X 64 + 100X 65 Subject to the constraints: (see next slide)

32. Flow Balance For Each Node Node (X 21 + X 31 ) – (X 12 + X 13 ) = -1 1 (X 12 +X 32 +X 42 +X 52 )–(X 21 +X 23 +X 24 +X 25 )=0 2 (X 13 + X 23 + X 53 ) – (X 31 + X 32 + X 35 ) = 0 3 (X 24 + X 54 + X 64 ) – (X 42 + X 45 + X 46 ) = 0 4 (X 25 +X 35 +X 45 +X 65 )–(X 52 +X 53 +X 54 +X 56 )=0 5 (X 46 + X 56 ) – (X 64 + X 65 ) = 1 6 Go to file 5-7.xls

33. Minimal Spanning Tree For connecting all nodes with a minimum total distance Decision: Which arcs to choose to connect all nodes? Objective: Minimize the total distance of the arcs chosen

34. Lauderdale Construction Example Building a network of water pipes to supply water to 8 houses (distance in hundreds of feet)

35. Characteristics of Minimal Spanning Tree Problems Nodes are not pre-specified as origins or destinations So we do not formulate as LP model Instead there is a solution procedure

36. Steps for Solving Minimal Spanning Tree 1.Select any node 2.Connect this node to its nearest node 3.Find the nearest unconnected node and connect it to the tree (if there is a tie, select one arbitrarily) 4.Repeat step 3 until all nodes are connected

37. Steps 1 and 2 Starting arbitrarily with node (house) 1, the closest node is node 3

38. Second and Third Iterations

39. Fourth and Fifth Iterations

40. Sixth and Seventh Iterations After all nodes (homes) are connected the total distance is 16 or 1,600 feet of water pipe