1 1 Slides by John Loucks St. Edwards University Modifications by A. Asef-Vaziri

2 2 Shortest-Route Problem n The shortest-route problem is concerned with finding the shortest path in a network from one node (or set of nodes) to another node (or set of nodes). n If all arcs in the network have nonnegative values then a labeling algorithm can be used to find the shortest paths from a particular node to all other nodes in the network. n The criterion to be minimized in the shortest-route problem is not limited to distance even though the term "shortest" is used in describing the procedure. Other criteria include time and cost. (Neither time nor cost are necessarily linearly related to distance.)

3 3 n Linear Programming Formulation Using the notation: Using the notation: x ij = 1 if the arc from node i to node j x ij = 1 if the arc from node i to node j is on the shortest route is on the shortest route 0 otherwise 0 otherwise c ij = distance, time, or cost associated c ij = distance, time, or cost associated with the arc from node i to node j with the arc from node i to node j continued Shortest-Route Problem

4 4 n Linear Programming Formulation (continued) Shortest-Route Problem

5 5 Susan Winslow has an important business meeting Susan Winslow has an important business meeting in Paducah this evening. She has a number of alternate routes by which she can travel from the company headquarters in Lewisburg to Paducah. The network of alternate routes and their respective travel time, ticket cost, and transport mode appear on the next two slides. If Susan earns a wage of $15 per hour, what route If Susan earns a wage of $15 per hour, what route should she take to minimize the total travel cost? Example: Shortest Route

6 6 6 A B C D E F G H I J K L M Paducah Lewisburg n Network Representation

7 7 Example: Shortest Route Transport Time Time Ticket Total Transport Time Time Ticket Total Route Mode (hours) Cost Cost Cost 1-2Train 4 $60 $ 20 $ Train 4 $60 $ 20 $ Bus 2 $30 $ 10 $ Bus 2 $30 $ 10 $ Train 3 $50 $ 30 $ Train 3 $50 $ 30 $ Plane 1 $15 $115 $ Plane 1 $15 $115 $ Taxi 6 $90 $ 90 $ Taxi 6 $90 $ 90 $ Bus 3 $45 $ 15 $ Bus 3 $45 $ 15 $ Taxi 3 $50 $ 50 $ Taxi 3 $50 $ 50 $ Taxi 1 $15 $ 15 $ Taxi 1 $15 $ 15 $ Bus 4 $70 $ 20 $ Bus 4 $70 $ 20 $ Bus 6 $95 $ 25 $ Bus 6 $95 $ 25 $ Train 2 $35 $ 15 $ Train 2 $35 $ 15 $ Bus 4 $70 $ 20 $ Bus 4 $70 $ 20 $ Train 1 $20 $ 10 $ Train 1 $20 $ 10 $ 30

8 8 Example: Shortest Route n LP Formulation Objective Function Objective Function Min 80 x x x x x x 25 Min 80 x x x x x x x x x x x x x x x x x x x x x x x x x x x x 56 Node Flow-Conservation Constraints Node Flow-Conservation Constraints x 12 + x 13 + x 14 + x 15 + x 16 = 1 (origin) x 12 + x 13 + x 14 + x 15 + x 16 = 1 (origin) – x 12 + x 25 + x 26 – x 52 = 0 (node 2) – x 12 + x 25 + x 26 – x 52 = 0 (node 2) – x 13 + x 34 + x 35 + x 36 – x 43 – x 53 = 0 (node 3) – x 13 + x 34 + x 35 + x 36 – x 43 – x 53 = 0 (node 3) – x 14 – x 34 + x 43 + x 45 + x 46 – x 54 = 0 (node 4) – x 14 – x 34 + x 43 + x 45 + x 46 – x 54 = 0 (node 4) – x 15 – x 25 – x 35 – x 45 + x 52 + x 53 + x 54 + x 56 = 0 (node 5) – x 15 – x 25 – x 35 – x 45 + x 52 + x 53 + x 54 + x 56 = 0 (node 5) x 16 + x 26 + x 36 + x 46 + x 56 = 1 (destination) x 16 + x 26 + x 36 + x 46 + x 56 = 1 (destination)

9 9 Excel Solution