NETWORK FLOW MODELS

Shortest Route ProblemSRP DefinitionDirected network G = (N, A)Arc length (cost): arc (i,j) ANode s as sourceFind the shortest length from node s to node IThe length = sum of the edges on the path 3Problem: Determine the shortest routes from the origin to all destinations.

Shipping Routes from Los AngelesExample Problem 4Solution MethodDijkstras AlgorithmSelect the node with the shortest direct route from the origin.Establish a permanent set with the origin node and the node that was selected in step 1.Determine all nodes directly connected to the permanent set nodes.Select the node with the shortest route (branch) from the group of nodes directly connected to the permanent set nodes.Repeat steps 3 and 4 until all nodes have joined the permanent set. 5

Network of Shipping RoutesThe Shortest Route ProblemDefinition and Example Problem Data (2 of 2) 6

Network with Node 1 in the Permanent SetThe Shortest Route ProblemSolution Approach (1 of 8)Determine the initial shortest route from the origin (node 1) to the closest node (3). 7

Network with Nodes 1 and 3 in the Permanent SetThe Shortest Route ProblemSolution Approach (2 of 8)Determine all nodes directly connected to the permanent set.

8Network with Nodes 1, 2, and 3 in the Permanent SetRedefine the permanent set.

The Shortest Route ProblemSolution Approach (3 of 8) 9

Network with Nodes 1, 2, 3, and 4 in the Permanent SetThe Shortest Route ProblemSolution Approach (4 of 8)Continue 10The Shortest Route ProblemSolution Approach (5 of 8)

Network with Nodes 1, 2, 3, 4, and 6 in the Permanent SetContinue

11The Shortest Route ProblemSolution Approach (6 of 8)

Network with Nodes 1, 2, 3, 4, 5, and 6 in the Permanent SetContinue 12

The Shortest Route ProblemSolution Approach (7 of 8)Network with Optimal Routes from Los Angeles to All DestinationsOptimal Solution 13Shortest Travel Time from Origin to Each Destination

The Shortest Route ProblemSolution Approach (8 of 8)Solution SummaryUsing LP for SPPWant to find the shortest path from the factory to the warehouseSupply of 1 at factoryDemand of 1 at warehouse

Decision VariablesXij = 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

Shortest Path Problem LP Formulation

Minimize

s.t.Total CostExample

Objective Function (in distance)Min 100X12 + 200X13 + 100X21 + 50X23 + 200X24 + 100X25 + 200X31 + 50X32 + 40X35 + 200X42 + 150X45 + 100X46 + 40X53 + 100X52 + 150X54 + 100X56 + 100X64 + 100X65

Subject to the constraints:

(see next slide)

Constraints: Flow Balance For Each NodeNode(X21 + X31) (X12 + X13) = -1 1(X12+X32+X42+X52)(X21+X23+X24+X25)=0 2(X13 + X23 + X53) (X31 + X32 + X35) = 0 3(X24 + X54 + X64) (X42 + X45 + X46) = 0 4(X25+X35+X45+X65)(X52+X53+X54+X56)=0 5(X46 + X56) (X64 + X65) = 1 6