Chapter 5: Transportation, Assignment and Network Models © 2007 Pearson Education Network Flow Models Consist of a network that can be represented with nodes and arcs Transportation Model Transshipment Model Assignment Model Maximal Flow Model Shortest Path Model Minimal Spanning Tree Model Characteristics of Network Models • • • A node is a specific location An arc connects nodes Arcs can be 1-way or 2-way • • • Types of Nodes Origin nodes Destination nodes Transshipment nodes Decision Variables XAB = amount of flow (or shipment) from node A to node B Flow Balance at Each Node (total inflow) – (total outflow) = Net flow Node Type Origin Destination Transshipment Net Flow 0 =0 The Transportation Model Decision: How much to ship from each destination? Objective: Minimize shipping cost origin to each Data Decision Variables Xij = number of desks shipped from factory i to warehouse j Objective Function: (in $ of transportation cost) Min 5XDA + 4XDB + 3XDC + 8XEA + 4XEB + 3XEC + 9XFA + 7XFB + 5XFC Subject to the constraints: Flow Balance For Each Supply Node (inflow) - (outflow) = Net flow - (XDA + XDB + XDC) = -100 (Des Moines) OR XDA + XDB + XDC = 100 (Des Moines) Other Supply Nodes XEA + XEB + XEC = 300 (Evansville) XFA + XFB + XFC = 300 (Fort Lauderdale) Flow Balance For Each Demand Node XDA + XEA + XFA = 300 (Albuquerque) XDB + XEB + XFB = 200 (Boston) XDC + XEC + XFC = 200 (Cleveland) Go to File 5-1.xls 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) Objective Function Max X61 Subject to the constraints: Flow Balance At Each Node Node (X61 + X21) – (X12 + X13 + X14) =0 (X12 + X42 + X62) – (X21 + X24 + X26) =0 (X13 + X43 + X53) – (X34 + X35) =0 (X14+ X24 + X34 + X64)–(X42+ X43 + X46) =0 (X35) – (X53 + X56) (X26 + X46 + X56) – (X61 + X62 + X64) =0 =0 Flow Capacity Limit On Each Arc Xij < capacity of arc ij Go to File 5-6.xls 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 Ray Design Inc Example • • • Want to find the shortest path from the factory to the warehouse Supply of at factory Demand of at warehouse Decision Variables Xij = flow from node i to node j Note: “flow” on arc ij will be if arc ij is used, and if not used Roads are bi-directional, so the roads require 18 decision variables Objective Function (in distance) Min 100X12 + 200X13 + 100X21 + 50X23 + 200X24 + 100X25 + 200X31 + 50X32 + 200X42 + 150X45 + 100X46 + 40X53 + 100X52 + 150X54 + 100X56 + + 100X65 Subject to the constraints: (see next slide) 40X35 + 100X64 Flow Balance For Each Node Node (X21 + X31) – (X12 + X13) = -1 (X12+X32+X42+X52)–(X21+X23+X24+X25)=0 (X13 + X23 + X53) – (X31 + X32 + X35) =0 (X24 + X54 + X64) – (X42 + X45 + X46) =0 (X25+X35+X45+X65)–(X52+X53+X54+X56)=0 (X46 + X56) – (X64 + X65) =1 Go to file 5-7.xls 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 Lauderdale Construction Example Building a network of water pipes to supply water to houses (distance in hundreds of feet) Characteristics of Minimal Spanning Tree Problems • • • Nodes are not pre-specified as origins or destinations So we not formulate as LP model Instead there is a solution procedure Steps for Solving Minimal Spanning Tree Select any node Connect this node to its nearest node Find the nearest unconnected node and connect it to the tree (if there is a tie, select one arbitrarily) Repeat step until all nodes are connected Steps and Starting arbitrarily with node (house) 1, the closest node is node Second and Third Iterations Fourth and Fifth Iterations Sixth and Seventh Iterations After all nodes (homes) are connected the total distance is 16 or 1,600 feet of water pipe ... (demand) 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... Net Flow 0 =0 The Transportation Model Decision: How much to ship from each destination? Objective: Minimize shipping cost origin to each Data Decision Variables Xij = number of desks shipped... constraints: Supply Nodes (with outflow only) - (XDA + XDB + XDC + XDE) = -100 (Des Moines) - (XFA + XFB + XFC + XFE) = -300 (Ft Lauderdale) Evansville (a supply node with inflow) (XDE + XFE) –