chapter 5: transportation, assignment and network models

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

1. Transportation Model2. Transshipment Model3. Assignment Model4. Maximal Flow Model5. Shortest Path Model6. Minimal Spanning Tree Model

Characteristics of Network Models

• A node is a specific location• An arc connects 2 nodes• Arcs can be 1-way or 2-way

Types of Nodes• Origin nodes• Destination nodes• Transshipment nodes

Decision VariablesXAB = amount of flow (or shipment) from

node A to node B

Flow Balance at Each Node

(total inflow) – (total outflow) = Net flow

Node Type Net FlowOrigin < 0

Destination > 0Transshipment = 0

The Transportation ModelDecision: How much to ship from each

origin to each destination?

Objective: Minimize shipping cost

Data

Decision VariablesXij = 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 NodesXEA + 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)

Transportation Models WithMax-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 Xij values

Go to File 5-3.xls

The Transshipment Model• Similar to a transportation model• Have “Transshipment” nodes with both inflow

and outflow

Node Type Flow BalanceNet Flow

(RHS)Supply inflow < outflow Negative

Demand inflow > outflow PositiveTransshipment inflow = outflow Zero

Revised Transportation Cost Data

Note: Evansville is both an origin and a destination

Objective Function: (in $ of transportation cost)

Min 5XDA + 4XDB + 3XDC + 2XDE + 3XEA + 2XEB + 1XEC + 9XFA + 7XFB + 5XFC + 2XFE

Subject to the 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) – (XEA + XEB + XEC) = -300

Demand NodesXDA + XEA + XFA = 300 (Albuquerque)

XDB + XEB + XFB = 200 (Boston)

XDC + XEC + XFC = 200 (Cleveland)

Go to File 5-4.xls

Assignment Model

• For making one-to-one assignments• Such as:

– People to tasks– Classes to classrooms– Etc.

Fit-it Shop Assignment ExampleHave 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

Estimated Wages Cost of Possible Assignments

Can be Represented as a Network Model

The “flow” on each arc is either 0 or 1

Decision Variables

Xij = 1 if worker i is assigned to project j

0 otherwise

Objective Function (in $ of wage cost)Min 11XA1 + 14XA2 + 6XA3 + 8XB1 + 10XB2 +

11XB3 + 9XC1 + 12XC2 + 7XC3

Subject to the constraints:(see next slide)

One Project Per Worker (supply nodes)- (XA1 + XA2 + XA3) = -1 (Adams)- (XB1 + XB2 + XB3) = -1 (Brown)- (XC1 + XC2 + XC3) = -1 (Cooper)

One Worker Per Project (demand nodes)XA1 + XB1 + XC1 = 1 (project 1)XA2 + XB2 + XC2 = 1 (project 2)XA3 + XB3 + XC3 = 1 (project 3)

Go to File 5-5.xls

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

Road Network Example

Need 2 arcs for 2-way streets

Modified Road Network

Decision Variables

Xij = number of cars per hour flowing from node i to node j

Dummy ArcThe X61 arc was created as a “dummy” arc

to measure the total flow from node 1 to node 6

Objective FunctionMax X61

Subject to the constraints:

Flow Balance At Each Node Node

(X61 + X21) – (X12 + X13 + X14) = 0 1(X12 + X42 + X62) – (X21 + X24 + X26) = 0 2 (X13 + X43 + X53) – (X34 + X35) = 0 3(X14+ X24 + X34 + X64)–(X42+ X43 + X46) = 0 4(X35) – (X53 + X56) = 0 5(X26 + X46 + X56) – (X61 + X62 + X64) = 0 6

Flow Capacity Limit On Each Arc

Xij < capacity of arc ij

Go to File 5-6.xls

The Shortest Path ModelFor 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 1 at factory• Demand of 1 at warehouse

Decision Variables

Xij = 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

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)

Flow Balance For Each Node Node

(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

Go to file 5-7.xls

Minimal Spanning TreeFor 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 8 houses (distance in hundreds of feet)

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

Steps for Solving Minimal Spanning Tree

1. Select any node2. Connect this node to its nearest node3. 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

Steps 1 and 2Starting arbitrarily with node (house) 1, the

closest node is node 3

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