OPS 301 Module C Transportation Models and Network Models

1
OPS 301Module CTransportation Modelsand
Network Models

• Dr. Steven Harrod

2
Topics

• Definition of Transportation Models
• Transportation
• Assignment
• Transshipment
• Formulation Tips
• Word Problem Practice
• CAUTION Significant material not in your text!

3
Network Models

• Model the flow or transportation of items as
  arrows between nodes or locations.
• Can be formulated as a linear program
• Related to graph theory in mathematics

4
Three Models

• Transportation Model
• Minimize cost of transportation
• Simple origin to destination transportation
• Transshipment Model
• Add intermediate junctions to transportation
• Flow conservation constraints at junctions
• Assignment Model
• Minimize cost, time, etc. of resource assignment
• Binary variables (0,1 integer)

5
Transportation Model

• A special class of linear programming
• Procedure that finds the least costly means of
  moving products from a series of sources to a
  series of destinations
• Can be used to help resolve distribution and
  location decisions

6
Need to Know

• The origin points and the capacity or supply at
  each
• The destination points and the demand per period
  at each
• The cost of shipping one unit from each origin to
  each destination

7
Transportation Problem
Figure C.1
8
Example Cost Data
Table C.1
9
Formulate

• What is the goal?
• Serve all customers at minimum cost
• What do we control?
• Quantities shipped on each route
• What are the limits?
• Quantities available at sources
• Quantities demanded at destinations

10
Variables

• Amount shipped from origin to destination
• Designate with serial index

11
Objective Function

• Minimize
• 5 X1 4 X2 3 X3
• 8 X4 4 X5 3 X6
• 9 X7 7 X8 5 X9

12
Limits Supply and Demand
X2
X5
X3
X6
X1
X8
X9
X4
X7
Figure C.1
13
Subject to

• X1 X2 X3 lt 100 (supply)
• X4 X5 X6 lt 300
• X7 X8 X9 lt 300
• X1 X4 X7 300 (demand)
• X2 X5 X8 200
• X3 X6 X9 200

14
Solve

• 9 variables
• 6 constraints

15
Answer

• Objective 3900

16
Transshipment

• Multi-step shipment route
• Requires flow conservation constraints

17
Change Prior Example

• Suppose cheap transportation available Ft.
  Lauderdale to Des Moines
• Suppose Ft. Lauderdale preferred manufacturing
  location
• Convert Des Moines into warehouse
• Increase production at Ft. Lauderdale

18
Real World

• Tropicana juice unit train
• Solid train of orange refrigerated boxcars
• Frozen concentrate juice
• Much cheaper than truck

19
Add Transfer Station
Boston (200 units demand)
Cleveland (200 units demand)
Des Moines (transfer)
X2
X5
X3
X6
Albuquerque (300 units demand)
X1
Evansville (300 units supply)
X8
X9
X4
X7
Fort Lauderdale (400 units supply)
Figure C.1
X10
20
New Objective Function

• Minimize
• 5 X1 4 X2 3 X3
• 8 X4 4 X5 3 X6
• 9 X7 7 X8 5 X9
• 2 X10

21
Subject to

• X1 X2 X3 X10 (Des Moines flow conservation)
• X4 X5 X6 lt 300
• X7 X8 X9 X10lt 400 (new Ft. Lauderdale)
• X1 X4 X7 300 (demand)
• X2 X5 X8 200
• X3 X6 X9 200

22
Modify for POM

• Rewrite X1 X2 X3 X10
• as X1 X2 X3 - X10 0
• X4 X5 X6 lt 300
• X7 X8 X9 X10lt 400 (new Ft. Lauderdale)
• X1 X4 X7 300 (demand)
• X2 X5 X8 200
• X3 X6 X9 200

23
Subject to (revised)

• X1 X2 X3 - X10 0 (Des Moines flow
  conservation)
• X4 X5 X6 lt 300
• X7 X8 X9 X10lt 400 (new Ft. Lauderdale)
• X1 X4 X7 300 (demand)
• X2 X5 X8 200
• X3 X6 X9 200

24
Solve

• 10 variables
• 6 constraints
• New trick FormatInsert/Delete

25
Answer

• Objective 3700

26
Assignment Model

• Time available?

27
Problem

• Each Leader can manage one project
• What assignments?
• Estimated time to complete in days

28
Objective Function

• Minimize
• 10 X1 15 X2 9 X3
• 9 X4 18 X5 5 X6
• 6 X7 14 X8 3 X9

29
Subject to

• X1 X2 X3 lt 1
• X4 X5 X6 lt 1
• X7 X8 X9 lt 1
• X1 X4 X7 1
• X2 X5 X8 1
• X3 X6 X9 1
• All Xs binary, 0 or 1

30
Answer

• Would you have guessed this answer?

31
Conclusion

• Define 3 Network Models
• Transportation
• Transshipment
• Assignment
• Formulate Problems
• Solve in POM