Implementation of the Classic Transportation
Problem with
Geographic Information Systems
By : Uchit Patel
Master’s Degree in Geographic Information Sciences
University of Texas at Dallas
Summer 2006
Introduction – Network Models
Important – Special case of linear optimization models
Many Real – world problems can be modeled by using networks e.g. Transportation,
Distribution, Scheduling
Visual Interpretation in addition to the mathematical formulation
Corresponding Mathematical formulation has a special structure that allows
extremely large problem to be solved very quickly
Introduction – Classic Transportation Problem (CTP)
- Bunch of stuff at a bunch of Places, Deliver the stuff to a bunch of Places - Minimize the
cost
- Each Supply and Demand have constraints (no. on the pictures)
- Total Supply = Total Demand
- Cost of shipping from Supply to Demand (no. on the lines)
Source:Tapojit Kumar and Susan M. Schilling. Comparison of Optimization Techniques In Large Scale Transportation Problems
Introduction
Well developed methods in Network Analysis – Solve problems regarding
Management of products, facilities, vehicles etc.
Few Implemented in the GIS environment
CTP as a Network Analysis Problem in GIS
Network Analysis functions available in ArcGIS
Research Questions
Main research question of the project is answer to following question
Can we implement CTP in GIS environment ?
Implement different methods for solving Initial Basic Feasible solution
Implement method for solving Optimal solution
Sub research questions are answers to following questions
Which Initial and Optimal solution gives faster result ?
How many Iterations takes place for Optimal solution in each method ?
Literature Review
Fields of
Operations Research
Mathematical Programming
Computing Machinery
Formulations of the CTP
Solution Methods for CTP – Initial Basic feasible Solution and Optimal
Solution
Implementing CTP Solution procedures in Computer Software
Literature Review
Formation of the CTP
- Attributions are to the 1940’s and later
- CTP Formulation (Hitchcock 1941)
- Application to the Network (Tolstoi 1939)
- Application of the Simplex Method to a Transportation Problem (Dantzig 1951)
Methods of finding Initial Basic Feasible Solution and Optimal Solution
- Variant's of Vogel’s approximation method (Mathirajan, Meenakshi 2004)
- Heuristic better than Vogel’s approximation method (Sharma, Prasad 2003)
- Heuristic for initial basic feasible solution (Adlakha and Kowalski 2003)
- Vogel’s better than Northwest corner method (Totschek and Wood 1960)
Literature Review
- Dual forest exterior point algorithm is faster than Transportation
Simplex (Paparrizos, Samaras 2004)
- Backward Decomposition gives improvement (Poh, Choo and Wong 2005)
- The most efficient method for solving TP arises by coupling a primal
transportation algorithm with a modified row minimum start rule and a modified
row first negative evaluator rule (Glover, Karney, Kligman,Napier 1974)
Implementation in Computer Software
- MODI algorithm was coded in FORTRAN V (Srinivasan, Thompson 1973)
- 20-to-1 time reduction possible with assembler language (Zimmer 1970)
Formulation
Shipping costs, Supply, and Demand for Power Co - Example
From
City
1
City
2
To
City City
3
4
Plant 1
Plant 2
Plant 3
$8
$9
$14
$6
$12
$9
$10
$13
$16
$9
$7
$5
Demand
(Million
kwh)
45
20
30
30
http://www.engr.sjsu.edu/udlpms/ISE%20265/set2_transport%20&%20network.ppt
Supply
(Million
kwh)
35
50
40
Formulation
Decision Variable
We have to determine how much electricity is sent from each plant
to each city
Xij = Amount of electricity produced at plant i and sent to city j
X14 = Amount of electricity produced at plant 1 and sent to city 4
Objective function
Minimize Z = 8X11+6X12+10X13+9X14
+9X21+12X22+13X23+7X24
+14X31+9X32+16X33+5X34
Formulation (Supply, Demand and Sign Constraints)
Each supply point has a limited production capacity
X11+X12+X13+X14 <= 35
X21+X22+X23+X24 <= 50
X31+X32+X33+X34 <= 40
Each destination point has a limited demand capacity
X11+X21+X31 >= 45
X12+X22+X32 >= 20
X13+X23+X33 >= 30
X14+X24+X34 >= 30
Sign Constraints
A negative amount of electricity can not be shipped all Xij’s must be non
negative
Xij >= 0 (i= 1,2,3; j= 1,2,3,4)
Formulation - LP
Min Z = 8X11+6X12+10X13+9X14+9X21+12X22+13X23+7X24
+14X31+9X32+16X33+5X34
X11+X12+X13+X14 <= 35
X21+X22+X23+X24 <= 50
X31+X32+X33+X34 <= 40
(Supply Constraints)
X11+X21+X31 >= 45
X12+X22+X32 >= 20
X13+X23+X33 >= 30
X14+X24+X34 >= 30
(Demand Constraints)
Xij >= 0 (i= 1,2,3; j= 1,2,3,4)
Formulation - CTP
1. A set of m supply points from which a good is shipped. Supply point i can
supply at most si units
2. A set of n demand points to which the good is shipped. Demand point j must
receive at least di units of the shipped good
3. Each unit produced at supply point i and shipped to demand point j incurs a
variable cost of cij
Xij = number of units shipped from supply point i to demand point j
i m j n
min
c
Xij
ij
i 1
j 1
j n
s.t. Xij  si (i  1,2,..., m)
j 1
i m
X
ij
 dj ( j  1,2,..., n)
i 1
Xij  0(i  1,2,..., m; j  1,2,..., n)
Balanced Transportation Problem
Total supply equals to total demand, the problem is said to be a
balanced transportation problem
j n
i m
s  d
i
i 1
j
j 1
CTP - Applications
Goods Supply
Transportation
Communication
Utility
Cash flow Models
Inventory to Machines
Agriculture
Military
Solution Procedure - Linear Programming (LP)
LP - Problem be reduced to a set of Linear functions
LP – Objective function is to be maximized or minimized
Solution LP
- Graphically
- Specialized Simplex Method
Graphical Method
(1)
Find Feasible Solution Space
- Non Negativity Constraints
- X – Exterior Point Variables
Y – Interior Point Variables
- Inequality is removed and graphed
- Constraints are accounted and
solution space is shaded – Feasible
solution
(2)
Find Optimal Solution
- Exists at Corner Points
- All Corner Points values are measured
- Insert in the Objective Function
- Minimization Problem – Lowest value
Optimal solution
Source:Tapojit Kumar and Susan M. Schilling. Comparison of Optimization Techniques In Large Scale Transportation Problems
Specialized Simplex Method (Solution Process)
Set up Transportation Simplex Tableau
Initialize Problem with any Basic Feasible Solution
Iterate
(1) Find Optimal Solution
(2) Test for Optimality
(a) If Optimal, Stop
(b) Not Optimal, Make changes to the solution and go to Step 1
http://www.utdallas.edu/~scniu/OPRE-6201/6201.htm
The Transportation Tableau
Matrix form of CTP
http://www.utdallas.edu/~scniu/OPRE-6201/6201.htm
Find Basic Feasible Solution
Implemented
- Northwest Corner Method (NWCM)
- Least Cost Method (LCM)
Find Optimal Solution
Implemented
- Stepping Stone Method
NWCM (Flow Chart)
Allocate minimum value of first row or column to north west corner square
Eliminate row or column whose capacity has been exhausted
Adjust corresponding supply and demand value
No
No
Check column capacity is
exhausted ?
Yes
Move horizontally one square
Allocate as much as possible
Adjust corresponding supply and demand value
Move vertically one square
Allocate as much as possible
Adjust corresponding supply and demand value
If each each row and column are
traversed or Total Allocation =
Total supply value = Total
source value
Yes
End
NWCM
LCM (Flow Chart)
Find square with minimum cell value
Allocate maximum as possible
Eliminate row or column whose capacity is exhausted
Adjust corresponding supply and demand value
Find square with minimum cell value from the remaining rows and column
Allocate maximum as possible
Eliminate row or column whose capacity is exhausted
Adjust corresponding supply and demand value
No
If each each row and column are
eliminated or Total Allocation =
Total supply value = Total
source value
Yes
End
LCM
Optimal Solution
Stepping Stone Method (Flow Chart)
Start with any initial basic feasible solution
Determine closed path starting from each empty square
Beginning at the first square assign ‘+’ and alternate ‘-’ at the corner squares
Sum the unit costs in squares with ‘+’ sign and subtract with ‘-’ sign
Transfer allocations
No
Test for optimality ?
Yes
End
Stepping Stone Method
Data
Network : 1043 Nodes and 1596 Arcs
Original Source – Tiger formatted set of Streets of DFW Metropolitan 2000 U. S.
Census Bureau dataset
Converted to ESRI Network Dataset
Sources and Destinations :
Different Nodes from the Network and Converted to ESRI Shapefile format
Test Network
Test Network with Sources and Destinations
Methodology
The project is implemented in VBA and ArcObjects in ArcGIS environment
Final application is .mxd form ArcMap Document
Methodology : Software Application
Welcome Screen
Methodology : Software Application
Select Network Dataset
Methodology : Software Application
Select Sources
Methodology : Software Application
Select Destinations
Methodology : Software Application
Execute a Model
Methodology : Software Application
Execute a Model
Methodology : Software Application
Select Any One Button from a CTP Toolbar
Methodology : Software Application
Printed Final Result
Methodology : Procedure
Data Input
- User selects the Network, Sources and Destinations (Network must be ESRI Network Dataset
format)
- Model is executed and depending of the cost field it finds the shortest path cost between all Sources and
Destinations
Initial Basic Feasible Solution
- Using OD Cost matrix values, values of Sources and Destinations two methods (algorithms) – NWCM
and LCM gives initial basic feasible solution
- User can select any one algorithm
Optimal Solution
- Based on the any one initial basic feasible solution (NWCM or LCM) Stepping
Stone Method gives Optimal Solution
- Check for Optimality if Optimal Print Results else iterate the same method for another time
Output
- Output will be printed in a .txt file
Methodology : Procedure (Flow Chart)
Cost field
Sources and
destinations
Network Dataset
OD Cost matrix generation
Find initial basic feasible solution by
LCM
Find initial basic feasible solution by
NWCM
Find Optimal solution by Stepping
Stone method
No
Test for
optimality ?
Yes
End
Analysis
NWCM Vs. LCM
- NWCM
- Quick solution
- Ignores any Cost Information
- Gives solution very far from Optimal
- LCM
- Tries to match Supply and Demand with consideration of cost
- Select the square with smallest cij value
- Gives solution near to optimal compare to NWCM
Testing Environment
- Test Network is tested for different sets of Sources and Destinations
- Tested for different 33 sets of Sources and Destinations
- Network is tested for very small, moderate and large scale of Sources and
Destinations (n + m)
- Maximum Sources 67 and Destinations 84 , ( 67 + 84) tested on the network
- System Configurations
Intel(R) Pentium(R) 4 CPU 2.80 GHz 1 GB RAM
Operating System - Microsoft Windows XP Professional
- Response Time
Time spent to find the initial basic feasible solution and optimal solution
Time is calculated after the data is initialized
Results : Time
Comparision of Time for Two Methods
2500000
Milliseconds
2000000
1500000
NWCM + OPTIMAL
LCM + OPTIMAL
1000000
500000
0
0
50
100
N+M
150
200
Results : Iterations
Comparision of Iterations for Two Methods
60
No. of Iterations
50
40
NWCR + OPTIMAL
30
LCM + OPTIMAL
20
10
0
0
50
100
N+M
150
200
Results
Yes, we can implement CTP in GIS environment
Total 33 different sets of Sources and Destinations tested
- 30 out of 33 results give faster solution by LCM than NWCM
- 30 out of 33 results take less no. of iterations by LCM than NWCM
Conclusions
Time
LCM yields appreciable savings over a period of time
Small Size Problem up to(10 + 15) – NWCM is accepted
As size (n + m) increases (moderate to large scale) NWCM takes very high time
compare to LCM
No. of Iterations
LCM yields comparatively less no. of iterations than NWCM
Small Size Problem up to(10 + 15) – NWCM is accepted
Future Research
Initial Feasible Solution
Vogel’s Approximation method gives better result than the two (NWCM and LCM)
Heuristic Method
As size of the problem (n + m) increases time is increases
Need a heuristic which gives faster result
Application Software
Parallel version of algorithm will give faster result
Time is consumed in searching square and path
Improved searching technique gives improvement – BFS (Graph Theory)
Zimmer 1970 reported 20 – t0 – 1 reduction in time if use assembler language
References
Hitchcock, F.L. 1941. The distribution of a product from several sources to numerous localities. Journal of Mathematics and Physics 20, 224-230
Schrijver, Alexander. 2002. On the history of the transportation and maximum flow problems. Math. Program., Ser. B 91 437-445.
Tapojit Kumar and Susan M. Schilling. Comparison of Optimization Techniques In Large Scale Transportation Problems.
R. Totschek and R. C. Wood.1960. An Investigation of Real-Time Solution of the Transportation Problem.
Jack B. Dennis. A High-Speed Computer Technique for the Transportation Problem.
V. Srinivasan, G. L. Thompson. Benefit-Cost Analysis of Coding Techniques for the Primal Transportation Algorithm. Journal of the Association for
Computing Machinery, Vol. 20, No. 2, 1973, 194-213.
Fred Glover, D. Karney, D. Klingman, A. Napier. A Computation Study on Start Procedures, Basic Change Criteria and Solution Algorithms for
Transportation Problems.
Management Science, Theory Series, Mathematical Programming Vol. 20, No. 5, Jan., 1974, pp 793 – 813.
Faulin Javier 2003. Combining Linear Programming and Heuristics to Solve a transportation Problem for a Canning Company in Spain. International
Journal of Logistics: Research and Applications Vol. 6, No. 1–2, 2003.
Poh K L, Choo K W, Wong C G 2005. A heuristic approach to the multi-period
multi-commodity transportation problem. Journal of the Operational Research Society (2005) 56, 708–718.
Papamanthou Charalampos, Paparrizos Konstantinos, Samaras Nikolaos 2004. Computational experience with exterior point algorithms for the
transportation problem.
Applied Mathematics and Computation 158(2004) 459-475.
Prasad Saumya, Sharma R.R.K 2003. Obtaining a good primal solution to the uncapacitated transportation problem. European Journal of Operations
Research 144(2003) 560-564.
Adlakha Veena, Kowalski Krzysztof 2003. A Simple heuristic for solving small fixed-charge transportation problems. Omega 31 (2003) 205-211.
Mathirajan, M. & Meenakshi, B. 2004. Experimental Analysis of Some Variants of Vogel’s Approximation Method. Asia – Pacific Journal of Operations
Research 21(4): 447-462.
Gottlieb, E. S. 2002. Solving Generalized Transportation Problems via Pure Transportation Problems. Naval Research Logistics 49 (7): 666-85.
Horner, Mark W. & O’Kelly, Morton E. 2005. A Combined Cluster and Interaction Model: The Hierarchical Assignment Problem. Geographical Analysis
37 :315-335, The Ohio State University.
Appelrath, Hans-Jurgen & Sauer Jurgen 2000. Integrating Transportation in a Multi-Site Scheduling Environment. Proceedings of the Hawai'i
International Conference On System Sciences, Maui, Hawaii.
http://www.utdallas.edu/~scniu/OPRE-6201/6201.htm
http://www.utdallas.edu/~curtin/classes/GISC6379/GISC6379.html
http://www.me.utexas.edu/~jensen/ORMM/models/unit/network/subunits/special_cases/t ransportation.html
http://www.mnsu.edu/research/URC/OnlinePublications/URC2004Articles/Kumar.pdf
http://skumar.mitindia.net/becse/or.htm
http://perso.orange.fr/jean-pierre.moreau/c_linear.html
http://www.engr.sjsu.edu/udlpms/ISE%20265/set2_transport%20&%20network.ppt
Questions ?
Thank You !