Implementation of the Classic Transportation Problem with Geographic Information Systems

1
Implementation of the Classic Transportation
Problem with Geographic Information Systems

• By Uchit Patel
• Masters Degree in Geographic Information
  Sciences
• University of Texas at Dallas
• Summer 2006

2
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

3
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)
• SourceTapojit Kumar and Susan M. Schilling.
  Comparison of Optimization Techniques In Large
  Scale Transportation Problems

4
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

5
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 ?

6
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

7
Literature Review

• Formation of the CTP
• - Attributions are to the 1940s 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 Vogels approximation
  method (Mathirajan, Meenakshi 2004)
• - Heuristic better than Vogels
  approximation method (Sharma, Prasad 2003)
• - Heuristic for initial basic feasible
  solution (Adlakha and Kowalski 2003)
• - Vogels better than Northwest corner
  method (Totschek and Wood 1960)

8
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)

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Formulation
Shipping costs, Supply, and Demand for Power Co -
Example
http//www.engr.sjsu.edu/udlpms/ISE20265/set2_tra
nsport2020network.ppt
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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 8X116X1210X139X14
• 9X2112X2213X237X24
• 14X319X3216X335X34

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Formulation (Supply, Demand and Sign Constraints)

• Each supply point has a limited production
  capacity
• X11X12X13X14 lt 35
• X21X22X23X24 lt 50
• X31X32X33X34 lt 40
• Each destination point has a limited demand
  capacity
• X11X21X31 gt 45
• X12X22X32 gt 20
• X13X23X33 gt 30
• X14X24X34 gt 30
• Sign Constraints
• A negative amount of electricity can not be
  shipped all Xijs must be non negative
• Xij gt 0 (i 1,2,3 j 1,2,3,4)

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Formulation - LP

• Min Z 8X116X1210X139X149X2112X2213X237X24
  
• 14X319X3216X335X34
• X11X12X13X14 lt 35 (Supply Constraints)
• X21X22X23X24 lt 50
• X31X32X33X34 lt 40
• X11X21X31 gt 45 (Demand Constraints)
• X12X22X32 gt 20
• X13X23X33 gt 30
• X14X24X34 gt 30
• Xij gt 0 (i 1,2,3 j 1,2,3,4)

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Formulation - CTP

• A set of m supply points from which a good is
  shipped. Supply point i can supply at most si
  units
• 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
• 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

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Balanced Transportation Problem

• Total supply equals to total demand, the problem
  is said to be a balanced transportation problem

15
CTP - Applications

• Goods Supply
• Transportation
• Communication
• Utility
• Cash flow Models
• Inventory to Machines
• Agriculture
• Military

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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

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Graphical Method

• 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
• Find Optimal Solution
• - Exists at Corner Points
• - All Corner Points values are measured
• - Insert in the Objective Function
• - Minimization Problem Lowest value
• Optimal solution

• SourceTapojit Kumar and Susan M. Schilling.
  Comparison of Optimization Techniques In Large
  Scale Transportation Problems

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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
19

• The Transportation Tableau
• Matrix form of CTP

http//www.utdallas.edu/scniu/OPRE-6201/6201.htm
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Find Basic Feasible Solution

• Implemented
• - Northwest Corner Method (NWCM)
• - Least Cost Method (LCM)

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Find Optimal Solution

• Implemented
• - Stepping Stone Method

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NWCM (Flow Chart)
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NWCM
24
LCM (Flow Chart)
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LCM
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Optimal Solution

• Stepping Stone Method (Flow Chart)

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Stepping Stone Method
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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

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Test Network
30
Test Network with Sources and Destinations
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Methodology

• The project is implemented in VBA and ArcObjects
  in ArcGIS environment
• Final application is .mxd form ArcMap Document

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Methodology Software Application

• Welcome Screen

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Methodology Software Application

• Select Network Dataset

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Methodology Software Application

• Select Sources

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Methodology Software Application

• Select Destinations

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Methodology Software Application

• Execute a Model

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Methodology Software Application

• Execute a Model

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Methodology Software Application

• Select Any One Button from a CTP Toolbar

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Methodology Software Application

• Printed Final Result

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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

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Methodology Procedure (Flow Chart)
42
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

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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

44
Results Time
45
Results Iterations
46
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

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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

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Future Research

• Initial Feasible Solution
• Vogels 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)

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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. 12, 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,
  708718.
• 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 Vogels
  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. OKelly, 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.

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• Questions ?
• Thank You !