Solving the Classic Transportation Problem with the Geographic Information
Systems:
A Test of Starting Procedures
Uchit Patel
Master’s Degree in Geographic Information Science
Summer 2006
07/31/2006
Abstract:
Intro: Simplex algorithms can be specialized to solve several linear programming models
that arise from network flow problems.
Problem Statement: Solve Classic Transportation Problem in GIS Environment.
Literature Review: Formulation of Classic Transportation Problem ,Different methods to
solve the Classic Transportation Problem.
Methods: Different algorithms will be implemented for solving initial basic feasible
solutions and optimal solution in ArcGIS software environment using VBA and
ArcObjects.
Results: Application in ArcGIS environment from which user can select any network,
sources and destinations and get transportation solution for different initial solutions and
optimal solution.
Keywords:
Classic transportation problem(CTP), Linear Programming(LP),Northwest corner
method(NWCM), Least cost method(LCM), Stepping Stone Method, Geographic
information systems
1
Introduction:
Transportation is one of the very important, big and growing fields of GIS. Network
analysis is an important sub-discipline within the field of Transportation GIS. There are
well-developed methods within network analysis to determine solutions to a wide range
of problems regarding the management of products, facilities or vehicles over a network.
But these methods are not used in the Network GIS. Unfortunately very few of the
methods that have been developed in Network GIS have been implemented in the
software of GIS. In order to address one such deficiency, this project seeks to implement
Classic Transportation Problem within ArcGIS environment.
One important application of Linear Programming has been in the area of physical
distribution (Transportation) of resources from one place to another, to meet a specific set
of requirements. It is easy to express a transportation problem mathematically in terms of
an LP model, which can be solved by the Simplex Method. Since transportation problem
involves a large number of variables and constraints, it takes a very long time to solve it
by simple Simplex Method. Simplex algorithms can be specialized to solve several linear
programming models that arise from Network flow problems.
2
What is the Classic Transportation Problem (CTP)?
Suppose we have bunch of stuff at a bunch of places (supply constraints) and we have to
deliver the stuff to a bunch of places (demand constraints). We have to minimize our cost
in delivering stuff while delivering across a network.
Applications of the CTP
Factories to Warehouses (Nabisco)
Warehouses to Retail Stores
Cash flow models (GM)
Cloth inventory to Sewing line (Farah Manufacturing)
Logs to Mills to Finishing Lines to Markets (Weyerhauser Lumber)
Cotton planting, picking, storing, distribution to gins (NM Dept. of Ag.)
Assigning personnel to sea and shore duty (U.S. Navy)
In the problem statement section, Introduction to CTP, How to formulate CTP as Linear
Programming Problem, How to solve Linear Programming Problem is covered in detail.
This section also includes a description of the basic steps of the algorithms to be
implemented and an explanation of how they allocate resources to demands. In the
Literature Review section previous research in the field of CTP is explained. The Data
Description section shows the data that has been used to test the performance of the final
application. The Methods of Analysis section explains how the analysis and the
implementation took place. The Results section shows the results from the analysis and
3
finally the Conclusion and Future Research section states the additional research question
and extensions to the present work.
Problem Statement:The Classic Transportation problem applies to situations in which a single product is to
be transported from several sources to several sinks.
Let there be m sources s1, s2, _ _, sm having ai (i = 1,2,_ _, m) units of supplies or capacity
respectively to be transported among n destinations having bj (j = 1,2,_ _,n) units of
requirements respectively.
We can set this up as a linear programming problem using the same (or similar)
language that we used for the Traveling Salesman Problem
Since we are trying to minimize the transport cost we know that the “sense of
optimization” is to minimize.
–
Let Z denote total transportation cost
–
Let xij denote the no. of truckloads to be shipped from cannery i to
warehouse j.
–
Let cij denote the cost of shipping a truckload from cannery i to warehouse
j.
–
The general form of the objective function is then
m
n
Minimize Z   cij xij
i 1 j 1
4
–
Where m is the number of supply centers and n is the number of demand
points
–
The specific instance of the objective function for our example is
n
x
j 1
ij
m
x
i 1
ij
 si for all i  1,2,..., m
 d j for all j  1,2,..., n
xij  0 for all i and j
We also have constraints on our supplies and demands
- For each demand location we have to deliver exactly the amount
Demanded
-
In our example we would have four of these demand constraints:
(1) We can’t truck negative truckloads of merchandise
Although this isn’t part of the formulation we must ensure that our supply
equals our demand.
-
If this isn’t really true we can fudge it:
Add “dummy” locations
Take up the extra supply or give the extra demand
The cost to get to dummy locations is VERY high
(2) Integer Solutions Property
- For transportation problems where every si and dj has an integer value, all
the basic variables (allocations) in every basic feasible solution (including
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an optimal one) also have integer values.
- We don’t need integer constraints
(3) Feasible solutions property
- As long as supply equals demand (as stated above) there will be feasible
solutions
(4) The CTP formulation is a linear program
- Each function is a linear function
- There are many variables
Each variable is plotted along an axis that is perpendicular to all other axes
More than 3 axes can’t exist in practical reality, only as a theoretical
construct
- The constraint lines form the boundary of the feasible region
-
The optimal solution lies within the feasible region
Linear Programming problems can be solved either graphically or algebraically.
However, a problem with more than three dimensions is cumbersome to graph. In
general, LP problems are solved by a technique called the Simplex Method.
A graphical method of solving an LP problem has the following steps:
1. Determination of the Feasible Solution Space:
First, the non-negativity constraints are accounted for. In figure , the horizontal axis
X and the vertical axis Y represent the exterior and interior- point variables,
respectively. Therefore, it restricts the solution space to the first quadrant.
Figure
6
To account for the remaining constraints, first, the inequality is removed by an
equation and then the equation is graphed. After all the constraints are accounted
for, a solution space is located and is shaded. This solution space defined by the
constraints is called the set of feasible solutions.
Source:Tapojit Kumar and Susan M. Schilling. Comparison of Optimization Techniques In Large Scale Transportation Problems
2. Determination of the Optimum Solution:
An optimum solution exists at a corner point of the solution space. In figure, the
corresponding corner points are (0, 0), (X1, Y1), (X2, Y2), and (X3, Y3). To find the
optimal value, first, all these corner points are measured and the values are inserted
in the objective function. Then, the obtained values from the objective function are
compared and the lowest value (since it is a minimization problem) is determined to
be the optimum solution.
The Simplex method is a method of solving linear programs
- Starts with a basic feasible solution
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i. Give values to variables that don’t violate constraints
- Iteratively improves on (makes changes to) that solution until the optimal
solution is found
- The transportation simplex method is a modification of the simplex method that
takes advantage of certain characteristics of the CTP and related problems.
A typical transportation problem is shown in Fig. 1. It deals with sources where a supply
of some commodity is available and destinations where the commodity is demanded. The
classic statement of the transportation problem uses a matrix with the rows representing
sources and columns representing destinations. The algorithms for solving the problem
are based on this matrix representation. The costs of shipping from sources to
destinations are indicated by the entries in the matrix. If shipment is impossible between
a given source and destination, a large cost of M is entered. This discourages the solution
from using such cells. Supplies and demands are shown along the margins of the matrix.
As in the example, the classic transportation problem has total supply equal to total
demand.
Figure 1. Matrix model of a transportation problem.
8
The network model of the transportation problem is shown in Fig. 2. Sources are
identified as the nodes on the left and destinations on the right. Allowable shipping links
are shown as arcs, while disallowed links are not included.
Figure 2. Network flow model of the transportation problem.
Only arc costs are shown in the network model, as these are the only relevant parameters.
All other parameters are set to the default values. On each supply node the positive
external flow indicates supply flow entering the network. On each destination node a
demand is a negative fixed external flow indicating that this amount must leave the
network.
Variations of the classical transportation problem are easily handled by modifications of
the network model. If links have finite capacity, the arc upper bounds can be made finite.
If supplies represent raw materials that are transformed into products at the sources and
the demands are in units of product, the gain factors can be used to represent
transformation efficiency at each source. If some minimal flow is required in certain
links, arc lower bounds can be set to nonzero values.
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Source:
http://www.me.utexas.edu/~jensen/ORMM/models/unit/network/subunits/special_cas
es/t ransportation.html
- Overview of the solution process
-
Set up the transportation simplex tableau
-
Initialize the problem with any feasible solution by Northwest corner
Method or Least cost method.
-
Iterate
(1) Compute optimal solution by Stepping Stone Method
(2) Test for optimality
(A) If optimal, Stop.
(B) If not optimal, make changes to the solution, and go to
Step 1
- What is the Simplex Tableau?
- It is a way of visualizing our problem to assist in finding the optimal
solution
- Problem Title (Rental Car Problem)
- One row for each Supply location
- One column for each Destination location
- Supply and Demand Totals
- Supply = Demand
- What are the empty boxes in the middle?
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-
Each empty box in the center represents a decision variable xij
-
The empty box holds two things:
Cost and
Either a Basic or a Non-Basic Variable value
- Costs are easy…they are constants
-
-
Make little boxes in the upper left corner of the ij cells
-
Put the given cij values in the little boxes
The variable values are harder
- xij variables with an assignment are called basic variables
- A basic variable is being used to send some stuff from a supply to a
demand
- Basic variables ALWAYS have a circle around them
- You will (MUST) always have m+n-1 basic variables
-
xij variables without an assignment are called non-basic variables
Non-Basic variable values represents the rate at which the objective
function would change IF some stuff went from this supply to this
demand
Non-Basic variables values NEVER have a circle around them
-
Use the Northwest corner method or Least Cost method to find an initial basic
feasible solution:
- What is a basic solution?
A solution that makes positive assignments to xij variables
11
xij variables with an assignment are called basic variables
xij variables without an assignment are called non-basic variables
-
What is a feasible solution?
One that doesn’t violate any constraints
This is a linear programming problem. The special structure of Transportation Problem
will allow us to take a number of shortcuts. The Simplex method suggests first find the
initial solution and then look for a simple “Pivot” to improve the solution. If no such
improvement can be found then current solution must be optimal.
The first step in the transportation algorithm is to select an initial set of basic variables.
There are (m + n -1) independent equations and hence (m + n -1) nonzero xij values in a
nondegenerate basic feasible solution. There are many ways of making the choice for an
initial basic.
We used following two methods for finding initial basic feasible solution.
North West Corner Method (NWCM)
If we ignore the total cost then it is trivial to find an initial feasible solution. We simply
assign the first group of demand to the first supply until the capacity is exhausted, and
then start assigning demand to the second supply until it too is at its capacity, and so on.
12
The northwest-corner method begins at the north west corner of the array. The maximum
numbers of units possible are assigned to X11 consistent with row and column
restrictions. That is,
X11 = min(first row constraint, first column constraint)
The source or destination whose capacity has been exhausted is eliminated from further
calculation. The available capacity of the remaining source or destination involved in
determining X11 is reduced by that amount. One then moves either horizontally or
vertically one cell to determine the value of the second basic variable. The direction of
the move is governed by whether the source or destination at the last assignment was
eliminated. In the event that a source and destination are simultaneously satisfied, the
next basic variable with a value zero can come from either the adjacent source or
destination entry. In that case we have an initial degenerate basic feasible solution.
Flow Chart – NWCM
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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
Least Cost Method (LCM)
The NWCM is a quick solution to find a feasible solution. The method ignores any cost
information. LCM is very similar to NWCM in that it selects one cell, saturates it and
deletes a row or column. This method tries to match demand and supply with some
consideration of costs.
The only difference between the least-cost method and the northwest-corner method is in
the choice of entering variables. Here, the strategy is to always select the cell with the
smallest cij value among all remaining cells as the entering cell. Ties are, as usual, broken
arbitrarily.
Flow Chart – LCM
14
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
Several other methods for constructing initial basic feasible solutions can be found. These
methods offer some differences in terms of total computational effort and in terms of the
quality of the produced initial basic feasible solutions. In general, it is difficult
to achieve a perfect balance between effort and quality. In fact, it may not even be
desirable to do so, since constructing an initial basic feasible solution is only the first
phase in the solution of a problem. In other words, it is the total solution effort for a
problem that matters in the end. We will, therefore, not attempt to dwell upon a detailed
discussion of these other methods.
From initial solution we check its optimality. An optimal solution is one in which there is
no opportunity cost. That is, there is no other set of transportation routes that will reduce
the total transportation cost. Stepping Stone Method is used for this.
15
A little bit of reflection should convince us that the present initial feasible scenario is
essentially the same as that at the start of Phase II of the standard Simplex method. There
is, however, a logistical difference, namely that the standard Simplex tableau associated
with the current solution is not explicitly available to us. Therefore, we need to develop a
different set of procedures for generating in formations that are necessary for the
execution of the Simplex algorithm. In particular, it is critical that we have corresponding
mechanisms for conducting optimality tests and for performing pivots.
We begin with the question of whether or not the current solution is optimal. In the
standard Simplex method, the optimality test is based on a reading of the coefficients of
the nonbasic variables in the zeroth row of the Simplex tableau; that is, it is based on a
reading of the reduced costs. Since the reduced costs are not explicitly available in a
given transportation tableau, our first task is to develop a method for (re)constructing
them.
Source:
http://www.me.utexas.edu/~jensen/ORMM/models/unit/network/subunits/special_cases/t
ransportation.html
The reduced cost associated with a nonbasic variable is defined to be the amount
by which the objective-function value degrades if we increase (nominally) the value of
that nonbasic variable by 1 (while holding all other nonbasic variables at 0). We will
16
apply this definition to constructively generate the reduced costs associated with all
nonbasic variables.
The detailed steps involved in Stepping Stone are summarized below.
(1) Determine a closed path, starting at the non basic variable being evaluated and
“stepping” from boxes with assignments back to the original empty box. Right
angle turns in this path are permitted only at basic variables (boxes with
assignments) and at the original non basic variable. Since only the boxes at the
turning points are considered to be on the closed path, both empty and assigned
boxes may be skipped over. The boxes at the turning points are often called the
“Stepping Stone” on the path.
(2) Beginning at the non basic variable cell being evaluated, we assign a + , and then
alternate minus and plus signs at the basic variables on the corner points of the
path.
(3) Sum the unit costs in the boxes with plus signs, and subtract the unit costs in the
boxes with minus signs. If we are minimizing costs the result is the net change in
the cost per unit from the changes made in the assignments.
(4) Repeat this procedure for each empty box in the transportation table.
If the net changes are all grater than or equal to 0, and if we are minimizing costs, we
have found an optimal solution. Otherwise, we could shift units into an empty box
and reduce the cost.
Iterate the same process until net changes are all grater than or equal to zero.
17
Flow Chart – Stepping Stone Method
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
These types of problems are arises when we are solving Network flow problems in GIS.
There is no implementation of them in any GIS software. This project serves to address
this problem by implementing and analyzing Classic transportation problems with GIS
software.
18
As we are implementing different starting solution procedures and optimal solution
procedure we are interested in which combination of starting solution procedure and
optimal solution procedure gives faster result.
Sub research question is which combination of starting solution procedure and optimal
solution procedure takes less no. of iterations.
Literature Review:-
Formulation of the CTP
The transportation problem and cycle canceling methods are classical in optimization.
The usual attributions are to the 1940’s and later. The transportation problem was
formulated by Hitchcock 1941. Similar results are described by Tolstoi 1939. He studied
TP and a negative cycle criterion is developed and applied to solve a large scale (10X68)
TP to optimality.( Schrijver 2002).
Methods of finding Initial Basic Feasible Solution
There are many methods available to find initial basic feasible solution. NWCM, LCM
and Vogel’s Approximation method are common to find initial basic feasible solution.
Some heuristics give better performance than given common methods. NWCM gives
solution very far from optimal solution and LCM and Vogel’s Approximation method
tries to give result near to optimal solution.
Proposed heuristic gives significantly better solutions than the well-known Vogel’s
approximation method (Sharma, Prasad 2003). This is a best heuristic method than
19
northwest, least cost and Vogel’s to get initial solution to incapacitated transportation
problem.
In a real-time application, Vogel’s Approximation Method (VAM) will yield an
appreciable savings over a period of time. On the other hand, if ease of programming and
memory space are prime considerations, the NWCM is still acceptable for moderate
matrix sizes ( up to 50 X 50). However, the difference in times between the two loading
techniques increases exponentially.(Totschek and Wood).
Another work presents a variant of Vogel’s Approximation Method for TP and this
method is more efficient than traditional Vogel’s Approximation Method (Mathirajan,
Meenakshi 2004).
Methods for finding Optimal Solution
There are two common methods for finding optimal solution. MODI (Modified
Distribution) method and Stepping Stone method. Some heuristics are generated to
getting better performance.
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).
Different methods are compared for speed factor. Transportation Simplex Method and
Genetic Algorithms are compared in terms of accuracy and speed when a large-scale
problem is being solved. Genetic Algorithms prove to be more efficient as the size of the
problem becomes greater (Kumar and Schilling).
20
Proposed digital computer technique for solving the CTP by stepping stone method. The
average time required to perform an iteration using the method described here depends
linearly on the size of the problem, m + n.(Dennis)
The solution of a real world problem to efficiently transport multiple commodities from
multiple sources to multiple different destinations using a finite fleet of heterogeneous
vehicles in the smallest number of discrete time periods gives improvement by backward
decomposition ( Poh, Choo and Wong 2005).
A dual forest exterior point algorithm is on average up to 4.5 times faster than network
simplex algorithm on TPs of size 300 X 300 and for all classes (Papamanthou,
Paparrizos, Samaras 2004). When problem size increases exterior point algorithm get
relatively faster.
Simple, efficient heuristic procedure proposed for solving small fixed-charged TP. The
proposed method obtains the best initial solution in Part I and uses simplex like iterations
in the Part II to improve that solution and to verify its optimality (Adlakha and Kowalski
2003).
The complexity of the problem of the transport routes with many delivery centres, linear
methods are only useful as a support technique. The development of models for
decisionmaking concerning routing scheduling based on a multi-phase algorithm with a
heuristic and an exact facet ( Faulin 2003).
Application Software Language
MODI Algorithm was coded in FORTRAN V, and further substantial time reductions
may result by a professional coding of the algorithm in Assembler language. Zimmer
21
reported that a 20-to-1 time reduction was possible by using Assembler rather than
FORTRAN in coding minimum path algorithms.(Srinivasan, Thompson 1973).
One work investigated generalized network problems in which flow conservation is not
maintained because of cash management, fuel management, capacity expansion etc
(Gottlieb 2002). Optimal solution to the pure problem could be used to solve the
generalized network problem.
One work introduces a generalized formulation that addresses the divide between
spatially aggregate and disaggregate location modeling (Horner, O’Kelly 2005).
It has been shown that, especially within a multi-site scheduling environment, it is
possible to treat transportation problems as scheduling problems (Appelrath, Hans-Jurgen
& Jurgen 2000).
Data Sources:-
22
The test network used to test and evaluate the application is a network with 1043 nodes
and 1596 arcs. The original source is a TIGER – formatted set of Streets in DFW
metropolitan from the 2000 U. S. Census Bureau dataset. This data is converted to ESRI
Network Dataset format. Sources and Destinations are randomly selected nodes from the
Network and converted to the ESRI Shapefile format.
This network is tested for very small, moderate and large scale of Sources and
Destinations. Maximum no. of Sources and Destinations are 67 and 84 (67 X 84) tested
in this network.
23
Analysis and Methodology:-
This study began with a review of literature – different methods proposed to solve the
CTP. Based on the review two algorithms were implemented to find the initial basic
feasible solution and one algorithm for finding optimal solution in ArcGIS software using
VBA and ArcObjects.
ESRI provides the models of all objects involved in ArcObjects. For this specific
implementation the model used uses the Network Analyst Object Model. VBA stands for
24
Visual Basic for Applications and it is the language used to implement the algorithms
within ArcGIS.
As explained before, the algorithms which used for getting initial feasible solution differs
in a result. The NWCM gives initial result which is very far from optimal solution. LCM
tries to give result near to optimal solution. NWCM is not using any cost information in
finding the basic solution, While LCM gives solution by considering cost information.
Optimal solution is found by Stepping Stone Method by using any one initial feasible
solution. The same optimal result is obtains for both the initial basic feasible solution
method and optimal solution method.
The objective of the project was to implement different combination of initial basic
feasible solution algorithm and optimal algorithm to compare the performance, no. of
iterations and response time for each of them. Based on that objective, the
implementation will include the NWCM, LCM and Stepping Stone Algorithm. To
analyze the no. of iterations and response time of each algorithm a counter and timer was
embedded within the implementation to calculate the time and iterations to obtain the
optimal results.
The following simplified flow chart explains how the implementation of algorithms took
place.
25
Classic Transportation Problem Implementation
Create OD Cost Matrix
-
Select Network Data Set, Sources and Destinations
-
Execute a Model to generate OD Cost matrix
-
OD Cost Matrix is generated based on the impedance attribute
value and gives shortest path cost for each pair of Sources and
Destinations
Find Initial Basic Feasible Solution
-
Using the OD Cost matrix values, values of Sources and
Destinations two algorithms NWCM, LCM is executed
and it gives initial basic feasible solution
-
Timer was enabled at the starting of each algorithm
Find Optimal Solution
-
Use any one initial basic feasible solution and execute Stepping
Stone algorithm to find the optimal solution
-
Iterate until optimal solution was found
-
Counter was enabled at the starting of Stepping stone algorithm
and incremented for each iteration
26
General Flow
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
27
Response Time was defined as follows
Time spent to find the initial basic feasible solution and optimal solution. Time is
calculated after the data is initialized.
Results:-
The implementation of two algorithms for finding basic initial feasible solution and one
algorithm for finding optimal solution were able to solve CTP. The response time and
no. of iterations are higher in NWCM + Stepping Stone than LCM + Stepping Stone.
The algorithms were implemented within ArcGIS using VBA. A toolbar was created with
two buttons to execute the algorithm. The first button executes the NWCM and Stepping
Stone Method and second button executes the LCM and Stepping Stone Method.
Every algorithm creates a text file with the results. Appendix 1 shows the screen shots of
application development.
Test Network was tested for different sets of Sources and Destinations. Tested for
different 33 sets of Sources and Destinations. Network were tested for very small,
moderate and large scale of sources and destinations. Maximum scale of sources 67 and
destinations 84 were tested on the network.
28
System Configurations
Intel(R) Pentium(R) 4 CPU 2.80 GHz 1 GB RAM
Operating System - Microsoft Windows XP Professional
Results – Time
Comparision of Time for Two Methods
2500000
Milliseconds
2000000
1500000
NWCM + OPTIMAL
LCM + OPTIMAL
1000000
500000
0
0
50
100
150
200
N+M
29
Results – No. of Iterations
Comparision of Iterations for Two Methods
60
No. of Iterations
50
40
NWCR + OPTIMAL
30
LCM + OPTIMAL
20
10
0
0
50
100
150
200
N+M
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.
30
Conclusions:-
After testing the algorithms performance, it is clear that time response increases when
using NWCM compare to LCM. The LCM is more efficient than NWCM. LCM takes
less no. of iterations for finding optimal solution compare to NWCM.
In general for small size problem up to 10 sources and 15 destinations time is not
affecting performance. So, for small size problem NWCM is still accepted. As size of the
problem (n + m) increases NWCM takes very high response time compare to LCM. For
moderate to large size of problem LCM is more effective.
Future Research:-
Initial Basic Feasible Solution
Vogel’s Approximation method (VAM) gives initial solution very near to optimal
solution. We can implement VAM and improve our performance.
Heuristic Method
As size of the problem increases response time is also increases so there is a need for
heuristic implementation. We can implement heuristic method and improve performance.
We have to select heuristic which gives optimal solution or nearly optimal solution.
31
Application Software
We can generate parallel version of the algorithm and it will give us better solution time
compare to method implemented.
Here major time is consumed in searching and finding paths so there is a need to find
quick searching techniques. BFS (Breath First Search) technique implemented in graph
data structure might be useful in our problem and gives improvement in response time.
Zimmer reported 20-to-1 reduction in response time is assembler language is used.
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. 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.
32
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.
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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
34
Appendix:Screen Shots of Application Software
Welcome Screen
Select Network Dataset
35
Select Sources
Enter Field which gives Supply Value
36
Select Destinations
Enter Field which gives Demand Value
37
Execute Model
38
Select Any One Button from a CTP Toolbar
39
Printed Final Result
40
All Tested Results
N+M
3+3
3+3
3+4
3+4
3+4
3+4
3+4
3+4
3+4
3+4
3+4
4+3
4+3
4+4
3+5
3+5
10 + 10
10 + 11
15 + 19
20 + 21
12 + 19
15 + 23
15 + 23
25 + 33
25 + 34
30 + 37
35 + 44
38 + 49
44 + 57
44 + 57
48 + 58
58 + 69
67 + 84
Time (Milliseconds)
NWCM + OPTIMAL LCM + OPTIMAL
0
0
0
0
15.625
0
15.625
0
15.625
0
0
0
0
15.625
15.625
0
296.875
46.875
1359.375
7453.125
828.125
2765.25
2750
22820
17812.5
510468.875
112203.125
209671.875
262468.75
262328.125
552062.5
743421.875
1966031.25
15.625
0
0
0
0
0
0
0
0
0
0
0
0
15.625
0
15.625
140.625
93.75
1796.875
5000
531.25
3187.5
3390.625
54828.125
14829.625
75390.625
64656.25
106468.75
6640.625
6421.875
25765.6
438984.3
80453.125
No. of Iterations
NWCR + OPTIMAL
LCM + OPTIMAL
3
2
5
2
5
3
4
4
2
3
4
4
4
5
5
4
14
2
11
26
10
15
15
21
17
30
32
42
29
29
51
33
45
1
1
2
2
1
2
2
2
1
2
2
1
2
2
3
4
8
5
13
18
7
16
16
52
12
42
19
16
1
1
3
21
2
41