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Transportation
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A brief:
1941: F.L. Hitchcock
The distribution of a product from severalsources to numerous localities
1947: T.C.Koopmans
Optimal utilization of the TransportationSystem
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What transportation model is:Typically arise in situations involving physical
movement of goods from plants to warehouses,
warehouses to wholesalers, wholesalers to retailers and
retailers to customers.
Solution requires the determination of how many units
should be transported from each supply origin to eachdemand destination in order to satisfy all the
destination demands while minimizing the total
associated cost of transportation.
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Contd. The same problem could be of
maximizationtoo. Example given:Distributionof financial resources tothevarious available options.
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Mathematical model of
Transportation Problem
The mathematical model represents
the formulationoftransportationproblem oftransporting singlecommodity from various sources of
supplyto various demanddestinations.
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Solution of Transportation problem
The solution is carried out in two phases.
1. Initial solution is found by variousmethods
2. Test the solution for optimality by using
MODI method or the Stepping stone
method.
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Solution Procedure:The problem is solved, in general, by using the step-by-step
procedure:
1. Define the objective function to be minimized with the
constraints imposed on the problem.
2. Set up the transportation table with m rows representing the
sources(plants, factories, etc) and n columns representing the
destinations (warehouses, stores, markets, etc)
3. Develop an initial feasible solution to the problem.
4. Examine whether the initial solution is feasible or not. The
solution is said to be feasible if the solution has allocations in
(m+n-1) cells with independent positions.
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The cells having allocations are known as occupied
cells and the remaining cells are known as empty or
unoccupied cells.
Test whether the solution is optimal or not. This is
done by computing opportunity costs associated with
the empty cells. Positive opportunity costs for all theempty cells signifies optimal solution.
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Finding an Initial Feasible Solution:There are several methods available to obtain
an initial solution.
The North-West Corner Method(NWCM) Lowest Cost Entry Method
Vogels Approximation Method.
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North WestCorner Method(NWCM)
It is so called because we begin with the
north-west corner or upper left corner cell
of our transportation table.
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Contd.
Area of Application:
1.
It is used in case of transportation withinthe campus of an organization as cost are
not significant.
2. It is used for transportation to satisfy such
obligations where cost is not the criteria.
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LeastCostMethod(LCM)
The allocation according to this method is
very useful as it takes into consideration the
lowest cost and therefore, reduces the
computation as well as the amount of time
necessary to arrive the optimal solution.
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V
ogels Approximation Method(VAM):
Vogels approximation method (penalty or regret
method) is a heuristic method and is preferred to
the other two methods, because it gives an initialsolution which is nearer to an optimal solution or
is the optimal solution itself. Each allocation is
made on the basis of the opportunity cost that
would have incurred if allocation in certain cellswith minimum unit transportation cost were
missed.
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Contd.Area of Application:
1. It is used to compute transportation routes in
such a way as to minimize transportation costfor finding out locations of warehouses.
2. It is used to find out locations of transportation
corporations depots where insignificant total
cost difference total cost difference may notmatter.
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The AssignmentProblemThe objective is to assign a number of
resources(items) to an equal number of
activities on a one to one basis so as to
minimize total costs of performing the tasks
at hand or maximize total profit of
allocation.
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Applications of Assignment Assign salesperson to sales territories
Assign vehicles to routes
Assign accountants to client accounts
Assign contracts to bidders through systematic evaluationof bids from competing suppliers.
Assign development engineers to several constructionsites.
Schedule teachers to classes. Teams are matched to projects by the expected cost of each
team to accomplish each project.
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Complete Enumeration Method
Simplex Method Transportation Method
Hungarian Method (Minimization Case)
Solutions of AssignmentProblem
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Complete Enumeration MethodAll possible assignments are listed out and
the assignment involving the minimum cost,
time or distance (or maximum profit)is
selected. It represents the optimal solution.
In case two or more assignments have the
same minimum cost, time or distance(ormaximum profit), then the problem has
multiple optimal solutions.
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Simplex Method An assignment problem can be formulated
as a linear programming problem an as such
can be solved by the simplex algorithm.
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Transportation Method Assignment problem could be solved as a special
case of Transportation problem but the optimality
condition for the solution violates as m+n-1 basicvariables in transportation that will be n+n-1 or
2n-1 basic variables of assignment will not be
there as only n basic variables will be available in
an assignment solutions. So a large number ofdummy rows need to be introduced and that will
make transportation problem conceptually
inefficient.
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Step 1In a given matrix subtract the smallest element
in each row from every element of that row and
do the same in the column.
Step2In the reduced matrix obtain from step 1,
subtract the smallest element in each column
from every element of that column
Hungarian Method
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Step 3
Make the assignment for the reduced matrix obtainedfrom step 1 and step 2
(all the zeros in rows/columns are either marked()or (x) andthere is exactlyone assignment in eachrow and each column. In such a case optimumassignmentpolicy for the given problem is obtained.
Ifthere is row or column withoutanassignmentgotothe nextstep.
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Step 4
Draw the minimum number of vertical andhorizontallines necessaryto cover all the zeros inthe reducedmatrix obtained from step 3 by adopting thefollowing procedure.
(i) mark() all rows thatdonothave assignments (ii) Mark () all columns (notalready marked) which
have zeros inthe marked rows (iii) Mark () all rows (notalready marked) thathave
assignments in marked columns (iv) Draw straight lines through all unmarked rows
and marked columns
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Step 5
Ifthe number of lines drawn are equal tothe
number of rows or columns, then it is anoptimum solution ,otherwise goto step 6Step 6
Selectthe smallestelementamong all the
uncovered elements. Subtractthis smallestelementfrom all the uncovered elements an addittothe elementwhich lies atthe intersectionoftwo line. Thus we obtain another reducedmatrix for fresh assignments.
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Step 7
Gothe step 3 and repeatthe procedure until
the umber of assignmentbecome equal tothe number of rows or columns. In such acase, we shall observe thatrow/columnhasan assignment.Thus, the currentsolution isanoptimum solution.