assignment transportation

8/6/2019 Assignment Transportation

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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.

assignment transportation

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