03a transportation problem

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Transportation Problem

1Sasadhar Bera, IIM Ranchi


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What is Transportation Problem?


The transportation model is a special class of the linear

programming problem.

Transportation deals with the situation in which a

commodity is shipped from sources to destinations.

The objective is to determine the amount of shipment

from sources to each destination that minimize the total

shipping cost while satisfying both the supply limits andthe demand requirement.


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Two Origins and Three Destinations



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Tabular Form

5Sasadhar Bera, IIM RanchiTS: Total Supply, TD: Total Demand


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LP formulation of Transportation Problem

xij : Amount transported from ithorigin to jthdestination

cij= unit transportation cost or shipping cost per unit item from ith origin to

jth

destination.Si supply amount available at i

thorigin, i =1 , 2, . .,m

Dj demand for jthdestination, j = 1, 2, . . .,n

m may be = n, < > n

Total supply = TS = Total Demand = TD =

Objective function = Zmin=

subject to , , , 0

Total number of variables = mn

Total number of constraints =(m+n). However, because the transportation

model is balanced (TD = TS), one of these equations must be redundant.

Thus, the model has (m+n-1) independent constraint equations i.e. (m+n-1)

numbers of restrictions. 6Sasadhar Bera, IIM Ranchi


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Transportation Problem: Example


The Navy has 9,000 pounds of material in Albany, Georgia that

it wishes to ship to three installations: San Diego, Norfolk, and

Pensacola. They require 4,000, 2,500, and 2,500 pounds,respectively. Government regulations require equal

distribution of shipping among the three carriers.

The shipping costs ($) per pound for truck, railroad, and

airplane transit are shown below. Formulate and solve a linear

program to determine the shipping arrangements (mode,

destination, and quantity) that will minimize the total shipping

cost.

DestinationMode San Diego Norfolk Pensacola

Truck 12 6 5

Railroad 20 11 9

Airplane 30 26 28


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Transportation Problem: Example (contd.)

Define the Decision Variables

We want to determine the pounds of material,xij , to be shippedby mode i to destinationj. The following table summarizes the

decision variables:

San Diego Norfolk Pensacola

Truck x11 x12 x13Railroad x21 x22 x23

Airplane x31 x32 x33

Min: (shipping cost per pound for each mode per destination pairing)*(number of pounds shipped by mode per destination pairing).

ZMin: 12x11+ 6x12+ 5x13 + 20x21+ 11x22+ 9x23 + 30x31+ 26x32+ 28x33



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Define the Constraints

Equal use of transportation modes:

(1)x11+x12+x13 = 3000

(2) x21+x22+x23 = 3000

(3) x31+x32+x33= 3000

Destination material requirements:(4)x11+x21+x31 = 4000

(5) x12+x22+x32 = 2500

(6) x13+x23+x33= 2500

Non-negativity of variables:

xij> 0, integer i= 1,2,3 and j= 1,2,3



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OBJECTIVE FUNCTION VALUE = 142000

Variable Value Reduced Costx11 1000.000 0.000x12 2000.000 0.000x13 0.000 1.000x21 0.000 3.000x22 500.000 0.000

x23 2500.000 0.000x31 3000.000 0.000x32 0.000 2.000x33 0.000 6.000


Solution SummarySan Diego will receive 1000 lbs. by truck and 3000 lbs. by airplane.

Norfolk will receive 2000 lbs. by truck and 500 lbs. by railroad.

Pensacola will receive 2500 lbs. by railroad.

The total shipping cost will be $142,000.

San Diego Norfolk Pensacola

Truck x11 x12 x13

Railroad x21 x22 x23

Airplane x31 x32 x33


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Special Purpose Algorithm to Solve

Transportation Problem


Transportation problem can be solved using LP. Due to

special structure of the transportation problem

special-purpose algorithms are developed to find out the

optimal solution. These special purpose algorithms are

computationally efficient (less complicated and less

computer memory required) than simplex method.

Two such types of solution algorithm are given below.

Vogel approximation method (VAM)

Modified distribution method (MODI)

VAM is used to generate the initial solution. Next, MODI is

used to find out optimal solution.


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Categories of Transportation Model


In real life problems, it is observed that total supply is not

equal to total demand. That is why we divide transportation

model into two categories:

Balanced transportation

Unbalanced transportation

Balanced transportation: A transportation problem in

which the total supply available at origins exactly satisfies

the total demand required at the destinations. In this case

Total supply (TS) = Total demand (TD).

Unbalanced transportation: In real life problems, total

supply is not equal to total demand. For solution of non-

balanced transportation problem we make it balanced by

adding dummy variable.


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Unbalanced Transportation Model

Total supply (TS) > Total demand (TD), surplus available of resources.



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Unbalanced Transportation Model (contd.)


Total supply (TS) > Total demand (TD), surplus available of resources.

By adding dummy demand variable we make it balanced.


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Unbalanced Transportation Model (contd.)


Total supply (TS) < Total demand (TD) , Shortage of demand.

In such a situation, we add dummy origin variable to make it

balanced


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Degeneracy in Transportation Problem


A feasible solution to transportation problem has mn

decision variables where m is number of origins and n is

number of destinations.

The solution is said to be degenerate if the number of

occupied cells is less than (m+n-1) at any stage of solution.

Degeneracy can occur at two stages:

At initial solution stage

During testing of the optimal solution


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How to Handle Degeneracy?


To handle degeneracy, we make use of an artificial quantity,

denoted by (epsilon). If is placed in the unoccupied cell

then the cell is considered occupied. The quantity is so

small that it does not affect the supply and demand

constraints.

For calculation purpose, the value of is assumed to bezero and we try to place at lowest unallocated cost cell.

Once is introduced into the solution, it will remain there

until degeneracy is removed or a final solution is arrived at,whichever occurs first.


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Degeneracy Example


Total number of restrictions = 4+4-1 = 7 but occupied cells = 6.

Hence to remove degeneracy, is allocated to lowest

unallocated cost cell.


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03a transportation problem

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