Transportation%20problems%20Operational%20Research%20Level%204

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Transportation problemsOperational Research
Level 4

• Prepared by T.M.J.A.Cooray
• Department of Mathematics

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Introduction

• Transportation problem is a special kind of LP
  problem in which goods are transported from a set
  of sources to a set of destinations subject to
  the supply and demand of the source and the
  destination respectively, such that the total
  cost of transportation is minimized.

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Examples

• Sources
• factories,
• finished goods warehouses ,
• raw materials ware houses,
• suppliers etc.

• Destinations
• Markets
• Finished goods ware house
• raw materials ware houses,
• factories,

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A schematic representation of a transportation
problem is shown below
D1
S1
a1 a2 ai am
b1 b2 bj bn
D2
Si
Dj
Dn
Sm
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• m- number of sources
• n- number of destinations
• ai- supply at source I
• bj demand at destination j
• cij cost of transportation per unit from
  source i to destination j
• Xij number of units to be transported from
  the source i to destination j

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• Destination j

1 2 j
n
Supply a1 a2 ai am
1 2 i m
c11 c12 c1j c1n




ci1 ci2 cij cin



cm1 cm2 cmn
S O U R C E i
Demand b1 b2
bj bn
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Transportation problem represented as a LP
model
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The ideal situation is shown below.,with
equalities instead of inequalities. There are
mn unknown variables and mn-1 independent
equations.
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• When solving the transportation problem ,the
  number of possible routes should be ? mn-1.
• If it is ltmn-1, it is called a degenerate
  solution.
• In such a case evaluation of the solution will
  not be possible.
• In order to evaluate the cells /routes (using
  the u-v method or the stepping stone method ) we
  need to imagine/introduce some used cells/routes
  carrying / transporting a very small quantity,
  say ?. That cell should be selected at the
  correct place.

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Example Consider a transportation problem
involving 3 sources and 3 destinations.
Source 1 2 3 Demand Destination 1 2 3 Destination 1 2 3 Destination 1 2 3 Supply 200 300 500 1000
Source 1 2 3 Demand 20 10 15 Supply 200 300 500 1000
Source 1 2 3 Demand 10 12 9 Supply 200 300 500 1000
Source 1 2 3 Demand 25 30 18 Supply 200 300 500 1000
Source 1 2 3 Demand 200 400 400 Supply 200 300 500 1000
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Types of transportation problems

• Balanced transportation problems
• Unbalanced transportation problems

Include a dummy source or a dummy destination
having a supply d or demand d to convert it
to a balanced transportation problem. Where d
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Example
Plant
1 2 3 4 5 Demand
10 2 3 15 9 25
5 10 15 2 4 30
15 5 14 7 15 20
20 15 13 - 8 30
20 20 30 10 25
W A 1 R E 2 H O 3 U S 4 E
Supply
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Solution of transportation problems

• Two phases
• First phase
• Find an initial feasible solution
• 2nd phase
• Check for optimality and improve the solution

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Find an initial feasible solution

• North west corner method
• Least cost method
• Vogels approximation method

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Checking for optimality

• U-V method
• Stepping-Stone method

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Example-( having a degenerate solution)
Introduce ? to for phase 2..
Destinations
1 2 3 Supply
3 2 3 25
5 6 5 15
1 3 4 20
2 5 7 10
20 20 30
Sources S1
S2 S3 S4
Demand
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Transshipment models.

• In transportation problems ,shipments are sent
  directly from a particular source to a particular
  destination to minimize the total cost of
  shipments.
• It is sometimes economical if the shipment passes
  through some transient nodes in between the
  sources and destinations.
• In transshipment models it is possible for a
  shipment to pass through one or more intermediate
  nodes before it reaches its destination.

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Transshipment problem with sources and
destinations acting as transient nodes

• Number of starting nodes as well as the number
  of ending nodes is the sum of number of sources
  and the number of destinations of the original
  problem.
• Let B
• be the buffer stock and it is added to all the
  starting nodes and all the ending nodes.?

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• ..
  ..
• 
  

a1B ajB amB B B
S1
B B B b1B bnB
S1
Sj
Sj
Sm
Sm
D1
D1
Dn
Dn
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• Destinations D1,D2,.Dn are included as
  additional starting nodes mainly to act as
  transient nodes.they dont have any original
  supply and the supply of these nodes should be
  at least B.
• The sources S1,S2,.Sm are included as
  additional ending nodes mainly to act as
  transient nodes.these nodes are not having any
  original demand.But each of these transient nodes
  is assigned with B units as the demand value.

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• We need to know the transshipment cost between
  the sources ,between the destinations and between
  sources and destinations .

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Example

• Supplies at the sources are 100,200,150 and 350
  and Demand at the destinations are 350 and 450
  respectively.

S1 S2 S3 S4 D1 D2
S1 0 4 20 5 25 12
S2 10 0 6 10 5 20
S3 15 20 0 8 45 7
S4 20 25 10 0 30 6
D1 20 18 60 15 0 10
D2 10 25 30 23 4 0
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S1 S2 S3 S4 D1 D2
S1 0 4 20 5 25 12 800100900
S2 10 0 6 10 5 20 8002001000
S3 15 20 0 8 45 7 800150950
S4 20 25 10 0 30 6 8003501150
D1 20 18 60 15 0 10 800
D2 10 25 30 23 4 0 800
800 800 800 800 8003501150 8004501250
Same algorithms can be used to solve this
transshipment problem.
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Transportation problem with some transient nodes
between sources and destination.

• Consider the case where the shipping items are
  first sent to intermediate finished goods ware
  houses from the supply points/factories and then
  to the destinations.
• To solve these problems the capacity at each
  transient node is made equal to B.
• Where B

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Example

• Multi plant organization has 3 plants and three
  market places.
• The goods from the plants are sent to market
  places through two intermediate finished goods
  warehouses.
• Cost of transportation per unit between plants
  and warehouses and warehouses to market places
  and also supply values of plants and demand
  values of the markets are shown in the table.

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M1 M2 M3 W1 W2 SUPPLY
P1 ? ? ? 15 30 200
P2 ? ? ? 28 10 300
P3 ? ? ? 30 15 400
W1 10 40 30 0 20
W2 25 15 35 25 0
DEMAND 100 400 400
900
900
900
900
900
Solution of the problem is same as Ordinary
transportation Problems.
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