transportation problems operational research level 4

MA 4020-Transportation problemsMA 4020-Transportation problems 11

Transportation problemsTransportation problemsOperational Research Level 4Operational Research Level 4

Prepared by T.M.J.A.CoorayPrepared by T.M.J.A.CoorayDepartment of MathematicsDepartment of Mathematics


Introduction Introduction

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


Examples:Examples:

SourcesSourcesfactories,factories,finished goods finished goods

warehouses ,warehouses ,raw materials ware raw materials ware

houses,houses, suppliers etc.suppliers etc.

Destinations Destinations MarketsMarketsFinished goods ware Finished goods ware

house house raw materials ware raw materials ware

houses,houses, factories,factories,


S1

Sm

Si

D1

D2

Dj

Dn

a1

a2

ai

am

b1

b2

bj

bn

A schematic representation of a transportation problem is shown below


m- number of sourcesm- number of sources n- number of destinationsn- number of destinations aaii- supply at source I- supply at source I bbj – j – demand at destination jdemand at destination j

ccij – ij – cost of transportation per unit from cost of transportation per unit from source i to destination jsource i to destination j

XXij – ij – number of units to be transported number of units to be transported from the source i to destination jfrom the source i to destination j


Destination jDestination j

cc1111 cc1212 cc1j1j cc1n1n

cci1i1 cci2i2 ccijij ccinin

ccm1m1 ccm2m2 ccmnmn

SOURCE i

12

i

m

1 2 j n

Demand b1 b2 bj bn

Supply a1

a2

ai

am


Transportation problem: represented as Transportation problem: represented as a a LP modelLP model

njandmiforX

njbX

miaXtosubject

XcZMinimize

ij

j

m

iij

i

n

jij

ij

m

i

n

jij

,..1,...10

,.....,2,1

,....,2,1

:

1

1

1 1


njandmiforX

njbX

miaXtosubject

XcZMinimize

ij

j

m

iij

i

n

jij

ij

m

i

n

jij

,..1,...10

,.....,2,1

,....,2,1

:

1

1

1 1

The ideal situation is shown below.,with equalities instead of inequalities. There are “mn” unknown variables and m+n-1 independent equations.


When solving the transportation problem ,the When solving the transportation problem ,the number of possible routes should be number of possible routes should be m+n-1. m+n-1.

If it is <m+n-1, it is called a degenerate solutionIf it is <m+n-1, it is called a degenerate solution.. In such a case evaluation of the solution In such a case evaluation of the solution

will not be possible.will not be possible. In order to evaluate the cells /routes (using In order to evaluate the cells /routes (using

the u-v method or the stepping stone the u-v method or the stepping stone method ) we need to imagine/introduce method ) we need to imagine/introduce some used cells/routes carrying / some used cells/routes carrying / transporting a very small quantity, say transporting a very small quantity, say . . That cell should be selected at the correct That cell should be selected at the correct place.place.


Example: Consider a transportation problem Example: Consider a transportation problem involving 3 sources and 3 destinations.involving 3 sources and 3 destinations.

SourcSourcee

11

22

33

DemanDemandd

DestinationDestination 1 2 31 2 3

SupplySupply

200200

300300500500

10001000

2020 1010 15151010 1212 992525 3030 1818

200200 400400 400400


Types of transportation Types of transportation problemsproblems

Balanced transportation problemsBalanced transportation problems

Unbalanced transportation problemsUnbalanced transportation problems

m

i

n

jji ba

1 1

m

i

n

jji ba

1 1

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= .1 11 1

lyrespectivebaorabm

i

n

jji

n

j

m

iij


Example Example

11 22 33 44 55 DemanDemand d

1010 22 33 1515 99 2525

55 1010 1515 22 44 3030

1515 55 1414 77 1515 2020

2020 1515 1313 -- 88 3030

2020 2020 3030 1010 2525Supply

W A 1RE 2HO 3US 4E

Plant


Solution of transportation Solution of transportation problemsproblems

Two phases:Two phases: First phase:First phase: Find an initial feasible solutionFind an initial feasible solution 22ndnd phase: phase: Check for optimality and improve the Check for optimality and improve the

solution solution


Find an initial feasible solutionFind an initial feasible solution

North west corner methodNorth west corner method Least cost methodLeast cost method Vogel’s approximation methodVogel’s approximation method


Checking for optimality Checking for optimality

U-V methodU-V method Stepping-Stone methodStepping-Stone method


ExampleExample-( having a degenerate solution)-( having a degenerate solution) Introduce Introduce to for phase 2.. to for phase 2..

11 22 33 SupplySupply

33 22 33 2525

55 66 55 1515

11 33 44 2020

22 55 77 1010

2020 2020 3030

Sources S1 S2 S3

S4

Demand

Destinations


Transshipment modelsTransshipment models.. In transportation problems ,shipments are In transportation problems ,shipments are

sent directly from a particular source to a sent directly from a particular source to a particular destination to minimize the total particular destination to minimize the total cost of shipments.cost of shipments.

It is sometimes economical if the shipment It is sometimes economical if the shipment passes through some transient nodes in passes through some transient nodes in between the sources and destinations.between the sources and destinations.

In transshipment models it is possible for In transshipment models it is possible for a shipment to pass through one or more a shipment to pass through one or more intermediate nodes before it reaches its intermediate nodes before it reaches its destination.destination.


Transshipment problem with sources and Transshipment problem with sources and

destinations acting as transient nodesdestinations acting as transient nodes Number of starting nodes as well as the Number of starting nodes as well as the

number of ending nodes is the sum of number number of ending nodes is the sum of number of sources and the number of destinations of of sources and the number of destinations of the original problem.the original problem.

Let Let B=B=

be the buffer stock and it is added to all the be the buffer stock and it is added to all the starting nodes and all the ending nodes.starting nodes and all the ending nodes.

n

jj

m

ii ba

11


S1 .. .... ..

… … … …

Dn

D1

Dn

D1

Sj

Sm Sm

Sj

S1a1+B

aj+B

am+B

B

B

B

B

B

b1+B

bn+B


Destinations D1,D2,….Dn are included as Destinations D1,D2,….Dn are included as additional starting nodes mainly to act as additional starting nodes mainly to act as transient nodes.they don’t have any original transient nodes.they don’t have any original supply and the supply of these nodes supply and the supply of these nodes should be at least B.should be at least B.

The sources S1,S2,….Sm are included as The sources S1,S2,….Sm are included as additional ending nodes mainly to act as additional ending nodes mainly to act as transient nodes.these nodes are not having transient nodes.these nodes are not having any original demand.But each of these any original demand.But each of these transient nodes is assigned with B units as transient nodes is assigned with B units as the demand value.the demand value.


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


Example Example Supplies at the sources are 100,200,150 and 350 and Supplies at the sources are 100,200,150 and 350 and

Demand at the destinations are 350 and 450 Demand at the destinations are 350 and 450 respectively.respectively.

S1S1 S2S2 S3S3 S4S4 D1D1 D2D2

S1S1 00 44 2020 55 2525 1212

S2S2 1010 00 66 1010 55 2020

S3S3 1515 2020 00 88 4545 77

S4S4 2020 2525 1010 00 3030 66

D1D1 2020 1818 6060 1515 00 1010

D2 D2 1010 2525 3030 2323 44 00


S1S1 S2S2 S3S3 S4S4 D1D1 D2D2

S1S1 00 44 2020 55 2525 1212 800+100=900800+100=900

S2S2 1010 00 66 1010 55 2020 800+200=1000800+200=1000

S3S3 1515 2020 00 88 4545 77 800+150=950800+150=950

S4S4 2020 2525 1010 00 3030 66 800+350=1150800+350=1150

D1D1 2020 1818 6060 1515 00 1010 800800

D2 D2 1010 2525 3030 2323 44 00 800800

800800 800800 800800 800800 800+35800+350=11500=1150

800+45800+450=12500=1250

Same algorithms can be used to solve this transshipment problem.


Transportation problem with some transient Transportation problem with some transient nodes between sources and destination.nodes between sources and destination.

Consider the case where the shipping items Consider the case where the shipping items are first sent to intermediate finished goods are first sent to intermediate finished goods ware houses from the supply points/factories ware houses from the supply points/factories and then to the destinations. and then to the destinations.

To solve these problems the capacity at each To solve these problems the capacity at each transient node is made equal to B. transient node is made equal to B.

Where B =Where B =

n

jj

m

ii ba

11


Example Example Multi plant organization has 3 plants and Multi plant organization has 3 plants and

three market places. three market places. The goods from the plants are sent to market The goods from the plants are sent to market

places through two intermediate finished places through two intermediate finished goods warehouses. goods warehouses.

Cost of transportation per unit Cost of transportation per unit between plants between plants and warehousesand warehouses and and warehouses to market warehouses to market placesplaces and also supply values of plants and and also supply values of plants and demand values of the markets are shown in demand values of the markets are shown in the table.the table.


M1M1 M2M2 M3M3 W1W1 W2W2 SUPPLYSUPPLY

P1P1 1515 3030 200200

P2P2 2828 1010 300300

P3P3 3030 1515 400400

W1W1 1010 4040 3030 00 2020W2W2 2525 1515 3535 2525 00DEMANDDEMAND 100100 400400 4040

00900

900

900 900

900

Solution of the problem is same as Ordinary transportation Problems.

transportation problems operational research level 4

Documents