lecture notes on transportation models management science

7/24/2019 lecture notes on transportation models management science

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1

Transportation Problems

Transportation is considered as a specialcase of LP

Reasons?

it can be formulated using LP technique so isits solution

(to p!


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"ere# $e attempt to firstl% define $hatare them and then stud%ing their solutionmethods& (to p'!


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Transportation Problem

e ha)e seen a sample of transportationproblem on slide * in lecture

"ere# $e stud% its alternati)e solutionmethod

+onsider the follo$ing transportationtableau (to p,!

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Re)ie$ of Transportation Problem

Warehouse supply of televisions sets: Retail store demand for television sets:

1- Cincinnati 300 A - New Yor 1!0

"- Atlanta "00 # - $allas "!0

3- %itts&ur'h "00 C - $etroit "00total (00 total )00

*o +tore,romWarehouse

A # C

1 1) 1. 11" 1/ 1" 13

3 13 1! 1(

LP formulation(to p.!


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.

/ Transportation 0ample ( of '!2odel 3ummar% and +omputer 3olution $ith 0cel

inimie 2 1)41A5 1.41#5 1141C5 1/4"A5 1"4"#5 134"C5 1343A5 1!43#5 1(43C

su&6ect to

41A 5 41#5 41 300

4"A5 4"#5 4"C"00

43A5 43#5 43C"00

41A5 4"A5 43A 1!0

41#5 4"#5 43# "!0

41C5 4"C5 43C "00

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Exhibit 4.15

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Transportation Tableau

e 4no$ ho$ to formulate it using LP technique Refer to lecture note

"ere# $e stud% its solution b% firstl% attempting todetermine its initial tableau 5ust li4e the first simple tableau6

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7

3olution toa transportation problem

8nitial tableau

9ptimal solution

8mportant :otes

Tutorials

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initial tableau

Three different $a%s&

:orth$est corner method

The 2inimum cell cost method


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*

:orth$est corner method

3teps&

1> assign largest possible allocation to the cellin the upper lefthand corner of the tableau

> Repeat step 1 until all allocations ha)e beenassigned

'> 3top> 8nitial tableau is obtained

0ample(to p1@!


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1@

:ortheast corner

1.@

3tep 1

2a (1.@#@@!

3tep

1.@

.@

@

1.@

@

@

1.

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8nitial tableau of : corner method

Repeat the abo)e steps# $e ha)e the follo$ingtableau>

3top> 3ince all allocated ha)e been assigned

0nsure that all columns and ro$s added up to its respecti)e totals>

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1

The 2inimum cell cost method

"ere# $e use the follo$ing steps&

3teps&

3tep 1 Aind the cell that has the least cost

3tep & /ssign as much as allocation to this cell

3tep '& Bloc4 those cells that cannot be allocated

3tep -& Repeat abo)e steps until all allocation ha)ebeen assigned>

0ample& (to p1'!


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1'

3tep 1 Aind the cell that has the least cost3tep & /ssign as much as allocation to this cell

3tep 1&

The min cost# so allocate as much resource as possible here

@@

3tep '

3tep &

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1-

3tep '& Bloc4 those cells that cannot be allocated3tep -& Repeat abo)e steps until all allocationha)e been assigned>

3tep '&

@@

@

7.

3econd iteration# step -

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1.

The initial solution

3top> The abo)e tableau is an initial tableaubecause all allocations ha)e been assigned

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1,


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subtracting their t$o of theirleast cost

3tep 1

(;,!

(117!

(.-!

(,-! (;.! (111@!

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1;

3teps D '

"ighest penalt%cost

Step 2:

Step 3:this has the least cost

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1*

3tep -

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@

3tep .3econd 8teration

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1

'rd8teration of


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8nitial tableau for


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9ptimal solution?

8nitial solution from&

:ortheast cost# total cost EF.#*.

The min cost# total cost EF-#..@


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3olution methods

e need a method# li4e the simplemethod# to chec4 and obtain the optimalsolution

T$o methods&

1> 3teppingstone method> 2odified distributed method (29G8!

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3teppingstone method

Let consider the follo$ing initial tableau from the 2in +ost algorithm

These are basic)ariables

There are:onbasic )ariables

Huestion& "o$ can $e introducea nonbasic )ariable into basic )ariable?

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8ntroducea nonbasic )ariable into basic )ariables

"ere# $e can select an% nonbasic )ariable asan entr% and then using the I and steps toform a closed loop as follo$s&

let consider this nonbasic )ariable

Then $e ha)e

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3tepping stone

I

I

The abo)e sa%ing that# $e add min )alue of all )e cells into cell that has I sign# and subtractsthe same )alue to the )e cellsThus# ma )e is min (@@#.! E .# and $e add . to cell /1 and /'# and subtract it from B1 and /'

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3tepping stone

The abo)e tableaus gi)e min cost E .J, I 1@J1@ I 17.J11 17.J- I 1@@J . E F-..

e can repeat this process to all possible nonbasic cells in that abo)etableau until one has the min cost6 :9T a Kood solution method

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Ketting optimal solution

8n such# $e introducing the netalgorithm called 2odified Gistribution(29G8!

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2odified distributed method (29G8!

8t is a modified )ersion of stepping stone method

29G8 has t$o important elements&

1>8t determines if a tableau is the optimal one

>8t tells %ou $hich nonbasic )ariable should be firstl%considered as an entr% )ariable

'>8t ma4es use of steppingstone to get its ans$er ofnet iteration

"o$ it $or4s?(to p'1!


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Procedure (29G8!3tep @& let ui#) # ciL )ariables represent ro$s# columns# and cost in the

transportation tableau# respecti)el%

3tep 1& (a! form a set of equations that uses torepresent all basic )ariables

uiI ) E ciL

(b! sol)e )ariables b% assignone )ariable E @

3tep& (a! form a set of equations use torepresent nonbasic )ariable (orempt% cell! as such

ciL ui ) E 4iL

(b! sol)e )ariables b% using step 1b information

3tep '& 3elect the cell that has the most )e )alue in b3tep -& Mse steppingstone method to allocate resource to cell in step '3tep .& Repeat the abo)e steps until all cells in a has no negati)e

)alue0ample (to p-!(to p'!


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29G8

+onsider to this initial tableau&

3tep @& let ui# ) # ci )ariables represent ro$s# columns# and cost in thetransportation tableau# respecti)el% (to p''!


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''

3tep @

+'/

3tep 1& (a! form a set of equations that uses torepresent all basic )ariables

ui I ) E ci

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'-

ui I ) E ci

(b! sol)e )ariables b% assignone )ariable E @

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3et one )ariable E @

Because $eadded annonbasic )ariable

3tep& (a D b!

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3tep& (a D b!

:ote this ma% loo4 difficult and complicated# ho$e)er# $e can add these


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3tep& (a D b!# alternati)e

1

,@7

1 I

I.

3tep '& 3elect the cell that has the most )e )alue in b

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3tep'

1

1 I

I.

3elect either one# (h%?!These cells mean# introduce it $ill reduce the min N to 1 cost unit

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3tep -& Mse steppingstone method

I

I

Arom here $e ha)e O> (to p-@!


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3tep -& Mse steppingstone method

3tep .& $e repeat steps 1- again for the abo)e tableau# $e ha)e

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3tep .

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3tep . cont

/ll positi)e3T9P

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8mportant :otes

hen start sol)ing a transportationproblem using algorithm# $e need toensure the follo$ing&

1> /lternati)e solution

> Total demand total suppl%

'> Gegenerac%

-> others

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/lternati)e solution

hen on the follo$ing 4 E @

ciL ui ) E 4iL

h%?

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Total demand total suppl%

:ote that# total demandE,.@# and total suppl% E ,@@

"o$ to sol)e it?

e need to add a dumm% ro$ and assign o cost to each cell as such >>

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-,

Gdss

0tra ro$# since Gemand Q suppl%

9ther eample OO (to p-7!


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Gdss

0tra column is added(to p-'!


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Gegenerac%

0ample O>> ie# basic )ariablesin the tableau

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Gegenerac%

m ro$s I n column 1 E the number of cells $ith allocations' I ' 1 E .

8t satisfied>

8f failed? OO considering O>>

(fi)e basic )ariables# and abo)e has . as $ell6!

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Gegenerac%

8f not matched# then $e select an nonbasic )ariable $ith least cheapestcost and considered it as a ne$ basic )ariable $ith assigned @ allocation to it

(note abo)e has onl% - basic )ariableonl%6!

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Gegenerac%

/dded this:ote& $e pic4 this o)er others because it has the least cost for the 2in problem6

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others

1> hen one route cannot be used

/ssign a big 2 cost to its cell

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8f 1@ changed to+annot deli)eredThen $e assigned2 )alue here


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Tutorials

2odule B

1# .# ;# 1'# 1# '-

these questions are attached in the follo$ing

slides


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1


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.,


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.7


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.;


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.*


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lecture notes on transportation models management science

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