Decision Analysis Individual_Project-Transportation SIMPLEX METHODOLOGY

8/11/2019 Decision Analysis Individual_Project-Transportation SIMPLEX METHODOLOGY

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#. Problem $e%inition

*.1 4b-ectie of the study

he ob-ectie is to determine the shortest or feasible routes to be used and the

optimize uantity to be shipped ia each route that $ill proide the minimum total

transportation cost altogether.

*.* %pecific description of the problem

he problems specifically happened at a company called Focker Generators Co. at

5%,. hough specific at the company under certain conditions (limitations

e'isted and certain assumptions and constrains are assumed during planning)! it

can still be apply at larger problems eery$here. %ome limitations of models and

certain assumptions and constraints hae to be made to simplify matters so that

forecasting can be done.

*.0 %copes of the study

his study inoles broad scopes consisting of operational research! decision

science! net$ork flo$ problems! linear programming! transportation! assignment

models! simple' method! stepping2stone method! /odified 6istribution (/46#)

method! heuristic method! 7orth$estern Corner method! /inimal+least cost

method! etc.

. 'o!el Constru"tion

0.1 he techniues used

/ethods like 7orth$estern Corner! /inimum2cost! 3euristic! %/!

%tepping2stone! /46#! etc.

0.* he ob-ectie function

o get the most minimum ob-ectie function alue.

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0.0 he constraints+criteria inoled

Constraints are needed because origins hae limited supply and destinations

hae specific demand. he 1*2ariables belo$ must eual to 8 if one $ants

minimization. , criterion needed is that %/ can only be applied to balanced

problem (total unit of demand must eual to total unit of supply)! if not a

dummy origin or dummy destination $ill be added.

otal ariables 9 m ' n 9 0 ' : 9 1*

otal constraints 9 m ; n 9 0 ; : 9 88 Cleeland %upply

'*1; '**; '*0; '*:= ?88 Bedford %upply

'01; '0*; '00; '0:= >88 "ork %upply

'11; '*1; '01 = ?88 Boston 6emand

'1*; '**; '0* = :88 Chicago 6emand

'10; '*0; '00 = *88 %t. &ouis 6emand

'1:; '*:; '0: = 1>8 &e'ington 6emand

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(. m)lementation an! results

(.1 $etaile! )resentation o% !ata*in%ormation

Figure@ 7et$ork representation of Focker Generators transportation problems

able 1@ ransportation Cost (per unit) for the Focker A ransportation roblem

4rigin 6estination

Boston Chicago %t. &ouis &e'ington

Cleeland 0 * < ?

Bedford < > * 0

"ork * > : >

(.# Results or %in!ings

%/ is a t$o2phase procedure. ,t phase #! ogels ,ppro'imation /ethod

(,/)! 7orth$est Corner method or the >2steps /inimum Cost /ethod (/C/)

can be used. ,/ $ill not be illustrated in this study though it is the method that

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'C'

#n tableau *! u12*! u*20and u021hae the lo$est alue in small bo' ie D*. Whenties bet$een arcs occur! $e follo$ the conention of selecting arc to $hich the

most flo$ can be allocated. #n this case it is u12*or Cleeland2Chicago. ertical

or horizontal at this cell can be choosen! but the lo$est alue $ill be choosen (and

$ritten in the cell) $hich is the ertical :88. his selection reduces >88 to 188E

and eliminates the column by dra$ing a line.

he ne't one is u021because more units of flo$ can be allocated to "ork2

Boston route. Bet$een ?88 or *>8! the lo$est one $ill be zeroedE line $ill be

dra$n. &ike usual ?88 is reduced to 0>8. Continuing at cell u *20! the result of 0

dra$n line is sho$n at tableau 0.

ableau *@ ransportation tableau after one iteration of the /C/

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ableau 0@ ransportation tableau after three iteration of the /C/

We hae no$ t$o arcs that ualify for minimum cost arc $ith alue D0@

u121and u*2:or Cleeland2Boston and Bedford2&e'ington respectiely. We can

allocate a flo$ of 188 units to u121route and a flo$ of 1>8 to u*2:route! so $e

allocate 1>8 units to u*2:route. ,gain! zeroed the lo$est one ie 1>8E dra$ a

ertical line to &e'ington column. he :88 is reduced to *>8 no$. 7e't is the u 12

1routeE ro$ Cleeland is dra$n a line. esult is sho$n at tableau :. ableau ? is

constructed using information from tableau >. From tableau >! total cost is

constructed at table 11.


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ableau :@ ransportation tableau after fie iteration of the /C/

ableau >@ Final tableau using /C/ during hase #

hase ##@ #terating to 4ptimal %olution


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he first step is to identify incoming arc through /46# computation. he

incoming arc is the currently unused route (unoccupied cell) $here making a flo$

allocation $ill cause the largest per2unit reduction in total cost. he # is source

and - is destination. euiring that ui; -9 ci-all for all the occupied cells in the

initial feasible solution leads to a system of si' euations and seen inde'es! or

ariables (belo$). We $ill al$ays choose u19 8! therefore 19 0 and *9 *.

O""u)ie! Cell ui v/ "i

Cleeland H Boston u1; 19 0

Cleeland H Chicago u1; *9 *

Bedford H Boston u*; 19 2(21)2* 9 :!

e0:9 c0:2 u02 :9 >2(21)2(21) 9


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%uppose that $e allocate alue of one unit (21) of flo$ to the incoming

cell (Bedford2Chicago route). o maintain feasibility $e $ould hae to reduce the

flo$ assigned to Cleeland2Chicago by 1! that is into 0II. But then $ed hae to

increase Cleeland2Boston to 181! and finally Bedford2Boston to *:I. %ee tableau

< to . hese : cells form a stepping2stone (cylindrical shape) path $ith the

tableau as pond. lus (;) or negatie (2) sign is placed on those 0 outgoing cells. ,

minus sign indicates that allocation to that cell $ill decrease by the amount

allocated to incoming cell. hus to determine the ma'imum amount allocated to

incoming cell! $e simply look to cells $ith minus sign. Because no cell can hae

a negatie flo$! the minus2sign cell $ith smallest-amount $ill determine the

ma'imum amount that can be allocated to incoming cell. 7e't all the ad-ustments

necessary are made to maintain feasibility. he incoming cell becomes an

occupied cell and outgoing cell is dropped from the solution. Bet$een the minus

sign! 250 unitsis less than :88 units! so $e identified Bedford2Boston as outgoing

arc. We then obtained ne$ solution by allocating *>8 units to Bedford2Chicago

arcE and making appropriate ad-ustments on first ro$ accordingly. Bedford2

Boston has been dropped from solution (its allocation has been drien to zero) at

tableau I.

ableau


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here are > unused cells aboe. #mproement inde' K*L for one of the

unused cell in tableau aboe is represented by #BiCii. #BiCii9 BiCiiH CiCii; CiBiiH

BiBii9 >2*;02< 9 21. 7e't unused cell no need to find anymore as this is not

optimal solution. here $ill be optimal solution only if the all of the > unused

cells hae zero or positie improement inde'. 4ther$ise! iterations hae to be

done again. #t can be easily done using computer.

ableau I@ 7e$ %olution after 4ne #teration in hase ##

'O$

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We no$ try to improe on the current solution from tableau I. ,gain the

first step is to apply /46# method to find best incoming arc! so recomputed ro$

and column inde'es by ci-9 ui; -for all occupied cells. %etting u19 8! c11and

c1*is 1and *$hich is 0 and * respectiely. hus u*; *9 > (u*is 0)! 09 * Hu*9 21 and :9 0 H u*9 8. %ee tableau 18.

u19 8 19 0

u*9 0 *9 *

u19 21 09 21

:9 8

For each unoccupied cell at tableau 18! ei-9 ci-2 ui2 -. hus e109 c102

u12 09 2(21)28 9 ?! and so on. 7ote that

net ealuation inde' for eery occupied cell is no$ eual+greater than zero. his

condition sho$s that if current unoccupied cells are used! the cost $ill actually

increase. Without an arc to $hich flo$ can be assigned to decrease the total cost!

$e hae reached optimal solution. Finally ob-ectie function alue of D0I>8 is

obtained at last (see tableau 1*). ,s e'pected! this solution is e'actly the same as

the one using the linear programming solution approach.

ableau 18@ /46# Jaluation of Jach Cell in %olution

able 11@ otal Cost of #nitial Feasible %olution 4btained 5sing /C/

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oute 5nits %hipped Cost er 5nit

(D)

otal Cost

(D)From o

Cleeland Boston 188 0 300

Cleeland Chicago :88 * 800

Bedford Boston *>8 < 1750

Bedford %t. &ouis *88 * 400

Bedford &e'ington 1>8 0 450

"ork Boston *>8 * 500

otal 10>8 1I :*88

able 1*@ 4ptimal %olution to Focker Generators ransportation ,lgorithms

oute 5nits %hipped Cost er 5nit

(D)

otal Cost

(D)From o

Cleeland Boston 0>8 0 18>8

Cleeland Chicago 1>8 * 088Bedford Chicago *>8 > 1*>8

Bedford %t. &ouis *88 * :88

Bedford &e'ington 1>8 0 :>8

"ork Boston *>8 * >88

otal 10>8 1< 0I>8

4b-ectie function alues of both methods in #nitial %olution for 7$C and /C/

method is D>?>8 and D:*88 respectiely! roughly close to the alue of D0I>8 optimal

solution. /C/ is proen to gie better and more accurate total cost than 7$Cmethod.

3. Con"lusions

3.1 A!vantages an! !isa!vantages o% the # te"hni4ues use!

4ne important characteristic of assignment problems is that only one

supply! -ob or $orker is assigned to one demand! machine or pro-ect. Whilst in

real2life! things can be more complicated and comple'. K*L ,dantages of

&east Cost /ethod are (1) his method proides accurate solution as

transportation cost is consider $hile making allocationE (*) #t is ery simple

and easy to calculate optimum solution under this method. 6isadantages of

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&east Cost /ethod are (1)@ his method does not follo$ steps by step rule

for obtaining optimal solutionE (*) his method is based on the selection

through personnel obseration $hen there is a tie in the minimum cost it

does not follo$ any systematic rule. K:L ,dantages of north$est corner

method are it is simple compare to ,/ or /C/. he disadantages are it

only consider and start from cell at north$est corner! $hich doesnt happen in

all of the cases! anytime. Both /46# and %tepping2stone method need to be

used in phase ##! so no disadantages can be ruled out.

3.# 5ene%its o% the usage*a))li"ation to the bene%i"iaries

7$C is easy to understand and use for layman $ithout dealing $ith too much

-argon! steps and technicalities. /C/ can be used for more comple' issueE

usually mimic the total cost closer to optimal solution rather than 7$C. ,/

approach gies the closest alue of cost to optimal solution.

3. Re"ommen!ation

But all this difficulties in phase # can be alleiate if one using computer

soft$ares like / or modeler like ,rena to sole comple' issues that need

many iterations! $hich can impossibly happen if done manually. #t is highly

suggested for firm to use and familiarize $ith these soft$ares rather than

manual calculations. #t is highly recommend that ,/ is used during phase #

to get more accurate cost close to optimal alue. /ore and further studies

need to be done to modify the models! limit the gaps bet$een simplified

solutions $ith real2$orld cases. ,ssumptions need to be more defined and

made as fe$ as possible.

%uggestions on oercoming 6egeneracy

, solution to a transportation problem that has less than m;n21 cells $ith

positie allocations is said to be degenerate. K1L o handle degenerate

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problems! create an artificially occupied cell. hat is! place a zero

(representing a fake shipment) in one of the unused suares and then treat that

suare as if it $ere occupied. he suare chosen must be in such a position as

to allo$ all stepping2stone paths to be closed. here is usually a good deal of

fle'ibility in selecting the unused suare that $ill receie the zero. K*L

6egeneracy can still happens during later solution stages. , transportation

problem can become degenerate after the initial solution stage if the filling of

an empty suare results in t$o or more cells becoming empty simultaneously.

his problem can occur $hen t$o or more cells $ith minus signs tie for the

lo$est uantity. o correct this problem! place a zero in one of the preiously

filled cells so that only one cell becomes empty. K*L

3.( 6imitations

#n real2$orld! there are many special cases in transportation algorithm that present

limitations like belo$@

i) %upply uneual to demand

/ost of the time supply does not eual to demand. #f total supply e'ceeds total

demand! no modification in linear programming formulation is necessary. #f total

supply is less than total demand! the linear programming model of a

transportation model $ill not hae a feasible solution. , dummy origin $ill be

added. #n either case! shipping cost coefficients of zero are assigned to each

dummy location or route as no goods $ill actually be shipped. ,ny units assigned

to a dummy destination represent e'cess capacity. ,ny units assigned to a dummy

source represent unmet demand. K1L

ii) 6egeneracy

6egeneracy occurs $hen the number of occupied suares or routes in a

transportation table solution is less than the number of ro$s plus the number of

columns minus 1. %uch a situation may arise in the initial solution or in any

subseuent solution. 6egeneracy reuires a special procedure to correct the

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problem since there are not enough occupied suares to trace a closed path for

each unused route and it $ould be impossible to apply the stepping2stone method.

K*L

iii) /ore han 4ne 4ptimal %olution

#t is possible for a transportation problem to hae multiple optimal solutions. his

happens $hen one or more of the improement indices are zero in the optimal

solution. his means that it is possible to design alternatie shipping routes $ith

the same total shipping cost. he alternate optimal solution can be found by

shipping the most to this unused suare using a stepping2stone path. #n the real

$orld! alternate optimal solutions proide management $ith greater fle'ibility in

selecting and using resources. K*L

i) /a'imization 4b-ectie Function

#n some transportation problems! the ob-ectie is to find a solution that ma'imizes

profits. 5sing the alues for profits per unit as coefficients in the ob-ectie

function! a ma'imization is simply soled rather than a minimization linear

program. his change does not affect the constraints. K1L

) 5nacceptable oute

,t times there are transportation problems in $hich one of the sources is unable to

ship to one or more of the destinations. he problem is said to hae an

unacceptable or prohibited route. #n a minimization problem! such a prohibited

route is assigned a ery high cost (;/) to preent this route from eer being used

in the optimal solution. #n a ma'imization problem! the ery high cost (2/) used

in minimization problems is gien a negatie sign! turning it into a ery bad

profit. K*L Jstablishing a route from eery origin to eery destination may not be

possible. o handle it! $e -ust drop the corresponding arc or branch from the

net$ork and remoe the corresponding ariable from the linear programming

formulation. #f applied aboe! there $ill be resulting 112ariable!


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function coefficient (2/ for profit $hile ;/ for cost) into unacceptable arc. #f the

problem has already been formulated! other option is to add a constraint to the

formulation that sets the ariable you $ant to remoe eual to zero. K1L

3.3 uture ,or7s

/any future $orks need to be done oercome the limitations! constraints!

etc described aboe. heories learned need to be applied. ,s said! more

studies need to be done! especially in local conte'ts. ecommendations aboe

are adised to be constantly applied. ,ssumptions need to be more precise and

should be made as fe$ as possible. /ore comparisons among the techniues

used need to be studied. 7e$er models should be created too.

8. Re%eren"es

K1L ,nderson! 6. .! %$eeney! 6. M.! Williams! . ,.! N /artin! O. (*88). An introduction to

management science: Quantitative approaches to decision making (12th Ed.). 4hio@

%outh2Western College ublishing.

K*L ender! B.! %tair Mr! . /.! N 3anna! /. J. (*81*). Quantitative analsis !or management

(11thEd.).rentice 3all.

K0L ,ailable from http:""###.e$pertsmind.com"%uestions"advantage-o!-least-cost-method-

&01&&'15.asp$. ,ccessed on *< Mune *81:.

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Decision Analysis Individual_Project-Transportation SIMPLEX METHODOLOGY

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