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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