Int. J.IndustrialMathematisVol. 2,No. 3(2010)231-235
Prodution Planning Using Fixed Levels of
Inputs: A DEA-based Approah
A.Amirteimoori a
,S.Kordrostami b
(a)Departmentof AppliedMathematis,Rasht Branh,IslamiAzadUniversity,Rasht,Iran.
(b)DepartmentofAppliedMathematis,LahijanBranh,IslamiAzadUniversity,Lahijan,Iran.
Reeived15July2010; revised10September2010; aepted 18September2010.
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Thetraditionalprodutionplanningbasedonlinearprogramminghasbeenstudied
exten-sivelyintheliterature. Inthispaper,aDEA-based produtionplanningideaisproposed.
Itisassumed thatsuppliesfortheinputsof thenew operationalunitan beforeastedin
thenextprodutionseason. Moreover,theinput-outputlevelsoftheexistingunitsremain
unhanged. With these foreasted inputs, the paper develops a DEA-based prodution
planning approah to determine themost favorable produtionplan for new operational
unit.
Keywords: DEA;EÆieny;Produtionplanning.
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1 Introdution
Data Envelopment Analysis (DEA), is urrently a populartehnique for analyzing
teh-nialeÆienyofasetofomparabledeisionmakingunits(DMUs)withmultipleinputs
and outputs and it has had a number of appliations. In DEA, we assume that the
as-sessedunitsarehomogeneous,i.e. theyperformthesametaskswithsimilarobjetivesand
onsume similarinputs and produe similaroutputs. Moreover, they operate in similar
operationalenvironments. Duringthereent years, manyappliationsof DEAhave been
studied. Animportantappliationofthistehniqueisinsolvingtheproblemofprodution
planning. Many authorshave studied theprodutionplanningfrom various perspetives.
Chazalet al. (2008)studiedtheprodutionplanningandinventorymanagement problem
basedontheassumptionthatthermunderdisussionatsinontinuoustimeonanite
periodinorder to dynamiallymaximizeits instantaneousprot. UsingDEA in
produ-tion planning or similar studies is notnew. Golany (1988) was the rst that used DEA
in prodution planning. He presentedan interative multi-objetive linear programming
proedure to generate a set of alternative eÆient points for DMU to onsider. Du et
al.(2010) lookedat theprodutionplanningproblem from theprodutivityand eÆieny
perspetiveusingdataenvelopmentanalysis. Theyproposedtwo planningideasina
en-tralizeddeision making environment when demandhanges an be foreasted. One was
optimizingthe average oroverall produtionperformaneof the entire organization,and
theother one wassimultaneouslymaximizingtotal outputsproduedand minimizing
to-tal inputsonsumedbyall units. Intheurrentstudy, we present a produtionplanning
approah foranew operationalunit byusingaxed levelof inputs. We onstrut a new
operationalunitandsuppliesfortheinputsofthisnewoperationalunitan beforeasted
in the next prodution season. Moreover, it is assumed that the input-output levels of
theexistingunitsremain unhanged. Withtheseforeasted resouresfornewoperational
unit, the paper develops a DEA-based prodution planning approah to determine the
mostfavorable produtionplan. A prinipal idea behind theproedure isto usethe
em-pirialprodutionfuntiondened bythe observed inputsand outputs to make the new
operationalunitasmostprodutivesalesize(MPSS).Thepaperisstruturedasfollows:
A briefbakgroundof DEAis presentedin setion2. The proposed produtionplanning
approah is given in setion 3. Setion 4 gives a simple numerial example. Conlusions
appearinsetion 5.
2 Preliminaries
ThetehnologysetTisextrapolatedfromtheobserveddataoninput-outputpairsf(x
j ;y
j )jj=
1;:::;ng in whih x
j = (x
1j ;x
2j ;:::;x
mj
) 0 and y
j = (y
1j ;y
2j ;:::;y
sj
) 0 are the
nonzero inputand output vetors, respetively, orresponding to DMU
j
. Charnes et al.
(1978) onstrutedthetehnologyset T
as
T
=
8
<
:
(x;y)j x n
X
j=1
j x
j
; y n
X
j=1
j y
j
;
j
0; j=1;:::;n 9
=
;
The lassialDEA radialinputmeasure ofeÆieny isalulatedas
Min f j(x
o ;y
o )2T
g
theabove formulation an berestated as
Min
s:t: P
n
j=1
j x
ij x
io
; i=1;:::;m
P
n
j=1
j y
rj y
ro
; r=1;:::;s
j
0 for allj
(2.1)
Bankeret al. (1984)took thenatureof returns to saleinto aount and onstrutedthe
tehnology setT
as
T
=
8
<
:
(x;y) j x n
X
j=1
j x
j
; y n
X
j=1
j y
j ;
n
X
j=1
j
=1; j =1;:::;n 9
=
Denition 2.1. A surfae
H=f(x;y) j u t
y v t
x=0; u0; v0g\T
is alled an eÆient supporting surfae of T
if for eah extreme eÆient observation j,
u t
y
j v
t
x
j 0.
Banker (1984)introduedtheonept ofMPSS asfollows:
Denition2.2. Aprodutionpossibility(x;y)2T
isBanker'sMPSSifforall(x;y)2
T
, we have
1.
This denition shows that Banker's MPSS must be assoiated with boundary points
and maximizes, average produtivity,
, for the DMU(x;y). Cooper, Thompson and
Thrall(1996)proposedthefollowinglinearfrationalprogrammingproblemto determine
theMPSS inT
:
Max
s:t: y
o
P
n
j=1
j y
rj
; r=1;:::;s
x
o
P
n
j=1
j x
ij
; i=1;:::;m
P
n
j=1
j =1;
;;
j
0 for all j
(2.2)
Theystatedthat DMU
o
is MPSSifthefollowing two onditionsaresatised:
(i)
(ii) Allslakvariablesmustbe zeroinanyoptimalsolution.
3 Prodution planning model
We assume that there are nDMUs indexedbyfDMU
j
j j=1;:::;ng. The i th input
andr thoutputofDMU
j
aredenotedbyfx
ij
j(i=1;:::;m)gandfy
rj
j(r=1;:::;s)g,
respetively. Supposethat thesupply forinputfij i=1;:::;mg in thenext prodution
season an be foreasted as fx
i
j i = 1;:::;mg. For these foreasted supplies, we will
determinethemostfavorableoutputplansforthenewoperationalunit. Werstsolvethe
followinglinearprogrammingproblem inorder to ndthemostfavorableeÆientsurfae
of the produtionpossibilityset T
that the new operational unit DMU
n+1
is projeted
to:
Min P
n
j=1 s
j
s:t: P
s
r=1 u
r y
rj P
m
i=1 v
i x
ij +s
j
=0; j=1;:::;n
P
m
i=1 v
i x
i =1;
u
r ;v
i
0 for alli;r
(3.3)
Letu
, v
and s
be an optimalsolutionto (3.3). It iseasy to show thats
j
=0forsome
j2f1;2;:::;ng. Moreover, theoptimalsolutionto (3.3) givesa supportingsurfaeof T
asfollows:
H=f(x;y) j u
y v
DMUsthatdo belongto H arelearlyMPSS inT
.
Now, eah produtionplanthat lieson H, makes thenew operationalunit asMPSS.
The setof all optimalprodutionplansthatmake DMU
o
asMPSS isdened asfollows:
P
n+1 =
(
(y
1 ;y
2 ;:::;y
s ) j
s
X
r=1 u
r y
r
=1; y
r l
r
; r =1;:::;s )
l
r
0 is a user- dened onstant to reet a lower bound on the r th output. One
prodution plan, for example, an be determined as y
r =
1
su
r
, r = 1;:::;s. Clearly,
P
s
r=1 u
r y
r
=1and hene DMU
n+1
isCCR eÆient.
Amulti-objetivelinearprogramming(MOLP)modelisdevelopedasfollowsto
simul-taneouslymaximize alloutput produtions:
Max fy
1 ;y
2 ;:::;y
s g
s:t: P
s
r=1 u
r y
r =1;
y
r l
r
; r=1;:::;s
(3.4)
Taking the priorityof the outputsinto aount, we an develop thefollowing linear
pro-grammingmodelasequivaleneto model(3.4):
Max
s:t: P
s
r=1 u
r y
r =1;
r y
r
; r=1;:::;s
y
r l
r
; r=1;:::;s
(3.5)
where
r
s represent positivevaluesthat reettheimportaneof y
r s with
P
s
r=1
r =1.
Theorem 3.1. Planned by the foregoing planning proedure, the new operational unit
DMU
n+1
an beome a MPSS when evaluated under the original prodution possibility
set.
Proof: SineDMU
n+1
2H, theproof islear.
4 Numerial example
Suppose thet we have data on 10 DMUswith two outputs and inputsas given inTable
1 and suppose that we are given a resoure alloation vetor (x
1 ;x
2
) = (6:5;6:2) (This
exampleistaken from Golany(1988)). Solvingtheprogram(3.3) numeriallyyields
u
1
=0:202808
u
2
=0:062402
u
3
=0:109204
u
Thissolutiongivesthe followingsupportingsurfaeof T
:
H=f(x
1 ;x
2 ;y
1 ;y
2
) j 0:202808y
1
+0:062402y
2
0:109204x
1
0:046802x
2
=0g\T
The setof all optimalprodutionplansisdetermined as
P =f(y
1 ;y
2
) j 0:202808y
1
+0:062402y
2
=1; y
1 ;y
2 0g
Clearly,
y
1 =
1
2(0:202808)
=2:465386; y
2 =
1
2(0:062402)
=8:012564
is an optimal prodution plan for DMU
11
in the next prodution season and with this
plan,DMU
11
isaMPSS.Ifweinorporatethepriorityofoutputsas
1
=0:3and
2 =0:7
and applytheLP model (3.5),theoptimalplan isdetermined as:
y
1
=0:4356317; y
2
=1:866993
Clearly, DMU
11
with(y
1 ;y
2
)=(0:4356317;1:86 69 93 ) isa MPSS.
5 Conlusion
In this paper, a DEA-based approah has been developed for making future prodution
plans for a new operational unit. The proedure proposed in this paper, is aimed at
assisting a new deisionmakerin setting themostfavorable produtiongoals. The
prin-ipalideabehindtheproedureisto usetheempirialprodutionfuntiondenedbythe
observedinputs andoutputsto make thenew operationalunit asMPSS.
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