Seediscussions,s a s,andau ho p o iles o  hispublica iona :h ps://www. esea chga e.ne /publica ion/39427516
Capi albudge ingp oblemswi h uzzy
cash lows
ARTICLE·SEPTEMBER1999
Sou ce:OAI
CITATIONS
16
READS
37
2AUTHORS:
Ch is e Ca lsson
ÅboAkademiUni e si y
307PUBLICATIONS4,332CITATIONS
SEEPROFILE
Robe Fulle
ÓbudaiEgye em
207PUBLICATIONS3,618CITATIONS
SEEPROFILE
A ailable om:Robe Fulle
Re ie edon:24Feb ua y2016
Capi al budge ing p oblems wi h uzzy cash lows∗
Ch is e Ca lsson
ch is e .ca lsson@abo. i
Robe Full´e
ulle @abo. i
Abs ac
We conside he in e nal a e o e u n (IRR) decision ule in capi al budge -
ing p oblems wi h uzzy cash lows. The possibili y dis ibu ion o he IRR a
any ≥0, is de ined o be he deg ee o possibili y ha he ( uzzy) ne p esen
alue o he p ojec wi h discoun ac o equals o ze o. Gene alizing ou ea -
lie esul s on uzzy capi al budege ing p oblems [5] we show ha he possibili y
dis ibu ion o he IRR is a highly nonlinea unc ion which is ge ing mo e and
mo e unbalanced by inc easing imp ecision in he u u e cash low. Howe e ,
i is s able unde small changes in he membe ship unc ions o uzzy numbe s
ep esen ing he lingusi ic alues o u u e cash lows.
Keywo ds: Capi al budge ing p oblem, in e nal a e o e u n, possibili y dis ibu ion,
sensi i i y analysis
1 In oduc ion
Many decision making p oblems conce n p ojec s in which he cos and bene i s acc ue
o e a numbe o yea s. In his pape we conside only cases in which he cos s and
bene i s a e en i ely mone a y, such as he capi al budege ing o capi al in es men
decisions a ising in comme ce and indus y. Au ho s conside wo kinds o decision
p oblems in capi al budge ing: accep -o - ejec and anking. In accep -o - ejec deci-
sions, each p ojec is conside ed independen ly o all o he p ojec s. Thus a po olio o
accep ed p ojec s is buil up om se e al independen decisions. In anking decisions,
all he a ailable p ojec s a e compa ed and anked in o de o a ou abili y wi h he
in en ion o adop ing a single p ojec : he mos a ou able. I should be no ed ha i
is o en impo an o include a null p ojec ep esen ing he s a us quo; all he p ojec s
may be un a ou able compa ed wi h he al e na i e o adop ing none o hem (i his
is possible). Se e al decision ules ha e been sugges ed [1, 6, 9] o help decision mak-
e s ank p ojec s which in ol e imes eams o cos s and bene i s, such as he payback
pe iod,accoun ing a e o e u n (ARR), in e nal a e o e u n (IRR) and ne p esen
alue (NPV).
We shall b ie ly desc ibe jus he IRR decision ule. Le {a0, a1, . . . , an}be a gi en ne
cash low o a p ojec ao e npe iods. We assume ha a0<0 as he p ojec s a s
∗appea ed in: C.Ca lsson and R.Full´e , Capi al budge ing p oblems wi h uzzy cash lows, Ma h-
wa e and So Compu ing, 6(1999) 81-89. [Zbl 0971.68148]
1
wi h an ini ial in es men . The IRR, deno ed by ∗∗, is de ined o be he alue o
such ha he NPV o he p ojec is ze o. Thus ind he IRR o awe need o sol e
S(a, ) := a0+a1
1 + +· · · +an
(1 + )n= 0 (1)
I is well-known ha , i he e is ein es men in a p ojec (ai<0 o some i≥1) hen
i s IRR may become ill-de ined, i.e. equa ion (1) may ha e mo e han one solu ion. I
he IRR o a p ojec is ill-de ined, i is no a sui able c i e ion o use in ei he accep -
o - ejec o anking decisions. Suppose, howe e , ha no p ojec conside ed in ol es
any ein es men . Then NPV is a s ic ly mono one dec easing unc ion o and he
equa ion (1) has a unique solu ion, mo eo e , he discoun a e can be in e p e ed
in s ic ly inancial e ms as an in e es a e. Now in an accep -o - ejec decision i is
clea ha , i he ma ke a e o in e es is 0, he p ojec should be accep ed i ∗∗ > 0
because his implies he ha NPV a 0is posi i e. In compa ing wo p ojec s, he
one wi h he highe IRR should be p e e ed.
2 IRR wi h uzzy cash lows
Mo e o en han no u u e cash lows (and in e es a es) a e no known exac ly, and
we ha e o wo k wi h hei es ima ions, such as ’a ound 5,000 in he nex ew yea s’ (o
’close o 3 %’). Fuzzy numbe s appea o be an adequa e ool o ep esen imp ecisely
gi en cash lows [3, 4, 7, 13, 14].
De ini ion 2.1 A uzzy numbe Ais a uzzy se o he eal line wi h a no mal, ( uzzy)
con ex and con inuous membe ship unc ion o bounded suppo . The amily o uzzy
numbe s will be deno ed by F.
A uzzy se Ais called a symme ic iangula uzzy numbe wi h cen e aand wid h
α > 0 i i s membe ship unc ion has he ollowing o m
A( ) = 


1−|a− |
αi |a− | ≤ α
0 o he wise
and we use he no a ion A= (a, α). I α= 0 hen Acollapses o he cha ac e is ic
unc ion o {a} ⊂ IR and we w i e A= ¯a.
We will use symme ic iangula uzzy numbe s o ep esen he alues o he linguis ic
a iable [16] cash.
I A= (a, α) and B= (b, β) a e uzzy numbe s o symme ic iangula o m and
λ∈IR hen A+B,A−Band λA a e de ined by he ex ension p inciple in he usual
way:
A+B= (a+b, α +β), A −B= (a−b, α +β), λA = (λa, |λ|α).
Fu he mo e, i Ai= (ai, αi) and λi= 1/(1 + )i,i= 0,1, . . . , n, hen we ge
A0+
n
X
i=1
Ai
(1 + )i=a0+a1
1 + +· · · +an
(1 + )n, α0+α1
1 + +· · · +αn
(1 + )n.(2)
2
Le Aand B∈ F be uzzy numbe s. The deg ee o possibili y ha he p oposi ion ”A
is equal o B” is ue deno ed by Pos[A=B] and de ined by he ex ension p inciple as
Pos[A=B] = sup
x∈IR
min{A(x), B(x)}= (A−B)(0),(3)
The Hausdo dis ance o Aand B, deno ed by D(A, B), is de ined by [12]
D(A, B) = max
θ∈[0,1] max {|a1(θ)−b1(θ)|,|a2(θ)−b2(θ)|}
whe e [a1(θ), a2(θ)] and [b1(θ), b2(θ)] deno e he θ-le el se s o Aand B, espec i ely.
Fo example, i A= (a, α) and B= (b, α) a e uzzy numbe s o symme ic iangula
o m wi h he same wid h α > 0 hen
D(A, B) = |a−b|.
Lemma 2.1 [10] Le δ > 0be a eal numbe , and le A= (a, α)and B= (b, β)be
symme ic iangula uzzy numbe s. Then om he inequali y D(A, B)≤δi ollows
ha
sup
∈IR
|A( )−B( )| ≤ max (δ
α,δ
β).(4)
Le {A0= (a0, α0), A1= (a1, α1), . . . , An= (an, αn)}be a gi en ne uzzy cash low o
a p ojec Ao e npe iods. By eplacing he c isp cash low alues wi h uzzy numbe s
in (1) we ge
A0+A1
1 + +· · · +An
(1 + )n=¯
0 (5)
whe e he equa ion is de ined in possibilis ic sense, and ¯
0 deno es he cha ac e is ic
unc ion o ze o. Tha is, he uzzy solu ion [2] o (5) is compu ed by
µIRR( ) = PosA0+
n
X
i=1
Ai
(1 + )i=¯
0=A0+
n
X
i=1
Ai
(1 + )i(0).
o each ≥0. Using he de ini ion o possibili y (3) and ep esen a ion (2) we ind
µIRR( ) = 




1−|S(a, )|
S(α, )i |S(a, )| ≤ S(α, ),
0 o he wise
whe e we used he no a ions
S(a, ) = a0+a1
1 + +· · · +an
(1 + )n, S(α, ) = α0+α1
1 + +· · · +αn
(1 + )n
We assume ha a0<0 ( he p ojec s a s wi h an ini ial in es men ), a0≤a1+· · ·+an
( he p ojec is a leas as good as he null p ojec ), and ai≥0, i = 1, . . . , n, (no
ein es men ). In his case we always ge quasi- iangula uzzy numbe s o IRR in
IR+
0and equa ion (5) has a unique maximizing solu ion, ∗, such ha ,
µIRR( ∗) = max
≥0µIRR( ) = 1,
3
and ∗coincides wi h ∗∗, which is he in e nal a e o e u n o he (c isp) p ojec
a= (a0, a1, . . . , an). Really, i ≥0 hen µIRR( ) = 1 i and only i S(a, ) = 0.
As an example conside a 4-yea p ojec
A={(−5, α),(3, α),(4, α),(6, α),(10, α)},
wi h uzzy IRR,
µIRR( ) = 












1−
−5 + 3
1 + +4
(1 + )2+6
(1 + )3+10
(1 + )4
α1 + 1
1 + +1
(1 + )2+1
(1 + )3+1
(1 + )4i |S(a, )| ≤ S(α, ),
0 o he wise
I is easy o compu e ha µIRR(0.781) = 1 o all α≥0, so he maximizing solu ion
o possibilis ic equa ion (5) is independen o α.
Howe e , he possibili y dis ibu ion o he IRR is ge ing mo e and mo e unbalanced
as he wid hs o he uzzy numbe s a e g owing. This means ha when compa ing
he uzzy IRR wi h he ma ke in e es a e 0in an accep -o - ejec decision, he
de uzzi ied alue o µIRR will de ini ely di e om ∗whene e he p ocess o de uzzi-
ica ion akes in o accoun all poin s wi h posi i e membe ship deg ees (and no only
he maximizing poin ).
Fo example, all p ojec s in Figs.1-3, ha e he same maximizing solu ion ∗= 0.781,
bu i we employ he cen e -o -g a i y me hod hen he de uzzi ied alue o he p ojec
wi h α0=α1=· · · =αn= 5 is a ound 0.84, which is esen ially bigge (in e ms o
a es o e u n) han 0.781.
In anking decisions we ha e o compa e possibili y dis ibu ions o a non-symme ic
quasi- iangula o m.
3 Sensi i i y analysis in uzzy capi al budge ing
Conside wo p ojec s A={A0, A1, . . . , An}and Aδ={Aδ
0, Aδ
1, . . . , Aδ
n}wi h uzzy
cash lows Ai= (ai, αi) and Aδ
i= (aδ
i, αi), i= 0,1, . . . , n. The uzzy IRR o p ojec
Aδ, deno ed by µδ
IRR, is compu ed by
µδ
IRR( ) = PosAδ
0+
n
X
i=1
Aδ
i
(1 + )i=¯
0=Aδ
0+
n
X
i=1
Aδ
i
(1 + )i(0).
o each ≥0. Using he de ini ion o possibili y (3) and ep esen a ion (2) we ind
µδ
IRR( ) = 




1−|S(aδ, )|
S(α, )i |S(aδ, )| ≤ S(α, ),
0 o he wise
whe e we used he no a ion
S(aδ, ) = aδ
0+aδ
1
1 + +· · · +aδ
n
(1 + )n.
4

Le ∗∗(δ) deno e he IRR o he c isp p ojec aδ= (aδ
0, aδ
1, . . . , aδ
n). Tha is,
S(aδ, ∗∗(δ)) = aδ
0+aδ
1
1 + ∗∗(δ)+· · · +aδ
n
(1 + ∗∗(δ))n= 0 (6)
In he ollowing we suppose ha ∗∗(δ) is he only solu ion o equa ion (6), i.e. aδ
0<0
and aδ
i>0 o i= 1, . . . , n.
The nex heo em shows ha i he cen e s o uzzy numbe s Aiand Aδ
iin p ojec s
Aand Aδa e close o each o he s, hen he e can only be a small de ia ion in he
possibili y dis ibu ions o hei uzzy IRR.
Theo em 3.1 Le δ > 0be a eal numbe . I
max{|a0−aδ
0|,|a1−aδ
1|,...,|an−aδ
n|} ≤ δ
hen
max
≥0|µIRR( )−µδ
IRR( )| ≤ min 1,δ
αmax.(7)
whe e
αmax = max{α0, α1, . . . , αn}
µIRR and µδ
IRR a e he possibili y dis ibu ions o IRR o p ojec s Aand Aδ, espec i ely.
P oo . I is su icien o show ha
|µIRR( )−µδ
IRR( )|=

PosA0+
n
X
i=1
Ai
(1 + )i=¯
0−PosAδ
0+
n
X
i=1
Aδ
i
(1 + )i=¯
0
=
A0+
n
X
i=1
Ai
(1 + )i(0) −Aδ
0+
n
X
i=1
Aδ
i
(1 + )i(0)
≤min 1,δ
α(8)
o any ≥0, because (7) ollows om (8). Using ep esen a ion (2) and applying
Lemma 2.1 o
A0+
n
X
i=1
Ai
(1 + )n= (S(a, ), S(α, )),
and
Aδ
0+
n
X
i=1
Aδ
i
(1 + )n= (S(aδ, ), S(α, )),
we ind
DA0+
n
X
i=1
Ai
(1 + )i, Aδ
0+
n
X
i=1
Aδ
i
(1 + )i=|(S(a, )−(S(aδ, )|=

a0+a1
1 + +· · · +an
(1 + )n−aδ
0+aδ
1
1 + +· · · +aδ
n
(1 + )n
≤
5
|a0−aδ
0|+1
1 + × |a1−aδ
1|+1
(1 + )n× |an−aδ
n| ≤ (n+ 1) ×δ,
o any ≥0, and
A0+
n
X
i=1
Ai
(1 + )i(0) −Aδ
0+
n
X
i=1
Aδ
i
(1 + )i(0)
≤
sup
∈IR A0+
n
X
i=1
Ai
(1 + )i( )−Aδ
0+
n
X
i=1
Aδ
i
(1 + )i( )
≤
max (n+ 1)δ
α0+α1+· · · +αn≤max (n+ 1)δ
(n+ 1) max{α0, α1, . . . , αn}=δ
αmax.
Which ends he p oo .
Theo em 3.1 can also be ex ended o uzzy cash lows wi h a bi a y (con inuous) uzzy
numbe s.
Theo em 3.2 Le δ > 0be a eal numbe . I
max{D(A0, Aδ
0), D(A1, Aδ
1), . . . , D(An, Aδ
n)} ≤ δ
hen
max
≥0|µIRR( )−µδ
IRR( )| ≤ min{1, ω(δ)}.
whe e ω(δ)deno es he maximum o moduli o con inui y o all he uzzy numbe s in
p ojec s Aand Aδa poin δ.
The p oo o his heo em is ca ied ou analogously o he p oo o Theo em 3.1 in [8].
4 Concluding ema ks
In his pape we ha e shown ha he uzzy IRR has a s abili y p ope y unde small
changes in he membe ship unc ions ep esen ing he uzzy cash lows. Ne e heless,
he beha io o he maximizing solu ion, ∗(δ), o possibilis ic equa ion
Aδ
0+
n
X
i=1
Aδ
i
(1 + )i=¯
0,
owa ds small pe u ba ions in he membe ship unc ions o he uzzy coe icien s can
be e y o ui ous. Tha is, he dis ance
| ∗− ∗(δ)|,
(which coincides wi h | ∗∗ − ∗∗(δ)|, he dis ance be ween he in e nal a es o e u ns
o c isp p ojec s a= (a0, a1, . . . , an) and aδ= (aδ
0, aδ
1, . . . , aδ
n) i Ai= (ai, αi) and
Aδ
i= (aδ
i, αi), i= 0,1, . . . , n) can be e y big e en o e y small δ.
6
In his manne , he uzzy model can be conside ed as a well-posed ex ension [11, 15]
o he (gene ally) ill-posed c isp in e nal a e o e u n decision ule.
I he uzzy numbe s in p ojec s Aand Aδa e no s ic ly unimodal ( o example
apezoidal) hen he se o maximizing solu ions o he uzzy IRR is a segmen o he
eal line. In his case any IRR ob ained om a c isp p ojec , in which he u u e cash
alues a e chosen om he co es o he co esponding uzzy numbe s, belongs o he
co e o he uzzy IRR.
5 Acknowledgemen
The second au ho has been pa ially suppo ed by he Hunga ian Resea ch Fund
OTKA T 019455.
Re e ences
[1] J. Ba on, Thinking and Deciding, 2nd edi ion, (Camb idge Uni e si y P ess,
1994).
[2] R.E. Bellman and L.A. Zadeh, Decision-making in a uzzy en i onmen , Man-
agemen Sciences, Se . B 17(1970) 141-164.
[3] J.J. Buckley, Po olio analysis using possibili y dis ibu ions, in: E. Sanchez
and L.A. Zadeh eds., App oxima e Reasoning in In elligen Sys ems, Decision
and Con ol, Pe gamon P ess, New-Yo k, 1987 69-76.
[4] J.J. Buckley, Fuzzy ma hema ics o inance, Fuzzy Se s and Sys ems, 21(1987)
257-273.
[5] C. Ca lsson and R. Full´e , On uzzy capi al budge ing p oblem, in: P oceedings
o he In e na ional ICSC Symposium on So Compu ing in Financial Ma ke s,
Roches e , New Yo k, USA, June 22-25, 1999, ICSC Academic P ess, 1999 ( o
appea ).
[6] Robe T. Clemen, Making Ha d Decisions: An In oduc ion o Decision Analy-
sis, 2nd edi ion, (Duxbu y P ess, An Imp in o Wadswo h Publishing Company,
Belmon , Cali o nia, 1996).
[7] C. Chiu and C.S. Pa k, Fuzzy cash low analysis using p esen wo h c i e ion,
The Enginee ing Economis , 39(1994) 113-138.
[8] M.Fed izzi and R. Full´e , S abili y in possibilis ic linea p og amming p oblems
wi h con inuous uzzy numbe pa ame e s, Fuzzy Se s and Sys ems, 47(1992)
187-191.
[9] Simon F ench, Readings in Decision Analysis, (Chapman and Hall, London,
1990).
[10] R.Full´e , On s abili y in possibilis ic linea equali y sys ems wi h Lipschi zian
uzzy numbe s, Fuzzy Se s and Sys ems, 34(1990) 347-353.
7
[11] R. Full´e , Well-posed uzzy ex ensions o ill-posed linea equali y sys ems, Fuzzy
Sys ems and Ma hema ics, 5(1991) 43-48.
[12] R.Goe schel and W.Voxman, Topological p ope ies o uzzy numbe s, Fuzzy Se s
and Sys ems, 10(1983), 87-99.
[13] E.E. Ha en, The use o uzzy mul ic i e ia analysis in e alua ing p ojec s wi h
iden ical ne p esen alues and isk, in: M. Ma es e al eds., P oceedings o
Se en h IFSA Wo ld Cong ess, June 25-29, 1997, P ague, Academia P ague,
Vol. III, 1997 154-157.
[14] C. Kah aman and Z. Ulukan, Fuzzy cash lows unde in la ion, in: M. Ma es e
al eds., P oceedings o Se en h IFSA Wo ld Cong ess, June 25-29, 1997, P ague,
Academia P ague, Vol. IV, 1997 104-108.
[15] M. Ko ´acs, A s able embedding o ill-posed linea sys ems in o uzzy sys ems,
Fuzzy Se s and Sys ems, 45(1992) 305–312.
[16] L.A.Zadeh, The concep o linguis ic a iable and i s applica ions o app oxima e
easoning, Pa s I,II,III, In o ma ion Sciences, 8(1975) 199-251; 8(1975) 301-357;
9(1975) 43-80.
8