Ma hwa e & So Compu ing 7 (2000) 229-241
229
The Logis ic Decision Making in Managemen
Accoun ing wi h Gene ic Algo i hms
and Fuzzy Se s
E. López-González, M.A. Rod íguez-Fe nández and C. Mendaña-Cue o
Dep . o Economy and Business Managemen , Uni . o León
Campus de Vegazana, s/n. (24071) León, Spain
dde{elg/m /cmc}@unileon.es
Abs ac
The logis ics p oblems in business en i onmen s deal wi h assigna ion om a
numbe o sou ces o a numbe o des ina ions. Each sou ce o e s amoun s o
goods, while each des ina ion demands quan i ies o hese goods. The objec is o
ind he cheapes anspo ing schedule ha sa is ies he demand wi hou iola ing
supply es ain s. In his pape we p opose o use Fuzzy Se s o ep esen s he
p e isional in o ma ion ela ed o cos s, demands and o he a iables. Mo eo e , we
sugges including he p oblem o sho es ou e o he dis ibu ion ehicles. Finally,
o sol e his complex p oblem we p opose o use a Gene ic Algo i hm wi h a Fuzzy
Fi ness Func ion.
Keywo ds: anspo a ion p oblem, sho es ou e p oblem, logis ic, minimising
cos s, imp ecise in o ma ion, uzzy se s and gene ic algo i hms.
1 In oduc ion
Many di e en op imisa ion p oblems can be o mula ed on ne wo ks: Fo ins ance,
we migh be in e es ed in inding he sho es pa h om one node o ano he , inding
he cheapes way o connec all he nodes, o inding he bes way o mo e objec s
h ough a ne wo k.
Ne wo ks op imisa ion ep esen s one o he mos impo an a eas o
managemen science o se e al easons [3] [5]. Fi s , ne wo ks can be used o
model many di e se applica ions such as anspo a ion sys ems, communica ion
sys ems, ehicle ou ing p oblems, p oduc ion planning and cash low analysis.
Second, manage s accep ne wo ks mo e eadily because hey p o ide a isual
pic u e o he p oblem unde s udy. Finally, ne wo ks ha e ce ain ma hema ical
E. López-González, M A. Rod íguez-Fe nández & C.Mendaña-Cue o
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p ope ies ha allow managemen scien is s o de elop special algo i hms ha a e
able o sol e much la ge p oblems han o he op imisa ion me hods.
In his pape we p e end o show a new app oach o a classical ne wo k
op imisa ion p oblem: he anspo a ion p oblem. Fi s ly, allowing he ehicles ha
can es ablish ou es, and, also, pe mi ing o manage wi h imp ecise o ague
in o ma ion. We p opose o use he Fuzzy Se s Theo y [15] [8] [16], wi h he aim o
being able o handle he unce ain y, which is a cha ac e is ic o decision-making
p ocesses in dis ibu ion p oblems.
Mo eo e , o op imise he dis ibu ion ne wo k we sugges o use a Gene ic
Algo i hm (GA) [4] [1] [10] [6]. The main eason o his is ha he GAs a e
heu is ic op imisa ion me hods which don’ impose es ic ions o he posing o he
p oblem. In his s udy, he algo i hm is cha ac e ised by i s use o a Fuzzy Fi ness
Func ion ha allows he e alua ion o imp ecise in o ma ion.
In he ligh o he abo e, he nex sec ion shows a desc ip ion o he p oblem o
be sol ed. Thi d sec ion p esen s he uzzy app oach o his p oblem. In ou h
sec ion we in oduces he gene ic algo i hm used o sol e he a o emen ioned
app oach. Fi h sec ion o e s a p ac ical example o he dis ibu ion p oblem and,
inally, we include some concluding ema ks.
2 The Logis ic Business P oblem
We conside he p oblem o inding he leas cos means o shipping supplies
om se e al o igins o se e al des ina ions. O igin poin s o sou ces can be
ac o ies, wa ehouses, o any o he poin s om which goods a e shipped.
Des ina ions a e any poin s ha ecei e goods. To sol e his p oblem we mus ind
how many goods a e supplied om each sou ce o each des ina ion and wha is he
ou e ha he dis ibu ion ehicle mus ollow in o de o minimize he o al
dis ibu ion cos s.
T adi ionally his p oblem has been analysed in wo di e en decisions. Fi s ly,
ob aining he quan i ies o be shipped om each o igin o each des ina ion
minimising he cos s and sa is ying he demand, o T anspo a ion P oblem (Figu e
1). On he o he hand, inding he sho es ou e ha one ehicle mus ollow o
each all he des ina ions, o Sho es Rou e P oblem, so known as T a elling
Salesman P oblem. In his pape we sugges o combine bo h p oblems in o de o
ob ain an op imal solu ion ha minimises he dis ibu ion cos s o he quan i ies
supplied and he ou es ollowed.
A logis ic p oblem has a e y la ge, some imes in ini e, numbe o easible
solu ions. T adi ionally, his p oblem has pa ially educed no pe mi ing shipmen s
among des ina ions. This kind o p oblems can be sol ed wi h Linea P og amming
and o he me hods as S epping S one, MODI, Hou h ake Me hod o Hamme Balas
Me hod. Bu he global app oach, conside ing he ehicles dis ibu ion ou es is
poo ly s udied [13] [5].
The Logis ic Decision Making in Managemen Accoun ing....
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1
Figu e 1. T anspo a ion P oblem
Mo eo e , some o he in o ma ion conside ed is imp ecise. Cos s, supplies and
demands may be no known in a c isp nume ical way [2]. The e o e, in his pape we
sugges o use Fuzzy Se s o ep esen hei ague knowledge.
3 Fuzzy Logis ic Model
Wi h his me hod he companies mus know how many commodi ies supply
om each o igin o each des ina ion when he in o ma ion on held is no p ecise and
a uzzy ep esen a ion in mo e accu a e.
As we show below, he in o ma ion necessa y is ela ed o:
• Des ina ion o clien demands. The company mus sa is y he clien demands.
Some imes his in o ma ion is ague; he e o e we conside a mo e ealis
ep esen a ion using uzzy se s. So, o n des ina ions we could ha e:
{
}
n
CCC
C
~
,
,
~
,
~
~
2
1

=
• Sou ces numbe and i s supply capaci y. We suppose ha each sou ce has a
supply capaci y, no mally known in a s ic way. So o m sou ces we could
ha e:
{
}
m
FFF
F
,
,
,
2
1

=
• Vehicle Capaci y. We conside ha in each sou ce he company has a ehicle
used o dis ibu e he commodi ies. The e o e, we mus ake in o accoun he
ehicle capaci y, because he numbe o a els will depend o i . Fo he
a o emen ioned m sou ces, he ehicle capaci ies could be:
{
}
m
VVVV
,
,
,
2
1

=
• T anspo ing cos s o he emp y ehicles. Some imes, he ehicles mus e u n o
hei espec i e sou ces due o ha each one has dis ibu ed hei capaci ies.
Mo eo e , he las a el is made be ween he las clien supplied and he sou ce
o he ehicle. Thus, we mus ake in o accoun wha is he shipping cos when
E. López-González, M A. Rod íguez-Fe nández & C.Mendaña-Cue o
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he ehicle is emp y. This, no mally depends o he dis ance ( ou ed kilome es).
So, o m sou ces, he anspo ing cos o emp y ehicles could be:
{
}
m
c c c
c
,
,
,
2
1

=
• Inc easing cos o each uni o commodi ies shipped. When he ehicles
dis ibu e he commodi ies he anspo ing cos inc ease wi h each uni shipped.
As well, his inc easing cos depends o he dis ance a elled. Mo eo e , he
knowledge o his cos could no be p ecise and hen ep esen ing i using uzzy
se s could imp o e he esul s ob ained. Thus, o m ehicles he inc easing cos
could be:
{
}
m
c c c c ∆∆∆=∆
~
,
,
~
,
~
~
2
1

• Dis ance among sou ces and des ina ions. The dis ibu ion is made om he
sou ces o he des ina ions o clien s. So, i is necessa y o know he dis ance
among hem.














=
mnmm
n
n
DDD
DDD
DDD
D
,
,
,
,
,
,
,
,
,
2
1
2
22
21
1
12
11




• Dis ance among des ina ions. As we men ioned be o e, he ehicles can
es ablish ou es om he sou ce o he se e al des ina ions ha his sou ce
supplies. Thus, i is necessa y o know he dis ances among he des ina ions














−
−
−
=
,
,
'
,
'
'
,
,
,
'
'
,
,
'
,
'
2
1
2
21
1
12




nn
n
n
DD
DD
DD
D
Acco ding wi h his app oach, we a e ying o each he eal dis ibu ion
p oblem. The ea e , we need some ool capable o sol e his complex p oblem o
able o gi e a easonable solu ion. In his pape we p opose o use a Gene ic
Algo i hm wi h a Fuzzy Fi ness Func ion.
Al hough we desc ibe o use uzzy se s, we sugges ep esen ing all he uzzy
in o ma ion as T apezoidal Fuzzy Numbe s (TFN).
4 Gene ic Algo i hms and Fuzzy Logis ic P oblems
Some au ho s ha e applied Gene ic Algo i hms o anspo a ion p oblems [14]
[12]. In many cases hey sol e he p oblem educing hei complexi y o using
p ecise knowledge o he a iables.
In ou app oach, we sugges a uzzy ep esen a ion (TFN) o he in o ma ion on
held and a less es ic i e model. Due o his, he GA mus de e mine he quan i ies
supplied om each sou ce o each des ina ion and he ou es a elled o he
ehicles. Fo bo h objec i es he c i e ia is he same: minimising he o al
dis ibu ion cos s.
The Logis ic Decision Making in Managemen Accoun ing....
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4.1. Gene ic Algo i hms
GAs a e sea ch algo i hms which use p inciples inspi ed by na u al gene ics o
e ol e solu ions o p oblems [7]. The basic idea is o main ain a popula ion o
ch omosomes, which ep esen s candida e solu ions o he conc e e p oblem being
sol ed, which e ol es o e ime h ough a p ocess o compe i ion and con olled
a ia ion. GAs ha e go a g ea measu e o success in sea ch and op imisa ion
p oblems.
A GA s a s o wi h a popula ion o andomly gene a ed ch omosomes
(solu ions), and ad ances owa d be e ch omosomes by applying gene ic ope a o s
modelled on he gene ic p ocesses occu ing in na u e. Du ing successi e i e a ions,
called gene a ions, ch omosomes in he popula ion a e a ed o hei adap a ion as
solu ions, and on he basis o hese e alua ions, a new popula ion o ch omosomes is
o med using a selec ion mechanism and speci ic gene ic ope a o s such as
c osso e and mu a ion. An e alua ion o i ness unc ion mus be de ised o each
p oblem o be sol ed. Gi en a pa icula ch omosome, a possible solu ion, he
i ness unc ion e u ns a single nume ical i ness, which is supposed o be
p opo ional o he u ili y o adap a ion o he solu ion ep esen ed by ha
ch omosome.
GAs may deal success ully wi h a wide ange o p oblem a eas, pa icula ly in
managemen applica ions. The main easons o his success a e: 1) GAs can sol e
ha d p oblems quickly and eliably, 2) GAs a e easy o in e ace o exis ing
simula ions and models, 3) GAs a e ex endible and 4) GAs a e easy o hyb idise. All
hese easons may be summed up in only one: GAs a e obus . GAs a e powe ul in
di icul en i onmen s whe e he space is usually la ge, discon inuous, complex and
poo ly unde s ood. They a e no gua an eed o ind he global op imum solu ion o a
p oblem, bu hey a e gene ally good a inding accep ably good solu ions o
p oblems quickly. These easons ha e been behind he ac ha , du ing he las ew
yea s, GAs applica ions ha e g own eno mously in many ields.
The basic p inciples o GAs we e i s laid down igo ously by Holland [7], and
a e well desc ibed in many books, such as [4] [11].
4.2 A Gene ic Algo i hm o Fuzzy Logis ic P oblems
To sol e he uzzy logis ic p oblem we p opose o use a GA wi h he ollowing
componen s:
Gene ic Rep esen a ion
The solu ions o he p oblems a e a se o dis ibu ion quan i ies o be shipped
om each sou ce o each des ina ion and he ou es ollowed by he ehicles o each
sou ce. To codi y his solu ions we p opose o use wo ma ixes: one ha con ains
he quan i ies dis ibu ed om each o igin o each des ina ion, and, o he , ha
ep esen s he pa h ollowed o he se e al ehicles.
The codi ica ion o quan i ies ma ix, o an example o i e sou ces ha mus
supply o ou des ina ions could be:

E. López-González, M A. Rod íguez-Fe nández & C.Mendaña-Cue o
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SSou ce 1 Sou ce 2 Sou ce 3 Sou ce 4 Sou ce 5 To al
Des ina ion 1
15
0
0
70
0
85
Des ina ion 2
0
40
0
0
0
40
Des ina ion 3
25
10
40
0
50
125
Des ina ion 4
0
0
60
0
5
65
To al
40
50
100
70
55
315
ha indica es he quan i ies supplied om each sou ce o each des ina ion.
On he o he hand, he codi ica ion o ehicles ou ed could be:
2
1
SSou ce 1Sou ce 2Sou ce 3Sou ce 4Sou ce 5
Des ina ion 1
1
2
1
4
3
Des ina ion 2
4
3
2
3
2
Des ina ion 3
2
1
3
2
1
Des ina ion 4
3
4
4
1
4
ha indica es he posi ion o each des ina ion in he ehicle ou es.
Mo eo e , in o de o gene a e use ulness solu ions, he company mus decide
he co e ing le el o each clien uzzy demand. The TFN can be iden i ied as
possibili y dis ibu ions. So, he company es ablish he isk o no co e demand o
each clien . Acco ding o his, he quan i y ma ix ep esen s a ailable solu ions o
he decision.
Fuzzy Fi ness Func ion
The i ness unc ion mus assign mo e alue o hese solu ions ha a e good
solu ions o he dis ibu ion p oblem. To his we p opose o calcula e he o al
dis ibu ion cos ha he wo ma ixes suppose. The s eps a e as ollows:
A he beginning, each ehicle depa s om i s sou ce ull o commodi ies. The
i s des ina ion, es ablished by he ou es ma ix, de e mines he cos o his i s
a el. Once he i s des ina ion is sa is ied (wi h one o mo e a els) he second
des ina ion is a emp ed. I he ehicle is emp y du ing he ou e, i e u ns emp y o
he sou ce o ull i s capaci y. As well, when all he des ina ions a e sa is ied hen
he ehicle go back o i s o igin, comple ing he ou e. The sum o all hese
anspo ing cos s (depa om o igin ull, a emp i s des ina ion, a emp second,
go back o o igin when emp y, e c.) would be a pa o he solu ion. To ob ain he
o al cos we mus add he cos o he o he dis ibu ion ou es con ained in he
ma ixes and es ablished o o he ehicles.
Due o he ac ha some o he in o ma ion can be ep esen ed by uzzy se s, we
sugges o use he ope a ions designed o hem [3]. Mo eo e , in he cases ha he
ope a ions could no p oduce a TFN, we app oxima e he esul as one o his [9],
assuming a li le e o .
To se up a hie a chy among he solu ions, he p oposal is o use he uzzy
dis ance [9], based on Hamming Dis ance. Doing i , we calcula e he dis ance om
he o igin
B
~
(single on 0) o each solu ion uzzy cos , which is de ined as ollows:
α
α
αααα
dBABABA
d
)
(
)
~
,
~
(
1
0
2
2
1
1
∫−+−= =
The Logis ic Decision Making in Managemen Accoun ing....
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3
5
whe e
[
]
2
1
,
αα
AA is con idence in e al o
A
~
a he signi ica ion le el
α
.
The mo e accu a e solu ions will ha e a lowe dis ance. Thus o ob aining he
i ness alue we p opose o calcula e he in e se o he dis ance o each solu ion.
Selec ion p ocess
We p opose o use Roule e Wheel Ranking [4] o selec he indi iduals ha will
be he “pa en s” o he nex gene a ion.
C osso e ope a o
T adi ional c osso e s canno be used, because he solu ions a e wo ma ixes
ha ha e o accomplish some cons ain s. Due o his we p opose wo di e en
c osso e s o each ma ix, as ollows:
♦ Dis ibu ion quan i ies ma ixes. To combine he in o ma ion o hese
ma ixes ob ained om wo “pa en s” we ha e used he c osso e p oposed o
Vignaux and Michalewicz o T anspo a ion P oblem [14].
♦ Vehicle ou es ma ixes. As well, we mus c oss he ou e ma ixes o bo h
“pa en s”. To his, we selec one sou ce and in e change be ween he “pa en s”
he ou es ha he ehicle o his o igin mus ollow.
I we ha e o c oss he ollowing ou e ma ixes o he a o emen ioned p oblem,
i s we andomly selec one sou ce (sou ce 2):
2
1
SSou ce 1 Sou ce 2 Sou ce 3Sou ce 4Sou ce 5
Des ina ion 1
1
2
1
4
3
Des ina ion 2
4
3
2
3
2
Des ina ion 3
2
1
3
2
1
Des ina ion 4
3
4
4
1
4
2
2
SSou ce 1 Sou ce 2 Sou ce 3Sou ce 4Sou ce 5
Des ina ion 1
3
4
4
1
2
Des ina ion 2
4
3
2
2
4
Des ina ion 3
1
2
3
3
3
Des ina ion 4
2
1
1
4
1
A e ha , we in e change he lis o ou es ollowed, being he ma ixes
esul ing.
'2
1
SSou ce 1Sou ce 2Sou ce 3Sou ce 4Sou ce 5
Des ina ion 1
1
4
1
4
3
Des ina ion 2
4
3
2
3
2
Des ina ion 3
2
2
3
2
1
Des ina ion 4
3
1
4
1
4
E. López-González, M A. Rod íguez-Fe nández & C.Mendaña-Cue o
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'2
2
SSou ce 1Sou ce 2Sou ce 3Sou ce 4Sou ce 5
Des ina ion 1
3
2
4
1
2
Des ina ion 2
4
3
2
2
4
Des ina ion 3
1
1
3
3
3
Des ina ion 4
2
4
1
4
1
Mu a ion ope a o
The in en ion o his ope a o is o in oduce di e si y in o he solu ions. T ying
o his we mus make di e ences among he wo ypes o ma ixes.
• Dis ibu ion quan i ies mu a ion. We p opose o use a special mu a ion
me hod designed o keep he solu ions esul ing as easible ones. The s eps a e
as ollow:
• S ep 1. We selec one sou ce (Sou ce A) and one des ina ion (Des ina ion A)
o he ma ix. This des ina ion mus ecei e a non-ze o quan i y om he
sou ce selec ed. Fo he example, i we mu a e he i s ma ix ob ained a e
he c ossing p ocess, he sou ce and des ina ion could be:
1
1
SSou ce 1 Sou ce 2 Sou ce 3
Sou ce 4
Sou ce 5 To al
Des ina ion 1
8
0
25
35
17
85
Des ina ion 2
0
40
0
0
0
40
Des ina ion 3
15
5
45
35
25
125
Des ina ion 4
17
5
30
0
13
65
To al
40
50
100
70
55
315
• S ep 2. We gene a e a andom numbe be ween 0 and he quan i y assigned
o his des ina ion. Fo he example could be 20.
• S ep 3. We sea ch in he emaining sou ces one quan i y assigned o a
di e en des ina ion which amoun is bigge han he andom numbe
gene a ed be o e (Sou ce B and Des ina ion B). Fo he example could be:
1
1
SSou ce 1
Sou ce 2
Sou ce 3Sou ce 4Sou ce 5 To al
Des ina ion 1
8
0
25
35
17
85
Des ina ion 2
0
40
0
0
0
40
Des ina ion 3
15
5
45
35
25
125
Des ina ion 4
17
5
30
0
13
65
To al
40
50
100
70
55
315
• S ep 4. We discoun he andom numbe o he assigned amoun s om he
Sou ce A o Des ina ion A and om he Sou ce B o Des ina ion B. A e
ha , we add he andom numbe o he assigned amoun s om Sou ce A o
Des ina ion B and om Sou ce B o Des ina ion A. Acco ding o his, o
he example, he mu a ed ma ix could be:
The Logis ic Decision Making in Managemen Accoun ing....
2
3
7
'1
1
SSou ce 1
Sou ce 2
Sou ce 3
Sou ce 4
Sou ce 5 To al
Des ina ion 1
8
0
25
35
17
85
Des ina ion 2
0
20
20
0
0
40
Des ina ion 3
15
25
25
35
25
125
Des ina ion 4
17
5
30
0
13
65
To al
40
50
100
70
55
315
Finally, wi h he a o emen ioned p ocess we can mu a e he quan i y ma ixes
main aining hem as easible ones.
• Vehicles ou es mu a ion. On he o he hand, o mu a ing he ou e ma ix we
p opose and in e change me hod. Fi s , we selec andomly one sou ce. A e
ha , we selec wo des ina ions and in e change hei posi ion in he ou e
ollowed o he sou ce ehicle.
Hal c i e ia o he bes solu ion sea ch
The p oposal is o he algo i hm o go h ough a numbe o gene a ions
speci ied by he use un il he bes solu ion is ound. Mo eo e , in o de no o lose
good solu ions, he cha ac e is ic e med eli ism [4] has been in oduced. This
p ocedu e consis s o keeping he bes indi idual om a popula ion in successi e
gene a ions unless and un il some o he indi idual succeeds in doing be e in
espec o sui abili y. In his way, he bes solu ion o a p e ious popula ion is no
los un il ou classed by a mo e sui able solu ion.
As explained, applica ion o he model p oposed he e allows sol ing he
dis ibu ion p oblem unde unce ain en i onmen al condi ions.
5 Expe imen : an Example o P ac ical Applica ion
In his sec ion we p esen an example ha deals wi h he logis ic p oblem o a
p oduc ion u ni u e company. To do ha we di ide his sec ion in subsec ions: one
wi h he posing o he p oblem and o he wi h he solu ion using he GA p oposed.
5.1 In oduc ion o he p oblem
Le i be imagined ha a u ni u e company supplies desks o se e al clien s. To
ob ain he desks ha e ou ac o ies and each one wi h i s own dis ibu ion ehicle.
In Cha 1 we show he p oduc ion capaci y o each ac o y and he capaci y,
anspo ing cos emp y and inc ease o anspo ing uni o hei espec i e
ehicles.
Fac o
y
P oduc ion
capaci y
Vehicle
capaci y
T anspo ing
cos emp y
($/Km.)
T anspo ing cos
Inc easing o p oduc uni
($/Km.)
A2000 1000 60 (4; 8; 8; 10)
B4000 1400 40 (8; 10; 10; 12)
C6000 200 80 (2; 4; 4; 6)
D2000 400 20 (6; 6; 6; 6)
Cha 1