CIT2016 – XII Cong eso de Ingenie ía del T anspo e
València, Uni e si a Poli ècnica de València, 2016.
DOI: h p://dx.doi.o g/10.4995/CIT2016.2016.3163
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METAHEURISTIC ANALYSIS IN
REVERSE LOGISTICS OF WASTE
An onio Se ano Elena
Enginee PhD, Uni e si y o Málaga, Spain
ABSTRAT
This pape ocuses in he use o sea ch me aheu is ic echniques on a dynamic and
de e minis ic model o analyze and sol e cos op imiza ion p oblems and loca ion in
e e se logis ics, wi hin he ield o municipal was e managemen o Málaga (Spain). In
his wo k we ha e selec ed wo me aheu is ic echniques ha ing ele ance in p esen
esea ch, o es he alidi y o he p oposed app oach: an impo an echnique o i s
in e na ional p esence as is he Gene ic Algo i hm (GA) and ano he in e es ing echnique
ha wo ks wi h swa m in elligence as is he Pa icles Swa m Op imiza ion (PSO). These
me aheu is ic echniques will be used o sol e cos op imiza ion p oblems and loca ion o
MSW eco e y acili ies ( ans e cen e s and ea men plan s).
Keywo ds: Re e se logis ics, Op imiza ion, Me aheu is ics, Gene ic Algo i hm, Pa icles
Swa m Op imiza ion.
1 INTRODUCTION
The main objec i e o his pape is o de elop he me hodology and apply ools o
modeling and sol ing he op imiza ion o ans e cos s o municipal solid was e (MSW)
and i s p ac ical applica ion in he op imiza ion eal p oblem and loca ing acili ies in 90
municipali ies in he p o ince o Málaga. The scien i ic me hod is used o achie e hese
objec i es, consis ing o a e iew he s a e o a on e e se logis ics, he selec ion o
sui able me aheu is ic echniques and applica ion o modeling compu e ool and selec ed
esolu ion. The hypo hesis o be p o ed is ha he use o me aheu is ics allows o sol e his
ype o eal p oblems whe e he exac algo i hms can no .
2 MATHEMATICAL MODEL DESIGN. OBJECTIVE FUNCTION
Fo he design o he objec i e unc ion we ha e been used some o models included in he
classi ica ion gi en by Klose and D exl (2000), he plan loca ion wo k and heu is ic use
by Ma in and Peleg in (1991) and he con ibu ions o O ega Mie (2008). In Tables 2.1,
2.2 and 2.3 a e indica ed indexes, pa ame e s and a iables used in he p oposed
ma hema ical model.
CIT2016 – XII Cong eso de Ingenie ía del T anspo e
València, Uni e si a Poli ècnica de València, 2016.
DOI: h p://dx.doi.o g/10.4995/CIT2016.2016.3163
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This wo k is licensed unde a C ea i e Commons A ibu ion-NonComme cial-NoDe i a i es 4.0 In e na ional License (CC BY-NC-
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Table 2.1 – Indexes o he ma hema ical model.
Table 2.2– Pa ame e s o he ma hema ical model.
Table 2.3 – Va iables o he ma hema ical model.
The objec i e unc ion is based on a dynamic model o mul iple pe iods, wi h de ined
capaci y and mul iple o igins, co esponding o a combina o ial op imiza ion p oblem o
minimize he o al cos s. This i ness unc ion can be exp essed as:
Fo= min (Fx + Fy + F + Fz) (2.1)
Being:
Fx= ∑
𝑖𝑗𝑡 xij dij CMT (2.2)
CIT2016 – XII Cong eso de Ingenie ía del T anspo e
València, Uni e si a Poli ècnica de València, 2016.
DOI: h p://dx.doi.o g/10.4995/CIT2016.2016.3163
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Fy= ∑
𝑗𝑘𝑡 yjk djk CTP (2.3)
F = ∑
𝑗𝑡 𝐶Fj j (2.4)
Fz= ∑
𝑘𝑡 𝐶Fk zk (2.5)
Subjec o:
R1: ∑−
𝑗xij ≤ −Pi
∀
i,
∀
(2.6)
R2: xij ≤ Pi j
∀
i,
∀
j,
∀
(2.7)
R3: ∑
𝑖𝑗 xij ≤ ∑
𝑗QTj j
∀
(2.8)
R4: ∑
𝑘yjk ≤ ∑
𝑖xij
∀
j,
∀
(2.9)
R5: yjk ≤ zk QPk
∀
j,
∀
k,
∀
(2.10)
R6: ∑
𝑘− zk ≤ −1 a −90
∀
(2.11)
R7: ∑
𝑖𝑗 xij ≤ ∑
𝑘QPk
∀
(2.12)
Wi h limi s:
xij ≥ 0
∀
i,
∀
j,
∀
(2.13)
yjk ≥ 0
∀
j,
∀
k,
∀
(2.14)
j
∈
{0, 1}
∀
j,
∀
(2.15)
zk
∈
{0, 1}
∀
k (2.16)
The design o cos unc ion (2.1) is de ined by op imiza ion (minimiza ion) o he sum o
anspo cos s (2.2) o all MSW collec ed om each municipali y o ans e cen e s,
anspo cos s (2.3) om each ans e cen e o ea men plan s and ixed cos s (2.4 and
2.5) ans e cen e s and ea men plan s. Cons ain s (2.6) o (2.12) allow o bound he
sea ch o op imal cos o a mo e educed con ex space and indica e i should collec all
MSW gene a ed in each municipali y and each pe iod o ime, ensu ing ha he was e
collec ed and anspo ed be ween municipali ies, ans e cen e s and ea men plan s do
no exceed he capaci y o acili ies open du ing he pe iod conside ed and gua an eeing a
minimum numbe o ea men plan s. The limi s (2.13) o (2.16) bound he sea ch o
posi i e in ege alues o a iables and indica e whe he he acili ies a e open o closed.
I is a de e minis ic linea p og amming p oblem wi h mixed a iables, in which he sea ch
o solu ions becomes mo e complica ed as inc ease he a iables in oduced being a NP-
ha d p oblem (Ga ey and Johnson (1979)), o which a good sol ing policy is o use
me aheu is ics, since exac me hods a e unable o deli e ing any esul s a all in easonable
imes (i.e. less han one yea ).
3 MODELING AND RESOLUTION
Gene ic Algo i hm (GA) se by Holland (1975) and pa icle swa m op imiza ion (PSO)
p oposed by Kennedy and Ebe ha (1995) a e aplied on a eal case o loca ing acili ies o
90 municipali ies in he p o ince o Málaga. The code o hese algo i hms has been
p og ammed wi h he MATLAB language allowing uni y bo h exac and me aheu is ic
CIT2016 – XII Cong eso de Ingenie ía del T anspo e
València, Uni e si a Poli ècnica de València, 2016.
DOI: h p://dx.doi.o g/10.4995/CIT2016.2016.3163
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me hods. Table 3.1 shows he da a ob ained wi h he di e en echniques used. In bold
ype a e indica ed he bes imes and i ness o ins ance. Acco ding o he esul s e lec ed,
he exac me hods used a e insu icien o sol e op imiza ion p oblems wi h a ays o
90×90×5.
Table 3.1 – Summa y o esul s ob ained wi h di e en echniques and municipali ies.
The eal p oblem o be sol ed is he op imiza ion o a MSW collec ion ne wo k o 90
municipali ies in he p o ince o Málaga. MSW collec ed om di e en municipali ies a e
anspo ed o i e ans e cen e s (Vélez-Málaga, Ronda, Cá ama, A chidona and
Campillos) and inally, alued and emo ed (i applicable) in ea men plan s (An eque a
and Casa abonela). On he desc ibed eal scena io wi h he cu en loca ion o each acili y
he hypo hesis a ises o es uc u e he ne wo k o was e collec ion, so ha o minimize
managing cos o hese was es applying equa ions designed and p oposal me aheu is ics.
PSO GA
CIT2016 – XII Cong eso de Ingenie ía del T anspo e
València, Uni e si a Poli ècnica de València, 2016.
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Figu e 3.1 – Resul s o he bes alues o he i ness unc ion o 1 execu ion PSO and
GA o 30 gene a ions and 90 municipali ies.
Figu e 3.1 shows he esul s ob ained a e a implemen a ion wi h each me aheu is ics o
90 municipali ies. We ha e s a ed wi h a popula ion o 20 indi iduals and has es ablished
30 gene a ions be o e s opping he execu ion o algo i hms.
PSO GA
Figu e 3.2 – Resul s o he bes alues o i ness unc ion o 30 independen
execu ions PSO and GA o 30 gene a ions and 90 municipali ies.
Figu e 3.2 shows he e olu ion o he bes alues a e 30 execu ions o each
me aheu is ics. The a e age alues o hese las execu ions co espond o he da a shown in
Table 3.1. Al hough he esul s ob ained wi h bo h echniques a e simila , he PSO
algo i hm is mo e e icien han he GA, o sol e big p oblems (ma ix o 90×90×5
a iables) wi h conside ably less compu a ional ime and wi h he ad an age o ha ing a
smalle numbe o pa ame e s o con igu e.
A s a is ical e i ica ion o he abo e esul s has been pe o med by applying he es " -
s uden " o a sample. The es applies o sizes o 40 and 90 municipali ies o be mo e
signi ican . The con as alue in he case o 40 municipali ies, is be ween he op imum

CIT2016 – XII Cong eso de Ingenie ía del T anspo e
València, Uni e si a Poli ècnica de València, 2016.
DOI: h p://dx.doi.o g/10.4995/CIT2016.2016.3163
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eached wi h exac me hods (30 samples equals) and he wo me aheu is ics. The esul s
ob ained wi h he " - es " o MATLAB can be seen in Table 3.2.
Table 3.2 – Resul s “ -s uden ” o 40 and 90 municipali ies wi h GA, PSO and Exac .
They a e sampled 30 i ness alues o each me aheu is ic echnique. The alue o he null
hypo hesis in he h ee compa isons is 1, which means han he con as ed samples a e
di e en om each o he wi h an e o p obabili y lowe o 5%. The lowe dispe sion and
a e age alue shown by PSO in his s a is ical s udy con i m a be e pe o mance o his
algo i hm compa ed o GA. You can also see ha he alue o s anda d de ia ion (sd)
ob ained o GA is g ea e han ha ob ained wi h PSO and he con idence in e al (ci) o
GA is much mo e open han ha o PSO. This means ha PSO is mo e obus and eliable
in p ac ice o decision making.
Boxplo 6 municipali ies Boxplo 20 municipali ies
Boxplo 40 municipali ies Boxplo 90 municipali ies
CIT2016 – XII Cong eso de Ingenie ía del T anspo e
València, Uni e si a Poli ècnica de València, 2016.
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Figu e 3.3 – Resul s "Boxplo " 6, 20, 40 and 90 municipali ies wi h GA, PSO, Exac .
Figu e 3.3 shows a g aphical o e iew o he s a is ical esul s ob ained by he "boxplo ".
The igu e shows a compa ison be ween he di e en me aheu is ics echniques and he
exac alues o he di e en sizes o municipali ies (6, 20 40 and 90).
The esul s o s a is ical s udies con i m he alidi y o wo echniques used wi h alues
close o op imal. In gene al, he esul s ob ained wi h GA a e wo se han hose ob ained
wi h PSO because o highe dispe sion de ined by highe alues o RIC (in e qua ile
ange) and also by s anda d de ia ion and con idence in e als g ea e . The esul s show
ha he PSO algo i hm is mo e e icien sea ching he global op imum in a smalle easible
space in compa a ion wi h GA.
Table 3.3 shows cu en and ac ual MSW anspo o al cos s since 90 municipali ies un il
ans e cen e s and ea men plan s. They a e calcula ed acco ding o he ma hema ical
model designed and da a p o ided by Depu a ion o Málaga. The calcula ions we e
pe o med o 2014 o acili a e compa ison wi h he da a ob ained wi h me aheu is ics.
Table 3.3 – Cos 2014 o 90 municipali ies acco ding o cu en dis ibu ion.
The esul s ob ained wi h me aheu is ics used in his wo k and o he pe iod 2014 can be
seen in Table 3.4. These esul s demons a e a sa ings o a leas 740.000 eu os (a
educ ion o 20% e e y yea ) in 2014 and a inal sa ings o he analyzed pe iod be ween
2010 and 2014, close o 3,7 million eu os.
CIT2016 – XII Cong eso de Ingenie ía del T anspo e
València, Uni e si a Poli ècnica de València, 2016.
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Table 3.4 – 2014 cos by applying algo i hms PSO and GA o 90 municipali ies.
Figu e 3.4 – Compa ison be ween he cu en dis ibu ion and me aheu is ic
p oposed, ans e cen e s and ea men plan s in he p o ince o Málaga.
Figu e 3.4 shows as would be he new dis ibu ion o ans e cen e s (o ange ci cle) and
ea men plan (g een ci cle) acco ding o he esul s ob ained wi h he bes me aheu is ic
(blue line). he cu en s a us o ans e cen e s (blue ci cle), ea men plan s ( ed ci cle)
and was e ans e s ( ed line) is also shown.
4. CONCLUSIONS
This esea ch ep esen s a b eak h ough o he use o cu en me aheu is ic as PSO unused
so a in acili y loca ions in e e se logis ics. Likewise, he PSO algo i hm has been mo e
e ec i e, as e and easie o p og am han GA. A e in oduced conside a ions mo e
ealis ic cha ac e han has gene ally been epo ed in he li e a u e. This ac is shown in
he esul s ob ained wi h me aheu is ics o he ac ual case o he 90 municipali ies in he
p o ince o Málaga. Al hough he e a e many esea ch abou acili y loca ions heo y, we
ha e ound e y ew ha p opose, wi hin he amewo k o e e se logis ics, quan i a i e
models o acili y loca ions and use o me aheu is ics echniques o sol ing an NP-ha d
CIT2016 – XII Cong eso de Ingenie ía del T anspo e
València, Uni e si a Poli ècnica de València, 2016.
DOI: h p://dx.doi.o g/10.4995/CIT2016.2016.3163
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p oblem accoun ing wi h eal da a.
ACKNOWLEDGEMENTS
This esea ch is pa o he PhD hesis de eloped in he Depa men o Economics and
Business Adminis a ion and di ec ed by PhDs El i a Lopez Maeso and En ique Alba
To es.
REFERENCES
Ga ey, M. R., y Johnson, D.S. (1979). Compu e s and In ac abili y. A Guide o he Theo y
o NP Comple eness. W.H.F eeman, New Yo k, NY.
Holland, J. (1975). Adap a ion in Na u al and A i icial Sys ems. MIT P ess, Camb idge,
MA, Es ados Unidos.
Kennedy, Y.S.J. and Ebe ha , R. (1995). Pa icle Swa m Op imiza ion. Neu al Ne wo ks,
1995. P oceedings. IEEE In e na ional Con e ence on Vol. 4, pp. 1942–1948.
Klose, A. y D exl, A. (2005). Lowe bounds o he capaci a ed acili y loca ion p oblem
based on column gene a ion. Managemen Science, 51, 1689-1705. Doi: 10.1287/mnsc.
1050.0410.
Ma ín, A., Peleg ín, B. (1991). Heu ís icas de descomposición lag angiana pa a algunos
p oblemas de localización disc e a. T abajos de In es igación Ope a i a, Vol. 7. 1, pp. 3–
15.
O ega Mie , M.A. (2008). U ilización de mé odos cuan i a i os pa a el análisis de
p oblemas de localización en Logís ica In e sa. Tesis Doc o al, Uni e sidad Poli écnica de
Mad id. h ps://www.educacion.gob.es/ eseo/mos a Re .do? e =559755.