a multicriteria facility location model

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A multicriteria facility location model for municipalsolid waste management in North Greece

Erhan Erkut a,*, Avraam Karagiannidis b, George Perkoulidis b,Stevanus A. Tjandra c

a Faculty of Business Administration, Bilkent University, Ankara, Turkeyb Laboratory of Heat Transfer and Environmental Engineering, Department of Mechanical Engineering,

Box 483, Aristotle University, GR-54124 Thessaloniki, Greece

c School of Business, University of Alberta, Edmonton, Alberta, Canada T6G 2R6

Available online 17 November 2006

Abstract

Up to 2002, Hellenic Solid Waste Management (SWM) policy specified that each of the country’s 54 prefectural gov-ernments plan its own SWM system. After 2002, this authority was shifted to the country’s 13 regions entirely. In thispaper, we compare and contrast regional and prefectural SWM planning in Central Macedonia. To design the prefecturalplan, we assume that each prefecture must be self-sufficient, and we locate waste facilities in each prefecture. In contrast, inthe regional plan, we assume cooperation between prefectures and locate waste facilities to serve the entire region. We pres-

ent a new multicriteria mixed-integer linear programming model to solve the location–allocation problem for municipalSWM at the regional level. We apply the lexicographic minimax approach to obtain a ‘‘fair’’ nondominated solution, asolution with all normalized objectives as equal to one another as possible. A solution to the model consists of locationsand technologies for transfer stations, material recovery facilities, incinerators and sanitary landfills, as well as the wasteflow between these locations. 2006 Elsevier B.V. All rights reserved.

Keywords: Environment; Location; Municipal solid waste; Multiple criteria analysis

1. Introduction

Municipal solid waste (MSW), commonly knownas trash or garbage, consists of everyday items such

as product packaging, grass clippings, furniture,

clothing, bottles, food scraps, newspapers, appli-ances, paint and batteries. If not dealt with prop-erly, waste can create serious environmental andhealth problems. In this paper, we focus on MSWmanagement in the region of Central Macedoniain Greece. We develop a mixed-integer linear pro-gramming model with multiple objectives to solvethe location–allocation problem, including the tech-nology selection of transfer stations, material recov-ery facilities (MRFs), incinerators, and sanitary

0377-2217/$ - see front matter 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.ejor.2006.09.021

* Corresponding author. Tel.: +1 780 492 3068; fax: +1 780 4923325.

E-mail addresses: [email protected] (E. Erkut), [email protected] (A. Karagiannidis), [email protected](G. Perkoulidis), [email protected] (S.A. Tjandra).

Available online at www.sciencedirect.com

European Journal of Operational Research 187 (2008) 1402–1421

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mailto:makis@%20aix.meng.auth.gr











landfills. Our model is more comprehensive andcomplex than the models in the literature, as weconsider different waste processing technologies inMRFs and incinerators. We take economic andenvironmental criteria into account.

Our research does not involve real decision mak-ers (DMs) from prefectures of the region. Weassume that all objectives are equally important,and we aim for a ‘‘fair’’ solution, i.e., a solution withall normalized objective function values as close toone another as possible. A popular fairness-orientedapproach in the literature is the lexicographic mini-

max (see e.g. [17,19]). Lexicographic minimax solu-tions are also called lexicographic max-ordering [9],lexicographic centers in location problems [20], andnucleolar solutions in game theory [18]. A variantof this approach, lexicographic maximin, is used

for telecommunication network design (see e.g.[21,23]). Following Ogryczak et al. [21], we discussthe conversion of the original lexicographic mini-max problem to a lexicographic minimization prob-lem. This enables us to use the standard sequentialalgorithm for lexicographic minimization. To thebest of our knowledge, this paper presents the firstattempt to apply the fairness concept of multiobjec-tive optimization to solid waste management.

In Greece each region consists of a number of prefectures. Legislation in Greece specified that each

prefecture of a region should plan its own solidwaste management system. These plans were finan-cially supported by the region. Recently the plan-ning authority shifted to the regions. We applyour model to compare and contrast regional andprefectural solid waste management planning. Inthe prefectural planning, we look for the optimallocation and allocation decision of waste facilitiesper prefecture taking into account multiple criteria.In the regional planning, we unite all prefectures inthe Central Macedonia region and look for the opti-mal location and allocation decision of waste facili-ties that cover the needs of all seven prefectures inthis region.

The paper is organized as follows. Section 2 con-tains an introduction to the municipal solid wasteproblem in Greece. Section 3 discusses the relevantliterature. In Section 4, we provide the formulationof the model (the notation is in the Appendix). InSection 5, we provide a short review of multiobjec-tive mathematical programming and lexicographicminimax approach. We also discuss the fair solutionin this section. Section 6 contains our empirical

results; namely an analysis of prefectural versus

regional MSW management planning in CentralMacedonia. Finally, we conclude the paper with adiscussion of possible further research in Section 7.

2. Municipal solid waste management in Greece

As a country’s population increases and its stan-dard of living improves, its amount of waste pro-duction increases and its landfill space becomesscarce. The MSW production in Greece increasedfrom 3900 thousand tons (kt) in 1997 to 4559 kt in2001 (see Fig. 1). The Hellenic MSW consistsmainly of organics (47%) and paper (20%), as illus-trated in Fig. 2.

MSW management in Greece is different thanthat of most European Union (EU) countries. Thequantity of waste in Greece continues to be some-

what lower than in other European countries,reflecting a less intense consumption pattern. Thecomposition of waste in Greece is also different,being high in biodegradable materials and low inpackaging materials. These positive characteristicsare balanced by certain negative features in thewaste management sector. The high number of opendump sites (reduced from over 5500 in 1990 to 1260in 2004) constitutes the most negative element, andthe percentage of useful material recovered is low.

3900

4082

4264

4447

4559

3500

4000

4500

5000

1996 1998 2000 2002

M S W

p r o d u c e d i n k t

Fig. 1. Trends in MSW generation in Greece 1997–2001 [6].

Fig. 2. Waste fractions of MSW in 2001 in Greece [6].

E. Erkut et al. / European Journal of Operational Research 187 (2008) 1402–1421 1403

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Greece, within the framework of the EU, is plan-ning its solid waste policy with the goals of protect-ing human health and preserving the naturalenvironment. Greece has developed a comprehen-sive legislative system that harmonizes with the

European legal framework. Various measures havebeen put in place for the integrated solid waste man-

agement (ISWM) policy. This policy promoteswaste reduction programs, recycling, and energyrecovery rather than landfills for waste disposal.The recycle-at-the-source programme, as part of the ISWM policy, has the following quantitativegoals by the end of 2005: utilising 50–65% of theweight of packaging waste; recycling 25–45% of the weight of packaging waste; and recycling 15%of the packaging material [25]. In order to reachthese goals, facilities for material recovery are con-

structed in various municipalities. In the future,Hellenic national legislation will continue to complywith EU regulations and directives on waste man-agement. Greece and the EU aim for a significantdecrease in the amount of waste generated, throughnew waste prevention initiatives, better use of resources, and a shift to more sustainable consump-tion patterns. Specific EU targets include reducingthe quantity of waste disposed by 20% by 2010and by 50% by 2050, with special emphasis on haz-ardous waste.

Greece consists of 13 administrative regions,which are further subdivided into 54 prefectures.We focus our discussion on the MSW management

system of the Central Macedonia region situated inNorth Greece (Fig.3). This region has the largest areaand the second largest population among all regions.It comprises 14.2% of the country’s total area(18,779 km2) and consists of seven prefectures:

Pieria, Imathia, Pella, Kilkis, Thessaloniki, Chalki-diki, and Serres, as shown in Fig. 3. The areas andpopulations of these prefectures are shown in Table 1.

In the region of Central Macedonia, 713 kt of waste was generated in 2001, from which only10.5% ended up in sanitary landfills, whereas39.6% and 49.9% ended up in uncontrolled andsemi-controlled landfills, respectively.

The trend in total waste generation is shown inFig. 4. According to Greek legislation, until 2020,50% (107 kt) of the estimated 214.5 kt of packagingmaterial and 78.9% (450 kt) of the 570 kt of organ-

ics should be diverted from landfills to materialrecovery facilities.

The ISWM system is structured into four phasesof collection, transportation, processing , and disposal

[3,13]. MSW collection initially involves picking uprefuse at the sources via collection vehicles. Betweenthe waste collection and the waste disposal stages,some processing operations such as separation,

Table 1The distribution of population and size of seven prefectures in theregion of Central Macedonia

Prefecture Area (km2) (% of thearea of Greece)

Population (% of thepopulation of Greece)

Pieria 1516 (1.1%) 129,846 (1.2%)Imathia 1701 (1.3%) 144,172 (1.3%)Pella 2506 (1.9%) 144,340 (1.3%)Kilkis 2519 (1.9%) 89,611 (0.8%)Thessaloniki 3683 (2.8%) 1,046,851 (9.5%)Chalkidiki 2886 (2.2%) 92,117 (0.8%)Serres 3968 (3.0%) 200,916 (1.8%)

966901

875

582713

0

200

400

600

800

1000

1200

1995 2001 2010 2013 2020

Fig. 4. Trend in total waste generation in Central Macedonia

(kt).

Fig. 3. Administrative region of Central Macedonia in NorthGreece and its seven prefectures: (1) Pieria, (2) Imathia, (3) Pella,

(4) Kilkis, (5) Thessaloniki, (6) Chalkidiki, and (7) Serres.

1404 E. Erkut et al. / European Journal of Operational Research 187 (2008) 1402–1421



compaction, composting, and incineration, might beemployed either to reduce the space needed to storethe waste, recycle material, or recover energy. Theconceptual framework of the waste flow system ispresented in Fig. 5. The waste flow starts from thewaste producers and continues either to the transfer

station, material recovery facility (MRF), incinerator

(i.e., waste-to-energy facility), or sanitary landfill . Ata transfer station, the MSW is unloaded from col-lection vehicles, held briefly, and reloaded ontolong-distance transport vehicles for shipment to

landfills or to other treatment or disposal facilities.An MRF processes recyclables in order to

recover commodity-grade materials for sale, or amixed-material fraction for processing or conver-sion, for example, into refuse-derived fuel (RDF)or compost. Composting is a biological process thatconverts organic material into a stable humus-likeproduct called compost. Three MRF types are con-sidered in this work: composting (aerobic digestionwith material recovery), RDF-producing (aerobicdigestion with material and energy recovery), andanaerobic digestion (material and energy recovery).Anaerobic digestion is a biological process thatdecomposes organic waste in order to producebiogas (CH4 and CO2) and fertilizer. It producesmaterial and renewable energy, while reducinggreenhouse gas emissions and the volume of wastegoing to sanitary landfills.

Incineration involves the destruction of organicand combustible waste at high temperatures (650– 1100 C). Three incinerator types are considered,mass-burn, rotary kiln, and combined pyrolysis and

gasification. The most common technology is

mass-burn. It involves the combustion of unpro-

cessed or minimally processed refuse. Mass-burnfacilities process raw waste, which is not shredded,sized, or separated before combustion. Large itemssuch as refrigerators and stoves, batteries, and haz-ardous waste materials are removed before combus-

tion. A rotary kiln incinerator is beneficial when themunicipal waste has high moisture content. Itmoves the trash through the combustion processand finally into the ash quenching pit. Pyrolysisand gasification are thermal processes that use hightemperatures to break down waste containing car-bon. Three phases are obtained after pyrolysis: solid(char), liquid (water and oils) and gas (light hydro-carbons, H2, CO and CO2). The gasification processthen breaks down the remaining hydrocarbons intolow calorific fuel gas, which can be used as fuel forpower and heat generation.

According to the waste flow shown in Fig. 5,MSW management involves a number of issues,such as the selection of solid waste treatment tech-nologies, the location of solid waste treatment facil-ities and landfills, the capacity-expansion strategiesof the sites, waste flow allocation to processing facil-ities and landfills, partitioning the service territoryinto districts, selecting collection days for each dis-trict and waste type, determining fleet composition,routing, scheduling and monitoring collection vehi-cles, benchmarking waste collection services, etc.

3. Operational research and solid waste management

Computerised systems based on OR techniquescan help decision makers achieve remarkable costsavings. As the waste problem gets more acute, siteselection for waste facilities becomes conflict-ridden,and the decision is usually met with considerablelocal opposition, e.g., with BANANA (Built Abso-lutely Nothing Anywhere Near Anyone), LULU(Locally Unwanted Land Use), NIMBY (Not inMy Back Yard), NOPE (Not On Planet Earth), orNOTE (Not Over There Either) arguments. There-fore, the solution should not be only cost effectivebut also environmentally and socially acceptable.Hence, the waste management facility location–allo-cation problem is characterised by multiple, oftenconflicting objectives. This condition has led severalauthors to propose multicriteria decisionapproaches to the problem. Multiobjective wastefacility location–allocation models take environ-mental and economic criteria into account [4,5,12– 14], whereas single objective models consider only

economic criteria [1,2,10,11,15].

Fig. 5. The flow of MSW.




Kirca and Erkip [15] proposed a location modelof transfer stations to minimize the total transporta-tion cost. This model accounted for the technologyselection of loading–unloading facilities and for thetype and number of transfer vehicles. Although the

model is static, by experimenting with data from dif-ferent prospected years, one can determine the tim-ing of investment for transfer stations.

Caruso et al. [4] developed a location–allocationmodel for planning process plants and sanitarylandfills for urban solid waste. They consideredtechnologies of incineration, composting, and recy-cling for a process plant. Their model minimizedthe total cost (i.e., opening and transportationcosts), amount of final disposal to the sanitary land-fill, and environmental impact. They proposed aniterative heuristic method consisting of six (also iter-

ative) heuristic procedures and run hierarchically toproduce a subset of approximate efficient solutionsusing the weighted sum technique. Initially, the heu-ristic method considers the transport from wasteproducers to process plants. Once the plants for thisfirst phase are located, transport to sanitary landfillsis considered. The main cycle of the heuristicmethod is due to weight recomputation. The refer-ence point method is then used to help the decisionmaker identify the final solution.

Karagiannidis et al. [14] proposed a set of multi-

ple criteria, which cover social, environmental,financial, and technical aspects, for dealing withoptimization of regional solid waste management.Karagiannidis and Moussiopoulos [13] proposed amodeling framework for regional solid waste man-agement that accounted for the four level wastefacility hierarchy: transfer station, material recoveryfacility, thermal treatment plant (i.e., incinerator),and sanitary landfill.

Hokkanen and Salminen [12] used the decision-aid method ELECTRE III to select a solid wastemanagement system in the Oulu district in NorthernFinland, with the following eight criteria: cost perton, technical reliability, global effects, local andregional health effects, acidic releases, surface waterdispersed releases, number of employees, andamount of recovered waste. Twenty-two alterna-tives under either decentralized or centralized man-agement systems were examined, with varioustreatment methods such as composting, RDF-com-bustion, and landfill.

Antunes [2] developed a mixed-integer optimiza-tion model to determine the location and size of

transfer stations and sanitary landfills. The model

has as a single objective to minimize total transpor-tation and opening costs, and was applied to centralPortugal.

Chambal et al. [5] developed a multiple-objectivedecision analysis model to select the best MSW

management strategy. This model is based uponthe hierarchy of waste management objectivesexpressed by the decision maker. The value-focused

thinking method helped to create the decisionmaker’s fundamental objectives hierarchy. Thishierarchy consists of a single overall (top-tier) objec-tive, which is separated into an appropriate numberof bottom-tier objectives. Each bottom-tier objec-tive is then quantified into an evaluation measurescore. The preferences of the decision maker, repre-sented by weights associated with each objective,determine the conversion from evaluation measure

scores to value units.

4. Model formulation

The proposed model is a mixed-integer linearprogramming model with multiple objectives withrespect to economic and environmental criteria.The notations used for the formulation of the modelare given in the Appendix. There are two kinds of variables used in the mathematical formulation of this model:

• 0–1 facility location variables u, v, w, x, wherethe variable has value one if the correspondingnew facility is opened and zero otherwise,

• continuous waste flow variables a, e, 1, f, h, g, i,j,d representing the quantity of flow betweenfacilities.

We denote x as the vector of variables used in themodel, i.e.,

x :¼ ðu; a; v; e; 1;w; f; h;x; g; i;j; dÞ:

The proposed mathematical model can be stated asfollows: Given a set of waste producers and the setof potential locations for transfer stations, MRFs,incinerators, and sanitary landfills, find the locationand typology of those waste facilities so as to satisfythe following five objectives, which are usually inconflict.

1. Minimize the greenhouse effect (GHE). Thegreenhouse effect describes how greenhouse gases,including carbon dioxide (CO2), methane (CH4),

nitrous oxide (N2O) and chlorofluorocarbons


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(CFCs) in the earth’s atmosphere absorb theamount of heat escaping from the earth intothe atmosphere, making the earth’s surface war-mer. Waste processing in MRF (anaerobic diges-tion), incinerators and landfills is considered to

be the source of greenhouse gases. We definethe greenhouse effect as a product of the amountof waste in the facility, and the greenhouse emis-sion coefficient associated with the facility or itstypology. It is represented in ton of CO2-equiva-lent and CH4 per year. The GHE objective func-tion is formulated as

min GHEð xÞ :¼Xm2V

Xq2 P

AGHEm cmq þ

Xn2 E

Xr2 D

CGHEn d nr

þXo2O

Xs2T

E GHEo eos :

2. Minimize the final disposal to the landfill (FIDI),i.e., the total amount (in tons/year) of waste and/or residue brought to all landfills from all wasteproducers and other facilities

min FIDIð xÞ :¼Xi2 I

Xo2O

Xs2T

goisw

os

þXl2 M

Xp2P

Xo2O

Xs2T

ilopswos

þ

Xq2 P Xo2O Xs2T

jmoqswos

þXn2 E

Xr2 D

Xo2O

Xs2T

dnorswos

or for short

min FIDIð xÞ :¼Xo2O

Xs2T

eoswos :

Thus, we wish to minimize the amount of wastethat cannot be recovered or converted further.Such waste occupies valuable landfill space,reducing the site’s life.

3. Maximize the energy recovery (ER) (in MW h/year) from MRFs, incinerators, and sanitarylandfills

maxERð xÞ :¼Xv2V

Xq2 P

AEv c

vq þ

Xn2 E

Xr2 D

CEn d

nr

þXo2O

Xs2T

E Eo eos :

4. Maximize the material recovery (MR) (in ton/year) from MRFs

maxMRð xÞ :¼ Xm2V

Xq2 P

AMm c

mq þ X

n2 E Xr2 D

CMn d

nr:

5. Minimize the total cost (TC) (in Euro/day),which includes the installation or opening costs,transportation costs, and treatment costs. Hence,

minTCð xÞ :¼ ICð xÞ þ TransC ð xÞ þ TreatC ð xÞ;

where

• installation cost IC(x): As installation cost,we consider the investment cost per tonne of waste

ICð xÞ : ¼Xl2 M

Xp2P

CFFlupb

lp þ

Xm2V

Xq2 P

CFFmvqc

mq

þXn2 E

Xr2 D

CFFnwrd

nr þ

Xo2O

Xs2T

CFFoxse

os ;

• transportation costs Trans C (x): In defining thetransportation cost, the maximum distancebetween a waste producer and either a transferstation or sanitary landfill (25 km, based on themaximum one-way distance of collectiontrucks in daily trips) and the 100 km betweena transfer station and a landfill are taken intoaccount.

TransC ð xÞ :¼Xi2 I

Xl2 M

Xp2P

CFVtp:lauipa

lip

þXi2 I

Xm2V

Xq2 P

CFVtp:maviqe

:miq

þXi2 I

Xn2 E

Xr2 D CFVtp

:n

awirf

n

ir

þXi2 I

Xo2O

Xs2T

CFVtp:oaxisg

:ois

þXl2 M

Xp2P

Xm2V

Xq2 P

CFVtplmuvpq1

lmpq

þXl2 M

Xp2P

Xn2 E

Xr2 D

CFVtplnuwprh

lnpr

þXl2 M

Xp2P

Xo2O

Xs2T

CFVtplouxpsi

lops

þXm2V

Xq2 P

Xo2O

Xs2T

CFVtpmovxqsj

moqs

þXn2 E

Xr2 D

Xo2O

Xs2T

CFVtpnowxrsd

nors;

• treatment costs Treat C (x)

TreatC ð xÞ :¼Xl2 M

Xp2P

CFVtrlupblp

þXm2V

Xq2 P

CFVtrmvqcmq

þXn2 E

Xr2 D

CFVtrnwrd nr þX

o2OXs2T

CFVtroxseos :




The model includes the following constraints,which construct a feasible set denoted by X :

1. Service demand constraints, i.e., the amount of waste produced at a waste producer is equal to

the sum of waste flow to other possible facilitiesai ¼

Xl2 M

Xp2P

alip þ

Xm2V

Xq2 P

emiq þXn2 E

Xr2 D

fnir

þXo2O

Xs2T

gois 8i 2 I :

2. Mass input–output relation constraints• No transfer station may keep the waste

blp ¼Xm2V

Xq2 P

1lmpq þXn2 E

Xr2 D

hlnpr þXo2O

Xs2T

ilops

8l 2 M ;p 2 P:

• Reduction on the output of an MRF and incin-erator determined by the mass preservation rate

of the MRF and incinerator

f vcmq ¼Xo2O

Xs2T

jmoqs 8m 2 V ; s 2 T ;

f nd nr ¼Xo2O

Xs2T

dno

rs 8n 2 E ;r 2 D:

3. Minimum amount requirement constraints,which ensure that a facility is opened, only if the minimum amount of waste processed by that

facility is available• Transfer stations: blp P k lpu

lp; 8l 2 M ;p 2 P.

• MRF facilities: cmq P g mqvmq; 8m 2 V ; q 2 P .

• Incinerators: d nr P hnrwnr; 8n 2 E ;r 2 D.

• Landfills: eos P uosx

os ; 8o 2 O; s 2 T .

4. Capacity constraints• Transfer stations: blp 6

k lpulp; 8l 2 M ;p 2 P.

• MRF facilities: cmq 6 g mqvmq; 8m 2 V ; q 2 P .

• Incinerators: d nr 6hnrw

nr; 8n 2 E ;r 2 D.

• Landfills: eos 6 uosx

os ; 8o 2 O; s 2 T .

5. Constraints on the maximum number of openedfacilities• Transfer stations:

Pl2 M

Pp2Pu

lp 6 p u.

• MRFs: P

m2V

Pq2 P v

mq 6 p v.

• Incinerators:P

n2 E

Pr2 Dw

nr 6 p w.

• Landfills: P

o2O

Ps2T x

os 6 p x.

6. Nonnegativity constraints for flow variables andbinary variables on location decision variables.

The purpose of the multicriteria model is to finda nondominated solution. In the next section, we

provide a brief review of the nondominated solu-

tions, and describe how a particular nondominatedsolution can be generated using this formulation.We are interested in a ‘‘fair’’ solution, where thenormalized objective functions are as close to oneanother as possible.

5. Lexicographic minimax approach to find a fair

solution

The multiple-objective mathematical program-ming (MOMP) with K conflicting objectives canbe formulated as

Min f ð xÞ ¼ ð f 1ð xÞ; . . . ; f K ð xÞÞ

s:t: x 2 X Rn;ð1Þ

where x is an n-dimensional vector of decision vari-ables, X is the decision space, and f (x) is a vector of K real-valued functions. The objective (or criterion)space, denoted by Y , is defined as Y :¼ { y : y = f (x),x 2 X } and Y RK . We refer to the elements of theobjective space as outcome vectors. In general, thereexists no solution that simultaneously optimizes allobjectives in MOMP. Instead, we focus on Pareto

optimal solutions. If f i (x) 6 f i (x 0) for alli = 1, . . . , K and f j (x) < f j (x 0) for at least one j , thenwe say x dominates x 0. A solution that is not domi-nated by any other is called Pareto optimal. A Par-

eto optimal solution cannot be improved in allobjectives simultaneously, i.e., x* 2 X is Pareto opti-mal if and only if there exists no x 2 X such that f (x) 6 f (x*) and f (x)5 f (x*). If x* is a Pareto opti-mal solution, f (x*) is called efficient and both x*

and f (x*) are called nondominated . Hence, Paretooptimality is defined in the decision space and effi-ciency is defined in the objective space.

The concept of a ‘‘fair’’ efficient solution is arefinement of the Pareto optimality. Suppose thatall objectives f 1, . . . , f K are in the same scale (if not,

a priori normalization should be applied). A feasiblesolution x 2 X is called the minimax solution of theMOMP (1) with K objectives, if it is an optimalsolution to the problem

min x

maxi¼1;...; K

f ið xÞ : x 2 X

: ð2Þ

Hence, the minimax solution is the solution thatminimizes the worst objective value. Moreover, theminimax solution is regarded as maintaining equityas described by the following theorem (the maximin

version of this theorem appeared in [21]).




Theorem 1. If there exists an efficient outcome vector

y 2 Y with perfect equity y 1 ¼ y 2 ¼ ¼ y K , then y

is the unique optimal solution of the minimax problem

min maxi¼1;...; K

y i : y 2 Y : ð3Þ

The optimal set of the minimax problem (2)always contains an efficient solution of the originalmultiple criteria problem (1). However, if the opti-mal solution is not unique, some of them may notbe efficient (see e.g. [20]). This is of course a disad-vantage, despite of the equity property of the mini-max solution. To resolve this problem, a refinementtechnique is needed to guarantee that only efficientsolutions are selected in case there are multiple opti-mal minimax solutions. We discuss in the following

a lexicographic minimax problem as a refinement of this minimax problem.

If we consider minimizing also the second worstobjective value, the third worst objective value,and so on, then we will obtain a lexicographic mini-max solution. Let a = (a1, a2, . . . , aK ) andb = (b1, b2, . . . , bK ) be two K -vectors. We say vectora is lexicographically smaller than b, a <lex b, if thereis index j 2 {1, . . . , K 1} such that ai = bi for alli 6 j and a j +1 < b j +1. And we say a is lexicographi-cally smaller or equal b, a 6lex b, if a <lex b or

a = b. Furthermore, let H : RK

! RK

a map whichorders the components of vectors in a nonincreasingorder, i.e., Hð y 1; y 2; . . . ; y K Þ ¼ ð y h1i; y h2i; . . . ; y h K iÞwith yh1iP yh2iP P yhK i, where yhi i denotesthe i th component of H( y). The lexicographic mini-max problem is formalized as

lex min x

fHð f 1ð xÞ; . . . ; f K ð xÞÞ : x 2 X g: ð4Þ

A feasible solution x 2 X is a lexicographic minimaxsolution if its outcome vector is lexicographicallyminimal with respect to H( f (x)), i.e.,

Hð f 1ð xÞ; . . . ; f K ð xÞÞ6lexHð f 1ð x0Þ; . . . ; f K ð x0ÞÞ;

x0 2 X :

In the following we state some properties of the lex-icographic minimax problem (see [7] for the proofs):

(a) The lexicographic minimax solution is an effi-cient solution of the MOMP (1).

(b) Every lexicographic minimax solution is alsoan optimal solution of the minimax problem.

(c) The value of a lexicographic minimax solution

is uniquely defined.

(d) The numbering of objective is irrelevant forthe lexicographic minimax solution, i.e., if f pð xÞ :¼ ð f pð1Þð xÞ; . . . ; f pð K Þð xÞÞ with p a permu-tation of {1, . . . , K }, then the set of lexico-graphic minimax solution with respect to an

objective vector f

is the same as that withrespect to the objective vector f p.(e) The set of lexicographic minimax solutions is

invariant under monotone transformations.

Property (b) together with Theorem 1 guaranteesthat the lexicographic minimax model generates effi-cient solutions with perfect equity, whenever suchan efficient solution exists. Property (c) implies thatall optimal solutions have the same H-objective val-ues, but the order might be different. For example, y 0 = (4, 1,2) and y00 = (2,4, 1) have different order

but H( y 0) = H( y00) = (4,2,1). Moreover, the set of lexicographic minimax solutions itself might belarge. This property enables us to present somealternative solutions to the DMs without worryingabout the differences in the objective values. Proper-ties (d) and (e) suggest that the lexicographic mini-max approach is an appropriate tool when thedecision maker is totally indifferent to the criteriaand the performance is measured at least on ordinalscale [7]. The multiobjective problem (1), therefore,is replaced with the lexicographic minimization

problem (4).The concept of lexicographic minimax solution is

known in the game theory as the nucleolus of amatrix game. The solution concept of the lexico-graphic minimax is a refinement of the minimaxsolution concept. Therefore, a natural idea to solvethe lexicographic minimax problem is to identify allthe minimax solutions and to sort their objectivevalue vector in a nonincreasing order to find the lex-icographically minimal one (see [7] for discreteproblems). For the case of multiobjective linear pro-gramming, one can use the sequential optimizationwith elimination of the dominating functions([16,17,21] for the lexicographic maximin). Thismethod depends heavily on the convexity of the fea-sible set. Ogryczak et al. [21] proposed a formula-tion that transfers any lexicographic maximinproblem (either convex or nonconvex) to a lexico-graphic maximization problem. Following this idea,we now describe how to transfer any lexicographicminimax problem (either convex or nonconvex) toa lexicographic minimization problem. We definean aggregated criterion #ið y Þ ¼ Pi

j¼1 y h ji expressing

the total of the i -worst (the largest) outcomes. For




a given outcome vector y, # i ( y) for i = 1, . . . , K , canbe found as the optimal value of the following inte-ger programming (IP) problem:

#ið y Þ ¼ max X K

j¼

1

y juij ð5Þ

s:t:X K

j¼1

uij ¼ i; ð6Þ

uij 2 f0; 1g; j ¼ 1; . . . ; K : ð7Þ

Since the coefficient matrix of (6) is totally unimod-ular, we can relax (7) to 0 6 uij 6 1 that results anLP formulation for a given outcome vector y. Thisproblem becomes nonlinear when y is consideredas a variable. To overcome this difficulty, we canuse the dual formulation of the LP version of (5)–

(7) as follows:

#ið y Þ ¼ min iki þX K

j¼1

d ij ð8Þ

s:t: ki þ d ij P y j; j ¼ 1; . . . ; K ; ð9Þ

d ij P 0; j ¼ 1; . . . ; K : ð10Þ

Consequently, we obtain

#ið f ð xÞÞ ¼ min iki þX K

j¼1

d ij : x 2 X ; ki þ d ij

(

P f jð xÞ; d ij P 0; j ¼ 1; . . . ; K )

; ð11Þ

which can be re-formulated as

#ið f ð xÞÞ ¼ min ki þ1

i

X K

j¼1

d ij : x 2 X ; ki þ d ij

(

P f jð xÞ; d ij P 0; j ¼ 1; . . . ; K

); ð12Þ

where #ið f ð xÞÞ :¼ #ið f ð xÞÞi

can be interpreted as the

worst conditional means [21]. Furthermore, forany two vectors y 0, y00 2 Y one can easily show that

Hð y 0Þ6lexHð y 00Þ if and only if #1ð y 0Þ; . . . ; # K ð y 0Þ

6lex #1ð y 00Þ; . . . ; # K ð y

00Þ

:

Consequently, the following assertion is valid.

Theorem 2. A feasible vector x 2 X is an optimal

solution of problem (4) if and only if it is the optimal

solution of the aggregated lexicographic problem

lex min #1ð f ð xÞÞ; . . . ; # K ð f ð xÞÞ

: x 2 X

: ð13Þ T a b l e 2

N

u m b e r o f w a s t e p r o d u c e r s a n d p o s s i b l e

s i t e s f o r w a s t e f a c i l i t i e s o f e a c h p r e f e c t u r e i n t h e r e g i o n o f C e n t r a l M a c e d o n i a

P r e f e c t u r e

N u m b e r o f w a s t e

p r o d u c e r s

A

m o u n t o f w a s t e

p

r o d u c t i o n ( t o n / d a y )

N u m b e r o f p o

s s i b l e s i t e s

f o r t r a n s f e r s t a t i o n s

N u m b e r o f p o s s i b l e

s i t e s f o r M R F s

N u m b e r o f p o s s i b l e s i t e s f o r

i n c i n e r a t o r s ( I C R T s )

N u m b e r o f p o s s i b l e s i t e s

f o r s a n i t a r y l a n d fi l l s

P i e r i a

1 3

1 2 9 . 7

1

1 a , b

0

3

I m

a t h i a

1 2

1 4 3 . 6

2

1 a

+ 1 a

0

2

P e l l a

1 1

1 5 8

2

2 b

0

2

K

i l k i s

1 2

8 9

3

1 a , b

0

2

T h e s s a l o n i k i

4 5

1 4 0 1 . 4

5

8

1 b , c

+ 1 a ,

b

2 d , e , f

9

C h a l k i d i k i

1 4

1 0 7

6

1 a

+ 1 b

0

2

S e

r r e s

2 7

2 0 1 . 6

6

1 a

+ 1 b

0

2

a

A n a e r o b i c d i g e s t i o n .

b

C o m p o s t i n g .

c

P r o d u c t i o n o f R D F .

d

M a s s b u r n .

e

R o t a r y k i l n .

f

P y r o l y s i s a n d G a s i fi c a t i o n .


http://-/?-

http://-/?-

http://-/?-

http://-/?-



Following Theorem 2, the lexicographic minimaxproblem (4) can be solved as a lexicographic mini-mization problem with linear objectives

lex min k1 þ X K

j¼1

d 1 j; . . . ; k K þ 1

K X K

j¼1

d Kj !(: x 2 X ; ki þ d ij P f jð xÞ; d ij P 0; i; j ¼ 1; . . . ; K

:

ð14Þ

This formulation is valid either for convex or non-convex X .

Despite of its interesting properties, there areonly a few applications of lexicographic minimax(or lexicographic maximin) approach. It originatedin game theory in early 60s, which has been later

refined to the formal nucleolus definition (see thereview in [18]). This approach has been generalizedto an arbitrary number of objective functions [24]and used for linear programming problems relatedto multiperiod resource allocation [16], for linearmultiple criteria problems [18], and for discreteproblems [7,8]. The lexicographic minimax turnsout to be a special case of the so-called orderedweighted averaging (OWA) aggregation for the mul-ticriteria problems, introduced by Yager in late 80s(see [21,26]). Furthermore, the fairness concept of

the lexicographic minimax has been investigated in[18] and used for several applications such as equita-ble resource allocation problems [16] and telecom-munication network design [21–23].

6. MSW planning in the Central Macedonia region

In this section, we contrast regional and prefec-tural solid waste management planning. In the pre-fectural planning (Section 6.1), we look for theoptimal location and allocation decision of waste

facilities per prefecture taking into account the five

objectives discussed in Section 4. In contrast, inthe regional planning (Section 6.2), we unite all pre-fectures in the region of Central Macedonia andlook for the optimal location and allocation deci-sion of waste facilities that cover the needs of all

seven prefectures in this region.

6.1. Prefectural planning

The number of waste producers and possible sitesfor waste facilities are different for each prefecture,as shown in Table 2. The maximum distancebetween a waste producer and a destination facility(transfer station or landfill) should be 25 km. Thisdistance limit, however, can be increased whentransportation of waste takes place among transfer

stations and material recovery facilities or incinera-tors, because such facilities cannot be densely placedin one area. There are already two transfer stationsin the south east of the Greater Thessaloniki areaand in Nea Michaniona, in the prefecture of Thessa-loniki. Moreover, to avoid social problems, we stip-ulate that there should be no more than twosanitary landfills in each prefecture.

By multiplying all maximization objective func-tions by 1, we get the following multiobjectiveminimization model:

Min ð f 1ð xÞ; f 2ð xÞ; f 3ð xÞ; f 4ð xÞ; f 5ð xÞÞs:t: x 2 X ;

where f 1, f 2, f 3, f 4, f 5 associate with GHE, FIDI, ER,MR, and TC, respectively. To avoid dimensionalinconsistency among various objectives, we scalethe values of GHE, FIDI, ER, MR, and TC intothe interval [0,1]. We define f 1, f 2, f 3, f 4, f 5 as thenormalized objective functions of GHE, FIDI,ER, MR, and TC, respectively. We define f ið xÞ :¼ f ið xÞ f min

i

f maxi

f mini

if the original objective f i (x) is minimization

and f ið xÞ :¼

f maxi

f ið xÞ

f maxi

f mini otherwise, where f

maxi and f

mini

Table 3Optimal values obtained by individual optimization of the objectives for each prefecture

Prefecture GHE FIDI ER MR TC

f min1 f max

1 f min2 f max

2 f min3 f max

3 f min4 f max

4 f min5 f max

5

Pieria 821.286 1592.72 58.18 129.7 2855.78 7490.69 0 14,304 593.09 13,097.2Imathia 705.363 1763.41 57.44 143.6 2452.69 8118.61 0 14,556 1883.84 15,549.7Pella 775.605 1939.01 63.16 157.9 2696.93 6742.33 0 4737 3229.52 13,776.8Kilkis 1092.92 1092.92 89 89 3800.3 3800.3 0 0 1175.32 3063.07Thessaloniki 6955.39 132,213 566.4 1401.45 24,185.3 186,603 0 167,010 55,037.1 203,660Chalkidiki 525.584 1313.96 42.8 107 1827.56 6249.49 0 12,312 2173.93 14,469.3

Serres 1109.62 2475.65 90.36 201.6 3858.37 11,645.2 0 22,248 3213.66 24,456.9




SanitaryLandfill

Pieria Katerini, Kolindros Kolindros Kolindros, Litohoro Kolindros Imathia Makrochorion Makrochorion Makrochorion, Meliki Makrochorion Pella Giannitsa Giannitsa Pella, Giannitsa Pella, GiannitsaKilkis Axioupoli, Kilkis Axioupoli, Kilkis Axioupoli, Kilkis Axioupoli, KilkThessaloniki Agios Antonios,

MavrorachiAgios Antonios,Mavrorachi

Mavrorachi, Scholari Langadas

Chalkidiki Poligiros Poligiros Poligiros Poligiros Serres Serres Serres Zevgolation Serres

++Rotary kiln.

a Anaerobic digestion.b Composting.c Production of RDF.d Mass burn.e Pyrolysis and Gasification.

Table 5Optimal values of model (21) at iteration 5 for each prefecture

Pieria Imathia Pella Kilkis Thessaloniki

ki þ 1i

P5 j¼1d ij i = 1 0.703877 0.680958 0.50665 – 0.500701

i = 2 0.527138 0.603665 0.5 – 0.500701 i = 3 0.433081 0.526863 0.497784 – 0.500701 i = 4 0.386052 0.459657 0.496675 – 0.462996 i = 5 0.329481 0.385823 0.488371 – 0.377118

f jð x5Þ j = 1 0.350399 0.526371 0.493351 0 0.033602 j = 2 0.244966 0.373259 0.49335 0 0.349882 j = 3 0.103195 0.090487 0.50665 1 0.500701 j = 4 0.244966 0.258038 0.49335 1 0.500701 j = 5 0.703877 0.680958 0.455154 0 0.500701



Table 6

Fair locations for MSW facilities in each prefecture in the region of Central MacedoniaWaste facilities Prefecture

Pieria Imathia Pella Kilkis Thessaloniki Chalkidiki

#WP 13 12 11 12 45 14 Transfer

stationAnatolikosOlympos

Andigonides(Kavasila),Macedonia(Rizomata)

Edessa,Exaplatanos,Leianovergi

Livadia,Mouriai,Polikastro

Agios Athanasios(Gefira), NeaMichaniona, Profitis,SE GTA, Sindos,Vrasna, West GTA

Arnea, Stageira(Ierisos), NeaKalliktratia, NeaMoudania, NikitasSithonia, Toroni (Sikia

MRF Katerinia Melikia Giannitsab Thermi IAa,c Kasandriaa

ICRT – – – – – –

SanitaryLandfill

Kolindros,Litohoro

Makrochorion,Meliki

Pella,Giannitsa

Axioupoli,Kilkis

Agios Antonios,Langadas

Poligiros

GHE f 1 x5

1091.6

(0.350399)1262.29(0.526371)

1349.57(0.493351)

1092.92 (0) 11,164.3 (0.0336024) 812.84 (0.364364)

FIDI f 2ð x5Þ

75.7(0.244966)

89.6 (0.373259) 109.9(0.49335)

89 (0) 858.569 (0.349882) 53 (0.158879)

ER f 3ð x5Þ

7012.39(0.103195)

7605.92(0.090487)

4692.73(0.50665)

3800.3 (1) 105,280 (0.500701) 6043.1 (0.0466742)

MR f 4ð x5Þ

10,800(0.244966)

10,800(0.258038)

2400 (0.49335) 0 (1) 83,387.9 (0.500701) 10,800 (0.122807)

TC f 5ð x5Þ

9394.45(0.703877)

11,189.7(0.680958)

8030.15(0.455154)

1175.32 (0) 129,453 (0.500701) 11,882.5 (0.789614)

+Mass burn; ++rotary kiln; +++pyrolysis and gasification.

a Anaerobic digestion.b Composting.c Production of RDF.



are the minimum and the maximum value of f i (x),respectively. Table 3 summarizes f max

i and f mini of

all prefectures and Table 4 gives the optimal solu-tions obtained by individual optimization of theobjectives for each prefecture. Table 2 shows thatthere is no incinerator location in Kilkis. Moreover,the minimum capacity of MRF facilities (compo-

sting and anaerobic digestion) in Kilkis exceedsthe waste supply. Therefore, no MRF facility is con-sidered in a feasible solution for this prefecture.Consequently, the only source of the greenhouse ef-fect and energy recovery is the sanitary landfill. Aswe have only sanitary landfill and transfer station,the values of greenhouse effect, energy recovery,and final disposal are the same for every feasiblesolution as shown in Table 3. Moreover, since mate-rial recovery is contributed only by the MRF facil-ity, its value is always zero. Hence we can consider asingle objective, namely minimizing the total cost,for this prefecture.

Now, we implement (14) with objective functions f ið xÞ, i = 1, . . . , K :¼ 5 for each prefecture to obtaina fair solution. We use CPLEX 8.1 to solve themixed-integer linear programming problem.

In Table 5, we summarize the results for eachprefecture (we show only the optimal objective val-ues and f ið xÞ for iteration 5). Note that it is not nec-essary to implement (14) for Kilkis since it has asingle objective to be minimized. There is no prefec-ture which has normalized objective functions satis-

fying perfect equity f 1ð xÞ ¼ ¼ f 5ð xÞ. However,

Pella has nearly perfect equity. The fair solutionsfor each prefecture are summarized in Tables 6and 7. In total, 63.3% (see Table 7) of waste goesto the final disposal at the sanitary landfills while23.3% of it goes directly from the waste producers.Moreover, building an incinerator (of any type) isnot recommended.

6.2. Regional planning

In the regional planning, we unite all prefecturesin Central Macedonia and look for the optimallocation and allocation decision of waste facilitiesthat cover the needs of all seven prefectures in thisregion. Therefore, we have 134 waste producerswith a total of 2230 tonnes of waste per day, 29 pos-sible sites for locating transfer stations, 9 possiblesites for locating material recovery facilities, 6 possi-ble sites for locating incinerators, and 22 possiblesites for locating sanitary landfills. Moreover, thereshould be, at most, 14 opened sanitary landfills.This combination implies that the mixed-integerprogramming model (in Section 4) has 337 con-straints and 942 variables with 73 binaries amongthem. Each linear programming problem in eachiterative process can be solved by using CPLEX8.1. Table 8 gives the optimal solutions obtainedby individual optimization of each objective.

In single-objective optimization, Tables 4 and 8show that the regional planning is only slightly bet-

ter (about 1%) than the prefectural one with respect

Table 7Fair waste flow distributions for each prefecture in the region of Central Macedonia

Waste flow Prefecture

Pieria Imathia Pella Kilkis Thessaloniki Chalkidiki Serres Sum over all prefectures

WP–TS 0 16.3 66.2 16.5 449.349 70.6 65 683.949

WP–MRF

a

90 73.7 0 0 345.062 23.8 45.7344 578.2964WP–MRFb 0 0 80 0 0 0 0 80WP–MRFc 0 0 0 0 559.739 0 0 559.739WP–ICRT 0 0 0 0 0 0 0 0WP–SL 39.7 53.6 11.7 72.5 47.3 12.6 90.8656 328.2656TS–MRFa 0 16.3 0 0 0 66.2 65 147.5TS–MRFb 0 0 0 0 0 0 0 0TS–MRFc 0 0 0 0 0 0 0 0TS–ICRT 0 0 0 0 0 0 0 0TS–SL 0 0 66.2 16.5 449.349 4.4 0 536.449MRFa –SL 36 36 0 0 138.025 36 44.2938 290.3188MRFb –SL 0 0 32 0 0 0 0 32MRFc –SL 0 0 0 0 223.896 0 0 223.896ICRT–SL 0 0 0 0 0 0 0 0

a Anaerobic digestion.b Composting.c Production of RDF.




Table 8The single-objective optimal solutions of the regional planning

GHE (min) FIDI (min) ER (max) MR (max)

Objective

value

11,982.8 967.1 231,649 236,979

Transfer

station

Achinos, Achladochori, Alistrati,Amfipolis, Ano Vrondou, Andigonides(Kavasila), Arnea, Agios Athanasios(Gefira), Edessa, Exaplatanos, Ierisos,Anatolikos Olympos (Leptokaria),Leianovergi, Livadia, Nea Michaniona,Mouriai, Nea Kallikratia, Nea Moudania,Neo Petritsi, Nikitas, Polikastro, Profitis,Makedonida (Rizomata), SE GTA,Toroni (Sikia), Sindos, Vrasna, WestGTA.

(as in the solution of min.GHE)

(as in the solution of min.GHE)

(as in the solutioGHE)

MRF Katerinib, Poligirosb, Serresb, Sindos IAb,Thermi IAb, Veroiab, Giannitsab

Katerinia, Poligirosb,Serresb, Sindos IAb, ThermiIAa,, Veroiab, Giannitsab,Zevoglationa

Kassandriaa, Katerinia,Melikia, Zevgolationa

Kassandriaa, KaMelikia, ThermiGiannitsab, Zevg

ICRT – Sindos IAc,d Sindos IAc – Landfill Axioupoli, Katerini, Kilkis, Kolindros,

Koufalia, Lachanas (Mavrorachi),Makrochorion, Poligiros, Serres,Giannitsa

Axioupoli, Kilkis,Kolindros, Koufalia,Lachanas

Xirochori, Axioupoli,Kilkis, Kolindros, Lachanas

Axioupoli, KilkiKolindros, LangMakrochorion, Poligiros, Zevgo

(Mavrorachi),Makrochorion, Poligiros,Serres, Giannitsa

(Mavrorachi), Litohoro,Makrochorion, Meliki,Pella, Poligiros, Giannitsa,Zevgolation

*** Production of RDF; +++

Pyrolysis and Gasification.a Anaerobic digestion.b Composting.c Mass burn.d Rotary kiln.



to all objectives: energy recovery (0.43%), materialrecovery (0.76%), total cost (1.08%), greenhouseemission (0.02%), and final disposal (0.02%). Alow level of collaboration between the prefecturesis the main reason for these insignificant differences.

In a problem with higher collaboration between pre-fectures, the regional plan would be more clearlysuperior to the prefectural plan.

A fair solution is summarized in Table 9 for theregion. Table 10 suggests that 66.86% of waste goesto the final disposal at the sanitary landfills while41.8% of it goes directly from the waste producers.Moreover, building an incinerator (of any type) isnot recommended. We also show the detailed solu-tion for the Pella prefecture in Fig. 6. This figureshows the inter-prefectural collaboration betweenPella and Imathia, where waste flows from Leiano-

vergi (in Imathia) and Krya Vrisi (in Pella) toGiannitsa (in Pella) and Makrochorion (inImathia).

Table 11 contrasts the fair solutions for prefec-tural and regional solid waste management plan-ning. This table shows that regional plan does notdominate prefectural plan (in fact, the regional planis better only in the total cost objective). The regio-nal plan has 5.4% more disposal to sanitary landfillsthan the prefectural plan. Consequently the regional

plan requires more sanitary landfills and produces4.8% more greenhouse emissions. Less waste flowto MRF facilities also explains why the regionalplan recovers less material and energy. Note that acomposting MRF does not recover energy and the

material recovery coefficient of an anaerobic diges-tion MRF is three times higher than its energy coef-ficient. Therefore, the prefectural plan recovers onlyslightly (0.31%) more energy but recovers signifi-cantly (5.7%) more materials than the regional plan.Nevertheless, the smaller number of MRFs resultsin the regional plan costing 7.1% less than the pre-fectural plan.

However, note that the analysis so far is based onthe fairness concept. If we ignore the fairness issue,it is possible to generate a regional solution thatcosts more, but recovers more energy and/or mate-

rial than the current one. For example, the solutionthat maximizes the material recovery in Table 8reduces the greenhouse emission from 19,573.6 to15,565.2 (by 20.5%), reduces disposal from 66.9%to 44.1%, and increases material recovery from123,992 to 236,976 (by 47.7%), in comparison tothe solution from Table 11. Yet these benefits comeat a steep increase in cost from 172,726 to 280,377(by 38.4%) and decrease in energy recovery from144,403 to 123,470 (by 14.5%). To allow for a thor-

Table 9Fair locations for MSW facilities in the region of Central Macedonia

GHE FIDI ER MR TC

Objective value 19,573.6 1,491.04 144,403 123,992 172,726Location decisions TS: Achinos, Achladochori, Alistrati, Amfipolis, Ano Vrondou, Andigonides (Kavasila), Arnea, Agios Athanasios

(Gefira), Edessa, Exaplatanos, Ierisos, Anatolikos Olympos (Leptokaria), Leianovergi, Livadia, Nea Michaniona,Mouriai, Nea Kallikratia, Nea Moudania, Neo Petritsi, Nikitas, Polikastro, Profitis, Makedonida (Rizomata), SEGTA, Toroni (Sikia), Sindos, Vrasna, West GTA.MRF: Katerinia, Thermi IAa,b

ICRT:

Sanitary landfill: Agios Antonios, Axioupoli, Kassandria, Kilkis, Kolindros, Koufalia, Langadas, Makrochorion,Meliki, Poligiros, Serres, Vrasna, Giannitsa, Zevgolation

**Composting; +mass burn; ++rotary kiln; +++pyrolysis and gasification.a Anaerobic digestion.b Production of RDF.

Table 10Fair waste flow distribution among the MSW facilities in the region of Central Macedonia

WP–TS WP–MRF* WP–MRF** WP–MRF*** WP–ICRT WP–SL TS–MRF* TS–MRF** TS–MRF***

374.881 702.019 0 530 0 623.35 0 0 0TS–ICRT TS–SL MRF* –SL MRF** –SL MRF*** –SL ICRT–SL

0 374.881 280.808 0 212 0




ough analysis of tradeoffs such as these, as well asdeciding whether or not prefectural cooperation is

worth the effort, participation by the DM isnecessary.

Fig. 6. MSW management of the prefecture of Pella in accordance with the best compromise solution for the region of CentralMacedonia, where the inter-prefectural collaboration of the prefecture of Pella with the prefecture of Imathia (with waste flows from Platyto Leianovergie and from Krya Vrisi to Makrochorion) is also illustrated.

Table 11The fair solutions for Prefectural vs. Regional MSW planning

Sum over all prefectures under prefectural planning Region of Central Macedonia

# WP 134 134# TS 28 28# MRF 5a + 1b + 1c 2a + 1c

# ICRT 0 0# SL 13 14Total waste flow to MRF* 725.7964 702.019Total waste flow to MRF** 80 0Total waste flow to MRF*** 559.739 530Total waste flow to ICRT 0 0Total waste flow to Sanitary landfill 1,410.93 1491.04GHE 18,632.60 19,573.6FIDI 1,410.93 1,491.04ER 144,856.54 144,403MR 131,476.00 123,992TC 185,993.72 172,726

+Mass burn; ++rotary kiln; +++pyrolysis and gasification.a Anaerobic digestion.b Composting.c Production of RDF.




7. Concluding remarks

We presented a new mixed-integer multiple-objective linear programming model, which helpsto solve the location–allocation problem of munici-

pal solid waste management facilities in the CentralMacedonia region in North Greece. We consideredfive objectives: (1) minimize the greenhouse effect,(2) minimize the amount of final disposal, (3) maxi-mize the amount of energy recovery, (4) maximizethe amount of material recovery, and (5) minimizethe total opening, transportation, and processingcosts. The multiobjective problem is formulated asa lexicographic minimax problem in order to find afair nondominated solution, a solution with all nor-malized objectives as equal as possible. We discussedhow to replace the original lexicographic minimax

problem with the lexicographic minimum problem.The model is applied to compare and contrast the

prefectural and regional planning for MSW man-agement. Obviously, it is possible to get betterresults with the regional plan than the prefecturalplan since the feasible region of the regional planis a relaxation of that of the prefectural plan. How-ever, our computational experiments with data fromCentral Macedonia show that the gains achieved bymoving from prefectural to regional plan are mini-mal since the waste flow between prefectures is

small. Of course, this depends on the data and theremay be other instances where regional plans domi-nate prefectural plans by a wider margin. Assumingthat all objective functions are equally important,the regional plan we generated is superior to ourprefectural plan only on the total cost objective.

An immediate extension of this research wouldbe to involve DMs in finding the best compromisesolution. In this case, more attractive solutions thanthose presented in this paper may be achieved. Aswell, other interactive approaches to solving theMOMP problem can be applied and compared.Another possible extension is to apply the modelto solve the problem of regional hazardous wastemanagement, which might require only a mild mod-ification on the set of constraints. However, in thatcase transportation and disposal risks as well associal opposition must be taken into account.

Acknowledgements

This research was supported by Natural Sciences

and Engineering Research Council of Canada and

the Greek Ministry of Development. The authorsare also grateful to three anonymous referees andthe guest editor for their comments which helpedimprove this manuscript.

Appendix

Waste producer

I set of locations of waste producersai quantity of waste (in ton/day) produced by

a waste producer i 2 I

Transfer station

P set of possible sites for the location of atransfer station

M set of possible typologies for a transfer sta-

tionulp binary variable for locating a transfer sta-

tion at site p 2 P with typology l 2 M

alip quantity of incompact waste (in ton/day)

generated by a waste producer i 2 I andcarried to a transfer station located at sitep 2 P with typology l 2 M

blp waste flow (in ton/day) variable from allwaste producers to transfer station p at sitel, i.e., blp ¼

Pi2 I a

lip

k lp lower limit capacity (in ton/day) of local

transfer station at site p 2 P with typologyl 2 M

k lp upper limit capacity (in ton/day) of localtransfer station at site p 2 P with typologyl 2 M

pu maximum number of opened transfer sta-tions

Material Recovery Facility (MRF)

P set of possible sitesfor the location of a MRFV Set of possible typologies for a MRFvm

q

binary variable for locating a MRF at siteq 2 P with typology m 2 V

emiq quantity of waste (in ton/day) variable gen-erated by a waste producer i 2 I and carriedto a MRF located at site q 2 P with typol-ogy m 2 V

1lmpq waste flow (in ton/day) variable from atransfer station located at site p 2 P withtypology l 2 M to a MRF at site q 2 P

with typology m 2 V

cmq waste flow (in ton/day) variable to a MRFat site q 2 P with typology m 2 V , i.e., waste

flow from all waste producers and transfer




stations to a MRF at site q 2 P with typol-ogy m 2 V , i.e. cmq ¼

Pi2 I e

miq þ

Pl2 M

Pp2P1

lmpq

g mq lower limit capacity (in ton/day) of a MRFat site q 2 P with typology m 2 V

g mq upper limit capacity (in ton/day) of a MRF

at site q 2 P

with typology m 2 V

smq waste and residue (in ton/day) variablefrom a MRF at site q 2 P with typologym 2 V , i.e.

smq ¼Xo2O

Xs2T

jmoqs

See the definition of jmoqs below AGHE

m emission coefficients for greenhouse effects(in ton of CO2-equivalent of CO2 andCH4 day/ton of waste year) from anMRF with typology m 2 V

AEm energy recovery coefficient (in MW h day/ton year) from an MRF with typology m 2 V

AMm coefficient for calculating the material

recovery (ton.day/ton of waste year) froman MRF with typology m 2 V

f v efficiency degree of MRF. It is assumedthat f v = 0.4

pv maximum number of opened MRFs

Incinerator (waste to energy facility)

D set of possible sites for the location of an

incineratorE set of possible typologies for an incineratorwnr binary variable for locating an incinerator

at site r 2 D with typology n 2 E

fnir quantity of waste (in ton/day) variable gen-erated by a waste producer i 2 I and carriedto an incinerator located at site r 2 D withtypology n 2 E

hlnpr waste flow (in ton/day) variable from atransfer station located at site p 2 P withtypology l 2 M to an incinerator at siter 2 D with typology n 2 E

d nr waste flow (in ton/day) variable to an incin-erator at site r 2 D with typology n 2 E ,i.e., waste flow from all waste producersand transfer stations to an incinerator atsite r 2 D with typology n 2 E , i.e.d nr ¼

Pi2 I f

nir þ

Pl2 M

Pp2Ph

lnpr

hnr lower limit capacity (in ton/day) of inciner-ator at site r 2 D with typology n 2 E

hnr upper limit capacity (in ton/day) of inciner-ator at site r 2 D with typology n 2 E

qnr waste flow (in ton/day) from an incinerator

at site r 2 D with typology n 2 E

CGHEn emission coefficients for greenhouse effects

(in ton of CO2-equivalent of CO2 andCH4 day/ton of waste year) from facilitiesin an incinerator with typology n 2 E

CEn Energy recovery coefficients (in MW h day/

ton year) from an incinerator with typologyn 2 E

CMn Material recovery coefficients (ton.day/ton

of waste.year) from an incinerator withtypology n 2 E

f n Efficiency degree of an incinerator. It isassumed that f n = 0.4

pw Maximum number of opened incinerators

Sanitary landfill

T Set of possible sites for the location of a

landfillO Set of possible typologies for a landfillxos Binary variable for locating a landfill at site

s 2 T with typology o 2 O

gois Quantity of waste (in ton/day) variable gen-erated by a waste producer i 2 I and carriedto a landfill located at site s 2 T with typol-ogy o 2 O

ilops Compacted waste flow (in ton/day) variablefrom a transfer station at site p 2 P withtypology l 2 M to a landfill at site s 2 T

with typology o 2 O

jmoqs Waste residue flow (in ton/day) variablefrom a MRF at site q 2 P with typologym 2 V to a landfill at site s 2 T with typol-ogy o 2 O

dnors Waste residue flow (in ton/day) variablefrom an incinerator at site r 2 D withtypology n 2 E to a landfill at site s 2 T

with typology o 2 O

eos Waste flow (in ton/day) variable to a land-fill at site s 2 T with typology o 2 O, i.e.waste flow from all waste producers, all

transfer stations, all MRFs, and all inciner-ators to landfill at site s 2 T with typologyo 2 O, i.e., eos ¼

Pi2 I g

ois þ

Pl2 M

Pp2Pi

lopsþP

m2V

Pq2 P j

moqs þ

Pn2 E

Pr2 Dd

nors

uos Lower limit capacity (in ton/day) of a land-

fill at site s 2 T with typology o 2 O

uos Upper limit capacity (in ton/day) of a land-

fill at site s 2 T with typology o 2 O

E GHEo Emission coefficients for greenhouse effects

(in ton of CO2-equivalent of CO2 andCH4. day/ton of waste.year) from facilities

in a landfill with typology o 2 O




E Eo Energy recovery coefficients (in MW h day/ton year) from a landfill with typologyo 2 O

px Maximum number of opened sanitary land-fills

CFF

i

jk Installation cost (in Euro/ton) of facility j

at site k with typology i where j 2 {u,v,w,x}, k 2 {p,q,r,s}, and i 2 {l,m,n, o}

CFVtpii0

jj0kk 0 Transportation cost (in Euro/ton) fromfacility j at site k with typology i to facility j 0 at site k 0 with typology i 0 where j 5

j 0 2 {u,v,w,x}, k 5 k 0 2 {p,q,r,s}, andi 5 i 0 2 {l,m,n, o}. The transportation costmay not dependent on typology

CFVtri jk Treatment cost (in Euro/ton) of thewaste/residue of facility j at site k withtypology i where j 2 {u,v,w,x}, k 2 {p,q,

r,s}, and i 2 {l,m,n, o}

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a multicriteria facility location model

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