European Journal of Operational Research 235 (2014) 187–194

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European Journal of Operational Research

journal homepage: www.elsevier .com/locate /e jor

Decision Support

A bi-objective model for the location of landfills for municipal solidwaste

0377-2217/$ - see front matter � 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.ejor.2013.10.005

⇑ Corresponding author. Tel.: +1 506 453 4869; fax: +1 (506)453 3561.E-mail addresses: haeiselt@unb.ca (H.A. Eiselt), marianov@ing.puc.cl

(V. Marianov).

H.A. Eiselt a,⇑, Vladimir Marianov b

a Faculty of Business Administration, University of New Brunswick, P.O. Box 4400, Fredericton, NB E3B 5A3, Canadab Department of Electrical Engineering, Pontificia Universidad Católica de Chile, Vicuña Mackenna 4860, Santiago, Chile

a r t i c l e i n f o a b s t r a c t

Article history:Received 17 March 2013Accepted 1 October 2013Available online 11 October 2013

Keywords:Landfill locationTransfer stationsCapacity allocationPollution decayCase studySolid waste management

This paper models the locations of landfills and transfer stations and simultaneously determines the sizesof the landfills that are to be established. The model is formulated as a bi-objective mixed integer opti-mization problem, in which one objective is the usual cost-minimization, while the other minimizes pol-lution. As a matter of fact, pollution is dealt with a two-pronged approach: on the one hand, the modelincludes constraints that enforce legislated limits on pollution, while one of the objective functionsattempts to minimize pollution effects, even though solutions may formally satisfy the letter of thelaw. The model is formulated and solved for the data of a region in Chile. Computational results for a vari-ety of parameter choices are provided. These results are expected to aid decision makers in the choice ofexcluding and choosing sites for solid waste facilities.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction people would not want to have them very close to their home),

For a long time, locations of facilities have been discussed bygeographers, mathematicians, computer scientists, economists,and, last, but not least, operations researchers. However, it wasonly in the 1970s that researchers formally realized that not allfacilities are desirable and customers find it beneficial to be closeto them. In particular, the contribution of Church and Garfinkel(1978) was the introduction of the concept of ‘‘obnoxious’’ facili-ties. Later discussions realized that most facilities are neither en-tirely desirable nor entirely undesirable: a nearby supermarket iscertainly beneficial to the people living in its proximity, but thosevery close by will not be too pleased by the early morning deliver-ies that are inevitably accompanied by noise and other sorts ofpollution.

Similarly, not all polluting facilities are the same. Early contri-butions to the field distinguished between noxious and obnoxiousfacilities, even though the distinction was purely semantic and wasnot reflected in the model. Erkut and Neuman (1989) suggested in-stead the more general term ‘‘undesirable facilities,’’ which hasbeen more or less universally accepted today. However, such uni-fication of models may not necessarily be warranted. As an exam-ple, consider a nuclear power plant and a truck reloading station.Both are arguable undesirable facilities (in the sense that most

but they differ to a large degree by the actual risk they pose. Whilethe nuclear power plant poses a tiny, but disastrous risk, the trans-fer station does not. From a modeling point of view, a mathemati-cal model that locates a truck reloading station will have to addresspollution, whereas it is typically not required to consider risk,which would not only be minimal, but also marginal. On the otherhand, ignoring risk in a model that locates nuclear power plantswould be foolish.

Landfills that accept municipal solid waste, the subject of thisinvestigation, are facilities that pose some environmental risks.Other than freak accidents such as the explosion of the Istanbullandfill in 1993 that claimed 39 lives, (Harriyet Daily News,2012), risk is mostly due to pollution of ground and surface water,and, to a lesser degree, air and noise pollution. These issues aretypically taken care of in constraints that simply do not allow thelocation of landfills in areas that do not have appropriate soil types(compacted clay with low hydraulic conductivity) or are in100-year flood plains, and those that specify smallest acceptabledistances between landfills and populated areas. Similarly, if apotential location does not have appropriate access to the existinghighway system, does not have desirable topographical features(e.g., the terrain is too steep), or does not have required soil prop-erties, overlays in GIS systems can be used to simply not considerany such location.

However, residents today are no longer happy with enduringindustrial or commercial facilities, even if they are not in directvicinity of their property. They will complain about health hazards,environmental pollution, noise, truck traffic, and decrease

188 H.A. Eiselt, V. Marianov / European Journal of Operational Research 235 (2014) 187–194

in property values. The NIMBY (not in my back yard), NIMTO (not inmy term of office), LULU (locally undesirable land use), and BANANA(build absolutely nothing anywhere near anything) and other, sim-ilar, syndromes are a clear testament to that sentiment. The modelpresented in this paper will add one more item to the alphabetsoup: the L-SOUP, i.e., the location of socially undesirable premises.

The duality of regulations and general sentiment in the popula-tion has caused us to use an approach that deals with pollution ontwo levels. The ‘‘hard’’ side of pollution will be dealt with by meansof constraints, while the ‘‘soft’’ side is put into an objective func-tion. For a discussion of soft and hard requirements, see e.g., Eiseltand Laporte (1987). The constraints will reflect what laws, bylaws,and other regulations prescribe in terms of maximal allowable pol-lution levels. In addition to that, the objective function that dealswith pollution will attempt to minimize the detrimental effectsgenerated by landfills and transfer stations.

Even though most people would consider a landfill a necessary,but undesirable facility, it is rarely treated as such from a formalpoint of view. As Eiselt and Marianov (2012) demonstrate, earlycontributions mostly use cost minimization to arrive at optimallandfill locations (Marks & Liebman, 1971; Fuertes, Hudson, &Marks, 1974; Gottinger, 1988 and others). The idea behind this isthat while landfills are inherently undesirable to neighbors closeby, the undesirable effects diminish fairly rapidly with distance.On the other hand, since the costs of collection, treatment, and dis-posal are eventually borne by the population at large, their interestis to have the treatment and disposal sites not too close, but suffi-ciently close to be reasonably cost-efficient, making it a model witha ‘‘pull’’ objective (see Eiselt & Laporte, 1995, for different classes ofobjectives) and ‘‘forbidden areas’’, i.e., a areas near the populationcenter, in which it is prohibited to locate a facility. An interestingcontribution that does take the populations sentiment directly intoaccount is by Fernández, Fernández, and Pelegrín (2000), who lo-cate a single facility in such a way as to minimize population oppo-sition, which is modeled by a sigmoid repulsion function.

Most models that site solid waste facilities have a few featuresin common. They feature population centers at known and fixedlocations with fixed and known numbers of people living there, afinite or infinite number of potential locations at which landfillsand/or transfer stations can be located, and distances betweenthem. As far as the choice of metric is concerned, many networkmodels use shortest distances between customers and facilities,while models in the plane use Euclidean or some other Minkowskidistance. In reality, we will have to distinguish between the dis-tance between points when it comes to transportation (for whichroad distances are relevant), and the straight line or Euclidean dis-tances representing how the pollution more likely moves. In orderto avoid having multiple distance measures in the same model,many authors use Euclidean distances for both, transportationand pollution. This is not only an approximation for transportation,but it also approximates pollution, as the propagation of pollutantsdepends on the medium: models that involve air pollution typi-cally use Gaussian plumes, while pollution in ground water followsthe aquifer (see, e.g., Daly & Zannetti, 2007, chap. 2 for air pollu-tion, and Persson & Destouni, 2009, or Stuart, Lapworth, Crane, &Hart, 2012 for groundwater pollution).

The last two decades have seen many new models and ap-proaches. However, a common theme is to use mathematical(cost-) optimization to find approximate locations of the proposedlandfill, and then employ tools from the toolkit of multicriteriadecision making to incorporate different and, at least partially, con-tradicting criteria. Examples for such approaches are Lahdelma,Salminen, and Hokkanen (2002), Kontos, Komilis, and Halvadakis(2003), Sumathi, Natesan, and Sarkar (2008), and Xi et al. (2010).

From a macro point of view, the number of landfills has de-creased dramatically throughout the last decades. For instances,

while there were close to 8000 landfills and dumps in the UnitedStates in 1988, there were only 1908 in 2010 (van Haaren, Theme-lis, & Goldstein, 2010). This means that existing and new landfillswill have larger capacities (thus increasing the level of undesirabil-ity) and many population centers will no longer have a landfill intheir direct vicinity, thus necessitating long transportation routes.In order to mitigate the latter effects, waste transfer stations havebeen established, to which garbage is hauled in collection vehicles,where the waste is compacted and reloaded onto larger and moreefficient transfer truck, which haul the waste to the treatmentfacility or the landfill. This means that landfills and treatment cen-ters on the one hand, and transfer stations on the other, have to beincluded in one comprehensive model, as the location of one suchfacility will influence the location of the other.

There are very few contributions that plan the location and thesize of a landfill in one comprehensive model. One of the difficul-ties of such a plan is that the model tends to become not only inte-ger, but also nonlinear, thus tremendously increasing the degree ofdifficulty. To see this, suppose that Qj denotes the size of a landfillat potential site j, and yj is a binary variable that indicates whetheror not a landfill is going to be constructed at site j, both Qj and yj arevariables. The capacity of the landfill at site j is then Qjyj, a nonlin-ear expression. One possibility is to use only a finite number of po-tential landfill sizes, i.e., discretize the landfill size. Although itincreases the size of the model, this is the approach we are using.

One of the few contributions that simultaneously optimize loca-tion and capacity is by André, Velasco, and González-Abril (2009).That paper locates a sequence of facilities on a plane, one at a time,over a time horizon, avoiding forbidden regions around populationnodes, and precluding location of a new landfill within a preset dis-tance of a closing landfill.

Our contribution is a linear integer model that addresses thelocation and sizing of landfills and, simultaneously, locates transferstations. We minimize costs and pollution, the latter emanatingfrom all facilities that are located and measured at all populatedcenters. The model considers economies of scale, i.e., larger land-fills are less expensive per unit of received waste. By using discretesizes, we can deal with economies of scale in a linear model. Wealso consider a policy consisting of defining saturation zones (orexclusion zones) around each facility, so that a populated point can-not fall in more than one such exclusion zone. The idea is that acustomer, being relatively close to a polluting facility, is ‘‘saturatedwith pollution’’, and cannot be subjected to additional pollutionfrom any facility that belongs to the system that handles municipalsolid waste. While the size of an exclusion zone for a landfill de-pends on the landfill’s capacity (assuming that a larger landfill willhave more traffic and thus more pollution), the exclusion zonearound a transfer station is assumed to be constant. Finally, sincea common policy is to limit the allowable pollution imposed onany populated center, we add a constraint that establishes upperbounds on pollution at every such populated center.

The remainder of this paper is organized as follows. Section 2presents our model, which includes the location of landfills andtransfer stations, as well as capacities of landfills. Section 3 intro-duces the region of study and delineates our computational resultswith a variety of sensitivity analyses. Section 4 summarizes ourfindings and provides some thoughts regarding future research.

2. The model

Consider a region, in which population centers are located atknown sites i = 1, . . . ,m. Typically, population is aggregated at thesecenters, so as to keep the size of the model manageable. We as-sume wi customers are located at site i, each one generating c tonsof garbage per year, so that we have to deal with cwi tons of

H.A. Eiselt, V. Marianov / European Journal of Operational Research 235 (2014) 187–194 189

garbage at site i. Furthermore, the search for suitable sites has nar-rowed down the choices for landfills to a finite number of sites atj = 1, . . . ,n, while transfer stations may be located at candidate sites‘ = 1, . . . ,s. While a site may be suitable for both, a transfer stationand a landfill, it can only house at most one of these facilities.

The remaining parameters of the model are as follows. When-ever it applies, the capacity units are tons in one year, written sim-ply as (ton):

Qjk

capacity (ton) of landfill at site j with capacity level k f Lj

fixed annualized cost to establish and operate alandfill at site j

Df Ljk

capacity-dependent annual cost ($/ton) to establishand operate a landfill of capacity level k at site j. Thiscost is a nonlinear, decreasing function of thecapacity

f T‘

fixed annualized cost to establish & operate atransfer station at site ‘

Df T‘

volume-dependent annual cost ($/ton) to establishand operate a transfer station at site ‘

pL, pT

pollution factors for landfills and transfer stations(kilometers2/ton)

Pc

maximum allowable pollution at any populatedcenter

tij, ti‘

unit transportation costs ($/ton) from customer idirectly to landfill j, and from customer i to transferstation ‘, along any predetermined route. These twounit costs should be the same per distance unit, aswe haul by collection truck in both cases

t‘j

Unit transportation costs (by transfer truck) ($/ton)between transfer station ‘ and landfill j. Note that alltransportation costs are expressed in terms of ($/ton), i.e., they indicate what it costs to ship one tonon the entire route

dij, di‘

Euclidean distances between customer i, transferstation ‘, and landfill j. Note that these distances arenot used for cost computations, but for theassessment of pollution at individual customer sites

The location variables in this model are

yjk

a zero-one variable that assumes a value of one, if weopen a landfill at site j with capacity level k, and

v‘

a zero-one variable that assumes the value of one, if weopen a transfer station at site ‘

The assignment or shipment variables we use in this model are

zij

a zero-one variable that equals 1, if all garbage fromcustomer i is shipped directly to landfill j, and

xi‘

a zero-one variable, which equals 1, if all waste fromcustomer i is shipped to transfer station ‘.

As far as the shipment variables are concerned, we first make thesingle-destination assumption. In other words, we assume that gar-bage collected at one customer site will be hauled in its entiretyto one landfill or to one transfer station, rather than being split. Thisappears to be a reasonable assumption regarding the practicality ofthe collection operation.

We also define a variable for the shipments between transferstations and landfills:

u‘j

this continuous variables measures the garbage quantity(ton) that is shipped from transfer station ‘ to landfill j

We can now formulate the mathematical model that involves twoobjective functions, viz., the costs of establishing waste treatmentfacilities and hauling garbage from the customers to the landfill,and a second objective that minimizes pollution at the customersites.

First consider the cost objective. The total cost zc can be writtenas

Minzc ¼X

j

Xk

f Lj þ Df L

jkQ jk

� �yjk þ

X‘

f T‘ m‘ þ

Xi

Df T‘ cwixi‘

!

þX

i

X‘

ti‘cwixi‘ þX

i

Xj

tijcwizij þX‘

Xj

t‘ju‘j; ð1Þ

where the costs in expression (1) comprise

� The annualized costs to establish and operate a landfill at site j.These costs have a fixed component and a capacity dependentcomponent. As the capacity of the landfill increases, the costDf L

jk per capacity unit decreases according to a negative expo-nential curve, as explained in the Section 3.� The annualized costs to establish and operate a transfer station

at any site ‘. These costs have a fixed component and a compo-nent related to the volume of garbage processed per year,� The transportation costs from customer i to transfer station ‘

(via collection truck),� The transportation costs for shipments between all customer-

landfill pairs (also by collection truck), and� The transportation costs of compacted garbage between all

pairs of transfer stations ‘ and landfills j.

The second objective concerns pollution. In our model, we use agravity expression that assumes that the ill effects of pollution de-crease with the square of the distance to the polluting facility,while it increases linearly with increasing amounts of waste. Thispaper only considers the part of the pollution at customer site ithat derives from the landfills and the transfer stations. The pollu-tion emitted from a landfill may be expressed as a function of thequantity of waste deposited at that landfill. In our model, the totalflow of waste to landfill j is

Picwizij þ

P‘u‘j, so that the pollution

at customer k’s site, due to a landfill located at j, ispL

Picwizijþ

P‘u‘jð Þ

ðdkjþeÞ2and the pollution at that customer, from all land-

fills, isP

jpL

Picwizijþ

P‘u‘jð Þ

ðdkjþeÞ2. Similarly, given that the amount of gar-

bage processed by a transfer station isP

icwixi‘, the pollution that

reaches customer k’s site from a transfer station at ‘ ispT

Picwixi‘

ðdk‘þeÞ2,

and the pollution at that customer, from all transfer stations isP‘

pT

Picwixi‘

ðdk‘þeÞ2. Then, the total pollution a customer k is exposed to

isP

jpL

Picwizijþ

P‘u‘jð Þ

ðdkjþeÞ2þP

‘

pT

Picwixi‘

ðdk‘þeÞ2. Given that there are wk cus-

tomers at site k, the total population-weighted pollution at site kcan obtained by multiplying this expression by wk, and the totalpollution that all customers are exposed to can then be defined as

Min zp ¼X

k

wk

Xj

pLP

icwizij þP

‘u‘j� �ðdkj þ eÞ2

þX‘

pTP

icwixi‘

ðdk‘ þ eÞ2

" #ð2Þ

This is the second objective function (the pollution objective).Consider now the constraints. In addition to the pollution objec-

tive, we assume that around each landfill site j, and around eachtransfer station ‘, there is an exclusion zone. Defining such

190 H.A. Eiselt, V. Marianov / European Journal of Operational Research 235 (2014) 187–194

exclusion zone allows avoiding population centers being in thevicinity of more than one undesirable facility. In other words,locating a facility ‘‘saturates’’ a surrounding area, as defined, forexample, in Chilean regulation (CONAMA, 2007), and populationcenters cannot belong to more than one such area.

This exclusion zone could have any shape, as long as it can bedefined by a level curve of an ‘‘undesirability’’ function that isnon-increasing with distance from the landfill. The size of theexclusion zone is usually larger than the size of the area directly af-fected by pollution, and besides the pollution itself, it has a rela-tionship to public perception of obnoxiousness or nuisance.Furthermore, we assume that the larger the capacity of the landfill,the larger the size of the exclusion zone.

The first set of constraints ensures that no customer is locatedin more than one exclusion zone generated by either a landfill ora transfer station. For that purpose, define NL

i as the set of all (land-fill site j, capacity level k) pairs that pollute customer i beyond thepermissible level. Similarly define NT

i as the set of transfer stationsthat pollute customer i beyond what is permitted by law. The set ofconstraints that states that each customer i can be in at most oneexclusion zone that individually pollutes him beyond what is de-sired, can be written as

Xðj;kÞ2NL

i

yjk þX‘2NT

i

m‘

0@

1A 6 18 i ð3Þ

We also include a constraint that limits the pollution at any popu-lated center:

Xj

pLP

kcwkzkj þP

‘u‘j� �ðdij þ eÞ2

þX‘

pTP

kcwkxk‘

ðdi‘ þ eÞ26 Pc 8 i ð4Þ

The next two sets of constraints guarantee that shipments are onlymade to facilities, if these actually exist. In particular, we formulate

zij 6X

k

yjk 8 i; j; ð5Þ

which states that a customer at site i can only choose a direct routeto landfill j, if a landfill of any capacity exists at site j, and

xi‘ 6 m‘ 8 i; ‘; ð6Þ

which guarantees that a customer i can only choose a route to atransfer station ‘, if a transfer station at site ‘ actually exists. Con-straints that ensure that shipments from transfer stations are direc-ted only to existing landfills are formulated below.

The next set of constraints guarantees that the aforementionedsingle-destination assumption is satisfied. In particular, considerone customer i at a time. This constraint allows exactly one routeto be chosen, either to a transfer station or directly to a landfill.X

j

zij þX‘

xi‘ ¼ 1 8 i ð7Þ

Now the conservation equation. It considers transfer station ‘ andstates that the flow from transfer station ‘ to all landfills (the left-hand side) equals the quantity of the inflow into transfer station ‘

from all customers. In particular, we can writeXj

u‘j ¼X

i

wicxi‘ 8 ‘ ð8Þ

The next set of constraints comprises capacity constraints at land-fills. More specifically, consider the landfill at candidate site j. Theinflow to the landfill is the flow from customer i to landfill j, andthe waste flow from transfer station ‘ to landfill j. This total inflowto landfill j cannot exceed the capacity available at landfill site j,which is expressed as the sum of all capacities established at sitej. The fact that only a single facility may exist at any candidate site

is formulated below. Note that if no landfill exists at site j at all, theright-hand side of the expression is zero, so that no flows are hauledto site j.X

i

wiczij þX‘

u‘j 6X

k

Q jkyjk 8 j ð9Þ

The last set of constraints guarantees that at each candidate site j,there can be no more than one transfer station or a landfill of onecapacity. Note that sites that do not allow either landfill or transferstation do not generate a constraint. Sites j that only allow landfills,have vj = 0, while sites j that allow only transfer stations have yjk = 0"j, k.

mj þX

k

yjk 6 1 8 j ð10Þ

What follows are specifications of the variables. Note that all vari-ables are binary, except for the flow variables from transfer stationsto landfills.

yjk; m‘; zij; xi‘ 2 f1;0g; u‘j � 0 8 i; j; k: ð11Þ

Once this problem is solved—more about that in Section 3—it will,in addition to the values of the variables, also provide numericalvalues for p, the number of landfills, and q, the number of transferstations.

3. Computational results

The model was applied to the location of landfills and transferstations for serving the Biobío region in the south of Chile, whichsize is around 300 kilometers by 300 kilometers square. In this re-gion, there are 128 cities, towns and villages with a population lar-ger than 500 inhabitants, which we consider as the population tobe served. These population concentrations account for over a90% of the total population. The remaining 10% of the populationis distributed in small, scattered hamlets (Chilean Census, 2002).The total population under consideration is 1,565,227 people. Inorder to find good candidates for the location of the landfills, weconsidered 45 points, scattered across the region. Graphs withthe distribution of populated areas and candidate locations areshown in Fig. 1. Fig. 1a shows the distribution of towns and villagesin the region, where the size of the dot is related to the populationof the town. Fig. 1b shows the 45 candidate landfill and transferstation locations.

We have chosen to use the following parameters. The daily per-capita waste generation c is about 3 lbs per day, which amounts toabout half a ton annually. The weights wi are the population of thecity or village multiplied by half a ton in the objective, thus indicat-ing the demand for service at a customer point. Similarly, the max-imal annual capacity of the system is half a ton multiplied by thetotal population of 1,565,227 for the amount of 782,614 tons.Landfills can have one of 20 possible discrete capacities, uniformlydistributed within the range 39,131 to 782,614 tons. The transpor-tation cost per ton per km, from customers to landfills or transferstations, is $0.4286, while the transportation cost from transferstations to landfills, per ton per km is $0.1429 (EPA, 2002). Thefixed annualized cost of establishing and operating a transfer sta-tion is $100,000, while the variable cost per annual ton is $10.The fixed cost per landfill is $300,000, while the variable compo-nent is given by the curve in Fig. 2.

This curve was elaborated using information data from actualcosts in Australia (BDA Group Economics, 2009). The functionrelating cost and capacity, for each possible capacity isDf L

jk ¼ 586:38ðQjkÞ�0:209 ($/ton). Finally, pL = 0.1 and pT = 0.025.As far the solution method is concerned, we use a two-pronged

approach. To get a general idea about the tradeoffs between costand pollution, we employ Cohon’s (1978) weighting method with

(b)(a)Fig. 1. (a) Geographical distribution of population. The size of the points shows population; and (b) candidate locations.

y = 586.38x -0.209

0

20

40

60

80

100

120

0 50,000 100,000 150,000 200,000 250,000 300,000 350,000 400,000 450,000 500,000

$/T

on

Landfill annual capacity

Cost/ton vs Capacity (Tons)

Actual cost/tonPower (Actual cost/ton)

Fig. 2. Cost per ton as a function of capacity of a landfill.

4

100

Capacity (ton)

Radius (km)

Max Cap

Fig. 3. Radius of the exclusion zone as a function of the capacity.

H.A. Eiselt, V. Marianov / European Journal of Operational Research 235 (2014) 187–194 191

weights on cost and pollution kckp > 0 and kc ¼ 1� kp, respec-tively. The composite objective in this approach is then

Min z ¼ kczc þ kpzp;

where zc and zp are as shown in relations (1) and (2).It is well known, though, that the weighting method, if applied

to non-convex optimization problems (as is the case here), willfind only supported non-dominated points. In other words, non-supported non-dominated points will not be found with thismethod. Remedies have been suggested by Soland (1979), Cohon(1978) and, more recently, Özlen and Azizoglu (2009), Coutinho-Rodrigues, Tralhão, and Alçada-Almeida (2012), and Gomes da Sil-va and Clímaco (2013). After having found supported non-domi-nated points with the regular weighting method, we ‘‘zoom in’’and investigate a small interval further. This approach, referredto as the epsilon-constraint method, was suggested by Soland(1979). It optimizes one of the objectives (the pollution, in thiscase) and constrains the remaining objective (total cost) to a va-lue Ck, viz.,

Xj

Xk

f Lj þ Df L

jkQ jk

� �yjk þ

X‘

f T‘ m‘ þ

Xi

Df T‘ cwixi‘

!þX

i

X‘

ti‘cwixi‘

þX

i

Xj

tijcwizij þX‘

Xj

t‘ju‘j 6 Ck

This model is then run a number of times, so as to further investi-gate the specific area of the tradeoff curve.

As far as exclusion zones are concerned, we define a circle-shaped exclusion zone around each transfer station and landfill.Furthermore, since our model includes multiple types of pollu-tion, we use Euclidean distances as an approximation. As al-ready mentioned earlier, the exclusion zone for a transferstation was chosen to be 4 kilometers irrespective of the trafficthe station encounters. For landfills, we assume a minimal ra-dius of the exclusion zone is also 4 kilometers, regardless ofthe size of the landfill. With increasing capacity, the radius ofthe exclusion zone is assumed to increase linearly as shownin Fig. 3 to a maximum value of 100 kilometers for a full sizedlandfill.

The model was run on a 64-bit PC Intel Core i7 machine with2.8 gigahertz and 4 gigabyte RAM. The solver of choice was CPLEX12.5 (IBM, 2012). Our benchmark problems are those that allow amaximal pollution of 10,000 and have a maximal exclusion radiusof 100. Given those parameter settings, Table 1 shows the numberof landfills and transfer stations in the solutions, the fixed and var-iable costs, as well as the pollution levels for a variety of weightson the cost and pollution objectives.

It is noteworthy that the number of landfills is very stablethroughout the range of observations. Ignoring the extremes, wecould recommend a single landfill to be located. On the other hand,the number of transfer stations varies quite a bit with changing

Table 1Costs and pollution for various weight combinations.

(kc,kp) p q Fixed cost Variable cost Total cost Pollution Computing time (minute:second)

(.999, .001) 2 2 32,074,189 8,019,851 40,094,040 1,023,288,685 2:45(.99, .01) 2 2 32,063,989 9,177,721 41,241,710 272,290,136 2:24(.9, .1) 1 3 28,964,984 15,612,046 44,577,031 73,886,873 1:18(.8, .2) 2 1 31,112,944 16,002,837 47,115,781 57,767,374 2:13(.75, .25) 1 1 27,644,874 22,980,753 50,625,627 46,393,246 2:14(.7, .3) 1 4 33,686,539 24,940,103 58,626,643 22,998,799 0:53(.6, .4) 1 4 33,643,984 25,094,034 58,738,019 22,803,707 0:29(.5, .5) 1 3 28,997,094 38,516,568 67,513,662 13,124,999 0:26(.4, .6) 1 1 27,795,504 42,285,634 70,081,138 10,616,400 0:23(.3, .7) 1 1 27,644,874 42,591,079 70,235,953 10,541,777 0:22(.2, .8) 1 1 27,644,874 42,591,079 70,235,953 10,541,777 0:20(.1, .9) 1 0 27,213,804 43,888,017 71,101,822 10,377,762 0:20

0.E+00

2.E+08

4.E+08

6.E+08

8.E+08

1.E+09

1.E+09

4.E+07 5.E+07 5.E+07 6.E+07 6.E+07 7.E+07 7.E+07 8.E+07

Pollution vs cost

Fig. 4. Tradeoff between pollution and total cost.

192 H.A. Eiselt, V. Marianov / European Journal of Operational Research 235 (2014) 187–194

importance of costs and pollution. In the entire range, though, theproblem is solvable to optimality without difficulties.

The resulting tradeoff curve is shown in Fig. 4.While the shape of the tradeoff curve is standard, its very steep

shape was unexpected. In other words, starting with very highweights on the cost objective (i.e., practically ignoring pollution),the pollution level is very high, which, again, is not unexpected.However, solutions with slightly lower weight of the cost objectiveincrease the costs only marginally, while dramatically reducingpollution. For instance, starting with the solution that results fromkc = .999, a cost increase of less than 3% will reduce pollution byclose to 75%. After that, an additional 8% cost increase buys another73% decrease in pollution. Beyond this point, the pollution de-creases are somewhat more moderate, even though another 5.7%in costs result in another 22% decrease of pollution.

Next, consider the actual location patterns of the landfills andtransfer stations. Fig. 5a–d shows the patterns for the above seriesof tests. Customers are shown by small dots, transfer stations aredenoted by bigger dots, and landfills are shown as dots with a cir-cle around them. The size of the circle indicates the size of thelandfill. The lines show the allocation of the customers to landfillsor transfer stations.

Fig. 5a shows the solution that essentially minimizes costs andignores pollution. There are one large and one small landfill, alongwith two transfer stations. Note that the landfills and transfer sta-tions are located pretty much central in relation to the customers, aclear result of the almost pure cost minimization (pull) objective.As the weight on the cost objective decreases and thus pollutionis deemed more important, the landfill moves farther away fromthe populated areas, and the cost is maintained low by openingseveral transfer stations (Fig. 5c). From then on, the number offacilities decreases and they stay far away from the populatedareas. More specifically, for kc = 0.75 and lower, the facilities arefar outside of the population agglomerations, so that the circle ofaffected areas includes only a 90 degree cone that includes all

customers in the circle. Further reductions in the importance ofcosts do not even mitigate the loss of proximity of customers tolandfills by any significant number of transfer stations. In practice,decision makers will probably want to avoid such extreme situa-tions that are good on pollution, but are very costly and are notlikely to be realized.

For additional computational tests, we first fixed the weight ofcosts at the level kc = 0.75 and considered decreasing limits onallowable pollution per customer. Starting with very high limitson allowable pollution, the optimal solution includes only a singlelandfill, apparently due to the strong economies-of-scale. As pol-lution limits are tightened, transfer stations are introduced (thisis similar to the present situation that can be observed in manydeveloped nations), the landfill moves away from the most popu-lated area, and, given that the pollution limits are tightened evenfurther, the single, large, very polluting landfill is replaced bysmall landfills, since a large landfill pollutes close by populatedareas beyond the allowed limits, no matter where it is located. Gi-ven the costs of the environmental controls required for landfills,solutions in this last part do not appear practical and implement-able. A similar, but even more pronounced, trend prevails forkc = .1.

At this point, we ran the model several times, allowing theparameter Ck to move in a range in which new non-dominatednon-supported solutions are expected to exist. In our case, we ex-plored the range between the last two solutions in Table 1, i.e., theparameter Ck took values in [70,235,953;71,101,822]. We sweptthe range by taking the largest cost (71,101,822) and multiplyingit each time by 0.999, until we recovered the least cost solutionin the range (70,235,953). We found 12 solutions, shown inTable 2.

All solutions in this range locate one landfill and one transferstation. In all these solutions, the location of the facilities is thesame: they differ only in the allocation of the demand points tofacilities. In other words, the solutions more or less look like thesolution shown in Fig. 5d.

As Table 2 shows, the running time of the model for each newsolution is a few seconds, so even if non-supported solutions haveto be found, the method is still fast and should not present a stum-bling block to analysts or planners (most certainly in view of thefact that we locate a multimillion dollar waste management sys-tem). The average time that it took for each run was slightly over29 seconds.

4. Summary and outlook

This paper proposes a model that, in addition to the usual cost-minimizing objective, includes a second objective that minimizesthe pollution any member of the population is exposed to. In addi-tion, constraints ensure that nobody is exposed to pollution levels

(a) Solution for λc = .999

(c) Solution for λc = .70

(b) Solution for λc = .9.

(d) Solution for λc = .4

Fig. 5. Location patterns for individual solutions.

Table 2Nondominated unsupported solutions in a narrow interval.

Ck Total cost Pollution Computing time (second)

71,030,720 71,030,383 10,397,061 3570,959,689 70,949,881 10,420,444 4770,888,730 70,876,288 10,426,904 2670,817,841 70,815,579 10,433,116 2470,747,023 70,711,324 10,444,781 2470,676,276 70,673,208 10,453,101 3570,605,600 70,599,303 10,480,193 3570,534,994 70,529,793 10,487,242 2670,464,459 70,424,623 10,501,168 2470,393,995 70,386,508 10,509,488 2870,323,601 70,284,395 10,531,542 2470,253,277 70,235,953 10,541,777 25

H.A. Eiselt, V. Marianov / European Journal of Operational Research 235 (2014) 187–194 193

beyond some maximum acceptable level by more than one facility.Results for the Biobío region in Chile demonstrate that

� practical problems of this type can be solved to optimalitywithin a fairly short period of time� tremendous reductions of pollution are possible at very reason-

able cost, and

� location patterns for reasonable choices of weights will includefew large landfills and some transfer stations.

Potential extensions of the model include the incorporation ofpollution along the transportation routes. This feature is reminis-cent of transportation risks in models that deal with the manage-ment of hazardous materials. Other extensions include theinclusion of different waste management systems, such as inciner-ation facilities (including waste-to-energy technologies) and recy-cling centers. One pertinent question concerns collection,particularly whether or not recyclables should be collected simul-taneously with regular waste.

Acknowledgments

This research was in part supported by a grant from the NaturalSciences and Engineering Council of Canada under Grant Number0009160, by Grant FONDECYT 1130265 and by Institute ComplexEngineering Systems, through grants ICM-MIDEPLAN P-05-004-Fand CONICYT FBO16. This support is gratefully acknowledged.We also like to express our appreciation to two anonymousreferees and an associate editor for their constructive comments

194 H.A. Eiselt, V. Marianov / European Journal of Operational Research 235 (2014) 187–194

that helped to improve the presentation and streamline theexposition.

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