The impact of an efficient collection sites location on the zoning phase in municipal solid waste management

Waste Management 34 (2014) 1949–1956

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Waste Management journal homepage: www.elsevier.com/locate/wasman

The impact of an efficient collection sites location on the zoning phase in municipal solid waste management Gianpaolo Ghiani, Andrea Manni, Emanuele Manni ⇑, Massimiliano Toraldo Dipartimento di Ingegneria dell’Innovazione, Università del Salento, via per Monteroni, 73100 Lecce, Italy

a r t i c l e

i n f o

Article history: Received 29 November 2013 Accepted 27 May 2014 Available online 21 June 2014 Keywords: Municipal waste management Location Zoning Heuristics

a b s t r a c t In this paper, we study two decisional problems arising when planning the collection of solid waste, namely the location of collection sites (together with bin allocation) and the zoning of the service territory, and we assess the potential impact that an efficient location has on the subsequent zoning phase. We first propose both an exact and a heuristic approach to locate the unsorted waste collection bins in a residential town, and to decide the capacities and characteristics of the bins to be located at each collection site. A peculiar aspect we consider is that of taking into account the compatibility between the different types of bins when allocating them to collection areas. Moreover, we propose a fast and effective heuristic approach to identify homogeneous zones that can be served by a single collection vehicle. Computational results on data related to a real-life instance show that an efficient location is fundamental in achieving consistent monetary savings, as well as a reduced environmental impact. These reductions are the result of one vehicle less needed to perform the waste collection operations, and an overall traveled distance reduced by about 25% on the average. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction As defined by the European Union’s Landfill Directive, Municipal Solid Waste (MSW), or simply municipal waste, includes ‘‘waste from households, as well as other waste which, because of its nature or composition, is similar to waste from households’’ (E.U., 1999). The amount of solid waste generated each year is rapidly increasing, thus making MSW management (MSWM) one of modern society’s most relevant issues, driven by the need to face a total waste production that since 1995 has increased considerably, as reported in Table 1 (OECD, 2012, 2013). There are many activities that must be performed for an efficient MSWM, involving several stages of the waste life-cycle: collection, (possibly) transformation, and disposal. Each of these activities involves taking many decisions at the strategic, tactical, and operational levels, like the selection of waste treatment sites and landfills (Wang et al., 2009), the future capacity expansion strategies of the sites waste flow allocation for transformation facilities and landfills (He et al., 2009), service territory zoning into districts (Mourão et al., 2009), collection days selection for each zone and for each waste type (Chu et al., 2006), fleet composition determination ⇑ Corresponding author. Tel.: +39 0832297737. E-mail addresses: [email protected] (G. Ghiani), andrea.manni@ unisalento.it (A. Manni), [email protected] (E. Manni), massimiliano. [email protected] (M. Toraldo). http://dx.doi.org/10.1016/j.wasman.2014.05.026 0956-053X/Ó 2014 Elsevier Ltd. All rights reserved.

and collection vehicles’ routing and scheduling (Ghiani et al., 2013). The reader is referred to Ghiani et al. (2014) for a recent survey of the most relevant issues. In this paper we focus on the waste collection stage. In particular, we first consider the problem of locating waste collection sites for road or curbside collection in a residential town and then the problem of zoning the service territory, so that each zone can be served by a single collection vehicle. For the collection sites location problem we propose both a mathematical programming model, based on that proposed in Ghiani et al. (2012), suitably modified to overcome some operational issues, and a fast heuristic approach. In both approaches, the goal is to choose where to locate the garbage collection bins, as well as the characteristics of such bins, such that two bin types that cannot be served contemporary by the same vehicle are not placed in the same collection area. For the zoning phase, given the fleet of collection vehicles that can be used, we employ a cheapest-insertion-based heuristic procedure that determines how many vehicles (zones) are necessary, along with the collection points to be served by each vehicle. The reminder of the paper is organized as follows. In Section 2 we review relevant literature. In Section 3 we describe both an exact and a heuristic approach to the problem of locating collection sites, whereas in Section 4 we present a fast heuristic procedure for determining collection vehicles’ routes to zone the service territory. In Section 5 we illustrate our computational results on a real-life test case. Finally, concluding remarks follow in Section 6.

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Table 1 Total waste production in the U.S., Europe, and Italy (tons/year).

U.S. Europe Italy

Table 2 Literature related to collection sites location.

1995

2010

Increase (%)

197,113,000 243,147,000 25,780,000

226,669,000 277,520,000 32,479,000

14.99 14.14 25.99

2. Literature review In this section, we review the main contributions related to both the location of collection sites and the zoning of the service territory.

2.1. Collection sites location As is well known, the problem of locating collection sites in residential areas is recognized to be a semi-obnoxious problem. Typically, citizens prefer having a waste collection site as close as possible to his/her home, but, at the same time, they aim at paying as less taxes as possible to have this service guaranteed. On the other hand, the community is interested not only in reducing the expenses related to the collection stage, but also in limiting the visual impact due to the presence of the collection bins close to the residential sites. Given that both these factors are related to the number of collection sites, a key objective is to provide a good quality of service with the least number of collection sites. Among the most recent literature on this topic, Bautista and Pereira (2006) model the problem of locating waste collection areas by proposing two mathematical formulations. The first is based on a set-covering formulation that is solved by a genetic algorithm. In the second approach, the authors propose a GRASP-based procedure to solve a Max-Sat problem. Both procedures are assessed on a test-case related to a city in the Barcelona metropolitan area. Badran and El-Haggar (2006) present a model to determine the best location for collection sites in Port Said (Egypt). The goal is to minimize the cost of municipal waste management. Erkut et al. (2008) use a multi-objective programming approach to tackle the location–allocation problem of waste management facilities, and employ it in the Central Macedonia region. Tralhão et al. (2010) study the problem of locating multi-compartment sorted waste containers, with the objective to determine which facilities to open, and the waste generation sources to assign to each facility. The approach is based on multi-objective programming and is tested on a case study represented by Coimbra (Portugal). CoutinhoRodrigues et al. (2012) present a bi-objective programming model to locate semi-obnoxious facilities and determine their capacities. The proposed approach tries to trade-off between the minimization of the total investment cost and the minimization of citizens’ inconvenience. The methodology is tested on a case study regarding the location of sorted waste containers. In the same year, Ghiani et al. (2012) face the problem of minimizing the total number of collection sites to be located in a MSWM system, chosen among a set of candidate locations. The proposed model determines also the optimal allocation of citizens to collection sites by ensuring that each citizen is serviced by a collection site which is within a threshold distance from his/her home. As will be highlighted in Section 3, the proposed model presents some drawbacks that represented the initial motivation for the present paper. Very recently, Toso and Alem (2014) face the problem of locating sorting centers in a medium-sized Brazilian city. The authors propose a deterministic version of the classical Capacity Facility Location Problem, as well as a two-stage recourse formulation and a risk-averse model. In Table 2 we summarize the main features about the analyzed literature.

Article

Uncertainty Multiple Exact Heuristic objectives method method

Bautista and Pereira (2006) Badran and El-Haggar (2006) Erkut et al. (2008) Tralhão et al. (2010) U Coutinho-Rodrigues et al. (2012) Ghiani et al. (2012) Toso and Alem (2014) U

U U U U

U U U U U U

U

2.2. Zoning The purpose of the zoning (or districting) phase is to determine collection districts. The districts must be such that the sum of solid waste loads within each district does not exceed the capacity of the vehicles that will perform the operations. The problem of districting, especially in the context of location and (arc) routing, is not widely addressed in the literature, or in many cases it is assumed to be solved a priori, de facto neglecting the positive impact it could have on the subsequent routing phase, or included in the strategic issues. Indeed, the zoning problem is strictly related to the Vehicle Routing Problem (VRP), which amounts to determine a set of routes for a heterogeneous fleet of vehicles, such that some customer requests are satisfied. In the literature, there are many variants that have been proposed, including time windows, accounting for dynamic requests, or combining the VRP with other problems. However, in this paper we do not intend to give an extensive review of the literature related to VRPs. Instead, we focus on the main contributions in which VRP approaches are used in order to determine the zones. A seminal contribution is that by Eisenstein and Iyer (1997), in which the authors study the zoning problem for the city of Chicago, by devising flexible schedules. They make use of data estimated from the field to model as random variables the weight and time required to collect garbage from a single block. Labelle et al. (2002) study the problem of partitioning a city into sectors, with respect to snow disposal operations, and for assigning the sectors to disposal sites. They first solve a districting problem in order to define the boundaries of each sector. Then, they solve an assignment problem to associate sectors with disposal sites. Sahoo et al. (2005) propose a mathematical model and a heuristic algorithm made up of two phases in order to divide the collection area into district, to make the problem more tractable. Nuortio et al. (2006) face the problem of zoning the service territory in Eastern Finland. In particular, they determine routes and schedules for the vehicles involved in the municipal solid waste operations, by modeling the problem as a Stochastic Periodic Vehicle Routing Problem with time windows, which is solved by the Guided Variable Neighborhood Thresholding metaheuristic. Operational aspects related to the route compactness and workload balancing of solid waste applications are considered in the work of Kim et al. (2006). They study a real-life waste collection Vehicle Routing Problem with time windows and present a route construction algorithm that is an extension of the insertion algorithm by Solomon (1987). Mourão et al. (2009) study the sectoring arc routing problem, applied to waste collection. The aim is to partition the service territory into a number of sectors, so that each sector can be covered by a set of vehicle trips. The authors propose three heuristics to face this problem. Benjamin and Beasley (2010) heuristically solve a routing problem, where empty vehicles based the (unique) depot and collect waste from customers, emptying themselves at the waste disposal facilities as and when necessary. Operative constraints as the time windows at customers, disposal facilities and depot, and driver rest periods are taken into account. More recently, Faccio et al. (2011) present a multi-objective routing model in which a heuristic is

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G. Ghiani et al. / Waste Management 34 (2014) 1949–1956 Table 3 Literature related to the zoning phase. Article

Uncertainty

Eisenstein and Iyer (1997) Labelle et al. (2002) Sahoo et al. (2005) Nuortio et al. (2006) Kim et al. (2006) Mourão et al. (2009) Benjamin and Beasley (2010) Faccio et al. (2011) Wy et al. (2013)

U

Multiple objectives

Exact method U U U

U

U U

Heuristic method

U U U U U U U U

integrated with traceability systems. By exploiting real-time information on the replenishment level and the position of the fleet of collection vehicles, as well as on the fullness of bins, the authors are able to optimize the route plan, deciding which bins should be emptied and which should not. Their goal is to minimize the distance covered by vehicles as well as the number of vehicles needed. In turn, this allows to achieve other goals, such as the minimization of travel time, number of stops, emissions and traffic congestions. The approach is assessed using simulation. Wy et al. (2013) study a rollon–rolloff waste collection VRP involving large containers that accumulate huge amounts of garbage at construction sites and shopping districts. The problem is complicated by the presence of constraints involving multiple disposal facilities, several types of customer demands, time windows and lunch breaks of drivers, as well as other real-world issues. The approach proposed by the authors consists in an iterative method based on large neighborhood search and is tested on a number of benchmark data. The main characteristics of the above described papers are reported and summarized in Table 3. 3. Collection sites location As briefly reported in Section 1, the collection sites location problem we study in this paper is similar to that of Ghiani et al. (2012). Indeed, the initial motivation for the present work arose when trying to implement in the real-world the solution obtained with the approaches proposed in Ghiani et al. (2012). In fact, the constraints reported in both the optimization model and the heuristic admitted feasible solutions without any restriction on the types of the bins allocated to a collection site, thus leading to inefficiencies in the subsequent phase of service territory zoning. In that case, in a feasible solution a collection site may host bins of types that cannot be served by the same vehicle (e.g., because of the presence of bins requiring side-loading compactors and other bins needing rear-loading compactors). Thus, at least two distinct vehicles must visit that collection site, with evident economic (on the side of the company in charge of performing the waste collection operations) and environmental (a higher number of vehicles in turn means, among other issues, higher emissions) inefficiencies. Moving on from these observations, in the present paper we propose a modification of both the optimization model and the constructive heuristic presented in Ghiani et al. (2012), to take into account additional constraints imposing that bins that cannot be served contemporary by the same vehicle type are not allocated to the same collection area. In the following, we first describe the optimization model including such new constraints and, then, the constructive heuristic modified accordingly. 3.1. Problem formulation As in Ghiani et al. (2012), given a set of candidate locations for unsorted waste collection, the objective of the optimization model

is the minimization of the total number of collection sites to be located. The model also determines the optimal allocation of citizens to collection sites, as well as which bins to place at those sites. The citizens allocation must be performed in such a way that the capacity of collection areas is not exceeded, and each citizen is serviced by a site which is within a threshold distance from his/her home. Moreover, each bin type is characterized by a length, a capacity, and a vehicle type (or a set of vehicle types) that has the capability of unloading it. In fact, in the problem we consider, a vehicle type can serve several (but not necessarily every) bin types, and a bin type can be served by several (but not necessarily every) vehicle types. Thus, the bin allocation phase must ensure that two bins that cannot be unloaded contemporary by the same vehicle type are not placed at the same collection site. The problem is formulated on a bipartite directed graph GðV 1 [ V 2 ; AÞ. Here, vertices in V 1 represent the waste generation sources (i.e., the citizens grouped in clusters according to their position), whereas V 2 are the potential sites where to collect the waste. According to Ghiani et al. (2012), we refer to each element of V 1 as a centroid. The set A is made up of arcs denoting waste flows between centroids and potential sites. Each arc is characterized by a distance dij between the centroid i 2 V 1 and the potential site j 2 V 2 . Denoted by Di the maximum distance between centroid i 2 V 1 and the associated collection site, U i # V 2 represents the set of potential sites such that U i ¼ fj 2 V 2 : dij 6 Di g. In addition, qi is the daily waste generation of centroid i 2 V 1 , and Lj is the linear length associated with potential collection site j 2 V 2 . Moreover, let K be the set of different bin types available for allocation to collection sites. Each type k 2 K is characterized by a capacity Q k , a linear length lk , and a total number bk of bins available for allocation. Finally, we consider a bin compatibility matrix C, in which, for each 0 0 k; k 2 K; ckk0 represents whether or not k and k are compatible (in that they can be unloaded by at least one common vehicle type). In 0 particular, ckk0 ¼ 1 if k and k are compatible, 0 otherwise. Decision variables are zj , binary variable that takes value 1 if the potential collection site j 2 V 2 (at least one bin type) is activated, 0 otherwise; xij , binary variable that takes value 1 if centroid i 2 V 1 is allocated to collection site j 2 V 2 , 0 otherwise; ykj , integer variable that represents the number of bins of type k 2 K to be allocated to collection site j 2 V 2 ; t kj , binary variable that takes value 1 if at least one bin of type k 2 K is allocated to collection site j 2 V 2 , 0 otherwise. Our mathematical model is formulated as follows:

Minimize

X zj

ð1aÞ

j2V 2

subject to X X qi xij 6 Q k ykj ; i2V 1

8 j 2 V2

ð1bÞ

k2K

X lk ykj 6 Lj zj ;

8 j 2 V2

ð1cÞ

k2K

X xij ¼ 1;

8 i 2 V1

ð1dÞ

j2U i

X

xih 6 1  zj ;

8 i 2 V 1; 8 j 2 V 2; d > 0

ð1eÞ

h:dij þd
X ykj 6 bk ;

8k2K

ð1fÞ

j2V 2 0

ykj 6 bk þ bk t k0 j ðckk0  1Þ; 8 j 2 V 2 ; 8 k; k 2 K; k – k ykj 6 tkj 6 ykj ; 8 j 2 V 2 ; 8 k 2 K bk zj 2 f0; 1g; 8 j 2 V 2

8 i 2 V 1; 8 j 2 V 2 ykj P 0; integer; 8 j 2 V 2 ; 8 k 2 K xij 2 f0; 1g;

tkj 2 f0; 1g;

8 k 2 K; 8 j 2 V 2

0

ð1gÞ ð1hÞ ð1iÞ ð1jÞ ð1kÞ ð1lÞ

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The objective function aims at minimizing the total number of activated collection sites. Constraints (1b) impose that the total waste directed by the centroids assigned to a collection site does not exceed its maximum capacity. Constraints (1c) ensure that the bins placed at a collection site fit in it, with respect to length. Constraints (1d) allocate each centroid i 2 V 1 to exactly one collection site, among those that are activated within the threshold distance. We note that in this constraints’ formulation we intentionally do not necessarily allocate a centroid to the closest activated site, because our goal is minimizing the number of activated sites, rather than minimizing the average distance between centroids and collection locations. Constraints (1e) force a centroid to be allocated to the closest activated collection site, within the threshold distance, allowing a certain tolerance through parameter d. This tolerance is fundamental in order to obtain feasible solutions. In principle, it may happen that the activated site that is the closest to a centroid has not enough capacity. Thus, the centroid is allocated to the second activated closest, and so on. Constraints (1f) avoid using more bins than those available. Constraints (1g) and (1h) avoid placing at a given site j 2 V 2 bin types that are not compatible each other, by tying variables ykj to variables tkj . These two sets of constraints are the novelty of this model formulation with respect to Ghiani et al. (2012). Typically, program (1a)–(1l) may be solved to optimality through a general-purpose solver for small instances only. For this reason, in the following section we develop a fast and efficient heuristic method to tackle real-world-sized instances. Algorithm 1. A constructive heuristic for the location of collection sites 1: procedure CONSTRUCTHEURISTICðV 1 ; V 2 Þ 2: V  ALLOCATECENTROIDSðV 1 ; V 2 Þ 3: if V  ¼ £ then 4: return null 5: else 6: for all j 2 V  do ALLOCATECENTROIDSðV  n fjg; V 1 Þ 7: V0 8: if V 0 – £ then V0 9: V 10: end if 11: end for 12: end if 13: return V  14: end procedure

3.2. A heuristic approach The complexity of the optimization model presented in Section 3.1 makes it possible to use it for small instances only. Thus, we propose a fast and efficient two-phases heuristic approach that (i) heuristically solves a capacitated allocation problem; (ii) reduces as much as possible the number of activated collection sites; (iii) decides how many bins of the different types to allocate to the collection sites chosen at the end of the first phase. The first phase of the procedure is depicted in Algorithms 1 and 2, whereas the second phase is illustrated in Algorithm 3. We remark that Algorithms 1 and 2 are the same as those of Ghiani et al. (2012), and are reported here for the sake of clarity. On the other hand, Algorithm 3 has been suitably modified in order to take into account the compatibility between different types of bins.

Algorithm 2. The procedure for allocating centroids to collection sites. 1: procedure ALLOCATECENTROIDSðV 1 ; V 2 Þ 2: Q max 2 RjV 2 j // Q max is the maximum amount of waste that j can be directed to j 3: R Q max ==R 2 RjV 2 j , where Rj is the residual capacity of site j £ 4: V  5: for all i 2 V 1 do V2 6: V0 7: check false 8: while check ¼ false do 9: j CLOSESTSITEði; V 0 ; dÞ 10: if j – null then 11: if qi 6 Rj then 12: hallocate i to j i 13: Rj Rj  qi 14: V V  [ fjg 15: check true 16: else 17: V0 V 0 n fjg 18: end if 19: else 20: return null 21: end if 22: end while 23: end for 24: return V  25: end procedure

In detail, the constructive heuristic used in the first phase is inspired by the drop heuristic for facility location (Feldman et al., 1966). We first start with a large number of activated sites; then, we try to remove one site at a time, only if such a removal is feasible and is associated with a cost reduction. In particular, we first assign each centroid to the potential collection site which is the closest, with a tolerance of d, having a residual capacity greater than the daily generation of wastes of the centroid. Each site j is characterized by a maximum amount of waste that can be directed to it, computed using the maximum length Lj , the capacities of bins, as well as the bin compatibility matrix C. In this phase, we do not take into account the objective of minimizing the number of activated collection sites (Algorithm 1, line 2). If the previous conditions are not satisfied for a particular centroid, then it cannot be assigned to any site. Afterwards, we temporarily drop one activated collection site at a time, allocating its centroids to the remaining activated sites. If this operation is successful, we permanently drop the collection site, and we iterate (Algorithm 1, lines 6–11). The output of this phase is represented by the subset of collection sites to be activated. The second phase of our approach (Algorithm 3) involves the choice of which bins of which type to allocate to the collection sites selected at the end of the first phase. Here, as in the mathematical model, the generic element ykj represents the numbers of bins of type k allocated to the collection site j. Thus, for each site j, we first check whether the sum of the waste generation of the centroids allocated to it can be satisfied by a single bin of type k, such that lk 6 Lj , its capacity Q k is greater than the total waste to be directed to j and the type k is compatible with the bins previously allocated (Algorithm 3, line 7, MINIMUMCAPACITYBIN method). If we do not succeed, we choose a bin type k having the maximum capacity among those available, as long as this bin type is compatible with the bin types already allocated to collection site j (Algorithm 3, line 9,

G. Ghiani et al. / Waste Management 34 (2014) 1949–1956

MAXIMUMCAPACITYBIN method). Finally, we increment by one the corresponding ykj (Algorithm 3, line 11). We iterate until the sum of the capacities of the bins allocated to j is greater than the total waste to be directed to it. Algorithm 3 The procedure for allocating bins to collection sites 1: procedure ALLOCATEBINSðV; K; CÞ 2: ykj ¼ 0; k 2 K; j 2 V 3: for all j 2 V do 4: residualCapacity htotal waste to be directed to j i 5: residualLength Lj 6: while residualCapacity > 0 and residualLength > 0 do 7: k MINIMUMCAPACITYBIN ðresidualLength; residualCapacity; K; C; yÞ 8: if k ¼ null then 9: k MAXIMUMCAPACITYBINðresidualLength; K; C; yÞ 10: end if 11: ykj ykj þ 1 12: residualCapacity residualCapacity  Q k 13: residualLength residualLength  lk 14: end while 15: end for 16: return A 17: end procedure

4. Zoning Given the set of collection sites V  # V 2 and the bins allocation resulting from the previous phase, the purpose of zoning (or districting) is to determine collection zones. These zones must be such that the sum of solid waste loads of collection sites within each of them does not exceed the capacity of the vehicles that will perform the operations. In this section we describe how we solve this problem, with the goal to minimize the number of vehicles utilized to perform the operations, as well as the service time. This results in (i) determining the least number of vehicles to visit the collection points; (ii) minimizing the traveled distance as a proxy of the service time. Each site j 2 V  to visit is characterized by a waste production qj P 0, as well as an average time sj to serve it. The fleet F is made up of heterogeneous collection vehicles. Each vehicle f 2 F has an associated depot df chosen from a set D from/to which it must start/conclude its route. Moreover, we denote by Q f the capacity of vehicle f 2 F. Each route has a maximum duration T and can be made up of a number of portions (not exceeding the vehicle’s capacity), each of which ends at a landfill in order to unload the waste. A vehicle f is allowed to visit site j only if the bins allocated to j can be loaded into f. This restriction is obtained by using a compatibility matrix M, in which entry mfj ¼ 1 if vehicle f 2 F can load the bins allocated to collection site j 2 V  , 0 otherwise. Algorithm 4. A constructive heuristic for determining collection zones. 1: procedure ZONINGHEURISTICðV  ; F; MÞ 2: for all f 2 F do hdf ; l; df i 3: routef 4: end for 5: for all j 2 V  do 6: hf ; pi FindRouteðj; F; MÞ 7: hinsert j in routef after position p i 8: end for 9: end procedure

1953

Our heuristic approach is depicted in Algorithms 4 and 5, and is based on the cheapest-insertion paradigm (Mosheiov, 1994) in order to determine the routes, by successively inserting customers into growing routes. In each step one customer is inserted. More specifically, for each vehicle f of the fleet, its route is initialized as the sequence hdf ; l; df i, where l denotes the landfill (Algorithm 4, lines 2–4). Then, for each collection site j to visit, we determine which is the vehicle f that can perform this task at the least additional cost, as well as the position of the route where to insert the visit to the collection site. Finally, the route is updated. The cheapest-insertion logic is implemented by means of the method FindRouteð; ; Þ, which is responsible of detecting the best route where the current collection site j can be inserted. For this purpose, for each vehicle f that is able to load the bins allocated to j, we compute the additional cost of allocating j to each position p of f as the travel time from p to j, plus the time needed to serve j, plus the travel time from j to p þ 1, minus the travel time from p to p þ 1 (Algorithm 5, line 8). Then, if the new route duration does not exceed the threshold T and the additional insertion cost is lower than the best cost found so far, we update the best vehicle, the best cost, and the best position (Algorithm 5, lines 10–14). It is worth noting that Algorithms 4 and 5 can be suitably modified in order to manage particular cases that, although very unlikely to happen, could potentially lead to unallocated collection sites. Suppose that the fleet of collection vehicles includes (among others) vehicles of two different types, say type one and two. Moreover, also suppose there are two kinds of bins such that bins of type one can be unloaded by vehicles of type one only, whereas bins of type two can be unloaded by both vehicle types. If the ‘‘best’’ vehicle to unload bins of type one is systematically of type two, then vehicles of type two would become full and there would not be any compatible vehicle for bins of type two. This case could be easily addressed by slightly modifying Algorithms 4 and 5, in such a way that more constrained bins are allocated first. However, we remark that this situation is very unlikely to happen and, in particular, it never happened in our computational experiments. Algorithm 5. A constructive heuristic for determining collection zones: FINDROUTE procedure. 1: procedure FINDROUTEðj; F; MÞ 2: bestPosition null 3: bestRoute null 4: bestCost þ1 5: for all f 2 F: mfj ¼ 1 do 6: for all p 2 routef do 7: // t p;j is the travel time from p to j 8: insertionCost tp;j þ sj þ tj;pþ1  t p;pþ1 9: hupdate the route duration T f i 10: if (T f 6 T and insertionCost < bestCost) then 11: bestPosition p 12: bestRoute f 13: bestCost insertionCost 14: end if 15: end for 16: end for 17: return hbestRoute; bestPositioni 18: end procedure

5. Computational results In this section, we illustrate the computational experiments we have performed in order to test the optimization model presented

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G. Ghiani et al. / Waste Management 34 (2014) 1949–1956

in Section 3.1, as well as the constructive heuristics we have developed to face the collection sites location and zoning phases. The heuristics are coded in Java, whereas the mathematical models are solved by means of IBM ILOG CPLEX 12.3. The experiments are performed on a machine with an Intel processor clocked at 2.8 GHz and equipped with 2 GB of RAM. The instance used is the same utilized by Ghiani et al. (2012), and is related to the city of Nardò (Apulia region, Italy), which is characterized by an average per-capita waste production of about 1.3 kg/day. The aim of our computational experiments is to show that an efficient collection sites location phase, performed by suitably modifying the optimization model and the constructive heuristic of Ghiani et al. (2012) in order to take into account the compatibility among different bin types, allows to achieve consistent savings in the subsequent zoning phase. For this purpose, we consider a set of potential collection sites such that jV 2 j ¼ 560. The population is grouped in 1163 centroids (jV 1 j ¼ 1163), and, for each i 2 V 1 , the parameter qi is set equal to the number of inhabitants associated to i multiplied by the per-capita daily generation of waste. Moreover, K ¼ 3 and the characteristics of the bins are as reported in Table 4, whereas the bin compatibility matrix is shown in Table 5. Finally, we consider three different vehicle types, whose capability to unload the various bin types is reported in Table 6. To test the performance of our approaches under different conditions, different values for the per-capita daily generation of waste are considered, namely 1.3, 1.4, and 1.5 kg/day. Similarly to Ghiani et al. (2012), for each centroid i 2 V 1 , we consider two values for the threshold distance from the closest activated site (140 and 150 m), and a tolerance parameter d ¼ 0:1Di . We note that the constructive heuristics are characterized by negligible execution times. In the first part of our experiments, we assess the quality of the solution of the collection sites location heuristic with respect to that of the optimization model (1a)–(1l). However, given that the optimization model is untractable on the whole Nardò instance, it is necessary to partition it, and compare the heuristic and exact solutions on these smaller instances. More precisely, we consider two separate areas, which we call A and B. This comparison is performed by considering the case with a per-capita daily generation

of 1.3 kg/day, and a value of Di ¼ 150 m for each centroid i 2 V 1 . Moreover, we impose a time limit of 3600 s to the MIP solver. The results are reported in Tables 7 and 8, and show that the solutions obtained by the optimization model when the time limit is reached are sub-optimal, with an optimality gap of about 8% for area A and 5% for area B. The average percentage deviation of the results obtained by using the heuristic with respect to the exact approach is 15.56% for the number of activated collection sites, and 6.44% for the number of bins allocated. All in all, this first experimentation allows to conclude that the results provided by the heuristic are not very far from those of the optimization model, having the great advantage of nearly instantaneous computing times. This would allow using the proposed heuristic in a what-if fashion, by launching it many times with different inputs and settings. Next, we compare the performance of the collection sites allocation heuristic we devise here, with that proposed in Ghiani et al. (2012), in terms of both the number of activated collection sites, and the number of bins allocated to such sites. The results of this comparison are reported in Tables 9 and 10. As the numbers highlight, the new heuristic approach allows to achieve a reduction in the number of activated collection sites, ranging between 12.67% and 18.42%, whereas the two approaches provide comparable performance in terms of overall number of bins utilized. The main difference lies in the choice of the bins type, with a greater number of bins of type k ¼ 3 utilized with respect to the previous approach.

Table 4 Bins available in the area of Nardò.

Table 8 Comparison between the optimization model and the constructive heuristic: bins allocation.

Type

Capacity (kg)

Length (m)

1 2 3

72 154 480

0.62 1.37 1.88

Table 7 Comparison between the optimization model and the constructive heuristic: activated collection sites Zone

A B

Zone

Opt. gap (%)

55 61

7.8 5.4

Percentage deviation (%)

64 70

16.36 14.75 15.56

Optimization model Bins allocated

Opt. gap (%)

Heuristic approach Bins allocated

92 94

7.8 5.4

97 101

Average

Table 5 Compatibility between bin types. Bin type

1

2

1 2 3

U U

U U

Percentage deviation (%)

5.43 7.45 6.44

3 Table 9 Results obtained by the constructive heuristic: collection sites. U

Threshold distance (m)

Per-capita daily generation (kg)

Collection sites Ghiani et al. (2012)

Collection sites heuristic approach

Percentage deviation (%)

140

1.3 1.4 1.5

221 237 239

193 206 203

12.67 13.08 15.06

3

150

1.3 1.4 1.5

212 224 228

181 185 186

14.62 17.41 18.42

U

Average

Table 6 Compatibility between vehicle types and bin types.

1 2 3

Collection sites

Heuristic approach collection sites

Average

A B

Vehicle type

Optimization model

Bin type 1

2

U U

U 15.21

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G. Ghiani et al. / Waste Management 34 (2014) 1949–1956 Table 10 Results obtained by the constructive heuristic: bins allocation. Threshold distance (m)

Per-capita daily generation (kg)

Ghiani et al. (2012)

Heuristic approach

k¼1

k¼2

k¼3

k¼1

k¼2

k¼3

140

1.3 1.4 1.5

58 63 64

36 37 38

142 152 152

25 24 27

24 22 20

151 162 161

150

1.3 1.4 1.5

52 45 50

32 36 36

143 153 155

19 21 23

24 18 18

153 170 173

Table 11 Results obtained by the zoning heuristic. Threshold distance (m)

Per-capita daily generation (kg)

No. of zones Ghiani et al. (2012)

No. of zones heuristic approach

Overall distance (m) Ghiani et al. (2012)

Overall distance (m) heuristic approach

Percentual deviation (%)

140

1.3 1.4 1.5

5 5 5

4 4 4

666,654.80 671,574.40 798,033.60

535,207.40 544,096.00 557,218.20

19.71 18.98 30.18

150

1.3 1.4 1.5

5 5 5

4 4 4

655,531.80 780,707.20 800,885.40

521,418.80 541,940.00 554,141.00

20.46 30.58 30.81

Average

This behavior can be explained as the result of the new feature introduced for the collection sites location heuristic, aiming at choosing bin types that implicitly allow to determine homogeneous zones. In fact, this choice has a great impact on the subsequent zoning phase, whose results are depicted in Table 11. In this table, we show the performance of the heuristic procedure illustrated in Section 4, in order to determine the number of zones in which the service territory should be partitioned. This partition is based on the collection sites obtained by considering the previous approach by Ghiani et al. (2012), as well as our new approach that takes into account bins compatibility when choosing which bins to allocate to the different sites. The performance indicators used to compare the two alternatives are the number of zones determined by the cheapest-insertion procedure, as well as the overall distance that should be covered by the vehicles in order to perform the tasks. As Table 11 shows, by applying the cheapest-insertion heuristic on the set of collection sites obtained with the new heuristic approach, we are able to identify a lower number of zones than the previous approach. In particular, we obtain one zone less than before. This obviously implies that one vehicle less than before is needed in order to perform the collection operations. In turn, this vehicle reduction determines a reduction of the overall distance traveled by the vehicles, that can be used as a proxy of the collection cost. As can be observed from the table, the traveled distance reduction ranges between 18.98% and 30.81%, which can result in consistent monetary savings, as well as a reduced environmental impact.

6. Conclusions In this paper we have faced two decisional problems arising when planning the collection of solid waste in a municipal waste management system. In particular, we have studied the problem of locating waste collection areas in a residential town, as well as the problem of zoning the service territory, in order to asses the impact of an efficient collection sites location on the subsequent zoning phase. For the first problem, we have proposed a modification of both an exact and a heuristic approach recently proposed in the literature, in order to take into account the compatibility among

25.12

bins when allocating them to collection sites. The main objective was to avoid inefficiencies deriving from the fact that two (or more) different vehicles must visit a given collection site, because of the presence of bins that cannot be unloaded by the same vehicle type. This has been obtained by suitably adding constraints in the optimization model, as well as modifying the heuristic approach. In order to assess the impact of these modifications on the zoning phase, we have devised a fast and efficient cheapest-insertionbased heuristic procedure to determine the number of zones in which to partition the service territory. Computational results on data related to a real-life problem have shown the effectiveness of the proposed procedures, resulting in one vehicle less needed to perform the collection operations, and an overall traveled distance that can be reduced by about 25% on the average. These reductions could determine consistent monetary savings in the waste collection operations, in addition to a reduced environmental impact. Acknowledgements This research was partially supported by the Ministero dell’Istruzione, dell’Università e della Ricerca Scientifica (MIUR) of Italy. This support is gratefully acknowledged. The authors also thank three anonymous referees for their useful comments, which helped to improve the paper. References Badran, M.F., El-Haggar, S.M., 2006. Optimization of municipal solid waste management in Port Said – Egypt. Waste Manage. 26, 534–545. Bautista, J., Pereira, J., 2006. Modeling the problem of locating collection areas for urban waste management. An application to the metropolitan area of Barcelona. Omega 34, 617–629. Benjamin, A.M., Beasley, J.E., 2010. Metaheuristics for the waste collection vehicle routing problem with time windows, driver rest period and multiple disposal facilities. Comput. Oper. Res. 37 (12), 2270–2280. Chu, F., Labadi, N., Prins, C., 2006. A scatter search for the periodic capacitated arc routing problem. Eur. J. Oper. Res. 169 (2), 586–605. Coutinho-Rodrigues, J., Tralh ao, L., Alçada-Almeida, L., 2012. A bi-objective modeling approach applied to an urban semi-desirable facility location problem. Eur. J. Oper. Res. 223 (1), 203–213. Eisenstein, D.D., Iyer, A.V., 1997. Garbage collection in Chicago: a dynamic scheduling model. Manage. Sci. 43 (7), 922–933.

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