A novel methodology for designing a household waste collection system for insular zones

Transportation Research Part E 77 (2015) 227–247

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Transportation Research Part E journal homepage: www.elsevier.com/locate/tre

A novel methodology for designing a household waste collection system for insular zones Pablo A. Miranda a,⇑, Carola A. Blazquez b,1, Rodrigo Vergara c, Sebastian Weitzler c a

School of Industrial Engineering, Pontificia Universidad Catolica de Valparaiso, Av. Brasil 2241, Valparaiso, Chile Department of Engineering Science, Universidad Andres Bello, Sazie 2315, Santiago, Chile c Graduate Student, School of Industrial Engineering, Pontificia Universidad Catolica de Valparaiso, Av. Brasil 2241, Valparaiso, Chile b

a r t i c l e

i n f o

Article history: Received 26 March 2014 Received in revised form 22 January 2015 Accepted 25 February 2015 Available online 27 March 2015 Keywords: Insular rural household waste Waste Collection System Design Waste collection site selection Periodic vehicle routing problem

a b s t r a c t This paper addresses the problem of designing a household waste collection system for rural insular areas using a barge for transportation, based on a novel mixed integer programming model that simultaneously integrates decisions of waste collection sites selection within the islands to be served, visit schedule for each selected collection site, and multi-period vehicle routing. An application to a real-world instance consisting of small rural islands located in the south of Chile shows the effectiveness and complexity of the model, along with the advantages of using a waste compactor instead of transporting the waste using bins onboard a barge. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction In recent decades, researchers and practitioners have shown a significant interest in problems involving the design, planning, and operation of an industrial or household solid waste collection system. This is mainly explained by a growth in the generated waste volume and its environmental impacts, and by an increase of environmental awareness and sustainability requirements in modern societies. Therefore, there has been a strong development of models and tools to support planning and design of urban solid waste management systems (Gottinger, 1988; Kirca and Erkip, 1988; Angelelli and Speranza, 2002a; Viotti et al., 2003; Kim et al., 2006; Karadimas et al., 2007; Sharholy et al., 2007; El-Hamouz, 2008; He et al., 2009; Arribas et al., 2010; Xu et al., 2010; De Figueiredo and Mayerle, 2008; Parker et al., 2010) based mainly on operations research techniques, as presented in this paper. Different modeling structures have been developed to address the Waste Collection System Design (WCSD) depending on the specific features of the problem. The following paragraphs describe some relevant examples of these features along with studies and approaches found in the literature. When assuming that the waste volume generated between two consecutive visits for each collection site or point is insufficient to fill a vehicle (Less than Truck Load, LTL), the WCSD problem will face the structure of a Vehicle Routing Problem, VRP (i.e., one destination/depot node and multiple origin nodes within a same route/vehicle). See Laporte and Osman (1995), and Ando and Taniguchi (2006) for a review of VRPs. In the case of hazardous substances and industrial waste, planners and modelers should regard ecological issues such as environmental and community risk minimization jointly with transportation costs by employing multi-objective methodologies. Furthermore, industrial waste collection systems usually are Full Truck Load (FTL), in which the entire load is a single ⇑ Corresponding author. Tel.: +56 (32) 227 3701. 1

E-mail addresses: [email protected] (P.A. Miranda), [email protected] (C.A. Blazquez). Tel.: +56 (32) 284 5579.

http://dx.doi.org/10.1016/j.tre.2015.02.019 1366-5545/Ó 2015 Elsevier Ltd. All rights reserved.

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consignment. Therefore, instead of a traditional VRP, several studies have considered a network flow-based routing problem between multiple origin–destination pairs for transporting hazardous waste (List et al., 1991; List and Mirchandani, 1991; Revelle et al., 1991; Saccomanno and Shortreed, 1993; Patel and Horowitz, 1994; Erkut, 1995; Sivakumar et al., 1995; Erkut and Verter, 1998; Giannikos, 1998; Zhang et al., 2000; Huang and Cheu, 2004; Meng et al., 2005; Dadkar et al., 2008). In the household waste case, the waste generated at each collection site is deemed as a LTL case since this volume is too low to be considered a FTL case when compared to vehicle capacities. Therefore, a household WCSD problem can be formulated as a combination of a visit scheduling problem and a VRP for each operation day, which is equivalent to the Periodic VRP, PVRP (Angelelli and Speranza, 2002b; Francis et al., 2008; Hemmelmayr et al., 2013), Inventory Routing Problem, IRP (Bertazzi et al., 2002, 2008; Campbell and Savelsbergh, 2004) or Cyclic Inventory Routing Problem, CIRP (Raa and Aghezzaf, 2007; Raa et al., 2007; Zhong and Aghezzaf, 2009; Zheng et al., 2009; Aksen et al., 2012). In the latter, the visit schedule is defined for a bounded, short period (e.g., one week or one month) that is repeated over time. Notice that the IRP and the CIRP may be defined as extensions of the PVRP, in which inventory management for customers or demand points is part of the optimization problem along with PVRP decisions. Finally, when the planner determines the set of waste collection sites to be served, the household WCSD problem may present a structure similar to the selective VRPs, as in Gendreau et al. (1998), Laporte and Martello (1990), Süral and Bookbinder (2003), Gribkovskaia et al. (2007), GutierrezJarpa et al. (2010), and Aksen et al. (2012). Several studies have addressed the ship routing and scheduling problem (Ronen, 1993; Desrosiers et al., 1995; Sherali et al., 1999; Christiansen et al., 2003; Bronmo et al., 2007; Agarwal and Ergun, 2008; Gatica and Miranda, 2011) by applying and extending VRP and IRP to the case of maritime transportation. Some specific studies have focused on ship routing problems distributing liquid bulk to nearby islands or archipelagos (Al-Khayyal and Hwang, 2007 and Agra et al., 2013). However, these studies do not address site selection or waste management issues, as in this research. The main objective and contribution of this paper is to develop a novel Mixed Integer Programming (MIP) model to support the system design for collecting rural insular household waste, and its application to an archipelago located in the south of Chile. In this case study, the waste treatment at the islands is currently performed by the inhabitants without any type of waste collection system. The continuous increase in the waste generation warrants a system for collecting this waste efficiently and environment-friendly. The insular household WCSD problem studied in this paper addresses simultaneously three main set of decisions in an integrated manner: waste disposal and collection site selection among the available sites at each island, site collection frequency, and visit sequence of a barge for each operation period (e.g., day). The proposed model deals with collection frequency decisions for each site based on a visit pattern selection scheme, simultaneously with daily vehicle routing decisions using a single barge. A visit pattern defines the days of the week that a collection site will be visited (e.g., Wednesday and Saturday, or Monday, Wednesday, and Friday). At the same time, the model optimizes the selection of collection sites among a set of potential sites for each island. Hence, a relevant contribution of this research is the novel integrative modeling approach of these three set of decisions to solve the WCSD problem for a set of islands. Note that there is a similarity of the household WCSD problem analyzed in this paper with the IRP, and particularly with the case of constant frequency in time such as CIRP found in the revised literature. Thus, the proposed model may be deemed as an extension of the IRP and CIRP, in which visit schedule and daily routes are determined. Unlike PVRP, IRP, and CIRP, the proposed model optimizes the selection of waste collection site at each island, similarly to selective VRPs. In addition, unlike selective VRPs, the demand for each potential collection site is unknown in advance since it depends on the number of sites selected for each island. Accordingly, to the authors’ knowledge, the problem studied in this research cannot be resolved directly with the existing models observed in the related literature (i.e. PVRP, IRP and CIRP), which highlights one significant contribution of this research. The proposed model is used to represent and optimize two operating strategies of the waste collection system by simply modifying parameter values: (i) employing a waste compacting machine on board a barge without the need of transporting full/empty bins, and (ii) transporting waste bins on board a barge. The paper is organized as follows. Section 2 describes the insular household WCSD problem studied in this paper. Section 3 presents the MIP formulation proposed for solving the problem. Section 4 presents the application of the model on a real-world case study along with the results and discussion. Finally, conclusions and future research are provided in Section 5.

2. Problem description According to the current household waste situation described previously, the problem addressed in this paper consists of designing a waste collection system to serve a set of nearby, small islands belonging to the same political or administrative district. Waste collection and removal locations among potential available sites at each island (e.g., quays, docks, piers, or ports) are selected. Barges for waste collection and transportation will depart and return to a depot, thus determining waste collection tours. Note that time windows for each collection site are observed based on the real world waste collection problem studied in this paper. This study focuses on the WCSD problem for a specific year, which is considered as the initial operation period. However, if a route reaches the maximum volume capacity of the barge during one of the following years, then the system should be

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reoptimized using the proposed methodology with the updated system information (population, waste generation, costs, etc.) for that year. Thus, the model and its parameters and decisions variables are defined for one year. Potential waste collection points represent places with some basic type of pier infrastructure that allow barges (but not vehicles) to access the island. Accordingly, any aspects of waste transportation within each island are not considered in this study, and are required as future research. Two waste collection strategies are identified for collecting and transporting the generated waste. The first strategy (strategy 1) considers the usage of compacting equipment on the barge, which loads, compacts, and transports the waste stored in bins at the collection sites or piers. The second strategy (strategy 2) consists of picking up and transporting full bins on the barge, and replacing them with empty bins at the collection sites. Both strategies employ the same typical bins for waste storage. These bins are located beside the piers (defining the waste disposal and collection sites) to which the island inhabitants transport and dispose their household waste. This waste transportation and disposal process is not included in this research. In order to simplify the operation of the system, visit frequency and collection periods present the following restrictions: collection sites should be visited at least once a week, visit days must be uniformly distributed throughout the week, and site visits should be repeated every week. This schema ensures that the waste will be stored at collection sites for short time periods (e.g., at most 7 days), in order to prevent an inadequate waste exposure to the environment and the community. These assumptions yield a waste collection policy similar to standard systems observed in urban regions or cities. Table 1 presents the collection site frequencies and collection days for each visit pattern schema. Note that waste is not collected on Sundays. The model selects one visit pattern from this list of feasible patterns for each chosen collection site. Additionally, the visit sequence or routing decisions for each period must be achieved according to the selected visit patterns, and chosen collection sites. When more than one potential site exists on an island, the generated waste will be divided homogeneously among the selected sites. 3. Model formulation The island household WCSD problem described in the previous section is formulated as a MIP model, which combines decisions of visit pattern selection from the group of feasible patterns, visit sequence for each operation period, and waste collection sites selection for each island. In addition, time windows constraints must be fulfilled for each selected collection site. These decisions should be optimized by minimizing total transportation costs. 3.1. Parameters and decision variables The model is built upon the definition of visit patterns and the parameters rpt that are computed based on Table 1. For each pattern p (p = 1, . . ., P with P = 12), the parameter rpt is equal to 1 if pattern p includes a visit on day t (t = 1, . . ., 6) and equal to zero, otherwise. For example, if the visit pattern p = 1 (collection day is on Mondays) is selected, then rp1 = 1 and rpt = 0 for t = 2, 3, 4, 5, 6. Subsequently, the model assigns a specific visit pattern to each selected collection site. Indexes and sets H set of islands to be served N set of nodes, including potential waste collection sites and the depot p index for visit pattern (p = 1, . . ., P, P = 12) t index for time period or day (t = 1, . . ., T, T = 6) i,j indexes for nodes (i e |N|), where i0 is the depot v index for island (v e H). Xv set of potential locations for waste collection on island v k index indicating the number of selected locations for waste disposal on an island (k = 1, . . ., |Xv|, for each island v)

The decision variables of the model, along with their dimensions in parenthesis, are the following: Zi X ip Uv k Y ijt W it

binary variable indicating if node i is selected as a collection site ((|N|  1)2 binary variables) binary variable denoting if the collection site at node i is visited according to the visit pattern p ((|N|  1)  P binary variables) binary variable indicating if island v is served using k collection sites (|N|  1 binary variables) binary variable representing if node j is visited immediately after node i during operation period or day t (|N|2  T binary variables) amount of waste to be collected from the collection site at node i on period or day t ((|N|  1)  T continuous variables) (continued on next page)

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T it

arrival time to the node i or the depot on period t (|N|  T continuous variables)

Model parameters Cij transportation cost from node i to node j rpt binary parameter specifying whether pattern p (p = 1, . . . , P) includes a visit on period or day t (t = 1, . . ., T) qkv pt amount of waste to be collected on period t (t = 1, . . ., T) from any collection site on island v, in the case that pattern p is selected (p = 1, . . ., P), and the island is served by using k waste collection sites (k = 1, . . ., |Xv|). This parameter is computed for each possible value of k and for each visit pattern p, as a function of the total weekly generated waste for island v. If Dv is the weekly generated waste for island v, then qkv pt ¼  PDTv   r pt k r t¼1 pt ai, bi earliest and latest times for arrivals at each node i M a large constant d operating time for pick-up at each collection site tij travel time from node i to node j CAP capacity of the barge (m3)

3.2. The optimization model According to the previous notation, definitions, and assumptions, the proposed MIP formulation for the studied insular household WCSD problem is the following: T X X

Min

C ij  Y ijt

ð1Þ

8i 2 N=i – i0

ð2Þ

t¼1 i;j2N;j–i

Subject to:

Zi ¼

P X

X ip

p¼1

X

Zi ¼

i2Xv

jXv j X

k  Uv k

8v 2 H

ð3Þ

k¼1

jXv j X Uv k ¼ 1

8v 2 H

ð4Þ

k¼1

W it P

P X qkv pt  X ip þ CAP  ðU v k  1Þ 8t ¼ 1; . . . ; T; 8v 2 H; 8i 2 Xv ; 8k ¼ 1; ::; jXv j

ð5Þ

p¼1

CAP P

X

W it

8t ¼ 1; . . . ; T

ð6Þ

i2Nfi0 g

Table 1 Waste collection visit patterns. Visit pattern

Weekly frequency

Potential visit days Monday

1 2 3 4 5 6 7 8 9 10 11 12

1 1 1 1 1 1 2 2 2 3 3 6

Tuesday

Wednesday

Thursday

Friday

Saturday

x x x x x x x

x x

x x

x x x

x x

x

x x x x

x

x x

P.A. Miranda et al. / Transportation Research Part E 77 (2015) 227–247 P X X X ip  r pt ¼ Y ijt p¼1

X

8t ¼ 1; . . . ; T; 8i 2 N=i – i0

231

ð7Þ

j2N;j–i

Y i0 it 6 1

8t ¼ 1; . . . ; T

ð8Þ

i2Nfi0 g

X X Y jit ¼ Y ijt j2N i–j

8i 2 N; 8t ¼ 1; . . . ; T

ð9Þ

j2N i–j

ai 6 T it 6 bi

8t ¼ 1; . . . ; T; 8i 2 N

T jt P ai0 þ t i0 J  ð1  Y i0 jt Þ  M

8t ¼ 1; . . . ; T; 8j 2 N=j – i0

T jt P T it þ t ij þ d  ð1  Y ijt Þ  bj

8t ¼ 1; . . . ; T; 8i; j 2 N=i – i0

ð10Þ ð11Þ ð12Þ

Z i ; X ip 2 f0; 1g 8p ¼ 1; . . . ; P; 8i 2 N=i – i0

ð13Þ

W it ; T it P 0 8i 2 N; 8t ¼ 1; . . . ; T

ð14Þ

Y ijt 2 f0; 1g 8i; j 2 N; j – i; 8t ¼ 1; . . . ; T

ð15Þ

U v k 2 f0; 1g 8v 2 H; k ¼ 1; ::; jXv j

ð16Þ

The objective function (1) minimizes total transportation costs within the planning horizon (t = 1, . . ., T). Constraint (2) indicates that if the node i is selected as a waste collection site (Zi = 1), then a single visit pattern must be selected from P feasible visit patterns (|N|  1 constraints). Constraints (3) relate the number of selected collection sites at each island v based on variables Uvk with the number of sites on the island that are selected to be served based on variables Zi (|H| constraints). Eq. (4) indicates that one value of k is determined as the number of selected collection sites that must be visited at each island v with a maximum number of jXv j waste collection sites (|H| constraints). Constraints (5) state lower bounds to the waste volume collected from each collection site i and for each period t, depending on the selected visit pattern p (Xip). These lower bound constraints are established for each period t, for each island v, for each collection site i belonging this island, and P for each possible number of collection sites k, k = 1, . . ., jXv j selected on this island (T  v 2H jXv j2 constraints). Constraint (6) ensures that the amount of waste removed on day t for all visited collection sites does not exceed the capacity of the barge (T constraints). Restrictions (7) indicate that if the collection site i is assigned to a pattern that includes a visit on period or day t, then the barge must travel from node i to another node j (T  (|N|  1) constraints). Expression (8) allows at most one collection route per day, i.e., only one departure from the depot i0 (T constraints). Constraint (9) indicates that if the barge arrives on period or day t to node i, then the barge must depart from the node i to another node j on the same day t (|N|  T constraints). Constraint (10) ensures that the earliest and latest times to arrive at each node are fulfilled (2  |N|  T constraints). Constraints (11) and (12) bound the arrival time at each node j as a function of the accumulated time on the navigation route, the waste loading times at previously visited collection points, and the activation of routing variables Yijt ((|N|  1)  T and |N|  (|N|  1)  T constraints, respectively). Finally, expressions (13)–(16) are domain constraints for decision variables of the model. An illustration example is presented in Appendix A for a better understanding of the model, particularly for Constraints (2), (3), (4), and (5). Appendix B highlights the structure of the solutions provided by the proposed model for a specific small instance. This appendix shows that when a restrictive instance is addressed, the model solution tends to divide islands in more than one collection site, and to visit more than once a week. 3.3. Valid inequalities The proposed model for the insular household WCSD problem is solved by the well-known Branch and Bound (B&B) algorithm using a standard optimization solver (CPLEX 10.0). Given the high complexity of the model, some constraints were reformulated to include them as valid inequalities, and thus, provide a stronger formulation. Valid inequalities consist of additional constraints for the model that are redundant in the original integer formulation. However, they are not redundant for the linear relaxation of the problem, as it is achieved in standard B&B algorithms. According to these facts, when the integer formulation is solved including valid inequalities, the same integer solutions are obtained. Nevertheless, when the linear relaxation is solved including valid inequalities, solutions obtained may probably be very different, in terms of decision variables values and/or the objective function. Good valid inequalities may reduce the time required for solving the problem, particularly when B&B is employed (See Nemhauser and Wolsey, 1988; Wolsey, 1998). The following equations present the proposed valid inequalities:

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P.A. Miranda et al. / Transportation Research Part E 77 (2015) 227–247 P X X X ip  rpt ¼ Y i0 jt p¼1

P X

8t ¼ 1; . . . ; T; 8i 2 N=i – i0

ð17Þ

j2N;j–i

Zi 6 3  2  Uv 1

8v 2 H

ð18Þ

i2Xv

W it P qkv pt  X ip þ CAP  ðU v k  1Þ 8v 2 H; i 2 Xv ; 8k ¼ 1; . . . ; jXv j; 8t ¼ 1; . . . ; T; 8p ¼ 1; ::; P

ð19Þ

Constraints (17) ensure that if some collection site i (i e N, i – i0) requires a visit on a specific period t (t = 1, . . ., T), then a route should depart from depot i0. Constraints (18) indicate that if an island v is decided to be visited through a single collection site i (Uv1 = 1), then at most one collection site of the island should be selected, in terms of collection site selection variables Zi. Inequalities (19) have the same aim as constraints (5), which is to compute the waste volume collected from each collection site i for each period t. However, in this case, the constraints are established for each potential visit pattern p, contrary to constraints (5) that include all patterns in the same constraints. Although these alternative constraints can be considered a reformulation of constraints (5), the numerical implementation showed that the B&B algorithm was faster when both sets of constraints are included in the formulation. Thus, constraints (19) were considered as valid inequalities instead a reformulation of constraints (5). In-depth analyses of reformulations, valid inequalities, and related solution approaches (e.g., Cutting Planes or Branch and Cut) remain as relevant future research, and are out of the scope of this paper.

4. Case study This section presents a real-world application of the model, in order to show the effectiveness of the proposed methodology. In addition, a comparison is performed between the solutions of each of the two waste collection strategies described in Section 2.

4.1. Description of the case study The case study consists of an archipelago of 20 islands and one inaccessible, isolated continental area (also referred to as an ‘‘island’’ for the remainder of the paper) located in the south of Chile, as shown in Fig. 1. This figure illustrates the depot location of the collection system named Dalcahue, where routes initiate and terminate. This study assumes that full bins are emptied at the depot and employed again for the next collecting routes. The process of transporting waste from Dalcahue to a final dumping site for subsequent treatment and final disposal is assumed to be accomplished by an external system. This process is not analyzed in this study. The criterion employed to select the islands to be included in the WCSD problem is based on the number of inhabitants projected to the year 2013. The system only involves islands with at least 300 inhabitants. On the contrary, if an island has a population less than 300, then it is assumed that the residents within the island may treat their waste employing recycling actions or dump sites, and thus, it is not considered in this study. Each island has one or more potential collection sites (piers), as shown in Table 2. Based on this information, the 21 islands contain 33 potential available sites for waste storage and collection. The time window for each potential collection site is set to [7:00 am–6:30 pm], and the barge must depart the depot at least at midnight and return within 24 h. The population of the islands was estimated based on the 1992 and 2002 censuses provided by the Chilean National Statistics Institute (INE). This data was employed to compute the expected population growth rate for each island. Hence, the population was projected to a 10 year horizon. Table 2 shows the expected waste generation growth rate based on the projected population, which is also employed to estimate the waste generation for each year until 2022 (10-year period). This projection is employed in the application and evaluation of the proposed model. The Solid Waste Group at the Pontificia Universidad Catolica de Valparaiso (PUCV) developed a diagnosis study for the solid waste management of the Chiloé and Palena provinces, which established an approximated waste generation rate of 0.5 kg/inhabitant-day. This value is assumed constant for all insular and isolated zones. Additionally, the diagnosis determined a waste density for the archipelago equal to 297 kg/m3 (GRS, 2009). Based on the diagnosis of the Solid Waste Group, this study employs a barge with a volume capacity of 30 m3, a weight capacity of 40 tons, a standard navigation speed between 9 and 10 nautical miles per hour or knots (kts), 90 h of autonomy, and an investment value of $192,780,000 Chilean pesos (CLP) 2. In this study, both strategies assume that each waste bin situated at the collection sites has a capacity of 770 liters with an investment unit cost of $210,700 (CLP). In addition, strategy 1 considers a compacting machine that compress the waste at a 3:1 ratio, and has an investment of $30,857,959 (CLP). Given that the same barge is considered for both strategies, other costs such as salary, maintenance, insurance, handling, etc. remain constant throughout different solutions. 2

US$ 1 = CLP$ 570 as of March 2014.

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P.A. Miranda et al. / Transportation Research Part E 77 (2015) 227–247

1

Aulén Island

2

Caucahué Island

3

Añihué Island

4

Mechuque Island

5

Butachauques Island

6

Ayacara Buill (continent)

7

Tac Island

8

Lin-Lin Island

9

Meulín Island

10

Llingua Island

11

Quenac Island

12

Caguache Island

13

Chelín Island

14

Alao Island

15

Apiao Island

16

Quehui Island

17

Chaulinec Island

18

Tranqui Island

19

Cailín Island

20

Laitec Island

21

Coldita Island

Fig. 1. Islands and isolated regions involved in the study.

Table 2 Waste generation for islands. Islands Alao Añihué Apiao Aulén Ayacara Buill Butachauques Caguache Cailín Caucahué Chaulinec Chelín Coldita Laitec Lin Lin Llingua Mechuque Meulín Quehui Quenac Tac Tranqui Total

Number of potential collection site

Waste generation for 2013 (kg/day)

Annual rate increase (%)

Three zone division

Two zone division

South North Central North Central North Central South North South Central South South Central Central North North Central North South North

South North South North North North South South North South South South South North North North North South South South North

1 1 1 1 2 3 1 2 2 1 2 1 2 2 1 2 1 1 2 1 3

263 283 406 164 540 453 257 412 291 371 228 185 394 299 225 283 420 606 256 177 372

2 1.6 2 1.2 2 1.6 2 5.5 1.6 2 4.2 5.5 5.5 2 2 1.6 2 4.2 2 1.6 1.4

33

6885

2.8

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For this case study, the model presents 8256 binary variables and 402 continuous variables, of which the most relevant binary variables are the 6936 the routing variables. In addition, the case study yields 8193 constraints, where the most significant is the set of 6.732 restrictions employed for computing arriving times at each node (Expression 12). Given the high complexity of the problem and the expected significant computational time in its resolution, three set of instances are defined by arbitrarily dividing the study area in zones as follows. Notice that this division procedure is not aimed to be proposed as a solution approach, but it is employed for (i) testing the model in smaller instances; and (ii) comparing both strategies (with and without compacting machine). (i) Three zones: The set of islands was divided into three subset zones, where each zone is solved independently (See Table 2 for islands contained in each zone). (ii) Two zones: The set of islands was divided into two subset zones, where each zone is solved independently (See Table 2 for islands contained in each zone). (iii) One zone: The problem was solved considering all islands simultaneously in one zone. The quality of the solution and the execution time for the three division scenarios are evaluated. Due to the complexity of the problem, the model was reformulated with valid inequalities as stated in Section 3.3, prior to the computational implementation with CPLEX solver, using an embedded standard B&B algorithm. Note that the resolution of the problem was not feasible without the described reformulation techniques. A processing and subsequent analysis of the model results was developed to determine the system configuration for waste removal with minimum operation costs and investments. For the analysis, if two or more collection routes are feasible to complete in one day according to the time constraints of the problem, then these routes are scheduled for the same day. Additionally, the analysis takes into consideration the life span of the solution for the waste collection system, based on barge capacity saturation for each route. Note that life span of the solution is different from life span of the system, where the latter is related to system capacity saturation when the barge is not capable of collecting and transporting the overall system waste generated in a given week. In other words, when the system life span is reached, no alternative routes or solutions exist. Whereas, when the solution life span is reached, alternative routes or solutions may be obtained.

4.2. Results for the three-zone Scenario When solving the WCSD problem using this zone division configuration, visit patterns were assigned to collection sites that contemplate only one visit each week for both strategies (i.e., bins and waste compactor). The waste collection weekly schedule for strategies 1 and 2 was constructed with the selected visit pattern and sequence analysis, as shown in Tables 3 and 4, respectively. Table 3 indicates that completing more than one route in one day is not feasible for strategy 1 due to time window restrictions. On the contrary, some pairs of routes for strategy 2 can be completed in one day by following the visit pattern selected for all collection sites, as shown in Table 4 (e.g., routes 1 and 2 for Wednesday, Thursday, and Friday). Fig. 2a and b show the associated waste collection routes for this scenario. Given a selected visit pattern, the number of bins at each collection site for both strategies is equal to the upper integer of the ratio between the accumulated waste volume since the last visit and the bin capacity. In strategy 2, empty bins are transported to the site for exchanging with full bins. Therefore, the number of required empty bins to be transported is equal to the quantity of bins at each site for the most saturated route. In other words, the number of bins onboard the barge is computed based on the route that contains the largest number of bins at each collection site. Table 5 shows the projected barge utilization for strategies 1 and 2 in the three-zone scenario with the maximum capacities (in volume or in number of bins) highlighted in bold. According to the capacities of the barge, bins, and waste compactor, a maximum compacted waste volume of 90 m3 for strategy 1 and a maximum of 36 bins are transported for strategy 2. In this scenario, the solutions obtained for both strategies require the same number of bins at the collections sites since only one collection site is selected at each island, and all sites are visited once a week. The generated waste will exceed the barge capacity for one route (Tuesday) in the year 2016 with strategy 2, while no route exceeds the barge capacity with strategy 1. Therefore, the results for strategies 1 and 2 are valid for 10 years and 3 years, respectively, considering that the waste collection system starts operating in 2013. As shown in Table 11, the required features for waste removal using strategy 1 are 220 bins at collection sites, one waste compactor, and one barge, which must travel a distance of 470 miles per week. Strategy 2 requires 220 bins, 33 on board empty bins used for replacing full bins, and one barge that will navigate 641 miles per week. Given the investment and costs presented in this table, strategy 1 is the most economical alternative. This strategy presents lower annual transportation costs, which compensates the additional investment of the waste compactor. Moreover, strategy 1 presents a larger system life span because this strategy provides a higher transportation capacity. The Net Present Value (NPV) is equal to $317.196.863 (CLP) for this zone division scenario when comparing the savings of using strategy 1 versus strategy 2, assuming a discount rate of 8% and a time horizon of 10 years. More details on the economic comparison are discussed in Section 4.5 and in Appendix C.

Monday

Tuesday

Wednesday

Thursday

Island

Waste collection site

Arrival time

Island

Waste collection site

Arrival time

Island

Waste collection site

Arrival time

Island

Waste collection site

Arrival time

Quenac Meulín Añihué Mechuque Dalcahue (Depot)

1 1 1 2 –

7:00 am 8:01 am 9:15 am 10:09 an 12:49 pm

Caucahue Aulen Butachauques Tac Dalcahue (Depot)

2 1 1 1 –

7:00 am 11:52 am 4:09 pm 5:45 pm 8:50 pm

Cailin Coldita Laitec Tranqui Chaulinec

1 1 2 3 1

7:42 am 9:06 am 9:54 am 12:42 pm 5:24 pm

Lin Lin Llingua Quehui Chelin Apiao

2 1 2 2 1

7:00 am 8:00 am 10:29 am 11:06 am 1:10 pm

Alao Dalcahue (Depot)

1 –

6:09 pm 9:25 pm

Caguache Ayacara Buill

1 1

2:16 pm 6:01 pm

Dalcahue (Depot)

–

10:49 pm

P.A. Miranda et al. / Transportation Research Part E 77 (2015) 227–247

Table 3 Visit sequences and schedule for the three zone division using strategy 1.

235

236

Monday

Tuesday

Wednesday

Island

Site

Arrival time

Island

Site

Arrival time

Island

Caucahue Aulen Butachauques Dalcahue (Depot)

2 1 1 –

7:00 am 11:52 am 4:09 pm 7:53 pm

Cailín Coldita Laitec Dalcahue (Depot)

1 1 2 –

7:47 am 9:13 am 10:01 am 7:01 pm

Route 1 Mechuque Añihué Tac

Thursday Site

Arrival time

Island

2 1 1

7:00 am 7:53 am 8:57 am

Route 1 Apiao Caguache Llingua

Dalcahue (Depot) Route 2

–

12:03 pm

Quenac Meulín Dalcahue (Depot)

1 1 –

2:56 pm 3:56 pm 6:13 pm

Friday Site

Arrival time

Island

Site

Arrival time

1 1 1

7:00 am 8:06 am 9:51 am

Route 1 Tranqui Chaulinec Alao

1 1 1

7:00 am 10:14 am 10:54 am

Dalcahue (Depot) Route 2

–

11:47 am

Dalcahue (Depot) Route 2

–

2:01 pm

Ayacara Buill Lin Lin Dalcahue (Depot)

1 1 –

2:40 pm 3:40 pm 5:57 pm

Quehui Chelín Dalcahue (Depot)

2 2 –

5:13 pm 5:49 pm 8:28 pm

P.A. Miranda et al. / Transportation Research Part E 77 (2015) 227–247

Table 4 Visit sequence and schedule for the three zone scenario using strategy 2.

P.A. Miranda et al. / Transportation Research Part E 77 (2015) 227–247

(a) Strategy 1 for Three Zone Scenario

(b) Strategy 2 for Three Zone Scenario

(d) Strategy 2 for Two Zone Scenario

237

(c) Strategy 1 for Two Zone Scenario

(e) Strategy 1 for One Zone Scenario

Fig. 2. Optimal navigation routes for each zone scenario and operation strategy.

4.3. Results for the two-zone scenario Tables 6 and 7 present the visit sequence and schedule for strategies 1 and 2 in the two-zone scenario, respectively. Similar to the three-zone scenario, some collection routes are organized in one day when feasible in terms of time windows. In this scenario, strategy 1 requires a total of 220 bins and one waste compactor onboard a barge that will travel 426 miles per week. Strategy 2 requires 220 bins, 34 onboard empty bins, and one barge that will navigate 597 miles per week. Fig. 2c and d depict the waste collection routes for each strategy. Table 8 presents the waste volume and percentage of the utilized barge capacity for strategy 1, and the number of bins and percentage of utilized barge capacity per route for strategy 2. This table shows that the generated waste will exceed the

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Table 5 Projected barge utilization for the three zone scenario. Year

Strategy 1 (m3) Mon

2013 2014 2015 2016 2017 2018 2019 2020 2021 2022

29.27 29.79 30.33 30.88 31.44 32.03 32.62 33.21 33.82 34.43

Tue

25.57 26.00 26.40 26.82 27.22 27.65 28.09 28.52 28.94 29.39

Strategy 2 (bins) Wed

47.07 48.76 50.53 52.44 54.35 56.47 58.59 60.86 63.21 65.71

Thu

60.36 62.01 63.73 65.47 67.22 69.10 71.01 72.97 75.04 77.14

Mon

29 31 31 31 31 32 32 32 33 34

Tue

32 33 36 37 39 41 43 46 48 51

Wed

Thu

Fri

Total number of bins

Route 1

Route 2

Route 1

Route 2

Route 1

Route 2

24 24 24 26 26 26 26 27 27 27

21 22 23 23 23 24 24 24 26 26

28 30 30 31 31 31 32 33 34 34

27 27 28 28 28 30 30 30 31 31

33 33 33 34 34 35 36 37 37 37

26 28 29 29 31 32 33 35 36 38

220 228 234 239 243 251 256 264 272 278

Number in bold indicate that the maximum number of bins (30) has been exceeded.

Table 6 Visit sequence and schedule for two-zone division based on strategy 1. Monday

Tuesday

Wednesday

Thursday

Island

Sites

Arrival times

Island

Sites

Arrival times

Island

Sites

Arrival times

Island

Sites

Arrival times

Ayacara Buill Aulén Caucahué Butachauques Dalcahue (Depot)

1 1 2 1 –

7:00 am 11:12 am 4:06 pm 6:01 pm 9:45 pm

Alao Chaulinec Apiao Caguache Quenac

1 1 1 1 1

7:00 am 7:41 am 8:38 am 9:44 am 10:53 am

Llingua Meulín Tac Añihue Mechuque

1 1 1 1 2

7:00 am 8:32 am 10:14 am 11:18 am 12:11 pm

Cailin Coldita Laitec Tranqui Quehui

1 1 2 3 2

7:48 am 9:12 am 10:00 am 12:50 pm 5:37 pm

Dalcahue (Depot)

–

1:16 pm

Lin Lin

1

1:42 pm

Chelin

2

6:13 pm

Dalcahue (Depot)

–

3:17 pm

Dalcahue (Depot)

–

8:52 pm

barge capacity for some routes in 2016 using strategy 2, while barge capacity is not exceeded within the 10-year horizon for routes using strategy 1. Table 11 presents the investment and costs for each strategy. Similarly to the three-zone division configuration, strategy 1 provides the most economical solution for the waste collection problem, although there is a higher investment involved with the compacting equipment. Additionally, this strategy is more attractive in terms of system life span. The NPV is equal to $ 367.260.057 (CLP) for this zone division scenario when comparing the savings of using strategy 1 versus strategy 2, assuming a discount rate of 8% and a time horizon of 10 years. More details on the economic comparison are discussed in Section 4.5 and in Appendix C. 4.4. Results for the one-zone scenario In terms of life span and economic criteria, both two- and three-zone scenarios discussed in previous subsections showed that strategy 1 is the best option for the waste collection system of the islands. Given the size of the instance involved, the one-zone scenario requires a large amount of computational time for reaching optimal solutions. Accordingly, the one-zone scenario implementation is aimed to obtain the optimal solution (without any arbitrary zone division) for the system only with strategy 1. Thus, no economic comparison was performed for this scenario with strategy 2. The assigned visit patterns for this scenario considered one waste collection visit per week for each selected site. Table 9 presents the schedule for the waste removal visits to each island. This strategy requires 220 bins, one waste compactor, and one barge with a traveling distance of 412 miles per week. Additionally, this scenario presents a solution life span of six years before any route is saturated (See Table 10). Fig. 2e illustrates the waste collection routes for this scenario. Table 11 presents the investment and costs for strategy 1. 4.5. Discussion A summary of the most relevant results for each zone division and strategy are presented in Table 11. Given the same strategy, the annual transportation cost diminishes as the problem is solved with less number of zones since the weekly traveled distances are reduced. For example, there is a 9.4% decrease in the transportation costs when using a two-zone division

Monday

Tuesday

Wednesday

Thursday

Friday

Saturday

Island

Sites

Arrival times

Island

Sites

Arrival times

Island

Sites

Arrival times

Island

Sites

Arrival times

Island

Sites

Arrival times

Island

Sites

Arrival times

Ayacara Buill Aulén Caucahué Dalcahue (Depot)

1

7:00 am

Tranqui

1

7:00 am

Cailin

1

7:47 am

Lin Lin

1

7:00 am

Añihue

1

7:00 am

Route 1

1 2 –

11:33 am 4:25 pm 7:42 pm

Chaulinec Alao Dalcahue (Depot)

1 1 –

10:14 am 10:55 am 2:01 pm

Coldita Laitec Dalcahue (Depot)

1 2 –

9:13 am 10:01 am 7:01 pm

Mechuque Butachauques Tac

2 1 1

8:31 am 10:11 am 11:47 am

Meulín Lin Lin Llingua

1 2 1

8:15 am 9:21 am 10:21 am

Quenac Caguache Apiao

1 1 1

7:00 am 8:09 am 9:15 am

Dalcahue (Depot)

–

2:52 pm

Dalcahue (Depot)

–

12:17 pm

Dalcahue (Depot) Route 2 Quehui Chelin Dalcahue (Depot)

–

12:42 pm

2 2 –

3:53 pm 4:29 pm 7:05 pm

P.A. Miranda et al. / Transportation Research Part E 77 (2015) 227–247

Table 7 Visit sequences and schedule for two-zone scenario based on strategy 2.

239

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Table 8 Projected barge utilization for the two-zone scenario. Year

Strategy 1 (m3) Mon

2013 2014 2015 2016 2017 2018 2019 2020 2021 2022

34.13 34.72 35.33 35.94 36.53 37.17 37.83 38.46 39.12 39.81

Tue

Strategy 2 (bins) Wed

36.60 37.36 38.06 38.89 39.62 40.44 41.25 42.07 42.92 43.77

39.76 40.49 41.27 41.98 42.75 43.56 44.36 45.16 45.98 46.83

Thu

51.78 54.00 56.33 58.80 61.33 64.08 66.89 69.86 72.99 76.27

Mon

32 33 34 34 34 35 35 35 37 37

Tue

33 33 33 34 34 35 36 37 37 37

Wed

32 33 36 37 39 41 43 46 48 51

Thu

34 35 35 36 36 38 38 39 39 40

Fri

34 36 36 37 37 39 39 39 41 41

Sat

Total number of bins

Route 1

Route 2

29 30 31 32 32 32 33 34 35 35

26 28 29 29 31 32 33 35 36 38

220 228 234 239 243 252 257 265 273 279

Number in bold indicate that the maximum number of bins (30) has been exceeded.

Table 9 Visit sequence for one-zone scenario. Monday

Tuesday

Wednesday

Island

Site

Arrival time

Island

Site

Arrival time

Island

Site

Arrival time

Ayacara Buill Aulén Caucahué Mechuque Dalcahue (Depot)

1 1 2 2 –

7:00 am 11:33 am 4:25 pm 6:06 pm 8:46 pm

Cailin Coldita Laitec Tranqui Quehui Chelín Dalcahue (Depot)

1 1 2 3 2 2 –

7:47 am 9:13 am 10:01 am 12:51 pm 5:38 pm 6:15 pm 8:53 pm

Alao Chaulinec Apiao Caguache Quenac Llingua Lin Lin Meulín Añihué Tac Butachauques Dalcahue (Depot)

1 1 1 1 1 1 2 1 1 1 1 –

7:00 am 7:41 am 8:38 am 9:44 am 10:53 am 12:01 pm 1:01 pm 2:07 pm 3:22 pm 4:26 pm 6:01 pm 9:45 pm

Table 10 Projected barge utilization for the one-zone scenario. Year

2013 2014 2015 2016 2017 2018 2019 2020 2021 2022

Strategy 1 (m3) Mon

Tue

Wed

30.12 30.62 31.18 31.70 32.24 32.81 33.40 33.96 34.55 35.14

51.78 54.00 56.33 58.80 61.33 64.08 66.89 69.86 72.99 76.27

80.37 81.95 83.48 85.11 86.66 88.36 90.03 91.73 93.47 95.27

Number in bold indicate that the maximum volume (90) in cubic meters has been exceeded.

instead of a three-zone division, and a 3.3% decrease in these costs with the one zone scenario instead of the two-zone scenario using strategy 1. These results are attributed to that the optimal solution of the problem can be achieved only with the one-zone scenario. In contrast, two- and three-zone scenarios provide solutions that are restricted to obeying predefined arbitrary divisions yielding longer travel distances. Notice that it is highly improbable or perhaps impossible that division boundaries established for the two- or three-zone scenarios coincide exactly with the optimal solution obtained in the one-zone scenario. Naturally, obtaining an optimal solution for one-zone scenario is more time consuming when compared to the two- or three-zone scenarios. Table 11 shows that the annual transportation cost for strategy 1 is lower than strategy 2 for the two- and three-zone scenarios (28% on average), since the latter strategy has a smaller volume capacity for waste transportation. This is due to three reasons. First, strategy 2 does not compact the waste; second, strategy 2 assumes that the barge transports bins with less than maximum capacity in most cases; and third, bins are not transported in strategy 1 making a better use of the barge capacity. Hence, strategy 2 presents less volume capacity on the barge and a larger number of routes for waste collection.

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P.A. Miranda et al. / Transportation Research Part E 77 (2015) 227–247 Table 11 Summary of results for 2013. Item

Three zone scenario

Barge Bins on board the barge Bins at collection sites Total bins required Bins Waste compactor Total investment Annual transportation cost Number of waste collection days Number of collection routes Weekly travel distance (miles) Solution life span (years)

Two zone scenario

One zone scenario

Strategy 1

Strategy 2

Strategy 1

Strategy 2

Strategy 1

$192,780,000 – 220 220 $46,574,000 $30,857,959 $270,211,959 $93,909,504 4 4 470 10

$192,780,000 33 220 253 $53,560,100 – $246,340,100 $128,094,720 5 8 641 3

$192,780,000 – 220 220 $46,574,000 $30,857,959 $270,211,959 $85,043,712 4 4 426 10

$192,780,000 34 220 254 $53,771,800 – $246,551,800 $119,149,056 6 7 597 3

$192,780,000 – 220 220 $46,574,000 $30,857,959 $270,211,959 $82,268,160 3 3 412 6

All zone division scenarios with strategy 1 present the same amount of investment. All scenarios use one barge with the same compacting equipment on board, and the number of bins at each island remains constant. For strategy 2, there is an increase of one unit in the total number of empty bins transported for exchanging with full bins at each site, when solving the problem with less number of zone divisions. Specifically, strategy 2 requires 33 and 34 empty bins on board the barge for three- and two-zone scenarios, respectively. These values correspond to the maximum number of bins needed for all routes. In terms of investment, strategy 1 has to invest on compacting equipment, while strategy 2 has to provide bins for transportation and swapping full bins with empty bins at each collection site. Table 12 presents a summary of the barge capacity usage for different zone scenarios and strategies per day of the week. This table reveals that on average the barge capacity utilization for different routes increases affecting the solution life span as the problem is solved with an inferior number of zones. Notice that this solution life span is reached when at least one route exceeds the maximum barge capacity, while the barge is still capable of serving the entire system. In other words, additional days are available for waste collection/removal. In this case, the proposed methodology should be employed to determine new optimal routes increasing the system life span. Appendix C presents the system re-optimization analysis and an economic evaluation to compare different solutions for the two- and three-zone division scenarios using strategies 1 and 2. This appendix shows that strategy 1 is by far the most convenient option for waste collection when compared to strategy 2, which is denoted by a significant difference in the NPV. For the three-zone division, the comparison yields a NPV of $381.025.488 (CLP) for a 4% of interest rate, $317.196.863 (CLP) for a 8% of interest rate, and $268.242.284 (CLP) for a 12% of interest rate, while the two-zone division yields a NPV of $442.787.823 (CLP) for a 4% of interest rate, $367.260.057 (CLP) for a 8% of interest rate, $309.330.031 (CLP) for a 12% of interest rate. Both evaluations consider a 10-year horizon period. This economic evaluation for comparing strategies was not possible for the one-zone scenario due to the required high computational time with strategy 2, and additionally, previous results indicate that this strategy is strongly not recommended. Obviously, it is preferable to optimize the system considering the whole instance without predefined zone divisions. When comparing annual transportation costs for the year 2013, the one-zone division solution presents a saving of 3.3% with respect to the two-zone division solution, and a 12.4% with respect to the three-zone division solution. Note that all solutions yield the same total investments (bins, compactor machine, and barge) for strategy 1. Given the above summary, it is clearly concluded that the solid waste removal system for the islands under study should employ strategy 1 with one barge, one waste compactor, and 220 bins. According to the solution obtained for the one-zone scenario, the barge should operate according to the weekly schedule presented in Table 9 and navigation routes shown in Fig. 2e. Although this system configuration presents a minimum solution life span of six years when comparing to other zone Table 12 Summary of barge capacity utilization. Day

Three zone scenario Strategy 1

Two zone scenario

Strategy 2 Route 1

Monday Tuesday Wednesday Thursday Friday Saturday

Strategy 1 Route 2

One zone scenario Strategy 1

Strategy 2 Route 1

Route 2

m3

% Usage

Bins

% Usage

Bins

% Usage

m3

% Usage

Bins

% Usage

Bins

% Usage

m3

% Usage

29.3 25.6 47.1 60.4 –

32.5% 28.4% 52.3% 67.1% –

29 32 24 28 33

80.6% 88.9% 66.7% 77.8% 91.7%

– – 21 27 26

– – 58.3% 75.0% 72.2%

34.1 36.6 39.8 51.8 – –

37.9% 40.7% 44.2% 57.5% – –

32 33 32 34 34 29

88.9% 91.7% 88.9% 94.4% 94.4% 80.6%

– – – – – 26

– – – – – 72.2%

30.1 51.8 80.4 – –

33.5% 57.5% 89.3% – –

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Table 13 Computation times for the problem resolution. Number of zones

Geographic zone

Strategy 1 (min)

Strategy 2 (min)

3 Zones

North Center South North South –

265.9 15.2 26.7 700.2 1095.5 1609.8

431.9 269.4 220.9 1.185.3 1566.8 –

2 Zones 1 Zone

division scenarios and strategies, it presents the most economic operation conditions with the lowest annual transportation costs of approximately $82 million Chilean pesos. Moreover, this configuration may be re-optimized once the solution life span is reached (year 2019), increasing the duration of the system solution without additional investment in barges and waste compactors. Regarding the structure of the model solutions, optimal system design tends to visit each island at a single collection site and only once a week (e.g., see Table 9 for the solutions obtained for the one-zone scenario), as can be expected, since this structure yields the minimum transportation costs or distances. However, when the barge capacity constraint is effectively restricted (See Eq. (6)), given a limited barge capacity or a large waste generation, the model solution tends to: (i) divide the waste generated in some islands in more than one collection site and/or (ii) visit some collection sites at least twice a week. For example, Table 7 shows the results for two-zone scenario based on strategy 2, in which Lin-Lin island is divided into two collection points that are visited on different days (Thursday and Friday). Appendix B presents another example with increased generated waste for Ayacara Buill Island, where its waste is collected through two collection sites, and each site is visited twice a week (Monday–Thursday and Wednesday–Saturday). These results manifest a relevant practical contribution of the proposed model through its flexibility for dealing with different instances in terms of barge capacity utilization. This flexibility is reached by effectively employing two logistic approaches: visit frequency selection, and waste collection using multiple collection sites for each island. Regarding computational time, the results for the different instances are summarized in Table 133. This table shows that the computational time decreases with the number of zones, and the computational time for strategy 2 is greater than strategy 1 for all zone divisions. In order to solve the one zone scenario with strategy 1, the algorithm was executed for more than a day (approximately 27 h). Given this result, and observing the favorable economic results obtained in the three- and two-zone divisions for strategy 1, strategy 2 was not implemented for this scenario. The results observed in Table 13, shows the very high complexity of the problem, which is inherited from the combination of well-known complex problems such as facility location, vehicle routing, pattern selection, and inventory routing problems. Notice that the time consumption is very high even for the small real instances addressed in this paper. Hence, the model is not solvable for larger instances using standard solvers, and therefore, other ad hoc methodologies are required for solving the proposed model. 6. Conclusions This paper presents a novel model for addressing the problem of designing a household waste collection system for serving a rural archipelago. The proposed model was employed to solve a real instance of small islands located in the provinces of Chiloé and Palena, Chile. The instance consists of 21 islands (20 islands and one isolated, continental area with a total of 33 potential waste collection sites) with a population between 300 and 1200 inhabitants per island. The application considered three scenarios: two scenarios with the division of the study area into three and two smaller zones, and one scenario with no division. Thus, the model was solved separately for the set of islands included in each zone. In addition, the model parameters were adapted to deal with two different waste collection strategies, which consists of (i) transporting bins with a barge, and (ii) using a waste compactor on board a barge. The application of the proposed model to these two strategies indicate that the use of a waste compactor on board a barge for transporting solid waste is more efficient than transporting full/empty bins. The amount of transported waste volume is strongly reduced when implementing the former strategy yielding a significant transportation cost reduction, and allowing a rapid investment cost recovery. The main contribution of this paper is the development and implementation of a novel mixed integer linear optimization model, which integrates decisions of waste collection site selection, visit pattern assignment, and multi period vehicle routing. In addition to dynamic routing and inventory features of the PVRP, IRP, and CIRP (well-known NP-Hard problems), the proposed model addresses waste collection site selection for each island to be served. It is noteworthy to mention that, in the proposed model, the demand for each collection site at each island is not known in advance since it depends on: (i) whether this site is selected to operate as a collection site; and (ii) the number of selected sites to become a collection site from the existing potential sites. This observation relies on the model assumption that the waste generated at each island is split among the selected collection sites. 3

All instances were solved in a standard desktop computer with an AMD Phenom(tm) 9650 Quad-Core processor of 2.36 GHz, and 3.25 Gb RAM.

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The results of the computational implementation denote the high complexity of the proposed model, which are related to the excessive computational time for solving small real world instances. This complexity, independent of the modeling structure of the proposed methodology, is inherited from the design problem analyzed in this paper, which involves three sets of decisions: selection of waste collection sites at each island, visit frequencies for each collection point, and daily visit sequences. Future research focuses on computational time performance, potentially involving solution approaches and model reformulation techniques. In terms of solution approaches, future research includes the use of Branch & Cut algorithms for this type of model since it can benefit from its modeling structure and previous applications for standard and selective vehicle routing problems. In addition, the zone division procedure employed in this paper can be embedded into a type of decomposition algorithm for solving the problem such as Lagrangian Relaxation, Benders, or Dantzig-Wolfe Decomposition. In terms of model re-formulation, the VRP literature provides several modeling approaches that may be analyzed as further research such as network flow and multi-stage based formulation, among others. Regarding extensions of the proposed model, the WCSD model may also be addressed with the use of multiple barges, multiple visit routes per day, and waste transportation and division within the islands (costs and decisions). Additionally, recycling strategies may be considered at the islands and/or at the depot (de Figueiredo and Mayerle, 2008; Parker et al., 2010; Ramos et al., 2014). Acknowledgments The authors would like to thank the Grupo de Residuos Solidos (Solid Waste Group) of the Pontificia Universidad Católica de Valparaíso (http://www.grs-pucv.cl) for their valuable help and for providing data employed in this research. This research was financially supported by Chilean National Fund for Scientific & Technological Development (FONDECYT Project N° 1140811). Appendix A A.1. An illustrative example In order to understand the model, particularly regarding the computation of the amount of waste removed from each collection site on each period t computed with Eq. (8), a small example of a generic island v with three potential collection sites is presented in Fig. 3i. Accordingly, the model determines the number of collection sites to be visited on each island (i.e., k = 1, 2, 3, as shown in Fig. 3ii) assuming a total waste generation of 60 m3 during a period, two operation days (t = 1, 2), and three visit patterns (See Fig. 3iii). The values for the waste collection parameters (qkvpt) involved in Eq. (8) are presented in Fig. 3iv. For example, pattern p = 1 indicates a visit on period t = 1, p = 2 denotes a visit on period t = 2, and p = 3 indicates visits on periods t = 1 and 2. In addition to the selected pattern for each collection site, the collected waste volume depends on the number of selected collection sites of island v (k = 1, 2, 3). For example, when selecting a collection point (Uv1 = 1, Uv2 = 0, Uv3 = 0),

(i) Graphic Representation of Island v

(ii) Number of Collection

(iii) Set of Visit Patterns

Fig. 3. Example of model decision variables and parameters.

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P.A. Miranda et al. / Transportation Research Part E 77 (2015) 227–247 Table A.1 Waste collection volumes q for k = 3 with X13 = 1, X21 = 1, and X31 = 1. Collection sitenoperation period

t = 1 (m3)

t = 2 (m3)

i=1 i=2 i=3

20 40 40

20 0 0

the total generated waste (60 m3 per period) will be assigned to that point and will be collected according the selected pattern. If pattern p = 1 is chosen, then the waste generated in two periods (120 m3) will be collected in period t = 1; if pattern p = 2 is selected, then the amount of waste produced in two periods (120 m3) will be collected in period t = 2; and if pattern p = 3 is chosen, then the generated waste for each period will be collected in the same period (60 m3 for each period t = 1 and t = 2). Conversely, if the three collection points are selected (Uv1 = 0, Uv2 = 0, Uv3 = 1), then the total waste generated on the island will be distributed homogeneously among the three sites (20 m3 per period). This waste volume will be collected on different periods or days depending on the visit pattern assigned to each point. For example, if site i = 1 is allocated to pattern p = 3 (X13 = 1), site i = 2 is assigned pattern p = 1 (X21 = 1), and site i = 3 is designated to pattern p = 1 (X31 = 1), then the waste will be collected at the three sites according to Table A.1. In this case, constraints (8) will be activated only for k = 3 since Uv1 = Uv2 = 0, and waste volumes are computed depending on visit pattern variables given by equations in Expression (20). These equations are equivalent to constraints (8) applied to this specific example.

i¼1

8 3 X > > > W q3v p1  X 1p ¼ q3v 11  X 11 þ q3v 11  X 12 þ q3v 31  X 13 11 P > > < p¼1

t¼1

> 3 X > > > > q3v p2  X 1p ¼ q3v 12  X 11 þ q3v 22  X 12 þ q3v 32  X 13 : W 12 P

t¼2

8 3 X > > > q3v p1  X 2p ¼ q3v 11  X 21 þ q3v 21  X 22 þ q3v 31  X 23 > W 21 P > < p¼1

t¼1

> 3 X > > > > q3v p2  X 2p ¼ q3v 12  X 21 þ q3v 22  X 22 þ q3v 32  X 23 : W 22 P

t¼2

8 3 X > > > W q3v p1  X 3p ¼ q3v 11  X 31 þ q3v 21  X 32 þ q3v 31  X 33 31 P > > < p¼1

t¼1

> 3 X > > > > q3v p2  X 3p ¼ q3v 12  X 31 þ q3v 22  X 32 þ q3v 32  X 33 : W 32 P

t¼2

p¼1

i¼2

ð20Þ

p¼1

i¼3

p¼1

Appendix B B.1. A solution example This appendix shows the results of model applied to the central zone instance of the three-zone division for strategy 1 (See Table 4). In this example, the daily waste generation of Ayacara Buill Island was increased from 1.8 m3 to 12 m3. Table B.1 presents the selected visit pattern and visit days for each collection site according to Table 1. Table B.1 Results with increased waste generation for island Ayacara Buill. Island

Collection site

Selected visit pattern

Visit days

Llingua Caguache Apiao Chelín

1 2 3 4 5 6 7 8 9 10 11

5 4 6 – 5 – 5 1 4 7 9

Friday Thursday Saturday – Friday – Friday Monday Thursday Monday, Thursday Wednesday, Saturday

Quehui Lin Lin Ayacara Buill

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Note that each of the two collection sites of Ayacara Buill Island is visited twice a week, and the two collection sites of Lin Lin Island are visited once a week. These results show that the proposed model employs i) visit frequency optimization through pattern selection, and ii) collection site selection for each island in an efficient and effective manner, in order to minimize transportation costs while satisfying capacity constraints for each route. These findings highlight the contribution and applicability of the proposed model. Appendix C C.1. Years re-optimization and economic evaluation This appendix provides an economic evaluation analysis to compare Strategy 1 (compacting machine) and 2 (bin transportation). This comparison is performed only for two- and three-zone scenarios, in which was computationally feasible to solve the model for both strategies and projected periods. Tables C.1 and C.2 present projected investment and transportation costs for three- and two-zone division scenarios, respectively. In addition, NPV values are shown for each strategy and for the difference between strategies. This difference represents the cost increment between strategy 2 and strategy 1. The investment for the year 2013 includes barge, compactor, and bins, while the investment for projected years after 2013 solely includes the additional bins required for each year. Note that the model was re-optimized every year when the barge capacity was reached, in order to increase the solution life span. Accordingly, these tables only consider solutions that respect barge capacity, as opposed to some saturated routes

Table C.1 Projected cash flows: investment in barge, compactor, and bins; and transportation costs in Chilean Pesos. Year

Three zone scenario Strategy 1

Strategy 2

Diff. Strategies (2–1)

Investment

Transportation costs

Total

Investment

Transportation costs

Total

2013 2014 2015 2016 2017 2018 2019 2020 2021 2022

$ $ $ $ $ $ $ $ $ $

$ $ $ $ $ $ $ $ $ $

$ $ $ $ $ $ $ $ $ $

$ $ $ $ $ $ $ $ $ $

$ $ $ $ $ $ $ $ $ $

$ $ $ $ $ $ $ $ $ $

NPV (4%) NPV (8%) NPV (12%)

$ 1.072.482.933 $ 959.241.728 $ 871.716.498

270.211.959 1.693.600 1.270.200 1.058.500 846.800 1.693.600 1.058.500 1.693.600 1.693.600 1.270.200

93.909.504 93.909.504 93.909.504 93.909.504 93.909.504 93.909.504 93.909.504 93.909.504 93.909.504 93.909.504

364.121.463 95.603.104 95.179.704 94.968.004 94.756.304 95.603.104 94.968.004 95.603.104 95.603.104 95.179.704

246.340.100 1.693.600 1.905.300 211.700 846.800 1.905.300 1.058.500 2.117.000 1.905.300 0

128.094.720 128.094.720 128.094.720 148.243.630 148.243.630 148.243.630 148.243.630 148.243.630 148.243.630 154.275.364

374.434.820 129.788.320 130.000.020 148.455.330 149.090.430 150.148.930 149.302.130 150.360.630 150.148.930 154.275.364

$ 1.452.615.996 $ 1.275.803.175 $ 1.139.500.735

$ $ $ $ $ $ $ $ $ $

10.313.357 34.185.216 34.820.316 53.487.326 54.334.126 54.545.826 54.334.126 54.757.526 54.545.826 59.095.660

$ 381.025.488 $ 317.196.863 $ 268.242.284

Table C.2 Projected cash flows: investment in barge, compactor and bins, and transportation costs in Chilean pesos. Year

Two zone scenario Strategy 1

Strategy 2

Diff. Strategies (2–1)

Investment

Transportation costs

Total

Investment

Transportation costs

Total

2013 2014 2015 2016 2017 2018 2019 2020 2021 2022

$ $ $ $ $ $ $ $ $ $

$ $ $ $ $ $ $ $ $ $

$ $ $ $ $ $ $ $ $ $

$ $ $ $ $ $ $ $ $ $

$ $ $ $ $ $ $ $ $ $

$ $ $ $ $ $ $ $ $ $

NPV (4%) NPV (8%) NPV (12%)

$ 997.697.038 $ 894.992.327 $ 815.611.552

270.211.959 1.693.600 1.270.200 1.058.500 846.800 1.693.600 1.058.500 1.693.600 1.693.600 1.270.200

85.043.712 85.043.712 85.043.712 85.043.712 85.043.712 85.043.712 85.043.712 85.043.712 85.043.712 85.043.712

355.255.671 86.737.312 86.313.912 86.102.212 85.890.512 86.737.312 86.102.212 86.737.312 86.737.312 86.313.912

246.551.800 2.117.000 1.270.200 423.400 635.100 2.117.000 1.058.500 2.540.400 1.693.600 1.481.900

$ 1.440.484.861 $ 1.262.252.385 $ 1.124.941.583

119.149.056 119.149.056 119.149.056 151.217.664 151.217.664 151.217.664 151.217.664 151.217.664 151.217.664 151.217.664

365.700.856 121.266.056 120.419.256 151.641.064 151.852.764 153.334.664 152.276.164 153.758.064 152.911.264 152.699.564

$ $ $ $ $ $ $ $ $ $

10.445.185 34.528.744 34.105.344 65.538.852 65.962.252 66.597.352 66.173.952 67.020.752 66.173.952 66.385.652

$ 442.787.823 $ 367.260.057 $ 309.330.031

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