Available online at ScienceDirect. Transportation Research Procedia 12 (2016 )

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1 Available online at ScienceDirect Transportation Research Procedia 12 (2016 ) The 9th International Conference on City Logistics, Tenerife, Canary Islands (Spain), June 2015 How many urban recycling centers do we need and where? A continuum approximation approach Javier Soto, Juan Carlos Muñoz, Ricardo Giesen* Department of Transport Engineering and Logistics, Pontificia Universidad Católica de Chile, , Chile Abstract This article presents a methodology that uses the continuum approximation approach to formulate the logistical problem of determining the number and location of recycling centers to which individuals in a region can deliver solid wastes to be recycled. To solve the proposed problem a two-stage approach is proposed. First, we solve an optimization problem whose main decision variable is the density of recycling sites per unit area. This problem is solved assuming that local conditions are similar throughout the region of interest. So, this problem is solved for each point of a grid that divides the region under study, yielding an optimal solution that reflects an approximation of the local optimum density. In the second stage, the solution is refined by identifying specific locations in which recycling centers could be installed, in a manner consistent with the approximate solution obtained in the first stage. Finally, this proposed approach is tested in a case study using data from Santiago, Chile Published The Authors. by Elsevier Published B.V. This by Elsevier is an open B.V. access article under the CC BY-NC-ND license ( Peer-review under responsibility of the organising committee of the 9th International Conference on City Logistics. Peer-review under responsibility of the organising committee of the 9th International Conference on City Logistics Keywords: Recycling Centers Location; Waste Collection; Reverse Logistics; Continuum Approximation method; Logistics Network Design; Urban logistics. * Corresponding author. Tel.: ; fax: address: Giesen@ing.puc.cl Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license ( Peer-review under responsibility of the organising committee of the 9th International Conference on City Logistics doi: /j.trpro

2 852 Javier Soto et al. / Transportation Research Procedia 12 ( 2016 ) Introduction Here introduce the paper, and put a nomenclature if necessary, in a box with the same font size as the rest of the paper. The paragraphs continue from here and are only separated by headings, subheadings, images and formulae. The section headings are arranged by numbers, bold and 10 pt. Here follows further instructions for authors. In this paper we present a methodology that uses the continuum approximation approach (see for example, Daganzo, 2005) to determine the number and location of recycling centers (or distribution centers) to which individuals and/or households in a region can deliver solid wastes to be recycled and then applied to the case of Santiago, Chile. We consider a company that operates recycling centers to which users can deliver their items to be recycled. Each of these recycling centers serves an area of influence, in which users in the region can deliver their waste items to be recycled. After these items are received in a center, they are classified and consolidated, and then transported to external recycling plants. One of the key trade-off that occurs in choosing the number of recycling centers is between minimizing the cost per kilogram of recycled material and maximizing the quantity of material attracted. The proposed methodology explicitly recognizes that the demand attracted by each recycling center depends on the number of households in its area of influence and the distance of each of them to the center. The logistics costs associated to this recycling system can be grouped into two categories: i. Leasing, operation and maintenance costs associated with the process of preparation and delivery of products in each of the recycling centers. The leasing cost is related to the rental fee paid for occupying each square-meter of land during a time horizon. The operating cost includes the salary paid to each employee of the recycling center. The maintenance cost includes a fixed rate for each period. This rate consolidates the costs associated with cleaning, maintenance and care of the recycling center. ii. Transportation costs associated with the movement of products from recycling centers to external recycling plants. Shipments are made in full truckloads, since vehicles are paid by distance regardless of the amount transported, and given the characteristics of the products which have extremely low inventory costs and do not lose their value. Accordingly every kilogram of material will be transported at a lower cost if we wait until filling the capacity of the vehicles. Also, from a social perspective, it makes sense to consider as part of logistics costs the costs incurred by users on their trips to the recycling centers. This adds a third category: iii. User costs associated with transportation of products from their homes to the recycling centers. This category consists mainly by the additional cost that may involve deliver waste to recycling centers. In this case, it is possible to differentiate in two user categories because some may travel to recycling centers in a private vehicle while others may simply walk to the nearest place. These differences strongly influence the cost incurred by users. Furthermore, the recovery of recyclable material has a value associated to the recycling centers, thus we can consider income associated with each of the products recovered at recycling plants: iv. Income (benefit) associated to received products in recycling plants. For each product recycled the company that owns the recycling centers is paid a fixed rate depending on the type of product. This income is a transfer from the recycling plant to the distribution centers. From a social standpoint, these values could represent the social or shadow prices of recycling these products. These categories are added to obtain an objective function that includes the costs and revenues listed above. This objective function is used to find the density of distribution centers in the region that maximizes the benefit of the company in the private case, and the social benefit in the social case. The main decision variable in each case is the density of recycling centers around each point of the region of interest. In this paper, we propose a methodology that uses continuum approximation (Daganzo, 2005) to determine the amount and location of recycling centers in a region. This type of problem has been studied and solved in the literature usually with integer programming methods in which you try to find the exact solution. Unfortunately, these solutions require a large amount of information about the instances to be solved and the computational requirements to find an optimal solution can be even today extremely time-consuming. A detailed review of integer programming models for localization issues can be found at Brandeau (1989) and Daskin (1995).

3 Javier Soto et al. / Transportation Research Procedia 12 ( 2016 ) In the literature there are some examples that use the Continuum Approximation (CA) approach for tackling logistics systems design problems. For example, Clarens and Hurdle (1975) design a bus network system considering user and operating costs. Erera (2000) used the CA approach for the design of large-scale logistics systems considering uncertainty. Ouyang and Daganzo (2004) develop an application for a terminal system design. Mangotra et al (2012) developed a joint solution for network design problem in which they solved for location and inventory in two stages. Medina et al (2012) propose a model for locating public transport stops in a corridor. Pulido et al (2014) used this approach on a combined problem of localization and distribution with time windows under conditions of fixed demand. However, we are not aware of previous research trying to address the problem of defining the number and location of recycling center using the continuum approximation approach. The novelty of this paper lies in the proposition of sections that distinguish between different attraction levels according to geographical location. This allows not only approximate the optimal number of facilities, but also to approximate the optimal location of every facility. The proposed methodology recognizes that demand attracted to each recycling center depends on the number of households in their area of influence and the distance of each household to that center. As Beckmann (1968) explains, when a user can access a good at the same price in different locations, the only relevant cost on the decision about where to consume it is the transportation cost. This is further emphasized in the case of recycling centers because users are accessing a free service and the only cost that they must pay is the transportation costs. Consequently, as the distance between a home and a recycling center increases, so does the cost per use. In the case of using a Euclidean metric, a good approximation of the area of influence of each center is as round as possible (Daganzo, 2005). The proposed formulation was then applied to the study and analysis of the B-corp Triciclos, a renowned multinational firm of Chilean origin created in 2009 with the main objective of encouraging sustainability in the use of resources. The company works in the area of collecting household solid waste that could be potentially recovered. The operation is based on the installation of several recycling centers (in Spanish they are called "Puntos Limpios") in the city in which they receive for free products that can be recycled. In these centers recyclable products are accumulated before being shipped to recycling plants. The case study of the proposed method will be centered in the city of Santiago, Chile, in areas where the company has demonstrated greater activity. In what follows, Section 2 presents a description of the proposed methodology and the development of the logistics cost of the system. Section 3 describes the case study with the results of the application for Santiago. Section 4 summarizes the main conclusions of this work. 2. Methodology The aim of this work is to obtain an approximation of the optimal number of recycling centers and its location in order to maximize the benefit to the logistics system, either from a private or social perspective. In each of these two cases, the objective function changes. While in the private perspective, the goal is to maximize the recycling firm s profits, when the social perspective is addressed, we must give a social value to each recycled material and also include a cost for those trips made by recyclers to bring their materials to the recycling center. We define a function that adds costs described by the density of demand in the region. Demand for recycling is modeled directly from the population in the area of interest. Generally, in the literature, the demand density is considered homogeneous, which facilitates the approximation of a solution by an explicit formula. The approach used in this work consists of dividing the area into a grid with sections of equal size and uniform density within each, but allowing different grid sections to have different densities. This approach makes it possible to account for local demand conditions in specific sectors of the region, as the density of demand for recycling could vary greatly from one place to another. This is a reasonable way to improve the approximation of the locations without losing the simplicity in modeling the continuum approximation. The variable of interest in the formulation is the optimal density of recycling centers for the region. It is shown that these solutions are quite robust because moderate variations of the input data perturb the solution of maximum benefit in a small way Variables and Parameters Let be a set of coordinates defining the position of each of the sections of the grid over the region. In each of the cells, we assume that there is a density of recyclable products generated per unit of surface and known equivalent

4 854 Javier Soto et al. / Transportation Research Procedia 12 ( 2016 ) to. Also, we assume that the attraction of products to the center depend on the distance of each household to the nearest recycling center. So, for a given distance we can compute a ratio of attracted items versus items produced from a distribution center, which we call. Thus, the demand that reaches a center from a given distance should grow for short distances, as the number of households is increasing, but should decrease for long distances as the rate of items attracted also decreases. The parameters used in the model are: Density of recyclable products generated per unit of area in the cell Average fixed income obtained per kilogram of recycled material Fixed leasing cost for each recycling center Fixed salary paid to each of the workers at the recycling centers Maximum number of kilograms processed in the recycling center per operator per month Fixed fare paid per kilometer for vehicles moving load from the recycling centers to the recycling plants Maximum load of freight vehicles Parameter indicating what percentage of trips is made for the sole purpose of recycling products. We also assume as known two functions that describe how people, living at a common distance to the closest recycling terminal, behave regarding recycling habits. We assume that this aggregated behavior depends only on the common distance from their homes to the closest recycling center. Is a function that determines the fraction of products that are attracted from a fixed distance. Is a function that relates the items attracted on each trip to the distance to the recycling center. The variable used in the model is: Local Density recycling centers in cell 2.2. Attraction After setting the function relating the rate of attraction with distance, we can obtain the total number of items attracted to any position on the grid. The function that gives the volume of products is given by: (1) In this expression, the value corresponds to an approximation of the radius of the zone of influence of a recycling center located at in which the local density is. Since the surface of this area can be expressed as, it is possible to estimate the radius assuming for example that the area is circular, following a similar method as Pulido et al (2014) Private Optimization The objective function is to maximize the total profit of the company. The income obtained by each of the processed products is. The cost incurred by the company for each product is composed of operational and transportation costs. These are modeled as presented next.

5 Javier Soto et al. / Transportation Research Procedia 12 ( 2016 ) Operational Costs The operational costs of the centers include a maintenance cost plus the monthly salary received by operators. It is assumed that the number of operators required to operate the center grows linearly with the number of products to be processed, adding a new operator for every processed items. The operation cost is determined by: (2) Where represents the monthly cost of renting each distribution center and is the salary paid to the operators. Transportation cost The transportation cost depends on the distance from distribution centers to the plants and the number of trips required to move all the received items. The number of trips needed is determined by the capacity of freight vehicles that we assume constant. Let be the distance between the center located at (i, j) and the recycling plant used for that center. Thus, if we define as the fare paid per kilometer, the transportation cost can be expressed as: (3) Objective function Finally, the function to maximize is modeled as: (4) 2.4. Social Optimization If instead of maximizing the private profit for this company, we would like to maximize the social benefit of the recycling network, we should redefine () as the social benefit of recycling a product (). We should expect that since the social value should include not just the alternative production cost from using raw materials (instead of from recyclable goods), but also the environmental externalities generated by extracting these raw materials and by disposing the recyclable goods. Moreover, we should add an additional term to the objective function to represent users hauling costs from their homes to the recycling center. Cost of users To compute users cost we add the number of visits weighted by the probability that these users make their trip only for the purpose of recycling. If users take advantage of another trip to recycle, we assume that there is no additional cost for users. Thus the cost of users can be computed as: (5) In this expression, is a function that relates the items carried on each trip to the distance to the recycling center. In addition, a parameter is included which indicates the percentage of trips made to the recycling center for the only purpose of recycling products, since some user take advantage of other trips to do their recycling. Objective function Finally, the new objective function is to maximize:

6 856 Javier Soto et al. / Transportation Research Procedia 12 ( 2016 ) (6) In both approaches, the only constraint being considered is the non-negativity of the single family variables of this problem,. 3. Solution Algorithm As a first step, the model is solved analytically. This give us the approximate optimal density for recycling centers for each point (i, j) of the region of interest. From this approximate solution, we proceed to identify specific locations for recycling centers to be installed using numerical methods. The following solution algorithm is applied to find jointly the number of recycling centers to install and their location in order to maximize the benefit of the company. i. Initial Solution: Depending on the density found in step 1, we obtain the first approximation of the number of recycling centers as, which is used as initial solution for the algorithm. ii. Influence Area: The influence area of each center is estimated as. Then, the radius of influence is approximated as. iii. Location of recycling centers: The recycling center is located in the position that maximizes the expected income based on the attracted demand at (i, j). Once located, the items drawn on the density of demand in each cell within the area of influence are discounted. iv. Step (iii) is repeated until all recycling centers had been located. v. Income maximization: We define the vector where each position contains the location of the N recycling centers that maximize the income of the company. Thus, the total income of the system with centers can be computed as. vi. Approximation to the optimum: The process is repeated from step (i) perturbing the initial number of recycling centers by adding or subtracting a recycling center to the solution. The algorithm terminates if the revenue obtained with recycling centers is greater than the income obtained using or centers, i.e.. Otherwise, we update the number of centers to the one that deliver higher revenues and repeat the algorithm. 4. Case Study The proposed methodology was applied to the eastern part of Santiago, Chile, for the operation of the recycling B- corp Triciclos. This company works in the area of collecting residential solid waste and recycling those that potentially could be recovered. They install and operate recycling centers, or "clean points" as they call them, which receive deliveries from individuals. In this centers deliveries are accumulated and then shipped to recycling plants. The type of waste considered in the analysis includes categories as cellulose, glass, plastic and metal. The eastern part of Santiago, defined as a region of analysis in this case study, has surface km2. Over this region a grid of 10x10 was placed. For each cell in this grid, a density of kilograms of household solid waste per km2 was estimated based on average waste generation per type of individual and projections of population per cell on the grid. As input for future waste generation we reviewed public information from government agencies to estimate generations per type of individual. In addition, census population projections were used to estimate future population in the region of analysis. In our case we define the type of individual only depending on the city where the person lives (the region of analysis is composed of seven adjacent cities). The information about waste generation was obtained from a governmental agency in terms of waste per capita in each city (CONAMA, 2006). From the data it s clear that cities with more wealthy people generate more waste. Table 1. Waste generation by income level Income Quintile Average waste generation (kg person/day)

7 Javier Soto et al. / Transportation Research Procedia 12 ( 2016 ) (Top) The company also agreed to provide historical information on the monthly kilograms recycled during the years 2012 and This was used to validate projections of kilograms attracted to new recycling centers. Parameters are also set to calculate the various costs and revenues of the proposed formulation; for example the income per each kilogram of material received is on average is 80 CLP (Approximately 0.13 USD). Furthermore, user surveys were conducted at two current recycling centers operated by Triciclos in the area. In these surveys data on kilograms of material each user brought to the recycling center, the distance from the origin to the point of recycling were taken and each user was asked if the visit was part of another trip or if the primary motive of the trip was to deliver products to the recycle center. A total of 889 valid surveys were obtained. Attraction curve of products for different distances from each recycling center was calibrated according to information reported by users in our surveys. This curve is used to estimate how much of the material generated finally come to one of the recycling centers. The curve should be able to reflect that at small distances the attraction will be high but the proportion of people will be small compared to the total in the area of influence, while long range attraction ought to be shrinking on a growing population. The information collected was used to calibrate the curve shown in Figure 1. Figure 1. Curve fraction of attracted material The proposed method was first applied using the private objective and the information provided by the company. The result obtained in this case was that the optimal number of recycling centers in the eastern part of Santiago is four. Figure 2 shows how the operating income varies for the company as we change the density of recycling centers in the region, the optimum occurs when an equivalent density is achieved at four recycling centers.

8 858 Javier Soto et al. / Transportation Research Procedia 12 ( 2016 ) Figure 2. Optimal recycling center density using a private objective Figure 3 shows the locations obtained using the private objective according to the grid that was built on the region. The proposed centers are concentrated in the south west of the region, which corresponds to considerably denser areas. Figure 3. Optimal location of recycling centers using a private objective The same analysis was made from a social perspective (social objective). To calibrate the parameter (), which represents the percentage of trips just to recycle, we used the percentage of users surveyed that recycled under this condition. In addition, the curve reflecting the relation between the number of items attracted on each trip to the distance from the recycling center was calibrated based on the surveyed data. Moreover, the parameter on social the benefit of recycling was redefined from the private case. This was done using the values proposed by Craighill and Powell (1995), in which they estimate the social value of recycling different product families. We use these values to weight the historical share of each product family reported by Triciclos. Following this procedure, we obtained that the average social value per kilogram of recycled material is CLP(Approximately 0.23 USD). Taking into account the social objective, i.e. using these new set of parameters; the procedure presented in Section 3 was repeated. In this case, according to the social perspective, we found an increase of nearly twice the earned income. However, the new solution suggests to add only one new recycling center while keeping the same four found in the private solution. That is, the optimum density is achieved with five recycling centers in eastern Santiago.

9 Javier Soto et al. / Transportation Research Procedia 12 ( 2016 ) Figure 4 shows how the social benefit varies as function of the density of recycling centers. Figure 4. Optimal recycling center density using a social objective. Figure 5 shows the locations obtained using the social objective on the grid that was built on the region. 5. Conclusions Figure 5. Optimal location of recycling centers using a social objective We proposed a model that uses the continuum approximation methodology to solve the network design problem of location and number of recycling centers to attend a region. The proposed model considers an objective function that incorporates the different costs of the system, both from a private perspective and a social perspective. The primary endpoint of the problem is the density of recycling centers in the defined area. This problem is solved testing combinations of many recycling centers together with their location. The proposed methodology was applied to a case study of the operation of the recycling B-corp Triciclos ( on the eastern part of Santiago, Chile. The results confirm again that the solutions provided by the continuum approximation approach are robust. Note that an approximate 81% increase in the perceived benefit for every kilogram of material only alters the optimal solution by adding a new recycling center. Furthermore, the location of the centers is not altered except for the new recycling center, which is added in the social solution. In the case study, the solutions tend to concentrate in the south west of the region. This coincides with the grid sections in this sector are areas where the population per unit area is considerably denser. Consequently, it could attract more material considering a fixed area of influence assuming that the conditions given by the curve of attraction are maintained for each grid section.

10 860 Javier Soto et al. / Transportation Research Procedia 12 ( 2016 ) The benefit received from a private perspective is relatively low for the cost structure of the company. However, the social margins obtained by the work of the recycling centers are considerably higher in the previous case. Accordingly, it is an investment with low profitability in private terms, but of great social contribution. In this regard, consideration should be subsidizing part of the difference between private income and social income, so that it can encourage the reuse of household solid waste. Acknowledgements The authors are grateful for funding provided under FONDECYT project This research also benefited from the support of the Centro de Desarrollo Urbano Sustentable (CEDEUS), CONICYT/FONDAP/ References Beckmann, M. (1968). Location Theory. Brandeau, M. L., & Chiu, S. S. (1989). An Overview of Representative Problems in Location Research. Management Science, 35(6), doi: /mnsc Clarens, G. and Hurdle V.F. (1975). An operating strategy for a commuter bus system. Transportation Science, 9, Craighilla, A. L., & Powellavb, J. C. (1996). Lifecycle assessment and economic evaluation of recycling: A case study. Resources Conservation and Recycling, 7, doi: / (96) Comisión Nacional del Medio Ambiente (CONAMA) (2006). Caracterización de residuos sólidos domiciliarios en la Región Metropolitana. Valparíso, Chile. Daganzo, C.F. (2005). Logistics Systems Analysis. Fourth Edition. Springer-Verlag, Heidelberg, Germany. Daskin, M.S. (1995) Network and Discrete Location: Models, Algorithms and Applications. Wiley, New York, USA. Erera, A. L. (2000) A Design of large-scale logistics systems for uncertain environments. Ph.D. Thesis, Dept. of Industrial Engineering and Operations Research, University of California, Berkeley, CA Medina, M., Giesen, R., & Muñoz, J.C. (2013). Model for the Optimal Location of Bus Stops and Its Application to a Public Transport Corridor in Santiago, Chile. Transportation Research Record: Journal of the Transportation Research Board, 2352 / Transit 2013, 2, Ouyang, Y., & Daganzo, C. F. (2006). Discretization and Validation of the Continuum Approximation Scheme for Terminal System Design. Transportation Science. doi: /trsc Pulido, R. J.C. Munoz and P. Gazmuri (2014) A continuous approximation model for locating warehouses and designing physical and timely distribution strategies for home delivery. Forthcoming in EURO Journal on Transportation and Logistics. DOI: /s z Tsao, Y. C., Mangotra, D., Lu, J. C., & Dong, M. (2012). A continuous approximation approach for the integrated facility-inventory allocation problem. European Journal of Operational Research, 222, doi: /j.ejor