A Cost Optimization Model for Hazardous Medical Waste Management in Portugal

Transcription

1 A Cost Optimization Model for Hazardous Medical Waste Management in Portugal João M. S. C. Nunes de Almeida Department of Civil Engineering, Instituto Superior Técnico ABSTRACT The hazardous medical waste management (HMW) is a very particular field of solid waste management as it requires following specific rules in order to correctly eliminate and treat the hazardous waste. According to Portuguese and European legislation the waste management operations should occur preferably in national territory, thus reducing inter-borders waste-flows. This poses a problem as the Portuguese incineration capacity for HMW is lower than the actual production; as a result there is an urgent need to expand such capacity. HMW in Portugal can be divided into two groups, roughly those that have to be incinerated and those that need to be decontaminated before being transported to landfills. The HMW companies work with both groups at the same time; therefore the HMW management system should be treated as one whole system instead of two separate ones. In this work a tool is proposed (Mixed Integer Linear Programming - MILP - Model) which will optimize in terms of cost, both the location of the facilities related to HMW management as well as the allocation of waste between the different nodes. The aim of this tool is to present the decision maker, in a reasonable amount of time, the optimal solution to the problem. This model will be applied to two different scenarios. The first one will consider none of the existing infrastructures; the locations of all the facilities will be treated as model variables. The second one will only locate the incinerator(s); all the other facilities location will be considered as model parameters. Keywords: Mixed Integer Linear Programming (MILP), Hazardous medical waste management, Facility location model and Cost optimization. 1. Introduction Medical waste is a specific type of solid waste produced in the activities related to healthcare, both of humans and animals. The nature of such waste makes it most of the time hazardous to human health and the environment. Therefore, in developed countries there is a strict legislation regarding the management operations (handling, storage, transportation and disposal) of such waste in order to prevent possible contamination of the surrounding environment. In Portugal the first legislation regarding medical waste management appeared only in 1990 and split medical waste into two groups: non-hazardous and hazardous medical waste (HMW). That legislation specified that HMW had to be incinerated, and so hospitals (the mains producers of HMW) began to install small incineration units in their facilities. In 1996 the legislation changed the classification of medical waste into four groups: I, II, III and IV. The first two are considered non-hazardous medical waste and consequently can be treated like urban solid waste (USW). The last two are considered HMW and as result need special requirements in terms of the management operations. Waste from groups III and IV must be stored in different containers and put in a storage area where they have to be collected within three days or seven days if the storage area is equipped with a refrigeration system. They must be transported from the collection point to the disposal point in 1

2 trucks equipped with isothermal containers. Group III waste can be eliminated the same way as Group IV or it can be decontaminated and then disposed as USW. Group IV waste has to be eliminated and the ashes produced have to be disposed at a non-hazardous industrial landfill; the only elimination method available in Portugal is incineration. As can be seen in figure 1 the path options for HMW are quite limited. Figure 1 Possible waste paths for HMW in Portugal With this new set of rules the HMW management market shifted, and several Group III waste treatment facilities, held by private companies, were opened. In 1999 due to poor environmental conditions, the hospital incineration units started to close leading to the current situation of only one working HMW incinerator located in Lisbon. Again, with this change, a new market was created as Group IV waste had now to be transported from the producing nodes to the only HMW incinerator in Portugal. Due to the characteristics of this transportation (small amounts, high collection frequency) the best competitors offering their services in transporting Group IV waste were the already established Group III waste treatment companies. Their solution was to collect both groups of waste at the same time, transport them to the Group III waste treatment unit where they built transfer stations (TS) storage units for waste, and finally by concentrating big amounts of Group IV waste in the TS, it was possible for them to transport the Group IV waste at lower cost to the Lisboa incinerator. This evolution in the legislation conditioned the growth of the HMW management market. When comparing all the possible paths for HMW (figure 1) and what happens in reality, one can observe that the waste management companies slowly adapted to the legal requirements imposed. Nowadays the HMW management system is not a system based in TS used by both groups of waste, in order to lower transportation costs, but an adaption of the existing Group III waste treatment layout to include Group IV waste collection and transportation. This last affirmation is justified by the disperse layout of infrastructure for Group III waste treatment (13 treatment facilities in 8 districts) high number of treatment facilities suggest lower use of TS by Group III waste and the concentrated layout of infrastructure for Group IV waste elimination (1 facility only) low number of incinerators suggests high use of TS. Another important issue that has to be taken into account is the cost inefficiency in the Group III waste treatment, which is quite a recent market that has grown quickly but not efficiently. Nowadays, as it can be seen in table 1, the installed treatment capacity of Group III waste is almost twice the predicted production for 2016, and since the analysis of the data available in CESUR (2009c) points to a stabilization in HMW production in a near future, such excess in capacity will not be necessary in the coming years. Table 1 Production vs. Treatment capacity Group III (ton/year) Group IV (ton/year) Lower bound Higher bound Lower bound Higher bound Treatment Capacity Estimated production for Differential Source: (CESUR, 2009c) Finally, also from table 1, it is possible to say that there is an urgent need to expand Group IV waste elimination capacity. According to Portuguese legislation all waste produced should be treated 2

3 within the Portuguese territory. In a best case scenario in 2016 there will be a deficit in elimination capacity of 360 ton/year. One can therefore conclude that the HMW management in Portugal is far from being optimal. In this work a mathematical optimization model (MILP) is proposed to address this problem. In section two a literature review is presented and in section three the model formulation detailed. In section four the input data considered for the case study is summed up followed, in section five, by results presentation and discussion. Finally in section six conclusions of this study are drawn and future research suggested. 2. Literature review The problem that is proposed to be solved has been extensively characterized in the available literature as the facility location problem. In these types of problems the objective is to optimally locate a certain type of facilities in order to best fulfill the model purpose. One can distinguish between two big categories of objectives: equity objectives, where the purpose is to minimize the maximal distance between the users and the facility and efficient objectives, where the purpose is to minimize the total distance (time/cost) of the whole system (Current, Daskin, & Schilling, 2002). The model proposed will have an efficient objective. Moreover this is a location-allocation problem which according to Owen and Daskin (1998) is normally solved using median or fixedcharge approaches. The median approach can be defined as minimizing the total demand-weighted travel cost (distance or time). Only variable costs, such as the transportation costs are taken into account in this approach. General formulations can be found in Owen and Daskin (1998), Current et al. (2002), ReVelle and Eiselt (2005) and ReVelle, Eiselt and Daskin (2008). The fixed-charge approach is similar to the median approach but it also includes the construction, capital or expansion costs. Therefore demand may not be assigned to the closest facility but to a farther facility whose fixed costs are lower. The nature of the solid waste management system makes it very suitable to be called a hierarchical facility system. It is composed by first level nodes which are the waste producers, followed by intermediate nodes that can be TS or treatment facilities and finally there are sinking nodes which are the waste disposal sites. Due to the predicted evolution of HMW production in Portugal, and due to the purpose of this model, which is to present the optimized layout of the HMW system in a near future (production data used refers to 2016), the approach considered will be a static deterministic mixed integer linear programming model. The literature on HMW optimization models is quite inexistent. Still there are a lot of articles on solid waste optimization models that are valuable to the HMW case. The first location-allocation models seen in the literature (1971 to 1988) had the objective of locating a few number of facilities, mainly intermediate nodes such as TS, and sinking nodes such as landfills. The big difficulty at that time was to solve the model since the computational capacities were not as evolved as nowadays. Therefore several of the articles found are focused on presenting heuristics that would help find a solution to the solid waste model presented (Walker W., 1976; Jenkins, 1982). In that same period, the articles focused on the model formulation and its application, propose mostly static fixed-charge approaches to locate the facilities stated above (Helms and Clark, 1971; Marks and Liebman, 1971; Harvey and O'Flaherty, 1973). As most of the problems were computer intractable some original procedures were found to make it possible to reach a solution. For instance (Greenberg, Caruana, & Krugman, 1976) propose a linear programming model for managing a three level hierarchical layout only considering hauling costs. The objective of this model is not to yield the optimal solution by itself (which facilities to open and where) but to test different solid waste management strategies, e.g. centralized landfills vs. disperse landfills. In order to reach a good solution the authors state that due to the added complexity of solving a MILP model (fixed-charge approach) it is preferable to include economies of scale by manually executing repetitive test runs of different scale plants (p.4) which is inexpensive to run, flexible and works (p.4). Walker, Aquilina and Schur (1974), also present a fixed-charge approach applied to a three level hierarchical system which purpose is to locate both the intermediate facilities and the sinking nodes facilities. These authors apart from considering both transportation costs and fixed costs of opening a facility at a certain node also consider the scale economies on variable treatment costs. Increasing a facility s capacity factor leads to the decrease of treatment costs per unit. Such costs are considered in 3

4 the model by including concave piece-wise linear cost functions. In this paper a heuristic method (SWAM) to solve the model is presented as well. In the later literature models one can distinguish Kulcar (1996), who proposes a fixed-charge model to locate, in a first part the transfer stations, and in a second part the collection trucks depots in the region of Brussels, as well as Badran and El-Haggar (2006), whose model determines the optimal location of collection stations and the optimal waste flows allocation considering only cost as a factor. Finally Komilis (2008) presents two different approaches for optimizing the haul and transfer of municipal solid waste. The first one is based on time, rather practical when information on cost is not available, and the second one based on cost, which is more appropriate when different types of vehicles with different costs are involved. In this article the chosen path for waste between nodes is given by a binary variable which is equal to one when a certain path is used. 3. Model formulation 3.1 Model objective and considerations The model purpose is to optimize the entire layout of the HMW management system in a cost point of view. For that purpose this model will optimize both the location of the facilities, i.e. TS, Group III waste treatment facilities (from now on referred simply as DS) and incinerators (INC), but also the allocation of waste to the different nodes. As it was seen, in reality the HMW cycle does not end in the DS, for Group III waste, or in the INC, for Group IV waste. However it was concluded that the exclusion of landfill locations would be a simplification to the model whose effects could be disregarded. Truth is, USW landfills are present all over the Portuguese territory and so the location of the DS will not have any relevant influence in the costs of transporting the refuse to the landfill. In what concerns the Group IV ashes, its transportation cost to non-hazardous industrial waste landfills is not significant enough to alter the incinerator location choice. Consequently the possible waste paths considered are as shown in figure 2. The Group III waste can go from the generating node to an intermediate node (TS) and then to the DS (GIII-1) or directly to the DS (GIII-2). The Group IV waste can go directly from the generating node to the INC (GIV-1) or, it can go through a TS between the initial node and the final node. The difference is that, as Group IV waste represents on average 10% of the HMW produced at each node there will be a difference in cost when this type of waste is transported together with Group III to the same TS (GIV-3) or otherwise (GIV-2). Figure 2 Possible waste flows considered by the model In terms of the costs considered, the model will include the different transportation costs between the several facilities, the fixed costs of opening a facility (included as an annuity in the model) and the variable costs of treatment for DS and elimination for INC. These last costs will be included in the model recurring to a similar formulation as the one presented by Walker et al. (1974). The choice of the path taken for each type of waste in each node will be done with binary variables as the ones used by Komilis (2008). 4

5 3.2 Indices The indices are: i {1,,m} for Generating nodes; j {1,,n} for Transfer Station nodes (TS); k {1,,p} for Disposal Site nodes (DS); l {1,,q} for Incinerator nodes (INC) and x {1,,r} for the Cost function. 3.3 Parameters Cs Ct Cts PG4 i PG3 i FixTS j FixDS kx FixINC lx VarDS kx VarINC lx δ β d ij, d ij, d ij, d ij, d ij, Cost per ton*km of transporting Group IV waste from a generating node to a TS or INC individually in a small capacity vehicle Cost per ton*km of transporting Group IV waste together with Group III waste from a generating to a TS in a small capacity vehicle, or Group III form a generating node to a TS or directly to the DS Cost per ton*km of transporting all types of waste from the TS to a DS or INC in big capacity vehicles Group IV waste production at node i (ton per year) Group III waste production at node i (ton per year) Fixed cost of opening a TS at location j Fixed cost of opening a DS at location k, associated to cost curve x Fixed cost of opening an INC at location l, associated to cost curve x Variable cost (per ton) of treating waste at a DS in location k, for cost curve x Variable cost (per ton) of eliminating waste at an INC in location l, for cost curve x Control parameter equal to the highest ratio Group IV waste produced at node i divided by Group III waste produced at the same node plus 1 Control parameter which has to be greater or equal than the total amount of HMW in the system Euclidean distance between the different set of nodes (km) 3.4 Variables QtDS kx Quantity of waste being treated at DS (k) with cost curve (x) QtINC lx Quantity of waste being treated at DS (k) with cost curve (x) Z1 il (1) if path GIV-1 is chosen, (0) otherwise Z2s ijl (1) if path GIV-2 is chosen, (0) otherwise Z2t ijl (1) if path GIV-3 is chosen, (0) otherwise Z3 ik (1) if path GIII-2 is chosen, (0) otherwise Z4 ijk (1) if path GIII-1 is chosen, (0) otherwise Y1 j (1) If TS at node j is open, (0) otherwise Y2 kx (1) If DS at node k is open, (0) otherwise Y3 lx (1) If INC at node l is open, (0) otherwise Δ1 jk (1) path chosen between TS (j) and DS (k), (0) otherwise Δ2 jl (1) path chosen between TS (j) and INC (l), (0) otherwise 5

6 3.5 Objective function The objective function represents the sum of all costs considered: Min Cost = [Z1 PG4 d Cs] + Z2s PG4 d Cs + d Cts + Z2t PG4 d Ct + d Cts + [Z3 PG3 d Ct] + Z4 PG3 d Ct + d Cts + FixTS Y1 + [QtDS VarDS + FixDS Y2 ] + [QtINC VarINC + FixINC Y3 ] (1) In this sum, the first five terms are related to transportation costs and the last three to facility costs (variable and fixed). 3.6 Constraints Z4 PG3 + Z2t PG4 δ Z4 PG3, i = 1,, m; j = 1,, n (2) [Z1 ] + Z2s + Z2t = 1, i = 1,, m (3) [Z3 ] + Z4 = 1, i = 1,, m (4) PG4 Z2s + Z2t + PG3 Z4 β Y1, j = 1,, n (5) PG3 Z4 + [PG3 Z3 ] β Y2, k = 1,, p (6) PG4 Z2s + Z2t + [PG4 Z1 ] β Y3, l = 1,, q (7) QtDS β Y2, k = 1,, p; x = 1,, r (8) QtINC β Y3, l = 1,, q; x = 1,, r (9) Y2 1, k = 1,, p (10) Y3 1, l = 1,, q (11) [PG3 Z3 ] + PG3 Z4 QtDS = 0, k = 1,, p (12) [PG4 Z1 ] + P43 Z2s + Z2t QtINC = 0, l = 1,, q (13) Z4 1 0, i = 1,, m; j = 1,, n; k = 1,, p (14) Z2s + Z2t 2 0, i = 1,, m; j = 1,, n; l = 1,, q (15) 1 = 1, j = 1,, n (16) 2 = 1, j = 1,, n (17) 6

7 Constraint (2) allows the flow of Group IV waste to be transported at reduced cost only when it goes from the same generating node, i, to the same transfer station, j, along with Group III waste. The coefficient δ serves as a control parameter, i.e. whenever the flow of Group III waste in a certain arch is equal to 0 then δ*0=0, which means that the Group IV waste flow in the same arch has to be 0 (since the right hand side of (2) cannot be negative). If the Group III waste flow is 0 then δ, which is a big number, allows Group IV waste flow to travel along this path. Constraints (3) and (4) confirm that for each node i only one path is taken, regarding respectively Group IV and III waste collection. This means that the waste produced at a certain node can only be transported through one path (method of transportation), avoiding having more waste transported than the one produced. Constraints (5), (6), (7), (8) and (9) can be considered as capacity constraints, though in this case they prevent waste flow to go through facilities that do not exist (when Y=0). The first three constraints of this group ensure that the path chosen for a certain node i, (Z=1) is going through existing facilities (Y=1). The last two constraints ensure that if a certain amount of waste is billed with a certain cost curve at a facility then the fixed cost associated with that curve must also be taken into account. The coefficient β serves has a control parameter. This coefficient can quickly transform these constraints into capacity constraints being β the capacity limit. The role of β in these constraints is analog to the role of δ previously mentioned. Constraints (10) and (11) ensure that only one cost curve is used per location, meaning that all the treatment costs of one location (k or l) are taken from the same cost curve. Finally constraints (12) and (13) guarantee that all the waste coming to a certain facility is charged with the according variable treatment cost. Constraints (2) to (17) are control constraints, to ensure that all waste of the same group flowing through a TS has the same final destination. 4. Input data for the model To run the previously formulated model to the Portuguese case study, one must have defined: (1) the geographical distribution of the HMW producers, (2) the production of each node, (3) the location where it will be possible to build the infrastructure (TS, DS and INC) and (4) the different types of costs. The first requisite was fulfilled by assuming a simplification. The producers, which are healthcare facilities, such as hospitals, public healthcare clinics (PHC), etc., are numerous and their precise location is difficult to obtain. Therefore it was considered that there would be one point per municipality to represent that area s producers. By observing the data available, it was possible to distinguish two types of healthcare facilities. Those that are use in a day-to-day basis (PHC, pharmacies, etc.) which account for 29% of total HMW production, and the necessity facilities, where the user goes if he needs special healthcare (hospitals) and which account for 71% of total HMW production. So in each node the production was dependent on the resident population (day-to-day share) and on the number of hospital beds in that municipality (necessity share). As the facilities in the HMW management sector are expensive to build, but also as the purpose of this application is to present a rough idea of the optimal locations for HMW management facilities it was decided that only one node per district would be considered for each of the three facilities type. Therefore based on the production of each municipality of a certain district, the gravitational centers were calculated and used as the possible location for opening the facilities. The different costs were estimated based on the available literature and are presented in table 2. Ct ( /ton/km) Cs ( /ton/km) Table 2 Considered costs for the case study Group III FixDS k1 ( /year) VarDS k1 ( /ton) FixDS k2 ( /year) VarDS k2 ( /ton) 15, , ,30 Cts ( /ton/km) FixTS j ( /year) Group IV FixINC l1 ( /year) VarINC l1 ( /ton) FixINC l2 ( /year) VarINC l2 ( /ton) 5, , ,98 5. Case study results The model was programmed using the GAMS software. The solver chosen was the CPLEX. As it is a MILP model, there is the need to define an optimization margin. The default value of 10% was 7

8 chosen for that margin, since it was considered that the improvements gained after attaining this value, in most of the cases were not worth the extra time spent. The objective of this case study is to observe, with the available data, what could be the optimized scenario for the HMW management system. Although the purpose of this model would be to create an interaction between the user and the results, allowing the user to adapt the technology choices and consequently converge into the optimal solution, due to the limited data available the case study presented will be pointing to the possible savings made by rearranging the number and location of such infrastructure while using the same technology available (cost estimated before). The first solution found had an optimization margin of 4,59% and an objective function value of 14,72 millions of Euros per year (total HMW management cost). After some analysis to the results it was understood that the solution found could be improved. By altering the model constraints to limit the location of the facilities a new solution was found with a total cost of 14,71 million of Euros, which is equivalent to an improvement of about 0,044%, compared to the first solution found. To try and reach the same solution with the original formulation the model was set to run with CPLEX option Objllim set to 14,713. After running for over 20 hours the model did not reach the solution. Still the analyzed results will focus on the best available results. The estimation for the total cost of HMW management, under the current real scenario and for the time period of the case study, is 19 millions of Euros per year. This means the optimized solution would reduce the total costs by 22,5%. In table 3, it is presented the location of the several facilities opened by the model, as well as the capacities used by the different DS and INC. First of all, one can see that most of the DS are working at a low capacity, with the exception of Porto and Lisboa the other DS are working at an average capacity of 18%. This is the result of very high transportation costs, which account for around 70% of the total costs. Therefore, under these conditions it is preferable to work at lower capacities than to transport the waste to farther facilities. In terms if the incinerators, although they are not working at a high capacity (50% of max. cap.), their usage is not as worrying as the ones seen for the DS. LOCATION Table 3 Location of infrastructure by model and existing facilities T. STATION DISP. SITES INCINERATOR LOCAT. EXISTING LOCAT. EXISTING % USED LOCAT. EXISTING % USED AVEIRO X X X - 18, BEJA X X - X BRAGA X X X X 31, BRAGANÇA X - X - 4, CASTELO BRANCO X X X X 10, COIMBRA X - X - 38, ÉVORA X - X - 12, FARO X X X X 12, GUARDA LEIRIA - X - X LISBOA X X X X 125,81 X X 48,22 PORTALEGRE X PORTO X X X X 63,64 X - 51,60 SANTARÉM X - X - 16, SETÚBAL X X - X VIANA DO CASTELO X VILA REAL X - X - 10, VISEU Total A sensitivity analysis to the relation between transportation and treatment costs led to a validation of the model. The higher the impact of treatment costs the more concentrated the layout became. However it must be stated that the number of incinerators opened by the model was constant, only varying for transportation costs much higher than the ones used (200%). This led to the conclusion that the location of incinerators was not only dependent on the direct costs associated with Group IV waste, but also with the layout of the Group III infrastructure (common transportation). 8

9 The results presented point out to the current inefficiency of the set up infrastructure, in terms of the DS units capacity. As it was said before Portugal has already almost twice the capacity needed to treat Group III waste. This leads to the understanding that the best solution for the HMW infrastructure location would be a high number of low capacity facilities well distributed over Portugal. In order to extract the reason behind the cost savings seen it is also possible to observe in table 3 the existing facilities in Portugal. One must be aware that in reality, each district might include more than one facility. At this moment, there are 13 DS equipped with TS and only two TS, in Portugal. When comparing to the results given by the model to reality it is possible to see that both situations are quite similar; in the model results there are 11 DS equipped with TS and 4 TS. So, one can conclude that the big share of savings observed in the optimization results, is due to a better distribution of the resources. Instead of positioning 13 DS in 8 districts the model equips 11 districts with a DS. Although the technology choice can be improved, the results of the model can be considered as more than satisfactory, since that any improvements in the technology choice will only affect a small portion of the total costs (treatment costs, around 30%). Therefore the total cost value given by this model is a very good approximation to what would be the lowest possible total cost of HMW management in Portugal, so considering other technologies or capacity for the used infrastructure would not lead to much lower values than the annual 14 millions of Euros. In terms of the INC case, as it was said at the beginning there is an urgent need of expanding its total capacity. The model presented a North-South division which is not only accurate since it is the less costly solution, but also because it is advisable that the HMW incineration market does not constitute a monopolist market. In the current situation only one operating company controls the whole share of HMW incineration. If fees are not regulated then the incentive of that company is to charge a little less than the second cheapest solution, which is to export HMW. This leads to the practice of higher and non-competitive fees by the solo service provider. Just to give an illustration of the solution presented by the model, it is included in appendix the influence areas of the different facilities. 6. Conclusion and further research Finally one can conclude from the results presented that the future of HMW management in Portugal will pass through the downsizing of treatment facilities in the Group III waste case, in order to present more competitive prices, but it will also include a better distribution of resources among the territory. In the incinerators case it is very important that a two operator solution is found quickly. Firstly due to the unacceptable location of the current incinerator, in central Lisbon, and also due to its capacity, already under the waste production, secondly due to the monopolist market which results in inflated fees for the user, and finally because of the high transportation costs which point to savings if there is a better distribution of incineration resources along the Portuguese territory. In terms of the transfer stations location and number, one can say that although they might generally seem the pillar of the HMW system, and so one would expect to see a high number of transfer stations serving both Group III and IV waste, due to the high transportation costs the benefits created by such facilities are easily overcome. So the real role of this type of infrastructure in the Portuguese case is to serve mostly as transfer stations for Group IV waste. Regarding future research two main aspects would be important to discuss. The first one concerns the application of the model to the same case study, considering the different cost curves associated with facilities of several types and capacities; as it was said before the savings described in section 5 are the biggest share of cost improvements possible to obtain by optimizing the HMW management system, still one could study this problem in a bigger detail by including such data into the model to quantify the gains of a more capacity efficient infrastructure. The second aspect is also related to studying this problem with more detail, but in a different approach. It consists in considering each of the several zones where the model located facilities as new case studies. Taking into account other factors, such as social, environmental, political, etc., it would be interesting to define inside of each of these new areas the best precise location of the facilities. 9

10 Appendix Influence areas of (1) TS, (2) DS and (3) INC (1) (2) (3) References Badran, M., & El-Haggar, S. (2006). Optimization of municipal solid waste management in Port Said - Egypt. Waste Management, 26, CESUR. (2009c). Plano Estratégico dos Resíduos Hospitalares Agência Portuguesa do Ambiente - DGS. Current, J., Daskin, M., & Schilling, D. (2002). Discrete Network Location Models. In Z. Drezner, & H. Hamacher, Facility Location: Applications and Theory (pp ). Berlin: Springer-Verlag. Greenberg, M., Caruana, J., & Krugman, B. (1976). Solid-waste management: A test of alternative strategies using optimization techniques. Environment and Planning A, 8, Harvey, D., & O'Flaherty, T. (1973). An analysis of solid waste transportation and disposal alternatives. INFOR, 3, Helms, B., & Clark, R. (1971). Selecting solid waste disposal facilities. Journal of the Sanitary Engineering Division, 97-4, Jenkins, L. (1982). Parametric mixed integer programming: An application to solid waste management. Management Science, 28-11, Komilis, D. (2008). Conceptual modeling to optimize the haul and transfer of municipal solid waste. Waste Management, 28, Kulcar, T. (1996). Optimizing solid waste collection in Brussels. European Journal of Operational Research, 90, Marks, D., & Liebman, J. (1971). Location models: Solid waste collection examples. Journal of the Urban Planning and Development Division, 97-1, Owen, S. H., & Daskin, M. S. (1998). Strategic facility location: A review. European Journal of Operational Research, 111, ReVelle, C., & Eiselt, H. (2005). Location analysis: A synthesis and survey. European Journal of Operational Reshearch, 165, ReVelle, C., Eiselt, H., & Daskin, M. (2008). A bibliography for some fundamental problem categories in discrete location science. European Journal of Operational Research, 184, Walker, W. (1976). A heuristic adjacent extreme point algorithm for the fixed charge problem. Management Science, 22-5, Walker, W., Aquilina, M., & Schur, D. (1974). Development and use of a fixed charge programming model for regional solid waste planning. Santa Monica, CA: Rand Corp. Paper. 10