key: cord-0747457-f8owqbdw
authors: Govindan, Kannan; Nasr, Arash Khalili; Mostafazadeh, Parisa; Mina, Hassan
title: Medical waste management during coronavirus disease 2019 (COVID-19) outbreak: A mathematical programming model
date: 2021-09-16
journal: Comput Ind Eng
DOI: 10.1016/j.cie.2021.107668
sha: eac8b4f3879f3e2a79a81aa73926b7296db6f343
doc_id: 747457
cord_uid: f8owqbdw

Municipal solid waste (MSW) directly impacts community health and environmental degradation; therefore, the management of MSW is crucial. Medical waste is a specific type of MSW which is generally divided into two categories: infectious and non-infectious. Wastes generated by coronavirus disease 2019 (COVID-19) are classified among infectious medical wastes; moreover, these wastes are hazardous because they threaten the environment and living organisms if they are not appropriately managed. This paper develops a bi-objective mixed-integer linear programming model for medical waste management during the COVID-19 outbreak. The proposed model minimizes the total costs and risks, simultaneously, of the population’s exposure to pollution. This paper considers some realistic assumptions for the first time, including location-routing problem, time window-based green vehicle routing problem, vehicles scheduling, vehicles failure, split delivery, population risk, and load-dependent fuel consumption to manage both infectious and non-infectious medical waste. We apply a fuzzy goal programming approach for solving the proposed bi-objective model, and the efficiency of the proposed model and solution approach is assessed using data related to 13 nodes of medical waste production in a location west of Tehran.

Municipal solid waste (MSW) has increased along with growth in the urban population (Cheng et al., 2020) . The waste management organization of Tehran municipality published a report that demonstrates more than three million tons of MSW were produced in Tehran during 2017 (http://statistics.tehran.ir). Plenty of definitions and categories are introduced in the literature for MSW, and the large number of definitions leads to confusion (Buenrostro et al., 2001) . The most appropriate MSW definition depends on the location of the municipality and the time period (Zhou et al., 2015) . As an example, Heinen (1995) once declared that MSWs must contain domestic residues and cannot have commercial or hazardous components. On the other hand, United States Environmental Protection Agency (EPA) categorizes municipal solid waste as that generated by commercial, institutional, and residential sources. In another example, Bruner and Ernst (1986) define MSW as the waste collected by authorized organizations and municipalities.

An efficient summary of MSW was introduced by Buenrostro et al. (2001) , in which the term MSW refers to the following seven categories: residential, institutional, commercial, construction (demolition), industrial, agricultural-animal husbandry, and special.

MSW management is a public service that incorporates an efficient economic and environmental waste disposal system for the citizens. Traditional MSW disposal systems such as landfills are no longer effective due to the lack of adequate land and/or the consequences of soil and water pollution (Abdel-Shafy and Mansour, 2018) . Thus, alternative solutions such as incineration, composting, or recycling are proposed (Zhou et al., 2015; Mohammadi et al., 2019) , but their implementation requires scarce and complicated technologies. The selection of a proper MSW disposal system should be based on the geographical location and climate as well as on the population's lifestyle (Buenrostro et al., 2001) . Choosing a MSW disposal system without having a complete understanding of its attributes is likely to result in unnecessary costs and environmental damages. Iran is no exception; Tehran's officials report that up to 20% of the city's municipality budget for collecting and disposing MSWs has been wasted because a proper MSW disposal system was not selected in a scientific way (http://pasmand.tehran.ir).

Recently, a lot of studies are devoted to MSW management using mathematical programming approaches with various assumptions and specifications, including capacitated locationallocation (Yu and Solvang, 2017; Tirkolaee et al., 2020) , vehicle routing problem (VRP) (Hannan et al., 2018; Louati and Chabchoub, 2019; Höke and Yalcinkaya, 2021) , location-routing problem (Rabbani et al., 2018; Asefi et al., 2019) ; uncertainty (Yadav et al., 2017; Tirkolaee et al., 2018; Singh, 2019; Wang et al., 2021) , and so on.

This study is conducted to introduce a new mixed-integer linear programming (MILP) model with the purpose of MSW management considering risks related to the population exposed to pollution, location-allocation, and green VRP. Multiple vehicles of different types are supposed to start their tour from separation centers and return in time after collecting MSWs from wastegenerating nodes. In separation centers, collected MSWs will be separated and sorted before moving into recycling, compost, incineration, or landfill centers. In addition, the vehicles' fuel consumption with respect to the extra weight of MSWs are considered as a part of the total cost of the supply chain. This has an indirect impact on reducing the pollution generated by vehicles while collecting MSWs. The main contributions of this paper are listed as follows:

 Formulating a novel multi-product, multi-period and bi-objective MILP model for MSW management with location-allocation, time window-based green VRP, population risks, vehicles scheduling and failure, split delivery, and load-dependent fuel consumption minimization considerations  Developing a novel fuzzy goal programming for converting the proposed bi-objective model into a single objective one under uncertainty  Using Tehran municipality actual data as a case study to measure efficiency and practicality of the proposed model and solution approach.

The remainder of this paper is structured as follows. In Section 2, we investigate the related literatures. Section 3 presents the proposed model. In Section 4, we implement a case study in Iran to represent the applicability of the proposed model. Discussions are presented in Section 5.

Sections 6 and 7 are allocated to sensitivity analysis and managerial implications, respectively.

Finally, the conclusion is presented in Section 8.

Health and economic considerations for the disposal of environment polluting materials, such as solid wastes, has attracted growing research attention, especially during recent years (Chowdhury et al., 2019; Rani et al., 2020) . Considering the undeniably destructive consequences of applying improper waste collection and disposal systems in both rural and urban area, this is definitely an inevitable part of public healthcare and prevention that must be acknowledged before treatment strategies are pursued (Darmian et al., 2020) . Due to basic environmental principles, the time to collect and dispose waste should be minimized . All residents have to make sure they do not violate green waste disposal codes that have been in place since the 19 th century. These basic notions and obligations have led to the introduction of MSW management. This field has been studied extensively in the literature of waste management, for which mathematical programming is observed to be very popular among the contributing researchers. Huang et al. (2001) developed a MILP model for MSW management purposes. They applied a chance constrained and fuzzy theory based integrated approach to solve their proposed model under uncertain circumstances. Li and Huang (2006) presented an interval-parameter two-stage MILP model to support the long-term of MSWs management in Regina. In order to accommodate uncertainty in their proposed model, they applied both two-stage stochastic programming and interval linear programming in parallel.

Another optimization approach has been introduced by Badran and El-Haggar (2006) for MSW management in Said port. They applied a MILP model to do waste management while minimizing total costs. Eiselt and Marianov (2014) developed a bi-objective MILP model to locate optimum landfill places for MSWs. The objective of their model was to simultaneously minimize both total costs and pollution. Xu et al. (2014) developed a fuzzy chance constrained mixed-integer programming (MIP) model to manage MSW under circumstances with multiple uncertainty. Their proposed model involves strategic to tactical decisions that lead to minimizing cost of supply chain. Yu et al. (2015) devoted their work to optimizing long-term performance of MSW management systems via formulating a bi-objective mathematical model. They proposed a dynamic programming approach for solving the bi-objective linear model which leads in optimizing long-term performance of MSW systems. Their model simultaneously evaluates both economic efficiency and environmental pollution emissions of a MSW management system in multiple consecutive time periods. An optimal collection and transportation scheme for MSW management is studied by Das and Bhattacharyya (2015) that aims to optimize the transportation route. They formulated an MIP model to represent the highlighted MSW collection and transportation problem. Then, they applied a heuristic method to solve this problem. Obtained results demonstrated that application of the proposed approach leads into decreasing more than 30% of the distance that shall be undertaken for MSW collection. Lee et al. (2016) installing temporary incinerators could help in managing medical wastes. Kargar et al. (2020) developed a MILP model to minimize costs, transportation risk, and the quantity of medical waste in generation centers. The model was validated by using a case study from Iran and demonstrated a trade-off between objectives for better managing waste flows. Nikzamir and Baradaran (2020) presented a bi-objective model considering costs and emissions of contamination that could be intensified during the COVID-19 outbreak. They modeled the transferring time between nodes as random variables and used a meta-heuristic algorithm for solving that complex model. showed how cooperation between municipal waste collectors could reduce cost as well as residual waste during COVID-19 pandemic. In another paper, showed the role of the government in the outbreak using a robust mathematical model. They used a stochastic programming approach for modeling the condition and solving the model by the Benders decomposition with Karush-Kuhn-Tucker conditions. Tirkolaee et al. (2021) analyzed the case study using an MIP model with time windows in the COVID-19 pandemic era. They minimized traveling time, environmental risk, violation from time windows, and priorities in provided services. They used a fuzzy chanceconstrained approach for solving these conditions. 

Commercial solvers, MH: Meta-heuristics, E: Exact methods As shown in Table 1 , the location-routing problem in medical waste management has been considered by researchers, and most of the papers presented are multi-objective and examine population risk. With the outbreak of the COVID-19 pandemic, some medical waste management papers have moved in this direction. In this paper, for the first time, a bi-objective MILP model takes into account the assumptions of location-routing problem, time window-based green vehicle routing problem, vehicles scheduling, vehicles failure, split delivery, population risk, and load-dependent fuel consumption for medical waste management during COVID-19 outbreak under uncertainty.

In this section, a bi-objective MILP model is developed for the management of medical wastes in the event of a pandemic COVID-19 outbreak. The proposed model is a multi-product, multiperiod and bi-echelon one including waste production nodes and potential collection centers; total cost and pollution risk are simultaneously minimized in this model. Waste production nodes include laboratories, hospitals, temporary hospitals, clinics, and health centers, which fall into two categories, namely nodes that produce COVID-19 wastes and are considered hazardous wastes and nodes that produce both infectious (hazardous) and non-infectious wastes. Vehicles move from collection centers to waste generation nodes and return to collection centers before the allowed time window following the collection of wastes. In Figure 1 , the overall structure of the network under study has been illustrated.

Waste production nodes Collection centers

The following points constitute the assumptions of the proposed model for an accurate statement of the problem:

 The network under study is a bi-echelon supply chain including waste production nodes and potential collection centers.  The supply chain of multiple products in multiple time periods is modeled through this study.  Whereas the location of the medical waste production nodes is assumed to be fixed, the solver is free to propose the optimum location(s) for the collection centers.  There is a finite number of vehicles with limited capacity, among which the solver is tasked to find the optimum set of vehicles to be supplied.  Vehicles are heterogeneous.

 Split delivery is possible.

 The time required to move between each pair of nodes in deterministic and an input to the model.  This study aims to find the optimum route to visit all medical waste production and collection centers through solving a multi-depot VRP with capacitated distribution centers.  Vehicles' fuel consumption depends on the weight of the waste with which the vehicle is loaded.  Hard time window constraints are considered for vehicles returning to distribution centers.  The vehicles' failure is considered.

The distance between nodes m and n mc 

The distance between node m and collection point c 

(1 ) , , 1,

(1 ) , 1, ,

(1 ) , , , , 1,

, , ,

, 1,

The objective of this model is to minimize both the supply chain's cost and population exposure risk at the same time. Considered cost items are as follows: vehicle hiring cost, collection centers' set-up cost, waste processing cost, and vehicle's fuel cost with respect to the extra load being carried via vehicles. On the other hand, minimizing the risk of population exposure to pollution implies the necessity of selecting low-populated routes that are about to be covered by the vehicles carrying hazardous wastes. The point here is that, once a vehicle picks up hazardous waste (either its type is hazardous or it is being picked up from an exposed location), then the vehicle is exposed and the risk of population exposure increases accordingly. This is formulated through equations (2) and (20).

According to VRP conditions, each vehicle is allowed to visit each waste production node at most once in each time period; no vehicle may visit a waste production node more than once in a single time period. Also, if a vehicle enters a node, it should leave it after service. These conditions are considered in constraints (3) and (4), respectively. Capacity limitations for vehicles are considered in constraint (5). In other words, each purchased vehicle is allowed to collect a maximum amount of waste as much as its capacity. Capacity limitations for collection centers to prevent overloading are guaranteed by constraint (6). It means that the total amount of waste collected by vehicles and transferred to a collection center should not exceed the capacity of the collection center. The total waste produced in each waste production node in each time period should be collected. Constraint (7) provides these conditions. If a vehicle is not given to a collection center, it is not possible to carry waste from waste production nodes to collection centers. It is ensured in constraint (8). Vehicles can be assigned to collection centers if they are purchased. Constraint (9) indicate this prerequisite. Constraint (10) states that the condition for collecting waste from waste production nodes is to visit nodes by vehicles. The sub-tour elimination constraint is presented in constraints (11) and (12). In addition, constraint (12) ensures that all vehicles leaving the collection center should return to the collection center before the specified time window. Each vehicle should not be allocated to more than one collection center. This requirement is considered in constraint (13). Vividly, a collection center that is not set-up and prepared can neither accept nor process any waste, which is demonstrated through constraint (14). Weights of the loads of waste being collected by vehicles and processed in collection centers are calculated through constraints (15) and (16). The cumulative amount of wastes being carried by a vehicle into each node is calculated via constraint (17). Application of constraints (18) and (19) results in no vehicles being allowed to collect waste from the same location at the same time.

The only non-linear term is the product of two binary variables of and in the first 

Previous researchers have eagerly attempted to find the appropriate methods to solve multi-objective problems (Mardan et al., 2019; Pinto and Li, 2020; Sosa and Dhodiya, 2021) . The goal programming method is one of the most popular methods in this regard (Zandkarimkhani et al., 2020) . Every method consists of its advantages and merits. Regarding the goal programming method, it has been found that this method would be more efficient when there are several objective functions in this model; the behind it is for each one of the objective functions, one aspiration will be targeted. Considering this method, there will be an attempt to change the undesirable deviations regarding each objective function into the minimum level (Nasr et al., 2021) . One important point that should be clearly considered in this method emphasizes that the aspiration levels must be exactly determined regarding the deterministic goal programming model; however, this parameter cannot be accurately determined for real world problems, mostly due to the uncertainty variable. One solution to this problem is employing the fuzzy theory, which incorporates uncertainty (Kim et al., 2000) . Thus, we can consider the goal programming method be incorporated with the fuzzy theory to provide an efficient and practical approach in order to solve the problem goals in the presence of uncertainty (Nasr et al., 2021) . The multiobjective solution approach, which is utilized in the current study for solving the multi-objective problem, is inspired by the method proposed by Nasr et al. (2021) . The considered multiobjective solution approach is presented in the following sections.

Step 1: Defining the goals of the objective functions

At first, all decision-makers are asked to determine accurately the amounts and domains of each goal. In this regard, the optimal values of each objective function, obtained as the result of the optimization of the model regarding each individual objective function, are provided to decisionmakers and then they are asked to assign each goal to its correspondent objective function in accordance with its obtained optimal value. Here, it is assumed that and indicate the Moreover, and refer to their corresponding goals as the following:

There are two objective functions in the proposed model; both are of the minimization type, so there will be:

Step 2: Implementation of the goals of the programming model In this step, the proposed model is indicated in the form of the model of the goal programming in the following way:

System constraints in which, shows the goals' positive deviations, and indicates the corresponding negative

Step

Here, for converting the bi-objective model into a single-objective one, the method proposed by Tavana et al. (2020) was utilized in the current study.

up i   Upper bound of i   low i   Lower bound of i   i  Membership function of i   i w Weight of i  Mathematical model i i i Max w    (28) . . s t up i i i up low i i i              (29) i i i i Z i         (30)

Therefore, using a newly developed fuzzy goal programming approach, the multi-objective model, which has been proposed in the current study, was converted into a single-objective model.

It has been almost 18 months since the first patient was diagnosed with COVID-19 in Wuhan.

During this time, many efforts have been made to prevent the outbreak of this virus Hellewell et al., 2020; Chowdhury et al., 2021) . countries. In Iran, this issue has resulted in an increase in the number of patients, as well. In this regard, in cities where the number of patients has increased sharply, the government has set up temporary hospitals to provide COVID-19 patients with necessary services to supplement the work of existing hospitals. A lack of management of wastes produced in these temporary hospitals can occasion more acute conditions because these hospitals have been set up in residential areas. Accordingly, the pollution caused by their wastes is a serious danger to the residents of these areas.

The performance of the proposed model is evaluated in this section using the data related to 13 hospitals, temporary hospitals, and healthcare centers in the west of Tehran. Among the 13 waste generating nodes, 8 provide healthcare services to patients with COVID-19. Hospital wastes are also divided into 5 categories, including paper waste, non-infectious plastic waste, infectious plastic waste, metal waste, and glass waste, out of which infectious plastic and metal waste are included in the category of hazardous wastes. It should be noted that the amount of generated waste has been collected over a period of one month in this study. Each week is considered as a time period and waste collection from production centers was done on a daily basis. To collect data every day during each week, a uniform distribution was used wherein the lower limit represents the lowest amount and the upper limit represents the highest amount of waste generated per week. Also, the loading time of the waste in the waste generation nodes has been considered as a function of the amount of generated waste. Since the proposed model is a strategic-operational model, it has been implemented at the beginning of each time period (beginning of each week). The results are then applied to all days of the week, and the model must be run again with new data every week. Figure 2 shows the geographical location of 12 waste generating nodes and 3 potential collection centers. The set-up costs of collection centers are presented in Table 2 . For example, the number 265,000

in the first row and first column of this table indicates that the cost of setting up collection center 1 (Shamsabad) is $265,000. The processing costs of waste in the collection centers for each time period are presented in Table 3 . The capacity of all three potential collection centers has been considered large enough that each center alone has the capacity to collect all the waste. However, the distance of these centers from the waste-generating factories, the processing cost, and the cost of setting up the centers can distinguish them from each other. For example, as it can be seen, Chitgar collection center has a higher operating cost than Makhsoos collection center, but it has lower transportation and processing costs due to its proximity to waste generating plants and proximity to the city. Table 4 states that the features provided by maps.google.com are used to calculate the geographical and time distances and resident population between nodes. For example, the distance between Omid and Atieh hospitals is 8.5 km, which maps.google.com calculates. Vehicle characteristics, including capacity, fuel consumption, and purchase cost, are shown in Table 5 . For example, the number 4.5 in the first row and first column of this table indicates that the capacity of vehicle 1 is equal to 4.5 tons. Tables 6 to 10 show the uniform distribution function to simulate the amount of waste produced by each node. For example, U(28,35) in the second row and the first column of Table 6 represents the amount of paper waste produced by node 2 (Atieh hospital). Because node 1 is considered the collection center, the amount of waste produced by this node is considered zero.

In this section, using case study data, the performance of the proposed model is evaluated. For this purpose, we convert the bi-objective model to a single objective one by the proposed fuzzy goal programming approach. To this end, goal 1 is considered $500,000 and goal 2 is 800,000

people, respectively. The single objective model is as follows: 

(3) (20)

The optimal values of decision variables and objective functions are obtained by running the single objective model in GAMS software using CPLEX solver, which are given below ( and Table 11 . shortened, and this will lead to reduced emissions and chain costs. However, it increases the waiting time of vehicles in the nodes, and increasing the waiting time of vehicles carrying hazardous waste increases the risk of hazardous waste emissions, so considering scheduling constraints, while it may lead to increased costs, the health of the community people and the protection of the environment is more assured.

This paper presented a novel bi-objective mathematical programming model for medical waste management during the COVID-19 outbreak. The investigation of the literature shows that although this problem has been studied previously in papers such as Kargar et al. (2020) and Tirkolaee et al. (2021) , this paper pursues a different vision of the issue. Hence, according to the best of our knowledge, for the first time, this paper has considered factors such as separation of infectious wastes from non-infectious wastes in the process of collection by vehicles, reduction of waiting time for vehicles to enter waste production nodes, and failure of vehicles carrying hazardous (infectious) wastes for managing medical waste during COVID-19 outbreak.

Considering these assumptions leads to results that can be useful for decision-makers. For example, the results showed that vehicles with a lower probability of failure (higher reliability) were assigned to collect infectious wastes, and vehicles with lower reliability were allocated to collect non-infectious wastes. The second objective function also caused infectious and noninfectious wastes to be separated from each other and not loaded into a vehicle simultaneously.

Another noteworthy point is that vehicles' waiting time to enter the wastes production nodes for vehicles carrying infectious wastes was zero, which also indicates the effectiveness and efficiency of the proposed model. In order to increase robustness of the optimum solution, the following points should be considered:

 In the proposed model, a parameter called the failure probability of vehicles is defined and included in the second objective function to increase the robustness of the understudy supply chain network. This leads to allocate vehicles that have higher reliability to collect infectious waste.  One factor that increases the probability of disruption in the network is assigning loads in excess of the capacity of the centers and vehicles. For this purpose, the capacity of centers and vehicles is defined deterministically to increase the robustness of the supply chain network under consideration.  One of the practical strategies in collecting hazardous waste is collecting waste during low traffic hours. Observations made in the understudy supply chain show that the transfer time between the two desired hospitals, during low traffic hours, is almost the same in different periods and has a maximum variance of 4%. Accordingly, in the proposed model, to provide service at certain times (low traffic hours), a parameter called time window is defined, which leads to increased robustness of the supply chain network.

This section aims to investigate the accuracy of the behavior and performance of the proposed model and create a Pareto frontier using different scenarios. A sensitivity analysis of objective function coefficients allows decision-makers to access compromise solutions without relying on extreme points. It is expected that by increasing and decreasing the coefficient of an objective function, this objective function's value does not get worse or better, respectively. Table 12 represents the scenarios and objective functions values for each scenario. Also, in Figures 3 and   4 , the first and second objective functions values for each scenario are illustrated, respectively. The Pareto frontier obtained from different scenarios is also shown in Figure 5 . Pareto frontier presented in Figure 5 also allows decision-makers to make optimal decisions according to the real world's requirements. Decision-makers who care about the health of people in the community do not pay attention to costs and choose scenarios that have the lowest risk to the population at risk of infection, but uncommitted decision-makers, regardless of the population at risk, choose the scenarios that impose the least costs to the chain.

This paper presents a flexible and practical model for medical waste management during the COVID-19 pandemic. The results of the proposed model help managers and decision-makers to adopt a scenario that imposes the least risk to the network given the budget. One of the unique features of this model is the consideration of reliability for vehicles. This issue leads to transfer infectious waste by highly reliable vehicles; in this case, the risk of population exposure to pollution is reduced. Also, to make appropriate decisions, decision-makers can identify low traffic hours and set a specific time window for collecting waste to increase the chain's efficiency and to reduce the fuel consumption of vehicles because traffic leads to increased fuel consumption. In addition, vehicle scheduling reduces vehicle waiting time on waste generation nodes, reducing population risk and vehicle fuel consumption. Due to its comprehensiveness, the proposed model is highly flexible and can be implemented in other supply chain networks by applying minor changes. For example, the proposed model can be used to collect hazardous industrial materials. Also, by eliminating the second objective function and constraint (20), the proposed model becomes a more general and flexible model that can be used to manage MSW and even purchase products from suppliers or collect returned products from customers to be used.

Medical waste management during COVID-19 outbreak is of great importance; its inefficient management, in addition to harming the environment, also endangers human lives. Therefore, this paper presents a bi-objective MILP model for managing both infectious and non-infectious waste under uncertainty. The proposed model aims to minimize total costs and risk of population exposure to pollution simultaneously, and a fuzzy goal programming approach was used to solve this model. The location-routing problem, vehicles scheduling, load-dependent fuel consumption, split delivery, population risk, and time window are some of the practical assumptions of the proposed model. For better management of medical wastes, factors such as separation of infectious wastes from non-infectious wastes in the process of collection by vehicles, reduction of waiting time for vehicles to enter wastes production nodes, and failure of vehicles carrying infectious wastes for the first time in this paper was considered. Finally, the validation of the proposed model and solution approach was evaluated using Tehran municipality actual data.

In addition to its advantages, each paper suffers from some limitations that may pave the way for future research. Accordingly, some of the following researches are suggested:  Adding treatment, recycling, and disposal centers to the proposed network;

 Considering social issues such as job creation in order to design a sustainable network for medical waste management;  Developing a meta-heuristic or heuristic algorithm for solving proposed problem in large scales and using big data (Govindan and Gholizadeh, 2021) .

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The routes traveled by each vehicle in each time period are presented in Table 11 . For example, the first row of this table states that vehicle 1, in time period 1, starts from the Chitgar collection center and first goes to the West health center. After collecting waste from this node, it visits West Nikan hospital, Eram clinic, Peyman clinic, and Treata hospital, respectively, and finally returns to Chitgar collection center. It should be noted that this vehicle only collects hazardous