key: cord-299889-x6c3p195
authors: Tirkolaee, Erfan Babaee; Abbasian, Parvin; Weber, Gerhard-Wilhelm
title: Sustainable fuzzy multi-trip location-routing problem for the epidemic outbreak of the novel Coronavirus (COVID-19)
date: 2020-11-10
journal: Sci Total Environ
DOI: 10.1016/j.scitotenv.2020.143607
sha: 
doc_id: 299889
cord_uid: x6c3p195

The performance of waste management system has been recently interrupted and encountered a very serious situation due to the epidemic outbreak of the novel Coronavirus (COVID-19). To this end, the handling of infectious medical waste has been particularly more vital than ever. Therefore, in this study, a novel mixed-integer linear programming (MILP) model is developed to formulate the sustainable multi-trip location-routing problem with time windows (MTLRP-TW) for medical waste management in the COVID-19 pandemic. The objectives are to concurrently minimize the total traveling time, total violation from time windows/service priorities and total infection/environmental risk imposed on the population around disposal sites. Here, the time windows play a key role to define the priority of services for hospitals with a different range of risks. To deal with the uncertainty, a fuzzy chance-constrained programming approach is applied to the proposed model. A real case study is investigated in Sari city of Iran to test the performance and applicability of the proposed model. Accordingly, the optimal planning of vehicles is determined to be implemented by the municipality, which takes 19.733 hours to complete the processes of collection, transportation and disposal. Finally, several sensitivity analyses are performed to examine the behavior of the objective functions against the changes of controllable parameters and evaluate optimal policies and suggest useful managerial insights under different conditions.

The most recent epidemic outbreak that caused a pandemic and public health emergencies is the spread of the novel coronavirus disease , threatening the health of the world population (Kargar et al., 2020) . The COVID-19 has quickly spread from Wuhan to other regions and influenced more than 200 countries throughout the world by the end of March, 2020 (Ahmadi et al., 2020 Govindan et al., 2020) . It is now a pandemic due to the rapid rise in the number of infected people and the lack of initial attention to the COVID-19 pandemic from world's leaders (Mardani et al., 2020) . So, we have encountered a critical and challenging period for the control and prevention of the pandemic. One of the serious concerns about the outbreak of COVID-19 is the incubation period, which fluctuates from 3 to 14 days (Issa and Elaziz, 2020) . It is obvious that the spread of COVID-19 may be increased through an inefficient waste management system (Sarkis et al. 2020 ).

In line with the significance of providing medical items, another substantial issue is handling the infectious medical waste related to COVID-19 generated in diagnosing and treating patients at health centers including hospitals and infirmaries. With the growing rate of confirmed cases, the amount of medical waste related to COVID-19 increases significantly, which is now considered as critical hazardous materials (HAZMAT). In other words, medical waste disposal is regarded as a significant way to handle the source of infection, strict establishment and standardization of the waste management of COVID-19 (Peng et al. 2020 ). hospitals and infirmaries), transportation (through the road network) and disposal of the COVID-19 related waste at pre-established disposal sites.

The main idea is to provide a decision support system (DSS) ensuring that COVID-19 related medical waste is timely, regularly, harmlessly and effectively is disposed by considering sustainable development. To this end, the sustainable multi-trip location-routing problem with time windows (MTLRP-TW) is introduced to address the collection, transportation and disposal processes considering the priorities of services and available budget of the system.

Here to address the sustainable development, the objectives are defined to concurrently minimize the total traveling time of waste-collection vehicles, total violation from time windows (service priorities) and the number of people live around disposal sites.

Accordingly, a novel MOMILP model is developed to formulate the problem and then to be validated using a real-life case study problem.

The location-routing problem (LRP) is an extension of the classic routing problem that integrates the strategic and operational decisions by facility location problem (FLP) and vehicle routing problem (VRP), respectively. Each of these problems has been frequently investigated in the literature as can be seen in Erkut et al. (2008) and Tirkolaee et al. (2018 Tirkolaee et al. ( , 2019 Tirkolaee et al. ( , 2020a . Due to the high application of LRP in supply chain and waste management systems, it has been studied by many researchers in different cases (Drexl and Schneider, 2015) . To be more specific, Zografros and Samara (1989) , as one of the pioneering research, suggested an LRP model for transportation and disposal of HAZMAT considering three minimization-type objectives of routing risk, disposal risk and travel time. They applied a goal programming (GP) approach to solve the model. (2007) Das et al. (2012) designed a multi-objective framework for routing of HAZMAT between generating nodes and disposal sites with the aim of total transportation cost and risk minimization. They conducted a real case study using posteriori technique with multi-objective programming approach to provide non-dominated solutions for the waste management system. A multi-objective mixed-integer linear programming model (MOMILP) was designed by Samanlioglu (2013) to address the industrial HAZMAT LRP. The objectives were to minimize total cost, total transportation risk and total risk for the population around treatment facilities. They investigated a real case study in Turkey using lexicographic weighted Tchebycheff formulation and CPLEX software.

Zhao and Ke (2017) analyzed the incorporation of inventory risks in LRP for explosive waste management. They developed a bi-objective model to concurrently minimize total cost and total risk. They investigated some numerical experiments using real-world data in China.

Aydemir-Karadag (2018) offered a profit-oriented model for HAZMAT LRP considering energy recovery and the application of polluter pays principle. She tested the applicability of the proposed model using hypothetical problem instances based on a real-life case study. Two meta-heuristic algorithms were proposed by Rabbani et al. (2018) to tackle an industrial HAZMAT LRP considering incompatible waste types. The objectives were to simultaneously minimize total cost, total site risk for people and total transportation risk. Beneventti et al. To the best of our knowledge, there is no study yet dealing with the efficient treatment of COVID-19 related medical waste at the operational level; i.e., in terms of timely collection, J o u r n a l P r e -p r o o f transportation and disposal within a waste management system. Moreover, this is the first study that introduces the MTLRP-TW under uncertain conditions. Therefore, due to the instability and uncertainty of the demand parameter, fuzzy chance-constrained programming approach is applied. Furthermore, a weighted goal programming (WGP) technique is then implemented to deal with the multi-objectiveness of the model.

This section describes the suggested methodology of the study to establish an efficient waste management system during the COVID-19 pandemic. To have an overall view, Figure 1 represents the execution steps.

Consider a network including parking site, demand nodes (hospitals and infirmaries) and preestablished and post-established disposal sites. The aim is to make locational and routing decisions under a specific situation imposed by the COVID-19 pandemic. Accordingly, at the first stage, the required additional disposal sites are established at the beginning of the time horizon, which are called post-established disposal sites. Vehicles routing plan is made at the second stage such that a fleet of vehicles is considered to start their first trip from the parking and end it at one of the available disposal sites. Furthermore, the next possible trips of these vehicles start from that disposal site and end at available disposal sites. It means that the destination and departure can be different, so the concept of "intermediate depots" is associated. For more information, see Tirkolaee et al. (2020b) .

According to this pandemic situation, it is necessary that the waste should be instantly collected, transported and disposed. Each demand node has its own service time window that should be served. So, the objective functions are to simultaneously (i) minimize the total traveling time, (ii) minimize total violation from time windows and (iii) minimize total infection/environmental risk imposed on the population around disposal sites. III. Each vehicle has a maximum service time.

IV. Demand parameters are considered as triangular fuzzy numbers.

V. There are both pre-and post-established disposal sites that have limited capacity.

VI. Candidate disposal site can be established just at the beginning of the planning periods.

VII. Each demand node should be served only by one vehicle.

VIII. From second trips onwards, vehicles may unload the collected waste in a different disposal site from which the trip has been already started. Now, the mathematical notations of the proposed model including sets and indices, parameters and variables are listed as follows.

Set of nodes ( ); ; here, 1 represents the parking, (1)

J o u r n a l P r e -p r o o f (10) ,

Objective function (1) minimizes the total traveling time of vehicles. Objective function (2) minimizes the total violation from time windows defined by demand nodes. Objective function (3) minimizes the disposal sites risk; i.e., the number of people around disposal sites is minimized. Constraint (4) represents the flow balance equation for each node. Constraint (5) guarantees that each demand node should be served in each period. Constraint (6) indicates the capacity limitation of each vehicle in each trip. Constraints (7) and (8) (19) and (20) guarantee that vehicles should start the first trip from parking and end it at a disposal site, respectively. Constraints (21) and (22) ensure that vehicles start the potential next trips from the disposal site (as the final node of its J o u r n a l P r e -p r o o f Journal Pre-proof first trip) and end it at a disposal site, respectively. Constraints (23) and (24) guarantee that vehicles should move back to the parking in their last trip to complete their tour in each period. Constraint (25) indicates the budget limitation of the waste management system. Constraint (26) expresses that vehicles can construct a route only when they are already assigned. Constraints (27) and (28) show the types of the variables. 

To deal with the uncertain nature of parameters and to develop a more realistic model, fuzzy mathematical programming is applied as an efficient approach (Tirkolaee et al., 2020c) . Here, constraints. The triangular fuzzy numbers have the appropriate applicability to cope with data which suffer from accuracy or information (Li et al., 2012) . If ̃ is taken into account as a triangular fuzzy number and the confidence level (ρ) is greater than 0.5, considering the confidence level of the fuzzy number versus the random number r, we have:

Eqs. (35) and (36) In the proposed model, demand parameter ̃ is the uncertain parameter of the model as an independent triangular fuzzy number. Now, the constraints that include this parameter are rewritten based on the fuzzy chance-constrained programming model. Therefore, Eqs. (6) and (7) are reformulated based on the chance-constrained planning approach.

Now, based on Eqs. (35) and (36), Eqs. (37) and (38) are defuzzified. It is noticeable that Eq.

(38) should be regarded as two inequalities: 

One of the most attractive multi-objective programming approaches is GP that was introduced by Charnes and Cooper (1977) . This method addresses optimization problems with multiple conflicting objectives. The main advantage of GP over other multi-objective programming techniques is the concurrent consideration of different objectives, and also permissibility of deviation from ideal objectives (goals) which makes the decision-making process flexible.

On the other hand, since we usually encounter multiple objectives with multiple units and importance degrees, it is required to normalize the objective function of GP and assign weights to the objectives to tackle the importance levels. Accordingly, WGP is proposed with the following mathematical structure: 

This section investigates the validation of the proposed model using a real case study problem in Sari, the capital of Mazandaran province, Iran. According to Figure 3 , 1 pre-established disposal site (node number 1) and 3 candidate locations (nodes numbers 2, 3 and 4) are given in the map. It should be noted that the candidate locations are adapted from the study conducted by Lahmian (2018). He evaluated the potential locations to establish disposal sites considering the criteria of i. geology, ii. land use, iii. slope, iv. vegetation, v. access roads, and vi. distance from towns of Sari city. Finally, the land use and distance from towns of Sari city were determined as the main effective criteria.

Moreover, based on Figure 4 , 16 hospitals and infirmaries are distributed within the city networks which generate COVID-19 related medical waste. Moreover, 4 vehicles are available at the parking site. The confidence levels of Eqs. (6) and (7); i.e., and are set to 0.7. The maximum available time for vehicles and available budget are set to 480 minutes and 1 million USD, respectively. Furthermore, = 0.8 and = 1.2 .

Sciences (2020).

To obtain the values of the triple goals, the single-objective model is separately solved by each objective function. The proposed model is implemented using CPLEX solver/GAMS software. Table 1 represents these values. Now, the proposed MILP model in Section is implemented to attain the optimal policy for the case study problem and evaluate its performance and complexity. vehicles. Moreover, the total violation time is equal to 2.982 hrs within a working weak and total infection risk size is 13214 people. Since the pre-established disposal site is also used for household waste, the 4 th candidate location is established. Table 3 represents the optimal routing plans of vehicles in the first time period. Now, to investigate the effects of key parameters on the objective functions, a sensitivity analysis is performed on the confidence levels ( , ) and budget level of the waste management system (ϒ). The obtained results are given in Tables 4 and 5 and Figures 5 and   6 .

As can be seen in Figure 5 , the objective functions reflect a direct behavior against the increase of the confidence levels. In other words, the objective functions grow with a different range of fluctuations in various change intervals. It means that reaching high confidence levels needs more resources and management should analyze and consider these behaviors to prevent any potential failures in the system.

On the other hand, according to Figure 6 , the objective functions show an indirect behavior against the increase of budget level. As an interesting point, the problem becomes infeasible J o u r n a l P r e -p r o o f for the 20% decrease in the budget level and the waste management system cannot provide a solution. So, it is indispensable that management considers the amount of available budget as an effective and critical parameter, and provides the required level under different real-world conditions, particularly for the COVID-19 pandemic.

The COVID-19 pandemic is still far from the over, and it is not certain that the future trends would confront any region of the world. Getting closer to the peak of the pandemic, waste management has not been receiving the priority which is required to minimize the detrimental impacts on the health and environment. The collected information from Mazandaran University of Medical Science and Sari Municipality demonstrates a variable trend on the decreasing or increasing in COVID-19 related medical waste amount. Designing and establishing an efficient DSS for controlling this problem can help the Sari city and provide a useful example for other cities and countries so that they can handle the spread of the disease using the same DSS. Hence, this study tried to efficiently collect, transport and dispose the COVID-19 related medical waste by modeling the problem as an MTLRP-TW.

The following main conclusions are yielded according to the numerical results of this study:

(1) A novel MOMILP formulation was proposed for the MTLRP-TW considering real-life assumptions for the waste management application, such as multiple planning periods and separate locations for the disposal sites and parking,

(2) Fuzzy chance-constrained programming was utilized to study the significant uncertainty of demand parameter that denotes the amount of COVID-19 related medical waste generated at hospitals and infirmaries, Parking-3-5-10-9-DS1 DS1-1-11-2-4-6-DS4 J o u r n a l P r e -p r o o f 

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