Reverse Logistics Network Design to Estimate the Economic and Environmental Impacts of Take-back Legislation: A Case Study for E-waste Management System in Washington State

Due to the increasing environmental concerns, resource reduction, and depleting landfill capacities in many countries, optimizing reverse logistics operations for returned and end-of-life products receive growing attention in the last decades [44]. Effective implementation requires the establishment of suitable logistical systems for product flow from consumers to producers [15]. The design of product recovery networks is one of the important and challenging issues in RL. In a RL network, used products originate from multiple sources and are transported to various types of recovery facilities. Typically, these networks are represented as mixed integer programs and the models are solved either to optimality using commercial solvers or through heuristic approaches. Examples of papers following this approach include [15, 52, 24, 33]. The physical location of facilities and transportation links need to be chosen to convey used products from their former users to a producer and to future markets again [15]. developed a model for optimizing the collection and recycling processes of end-of-life (EOL) computers and home appliances in Taiwan. [22] proposed a cost-minimization model for a multi-time-step, multi-type hazardous-waste RL system. They presented application cases to demonstrate the feasibility of their proposed approach. [27] presented a stochastic programming-based approach by which a deterministic location model for product recovery network design may be extended to account explicitly for uncertainties. They applied it to a representative real case study on recycling hazardous products, reusing products, and full recycling in Turkey. [33] determined the number and location of centralized return centers using a nonlinear mixed-integer programming model and a genetic algorithm that solved the RL problem involving product returns. [30] developed a location-allocation model to determine the number and locations of the waste disposal plants in the region of Guangzhou (China) while [7] focused on selecting the locations of municipal waste collection points in Barcelona.[26] presented a development of genetic algorithm (GA) model for recycling spent batteries.[17] developed a mixed-integer linear programming (MILP) model to maximize the profit of a reverse logistics network and presented a case of mobile phones and digital camera remanufacturing. [48] developed a mixed integer nonlinear programming (MINLP) model to maximize the profit of reverse logistics network and presented a case of truck tire remanufacturing. [57] developed a multi-objectives MILP model for network design for regional hazardous waste classified by certain properties, including treatment technology, industrial distribution, and hazardous characteristics. [38] proposed a mathematical model for the design of a reverse logistics network that consisted of retailers, warehouses, reprocessing centers, remanufacturing factories, distribution and recycling centers, spare markets, disposal sites, and suppliers. They considered modular product structures with different disposal and recycling fractions for each module of each product in the model. [46] analyzed the feasibility of a sustainable RL adopted by phone manufacturing industries, their results show that the refurbishing process stops waste disposal of 4.5 tons of Electrical and electronic waste (EEW) in Brazil. [55] proposed a multi-period multi-objective MILP for a medical waste reverse logistics network to responsively address the rapid growth of medical waste within the planning window during the COVID-19 in keeping with reducing the risk of epidemic spread.

In addition to the studies provided in this section, readers are referred to [41] for a comprehensive survey on reverse logistics network design models.

[3] studied the solid WMS of Central Portugal and develops a MIP model, combining elements of a p-median model and of a capacitated facility location model with transshipment, in which a maximum distance between sources and transfer stations, and between transfer stations and landfills is imposed. [34] studied a three-echelon network design problem with central treatment facilities, transfer stations and landfills, where the goal is to minimize the total cost of the solid WMS. [18] studied the problem of minimizing the total number of collection sites to be located in a SWM system, chosen among a set of candidate locations. Such an objective ensured not only the reduction of the impact due to the presence of the collection bins close to the residential sites, but also the reduction of the overall cost related to the collection phase. [28] proposed a life-cycle optimization framework for municipal solid waste management. Their developed framework was built on a mixed-integer programming problem of the waste management supply chain and was capable of identifying the optimal waste management infrastructure under a variety of objective functions and constraints. [47] illustrated a three-stage MINLP-based approach helping companies in Mexico to locate plant capacities and material flows. [4] evaluated the location planning of a centralized WMS in rural areas of developing countries to optimize material and energy recovery costs. The configuration in this study includes technologies such as reuse (or recycling), landfilling, refuse-derived fuel, and composting. [53] presented a system dynamics model for municipal solid WMS in Vietnam by proposing a model aiming at analyzing the contribution and activities of the informal sector, focusing on its roles and impacts. [45] presented a novel real-time WMS in smart cities based on the Internet of things aiming at minimizing pollution, and optimization of value recovery with considering the uncertainty of waste flow in separation centers. [51] presented a model for municipal solid WMS in Fayoum Governorate (Egypt) by proposing a MILP model aiming at minimizing the net daily cost, determining the best location for collection sites among a given set of candidate locations, and optimal flow of waste of the system.

In this category, [6] holistically identified efficiency conditions for extended producer responsibility types of legislation and discusses the economic and environmental impacts while characterizing the right policy for take-back. [54] develop a decentralized coordination mechanism for allocation of WEEE and choice of disassembly levels to independent WEEE recycling companies. [19] developed an approach providing an efficient assignment for all involved actors in WEEE program (e.g. the producers, the municipalities, and the collective schemes) by presenting an optimization-based decision support tool for the coordinating government institution in Denmark. [10] formulated an optimization model for optimal production and carbon emission reduction (CER) rate decisions. They analyzed how carbon taxes and take-back legislation affect firms’ production and CER decisions. [29] presented a single-period stylized model for manufacturers to maximize their profit, considering mandated collection and recycling target rates for production, and recycling strategies. Their study shows that strict mandated collection legislation may not always have a negative impact on profits. However, more stringent mandated legislation may bring undesirable outcomes, because the recycling rate is not affected by extreme recycling benefits. [39] developed a multitier e-cycling network model with a variational inequality formulation in a Cournot game. The formulation provides endogenous equilibrium prices and material flows between tiers. [21] extended the work of [39] and proposed a closed-loop supply chain model for WEEE in a Cournot pricing game with perfect information. They numerically found that minimum recovery targets stimulate manufacturers’ reverse chain activities, which contradicts later findings by [29], regarding disincentivization of manufacturers to implement green product design whose maximum recycling benefits are either too low or too high.

Policy makers aim to design public systems that operate cost-effectively to perform a necessary function. This translates to a perspective in which optimizing the entire system with respect to some criteria defined over all components of the system is a logical goal, if a reasonable representation of the system under investigation could be modeled. In other words, the policy makers prefer to have a system with all components operating optimally. To meet such a requirement, we develop a mixed integer linear programming (MILP) model that optimizes the total cost incurred and total emission generated through the e-waste reverse logistics network illustrated in Figure 6. Under the system-optimum model, all the components in the network are committed to execute the optimal decision determined by the policy maker. Recall that, in this system, residents drop off e-waste items at one of the open drop-off sites using their personal vehicles. E-waste items are then transported from the respective drop-off sites to one of the available primary processors. Afterward, recovered materials and waste streams are sent from each primary processor to the appropriate secondary processing facilities or recovered material re-processing facilities.

The system optimum model developed in the previous section provides optimal device and material throughputs for each node in the network for the defined system in terms of the selected objective (i.e., least cost, least emissions). The least environmental impact solution may be optimal from the policy maker standpoint, whereas from the perspective of manufacturers paying for an EPR program, the least cost solution is likely to be the preferred solution.

Regardless of the objective function used, in order to minimize total cost or emission of the network in the system optimum model, the residents of an area are assigned to a drop-off site to deliver e-waste products. However, there is no guarantee that the residents necessarily follow what the policy-maker enforces in practice. A resident may decide to go to the closest drop-off site in reality regardless how much the total cost and emission of the reverse logistics network are impacted by such a decision. However, the drop-off sites that are the closest to residence areas may not be assigned by the model when minimizing total cost and emission through the entire logistics network. In order to address this situation, a user-optimum model is designed in this section, which consists of the following two separate programming models:

• 1.

  Model (User, I) – In this model, the residence areas are assigned to the closest drop-off sites to deliver their e-waste products.

• 2.

  Model (User, II) – The collected e-waste products in drop-off sites from Model I are transported to the primary and secondary facilities for further processing.

Clearly, the optimal solutions obtained from these two separate models together form a sub-optimal solution to the system optimum model.

The multi-objective problem programming model presented in section 4.2 aims to find a balance between economic, environmental, and social sustainability. To address this multi-objective problem, the $\epsilon$-constraint method is chosen for its simplicity and wide use in similar problems. This method has been successfully applied in various multi-objective scenarios in the past [13, 20, 36].

The proposed user model aims to minimize both total cost and $CO_{2}$ emissions and can be summarized as follows:

The proposed user model involves minimizing user total cost ($TC_{user}$) and $CO_{2}$ emissions ($Em_{user}$), represented by the decision variables in the vector $x$, within the feasible solution space ($S$). Using the $\epsilon$-constraint method, the multi-objective problem is transformed into a single objective programming model by selecting one of the objective functions as the primary objective and expressing the remaining objectives as constraints with defined bounds [32]. If the total user cost objective (36) is chosen as the primary objective, the resulting single objective programming model includes the total $CO_{2}$ emission objectives as the constraint.

Where $v=0,1,...,V$ and $\Delta\epsilon_{Em}=\frac{Em_{user}^{max}-Em_{user}^{min}}{V}$.

To determine the minimum values for the total $CO_{2}$ emission and objective function, the following steps are followed:

1. 1.

   Find the optimal solution for each objective function in $S$. Then, create a set containing the optimal solutions for the $TC_{user}$, and $Em_{user}$ objective functions, represented as $X_{TC}^{*}$, and $X_{Em}^{*}$ respectively, we call the solution space $\Re=\{X_{TC}^{*},X_{Em}^{*}\}$.

2. 2.

   Find the values of objective function $TC_{user}$ for $X_{Em}^{*}$, and $Em_{user}$ for $X_{TC}^{*}$.

3. 3.

   Find the minimum and maximum value of $Em_{user}$ as $Em_{user}^{min}=min\{Em_{user}(x),x\in\Re\}$ and $Em_{user}^{max}=max\{Em_{user}(x),x\in\Re\}$

This paper employs the modification suggested by [31] to ensure that the optimal solution of equation (5.1) is also a Pareto optimal solution for the original multi-objective problem outlined in equation (5.1). This is achieved by converting the constraints associated with the added objective functions into equalities by introducing slack variable ($S$) and then incorporating them as penalties in the single objective function. As a result, equation (5.1) is transformed into the following model:

$s.t.$  (10)-(24)

The value of $\theta$ is set to a sufficiently small number (typically between $10^{-3}$ and $10^{-6}$) that does not impact the objective function. By solving the current single-objective programming model with a particular value of $v$, a Pareto optimal solution for the original multi-objective problem (5.1) is obtained. This process is repeated for multiple values of $v$, resulting in a set of $v+1$ Pareto optimal solutions that defines the Pareto front.

In this section, an overview of the robust optimization approach proposed by Bertsimas and Sim (2004) is presented. To do so, the following linear programming model is considered:

where the technological coefficients $\tilde{a}_{ij}$ are assumed to be uncertain. In other words, each coefficient $\tilde{a}_{ij}$ is regarded as an independent, symmetric and bounded parameter, which can take values in $[a_{ij}-\hat{a}_{ij},a_{ij}+\hat{a}_{ij}]$, i.e. $\tilde{a}_{ij}\in[a_{ij}-\hat{a}_{ij},a_{ij}+\hat{a}_{ij}]$. In this definition, $a_{ij}$ and $\hat{a}_{ij}$ denote the nominal value and the maximum deviation from the nominal value, respectively. Associated with each row i in problem (1) is $J_{i}$, which is defined as the set of all coefficients in row i that are subject to uncertainty. Furthermore, a scaled deviation $\eta_{ij}\in[-1,1]$ is defined for each uncertain coefficient $\tilde{a}_{ij}$ as $\eta_{ij}=\dfrac{\tilde{a}_{ij}-{a}_{ij}}{\hat{a}_{ij}}$ that represents the scaled perturbation of $\tilde{a}_{ij}$ from its nominal value $a_{ij}$.

Bertsimas and Sim (2004) also introduced a parameter $\Gamma_{i}\in[0,|J_{i}|]$ as the budget of uncertainty for each constraint i, where $|J_{i}|$ denotes the number of elements of set $J_{i}$. In fact, $\Gamma_{i}$ is the maximum number of parameters that can really deviate from their nominal values for each constraint i. The parameter $\Gamma_{i}$ that bounds the total scaled deviation of uncertain parameters as $\sum_{j\in J_{i}}|\eta_{ij}|\leq\Gamma_{i}$ adjusts the robustness of the proposed method against the level of solution conservatism. In particular, $\Gamma_{i}=0$ represents the nominal or deterministic formulation, whereas $\Gamma_{i}=|\eta_{ij}|$ relates to the worst-case formulation in which all uncertain parameters are fixed at their worst-case values from the uncertainty set. However, decision maker can make a trade-off between the protection level of constraint i and the degree of conservatism of the solution if $\Gamma_{i}\in(0,|J_{i}|)$. Therefore, the budget of uncertainty $\Gamma_{i}$ that is an input to the robust optimization model can specify how risk averse the decision-maker is.

Bertsimas and Sim (2004) proposed a nonlinear programming model as follows, which is equivalent to the the uncertain model (1):