Sustainable Hybrid Manufacturing and Refurbishing Systems with Substitution

Abstract

This study considers the planning decisions in a production facility that receives returned products as a contribution to sustainability through the reduction in waste from used products. It involves three processes: the refurbishing and disassembly of returned items and the manufacturing of new items. The process is driven by the demands of new items and collected secondhand items. In this study, we consider a downward substitution, in which new items could be used to meet the demand for secondhand items. The objective is to determine the best production planning schedule to satisfy all demands while minimizing the total costs of production and inventory. We propose and analyze different configurations distinguished by their level of integration and whether they allow substitution. Integration is mainly achieved through the sharing of the inventory of returns between the refurbishing line and the disassembly line, which responsible for feeding the manufacturing process with “as good as new” parts. The best configuration is identified based on the total cost and environmental impact. Four heuristics based on combining relax-and-fix and fix-and-optimize approaches are proposed. Numerical experiments have shown that the heuristics are very efficient, achieving gaps of less than 1% from the optimum in short CPU times for most instances. Numerical experiments have also shown that integration through the sharing of returns inventories leads to considerable cost and environmental benefits.

1. Introduction

When products have a short useful life, especially electronic products, often driven by rapid technological advancements, they are quickly rendered obsolete as newer, more advanced models become available [1]. This obsolescence accelerates the disposal of older products, leading to increased waste generation. The rapid pace of technological innovation encourages consumers to frequently upgrade to the latest versions, contributing to a cycle of continuous consumption and disposal. Consequently, this cycle not only generates substantial amounts of electronic waste but also strains natural resources due to the constant production of new devices. The environmental impact is further compounded by the challenges in recycling and properly disposing of complex electronic components. Therefore, there is a critical need for sustainable practices to mitigate environmental damage. One of the impactful references in sustainable manufacturing practices concept is [2], which reviews and evaluates the new sustainability assessment techniques for manufacturing and makes recommendations based on the concept of sustainability.

Product recovery is one of the key strategies in sustainable manufacturing and waste management. It encompasses processes such as recycling, refurbishing, remanufacturing, and reusing products at the end of their life cycle. By recovering products, manufacturers can reclaim valuable materials and components, reducing the need for raw material extraction and minimizing environmental impact. This approach reduces landfill waste, lowers production costs, and conserves energy, resulting in better financial and environmental sustainability. Sustainable IT (also known as Green IT or Circular Computing) is one of the most environmentally friendly methods of reusing discarded user electronics [3].

Refurbishing is the process of restoring end-of-life products to a like-new condition through a series of actions such as cleaning, repairing, replacing damaged parts, and updating software. This practice extends the useful life of products, making them suitable for reuse and reducing the demand for new products. Some original equipment manufacturers (OEMs) may transform traditional production systems into hybrid production systems that combine manufacturing and refurbishing activities. By integrating refurbishing within manufacturing, companies can reduce costs, conserve resources, and meet sustainability goals while providing high-quality, reliable products to consumers. This synergistic method leverages the strengths of both refurbishment, which emphasizes reuse and sustainability, and manufacturing, which focuses on innovation and efficiency, to create a more sustainable and economically viable production system.

Integrating refurbishing and manufacturing in hybrid systems brings greater difficulties than traditional manufacturing systems. One primary difficulty is the complexity of coordinating and managing two distinct processes within a single operational framework. Another challenge is the need for a skilled workforce capable of performing both refurbishing and manufacturing tasks, which often require different sets of expertise and training. Lastly, market acceptance can be a barrier, as customers may have varying perceptions of the value and reliability of refurbished versus manufactured products, necessitating clear communication and marketing strategies to build trust and demand.

This research is motivated by real cases at a large company specializing in the production of electronic products. Its products range from smartphones and TV sets to large household appliances. The authors had several meetings with the management of the company to explore potential collaboration projects. These meetings have led to the consideration of the current project as a priority. The collaboration was limited to sharing data about the processes and some strategic plans without any funding or particular deliverables from the authors. Other projects discussed with the company concerned mainly the analysis and improvement of their management information system. The company invested in a large new assembly line of smartphones. However, due to a shortage of imported components, and to avoid under-utilization of the line, the company seeks to use it in order to develop a sustainable supply chain through the recovery of used phones. They aim to establish an integrated hybrid manufacturing and refurbishing system, although this approach significantly complicates the process and necessitates sophisticated planning systems. If a key part of the phone is used in the assembly of a new phone, it entails a thorough check of the part. In the case of smartphones, the key part is the screen assembly, which can have a value as high as 80% of the total price of the phone. The price of a high-end phone from the company is about USD 250 while the price of its screen assembly can be as high as USD 200. Thus, the purpose of our collaboration with the company is to design an efficient sustainable supply chain by proposing and modeling several configurations. We started with a basic configuration, as shown in Figure 1, which is the current idea of the company, where the new line will be dedicated to refurbishing. This configuration is then extended by proposing several configurations with different levels of integration between the refurbishing, disassembling, and manufacturing processes, and whether allowing the downward substitution of used products with new ones.

The objective is to satisfy the deterministic dynamic demands (for both used and new products) over the time horizon in a way that minimizes the total costs. These costs consist of processing/purchasing, holding, and setup costs throughout the hybrid system, including the following processes: collection, disassembling, manufacturing, and refurbishing. This is achieved by deciding, in each period, the output quantities of each process, as well as the quantities to be purchased and substituted.

The first challenge addressed in this study is the need to propose several realistic process configurations to discuss with the company and to assess based on economic and ecological criteria. The second challenge is the development of efficient optimization techniques to solve the corresponding planning problems. We made the decision to start with the existing company configuration and gradually increase its complexity by adding the substitution and integration of refurbishing within the manufacturing process. For the development of solution approaches, the focus was mainly on the design of MILP-based heuristics, which are easier to understand for the practitioner and are easy to parameterize. We show that the configurations with the allowed substitution and the integration of refurbishing within the manufacturing process have a higher value from an economic and ecological point of view, resulting in higher sustainable manufacturing processes. The proposed solution approaches, based on the relax-and-fix and fix-and-optimize ideas, are all efficient, but the time-based decomposition approach seems to be the best compromise between CPU time and solution quality.

In this study, several managerial insights are provided from different configurations, focusing on their environmental impact and total costs. The study also analyzes the economic and sustainability advantages of the substitution between used and new products and the integration of returns in both the manufacturing and refurbishing processes. Additionally, efficient heuristics are proposed to solve the more complicated configurations.

The rest of the paper is structured as follows. Section 2 presents the main recent research related to our study. The description and the MILP formulation for different configurations are presented in Section 3. Section 4 presents variants of the full configuration. Section 5 details the proposed heuristics. Numerical experiments are presented in Section 6. Finally, Section 7 presents our concluding remarks.

2. Literature Review

This section provides the main recent research related to our study, which is an extension of the lot-sizing problem with returns and substitution. The lot-sizing problem is one of the most famous problems in production planning and inventory management and has been extensively studied over the years to address new realities and challenges. This literature review attempts to discuss three main research streams: lot-sizing problems with returns, lot-sizing problems with substitution, and studies with returns and substitution.

This study is an extension of the classical lot-sizing problem. The earliest study on the dynamic lot-sizing problem is the seminal paper by [4]. Over the years, various variants, such as lot-sizing models that include limited capacity, multi-item, multi-level, parameter uncertainty, pricing, non-linear costs, and backlogging, have been developed to address more complex and realistic scenarios encountered in various industries. For a comprehensive review, we refer the reader to [5,6].This extension addresses problems where returned products need to be reintegrated into the production system. It involves optimizing the handling of returns, refurbishing or remanufacturing returned items, and reintroducing them into the production and distribution cycle. This adds a layer of complexity and requires careful planning to manage inventory levels and costs effectively. Lot-sizing problems involving returns can be classified into continuous and discrete models. Continuous models were developed by [7,8,9,10,11]. The discrete models, which include returns, were investigated by [12,13,14,15,16].

Companies manufacturing new products can adapt their manufacturing facilities to handle the disassembly, remanufacturing, or refurbishing of used items. These hybrid systems can vary in their configuration. One approach might involve using the same production line for both new and returned products, necessitating the consideration of the associated changeover times and costs. Alternatively, distinct lines might designate new and used products. Hybrid systems present greater management challenges compared to traditional new product assembly lines, due to complex design and planning decisions [12,17,18]. Another source of complexity is the uncertainty on demands, returns, and refurbishing times [11,19,20].

Downward substitution has been applied in the automotive sector, as highlighted in the case study presented in [21]. Early research on substitution includes work by [22,23,24]. In the context of dynamic demand models, notable contributions include studies by [25,26,27,28], as well as the book by [29]. A comprehensive survey and literature classification on the topic was conducted by [30].

Most solutions in the literature for the lot-sizing problem with remanufacturing are heuristic approaches. To the best of our knowledge, the only exceptions are [12,31], which proposed exact algorithms. Lot-sizing problems with returns are NP-hard in general [32]. A few exceptions of cases that can be solved in polynomial times are discussed by [12]. Heuristic solutions to NP-hard versions include extensions of classical lot-sizing heuristics like Silver and Meal [33,34]; a dynamic programming-based heuristic by [35]; metaheuristics such as variable neighborhood search by [36], tabu search by [37], a memetic algorithm, and a hybrid algorithm by [38]. For the multi-item problem, [39] developed a column generation approach with specific capacity constraints, while [40] tackled a variant of the multi-item lot-sizing problem with returns using a variable neighborhood descent heuristic.

Uncertainty on demand is one of the criteria that can be used to classify models presenting hybrid systems with substitution. Stochastic demand models include [7,41,42,43]. Some studies ignored the storage of returns despite its importance. For example, ref. [44] assumed that there is a continuous flow of returns with a fixed cost that can be included directly in the remanufacturing cost. Based on the discussions by the authors with some refurbishing companies, the storage of returns should be highly considered. In particular, many expensive storage spaces are occupied by items (returns), which may not be exploitable and are quickly discarded after checking. Processing capacities are often considered infinite [7,37,45] or a finite capacity without any sharing of resources between different processes such as manufacturing and remanufacturing [46,47]. Capacity restrictions and resource capacity sharing are considered in our work. With respect to setup costs and time, to the best of our knowledge, there are no studies on joint setups between manufacturing and remanufacturing/refurbishing except in [48].

Substitution can be either downward (in which the higher-value product is used to satisfy the low-value product) or upward. To the best of our knowledge, all studies in literature allow downward substitution, where the shortage of remanufactured products is compensated by the higher value of the new product but at the same price as the used one.

The configurations and variants presented in this work can be compared to [7,10,46,49,50,51], which focused on the integration of two processes, manufacturing and remanufacturing, with returns and substitution. However, our work integrates five processes (collection, refurbishing, disassembling, purchasing, and manufacturing). In previous works, returns were considered as parameters, while we treat returns as a decision variable. In addition to the two contributions mentioned above, this work differs from previous studies by investigating the environmental impacts.

The following is a summary of the paper’s contributions:

• It suggests and analyzes the effectiveness and sustainability of several configurations that integrate the processes of collection, refurbishing, disassembling, manufacturing, and substitution.
• It discusses and justifies the usefulness of downward substitution and analyzes its value from an economic and sustainability point of view.
• It analyzes the possibility that the two processes, manufacturing and refurbishing, share the same line.
• It proposes and compares four efficient heuristics based on relax-and-fix and fix-and-optimize approaches to solve the corresponding optimization problems. The heuristics are capable of obtaining solutions close to optimum in less than 10 s for reasonable-sized instances.

3. Configurations Description and Formulation

This section describes several possible configurations of a production unit capable of producing both refurbished and new products. To analyze the value of integrating these processes together with substitution, we present five configurations. We begin with a reference basic configuration (Configuration 1) presented in Figure 1 that is extended successively to reach the most comprehensive configuration containing all processes involved in the disassembling, refurbishing, and manufacturing of new products, considering the substitution flow from new products to refurbished products. This configuration is illustrated in Figure 2. The common assumptions of the configurations are cited below:

• In all configurations, the production facility receives returned products. Some of the returns will be discarded and sent directly to an independent recycling process (not shown here). The accepted items can be refurbished (Path 1 in all configurations) and used to satisfy the demand for second-hand products or disassembled (Path 3 in Configurations 2–5).
• The key parts extracted from accepted returns (Path 3) are considered as good as new (by undergoing some processes) as in most prior studies.
• We suppose a simple bill of material with a one-to-one relationship between the key purchased part and the manufactured product.
• The manufacturing of the new products can use new purchased parts (Path 2 in configurations 1, 3 and 5).
• The remaining components resulting from disassembling can be recycled or disposed of.
• All configurations are evaluated with and without the substitution of secondhand products with new products for the same price, and with and without capacity.
• The aim of the configurations is to assess the potential economic and environmental impact when incorporating returns and substitution into the system.
• The objective is to meet the deterministic dynamic demands of both used (refurbished) and new products while minimizing the total costs (which include disassembly, manufacturing, and inventory holding costs) and to decide, for each period, the quantity to be collected, refurbished, disassembled, purchased, manufactured and substituted.

The notations used in the formulations are described below. Note that we have used the same letter to represent the same group of decision variables. For example, y is used for binary setup variables, q for quantities, and I for inventory. The same approach is used for the parameters. Then, we distinguish between the processes using superscript, while subscript represents the time index of the variable. For example, $y_t^r$ represents the binary setup variable for refurbishing in the time period t.

Sets and parameters

• T: the length of planning horizon that discretizes in time periods $t=1\dots T$.
• $d_t^a$, $d_t^r$: demands for manufactured and refurbished products in period t respectively.
• $h^c$, $h^r$, $h^d$, $h^n$, and $h^a$: unit holding costs for collected, refurbished, disassembled, newly purchased, and manufactured items, respectively.
• $c^r$, $c^d$, $c^n$, and $c^a$: unit (purchasing or processing) costs for refurbishing, disassembling, new parts, and manufacturing, respectively.
• $C^{sc}$, $C^{sr}$, $C^{sd}$, $C^{sn}$, and $C^{sa}$: setup costs for collection, refurbishing, disassembling, supply of new parts, and manufacturing, respectively.
• $Q^r$, $Q^a$ and Q: capacity of refurbishing, manufacturing and shared lines, respectively.
• $T^M$, $T^a$ and $T^r$: major setup time, minor setup time for manufacturing and minor setup time for refurbishing, respectively.
• $t^a$ and $t^r$: unit time for manufacturing and refurbishing, respectively.

Decision variables

• $q_t^c$, $q_t^r$, $q_t^d$, $q_t^a$, $q_t^{pn}$ and $q_t^s$: quantity to be collected, refurbished, disassembled, manufactured, purchased and substituted, respectively, in period t.
• $q_t^{cr},q_t^{cd}$: quantity collected for refurbishing and disassembling, respectively, in period t.
• $q_t^{rd}$: quantity of demand satisfied from refurbishing in period t.
• $q_t^u,$$q_t^n$: quantity manufactured using disassembled and purchased parts in period t.
• $I_t^{cr}$, $I_t^{cd}$, $I_t^c$, $I_t^r$, $I_t^d$, $I_t^n$, and $I_t^a$: inventory levels of collected items for refurbishing, collected items for disassembly, collected items for both, refurbished items, disassembled items, new parts, and manufactured new items, respectively.
• $y_t^{cr}$, $y_t^d$, $y_t^r$, $y_t^d$, $y_t^n$, $y_t^a$, and $y_t^M$: binary variables indicating if collection (for refurbishing), disassembled, purchasing, refurbishing, manufacturing, and refurbishing in shared line, is incurred, respectively, in period t.

Configurations without substitution in Figure 3 were obtained by simply removing the variable $q^s$ from the model.

3.1. Configuration 1

This is a reference configuration as presented in Figure 1, and it is composed of two independent lines: Path 1 and Path 2. In this configuration, all collected products are refurbished (after selection), none of them are disassembled, and newly manufactured items use only purchased new parts. Path 1 makes refurbished products satisfy the demand for refurbished products. Path 2 independently receives new parts and manufactures new items. This reference configuration does not consider the integration of refurbishing into the production of new items. It is supposed to be the least attractive configuration from the economic and ecological points of view. To reduce the number of models (model for path 1 and model for path 2), we formulated this configuration using a single model called Model MAR using a mixed integer linear program (MILP) described below.

The MILP formulation of the reference problem is referred to as the manufacturing and refurbishing (MAR) model. This represents the configuration shown in Figure 1, without capacity constraints.

$$\begin{array}{c}\hfill \mathbf{Model}\mathbf{MAR}\mathbf{:}\mathrm{Minimize}\sum _{t=1}^T(h^c{I}_t^{cr}+h^r{I}_t^r+h^n{I}_t^n+h^a{I}_t^a)\\ \hfill +\sum _{t=1}^T(C^{sc}{y}_t^{cr}+C^{sr}{y}_t^r+C^{sn}{y}_t^n+C^{sa}{y}_t^a)\\ \hfill +\sum _{t=1}^T(c^r{q}_t^r+c^n{q}_t^{pn}+c^a{q}_t^a)\end{array}$$

(1)

$$\mathrm{Subject}\mathrm{to}:I_t^{cr}=I_{t-1}^{cr}+q_t^{cr}-q_t^{r}t=1\dots T$$

(2)

$$I_t^r=I_{t-1}^r+q_t^r-q_t^{rd}t=1\dots T$$

(3)

$$q_t^{rd}=d_t^{r}t=1\dots T$$

(4)

$$I_t^n=I_{t-1}^n+q_t^{pn}-q_t^{n}t=1\dots T$$

(5)

$$q_t^a=q_t^{n}t=1\dots T$$

(6)

$$I_t^a=I_{t-1}^a+q_t^a-d_t^{a}t=1\dots T$$

(7)

$$q_t^a\le M\times y_t^{a}t=1\dots T$$

(8)

$$q_t^r\le M\times y_t^{r}t=1\dots T$$

(9)

$$q_t^{cr}\le M\times y_t^{cr}t=1\dots T$$

(10)

$$q_t^{pn}\le M\times y_t^{n}t=1\dots T$$

(11)

$$q_t^{cr},q_t^r,q_t^{rd},q_t^u,q_t^a,q_t^n,q_t^{pn},I_t^{cr},I_t^r,I_t^n,I_t^a\ge 0t=1\dots T$$

(12)

$$y_t^{cr},y_t^r,y_t^a,y_t^n\in \{0,1\}t=1\dots T$$

(13)

The objective function (1) consists of the minimization of the total costs due to setup, purchasing/processing, and inventory costs. Constraints (2), (3), (5), and (7), represent the inventory balance equations for collected items, refurbishing, new parts, and the manufacturing of new items, respectively. Constraints (4) and (6) are trivial in this model because disassembly and substitution are not allowed. They indicate that the demand for refurbished items is fully satisfied using the output of refurbishing and that the manufacturing of new items uses only purchased new parts. These two equations are transformed into extensions (other configurations) of the model. Constraints (8)–(10) link the continuous variables to the corresponding setup variables. This is the same as constraints (11) for purchased new parts. Big-M represents an upper bound on the variables on the right-hand side of the constraints. Finally, constraints (12) and (13) define the domains of the decision variables.

3.2. Configuration 2

The new items use only parts resulting from the disassembly of returns. However, the returns of items to be disassembled and those to be refurbished are completely independent. This configuration can be used to show the limitations of partial integration, where returns are not shared between new and refurbished items processes. To model this configuration, variables $q^n$, $p^n$, $y^n$, and $I^n$ are removed from the MAR model of Configuration 1 together with constraints (5), (6) and (11). Constraints (14)–(18) are added:

$$I_t^{cd}=I_{t-1}^{cd}+q_t^{cd}-q_t^{d}t=1\dots T$$

(14)

$$I_t^d=I_{t-1}^d+q_t^d-q_t^{u}t=1\dots T$$

(15)

$$q_t^{cd}\le \sum _{t^{\prime }=t}^T{d}_{t^{\prime }}^a\times y_t^{cd}t=1\dots T$$

(16)

$$q_t^d\le \sum _{t^{\prime }=t}^T{d}_{t^{\prime }}^a\times y_t^{d}t=1\dots T$$

(17)

$$q_t^a=q_t^{u}t=1\dots T$$

(18)

Constraints (14) and (15) represent the new inventory balance equations for collected and disassembled items, respectively. Constraints (16) and (17) link the continuous variables of the quantities collected and disassembled to the corresponding binary variables. Constraint (18) indicates that manufacturing only uses parts from disassembly.

3.3. Configuration 3

This is a combination of Configuration 1 and Configuration 2. New products can be manufactured using disassembled or purchased parts. The inventories of the collected items for disassembly and refurbishment were separate. In other words, Paths 1 and 2 in Figure 2 are separated. Constraint (18) is removed from the model of Configuration 2 and replaced by (5), (11), and (19).

$$q_t^a=q_t^u+q_t^{n}t=1\dots T$$

(19)

3.4. Configuration 4

This is similar to Configuration 2 except that the inventory of collected items is shared between the refurbishing and disassembly processes. In other words, Path 2 (Figure 2) is deactivated in this configuration. Configuration 4 is used to highlight the importance of sharing returns, but it is limited in terms of flexibility as it does allow the use of purchased parts. To model Configuration 4, the formulation of Configuration 2 was modified as follows. The variables $I_t^{cr}$, $q_t^{cr}$, and $y_t^{cr}$ are replaced by $I_t^c$, $q_t^c$, and $y_t^c$, respectively. Constraints (2), (10) and (14) are replaced by (20) and (21), respectively.

$$I_t^c=I_{t-1}^c+q_t^c-q_t^r-q_t^{d}t=1\dots T$$

(20)

$$q_t^c\le M\times y_t^{c}t=1\dots T$$

(21)

3.5. Configuration 5

This is a fully integrated configuration, as illustrated in Figure 2. We refer to this as the process of integrated manufacturing and refurbishing with substitution (IMARS)). The purpose of this configuration is to compare the computational performances of the proposed heuristics. The MILP formulation is as follows:

$$\begin{array}{c}\hfill \mathbf{Model}\mathbf{IMARS}\mathbf{:}\mathrm{Minimize}Z=\sum _{t=1}^T(h^c{I}_t^c+h^r{I}_t^r+h^d{I}_t^d+h^n{I}_t^n+h^a{I}_t^a)\\ \hfill +\sum _{t=1}^T(C^{sc}{y}_t^c+C^{sr}{y}_t^r+C^{sd}{y}_t^d+C^{sn}{y}_t^n+C^{sa}{y}_t^a)\\ \hfill +(\sum _{t=1}^T{c}^r{q}_t^r)+\sum _{t=1}^T{c}^d{q}_t^u+\sum _{t=1}^T{c}^n{q}_t^n+\sum _{t=1}^T{c}^a{q}_t^a)\end{array}$$

(22)

$$\mathrm{Subject}\mathrm{to}:I_t^d=I_{t-1}^d+q_t^d-q_t^{u}t=1\dots T$$

(23)

$$I_t^a=I_{t-1}^a+q_t^a-q_t^s-d_t^{a}t=1\dots T$$

(24)

$$q_t^{rd}+q_t^s=d_t^{r}t=1\dots T$$

(25)

$$q_t^d\le \sum _{t^{\prime }=t}^T(d_{t^{\prime }}^a+d_{t^{\prime }}^r)\times y_t^{d}t=1\dots T$$

(26)

$$q_t^c,q_t^r,q_t^{rd},q_t^d,q_t^u,q_t^a,q_t^n,q_t^s,I_t^c,I_t^r,I_t^d,I_t^n,I_t^a\ge 0t=1\dots T$$

(27)

$$y_t^c,y_t^r,y_t^d,y_t^a,y_t^n\in \{0,1\}t=1\dots T$$

(28)

$$\mathrm{and}\mathrm{Constraints}\left(3\right),\left(5\right),\left(8\right),\left(9\right),\left(11\right),\left(19\right)–\left(21\right)$$

(29)

4. Variants of Full Configuration

To compare the performances of the proposed heuristics, three variants of full configuration (Configuration 5) are described by extending the model IMARS presented above. The following main characteristics are considered:

• Capacity restrictions in variant1.
• Resource sharing between manufacturing and refurbishing in variant2.
• Setup times in case of sharing resources in variant3.

The required changes to the IMARS model are also described. The choice of characteristics considered was based on the lot-sizing literature indicating that these parameters usually have an impact on the complexity of lot-sizing problems (e.g., Chen and Thizy [52]; Trigeiro et al. [53]; Bayley et al. [54]).

4.1. Variant1

This variant considers the limited processing capacities for the refurbishing and manufacturing process. Capacity restrictions make lot-sizing problems difficult to solve in general [52]. While many uncapacitated lot-sizing and production-planning problems can be solved in polynomial time, their capacitated counterparts are NP-hard [6]. In our case, the capacitated problem is obtained by replacing Constraints (8) and (9) with Constraints (30) and (31). Capacity restriction is applied only to manufacturing and refurbishing processes, though the study can be easily extended by putting restrictions on other resources such as storage capacities. Such restrictions can be the subject of future extensions of our work.

$$t^a\times q_t^a\le Q^a\times y_t^{a}t=1\dots T$$

(30)

$$t^r\times q_t^r\le Q^r\times y_t^{r}t=1\dots T$$

(31)

4.2. Variant2

Here, the shared line is used to refurbish used products and manufactured new products. This approach is suitable if refurbishing and manufacturing operations are performed on the same production line using the same production resources to enhance operational efficiency and sustainability by optimizing the use of equipment, facilities, and materials across both processes [12]. The challenge introduced by this approach is the balanced resource allocation between the two processes. To formulate this variant, the IMARS model was modified by adding constraints (32), where Q is the capacity of the shared line, which is expressed in time units.

$$t^a\times q_t^a+t^r\times q_t^r\le Qt=1\dots T$$

(32)

4.3. Variant3

In this case, in addition to the capacity sharing between refurbishing and manufacturing, there are two types of setup times to be considered: a major setup time $T^M$ is incurred when at least one type of production takes place, a minor setup time for manufacturing new products ($T^a$), and a minor setup time for refurbishing ($T^r$). Introducing setup time is crucial for accurately reflecting real-world conditions. Setup time represents the time required to prepare equipment or machinery for a new production run. For further details about introducing setup time in a lot-sizing problem, the reader may refer to [53] where detailed related theoretical developments and experiments are provided.

The IMARS model is modified by adding constraints (33)–(35), where $y_t^M$ is a binary variable equal to 1 if manufacturing and refurbishing are carried out in a shared line.

$$t^a\times q_t^a+t^r\times q_t^r+T^a\times y_t^a+T^r\times y_t^r+T^M\times y_t^M\le Qt=1\dots T$$

(33)

$$y_t^a\le y_t^{M}t=1\dots T$$

(34)

$$y_t^r\le y_t^{M}t=1\dots T$$

(35)

5. Heuristic Solution Methods

The IMARS model and its variants can be considered extensions of the One Warehouse Multi Retailer (OWMR) problem, which is classified as NP-hard by [55]. Therefore, heuristic techniques are developed to address these MILP models. Heuristics-based MILPs have been widely used to solve MILP models due to the enhanced performances of the available MILP solvers. Among these, we have selected relax-and-fix heuristics (RFHs) and fix-and-optimize heuristics (FOs). RFHs aim to decompose large, complex optimization problems into smaller, more manageable subproblems. This is achieved by relaxing some decision variables (treating them as continuous instead of discrete) while fixing others at specific values. The process is iterative, where each iteration solves a subproblem with a subset of variables relaxed, and the solution is progressively refined by fixing more variables based on the previous iteration’s results. This approach balances the need for solution accuracy with computational efficiency, making it suitable for solving large-scale mixed-integer linear programming (MILP) problems [56].

In lot-sizing problems, the decision variables are often associated with different time periods. The RFH takes advantage of this structure by decomposing the problem into smaller subproblems, each corresponding to a subset of time periods. This approach simplifies the problem-solving process. The RFH has been applied by [57] to the uncapacitated single-item lot-sizing problem, while [58] employed it for the capacitated single-level multi-item lot-sizing problem. Additionally, [59,60] applied the RFH to address a multi-level lot-sizing problem.

Fix-and-optimize heuristics (FOs) focus on iteratively improving a solution to a mixed-integer linear programming (MILP) problem. This method starts with an initial feasible solution to the MILP problem. In each iteration, most of the variables are fixed at their current values from the initial or previous solutions. A small subset of variables is selected and optimized while keeping the others fixed. This subset can be chosen based on various criteria, such as their potential to improve the objective function or their role in the problem’s structure. This process is repeated until no further significant improvements can be made. More details regarding the FO heuristic can be found in [56]. The use of the FO heuristic in solving lot-sizing problems can be found in Sahling et al. [61,62] for a multi-level capacitated problem.

The combined use of the two approaches RFH and FO heuristics can be found in [63] with an application to a multi-stage lot-sizing and scheduling problem, in [64] with an application to multi-level lot-sizing problems, and in [48] with an application to multi-level lot sizing with returns and substitution.

The next sections present four heuristics. The strategy of the first one is applied as in [48], but the strategies of the three others are adapted. The most original ideas are those implemented in the second and third approaches (value-based and process-based relax-and-fix and fix-and-optimize heuristics). Each heuristic has two main phases; the first phase consists of constructing a good initial solution by using the relax-and-fix heuristic, which will be enhanced in the second phase by utilizing fix-and-optimize heuristic. The fourth heuristic is a “pure” fix-and-optimize heuristic, which starts with a simple L4L solution of the lot-sizing decisions.

5.1. Time-Based Relax-and-Fix and Fix-and-Optimize Heuristic (TRF-FO)

The TRF-FO heuristic has two main phases. In the first one, a good initial solution is constructed using the TRF heuristic. In the second one, the FO heuristic executes an improvement process on the initial solution.

5.1.1. Time-Based Relax-and-Fix Heuristic (TRF)

The TRF heuristic is used as the first-phase TRF-FO to generate a good initial solution. Its strategy revolves around dividing the planning period into three sets (windows) of time periods. In the first set, called $W^F$, the optimal values of binary variables are obtained from the solution of the MILP sub-problem, which has been solved in the preceding iteration. In the second set of periods, called $W^I$, the variables should be binary.

To provide a good first solution, the TRF heuristic is used as the first phase of TRF-FO. The procedure is illustrated in Figure 4. Its main strategy is to split the planning period into several windows of time periods. The setup variables in the first window, which we refer to as $W^F$, are fixed at the preceding optimal values obtained by solving a MILP sub-problem in the earlier iterations. The setup variables are forced to be binary in the second window, $W^I$. The setup variables in the last window $W^R$ are relaxed. The window length is determined by the parameters $\alpha$ and $\beta$, where $\beta <\alpha$, $\alpha$ determines the window length of $W^I$, and $\beta$ denotes the sliding step of $W^I$. Then, at each iteration, $W^F$, $W^I$, and $W^R$ are updated as indicated in Figure 4. In the first iteration and in the first $\alpha$ periods (where $\alpha <T$), the setup variables in a MILP sub-problem (called the relax-and-fix lot-sizing $RFLS$ model), are forced to be binary, and all other setup variables are relaxed. In the second iteration, setup variables from periods where $t=1$ to $\beta$ are taken to be their optimal recorded values, which have been obtained by solving the MILP sub-problem in the preceding iteration. The variables from periods $\beta +1$ to $\beta +\alpha$ are then forced to be binary and relaxed from periods $\beta +\alpha +1$ to T. In general, from iteration k to iteration $k+1$, the following $\beta$ periods are added to the window $W^F$. Window $W^I$ is shifted forward by $\beta$ periods, and the first $\beta$ periods are eliminated from window $W^R$.

Let ${\overline{y}}_t^{\left(k\right)}$ be the binary value of $y_t$ found at iteration k.

For each iteration k, the following $RFLS$ model is resolved using a MILP solver:

Model TRFLS

Minimize Z (see Equation (22))

subject to (20), (23)–(27), (29) and

$$y_t^b={\overline{y}}_t^{b(k-1)}t\in W^F,b=\{c,r,d,n,a\}$$

(36)

$$y_t^b\in \{0,1\}t\in W^I,b=\{c,r,d,n,a\}$$

(37)

$$y_t^b\in [0,1]t\in W^R,b=\{c,r,d,n,a\}$$

(38)

The pseudo code of the TRF heuristic is presented in Algorithm 1.

Algorithm 1 Pseudocode of Time-Based Relax-and-Fix Heuristic (TRF)

$k=0$ $W^F=\varnothing$ $W^I=[1,\alpha ]$ $W^R=[\alpha +1,T]$ $Solve$ $RFLS$ $k=k+1$ while $k\times \beta +\alpha <T$ do    $W^F=[1,k\times \beta ]$    $W^I=[k\times \beta +1,k\times \beta +\alpha ]$    $W^R=[k\times \beta +\alpha +1,T]$    $Solve$ $RFLS$    $k=k+1$ end while $W^F=[1,k\times \beta ]$ $W^I=[k\times \beta +1,T]$ $W^R=\varnothing$ $Solve$ $RFLS$

5.1.2. Fix-and-Optimize Heuristic (FO)

As summarized in Algorithm 2, the FO heuristic is the second phase of heuristics. It typically starts with good solution found in the first phase and iteratively improves it. The basic idea is to partition the planning horizon into two sets of time periods, $W^F$ and $W^I$, which exchange periods between them. For each period in $W^F$, the corresponding setup variables are fixed to their binary values in the previous solution, and $W^I$ is the subset of periods, where the setup variables are to be re-optimized. The length of the $W^I$ set is equal to $\gamma$, which can take its values arbitrarily between [$\gamma min$, $\gamma max$]. In the initial iteration ($k=0$), we start by fixing the setup variables of the first $\gamma$ periods, i.e.,  $W^I=[1,\gamma ]$ and $W^F=[\gamma +1,T]$. For the subsequent iterations, $W^I$ will shift to the next $\gamma$ periods till the end of the planning horizon. Note that the choice of $\gamma$ is important. A small value of $\gamma$ will result in more CPU time and higher-quality solutions. On the other hand, larger values of $\gamma$ will require less CPU time and often will result in low-quality solutions. At each iteration, if there is an improvement in the objective function, we restart the $W^I$ set from the beginning of the horizon and reset the number of iterations ($k=0$). Otherwise, the value of $\gamma$ is regenerated randomly in [$\gamma min$, $\gamma max$], and $W^I$ and $W^F$ are updated by sliding them forward by $\gamma$ periods and solving the FOLS model (TRFLS model without constraints (38)). When there is no improvement and the $W^I$ window hits the end of the horizon, the FO heuristic is stopped. The proceduce is illustrated in Figure 5.

Algorithm 2 Pseudocode of the Fix-and-Optimize Heuristic

$S^{*}=S^0$ $\gamma =U[\gamma min,\gamma max]$ $k=0$ while $k\times \gamma <T$ do    $W^I=[k\times \gamma +1,(k+1)\times \gamma ]$    $W^F=[1,T]\setminus W^I$    $Solve$ FOLS model    if $f\left(S\right)<f\left(S^{*}\right)$ then      $S^{*}=S$      $\gamma =U[\gamma min,\gamma max]$      $k=0$    else       $k=k+1$    end if end while

5.2. Value-Based Relax-and-Fix and Fix-and-Optimize Heuristic (VRF-FO)

The VRF-FO heuristic, like TRF-FO, has two main phases. The first phase consists of building a good feasible solution using value-based relax-and-fix heuristic (VRF) (described below), while the second phase aims to improve this solution using the fix-and-optimize heuristic (FO) (described previously in Section 5.1.2).

The main idea of the VRF heuristic is to iteratively fix the variables that have a fractional value closest to 0.5 until all binary variables are fixed or integers. A similar approach was used by [64]. This can be conducted partially or completely independently of the time index of the decision variables. More specific to our problem, for each binary variable $y_t^b$:$b=\{c,r,d,n,a\}$ (corresponding to collection, refurbishing, disassembly, replenishment, and manufacturing), we decompose the planning horizon into three subsets of periods, $W^{bF}$, $W^{bI}$ and $W^{bR}$; i.e., each process has its own subsets. This is different from TRF heuristic, where all setup variables ($y_t^c$, $y_t^r$, $y_t^d$, $y_t^n$, $y_t^a$) were simultaneously decomposed on the same set of time periods. The variables $y_t^b$ are fixed to their optimal value in $W^{bF}$, forced to be binary in $W^{bI}$ and relaxed in $W^{bR}$. The size of each subset is determined using two parameters, $\alpha$ and $\beta$, where $\beta <\alpha$.

The number of periods in $W^{bI}$ is $\alpha$. At each iteration, $\beta$ new periods are added to $W^{bF}$, we change $\beta$ periods in $W^{bI}$, and the remaining periods belong to $W^{bR}$. The pseudocode of the VRF heuristic is summarized in Algorithm 3. Figure 6 illustrates an example of VRF iterations for $T=8$, $\alpha =4$ and $\beta =2$. The example illustrates the VRF heuristic for one value of “b” in $W^{bF}$, $W^{bI}$ and $W^{bR}$. The behavior of VRF will be the same on any other values of b, but not necessarily on the same time periods. That is, in a given iteration and for $b,b^{\prime }\in \{c,r,d,n,a\}$, we might have $W^{bF}\ne W^{b^{\prime }F}$, $W^{bI}\ne W^{b^{\prime }I}$ and $W^{bR}\ne W^{b^{\prime }R}$. In iteration 0, the model is solved with all variables relaxed. As $\alpha =4$, we identify the four binary variables whose values are closest to 0.5 (${\overline{y}}_1^b,{\overline{y}}_3^b,{\overline{y}}_4^b,{\overline{y}}_6^b$). Integrality is imposed on these four variables and relaxed for all remaining variables ($W^{bI}=\{1,3,4,6\}$, $W^{bR}=\{2,5,7,8\}$. Once this model is solved, two variables (since $\beta =2$) with the smallest indexes in $W^{bI}$ are fixed (here $t=1$ and $t=3$) and moved to $W^{bF}$, and two relaxed variables with the closest value to 0.5 enter $W^{bI}$ (here $t=2$ and $t=7$). In the third iteration of the example, all variables respect integrality constraints. The procedure stops.

Algorithm 3 Pseudocode of the value-based relax-and-fix heuristic (VRF)

$k=0$ $W^{bF}=\varnothing$, $b=c,r,d,n,a$ $W^{bI}=\varnothing$, $b=c,r,d,n,a$ $W^{bR}=\{1,T\}$, $b=c,r,d,n,a$ $Solve$ VRFLS $k=1$ while $k\times \beta +\alpha <T$ and $W^{bR}\ne \Phi$ do    Rank periods in $W^{bF}$ according to $|{\overline{y}}_t^b-0.5|$    Move the first $\beta$ periods from $W^{bI}$ to $W^{bF}$, if any    Move the first $\beta$ periods from $W^{bR}$ to $W^{bI}$, if any    $Solve$ VRFLS    $k=k+1$ end while

5.3. Process-Based Relax-and-Fix and Fix-and-Optimize Heuristic (PRF-FO)

Similar to the previous approaches, the PRF-FO heuristic uses a fix-and-optimize procedure to improve a solution obtained using a variant of the relax-and-fix heuristic. This variant is called the process-based relax-and-fix heuristic (PRF) and is detailed below.

The processes in our problem are collecting, refurbishing, disassembling, manufacturing, and purchasing. Setup variables related to these processes are, respectively, $y_t^c$, $y_t^r$, $y_t^d$, $y_t^a$ and $y_t^n$. The basic idea of the PRF heuristic is to start fixing setup variables of processes with the largest contribution to the objective function Z. We introduce costs $Z^c$, $Z^r$, $Z^d$, $Z^a$, and $Z^n$ corresponding, respectively, to the costs of collecting, refurbishing, disassembling, manufacturing and purchasing.

$$Z^c=\sum _{t=1}^T(h^c{\overline{I}}_t^c+C^{sc}{\overline{y}}_t^c)$$

$$Z^r=\sum _{t=1}^T(h^r{\overline{I}}_t^r+C^{sr}{\overline{y}}_t^r+c^r{\overline{q}}_t^r)$$

$$Z^d=\sum _{t=1}^T(h^d{\overline{I}}_t^d+C^{sd}{\overline{y}}_t^d+c^d{\overline{q}}_t^u)$$

$$Z^a=\sum _{t=1}^T(h^a{\overline{I}}_t^a+C^{sa}{\overline{y}}_t^a+c^a{\overline{q}}_t^a)$$

$$Z^n=\sum _{t=1}^T(h^n{\overline{I}}_t^n+C^{sn}{\overline{y}}_t^n+c^n{\overline{q}}_t^n)$$

The algorithm starts by solving the model with all setup variables being relaxed. The values of $Z^c$, $Z^r$, $Z^d$, $Z^a$, and $Z^n$ are calculated and sorted in decreasing order. The relax-and-fix procedure is applied to the vectors of the setup variables sequentially while respecting that order. For example, if the initial step resulted in $Z^a\ge Z^c\ge Z^d\ge Z^r\ge Z^n$, then the binary variables are considered in the following order: Y = [$y_1^a$,$y_2^a$,…,$y_T^a$,$y_1^c$, $y_2^c$,…,$y_T^c$, $y_1^d$, $y_2^d$,…, $y_T^d$, $y_1^r$, $y_2^r$,…, $y_T^r$, $y_1^n$, $y_2^n$,…,$y_T^n$]. It can be written as Y = [$y_1^{\prime }$, $y_2^{\prime }$,…,$y_T^{\prime }$,$y_{T+1}^{\prime }$,…,$y_{2T}^{\prime }$, $y_{2,T+1}^{\prime }$,…, $y_{3T}^{\prime }$, $y_{3,T+1}^{\prime }$,…, $y_{4T}^{\prime }$, $y_{4,T+1}^{\prime }$,…, $y_{5T}^{\prime }$] after variable changing. This means that the fixing of setup variables related to each process is carried out in a sequential manner instead of in parallel manner compared to VRF and more particularly to TRF. This list of $5\times T$ variables is decomposed into three intervals: $W^{bF}$, $W^{bI}$ and $W^{bR}$. the variables are fixed in the first interval, forced to be binary in the second interval, and relaxed in the last interval. The size of each subset is determined by the parameters $\alpha$ and $\beta$ and according to the order of the processes as described in Algorithm 4. The sub-problem to solve in the PRF heuristic, which we call PRFLS, is the same as TRFLS except that the subsets $W^F$, $W^I$, and $W^R$ are replaced, respectively, by $W^{bF}$, $W^{bI}$, and $W^{bR}$.

Algorithm 4 Pseudocode of process-based relax-and-fix heuristic (PRF)

$\xi =\{1,2,\dots ,T\}$ $k=0$ $W^{bF}=\varnothing$, $b=c,r,d,n,a$ $W^{bI}=\varnothing$, $b=c,r,d,n,a$ $W^{bR}=\{1,T\}$, $b=c,r,d,n,a$ $Solve$ $PRFLS$ Calculate: $Z^r$, $Z^c$, $Z^d$, $Z^a$ and $Z^n$ Rank process according to values of $Z^c$, $Z^r$, $Z^d$, $Z^a$ and $Z^n$ Let j the order of process b while $k\times \beta +\alpha <jT$ do    $W^{bF}=\{t\in \xi |t\le k\beta -(j-1)T\}$    $W^{bI}=\{t\in \xi |\le k\beta -(j-1)T<t\le \alpha +k\beta -(j-1)T)\}$    $W^{bR}=\xi \setminus (W^{bF}\cup W^{bI})$    $Solve$ $PRFLS$    $k=k+1$ end while

5.4. The Lot-for-Lot and Fix-and-Optimize Heuristic (L4L-FO)

The L4L-FO heuristic is similar to TRF-FO, VRF-FO, and PRF-FO in the sense that it uses a fix-and-optimize procedure to improve an initial solution. The initial solution in this case is obtained using the basic lot-for-lot (L4L) heuristic. The latter satisfies each order in its due date, which results in zero stock over the whole planning horizon.

6. Numerical Experiments

This section has three objectives:

• It compares the five configurations presented in Section 4 from an economic and environmental point of view and provides managerial insights for companies interested in refurbishing and manufacturing.
• It briefly discusses the complexity of the problem for the different variants described in Section 4. This is carried out by comparing the CPU time required to obtain the optimal solutions using the commercial MILP solver.
• It evaluates the numerical performance of the proposed heuristics by comparing their best solutions with the optimal solutions obtained for small instances. For larger instances, where no optimal solutions were obtained, a comparison is made only between the heuristics.

The heuristics were implemented in GAMS with the commercial solver CPLEX 12.4.

All experiments were run using a PC with an Intel Core i5 2.4 GHz CPU and 4 GB RAM under a Windows 10 professional operating system.

6.1. Data Generation

The parameters listed in

Table 1. Range of values for different parameters. The nominal or default values are Medium.

Parameters	Dem. Variability (${\mathit{d}}_{\mathit{t}}^{\mathit{a}}$ and ${\mathit{d}}_{\mathit{t}}^{\mathit{r}}$)	Average Dem. (${\mathit{d}}_{\mathit{t}}^{\mathit{a}}$ and ${\mathit{d}}_{\mathit{t}}^{\mathit{r}}$)	${\mathit{C}}^{\mathit{sc}}$	${\mathit{C}}^{\mathit{sr}}$	${\mathit{C}}^{\mathit{sd}}$	${\mathit{C}}^{\mathit{sn}}$	${\mathit{C}}^{\mathit{sa}}$	${\mathit{Q}}^{\mathit{a}}$	${\mathit{Q}}^{\mathit{r}}$
Low	U (50,70)	U (0,60)	40	150	100	37.5	200	Dmax	Dmax
Medium (def.)	U (30,90)	U (30,90)	80	300	200	75	400	1.25 × Dmax	1.25 × Dmax
High	U (0,120)	U (90,150)	120	450	300	112.5	600	1.5 × Dmax	1.5 × Dmax

and

Table 2. Nominal (Medium) values of unit costs (low values: $Lv=0.5\times Mv$; high values: $Hv=1.5\times Mv$).

Parameters	${\mathit{h}}^{\mathit{a}}$	${\mathit{h}}^{\mathit{r}}$	${\mathit{h}}^{\mathit{d}}$	${\mathit{h}}^{\mathit{n}}$	${\mathit{h}}^{\mathit{c}}$	${\mathit{c}}^{\mathit{a}}$	${\mathit{c}}^{\mathit{r}}$	${\mathit{c}}^{\mathit{d}}$	${\mathit{c}}^{\mathit{n}}$
Medium/nominal Values ($Mv$)	2	2	1	1	1	4	3	1.3	2.5

can take three levels, low value ($Lv$), medium value ($Mv$), and high value ($Hv$), where $Lv=0.5\times Mv$ and $Hv=1.5\times Mv$. We generate the values of demand $d_t^a$ and $d_t^r$ from a uniform distribution U(min,max). The lengths of the planning horizons were the $T=24$, $T=48$, and $T=96$ time periods. Demands $d^a$ and $d^r$ are characterized by variability and level (average value). The variability measures the difference between the $maximum$ and $minimum$ values of demand, whereas the level of demand is the average demand, which is ($max-min$)/2. As indicated in Table 1, the variability and level of demand can be low, medium, or high. When variability varies, the level is kept constant, and vice versa.

For each parameter, the default values (nominal values) are medium unless indicated otherwise. We have carefully chosen the medium values suchthat all processes (collection, manufacturing, disassembling, purchasing, refurbishing, and substitution) are activated. For each instance, five instances were generated randomly by varying the demand, as indicated in Table 1.

When considering the variant with shared capacity and joint setup times (Variant3, described in Section 4), two types of setup times are considered: a major setup time $T^M$ must be incurred when at least one type of production is produced, a minor setup time ($T^a$) for manufacturing, and a minor setup time ($T^r$) refurbishing. To maintain the same level of capacity as in the other tests, we consider that the unit time $t^a$ and $t^r$, for manufacturing and refurbishing, respectively, are equal to 1. To avoid infeasibility, we have added the setup time to the capacity of the shared line. The values for setup times are chosen as follows: $T^M=U(15\%,25\%)\times 2Dmax$, $T^a=U(30\%,40\%)\times T^M$ and $T^r=(15\%,30\%)\times T^M$. The different values of the setup times and shared capacities are listed in

Table 3. Setup times and shared capacity.

Parameters	${\mathit{t}}^{\mathit{a}}$	${\mathit{t}}^{\mathit{r}}$	${\mathit{T}}^{\mathit{a}}$	${\mathit{T}}^{\mathit{r}}$	${\mathit{T}}^{\mathit{M}}$	Q
Low	1	1	10	5	30	$t^a\times$$Dmax+T^a+T^r+T^M$
Medium	1	1	15	10	40	$t^a\times$ 1.25$Dmax+T^a+T^r+T^M$
High	1	1	20	15	50	$t^a\times$ 1.5$Dmax+T^a+T^r+T^M$

.

6.2. Analysis of Configurations and Managerial Insights

We compare the five configurations presented in Section 4 by analyzing their performance based on two criteria: (i) cost reduction (gain) compared to the reference MAR model and (ii) collection efficiency. Collection efficiency is the ratio of the total quantity collected to the total demand for both refurbished and new items and is used as a sustainability indicator.

$$\mathit{Collection}\mathit{efficiency}=100\times {\displaystyle \frac{q^c}{d^a+d^r}}$$

Configurations with higher collection efficiency are preferred.

The Gain is defined as follows:

$$Gain(\%)=100\times {\displaystyle \frac{Cost1-Cost}{Cost1}}$$

where $Cost1$ is the cost of the reference configuration (Configuration 1), and $Cost$ is the total cost of the configuration for which the $gain$ is measured.

The manufacturing cost varied between a minimum value of ($c^a$ = 1) and a maximum value of ($c^a$ = 12). This is the interval within which the total cost of a new product (including the cost of purchased or used parts) is lower than the cost of refurbished items. The results are summarized in Figure 7. The first column is for the configuration number. The second and third columns represent the gain curves, while the last two columns illustrate collection efficiency. Both capacitated and uncapacitated cases are considered.

6.2.1. Gain

a.

Value of substitution

The curves in the second and third columns of Figure 7 show that substitution always results in cost reduction compared to Configuration 1 (reference configuration). This is a trivial result. However, this analysis allows us to highlight which configurations would result in higher cost reduction and to check whether this cost reduction will be higher in capacitated or in uncapacitated models. In the base case, substitution results in a maximum gain of 14%. The maximum gain from Configuration 3 without substitution is 1.5%. When substitution is allowed, the maximum gain increases to 20%. These results also show that gains are more important with uncapacitated models (orange lines) compared to capacitated cases. This could be explained by the higher flexibility in uncapacitated models resulting in a higher impact of substitution on the total cost.

b.

Value of integration

We start by comparing the gains of Configuration 2 and Configuration 4 without substitution in the uncapacitated case (orange curves in the second column of Figure 7). The main difference between the two configurations is the fact that Configuration 4 links Path 1 and Path 3 through the sharing of returns inventory. The maximum gain in Configuration 2 is almost 1%, while it is close to 3% in Configuration 4. A similar comparison can be made between Configuration 3 and Configuration 5, where both include Path 1, Path 2, and Path 3 but only Configuration 5 allows the sharing of the inventory of returns between Path 1 and Path 3. Again, Configuration 5 has higher gains compared to Configuration 3. The same remarks can be made when comparing capacitated cases (blue curves). It has been shown in the literature that integrated processes are economically more interesting. The main questions that arise when we want to implement these models are whether the gain from integration counterbalances the fixed cost of installation. In the case of our models and in the context of our collaboration with the electronics company, the cost of switching to an integrated configuration is almost negligible. The same facility hosts the returns storage together with both lines of new and used products.

As expected, the gain decreases as the manufacturing cost increases. When this happens, the difference between the manufacturing and refurbishing costs becomes too high. The total cost of setting up the refurbishing line and the production of refurbished items becomes more interesting in this case.

6.2.2. Collection Efficiency

a.

Value of substitution

If substitution is allowed, the upper right curve representing Configuration 1 shows that when the manufacturing cost is very small, the model prefers substitution instead of the higher cost of collection and refurbishing. As $c^a$ increases, refurbishing becomes more interesting and cheaper, increasing the number of collected items. When capacity is limited, the manufacturing production capacity limits the amount produced and, consequently, the amount that can be substituted. In Configuration 1 with substitution, if $c^a$ is low, then substitution is high, and since the substituted quantities $q^s$ are manufactured only from new parts, the collection is low. As $c^a$ increases, $q^s$ decreases, and collection therefore increases, which explains why the curves are increasing. The capacitated model shows more collection because the manufacturing process cannot satisfy both demands simultaneously, so $q^s$ will be lower compared to the case without capacity, and hence collection will be high in the capacitated case. Configurations 2 and 4 are trivial because all demands ($d^r$ and $d^a$) are satisfied through collection. In Configurations 3 and 5, when $c^a$ is small, it is preferable to use parts from the disassembly of the collected items. These are cheaper than the purchased new parts. The manufactured items are used to satisfy the demand for new items and refurbished items (through substitution); as $c^a$ increases, substitution becomes more expensive and less attractive, resulting in less disassembly and less collection efficiency. This is further supported by the results discussed in Section 6.3 and Figure 8.

b.

Value of integration

The most important impact of integration on collection efficiency can be observed between Configuration 3 (no integration) and Configuration 5 (integration through an inventory of shared returns). In the fourth column of Figure 7, we can observe that collection efficiency increases from 85% in the uncapacitated case of Configuration 3 (55% in the capacitated case) to 99% in the uncapacitated case of Configuration 5 (85% in the capacitated case). When the capacity is finite in Configuration 3, this results in more generated setups, and because of the higher collection setup cost, the model prefers purchasing new items that have lower fixed costs.

Below is more discussion about Figure 7. In Configuration 1, all collected items are used to satisfy the demand for the refurbished items. The collection efficiency being at 50% is because the average demands for refurbished and new items are equal. The latter are fully satisfied using the purchased new parts because there is no disassembly. Configuration 2 does not allow purchased items; hence, all demands are satisfied through collection, resulting in 100% collection efficiency. When substitution is not allowed, the level of collection efficiency is independent of the assembly cost $c^a$. We also notice a positive correlation between gain and collection efficiency. As we increase $c^a$ in models with substitution, both collection efficiency and gain decrease. The reasons for these behaviors were explained before. The particular case of configuration 1 was also explained in the beginning of this section.

We conclude from these results that the integration of processes through the sharing of returns and capacities, together with substitution between refurbished and new items (which leads to Configuration 5), resulted in a decrease of up to 20% in total costs (economic indicator) and an increase in collection efficiency (environment-related indicator) of up to 15%. This illustrates the added value of the integration of processing in refurbishing and substitution in hybrid systems. Our first recommendation for the decision maker is to consider downward substitution as this might not incur high investment costs from the company. If the company chooses to use the three processes (refurbishing, disassembly, and optional use of used or new parts), we recommend, based on the above results, choosing Configuration 5 (integrated configuration) with substitution).

6.3. Detailed Analysis of the Integrated Model

This section focuses on Configuration 5 with substitution, which is the best configuration among those presented previously in terms of gain and collection efficiency. The processing capacity is assumed to be infinite. This section analyzes the characteristics of optimal solutions when different parameters are varied, and it provides useful insights. In particular, while varying the costs, we analyze the following.

• The ratio of the collected quantity to the total demands for new and refurbished products (Collection efficiency $=100\times q^c/(d^a+d^r)$). Collection efficiency was defined in the previous section. Its purpose is to measure the environmental impact of the model and how it is affected by the variation of different costs.
• The percentage of used parts in the manufacturing of new products (Used parts in new items $=100\times q^u/q^a$). This can be considered as another "sustainability" indicator showing the percentage of old (but as good as new) extracted parts from the disassembly process.
• The ratio of the substitution quantity to the demand of refurbished products (Substitution percentage $=100\times q^s/d^r$). This indicator shows the cost ranges under which substitution is economically viable.

All parameters were set to their default values unless indicated otherwise.

Three demand cases were considered in each simulation. In the first case (Case A), the average demands for new and refurbished products were equal ($d^a$Low, $d^r$Low). In Case B, the average $d^r$ was twice the average $d^a$ ($d^a$Low, $d^r$Medium). Finally, in Case C, the average $d^r$ was four times the average $d^a$ ($d^a$Low, $d^r$ High).

6.3.1. Effect of Varying the Manufacturing Cost of New Products $\left(c^a\right)$

As previously discussed, variations in the manufacturing cost $c^a$ have an impact on the total cost and on the collection efficiency. We previously noticed that collection efficiency decreases in the full configuration (Configuration 5) when $c^a$ increases. This can be explained by the decrease in the attractiveness of substitution, as shown in Figure 8. The substitution cost becomes higher than the fixed costs that can be incurred if the refurbishing line is activated. We also notice that, even when substitution decreases to zero, the collection efficiency is above 80% on average (blue curve). This is due to the fact that the model prefers using parts from disassembled returns instead of purchased parts (green curve).

6.3.2. Effect of Varying Unit Refurbishing Cost $\left(c^r\right)$

Figure 9 shows that. starting from a certain level of refurbishing cost $c^r$, the entire demand $d^r$ is satisfied through substitution (the ratio of substitution quantity to demand $d^r$ equals 100%). The threshold value depends on the default value of the manufacturing cost $c^a$ and the ratio $d^a/d^r$. The maximum value of collection efficiency reaches 100% when $d^r$ is four times that of $d^a$. It is interesting to note that collection efficiency is approximately 95% even if $d^r$ is "just" twice the value of $d^a$.

6.3.3. Effect of Varying Purchasing Cost of New Parts $\left(c^n\right)$

As the purchasing price of new parts increases, the level of substitution decreases, as shown in Figure 10. For larger values of the refurbished item demand, the ratio of the substitution level ($q^s$) to the demand for refurbished items decreases significantly. In fact, after the threshold value of $c^n$, both new and refurbished items are made using collected items ($q^c/(d^a+d^r)=100\%$), some of which are refurbished, and the others are disassembled so that the resulting key part is used in the manufacturing of new items. A percentage of these items is used to satisfy the demand for new items, and the remainder is used as a substitute for the demand for refurbished items.

6.3.4. Effect of Varying Collection Setup Cost $\left(C^{sc}\right)$

Figure 11 shows the impact of varying the collection setup cost between ($\left(C^{sc}\right)$ = 10) and ($\left(C^{sc}\right)$ = 160) for the three previously presented ratios. It seems that the substitution ratio was not significantly affected by the variation in $C^{sc}$. This is more visible in the right-most figure (Case C), and, as expected, increasing the $C^{sc}$ results in considerably smaller amounts of collected and disassembled items.

6.3.5. Effect of Varying Manufacturing Setup Cost $\left(C^{sa}\right)$

Figure 12 shows the experimental results when the fixed manufacturing cost $\left(C^{sa}\right)$ varied between 300 and 420. For a low demand for refurbished items (Case A in Figure 12), substitution is not attractive, although it is not eliminated, even for relatively large values of $\left(C^{sa}\right)$. However, substitution is not used for very large values of $d^r$. In fact, the high demand for refurbished items justifies relying fully on the collection of such items as it exploits economies of scale.

6.4. Performance of the Heuristics

The computational performance of the four heuristics described above (VRF-FO, PRF-FO, TRF-FO, and L4L-FO) was performed on the fully integrated configuration (Configuration 5 with substitution).

For small instances (with $T=24$ periods), the performance is evaluated by measuring the $GAP$ between the solution found by the heuristics ($z^H$) and the optimal solution ($z^{OPT}$) obtained by CPLEX. The GAP was computed using the following formula:

$$GAP=100\times {\displaystyle \frac{z^H-z^{OPT}}{z^{OPT}}}$$

For large instances ($T=48$ and $T=96$ time periods), the performance of each heuristic was compared to the best known solutions (see Section 6.4.2).

In these heuristics, the choice of window sizes is a crucial parameter that significantly affects the algorithm’s performance. For constructive heuristics TRF, VRF, and PRF, big $\alpha$ and small $\beta$ typically allow for more detailed and accurate local optimizations but may lead to longer overall computation times as more iterations are needed to cover the entire planning horizon. To find a good compromise between the CPU and the GAP, we tested the following combinations: (6,4), (8,4), (10,4), (12,4), (16,4), (8,6), (10,6), (12,6), (16,6), (10,8), (12,8), (16,8). The combination ($\alpha =10$, $\beta =4$) gave good compromise in the majority of tests. For the FO heuristic, the good interval found for the variation of $\gamma$ is $\gamma min=6$ and $\gamma max=10$. For large instances, we may not be able to solve each sub-problem to optimality within a reasonable CPU time. We set the relative gap of CPLEX to $0.1\%$. Other simulations consisted of forcing the integrality of binary variables of some processes (e.g., $y_n$ for the purchasing of new parts from suppliers) and applying the RFH on the other binary variables. All these simulations yielded poorer results compared to the approach where RFH is applied to all binary variables.

Given the structure of Configuration 5 with substitution, we conducted a numerical study to evaluate the following cases, which were already detailed in Section 4: (i) Base case: manufacturing and refurbishing with infinite capacity; (ii) Variant 1: manufacturing and refurbishing with finite capacity, (iii) Variant 2: a problem with finite and shared capacity; and (iv) Variant 3: a problem with shared capacity and joint setup. The tests included varying the demands (variability and volume), setup costs, capacities, and setup times for the three variants, if applicable. The tested time horizon lengths were $T=24$, $T=48$, and $T=96$.

6.4.1. Small Instances

Table 4. Performanceof CPLEX and the four heuristics on instances with $T=24$ time periods with a nominal value of parameters.

Variant	CPLEX	VRF-FO		PRF-FO		TRF-FO		L4L-FO
	CPU (s)	CPU (s)	GAP (%)	CPU (s)	GAP (%)	CPU (s)	GAP (%)	CPU (s)	GAP (%)
Infinite capacity (base case)	1087.39	17.54	0.06	19.25	0.14	20.71	0.01	12.54	0.33
No shared capacity	400.23	9.36	0.34	10.44	0.12	10.87	0.04	5.52	0.54
Shared capacity	796.95	9.20	0.03	12.03	0.00	14.31	0.09	7.29	0.17
Shared capacity and	391.91	10.15	0.25	8.92	0.03	12.58	0.11	7.18	0.44
joint setup
Average	669.12	11.56	0.17	12.66	0.07	14.62	0.06	8.13	0.37

presents a summary of the results obtained for $T=24$ time periods on nominal (medium) values of the parameters described in Section 6.1. The first column indicates the names of the variants including the base case (infinite capacity). The second column shows the average CPU time for CPLEX to obtain optimal solutions. The other columns indicate the average CPU time and average GAP for each of the four heuristics. These results show that the hardest problem to solve is the base case with infinite capacity. On average, CPLEX took more than 15 min to solve the instances optimaly. Uncapacitated lot-sizing problems are known to be relatively easy to solve. Removing capacity in such cases allows the decomposition of the problem into independent uncapacitated single-item lot-sizing problems, which can be efficiently solved using polynomial-time dynamic programming algorithms (see, for example, [52,65]). In our case, the process is more complex and integrated through the sharing of the inventory of returns and through substitution. Infinite capacities result in a larger solution space and problems that are more complex to solve. Adding capacity constraints in the three variants makes the problem easier to solve as the solution space becomes smaller. Tighter capacities can lead to problems that are even easier problems solve for certain variants, as discussed later in

Table 5. Performance of the heuristics for instances with $T=24$ time periods for low and high parameter values.

Variant	Parameter		CPLEX	VRF-FO		PRF-FO		TRF-FO		L4L-FO
			CPU (s)	CPU (s)	GAP (%)	CPU (s)	GAP (%)	CPU (s)	GAP (%)	CPU (s)	GAP (%)
Infinite capacity	D_Var	Low	14,557.32	21.23	0.08	42.68	0.03	33.39	0.10	8.76	0.12
		High	44.53	7.76	0.03	9.87	0.05	9.71	0.03	7.14	0.05
	D_Val	Low	163.40	11.79	0.21	11.81	0.00	14.55	0.01	8.68	0.27
		High	4.95	6.24	0.07	6.54	0.08	6.30	0.01	5.23	0.16
	Setup	Low	5.03	5.64	0.00	5.86	0.00	5.11	0.00	5.17	0.16
		High	4801.91	19.64	0.00	36.40	0.01	25.57	0.01	17.70	0.05
No shared capacity	D_Var	Low	1888.88	8.49	0.06	13.07	0.00	13.79	0.02	6.44	0.15
		High	36.20	5.52	0.00	6.44	0.08	5.39	0.00	5.16	0.26
	D_Val	Low	363.62	9.15	0.39	10.66	0.09	9.26	0.09	6.44	0.13
		High	89.95	4.90	0.21	6.99	0.08	5.95	0.07	4.52	0.39
	Setup	Low	14.10	4.70	0.18	4.12	0.05	4.77	0.10	3.68	0.28
		High	845.75	11.55	0.46	10.74	0.12	13.04	0.22	6.89	0.37
	Capacity	Low	229.71	8.19	0.11	9.25	0.00	5.59	0.04	5.04	0.15
		High	360.50	6.84	0.20	9.41	0.06	8.29	0.03	5.11	0.24
Shared capacity	D_Var	Low	1406.94	9.66	0.23	5.90	0.00	13.69	0.02	8.30	0.19
		High	49.73	5.85	0.27	6.39	0.05	6.35	0.03	4.86	0.07
	D_Val	Low	66.35	7.82	0.05	6.66	0.34	7.03	0.14	6.59	0.45
		High	601.28	8.22	0.13	10.23	0.08	11.39	0.00	5.26	0.18
	Setup	Low	125.09	6.67	0.07	6.06	0.10	8.23	0.00	6.25	0.17
		High	1575.63	10.24	0.01	7.74	0.08	12.48	0.06	7.98	0.13
	Capacity	Low	1522.29	9.96	0.15	12.56	0.20	15.65	0.04	9.89	0.13
		High	1010.45	9.74	0.08	8.28	0.00	12.78	0.09	5.65	0.36
Shared capacity and joint setup	D_Var	Low	4023.95	10.39	0.11	8.20	0.12	18.58	0.05	9.73	0.30
		High	50.61	5.67	0.03	6.21	0.08	5.88	0.03	4.43	0.07
	D_Val	Low	75.42	5.80	0.17	5.40	0.09	5.74	0.13	5.25	0.08
		High	426.37	9.26	0.04	9.84	0.04	11.87	0.01	5.55	0.21
	Setup	Low	126.61	6.75	0.13	6.91	0.12	7.96	0.00	5.86	0.18
		High	902.09	9.93	0.26	6.39	0.00	12.01	0.22	7.86	0.49
	Capacity	Low	987.06	6.93	0.12	11.08	0.00	16.47	0.09	9.46	0.22
		High	723.46	9.89	0.03	8.49	0.00	13.28	0.11	6.72	0.35
	Setup Time	Low	438.68	10.10	0.26	8.47	0.02	13.89	0.11	6.74	0.45
		High	387.04	7.65	0.11	8.16	0.02	12.75	0.00	4.68	0.72
Average			1184.53	8.82	0.13	10.21	0.06	11.46	0.06	6.78	0.24

.

All four heuristics provide excellent results with average gaps of less than 0.5% in reasonable CPU times (less than 15 s on average). The highest-quality heuristics are the process-based RF-FO (PRF-FO), and the time-based RF-FO (TRF-FO). While TRF-FO performs slightly better than PRF-FO (0.06% and 0.07%, respectively), the latter needs less CPU time on average (12.66 vs. 14.62 s). Figure 13 compares the four methods by plotting their CPU time vs. their percentage gap from optimum. It is clear that no method is dominated by others. This figure helps the decision maker to choose between methods based on his/her preference for faster vs. higher-quality solutions.

Further analysis was carried out on small instances with $T=24$ by varying the parameters between their low values and high values. The results are shown in Table 5. The first column indicates the variant. The second column shows the parameter being changed. The third column indicates whether the parameter is at a low or high value (see Table 1 and Table 2). The rest of the columns are similar to Table 4 and show the CPU times and gaps of CPLEX and the heuristics.

The first result from Table 5 is that all heuristics still show excellent performance with an average $GAP$ smaller than 0.5% and average CPU of less than 15 s. The second result is the fact that some heuristics are now dominated when compared based on their $GAP$ and CPU time. This is the case, for example, for the PRF-FO heuristic when the average demand and variability are high in the base case with infinite capacity. This is also the case for the L4L-FO heuristic on instances with small setup costs in the same base case. The boldface values indicate the situations where one heuristic was dominating all other approaches. This happened once with the TRF-FO heuristic and twice with the PRF-FO heuristic.

It is also possible to observe from the table that problems with lower demand variability are the hardest to solve, especially when demand is infinite. For some exceptional instances, CPLEX has taken several hours to find the optimal solution. As expected, and as reported in the production-planning literature, one can observe higher values of setup costs and tighter processing capacities make the problems harder to solve. CPLEX needs more than 10 times the CPU time with high setup cost values compared to low setup costs.

The L4L-FO heuristic, despite its simplicity, is rarely dominated because of its high speed and a competitive gap of 0.24% on average. CPLEX is very sensitive to variations in parameters, such as demand variability, demand volume, and setup costs, while the heuristics are more stable in both criteria.

6.4.2. Large Instances

As was observed on instances with $T=24$ time periods, CPLEX takes more than half an hour and several hours in some cases to find the optimal solution for several instances. Hence, for larger instances with $T=48$ and $T=96$, CPLEX was not run on these instances, and the analysis is conducted by comparing the heuristics with each other. In

Table 6. Comparing heuristics for instances with $T=48$ time periods for low and high parameter values.

Variant	Parameter		VRF-FO		PRF-FO		TRF-FO		L4L-FO
			CPU (s)	GAP’ (%)	CPU (s)	GAP’ (%)	CPU (s)	GAP’ (%)	CPU (s)	GAP’ (%)
Infinite capacity	D_Var	Low	34.48	0.59	159.67	0.26	56.95	0.00	25.89	0.25
		High	24.08	0.77	9.31	0.95	19.88	0.00	15.85	0.72
	D_Val	Low	39.67	0.50	27.23	0.45	36.77	0.00	19.70	0.14
		High	10.19	1.20	6.66	1.10	10.15	0.00	11.93	1.11
	Setup	Low	10.19	1.27	6.34	0.31	9.45	0.00	9.87	1.20
		High	55.64	0.61	115.23	0.63	59.09	0.00	46.88	0.10
No Shared capacity	D_Var	Low	8.42	1.24	9.80	0.96	9.91	0.00	12.97	0.77
		High	13.14	1.10	5.08	0.57	8.56	0.00	10.87	0.60
	D_Val	Low	17.72	1.19	8.62	0.48	11.38	0.00	11.47	0.96
		High	7.07	1.09	4.77	0.26	5.60	0.00	5.93	0.61
	Setup	Low	8.28	1.11	4.26	0.14	4.80	0.00	8.29	0.58
		High	14.27	1.34	9.47	0.19	11.55	0.00	12.13	1.21
	Capacity	Low	9.02	1.51	6.79	1.00	7.19	0.00	11.13	1.05
		High	12.88	1.37	7.82	0.84	8.56	0.00	8.78	1.78
Shared capacity	D_Var	Low	12.76	1.46	5.26	0.10	11.84	0.00	15.74	1.67
		High	21.16	0.40	6.58	0.35	9.50	0.00	10.48	0.76
	D_Val	Low	17.37	0.98	9.18	0.11	10.37	0.00	10.71	1.43
		High	9.59	0.98	5.23	0.32	7.43	0.00	9.72	1.01
	Setup	Low	12.75	1.04	4.13	0.54	6.32	0.00	9.81	1.15
		High	19.51	1.18	9.90	0.38	14.85	0.00	15.56	0.50
	Capacity	Low	11.87	1.87	6.99	0.87	12.25	0.00	14.30	1.26
		High	18.54	1.34	6.72	0.71	13.19	0.00	9.61	1.29
Shared capacity Joint setup	D_Var	Low	5.78	0.00	7.30	4.63	15.24	4.47	14.32	5.78
		High	15.93	1.79	5.45	0.47	9.69	0.00	12.15	0.48
	D_Val	Low	23.25	1.09	10.06	0.08	10.60	0.00	14.44	0.51
		High	9.45	1.22	5.77	0.65	8.36	0.00	11.96	1.38
	Setup	Low	13.50	0.83	5.46	0.00	7.43	0.15	12.13	0.86
		High	16.98	1.74	10.93	0.71	16.78	0.00	15.68	0.99
	Capacity	Low	13.69	1.22	8.53	0.97	14.89	0.00	14.86	0.96
		High	15.21	1.25	7.16	0.31	14.80	0.00	12.94	1.04
	Setup Time	Low	16.59	1.80	9.37	0.90	14.61	0.00	14.23	1.33
		High	16.96	0.79	8.32	0.98	12.68	0.00	16.13	1.12
Average			16.75	1.12	16.04	0.66	14.71	0.14	13.95	1.08

and

Table 7. Comparing heuristics for instances with $T=96$ time periods for low and high parameter values.

Variant	Parameter		VRF-FO		PRF-FO		TRF-FO		L4L-FO
			CPU (s)	GAP’ (%)	CPU (s)	GAP’ (%)	CPU (s)	GAP’ (%)	CPU (s)	GAP’ (%)
Infinite capacity	D_Var	Low	30.70	2.72	300.96	0.30	92.38	0.00	42.12	2.02
		High	47.75	3.44	15.24	0.00	39.95	0.24	29.58	3.45
	D_Val	Low	79.01	1.93	70.89	0.34	89.69	0.00	65.73	1.29
		High	14.95	2.56	11.69	0.12	24.01	0.00	21.71	3.05
	Setup	Low	15.79	2.87	10.06	0.00	23.17	0.06	19.51	4.60
		High	80.94	1.93	235.00	0.23	118.07	0.00	47.98	1.41
No Shared capacity	D_Var	Low	10.63	1.18	12.61	0.16	18.36	0.00	13.49	4.75
		High	22.49	3.38	12.21	0.00	16.42	0.12	29.88	4.04
	D_Val	Low	19.38	3.85	14.04	0.00	21.33	0.01	32.61	3.96
		High	10.42	2.93	9.70	0.08	10.34	0.00	13.44	7.13
	Setup	Low	9.87	2.45	11.27	0.00	11.81	0.23	14.91	6.06
		High	17.31	4.25	14.70	0.00	20.75	0.51	27.53	5.02
	Capacity	Low	11.77	2.77	12.19	0.93	17.49	0.00	19.12	4.78
		High	15.24	4.14	13.17	0.00	18.05	0.40	28.03	3.96
Shared capacity	D_Var	Low	20.05	4.95	11.57	0.00	17.89	1.18	25.18	5.46
		High	28.00	4.47	12.10	0.00	17.87	0.65	27.17	2.92
	D_Val	Low	32.28	4.30	15.18	0.00	19.92	1.09	37.04	3.17
		High	11.94	2.45	9.82	0.00	14.14	0.52	17.37	5.84
	Setup	Low	10.73	2.54	11.11	0.00	14.36	0.38	17.20	6.23
		High	33.75	3.35	17.29	0.00	21.55	0.06	42.84	3.26
	Capacity	Low	19.66	4.03	12.04	0.32	18.08	0.00	28.32	3.93
		High	26.09	3.84	16.29	0.00	19.68	0.60	34.63	2.84
Shared capacity Joint setup	D_Var	Low	17.43	2.82	14.96	0.00	22.13	0.15	33.36	3.66
		High	43.43	3.22	13.07	0.00	25.30	0.29	32.11	3.46
	D_Val	Low	54.91	4.06	20.20	0.00	23.67	0.57	38.87	3.50
		High	13.07	2.49	10.16	0.00	15.53	0.67	20.20	5.87
	Setup	Low	12.91	3.39	11.04	0.00	15.26	1.04	21.38	6.48
		High	44.86	3.86	19.67	0.00	26.86	0.76	36.27	3.89
	Capacity	Low	30.70	3.97	15.58	0.00	19.83	0.01	27.47	3.12
		High	32.03	3.59	16.92	0.00	20.49	0.23	29.97	3.12
	Setup Time	Low	35.10	3.17	16.81	0.00	20.82	0.40	34.48	2.65
		High	27.28	4.60	17.36	0.03	21.09	0.00	31.98	3.23
Average			27.51	3.30	31.40	0.08	27.38	0.32	29.42	4.00

, the new gap ($GA{P}^{\prime }$) is formulated as

$$GA{P}^{\prime }=100\times {\displaystyle \frac{z^H-z^{BEST}}{z^{BEST}}},$$

where $Z^H$ is the objective function value of the heuristic and $Z^{BEST}$ is the best known objective function value obtained by any one of the four heuristics. The values indicated in boldface characters indicate those cases where a given heuristic has obtained the best results among the four.

The heuristic TRF-FO provided the best results on all instances with $T=48$ (Table 6), except in Variant 3 (shared capacity and joint setups) when demand variability was low or when setup costs were small. In the former case, the VRF-FO heuristic performs much better than PRF-FO, TRF-FO, and L4L-FO. Nevertheless, on average, the worst performance is that of VRF-FO with a 1.12% gap from the best known solution. This makes it dominated by L4L-FO, which is still the fastest heuristic (as in instances with $T=24$) and with a gap of 1.08%.

For the largest instances with $T=96$ (Table 7), the best results were obtained using the PRF-FO heuristic in 75% of the cases (24 out of 32 in Table 7). The other 25% best results were obtained using the TRF-FO heuristic. Table 7 shows that these two heuristics (PRF-FO and TRF-FO) are the only ones that are not dominated. The decision maker can choose between a fast heuristic with relatively good performance (TRF-FO) and a slightly slower heuristic with the best performance (PRF-FO).

The average CPU time required by the heuristics is proportional to the length of the planning horizon. Figure 14 shows the CPU time of the heuristics for different variants with time horizons of $T=24,48,$ and 96 periods. These figures are based on the average CPU times calculated from Table 5, Table 6 and Table 7 for the infinite capacity (NC), limited capacity (C), shared capacity (SC), and shared capacity with joint setups (SCJ) variants. Again, this figure shows that the infinite—capacity instances are the hardest to solve. Also, heuristics TRF-FO and PRF-FO are the most sensitive to the increase in horizon length in this case. The CPU time for PRF-FO heuristic increased by almost six times when the horizon length increased from 24 to 96 time periods. On the other hand, we observe that the heuristics performance is comparable between the last three variants (C, SCNJ, and SCJ).

However, there are instances whose complexity is comparable to that of problems without the joint setup. This is the case for problems with medium values of the tested parameters. This could be because we adjusted the capacities to ensure feasibility (see Table 3). It can also be observed that the heuristic still provides very good-quality solutions with an average gap, for example, of less than 0.2 % in the case of average parameter values.

6.5. Discussion

The numerical experiments above show that integration through the sharing of inventories of returns leads to considerable cost and environmental benefits. The presented simulations indicate the relationship between the variation of parameters and their impact on cost and environmental indicators. This will allow the decision to identify the range of parameters within which substitution is effective. We have noticed, for example, that the environmental and cost indicators are very sensitive to variations in refurbishing costs.

The proposed heuristics give solutions very close to the optimum with gaps from the optimum of less than 0.5% on average. The four heuristics are rather fast. They take on average less than 11 s, 17 s, and 32 s to solve instances with time horizons of 24, 48, and 96 time periods, respectively. The advantage of such heuristics is that they are easy to implement and understand. However, they rely on the use of a solver (here a commercial one). Developing a solver-independent approach is often preferred by practitioners. Metaheursitics such as the Large Neighborhood Search or decomposition heuristics such as the Lagrangian relaxation can be good alternatives. Another limitation related to these results is the fact that the comparison was made on only one solver: CPLEX. Other commercial solvers can be tested for both the evaluation and the implementation of the heuristics. Finally, the heuristics have been designed for the single-product case. The extension to problems with multiple products or multi-level structures will require further developments and adaptation of the heuristics.

7. Conclusions

We studied a hybrid system in a facility that receives returned items and is capable of refurbishing, disassembling, and manufacturing new products. The company has purchased new assembly lines of smartphones but faced a shortage of spare parts. Hence, it wanted to use the new line for refurbishing returned products from the market. The first objective was to propose configurations with substitution and different levels of integration between these processes. The selection criteria for the configuration were the environmental impact and total processing costs. The second objective was to propose efficient approaches for solving large problem instances. Hence, the first research question was about which configuration is the most cost-effective and which is the most ecological. The second question was about how to formulate and solve the production planning problem in the most efficient way.

Five configurations were proposed. The most basic configuration assumed that returns are solely used for refurbishing, and the manufacturing line of new products uses only new parts. The other configurations were constructed gradually by allowing the use of disassembled parts in the manufacturing of new products and allowing the sharing of returns between the refurbishing and disassembly processes. With respect to our first objective, numerical experiments showed that integration through the sharing of inventories of returns leads to considerable cost and environmental benefits. In terms of substitution, we have presented several managerial insights that would allow the decision-maker to identify the range of parameters that would make substitution cost-effective. Our contribution to sustainable manufacturing is reflected in the suggestion of alternative process designs with different levels of integration and substitution to encourage collection and refurbishing.

The second objective of this study concerned the suggestion of an efficient solution approach for the associated production planning problem. We proposed four heuristics based on relax-and-fix and fix-and-optimize approaches. The heuristics give solutions that are very close to optimum with gaps from optimum of less than 0.5% on average. The four heuristics are rather fast, as they take on average less than 11 s, 17 s, and 32 s to solve instances with time horizons of 24, 48, and 96 time periods.

In this study, we limited our attention to problems with a single type of product. In practice, the company uses the same capacity to refurbish and manufacture different families and types of products. Evaluating the effects of multiple products on the presented configurations is an interesting topic for future work. This could concern products of the same category but with different finishing levels (high to low end) or completely different models that can be manufactured or refurbished on the same lines. Our models assume that prices are taken as parameters, and an important question in practice is how to set the price of refurbished items. Taking a revenue-management perspective, where lot sizes and prices are jointly optimized, will make the emerging problems more challenging to solve but will result in better system performance. It is well known in the literature that lot-sizing models with pricing decisions are mathematically challenging [6], while allowing more flexibility by setting the price as a decision variable will result in higher profit. Future research could also consider additional indicators to enhance the sustainability and efficiency of hybrid manufacturing and refurbishing systems. These indicators include measuring the carbon footprint reduction achieved through refurbishing, assessing the material recovery rate, and evaluating the energy efficiency in refurbishing processes compared to new manufacturing. Other important indicators are tracking the average product lifecycle extension, determining the waste diversion rate, and gauging customer satisfaction with refurbished products. Analyzing the economic viability of refurbishing versus new manufacturing, assessing supply chain resilience, monitoring regulatory compliance, and tracking the rate of innovation in refurbishing technologies are also crucial. Incorporating these indicators will provide a comprehensive understanding of the impacts and benefits of sustainable hybrid manufacturing and refurbishing systems, driving further advancements in the field.

Overall, sustainability in manufacturing units can be achieved through a better management of returns. At the strategic level, this involves the design of different processes to allow the sharing of resources for manufacturing and refurbishing/remanufacturing and to allow flexibility through substitution. The resulting challenging production-planning problem needs to be handled with efficient solution approaches using operations research tools.