On the dynamics of closed-loop supply chains with capacity constraints

Open Research Online The Open University’s repository of research publications and other research outputs On the dynamics of closed-loop supply chains with capacity constraints Journal Item How to cite: Dominguez, Roberto; Ponte, Borja; Cannella, Salvatore and Framinan, Jose M. (2019). On the dynamics of closed-loop supply chains with capacity constraints. Computers & Industrial Engineering, 128 pp. 91–103. For guidance on citations see FAQs. c [not recorded] Version: Accepted Manuscript Link(s) to article on publisher’s website: http://dx.doi.org/doi:10.1016/j.cie.2018.12.003 Copyright and Moral Rights for the articles on this site are retained by the individual authors and/or other copyright owners. For more information on Open Research Online’s data policy on reuse of materials please consult the policies page. oro.open.ac.uk Dominguez, R., Ponte, B., Cannella, S., Framinan, J.M. 2019. On the dynamics of closed-loop supply chains with capacity constraints. Computers & Industrial Engineering, 128, 91-103. On the dynamics of closed-loop supply chains with capacityconstraints Abstract In this paper, we investigate the dynamic behaviour of a closed-loop supply chain with capacity restrictions both in the manufacturing and remanufacturing lines. We assume it operates in a context of a twofold uncertainty by considering stochastic demand and return processes. From a Bullwhip perspective, we evaluate how the four relevant factors (specifically, the two capacities and two sources of uncertainty) interact and determine the operational performance of the system by measuring the variability of the manufacturing and remanufacturing lines and the net stock. Interestingly, while the manufacturing capacity only impacts on the forward flow of materials, the remanufacturing capacity affects the dynamics of the whole system. From a managerial viewpoint, this work suggests that capacity constraints in both remanufacturing and manufacturing lines can be adopted as a fruitful bullwhip-dampening method, even if they need to be properly regulated for avoiding a reduction in the system capacity to fulfil customer demand in a cost-effective manner. Keywords: Bullwhip effect; supply chain dynamics; reverse logistics; capacity limitations; simulation. 1. INTRODUCTION An almost-ubiquitous problem occurring in supply chains (SC) is the so-called bullwhip effect (BWE) (Lee et al. 1997), which refers to the fact that even small variations in customer demand may generate high alterations in upstream production for suppliers (Huang et al. 2017, Lin et al. 2017, Dominguez et al. 2014). This phenomenon has important consequences in real-life SC (see e.g. Zotteri 2013, Isaksson and Seifert 2016, Chiang et al. 2016, Trapero and Pedregal 2016, Jin et al. 2017, de Oliveira Pacheco et al. 2017, Lin et al. 2017, Pastore et al. 2017). Indeed, recent empirical works have shown that BWE may emerge in two-thirds of firms from USA (Bray and Mendelson 2012) and Chine (Shan et al. 2014) and that its consequences are, by nature, global and impact both developed and developing countries, as pointed out by the European Central Bank (Altamonte et al. 2012), the European Bank for Reconstruction and Development Working (Zavacka et al. 2012) and by the World Bank (Ferrantino and Taglioni 2014). Some of the consequences of BWE are excess of inventory, poor customer service and poor product forecasts (Trapero et al. 2012). In the last two decades, different efforts to explain and reduce the BWE have emerged and continue to grow (Wang and Disney 2016). However, even if a number of advances have been made for limiting BWE, there is still substantial room for improvement. More specifically, after conducting the most recent literature survey on the BWE, Wang and Disney (2016) identify several opportunities for Dominguez, R., Ponte, B., Cannella, S., Framinan, J.M. 2019. On the dynamics of closed-loop supply chains with capacity constraints. Computers & Industrial Engineering, 128, 91-103. future research such as BWE in complex system, with pricing considerations, in service chain, with research competition, etc. Among those, two stand out, i.e.: investigating the BWE under capacityconstrained and the dynamics of Closed-Loop SC (CLSC). Manufacturing firms are fundamental in supporting most modern economies (Trapero et al. 2015). Consequently, studying the impact of manufacturing capacity constraints in SC dynamics has been an issue in the past years. Capacity constraints usually refer to considering limits on the order sizes placed to suppliers, or limits on the orders’ acceptance channel. In this regard, literature has shown that such interpretation of capacity can stabilize the orders and generate a smoothing effect on production (see e.g., Evans and Naim 1994, Chen and Lee 2012, Shukla and Naim 2017, Ponte et al. 2017, Framinan 2017). However, these restrictions may negatively impact on inventory holding costs and customer service level (Cannella et al. 2008, Nepal et al. 2012, Spiegler and Naim 2014, Hussain et al. 2016). In general, works dealing with the implications of capacity limits on the dynamics performance of the SC are relatively scarce (Ponte et al. 2017) and, to the best of authors’ knowledge, their subject of study is a traditional forward SC as opposed to a CLSC. In a CLSC, recycling and remanufacturing activities – i.e., taking back products from customers and returning them to the original manufacturer for the recovery of added-value by reusing the whole product or part of it (Genovese et al. 2017) — are implemented (Jerbia et al. 2018). CLSC design is the desired business model for companies due to the potential value recovery, environmental sustainability, and special importance given by the customers (Jabbarzadeh et al. 2018). In the last decade, some works have been exploring the characteristic the CLSC, specifically by focusing on how some key factors of this structure (e.g., the percentage of return yields, the remanufacturing lead-time, and the adoption of different order policies) may impact on the performance in terms of BWE, inventory stability and customer service level. Particularly, most of studies have shown that increasing the percentage of return yields can reduce the BWE (see e.g., Tang and Naim 2004, Zhou and Disney 2006, Hosoda et al. 2015, Cannella et al. 2016, Zhao et al. 2018). However, to the best of the authors’ knowledge, these studies assume infinite production capacity. In the light of the above-mentioned results, we argue that exploring the dynamics behavior of a capacitated CLSC by understanding how a limitation in the capacity of the manufacturing and remanufacturing lines impacts on SC performance can be reasonably considered a major challenge for OM communities. Hence, in this work we aim to shed light on this topic and, to fulfil the research objective, we model a CLSC via difference equation modelling (Riddals et al. 2000) characterized by a limitation in both manufacturing and remanufacturing operations. Moreover, given the need of modern SCs for surviving and thriving in turbulent and volatile environments (Wikner et al. 2017), we consider stochasticity in both the return yield and the customer demand. Thus, we perform a rigorous Design of Experiment (DoE) considering four key Dominguez, R., Ponte, B., Cannella, S., Framinan, J.M. 2019. On the dynamics of closed-loop supply chains with capacity constraints. Computers & Industrial Engineering, 128, 91-103. factors, i.e., (1) the variability of the return yields, (2) the capacity factor of the manufacturer, (3) the capacity factor of the remanufacturer, and (4) the variability of the customer demand. The results of this works reveal that a low capacity in the remanufacturer may smooths the bullwhip effect in the fabrication of both new and remanufactured products while maintaining a good inventory performance. However, if capacity is reduced below certain threshold value, it can also generate detrimental consequences in terms of inventory holding costs and customer service level. From a managerial point of view, this work suggests that imposing capacity limits in both remanufacturing and manufacturing processes can be adopted as a bullwhip-dampening method. In order to the set suitable capacity of both nodes, managers should also take into account degree of uncertainty of both the market demand and the return yield. The rest of the paper is organized as follows: Section 2 presents a literature review of studies dealing with BWE, capacity constraints and CLSC. Section 3 details the model of the capacitated CLSC and the key performance indicators employed. Section 4 describes the experimental design, while Section 5 shows the results obtained from the simulations. Section 6 contains the summary of findings and managerial implications. Finally, Section 7 presents the main conclusions of the work. 2. LITERATURE REVIEW In this section, we provide an overview of the previous works investigating the BWE assuming capacity constraints, or in a CLSC. As discussed in the previous section, although a number of contributions have been produced in this areas separately, we are not aware of any work jointly investigating these two aspects. 2.1. The impact of capacity constraints on supply chains In BWE literature, the problem of capacity constraint has been addressing in few studies, usually developed by adopting methodologies based on the dynamics of the systems (i.e., control theory and what-if simulation, such as continuous and discrete-event simulations). Among these work, to the best of the authors’ knowledge, Evans and Naim (1994) can be considered the seminal work. Via differential equation modelling, the authors conclude that the capacity constraints may improve the behavior of SC in terms of bullwhip effect and inventory stability, but at the expense of reducing the inventory service levels. Essentially, Evans and Naim (1994), show for the first time that an unconstrained SC does not always produce the best response. Contrarily, De Souza et al. (2000), using system dynamics, conclude that SC performance can be seriously affected by capacity shortages. In this fashion, they suggest that capacity planning is central for the dynamics of the SC. Analogously, Helo (2000), also via system dynamics, suggests that a limited capacity negatively impact the responsiveness of the SC. Vlachos and Tagaras (2001), by adopting both analytical methods and simulation, show that imposing capacity limits reduces Dominguez, R., Ponte, B., Cannella, S., Framinan, J.M. 2019. On the dynamics of closed-loop supply chains with capacity constraints. Computers & Industrial Engineering, 128, 91-103. the system’s response, particularly for long production lead-time. Similarly to Evans and Naim (1994), Wilson (2007), through system dynamics modelling, finds out that short-term limitations on capacity may produce a poor customer service level; however, they can improve the SC behavior. Analogously, Cannella et al. (2008), via differential equations modelling, show that the BWE is reduced if capacity limits are imposed, but they also can create a significant stock-out phenomenon. Boute et al. (2009), via analytical method, demonstrate that inflexible limits on capacity generate stochastic lead times and thus they amplifies the desired inventory on-hand and, in general, the operational costs. Juntunen and Juga (2009), via discrete event simulation show, in line with previous studies, that the fill rate does not necessarily improve by increasing the capacity limitation in distribution. Contrarily, Hamdouch (2011), by adopting a network equilibrium method, shows that capacity limitations generate poor market response and SC behavior. Interestingly, Nepal et al. (2012), via differential equations modelling, report that capacity restriction does not have a significant impact on the order variability but, in contrasts, it can strongly affect the stability of the inventory. Chen and Lee (2012), via mathematical analysis, in line with those studies showing the benefits of capacity constraints in terms of BWE reduction, argue that considering a fixed capacity in SC reduces the BWE. Contrarily, Spiegler and Naim (2014), via system dynamics show that capacity restrictions have a negative effect on both inventory and service customer levels, even if it emerges a positive impact on the ‘backlash’ effect (i.e., BWE on transportation). In line with most of the previous studies, Hussain et al. (2016), using differential equations modelling, show that restrictions in the order size due to capacity limitation may avoid “phantom” large orders value, a similar conclusion to that by Shukla and Naim (2017) via system dynamics modelling. Finally, Framinan (2017) analytically demonstrates that if capacity refers to the rejection of orders in excess of a given threshold, then capacity dampens the BWE. In summary, most of the above-mentioned studies have reported contradictory results regarding the impact of capacity constraints on the dynamics of the SC. However, most of them, by adopting different methodological approaches, agree on the positive impact of the capacity limitations on BWE since these restrictions dampen order variability. Overall, none of the studies investigates how capacity limitations may impact the dynamics of a CLSC, particularly when these affect to both manufacturer and remanufacturer. 2.2. The dynamics of closed-loop supply chains In BWE literature, similarly to the capacity-constrained SC, the CLSC has not been receiving special attention. Only in the last few years, thanks to the challenges advocated by the sustainability principles, an increasing number of studies dealing with the impact of considering both forward and reverse flows has been detected. Historically, the work of Tang and Naim (2004) can be considered the first effort in Dominguez, R., Ponte, B., Cannella, S., Framinan, J.M. 2019. On the dynamics of closed-loop supply chains with capacity constraints. Computers & Industrial Engineering, 128, 91-103. analyzing the BWE in a CLSC. The authors, via difference equation modelling, study three ad-hoc order policies for hybrid manufacturing/remanufacturing systems. They conclude that increasing recollected products and operating with higher information transparency on the pipeline of the remanufacturer may strongly improve the performance of a CLSC. By using control theory, Zhou and Disney (2006) analyze the impact on the inventory variance and demand amplification of a combined “in-use” and remanufacturing lead-time and the return rate. Zanoni et al. (2006) use a discrete-event simulation model to carry out a comparative study between four different replenishment rules in terms of order amplification. They show how the BWE of the downstream (forward) flow in the SC can be reduced in the dual policy, while the BWE of the upstream flow (reverse) can be reduced in the shifted pull policy. Pati et al. (2010) use a statistical analysis on a six-stage reverse SC and conclude that the reverse flow does not experience a demand amplification. By means of agent-based simulation, Adenso-Díaz et al. (2012) analyze the impact of 12 factors in both forward and reverse SC and do not detect significant differences between the performances of the two SC structures in terms of order rate amplification. Turrisi et al. (2013), via difference equation modelling, propose a novel replenishment rule to coordinate the upstream and downstream flows in a CLSC and show that a reduction of BWE can be obtained by increasing return rate of recollected products. However, they do not find significant differences in terms of inventory variance. Analogously, Corum et al. (2014) employ a discrete-event simulation model to show that a CLSC allows reducing the demand amplification phenomenon. Hosoda et al. (2015), via analytical methods, study the impact of the correlation between demand and returns, and show that increasing the yield may have a negative effect in terms of inventory variability. Cannella et al. (2016) employ difference equation modelling to show that shifting from a forward SC to a CLSC always generates benefits in terms of inventory and order variances, both in stable and turbulence market scenarios. Dev et al. (2017), via difference equation modelling, conclude that, in a CLSC, continuous review policies outperform the periodic review policies in terms of BWE. Zhou et al. (2017) study the quality of recollected product in different levels of the SC using difference equation modelling, and show that a higher return yield decreases the BWE. The magnitude of this reduction depends on the combination of control parameters (i.e, the degree of return yield at each echelon and the lead-times in the CLSC). Hosoda and Disney (2017), via analytical method, explore the socalled “lead time paradox” in CLSCs, which refers to the scenarios in which increasing the remanufacturing lead-time sometimes decreases the cost. They show that shortening the remanufacturing lead time does not contribute to lower inventory costs but could generate some other benefits, such as lower capacity cost and in-transit inventory. Sy (2017) employs system dynamics to analyse a hybrid production-distribution system and show that, under three scenarios, and show that the centralization of the customer demand information reduces the BWE.. Similarly, Zhao et al. (2018) study, via system dynamics, the impact of three orders policies based on the degree of shared information in a CLSC. In line Dominguez, R., Ponte, B., Cannella, S., Framinan, J.M. 2019. On the dynamics of closed-loop supply chains with capacity constraints. Computers & Industrial Engineering, 128, 91-103. with literature on information sharing, they conclude that centralized demand information and a vendor managed inventory reduce both bullwhip and inventory variability. In summary, previous studies show a lack of consensus on the impact of BWE and inventory variance when a CLSC is the subject of research. As remarked by Cannella et al. (2016) and Zhao et al. (2018), these conflicting results may depend on different SC configurations and modelling assumptions, particularly with respect to the remanufacturing lead-time. However, it can be noticed a general agreement on the impact of the return yield, as most studies note that the BWE can be reduced by increasing the percentage of products recollected from the market for remanufacturing. However, to the best of the authors’ knowledge, there is no evidence on how a CLSC performs if capacity limitations are considered both in the forward and reverse production flows. 3. CLOSED-LOOP SUPPLY CHAIN MODEL Figure 1 provides an overview of the hybrid manufacturing/remanufacturing system considered in this research work, together with its main parameters. This CLSC is described in detail in the following paragraphs. Figure 1. Structure of the hybrid manufacturing/remanufacturing system. The CLSC modelled integrates both manufacturing and remanufacturing processes into the same SC and operates on a discrete-time basis, being the time unit t. We consider two sources of stochasticity, i.e. the consumer demand (𝑑𝑡) and the returns (𝑟𝑡). As usually assumed in this field, the demand is an independent and identically distributed (i.i.d.) random variable (𝑥𝑡) following a normal distribution with mean μ and standard deviation σ, being the coefficient of variation 𝐶𝑉𝑑= 𝜎/𝜇, which is constrained to only positive values. That is, Dominguez, R., Ponte, B., Cannella, S., Framinan, J.M. 2019. On the dynamics of closed-loop supply chains with capacity constraints. Computers & Industrial Engineering, 128, 91-103. 𝑑𝑡= max{𝑥𝑡,0},𝑥𝑡→ 𝑁(𝜇,𝜎2). (1) In order to account for the stochasticity of the returns, we model the return yield (𝑧𝑡), i.e. the percentage of sold products that come back to the SC after consumption, as a i.i.d. random variable (𝑦𝑡) following a normal distribution with mean β and standard deviation ξ, being the coefficient of variation 𝐶𝑉 𝑟= 𝜉/𝛽, which has been constrained to values between 0 and 1. This approach allows us to model the returns as the product of the yield and the demand before a constant consumption lead time 𝑇𝑐. Similarly, this variable has been constrained to prevent negative values from happening, which would be meaningless in practice. 𝑟𝑡= 𝑦𝑡𝑑𝑡−𝑇𝑐,𝑦𝑡=min⁡{max{𝑧𝑡,0},1},𝑧𝑡→ 𝑁(𝛽,𝜉2). (2) Each period t, the operation of the hybrid manufacturing/remanufacturing system can be divided into three sequential stages, which are detailed below, including the associated mathematical formulation. 3.1. Stage I: Reception, settling and feeding At the beginning of each period t, the serviceable inventory receives the product from both the manufacturer (new products) and remanufacturer (assuming as-good-as-new products) processes, once these have been completed after the respective constant lead times 𝑇𝑚 and 𝑇𝑟. In this sense, the serviceable inventory is ready for facing the consumer demand that will be received during this period. Moreover, the raw material inventory provides the manufacturing equipment with the quantity required according to the order issued at the end of the previous period. Similarly, the returns collected during the previous period are fed into the remanufacturing process, which hence operates according to a push policy — Hosoda and Disney (2017) justifies that this common assumption fits well with the ethics of sustainability. In this regard, we note that the capacity constraints of the manufacturing and remanufacturing process, respectively 𝜓𝑚 and 𝜓𝑟, play a key role. If we take into consideration the capacity required for both processes under a stability situation defined by the mean values 𝜇 (for the demand) and 𝛽 (for the returns), that is, (1 − 𝛽)𝜇 for the manufacturing process (i.e. the average net demand) and 𝛽𝜇 for the remanufacturing process (i.e. the average returns); we define the coefficients of capacity as 𝐶𝑜𝐶𝑚= 𝜓𝑚/[(1 − 𝛽)𝜇] and 𝐶𝑜𝐶𝑟= 𝜓𝑚/[𝛽𝜇]. Note that these coefficients inform about the excess capacity available in relative terms. We note that to ensure the stability of the system, both must be greater than the unity. Under these circumstances, the manufacturing completion rate responds to the order placed 𝑇𝑚+ 1 periods ago, as long as there is capacity available, by Dominguez, R., Ponte, B., Cannella, S., Framinan, J.M. 2019. On the dynamics of closed-loop supply chains with capacity constraints. Computers & Industrial Engineering, 128, 91-103. 𝑚𝑐𝑡=min{𝑜𝑡−𝑇𝑚−1 +𝑚𝑏𝑡−1,𝜓𝑚}. (3) As Equation (3) illustrates, it is also necessary to consider the manufacturing backlog (𝑚𝑏𝑡) which measures the pending orders that could not be processed when required and will be considered as soon as capacity becomes available. This variable can be expressed by 𝑚𝑏𝑡= max{𝑜𝑡−𝑇𝑚−1 +𝑚𝑏𝑡−1 − 𝜓𝑚,0}. (4) It can be easily checked that if 𝑜𝑡−𝑇𝑚−1 +𝑚𝑏𝑡−1 ≥ 𝜓𝑚, the manufacturing system has no pending work, i.e. 𝑚𝑏𝑡= 0, while 𝑜𝑡−𝑇𝑚−1 +𝑚𝑏𝑡−1 < 𝜓𝑚 would result in pending orders, i.e. 𝑚𝑏𝑡> 0. The rationale employed for modelling the remanufacturing line is similar if assuming that it operates according to a push policy. For this reason, the remanufacturing completion rate (𝑟𝑐𝑡) corresponds to the returns collected 𝑇𝑟+ 1 periods ago, as long as the remanufacturing capacity allows it, by 𝑟𝑐𝑡=min{𝑟𝑡−𝑇𝑟−1 +𝑟𝑏𝑡−1,𝜓𝑟}; (5) while the remanufacturing backlog (𝑟𝑏𝑡) would be expressed as 𝑟𝑏𝑡= max{𝑟𝑡−𝑇𝑟−1 +𝑟𝑏𝑡−1 − 𝜓𝑟, 0}. (6) Overall, the on-hand serviceable stock, or initial stock (𝑖𝑠𝑡), which is available for fulfilling the demand received during the period can be expressed as a function of the net stock (𝑛𝑠𝑡), or excess on-hand inventory at the end of the previous period, or, by 𝑖𝑠𝑡=𝑛𝑠𝑡−1 +𝑚𝑐𝑡+𝑟𝑐𝑡. (7) 3.2. Stage II: Manufacturing, serving, and returns collection During period t, orders from consumers are received. These are satisfied as long as on-hand inventory is available. In this sense, the final position of this inventory can be expressed as 𝑛𝑠𝑡=𝑖𝑠𝑡− 𝑑𝑡, (8) where positive values of this variable refer to holding and negative values indicate stock-outs, which will be satisfied as soon as possible (ideally, at the beginning of the next period). In this regard, the on-order inventory, or work-in-progress (𝑤𝑡), at the end of the period can be obtained by 𝑤𝑡= 𝑤𝑡−1 +(𝑜𝑡−1 −𝑚𝑐𝑡)+(𝑟𝑡−1 −𝑟𝑐𝑡). (10) Dominguez, R., Ponte, B., Cannella, S., Framinan, J.M. 2019. On the dynamics of closed-loop supply chains with capacity constraints. Computers & Industrial Engineering, 128, 91-103. Figure 3. Main effects plot for 𝛴𝑟𝑐. Figure 4. Main effect plots for 𝛴𝑛𝑠. 5.2. Interactions The first order interactions are shown in Figures 5 to 7. Firstly, looking into Figure 5, we observe significant interactions between 𝐶𝑜𝐶𝑚 and the other three factors. These interactions are particularly strong for the factors 𝐶𝑉𝑑 and 𝐶𝑉 𝑟 (see also F-Values in Appendix A), i.e., the reduction obtained for 𝛴𝑚𝑐 by reducing 𝐶𝑜𝐶𝑚 is more significant for higher values of 𝐶𝑉𝑑 and 𝐶𝑉 𝑟. The interaction between 𝐶𝑜𝐶𝑚 and 𝐶𝑜𝐶𝑟 is only observed for very low values of 𝐶𝑜𝐶𝑟 (𝐶𝑜𝐶𝑟=1,1), where 𝛴𝑚𝑐 is less sensitive to 𝐶𝑜𝐶𝑚. 𝐶𝑜𝐶𝑟 shows less Dominguez, R., Ponte, B., Cannella, S., Framinan, J.M. 2019. On the dynamics of closed-loop supply chains with capacity constraints. Computers & Industrial Engineering, 128, 91-103. interaction with 𝐶𝑉𝑑 and 𝐶𝑉 𝑟. This result implies that reducing 𝐶𝑜𝐶𝑟 from 2.1 to 1.1 produces a higher reduction of 𝛴𝑚𝑐 for lower/higher values of 𝐶𝑉𝑑 and 𝐶𝑉 𝑟, respectively, being the former interaction more significant than the latter (see also F-Values in Appendix A). Finally, there is also a significant interaction between 𝐶𝑉𝑑 and 𝐶𝑉 𝑟. Thus, we can conclude that 𝛴𝑚𝑐 is more sensitive to 𝐶𝑉 𝑟 for lower values of 𝐶𝑉𝑑. In Figure 6 the interaction plots for 𝛴𝑟𝑐 are shown. Assuming that 𝐶𝑜𝐶𝑚 has no impact on 𝛴𝑟𝑐, the most significant interactions take place between 𝐶𝑜𝐶𝑟 and the other two factors, 𝐶𝑉𝑑 and 𝐶𝑉 𝑟, being more significant the interaction with the latter factor (see F-Values in Appendix A). More specifically, the reduction of 𝛴𝑟𝑐 resulting from reducing 𝐶𝑜𝐶𝑟 is more significant for higher values of 𝐶𝑉𝑑 and 𝐶𝑉 𝑟. Finally, we analyze the interaction plots for σ_NSt (see Figure 7). The most significant interactions take place between 𝐶𝑜𝐶𝑚 and 𝐶𝑉𝑑, and between 𝐶𝑜𝐶𝑟 and 𝐶𝑉 𝑟, being the former more significant than the latter (see F-Values in Appendix A). More specifically, it can be observed that, when 𝐶𝑜𝐶𝑚 is reduced below 2.1, the increase in 𝛴𝑛𝑠 is higher for higher values of 𝐶𝑉𝑑. Similarly, when 𝐶𝑜𝐶𝑟 is reduced below 1.6 the increase in 𝛴𝑛𝑠 is higher for higher values of 𝐶𝑉 𝑟. Figure 5. Interaction plot for 𝛴𝑚𝑐. Dominguez, R., Ponte, B., Cannella, S., Framinan, J.M. 2019. On the dynamics of closed-loop supply chains with capacity constraints. Computers & Industrial Engineering, 128, 91-103. Figure 6. Interaction plot for 𝛴𝑟𝑐. Figure 7. Interaction plot for 𝛴𝑛𝑠. Dominguez, R., Ponte, B., Cannella, S., Framinan, J.M. 2019. On the dynamics of closed-loop supply chains with capacity constraints. Computers & Industrial Engineering, 128, 91-103. 6. SUMMARY OF FINDINGS AND MANAGERIAL IMPLICATIONS We now summarize the main findings and contributions of our work. These are summarized in three different findings, referring to the different performance metrics (i.e., the BWE of the manufacturing and the remanufacturing line, and the variability of the net stock). We present also interesting implications for managers, suggesting different ways to improve the dynamic performance of a capacitated CLSC. (1) The capacity restriction in the manufacturing line of a CLSC limits the BWE placed by the manufacturer (as in a traditional forward SC). This limitation is more significant when the return yield or the customer demand present higher variability. However, the capacity constraints of the manufacturing line has no significant impact on the BWE of the remanufacturing line. Firstly, we reassert the evidence that capacity constraints may improve the dynamic performance of a SC by reducing the BWE of the manufacturer. As a practical implication, managers may consider to smooth the manufacturing process by limiting its maximum capacity, obtaining a higher performance improvement as the capacity limit is lower. This effect is especially important when there is a turbulent market demand or when the return yield is very uncertain. However, the limitation in the capacity of the manufacturing line does not affect the BWE of the remanufacturing line. (2) The capacity restriction of the remanufacturing line in a CLSC limits the BWE of the remanufacturing line, especially when the return yield or the customer demand present higher uncertainty. In addition, the capacity constraints of the remanufacturing line may limit the BWE of the manufacturing line, especially for high variability of the return yield or low variability of the customer demand. This novel finding allows to understand the impact of the capacity constraints of the remanufacturing line on the dynamic behavior of capacitated CLSCs. Limiting the capacity of the remanufacturing line has a positive impact on the stability of the remanufacturing process, also obtaining higher improvements as the capacity limits gets to lower values. As in the previous case (1), this impact is especially important when the market demand or the return yield present very variable conditions. In addition, the manufacturing line maybe also be benefited from the limitation of the remanufacturing capacity, but in a lower magnitude. More specifically, this benefit could be only appreciated when the capacity of the remanufacturing line is below a certain threshold value (see Figure 5). Furthermore, this effect is exacerbated when there is a high uncertainty of the return yield and (contrarily to the previous case) a more stable market demand. In summary, in capacitated (real-life) CLSCs, a further method for improving the dynamic behavior of both manufacturing and remanufacturing lines is to limit the capacity of the remanufacturing line, which is able to indirectly smooth instabilities in the manufacturing line. Dominguez, R., Ponte, B., Cannella, S., Framinan, J.M. 2019. On the dynamics of closed-loop supply chains with capacity constraints. Computers & Industrial Engineering, 128, 91-103. (3) Reducing either the capacity of the manufacturing line (particularly in case of a high variability of customer demand) or the remanufacturing line (particularly in case of a high variability of the return yield) below a certain threshold value has a negative impact on the variability of the net stock. However, this negative impact is more sensitive to the capacity of the manufacturing line than to the capacity of the remanufacturing line. This finding goes in countertendency with the previous findings, since it highlights the negative impact of capacity limitation of both lines on the variability of the net stock. While reducing capacity of the manufacturing/remanufacturing lines produce a continuous improvement in terms BWE, the variability of the net stock does not present significant changes until a threshold capacity value is reached. From that point on, the variability of the net stock suddenly increases as the capacity is smaller (see Figure 7). Interestingly, such threshold value seems to be very similar for both lines (around 160% of the mean customer demand). In the light of the above findings, we would recommend managers of a CLSC to take cautious decisions on the capacity planning of both manufacturing and remanufacturing processes. In fact, while the capacity of the manufacturing line has a major effect on the dynamic of the SC, the capacity of the remanufacturing line may also play an important role. By limiting both capacities (i.e., avoiding over-capacitated manufacturing/remanufacturing processes), it is possible to smooth the production of both new and remanufactured products. This decision needs to be taken carefully, since reducing capacity limits over a capacity threshold may have a negative impact on the dynamic of the net stock, thus increasing costs related to inventory holding costs and stock-outs. In this sense, and considering that both lines may share a common capacity threshold, it would be advisable to reduce the capacity of both processes until such threshold, thus smoothing both processes while maintaining a good performance of the net stock. Additional capacity reduction over such capacity threshold could be recommended only after a proper trade-off analysis between inventory holding costs/customer service level and production and remanufacturing costs. Finally, uncertainty in market demand and return yield accentuates the positive and negative effects discussed above. Thus, if the SC is characterized by uncertainty in both market demand and return yields, managers would be more willing to reduce capacity of both processes in order to alleviate the negative consequences of such uncertainties, while in the other hand, more consideration should be given to overstepping the capacity threshold. 7. CONCLUSIONS In this paper we explore the dynamic behavior of a capacitated CSLC. To do so, we modelled a monoechelon SC with reverse flow characterized by a capacity limitation in both manufacturing and Dominguez, R., Ponte, B., Cannella, S., Framinan, J.M. 2019. On the dynamics of closed-loop supply chains with capacity constraints. Computers & Industrial Engineering, 128, 91-103. remanufacturing processes. We adopted difference equation modelling approach and a rigorous DoE for assessing the impact of four key factors, i.e., the variability of the return yields, the capacity factor at manufacturing line, the capacity factor at the remanufacturing line and the variability of the customer demand. The most interesting results concerns the impact on the BWE of the remanufacturer capacity, which may influence the dynamics of the manufacturer. More specifically, a low capacity in the remanufacturer line may create a smoothing effect in the fabrication of both new and remanufactured products, but it can also generate detrimental consequences in terms of inventory holding costs and customer service level. From a managerial viewpoint, this work suggests that capacity constraints in both remanufacturing and manufacturing processes can be adopted as a BWE-dampening method. However, a proper tuning of these constraints should take into account the market environment and degree of uncertainty of the return yield. As this work is the first attempt to explore the dynamics of a CLSC with capacity constraints in both manufacturing and remanufacturing processes, it is clear that future works is needed to deepening our analysis. Firstly, more complex and real-life CLSC structures need to be analysed (e.g., multi-echelon, divergent structure, Dominguez et al. 2018, Cabral and Grilo 2018). As we do not focus on the impact of remanufacturing and manufacturing lead times, further studies may explore the effect of the interaction between these variables, such as the “lead-time paradox” advocated by Hosoda et al. 2015. Also, we assumed an i.i.d. demand, but other demand structures can be studied, such as the auto-correlated demand (see e.g. Babai et al. 2016). Furthermore, modelling the capacity is still an issue, since all the complexities of a real manufacturing system cannot be captured by considering a limitation in order quantity limitation to the orders placed to suppliers or limitation to the orders’ acceptance channel. Thus, further studies modelling load-dependent lead time, by adopting empirical from scheduling theory (see e.g. CT-TP curve, clearing functions, etc. Orcun, 2009, Mönch, 2013). Finally, the impact of inventory obsolescence (Babai et al. 2018) should also be explored in capacitated CLSCs. 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The impact of product returns and remanufacturing uncertainties on the dynamic performance of a multi-echelon closed-loop supply chain. International Journal of Production Economics, 183, 487-502. Zotteri G. 2013. An empirical investigation on causes and effects of the Bullwhip-effect: Evidence from the personal care sector. International Journal of Production Economics 143, 489-498. Dominguez, R., Ponte, B., Cannella, S., Framinan, J.M. 2019. On the dynamics of closed-loop supply chains with capacity constraints. Computers & Industrial Engineering, 128, 91-103. Appendix A Table A.1. Analysis of Variance for 𝛴𝑚𝑐. Table A.2. Analysis of Variance for 𝛴𝑟𝑐