Dynamic multi-period recycling collection routing with uncertain material quality

Cuellar-Usaquén, Daniel; Ulmer, Marlin W.; Antons, Oliver; Arlinghaus, Julia C. Article — Published Version Dynamic multi-period recycling collection routing with uncertain material quality OR Spectrum Provided in Cooperation with: Springer Nature Suggested Citation: Cuellar-Usaquén, Daniel; Ulmer, Marlin W.; Antons, Oliver; Arlinghaus, Julia C. (2025) : Dynamic multi-period recycling collection routing with uncertain material quality, OR Spectrum, ISSN 1436-6304, Springer, Berlin, Heidelberg, Vol. 47, Iss. 3, pp. 699-742, https://doi.org/10.1007/s00291-025-00808-z This Version is available at: https://hdl.handle.net/10419/330555 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Vol.:(0123456789) OR Spectrum (2025) 47:699–742 https://doi.org/10.1007/s00291-025-00808-z ORIGINAL ARTICLE Dynamic multi‑period recycling collection routing withuncertain material quality DanielCuellar‑Usaquén1· MarlinW.Ulmer2 · OliverAntons3,4· JuliaC.Arlinghaus3,4 Received: 16 February 2024 / Accepted: 6 January 2025 / Published online: 8 February 2025 © The Author(s) 2025 Abstract We consider the problem of collecting and processing waste material. At a production facility, a known amount of inventory is required for production (e.g., paper) for every period. Instead of new material, the facility relies on collected and processed waste material (e.g., paper waste). This material is collected from regional waste collection locations. The amount of waste material per location is uncertain, as is the quality of the collected waste, i.e., the resulting inventory when processing the material. If the inventory is insufficient at the end of a period, costly new material must be bought. Each period, decisions are made about how much waste material to collect from which location and how to route the collection vehicles accordingly. Ideally, inventory is built to hedge against quality uncertainty and to ensure efficient routing operations in future periods. We propose a stochastic lookahead method that samples a set of scenarios and solves a simplified two-stage stochastic program in every period. We show the value of our method for two case studies, one based on realworld data from Sachsen-Anhalt, Germany, and one from the literature with data from the United Kingdom. We further conduct a detailed analysis of our method and the problem characteristics. The results show that our method effectively anticipates all sources of uncertainty, reducing cost significantly compared to benchmark policies. This superior performance is due to appropriate state-dependent supplier selection that considers the percentage of material loss, available material, and routing cost for current and future periods. Keywords Routing· Circular economy· Sequential decision process· Stochastic lookahead Extended author information available on the last page of the article 700 D.Cuellar-Usaquén et al. 1 Introduction The circular economy proposes replacing linear sourcing of production materials via global supply chains with recycling local waste materials (such as paper, plastics, glass, metals, electronic parts, etc.). These materials are collected from local waste suppliers (e.g., waste facilities), cleaned, sorted, and then integrated into the production process. Circular economy projects kick off worldwide, and the European Union recently issued a Circular Economy Action Plan (https:// envir onment. ec. europa. eu/ strat egy/ circu larecono myactionplan_ en). In the German state of Sachsen-Anhalt, a large research consortium of Universität Magdeburg, Fraunhofer Institute for Factory Operation and Automation IFF Magdeburg, and Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg work on improving operations in the circular economy in their research cluster SmartProSys (https:// www. smart prosys. ovgu. de). There are many reasons for the surge in circular economy projects, not only the high cost of new material and the increasing uncertainty in global supply chains due to political conflicts and natural disasters but also the commitment to more environmentally sustainable production. Replacing classical linear sourcing of new materials with recycled local materials brings several new challenges in production planning and logistics. In this work, we focus on four of these challenges related to the collection of materials from the suppliers to ensure a steady availability of materials for production. First, in contrast to large shipments of new goods, collecting smaller batches of waste materials from local suppliers requires detailed routing considerations and the corresponding transportation cost. Second, the available quantities of materials at local suppliers depend on local consumption for recycling, making them uncertain. Third, the collected waste materials require cleaning and sorting. As the quality of the waste materials is uncertain, the final quantity that can be used is also uncertain. Fourth, in contrast to operations with less uncertainty, decisions are interconnected and must anticipate potential future events to ensure effective utilization of resources, such as processing and fleet capacity. These four challenges combined lead to a complex planning problem for companies relying on circular economy sourcing. The problem can be modeled as a stochastic dynamic multi-period inventory routing problem. It is a dynamic and multi-period problem because a sequence of decisions must be made period by period over the planning horizon. These decisions are connected as every decision impacts the states subsequent decisions are made in. It is stochastic because there are two stochastic components: first, the available quantities at each supplier in the next period(s), and second, the net quantity of materials that can eventually be used for production. Inventory of net material at the production facility can be built with a negligible holding cost. However, the quantity of waste materials that can be processed every period is limited. Every period, decisions are made about the suppliers to visit for collection, the quantity to collect from each supplier, and the routing of transportation trucks for collections. Besides transportation costs, linear backorder sourcing costs can occur if the net quantity does not satisfy the material demand at the 701 Dynamic multi‑period recycling collection routing with… production facility and new material has to be used instead. The challenge is now to determine cost-efficient collection quantities and routes, hedge against potential backorder sourcing costs due to insufficient net quantities, and ideally, build some inventory for future periods. We employ the sequential decision process framework proposed by (Powell 2021) to model the dynamic and stochastic components of our problem through states, decisions, realization of cost, observation of exogenous information, and transitions to next states. In our solution approach, we approach the challenges as follows. First, we generate a set of multi-period scenarios in every period to capture the two sources of uncertainty. Each scenario contains realizations of waste quantities at the suppliers in the following periods and, for each supplier and the current and future periods, the percentages of collected quantities that can be used for production (i.e., the net quantities). Based on the scenarios, we solve an auxiliary model, a two-stage stochastic mixed-integer program, to determine the suppliers to visit and the quantities to collect. Instead of integrating routing decisions explicitly, routing is approximated in the stochastic program. Once the suppliers visit and the quantities to collect are determined, the detailed routing solution is determined via a routing heuristic from the literature. We apply our method to a real-world case in Sachsen-Anhalt with real geographical data of production and waste facilities as well as real supply data for paper waste material. We also test our method for the waste collection data provided by Keskin etal. (2023). In a comprehensive computational study, we derive two main managerial insights. First, anticipatory decision-making reduces costs but requires effective management of dynamism and all sources of uncertainty throughout the planning horizon. Second, supplier selection should balance quality loss, supply volumes, and routing efficiency, considering the factors independently can lead to cost overruns. Our paper makes the following contributions: • We introduce a new and important problem. To the best of our knowledge, we are the first to address a dynamic routing problem with stochastic yield or quality loss. We present the modeling of the problem as a sequential decision process and provide a comprehensive literature survey of the problem’s components. • We propose a tailored solution method named Stochastic Lookahead over Multiple periods (STM) that leverages a two-stage stochastic program and a routing heuristic to tackle the problem’s main challenges: a complex decision space, dynamic multi-period decision-making, and uncertain supply volumes and quality. Our method follows the general solution framework proposed by Cuellar-Usaquén etal. (2024) but is tailored to our new problem, particularly with respect to stochastic quality loss and processing capacity. • We present a new set of instances based on real volumes of paper collection in Sachsen-Anhalt, Germany. Furthermore, we evaluate several heuristics, discussing their advantages and disadvantages using both real data and existing data from the literature. The created datasets are available online (https:// www. ms. ovgu. de/ Resea rch. html). • We present a comprehensive analysis of the method and problem characteristics and further derive valuable insights into both the problem and the model. 702 D.Cuellar-Usaquén et al. The paper is outlined as follows. In Sect.2, the relevant literature is presented. The problem is defined in Sect.3. We present our method in Sect.4, and the computational study in Sect.5. The paper concludes with a summary and outlook in Sect.6. 2 Literature Our problem comprises several components: inventory routing, multi-period routing, raw material collection, and stochastic dynamic decision making. To the best of our knowledge, there is no work considering all four components yet. Thus, in the following, we will discuss the most related work considering two or three of them. There is also extensive work on the individual components, which we discuss in the second part of the literature review. 2.1 Most related work While related problems have been explored in the literature, they often lack the dynamic aspect over a multi-period decision horizon. Habibi et al. (2017, 2019) address decision-making challenges in third-party reverse logistics, focusing on integrating End-of-Life product collection and disassembly processes. The deterministic variant is solved in Habibi etal. (2017), and the stochastic version, considering uncertainties in supply and quality of raw materials, is addressed in Habibi etal. (2019). The authors propose solutions in the form of Two-Phase Iterative Heuristics and a method based on sample average approximation, respectively. In contrast to prior work, our approach involves addressing inventory management and collection routing decisions on a period-by-period basis, anticipating uncertainty values and decisions for future periods while considering the realization of period-specific uncertainties. For comparative purposes, we adapt their intra-period approach in our benchmark policy STS. We confirm that explicit consideration of quality uncertainties is essential for single-period decision-making; however, for our multi-period setting, a combined intraand inter-period anticipation is substantially more effective. Markov etal. (2020) address a multi-period solid waste collection problem using a heterogeneous fleet to collect waste from containers with uncertain supply. They aim to minimize costs associated with container overflows by developing collection routes on a period-by-period basis. They propose a metaheuristic combined with forecasting within a rolling horizon, significantly outperforming deterministic policies. Their approach incorporates dynamic, probability-based costs of container overflows and route failures. While both papers involve dynamic, period-by-period decision-making, our paper additionally focuses on deciding the collection routes and the inventory of material processed at the depot to avoid cost overruns due to unsatisfied demand. Besides the uncertain volumes, our decisions must also anticipate the uncertainty in yield of the recycled material. Keskin etal. (2023) address a multi-period dynamic vehicle routing problem for a waste collection company in the United Kingdom. They introduce the concept of “touting”, a demand management technique in which customers are encouraged to 703 Dynamic multi‑period recycling collection routing with… place orders earlier than usual, allowing the company to combine stops at existing customers with stops at nearby “touted” customers. The supply of waste material from customers is known once the customer is contacted. A rolling horizon and simulation model are used to solve the problem; the authors propose rules to identify promising facilities to call based on the expected supply volumes and how they could be integrated into the routes. Empirical results indicate substantial improvements in routing efficiency compared to the no-touting strategy. Unlike our work, the authors do not plan inventory over the periods. Following the approach presented in Keskin etal. (2023), we test several supplier selection rules as benchmark policies. Our results indicate that explicit anticipation of current and future routing decisions and uncertainties is superior to the practically-inspired decision rules. The consideration of raw material/product quality from suppliers is explored in other contexts. The emphasis on quality loss in supply chains often revolves around the freshness and perishability of products (Rohmer etal. 2019). Addressing a vehicle routing problem, Stellingwerf et al. (2021) employ a timeand temperaturedependent kinetic model to simulate the degradation of products over time. The quality of products changes after each customer visit, attributed to fluctuations in the vehicle’s temperature during transportation. A similar perspective is provided by Alvarez etal. (2022), which tackles a production routing problem specifically concerning perishable goods. In this context, the quality of products is characterized by a decay rate over time. On the other hand, in a manufacturing and disassembly context, Laouini etal. (2023) address the material yield of collected products, seeking to satisfy demand for finished products from the collection of recycled product resources. In contrast to our work, the mentioned studies emphasize that raw material quality/yield is affected over time and does not have a direct impact on collection routes before reaching the depot, as is the case with perishable products. Additionally, while uncertainty in raw material quality loss is addressed in manufacturing contexts, routing decisions are not involved, and the uncertainty is not related to the quantity to be replenished in the processing center. Alvarez et al. (2021) address the stochastic inventory routing problem (SIRP) under uncertainty in both product supply and customer demands. The authors propose a heuristic solution method based on the progressive hedging algorithm, delivering high-quality solutions within reasonable running times for problems with a large number of scenarios. In contrast to our work, the authors contemplate uncertainty in the supply of a single supplier and generate anticipation for a single period going forward. Our proposed methodology explicitly considers future uncertainties and routing for several suppliers. Methodologically, some works share similarities with our work. Elbek et al. (2015) solve a waste collection problem with uncertainty in the amount available in the containers. The problem is modeled as a two-stage stochastic program and solved using a lookahead algorithm. As in our work, the authors dynamically solve the collection policy for each day of the planning horizon but do not consider the product quality, inventory, or processing capacity at the depot. Brinkmann etal. (2019, 2020) focus on dynamic bicycle transportation to maintain optimal inventories at bike share stations. The authors propose a lookahead approach to evaluate inventory decisions, which involves sampling future demand 704 D.Cuellar-Usaquén et al. and selecting inventory to minimize unmet demand. The time-influenced lookahead horizon is obtained using the value function approximation (VFA). Notably, this horizon does not extend to future routing decisions. In parallel, our research shares similarities that span both problem formulation and methodological aspects, as we also use future samples to inform inventory decisions. However, our approach explicitly integrates routing and associated cost considerations into the forecasting model. Finally, Cuellar-Usaquén etal. (2024) solve a dynamic stochastic multi-period problem encompassing purchasing, inventory, and routing decisions for a first-mile operation. The lookahead algorithm proposed samples suppliers’ purchase prices as well as supply and demand at the depot. In addition, an adaptive algorithm is introduced to capture the consolidation behavior of suppliers in routing decisions. Cuellar-Usaquén etal. (2024) introduce the general framework of combining twostage stochastic programming with routing cost approximation for dynamic decision problems. For our work, we use this framework for a different problem. In contrast to Cuellar-Usaquén etal. (2024), we model the behavior of material quality loss upon arrival at the depot, considering processing capacity and cumulative inventory at suppliers. This changes the design of the scenarios, and the two-stage stochastic program. 2.2 Work onindividual problem components In addition to the most relevant research discussed, there is related work in recycling collection routing, inventory routing, and the uncertain supply and quality loss of raw materials in manufacturing and supply chains. These studies are discussed in detail in the following sections and summarized in the subsequent table. Table1 presents a summary of the relevant literature of the fields of Inventory Routing Problems (IRPs), Reverse Materials Requirements Planning (RMRP), Supply Chain Management (SCM), Vehicle Routing Problems (VRPs) and MultiPeriod Routing (MP-R). They are categorized as follows: Problem features indicate whether the problem considers dynamic decisions, stochastic parameters, or multi-period aspects, reflecting the impact of decisions over a time horizon longer than one period; Decisions specify whether the problem involves routing, inventory management decisions, or both; Uncertainty highlights whether the work considers uncertainty in at least one component of the problem; and Anticipation identifies whether the proposed method anticipates the impact of a decision on future costs. For deterministic problems, the Anticipation column is marked as ’n/a,’ as anticipation is only applicable in stochastic problems. 2.2.1 Vehicle routing inthecontext ofrecycling collection In the 1980s, the difficulties of routing municipal waste disposal were recognized by Raff (1983) as its own category of Vehicle Routing Problems (VRP). Its global impact and relevance to different societal dimensions, including resource management, energy utilization, ecological damages, and monetary costs, are acknowledged 705 Dynamic multi‑period recycling collection routing with… Table 1 Literature classification, highlighting similarities and differences of our proposed problem and the literature Paper Problem features Decisions Uncertainty Anticipation Dynamic Stochastic Multi-period Routing Inventory Supply Quality IRP Habibi etal. (2017) ✓ ✓ ✓ n/a Rohmer etal. (2019) ✓ ✓ ✓ n/a Alvarez etal. (2022) ✓ ✓ ✓ n/a Moin etal. (2011) ✓ ✓ ✓ n/a Mjirda etal. (2014) ✓ ✓ ✓ n/a Chitsaz etal. (2019) ✓ ✓ ✓ n/a Bertazzi etal. (2020) ✓ ✓ ✓ n/a Chitsaz etal. (2020) ✓ ✓ ✓ n/a Chi and He (2023) ✓ ✓ ✓ n/a Jieyu etal. (2024) ✓ ✓ ✓ n/a Habibi etal. (2019) ✓ ✓ ✓ ✓ ✓ ✓ ✓ Alvarez etal. (2021) ✓ ✓ ✓ ✓ ✓ ✓ Brinkmann etal. (2019) ✓ ✓ ✓ ✓ ✓ ✓ Brinkmann etal. (2020) ✓ ✓ ✓ ✓ ✓ ✓ ✓ Nolz etal. (2014) ✓ ✓ ✓ ✓ ✓ ✓ Mes etal. (2014) ✓ ✓ ✓ ✓ ✓ ✓ ✓ Markov etal. (2020) ✓ ✓ ✓ ✓ ✓ ✓ ✓ Liu etal. (2021) ✓ ✓ ✓ ✓ ✓ ✓ Frifita etal. (2022) ✓ ✓ ✓ ✓ ✓ Hasturk etal. (2024) ✓ ✓ ✓ ✓ ✓ ✓ RMRP/SCM Laouini etal. (2023) ✓ ✓ ✓ ✓ ✓ 706 D.Cuellar-Usaquén et al. Table 1 (continued) Paper Problem features Decisions Uncertainty Anticipation Dynamic Stochastic Multi-period Routing Inventory Supply Quality Inderfurth and Langella (2006) ✓ ✓ ✓ ✓ Rickli and Camelio (2014) ✓ ✓ ✓ Keyvanshokooh etal. (2016) ✓ ✓ ✓ ✓ ✓ Üster and Hwang (2017) ✓ ✓ ✓ ✓ Üster and Memişoğlu (2018) ✓ ✓ ✓ ✓ Liu and Zhang (2018) ✓ ✓ ✓ ✓ ✓ Memişoğlu and Üster (2021) ✓ ✓ ✓ ✓ ✓ ✓ Zhou etal. (2022) ✓ ✓ ✓ ✓ ✓ Li etal. (2023) ✓ ✓ ✓ ✓ ✓ ✓ VRP Stellingwerf etal. (2021) ✓ n/a Kim etal. (2009) ✓ n/a De Bruecker etal. (2018) ✓ n/a Ismail and Loh (2009) ✓ ✓ ✓ Gruler etal. (2017) ✓ ✓ ✓ Cook and Lodree (2017) ✓ ✓ ✓ ✓ Jammeli etal. (2021) ✓ ✓ ✓ Kyriakidis etal. (2020) ✓ ✓ ✓ ✓ Marković etal. (2020) ✓ ✓ ✓ Sasha Dong etal. (2022) ✓ ✓ ✓ ✓ MP-R Bogh etal. (2014) ✓ ✓ n/a Archetti etal. (2015) ✓ ✓ n/a 713 Dynamic multi‑period recycling collection routing with… EUR. The top center of the figure now shows the realization of the stochastic information: the realized loss and the realized increase in supply for the next period. As suppliers 1 and 2 are not visited, the loss is not observed, but only the increase is shown, 20 units for supplier 1 and 30 units for supplier 2. For supplier 3, the realized loss is 30% , higher than expected. The increase in supply is 50. The higher-than-expected loss leads to an inventory increase of only 70. Given the demand of 80, ten units need to be purchased through the costly backorder option to reduce the net inventory to zero. Thus, a backorder cost of 1000 EUR incurs leading to a total cost of 1195 EUR at period t=1 . The next state is then shown on the right. The inventory value is −80 , and the supply values are updated according to the realized increase. Another potential decision is presented at the bottom of Fig.1. This decision invests routing costs to increase the depot’s expected inventory and avoid orders from the backup option. Two vehicles are employed. The first one collects all supplies from supplier 3, similar to the first decision. The second vehicle first visits supplier 1 and collects all 80 available supply units, indicated by the value (80) next to the vehicle when leaving supplier 1. Then, the vehicle visits supplier 2 and collects an additional 20 units, returning full-back to the depot. The route of the second vehicle has a duration of 250 minutes and a cost of 375 EUR. As 200 units are collected overall, the production capacity limit is not exceeded. The bottom center shows the realization of stochastic information, similar to before, but now also for the loss for suppliers 1 ( 40% ) and 2 ( 50% ). This collection leads to an inventory increase of 48 +10 +70 =128 and 128 −80 =48 inventory units remaining at the end of the period without incurring the cost of backorders (0 EUR), saving 820 EUR compared to the previous decision. In the new state on the right, the net inventory is therefore −32 . 3.3 Sequential decision process The problem at hand is a stochastic and dynamic decision problem. It is stochastic because the amount of waste material available at the suppliers is known at the beginning of each period, and the amount of supply that can be used to generate product inventory is revealed when it arrives at the depot. It is dynamic because a sequence of decisions must be made, one per period, with each current decision affecting the inventory volumes of future periods and consequently influencing subsequent decisions. A dynamic stochastic decision problem can be modeled as a sequential decision process (Powell 2021), modeling the problem as a sequence of states. In each state, a decision is made, and the cost is observed. Next, stochastic information is revealed (resulting in further cost), and a transition leads to the next state. In the following, we define the states, the decisions, the cost functions, the stochastic information, and the transition function of our problem. For an overview of the notation used, refer to Table2. 714 D.Cuellar-Usaquén et al. Table 2 Overview of notation Notations Definitions Sets MSet of suppliers TSet of periods FSet of vehicles V Set of vertex, V∶= M∪{0} A Set of arcs, A= {(i,j)∶i,j∈V|i≠j} U A set of nodes where U⊂V 𝛿+(U) Set of arcs (i,j) with i∈U and j∈V⧵U 𝛿−(U) Set of arcs (i,j) with j∈U and i∈V⧵U T′ Set of periods in the lookahead horizon Ω Set of scenarios Parameters d, 𝛽 Demand of new product and processing capacity at the depot QVehicle capacity 𝜏ij Travel time between i and j for (i,j)∈A with service time included cCost for every time unit traveled fBackorder cost per unit lmax Maximum working time per vehicle and period 𝜇r m,𝜎r m Mean and standard deviation of probability distribution for supply increase value for supplier m∈M 𝜇𝜙 m ,𝜎 𝜙 m Mean and standard deviation of probability distribution for quality loss percentage value for supplier m∈M rmt ′ 𝜔,𝜙mt ′ 𝜔 Realization of increase in supply and quality loss percentage for supplier m∈M at period t�∈T� in scenario 𝜔∈Ω It Net inventory at the depot in period t∈T qmt Amount of supply available at supplier m∈M at period t∈T hSize of the lookahead horizon 𝜏 m Direct trip cost from the depot to every supplier m∈M 𝛾 Discount parameter for routing cost estimation 𝜃stock Additional percentage collected over expected supply for safety stock Variables zmvt Amount of supply to collect from supplier m∈M by vehicle v∈F at period t∈T xijvt Equal to 1 if the arc (i , j )∈ A is activated in the route of vehicle v∈F at period at period t∈T , 0 otherwise emft Equal to 1 if supplier m∈M is visited by vehicle v∈F at period t∈T 0 otherwise z′ mt ′ 𝜔 Amount of supply to be collected from supplier m∈M in period t�∈T� in the scenario 𝜔∈Ω e′ mt ′ 𝜔 Equal to 1 if supplier m∈M is visited at period t�∈T� in the scenario 𝜔∈Ω I′ t ′ 𝜔 Inventory level of new product at the end of period t�∈T� in the scenario 𝜔∈Ω yt ′ 𝜔 Backorder amount variable at period t�∈T� in scenario 𝜔∈Ω kmt ′ 𝜔 Inventory level of waste material from supplier m∈M at the end of period t�∈T� in the scenario 𝜔∈Ω 715 Dynamic multi‑period recycling collection routing with… 3.3.1 Global notation We denote the set of suppliers as m∈M . We define the collection network as a complete, directed graph G=(V,A) . Let V∶= M∪{0} be the set of vertices, where 0 represents the depot. Let A be the set of arcs where A= {(i,j)∶i,j∈V|i≠j} . For the sub-tour elimination constraints, given a set U⊂V , we define 𝛿+(U) as the set of arcs (i,j) with i∈U and j∈V⧵U , and 𝛿−(U) as the set of arcs (i,j) with j∈U and i∈V⧵U . The periods are denoted as t∈T with T={1, 2, …,|T| }. In each period, the demand is the same, denoted as d. For each supplier, the increase in supply per period and the quality loss follow known probability distributions with mean and standard deviation ( 𝜇r m,𝜎r m) for supply increase value and ( 𝜇 𝜙 m ,𝜎 𝜙 m) for quality loss percentage. We assume a sufficiently large available set of vehicles F (e.g., |F|=|M| ). Vehicles have a maximum load capacity Q, and a maximum working time per vehicle and period lmax . The travel time between i and j for (i,j)∈V is denoted by 𝜏ij , and for every time unit traveled, there is a cost of c. The service time to load the waste material on the vehicles is the same for all suppliers. We include the service times in the travel times 𝜏ij , leading to asymmetric travel times from/to the depot. At the end of the collection, when the vehicles arrive at the depot, the sum of the quantities collected cannot exceed the processing capacity defined as a factor of the demand, 𝛽×d , with 𝛽 being the processing capacity scaling parameter. The backorder cost per unit is defined as f. 3.3.2 State A decision is made in every period t∈T . The state comprises all information available to make a decision. We denote the state in period t∈T as St . For our problem, the state St consists of two components, one related to the depot and one related to the suppliers: The first is the net inventory at the depot in period t, denoted by It . The second is the amount of supply available at supplier m∈M at period t, denoted by qmt . We note that the net inventory already captures the demand for the day and, therefore, can be negative. State St can be summarized as St=(It,qt) , where It is a scalar and qt is a |M|-dimensional vector. At the beginning of the process, no inventory is available at the depot, I0=−d . The initial amount of supply of supplier m∈M is denoted qm0 . 3.3.3 Decision We denote a decision at period t∈T as at . A decision at=(zt,xt) has two components that reflect collection and routing parts. The collection part is modeled via decision matrix zt=(zmvt)m∈M,v∈F . It determines the amount of waste material to collect from each supplier m by each vehicle v. The second part of the decision is the definition of collection routes, modeled via xt=(xijvt)i,j∈M,i≠j,v∈F . The variable xijvt ∈{0, 1} 716 D.Cuellar-Usaquén et al. indicates if the arc from supplier i to supplier j is activated in the route of vehicle v at period t. In the following, we summarize the decision space using a mixed-integer formulation. For the formulation, we use the auxiliary variable emft , which takes the value of one if supplier m∈M is visited by vehicle v∈F at period t and 0 otherwise. A decision at=(zt,xt) at period t∈T is feasible if the constraints(1–10) are satisfied. Equation (1) ensures that the processing capacity 𝛽×d in the depot is not exceeded. Equation(2) ensures that the collection should not exceed the available supply of waste material of the suppliers and that a vehicle can only collect supply if a supplier is visited. Equation(3) ensures that the vehicle capacity Q is not exceeded if one or more suppliers are visited. Eq.(4) imposes non-split visit constraints that ensure that each supplier is visited by at most one vehicle. Eq.(5) imposes that, for each visited supplier, exactly one arc must enter and leave the relative node. Equation(6) and Eq.(7) are the sub-tour elimination constraints and maximum travel time constraints (Manerba and Mansini 2016). Finally, Eqs.(8–10) define the domain of the variables. (1) ∑ v∈F ∑ m∈M zmvt ≤𝛽 d (2) zmvt ≤qmtemvt,∀m∈M,∀m∈F (3) ∑ m∈M z mvt ≤Q,∀v∈F (4) ∑ v∈F e mvt ≤1, ∀m∈M (5) ∑ (i,j)∈𝛿 − ({b}) x ijvt =∑ (i,j)∈𝛿 + ({b}) x ijvt =e bvt ,∀b∈M,∀v∈F (6) ∑ ( i , j )∈𝛿 + () x ijvt ≥e bvt ,∀ ⊆ M,∀b∈,∀v∈F (7) ∑ (i,j)∈ 𝜏ij x ijvt ≤lmax,∀v∈F (8) zmvt ≥ 0, ∀m∈M,∀v∈F (9) emvt ∈{0, 1},∀m∈M,∀v∈F (10) xijvt ∈{0, 1},∀(i,j)∈,∀v∈F 717 Dynamic multi‑period recycling collection routing with… The routing cost associated with decision at in state St is determined by multiplying the duration of each route by the cost per unit of time. The routing cost Cr(St,at) can be formally defined as: 3.3.4 Stochastic information andtransition function After a decision at is taken in state St , the state is transferred to post-decision state Sa t=(It,qa t,zt) . The collection decision, zt , induces changes in the amount of supply available at the suppliers. The final inventory of supply at the suppliers at period t is the difference between the amount of supply on hand and the amount collected: The exogenous information 𝜔t+1=(z𝜔 t+1,r𝜔 t+1) reveals the inventory that can be generated from the supply collected and the additional supply available at the suppliers. The realization of generated inventory, z𝜔 t+1 , impacts both the satisfaction of demand and the final inventory at the depot. We define I𝜔 t+1 as the difference between the revealed amount of material that can be used and the net inventory level at period t: In the case that the inventory at the depot is not enough to meet the demand in period t, a backorder cost is incurred for the purchase of material to meet the demand. The realized backorder cost is defined as: After applying the transition function T( S a t ,𝜔 t+1) , a new state St+1=(It+1,qt+1) is reached. Firstly, the net inventory It+1 is updated, considering the realization of the final inventory level of the new product at the depot and the demand of period t+1 . Secondly, the supply of waste material available qmt+1 is updated based on the final inventory at each supplier and the newly generated amount. We define this update as follows: (11) C r(St,at)=c⋅ ∑ v∈F ∑ (i,j)∈A 𝜏ijxijvt . (12) q a mt =qmt − ∑ v∈F zmvt,∀m∈M . (13) I𝜔 t+1=It+z𝜔 t+1. (14) Ce(St,at,𝜔t + 1)=f ⋅ max(0, −I𝜔 t+1). (15) It+1=max(0, I𝜔 t+1)−d (16) qmt+1= q a mt + r 𝜔 mt+1 , ∀ m ∈ M . 718 D.Cuellar-Usaquén et al. 3.3.5 Policy A solution for a sequential decision process is a policy 𝜋 . A policy assigns a decision at=A𝜋(St) to every state St . The overall set of policies is defined as Π . An optimal solution 𝜋∗∈Π minimizes the expected cost that is composed of the routing cost and the backorder cost: starting from state S0 . Ter m Ce(St,A𝜋(St)) indicates the expected backorder cost. 4 Method Even the small example in Sect.3 already highlights the challenges in decision-making for this problem. First, uncertainty manifests not only in the supply per period but also in the loss, i.e., the percentage of collected material that cannot be used. Second, a state’s decision comprises a complex inventory routing problem. Thus, solution methodology must account for the two sources of uncertainty within the period and for the following periods and must be able to tackle the complex decision space thoroughly. To this end, we propose a stochastic lookahead method based on the general framework of Cuellar-Usaquén et al. (2024). Our method is denoted STochastic lookahead over Multiple periods (STM). In the following, we give an overview of the functionality of our method. For details and a pseudo-code representation, we refer to AppendixA.1. The idea of STM is to capture current and future stochasticity by solving a stochastic lookahead model based on a set of sampled multi-period scenarios. Each scenario comprises the realization of loss in the current period and the realizations of supply and loss in future periods. Then, a decision is taken that minimizes the average cost while considering all scenarios generated. Mathematically, the stochastic lookahead can be modeled as a two-stage stochastic program. Solving the full stochastic program explicitly is computationally intractable since routing decisions in the current and future periods must be considered. Instead, we assume direct trips but approximate routing consolidation via a discount parameter. Cuellar-Usaquén etal. (2024) have shown that this procedure works very well for routing problems with a limited number of stops per vehicle due to capacity constraints, as given for our problem. After the collection decisions are determined, the detailed routing is done via a heuristic. Fig.2 presents the sequence for obtaining a decision using STM, following the framework introduced in Cuellar-Usaquén etal. (2024). In a state, the state information is observed and a set of multi-period scenarios are generated. With the scenarios and the routing discount factor 𝛾 , a reduced two-stage stochastic program is created and solved (Step I). The solution prescribes the selection of suppliers to be visited and the quantities to be collected. Based on this information, (17) 𝜋 ∗=argmin 𝜋∈Π 𝔼 [∑ t∈T (C(St,A𝜋(St)) + Ce(St,A𝜋(St)) | S0) ], 719 Dynamic multi‑period recycling collection routing with… the explicit collection routes are determined (Step II). Steps I and II of Fig.2 are illustrated in more detail in Fig.3 for the state previously introduced in the example (Sect.3.2). Here, at time t, we assume a forward period horizon of three periods and two scenarios, one at the top and one at the bottom ( 𝜔1,𝜔2) . In the first step, shown on the left, STM solves a two-stage stochastic program with one joint decision for time t (first-stage decisions) and individual decisions in the scenarios for times t+1 , t+2 , t+3 (second-stage decisions). All decisions assume direct trips and are evaluated via discounted direct trip costs. In this example, for period t, two suppliers (and their respective collection volumes) are selected. Then, in the second step, shown on the right, the decision at time t is transferred to a routing decision. In the following, we first present the stochastic lookahead model and then the routing heuristic. 4.1 The stochastic lookahead model In each state St during period t, the stochastic lookahead model solves an auxiliary model (Fleckenstein et al. 2023), a two-stage stochastic program to determine the amount of supply to collect, inventory levels, and the suppliers to visit from the depot. This lookahead model solves the problem over a shorter planning horizon by combining an approximation of future information with an approximation of future decisions. Fig. 2 Flowchart for decision making in a given state using STM (adapted from Cuellar-Usaquén etal. 2024) Fig. 3 Illustration of the STM-policy with simplified two-stage stochastic program on the left and resulting routing decision on the right 720 D.Cuellar-Usaquén et al. Therefore, in the lookahead model, a forward period horizon T′ is composed for the subsequent periods: t,t+1, …,t+h . Notably, decisions related to t′>t are employed to evaluate the quality of decisions to be made in period t. For each period t�∈T� , a set of scenarios, i.e., sample paths 𝜔∈Ω , is constructed. Each sample path determines the realized amount of supply ( rmt ′ 𝜔 ) and the quality loss percentage ( 𝜙mt ′ 𝜔 ) for every period t�∈T� and every supplier m∈M . The two-stage stochastic program is presented in Eqs.(18–29). Derived from the model defined in Sect.3.3, this stochastic program models multiple scenarios and time periods. The stochastic program simplifies the routing decision by assuming discounted direct trips. To this end, it introduces two additional parameters, 𝜏 m and 𝛾 . The parameter 𝜏 m is the direct trip cost from the depot to every supplier m if we collect something from the supplier, with 𝜏 m∶= 𝜏0m+𝜏m0,∀m∈M . The parameter 𝛾 serves as a discount parameter, estimating the routing cost, as direct trips may overestimate the actual routing cost. It takes values in the range [0,1]. A weight close to zero indicates consolidation potential and a weight close to one indicates that the supplier is usually visited by direct travel. The parameter is tuned via enumeration. The objective function (18) minimizes the expected total cost of collection routing and backorder cost. Equation(19) guarantees the inventory flow and ensures demand satisfaction at the depot, considering potential future values of the quality loss percentage. Equation(20) ensures suppliers’ flow of inventory and new supply. Equation(21) and Eq.(22) ensure not exceeding the capacity of the vehicles, along with the constraints of no split visits and not exceeding the processing capacity of the depot. Nonanticipativity constraints, represented by Eq.(23) and Eq.(24), ensure that the firstperiod decisions on the horizon T′ , corresponding to state St , remain consistent across sample paths. Finally, Eqs.(25–29) define the variable domains. Compared with the model presented in Sect.3.3, the two-stage stochastic program does not take into account the vehicle-specific routing decision, xijvt being replaced by supplier selection decisions e′ mt ′ 𝜔 . A backorder amount variable yt ′ 𝜔 is added. The constraints (1–4) are adjusted to constraints (20–22); and the constraints (5–7) are removed. The stochastic program can be solved using a standard solver setting a maximum gap of 10% for the instances considered in this paper. Let e and z represent the computational results for the decision variables e′ and z′ . The values of e and z for the period t serve as input information for the routing heuristic. s.t. (18) min ∑ 𝜔∈Ω 1 | Ω |(∑ t�∈T� c ∑ m∈M 𝛾 𝜏 me� mt�𝜔+f⋅yt�𝜔 ) (19) I � t�𝜔=I� t�−1𝜔+ ∑ m∈M z� mt�𝜔(1−𝜙mt�𝜔)−d+yt�𝜔,∀t�∈T�,∀𝜔 ∈Ω (20) kmt � 𝜔 =k mt � −1𝜔 +r mt � 𝜔 −z � mt � 𝜔 ,∀m∈M,∀t � ∈T � ,∀𝜔 ∈Ω 721 Dynamic multi‑period recycling collection routing with… 4.2 Routing heuristic After solving the two-stage stochastic program, which returns the values of the supply collected ( z ) and supplier selection ( e ) at a state St , the routing heuristic proceeds to determine the routing decision, xijvt , by solving a Distance Constrained Capacitated Vehicle Routing Problem (DCVRP). The conceptual process is outlined below, with further details available in Cuellar-Usaquén etal. (2024). First, a giant tour is created using the nearest neighbor algorithm based on the selected suppliers, following the approach outlined in Cuellar-Usaquén etal. (2021). Subsequently, an augmented graph is constructed using the giant tour and the supply collected ( z ). The split procedure from Prins (2004) extracts the pool of routes from the augmented graph, respecting vehicle capacities and maximum travel time constraints. The construction of the augmented graph follows a Directed Acyclic Graph (DAG) structure. Subsequently, a single-source shortest path problem is solved to find the set of routes that minimizes travel time (Cormen etal. 2022). This involves using a topological ordering of vertices, resulting in a complexity of O(|A|). The resulting routes are then implemented in the decision variable xijvt , where each route corresponds to a vehicle assignment. (21) z� mt � 𝜔 ≤Qe � mt � 𝜔 ,∀m∈M,∀t � ∈T � ,∀𝜔 ∈Ω (22) ∑ m∈M z� mt�𝜔≤𝛽d,∀t�∈T� ,∀𝜔 ∈Ω (23) z ′ mt𝜔 = ∑ 𝜔 ′ ∈Ω z′ mt𝜔′ | Ω | ,∀m∈M,∀𝜔 ∈Ω (24) e ′ mt𝜔 = ∑ 𝜔 ′ ∈Ω e′ mt𝜔′ | Ω | ,∀m∈M,∀𝜔 ∈Ω (25) e′ mt′𝜔∈{0, 1},∀m∈M,∀t′∈T′,∀𝜔∈Ω (26) z′ mt ′ 𝜔≥0, ∀m∈M,∀t′∈T′,∀𝜔∈Ω (27) I′ t′𝜔≥0, ∀t′∈T′,∀𝜔∈Ω (28) kmt′𝜔≥0, ∀m∈M,∀t′∈T′,∀𝜔∈Ω (29) yt′𝜔≥0, ∀t′∈T′,∀𝜔∈Ω 722 D.Cuellar-Usaquén et al. 5 Computational study In this section, we present our computational study. We first describe the test instances and the benchmark policies. Next, we provide the implementation details and parameter tuning. We then analyze the value of single-period and multi-period anticipation of our method. Finally, we investigate the decision-making of our method in detail. 5.1 Instances We present two main instance settings for geography, supply and demand. One setting is based on the recycling collection data from the German state of SachsenAnhalt, and the second one is adapted from the United Kingdom (UK) waste collection data presented in Keskin etal. (2023). The instances differ in the number of suppliers and their geographical distribution. • Sachsen-Anhalt: The data comprises locations and supply data of nine paper collection facilities, as well as a paper processing facility spread over the entirety of Sachsen-Anhalt. The travel distances between the locations are calculated via Google Maps on free roads and multiplied by a factor of 1.3 to mimic potential traffic. We have access to the monthly supply data in tons for each supplier from January 2021 to December 2022. Based on the data, we calculate the expected supply per supplier and period as the average weekly supply. We assume the supply follows a normal distribution and fits the mean and standard deviation of the data accordingly. The expected values range in the interval 𝜇∈[0.3, 7.2] tons and the standard deviation in the interval 𝜎∈[0.07, 0.9] . • UK-instances from Keskin etal. (2023): The data comprises travel times and supply data in liters from a waste collection company operating in a smaller region of the United Kingdom. The travel time is obtained using the coordinates of the customers (suppliers). The supply distribution is calculated from a real-world dataset that covers three months of waste collections for two drivers operating from one depot. For these instances, we focused on the customers with a higher waste volume. To this end, we sorted the suppliers by expected value in descending order and selected the first 30 customers. The supply follows a normal distribution with 𝜇 ranging in interval [40.23,278.64] liters for the different suppliers and the standard deviation 𝜎∈[50.04, 512.96] . Even though negative values are highly unlikely, we truncate the supply distributions at 0 liters (and 2 ⋅ 𝜇 liters to ensure symmetry). For both data sets and our main experiments, we assume a time horizon of 20 periods (e.g., reflecting weekly collections). We set the demand per period to 50% percent of the expected supply per period. We further set the processing capacity as two times the demand, and there is no initial inventory of new product at the depot. In practice, the average quality loss for paper is around 20% (AF&PA 2024). Consequently, the 729 Dynamic multi‑period recycling collection routing with… the suppliers with higher supply volumes is postponed to the second period. Here, our method anticipates the supply increase from period one to period two. Instead of sending a vehicle to the suppliers in the first period and returning it to the depot half-empty, it satisfies the demand of nearby suppliers in the first period. In the second period, suppliers 3 and 47 are visited individually, each consuming an entire vehicle capacity. As before, we can also observe a time-consistent pattern for many suppliers, e.g., for suppliers 13, 31, 46, and 45. Fig. 7 Supplier selection percentage over the planning horizon 730 D.Cuellar-Usaquén et al. 5.5.2 Problem dimensions Next, we analyze the impact of changes in loss volatility, demand uncertainty and processing capacity. For the following analysis, we increase the number of instance realizations to 100 for smoother values. Loss volatility One important feature considered in the problem is uncertainty in quality loss. To investigate the impact of loss uncertainty further, we vary the volatility. Besides our original variation with a coefficient of variation (COV) of 0.1, COVs of 0, 0.2, and 0.5 are tested for the Sachsen-Anhalt case and policies Myopic, STS, and STM. We further apply a variant of STM where loss is integrated into the scenarios via expected values only, policy STM(q). The results are shown in Fig.8. The x-axis depicts the COV, and the y-axis represents the objective values of the different policies. We observe that the Myopic policy performs increasingly worse with increasing loss volatility. This result can be expected since this policy only operates on expected values. In contrast, policy STS explicitly considers loss volatility in the scenarios. Its performance does not only stay constant but even increases with increasing volatility. The reason is again that with different loss values in the scenarios, STS builds inventory. We further observe that policy STM is not affected much by increasing COVs. Thus, even when volatility is very uncertain, our proposed policy proves to be quite effective. Finally, we observe that the difference between STM and STM(q) is rather small, about 3% for the case of 0.5 COV. Demand uncertainty In our main experiments, we assume that the demand per period is static and deterministic. We now analyze the impact of relaxing this assumption. In the new Fig. 8 Changing the coefficient of variations of the loss distribution 731 Dynamic multi‑period recycling collection routing with… experiments, we assume that the demand in a period only reveals after all collections were made (In practice, the demand in a period reflects the amount of product needed for production in the beginning of the next period. Therefore, it might only become known at the end of the current period). In our new experiments, demand per period follows a normal distribution. The expected value is the same as the fixed demand in our main experiments, 50% of the expected supply. We test coefficient of variations (COVs) of 0, 0.1, 0.2, 0.3, 0.4, and 0.5, where a COV of 0 represents the original variation. We apply the STM, STS, and Myopic policies to this new set of instances. In the generation of scenarios, we now also sample demand. The results for the Sachsen-Anhalt case are presented in Fig.9. The x-axis represents the COV values, and the y-axis shows the objective function of the policies. Uncertainty in demand significantly affects the performance of the Myopic policy, whose objective is to meet demand at its expected value without accumulating inventories, leading to high backorder costs when demand exceeds the average. In contrast, the STS and STM policies demonstrate rather stable performances as the COV increases, indicating that uncertain demand can be captured well via the scenarios. As expected, STM achieves superior results, because it captures future periods as well. Processing capacity In our main experiments, we assume a processing capacity of 𝛽=2 times the daily demand. Since the (expected) expected loss over all suppliers is 0.2 in our instances, this means that inventory of about 1.6 times the daily demand can be produced per period. We now analyze the impact of processing capacity by testing 𝛽=1, 1.5, 2, 2.5, 3 times the daily demand. We apply policies STM, STS, and Myopic to the set of new instances. The results for Sachsen-Anhalt are shown in Fig.10. The x-axis shows the maximum processing capacity. The y-axis shows the relative difference to STM in our main setting with a capacity of 𝛽=2 . We observe that with more capacity, improvements are marginal. Thus, for our main setting, production capacity is not a bottleneck. Once capacity decreases, we see an increase in cost. The reasons for the cost are threefold. First, the likelihood of costly backorders increases with loss uncertainty and reduced capacity. Second, the policies may decide to travel longer routes to collect a supply of higher quality, i.e., the smaller expected loss. Fig. 9 Changing the coefficient of variations of the demand distribution 732 D.Cuellar-Usaquén et al. Third, with limited capacity, inventory cannot be built to hedge against supply uncertainty in future periods. We now investigate the impact of the capacity constraint on decision-making in more detail. To this end, we plot the average processing capacity utilization over the time horizon. The results are depicted in Fig.11. The x-axis shows the period, and the y-axis shows the capacity utilization in tons (We recall that the net inventory demand is 10 tons per day). We make two main observations. First, with a decreasing capacity limit, the usage becomes more level. This behavior can be expected since, with limited processing capacity, inventory cannot be built, and every day, (nearly) all capacity is used to avoid costly backorders. However, we also observe some patterns in cases with high capacity limits. Given that the values are calculated over 100 runs, the differences cannot be explained by noise only. For example, we observe, that for higher capacities 𝛽=2 and 𝛽=2.5 , in the first, fourth and seventh period, significantly more processing capacity is used while in other periods (e.g., two, five, and eight), less supply is processed. In the first period, no initial inventory is given, and the policies decide to collect significantly more than is needed to build an inventory for future periods. Consequently, Fig. 10 Performance for changing processing capacity parameter 𝛽 (baseline 𝛽 = 2 ) Fig. 11 Processing capacity utilization over time for different capacity limits 𝛽 733 Dynamic multi‑period recycling collection routing with… the policy is flexible in the second period and may decide to save routing costs by collecting less. The same repeats over time, e.g., with periods 4 and 5 or 16 and 17. These results indicate that there might be value in having more flexibility concerning the processing capacity, e.g., by scheduling working shifts dynamically. 6 Conclusion andoutlook In this paper, we have presented a dynamic multi-period recycling collection problem with uncertainty in the available future supply and the supply’s quality. We have addressed this challenge using a tailored stochastic lookahead method, which employs a two-stage stochastic program to make decisions on supplier selection and quantities to be collected, while routing is solved heuristically in a second step. We have used an approximate routing cost with a parameterized discount factor to make the stochastic program tractable and incorporate routing decisions, future scenarios, and periods. Computational results, using instances from a real collection operation in Germany and instances from the literature, illustrate that our method effectively anticipates sources of uncertainty and optimizes the use of processing capacity at the depot and available supply from suppliers. There is a variety of future research opportunities. In our work, we assumed that each supplier’s quality loss probability distribution is known and static. Future work may relax this assumption. For example, quality loss distributions may be unknown initially and could be approximated based on observed qualities via online learning. Future work may face a challenging trade-off between effective routing (exploitation) and effective learning (exploration). In addition, we have assumed fixed service times to collect material from the suppliers. Addressing stochastic and quantity-dependent service times would be an interesting avenue for further study. This would introduce new challenges in collection decisions with respect to cost and volumes. Another interesting avenue is a closer consideration of the processing of the supply. For example, future work may add another inventory for non-processed supply instead of assuming only the processed inventory. Furthermore, in our work, we assumed processing capacity is a constraint. We further have shown that the capacity is not consumed equally per period but at distinct peak periods. Based on this insight, future research may add flexibility to processing capacity, e.g., allowing the processing of more material in one period but less in a subsequent period. Furthermore, processing capacity may be stochastic because of uncertain processing time or varying energy costs per period. Further, we observed consistency patterns in the periods specific suppliers were visited. Such consistency may allow for better planning and cooperation with the waste collection facilities. Future research may investigate how consistency may be integrated into decision-making explicitly and to what cost. Finally, our literature survey revealed that work on routing problems with quality or yield uncertainty is somewhat limited. However, such uncertainty is present in various problems, not only in recycling but also in collecting farming products or reverse logistics. 734 D.Cuellar-Usaquén et al. Appendix A In the Appendix, we describe the logic of the dynamic components, the revelation of stochasticity, and the decision-making process of our method over the planning horizon. We then present details on the fixed route policy and the results of additional experiments. A.1 Decision making ontheplanning horizon Algorithm1 presents the period-by-period flow of information and decision making using the STM-policy. The algorithm receives two types of input. First, it processes problem information such as the planning horizon (T), supplier details (M), behavior of the uncertainty sources ( 𝜇r , 𝜎r , 𝜇𝜙 , 𝜎𝜙 ), travel time and vehicle characteristics ( 𝜏,Q,lmax ), and cost information (c,f). Second, it uses the parameters of the proposed stochastic lookahead method such as the size of the lookahead horizon (h), the number of sample paths within the lookahead ( |Ω| ), and the discount parameter for routing approximation ( 𝛾 ). The algorithm returns the cumulative cost over the entire planning horizon ( TotalCost ). Algorithm1 starts by initializing the net inventory at the depot ( I1 ) and supply at the suppliers ( q1 ) for the first state and sets the total cost of the planning horizon to zero, this value is stored in the variable TotalCost (lines 1 - 4). In every period (lines 5 - 17), the function SampledMultiperiodScenarios (line 6) constructs the supply and percentage loss realizations to be evaluated within the stochastic program. This function takes as input the information from each uncertainty source ( 𝜇r , 𝜎r , 𝜇𝜙 , 𝜎𝜙 ) and generates a number |Ω| of sample paths associated with the forward period horizon of size h respective to the current period t. With the state information ( St ), the generated scenarios ( Ω ), the discount parameter ( 𝛾 ), and additional problem information, the two-stage stochastic program is solved by calling the function StochasticLookahead (line 7). The solution to the stochastic program returns the selected suppliers ( et ) and the quantities to be collected at each of them ( zt ). This information, along with the travel time ( 𝜏 ) and the cost per minute travelled (c), is given to the RoutingHeuristic function (line 8), which returns the explicit collection routes ( xt ). With the routes and quantities to be collected decided, the decision at for the period t is completed (line 9). Then, the routing cost ( Cr ) is calculated by considering the travel time of the routes and the cost per minute (line 10). Having the action at for the state St , the post-decision state Sa t is generated (line 11). Then, the uncertainty related to the inventory that can be used from the material collected ( z𝜔 t+1 ) and the additional supply available from suppliers ( r𝜔 t+1 ) is revealed (line 12), and the net new inventory ( I𝜔 t+1 ) is calculated based based on material generated (line 13). If the net inventory is negative, the cost of backorders ( Ce ) is incurred to meet demand (line 14). Finally, the costs of routing and backorders for the current period are accumulated within the variable TotalCost (line 15), and the new state St+1 is calculated using the transition function (line 16). This procedure continues until there are no more periods in the planning horizon. 735 Dynamic multi‑period recycling collection routing with… Algorithm1 Decision-making of the STM-policy over the planning horizon A.2 Solution approach based onfixed collection routes This section presents the Fixed-policy, which utilizes fixed, period-dependent collection routes based on historical routes observed. We first explain how we simulate the historical routes and then describe how the policy constructs a solution. Next, we detail how we determine the number of routes per period. We create 100 tuning instances to identify promising routes following the setting described in Sect.5.1. Each instance is solved using the STM-policy. We extract the different routes for each period and count the number of times each route was used. Route symmetry is excluded in this count; for example, route 0−1−2−0 is considered equivalent to route 0−2−1−0 . The routes for each period are then sorted in descending order based on their frequency of use and subsequent routes are removed in case they contain already covered suppliers to ensure no repeated suppliers. Starting with the route with the highest frequency of use in each period, the policy collects material by following the order of the suppliers on the route, ensuring the vehicle capacity is not exceeded. If not all suppliers on a route are selected due to capacity constraints, the routing and its cost are re-adjusted based on the selected 736 D.Cuellar-Usaquén et al. suppliers. This process is repeated with the next route until the processing capacity at the depot is full or all available routes for the period have been used. The number of routes per period should be enough to avoid high backorder costs but not so large as to generate cost overruns for unnecessary collections. The maximum number of routes per period t∈T is defined as Bt =min( ⌈ r t ⋅(1+𝜃Fixed) ⌉ , | K t|) where rt is the average number of routes used in period t over the 100 tuning instances, 𝜃Fixed is a control parameter that allows for varying the fleet size, and Kt is the ordered routes available for the period t. The parameter 𝜃Fixed is tuned via enumeration. We tested 𝜃Fixed values in {0.0, 0.1, 0.25, 0.5, 1.0, 1.5, 2.0} . The values that minimize the expected objective function were selected using 30 tuning instance realizations. For the Sachsen-Anhalt case, 𝜃Fixed was fixed at 0; for the UK-instance case, it was fixed at 1.5. A.3 Difference betweenSTS andmyopic Figure12 illustrates the differences in decision-making between the STS and Myopic policies during the first period of the planning horizon, starting with an identical initial state for both policies. The figure shows the difference in routing cost, supply collected, resulting inventory, and backorders. The values are calculated as the difference between the average value for STS and the average value for Myopic. We observe that the routing cost is the same for both policies. However, the STS policy collects a larger amount than the Myopic policy (0.5 tons more on average). This additional amount results from the different loss scenarios, leading to higher inventory and fewer backorders. A.4 Multi‑period policy decision making In this section, we delve into the intra-period behavior of policies in more detail. Table3 provides insights into various indicators for each policy. At the top are the indicators for the Sachsen-Anhalt case, and at the bottom are the indicators for the Fig. 12 Comparison of decisions made by STS and Myopic policies in the single period version 737 Dynamic multi‑period recycling collection routing with… UK-instances case. The Routing indicator refers to the average accumulated routing cost over the planning horizon. Then, we present the average number of suppliers visited per period. The Backorders indicator reflects the average accumulated backorder cost over the planning horizon. Backorder periodicity indicates the average number of periods until the backorder cost is incurred again. Finally, the Loss indicator shows the average loss incurred by each policy. We observed that the superior performance of STM is due to a balanced approach across all indicators. Single-focus policies negatively impact other indicators, affecting overall performance, particularly the percentage of loss and the number of suppliers visited (Quality+M, Volume+M, Distance+M). The Fixed-policy offers advantages by capturing supplier visit behavior but lacks flexibility in some periods, leading to increased routing and backorder costs. Although the policies of the Myopic+M and STS+M do not visit many suppliers, they incur higher routing costs, indicating sporadic visits to unsuitable suppliers. The STM-policy has the fewest supplier visits but maintains stable periodicity and backorder costs. These findings suggest that anticipating potential supplier events allows for better supplier selection, effectively balancing available material and losses. A.5 Vehicle parameters In the following, we analyze the performance of the policies in case vehicle parameters change. First, we investigate the impact of the vehicle capacity by test capacities of 5 tons (small truck) and 20 tons (truck with trailer). We compare the changes of the policies with respect to STM and our basis Sachsen-Anhalt setting with 10 tons capacity. The results are shown in Fig.13. We observe that our policy becomes even more effective if capacity is increased. Notably, for the 20 tons instances, policy Myopic achieves similar results as our policy with 10 tons capacity. In the case of very limited truck capacities of 5 tons, we observe a substantial increase in cost. Thus, having sufficiently large trucks is very important for the company. Table 3 Analysis of multi-period policy indicators Indicators Quality+M Volume+M Distance+M Fixed Myopic+M STS+M STM Sachsen-Anhalt Routing (€) 29045 17255 17670 15349 15795 12806 10635 Suppliers visited 4.5 2.0 2.8 1.9 2.0 1.5 1.3 Backorders (€) 0.3 0.0 51.7 45.0 0.0 0.0 10.9 Backorder periodicity 7.0 – 9.2 20.0 – – 11.5 Loss (%) 17.9 22.6 24.8 24.3 24.6 21.8 23.3 UK-instances Routing (€) 19902 8177 20527 15380 6841 6908 5861 Suppliers visited 8.6 2.4 9.3 4.6 3.1 3.2 2.7 Backorders (€) 1.3 28.2 5.7 636.1 7.8 60.4 22.8 Backorder periodicity 11.0 12.5 2.5 10.7 7.0 11.3 14.8 Loss (%) 14.7 19.5 20.5 19.5 19.7 19.4 19.3 738 D.Cuellar-Usaquén et al. Next, we analyze the routing cost. In our main experiments, we assume a cost of 1.5 Euros per minute of travel. Now, we vary the cost from 0.5 to 1.0 up to 3 Euros per minute. The results are shown in Fig.14, again relative to the base setting. Regardless of the routing cost, STM performs superior. The policy is particularly effective when the routing cost is small. A.6 Backorder cost In this section, we show the results for varying backorder cost in Fig.15. We run the three policies for backorder costs of 50, 100, 250, 500, 1000, and 2000 per ton. We observe that the results remain comparably stable. Ideally, backorder cost is avoided. Only for cases with (unrealistically) cheap backorder cost of 50, all policies switch from collection routing to backorder. Fig. 13 Changing of vehicle capacity Fig. 14 Changing of routing cost