## 2 Models for Sorting and Logistics Operators  
In this section, we model collection and sorting planning in an industrial solid waste management system. The processes of the case studied are sketched in Figure 1 for transport and in Figure 2 for sorting. Transport is carried out by a specialized logistics operator that serves customers by transporting empty containers to their location and picking full containers to be delivered to the sorting plant. Several industries produce mixed waste that must be sorted. In particular, warehouses and transport operators are a source of recyclable waste where different types of plastic and paper are grouped. In the analyzed sorting plant, waste goes through a sorting cabin where it is visually inspected and separated into its components. After several other sequential operations, the separated waste is transformed into secondary raw material and can be sent to other processing plants. Different technologies can be used to improve the automation and digitization phases of the collection and sorting processes both at the operational and planning levels. [PERSON]  
Figure 1: Diagram of the process operated by the carriers. et al. [28] surveyed how tracking technologies, such as RFID and GPS, could support waste management. In particular, RFID and image recognition applied to bins are used to track waste production and improve planning processes. Recently, [PERSON] et al. [29] surveyed the research literature considering artificial intelligence applications to municipal waste management. Artificial intelligence and computer vision can be applied to sorting to automatically separate waste [30]. In this context, our case study is related to an industrial waste recycling network, where RFID is used to track the position of containers located in customer locations or in a sorting plant. The sorting process is still not automated, and visual inspection coupled with manual sorting at the cabin is performed. Customers, the logistics operator, and the sorting operator are connected through email and the Enterprise Resource Planning (ERP) system. The planning of the two processes is scheduled every week/month independently by the sorting and the logistics operators. This leads to peaks in daily sorting operations because the customers interact directly only with the logistics operator. In the studied case, only a not formally specified interaction between logistics and sorting operator occurs, mostly by phone call, to synchronize transports with sorting operations. Starting with these premises, we decided to focus on negotiation as a way to manage our waste management problem. In particular, the decision process that needs optimization and is studied in this paper is as follows. Business customers establish contracts with recycling operators. They generate waste during their business processes, which is stored in local buffers (e.g., containers). Once the buffers are full, a recycling request is generated to the logistics operator in charge to define a transportation plan. Once the transportation schedule is established and communicated to the sorting operator, the latter can plan the sorting operations at the plant. The plant operates sequential sorting stations that require setup and staffing. Additionally, waste buffers have limited capacity and incur occupancy costs. After the sorting/recycling operations, the recycled material can be directed to secondary raw material markets. Although the planning of these two subproblems (collection and sorting) is usually completed sequentially and separately, this study proposes an effective method to optimize and integrate both planning phases. We use an improved auction-based scheme to solve the sorting and collection problems in an integrated way and obtain an optimal multiperiod schedule for the two sub-processes. In particular, concerning the approach described in [1], the bid generation scheme is designed to produce a more robust transport plan and thus greater opportunities for the sorting operators to optimize the sorting plan.