## 4 Experimental Results  
The algorithm and the models have been implemented in Python 3.12.0 and solved with GurobiTM on a Windows machine equipped with an Intel CoreTM i7-12700K @ 3.60 GHz, 12 core and 32 GB RAM. The instance used for experimentation is adapted from a real recycling plant composed of two stations. The number of customers is 30. The collection time horizon is 25 days; the sorting problem time horizon is 30 days. The cost of collection is set to a uniform distribution such that \(c_{i}^{c}\in[5,15]\), while quantities are chosen uniformly at random in the interval \([3,5]\), and the maximum capacity for trucks is fixed and equal to 10. The variable selection cost \(c_{i}^{s}\) is set to 3, while the setup cost is 5 and the inventory storage cost per unit of quantity and time is set to 1. The inventory capacity is 50, while the minimum and maximum production capacity for stations for a unit of time are 0 and 10, respectively. No loss is considered, i.e., \(\alpha=1\). Figure 4 reports the results when the algorithm is run with the number of bids generated ranging from 1 to 20. If we have only one bid, the logistics operator proposes its optimal solution to the recycling plant that is used to solve the plant planning problem. In that case, we have the minimum cost for the collection process and the maximum cost for the sorting process. In the other cases, the winner determination problem defines the bid to be selected. Figure 4a shows the number of the selected bids. The objective values of the collection and sorting problems are reported in Figures 4b and 4c, respectively. The total cost, that is, the sum of the two objectives, is reported in Figure 4d. As can be seen by analyzing the results, as the number of bids generated by the algorithm increases, the collection cost tends to increase as well (see Figure 4b), while the sorting cost shows a nonincreasing behavior (see Figure 4c). However, in general, they produce a decreasing total cost. In fact, even though the percentage of increase and the percentage of reduction of the two cost functions are substantially the same, that is, about 10%, the total cost exhibits a 5% overall reduction (see Figure 4d). For comparison, in Figure 5, we report the results obtained by implementing a competing approach [1] on the same testbed. As can be seen, while the sorting cost decreased with the increasing number of bids and the collection cost was reduced instead, similarly to what happened with our approach, the overall total cost increased by about 2%. This shows the effectiveness of the ingredients used in the proposed algorithm and the models to improve the overall efficiency of the system. Figure 4: Results obtained by the proposed approach. (**a**) shows the number of bids generated and the number of selected bids; (**b**) shows the number of bids generated and the collection cost; (**c**) shows the number of bids generated and the sorting cost; (**d**) shows the number of bids generated and the total cost. Further Comparison with an Integrated Model and Increasing Instance Size  
In this section, we compare the proposed model with a reference integrated model defined in a single decision-maker environment where the decision-maker optimizes simultaneously the two (i.e., collection and sorting) problems, minimizing the objective of the sorting problem. The model (denoted P3) is formed by the constraints of both model P1 and model P2 and is detailed in the following. \[\min Z_{c}=\sum_{i\in I}\sum_{h\in H}\sum_{t\in T_{c}}c_{ii}^{c} \cdot z_{iht}\] s.t.