The model is formulated under the following key assumptions which are also made in the study 180 conducted by Özçelik et al. (2020) .
Assumptions:
182 -All collected end-of-life products are recovered in the recycling centers.
-One year is considered to be a planning period.
-Both distances and collection center capacities are affected by the ripple effect.
-All temporary facilities are considered to be identical. 186 In section 2.2, the formulated optimization model is represented. and an approach can be implemented to obtain the optimal or near-optimal Pareto-front.
Moreover, all possible route combinations can be considered; however, it can cause some 201 unrealistic circumstances.
In the following, the proposed stochastic optimization model is presented after the information is 203 provided regarding the model.
Indices and sets: 205 : index defined for the primary collection center ∈
: index defined for the secondary collection center ∈
: index defined for the recycling center ∈ 208 : index defined for the product type represents the occurrence probability of the scenario.
First stage constraints: 
Constraint sets (6)-(9) are employed to control capacities by considering the capacity loss caused 283 by disruptions in each scenario. Here, whilst constraint sets (6) and (7) implies that the total 284 number of products delivered from the permanent primary collection center to the secondary collection center for the product cannot exceed the capacities of the and 286 facilities respectively. Constraint set (8) ensures that the total number of products transported 287 from secondary collection center to recycling center for the product cannot exceed the 288 capacity of the facility. Similarly, Constraint set (9) guarantees that the total number of 289 products delivered from the temporary primary collection center to the secondary 290 collection center for the product cannot exceed the capacity of the facility if the 291 temporary primary collection center is used.
Constraint set (10) is used to balance the number of delivered products between the stages on the 294 RSC network, thereby; it prevents the products to be handled in the secondary collection 295 center.
Chance constraints ( -reliability):
The chance-constraints are added to ensure a certain service level in the presence of the ripple 298 effect. In other words, the main reason for adopting the -reliability constraint in the developed 299 model is to analyze the impact of various service levels on the system performance in terms of 300 the operational cost and CO 2 emission amounts. The -reliability approach, also known as chance 301 constraint, was first presented by Daskin et al. (1997) for the facility location problem. Since 302 then, it has been applied to SC design studies (Oksuz and Satoglu, 2020).
Constraint set (11) denotes the reliability of the system and ensures that the pre-determined -307 reliability level is satisfied by considering the scenarios. The -reliability level is satisfied by 308 summing the occurrence probability of scenarios. A higher service level means more scenarios 309 are addressed. Constraint sets (12) and (13) are either/or constraints and reveal that the service is 310 provided or not. These constraints guarantee that the collected products are recovered by meeting 311 the mandatory limit for each product and scenario. In other words, the scenario is included to the reliability set, if the product p is delivered from the facility to the facility by satisfying 313 the limit. (15) and (16) 320 imply the nonnegativity for integer decision variables.
The optimization model including both stages is coded together by using CPLEX solver. The In section 4.2, the experimental settings are provided. Table 2 ) to address the uncertainty regarding the scenarios generated based on two 370 main disruption levels (low-and high-impact). As an example, the disruption levels on the areas 371 are illustrated in Figure 5 , in which (a), (b), and (c) correspond to the "low-impact on R2", "high-372 impact on R3 and low-impact on R1", and "low-impact on R2-R3 and high-impact on R1", 373 respectively.
When the disaster occurs, the distances between facilities and the facility capacities are affected.
Thus, using increase ratios for the distances and capacities is plausible to represent the influence 376 of the ripple effect on the network. The key parameters used in the industrial application are 377 given in Table 3 . Accordingly, the ratios are randomly generated with a 383 continuous uniform distribution from these intervals.
The cases differ from each other in terms of the determined distribution intervals for damage Insert Table 2 The parameter values for the scenarios (considered for case 1)
Insert Table 3 The parameter values representing the disruption levels with respect to the cases values, the total number of temporary facilities, expected emission levels, and CPU times (in 414 seconds). The last column of Table 4 represents the tractability of the proposed model.
From Table 4 , it is deduced that a decrease in -reliability levels leads to better results in terms of 416 the objective function values and expected emission levels for each case. A similar trend is also 417 observed for the total number of temporary facilities. The change in total number of temporary 418 facilities with respect to service levels for cases 1-3 is visualized in Figure 6 .
When the service level varies from 80% to 90%, the total number of temporary facilities is equal 420 to two for cases 1-3. On the other side, when the service level varies from 95% to 99%, the total 421 number of temporary facilities is equal to three, except for = 0.99 in case 3 (See : Table 4 and 422 Figure 6 ). The reason behind the exception is that the transportation cost is larger than the facility 423 utilization cost for case 3, thus; opening a facility is plausible to decrease the overall cost.
Insert Table 4 The summary of computational results for cases 1-3 with respect to the - Tables 5-7 present all results regarding the emission levels and the amount of the collected 449 products from the sub-regions (i.e. R1, R2, and R3) for each scenario with respect to each -450 level. In these tables, the scenarios, which do not affect the results, are represented by zero. Insert Table 6 The computational results for case 2 464 465 Insert Table 7 The computational results for case 3 In these figures, low-and high-impacts are highlighted in grey and red colors, respectively (See:
478 Figure 5 and Table 2 ). The changes in emission amounts along with the scenarios are measured investigated regarding R1, 50% of the temporary collection centers are located in this region to 501 ensure the determined service level. That is to say, a large ratio of temporary centers belongs to 502 this region. Moreover, the centers located in this region are far away from the recycling centers.
For these reasons, it can be deduced that the fluctuations are occurred due to the number of 504 temporary collection centers and the high distance to the recycling centers.
Notably, the number of collected products shows an upward trend by increasing the -reliability 506 level in R2 and R3 whereas there are some fluctuations in results regarding R1. But it can be still 507 said that the value ranges are close for each service level. Overall, these results provide insights 508 into the change in the -reliability level that does not yield a noteworthy effect.
The results contrasted in Figure 10 also allow comparison of the centralized and decentralized 510 distribution strategies. When the amounts of delivered products are explored with respect to the 511 sub-regions, it is observed that the centralized strategy is more preferable than a decentralized 512 strategy in the considered RSC network. The amount of product collected from the R1 is quite 513 low compared to the R2 and R3 sub-regions and this situation promotes the centralized strategy.
Dividing the handled region to the sub-region provides more sensible results regarding the 515 information on how the considered sub-regions are affected individually under disruptions. As 516 can be also seen in Figure 10 suggested that the RSC managers should be aware of the fact that utilizing uniformly distributed 539 and consistent data is crucial to avoid biased decisions when interpreting the emission levels.
To provide a better understanding regarding the extraordinary situations, Table 8 is presented for 541 the results of cases 4-6, which are designed to represent these situations. Additionally, the change 542 in the total number of temporary facilities and the overall cost with respect to service levels for 543 cases 4-6 is visualized in Figure 12 (a) and (b), respectively. It is observed from the results that as 544 the service level increases, the number of temporary facilities increases as well. This interrelation 545 is observed more prominently in cases 4-6 since they correspond to the strong disruptions (See:
546 Table 8 and Figure 12 (a) ). Overall, it can be interpreted that the total number of temporary 547 facilities is larger for cases 4-6 compared to cases 1-3 under strong disruptions.
From the feasibility aspect, when considering cases 4-6, the feasible solutions are only attained 550 for = 0.94, = 0.84, = 0.80, respectively. Therefore, it can be deduced from Nikzad, E., Bashiri, M., Oliveira, F., 2019. Two-stage stochastic programming approach for the 