# Optimizing a Reverse Supply Chain Network for Electronic Waste under Risk and Uncertain Factors
## 3 A Developed Mathematical Model
Total cost includes the sum of collection costs, set-up costs, processing costs, transportation costs, disposal costs, and risk costs and minus the profit obtained from selling used items and recovery materials (as seen in Equation (1)). The explanations and mathematical formulations of these components are presented in Equations (2)-(8). Objective function  
\[\begin{split}\text{Total cost (Z) = collection costs (T1) + set-up costs (T2) + processing costs (T3)}\\ + transportation costs (T4) + disposal costs (T5) + risk costs (T6)}\\ - profit received from selling recovery materials and used items (P)Collection costs (\(T1\)) are to collect EoL products from consumers, end-users, or retailers, which can be described in Equation (2).  
\[T1=\sum_{c=1}^{C}\sum_{d=1}^{D}\sum_{p=1}^{P}Q1_{cdp}\times\widetilde{CL}_{p} \tag{2}\]  
Note that \(\widetilde{CL}_{p}\) is an uncertain parameter because it is challenging to estimate the collection cost exactly, which depends on the quality and volume of return products. Set-up costs (\(T2\)) are the cost for building dismantling, recycling, and repairing centers, which can be calculated using Equation (3). \[T2=\sum_{d=1}^{D}X_{d}\times\widetilde{Sd}_{d}+\sum_{e=1}^{E}X_{e}\times \widetilde{Se}_{e}+\sum_{r=1}^{R}X_{r}\times\widetilde{Sr}_{r} \tag{3}\]  
Processing costs at the treatment centers can be defined as Equation (4). \[T3=\sum_{c=1}^{C}\sum_{d=1}^{D}\sum_{p=1}^{P}Q1_{cdp}\times\widetilde{Od}_{pd} +\sum_{d=1}^{D}\sum_{e=1}^{E}\sum_{w=1}^{W}Q3_{dew}\times\widetilde{Oe}_{uw}+ \sum_{d=1}^{D}\sum_{r=1}^{R}\sum_{i=1}^{I}Q4_{dri}\times\widetilde{Or}_{ir} \tag{4}\]  
Transportation costs for shipping different types of parts from one center to another center are presented in Equation (5). \[T4=\sum_{c=1}^{C}\sum_{d=1}^{D}\sum_{p=1}^{P}Q1_{cdp}\times B1_{ cd}\times\widetilde{Tp}_{p}+\sum_{d=1}^{D}\sum_{s=1}^{S}\sum_{u=1}^{w}Q2_{ dsu}\times B2_{ds}\times\widetilde{Tu}_{u} \tag{5}\] \[+\sum_{d=1}^{D}\sum_{e=1}^{E}\sum_{w=1}^{W}Q3_{dew}\times B3_{dew }\times\widetilde{Tw}_{w}+\sum_{d=1}^{D}\sum_{r=1}^{R}\sum_{i=1}^{I}Q4_{dri} \times B4_{dr}\] \[\times\widetilde{T}_{i_{1}}+\sum_{d=1}^{D}\sum_{i=1}^{L}\sum_{h= 1}^{H}Q5_{dh}\times B5_{al}\times\widetilde{T}\bar{h}_{h}+\sum_{e=1}^{E}\sum_ {s=1}^{S}\sum_{w=1}^{W}Q6_{esw}\] \[\times B6_{es}\times\widetilde{Tw}_{w}+\sum_{r=1}^{R}\sum_{m=1}^{ M}\sum_{i=1}^{I}Q7_{rmi}\times B7_{rm}\times\widetilde{T}_{i_{l}}\] \[+\sum_{r=1}^{R}\sum_{l=1}^{L}\sum_{h=1}^{H}Q8_{rth}\times B8_{rl} \times\widetilde{Th}_{h}\]  
The disposal costs for toxic substances can be found by Equation (6). \[T5=\left(\sum_{d=1}^{D}\sum_{l=1}^{L}\sum_{h=1}^{H}Q5_{dth}+\sum_{r=1}^{R}\sum _{l=1}^{L}\sum_{h=1}^{H}Q8_{rth}\right)\times\widetilde{D}_{h} \tag{6}\]  
Risk costs (\(T6\)) are the cost incurred from the likelihood of any disruptive occurrences that might have an influence on the part of processing costs at treatment centers and transportation costs. In a supply chain system, the risk score is calculated by multiplying the probability and the loss of occurrence [59]. In this study, the first three terms of risk costs in Equation (7) represent the risk resulting from unexpected events such as accidents, technological issues, or less manpower during the treatment of EoL products or parts at dismantling, repairing, and recycling centers. The rest of Equation (7) presents the risk during transportation activities within an RSC network.