## 2 Materials and Methods
### Stage 2: Inputs and Outputs Determination
#### 2.2.2 Input Costs Functions and Outputs Variable  
Now, using the picking rate and the batching model proposed by [27], we were able to generate the number of daily fulfilled orders and an estimation of the associated costs incurred by the warehouses. The outputs are defined as followed:  
\[M^{\text{unij}}=number\ of\ items\ fulfilled \tag{3}\]  
\[L^{\text{unij}}=\frac{M^{\text{unij}}}{Q^{\text{unij}}} \tag{4}\]\[A^{wij}=\frac{Q^{wij}}{O^{wij}} \tag{5}\]  
where \(M^{wij}\) is the sum of daily items fulfilled from each order and batch given the warehouse, the routing policy, and the type of paperless technology. We assumed that there is an 8-h shift and a maximum of 2 h overtime per picker (if all the orders are not fulfilled during the 8-h shift). If an order is not picked in either time period, it is considered unfulfilled and the service level for that day will be less than 1 (the highest value). \(L^{wij}\) is the service level of the daily picking process and takes values between 0 and 1, where \(Q^{wij}\) is the total daily ordered items. A value of 1 indicates that the total order made in a day has been fulfilled and ready to be shipped on schedule. \(A^{wij}\) is the accumulation level, where \(O^{wij}\) represents the total orders. It reflects the level of effort required of labor inputs to fulfill the sum of orders in a working day. We present a modified cost function \(C_{d}^{wij}\) of [15], which comprised four main daily cost components:  
- Daily cost depending on the number of stock locations \(C_{d,SL}^{wij}\). - Daily cost depending on the number of pickers \(C_{d,P}^{wij}\). - Daily cost depending on the picking errors \(C_{d,E}^{wij}\). - Daily fixed cost \(C_{d,F}^{wij}\). \[C_{d}^{wij}=C_{d,SL}^{wij}+C_{d,P}^{wij}+C_{d,E}^{wij}+C_{d,F}^{wij} \tag{6}\]  
The cost function in Equation (6) can be written based on the notation in Table 3 as follows:  
\[C_{d}^{wij}=\left(\frac{n_{SL}^{w}C_{SL}^{j}}{h_{SL}}+C_{h,P}+\sum_{x}\frac{C_ {D,P}^{w}}{h_{Dx,P}}+\frac{C_{F}^{w}}{h_{F}}\right)t_{tot}^{wij}+C_{h,P}\left( \sum_{k}p_{ck}^{wij}T_{ck}^{j}\right) \tag{7}\]  
The total daily required time \(\left(T_{ck}^{j}\right)\) to fix errors is subject to the paperless technologies used during the picking process. For example, for barcodes handheld, if the error \(e_{1}\) occurs, the picker will need some more time to confirm the right item and will therefore spent \(t_{c}^{j}\) unit of time to do so. If the \(e_{2}\) occurs the picker will spend \(2.t_{tot}^{j}\) time to correct the error. This is because the picker will need twice as much time to get information about both the right and wrong product, to search, to pick, \begin{table}
\begin{tabular}{c c c} \hline \hline
**Cost** & **Expression** & **Notation** & **Description** \\ \hline \(C_{d,SL}^{wij}\) & \(\frac{n_{SL}^{w}C_{SL}^{j}}{h_{SL}}t_{tot}^{wij}\) & \(C_{SL}^{j}\) [5] & Storage location unitary cost \({}^{\mathrm{a}}\) \\  & & \(n_{SL}^{w}\) & Number of available storage locations \\  & & \(h_{SL}\) [h] & Storage location devices total usage hours \\ \(C_{d,P}^{wij}\) & \(\left(C_{h,P}+\sum\limits_{x}\frac{C_{D,P}^{e_{1}}}{h_{Dx,P}}\right)t_{tot}^{wij}\) & \(C_{h,P}\) [5/h] & Picker hourly cost \\  & & \(C_{D,P}^{i}\) [5] & Cost of picker device type \(x\) \\  & & \(h_{Dx,P}\) [h] & Picker device total usage hours \\ \(C_{d,E}^{wij}\) & \(C_{h,P}\left(\sum\limits_{k}p_{ck}^{wij}T_{ck}^{j}\right)\) & \(p_{ck}^{wij}\) & Occurrence probability of an error \\  & & \(T_{ck}^{j}\) & Total required time to fix daily total errors \(ek\) \\ \(C_{d,F}^{wij}\) & \(\frac{C_{F}^{w}}{h_{F}}t_{tot}^{wij}\) & \(C_{F}^{w}\) [5] & Fixed costs \({}^{\mathrm{a}}\) \\  & & \(h_{F}\) [h] & Fixed elements total usage hours \\ \hline \hline \end{tabular} Note. \({}^{\mathrm{a}}\) Variable according to the considered technology \(j\) or warehouse \(w\).  
\end{table} 
Table 3: Cost function components and notations. and to store. On the other hand, if \(e_{3}\) occurs, additional \(2\cdot t_{net}^{j}+t_{traw}\) time will be needed.