## 2 Bi-Objective Model for APP with Production in Advance and Backordering
### Analysis of the Impact of Production in Advance and Stockout  
(1) Production in advance  
Manufacturing enterprises make use of the remaining production capacity to carry out production in advance, which can make balanced use of the effective production capacity of the enterprise in each period and avoid the impact caused by the change of workers, equipment maintenance or damage, holidays, and other conditions [9]. However, production in advance will increase the inventory cost of production, which can be expressed by Equation (1). \[\sum_{t=1}^{T}\sum_{i=1}^{I}C_{lit}\cdot CK_{i} \tag{1}\]  
(2) Stockout  
Due to the decline of actual production capacity or the shortage of production materials, production enterprises may be backordered [2]. Stockout will lead to a decline in customer satisfaction or loss of sales. Generally, the longer the replenishment time and the greater the backorder quantity, the greater the possibility of a loss of sales [24]. Therefore, the stockout cost, which includes backorder cost (BC) and LSC, depends on the replenishment time and the backorder quantity. In order to determine the stockout cost, it is necessary to determine firstly the critical value of the proportion of backorder quantity to the total demand for different replenishment times that customers can tolerate. \begin{table}
\begin{tabular}{l l} \hline \hline \multicolumn{1}{c}{**Parameters**} & \multicolumn{1}{c}{**Description**} \\ \hline \(C_{2t}\) & Total raw material cost in period \(t\) \\ \(PW_{ik}\) & Labor hours required for a unit product \(i\) of worker \(k\) \\ \(PN_{it}\) & The production capacity for product \(i\) in period \(t\) \\ \(C_{lit}\) & The inventory of product \(i\) in period \(t\) \\ \(CK_{i}\) & Inventory cost of product \(i\) \\ \(CN_{i}\) & The inventory capacity of product \(i\) \\ \(C_{3t}\) & Total inventory cost in period \(t\) \\ \(R_{ij}\) & The demand of raw material \(j\) for producing unit \(i\) \\ \(RM_{jl}\) & Total demand of raw material \(j\) in period \(t\) \\ \(RC_{jl}\) & The price of raw material \(j\) in period \(t\) \\ \(WH_{tk}\) & The worker number for \(k\) type in period \(t\) \\ \(WI_{k}\) & The basic salary of \(k\) type worker in a planning period \\ \(WL_{ik}\) & The number of laid-off workers for \(k\) type in period \(t\) \\ \(WHC\) & Training cost for a new worker \\ \(WR\) & Maximum regular labor hours in a period \\ \(WO\) & Maximum overtime labor hours in a period \\ \(WRT_{lk}\) & Total regular labor hours for \(k\) type worker of period \(t\) \\ \(WOT_{ik}\) & Total overtime labor hours for \(k\) type worker of period \(t\) \\ \(WRC_{k}\) & Unit regular time labor cost for \(k\) type worker \\ \(WOC_{k}\) & Unit overtime labor cost for \(k\) type worker \\ \(C_{dt}\) & Total labor cost in period \(t\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: \(Cont.\) \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{1}{c}{**Variables**} & \multicolumn{1}{c}{**Description**} \\ \hline \(PP_{it}\) & Production quantity of \(i\) in period \(t\) \\ \(W_{ik}\) & The number of \(k\) type workers employed in period \(t\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Decision variables. The threshold for customer loss: Let the threshold for customer loss be the critical value. Referring to the definition of exponential partial backlogging rate in the inventory model [25], the threshold for customer loss is defined as Equation (2). \[\beta(t^{\prime}-t)=k_{0}\cdot e^{-k_{1}(t^{\prime}-t-1)} \tag{2}\]  
where \(t^{\prime}-t\) is the customer waiting time (\(t^{\prime}>t\)), \(t\) represents the planning period of customer demand, and \(t^{\prime}\) is the period of replenishment. \(k_{0}\) is the backordering intensity coefficient, that is, the maximum backorder rate acceptable to the customer for one planned period of delayed delivery. \(k_{1}\) is the waiting time resistance, and \(0<k_{0}<1,k_{1}>0\).