## 3 Problem Formulation  
The APP problem can be described as follows. It is assumed that a firm produces a single product to satisfy market demand over the planning horizon. The problem is to determine the most effective manufacturing mix that satisfies the projected demand by adjusting production rates, workforce sizes and loads, inventory levels, order status, supplier collaboration, and other controllable variables. The objective functions of this APP decision problem are to maximize sales revenue, to minimize total production costs, and to minimize repair costs as well. Thus, the APP problem can be derived into a mathematical model under the following assumptions:  
Notations  
The notations used in the model are given as follows. Parameters  
\(sr_{i}=\) sales revenue for product \(i\) ($/unit)  
\(pc_{i}=\) production cost of regular time for product \(i\) ($/unit)  
\(oc_{i}=\) production cost of overtime for product \(i\) ($/unit)  
\(cc_{i}=\) inventory carrying costs for product \(i\) in each period ($/unit period)  
\(bc_{i}=\) stock-out cost for product \(i\) in period \(t\) ($/unit period)  
\(lc_{t}=\) regular payroll cost per worker in period \(t\) ($/man hour)\(hc_{t}=\) cost of hiring one worker in period \(t\) ($/man-day)  
\(fc_{t}=\) cost of firing one worker in period \(t\) ($/man-day)  
\(rc_{i}=\) repair cost for product \(i\) ($/unit)  
\(\rho_{i}=\) defect rate for product \(i\)  
\(k_{i}=\) labor time for product \(i\) (man-hour/unit)  
\(\delta=\) regular working hours per worker per day  
\(\beta\)= fraction of regular workforce available for overtime use in period \(t\)  
\(W_{o}=\) initial workforce level (man-day)  
\(W_{tmax}=\) maximum workforce level available in period \(t\)  
\(t=\) planning horizon or number of periods  
\(I_{o}=\) initial inventory level (units)  
\(B_{o}=\) initial backorder level (units)  
\(D_{itmax}=\) maximum demand for product \(i\) in period \(t\) (units)  
\(D_{itmin}=\) minimum demand for product \(i\) in period \(t\) (units)  
\(M_{t}=\) regular time machine capacity in period \(t\) (machine-hour)  
\(\theta_{Mt}=\) fraction of regular machine capacity available for overtime use in period \(t\)  
\(b_{i}=\) machine time for product \(i\) (machine-hour/unit)  
\(\bar{I}_{t}=\) the storage space limitation in period \(t\)  
\(\alpha_{i}=\) fraction of subcontracting output available for product \(i\)  
\(C_{tmax}=\) maximum capacity limitation of subcontracting in period \(t\)  
\(Q_{tMOQ}=\) minimum order quantity for subcontracting in period \(t\)  
Decision Variables  
\(I_{it}=\) inventory level for product \(i\) in period \(t\) (units)  
\(B_{it}=\) backorder level for product \(i\) in period \(t\) (units)  
\(RP_{it}=\) the unit of regular time production for product \(i\) in period \(t\)  
\(OP_{it}=\) the unit of overtime production for product \(i\) in period \(t\)  
\(OS_{it}=\) the unit of subcontracting production for product \(i\) in period \(t\)  
\(W_{t}=\) the number of workers in period \(t\)  
\(H_{t}=\) the number of workers hired in period \(t\)  
\(L_{t}=\) the number of workers laid off in period \(t\)  
Objective Functions  
This proposed APP problem considers three objectives: maximizing sales revenue, minimizing total production cost, and minimizing repair cost. The sales revenue indicates the market share. When market share is a main concern for a company, maximizing sales revenue becomes a primary objective in the APP problem. Repair costs can represent inner failure costs in the production process. In many cases, inner failure costs may be a chief goal because they impact the utilization of production resources. When inner failure costs are high, a company may be forced to use overtime production or more resources. In these cases, repair costs should be considered a separate objective, instead of a component of production costs. In this case, minimizing total production costs, maximizing sales revenue, and minimizing repair costs are all important considerations for the case company.