## 3 Materials and Methods: Workforce Capacity and Workload  
### A Few Definitions  
In LM, the first step to balance a single-model assembly line is to compute customer _takt_ time (_TT_) as in Equation (1). _Takt_ time is the ratio of the effective working time (or net available time _AT_) to the number of units (_q_) to be produced during that time [5; 39]. This conveys the lean principle that tasks on the shop floor are synchronized with a beat set by the customer: the _takt_ time is the maximum amount of time that can elapse between two consecutive units in order to meet demand. \[TT=\frac{AT}{q} \tag{1}\]  
In this paper, the term \"cycle time of a process\" (such as an assembly line) refers to the time elapsed between two consecutive unit completions (at the end of the line) and \"cycle time of a workstation\" is the amount of time elapsed between two consecutive units exiting the station. Since the cycle time of a serial line process equals the longest cycle time of the workstations [12], the longest station time (sum of the task times assigned to the station) of the assembly line cannot be above the _takt_ time [39] or desired cycle time. In practice, in order to compensate unexpected stoppages, the desired cycle time may be set below the _takt_ time. When the _takt_ time (or desired cycle time) is known, the theoretical minimum number of workstations (_w_) is simply computed according to Equation (2) [5], where _OT_ is the total operation time per workpiece (the sum of task times \(t_{i}\)). \[w=\frac{OT}{TT}=\frac{\sum_{i}t_{i}}{TT} \tag{2}\]  
Finally, tasks are assigned to workstations so that the resulting station times are never above the _takt_ time. As an elemental task cannot be split between two workstations, the resulting number of workstations may be greater than \(w\). Since we assume that, in a multi-model line, the desired number of units of each model is known, because it is related to customer demand, but the time the line will be devoted to each model is unknown, in the following sections, we report a new framework to staff production systems that relies on the concepts of workload and capacity. When an operator, a workstation or an assembly line are assigned a job to do, we can say that they are given a certain workload. We can define the workload (e.g., of a workstation) as the amount of work assigned to such workstation per time unit (e.g., production in parts per day or man-hours per day). The capacity of a workstation is defined as the amount of work that a workstation can process per time unit [34]. In this paper, without loss of generality, we assume that a workstation is operated by one person at a time. Equation (3) describes the capacity that a given workstation requires to perform operation \(i\) on model \(k\) (_CPik_) in terms of how many times an operation that requires \(t_{ik}\) time units can be repeated in the available time (_AT_). \[CP_{ik}=\frac{AT}{t_{ik}} \tag{3}\]  
While extant line balancing methods are aimed at achieving similar station times for all workstations with values close to, but always below, the desired cycle time, in our approach, since the cycle time is not known, we balance capacity against workload in search of similar load/capacity ratios for each workstation, with station workload being less than or equal to station capacity. For this reason, the load/capacity ratio is defined as a dimensionless magnitude that we name unit workload of operation \(i\) on model \(k\) (_Wuik_) or, in consequence, the unit workload that bears a certain workstationthat has to perform task \(i\) on model \(k\). This is the ratio of the desired production of model \(k\) (\(q_{k}\)) to the capacity \(CP_{ik}\) of the workstation, both \(q_{k}\) and \(CP_{ik}\) being measured in parts per day (Equation (4)).