M unit-procedures and τj is the total duration for the jth unit-procedure. The unit-procedure with the maximum duration is sometimes referred to as the bottleneck. This relationship applies when each unit-procedure has a single dedicated equipment unit. If redundant equipment units are available for at least one unit-procedure, the minimum cycle-time becomes: Where is the number of redundant equipment for unit procedure . If equipment is reused within a process, the minimum cycle-time becomes more dependent on particular process details. For example, if the drying procedure in the current example is replaced with another reaction in the reactor, the minimum cycle time depends on the operating policy and on the relative durations of other procedures. In the cases below, an increase in the hold time in the tote can decrease the average minimum cycle time. Visualization Various charts are used to help schedulers visually manage schedules and constraints. The Gantt chart is a display that shows activities on a horizontal bar graph in which the bars represent the time of the activity. Below is an example of a Gantt chart for the process in the example described above. Another time chart which is also sometimes called a Gantt chart shows the time during which key resources, e.g. equipment, are occupied. The previous figures show this occupancy-style Gantt chart. Resources that are consumed on a rate basis, e.g. electrical power, steam or labor, are generally displayed as consumption rate vs time plots. Algorithmic methods When scheduling situations become more complicated, for example when two or more processes share resources, it may be difficult to find the best schedule. A number of common scheduling problems, including variations on the example described above, fall into a class of problems that become very difficult to solve as their size (number of procedures and operations) grows. A wide variety of algorithms and approaches have been applied to batch process scheduling. Early methods, which were implemented in some MRP systems assumed infinite capacity and depended only on the batch time. Such methods did not account for any resources, and would produce infeasible schedules. Mathematical programming methods involve formulating the scheduling problem as an optimization problem where some objective, e.g. total duration, must be minimized (or maximized) subject to a series of constraints which are generally stated as a set of inequalities and equalities. The objective and constraints may involve zero-or-one (integer) variables as well as nonlinear