One of the most valuable commodities in today’s society is time. The faster something can be done, the better. To discuss the value of time, Jens Schultz, a doctorate student at the University of Berlin, offered Mines a guest lecture on Conflict Analysis in Cumulative Scheduling. Schultz’ research searches for ways to help businesses maximize their output while minimizing the time required by using mathematical analysis.

Schulz has been studying how to use math to efficiently assign and accomplish jobs. She defines a job as a task or a set of tasks that must be completed. These tasks can range from building a single part to assembling multiple parts together.

Jobs typically vary by two factors, the number of workers and the amount of time necessary to complete them. By examining these two factors and using mathematical equations, it is possible to minimize the amount of time necessary to complete a job.

This analysis can be done graphically through a process called time tabling. The required number of workers is placed on the y-axis and the time a task requires is placed on the x-axis. Each task is made into a box, with the height being the number of workers and the width being the time.

Then, a limit is placed on the x-y graph. This limiting factor is equal to the number of workers for the total job. As Schulz said, “Limited resources (workers) can only do so much at once.” Next, each worker by time box is fit in under the limiting factor. Fit together like a puzzle, the boxes cannot be stacked above the limiting factor because the job only has so many workers available.

An example of this would be a job requiring task one and task two to be completed with only four workers. Task one requires two workers and takes four hours to complete. Task two requires three workers and takes three hours to complete. Due to the limiting factor of four workers, both tasks cannot be going on at the same time. Thus the total amount of time to complete the tasks would be found by putting the boxes next to each other on the graph and adding the total time it takes, which is seven hours in this example.

This is a very simple time tabling exercise, but the process becomes much more complicated when other factors are added. For example, if task two must be completed at the same time as task one, it means that either the number of workers required for the task or the limiting factor

must change. Limiting factors are relatively constant, which means the tasks themselves must be altered. If this exercise is repeated with a wide range of steps and tasks that have to be completed in a certain order, it can become very confusing.

These table analyses can be applied to assembly lines, offices, or anything that can be

quantified by the number of workers and the time required for completion. Conflict analysis, particularly time tabling, plays a major role in solving challenging scheduling instances efficiently.