Finding Minimum Production Time and Optimal Procedure for a Project When Step Times are Given by Constructive Real Numbers

In the production process, some jobs can only be done sequentially, while others can be done synchronously. Given the time needed to complete each job, an oriented graph with a source vertex and a sink vertex can be generated. Each route starting from the source vertex and ending at the sink vertex represents a way of completing the project. It is possible to find the minimum production time and the optimal production path when time for each job is given by a rational number. However, this paper proves that there does not exist a computer program that can always generate the minimum time and optimal path when the time needed for each job is a constructive real number.