#Python Code include: #Problem Narration #Define Gurobi model using python #Set and their explanation #Parameter / data and their description #decision variabel and their explanation #Contsraint and their explanation #run optimization and their explanation #Print the result and explanation

7.4 Sequencing jobs on a bottleneck machine In workshops it frequently happens that a single machine determines the throughput of the entire pro- duction (for example, a machine of which only one is available, the slowest machine in a production line, etc.). This machine is called the critical machine or the bottleneck. In such a case it is important to schedule the tasks that need to be performed by this machine in the best possible way. The aim of the problem is to provide a simple model for scheduling operations on a single machine and that may be used with different objective functions. We shall see here how to minimize the total processing time, the average processing time, and the total tardiness. A set of tasks (or jobs) is to be processed on a single machine. The execution of tasks is non-preemptive (that is, an operation may not be interrupted before its completion). For every task i its release date and duration are given. For the last optimization criterion (total tardiness), a due date (latest completion time) is also required to measure the tardiness, that is, the amount of time by which the completion of jobs exceeds their respective due dates. The following table lists the data for our problem. What is the optimal value for each of the objectives: minimizing the total duration of the schedule (= makespan), the mean processing time or the total tardiness? Table 7.6: Task time windows and durations Job 1 2 3 4 5 6 7 Release date 2 5 4 0 0 8 9 Duration 5 6 8 4 2 4 2 Due date 10 21 15 10 5 15 22