Mathematics of scheduling
The mathematics of scheduling involves developing and implementing optimal plans to manage time, resources, and costs effectively in various operational contexts. Rooted in historical practices dating back to the Industrial Revolution, scheduling has evolved significantly with advancements such as assembly line production. In manufacturing, scheduling is crucial as it coordinates multiple tasks and processes across various machines, each with its own processing times. This complexity is further amplified in scenarios requiring the transport of materials between facilities.
Mathematical tools from operations research, such as simulation, linear programming, and statistical analysis, play a vital role in crafting these schedules. Scheduling is not limited to manufacturing; it also applies to service industries, such as allocating personnel for emergency services while adhering to legal work regulations. The intricacies of scheduling are heightened by factors like diverse product types and variable processing times, necessitating a deep understanding of resources and operational flow.
Schedulers often rely on two primary modeling approaches: deterministic models, which assume consistent conditions, and stochastic models, which account for variability and unexpected events. Each model has its applications depending on the predictability of the manufacturing process. Overall, the mathematics of scheduling is fundamental to enhancing operational efficiency and minimizing waste across various industries.
Mathematics of scheduling
Summary: Scheduling can be a complex mathematical exercise and is necessary to keep businesses and supply chains running efficiently.
Intense competitiveness forces companies to optimize performance in terms of cost, time, and resources. Scheduling is the process of developing and implementing optimal operational plans. Formal concepts of scheduling date to the Industrial Revolution and innovations like Henry Ford’s assembly line, although the basic ideas probably existed from antiquity in any society where people manufactured goods.
In manufacturing, multiple tasks are carried out in sequence to produce a final output from raw materials. Further, steps in a manufacturing process may be performed on different machines that require variable time to deliver outputs and it is possible that materials will be transported between facilities. A mathematically determined schedule that takes into account all relevant variables in the process serves to optimally allocate resources with respect to demand of the tasks, including shortening time intervals to reduce unproductive time and minimizing costs from wasted time and materials. Operations research is a field of applied mathematics and science that uses mathematical tools, such as simulation and modeling, linear programming, numerical analysis, graph theory, and statistical analysis, to arrive at optimal or near-optimal solutions to complex problems like scheduling. It may also tackle problems in which the resources are not materials but people. The scheduling of airplane crews is a highly constrained and difficult problem because of legal limits on work and rest times as well as the need for crews to return to a home base. Allocation of police, fire, and ambulance services is also a widely used and very important application of scheduling theory.
Production Management
As a part of production management, scheduling interferes with many different aspects of business such as the supply chain, inventory maintenance, and accounting. For example, consider a paint company that makes provisions of sales for the next month by analyzing previous data. In light of these provisions, schedulers determine the expected arrival time and amount of different types of chemicals, which have different delivery times.
The supply chain should be able to deliver the correct amounts of chemicals in time. In a similar way, accounting of the cost of supply and inventory should be accessible for the schedulers. Because of the number of operational parts of business that scheduling is related with, it is apparent that scheduling is a very complex process. It gets more complex with larger variation in types of products and larger numbers of machines varying in processing times. Thus, schedulers demand thorough knowledge of factors such as the processing time of each machine, delivery time, the amount of resources to allocate among machines, and the size and flow of operations for each product.
Manufacturing
In many manufacturing processes, different machines might share the same input, or inputs of a machine might consist of outputs from multiple machines. Scheduling operations in these type of cases requires extensive mathematical modeling. Two basic types of modeling for production scheduling are distinguished by the presence of randomness within. Deterministic models do not include the probability of faults in processes or critical changes in capacity or resource availability. They are based on previous averages of production figures and output rates, so they do not easily adapt to changes in demand or capacity constraints. In these cases, rescheduling is needed, which causes time and resource loss if repeated too many times. They are best suited to manufacturing productions that involve less risk of defects. Stochastic models, on the other hand, involve the probability of unexpected malfunctions or critical changes by distributing probability analytically to individual steps of the schedule. Usually, they are appropriate for processes consisting of many individual operations. For example, these models examine machine failure rates and aim to provide options for when a breakdown occurs. Also, these models maintain an inventory of materials, which may prove critical in maintaining production. Simulations of models provide schedulers an environment to test possibilities that can obstruct the flow of production.
Bibliography
Conway, Richard W., William L. Maxwell, and Louis W. Miller. Theory of Scheduling. New York: Dover Publications, 2003.
Pinedo, Michael. Scheduling: Theory, Algorithms, and Systems. New York: Springer, 2008.
Blazewicz, J., K. H. Ecker, E. Pesch, G. Schmidt, and J. Weglarz. Scheduling Computer and Manufacturing Processes. New York: Springer, 2001.
Kogan, K., and E. Khmelnitsky. Scheduling: Control-Based Theory and Polynomial-Time Algorithms. New York: Springer, 2000.