FINDING THE OPTIMAL SOLUTION FOR A SPECIAL CASE OF THE FIXED INTERVAL JOB SCHEDULING PROBLEM WITH DIFFERENT PROCESSOR SIZE REQUIREMENTS AND A HIERARCHICAL STRUCTURE

Abstract

ABSTRACT In many operational scheduling problems, we encounter situations wherein a resource with a higher capacity can undertake a job that requires a resource with a lower capacity, but not vice versa. For example, a crane with a lifting capacity of 40 tons can be used, if necessary, to lift a load of 27 tons at a construction site, but a crane with a lifting capacity of only 30 tons cannot be used to lift a load of 35 tons. Another familiar example is the assignment of airplanes to different flights. In almost all such cases, a resource that has a higher capacity is a lot more expensive than the one with a lower capacity. Further, each task has to be completed in a time interval that may be fixed or flexible. In the literature, job scheduling problems like these are called Hierarchical Class Scheduling Problems (HCSP), because of the hierarchical structure of their capacity requirements. For problems like these, currently no viable algorithmic techniques are available to determine the optimal mix of resources or processors needed to complete all the jobs on schedule, in spite of the availability of modern powerful computers. Scheduling problems of this type, among many others, are categorized as NP-hard problems in the specialized field of computer science that deals with the subject of computational complexity and solvability of problems. In this paper, we consider a special case of the Hierarchical Class Scheduling Problem (HCSP) and present some propositions that help us find the optimal solution for the special case. Keywords Fixed Interval Scheduling, Hierarchical Class Scheduling, NP-hard Problems