# On The Comparison Of Plans: Proposition Of An Instability Measure For Dynamic Machine Scheduling

## Martin Josef Geiger

Helmut Schmidt University, Logistics Management Department, Holstenhofweg 85, 22041 Hamburg, Germany, m.j.geiger@hsu-hh.de ABSTRACT. On the basis of an analysis of previous research, we present a **generalized approach**
for measuring the difference of plans with an exemplary application to machine scheduling. Our work is motivated by the need for such measures, which are used in dynamic scheduling and planning situations. In this context, quantitative approaches are needed for the assessment of the robustness and stability of schedules. Obviously, any 'robustness' or 'stability' of plans has to be defined w. r. t. the particular situation and the requirements of the human decision maker. Besides the proposition of an instability measure, we therefore discuss possibilities of obtaining meaningful information from the decision maker for the implementation of the introduced approach.

KEYWORDS. **Robust scheduling; dynamic scheduling; rescheduling; stability measure.**

## 1 Scheduling In Dynamic Environments

Planning and scheduling generally considers a set of activities (operations) for which (i) assignments to resources, such as machines, and (ii) definition of starting times must be found, thus defining a schedulex **for a given problem. While the assignment of operations to machines is often given by**
technical side constraints, the processing order on the machines, and thus the precise starting times of the jobs is determined by the planning procedure/ algorithm. W. r. t. the terminology used in machine scheduling, operations, denoted as Ojk**, are often grouped into 'jobs'** Jj = {Oj1, . . . , Ojoj }**, and**
precedence constraints among the operations belonging to a **particular job implement the technical**
requirements of processing.

Besides these general, static considerations, practical problems of scheduling in manufacturing environments are in most cases of dynamic nature. In these situations, a schedule needs to be found that is modified during the actual execution, adapting to the dynamically changing conditions. Changes of relevance to the scheduling are referred to as rescheduling factors **(Dutta, 1990). Along with** these issues come modifications of the relevant data and the characteristics of the problem. Depending on the changes and on the initial schedulext at timet, a revision ofxt **might be necessary.**
Such dynamic adaptations raise stability and robustness issues. Obviously, an impact on the organization has to be expected when making changes to once defined and committed plans. Therefore, quantitative approaches of measuring the difference of plans, and thus the robustness of schedules, are needed for such practical applications.

## Rescheduling Factors

Changes to the machine environment are the most commonly discussed factors of dynamic scheduling in the literature. Machine failures are reported to play a role **in dynamic situations (Abumaizar and Svestka,** 