
$$(1)$$
$$\left({2}\right)$$

times sjk **and the revised schedule** x
′ **with starting times** s
′

jk **of operations** Ojk.

$$s t a b(x,x^{\prime})=\sum|s_{j k}-s_{j k}^{\prime}|$$

jk| (1)
The stability measure of Wu et al. (1993) comes with the implicit assumption, that a discrimination between operations being shifted forward or **backward in time has an equal impact on the stability.**
It therefore is only applicable in situations in which this assumption holds. In many production situations however, a difference between delayed operations **and earlier executed operations can be** observed. Delaying operations does not necessarily result **in a higher organizational overhead, but it** has an obvious effect on the completion of the jobs, and is in this sense already captured by other optimality criteria. An earlier execution of operations however often results in a considerable effort. For example, the required material and tools have to be made available, computerized numerical control (CNC) programs need to be completed upon start of production, etc.

In the work of Lin et al. (1994), only operations with earlier **starting times are considered in the**
proposed stability measure. This implies that delays do not **affect the schedules stability while earlier**
starting times do, see Expression (2).

$$s t a b(x,x^{\prime})=\sum\operatorname*{max}(0,s_{j k}-s_{j k}^{\prime})$$

jk**) (2)**
To some extent, the approach of Lin et al. (1994) can be seen as **an answer to the criticism mentioned** above with regard to the work of Wu et al. (1993). The stability measure is therefore suitable for manufacturing environments where only earlier executions **of operations present a relevant change to**
the production plan. The approaches of Wu et al. (1993) and Lin et al. (1994) are combined in the work presented by Rangsaritratsanee et al. (2004) into the definition of an overall cost function. As a result, a more general way of measuring the stability of schedules is derived, making it possible to individually discriminate between the effect of earlier and later scheduled operations. As opposed to starting times of operations, Cowling and Johansson (2002) propose a combination of start and completion times of the entire job, leading to an overall cost function. In a later work of Cowling et al. (2004), only the completion times Cj **of jobs are relevant for the proposed stability**
measure. As in this stability measure it is not the operations that are considered but the completion of the job as a whole, this concept differs significantly from other approaches of measuring the stability of schedules. Its use can be seen in manufacturing environments where changes of the completion of the jobs are more important than the operations themselves.

The approach of Watatani and Fujii (1992), later also used by **Iima (2005), is based on the sequences**
of the operations. Here, schedules x and x
′ **are considered to be different if the sequences of the**
operations differ. This is the case for operations Ojk, Olo **for which** sjk < slo ∧ s
′

jk > s′lo**. This measure**
is useful when changing the sequence of operations is difficult with respect to the organization of the manufacturing environment and therefore results in a necessary organizational effort. An example of such a production situation would be a flow-oriented manufacturing environment where jobs are transported in a fixed sequence, e. g. by means of a belt. Changes to the sequence through jobs overtaking others impose problems here. Stability measures are implemented in rescheduling strategies either in a predictive-reactive way, where the problem is treated as a multi-objective optimization problem maximizing both utility and stability, or in concepts of robust scheduling, where they are used to compute a quantitative measure for a schedule robustness with respect to changes of the manufacturing situation.