# Computing Idle Times in Fuzzy Flexible Job Shop Scheduling
## 1 Introduction
More generally, idle times can be relevant indicators of the efficiency of production systems, especially in those where expensive resources are part of the considered production layout since idle times are directly related to resource utilisation. Also, minimising the total core idle time (i.e., the idle time within the system) is strongly aligned with minimising the makespan, a classical objective in scheduling. Beyond manufacturing, in service environments where the resources are human workers (e.g., in healthcare), it may be desirable to minimise idle times not only to reduce costs and gain efficiency but also to maximise workers' satisfaction [32; 33; 34]. Idle times have garnered increasing attention with the growth of energy-aware or green scheduling. According to [35], studies on energy-efficient scheduling are classified into four groups based on energy-saving methods. The first is to reduce unnecessary idle time: it is estimated that, in actual manufacturing processes, machine tools stay idle most of the time and consume about 80% of energy while being in this idle state; in consequence, over 30% of the articles reviewed in [36] consider idle energy consumption as part of the objective function to be minimised. The second approach is to directly turn off the resources when the idle time is long enough and the energy savings compensate for the cost of turning on/off. The third approach is to adjust the machine speed when energy consumption depends on it. Finally, the fourth one is to favour off-peak production: given that, in peak time, electricity costs (and associated pollution) are high, scheduling part of the production at a non-peak time can save a significant amount of electricity cost and mitigate pollution. We find in the literature many recent examples of green scheduling (many flexible job shop problems) where idle times play an important part. In [35], they consider an energy-efficient flexible job shop under time-of-use pricing and scheduled downtime. With prices of electricity varying over discrete time intervals, the key is to take advantage of idle times and shift the inevitable idle time to peak periods and the production to off-peak periods in order to minimise the energy cost. The flexible job shop problem is also addressed in [37; 38]. In the former, the objective is to minimise the total energy consumption by adopting an on/off strategy that takes into account idle times. In the latter, energy consumption is composed of processing energy consumption, idle energy consumption, setup energy consumption, and common energy consumption. A multiobjective dynamic version of the flexible job shop scheduling problem minimising both energy and completion time is tackled in [39] through idle machine time arrangement and machine speed level selection. Inspired by real-world problems, in [40], they present an approach to minimising idle energy consumption and apply it to a heat-intensive process employing a steel-hardening furnace, which requires a high power input to reach and maintain the specific operating temperature, so a considerable amount of energy is consumed even during the periods when no material is being processed. Ref. [41] addresses scheduling in virtual network functions, modelled as a job shop problem, with the goal of simultaneously minimising the makespan and the idle energy loss. In [42] we find another multiobjective scheduling problem, here applied to bakery production, taking into account the reduction in oven idle times as a means of reducing energy consumption. The results show that, by penalising the best makespan by a marginal amount, alternative optimal solutions obtain a considerable reduction in oven idle time by up to 61%, resulting in a strategy that allows small and medium-sized bakeries to lower production costs and reduce CO2 emissions. Clearly, to enhance its scope of application and come closer to solving real-world problems, fuzzy scheduling also needs to handle and take into account idle times.